Bone Remodeling and Osseointegration of Implants 9819914248, 9789819914241

Table of contents :
Preface
Contents
1 Introduction
1.1 Concluding Remarks
References
2 The Finite Element Approach
2.1 History Behind the Approach
2.2 Mesh and Mesh-Free
2.3 General Principles
2.4 Constructing the Model
2.5 Assigning Materials Properties of Bone
2.6 Issue with Non-linearity
2.7 Accuracy of FEM
2.8 Concluding Remarks
References
3 Understanding the Biomechanics
3.1 Biomechanics of the Mandible
3.1.1 Hypotheses Based on Non-lever Action
3.2 Biomechanics of the Temporomandibular Joint
3.3 Concluding Remarks
References
4 Mathematical Analysis of the Mandible
4.1 Tooth Contact and the Biting Force
4.2 Calculating Forces at the TMJ
4.3 Barbenel’s Analysis
4.4 Pruim, De Jongh, Ten Bosch’s Analysis
4.5 Throckmorton and Throckmorton’s Analysis
4.6 Concluding Remarks
References
5 Understanding Bone Structures
5.1 Bone Healing Process
5.2 Fracture Healing and Numerical Simulation
5.3 Bone Remodeling Around Dental Implants
5.3.1 Strain Energy Density
5.3.2 Mechanobiological Approach
5.3.3 Stanford Theory
5.4 Concluding Remarks
References
6 Patient-Specific Modeling
6.1 Mechanical Properties of Bone Tissue from CT Scans
6.1.1 Hounsfield Units to Density
6.1.2 Cortical Bone
6.1.3 Cancellous Bone
6.2 Muscle Forces and other Boundary Conditions
6.3 Concluding Remarks
References
7 Artificial Intelligence, Machine Learning, and Neural Network
7.1 Fuzzy Logic
7.2 Machine Learning and Neural Network
7.3 Predicting the Osseointegration Process
7.4 Design Optimization
7.5 Bone Healing and Remodeling
7.6 Concluding Remarks
References

Citation preview

Tissue Repair and Reconstruction Andy H. Choi

Bone Remodeling and Osseointegration of Implants

Tissue Repair and Reconstruction Series Editors Andy H. Choi, Carlingford, NSW, Australia Besim Ben-Nissan, Sydney, NSW, Australia

SpringerBriefs in Tissue Repair and Reconstruction provides a unique perspective and in-depth insights into the latest advances and innovations contributing to improved and better treatments for patients with damaged soft and hard tissues as a result of diseases, trauma, and implantations. The book series consists of volumes that offer biomedical researchers better insights into the advancements of biomaterials science and their translation from the laboratory to a clinical setting. Similarly, the series provides information to surgeons and medical practitioners on novel ideas in biomedical science and engineering on top of disseminating new ideas and know-hows in diagnostics and treatment options for patients from head to toe. The series will cover a number of key topics: Fundamental Concepts and Surface Modifications: The topic will provide detailed information on the discovery and advancements of biomaterials surface modification approaches and their use within the human body in a safe manner and without provoking any negative tissue response. Computational Simulations and Biomechanics: Anatomically accurate computational models are being in all fields of medicine particularly in orthopedics and dentistry to reveal the biomechanical functions and behaviors of bones and joints when damaged, diseased, and in the health state. They also contribute to our understanding during the design and applications of implants and prosthetics subjected to functional loadings and movements. Surgical Advances and Treatment Options: Discusses how surgical techniques are revolutionized by our deeper understanding into biomaterials science and tissue engineering. The section also focuses on the latest innovations and surgical advancements currently being used to treat patients with damaged tissues. Post-Operative Treatment and Rehabilitation Engineering: Expands the independence and functionality of the patient after surgery while at the same time reducing the chance of complications such as wound infections and dislocations. Advances in technologies are creating new opportunities in how physiotherapy rehabilitations are delivered.

Andy H. Choi

Bone Remodeling and Osseointegration of Implants

Andy H. Choi School of Life Sciences University of Technology Sydney Sydney, NSW, Australia

ISSN 2731-9180 ISSN 2731-9199 (electronic) Tissue Repair and Reconstruction ISBN 978-981-99-1424-1 ISBN 978-981-99-1425-8 (eBook) https://doi.org/10.1007/978-981-99-1425-8 © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

Originally introduced as a technique used to solve structural mechanics problems, finite element analysis (FEA) was quickly recognized as a universal procedure of numerical approximation to all physical problems that can be modeled by a differential equation description. It has been accepted that the theory behind the bone remodeling sequence is to maintain the integrity of the skeleton. Key phases of fracture healing have previously been examined using mathematical modeling and FE simulations. The application of numerical models to simulate the fracture healing process may prove to be advantageous in defining the optimal mechanical-based treatment or reconstruction after an accident or illness. Furthermore, FEA is also widely utilized to simulate the bone remodeling process as well as the osseointegration at the boneimplant interface following implantations. During the first year of function, bone remodeling will occur in response to occlusal forces and to establish normal dimensions of the peri-implant soft tissues. This is of vital significance as the application of implants may alter the mechanical environment of the surrounding bone tissue. Changes in the internal stress state will govern whether constructive or destructive remodeling will take place in the bone tissue surrounding the implant. Consequently, it is imperative that the effect of bone remodeling is taken into consideration when determining the performance and efficacy of implants and prostheses. On the other hand, biomechanical simulations based simply on FEA will need substantial amounts of time and powerful computing hardware thus making it not as ideal in clinical settings where real-time responses are required. Furthermore, an insufficient quantity of mathematical models is currently available that incorporates bone structural information at various scales. Hence, a more thorough examination of the bone remodeling process can be achieved only if bone structures on multiple levels (macro, micro, and mesoscale) are included into the computational models. It has been suggested that the utilization of artificial intelligence can address the issue associated with the extensive computational resources needed during biomechanics and bone remodeling simulations. With proper training and testing, FEA combined with artificial intelligence can be used in applications such as predicting the survivability of a particular dental implant design employed to restore the functional needs of an individual patient or the bone healing and remodeling process v

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Preface

within synthetic bone tissue engineering scaffolds used to treat large bone defects. The clinical significance with such approach is that it could ease the computational costs associated with the utilization of FEM in applications such as design optimization where testing every possible design scheme is impractical. This is particularly true when it comes to dental implants in which the stress developed at the boneimplant interface is governed by a number of design variables such as materials used and the implant geometry. It is envisaged that this book will provide an in-depth examination of the latest research and advances in bone remodeling algorithm and their incorporation into FE models and the use of artificial intelligence to predict the osseointegration and bone healing process. The book is also intent to offer a fundamental insight into the biomechanical behavior of the mandible, which is crucial if appropriate prosthetic devices and surgical techniques are to be further refined. Finally, I would like to express my deepest gratitude to my friend, mentor, and series co-editor Prof. Besim Ben-Nissan for his advice and support for more than two decades. Also, I would like to give special thanks to Dr. Ramesh Premnath for making the Tissue Repair and Reconstruction book series possible. I would also like to acknowledge the team at Springer Publishing, especially Ramesh Kumaran, for their help during the entire process of publication. Sydney, Australia

Andy H. Choi

Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 4 4

2 The Finite Element Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 History Behind the Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Mesh and Mesh-Free . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 General Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Constructing the Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Assigning Materials Properties of Bone . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Issue with Non-linearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Accuracy of FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7 7 9 11 12 13 17 18 18 19

3 Understanding the Biomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Biomechanics of the Mandible . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Hypotheses Based on Non-lever Action . . . . . . . . . . . . . . . . . . 3.2 Biomechanics of the Temporomandibular Joint . . . . . . . . . . . . . . . . . . 3.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 23 25 25 27 27

4 Mathematical Analysis of the Mandible . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Tooth Contact and the Biting Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Calculating Forces at the TMJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Barbenel’s Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Pruim, De Jongh, Ten Bosch’s Analysis . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Throckmorton and Throckmorton’s Analysis . . . . . . . . . . . . . . . . . . . . 4.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

29 29 30 32 34 37 39 39

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5 Understanding Bone Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Bone Healing Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Fracture Healing and Numerical Simulation . . . . . . . . . . . . . . . . . . . . . 5.3 Bone Remodeling Around Dental Implants . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Strain Energy Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Mechanobiological Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Stanford Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 44 49 52 54 57 62 63 64

6 Patient-Specific Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Mechanical Properties of Bone Tissue from CT Scans . . . . . . . . . . . . 6.1.1 Hounsfield Units to Density . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 Cortical Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Cancellous Bone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Muscle Forces and other Boundary Conditions . . . . . . . . . . . . . . . . . . . 6.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

71 73 73 75 77 79 80 81

7 Artificial Intelligence, Machine Learning, and Neural Network . . . . . . 7.1 Fuzzy Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Machine Learning and Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Predicting the Osseointegration Process . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 Design Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 Bone Healing and Remodeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

83 83 86 88 89 91 92 93

Chapter 1

Introduction

Dental and orthopedic implant systems with predictable long-term results backed by sound scientific research in addition to clinical trials are currently available. The manner in which mechanical stresses are transferred from the implant to the surround bone structure without creating forces of an extent that could potentially threaten the longevity of the implant and/or prosthesis will determine its success from a clinical perspective [1, 2]. For more than forty years, there has been a steady increase in the application of dental implants. This interest has motivated the development of a variety of implant designs and several new dental systems. Designing different categories of implants is now made possible through the application of modern engineering techniques as well as the utilization of computational modeling and simulations. Replacing missing teeth successfully is one of the biggest challenges facing dental professionals. Conventional full dentures may not stress bone properly resulting in inflammation of soft tissues and a reduction in alveolar bone height [3–10]. Despite the fact that implants have been used to support dental prostheses for a number of years, they have not always enjoyed a favorable reputation. However, this situation has dramatically changed with the development of endosseous osseointegrated dental implants. They are the nearest replacement equivalent to the natural tooth and are a useful addition in the management of patients who have missing teeth as a consequence of diseases, development anomalies, or trauma [11]. It is of paramount importance to determine the conditions of success for implant systems as well as testing implants in well-controlled clinical trials. The most obvious sign when an implant system is failing is mobility and its impact on the surrounding bone tissues. The restoration of dentition by providing a way of transmitting masticatory forces to the maxillary or mandibular bone is the primary function of a dental implant system. The significance of gaining an insight into the way in which the stresses acting in a dental implant and how it is distributed into the surrounding bone tissues is of vital importance in the field of prosthetic replacement where the primary intent is to replace a damaged tooth so that the patient can effectively function.

