Basic Pharmacokinetics [3 ed.] 0367752158, 9780367752156

Table of contents :
Cover
Half Title
Title Page
Copyright Page
Dedication
Contents
List of Figures
1. Introduction to Pharmacokinetics
1.1. Introduction
1.2. Pharmacokinetics and Its Related Fields
1.2.1. Biopharmaceutics
1.2.2. Pharmacokinetics
1.2.3. Clinical Pharmacokinetics
1.2.4. Pharmacodynamics
1.2.5. Population Pharmacokinetics
1.2.6. Toxicokinetics
1.2.7. Pharmacogenetics
1.3. Application of the Pharmacokinetic Principles in the Biomedical Fields
1.3.1. Design and Evaluation of Dosage Forms
1.3.2. Evaluation of Generic Drug Products
1.3.3. Pharmacological Testing
1.3.4. Toxicological Testing
1.3.5. Evaluation of Organ Function
1.3.6. Therapeutic Drug Monitoring
1.4. The Blood Drug Concentration-Time Profile
1.5. Linear and Nonlinear Pharmacokinetics
1.5.1. Linear Pharmacokinetics
1.5.2. Nonlinear Pharmacokinetics
1.6. Pharmacokinetic Modeling
1.6.1. Compartmental Modeling
1.6.2. Physiological Modeling
1.6.3. Population Pharmacokinetic Modeling
1.6.4. Noncompartmental Data Analysis Approach
1.6.5. Pharmacokinetic-Pharmacodynamic Modeling
1.7. Pharmacokinetic Simulations
1.8. Essential Graphical, Mathematical, and Statistical Fundamentals Used in Pharmacokinetics
1.8.1. Graphs
1.8.2. Curve Fitting
1.8.3. Determination of the Straight-Line Parameters
1.8.3.1. Graphical Determination of the Straight-Line Parameters
1.8.3.2. The Least Squares Method
1.8.4. Application of Basic Calculus Principles in Pharmacokinetics
2. Drug Pharmacokinetics Following Single Intravenous Bolus Administration: Drug Distribution
2.1. Introduction
2.2. Drug Distribution
2.2.1. The Rate and Extent of Drug Distribution
2.3. The Volume of Distribution
2.4. Drug Distribution after Single IV Bolus Drug Administration
2.5. Drug Protein Binding
2.5.1. Effect of Changing the Plasma Protein Binding
2.5.2. Determination of Plasma Protein Binding
2.6. Drug Partitioning to Blood Cells
2.7. Summary
3. Drug Pharmacokinetics Following Single IV Bolus Administration: Drug Clearance
3.1. Introduction
3.2. Drug Clearance
3.2.1. The Total Body Clearance
3.2.2. Physiological Approach to Drug Clearance
3.2.3. The Plasma Drug Concentration-Time Profile
3.3. Total Body Clearance and Volume of Distribution Are the Independent Pharmacokinetic Parameters
3.4. Determination of the Total Body Clearance
3.5. Summary
4. Drug Pharmacokinetics Following Single IV Bolus Administration: The Rate of Drug Elimination
4.1. Introduction
4.2. Drug Elimination
4.3. The Kinetics of the Drug Elimination Process
4.3.1. Zero-Order Elimination
4.3.1.1. The Zero-Order Elimination Rate Constant
4.3.1.2. The Half-Life In Zero-Order Elimination
4.3.2. First-Order Elimination
4.3.2.1. The First-Order Elimination Rate Constant
4.3.2.2. Determination of the First-Order Elimination Rate Constant
4.3.2.3. The Half-Life in First-Order Drug Elimination
4.4. The Mathematical Expressions for Plasma Drug Concentrations after Single IV Bolus Dose when the Elimination Process Follows First-Order Kinetics
4.5. The Relationship between the First-Order Elimination Rate Constant, Total Body Clearance, and Volume of Distribution
4.6. The Area under the Drug Concentration-Time Curve
4.7. Calculation of Pharmacokinetic Parameters after Single IV Bolus Dose
4.8. The Effect of Changing the Pharmacokinetic Parameters on the Plasma Drug Concentration-Time Profile after Single IV Bolus Dose
4.8.1. Dose
4.8.2. Volume of Distribution
4.8.3. Total Body Clearance
4.9. Summary
5. Drug Absorption Following Extravascular Administration: Biological, Physicochemical, and Formulation Considerations
5.1. Introduction
5.2. The Drug Absorption Process
5.2.1. The Absorption Barriers
5.2.2. Mechanisms of Drug Absorption
5.2.2.1. Passive Diffusion
5.2.2.2. Carrier-Mediated Transport
5.2.2.3. Paracellular
5.2.2.4. Other Mechanisms
5.3. Molecular and Physicochemical Properties Affecting Drug Absorption
5.3.1. Molecular Structure Features Affecting Drug Absorption
5.3.2. The Physicochemical Drug Properties
5.3.2.1. Drug Solubility
5.3.2.2. Drug Dissolution Rate
5.3.3. Drug Stability
5.4. Physiological Factors Affecting Drug Absorption After Different Routes of Administration and Formulation Strategies to Accommodate These Factors
5.4.1. Parenteral Drug Administration
5.4.2. Oral Drug Administration
5.4.3. Rectal Drug Administration
5.4.4. Intranasal Drug Administration
5.4.5. Pulmonary Drug Administration
5.4.6. Transdermal Drug Administration
5.5. Integration of the Physical, Chemical, and Physiological Factors Affecting Drug Absorption
5.5.1. The Biopharmaceutics Classification System
5.5.2. The Biopharmaceutics Drug Disposition Classification System (BDDCS)
5.6. Summary
References
6. Drug Pharmacokinetics Following Single Oral Drug Administration: The Rate of Drug Absorption
6.1. Introduction
6.2. Drug Absorption after Oral Administration
6.2.1. Zero-Order Drug Absorption
6.2.2. First-Order Drug Absorption
6.3. The Plasma Concentration-Time Profile After Single Oral Dose
6.4. Determination of the Absorption Rate Constant
6.4.1. The Method of Residuals
6.4.1.1. Lag Time
6.4.1.2. Flip-Flop of ka and k
6.4.2. Wagner-Nelson Method
6.4.2.1. Application of the Wagner-Nelson Method
6.5. Summary
References
7. Drug Pharmacokinetics Following Single Oral Drug Administration: The Extent of Drug Absorption
7.1. Introduction
7.2. Causes of Incomplete Drug Bioavailability
7.2.1. The First-Pass Effect
7.2.2. The GIT Drug Transporters
7.2.3. Intestinal Drug Metabolism
7.3. The Rationale for Bioavailability Determination
7.4. Determination of the Drug In Vivo Bioavailability
7.4.1. Drug Bioavailability
7.4.1.1. Absolute Bioavailability
7.4.1.2. Relative Bioavailability
7.4.2. Calculation of the Drug Bioavailability
7.4.3. Determination of the Drug Bioavailability from Urinary Excretion Data
7.5. In Vivo Bioavailability Basic Study Design
7.6. Calculation of the AUC Using the Linear Trapezoidal Rule
7.7. The Effect of Changing the Pharmacokinetic Parameters on the Plasma Drug Concentration-Time Profile after Single Oral Dose
7.7.1. Dose
7.7.2. Bioavailability
7.7.3. Total Body Clearance
7.7.4. Volume of Distribution
7.7.5. Absorption Rate Constant
7.8. Summary
References
8. Bioequivalence
8.1. Introduction
8.2. General Definitions
8.3. Regulatory Requirement for Bioequivalence
8.4. Criteria for Requesting a Waiver of the In Vivo Bioequivalence Determination
8.5. Approaches for Demonstrating Product Bioequivalence
8.5.1. In Vivo Pharmacokinetic Studies
8.5.2. In Vitro Test Predictive of In Vivo Human Bioavailability
8.5.3. Acute Pharmacodynamic Effect
8.5.4. Comparative Clinical Studies
8.5.5. In Vitro Dissolution Testing
8.6. Pharmacokinetic Approach to Demonstrate Product Bioequivalence
8.6.1. Planning for the In Vivo Bioequivalence Study
8.6.2. Selection of the Reference Drug Product
8.6.3. In Vitro Testing of the Study Products
8.6.4. In Vivo Bioequivalence Study Design
8.6.4.1. Basic Principles
8.6.4.2. Ethical Approval
8.6.4.3. The Study Subjects
8.6.4.4. Number of Volunteers
8.6.4.5. Drug Administration
8.6.4.6. Experimental Protocol
8.6.4.7. Collection of Blood Samples
8.6.4.8. Analysis of Bioequivalence Study Samples
8.6.4.9. Pharmacokinetic Parameter Determination
8.6.4.10. Statistical Analysis
8.6.4.11. Documentation and Reporting
8.7. Special Issues Related to Bioequivalence Determination
8.7.1. Multiple-Dose Bioequivalence Studies
8.7.2. Food-Effect Bioequivalence Studies
8.7.3. Drugs with Long Half-Lives
8.7.4. Determination of Bioequivalence from the Drug Urinary Excretion Data
8.7.5. Fixed-Dose Combination
8.7.6. Measuring Drug Metabolites in Bioequivalence Studies
8.7.7. Highly Variable Drugs
8.7.8. Drugs Following Nonlinear Pharmacokinetics
8.7.9. Endogenous Substances
8.7.10. Enantiomers versus Racemates
8.7.11. Narrow Therapeutic Range Drugs
8.7.12. Oral Products Intended for the Local Effect of the Drug
8.7.13. First Point Cpmax
8.7.14. Biological Products
8.8. Summary
References
9. Drug Pharmacokinetics during Constant Rate IV Infusion, the Steady-State Concept
9.1. Introduction
9.2. The Steady State
9.3. The Time Required to Achieve Steady State
9.3.1. Changing the Drug Infusion Rate
9.4. Loading Dose
9.5. Termination of the Constant Rate IV Infusion
9.6. Determination of the Pharmacokinetic Parameters
9.6.1. Total Body Clearance
9.6.2. Elimination Rate Constant
9.6.3. Volume of Distribution
9.7. Dosage Forms with Zero-Order Input Rate
9.8. The Effect of Changing the Pharmacokinetic Parameters on the Plasma Drug Concentration-Time Profile during Constant Rate IV Infusion
9.8.1. Infusion Rate
9.8.2. Total Body Clearance
9.8.3. Volume of Distribution
9.8.4. Loading Dose
9.9. Summary
10. Steady State during Multiple Drug Administration
10.1. Introduction
10.2. The Plasma Drug Concentration-Time Profile during Multiple Drug Administration
10.3. The Time Required to Achieve Steady State
10.4. The Loading Dose
10.4.1. IV Loading Dose
10.4.2. Oral Loading Dose
10.5. The Average Plasma Concentration at Steady State
10.6. Drug Accumulation
10.7. Controlled Release Formulations
10.8. The Effect of Changing the Pharmacokinetic Parameters on the Steady-State Plasma Drug Concentration during Multiple Drug Administration
10.8.1. Dosing Rate
10.8.2. Total Body Clearance
10.8.3. Volume of Distribution
10.8.4. Absorption Rate Constant
10.9. Dosing Regimen Design
10.9.1. Factors to Be Considered
10.9.1.1. The Therapeutic Range of the Drug
10.9.1.2. The Required Onset of Effect
10.9.1.3. The Drug Product
10.9.1.4. Progression of the Patient Disease State
10.9.2. Estimation of the Patient Pharmacokinetic Parameters
10.9.3. Selection of Dose and Dosing Interval
10.9.3.1. Multiple Controlled Release Oral Formulation
10.9.3.2. Multiple IV or Fast-Release Oral Formulations
10.9.4. Selection of the Loading Dose
10.10. Summary
11. Renal Drug Excretion
11.1. Introduction
11.2. Studying Drug Elimination through a Specific Pathway
11.3. The Renal Excretion of Drugs
11.4. Determination of the Drug Renal Excretion Rate
11.4.1. Experimental Determination of the Renal Excretion Rate
11.4.2. The Drug Renal Excretion Rate-Time Profile
11.5. The Renal Clearance
11.6. The Cumulative Amount of the Drug Excreted in Urine
11.7. Determination of the Pharmacokinetic Parameters from the Renal Excretion Rate Data
11.7.1. The Elimination Rate Constant and Half-Life
11.7.2. The Renal Excretion Rate Constant
11.7.3. The Volume of Distribution
11.7.4. The Renal Clearance
11.7.5. The Fraction of Dose Excreted Unchanged in Urine
11.7.6. Bioavailability
11.8. The Effect of Changing the Pharmacokinetic Parameters on the Urinary Excretion of Drugs
11.8.1. Dose
11.8.2. The Total Body Clearance
11.8.3. The Renal Clearance
11.9. Summary
References
12. Metabolite Pharmacokinetics
12.1. Introduction
12.2. Drug Metabolism
12.2.1. Metabolizing Enzymes
12.2.2. Formation of Active Metabolites
12.2.3. Formation of Toxic Metabolites
12.2.4. Metabolic Activation of Prodrugs
12.3. Metabolite Pharmacokinetics
12.4. A Simple Model for Metabolite Pharmacokinetics
12.4.1. Metabolite Concentration-Time Profile
12.5. The General Model for Metabolite Kinetics
12.6. Determination of the Metabolite Pharmacokinetic Parameters
12.6.1. Metabolite Elimination Rate Constant, k(m)
12.6.2. Fraction of the Parent Drug Dose Converted to a Specific Metabolite, fm
12.6.3. Metabolite Clearance, CL(m)
12.6.4. Metabolite Volume of Distribution, Vd(m)
12.6.5. Metabolite Formation Clearance, fm CLT
12.7. Steady-State Metabolite Concentration during Repeated Administration of the Drug
12.8. Metabolite Pharmacokinetics after Extravascular Administration of the Parent Drug
12.9. Kinetics of Sequential Metabolism
12.10. The Effect of Changing the Pharmacokinetic Parameters on the Drug and Metabolite Concentration-Time Profiles after Single IV Drug Administration and during Multiple Drug Administration
12.10.1. Drug Dose
12.10.2. Drug Total Body Clearance
12.10.3. Drug Volume of Distribution
12.10.4. Fraction of the Drug Dose Converted to the Metabolite
12.10.5. Metabolite Total Body Clearance
12.10.6. Metabolite Volume of Distribution
12.11. Summary
References
13. Nonlinear Pharmacokinetics
13.1. Introduction
13.2. Causes of Nonlinear Pharmacokinetics
13.2.1. Dose-Dependent Drug Absorption
13.2.2. Dose-Dependent Drug Distribution
13.2.3. Dose-Dependent Renal Excretion
13.2.4. Dose-Dependent Drug Metabolism
13.2.5. Other Conditions That Can Lead to Nonlinear Pharmacokinetics
13.3. Pharmacokinetics of Drugs Eliminated by Dose-Dependent Metabolism, Michaelis-Menten Pharmacokinetics
13.3.1. Michaelis-Menten Enzyme Kinetics
13.3.2. The Pharmacokinetic Parameters
13.3.3. Drug Concentration-Time Profile after Administration of a Drug Which Is Eliminated by Single Metabolic Pathway That Follows Michaelis-Menten Kinetics
13.3.3.1. After Single IV Bolus Administration
13.3.3.2. During Multiple Drug Administration
13.4. Determination of the Pharmacokinetic Parameters for Drugs with Elimination Process that Follows Michaelis-Menten Kinetics
13.4.1. The Volume of Distribution
13.4.2. The Total Body Clearance
13.4.3. The Half-Life
13.5. Oral Administration of Drugs that Are Eliminated by a Process that Follows Michaelis-Menten Kinetics
13.6. Determination of the Michaelis-Menten Parameters and Calculation of the Appropriate Dosage Regimens
13.6.1. Mathematical Method
13.6.2. The Direct Linear Plot
13.6.3. The Linear Transformation Method
13.7. Multiple Elimination Pathways
13.8. The Effect of Changing the Pharmacokinetic Parameters on the Drug Concentration-Time Profile
13.8.1. The Dose
13.8.2. The Vmax
13.8.3. The Km
13.9. Summary
References
14. Multicompartment Pharmacokinetic Models
14.1. Introduction
14.2. Compartmental Pharmacokinetic Models
14.3. The Two-Compartment Pharmacokinetic Model
14.4. The Parameters of the Two-Compartments Pharmacokinetic Model
14.4.1. Definition of the Pharmacokinetic Parameters
14.4.2. The Mathematical Equation That Describes the Plasma Concentration-Time Profile for Drugs That Follow Two-Compartment Pharmacokinetic Models
14.5. Determination of the Two-Compartment Pharmacokinetic Model Parameters
14.5.1. The Method of Residuals
14.5.2. Determination of the Other Model Parameters
14.5.2.1. Volume of the Central Compartment, Vc
14.5.2.2. The Area under the Plasma Concentration-Time Curve, AUC
14.5.2.3. The Total Body Clearance, CLT
14.5.2.4. The First-Order Elimination Rate Constant from the Central Compartment, k10
14.5.2.5. The First-Order Transfer Rate Constant from the Peripheral Compartment to the Central Compartment, k21
14.5.2.6. The First-Order Transfer Rate Constant from the Central Compartment to the Peripheral Compartment, k12
14.5.3. Determination of the Volumes of Distribution for the Two-Compartment Pharmacokinetic Model
14.5.3.1. The Volume of Distribution at Steady State, Vdss
14.5.3.2. The Volume of Distribution during the Elimination Phase, Vdβ
14.6. Pharmacokinetic Behavior of Drugs that Follow the Two-Compartment Pharmacokinetic Model
14.6.1. Oral Administration of Drugs that Follow the Two-Compartment Pharmacokinetic Model
14.6.2. Constant Rate IV Administration of Drugs That Follow the Two-Compartment Pharmacokinetic Model
14.6.3. Multiple Administration of Drugs That Follow the Two-Compartment Pharmacokinetic Model
14.6.4. Renal Excretion of Drugs That Follow the Two-Compartment Pharmacokinetic Model
14.7. Effect of Changing the Pharmacokinetic Parameters on the Concentration-Time Profile of Drugs That Follow Two-Compartment Pharmacokinetic Model
14.7.1. Dose
14.7.2. Total Body Clearance
14.7.3. Volume of the Central Compartment
14.7.4. The Hybrid Distribution and Elimination Rate Constants
14.7.5. The Inter-Compartmental Clearance
14.8. The Three-Compartment Pharmacokinetic Model
14.9. Compartmental Pharmacokinetic Data Analysis
14.9.1. Construction of the Compartmental Model
14.9.2. Mathematical Description of the Model
14.9.3. Fitting the Model Equation to the Experimental Data
14.9.4. Evaluation of the Pharmacokinetic Model
14.10. Summary
References
15. Drug Pharmacokinetics Following Administration by Intermittent Intravenous Infusions
15.1. Introduction
15.2. The Drug Concentration-Time Profile after Administration by Intermittent IV Infusions
15.2.1. After the First Dose
15.2.2. After Repeated Administration Before Reaching Steady State
15.2.3. At Steady State
15.3. The Effect of Changing the Pharmacokinetic Parameters on the Steady-State Plasma Concentration during Repeated Intermittent IV Infusions
15.3.1. Dose
15.3.2. Infusion Time
15.3.3. Total Body Clearance
15.3.4. Volume of Distribution
15.4. Application of the Pharmacokinetic Principles for Intermittent IV Infusion in Clinical Practice
15.4.1. Pharmacokinetic Characteristics of Aminoglycosides
15.4.2. Guidelines for Aminoglycoside Plasma Concentration
15.4.3. The Extended-Interval Aminoglycoside Dosing Regimen
15.5. Individualization of Aminoglycoside Therapy
15.5.1. Estimation of the Patient Pharmacokinetic Parameters
15.5.1.1. Estimation of the Patient Pharmacokinetic Parameters Based on the Patient Information
15.5.1.2. Estimation of the Patient’s Specific Pharmacokinetic Parameters from Aminoglycoside Blood Concentrations
15.5.2. Determination of the Dosing Regimen Based on the Patient’s Specific Parameters
15.5.2.1. Selection of the Dosing Interval (τ)
15.5.2.2. Selection of Dose
15.5.2.3. Selection of the Loading Dose
15.6. Summary
References
16. Physiological Approach to Hepatic Clearance
16.1. Introduction
16.2. The Organ Clearance
16.3. Hepatic Extraction Ratio
16.4. Intrinsic Clearance
16.5. Systemic Bioavailability
16.6. The Effect of Changing Intrinsic Clearance and Hepatic Blood Flow on the Hepatic Clearance, Systemic Availability, and Drug Concentration-Time Profile
16.6.1. Low Extraction Ratio Drugs
16.6.1.1. Assume that the Drug CLint Increases to Double Its Original Value Due to Enzyme Induction and Q Stays the Same
16.6.1.2. Assume that Q Decreases by 50% (i.e., New Q = 0.75 L/min) without Affecting CLint
16.6.2. High Extraction Ratio Drugs
16.6.2.1. Assume that the Drug CLint Increases to Double Its Original Value Due to Enzyme Induction and Q Stays the Same
16.6.2.2. Assume that Q Decreases by 50% (New Q = 0.75 L/min) without Affecting CLint
16.7. Protein Binding and Hepatic Extraction
16.8. Summary
References
17. Pharmacokinetics in Patients with Eliminating Organ Dysfunction
17.1. Introduction
17.2. Patients with Renal Dysfunction
17.2.1. Dosing Regimens in Renal Dysfunction Patients Based on the Creatinine Clearance
17.2.2. A General Approach for Calculation of Dosing Regimens in Renal Dysfunction Patients
17.3. Patients Receiving Renal Replacement Therapy
17.3.1. The Principle of Dialysis
17.3.2. Factors Affecting the Drug Clearance during Dialysis
17.3.3. Dose Adjustment during Dialysis
17.4. Patients with Hepatic Insufficiency
17.4.1. Pharmacokinetic and Pharmacodynamic Changes in Hepatic Dysfunction
17.4.2. Dose Adjustment in Hepatic Dysfunction
17.5. Other Patient Populations
17.6. Summary
References
18. Noncompartmental Approach in Pharmacokinetic Data Analysis
18.1. Introduction
18.2. The Principles of Noncompartmental Data Analysis Method
18.3. The Mean Residence Time after IV Bolus Administration
18.3.1. Calculation of the AUC
18.3.2. Calculation of the AUMC
18.4. The MRT after Different Routes of Administration
18.4.1. The MRT after Extravascular Administration
18.4.2. The MRT after Constant Rate IV Infusion
18.5. Other Pharmacokinetic Parameters that Can Be Determined Using the Noncompartmental Approach
18.6. Determination of the MRT for Compartmental Models
18.7. Summary
References
19. Pharmacokinetic-Pharmacodynamic Modeling
19.1. Introduction
19.2. Pharmacokinetic-Pharmacodynamic Modeling
19.2.1. The Pharmacokinetic Model
19.2.2. Measuring the Response
19.2.3. The Pharmacodynamic Model
19.2.3.1. The Fixed Effect Model
19.2.3.2. The Linear Model
19.2.3.3. The Log-Linear Model
19.2.3.4. The Emax Model
19.2.3.5. The Sigmoid Emax Model
19.3. Integrating the Pharmacokinetic and Pharmacodynamic Models
19.3.1. Direct Response versus Indirect Response
19.3.2. Direct Link versus Indirect Link
19.3.3. Time-Variant versus Time-Invariant
19.4. Direct Link PK/PD Models for Drugs with Direct Response
19.5. Indirect Link PK/PD Models for Drugs with Direct Response
19.5.1. The Effect Compartment Approach
19.6. PK/PD Models for Drugs with Indirect Response
19.7. Other PK/PD Models
19.8. The PK/PD Modeling Process
19.8.1. Stating the Objectives, Proposing the Model and Designing the Study
19.8.2. Initial Data Exploration and Data Transformation
19.8.3. Refining and Evaluation of the PK/PD Model
19.8.4. Validation of the PK/PD Model
19.9. Applications of the PK/PD Modeling in Drug Development and Clinical Use of Drugs
19.10. Summary
References
20. Pharmacogenetics: The Genetic Basis of Pharmacokinetic and Pharmacodynamic Variability
20.1. Introduction
20.2. Gene Structure
20.3. Genetic Background Information
20.3.1. Gene Variants, Alleles
20.3.2. Polymorphisms
20.3.3. Gene Nomenclature
20.3.4. Genotype versus Phenotype
20.3.5. Monogenic versus Polygenic
20.3.6. Homozygous versus Heterozygous Genotype
20.4. Genetic Polymorphism in Pharmacokinetics
20.4.1. Cytochrome P450 Enzymes
20.4.2. Thiopurine Methyltransferase (TPMT)
20.4.3. N-acetyltransferase
20.4.4. UDP-Glucuronosyltransferase (UGT)
20.4.5. Drug Transporters
20.5. Genetic Polymorphism in Pharmacodynamics
20.6. Implementation of Pharmacogenetic Testing in Clinical Practice
20.6.1. Pharmacogenetic Training for Healthcare Providers
20.6.2. The Pharmacogenetic Tests
20.6.3. Interpretation of the Pharmacogenetic Test Results
20.6.4. Guidelines for Applying the Pharmacogenetic Testing
20.6.5. Enablers for the Implementation of Pharmacogenetics in Clinical Practice
20.7. Summary
References
21. Therapeutic Drug Monitoring
21.1. Introduction
21.2. General Principles of Initiation and Management of Drug Therapy
21.2.1. The Use of Therapeutic Drug Monitoring in the Management of Drug Therapy
21.3. Drug Blood Concentration versus Drug Dose
21.4. The Therapeutic Range
21.5. Drug Candidates for Therapeutic Drug Monitoring
21.5.1. Drugs with Low Therapeutic Index
21.5.2. Drugs with Large Variability in Their Pharmacokinetic Behavior
21.5.3. Drugs Used in High-Risk Patients or Patients with Multiple Medical Problems
21.6. Determination of the Drug Concentration in Biological Samples
21.6.1. The Biological Samples
21.6.2. The Time of Sample
21.6.3. The Measured Drug Moiety
21.6.4. The Analytical Technique
21.7. Establishing a Therapeutic Drug Monitoring (Clinical Pharmacokinetic) Service
21.7.1. Major Requirements
21.7.2. Dosage Regimen Recommendation
21.7.2.1. Determination of the Initial Dosing Regimen
21.7.2.2. Determination of the Patient’s Specific Pharmacokinetic Parameters
21.7.2.3. Calculation of the Dosage Requirements Based on the Patient’s Specific Pharmacokinetic Parameters of the Drug
21.7.3. The Pharmacoeconomics of Therapeutic Drug Monitoring
21.8. Summary
References
22. Pharmacometric Applications in Drug Development and Individualization of Drug Therapy
22.1. Introduction
22.2. Pharmacometric Applications during the Preclinical Phase of Drug Development
22.2.1. Physiologically Based Pharmacokinetic Models
22.2.1.1. Physiologically Based Pharmacokinetic Model Development
22.2.1.2. Applications of the PBPK Models
22.3. Pharmacometric Applications during the Clinical Phases of Drug Development
22.3.1. Population Pharmacokinetic Analysis
22.3.1.1. Data Consideration for Population Analysis
22.3.1.2. The Population Pharmacokinetic Models
22.3.1.3. Statistical Analysis and Parameter Estimation
22.3.1.4. Model Evaluation and Diagnostics
22.3.1.5. Reporting of the Population Pharmacokinetic Analysis Results
22.3.1.6. Application of Population Pharmacokinetic Analysis in Drug Development
22.3.1.7. Application of Population Pharmacokinetic Analysis for Drug Use Decisions in Drug Labeling
22.4. Pharmacometric Applications in Clinical Drug Use
22.4.1. Model-Based Therapeutic Drug Monitoring
22.4.1.1. Model Development
22.4.1.2. Monitoring Drug Concentration
22.4.1.3. Dosage Regimen Design
22.5. Summary
References
23. Answer for the Practice Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 6
Chapter 7
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
Chapter 16
Chapter 17
Chapter 18
Glossary
Index

Citation preview

Basic Pharmacokinetics

This book introduces basic pharmacokinetic concepts to beginner learners to help them understand the absorption, distribution, metabolism, and excretion of drugs. After a basic introduction to pharmacokinetics and its related fields, the book provides a clear introduction to quantitative pharmacokinetic relations and the interplay between pharmacokinetic parameters after different routes of drug administration. Emphasizing the application and importance of pharmacokinetic concepts in clinical practice throughout, the book features:

• A clear, simple, and concise style with the use of graphs and simulations to aid learning. • Bullet point summaries of each concept to demonstrate applications in clinical practice. • Practice problems and solved examples to help the reader understand the best approach for calculating pharmacokinetic parameters.

• A glossary of key words and acronyms.

This book is an essential read for undergraduate and graduate pharmacokinetic students in pharmacy, pharmacology, and pharmaceutical science programs worldwide. Accompanying the book is a website with self-instructional tutorials and pharmacokinetic simulations, allowing visualization of concepts for enhanced comprehension. This learning tool received an award from the American Association of Colleges of Pharmacy for innovation in teaching, making it a valuable supplement to this textbook. Mohsen A. Hedaya, PharmD, PhD, is a professor at the Faculty of Pharmacy, Tanta University, Egypt, and is currently on leave to work at the Faculty of Pharmacy, Kuwait University, Kuwait. He received his Bachelor of Science in Pharmacy degree from Tanta University, and his Doctor of Pharmacy and Doctor of Philosophy degrees from the University of Minnesota, USA. He joined the College of Pharmacy, Washington State University, USA, in 1993 as an Assistant Professor of Pharmaceutical Sciences. After returning to Egypt in 1999, he was promoted to the rank of associate and then full professor. He served as the Chair of the Clinical Pharmacy Department and Vice Dean for Academic Affairs at the Faculty of Pharmacy, Tanta University. Dr. Hedaya’s area of interest is pharmacokinetics. He has taught basic and advanced pharmacokinetic classes to professional pharmacy students and graduate students in Egypt, the United States, and Kuwait. His research interest includes pharmacokinetic drug interactions, drug delivery to the brain, pharmacokinetic computer simulations, and bioequivalent study design and data analysis. The computer-based and online educational materials developed by Dr. Hedaya in the area of pharmacokinetics are currently being used by more than 350 educational, research, and industrial institutions around the world.

Basic Pharmacokinetics Third Edition

Mohsen A. Hedaya

Cover image: © Getty Images Third edition published 2024 by Routledge 605 Third Avenue, New York, NY 10158 and by Routledge 4 Park Square, Milton Park, Abingdon, Oxon OX14 4RN Routledge is an imprint of the Taylor & Francis Group, an informa business © 2024 Mohsen A. Hedaya The right of Mohsen A. Hedaya to be identified as author of this work has been asserted in accordance with sections 77 and 78 of the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. Trademark notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. First edition published by CRC Press 2007 Second edition published by Routledge 2012 Library of Congress Cataloging-in-Publication Data Names: Hedaya, Mohsen A, author. Title: Basic pharmacokinetics / Mohsen A Hedaya. Description: Third Edition. | New York : Routledge, 2023. | Revised edition of the author’s Basic pharmacokinetics, c2012. | Includes bibliographical references and index. Identifiers: LCCN 2023005241 | ISBN 9780367752149 (hardback) | ISBN 9780367752156 (paperback) | ISBN 9781003161523 (ebook) Subjects: LCSH: Pharmacokinetics. Classification: LCC RM301.5 .H43 2023 | DDC 615/.7--dc23/eng/20230403 LC record available at https://lccn.loc.gov/2023005241 ISBN: 978-0-367-75214-9 (hbk) ISBN: 978-0-367-75215-6 (pbk) ISBN: 978-1-003-16152-3 (ebk) DOI: 10.4324/9781003161523 Typeset in Sabon LT Pro by KnowledgeWorks Global Ltd. A companion website hosted by Mohsen Hedaya is available at: http://e-pharmacokinetics.com/epharma/

To the memory of my parents, may God have mercy on their souls.

Contents

List of Figures

xxv

1 Introduction to Pharmacokinetics1 1.1 Introduction 1 1.2 Pharmacokinetics and Its Related Fields 2 1.2.1 Biopharmaceutics 2 1.2.2 Pharmacokinetics 2 1.2.3 Clinical Pharmacokinetics 3 1.2.4 Pharmacodynamics 4 1.2.5 Population Pharmacokinetics 4 1.2.6 Toxicokinetics 5 1.2.7 Pharmacogenetics 5 1.3 Application of the Pharmacokinetic Principles in the Biomedical Fields 5 1.3.1 Design and Evaluation of Dosage Forms 5 1.3.2 Evaluation of Generic Drug Products 6 1.3.3 Pharmacological Testing 6 1.3.4 Toxicological Testing 6 1.3.5 Evaluation of Organ Function 6 1.3.6 Therapeutic Drug Monitoring 6 1.4 The Blood Drug Concentration-Time Profile 7 1.5 Linear and Nonlinear Pharmacokinetics 8 1.5.1 Linear Pharmacokinetics 8 1.5.2 Nonlinear Pharmacokinetics 8 1.6 Pharmacokinetic Modeling 9 1.6.1 Compartmental Modeling 10 1.6.2 Physiological Modeling 10 1.6.3 Population Pharmacokinetic Modeling 11 1.6.4 Noncompartmental Data Analysis Approach 12 1.6.5 Pharmacokinetic-Pharmacodynamic Modeling 12

viii Contents 1.7 Pharmacokinetic Simulations 12 1.8 Essential Graphical, Mathematical, and Statistical Fundamentals Used in Pharmacokinetics 14 1.8.1 Graphs 14 1.8.2 Curve Fitting 16 1.8.3 Determination of the Straight-Line Parameters 16 1.8.3.1 Graphical Determination of the Straight-Line Parameters 17 1.8.3.2 The Least Squares Method 17 1.8.4 Application of Basic Calculus Principles in Pharmacokinetics 18

2 Drug Pharmacokinetics Following Single Intravenous Bolus Administration: Drug Distribution21 2.1 Introduction 21 2.2 Drug Distribution 21 2.2.1 The Rate and Extent of Drug Distribution 22 2.3 The Volume of Distribution 23 2.4 Drug Distribution after Single IV Bolus Drug Administration 26 2.5 Drug Protein Binding 28 2.5.1 Effect of Changing the Plasma Protein Binding 29 2.5.2 Determination of Plasma Protein Binding 30 2.6 Drug Partitioning to Blood Cells 31 2.7 Summary 32

3 Drug Pharmacokinetics Following Single IV Bolus Administration: Drug Clearance34 3.1 Introduction 34 3.2 Drug Clearance 34 3.2.1 The Total Body Clearance 35 3.2.2 Physiological Approach to Drug Clearance 35 3.2.3 The Plasma Drug Concentration-Time Profile 37 3.3 Total Body Clearance and Volume of Distribution Are the Independent Pharmacokinetic Parameters 38 3.4 Determination of the Total Body Clearance 39 3.5 Summary 40

4 Drug Pharmacokinetics Following Single IV Bolus Administration: The Rate of Drug Elimination41 4.1 Introduction 41 4.2 Drug Elimination 41

Contents ix 4.3 The Kinetics of the Drug Elimination Process 42 4.3.1 Zero-Order Elimination 42 4.3.1.1 The Zero-Order Elimination Rate Constant 42 4.3.1.2 The Half-Life In Zero-Order Elimination 44 4.3.2 First-Order Elimination 46 4.3.2.1 The First-Order Elimination Rate Constant 46 4.3.2.2 Determination of the First-Order Elimination Rate Constant, k 49 4.3.2.3 The Half-Life in First-Order Drug Elimination 51 4.4 The Mathematical Expressions for Plasma Drug Concentrations after Single IV Bolus Dose when the Elimination Process Follows First-Order Kinetics 52 4.5 The Relationship between the First-Order Elimination Rate Constant, Total Body Clearance, and Volume of Distribution 54 4.6 The Area under the Drug Concentration-Time Curve 54 4.7 Calculation of Pharmacokinetic Parameters after Single IV Bolus Dose 55 4.8 The Effect of Changing the Pharmacokinetic Parameters on the Plasma Drug ConcentrationTime Profile after Single IV Bolus Dose 58 4.8.1 Dose 58 4.8.2 Volume of Distribution 58 4.8.3 Total Body Clearance 59 4.9 Summary 59

5 Drug Absorption Following Extravascular Administration: Biological, Physicochemical, and Formulation Considerations65 5.1 Introduction 65 5.2 The Drug Absorption Process 66 5.2.1 The Absorption Barriers 66 5.2.2 Mechanisms of Drug Absorption 66 5.2.2.1 Passive Diffusion 67 5.2.2.2 Carrier-Mediated Transport 68 5.2.2.3 Paracellular 69 5.2.2.4 Other Mechanisms 69 5.3 Molecular and Physicochemical Properties Affecting Drug Absorption 69 5.3.1 Molecular Structure Features Affecting Drug Absorption 69

x Contents 5.3.2 The Physicochemical Drug Properties 70 5.3.2.1 Drug Solubility 70 5.3.2.2 Drug Dissolution Rate 72 5.3.3 Drug Stability 73 5.4 Physiological Factors Affecting Drug Absorption After Different Routes of Administration and Formulation Strategies to Accommodate These Factors 74 5.4.1 Parenteral Drug Administration 74 5.4.2 Oral Drug Administration 74 5.4.3 Rectal Drug Administration 77 5.4.4 Intranasal Drug Administration 78 5.4.5 Pulmonary Drug Administration 78 5.4.6 Transdermal Drug Administration 79 5.5 Integration of the Physical, Chemical, and Physiological Factors Affecting Drug Absorption 80 5.5.1 The Biopharmaceutics Classification System 81 5.5.2 The Biopharmaceutics Drug Disposition Classification System (BDDCS) 83 5.6 Summary 84 References 85

6 Drug Pharmacokinetics Following Single Oral Drug Administration: The Rate of Drug Absorption87 6.1 Introduction 87 6.2 Drug Absorption after Oral Administration 88 6.2.1 Zero-Order Drug Absorption 89 6.2.2 First-Order Drug Absorption 89 6.3 The Plasma Concentration-Time Profile After Single Oral Dose 91 6.4 Determination of the Absorption Rate Constant 94 6.4.1 The Method of Residuals 95 6.4.1.1 Lag Time 97 6.4.1.2 Flip-Flop of ka and k 98 6.4.2 Wagner-Nelson Method 100 6.4.2.1 Application of the Wagner-Nelson Method 101 6.5 Summary 104 References 107

Contents xi

7 Drug Pharmacokinetics Following Single Oral Drug Administration: The Extent of Drug Absorption108 7.1 Introduction 108 7.2 Causes of Incomplete Drug Bioavailability 109 7.2.1 The First-Pass Effect 109 7.2.2 The GIT Drug Transporters 110 7.2.3 Intestinal Drug Metabolism 111 7.3 The Rationale for Bioavailability Determination 112 7.4 Determination of the Drug In Vivo Bioavailability 113 7.4.1 Drug Bioavailability 113 7.4.1.1 Absolute Bioavailability 114 7.4.1.2 Relative Bioavailability 115 7.4.2 Calculation of the Drug Bioavailability 115 7.4.3 Determination of the Drug Bioavailability from Urinary Excretion Data 117 7.5 In Vivo Bioavailability Basic Study Design 119 7.6 Calculation of the AUC Using the Linear Trapezoidal Rule 120 7.7 The Effect of Changing the Pharmacokinetic Parameters on the Plasma Drug ConcentrationTime Profile after Single Oral Dose 124 7.7.1 Dose 125 7.7.2 Bioavailability 125 7.7.3 Total Body Clearance 126 7.7.4 Volume of Distribution 126 7.7.5 Absorption Rate Constant 127 7.8 Summary 127 References 131

8 Bioequivalence132 8.1 8.2 8.3 8.4

Introduction 132 General Definitions 133 Regulatory Requirement for Bioequivalence 133 Criteria for Requesting a Waiver of the In Vivo Bioequivalence Determination 134 8.5 Approaches for Demonstrating Product Bioequivalence 135 8.5.1 In Vivo Pharmacokinetic Studies 136 8.5.2 In Vitro Test Predictive of In Vivo Human Bioavailability 136

xii Contents 8.5.3 Acute Pharmacodynamic Effect 136 8.5.4 Comparative Clinical Studies 137 8.5.5 In Vitro Dissolution Testing 137 8.6 Pharmacokinetic Approach to Demonstrate Product Bioequivalence 138 8.6.1 Planning for the In Vivo Bioequivalence Study 138 8.6.2 Selection of the Reference Drug Product 138 8.6.3 In Vitro Testing of the Study Products 138 8.6.4 In Vivo Bioequivalence Study Design 139 8.6.4.1 Basic Principles 139 8.6.4.2 Ethical Approval 139 8.6.4.3 The Study Subjects 139 8.6.4.4 Number of Volunteers 140 8.6.4.5 Drug Administration 140 8.6.4.6 Experimental Protocol 140 8.6.4.7 Collection of Blood Samples 141 8.6.4.8 Analysis of Bioequivalence Study Samples 141 8.6.4.9 Pharmacokinetic Parameter Determination 142 8.6.4.10 Statistical Analysis 142 8.6.4.11 Documentation and Reporting 143 8.7 Special Issues Related to Bioequivalence Determination 143 8.7.1 Multiple-Dose Bioequivalence Studies 143 8.7.2 Food-Effect Bioequivalence Studies 144 8.7.3 Drugs with Long Half-Lives 144 8.7.4 Determination of Bioequivalence from the Drug Urinary Excretion Data 145 8.7.5 Fixed-Dose Combination 145 8.7.6 Measuring Drug Metabolites in Bioequivalence Studies 146 8.7.7 Highly Variable Drugs 146 8.7.8 Drugs Following Nonlinear Pharmacokinetics 147 8.7.9 Endogenous Substances 147 8.7.10 Enantiomers versus Racemates 147 8.7.11 Narrow Therapeutic Range Drugs 148 8.7.12 Oral Products Intended for the Local Effect of the Drug 148 8.7.13 First Point Cpmax 148 8.7.14 Biological Products 148 8.8 Summary 149 References 149

Contents xiii

9 Drug Pharmacokinetics during Constant Rate IV Infusion, the Steady-State Concept151 9.1 Introduction 151 9.2 The Steady State 152 9.3 The Time Required to Achieve Steady State 155 9.3.1 Changing the Drug Infusion Rate 156 9.4 Loading Dose 157 9.5 Termination of the Constant Rate IV Infusion 159 9.6 Determination of the Pharmacokinetic Parameters 159 9.6.1 Total Body Clearance 160 9.6.2 Elimination Rate Constant 160 9.6.3 Volume of Distribution 160 9.7 Dosage Forms with Zero-Order Input Rate 162 9.8 The Effect of Changing the Pharmacokinetic Parameters on the Plasma Drug Concentration-Time Profile during Constant Rate IV Infusion 163 9.8.1 Infusion Rate 163 9.8.2 Total Body Clearance 163 9.8.3 Volume of Distribution 164 9.8.4 Loading Dose 164 9.9 Summary 164

10 Steady State during Multiple Drug Administration167 10.1 Introduction 167 10.2 The Plasma Drug Concentration-Time Profile during Multiple Drug Administration 168 10.3 The Time Required to Achieve Steady State 172 10.4 The Loading Dose 173 10.4.1 IV Loading Dose 173 10.4.2 Oral Loading Dose 174 10.5 The Average Plasma Concentration at Steady State 175 10.6 Drug Accumulation 178 10.7 Controlled Release Formulations 179 10.8 The Effect of Changing the Pharmacokinetic Parameters on the Steady-State Plasma Drug Concentration during Multiple Drug Administration 180 10.8.1 Dosing Rate 180 10.8.2 Total Body Clearance 181 10.8.3 Volume of Distribution 181 10.8.4 Absorption Rate Constant 181

xiv Contents 10.9 Dosing Regimen Design 182 10.9.1 Factors to Be Considered 182 10.9.1.1 The Therapeutic Range of the Drug 182 10.9.1.2 The Required Onset of Effect 182 10.9.1.3 The Drug Product 182 10.9.1.4 Progression of the Patient Disease State 183 10.9.2 Estimation of the Patient Pharmacokinetic Parameters 183 10.9.3 Selection of Dose and Dosing Interval 183 10.9.3.1 Multiple Controlled Release Oral Formulation 184 10.9.3.2 Multiple IV or Fast-Release Oral Formulations 184 10.9.4 Selection of the Loading Dose 185 10.10 Summary 186

11 Renal Drug Excretion190 11.1 11.2 11.3 11.4

Introduction 190 Studying Drug Elimination through a Specific Pathway 191 The Renal Excretion of Drugs 192 Determination of the Drug Renal Excretion Rate 193 11.4.1 Experimental Determination of the Renal Excretion Rate 194 11.4.2 The Drug Renal Excretion Rate-Time Profile 195 11.5 The Renal Clearance 196 11.6 The Cumulative Amount of the Drug Excreted in Urine 199 11.7 Determination of the Pharmacokinetic Parameters from the Renal Excretion Rate Data 201 11.7.1 The Elimination Rate Constant and Half-Life 201 11.7.2 The Renal Excretion Rate Constant 201 11.7.3 The Volume of Distribution 201 11.7.4 The Renal Clearance 201 11.7.5 The Fraction of Dose Excreted Unchanged in Urine 201 11.7.6 Bioavailability 201 11.8 The Effect of Changing the Pharmacokinetic Parameters on the Urinary Excretion of Drugs 204 11.8.1 Dose 204 11.8.2 The Total Body Clearance 204 11.8.3 The Renal Clearance 205 11.9 Summary 205 References 207

Contents xv

12 Metabolite Pharmacokinetics208 12.1 Introduction 208 12.2 Drug Metabolism 209 12.2.1 Metabolizing Enzymes 209 12.2.2 Formation of Active Metabolites 210 12.2.3 Formation of Toxic Metabolites 210 12.2.4 Metabolic Activation of Prodrugs 211 12.3 Metabolite Pharmacokinetics 211 12.4 A Simple Model for Metabolite Pharmacokinetics 213 12.4.1 Metabolite Concentration-Time Profile 215 12.5 The General Model for Metabolite Kinetics 216 12.6 Determination of the Metabolite Pharmacokinetic Parameters 218 12.6.1 Metabolite Elimination Rate Constant, k(m) 218 12.6.2 Fraction of the Parent Drug Dose Converted to a Specific Metabolite, fm 218 12.6.3 Metabolite Clearance, CL(m) 219 12.6.4 Metabolite Volume of Distribution, Vd(m) 219 12.6.5 Metabolite Formation Clearance, fm CLT 219 12.7 Steady-State Metabolite Concentration during Repeated Administration of the Drug 222 12.8 Metabolite Pharmacokinetics after Extravascular Administration of the Parent Drug 225 12.9 Kinetics of Sequential Metabolism 226 12.10 The Effect of Changing the Pharmacokinetic Parameters on the Drug and Metabolite Concentration-Time Profiles after Single IV Drug Administration and during Multiple Drug Administration 227 12.10.1 Drug Dose 227 12.10.2 Drug Total Body Clearance 228 12.10.3 Drug Volume of Distribution 229 12.10.4 Fraction of the Drug Dose Converted to the Metabolite 229 12.10.5 Metabolite Total Body Clearance 230 12.10.6 Metabolite Volume of Distribution 230 12.11 Summary 231 References 234

13 Nonlinear Pharmacokinetics236 13.1 Introduction 236 13.2 Causes of Nonlinear Pharmacokinetics 236 13.2.1 Dose-Dependent Drug Absorption 237 13.2.2 Dose-Dependent Drug Distribution 238

xvi Contents 13.2.3 Dose-Dependent Renal Excretion 239 13.2.4 Dose-Dependent Drug Metabolism 239 13.2.5 Other Conditions That Can Lead to Nonlinear Pharmacokinetics 240 13.3 Pharmacokinetics of Drugs Eliminated by Dose-Dependent Metabolism, Michaelis-Menten Pharmacokinetics 240 13.3.1 Michaelis-Menten Enzyme Kinetics 240 13.3.2 The Pharmacokinetic Parameters 242 13.3.3 Drug Concentration-Time Profile after Administration of a Drug Which Is Eliminated by Single Metabolic Pathway That Follows Michaelis-Menten Kinetics 243 13.3.3.1 After Single IV Bolus Administration 243 13.3.3.2 During Multiple Drug Administration 245 13.4 Determination of the Pharmacokinetic Parameters for Drugs with Elimination Process that Follows Michaelis-Menten Kinetics 246 13.4.1 The Volume of Distribution 246 13.4.2 The Total Body Clearance 246 13.4.3 The Half-Life 246 13.5 Oral Administration of Drugs that Are Eliminated by a Process that Follows Michaelis-Menten Kinetics 247 13.6 Determination of the Michaelis-Menten Parameters and Calculation of the Appropriate Dosage Regimens 247 13.6.1 Mathematical Method 248 13.6.2 The Direct Linear Plot 249 13.6.3 The Linear Transformation Method 251 13.7 Multiple Elimination Pathways 252 13.8 The Effect of Changing the Pharmacokinetic Parameters on the Drug Concentration-Time Profile 253 13.8.1 The Dose 253 13.8.2 The Vmax 254 13.8.3 The Km 254 13.9 Summary 254 References 256

14 Multicompartment Pharmacokinetic Models258 14.1 14.2 14.3 14.4

Introduction 258 Compartmental Pharmacokinetic Models 260 The Two-Compartment Pharmacokinetic Model 260 The Parameters of the Two-Compartments Pharmacokinetic Model 263 14.4.1 Definition of the Pharmacokinetic Parameters 263 14.4.2 The Mathematical Equation That Describes the Plasma Concentration-Time Profile for Drugs That Follow Two-Compartment Pharmacokinetic Models 264

Contents xvii 14.5 Determination of the Two-Compartment Pharmacokinetic Model Parameters 266 14.5.1 The Method of Residuals 266 14.5.2 Determination of the Other Model Parameters 268 14.5.2.1 Volume of the Central Compartment, Vc 268 14.5.2.2 The Area under the Plasma Concentration-Time Curve, AUC 268 14.5.2.3 The Total Body Clearance, CLT 268 14.5.2.4 The First-Order Elimination Rate Constant from the Central Compartment, k10 268 14.5.2.5 The First-Order Transfer Rate Constant from the Peripheral Compartment to the Central Compartment, k 21 268 14.5.2.6 The First-Order Transfer Rate Constant from the Central Compartment to the Peripheral Compartment, k12 269 14.5.3 Determination of the Volumes of Distribution for the Two-Compartment Pharmacokinetic Model 269 14.5.3.1 The Volume of Distribution at Steady State, Vdss 269 14.5.3.2 The Volume of Distribution during the Elimination Phase, Vd β 270 14.6 Pharmacokinetic Behavior of Drugs that Follow the Two-Compartment Pharmacokinetic Model 272 14.6.1 Oral Administration of Drugs that Follow the Two-Compartment Pharmacokinetic Model 272 14.6.2 Constant Rate IV Administration of Drugs That Follow the Two-Compartment Pharmacokinetic Model 273 14.6.3 Multiple Administration of Drugs That Follow the Two-Compartment Pharmacokinetic Model 273 14.6.4 Renal Excretion of Drugs That Follow the Two-Compartment Pharmacokinetic Model 274 14.7 Effect of Changing the Pharmacokinetic Parameters on the Concentration-Time Profile of Drugs That Follow Two-Compartment Pharmacokinetic Model 274 14.7.1 Dose 275 14.7.2 Total Body Clearance 275 14.7.3 Volume of the Central Compartment 275 14.7.4 The Hybrid Distribution and Elimination Rate Constants 275 14.7.5 The Inter-Compartmental Clearance 275

xviii Contents 14.8 The Three-Compartment Pharmacokinetic Model 276 14.9 Compartmental Pharmacokinetic Data Analysis 278 14.9.1 Construction of the Compartmental Model 278 14.9.2 Mathematical Description of the Model 278 14.9.3 Fitting the Model Equation to the Experimental Data 279 14.9.4 Evaluation of the Pharmacokinetic Model 280 14.10 Summary 282 References 284

15 Drug Pharmacokinetics Following Administration by Intermittent Intravenous Infusions286 15.1 Introduction 286 15.2 The Drug Concentration-Time Profile after Administration by Intermittent IV Infusions 287 15.2.1 After the First Dose 288 15.2.2 After Repeated Administration Before Reaching Steady State 289 15.2.3 At Steady State 290 15.3 The Effect of Changing the Pharmacokinetic Parameters on the Steady-State Plasma Concentration during Repeated Intermittent IV Infusions 291 15.3.1 Dose 291 15.3.2 Infusion Time 291 15.3.3 Total Body Clearance 291 15.3.4 Volume of Distribution 291 15.4 Application of the Pharmacokinetic Principles for Intermittent IV Infusion in Clinical Practice 292 15.4.1 Pharmacokinetic Characteristics of Aminoglycosides 292 15.4.2 Guidelines for Aminoglycoside Plasma Concentration 292 15.4.3 The Extended-Interval Aminoglycoside Dosing Regimen 293 15.5 Individualization of Aminoglycoside Therapy 293 15.5.1 Estimation of the Patient Pharmacokinetic Parameters 294 15.5.1.1 Estimation of the Patient Pharmacokinetic Parameters Based on the Patient Information 294 15.5.1.2 Estimation of the Patient’s Specific Pharmacokinetic Parameters from Aminoglycoside Blood Concentrations 295

Contents xix 15.5.2 Determination of the Dosing Regimen Based on the Patient’s Specific Parameters 298 15.5.2.1 Selection of the Dosing Interval (τ) 299 15.5.2.2 Selection of Dose 299 15.5.2.3 Selection of the Loading Dose 299 15.6 Summary 303 References 305

16 Physiological Approach to Hepatic Clearance306 16.1 16.2 16.3 16.4 16.5 16.6

Introduction 306 The Organ Clearance 306 Hepatic Extraction Ratio 307 Intrinsic Clearance 308 Systemic Bioavailability 308 The Effect of Changing Intrinsic Clearance and Hepatic Blood Flow on the Hepatic Clearance, Systemic Availability, and Drug ConcentrationTime Profile 309 16.6.1 Low Extraction Ratio Drugs 310 16.6.1.1 Assume that the Drug CLint Increases to Double Its Original Value Due to Enzyme Induction and Q Stays the Same 310 16.6.1.2 Assume that Q Decreases by 50% (i.e., New Q = 0.75 L/min) without Affecting CLint 311 16.6.2 High Extraction Ratio Drugs 312 16.6.2.1 Assume that the Drug CLint Increases to Double Its Original Value Due to Enzyme Induction and Q Stays the Same 313 16.6.2.2 Assume that Q Decreases by 50% (New Q = 0.75 L/min) without Affecting CLint 314 16.7 Protein Binding and Hepatic Extraction 316 16.8 Summary 317 References 318

17 Pharmacokinetics in Patients with Eliminating Organ Dysfunction319 17.1 Introduction 319 17.2 Patients with Renal Dysfunction 319 17.2.1 Dosing Regimens in Renal Dysfunction Patients Based on the Creatinine Clearance 320 17.2.2 A General Approach for Calculation of Dosing Regimens in Renal Dysfunction Patients 321

xx Contents 17.3 Patients Receiving Renal Replacement Therapy 324 17.3.1 The Principle of Dialysis 324 17.3.2 Factors Affecting the Drug Clearance during Dialysis 325 17.3.3 Dose Adjustment during Dialysis 325 17.4 Patients with Hepatic Insufficiency 325 17.4.1 Pharmacokinetic and Pharmacodynamic Changes in Hepatic Dysfunction 326 17.4.2 Dose Adjustment in Hepatic Dysfunction 326 17.5 Other Patient Populations 328 17.6 Summary 328 References 330

18 Noncompartmental Approach in Pharmacokinetic Data Analysis332 18.1 Introduction 332 18.2 The Principles of Noncompartmental Data Analysis Method 333 18.3 The Mean Residence Time after IV Bolus Administration 333 18.3.1 Calculation of the AUC 334 18.3.2 Calculation of the AUMC 334 18.4 The MRT after Different Routes of Administration 337 18.4.1 The MRT after Extravascular Administration 337 18.4.2 The MRT after Constant Rate IV Infusion 338 18.5 Other Pharmacokinetic Parameters that Can Be Determined Using the Noncompartmental Approach 339 18.6 Determination of the MRT for Compartmental Models 340 18.7 Summary 341 References 342

19 Pharmacokinetic-Pharmacodynamic Modeling344 19.1 Introduction 344 19.2 Pharmacokinetic-Pharmacodynamic Modeling 344 19.2.1 The Pharmacokinetic Model 345 19.2.2 Measuring the Response 345 19.2.3 The Pharmacodynamic Model 346 19.2.3.1 The Fixed Effect Model 346 19.2.3.2 The Linear Model 346 19.2.3.3 The Log-Linear Model 348 19.2.3.4 The Emax Model 348 19.2.3.5 The Sigmoid Emax Model 349

Contents xxi 19.3 Integrating the Pharmacokinetic and Pharmacodynamic Models 350 19.3.1 Direct Response versus Indirect Response 350 19.3.2 Direct Link versus Indirect Link 351 19.3.3 Time-Variant versus Time-Invariant 351 19.4 Direct Link PK/PD Models for Drugs with Direct Response 351 19.5 Indirect Link PK/PD Models for Drugs with Direct Response 352 19.5.1 The Effect Compartment Approach 353 19.6 PK/PD Models for Drugs with Indirect Response 355 19.7 Other PK/PD Models 357 19.8 The PK/PD Modeling Process 357 19.8.1 Stating the Objectives, Proposing the Model and Designing the Study 358 19.8.2 Initial Data Exploration and Data Transformation 358 19.8.3 Refining and Evaluation of the PK/PD Model 358 19.8.4 Validation of the PK/PD Model 358 19.9 Applications of the PK/PD Modeling in Drug Development and Clinical Use of Drugs 359 19.10 Summary 359 References 360

20 Pharmacogenetics: The Genetic Basis of Pharmacokinetic and Pharmacodynamic Variability361 20.1 Introduction 361 20.2 Gene Structure 362 20.3 Genetic Background Information 362 20.3.1 Gene Variants, Alleles 362 20.3.2 Polymorphisms 363 20.3.3 Gene Nomenclature 363 20.3.4 Genotype versus Phenotype 363 20.3.5 Monogenic versus Polygenic 364 20.3.6 Homozygous versus Heterozygous Genotype 364 20.4 Genetic Polymorphism in Pharmacokinetics 365 20.4.1 Cytochrome P450 Enzymes 365 20.4.2 Thiopurine Methyltransferase (TPMT) 367 20.4.3 N-acetyltransferase 367 20.4.4 UDP-Glucuronosyltransferase (UGT) 368 20.4.5 Drug Transporters 368

xxii Contents 20.5 Genetic Polymorphism in Pharmacodynamics 369 20.6 Implementation of Pharmacogenetic Testing in Clinical Practice 370 20.6.1 Pharmacogenetic Training for Healthcare Providers 371 20.6.2 The Pharmacogenetic Tests 371 20.6.3 Interpretation of the Pharmacogenetic Test Results 371 20.6.4 Guidelines for Applying the Pharmacogenetic Testing 372 20.6.5 Enablers for the Implementation of Pharmacogenetics in Clinical Practice 372 20.7 Summary 373 References 373

21 Therapeutic Drug Monitoring375 21.1 Introduction 375 21.2 General Principles of Initiation and Management of Drug Therapy 376 21.2.1 The Use of Therapeutic Drug Monitoring in the Management of Drug Therapy 377 21.3 Drug Blood Concentration versus Drug Dose 378 21.4 The Therapeutic Range 379 21.5 Drug Candidates for Therapeutic Drug Monitoring 380 21.5.1 Drugs with Low Therapeutic Index 380 21.5.2 Drugs with Large Variability in Their Pharmacokinetic Behavior 381 21.5.3 Drugs Used in High-Risk Patients or Patients with Multiple Medical Problems 381 21.6 Determination of the Drug Concentration in Biological Samples 381 21.6.1 The Biological Samples 381 21.6.2 The Time of Sample 382 21.6.3 The Measured Drug Moiety 382 21.6.4 The Analytical Technique 384 21.7 Establishing a Therapeutic Drug Monitoring (Clinical Pharmacokinetic) Service 384 21.7.1 Major Requirements 385 21.7.2 Dosage Regimen Recommendation 385 21.7.2.1 Determination of the Initial Dosing Regimen 385 21.7.2.2 Determination of the Patient’s Specific Pharmacokinetic Parameters 385

Contents xxiii 21.7.2.3 Calculation of the Dosage Requirements Based on the Patient’s Specific Pharmacokinetic Parameters of the Drug 386 21.7.3 The Pharmacoeconomics of Therapeutic Drug Monitoring 386 21.8 Summary 386 References 387

22 Pharmacometric Applications in Drug Development and Individualization of Drug Therapy388 22.1 Introduction 388 22.2 Pharmacometric Applications during the Preclinical Phase of Drug Development 389 22.2.1 Physiologically Based Pharmacokinetic Models 389 22.2.1.1 Physiologically Based Pharmacokinetic Model Development 390 22.2.1.2 Applications of the PBPK Models 395 22.3 Pharmacometric Applications during the Clinical Phases of Drug Development 397 22.3.1 Population Pharmacokinetic Analysis 397 22.3.1.1 Data Consideration for Population Analysis 398 22.3.1.2 The Population Pharmacokinetic Models 398 22.3.1.3 Statistical Analysis and Parameter Estimation 399 22.3.1.4 Model Evaluation and Diagnostics 400 22.3.1.5 Reporting of the Population Pharmacokinetic Analysis Results 400 22.3.1.6 Application of Population Pharmacokinetic Analysis in Drug Development 400 22.3.1.7 Application of Population Pharmacokinetic Analysis for Drug Use Decisions in Drug Labeling 401 22.4 Pharmacometric Applications in Clinical Drug Use 402 22.4.1 Model-Based Therapeutic Drug Monitoring 402 22.4.1.1 Model Development 402 22.4.1.2 Monitoring Drug Concentration 403 22.4.1.3 Dosage Regimen Design 403 22.5 Summary 403 References 404

xxiv Contents

23 Answer for the Practice Problems

406

Chapter 1 406 Chapter 2 406 Chapter 3 406 Chapter 4 407 Chapter 6 409 Chapter 7 410 Chapter 9 411 Chapter 10 412 Chapter 11 414 Chapter 12 414 Chapter 13 415 Chapter 14 416 Chapter 15 417 Chapter 16 418 Chapter 17 419 Chapter 18 420

Glossary Index

421 425

Figures

1.1 1.2 1.3

1.4 1.5 1.6 1.7 1.8 2.1

2.2

2.3

2.4 2.5

A representative example of the blood drug concentration-time profile after single oral dose of a drug 7 Representative examples of compartmental pharmacokinetic models. (A) one-compartment model, (B) two-compartment model, and (C) three-compartment model 10 A representing example of a physiologically based pharmacokinetic model which includes the heart, brain, intestine, liver, kidney, and other tissues. The term Q represents the blood flow, and the subscript indicates the organ 11 Simulation of the blood concentration-time profiles after single oral drug administration when the elimination rate constant is different 13 An example of (A) the Cartesian (linear) scale and (B) the semilog scale14 The data plotted on the semilog scale 16 Estimation of the slope and y-intercept of a straight line graphically on the linear (A) and semilog (B) scales 17 The least squares method determines the line that can minimize the sum of the squared distance between the experimental data and the line18 Adding an amount of a drug to a beaker with coated wall that can adsorb the drug (A) produces drug concentration in the liquid similar to the concentration produced if the same amount of the drug is added to a beaker with larger volume (B) 23 The plasma drug concentration-time profile (A) when different doses of the same drug are administered to the same individual, and (B) when the same dose of a drug is administered to different individuals who have different Vd 24 Diagrams representing (A) one-compartment pharmacokinetic model, where the drug distributes rapidly to all parts of the body, and (B) two-compartment pharmacokinetic model, where the drug distributes rapidly to the organs forming the central compartment and slowly to the organs forming the peripheral compartment 26 A hypothetical example of the drug concentration-time profiles in different organs on the linear scale for a drug that follows the one-compartment pharmacokinetic model and eliminated by first-order process 27 A hypothetical example of the drug concentration-time profiles in different organs on the semilog scale for a drug that follows the two-compartment pharmacokinetic model and eliminated by first-order process 27

xxvi Figures 2.6 3.1 3.2 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 5.1 5.2 5.3 5.4 5.5 5.6 5.7

A diagram representing the distribution of the drug between the plasma and the tissues28 Higher clearance does not always mean faster elimination, because the rate of decline in plasma drug concentration depends on both CLT and Vd. Drug B has lower clearance and is eliminated faster than Drug A37 The decrease in the function of the eliminating organ of a drug results in slower rate of decline in plasma drug concentration, i.e., slower rate of elimination38 A plot of (A) the amount of the drug in the body and (B) the plasma drug concentration versus time on the linear scale when the drug elimination follows zero-order kinetics 43 A plot of the plasma drug concentration versus time on the linear scale44 The half-life of the drugs eliminated by zero-order processes is concentration dependent45 A plot of the natural logarithm of the plasma drug concentration versus time on the linear scale when the elimination of the drug follows first-order kinetics47 A plot of the logarithm of the plasma drug concentration versus time on the linear scale when the elimination of the drug follows first-order kinetics48 A plot of the plasma drug concentration versus time on the linear scale when the elimination of the drug follows first-order kinetics48 A plot of the plasma drug concentration versus time on the semilog scale when the elimination of the drug follows first-order kinetics48 Drugs with larger first-order elimination rate constant are eliminated at a faster rate49 A plot of the plasma drug concentration versus time on the semilog scale50 The half-life of the drug is independent of the drug concentration when the drug elimination process follows first-order kinetics51 Graphical determination of the half-life52 Plasma concentration-time profiles for different drugs with different half-lives52 Plasma concentration-time profiles (A) after administration of different doses of the same drug and (B) after administration of the same dose of drugs that have different clearances 55 Calculation of the pharmacokinetic parameters after single IV drug administration56 A plot of the plasma concentration-time curve57 A diagram showing the basic structure of the cellular membrane66 The common mechanisms for drug transport across biological membranes67 A model that describes the dissolution of drug from a spherical particle72 The different structure features that increase the surface area of the drug absorption surface in the small intestine76 Schematic presentation of the different skin layers showing the different drug transport mechanisms across the skin79 The biopharmaceutical classification system as described by Amidon et al based on the drug solubility and permeability81 The biopharmaceutical drug disposition classification system as described by Wu and Benet based on the drug solubility and extent of drug metabolism83

Figures xxvii 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8

6.9 6.10 6.11 6.12 6.13 6.14 7.1 7.2 7.3 7.4 7.5 7.6 8.1 9.1

The relationship between the plasma concentration-time profile and the onset, intensity, and duration of the drug effect88 The processes involved in the absorption of drugs administered as solid dosage forms88 The drug amount-time profile in the GIT and in the body after administration of single oral dose of a drug that follows first-order absorption and elimination89 A diagram representing the pharmacokinetic model that describes the drug absorption from the GIT90 The plasma drug concentration-time profile after administration of a single oral dose of the drug92 The plasma drug concentration-time profile after administration of single oral dose of the drug plotted on the semilog scale93 The plasma drug concentration-time profile during the elimination phase after administration of a single oral dose declines at a rate dependent on the first-order elimination rate constant95 The method of residuals: The residuals versus time plot on the semilog scale (the dashed line) declines at a rate dependent on the first-order absorption rate constant. The first-order absorption rate constant is calculated from the slope of this plot, slope = −ka/2.30396 The method of residuals when there is lag time for drug absorption after oral administration 97 Simulation of the plasma drug concentration-time profile of A drug when administered in the form of two different formulations (A) ka > k and (B) ka < k 98 The method of residuals to calculate the first-order absorption rate constant 99 The fraction of dose remaining to be absorbed versus time plots, (A) when the drug absorption follows first-order kinetics, and (B) when the drug absorption follows zero-order kinetics 101 A plot of the fraction of dose remaining to be absorbed versus time on the semilog scale 103 The fraction of dose remaining to be absorbed versus time is not linear on the linear scale (A) and is linear on the semilog scale (B) 104 A diagram illustrating the different causes of drug loss during the absorption process after oral administration110 A typical plasma concentration-time profile after administration of a single oral dose of the drug114 The plasma concentration-time profile after administration of the three products for the same drug114 The plasma drug concentration-time profiles after a single administration of the same dose of a drug from different drug products that have different absolute bioavailability119 Calculation of the area of a trapezoid120 The AUC is the sum of the area of all trapezoids and the area under the tail of the curve121 A diagram illustrating the crossover experimental design for the bioequivalence study141 The drug plasma concentration-time profile after starting the constant rate IV drug infusion152

xxviii Figures 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 11.1 11.2 11.3 11.4

The steady-state drug concentration is proportional to the infusion rate if the total body clearance is the same153 The steady-state drug concentration is inversely proportional to the total body clearance of the drug if the drug is administered at the same infusion rate154 The time required to achieve steady state during constant rate IV infusion is dependent on the half-life of the drug156 The change in the IV infusion rate results in changing the plasma drug concentration until the new steady state is achieved156 The plasma drug concentration-time profile after administration of a bolus IV loading dose and constant rate IV infusion simultaneously157 The plasma drug concentration-time profile after administration of different loading doses followed immediately by the same continuous IV infusion rate of the drug158 The plasma drug concentration-time profile during and after termination of the continuous IV infusion of the drug159 A plot of the plasma drug concentrations obtained after termination of the infusion versus time on the semilog scale161 Plasma drug concentration-time profile after administration of a dosage form that releases the drug at zero-order rate every 8 hr to achieve relatively constant plasma concentration162 The plasma drug concentration-time profile during multiple IV administration of a fixed dose of the drug every fixed dosing interval168 The plasma drug concentration during multiple administration of a rapidly absorbed drug170 The plasma drug concentration during repeated administration of different drugs that have different half-lives172 Administration of different doses of the same drug will achieve steady-state concentrations that are proportional to the administered dose173 The plasma drug concentration-time profiles after administration of different loading doses followed by the same maintenance dose174 The steady-state plasma drug concentration fluctuates around the average plasma drug concentration during multiple drug administration175 The AUC from time zero to ∞ after administration of single dose of the drug is equal to the AUC from time zero to τ at steady state during multiple administration of the same dose of the drug177 The plasma concentration-time profile after multiple administration of the same dose of immediate release and controlled release oral formulations180 The diagram represents a drug that is eliminated by renal and nonrenal elimination pathways194 The slope of the drug renal excretion rate versus time plot on the semilog scale is equal to −k/2.303 and the y-intercept is equal to ke dose196 The drug renal excretion rate versus plasma drug concentration at the middle of the urine collection interval plot198 The cumulative amount of the drug excreted in urine after single IV dose increases exponentially until it reaches a plateau199

Figures xxix 11.5 The half-life and the elimination rate constant are determined from the renal excretion rate versus time plot203 11.6 The renal excretion rate versus drug plasma concentration at the middle of the urine collection interval203 12.1 The plasma drug and metabolites concentration-time profiles after single IV administration of the drug, which is metabolized to two different metabolites212 12.2 Schematic presentation of the parallel and sequential drug metabolism212 12.3 Schematic presentation of a simple pharmacokinetic model for drug metabolism213 12.4 The plasma drug and metabolite concentration-time profiles after IV administration of single dose of the drug215 12.5 The plasma drug and metabolite concentration-time profiles after IV administration of single dose of the drug when the metabolite elimination rate constant is smaller than the drug elimination rate constant216 12.6 The drug and metabolite concentration-time profiles after IV administration of single dose of the drug when the metabolite elimination rate constant is larger than the drug elimination rate constant216 12.7 Schematic presentation of the general model for drug metabolism217 12.8 Schematic presentation for the different pathways involved in the elimination of the drug and its two metabolites in the practice problem223 12.9 The drug and metabolite concentration-time profiles after oral administration of single dose of the drug226 12.10 The drug and metabolites concentration-time profiles after IV administration of single dose of a drug that is metabolized to metabolite 1, which is then metabolized to metabolite 2227 12.11 Schematic presentation for the different pathways involved in the elimination of the parent drug and its two metabolites in problem 12.2232 13.1 The relationship between the administered dose and AUC or steady-state concentration for drugs that follow linear and nonlinear pharmacokinetics237 13.2 The relationship between the drug metabolic rate and the drug concentration241 13.3 Enzyme induction causes increase in the Vmax without affecting Km242 13.4 Competitive inhibition of the metabolizing enzyme increases Km without affecting Vmax243 13.5 Schematic presentation of the pharmacokinetic model for single IV bolus dose of a drug which is eliminated by single metabolic pathway that follows Michaelis-Menten kinetics244 13.6 The plasma drug concentration-time profile after single IV bolus dose of a drug, which is eliminated by single metabolic pathway that follows Michaelis-Menten kinetics244 13.7 The relationship between the steady-state drug concentration and the dosing rate during multiple administration of a drug when the drug elimination process follows Michaelis-Menten kinetics245

xxx Figures 13.8 The direct linear plot249 13.9 Estimation of the Vmax, Km, and the dose from the direct linear plot250 13.10 A plot of the dosing rate versus the dosing rate divided by the steady-state drug concentration251 13.11 Schematic presentation of (A) the pharmacokinetic model for a drug eliminated by two parallel saturable elimination pathways that follow Michaelis-Menten kinetics252 14.1 A model representing the distribution of the dye from the liquid in the center of the beaker to the beaker wall coating material259 14.2 The dye concentration-time profile in the liquid with the decrease in the dye concentration representing the slow distribution of the dye from the liquid in the center of the beaker to the beaker wall coating material259 14.3 The diagram represents different compartmental pharmacokinetic models261 14.4 The plasma concentration-time profile of a drug that follows the two-compartment pharmacokinetic model after single IV bolus administration262 14.5 The drug concentration-time profile in the central and peripheral compartments after administration of single IV bolus dose of a drug that follows two-compartment pharmacokinetic model262 14.6 A block diagram that represents the two-compartment pharmacokinetic model with first-order transfer between the central and peripheral compartments and first-order drug elimination from the central compartment263 14.7 The method of residuals is applied to separate the two exponential terms of the equation that describes the plasma concentration-time profile of drugs that follow the two-compartment pharmacokinetic model after single IV dose267 14.8 Application of the method of residuals in solving the practice problem271 14.9 The plasma concentration-time profile for a drug that follows two-compartment pharmacokinetic model after administration of single oral dose273 14.10 The plasma concentration-time profile for a drug that follows threecompartment pharmacokinetic model after administration of single IV bolus dose277 14.11 A block diagram representing the three-compartment pharmacokinetic model with the two peripheral compartments connected to the central compartment and drug elimination from the central compartment 277 14.12 Examples of the diagnostic graphs for compartmental pharmacokinetic data analysis 281 14.13 Examples of the residual plots used as diagnostic tests for evaluating curve fitting 282 15.1 The plasma drug concentration-time profile during repeated administration of intermittent IV infusions287 15.2 The plasma drug concentration-time profile during and after administration of the first dose of a drug by constant rate IV infusion over a period of t′ on the semilog scale289 15.3 The plasma concentration-time profile during repeated administration of intermittent IV infusion before reaching steady state289

Figures xxxi 15.4 The plasma concentration-time profile during repeated administration of intermittent IV infusion at steady state290 15.5 Graphical determination of aminoglycoside half-life and the concentration at the end of drug administration from the plasma concentration-time profile after administration of the first dose as short IV infusion of duration t′296 15.6 Graphical determination of aminoglycoside half-life and concentration at the end of drug administration from the drug concentrations versus time plot after the end of drug administration296 15.7 The plasma samples obtained for the determination of the pharmacokinetic parameters during repeated intermittent IV infusion before reaching steady state297 15.8 The plasma samples obtained for the determination of the pharmacokinetic parameters during repeated intermittent IV infusion at steady state298 15.9 Graphical determination of the drug half-life and drug concentration at the end of drug administration from a plot of the drug concentrations versus time after the end of drug administration300 16.1 The plasma concentration-time profile for a low extraction ratio drug after administration of (A) single IV bolus dose and (B) single oral dose, before (_______) and after (------) enzyme induction 311 16.2 The plasma concentration-time profile for a low extraction ratio drug after administration of (A) single IV bolus dose and (B) single oral dose, when the hepatic blood flow is 1.5 L/min (_______) and after the reduction of the hepatic blood flow to 0.75 L/min (------) 312 16.3 The plasma concentration-time profile for a high extraction ratio drug after administration of (A) single IV bolus dose and (B) single oral dose, before (_______) and after (------) Enzyme induction 313 16.4 The plasma concentration-time profile for a high extraction ratio drug after administration of (A) a single IV bolus dose and (B) a single oral dose, when the hepatic blood flow is 1.5 L/min (_______) and after the reduction of the hepatic blood flow to 0.75 L/min (------) 314 17.1 Schematic presentation of the principle of hemodialysis324 19.1 Schematic presentation of the relationship between pharmacokinetics and pharmacodynamics345 19.2 The drug concentration-effect relationship for a drug that follows the linear pharmacodynamic model347 19.3 The drug concentration-effect relationship for a drug that follows the linear pharmacodynamic model in presence of a baseline effect347 19.4 The drug concentration-effect relationship for a drug that follows the log-linear pharmacodynamic model348 19.5 The drug concentration-effect relationship for a drug that follows the Emax pharmacodynamic model349 19.6 The drug concentration-effect relationship for a drug that follows the sigmoid Emax pharmacodynamic model 350 19.7 A plot of the plasma drug concentration versus drug effect with the arrows representing the chronological order of the data points 353 19.8 A diagram representing one-compartment pharmacokinetic model linked to an effect compartment354

xxxii Figures 19.9 Diagrams representing two-compartment pharmacokinetic model linked to an effect compartment355 19.10 Schematic presentation of the four basic indirect response models and the differential equations that describe the rate of change in the drug response with respect to time for each model356 21.1 The basic steps in the process of initiation and management of drug therapy376 21.2 A schematic presentation of the relationship among the dosage regimen, the blood concentration, the concentration at the site of action, and the drug effect378 21.3 The therapeutic range is chosen to produce the highest probability of the therapeutic effect and the lowest probability of toxicity379 22.1 A representative example of the whole-body PBPK models consists of nine different compartments 390 22.2 Sections of the whole-body PBPK models that demonstrate the presentation of (A) lung inhalation and excretion of drugs, (B) enterohepatic recycling, and (C) renal and hepatic drug elimination 391

1

Introduction to Pharmacokinetics

Objectives After completing this chapter, you should be able to:

• • • • • •

Compare the scope of knowledge covered in pharmacokinetics and its related fields. Discuss the different applications of pharmacokinetics in the biomedical fields. List the major differences between linear and nonlinear pharmacokinetics. Describe the general approaches utilized in pharmacokinetic modeling. Explain the rationale for pharmacokinetic-pharmacodynamic modeling. Discuss how pharmacokinetic simulations can be used to demonstrate the basic pharmacokinetic concepts. • Present the pharmacokinetic results using graphical method. • Describe the basic mathematical and statistical principles utilized in pharmacokinetics. 1.1 Introduction Pharmacokinetics is the field of science that deals with the kinetics of drug absorption, distribution, metabolism, and elimination (ADME) processes. Together with its related fields, pharmacokinetics has numerous applications from the early stages of new drug development to the clinical use of drugs. Potential new drug candidates should be adequality absorbed into the systemic circulation from their site of administration, distributed in sufficient quantities to the site of action, and maintain effective concentrations for enough time to sustain drug effect when the drug is administered repeatedly. During the early phases of drug development, pharmacokinetic studies in animal species are used to predict the drug pharmacokinetic behavior in humans. The results of these animal studies are used to estimate the appropriate human dose of this candidate drug to be used in the clinical phases of drug development. As part of the clinical trials, the drug concentration-effect relationship and the range of drug concentrations in systemic circulation that are associated with the optimal therapeutic effect with minimum toxicity in humans are determined. Also, the factors that affect the drug pharmacokinetic behavior in different patient populations are identified and utilized for dosage recommendation in different patient populations to ensure optimal therapeutic outcome in all patients. During clinical use, the blood drug concentrations can be used as a guide for individualization of drug therapy and to guard against toxicity when using drugs with narrow therapeutic window.

DOI: 10.4324/9781003161523-1

2  Introduction to Pharmacokinetics 1.2  Pharmacokinetics and Its Related Fields Pharmacokinetics is the field of science that deals with the kinetics of drug absorption, distribution, metabolism, and excretion processes. These processes usually determine the drug concentration-time profile in the body, and hence the time course of the drug effect after administration. Several other related disciplines deal with these processes from different perspectives to characterize the drug dose-concentration-effect relationship. 1.2.1 Biopharmaceutics

It is the field that involves using the drug physicochemical properties, and the biological, physiological, and pathological characteristics of the human body to formulate appropriate drug dosage forms. These dosage forms are designed to provide the maximum drug availability at the site of drug action. The drug physicochemical properties determine the drug solubility, stability, membrane permeability, and affinity to different tissue components. Optimal formulation of compounds with balanced hydrophilic and lipophilic properties helps to achieve adequate drug dissolution in biological fluids and absorption through biological membranes. Also, the stability of drugs at the different sites of drug administration is important in determining the possible routes of drug administration and the selection of the appropriate dosage form for each route of administration. Furthermore, specialized dosage forms can be designed to control the rate of drug release from the dosage form. Controlled release dosage forms can affect the time course of the drug concentration in all parts of the body, including the site of action which is an important factor in determining the time course of the drug effect. Additionally, targeted drug delivery systems have been developed to preferentially increase the drug distribution to the intended site of drug effect which improves the therapeutic outcome and lowers the drug adverse effects. So, integration of the drug properties, and the physiological features of the different sites of drug administration with the characteristics of the different dosage forms, is important to achieve the best therapeutic outcome from using the drugs. 1.2.2 Pharmacokinetics

It is the field that involves quantitative investigation of the kinetics of drug absorption, distribution, metabolism, and excretion, including the rate and extent of each of these processes. After drug administration, these processes are the main determinants of the time course of the drug profile in the body, which determines the time course of the drug effect. The pharmacokinetic principles are usually applied to the disposition of drugs; however, the same principles can be applied to any compound such as nutrients, hormones, toxins, pollutants, pesticides, and others. Knowledge about the rate of drug absorption is important because faster drug absorption leads to faster onset of drug effect, which is essential in the treatment of acute conditions and in emergency situations. However, during multiple drug administration for the management of chronic diseases such as hypertension or diabetes, the use of drug products with slow rate drug absorption may be more appropriate. This is because slow drug absorption causes small fluctuation in the blood drug concentrations leading to steady therapeutic effect. Also, the extent of drug absorption is important because the effective drug amount is not the drug content of the dosage form, but it is the amount of drug absorbed into the systemic circulation.

Introduction to Pharmacokinetics 3 Drug distribution to the site of action is necessary for the drug to produce its desired effect. The rate of drug distribution to its site of action affects the onset of effect and the extent of distribution determines the intensity of the effect. Also, drug distribution to tissues other than the site of action of the drug can cause adverse drug effects. For example, an antibiotic, which is very effective in the eradication of the bacteria causing meningitis in vitro, can be useful in the treatment of meningitis only if it is distributed to the central nervous system (CNS) in enough quantities to achieve concentrations above the minimum inhibitory concentration of the causative organism. Also, an anticancer drug is effective only when it is distributed in enough quantities into the cancerous tissues. Furthermore, flue and cold medicine containing antihistaminic drugs that distribute to the CNS produce dizziness more than those that do not distribute to the CNS. The rate of drug elimination from the body is important in determining the duration of drug effect after a single drug administration, and the required frequency of drug administration during multiple drug administration. Drugs that are eliminated faster will have shorter duration of effect after single administration because the drug concentration will decline below the minimum effective concentration faster. While during multiple administration of rapidly eliminated drugs, more frequent administration will be required to maintain effective drug concentrations all the time. Also, the pathways involved in drug elimination are important in determining the required dose adjustment in patients with compromised function of the eliminating organ. For example, patients with reduced kidney function require lower doses of drugs that are mainly eliminated by the kidney, while patients with liver diseases usually require lower doses of the drugs that are mainly eliminated by hepatic metabolism. The above-mentioned processes influence the drug pharmacokinetic characteristics that determine the drug concentration-time profile in the body after drug administration. Each of these processes is associated with one or more parameters that are usually dependent on the drug, the patient, and some parameters are dependent on the drug product characteristics. These are the pharmacokinetic parameters that determine the rate and the extent of ADME. Different drugs have different pharmacokinetic parameters in the same patient population. Also, there is usually some random variation in the pharmacokinetic parameters of the same drug in the same population, but the variation in the pharmacokinetic parameters of the same drug can be large in different patient populations. Furthermore, administration of the same drug to the same patient population using different drug products or different dosage forms may produce different drug concentration-time profiles due to variation in pharmacokinetic parameters. The study of pharmacokinetics usually involves determination of the drug pharmacokinetic parameters and the factors that can affect these parameters to predict the pharmacokinetic behavior of drugs under different conditions. 1.2.3  Clinical Pharmacokinetics

It is the field that applies the basic pharmacokinetic principles in clinical practice. The optimal dosage regimen is selected based on the patient’s specific drug pharmacokinetic parameters to produce the maximum therapeutic effect with minimum adverse effects. The patient’s specific characteristics, such as age, weight, gender, kidney function, liver function, diseases, hydration state, concomitant drug use, and any other factors that can affect the pharmacokinetics of the drug are used to predict the drug pharmacokinetic parameters in that particular patient. Then the basic pharmacokinetic principles are used to

4  Introduction to Pharmacokinetics calculate the initial dosage regimen that should achieve therapeutic drug concentrations for each patient based on the patient’s specific information. When the patient starts taking the drug, determination of the blood drug concentrations can be utilized to determine the patient’s specific pharmacokinetic parameters. The optimal dosage regimen for that individual patient can be calculated based on the patient’s specific pharmacokinetic parameters. If the patient continues taking the drug, the blood drug concentration together with the clinical monitoring parameters are used to modify the dosage regimen whenever it is necessary. 1.2.4 Pharmacodynamics

It is the field that deals with the quantitative relationship between the drug concentration at the site of action and the drug effect. The existence of a relationship between the drug concentration in the body and the resulting drug therapeutic and toxic effects is obvious; however, this relationship can be different for different drugs. This is because different drugs produce their effects by different mechanisms. The effect of some drugs results from direct interaction between the drug and its target which can be a receptor, an enzyme, or a specific tissue component. The intensity of effect is not always proportional to the drug concentration at the site of action since maximum drug effect is achieved when all available receptors or enzymes are affected by the drug. Also, the rate at which the equilibrium between the drug in the systemic circulation and the drug at its site of action is achieved determines the relationship between the drug blood concentration-time profile and the drug effect-time profile. Furthermore, some drugs produce their therapeutic effect by initiation of a sequence of events that leads to the therapeutic effect. In this case, the drug effect may appear after the drug is eliminated from the body such as in the case of oral anticoagulants and some anticancer drugs. Studying the drug concentration-effect relationship in the range of concentrations observed during the clinical use of drugs can be useful in determining the expected changes in the drug effect due to the change of the time course of the drug in the body. Also, characterization of the concentration-effect relationship is useful for designing better drug formulations and more effective dosage regimens to achieve the drug blood concentration-time profile that produces the optimal therapeutic effect. 1.2.5  Population Pharmacokinetics

It is the field that deals with the relationship between the drug dosage regimen and the resulting drug profile in the body in different patient populations. Population pharmacokinetic analysis has been applied during the drug development process and also during clinical drug use. This approach can be applied to analyze the results of data-rich studies performed in small number of patients, and also to analyze sparse data obtained from large number of patients with different characteristics while receiving the drug of interest under different clinical protocols. Using the appropriate modeling approaches, the drug pharmacokinetic behavior, the between patient variability and the random variability which cannot be explained by any factors can be determined. The information about pharmacokinetic variability is very important in supporting safe and effective dosage recommendation for the different patient populations. This is because using the average dose of drugs with narrow therapeutic range that have high variability in their pharmacokinetic behavior will result in high probability of toxic and/or subtherapeutic concentrations.

Introduction to Pharmacokinetics 5 1.2.6 Toxicokinetics

It is the field that applies the pharmacokinetic principles to determine the systemic exposure level of compounds that are associated with observing signs of toxicity. Toxicokinetic studies in general are pharmacokinetic studies performed at toxic doses and are usually performed during the preclinical toxicity studies. This is important because the pharmacokinetic behavior of the drugs can be different at high doses due to the saturation of one or more of the absorption, distribution, metabolism, and excretion processes. The results of the toxicokinetic studies in laboratory animals are extrapolated to establish the drug concentration-toxic effect relationship in humans. This is important in guiding the dosage recommendation for the initial clinical testing of drugs. 1.2.7 Pharmacogenetics

Pharmacogenetics and pharmacogenomics are the fields that deal with the genetic basis of variability in drug effect, which usually results from pharmacokinetics and/or pharmacodynamics variation. It is well documented that variation in the genes encoding the drugmetabolizing enzymes, drug transporters, or drug targets can lead to variation in the drug response. This may result from genetically mediated variation in the activity of the drugmetabolizing enzymes in different patients. Also, genetic variation in the expression of some drug targets leads to variation in the intensity of the drug therapeutic/adverse effects in different patients. Furthermore, some drugs are effective only against specific genotype of some diseases. With the advancement in the field of genetics, pharmacogenetic testing has been developed to identify patients’ specific genetic characteristics for optimizing dosage selection, avoiding adverse effects, and identifying responders from non-responders. Depending on their clinical significance, regulatory agencies usually classify pharmacogenetics tests as required, recommended, or may be performed when using the drugs. 1.3  Application of the Pharmacokinetic Principles in the Biomedical Fields Pharmacokinetics has numerous applications from the early stages of drug development to the clinical use of drugs. The pharmacokinetic principles are also applied in areas other than the drug development, such as risk assessment for determining the acceptable limit of exposure to chemical pollutant, and in health physics to study the distribution and fate of radioactive materials from the body after exposure. The following are few examples of the applications of pharmacokinetics in the biomedical fields. 1.3.1  Design and Evaluation of Dosage Forms

The rate and extent of drug absorption can be different when different products for the same active drug are administered to the same group of patients. So, formulation optimization is important to manufacture products with the desired absorption characteristics. The formulation studies are usually guided by the in vitro characteristics of the developed formulations. However, pharmacokinetic studies are usually necessary to evaluate and compare the in vivo performance of the developed formulations and the selection of the optimized formulation. Pharmacokinetic studies are also used to evaluate and compare the in vivo performance of different dosage forms administered by different routes of administration. This is in addition to evaluation of specialized targeted drug delivery systems that are designed and formulated to preferentially deliver the drug to specific site in the body such as a tumor.

6  Introduction to Pharmacokinetics 1.3.2  Evaluation of Generic Drug Products

Generic products containing an active drug can be marketed after the patent protection for that active drug expiries. It is necessary to prove that the generic products are therapeutically equivalent to the originally approved product for the drug, which is usually known as the innovator’s products. This comparison is commonly performed using pharmacokinetic studies that are known as the bioequivalence studies. The bioequivalence studies are performed to compare the rate and extent of drug absorption after administration of the same dosage form of the generic and innovator’s products containing the same dose of the same active drug to normal volunteers. Drug products for the same active drug that have similar rate and extent of absorption should produce similar blood drug concentration-time profiles, and hence similar drug effects. 1.3.3  Pharmacological Testing

The drug concentration-effect relationship can be characterized by using an appropriate in vitro experimental model, or by performing an in vivo pharmacokinetic study that involves measuring the drug concentration and drug effect, simultaneously. Characterization of the drug concentration-effect relationship can help in predicting the therapeutic effect of the drug that should result from different drug concentration-time profiles. This is important for determining the dosage regimen that should achieve the drug concentrationtime profile required to produce the desired drug effect. 1.3.4  Toxicological Testing

Pharmacokinetic studies are performed to determine different tissue drug exposure and accumulation after single and multiple drug administration. The drug tissue exposure and accumulation are usually correlated with the observed toxicity of the drug. This information is useful in determining the tolerable dose of the drug. 1.3.5  Evaluation of Organ Function

Determination of the pharmacokinetic parameters for some markers can be used to evaluate the function of the eliminating organ of this marker. For example, creatinine is an endogenous compound that is completely excreted by the kidney. The creatinine clearance, which reflects the efficiency of the kidneys in eliminating creatinine, is used as a measure of the glomerular filtration rate and hence, the kidney function. Reduction of the kidney function leads to proportional reduction in the creatinine clearance. Similarly, the clearance of markers that are completely eliminated by the liver such as indocyanine green can be used to evaluate the liver function. 1.3.6  Therapeutic Drug Monitoring

Many drugs have a well-defined therapeutic range, which is the range of blood drug concentrations associated with the maximum therapeutic effect and minimum toxicity. The patient’s specific pharmacokinetic parameters of a drug can be determined by measuring the drug concentrations while the patient is taking the drug or can be estimated from the pharmacokinetic parameters of the drug in a population with characteristics similar to that of the patient. Using the estimated drug pharmacokinetic parameters,

Introduction to Pharmacokinetics 7 the pharmacokinetic principles are applied to calculate the appropriate dose and dosing interval of a specific drug product that should achieve drug concentrations in the desired range in each patient. 1.4  The Blood Drug Concentration-Time Profile After extravascular administration, the drugs used for their systemic effects have to be absorbed into the systemic circulation where they are distributed to all parts of the body, including eliminating and non-eliminating organs. The drug distributed to eliminating organs has the chance to be eliminated from the body, while the drug distributed to the non-eliminating organs is redistributed back to the systemic circulation until all the absorbed drug is eliminated from the body. Drugs administered intravenously are introduced directly to the systemic circulation, so they undergo distribution and elimination only. The blood drug concentration-time profile, which can be characterized by determination of the drug concentrations in serial blood samples obtained after drug administration, reflects the drug pharmacokinetic behavior in the body. This is because the drug concentration-time profile depends on the rate and extent of drug absorption, distribution, and elimination processes that occur simultaneously after drug administration. This profile can be described by mathematical equations that include the pharmacokinetic parameters for the different processes. Administration of the same dose of different drugs produces different blood concentration-time profiles because of the differences in the pharmacokinetic characteristics of the different drugs. Also, administration of the same dose of a drug to a group of individuals produces different blood concentration-time profiles because of the random variation in the way everyone absorbs, distributes, and eliminates the drug. The blood drug concentration-time profile is usually correlated with the drug effecttime profile with higher drug concentrations producing more intense effect. Factors that can affect the blood concentration-time profile of a drug can alter the therapeutic and/ or adverse effects of the drug. The blood concentration-time profile can be affected by factors such as the drug dose, frequency of administration, dosage form, patient characteristics, and concurrent drugs used by the patient. Figure 1.1 is an example of the blood concentration-time profile after administration of a single oral dose of a drug. The profile

Figure 1.1 A representative example of the blood drug concentration-time profile after single oral dose of a drug.

8  Introduction to Pharmacokinetics shows an initial increase in drug concentration during the absorption phase of the drug, followed by a decline in the drug concentration until the drug is completely eliminated from the body. Clinical Importance:

• Variation in the drug pharmacokinetic behavior can produce different drug concen-

tration-time profiles after administration of similar dose of the same drug in different patients and can explain part of the variability in the observed drug effects. • Clinically significant pharmacokinetic drug-drug interactions usually result from modification of the pharmacokinetic behavior, and hence the therapeutic effect, of one or both drugs when used together. • Controlled release formulations and specialized delivery systems can be used to achieve the drug blood concentration-time profile that can produce the desired drug effect-time profile. • For many drugs, the optimal therapeutic outcome is observed when the drug blood concentrations are within a certain range, which is known as the therapeutic range. So, the dose, dosing interval, and dosage form of drugs used to manage chronic conditions should be selected to maintain the blood drug concentration within this range all the time. 1.5  Linear and Nonlinear Pharmacokinetics 1.5.1  Linear Pharmacokinetics

Linear pharmacokinetics is also known as dose-independent and concentrationindependent pharmacokinetics. When a drug follows linear pharmacokinetics, its pharmacokinetic parameters such as half-life, elimination rate constant, total body clearance, and volume of distribution are constant and do not change with the change in drug concentration or the drug amount in the body. In linear pharmacokinetics, the drug absorption, distribution, and elimination processes, which govern the blood drug concentration-time profile, follow first-order kinetics. Also, the change in dose results in proportional (linear) change in the drug concentration-time profile in the body after single and multiple drug administration. 1.5.2  Nonlinear Pharmacokinetics

Nonlinear pharmacokinetics is also known as dose-dependent and concentrationdependent pharmacokinetics. When a drug follows nonlinear pharmacokinetics, its pharmacokinetic parameters such as the half-life, total body clearance, or volume of distribution are dependent on the drug concentration or the drug amount in the body. In nonlinear pharmacokinetics, at least one of the absorption, distribution, and elimination processes, which affect the blood drug concentration-time profile, is saturable and does not follow first-order kinetics. The change in dose results in disproportional (more than or less than proportional) change in the blood drug concentration-time profile after single and multiple drug administration. Table 1.1 summarizes the differences between linear and nonlinear pharmacokinetics.

Introduction to Pharmacokinetics 9 Table 1.1  The differences between linear and nonlinear pharmacokinetics Linear pharmacokinetics

Nonlinear pharmacokinetics

• Also, known as dose-independent or concentration-independent pharmacokinetics. • The absorption, distribution, and elimination of the drug follow first-order kinetics. • All the pharmacokinetic parameters such as the half-life, total body clearance, and volume of distribution are constant and do not depend on the drug concentration. • The change in drug dose results in proportional change in the drug concentration.

• Also, known as dose-dependent, or concentration-dependent pharmacokinetics. • At least one of the pharmacokinetic processes (absorption, distribution, or elimination) is saturable. • One or more of the pharmacokinetic parameters such as the half-life, total body clearance, or volume of distribution are concentration-dependent. • The change in drug dose results in more than or less than proportional change in drug concentration.

Clinical Importance:

• The change in the dosing rate of the drugs that follow linear pharmacokinetics usu-

ally results in proportional change in the drug blood concentrations. So, it is easier to predict the change in drug concentrations and therapeutic outcome resulting from changing the dose when the drug follows linear pharmacokinetics. • Drugs that follow nonlinear pharmacokinetics should be used with caution since small change in dose may result in disproportional change in the plasma drug concentration and drug effect. 1.6  Pharmacokinetic Modeling Pharmacokinetic models are usually constructed to describe the relationship between the administered drug dose and the resulting drug profile in the body that reflects drug exposure. While in pharmacokinetic-pharmacodynamic models, the drug response is also monitored and included in the models. The common feature of these models is that the relationships between the different model components are described by mathematical equations that collectively describe the pharmacokinetic behavior of the drug. These mathematical equations include the pharmacokinetic parameters that govern the drug absorption, distribution, and elimination processes, in addition to some constants representing the dose, time of the dose, number of doses, and the frequency of drug administration. Some pharmacokinetic models include additional anatomical, physiological, and biochemical parameters. The choice of the modeling strategy depends on the availability of the data needed to construct the model, and the model “fitness of purpose”. Pharmacokinetic modeling utilizes all the available information about the drug and the experimentally generated data to estimate the pharmacokinetic model parameters that govern the drug pharmacokinetic behavior under the experimental conditions. Then the estimated model parameters and the mathematical equations are used to predict the drug pharmacokinetic behavior under different conditions such as after administration of different doses by different routes of administration using different dosage forms. Some models are constructed to evaluate the factors affecting the drug pharmacokinetic

10  Introduction to Pharmacokinetics behavior, to assess the variability in the drug pharmacokinetic behavior, or to determine specific organ exposure to the drug. The following are the common modeling strategies used in pharmacokinetics. 1.6.1  Compartmental Modeling

In compartmental modeling, the body is presented by one or more kinetically homogeneous compartments depending on the rate of drug distribution to the different parts of the body. When the drug is rapidly distributed to all parts of the body and rapid equilibrium is established between the drugs in all parts of the body, the body is presented by one compartment and the drug follows the one-compartment pharmacokinetic model. When the drug distribution to some organs/tissues is faster than its distribution to others, the body can be presented by two different compartments and the drug follows the two-compartment pharmacokinetic model. The model usually includes parameters that describe drug absorption, distribution between the compartments, and elimination from one or more of the compartments. These models differ in the number of compartments and the arrangement of the compartments relative to each other as shown in Figure 1.2. Compartmental modeling is data-based modeling, meaning that the blood drug concentration-time data obtained after drug administration is used to choose the best compartmental pharmacokinetic model that can describe the drug pharmacokinetic behavior. 1.6.2  Physiological Modeling

In physiological modeling, the body is divided into a series of organs or tissue spaces and the model describes the distribution and elimination of the drug in the different organs. The model is usually constructed to include the organs and tissues that have the drug site of action, drug toxicity, drug elimination, and any other organ that have significant effect on the drug pharmacokinetic behavior. Anatomical, physiological, biochemical, and experimentally determined information such as organ size, organ blood flow, drug uptake to tissues, and drug elimination rates by different organs are the parameters used to build the model. The model equations utilize these parameters to simulate the drug profile in the blood and different organs. Then the model structure and parameter values are refined until the model predicted drug blood and tissue concentrations are in agreement with the experimentally obtained data. After validation of the model, it can be used to predict the drug pharmacokinetic behavior under different scenarios. Physiological

Figure 1.2  Representative examples of compartmental pharmacokinetic models. (A) onecompartment model, (B) two-compartment model, and (C) three-compartment model.

Introduction to Pharmacokinetics 11

Figure 1.3 A representing example of a physiologically based pharmacokinetic model which includes the heart, brain, intestine, liver, kidney, and other tissues. The term Q represents the blood flow, and the subscript indicates the organ.

models are very useful in toxicological testing to determine specific organ exposure to drugs. This modeling approach is also useful for interspecies scaling to predict the drug pharmacokinetic behavior in different animal species and in humans by using the organ size, organ blood flow, and drug tissue uptake and elimination parameters specific to the species of interest. These models can also be used to predict the change in drug pharmacokinetic behavior due to physiological and pathological changes. A representative example of a physiological pharmacokinetic model is presented in Figure 1.3. 1.6.3  Population Pharmacokinetic Modeling

This involves modeling the relationship between the drug intake and the resulting body exposure to the drug by characterization of the drug profile in the body. Additionally, population modeling determines the magnitude of the variability in the drug pharmacokinetic parameters. Population modeling can utilize sparse data from large number of patients with different characteristics receiving the drug using different clinical protocol. The population modeling process involves using three types of models. The model that describes the pharmacokinetic behavior of the drug in the body. Also, the model that describes the between patient variability and the random variability which cannot be explained by any factors. Additionally, the model that describes how the patients’ specific characteristics “covariates”, such as age, weight, demographics, or diseases, can affect the drug behavior in the body. So, in addition to estimation of the pharmacokinetic parameters, the random variability and the factors affecting the variability in the pharmacokinetic behavior are determined. This is very important to support dosage recommendation decision in different patient populations during the drug developing process.