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. H. Choi, Bone Remodeling and Osseointegration of Implants, Tissue Repair and Reconstruction, https://doi.org/10.1007/978-981-99-1425-8_1

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Bone remodeling will occur during the first year of function in response to occlusal forces in addition to creating normal dimensions of the peri-implant soft tissues. The determination of whether destructive or constructive bone remodeling that will take place lies in the changes in the internal state of stress in bone due to occlusal forces. Abnormally high concentrations of stress in the supporting tissues can lead to patient discomfort, pressure necrosis, and eventually failure of the implant system. On the other hand, disuse atrophy similar to the loss of alveolar crest after the removal or natural teeth could be the consequence of low stress levels around a dental implant [12]. Typically, a definite landmark on the implant such as the implant-abutment junction is used to determine the “ideal” bone level. As a result, it may differ between implant systems. Furthermore, quantifying minute changes in bone levels such as 0.2 mm per year is impossible using conventional radiographs, and it is generally believed that bone levels are to be more or less stable. Therefore, these specified changes are appropriate in defining the average or mean changes across a large number of implants instead of an individual implant. For example, an obvious change of one mm or more can happen in a very small number of implants compared to the majority that remained unchanged or in a steady state. In addition, it is also problematic in specifying the conditions required to signify failure for an individual implant based on the levels of change over a given period. An accelerated change in bone level may result in stability for an extended period. Conversely, progressive or continuous bone loss is a troublesome sign of imminent failure. Hence, it can be stated that an implant with obvious loss of bone may be considered as “surviving” instead of “successful.” The design of the implant will greatly influence its initial stability and subsequent function [3]. The key design parameters are: 1. Shapes of the implants such as solid or hollow cylinders and solid or hollow screws are designed to offer adequate stability during the early stages of implantation and maximizing the potential area for osseointegration. Osseointegration, by definition, is the direct structural and functional connection between the ordered, living bone and the surface of a load-carrying implant. Screw-shaped implants also provide sound distribution of load characteristics during function. Even minor variations in the size and pitch of the threads can improve the initial stability. 2. Surface feature is another design parameter that will influence the degree of osseointegration of the dental implant. There is the option of increasing surface contact with bone through an increase in surface roughness. Yet, one has to keep in mind that the tradeoff for this increase is the amount of bacterial contamination on the implant surface if it becomes exposed within the oral cavity as well as the increase likelihood of surface corrosion and ionic exchange. 3. Implant lengths are available from 6 mm up to 20 mm. Implant lengths between 8 and 15 mm are frequently used as they correspond quite closely to normal root lengths.

1 Introduction

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4. The diameter of an implant can be as high as 6 mm, which are considerably stronger and are available for clinical applications. However, they are not used extensively due to the fact that sufficient bone width is not encountered regularly. A minimum of 3.25 mm in diameter is required to ensure adequate implant strength. Most implants are approximately 4 mm in diameter. Implant stability is extremely important at the time of implantation, and variables such as the design of the implant in addition to the quantity and quality of the surrounding bone tissue will govern this stability. Following the loss of a tooth, the width and height of the alveolar bone will begin to resorb. The most beneficial situation for implant treatment would ideally consists of a well-established cortex and densely trabeculated medullary spaces with a good blood supply. Good initial stability may also be offered by bone tissues that are predominately composed of cortical bone. In this case, extra precaution must be taken to avoid overheating during the drilling process in particular for sites that are more than 10 mm in depth as the bone can be damaged easily. On the other hand, bone with a thin or absent cortical layer and sparse trabeculation will offer very poor initial implant stability. Such bone offers very few cells with adequate osteogenic potential for promoting osseointegration. Furthermore, the quality of bone is compromised by factors such as infection, heavy smoking, and irradiation. The rate of success for dental implantation can also be improved by using surgical approaches that avoids heating the bone tissue. Damage to bone tissue occurs after one minute at a temperature of approximately 47 °C, but this can be avoided by utilizing slow drilling speed, successive incrementally larger sharp dills, and copious saline irrigation. Measuring the cutting torque during the preparation of the implant site can be used to determine the bone quality. The stability of an implant as well as improving the bone-to-implant contact has been examined using resonance frequency analysis [3]. This non-invasive tool evaluates the stiffness across the interface between bone and implant. Moreover, it has been demonstrated that immediate loading is consistent with subsequent successful osseointegration under certain conditions such as adequate bone quality and if the functional forces can be properly controlled [3]. Abutments are connected to the implant after the recommended healing period of around three months to allow the construction of prosthesis. Subsequently, this procedure prevents further surgery to uncover the implants. It is important to note that the implant should not be excessively loaded during the early healing phase following its installation. Currently, we cannot quantify accurately or precisely determine the optimum period of healing before loading can begin. Any movement of the implant within the bone during this stage will result in fibrous tissue encapsulation instead of osseointegration. Similarly, this is somewhat comparable to the healing of a fracture where stabilization of the bone fragments is of vital importance in promoting union. Increases in bone-to-implant contacts as well as sustaining osseointegration are possible through carefully planned occlusal loading during functional movements. In contrast, insufficient load transfer or too much loading can result in bone loss

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and/or component failure. Parafunctional activities as well as the design of the prosthesis are two primary factors that can lead to implant overload. Excessive occlusal forces created in either or both circumstances present opportunities for loosening and/or fracture of the screws through bending overload [13]. Bending overload can be described as a situation where occlusal forces on an implant-supported prosthesis exert a bending moment resulting from non-axial loading on the implant cross-section at the crestal bone level [14].

1.1 Concluding Remarks Currently, one of the major drawbacks of synthetic implants is their inability to adapt to the local tissue environment. The primary factors in determining the potential applications of a biomaterial are its functionality and biocompatibility. Additional considerations such as its aesthetics and mechanical capability are needed when choosing a biomaterial to be used as a dental implant. Dental implants of today need to demonstrate effective osseointegration and maintain long-term stability of their favorable properties to preserve both the structure of the implant and the integrity of surrounding bone tissues. Based on their chemical and physical properties, titanium and its ternary alloys such as Ti-6Al-4 V and more recently TiZr appear to be particularly ideal for dental implants and prostheses. The passivating oxide on the implant surface permits intimate apposition of hard and soft tissues, physiological fluids, and proteins to the titanium surface. To address the concerns centered on aesthetics, dental implants manufactured from the bioceramic zirconia (partially stabilized zirconia) are a non-metal alternative to titanium implants.

References 1. Semisch-Dieter OK, Choi AH, Ben-Nissan B et al (2021) Modifying an implant: a mini-review of dental implant biomaterials. BIO Integration 2:12–21. https://doi.org/10.15212/bioi-20200034 2. Choi AH, Ben-Nissan B (2018) Anatomy, modeling and biomaterial fabrication for dental and maxillofacial applications. Bentham Science Publishers, United Arab Emirates 3. Palmer R (1999) Introduction to dental implants. Brit Dent J 187:127–132 4. Palmer R, Howe L (1999) Assessment of the dentition and treatment options for the replacement of missing teeth. Brit Dent J 187:247–255 5. Palmer P, Palmer R (1999) Implant surgery to overcome anatomical difficulties. Brit Dent J 187:532–540 6. Palmer R, Palmer P, Floyd P (1999) Basic implant surgery. Br Dent J 187:415–421 7. Albrektsson T, Sennerby L (1991) State of the art in oral implants. J Clin Periodontol 18:474– 481 8. Atkinson PJ, Woodhead C (1979) Remodeling in the aging mandible. J Implantol 8:372–383 9. Hudis MM (1978) The mouth: medical consideration in the construction and design of oral prostheses. Ann Dent 37:45–56

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10. Schnitman A, Shulman LB (1977) Dental implants: their current status. J Mass Dent Soc 26:278–286 11. Skalak R (1983) Biomechanical considerations in osseointegrated prostheses. J Prosthet Dent 49:843–848 12. Grenoble DE (1974) Design criteria for dental implants. Oral Implantol 5:44–64 13. Haack JE, Sakaguchi RL, Sun T et al (1995) Elongation and preload stress in dental implant abutment screws. Int J Oral Maxillofac Implants 10:529–536 14. Rangert BR, Sullivan RM, Jemt TM (1997) Load factor control for implants in the posterior partially edentulous segment. Int J Oral Maxillofac Implants 12:360–370

Chapter 2

The Finite Element Approach

The finite element technique is a numerical method used to investigate structures and continua. Generally, the problem at hand is too difficult to be resolved in an acceptable manner using classical analytical means. Initially introduced as an approach applied to describe structural mechanics problems, finite element analysis (FEA) was quickly recognized as a universal technique of mathematical approximation to all physical problems that can be modeled by a differential equation description. The FE approach was introduced in orthopedic biomechanics in the 1970s in an effort to examine the stresses, strains, and deformation in human bones during functional loadings. In dentistry, the utilization of FEA in areas such as the design and analysis of implants and deformations during functional loadings accelerated after the 1980s. Since then, it has been applied frequently in oral and maxillofacial surgery in addition to orthopedics and spinal research to analyze problems such as implant design, fracture healing, bone remodeling, and interactions at the bone-implant interface. It has also been used to study the mechanical properties of biomedical nanocoatings. More recently, it has been applied in nanomedicine to examine the mechanics of a single cell and to gain fundamental understandings into how the particulate nature of blood influences nanoparticle delivery [1, 2].

2.1 History Behind the Approach A “lattice analogy” intended for stress analysis was first suggested by researchers in 1906. The continuum was replaced by a methodical pattern of elastic bars. Properties of the bars were chosen in a way that produced displacements of the joints to estimated displacements of points in the continuum. This approach worked toward exploiting on well-known methods of structural analysis. The finite element method (FEM) that we are all familiar today was initially suggested by a German-American mathematician by the name of Richard L. Courant [3]. In a lecture on mathematics in 1941 that was later published in 1943, piecewise © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. H. Choi, Bone Remodeling and Osseointegration of Implants, Tissue Repair and Reconstruction, https://doi.org/10.1007/978-981-99-1425-8_2

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polynomial interpolation over triangular subregions and the principle of stationary potential energy was applied by Courant to investigate the Saint–Venant torsion problem. The work of Courant was overlooked until it was further expanded afterward by engineers independently [4]. None of the aforementioned work was of much practical value at the time as computer power was not available to generate and resolve large series of simultaneous algebraic equations. It was not accidental that major advances in digital computers and programming language overlapped with the expansion of finite element analysis. Engineers by 1953 had created and ascertained stiffness equations in matrix format using digital computers. A large quantity of this work was conducted within the aerospace sector. During that time, a large finite element (FE) problem consisted of hundred degrees of freedom. In 1953 at the Boeing Airplane Company, Turner suggested that a triangular plane stress element could be utilized to model the skin of a delta wing. Published almost at the same time with similar work performed in England, this work signifies the beginning of the widespread application of finite element analysis. As a consequence of company policies against publication, much of this early work went unrecognized [5–7]. The phrase “finite element method” was invented by Clough in 1960, and it was becoming clear the practical value of the method. New elements designed for the purpose of stress analysis were created mainly by intuition and physical argument. In 1963, the method gained respectability when it was recognized as having a sound numerical foundation, and it can be considered as the explanation of a variational problem through the minimization of a function. Subsequently, the method was being applied to all field problems that can be modeled in a variational form [8, 9]. Large general-purpose FE computer programs emerged during the late 1960s and early 1970s, and examples of such programs include ANSYS, ASKA, and NASTRAN. Every one of these programs contained many different forms of elements and can carry out heat transfer, dynamic, and static investigations. Extra capabilities were quickly incorporated. In addition, pre-processors (for data input) and post-processors (for result evaluation) were also included. These processors rely on graphics; hence, performing finite element analysis was cheaper, quicker, and easier. During the early 1980s, the development of graphics became intensive as software and hardware for interactive graphics became affordable and accessible. Typically, a general-purpose FE program contains over hundred thousand lines of codes and normally resides on a superminicomputer or a mainframe. Adaptations of general-purpose programs began to appear on personal computers during the mid1980s, and FE calculations can now be carried out on personal computers, laptops, mainframes, and cloud-based computing services [10]. Hundreds of analyses and analysis-related programs are now available, for purchase or for lease, cheap or expensive, general or narrow, large or small. One has to keep in mind that the results obtained from FE simulations are seldom precise. However, by processing more equations, errors are reduced and results sufficiently accurate for engineering applications are attainable at reasonable costs.

2.2 Mesh and Mesh-Free

9

2.2 Mesh and Mesh-Free Finite element analysis examines a complex problem by redefining it as the summation of the solutions of a series of interrelated simpler problems. The first step involves subdividing (or discretize) the complex geometry into a suitable set of smaller “elements” of “finite” dimensions. This creates the “mesh” model of the investigated structure when combined (Fig. 2.1). Each element can take on a specific geometric shape (for instance, cube, triangle, tetrahedron, square, etc.) with an explicit internal strain function. The equilibrium equations between the external forces acting on the element and the displacements taking place at its corner points or “nodes” can be described using these functions, the actual geometry of the element, and a suite of boundary conditions such as constrain points. Each node of the element will generate one equation for each degree of freedom. Typically, these equations are properly written in matrix form for use in a computer algorithm. From the example in Fig. 2.1, and as a whole, the FE approach models a structure as an assembly of small parts or elements. A simple geometry is utilized to define each element, and hence, it is much easier to investigate than the actual structure. Fundamentally, FEM approximates a complex situation using a model that comprises of piecewise-continuous simple solutions. Elements are called “finite” to distinguish them from differential elements used in calculus. The creation of a three-dimensional (3-D) FE mesh has been widely acknowledged as a task that is the most labor intensive and time consuming. Even with the use of the most sophisticated mesh-generator, time remains limitless. It is possible to obtain a mesh very quickly for a given distribution of points. However, low quality or distorted meshes result in greater chances of producing errors in the analysis; hence, remeshing is necessary, and this might not be feasible for complex 3-D geometries in finite time. Furthermore, mesh-based FE approach is not ideal to solve problems with discontinuities that do not align with element edges. Again, one approach in dealing with this issue is remeshing [11, 12].

Fig. 2.1 a Solid model and b finite element mesh model (b) of a dental implant

10

2 The Finite Element Approach

As a result, meshless (or mesh-free) approach was introduced aimed at removing some of the difficulties related to the need of a mesh to construct the approximation. Discrete points instead of tetrahedron mesh are used in the meshless method for representing continuum, and the partial differential equations are solved by taking the advantages of interpolation methods [12]. In the meshless approach, large deformation can be managed more robustly and issues with moving discontinuities such as crack propagation can be handled with relative ease. Above all, mesh alignment sensitivity is negligible [11]. A number of meshless techniques have been suggested and they can be categorized according to the definition of the shape functions and/or based on the minimization method of approximation. The minimization may be via a weak form (as in the Galerkin methods) or a strong form (as in the Point Collocation approach) [12]. Nonetheless, there are a number of disadvantages associated with the meshless method. The shape functions are rational functions that involves highorder integration scheme to be computed accurately. Furthermore, handling important boundary conditions is not as straightforward as in mesh-based approaches since the shape functions in the meshless method are not interpolants and do not satisfy the Kronecker delta property [11, 13]. This results in an increase in the computational cost of the analysis as the essential boundary conditions cannot be imposed directly in the system of equations [14]. On the other hand, the remeshing efficiency in the meshless methods enables the analyses of soft materials such as muscles and the resultant large distortions in comparison to the finite element approach. Furthermore, the smoothness and accuracy of the stress fields obtained with the meshless methods are extremely useful in predicting the remodeling process of biological tissues [15–18]. The natural element method (NEM), a meshless method based on the natural neighbor concept to define the shape functions [19], has been suggested to be more advantageous over the FEM when it comes to problems that involves complex geometries or large mesh distortions where the design of the mesh is costly. Another advantage of NEM is its ability to be easily coupled with finite element and implemented into any FEA frameworks [20]. This approach was applied by Doweidar et al. to examine aspects where NEM is competitive with FEA such as modeling of human articular joints like the knee [20]. Another meshless method that can be used to study the bone remodeling process is based on the combination of NEM and the radial point interpolation method (RPIM) [14, 21]. RPIM is based on the Galerkin weak form formulation utilizing 3-D mesh-free shape functions generated using radial basis functions. More importantly, essential boundary conditions can be enforced as easily as in the FE approach since the RPIM shape functions have the Kronecker delta functions property [22]. The proposed method known as natural neighbor radial point interpolation method (NNRPIM) has been suggested to be useful in the micro and macroscale bone tissue remodeling analysis [17, 23]. According to the authors, a nodal distribution is only needed in NNRPIM. The initial nodal distribution is used to construct all the other numerical structures needed to solve the problem. This approach is more suitable to discretize highly irregular domains such as biological structures

2.3 General Principles

11

compared to conventional mesh-dependent numerical approaches [18, 21]. Additionally, NNRPIM allows a total random node distribution for the discretized problem unlike in FEM where geometrical restrictions on elements are enforced for the convergence of the method [16].

2.3 General Principles The FE method is an approach of piecewise calculation in which the approximating function φ is generated by linking simple functions, each defined over a small region (element). A finite element is a region in space where a function φ is introduced from nodal values of φ on the boundary of the region in a fashion that interelement continuity of φ to be preserved in the assembly. In general, a FE analysis consists of the following steps. Certain stages during the analysis will require decisions by the analyst as well as providing input data for the computer program (steps 1, 4, and 5), while others are carried out automatically (steps 2, 3, 6, and 7) [1]: 1. Divide the continuum or structure into finite elements. Built-in mesh generation function or external meshing programs such as MeshLab will assist the user in performing this task. 2. Formulate the properties for each element. In stress analysis, this means determining nodal loads associated with all element deformation states that are allowed. 3. Assemble elements to obtain the finite element model of the structure. 4. Application of known loads (nodal forces and/or moments in stress analysis). 5. Specify how the structure is supported. In stress analysis, this stage involves fixing several nodal displacements to known values, which are often zero. 6. Solve simultaneous linear algebraic equations to determine nodal degrees of freedom. In stress analysis, it is nodal displacements. 7. In stress analysis, compute element strains from the nodal degrees of freedom and the element displacement field interpolation and finally calculate stresses from strains. Outputs are sorted and displayed in graphical form by the software’s post-processor. In the meshless method, Belinha has summarized the typical procedure used to solve problems such as bone tissue remodeling analysis [24]: 1. Discretize the problem domains into nodes and the accuracy and performance will be strongly influenced the manner in which this is carried out. 2. Utilizing a Gaussian integration scheme or other integration numerical techniques, construct the background integration mesh to amalgamate the weak form equations. 3. Impose nodal connectivity via influence domains. Every integration point will have its own influence domain that is formed by the set of nodes near that point.