12  Introduction to Pharmacokinetics 1.6.4  Noncompartmental Data Analysis Approach

All the models used to describe the pharmacokinetic behavior of drugs usually have some assumptions that must be met to ensure the validity of the model. When the objective of the pharmacokinetic study can be achieved without making any model assumption, the noncompartmental (model independent) data analysis approach can be used. This approach uses some pharmacokinetic parameters that can be calculated without making any model assumption. For example, the maximum blood drug concentration achieved after drug administration, the time to achieve the maximum blood drug concentration, and the area under the plasma concentration-time curve can be determined without making any assumption of a specific pharmacokinetic model. Statistical comparison of these three parameters determined after administration of two different products for the same active drug to normal volunteers can be used to compare the rate and extent of drug absorption from the different products. So, the noncompartmental data analysis approach is the appropriate method for analyzing bioequivalence study results. 1.6.5  Pharmacokinetic-Pharmacodynamic Modeling

These models include a pharmacokinetic and a pharmacodynamic components. The pharmacokinetic part of the model describes the relationship between the dosage regimen and the drug concentration-time profile in the body, including the drug site of action. While the pharmacodynamic part describes the relationship between the drug concentration-time profile at the site of action and the resulting drug response. Together, both parts describe the drug dosage regimens-drug concentration-drug response relationship. This relationship is described by a set of mathematical equations that include the model pharmacokinetic and pharmacodynamic parameters. This modeling approach is important for selecting the appropriate dosing regimen that should achieve the desired therapeutic outcome. Clinical Importance:

• Pharmacokinetic models can be developed to predict the drug profile in the body and hence the expected drug effect after administration of different doses by different routes of administrations using different dosing forms. • Physiological models can be developed to determine the different tissue exposure and fetal exposure to drugs after administration of different doses that are important in assessing the adverse effects of drugs. • Population pharmacokinetic models developed during the clinical phases of drug development are important in making dosing regimen recommendation for the different patient populations. • Pharmacokinetic-pharmacodynamic models are developed to help in selecting the appropriate drug dosing regimens for obtaining the desired therapeutic outcome in different patient populations. 1.7  Pharmacokinetic Simulations Pharmacokinetic simulations usually involve simulation of the plasma concentration-time profile of the drug under different conditions. Simulations have numerous applications in teaching, training, and in research. The idea of the pharmacokinetic simulations is to use the mathematical equation that describes the blood drug concentration-time profile

Introduction to Pharmacokinetics 13

Figure 1.4 Simulation of the blood concentration-time profiles after single oral drug administration when the elimination rate constant is different.

and including the different pharmacokinetic parameter values to calculate the blood drug concentrations at different time points. Then one or more of the pharmacokinetic parameters that reflect the condition being examined are changed, and the new drug profile is recalculated. Comparing the profiles resulting from changing one of the parameters demonstrates how the changed parameter affects the blood drug concentration-time profile. For example: The drug plasma concentration-time profile after a single oral dose of the drug can be described by Eq. 1.1. Cp =

(

)

FD ka e− kt − e− ka t Vd(ka − k) 

(1.1)

where Cp is the drug plasma concentration at any time t, D is the dose of the drug, F is the drug bioavailability that is a measure of the extent of drug absorption, ka is the absorption rate constant, k is the elimination rate constant, and Vd is the volume of distribution. The plasma drug concentration can be calculated at different time points by including the value for each of the pharmacokinetic parameters and using different values for time. The calculated plasma concentrations at different time points are plotted to simulate the drug concentration-time profile. Assume that we want to simulate the effect of the decrease in the eliminating organ function on the drug concentration-time profile. Since the elimination rate constant (k) is the parameter that reflects the rate of the elimination process, Eq. 1.1 is used to calculate the blood concentration-time profiles using different values for k while keeping the values for all other parameters constant. Figure 1.4 shows the simulation of the blood drug concentration-time profiles using decreasing values of k to reflect decrease in the rate of elimination. The simulation indicates that the decrease in the eliminating organ function of the drug results in slowing the rate of decline in the drug concentration-time curve during the elimination phase. Clinical Importance:

• Pharmacokinetic simulations can be used to simulate the drug concentration-time profiles under different drug administration scenarios that are not investigated experimentally.

14  Introduction to Pharmacokinetics

• The effect of changing each of the pharmacokinetic parameters on the blood drug concentration-time profile and the interplay between the different pharmacokinetic parameters can be visualized using pharmacokinetic simulations. • The drug effect-time profile can be simulated if the equation for the pharmacodynamic model is included in the simulation. 1.8 Essential Graphical, Mathematical, and Statistical Fundamentals Used in Pharmacokinetics Mathematical expressions, graphs, and simulations are important in describing pharmacokinetic processes, constructing pharmacokinetic models, demonstrating pharmacokinetic concepts, and establishing the relationship between the different pharmacokinetic parameters. Knowledge of the basic mathematical principles can help in understanding how the mathematical expressions that describe the different pharmacokinetic concepts are derived. The following graphical, mathematical, and statistical methods are used frequently in pharmacokinetics. 1.8.1 Graphs

Graphs are commonly utilized in pharmacokinetics to present the drug concentrationtime or drug effect-time profiles. There are two types of scales that are commonly utilized in pharmacokinetics: the Cartesian (linear) scale and the semilog scale. In the linear scale, both the y-axis and the x-axis are linear, i.e., the values on each of the axes are equally spaced as shown in Figure 1.5A. The data presented on the linear scale usually give the feeling of the actual magnitude of the difference between data points or profiles. Both axes on the linear scale can have negative and positive values; however, only positive

Figure 1.5  An example of (A) the Cartesian (linear) scale and (B) the semilog scale.

Introduction to Pharmacokinetics 15 coordinate values including the origin (0,0 point) are used to present pharmacokinetic data such as plasma concentration versus time profile. While the semilog scale has a logarithmic scale on the y-axis and a linear scale on the x-axis, as in Figure 1.5B, this type of graphs is useful in plotting values that are changing exponentially, which is common in pharmacokinetics. Plotting data on the semilog scale (plasma concentration versus time) produce profiles like those obtained by plotting log-dependent variable values versus time (log plasma concentration versus time) on the linear scale. The semilog graph paper consists of repeated units called cycles. Each of these cycles covers one log10 unit, or ten-fold increase in number. The y-axis on the semilog scale does not have a zero point and the positive coordinate side of the x-axis is used to plot the plasma concentration-time data. The semilog graph paper is available in one, two, three, or more cycles per sheet. Figure 1.5B represents a three-cycle semilog scale. The choice of the number of cycles depends on the range of values to be plotted. To plot data on the semilog scale, the first step is to determine the range of values for both the dependent and independent variables. The values for the dependent variable are arranging in the ascending or descending order and the range of values is used to determine the number of cycles needed to plot the data. Using the graph paper with the correct number of cycles is important for good data presentation. Practice Problem: Question: Plot the following drug concentration-time data on the semilog scale. Time (hr)

Drug concentration (ng/mL)

1 3 6 10 14 18 24

500 303 143 53 19 7.1 1.6

Answer:

• The scale on the x-axis is chosen to make the time points cover the largest portion of the x-axis.

• The drug concentration values are used to determine the number of cycles needed 500

303

143

53

19

7.1

1.6

Three cycles are needed to plot these values. First cycle covers Second cycle covers Third cycle covers

1 10 100

– – –

10 100 1000

Figure 1.6 represents the drug concentration-time data plotted on a three-cycle graph paper.

16  Introduction to Pharmacokinetics

Figure 1.6  The data plotted on the semilog scale. 1.8.2  Curve Fitting

Curve fitting is the statistical procedures that estimate the best model equation that describes the relationship between a set of variables.

• The first step in the curve fitting procedures is to use the theoretical, biological, and

physiological background information to suggest a model that can describe the relationship between the variables. Then find the mathematical equation that can describe the model; linear, exponential, logarithmic, hyperbolic, or any other equation. • The different equations for different models are fitted to the obtained experimental data using statistical software. The equation that provides the best fit, i.e., provide the best description for the obtained experimental data, is the equation for the best model that describes the relationship between the variables. • Estimation of the model parameters during the curve fitting procedures provides the exact equation that describes the relationship between the variables. The exact equation can be used to predict the dependent variable values under different conditions that were not tested experimentally. • Curve fitting can be very simple as finding the straight-line equation that can be performed using a simple scientific calculator to estimate the slope and y-intercept of a linear relationship. Also, it can be complex procedures such as in case of fitting experimental data to multi-exponential equations, which usually require computers and specialized data analysis software. 1.8.3  Determination of the Straight-Line Parameters

Linear relationships between the dependent variable (y) and the independent variable (x) can be described by a straight-line equation (i.e., y = a + b x), where a is the y-intercept and b is the slope. The general principles used to estimate the slope and y-intercept for a linear relationship will be discussed briefly, since this is used frequently to determine some pharmacokinetic parameters in simple pharmacokinetic models. It is important to note that experimental observations that are used to estimate the equation parameters are usually associated with error with varying magnitude. So, statistical approaches are used to calculate the best estimate for the slope and y-intercept.

Introduction to Pharmacokinetics 17

Figure 1.7 Estimation of the slope and y-intercept of a straight line graphically on the linear (A) and semilog (B) scales. 1.8.3.1  Graphical Determination of the Straight-Line Parameters

The line of best fit is the line that passes as close as possible through all the observed data points, and these points fall randomly around the line. Although graphical determination of the best line is used frequently as it provides quick and easy estimate for the line parameters, this is not accurate because it is subjective and different individuals may draw different lines for the same set of data. When the line is determined graphically, the y-intercept is determined by back extrapolating the line to the y-axis. The slope of the line is Δy/Δx, which can be determined by taking two different points on the line and these two points are used to calculate the slope as in Figure 1.7. The slope of the line on the linear scale can be determined as in Eq. 1.2. Slope =

y 2 − y1 x 2 − x1 

(1.2)

where x1, y1 and x2, y2 are the x- and y-coordinates of the two points on the line (not the experimentally obtained points). While the slope of the line on the semilog scale can be determined as in Eq. 1.3. Slope =

log y 2 − log y1    x 2 − x1 

(1.3)

1.8.3.2  The Least Squares Method

The least squares method is a statistical approach used to calculate the slope and yintercept values for the best line that describes the experimental data. The idea is to choose the line that can minimize the sum of the squared deviations between each experimental data point and the line as shown in Figure 1.8. The calculated slope and y-intercept using the least squares method are usually more accurate than the graphical method. The linear relationship between two variables can be evaluated by calculating some parameters as the correlation coefficient (r) and the coefficient of determination (R2) which determine how much of the variability in the dependent variable results from the change in the independent variable. Values for r and R2 that are close to unity indicate

18  Introduction to Pharmacokinetics

Figure 1.8 The least squares method determines the line that can minimize the sum of the squared distance between the experimental data and the line.

strong linear relationship between the two variables. Most of the scientific calculators can perform this analysis and calculate the slope and y-intercept for a given set of data. 1.8.4  Application of Basic Calculus Principles in Pharmacokinetics

The different pharmacokinetic parameters are related together by mathematical equations. We will not discuss the detailed mathematical derivation of these equations in this book; however, describing how the equations were obtained can help in understanding the concepts presented by the equations. The pharmacokinetic processes can be described by differential equations that express the rate of change of the dependent variable which can be drug concentration, drug amount, or drug effect with respect to the independent variable, time. For example:

• The rate of drug elimination after single intravenous (IV) administration can be de-

scribed by (dA/dt), the rate of change of the amount of the drug in the body (A) with respect to time. • The rate of change of the amount of the drug in the body after single oral drug administration can be described by (dA/dt), the rate of change of the amount of the drug in the body (A) with respect to time. This is equal to the difference between the rate of drug absorption and the rate of drug elimination. Since the different pharmacokinetic processes follow first-order or zero-order kinetics, the rate of these processes can be expressed mathematically. Assuming first-order elimination, the rate of change of the amount of the drug in the body after single IV administration can be described as in Eq. 1.4. dA = −k A dt 

(1.4)

where k is the first-order elimination rate constant, A is the amount of the drug in the body, and the negative sign because the amount of the drug in the body is decreasing.

Introduction to Pharmacokinetics 19 The differential equations can be used to calculate the rate of drug elimination and the amount of the drug remaining in the body over a short period of time if we know the elimination rate constant and the amount of the drug in the body at this time. However, these equations cannot be used to calculate how much drug remaining in the body or the rate of elimination after a certain time. To do this, the differential equation will need to be solved. This is done by integrating the differential equation to obtain Eq. 1.5, which can be transformed to obtain Eq. 1.6 ln A = ln A0 − kt 

(1.5)

A = A0 e− kt 

(1.6)

where A is the amount of the drug in the body at any time t, A0 is the initial drug amount of the drug in the body, k is the first-order elimination rate constant. If the initial amount of the drug in the body after single IV administration (i.e., dose) and the first-order elimination rate constant are known, substitution for any value of time in Eq. 1.6 gives the amount of the drug in the body at that time. The same can be done if the elimination process follows zero-order kinetics. The rate of change of the amount of the drug in the body after single IV administration can be expressed as in Eq. 1.7: dA = −K0 dt 

(1.7)

where K0 is the zero-order elimination rate constant, and the negative sign because the amount of the drug in the body is decreasing. Again, solving this differential equation by integration produces Eq. 1.8. A = A0 − K0 t

(1.8)

where A is the amount of the drug in the body at any time t, A0 is the initial drug amount of the drug in the body, K0 is the zero-order elimination rate constant. If the initial amount of the drug in the body after single IV administration (i.e., dose) and the zeroorder elimination rate constant are known, substitution for any value of time in Eq. 1.8 gives the amount of the drug in the body at that time. Practice Problems 1.1 The following is the drug concentration in a liquid dosage form that was kept at room temperature for 16 days. Time (days)

Drug concentration (mg/mL)

2 4 8 12 16

89.4 80.5 59.1 41.0 20.1

20  Introduction to Pharmacokinetics Plot the antibiotic concentration versus time data on the linear scale, then draw the best line that goes through the data points. Then from the graph: a Calculate the slope of the plotted line. b Calculate the y-intercept. c From the graph, estimate the drug concentration after 10 days. 1.2 The following is the plasma drug concentration after IV administration of 500 mg of a drug. Time (hr)

Drug concentration (mg/L)

2 3 4 8 12 16 24

12.6 10.1 7.9 3.2 1.25 0.49 0.08

a How many cycles do you need to plot the data? Plot the drug concentration versus time data on the semilog scale, then draw the best line that goes through the data points. Then from the graph: b Calculate the slope of the plotted line. c Calculate the y-intercept. d From the graph, estimate the drug concentration after 10 hr.

2

Drug Pharmacokinetics Following Single Intravenous Bolus Administration Drug Distribution

Objectives After completing this chapter, you should be able to:

• • • • •

Describe the factors affecting the rate and extent of drug distribution. Explain the concept of volume of distribution and the factors affecting it. Calculate the drug volume of distribution after single IV administration. Calculate the appropriate IV dose required to achieve a specific drug concentration. Describe the relationship between the amount of drug in the body, plasma drug concentration, and volume of distribution. • Describe the effect of drug distribution rate to different organs on the pharmacokinetic behavior of the drug. • Discuss the clinical significance of the drug plasma protein binding and blood cell partitioning. • Analyze the effect of changing plasma protein binding and tissue binding on the drug distribution characteristics and volume of distribution. 2.1 Introduction Intravenous (IV) bolus drug administration involves direct administration of drug into the systemic circulation over a short period of time. In the systemic circulation, drugs can reversibly bind to plasma proteins and partition into blood cells, leaving a fraction of the drug as free unbound molecules. The drug can be distributed with the systemic circulation to all parts of the body, including the sites of drug effect and elimination. The free drug can leave the circulation by permeation across the capillary membrane to reach the interstitial space of different tissues, where it can cross the cellular membrane and be distributed intracellularly. The drug in different tissues can distribute back to the systemic circulation to maintain the drug distribution equilibrium between the systemic circulation and tissues. 2.2  Drug Distribution The rate and extent of drug distribution to different parts of the body can be different. This is due to tissue-related factors such as tissue perfusion, tissue composition, and the presence or absence of anatomical barriers and also due to the drug-related factors such as the physicochemical properties of drug and drug binding to plasma proteins and tissue components. DOI: 10.4324/9781003161523-2

22  Single IV Bolus Administration, Drug Distribution 2.2.1  The Rate and Extent of Drug Distribution

Lipophilic drugs can readily cross the capillary membrane and the cellular membrane, making tissue perfusion to be the rate-determining step for their distribution to tissues. Generally, the rate of distribution of lipophilic drugs is faster for highly perfused tissues such as lung, liver, and kidney compared to the poorly perfused tissues such as fat. Therefore, for lipophilic drugs, faster equilibrium is achieved between the blood and highly perfused tissues compared to poorly perfused tissues. On the other hand, hydrophilic drugs cannot cross the lipophilic biological membranes easily. This makes the membrane permeability, which depends on the drug physicochemical properties, the rate-determining step for tissue distribution of these drugs. Therefore, hydrophilic drugs usually distribute slowly to tissues. The extent of drug distribution to a particular tissue depends on its affinity to that tissue, which is determined by the drug and tissue characteristics. Quantitatively, the extent of drug distribution to a particular tissue is expressed as the tissue distribution coefficient (KpT), which is the ratio of the total (free + bound) drug concentration in the tissue (CT) to the total (free + bound) blood drug concentration (CB) at equilibrium, as shown in Eq. 2.1. Higher KpT indicates higher tissue to blood concentration ratio at equilibrium. A drug can have different KpT values for different tissues, and different drugs may have different KpT values for the same tissues. Determination of the amount of drug that distributes to a particular tissue (AmountT) is important because it is a measure of the organ exposure to the drug. In this case, the volume of the tissue (VT) should be considered, as shown in Eq. 2.2. KpT =

CT (2.1) CB

Amount T = C T VT = KpT CB VT (2.2) The factors affecting drug distribution to tissues include drug characteristics with low molecular weight drugs distributing to tissues easier than high molecular weight drugs, lipophilic drugs better than hydrophilic drugs, and unionized drugs at physiological pH more than ionized drugs. Also, the tissue characteristics and tissue composition affect the extent of drug distribution to different tissues, with lipophilic drugs distributing to fat-rich tissues more than hydrophilic drugs. Additionally, the extent of drug binding to blood and tissue constituents can affect the extent of drug binding to different tissues. Clinical Importance:

• The rate of drug distribution is important when a drug is administered to treat an

acute condition. If the drug is distributed rapidly to the site of action, the onset of drug effect will be faster. • During multiple drug administration for the treatment of chronic diseases, equilibrium is established between the drug in the systemic circulation and at the site of action, so the rate of drug distribution to tissues may not have significant effect. • The extent of drug distribution to a particular organ determines the drug concentration and hence the organ exposure to the drug. If this organ is the site of drug action, higher extent of distribution results in higher drug concentration and more intense therapeutic effect. While if this organ is the site of drug toxicity, higher extent of distribution results in more toxicity.

Single IV Bolus Administration, Drug Distribution 23 2.3  The Volume of Distribution After IV administration, the drug in the systemic circulation can be distributed to tissues, leaving a fraction of the administered drug dose in blood. The drug is not distributed homogeneously all over the body since it has different affinities for different tissues. Also, different drugs have different distribution characteristics because their affinities to different tissues are different. This means that the drug amount in blood, and hence the drug concentration, will be different after administration of different drugs depending on the administered dose and the drug distribution characteristics. Extensive drug distribution to tissues leaves small amount of drug in blood and the blood drug concentration will be low. This low blood drug concentration seems as if the drug is distributed in a large volume. While limited tissue distribution results in higher blood drug concentration, which looks as if the drug is distributed in a small volume. However, this is an imaginary volume because the drug is not distributed homogeneously in all parts of the body. This imaginary volume is the pharmacokinetic parameter that quantitatively describes the distribution process and is known as the volume of distribution (Vd). Different drugs have different Vd because of the differences in their affinities to different tissues. Also, there are variations in the Vd of a given drug in different patients due to differences in body mass and body composition. Consider the situation when a drug is added to a beaker containing a liquid and the beaker wall is coated with an adsorbing material. The drug is distributed to the coating material, leaving a fraction of the added drug amount dissolving in the liquid in the middle of the beaker. If the drug is extensively distributed to the coating materials, the drug concentration in the liquid will be similar to the concentration produced if the same amount of the drug is added to an uncoated beaker with larger volume. In this case, and because of the unequal distribution, we can say that the beaker with the coated wall behaves as if it has volume similar to that of the larger beaker, as illustrated in Figure 2.1. The drug concentration observed in the liquid inside the uncoated beaker is dependent on the amount of the drug added and the volume of the liquid, a relationship that can be expressed by Eq. 2.3: Concentration =

Drug amount (2.3) Volume

This equation describes the relationship between drug amount, concentration, and volume when the drug is distributed homogeneously in the content of the beaker. However,

Figure 2.1 Adding an amount of a drug to a beaker with coated wall that can adsorb the drug (A) produces drug concentration in the liquid similar to the concentration produced if the same amount of the drug is added to a beaker with larger volume (B).

24  Single IV Bolus Administration, Drug Distribution it can also be used in the situation when drug distribution in the liquid and the coating material is different. In this case, the calculated volume is not a real volume. After administration of an IV bolus dose of a drug, a fraction of the dose is distributed to the tissues and the remaining drug stays in the blood. Using Eq. 2.3 to describe the relationship between the dose (D), initial blood concentration (concentration at time zero, Cp0), and volume of distribution (Vd), Eq. 2.4 can be obtained. Vd =

Dose (2.4) Cp0

The Vd is the pharmacokinetic parameter that relates the drug amount in the body and the blood drug concentration. As mentioned previously, the Vd is not a real volume; it is known as the apparent volume of distribution. In other words, we can say that the body behaves as if it has a volume equal to Vd. The Vd has units of volume (e.g., liters, L), or volume/weight (e.g., liter/kilogram, L/kg). Clinical Importance:

• Drugs with high affinity to tissues usually have large Vd. • Administration of increasing doses of the same drug to the same patient should produce proportional increase in blood and tissue drug concentrations, as shown in Figure 2.2A.

• Administration of the same dose of a drug to different patients who have different Vd should produce blood and tissue concentrations that are inversely proportional to the Vd, as shown in Figure 2.2B. • The extent of drug distribution to the site of drug action is correlated with the intensity of the drug effect, and the distribution of the drug to tissues other than the site of action may result in the development of adverse effects. • The Vd is dependent on the body mass. So after administration of the same dose to patients with different body mass, e.g., 30-kg child, 60-kg female, or 100-kg male patients, the initial blood concentration will be higher in the 30-kg child > 60-kg female > 100-kg male patients. • Since most drugs have specific therapeutic range where the drug is most likely to produce the optimal therapeutic effect with minimum adverse effects, the Vd allows

Figure 2.2 The plasma drug concentration-time profile (A) when different doses of the same drug are administered to the same individual, and (B) when the same dose of a drug is administered to different individuals who have different Vd.

Single IV Bolus Administration, Drug Distribution 25 calculation of the IV dose required to achieve specific blood drug concentration. Also, it allows the calculation of the blood concentration achieved after IV administration of any dose of the drug. Practice problems: a  Question: What is the IV bolus dose required to achieve initial blood concentration of 15 mg/L for an antibiotic that has Vd of 25 L? Answer:  Using Eq. 2.4: Dose = Vd × Cp0 Dose = 25 L × 15 mg/L = 375 mg b  Question: What is the initial blood drug concentration achieved in a 60-kg patient after administration of 500 mg of a drug that has a volume of distribution of 0.4 L/kg? Answer:  Vd = 0.4 L/kg × 60 kg = 24 L Using Eq. 2.4: Cp0 = Dose/Vd Cp0 = 500 mg/24 L = 20.83 mg/L c  Question: A 30-kg child received a full vial of 1000 mg of an antibiotic as IV bolus dose. Calculate the initial blood drug concentration if the volume of distribution of this antibiotic is 0.6 L/kg. Answer: Vd = 0.6 L/kg × 30 kg = 18 L Using Eq. 2.4: Cp0 = Dose/Vd Cp0 = 1000 mg/18 L = 55.56 mg/L d  Question: After IV administration of 600 mg of a drug, the initial blood drug concentration achieved was 20 mg/L. Calculate the volume of distribution of this drug in this patient. Answer:  Using Eq. 2.4: Vd = Dose/Cp0 Vd = 600 mg/20 mg/L = 30 L The distribution of drugs from the systemic circulation to tissues involves several general steps that depend on the drug and tissue characteristics. Low molecular weight and relatively nonpolar drugs can cross the leaky membrane of the blood capillaries to reach the extracellular space of the tissues. Nonpolar drug molecules in the tissue extracellular space have the ability to cross the cellular membrane, distribute in the intracellular fluid, and can bind to cellular constituents. The drug physicochemical properties that affect the extent of tissue distribution can be used to make rough prediction for the Vd for some drugs.

• Highly polar drugs with high molecular weight like heparin and insulin have limited

ability to cross the capillary membrane. These drugs distribute mainly in the vascular volume and usually have very small Vd. • Polar drugs with small molecular weight can cross the capillary membrane and distribute to the extracellular space of tissues without crossing the cellular membrane. The aminoglycoside antibiotics are hydrophilic drugs that distribute mainly in the extracellular space. The Vd of these drugs is in the range of 0.2 L/kg, which is roughly equal to the volume of the body extracellular fluids.

26  Single IV Bolus Administration, Drug Distribution

• Some nonpolar drugs can distribute to the extracellular space, cross the cellular mem-

brane, and stay in the intracellular water without binding to any cellular components. Aminopyrine is a compound that is distributed mainly into the extracellular and intracellular fluids, i.e., the total body water, and has Vd in the range of 0.6 L/kg, which is nearly equal to the volume of the total body water. • Other nonpolar drugs can distribute intracellularly and bind to specific cell components of different tissues and usually have higher Vd depending on their extent of binding to cellular constituents. • The estimated average Vd in a 70-kg individual for the antiepileptic drug phenytoin, the antibiotic ciprofloxacin, the cardiac glycoside digoxin, and the tricyclic antidepressant imipramine are 50, 120, 500, and 1600 L, respectively. Remember, Vd is not a real volume, and large Vd indicates extensive drug distribution to the tissues. 2.4  Drug Distribution after Single IV Bolus Drug Administration When the rate of drug distribution to all tissues after IV bolus administration is fast, rapid equilibrium is established between the drug in blood and the drug in all tissues and the body behaves as one kinetically homogeneous compartment. The drug in this case follows the one-compartment pharmacokinetic model, which is presented by the diagram, as shown in Figure 2.3A. The decrease in the drug concentration in blood due to drug elimination results in immediate and proportional decrease in the drug concentration in all tissues as presented in Figure 2.4. However, when the rate of drug distribution is rapid to some tissues and slow to others, the drug concentrations in blood and some tissues are not proportional initially right after drug administration. This is because equilibrium between the drug in blood and all tissues is not established at this time. However, once the distribution equilibrium is established, the decrease in drug concentration in blood due to drug elimination will cause proportional decrease in the drug concentration in all tissues, as shown in Figure 2.5. In this case, the body behaves as two kinetically homogeneous compartments, and the drug follows the two-compartment pharmacokinetic model as presented by the diagram in Figure 2.3B. The first compartment includes the blood and the highly perfused tissues

Figure 2.3 Diagrams representing (A) one-compartment pharmacokinetic model, where the drug distributes rapidly to all parts of the body, and (B) two-compartment pharmacokinetic model, where the drug distributes rapidly to the organs forming the central compartment and slowly to the organs forming the peripheral compartment.

Single IV Bolus Administration, Drug Distribution 27

Figure 2.4 A hypothetical example of the drug concentration-time profiles in different organs on the linear scale for a drug that follows the one-compartment pharmacokinetic model and eliminated by first-order process. Rapid equilibrium is established between the drug in plasma and all tissues after administration, so the drug concentrations decline at the same rate in all tissues. The differences in drug concentration in different organs are due to the differences in drug affinity to different organs.

and is known as the central compartment. Rapid equilibrium is established between the drug in blood and the organs/tissues included in this compartment. The second compartment is known as the peripheral compartment and includes the organs/tissues where the drug is slowly distributed. In this book, the discussion of different pharmacokinetic concepts assumes that the drugs follow the one-compartment pharmacokinetic model unless stated otherwise. The concept of multicompartment pharmacokinetic models is discussed in Chapter 14.

Figure 2.5 A hypothetical example of the drug concentration-time profiles in different organs on the semilog scale for a drug that follows the two-compartment pharmacokinetic model and eliminated by first-order process. The drug profiles in the organs forming the central compartment (plasma, liver, and lung) are not parallel to the profiles in the organs forming the peripheral compartment (muscles and brain) initially. However, after establishing the distribution equilibrium between all organs, the drug profiles in all organs decline at the same rate as indicated by the parallel profiles.

28  Single IV Bolus Administration, Drug Distribution Clinical Importance:

• One of the useful applications of pharmacokinetics modeling is to predict the drug behavior after different drug administration scenarios. So, selection of the correct pharmacokinetic model is important for obtaining accurate predictions. • Studying the rate and extent of drug distribution to different tissues is important because the drug will be effective only if it can distribute in enough quantities to the tissues that have the drug site of action. • When the drug is distributed slowly to its site of action, the maximum drug effect is not observed when the blood drug concentration is the highest. Usually there is delay in the drug effect after IV bolus drug administration. 2.5  Drug Protein Binding The drug in the systemic circulation can bind to plasma proteins, such as albumin, alpha1-acid glycoprotein, globulins, and lipoprotein, and can partition inside the blood cells to bind to blood cell constituents. The extent of drug plasma protein binding can be expressed as the fraction or percent bound, which can range from 0.0 or 0.0% to > 0.99 or >99%. This corresponds to free fraction or percent free, which can range from 1.0 or 100% to emulsion > oily suspension > solid implant, due to the drug release rate from the formulation. Drugs administered as aqueous solutions are absorbed rapidly, while drugs administered in the form of oily suspensions may be absorbed over a period of several days or weeks, whereas the absorption of drugs administered as solid implants may take several months. Other strategies used to control the rate of drug release from the formulations and drug absorption after parenteral administration include encapsulation, using esters of the active drugs as prodrugs, and in situ solidifying depot (11). The extent of drug absorption is also different for the different formulations and can be affected also by the stability of the drug at the site of administration. 5.4.2  Oral Drug Administration

The oral route is the most used route for drug administration. The major components of the GIT are the buccal cavity, esophagus, stomach, small intestine, large intestine (colon), and the rectum. These GIT segments differ from each other with respect to the anatomical structure, transit time, secretions, and pH, which can affect the rate and extent of drug absorption. Understanding the physiological nature of the different GIT regions can help in developing strategies to control oral drug absorption. The buccal cavity: The buccal cavity is very rich in blood supply with thin epithelial layer allowing rapid drug absorption and rapid onset of action. Absorption from the buccal cavity is mainly by passive diffusion and the absorbed drug directly reaches the systemic circulation bypassing the first-pass hepatic metabolism. However, the small surface area and short transit time can limit drug absorption from the buccal cavity. So, formulations designed for buccal drug delivery have to dissolve rapidly and/or have to be retained in the mouth cavity for a period of time to ensure maximum drug absorption.

Drug Absorption Following Extravascular Administration 75 Fast-dissolving tablets are usually taken without water and are expected to disperse, and dissolve once placed in the tongue allowing large fraction of the administered dose to be absorbed from the buccal cavity. Also, sublingual tablets are retained under the tongue and dissolve rapidly, which provide rapid drug absorption. Furthermore, increasing the drug resident time in the buccal cavity has been achieved by incorporation of drugs in chewing gums to slowly release the drug during chewing, and by using bioadhesive formulations to retain the dosage forms in the buccal cavity and release the drug over an extended period of time (12). Esophagus: Drugs are not absorbed from the esophagus because of its very short transit time, very thick mucosal lining, and very small surface area. Stomach: The stomach is not a major site for drug absorption because of its limited surface area and short transit time. The average gastric pH is about 1–3, which can cause decomposition of acid-labile drugs. This acidic medium of the stomach enhances the dissolution of basic drugs and maintains the dissolved acidic drugs in the more absorbable unionized form. One of the important factors affecting the rate of drug absorption is the gastric emptying rate which determines the rate at which the drug reaches the small intestine, the main site for drug absorption. The gastric emptying rate is delayed by food with large, high viscosity, and fat rich meals delaying the gastric emptying. Drugs such as narcotic analgesics and anticholinergic drugs delay the gastric emptying rate, while prokinetic drugs such as metoclopramide usually speed the gastric emptying rate. Diseases such as pyloric stenosis, gastroenteritis, and gastroesophageal reflux can slow the gastric emptying rate. Formulation strategies that can increase the gastric retention of dosage forms are useful when the formulation is intended to deliver the drugs to the stomach such as in the treatment of Helicobacter Pylori infection. Also, this approach can be used to develop controlled release formulations for acid soluble drugs or for drugs with site specific absorption from the upper part of the small intestine to increase the extent of their absorption. Gastric retention can be achieved using floating tablets that are low-density formulations that can absorb water, swell and float above the gastric contents delaying their gastric emptying (13). Mucoadhesion is another technique used to increase the formulation gastric retention (14). Furthermore, formulations that can expand or unfold in the stomach to become too large to exit through the pylorus have been tried to increase the formulation retention in the stomach. The effect of increasing the gastric transit time on the extent of drug absorption depends on the drug stability in the stomach and the mechanism of drug absorption whether it is passive or active. Small intestine: The small intestine is about 6 m in length with the first 20–30 cm represent the duodenum, the following 2.5 m constitute the jejunum, and the rest of the small intestine is the ileum. It is the main site of absorption for most drugs because of its high surface area due to the presence of folds of Kerckring, villi, and microvilli, as illustrated in Figure 5.4. The jejunum and ileum are supplied by branches from the superior mesenteric artery and drain to the portal vein that takes the blood to the liver where a fraction of the absorbed drug can be metabolized before reaching the systemic circulation. The intestinal transit time is about 6–8 hr, which is an important factor in determining the absorption of drugs from different dosage forms. It depends on the presence of food, pathological conditions, drugs, and the dosage from characteristics. The presence of food usually prolongs the intestinal transit time. Conditions such as diarrhea and irritable bowel syndrome shorten the transit time, while constipation, intestinal obstruction, and autonomic neuropathy usually delay the intestinal transit time. Narcotic analgesics and

76  Drug Absorption Following Extravascular Administration

Figure 5.4 The different structure features that increase the surface area of the drug absorption surface in the small intestine.

drugs with anticholinergic activity prolong the intestinal transit time, while prokinetic drugs shorten the intestinal transit time. When designing dosage forms release the drug over > 8 hr, drug stability and colon absorption are necessary for complete absorption. The small intestine has high expression of the p-glycoprotein (P-gp) transport system that can transport a wide variety of drugs outside the absorptive epithelial cells. The P-gp transport system is localized mainly on the apical side of the epithelial cells and can transport the drug outside the epithelial cell back to the intestinal lumen. Drugs that are substrates for P-gp usually have reduced bioavailability and inhibition of this transport system can affect the extent of absorption of these drugs. Another important factor that affects the extent of drug absorption is the drug metabolism in the small intestine and liver before reaching the systemic circulation. The small intestine and liver contain a variety of drug-metabolizing enzymes, including the Cytochrome P450 (CYP 450) enzymes that can metabolize many drugs before reaching the systemic circulation. Drug metabolism in the small intestine and liver during the absorption process is called the presystemic metabolism and can reduce the bioavailability of many drugs. Administration of drugs that can inhibit these enzymes can affect the bioavailability of many drugs, especially those that undergo extensive presystemic metabolism. Enteric coated formulations have been used to deliver acid-labile drugs to the small intestine by coating the formulation with an acid resistant film that can protect the formulation from the acidic pH of the stomach. Once in the alkaline environment of the small intestine, the film dissolves releasing the drug in the small intestine. Drug absorption in this case is delayed depending on the gastric emptying rate of the formulation. This delay in drug absorption, which is also known as the lag time for drug absorption, can be short when the drug is administered in the fasted state or long when the drug is administered after meal. Controlled release formulations are usually developed to have a slow and constant rate of drug release over an extended period of time. This slow rate of drug release from the formulation results in slow and relatively constant rate of drug absorption to the systemic circulation, reducing the fluctuations in plasma drug concentrations during repeated drug administration. The most common formulation approaches for controlling drug release include the matrix systems, the membrane-controlled systems, and the osmotic pump systems (15). The matrix-controlled release delivery systems are prepared by dispersing the

Drug Absorption Following Extravascular Administration 77 drug in a water-soluble matrix, then the drug is released from the delivery system when the matrix dissolves in the GIT contents. While in the membrane-controlled systems, the tablets or pellets are coated with a membrane and act as drug reservoir. Water diffuses through the membrane into the delivery system, dissolves the drug and the membrane controls the rate of drug diffusion out of the dosage form. Whereas the osmotic pump systems are similar to the membrane-controlled systems where the tablet is coated with a membrane that allows water to pass to the core and dissolves the tablet contents. The dissolved drug is released from the system through holes drilled in the membrane and the drug release rate can be controlled by controlling the rate of water diffusion across the membrane into the core or by controlling the viscosity of the solution formed inside the system. Large intestine: It is approximately 130 cm in length with a diameter larger than that of the small intestine. It consists of the cecum, ascending, transverse, descending, sigmoid colon, rectum, and anus. The surface area per unit length of the large intestine is much less than that of the small intestine because of the lack of villi. The colonic wall has limited metabolic activity compared to the small intestine. The large intestine has many aerobic and anaerobic bacteria that have digestive and metabolic functions and can cause drug degradation. Drug absorption occurs mainly in the cecum and the ascending colon because as you go further in the large intestine, the contents increase in viscosity due to water absorption causing less mixing, slower solubility, and lower chance for the drug to get in contact with the absorption surface. The reduced surface area of the large intestine is balanced by the prolonged transit time, which can lead to significant drug absorption which occurs mainly by passive diffusion and via the aqueous pores. The drugs absorbed from the large intestine, except for the lower part of the rectum, reach the liver with the portal circulation where it can undergo presystemic hepatic metabolism. Drug delivery to the colon can be useful for treating colonic diseases such as inflammatory diseases and infections, or for the systemic effect of drugs that can be absorbed from the colon. The absorption of drugs from the colon is usually delayed and depends on the gastric and intestinal transit time. The formulation strategies used for delivering drugs to the colon must protect the formulation in the stomach and small intestine and then release the drug in the ascending colon. The use of acid resistance coating with controlling the drug release to start 4–6 hr after leaving the stomach allows the delivery of the drug to the ascending colon (16). Also, formulations that contain a prodrug, which can be hydrolyzed by the colonic bacteria to liberate the active drug in the colon, have been used in formulations containing salicylazosulphapyridine for the treatment of inflammatory bowel disease (17). Additionally, coating of the formulations with polymers that undergo degradation by the colonic bacteria can release the drug in the colon. 5.4.3  Rectal Drug Administration

The rectum is the last 15–20 cm of the large intestine, which can be used for rectal drug administration in the form of suppositories and enemas. Rectal administration has several advantages, including bypassing the presystemic hepatic metabolism when the drug is absorbed from the lower part of the rectum. It is also convenient in some patient populations such as elderly, children, and in unconscious patients when the oral route is not possible and parenteral formulations are not available. Rectal drug absorption is limited

78  Drug Absorption Following Extravascular Administration by the short rectal transit time, and the small surface area of the rectum. So, rapid drug release from the rectal formulation is required. Drug solubility in the base of the dosage form is important in determining the rate of drug release, with lipophilic drugs released faster from hydrophilic bases and hydrophilic drugs released faster from lipophilic bases (18). The rapid release produces high drug concentration due to the limited spreading of the rectal contents to the colon, and the small dilution of the released drug in the limited volume of rectal contents. This high drug concentration produced in the rectum is the driving force for drug absorption, which is mainly by passive diffusion. Absorption by passive diffusion requires balanced drug hydrophilic and lipophilic properties to ensure adequate solubility in the rectal contents and permeability across the biological membrane. 5.4.4  Intranasal Drug Administration

The nasal cavity starts with the nostrils that open at the back into the nasopharynx and lead to the trachea and esophagus. The mucosal lining of the nasal cavity varies in thickness and vascularity and is covered by an epithelial cell layer covered by microvilli, which significantly increases the surface available for absorption. The cells and glands of the mucosa secrete mucus that forms a continuously renewed layer covering the nasal cavity. The anterior part of the nasal cavity is covered by ciliated epithelial cells that together with the mucus layer play an important role in clearing particles deposited in the nasal cavity. The mucociliary mechanism is important because it can also clear drugs administered to the nasal cavity either backward to the nasopharynx where they can be swallowed or forward where they can be removed by sneezing. Drugs administered by the intranasal route can be absorbed rapidly to the systemic circulation bypassing the hepatic first-pass metabolism and producing high plasma drug concentration. The main mechanisms of drug absorption from the nasal cavity are passive diffusion and through the aqueous pores. The rate of mucus flow and the drug retention in the nasal cavity are important factors that affect drug absorption from the nasal cavity. Drug formulations administered into the nasal cavity may contain the drug in the form of solution, suspension, emulsion, and dry powder. Small molecular weight drugs have relatively good systemic absorption with the absorption rate decreasing sharply when the molecular weight exceeds 1000 Daltons. The absorption of drugs after intranasal administration can be enhanced by buffering the formulation to the pH that provides the optimum nasal drug absorption (19). Other strategies to enhance nasal drug absorption include the use of penetration enhancer to increase the paracellular drug absorption. Also, increasing the formulation viscosity and using mucoadhesive have been used to prolong the nasal residence time to allow more time for drug absorption. 5.4.5  Pulmonary Drug Administration

The respiratory system starts with the nose and includes the pharynx, larynx, trachea, bronchi, and bronchioles and ends up with the alveoli. The main function of the respiratory system is oxygenation of the blood and the removal of carbon dioxide, which occurs mainly in the lobules that contain the alveoli. The presence of ciliated epithelial layer and mucous in the upper respiratory tract, bronchi, and bronchioles provides an efficient cleansing mechanism to protect the lung from any inhaled foreign particles. The alveolar

Drug Absorption Following Extravascular Administration 79 epithelium and the capillary endothelium are highly permeable to water, most gases, and lipophilic compounds. However, they are impermeable to hydrophilic substances, ionized compounds, and large molecular weight molecules. The large surface area and the high blood flow allow rapid absorption of any substance, which can permeate through the alveoli-capillary membrane (20). Drugs are usually delivered to the pulmonary system as very fine liquid or solid particles using different types of inhalation devices, including metered-dose inhalers, dry powder inhalers, and nebulizers. Pulmonary drug delivery for local drug effects, as in case of using corticosteroids and α-agonists in the management of bronchial asthma, has the advantages of lower systemic adverse effects, rapid onset of action, and the use of smaller doses of the drugs. While for the systemic effect, the inhaled drugs must be delivered to the sites where they can be efficiently absorbed to reach the systemic circulation. A fraction of the inhaled drug dose is usually deposited in the respiratory tract and expelled by the mucociliary system where the drug can be swallowed and absorbed from the GIT. Several factors can affect the site of deposition of the aerosol particles, including particle size, speed of delivery, particle charge, and density. Particles in the range from 0.5 to 5 µm can travel longer distances down the respiratory tract, while larger particle size particles can collide with the mucus layer and then removed by the mucociliary clearance. Other factors affecting the site of particle deposition include obstructive airway diseases and rate of respiration. Rapid respiration rate causes premature deposition of inhaled particles and slow rate of respiration allows more time for particles to deposit in distal sites of the lung. 5.4.6  Transdermal Drug Administration

The main function of the skin is to provide sensation and protection from the surrounding environment. The skin consists of three layers; the outer epidermis layer, followed by the dermis, and the inner SC fat layer, as illustrated in Figure 5.5. The epidermis is a dry and tough layer that constitutes a barrier for the penetration of substance from the

Figure 5.5 Schematic presentation of the different skin layers showing the different drug transport mechanisms across the skin.

80  Drug Absorption Following Extravascular Administration environment into the body and prevents the loss of water, electrolytes, and nutrients from the body. The outer layer of the epidermis is the stratum corneum that is formed of layers of keratinized dead cells and provides the main barrier for drug absorption across the skin. The dermis is a fibrous layer that provides support for the epidermis and is supplied by blood vessels, lymph vessels, and nerves and contains hair follicles, sweat and sebum gland. The fat layer provides an insulation layer and provides storage area for fat and nutrients. Drug application to the skin can be used to treat local dermatological conditions and to deliver drugs that produce systemic effects over an extended period of time. Drug absorption after transdermal application starts with drug release from the delivery system, then diffusion across the stratum corneum, epidermis, and dermis where it can reach the systemic circulation through the existing capillary network. Drugs applied to the skin can be absorbed by the transcellular and paracellular pathways with lipophilic drugs absorbed mainly by the transcellular pathway. Additional minor pathways include absorption through the hair follicles and sebaceous glands. Factors that can affect the rate of drug absorption across the skin include the surface area of drug application, skin thickness at the site of application, hydration, and occlusion of the site of application (21). Transdermal drug delivery provides an alternative route of administration for some drugs when the oral administration is not possible, and the drugs absorbed bypass the first-pass hepatic metabolism. The rate of drug absorption after transdermal application can be controlled to provide sustained plasma drug concentrations for extended period, which is useful for administration of drugs with short half-life. However, transdermal delivery systems can cause immunological sensitization and irritation at the site of application. These delivery systems are suitable only for potent drugs because of the limited drug absorption across the skin and are more expensive than conventional dosage forms. The general features of these transdermal delivery systems include a drug reservoir that contains the drug and a mechanism that controls the rate of drug release from the delivery system. These delivery systems are designed to have slow rate of drug release, so the release rate becomes the rate limiting step in the drug absorption process. This will limit the effect of the physiological variables on the drug absorption rate. Several formulation strategies have been used to improve the absorption of the drugs applied to the skin including hydration of the stratum corneum by using occlusive devices such as patches or by using moisturizing additives, also, using penetration enhancers and the use of prodrugs that can liberate the active drug in vivo. Nitroglycerin transdermal patches have been used by ischemic heart disease patients and scopolamine transdermal patches are used to prevent nausea and vomiting caused by motion sickness. 5.5 Integration of the Physical, Chemical, and Physiological Factors Affecting Drug Absorption Studying drug absorption requires consideration of the drug physicochemical properties in conjunction with the physiological conditions at the different sites of drug administration. A balance in the hydrophilic and lipophilic drug properties is important because drug absorption requires drug dissolution in the aqueous environment at the site of administration and permeation across the lipoid membrane. Attempts have been made to utilize the physicochemical properties and physiological behavior to classify the drugs to

Drug Absorption Following Extravascular Administration 81

Figure 5.6 The biopharmaceutical classification system as described by Amidon et al based on the drug solubility and permeability.

different classes with similar in vivo oral absorption characteristics. This effort resulted in the development of the Biopharmaceutics Classification System (BCS), and the Biopharmaceutics Drug Disposition Classification System (BDDCS). 5.5.1  The Biopharmaceutics Classification System

Drug absorption from the GIT is dependent primarily on the solubility of the drug in the GIT lumen and the ability of the dissolved drug to permeate across the GIT membrane. The BCS was proposed in 1995 by Amidon et al who utilized the solubility and permeability data to predict the drug in vivo absorption characteristics after administration in the form of immediate release oral solid dosage forms (22, 23). The BCS categorizes the drug solubility and permeability as either high or low to produce four different categories, as presented in Figure 5.6. Rapid dissolution: An immediate release product is considered rapidly dissolving when at least 85% of the labeled amount of the drug dissolves within 15 min using standard dissolution apparatus in 900 mL or less of 0.1 N HCl, simulated gastric fluid or pH 4.5, and simulated intestinal fluid or pH 6.8. Solubility: The drug is considered highly soluble if the drug amount in the highest strength of the fast-release product is soluble in 250 mL or less of aqueous media in the entire pH range of 1.0–7.5 at 37°C. Permeability: There is no strict categorization of the drug permeability because of the many factors that affect the process. Different methods have been used by different researchers to assess drug permeability, which resulted in the assignment of some drugs to different classes depending on the method used. The following are different ways to assess the drug permeability.

• The permeability can be assessed from the extent of absorption determined in human

pharmacokinetic studies. High permeability is concluded if the drug absolute bioavailability is ≥90% of the administered dose, or ≥90% of the administered dose is recovered in urine as unchanged drug or as the sum of the unchanged drugs and oxidative and conjugated metabolites.

82  Drug Absorption Following Extravascular Administration

• Drugs with predicted human absorption of ≥90% of the administered dose using ani-

mal models such as the rat perfusion model or in vitro models such as the Caco-2 cell monolayer model are considered highly permeable. • Metoprolol has been used as a reference standard for the high/low permeability boundary since it is 95% absorbed from the GIT. Drugs with log P values greater than that of metoprolol (log P = 1.72) are considered highly permeable. The BCS has been adopted by the US Food and Drug Administration (FDA) to allow waiver of the in vivo bioequivalence testing for immediate release rapidly dissolving solid oral dosage forms of BCS class I drugs. The in vivo bioequivalence study is a study performed in normal volunteers to compare the rate and extent of drug absorption from a new generic product and the marketed innovator’s product that contain the same active drug. Different immediate release products of BCS class I drugs should have rapid in vivo dissolution and high permeability, based on the properties of class I drugs, which should lead to similar rate and extent of drug absorption. This means that these products are bioequivalent because the rate and extent of absorption of BCS class I drugs from immediate release formulations are not dependent on the formulation characteristics. So, for regulatory purposes, the in vivo bioequivalent study for immediate release solid oral formulation of BCS class I drugs can be waived and replaced by in vitro experiments. These in vitro experiments should demonstrate that the product fulfills the criteria for immediate release formulations, and the solubility and permeability of the drug fulfill the BCS class I requirements. Also, different immediate release formulations of BCS class III drugs are expected to have rapid in vivo drug dissolution and the drug absorption will be limited by the low permeability of the drug. This indicates that the absorption of BCS class III drugs is also not affected by the formulation characteristics unless the formulation contains additives that can affect the drug permeability across the membrane. Based on this information, the in vivo bioequivalence studies for immediate release solid oral dosage forms of BCS class III drugs can be waived when no additives that can affect the drug permeability across the GIT membrane are included in the formulation. Again, in vitro testing will be required to prove that the product fulfills the immediate release criteria, and the drug solubility and permeability fulfill the BCS Class III requirements. The BCS has been used extensively by the pharmaceutical industry throughout the drug discovery and development processes. This is because this classification system allows assessment of the effect of formulation factors on the rate and extent of drug absorption after oral administration of immediate release formulations. Drugs in the BCS class I are highly soluble and highly permeable in the entire pH range and their absorption should not be sensitive to formulation factors when administered as rapidly dissolving dosage forms, while drugs in the BCS class III are highly soluble and their absorption is limited by the permeability across the membrane. This makes BCS class III drugs also not sensitive to formulation factors unless the formulation is designed to promote the drug permeability across the GIT membrane. Conversely, formulations containing BCS class II drugs usually require strategies that can enhance the drug solubility and/or dissolution rate. While formulations for BCS Class IV drugs are the most challenging because these drugs require strategies for enhancing the drug solubility, dissolution rate, and membrane permeability.

Drug Absorption Following Extravascular Administration 83

Figure 5.7 The biopharmaceutical drug disposition classification system as described by Wu and Benet based on the drug solubility and extent of drug metabolism. 5.5.2  The Biopharmaceutics Drug Disposition Classification System (BDDCS)

The main objective of the BCS described previously is to use the drug solubility and permeability information to predict the in vivo absorption behavior of drugs administered in the form of immediate release oral solid dosage forms. The BDDCS is a classification system that utilizes the drug solubility and the extent of drug metabolism as a tool to predict the absorption, disposition characteristics, and the drug-drug interaction potential of drugs. In 2005, Wu and Benet reviewed a large number of drugs that were classified in the literature according to the four BCS categories. They recognized that the major route of elimination for the high permeability class I and class II drugs in human is via enzymatic metabolism, while the major route of elimination for the low permeability class III and class IV drugs is through the renal and biliary excretion of the unchanged drugs. So, the BDDCS was proposed which classifies the drugs according to their solubility and extent of metabolism, rather than the solubility and permeability used by the BCS, as in Figure 5.7 (24, 25). Since the extent of drug metabolism can be estimated better than the permeability, this new classification system decreases the number of drugs that are classified in more than one class due to the uncertainty of permeability determination using different techniques. The correlation between the extent of drug metabolism and the drug permeability arises from the ability of the highly permeable drugs to cross the biological membranes and get access to the metabolizing enzymes, while low permeability drugs usually have limited access to the metabolizing enzymes and are excreted unchanged from the body. Extensive metabolism was defined as ≥90% metabolism, which means that after a single oral administration of the highest drug strength, a mass balance study can account for ≥90% of the administered dose in the form of metabolites. The objectives of the BDDCS were to predict the role of GIT and liver transporters following oral administration and to predict the potential for drug-drug interactions. Role of transporters:

• Class I drugs are highly soluble, highly permeable, and extensively metabolized. These drugs dissolve rapidly in the GIT, have easy access to the metabolizing enzymes because of their high permeability, and then undergo extensive metabolism. So, the

84  Drug Absorption Following Extravascular Administration transporters do not have a significant role in the disposition of these drugs in the intestine and liver. However, the in vivo pharmacokinetic behavior of these drugs may be influenced by the transporters in other organs such as the kidney and blood brain barrier. • Class II drugs are poorly soluble, highly permeable, and extensively metabolized. The amount of these drugs available in the GIT for absorption is limited by their low solubility. However, once in solution they can readily cross the GIT membrane because of their high permeability. The drug in the epithelial layer of the GIT can be metabolized, effluxed back to the gut lumen by the transporter, or crosses the basolateral membrane and reaches the portal circulation. Efflux transport of the drugs back to the gut lumen can significantly reduce the bioavailability of many drugs. The drug molecules that are effluxed to the gut lumen can be absorbed again and will have another chance to be metabolized, which demonstrates the interplay between the transporters and the metabolizing enzymes. So, manipulation of the efflux transporter activity can affect the extent of drug absorption and extent of metabolism in the intestinal wall. The transporters that control the drug uptake and efflux in the liver can have similar effect on the extent of hepatic drug metabolism. So, for class II drugs, the role of efflux transporter dominates the drug disposition in the intestine, but both uptake and efflux transporters can affect the drug disposition in the liver. • Classes III and IV drugs have low permeability and poor metabolism. The uptake transporters play an important role in the intestinal absorption and liver entry of these poorly permeable drugs. Once inside the cells, the efflux transporters can also affect the disposition of these drugs. So, both uptake and efflux transporters can play a significant role in the disposition of these drugs. Drug-drug interaction potential: The BDDCS cannot predict the significant drug-drug interactions for all drugs; however, it can point out the important drug-drug interactions that should be investigated further.

• Class I drugs are expected to be affected significantly by metabolic interactions in the intestine and liver.

• Class II drugs are most likely affected by metabolic, efflux transporter, and efflux trans-

porter-enzyme interplay drug interactions in the intestine. Class II drugs can also be affected by metabolic, uptake transporter, efflux transporter, and transporter-enzyme interplay drug interactions in the liver. • Classes III and IV drugs are likely affected by drug interactions that involve the uptake transporter, efflux transporter, and uptake-efflux transporter interplay. 5.6 Summary

• The absorption of drugs is a complex process affected by factors related to the drug

physicochemical properties, the biological and physiological factors related to the site of administration, and the dosage form characteristics. • The physiological conditions and absorption barriers are different for the different sites of administration, which can be favorable or unfavorable for drug absorption.

Drug Absorption Following Extravascular Administration 85

• Different drugs have different structure features and physicochemical properties,

which can influence the dissolution and permeation across the absorptive membrane at the different sites of administration. • The formulation of the dosage forms utilizes the drug physicochemical properties and the physiological conditions at the intended site of administration to control the rate of drug release after administration. • The discussion of the different factors affecting drug absorption is a very simple description of a very complex process. References 1. Singer SJ and Nicholson GL “The fluid mosaic model of the structure of cell membrane” (1972) Science; 153:1010–1012. 2. Higuchi WI and Ho NFH “Membrane transfer of drugs” (1988) Int J Pharm; 2:10–15. 3. Oh DM, Han HK and Amidon GL “Drug transport and targeting: Intestinal transport” (1999) Pharm Biotechnol; 12:59–88. 4. Lipinski CA, Lombardo F, Dominy BW and Feeney PJ “Experimental and computational approaches to estimate solubility and permeability in drug discovery and development settings” (1997) Adv Drug Delivery Rev; 23:4–25. 5. Taniguchi C, Kawabata Y, Wada K, Yamada S and Onoue S “Microenvironmental pH-modification to improve dissolution behavior and oral absorption for drugs with pH-dependent solubility” (2014) Expert Opin Drug Deliv; 11:505–516. 6. Serajuddin ATM “Salt formation to improve drug solubility” (2007) Adv Drug Deliv Rev; 59:603–616. 7. Singhal D and Curatolo W “Drug polymorphism and dosage form design: A practical perspective” (2004) Adv Drug Deliv Rev; 56:335–347. 8. Laza-Knoerr AL, Gref R and Couvreur P “Cyclodextrins for drug delivery” (2010) J Drug Target; 18:645–656. 9. Horter D and Dressman JB “Influence of physicochemical properties on dissolution of drugs in the gastrointestinal tract” (1997) Adv Drug Deliv Rev; 25:3–14. 10. Johnson KC and Swindell AC “Guidance in the setting of drug particle size specifications to minimize variability in absorption” (1996) Pharm Res; 13:1795–1798. 11. Chaudhary K, Patel MM and Mehta PJ “Long-acting injectables: Current perspectives and future promise” (2019) Crit Rev Ther Drug Carrier Syst; 36:137–181. 12. Nagai T and Konishi R “Buccal/gingival drug delivery systems” (1987) J Cont Rel; 6:353–360. 13. Ingani HM, Timmermans J and Moes AJ “Concept and in vivo investigation of peroral sustained release floating dosage forms with enhanced gastrointestinal transit” (1987) Int J Pharmacut; 35:157–164. 14. Burton S, Washington N, Steele RJC, Musson R and Feely L “Intragastric distribution of ionexchange resins: A drug delivery system for the topical treatment of the gastric mucosa” (1995) J Pharm Pharmacol; 47:901–906. 15. Collett JH and Moreton RC “Modified-release peroral dosage forms” in “Aulton’s pharmaceutics, the design and manufacture of medicines”, 3rd Edition, Edited by Aulton ME (2007) Churchill Livingstone Elsevier, Edinburgh, pp 483–499. 16. Dew MJ, Hughes PJ, Lee MG, Evans BK and Rhodes J “An oral preparation to release drugs in the human colon” (1982) Br J Clin Pharmacol; 14:405–408. 17. Rubinstein A “Microbially controlled drug delivery to the colon” (1990) Biopharm Drug Disposition; 11:465–487. 18. DeBoer AG, Moolenaar F, deLeede LGJ and Breimer DD “Rectal drug administration: Clinical pharmacokinetic considerations” (1982) Clin Pharmacokinet; 7:285–311.

86  Drug Absorption Following Extravascular Administration 19. Chien YW, Su KSE and Chang S “Nasal systemic drug delivery” in “Drugs and the pharmaceutical sciences”, Edited by Chien YW (1989) Marcel Dekker, New York, NY, pp 1–26. 20. Siekmeier R and Scheuch G “Systemic treatment by inhalation of macromolecules: Principles, problems and examples” (2008) J Physio Pharmacol; 59:53–79. 21. Ranade VV “Drug delivery systems: 6. Transdermal drug delivery” (1991) J Clin Pharmacol; 31:401–418. 22. Amidon GL, Lennernas H, Shah VP and Crison JR “A theoretical basis for a biopharmaceutic drug classification: The correlation of in vitro drug product dissolution and in vivo bioavailability” (1995) Pharm Res; 12:413–420. 23. Dahan A, Miller JM and Amidon GL “Prediction of solubility and permeability class membership: Provisional BCS classification of the world’s top oral drugs” (2009) AAPS J; 11:740–746. 24. Wu C-Y and Benet LZ “Predicting drug disposition via application of BCS: Transport⁄absorption⁄ elimination interplay and development of a biopharmaceutics drug disposition classification system” (2005) Pharm Res; 22:11–23. 25. Benet LZ “Predicting drug disposition via application of a biopharmaceutics drug disposition classification system” (2009) Basic Clin Pharmacol Toxico; 106:162–167.

6

Drug Pharmacokinetics Following Single Oral Drug Administration The Rate of Drug Absorption

Objectives After completing this chapter, you should be able to:

• Describe the general shape of the plasma drug concentration-time profile after single oral dose.

• Explain why the rate and the extent of drug absorption after oral administration affect the onset and the duration of drug effect.

• Evaluate the rate of drug absorption by comparing the plasma drug concentrationtime profiles.

• Calculate the drug pharmacokinetic parameters after oral drug administration. • Discuss the clinical importance of the rate of drug absorption after oral drug administration.

• Analyze how the change in the absorption rate constant affects the plasma concentration-time profile after single oral administration.

• Apply the method of residuals and the Wagner-Nelson method to calculate the absorption rate constant after single oral administration.

6.1 Introduction Drugs administered by any extravascular route have to be absorbed into the systemic circulation before producing their systemic effects. The oral route is commonly used because of its convenience, especially when the drug is used to treat chronic diseases. The rate of absorption affects the rate at which the drug appears in the systemic circulation and is determined by the rate constant for the absorption process. While the extent of drug absorption reflects the fraction of the administered dose that reaches the systemic circulation, and it determines the bioavailable amount of the drug. The pharmacokinetic parameters that describe the rate and extent of drug absorption together with the clearance and volume of distribution determine the plasma drug concentration-time profile and hence the onset and intensity of the drug effect for most drugs are illustrated in Figure 6.1. The assumption is that the plasma drug concentration must exceed the minimum effective concentration (MEC) to produce the effect, and the intensity of effect is proportional to the plasma drug concentration. After single oral administration, the rate of drug absorption affects the onset of drug effect with rapidly absorbed drugs having a faster onset of effect. Also, the extent of drug absorption affects the amount of the drug that reaches the systemic circulation, with a larger extent of drug absorption producing higher plasma drug concentration and more intense effect. The period the plasma drug concentration stays above the MEC, which is DOI: 10.4324/9781003161523-6

88  Single Oral Drug Administration, Rate of Drug Absorption

Figure 6.1 The relationship between the plasma concentration-time profile and the onset, intensity, and duration of the drug effect.

determined from the entire drug concentration-time profile, is important in determining the duration of drug effect. The following discussion deals with the kinetics of drug absorption after oral administration. However, the same principles can be applied to drug absorption after administration by the other extravascular routes. 6.2  Drug Absorption after Oral Administration Drug absorption is a complex process that involves a series of steps starting from the disintegration of solid dosage forms to small particles, then dissolution of the drug from these particles in the gastrointestinal tract (GIT), and then permeation of the dissolved drug across the biological membrane as illustrated in Figure 6.2. Drugs administered as powders or suspensions will have to dissolve and then permeate across the membrane, while drugs administered in the form of solutions are ready to be absorbed once they reach their site of absorption. In general, the rate of drug absorption is usually faster from solution > suspension > capsules > tablets. So, the characteristics of the dosage forms are important in determining the rate and the extent of drug absorption. The rate constant that reflects the rate of disintegration, dissolution, and absorption processes and describes the rate of the overall absorption process is the absorption rate constant (ka). The absorption rate constant after administration of different dosage forms for the same active drug can be different because of differences in the formulation characteristics. Some dosage forms are formulated to control their rate of drug release to control the rate of drug absorption. The absorption of most drugs follows first-order kinetics; however, in some cases, the absorption follows zero-order kinetics.

Figure 6.2  The processes involved in the absorption of drugs administered as solid dosage forms.

Single Oral Drug Administration, Rate of Drug Absorption 89 6.2.1  Zero-Order Drug Absorption

When the absorption process follows zero-order kinetics, the rate of drug absorption is constant and is independent of the drug amount available for the absorption process. This usually occurs when the drug is absorbed only by a carrier-mediated transport system and the transport system is saturated. In this case, the rate of drug absorption becomes constant and independent of the drug concentration at the site of drug absorption. Also, modified-release formulations can be designed to release the drug at a constant rate over an extended period. When the rate of drug release from the dosage form is the ratedetermining step for the absorption process, the rate of drug absorption will be constant and will follow zero-order kinetics. The zero-order absorption rate constant has units of amount/time. After administration of single dose of a drug that follows zero-order absorption and first-order elimination, the plasma drug concentration-time profile will have a slow rise in the drug concentration during the drug absorption phase, followed by a slow decline in the drug concentration during the elimination phase. 6.2.2  First-Order Drug Absorption

After oral administration of an immediate release formulation, it is assumed that all the administered dose becomes ready to be absorbed after a short time. The amount of drug in the GIT (Ai) after oral administration declines exponentially at a rate dependent on the absorption rate constant, ka, when the absorption process follows first-order kinetics as in Figure 6.3. The amount of the drug in the GIT at any time can be described by Eq. 6.1, assuming that the decline in drug amount in the GIT is only due to drug absorption which follows first-order kinetics. The first-order absorption rate constant has units of time−1. A i = Dose e − kat 

(6.1)

The block diagram in Figure 6.4 represents the pharmacokinetic model for oral drug administration assuming first-order absorption and elimination. The rate of change of

Figure 6.3 The drug amount-time profile in the GIT and in the body after administration of single oral dose of a drug that follows first-order absorption and elimination.

90  Single Oral Drug Administration, Rate of Drug Absorption

Figure 6.4 A diagram representing the pharmacokinetic model that describes the drug absorption from the GIT. Ai is the amount of the drug in the GIT, A is the amount of the absorbed drug in the body, ka is the first-order absorption rate constant, and k is the first-order elimination rate constant.

the amount of the drug in the body (A) depends on the rate of drug absorption (ka Ai) and the rate of drug elimination (k A) as in Eq. 6.2. dA = ka A i − kA dt 

(6.2)

Equation 6.2 indicates that the amount of the drug in the body increases if the rate of drug absorption is larger than the rate of drug elimination and decreases if the rate of drug elimination is larger than the rate of drug absorption.

• Initially, the rate of drug absorption is at its highest rate because the total dose is in • • • •

the GIT. Also, the rate of drug elimination is at its lowest rate because no drug is in the body at the time of drug administration. So, the drug amount in the body increases. With time, the rate of drug absorption decreases because of the decrease in the amount of the drug remaining to be absorbed. Also, the rate of elimination increases due to the increase in drug amount in the body. When the rate of drug absorption becomes equal to the rate of drug elimination, the maximum amount of the drug in the body is achieved. After that, the rate of drug elimination becomes larger than the rate of drug absorption and the amount of the drug in the body decreases. So, after administration of single oral dose of a drug that follows first-order absorption and elimination, the drug amount-time profile in the body increases initially and then decreases as in Figure 6.3.

Integrating Eq. 6.2 from time zero to ∞ yields Eq. 6.3 which describes the drug amounttime profile at any time. A = 

(

)

A i0 ka − kt e − e − kat ka − k 

(6.3)

where Ai0 is the initial amount of the drug in the GIT, which is equal to the dose (D) of the drug. The orally administered dose is not always absorbed completely; so the term (F) can be included in the equation to account for the fraction of administered dose that will be absorbed. Dividing Eq. 6.3 by Vd gives Eq. 6.4, which is the general equation that describes the plasma drug concentration-time profile after administration of single oral dose. Cp =  

(

)

FDka e − kt − e − kat Vd(ka − k) 

(6.4)

Single Oral Drug Administration, Rate of Drug Absorption 91 This equation has two exponents; one includes the rate constant for the absorption process and the other includes the rate constant for the elimination process, indicating that the plasma concentration-time profile for the drug after oral administration is dependent on the absorption and elimination processes. Clinical Importance:

• The rate of drug absorption affects the rate of its appearance in the systemic circulation and the onset of drug effect.

• The rate of drug absorption can be different after administration of different products

for the same active drug. So, evaluation of the rate of drug absorption after administration of different drug products is important. • Fast absorption is usually required when immediate therapeutic effect is needed such as in emergency cases and for the treatment of acute symptoms. For example, administration of rapidly absorbed coronary vasodilators is required for the management of acute ischemic heart attack. • Rapid drug absorption is not always better than slow absorption. Rapid drug absorption during multiple drug administration results in large fluctuations in plasma drug concentrations within each dosing interval, which may lead to fluctuation in the drug effect during the dosage interval. • Slow drug absorption is preferred during multiple drug administration for the management of chronic diseases. This is because it produces small fluctuations in drug concentrations and produces a steady drug effect. The slowly absorbed formulations of antihypertensive medications, such as indapamide, diltiazem, and verapamil, and antidiabetic medications, such as phenformin and glimepiride, are commonly used by hypertensive and diabetic patients. 6.3  The Plasma Concentration-Time Profile After Single Oral Dose The plasma drug concentration-time profile presented in Figure 6.5 is an example of the drug profiles obtained after single oral drug administration. The profile includes an initial absorption phase when the plasma drug concentration is increasing, then a maximum plasma drug concentration (Cpmax) is achieved at a time equal to tmax. After the Cpmax, drug absorption continues but the plasma drug concentration decreases. When the absorption process is completed, the elimination phase starts, and the plasma drug concentration decreases at a rate dependent on the rate of the elimination process. The initial phase is called the absorption phase; however, both absorption and elimination occur simultaneously during this time. While during the elimination phase the absorption process is completed, so the plasma drug concentration declines at a rate dependent on the rate of the elimination process. The time to achieve the maximum plasma concentration, tmax, reflects the rate of drug absorption with shorter tmax indicating faster drug absorption. It is dependent on the absorption rate constant and the elimination rate constant, as in Eq. 6.5. t max =  

lnka − lnk ln(k a /k)  = ka − k ka − k 

(6.5)

92  Single Oral Drug Administration, Rate of Drug Absorption

Figure 6.5 The plasma drug concentration-time profile after administration of a single oral dose of the drug.

The Cpmax can be calculated by substituting for the time in Eq. 6.4 by tmax: Cpmax =  

(

)

FDka e − ktmax − e − kat max Vd(ka − k) 

(6.6)

Larger ka that indicates faster drug absorption will make tmax shorter and Cpmax higher, if the other parameters (k and Vd) were kept constant. While larger k that indicates faster drug elimination will make tmax shorter and Cpmax lower, if the other parameters (ka and Vd) were kept constant. Clinical Importance:

• The drug pharmacokinetic parameters such as CLT, Vd, k, and t1/2 for the same ac-

tive drug should be similar after administration of different products by the same or by different routes of administration. Also, these parameters should be similar after administration of different doses of the same drug if the elimination follows first-order kinetics. However, the rate of drug absorption can be different after administration of different products for the same drug. • The time to achieve the maximum plasma drug concentration, tmax, can be used to compare the rate of drug absorption after administration of different products for the same active drug. • The maximum plasma drug concentration, Cpmax, cannot be used to compare the rate of drug absorption because it is affected by both the rate and extent of drug absorption. Practice Problems: Question: Calculate the plasma concentrations of a drug 2, 10, and 24 hr after administration of single 500-mg tablet, if Vd is 30 L, F is 0.85, ka is 0.23 hr−1, and k is 0.10 hr−1. Answer:  The plasma drug concentration at any time is described by Eq. 6.4.

Single Oral Drug Administration, Rate of Drug Absorption 93 Substitute in Eq. 6.4 for the values of pharmacokinetic parameters and different time values. Cp2hr =  

−1 0.85500 mg 0.23hr −1 −0.1hr −1 2hr −  e −0.23hr 2hr ) =  4.7 mg/L −1 −1 (e 30L(0.23hr −  0.1hr )

Cp4hr =  

−1 −1 0.85500 mg 0.23hr −1 (e −0.1hr 4hr −  e −0.23hr 4hr ) =  6.7 mg/L 30L(0.23hr −1 −  0.1hr −1)

Cp10hr =  

−1 0.85500 mg 0.23hr −1 −0.1hr −1 10hr −  e −0.23hr 10hr ) =  2.17 mg/L −1 −1 (e 30L(0.23hr −  0.1hr )

Question: Calculate the tmax and Cpmax in the previous problem. Answer:  t max =

Cpmax =

lnka − lnk ln0.23 − ln0.1 = = 6.4 hr ka − k 0.23 − 0.1

−1 −1 0.85500 mg 0.23hr −1 (e −0.1hr 6.4hr − e −0.23hr 6.4hr ) = 7.46 mg/L 30L(0.23hr −1 −  0.1hr −1)

Question: Do you expect tmax and Cpmax to be different if the administered dose was 1000 mg in the previous example? If different, calculate the new values. Answer:  From Eq. 6.6, Cpmax is proportional to the administered dose, so Cpmax will be 14.92 mg/L if the dose was 1000 mg. While tmax depends on k and ka, so it should be the same. When the plasma drug concentration-time curve is plotted on the semilog scale, the elimination phase of the profile will be linear because the plasma drug concentration decreases at a rate dependent on the elimination rate constant as in Figure 6.6. The

Figure 6.6 The plasma drug concentration-time profile after administration of single oral dose of the drug plotted on the semilog scale. The half-life and the first-order elimination rate constant can be calculated during the elimination phase of the drug.

94  Single Oral Drug Administration, Rate of Drug Absorption elimination rate constant and the half-life can be determined from the terminal elimination phase of the plasma drug concentration-time profile plotted on the semilog scale. However, this is only possible when the absorption process is completed, and the only process that affects the drug concentration is the elimination process. The elimination rate constant can be determined from the slope of the linear part of the curve on the semilog scale. The slope of the line is equal to −k/2.303. The half-life of the drug can also be determined by measuring the time required for any drug concentration in the linear part of the profile to decrease by 50%. When the linear part of the curve is back extrapolated to the y-axis as in Figure 6.6, the y-intercept is equal to the coefficient in the general equation for the drug concentration-time profile after oral administration as in Eq. 6.7. y-intercept  =

FDka Vd(ka − k) 

(6.7)

The value of the y-intercept can be determined graphically from the plasma drug concentration-time plot on the semilog scale. So, the y-intercept value can be used to calculate the Vd if the other parameters, F, D, k, and ka are known. The AUC after single oral drug administration can be expressed in terms of the other pharmacokinetic parameters as in IV drug administration. However, after oral administration, the fraction of dose that reaches the systemic circulation should be considered when calculating the AUC from the other pharmacokinetic parameters. So, the AUC from time zero to infinity after single oral dose can be expressed as in Eq. 6.8. AUCoral |tt ==∞0 =

FD F Dose = k Vd CL T 

(6.8)

where F is the oral bioavailability, and CLT is the total body clearance of the drug. Experimentally the AUC can be calculated from serial drug concentrations measured after drug administration using the trapezoidal rule, which will be discussed in detail in the next chapter. Knowing the AUC, F, and dose, the CLT can be determined. When the bioavailability is not known, the combined term CLT/F that is referred to as the oral clearance of the drug is calculated. 6.4  Determination of the Absorption Rate Constant Determination of the absorption rate constant is important because it governs the rate of drug absorption after oral administration. Evaluation of the clinical usefulness of a dosage form must include the rate of drug absorption especially when the drug is used for treating acute symptoms or emergency conditions. The superiority of one dosage form over the other can be based on the rapid absorption of the active drug. There are several graphical methods that can be used to calculate ka, the drug absorption rate constant. The method of residuals and the Wagner-Nelson method are commonly used to estimate the absorption rate constant.

Single Oral Drug Administration, Rate of Drug Absorption 95 6.4.1  The Method of Residuals

The method of residuals is a graphical method used to determine the drug absorption rate constant and has the following assumptions: a The absorption rate constant is larger than the elimination rate constant, i.e., ka > k. b Both drug absorption and elimination follow first-order kinetics. c The drug pharmacokinetics follow one-compartment model. The idea of the method of residuals is to characterize the drug elimination rate from the terminal elimination phase of the plasma concentration-time profile after single oral administration. Then the contribution of the drug absorption rate and the drug elimination rate to the plasma concentration-time profile during the absorption phase is separated. This allows the estimation of the first-order absorption rate constant. The plasma drug concentration-time profile after oral administration is described by Eq. 6.4, a biexponential equation that describes the absorption and elimination processes. As the time (t) increases after drug administration, the values for the two exponential terms e − kat and e−kt in Eq. 6.4 decrease because the power in both terms has negative sign. However, the exponential term e − kat decreases faster than e−kt because ka > k. As time gets longer, the term e − kat approaches zero indicating that the absorption process is complete, and Eq. 6.4 is reduced to: Cp =  

FDka e − kt   Vd(ka − k) 

(6.9)

Eq. 6.9 indicates that when the absorption process is completed, the plasma drug concentration declines at a rate dependent only on the rate of drug elimination, which is dependent on k. This represents the elimination phase of the drug concentration-time profile after oral administration. The drug t1/2 can be calculated from the elimination phase of the curve as illustrated in Figure 6.7. Also, k can be calculated from the slope of the linear elimination phase on the semilog scale, where the slope is equal to −k/2.303.

Figure 6.7 The plasma drug concentration-time profile during the elimination phase after administration of a single oral dose declines at a rate dependent on the first-order elimination rate constant.

96  Single Oral Drug Administration, Rate of Drug Absorption If the linear part of the plasma drug-concentration-time curve that represents the elimination phase is identified and back extrapolated to time zero, the y-intercept is equal to: FDka as mentioned in Eq. 6.7, and as illustrated in Figure 6.7. During the absorption   Vd(ka − k) phase, the absorption and elimination processes occur simultaneously. The contribution of the absorption rate and elimination rate of the drug on the plasma concentration-time profile during the absorption rate is separated by calculating the residuals. The residuals are calculated from the difference between the y-coordinate values on the extrapolated line from the elimination phase and the y-coordinate values on the plasma drug concentration-time profile (the measured drug concentration) at the same time points as follows: Residuals =  

FDka e − kt − Cp Vd(ka − k) 

(6.10)

Residuals =  

FDka FDka e − kt −     (e − kt − e − kat ) Vd(ka −  k) Vd(ka − k) 

(6.11)

Residuals =  

FDka e − kat Vd(ka − k) 

(6.12)

This means that a plot of the residuals versus time on the semilog scale declines at a rate dependent on the absorption rate constant, ka. The absorption rate constant can be determined from the slope of the residuals versus time plot (slope = −ka/2.303). Since the slope of this plot is always negative, ka always has a positive value. The first-order absorption rate constant can be determined using the method of residuals as illustrated in Figure 6.8 by the following steps:

• The plasma drug concentrations are plotted against their corresponding time values on the semilog scale.

Figure 6.8 The method of residuals: The residuals versus time plot on the semilog scale (the dashed line) declines at a rate dependent on the first-order absorption rate constant. The firstorder absorption rate constant is calculated from the slope of this plot, slope = −ka/2.303.

Single Oral Drug Administration, Rate of Drug Absorption 97

• The terminal elimination phase is identified from the linear part of the plasma profile, • •

• • •

and the best line that represents the elimination phase is drawn and is back extrapolated to the y-axis. The slope of the line that represents the elimination phase is calculated from the relationship, slope = −k/2.303. At least three points on the extrapolated line at three different time values during the absorption phase of the drug are chosen. Vertical lines from the points on the extrapolated line are dropped to determine the corresponding points (at the same time values) on the plasma drug concentration-time curve. The differences between the y-coordinate values of the points on the extrapolated line and the corresponding y-coordinate values on the plasma drug concentration-time curve are calculated. These differences are the residuals. The values of the residuals are plotted versus their corresponding time values on the same graph. A straight line should be obtained with a slope of −ka/2.303. The extrapolated line representing the elimination phase and the residuals versus time plot should have the same y-intercept as in Figure 6.8. This is because the equations that describe the two lines have the same coefficient; so substituting for time by zero in the two equations should give the same value.

6.4.1.1  Lag Time

Under some conditions, the absorption of drugs after oral administration does not start immediately due to some physiological factors such as delayed gastric emptying or formulation factors as delay in tablet disintegration. This delay time before starting drug absorption is known as the lag time for drug absorption. In some cases, the lag time for drug absorption is short and does not significantly affect the plasma drug concentration-time profile. However, the lag time can be long such as after administration of enteric coated formulations. In this case, the dosage form disintegration, drug dissolution, and absorption start after the dosage form reaches the small intestine, which may take hours, especially if the drug is administered with food. The plasma drug concentration-time profile will be shifted to the right on the time scale because no drug will be detected in plasma before the start of drug absorption as in Figure 6.9.

Figure 6.9 The method of residuals when there is lag time for drug absorption after oral administration. The lag time is calculated from the x-coordinate of the point of intersection of the two lines representing the elimination and absorption.

98  Single Oral Drug Administration, Rate of Drug Absorption The plasma drug concentration-time profile after single dose of an oral dosage form that has lag time can be described by Eq. 6.13. Cp =  

FDka (e − k(t − t0 ) − e − ka (t − t0 ) ) Vd (ka − k)

(6.13)

where t is the time of drug administration and t0 is the lag time for drug absorption. In Eq. 6.13, t0 is equal to t during the lag time (when t ≤ t0), while t0 stays constant when t ≥ t0. So, the plasma drug concentration will be zero during the lag time and starts to rise after the lag time. When the method of residuals is applied to estimate ka in the presence of absorption lag time, the extrapolated line representing drug elimination and the line resulting from the residual versus time intersect at a point with x-coordinate value equals to the lag time for drug absorption as in Figure 6.9. 6.4.1.2  Flip-Flop of ka and k

One of the assumptions of the method of residuals is that ka > k. In this case, the terminal elimination phase represents the elimination process and the line resulting from the residuals versus time represents the absorption process. However, when ka < k, the terminal elimination phase will represent the slower process (drug absorption), while the line resulting from the residuals versus time will represent the elimination process. This behavior is known as flip-flop of ka and k. The existence of this behavior will make the calculated k after oral drug administration to be different from that determined after IV administration. Figure 6.10 demonstrates an example of the plasma drug concentrationtime profile of a drug when administered in the form of two different formulations one has ka > k and the other has ka < k. When ka > k, the terminal elimination phase reflects the elimination rate of the drug and the residuals versus time line reflects the rate of absorption. While when ka < k, the terminal elimination phase reflects the absorption rate of the drug and the residuals versus time line reflects the rate of elimination. The flip-flop of ka and k is usually suspected in drugs that have short elimination half-life (large k), and also after administration of extended release drug products that usually have a slow rate of drug absorption (small ka).

Figure 6.10 Simulation of the plasma drug concentration-time profile of A drug when administered in the form of two different formulations (A) ka > k and (B) ka < k.

Single Oral Drug Administration, Rate of Drug Absorption 99 Practice Problems: Question: After oral administration of a single dose of an antibiotic, the following concentrations were measured. Time (hr)

Drug concentration (µg/L)

0 0.2 0.5 1.0 2.0 2.5 4.0 6.0 8.0

0 88 185 277 321 311 246 161 102

Calculate the first-order absorption rate constant. Answer: Plot the plasma concentration-time profile and follow the procedures for the method of residuals as in Figure 6.11. Calculate the residuals: Time (hr)

Residuals (µg/L)

0.2 0.5 1.0

660 − 88 = 572 560 − 185 = 375 420 − 277 = 143

The line resulting from plotting the residuals versus time has a slope of −0.43 hr−1. Slope = −0.43hr −1 =

− ka 2.303

ka = 0.99 hr −1

Figure 6.11  The method of residuals to calculate the first-order absorption rate constant.

100  Single Oral Drug Administration, Rate of Drug Absorption 6.4.2  Wagner-Nelson Method

The Wagner-Nelson method is a method that can be used to determine the absorption rate constant for drugs when their absorption follows zero-order kinetics or first-order kinetics (1, 2). The requirement for this method is that the drug follows one-compartment pharmacokinetic model, and the drug follows linear pharmacokinetics. A modification of this method, the Loo-Reigelman method, can be used to determine the absorption rate constant when the drug pharmacokinetics follow two-compartment model (3). The Wagner-Nelson method uses the relationship between the fraction of the administered dose remaining to be absorbed at different time points to determine the absorption rate constant. A mass balance equation can be written to determine the amount of the drug absorbed as follows: The amount of drug absorbed up to time t ( A at ) = Amount of drug in the body ( at time t ) + Amount of drug excreted ( up to time t ) 

(6.14)

where amount of the drug in the body (t) = Cpt Vd and amount of drug excreted(up to t) = k Vd  ∫ tt == 0t Cp dt t= t

So, A at = Cpt Vd + k Vd

∫ Cp dt

t=0

(6.15)



The total amount of drug absorbed up to time  ∞ ( A a∞ ) = 0 + total amount of drug excreted ( up to time  ∞ )



(6.16)

where amount of the drug in the body (t = ∞) = zero and Amount of the drug excreted(t = ∞) = k Vd  ∫ tt =∞ = 0 Cp dt t =∞

So, A a∞ = k Vd

∫ Cp dt

t=0

(6.17)



This means that the fraction of the administered dose already absorbed at any time is determined by dividing Eq. 6.15 by Eq. 6.17. t= t

Cpt Vd + k Vd Fraction of dose absorbed(t) =

t =∞

k Vd

∫ Cp dt

t=0

∫ Cp dt

t=0

(6.18) 

Single Oral Drug Administration, Rate of Drug Absorption 101 Dividing both sides of the equation by Vd yields Eq. 6.19: t=t

A Fraction of dose absorbed(t) = at = A a∞

Cpt + k t =∞

k

∫ Cp dt

t=0

∫ Cp dt

(6.19)

t=0

The fraction of the administered dose remaining to be absorbed can be determined by Eq. 6.20. Fraction of dose remaining to be absorbed(at time t)  = 1 −  

A at A a∞

(6.20)

The calculated fraction of dose remaining to be absorbed at different time points can be used to determine if the absorption process follows zero-order or first-order kinetics, and to determine the rate constant for the absorption process. If the absorption process is first order, a plot of the fraction remaining to be absorbed versus time should give a straight line on the semilog scale as in Figure 6.12A. The slope of this line is equal to −ka/2.303. While if the absorption process is zero-order, a plot of the fraction remaining to be absorbed versus time should give straight line on the linear scale as in Figure 6.12B. The slope of this line is equal to −ka. The fraction remaining to be absorbed can be expressed as percent remaining to be absorbed by multiplying the fraction by 100. 6.4.2.1  Application of the Wagner-Nelson Method

The absorption rate constant can be determined using the Wagner-Nelson method by going through the following steps:

• Plot the plasma drug concentration-time data to calculate the elimination rate constant, k.

Figure 6.12 The fraction of dose remaining to be absorbed versus time plots, (A) when the drug absorption follows first-order kinetics, and (B) when the drug absorption follows zeroorder kinetics.

102  Single Oral Drug Administration, Rate of Drug Absorption

• Construct a table (similar to the table in the solved example) to calculate all the necessary information to calculate the fraction of the drug remaining to be absorbed.

• Use the plasma drug concentration at different time points to calculate the AUC be• • •

• • •

tween each pair of plasma concentrations, partial AUC (AUC |tt == nn−1). Calculate the cumulative AUC from time zero to time t, for each time point (AUC |tt == 0t ). Multiply each cumulative AUC value by the elimination rate constant (k ⋅ AUC |tt == 0t ). Add the value of the plasma drug concentration to the product of k and the cumulative AUC at each time point (Cp + k ⋅ AUC |tt == 0t ). This is the amount of the drug absorbed up to time t divided by Vd. The last value in this column when the plasma drug concentration is not detected is the total amount of drug absorbed up to t = ∞ divided by Vd. Calculate the fraction of dose absorbed at each time point by dividing the amount absorbed at each time point by the amount absorbed at t = ∞, (Aa t/Aa ∞). Calculate the fraction of dose remaining to be absorbed at each time point [1 − (Aa t/Aa ∞)]. Plot the fraction of dose remaining to be absorbed at different time, on the linear and semilog scales to determine the absorption rate constant.

Practice Problems: Question: The following plasma concentration-time data were obtained after a single oral dose of 1000 mg of a drug. Using the Wagner-Nelson method, determine the order of the drug absorption of process, and estimate the absorption rate constant, if the firstorder elimination rate constant is 0.33 hr−1.

Time (hr)

Conc (mg/L)

Time (hr)

Conc (mg/L)

0 0.2 0.5 1.0 2.0 3.0

0 10.0 21.5 33.4 40.7 37.6

4.0 5.0 8.0 12.0 20.0

31.1 24.5 10.2 2.9 0.2

Table 6.1  The calculation of the fraction of dose remaining to be absorbed t =t t =t t = t A /A Time Conc AUC |t = n at a∞ t = n −1 AUC  |t = 0 k ⋅ AUC  |t = 0 Cp + k ⋅ AUC  |t = 0 (hr) (mg/L)

1 − (Aat/Aa∞)

0 0.2 0.5 1.0 2.0 3.0 4.0 5.0 8.0 12.0 20.0

1 0.8863 0.7424 0.5616 0.3467 0.2384

0 10.0 21.5 33.4 40.7 37.6 31.1 24.5 10.2 2.9 0.2

0 1.0 4.73 13.7 37.0 39.2 34.4 27.8 52.0 42.4 12.4 0.6

0 1.0 5.73 19.43 56.43 95.63 130.03 157.83 209.83 262.23 274.63 275.23

0 0.33 1.89 6.41 18.62 31.56 42.9 52.1 69.24 86.54 90.6 90.8

0 10.33 23.39 39.81 59.32 69.16 74.0 76.6 79.44 89.44 90.8 90.8 = Aa∞

0 0.1137 0.2576 0.4384 0.6533 0.7616

Single Oral Drug Administration, Rate of Drug Absorption 103

Figure 6.13  A plot of the fraction of dose remaining to be absorbed versus time on the semilog scale.

Answer: Use the drug concentration-time data to construct the table that allows calculation of the fraction of dose remaining to be absorbed as described above. Table 6.1 includes the stepwise calculation of the information needed to calculate the fraction of dose remaining to be absorbed. a A plot of the fraction remaining to be absorbed versus time on the semilog scale is linear indicating that the absorption of the drug follows first-order kinetics, Figure 6.13. b The slope of the line on the semilog scale = −0.2076 hr−1. slope = − ka / 2.303, so ka = 0.478 hr −1 . Practice Problems: Question: When applying the Wagner-Nelson method to determine the absorption rate constant for an oral hypoglycemic drug, the following data were obtained: Time (hr)

Fraction remaining to be absorbed

1 2 3 5

0.63 0.40 0.25 0.10

a What is the order of the absorption process of this oral hypoglycemic drug? b Calculate the absorption rate constant Answer: Plot the fraction remaining to be absorbed on the linear scale and semilog scale as in Figure 6.14. The relationship is linear on the semilog scale indicating first-order absorption. Slope  =   −0.2 hr −1 =   ka = 0.46 hr −1

− ka 2.303

104  Single Oral Drug Administration, Rate of Drug Absorption

Figure 6.14 The fraction of dose remaining to be absorbed versus time is not linear on the linear scale (A) and is linear on the semilog scale (B).

6.5 Summary

• The absorption rate constant is the rate constant that determines the rate of drug absorption from the site of administration.

• Most drugs are absorbed by the first-order process; however, the absorption of some drugs can follow zero-order kinetics.

• The absorption rate constant determined after drug administration is an operative rate constant that accounts for all the steps required for drug absorption into the systemic circulation, including disintegration, dissolution, and absorption. • The absorption rate constant can be different after administration of different drug products for the same active drug. • The first-order absorption rate constant has units of time−1. Practice Problems 6.1 The plasma drug concentration in a patient who had received a single oral dose of a drug (10 mg/kg) was determined as follows: Time (hr)

Drug concentration (µg/L)

0 1 2 3 4 6 8 10 15 20 25 30

0 84 141 177 199 207 188 176 111 63 33.5 17.3

Plot the plasma concentration-time profile on a semilog graph paper and then find the following parameters directly from the graph. (Do not use the method of residuals.) a The elimination rate constant and the elimination half-life b The tmax c The Cpmax

Single Oral Drug Administration, Rate of Drug Absorption 105 6.2 Two drugs have the following pharmacokinetic parameters; after a single oral dose of 500 mg. Both drugs are completely absorbed: Drug

ka (hr−1)

k (hr−1)

Vd (L)

A B

1.0 0.2

0.2 1.0

10 20

a Calculate tmax for each drug. b Calculate Cpmax for each drug. 6.3 The plasma concentration-time profile after a single oral dose can be expressed as follows: Cp =  

(

FDka e − kt − e − kat Vd(ka − k)

)

After administration of a single 500-mg oral dose of a drug, the plasma concentration-time profile can be expressed as follows:

(

Cp = 75mg/L e −0.1t − e −0.9t

)

where Cp is in mg/L and t is in hr, and

FDka = 75mg/L  Vd(ka − k)

a Calculate the tmax after administration of the 500-mg oral dose b Calculate the Cpmax after administration of the 500-mg oral dose c Do you expect tmax and Cpmax to be different if the dose was 100 mg 6.4 A single dose of 500 mg of an antibiotic was given to a 70-kg patient as an oral tablet. The plasma concentration-time profile can be described by the following equation:

(

Cp = 21 mg /L e−0.115t − e−0.624 t

)

where Cp is in mg/L and t is in hours. The drug has 100% bioavailability (F = 1). a b c d e f g

Calculate the plasma concentration 6 hr after drug administration. Calculate the elimination half-life of this drug. Calculate the first-order absorption rate constant of this drug. Calculate the volume of distribution of this drug. Calculate tmax. Calculate Cpmax. What is the slope of the terminal elimination phase for the plasma concentrationtime plot (on semilog scale) after administration of the 500-mg dose?

6.5 A single dose of 500 mg of an antibiotic was given to a 70-kg patient as an oral tablet. The plasma concentration-time profile can be described by the following equation:

(

Cp = 10 e−0.154 t − e−0.954 t

)

106  Single Oral Drug Administration, Rate of Drug Absorption

where Cp is in mg/L and t is in hours, and the drug is completely absorbed.

a Calculate the elimination half-life of this drug. b Calculate the volume of distribution of this drug. c Calculate the total body clearance of this drug. d Calculate tmax and Cpmax. e Calculate the AUC after administration of 500-mg oral dose. 6.6 A patient received a single oral dose of 5 mg of a bronchodilator that is completely absorbed after oral administration. The following plasma concentrations time data were obtained: Time (hr)

Conc (µg/L)

Time (hr)

Conc (µg/L)

0.0 0.25 0.50 0.75 1.0 1.5 2.0

0.0 33 60 81 97 117 127

3.0 4.0 6.0 8.0 10 12 14

125 111 75 47 28 16 9.5

a Plot the plasma concentration-time curve and determine the absorption rate constant using the method of residuals. b What is the equation that describes the plasma concentration-time profile of this drug after oral administration? c Calculate tmax and Cpmax for this drug after administration of a single 5-mg dose. d Calculate Vd for this drug 6.7 The following results were obtained when the Wagner-Nelson method was used to calculate the absorption rate constant after an oral administration: Time (hr)

Fraction remaining to be absorbed

1 2 3 5

0.7 0.5 0.35 0.17

a What is the order of absorption of this drug? b Calculate the absorption rate constant. 6.8 A 60-kg patient received a single oral dose of 25 mg of an antibiotic that is completely absorbed after oral administration. Serial samples were drawn and the drug plasma concentration was determined using a sensitive analytical method. The plasma concentrations were as follows: Time (hr)

Conc (ng/mL)

Time (hr)

Conc (ng/mL)

0.0 0.2 0.5 1.0 2.0 3.0

0.00 88.5 184.9 276.9 321.6 292.8

4.0 6.0 8.0 10.0 12.0 14.0

246.1 161.0 102.2 64.5 40.66 25.61

Single Oral Drug Administration, Rate of Drug Absorption 107 a Calculate the absorption rate constant of this drug using the method of residuals. b Calculate the half-life and the volume of distribution of this drug. 6.9 A 60-kg patient received a single oral dose of 500 mg of an antibiotic which is completely absorbed after oral administration. Serial samples were drawn, and the drug plasma concentration was determined using a sensitive analytical method. The plasma concentrations were as follows: Time (hr)

Conc (mg/L)

Time (hr)

Conc (mg/L)

0.0 0.2 0.4 0.7 1.0 1.5

0.00 3.1 5.2 7.1 7.9 7.9

2.0 3.0 4.0 5.0 6.0 7.0

7.2 5.1 3.4 2.2 1.4 0.88

a Calculate the absorption rate constant of this drug using the method of residuals. b What is the equation that describes the plasma concentration-time profile of this drug after oral administration? c Calculate tmax and Cpmax for this drug after administration of a single 5-mg dose. d Calculate the half-life and Vd for this drug. References 1. Wagner JG and Nelson E “Kinetic analysis of blood levels and urinary excretion in the absorptive phase after single doses of drug” (1964) J Pharm Sci; 53:1392–1403. 2. Wagner JG and Nelson E “Percent absorbed time plots derived from blood level and/or urinary excretion data” (1963) J Pharm Sci; 52:610–611. 3. Loo JCK and Riegelman S “New method for calculating the intrinsic absorption rate of drugs” (1968) J Pharm Sci; 57:918–928.

7

Drug Pharmacokinetics Following Single Oral Drug Administration The Extent of Drug Absorption

Objectives After completing this chapter, you should be able to:

• Define the bioavailability of the drug after oral administration. • Discuss the different factors contributing to the “first-pass effect” after oral drug administration.

• Discuss the importance of drug transporters and drug metabolism in determining drug bioavailability.

• Explain the importance of determining the in vivo bioavailability of drugs after oral administration.

• Calculate the area under the curve using the trapezoidal rule. • Calculate the absolute bioavailability and the relative bioavailability of drugs after oral administration.

• Analyze the effect of changing the drug pharmacokinetic parameters on the plasma concentration-time profile after single oral drug administration.

7.1 Introduction Drugs administered to produce systemic effects must be absorbed into systemic circulation to reach their site of action and produce the desired effects. This means that the amount of drug absorbed to the systemic circulation is what produces therapeutic effect. The concept of bioavailability, which is a measure of the extent of drug absorption to the systemic circulation, is important during the development of new drugs and for evaluating different products for the same active drug. As discussed previously, numerous factors can affect the rate and extent of drug absorption after extravascular administration. So, it is possible to see variation in drug effects after administration of different drug products for the same drug, which results from variation in the rate and extent of drug absorption from the different products. This has added a new dimension to evaluating drug products that includes determination of the bioavailability of the active drug and the demonstration of the bioequivalence of different marketed drug products for the same active drug. This is important to ensure the efficacy of drug products and the consistency of the therapeutic effect when the patient switches between marketed products for the same drug.

DOI: 10.4324/9781003161523-7

Single Oral Drug Administration, Extent of Drug Absorption 109 7.2  Causes of Incomplete Drug Bioavailability Intravenous drug administration involves a direct administration of the drug to the systemic circulation, which guarantees 100% bioavailability. Drugs administered by extravascular routes usually have incomplete bioavailability. Generally, this can result from losing the intact drug molecules during the absorption process and/or inability of the drug to cross the biological membrane. In addition to the factors affecting drug absorption discussed in Chapter 5 of this book, the first-pass effect that results in drug loss due to degradation, metabolism, and transport during the absorption process can affect the extent of drug absorption for many drugs. The first-pass effect significantly reduces the bioavailability of many drugs, and the modulation of any of its components can significantly alter the drug bioavailability. 7.2.1  The First-Pass Effect

Drug absorption from the gastrointestinal tract (GIT) involves the passage of the drug from the GIT lumen through the gut wall to reach the hepatic portal vein. The absorbed drug is then delivered via the portal vein to the liver before it can reach the systemic circulation. During the absorption process, part of the administered dose may be lost in the GIT lumen by degradation, metabolism, and/or excretion in feces due to inability to cross the GIT membrane, leaving a fraction of dose (FF) to reach the gut wall. The gut wall contains metabolizing enzymes and efflux transporters, allowing a fraction of the drug amount reaching the gut wall to get into the portal vein (FF × FG). The drug delivered to the liver via the portal vein can be metabolized by the hepatic metabolizing enzymes or excreted in bile, permitting a fraction of the drug delivered to the liver to reach the systemic circulation (FF × FG × FH) (1). This means that the overall oral drug bioavailability (F), which is the fraction of dose that reaches the systemic circulation after oral administration, can be expressed as in Eq. 7.1. F = FF × FG × FH



(7.1)

The loss of the drug during the sequence of steps involved in the absorption process is known as the first-pass effect that can be illustrated as in Figure 7.1. Based on this, the bioavailability of the drug is determined by the different processes affecting the amount of the drug that reaches the systemic circulation. This includes chemical hydrolysis of the drug, complexation leading to the formation of nonabsorbable complex, adsorption of nonabsorbable compound, inability of the drug to cross the GIT membrane due to poor solubility or poor permeability, and efflux transporters that decrease the absorption of drugs. Also, enzymatic drug metabolism in the gut lumen, gut wall, and the liver decreases the amount of the drug that reaches the systemic circulation. This is in addition to drug excretion in bile as the parent drugs or as metabolites. The loss of the drug during the absorption process before it reaches the systemic circulation is also referred to as presystemic elimination. Clinical Importance:

• Drugs that undergo extensive first-pass effect usually have low drug bioavailability after oral administration.

110  Single Oral Drug Administration, Extent of Drug Absorption

Figure 7.1 A diagram illustrating the different causes of drug loss during the absorption process after oral administration. The bioavailable drug is the amount of drug that escapes degradation and metabolism in the gut lumen, gut wall metabolism, efflux transport, hepatic metabolism, and biliary excretion and reaches the systemic circulation.

• Changes in the gut and/or hepatic metabolic activity due to drug interactions or change

in the hepatic function can cause significant changes in the bioavailability of drugs that undergo extensive first-pass effect. The change in the drug eliminated during the first pass from 90 to 80% results in increasing the bioavailability from 10 to 20%, which represent 100% increase in bioavailability (2). • Drugs that are extensively metabolized in the gut wall and/or liver can undergo saturable first-pass metabolism resulting in dose-dependent bioavailability. This is because after administration of large doses, larger fraction of the drug dose can escape the firstpass metabolism compared with after administration of smaller doses. Propranolol and verapamil are examples of drugs that show saturable first-pass metabolism. • The rate of drug absorption is important in observing the saturable first-pass metabolism because when the drug is rapidly absorbed, a large quantity of the drug is delivered to the metabolizing enzymes at the same time increasing the chance of enzyme saturation. 7.2.2  The GIT Drug Transporters

There are several classes of drug transporters that are different in their distribution pattern, substrate specificity, capacity, and cellular expression pattern. Drug transporters are expressed in various organs and tissues throughout the body; however, our discussion will focus on the transporters that can affect drug absorption from the GIT. In the GIT, transporters are expressed on the apical side of the absorptive epithelial cells facing the lumen, and on the basolateral side facing the blood capillaries. Different drug transporters are involved in the uptake of drug molecules inside the epithelial cells and efflux of drug molecules out of the epithelial cells. At the apical side, efflux transporters promote the transfer of the absorbed molecules back to the GIT lumen, hindering the absorption process, while the uptake transporters help the absorption of drug molecules into the epithelial cells. However, at the basolateral side, the efflux transporters promote drug transfer to the blood, and the uptake transporters remove the drug molecules from the blood to the epithelial calls.

Single Oral Drug Administration, Extent of Drug Absorption 111 An example of drug transporters that have been shown to significantly affect the bioavailability of numerous orally administered drugs is the P-glycoprotein (P-gp) transporter. This transporter is expressed in different organs, including kidney, small intestine, liver, lungs, colon, and blood brain barrier. In the small intestine, P-gp is localized in the apical surface of intestinal epithelial mucosa with the primary function of promoting the excretion of toxic compounds in the epithelial cells back to the intestinal lumen. Substrates for P-gp include many anticancer drugs, antibiotics, antivirals, calcium channel blockers, and immunosuppressive agents. The absorption of drugs that are P-gp substrates in the GIT is reduced by the efflux function of this transporter leading to the reduction of their systemic bioavailability. Modulation of the P-gp function can lead to changes in the bioavailability of the drugs that are P-gp substrates. Clinical Importance:

• Drugs like rifampicin, steroid hormones, some anticancer drugs, retinoic acid, sodium

butyrate, and natural products such as St John’s Wort can upregulate the expression of P-gp leading to enhancing the efflux function of this transporter and reducing the absorption and bioavailability of drugs that are P-gp substrates. • Drugs such as calcium channel blockers, immunosuppressive agents, natural products as grapefruit juice, and some compounds developed specifically as specific P-gp inhibitors can inhibit the efflux effect of P-gp leading to increase in the absorption and bioavailability of the drugs that are P-gp substrates. • Drug-drug interactions that affect P-gp function can lead to significant change in the drug therapeutic effect (3). For example, the calcium channel blocker valspodar can significantly increase the plasma digoxin concentrations causing digoxin adverse effects. • It has been reported that due to genetic factors, different individuals can have different expressions of P-gp. This can cause large variability in the bioavailability of some P-gp substrates. 7.2.3  Intestinal Drug Metabolism

The involvement of the small intestine in the first-pass metabolism has been well recognized for many drugs. The CYP3A4 makes up about 80% of the total Cytochrome P450 (CYPs) present in the small intestine and is involved in the metabolism of more than 40% of the drugs used orally. While CYP2C9 comprises the second most abundant CYPs enzyme in the small intestine. UDP-glucuronosyltransferase and sulfotransferases conjugation enzymes are also expressed in the small intestine and are involved in the presystemic metabolism of many drugs. The CYP3A4 expressed in the small intestine is less than that in the liver; however, the contribution of the intestinal CYP3A4 to the presystemic metabolism of many orally administered drugs is significant. Many of the CYP3A4 substrates are also substrates for the P-gp efflux transport protein present in the apical surface of the GIT epithelial cell membrane. When the drugs that are CYP3A4 and P-gp substrates cross the epithelial cell membrane, they can be excreted back to the intestinal lumen by P-gp, metabolized by the CYP3A4 in the cells, or cross the basolateral membrane to reach the blood. The drug molecules that are excreted to the intestinal lumen by the P-gp can be reabsorbed

112  Single Oral Drug Administration, Extent of Drug Absorption and become exposed to the CYP3A4 again, which increases the chance for the drug to be metabolized. So, variation in the P-gp activity during the absorption process will be accompanied by changes in the GIT CYP3A4 drug metabolism. Also, variation in the GIT CYP3A4 metabolic activity can lead to differences in the contribution of the P-gp to drug absorption. The interplay between P-gp and CYP3A4 and the difference in their expression between individuals contribute to the variability in drug absorption and bioavailability. Clinical Importance:

• Drug interactions causing induction or inhibition of the intestinal metabolizing en-

zymes can significantly alter the bioavailability of drugs that undergo extensive intestinal metabolism. • Modulation of the P-gp activity can affect the extent of drug metabolism in the small intestine due to the interplay between P-gp and CYP3A4. 7.3  The Rationale for Bioavailability Determination Determination of drug bioavailability is necessary during all stages of drug development with different rationales for the different stages:

• During the early stage of new drug development, evaluation of drug bioavailability

•

•

•

•

is important to determine if the drug can be administered orally. Also, bioavailability studies can help in the selection of the active drug derivative with better absorption properties. This is essential for drugs that are developed for the management of chronic diseases that are preferred to be administered orally, especially in the presence of effective oral alternatives. During the clinical phases of drug development, comparing the bioavailability of different formulations can be used for formulation optimization. This is important for selecting the drug formulation that will be used in clinical trials and then marketed after approval. Determination of the in vivo bioavailability of the active drug in human is usually required by regulatory authorities as part of all new drug applications for approving a new drug entity for marketing. The drug bioavailability after administration of the marketed product is important for calculating the IV to oral dose conversion. Determination of the drug in vivo bioavailability after administration of the marketed drug product in human is usually required by regulatory authorities as part of supplemental application. For example, when the supplemental applications propose changes in the manufacturing site, manufacturing process, product formulation, or dosage strength, this is important to ensure that the proposed changes in the formulation do not affect the in vivo drug bioavailability. Under certain conditions, determination of the in vivo bioavailability after administration of the drug product can be waived by regulatory authorities. Instead, information should be provided to prove that the drug product falls in one of the categories that do not require measuring the in vivo bioavailability. The criteria for waiving the in vivo bioavailability determination are similar to those for waiving the in vivo bioequivalence and will be discussed in the next chapter.