12

4. 5. 6. 7.

2 The Finite Element Approach

Again, the performance of the meshless method will be affected greatly by the shape and size of the influence domain. The influence domain of that integration point is used to create the shape functions. Approximation functions are used to obtain the field variables. The nodal-based system of equations is created through the assembly of local nodal matrix into the global equation system matrix. Finally, solve the discrete system using an appropriate solver.

2.4 Constructing the Model The projected response of the FE model depends on choosing the most appropriate type of elements and therefore achieving the goals set out for the analysis such as stress analysis. Subsequently, a wide diversity of different types of elements is offered by FEA, and they can be categorized according to family, order, and topology. The element family refers to the characteristics of displacement and geometry that the element is attempting to model. Among the most common families used for typical structural models are one-dimensional (1-D) beam elements, two-dimensional (2-D) plane stress and plane strain elements, axisymmetric elements, and 3-D solid and shell elements. Beam elements are helpful in modeling beam-like structures where length is much greater than other dimensions and the overall deflection and bending moments can be calculated. This type of model however will not be capable of predicting the local stress concentrations at the point of application of a load or at joints. Plane stress elements are ideal for thin 2-D structures where stresses out of the plane can be ignored. Plane strain elements simulate a special 3-D stress state taking place when out-of-plane deformation is constrained (for instance in relative thick plates). By utilizing well-defined characteristics of an axisymmetric geometry, 3-D stress fields can be modeled and simulated under 2-D conditions through the use of axisymmetrical elements. Shell elements can be used effectively for 3-D structures that are thin in comparison to other dimensions, such as sheet metal parts in which bending and in-plane forces are vital. These elements on the other hand will not predict stresses that changes through the thickness of the shell due to local bending effects. All 3-D conditions should ideally be modeled using solid elements. However, the order of magnitude of such a solid model may impose practical limits on the choice of those elements due to the computational effort of most FE solvers is roughly proportional to the number of equations and the square of the bandwidth. Thus, an acceptable reduction of a 3-D situation should be considered seriously in each FE analysis. Elements can also be classified according to order. Along each edge, linear elements have two nodes, while parabolic have three. Lower-order elements are stiffer than higher-order elements and this is related to bending as having more nodes will provide more degrees of freedom. A degree of freedom signifies the liberty of rotational or translatory motion of a particular node in space. For instance,

2.5 Assigning Materials Properties of Bone

13

shell elements have six degrees of freedom at each of their unrestrained nodes (three translations [x, y, and z] and three rotations around the x-, y-, and z-axes). Conversely, the unrestrained nodes in 3-D (solid) elements have only three translational degrees of freedom, and 2-D elements have only two. An increase in the number of degrees of freedom results in more variables in the stiffness formulation, and it is more computer intensive. Higher-order elements produce more accurate results for an equal mesh grid, but a finer grid of lower-order elements can end up to be more efficient with the same accuracy. Element topology refers to the general shape of the element, for example, quadrilateral or triangular. The topology is also determined by the family of the element (i.e., 2-D or 3-D). Typically, quadrilateral elements could be considered more ideal than triangular elements in complex structural models as the quadrilateral can match the true displacement functions more accurately due to a higher number of degrees of freedom. Moreover, the number of elements in meshes tends to be higher if constructed from triangular elements. The simplicity of triangular elements on the other hand makes them very attractive particularly in automatic mesh generation. Triangular-shaped elements are easier to fit into geometrically complex structures. The combination of various element topologies and element orders (such as triangular and parabolic) can improve the accuracy of a topologically lesser element. Since any deviation in shape from the “ideal” internal elemental strain function will contribute to mathematical inaccuracies, the predictive accuracy of the FE model will be determined by the shape of the elements. Since FEA offers an approximation to the exact solution, a numerical result closer to this true value can be achieved if the displacements in a FE model becomes increasingly continuous. A vital component in FEA is the behavior of the individual elements. A few good elements may generate better results than many poorer elements.

2.5 Assigning Materials Properties of Bone The predictive accuracy of a FE model also lies in the proper assignment of material properties, as stresses and strains within a structure are calculated based on the materials properties. These properties can be categorized as isotropic, anisotropic, orthotropic, and transversely isotropic. The mechanical properties are the same in all directions for an isotropic material, and given the fact that the Young’s modulus, Poisson’s ratio, and the shear modulus are interrelated, only two out of the three variables need to be determined for the elastic behavior to be characterized completely. A material is classed as anisotropic if its properties are different when evaluated in different directions. An orthotropic material is an anisotropic material that demonstrates extreme values of stiffness in mutually perpendicular directions. These directions are referred to as principal directions of the material. The Young’s modulus for an orthotropic material contains only nine independent coefficients. A material is considered as transversely isotropic if it behaves in the same way in every direction about a single axis of symmetry.

14

2 The Finite Element Approach

A constitutive relationship exists that connects the components of stress to the components of strain. The constitutive relationship seems to be sufficient for bone if it is believed as an anisotropic, inhomogeneous, and linearly elastic material [25]. Information on the mechanical properties of the mandible has been determined on small specimens obtained from human tibiae and from the mandibles of cadavers and humans using materials testing methods [26–32] and ultrasonic wave techniques [33–39]. Values of elastic constants reported in a number of studies are summarized in Table 2.1. Using a material testing approach, Arendts and Sigolotto noticed that the mandible is stiffer and stronger in the longitudinal direction than the radial and tangential directions after examining specimens obtained at various regions in three mandibles. They suggested that this could be the result of the orientation of osteons, apatite crystals, and collagen fibers [31, 32]. Similar observations were made when the relationship between muscle attachments in the mandible, and the orientations of apatite crystals were reported [40]. Using a continuous-wave ultrasonic approach, parallelepiped specimens taken at ten different locations in the mandible were tested [41]. Data on the elastic properties revealed that the human mandibular bone is elastically homogeneous but anisotropic and the mandible seems like a slightly less stiff long bone bent into the shape of a horseshoe. Ultrasonic wave techniques were also used in another study in which the Young’s modulus, Poisson’s ratio, and the shear modulus of 17 mandibles were determined [42]. By utilizing the Archimedes principle of Buoyancy technique, the density of each sample was calculated and an average density of 1.768 g/cc was recorded from the mandible samples. Table 2.1 Young’s modulus (E), shear modulus (G) in GPa, and Poisson’s ratio (υ) for the human mandible. The 1-direction is radial, the 2-direction is circumferential, and the 3-direction is longitudinal directions [31, 32, 41–43] Arendts and Sigolotto [31, 32] E1

6.9

Ashman and Van Buskirk [41]

Dechow et al. [42]

Carter [43]

10.8

11.3

13.0

E2

8.2

13.3

13.8

13.0

E3

17.3

19.4

19.4

19.0

4.5

5.3

G12

3.81

G13

4.12

5.2

5.9

G23

4.63

6.2

5.9

0.309

0.274

0.22

υ12

0.270

υ13

0.125

0.249

0.237

0.29

υ21

0.150

0.381

0.317

0.22

υ23

0.325

0.224

0.273

0.29

υ31

0.310

0.445

0.405

0.42

υ32

0.315

0.328

0.376

0.42

2.5 Assigning Materials Properties of Bone

15

Table 2.2 Young’s modulus of cortical bone (in GPa) determined using acoustic microscopy and nanoindentation technique (data are reported as mean ± standard deviation) [29, 44] Turner et al. [29] Direction

Longitudinal

Rupin et al. [44] Transverse

Acoustic

20.55 ± 0.21

14.91 ± 0.52

Nanoindentation

23.45 ± 0.21

16.58 ± 0.32

Transverse 35 ± 11 19.5 ± 3

Nanoindentation is another technique that can be employed to estimate the Young’s modulus of human bone at a microstructural level, but the results obtained may not be accurate due to the method determines the Young’s modulus with the assumption that the material behaves in an elastically isotropic manner [29]. Despite the capability of nanoindentation in providing elastic and post-yield properties of bone, it is a destructive method. On the other hand, scanning acoustic microscopy is an attractive non-destructive and non-contact quantitative alternative to map linear elastic properties [37]. A face-to-face comparison was carried out in a study by Rupin et al. between scanning acoustic microscopy and nanoindentation estimates of Young’s modulus of cortical bone transverse cross-sections extracted from the middiaphysis of the right femur [44]. They noticed that the Young’s modulus obtained from acoustics was nearly two times higher than those obtained from nanoindentation, suggesting the difference could be due to the fixed assumed value of Poisson’s ratio used in determining the acoustic Young’s modulus. On the contrary, the study by Turner et al. showed the elastic moduli estimated using the nanoindentation approach was between 4 to 14% higher than using acoustic microscopy [29] (Table 2.2). It was previously suggested that the Young’s modulus of bone varies between different anatomical regions within the human maxilla and mandible. Using nanoindentation, bone samples from maxillary anterior, maxillary posterior, mandibular anterior, and mandibular posterior were tested. Results of their study revealed that the Young’s modulus of the mandible (18.3 GPa) was higher than the maxilla (14.9 GPa), and the Young’s modulus of the posterior jawbone (17.5 GPa) was higher than anterior jawbone (15.7 GPa) [45]. In addition to the studies carried out on cortical bone, the mechanical properties of cancellous bone have also been ascertained on various locations throughout the human mandible as well as on other human bones such as the tibiae in a number of investigations using a variety of mechanical testing techniques [28, 46–50]. Using the indentation approach with a stainless-steel indenter, regional variations in the Young’s modulus of proximal tibia (both over the transverse plane and through the depth of the epiphysis) were converted from stiffness measurements and the values ranged from 4.2 to 1196.6 MPa with a mean and standard deviation of 331.4 MPa and 312.6 MPa, respectively [46]. The orthotropic elastic properties of cancellous bone were determined using 75 specimens from three human proximal tibiae. A modified ultrasonic technique was used to determine the Young’s moduli and shear moduli. A comparison was also performed between the moduli measured ultrasonically to those measured in a tensile