Single Oral Drug Administration, Extent of Drug Absorption 113 7.4  Determination of the Drug In Vivo Bioavailability Determination of the drug bioavailability after the oral or other extravascular route of administration utilizes the same approach. Usually, a single dose of the drug is administered, then the drug concentration in the systemic circulation as a function of time is determined. The plasma concentration-time profile shown in Figure 7.2 represents a typical drug profile obtained after single oral drug administration. The Cpmax, tmax, and area under the plasma concentration-time curve (AUC) are the three parameters determined from the drug concentration-time profile and reflect the rate and extent of drug absorption. Shorter tmax indicates a faster rate of drug absorption, because tmax is not affected by the extent of drug absorption. While higher AUC indicates a higher extent of absorption, it is not affected by the rate of drug absorption. However, higher Cpmax indicates a faster rate and/or a higher extent of drug absorption. Practice Problems: Single doses of three different products for the same drug were administered to the same individuals in three different occasions. The tmax, Cpmax, and AUC were determined, and the plasma drug profiles are presented in Figure 7.3.

Product 1 Product 2 Product 3

tmax (hr)

Cpmax (mg/L)

AUC (mg hr/L)

1.75 4.0 3.0

14.2 11.8 7.7

100 133 67

Questions: a Rank the three products according to their rate of absorption. b Rank the three products according to their extent of absorption. c Comment on the Cpmax observed after administration of three products. Answers: a By comparing tmax, product 1 is absorbed faster than product 3, and both are faster than product 2. b By comparing AUCs, the extent of absorption of product 2 is larger than that for product 1, and both are larger than product 3. c Cpmax is affected by the rate and/or the extent of drug absorption. So, it is not possible to use Cpmax only to compare the rate and extent of absorption for different products. Product 2 has a higher extent of absorption compared to product 1 (higher AUC); however, the Cpmax for product 1 is higher than that for product 2 because of the faster rate of absorption. Product 3 has faster absorption compared to product 2 (shorter tmax), but the Cpmax for product 2 is higher than that for product 3 because of the higher extent of absorption. 7.4.1  Drug Bioavailability

Drug bioavailability can be defined as the rate and extent to which the active drug ingredient is absorbed from a drug product and becomes available at the site of drug action.

114  Single Oral Drug Administration, Extent of Drug Absorption

Figure 7.2 A typical plasma concentration-time profile after administration of a single oral dose of the drug.

Since it is not possible to measure the drugs at its site of actions, the drug bioavailability is assessed by measuring the amount of the drug that reaches the systemic circulation. This is because the drug in the systemic circulation will have access to the drug site of action, and changes in the drug amount that reaches the systemic circulation will result in proportional changes in the drug availability at the site of action. There are two types of drug bioavailability: absolute and relative. 7.4.1.1  Absolute Bioavailability

It is the fraction of the administered dose that reaches the systemic circulation after oral (or any extravascular) administration, which is a measure of the extent of drug absorption. The absolute bioavailability is determined by comparing the amount of drug that reaches the systemic circulation after oral administration and after IV administration of the same dose of the same drug. This is because after IV administration, 100% of the dose reaches the systemic circulation. The absolute bioavailability can have values

Figure 7.3 The plasma concentration-time profile after administration of the three products for the same drug.

Single Oral Drug Administration, Extent of Drug Absorption 115 between zero and one. When the drug is not absorbed at all after oral administration, the absolute bioavailability is equal to zero, while when the entire oral dose reaches the systemic circulation, the absolute bioavailability is equal to one. The absolute bioavailability can also be expressed in terms of percentage (0–100%). 7.4.1.2  Relative Bioavailability

It is the absolute bioavailability of the drug from a drug product relative to the absolute bioavailability of the same drug from a second drug product. The relative bioavailability is determined by comparing the amount of the drug that reaches the systemic circulation after administration of two different oral drug products (a test product and a reference product) that contain the same active drug. The relative bioavailability can have any positive value and can be more than one when the drug bioavailability from the test product is higher than that of the reference product. 7.4.2  Calculation of the Drug Bioavailability

When drug elimination follows first-order kinetics, the AUC after IV administration of single dose of the drug is given by Eq. 7.2. AUC IV =  

Dose k Vd 

(7.2)

The AUC after oral administration of a single dose of the same drug is given by Eq. 7.3. AUCoral =  

F Dose k Vd 

(7.3)

where F is the absolute bioavailability of the drug after administration of the oral dosage form. If the same dose of the drug is used for the oral and IV administration, and because the drug CLT (k Vd) is the same after the different routes of administration, dividing Eq. 7.3 by Eq. 7.2 gives Eq. 7.4. AUCoral F Doseoral /k Vd = =F AUCIV Dose IV /k Vd 

(7.4)

This means that the drug absolute bioavailability can be determined from the ratio of the AUC after oral administration to that after IV administration of the same drug to the same individual (to ensure similar CLT). Equation 7.5 is the general equation for calculating the absolute bioavailability of the drug after oral administration if the oral and IV doses are different. F=

  AUCoral Dose IV AUC IV Doseoral 

(7.5)

116  Single Oral Drug Administration, Extent of Drug Absorption Similarly, the relative bioavailability for different formulations or products of the same drug can be determined by comparing the AUCs after administration of these products to the same individuals. Usually, the new oral product under investigation is called the test product and the other oral product used in the comparison is called the reference product as in Eqs. 7.6 and 7.7. AUC test =

Ftest Dose test k Vd 

AUCstandard =

(7.6)

Fstandard Dosestandard   k Vd 

(7.7)

where Ftest is the absolute bioavailability of the drug from the test drug product, and Fstandard is the absolute bioavailability of the drug from the standard drug product. If the same dose of the two drug products is administered, and because the drug CLT is similar after administration of the different products, dividing Eq. 7.6 by Eq. 7.7 gives Eq. 7.8. AUC test Ftest Dose/k Vd Ftest = = AUCstandard Fstandard Dose/k Vd Fstandard 

(7.8)

This means that the drug bioavailability from the test drug product relative to the drug bioavailability from the standard drug product can be determined from the ratio of the calculated AUC after administration of the two products to the same individuals, as in the general equation, Eq. 7.9. Frelative =

Ftest Fstandard

=

AUC test Dosestandard  AUCstandard  Dose test 

(7.9)

where the relative bioavailability (Ftest/Fstandard) is the ratio of the absolute bioavailability of the test product to the absolute bioavailability of the standard product. Calculating the relative bioavailability does not give the value of the absolute bioavailability for the individual products, but it gives the ratio of the absolute bioavailability for the two products. For example, assume that the bioavailability of product A relative to the bioavailability of product B is 0.8. This means that the ratio of the absolute bioavailability of product A to that of product B is 0.8 without identifying the exact value of the absolute bioavailability of the two products. It can be any combination of the absolute bioavailability that gives a ratio of 0.8 (e.g., Ftest/Fstandard can be 0.8/1.0, 0.6/0.75, 0.4/0.5, and 0.2/0.25). Practice Problems: Question: In a study to evaluate the bioavailability of an antibiotic, the calculated AUCs were as follows: 250-mg IV bolus dose 500-mg suspension 500-mg capsule

AUC = 69.26 mg hr/L AUC = 61.93 mg hr/L AUC = 50.90 mg hr/L

Single Oral Drug Administration, Extent of Drug Absorption 117 a Calculate the absolute bioavailability of the oral suspension and the capsule of this antibiotic. b Calculate the bioavailability of the oral capsule relative to the oral suspension of this antibiotic. Answer: a The dose of the IV dose is different from the dose for the suspension and capsule. The bioavailability should be calculated from AUCs obtained from similar doses. So, the difference in doses should be considered when calculating the bioavailability. The absolute bioavailability of the suspension Fsuspension =

AUCsuspension Dose IV 61.93mg hr/L   250 mg × = × = 0.445 AUC IV Dosesuspension 69.26 mg hr/L   500 mg

The absolute bioavailability of the capsule Fcapsule =

AUC capsule Dose IV 50.9 mg hr/L 250 mg × = × = 0.365 AUC IV Dose capsule 69.26 mg hr/L 500 mg

b The bioavailability of the capsule relative to the suspension Frelative =

AUC capsule Dosesuspension 50.9 mg hr/L 500 mg × = × = 0.82 AUCsuspension Dose capsule 61.93 mg hr/L 500 mg

The relative bioavailability can also be determined from the ratio of the absolute bioavailability of the two oral products Frelative =

Fcapsule 0.365 = = 0.82 Fsuspension 0.445

7.4.3  Determination of the Drug Bioavailability from Urinary Excretion Data

The drug that reaches the systemic circulation is excreted by one or more elimination pathways such as metabolism and urinary excretion. The fraction of the drug in the systemic circulation that is excreted by each pathway is usually constant for a particular drug in an individual patient. So, for a drug that is partially excreted unchanged in urine, administration of larger doses to the same individual results in proportional increase in the amount of the drug recovered unchanged in urine. Based on this, it is possible to use the drug urinary excretion information after a single oral administration to calculate the drug bioavailability. For example, if the total amount of a drug recovered unchanged in urine after administration of single IV dose of 100 mg is 60 mg, this means that 60% of the drug that reaches the systemic circulation is excreted unchanged in urine. Assume that the same dose of this drug is administered orally to the same individual and the total amount of the drug excreted unchanged in urine is 30 mg. This means that the 30 mg of the drug

118  Single Oral Drug Administration, Extent of Drug Absorption excreted in urine after the oral dose represents 60% of the amount of the drug that reached the systemic circulation. This indicates that only 50 mg of the drug reached the systemic circulation after oral administration, and the absolute oral bioavailability of this drug is 50%. The use of urinary excretion data to calculate the drug bioavailability requires that a significant amount of the drug is excreted unchanged in urine after IV and oral administration. Single IV and oral doses of the drug are administered to the same group of individuals in two different study periods, then urine is collected until all the drug is excreted from the body. The total amount of the drug excreted in urine (Ae∞) is calculated from the volume of the total urine sample and the concentration of the drug in the total urine sample as in Eq. 7.10. A e∞ = volume of sample × drug concentration in sample 

(7.10)

The absolute bioavailability of the oral product can be calculated as in Eq. 7.11. F=

A e∞ oral Dose IV A e∞ IV Doseoral 

(7.11)

Similarly, the relative bioavailability can be determined from the urinary excretion data after administration of two oral products for the same drug as in Eq. 7.12. Frelative =

A e∞ test Dose reference A e∞ reference Dose test 

(7.12)

Practice Problems: Question: In a study to evaluate the bioavailability of an antibiotic using the urinary excretion data, the amount of the antibiotic excreted unchanged in urine after administration of different formulations were as follows: Dose/formulation

Amount recovered unchanged in urine

500-mg IV bolus 500-mg capsule 250-mg suspension

350 mg 225 mg 150 mg

a Calculate the absolute bioavailability of the antibiotic from the capsule and the suspension. b Calculate the bioavailability of the suspension relative to the capsule. Answer: a Capsule:F =

A e∞ capsule Dose IV 225mg 500 mg = = 0.643 A e∞ IV DoseCapsule 350 mg 500 mg

Suspension:F =

A e∞ suspension Dose IV 150 mg 500 mg = = 0.857 A e∞ IV Dosesuspension 350 mg 250 mg

Single Oral Drug Administration, Extent of Drug Absorption 119 b Bioavailability of suspension relative to the capsule Frelative =

A e∞ suspension Dose capsule 150 mg 500 mg = = 1.333 A e∞ capsule Dosesuspension 225mg 250 mg

Clinical Importance:

• The bioavailable dose, which is the amount of the drug that reaches the systemic circulation, is what produces the therapeutic effect.

• Administration of the same labeled dose of products with lower bioavailability pro-

duces lower Cpmax and AUC as illustrated in Figure 7.4. The lower plasma concentrations achieved by the products with lower bioavailability decrease the systemic exposure to the drug and may decrease the intensity and shorten the duration of therapeutic effect after single-dose administration. • Some food supplements and nutritional products may not have any benefits because the active ingredients they contain are poorly absorbed. • Patients taking drugs for the management of chronic diseases may experience changes in therapeutic effect when they switch to drug products that contain the same active ingredient but have lower oral bioavailability. • The oral bioavailability of drugs is important for calculating the IV-to-oral dose conversion. For a drug that is 50% bioavailable after oral administration, the oral dose required to produce the same effect should be twice the IV dose. This can also explain the large difference between the IV and oral dose of drugs with low oral bioavailability such as morphine, acyclovir, and propranolol. 7.5  In Vivo Bioavailability Basic Study Design The in vivo bioavailability studies for most drugs are usually performed in normal adult volunteers. This is because the objective of these studies is to compare the extent of absorption of two different formulations that can be investigated in normal volunteers. The study usually involves the administration of the drug product under investigation and an appropriate reference product to the study participants in two different occasions in a crossover experimental design. This means that each volunteer will receive the two

Figure 7.4 The plasma drug concentration-time profiles after a single administration of the same dose of a drug from different drug products that have different absolute bioavailability.

120  Single Oral Drug Administration, Extent of Drug Absorption products under investigation, and each volunteer will act as his/her own control. The choice of the reference products can be different depending on the objective of the different studies. In these studies, serial blood samples are obtained after each drug administration, and the AUCs are calculated and compared as described before.

• Determination of the absolute bioavailability for a new drug moiety requires admin-

istration of the solution of the active drug intravenously, and the drug formulation proposed for marketing. • Determination of the bioavailability of a new dosage form or formulation of an approved drug requires administration of the new formulation and one of the approved and marketed drug products as the reference standard. • Bioavailability studies conducted to investigate a new extended-release product require comparing the new drug product with a reference product, which can be drug solution, drug suspension, or marketed immediate release or extended-release products containing the same drug. • Bioavailability studies designed to investigate the bioavailability of individual drug ingredients used in combination products should compare the extent of absorption of each active drug ingredient after administration of the combination product and after administration of the individual ingredients concurrently as separate single-ingredient products. The reference products can be two or more currently marketed, single-ingredient drug products each of which contains one of the active drug ingredients in the combination product. It is also possible to use an approved product that contains the same drug combination as the product under investigation. 7.6  Calculation of the AUC Using the Linear Trapezoidal Rule The principle of the linear trapezoidal rule for calculating the AUC is based on dividing the plasma drug concentration-time profile to several trapezoids. The area of each trapezoid is calculated as presented in Figure 7.5 and in Eq. 7.13. Then the total AUC is

Figure 7.5  Calculation of the area of a trapezoid.

Single Oral Drug Administration, Extent of Drug Absorption 121

Figure 7.6  The AUC is the sum of the area of all trapezoids and the area under the tail of the curve.

calculated from the sum of the areas of these trapezoids in addition to the area under the tail of the curve as in Figure 7.6.  a  +  b  Area =  ⋅W  2 

(7.13)



Equation 7.14 is the general equation for calculating the area of a trapezoid,  C   +  C n+1  Area =  n  ⋅ (t n+1 − t n )  2

(7.14) 

while Eqs. 7.15 and 7.16 are the equations for calculating the area of the tail of the curve. Area of the tail =

Clast k 

(7.15)

Area of the tail =

Clast t1/2 = 1.44Clast t1/2 0.693 

(7.16)

or

where Clast is the last measured concentration, and k is the first-order elimination rate constant. The total AUC is the sum of all trapezoids plus the area of the tail as in Eq. 7.17. AUC = (1 / 2)(C0 + C1 )( t1 − t 0 ) + (1 / 2)(C1 + C 2 )( t 2 − t1 ) +  + area of tail



(7.17)

122  Single Oral Drug Administration, Extent of Drug Absorption Practice Problems: A patient received single doses of 250-mg IV, 500-mg oral suspension, and a 500-mg oral capsule on three separate occasions, and the following plasma concentrations were obtained: Time (hr)

1 2 3 4 6 8 12

IV bolus

Oral suspension

Oral capsule

Concentration (mg/L)

Concentration (mg/L)

Concentration (mg/L)

6.3 5.0 4.0 3.2 2.0 1.3 0.5

5.0 7.0 7.4 7.0 5.4 3.7 1.6

3.1 4.7 5.2 5.3 4.5 3.4 1.7

The half-life of this drug is 3 hr, and the absorption process is completed after 12 hr of oral administration. Questions: a Calculate the AUC after each drug administration. b Calculate the absolute bioavailability of the oral suspension and the oral capsule of this antibiotic. c Calculate the bioavailability of the oral capsule relative to the oral suspension of this antibiotic. Answer: a Calculation of the AUC • For the IV dose: Plot the plasma concentration-time profile after IV administration on the semilog scale to determine Cp0. Graphically, Cp0 is 8 mg/L. The elimination rate constant k = 0.693/3 hr = 0.231 hr−1. AUC IV =

Cp0 8mg/L = = 34.63mg hr/L k 0.231hr −1

• For the 500-mg oral suspension use the trapezoidal rule:  C + C n+1  Area of a trapezoid =  n  ⋅ (t n+1 − t n )  2  0 + 5 Area of trapezoid 1 =  ⋅ (1 − 0) = 2.5mg hr/L  2 

Single Oral Drug Administration, Extent of Drug Absorption 123  5+ 7 Area of trapezoid 2 =  ⋅ (2 − 1) = 6 mg hr/L  2   7 + 7.4  Area of trapezoid 3 =  ⋅ (3 − 2) = 7.2 mg hr/L  2   7.4 + 7  Area of trapezoid 4 =  ⋅ (4 − 3) = 7.2 mg hr/L  2   7 + 5.4  Area of trapezoid 5 =  ⋅ (6 − 4) = 12.4 mg hr/L  2   5.4 + 3.7  Area of trapezoid 6 =   ⋅ (8 − 6) = 9.1mg hr/L  2  3.7 + 1.6  Area of trapezoid 7 =   ⋅ (12 − 8) = 10.6 mg hr/L  2 Area of the tail =

Cplast 1.6 mg/L = = 6.93mg hr/L k 0.231hr −1

Total AUC = 2.5 + 6.0 + 7.2 + 7.2 + 12.4 + 9.1 + 10.6 + 6.93 = 61.93mg hr/L • For the 500 mg capsule use the trapezoidal rule:  0 + 3.1  Area of trapezoid 1 =   ⋅ (1 − 0) = 1.55mg hr/L  2   3.1 + 4.7  Area of trapezoid 2 =   ⋅ (2 − 1) = 3.9mg hr/L   2  4.7 + 5.2  Area of trapezoid 3 =   ⋅ (3 − 2) = 4.95mg hr/L  2  5.2 + 5.3  Area of trapezoid 4 =   ⋅ (4 − 3) = 5.25mg hr/L  2  5.3 + 4.5  Area of trapezoid 5 =   ⋅ (6 − 4) = 9.8mg hr/L  2

124  Single Oral Drug Administration, Extent of Drug Absorption  4.5 + 3.4  Area of trapezoid 6 =   ⋅ (8 − 6) = 7.9 mg hr/L  2  3.4 + 1.7  Area of trapezoid 7 =   ⋅ (12 − 8) = 10.2 mg hr/L  2 Area of the tail =

Cplast 1.7 mg/L = = 7.36 mg hr/L k 0.231hr −1

Total AUC = 1.55 + 3.9 + 4.95 + 5.25 + 9.8 + 7.9 + 10.2 + 7.36 = 50.9 mg hr/L b The absolute bioavailability of the suspension (note that the doses for the IV and the oral suspension are different): Fsuspension =

AUCsuspension Dose IV 61.93 mg hr/L 250 mg × = × = 0.89 AUC IV Dosesuspension 34.63 mg hr/L 500 mg

The absolute bioavailability of the capsule (note that the doses for the IV and the oral capsule are different): Fcapsule =

AUC capsule Dose IV 50.9 mg hr/L 250 mg × = × = 0.73 AUC IV Dose capsule 34.63 mg hr/L 500 mg

c The bioavailability of the capsule relative to the suspension (the dose is similar): Frelative =

AUC capsule Dosesuspension 50.9 mg hr/L 500 mg × = × = 0.82 AUCsuspension Dose capsule 61.93 mg hr/L 500 mg

The relative bioavailability can also be determined from the ratio of the absolute bioavailability of the two oral products Frelative =

Fcapsule 0.73 = = 0.82 Fsuspension 0.89

7.7 The Effect of Changing the Pharmacokinetic Parameters on the Plasma Drug Concentration-Time Profile after Single Oral Dose After administration of single oral dose, the plasma drug concentration-time profile depends on the dose, F, CLT, Vd, and ka of the drug. The CLT and the Vd are constants for a given drug in each patient and are independent of the dose, route of administration, and the formulation characteristics if the elimination process follows first-order kinetics. However, F and ka are dependent on the formulation characteristics and can be different for the different products for the same active drug. The drug effect-time profile is dependent on the plasma drug concentration-time profile for many drugs. The onset of drug effect usually depends on the time required to achieve the minimum effective concentration,

Single Oral Drug Administration, Extent of Drug Absorption 125 and it is dependent on the rate of drug absorption. The intensity of drug effect is related to the maximum drug concentration achieved, while the duration of effect is related to the period of time the plasma drug concentrations stay above the minimum effective concentration. This means that changes in the plasma drug concentration-time profile usually cause a modification of the therapeutic drug effect. So, it is important to examine how the change in the different pharmacokinetic parameters affects the plasma drug concentration-time profile after single oral administration. The following is a discussion of the effect of changing the different parameters on the plasma drug concentration-time profile, assuming first-order elimination. 7.7.1 Dose

The effect of administration of increasing doses of the drug:

• Administration of increasing oral doses of the same drug product to the same individ-

ual produces proportional increase in the plasma drug concentrations, including Cpmax and AUC due to the increase in the drug amount that reaches the systemic circulation. • If larger doses of the same product are used, the amount of the drug absorbed per unit time will be larger leading to proportional increase in Cpmax. However, ka and tmax that describe the rate of the absorption process will be the same. • Changing the dose does not affect k, t1/2, Vd, or CLT, so the rate of decline in plasma drug concentration during the terminal elimination phase does not change. Clinical Importance:

• Patients taking extra doses of their medications will have higher plasma drug concentrations, which usually produce more intense therapeutic effect.

• Higher plasma drug concentrations increase the systemic exposure that increases the

intensity of drug effect and increases the chance of producing adverse effects in low therapeutic index drugs. • Administration of larger oral doses of the drug produces drug concentrations-time profiles that stay above the drug minimum effective concentration for longer duration of time and can prolong the drug effect. 7.7.2 Bioavailability

The effect of administration of different products for the same drug that have different bioavailability:

• The bioavailable dose, which is the amount of the drug responsible for producing the

drug effect after oral drug administration, is dependent on the bioavailability of the drug from the drug product used. Using products with lower bioavailability decreases the bioavailable dose. • Oral administration of different drug products for the same drug that have different bioavailability should produce different Cpmax and AUCs and are expected to produce different therapeutic effects. • The different Cpmax and AUCs observed after administration of products for the same active drug with different oral bioavailability should not affect CLT, Vd, ka, and k.

126  Single Oral Drug Administration, Extent of Drug Absorption Clinical Importance:

• Patients who switch between oral drug products for the same drug that have different

oral bioavailability without considering the difference in bioavailability can experience alteration in drug effect and loss of disease control. • Regulatory authorities put regulations to ensure that marketed drug products for the same drug have a similar rate and an extent of absorption to allow patients to switch between marketed drug products for the same drug safely without compromising therapeutic effect. 7.7.3  Total Body Clearance

The effect of the change in CLT of the drug:

• Administration of the same dose of a drug using the same product (the same F and ka)

to a group of patients who have different CLT, due to different degrees of eliminating organ dysfunction, results in different rates of decline in the blood drug concentration during the elimination phase, if Vd is similar. • Patients with higher CLT will have a faster rate of drug elimination during the elimination phase (larger k, and shorter t1/2), while patients with lower CLT will have a slower rate of drug elimination (smaller k, and longer t1/2), if Vd is similar in these patients. • The AUC will be smaller in the patients with higher CLT and larger in the patients with lower CLT, because the AUC is inversely proportional to CLT. Clinical Importance:

• Administration of the same dose of a drug to all patients should produce different drug AUCs because of the variation in drug CLT. This will result in variation in the drug effect, so the drug dose should be individualized for different patients. • Patients with eliminating organ dysfunction have lower drug CLT and eliminate the drug at slower rate and usually require less than the average doses to produce the desired drug effect. 7.7.4  Volume of Distribution

The effect of the change in Vd of the drug:

• Administration of the same dose of a drug from the same product (the same F and ka)

to patients with different Vd, for example, patients with different body weights, but have similar CLT, will cause change in the rate of decline in the drug concentration during the elimination phase without affecting the AUC. • Assuming similar CLT, patients with smaller Vd will eliminate the drug at a faster rate (larger k, and shorter t1/2), while patients with larger Vd will eliminate the drug at slower rate (smaller k, and longer t1/2). • The AUC is directly proportional to the bioavailable dose (FD), and inversely proportional to the CLT and both were assumed to be constant. • In this discussion, it is assumed that the change is only in Vd while CLT is constant. Although Vd and CLT are independent pharmacokinetic parameters, it is possible that the change in Vd is accompanied by change in CLT. For example, patients with small body weight usually have lower Vd, and the lower body weight may be accompanied by lower CLT. So, there are some conditions that can lead to change in both CLT and Vd.

Single Oral Drug Administration, Extent of Drug Absorption 127 Clinical Importance:

• The drug dose should be calculated based on body weight rather than administration of a fixed dose to all patients, especially for low therapeutic index drugs.

• For many drugs, the doses for obese patients are calculated based on their ideal body weight rather than total weight, because their body composition is different from that of the average weight patients.

7.7.5  Absorption Rate Constant

The effect of administration of different products for the same drug that have different rates of absorption:

• Larger absorption rate constant results in faster drug absorption that is usually reflected in shorter tmax.

• Faster drug absorption produces higher Cpmax if the elimination rate is similar (similar CLT and Vd).

• The change in the rate of drug absorption (different ka) does not affect CLT, Vd, F, AUC, and k when the drug elimination follows first-order kinetics.

Clinical Importance:

• Faster drug absorption after oral administration usually results in a rapid onset of effect when there is direct correlation between plasma drug concentration and drug effect.

• Rapid drug absorption is usually preferred when the drug is used for the treatment of acute conditions.

• Slow drug absorption is usually preferred during multiple drug administration for the

management of chronic diseases to ensure steady drug effect between drug administrations.

7.8 Summary

• Orally administered drugs are not absorbed completely to the systemic circulation • •

• •

because of many physiological, physicochemical, and formulation factors, which can all contribute to the presystemic drug elimination. The absolute bioavailability of an oral product is the fraction of the administered dose that reaches the systemic circulation, while the relative bioavailability is the bioavailability of an oral product relative to the bioavailability of a second oral drug product. The absolute bioavailability after oral administration can be determined by comparing the AUCs calculated after oral and IV drug administration in the same individuals. While the relative bioavailability of oral products is determined from the comparison of AUCs calculated after oral administration of the two products under investigation. Similarly, the absolute and relative bioavailability of drug products administered by any other extravascular route such as intranasal, rectal, or transdermal can be determined. The use of drug products with lower bioavailability results in the reduction of the bioavailable dose of the drug that may reduce the intensity and the duration of therapeutic effect after single drug administration and can decrease the therapeutic effect during multiple drug administration.

128  Single Oral Drug Administration, Extent of Drug Absorption Practice Problems 7.1 The absolute bioavailability of ampicillin capsules is 50%. What does this mean? 7.2 The bioavailability of ofloxacin suspension relative to ofloxacin tablet is 125%. What does this mean? 7.3 After IV administration of gentamicin, all the dose is excreted unchanged in urine, while after oral gentamicin, administration less than 1% of the oral dose is excreted unchanged in urine. Explain. 7.4 Different formulations of the same drug were administered to the same individuals and the following data were obtained: • After IV bolus administration of 10 mg, the AUC was 5 mg hr/L. • After intramuscular (IM) administration of 20 mg, the AUC was 9 mg hr/L. • After oral administration of 50 mg capsule, the AUC was 10 mg hr/L. • After oral administration of 20 mg oral suspension, the AUC was 5 mg hr/L. a Calculate the absolute bioavailability of the drug from the capsules. b Calculate the absolute bioavailability of the drug after IM administration. c Calculate the bioavailability of the oral suspension relative to the capsules. d Excluding the IV, which formulation has the highest absolute bioavailability? 7.5 A single dose of two products of paracetamol and two products of ibuprofen were administered for the same group of individuals in four different occasions and tmax, Cpmax, and AUC were determined. Dose (mg) tmax (hr) Cpmax (mg/L) AUC (mg hr/L) Paracetamol product A Paracetamol product B Ibuprofen product A Ibuprofen product B

500 500 250 250

2.0 2.5 3.0 3.5

4.1 6.1 13.7 11.6

22.2 26.6 67.2 58.5

a What is the bioavailability of paracetamol product A relative to paracetamol product B? b What is the bioavailability of ibuprofen product A relative to ibuprofen product B? c What is the bioavailability of paracetamol product A relative to ibuprofen product A? d Which one of the four products has the highest absolute bioavailability? 7.6 After administration of a single-dose IV bolus dose of acyclovir 300 mg to a patient, the total AUC was 16 mg hr/L, and the elimination half-life was 2.4 hr. After an oral administration of a 300-mg tablet to the same patient, the following plasma concentrations were obtained: Time (hr)

Drug concentration (mg/L)

0 1 3 5 7 10 15 20

0 0.364 0.716 0.787 0.730 0.569 0.318 0.162

a Calculate the absolute bioavailability of acyclovir in this patient.

Single Oral Drug Administration, Extent of Drug Absorption 129 7.7 After IV administration of 500 mg of an antibiotic, the plasma conc-time profile was linear on semilog scale with y-intercept of 20 mg/L, and a slope of −0.043 hr−1. After an oral administration of an 800-mg capsule of the same antibiotic in the same patient, the following data were obtained. Time (hr)

Concentration (mg/L)

1 3 6 13

7.0 13.0 9.0 4.5

Calculate the absolute bioavailability of the antibiotic from the oral capsule. Show your work. 7.8 Different ibuprofen products were administered to the same group of individuals and the following data were obtained after: Ibuprofen intramuscular, 100 mg Ibuprofen capsule, 400 mg Ibuprofen suspension, 200 mg Ibuprofen rectal suppository, 200

The AUC was 40 mg hr/L The AUC was 180 mg hr/L The AUC was 120 mg hr/L The AUC was 100 mg hr/L

a What is the bioavailability of the ibuprofen suspension relative to the capsule? b What is the bioavailability of the ibuprofen intramuscular injection relative to the capsule? c Which product has the highest absolute bioavailability? d What is the absolute bioavailability of ibuprofen intramuscular injection? 7.9 A group of volunteers received a single oral dose of two different formulations of an oral hypoglycemic drug. Each tablet from the two formulations contains 500 mg of the drug. The average plasma drug concentrations obtained at different time points after administration of the two formulations are tabulated below. Time (hr) Formulation A

0 1 2 4 6 8 10 12 20 30

Formulation B

Concentration (mg/L)

Concentration (mg/L)

0 12.9 18.6 20.9 18.8 15.9 13.2 10.8 4.87 1.79

0 6.89 11.9 17.7 19.8 19.8 18.6 16.8 9.36 3.78

a Which formulation is absorbed faster? Why? b What is the bioavailability of formulation A relative to the bioavailability of formulation B? How can you interpret this result?

130  Single Oral Drug Administration, Extent of Drug Absorption 7.10 Different formulations of the same drug were administered to a group of individuals and the plasma concentrations were measured after administration of each formulation. The results are tabulated below.

a b c d e

Time (hr)

Concentration Concentration Concentration Concentration (mg/L) (mg/L) (mg/L) (mg/L)

Dose

200-mg IV

0 1 2 3 5 7 10 15 20

7.40 5.5 4.56 2.23 1.22 0.50 0.111 0.0245

500-mg Syrup 500-mg Capsule

500-mg Tablet

0 9.22 11.9 11.6 8.32 5.16 2.27 0.527 0.119

0 4.22 5.99 6.32 5.27 3.70 1.89 0.518 0.129

0 6.0 8.14 8.25 6.35 4.15 1.94 0.475 0.11

Calculate the half-life and volume of distribution of this drug Which oral formulation has the fastest absorption rate? Calculate the absolute bioavailability of the three oral formulations Calculate the bioavailability of the oral capsule relative to the syrup Calculate the bioavailability of the tablet relative to the bioavailability of the capsule

7.11 The following plasma concentrations were obtained after administration of a single 500 mg tablet of an antihypertensive drug to a patient. Time (hr)

Concentration (mg/L)

1 3 6 10 15 20 26

3.75 6.1 5.2 3.14 1.32 0.56 0.197

When a single IV bolus dose of 250 mg was administered to the same patient, the total AUC was 60 mg hr/L. a What is the half-life of this drug in this patient? b What is the total AUC of the drug after administration of a 500-mg oral dose in this patient? c What is the bioavailability of the oral tablet in this patient? d What is the CLT and the Vd of this drug in this patient? 7.12 A patient received a single 300-mg IV bolus dose and a single 500-mg oral dose of an antibiotic on two separate occasions, and the following plasma concentrations were obtained.

Single Oral Drug Administration, Extent of Drug Absorption 131 IV administration

Oral administration

Time (hr) 2 6 12 24

Time (hr) 0 1 3 5 7 10 15 20 30

Concentration (mg/L) 12.86 9.45 5.95 2.36

Concentration (mg/L) 0 2.24 4.69 5.54 5.58 5.0 3.68 2.57 1.20

a Calculate the half-life, volume of distribution of this drug in this patient. b Calculate the fraction of the drug absorbed after oral administration. 7.13 After oral administration of a single 500-mg tablet of an antihypertensive drug to a normal volunteer, the total calculated AUC was 70 mg hr/L. After an IV bolus administration of the same drug to the same volunteer, the drug half-life was 3 hr, and the volume of distribution was 25 L. a Calculate the bioavailability of this oral tablet in this volunteer. b What is the expected AUC after administration of a single IV bolus dose of 250 mg of this drug to the same volunteer? 7.14 A patient received a single 200-mg IV bolus dose and a single 500-mg oral dose of an antibiotic on two separate occasions, and the following plasma concentrations were obtained. The plasma drug concentrations obtained after the IV and the oral doses are tabulated below. IV administration (200 mg)

Oral administration (500 mg)

Time (hr) 2 4 8 12

Time (hr) 0 2 4 8 12

Concentration (mg/L) 6.30 5.16 3.46 2.32

Concentration (mg/L) 0 6.9 5.5 3.7 2.5

a What is the AUC after the 200-mg IV dose? b What is the AUC after the 500-mg oral dose? c What is the fraction of the oral dose absorbed to the systemic circulation? References 1. Rowland M, Benet LZ and Graham GG “Clearance concepts in pharmacokinetics” (1973) J Pharmacokinet Biopharm; 1:123–136. 2. Lilja JJ, Neuvonen M and Neuvonen PJ “Effect of regular consumption of grapefruit juice on the pharmacokinetics of simvastatin” (2004) Br J Clin Pharmacol; 58:56–60. 3. Kunta JR and Sinko PJ “Intestinal drug transporters: In vivo function and clinical importance” (2004) Curr Drug Metab; 5:109–124.

8

Bioequivalence

Objectives After completing this chapter, you should be able to:

• Define bioavailability, bioequivalence, pharmaceutical equivalence, and therapeutic equivalence.

• Discuss the situations when in vivo bioequivalence studies are required. • Discuss the situations when in vivo bioequivalence studies can be waived. • Describe the different approaches that can be used to demonstrate bioequivalence of drug products.

• Describe the general guidelines for study design, study execution, and data analysis of in vivo bioequivalence studies.

• List the different components of the analytical assay validation. • Explain the different approaches to demonstrate product bioequivalence in special conditions, such as in presence of active metabolites, fixed-dose combinations, highly variable drugs, endogenous compounds, and long half-life drugs.

8.1 Introduction Drug approval process can be different from one country to the other; however, the process follows the same general steps. Approval of a new molecular entity that has never been marketed before requires the drug company that develops the drug to submit a new drug application (NDA), which includes evidence to prove that the new drug is safe and effective for the treatment of some specific indication. If the NDA is approved, the drug can be marketed as an “innovator product” exclusively by that drug company for the remaining of the patent period. When the drug patent expires, other pharmaceutical companies can request marketing “generic products” for the same active drug. The generic drug products are similar to the innovator drug product with regard to the active ingredient, dosage form, route of administration, strength, quality, performance, and intended use. Marketing approval of a generic drug product requires pharmaceutical companies to file abbreviated NDA (ANDA). The ANDA usually should not include evidence of drug safety and effectiveness, because these have been proven previously. Instead, the ANDA should provide evidence of bioequivalence (BE) of the generic and innovator products. This means that the ANDA should provide evidence to prove that after administration of the generic product, the resulting therapeutic effect and safety profile are similar to those of the innovator product. DOI: 10.4324/9781003161523-8

Bioequivalence 133 Once approved, applicant(s) can manufacture and market the generic drug products to provide safe, effective, and low-cost alternative drug products. Also, supplemental drug applications filed to propose changes in the manufacturing process or formulation of approved drugs should include evidence of bioequivalence to prove that the proposed changes will not affect the in vivo performance of the original product. The primary objective of the ANDA is to demonstrate that the safety and efficacy of the generic drug product are comparable to that of the approved and marketed innovator drug product. This should allow patients who are using the innovator product to switch to the generic product for the same drug without compromising the safety and therapeutic efficacy. The BE studies represent the bases for evaluating the “interchangeability” between the generic and innovator products. Regulatory drug authorities around the world develop guidance for pharmaceutical industry for the design, execution, data analysis, data reporting, and submission of BE study results. This helped in the standardization of the approaches used to demonstrate the BE of pharmaceutical products and resulted in the approval of a large number of generic drug products every year. 8.2  General Definitions Drug products: The finished dosage forms such as tablets, capsules, or solutions that contain the active drug ingredient(s) with or without inactive ingredients. Pharmaceutical alternatives: Drug products that contain the same therapeutic moiety but differ in dosage form, strength, salt, or ester of the active therapeutic moiety. Pharmaceutical equivalents: Drug products that have the same dosage form for the same route of administration and contain the same strength of the same salt or ester for the same active ingredient but differ in shape, release mechanism, labeling, scoring, and excipients, including color, flavor, and preservative. Bioequivalent products: Pharmaceutical equivalent products that have no significant difference in the rate and extent to which the active ingredient becomes available at the site of action when administered under similar conditions in an appropriately designed study. Therapeutic Equivalents: Drug products for the same active ingredient that can be substituted with the full expectation of producing the same clinical effect and safety profile as the prescribed product. Drug products are considered to be therapeutically equivalent if they are pharmaceutical equivalents and bioequivalent. Reference listed drug products: An approved drug product to which new generic versions are compared to prove that they are bioequivalent. 8.3  Regulatory Requirement for Bioequivalence Based on the previous definition, bioequivalent products for the same active drug have similar rates and extent of drug availability at the site of action. This can happen if administration of the two products produces similar drug concentration-time profiles in the systemic circulation. Similar drug profiles in the systemic circulation can be achieved only when the rate and extent of drug absorption after administration of the two products are similar. So, demonstration of product bioequivalence can be based on comparing the rate and extent of drug absorption after administration of the two products. Pharmaceutical companies are required to prove that the generic drug product

134 Bioequivalence and the approved reference standard product are bioequivalent before marketing the generic product. This is to ensure that the generic products for the same active drug will produce therapeutic effects similar to that of the reference listed drug product. Demonstration of in vivo bioequivalence for drug products is required in the following applications (1): a All ANDA-requesting approval of a generic drug product should include evidence demonstrating that the generic drug product and the reference listed drug are bioequivalent. b Supplemental applications that propose change in the manufacturing site, change in the manufacturing process, change in product formulation, or change in dosage strength should include evidence demonstrating that the generic drug product after the proposed changes and the reference listed drug are bioequivalent. c Supplemental application containing new evidence demonstrating in vivo bioequivalence of an approved generic drug product should be submitted by pharmaceutical companies if requested by the drug regulatory authorities. New bioequivalence studies can be requested in case of the availability of data demonstrating that the recommended drug dosage regimen is based on incorrect pharmacokinetics information, or data showing significant (>25%) intra-batch and batch-to-batch variability in the bioavailability of the drug product. The in vivo bioequivalence study is not performed if the characteristics of the drug product meet at least one of the criteria that permit waiver of in vivo bioequivalence determination. Instead, the application should contain evidence to prove that the drug product falls into one of the product categories that do not require performing in vivo bioequivalence study. Clinical Importance:

• The prove of product bioequivalence ensures that patients continue to have the same

therapeutic effect when they switch between marketed generic products for the same active drug. • The prove of product bioequivalence ensures that any changes in the formulation and manufacturing processes do not affect the therapeutic effect of the drug product. • The prove of product bioequivalence ensures that drug dosage recommendations are based on accurate data and that the drug product continues to have the same in vivo performance and is expected to produce the same therapeutic effect. 8.4 Criteria for Requesting a Waiver of the In Vivo Bioequivalence Determination Under certain circumstances when there is sufficient evidence that in vivo bioequivalence studies are not required for a particular product, the drug manufacturer may apply for waiver of the in vivo bioequivalence determination requirements. The same criteria can be used for waiving the in vivo bioavailability requirement for drug products. The drug manufacturer has to submit evidence that the product meets at least one of the criteria that allow granting waiver of the in vivo bioequivalence determination (2).

Bioequivalence 135 The following are the criteria for requesting waiver of the in vivo bioequivalence determination: a Drug products when bioequivalence is self-evidence, as in the case of: • Parenteral solutions intended for administration by injection or ophthalmic and otic solutions that contain the same active and inactive ingredients in the same concentration as an approved drug product. • Products administered by inhalation such as inhalation anesthetics that contain the same active ingredient in the same dosage form as an approved drug product. • Solutions for application to the skin, oral solutions, elixirs, syrups, tinctures, solutions for aerosolization or nebulization, nasal solutions, or similar other solubilized forms that contain an active drug ingredient in the same concentration and dosage form as an approved drug product. The product should not contain inactive ingredients that can affect the in vivo bioavailability of the active ingredient. b Drug products when bioavailability can be determined or bioequivalence can be demonstrated by evidence obtained in vitro, such as: • The drug product is in the same dosage form, but in a different strength, and is proportionally similar in its active and inactive ingredients to another approved drug product for the same manufacturer. This condition does not apply to delayedrelease or extended-release products. • The drug product is a reformulated product that is identical to another approved drug product by the same manufacturer, except for a different color, flavor, or preservative that should not affect the bioavailability of the active ingredient of the reformulated product. • The drug product is shown to meet an in vitro test that has been correlated with in vivo product performance. For example, immediate release solid dosage forms of the Biopharmaceutics Classification System (BCS) Class I drugs and Class III drugs under certain conditions. The same criteria are used by drug manufacturers to apply for requesting waiver of the in vivo bioavailability requirement for drug products. Clinical Importance:

• When the dosage form, formulation characteristics, and/or route of administration do not affect the drug absorption process, in vivo bioequivalence studies are not required.

• The availability of alternative simple methods that can demonstrate product bioequiv-

alence can save the effort and high cost needed to perform in vivo bioequivalence studies.

8.5  Approaches for Demonstrating Product Bioequivalence Several in vivo and in vitro methods are available to establish the bioequivalence of specific drug products. Regulatory drug authorities require the use of the most accurate, sensitive, and reproducible method available to demonstrate product bioequivalence. The

136 Bioequivalence selection of the method depends on the objective of the study, the analytical methods available, and the nature of the drug product (3). 8.5.1  In Vivo Pharmacokinetic Studies

The most acceptable approach for assessing the bioequivalence of orally administered products for systemically acting drugs involves administration of the drug product and then measuring the active drug concentration as a function of time in the systemic circulation. Characterization of the plasma drug concentration-time profiles and comparison of pharmacokinetic parameters, such as AUC, tmax, and Cpmax, have been shown to be the most acceptable approach (4). This is because when two products for the same active drug have similar rates and extent of drug absorption, the drug profile in the systemic circulation after administration of single dose of the two products should be comparable. Drug products that produce similar drug concentration-time profiles are expected to produce similar therapeutic effects. The pharmacokineticbased approach for demonstrating bioequivalence will be discussed in detail in the next section. 8.5.2  In Vitro Test Predictive of In Vivo Human Bioavailability

In vitro dissolution test for solid dosage forms can be utilized as a quality control (QC) tool to evaluate the product quality and to predict the in vivo product performance. The development of in vitro dissolution testing to predict the in vivo product performance usually involves correlation of the in vitro rate and extent of drug dissolution from the dosage form with the in vivo drug absorption, which is known as in vitro-in vivo correlation (IVIVC). Regulatory drug authorities define four different levels of correlation between the in vitro dissolution test results and the in vivo rate and extent of drug absorption. Level A correlation is the highest degree of correlation, which shows point-to-point correlation between the in vitro dissolution rate and the in vivo absorption rate from the dosage form, while in level B correlation, the mean in vitro dissolution time is correlated with the in vivo mean residence time (MRT) or mean in vivo dissolution time. However, in level C and multiple level C correlation, one or more dissolution time points such as time for 50% or 90% dissolution are correlated with one or more mean pharmacokinetic parameters, such as AUC, tmax, or Cpmax. Only level A correlation is accepted by regulatory authorities as a justification for using the in vitro dissolution testing to predict the in vivo absorption of the drug. This level of IVIVC can use in vitro dissolution parameters for a dosage form to predict the expected AUC and Cpmax after administration of that dosage form within 10% of the experimentally observed values of these parameters. Once level A IVIVC is established and validated, the in vitro test can serve as a surrogate for bioequivalence testing, and for evaluating new drug formulations. The IVIVC has important applications in the rational development and evaluation of extended-release dosage forms (5). 8.5.3  Acute Pharmacodynamic Effect

Studies that use acute pharmacodynamic measures to establish bioequivalence for two drug products can be used only in situations when the drug and/or metabolite concentrations in biological fluids cannot be determined with sufficient accuracy. Also, when

Bioequivalence 137 the drug concentration in the systemic circulation is not correlated with the drug therapeutic effect, such as in the case of topically applied drugs or drugs administered by inhalation that is not intended to produce systemic effects. When the drug pharmacodynamic effect is used to demonstrate bioequivalence, several important issues should be considered:

• The pharmacodynamic effect measured should be relevant to the efficacy and/or safety of the drug.

• The approach used for measurement should be validated for accuracy, precision, specificity, and reproducibility.

• The effect produced after administration of the test and reference products should not reach the maximum effect to allow for the detection of formulation differences.

• The drug effect should be evaluated under double-blind conditions; meaning that nei-

ther the subject nor the evaluator knows which product is being evaluated to eliminate personal bias. • The experiment should be performed using the crossover experimental design if possible. If a placebo effect can occur, a placebo treatment can be added as a third phase in the study design. • The underlying disease state and the natural history of the condition should be considered in the study design if the drug effect is evaluated in patients. • The effect-time profile can be constructed by repeating the measurement at different time points if the drug effect changes with time. 8.5.4  Comparative Clinical Studies

In some instances when measuring the drug in the systemic circulation is not possible and when there is no meaningful acute drug effect that can be measured, clinical trials can be used to compare different products and to demonstrate bioequivalence. In this case, several important issues should be considered.

• The target parameter should be a relevant clinical endpoint from which the onset and intensity of response can be defined.

• A placebo treatment should be included as a third phase in the study protocol. • A safety endpoint should be included in the final comparative assessment, if possible. • Appropriate statistical procedures should be used to compare the study results. The acceptance limit has to be defined on case-by-case basis depending on variability in the target endpoint, and the specific clinical condition.

8.5.5  In Vitro Dissolution Testing

Under certain conditions, bioequivalence can be demonstrated using in vitro approaches. The rate and extent of drug absorption from rapidly dissolving oral drug products for drugs that are highly soluble and highly permeable (BCS Class I drugs) do not depend on the drug formulation. In this case, documentation of product bioequivalence using in vitro approaches is appropriate. Evidence of the drug high solubility, high permeability, and the rapidly dissolving product as defined by regulatory authority should be included as part of the study report (6).

138 Bioequivalence 8.6  Pharmacokinetic Approach to Demonstrate Product Bioequivalence Several regulatory guidance documents are available to outline the general requirements for the design, execution, and reporting of the in vivo bioequivalence studies (2, 3, 7, 8). The primary objectives of these guidance documents are to guide pharmaceutical companies to fulfill all the requirements for performing in vivo bioequivalence studies and increase the likelihood of their acceptance and approval. The general guidelines can be applied for most drug products; however, modification of these guidelines may be necessary for some drug products with special characteristics. The bioequivalence study consists of a clinical phase that involves drug product administration to the subjects participating in the study and the laboratory phase that involves sample analysis, data analysis, and preparation of the report. In general, it is required to apply the principles of Good Clinical Practices (GCP) while performing the clinical phase of the bioequivalence study and the principles of Good Laboratory Practices (GLP) during all the laboratory activities of the study. 8.6.1  Planning for the In Vivo Bioequivalence Study

Planning for the in vivo bioequivalence study involves collecting information about the following:

• Information about the drug/drug products to see if it falls under any of the categories

that are eligible for requesting waiver from performing the in vivo bioequivalence study.

• Information about the test product to help in the selection of the reference drug product to use in the in vivo bioequivalence study.

• Information about the general pharmacokinetic behavior of the drug and the expected

range of plasma drug concentrations after administration of single dose of the drug product under investigation. • Information about the available analytical methods that are suitable for the analysis of the drug in biological fluids in the expected range of drug concentration. • Information about any expected adverse effects after administration of the drug and any special precautions that should be considered after administration of the drug. 8.6.2  Selection of the Reference Drug Product

An approved drug product in the same dosage form containing the same active drug moiety in the same strength as the test product is selected as the reference drug product. The reference drug product is normally the innovator product for which efficacy, safety, and quality have been established (9). 8.6.3  In Vitro Testing of the Study Products

Before the start of the vivo bioequivalence study, the test and the reference drug products usually undergo in vitro evaluation. This usually includes physical description, dimensions, mean weight, weight uniformity, content uniformity, and in vitro dissolution testing. This is important because differences in drug contents or in vitro dissolution rate between the test and reference products most likely will lead to differences in the rate and extent of drug absorption. So, it is important to ensure similarity in the drug contents and/or in vitro dissolution rate for the test and reference products before starting the clinical phase of the in vivo bioequivalence study.

Bioequivalence 139 The in vitro dissolution testing is an essential part of the in vitro assessment for generic drug products. The in vitro dissolution of the test and reference drug products can be compared using single-point estimate or by comparing the dissolution profiles at different time points. For example, rapidly dissolving products are defined as having at least 85% dissolution of the drug contents within 30 min in 900 mL at pHs 1.2, 4.5, and 6.8. So, rapidly dissolving products can be compared by determining the % dissolved at 30 min at the different pHs, while the dissolution profile can be compared by determining the similarity factor (f2) at the three pHs, which can be more accurate than relying on a single-point dissolution test. The similarity factor can be calculated according to Eq. 8.1.    100 F2 = 50 log  n  1  (R t − Tt )2  1+ n  t =1

∑

       

(8.1)

where Rt and Tt are the cumulative percentages dissolved from the reference and test products, respectively, at each of the n sampling time points during the dissolution experiment. When the dissolution profiles for the reference and test products are identical, F2 will be equal to 100, while when the difference is 10% at each time point, F2 will be equal to 50. So, values for F2 in the range of 50–100 indicate that the two dissolution profiles are similar (10). All the results of the in vitro evaluation of the test and reference products are included in the final bioequivalence report. 8.6.4  In Vivo Bioequivalence Study Design 8.6.4.1  Basic Principles

The in vivo bioequivalence study involves single-dose administration of the test drug product and a suitable reference product on two different occasions to normal adult volunteers in the fasting state, followed by comparing the rate and extent of drug absorption from the two products (8, 9). 8.6.4.2  Ethical Approval

The bioequivalence study must be performed in accordance with the ethical principles included in the current version of the Declaration of Helsinki for involving human subjects in research. An independent institutional review committee must review the study protocol to confirm that the study complies with the ethical standards for using human subjects in research. The study can be performed only after approval of the study protocol by the independent review committee. 8.6.4.3  The Study Subjects

The objective of the in vivo bioequivalence study is to compare the rate and extent of drug absorption after administration of the test and reference products, without evaluating the

140 Bioequivalence drug therapeutic effect. So, the study can be performed in normal volunteers with the following characteristics:

• • • • • •

Normal healthy male and female volunteers preferably nonsmokers. Between 18 and 50 years of age. Within 10% of their ideal body weight. No history of serious chronic diseases. No history of adverse reaction to the drug in the study or any other drug in its class. Participants are subjected to physical examination and routine laboratory tests to ensure normal renal, hepatic, and hematological functions. • The volunteers are not allowed to take any prescription or over-the-counter drugs for the two weeks before the study and should not use alcohol and caffeine or other xanthine-containing beverages during the two days before the study. • After ethical approval of the study protocol, each participating volunteer should sign an informed consent form that contains detailed information about the study. 8.6.4.4  Number of Volunteers

The number of volunteers required for the bioequivalence study depends on the variability in the pharmacokinetic parameters of the drug, the acceptable significance level (α = 0.05), and the acceptable deviation level between the products being compared (±20%). Since the variation in the pharmacokinetic parameters for most drugs is < 30% coefficient of variation (CV), enrollment of 24 volunteers should be sufficient to ensure adequate statistical results in most bioequivalence studies. Drugs with larger variability in their pharmacokinetic parameters require a larger number of volunteers (11). 8.6.4.5  Drug Administration

The dose of the test product and the reference product should be similar. Single dose of the drug is administered with 250 mL of water after overnight fasting (8 hr) and fasting usually continues for at least 4 hr after drug administration. The volunteers are monitored clinically throughout the study period, and any complains must be recorded in the clinical data sheet for each volunteer (8, 9). 8.6.4.6  Experimental Protocol

The bioequivalence study protocol used for most drug products is as follows:

• Single-dose: each volunteer receives one dose of one product in each study period. • Two-treatments: each volunteer receives the test product and reference product on two different study periods.

• Two-periods: the two products are administered in two different time periods. • Two-sequences: some volunteers receive the test product first, then the reference product, and other volunteers receive the reference product first then the test product.

• Crossover: each volunteer receives the two products so each volunteer act as his/her own control.

The diagram in Figure 8.1 represents the experimental design for the bioequivalence study. The volunteers are randomly assigned to one of two groups that will receive the

Bioequivalence 141

Figure 8.1  A diagram illustrating the crossover experimental design for the bioequivalence study.

drug products under investigation in two different sequences (8, 9). The period between treatments, known as the washout period, should allow complete elimination of the drug before administration of the second drug product. The washout period should be at least five times the elimination half-life of the drug. 8.6.4.7  Collection of Blood Samples

Comparison of the test product and the reference product in bioequivalence studies with pharmacokinetic endpoints is based on characterization of the plasma drug concentration-time curve. Enough blood samples should be obtained during the absorption phase of the drug to allow good estimation of Cpmax, and sampling should continue for at least three drug elimination half-lives to permit accurate determination of the total AUC. The sampling schedule is usually determined from a pilot study performed before the actual bioequivalence study or from the information available about the pharmacokinetic behavior of the drug in the literature. After administration of the test and reference drug products, the sampling times should be identical. Samples should be stored frozen until analysis by a suitable analytical technique. 8.6.4.8  Analysis of Bioequivalence Study Samples

The samples obtained after drug administration are analyzed to determine the concentration of the active drug or its metabolite using a validated analytical method. Analytical method validation is important to demonstrate that the method used for quantitative measurement of the concentration of a given analyte in the biological matrix is reliable and reproducible. The main parameters for validation include selectivity, accuracy, precision, sensitivity, reproducibility, and stability (12). Selectivity: It is the ability of the analytical method to differentiate and quantify the analyte of interest in the presence of other compounds in the samples. Accuracy: It is the closeness of the mean concentration results obtained by the analytical method and the true value of the analyte concentration. Precision: It is the closeness of individual measurements when several aliquots of the same sample are analyzed repeatedly. Extraction efficiency or recovery: It is the amount of the analyte extracted from the sample compared to the total amount of the analyte in the sample.

142 Bioequivalence The calibration curve: It is constructed from the relationship between the detector response and the analyte concentration, and it is used to estimate the analyte concentration in the study samples. Stability: It is the drug stability in biological matrix during sample handling, sample preparation, and sample storage. Once the analytical method is validated, the bioequivalence study samples are analyzed using the same procedures utilized during the validation. The detector response of the unknown study sample is used to estimate the drug concentration in the sample utilizing the mathematical equation generated from the calibration standards analyzed on the same run. QC samples containing known analyte concentrations are included with each run and are treated as unknown samples. The estimated analyte concentrations in these QC samples are compared with the nominal concentrations of the samples, and the results are used to accept or reject the run results. 8.6.4.9  Pharmacokinetic Parameter Determination

The bioequivalence of two different products for the same active drug is demonstrated if the rate and extent of drug absorption after administration of the two products are comparable. 8.6.4.9.1  THE EXTENT OF DRUG ABSORPTION

The extent of drug absorption is generally determined from the drug AUC. The AUC is calculated using the trapezoidal rule until the last measured concentration (AUC0-t), then extrapolated to calculate the total area (AUC0-∞). Enough samples should be obtained to make AUC0-t cover at least 80% of the total AUC0-∞. 8.6.4.9.2  THE RATE OF DRUG ABSORPTION

The rate of drug absorption is determined from Cpmax and tmax. The Cpmax is the highest drug concentration measured after drug administration, and tmax is the time of the sample that has the highest concentration. Other pharmacokinetic parameters such as the terminal elimination rate constant and the elimination half-life are calculated, and they are used in the extrapolation of the AUC. 8.6.4.10  Statistical Analysis

The analysis of variance (ANOVA) is usually performed on the pharmacokinetic parameters that reflect the rate and extent of drug absorption, AUC, Cpmax, and tmax. Appropriate statistical model relevant to the study design is applied. The statistical model includes factors that explain the different sources of errors in the calculated pharmacokinetic parameters including:

• • • •

Sequence effect (order effect) Subjects nested in sequence Period effect (phase effect) Treatment effect (product effect)

Bioequivalence 143 All these effects that contribute to the variation in the calculated parameters are analyzed by the ANOVA to test the assumptions of the study design (11). However, the decision about product bioequivalence is not usually made based on the ANOVA results. The statistical method used to test for bioequivalence is the two one-sided t-test procedures (13). These procedures are based on calculation of the 90% confidence interval for the ratio of the average log-transformed pharmacokinetic parameters for the test and reference products. The log-transformed values of AUC and Cpmax are used because these parameters are not normally distributed around their mean values, but the log-transformed values are normally distributed (log-normal distribution). To establish bioequivalence, the calculated 90% confidence intervals for the ratio of the average log-transformed AUC and Cpmax should fall within the bioequivalence limit of 80–125%. Products with confidence intervals that fall outside this range are not considered bioequivalent. 8.6.4.11  Documentation and Reporting

All information related to the bioequivalence study has to be documented and included in the bioequivalence study report. This includes information about the drugs under investigation, volunteers, study design, analytical technique, pharmacokinetic analysis, in vitro testing, and statistical analysis. Also, the volunteers’ clinical data sheets, the detailed results of the measured drug concentrations, the estimated pharmacokinetic parameters, analytical technique validation report, statistical analysis report, examples of the analytical instrument output, e.g., chromatograms for the analysis of bioequivalence samples, and the bioequivalence decision based on the obtained results should be included in the report (14). 8.7  Special Issues Related to Bioequivalence Determination The general guidelines for conducting in vivo bioequivalence studies can be applied for most drugs and drug products. However, modification of these general guidelines may be necessary to demonstrate bioequivalence of drug products with special characteristics. These modifications can be as simple as changing the acceptance limit for bioequivalence or can be significant such as using different study protocols. The following are a few suggested modifications of the general bioequivalence study guidelines while performing bioequivalence studies for drugs and drug products with special characteristics. The applicability of these suggestions must be evaluated for each drug. 8.7.1  Multiple-Dose Bioequivalence Studies

Bioequivalence studies can be conducted to compare the performance of the test and reference drug products during multiple administration. The crossover experimental design is usually used in multiple-dose bioequivalence studies. Whenever a multiple-dose study is conducted, sufficient doses of the test and reference products should be administered to achieve steady state. Samples should be obtained to completely characterize the blood concentration-time profile during one dosing interval at steady state. The pharmacokinetic parameters calculated in multiple-dose bioequivalence study are: AUC0-τ:  The AUC during one dosing interval at steady state. Cpmax ss:  The maximum drug concentration at steady state. It is the highest measured concentration during one dosing interval at steady state.

144 Bioequivalence Cpmin ss:  The minimum drug concentration at steady state. It is the drug concentration just before drug administration at steady state. Cpaverage ss:  The average drug concentration at steady state. It is calculated as: Cpaverage ss =

AUC0-τ τ 

(8.2)

where τ is the length of the dosing interval. tmax ss:  The time to achieve the maximum drug concentration within the dosing interval at steady state. %Swing: 100 (Cpmax ss − Cpmin ss ) /Cpmin ss %Fluctuation: 100 (Cpmax ss − Cpmin ss ) /Cpaverage ss Both % swing and % fluctuation are measures of the variation in the drug concentrations during repeated drug administration at steady state. In multiple-dose bioequivalence studies, Cpmax ss, Cpmin ss, and tmax ss are determined directly from the drug concentrations, AUC0-τ is calculated by the trapezoidal rule, and % swing and % fluctuation are calculated as mentioned above. The % swing and % fluctuation are used as the primary parameters that describe the rate of drug absorption while AUC0-τ is a measure of the extent of drug absorption (3, 7). The decision regarding the bioequivalence of the test and reference products is based on the statistical comparison of the parameters calculated for the two products. 8.7.2  Food-Effect Bioequivalence Studies

Coadministration of food with oral products can affect the rate and extent of drug absorption. This can be due to many factors, including delay in gastric emptying, change in GIT pH, stimulation of bile secretion, change in luminal drug metabolism, and direct interaction between the food constituents and the drug. So, bioequivalence studies in fed subjects are usually recommended for immediate release dosage forms that contain drugs with narrow therapeutic range, drugs that exhibit nonlinear pharmacokinetic behavior, drugs recommended to be given with food, and for all modified-release oral dosage forms (7). The experimental designs of these studies are usually single-dose, two-treatment, two-period, two-sequence, and crossover designs. The highest strength of each drug product is administered within 5 min of completion of a meal, which is expected to produce the maximum effect on drug absorption. Breakfast with high fat and high calories is usually used as the test meal for bioequivalence studies in fed subjects. All the study procedures and data analysis are usually similar to those of the bioequivalence studies in the fasting state. 8.7.3  Drugs with Long Half-Lives

Bioequivalence studies for oral products of drugs that have long half-lives should involve prolonged period of sampling to ensure adequate characterization of the drug terminal elimination half-life. For these drugs, the crossover experimental design may not be

Bioequivalence 145 practical because of the prolonged washout period required to ensure complete elimination of the drug. In this case, the parallel experimental design may be used, where the test and reference drug products are administered to two different groups of volunteers (8). After each drug administration samples should be obtained for at least two to three days to make sure that the drug in the GIT is completely absorbed and to obtain good estimates for Cpmax and tmax. Testing the bioequivalence of products for drugs with long half-lives that have low intrasubject variability can utilize the AUC truncated at 72 hr in the data analysis. 8.7.4  Determination of Bioequivalence from the Drug Urinary Excretion Data

When significant amount of the drug is excreted unchanged in urine, the drug urinary excretion data can be used to test the bioequivalence of different products. Single oral doses of the test and reference drug products are administered to the volunteers enrolled in the study on two different periods. Urine samples are collected after drug administration at predetermined time intervals until all the dose of the drug is eliminated from the body. The urinary excretion rate calculated for each urine collection interval is plotted at its corresponding time (see Chapter 11). The urinary excretion rate versus time profile should have the same shape as the plasma drug concentration-time profile. The highest value on the urinary excretion rate-time profile is proportional to Cpmax, and the interval when this highest urinary excretion rate is obtained reflects the tmax. However, because frequent urine samples cannot be obtained, the urinary excretion rate data cannot be used to detect small changes in the rate of drug absorption after administration of different products. The total amount of the drug excreted unchanged in urine (Ae∞) is a good measure of the extent of drug absorption after oral drug administration. So, Ae∞ can be used to compare the extent of drug absorption from different products. In bioequivalence studies and when the same dose is administered, the bioavailability of the drug from the test product relative to that of the reference product can be determined as in Eq. 8.3. Frelative =

A e∞ test A e∞ reference 

(8.3)

8.7.5  Fixed-Dose Combination

Fixed-dose combinations (FDCs) are products that contain fixed amounts of multiple drugs that are used in the treatments of a specific disease state. For example, FDC for the treatment of tuberculosis (TB), which contains rifampicin, isoniazid, ethambutol, and pyrazinamide. Also, FDC for the treatment of human immunodeficiency virus (HIV) usually contains a combination of drugs from the nucleoside reverse transcriptase inhibitors, the non-nucleoside reverse transcriptase inhibitors, and the protease inhibitors. The rationale is that these diseases have to be managed using combination therapy, so having this combination of drugs in one product should improve the patient compliance and enhance the therapeutic outcome. Including this combination of drugs in the same dosage form makes it challenging during the formulation and manufacturing to maintain the stability of all drugs in the combination. Bioequivalence testing for FDC is performed utilizing single-dose, twoformulation, and two-period crossover study designs. The volunteers receive the FDC as the test product, and the single drug products for all drugs in the combination administered

146 Bioequivalence together as the reference product, with suitable washout period between treatments. The AUC and Cpmax obtained for each drug after administration of the FDC and the individual drug products are compared to determine if they are bioequivalence. All the drugs in the FDC must be bioequivalent to the drug products containing the individual drugs (15). 8.7.6  Measuring Drug Metabolites in Bioequivalence Studies

Measuring the parent drug released from the dosage form is generally recommended in the bioequivalence studies since the parent drug concentration in the systemic circulation is more sensitive to changes in the formulation performance. However, measuring the metabolite may be necessary when the parent drug concentrations are too low and measuring the drug concentration in biological fluids for adequate period of time is not possible. In this case, the metabolite Cpmax and AUC are calculated and the confidence intervals for the metabolite parameters are used to demonstrate bioequivalence. Also, when the metabolite contributes significantly to the therapeutic and/or adverse effects of the drug, it is recommended that both the parent drug and the metabolite should be measured (8). In this case, the pharmacokinetic parameters of the parent drug determined after administration of the test and reference products are compared to demonstrate the bioequivalence of the products. The metabolite data provide additional evidence for comparable therapeutic/adverse effects. 8.7.7  Highly Variable Drugs

Highly variable drugs are those that have within-subject variability of ≥ 30% CV in the AUC and/or Cpmax. This means that when the same volunteer receives the same product for highly variable drugs at different times, large variation in the AUC and/or Cpmax can be observed. So, a test drug product that is bioequivalent to the reference product may fail to meet the bioequivalence acceptance criteria. This is because the width of the 90% confidence interval for the ratio of the average log-transformed AUC and Cpmax values for the test and reference products is directly proportional to the variability in the pharmacokinetic parameters and is inversely proportional to the number of the volunteers participating in the study. Improving the power of the bioequivalence study for highly variable drugs while using two-treatment crossover design may require increasing the number of volunteers to 72 or 120 depending on the within-subject variability. Performing the bioequivalence study with this large number of volunteers can be troublesome and may not be justified specially if they may develop adverse effects after taking the drug (16). Several approaches have been suggested for testing the bioequivalence of products for highly variable drugs without the need for using a large number of volunteers. These approaches include increasing the bioequivalence acceptance limit, for example, from 80–125% to 70–143% when testing the bioequivalence of products for highly variable drugs (17). Based on this wider acceptance limit, the number of volunteers needed to be enrolled in the bioequivalence study will not be very large. Another approach involves extension of the bioequivalence acceptance limit to the limit that will make using 24 volunteers in the bioequivalence study sufficient to demonstrate product bioequivalence. An additional approach involves extension of the acceptance limit based on the withinsubject variability in the observed parameters after administration of the reference product. In this case, a replicate study design is used where each volunteer receives the reference drug product twice at two different periods in addition to the test product, which is

Bioequivalence 147 administered once or twice. This replicate design allows calculation of the within-subject variability, which is used to calculate the new wider bioequivalence acceptance limit. 8.7.8  Drugs Following Nonlinear Pharmacokinetics

Drugs are considered following nonlinear pharmacokinetics behavior when the change in dose causes disproportional change in AUC after single administration or disproportional change in steady-state concentration during multiple administration. These drugs can be treated as those following linear pharmacokinetics if the dose-normalized AUC deviates (increase or decrease) by less than 25%. The in vivo bioequivalence studies should be performed in the fed and fasted states except if taking the drug with food or while fasting is contraindicated. Also, if there is evidence that nonlinearity occurs after the drug reaches the systemic circulation, bioequivalence study in the fed state may not be performed. When the increase in dose results in more than proportional increase in the AUC, such as in case of nonlinear elimination, the bioequivalence study should be performed using single dose of the highest strength of the drug product. This is because variation in the AUC and Cpmax after administration of products of these drugs results from differences in the rate and extent of drug absorption and/or differences in drug clearance resulting from the nonlinearity of the elimination mechanism. Variation in AUC and Cpmax after administration of larger doses will reflect differences in drug absorption more than differences in the drug clearance. While the nonlinearity is observed only during multiple drug administration, multiple-dose studies using the highest strength in the nonlinear range should be performed. In this case, single-dose studies are not required. Drugs that have nonlinear absorption, the increase in dose results in less than proportional increase in the AUC. For these drugs, the bioequivalence study should be performed using the lowest strength of the drug product, which should produce the lowest nonlinear absorption (18). In this case, comparing AUC and Cpmax will reflect the difference of the rate and extent of drug absorption after administration of the two products. 8.7.9  Endogenous Substances

Drug products, the active ingredients of which are naturally occurring in the body, represent a challenge for bioequivalence evaluation. Examples of these products include potassium supplements, iron salts, some steroid formulations, and insulin. This is because, after administration of products containing such compounds, it will not be possible to differentiate between the endogenous compound and the exogenous compound in the measured concentrations. The protocol for the bioequivalence study can be modified according to the product under investigation. For example, insulin products can be evaluated by comparing the decrease in blood glucose concentration after administration of different insulin products, while iron products can be evaluated by clinical trials, whereas products containing potassium salts can be evaluated by determination of the urinary potassium excretion under controlled conditions (19). 8.7.10  Enantiomers versus Racemates

It is well documented that different enantiomers of the same optically active drug can differ in their therapeutic effect and in their pharmacokinetic characteristics. However, because of the difficulties associated with separation of individual enantiomers, most of

148 Bioequivalence the optically active drugs are marketed as the racemate mixture of the individual enantiomers. The bioequivalence studies for optically active drugs are usually performed using the racemate mixture of the drug. However, measuring the individual enantiomers in bioequivalence studies is recommended only when the enantiomers have different pharmacological activities, the enantiomers have different pharmacokinetic characteristics, the primary drug safety and activity reside with the minor enantiomer, and at least one of the enantiomers exhibits nonlinear absorption. All these conditions have to be met for recommending the measurement of the individual enantiomers in bioequivalence studies for optically active drugs (8). 8.7.11  Narrow Therapeutic Range Drugs

Narrow therapeutic range drugs are drugs that have small differences between their minimum effective concentration and minimum toxic concentration. So, these drugs are usually subjected to therapeutic drug monitoring. It is recommended that drug manufacturers consider additional testing and/or control for the quality of drug products containing narrow therapeutic range drugs. This additional testing should provide increased assurance of interchangeability of these products. The usual acceptance bioequivalence limit of 80–125% is applied for products that contain narrow therapeutic range of drugs (8). 8.7.12  Oral Products Intended for the Local Effect of the Drug

Bioequivalence studies to compare oral drug products that are intended to produce local GIT effects should be based on clinical efficacy and safety endpoints. In these studies, the clinical effect observed after administration of the two products is compared. As part of safety assessment, studies can be designed to compare the degree of systemic exposure that occurs following administration of products that are intended for local effect. The clinical activity of products of some locally acting drugs can be assessed and compared using in vitro testing instead of the in vivo testing (8). Cholestyramine, which is a bile acid sequestering antilipemic agent that adsorbs anions of bile acids, conjugates in the small intestine and forms nonabsorbable complex, which is excreted in the feces. In vivo bioequivalence studies are not required to document bioequivalence of different cholestyramine products. Instead, in vitro bile acid salt binding studies with cholestyramine are recommended to demonstrate the bioequivalence of cholestyramine generic and innovator products. 8.7.13  First Point Cpmax

The first measured drug concentration in the bioequivalence study is sometimes the highest drug concentration due to insufficient early sampling, which raises questions regarding the accuracy of estimating Cpmax. A pilot study can help in determining the approximate time for achieving the highest concentration. Extensive sampling during the early time after drug administration (e.g., 5 min, and 15 min) may be sufficient to determine Cpmax even if the highest concentration occurs in the first measured sample (8). 8.7.14  Biological Products

Biological medical products are usually produced from living organisms and exhibit complex molecular structures. The complex chemical structure is crucial for their biological activities and can be very sensitive to the conditions used during manufacture

Bioequivalence 149 and storage. Biosimilars are biological medical products that are similar to approved biological reference products. They must be similar to the reference product with regard to molecular structure, purity, and biological activity. Regulatory drug authorities developed special guidelines to demonstrate similarity of biological products that include analytical studies, animal studies to compare efficacy and toxicity, and clinical studies to assess pharmacokinetics, pharmacodynamics, and immunogenicity (20). When approved by regulatory authorities, the generic biological product becomes interchangeable with the reference product. 8.8 Summary

• Bioequivalent drug products are products containing the same active drug, the same • •

• •

strength, the same dosage form for the same route of administration and have the same rate and extent of drug absorption. Bioequivalent drug products are expected to produce the same blood concentrationtime profiles and hence similar therapeutic and adverse effects after administration. Generic product manufacturers must demonstrate that their products are bioequivalent with the innovator products for the same active drugs. This is important to ensure that switching from the innovator product to the generic product does not compromise the drug therapeutic efficacy and safety. The design, execution, data handling, and reporting of in vivo bioequivalence studies have to follow the general guidelines developed by the drug regulatory authorities. It is required that the principles of good clinical practices be applied to the clinical phase of the bioequivalence study and the principles of good laboratory practices be applied to all the laboratory activities during the study.

References 1. U.S. Code of Federal Regulations. Title 21 – Food and drugs, Chapter I, Food and drug administration, department of health and human services, Sub-chapter D, Drugs for human use, Part 320, Bioavailability and bioequivalence requirements (April 2010). 2. Food and Drug Administration. Draft guidance for industry: “Waiver of in vivo bioavailability and bioequivalence studies for immediate release solid oral dosage forms based on biopharmaceutical classification system” (2001). 3. Food and Drug Administration. Guidance for industry: “Bioavailability and bioequivalence studies submitted in NDAs or INDs – General considerations” (2014). 4. Food and Drug Administration. Draft guidance for industry: “Bioequivalence studies with pharmacokinetic endpoints for drugs submitted under an ANDA” (2021). 5. Food and Drug Administration. Draft guidance for industry: “Extended-release oral dosage forms: Development, evaluation and application of in vitro/in vivo correlation” (1997). 6. Amidon GL, Lennernas H, Shah VP and Crison JR “A theoretical basis for a biopharmaceutic drug classification: The correlation of in vitro drug product dissolution and in vivo bioavailability” (1995) Pharm Res; 12:413–420. 7. Food and Drug Administration. Draft guidance for industry: “Bioavailability and bioequivalence studies for orally administered drug products – General considerations” (2002). 8. European Medicines Agency. Guidelines on the investigation of bioequivalence (2010). 9. Food and Drug Administration. Draft guidance for industry: “Referencing approved drug products in ANDA submissions” (2020). 10. Shah VP, Tsong Y and Sathe P “In vitro dissolution profile comparison – Statistics and analysis of the similarity factor, f2” (1998) Pharm Res; 15:889–896.

150 Bioequivalence 11. Food and Drug Administration. Draft guidance for industry: “Statistical approaches to establish bioequivalence” (2001). 12. Food and Drug Administration. Draft guidance for industry: “Bioanalytical method validation” (2018). 13. Schuirmann DJ “A comparison of the two one-sided tests procedure and the power approach for assessing the equivalence of average bioavailability” (1987) Pharmacokinet Biopharm; 15:657–680. 14. Henney JE “Review of generic bioequivalence studies” (1999) JAMA; 282:1995. 15. Food and Drug Administration. Draft guidance for industry: “Fixed dose combination and copackaged drug products for treatment of HIV” (2004). 16. Shah VP, Yacobi A, Barr WH and Benet LZ et al “Evaluation of orally administered highly variable drugs and drug formulation” (1996) Pharm Res; 13:1590–1594. 17. Boody AW, Snikeris FC, Kringle RO, Wei GC, Oppermann JA and Midha KK “An approach for widening the bioequivalence acceptance limits in the case of highly variable drugs” (1995) Pharm Res; 12:1865–1868. 18. Health Canada. Report of expert advisory committee on bioavailability and bioequivalence, “Bioequivalence requirements: drugs exhibiting non-linear pharmacokinetics” (2003). 19. Food and Drug Administration. Draft guidance for industry: “Potassium chloride modified release tablets/capsules: In vivo bioequivalence and in vitro dissolution testing” (2002). 20. Food and Drug Administration. Draft guidance for industry: “Biosimilars and interchangeable biosimilars: Licensure for fewer than all conditions of use for which the reference product has been licensed” (2020).

9

Drug Pharmacokinetics during Constant Rate IV Infusion, the Steady-State Concept

Objectives After completing this chapter, you should be able to:

• Define the steady state during constant rate IV infusion. • List the factors that affect the steady-state plasma drug concentration during constant rate IV infusion.

• Calculate the steady-state drug concentrations and the pharmacokinetic parameters during constant rate IV infusion.

• Explain the rationale for using the loading dose to achieve faster approach to steady state.

• Analyze the effect of changing drug pharmacokinetic parameters on the steady-state plasma concentration during constant rate IV infusion.

• Recommend an appropriate IV loading dose and IV infusion rate of drugs to achieve specific steady-state plasma concentrations.

9.1 Introduction Drugs can be administered as a single dose for the management of acute medical conditions or transient symptoms. The effect of drugs after administration of a single dose subsides with time due to drug elimination and repeated drug administration may be required if the symptoms persist. Also, management of chronic diseases requires multiple drug administration. In hospitalized patients, drugs can be administered by constant rate IV infusions to maintain steady drug concentrations, and hence steady drug effect. Constant rate IV infusion involves direct administration of the drug into the systemic circulation continuously at constant rate with the aid of an infusion pump. The drug solution for IV administration is prepared in concentration (amount/volume) depending on the drug solubility, the average drug dose, and the fluid volume that can be administered to the patient. Then the infusion rate of the drug solution (volume/time) is selected to administer the drug at a constant rate (amount/time). This constant rate of drug administration should maintain constant plasma drug concentration at steady state if the drug administration continues. The rate of drug administration can be changed by increasing or decreasing the rate of infusion. The major drawback of this route of drug administration is that it can only be used in hospitalized patients because of the need for an infusion pump, and the special precautions needed for the preparation and administration of the IV solutions. Many drugs can be administered by this route of administration such as heparin, vasodilators, bronchodilators, inotropic agents, and general anesthetics. The DOI: 10.4324/9781003161523-9

152  Drug Pharmacokinetics during Constant Rate IV Infusion therapeutic effect of these drugs is highly correlated with their concentrations in the body. So, the rate of infusion can be titrated up and down to achieve the drug concentration that produces the desired effect. 9.2  The Steady State The rate of drug administration during IV infusion is constant. When drug elimination follows first-order kinetics, the rate of drug elimination is equal to the product of the elimination rate constant (k) and the amount of the drug in the body (A). Shortly after starting the IV infusion, the rate of drug elimination is very low because the amount of the drug in the body is small. So, the amount of the drug in the body increases because the rate of drug administration is larger than the rate of drug elimination. As the IV infusion continues the amount of the drug in the body increases and the rate of drug elimination increases. The rate of drug elimination continues to increase until it becomes equal to the rate of drug administration. At this point, the rate of drug administration is equal to the rate of drug elimination and the drug amount in the body and plasma drug concentration remain constant while the drug is being administered as illustrated in Figure 9.1. The state when the rate of drug administration is equal to the rate of drug elimination is known as the steady state. Mathematically, the rate of change of the amount of the drug in the body at any time is the difference between the rate of drug administration and the rate of drug elimination as expressed in the differential equations, Eqs. 9.1 and 9.2. dA = Rate of drug administration − Rate of drug elimination dt 

(9.1)

dA = K0 − kA dt 

(9.2)

where A is the amount of the drug in the body, K0 is the rate of drug administration (amount/time), and k is the first-order elimination rate constant. Integrating Eq. 9.2 and

Figure 9.1 The drug plasma concentration-time profile after starting the constant rate IV drug infusion.

Drug Pharmacokinetics during Constant Rate IV Infusion 153 dividing by Vd, Eq. 9.3 that is the equation for the plasma drug concentration at any time during the infusion is obtained. Cp =

K0 (1 − e − kt ) k Vd 

(9.3)

where t is the time after the start of the infusion. Equation 9.3 indicates that after starting the IV infusion, the plasma drug concentration increases and exponentially approaches the steady-state concentration. The steady state is achieved after administration of the IV infusion for a period of time. So, substitution for time in Eq. 9.3 by a large value, Eq. 9.4 that describes the steady-state concentration is obtained. Cpss =

K0 K = 0 k Vd CL T 

(9.4)

Examining Eq. 9.4 shows that the plasma drug concentration at steady state is directly proportional to the rate of the drug infusion and inversely proportional to the drug CLT. This means that if a patient is receiving a drug as a constant rate IV infusion, changing the rate of drug infusion leads to proportional change in the steady-state drug concentration as in Figure 9.2. Also, the drug steady-state concentration will be higher in patients with lower clearance as in Figure 9.3. This illustrates the importance of adjusting the rate of drug administration based on the patient’s specific characteristics. Clinical Importance:

• The drug infusion rate can be changed up and down until the desired plasma drug concentration that produces the desired therapeutic effect is achieved.

• Patients with eliminating organ dysfunction require lower drug infusion rate compared to patients with normal eliminating organ function to achieve the same drug concentration.

Figure 9.2 The steady-state drug concentration is proportional to the infusion rate if the total body clearance is the same.

154  Drug Pharmacokinetics during Constant Rate IV Infusion

Figure 9.3 The steady-state drug concentration is inversely proportional to the total body clearance of the drug if the drug is administered at the same infusion rate.

• The drug IV infusion rate used to stabilize the patient’s condition, while in the hospi-

tal, is used to calculate the appropriate dosage regimen for the patient to be used after discharge from the hospital.

Practice Problems: a  Question: A patient received constant rate IV infusion of 30 mg/hr until steady-state drug concentration of 10 mg/L was achieved.

• What is the elimination rate of the drug at steady state? • What is the infusion rate required to achieve steady state of 15 mg/L? Answer:

• At steady state, the rate of administration is equal to the rate of elimination. So the rate of drug elimination at steady state is equal to 30 mg/hr.

• Increasing the plasma drug concentration by 50% requires increasing the infusion rate by 50%. So, the infusion rate should be increased to 45 mg/hr.

b  Question: What is the IV infusion rate of the same drug required to achieve steadystate plasma drug concentration of 12 mg/L in two patients who have CLT of 3.0 L/ hr and 5 L/hr, respectively? Answer: Cpss =

K0 CL T K

0 • Patient 1:   12 mg/L = 3.0 L/hr   K0 = 36 mg/hr

K

0 • Patient 2:   12 mg/L = 5.0 L/hr   K0 = 60 mg/hr

Drug Pharmacokinetics during Constant Rate IV Infusion 155

• The patient with lower CLT requires lower infusion rate to achieve the same steadystate concentration.

9.3  The Time Required to Achieve Steady State During constant rate IV infusion, the drug plasma concentration increases gradually until it reaches steady state as in Figure 9.1. The infusion rate is usually selected to achieve the desired plasma drug concentration at steady state. So, before reaching steady state, the drug is not expected to produce its full therapeutic effect, which makes it important to know the time required to achieve steady state. Equation 9.3 describes the plasma drug concentration at any time during the infusion, which indicates that the drug concentration exponentially approaches steady state after the start of the infusion. So, theoretically it should take long time for the plasma concentration to reach steady state. Practically, it can be assumed that steady state is achieved when the drug concentration is about 98% of the true steady-state concentration. So, we can assume that the time to archive steady state, tss, is the time when the plasma drug concentration is equal to 0.98 Cpss, and Eq. 9.3 can be rewritten as follows: 0.98Cpss =

K0 (1 − e − ktss ) k Vd 

(9.5)

Substitution for Cpss from Eq. 9.4 yields as follows: 0.98

K0 K =   0 (1 − e − ktss ) k Vd k Vd 

(9.6)

The time needed to achieve steady state is determined by solving for tss. 0.98 = 1 − e − ktss 

(9.7)

ln 0.02 = l e − ktss

t ss = 3.91/k = 3.91t1/2 /0.693 = 5.6 t1/2 

(9.8)

This means that the time required to achieve steady state is dependent on the half-life of the drug. It takes 5.6 times the drug elimination half-life of continuous infusion to achieve drug concentration about 98% of the true steady-state concentration. Generally, it takes five to six elimination half-lives of continuous IV infusion of the drug to reach steady state. The longer the drug half-life the longer it takes to reach steady state. Figure 9.4 shows the time required to achieve steady state during administration of different drugs that have different half-lives. Administration of the same drug to the same patient should achieve steady-state concentrations that are proportional to the IV infusion rate as in Figure 9.2. The time to achieve steady state in this case will be similar since it is the same drug that has the same half-life during the different infusions. Clinical Importance:

• After starting the IV infusion, a period that is equal to 5–6 times the drug elimination half-life should be allowed before evaluating the drug therapeutic effect.

156  Drug Pharmacokinetics during Constant Rate IV Infusion

Figure 9.4 The time required to achieve steady state during constant rate IV infusion is dependent on the half-life of the drug. The longer the half-life the longer it takes to achieve steady state.

• When immediate drug effect is needed in emergency cases, starting the IV infusion

only may not be enough to obtain the desired rapid drug effect. This is because it takes time for the drug to accumulate in the body to achieve therapeutic drug concentration specially for drugs with long half-lives.

9.3.1  Changing the Drug Infusion Rate

As mentioned previously, one of the advantages of using IV infusion as a route of drug administration is the ease of changing the drug infusion rate to achieve the desired therapeutic effect. Assume that a patient is receiving constant rate IV infusion of a drug until steady state is achieved. If the infusion rate is changed, a new steady state drug concentration will be achieved. The new steady-state concentration is not achieved immediately, but the drug concentration changes to exponentially approach the new steady state. When the drug infusion rate is changed, it takes five to six half-lives of continuous drug infusion for the drug concentration to achieve the new steady-state concentration as illustrated in Figure 9.5.

Figure 9.5 The change in the IV infusion rate results in changing the plasma drug concentration until the new steady state is achieved.

Drug Pharmacokinetics during Constant Rate IV Infusion 157 9.4  Loading Dose During constant rate IV administration, the drug accumulates until steady state is achieved after five to six half-lives. This may not be acceptable especially for drugs with long half-lives when immediate achievement of therapeutic drug concentrations is necessary such as in emergency situations. In this case, administration of a loading dose will be necessary. The loading dose is an IV bolus dose administered at the time of starting the IV infusion to immediately achieve plasma drug concentration as close as possible to the desired steady state. When the loading dose is accurately calculated, plasma drug concentration close to the desired steady-state concentration will be achieved immediately after administration of the loading dose. After administration of the loading dose and starting the IV infusion, the plasma drug concentration at any time will be the sum of the drug concentration resulting from the IV infusion, and the drug concentration remaining from the IV bolus dose as mentioned in Eq. 9.9 and illustrated by Figure 9.6. The drug concentration resulting from the IV infusion is increasing and is exponentially approaching the steady state, and the drug concentration remaining from the IV bolus loading dose is declining exponentially. So, when the loading dose is accurately calculated and administered, drug concentration very close to the steady-state drug concentration is achieved immediately after initiation of drug administration. Cptotal = Cpfrom infusion + Cpfrom IV bolus

(9.9)

The steady-state plasma drug concentration is dependent only on the infusion rate and the drug CLT as indicated from Eq. 9.4. So, the loading dose does not affect the steadystate concentration. If the loading dose produces initial plasma concentration higher or lower than the steady-state concentration that should be achieved by the infusion, the drug concentration will decline or increase slowly to reach the steady state. It takes five to six half-lives for the initial drug concentration produced by the loading dose to reach steady state. So, administration of the right loading dose is important to achieve drug plasma concentration very close to the steady-state concentration immediately after the start of drug administration as in Figure 9.7. The IV loading dose required to achieve a

Figure 9.6 The plasma drug concentration-time profile after administration of a bolus IV loading dose and constant rate IV infusion simultaneously.

158  Drug Pharmacokinetics during Constant Rate IV Infusion

Figure 9.7 The plasma drug concentration-time profile after administration of different loading doses followed immediately by the same continuous IV infusion rate of the drug.

certain concentration can be calculated from the relationship between the dose, Vd, and drug concentration as in Eq. 9.10. Loading dose ( IV bolus ) = Desired Cp × Vd



(9.10)

Clinical Importance:

• Administration of an IV bolus loading dose at the same time of starting the constant

rate IV infusion leads to faster approach to the desired steady-state drug concentration.

• Administration of loading dose is necessary when immediate therapeutic effect is needed as in emergency cases.

• The steady-state drug concentration is dependent on the infusion rate and drug CLT and is not affected by the loading dose.

Practice Problems: A  Question: A patient was admitted to the emergency room with severe shortness of breath and the physician wanted to start the patient on IV infusion of a bronchodilator. The drug has half-life of 6 hr, and Vd of 50 L. Calculate the IV loading dose and the IV infusion rate that should achieve plasma concentration of 12 mg/L. Answer:

• Loading dose = 50 L × 12 mg/L = 600 mg 12 mg/L =

 K0 50L × 0.231hr −1

• K0 = 69.3 mg/hr

Drug Pharmacokinetics during Constant Rate IV Infusion 159 B  Question: A patient received an IV loading dose of 120 mg followed by 24 mg/hr of an antiarrhythmic drug that has half-life of 3 hr and Vd of 25 L. Calculate the initial plasma drug concentration after administration of the loading dose, and the steady-state plasma drug concentration. Answer:

• Initial drug concentration = 120 mg/25 L = 4.8 mg/L • Cpss =

 K0 24 mg/hr = = 4.16 mg/L k Vd 25L × 0.231hr −1

9.5  Termination of the Constant Rate IV Infusion Drug administration by constant rate IV infusion is usually used in hospitalized patients to produce the drug’s therapeutic effect for a period of time such as in case of anesthetic drugs. It is also used to stabilize the patient’s condition such as in case of administration of vasodilators, bronchodilators, or inotropic drugs. If the drug administration continues for more than 5–6 times the drug elimination half-life, steady state is achieved. Once the desired therapeutic outcome is achieved or the patient condition is stabilized, the IV infusion is terminated. After termination of the IV infusion at any time whether steady state is achieved or not, the drug concentration declines at a rate dependent on the elimination rate constant of the drug as illustrated in Figure 9.8. When the drug elimination follows first-order kinetics, the plasma drug concentration after termination of the infusion declines exponentially on the linear scale, and linearly on the semilog scale. The drug elimination rate constant and half-life can be estimated from the post-infusion drug concentration-time profile. 9.6  Determination of the Pharmacokinetic Parameters The drug CLT can be determined during the administration of the IV infusion if steady state is achieved. While determination of the elimination rate constant and volume of distribution requires measuring plasma drug concentrations after termination of the infusion.

Figure 9.8 The plasma drug concentration-time profile during and after termination of the continuous IV infusion of the drug.

160  Drug Pharmacokinetics during Constant Rate IV Infusion 9.6.1  Total Body Clearance

The CLT of the drug can be determined from the IV infusion rate and the steady state concentration by rearrangement of Eq. 9.4. CL T =

K0 Cpss 

(9.11)

The steady-state plasma concentration is determined by measuring the plasma drug concentration during the IV infusion at steady state or can be estimated from the plasma concentrations during the post-infusion phase. In this case, the plasma concentrations measured after termination of the IV infusion are plotted on the semilog scale. The postinfusion plasma concentration-time profile is back extrapolated to determine the plasma concentration at the time of termination of the IV infusion which is equal to the steadystate concentration. 9.6.2  Elimination Rate Constant

The first-order elimination rate constant, k, can be determined from the post-infusion plasma drug concentration-time profile plotted on the semilog scale. Slope =

−k 2.303 

(9.12)

The drug half-life can be determined from k (half-life = 0.693/k). Also, the half-life can be determined graphically from the post-infusion drug concentration-time profile by determining the time required for any concentration during the post-infusion phase to decline by 50%. 9.6.3  Volume of Distribution

The volume of distribution is determined from CLT and k. Vd =

CL T k 

(9.13)

Practice Problems: Question: A patient is receiving an antiarrhythmic drug as a constant rate IV infusion of 24 mg/hr. After three days, the drug infusion was terminated, and three plasma samples were obtained. Time after the end of infusion

Drug concentration

(hr) 1 3 6

(mg/L) 3.3 2.2 1.2

Drug Pharmacokinetics during Constant Rate IV Infusion 161 a Calculate the CLT and Vd of this drug in this patient. b The patient experienced irregular heart rhythm again and the physician wanted to start this antiarrhythmic again. Recommend an IV loading dose and a constant rate IV infusion that should achieve a steady state of 10 mg/L. Answer: a The steady-state drug plasma concentration and the half-life (or the elimination rate constant) from the plot of the plasma drug concentrations obtained after termination of the infusion versus time on semilog scale as in Figure 9.9. Time zero represents the time when the infusion was terminated, so the y-intercept is equal to the steady-state drug concentration. Cpss = 4 mg/L Cpss =

K0 CL T

CL T =

24 mg/hr = 6 L/hr 4 mg/L

t1/2 = 3.5 hr   k  = 0.198 hr −1 Vd =

CL T 6 L/hr = = 30.3L k 0.198 hr −1

b LD = Cp × Vd = 10 mg/L × 30.3L = 303 mg K0 = Cpss × CL T = 10 mg/L × 6 L/hr = 60 mg/hr

Figure 9.9 A plot of the plasma drug concentrations obtained after termination of the infusion versus time on the semilog scale. The y-intercept is equal to the steady-state concentration and the half-life can be calculated graphically.

162  Drug Pharmacokinetics during Constant Rate IV Infusion Recommendation: 300 mg administered as an IV bolus dose (loading dose) followed immediately by 60 mg/hr as a constant rate IV infusion. This infusion should achieve steady-state concentration of 10 mg/L. 9.7  Dosage Forms with Zero-Order Input Rate The steady therapeutic effect achieved during constant rate IV drug administration prompted scientists to develop dosage forms for extravascular administration that can deliver the drug at a constant rate. The most common strategy utilized to achieve this goal is to design dosage forms that can release the drug at slow zero-order rate. If the drug release rate from the dosage form becomes the rate-determining step in the drug absorption process, the drug is delivered to the systemic circulation at a constant rate. In this case, the systemic drug concentration is maintained relatively constant for a period of time and the therapeutic effect achieved is usually stable. Examples of these dosage forms include modified release oral dosage forms which can release the drug at zero-order rate over an extended period of time. Also, transdermal patches have been developed to release the drug at a constant rate over a period of days as in case of clonidine and scopolamine patches. Besides, intramuscular hormonal injections of sustained release formulations that can deliver hormones at a relatively constant rate for weeks, and subcutaneous contraceptive implants developed to deliver the drugs at constant rate for months. Additionally medicated devices have been developed to release therapeutic agents included in the components of the device over long time such as in progesterone medicated intrauterine devices, and drug-eluting stent. The plasma drug concentration achieved after multiple administration of these extravascular dosage forms with zero-order release rate is not identical to the profile achieved during administration of constant rate IV infusion as shown in Figure 9.10. This is because of many factors related to the formulation design, and the complex nature of the absorption process. However, many well-designed dosage forms have been shown to maintain relatively constant plasma drug concentration and produce steady therapeutic effect over the manufacturer claimed period of time.

Figure 9.10 Plasma drug concentration-time profile after administration of a dosage form that releases the drug at zero-order rate every 8 hr to achieve relatively constant plasma concentration.

Drug Pharmacokinetics during Constant Rate IV Infusion 163 9.8 The Effect of Changing the Pharmacokinetic Parameters on the Plasma Drug Concentration-Time Profile during Constant Rate IV Infusion During constant rate IV infusion, the plasma drug concentration increases gradually until steady state is achieved. The steady-state drug concentration is dependent on the infusion rate and CLT of the drug. The time to achieve steady-state concentration is dependent on the half-life of the drug and administration of an IV loading dose can lead to faster approach to steady state. Using the reported drug pharmacokinetic parameters, the infusion rate is calculated to achieve steady-state drug concentration which is expected to produce the desired therapeutic effect. It is important to examine how the change in each of the pharmacokinetic parameters affects the steady-state concentration achieved during constant rate IV infusion, if the drug elimination follows first-order process. 9.8.1  Infusion Rate

The effect of changing the infusion rate of the drug:

• The steady-state plasma drug concentration is proportional to the drug IV infusion rate, when CLT is constant.

• Changing the infusion rate does not affect the drug k, t1/2, Vd, or CLT, so the time to

achieve steady state will not change, despite the proportional change in steady-state drug concentration. • The drug concentration-time profile declines at a rate dependent on the drug elimination rate constant after termination of the IV infusion. The slope of the drug concentration-time profile after termination of the IV infusion on the semilog scale is equal to −k/2.303, when drug elimination follows first-order kinetics. Clinical Importance:

• The IV infusion rate can be changed up and down until the drug concentration that produces the desired therapeutic effect with minimum adverse effects is achieved.

• The IV infusion rate that produces the desired effect is used to calculate the drug dose

to be used by the patient after discharge from the hospital, when continued therapy with the same drug is required. The IV infusion rate is multiplied by the dosing interval to calculate the IV dose to be given repeatedly. • For example, if the IV infusion rate is 50 mg/hr of the drug, the patient should get 300 mg of the same drug IV every 6 hr. To calculate the oral dose, the drug oral bioavailability should be considered. 9.8.2  Total Body Clearance

The effect of changing CLT of the drug:

• When the same drug is administered by the same IV infusion rate to different individuals, the steady-state concentration will be higher in patient with lower CLT and lower in those with higher CLT.

164  Drug Pharmacokinetics during Constant Rate IV Infusion

• Patients with different CLT but have the same Vd will have different k and t1/2. This

means that the time to reach steady state will be different with drugs that have smaller k and longer t1/2 reaching steady state after longer time.

Clinical Importance:

• Administration of the same drug at the same IV infusion rate to all patients is not ap-

propriate and usually results in large variation in the steady-state drug concentrations because of differences in the patients CLT. The infusion rate should be individualized based on the patients’ conditions. • Patients with eliminating organ dysfunction, i.e., have lower drug CLT, require IV infusion rate lower than the infusion rate required in patients with normal eliminating organ function. 9.8.3  Volume of Distribution

The effect of changing Vd of the drug:

• The steady-state drug concentration is not affected by changes in Vd if both the infu-

sion rate and drug CLT are the same. However, changes in Vd in some cases are accompanied by changes in CLT also. • Patients with different Vd but have the same CLT will have different k and t1/2. This means that the time to reach steady state will be different, with drugs that have smaller k and longer t1/2 reaching steady state after longer time. Clinical Importance:

• Calculation of the drug loading dose and IV infusion rate should be based on the pa-

tients’ characteristics to accurately achieve the desired therapeutic drug concentration.

9.8.4  Loading Dose

The effect of administration of a loading dose of the drug:

• Administration of an IV loading dose at the time of starting the IV infusion leads to faster approach to steady state.

• The loading dose does not affect the steady-state drug concentration which is dependent on the infusion rate and the drug clearance.

Clinical Importance:

• Administration of loading dose is necessary in emergency situations when immediate drug therapeutic effect is required specially for drugs with long half-lives.

9.9 Summary

• Constant rate IV infusion of drugs is only used in hospitalized patients because of the

special precautions required for IV drug administration and the need for infusion pump.

• The drug IV infusion rate can be changed by changing the flow rate of the drug solution while monitoring the patient, until the desired drug effect is obtained.

Drug Pharmacokinetics during Constant Rate IV Infusion 165

• When immediate drug effect is required, an IV bolus dose of the drug is administered

at the same time of starting the IV infusion to achieve therapeutic drug concentrations immediately after starting drug therapy. • The drug concentration-time profile declines exponentially after termination of the IV infusion, and the rate of decline is dependent on the drug first-order elimination rate constant. • When continued drug therapy is needed after discharge of the patient from the hospital, the drug infusion rate is used to guide the calculation of the drug dose. Practice Problems 9.1 A physician asked you to recommend a loading dose and a constant rate IV infusion of a drug to rapidly achieve a steady-state concentration of 15 mg/L. The half-life of this drug is 6 hr, and the volume of distribution is 30 L. What is your recommendation? 9.2 A patient received an antiasthmatic medication as 200-mg IV loading dose followed immediately by constant rate IV infusion of 70 mg/hr. The IV infusion continued for five days and after termination of the infusion, the half-life was found to be 3 hr and the volume of distribution was 30 L. a b c d

What is the elimination rate of this drug during the infusion at steady state? What is the steady-state concentration during the infusion? What is the elimination rate constant of this drug? What is the steady-state concentration if a loading dose of 400 mg was administered followed by the 70 mg/hr infusion? e What is the time to reach steady state if no loading dose was administered? f What is the steady-state concentration if an infusion rate of 140 mg/hr was administered? 9.3 A patient was admitted to the hospital because of cardiac arrhythmia. He received antiarrhythmic drug as an IV loading dose injection followed by constant rate infusion of 30 mg/hr. The infusion continued for three days and then the physician decided to stop the infusion. Samples were obtained after the end of infusion. A plot of the plasma concentrations obtained after stopping the infusion versus time was linear on semilog graph paper and has a slope of −0.043 hr−1 and an intercept of 7.5 mg/L. a Calculate the half-life and the volume of distribution of the antiarrhythmic drug in this patient. b Recommend an IV loading dose and a constant rate infusion that can achieve steady-state plasma concentration of 12 mg/L. 9.4 A patient received an antiasthmatic medication as 300-mg IV loading dose followed immediately by constant rate IV infusion of 100 mg/hr. The IV infusion continued for five days and after termination of the infusion, three plasma samples were obtained. A plot of the plasma concentrations obtained after termination (stopping) of the infusion has a y-intercept of 16 mg/L and a slope of −0.06 hr−1. a b c d

What is the half-life of this drug? What is the volume of distribution of this drug? What is the steady-state concentration if no loading dose was administered? What is the steady-state drug concentration if the infusion rate was 200 mg/hr?