16 Table 2.3 Young’s modulus (E), shear modulus (G) in MPa, and Poisson’s ratio (υ) of human cancellous bone obtained using the ultrasonic approach and uniaxial compression testing. The works of Rho [49] and Turner et al. [50] centered on human proximal tibiae, while the work of Nikodem [47] is based on the human femur. The 1-direction is radial, the 2-direction is circumferential, and the 3-direction is longitudinal directions (Note Standard deviation in parentheses)

2 The Finite Element Approach Rho [49]

Turner et al. [50]

Nikodem [47]

E1

202 (154)

316.7

147.41 (67.60)

E2

232 (180)

385.5

157.36 (118.81)

E3

769 (534)

959.3

174.26 (131.96)

G12

88.7

G13

125.6

G23

165.7

υ12 υ13 υ21 υ23

0.3

υ31

0.3

υ32

0.3

test. Ultrasonic specimens were cut from the center of each tensile specimen and the ultrasonic Young’s modulus was calculated [50]. A similar study was also conducted by Rho in which the Young’s modulus of cancellous structures was measured using ultrasonic techniques [49]. One hundred and forty-four cancellous bone specimens were taken from the right proximal tibiae of eight frozen human cadavers. The study showed the Young’s modulus of cancellous bone exhibited inhomogeneity and some consistency pattern along both the length and the circumference of the proximal tibiae (Table 2.3). Non-destructive compressive mechanical testing was used in the study by Nicholson et al. [48] to estimate the structural Young’s modulus of the cancellous framework of 48 human vertebral bone cubes in the three orthogonal axes. Based on their findings, significant mechanical anisotropy was observed with mean structural Young’s modulus varied from 165 MPa in the supero-inferior direction to 43 MPa in the lateral direction. Later, the Young’s modulus of cancellous bone was determined in three orthogonal directions using bone tissues coming from the area of four human femoral heads without lesions [47]. Uniaxial compression tests were performed on the tissue specimens where the Young’s modulus in three orthogonal directions were determined (Table 2.3). A failure test was also performed on the tissue samples in the direction perpendicular to the neck-shaft angle of the hip joint with the purpose of determining the ultimate compressive strength (11.36 ± 3.96 MPa). The study by O’Mahony et al. used a fresh edentulous human mandible to determine the physical and mechanical properties in three orthogonal directions [28]. Small cubes of cancellous bone aligned with the infero-superior, bucco-lingual, and mesio-distal axes were obtained. Samples were excised from seven anatomic sites (pre-existing tooth positions) and tested non-destructively using compressive testing to obtain a total of 21 Young’s moduli (Table 2.4). Prior to testing, cubes cut with faces aligned with the anatomic axes were kept moist with buffered saline.

2.6 Issue with Non-linearity

17

Table 2.4 Mechanical and physical properties of cancellous bone based on anatomical sites according to O’Mahony et al. [28] Anatomical Sites

Young’s modulus (MPa) Infero-Superior

Bucco-Lingual

Mesio-Distal

Apparent Density (g/cm3 )

Left lateral incisor

105

160

312

0.40

Left central incisor

56

126

52

0.30

Right central incisor

123

1113

2283

0.98

Right canine

48

71

140

0.23

Interior right 1st premolar

262

1464

1768

0.75

Superior right 2nd premolar

158

553

805

0.78

Right 2nd premolar

47

91

998

0.39

As mentioned previously, accurate data are needed to perform any FE modeling and analyses related to bone tissue (such as bone remodeling, stress analysis at the implant-bone interface). The use of medical imaging such as computed tomography (CT) scans can offer a more appropriate approximation of the mechanical properties, and this approach will be discussed further in the Patient-Specific Modeling chapter.

2.6 Issue with Non-linearity In FEA, a problem is non-linear if the relationship between force and displacement is dependent on the current state of the force, displacement, and stress–strain relations [51]. Non-linearity in a problem or structure can be categorized as geometric nonlinearity, material non-linearity, and non-linearity due to boundary conditions. Geometric non-linearity occurs when the relationships of displacements and strains are non-linear with forces and stresses. This results in changes in structural behavior and loss of structural stability. Examples are large displacement problems and buckling. Material non-linearity arises if stresses and strains are related by a straindependent matrix instead of a matrix of constants. Consequently, the computational difficulty is that equilibrium equations must be written using materials properties that depend on strains, but strains are not known in advance. Plastic flow is frequently a cause of material non-linearity. Non-linearity induced by boundary conditions occurs when the load and the resistance to the deformation induced by the loads that represent the effects of the surrounding environment on the model. Boundary conditions can induce nonlinearity if they vary with displacement of the structure. A discontinuous character is displayed in many of these non-linear boundary conditions, making them some

18

2 The Finite Element Approach

of the most severe non-linearities in mechanics. Friction slip effects and contact are some examples of non-linearity induced by boundary conditions. The key difference between non-linear and linear FEA rests in the solution of the algebraic equations. Compared to linear analysis, non-linear analysis is typically more expensive and complex. Furthermore, results from non-linear analysis are not always unique and normally require an iterative incremental solution strategy to ensure that equilibrium is satisfied at the end of each step. Hence, this is completely different to linear problems. The categorizations “non-linear” and “linear” are artificial in that physical reality presents different problems, some of which can be approximated satisfactorily by linear equations. For many problems of stress analysis, it is fortunate that linear approximations are quite adequate. Non-linear approximations are more challenging to formulate, and solving the resulting equations could cost ten to one hundred times as much as a linear approximation having the same number of degrees of freedom. Numerous physical situations display non-linearities too great to be overlooked. Stress–strain relations may be non-linear in either a time-independent or a timedependent way. An alternation in configuration may result in loads to change their magnitude or distribution or cause gaps to open or close. Mating components may slip or stick. Casting and welding processes cause the material to change in conductivity, phase, and modulus. The generation and shedding of vortices in fluid flow past a structure create oscillatory loads on the structure. Pre-buckling rotations change the effective stiffness of a shell and alter its buckling load. Hence, it can be realized that non-linear effects may vary in type and may be mild or severe. The analyst performing any FE simulation and analysis must understand the physical problem and must be familiar with different solution strategies. A single strategy will not always work well or may not work at all for certain problems. A number of attempts may be required in order to obtain a satisfactory outcome.

2.7 Accuracy of FEM The accuracy of the FEM can only be objectively established via a convergence test. It is used to determine how well the mathematics has been approximated. The approximate solution tends to improve as a series of meshes get more and more refined in a process commonly referred to as h-convergence. As more elements and nodes are used, the calculated displacement at any given node approaches the exact but generally unknown displacement solution.

2.8 Concluding Remarks FEA is widely used in all biomechanical fields, especially for assessing stresses and strains in dental implants and the surrounding bone structures as well as in normal

References

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bone. The predictive accuracy of the FE model is influenced by the geometric detail of the object to be modeled, the applied boundary conditions and the material properties assigned. In recent years, a number of experimental investigations have been carried out to determine the mechanical properties of both cortical and cancellous bone. Due to the fact that bone is a heterogeneous, anisotropic material, the anatomical location of the test specimen will greatly influence the experimental results. Testing parameters such as the vibration frequency in ultrasonic tests, or the deformation rate and the duration of load in mechanical tests will also play an important role in governing the accuracy and reliability of the test results. Features also change based on the type of bone, biological variables such as age, level of activity of the living bone and its possible state of pathological degradation (such as osteoporosis), in conjunction with the preservation conditions of the specimen until it is used for experimentation, all resulting in an irregular distribution of its mechanical properties.