9.5 A patient received an antiasthmatic medication as 300-mg IV loading dose followed immediately by constant rate IV infusion of 100 mg/hr. The IV infusion continued

166  Drug Pharmacokinetics during Constant Rate IV Infusion for five days and after termination (stopping) of the infusion, three plasma samples were obtained. Time after termination of the infusion(hr) Plasma drug conc (mg/L) 3 6 12

a b c d e

10.6 7.0 3.0

Calculate the plasma drug concentration at steady state graphically. What is the elimination rate of the drug at steady state? What is the half-life of this drug? What is the steady-state concentration if the loading dose was changed to 600 mg? What is the steady-state drug concentration if the infusion rate was 200 mg/hr?

9.6 A 60-kg male was admitted to the hospital with chief complain of shortness of breath. He was diagnosed as having acute bronchial asthma. He received a single IV loading dose of theophylline, followed by a constant rate IV infusion of 60 mg/hr of the same drug. On the fourth day after starting the IV infusion, the patient complained of nausea and vomiting which are the most common adverse effects of this drug. For this reason, the continuous IV infusion was stopped, and three plasma samples were obtained 1, 3, and 9 hr after termination of the IV infusion. Time after termination of the IV infusion Concentration (mg/L) 1 3 9

18.3 15.0 8.3

a What is the rate of elimination of this anti-asthmatic drug during the constant rate IV infusion at steady state? b Calculate the half-life of this drug graphically. c Calculate the volume of distribution of this drug. d Recommend an IV loading dose and an IV infusion rate that should achieve steady-state theophylline concentration of 15 mg/L in this patient. 9.7 A patient received an anticancer drug by constant rate IV infusion of 20 mg/hr. The IV infusion continued for three days and after termination of the infusion, the halflife was found to be 6 hr and the volume of distribution was 35 L. a b c d

What is the elimination rate of this drug during the infusion at steady state? What is the steady-state concentration during the infusion? What is the elimination rate constant of this drug? What is the steady-state concentration if a loading dose of 200 mg was administered followed by the 20 mg/hr infusion? e What is the time to reach steady state if the loading dose was not administered? f What is the steady-state concentration if the infusion rate was 40 mg/hr?

10 Steady State during Multiple Drug Administration

Objectives After completing this chapter, you should be able to:

• Define the Cpmax ss, Cpmin ss, and Cpaverage ss during multiple drug administration. • List the factors that affect the steady-state plasma concentration during multiple drug administration.

• Explain the rationale for administration of loading dose for faster approach to steady state.

• Calculate the steady-state drug concentrations and patients’ pharmacokinetic parameters during multiple drug administration.

• Analyze the effect of changing each of the pharmacokinetic parameters on the steadystate plasma concentration during multiple drug administration.

• Recommend IV and oral dosing regimens to achieve specific plasma concentrations at steady state in patients.

• Evaluate the appropriateness of certain dosing regimens for a particular patient based on the patient’s specific pharmacokinetic parameter.

10.1 Introduction Constant rate IV infusions are used to maintain relatively constant drug concentrations to produce steady therapeutic drug effect in hospitalized patients. When the patient condition is stabilized and continued drug use is required, the patient is shifted to multipledose regimen, which involves IV or oral administration of a fixed dose of the drug (D) every fixed dosing interval (τ). Multiple-dose regimens are used in the treatment of transient disease conditions that require drug administration for a period of time such as in treating infections, and also in the management of chronic diseases. After initiation of the multiple-dose regimens, subsequent doses of the drug are administered before complete elimination of the previous doses of the drug. This results in drug accumulation in the body during repeated drug administration. When the elimination process follows firstorder kinetics, the rate of drug elimination is proportional to the amount of the drug in the body. As the drug accumulates in the body, the amount of the drug eliminated during each dosing interval increases with time. This continues until the rate of drug administration becomes equal to the rate of drug elimination during each dosing interval and steady state is achieved. At steady state, the average amount of the drug in the body and the average plasma drug concentration remain constant as long as the dosing rate of the drug is constant. DOI: 10.4324/9781003161523-10

168  Steady State during Multiple Drug Administration 10.2 The Plasma Drug Concentration-Time Profile during Multiple Drug Administration The plasma drug concentration is changing within each dosing interval during multiple drug administration at steady state. If a fixed IV dose of the drug is administered at every fixed dosing interval, the plasma drug concentration just before drug administration is the minimum plasma drug concentration at steady state (Cpmin ss), while the plasma drug concentration right after drug administration is the maximum plasma drug concentration at steady state (Cpmax ss). At steady state, the rate of drug administration is equal to the rate of drug elimination, which indicates that the amount of the drug eliminated during each dosing interval is equal to the administered IV dose. This means that when the dose of the drug is administered at the beginning of the dosing interval, the drug concentration increases from Cpmin ss to Cpmax ss. During the dosing interval, the drug is eliminated, and the drug concentration has to return back to Cpmin ss at the end of the dosing interval. So, the plasma drug concentration-time profiles during the successive dosing intervals at steady state are identical if the same dose is administered at equally spaced dosing interval as shown in Figure 10.1. For a drug that is eliminated by first-order process, during multiple IV administration of equal doses, D administered every fixed dosing interval τ, the amount of the drug in the body (A) after administration of the first dose is equal to: A=D The amount of the drug in the body before administration of the second dose is equal to: A = De − kτ The amount of the drug in the body after administration of the second dose is equal to: A = D + De − kτ

Figure 10.1 The plasma drug concentration-time profile during multiple IV administration of a fixed dose of the drug every fixed dosing interval.

Steady State during Multiple Drug Administration 169 to:

The amount of the drug in the body before administration of the third dose is equal

(

)

A = D + De − k τ e − k τ = De − k τ + De −2k τ

to:

The amount of the drug in the body after administration of the third dose is equal

A = D + De − k τ + De −2k τ At steady state, the amount of the drug in the body after drug administration, Amax ss, is equal to:

(

)

A max ss = D 1 + e − kτ + e −2 kτ + e −3kτ + e −4 kτ + 

This series can be solved mathematically as in Eq. 10.1: A max ss =

D 1 − e − kτ

(

(10.1)

)

where k is the first-order elimination rate constant. Dividing Eq. 10.1 by Vd, Eq. 10.2 which describes the maximum plasma drug concentration at steady state is obtained. Cpmax ss =

D Vd 1 − e − kτ

(

)

(10.2)

Since Cpmax ss declines to become Cpmin ss at the end of the dosing interval, Eq. 10.3 that describes the relationship between Cpmax ss and Cpmin ss can be obtained as follows: Cpmin ss = Cpmax ss e − kτ



(10.3)

At steady state, drug administration of the IV dose causes an increase in the plasma drug concentration from Cpmin ss to Cpmax ss. The change in the plasma drug concentration depends on the dose of the drug and Vd as in Eq. 10.4: Cpmax ss − Cpmin ss =

Dose Vd 

(10.4)

Rapid drug absorption after extravascular drug administration allows most of the drug to reach the systemic circulation before significant amount of the drug is eliminated as in Figure 10.2. Also, Cpmax ss is achieved rapidly making the time difference between Cpmax ss and Cpmin ss approximately equal to the dosing interval. So, Eqs. 10.3, 10.5, and 10.6 can be written to approximate the Cpmax ss and Cpmin ss during multiple extravascular

170  Steady State during Multiple Drug Administration

Figure 10.2 The plasma drug concentration during multiple administration of a rapidly absorbed drug. At steady state, the Cpmax ss and Cpmin ss will be similar during each dosing interval as long as the same dose is administered at fixed dosing interval.

drug administration when the drug absorption is rapid. In these equations, the bioavailability of the drug must be considered because rapid absorption does not mean complete absorption. Cpmax ss =

FD Vd 1 − e − kτ

(

)

(10.5)

and Cpmin ss = Cpmax ss e − kτ and Cpmax ss − Cpmin ss =

F Dose Vd 

(10.6)

When the drug is slowly absorbed, it takes time for the maximum drug concentration to be achieved. So, the difference between the Amax ss and Amin ss is not equal to the dose because significant amount of the drug is eliminated during the absorption process. Also, the time difference between Cpmax ss and Cpmin ss is not equal to τ. So, Cpmax ss and Cpmin ss after oral administration of slowly absorbed products cannot be calculated using Eqs. 10.5 and 10.3. Clinical Importance:

• During multiple IV or rapidly absorbed oral administration, Cpmax ss is proportional to the bioavailable dose. So, the dose can be changed to achieve the desired Cpmax ss.

• The maximum drug concentration is achieved shortly after drug administration, then

the drug concentration decreases during the remaining of the dosing interval. The longer the dosing interval while administration of the same drug, the larger the difference between Cpmax ss and Cpmin ss.

Steady State during Multiple Drug Administration 171

• When the dosing intervals are not equal, such as administration of drugs after

meals, Cpmax ss and Cpmin ss will not be equal after administration of each dose at steady state.

Practice Problems: a  Question: A patient is taking 400 mg IV every 8 hr from a drug that has Vd of 25 L and elimination rate constant of 0.21 hr−1.

• Calculate Cpmax ss and Cpmin ss of this drug in this patient. • Calculate Cpmax ss and Cpmin ss of this drug if the patient was shifted to 600 mg oral every 8 hr using an oral product that has bioavailability of 75%.

Answer:

• For 400 mg IV every 8 hr: Cpmax ss =

FD 400 mg = = 16.7 mg/L −1 − kτ Vd(1 − e ) 25L(1 − e −0.21hr  8 hr )

Cpmin ss = Cpmax ss e − kτ = 16.7e −0.21hr

−1

8 hr

= 3.11mg/L

• For 600 mg oral every 8 hr: Cpmax ss =

FD 0.75 × 600 mg = 22.1mg/L = −1 − kτ Vd(1 − e ) 25L(1 − e −0.21hr  8 hr )

Cpmin ss = Cpmax ss e − kτ =  22.1e −0.21hr

−1

8 hr

= 4.12 mg/L

b  Question: A patient is taking 240 mg IV of a drug every 12 hr. At steady state, Cpmax ss and Cpmin ss were found to be 8.0 and 4 mg/L, respectively. Calculate the Vd and halflife of this drug. Answer: Cpmax ss − Cpmin ss =

F Dose Vd

8.0 mg/L − 4.0 mg/L = Vd =

240 mg = 60 L 4 mg/L

Cpmin ss = Cpmax ss e − kτ

240 mg Vd

172  Steady State during Multiple Drug Administration 4 mg/L = 8mg/L e − k12hr ln

4 mg/L = ln e − k12 hr 8 mg/L

−0.693 = − k × 12 hr k = 0.0578hr −1 half-life = 12 hr 10.3  The Time Required to Achieve Steady State During multiple drug administration, the drug accumulates in the body until steady state is achieved. An approach similar to that used for constant rate IV infusion can be used to calculate the time required to achieve steady state during multiple drug administration. The time required to reach steady state is dependent on the half-life of the drug. Generally, the drug must be administered repeatedly for a period equal to five to six times the elimination half-life of the drug to reach steady state. The longer the drug half-life, the longer it takes to reach steady state. Figure 10.3 represents the plasma drug concentration during repeated administration of different drugs that have different elimination half-lives. Administration of larger doses of the same drug will produce higher steady-state drug concentrations; however, the time to achieve steady state is similar as illustrated in Figure 10.4. Clinical Importance:

• After initiation of drug therapy, the full drug therapeutic effects are observed when the drug reaches steady state. This may take 7–10 days for drugs with relatively long half-lives such as phenytoin, carbamazepine, and digoxin.

Figure 10.3 The plasma drug concentration during repeated administration of different drugs that have different half-lives. Drugs with shorter half-lives will reach steady state faster than the drugs with longer half-lives.

Steady State during Multiple Drug Administration 173

Figure 10.4 Administration of different doses of the same drug will achieve steady-state concentrations that are proportional to the administered dose. However, the time to achieve steady state will be the same.

• After initiation of drug therapy, the therapeutic effect of the drugs should be evaluated after allowing enough time for the drug to reach steady state.

• The longer the drug half-life, the longer it takes to observe the full drug therapeutic effect. When immediate drug effect is needed such as in emergency situations, administration of a loading dose may be necessary.

10.4  The Loading Dose The loading dose is a dose larger than the maintenance dose administered at the time of initiation of therapy to achieve plasma drug concentrations as close as possible to the desired steady-state concentration as illustrated in Figure 10.5. Administration of the loading dose is necessary when waiting for five to six half-lives to achieve steady state is not appropriate because the patient condition requires immediate therapeutic intervention. The loading dose can be given whether the drug is administered by multiple IV or multiple oral doses. The loading dose is calculated to immediately achieve drug concentration within the therapeutic range. Then, the maintenance dose is chosen to maintain the steady-state concentration within the therapeutic range. The loading dose does not affect the steady-state concentration and similar maintenance doses should achieve similar steady-state concentrations whether a loading dose is administered or not as in Figure 10.5. 10.4.1  IV Loading Dose

The loading dose is calculated from the desired plasma drug concentration and the Vd of the drug. The desired drug concentration is usually a drug concentration within the therapeutic range. It can be the upper limit of the therapeutic range or a concentration in the middle of the therapeutic range to be more conservative. Loading dose = Cpdesired × Vd 

(10.7)

174  Steady State during Multiple Drug Administration

Figure 10.5 The plasma drug concentration-time profiles after administration of different loading doses followed by the same maintenance dose. The steady-state plasma drug concentration is similar because the maintenance dose is similar. 10.4.2  Oral Loading Dose

Oral loading dose can be calculated similarly; however, the bioavailability of the drug has to be considered. Loading dose =

Cpdesired Vd F 

(10.8)

Clinical Importance:

• The steady-state concentration is dependent on the maintenance dose and the drug

pharmacokinetic parameters. The loading dose does not affect the Cpmax ss and Cpmin ss achieved during multiple drug administration.

Practice Problems: a  Question: An antiarrhythmic drug has Vd of 45 L, half-life of 12 hr, and its therapeutic range is 6–12 mg/L. Recommend IV and oral loading doses of this drug that should achieve a plasma drug concentration of about 10 mg/L. The oral drug product is rapidly absorbed and has bioavailability of 60%. Answer:

• For IV LD: LD = 10 mg/L × 45L = 450 mg IV dose

• For oral LD: LD =

10 mg/L × 45L = 750 mg 0.6

Steady State during Multiple Drug Administration 175 10.5  The Average Plasma Concentration at Steady State During multiple drug administration, the plasma drug concentration at steady state and the Cpmax ss and Cpmin ss fluctuate around an average drug concentration as in Figure 10.6. The average steady-state concentration is not the arithmetic mean of Cpmax ss and Cpmin ss. At steady state, the rate of drug administration is equal to the rate of drug elimination. The rate of drug administration is equal to FD/τ, where F is the bioavailability, D is the dose, and τ is the dosing interval. The rate of drug elimination is changing with time within the dosing interval at steady state because the drug concentration is continuously changing within the dosing interval. However, the average rate of drug elimination during the entire dosing interval can be calculated from the product of the elimination rate constant and the average amount of the drug in the body at steady state as in Eq. 10.9. FD   = kA average ss    τ    

FD   = k VdCpaverage ss τ 

Cpaverage ss =

FD k Vd τ 

(10.9)

(10.10)

(10.11)

This means that the average plasma concentration at steady state is dependent on the dosing rate (FD/τ) and the total body clearance (CLT, k Vd). Different dosing regimens can provide the same dosing rate if the bioavailability is constant. For example, consider the following dosing regimens: 150 mg q 6 hr 200 mg q 8 hr 300 mg q 12 hr 600 mg q 24 hr

Figure 10.6 The steady-state plasma drug concentration fluctuates around the average plasma drug concentration during multiple drug administration.

176  Steady State during Multiple Drug Administration All these dosing regimens from the same drug product have the same dosing rate. According to Eq. 10.11, the average drug concentration achieved at steady state while administration of any of these regimens should be similar, assuming that the CLT is constant. Although the average steady-state concentration is similar, the fluctuation in the plasma concentration around this average concentration is different for the different dosing regimens. This means that the maximum and the minimum concentrations at steady state are different for the different dosing regimens. The regimen with the largest dose and longest dosing interval usually has the largest fluctuations in the plasma concentration. Large fluctuations in the drug concentration at steady state produce high drug concentration, which may produce adverse effects at the beginning of the dosing interval and low drug concentration that may be subtherapeutic toward the end of the dosing interval. So, the optimal dosing regimen must maintain the drug concentration within the therapeutic range during the entire dosing interval. The total AUC observed after administration of a single dose of the drug is dependent on the bioavailable dose (FD) and CLT as in Eq. 10.12. AUC =

FD k Vd 

(10.12)

By substitution for the value of AUC in Eq. 10.11, Eq. 10.13 is obtained which is an expression for the average drug concentration at steady state. Cpaverage ss =

AUC τ 

(10.13)

After administration of a single dose of a drug, the AUC calculated from time zero to ∞ is equal to the AUC from time 0 to τ during multiple administration of the same dose of the drug at steady state as presented in Eq. 10.14 and as shown in Figure 10.7. AUC0−∞ after a single dose = AUC0−τ at steady state 

(10.14)

This means that the average plasma concentration that should be achieved at steady state can be determined from the AUC calculated after administration of single dose of the drug. Clinical Importance:

• Different dosage regimens that deliver the drug at the same dosing rate should achieve

the same average drug concentration at steady state. However, the appropriate dose and dosing interval should be selected to maintain the plasma drug concentration within the therapeutic range all the time. • Multiple drug administration at unequal dosing interval, such as administration of the drug three times a day after meals, will not achieve equal Cpmax ss and Cpmin ss at steady state after each dose. However, Cpaverage ss during the day will be the same during successive days.

Steady State during Multiple Drug Administration 177

Figure 10.7 The AUC from time zero to ∞ after administration of single dose of the drug is equal to the AUC from time zero to τ at steady state during multiple administration of the same dose of the drug.

Practice Problems: a  Question: A patient is receiving 1000 mg of an antibiotic as multiple IV bolus doses every 12 hr to treat her lung infection. The maximum and minimum steady-state plasma concentrations of this antibiotic were 48 and 5 mg/L, respectively.

• Calculate the volume of distribution and the total body clearance of this antibiotic. • Calculate the average plasma concentration of this antibiotic at steady state while receiving 1000 mg every 12 hr.

Answer: Cpmax ss − Cpminss =

Dose Vd

48 mg/L − 5 mg/L = Vd =

1000mg Vd

1000 mg = 23.3L 43mg/ L

Cpmin ss = Cpmax ss e − kτ 5mg/L = 48mg/L e − k12 hr ln

5mg/L =   − k12 hr 48mg/L

−2.262 = − k12 hr

178  Steady State during Multiple Drug Administration k = 0.188hr −1 CL T = 0.188hr −1 × 23.3L = 4.38L/hr Cpaverage ss =

FD k Vd τ

Cpaverage ss =

1000 mg = 19 mg/L 0.188hr −1 23.3L 12 hr

b  Question: A hospitalized patient was stabilized while receiving constant rate IV infusion of 30 mg/hr of a bronchodilator.

• Recommend an IV dose of the same drug to be administered every 8 hr. • Recommend an oral dose of the same drug to be administered every 8 hr if the bioavailability of the drug from the oral product is 60%.

Answer:

• The IV dose:

IV dose = 30 mg/hr × 8 hr = 240 mg every 8 hr

• The oral dose:

Oral dose = (30 mg/hr × 8 hr)/0.6 = 400 mg every 8 hr Both IV and oral doses should achieve Cpaverage ss similar to the steady-state drug concentration achieved during the IV infusion.

10.6  Drug Accumulation During multiple drug administration, the drug is accumulated in the body until steady state is achieved. The degree of drug accumulation is different from a drug to the other. Several approaches have been used to describe drug accumulation during multiple drug administration. One of these approaches that can be applied for IV and oral dosing regimens is the calculation of the accumulation ratio (Raccum). The accumulation ratio is defined as the amount of drug in the body at steady state relative to the dose of the drug. The larger the accumulation ratio, the higher is the degree of drug accumulation. R accum =

A average ss FD 

(10.15)

R accum =

FD/kτ 1 =  FD kτ 

(10.16)

Drugs that are eliminated slowly from the body (small k) and are administered frequently (short τ) usually have large accumulation ratio. In these drugs, the average amount of the drug in the body at steady state is much larger than the dose of the drug, indicating high accumulation.

Steady State during Multiple Drug Administration 179 Clinical Importance:

• When patients develop toxicity while receiving drugs which are not accumulated in the

body to large extend, discontinuation of therapy in most cases will be enough for the patient to recover from this toxicity. • Patients who develop toxicity while taking drugs with high degree of accumulation will continue to experience the signs and symptoms of toxicity for prolonged period even after discontinuation of the drug therapy. Additional measures must be installed to accelerate the rate of drug elimination and hasten the recovery from this toxicity. • Missing a dose during multiple drug administration of a drug which has low accumulation leads to elimination of most of the drug in the body which may significantly affect the drug therapeutic effect. In this case, a replacement dose should be administered once the patient remembers that he/she missed the dose. • While missing a dose of a drug which has high degree of accumulation may not affect the amount of the drug in the body significantly. In this case the patient can be instructed to take the drug dose at the time of the following dose unless the drug has very narrow therapeutic index. 10.7  Controlled Release Formulations Controlled release formulations are specially designed formulations that can slowly release the drug from the dosage forms over an extended period of time. The rate of drug absorption from these formulations usually depends on the rate of drug release from the formulation. This means that if the formulation is designed such that it releases the drug at a slow and constant rate over an extended period, the rate of drug absorption becomes slow and relatively constant over the drug release period. This slow absorption rate that usually covers the entire dosing interval causes slow rise and slow decline in the plasma drug concentration within each dosing interval. This causes small fluctuations in the plasma drug concentration at steady state while taking these formulations. Controlled release formulations allow administration of larger doses of the drug less frequently without increasing the fluctuations in the drug concentration during the dosing interval. Administration of drugs with short half-lives in the form of immediate release formulations results in rapid increase followed by rapid decline in the plasma drug concentration and large fluctuation in the plasma drug concentration. This large fluctuation in the plasma drug concentration causes variation in the drug therapeutic effect within the dosing interval when direct plasma drug concentration-effect relationship exists. Controlled release formulations are ideal for these drugs to decrease the frequency of their administration, decrease the fluctuation in the plasma drug concentration, and decrease variation in therapeutic effect within the dosing interval. Figure 10.8 represents the plasma drug concentration during multiple administration of immediate release and controlled release formulations. Clinical Importance:

• Controlled release formulations are ideal for drugs with short half-lives that have direct plasma drug concentration-effect relationship.

• Controlled release formulations are not always the preferred formulations for all

drugs. These formulations may not add any benefits for drugs with long half-lives

180  Steady State during Multiple Drug Administration

Figure 10.8 The plasma concentration-time profile after multiple administration of the same dose of immediate release and controlled release oral formulations.

like digoxin, which produce small fluctuations in drug concentration during repeated administration, and for drugs that do not have direct plasma concentration-effect relationship such as the oral anticoagulants. Also, drugs that undergo saturable first-pass effect such as propranolol have usually lower oral bioavailability when administered as controlled release products. 10.8 The Effect of Changing the Pharmacokinetic Parameters on the Steady-State Plasma Drug Concentration during Multiple Drug Administration During multiple drug administration, the drug is accumulated in the body until steady state is achieved. The Cpaverage ss is directly proportional to the drug administration rate, FD/τ and inversely proportional to the drug CLT. While the time required to achieve steady state is dependent on the drug half-life. It is important to examine how the change in each of the pharmacokinetic parameters affects the steady-state drug concentration achieved during multiple drug administration. 10.8.1  Dosing Rate

The effect of changing dosing rate by changing the dose, the bioavailability, or the frequency of administration of the drug:

• Higher dosing rate is achieved by administration of larger dose, using products with higher bioavailability, or administration of the drug more frequently. Increasing the dosing rate causes proportional increase in the average steady-state concentration if the CLT remains constant. • Increasing the dosing rate increases the average steady-state concentration; however, the time to achieve steady state is not affected. Clinical Importance:

• Larger doses produce higher average plasma concentrations leading to more intense therapeutic effect and may produce adverse effects in narrow therapeutic index drugs.

Steady State during Multiple Drug Administration 181

• Switching patients from one product to the other for the same active drug may cause

change in the average steady-state concentration, and hence the therapeutic effect, if the products have different bioavailability.

10.8.2  Total Body Clearance

The effect of changing CLT of the drug:

• The average steady-state plasma concentration is inversely proportional to CLT. Lower

CLT results in higher steady-state concentration when the dosing rate is kept constant.

• If CLT is different in patients with the same Vd, the plasma drug profile in the patient

with higher CLT will reach steady state faster than in patients with lower CLT. This is because higher CLT means larger k and shorter t1/2, when Vd is constant.

Clinical Importance:

• Patients with eliminating organ dysfunction usually require lower than average doses to achieve therapeutic drug concentration. Administration of the average recommended doses to these patients may cause toxicity.

10.8.3  Volume of Distribution

The effect of changing Vd of the drug:

• If the dosing rate and CLT are constant, the same steady state should be achieved in patients with different Vd. Patients with similar CLT and different Vd have different t1/2 and k, so the time to achieve steady state will be different. • In most cases, differences in Vd are accompanied by differences in CLT, so the change in the dependent parameters (t1/2 and k) will depend on the relative changes in CLT and Vd. Clinical Importance:

• The dose should be based on the body weight, or ideal body weight in obese patients, rather than administration of a fixed dose to all patients, especially for narrow therapeutic index drugs.

10.8.4  Absorption Rate Constant

The effect of administration of different products of the drug which have different rates of absorption:

• The absorption rate of the drug does not affect the average steady-state drug concentration. • Fast rate of drug absorption produces large fluctuations, while slow absorption produces small fluctuations in drug concentrations at steady state.

Clinical Importance:

• When direct relationship exists between the drug concentration and the drug effect,

slowly absorbed drug products usually produce steady therapeutic effect, which is useful in the management of chronic diseases.

182  Steady State during Multiple Drug Administration 10.9  Dosing Regimen Design The dosage requirement for different patients is different because of the differences in the patients’ characteristics. Optimization of a dosing regimen involves selection of the appropriate dose and dosing interval for each individual patient. The appropriate dosing regimen is the regimen that maintains the plasma drug concentration within the therapeutic range all the time while using the drug. 10.9.1  Factors to Be Considered

Few factors must be considered before calculating the dosing regimen. These factors should help in the selection of the proper dosing regimen. 10.9.1.1  The Therapeutic Range of the Drug

The therapeutic range of the drug is the range of plasma drug concentrations associated with the maximum probability of producing the desired drug therapeutic effect and the minimum probability of producing adverse effects. This range of concentrations is different for different drugs. Some drugs may have narrow therapeutic range and others may have wide therapeutic range. The optimal dosing regimens for drugs with wide therapeutic range are easier to determine because wide range of dosing rates can maintain the plasma concentration within the therapeutic range at steady state. This means that drugs with wide therapeutic range are safer drugs. However, drugs with narrow therapeutic range require accurate calculation of the appropriate dosing regimen because small change in dose can result in toxic or subtherapeutic effects. 10.9.1.2  The Required Onset of Effect

It is important to determine if an immediate drug effect is required or not. Immediate drug effect is usually required in emergency cases and when the patient condition is severe enough to necessitate rapid therapeutic intervention. In this case, the dosing regimen should include an initial loading dose to achieve faster approach to steady state and faster drug effect. On the other hand, when the patient condition is stable and the dosing regimen is used to initiate the treatment of a chronic condition, administration of loading dose may not be necessary. 10.9.1.3  The Drug Product

It is important to consider the availability of controlled release products for the drug of interest and the suitability of these products for the patient condition. This is because slow-release formulations produce small fluctuations in drug concentration within each dosing interval, while fast release formulations produce large fluctuations in drug concentration. When controlled release products are used, the goal of the dosing regimen design is to maintain the average plasma drug concentration within the therapeutic range. When fast release products are used, the dosing regimen should be selected to ensure that the Cpmax ss does not exceed the upper limit of the therapeutic range and the Cpmin ss does not go below the lower limit of the therapeutic range.

Steady State during Multiple Drug Administration 183 10.9.1.4  Progression of the Patient Disease State

The progression of the patient disease state and the response of the patient to the administered drugs must be considered when selecting the dosing regimens. For example, progression of the renal disease with time may necessitate periodic modification of the dose of the drugs excreted mainly by the kidney. Also, the dose of some drugs is increased gradually after initiation of therapy because of the increase in their metabolic rate such as in case of carbamazepine autoinduction. So, with the initiation of the dosing regimen, a plan should be installed to periodically evaluate the dosing regimen and to make the necessary modification. 10.9.2  Estimation of the Patient Pharmacokinetic Parameters

The dose of the drug in each patient must be individualized according to the patient’s specific pharmacokinetic parameters. The drug pharmacokinetic parameters can depend on many factors such as age, weight, sex, eliminating organ function, concomitant drugs, genetic factors, and many other factors. So, selection of the appropriate dosing regimen for a drug in a particular patient requires knowledge of the drug pharmacokinetic parameters in this patient, which can be estimated with different levels of accuracy depending on the available information about the patient.

• If no information is known about the patient’s medical history, the average value for

the drug pharmacokinetic parameters in the general population is the best estimate for the patient’s pharmacokinetic parameters. These estimates can be used to calculate the dosing regimen for the patient. The dosing regimen calculated this way has to be used with caution because it is based on an approximate estimate for the patient’s pharmacokinetic parameters. • If information is available about the patient’s medical history, the best estimate for the patient’s specific pharmacokinetic parameters is the parameters of a patient population similar to that of the patient with regard to age, weight, gender, and organ function, in addition to other medical conditions and other drugs that can affect the pharmacokinetics of the drug of interest. These parameter estimates are better approximation of the drug pharmacokinetic parameters in that particular patient compared to using the average population parameters and can be used to calculate the dosing regimen for the patient. • If the patient has history of using the drug under consideration, the previous dose, drug concentration if measured, and the therapeutic outcomes should be used to estimate the patient’s specific pharmacokinetic parameters and to design the new drug dosing regimen. Any changes in the patient’s condition that can alter the drug pharmacokinetics should be considered in the design of the new dosing regimen. 10.9.3  Selection of Dose and Dosing Interval

The dose and the dosing interval of the drug should be selected to maintain plasma drug concentration within the therapeutic range all the time. The selection of dose and dosing interval depends on the drug pharmacokinetic parameters and therapeutic range of the drug. Also, dosing regimens that involve using modified release formulations are designed to accommodate the dosing intervals recommended by the manufacturer.

184  Steady State during Multiple Drug Administration 10.9.3.1  Multiple Controlled Release Oral Formulation

Controlled release formulations are slowly absorbed and produce small fluctuations in the plasma drug concentration at steady state. The choice of the dosing interval of controlled release formulations usually depends on the formulation characteristics and is usually recommended by the product manufacturer. The estimates for the drug pharmacokinetic parameters should be used to calculate the dose required to achieve an average plasma drug concentration within the therapeutic range according to: Cpaverage ss =

FD k Vd τ

10.9.3.2  Multiple IV or Fast-Release Oral Formulations

During repeated administration of IV and fast-release oral formulations, large fluctuations in the plasma concentrations are observed. The dose and dosing intervals should be selected to keep Cpmax ss and Cpmin ss within the therapeutic range all the time. Usually, the dosing interval is selected based on the desired maximum and minimum plasma drug concentrations, then the dose to achieve the maximum concentration is calculated. 10.9.3.2.1  SELECTION OF THE DOSING INTERVAL

The target maximum and minimum plasma drug concentrations are selected to be within the therapeutic range of the drug. Then the dosing interval is calculated from the time required for the maximum plasma concentration to decline to the minimum plasma concentration. The target maximum and minimum concentrations and the drug elimination rate constant are substituted in the following equation and solved for the dosing interval. Cpmin ss = Cpmax ss e − kτ The calculated dosing interval should be rounded to the nearest practical dosing interval. The practical dosing interval is the interval that allows drug administration at the same time every day (e.g., 6, 8, 12, and 24 hr). 10.9.3.2.2  SELECTION OF DOSE

The dose required to achieve the target maximum steady-state drug concentration when given every τ (practical τ calculated before) can be determined from the following relationship: Cpmax ss =

FD Vd(1 − e − kτ )

Again, it may be necessary to round the selected dose to the nearest practical dose.

Steady State during Multiple Drug Administration 185 10.9.4  Selection of the Loading Dose

When immediate effect of the drug is required, the dosing regimen should include a recommendation for a loading dose. The loading dose is calculated from the desired plasma concentration and the drug Vd. Loading dose  =

Cpdesired Vd F

Clinical Importance:

• The drug dosing regimen, dose, and dosing interval should be selected for each individual patient to ensure that the plasma drug concentration remains within the therapeutic range all the time. • The dosing regimen should be individualized based on the patient specific pharmacokinetic parameters. Practice Problems: a  Question: A patient was admitted to the hospital after experiencing an episode of ventricular arrhythmia. The physician decided to start the patient on IV regimen of an antiarrhythmic drug. The Vd of this antiarrhythmic drug is 45 L, the t1/2 is 7 hr, and the therapeutic range is 4–12 mg/L.

• Recommend an IV loading dose and the IV dosing regimen (dose and dosing interval) that should maintain the drug concentration within the therapeutic range all the time.

• The patient was stabilized on the regimen you recommended, and she is ready to go home. The physician wants to shift the patient to oral formulation of the same drug. What will be the dosing regimen (dose and dosing interval) from the oral formulation if the oral formulation of this drug is rapidly absorbed and has bioavailability of 75%?

Answer: The loading dose is the dose that should achieve the maximum plasma concentration Loading dose = Cpdesired × Vd LD = 12 mg/L × 45L = 540 mg Dosing interval 4 mg/L = 12 mg/L e −0.099 hr ln

−1

4 mg/L = −0.099 hr −1 τ 12 mg/L

−1.0986 = −0.099 hr −1 τ τ = 11hr   

τ

186  Steady State during Multiple Drug Administration So, an appropriate dosing interval will be 12 hr. Selection of dose Cpmax ss =

D Vd 1 − e − kτ

(

12 mg/L =  

(

)

D

45L 1 − e

−0.099 hr −1 12 hr

)

Dose = 375mg Recommendation: The patient should receive an IV loading dose of 540 mg and then the maintenance dose is 375 mg IV administered every 12 hr.

• The dosing regimen from the oral formulation • The dosing interval is dependent on the rate of elimination of the drug, so this rapidly

absorbed oral formulation should also be administered every 12 hr. However, the dose will be different because of the incomplete bioavailability. Cpmax ss =

12 mg/L =

FD Vd(1 − e − kτ )

(

0.75D

45L 1 − e −0.099 hr

−1

12 hr

)

Dose = 500 mg Recommendation: 500 mg of the oral formulation every 12 hr 10.10 Summary

• Multiple drug administration is used to treat transient medical conditions that require drug use for a period of time and in the management of chronic diseases.

• It takes multiple drug administration for a period equal to five to six times the drug elimination half-life to reach steady state. At steady state, the rate of administration is equal to the rate of elimination. • The average plasma drug concentration achieved at steady state is directly proportional to the doing rate and inversely proportional to the drug clearance. • At steady state, the plasma drug concentration is fluctuating around an average drug concentration. The dosing regimen should be selected to maintain plasma drug concentration within the therapeutic range all the time. • Individualization of drug therapy by selecting the optimum drug dose and doing regiment based on the pharmacokinetic parameters for each patient is necessary specially for drugs with narrow therapeutic range.

Steady State during Multiple Drug Administration 187 Practice Problems 10.1 An antibiotic has a volume of distribution of 20 L and an elimination rate constant of 0.1 hr−1. This antibiotic is administered by repeated IV bolus doses of 200 mg every 8 hr. a Calculate the maximum and the minimum plasma drug concentration at steady state. b Calculate the average steady-state concentration of this antibiotic. c What is the dose that should be given every 8 hr to achieve steady-state average plasma drug concentration of 25 mg/L? 10.2 A 72-year-old, 85 kg, male patient was admitted to the hospital for the treatment of severe lung infection. His dosing regimen of the antibiotic was 500 mg daily given at 8:00 AM by IV bolus administration. At steady state, the maximum and the minimum plasma concentration of this antibiotic were 35 and 10 mg/L. a Calculate the half-life of this antibiotic in this patient. b Calculate the volume of distribution of this antibiotic in this patient. c Calculate the average steady-state plasma concentration of the drug. 10.3 After a single oral dose of 200 mg theophylline to a 10-year-old patient, the total AUC was 100 mg hr/L. (Assume that theophylline is completely absorbed.) a Calculate the average theophylline plasma concentration at steady state if this patient is taking 200 mg theophylline every 6 hr. b Calculate the average theophylline plasma concentration at steady state if this patient is shifted to 400 mg every 12 hr from a controlled release formulation. c Calculate the dose of this controlled release formulation that can achieve an average steady-state concentration of 10 mg/L in this patient. 10.4 An antihypertensive drug is administered as 200 mg daily for a controlled release formulation, which is known to have 75% bioavailability. The drug has a half-life of 24 hr and a volume of distribution of 24 L. a Calculate the average steady-state concentration of this drug. b What is the dose required to achieve an average steady-state concentration of 20 mg/L using the same formulation? c What is the daily IV dose required from an IV formulation of the drug to achieve an average steady-state concentration of 20 mg/L? d Calculate the maximum and the minimum plasma drug concentration at steady state while taking the regimen you recommended in c. 10.5 A patient is receiving his antihypertensive medication as 500 mg every 8 hr in the form of an oral capsule. After administration of this oral capsule, only 80% of the active ingredient in the capsule reaches the systemic circulation. The CLT of this drug in this patient was 2.5 L/hr. a What is the average steady-state plasma concentration of this drug in this patient? b What will be the average steady-state plasma concentration if the patient was taking 1000 mg every 8 hr from the same oral drug product? c What will be the CLT of the drug in this patient while taking the 1000 mg every 8 hr oral dose?

188  Steady State during Multiple Drug Administration d What will be the oral dose required to achieve an average plasma concentration of 30 mg/L from this drug in this patient? e Because the patient had problems swallowing the oral capsules, he was shifted to 500 mg IV every 8 hr regimen. What will be the average steady-state concentration of this drug in this patient? 10.6 A patient received a single IV bolus dose of 300 mg of an antibiotic and the following plasma concentrations were obtained. Time (hr)

Drug concentration (mg/L)

2 6 12 24

12.86 9.45 5.95 2.36

a What is the maximum and the minimum steady-state drug concentration in a patient who was taking 250 mg every 12 hr from an oral formulation that is rapidly absorbed and has bioavailability of 80%? b What is the average steady-state concentration of this drug in the patient while taking the regimen in a? c What will be the maximum and the minimum steady-state drug concentration if the dose was increased to 500 mg every 12 hr from this oral formulation? 10.7 A patient is taking 450 mg of an antibiotic every 12 hr as IV bolus doses. At steady state, the maximum and minimum plasma antibiotic concentrations were 40 and 10 mg/L, respectively. a b c d e

Calculate the half-life of this antibiotic in this patient. Calculate the volume of distribution of this antibiotic in this patient. Calculate the average steady-state plasma concentration of the drug. What is the approximate time for this drug to achieve steady state? Calculate the amount of the drug eliminated from this antibiotic during one dosing interval at steady state.

10.8 A patient is receiving IV theophylline to control an acute episode of bronchial asthma. The maintenance dose is 320 mg IV bolus of theophylline every 8 hr. At steady state, the maximum and minimum theophylline plasma concentrations were 18 mg/L and 10 mg/L, respectively. a What is the half-life and the volume of distribution of theophylline in this patient? b What is the daily dose of an oral theophylline preparation that is required to achieve an average plasma theophylline concentration of 18 mg/L? (The bioavailability of the oral preparation is 100%.) 10.9 A patient was admitted to the hospital because of severe attack of ventricular arrhythmia. The patient’s acute condition was treated, and the physician wanted to start the patient on multiple IV bolus administration on an antiarrhythmic drug. The physician asked you to recommend a loading dose and the dosing regimen that should maintain the maximum and the minimum steady-state drug concentration, approximately 20 and 10 mg/L, respectively. This drug has an elimination rate constant of 0.06 hr−1, and volume of distribution of 30 L.

Steady State during Multiple Drug Administration 189 a What is your recommendation for the IV loading dose that should immediately achieve 20 mg/L? b What is the most appropriate dosing interval for this drug? c What is the appropriate maintenance dose for this drug? d What is the average steady-state concentration of this drug achieved from the regimen you recommended? e The patient was stabilized on the regimen you recommended, and he is ready to go home. The physician asked you to recommend the dosing regimen of the oral formulation of the same drug to achieve the same average steady-state concentration if this oral formulation is known to have bioavailability of 75%. What is your recommendation? 10.10 The therapeutic range of an antihypertensive drug is 8–25 mg/L. The drug has halflife of 7 hr and volume of distribution of 30 L. After oral administration, this drug is rapidly absorbed and only 85% of the dose reaches the systemic circulation. a What is the most appropriate dosing interval for this drug? b What is the appropriate maintenance dose for this drug? c What is the average steady-state concentration achieved from the regimen you recommended?

11 Renal Drug Excretion

Objectives After completing this chapter, you should be able to:

• • • • • • • •

Discuss the importance of studying the drug elimination through a specific pathway. Describe the processes involved in renal drug excretion. Calculate the drug pharmacokinetic parameters from the urinary excretion data. Explain the rationale for determination of the drug bioavailability from the renal excretion data. Calculate the renal clearance of drugs from the urinary excretion data. Describe the method used for determination of the creatinine clearance. Analyze the effect of changing the dose, total body clearance, and renal clearance on the urinary excretion of drugs. Predict the change in the overall elimination rate of drugs due to the change in drug renal excretion.

11.1 Introduction Drugs are eliminated from the body by excretion of the unchanged drug from the body, or by metabolism to form one or more metabolites that are then excreted from the body. The major excretion pathways include renal excretion of drugs in urine, biliary excretion of drugs to the gastrointestinal tract, and lung excretion of volatile drugs during expiration. This is in addition to other minor excretion pathways such as excretion of drugs in sweat and milk. Drug metabolism involves enzymatic modification of the drug chemical structure to form a new chemical entity known as the metabolite. Most of the metabolites are pharmacologically inactive; however, few metabolites produce pharmacological activity and some metabolites can cause adverse effects. Most drugs are eliminated from the body by one or more elimination pathways. The CLT is the sum of the clearances associated with each of the drug elimination pathways. If a drug is eliminated by renal excretion and hepatic metabolism, the CLT for this drug is the sum of its renal clearance and metabolic clearance. Also, drugs that are metabolized by different metabolic pathways to form different metabolites, each of these metabolic pathways has its own clearance. A general relationship that relates the drug total body clearance to the clearances for the different elimination pathways can be written as in Eq. 11.1. CL T =  CL R + CL M + CL L + CL B + .  DOI: 10.4324/9781003161523-11

(11.1)

Renal Drug Excretion 191 where CLR is the renal clearance, CLM is the metabolic clearance, CLL is the lung clearance, and CLB is the biliary clearance. If a drug is not eliminated through a particular pathway, the clearance associated with this pathway is equal to zero. If a drug is completely eliminated by a single elimination pathway, the CLT becomes equal to the clearance of this elimination pathway. So, if a drug is completely eliminated by metabolism to form one metabolite, the CLR of this drug is equal to zero and its CLT is equal to CLM. Also, when the drug elimination processes follow first-order kinetics, each of the elimination pathways should have a rate constant that is dependent on the clearance of each elimination pathway and the Vd of the drug as in Eqs. 11.2 and 11.3. CL T CL R CL M CL L CL B = + + + + Vd Vd Vd Vd Vd 

(11.2)

k = k e + k m + k l + kb +  

(11.3)

where ke is the first-order rate constant for the renal excretion, km is the first-order rate constant for the metabolic process, kl is the first-order rate constant for the lung excretion, and kb is the first-order rate constant for the biliary excretion. This means that the first-order elimination rate constant for the overall elimination process is the sum of the rate constants for the different elimination pathways. 11.2  Studying Drug Elimination through a Specific Pathway In the previous chapters, drug elimination was viewed as one elimination process that decreases the amount of the active drug in the body. The rate of decline in the plasma drug concentration that reflects the rate of drug elimination by all elimination pathways was used to determine the rate constant and half-life of the overall elimination of the drug. Also, the total body clearance was a measure of the drug clearance by all elimination pathways. The contribution of each elimination pathway to the overall elimination process determines how much drug is eliminated by each pathway. This means that the fraction of the administered IV dose of a drug excreted by a specific pathway is determined from the magnitude of the drug clearance or rate constant associated with this specific pathway relative to the CLT or k, respectively. For example, if the CLR of a drug is 60% of its CLT (this means that ke is 60% of k), 60% of the administered IV dose will be excreted unchanged in urine and the rest of the dose is excreted by the other elimination pathways. Studying the rate of drug excretion through a specific elimination pathway is not easy because it requires determination of the rate of elimination through this pathway. The easiest excretion pathway that can be studied is the renal drug excretion which can be achieved by collecting urine samples and measuring the amount of drug excreted unchanged in urine during a certain period of time. Also, the rate of drug metabolism can be studied by measuring the rate of metabolite appearance in the systemic circulation after drug administration. Furthermore, determination of the rate of lung excretion of drugs can be studied by collecting the expired air over a period of time and measuring the drug concentration in this expired air. Moreover, measuring the rate of biliary drug excretion requires collection of bile over certain time intervals and measuring the drug concentration in the collected bile. Some of these experiments can only be performed in experimental animals. The kinetics of the renal drug excretion will be discussed in this chapter, while the kinetics of drug metabolism will be discussed in the next chapter.

192  Renal Drug Excretion Clinical Importance:

• Determination of the contribution of the different elimination pathways to drug elimi-

nation is important to determine the effect of changing the eliminating organ function on the overall drug elimination rate. • The clearance of drugs that are eliminated mainly by the kidney will be affected by the decrease in renal function. While the clearance of drugs that are eliminated mainly by hepatic metabolism will be affected by the decrease in hepatic function. • Renal failure patients require smaller doses of digoxin and drugs that are excreted mainly by the kidney, while hepatic failure patients require lower doses of lidocaine and drugs that are eliminated mainly by metabolism. The higher the degree of eliminating organ dysfunction the higher the reduction in the drug dose. • The clinical significance of drug-drug interactions that involves drug elimination processes is dependent on the contribution of the affected pathway to the overall drug elimination. For example, metronidazole that is an inhibitor of the metabolizing enzymes can significantly slow the rate of elimination of warfarin that is eliminated mainly by metabolism, while probenecid inhibits the renal active secretion of penicillin G leading to significant reduction in penicillin G elimination rate that is eliminated mainly by the kidney. 11.3  The Renal Excretion of Drugs The renal excretory function starts with the filtration of the blood reaching the glomeruli which occurs normally at a rate of 125 mL/min and ends with urine formation at a rate of 1–2 mL/min. This means that about 98–99% of the water in the glomerular filtrate is reabsorbed in the renal tubules. When a solute is filtered in the glomeruli, the concentration of that solute in the glomerular filtrate is equal to its concentration in plasma. If the solute is not secreted or reabsorbed in the renal tubules, the concentration of this solute increases as it moves through the renal tubules due to water reabsorption. Water reabsorption makes the solute concentration gradient between the tubular lumen and blood in favor of passive tubular reabsorption if the solute can permeate across the tubular membrane. Some compounds can also undergo carrier-mediated reabsorption from the renal tubules. However, solute secretion from the blood to the renal tubules must occur by active transport because it is against the concentration gradient. So, the three processes involved in the renal excretion of drugs are the glomerular filtration, active tubular secretion, and tubular reabsorption (1).

• Glomerular filtration: The blood is filtered in the glomeruli and the rate of this pro-

cess is called the glomerular filtration rate (GFR). All small molecules with molecular weight below 2000 g/mole can be filtered freely in the glomeruli. However, large protein molecules such as albumin and drug molecules that are bound to plasma proteins cannot be filtered. • Active tubular secretion: The drug molecules can be actively secreted from the plasma to the lumen of the proximal tubules by several transport systems that are specific for different classes of compounds. These transport systems are saturable and different drugs in the same chemical class can compete for the same transporter. Drugs with higher affinity for the transporter can inhibit the secretion of the drugs with lower affinity.

Renal Drug Excretion 193

• Tubular reabsorption: Drugs and solutes filtered in the glomeruli and actively secreted in the proximal tubules can be reabsorbed throughout the tubules by passive or carrier-mediated reabsorption. Because of water reabsorption in the renal tubules, lipophilic molecules can be reabsorbed by passive diffusion.

The rate of renal excretion of a drug is the sum of the rate of filtration and active secretion minus the rate of reabsorption from the renal tubules: Renal excretion rate = rate of filtration + rate of active secretion − rate of reabsorption. Clinical Importance:

• Impaired kidney function is usually associated with fewer number of functioning

• •

• • •

nephrons leading to lower GFR. Also, some diseases such as nephrotic syndrome can affect the glomerular function leading to excretion of large protein molecules such as albumin and globulins in urine. Probenecid has been shown to inhibit the active renal excretion of penicillin G, zidovudine, gemifloxacin, and methotrexate. While the active renal excretion of metformin can be inhibited by cimetidine, trimethoprim, and pyrimethamine. Glucose is a hydrophilic molecule that undergoes carrier-mediated reabsorption in the renal tubules. In diabetic patients, the glucose concentration in the glomerular filtrate and the renal tubular lumen is high. This causes saturation of the glucose carriermediated reabsorption which leads to incomplete reabsorption of glucose from the renal tubules and glucosuria. Ionizable molecules can be reabsorbed in the renal tubules when they are in their unionized form. While hydrophilic and ionized molecules have very limited reabsorption from the renal tubules. Since ionizable molecules can be passively reabsorbed in the renal tubules when they are in their unionized form, changing the urinary pH can be used to manipulate the urinary secretion of ionizable drugs. The urinary excretion of acidic compounds such as salicylates, methotrexate, and barbiturates can be increased by urine alkalinization with sodium bicarbonate, while the urinary excretion of basic compounds such as amphetamines and phencyclidine can be increased by urine acidification with ammonium chloride.

11.4  Determination of the Drug Renal Excretion Rate The drugs excreted by the kidney are collected in the urinary bladder, then excreted with urine when the bladder is evacuated. Urine samples are usually collected over predetermined time intervals after making sure that the bladder is empty at the beginning of the interval. Then the sample is collected at the end of the interval by completely emptying the bladder to make sure that the drug in the sample represents the total drug excreted during the urine collection interval. The amount of the drug excreted in urine over each collection interval is calculated from the volume of urine sample and the drug concentration in the sample. The average renal excretion rate of the drug over the urine collection interval is determined by dividing the amount of drug excreted in urine by the length of the using collection interval.

194  Renal Drug Excretion

Figure 11.1 The diagram represents a drug that is eliminated by renal and nonrenal elimination pathways. A is the amount of the drug in the body, k is the first-order overall elimination rate constant, and ke is the first-order renal excretion rate constant.

Consider a drug that is eliminated by different elimination pathways, including renal excretion that follows first-order kinetics, as in Figure 11.1. The renal excretion rate of the drug at any time can be described by Eq. 11.4. dA e = ke A dt 

(11.4)

where Ae is the amount of the drug excreted in urine, ke is the first-order renal excretion rate constant, and A is the amount of the drug in the body. However as mentioned above, urine samples are collected over certain time intervals, so the amount of the drug excreted in urine, ΔAe over a certain urine collection interval Δt, can be expressed as in Eq. 11.5.   

∆A e = ke A average ∆t 

(11.5)

where ΔAe/Δt is the average renal excretion rate over the urine collection interval, ke is the first-order renal excretion rate constant, and Aaverage is the average amount of the drug in the body during the urine collection interval. 11.4.1  Experimental Determination of the Renal Excretion Rate

While planning a renal excretion rate experiment for a drug, the total period of sampling and the duration of each urine collection interval must be specified based on the drug pharmacokinetic characteristics. In general, frequent samples are obtained when the monitored variable is changing rapidly, and less frequent samples are obtained when the variable is changing slowly. After single IV bolus drug administration, frequent samples are obtained initially and then the sampling frequency decreases with time. For example, if the drug has an elimination half-life of 6 hr, the total sampling duration can be chosen as 18–24 hr. This sampling duration represents about 3–4 elimination half-lives for the drug which is enough to eliminate more than 90% of the administered drug dose. The urine collection intervals can be chosen as: 0–2, 2–4, 4–8, 8–12, 12–18, and 18–24 hr. This is an example of the sampling schedule; however, the sampling schedule can be changed as needed to obtain the required information from the study. In human subjects, urine samples obtained over intervals shorter than 1 hr cannot be obtained accurately since complete collection of the urine samples is required for accurate determination of the renal drug excretion rate. The following steps are usually followed for the determination

Renal Drug Excretion 195 of the drug renal excretion rate during each urine collection interval after single drug administration:

• The subjects are asked to empty their bladders before drug administration. The drug is administered, and the time of drug administration is taken as time zero.

• Urine samples are collected by completely emptying the bladder at the end of each

urine collection interval. When multiple collections are obtained during the collection interval which is common during long intervals, a final urine collection is obtained at the end of the interval and all the collected urine is combined together as one sample. • The volume of the total urine sample collected over the entire interval is determined accurately and recorded. An aliquot of the urine sample is kept for drug analysis. • The drug concentration is determined in each urine sample. • The total amount of the drug excreted during each urine collection interval is calculated from the volume and drug concentration of each urine sample as follows: Amount of drug excreted = Drug concentration × Sample volume 

(11.6)

• The average renal excretion rate during each urine collection interval is determined

from the amount of the drug excreted during the interval and the length of the collection interval as follows: Renal excretion rate =

Amount excreted Time of urine collection 

(11.7)

11.4.2  The Drug Renal Excretion Rate-Time Profile

The renal excretion rate of the drug at any time is determined from the renal excretion rate constant and the amount of the drug in the body according to Eq. 11.4, if the renal excretion process follows first-order kinetics. Since the amount of the drug in the body after single IV administration decreases with time due to drug elimination, the renal excretion rate of the drug decreases also with time. The drug renal excretion rate after single IV bolus administration declines at the same rate of decline of the amount of the drug in the body. Eq. 11.8 describes the amount of the drug in the body at any time after single IV bolus dose. A = A 0 e − kt



(11.8)

where A is the amount of the drug in the body at any time t, A0 is the initial amount of the drug in the body after the IV bolus administration which is equal to the dose, and k is the first-order elimination rate constant. Since the renal excretion rate of the drug at any time is the product of the renal excretion rate constant and the amount of the drug in the body, the drug renal excretion rate (ΔAe/Δt) at any time after a single IV bolus dose can be presented as in Eq. 11.9. ∆A e = ke A 0e− kt ∆t 

(11.9)

196  Renal Drug Excretion

Figure 11.2 The slope of the drug renal excretion rate versus time plot on the semilog scale is equal to −k/2.303 and the y-intercept is equal to ke dose. The elimination rate constant, k, can be determined from the slope of the plot, the elimination half-life can be calculated from k or estimated graphically by determining the time required for any value on the line to decrease by 50%, and ke can be determined from the y-intercept and the drug dose.

Equations 11.8 and 11.9 indicate that both the amount of the drug in the body and the renal excretion rate of the drug after single IV bolus dose decline at a rate dependent on k, the overall elimination rate constant of the drug. To construct the drug renal excretion rate versus time plot, the experimentally determined renal excretion rate for the drug during each urine collection interval is plotted against the time corresponding to the middle of the urine collection interval (tmid). This is because the calculated drug renal excretion rate represents the average rate of renal drug excretion during the urine collection interval. After single IV bolus drug administration, a plot of the drug renal excretion rate versus time on the semilog scale is a straight line with y-intercept that equals ke A0 (or ke Dose) and the slope is equal to −k/2.303. This means that the elimination rate constant, the half-life, and the renal excretion rate constant of the drug can be determined from the drug renal excretion rate versus time plot as in Figure 11.2. Since the drug renal excretion rate at any time is the product of ke and the amount of the drug in the body, the drug renal excretion rate-time profile is always parallel to the drug amount-time profile and drug concentration-time profile. So, after extravascular drug administration, the drug renal excretion rate-time profile increases initially during the drug absorption phase and then declines during the elimination phase. The drug elimination rate constant and half-life can be determined from the renal excretion ratetime profile during the elimination phase after extravascular drug administration. 11.5  The Renal Clearance The CLR for a drug represents the contribution of the renal excretion pathway to CLT. The CLT is defined as the volume of the blood or plasma which is completely cleared from the drug per unit time. Similarly, the CLR is defined as the volume of the blood or plasma which is completely cleared from the drug per unit time by the kidney. As mentioned previously, the drug renal excretion rate is determined by: ∆A e = ke A average = ke A t-mid ∆t 

(11.10)

Renal Drug Excretion 197 where Aaverage is the average amount of the drug during the urine collection interval, which is approximately equal to At-mid, the amount of drug in the body at the mid-point of the urine collection interval. Since the amount of the drug is the product of the drug concentration and Vd, Eqs. 11.11–11.13 can be obtained. ∆A e = ke VdCpt-mid ∆t 

(11.11)

∆A e ∆t = k Vd e Cpt-mid 

(11.12)

∆A e ∆t = Renal clearance Cpt-mid 

(11.13)

and

or

where Cpt-mid is the plasma drug concentration at the mid-point of the urine collection interval. This means that the drug CLR can be determined by collecting urine over a certain time interval and also obtaining a plasma sample at the middle of the urine collection interval. The drug renal excretion rate is determined as mentioned previously and the CLR is calculated from the drug renal excretion rate and the plasma drug concentration as in Eq. 11.13. Multiple urine collections and plasma samples at the middle of each urine collection can be obtained after a single drug administration to calculate the CLR during the different intervals. When the drug elimination follows first-order kinetics, the drug CLR estimated during the different intervals should be similar (2). Eq. 11.11 is an equation of a straight line that passes through the origin. A plot of the drug renal excretion rate versus the plasma drug concentration at tmid on the linear scale is a straight line that passes through the origin and the slope of the line is equal to the renal clearance as in Figure 11.3. This graphical method requires multiple urine and plasma samples and can be used to obtain an accurate estimate for the CLR. Clinical Importance:

• Determination of the CLR of a drug is important to determine the contribution of the renal excretion pathway to the overall elimination of the drug.

• The estimate for the CLR of a drug can suggest the mechanism of renal excretion for

some drugs. Drugs that are not bound to plasma protein and are filtered in the glomeruli and neither secreted not reabsorbed should have CLR equals to the GFR. When the estimated drug CLR is larger than the average GFR, this means that the drug is actively secreted in the renal tubules. When the drug CLR is lower than the average GFR, this means that the drug is filtered in the glomeruli and then reabsorbed in the renal tubules, or the drug is filtered in the glomeruli, actively secreted, and then reabsorbed in the renal tubules.

198  Renal Drug Excretion

Figure 11.3 The drug renal excretion rate versus plasma drug concentration at the middle of the urine collection interval plot. The renal clearance is determined from the slope of the line.

• Creatinine is an endogenous compound that is completely eliminated from the body

by the kidney. It is filtered in the glomeruli and neither secreted nor reabsorbed significantly in the renal tubules. So, the renal clearance of creatinine represents an accurate measure of the GFR that is correlated with the number of functioning nephrons and the kidney function. The renal clearance of creatinine that is known as the creatinine clearance is used as a diagnostic test to evaluate the kidney function. • Creatinine clearance is determined by collecting urine over a period of 24 hr. The average creatinine renal excreted rate is calculated from the creatinine concentration in urine sample, the volume of urine sample, and the collection interval. Then the creatinine clearance is estimated from the renal excretion rate and one serum creatinine concentration obtained at any time during the urine collection interval because the plasma creatinine concentration is relatively constant within an individual. Practice Problems: a  Question: Urine was collected in a patient over a 24-hr period to determine the creatinine clearance. The total volume of urine collected was 1600 mL and the creatinine concentration in urine was 0.25 mg/mL. If the serum creatinine determined in this patient was 1.4 mg/dL (1 mg/100 mL).

• Calculate the total amount of creatinine excreted during the 24 hr. • Calculate the average renal excretion rate of creatinine during the 24 hr. • Calculate the creatinine clearance in this patient. Answer:

• Amount of creatinine = Volume of sample × creatinine concentration Creatinine amount = 1600 mL × 0.25 mg/mL = 400 mg

• Creatinine renal excretion rate = Amount = Time

400 mg = 0.2778mg/min 24 hr  ×  60 min/hr

Renal Drug Excretion 199

• Creatinine clearance =

Renal Exc Rate 0.2778mg/min = = 19.8mL/min S.Creatinine 1.4 mg/100 mL

11.6  The Cumulative Amount of the Drug Excreted in Urine After single IV drug administration, the renal excretion rate of the drug represents the average rate of drug excretion in urine during a certain time interval. While the cumulative amount of the drug excreted in urine represents the total amount of the drug excreted in urine from the time of drug administration up to a certain time. It is calculated by adding the amount of the drug excreted in urine during the different urine collection intervals. When collection of urine continues until all the administered drug is eliminated from the body, the cumulative amount of the drug excreted in urine increases exponentially until it reaches a plateau as in Figure 11.4. The drug renal excretion rate at any time can be described by Eq. 11.9. The total amount of the drug excreted in urine is determined by integrating this equation and substituting for time by infinity to obtain Eq. 11.14. Ae ∞ =  

ke k A o =   e Dose IV k k 

(11.14)

where Ae∞ is the total amount of the drug excreted in urine when all the drug is eliminated from the body. This means that after an IV dose of a drug the total amount of the drug excreted unchanged in urine is a fraction of the administered dose. This fraction (f) is equal to the ratio of ke to k, and the ratio of CLR to CLT as in Eqs. 11.15 and 11.16. Ae ∞ =

Renal clearance ke Vd Dose IV =   Dose IV Total body clearance k Vd 

(11.15)

and f = 

Ae ∞ k CL R =  e = Dose IV k CL T 

(11.16)

Figure 11.4 The cumulative amount of the drug excreted in urine after single IV dose increases exponentially until it reaches a plateau. The plateau represents the total amount of the drug excreted in urine, Ae∞.

200  Renal Drug Excretion When the entire dose of the drug is excreted unchanged in urine, this means that CLR is equal to CLT and ke is equal to k. When no drug is excreted unchanged in urine, the CLR and ke will be equal to zero. The same principle can be applied for the renal drug excretion after extravascular drug administration. However, for extravascular drug administration, the fraction of the bioavailable dose, which is excreted unchanged in urine, f is equal to the ratio of the ke to k, and the ratio of CLR to CLT as in Eq. 11.17. f = 

A e ∞ ke CL R = = FD k CL T 

(11.17)

Rearrangement of Eq. 11.17, and because FD/CLT is equal to the AUC, Eq. 11.18 can be obtained. CL R = CL T

Ae ∞ Ae ∞ = FD AUC |o∞ 

(11.18)

This means that the renal clearance can be determined from the cumulative amount of the drug excreted unchanged in urine and the drug AUC (2). This relationship can be applied for both IV and extravascular drug administration since the drug bioavailability affects both the amount excreted unchanged in urine and the AUC to the same extent. Clinical Importance:

• The fraction of the bioavailable dose excreted unchanged in urine is constant for the

same drug in the same patient. So the cumulative amount of the drug excreted unchanged in urine after administration of the same dose of different products for the same drug can be used to compare the drug bioavailability after administration of different products and to assess the bioequivalence of different products. • The absolute bioavailability of a drug can be determined from the ratio of the cumulative amount of the drug excreted unchanged in urine after oral and IV administration of the same dose of the same drug. Fabsolute =

A e∞ oral A e∞ IV

(11.19) 

• Also, the relative bioavailability of different drug products for the same active drug can be determined from the ratio of the cumulative amount of the drug excreted unchanged in urine after administration of the test and standard drug products (3). Frelative =

A e ∞ test A e ∞ standard

(11.20) 

Renal Drug Excretion 201 11.7 Determination of the Pharmacokinetic Parameters from the Renal Excretion Rate Data 11.7.1  The Elimination Rate Constant and Half-Life

The elimination rate constant of the drug is determined from the plot of the drug renal excretion rate versus time after single IV administration on the semilog scale. The slope of the resulting straight line is equal to −k/2.303. The half-life of the drug can be determined graphically from the same plot by determining the time required for the renal excretion rate at any time to decrease by 50% as in Figure 11.2. 11.7.2  The Renal Excretion Rate Constant

The renal excretion rate constant, ke, can be determined from the y-intercept of the renal excretion rate versus time plot on the semilog scale, which is equal to ke A0. Since A0 is equal to the dose, ke can be determined. Also, ke can be determined if both CLR and Vd are known (CLR = ke Vd). Furthermore, ke can be determined if the fraction of dose excreted unchanged in urine after IV administration, f and k are known (f = ke/k). 11.7.3  The Volume of Distribution

The Vd of the drug cannot be determined from the renal excretion rate versus time plot. Plasma samples after drug administration are required to estimate Vd. The Vd can be determined if CLR and ke are known (CLR = ke Vd). 11.7.4  The Renal Clearance

The CLR can be determined from ke and Vd (CLR = ke Vd). Also, the CLR during each urine collection interval is determined from the ratio of the renal excretion rate and the average plasma concentration during the urine collection interval. When serial urine and plasma samples are obtained, CLR can be determined from the slope of the plot of the renal excretion rate versus the plasma concentration at the mid-point of the urine collection interval on the linear scale. Also, the CLR can be determined from the total amount of the drug excreted unchanged in urine and the drug AUC (CLR = Ae∞/AUC). Furthermore, the CLR can be calculated if the fraction of the IV dose excreted unchanged in urine and the CLT are known (f = CLR/CLT). 11.7.5  The Fraction of Dose Excreted Unchanged in Urine

The fraction of the bioavailable dose that is excreted unchanged in urine is determined from the ratios of ke/k, CLR/CLT, or Ae∞/FDose. 11.7.6 Bioavailability

The drug absolute bioavailability is determined from the ratio of Ae∞ after oral and IV administration of the same dose of the same drug (F = Ae∞ oral/Ae∞ IV). While the relative bioavailability is determined from the ratio of Ae∞ after administration of the same dose of the test and reference products for the same drug.

202  Renal Drug Excretion Practice Problems: a  Question: After single IV dose of 500 mg of a new drug to a patient, the following data were obtained: Collection interval (hr)

Urine volume (mL) Urine concentration Cpt mid (mg/L) (mg/mL)

0–2 2–4 4–8 8–12 12–18 18–24

119 81 160 220 284 212

0.60 0.70 0.50 0.23 0.15 0.10

22.3 17.7 12.5 7.88 4.42 2.21

• Estimate the biological half-life of this drug in this patient using the urinary excretion data. • Estimate the renal clearance of this drug in this patient. • Calculate the fraction of the administered dose excreted unchanged in the urine from the available data.

Answer:

• Use the available information to calculate the renal excretion rate as follows: Collection Urine volume Urine Amount Renal Cpt mid (mg/L) interval (hr) (mL) concentration excreted excretion rate (mg/mL) (mg/L) (mg/hr) 0–2 2–4 4–8 8–12 12–18 18–24

119 81 160 220 284 212

0.60 0.70 0.50 0.23 0.15 0.10

71.4 56.7 80.0 50.6 42.6 21.2

35.7 28.35 20.00 12.65 7.10 3.53

22.3 17.7 12.5 7.88 4.42 2.21

• The half-life can be estimated from the renal excretion rate versus time plot as in Figure 11.5.

The half-life is = 6 hr

k = 0.1155hr −1

The elimination rate constant can also be determined from the slope of the line.

• The CLR can be estimated from the slope of the renal excretion rate versus plasma concentration plot as in Figure 11.6. The renal clearance = 1.6L/hr

• The fraction excreted unchanged in urine can be calculated from the ratio of the (ke/k) or (CLR/CLT).

Renal Drug Excretion 203

Figure 11.5 The half-life and the elimination rate constant are determined from the renal excretion rate versus time plot.

The plasma concentration-time data can be used to determine the CLT. Plot the plasma concentration-time data and calculate the Vd, k, and CLT. CL T = 2.31L /hr Fraction =

CL R 1.6 L/hr = = 0.69 CL T 2.31 L/hr

Also, the renal excretion rate constant can be calculated from the y-intercept of the renal excretion rate-time profile.  y-intercept = ke Dose = ke 500mg = 40mg/hr ke = 0.08hr −1 Fraction =

ke 0.08hr −1 = = 0.69 k 0.1155hr −1

Figure 11.6 The renal excretion rate versus drug plasma concentration at the middle of the urine collection interval.

204  Renal Drug Excretion 11.8 The Effect of Changing the Pharmacokinetic Parameters on the Urinary Excretion of Drugs The renal excretion rate-time profile of a drug after single IV dose declines at a rate dependent on the drug elimination rate constant. The y-intercept for the drug renal excretion rate-time profile on the semilog scale is equal to ke Dose. While the fraction of the IV dose excreted unchanged in urine is determined from the ratio of ke/k and CLR/CLT. 11.8.1 Dose

The effect of changing the dose of the drug:

• After administration of increasing doses of the drug, the drug renal excretion rate-time

plots on the semilog scale will be parallel, i.e., decline at the same rate when the drug elimination processes follow first-order kinetics. This is because k, ke, CLT, and CLR are dose independent. • In this case, administration of increasing doses of the drug results in proportional increase in the cumulative amount of the drug excreted unchanged in urine because the fraction of the dose excreted unchanged in urine is constant. Clinical Importance:

• The cumulative amount of the drug excreted unchanged in urine can be used to assess the bioavailability and bioequivalence of oral drug products.

11.8.2  The Total Body Clearance

Here it is assumed that the CLR does not change, the change in the drug nonrenal clearance does not affect the renal clearance, and the change in the CLT is due to change in the nonrenal clearance.

• The decrease in CLT results in prolongation of the drug half-life which is manifested

by the slower rate of decline of the renal excretion rate-time plot on the semilog scale, when Vd is the same. • The change in CLT because of the change in nonrenal clearance depends on the magnitude of the change in nonrenal clearance and the contribution of the nonrenal clearance to CLT. • Although the CLR does not change, the fraction of the administered dose excreted unchanged in urine will be higher when the nonrenal clearance is lower. This is because the decrease in the nonrenal clearance will increase in the contribution of the CLR to the CLT. Clinical Importance:

• The metabolic clearance and CLT can be reduced without affecting the drug CLR in case of patients with hepatic dysfunction and drug interactions causing inhibition of the drug-metabolizing enzymes. • Patients with reduced CLT usually require lower than average doses of the drugs to produce the desired effect, with the reduction in dose depending on the reduction in CLT.

Renal Drug Excretion 205

• Drug-drug interactions causing metabolizing enzyme induction can increase CLT with-

out affecting CLR which increases the dosage requirements of patients and decrease the cumulative drug excretion in urine.

11.8.3  The Renal Clearance

Here it is assumed that the CLR changes without affecting the drug nonrenal clearance. The CLT will change because it is the sum of all eliminating organ clearances.

• The decrease in CLR and the accompanied decrease in CLT result in prolongation of

the drug half-life which is manifested by the slower rate of decline of the renal excretion rate-time plot on the semilog scale, when Vd is the same. • The decrease in drug CLR as in patients with renal dysfunction usually causes less than proportional decrease in the CLT, unless the drug is excreted completely unchanged in urine. The change in CLT depends on the magnitude of the change in CLR and the contribution of the CLR to CLT. Clinical Importance:

• Patients with renal dysfunction usually have lower CLT and require less that average

dose of the drugs. The reduction in dose depends on the degree of kidney dysfunction and the fraction of the drug dose excreted unchanged in urine. • The fraction of the administered dose excreted unchanged in urine will be lower in patients with renal dysfunction, because CLR will represent a smaller fraction of the drug CLT. 11.9 Summary

• Most drugs are eliminated from the body by multiple elimination pathways, with the

pharmacokinetic parameters k and CLT for the drugs equal to the sum of the rate constants and clearances for all elimination pathways. • Studying the renal excretion of drugs by collecting urine samples over predetermined urine collection intervals after drug administration can be used to study drug pharmacokinetics and obtain estimates for the drug pharmacokinetic parameters. • Determination of the renal clearance of endogenous markers such as creatinine is used to assess the excretory function of the kidney. • Patients with reduced kidney function have lower renal clearances for drugs that are excreted by the kidney and require lower than the average doses to achieve the desired drug therapeutic effect. Practice Problems 11.1 The urine was collected in a patient over 24-hr period to determine the creatinine clearance. The total volume of urine collected was 1800 mL and the creatinine concentration in urine was 0.4 mg/mL. If the serum creatinine determined in this patient was 1 mg/dL (1 mg/100 mL). a Calculate the amount of creatinine excreted (eliminated) during the 24 hr. b Calculate the average renal excretion rate of creatinine during the 24-hr period.

206  Renal Drug Excretion c Calculate the creatinine clearance in this patient. d Calculate the renal clearance of creatinine in this patient. 11.2 A patient received a single 1000-mg IV dose of an antibiotic. Urine and plasma samples were collected, and the following results were obtained: Interval (hr) Urine volume (mL) Urine conc (µg/mL) Plasma conc (Cpt mid) (µg/mL) 0–1 1–2 2–4 4–8

67 70 100 250

2.1 1.01 0.5 0.05

Not determined Not determined 0.5 µg/mL at time 3 hr Very low

a Calculate the average renal excretion rate of the drug during the first urine collection interval (0–1 hr). b Calculate the average renal excretion rate of the drug during the third urine collection interval (2–4 hr). c Calculate the renal clearance of this drug in this patient. d What is the slope of the renal excretion rate versus time plot on a semilog graph paper? 11.3 A new antihypertensive drug is rapidly but incompletely absorbed. • When 800 mg is given intravenously to normal volunteers: Only 800 mg is recovered in urine unchanged from time 0 to ∞. The AUC obtained from time zero to infinity is 400 mg hr/L. • When 400 mg is given orally: Only 350 mg of the drug is recovered unchanged in urine from time 0 to ∞. The elimination half-life is 8 hr. a What is the bioavailability of this drug after oral administration? 11.4 After an IV injection of 10 mg of a new drug to a patient, the following data were obtained: Collection interval (day) Urine volume (mL) Urinary drug conc. (μg/mL) Cpt mid (μg/L) 0–1 1–2 2–3 3–4 4–5 5–7

1250 1500 1750 1380 1630 3130

1.8 0.984 0.544 0.448 0.248 0.136

16.0 10.4 6.8 4.4 2.9 1.5

a Using a graphical method, estimate the biological half-life of this drug in this patient. b Calculate the renal clearance and the total body clearance of this drug in this patient. c Calculate the fraction of the administered dose excreted unchanged in the urine from the above data. d Estimate how much drug is in the body 5 days after the dose was administered. e Assuming that the unexcreted portion of this drug is metabolized, determine its metabolic rate constant (km) in this patient.

Renal Drug Excretion 207 11.5 After an IV injection of 1000 mg of a new drug to a patient, the following data were obtained: Collection interval (hr) Urine volume (mL) Urinary conc (mg/mL) Cpt mid (mg/L) 0–2 2–4 4–6 6–8 8–10 10–14

125 150 175 138 163 313

1.776 0.933 0.504 0.403 0.215 0.112

15.9 10.0 6.3 4.97 2.51 1.24

a Using a graphical method, estimate the biological half-life of this drug in this patient. b Calculate the renal clearance of this drug in this patient. c Calculate the fraction of the administered dose excreted unchanged in the urine from the above data. d Assuming that the unexcreted portion of this drug is metabolized, determine its metabolic rate constant (km) in this patient. References 1. Cafruny EJ “Renal tubular handling of drugs” (1977) Am J Med; 62:490–496. 2. Tucker GT “Measurement of the renal clearance of drugs” (1981) Br J Clin Pharm; 12:761–770. 3. U.S. Code of Federal Regulations (April 2010), Title 21-Food and Drugs, Chapter I, Food and Drug Administration, Department of Health and Human Services, Sub-chapter D, Drugs for human use, Part 320, Bioavailability and Bioequivalence Requirements, Sec. 24.

12 Metabolite Pharmacokinetics

Objectives After completing this chapter, you should be able to:

• Describe the characteristics of the different metabolic pathways involved in the formation of the metabolites after drug administration.

• Compare the models that can be used to quantitate the formation and elimination of metabolites after the administration of the parent drug.

• Discuss the importance of studying metabolite pharmacokinetics after single and multiple drug administration.

• Discuss the factors that affect the rate and amount of metabolite formation after the administration of the parent drug.

• Apply the different pharmacokinetic equations to solve problems that involve the calculation of the metabolite pharmacokinetic parameters.

• Analyze the effect of changing each of the drug and metabolite pharmacokinetic param-

eters on the drug and metabolite profiles after single and multiple drug administration.

12.1 Introduction The elimination of drugs includes the excretion of the unchanged drug from the body by different routes of excretion and drug metabolism via different metabolic pathways. Drug metabolism involves the modification of the drug’s chemical structure, usually by specialized enzyme systems, to form a new chemical entity called the metabolite. The metabolites have pharmacokinetic behavior different from that of the parent drugs because of the differences in the chemical structure. Most metabolites are pharmacologically inactive; however, some metabolites possess pharmacological and/or toxicological activities. So, studying the factors that affect the metabolite formation and elimination is necessary for accurate prediction of the therapeutic and the adverse effects produced by the parent drug and its metabolite(s). Drug-drug interactions that involve induction or inhibition of the drug-metabolizing enzymes can affect the rate of drug metabolism. Also, many nutritional, environmental, and genetic factors in addition to alcohol and smoking can affect the activity of the drug-metabolizing enzymes. So, alteration of the drug metabolic rate is one of the important mechanisms by which clinically significant drug-drug interactions occur.

DOI: 10.4324/9781003161523-12

Metabolite Pharmacokinetics 209 12.2  Drug Metabolism Drugs can be metabolized by various enzyme systems through different metabolic pathways, which can be classified to phases I and II metabolic reactions (1). Phase I metabolic reactions include oxidation, reduction, and hydrolysis leading to the introduction of polar function groups to form more polar metabolites, while phase II metabolic reactions involve conjugation between the drug/metabolite and other compounds such as glucuronic acid, glutathione, and amino acids. Phase I metabolic reactions do not have to occur before phase II reactions because some drugs are metabolized directly by phase II reactions, and a few drugs are metabolized by phase II reactions followed by phase I reactions. 12.2.1  Metabolizing Enzymes

The cytochrome P450 (CYPs) is a superfamily of metabolizing enzymes that are responsible for approximately 75% of the total drug metabolism in humans (2, 3). The CYPs-metabolizing enzymes are present in many organs, including liver, kidney, lung, gastrointestinal tract, brain, nasal mucosa, and skin. However, the importance of the role of hepatic CYPs enzymes in drug metabolism arises from the existence of a large number of enzymes in the liver and the high hepatic blood flow that exposes a large amount of the drug in the systemic circulation to the hepatic CYPs enzymes. Also, orally administered drugs are usually absorbed into the portal vein and have to pass through the liver before reaching the systemic circulation, which exposes orally administered drugs to the intestinal and hepatic metabolizing enzymes. Since the drugs have to cross the cell membrane to come in contact with the metabolizing enzymes, lipophilic drugs are usually eliminated by metabolism, while hydrophilic drugs, which cannot penetrate the cell membrane, are mainly excreted unchanged from the body. Enzyme systems other than CYPs that are involved in drug metabolism include the flavin-containing monooxygenases, dehydrogenases, reductases, and hydrolases. This is in addition to the enzymes involved in conjugation reactions such as methyltransferases, glutathione S-transferases, sulfotransferases, N-acetyltransferases, amino acid N-acyl transferases, and UDP-glucuronosyltransferases. The different drug-metabolizing enzymes have different specificities for the substrates and the metabolic reaction that they can catalyze. So, identification of the specific enzymes involved in the metabolism of a new drug is important in identifying potential drug-drug interactions for that drug. Also, the capacity of the different metabolic pathways, which is dependent on the expression of the different metabolizing enzymes, can vary leading to a variation in the drug metabolic rate and the overall drug elimination rate in different individuals. Clinical Importance:

• The metabolizing enzymes can be induced by other drugs or environmental factors lead-

ing to an increase in the activity of the enzymes and an increase in the rate of drug metabolism through the pathway catalyzed by that enzyme. If the induced metabolic pathway is a major elimination pathway for the drug, the drug clearance and the drug elimination rate will increase leading to therapeutic failure if the drug dose was not adjusted. • The metabolizing enzymes can be inhibited by drugs, and their activity can be reduced in diseases such as in liver diseases. When the drug is eliminated mainly by metabolism, unexpected drug toxicity may occur in patients with reduced liver function and patients taking enzyme inhibitors, if the drug dose was not adjusted.

210  Metabolite Pharmacokinetics

• Variation in the metabolizing enzyme activity due to genetic factors can lead to a sig-

nificant variation in the rate of drug metabolism between individuals. This can lead to a significant variation in drug response in different patients after the administration of the same dose of the same drug.

12.2.2  Formation of Active Metabolites

Drug metabolism is usually viewed as an elimination pathway for drugs since the formed metabolites in most cases are pharmacologically inactive. However, in some cases, the formed metabolites possess pharmacological activity, which can be similar or different from that of the drug. For these drugs, evaluation of the observed therapeutic effect should include the effect of the drug and the effect of the pharmacologically active metabolite. Also, the factors that can increase or decrease the active metabolite formation should be determined. Clinical Importance:

• The antiarrhythmic drug procainamide (PA) is partially metabolized to n-acetyl pro-

cainamide (NAPA) that also possesses antiarrhythmic activity that contributes to the therapeutic effect observed in patients taking PA. Monitoring PA therapy requires monitoring of the plasma drug and metabolite concentrations. • The metabolism of several psychotropic drugs results in the formation of active metabolites, such as nortriptyline, the metabolite of amitriptyline; nordoxepin, the metabolite of doxepin; desipramine, the metabolite of imipramine, norfluoxetine, the metabolite of fluoxetine, and 9-hydroxy-risperidone, the metabolite of risperidone. Evaluation of the observed therapeutic effect after the administration of these drugs should consider the differences in plasma concentration, CNS distribution, and potency of the parent drugs and their metabolites. • Codeine is used as an antitussive in cough suppressant products. It is also used in analgesic products, because 5–10% of codeine dose is metabolized to morphine, which has a strong analgesic effect. Genetic variability can significantly affect the amount of morphine formed in vivo and the observed analgesic effect after codeine administration. 12.2.3  Formation of Toxic Metabolites

Drug metabolism sometimes results in the formation of toxic metabolites that can contribute to the drug’s adverse effects. For these drugs, it is important to monitor the accumulation of this toxic metabolite in the body if the drug will be used for a period of time. Also, the factors that can affect the amount of this toxic metabolite after the administration of the drug should be determined. Clinical Importance:

• The major metabolite of the antiepileptic drug carbamazepine (CBZ) is carbamaz-

epine epoxide that has antiepileptic effect and contributes to CBZ toxicity. Monitoring the patients taking CBZ requires monitoring of the plasma drug concentration of CBZ and its epoxide metabolite to avoid toxicity. The therapeutic range of CBZ is 4–12 μg/mL and the suggested acceptable range of the epoxide metabolite concentration is between 0.4 and 4 μg/mL.

Metabolite Pharmacokinetics 211

• Cocaethylene is one of the cocaine metabolites formed only when cocaine is abused

simultaneously with alcohol. This metabolite has CNS stimulant activity similar in potency to that of cocaine and can cause serious cardiovascular complications. This metabolite has been implicated with the development of cocaine toxicity because it was detected in the blood of most of the patients presented to the emergency room with cocaine overdose.

12.2.4  Metabolic Activation of Prodrugs

Prodrugs are pharmacologically inactive chemical derivatives of the active drug that can undergo enzymatic metabolism inside the body, liberating the active drug moiety. Lipophilic prodrugs are used to improve the absorption and tissue distribution of highly hydrophilic pharmacologically active compounds. Also, chemically stable prodrugs are used to improve the stability of active compounds during manufacturing, storage, and administration, then undergo enzymatic metabolism inside the body to liberate the active drug moiety. Clinical Importance:

• Enalapril is an oral prodrug for the ACE inhibitor enalaprilat, which is usually ad-

ministered intravenously because of its poor absorption. Once absorbed, enalapril is metabolized by the esterase enzymes liberating the active moiety enalaprilat. • Simvastatin is a statin prodrug with a six-membered lactone ring, which is rapidly hydrolyzed in vivo to generate the active metabolite. Most of simvastatin activation occurs during the absorption process producing the active metabolite that inhibits HMG-CoA reductase producing its antilipidemic effect. • L-dopa is a prodrug for dopamine that is used for the treatment of Parkinson’s disease. Unlike dopamine, after oral administration, L-dopa is absorbed and can be distributed to the CNS. In the brain, L-dopa is decarboxylated by the enzyme dopa-decarboxylase leading to the formation of dopamine. 12.3  Metabolite Pharmacokinetics Drugs can be metabolized through different metabolic pathways to one or more metabolites that can be detected simultaneously in the body as in Figure 12.1. Some drugs are metabolized to different metabolites through parallel metabolic pathways, then the formed metabolites are eliminated from the body. Other drugs are metabolized to one metabolite that is then metabolized to another metabolite that can be eliminated from the body as illustrated in Figure 12.2. When a drug is eliminated via parallel metabolic pathways, the amount of the drug metabolized through each pathway is proportional to the clearance (and the rate constant) associated with each metabolic pathway, assuming first-order elimination. Taking the parallel metabolism example in Figure 12.2, the drug CLT is the sum of the metabolic clearances responsible for the formation of metabolite 1, metabolite 2, and the renal clearance of the drug as in Eq. 12.1. CL T = CL m1 + CL m2 + CL R (12.1)

212  Metabolite Pharmacokinetics

Figure 12.1 The plasma drug and metabolites concentration-time profiles after single IV administration of the drug, which is metabolized to two different metabolites.

Also, the overall elimination rate constant, k, for the drug is the sum of the rate constants associated with the three pathways as in Eq. 12.2. k = km1 + km2 + ke(12.2) The fraction of the drug dose metabolized to metabolite 1, fm1 can be determined from the ratio of the metabolic rate constant for the formation of metabolite 1 to the overall elimination rate constant as in Eq. 12.3. fm1 =

km1 k = m1 (12.3) km1 + km2 + ke k

Similarly, Eq. 12.4 can be used to determine the fraction of drug dose metabolized to metabolite 2, fm2. fm2 =

km2 k = m2 (12.4) km1 + km2 + ke k

Figure 12.2 Schematic presentation of the parallel and sequential drug metabolism. The rate constants km1 and km2 are the drug metabolic rate constant for the formation of metabolite 1 and metabolite 2, respectively, ke is the drug renal excretion rate constant, and km2(m1) is the metabolic rate constant for metabolite 1 that is responsible for the formation of metabolite 2 in the sequential metabolism example.

Metabolite Pharmacokinetics 213 Also, the fraction of drug dose excreted unchanged in urine, f can be determined from Eq. 12.5. f=

ke k = e (12.5) km1 + km2 + ke k

Clinical Importance:

• Enzyme induction and enzyme inhibition can affect a specific metabolic pathway, leading

to a change in the rate constant associated with this metabolic pathway. The effect on the overall elimination rate depends on the contribution of the affected pathway to the overall drug elimination rate. Also, the fraction of drug dose eliminated by the different pathways will be affected and will depend on the ratio of the elimination rate constant of each pathway to the overall elimination rate constant in the induction or inhibition state. • The overall rate of drug elimination rate, and the fraction of drug dose eliminated through each pathway, can change because of the change in the function of the eliminating organ responsible for drug elimination through one of the pathways. The magnitude of the change depends on the contribution of the affected pathway to the overall drug elimination. • For example, 70% of digoxin dose is excreted unchanged in urine and the rest of the dose is eliminated by hepatic metabolism. If the kidney function decreases by 50% and if it is assumed that the decrease in kidney function does not affect digoxin hepatic metabolism, the total body clearance of digoxin will decrease by 35% (50% of the renal clearance) to become 65% of the original total body clearance. After the decrease in kidney function, (35/65) 53.8% of digoxin dose will be excreted unchanged in urine and (30/65) 46.2% of the dose will be eliminated by hepatic metabolism. 12.4  A Simple Model for Metabolite Pharmacokinetics To differentiate between the pharmacokinetic parameters of the drug from those of the metabolite in this chapter, the metabolite parameters are designated by the subscript letter “m” in brackets. For example, Cp, CLT, Vd, k, ke, km, t1/2, and AUC are the parameters for the drug, while Cp(m), CLT(m), Vd(m), k(m), ke(m), km(m), t1/2(m), and AUC(m) are the parameters for the metabolite. The simple pharmacokinetic model for drug metabolism described by the scheme in Figure 12.3 has the following assumptions (4):

• • • •

The drug is administered as single IV bolus dose. The drug is completely metabolized to one metabolite by a first-order process. The metabolite is completely excreted in urine by a first-order process. The metabolite is not converted back to the parent drug.

Figure 12.3 Schematic presentation of a simple pharmacokinetic model for drug metabolism. In this model, A is the amount of drug in the body, k is the first-order elimination rate constant for the drug, A(m) is the amount of metabolite, k(m) is the first-order elimination rate constant for the metabolite, and Ae(m) is the amount of the metabolite excreted from the body.

214  Metabolite Pharmacokinetics

• The metabolite and the drug follow one-compartment pharmacokinetic model. • The amount and concentration of the drug and the metabolite are expressed in moles and moles/volume to accommodate the difference in molecular weight of the drug and the metabolite.

After single IV dose of the drug, the rate of drug elimination is equal to the rate of formation of the metabolite because the drug is eliminated by one elimination pathway. The rate of metabolite formation and elimination can be expressed as follows: The rate of metabolite formation = Ak The rate of metabolite elimination = A (m) k(m) So, the rate of change of the amount of metabolite in the body presented in Eq. 12.6 is the difference between the rate of metabolite formation and the rate of metabolite elimination. dA(m) = Ak − A(m) k(m) (12.6) dt Integration of Eq. 12.6 gives Eq. 12.7. A(m) =

kD (e− kt − e− k(m)t )(12.7) k(m) − k

Dividing Eq. 12.7 by the volume of distribution for the metabolite, Vd(m), Eq. 12.8, which describes the metabolite concentration-time profile at any time after the administration of single IV dose of the drug, is obtained. Cp(m) =

kD (e− kt − e− k(m)t )(12.8) Vd(m) (k(m) − k)

Equation 12.8 describes the formation and the elimination of the metabolite. The two exponential terms in this equation include the drug elimination rate constant that is the same as the metabolite formation rate constant in the model, and the metabolite elimination rate constant. At the time of drug administration, there is no metabolite in the body and Cp(m) is equal to zero. Then Cp(m) starts to increase because the rate of metabolite formation is more than the rate of its elimination. The maximum metabolite plasma concentration, Cpmax(m), is achieved when the rate of metabolite formation becomes equal to the rate of its elimination. The time when Cpmax(m) is achieved is called tmax(m). After tmax(m), the rate of metabolite elimination is higher than the rate of metabolite formation and the metabolite plasma concentration declines. An example of the plasma drug and metabolite concentration-time profiles after IV administration of single dose of the drug is illustrated in Figure 12.4. The time required to reach the maximum metabolite concentration is dependent on the rate constant for the elimination of the drug and the metabolite as in Eq. 12.9: t max(m) =

ln(k(m) /k) (12.9) k(m) − k

Metabolite Pharmacokinetics 215

Figure 12.4 The plasma drug and metabolite concentration-time profiles after IV administration of single dose of the drug.

The maximum metabolite concentration, Cpmax(m), is determined by substitution for tmax(m) in Eq. 12.8. 12.4.1  Metabolite Concentration-Time Profile

After drug administration, the formed metabolite has pharmacokinetic parameters that are different from those of the parent drug. In the simple model described previously, although the molar amount of the metabolite formed in vivo is equal to the molar amount of the drug administered, the observed AUCs for the drug and the metabolite are always different. The AUC of any compound is dependent on the amount of compound in the systemic circulation and the total body clearance of that compound. So, the metabolite plasma concentrations can be higher or lower than those for the drug depending on the clearances of the drug and the metabolite. However, the decline in the terminal metabolite concentration-time profile after the administration of the drug depends on the magnitude of the metabolite elimination rate constant relative to the drug elimination rate constant (4). When k(m) < k: This means that the metabolite is eliminated at a slower rate compared to that for the parent drug, and the half-life of the metabolite is longer than the half-life of the drug. After single IV dose of the drug, the slope of the terminal phase of the metabolite concentration-time profile on the semilog scale reflects the elimination rate of the metabolite and is equal to −k(m)/2.303, as in Figure 12.5. The profile of the metabolite in this case is described as elimination rate limited. When k(m) > k: This is the situation when the metabolite is eliminated faster than the parent drug, and the half-life of the metabolite is shorter than the half-life of the drug. After IV bolus dose of the drug, the slope of the decline phase of the metabolite concentration-time profile on the semilog scale is like that of the drug as in Figure 12.6. The slope of the decline phase of the metabolite concentration-time profile does not reflect the rate of metabolite elimination, but it reflects the rate of drug elimination and is equal to −k/2.303. The profile of the metabolite in this case is described as formation rate limited.

216  Metabolite Pharmacokinetics

Figure 12.5 The plasma drug and metabolite concentration-time profiles after IV administration of single dose of the drug when the metabolite elimination rate constant is smaller than the drug elimination rate constant.

Clinical Importance:

• The screening tests for drugs of abuse in biological samples depend on the detection of the metabolites of illicit drugs with long half-lives. These metabolites can be detected for some time after the complete elimination of the drug. Benzoylecgonine, the metabolite of cocaine, can be detected in urine for up to two to four days after cocaine use. Also, the metabolite delta-9-tetrahydrocannabinol-9-carboxylic acid can be detected in urine for up to 3 days after a single marijuana use and for up to 10–15 days after daily use.

12.5  The General Model for Metabolite Kinetics The general model for drug metabolism is described by the scheme in Figure 12.7 (4). The difference between the general model and the simple model presented previously is that the general model can be applied for most drugs when only a fraction of the

Figure 12.6 The drug and metabolite concentration-time profiles after IV administration of single dose of the drug when the metabolite elimination rate constant is larger than the drug elimination rate constant.

Metabolite Pharmacokinetics 217

Figure 12.7 Schematic presentation of the general model for drug metabolism. Cp and Cp(m) are the plasma drug concentration and metabolite concentration, A and A(m) are the amount of the drug and metabolite, k is the first-order elimination rate constant for the drug, fm is the fraction of the drug dose that is metabolized to the metabolite of interest, (1 − fm) is fraction of the drug dose that is eliminated by the other pathways, Vd and Vd(m) are the volume of distribution of the drug and the metabolite, k(m) is the first-order elimination rate constant for the metabolite, Ax is the amount of the parent drug eliminated by pathways other than metabolism, and Ax(m) is the amount of metabolite eliminated from the body by different pathways.

drug dose is metabolized to a metabolite and the remaining of the dose is eliminated by other elimination pathways. In this model, drug metabolism is only a fraction of the overall drug elimination. So, the rate of metabolite formation represents a fraction (fm k) of the total rate of drug elimination. Using an approach similar to that used in the simple model, Eq. 12.10, which describes the metabolite concentration at any time after a single IV dose of the drug when only a fraction (fm) of drug dose is metabolized, can be obtained. Cp(m) =

fm kD (e− kt − e− k(m)t )(12.10) Vd(m) (k(m) − k)

In this general model, the metabolite formation clearance is the fraction of the drug total body clearance, which is responsible for the formation of the metabolite (fm CLT). The metabolite can have longer or shorter half-life compared to that of the drug. The metabolite can follow formation-rate-limited or elimination-rate-limited profile depending on the relationship between k(m) and k, as described previously. The metabolite area under the plasma concentration-time curve, AUC(m), is dependent on the amount of the metabolite formed in vivo, fm Dose, and the total body clearance of the metabolite, CLT(m), as in Eq. 12.11. AUC(m) =

fm D (12.11) CL T(m)

Based on Eq. 12.11, the metabolite AUC is directly proportional to the amount of metabolite formed in vivo and is inversely proportional to the metabolite clearance, while the drug AUC after IV administration of the drug is determined from the dose and CLT of the drug as in Eq. 12.12. AUC =

D (12.12) CL T

218  Metabolite Pharmacokinetics Dividing Eq. 12.11 by Eq. 12.12 gives the ratio of the AUC for the metabolite to that of the drug. Equation 12.13 indicates that the ratio of the AUC of the metabolite to that of the drug after a single IV dose of the drug is the ratio of the metabolite formation clearance to the metabolite elimination clearance. AUC(m) fm CL T = (12.13) AUC CL T(m) 12.6  Determination of the Metabolite Pharmacokinetic Parameters When the drug metabolite is pharmacologically active or can cause adverse effects, the determination of the metabolite pharmacokinetic parameters becomes very important. Parameters such as the metabolite elimination rate constant, the amount of the metabolite formed, metabolite clearance, and metabolite accumulation during multiple drug administration are very important for the characterization of the metabolite pharmacokinetics behavior. Some metabolite parameters can be determined from the metabolite profile after the administration of the drug. However, some other metabolite pharmacokinetic parameters require the administration of the preformed purified metabolite. This is possible if the metabolite is available in sufficient quantities, and it is safe to administer the preformed metabolite experimentally. 12.6.1  Metabolite Elimination Rate Constant, k(m)

When k(m) < k, the metabolite half-life (t1/2(m)) is longer than that for the drug. After IV drug administration, the terminal phase of the metabolite profile declines at a rate dependent on k(m), which can be determined from the slope of the linear part of the metabolite concentration-time profile after drug administration on the semilog scale (slope = −k(m)/2.303). Also, t1/2(m) can be determined graphically from the same plot. However, when k(m) > k and t1/2(m) < t1/2, the decline in the metabolite profile does not reflect the rate of metabolite elimination. Indirect methods can be used to determine k(m) after the administration of the drug. Alternatively, k(m) and t1/2(m) can be determined after the administration of the preformed metabolite. 12.6.2  Fraction of the Parent Drug Dose Converted to a Specific Metabolite, fm

After IV administration of the drug, AUC(m) is dependent on the amount of the metabolite formed in vivo and CLT(m) as in Eq. 12.11 (AUC(m) = (fmD/CLT(m))). Since both fm and CLT(m) are not known, the amount of metabolite formed after drug administration cannot be determined from the information obtained after the administration of the drug. So, CLT(m) should be determined after IV administration of the preformed purified metabolite as follows: AUC(m) ′ =

M (12.14) CL ′T(m)

where AUC′(m) is the metabolite AUC after metabolite administration, M is the dose of the metabolite, and CL′T(m) is the total body clearance of the metabolite after metabolite administration. If equimolar doses of the drug and metabolite are administered (i.e., D = M) and

Metabolite Pharmacokinetics 219 because the metabolite clearance should be the same after drug administration and after metabolite administration (i.e., CL′T(m) = CLT(m)), the fraction of the drug dose converted to the metabolite can be determined from the ratio of the metabolite AUC after IV administration of equimolar doses of the drug and the metabolite as in Eq. 12.15. AUC(m)   fm D CL ′T(m) = fm (12.15) = AUC(m) M CL T(m) ′ Also, urinary recovery of the metabolite after IV administration of the drug can be used to determine the fraction of drug dose converted to a particular metabolite under certain conditions. This is possible only when the formed metabolite is completely excreted unchanged in urine without undergoing sequential metabolism, the metabolite is stable in urine after urinary excretion, and urine is collected completely until all the drug and metabolite are completely eliminated. 12.6.3  Metabolite Clearance, CL(m)

The total body clearance of the metabolite can be determined after the administration of the drug only if fm is known. This is determined by administration of a dose of the drug and calculating the AUC(m). The metabolite clearance, CLT(m), is then calculated by substituting the values for fm, D, and AUC(m) in Eq. 12.11. When fm is unknown, administration of the preformed metabolite is necessary for the determination of CLT(m) as in Eq. 12.14. 12.6.4  Metabolite Volume of Distribution, Vd(m)

Determination of the metabolite volume of distribution requires IV administration of the preformed metabolite. The metabolite volume of distribution can be determined as in Eq. 12.16. Vd(m) =

M (12.16) Cp0(m)

where Vd(m) is the metabolite volume of distribution, M is the dose of the metabolite, and Cp0(m) is the plasma metabolite concentration at time zero. Also, Vd(m) can be determined if the CLT(m) and k(m) are known as in Eq. 12.17. Vd(m) =

CL T(m) (12.17) k(m)

12.6.5  Metabolite Formation Clearance, fm CLT

The metabolite formation clearance is the fraction of the CLT of the drug that is responsible for the formation of a particular metabolite. This can be determined from the CLT of the drug and the fraction of the drug dose converted to that metabolite as in Eq. 12.18. Metabolite formation clearance = fm CL T (12.18)

220  Metabolite Pharmacokinetics Practice Problems: Question: Cocaine (Coc) is metabolized to two major metabolites, benzoylecgonine (BE) and ecgonine methyl ester (EME), in addition to several minor metabolites. After concurrent abuse of cocaine and ethanol, cocaethylene (CE), which is a pharmacologically active and more toxic metabolite, is formed. Concurrent abuse of cocaine and alcohol has been associated with very high incidence of sudden death, which has been attributed to the effect of ethanol on the rate of cocaine metabolism and the formation of CE. In an experiment to study the pharmacokinetic interactions between cocaine and ethanol in experimental animals, a group of rats received cocaine, cocaine + ethanol, BE, BE + ethanol, and CE + ethanol by intraperitoneal (IP) administration in a crossover experimental design. The following results were obtained: Treatment

Dose

AUC

Coc

100 μmol/kg

BE Coc + ethanol

50 μmol/kg 100 μmol/kg

BE + ethanol CE + ethanol

50 μmol/kg 50 μmol/kg

Coc BE CE BE Coc BE CE BE CE

4.2 μmol hr/L 8.15 μmol hr/L 0.0 μmol hr/L 9.32 μmol hr/L 6.78 μmol hr/L 4.12 μmol hr/L 0.432 μmol hr/L 9.93 μmol hr/L 1.02 μmol hr/L

Assuming that the bioavailability of cocaine and its metabolites with and without alcohol after IP administration is 100%, answer the following questions: a What is the effect of ethanol administration on cocaine CLT? b What is the fraction of cocaine dose metabolized to BE when cocaine is administered alone and in combination with ethanol? c What is the fraction of cocaine dose metabolized to CE when cocaine is administered with alcohol? d What is the formation clearance of BE and CE when cocaine is administered alone and in combination with ethanol? e Comment on the effect of ethanol on cocaine metabolic profile. Answer: a When cocaine was given alone: CL T coc =

F Dose 100 µmol = = 23.8 L/hr AUC 4.2 µmol-hr/L

When cocaine was given with ethanol: CL T coc =

F Dose 100 µmol = = 14.7 L/hr AUC 6.78 µmol-hr/L

Ethanol decreased CLT Coc by approximately 40%.

Metabolite Pharmacokinetics 221 b When BE was administered alone: F Dose 50 µmol = = 5.36 L/hr AUC 9.32 µmol-hr/L

CL T BE =

After cocaine administration: AUCBE =

BE (amount) CL T BE

8.15 µMole-hr/L =

BE (amount) 5.36 L/hr

BE amount = 43.7 μmol = 43.7% of the cocaine dose. BE total body clearance when BE was administered with ethanol:

CL TBE =

F Dose 50 µmol = = 5.035 L/hr AUC 9.93 µmol-hr/L

After cocaine administration with ethanol: AUCBE =

BE (amount) CL T BE

4.12 µmol-hr/L =

BE (amount) 5.035 L/hr

BE amount = 20.7 μmol = 20.7% of cocaine dose when administered with ethanol. The fraction of cocaine dose metabolized to BE decreased by approximately 50% when cocaine was administered with ethanol. CL T CE =

F Dose 50 µmol = = 49 L/hr AUC 1.02 µmol-hr/L

c After cocaine administration with ethanol: AUCCE =

CE (amount) CL T CE

0.432 µmol-hr/L  =

CE (amount) 49L/hr

CE amount = 21.2 μmol = 21.2% of cocaine dose when given with ethanol. d When cocaine is administered alone: Formation clearance of BE = fm BE.CL T coc = (0.437 ) 23.8 L/hr = 10.4 L/hr Formation clearance of CE = fm CE. CL T coc = (0) 23.8 L/hr = 0 L/hr

222  Metabolite Pharmacokinetics When cocaine is administered with ethanol: Formation clearance of BE = fm BE.CL T coc = (0.207 ) 14.7 L/hr = 3.04 L/hr Formation clearance of CE = fm CE. CL T coc = (0.212) 14.7 L/hr = 3.12 L/hr e Ethanol inhibits the CLT of cocaine by approximately 40% and results in the formation of a new metabolite, cocaethylene. Also, when cocaine is administered with ethanol, the amount of BE formed will be lower than when cocaine is administered alone. 12.7 Steady-State Metabolite Concentration during Repeated Administration of the Drug The drug concentration in the body increases gradually during multiple drug administration until steady state is achieved. As the drug accumulates, the rate of metabolite formation increases, which increases the metabolite concentration in the body. When the drug steady state is achieved, the rate of metabolite formation becomes constant, and gradually the rate of metabolite formation becomes equal to the rate of metabolite elimination and steady state is achieved. At steady state during multiple drug administration, the metabolite concentration and the metabolite-to-drug concentration ratio become constant. Monitoring the metabolite concentration in addition to the drug concentration during multiple drug administration is important when the metabolite contributes significantly to the drug pharmacological and/or adverse effects. At steady state, the rate of metabolite formation is determined from the fraction of the drug clearance responsible for metabolite formation and the average drug steady-state concentration (fm CLT Cpss). However, the rate of metabolite elimination is determined from the metabolite clearance and the average metabolite steady-state concentration (CLT(m) Cp(m) ss). Equation 12.19 describes the rate of metabolite formation and the rate of metabolite elimination at a steady state. fm CL T Cpss = CL T(m) Cp(m)ss (12.19) By rearrangement of Eq. 12.19, Eq. 12.20 is obtained. Cp(m)ss fm CL T = (12.20) Cpss CL T(m) This means that the metabolite-to-drug concentration ratio at a steady state during multiple drug administration can be determined from the ratio of the metabolite formation clearance to the metabolite elimination clearance. Also, it has been shown that the metabolite-to-drug AUC ratio after single IV dose of the drug can be determined from the ratio of the metabolite formation clearance to the metabolite elimination clearance according to Eq. 12.13. Combining Eqs. 12.13 and 12.20, Eq. 12.21 can be obtained. Cp(m)ss fm CL T AUC(m) = = (12.21) Cpss CL T(m) AUC

Metabolite Pharmacokinetics 223 This relationship suggests that the ratio of the average metabolite concentration to the average drug concentration at steady state during multiple administration of the drug can be predicted from the ratio of metabolite-to-drug AUC measured after a single IV bolus dose of the drug. Also, the metabolite-to-drug concentration ratio during constant rate IV infusion of the drug can be predicted from the ratio of metabolite to parent drug AUC after single IV bolus dose of the drug. Clinical Importance:

• PA is an antiarrhythmic drug that is eliminated 50–60% unchanged in urine and

40–50% by metabolism through the acetylation pathway to form an active metabolite NAPA that is also eliminated mainly unchanged in urine. The activity of the acetylation pathway depends on the acetylation phenotype with rapid acetylator having higher activity compared to slower acetylators. During multiple administration of PA, the steady-state NAPA/PA concentration ratio is 1.2 or greater in rapid acetylator and is less than 0.8 in slow acetylator. This is because the formation clearance of NAPA is higher in rapid acetylator compared to slower acetylators. • During multiple administration of the antiepileptic drug CBZ, its epoxide metabolite is accumulated to achieve metabolite-to-drug steady-state concentration ratio of 0.12 when CBZ is used as monotherapy. The epoxide metabolite to CBZ ratio is increased to 0.14 when CBZ is taken with phenobarbital, 0.18 when CBZ is taken with phenytoin, and about 0.25 when CBZ is taken with both phenytoin and phenobarbital. Coadministration of the enzyme inducers, phenobarbital, and phenytoin increased the metabolite formation clearance without affecting its elimination clearance, leading to the increase in the metabolite-to-drug steady-state concentration ratio. Practice Problems: Question: A drug is eliminated by renal excretion of the unchanged drug and metabolism to a pharmacologically active metabolite after administration to humans. This metabolite is eliminated in the urine or metabolized to a second metabolite that is completely excreted in urine according to the scheme in Figure 12.8. After the administration of 100 μmol of the parent drug to a normal volunteer, the total amounts of the parent drug, metabolite 1, and metabolite 2 recovered in the urine were 20, 30, and 50 μmol, respectively. The half-life of the parent drug is 4 hr, and its volume of distribution is 25 L.

Figure 12.8 Schematic presentation for the different pathways involved in the elimination of the drug and its two metabolites in the practice problem.

224  Metabolite Pharmacokinetics a Calculate the CLT and the AUC of the parent drug after the administration of the 100-μmol dose. b What is the renal clearance of the parent drug? c What is the fraction of the administered dose of the parent drug that is metabolized to metabolite 1? d If the observed AUC of metabolite 1 was 10 μmol hr/L, what is the CLT of metabolite 1? e What is the renal clearance of metabolite 1? f What is the formation clearance of metabolite 1 and metabolite 2? g If a dose of 300 μmol of the parent drug is administered to the same volunteer, what are the expected AUCs of the parent drug and metabolite 1? h If 100 μmol of metabolite 1 is administered to the same volunteer, what are the total amounts of metabolite 1 and metabolite 2 that will be excreted in urine? i If a constant rate IV infusion of the parent drug is administered to this volunteer, what is the expected ratio of the metabolite concentration to that of the parent drug at steady state? j From the information provided above, can you determine whether metabolite 1 and metabolite 2 follow formation-rate-limited or elimination-rate-limited profile? Answer: The fraction of the parent drug dose excreted in urine: Fraction (parent drug) = a CL T = k Vd = AUC =

Amount excreted in urine 20 µmol = = 0.2 Dose of parent drug 100 µmol

0.693  25 L = 4.33 L/hr 4 hr

100 µmol = 23 µmol-hr/L 4.33 L/hr

b Renal clearance = Fraction excreted unchanged in urine x CLT CL R = (0.2) 4.33 L/hr = 0.866 L/hr c Fraction metabolized to metabolite 1 = 1 − 0.2 = 0.8 80 µmol d CL T (m 1) = = 8 L/hr 10 µmol-hr/L Amount of metabolite 1 excreted in urine e Fraction excreted in urine of metabolite 1 = Total amount of metabolite 1 formed Fraction(m 1) =

30 µmol = 0.375 80 µmol

Renal clearance of metabolite I = fraction excreted in urine × CLT of metabolite 1 CL R (m1) = 0.375 × 8 L/hr = 3 L/hr

Metabolite Pharmacokinetics 225 f Formation clearance (metabolite 1) = fm (metabolite 1) × CLT (parent drug)   = 0.8 × 4.33 L/hr = 3.46 L/hr Formation clearance ( metabolite 2) = fm (metabolite 2) × CL T (metabolite 1)   = (1 − fraction excreted in urine of metabolite 1) × CL T (metabolite 1)   = (1 − 0.375) × 8 L/hr = 3.46 L/hr g Increasing the dose of the parent drug will cause proportional increase in the drug AUC and the metabolite AUC: AUC parent  drug = 3 × 23 µmol-hr/L = 69 µmol-hr/L AUC m 1 = 3 × 10 µmol-hr/L = 30 µmol-hr/L h After the administration of 100 μmol of metabolite 1: Amount of metabolite 1 excreted in urine = (30/80)100 µmol = 37.5 µmol Amount of metabolite 2 excreted in urine = 100 µmol − 37.5 µmol = 62.25 µmol i Ratio =

Cpss(m 1) fm1 CL T 0.8 × 4.33 L/hr = = = 0.433 Cpss CL T (m1) 8 L/hr

j No because we do not have any information about the half-life and the excretion rate constant of the parent drug and the metabolites. 12.8 Metabolite Pharmacokinetics after Extravascular Administration of the Parent Drug After IV bolus administration, the maximum drug concentration is achieved immediately, which produces the highest rate of metabolite formation. Then, the rate of metabolite formation decreases as the amount of drug in the body decreases due to drug elimination. While after oral administration, the plasma drug concentration increases gradually with time until it reaches a maximum value, then it declines due to drug elimination. Since the rate of metabolite formation is dependent on the amount of the drug in the body, the rate of metabolite formation is very low initially. Then the rate of metabolite formation increases as the amount of the drug in the body increases reaching the maximum rate of metabolite formation when the drug amount in the body is the highest. After this point, the metabolite formation rate decreases. This makes the time to achieve the maximum metabolite concentration (tmax(m)) usually longer than the time to achieve the maximum drug concentration tmax. The drug and the metabolite plasma concentration-time profiles after oral administration of the parent drug are illustrated in Figure 12.9. The plasma metabolite concentration-time profile after oral drug administration is described by a triexponential equation that describes drug absorption, metabolite formation (or drug elimination to form the metabolite of interest), and metabolite elimination. The example presented here assumes that during the absorption process, the drug does

226  Metabolite Pharmacokinetics

Figure 12.9 The drug and metabolite concentration-time profiles after oral administration of single dose of the drug.

not undergo first-pass metabolism. However, in many cases, the drug can be metabolized during the absorption process. This will usually result in the appearance of the metabolite and the drug simultaneously in the systemic circulation. So, when the drug undergoes presystemic metabolism, the resulting metabolite concentration-time profile may be different from the profile described in Figure 12.9. 12.9  Kinetics of Sequential Metabolism In sequential metabolism, the drug is metabolized to form a metabolite that is metabolized further to form a second metabolite. After IV administration of the drug, metabolite 1 plasma concentration increases initially until it reaches Cpmax(m1), then it decreases due to metabolite 1 elimination. The rate of formation of metabolite 2, which is dependent on metabolite 1 concentration, is low initially and gradually increases until it reaches its maximum rate of formation when metabolite I concentration is equal to Cpmax(m1). Beyond this point, the rate of formation of metabolite 2 decreases. So, Cpmax(m1) is achieved first, then Cpmax(m2). The drug and metabolites concentration-time profiles after IV administration of a drug that is sequentially metabolized to two metabolites are illustrated in Figure 12.10. Assuming first-order kinetics, the plasma metabolite 2 concentration-time profile is described by a triexponential equation. This equation describes the formation of metabolite 1 (or parent drug elimination to form metabolite 1), the elimination of metabolite 1 (or the formation of metabolite 2), and the elimination of metabolite 2. Clinical Importance:

• The antiepileptic drug CBZ is metabolized through different pathways, one of which

results in the formation of its major metabolite CBZ-epoxide. The epoxide metabolite is further metabolized to CBZ-diol that is then excreted in urine. • Many drugs are metabolized through phase I metabolic reactions, including oxidation, reduction, and hydrolysis. The formed metabolites can be metabolized further through phase II metabolic reactions by conjugation with glucuronic acid, glutathione, amino acids, etc.

Metabolite Pharmacokinetics 227

Figure 12.10 The drug and metabolites concentration-time profiles after IV administration of single dose of a drug that is metabolized to metabolite 1, which is then metabolized to metabolite 2.

12.10 The Effect of Changing the Pharmacokinetic Parameters on the Drug and Metabolite Concentration-Time Profiles after Single IV Drug Administration and during Multiple Drug Administration After IV bolus administration, the plasma drug concentration-time profile depends on the drug pharmacokinetic parameters. During the drug elimination process, the metabolite is formed, and its plasma concentration increases initially and then decreases. The rate of increase in metabolite concentration is dependent on the rate of drug elimination, with rapidly eliminated drugs producing rapid increase in metabolite concentration. The rate of decline in the metabolite concentration depends on the metabolite elimination rate constant, when k(m) < k, and depends on the drug elimination rate constant, when k(m) > k. The metabolite-to-drug AUC ratio after single IV drug administration and the metabolite-todrug steady-state concentration ratio during multiple drug administration are dependent on the ratio of the metabolite formation clearance to the metabolite elimination clearance. The following discussion deals with the effects of changing the different drug and metabolite pharmacokinetic parameters on the drug and metabolite plasma concentrationtime profiles. This discussion is intended to show how each of the parameters affects the drug and metabolite concentration-time profiles. However, the change in one pharmacokinetic parameter may be accompanied by a change in some other parameters. For example, the change in the drug metabolic clearance due to liver dysfunction is usually accompanied by a change in the metabolite clearance when the metabolite is further metabolized to a second metabolite by the liver. Also, the change in the drug volume of distribution most probably will be accompanied by a change in the metabolite volume of distribution. 12.10.1  Drug Dose

The effects of changing the dose of the drug without changing the drug and metabolite parameters:

• After single drug administration, the initial drug concentration and drug AUC will be

proportional to the administered dose, while the slope of the drug concentration-time profile on the semilog scale will be the same.

228  Metabolite Pharmacokinetics

• Increasing the drug dose causes proportional increase in the amount of metabolite formed in vivo and metabolite AUC, because fm is the same.

• The metabolite-to-drug AUC ratio after single dose and the steady-state metaboliteto-drug concentration ratio during multiple administration do not change because the metabolite formation clearance and elimination clearance are the same.

Clinical Importance:

• Increasing the metabolite formation due to the increase in drug dose can lead to serious

consequences when the metabolite is toxic. Paracetamol overdose can cause liver failure due to the formation of its toxic metabolite, N-acetyl-p-benzoquinoneimine. This metabolite is normally eliminated by conjugation with glutathione. However, when an excessive amount of this metabolite is formed after a large paracetamol dose, the liver’s glutathione is depleted, and the toxic metabolite attaches the liver causing liver damage.

12.10.2  Drug Total Body Clearance

The effects of changing the drug clearance without change in the drug dose, Vd, fm, or any of the metabolite parameters:

• The rate of drug elimination will change leading to a change in the rate of metabolite formation, since Vd is the same. Higher CLT increases the rate of drug elimination and metabolite formation rate, when Vd is constant. • The drug AUC is inversely proportional to the CLT, so higher clearance leads to smaller AUC. • The metabolite-to-drug AUC ratio after single dose, and the metabolite-to-drug steadystate concentration ratio during multiple administration, will change. This is because the metabolite formation clearance, fm, CLT will change, while the metabolite elimination clearance remains the same. Higher CLT increases the metabolite formation clearance and increases the metabolite-to-drug AUC and steady-state concentration ratios. Clinical Importance:

• The clinical consequences of enzyme induction and inhibition that affect drug metabolism

depend on the contribution of the metabolic pathway to the overall drug elimination process and the therapeutic and toxic effects of the drug and its metabolite. When the drug is the active compound and the metabolite is pharmacologically inactive, enzyme induction accelerates drug elimination and decreases the body exposure to the drug. While if the drug is a prodrug, enzyme induction speeds the rate of formation of the active moiety and increases the drug effect. Whereas, if the drug is metabolized to toxic metabolite, enzyme induction increases the rate of formation of the toxic metabolite and increases toxicity. • After the administration of the antiarrhythmic drug PA, 50–60% of the dose is excreted unchanged in urine and 40–50% by metabolism to the active metabolite NAPA that is eliminated mainly unchanged in urine. Reduced renal function decreases the CLR and CLT of PA and NAPA. So, in patients with renal dysfunction, the fraction of PA dose excreted unchanged in urine decreases and AUCPA increases because of the decrease in CLT PA. Also, the fraction of dose metabolized to NAPA increases causing increase in AUCNAPA which increases further due to the decrease in CLT NAPA. So, PA dose reduction is required in patients with reduced kidney function. • In the PA example above, the change in renal function affects fm, CLT PA (metabolite formation clearance), and CLT NAPA (metabolite elimination clearance). So, NAPA to

Metabolite Pharmacokinetics 229 PA AUC ratio after a single dose and steady-state concentration during multiple PA administration depend on the relative magnitude of the change in formation and elimination clearances of NAPA. 12.10.3  Drug Volume of Distribution

The effects of changing the drug volume of distribution without change in the drug dose, CLT, fm, or any of the metabolite parameters:

• After single IV dose of the drug, the initial drug concentration will be inversely proportional to Vd; however, the drug AUC remains the same because CLT is the same.

• The rate of drug elimination will change, with larger Vd causing a slower rate of drug elimination, because CLT is the same.

• The amount of metabolite formed will not be affected by the change in the drug Vd, if

fm is kept constant. However, the rate of metabolite formation will be affected by the rate of drug elimination. • The change in drug Vd does not affect the amount of the metabolite formed in vivo and the metabolite AUC. • The metabolite-to-drug AUC ratio after single dose and the steady-state metaboliteto-drug concentration ratio during multiple administration do not change since the metabolite formation clearance and elimination clearance are the same. 12.10.4  Fraction of the Drug Dose Converted to the Metabolite

The effects of changing the fraction of drug dose converted to the metabolite, fm, without change in the drug dose, CLT, Vd or, any of the metabolite parameters:

• After IV drug administration, the drug concentration-time profile will not be affected by the change in fm since the drug clearance and Vd are the same.

• The amount of metabolite formed after the administration of the same dose of the

drug will change due to the change in the metabolite formation clearance, with larger fm resulting in the formation of a larger amount of the metabolite. • The metabolite AUC will be proportional to the amount of the metabolite formed since the metabolite clearance is the same. • The metabolite-to-drug AUC ratio after single dose and the steady-state metabolite-todrug concentration ratio during multiple administration will be affected by the change in fm. This is because the change in fm causes proportional change in the formation clearance of metabolite, while the metabolite elimination clearance is the same. Clinical Importance:

• Enzyme induction and enzyme inhibition can affect the rate of some specific elimina-

tion pathways resulting in a change in the fraction of drug dose eliminated by those pathways. The effect on the overall drug clearance depends on the contribution of the induced or inhibited pathway to the overall elimination of the drug. Examples of commonly prescribed enzyme inducers are carbamazepine, phenytoin, efavirenz, rifampicin, phenobarbital, and St. John’s wort. However, examples of commonly used enzyme inhibitors include amiodarone, clarithromycin, erythromycin, diltiazem, verapamil, fluoxetine grapefruit juice, metronidazole, terbinafine, cimetidine, ciprofloxacin, fluconazole, itraconazole, fluoxetine, and ritonavir.

230  Metabolite Pharmacokinetics

• Genetic polymorphism in drug metabolism can cause large differences in the activity

of some specific metabolic pathways. This can lead to variation in the elimination rate of drugs metabolized by this pathway and variation in the fraction of drug dose eliminated by the different pathways. For example, genetic variation in the acetylation pathway results in variation in the metabolic rate of the antiarrhythmic drug procainamide and the antituberculosis drug isoniazid in slow and rapid acetylators. Also, genetic variation in the CYP 2D6 metabolic pathway results in variation in the metabolic rate of tramadol, dextromethorphan, risperidone, codeine, haloperidol, and metoprolol in poor, intermediate, extensive, and ultrarapid metabolizers.

12.10.5  Metabolite Total Body Clearance

The effects of changing the metabolite clearance, CLT(m), without change in the drug dose, drug parameters, and the other metabolite parameters:

• After the administration of the same dose, the drug concentration-time profile and the drug AUC will not change since the drug CLT and Vd are the same.

• The amount of the metabolite formed in vivo will be the same, because fm is the same. • The metabolite AUC will be inversely proportional to the metabolite clearance. • The rate of metabolite elimination rate will change due to the change in metabolite clearance if the metabolite Vd is the same. Higher metabolite clearance produces faster rate of metabolite elimination and lower metabolite clearance produces slower rate of metabolite elimination if the metabolite Vd is the same. • The metabolite-to-drug AUC ratio after single dose and the steady-state metabolite-todrug concentration ratio during multiple administration will be affected by the change in metabolite clearance. Higher metabolite clearance causes lower metabolite-to-drug AUC ratio after single dose and lower metabolite-to-drug steady-state concentration ratio. However, lower metabolite clearance results in larger metabolite-to-drug ratios, because these ratios are dependent on the ratio of the formation clearance to the elimination clearance of the metabolite. Clinical Importance:

• The decrease in kidney function when the active metabolite is eliminated mainly in

urine can lead to metabolite accumulation and serious clinical consequences. In uremic patients, accumulation of norpethidine in patients taking pethidine can cause irritability and twitching, while accumulation of the clofibric acid metabolite in patients taking clofibrate can lead to tenderness and muscle weakness. Also, accumulation of oxipurinol in patients taking allopurinol can lead to side effects, and peripheral neuritis can occur in patients taking nitrofurantoin due to the accumulation of toxic metabolites.

12.10.6  Metabolite Volume of Distribution

The effects of changing the metabolite volume of distribution, Vd(m), without change in the drug dose, drug parameters, and the other metabolite parameters:

• After the administration of the same drug dose, the drug concentration-time profile and the drug AUC will be the same since the drug CLT and Vd are the same.

Metabolite Pharmacokinetics 231

• The amount of the metabolite formed in vivo remains constant since fm and CLT are the same.

• The change in the metabolite Vd does not affect the metabolite AUC. However, the

change in metabolite Vd affects the rate of metabolite elimination rate if the metabolite clearance is the same. Larger metabolite Vd produces a slower rate of metabolite elimination, whereas smaller metabolite Vd produces a faster rate of metabolite elimination. • The metabolite-to-drug AUC after single dose and the steady-state metabolite-to-drug concentration ratio during multiple administration will not be affected by the change in the metabolite Vd. This is because the metabolite formation clearance and elimination clearance are the same. 12.11 Summary

• Drug metabolism involves enzymatic modification of the drug’s chemical structure to form a new chemical entity known as the metabolite.

• Most of the metabolites are pharmacologically inactive; however some metabolites have pharmacological activity, and some metabolites contribute to the drug’s adverse effects.

• The metabolite pharmacokinetic behavior is different from that of the parent drug. • Enzyme induction and inhibition can significantly alter the drug pharmacokinetics and change the fraction of the drug dose converted to a particular metabolite.

• Genetic polymorphism involving metabolizing enzymes can significantly alter the pharmacokinetic behavior and metabolic profile of drugs metabolized by these enzymes.

• Drugs that are metabolized to pharmacologically active or toxic metabolites should be monitored by measuring the drug and metabolite concentrations.

Practice Problems 12.1 A drug is metabolized to three different metabolites through three parallel pathways. After the administration of 1000 mg of the parent drug, plasma samples were obtained, and the concentrations of the parent drug and the metabolites were determined. Time (hr)

Drug (mg/L)

Metabolite I (mg/L)

Metabolite II (mg/L)

Metabolite III (mg/L)

0 1 2 4 8 12 16 24

50 36.6 26.8 14.4 4.1 1.2 0.34 0.03

0 21.0 26.3 21.3 7.7 2.33 0.678 0.056

0 25.7 42.1 56.9 54.3 40.9 28.6 13.1

0 6.43 10.5 14.2 13.5 10.2 7.1 3.28

a Plot the plasma concentration-time profiles for the parent drug and its metabolites and determine which of the metabolites follow formation-rate-limited profile and which follow elimination-rate-limited profile. 12.2 After the administration of a drug to humans, it is eliminated by renal excretion and metabolism to two different metabolites that are completely excreted in urine according to the scheme in Figure 12.11.

232  Metabolite Pharmacokinetics

Figure 12.11 Schematic presentation for the different pathways involved in the elimination of the parent drug and its two metabolites in problem 12.2.

After the administration of 100 μmol of the parent drug to a normal volunteer, the total amounts of the parent drug, metabolite 1, and metabolite 2 recovered in the urine were 20, 30, and 50 μmol, respectively. The half-life of the parent drug is 4 hr, and its volume of distribution is 25 L. a Calculate the CLT and the AUC of the parent drug after the administration of the 100-μmol dose. b What is the renal clearance of the parent drug? c What is the fraction of the administered dose of the parent drug that is metabolized to metabolite 1? d If the observed AUC of metabolite 1 was 10 μmol hr/L, what is the CLT of metabolite 1? e What is the renal clearance of metabolite 1? f What is the formation clearance of metabolite 1 and metabolite 2? g If a dose of 300 μmol of the parent drug is administered to the same volunteer, what are the expected AUCs of the parent drug and metabolite 1? h If 100 μmol of metabolite 1 is administered to the same volunteer, what are the total amounts of metabolite 1 and metabolite 2 that will be excreted in urine? i If a constant rate IV infusion of the parent drug is administered to this volunteer, what is the expected ratio of the metabolite concentration to that of the parent drug at steady state? j From the information provided above, can you determine whether metabolite 1 and metabolite 2 follow formation-rate-limited or elimination-rate-limited profile? 12.3 One mmol of a drug was administered by IV to a human volunteer and the following data were obtained. Time (hr)

Parent drug Concentration (μmol/L)

Metabolite Concentration (μmol/L)

0.5 1.5 3.0 5.0 7.0 9.0 12.5 17.5 23.5 30.0

13.0 10.8 9.5 7.5 5.6 4.5 2.8 1.5 0.74 0.32

0.56 1.15 2.5 4.7 6.0 6.4 4.3 2.3 1.1 0.48

Metabolite Pharmacokinetics 233 On a separate occasion, two weeks later, the same volunteer received 1 mmol of the metabolites IV and the following data were obtained. Time (hr)

Metabolite Concentration (μmol/L)

1.0 3.5 6.0 8.5 11.0 16.0 20.0 25.0 32.0

21 9.9 4.5 2.0 0.94 0.2 0.059 0.013 0.0014

a Calculate the following parameters:

• • • •

Fraction of parent drug converted to the metabolite Total body clearance of the metabolite Volume of distribution of the metabolite Formation clearance and formation rate constant of the metabolite

b If the parent drug is to be given as a constant rate infusion at 0.25 mmol/hr, what will be the steady-state concentration of the metabolite? c What will be the concentration ratio of metabolite to parent drug at steady state during a constant rate infusion of the drug? 12.4 An oral hypoglycemic drug is eliminated from the body by renal excretion or metabolism to a metabolite that is completely excreted in urine. After the administration of 1000 mg of this drug to a normal volunteer, plasma samples were obtained, and the concentrations of the parent drug and its major metabolite were determined. Time (hr)

Parent drug Concentration (mg/L)

Metabolite Concentration (mg/L)

1 2 3 4 6 8 10 12 16 20 24

2.97 5.1 6.87 7.56 8.42 8.37 7.82 7.04 5.34 3.847 2.69

0.166 0.61 1.27 1.65 2.44 2.85 2.95 2.85 2.35 1.75 1.24

234  Metabolite Pharmacokinetics On a separate occasion, after the administration of 1000 mg of the metabolite to the same volunteer, the following concentrations were observed: Time (hr)

Concentration (mg/L)

1 4 8 12 16

33.5 10.1 2.04 0.411 0.083

The bioavailability of this oral hypoglycemic drug is known to be 80% due to incomplete absorption from the GIT. Because the difference in molecular weight between the parent drug and the metabolite is small, assume that the molecular weights are equal. The elimination rate constant of the parent drug is 0.2 hr−1. a Calculate the CLT and Vd of the metabolite. b What is the fraction of the parent drug (in the systemic circulation) that is converted to the metabolite? c What are the formation clearance and the formation rate constant of the metabolite? d What is the total amount of the parent drug and its metabolite excreted in urine after the administration of the 1000 mg dose orally? e If the parent drug is administered as 1000 mg every 12 hr, what will be the average steady-state plasma concentrations of the parent drug and the metabolite? 12.5 After IV administration of an antihypertensive drug, 30% of the administered dose is excreted unchanged in urine and 70% of the dose is metabolized to inactive metabolite that is completely eliminated in the urine. After the administration of 50 μmol of this drug, its half-life was 1 hr, its volume of distribution was 20 L, and its AUC was 3.6 μmol hr/L. a b c d

What is the renal clearance of the parent drug? What is the metabolic clearance of the parent drug? What is the formation clearance of the metabolite? If the observed metabolite AUC was 5 μmol hr/L, what are the metabolite CLT, metabolic clearance, and renal clearance? e If the metabolite volume of distribution is 30 L, determine whether this metabolite follows formation-rate- or elimination-rate-limited behavior. f If 120 μmol of the drug is administered, what will be the expected parent drug and metabolite AUCs? g If the parent drug is administered by constant rate IV infusion, what will be the steady-state metabolite to parent drug concentration ratio? References 1. Meyer UA “Overview of enzymes of drug metabolism” (1996) J Pharmacokinet Biopharm; 24:449–459.

Metabolite Pharmacokinetics 235 2. Guengerich FP “Cytochromes p450, drugs, and diseases” (2003) Mol Intervent; 3:194–204. 3. Nelson DR, Koymans L, Kamataki T, Stegeman JJ, Feyereisen R, Waxman DJ, Waterman MR, Gotoh O, Coon MJ, Estabrook RW, Gunsalus IC and DW N “P450 superfamily: Update on new sequences, gene mapping, accession numbers and nomenclature” (1996) Pharmacogenetics; 6:1–42. 4. Houston JB “Drug metabolite kinetics” (1981) Pharmacol Ther; 15:521–552.

13 Nonlinear Pharmacokinetics

Objectives After completing this chapter, you should be able to:

• Discuss the most common causes of nonlinear pharmacokinetic behavior and the common characteristics of nonlinear pharmacokinetics.

• Define the Michaelis-Menten pharmacokinetic parameters. • Describe the basic characteristics of the pharmacokinetic behavior of drugs that follow Michaelis-Menten kinetics.

• Calculate the Michaelis-Menten pharmacokinetic parameters using mathematical and graphical methods.

• Analyze the effect of changing the Michaelis-Menten pharmacokinetic parameters on the steady-state plasma concentration after single and multiple drug administration.

• Recommend the dosing regimen required to achieve therapeutic plasma concentration of drugs that follow Michaelis-Menten kinetics in patients.

13.1 Introduction In the previous discussion of the pharmacokinetic behavior of drugs, it was assumed that the drugs follow linear pharmacokinetics. This means that drug absorption, distribution, and elimination processes follow first-order kinetics, and the drug pharmacokinetic parameters CLT, Vd, k, and t1/2 are constant irrespective of the administered dose. Also, the AUC of the drug after single administration and the steady-state drug concentration during multiple administration are proportional to the administered dose. However, in some drugs, one or more of the absorption, distribution, metabolism, and excretion processes do not follow first-order kinetics, which can make the drug pharmacokinetic parameters different after administration of different doses. Also, the AUC after single drug administration and the average steady-state drug concentration during multiple administration are not proportional to the administered dose. In this case, the drugs follow nonlinear pharmacokinetics, which is also known as dose-dependent or concentration-dependent pharmacokinetics as in Figure 13.1. 13.2  Causes of Nonlinear Pharmacokinetics Drugs with evidence of nonlinear pharmacokinetic behavior are studied further to determine the cause of this nonlinearity. The nonlinear behavior occurs because of dose-dependent absorption, distribution, excretion, and/or metabolism when at least one of these processes DOI: 10.4324/9781003161523-13

Nonlinear Pharmacokinetics 237

Figure 13.1 The relationship between the administered dose and AUC or steady-state concentration for drugs that follow linear and nonlinear pharmacokinetics.

does not follow first-order kinetics (1, 2). The dose-dependent metabolism, which is the most common cause of nonlinear pharmacokinetic behavior in the drugs used clinically, will be discussed in detail in this chapter. 13.2.1  Dose-Dependent Drug Absorption

Dose-dependent absorption may be observed in drugs with limited solubility, drugs absorbed by carrier-mediated transport, and drugs with saturable pre-systemic metabolism (3). Drugs with limited aqueous solubility especially those administered in large doses are not absorbed completely due to incomplete drug dissolution in the GIT. After administration of increasing doses of drugs with limited aqueous solubility, smaller fractions of the administered dose can dissolve in the GIT, leading to lower extent of absorption if the drug transit time in the GIT does not change. Partial dissolution does not represent a major cause of incomplete bioavailability in drugs that are used in small doses such as digoxin. Passive diffusion is the main mechanism of absorption for most drugs. However, some drugs are absorbed from the small intestine via a specific carrier-mediated transport. The rate of drug transport by the carrier system can be described by Eq. 13.1. Transport rate =

Tmax CGIT (13.1) K T + CGIT

where Tmax is the maximum rate of drug transport, CGIT is the drug concentration at the site of transport, and KT is a constant related to the affinity of the drug to the carrier system. Administration of increasing doses of drugs that are absorbed by carrier-mediated transport can saturate the transport system and the rate of drug absorption becomes constant, i.e., the absorption process follows zero-order kinetics. So, larger doses require longer time for their complete absorption by this transport system. When the drug transit time in the GIT segment containing the transport system is not long enough to allow complete absorption, increasing the dose will decrease the fraction of dose absorbed and will lead to less than proportional increase in the drug AUC. Saturation of the transport system after administration of large doses of the drug is more likely when the transport system is localized in a specific site in the GIT, and the drug is administered in the form of immediate-release formulation. Some drugs that undergo extensive presystemic metabolism after oral administration can saturate the drug-metabolizing enzymes when administered in large doses. This allows

238  Nonlinear Pharmacokinetics larger fraction of the administered dose to escape the presystemic elimination and increases the drug bioavailability with the increase in dose. Propranolol is a nonselective beta-adrenergic blocking agent that undergoes saturable presystemic metabolism. Administration of increasing doses of propranolol produces more than proportional increase in propranolol AUC due to the increase in its bioavailability. Saturable presystemic metabolism is more pronounced after administration of rapidly absorbed dosage forms where large amount of the drug is presented to the metabolizing enzymes over a short time, which increases the chance of enzyme saturation. All these conditions can lead to dose-dependent absorption and nonlinear pharmacokinetic behavior. Clinical Importance:

• The antifungal drug griseofulvin is used in doses of 100–600 mg/day and has aqueous

solubility of 10 μg/mL. After administration of increasing griseofulvin doses, incomplete dissolution decreases the extent of its absorption and produces less than proportional increase in the AUC. • Drugs such as L-dopa, methyldopa, cephalexin, amoxicillin, methotrexate, and cephradine are absorbed by carrier-mediated transport from the GIT and can have dosedependent bioavailability, when administered in large doses. • Propranolol, nicardipine, omeprazole, verapamil, and atorvastatin are examples of drugs that undergo saturable presystemic metabolism and can have dose-dependent bioavailability. 13.2.2  Dose-Dependent Drug Distribution

The volume of distribution of drugs is dependent on the drug binding to plasma and tissue proteins. The relationship between the drug Vd and the free fraction of the drug in plasma and tissues can be described by Eq. 13.2.  fu P  Vd = Vp + Vt  (13.2)  fu T  where Vp is the volume of plasma, Vt is the volume of tissues, fu p and fu T are the free fractions of the drug in plasma and tissues, respectively. Based on this relationship, the change in drug binding to plasma or tissue proteins, which affects the drug free fraction in plasma and tissues, can lead to change in the drug Vd. When the drug binding to plasma and tissue proteins changes by the same percentage, the drug Vd does not change. However, if the drug free fraction in plasma increases more than that in tissues, the drug Vd will increase. The extent of drug protein binding can change due to change in protein concentration such as in some diseases, and due to displacement from the binding sites in presence of a competing molecule. Also, some drugs have saturable plasma protein binding, which leads to higher free fraction at high plasma drug concentration. The change in the free fraction of the drug in plasma can affect the CLT of drugs with low intrinsic clearance. This is because the increase in the drug free fraction increases the free drug concentration in plasma, which increases the chance of the free drug molecules to encounter the metabolizing enzymes. This leads to approximately proportional increase in the drug CLT. So, administration of increasing doses of the drugs with saturable plasma protein

Nonlinear Pharmacokinetics 239 binding results in less than proportional increase in the drug AUC due to the increase in the drug CLT (4). Clinical Importance:

• Saturable plasma protein binding in naproxen, disopyramide, warfarin, valproic acid,

and tolbutamide is the main cause of concentration-dependent CLT and the nonlinear pharmacokinetic behavior. In these drugs, good correlation exists between the CLT and the unbound fraction in plasma.

13.2.3  Dose-Dependent Renal Excretion

Urinary drug excretion involves glomerular filtration, active and passive tubular reabsorption, and active tubular secretion. Active drug secretion in the renal tubules and active tubular reabsorption are saturable processes. When the drug concentration is low, the rate of tubular secretion increases proportional to the drug concentration and the renal clearance is constant. However, when the transport system is saturated at high drug concentration, the rate of tubular secretion becomes constant, and the drug renal clearance decreases. Drugs that are mainly eliminated by renal excretion and are actively secreted in the renal tubules can follow nonlinear pharmacokinetics when administered in doses that are high enough to saturate the renal tubular secretion. In this case, administration of increasing doses results in more than proportional increase in the drug AUC due to the decrease in the drug clearance. Clinical Importance:

• Examples of drugs that are mainly excreted in urine and are actively secreted in the

renal clearance include ampicillin, dicloxacillin, cephalexin, enalaprilat, methotrexate, cimetidine, and famotidine. However, in the range of doses used clinically, all these drugs do not saturate the renal tubular secretion and follow linear pharmacokinetics. • Ascorbic acid is reabsorbed from the renal tubules by an active transport system. Administration of large doses of ascorbic acid can cause saturation of ascorbic acid reabsorption and larger proportional of the dose is excreted in urine, leading to increase in its renal clearance. 13.2.4  Dose-Dependent Drug Metabolism

Drug metabolism involves enzymatic reactions that follow first-order kinetics at low drug concentration. However, when the drug concentration is high, the enzymes can be saturated and the metabolic clearance becomes dependent on the drug concentration, with higher drug concentration resulting in lower clearance. Administration of increasing doses of the drugs that are eliminated by saturable metabolism leads to more than proportional increase in the drug AUC and steady-state concentration. When the drug is eliminated by multiple parallel pathways in which one or more of these pathways can be saturated, the drug pharmacokinetic behavior usually depends on the relative contribution of the saturable pathway to the overall elimination of the drug. When the saturable metabolic pathway represents a major elimination pathway, the nonlinear behavior of the drug will be evident.

240  Nonlinear Pharmacokinetics Clinical Importance:

• Drugs like phenytoin and salicylates have been shown to have saturable drug metabolism and follow nonlinear pharmacokinetics in the range of doses that are used clinically (5, 6).

13.2.5  Other Conditions That Can Lead to Nonlinear Pharmacokinetics

Product inhibition can lead to nonlinear pharmacokinetic behavior. It occurs when the parent drug is metabolized to metabolites that can compete with the drug for the same metabolizing enzymes (7). Administration of different doses of the drug produces different amounts of the metabolites that inhibit the metabolism of the drug to different degrees. Also, the metabolite concentration after administration of the parent drug is always changing which makes its inhibitory effect on the rate of drug metabolism changing all the time. This causes dose-dependent variation in the drug pharmacokinetic behavior. Time dependent pharmacokinetics is observed when the drug pharmacokinetic behavior changes with time and can be considered a type of nonlinear pharmacokinetics. Some drugs like carbamazepine can induce their own metabolism in a phenomenon known as autoinduction (8). After initiation of carbamazepine therapy in patients, the drug accumulates to achieve an initial steady-state concentration. However, carbamazepine clearance increases gradually during the first four to five weeks of therapy, leading to gradual decline in the carbamazepine steady-state concentration if the dosing rate is constant. The new steady state depends on the dosing rate and the higher drug clearance reached after induction. Clinical Importance:

• Isosorbide dinitrate is metabolized by denitration to isosorbide-2-mononitrate and

isosorbide-5-mononitrate. These metabolites are further denitrated to isosorbide by the same enzymatic pathway. So, the metabolites compete with the parent drug for the same enzymes resulting in slower rate of parent drug metabolism. • Carbamazepine therapy is usually initiated with only a fraction of the recommended dose and then the dose is increased gradually every two to three weeks depending on the response and adverse effects. 13.3 Pharmacokinetics of Drugs Eliminated by Dose-Dependent Metabolism, Michaelis-Menten Pharmacokinetics Drugs that follow nonlinear pharmacokinetics due to saturable metabolism have to be used with caution since small change in dose can cause more than proportional increase in the steady-state drug concentration. Dose-dependent metabolism, also known as concentration-dependent metabolism and capacity-limited metabolism, is the main cause of nonlinearity in most drugs that show evidence of nonlinearity in the range of doses used clinically. 13.3.1  Michaelis-Menten Enzyme Kinetics

Enzymatic reactions involve interaction between the enzymes (E) and the substrate (D, the drug) to form enzyme-substrate complexes (ED). The ED complexes dissociate to give the product of the reaction (M, the metabolite) and the enzyme as in Eq. 13.3. E + D ↔ ED → E + M(13.3)

Nonlinear Pharmacokinetics 241 The enzyme can go back to react with the drug to form another molecule of the metabolite and so on. According to the Michaelis-Menten enzyme kinetic principles, a quantitative relationship between the enzymatic reaction rate (V) and the substrate concentration (S) can be expressed as in Eq. 13.4. V=

Vmax S (13.4) Km + S

where Vmax is the maximum rate of the enzymatic reaction, and Km is the MichaelisMenten constant (M-M constant) that is a measure of the affinity of the substrate to the enzyme (9, 10). The M-M constant, Km, is the substrate concentration when the reaction rate is ½ Vmax. Assuming that a drug is metabolized via a single metabolic pathway, Eq. 13.5 describes the relationship between the drug metabolic rate and the plasma drug concentration (Cp). Rate =

Vmax Cp (13.5) K m + Cp

Graphically, this relationship can be presented as in Figure 13.2. This hyperbolic relationship indicates that as the drug concentration increases the rate of drug metabolism increases until it reaches a plateau at high drug concentration. The plateau indicates that the drug-metabolizing enzymes are saturated, and the metabolic rate is at its maximum value. The change in the drug metabolic rate as a function of the drug concentration is different at different drug concentrations as illustrated in Figure 13.2. At low drug concentration (i.e., Km ≫ Cp), the drug metabolic rate is proportional to the drug concentration and the metabolic process follows first-order kinetic. In this case, Eq. 13.5 can be approximated by Eq. 13.6. V  Rate ≅  max  Cp (13.6)  Km 

Figure 13.2 The relationship between the drug metabolic rate and the drug concentration. Vmax is the maximum rate of drug metabolism, and Km is the M-M constant, and it is the drug concentration when the reaction rate is ½ Vmax.

242  Nonlinear Pharmacokinetics While, at high drug concentration (i.e., Km ≪ Cp), the metabolizing enzymes are saturated. The drug metabolic rate is constant and independent of the drug concentration and the metabolic process follows zero-order kinetic. In this case, Eq. 13.5 can be approximated by Eq. 13.7.  V Cp  ≅ Vmax(13.7) Rate ≅  max  Cp  Clinical Importance:

• Theoretically, all metabolic pathways involved in drug metabolism can be saturated

leading to nonlinear pharmacokinetics. However, the drug concentrations achieved by the clinically used doses are much lower than Km for the different metabolic pathways. This makes drug metabolism of most drugs in the range of doses used clinically follow first-order kinetics. • Saturable drug metabolism becomes clinically significant when the metabolizing enzymes are saturated in the range of doses used clinically, such as in case of phenytoin and salicylates. 13.3.2  The Pharmacokinetic Parameters

The Vmax of a metabolic pathway is the maximum rate of drug metabolism through this metabolic pathway. It has units of rate, i.e., amount/time. Each metabolic pathway has its own Vmax that depends on the amount of enzymes involved in the metabolic process. Enzyme induction increases the amount of the metabolizing enzymes which increases Vmax. The amount of the enzyme increases without affecting the affinity of the enzyme to the drug. This means that enzyme induction increases Vmax without affecting Km, as illustrated in Figure 13.3. The M-M constant, Km, is a qualitative characteristic of how an enzyme interacts with the drug, and it is independent of the enzyme concentration. This constant is equal to the substrate concentration when the metabolic rate is half of its maximum value, so it has units of concentration. In presence of a competitive inhibitor for the metabolizing

Figure 13.3  Enzyme induction causes increase in the Vmax without affecting Km.

Nonlinear Pharmacokinetics 243

Figure 13.4  Competitive inhibition of the metabolizing enzyme increases Km without affecting Vmax.

enzyme, the drug metabolic rate at any given drug concentration is slower than the rate in absence of the inhibitor. However, as the drug concentration increases relative to the inhibitor concentration, the maximum metabolic rate is achieved. This means that in presence of a competitive inhibitor, Km increases without affecting Vmax, as illustrated in Figure 13.4. Clinical Importance:

• Administration of enzyme inducers such as carbamazepine, phenytoin, efavirenz,

rifampicin, phenobarbital and St. John’s wort lead to increase in the amount of metabolizing enzyme and Vmax of the induced pathway. The rate of metabolism at any given drug concentration will be higher in the induction state, as presented in Figure 13.3. • Drugs that are known to competitively inhibit drug-metabolizing enzymes include cimetidine, valproic acid, amiodarone, chloramphenicol, isoniazid, and omeprazole. The rate of drug metabolism at any given drug concentration will be lower in the presence of the enzyme inhibitor, as presented in Figure 13.4. 13.3.3 Drug Concentration-Time Profile after Administration of a Drug Which Is Eliminated by Single Metabolic Pathway That Follows Michaelis-Menten Kinetics 13.3.3.1  After Single IV Bolus Administration

The model that describes the drug pharmacokinetic behavior after single IV dose when the drug is eliminated via single metabolic pathway that follows MichaelisMenten kinetics is presented in Figure 13.5. The initial drug concentration is dependent on the dose and the drug Vd as in Eq. 13.8. This means that administration of increasing doses produces initial drug concentrations (Cpo) that are proportional to the administered dose. Cpo =

Dose (13.8) Vd

244  Nonlinear Pharmacokinetics

Figure 13.5 Schematic presentation of the pharmacokinetic model for single IV bolus dose of a drug which is eliminated by single metabolic pathway that follows Michaelis-Menten kinetics.

The rate of decline in the amount of the drug in the body (A) is equal to the rate of drug metabolism. Eq. 13.9 is the differential equation that describes the rate of change of the amount of the drug in the body after single IV dose. −dA Vmax Cp = (13.9) dt K m + Cp Integration of Eq. 13.9 and dividing by the volume of distribution gives Eq. 13.10.  Cp  V t Cp = Cp0 + K m ln  0  − max (13.10) Vd  Cp  Equation 13.10 is not an explicit expression for the plasma concentration versus time relationship. This means that the equation cannot be used to calculate the plasma concentration at any time after drug administration by substituting for different values of time. This is because the term for the drug concentration presents in both sides of the equation and cannot be separated (9). Numerical integration methods can be used to calculate the plasma drug concentration at different time after drug administration. The plasma concentration-time profile after single IV bolus dose of a drug, which is eliminated by a single metabolic pathway that follows Michaelis-Menten kinetics, is presented in Figure 13.6.

Figure 13.6 The plasma drug concentration-time profile after single IV bolus dose of a drug, which is eliminated by single metabolic pathway that follows Michaelis-Menten kinetics.

Nonlinear Pharmacokinetics 245 13.3.3.2  During Multiple Drug Administration

At steady state during multiple drug administration, the rate of drug administration is equal to the rate of drug elimination. When the drug is eliminated by single metabolic pathway that follows Michaelis-Menten kinetics, the relationship between the dosing rate (FD/τ) and steady-state plasma drug concentration can be described by Eq. 13.11 (10). FD Vmax Cpss = (13.11) τ K m + Cpss This equation can be rearranged to obtain Eq. 13.12, which describes the steady-state plasma drug concentration. Cpss =

(FD/τ) K m (13.12) Vmax − (FD/τ)

This equation indicates that administration of increasing doses of the drug results in more than proportional increase in the steady-state plasma drug concentration, as presented in Figure 13.7. When the dosing rate is close to the Vmax of the drug, small change in the dose causes large increase in the steady-state drug concentration. Also, if the dosing rate (FD/τ) exceeds Vmax, the expected steady-state drug concentration will be a negative value according to Eq. 13.12. From its definition, Vmax is the maximum rate of drug elimination. So, if the dosing rate exceeds Vmax, steady state will never be achieved and continuous accumulation of the drug in the body will occur. Clinical Importance:

• The initial dose of drugs that are eliminated by a process that follows M-M kinetics

should be accurately calculated and changing the dose should be done with caution. This is because small change in dose can lead to significant change in drug concentrations. • For example, taking the average values of Vmax and Km for phenytoin, a daily phenytoin dose of 300 mg should achieve phenytoin Cpss of 7.5 mg/L, while a dose of 400 mg should achieve Cpss of 20 mg/L, and 450 mg should achieve Cpss of 45 mg/L.

Figure 13.7 The relationship between the steady-state drug concentration and the dosing rate during multiple administration of a drug when the drug elimination process follows Michaelis-Menten kinetics.

246  Nonlinear Pharmacokinetics 13.4 Determination of the Pharmacokinetic Parameters for Drugs with Elimination Process that Follows Michaelis-Menten Kinetics The pharmacokinetic behavior of the drugs that are eliminated by a process that follows Michaelis-Menten kinetics depends on the pharmacokinetic parameters of the drug. The Vd determines the relationship between the dose and the initial drug concentration after single drug administration. While the total body clearance determines the relationship between the dosing rate and the steady-state concentration during multiple drug administration, and the half-life determines the rate of decline in the drug concentration with time. So, determination of the drug pharmacokinetic parameters is important to predict the pharmacokinetic behavior of the drug. 13.4.1  The Volume of Distribution

Drug distribution is similar whether drug elimination follows first-order or MichaelisMenten kinetics. The Vd can be calculated from the dose of the drug and the initial drug concentration achieved after a single IV dose of the drug. The relationship between dose, Cp0, and Vd is described in Eq. 13.9 (Cp0 = Dose/Vd). 13.4.2  The Total Body Clearance

The total body clearance is determined by dividing the drug metabolic rate by drug concentration. Equation 13.13, which describes the drug clearance, assumes that the drug is eliminated by single process that follows Michaelis-Menten kinetics. In this case, CLT is dependent on the drug concentration. As the drug concentration increases, the CLT decreases. CL T =

Vmax K m + Cp

(13.13)

13.4.3  The Half-Life

The half-life is the time required to decrease the plasma drug concentration by 50%. It can be calculated from the drug CLT and Vd as in Eq. 13.14. t1/2 =

0.693 Vd (K m + Cp) Vmax

(13.14)

Equation 13.14 indicates that the half-life of drugs that are eliminated by a process that follows Michaelis-Menten kinetics is dependent on the drug concentration. Higher drug concentrations lead to longer half-lives. This is important because when a patient takes an overdose of a drug that follows Michaelis-Menten kinetics, high plasma drug concentration is achieved, which can produce drug toxicity. In this case, it takes long time for the drug concentration to decrease to therapeutic drug concentrations because of the long half-life. The CLT and the t1/2 are different at different drug concentrations. So, these parameters must be calculated at specific drug concentration. After single drug administration, the drug concentration will be changing with time so as the CLT and the t1/2. The CLT calculated from the dose and AUC (CLT = Dose/AUC) is an average value for the CLT. At

Nonlinear Pharmacokinetics 247 steady state, the plasma drug concentration is fluctuating around an average steady-state concentration. The CLT and the t1/2 are estimated using this average concentration. 13.5 Oral Administration of Drugs that Are Eliminated by a Process that Follows Michaelis-Menten Kinetics The rate of drug absorption significantly affects the plasma concentration-time profile after single oral dose when drug elimination is concentration dependent. After oral administration of the same dose of a drug, rapidly absorbed formulations usually produce larger AUC and produce higher body exposure to the drug, compared to slowly absorbed formulations. This is because rapidly absorbed formulations achieve higher maximum drug concentration. At higher drug concentration, the CLT is lower, and the t1/2 is longer when drug elimination is concentration dependent. During multiple drug administration, the steady-state plasma drug concentration is dependent on the dose, Vmax and Km. So, the rate of drug absorption does not significantly affect the average steady-state plasma concentration. However, faster drug absorption may result in larger fluctuations in drug concentration at steady state. Clinical Importance:

• The intensity of pharmacological effect of some drugs can be different after a single oral

administration due to the difference in their rate of absorption. Alcohol is eliminated from the body by a saturable metabolic process. Consumption of the same amount of alcohol can produce different plasma concentration-time profiles and different effects when the rate of absorption is different. • The blood alcohol concentration-time profile after consumption of a certain amount of alcohol over a short period of time (rapid absorption) is much higher than that achieved after consumption of the same amount of alcohol over a longer period (slow absorption). Also, alcohol intake on an empty stomach results in faster absorption and higher alcohol blood concentration when compared to alcohol intake on a full stomach which leads to slower alcohol absorption. Furthermore, faster absorption of alcohol after consumption of beverages with higher alcohol concentration may increase the bioavailability of alcohol due to saturable presystemic metabolism. The high alcohol blood concentrations achieved because of the rapid absorption and the higher bioavailability decreases alcohol clearance and prolongs its half-life, which leads to the higher alcohol concentration-time profile and the more intense effect. 13.6 Determination of the Michaelis-Menten Parameters and Calculation of the Appropriate Dosage Regimens Dosage recommendation for drugs that are eliminated by concentration-dependent elimination process should be done with caution especially for the drugs with narrow therapeutic range. This is because small change in dose can lead to significant change in the steady-state drug concentration. Also, the inter-patient variability makes individualization of dosage regimen for these drugs difficult based on the patient’s demographic information only. This emphasizes the importance of determination of the patient’s specific Michaelis-Menten parameters for calculating the proper

248  Nonlinear Pharmacokinetics dosage requirements for each individual patient. All the methods used to calculate the Michaelis-Menten parameter, Vmax and Km, have several assumptions that can be summarized as follows (11, 12):

• The drug is eliminated from the body by single elimination pathway that follows Michaelis-Menten kinetics.

• The drug is administered repeatedly in a fixed dose and dosing interval until steady state is achieved.

• It is required to have information about two different dosing regimens and the steadystate drug concentrations achieved while each regimen being administered.

13.6.1  Mathematical Method

The relationship between the dosing rate and the steady-state plasma drug concentration is described by Eq. 13.11. The two different dosing rates and their corresponding steadystate drug concentrations can be used to construct two different equations in the same form of Eq. 13.11 as follows: FD1 Vmax Cpss 1 = τ K m + Cpss 1

and

FD2 Vmax Cpss 2 = τ K m + Cpss 2

These two equations contain two different unknowns that are the two MichaelisMenten kinetic parameters, Vmax and Km. The values for the dosing rates and the corresponding steady-state concentrations are substituted in the equations and the two equations can be solved simultaneously to determine Vmax and Km. Practice Problems: Question: The steady-state phenytoin concentration achieved during administration of 125 mg every 12 hr was 4.4 mg/L, while the steady-state concentration achieved during administration of 250 mg every 12 hr was 20 mg/L. If phenytoin bioavailability is 100%, calculate phenytoin Vmax and Km in this patient, mathematically. Answer: Using the provided information about the daily doses and the corresponding steady-state concentrations, two equations can be written. (1)250 mg Vmax 4.44 mg/L = day K m + 4.44 mg/L

and

(1)500 mg Vmax 20 mg/L = day K m + 20 mg/L

These two equations have two unknowns and can be solved simultaneously. Rearranging the first equation:

(250 mg/day ) K m + 1110 mg2 /L/day = Vmax ( 4.44 mg/L ) Solving for Km: Km =

Vmax (4.44 mg/L) − 1110 mg 2 /L/day 250 mg/day

Nonlinear Pharmacokinetics 249 Substituting for Km in the second equation and solving for Vmax: 7780 mg 2 /L/day = 11.12 mg/L Vmax Vmax = 700 mg/day Solving for Km: Km =

700 mg/day (4.44 mg/L) − 1110 mg 2 /L/day = 8 mg/L 250 mg/day

13.6.2  The Direct Linear Plot

The direct linear plot is a graphical method which can be used to estimate Vmax and Km. The plot is constructed by plotting the dosing rate on the y-axis and the steady-state drug concentration on the left side of the x-axis.

• A line is drawn between each dosing rate and its corresponding steady-state concentration. • The two lines for the two doing rates are extrapolated until they intersect. • The x-coordinate and y-coordinate of the point of intersection are used to estimate Km and Vmax/F, respectively, as presented in Figure 13.8.

• The direct linear plot can be used to estimate the steady-state plasma drug concentra-

tion achieved after administration of a specific dosing rate by drawing a line between the point of intersection and the dosing rate. Extrapolation of the line meets the x-axis at a point corresponding to the expected steady state that should be achieved after administration of this dosing rate. • The direct linear plot can be used also to determine the dosing rate required to achieve a certain steady-state drug concentration by drawn a line between the point of intersection and the desired steady-state concentration. The line crosses the y-axis at a point corresponding to the dosing rate required to achieve the desired steady-state drug concentration.

Figure 13.8  The direct linear plot.

250  Nonlinear Pharmacokinetics Practice Problems: Question: A 32-year-old, 75-kg female has been taking 200 mg of phenytoin daily. Because her average phenytoin plasma concentration was only 6 mg/L, her phenytoin dose was increased to 350 mg/day. The average steady-state phenytoin concentration achieved was 21 mg/L. (Vd of phenytoin is 0.75 L/kg, and F = 1). a Using the direct linear plot, calculate phenytoin Vmax and Km in this patient. b Calculate the dose required to achieve an average steady-state phenytoin plasma concentration of around 15 mg/L. c Calculate phenytoin half-life at steady state while the patient was taking 350 mg/day. d Because of poor seizure control, phenobarbitone was added to the patient’s medications. After several weeks, phenytoin plasma concentration was 14 mg/L, while taking 360 mg/day phenytoin. Comment on the decrease in phenytoin plasma concentration. Answer: a From the direct linear plot in Figure 13.9: Vmax = 500 mg/day K m = 9 mg/L b The dose required to achieve steady-state concentration of 15 mg/L can be determined graphically and mathematically. Graphically from Figure 13.9, the dose is approximately 315 mg/day Mathematically:

FD Vmax Cpss = τ K m + Cpss

FD 500 mg/day 15 mg/L = = 312.5 mg/day τ 9 mg/L + 15 mg/L

Figure 13.9  Estimation of the Vmax, Km, and the dose from the direct linear plot.

Nonlinear Pharmacokinetics 251 c Steady state when the dose was 350 mg/day = 21 mg/L t1/2 =

0.693 Vd (K m + Cp) Vmax

t1/2 =

0.693(0.7 L/kg × 75 kg)  (9 mg/L + 21 mg/L) = 2.18 days 500 mg/day

d Phenobarbitone is an enzyme inducer, which can increase the rate of phenytoin metabolism due to the increase in its Vmax. The increase in Vmax due to enzyme induction results in decreasing phenytoin steady-state concentration. 13.6.3  The Linear Transformation Method

The linear transformation methods involve mathematically manipulating Eq. 13.11, which is a nonlinear equation, to obtain an equation in the form of the straight-line equation as in Eqs. 13.15 and 13.16 (13). FD Vmax Cpss = τ K m + Cpss FD FD Km + Cpss = Vmax Cpss(13.15) τ τ Rearrangement and dividing by F Cpss  D D/τ Vmax = −Km + (13.16) τ Cpss F Equation 13.16 is in the form of a straight-line equation. A plot of the dosing rate (D/τ) versus the dosing rate divided by the average steady-state concentration yields a straight line with a slope equal to −Km and Y-intercept equal to Vmax/F as in Figure 13.10.

Figure 13.10 A plot of the dosing rate versus the dosing rate divided by the steady-state drug concentration. The slope is equal to −Km, and the y-intercept is Vmax/F.

252  Nonlinear Pharmacokinetics The parameters Vmax and Km estimated from the slope and the y-intercept of the plot can be used to calculate the dosing rate required to achieve a certain Cpss or conversely, to calculate the expected Cpss from administration of a given dosing rate. 13.7  Multiple Elimination Pathways In the previous discussion, it was assumed that the drug is eliminated by single saturable elimination pathway that follows Michaelis-Menten kinetics. Many drugs are eliminated by more than one elimination pathway. If the drug is eliminated by two saturable elimination pathways that follow Michaelis-Menten kinetics, each pathway will have its own Vmax and Km as in Figure 13.11A. The rate of drug metabolism will be the sum of the rates for the two different pathways as in Eq. 13.17. Rate =

Vmax 1 Cp Vmax 2 Cp + (13.17) K m 1 + Cp K m 2 + Cp

As the drug concentration increases, one of the pathways is saturated first then the second pathway will be saturated at different drug concentration. The fraction of dose that is eliminated by each of the pathways is dependent on the relative magnitude of the average clearances associated with each pathway. Since the clearances for the two pathways are dependent on the drug concentration, the fraction of dose eliminated by each pathway is different after administration of different doses. The drug will always follow nonlinear pharmacokinetics.

Figure 13.11 Schematic presentation of (A) the pharmacokinetic model for a drug eliminated by two parallel saturable elimination pathways that follow Michaelis-Menten kinetics. While part (B) represents the pharmacokinetic model for a drug eliminated by two elimination pathways; one saturable pathway that follows Michaelis-Menten kinetics and the other follows first-order kinetics.

Nonlinear Pharmacokinetics 253 Some drugs are eliminated by two elimination pathways and one of them is saturable and the other follows first-order kinetics, as in Figure 13.11B. The rate of drug elimination in this case is the sum of the rates of the two pathways as in Eq. 13.18. Rate =

Vmax Cp + k Vd Cp(13.18) K m + Cp

Administration of increasing doses of the drug causes reduction in the clearance of the saturable elimination pathways, while the clearance of the pathway that follows firstorder kinetics remains constant. This makes larger fraction of the administered doses eliminated via the first-order pathway with the increase in dose. The increase in drug concentration makes the contribution of the saturable pathway to the CLT smaller, and larger fraction of the drug dose is eliminated by the first-order pathway. 13.8 The Effect of Changing the Pharmacokinetic Parameters on the Drug Concentration-Time Profile After single IV dose of a drug that is eliminated by single elimination pathway that follows Michaelis-Menten kinetics, initially the drug concentration declines slowly. Then the rate of decline increases as the drug concentration decreases because the drug elimination is concentration dependent, while, during multiple drug administration, the steadystate concentration is dependent on the dose and the drug CLT that is also dependent on the drug concentration. 13.8.1  The Dose

The effect of administration of increasing doses of the drug:

• The initial plasma drug concentration after single IV drug administration is proportional to the administered dose, since Vd is constant.

• The increase in dose causes more than proportional increase in AUC. • The drug CLT is lower, and the half-life is longer at higher drug concentration. So, the rate of decline in the plasma drug concentration is slower at higher drug concentration and faster at lower drug concentration. • At any given concentration, the rate of decline in the drug concentration is similar since Km and Vmax are constant. • During multiple administration, higher doses result in more than proportional increase in the steady-state concentration, and it takes longer time to achieve steady state because the half-life is longer at higher concentration. Clinical Importance:

• The dose of drugs that are eliminated by saturable metabolism that follows Michaelis-

Menten kinetics should be selected accurately. This is because small change in the dose of drugs such as phenytoin can lead to significant change in steady-state concentration during multiple drug administration.

254  Nonlinear Pharmacokinetics 13.8.2  The Vmax

The effect of changing the Vmax such as in case of enzyme induction:

• Higher Vmax results in higher CLT and shorter half-life. This leads to faster rate of decline in the drug concentration and smaller AUC.

• During multiple administration of the same dose, higher Vmax results in lower steadystate concentration because of the higher clearance, and shorter time to achieve steady state because of the shorter half-life.

Clinical Importance:

• Enzyme induction can lead to increase in drug clearance and decrease in steady-state drug concentration. Patients receiving phenytoin and started taking an enzyme inducer may need to increase their phenytoin doses to keep phenytoin concentrations within the therapeutic range.

13.8.3  The Km

The effect of changing Km as in case of the presence of competitive inhibitor to drug metabolism:

• Larger Km results in lower CLT and longer half-life, leading to slower rate of decline in the drug concentration and larger AUC after single drug administration.

• During multiple administration of the same dose, larger Km results in higher steady-

state concentration because of the lower clearance and the time to achieve steady-state concentration is longer because of the longer half-life.

Clinical Importance:

• The presence of a competitive inhibitor can slow the rate of drug metabolism at any

given drug concentration. Patients receiving phenytoin and started taking an enzyme inhibitor such as fluoxetine and chloramphenicol may need to decrease their phenytoin doses to keep phenytoin concentration within the therapeutic range.

13.9 Summary

• Nonlinear pharmacokinetic behavior of drugs is observed if at least one of the absorp-

tion, distribution, metabolism, or excretion processes does not follow first-order kinetics.

• Nonlinear pharmacokinetics is characterized by concentration-dependent pharmacoki-

netic parameters, and lack of linear relationship between dose and AUC after single dose, and between dosing rate and steady-state concentration during multiple administration. • Drugs that are metabolized by saturable elimination pathways have lower clearance and longer elimination half-lives at higher drug concentration. • Calculation of the dosing requirement and dose adjustment for the drugs that follow Michaelis-Menten kinetics can be very challenging because of the nonlinear relationship between the dose and steady-state concentration. • The use of plasma drug concentration as a guide for optimization of drug therapy for drugs that follow Michaelis-Menten kinetics is very useful to ensure optimal disease control.

Nonlinear Pharmacokinetics 255 Practice Problems 13.1 A 78-kg, 28-year-old man is receiving phenytoin for the treatment of seizures. When this patient was taking a daily oral dose of 250-mg phenytoin his steady state-plasma concentration was 7.1 mg/L. Because phenytoin plasma concentration was well below the therapeutic range, the patient’s daily dose was increased to 450-mg phenytoin, which resulted in steady-state plasma concentration of 30 mg/L. (Assume that the absolute bioavailability of oral phenytoin is 100%, and that the volume of distribution of phenytoin is 50 L) a Graphically calculate the patient’s Vmax and Km. b Calculate phenytoin half-life in this patient at steady state while taking 450 mg daily. c What is the steady-state concentration that should be achieved if the dose was 300 mg daily? d What is the daily phenytoin dose that should achieve a steady-state phenytoin plasma concentration of 20 mg/L in this patient? 13.2 A 65-kg male was started on 260-mg phenytoin daily at bed time to control his seizures. After three months of phenytoin therapy, his serum phenytoin concentration was found to be 5 mg/L. Because of the poor control of his seizures, his phenytoin dose was increased to 240 mg of phenytoin twice daily. One month after starting this new dosage regimen, the serum phenytoin concentration was found to be 23 mg/L. The volume of distribution of phenytoin is 0.7 L/kg. a Calculate the Vmax and Km using a graphical method. b Because the therapeutic range of phenytoin is 10–20 mg/L, recommend a dosage regimen to maintain steady-state serum phenytoin concentration around 15 mg/L. c What will be the half-life of phenytoin at steady state if a dose of 200 mg every 12 hr was given to this patient? d If the phenytoin dose was changed and the steady-state serum phenytoin concentration was found to be exactly equal to the value of Km, you determine in part a in this problem. What will be the phenytoin elimination rate during this dosage regimen? 13.3 A 65-kg, 21-year-old woman is receiving phenytoin for the treatment of seizures. When she was taking a daily oral dose of 300 mg phenytoin, her steady-state plasma concentration was 4.9 mg/L. Because phenytoin plasma concentration was well below the therapeutic range, the patient’s dose was increased to 500 mg phenytoin, which resulted in steady-state plasma concentration of 20 mg/L. (Assume that the absolute bioavailability of oral phenytoin is 100%, and the volume of distribution of phenytoin is 0.7 L/kg) a Graphically calculate the patient’s Vmax and Km. b Recommend a daily dose of phenytoin for this patient to achieve phenytoin plasma concentration of 15 mg/L at steady state. c Calculate the half-life and the total body clearance when the plasma concentration is 15 mg/L. d What will be the steady-state concentration that should be achieved if the dose was 450 mg daily?

256  Nonlinear Pharmacokinetics 13.4 A 70-kg, 34-year-old man is receiving phenytoin for the treatment of seizures. When this patient was taking a daily oral dose of 350-mg phenytoin, his steadystate plasma concentration was 4.5 mg/L. Because phenytoin plasma concentration was well below the therapeutic range, the patient’s dose was increased to 600-mg phenytoin, which resulted in steady-state plasma concentration of 17.7 mg/L. (Assume that the absolute bioavailability of oral phenytoin is 100%.) a Graphically calculate the patient’s Vmax and Km. b Although the patient’s phenytoin plasma concentration is in the therapeutic range, the seizures were not controlled. Phenobarbitone, another drug used in the treatment of seizures which is known to be an enzyme inducer, was added to the patient drug therapy. Four months later, while still taking 600-mg phenytoin/day in addition to the phenobarbitone, the patient phenytoin plasma concentration was found to be 15 mg/L. Calculate the patient’s new pharmacokinetic parameter responsible for this decrease in phenytoin plasma concentration (Vmax and Km). c Recommend a daily dose of phenytoin at this induction state for this patient in order to achieve steady-state plasma concentration of 20 mg/L. 13.5 A 25-kg, 14-year-old female was admitted to the hospital because of frequent episodes of seizures. She was diagnosed as having epileptic seizures and she was started on IV phenytoin. She received a loading dose of 5 mg/kg followed by 80-mg phenytoin IV given every 12 hr. At steady state, the average plasma phenytoin concentration was found to be 7.8 mg/L. The phenytoin dose was increased to 100-mg phenytoin IV given every 12 hr. At steady state, the plasma phenytoin concentration was 12.6 mg/L. Because the patient condition was stable, the IV phenytoin was replaced with phenytoin oral suspension, which is known to have 100% bioavailability. The patient went home with a prescription for 225-mg phenytoin suspension at bed time every day. (The volume of distribution of phenytoin is 0.8 L/kg.) a Graphically estimate phenytoin Vmax and Km in this patient. b What is the expected average steady-state phenytoin concentration in this patient while taking the 225-mg phenytoin suspension at night? c Calculate phenytoin CLT and half-life in this patient while taking the phenytoin suspension (225-mg phenytoin at bed time) at steady state. References 1. Evans WE, Schentag JJ and WJ J “Applied pharmacokinetics: Principles of therapeutic drug monitoring” 3rd Edition (1992) Lippincott Williams & Wilkins; Philadelphia, PA, USA, pp 2–33. 2. Ludden TM “Nonlinear pharmacokinetics. Clinical applications” (1991) Clin Pharmacokinet; 20:430–432. 3. Hsu FH, Prueksritanont T, Lee MG and Chiou WL “The phenomenon and cause of the dosedependent oral absorption of chlorothiazide in rats: Extrapolation to human data based on the body surface area concept” (1987) J Pharmacokinet Biopharm; 15:369–386. 4. Yu HY, Shen YZ, Sugiyama Y and Hanano M “Dose-dependent pharmacokinetics of valproate in guinea pigs of different ages” (1987) Epilepsia; 28:680–687. 5. Levy G “Pharmacokinetics of salicylate in man” (1979) Drug Metab Rev; 9:3–19. 6. Ludden TM, Hawkins DW, Allen JP and Hoffman SF “Optimum phenytoin dosage regimen” (1976) Lancet; 1(7954):307–308.

Nonlinear Pharmacokinetics 257 7. Perrier D, Ashley JJ and Levy G “Effect of product inhibition in kinetics of drug elimination” (1973) J Pharmacokinet Biopharm; 1:231–242. 8. Levy RH “Time-dependent pharmacokinetics” (1983) Pharmacol Ther; 17:383–392. 9. Wagner J “Properties of the Michaelis-Menten equation and its integrated form which are useful in pharmacokinetics” (1973) J Pharmacokinet Biopharm; 1:103–121. 10. Wagner JG, Szpunar GJ and Ferry JJ “Michaelis-Menten elimination kinetics: Areas under curves, steady-state concentrations, and clearances for compartment models with different types of input” (1985) Biopharm Drug Dispos; 6:177–177. 11. Mullen PW “Optimal phenytoin therapy: A new technique for individualizing dosage” (1978) Clin Pharmacol Ther; 23:228–228. 12. Vozeh S, Muir KT, Sheiner LB and Follath F “Predicting individual phenytoin dosage” (1981) J Pharmacokinenet Biopharm; 9:131–146. 13. Mullen PW and Foster RW “Comparative evaluation of six techniques for determining the Michaelis-Menten parameters relating phenytoin dose and steady-state serum concentrations” (1979) J Pharm Pharmacol; 31:100–104.

14 Multicompartment Pharmacokinetic Models

Objectives After completing this chapter, you should be able to:

• Describe the differences between the one-compartment and the two-compartment pharmacokinetic models.

• Define all the parameters of the two-compartment pharmacokinetic model. • Describe the plasma concentration-time profile after single IV and oral administration of drugs that follow the two-compartment pharmacokinetic model.

• Calculate all the pharmacokinetic parameters of the two-compartment pharmacokinetic model from plasma concentrations obtained after single IV administration.

• Analyze the effect of changing one or more of the pharmacokinetic parameters on the plasma concentration-time profile after administration of drugs that follow twocompartment pharmacokinetic model. • Describe the general steps for compartmental modeling. • Discuss the general approaches used to evaluate the goodness of compartmental model fit. 14.1 Introduction Drug distribution from the systemic circulation to the different parts of the body can be demonstrated by a beaker filled with liquid and its inner wall is covered with a coating material as in Figure 14.1. The liquid in the center of the beaker represents the systemic circulation and the coating material covering the beaker wall represents the tissues. If a drop of dye is added to the liquid, the dye is distributed in the liquid first and then to the coating material. If the dye is rapidly distributed to the coating material, the dye concentration in the liquid and the coating material will be different, but the distribution equilibrium is achieved rapidly. This means that the dye concentration in the liquid and the coating material becomes constant shortly after addition of the dye. In this case, the beaker behaves as single homogenous compartment despite the difference in dye concentration in the liquid and coating material. This is the model that we used previously to describe the drug distribution process since we assumed that the drug is rapidly distributed to all parts of the body once it enters the systemic circulation. When the drug is eliminated from the body, the drug concentrations in the systemic circulation and in all tissues decline at the same rate. Drugs exhibiting this behavior follow the one-compartment pharmacokinetic model. After IV bolus administration of drugs that follow this behavior and eliminated by first-order process, the plasma drug concentration DOI: 10.4324/9781003161523-14

Multicompartment Pharmacokinetic Models 259

Figure 14.1 A model representing the distribution of the dye from the liquid in the center of the beaker to the beaker wall coating material. The distribution process causes decrease in the dye concentration in the liquid and increase in the dye concentration in the coating material. When the dye distribution to the beaker wall coating material (A → B) is fast, the beaker behaves as one compartment, while when the dye distribution to the beaker wall coating material (A → B) is slow, the beaker behaves as if it consists of two different compartments.

declines exponentially, and the plasma drug concentration-time profile is linear on the semilog scale. The distribution of some drugs from the systemic circulation to the different tissues is slow. The beaker model described in Figure 14.1 can be applied to demonstrate slow distribution. After addition of the dye, it is distributed rapidly in the liquid. Then the dye starts to distribute slowly to the beaker wall coating material. The distribution of the dye to the coating material causes gradual decrease in the dye concentration in the liquid and increase in the dye concentration in the coating material. The dye concentration in the liquid reaches a constant value when equilibrium is established between the dye in the liquid and the coating material as illustrated in Figure 14.2. After IV administration of a drug that is slowly distributed to the tissues, the drug is immediately distributed in the systemic circulation and other highly perfused tissues. Initially, drug concentration in the systemic circulation decreases rapidly due to drug distribution to the tissues and drug elimination from the body. When equilibrium is established between the drug in the systemic circulation and all tissues, drug concentration in the systemic circulation declines at a rate dependent on the rate of drug elimination that is

Figure 14.2 The dye concentration-time profile in the liquid with the decrease in the dye concentration representing the slow distribution of the dye from the liquid in the center of the beaker to the beaker wall coating material.

260  Multicompartment Pharmacokinetic Models slower than the initial rate of decline. After IV bolus administration of drugs that follow this behavior and are eliminated by first-order process, the plasma drug concentrationtime profile has an initial rapid decline phase followed by a linear decline on the semilog scale. These drugs follow the two-compartment, or any other multicompartment pharmacokinetic model. These models consist of a central compartment connected to one or more peripheral compartments. 14.2  Compartmental Pharmacokinetic Models Compartmental pharmacokinetic modeling is an approach that describes drug distribution to different parts of the body by drug distribution to different compartments. The models differ in the number of compartments, the compartment(s) where drug elimination occurs, and the arrangement of these compartments. The number of compartments in the model depends on the rate of drug distribution to the different parts of the body. If the drug in the systemic circulation is distributed rapidly to all parts of the body, the body behaves as single compartment, and the drug pharmacokinetic behavior can be described by one-compartment pharmacokinetic model. If the drug is distributed rapidly to some tissues and slowly to other tissues, the two-compartment pharmacokinetic model will be appropriate to describe the pharmacokinetic behavior of this drug. When the drug is distributed to the different parts of the body at three distinguished rates, for example rapid, slow, and very slow, the three-compartment pharmacokinetic model can be used to describe this pharmacokinetic behavior. The compartmental models are data driven because the plasma drug concentrations obtained after drug administration are used to determine the rate of the distribution process and select the appropriate compartmental model. The pharmacokinetic behavior of most drugs can be described by one-, two-, or three-compartment pharmacokinetic models; however, models with more compartments can be used if the obtained data can support these complicated models (1). Pharmacokinetic models that have the same number of compartments can be different when drug elimination occurs from different compartments (2). The different twocompartment pharmacokinetic models presented in Figure 14.3A differ in the compartment where drug elimination takes place. For the three-compartment pharmacokinetic model, there are seven different possibilities for the compartment(s) where drug elimination takes place. Also, compartmental models can differ in the way the compartments are arranged. The models presented in Figure 14.3B are different examples of the threecompartment pharmacokinetic models. Clinical Importance:

• Pharmacokinetic modeling in general involves the development of models that can quantitatively describe the pharmacokinetic behavior of the drug in the body.

• Selection of the appropriate model can help in predicting the drug behavior in the body in different scenarios to optimize drug therapy and reduce drug toxicity.

14.3  The Two-Compartment Pharmacokinetic Model The model consisting of a central compartment connected to a peripheral compartment with elimination from the central compartment is the model commonly used to describe the pharmacokinetic behavior of drugs that follow two-compartment model. So, this

Multicompartment Pharmacokinetic Models 261

Figure 14.3 The diagram represents different compartmental pharmacokinetic models. (A) Examples of two-compartment pharmacokinetic models with elimination from compartment 1, compartment 2, or both compartments. (B) Examples of the possible three-compartment pharmacokinetic models that differ in the arrangement of the compartments.

model will be used in the following discussion. After IV bolus dose, the drug is distributed rapidly to the body spaces and tissues that are presented by the central compartment. Then the drug is distributed by a first-order process from the central compartment to the other body spaces and tissues that are presented by the peripheral compartment. Because the blood is usually part of the central compartment, the drug in the central compartment can be delivered to the eliminating organ(s). So, distribution and elimination occur simultaneously after drug administration that causes rapid decline in the drug concentration in the central compartment. After completion of the distribution process and establishing the equilibrium between the drug in the central compartment and the drug in the peripheral compartment, the drug concentration in the central compartment declines at a rate dependent on drug elimination. The rate of decline in the drug concentration in the central compartment due to drug elimination is slower than the initial rate of decline caused by distribution and elimination. So, the plasma drug concentration-time profile that represents the drug profile in the central compartment consists of two phases on the semilog scale. An initial distribution phase characterized by rapid decline in drug concentration, followed by a terminal elimination phase with slower rate of decline in drug concentration. The drug concentration-time profile during the terminal elimination phase is linear on the semilog scale since its decline is dependent only on the rate of drug elimination (3). A typical plasma concentration-time profile after IV bolus administration of drugs that follow two-compartment pharmacokinetic model is presented in Figure 14.4. After an IV dose of a drug that follows the two-compartment pharmacokinetic model, the model assumes that at time zero, the total dose is in the central compartment and there is no drug in the tissues presented by the peripheral compartment. The drug in the central compartment declines rapidly due to the distribution of drug from the central compartment to the peripheral compartment and, due to drug elimination, that occurs simultaneously. The drug in the central compartment is transferred to the peripheral compartment and can return back to the central compartment by first-order processes. Initially the amount of the drug in the central compartment is larger than the amount of

262  Multicompartment Pharmacokinetic Models

Figure 14.4 The plasma concentration-time profile of a drug that follows the two-compartment pharmacokinetic model after single IV bolus administration.

the drug in the peripheral compartment, so the net drug transfer is from the central compartment to the peripheral compartment. This means that initially the amount of drug in the peripheral compartment increases with time. As the amount of drug in the peripheral compartment increases, the rate of drug transfer from the peripheral to the central compartment approaches that from the central to the peripheral compartment. When these two rates become equal, the amount of drug in the peripheral compartment reaches a maximum value. Because the drug is continually eliminated from the central compartment, the amount of drug in the central compartment decreases and the rate of drug transfer from the peripheral to the central compartment becomes larger than that from the central to peripheral compartment. The net drug transfer is from the peripheral to the central compartment, and the amount of drug in the peripheral compartment starts to decline parallel to the decline in the amount of the drug in the central compartment. Figure 14.5 represents the drug concentration-time profiles in the central and peripheral compartments after administration of single IV bolus dose. The concentration of drug in the central compartment is determined by dividing the amount of

Figure 14.5 The drug concentration-time profile in the central and peripheral compartments after administration of single IV bolus dose of a drug that follows two-compartment pharmacokinetic model.

Multicompartment Pharmacokinetic Models 263

Figure 14.6 A block diagram that represents the two-compartment pharmacokinetic model with first-order transfer between the central and peripheral compartments and first-order drug elimination from the central compartment.

drug in the central compartment by the volume of the central compartment. Likewise, the concentration of drug in the peripheral compartment is determined by dividing the amount of drug in the peripheral compartment by the volume of the peripheral compartment. Clinical Importance:

• After IV administration of some drugs, the intensity of drug effect gradually increases

until the maximum intensity is achieved when the plasma drug concentration is decreasing. There are several possibilities for this behavior. One of them is that the site of drug effect is a tissue or organ that is part of the peripheral compartment. In this case, the maximum drug concentration at the site of action (part of the peripheral compartment), which is associated with the highest intensity of drug effect, is achieved after certain time of IV drug administration.

14.4  The Parameters of the Two-Compartments Pharmacokinetic Model The two-compartment pharmacokinetic model presented by the block diagram in Figure 14.6 assumes that the drug transport between the central and peripheral compartments follows first-order kinetics, and that the drug is eliminated from the central compartment by a firstorder process. 14.4.1  Definition of the Pharmacokinetic Parameters

The pharmacokinetic parameters used to derive the equations for the two-compartment pharmacokinetic model can be defined as follows: X Y k12

is the amount of drug in the central compartment and has units of mass. is the amount of drug in the peripheral compartment and has units of mass. is the first-order transfer rate constant from the central compartment to the peripheral compartment and has units of time-1.

264  Multicompartment Pharmacokinetic Models k21 k10 A and B α β t1/2 α t1/2 β Vc

is the first-order transfer rate constant from the peripheral compartment to the central compartment and has units of time-1. is the first-order elimination rate constant from the central compartment and has units of time-1. are the hybrid coefficients and have units of concentrations. is the hybrid first-order rate constant for the distribution process and has units of time-1. is the hybrid first-order rate constant for the elimination process and has units of time-1. is the half-life for the distribution process and has units of time. is the half-life for the elimination process and has units of time. is the volume of the central compartment, and it is the volume of distribution of the drug right after IV drug administration and has units of volume. This term relates the administered dose to the initial plasma drug concentration (central compartment concentration) after administration of IV bolus dose. Cp0 =

Vdss

Dose Vc

(14.1)

is the volume of distribution of the drug at steady state and has units of volume. This term relates the amount of the drug in the body and the plasma drug concentration at steady state. Amount of the drug in the body at steady state = Vdss Cpss

(14.2)

Vdβ or Vdarea is the volume of distribution of the drug during the elimination phase and has units of volume. This term relates the amount of the drug in the body and the plasma drug concentration during the elimination phase (β-phase). Amount of the drug in the body during the elimination phase (14.3) = Vdβ Cpβ-phase 14.4.2 The Mathematical Equation That Describes the Plasma Concentration-Time Profile for Drugs That Follow Two-Compartment Pharmacokinetic Models

After single IV bolus dose and based on the pharmacokinetic model in Figure 14.6, the rate of change of the amount of the drug in the central compartment (X) at any time is dependent on the rates of the three processes affecting the amount of drug in the central compartment. This is equal to the rate of drug transfer from the peripheral compartment to the central compartment minus the rate of drug transfer from the central compartment to the peripheral compartment, minus the rate of drug elimination. Similarly, the rate of change of the amount of drug in the peripheral compartment (Y) is equal to the rate of drug transfer from the central compartment to the peripheral compartment minus rate of drug transfer from the peripheral compartment to the central compartment. The differential equations that describe these two rates are as follows: dX = k21 Y − k12 X − k10 X dt

(14.4)

Multicompartment Pharmacokinetic Models 265 and dY = k12 X − k21 Y(14.5) dt Integrating the first differential equation yields the equation for the amount of drug in the central compartment (X) as a function of time after single IV bolus dose (D): X=

D(α − k21) −α t D(k21 − β) −β t e + e (14.6) (α − β) (α − β)

This equation contains two exponents, one exponent describes the distribution process and the other describes the elimination process. These exponents contain the hybrid rate constants for the distribution and elimination processes, α and β. While getting the integrated equation for the amount of the drug in the central compartment, the following two relationships were obtained: α + β = k12 + k21 + k10 (14.7) α β = k21 k10(14.8) where α=

1 (k12 + k21 + k10 ) + (k12 + k21 + k10 )2 − 4 k21k10 (14.9)    2

β=

1 (k12 + k21 + k10 ) − (k12 + k21 + k10 )2 − 4 k21 k10 (14.10)  2

and

Since the distribution process is usually faster than the elimination process, the larger hybrid rate constant α is the rate constant for the distribution process and the smaller hybrid rate constant β is the rate constant for the elimination process as in Eqs. 14.9 and 14.10. During the distribution phase, the drug distribution rate does not depend only on k12, the transfer rate constant from the central to peripheral compartment. This is because while the drug is distributing from the central to the peripheral compartment, there is drug returning back to the central compartment at a rate dependent on the rate constant k21, and also there is elimination from the central compartment affected by the rate constant k10. So, the observed rate of the distribution process is described by the hybrid rate constant α, which is dependent on the three rate constants k12, k21, and k10 as in Eq. 14.9. Similarly, the drug elimination rate does not depend only on k10, the elimination rate constant from the central compartment. This is because during the elimination of the drug from the central compartment, there is drug transfer from the central to the peripheral compartment at a rate dependent on the rate constant k12, and drug returning back to the central compartment at a rate dependent on the rate constant k21. So, the observed rate for the elimination process is described by the hybrid rate constant β, which is dependent on the three rate constants k12, k21, and k10 as in Eq. 14.10. The first-order rate constants k12, k21, and k10 are usually termed the micro rate constants, while α and β are termed the macro rate constants.

266  Multicompartment Pharmacokinetic Models Dividing Eq. 14.6 by the volume of the central compartment, Vc gives the equation for the drug concentration in the central compartment, and hence the plasma drug concentration, at any time after a single IV bolus dose. Cp =

D(α − k21) −αt D(k21 − β) −βt e + e (14.11) Vc (α − β) Vc (α − β)

which can be simplified to Cp = A e−α t + B e−β t (14.12) where A=

D(α − k21) (14.13) Vc (α − β)

B=

D (k21 − β) (14.14) Vc (α − β)

and

Equation 14.11 and its simplified form Eq. 14.12 are the equations that describe the plasma drug concentration at any time after single IV bolus dose of a drug that follows the two-compartment pharmacokinetic model (4). These equations include two exponents: one describes the distribution process and includes the larger hybrid rate constant α, and the other describes the elimination process and includes the smaller hybrid rate constant β. As time elapses after IV drug administration, the exponential term that has the distribution (larger) hybrid rate constant approaches zero and the plasma concentration declines at a rate dependent on the hybrid elimination rate constant β. So, the plasma drug concentration-time profile after single IV bolus dose on the semilog scale has a rapidly declining distribution phase and a linear terminal elimination phase. 14.5 Determination of the Two-Compartment Pharmacokinetic Model Parameters The pharmacokinetic parameters k21, Vc, α, and β, in Eq. 14.11, or the parameters A, B, α, and β in Eq. 14.12, can be estimated from the plasma drug concentrations obtained after single IV dose (D) of the drug using nonlinear regression analysis utilizing specialized data analysis software. The estimated parameters in both equations can be used to calculate all the other model parameters. The pharmacokinetic parameters allow prediction of the drug steady-state plasma concentration during repeated drug administration and determination of the dose required to achieve certain drug concentration at steady state. The pharmacokinetic parameters in Eq. 14.12, A, B, α, and β can be estimated graphically utilizing the method of residuals. 14.5.1  The Method of Residuals

The method of residuals is a graphical method used to estimate the two-compartment pharmacokinetic model parameters after single IV dose. The basic principle of the method of

Multicompartment Pharmacokinetic Models 267

Figure 14.7 The method of residuals is applied to separate the two exponential terms of the equation that describes the plasma concentration-time profile of drugs that follow the twocompartment pharmacokinetic model after single IV dose.

residuals is to separate the two exponential terms in Eq. 14.12 as illustrated in Figure 14.7. The method of residuals can be summarized by the following steps:

• The experimentally obtained plasma drug concentrations are plotted against their corresponding time values on the semilog scale.

• The plasma drug concentrations that decline linearly during the elimination phase are

identified. The best line that passes through these points is drawn and the line is back extrapolated to the y-axis. • This line is corresponding to the (B e−βt) term in the biexponential equation. The yintercept of the extrapolated line is equal to the coefficient B in the equation. The hybrid rate constant β can be determined from the slope of the line (slope = −β/2.303). Also, the t1/2 β can be determined directly from the line by calculating the time required for any concentration on the line to decrease by 50%. Then β is calculated as follows: β=

0.693 (14.15) t1/2 β

• The residuals are calculated from the difference between the observed plasma concen-

tration-time data and the corresponding values at the same time on the extrapolated line representing the elimination phase. • The residuals are plotted versus their corresponding time values. • The residual versus time plot is linear on the semilog scale and this line is corresponding to the (A e−αt) term in the biexponential equation. The y-intercept of this line is equal to the coefficient A in the equation. The hybrid rate constant α can be determined from the slope of the line (slope = −α/2.303). Also, the t1/2 α can be determined directly from the line by calculating the time required for any point on the line representing the α-phase to decrease by 50%. Then the hybrid rate constant α is calculated as follows: α=

 0.693 (14.16) t1/2 α

268  Multicompartment Pharmacokinetic Models

• A is the y-intercept of the faster process (the process with shorter t1/2, the distribution process), and B is the y-intercept of the slower process (the process with longer t1/2, the elimination process).

14.5.2  Determination of the Other Model Parameters

After estimation of A, B, α, and β using the method of residuals, the other model parameters can be calculated (1). 14.5.2.1  Volume of the Central Compartment, Vc

After IV bolus administration, the drug is distributed initially in the central compartment. So, the volume of the central compartment can be determined from the dose and the initial drug concentration in the central compartment that is the same as the initial plasma drug concentration. The plasma concentration at time zero is determined from Eq. 14.12 by substituting for the time by zero, and it is equal to (A + B). Vc =

Dose Dose = Cp0 A+B

(14.17)

14.5.2.2  The Area under the Plasma Concentration-Time Curve, AUC

The area under the plasma concentration-time curve is determined by Eq. 14.18, which is obtained by integrating Eq. 14.12, that describes the plasma concentration-time profile, from time 0 to ∞. AUC |tt =∞ =o =

A B + α β

(14.18)

14.5.2.3  The Total Body Clearance, CLT

The CLT is determined from the dose and the AUC similar to the one-compartment pharmacokinetic model. CL T =

Dose AUC |tt =∞ =o

(14.19)

14.5.2.4  The First-Order Elimination Rate Constant from the Central Compartment, k10

The CLT is the product of the first-order elimination rate constant k10 and Vc. When Vc and CLT are known, k10 can be calculated. CL T = k10 Vc k10 =

CL T Vc

(14.20) (14.21)

14.5.2.5 The First-Order Transfer Rate Constant from the Peripheral Compartment to the Central Compartment, k21

As a result of integrating the differential equation to obtain the integrated equation for the amount of the drug in the central compartment, the relationship in Eq. 14.8 has been obtained (α β = k21 k10). Based on this relationship, k21 can be calculated. k21 =

αβ k10

(14.22)

Multicompartment Pharmacokinetic Models 269 14.5.2.6 The First-Order Transfer Rate Constant from the Central Compartment to the Peripheral Compartment, k12

Also, while getting the integrated equation for the amount of the drug in the central compartment, the relationship in Eq. (14.7) has been obtained (α + β = k12 + k21 + k10). Based on this relationship, k12 can be calculated. k12 = (α + β) − (k21 + k10 )

(14.23)

14.5.3 Determination of the Volumes of Distribution for the Two-Compartment Pharmacokinetic Model

In the one-compartment pharmacokinetic model, the drug is distributed rapidly to all parts of the body, establishing the distribution equilibrium once the drug enters the systemic circulation. So, the body behaves as one homogenous compartment that has one volume of distribution. However, in the two-compartment pharmacokinetic model, there is more than one volume of distribution. Initially, after IV drug administration, the drug is distributed in the central compartment only and the drug volume of distribution is equal to Vc. Then the drug distributes from the central compartment to the peripheral compartments. During steady state, the drug volume of distribution is equal to Vdss, while during the elimination phase, the drug volume of distribution is equal to Vdβ. The volume of the central compartment, Vc, is the smallest volume, Vdβ is the largest volume, while Vdss is larger than Vc and smaller than Vdβ. Vc can be calculated from the dose of the initial drug concentration as in Eq. 14.17. 14.5.3.1  The Volume of Distribution at Steady State, Vdss

The Vdss relates the amount of the drug in the body and the plasma drug concentration during steady state when the drug is administered by constant rate IV infusion and the drug concentration in the body is constant. When the drug concentration is constant, the rate of drug transfer from the central to the peripheral compartment is equal to the rate of drug transfer from the peripheral to the central compartment. Also, a transient steady state is achieved for a moment after single IV administration when the drug concentration in the peripheral compartment reaches its maximum value, and the net rate of drug transfer between the central and peripheral compartments is equal to zero. The rate of drug transfer can be expressed by the drug transfer rate constant and the amount of the drug in each compartment. At steady state: k12 X = k21 Y Y=

(14.24)

X k12 Cpss Vc k12 = k21 k21

(14.25)

Since the amount of the drug in the central compartment is the product of the plasma drug concentration and Vc, and at steady state, Vdss relates the amount of the drug in the body to the drug concentration in plasma, Eq. 14.26 that describes Vdss can be derived. Vdss =

X + Y Cpss Vc + Cpss Vc (k12 /k21) = Cpss Cpss

Vdss = Vc + Vc

k12 k21

(14.26) (14.27)

270  Multicompartment Pharmacokinetic Models k   Vdss = Vc  1 + 12   k21 

(14.28)

14.5.3.2  The Volume of Distribution during the Elimination Phase, Vdβ

During the elimination phase, distribution equilibrium is established between the drugs in the central and peripheral compartments which makes the drug concentration in the central and peripheral compartments declines at the same rate. However, there is more drug transfer from the peripheral compartment to the central compartment to compensate for the drug elimination which occurs from the central compartment. The volume of distribution during the elimination phase can be calculated from the CLT and the firstorder hybrid elimination rate constant. Vdβ =

CL T Vc k10 = β β

(14.29)

Clinical Importance:

• Accurate determination of the model parameter requires obtaining serial plasma samples at different time points and accurate determination of the drug concentration in these samples. • Accurate determination of the model parameters allows good prediction of the drug behavior in the body to optimize drug therapeutic effect and reduce toxicity. Practice Problems: Question: After single IV bolus dose of 1000 mg of an antiarrhythmic drug, the following concentrations were obtained: Time (hr)

Concentration (mg/L)

0.2 0.5 1.0 2.0 4.0 6.0 8.0 12.0

120 84 53 29 18 15 12.5 8.8

a Using the method of residuals, calculate the following parameters: t1/2 α, t1/2 β, k12, k21, k10, Vc, Vdβ, Vdss, AUC, and CLT. b What will be the amount of drug remaining in the body after 15 hr? Answer: The method of residuals presented in Figure 14.8 can be performed by the following procedures:

• Plot the concentration versus time on the semilog scale. • Identify the best line that represents the drug elimination process. • The y-intercept is equal to B. The hybrid elimination rate constant (β) and β-half-life can be determined from the line representing the elimination process.

Multicompartment Pharmacokinetic Models 271

Figure 14.8  Application of the method of residuals in solving the practice problem.

• Calculate the residuals from the difference between the plasma drug concentrations and the values on the extrapolated line during the distribution phase.

• Plot the residuals versus time and draw the best line that goes through the points. • The y-intercept of this line is equal to A. The hybrid distribution rate constant α and α-half-life can be determined from this line.

a A = 120 mg/LB = 25 mg/L α = 1.38 hr −1

t1/2 α = 0.5 hr

β = 0.087 hr −1

t1/2 β = 8 hr

Vc =

1000 Dose = = 6.9 L A + B 120 mg/L + 25 mg/L

AUC =

A B 120 mg/L 25 mg/L + = + = 374.4 mg hr/L −1 α β 1.38 hr 0.087 hr −1

CL T =

Dose 1000 mg = = 2.67 L/hr AUC 374.4 mg hr/L

k10 =

CL T 2.67 L/hr = = 0.387 hr −1 Vc 6.9 L

k21 =

αβ 1.38 hr −1 × 0.087 hr −1 = = 0.310 hr −1 k10 0.387 hr −1

k12 = (α + β) − (k21 + k10 ) = (1.38 hr −1 + 0.087 hr −1) − (0.310 hr −1 + 0.387 hr −1) = 0.77 hr −1

Vdβ =

CL T 2.67 L/hr = = 30.7 L β 0.087 hr −1

272  Multicompartment Pharmacokinetic Models  k  0.77 hr −1   Vdss = Vc  1 + 21  = 6.7 L  1 + = 23.3 L  k21  0.310 hr −1   b Cp = A e−αt + B e−βt Cp15 hr = 120 mg/L e−1.38 hr

−1

 15 hr

+ 25 mg/L e−0.087  hr

−1

15 hr

= 6.78 mg/L

Amount of the drug in the body during the elimination phase = Cp × Vdβ Amount15 hr = Cp15 hr   × Vdβ = 6.78 mg/L × 30.7 L = 208 mg 14.6 Pharmacokinetic Behavior of Drugs that Follow the Two-Compartment Pharmacokinetic Model This section briefly discusses the pharmacokinetic behavior of drugs that follow the twocompartment pharmacokinetic model after oral administration, during constant rate IV infusion, and during multiple administration. This is in addition to the renal excretion of drugs that follow the two-compartment pharmacokinetic model. 14.6.1 Oral Administration of Drugs that Follow the Two-Compartment Pharmacokinetic Model

After oral administration of a drug that follows two-compartment pharmacokinetic model, the drug is absorbed into the systemic circulation that is part of the central compartment. Once in the central compartment, the drug can be distributed to the peripheral compartment or eliminated from the body. If the drug is rapidly absorbed, the decline in the plasma concentration-time profile after the end of the absorption phase will be biexponential reflecting the distribution and the elimination phases. The biexponential decline in the plasma drug concentration after drug absorption will be clear only if the absorption, distribution, and elimination processes proceed at three distinctive rates. However, if drug absorption is slow, the biexponential decline in the plasma concentration after the end of drug absorption may not be evident. Figure 14.9 represents the plasma concentration-time profile after a single oral dose of a drug that follows the two-compartment pharmacokinetic model. This drug profile can be described by a triexponential equation that represents the absorption, distribution, and elimination processes as in Eq. 14.30 (2). This equation can be fitted to the observed plasma drug concentrations after single oral administration to estimate the model parameters utilizing specialized pharmacokinetic data analysis software.  k FD   (k21 − α) e−αt (k − β) e−βt (k − ka ) e− ka t  Cp =  a + 21 + 21    Vc   (β − α)(k a − α) (ka − β)(α − β) (α − ka )(β − ka ) 

(14.30)

The absorption rate constant for drugs that follow linear kinetics, with zero-order or first-order absorption, and follow two-compartment pharmacokinetic model can be determined using the Loo-Reigelman method (5). This method is similar in principle to the Wagner-Nelson method that is used for the determination in the absorption rate

Multicompartment Pharmacokinetic Models 273

Figure 14.9 The plasma concentration-time profile for a drug that follows two-compartment pharmacokinetic model after administration of single oral dose.

constant as discussed in Chapter 6. The method depends on calculation of the fraction of the dose remaining to be absorbed at different time points [1−(fraction of dose absorbed)] to determine the order of the drug absorption process and to calculate the absorption rate constant. 14.6.2 Constant Rate IV Administration of Drugs That Follow the Two-Compartment Pharmacokinetic Model

During constant rate IV infusion, the plasma drug concentration-time profile increases gradually until it reaches the steady state. The steady state plasma concentration is dependent on the rate of the IV infusion and the CLT of the drug as in Eq. (14.31). This relationship is similar for drugs that follow the one- and two-compartment pharmacokinetic models. Cpss =

Infusion rate K = o (14.31) CL T CL T

The time to reach steady state during constant rate IV infusion is equal to 5–6 times the elimination half-life (t1/2 β). Termination of the IV infusion results in biexponential decline in the drug plasma concentration, a rapid distribution phase and a slow elimination phase. Administration of a loading dose may be necessary to achieve faster approach to steady state especially in emergency situations. Calculation of the loading dose is based on the desired steady-state concentration and an average value for Vc and Vdβ (Cpss × Volume). 14.6.3 Multiple Administration of Drugs That Follow the Two-Compartment Pharmacokinetic Model

Drugs that follow two-compartment pharmacokinetic model accumulate during repeated administration until steady state is achieved. At steady state, the plasma concentration will be changing during each dosing interval; however, the maximum and minimum plasma concentrations will be similar if the drug is administered as a fixed dose at equally

274  Multicompartment Pharmacokinetic Models spaced intervals. The average steady-state concentration is directly proportional to the dosing rate and inversely proportional to the CLT as in Eq. (14.32). Cpaverage ss =

F Dose (14.32) CL T τ

where τ is the dosing interval and F is the drug bioavailability. This relationship is similar for drugs that follow the one- and two-compartment pharmacokinetic models. It takes five to six elimination half-lives (t1/2 β) to reach the steady state during multiple drug administration. Administration of a loading dose may be necessary to achieve faster approach to steady state. 14.6.4 Renal Excretion of Drugs That Follow the Two-Compartment Pharmacokinetic Model

For a drug that follows two-compartment pharmacokinetic model, the amount of drugtime profile in the central compartment can be described by biexponential equation. Similarly, the renal excretion rate (dAe/dt) versus time profile can be described by biexponential equation as in Eq. 14.33. dA e = A′ e−αt + B′ e−βt (14.33) dt where A′ and B′ are the hybrid coefficients and have units of amount/time. The renal clearance can be determined from the renal excretion rate and the average plasma concentration during the urine collection interval as in Eq. 14.34. CL R =

∆A e /∆t (14.34) Cpt-mid

The renal clearance can also be determined from the total amount of the drug excreted in urine and the drug AUC. CL R =

Ae∞ (14.35) AUC |tt=∞ =o

The fraction of the drug IV dose excreted in urine is determined from the ratio of the renal clearance to the total body clearance. These relationships are similar to those for drugs that follow one-compartment pharmacokinetic model. 14.7 Effect of Changing the Pharmacokinetic Parameters on the Concentration-Time Profile of Drugs That Follow Two-Compartment Pharmacokinetic Model After single IV dose, the rate of drug distribution and elimination depend on the hybrid distribution rate constant and the elimination rate constant, respectively. During multiple drug administration, the steady-state concentration is directly proportional to the drug administration rate and inversely proportional to the CLT, and the time to reach steady state is dependent on the hybrid elimination rate constant (6). The following is a discussion of

Multicompartment Pharmacokinetic Models 275 how the change in each of the pharmacokinetic parameters affects the drug plasma concentration-time profile after single dose and the steady-state drug concentration during multiple administration of drugs that follow the two-compartment pharmacokinetic model. 14.7.1 Dose

The effect of administration of increasing doses of the drug:

• Administration of increasing dose results in proportional increase in the plasma drug concentrations, and the resulting drug concentration-time profiles are parallel.