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37. Raum K, Grimal Q, Varga P et al (2014) Ultrasound to assess bone quality. Curr Osteoporos Rep 12:154–162 38. Mahmoud A, Cortes D, Abaza A et al (2008) Noninvasive assessment of human jawbone using ultrasonic guided waves. IEEE Trans Ultrason Ferroelectr Freq Control 55:1316–1327 39. Pithioux M, Lasaygues P, Chabrand P (2002) An alternative ultrasonic method for measuring the elastic properties of cortical bone. J Biomech 35:961–968 40. Bacon GE, Bacon PJ, Griffiths RK (1980) Orientation of apatite crystals in relation to muscle attachment in the mandible. J Biomech 13:725–729 41. Ashman RB, Van Buskirk WC (1987) The elastic properties of a human mandible. Adv Dent Res 1:64–67 42. Dechow PC, Nail GA, Schwartz-Dabney CL et al (1993) Elastic properties of human supraorbital and mandibular bone. Am J Phys Anthropol 90:291–306 43. Carter R (1989) The elastic properties of the human mandible. Dissertation, Tulane University 44. Rupin F, Saïed A, Dalmas D et al (2009) Assessment of microelastic properties of bone using scanning acoustic microscopy: a face-to-face comparison with nanoindentation. Jpn J Appl Phys 48:07GK01–1–07GK01–6 45. Seong WJ, Kim UK, Swift JQ et al (2009) Elastic properties and apparent density of human edentulous maxilla and mandible. Int J Oral Maxillofac Surg 38:1088–1093 46. Vijayakumar V, Quenneville CE (2016) Quantifying the regional variations in the mechanical properties of cancellous bone of the tibia using indentation testing and quantitative computed tomographic imaging. Proc Inst Mech Eng H 230:588–593 47. Nikodem A (2012) Correlations between structural and mechanical properties of human trabecular femur bone. Acta Bioend Biomech 14:37–46 48. Nicholson PH, Cheng XG, Lowet G et al (1997) Structural and material mechanical properties of human vertebral cancellous bone. Med Eng Phys 19:729–737 49. Rho JY (1996) An ultrasonic method for measuring the elastic properties of human tibial cortical and cancellous bone. Ultrasonics 34:777–783 50. Turner CH, Cowin SC, Rho JY et al (1990) The fabric dependence of the orthotropic elastic constants of cancellous bone. J Biomech 23:549–561 51. Zienkiewicz OC, Taylor RL (1991) The finite element method, 4th ed. McGraw-Hill Book Company (UK) limited, London

Chapter 3

Understanding the Biomechanics

3.1 Biomechanics of the Mandible Primarily responsible for speaking, swallowing, and chewing, the masticatory system is a functional unit of the human body that consists of bones, teeth, muscles, ligaments, and joints. The mastication system is a highly complex and refined unit. In order to study occlusion, it is essential to gain an in-depth understanding of its biomechanics and functional anatomy. The mandible is a horseshoe-shaped bone that is suspended below the maxilla by muscles, ligaments, and other soft tissues, all of which provide the mandible mobility necessary to function with the maxilla. The mandible also supports the lower teeth and makes up the lower portion of the facial skeleton. The muscles provide the energy needed to move the mandible and allows function of the masticatory system. Four pairs of muscles situated on the left and right side of the mandible form a group commonly referred to as the muscles of mastication. The most frequently used biomechanical analogy for the mandible has been the Class III lever, in which the condyle acts as a fulcrum, the masticatory muscles as applied force, and the bite pressure as resistance (Fig. 3.1) [1]. The Class III lever requires that the bite force be less than the applied force (e.g., the masseter and temporalis) to reach mechanical equilibrium in order for the masticatory “machine” to have a mechanical advantage of less than one. The notion of the believe that the mandible acts like a Class III lever has however been questioned by other researchers based on the claim that the condyle and articular fossa are fundamentally not designed by nature to withstand the large resultant forces needed by the lever model [2, 3]. Furthermore, criticisms were also raised by researchers that the Class III lever model does not include a consideration of the issue regarding the center of mandibular rotation [2, 4]. Moreover, observations from two studies have also raised concerns with the idea that whether or not if there is a force at the condyle, rather than the condyle being the mechanisms of force generation [2, 5]. The study by Roberts in 1974 postulated that it is not possible for an additional reaction force to be present at the condyle [5]. A © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. H. Choi, Bone Remodeling and Osseointegration of Implants, Tissue Repair and Reconstruction, https://doi.org/10.1007/978-981-99-1425-8_3

23

24

3 Understanding the Biomechanics

Fig. 3.1 Mandible as a Class III lever [1]

closed triangle can be created using vectors of the bite force together with the muscles of masseter and temporalis, as each force can be described by a vector whose length is equivalent to the force magnitude. In addition, since accurate and complete data on the direction of the three forces as well as the information concerning the magnitude of the muscle forces generated by the masseter and temporalis were unavailable, Roberts argues that the creation of a triangle with these forces in equilibrium is completely theoretical [5]. Likewise, the study by Roberts and Tattersall also stated that their analysis is based on the simple assumptions that during the elevation of the mandible, no significant force is expended at the temporomandibular joint [2]. The study by Parrington in 1934 suggested that the mandible could be considered as a stationary beam when analyzing external forces on the mandible during mastication [6]. This approach was later used in a number of studies where the internal stresses and strains on the mandible during mastication were examined [7–10]. The study by Stern has pointed out that since under conditions of equilibrium, the moments about any point will be equal to zero; hence, the figures regarding the amount of force which is being applied to the dentition will be the same when calculating moments about the center of the mandibular condyle, the center of the chin, the instantaneous center of rotation, or any other point in space [11]. Despite the fact that it may be appropriate to examine mammalian jaws purely in the lateral projection for bilateral molar or incisor biting, Hylander postulated in his study in 1975 that such an analysis for unilateral biting is incomplete [12]. This is based on the idea that the projected bite point is never located in the midsagittal plane, and as a result, it is beneficial to investigate the human mandible in the frontal projection. In his later study, he determined that during mastication, lower levels of compressive stress would act across the ipsilateral condyle, while a large compressive reaction force will act across the contralateral mandibular condyle during both molar biting and unilateral mastication [13]. Furthermore, the ipsilateral condyle might even be free of stress or there may be tensile stress acting across

3.2 Biomechanics of the Temporomandibular Joint

25

it under certain circumstances during unilateral molar biting. Additionally, the total magnitude of force at the condyles might not be affected by the narrowing or widening of the dental arches, but only the proportion each condyle will sustain as previously suggested [12, 14]. The existence of condylar forces is determined by the nature of muscle positions in the sagittal plane.

3.1.1 Hypotheses Based on Non-lever Action Disagreement also exists regarding the legitimacy of the lever action hypothesis and the human mandible behaving as a Class III lever. Either directly or indirectly, several researchers have proposed that there are little or no reactive forces detected at either of the mandibular condyle. The first study that disagreed with the idea of lever action of human jaw mechanics was by Wilson in the 1920, claiming that there can be no reaction force at the mandibular condyle since the resultant force of the muscles of the masseter, medial pterygoid, and temporalis rests perpendicular to the occlusal plane. Consequently, the interpretation of the human mandible as a lever is incorrect [15]. The concept of non-lever action of mandibular mechanics was further reinforced in the study by Robinson, suggesting that the resultant adductor muscle force passes through the first molar. Hence, biting on the first molar in this projection would indeed result in a non-lever action of the mandible, that is all the resultant muscle force would be transmitted directly through the teeth assuming that the resultant force has been correctly determined [3]. The study by Frank [16] also supported the non-lever action of the mandible, and the reasons were based on radiographic analyses of the human mandibular condyle. Frank observed in each of the radiographs that the condyle was never in direct contact with the articular eminence. Based on these findings, it was concluded that the mandible could not function as a lever as the condyle was not functioning as a fulcrum. In 1971, Gingerich proposed that the mandible performs like a link between the bite force and the adductor muscle instead of a lever during biting, and subsequently reinforcing the non-lever action hypothesis of the human mandible [17]. A later study by Tattersall has also indicated that the human mandible does not function as a lever during chewing or biting [18].

3.2 Biomechanics of the Temporomandibular Joint As a compound joint, the function and structure of the temporomandibular joint (TMJ) can be divided into two distinct systems:

26

3 Understanding the Biomechanics

1. The first system centers on the tissues encapsulating the inferior synovial cavity. The condyle-articular disk complex is the joint structure responsible for the rotational movement within the TMJ. This is due to the fact that the articular disk is attached tightly to the condyle by discal ligaments; therefore, the only physiological movement that can occur between these surfaces is rotation of the disk on the articular surface of the condyle. 2. The second system centers on the function against the surface of the mandibular fossa and is interrelated to the condyle-articular disk complex as discussed above. Given the articular disk is not firmly fixed to the articular fossa, free sliding movement can occur between these surfaces in the superior cavity. This movement occurs as a result of the mandible being positioned forward (translation). Translation therefore occurs in the superior joint cavity between the superior surface of the articular disk and the mandibular fossa. As muscle activity increases, the condyle is forced more intensely against the articular disk and the articular disk against the mandibular fossa. This in turn causes an increase in the interarticular pressure of these joint structures. In essence, the interarticular pressure is the pressure between the articular surfaces within the TMJ. The presence of interarticular pressure is crucial as the joint will technically dislocate and the articular surfaces will separate if there is no interarticular pressure [19]. As the interarticular pressure increases, the condyle will position itself in the intermediate zone as a result, and this is where the articular disk is the thinnest. Conversely, a reduction in the interarticular pressure will cause the disk space to widen, and this space is filled through the rotation of the articular disk to the posterior or anterior borders. Moreover, the width of the articular disk space varies with interarticular pressure. For instance, the disk space widens if the mandible is at the closed rest position due to the interarticular pressure being low. On the other hand, high pressure due to the clenching of the teeth will lead to a narrowing of the articular space. It is also vital to gain an insight into the correlation between the activity of the mandible and the articular disk. There is minimal elastic traction on the articular disk when the teeth are bought together and the condyle is in the closed joint position. During the opening movement of the mandible, the condyle is pulled forward along the articular eminence and the force to retract the articular disk is increased. The superior retrodiscal lamina is being pulled gradually by this opening movement. During the protrusion of the mandible, the superior retrodiscal lamina is also involved, and the ligament will keep the articular disk rotated as far posteriorly on the condyle as permitted by the width of the articular space as the mandible is moved to and returned from a full forward position. As discussed above, four pairs of muscles make up the muscles of mastication, and the lateral pterygoid muscle is one of those muscles. The superior lateral pterygoid is activated only in combination with the action of the elevator muscles during the closing movement of the mandible [19]. This is also accurate during a power stoke. Theoretically, it can be stated that the articular disk is protracted by the superior

References

27

lateral pterygoid, and the disk is pulled anteriorly and medially as soon as the pterygoid muscle is active. However, this function does not happen during mandibular movements. Gaining a more thorough understanding into the significance of the function of the superior lateral pterygoid during the power stroke can be achieved by observing the mechanics of chewing. A reduction in the interarticular pressure on the biting side is observed once resistance is encountered such as when biting on hard food during the closing movement of the mandible. This phenomenon takes place due to the force of closure being applied to food instead of the joint [19].