• During multiple administration, the average steady-state drug concentration is directly proportional to the administered dose, if the CLT is constant.

14.7.2  Total Body Clearance

The effect of the change in CLT of the drug:

• Lower CLT results in slower rate of drug elimination and longer half-life if the volume

of distribution does not change. Also, the AUC is inversely proportional to the CLT after administration of the same dose. • During multiple administration of the same dose, the steady-state drug concentration is inversely proportional to the CLT. 14.7.3  Volume of the Central Compartment

The effect of the change of Vc of the drug:

• The change in Vc is accompanied by a proportional change in Vdss and Vdβ. Larger Vc

results in lower initial drug plasma concentration after administration of the same dose and similar AUC, if CLT is the same. • Multiple administration of the same dose should achieve the same average steady-state concentration because CLT is the same. However, the time to achieve steady state will be different. This is because when the volume of distribution is different and CLT is constant, the elimination half-life should be different. 14.7.4  The Hybrid Distribution and Elimination Rate Constants

The effect of the change in α and β:

• The change in the hybrid distribution rate constant results in change in the rate of completion of the distribution process, while the change in the hybrid elimination rate constant results in change in the rate of drug elimination.

14.7.5  The Inter-Compartmental Clearance

The effect of the change in the inter-compartmental clearance of the drug:

• The inter-compartmental clearance determines the central to peripheral compartment drug amount and concentration ratios when the distribution equilibrium is established

276  Multicompartment Pharmacokinetic Models during the elimination phase after single drug administration or at steady state during multiple drug administration. • The inter-compartmental clearance (Q) from the central to peripheral compartment is equal to that from the peripheral to central compartment. It is dependent on the first-order transfer rate constants between the compartments, Vc and Vp, which are the volume of the peripheral compartment as in Eq. 14.36. Q = k12 Vc = k21 Vp(14.36)

• The change in the transfer rate constants or the volumes of the two compartments

results in establishing a new distribution equilibrium that changes the central to peripheral drug amount and concentration ratios.

Clinical Importance: The relationship between the dosing rate, drug clearance, and average steady-state concentrations is similar for drugs that follow the one- or two-compartment pharmacokinetic models. So, the clinical significance for changing the drug pharmacokinetic parameters in these drugs is similar to what was discussed previously for drugs following the onecompartment pharmacokinetic model.

• Administration of increasing doses results in more intense drug effect. • Switching drug products for the same active drug may lead to change in therapeutic effect if the products have different bioavailability.

• Patients with reduced eliminating organ function may require lower doses than those with normal eliminating organ function.

• Products for that same active drug that have different absorption rate can affect the fluctuation in plasma drug concentration between doses.

14.8  The Three-Compartment Pharmacokinetic Model Development of sensitive analytical techniques made it possible to demonstrate that some drugs follow the three-compartment pharmacokinetic model. After drug administration into the central compartment, the drug is distributed slowly to the tissues. However, the distribution of the drug to some tissues is much slower than its distribution to other tissues. This results in two distinct rates of distribution and a biexponential distribution phase followed by a terminal elimination phase. Figure 14.10 is an example of the plasma concentration-time profile for a drug that follows three-compartment pharmacokinetic model. The diagram presented in Figure 14.11 is an example of a three-compartment pharmacokinetic model in which the elimination of the drug is from the central compartment, and the two peripheral compartments are connected to the central compartment. The equation that describes the plasma concentration-time profile after single IV bolus dose is triexponential with the three exponential terms describing the rapid and slow distribution processes and the elimination process. Eq. 14.37 is the mathematical expression that describes the plasma concentration-time profile for a drug that follows threecompartment pharmacokinetic model after administration of single IV bolus dose (2).  D   (k − α)(k31 − α) −αt (k21 − β)(k31 − β) −βt (k21 − γ )(k31 − γ ) − γt  Cp =    21 e + e + e  (14.37)  Vc   (β − α)(γ − α) (α − β)(γ − β) (α − γ )(β − γ ) 

Multicompartment Pharmacokinetic Models 277

Figure 14.10 The plasma concentration-time profile for a drug that follows three-compartment pharmacokinetic model after administration of single IV bolus dose.

This equation can be simplified to: Cp = A e−αt + B e−βt + C e− γt(14.38) The pharmacokinetic parameters for the three-compartment pharmacokinetic models are usually obtained by nonlinear regression analysis utilizing specialized pharmacokinetic data analysis software. Clinical Importance:

• The plasma concentration-time profile after administration of high dose methotrexate

can be described by three-compartment pharmacokinetic model. This model was used

Figure 14.11 A block diagram representing the three-compartment pharmacokinetic model with the two peripheral compartments connected to the central compartment and drug elimination from the central compartment.

278  Multicompartment Pharmacokinetic Models with other factors that can affect methotrexate pharmacokinetics like kidney function and body weight or surface area to personalize methotrexate therapy to reduce toxicity and improve treatment outcome. • After administration of aminoglycosides by short IV infusion, the plasma drug profile follows the three-compartment pharmacokinetic model. A rapid distribution phase, followed by elimination phase, and then the drug that was bound to different tissues is redistributed back to the systemic circulation producing low drug concentration over long time. The drug amount bound to tissues is small, but it may accumulate during multiple administration. 14.9  Compartmental Pharmacokinetic Data Analysis Compartmental pharmacokinetic modeling of experimentally obtained data involves construction of the model, mathematical presentation of the model, estimation of the pharmacokinetic model parameters, and evaluation of the obtained parameter estimates. 14.9.1  Construction of the Compartmental Model

The first step in compartmental pharmacokinetic data analysis is to construct the model that can describe the pharmacokinetic behavior of the drug. The choice of the compartmental pharmacokinetic model is usually based on the observed drug concentrations after drug administration. For example, when the plasma drug concentrations observed after single IV bolus dose of the drug decline as a straight line on the semilog scale, this suggests that the drug pharmacokinetic behavior can be described by the one-compartment pharmacokinetic model. If the decline in the drug concentrations is curvilinear on the semilog scale, the two-compartment pharmacokinetic model will be appropriate in this case. If the drug concentrations after single IV bolus dose on the semilog scale decline at three distinct rates, this pharmacokinetic behavior can be described by the three-compartment pharmacokinetic model. In addition to the number of the compartments, other model components such as the input function that depends on the route of drug administration and the output function that describes the drug elimination process should be included in the model. Drug input into the systemic circulation, which is usually part of the central compartment, can be instantaneous in case of IV bolus administration, first order for oral administration, or zero order when the drug is administered by constant rate IV infusion. While drug elimination can follow first-order process, Michaelis-Menten process or a combination of the two. The compartment where drug elimination occurs is usually the central compartment because most of the eliminating organs are highly perfused organs, unless there are evidence that drug elimination takes place in organs that are part of the peripheral compartments. Once all the model components are included, the model is defined mathematically. 14.9.2  Mathematical Description of the Model

A set of differential equations are used to describe the rate of change of the drug amount or concentration (dependent variable) with respect to time (independent variable) in the different compartments of the constructed model. These equations include the pharmacokinetic parameters that control the drug pharmacokinetic behavior in addition to constants like the administered dose. For each compartment, the rate of change in the drug amount is the difference between the rate of drug entering the compartment and the rate

Multicompartment Pharmacokinetic Models 279 of drug leaving the compartment. The differential equations describe the rate of change of the drug amount at a finite period of time. Solving the differential equations gives the integrated equations that can be used to calculate the drug amount in the different compartments at any time after drug administration. These integrated equations contain the pharmacokinetic model parameters that need to be estimated to allow calculation of the drug concentration in the different compartments at any time. 14.9.3  Fitting the Model Equation to the Experimental Data

Pharmacokinetic studies usually involve determination of the plasma drug concentrations at different time points. So, the model equation that describes the drug concentration-time profile in the central compartment is fitted to the observed plasma drug concentrations and their corresponding time values to estimate the pharmacokinetic model parameters. This fitting process is performed by nonlinear regression analysis utilizing specialized pharmacokinetic analysis or statistical software. The basic principle for estimation of the pharmacokinetic parameters involves selecting the best values for the pharmacokinetic parameters that will minimize the sum of the squared differences between the experimentally observed drug concentrations and the model-predicted drug concentrations (7). The model-predicted concentrations are determined by substituting the estimated pharmacokinetic parameters in the model equation and calculating the drug concentration at different time points. This process is not simple because the data analysis program has to check all possible combinations of the parameter values to find the combination of the parameter values that will minimize the sum of the squared error for all data points. So, most programs require the input of an initial estimate for each parameter to be used as the starting point for the parameter estimation process. Different programs utilize different algorithms to search for the best estimates for the pharmacokinetic parameters. For example, the pharmacokinetic parameters in the equation for the two-compartment pharmacokinetic model for drugs administered by IV bolus dose are A, B, α, and β, Eq. 14.12. The dose and the experimentally determined drug concentrations at different time points are used to estimate the pharmacokinetic parameters. The best estimates for these parameters are the values that should minimize the sum of the squared differences between all the observed and predicted concentrations. However, the pharmacokinetic parameters in the equation for the three-compartment pharmacokinetic model for drugs administered by a single IV bolus dose are A, B, C, α, β, and γ, Eq. 14.38. These parameters are estimated by selecting the best values for the six parameters that minimize the sum of the squared differences between all the observed and predicted concentrations. The drug concentrations in the experimentally obtained samples are used to estimate the pharmacokinetic parameters. So, accuracy of pharmacokinetic parameter estimation depends on the number of samples, the period over which the samples were obtained, and the accuracy of the analytical method used for determination of the drug concentration in the samples. In general, complicated models have more pharmacokinetic parameters and require larger number of samples for accurate parameter estimation. Enough samples should be obtained during each phase of the plasma drug concentration-time profile to obtain accurate estimation of all model parameters. There is no strict rule for the number and the timing of samples that should be obtained during pharmacokinetic experiments. However, the minimum number of samples required to obtain reasonable estimates for the pharmacokinetic parameters should not be less than three samples for each phase of the drug profile, and samples should be spread over the entire drug concentration-time profile. In addition to the number and timing of samples, the analytical procedures used

280  Multicompartment Pharmacokinetic Models for the determination of the drug concentration in the obtained samples must be accurate and precise to minimize the error in the experimental data. 14.9.4  Evaluation of the Pharmacokinetic Model

The primary goal of modeling in pharmacokinetics is to choose the best model that can describe the drug pharmacokinetic behavior in the body and to estimate the model parameters with acceptable accuracy. For example, in the previous discussion of the two-compartment pharmacokinetic model, it was mentioned that after IV bolus administration, the drug is distributed rapidly to the tissues of the central compartment and slowly to the tissues of the peripheral compartment. The drug may be distributed to the different tissues at different rates, but the drug distribution can be approximated by two rates and the two-compartment model can approximate the overall behavior of the drug in the body. So, the observed plasma drug concentrations can be described with reasonable accuracy by the equation for the two-compartment pharmacokinetic model. Using the three-compartment pharmacokinetic model equation can describe the observed data better and may decrease the sum of the squared deviations between the observed and the predicted concentrations. However, the use of complex models is not necessarily better because increasing the number of compartments increases the number of model parameters. When the same number of data points is used to estimate larger number of pharmacokinetic parameters, the accuracy of the parameter estimates is usually compromised. Evaluation of the goodness of fit of the experimental data to the pharmacokinetic model is an important step in compartmental modeling. There is no single diagnostic procedure or statistical test that can be used to determine the validity of the model to describe the observed data. However, there are several statistical and graphical methods that are generally used to evaluate how well the model fits the data set (7).

• Most data analysis programs calculate the standard error and the coefficient of vari-

ation for each parameter estimate. Coefficient of variation >20–30% for any parameter estimate indicates uncertainty of this parameter estimate that usually results from overparameterization of the model or insufficient data. • A scattered plot of the observed concentrations around the model-predicted drug concentration-time curve is used to determine how well the model fits the data. Small differences between the observed and model-predicted concentrations and random distribution of the observed values around the predicted profile indicate good fit. However, the existence of a trend in the data points, which is presented by a group of consecutive data points above or below the predicted profile, is an indication of inappropriate fit, as in Figure 14.12A. • The observed versus the model-predicted drug concentration plot is a useful diagnostic plot for evaluation of the goodness of fit. Good model fit can be concluded when the values are gathered uniformly, randomly, and closely around the line with slope equal to one. The presence of a trend in the data points is an indication of inappropriate fit, as in Figure 14.12B. • The residuals, which are the difference between the observed drug concentrations and the model-predicted drug concentrations, are the measure of error in each data point. The residuals versus predicted concentration plot is used to examine the error value at the different drug concentrations and determine if the model fit one end of the curve

Multicompartment Pharmacokinetic Models 281

Figure 14.12 Examples of the diagnostic graphs for compartmental pharmacokinetic data analysis. The scattered plot of the observed concentration and the model-predicted plasma drug profile (A). Random distribution of the data around the predicted profile indicates good data fitting (left), while the presence of a trend that appears as series of observations above or below the model-predicted values indicates inappropriate data fitting (right). The observed versus predicted plasma drug concentration (B). Uniform and random distribution of data around the line with slope of one indicates good data fitting (left), while the presence of trend indicates unacceptable data fitting (right).

better than the other. Also, the residual versus time plot is used to evaluate if the model accurately accounts for all the different phases in the drug concentration-time profile. • The plot of the weighed residuals versus predicted values and weighed residuals versus time are useful to evaluate the model fit as in Figure 14.13A and B. When the model fits the observed data properly, the weighted residual versus predicted value plot should be small, with approximately uniform magnitude, and randomly distributed around the zero residual line. The plot of weighted residuals versus time should have the same properties. • Some pharmacokinetic analysis programs calculate the Akaike Information Criterion to help in selecting the appropriate model. This is an approach used to determine if going to more complex model improves the fit without compromising the accuracy in model parameter estimation. Clinical Importance:

• Compartmental modeling and parameter estimation are usually performed based on the user-defined model and initial parameter estimates. So, evaluation of the obtained modeling results is important to ensure proper model selection and goodness of fit.

282  Multicompartment Pharmacokinetic Models

Figure 14.13 Examples of the residual plots used as diagnostic tests for evaluating curve fitting. The scattered plot of the weighted residuals versus predicted concentrations (A), and the scattered plot of the weighted residuals versus time (B). Small, uniform, and randomly distributed weighted residuals indicate good fitting of data (left), while the presence of a trend or unequal weighed residuals suggests improper data fitting (right).

• Accurate pharmacokinetic parameter estimation is necessary to utilize the pharmacoki-

netic modeling results to predict the drug pharmacokinetic behavior after administration of different doses by different routes of administration using different drug formulations.

14.10 Summary

• Compartmental models are used to describe the pharmacokinetic behavior of drugs by • • • •

presenting the body by one or more kinetically homogenous compartments based on the different rates of drug distribution to the different body organs and tissues. The obtained drug concentrations after drug administration are usually utilized to select the appropriate compartmental model that can describe the drug pharmacokinetic behavior. The use of the appropriate pharmacokinetic model to describe the drug pharmacokinetic behavior is important for accurate prediction of the drug concentration achieved by different doses and calculation of dose required to achieve a specific concentration. During multiple drug administration, the steady state is achieved after five to six times the elimination half-life, and the average steady-state concentration is directly proportional to the dosing rate and inversely proportional to the drug clearance. Compartmental pharmacokinetic modeling is usually performed using specialized data analysis software. So, evaluation of the obtained results is important to ensure the selection of the appropriate model and to estimate the model parameters accurately.

Multicompartment Pharmacokinetic Models 283 Practice Problems 14.1 A drug that follows a two-compartment pharmacokinetic model is given to a patient by rapid IV injection. Would the drug concentration in each tissue be the same after the drug equilibrates between the plasma and all the tissues in the body? Explain. 14.2 A drug follows two-compartment pharmacokinetic model. If a single IV bolus dose is given, what is the cause of the initial rapid decline in the plasma concentration (α-phase)? What is the cause of the slower decline in the plasma concentration (β-phase)? 14.3 A drug that follows two-compartment pharmacokinetic model was given as a single IV bolus dose of 5.6 mg/kg. The equation that describes the plasma concentrationtime data is as follows:

Cp ( mg/L ) = 18 e−2.8 t + 6 e−0.11 t a Calculate the plasma concentration after 0.5, 3, and 12 hr of drug administration. b What will be the equation that describes the plasma concentration-time profile if the dose given was 11.2 mg/kg?

14.4 After administration of a single IV bolus dose of 75 mg of a drug to a healthy volunteer, the pharmacokinetics of this drug followed two-compartment model. The following parameters were obtained:

A = 4.62 mg/L B = 0.64 mg/L



α = 8.94 hr −1

β = 0.19 hr −1

a Calculate the following parameters:

t1/ 2 α , t1/ 2 β , k12 , k21 , k10 , Vc , Vdβ , Vdss , AUC, and CL T . b What will be the amount of drug remaining in the body after 8 hr?

14.5 A single dose of 500 mg of a drug was administered by a rapid IV injection into a 70-kg patient. Plasma samples were obtained over a 7-hr period and assayed for the drug. The results are tabulated below: Time (hr)

Concentration (mg/L)

0.00 0.25 0.5 0.75 1.0 1.5 2.0 2.5 3.0 4.0 5.0 6.0 7.0

70.0 53.8 43.3 35.0 29.1 21.2 17.0 14.3 12.6 10.5 9.0 8.0 7.0

284  Multicompartment Pharmacokinetic Models a Calculate the following parameters:

t1/ 2 α , t1/ 2 β , k12 , k21 , k10 , Vc , Vdβ , Vdss , AUC, and CL T . b What will be the plasma concentration 12 hr after drug administration? c What will be the initial plasma concentration if a dose of 1500 mg is administered as an IV bolus? d Which of the pharmacokinetic parameters above will change with the increase in the dose to 1500 mg? e Calculate the new values for these parameters. f What will be the IV infusion rate required to achieve a steady-state plasma concentration of 10 mg/L?

14.6 The plasma concentration time data following a 50-mg IV bolus dose of lidocaine is tabulated below: Time (hr)

Concentration (mg/L)

2.0 4.0 10 15 20 40 60 90 120 180 240

1.51 1.20 0.796 0.639 0.462 0.329 0.271 0.242 0.179 0.112 0.081

a Using the method of residuals calculate the following parameters:

t1/ 2 α , t1/ 2 β , k12 , k21 , k10 , Vc , Vdβ , Vdss , AUC, and CL T . b Calculate the plasma concentration at the time of each sample from the equation that describes the plasma concentration-time profile, and comment on the differences between the measured and the calculated concentrations. c What will be the amount of lidocaine remaining in the body after 240 min? d What will be the plasma concentration when 90% of the administered dose is eliminated?

References 1. Gibaldi M and Perrier D “Pharmacokinetics” 2nd Edition (1982), Marcel Dekker, New York, NY, USA. 2. Wagner JG “Pharmacokinetics for the pharmaceutical scientist” (1993) Technomic Publishing Co., Lancaster, PA, USA. 3. Loughnan PM, Sitar DS, Ogilvie RI and Neims AH “The two-compartment open-system kinetic model: A review of its clinical implications and applications” (1976) J Pediatr; 88:869–873.

Multicompartment Pharmacokinetic Models 285 4. Mayersohn M and Gibaldi M “Mathematical methods in pharmacokinetics II: Solution of the two-compartment open model” (1971) Am J Pharm Ed; 35:19–28. 5. Loo JCK and Riegelman S “New method for calculating the intrinsic absorption rate of drugs” (1968) J Pharm Sci; 57:918–928. 6. Jusko WJ and Gibaldi M “Effects of change in elimination on various parameters of the two-compartment open model” (1972) J Pharm Sci; 61:1270–1273. 7. Bourne DWA “Mathematical modeling of pharmacokinetic data” (1995) Technomic Publishing Co., Lancaster, PA, USA.

15 Drug Pharmacokinetics Following Administration by Intermittent Intravenous Infusions

Objectives After completing this chapter, you should be able to:

• Describe the plasma drug concentration-time profile after administration by multiple intermittent IV infusions.

• Calculate the steady state maximum and minimum drug concentration after administration by intermittent IV infusions.

• Predict aminoglycoside half-life based on the patient’s kidney function. • Analyze the effect of changing the pharmacokinetic parameters on the maximum and minimum steady state drug concentration.

• Apply the Sawchuck-Zaske method for determination of the patient’s specific aminoglycoside pharmacokinetic parameters.

• Recommend the appropriate aminoglycoside dosing regimen based on the patient’s specific pharmacokinetic parameters.

15.1 Introduction Drug administration by intermittent IV infusion involves repeated drug administration by short IV infusion at every fixed dosing interval. Following repeated administration of these short IV infusions, the drug accumulates in the body until steady state is achieved. At steady state, the maximum and minimum plasma drug concentrations will be constant if the same dose is administered over the same infusion time at every fixed dosing interval. This route of administration is usually used for drugs that follow multicompartment pharmacokinetic models. This is because IV bolus administration of these drugs usually produces high plasma drug concentration, because of the slow drug distribution to the peripheral compartment. This high concentration can be associated with serious adverse effects. Examples of drugs administered by this route of administration are vancomycin and aminoglycosides. During multiple intermittent IV infusions, the infusion time is (t′) and the dose is repeated every fixed dosing interval (τ). The time between the maximum and the minimum drug concentration is (τ − t′) as illustrated in Figure 15.1. If the infusion time is neglected and it is assumed that the drug is administered by IV bolus administration, the calculation will ignore the amount of the drug eliminated during the infusion time. This error can be significant in drugs with short half-lives and can influence the calculation of the pharmacokinetic parameters in these drugs. DOI: 10.4324/9781003161523-15

Drug Administration by Intermittent IV Infusions 287

Figure 15.1 The plasma drug concentration-time profile during repeated administration of intermittent IV infusions. The dosing interval (τ) is the sum of the duration of the infusion (t′) and the post-infusion time (τ − t′).

Clinical Importance:

• Administration of vancomycin over a period of less than 30 min has been associated

with erythematous reactions, intense flushing known as the “red-neck” syndrome, tachycardia, and hypotension. These adverse effects have been well correlated with the rate of drug administration. Vancomycin is usually administered as a constant rate IV infusion over a period of 1 hr. These slow rates of administration allow distribution of the drug to all parts of the body and avoid the high plasma drug concentration achieved after rapid IV administration. • Rapid IV administration of aminoglycoside can produce high plasma concentration at the end of drug administration, which can increase the risk of aminoglycoside ototoxicity. Aminoglycosides are usually administered by IV infusion over a period of 0.5–1.0 hr. 15.2 The Drug Concentration-Time Profile after Administration by Intermittent IV Infusions Studying the drug pharmacokinetic behavior when the drug is administered by intermittent IV infusions requires the development of the equations that describe this profile during and after drug administration. There are three different situations that must be considered. These are when the drug is administered for the first time, when the drug is administered repeatedly before reaching steady state, and when the drug is administered repeatedly at steady state. Although it was mentioned that this route of drug administration is suitable for drugs that follow multiple-compartment pharmacokinetic models, most of the methods used for individualization of therapy for these drugs assume that they follow one-compartment pharmacokinetic model. This is because during the clinical use of these drugs, the obtained samples after administration are not enough to characterize their multicompartment pharmacokinetic behavior. Also, the use of the one-compartment model does not introduce significant errors. The following discussion assumes that the drug is eliminated by first-order process and that the drug follows onecompartment pharmacokinetic model (1, 2).

288  Drug Administration by Intermittent IV Infusions 15.2.1  After the First Dose

The rate of change of the amount of drug in the body during the constant rate IV infusion is equal to the difference between the rate of drug administration and the rate of drug elimination as in Eq. 15.1. dA = K0 − k A (15.1) dt where K0 is the rate of the IV infusion which is equal to (dose/infusion time and has units of mass/time), k is the first-order elimination rate constant, and A is the amount of drug in the body. Integrating Eq. 15.1 gives the equation that describes the change in the amount of drug in the body during a constant rate IV infusion. A=

K0 (1 − e− kt )(15.2) k

Dividing Eq. 15.2 by Vd gives Eq. 15.3 that describes the plasma concentration at any time during constant rate IV infusion. Cp =

K0 (1 − e− kt )(15.3) k Vd

If the drug infusion continues for long time, Eq. 15.3 is reduced to Eq. 15.4, which describes the steady state plasma concentration during constant rate infusion. Cpss =

K0 (15.4) k Vd

The plasma drug concentration at the end of the IV infusion when administered over a period of t′ is described by Eq. 15.5. Cp =

K0 (1 − e− kt ′ )(15.5) k Vd

After termination of the IV infusion at time t′, the drug concentration declines exponentially as shown in Figure 15.2. The plasma drug concentration during or after the IV infusion can be described by Eq. 15.6.  K  Cp =  0 1 − e− kt ′  e− k (t − t ′)(15.6) k Vd  

(

)

where t is the time from starting drug administration and t′ is the infusion time. This is the general equation that describes the drug concentration during and after a constant rate IV infusion. During the IV infusion, t is equal to t′ and Eq. 15.6 is reduced to Eq. 15.5. After termination of the IV infusion, t becomes larger than t′ and Eq. 15.6 describes the drug concentration at any time after termination of the drug infusion.

Drug Administration by Intermittent IV Infusions 289

Figure 15.2 The plasma drug concentration-time profile during and after administration of the first dose of a drug by constant rate IV infusion over a period of t′ on the semilog scale. 15.2.2  After Repeated Administration Before Reaching Steady State

During repeated intermittent IV infusion, the drug concentration during the IV infusion is the sum of the drug concentration resulting from drug infusion plus the drug concentration remaining in the body from previous doses. If we assume that the drug concentration just before drug administration is equal to Cpx as illustrated in Figure 15.3, the drug concentration at the end of the infusion can be described by Eq. 15.7: Cp =

K0 (1 − e− kt ′ ) + Cpx e− kt ′ (15.7) k Vd

When the IV infusion is terminated after time equal to t′, the drug concentration declines exponentially as shown in Figure 15.3. The drug concentration after termination of the IV infusion can be described by the following equation:  K  Cp =  0 (1 − e− kt ′ ) + Cpx e− kt ′  e− k (t − t ′)(15.8) k Vd  

Figure 15.3 The plasma concentration-time profile during repeated administration of intermittent IV infusion before reaching steady state.

290  Drug Administration by Intermittent IV Infusions Equation 15.8 is the general equation for drug administration as short constant rate IV infusion. During the infusion, t is equal to t′ and Eq. 15.8 is reduced to Eq. 15.7, while after the end of infusion, t becomes bigger than t′ and the equation can be used to calculate the drug concentration at any time after termination of the infusion. However, if the drug is administered for the first time, Cpx is equal to zero and the equation is reduced to Eq. 15.5 or 15.6. 15.2.3  At Steady State

Repeated drug administration by intermittent IV infusion leads to drug accumulation in the body until steady state is achieved. At steady state, the drug concentration right after the end of drug administration is equal to Cpmax ss and the drug concentration just before drug administration is equal to Cpmin ss, providing that the same dose is administered at a fixed dosing interval as shown in Figure 15.4. An expression like Eq. 15.7 can be written to describe the drug concentration after the end of the short IV infusion at steady state. However, at steady-state Cpx, which is the drug concentration just before drug administration, is equal to Cpmin ss as in Eq. 15.9. Cpmax  ss =

K0 (1 − e− kt ′ ) + Cpmin ss e− kt ′(15.9) k Vd

Since Cpmin ss is equal to the concentration remaining from Cpmax ss after the post-infusion period (τ − t′), Cpmin ss can be expressed by Eq. 15.10: Cpmin ss = Cpmax  ss e− k (τ− t ′)(15.10) By substitution for the value of Cpmin ss in Eq. 15.9 and solving for Cpmax ss, Eq. 15.11, which describe Cpmax ss, is obtained: Cpmax  ss =

K0 (1 − e− kt ′ ) (15.11) k Vd(1 − e− kτ )

Figure 15.4 The plasma concentration-time profile during repeated administration of intermittent IV infusion at steady state.

Drug Administration by Intermittent IV Infusions 291 15.3 The Effect of Changing the Pharmacokinetic Parameters on the SteadyState Plasma Concentration during Repeated Intermittent IV Infusions The steady-state drug concentration achieved during multiple intermittent IV infusion is directly proportional to the administered dose and inversely proportional to the CLT. It takes five-to-six elimination half-lives to achieve steady state. Administration of a loading dose leads to faster approach to steady state without affecting the steady-state concentration. 15.3.1 Dose

The effect of administration of increasing doses of the drug:

• Administration of increasing doses leads to proportional increase in the steady-state maximum and minimum plasma concentrations when the CLT is the same.

• The AUC within each dosing interval at steady state increases with the increase in dose. • The time to achieve steady state should be the same if CLT and Vd are the same. 15.3.2  Infusion Time

The effect of changing the IV infusion time:

• Administration of the same dose of the drug over longer duration of time results

in lower maximum drug concentration and higher minimum drug concentration at steady state, i.e., smaller fluctuations in the drug concentrations at steady state.

15.3.3  Total Body Clearance

The effect of the change in CLT of the drug:

• After administration of the same dose over the same infusion time, the steady-state plasma concentration is inversely proportional to CLT.

• Patients with lower CLT should achieve steady state slower than those with higher clearance, when Vd is the same. This is because of the longer t1/2.

15.3.4  Volume of Distribution

The effect of the change of Vd of the drug:

• The Vd does not affect the steady-state concentration if the CLT and the dose are the same.

• When Vd is different in patients with similar CLT, the t1/2 and k (the dependent parameters) will be different, and the time to achieve steady state will be different.

Clinical Importance: The relationship between the dose, CLT, steady-state concentrations, and time to achieve steady state when the drug is administered by intermittent IV infusion is similar to the relationship during multiple drug administration by other routes of administration. So,

292  Drug Administration by Intermittent IV Infusions the clinical significance for changing the drug pharmacokinetic parameters in these drugs is similar to what was discussed previously for multiple drug administration.

• Administration of increasing doses results in more intense drug effects and possible adverse effects.

• Patients with reduced eliminating organ function require lower doses than those with normal eliminating organ function.

• The infusion time should be selected carefully for each drug to avoid achieving toxic drug concentrations.

• The dosing regimens, including dose, dosing interval, infusion time, and loading dose if needed, for drugs administered by intermittent IV infusion and have narrow therapeutic index should be selected carefully to ensure optimal drug therapeutic effect with minimum adverse effects.

15.4 Application of the Pharmacokinetic Principles for Intermittent IV Infusion in Clinical Practice The classical example of drugs that are administered by intermittent IV infusion and are frequently monitored in patients are aminoglycosides. Aminoglycosides, such as gentamicin, tobramycin, amikacin, and netilmicin, are antibacterial agents that have bactericidal activity against Gram-negative aerobic bacteria. 15.4.1  Pharmacokinetic Characteristics of Aminoglycosides

Aminoglycosides are administered parenterally to treat systemic infections because they are not absorbed after oral administration. Drugs from this antibiotic group such as neomycin and streptomycin are administered orally to produce local gastrointestinal effects. After IM administration, the maximum aminoglycoside plasma concentration is achieved within 30–60 min in patients with normal kidney function and after 2–5 hr in patients with reduced kidney function depending on the severity of the kidney dysfunction. Aminoglycosides are distributed primarily in the extracellular fluid because of their poor lipid solubility. Their average Vd is 0.2–0.25 L/kg. After parenteral administration, aminoglycosides are completely eliminated unchanged in urine (3). 15.4.2  Guidelines for Aminoglycoside Plasma Concentration

The most common adverse effects of aminoglycosides are nephrotoxicity, ototoxicity, and neuromuscular blockade. The goal of conventional aminoglycoside dosing adjustment strategies is to maintain the steady-state maximum and minimum plasma concentration within a certain range of concentrations to ensure optimal efficacy with minimal toxicity. Table 15.1 lists the general guidelines for the desired maximum and minimum aminoglycoside plasma concentrations at steady state using the conventional dosing. Clinical Importance:

• Urinary tract or intrabdominal infections with gram-negative bacteria usually require

maximum aminoglycoside concentrations in the lower range of the recommended concentrations. While lung infection with pseudomonas aeruginosa usually requires

Drug Administration by Intermittent IV Infusions 293 Table 15.1  T he general recommendation for steady-state aminoglycoside concentrations while using the conventional dosing regimen Drug

Maximum (peak) concentration Infections by susceptible bacteria in easily accessible sites Infections by less susceptible bacteria in difficult to access sites Minimum (trough) concentration Serious infections Life-threatening infections

Gentamicin Tobramycin Netilmicin

Amikacin

6–8 mg/L 8–10 mg/L

20–25 mg/L 25–30 mg/L

0.5–1 mg/L 1–2 mg/L

1–4 mg/L 4–8 mg/L

aminoglycoside maximum concentration in the upper range of the recommended concentration. • Peak aminoglycoside concentrations above the recommended concentrations have been associated with increased risk of ototoxicity, which can be irreversible, while high trough aminoglycoside concentrations have been associated with increased risk of nephrotoxicity, which is usually reversible. 15.4.3  The Extended-Interval Aminoglycoside Dosing Regimen

The total daily dose of aminoglycosides administered once a day has been utilized effectively to treat systemic infections. Aminoglycosides have concentration-dependent bactericidal effect and concentration-dependent postantibiotic effect. This means that higher concentration produces faster rate of bacterial killing and the duration of the bactericidal effect of aminoglycosides extends for a period of time after their plasma concentrations fall below the minimum inhibitory concentration. Also, administration of larger doses while extending the dosing intervals does not significantly increase aminoglycoside toxicity. These factors provide the basis for the rationale of once daily aminoglycoside dosing. The effectiveness of once daily aminoglycoside dosing in all patients to treat all infections is yet to be determined. There are no known optimal maximum and minimum plasma drug concentrations when aminoglycosides are administered once daily (4, 5). The guidelines in Table 15.1 do not apply to the once-daily dosing method. The following discussion covers the conventional aminoglycoside dosing adjustment method only. 15.5  Individualization of Aminoglycoside Therapy Individualization of drug therapy in general is the process of selecting the dosing regimen for each patient based on his/her own specific pharmacokinetic parameters. The process starts by estimating the pharmacokinetic parameters of the drug in the patient and then using these parameters to calculate the appropriate dosing regimen for that patient. Individualization of drug therapy requires selecting the target steady-state concentration and the pharmacokinetic model that describes the drug pharmacokinetic behavior. The general recommendations for the maximum and minimum steady-state aminoglycoside concentrations presented in Table 15.1 are usually used as the target concentrations when calculating the appropriate dosing regimen. The Sawchuk-Zaske

294  Drug Administration by Intermittent IV Infusions method for individualization of aminoglycoside therapy will be used as the basis of the following discussion (2). It is one of the first methods developed to calculate the patient’s specific pharmacokinetic parameters from a few blood concentrations and utilizes these drug concentrations in calculating the appropriate dosing regimen for specific patients. The assumption that aminoglycosides follow one-compartment pharmacokinetic model makes this method relatively simple without significantly compromising its clinical utility. This method is widely used for the individualization of aminoglycoside therapy (2, 6, 7). 15.5.1  Estimation of the Patient Pharmacokinetic Parameters

The appropriate dosing regimen is calculated based on aminoglycoside pharmacokinetic parameters. Before the start of aminoglycoside therapy, it is not possible to determine the patient’s specific pharmacokinetic parameters. However, the initial dosing regimen is usually calculated using the aminoglycoside parameters estimated based on the patient medical information. The patient’s specific pharmacokinetic parameters can be estimated after starting aminoglycoside therapy. The estimated patient’s specific pharmacokinetic parameters are used to modify the dosing regimen if necessary. 15.5.1.1 Estimation of the Patient Pharmacokinetic Parameters Based on the Patient Information

The factors that affect the pharmacokinetic parameters of aminoglycosides are evaluated in the patient. Then aminoglycoside pharmacokinetic parameters in a population like that of the patient are used as the initial parameter estimate for the patient. 15.5.1.1.1 HALF-LIFE

Aminoglycoside half-life is highly correlated with the patient’s kidney function since aminoglycosides are eliminated completely unchanged by the kidney. The patient’s kidney function can be determined from the creatinine clearance (CrCL). The CrCL is either determined directly or estimated from the serum creatinine and some patient’s information. Direct CrCL determination requires 24-hr urine collection and determination of serum creatinine as discussed in Chapter 11. While the method of Cockroft and Gault, for example, can provide good estimate for the CrCL in mL/min from the age, weight, serum creatinine, and gender. According to this method, the CrCL can be estimated in males from Eq. 15.12, and in females from Eq. 15.13. The estimated CrCL can be used to estimate the half-life or the elimination rate constant of aminoglycosides: CrCL =

(140-age) (Wt in kg) (15.12) 72 (s. Cr. in mg/dL)

 (140-age) (Wt in kg)  CrCL = 0.85  (15.13)  72 (s. Cr. in mg/dL)  Since aminoglycosides are eliminated completely by the kidney, the half-life of aminoglycoside can be estimated empirically from the average aminoglycoside half-life and the

Drug Administration by Intermittent IV Infusions 295 patient’s kidney function. The patient’s kidney function (KF) is determined by comparing the patient’s CrCL with the normal CrCL, which is in the range of 120 mL/min as in Eq. 15.14. Patient KF =

Patient CrCL (15.14) 120 mL/min

Aminoglycoside half-life can be estimated assuming that the average half-life of aminoglycoside is 2.5 hr, as in Eq. 15.15: Estimated t1/2 =

Average normal t1/2  (2.5 hr) (15.15) Patient KF

When the estimated CrCL is more than 120 mL/min, the patient’s kidney function is considered equal to unity and the half-life in this case is considered equal to the average half-life of aminoglycosides. Studying the elimination rate of aminoglycosides in a large number of patients with different kidney functions resulted in the development of Eq. 15.16, which relates the elimination rate constant of aminoglycosides to the CrCL. So, the patient’s CrCL can be substituted in this equation to estimate the elimination rate constant of aminoglycoside (3):

(

)

k for aminoglycosides in hr −1 = 0.00293 (CrCL in mL/min ) + 0.014 (15.16) 15.5.1.1.2  VOLUME OF DISTRIBUTION

Aminoglycosides are hydrophilic drugs that are distributed mainly in the body extracellular space. Their average Vd is 0.25 L/kg based on the ideal body weight since adipose tissues have lower extracellular fluid contents. Conditions like ascites and overhydration can increase the Vd of aminoglycosides. The estimated aminoglycoside elimination rate constant and Vd are used to calculate the initial aminoglycoside dosing regimen. 15.5.1.2 Estimation of the Patient’s Specific Pharmacokinetic Parameters from Aminoglycoside Blood Concentrations

After starting aminoglycoside therapy, the patient’s specific parameters should be determined by designing an appropriate sampling schedule that allows pharmacokinetic parameter determination. The method assumes that aminoglycosides are administered by intermittent IV infusion over a period of t′, drug administration is repeated every τ, and that aminoglycosides follow one-compartment pharmacokinetic model. Although aminoglycosides follow multicompartment pharmacokinetic model, the use of one-compartment pharmacokinetic model can generally provide clinically acceptable estimates for the pharmacokinetic parameters. 15.5.1.2.1  IF THE PATIENT IS TO RECEIVE THE FIRST AMINOGLYCOSIDE DOSE

After administration of the first dose of the drug as constant rate IV infusion over a period of t′ and at rate equal to K0 (which is equal to dose/t′), the plasma drug concentration at the end of drug administration can be determined from Eq. 15.3. Cp =

(

K0 1 − e− kt ′ k Vd

)

296  Drug Administration by Intermittent IV Infusions

Figure 15.5 Graphical determination of aminoglycoside half-life and the concentration at the end of drug administration from the plasma concentration-time profile after administration of the first dose as short IV infusion of duration t′.

To estimate the pharmacokinetic parameters, three blood samples are obtained over a period equivalent approximately to the estimated aminoglycoside half-life. For example, for a patient with normal kidney functions, samples should be drawn over approximately 2.5 hr period (e.g., at 0.5, 1.5, and 3 hr) after the end of drug administration. Patients with reduced kidney function may require obtaining the samples over a longer period according to their estimated aminoglycoside half-lives. The samples are analyzed for aminoglycoside and the measure drug concentrations are plotted on the semilog scale as in Figure 15.5. The best line that fits the three points is drawn and back extrapolated to the time when the IV infusion was terminated to determine the drug concentration at the end of the drug administration (Cp). Figure 15.5 represents the entire aminoglycoside plasma concentrationtime profile during and after the end of drug administration. However, for determination of the drug half-life and the drug concentration right at the end of drug administration, a plot of the drug concentration versus time after the end of drug administration is sufficient as in Figure 15.6. In this plot, the y-intercept is equal to the plasma drug concentration at the end of drug administration. The elimination rate constant can be determined from the slope of

Figure 15.6 Graphical determination of aminoglycoside half-life and concentration at the end of drug administration from the drug concentrations versus time plot after the end of drug administration.

Drug Administration by Intermittent IV Infusions 297 the line, where the slope of the resulting line on the semilog scale is equal to −k/2.303. The half-life can be calculated from k or determined graphically by finding the time required for the drug concentration to decrease by 50% as in Figure 15.6. Knowing either the elimination rate constant or the half-life, the other parameter can be calculated (k = 0.693/t1/2). The drug elimination rate constant can also be determined mathematically by using any two plasma drug concentrations on the fitted line from Eq. 15.17. Cpt2 = Cpt1 e− kt(15.17) where Cpt1 and Cpt2 are two different plasma concentrations after the end of drug administration, Cpt1 is the sample obtained at earlier time and Cpt2 is the sample obtained at later time, k is the first-order elimination rate constant, and t is the time difference between the two samples. After determination of plasma drug concentration at the end of drug administration and the elimination rate constant, aminoglycoside Vd can be determined by substitution in Eq. 15.3. 15.5.1.2.2 IF THE PATIENT RECEIVED AMINOGLYCOSIDES BEFORE BUT STEADY STATE WAS NOT ACHIEVED

The plasma concentration after the end of administration of the second dose of the drug by constant rate IV infusion (or repeated administration before steady state is achieved) is the sum of the drug concentration resulting from the administered drug and the drug remaining from previous drug administration. In this case, the drug concentration at the end of drug administration can be expressed by Eq. 15.7 as described previously: Cp =

(

)

K0 1 − e− kt ′ + Cpx e− kt ′ k Vd

To determine aminoglycoside k, t1/2, and Vd, one blood sample should be obtained just before drug administration and two or three samples after the end of drug administration as shown in Figure 15.7. The plasma drug concentration obtained just before drug administration is equal to Cpx in Eq. 15.7. The samples should be obtained after the end of drug administration over a period of time equivalent approximately to the estimated aminoglycoside t1/2 in the patient. Aminoglycoside t1/2 or k can be determined as mentioned

Figure 15.7 The plasma samples obtained for the determination of the pharmacokinetic parameters during repeated intermittent IV infusion before reaching steady state.

298  Drug Administration by Intermittent IV Infusions

Figure 15.8 The plasma samples obtained for the determination of the pharmacokinetic parameters during repeated intermittent IV infusion at steady state.

before from the plasma concentrations obtained after the end of drug administration. The plasma concentration at the end of drug administration is determined graphically by back extrapolation to the time of termination of the IV infusion. By substitution for Cp, Cpx, and k in Eq. 15.7, aminoglycoside Vd can be estimated. 15.5.1.2.3 IF THE PATIENT RECEIVED AMINOGLYCOSIDES AND STEADY STATE HAS BEEN ACHIEVED

At steady state, Cpmax ss and Cpmin ss are constant in consecutive dosing intervals. So, the drug concentration in the sample obtained just before drug administration should be equal to the drug concentration at the end of the following dosing interval. So, two plasma samples are obtained, one sample just before drug administration and the other at 1 hr after the end of drug administration as shown in Figure 15.8. The Cpmax ss is expressed by Eq. 15.11: Cpmax  ss =

(

K0 1 − e− kt ′

(

k Vd 1 − e

)

− kτ

)

where K0 is the infusion rate (dose/t′), k is the first-order elimination rate constant, Vd is the volume of distribution, and τ is the dosing interval. Drug concentrations determined before and after the drug administration can be used to calculate k. The two different concentrations are known and the time difference between the two concentrations is known, so k can be calculated. Also, the drug concentration right after the end of drug administration can be determined graphically or mathematically. By substitution for k and the Cpmax ss in Eq. 15.11, Vd can be estimated. 15.5.2 Determination of the Dosing Regimen Based on the Patient’s Specific Parameters

The estimated aminoglycoside k and Vd are used to determine the dose (D) and the dosing interval (τ) that should achieve a specific Cpmax ss (peak) and Cpmin ss (trough).

Drug Administration by Intermittent IV Infusions 299 15.5.2.1  Selection of the Dosing Interval (τ)

After selecting the drug infusion time (t′) and the target Cpmax ss and Cpmin ss, τ can be calculated from Eq. 15.10. The calculated τ should be approximated to the nearest practical dosing interval, i.e., 4, 6, 8, 12, 24, 48, or 72 hr. Cpmin ss = Cpmax  ss e− k (τ− t ′) 15.5.2.2  Selection of Dose

The approximated value of τ, the infusion time t′, the target Cpmax ss, and the calculated pharmacokinetic parameters are used to calculate aminoglycoside dose from Eq. 15.11. The calculated dose should be approximated to the nearest practical dose (approximate to the nearest 5 mg for aminoglycosides). Cpmax  ss =

K0 (1 − e− kt ′ ) k Vd(1 − e− kτ )

15.5.2.3  Selection of the Loading Dose

When administration of a loading dose is necessary, the loading dose should be selected to achieve plasma concentration equal to Cpmax ss after the loading dose. The loading dose is calculated from Eq. 15.3 after substation for t′, k, Vd, and the Cp. The actual values of the Cpmax ss (peak) and Cpmin ss (trough) that should be achieved by using the approximated values of the dose and the dosing interval can be calculated from Eqs. 15.10 and 15.11. The patient should be monitored for the therapeutic and adverse effects of aminoglycoside therapy and the dosing regimen is modified if necessary. Reassessment of aminoglycoside regimen is necessary when the patient’s condition changes in a way that can alter aminoglycoside pharmacokinetics. Practice Problems: Question: A 43-year-old, 68-kg, female was admitted to the hospital to treat lung infection with gentamicin. She received her first dose of 140 mg administered by constant rate IV infusion over a period of 1 hr. Three blood samples were drawn after the end of drug administration, and the concentrations came as follows: Time after end of administration Gentamicin concentration (mg/L)

2 hr 8.4

4 hr 5.85

8 hr 2.84

a Estimate the t1/2 and Vd of gentamicin in this patient. b Recommend a dosing regimen (dose and dosing interval) that should achieve maximum and minimum steady-state plasma gentamicin concentration of 8 and 1 mg/L, respectively. Gentamicin is administered by constant rate IV infusion over 1 hr. Answer: a The t1/2 and the plasma concentration achieved at the end of drug administration can be determined graphically as in Figure 15.9: The half-life = 3.85 hr

300  Drug Administration by Intermittent IV Infusions

Figure 15.9 Graphical determination of the drug half-life and drug concentration at the end of drug administration from a plot of the drug concentrations versus time after the end of drug administration.

Plasma concentration at the end of drug administration = 12 mg/L Cp =

K0 (1 − e− kt ′ ) k Vd

12 mg/L =

(

−1 140 mg/hr 1 − e−0.18 hr 1 hr −1 0.18 hr Vd

)

Vd = 10.68 L b Dosing regimen: Cpmin ss = Cpmax  ss e− k (τ− t ′) 1 mg/L = 8 mg/L e−0.18 hr

−1

(τ−1 hr)

(τ − t′) = 11.55 hr An appropriate dosing interval = 12 hr Cpmax  ss =

8 mg/L =

K0 (1 − e− kt ′ ) k Vd(1 − e− kτ ) K0 (1 − e−0.18 hr −1

−1

0.18 hr 10.68 L (1 − e

1 hr

)

−0.18 hr −1 12 hr

)

K0 = 82.6 mg administered over 1 hr Recommendation: 85 mg of gentamicin administered as a constant rate IV infusion over 1 hr and repeated every 12 hr.

Drug Administration by Intermittent IV Infusions 301 Practice Problems: Question: A 38-year-old 68-kg male was admitted to the hospital because of acute pneumonia, and he is to receive tobramycin. His serum creatinine was found to be 1.6 mg/dL upon admission to the hospital. a Estimate the t1/2 of tobramycin in this patient based on his kidney function. b A loading dose of 2-mg/kg tobramycin given as constant rate IV infusion of 1 hr duration followed by 150-mg tobramycin every 8 hr given as constant rate IV infusion of 1 hr duration was prescribed. The patient received his first dose of tobramycin and before administration of the second dose, you were asked to determine the pharmacokinetic parameters of tobramycin in this patient. Recommend a sampling schedule around the second dose, which can allow the determination of tobramycin pharmacokinetic parameters in this patient (please give the exact number and time of samples). c Despite the pharmacokinetic consultation, the patient continued to receive 150-mg tobramycin every 8 hr given as constant rate IV infusion of 1 hr duration. At steady state, the maximum and the minimum plasma concentrations of tobramycin (measured immediately after the end of drug administration and just before drug administration at steady state) were 11.3 and 3.85 mg/L, respectively. Calculate the t1/2 and the Vd of tobramycin in this patient. d Recommend a dosing regimen (dose and dosing interval) that will achieve steady-state maximum and minimum plasma concentrations of 6 and 1 mg/L, respectively. Answer: a CrCL =

(140-age) (Wt in kg) 72 (s. Cr. in mg/dL)

CrCL =

(140-38 year) (68 kg) = 60 mL/min 72(1.6 mg/dL)

Fraction of KF =

Patient Cr CL 60 mL/min = = 0.5 120 mL/min 120 mL/min

Estimated tobramicin t1/2 =

Normal t1/2  (2.5 hr) 2.5 hr = = 5 hr Fraction of KF 0.5 

Also, using Eq. 15.16 the elimination rate constant can be estimated:

(

)

k for aminoglycosides in hr −1 = 0.00293 (CrCL in mL/min ) + 0.014

(

)

k hr −1 = 0.00293 (60) + 0.014 = 0.1898 hr −1 Half-life = 3.7 hr The half-life estimated by the two methods is slightly different. May be the average of the two estimates can be a better estimate.

302  Drug Administration by Intermittent IV Infusions b The blood samples will be obtained around the second sample (before reaching steady state). So, one sample should be obtained before administration of the second dose and two to three samples should be obtained after the end of drug administration over a period equivalent approximately to the estimated tobramycin half-life (5 hr). First sample: Second sample: Third sample: Fourth sample:

Just before drug administration 0.5 hr after the end of drug administration 2 hr after the end of drug administration 5 hr after the end of drug administration

c Cpmin ss = Cpmax  ss e− k (τ− t ′) 3.85 mg/L = 11.3 mg/L e− k (8 hr −1 hr) k = 0.154 hr −1 t1/2 = 4.5 hr Cpmax  ss =

K0 (1 − e− kt ′ ) k Vd(1 − e− kτ )

11.3 mg/L =

150 mg/hr (1 − e−0.154 hr −1

0.154 hr Vd(1 − e

−1

1 hr

)

−0.154 hr −1 8 hr

)

Vd = 17.4 L d Dosing regimen: Cpmin ss = Cpmax ss e− k (τ− t ′) 1 mg/L = 6 mg/L e−0.154 hr

−1

(τ−1 hr)

(τ − t′) = 11.6 hr An appropriate dosing interval = 12 hr Cpmax ss =

6 mg/L =

K0 (1 − e− kt ′ ) k Vd(1 − e− kτ ) K0 (1 − e−0.154 hr −1

−1

0.154 hr 17.4 L (1 − e

1 hr

)

−0.154 hr −1 12 hr

)

K0 = 95 mg administered over 1 hr Recommendation: 95 mg of tobramycin administered as constant rate IV infusion over 1 hr and repeated every 12 hr.

Drug Administration by Intermittent IV Infusions 303 15.6 Summary

• Drug administration by intermittent IV infusion is useful when IV bolus administra•

• • •

tion results in high plasma drug concentration that may increase the risk of adverse effects such as in case of aminoglycosides and vancomycin. Assuming that the drug is administered by IV bolus administration introduces significant error to the calculation of drug concentrations. This error results from ignoring the amount of drug eliminated during drug administration by short infusion and can have significant clinical sequences. During multiple drug administration by intermittent IV infusion, the steady state is achieved after five to six times the elimination half-life, and the steady-state concentration is directly proportional to the dosing rate and inversely proportional to the drug clearance. The methods used for individualization of aminoglycoside therapy assume that these drugs follow the one-compartment pharmacokinetic model and can calculate the appropriate dosage regimen with reasonable accuracy. The efficacy and safety of aminoglycosides have been associated with a certain range of plasma concentrations. So, using the plasma concentrations as a guide for individualization of aminoglycosides therapy is important to ensure optimal therapeutic effect with minimal toxicity.

Practice Problems 15.1 A 60 kg patient received a 1 hr infusion of gentamicin, and the pharmacokinetic study showed that gentamicin half-life is 2.7 hr and volume of distribution is 0.21 L/kg. The desired maximum and minimum steady-state concentrations for this patient are 6 and 1 mg/L, respectively. a Calculate the dosing regimen to achieve steady-state concentrations around these desired concentrations. b Calculate the actual steady-state maximum and minimum plasma concentrations for the regimen you recommended. 15.2 A patient is receiving 75-mg gentamicin every 4 hr as an infusion of 1 hr duration. The maximum and minimum plasma gentamicin concentrations achieved at steady state are 6 and 2.35 mg/L, respectively. a Calculate gentamicin half-life in this patient. b Calculate gentamicin volume of distribution in this patient. c Recommend a dosing regimen to achieve steady-state maximum and minimum plasma gentamicin concentrations around 8 and 1 mg/L, respectively. d Calculate the actual steady-state maximum and minimum plasma concentrations that should be achieved with the regimen you recommended in c. 15.3 A patient is receiving 50 mg every 6 hr of tobramycin as intermittent IV infusion of 1 hr duration. At steady state, three plasma samples were obtained and tobramycin concentrations were as follows: Time

Before the dose

0.5 hr after the infusion

2 hr after the infusion

Conc (mg/L)

1.26

3.56

2.52

304  Drug Administration by Intermittent IV Infusions a Calculate tobramycin half-life in this patient. b Calculate tobramycin volume of distribution in this patient. c Recommend a dosing regimen to achieve steady-state maximum and minimum plasma tobramycin concentrations around 10 and 1.5 mg/L, respectively. d Calculate the actual steady-state maximum and minimum plasma concentrations that should be achieved with the regimen you recommended in c. 15.4 A 60-year-old, 60-kg male was admitted to the hospital because of acute pneumonia. Gentamicin 75 mg every 12 hr given as a short infusion of 1 hr was prescribed. Because his serum creatinine was found to be 2.5 mg/dL, the pharmacokinetic service was consulted to adjust gentamicin dosing regimen. a Estimate the creatinine clearance in this patient. b What is the percentage renal function remaining in this patient? c What is the estimated half-life of gentamicin in this patient? Four plasma samples were drawn around the second dose, just before giving the second dose and 1, 4, and 10 hr after the second dose. Samples were analyzed for gentamicin and the results came back as follows: Time: just before

1 hr post

4 hr post

10 hr post

Conc (mg/L): 2.1

6.5

5.11

3.13

d Calculate the half-life and the volume of distribution of gentamicin in this patient. e What was the maximum plasma concentration after the first dose? f What would be the steady-state maximum and the minimum plasma concentrations if the 75 mg every 12 hr was continued? g Comment on the difference between the maximum plasma concentration after the first dose and the maximum plasma concentration at steady state (during the 75 mg every 12 hr regimen). h Recommend a dosing regimen (dose and dosing interval) to achieve steadystate maximum and minimum plasma concentrations of 7 and 1 mg/L, respectively. 15.5 A patient is receiving tobramycin as 100 mg every 8 hr as a short infusion over half an hour. His serum creatinine was 1 mg/dL. At steady state, two plasma samples were obtained, 0.5 before the administration of the dose and 1 hr after the end of the infusion, and the tobramycin plasma concentrations were 2.18 and 8.73 mg/L in the two samples, respectively. a Calculate the half-life and volume of distribution of tobramycin in this patient. b Recommend a dosing regimen to achieve steady-state maximum and minimum plasma concentrations of 9 and 1 mg/L, respectively. c Calculate the actual maximum and minimum plasma concentrations achieved at steady state with the regimen you recommended in b. d The patient came back to the hospital six months later and the serum creatinine was 2 mg/dL. What is the estimated new half-life of tobramycin in this patient?

Drug Administration by Intermittent IV Infusions 305 15.6 An 84-year-old, 50-kg male is admitted to the hospital and he is to receive gentamicin. His serum creatinine was 1.2 mg/dL. Gentamicin 65 mg every 12 hr given as short infusion of 1 hr duration was prescribed. a Estimate the half-life of gentamicin in this patient. b Recommend a sampling schedule around the second dose that will allow the determination of the pharmacokinetic parameters in this patient. Samples were obtained and a dose of 75 mg every 24 hr as short infusion (1 hr duration) was recommended. At steady state, two samples were obtained, 1 and 8 hr after the end of the dose, and the gentamicin plasma concentration was 9.25 and 5.4 mg/L, respectively. c Calculate the half-life and volume of distribution of gentamicin in this patient. d Recommend a dosing regimen to achieve steady-state maximum and minimum plasma concentrations of 9 and 1 mg/L, respectively. e Calculate the actual maximum and minimum plasma concentrations achieved at steady state with the regimen you recommended in d. References 1. Murphy JE and Winter ME “Clinical pharmacokinetic pearls: Bolus versus infusion equations” (1996) Pharmacother; 16:698–700. 2. Sawchuk RJ, Zaske DE, Cippolle JR, Wargin WA and Strate RG “Kinetic model for gentamicin dosing with the use of individual patient parameter” (1977) Clin Pharmacol Ther; 21:362–365. 3. Bauer LA “Applied clinical pharmacokinetics” 2nd Edition (2008) McGraw-Hill Companies, Inc., New York, USA. 4. Nicolau DP, Freeman CD, Belliveau PP, Nightingale CH, Ross JW and Quintiliani R “Experience with a once-daily aminoglycoside program administered to 2184 adult patients” (1995) Antimicrob Agents Chemother; 39:650–655. 5. Barclay ML, Begg EJ and Hickling KG “What is the evidence for once-daily aminoglycoside therapy? (1994) Clin Pharmacokinet; 27:32–48. 6. Zaske DE, Sawchuk RJ, Gerding DN and Strate RG “Increased dosage requirements of gentamicin in burn patients” (1976) J Trauma; 16:824–828. 7. Zaski DE, Sawchuk RJ and Strate RG “The necessity of increased doses of amikacin in burn patients” (1978) Surgery; 84:603–608.

16 Physiological Approach to Hepatic Clearance

Objectives After completing this chapter, you should be able to:

• Describe the physiological meaning of the organ clearance in terms of organ blood flow, intrinsic clearance, and extraction ratio.

• Describe the effect of changing the hepatic intrinsic clearance and blood flow on the efficiency of the liver in eliminating the drug (extraction ratio).

• Analyze the effect of changing the intrinsic clearance on the plasma concentration-time profile of high and low extraction ratio drugs after IV and oral administration.

• Analyze the effect of changing the liver blood flow on the plasma concentration-time profile of high and low extraction ratio drugs after IV and oral administration.

• Discuss the drug characteristics that will make the drug elimination rate sensitive to changes in the protein binding.

• Evaluate the potential for clinically significant drug interactions that involve enzyme

induction or inhibition, modification of the hepatic blood flow, and change in protein binding based on the characteristics of the interacting drugs.

16.1 Introduction The physiological approach to organ clearance is a simple approach that utilizes the organ’s ability to eliminate the drug, organ blood flow, and protein binding to describe the rate of drug elimination by the organ and the drug profile in the body. This approach allows prediction of the variations in drug disposition that occur due to physiological and pathological changes. For example, the hepatic clearance of drugs is dependent on the liver’s intrinsic ability to metabolize the drug, which is dependent on the metabolizing enzyme activity, and the rate of delivery of the drug to the liver, which is dependent on the liver blood flow. Enzyme induction and enzyme inhibition can alter the liver’s ability to metabolize the drug and can lead to changes in hepatic clearance. Also, diseases such as congestive heart failure and drugs such as beta-blockers can decrease the cardiac output and decrease the liver blood flow, which can affect the hepatic clearance of some drugs. 16.2  The Organ Clearance The organ clearance is defined as the volume of the blood completely cleared from the drug per unit time by that organ. It is a measure of the efficiency of the organ in eliminating the drug from the blood reaching the organ. Based on the definition above, the DOI: 10.4324/9781003161523-16

Physiological Approach to Hepatic Clearance 307 clearance can be estimated from the organ blood flow and the difference in the arterial and venous drug concentration across the organ. Eq. 16.1 describes the general relationship between the drug clearance, excretion rate, and concentration, which can be modified to describe the drug clearance by a particular organ as in Eq. 16.2. CL =

Drug excretion rate (16.1) Drug concentration

CL organ = 

Amount of drug delivered to the organ-Amount of drug leaving the organ Concentration of drug delivered the organ (16.2)

The amount of drug delivered to the organ is equal to the product of drug concentration in the arterial side and the organ blood flow, while the amount of drug leaving the organ is equal to the product of drug concentration in the venous side and the organ blood flow (1). Based on this definition, the hepatic clearance, CLH, can be expressed by Eq. 16.3: CL H =

QCpa − QCpv (16.3) Cpa

where Q is the hepatic blood flow, Cpa is the drug concentration in the arterial side of the liver (portal and arterial), and Cpv is the drug concentration in the venous side of the liver. The following two equations can be obtained from Eq. 16.3.  Cp − Cpv  CL H = Q  a  (16.4)  Cpa  CL H = QE (16.5) where E is the extraction ratio, which is a measure of the efficiency of the organ in eliminating the drug during single pass through the organ. 16.3  Hepatic Extraction Ratio The hepatic extraction ratio is the fraction of the amount of drug delivered to the liver, which is eliminated during single pass through the liver. When the drug that reaches the liver is completely eliminated, i.e., Cpv is equal to zero, then the hepatic extraction ratio becomes equal to one. On the other hand, when no drug is eliminated while passing through the liver, i.e., Cpv = Cpa, then the hepatic extraction ratio is equal to zero. The hepatic extraction ratio can take values from zero to one (0–1), and it is a measure of the liver elimination efficiency during single pass of the drug through the liver. This is different from the fraction of dose metabolized, which is the fraction of the administered dose that will be eliminated from the body through the metabolic pathway. There are examples of drugs that are completely metabolized but have low hepatic extraction ratio and drugs that have high hepatic extraction ratio and are not completely metabolized. The hepatic extraction ratio is a function of the intrinsic ability of the liver to eliminate the drug (intrinsic clearance, CLint) and the hepatic blood flow as in Eq. 16.6 (2). E=

CL int (16.6) Q + CL int

308  Physiological Approach to Hepatic Clearance Drugs are classified as low extraction ratio drugs if their hepatic extraction ratio is less than 0.3 (e.g., diazepam, warfarin, salicylic acid, phenytoin, and procainamide), or high extraction ratio drugs if their hepatic extraction ratio is larger than 0.7 (e.g., lidocaine, propranolol, morphine, verapamil, and propoxyphene), while if hepatic extraction ratio is between 0.3 and 0.7, the drugs are considered intermediate extraction ratio drugs. Drugs in the same hepatic extraction ratio class usually share some common characteristics that relate to alteration in the drug overall elimination process in response to changes in the physiological parameters. So, the effect of some physiological and pathological changes on the elimination of a particular drug can be predicted from the extraction ratio of the drug. Drugs eliminated by the kidney can also be classified according to their renal extraction ratio. Atenolol, digoxin, and methotrexate are examples of low extraction ratio drugs because their extraction ratio by the kidney is less than 0.3, while metformin and p-aminohippuric acid are examples of high extraction ratio drugs because their extraction ratio by the kidney is more than 0.7. 16.4  Intrinsic Clearance The organ CLint is the maximum ability of the organ to eliminate the drug in absence of any flow limitation. In the liver, for example, CLint is the inherent ability of the hepatic enzymes to metabolize a drug and it reflects the maximum capacity of the liver to metabolize the drug (3). On the other hand, the actual observed drug CLH depends on the probability of the drug molecules to get in contact with the metabolizing enzymes. This depends on the amount of enzymes available, which is reflected by CLint, and the ability of the drug to diffuse to the organ to get in contact with the enzymes, which is dependent on the organ blood flow. Equation 16.7 describes the relationship between CLH, Q, E, and CLint.  CL int  CL H = QE = Q   (16.7)  Q + CL int  By rearrangement, Eq. 16.8, which describes CLint, can be obtained. CL int =

QE (16.8) 1− E

This relationship indicates that CLint for high extraction ratio drugs is much higher than CLH, while for low extraction ratio drugs, CLint is close to CLH. 16.5  Systemic Bioavailability Orally administered drugs that are completely absorbed to the portal circulation can have incomplete bioavailability due to hepatic metabolism during the first pass through the liver. In this case, the bioavailability (F) is dependent on the hepatic extraction ratio as in Eq. 16.9 (4). F = 1 − E(16.9) This means that F for high hepatic extraction ratio drugs is much lower than F for low hepatic extraction ratio drugs when the drugs are absorbed completely to the portal

Physiological Approach to Hepatic Clearance 309 circulation. Factors that can alter the hepatic extraction ratio such as the change in Q and/or CLint can affect F for orally administered drugs, while F of drugs that are not absorbed completely from the gastrointestinal tract (GIT) to the portal circulation is determined from the fraction of dose absorbed to the portal circulation and the fraction of the dose that escapes the first-pass hepatic metabolism. F = FG (1 − E)(16.10) where FG is the fraction of drug dose absorbed to the portal circulation. 16.6 The Effect of Changing Intrinsic Clearance and Hepatic Blood Flow on the Hepatic Clearance, Systemic Availability, and Drug Concentration-Time Profile The drug CLint and Q can change due to some physiological and pathological conditions or because of using some drugs. Liver diseases such as cirrhosis usually decrease the CLint of drugs, while congestive heart failure decreases the hepatic blood flow. Also, some drugs can induce or inhibit the hepatic metabolizing enzymes that can affect the CLint for other drugs. However, drugs like beta-blockers decrease the cardiac output and the hepatic blood flow. In general, the effect of the change in CLint or Q on drug clearance, bioavailability, and elimination rate of low extraction ratio drugs and high extraction ratio drugs is different. However, within each class of drugs, the effect of the change in the CLint or Q is approximately similar (1). This is important because based on the drug extraction ratio, it is possible to expect the alteration in the drug pharmacokinetic behavior because of some physiological and pathological conditions and to predict the outcome of some drug-drug interactions. The CLH of any drug is dependent on the drug CLint and Q as presented in Eq. 16.7. For drugs that have CLint ≪ Q, Eq. 16.7 can be approximated by Eq. 16.11. CL H = Q

CL int = CL int(16.11) Q 

Equation 16.11 indicates that the CLH for the drugs that have CLint ≪ Q (i.e., low extraction ratio drugs) is approximately equal to the drug CLint. Any change in the enzyme activity by induction or inhibition that affects the CLint will affect the CLH of the low extraction ratio drugs. Changes in Q will not have significant effect on the CLH of low extraction ratio drugs. On the other hand, when CLint ≫ Q, Eq. 16.7 can be approximated by Eq. 16.12. CL H = Q(16.12) Equation 16.12 indicates that the CLH for the drugs with CLint ≫ Q (i.e., high extraction ratio) is approximately equal to Q. Any change in Q will affect the CLH of high extraction ratio drugs. Changes in the drug CLint will not have significant effect on CLH of high extraction ratio drugs. The previous discussion indicates that drugs within the same class based on their extraction ratio (low, high, or intermediate) are affected similarly by the change in CLint and Q. The following numerical examples demonstrate how the change in CLint and Q

310  Physiological Approach to Hepatic Clearance affects the CLH, F, and the overall plasma drug concentration-time profile, in low and high extraction ratio drugs. 16.6.1  Low Extraction Ratio Drugs

Assume that we have a low extraction ratio drug that is completely metabolized meaning that the CLH is equal to CLT. After oral administration, the drug is completely absorbed to the portal circulation indicating that the hepatic metabolism during the first pass through the liver is the cause of its incomplete bioavailability. The drug extraction ratio is equal to 0.05, and Q is 1.5 L/min. The CLint of this drug can be calculated from Eq. 16.8: CL int =

QE 1.5 L/min 0.05 = = 0.079 L/min 1− E 1 − 0.05

CLH can be calculated from Eq. 16.5: CL H = QE = 1.5 L/min × 0.05 = 0.075 L/min F can be calculated from Eq. 16.9: F = 1 − E = 1 − 0.05 = 0.95 16.6.1.1 Assume that the Drug CLint Increases to Double Its Original Value Due to Enzyme Induction and Q Stays the Same

The new CLint after induction: CL int = 0.079 L/min × 2 = 0.158 L/min The extraction ratio after induction: E=

CL int 0.158 L/min = = 0.095 Q + CL int 1.5 L/min + 0.158 L/min

CLH after induction: CL H = QE = 1.5 L/min × 0.095 = 0.143 L/min F after induction: F = 1 − E = 1 − 0.095 = 0.905 Based on these calculations, enzyme induction causes an increase in the extraction ratio from 0.05 to 0.095. This small change represents about 90% increase in the extraction ratio, which results in significant increase in the CLH from 0.075 to 0.143 L/min (about 90% increase). The change in extraction ratio also decreases F from 0.95 to 0.905 (about 5% decrease). So, enzyme induction significantly increases the CLH (and CLT in this example) of low extraction ratio drugs after IV and oral administration, which

Physiological Approach to Hepatic Clearance 311

Figure 16.1 The plasma concentration-time profile for a low extraction ratio drug after administration of (A) single IV bolus dose and (B) single oral dose, before (——) and after (_______) enzyme induction.

significantly increases the rate of decline in the drug concentration-time profile after IV and oral administration, if Vd does not change. However, enzyme induction does not significantly affect F of low extraction ratio drugs as illustrated in Figure 16.1. The pharmacokinetic parameters

Before induction

After induction

Q E CLint CLH and CLT F

1.5 L/min 0.05 0.079 L/min 0.075 L/min 0.95

1.5 L/min 0.095 0.158 L/min 0.143 L/min 0.905

16.6.1.2 Assume that Q Decreases by 50% (i.e., New Q = 0.75 L/min) without Affecting CLint

The new extraction ratio after the reduction in Q: E=

CL int 0.079 L/min = = 0.095 Q + CL int 0.75 L/min + 0.079 L/min

CLH after the decrease in Q: CL H = QE = 0.75 L/min × 0.095 = 0.0712 L/min F after the decrease in Q: F = 1 − E = 1 − 0.095 = 0.905 Based on these calculations, the decrease in Q results in an increase in the extraction ratio from 0.05 to 0.095, which represents about 90% increase in the extraction ratio. The drug CLH decreases from 0.075 L/min to 0.0712 L/min (about 5% decrease), and F decreases from 0.95 to 0.905 (about 5% decrease). So, the change in Q does not significantly affect the CLH (and CLT in this example) and F of low extraction ratio drugs,

312  Physiological Approach to Hepatic Clearance

Figure 16.2 The plasma concentration-time profile for a low extraction ratio drug after administration of (A) single IV bolus dose and (B) single oral dose, when the hepatic blood flow is 1.5 L/min (_______) and after the reduction of the hepatic blood flow to 0.75 L/min (——).

which should not affect the plasma concentration-time profile of the drug after IV and oral administration as illustrated in Figure 16.2. The pharmacokinetic parameters

Normal hepatic blood flow

Low hepatic blood flow

Q E CLint CLH and CLT F

1.5 L/min 0.05 0.079 L/min 0.075 L/min 0.95

0.75 L/min 0.079 0.079 L/min 0.0712 L/min 0.905

16.6.2  High Extraction Ratio Drugs

Assume that we have a high extraction ratio drug that is completely metabolized meaning that the CLH is equal to CLT. After oral administration, the drug is completely absorbed to the portal circulation, which means that the hepatic metabolism during the first pass through the liver is the cause of its incomplete bioavailability. The drug extraction ratio, E, is equal to 0.95, and Q is 1.5 L/min. The CLint of this drug can be calculated from Eq. 16.8: CL int =

QE 1.5 L/min 0.95 = = 28.5 L/min 1− E 1 − 0.95

CLH can be calculated from Eq. 16.5: CL H = QE = 1.5 L/min × 0.95 = 1.425 L/min F can be calculated from Eq. 16.9: F = 1 − E = 1 − 0.95 = 0.05

Physiological Approach to Hepatic Clearance 313 16.6.2.1 Assume that the Drug CLint Increases to Double Its Original Value Due to Enzyme Induction and Q Stays the Same

The new CLint after induction: CL int = 28.5 L/min × 2 = 57.0 L/min The extraction ratio after induction: E=

CL int 57 L/min = = 0.974 Q + CL int 1.5 L/min + 57 L/min

CLH after induction: CL H = QE = 1.5 L/min × 0.974 = 1.461 L/min F after induction: F = 1 − E = 1 − 0.974 = 0.026 Based on these calculations, enzyme induction results in the increase in the extraction ratio from 0.95 to 0.974 (about 2.5% increase), which results in a very small increase in CLH from 1.425 to 1.461 L/min (about 2.5% increase), and a significant decrease in F from 0.05 to 0.026 (about 50% decrease). So, enzyme induction does not significantly affect the CLH (and CLT in this example) of high extraction ratio drugs and will not significantly affect the rate of decline in the plasma concentration-time profile after single IV and oral drug administration. However, enzyme induction decreases F of high extraction ratio drugs, which results in significant decrease in the drug area unser the plasma concentration-time curve (AUC) as illustrated in Figure 16.3. The pharmacokinetic parameters

Before induction

After induction

Q E CLint CLH and CLT F

1.5 L/min 0.95 28.5 L/min 1.425 L/min 0.05

1.5 L/min 0.974 57.0 L/min 1.461 L/min 0.026

Figure 16.3 The plasma concentration-time profile for a high extraction ratio drug after administration of (A) single IV bolus dose and (B) single oral dose, before (——) and after (_______) Enzyme induction.

314  Physiological Approach to Hepatic Clearance 16.6.2.2 Assume that Q Decreases by 50% (New Q = 0.75 L/min) without Affecting CLint

The extraction ratio after the reduction in Q: E=

CL int 28.5 L/min = = 0.974 Q + CL int 0.75 L/min + 28.5 L/min

CLH after the decrease in Q: CL H = QE = 0.75 L/min × 0.974 = 0.7305 L/min F after the decrease in Q: F = 1 − E = 1 − 0.974 = 0.026 Based on these calculations, the decrease in Q results in the increase in the extraction ratio from 0.95 to 0.974 (about 2.5% increase), significant decrease in CLH from 1.425 to 0.7305 L/min (about 50% decrease), and significant decrease in F from 0.05 to 0.026 (about 50% decrease). So, the decrease in Q significantly decreases the CLH (and CLT in this example) of high extraction ratio drugs after IV and after oral administration, which is reflected by the slower rate of decline in the drug concentration-time profile. After oral administration, the decrease in clearance of the high extraction ratio drugs as a result of the reduction in Q increases the drug AUC. However, the decrease in Q also causes a reduction in F of this high extraction ratio drug that decreases the AUC. So, the increase in the AUC due to the reduction in drug clearance is compensated by the decrease in AUC due to the reduction in F and the drug AUC does not change significantly as illustrated in Figure 16.4. The pharmacokinetic parameters

Normal hepatic blood flow

Low hepatic blood flow

Q E CLint CLH and CLT F

1.5 L/min 0.95 28.5 L/min 1.425 L/min 0.05

0.75 L/min 0.974 28.5 L/min 0.7305 L/min 0.026

Figure 16.4 The plasma concentration-time profile for a high extraction ratio drug after administration of (A) a single IV bolus dose and (B) a single oral dose, when the hepatic blood flow is 1.5 L/min (_______) and after the reduction of the hepatic blood flow to 0.75 L/min (——).

Physiological Approach to Hepatic Clearance 315 Clinical Importance:

• The calcium channel blocker felodipine is high extraction ratio drug that has 15% bio-

• • • • • • •

availability due to presystemic metabolism. Inhibition of felodipine metabolism due to administration of grapefruit juice significantly increases its bioavailability resulting in approximately twofold increase in felodipine AUC (5). Examples of clinically significant drug interactions between low extraction ratio drugs and enzyme inducers and inhibitors (6): Valproic acid with the enzyme inducers phenytoin and carbamazepine. Ethinyl estradiol with the enzyme inducer carbamazepine. Simvastatin with the enzyme inhibitor clarithromycin. Prednisone with the enzyme inhibitor verapamil. Risperidone and tramadol with the enzyme inhibitor fluoxetine. Warfarin with the enzyme inhibitor metronidazole.

Practice Problem: Question: An experimental antiarrhythmic drug (MMH-98) is completely metabolized by the liver to inactive metabolites. In clinical trials, this drug was found to have hepatic extraction ratio of 0.89 in a congestive heart failure patient who has hepatic blood flow of 1.22 L/min. This drug is 96% bound to plasma protein. Although the oral bioavailability of this drug is less than 100%, when it is administered orally all the dose is absorbed from the GIT and reaches the portal circulation. a b c d

Calculate the CLT of this drug in this patient. Calculate the CLint of this drug in this patient. What is the AUC produced after administration of a single dose of 300 mg orally? In addition to the antiarrhythmic drug the patient started taking an inotropic drug which increases the cardiac output with no effect on any other organs. Due to this increase in cardiac output, the liver blood flow increased to 1.62 L/min.

How does the increase in the hepatic blood flow affect the value of each of the following parameters? Hepatic intrinsic clearance: Hepatic extraction ratio: Total body clearance: Oral bioavailability:

increase increase increase increase

decrease decrease decrease decrease

no change no change no change no change

Answer: a Since the drug is completely metabolized, CLH is equal to CLT CL T = CL H = QE = 1.22 L/min × 0.89 = 1.086 L/min b CL int =

QE 1.22 L/min × 0.89 = = 9.87 L/min 1− E 1 − 0.89

c AUC =

F Dose (1 − E)Dose 0.11 × 300 mg = = = 30.4 mg-min/L CL T CL T 1.086 L/min

316  Physiological Approach to Hepatic Clearance d The hepatic intrinsic clearance is dependent on the intrinsic ability of the liver to metabolize the drug and is independent of the hepatic blood flow. So, the hepatic intrinsic clearance will not change due to the increase in hepatic blood flow • Hepatic extraction ratio:  E =

9.87 L/min CL int     = = 0.859 Q + CL int 1.62 L/min + 9.87 L/min

The increase in hepatic blood flow decreases the extraction ratio. • Total body clearance: CL T = CL H = QE = 1.62 L/min × 0.859 = 1.392 L/min The CLT that is equal to CLH for this drug increases with the increase in the hepatic blood flow. • Oral bioavailability: Bioavailability ( F ) = 1 − E = 1 − 0.859 = 0.141 The oral bioavailability increases because the hepatic extraction ratio decreases due to the increase in hepatic blood flow. 16.7  Protein Binding and Hepatic Extraction The previous discussion has been concerned with the total drug concentration in the blood. However, it is known that most drugs are bound to blood constituents to some extent. So, it is important to consider the effect of drug protein binding on drug clearance and half-life. It is usually assumed that only the free (unbound) drug is available for elimination. However, many examples exist in which the hepatic extraction is sufficiently high that the drug can be removed from its binding sites during passage through the liver. Two different extraction processes are usually observed. One restricted or limited to the free drug molecules and one which is nonrestricted in which both free and bound drug molecules can be removed by the liver (7). When the free drug concentration is considered, the following relationship can be obtained:  f CL ′int  CL H = QE = Q  u (16.13) ′   Q + fu CL int where fu is the free fraction of the drug in the blood, and CL’int is the intrinsic clearance for the free drug. The effect of protein binding on the hepatic clearance of drugs is different depending on the drug characteristics. The clearance of high extraction ratio drugs is considered flow dependent. So, changes in the protein binding of these drugs do not affect their hepatic clearance. This is because during drug passage through the liver both free and bound drug molecules can be extracted and eliminated by the liver. The hepatic clearance of drugs that have low extraction ratio and are not bound to protein to large extent (3.5 60% metabolized), while patients with a score of 10 or more require a 50% reduction of their initial dose of the same group of drugs. It is important to note that this is a recommendation for the starting dose of drugs. After starting drug therapy, the therapeutic and adverse effects of the drugs should be monitored, and dose adjustment should be made when necessary. In addition to the Child-Pugh classification, dynamic liver function tests have been developed to assess the liver handling of drugs in liver dysfunction patients. However, these tests are only applied for research purposes. This is achieved by IV administration of exogenous compounds that are mainly eliminated by the liver and then following the disappearance of the compound from the patient systemic circulation, or the appearance of a metabolite in plasma, urine, or in the expired air as in the breath test. Examples of these chemical probes include indocyanine green, galactose, sorbitol, antipyrine, caffeine, and midazolam. The rate of metabolism of some drugs such as aminopyrine, erythromycin, and caffeine has been evaluated using the 14CO2 breath test. The 14C-labeled drugs are administered to the patients and the rate of 14CO2 excreted in expired air is used as a measure of the rate of drug metabolism. The results of the breath test have been shown to correlate well with the Child-Pugh classification. There is no approach that has been shown to be superior to the other in dose adjustment in patients with liver diseases. However, the ChildPugh score is used more in clinical practice to guide the dose adjustment in patients with liver diseases because the information required to apply this approach is readily available. Clinical Importance:

• Lidocaine is an antiarrhythmic drug that is eliminated mainly by hepatic metabo-

lism. So, dosage reduction is usually required when lidocaine is used in patients with liver dysfunction. Calculation of the initial lidocaine dose is usually guided by the Child-Pugh score. • The antiepileptic drugs carbamazepine, phenytoin, phenobarbital, and valproic acid are eliminated mainly by hepatic metabolism and their clearances are reduced in patients with reduced liver function. The effect of liver disease on the clearance of these drugs is variable but patients with reduced liver function usually require lower than average dose of these drugs. Practice Problems: Question: A 55-year-old female was admitted to the hospital after developing an episode of ventricular arrhythmia. The patient had a history of multiple medical problems, including liver cirrhosis, hypertension, and ischemic heart disease. Her laboratory values upon admission were serum creatinine 1.1 mg/dL, serum albumin 3.2 gm/dL, total bilirubin 4.5 mg/dL, and prothrombin time 8 seconds more than control. Physical

328  Pharmacokinetics in Patients with Eliminating Organ Dysfunction examination showed that the patient is alert without any signs of encephalopathy, and the patient has mild ascites. The physician wanted to start the patient on lidocaine and asked you to recommend an average starting dose of lidocaine. Answer: Lidocaine is an antiarrhythmic drug that is completely metabolized. So, the patient’s liver condition may affect the clearance of lidocaine. It is wise to calculate the Child-Pugh score for his patient according to Table 17.1 to determine if dose reduction is necessary. Calculation of the Child-Pugh score: Serum albumin Bilirubin Prothrombin time Ascites Encephalopathy

3.2 gm/dL 4.5 mg/dL 8 sec>control Mild Absent Total

score = 2 score = 3 score = 3 score = 2 score = 1 Child-Pugh score = 11

According to the Child-Pugh score, the starting dose of lidocaine in this patient should be 50% of the average recommended dose of lidocaine. Lidocaine dose can be adjusted after the start of therapy according to the therapeutic and the adverse effects of lidocaine. 17.5  Other Patient Populations Drug regulatory authorities around the world recommend studying the drug pharmacokinetic behavior in different patient populations during the drug development process. The objectives of these studies are to determine if any patient population requires special dosage recommendations to ensure optimal drug efficacy and safety and to guide these recommendations. Investigation of the pharmacokinetic behavior in special populations is usually carried out by population pharmacokinetic studies performed during phase III clinical trials when sparse samples are obtained from a large number of patients representing most of the populations that will be receiving the drug. When there is evidence that the drug pharmacokinetic behavior is different in any patient population, appropriate dosage recommendation should be developed for this patient population. The recommendation is made to ensure that most patients in this special population get drug exposure within the range of the desired target exposure. When dosing recommendation for a special population cannot be developed for any reason, a warning for compromised efficacy and/ or safety in this special population should be included in the drug label. Also, exclusion of this special population from the drug indication is also possible when there is great potential for serious clinical consequences associated with drug use in this special population. 17.6 Summary

• Eliminating organ dysfunction slows the rate of drug elimination and decreases the dosage requirements of drugs that are eliminated by the impaired pathway.

• Drugs with a higher extent of excretion by the impaired pathway require larger reduction in their dose.

• The higher the degree of eliminating organ impairment, the larger the required reduction of the drug dose.

Pharmacokinetics in Patients with Eliminating Organ Dysfunction 329

• Studying the drug pharmacokinetic behavior in different patient populations during

the drug development process is important to ensure proper dosage recommendation in all patient populations.

Practice Problems 17.1 A patient admitted to the hospital because of acute myocardial infarction was started on oral antiarrhythmic medication. He was given 600 mg every 12 hr from an oral formulation, which is known to be rapidly absorbed but only 80% bioavailable. At steady state, the maximum and minimum plasma concentrations were 30 and 10.5 μg/mL, respectively. a Calculate mathematically the half-life and the volume of distribution of this drug in this patient. b The patient suddenly developed acute renal failure and the kidney function dropped to 30% of its normal value. If the renal clearance of this drug is 75% of the total body clearance, recommend an appropriate dose for this patient after developing the renal failure. c Calculate the maximum and minimum plasma concentrations achieved at steady state from the regimen you recommended in part b. 17.2 A 65-year-old man was taking his antihypertensive medication as 500 mg every 12 hr. He was taking his medication in the form of an oral tablet that is rapidly and completely absorbed. After the administration of a single oral 500-mg dose, 375 mg is excreted unchanged in urine and the drug half-life is 8 hr. The average plasma concentration of this antihypertensive drug in this patient while taking 500 mg every 12 hr was 20 mg/L. a Calculate the volume of distribution of this drug in this patient. b This patient developed acute renal failure and his creatinine clearance decreased from the normal value of 120–30 mL/min. Calculate the new total body clearance for this drug in this patient after developing the renal failure if the deterioration in the kidney function does not affect the drug distribution characteristics. c Recommend a maintenance dose for this patient after developing the renal failure if the usual dose in patients with normal kidney function is 500 mg every 12 hr. d Calculate the maximum and minimum plasma concentrations achieved at steady state from the regimen you recommended in c. 17.3 A patient was admitted to the hospital because of severe pneumonia. An IV loading dose of an antibiotic followed by IV maintenance doses of 300 mg q12 hr was prescribed. The creatinine clearance was 120 mL/min, which is considered normal for this patient. After 7 days of therapy (300 mg q12 hr), the maximum and minimum plasma concentrations were 38 and 12.5 mg/L, respectively. a Calculate mathematically the half-life and the volume of distribution of this drug in this patient. b At steady state (while receiving 300 mg q12 hr), 180 mg of the drug is excreted unchanged in urine during one dosing interval. What is the renal clearance of this drug?

330  Pharmacokinetics in Patients with Eliminating Organ Dysfunction c After 15 days of antibiotic therapy, the kidney function of this patient deteriorated, and his creatinine clearance was found to be 40 mL/min. What is the new half-life of the drug after this change in the patient’s kidney function? d What will be the steady-state maximum and minimum plasma concentrations if this patient continues taking 300 mg q12 hr IV despite the change in his kidney function? 17.4 After IV administration of procainamide (an antiarrhythmic drug), 60% of the administered dose is excreted unchanged in urine. The rest of the dose is metabolized in the liver. The average volume of distribution of procainamide is 2 L/kg and the half-life is 3 hr in patients with normal kidney function. a Calculate the renal clearance of procainamide in patients with normal kidney function. b Calculate the half-life of procainamide in a patient with creatinine clearance of 30 mL/min (normal creatinine clearance is 120 mL/min). c Calculate the daily oral maintenance dose required in a patient with creatinine clearance of 30 mL/min (normal creatinine clearance is 120 mL/min). (The average daily oral maintenance dose in patients with normal kidney function is 3 gm/day.) d If a patient with creatinine clearance of 30 mL/min (normal creatinine clearance is 120 mL/min) receives 750 mg every 6 hr, calculate the maximum and minimum plasma concentrations at a steady state in this patient. 17.5 A new antihypertensive drug is rapidly but incompletely absorbed. • When 800 mg is given intravenously to normal volunteers: 600 mg can be recovered in the urine unchanged from time zero to infinity. The AUC obtained from time zero to infinity is 400 mg hr/L. The elimination half-life is 8 hr. • When 400 mg is given orally: The AUC from time zero to infinity is 125 mg hr/L. a If a patient with normal kidney function is receiving 800 mg every 12 hr orally, what will be the average plasma concentration at steady state? b Calculate the maximum and the minimum plasma concentrations achieved at steady state for the regimen in part a (800 mg every 12 hr orally) if the drug is rapidly absorbed after oral administration. c If the average maintenance dose for a patient with normal kidney function is 800 mg IV every 12 hr, what will be the maintenance dose for a patient whose creatinine clearance is only 10% of normal? d What will be the maximum and the minimum plasma concentrations at a steady state for the IV regimen you recommended in part c for the patient with 10% of the normal kidney function if the change in kidney function does not affect the volume of distribution. References 1. Vanholder R, DeSmet R, Glorleux G and Arglles A “Review in uremic toxins: Classification, concentration, and interindividual variability” (2003) Kidney Int; 63:1934–1943. 2. Sun H, Frassetto L and Benet LZ “Effect of renal failure on drug transport and metabolism” (2006) Pharmaco Ther; 109:1–11.

Pharmacokinetics in Patients with Eliminating Organ Dysfunction 331 3. Koup JR, Jusko WJ and Elwood CM “Digoxin pharmacokinetics: Role of renal failure in dosage regimen design” (1979) Clin Pharmacol Ther; 18:9–21. 4. Jusko WJ, Szefler SJ and Goldfarb AL “Pharmacokinetic design of digoxin dosage regimen in relation to renal function” (1974) J Clin Pharmacol; 14:525–535. 5. DiPiro JT, Spruill WJ and Wade WE “Concepts in clinical pharmacokinetics” 4th Edition (2005) American Society of Hospital Pharmacist, Inc., Bethesda, MD, USA. 6. Giusti DL and Hayton WL “Dosage regimen adjustments in renal impairment” (1973) Drug Intell Clin Pharm; 7:382–387. 7. MacKichan JJ “Influence of protein binding and use of unbound (free) drug concentration” in “Applied pharmacokinetics and pharmacodynamics- principles of therapeutic drug monitoring”, Edited by Burton ME, Shaw LM, Schentang JJ and Evans WE (2006) Lippincott Williams & Wilkins, Philadelphia, PA, USA. 8. Blaschke TF “Protein binding and kinetics of drugs in liver diseases” (1977) Clin Pharmacokinet; 2:32–44. 9. Frye RF, Zgheib NK, Matzke GR, Chaves-Gnecco D, Rabinovitz M, Shaikh OS and Branch RA “Liver disease selectivity modulates cytochrome P450-mediated metabolism” (2006) Clin Pharmacol Ther; 80:235–245. 10. Rossat J, Maillard M, Nussberger J, Brunner HR and Burnier M “Renal effects of selective cyclooxygenase-2 inhibition in normotensive salt-depleted subjects” (1999) Clin Pharmacol Ther; 66:76–84.

18 Noncompartmental Approach in Pharmacokinetic Data Analysis

Objectives After completing this chapter, you should be able to:

• Discuss the situations when the noncompartmental data analysis approach can be applied.

• Define the mean residence time (MRT) after different routes of administration. • Calculate the area under the plasma concentration-time curve (AUC), the area under

the first-moment curve (AUMC), and the MRT from the plasma concentration-time data using the noncompartmental approach. • Analyze the drug absorption characteristics after extravascular administration using the noncompartmental approach. • Describe how the MRT can be determined using the compartmental approach. 18.1 Introduction Different modeling approaches have been used to describe the pharmacokinetic behavior of the drug in the body and to analyze pharmacokinetic data. The process usually requires good knowledge and skills in pharmacokinetics and statistics to ensure proper model selection, testing the model assumptions, and evaluation of the obtained results. The noncompartmental approach in pharmacokinetic data analysis is an alternative method that can be applied without the need to propose any specific model and require fewer assumptions. There is no data analysis approach that is superior to the other. However, the choice of data analysis method usually depends on the objectives of the experiment and the suitability of each method to achieve these objectives. A classic example of important pharmacokinetic studies that are usually analyzed using the noncompartmental approach is the bioequivalence studies. Clinical Importance:

• The main objective of the bioequivalence studies is to compare the rate and extent

of drug absorption after administration of test and reference products for the same active drug. This can be achieved by comparing the tmax, Cpmax, and AUC observed in normal volunteers after administration of the two products. These parameters can be determined without assuming any model, which makes the noncompartmental data analysis approach appropriate for these studies.

DOI: 10.4324/9781003161523-18

Noncompartmental Approach in Pharmacokinetic Data Analysis 333 18.2  The Principles of Noncompartmental Data Analysis Method The noncompartmental approach in pharmacokinetic data analysis is usually applied for drugs that follow linear pharmacokinetics. The principles of the noncompartmental approach are based on the statistical moment theory that has been utilized in chemical engineering. This theory views the drug molecules in the body as randomly distributed and each molecule has certain probability to be eliminated at a certain time (t). So, according to this theory, the time course for the drug concentration in plasma can be regarded as a probability density function. This probability density function multiplied by time raised to a certain power (0, 1, or 2) and integrated over time yields the area under the moment curve. For example, the area under the zero-moment curve can be determined from Eq. 18.1, and it is equal to the AUC. Also, the area under the first-moment curve, AUMC can be determined as in Eq. 18.2. Only the area under the zero and first moments are utilized in pharmacokinetics since higher moments are subjected to large computational error. t =∞

Area under the zero-moment curve =

∫

t =∞

t0 Cp dt =

t =0

∫t

1

t =0

(18.1)

t =0

t =∞

Area under the first-moment curve =

∫ Cp dt = AUC

t =∞

Cp dt =

∫ t Cp dt = AUMC

(18.2)

t =0

The principles of moment analysis have been utilized to estimate the pharmacokinetic parameters such as the MRT, drug total body clearance (CLT), and the volume of distribution at steady state (Vdss) (1, 2). 18.3  The Mean Residence Time after IV Bolus Administration After IV administration, the drug molecules are distributed throughout the body and some drug molecules are eliminated faster than others. The difference in the residence time of different molecules in the body occurs by chance according to the statistical moment theory. The MRT of a drug in the body is defined as the average time for the residence of all drug molecules in the body (3). So, the drug MRT can be calculated from the total residence time for all drug molecules (by adding the residence time for all molecules) and dividing this by the number of the drug molecules as in Eq. 18.3. MRT =

Total residence time for all drug molecules (18.3) Number of drug molecules

Equation 18.3 explains the meaning of the drug MRT; however, practically it cannot be used in its calculation. The MRT can be calculated from the area under the zero- and first-moment curves according to Eq. 18.4. MRT =

AUMC (18.4) AUC

where AUMC is the area under the first-moment curve, and AUC is the area under the zero-moment curve, or the area under the plasma concentration-time curve. The MRT has units of time.

334  Noncompartmental Approach in Pharmacokinetic Data Analysis 18.3.1  Calculation of the AUC

The AUC can be calculated utilizing thetrapezoidal rule as described previously in details in Chapter 7. The plasma concentration-time profile is divided to trapezoids and the area of each trapezoid is calculated according to Eq. 18.5.  C + C n+1  Area of a trapezoid =  n  . ( t n+1 − t n )(18.5)  2 The AUC0-t is calculated by adding the area of all the trapezoids. While the AUCt-∞ is calculated by extrapolation to time ∞ as in Eq. 18.6. AUC t-∞ =

Cplast (18.6) λ

where Cplast is the last measured plasma drug concentration and λ is the rate constant for the terminal decline phase in the plasma drug concentration. The total AUC0-∞ is calculated by adding the two areas as in Eq. 18.7. AUC0-∞ = AUC0-t + AUC t-∞ (18.7) 18.3.2  Calculation of the AUMC

The AUMC can also be calculated utilizing the trapezoidal rule after dividing the curve to trapezoids. The area of each trapezoid is calculated according to Eq. 18.8. t C +t C  Area of a trapezoid =  n n n+1 n+1  . ( t n+1 − t n )(18.8)   2 The AUMC0-t is calculated by adding the area of all the trapezoids. The AUMCt-∞ is calculated as in Eq. 18.9. AUMC t-∞ =

t last Cplast Cplast + (18.9) λ λ2

where Cplast is the last measured plasma drug concentration and λ is the rate constant for the terminal elimination phase in the plasma drug concentration. The total AUMC0-∞ is calculated by adding both areas as in Eq. 18.10. AUMC0-∞ = AUMC0-t + AUMC t-∞ (18.10) Practice Problems: Question: After single IV bolus administration of 50 mg of a drug, the following plasma concentrations were obtained:

Noncompartmental Approach in Pharmacokinetic Data Analysis 335 Time (hr)

Concentrations (μg/L)

0 1 3 6 9 12 18 24

250 225 185 135 105 77 40 22

The terminal rate constant for the decline in the drug concentration = 0.1 hr−1. a Calculate the MRT Answer: The MRT is calculated from the AUC and AUMC.

• Calculation of the AUC by the trapezoidal rule:  250 + 225  Area of trapezoid 1 =   ⋅ (1 − 0) = 237.5 µg hr/L  2  225 + 185  Area of trapezoid 2 =   ⋅ (3 − 1) = 410 µg hr/L  2  185 + 135  Area of trapezoid 3 =   ⋅ (6 − 3) = 480 µg hr/L  2  135 + 105  Area of trapezoid 4 =   ⋅ (9 − 6) = 360 µg hr/L  2  105 + 77  Area of trapezoid 5 =   ⋅ (12 − 9) = 273 µg hr/L  2  77 + 40  Area of trapezoid 6 =   ⋅ (18 − 12) = 351 µg hr/L  2   40 + 22  Area of trapezoid 7 =   ⋅ (24 − 18) = 186 µg hr/L  2  AUC t-∞ =

22 = 220 µg hr/L 0.1

AUC0-∞ = 237.5 + 410 + 480 + 360 + 273 + 351 + 186 + 220 = 2517.5 µg hr/L

336  Noncompartmental Approach in Pharmacokinetic Data Analysis

• Calculation of the AUMC

To calculate the AUMC, it is easier to add a column to the available data that include the product of the time, and the plasma concentration as follows: Time (hr)

Concentrations (μg/L)

Time × Concentration (μg hr/L)

0 1 3 6 9 12 18 24

250 225 185 135 105 77 40 22

0 225 555 810 945 924 720 528

 0 + 225  Area of trapezoid 1 =   ⋅ (1 − 0) = 112.5 µg hr 2 /L  2   225 + 555  2 Area of trapezoid 2 =   ⋅ (3 − 1) = 780 µg hr /L  2  555 + 810  2 Area of trapezoid 3 =   ⋅ (6 − 3) = 2047.5 µg hr /L  2  810 + 945  2 Area of trapezoid 4 =   ⋅ (9 − 6) = 2632.5 µg hr /L  2  945 + 924  2 Area of trapezoid 5 =   ⋅ (12 − 9) = 2803.5 µg hr /L  2  924 + 720  2 Area of trapezoid 6 =   ⋅ (18 − 12) = 4932 µg hr /L  2  720 + 528  2 Area of trapezoid 7 =   ⋅ (24 − 18) = 3744 µg hr /L  2 AUMC t-∞ =

t last Cplast Cplast 528 22 + = + = 7480 µg hr 2 /L 2 λ λ 0.1 (0.1)2

AUMC0-∞ = 112.5 + 780 + 2047.5 + 2632.5 + 2803.5 + 4932 + 3744 + 7480 = 24532 µg hr 2 /L MRT =

AUMC 24532 µg hr 2 /L = = 9.74 hr AUC 2517.5 µg hr/L

Note that when calculating the MRT, there was no assumption of any compartmental model. The only needed information is the rate constant for the terminal elimination phase.

Noncompartmental Approach in Pharmacokinetic Data Analysis 337 18.4  The MRT after Different Routes of Administration 18.4.1  The MRT after Extravascular Administration

The MRT for a given drug in an individual should be constant after single IV bolus administration when the drug follows linear pharmacokinetics. However, after oral or any extravascular route of administration, the drug molecules spend additional time at the site of absorption. In this case, the observed MRToral is equal to the sum of the MRT in the body after reaching the systemic circulation (similar to the MRT after single IV dose) and the mean absorption time (MAT) that is the average time the drug molecules spend at the site of absorption (2, 4). MRToral = MRTIV + MAT(18.11) This means that the noncompartmental data analysis can be used to compare the rate of drug absorption after administration of different products for the same active drug. This is because if the products contain the same active drug, the MRTIV in the body should be similar in each patient. The difference in the observed MRToral for the different products results from the difference in the MAT. Longer MRToral indicates slower absorption. The MRT is calculated after oral administration from the plasma concentrationtime data after calculating the AUC and AUMC as described before. This approach allows comparison of the rate of drug absorption after administration of different drug products without the need to assume specific compartmental model or knowledge of the order of drug absorption. If the objective of the study is to compare the rate of absorption of two different oral products for the same active drug, the MRT is calculated after administration of the two products. The difference in the drug MRT in this case represents the difference in the MAT that reflects the rate of drug absorption after administration of the two products. However, if the objective of the study is to calculate the MAT, in this case, the drug must be administered by IV and oral administration in two difference occasions and the MAT is calculated from the difference in the MRToral and MRTIV as in Eq. 18.11. Practice Problems: Question: In an experiment to investigate the drug absorption from two different oral products of the same active drug, a normal volunteer received single IV bolus dose of 100 mg, single oral dose of 100 mg of formulation A, and single oral dose of 100 mg of formulation B in three different occasions. The following data were obtained: Formulation

AUC (μg hr/L)

AUMC (μg hr2/L)

IV bolus (100 mg) Formulation A (100 mg) Formulation B (100 mg)

352 264 299

1232 1056 1495

a How would you interpret these data?

338  Noncompartmental Approach in Pharmacokinetic Data Analysis Answer: The available information can be used to compare the rate and the extent of drug absorption from the two oral formulations: a The rate of absorption: MRTIV =

AUMC IV 1232 µg hr 2 /L = = 3.5 hr AUCIV 352 µg hr/L

MRTFormulation A =

AUMC Formulation A 1056 µg hr 2 /L = = 4.0 hr AUCFormulation A 264 µg hr/L

MRTFormulation B =

AUMC Formulation B 1495 µg hr 2 /L = = 5.0 hr AUCFormulation B 299 µg hr/L

The MAT can be calculated for the two oral formulations to compare their rate of absorption: MATFormulation A = MRTFormulation A − MRTIV = 4.0 hr − 3.5 hr = 0.5 hr MATFormulation B = MRTFormulation B − MRTIV = 5.0 hr − 3.5 hr = 1.5 hr This means that formulation A is absorbed faster than Formulation B b The extent of absorption (bioavailability): FFormulation A =

AUC Formulation A 264 µg hr/L = = 0.75 AUCIV 352 µg hr/L

FFormulation B =

AUC Formulation B 299 µg hr/L = = 0.85 AUC IV 352 µg hr/L

The absolute bioavailability of the drug from formulation A is 75% and the absolute bioavailability of the drug from formulation B is 85%. 18.4.2  The MRT after Constant Rate IV Infusion

When the drug is administered by constant rate IV infusion for a certain duration of time equal to (T), the MRT is usually longer than the MRT after IV bolus administration because the drug stays in the IV solution during the infusion for a period of time. So, the MRT after constant rate IV infusion can be determined as in Eq. 18.12 MRTIV  infusion = MRTIV +

T (18.12) 2

Where T is the duration of the IV infusion.