3.3 Concluding Remarks The shape of the mandible is designed by nature in such a way so that it can resist any shearing or bending stress as well as preventing any fractures during functional movements. The techniques typically used to record the biomechanical actions that accompany the functional loading of the human mandible are highly invasive; hence, a full and clear understanding into the events are still incomplete. In this regard, computer modeling and simulation offer a promising alternative approach with the added capacity to predict local stresses and strains in inaccessible positions. Mathematical approaches such as finite element analysis provide unparalleled precision in the representations of gradients, magnitudes, and directions of stresses and strains throughout the entire mandible. The biomechanical behavior of mammalian jaw could also be extrapolated through experimental studies, and to a certain extent, accuracy could be offered.

References 1. Smith RJ (1978) Mandibular biomechanics and temporomandibular joint function in primates. Am J Phys Anthropol 49:341–349 2. Roberts D, Tattersall I (1974) Skull form and the mechanics of mandibular elevation in mammals. Am Mus Novitates 2356:1–9 3. Robinson M (1946) The temporomandibular joint: theory of reflex controlled non-lever action of the mandible. J Am Dent Assoc 33:1260–1271 4. Grant PG (1973) Biomechanical significance of the instantaneous center of rotation: the human temporomandibular joint. J Biomech 6:109–113 5. Roberts D (1974) The etiology of the temporomandibular joint dysfunction syndrome. Am J Orthodont 66:498–515 6. Parrington FR (1934) On the cynodont genus galesaurus, with a note on the functional significance of the changes in the evolution of the theriodont skull. Ann Mag Nat Hist 13:38–67 7. Hylander WL (1977) In vivo bone strain in the mandible of galago crassicaudatus. Am J Phys Anthrop 46:309–326 8. Bock WJ, Kummer B (1968) The avian mandible as a structural grider. J Biomech 1:89–96 9. Badoux DM (1966) Statics of the mandible. Acta Morph Neerl Scand 6:251–256 10. Bock WJ (1966) An approach to the functional analysis of bill shape. Auk 83:10–51

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11. Stern JT Jr (1974) Biomechanical significance of the instantaneous center of rotation: the human temporomandibular joint. J Biomech 7:109–110 12. Hylander WL (1975) The human mandible: lever or link? Am J Phys Anthrop 43:227–242 13. Hylander WL (1979) An experimental analysis of temporomandibular joint reaction force in macaques. Am J Phys Anthropol 51:433–456 14. Hylander WL (1984) Stress and strain in the mandibular symphysis of primates: a test of competing hypotheses. Am J Phys Anthropol 64:1–46 15. Wilson GH (1920) The anatomy and physics of the temporomandibular joint. J Nat Dent Assoc 7:414–420 16. Frank L (1950) Muscular influence on occlusion as shown by x-ray of the condyle. Dent Digest 56:484–488 17. Gingerich PD (1971) Functional significance of mandibular translation in vertebrate jaw mechanics. Postilla 152:1–10 18. Tattersall I (1973) Cranial anatomy of archaeolmurinae (lemuroidea primates). Anthrop Pap Amer Mus Nat Hist 52:1–110 19. Okeson J (1985) Fundamentals of occlusion and temporomandibular disorders. The CV Mosby Co., USA

Chapter 4

Mathematical Analysis of the Mandible

During mastication, a bite force is produced by combining numerous muscles acting on the mandible at the same time. Presently, the biggest issue is defining the involvement of each muscle in creating this bite force, as the conditions of static equilibrium cannot be utilized in resolving this problem. Mastication can be defined as the process of chewing food and the food is crushed and grounded against the mandibular and maxillary teeth. This process is extremely complex that utilizes teeth, periodontal supportive structure, the muscles of mastication as well as the salivary glands, cheeks, tongue, palate, and lips.

4.1 Tooth Contact and the Biting Force Mastication consists of a rhythmic and well-controlled opening and closing of teeth in the mandible and the maxilla. Every opening and closing of the mandible signifies a chewing stroke and the movement pattern of a complete chewing stroke can be described by a tear-shaped pattern. The complete movement can be divided into an opening and a closing phase. Furthermore, the closing phase can be further divided into the crushing phase and the grinding phase [1]. The following sequence occurs if the mandible is traced in the frontal plane during a single chewing stroke: I. The mandible will drop downwards during the opening phase to a position in which the incisal edges of the teeth are approximately 16 to 18 mm apart [2]. The mandible then moves in a lateral direction 5 to 6 mm from the midline. The closing movement of the mandible begins at the end of this stage. II. The closing movement can be categorized into two distinct phases: crushing and grinding. Crushing is the first phase, and the food is trapped between the mandibular and maxillary teeth. The grinding phase is the second phase where the mandible is guided by the occlusal surfaces of the teeth back to the intercuspal © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. H. Choi, Bone Remodeling and Osseointegration of Implants, Tissue Repair and Reconstruction, https://doi.org/10.1007/978-981-99-1425-8_4

29

30

4 Mathematical Analysis of the Mandible

position, which causes the cuspal inclines of the teeth to pass across each other, permitting shearing and grinding of the mass of food. Slight anterior movement of the mandible can be observed throughout the opening phase during a regular chewing stroke if the movement of a mandibular incisor is drawn in the sagittal plane. The mandible on the other hand follows a somewhat posterior trace during the closing movement, finishing in an anterior movement returning to the starting maximum intercuspal location. Understanding the role of tooth contacts during mastication is vital. At the beginning, little contacts occur when food is introduced into the mouth. The incidence of tooth contacts increases as the food mass is broken down. Contacts will take place during every stroke in the final stages of mastication just before swallowing [3]. Two types of contact have been identified [4]: 1. Gliding, which occurs as the cuspal inclines pass by each other during the opening and grinding phases of mastication, and 2. Single, which occurs in the maximum intercuspal position. It has been noticed that around 194 ms is the average time length for tooth contact during mastication [5]. It is apparent that these contacts influence or even dictate the initial opening and final grinding phase of the chewing stroke based on these observations. Data related to the quality and quantities of tooth contacts as well as issues concerning the nature of the chewing stroke during mastication are constantly relayed back to the central nervous system. By utilizing this feedback mechanism, adjustments in the chewing stroke can then be carried out based on the kind of food being chewed. Typically, the maximum biting force that can be applied to the teeth varies between different individuals. The study by Brekhus and co-workers described the biting load for males vary from 53 to 64 kg while the maximum biting loads range from 35 to 44 kg for females [6]. Moreover, it has been observed that the magnitude of the maximum biting force appears to increase with age up to the point of adolescence [7, 8]. With exercise and practice, it has been shown that an individual can increase their maximum biting force over a period of time [6, 8, 9]. Observations from the study by Howell and co-workers suggested that the maximum force that can be applied to the central incisors was between 13 to 23 kg, and the maximum force that can be applied to the first molar was 41 to 90 kg [10]. The greatest amount of force is placed on the first molar region during chewing, and chewing mostly occurs on the first molar and second premolar area with harder foods [11]. Subsequently, an individual consuming a diet composed of large quantities of hard food will develop stronger biting force.

4.2 Calculating Forces at the TMJ As previously mentioned, defining the weight of each muscle in the creation of a bite force has been challenging, and one simple approach that has been hypothesized is

4.2 Calculating Forces at the TMJ

31

to assume each muscle will produce a force relative to its cross-sectional area [12]. For the temporomandibular joint (TMJ), the unknowns for bilateral static bite force would then be the magnitude of the proportionality constant and the magnitude and direction of the reaction force at the TMJ. These can be determined by balancing the horizontal and vertical components of the force projected onto the sagittal plane and the moments about an axis perpendicular to the sagittal plane [13]. Contradictory results were observed during calculations of the forces acting on the TMJ from mathematical models based on the disagreement centered on the role of the TMJ and its possible action as a load-bearing joint in spite of the amount of time spent by various researchers in addressing this issue [14–27]. With all vectors determined in three-dimensions, a complete model of all the forces contributing to the TMJ reaction force would be complex. The following information is needed to calculate the total joint reaction force: 1. The direction and magnitude of the bite force; and 2. The lengths of the moment arms in addition to the magnitude and direction of each muscle force. Force transducers can be quickly used to determine the extent and direction of the bite force, while the moment arm lengths can be established from lateral cephalograms of human subjects [15]. Anatomical considerations such as the total cross-sections of all muscle fibers or the physiological cross-sections as well as other physiological parameters such as the speed of contraction, level of neuronal activation, and the muscle length will determine the force magnitude each muscle can apply [28, 29]. The force magnitude can be estimated based on the data available on the physiological cross-sectional areas of various muscles (Table 4.1). The value of maximum muscle tension (┌), a correlation between the physiological cross-sectional area of a muscle, and the maximum force it can exert is given in Table 4.2. Two methods have been used to calculate the magnitudes of the muscle forces. The first approach utilizes the total cross-sectional area to estimate the force produced by each muscle [26, 28, 29, 33, 36]. The second approach, the force of each muscle is determined based on the integrated electromyogram (IEMG) from each muscle (Table 4.3) [20, 28, 37]. Table 4.1 Physiological cross-sectional areas of the muscles of mastication (cm2 ) [28, 29]

Muscles

Schumacher [29]

Pruim et al. [28]