Noncompartmental Approach in Pharmacokinetic Data Analysis 339 18.5 Other Pharmacokinetic Parameters that Can Be Determined Using the Noncompartmental Approach The noncompartmental approach can also be used to calculate other pharmacokinetic parameters such as the bioavailability, CLT, and Vdss. Estimation of the bioavailability and the clearance has been discussed previously and the methods used were similar for drugs that follow different compartmental models. The absolute bioavailability is determined from the ratio of the AUC after oral and IV administration as in Eq. 18.13, and the relative bioavailability of two oral products is determined from the ratio of the AUC for the two different oral products as in Eq. 18.14. FAbsolute =

AUCoral (18.13) AUC IV

FRelative =

AUC Product A (18.14) AUC Product B

While the drug clearance can be determined after IV administration from the dose and the AUC as in Eq. 18.15, and after oral administration as in Eq. 18.16. CL T =

DoseIV (18.15) AUC IV

CL T Doseoral = (18.16) F AUCoral The volume of distribution at steady state, Vdss, represents the proportionality constant that relates the amount of the drug in the body at steady state during continuous IV infusion and the resulting steady-state plasma drug concentration. Also, Vdss relates the average amount of the drug in the body at steady state during multiple drug administration and the resulting average steady-state plasma drug concentration. At steady state, the drug is in a state of equilibrium in all parts of the body regardless of the number of compartments. Estimation of Vdss does not require drug administration to reach steady state because according to the statistical moment theory, Vdss is the product of the clearance and MRT (3). So, after single IV bolus administration, Vdss can be estimated as follows: Vdss = CL T MRT(18.17) Vdss =

DoseIV AUMC DoseIV AUMC × = (18.18) AUC AUC AUC2

Vdss can also be determined after constant rate IV infusion as follows: T  Vdss = CL T MRTinfusion = CL T  MRTIV +  (18.19)  2 which can be rearranged to: Vdss =

Infused dose (AUMC) Infused dose (T) + (18.20) (AUC)2 2 (AUC)

340  Noncompartmental Approach in Pharmacokinetic Data Analysis where T is the duration of the IV infusion, and the infused dose is the total amount of drug administered during the IV infusion and can be calculated from the product of the infusion rate and the infusion duration. Vdss after oral drug administration cannot be determined this way unless we make some assumptions regarding the bioavailability and the MAT. Clinical Importance:

• The effect of ciprofloxacin and clarithromycin coadministration on sildenafil phar-

macokinetics was studies in normal volunteers by obtaining serial blood samples after administration of sildenafil alone and with each of the two drugs. The results were analyzed using the noncompartmental approach. Sildenafil Cpmax and AUC increased significantly without significant delay in its elimination due to ciprofloxacin and clarithromycin coadministration. This documented the clinically significant drug interaction between sildenafil and the two drugs that resulted from increasing sildenafil bioavailability due to inhibition of its presystemic metabolism (5). • The results of a study to improve the oral bioavailability of the anticancer drug paclitaxel by improving its aqueous solubility and inhibiting its presystemic metabolism in experimental animals were analyzed by the noncompartmental approach. Comparing paclitaxel Cpmax and AUC showed that using self-emulsifying drug delivery system to improve paclitaxel aqueous solubility can improve its bioavailability by more than 4.5 folds. However, coadministration of cyclosporin A as inhibitor of paclitaxel presystemic metabolism increases its bioavailability from this delivery system by 7.8 folds (6). 18.6  Determination of the MRT for Compartmental Models The MRT can be determined by the noncompartmental approach from AUC and AUMC. It can also be calculated when the pharmacokinetic model is known using the model equations. The following are the equations for calculating the MRT for selected compartmental models:

• For single IV bolus dose when the drug follows one-compartment pharmacokinetic model, Eq. 18.21 can be used to calculate the MRTIV. MRTIV =

1 (18.21) k

where k is the first-order elimination rare constant.

• For single oral dose when the drug follows one-compartment pharmacokinetic model, Eqs. 18.22 and 18.23 can be used to calculate the MRToral and MAT, respectively. MRToral =

1 1 + (18.22) k ka

and  MAT =

1 (18.23) ka

where k is the first-order elimination rate constant and ka is the first-order absorption rate constant. • For drugs administration by constant rate infusion when the drug follows onecompartment pharmacokinetic model, Eq. 18.24 can be used to calculate the MRTinfusion.

Noncompartmental Approach in Pharmacokinetic Data Analysis 341 MRTinfusion =

1 T + (18.24) k 2

where k is the first-order elimination rate constant and T is the duration of the IV infusion. • For single IV bolus dose when the drug follows two-compartment pharmacokinetic model, Eq. 18.25 can be used to calculate the MRTIV. MRTIV  2 comp =

1 1 1 + − (18.25) α β k21

where α and β are the first-order hybrid rate constant for the distribution and elimination, respectively, and k21 is the first-order transfer rate constant from the peripheral compartment to the central compartment. • For a single oral dose when the drug follows two-compartment pharmacokinetic model, Eq. 18.26 can be used to calculate the MRToral. MRToral 2 comp =

1 1 1 1 + + − (18.26) ka α β k21

18.7 Summary

• The noncompartmental approach for pharmacokinetic data analysis allows quantita-

tive description of the drug pharmacokinetic behavior without the need to assume any compartmental model. • This approach is useful when the objective of the pharmacokinetic analysis is to compare drug exposure that can be achieved by comparing the drug AUC and other parameters such as Cpmax, tmax, MRT, F, CLT, and Vdss. • This approach can be used for example to compare the rate and extent of drug absorption after administration of different products for the same drug. • The noncompartmental analysis can be used to compare the drug pharmacokinetic behavior in different disease states, or to analyze the results of pharmacokinetic drug interaction studies. Practice Problems 18.1 After administration of a single IV bolus dose of 1 mg, the following plasma concentrations were obtained: Time (hr)

Concentration (μg/L)

0 1 3 6 9 12 18 24

50 39 24 12 6.8 4.1 1.8 0.94

342  Noncompartmental Approach in Pharmacokinetic Data Analysis a Calculate the MRT, clearance and Vdss for this drug if the terminal rate for decline of the plasma dug concentration is 0.1 hr−1. 18.2 In a study to compare the absorption characteristics of three different oral formulations for the same active drug, a group of volunteers received a single dose of the 25 mg as IV solution and the three oral formulations in four different occasions. The following data were obtained: Formulation:

AUC (μg hr/L)

AUMC (μg hr2/L)

IV solution Formulation A Formulation B Formulation C

6250 3750 2500 4375

40625 31875 17500 39375

a Arrange formulations A, B, and C according to the extent of drug absorption. b Arrange formulations A, B, and C according to the rate of drug absorption. 18.3 A volunteers received a single dose of 20 mg of IV solution, capsule, and tablet for the same active drug on three different occasions. The following plasma concentrations were obtained:

a b c d

Time (hr)

IV solution Concentration (μg/L)

Oral capsule Concentration (μg/L)

Oral tablet Concentration (μg/L)

0 0.5 1 2 3 4 6 8 10 12 18 24

400 371 344 296 255 220 162 120 89 66 27 11

0 32 58 92 111 119 116 101 83 66 30 13

0 56 97 149 172 178 162 133 105 80 34 14

Calculate the MRT for this drug after IV administration. Calculate the MRT after the two oral formulations. Calculate the MAT for the two oral formulations. Calculate the clearance and the Vdss for this drug.

References 1. Yamaoka K, Nakagawa T and Uno T “Statistical moment in pharmacokinetics” (1978) J Pharmacokinet Biopharm; 6:547–558. 2. Cutler DJ “Theory of the mean absorption time, and adjunct to conventional bioavailability studies” (1978) J Pharm Pharmacol; 30:476–478.

Noncompartmental Approach in Pharmacokinetic Data Analysis 343 3. Benet LZ and Galeazzi RL “Noncompartmental determination of the steady-state volume of distribution” (1979) J Pharm Sci; 6:1071–1074. 4. Riegelman S and Collier P “The application of the statistical moment theory to the evaluation of in vivo dissolution time and absorption time” (1980) J Pharmacokinet Biopharm; 8:509–534. 5. Hedaya MA, El-Afifi DR and El-Maghraby GM “The effect of ciprofloxacin and clarithromycn on sildenafil absorption and pharmacokinetics” (2006) Biopharm Drug Dispos; 27:103–103. 6. Al-Kandari BM, Al-Soraj MH and Hedaya MA “Dual formulation and interaction strategies to enhance the Oral bioavailability of paclitaxel” (2020) J Pharm Sci; 109:3386–3393.

19 Pharmacokinetic-Pharmacodynamic Modeling

Objectives After completing this chapter, you should be able to:

• Describe the different components of the pharmacokinetic/pharmacodynamic (PK/PD) models.

• Discuss the approaches used to measure drug effect. • Describe the different pharmacodynamic models that can relate the drug concentration at the site of action and drug effect.

• Discuss the differences between the direct and indirect response models, the direct and indirect link models, and time variant and invariant response models.

• Describe the PK/PD modeling approaches for the different types of drug response. • List the general steps involved in developing PK/PD models. 19.1 Introduction Information about the drug pharmacokinetic behavior in the body becomes more valuable when the drug profile in the body is correlated with the observed drug effects. Also, the results of pharmacodynamic studies that involve monitoring drug effects are more useful when these effects are linked to the drug dose or concentration in plasma or at the site of action. The primary objective of PK/PD modeling is to construct a model that describes the relationship among the administered dose, the resulting drug concentration-time profile in the body, and the time course of the drug effects. PK/PD modeling can be useful in determining the dose required to produce the desired effect, predicting the effect produced by a certain dose, and identifying the temporal pattern of drug effect. 19.2  Pharmacokinetic-Pharmacodynamic Modeling The pharmacokinetic models describe the relationship between the dose of the drug and the drug concentration-time profile in all parts of the body, including the site of drug action, while the pharmacodynamic models describe the relationship between the drug concentration at the site of action and the time course of the drug effect, as illustrated in Figure 19.1. The PK/PD model combines the pharmacokinetic model and the pharmacodynamic model. The way these two models are linked together is different for different drugs depending on the site of drug effect, the mechanism of action of the drug, and if the DOI: 10.4324/9781003161523-19

Pharmacokinetic-Pharmacodynamic Modeling 345

Figure 19.1  Schematic presentation of the relationship between pharmacokinetics and pharmacodynamics.

concentration-effect relationship changes with time (1–3). These models have numerous applications in drug development and clinical drug use. 19.2.1  The Pharmacokinetic Model

The drug concentration-time profile in the body is a good measure for the exposure of the site of action to the drug. The exposure of the site of action to the drug can be described utilizing compartmental pharmacokinetic models or noncompartmental pharmacokinetic parameters. The choice usually depends on the study design and the nature of the drug exposure-effect relationship. The drug effect may change in response to the change in drug concentration, such as after administration of general anesthetics, bronchodilators, and antihypertensive drugs. The maximum information can be obtained from correlating multiple drug concentrations with their corresponding effect values. In this case, compartmental pharmacokinetic models can be useful to fully characterize the drug concentration-time profile, which can be used to characterize the concentration-effect relationship over time. However, the effect of other drugs can be determined once on a given day or every period of time, such as the effect of oral anticoagulants, cholesterol-lowering drugs, and anti-HIV drugs. For these drugs, the effect can be correlated with some noncompartmental pharmacokinetic parameters, such as the drug AUC, which represent the average drug concentration over the dosing intervals, or other measures of drug exposure such as Cpmax, Cpmin, or any drug concentration obtained during the dosing interval. The total drug concentration in plasma is used to characterize the drug concentrationeffect relationship for most drugs. However, the free drug concentration should be used when the drug binding in plasma or tissues is concentration dependent or in case of diseases that can significantly affect the drug protein binding. Also, when the drug is metabolized to a pharmacologically active metabolite, both the parent drug and the active metabolite concentrations should be included while characterizing the concentration-effect relationship. The difference in potencies for the parent drug and the metabolite should be considered. Furthermore, when a chiral drug is used as a racemate mixture, the difference in the pharmacokinetic and pharmacodynamic properties of individual enantiomers should be considered when characterizing the concentration-effect relationship for this drug. 19.2.2  Measuring the Response

The therapeutic and adverse effects of the drug can be quantified using a variety of efficacy measures. These efficacy measures must be predictive for the drug effect to increase the predictive power of the PK/PD model. They can be classified to biomarkers, surrogate

346  Pharmacokinetic-Pharmacodynamic Modeling markers, and clinical outcomes (4). Biomarkers are quantifiable physiological, pathological, biochemical, or anatomical measurements that are affected by the drug and may or may not be relevant for monitoring the clinical outcome of the drug treatment. Examples of biomarkers that reflect the drug action but are not indicative of the clinical outcome include ACE inhibition in response to ACE inhibitor therapy, and inhibition of ADPdependent platelet aggregation as the effect of antiplatelet therapy. Surrogate endpoints are biomarkers that are used in clinical trials as a substitute for clinical endpoints that are expected to predict the effect of therapy. Examples of surrogate markers are blood pressure measurement to assess the effect of antihypertensive drugs, cholesterol level to determine the effect of cholesterol lowing drugs, viral load to assess the efficacy of antiHIV drugs, and tumor marker as a measure for the response to anticancer drug therapy. While clinical endpoints are characteristics or variables that reflect how the patient feels or functions, such as cure or decreased morbidity. It is usually difficult to evaluate the clinical endpoints, so they are often predicted from surrogate markers. 19.2.3  The Pharmacodynamic Model

The pharmacodynamic models describe the relationship between the concentration at the site of action and the observed effect for drugs that produce direct effect. These drugs produce their effect by direct interaction with the drug target at the site of action without any time delay, and the effect disappears once the drug is eliminated from the site of action. To select the appropriate pharmacodynamic model that describes the concentration-effect relationship for a specific drug, the drug effect resulting from different drug concentrations is determined experimentally. Then the different equations for the different pharmacodynamic models are fitted to the concentration-effect data to determine the best model that can describe the concentration-effect relationship. The following are the different pharmacodynamic models that can describe the relationship between the drug concentration at the effect site and the observed drug effect (1, 5–7). 19.2.3.1  The Fixed Effect Model

This is the simplest pharmacodynamic model that relates a certain drug concentration to a fixed effect that is either present or not such as cure or no cure, sleep or awake. In this model, there is only one parameter that is the drug concentration at which the effect appears. This model has limited applications in PK/PD modeling because of the limited ability of the model to predict the effect at different time points. 19.2.3.2  The Linear Model

This model assumes a linear relationship between the drug concentration at the site of action and the observed effect over a certain range of drug concentrations. E = SC(19.1) where E is the intensity of effect, C is the drug concentration at the site of action, and S is slope parameter that determines the rate of change of the intensity of effect with the change in drug concentration. The slope of this linear relationship reflects the potency of the drug.

Pharmacokinetic-Pharmacodynamic Modeling 347

Figure 19.2 The drug concentration-effect relationship for a drug that follows the linear pharmacodynamic model.

The linear model expressed by Eq. 19.1 and presented in Figure 19.2 assumes that there is no effect in absence of the drug, when the drug concentration is equal to zero. If the effect has some measurable value in absence of the drug, such as when the measured effect is the blood pressure or heart rate, Eq. 19.1 can be modified to Eq. 19.2. E = E 0 + SC (19.2) where E0 is the effect in absence of the drug or the baseline effect. In this case, the concentration-effect relationship can be presented as in Figure 19.3. If a linear relationship exists between the drug concentration and the observed effect, linear regression can be performed between the measured drug effect at different concentrations to estimate the model parameter(s), the slope (S) and baseline effect (E0). The linear relationship between the drug concentration and effect usually exists only over a certain range of drug concentrations, and outside this range, this linear relationship is not certain. In this case, it has to be specified that the model should be used only over this range of concentrations and extrapolation outside this range is not appropriate.

Figure 19.3 The drug concentration-effect relationship for a drug that follows the linear pharmacodynamic model in presence of a baseline effect.

348  Pharmacokinetic-Pharmacodynamic Modeling 19.2.3.3  The Log-Linear Model

This model assumes a linear relationship between the drug effect and the logarithm of the drug concentration at the site of action over a certain range of drug concentrations. E = S Log C + I(19.3) where E is the drug effect, S is the slope of the line, C is the drug concentration at the effect site, and I is a constant. The slope of the line reflects the potency of the drug. As mentioned under the linear pharmacodynamic model, it is important to specify the range of drug concentration over which the log-linear relationship between the drug concentration and effect exists, as presented in Figure 19.4. This log-linear relationship may not exist outside this range of concentration and extrapolation outside this range is not appropriate. The measured drug effect should be in the range of 20–80% of the maximum response to obtain the log-linear relationship. However, the log-linear model does not allow prediction of the maximum drug effect, which should be observed if the drug concentration is allowed to increase indefinitely. This makes it difficult to ensure that all the effect measurements are in this range. The log-linear model has been applied extensively to the description of in vitro pharmacodynamic studies. When the concentration-effect relationship can be described by the log-linear model, linear regression is performed between log drug concentrations and the measured drug effect at each concentration to determine the slope of this log-linear relationship. The estimated slopes for different drugs can be used to compare the relative potencies of these drugs and to determine the combined effect of drugs. 19.2.3.4  The Emax Model

This model assumes that the effect of the drug increases as the drug concentration increases until it reaches a plateau or a maximum effect at very high drug concentration, as presented in Figure 19.5. The concentration-effect relationship can be described mathematically by a hyperbolic function in Eq. 19.4.

Figure 19.4 The drug concentration-effect relationship for a drug that follows the log-linear pharmacodynamic model.

Pharmacokinetic-Pharmacodynamic Modeling 349

Figure 19.5 The drug concentration-effect relationship for a drug that follows the Emax pharmacodynamic model.

E=

E max C (19.4) EC50 + C

where Emax is the maximum effect resulting from the drug that represents the intrinsic activity of the drug and EC50 is the drug concentration when the effect is 50% of the maximum effect that is a measure of drug potency. This model can be used to describe the concentration-effect relationship when a baseline effect is observed in absence of the drug. In this case, Eq. 19.4 can be modified to account for the baseline effect, E0 as in Eq. 19.5. E = E0 +

E max C (19.5) EC50 + C

The model can also be used to describe the concentration-effect relationship when the drug causes reduction in the value of the measured effect, such as lowering the blood pressure by antihypertensive drugs as in Eq. 19.6. E = E0 −

E max C (19.6) EC50 + C

In this case, the maximum effect of Emax (maximum reduction in blood pressure) can be calculated from the difference between the baseline blood pressure (blood pressure before using the drug) and the lowest blood pressure observed while using increasing doses of the drug. When the drug concentration-effect relationship follows the Emax model, the model equation is fitted to the different drug concentrations and their corresponding drug effect measurements with nonlinear regression to estimate the model parameters, Emax and EC50. 19.2.3.5  The Sigmoid Emax Model

When the concentration-effect relationship does not follow the typical hyperbolic function, the sigmoid Emax model can be used to describe the concentration-effect relationship.

350  Pharmacokinetic-Pharmacodynamic Modeling

Figure 19.6 The drug concentration-effect relationship for a drug that follows the sigmoid Emax pharmacodynamic model.

If the drug concentration-effect curve is S-shaped or if the response approaches Emax very slowly, then deviation from simple hyperbola can be described by Eq. 19.7. E=

E max C n (19.7) EC50 n + C n

where n is the parameter affecting the shape of the curve. If n is greater than one, then the curve will be S-shaped. While if n is less than one, then the initial portion of the curve has slope greater than the simple hyperbola, and beyond EC50, the slope is less than the simple hyperbola, as shown in Figure 19.6. The sigmoid Emax model is applied when the interaction between a drug molecule with the receptor affects the interaction of other drug molecules with the receptors (5, 7). For example, the interaction of one drug molecule with the receptor can facilitate the interaction of the receptor with other drug molecules. In this case, n is > 1 in the model equation. On the other hand, the drug-receptor interaction may hinder the receptor interaction with other drug molecules. In this case, n is < 1 in the model equation. If the interaction of one drug molecule with the receptor does not affect the interaction of the other drug molecules with the receptor, n is equal to 1 and Eq. 19.7 is reduced to the equation for the Emax model. The sigmoid Emax model should not be used if the maximum effect is not clearly defined. When the drug concentration-effect relationship follows the sigmoid Emax model, the model equation is fitted to the drug effect measurements and their corresponding drug concentrations with nonlinear regression to estimate the model parameters, Emax, EC50, and n. 19.3  Integrating the Pharmacokinetic and Pharmacodynamic Models The following three PK/PD model characteristics must be considered while selecting the appropriate strategy to combine the pharmacokinetic and pharmacodynamic information (8). 19.3.1  Direct Response versus Indirect Response

Drugs with direct response are the drugs that produce their effect due to direct effect at the site of action without any time delay. Any change in the drug concentration at the site of action is immediately reflected in the observed drug effect, with no drug effect observed once the drug disappears from the site of action. Examples of drugs that produce direct

Pharmacokinetic-Pharmacodynamic Modeling 351 response are general anesthetics which usually produce their anesthetic effect when they are present in the body in sufficient quantities. Once the anesthetic drugs are eliminated from the body, their effect disappears, and the patients start to gain conscious. Many antihypertensive drugs, bronchodilators, and antidiabetic drugs also produce direct response. While the drugs with indirect response produce their effect by inhibiting or stimulating a receptor, mediator, enzyme, precursor, or cofactor, this effect is followed by a cascade of events that leads to the production of the drug effect which appears later. In this case, the drug effect may be observed after the drug disappears from the body. Although the drug response is not produced directly by the drug, higher drug concentrations usually lead to more intense drug effect. Examples of drugs that produce their effect by indirect response include most anticancer drugs and oral anticoagulants. Oral anticoagulants inhibit vitamin K reductase leading to interference with synthesis of vitamin K-dependent coagulating factors. The effect of oral anticoagulants is usually observed after few days when the existing clotting factors disappear from the body. 19.3.2  Direct Link versus Indirect Link

The plasma concentration of the drugs that act by direct response can be directly or indirectly linked to the drug concentration at the site of action. When rapid equilibrium is established between the drug in plasma and at the site of action, any change in the plasma drug concentration immediately produces proportional change in the drug concentration at the site of action, which leads to change in drug effect. In this case, the plasma drug concentration is said to be directly linked to the drug concentration at the site of effect, and both plasma concentration and site of action concentration can be used to characterize the drug concentration-effect relationship. The existence of delayed equilibrium between the drug in plasma and at the site of action results in delay in observing the change in drug effect due to changes in plasma drug concentration. This delay in the change of drug effect is due to distributional delay and not due to indirect response resulting from the mechanism of action of the drug. In this case, the plasma drug concentration is considered indirectly linked to the site of effect concentration. 19.3.3  Time-Variant versus Time-Invariant

Most PK/PD models assume time invariant, meaning that the drug concentration-effect relationship does not change with time. However, there are instances when the drug concentration-effect relationship changes with time. In this case, the same drug concentration may produce lower or higher effects during repeated drug administration. Tolerance is the decrease in the drug effect over time, while sensitization is the increase in drug effect over time, when the drug concentration at the site of action is the same. The existence of tolerance or sensitization should be considered while developing the PK/PD models, and the models should be designed to account for the time-dependent change in the pharmacodynamic parameters. 19.4  Direct Link PK/PD Models for Drugs with Direct Response When the drug effect is caused by the direct drug interaction with the site of action and rapid equilibrium exists between the drug in plasma and at the site of action, the drug effect-time profile will have the same shape as the plasma concentration-time profile. This means that the maximum drug effect is observed at the same time of the maximum

352  Pharmacokinetic-Pharmacodynamic Modeling plasma drug concentration. Also, the intensity of drug effect increases when the plasma concentration increases and decreases when the plasma concentration decreases. However, the magnitude of the change in drug effect due to the change in the plasma drug concentration depends on the drug concentration-effect relationship, i.e., the pharmacodynamic model. In the direct link models, the plasma drug concentration is proportional to the drug concentration at the site of action, so it can be used as the input function in the pharmacodynamic models to characterize the drug concentration-effect relationship. The direct link models are also known as the steady-state models, because at steady state equilibrium is established between the drug in plasma and the drug at the site of action. For this reason, the therapeutic range for drugs during multiple administration is stated as the range of drug concentrations in plasma because of the equilibrium between the drug in plasma and at the site of action. When the plasma drug concentration is within the specified therapeutic range, the concentration at the site of action is not known. However, it is proportional to the plasma drug concentration, and it should produce the optimal therapeutic activity with minimal adverse effects. The PK/PD model in this case consists of the pharmacokinetic model and the pharmacodynamic model. The modeling involves using the plasma concentration-time data to estimate the drug pharmacokinetic parameters and to characterize the plasma concentration-time profile. Then the drug effect-time data and the plasma concentration-time profile are used to estimate the pharmacodynamic model parameters. For Example:

• The plasma drug concentration at any time after single IV bolus dose, when the drug follows one-compartment pharmacokinetic model, can be describe by Eq. 19.8: Cp =

D − kt e (19.8) Vd

• If we assume that the drug concentration-effect relationship follows the sigmoid Emax

pharmacodynamic model, the relationship between the drug concentration at the site of action and also in plasma (because of the rapid equilibrium) can be described by Eq. 19.9: E=

E max Cen E max Cpn = (19.9) n n EC50 + Cen EC50 + Cpn

The PK/PD model can be described mathematically by the two equations above. Estimation of the model parameters usually starts by fitting the pharmacokinetic model equation to the drug concentration-time data to estimate the pharmacokinetic parameters, Vd and k. Then the estimated pharmacokinetic parameters are used with the drug effect-time data to estimate the pharmacodynamic model parameters, Emax, EC50, and n. It is also possible to fit both the plasma concentration-time data and the effecttime data to the two equations simultaneously to estimate all the model parameters. 19.5  Indirect Link PK/PD Models for Drugs with Direct Response The drug effect may lag behind the plasma drug concentration as a result of delay in the drug distribution to its site of action. The existence of distributional delay can be manifested by the increase in drug effect when the plasma drug concentration is decreasing

Pharmacokinetic-Pharmacodynamic Modeling 353

Figure 19.7 A plot of the plasma drug concentration versus drug effect with the arrows representing the chronological order of the data points. In plot (A), rapid equilibrium exists between the drug in plasma and the drug at the site of action as in the direct link models, while in plot (B) the equilibrium between the drug in plasma and the drug at the site of action is delayed as in the indirect link models.

and observing the maximum effect when the plasma concentration is not at its maximum value. Also, observing a counterclockwise hysteresis when plotting the plasma drug concentration versus drug effect and connecting the points in chronological order is another indication of the delayed drug distribution to the site of action. This hysteresis is not observed when rapid equilibrium is achieved between the drug in plasma and the drug at the site of action as in Figure 19.7. The distributional delay can be demonstrated for example by drugs that act on the CNS when these drugs are slowly distributed to the brain. After IV administration, the plasma concentration of these drugs will always be decreasing, while the drug concentration in the brain increases initially until a maximum concentration is achieved and then declines parallel to the plasma concentration. The drug effect-time profile in this case shows an initial increase in effect until the maximum effect is achieved, followed by decline in effect. In this example, the brain drug concentration, which can be estimated from the pharmacokinetic model, should be used as the input concentration for the pharmacodynamic model. A more general approach that involves the introduction of an effect compartment to the PK/PD model has been used to describe the drug concentration-effect relationship in presence of the equilibrium delay. 19.5.1  The Effect Compartment Approach

This modeling approach involves introduction of a hypothetical effect compartment, which links the pharmacokinetic model and the pharmacodynamic model (9, 10). This approach assumes that the drug transfer to the effect compartment does not affect the pharmacokinetic behavior of the drug. It also assumes that the drug transfer into and out of the effect

354  Pharmacokinetic-Pharmacodynamic Modeling compartment follows first-order processes, and that the intensity of drug effect is related to the drug concentration in the effect compartment. In this case, the PK/PD model consists of the pharmacokinetic model, the pharmacodynamic model, and the effect compartment that links the pharmacokinetic and the pharmacodynamic models. The basic principles of this approach involve using the plasma concentration-time data to estimate the drug pharmacokinetic parameters. Then the estimated pharmacokinetic model parameters and the drug effect-time data are used to estimate the drug profile in the effect compartment. After that, the drug profile in the effect compartment is used to characterize the drug concentrationeffect relationship and to estimate the pharmacodynamic model parameters. For Example:

• The model in Figure 19.8 represents one-compartment pharmacokinetic model that

is linked to an effect compartment. The plasma drug concentration at any time after single IV bolus dose can be described by Eq. 19.8. Cp =

D − kt e Vd

• Also, the equation for the drug concentration in the effect compartment (Ce) can be derived as in Eq. 19.10. Ce =

(

)

D ke0 e− kt − e− ke0t (19.10) Vd (ke0 − k)

• If it is assumed that the drug concentration-effect relationship follows the sigmoid Emax pharmacodynamic model, the relationship between the drug concentration in the effect compartment, Ce, and the drug effect can be described by Eq. 19.11 (5–7): E=

E max Cen (19.11) EC50 n + Cen

Figure 19.8 A diagram representing one-compartment pharmacokinetic model linked to an effect compartment. The parameters, k is the first-order elimination rate constant; k1e and ke0 are the first-order transfer rate constant into and out of the effect compartment, respectively.

Pharmacokinetic-Pharmacodynamic Modeling 355

Figure 19.9 Diagrams representing two-compartment pharmacokinetic model linked to an effect compartment. (A) The effect compartment is linked to the central compartment, and (B) the effect compartment is linked to the peripheral compartment.

The PK/PD model can be described mathematically by the above three equations. Estimation of the model parameters usually starts by fitting the pharmacokinetic model equation to the drug concentration-time data to estimate the pharmacokinetic parameters, Vd and k. Then the estimated pharmacokinetic parameters are used with the drug effect-time data to estimate the pharmacodynamic model parameters, Emax, EC50, n, and ke0. It is also possible to fit the three PK/PD model equations simultaneously to the plasma concentration-time data and the effect-time data to estimate all the model parameters. Similar approach can be used if the drug follows two-compartment pharmacokinetic model where the effect compartment can be linked to the central or the peripheral compartment, as shown in Figure 19.9. The plasma drug concentrations and the drug effect data are used to estimate the pharmacokinetic parameters and the drug concentration in the effect compartment at different time points which is then substituted in the equation for the pharmacodynamic model to estimate the pharmacodynamic parameters. When the effect compartment is linked to the peripheral compartment in a multicompartment pharmacokinetic model, accurate characterization of the concentration-effect relationship and precise estimation of the pharmacodynamic parameters can only be achieved when information about the drug concentration in the peripheral compartment is available. 19.6  PK/PD Models for Drugs with Indirect Response The lag between the time course of the drug concentration and the observed drug response can result from indirect response mechanism. Indirect response results from the effect of the drug on an intermediary factor which leads to series of steps that cause the drug effect. The indirect response models have been used to describe the pharmacodynamic effect of many anticancer drugs and also oral anticoagulants as mentioned previously (11–14). Despite this temporal dissociation between drug concentration and effect, higher drug concentrations usually result in more intense drug effect. Several approaches have been used to model the PK/PD of drugs with indirect response. One of the commonly used approaches proposed four basic models to relate the time course of the drug

356  Pharmacokinetic-Pharmacodynamic Modeling concentration and the drug effect of these drugs. These models assume zero-order production of the drug response and first-order disappearance of the response. So, the rate of change in drug response in absence of the drug can be written as follows: dR = kin − kout R Baseline (19.12) dt where kin is the zero-order rate constant for the production of response R, kout is the first-order rate constant for the disappearance of response, and RBaseline is the value of the response in the absence of the drug. The approach assumes that all drugs based on their mechanism of action can produce their response by either stimulation or inhibition of the production or disappearance of effect. The degree of this stimulation or inhibition is related to the drug concentration at the effect site via the sigmoid Emax pharmacodynamic model. The indirect response models are also known as the mechanism-based models because they are based on the drug mechanism of action. The oral anticoagulants for example produce their effect by inhibition of the clotting factor synthesis. So, the indirect response model for the inhibition of the production of effect can be selected to describe the concentration-effect relationship of these drugs based on their mechanism of action. The four different basic models with the differential equation for each model are illustrated in Figure 19.10.

Figure 19.10 Schematic presentation of the four basic indirect response models and the differential equations that describe the rate of change in the drug response with respect to time for each model. The four models are (A) inhibition of the production of response model, (B) inhibition of the loss of response model, (C) stimulation of the production of response model, and (D) stimulation of the loss of response model.

Pharmacokinetic-Pharmacodynamic Modeling 357 The modeling process starts with the selection of the appropriate indirect response model based on the drug mechanism of action, and a pharmacokinetic model to describe the pharmacokinetic behavior of the drug.

• The pharmacokinetic model equation is fitted to the drug concentration-time data to estimate the pharmacokinetic model parameters.

• The drug concentration-time profile at the site associated with the drug effect, which

can be the plasma, the peripheral compartment, or a hypothetical effect compartment, is characterized. • The equation for the selected indirect response model is fitted to the drug effect-time data and the calculated drug concentrations at the site of action to estimate the pharmacodynamic model parameters, kin, kout, the parameter for the maximum (inhibition or stimulation) effect, Imax or Smax, and the parameter for the drug concentration at half maximum (inhibition or stimulation) effect, IC50 or SC50. • It is also possible to fit the PK/PD model equations simultaneously to the drug concentration-time data and the effect-time data to estimate all the parameters for the PK/PD model. 19.7  Other PK/PD Models The models described above are the general PK/PD models that have been utilized to analyze the drug concentration-effect relationship for many drugs. However, there are drugs with special response characteristics, which necessitate modification of these general models or development of new modeling approaches to suit these special drug effect features.

• When the concentration-effect relationship changes with time due to tolerance or sen-

•

•

• •

sitization, the pharmacodynamic parameters, Emax and EC50 in the Emax model, change with time. In this case, the pharmacodynamic model should account for the change in the parameters over time by including a time-dependent exponential term that can change the values of the PD parameters with time. When the drug effect on the target site is irreversible, the time course of the drug effect is dependent on the turnover rate of the target rather than the pharmacokinetics of the drug. The pharmacodynamic models used to describe the time course of the effect of these drugs should include the rate of formation and degradation of this target in addition to the drug concentration-effect relationship. The time delay from drug-receptor interaction until the generation of the pharmacological effect can be modeled using indirect response PK/PD models, or incorporation of the concept of the transit compartments to account for the delay in the drug pharmacological effect. The time course of the drug effect after administration of different doses can be used to characterize the dose-effect-time relationship when drug concentrations are not available. The effect of the drug treatment on the disease progression should be incorporated in the model especially if the drug treatment is intended to modify the disease progression.

19.8  The PK/PD Modeling Process The PK/PD modeling usually goes through several iterative steps starting usually with proposing a tentative model based on the available information about the drug and its effect. The mathematical relationships that describe the proposed model are fitted

358  Pharmacokinetic-Pharmacodynamic Modeling to the experimental data and the model is evaluated and modified. The process is repeated to refine the model until an appropriate model is developed. An appropriate model is the simplest model that can describe the experimental data and has good predictability. The following is a summary of the steps that are followed in the PK/PD modeling process (5). 19.8.1  Stating the Objectives, Proposing the Model and Designing the Study

The objectives of performing the modeling must be stated clearly because the PK/PD study must be designed to fulfill these objectives. All the available pharmacokinetic and pharmacodynamic information about the drug under investigation must be used to propose a preliminary model to describe the drug concentration-effect relationship. The PK/PD study is executed to obtain the necessary pharmacokinetic and pharmacodynamic data. 19.8.2  Initial Data Exploration and Data Transformation

This is important to determine how the experimentally obtained data agree with the proposed PK/PD model. The drug concentration-time curve is used to characterize the pharmacokinetic behavior of the drug. The drug effect-time profile, the drug concentration-time profile, and the drug mechanism of action are examined to determine if the drug effect can be classified as direct or indirect response as well as to determine if rapid equilibrium exists between the drug in plasma and at the site of action, or there is distributional delay. The effect can be expressed as the actual measured values, the change in the baseline effect, or the percentage change in effect. The drug concentration-effect plot is used to select the appropriate pharmacodynamic model. Based on this initial exploration of data, the proposed model is modified to concur with the observed results. 19.8.3  Refining and Evaluation of the PK/PD Model

The mathematical equations that describe the PK/PD model are fitted to the obtained experimental data to estimate the model parameters using specialized data analysis software. All available PK/PD data analysis software has diagnostic statistics to examine the model goodness of fit. Based on this evaluation, the PK/PD model may be refined, and the process is repeated until the model that can properly describe the data is obtained. 19.8.4  Validation of the PK/PD Model

The real model validity is determined from its ability to predict the drug pharmacokinetic and pharmacodynamic behavior in situations that were not used to develop or refine the model. The predictability of the PK/PD models is important especially when it provides evidence of safety, efficacy, support new doses, and dosage regimen in the target population or subpopulations. The predictability of the model can be evaluated by performing a separate validation study and comparing the obtained data with the predicted data using model simulations.

Pharmacokinetic-Pharmacodynamic Modeling 359 19.9 Applications of the PK/PD Modeling in Drug Development and Clinical Use of Drugs PK/PD modeling is useful for enhancing the effective utilization of resources in all stages of drug development. During the preclinical phase of drug development, studies that relate the pharmacokinetic information of the candidate drug with its pharmacodynamic properties usually provide important information that is necessary for subsequent development activities. During this early phase of drug development, biomarkers can be tested and evaluated for their potential use as surrogate marker for the drug clinical efficacy and toxicity. The development of mechanism-based PK/PD models to describe the dose-concentration-effect relationship in different animal species allow the prediction of the concentration-effect relationship in humans, which is very important in selecting the appropriate drug doses in phase I clinical studies. Also, the PK/PD models allow the comparison of the potencies and intrinsic activities of different drugs and prediction of drug potency in humans from the animal PK/PD data. Furthermore, the different physiological and pathological factors and the potential drug interactions that can alter the drug pharmacokinetic and pharmacodynamic properties can be investigated. Characterization of the drug concentration-effect relationship allows the selection and optimization of the dosage form and dosage regimen that can achieve the drug concentration-time profile required to produce the desired effect. During the different phases of clinical drug development, integration of the pharmacokinetic and pharmacodynamic information provides beneficial information to determine the range of doses that can be tolerated and to learn how to use the drug in the target patient population. The dose escalation studies provide the opportunity to examine the dose-concentration-effect relationship for therapeutic and adverse effects over a wide range of doses. Correlation of the drug concentration and drug effect profiles can be used to determine the impact of the route of administration and the rate of drug absorption on the drug effect, which is important for optimization of the dosage form characteristics. Population PK/PD modeling is frequently applied to examine the doseconcentration-effect relationship in patients and to identify the different factors that can contribute to the interindividual variability in drug response. Factors such as age, gender, disease, other drugs, and tolerance development are usually investigated in population PK/PD studies, and the dosage requirements in the different patient subpopulations are usually determined. Also, the PK/PD information can be used to examine the effect of ethnic and genetic factors on the dose-concentration-effect relationship, which can provide the necessary information for global drug development. 19.10 Summary

• PK/PD modeling is an approach to correlate the drug concentration-time profile in the body with the time course of the drug effect.

• The drug concentration-effect relationship can be different for different drugs depending on the drug site of action and mechanism of action.

• The measures used to quantify the pharmacodynamic effect of the drug should be

predictive for the drug therapeutic and adverse effects to increase the predictive power of the PK/PD model. • Properly constructed and validated PK/PD models can predict the drug effect after administration of different doses and can determine the dose required to achieve a specific effect. • PK/PD modeling has numerous applications in all stages of the drug development process.

360  Pharmacokinetic-Pharmacodynamic Modeling References 1. Holford NHG and Sheiner LB “Pharmacokinetic and pharmacodynamic modeling in vivo” (1981) CRC Crit Rev Bioeng; 5:273–322. 2. Meibohm B and Derendorf H “Basic concepts of pharmacokinetic/pharmacodynamic (PK/PD) modeling” (1997) Int J Clin Pharmacol Ther; 35:401–413. 3. Derendorf H and Meibohm B “Modeling of pharmacokinetic/pharmacodynamic (PK/PD) relationships: Concepts and perspectives” (1999) Pharm Res; 16:176–185. 4. US Food and Drug Administration. Draft guidance for industry: “Exposure-response relationships: Study design, data analysis, and regulatory applications” (2003). 5. Wagner JG “Kinetics of pharmacologic response” (1968) J Theor Biol; 20:173–201. 6. Holford NHG and Shiner LB “Understanding the dose-effect relationship: Clinical application of pharmacokinetic-pharmacodynamic models” (1981) Clin Pharmacokinet; 6:429–453. 7. Holford NHG and Shiner LB “Kinetics of pharmacologic response” (1982) Pharmacol Ther; 16:141–166. 8. Rohatagi S, Martin NE and Barrett JS “Pharmacokinetic/pharmacodynamic modeling in drug development” in “Applications of pharmacokinetic principles in drug development”, Edited by Krishna R (2004), Kluwer Academic/Plenum Publisher, New York, NY, USA. 9. Segre G “Kinetics of interaction between drugs and biological systems II” (1968) Farmaco; 23:906–918. 10. Sheiner LB, Stanski DR, Vozeh S, Miiller RD and Ham J “Simultaneous modeling of pharmacokinetics and pharmacodynamics: Application to d-tubocurarine” (1979) Clin Pharmacol Ther; 25:358–371. 11. Daneyka NL, Garg V and Jusko WJ “Comparison of four basic models of indirect pharmacologic response” (1993) J Pharmacokinet Biopharm; 21:457–478. 12. Jusko WJ and Ko HC “Physiologic indirect response models characterize diverse type of pharmacodynamic effects” (1994) Clin Pharmacol Ther; 56:406–419. 13. Levy G “Mechanism-based pharmacodynamic modeling” (1994) Clin Pharmacol Ther; 56:356–358. 14. Sharma A and Jusko WJ “Characterization of four basic models of indirect pharmacodynamic responses” (1996) J Pharmacokinet Biopharm; 24:611–635.

20 Pharmacogenetics The Genetic Basis of Pharmacokinetic and Pharmacodynamic Variability

Objectives After completing this chapter, you should be able to:

• Define the common terms used in the field of pharmacogenetics. • Describe the basic gene structure and the consequences of gene variation. • List few examples of clinically significant genetic polymorphism involving drugmetabolizing enzymes.

• List few examples of clinically significant genetic polymorphism involving drug pharmacodynamics.

• Evaluate the clinical significance of genetic polymorphism that involves drug pharmacokinetics and pharmacodynamics.

• Discuss the benefits of applying pharmacogenetic testing for optimization of drug therapy.

• Discuss the barriers and enablers for implementing pharmacogenetics in clinical practice. 20.1 Introduction The genetic bases of variation in drug response have been recognized since the 1950s when some patients experienced prolonged neuromuscular blockade after receiving the muscle relaxant suxamethonium chloride in preparation for surgical procedures. These patients were found to have atypical pseudocholinesterase enzyme with reduced cholinesterase activity, which led to the prolonged duration of effect after using this muscle relaxant. Also, the slow and rapid acetylators metabolize drugs through the acetylation pathway at two distinct rates. After administration of drugs that are metabolized by the acetylation pathway, such as the antituberculosis drug isoniazid and the antiarrhythmic drug procainamide, different drug profiles of these drugs are observed in the slow and rapid acetylators. Furthermore, glucose-6-phosphate dehydrogenase deficiency is a genetic abnormality that causes hemolytic anemia in patients after eating fava beans or taking some drugs such as the antimalarial drug primaquine, sulfonamides, or aspirin. These clinical observations resulted in the development of the field of pharmacogenetics that deals with the genetic bases of variations in drug pharmacokinetics and/or pharmacodynamics. Understanding the patient’s genetic makeup is the key for the development of personalized medications with greater efficacy and safety. The current clinical applications of pharmacogenetics are limited and are mainly directed to screening patients for variation in some of the cytochrome P450 (CYPs) genes and selecting the drug dose based on DOI: 10.4324/9781003161523-20

362  Pharmacogenetics the results of these tests. Few pharmacogenetic tests are currently used to screen patients before the start of therapy when the drug therapeutic or toxic effects are influenced by the patient’s genotype. However, the continuous progress in this field should provide additional tools that can help in clinical decisions for selecting the drug and the drug dose, and for predicting drug response based on the genetic makeup of the patient. 20.2  Gene Structure The deoxyribonucleic acid (DNA) is the molecule that contains the specific genetic information for an individual. It has a double helix structure that resembles a twisted ladder. The backbone of the ladder is formed by the 5-carbon sugar, deoxyribose linked together by phosphate groups, while the steps of the ladder are formed by two nitrogenous bases linked together by hydrogen bonds. The four nitrogenous bases involved in the formation of the DNA are the two purines, adenine (A) and guanine (G), and the two pyrimidines, cytosine (C) and thymine (T). Adenine always pairs with thymine (A-T) and guanine pairs with cytosine (G-C). The DNA molecule consists of about 3 billion base pairs. The sequence of the nitrogenous bases in the DNA structure determines the specific genetic code of an individual and gives the instructions to the body to produce specific proteins with the individual’s specific genetic characteristics. The DNA is housed in the nucleus of all cells except the red blood cells and platelets, and it is known as nuclear DNA (nDNA). Also, small amount of DNA is present in the mitochondria that are known as mitochondrial DNA (mtDNA). All cells of an individual have the same DNA, while different individuals have variations in the structure of their DNA. The Genes are short sections of DNA molecule that include the instructions for the formation of specific proteins that are essential for the characteristics, structure, regulation, and function of the body. These proteins can be enzymes, receptors, cell components, transporters, hormones, or proteins responsible for the color of the eye or the skin. Pharmacogenes are genes that are responsible for coding proteins involved in drug metabolism, drug action, or drug toxicity. The Chromosomes are present in the nucleus of the cells, where the DNA molecule is arranged to form 23 pairs of chromosomes. The genes inherited from the mother are present in 23 chromosomes and the genes inherited from the father are present in the other 23 chromosomes. So, humans are diploid, which means that the human cells contain DNA molecules that have two sets of genes, one set comes from the mother and the other comes from the father. The fertilized egg, formed by combination of the egg and the sperm, contains the entire set of genes included in the 23 pairs of chromosomes the offspring will have. This set of genes is passed to other cells with every cell division (1). 20.3  Genetic Background Information 20.3.1  Gene Variants, Alleles

Alleles are different variants of the same gene, which result from different sequences of the base pairs. Gene variations can be inherited from the parents, occur due to an error during replication, or because of exposure to environmental factors. The existence of different alleles of a gene that is responsible for coding a specific protein can result in the formation of different amounts or different structures of the proteins produced leading to variation in their function. Some genes are more prone to variation than others. The most

Pharmacogenetics 363 common cause of gene variation is the single-nucleotide polymorphism (SNP), which results from substitution of one base pair in the normal nucleotide sequence. Other causes of gene variation include copy number variation when a section on the DNA is repeated several times or deleted completely. The existence of multiple copies of the genes results in the formation of larger amount of the protein encoded with this gene, while genes with less than two copies can lead to deficiency in the encoded protein. For example, the existence of multiple copies of the gene encoding the formation of one of the CYPs metabolizing enzymes leads to the formation of larger amount of the enzyme and faster rate of metabolism through the metabolic pathway catalyzed by this enzyme. However, the existence of less than two copies of this gene results in reduction of the amount of enzyme, lower enzyme activity, and slower rate of metabolism by this enzyme. 20.3.2 Polymorphisms

Polymorphism refers to variation in gene structure that occurs in more than 1% of the population. Each change in the nucleotide sequence of a gene introduces a new variant form of the gene, which is called allele as mentioned previously. The alleles of the gene carry the specific codes required to instruct the body to build proteins with specific characteristics based on these codes. So, the specific alleles an individual carries determine the characteristics of the proteins produced, which can be the activity of an enzyme or hormone, the function of a receptor or transporter, susceptibility to diseases, or can carry some body features. These characteristics are different among individuals because of the genetic variations. 20.3.3  Gene Nomenclature

A standardized format for the nomenclature of the different alleles has been developed to help in communicating and cataloguing the specific attributes of specific genetic polymorphism. The allele name consists of the gene name followed by an asterisk and a numerical value (2). The numerical value takes the number 1 for the most common allele that encodes the active protein and is referred to as the wild type. Then the other alleles take numbers in the chronological order of their identification. For example, the alleles for the gene that encodes the CYP2D6 enzyme can have the following names: CYP2D6*1, CYP2D6*2, CYP2D6*3, CYP2D6*4, and so on. The name includes the supergene family [CYP], family [2], subfamily [D], and the specific gene [6] followed by * and the number that denotes the different alleles. 20.3.4  Genotype versus Phenotype

The genotype describes the specific variants of the genes an individual carries within his/ her genetic code, while phenotype is used to identify or classify the observable characteristics resulting from specific genotype. These characteristics can be morphological, developmental, biochemical, physiological, or behavioral. The phenotypes result from the expression of the genes and the influence of environmental factors and interaction between the two. Haplotype is a term used when a group of related genes usually present on the same chromosome are inherited together and the individuals who share these genes may have common line of descent. Genetic polymorphism resulting from deletion of a gene results in absence of the gene product and loss of function. The presence of the CYP2D6 allele associated with loss of

364  Pharmacogenetics gene products in an individual leads to the common phenotype of poor metabolizers, while genetic polymorphism resulting from gene duplication leads to higher formation rate of gene products such as in the CYP2D6 alleles associated with excessive gene product. In these individuals, excessive amount of the CYP2D6 enzyme is produced leading to higher metabolic activity and the common phenotype of ultrarapid metabolizers. Additionally, gene translocation, which occurs due to change in the location of the gene in the chromosomes, usually results in the formation of nonfunctioning gene and the absence of gene products. 20.3.5  Monogenic versus Polygenic

The mode of inheritance of a phenotypic characteristic is dependent on whether this characteristic is transmitted by single gene at single position (monogenic) or by multiple genes at different positions (polygenic) on the chromosome. The activity of CYP2D6 enzyme is inherited as a monogenic trait. The different CYP2D6 alleles are usually associated with the absence or excessive formation of the gene product. So, inheritance of the gene that encodes the formation of CYP2D6 metabolizing enzyme produces alleles that will lead to the formation of the same phenotypes. In this case, the genetic polymorphism causing variation in the rate of CYP2D6 metabolism is observed in the general population as three distinct clusters: poor, extensive, and ultrarapid metabolizers, i.e., polymodal frequency distribution (3). On the other hand, the different CYP2C9 alleles, which carry the genetic code for the formation of the CYP2C9 enzyme, produce alloenzymes that are different forms of the enzyme with the same function and unequal activities for the different substrates. In this case, genetic variation in the rate of CYP2C9 metabolism is observed in the general population as variation in the metabolic rate around a mean value, i.e., unimodal frequency distribution (4). Also, the therapeutic effect of warfarin has been linked to multiple genes. The strongest association was linked with the genes responsible for the formation of CYP2C9 enzyme that affects the rate of warfarin metabolism, and vitamin K reductase that is the therapeutic target for warfarin. So, warfarin response is considered a polygenic characteristic and the variation in warfarin dose required to achieve adequate anticoagulant effect is observed in the general population as unimodal frequency distribution (5). 20.3.6  Homozygous versus Heterozygous Genotype

Individuals who are homozygous for a specific gene are those who inherited the same allele from both biological parents, while heterozygous individuals have two different alleles of the same gene inherited from their parents. For example, the gene CYP2C9 has three different alleles: CYP2C9*1, CYP2C9*2, and CYP2C9*3. This means that there are six possible genotypes: CYP2C9*1/*1, CYP2C9*2/*2, CYP2C9*3/*3, CYP2C9*1/*2, CYP2C9*1/*3, and CYP2C9*2/*3. The first three genotypes are considered homozygous because they have similar alleles, and the last three genotypes are considered heterozygous because they have different alleles. The allele is considered dominant if it influences the phenotypic characteristic of the individual and recessive if it does not. Both dominant and recessive alleles of drugmetabolizing enzymes have been described. The phenotype is usually influenced by the number of alleles that encode the active protein. A good example to demonstrate the relationship between the number of alleles encoding the active metabolizing enzyme

Pharmacogenetics 365 and the metabolic phenotype has been described for CYP2D6. Homozygous individuals with two inactive alleles usually have absence or very little metabolizing enzymes and have the least metabolic activity (poor metabolizers). Also, homozygous individuals with two active alleles usually have duplication of the CYP2D6 on the same chromosome, excessive formation of the metabolizing enzymes, and the highest metabolic activity (ultrarapid metabolizers). Whereas heterozygous individuals with one active CYP2D6 allele have metabolizing enzymes and metabolic activity less than those with two active alleles (extensive metabolizers) (6). 20.4  Genetic Polymorphism in Pharmacokinetics Most of the clinically significant genetically based variabilities in pharmacokinetics are related to variation in drug metabolism. However, genetic variations in drug transporters have been described and changes in the pharmacokinetics behavior of substrates for these transporters have been reported. Most genetic polymorphisms in drug metabolism have been identified by observing some patients experiencing adverse effects or therapeutic failure while receiving the average doses of drugs. Further investigations are usually required to reveal the genetic basis of the variations in drug response. 20.4.1  Cytochrome P450 Enzymes

CYP2D6: The genetic variation in CYP2D6 is a well-characterized genetic polymorphism in the CYPs enzymes. It was discovered when some of the patients taking the antihypertensive drug debrisoquine (not used clinically now), which is metabolized by CYP2D6, experienced severe hypotension that was accompanied by higher plasma debrisoquine concentrations. This increased effect was explained by inherited deficiency in debrisoquine metabolizing enzymes. Approximately 25% of the prescribed drugs are metabolized by CYP2D6. Genetic polymorphisms in the CYP2D6 gene have been associated with large variations in the activity of CYP2D6, which range from complete deficiency, intermediate activity, extensive activity to ultrarapid activity (7). The prevalence of poor metabolizers is around 6–10% in Caucasians, slightly higher in African Americans, and around 2% in Asians. The ultrarapid metabolizers represent about 7% of the Caucasian population, while in the Middle East and North Africa the percentage of ultrarapid metabolizers is much higher and can reach up to 35%. The ethnic variation in the prevalence of the different metabolizers’ phenotypes is important and should be considered while developing global drug use recommendations. CYP2D6 deficiency can lead to augmentation in the pharmacological effect of the drug, if CYP2D6 is the major route of its elimination such as in case of fluoxetine and tricyclic antidepressants. While gene duplication of CYP2D6 can lead to inheritance of ultrarapid metabolizers phenotype that has been associated with reduced effect of some antidepressant and antipsychotic drugs due to their fast rate of metabolism (7). Also, the drug effect can be affected significantly if CYP2D6 is involved in the activation of the drug. For example, CYP2D6 deficiency can lead to lack of the analgesic effect of codeine because the metabolism of codeine to morphine is catalyzed by CYP2D6 (8). Genotyping to determine CYP2D6 activity has been recommended before using CYP2D6 substrates with narrow therapeutic index to protect poor metabolizers from developing adverse effects. This is specifically important in patients with personal or family history of adverse effects or reduced activity of drugs metabolized by CYP2D6.

366  Pharmacogenetics CYP2C9 is the main enzyme system involved in the metabolism of several drugs with narrow therapeutic range, including warfarin, phenytoin, losartan, glipizide, and some nonsteroidal anti-inflammatory drugs (NSAID). Many CYP2C9 alleles have been identified with three alleles that can encode alloenzymes with significantly different catalytic activities for the metabolism of S-warfarin. The allele expressing the wild-type protein is designated as CYP2C9*1/*1, while the CYP2C9*2/*2 allele encodes alloenzymes with about 30–40% of the metabolic activity of the wild type for S-warfarin metabolism, and the CYP2C9*3/*3 allele encodes alloenzymes with almost no activity for S-warfarin metabolism. Modest reduction in the metabolism of S-warfarin has been observed in the heterozygous genotypes, CYP2C9*1/*2, CYP2C9*1/*3, and CYP2C9*2/*3. So, one should expect that patients carrying one or more of the alleles associated with the reduced metabolic activity to have higher S-warfarin concentration for a given warfarin dose. These patients usually require lower doses of warfarin to produce therapeutic anticoagulant effect compared with the patients who do not have these alleles (9). Also, the incidence of gastrointestinal (GIT) bleeding in patients receiving NSAIDs and the incidence of hypoglycemia in patients treated with glimepiride or glibenclamide are significantly higher in patients with the CYP2C9 genotypes associated with the lower metabolic activity. Epidemiological studies demonstrated that individuals who are heterozygous for the inactive CYP2C9 allele with intermediate metabolic activity are about 35–40% among Caucasians and 3.5% in Japanese. However, subjects who are homozygous for the two inactive CYP2C9 alleles with presumed poor metabolizer phenotype are 4% among Caucasian and 0.2% among Japanese (10). CYP2C19 is involved in the metabolism of the S-enantiomer of the antiepileptic drug mephenytoin and other drugs such as diazepam, citalopram, propranolol, omeprazole, and proguanil. Similar to the CYP2D6 isoenzyme, specific gene variations in the gene encoding CYP2C19 lead to the poor metabolizer phenotype with respect to the therapeutic agents metabolized by these enzymes. However, the ultrarapid phenotype has not been identified for this enzyme isoforms. The prevalence of poor metabolizers is about 2–5% in Caucasian and African Americans, 10–23% in the Asian population, and about 70% in the South Pacific population. Higher frequency of adverse effects has been reported in poor metabolizers while receiving narrow therapeutic index drugs metabolized by the CYP2C19. A dose reduction is recommended for drugs that are completely or partially metabolized by CYP2C19 such as the tricyclic antidepressants and citalopram, fluoxetine, and sertraline in poor metabolizers. Studies have shown that the proton pump inhibitors metabolized by CYP2C19, such as omeprazole, rabeprazole, and lansoprazole, are more effective in the management of peptic ulcer in patients with one or more of the CYP2C19 alleles associated with the poor metabolizer phenotype. Also, the antimalarial prodrugs proguanil and chlorproguanil, which have to be activated by the CYP2C19, are not effective in patients carrying defective CYP2C19 allele because of their inability to activate the prodrug (11). Furthermore, clopidogrel is a prodrug that must be metabolized by CYP2C19 to form the active metabolite, which produces the antiplatelet activity. Increased risk of thrombotic events can result in patients with decreased CYP2D19 activity while receiving clopidogrel. Also, higher risk of bleeding is possible in patients with increased CYP2C19 activity (12). CYP3A4 is the predominant metabolizing enzyme present in the adult liver and small intestine. Large variation in the CYP3A4 activity is observed among individuals due to the variability in the isoenzyme expression and the overlap in substrate specificity. Isoenzymes are different forms of the enzymes with the same function, which are encoded by

Pharmacogenetics 367 different genes at different positions on the chromosomes. Numerous alleles have been identified for the CYP3A4 gene; however, most of these alleles occur at low frequencies and do not alter the phenotype to any appreciable extent. Despite the genetic influence on the activity of CYP3A enzymes, it does not significantly contribute to the variability in the metabolic clearance of drugs metabolized by these enzymes. The enzyme CYP3A5 is expressed in only 50% of African Americans and about 20% of Caucasians. Individuals who have both CYP3A4 and CYP3A5 usually have higher metabolic rate for the drugs that are metabolized by both enzymes (11). 20.4.2  Thiopurine Methyltransferase (TPMT)

The genetic polymorphism of the thiopurine methyltransferase (TPMT) is well documented. The enzyme TPMT catalyzes the S-methylation of the thiopurine drugs such as azathioprine, mercaptopurine, and thioguanine, which are used in patients with leukemia, rheumatoid diseases, inflammatory bowel disease, and organ transplantation. The thiopurine drugs are inactive prodrugs that require activation of thioguanine nucleotide to produce their cytotoxic effect. In the hematopoietic tissues, TPMT is the main metabolizing enzyme of these drugs. The activity of TPMT in the red blood cells is variable and polymorphic. About 90% of patients have high TPMT activity, 9.7% of patients have intermediate activity, and 0.3% of patients have very low or undetectable activity for this enzyme. This makes the patients with low TPMT activity at high risk of developing lifethreatening hematological toxicity while taking these thiopurine drugs. Determination of the functional TPMT status before the start of azathioprine or mercaptopurine therapy can be very useful in selecting the dose of these drugs to avoid the serious hematological toxicity in patients with low TPMT activity (13). TPMT-deficient patients who are at high risk of developing toxicity while taking thiopurine drugs may require dose reduction to 20–50% of the standard dose. The low or no functioning TPMT enzyme resulting from the reduced function TPMT variants has been associated with the development of ototoxicity and hearing loss in patients receiving the cytotoxic drug cisplatin. It is suggested that the reduced TPMT activity can enhance cisplatin cytotoxic activity due to reduced inactivation of the cisplatinDNA cross-linking. Pharmacogenetic testing can identify the patients with increased risk of cisplatin ototoxicity. It is recommended that patients who are tested positive for the TPMT*2, TPMT*3A, TPMT*3B, or TPMT*3C should receive alternative treatments if possible. If these patients must receive cisplatin, they should undergo frequent audiometric monitoring of their hearing function during and after treatment. 20.4.3 N-acetyltransferase

Acetylation is the main metabolic pathway for the elimination of drugs such as isoniazid, hydralazine, amrinone, and phenelzine. The bimodal distribution of the rate of isoniazid acetylation in different individuals was one of the first observed genetically based variation in drug metabolism. Isoniazid is metabolized by the enzyme N-acetyltransferase 2 (NAT2), which is encoded by different alleles. Genetically controlled differences in acetylation activity have been observed in individuals carrying the slow acetylation allele and exhibit the “slow acetylator” phenotype, and those with the heterozygous alleles and exhibit the “rapid acetylator” phenotype. Large ethnic differences exist in the distribution of the acetylation status. The percentage of slow acetylators in Caucasians and African Americans

368  Pharmacogenetics is between 50 and 60%, while the percentage is about 20% in Chinese, and 10–15% in Japanese and Eskimos. The genetic polymorphism in the acetylation pathway must be considered when using drugs that are eliminated mainly via this metabolic pathway. Smaller doses of isoniazid should be administered in slow acetylators to avoid the development of peripheral neuropathy, which usually results from the higher isoniazid concentrations achieved if they receive the standard doses. Also, rapid acetylators are more susceptible to isoniazid hepatotoxicity, which results from the faster rate of formation of the metabolite implicated in isoniazid-induced hepatotoxicity (14). Besides, slow acetylators are more susceptible to the development of systemic lupus erythematosus-like syndrome while using the antihypertensive drug hydralazine. This adverse effect is probably associated with the elevated plasma hydralazine concentration in this patient population. 20.4.4  UDP-Glucuronosyltransferase (UGT)

Glucuronidation is the major metabolic pathway for numerous drugs and endogenous compounds. Genetic polymorphism has been reported for the different UGT families, which can contribute to the large interindividual variability in the rate of the glucuronidation process. Unconjugated hyperbilirubinemia as in Gilbert’s and Crigler-Najjar syndromes has been associated with UGT1A1 polymorphism and occurs in individuals carrying the enzyme variant that produces slow glucuronidation (15). Another example of the UGT1A1 polymorphism involves the disposition of the anticancer drug irinotecan, which is a prodrug metabolized to the active metabolite SN-38. The active metabolite SN-38 is eliminated by metabolism through the glucuronidation pathway. Irinotecan can cause severe and sometimes fatal toxicities, including myelosuppression and diarrhea. It was suggested that patients with the UGT1A1 allele associated with the low glucuronidation activity are more susceptible to these toxicities (16). The association between the UGT1A1 polymorphism and irinotecan toxicity is included in the label of irinotecan; however, there is no clear recommendation for the need of pharmacogenetic screening for UGT1A1 before initiation of irinotecan therapy. 20.4.5  Drug Transporters

Drug transporters such as the ATP binding cassette (ABC) transporters and the solute carrier (SLC) transporters play a significant role in drug absorption, distribution, metabolism, and excretion. The key ABC transporters involved in drug disposition are ABCB1 (P-gp, MDR1), the ABCC (MRP 1-to-9), and the ABCG2 (breast cancer resistant protein). ABCB1 plays a significant role in drug disposition; however, the influence of polymorphism on its function is not clear. There have been more than 100 genetic variants for the ABCB1 gene identified. Some of these variants have been linked to the increased susceptibility to renal cell carcinoma, Parkinson’s disease, inflammatory bowel disease, refractory epilepsy, and response to HIV therapy. Clinical studies have shown an association between ABCB1 polymorphisms and amlodipine pharmacokinetics with higher clearance and lower area under the curve observed in patients with some haplotypes of ABCB1 (17). Also, some ABCC2 alleles have been associated with a two-fold increase in methotrexate concentrations specially after administration of high methotrexate dose, resulting in an increased need for the folate rescue in these patients (18). Higher concentrations of pravastatin during multiple administration and lower frequency of diarrhea in patients taking irinotecan have been associated with some specific ABCC2 alleles.

Pharmacogenetics 369 Also, ABCG2 polymorphism have been linked to variation in the pharmacokinetics of several drugs such as achieving higher plasma concentrations of diflomotecan, lower plasma concentrations of rosuvastatin, and alteration of irinotecan conversion to its active metabolite, SN-38. However, the clinical significance of these observations is yet to be determined and should be studied further in larger number of patients (11). The SLC transporters that are engaged in drug disposition include the organic cation transporter (OCT), organic anion transporter (OAT), nucleoside transporter, and the polypeptide transporters (OATPs). Polymorphisms in SLCO1B1, which is one of the members of the polypeptide transporters, has been linked to variation in pravastatin and fexofenadine pharmacokinetics, and increased exposure to repaglinide and nateglinide during multiple administration (19). The existence of genetic polymorphism in different transporters is well documented. However, the consequences of the transporter polymorphism are not clinically significant to necessitate the determination of transporter genetic status before using drugs that are substrates for these transporters. 20.5  Genetic Polymorphism in Pharmacodynamics Most drugs produce their therapeutic effects by interaction with a specific target that can be an enzyme or a receptor. Genetic polymorphism that alters the interaction between the drug and its specific target can increase or decrease the drug response. This can explain why some patients respond better than others to the same treatment. Also, some idiosyncratic drug reactions have been shown to occur more frequently in patients carrying some specific alleles. There are few documented examples that link genetic polymorphism to the drug response. Beta-blockers: Genetic polymorphism in the gene encoding the β1-adrenergic receptor can lead to variation in the β-blocker response. Studies have demonstrated that in hypertensive patients some specific haplotypes respond better to β-blockers and had greater reduction in blood pressure compared to patients with other haplotypes. These observations explain the greater blood pressure response to β-blockers in Caucasians compared to African Americans. This is because the frequency of the alleles associated with the higher β1-receptor activity is higher in Caucasians compared to African Americans (20). Warfarin: A well-documented example of genetic polymorphism that affects drug response is the variation in warfarin anticoagulant effect, which is linked to vitamin K epoxide reductase complex subunit 1 (VKORC1) gene polymorphism. This is the gene that encodes vitamin K reductase, which is inhibited by warfarin resulting in the anticoagulant effect. This inhibition interferes with carboxylation of vitamin K-dependent coagulating factors and proteins C and S. Two haplotypes have been linked to the daily warfarin dose required to maintain adequate anticoagulation effect. The different haplotypes produce different amounts of VKORC1, which is the target for warfarin effect. The haplotype associated with the lower VKORC1 amount requires lower doses of warfarin compared to the other haplotype. The higher frequency of the haplotype associated with lower VKORC1 amounts in the Asian population can explain the lower warfarin dosage requirement in these patients (21). Fluoropyrimidines: The antimetabolite fluoropyrimidines such as 5-fluorouracil (5-FU) and its prodrugs capecitabine and tegafur can cause severe toxicity, which can be fatal in some patients. This toxicity has been linked to the activity of the dihydropyrimidine dehydrogenase (DPD) enzyme because the majority of 5-FU dose is metabolized by this enzyme to inactive metabolite. The decrease in DPD activity increases the

370  Pharmacogenetics intracellular concentration of 5-FU and increases the risk of toxicity. Over 160 different allele variants of the gene, DPYD, which encodes the enzyme DPD can result in a wide range of DPD activity and risk of fluoropyrimidines toxicity. One of the approaches used to select the initial dose of fluoropyrimidines utilizes four different variants of DPYD to predict DPD activity and to assign PDP activity score that can range from 0 for no activity to 2 for normal activity (22). The PDP activity score is used to select the initial fluoropyrimidines dose. Patient with score of 0 should avoid fluoropyrimidines, while patients with score of 1 should start at 50% of the average dose, and patients with score of 2 should get the average dose. After initiation of 5-FU therapy, it is recommended that pharmacokinetic monitoring should be used for dose optimization. Abacavir: This is an antiretroviral drug used in combination with other drugs in patients with HIV infection. Patients receiving abacavir can develop severe and potentially fatal hypersensitivity reactions and experience symptoms, including rash, fever, and shortness of breath. The human leukocyte antigen B (HLA-B) gene plays an important role in how the immune system recognizes and responds to pathogens and mediates hypersensitivity reactions. The presence of the allele HLA-B*57:01 has been found to be associated with the development of abacavir hypersensitivity in patients with different ethnicities. So, it is recommended to screen for HLA-B*57:01 allele before the initiation of abacavir therapy. Abacavir is contraindicated in patients who are tested positive for the presence of HLA-B*57:01 and patients who experienced hypersensitivity reactions toward abacavir. Others: Polymorphism near the human interferon gene has been used to predict the response of hepatitis C to interferon treatment. Genotype 1 hepatitis C patients carrying certain genetic variants achieve sustained virological response after treatment with pegylated interferon alfa-2a or pegylated interferon alfa-2b, compared to other patients. Additional examples include the mutation in the angiotensin-converting enzyme (ACE) gene, which has been proposed as a possible cause of variation in response to ACE inhibitors. Furthermore, β2-adrenergic receptor mutations have been linked to the reduction of the bronchodilator effect of albuterol. 20.6  Implementation of Pharmacogenetic Testing in Clinical Practice The term personalized medicine has been associated with the use of pharmacogenetic information as a tool for optimization of drug therapy. This is because the genetic marker information obtained by the pharmacogenetic testing provides the prescriber with the patient’s specific characteristics related to the rate of drug metabolism, drug efficacy, and susceptibility to drug adverse effects. This information can help clinicians in identifying responders from non-responders to specific drug therapy, optimizing drug dosing regiments, and avoiding serious adverse effects. Despite the rapid development in the field of pharmacogenetics, implementation of the pharmacogenetic knowledge in clinical practice has been limited. Ideally, incorporation of pharmacogenetic testing as part of the routine clinical practice starts when a physician who is knowledgeable about pharmacogenetics orders the test. A blood or saliva sample is then taken from the patient and sent to the clinical laboratory for analysis. When the test results become available, the physician reviews and discusses the results with the patient to select the optimum treatment strategy according to the available clinical pharmacogenetic guidelines. The process seems to be simple, but there are numerous issues that need to be considered for the widespread implementation of this service.

Pharmacogenetics 371 20.6.1  Pharmacogenetic Training for Healthcare Providers

Pharmacogenetics have applications for guiding drug therapy in all medical specialties, which makes it necessary for all healthcare professionals to be knowledgeable about this field. All healthcare professionals have been exposed to pharmacogenetics during their training and they know that applying the pharmacogenetic information can be useful while developing the patient’s specific therapeutic plan. However, most of them do not know how to incorporate the available pharmacogenetic resources in their routine clinical practice. The difficulty arises from the lack of clear and consistent standard procedures for incorporating pharmacogenetics in routine workflow. This includes educating the patients about pharmacogenetics, selecting, and ordering the pharmacogenetic test, interpreting the test results, incorporation of the test results in the patient records, and using the pharmacogenetic information in clinical decisions. So, special pharmacogenetic education and training programs can be useful for those interested in this field. 20.6.2  The Pharmacogenetic Tests

The pharmacogenetic tests are usually evaluated based on their analytical validity, clinical validity, clinical utility, in addition to ethical, social, and economic considerations. Analytical utility is the ability of the test to detect the genotype of interest, while clinical validity is the ability of the detected genotype to predict the drug response. Whereas the clinical utility evaluates whether utilizing the test results to guide the selection of drug therapy provides improvement in the therapeutic outcome compared to empirical treatment. Pharmacogenetic test’s approval by regulatory agencies is usually based on the analytical validity and some clinical validity information only. This indicates that test approval by regulatory authorities does not indicate that the use of the test will improve the clinical outcome in patients. The test can be carried out to determine the variants of one specific gene, such as the variants of CYP2C19 for a patient who will start taking clopidogrel. Other tests can cover a panel of multiple genes, such as pharmacogenetic tests that cover all the genes that encode different metabolizing enzymes and drug targets for commonly used drugs. The panel test is more expensive than the single gene test, but the results of the panel test can be incorporated in patient’s records for future selection of other drugs while treating other medical conditions. The cost of the pharmacogenetic tests is an important issue to be considered for the implementation of these tests in clinical practice. This is because except for some limited number of tests, patients are usually required to pay for the cost of their pharmacogenetic tests. So, cost-effectiveness studies are very important to support the use of pharmacogenetic testing in clinical practice. 20.6.3  Interpretation of the Pharmacogenetic Test Results

The results of the pharmacogenetic tests may be provided as raw genetic information with a list of all identified variants of each tested gene. The raw results do not indicate if the identified gene variants produce low, intermediate, or high activity of the protein product of the gene. Some clinical laboratories interpret the genetic data and provide brief report of the patient phenotype, with drug use recommendations. While interpreting the pharmacogenetic test results, the overall patient’s condition should be considered. This should include other patient’s medical conditions, organ functions, and other concomitant drugs. So, the best drug based on the pharmacogenetic test may not be used if

372  Pharmacogenetics the patients have a condition that makes the use of this drug inappropriate. Also, a drug that produces acceptable therapeutic outcome should not be discontinued if the pharmacogenetic test results suggest that this drug may not be effective. 20.6.4  Guidelines for Applying the Pharmacogenetic Testing

Evidence-based guidelines are essential for implementing the pharmacogenetic testing in routine clinical practice to guide the selection of appropriate therapeutic intervention. Regulatory authorities, expert working groups, and medical networks have been developing and updating these evidence-based clinical pharmacogenetic guidelines. For example, the US Food and Drug Administration (FDA) provides an updated list of all FDA-approved drugs with pharmacogenomic information in their labels. This list includes whether the pharmacogenetic tests are required, recommended, or may be performed when using the drugs. This is in addition the drug use recommendation based on the patient genotype (23). Also, pharmacogenetic experts in the Dutch Pharmacogenetics Working Group (DPWG), the Clinical Pharmacogenetics Implementation Consortium (CPIC), the Canadian Pharmacogenomics Network for Drug Safety (CPNDS), the French National Network (Réseau) of Pharmacogenetics (RNPGx), and others have been very active in developing clinical pharmacogenetic guidelines. These different working groups use different methods for evaluating the source of pharmacogenetic information and the strength of the drug-gene-response association to develop their guidelines. This can explain why the different clinical pharmacogenetic guidelines for the same drug from the different working groups can be slightly different. Examples of the drugs that require pharmacogenetic testing before their use include cetuximab that is a recombinant monoclonal antibody used in the treatment of some types of cancer. Cetuximab binds specifically to the human epidermal growth factor receptor (EGFR). So, testing for this receptor should be performed before using cetuximab because its use should be restricted to tumors that express the EGFR. Also, treatment with the antiretroviral drug maraviroc is restricted to patients with confirmed infection with CCR5-tropic HIV-1 because this drug is effective only against this type of infection. Furthermore, the anti-leukemic drug dasatinib is only effective against Philadelphia chromosome-positive acute lymphoblastic leukemia (Ph+ ALL) and should be used only in patients with this disease. Moreover, the anticancer drug trastuzumab is effective against human EGFR 2 (HER2) overexpressing cancers, so it should only be used in tumors that are tested positive for HER2. For other drugs, the pharmacogenetic tests are recommended before using the drug to optimize drug therapy and avoid adverse effects. For example, testing for the allele associated with the increased incidence of carbamazepine-induced Stevens-Johnson syndrome (HLA-B*1502) is recommended in the Asian population when they start taking carbamazepine. Also, it is recommended to test patients taking irinotecan for the UGT allele associated with the lower elimination rate of its active metabolite SN-38, which has been implicated for the serious adverse effects observed after irinotecan therapy. 20.6.5  Enablers for the Implementation of Pharmacogenetics in Clinical Practice

Incorporation of pharmacogenetic information in routine clinical practice can be boosted by joint effort by all pharmacogenetic stakeholders. This includes formal training programs for all healthcare providers in pharmacogenetics. This training should include

Pharmacogenetics 373 interpretation of pharmacogenetic test results and utilization of these results in clinical decision-making. Also, collaboration between the healthcare professionals is necessary to implement practice model for utilizing pharmacogenetic testing, and to develop special tools for direct incorporation of the patients’ pharmacogenetic test results in their medical records. Furthermore, large-scale prospective randomized clinical trials are important to demonstrate the clinical utility and cost-effectiveness of pharmacogenetic testing in clinical practice. Additionally, the development of consistent clinical pharmacogenetic guidelines for different drugs helps in promoting safe and effective drug use and predicting the patient response to drug therapy. 20.7 Summary

• Genetic variation can cause alteration in drug pharmacokinetics and pharmacodynamics and hence, drug response.

• Genetic variation in drug response usually results from polymorphism in the genes encoding drug-metabolizing enzymes, drug transporters, or drug receptors.

• The patient-specific genotype cannot be predicted without performing a valid pharmacogenetic test.

• Pharmacogenetic information can be useful for selecting drug therapy, optimizing drug dosing regiments, and avoiding serious adverse effects.

• Evidence-based pharmacogenetic clinical guidelines are necessary for incorporating the pharmacogenetic information in drug prescription decisions.

• There are barriers and enablers that are important to be considered for implementation of pharmacogenetic information in routine clinical practice.

References 1. Orrico KB “Basic concepts in genetics and pharmacogenomics for pharmacists” (2019) Drug Target Insights; 13:1–7. 2. Sim SC and Ingelman-Sundberg M “The human cytochrome P450 (CYP) allele nomenclature website: A peer-reviewed database of CYP variants and their associated effects” (2010) Hum Genomics; 4:278–281. 3. Linder MW, Evans WE, and McLeod HL “Application of pharmacogenetic principles to clinical pharmacology” in “Applied pharmacokinetics and pharmacodynamics: Principles of therapeutic drug monitoring” Edited by Burton ME, Shaw LM, Schentag JJ and Evans WE (2006) Lippincott Williams and Wilkins, New York, NY, USA. 4. Lee CR, Goldstein JA and Pieper JA “Cytochrome P450 2C9 polymorphism: A comprehensive review of the in-vitro and human data” (2002) Pharmacogenetics; 12:251–263. 5. Krynetskiy E and McDonnell P “Building individualized medicine: Prevention of adverse reaction to warfarin therapy” (2007) J Pharmacol Exp Therap; 322:427–434. 6. Agundez JAG, Ledesma MC, Ladero JM and Benitez J “Prevalence of CYP2D6 gene duplication and its repercussion on the oxidative phenotype in a white population” (1995) Clin Pharmacol Ther; 57:265–269. 7. Meyer UA “Pharmacogenetics: The slow, the rapid, and the ultrarapid” (1994) Proc Natl Acad Sci USA; 91:1983–1984. 8. Sindrup SH and Brosen K “Pharmacogenetics of codeine hypoalgesia” (1995) Pharmacogenetics; 5:335–346. 9. Higashi MK, Veenstra DL, Kondo LM, Wittkowsky AK, Srinouanprachanh SL, Farin FM and Rettie AE “Association between CYP2C9 genetic variants and anticoagulationrelated outcomes during warfarin therapy” (2002) JAMA; 287:1690–1698.

374  Pharmacogenetics 10. Nasu K, Kubota T and Ishizaki T “Genetic analysis of CYP2C9 polymorphism in a Japanese population” (1997) Pharmacogenetics; 7:405–409. 11. Williams JA, Anderson T and Anderson T, et al. “PhRMA white paper on ADME pharmacogenomics” (2008) J Clin Pharmacol; 48:849–889. 12. Russmann S, Rahmany A and Niedrig D, et al. “Implementation and management outcomes of pharmacogenetic CYP2C19 testing for clopidogrel therapy in clinical practice” (2021) Eur J Clin Pharmacol; 77:709–716. 13. Evans WE, Hon YY and Bomgaars L, et al. “Preponderance of thiopurine S methyltransferase deficiency and heterozygosity among patients intolerant to mercaptopurine or azathioprine” (2001) J Clin Oncol; 19:2293–2301. 14. Bonicke R and Reif W “Enzymatic inactivation of isonicotinic acid hydrizide in human and animal organism” (1953) Naunyn Schmiedebergs Arch Exp Pathol Pharmakol; 220:321–323. 15. Kadakol A, Ghosh SS, Sappal BS, Sharma G, Chowdhury JR and Chowdhury NR “Genetic lesions of bilirubin uridine-diphosphoglucuronate glucuronosyltransferase (UGT1A1) causing Crigler-Najjar and Gilbert syndromes: Correlation of genotype to phenotype” (2000) Hum Mutat; 16:297–306. 16. Paoluzzi L, Singh AS, Price DK, Danesi R, Mathijssen RH, Verweij J, Figg WD and Sparreboom A “Influence of genetic variants in UGT1A1 and UGT1A9 on the in vivo glucuronidation of SN-38” (2004) J Clin Pharmacol; 44:854–860. 17. Leschziner GD, Andrew T, Pirmohamed M and Johnson MR “ABCB1 genotype and PGP expression, function and therapeutic drug response: A critical review and recommendations for future research” (2007) Pharmacogenomics J; 7:154–179. 18. Rau T, Erney B, Gores R, Eschenhagen T, Beck J and Langer T “High-dose methotrexate in pediatric acute lymphoblastic leukemia: Impact of ABCC2 polymorphisms on plasma concentrations” (2006) Clin Pharmacol Ther; 80:468–476. 19. Chung JY, Cho JY, Yu KS, Kim JR, Oh DS, Jung HR, Lim KS, Moon KH, Shin SG and Jang IJ “Effect of OATP1B1 (SLCO1B1) variant alleles on the pharmacokinetics of pitavastatin in healthy volunteers” (2005) Clin Pharmacol Ther; 78:342–350. 20. Shin J and Johnson JA “Pharmacogenetics of beta-blockers” (2007) Pharmacotherapy; 27:874–887. 21. Rieder MJ, Reiner AP, Gage BF, Nickerson DA, Eby CS, McLeod HL, Blough DK, Thummel KE, Veenstra DL and Rettie AE “Effect of VKORC1 haplotypes on transcriptional regulation and warfarin dose” (2005) N Engl J Med; 352:2285–2293. 22. Lunenburg CATC, van der Wouden CH and Nijenhuis M, et al. “Dutch Pharmacogenetics Working Group (DPWG) guideline for the gene-drug interaction of DPYD and fluoropyrimidines” (2020) Eur J Hum Genet; 28:508–517. 23. Table of pharmacogenomic biomarkers in drug labeling last accessed April 30, 2023.

21 Therapeutic Drug Monitoring

Objectives After completing this chapter, you should be able to:

• Describe the importance of therapeutic drug monitoring in the initiation and management of drug therapy.

• Compare the difference between dose-response and concentration-response relationships. • Define the therapeutic range of drugs. • Discuss the characteristics that make drugs good candidates for therapeutic drug monitoring.

• Describe the proper procedures that should be followed for obtaining and analyzing samples for therapeutic drug monitoring.

• Recommend the general requirements and resources needed to establish therapeutic drug monitoring service.

21.1 Introduction Therapeutic drug monitoring (TDM) involves the use of drug concentration in biological fluids as a tool for individualization of drug therapy and dosage adjustment. The rationale is based on the existence of strong correlation between the drug concentration in the systemic circulation and the therapeutic as well as adverse effects of many drugs. This correlation indicates that most drugs exert their desired therapeutic effects with minimal adverse effects in the majority of patients when their blood concentrations are maintained within a certain concentration range that is known as the therapeutic range. During multiple drug administration, the steady-state drug concentration achieved in a particular patient is dependent on the dosing rate of the drug and the drug total body clearance in this patient. When the same dose of a drug is administered to a group of patients, different drug concentrations are achieved in the different patients due to many factors related to the drug product and the patient’s specific characteristics. This explains why the correlation between the blood drug concentration and response is stronger than the correlation between the drug dose and response. TDM is a multidisciplinary activity that utilizes pharmaceutical, pharmacokinetic, and pharmacodynamic knowledge for optimization of drug therapy in a variety of clinical situations (1). The process starts with the selection of an initial dosage regimen based on the patient’s specific characteristics. After initiation of drug therapy, the blood drug concentrations are measured to estimate the patient’s specific pharmacokinetic parameters of DOI: 10.4324/9781003161523-21

376  Therapeutic Drug Monitoring the drug. These pharmacokinetic parameters are used to determine the dosage regimen that should maintain the blood drug concentration within the desired range of drug concentrations. The desired blood concentrations are usually the concentrations that have been associated with the optimal therapeutic effects and minimal adverse effects. In addition to monitoring the patient’s clinical condition, periodical determination of the blood drug concentration may be necessary during long-term drug use to monitor the change in drug concentration that might happen due to changes in the patient’s condition. The application of TDM became possible after the development of analytical procedures and techniques for rapid determination of the blood drug concentration with acceptable accuracy and precision. Also, this service became attractive because several studies have proven the cost-effectiveness of its application in a variety of clinical settings. 21.2  General Principles of Initiation and Management of Drug Therapy The process of initiating drug therapy, which is simplified in the scheme presented in Figure 21.1, usually starts with the diagnosis of the patient’s condition, based on the available information. Once the patient’s condition is diagnosed, one or more drugs are usually prescribed to treat all the patient’s medical problems, and the desired therapeutic outcomes for each of the patient’s problems are specified. The dosage regimens of the drugs are determined based on the patient’s demographic characteristics, and a set of clinical monitoring parameters are selected for evaluation of the therapeutic and adverse effects of each drug. After initiation of drug therapy and allowing enough time for the drugs to achieve steady state, the patient’s medical condition is monitored to evaluate if the desired therapeutic outcomes are achieved. For each of the patient’s medical problem, if

Figure 21.1  The basic steps in the process of initiation and management of drug therapy.

Therapeutic Drug Monitoring 377 the desired therapeutic outcome is achieved, the selected drug and dosage regimen should continue if continuous drug use is required, and the desired therapeutic goals are maintained. However, if the therapeutic outcome is not achieved, the drug dosage regimen should be modified, and the patient is monitored clinically after allowing enough time to achieve the new steady-state drug concentration. The process is repeated until the desired therapeutic outcome is achieved. If modification of the dosage regimen does not result in achieving the therapeutic goal, changing the drug or addition of another drug may be necessary. The process is repeated until the desired therapeutic outcome is achieved (2, 3). For example, when oral hypoglycemic drugs are used by diabetes mellitus patients, the desired therapeutic outcome is usually to maintain the blood glucose level within the normal range without experiencing any adverse effects. After using the oral hypoglycemic drug for at least five elimination half-lives, the blood glucose level is evaluated. Adequate control of the blood glucose without experiencing any adverse effects indicates that the desired therapeutic outcome of the drug is achieved, and the drug use should be continued using the same dosage regimen. However, inadequate reduction of the blood glucose level to the normal range may necessitate increasing the dose of the oral hypoglycemic drugs, and the appearance of adverse effects may require reduction of the drug dose. After using the modified dosage regimen for enough time to achieve the new steady-state concentration, the drug effect is evaluated to determine if the desired therapeutic outcome is achieved. The process is repeated until the desired outcome is achieved. When the blood glucose level cannot be maintained within the normal range using the selected oral hypoglycemic drug, the use of a different oral hypoglycemic drug or the addition of a second glucose-lowering drug may be necessary. The same process is repeated to evaluate if the new drug or drug combination can achieve the desired therapeutic goal. Different medical conditions usually have different desired therapeutic outcomes and different clinical monitoring parameters. 21.2.1  The Use of Therapeutic Drug Monitoring in the Management of Drug Therapy

For many drugs, it is possible to utilize the general approach described before for assessing the success or failure of therapy by monitoring the patient clinically. However, under certain medical conditions, complete relying on this approach may not be appropriate and can expose the patient to serious medical problems. Blood drug concentrations measurements can be a valuable guiding tool for avoiding these serious problems and for achieving the desired therapeutic outcomes in a timely manner. Also, measuring the blood drug concentration in the early course of starting the drug therapy can help in the speedy selection of the appropriate dosing regimen for each patient. This practice usually cut down the time required for making the decisions required to achieve the therapeutic outcomes. Measuring the blood drug concentration can be very useful for differentiating between therapeutic failure, undertreatment, and noncompliance. When the drug is administered and the desired therapeutic effects of the drug are not observed, this may result from the use of the wrong drug, the wrong dose, or due to noncompliance. If the drug concentration is in the upper end of the therapeutic range, this indicates that a different drug should be used, or an additional drug should be added to the patient’s therapy. If the drug concentration is in the lower end or below the therapeutic range, it may be appropriate to increase the dose and make sure that the patient is taking the drug correctly before attempting to change the drug, or to add a new drug. TDM is also useful when the drug is used for prophylaxis because in this case, relying on the drug therapeutic effect may not

378  Therapeutic Drug Monitoring be appropriate. Examples of the prophylaxis use of drugs such as the use of lithium for prevention of manic-depressive illness, the use of phenytoin to prevent the development of seizures after trauma or neurosurgical procedures, and the use of cyclosporine and tacrolimus for preventing transplant rejection. In these cases, developing the symptoms that indicate insufficient drug doses can have serious consequences. Measuring the blood drug concentrations can help in making sure that the drug is present in the body in sufficient quantities to produce its desired effects and to avoid the development of toxicity. Measurement of drug concentration can help in differentiating between drug toxicity and other medical conditions, and in making decisions regarding the drug therapy. For example, digoxin toxicity may result in the development of cardiac arrhythmias which mimics the symptoms of heart conditions that can develop in the patient population usually treated with digoxin. Also, the nephrotoxicity that is a common adverse effect of the immunosuppressive drug cyclosporine usually results in the increase in serum creatinine which is similar to the common symptoms of graft rejection in renal transplant patients. Furthermore, the nephrotoxicity of the aminoglycoside antibiotics may be hard to differentiate from the decrease in kidney function caused by severe generalized infection. TDM can also be useful when monitoring the therapeutic effect of the drug is not easy such as in case of antiepileptic drugs when the incidence of seizures is infrequent, or in psychiatric conditions that cannot be evaluated easily. Also, when the drug’s adverse effect is very serious or irreversible, it is not possible to use these adverse effects as monitoring parameters such as in case of the irreversible aminoglycosides’ ototoxicity. In these cases, the blood drug concentration provides additional information that can be utilized to achieve optimum patient care. 21.3  Drug Blood Concentration versus Drug Dose After drug administration, the drug concentration-time profile can be variable in different patients depending on the pharmacokinetic characteristics of the drug in individual patients. This means that administration of the same dosage regimen of the same drug to different individuals usually produces different drug profiles due to the interindividual variability in drug pharmacokinetics. The drug in the systemic circulation is distributed to all parts of the body to exert its therapeutic and adverse effects. The intensity of the therapeutic and toxic effects for most drugs usually depends on the drug concentrations at the site of action which is usually at equilibrium with the drug concentration in blood, as presented in Figure 21.2. Based on this, the correlation between the drug blood concentrations and effect is much stronger than the correlation between the drug dose and effect (1, 4). The inter-patient variability in drug pharmacokinetics and response usually weakens the correlation between the drug dose and response. So, the same dose of the drug should not be administered to all patients. The drug dose should be tailored according to the

Figure 21.2 A schematic presentation of the relationship among the dosage regimen, the blood concentration, the concentration at the site of action, and the drug effect.

Therapeutic Drug Monitoring 379 specific patient’s characteristics. The variability in drug pharmacokinetics and response results from many factors such as age, weight, gender, disease state, drug interactions, genetic factors, factors related to the drug formulations, nonlinear disposition, environmental factors, dietary factors, occupational exposure, stress, smoking, and seasonal variations. Some of these factors significantly contribute to the variability in drug disposition, and some factors only have minimal effect. However, it is important to note that the overall variability in the drug disposition result from the additive effect of all these factors. 21.4  The Therapeutic Range The correlation between the blood drug concentration and the drug therapeutic effect is well documented for many drugs, with higher drug concentrations usually produce more intense effect. This may imply that higher blood drug concentration should produce better therapeutic effect. However, studying the relationship between the blood drug concentration and the drug’s adverse effects showed higher incidence of adverse effects at higher blood drug concentrations. For example, most patients receiving phenytoin start showing evidence of therapeutic effect, as indicated by the decrease in seizure frequency when the blood concentration is above 10 mg/L. As phenytoin concentration increases above 20 mg/L, patients develop nystagmus, then ataxia in the concentration range of 30–40 mg/L, while mental changes occur when phenytoin concentration exceeds 40 mg/L. The appearance of toxic effects usually requires holding or decreasing the drug dose to decrease its serum concentration. So, the blood drug concentration should be kept within the range of concentrations that produce the maximum therapeutic effect and minimum adverse effects. This range of blood drug concentrations is known as the therapeutic range. The therapeutic range is defined as the range of drug concentrations in blood which is associated with the highest probability of the desired therapeutic effect and the lowest probability of the undesirable adverse effects (1). Figure 21.3 represents a hypothetical example which shows that at very low drug concentrations, the probability of achieving the desired therapeutic effect or toxicity is very low. As the blood drug concentration increases, the probability of achieving the therapeutic effect increases and then stays relatively constant. Over the same concentration range, the probability of developing toxicity

Figure 21.3 The therapeutic range is chosen to produce the highest probability of the therapeutic effect and the lowest probability of toxicity.

380  Therapeutic Drug Monitoring increases more slowly. With further increase in blood concentration, the probability of developing toxicity increases more rapidly. According to Figure 21.3, if one selects 10 mg/L as the lower end of the therapeutic range, then the minimum probability of achieving the desired therapeutic effect would be about 40%. If 20 mg/L was chosen as the upper end of the therapeutic range, then the maximum probability of achieving the desired response would be about 80%. Over the same concentration range, the probability of developing undesired toxicity would remain less than 10%. So, the therapeutic range is the range of blood drug concentrations within which most patients will have the desired therapeutic response with minimum incidence of adverse effects. It is important to note that few patients may have the desired therapeutic effect when the blood drug concentration is below the therapeutic range and few patients may not show any therapeutic effects even if the blood drug concentration is within the therapeutic range. Also, few patients may develop toxicity while their drug blood concentration is within the therapeutic range and some patients may not develop toxicity even if their blood drug concentration is slightly above the therapeutic range. So, the decision regarding dosage adjustment should be guided by the blood drug concentration in conjunction with the clinical evaluation of the patient. The therapeutic range for a drug is determined by measuring the steady-state blood drug concentration and evaluating the therapeutic and adverse effects of the drug at the same time in large number of patients who are taking the drug. From this information, the range of drug concentrations associated with the highest frequency therapeutic efficacy and lowest frequency of adverse effect is identified. This is the therapeutic range of the drug. So, the therapeutic range of drugs is expressed as a range of blood drug concentrations, even though the drug effect is correlated with the drug concentration at the site of action. This is because most drugs are monitored at steady state, and at steady state, equilibrium is established between the drug in blood and all parts of the body, including the site of action. This means that the ratio of the blood drug concentration to the drug concentration at the site of action is constant at steady state. The change in blood drug concentration is usually accompanied by a proportional change in the drug concentration at the site of action at steady state. So, when the drug concentration in blood is within the therapeutic range, the drug concentration at the site of action, which is not known, is always in the range that produces the optimal therapeutic effect with minimal toxicity. 21.5  Drug Candidates for Therapeutic Drug Monitoring Application of TDM is not recommended for all drugs. The use of blood drug concentration measurements for dosage optimization has been used for antibiotics such as aminoglycoside and vancomycin, antiepileptic drugs such as phenytoin, carbamazepine, valproic acid, phenobarbital, and ethosuximide, cardiovascular drugs such as digoxin, lidocaine, and procainamide, immunosuppressants such as cyclosporine, tacrolimus, and sirolimus, cytotoxic drugs such as methotrexate, bronchodilator such as theophylline, psychiatric medications such as lithium, benzodiazepines, and tricyclic antidepressants in addition to many other drugs. All these drugs fall at least into one of the following drug categories that make the drug a good candidate for TDM (5). 21.5.1  Drugs with Low Therapeutic Index

The drug has low therapeutic index when the ratio of the minimum toxic concentration to the minimum effective concentration is small. These drugs require accurate determination of the drug dosage regimen to avoid drug toxicity or subtherapeutic effects. This is because the

Therapeutic Drug Monitoring 381 therapeutic range for these drugs is narrow and any small deviation from the appropriate dosage regimen can produce drug concentrations outside of the therapeutic range during the dosing interval. TDM can serve as a guide for choosing the appropriate dosage regimen that can maintain the drug concentration within the therapeutic range all the time. While drugs that have high minimum toxic concentration to minimum effective concentration ratio have high therapeutic index and are considered safe drugs. This is because different dosage regimens from these drugs can achieve drug concentrations within the therapeutic range due to their wide therapeutic range. When these drugs are used clinically, an average dosage regimen is administered, and the resulting drug concentrations usually fall within the therapeutic range all the time despite the inter-patient variability in drug disposition. A wide range of dosing regimens should produce the desired therapeutic effect with minimal toxicity. So, application of TDM may not be necessary for high therapeutic index drugs. 21.5.2  Drugs with Large Variability in Their Pharmacokinetic Behavior

Drugs with large variability in their pharmacokinetic behavior have poor correlation between the administered dose and the resulting blood drug concentration-time profile. Administration of the same dosage regimen of these drugs to a group of patients can produce a wide range of drug concentrations at steady state. This makes it very difficult to choose the drug dosage regimen that can maintain blood drug concentration within the therapeutic range, based only on the patient’s information. For these drugs, TDM can be a good tool for calculating the appropriate dosage regimen required to achieve blood drug concentrations within the therapeutic range. 21.5.3  Drugs Used in High-Risk Patients or Patients with Multiple Medical Problems

Patients with multiple medical problems can face serious medical consequences if they develop drug toxicity or when there is delay in receiving the appropriate doses of the prescribed drugs. So, careful selection of the appropriate dosage regimens of drugs for these patients is very important. However, this is not an easy task because the multiple diseases in these patients may affect the drug pharmacokinetic behavior, and the multiple medications they receive can increase the probability of drug interactions. TDM can be very useful in guiding the selection of the appropriate dosage regimens of drugs for these high-risk patients. 21.6  Determination of the Drug Concentration in Biological Samples Application of TDM involves the use of the drug concentration in biological samples as a guide for selecting the appropriate dosage regimen for each individual patient. The right recommendation for dosage regimens cannot be made unless the measured drug concentration in biological fluid is reliable. The reliability of the drug concentration comes from the collection of the appropriate biological fluid sample at the right time using the proper procedures. Also, the appropriate compound should be determined in these samples utilizing accurate and precise analytical techniques. 21.6.1  The Biological Samples

The biological samples used in TDM for most drugs are whole blood, plasma, or serum. The whole blood is drawn directly from blood vessels. When an anticoagulant is added to the blood sample, plasma can be obtained by centrifugation. If the blood sample is left

382  Therapeutic Drug Monitoring to coagulate, centrifugation of the coagulated blood sample will give serum. Some drugs can also be measured in urine and saliva samples; however, the measured concentrations in these samples must be carefully interpreted. The drug properties and the techniques used in the analysis usually determine the appropriate sample matrix, the type of collection tubes, and the proper anticoagulant to be used when whole blood, plasma, or serum samples are needed (6). Adherence to the sample collection instruction for each drug is necessary to obtain reliable sample that can be used to determine the drug concentration accurately. Stability of the drug in the collected samples is an important issue that should be considered. Although most of the samples drawn for TDM in hospitals are analyzed shortly after collection, there are situations when this is not the case, such as when samples are drawn in the doctor’s office and then send to clinical laboratory for analysis. In this case, the samples should be stored in the condition that will ensure the maximum stability of the drug in the sample until analysis. 21.6.2  The Time of Sample

The blood drug concentration is continually changing during the dosing interval during multiple drug administration. Large fluctuations in the blood drug concentration are usually observed when the drug is rapidly absorbed and/or rapidly eliminated. So, the time of blood samples should be selected carefully to facilitate the interpretation of the measured drug concentration. There is usually optimal time for obtaining the blood samples for each drug. However, during multiple drug administration of the drug, the general role is that blood samples should be obtained after the drug steady-state concentration is achieved. This usually takes a period equivalent to five elimination half-lives of the drug after the start of drug therapy (7). There are exceptions to this role such as when multiple samples are obtained before reaching steady state to estimate the patient’s specific pharmacokinetic parameters and calculation of the proper dosage regimen for aminoglycoside and vancomycin. When drug efficacy is the main reason for blood concentration monitoring, the minimum drug concentration at steady state is usually determined by measuring the drug concentration just before drug administration. If toxicity is the main reason for drug concentration monitoring, the maximum drug concentration at steady state is usually determined by measuring the drug concentration at the time when the highest drug concentration is expected (6). When steady state is achieved while using aminoglycosides and vancomycin, the maximum and minimum blood concentrations are usually measured to make sure that the drug concentrations are in the range that produces the optimal effect with minimum toxicity all the time (8). For drugs with long elimination half-lives and follow multicompartment pharmacokinetic models such as lithium and digoxin, blood samples are usually obtained 6–8 hr after drug administration to reflect the average drug concentration at steady state. 21.6.3  The Measured Drug Moiety

The free drug molecules that are unbound to plasma or tissue proteins are responsible for the drug’s therapeutic and adverse effects. This is because the free drug molecules can cross the biological membranes to become available at the site of drug action. However, measuring the free drug concentration is not simple and is not routinely performed in clinical laboratories. When the percentage of the drug bound to plasma protein is constant, the total drug concentration (free + bound) represents a simple alternative to

Therapeutic Drug Monitoring 383 measuring the free drug concentration. The measured total drug concentration in biological samples can be used for dosage adjustment keeping in mind that the free drug concentration is always a constant fraction of the total drug concentration. For example, the plasma protein binding for the antiepileptic drug phenytoin is 90%, and the therapeutic range for the free phenytoin concentration is 1–2 μg/mL. So, we can consider the therapeutic range for phenytoin using the total phenytoin concentration to be 10–20 μg/mL, if phenytoin protein binding remains relatively constant. The therapeutic range for most drugs is usually reported as the total plasma drug concentration assuming that the drug protein binding is constant, since most laboratories analyze the total drug concentration in the samples. However, there are situations when the drug of interest is highly bound to plasma protein and the patient has conditions that can alter the drug protein binding. The conditions that can alter the drug protein binding include renal failure, liver disease, malnourishment, elderly, hyperbilirubinemia, cystic fibrosis, and concurrent administration of medications that can displace the drug from its binding sites. In these cases, the percentage of the drug bound to plasma protein is different from the average value and measuring the total drug concentration may not be appropriate. Under these conditions, measuring the free drug concentration is necessary (9, 10). Examples of drugs that are highly bound to plasma proteins and measuring the free drug concentration is necessary for conditions that can affect drug protein binding include phenytoin, carbamazepine, valproic acid, antiretroviral drugs, digoxin, and immunosuppressant drugs. Measuring the free drug concentration involves an additional step to separate the free drug from the bound drug in the biological matrix. The equilibrium dialysis, ultrafiltration, and ultracentrifugation techniques can be used; however, the ultrafiltration technique is commonly used by the clinical laboratories because it is the simplest and quickest. Recently, a solid phase microextraction technique has been developed that allows separation of the free drug from the bound drug in biological matrices. Once the free drug is separated, the same analytical technique can be used for determination of the free drug concentration. The therapeutic range for the free drug concentration has to be considered when interpreting the measured free drug concentrations in patients. Some drugs are metabolized to pharmacologically active metabolites that have therapeutic effects like that of the parent drug. The observed therapeutic effect usually results from the parent drug and the drug metabolite. In this case, TDM should involve measuring the parent drug and the metabolite concentration in the biological samples to account for all the drug moieties responsible for the observed therapeutic effect of the drug. Dosage adjustment recommendation should consider the contribution of both the parent drug and the metabolite to the therapeutic effect. The therapeutic range for these drugs should include recommendations for the parent drug and the active metabolite concentrations. Examples of such drugs include the antiarrhythmic drug procainamide and its active metabolite N-acetylprocainamide that has antiarrhythmic effect like that of the parent drug. The therapeutic range for procainamide is 4–10 μg/mL and many laboratories recommend the range of 10–30 μg/mL for the sum of procainamide and N-acetylprocainamide concentrations (11). Similarly, when the drug is metabolized to a metabolite that can contribute significantly to the drug’s adverse effect, both the parent drug and the toxic metabolite should be measured. Such as in case of the antiepileptic drug carbamazepine that is metabolized to carbamazepine-10,1-epoxide. This metabolite contributes significantly to the therapeutic and toxic effects of carbamazepine. The therapeutic range for carbamazepine is 4–12 μg/mL, and it is recommended to keep carbamazepine-10,1-epoxide concentration between 0.4 and 4 μg/mL (11).

384  Therapeutic Drug Monitoring 21.6.4  The Analytical Technique

The application of TDM has been facilitated by the development of rapid, selective, sensitive, accurate, and precise analytical techniques that are suitable for quantifying the drug concentration in biological fluids. Currently most of the drug assays performed in clinical settings utilize the commercially available immunoassays. This includes the fluorescence polarization immunoassay (FPIA), the enzyme multiplied immunoassay (EMIT), and enzyme-linked immunosorbent assay (ELISA). These methods are widely used because they are easily automated to ensure short turn-around time. The main disadvantage of these methods is their potential lack of selectivity due to cross-reactivity with compounds that are structurally related to the analyte of interest leading to overestimation or underestimation of the drug concentration. Also, these methods can only be applied for the analysis of drugs when the special analytical kits required for their analysis are commercially available. So, the immunoassays cannot be readily applied for the analysis of newly developed drugs, or the drugs that are not routinely monitored. The other commonly used analytical techniques include the high-performance liquid chromatography (HPLC) that can be applied to a variety of drugs with minimal sample preparation procedures. Also, the gas chromatography (GC) can be used for the analysis of volatile drugs or drugs that can be derivatized to form volatile derivative. The GC involves extensive sample preparation procedures, so it is usually used for the analysis of drugs that cannot be analyzed by the HPLC technique such as valproic acid. Coupling of the mass spectrometry (MS) with the HPLC technique led to the development of the LC-MS and LC-MS-MS techniques that can be utilized for the analysis of samples collected for TDM. The LC-MS-MS technique has the advantage of extreme sensitivity, the ability to analyze more than one drug simultaneously, and the ability to quantify the drug and its metabolite in the same run. The analytical assays developed utilizing the HPLC, GC, and LC-MS-MS techniques are usually selective, sensitive, accurate, and precise. However, using these techniques requires extensive analytical skills and shifting from one analytical condition to the other to analyze different drugs takes time, which prolongs the turn-around time compared to the immunoassays. These techniques are usually used in research and as the reference methods for the validation of the immunoassays (6). The other analytical techniques used for the analysis of drug in biological samples include the radioimmunoassay (RIA). This technique is not commonly used for the routine analysis of samples in clinical settings because of the complexity associated with handling of radioactive materials. Other techniques include the use of ion-selective electrode in the analysis of lithium in biological fluids. All the analytical techniques used for the analysis of drugs in biological fluids to support the TDM should be fully validated. The analytical techniques should be validated for accuracy, precision, selectivity, sensitivity, linearity, reproducibility, and robustness to ensure that the measured drug concentrations are reliable. 21.7 Establishing a Therapeutic Drug Monitoring (Clinical Pharmacokinetic) Service The TDM is a multidisciplinary service that involves cooperation among the clinicians, pharmacists, nurses, and analysts. It is well documented that application of this service can facilitate the achievement of the desired therapeutic outcomes and decrease the incidence of adverse effects. So, application of this service can be beneficial for the patients and can have pharmacoeconomic benefits.

Therapeutic Drug Monitoring 385 21.7.1  Major Requirements

Establishing TDM service in a healthcare institution requires the availability of trained healthcare professionals to run the service in addition to laboratory facility for drug analysis. The drug analysis facility to support the TDM service should have the capability to perform quantitative drug analysis in blood and other biological fluids in short turnaround time with acceptable accuracy and sensitivity. The pharmacy department that usually runs this service should have trained pharmacists who have the basic and clinical pharmacokinetic knowledge required to interpret the measured drug concentrations and examine the patient clinical data to recommend the appropriate dosage regimen for each patient. In addition to the clinical judgment, the pharmacists usually utilize computer software that can use the patient dosing history and the measured drug concentrations for estimating the patient individual pharmacokinetic parameters and calculating the recommended dose for each patient. A wide variety of software are available ranging from simple software for hand-held calculators to sophisticated software that stores the patient information every time it is used in a database and uses the stored database in subsequent recommendations. 21.7.2  Dosage Regimen Recommendation

Optimization of drug therapy can be achieved empirically sometimes at considerable expense, time, and occasional toxicity. However, drug concentration in biological fluids can serve as a therapeutic endpoint to guide and assess drug therapy, and as a prophylactic of toxicity. Application of TDM usually involves going through the following steps (11). 21.7.2.1  Determination of the Initial Dosing Regimen

The first step usually involves the selection of an initial dosing regimen for the patient to initiate drug therapy. The patient’s specific demographic information, including age, sex, weight, eliminating organ function, nutritional status, pharmacogenetic information, and other medications, are used to estimate the patient’s pharmacokinetic parameters for the drug under consideration. The estimated drug pharmacokinetic parameters for the patient are used to calculate the initial dosage requirements. So, the pharmacokinetic parameters used in the calculation of the initial dosing regimen for the patient are the pharmacokinetic parameters of the drug in a population like that of the patient being monitored. 21.7.2.2  Determination of the Patient’s Specific Pharmacokinetic Parameters

After initiation of drug therapy, the next step should be the determination of the patient’s specific pharmacokinetic parameters for the drug. This is usually achieved by planning a sampling strategy that will allow calculation of the patient’s specific drug pharmacokinetic parameters. The number of samples and the time of these samples can be different depending on the drug being monitored as described previously. Knowledge of the clinical pharmacokinetics of the drug used by the patient can help in determining the number and time of samples required to estimate the patient’s specific pharmacokinetic parameters. The obtained sample(s) are analyzed to determine the drug concentration using a validated analytical technique. The measured concentrations are used to estimate the patient’s specific pharmacokinetic parameters for the drug using an appropriate data analysis software.

386  Therapeutic Drug Monitoring 21.7.2.3 Calculation of the Dosage Requirements Based on the Patient’s Specific Pharmacokinetic Parameters of the Drug

After estimation of the patient’s specific pharmacokinetic parameters, the final step should be calculation of the dosage regimen that should achieve steady-state drug concentration within the therapeutic range. This calculated dosage regimen is specific for this patient because it is calculated based on the patient’s specific pharmacokinetic parameters. Recommendations should be made to adjust the dosing regimen to achieve the desired therapeutic concentration. If the patient continues to receive the drug for long time, such as in the treatment of chronic diseases like epilepsy and cardiovascular diseases, it is important to periodically check the blood drug concentration to make sure that the drug concentrations continue to fall within the therapeutic range. Further dosage adjustment can be made based on the patient’s clinical condition and the results of these follow up samples. 21.7.3  The Pharmacoeconomics of Therapeutic Drug Monitoring

It has been well documented that the application of TDM service improves the patient response to many drugs and decreases the incidence of adverse drug reactions. Application of this service usually utilizes many resources such as equipment, space, analysis cost, software, training of the clinical and laboratory staff, personnel effort in addition to other costs associated with applying this service. However, this cost is supposed to be counterbalanced by the positive patient outcomes such as decreased incidence of adverse drug reactions and decreased length of hospitalization. Pharmacoeconomic analysis of the impact of the TDM in adult patients with tonic-colonic epilepsy showed that application of TDM improved the seizure control, reduced the incidence of adverse reactions, increased the chance for remission, improved the patient’s earning capacity, and decreased the cost for patients due to lower hospitalization per seizures (12). Also, a meta-analysis of TDM studies showed that TDM is beneficial for patients taking digoxin and theophylline. This study also showed that TDM service run by clinical pharmacists had higher chance of achieving desirable drug concentrations and decreased the frequency of inappropriately collected samples (13). Several studies also examined the role of TDM in optimization of aminoglycoside therapy by targeting the maximum and minimum blood concentrations associated with increase efficacy and reducing the incidence of aminoglycoside toxicity. These studies demonstrated high cost-effectiveness of dose individualization of aminoglycosides using TDM (14). 21.8 Summary

• TDM is an important service that involves using blood drug concentration measure• • • • •

ment for optimization of drug therapy and ensuring that the blood drug concentration is within the therapeutic range all the time. The therapeutic range of a drug is the range of drug concentrations associated with the highest probability of therapeutic effect and lowest probability of adverse effect. Drugs with low therapeutic index, drugs with high variability in their pharmacokinetic behavior, and drugs used in high-risk patients are good candidate for TDM. The analytical technique used for sample analysis should be sensitive, accurate, and rapid. TDM can be provided by pharmacists who are trained in basic and clinical pharmacokinetics. Providing TDM service is cost-effective as it can speed identification of the optimum dosage regimen and decrease the incidence of drug adverse effects.

Therapeutic Drug Monitoring 387 References 1. Gross AS “Best practice in therapeutic drug monitoring” (2001) Br J Clin Pharmacol; 52(Suppl 1): 5S–10S. 2. Rowland M and Tozer TN “Clinical pharmacokinetics and pharmacodynamics: Concepts and applications” 4th Edition (2011), Lippincott Williams & Wilkins, Philadelphia, PA, USA. 3. Kang JS and Lee MH “Overview of therapeutic drug monitoring” (2009) The Korean J Intern Med; 24:1–10. 4. Aronson JK and Hardman M “ABC of monitoring drug therapy: Measuring plasma drug concentrations” (1992) BMJ; 305:1078–1080. 5. Reynolds DJ and Aronson JK “ABC of monitoring drug therapy: Making the most of plasma drug concentration measurements” (1993) BMJ; 306:48–51. 6. Llorente Fernández E, Parés L, Ajuria I, Bandres F, Castanyer B, Campos F, Farré C, Pou L, Queraltó JM and To-Figueras J “State of the art in therapeutic drug monitoring” (2010) Clin Chem Lab Med; 48:437–446. 7. Mann K, Hiemke C, Schmidt LG and Bates DW “Appropriateness of therapeutic drug monitoring for antidepressants in routine psychiatric inpatient care” (2006) Ther Drug Monit; 28:83–88. 8. Hammett-Stabler CA and Johns T “Laboratory guidelines for monitoring of antimicrobial drugs. National academy of clinical biochemistry” (1998) Clin Chem; 44:1129–1140. 9. Chan K and Beran RG “Value of therapeutic drug level monitoring and unbound (free) levels” (2008) Seizure; 17:572–575. 10. Sproule B, Nava-Ocampo AA and Kapur B “Measuring unbound versus total valproate concentrations for therapeutic drug monitoring” (2006) Ther Drug Monit; 28:714–715. 11. Bauer LA “Applied clinical pharmacokinetics” (2008) McGraw-Hill Companies Inc., New York, NY, USA. 12. Rane CT, Dalvi SS, Goptay NJ and Shah PU “A pharmacoeconomic analysis of the impact of therapeutic drug monitoring in adult patients with generalized tonic-clonic epilepsy” (2001) Br J Clin Pharmacol; 52:193–195. 13. Ried LD, Horn JR and McKenna DA “Therapeutic drug monitoring reduces toxic drug reactions: A meta-analysis” (1990) Ther Drug Monit; 12:72–78. 14. Streetman DS, Nafziger AN, Destache CJ and Bertino AS Jr. “Individualized pharmacokinetic monitoring results in less aminoglycoside-associated nephrotoxicity and fewer associated cost” (2001) Pharmacotherapy; 21:443–451.

22 Pharmacometric Applications in Drug Development and Individualization of Drug Therapy

Objectives After completing this chapter, you should be able to:

• Define the term pharmacometrics. • Describe the basic steps for developing a PBPK model. • Demonstrate how the PBPK models can be used in inter-dose, inter-route, inter-tissue, and interspecies extrapolation.

• Discuss some applications of the PBPK models in drug development and clinical practice. • Describe the basic components of the population pharmacokinetic model. • Discuss applications of the population pharmacokinetic models during the drug development process for drug use decisions in different patient populations.

• Describe the general approach for applying the model-based TDM for the individualization of drug therapy.

22.1 Introduction Pharmacometrics is the area of pharmacokinetics that involves the utilization of mathematical and statistical modeling approaches to understand, characterize, and predict the drug dose-concentration-effect relationship in different populations. This field has been evolving over the years due to the advancement in the data analysis approaches and the availability of specialized data analysis software. Initially, pharmacokinetic models were developed to describe the drug concentration-time profile after drug administration, and different pharmacodynamic models were developed to characterize the drug concentration-effect relationship. Then integration of the pharmacokinetic and pharmacodynamic models allowed the prediction of the time course of the drug effect after the administration of different dosing regimens. Introduction of the population analysis and modeling approaches allowed the estimation of the drug pharmacokinetic and pharmacodynamic parameters from sparse samples obtained from a large number of patients with different characteristics. In addition to the estimation of the drug parameters, the population methods allowed identification of the factors that can affect the drug pharmacokinetics and pharmacodynamics behavior, and the variability in the estimated parameters (1). The pharmacometrics principles can be applied during the drug development process and in optimization of patient pharmacotherapy. A large amount of data are usually generated during the drug development process. Application of innovative modeling approaches can extract very valuable knowledge from these data. This can result in speeding DOI: 10.4324/9781003161523-22

Pharmacometric Applications 389 and decreasing the cost of the drug development process and optimization of clinical drug use (2). The different pharmacokinetic-pharmacodynamic modeling approaches have been utilized during the preclinical and clinical phases of drug development to characterize the drug dose-exposure-response relationship. While the physiologically based pharmacokinetic (PBPK) models have been used during the preclinical phase of drug development to describe the drug exposure-response relationship in specific organ, to predict the drug pharmacokinetic behavior in humans, and to estimate the first-in-human drug dose. The knowledge gained from population modeling during clinical trials is used to develop dosage recommendations for the different patient populations. Also, population modeling can be applied to the data obtained during therapeutic drug monitoring (TDM) for individualization of drug therapy and optimization of therapeutic outcome. 22.2 Pharmacometric Applications during the Preclinical Phase of Drug Development The primary objective of the preclinical phase of drug development is to select safe and effective potential drug candidates to be tested in humans. These drug candidates should have favorable biopharmaceutical, pharmacokinetics and pharmacodynamic characteristics, and low cost of manufacturing. The pharmacokinetic-pharmacodynamic modeling approaches that integrate the pharmacokinetic and the pharmacodynamic models are usually utilized to characterize the exposure-response relationship. Also, the development of PBPK models during the preclinical phase provides valuable information about the drug exposure-effect relationship in the entire body and in specific organ. The PBPK models can be valuable in predicting the drug pharmacokinetic behavior in humans, predicting the expected variation in the drug pharmacokinetic behavior due to physiological and pathophysiological changes, and estimating the first dose in humans that is used in the early clinical trials. Furthermore, application of scale-up to human approaches to the knowledge generated from the preclinical pharmacokinetic studies can expedite the design and execution of the initial clinical trials. The principles of pharmacokinetic-pharmacodynamic modeling were discussed in Chapter 19, and the PBPK modeling will be discussed in this chapter. 22.2.1  Physiologically Based Pharmacokinetic Models

PBPK models consist of a series of compartments; each of them represents one or more organ, tissue, or body space (3). These models utilize organ blood flow, organ size, and drug affinity to determine the amount of drug delivered and distributed to each organ. Inclusion of information about the elimination efficiency of the eliminating organs allows the characterization of the time course of the drug concentration in the blood and different organs and the estimation of the exposure of different organs to the drug. The information needed to construct the PBPK models include anatomical and physiological parameters that are readily available in the literature for humans and different animal species and are independent of the drug under investigation. This is in addition to drugspecific information that can be determined experimentally, such as the physicochemical properties that can affect drug distribution and elimination. Figure 22.1 represents an example of the PBPK models that consists of different compartments representing different organs, and the last compartment represents all the remaining organs and tissues. The venous blood that returns from all organs to the heart is pumped entirely to the lung for oxygenation. The oxygenated blood returns to the heart

390  Pharmacometric Applications

Figure 22.1 A representative example of the whole-body PBPK models consists of nine different compartments. Each of the first eight compartments represents one organ with the ninth compartment representing all the remaining organs and tissues. The term Q represents the blood flow, with the subscript indicating a specific organ. The blood flow entering the organ is equal to the blood flow leaving the organ. The arrows coming out of the liver and the kidney indicate drug elimination by these organs, while the arrows in the closed circulation loop indicate the direction of blood flow (4).

where it is pumped to all organs with different organs receiving different blood flow rates, but the blood flow to each organ does not change with time. The blood flow into the organ is equal to the blood flow out of the organ. The model in Figure 22.1 assumes that the drug is eliminated by the liver and the kidney. The blood flow to the liver as presented in the model comes partially from the arterial blood and partially from the blood leaving the small intestine via the portal vein. So, the general structure of the model is based on the real anatomical and physiological information. 22.2.1.1  Physiologically Based Pharmacokinetic Model Development

The steps involved in the development of PBPK models include the selection of the components of the model and writing the mathematical expressions that describe the change in the drug concentrations in all components of the model. This is followed by model

Pharmacometric Applications 391 parameterization by including the anatomical and physiological parameters and parameters related to the drug properties. The model is then used to simulate the drug concentration-time profile in all model components and the simulated data are compared with the experimentally obtained information. Modification of the model may be necessary until the simulated drug concentration-time profiles in the different model components agree with the experimental data. 22.2.1.1.1  MODEL COMPONENTS

The general rule is to include all the relevant organs and body spaces while using the lowest number of organs. So, the model should include all organs and tissues where the drug can exert its pharmacological and adverse effects, eliminating organs, organs where drug absorption takes place, organs and tissues that receive a large amount of the drug, and body spaces that are easily sampled (4). For example, models developed for gaseous anesthetic agents should include the lung that is the main site for inhalation and excretion of these drugs as in Figure 22.2A. Adipose tissues should be included in the models for highly lipophilic drugs or chemicals because these drugs are extensively distributed in

Figure 22.2  Sections of the whole-body PBPK models that demonstrate the presentation of (A) lung inhalation and excretion of drugs, (B) enterohepatic recycling, and (C) renal and hepatic drug elimination.

392  Pharmacometric Applications fatty tissues. PBPK models for drugs that undergo enterohepatic recycling should account for this process as presented in Figure 22.2B (5). Orally administered drugs should include the intestine as part of the model while transdermally applied drugs should include the skin in their model. The kidney should be included in the models for renally excreted drugs, and the liver is included in the models for drugs that are eliminated by hepatic metabolism as in Figure 22.2C (6). 22.2.1.1.2  MATHEMATICAL PRESENTATION OF THE MODEL

Once the model general structure is defined, all the relevant pharmacokinetic processes that affect the pharmacokinetic behavior of the drug in all parts of the body are expressed by mathematical equations. This should include a set of differential equations that describe the rate of change of the concentration of the drug in all organs and tissues included in the model. The model equations should describe all the pharmacokinetic processes that affect the drug concentration-time profile in the body (4, 7). For example:

• Non-eliminating organs

The rate of change of the drug amount in a non-eliminating organ is the difference between the amount of drug reaching the organ and the amount of drug leaving the organ to the venous circulation as in Eq. 22.1. dA T = QTCa − QTC v, T(22.1) dt

where AT is the amount of the drug in the tissue, QT is the blood flow rate to the tissue, Ca and Cv,T are the drug concentration in the arterial blood and the venous blood for the tissue, respectively. Similarly, the differential equation for the rate of change in the drug concentration in the tissue can be expressed as in Eq. 22.2. VT

dC T = QTCa − QTC v, T (22.2) dt

where CT is the concentration of the drug in the tissue, VT is the volume of the tissue. One differential equation should be written for each non-eliminating organ. It should be noted that the drug concentration in the arterial blood for all organs is similar, but the drug concentration in the venous blood of different organs is different. This is because the venous blood concentration depends on the affinity of the drug to the organ constituents. When the drug distribution to the organ follows the flow-rate limited model, the drug concentration in the venous blood of each organ can be determined from the tissue to blood distribution coefficient (Kp) and the tissue drug concentration. The rate of change of the drug concentration in the tissue can be written as in Eq. 22.3. VT

C dC T = QTCa − QT T (22.3) Kp dt

For example, the differential equation that describes the rate of change of the drug concentration in the muscles can be written as in Eq. 22.4. VM

C dC M = QMCB − QM M (22.4) KpM dt

Pharmacometric Applications 393 where VM is the volume of the muscles, CM is the muscle drug concentration, QM is the muscle blood flow, CB is the blood drug concentration, and KpM is the muscle to blood distribution coefficient (KpM = CM/Cv,M). • Eliminating organs The rate of change of the amount of the drug in an eliminating organ is the difference between the amount of drug reaching the organ, and the sum of the amount of drug leaving the organ to the venous circulation and the amount of the drug eliminated by the organ as in Eq. 22.5. VT

dC T = QTCa − QTC v, T − rate of drug elimination(22.5) dt

So, the equation for an eliminating organ like the kidney can be written as in Eq. 22.6. VK

C dC K = QKCB − QK K − CL RC K(22.6) KpK dt

where VK is the volume of the kidney, CK is the kidney drug concentration, QK is the kidney blood flow, CB is the blood drug concentration, KpK is the kidney to blood distribution coefficient, and CLR is the renal clearance. In this equation, the two kidneys are considered one organ, so the volume and blood flow for both kidneys should be used. • Drug administration IV bolus drug administration involves a direct introduction of the entire dose to the venous blood, so the initial venous drug concentration in the model is determined by dividing the dose by the volume of the blood. While for constant rate IV infusion, the drug concentration in the venous blood is determined from the infusion rate and the drug concentration in the venous blood from the different organs. So, the rate of change in the drug concentration in the blood (CB) can be expressed by Eq. 22.7. VB

dCB = K0 + dt

∑Q C T

v,T

− QBCB (22.7)

where K0 is the rate of drug infusion, ΣQTCv,T is the sum of the products of the blood flow and the venous drug concentration for all organs, and QB is the total blood flow to all organs. While after oral drug administration, the rate of drug absorption can be included as in Eq. 22.8, assuming first-order drug absorption. dAG = ka FDe− kat (22.8) dt where AG is the amount of the drug in the GIT at the absorption site, ka is the first-order absorption rate constant, F is the fraction of dose absorbed to account for incomplete absorption, and D is the dose. The mass balance equation for the intestine in the model can be expressed as in Eq. 22.9. VI

C dC I = QIC I + ka FDe− kat − QI I (22.9) KpI dt

where VI is the volume of the intestine, CI is the drug concentration in the intestine, QI is the blood flow to the intestine, and KpI is the intestine to blood distribution

394  Pharmacometric Applications coefficient. Drug absorption by other routes of administration can be described by mathematical equations and included in the model. • Distribution and elimination processes Mathematical expressions can be included to account for drug protein binding, and drug distribution by passive diffusion or by carrier mediated transport. Also, firstorder or saturable elimination from different organs can be described by mathematical equations in the model. 22.2.1.1.3  MODEL PARAMETERIZATION

In this step, estimates for all the model parameters should be included in the equations that describe the model. Anatomical and physiological model parameters such as tissue and organ volumes, blood flow rate, cardiac output, and breathing rate are obtained from the literature (8, 9). Drug physicochemical parameters such as tissue partition coefficient, membrane permeability, biochemical parameters such as the enzymatic metabolic activity, plasma protein binding, and blood-to-plasma distribution ratio can be determined experimentally using in vitro or in vivo techniques. 22.2.1.1.4  MODEL SIMULATION AND PARAMETER ESTIMATION

Specialized software is used to simulate the time course of the drug concentration in all tissues and organs utilizing the mathematical equations that describe the PBPK model after including the estimated model parameters (7). The simulations are used for model evaluation by comparing the simulated concentrations-time profiles and the experimentally determined concentration-time profiles in the different organs. The model may be modified and refined based on the results of the simulations. Once the model presentation is optimized, it will be possible to simulate the drug pharmacokinetic behavior under different conditions that are difficult to duplicate experimentally, for example, when the experimental subjects are exposed to the risk of toxicity, when long-term exposure to the drug is investigated, or when the cost of performing the experiment is very high. PBPK models can also be used for parameter estimation. In this case, it is required to experimentally determine the drug concentration-time profile in several organs and tissues that are included in the model. Then the equations that describe the drug concentration in these organs are fitted to the experimental drug concentration-time data to estimate the model parameters (10). 22.2.1.1.5  MODEL VALIDATION

PBPK model validation is important to ensure that the model accounts for all the determinants and processes that affect the pharmacokinetic behavior of the drug. Agreement between the model predicted plasma drug concentrations, tissue concentrations, and urinary excretion rate data and experimental data indicates an adequacy of the model and increases confidence in the model for extrapolation and prediction. The appropriate diagnostic and statistical tests should be used in this comparison. It is also important to determine the extent of change in the model predictions due to the change in the value of each parameter to determine the sensitivity of the model to changes in the different parameters.

Pharmacometric Applications 395 22.2.1.2  Applications of the PBPK Models

The PBPK models have many applications that can be very useful in drug development and during therapeutic drug use. These models provide valuable information that can be used in determining the exposure-response relationship in different organs and tissues, interspecies scaling, and estimation of the first-in-human dose for use in phase I clinical trials. 22.2.1.2.1 PREDICTION OF THE DRUG PHARMACOKINETIC BEHAVIOR UNDER DIFFERENT CONDITIONS

The PBPK models can be used to describe general or specific aspects of the drug pharmacokinetic behavior under different conditions. This includes conditions other than those under which the model was developed. The models can be used for inter-dose extrapolation, where the model developed using a certain dose of the drug can be used to predict the drug tissue distribution and disposition after administration of different doses. Also, PBPK models developed following a certain route of administration can be used to predict the drug pharmacokinetic behavior after different routes of administration. Furthermore, the drug concentration-time profiles in the tissues and organs included in the PBPK models can be utilized to predict the drug profile in other tissues and organs. Similarly, the pharmacokinetic behavior of a drug can be predicted from PBPK models developed for other chemically related drugs (11). 22.2.1.2.2  INTERSPECIES SCALING

The general principles of this scaling approach involve the development of the PBPK model for the drug in one species usually called the test species, and to use this model to predict the drug pharmacokinetic behavior in different species. The goal is usually to use the drug pharmacokinetic properties in experimental animals to predict the drug pharmacokinetic behavior in humans. This usually involves replacing the physiological, anatomical, and biochemical parameters for the test species in the PBPK model by the parameters for the species of interest (12, 13). For example, if the goal is to scale the drug pharmacokinetic behavior to humans, the different organ volumes, blood flow, and cardiac output in humans are incorporated in the model. Also, drug-specific parameters related to drug absorption, metabolic rate, blood-to-plasma ratio, and protein binding in humans should be incorporated in the model. Once the PBPK model is scaled to the species of interest, it must be validated by comparing the model-predicted concentrations with the experimentally determined concentrations. Agreement between the predicted and experimentally determined blood drug concentration can be a good indication of successful model scaling. 22.2.1.2.3  ESTIMATING THE FIRST-TIME-IN-HUMAN DOSE

The first step in the clinical phases of drug development involves the selection of the starting dose to be administered to humans based on the information generated during the preclinical experiments. The objective usually is to select the starting dose that should be administered as a single escalating dose to healthy volunteers to evaluate the tolerability of the new compound and to obtain some pharmacokinetic information if possible. There are four main methods that are commonly used to select the starting dose

396  Pharmacometric Applications for noncytotoxic drugs (14). The pharmacokinetically guided approach is the method of choice since it utilizes the systemic exposure instead of dose for extrapolation from animals to humans. The desired systemic exposure, usually measured as the AUC or Cpmax, in humans is defined as the systemic exposure corresponding to the no-observed adverse effect level (NOAEL). When the AUC corresponding to the NOAEL is available for more than one species, the species with the lowest AUC is used. The clearance of the drug in humans is predicted using the physiological approach or any other approach from the estimated drug clearance in different animal species. Then the starting dose can be calculated as in Eq. 22.10. Starting dose = AUC preclinical × CL humans(22.10) where AUCpreclinical is the AUC associated with NOAEL determined in experimental animals, and CLhumans is the predicted drug clearance in humans using animal scale-up. There are few points that should be considered when utilizing the pharmacokinetically guided approach in calculating the starting dose in humans. This approach assumes that the concentration-effect relationship is similar in humans and the other species that may not be true for all drugs. When there is large difference in the protein binding between species, it will be more appropriate to use the free drug concentration to describe drug exposure. Also, the approach described above assumes linear pharmacokinetics, and the parent drug is the only active moiety. So, the estimated dose should be corrected in the case of evidence of nonlinearity or the presence of active metabolites. Moreover, the differences in the rate and extent of drug absorption from the animal and human formulations should be considered when calculating the dose. The other methods for estimating the starting doses in humans are empirical. These include the dose by a factor method that utilizes the animal dose associated with NOAEL multiplied by a safety factor. The other is the similar drug approach that can be used when clinical data are available about another compound from the same pharmacological class as the drug under investigation. However, the comparative approach utilizes two or more methods to estimate the starting dose for humans and then comparing the results to come to the optimal starting dose. 22.2.1.2.4  APPLICATIONS OF PBPK MODELS IN CLINICAL DRUG USE

PBPK models have been developed to describe anticancer drug distribution to normal and cancerous tissues. The primary aim of these models is to quantify the normal and cancerous tissue exposure to the drugs and predicting the therapeutic and adverse effects of the drug. This has been very useful for the optimization of the anticancer drug therapy by developing new approaches for maximizing the cancerous tissue drug exposure while minimizing the normal tissue drug exposure. Also, the drug distribution to the fetus from the mother through the placenta and to neonates through breast milk has been described by developing PBPK models for pregnant and lactating women. These models can quantify the fetal and neonatal drug exposure and characterize the time course of the drug concentration in the different tissues and organs of the fetus and the neonate. This is very important to assess the possible pharmacological and toxicological effects that can result from the fetal and neonatal drug exposure. Furthermore, PBPK models have been used to describe the tissue distribution of several antibiotics. This can be useful since the drug concentration in the infected tissue is the important determinant of the therapeutic effect

Pharmacometric Applications 397 of these drugs. The pharmacokinetic behavior of many other drugs such as anesthetics, sedatives, antibodies, and large molecular weight oligonucleotides has been described using PBPK models. 22.3 Pharmacometric Applications during the Clinical Phases of Drug Development The main objective of phase I clinical trial is to identify the tolerated dose of the drug. This is in addition to obtaining information about the general pharmacokinetic behavior of the drug after single and multiple doses. While, phase II clinical trials are performed to prove that the drug is effective and to identify the optimal effective dose of the drug that produces the desired therapeutic effect with minimum adverse effects. During these early phases of clinical development, other useful information about the drugs can be obtained such as the dose-concentration-response relationship in humans, drug-food interactions, drug-drug interactions, and pharmacokinetics in some patient subpopulations. However, the number of subjects enrolled in phases I and II clinical trials is usually not large. Phase III clinical trials which are usually controlled and randomized trials, are performed to establish the safety and efficacy of the drug in different patient populations. A large number of patients are usually enrolled in phase III clinical trials, and the results of these trials represent a crucial and essential component of the new drug application to regulatory authorities. Phase IV clinical trials are performed after drug marketing to explore possible use of drug combination, dose optimization in patient subpopulations, or to investigate possible new drug indication. A large amount of data are generated while performing the clinical trials with very valuable knowledge hidden within these data. The innovative data analysis, modeling, statistical, graphical, and simulation approaches utilized in pharmacometrics can be very valuable in extracting useful knowledge from these data. The application of the population data analysis approach represents significant advancement in pharmacometrics for analyzing the data obtained during the drug development process (1). The population approach in data analysis was first applied on sparse drug concentrations measured during TDM in patients receiving the drugs to estimate the individual pharmacokinetic parameters and to optimize dosing regimen (15). This data analysis approach was expanded to include models that describe the drug concentration-response relationship and then became an essential tool in the whole drug development process. Population data analysis can be applied on the results of data-rich studies and studies with sparse data for a precise estimation of the pharmacokinetic and pharmacodynamic parameters, evaluation of the parameters’ variability, and determination of the factors affecting this variability. This information can expedite the drug development process, support dosage recommendation for the different patient populations, and for individualization of drug therapy. 22.3.1  Population Pharmacokinetic Analysis

The population pharmacokinetic analysis approach involves analyzing all the data from individuals within a population simultaneously using complex statistical procedures utilizing specialized software. It is important to ensure that all the procedures used in this analysis are appropriate, well documented, and in compliance with the regulatory guidance for performing population pharmacokinetic studies. This includes data adequacy,

398  Pharmacometric Applications data handling, population pharmacokinetic model, modeling software, model validation, and reporting of the study results (16). 22.3.1.1  Data Consideration for Population Analysis

As mentioned previously, population data analysis can be used to analyze sparse data obtained from a large number of patients and to analyze data generated from pharmacokinetic studies in fewer number of individuals with extensive sampling. The data required for any pharmacokinetic study include the drug dose, time of dose, time of each sample, and drug concentration in each sample. For population pharmacokinetic studies, the data should also include information about the factors that can affect the drug pharmacokinetic behavior for each participant. The data used in the population analysis should be organized, tabulated, and the procedures followed in the case of using concentrations below the limit of quantitation of the analytical procedures, missing data, and outliers should be acceptable and documented (17). 22.3.1.2  The Population Pharmacokinetic Models

The population model consists of the structural model, the statistical model, and the covariant model. The structural model is the mathematical expression that describes the pharmacokinetic model. The statistical model describes all sources of explained and unexplained variability in the structural model parameters, while the covariate model describes the intrinsic and extrinsic factors that can influence the drug concentration-time profile in different patients (16, 17). The structural model: This is the mathematical expression that describes the pharmacokinetic behavior of the drug in the body. For example, one- or two-compartment pharmacokinetic model with the input function depending on the route of drug administration. This mathematical expression describes the relationship between time as the independent variable and drug concentration as the dependent variable. Equation 22.11 describes the drug concentration-time profile of a drug that follows one-compartment pharmacokinetic model with first-order elimination after single IV dose. Cp =

dose − e Vd

CL T t Vd

(22.11)

where Cp is the plasma drug concentration at any time, CLT is the total body clearance, and Vd is the volume of distribution. This equation describes an exponential relationship between drug concentration and time, as it assumes linear pharmacokinetic behavior which means that the pharmacokinetic parameters CLT and Vd are constant and do not change with time or with the change in dose. The choice of the model is usually based on prior information that may be available about the drug under investigation. Drug concentration-time plots constructed using the available data can also be used to suggest the initial pharmacokinetic model that should be used in the analysis. When information is available, it suggests that some patients’ factors can affect the drug pharmacokinetic parameters, and these factors can be incorporated in the structural model. For example, the creatinine clearance can affect the drug CLT of drugs that are eliminated mainly unchanged by the kidney, and the body weight can affect the drug Vd. So, the CLT term can be multiplied by a factor related to the

Pharmacometric Applications 399 kidney function estimated from the patient creatinine clearance, and Vd can be normalized for body weight in the structure model (16, 17). The statistical model: This is the model that describes all sources of variability in the pharmacokinetic parameters.

• The model assumes that there is random variability (random effect) in the estimates

for the individual pharmacokinetic parameters around the population mean value. It is also possible to assume that the estimates for some pharmacokinetic parameters in all subjects are the same (fixed effect), and they are equal to the population parameter estimates. • The between-subject variability is the random variation in the pharmacokinetic parameters between subjects within a given patient population. The model also describes the distribution (normal, log-normal, or other) of the random variation in the individual parameter estimates. • The within-subject variability describes the pharmacokinetic parameter variability in the same subject when the same subject receives the same drug in different occasions. This is included only when information is available about drug administration to the same subject in different occasions. • The residual unexplained variability that is determined from the difference between the observed and the model predicted concentrations. This can result from analysis error, model misspecification, and error in the sample or drug administration time. The model also describes the statistical distribution of the residual unexplained difference between the observed and predicted concentrations. The covariate model: The pharmacokinetic parameters that determine the drug pharmacokinetic behavior can be affected by factors, known as covariates. This includes intrinsic factors as age, weight, gender, kidney function, liver function, and pharmacogenetic factors, and extrinsic factors such as concomitant diseases and concurrent drug use. A graphical presentation of pharmacokinetic indicators of drug exposure such as AUC and Cpmax versus patients’ demographic factors, laboratory tests, or pathological conditions can give insight about the important covariates that can be incorporated in the covariates model. There are standard statistical procedures that should be followed to ensure the inclusion of all important covariates that can influence the drug pharmacokinetic parameters in the model. This covariate model component of the population pharmacokinetic model should quantitatively describe the relationship between the covariates and the pharmacokinetic parameters to account for part of the between-subject variability. The between-subject variability in the pharmacokinetic parameter estimate in the entire population is usually large. The introduction of covariates divides the study population to subpopulations with reduction in the between subject variability within each of the subpopulations. This is very important to support dosage recommendation decision in the different patient subpopulations (16, 17). 22.3.1.3  Statistical Analysis and Parameter Estimation

Specialized population pharmacokinetic data analysis software is usually utilized to estimate the model parameters that can minimize the deviation of the model-predicted drug concentrations from the observed concentrations. Statistically, the parameter values that

400  Pharmacometric Applications will minimize the maximum likelihood estimation objective function value (OFV) will be selected as the best estimates for the model parameters. Because of the complexity of the model, it is usually required to provide a set of initial values for the model parameters. This is to help the analysis software to reach the set of parameter values that will minimize the OFV. 22.3.1.4  Model Evaluation and Diagnostics

Model adequacy and model assumptions are usually evaluated at all stages of the population analysis procedures. All data analysis software calculates the standard error and the 95% confidence interval for each parameter estimate, which are important in determining the precision and certainty of the estimates. Small standard error and narrow confidence intervals indicate precise parameter estimation. Simulations are important to evaluate the developed model and for testing the model assumptions. Model evaluation can be performed by using a subset of the data used to develop the model or a new data set and compare the model simulated drug profile and the observed drug concentrations. Graphical diagnostics are also useful in model evaluation. Plots such as scattered plot of the observed drug concentrations around the model-predicted profile, the observed versus model-predicted drug concentrations, and residual plots such as weighted residuals versus predicted concentrations are examples of the graphical diagnostics (17). The measured drug concentrations should be randomly distributed around the model-predicted profiles. Plots of the observed versus model-predicted drug concentrations should be linear with slope equal to unity, while the weighted residuals should be randomly distributed and have relatively equal magnitude at all values of the predicted concentrations. 22.3.1.5  Reporting of the Population Pharmacokinetic Analysis Results

The final report of the population analysis should include detailed information about all the procedures used in the analysis. This includes the goal of the analysis, the source of data, the number of subjects, and the number of samples per subject, and handling of missing data and outliers. The methods section should include information about the analytical procedures used for sample analysis and its lower limit of quantitation, the software used in the population analysis, the rationale for choosing the structural model, and the method used to build the covariate model. Evidence of model validation that includes statistical tests, simulation runs, and graphical plots should be included. The discussion of the results should address the clinical relevance of the obtained results and covariates effect on drug pharmacokinetics, also, the agreements between the obtained results and the knowledge available about the drug under investigation (18). 22.3.1.6  Application of Population Pharmacokinetic Analysis in Drug Development

The following are some of these applications (19):

• Population pharmacokinetic analysis identifies the factors that can influence the drug pharmacokinetic behavior. This is useful for selecting the appropriate dosage regimens that are used in clinical trials to decrease the variability in drug response in different patients. For example, the existence of strong correlation between pharmacokinetic

Pharmacometric Applications 401

•

•

•

•

parameters that are indicators of drug exposure such as AUC or Cpmax and the body weight, usually suggests the calculation of the dosage regimen based on the patient body weight. Validated population pharmacokinetic models can be used for predicting drug exposure following drug administration using dosage regimens that have not been investigated previously in clinical trials. This includes different drug dose, different frequencies of drug administration, or administration of a loading dose. Population pharmacokinetic models and simulations provide information about the magnitude of effect of the different covariates on the drug pharmacokinetic behavior. This information can be used to determine the number of patients in each subpopulation that should be included in the study to provide sufficient power to detect covariate effect. The population pharmacokinetic analysis provides information about the parameters that are good indicators for drug exposure such as AUC, Cpmax, and minimum drug concentration during multiple drug administration. These exposure measures can be used for subsequent exposure-response analysis after including measures of an appropriate pharmacodynamic biomarker or clinical endpoint. This is useful in predicting the expected drug response after administration of different dosage regimens. Population pharmacokinetic models and simulations developed using adults’ data can be used to select the optimum dosage regimens to be used in investigations involving pediatric patients. This can be done by considering the developmental changes across different ages that affect the drug pharmacokinetic behavior and information about the absorption of the drug from the pediatric formulation. Since population pharmacokinetic analysis can be applied using sparse samples, this analysis can be appropriate for data obtained from pediatric patients when frequent sampling is not possible.

22.3.1.7 Application of Population Pharmacokinetic Analysis for Drug Use Decisions in Drug Labeling

The following are some of these applications (19):

• Results of population pharmacokinetic analysis provide information about the general

pharmacokinetic behavior of the drug, and the factors that can affect the drug pharmacokinetics. This is important for recommending safe and effective dosing regimens for the general population and for special populations. • The information about drug exposure-response relationship is used to establish a range of optimal drug exposure that is associated with the maximum therapeutic effect with minimum adverse effect. If the effect of a covariate on the drug pharmacokinetic behavior is small, and the drug exposure is maintained within the optimal range, recommendation of dosage adjustment will not be necessary, while if the effect of another covariant takes drug exposure outside this optimal range, recommendation of dosage adjustment in the presence of this covariate will be necessary. • Drug use recommendations related to concomitant drug use and the potential for clinically significant drug-drug interactions can be reached from the results of population pharmacokinetic studies. However, population pharmacokinetic studies used to evaluate drug-drug interactions should examine each individual drug not a class

402  Pharmacometric Applications of drugs, and the number of subjects receiving the interacting drug in the study should be adequate. The mechanism of drug interaction should be investigated by examining the effect of each drug on the different pharmacokinetic processes for the other drug. 22.4  Pharmacometric Applications in Clinical Drug Use TDM involves utilization of blood drug concentration as a guide for the optimization of dosage regimens in patients. Drug candidates for TDM are drugs with established exposure-effect relationship that have a narrow margin of safety or high variability in their pharmacokinetic parameters. The use of the average recommended dose of these drugs in all patients most likely will achieve drug concentrations that are different from the target concentrations in many patients. TDM can be as simple as measuring the steady-state plasma drug concentration during multiple drug administration followed by an empirical adjustment of dose to achieve the desired concentration target, while model-based TDM involves using pharmacokinetic models for the optimization of dosage regimens. 22.4.1  Model-Based Therapeutic Drug Monitoring

The use of pharmacokinetic models in TDM is appropriate because these models describe the drug dose-concentration relationship. The process starts with the development of a population pharmacokinetic model to estimate the drug pharmacokinetic parameters that can be used for the selection of the initial dosage regimens in each patient. Then plasma drug concentration can be measured after the initiation of drug therapy. Finally, the measured concentration together with the model parameter estimates can be used for adjusting drug dosage regimens to achieve the desired drug concentrations in individual patients. The following is a brief discussion of one of the approaches used in model-based TDM; however, there are other approaches that are used in clinical practice for dose individualization (20). 22.4.1.1  Model Development

The drug pharmacokinetic behavior is usually different in different patients due to variation in the individual pharmacokinetic parameters that can be explained at least partly by different covariates. Accumulating information about the behavior of a drug in different individuals within a patient population represents the bases for developing the population pharmacokinetic model. When caring for a new patient, the model can be used to predict the pharmacokinetic behavior in this new patient. This model prediction is based on information from the drug pharmacokinetic behavior in similar patients. The predicted individual pharmacokinetic parameters can be used to select the appropriate initial dosage regimen for the new patient that should achieve the desired drug concentration. Although it is not possible to determine the exact value for the individual pharmacokinetic parameters, the individual parameters can be estimated with different degrees of certainty. Several modeling approaches have been used to develop the population pharmacokinetic models. This includes the parametric methods that have some assumptions about the distribution of the random variation in the individual parameter estimates. Also, the nonparametric modeling method does not make any assumptions about the

Pharmacometric Applications 403 distribution of the random variation in the individual parameter estimates. Whatever the method used in creating the population model or estimating the individual parameters, accurate prediction of the individual pharmacokinetic parameter is important for an accurate selection of the appropriate dosage regimen. 22.4.1.2  Monitoring Drug Concentration

The initial dosage regimen is selected for a particular patient based on the drug pharmacokinetic parameters in patients with similar characteristics using the population model. After that, it is important to check the drug pharmacokinetic behavior in the patient and adjust the dosage regimen if needed. It is necessary to wait for five half-lives until steady state is achieved before evaluating the drug plasma concentration. The time during the dosing interval when the plasma sample is obtained should be determined. This can be different for different drugs but for most drugs, plasma sample that reflects the lowest drug concentration at the end of the dosing interval is usually obtained. The therapeutic range for the drug, which is the range of drug concentrations associated with the highest probability of achieving therapeutic effect with minimum adverse effect in most patients, is usually used as the target concentration range after the initiation of the drug therapy. The drug concentration measured, while the patient is taking the initial dosage regimen, should be evaluated in conjunction with the patient clinical condition. The patient clinical response and the measured drug concentration are used to identify a new target plasma concentration from the general range for this specific patient. The measured plasma drug concentration is used in conjunction with the pharmacokinetic population model parameter values to obtain better estimates for the pharmacokinetic parameters in the patient. Statistical methods such as the Bayesian nonlinear least squares regression are used to estimate the most likely individual pharmacokinetic parameter values. While doing this, the procedures consider the parameter estimates in similar patients and their associated variability, and the measured drug concentration and its variability. The pharmacokinetic parameters estimated using these statistical procedures are more precise and can predict the drug pharmacokinetic behavior in the patient better. These estimated pharmacokinetic parameters can be used for adjusting the dosage regimen in individual patients (20). 22.4.1.3  Dosage Regimen Design

The dosage regimen is designed using the individual pharmacokinetic parameters estimated from the measured drug concentration and the pharmacokinetic population model. This regimen should achieve the desired individual patient target plasma drug concentration, determined from the measured drug concentration and evaluation of the patient clinical condition. Periodic evaluation of the dosage regimen by measuring the plasma drug concentration and evaluation of the patient clinical condition is necessary to ensure that the patient therapeutic goals are achieved. 22.5 Summary

• Application of the pharmacometric principles can lead to speeding and decreasing the

cost of the drug development process and can also help in the optimization of clinical drug use.

404  Pharmacometric Applications

• Pharmacokinetic-pharmacodynamic modeling is used during the preclinical and clini• • •

•

cal phases of drug development to describe the drug exposure-response relationship in experimental animals and in humans. PBPK models can be used to determine the drug exposure-response relationship in the whole body as well as in specific organ. PBPK models can be used to predict the pharmacokinetic behavior of drugs in humans and to determine the first-in-human drug dose to be used in the initial clinical trials. Population pharmacokinetic models are used to extract valuable knowledge about the drug pharmacokinetic behavior, the factors affecting the variability in the drug pharmacokinetics, and the magnitude of the variability in the drug pharmacokinetic behavior in the different patient populations. Model-based TDM utilizes population pharmacokinetic models for a precise estimation of individual dosage requirements.

References 1. Williams PJ and Ette EI “Pharmacometrics: Impacting drug development and pharmacotherapy” in “Pharmacometrics: The science of quantitative pharmacology” (2007) John Wiley& Sons, Hoboken, NJ, USA, pp. 1–21. 2. Ette EI, Chu HM and Godfrey CJ “Data supplementation: A pharmacokinetic pharmacodynamic knowledge creation approach for characterizing an unexplored region of the response surface” (2005) Pharm Res; 22:523–531. 3. Bischoff KB “Physiological pharmacokinetics” (1986) Bull Math Biol; 48:309–322. 4. Nestorov I “Whole body pharmacokinetic models” (2003) Clin Pharmacokinet; 42:883–908. 5. Ploeger B, Mensinga T, Sips A, Meulenbelt J and DeJongh J “A human physiologically based model for glycyrrhizic acid, a compound subject to presystemic metabolism and enterohepatic cycling” (2000) Pharm Res; 17:1516–1525. 6. Clewell HJ, Gentry PR, Gearhart JM, Covington TR, Banton MI and Andersen ME “Development of a physiologically-based pharmacokinetic model of isopropanol and its metabolite acetone” (2001) Toxicol Sci; 63:160–172. 7. Espié P, Tytgat D, Sargentini-Maier ML, Poggesi I and Watelet JB “Physiologically based pharmacokinetics (PBPK)” (2009) Drug Metab Rev; 41:391–407. 8. Davies B and Morris T “Physiological parameters in laboratory animals and humans” (1993) Pharm Res; 10:1093–1095. 9. Leggert RW and Williams LR “Suggested reference values for regional blood volumes in humans” (1989) Health Phys; 60:139–154. 10. Ebling WF, Wada DR and Stanski DR “From piecewise to full pharmacokinetic modeling: Applied to thiopental disposition in the rat” (1994) J Pharmacokinet Biopharm; 22:259–292. 11. Nestorov I, Aarons L and Rowland M “Quantitative structure-pharmacokinetics relationships: II. Mechanistically based model for the relationship between the tissue distribution parameters and the lipophilicity of the compounds” (1998) J Pharmacokinet Biopharm; 26:521–546. 12. Mordenti J “Man versus beast: Pharmacokinetic scaling in mammals” (1986) J Pharm Sci; 75:1028–1040. 13. Ings RMJ “Interspecies scaling and comparisons in drug development and toxicokinetics” (1990) Xenobiotica; 11:1201–1231. 14. Reigner BG and Blesch KS “Estimating the starting dose for entry into humans: Principles and practice” (2002) Eur J Clin Pharmacol; 57:835–845. 15. Sheiner LB and Beal SL “Evaluation of methods for estimating population pharmacokinetics parameters I. Michaelis-Menten model: Routine clinical pharmacokinetic data” (1980) J. Pharmacokinet Biopharm; 8:553–571.

Pharmacometric Applications 405 16. Mouls DR and Upton RN “Basic concepts in population modeling, simulation, and modelbased drug development” (2012) CPT Pharmacometrics Syst Pharmacol; 1:e6,:1–14. 17. Mouls DR and Upton RN “Basic concepts in population modeling, simulation, and modelbased drug development—Part 2: Introduction to pharmacokinetic modeling methods” (2013) CPT Pharmacometrics Syst Pharmacol; 2:e38, 1–14. 18. Wade JR, Edholm M and Salmonson T “A guide for reporting the results of population pharmacokinetic analyses: A Swedish perspective” (2005) AAPS J; 7:Article 45. 19. Food and Drug Administration, Center for Drug Evaluation and Research, Center for Biologics Evaluation and Research, “Population pharmacokinetics, Guidance for Industry” (2022). 20. Goutelle S, Woillard JB, Neely M, Yamada W and Bourguignon L “Nonparametric methods in population pharmacokinetics” (2022) J Clin Pharmacol; 62:142–157.

23 Answer for the Practice Problems

Chapter 1 1.1 a 5 mg/mL/day b 100 mg/mL c 50 mg/mL 1.2 a b c d

4 cycles 0.10 hr−1 20 mg/L 2.0 mg/L

Chapter 2 2.4 a 30 L b 600 mg IV c 40 mg/L 2.5 a 1.25 L/kg b 15 mg/kg c 12.8 mg/L 2.6 0.6–1.5 mg/L 2.7 Drug therapeutic range free concentration 0.8–2.4 mg/L. Free drug concentration in first patient 2.0 mg/L, within the therapeutic range. Free drug concentration in second patient 3.0 mg/L, higher than the therapeutic range. Chapter 3 3.1 a Quinidine b Theophylline

DOI: 10.4324/9781003161523-23

Answer for the Practice Problems 407 c Theophylline CLT = 0.0495 L/kg/hr Ampicillin CLT = 0.18 L/kg/hr Quinidine CLT = 0.24 L/kg/hr 3.2 a Renal clearance = Q × E = 1.5 × 0.2 = 0.3 L/min b Hepatic clearance = 1.2 × 0.3 = 0.36 L/min c CLT = 0.3 + 0.36 = 0.66 L/min 3.3 The CLT = k Vd = before renal disease = 6.93 L/hr, and after renal disease = 3.47 L/hr. The renal disease decreased the efficiency of the kidney for eliminating the drug. This is reflected in the lower total body clearance and smaller elimination rate constant. Chapter 4 4.1 a b c d e

False True True False False

a b c d e f

Cefazolin Chlorpheniramine Cannot be determined Chlorpheniramine Ciprofloxacin Cannot be determined

4.2

4.3 a Drug A because the half-life is dose independent. b k = 0.1155 hr−1. c Drug A: Increasing the dose increases the initial rate of drug elimination, while elimination rate constant and half-life remain constant (dose independent). Drug B: Increasing the dose does not affect the elimination rate and the elimination rate constant, while the half-life is longer when the concentration is higher. 4.4 a A plot of the amount versus time is straight line on the linear scale, so the elimination follows zero-order kinetics. b The elimination rate constant = 11 mg/hr. c The half-life immediately after drug administration = 18.2 hr. d The elimination rate constant will not change with the change in dose, 11 mg/hr. e The amount of drug remaining 20 hr after administration of 400 mg = 180 mg. f No drug will be remaining 40 hr after administration of 400 mg.

408  Answer for the Practice Problems 4.5 a A plot of the amount versus time is straight line on the semilog scale, so the elimination follows first-order kinetics. b The elimination rate constant, k = 0.4617 hr−1. c The dose is equal to the y-intercept = 500 mg. d The equation: A = 500 mg e−0.4617 t. e The amount of the drug after 10 hr = 4.94 mg. f The half-life = 1.5 hr. 4.6 a half-life = 3 hr, k = 0.231 hr−1, Vd = 30 L, and CLT = 6.93 L/hr. b The AUC = 144.1 mg-hr/L. c If the dose is only 500 mg, Cp0 will be 16.65 mg/L, and the AUC will be 72.05 mg hr/L. Half-life, k, Vd, and CLT will not change. d The dose = 1500 mg. 4.7 a Half-life = 0.9 hr, k = 0.77 hr−1, Vd = 50 L, CLT = 38.5 L/hr, AUC = 5.195 mg hr/L. b The concentration will be 0.5 mg/L after 2.7 hr. c If the dose is only 400 mg, Cp0 will be 8 mg/L, and the AUC will be 10.39 mg hr/L. Half-life, k, Vd, and CLT will not change. d The plasma concentration after 10 hr = 0.0018 mg/L.   4.8 Calculate the plasma concentrations at different time points by substitution in the equation for different values for time. Plot the concentrations on semilog graph paper. 4.9 a Half-life = 3 hr, k = 0.2303 hr−1, Vd = 400 L, CLT = 92.12 L/hr, AUC = 4.342 mg hr/L. b The slope will be the same and the y-intercept will be 2.5 mg/L. c The concentration will be 0.25 mg/L after 6 hr. d The dose = 1600 mg. 4.10 a Half-life = 5 hr, k = 0.1382 hr−1, Vd = 0.5 L/kg, CLT = 0.0691 L/hr, AUC = 72.46 mg hr/L. b The slope will be the same and the y-intercept will be 40 mg/L. c The concentration will be 1 mg/L after 16.69 hr. d The initial concentration should be 5.238 mg/L, so the dose = 2.62 mg/kg. 4.11 a b c d e

Ampicillin, largest k Ampicillin, smallest Vd Propranolol, highest clearance (k × Vd) Propranolol, largest Vd Ampicillin, fastest rate of elimination

Answer for the Practice Problems 409 4.12 a Before renal disease: Half-life = 2 hr, k = 0.3465 hr−1, Vd = 20 L, CLT = 6.93 L/hr, AUC = 86.6 mg hr/L. After renal disease: Half-life = 4 hr, k = 0.1733 hr−1, Vd = 20 L, CLT = 3.47 L/hr, AUC = 173 mg hr/L. b The renal disease decreased the efficiency of the kidney for eliminating the drug. This is reflected in the lower total body clearance, smaller elimination rate constant, and longer half-life. 4.13 a b c d

Half-life = 7 hr, k = 0.099 hr−1, Vd = 40 L, CLT = 3.96 L/hr, AUC = 202 mg hr/L. The plasma concentration after 14 hr = 5 mg/L. The slope should be the same −0.043. The initial plasma concentration = 10 mg/L.

a b c d e

Half-life = 3 hr. The MEC is dose independent = 2 mg/L. Duration = 9 hr. Dose = 60 mg. The concentration will always be lower than the MEC, so no effect is observed.

4.14

Chapter 6 6.1 a k = 0.13 hr−1 Half-life = 5.33 hr b tmax = 6 hr c Cpmax = 2.7 mg/L 6.2 a tmax: Drug A = 2.0 hr   Drug B = 2.0 hr b Cpmax: Drug A = 33.4 mg/L   Drug B = 3.34 mg/L 6.3 a tmax = 2.746 hr b Cpmax = 50.66 mg/L c tmax will be the same, Cpmax = 10.1 mg/L 6.4 a b c d e f g

Cp after 6 hr = 10 mg/L Half-life = 6 hr ka = 0.624 hr−1 Vd = 29.2 L tmax = 3.32 hr Cpmax = 11.69 mg/L Slope = −0.05 hr−1

410  Answer for the Practice Problems 6.5 a b c d e

Half-life = 4.5 hr Vd = 59.6 L CLT = 9.18 L/hr tmax = 2.28 hr  Cpmax = 5.9 mg/L AUC = 54.5 mg hr/L

a b c d

ka = 0.6 hr−1 Equation: Cp = 464 µg/L (e−0.277 t − e−0.6 t) tmax = 2.39 hr Cpmax = 128.8 µg/L Vd = 20 L

6.6

6.7 a A plot of the fraction remaining to be absorbed versus time is linear on semilog graph paper indicating first-order absorption. b ka = 0.354 hr−1. 6.8 a ka = 1.0 hr−1 b Half-life = 3 hr   Vd = 50 L 6.9 a b c d

ka = 1.3 hr−1 Equation: Cp = 22.1 mg/L (e−0.46 t − e−1.3 t) tmax = 1.24 hr   Cpmax = 8.08 mg/L half-life = 1.5 hr Vd = 35 L

Chapter 7 7.1 After administration of ampicillin capsules, 50% of the dose reaches the systemic circulation. 7.2 The ratio of the absolute bioavailability of ofloxacin suspension to the absolute bioavailability of ofloxacin tablet is 1.25. 7.3 Gentamicin oral bioavailability is less than 1%. 7.4 a b c d

Fcapsule = 0.4 FIM = 0.9 Fsuspension/Fcapsule = 1.25 IV > IM > suspension > capsule

a b c d

0.83. 1.15. Cannot compare products for two different drugs. Absolute bioavailability of paracetamol product B > paracetamol product A. Absolute bioavailability of ibuprofen product A > ibuprofen product B. Cannot compare paracetamol and ibuprofen products.

7.5

Answer for the Practice Problems 411  7.6 F = 0.638  7.7 AUCIV = 202 mg hr/L CLT = 2.475 L/hr AUC oral = 3.5 + 20 + 33 + 47.25 + 45.45 = 149.2 mg hr/L F = 0.46  7.8 a b c d

1.33. 0.89. Ibuprofen suspension. Cannot be determined, IV data is required.

 7.9 a Formulation A, because it has shorter tmax. b FA/FB = 0.75. This means that the absolute bioavailability of formulation A is only 75% of the absolute bioavailability of formulation B. 7.10 a b c d e

Half-life = 2.31 hr   Vd = 20 L. The syrup has the fastest absorption rate. Absolute bioavailability: Syrup = 0.742, Capsules = 0.558, Tablets = 0.466. Fcapsule/Fsyrup = 0.75. Ftablet/Fcapsule = 0.835.

a b c d

Half-life = 4 hr AUC = 64.6 mg hr/L Ftablet = 0.54 CLT = 4.17 L/hr   Vd = 24 L

7.11

7.12 a Half-life = 9 hr   Vd = 20 L b Bioavailability = 0.36 7.13 a Bioavailability = 0.809 b AUC = 43.3 mg hr/L 7.14 a 77.0 mg hr/L b 75.1 mg hr/L c 0.39 Chapter 9 9.1 Loading dose = 450 mg   Infusion rate = 52 mg/hr 9.2 a At steady-state rate of elimination = rate of administration = 70 mg/hr. b Cpss = 10.1 mg/L.

412  Answer for the Practice Problems c d e f

k = 0.231 hr−1. Cpss is not affected by loading dose = 10.1 mg/L. Approximately 15 hr. Cpss = 20.2 mg/L.

9.3 a Half-life = 7 hr   Vd = 40.4 L b Loading dose = 485 mg   Infusion rate = 48 mg/hr 9.4 a b c d

Half-life = 5 hr Vd = 45.2 L Cpss = 16 mg/L Cpss = 32 mg/L

a b c d e

Cpss = 16 mg/L Elimination rate = 100 mg/hr Half-life = 5 hr Cpss = 16 mg/L Cpss = 32 mg/L

a b c d

Elimination rate = 60 mg/hr Half-life = 7 hr Vd = 30.3 L Loading dose = 455 mg   Infusion rate = 45 mg/hr

a b c d e f

Elimination rate = 20 mg/hr Cpss = 4.95 mg/L k = 0.1155 hr−1 Cpss = 4.95 mg/L Approximately 30 hr Cpss = 9.9 mg/L

9.5

9.6

9.7

Chapter 10 10.1 a Cpss max = 18.16 mg/L   Cpss min = 8.16 mg/L b Cpaverage = 12.8 mg/L c Dose = 400 mg every 8 hr 10.2 a Half-life = 13.3 hr b Vd = 20 L c Cpaverage = 20 mg/L

Answer for the Practice Problems 413 10.3 a Cpaverage = 16.67 mg/L b Cpaverage = 16.67 mg/L c Dose = 240 mg every 12 hr 10.4 a b c d

Cpaverage = 9 mg/L Dose = 444.4 mg Dose = 333.3.mg Cpss max = 27.8 mg/L   Cpss min = 13.9 mg/L

a b c d e

Cpaverage = 20 mg/L Cpaverage = 40 mg/L CLT = 2.5 L/hr Dose = 750 mg every 8 hr Cpaverage = 25 mg/L

10.5

10.6 a Cpss max = 16.58 mg/L   Cpss min = 6.58 mg/L b Cpaverage = 10.8 mg/L c Cpss max = 33.16 mg/L   Cpss min = 13.16 mg/L 10.7 a b c d e

Half-life = 6 hr Vd = 15 L Cpaverage = 21.6 mg/L Approximately 30 hr At steady-state rate of elimination = rate of administration = 450 mg

10.8 a Half-life = 9.43 hr   Vd = 40 L b Dose = 1268 mg 10.9 a b c d e

Loading dose = 600 mg Dosing interval = 11.5 hr ≅ 12 hr Dose = 308 mg ≅ 300 mg Cpaverage = 13.9 mg/L Dosing regimen = 400 mg every 12 hr

10.10 a Dosing interval = 11.5 hr ≅ 12 hr b Dose = 613.4 mg ≅ 600 mg c Cpaverage = 14.3 mg/L

414  Answer for the Practice Problems Chapter 11 11.1 a b c d

Amount = 720 mg Renal excretion rate = 30 mg/hr = 0.5 mg/min CrCL = 50 mL/min CLR = CrCL = 50 mL/min

a b c d

Renal excretion rate = 140.7 μg/hr Renal excretion rate = 25 μg/hr CLR = 50 mL/hr Slope = −k/2.303 = 0.3 hr−1

11.2

11.3 a Bioavailability = 0.875 11.4 a b c d e

Half-life = 1.61 days CLR = 140 L/day   CLT = 215 L/day Fraction = 0.65 Amount = 1.165 mg km = 0.15 day−1

a b c d

Half-life = 3 hr CLR = 7 L/hr Fraction = 0.6 km = 0.091 hr−1

11.5

Chapter 12 12.1 a Metabolite I: Formation rate limited Metabolite II and Metabolite III: Elimination rate limited 12.2 CLT = 4.33 L/hr   AUC = 23 μmol hr/L CLR = 0.866 L/hr Fraction = 0.3 CLT = 3.0 L/hr CLR = 3.0 L/hr Formation clearance metabolite 1 = 1.3 L/hr Formation clearance metabolite 2 = 2.165 L/hr g AUC of drug = 69 μmol hr/L AUC of metabolite 1 = 30 μmol hr/L h Amount of metabolite 1 = 100 μmol, amount of metabolite 2 = 0 μmol a b c d e f

Answer for the Practice Problems 415 i Ratio = 0.433 j No, because we do not have information about the elimination rate constants of the parent drug and the metabolite. 12.3 a Fraction metabolized = 0.924 CLT(m) = 10.87 L/hr Vd(m) = 35.7 L Metabolite formation clearance = 7.65 L/hr, formation rate constant = 0.1159 hr−1 b Cpss(m) = 21 μmol/L c Ratio = 0.82 12.4 a CLT(m) = 8 L/hr   Vd(m) = 20 L b Fraction = 0.546 c Metabolite formation clearance = 2.87 L/hr, formation rate constant = 0.109 hr−1 d Amount = 800 mg e Cpss = 12.7 mg/L   Cpss(m) = 4.55 mg/L 12.5 a b c d e f g

CLR = 4.16 L/hr CLm = 9.70 L/hr Formation clearance = 9.702 L/hr CLT(m) = 7 L/hr, CLm(m) = 0 L/hr, and CLR(m) = 7 L/hr k(m) = 0.233 hr−1, k(m) < k (i.e. elimination rate limited) AUC = 8.66 μmol hr/L  AUC(m) = 12 μmol hr/L Ratio = 1.39

Chapter 13 13.1 a b c d

Vmax = 600 mg/day   Km = 10 mg/L Concentration = 30 mg/L half-life = 2.31 day Cpss = 10 mg/L Dose = 400 mg/day

a b c d

Vmax = 630 mg/day   Km = 7.2 mg/L Dose = 425 mg/day Half-life = 0.986 day Elimination rate when Cp = Km is equal to ½ Vmax = 315 mg/day

a b c d

Vmax = 640 mg/day   Km = 5.6 mg/L Dose = 466 mg/day Half-life = 1.01 day   CLT = 31.1 L/day Cpss = 13.3 mg/L

13.2

13.3

416  Answer for the Practice Problems 13.4 a Vmax = 790 mg/day   Km = 5.6 mg/L b Enzyme induction will increase Vmax without affecting Km. New Vmax = 825 mg/day   Km = 5.6 mg/L c Dose = 645 mg/day 13.5 a Vmax = 340 mg/day   Km = 8.8 mg/L b Cpss = 17.2 mg/L c Half-life = 1.06 day CLT = 13.1 L/day Chapter 14 14.1 The drug concentration will be different in different tissues depending on the affinity of the drug to each tissue. 14.2 During the alpha-phase, the decline in drug concentration is due to the distribution and elimination processes, while during the beta-phase, the decline in drug concentration is due to the elimination process only. 14.3 a After 0.5 hr:   Cp = 10.1 mg/L After 3 hr:   Cp = 4.32 mg/L After 12 hr:   Cp = 1.60 mg/L b Cp (mg/L) = 36 e−2.8 t + 12 e−0.11 t 14.4 a t1/2 α = 0.0775 hr   t1/2 β = 3.65 hr Vc = 14.26 L   AUC = 3.885 mg-hr/L CLT = 19.3 L/hr   k10 = 1.353 hr−1 k21 = 1.255 hr−1  k12 = 6.52 hr−1 Vdss = 88.3 L   Vdβ = 101.6 L b Cp (8 hr) = 0.14 mg/L   Amount remaining = 14.2 mg 14.5 a A = 53.0 mg/L   B = 17 mg/L α = 1.34 hr−1   β = 0.13 hr−1 t1/2 α = 0.52 hr   t1/2 β = 5.3 hr Vc = 7.14 L   AUC = 170.0 mg hr/L CLT = 2.94 L/hr   k10 = 0.412 hr−1 k21 = 0.423 hr−1  k12 = 0.635 hr−1 Vdss = 17.8 L   Vdβ = 22.6 L b Cp (12 hr) = 3.57 mg/L c Cp0 = 210 mg/L d A, B and AUC e A = 159 mg/L   B = 51 mg/L   AUC = 510 mg hr/L f Infusion rate = 29.4 mg/hr

Answer for the Practice Problems 417 14.6 a A = 1.356 mg/L   B = 0.431 mg/L α = 0.124 min−1  β = 0.0072 min−1 t1/2 α = 5.59 min   t1/2 β = 96.0 min Vc = 28.0 L   AUC = 70.8 mg min/L CLT = 0.706 L/min   k10 = 0.0252 min−1 k21 = 0.0354 min−1  k12 = 0.0706 min−1 Vdss = 83.8 L   Vdβ = 98.0 L b The equation describes the best curve that describe the best profile. c Amount = 7.94 mg d The concentration when 5 mg is remaining = 0.051 mg/L Chapter 15 15.1 a Regimen: 75 mg every 8 hr administered as iv infusion of 1 hr duration b Cp max ss = 6.02 mg/L   Cpmin ss = 0.998 mg/L 15.2 a b c d

Half-life = 2.22 hr Vd = 15 L Regimen: 130 mg every 8 hr administered as IV infusion of 1 hr duration Cp max ss = 8.1 mg/L   Cpmin ss = 0.9 mg/L

a b c d

Half-life = 3.0 hr Vd = 15 L Regimen: 140 mg every 8 hr administered as iv infusion of 1 hr duration Cp max ss = 9.89 mg/L   Cpmin ss = 1.96 mg/L

a b c d e f g h

CrCL = 26.7 mL/min. 22%. Estimated half-life = 11.36 hr. Half-life = 8.5 hr   Vd = 14 L. Cp max = 5.15 mg/L. Cp max ss = 8.25 mg/L   Cpmin ss = 3.37 mg/L. Concentration increases due to drug accumulation. Regimen: 90 mg every 24 hr administered as iv infusion of 1 hr duration.

a b c d

Half-life = 3 hr   Vd = 5.09 L. Regimen: 90 mg every 12 hr administered as iv infusion of 1 hr duration. Cp max ss = 8.9 mg/L   Cpmin ss = 0.625 mg/L. Approximately, the half-life will be 6 hr.

15.3

15.4

15.5

418  Answer for the Practice Problems 15.6 a Estimated half-life = 9.2 hr. b Four samples should be obtained: Before drug administration, then 1, 4, and 10 hr after drug administration. c Half-life = 9 hr Vd = 8.57 L. d Regimen: 65 mg every 24 hr administered as iv infusion of 0.5 hr duration. e Cp max ss = 8.67 mg/L   Cpmin ss = 1.48 mg/L. Chapter 16 16.1 Small variability in the extraction ratio for high extraction ratio drugs results in significance variability in the drug bioavailability. 16.2 Enzyme induction significantly affects the hepatic clearance of low extraction ratio drugs and has little effect on the hepatic clearance for high extraction ratio drugs. 16.3 a CLH = 38.96 mL/min b CLH = 38.5 mL/min c CLH = 85.0 mL/min 16.4 a b c d

CLH = 1.33 L/min CLH = 0.923 L/min Extraction ratio: Before CHF = 0.89 After CHF = 0.92 Bioavailability: Before CHF = 0.11 After CHF = 0.08

16.5 Note: The arrows represent the significant change in the CLT, half-life, or bioavailability, while small or minor changes are considered no change. Effect of enzyme induction: Drug

CLT

Half-life

Bioavailability

A B C D E F G H

↑ ↔ ↔ ↑ ↔ ↑ ↑ ↔

↓ ↔ ↔ ↓ ↔ ↓ ↓ ↔

↔ ↔ ↓ ↔ ↔ ↔ ↔ ↓

Effect of enzyme inhibition: Drug

CLT

Half-life

Bioavailability

A B C D E F G H

↓ ↔ ↔ ↓ ↔ ↓ ↓ ↔

↑ ↔ ↔ ↑ ↔ ↑ ↑ ↔

↔ ↔ ↑ ↔ ↔ ↔ ↔ ↑

Answer for the Practice Problems 419 Effect of displacement from the plasma protein binding sites: Drug

CLT

Half-life

Bioavailability

A B C D E F G H

↑ ↔ ↔ ↑ ↔ ↔ ↔ ↔

↓ ↔ ↔ ↓ ↔ ↔ ↔ ↔

↔ ↔ ↔ ↔ ↔ ↔ ↔ ↔

Effect of the decrease in hepatic blood flow: Drug

CLT

Half-life

Bioavailability

A B C D E F G H

↔ ↔ ↓ ↔ ↔ ↔ ↔ ↓

↔ ↔ ↑ ↔ ↔ ↔ ↔ ↑

↔ ↔ ↓ ↔ ↔ ↔ ↔ ↓

Chapter 17 17.1 a Half-life 7.9 hr   Vd = 24.6 L b Dose = 285 mg every 12 hr c Half-life in failure = 16.6 hr Cpmax ss = 23.5 mg/L Cpmin ss = 14.2 mg/L The maximum and the minimum concentrations change due to the reduction in dose, but the average concentrations resulting from the full dose and the reduced dose before and after developing the renal failure are similar. 17.2 a b c d

Vd = 24 L CLT = 0.91 L/hr Dose = 219 mg every 12 hr Cp max ss = 57.0 mg/L   Cp min ss = 36.2 mg/L

a b c d

Half-life = 7.5 hr   Vd = 11.76 L CLR = 0.65 L/hr Half-life in failure = 12.5 hr Cp max ss = 52.8 mg/L   Cp min ss = 27.3 mg/L

17.3

420  Answer for the Practice Problems 17.4 a b c d

CLR = 0.277 L/hr/kg Half-life = 5.45 Dose = 1.65 g every day Cp max ss = 10.0 mg/L   Cp min ss = 4.67 mg/L

a b c d

Cpaverage = 20.9 mg/L Cp max ss = 33.6 mg/L   Cp min ss = 11.89 mg/L The IV dose = 260 mg every 12 hr Cp max ss = 39.4 mg/L   Cp min ss = 28.1 mg/L

17.5

Chapter 18 18.1 a MRT = 5.97 hr b CLT = 4.14 L/hr c Vdss = 24.7 L 18.2 a F for A = 0.6, B = 0.4, C = 0.7 The extent of absorption: C > A > B b MRT for IV = 6.5 hr, A = 8.5 hr, B = 7 hr, and C = 9 hr The rate of absorption: B > A > C 18.3 a b c d

MRTiv = 6.59 hr MRTcapsule = 9.9 hr   MRTtablet = 9.1 hr MATcapsule = 3.31 hr   MATtablet = 2.51 hr CLT = 7.41 L/hr Vdss = 48.9 L

Glossary

Absolute bioavailability  The fraction of the administered dose that reaches the systemic circulation. It can have values between 0 and 1, or 0 and 100%. Absorption  Drug absorption involves the transfer of the drug across the absorption membrane from the site of drug administration to the systemic circulation. Active transport  Carrier-mediated transport when the drug is transported against the concentration gradient. It requires energy. Alpha (α)  The hybrid first-order rate constant for the distribution process in the two-compartment pharmacokinetic model. Alpha half-life (t1/2 α)  The half-life for the distribution process in the two-compartment pharmacokinetic model. Beta (β)  The hybrid first-order rate constant for the elimination process in the two-compartment pharmacokinetic model. Beta half-life (t1/2 β)  The half-life for the elimination process in the two-compartment pharmacokinetic model. Bioequivalence study  The study performed according to the regulatory guidelines to test the bioequivalence of two different products for the same active drug. Bioequivalent products  Drug products for the same active drug with the same dosage form, route of administration, strength, salt, or ester and have no significant difference in the rate and extent to which the active ingredient becomes available at the site of action. Carrier-mediated transport  The transport of drugs across the biological membranes utilizing carrier proteins that are specific for a particular drug or a group of drugs. Compartmental models  A modeling approach to describe the pharmacokinetic behavior of drugs by dividing the body to different compartments based on the rate of drug distribution to the different parts of the body. The models differ in the number of compartments, the compartment(s) where drug elimination occurs, and the arrangement of these compartments. Controlled release formulation  Specially designed formulation that can slowly release the active ingredient(s) from the dosage form over an extended period of time. Cytochrome P450  A superfamily of metabolizing enzymes that are responsible for approximately 75% of the total drug metabolism in humans. Dissolution rate   The rate at which the drug molecules dissolve in a solvent under certain specific conditions. Dosage form  The forms in which the active drugs are marketed for use and typically include the active drug components and inactive additives.

422 Glossary Dosing interval (τ)  The time between doses during multiple drug administration. Extraction ratio (E)  The fraction of the drug amount presented to the organ that is eliminated during single pass through that organ. Facilitated diffusion  Carrier-mediated transport when the drug is transported with the concentration gradient. It does not require energy. First-order absorption rate constant (ka)  The first-order rate constant for the absorption process. It encompasses the rate of all the steps involved in drug absorption and it determines the rate of drug absorption. First-order drug elimination  It is characterized by drug elimination rate equals to the product of the amount of the drug in the body and the first-order elimination rate constant. The half-life and clearance are constant. First-order elimination rate constant (k)  The first-order rate constant for the elimination process. It determines the rate of drug elimination, and it is inversely proportional to the half-life (k = 0.693/t1/2). Generic drug products  Drug products manufactured by pharmaceutical companies after the drug patent expires and are similar to the innovator drug product with regard to the active ingredient, dosage form, route of administration, strength, quality, performance, and intended use. Generic products are marketed if they are bioequivalent to the innovator products. Half-life (t1/2)  The time required to eliminate 50% of the amount of the drug in the body, or to decrease the plasma drug concentration by 50%. Immediate release tablets  These are tablets that do not include any additives that can specifically affect the release of the active ingredient from the tablets. Intravenous bolus administration  Drug administration into the venous circulation over very short time. Intrinsic clearance (CLint)  The maximum ability of an organ to eliminate the drug in the absence of any flow limitation. Linear pharmacokinetics  The pharmacokinetic behavior of drugs when the absorption, distribution, and elimination processes follow first-order kinetics. The pharmacokinetic parameters are concentration independent, and linear relationship exists between the dose and AUC after single administration, and between the dose and steady-state concentration during multiple administration. Loading dose  The loading dose is a drug dose administered at the time of starting drug therapy to rapidly achieve drug concentration as close as possible to the desired steady-state concentration. Log P  The logarithm of the concentration ratio of a compound in n-octanol to that in water at equilibrium. Metabolism  Modification of the drug chemical structure, usually by specialized enzyme systems, to form a new chemical entity called the metabolite. Michaelis-Menten constant (Km)  The M-M constant is a measure of the affinity of the substrate to the enzyme, and it is equal to the substrate concentration when the reaction rate is half its maximum rate (1/2 Vmax). Nonlinear pharmacokinetics  The pharmacokinetic behavior of drugs when at least one of the absorption, distribution, and elimination processes does not follow firstorder kinetics. The pharmacokinetic parameters are concentration dependent, and the relationships between the dose and AUC after single administration and between the dose and steady-state concentration during multiple administration are not linear.

Glossary 423 Organ clearance  The volume of the plasma (or blood) completely cleared from the drug per unit time by a specific organ. The organ clearance is a measure of the efficiency of the organ in eliminating the drug. Paracellular  Absorption of small drug molecules through the junction gap between adjacent epithelial cells of the absorption surface. Passive diffusion  It is a process by which molecules are transported across biological membranes from higher concentration to lower concentration with the concentration gradient. Pharmacodynamics  The field that deals with the quantitative relationship between the drug concentration at the site of action and the drug effect. Pharmacogenetics  The field that deals with the genetic basis of variability in drug response. Pharmacokinetic model  A model constructed to quantitatively describe the relationship between the administered drug dose and the resulting drug concentration-time profile in the body. Pharmacokinetic-pharmacodynamic model  A model constructed to quantitatively describe the relationship between the dosing regimen, the drug concentration-time profile in the body, including the drug site of action, and the resulting drug response. Pharmacokinetics  The field that deals with the kinetics of drug absorption, distribution, metabolism, and excretion, and how these processes affect the drug concentrationtime profile in the body. Pharmacometrics  The use of mathematical and statistical modeling approaches to characterize and predict the drug dose-concentration-effect relationship in different populations. Physiologically based pharmacokinetic (PBPK) models  A modeling approach to describe the drug pharmacokinetic behavior by including a series of compartments, each of which represents one or more organ, tissue, or body space. These models utilize organ blood flow, organ size, and drug affinity to determine the amount of drug delivered and distributed to each organ. Rapidly dissolving tablets  These are immediate release tablets with at least 85% of the labeled amount of the drug dissolving within 15 minutes using standard dissolution apparatus in 0.1 N HCl, pH 4.5, and pH 6.8. Relative bioavailability  The absolute bioavailability of one product relative to the absolute bioavailability of another product. It can have any positive value. Solubility  The maximum amount of the drug that can dissolve in a certain volume of the solvent at a specific temperature. Steady state  The state when the rate of drug reaching the systemic circulation is equal to the rate of drug elimination. The area under the curve (AUC)  It is obtained by integrating the equation that describes the plasma drug concentration with respect to time from time zero to infinity. The Biopharmaceutics Classification System (BCS)  This is a system that classifies different drugs based on their solubility and permeability, to predict their in vivo absorption characteristics from immediate release oral solid dosage forms. The Biopharmaceutics Drug Disposition Classification System (BDDCS)  This is a system that classifies different drugs based on their solubility and extent of metabolism, to predict their in vivo absorption and disposition characteristics. This includes the role of transporters in the drug absorption, and the drug-drug interaction potential.

424 Glossary The cumulative amount of the drug excreted in urine  The total amount of the drug excreted in urine from the time of drug administration until the drug is completely eliminated from the body. The first-pass effect  This term refers to the different processes by which the administered drug is lost before reaching the systemic circulation. The lag time of drug absorption  The time between drug administration and the start of drug absorption. The maximum rate of enzymatic reaction (Vmax)  The maximum rate of drug metabolism through a certain metabolic pathway, and it is dependent on the amount of enzymes catalyzing this metabolic pathway. The mean residence time (MRT)  The average time for the residence of all drug molecules in the body. The therapeutic range  The range of plasma drug concentrations associated with the maximum probability of producing the desired drug therapeutic effect and the minimum probability of producing adverse effects. Therapeutic drug monitoring  The use of drug concentration in blood as a tool for individualization of drug therapy and dose adjustment. Total body clearance (CLT)  The volume of the plasma (or blood) completely cleared from the drug per unit time. The clearance is a measure of the efficiency of the body in eliminating the drug. Volume of distribution (Vd)  The volume where the drug is distributed in. It is an imaginary volume, and it relates the amount of the drug in the body to the plasma drug concentration (Vd = Drug amount in the body/plasma concentration). Volume of distribution at steady state (Vss)  The volume of distribution of the drug at steady state. It relates the amount of the drug in the body and the plasma drug concentration at steady state. Volume of distribution of the drug during the elimination phase (Vdβ)  The volume of distribution of the drug during the elimination phase in the two-compartment model. It relates the amount of the drug in the body and the plasma drug concentration during the elimination phase. Volume of the central compartment (Vc)  The volume of the central compartment in the two-compartment model, and it is the volume of distribution of the drug immediately after IV drug administration. Zero-order drug elimination  It is characterized by a constant rate of drug elimination, independent of the drug amount or concentration in the body. The half-life and clearance are concentration dependent.

Index

Note: Bold page numbers refer to tables and italic page numbers refer figures. abacavir 370 abbreviated new drug application (ANDA) 132–133 ABCB1 368 ABCC 368 ABCG2 368–369 absolute bioavailability 114–116 absorption 66–69; see also drug absorption process absorption, distribution, metabolism, and elimination (ADME) processes 1, 3 absorption rate constant 94, 127; residuals method 95–99, 95–99; Wagner-Nelson method 100–103, 101, 103–104 accuracy 141 acetylation 367–368 active metabolites 210 active transport 31, 68, 192 active tubular secretion 192 acute hepatitis 325 acute pharmacodynamic effect 136–137 adenine 362 Akaike Information Criterion 281 alleles of gene 362–363 aminoglycoside concentrations 292–293, 293, 325; dose selection 299; dosing interval 299; first dose of drug 295–297, 296; half-life 294–295; loading dose 299; second dose of drug 297, 297–298; steady state 298, 298; volume of distribution 295 analysis of variance (ANOVA) 142–143 area under the curve (AUC) 334, 368 ATP binding cassette (ABC) transporters 368 AUMC 334–336 BCS see Biopharmaceutics Classification System (BCS) BDDCS see Biopharmaceutics Drug Disposition Classification System (BDDCS)

benzoylecgonine 216 beta-blockers 306, 309, 322, 369 bioavailability see drug bioavailability bioequivalence study: drug metabolites in 146; issues 143–149; pharmacokinetic approach 138–143; product 135–137; regulatory requirement for 133–134; samples 141–142; urinary excretion data 145; in vivo 134–135 bioequivalent products 133 biological medical products 148–149 biomarkers 346 biopharmaceutics 2 Biopharmaceutics Classification System (BCS) 81, 81–82 Biopharmaceutics Drug Disposition Classification System (BDDCS) 83, 83, 84 blood alcohol concentration-time profile 247 blood drug concentration 377–378 blood drug concentration-time profile 7, 7–8 blood samples 141 buccal cavity 74–75 calculus principles 18–19 calibration curve 142 carbamazepine (CBZ) 210, 223, 226, 240 carrier-mediated transport 68–69 Cartesian scale 14, 14 cetuximab 372 Child-Pugh classification 326, 326–328 chirality 70 cholestyramine 148 chromosomes 362 chronic hepatitis 325 ciprofloxacin 340 clarithromycin 340 classes III drugs 84 classes IV drugs 84 class I drugs 83–84 class II drugs 84

426 Index clinical pharmacokinetics 3–4 cocaethylene 211 codeine 210 compartmental models 10, 10 compartmental pharmacokinetic models 260, 261; construction 278; evaluation 280–282, 281–282; fitting process 279–280; mathematical description 278–279 complexation 72 constant rate IV infusion 159 controlled release formulations 179–180, 180, 184 covariate model 399 creatinine 39, 198 creatinine clearance (CrCL) 294–295, 320–321 crossover product 140 crystal lattice 72 crystalline forms 72 cumulative amount of drug 199, 199–200 curve fitting 16 cyclodextrins 72 CYP2C9 enzyme 111, 364–366 CYP2C19 enzyme 366, 371 CYP2D6 enzyme 363–365 CYP3A4 enzyme 111–112, 366–367 cytochrome P450 (CYP 450) 76, 111, 209, 326, 365–367 cytosine 362 deoxyribonucleic acid (DNA) 362 diffusion coefficient 73 diffusion process 67–68 digoxin 320–321 dihydropyrimidine dehydrogenase (DPD) enzyme 369–370 direct linear plot 249, 249–251, 250 direct link PK/PD model 351–352 direct link vs. indirect link 351 direct response vs. indirect response 351–352 dissolution rate 72, 72–73 dosage forms 5 dose-dependent: drug absorption 237–238; drug distribution 238–239; drug metabolism 239–240; renal excretion 239 dosing interval 184, 299 dosing regimen factors 182–183 drug absorption process: administration and formulation strategies 74–80, 76, 79; after oral administration 88, 88–91; barriers 66; BCS 81, 81–82; BDDCS 83, 83; carrier-mediated transport 68–69; dissolution rate 72, 72–73; extent of 142; large intestine 77; molecular structure features 69–70;

paracellular 69; passive diffusion 67–68; rate of 142; small intestine 75–77, 76; solubility 70–72; stability 73–74; transporters, role of 83–84 drug accumulation 178–179 drug administration 393–394 drug bioavailability 113–114, 125–126; absolute 114–116; calculation of 115–117; causes of 109–112, 110; rationales for 112; relative 115–116; urinary excretion information 117–119; in vivo 113–120 drug clearance 34–38, 37–38 drug complexation 72 drug concentration-time curve (AUC) 54–55, 55, 113, 114, 115–116, 142, 147; linear trapezoidal rule for 120, 120–124, 121; multiple drug administration 176, 177 drug crystal form 72 drug distribution 21; after IV bolus administration 26, 26–28, 27; to blood cells 31–32; rate and extent of 22; volume of 23–26, 24, 38–39 drug-drug interaction potential 84 drug elimination process 34, 41–42; first-order elimination 46–47, 47–52, 49–52; zero-order elimination 42–45, 43–45; see also renal drug excretion drug metabolism 34, 209; active metabolites 210; enzymes 209–210; prodrugs 211; toxic metabolites 210–211 drug protein binding 28, 28–31 drug salts 71 drug solubility 70; complexation 72; crystal form 72; pH effect 70–71; salts 71; site of absorption 73 drug transporters 368–369 dye concentration-time profile 258–260, 260 effect compartment approach 353–355, 354–355 eliminating organs 393 Emax model 348, 348–349, 349 enalapril 211 enantiomers 147–148 endogenous substances 147 enzyme induction 254 enzyme-linked immunosorbent assay (ELISA) 384 enzyme multiplied immunoassay (EMIT) 384 enzyme-substrate complexes (ED) 240–241 epidermal growth factor receptor (EGFR) 372 equilibrium dialysis 30 esophagus 75, 78 extraction efficiency or recovery 141 extraction ratio 36

Index  427 facilitated diffusion 68 fast-release oral formulations 184 Fick’s law of diffusion 67 first-order absorption rate constant 89, 89–90, 90, 95–96, 96 first-order drug absorption 89, 89–91, 90 first-order drug elimination 51, 51–52, 52 first-order elimination process 46; half-life 51, 51–52, 52; rate constant 46–47, 47–50, 49–50 first-order elimination rate constant 54 first-order kinetics 18–19, 52, 52–54 first-pass effect 109–110, 110 first point Cpmax 148 first-time-in-human dose 395–396 fitting process 279–280 5-fluorouracil (5-FU) 369–370 fixed-dose combinations (FDCs) 145–146 fixed effect model 346 flip-flop 98, 98 fluorescence polarization immunoassay (FPIA) 384 fluoropyrimidines 369–370 Food and Drug Administration (FDA) 372 food-effect bioequivalence studies 144 IV bolus loading dose 157, 157–159, 158 IV infusion rate 155–156, 156, 163 IV loading dose 173 free drug molecules 382–383 gas chromatography (GC) 384 gastric retention 75 gastrointestinal tract (GIT) 66, 74, 84, 89–90; dose-dependent drug absorption 237; drug transporters 110–111 gene nomenclature 363 generic drug products 6 genes 362 gene structure 362 genetic polymorphism 230; in pharmacodynamics 369–370 genetic polymorphism in pharmacokinetics 363–365; cytochrome P450 enzymes 365–367; drug transporters 368–369; N-acetyltransferase 367–368; thiopurine methyltransferase 367; UDP-glucuronosyltransferase 368 gene variations 362–363 genotype 363–364 glomerular filtration 192 glomerular filtration rate (GFR) 39 glucose-6-phosphate dehydrogenase deficiency 361 glucuronidation 368 Good Clinical Practices (GCP) 138 Good Laboratory Practices (GLP) 138

graphs 14, 14–15 griseofulvin 238 guanine 362 half-life 41; aminoglycoside concentrations 294–295; first-order elimination process 51, 51–52, 52; MichaelisMenten pharmacokinetics 246–247; renal drug excretion 201–202, 203; zero-order elimination 44, 44–45, 45 haplotype 363 hemodialysis process 324, 324–325 Henderson-Hasselbalch equations 71 hepatic cirrhosis 326 hepatic dysfunction 325–326; dose adjustment in 326–328; pharmacokinetic and pharmacodynamic changes 326 hepatic extraction 316–317; ratio 307–308 hepatitis 325 heterozygous genotype 364–365 high extraction ratio drugs 312–316, 313–314 highly variable drugs 146–147 high-performance liquid chromatography (HPLC) 384 homozygous genotype 364–365 hydrogen-bond donor/acceptor 70 hydrophilic drugs 34 immediate release tablets 81–83 indirect link PK/PD model 352–355, 353–355 indirect response PK/PD model 355–357, 356 indocyanine green 40 initiating drug therapy 376, 376–377 inter-compartmental clearance 275–276 intermittent hemodialysis (IHD) 324–325 intermittent IV infusion 286–287, 287; aminoglycoside concentrations (see aminoglycoside concentrations); doses effect 291; first dose of drug 288, 289; infusion time 291; repeated administration of 289, 289–290; at steady state 290, 290; total body clearance 291; volume of distribution 291 interspecies scaling 395 intestinal drug metabolism 111–112 intramuscular (IM) administration 74 intranasal drug administration 78 intrinsic clearance 308 in vitro dissolution 137–139 in vitro-in vivo correlation (IVIVC) 136 in vivo bioequivalence 134–135, 138–143 in vivo drug bioavailability 113–120 in vivo human bioavailability 136 in vivo pharmacokinetic studies 136 isosorbide dinitrate 240

428 Index kidney dysfunction 320–321 kidney function (KF) 319–320; see also hepatic dysfunction; renal dysfunction; renal replacement therapy (RRT) lag time of drug absorption 97, 97–98 large intestine 77 LC-MS-MS technique 384 L-dopa 211 least squares method 17–18, 18 lidocaine 327 linear model 346–347, 347 linear pharmacokinetics 8, 9 linear transformation methods 251, 251–252 lipophilic drugs 22, 34 lithium 325 loading dose 164, 185–186 log-linear model 348 log P 69–70 long half-lives drugs 144–145 Loo-Reigelman method 272–273 low extraction ratio drugs 310–312, 311–312 mathematical method 248–249 matrix-controlled release delivery systems 76–77 maximum rate of enzymatic reaction 241, 241 mean residence time (MRT) 333; after constant rate IV infusion 338; after extravascular administration 337–338; AUC 334; AUMC 334–336; for compartmental models 340–341 metabolism: intestinal drug 111–112; sequential metabolism, kinetics of 226, 227; see also drug metabolism metabolite concentration-time profile 215–216, 216 metabolite kinetics 216–218, 217 metabolite pharmacokinetics 211–213, 212; clearance of metabolite 219–222; drug dose 227–228; fraction of drug dose 229–230; model for drug metabolism 213–216, 215–216; parent drug, oral administration of 225–226, 226; rate constant 218; sequential metabolism, kinetics of 226, 227; specific metabolite 218–219; steady-state metabolite concentration 222–225, 223; total body clearance 228–230; volume of distribution 219, 229–231 metoprolol 82 Michaelis-Menten constant 241 Michaelis-Menten pharmacokinetics: after single IV bolus administration 243–244, 244; direct linear plot 249, 249–251, 250; dose of drugs 253; enzyme kinetic principles 240–242, 241; half-life

246–247; Km, effect of 254; linear transformation methods 251, 251–252; mathematical method 248–249; multiple drug administration 245, 245; multiple elimination pathways 252, 252–253; oral administration 247; parameters 242, 242–243, 243; total body clearance 246; Vmax, effect of 254; volume of distribution 246 minimum effective concentration (MEC) 87–88 mitochondrial DNA (mtDNA) 362 molecular weight 69 monogenic characteristics 364 MRT see mean residence time (MRT) mucoadhesion 75 multiple-dose bioequivalence studies 143–144 multiple drug administration 168, 168–172, 170; absorption rate constant 181; controlled release formulations 179–180, 180, 184; dose and dosing interval 183–184; dosing rate 180–181; dosing regimen factors 182–183; drug accumulation 178–179; IV loading dose 173; loading dose 185–186; oral loading dose 174; patient pharmacokinetic parameters 183; plasma drug concentration at steady state 175, 175–178, 177; time to achieve 172, 172–173, 173; total body clearance 181; volume of distribution 181 multiple IV oral formulations 184 N-acetylprocainamide 383 N-acetyltransferase 367–368 N-acetyltransferase 2 (NAT2) enzyme 367 narrow therapeutic range drugs 148 nephrotoxicity 378 new drug application (NDA) 132 noncompartmental data analysis method 12, 333, 339–340; see also mean residence time (MRT) non-eliminating organs 392–393 nonlinear pharmacokinetics 8, 9, 147, 236–237; conditions 240; dose-dependent drug absorption 237–238; dose-dependent drug distribution 238–239; dosedependent drug metabolism 239–240; dose-dependent renal excretion 239 no-observed adverse effect level (NOAEL) 396 Noyes and Whitney model 72–73 oral anticoagulants 351 oral drug administration 74–77, 76 oral loading dose 174 organ clearance 35, 306–307 organ function 6

Index  429 paracellular absorption 69 parenteral drug administration 74 passive diffusion 67–68, 237 patient disease state 183 patient pharmacokinetic parameters 183 permeability 81–82 P-glycoprotein (P-gp) transporter 111–112 pharmaceutical alternatives 133 pharmaceutical equivalents 133 pharmacodynamic model 4, 344–346, 345; Emax model 348, 348–349, 349; fixed effect model 346; linear model 346–347, 347; log-linear model 348; sigmoid Emax model 349–350, 350 pharmacodynamics 4; genetic polymorphism in 369–370 pharmacogenetics 5, 371; in clinical practice 372–373 pharmacogenetic testing 370–373 pharmacokinetic model 344–345, 345 pharmacokinetic-pharmacodynamic model 12 pharmacokinetics 1–3; applications 5–7; calculus principles in 18–19; clinical 3–4; curve fitting 16; graphs 14, 14–15; linear 8, 9; modeling 9–12, 10–11; nonlinear 8, 9; population 4; simulations 12–14, 13; straight-line parameters 16–18, 17–18 pharmacological testing 6 pharmacometrics 388–389 phase II metabolic reactions 209 phase I metabolic reactions 209 pH effect 70–71 phenobarbitone 251 phenotype 363–364 phenytoin concentration 379 physiologically based pharmacokinetic (PBPK) models 389–391, 390; applications 400–402; behavior conditions 395; in clinical drug use 396–397; components 391, 391–392; covariate model 399; distribution and elimination process 394; drug administration 393–394; eliminating organs 393; first-time-inhuman dose 395–396; interspecies scaling 395; model evaluation and diagnostics 400; non-eliminating organs 392–393; parameterization 394; population analysis 397–402; simulation and parameter estimation 394; statistical analysis and parameter 399–400; statistical model 399; structural model 398–399; in TDM 402–403; validation 394 physiological modeling 10–11, 11 PK/PD model: applications of 359; direct link 351–352; drug effect features

357; indirect link 352–355, 353–355; indirect response 355–357, 356; process 357–358 plasma concentration-time profile for drugs 87–88, 88, 264–266; after single oral dose 91–94, 92–93 plasma drug concentration: after single IV bolus dose 52, 52–54; pharmacokinetic parameters on 58–59 plasma drug concentration time profile 37–38, 38, 124–125; absorption rate constant 127; bioavailability 125–126; doses 125; total body clearance 126; volume of distribution 126–127; see also multiple drug administration plasma protein binding: determination of 30–31; effect of changing 29–30 polygenic characteristics 364 polymorphism 363 population pharmacokinetic analysis 397–402 population pharmacokinetics 4, 11 precision 141 procainamide (PA) 210, 223, 228–229, 383 prodrugs 211 product bioequivalence 135–137 product inhibition 240 propranolol 238 protein binding 316–317 pulmonary drug administration 78–79 racemates 147–148 radioimmunoassay (RIA) 384 rapid dissolution 81 rapidly dissolving tablets 70, 72, 81–82, 139 rectal drug administration 77–78 reference drug product 138 relative bioavailability 115–116 renal drug excretion 190–193; bioavailability 201; cumulative amount in urine 199, 199–200; dose effect 204; elimination rate constant and half-life 201–202, 203; experimentation 194–195; fraction of dose 201; rate constant 201; rate of 193–196, 194, 196; renal clearance 196–199, 198, 201, 205; time profile 195–196; total body clearance 204–205; volume of distribution 201 renal dysfunction 319–320; creatinine clearance 320–321; dosing regimens in 321–324 renal replacement therapy (RRT) 324; dose adjustment 325; drug clearance factors 325; principle of dialysis 324–325 required onset of effect 182 residuals method 95–99, 95–99 rule of 5 69

430 Index Sawchuk-Zaske method 293–294 selectivity 141 semilog scale 14, 14–15 sequential metabolism, kinetics of 226, 227 sigmoid Emax model 349–350, 350 simvastatin 211 single-dose product 140 single IV bolus dose: pharmacokinetic parameters calculation 55–58, 56–57; plasma drug concentration 52, 52–54 single-nucleotide polymorphism (SNP) 363 sink condition 67 small intestine 75–77, 76 solubility 81; see also drug solubility solute carrier (SLC) transporters 368–369 stability 142 stability of drugs 73–74 statistical model 399 steady-state drug concentration 152–154, 152–155; constant rate IV infusion 159; elimination rate constant 160; IV bolus loading dose 157, 157–159, 158; IV infusion rate 155–156, 156, 163; loading dose 164; total body clearance 160, 163–164; volume of distribution 160–162, 164; with zero-order input rate 162, 162 steady-state metabolite concentration 222–225, 223 Stokes-Einstein relationship 69 stomach 75 straight-line parameters 16–18, 17–18 structural model 398–399 subcutaneous (SC) injections 74 surface area 73 surrogate markers 346 systemic bioavailability 308–309 therapeutic drug monitoring (TDM) 6–7, 375–376; analytical technique 384; biological samples 381–382; blood concentration vs. dose 378, 378–379; dosage regimen 385–386; for high-risk patients 381; initiation and management 376–378; large variability 381; low therapeutic index 380–381; measured drug moiety 382–383; pharmacoeconomics of 386; pharmacokinetic models in 402–403; requirements 385; therapeutic range 379, 379–380; time of sample 382 therapeutic equivalents 133 therapeutic range 182, 379, 379–380 thickness of the unstirred layer 73 thiopurine methyltransferase (TPMT) 367 three-compartment pharmacokinetic model 276–278, 277

thymine 362 time dependent pharmacokinetics 240 time-variant vs. time-invariant 351 total body clearance 35, 38–40, 54, 59; intermittent IV infusion 291; metabolite pharmacokinetics 228–230; MichaelisMenten pharmacokinetics 246; multiple drug administration 181; plasma drug concentration time profile 126; renal drug excretion 204–205; steady-state drug concentration 160, 163–164; two-compartment pharmacokinetic model 268, 275 toxic metabolites 210–211 toxicokinetics 5 toxicological testing 6 transdermal drug administration 79, 79–80 tubular reabsorption 193 two-compartment pharmacokinetic model 260–263, 262–263; constant rate IV administration of drugs 273; doses, effect of 275; first-order elimination rate constant 268; first-order transfer rate constant 268–269; hybrid distribution and elimination rate constants 275; inter-compartmental clearance 275–276; method of residuals 266–268, 267; multiple administration of drugs 273–274; oral administration of drugs 272–273, 273; pharmacokinetic parameters, definition of 263–264; plasma concentration-time curve 268; plasma concentration-time profile for drugs 264–266; renal excretion of drugs 274; total body clearance 268, 275; volume of distribution at steady state 269–270; volume of distribution during the elimination phase 270–272, 271; volume of the central compartment 268, 275 two-periods product 140 two-sequences product 140 two-treatments product 140 UDP-glucuronosyltransferase (UGT) 368 UGT1A1 polymorphism 368 ultrafiltration 30 urine, drug excretion in see renal drug excretion volume of distribution 23–26, 24, 38–39, 54, 58–59, 126–127; aminoglycoside concentrations 295; intermittent IV infusion 291; metabolite pharmacokinetics 219, 229–231; Michaelis-Menten pharmacokinetics

Index  431 246; multiple drug administration 181; plasma drug concentration time profile 126–127; renal drug excretion 201; steady-state drug concentration 160–162, 164 volume of distribution at steady state 269–270 volume of distribution of the drug during the elimination phase 270–272, 271 volume of the central compartment 268

Wagner-Nelson method 100–103, 101, 103–104 warfarin 369 zero-order drug absorption 89 zero-order drug elimination 42; half-life 44, 44–45, 45; rate constant 42–44, 43 zero-order input rate 162, 162 zero-order kinetics 19