Masseter

3.02

5.3*

Medial pterygoid

1.97

Lateral pterygoid

1.83

2.1

Temporalis

3.81

4.2**

Openers

–

1.0***

* Includes both masseter and medial pterygoid **Includes both anterior and posterior temporalis ***Only digastric was measured

32

4 Mathematical Analysis of the Mandible

Table 4.2 Average maximum muscle tension (┌) in N/m2 [28, 30–35] Author

Maximum muscle tension (┌)

Pruim et al. [28]

1.4 × 106

Ikai and Fukunaga [30]

0.7 × 106

Hettinger [31]

0.4 × 106

Carlsoo [32, 33]

1.1 × 106

Morris [34]

0.9 × 106

Fick [35]

1.0 × 106

Table 4.3 Averaged calculated maximum muscle forces (N) in each single muscle group and related standard deviations [28, 29, 32, 33, 38, 39] Muscle

Osborn and Baragar [38]

Prium et al. [28]

Schumacher [29]

Carlsoo [32, 33, 39]

Masseter

450

639 ± 176*

340

614 ± 107

Medial pterygoid

254

190

299 ± 46

Lateral pterygoid

382

378 ± 106

175

525

Temporalis (anterior)

264

362 ± 65

420**

519 ± 102

Temporalis (posterior)

323

197 ± 26

Openers

107

115 ± 40

305 ± 102 –

–

* Masseter + Medial Pterygoid **Anterior Temporalis + Posterior Temporalis

4.3 Barbenel’s Analysis The first analysis of the force actions at the TMJ was presented in 1972 by Barbenel [21]. The co-ordination system used in the study is shown in Fig. 4.1. During the biting phase, the force system imposed on the mandible when it is stationary comprise of the following three categories of forces: 1. Forces due to muscle action: The analyses were based on the mandible in the rest position, with the muscles of each side acting equally. The muscle forces considered in the analysis were those produced by the masseter (F masseter ), temporalis (F temporalis ) and the internal and external pterygoid (F Int Pterygoid and F Ext Pterygoid ) muscles (Fig. 4.2a). 2. Forces due to bite load: This load was arbitrarily considered acting equally on each side of the mandible, with a magnitude of L and directed at an angle θ to the y-axis. The moment of the load about the intercondylar axis:

4.3 Barbenel’s Analysis

33

Fig. 4.1 Co-ordinate axes used in joint force analysis. The z-axis is directed inwards along the intercondylar axis [21]

L(X cosθ − Y sinθ)

(4.1)

where X and Y is the co-ordinate of the point at which the line of action of the load intersects the occlusal plane (Figure 4.2b). 3. Force at the TMJ: The force was assumed to act at an angle of φ to the y-axis and have a magnitude R (Fig. 4.2c). Three equations of equilibrium for the mandible subjected to the force systems were derived by examining moments about the intercondylar axis and applying the determined muscle force components as shown in Table 4.4: a. Force component in the x-direction is equal to zero:

Fig. 4.2 a Muscles considered in the analysis, b bite load parameters and c temporomandibular joint force parameters [21]

34

4 Mathematical Analysis of the Mandible

Table 4.4 Muscle force parameters used in Barbenel’s analysis [21] Muscle

Moment arm about intercondylar axis (cm)

Force component x-axis

y-axis

z-axis −0.12

Masseter

2.7

0.55

0.84

Internal pterygoid

2.3

0.50

0.78

0.38

External pterygoid

0.0

0.91

−0.17

0.39

Temporalis

2.6

−0.46

0.84

−0.18

0.84Fmasseter + 0.84Ftemporalis + 0.77FInt Pterygoid − 0.17FExt Pterygoid − R cosφ − L cosθ = 0

(4.2)

− R sinφ + L sinθ = 0

(4.3)

b. Force component in the y-direction is equal to zero: 0.55Fmasseter − 0.46Ftemporalis + 0.50FInt Pterygoid + 0.91FExt Pterygoid

c. Sum of the moments of the forces about any axis is equal to zero: 2.7Fmasseter + 2.6Ftemporalis + 2.3FInt Pterygoid − L(X cosθ − Y sinθ ) = 0 (4.4)

4.4 Pruim, De Jongh, Ten Bosch’s Analysis In the 1980s, Prium and co-workers ascertained all muscle forces and the forces in the TMJs during bilateral biting at three separate locations on the human dentition using a mechanical approach [28]. Data on integrated electromyographic activity (IMEG) obtained using surface electrodes to bilaterally record the electromyographic activity in the main muscles as well as bite forces measured bilaterally through the application of two calibrated steel transducers together with strain gauges were fed into a calculation program as inputs in an effort to estimate these forces. It was also stated that the activities of the lateral pterygoid muscles were not recorded in their study. Furthermore, an agreement was also reached that the centroids of the mandibular condyle will be the origins of an x- and y-axis system in which the x-axis was oriented parallel to the mandibular occlusal plane, thus to the bite plane in their experimental set-up (Fig. 4.3). In their study, a number of assumptions were necessary: 1. A single vector can be used to signify the combined forces of the masseter and medial pterygoid muscles. The lateral pterygoid muscle acts in a direction

4.4 Pruim, De Jongh, Ten Bosch’s Analysis

35

Fig. 4.3 Mathematical model based on the mandible [28]. (M. + M. P.) Masseter + medial pterygoid; (B) bite force; (A. T.) anterior temporalis; (P. T.) posterior temporalis; (L. P.) lateral pterygoid; (O) opener; (J. R.) joint reaction force

parallel to the occlusal plane and consequently, the force the muscle produced has no y-component. 2. Forces exerted by contracting muscles can be represented by vectors and the direction of these vectors can be defined by the connecting lines between the centroids of the origins and the insertions of the muscles. 3. The EMG activity of the masseter is also representative for the medial pterygoid muscle. The activity in the floor of the mouth is illustrative for the mouth openers. There exists a muscle independent value ┌ (in N/m2 ) for each experimental subject, which correlates the maximum force a muscle can exert (F m(max) ) and the physiological cross-section of the muscle φm (in m2 ) as described in Eq. 4.5: Fm(max) = ┌ · φm

(4.5)

The values of the physiological cross-section of muscles for male subjects correspond to the values for test subjects as determined by Schumacher [29] and are given in Table 4.5. During isometric contraction, the force a muscle can exert (F m ) is directly proportional to the IEMG. Hence, Eq. 4.6 can be used to express the correlation between IEMG and muscle force: ( ) IEMGm(actual) (4.6) · Fm(max) Fm = IEMGm(max)

36

4 Mathematical Analysis of the Mandible

Table 4.5 Physiological cross-sections of the muscles (cm2 ) [28, 29]

Muscles

Schumacher [29]

Pruim et al. [28]

Masseter

3.4

5.3*

Medial pterygoid

1.9

Anterior temporalis

4.2

2.6

Posterior temporalis

4.2

1.6

Openers

–

1.0

*

Includes both masseter and medial pterygoid

where IEMGm(max) is the maximum value of IEMGm of a single muscle ever recorded during a complete experimental session from a pair of electrodes. To calculate all forces, the laws of static equilibriums were used: a. When the moments are referred to the origin and Eq. 4.6 is used, a new equation is derived: ∑ ( IEMGm(actual) ) (4.7) ┌ · φm · am + Fb · ab = 0 IEMGm(max) muscles Values of IEMGm and the sum of both left and right bite forces (F b ) were obtained for each bite, and the muscle and bite force lever arms (am and ab , respectively) were obtained from the cephalometric data. For each bite, ┌ was determined using Eq. 4.7 along with the data on the physiological cross-sections of muscles (φ) from Table 4.5. All forces in the elevator and antagonistic muscles during that bite were calculated after substituting the value of ┌ in Eq. 4.5. b. The sum of the moments = 0 ∑ Fm · am + Fb · ab = 0 (4.8) c. The sum of the x-component = 0 ∑

Fm · cos αm + F j · cos α j + F p = 0

(4.9)

muscles

d. The sum of the y-component = 0 ∑ muscles

Fm · sin αm + F j · sin α j − Fb = 0

(4.10)

4.5 Throckmorton and Throckmorton’s Analysis

37

4.5 Throckmorton and Throckmorton’s Analysis In the study by Throckmorton and Throckmorton [13], the magnitude of the total joint reaction force (F TMJ ) and its direction (φ) were determined using a 2-D model as shown in Fig. 4.4. The laws of statics were used to determine the magnitude and direction of the TMJ reaction force in two stages. The sum of all moments is zero and the sum of all forces is zero in static equilibrium. The first step is to determine the magnitude of the muscle forces from the sum of the moments: F(t) · a + F(m) · b + F(B) · c = 0

(4.11)

where F (t) , F (m) , and F (B) are the forces exerted by the temporalis, masseter, and bite force, respectively (Fig. 4.4). Determining the perpendicular distance between the muscle vector and the center of joint rotation is the most widely used approach to calculate the moment of the jaw muscles. The moment of a particular muscle is derived from multiplying the measured perpendicular distance with the magnitude of the muscle force vector.

Fig. 4.4 Schematic of the 2-D model used by Throckmorton and Throckmorton to determine the total joint reaction force [13]. F (t) is the temporalis muscle force; F (m) is the masseter muscle force; F (mB) is the molar bite force; F (iB) is the incisor bite force; F (TMJ) is the total joint reaction forces; (a) is the moment arm for the temporalis muscle; (b) is the moment arm for the masseter muscle force; (c) is the moment arm for the bite force; (θt ) is the angle with the occlusal plane for the temporalis muscle force; (θm ) is the angle with the occlusal plane for the masseter muscle force; (θB ) is the angle with the occlusal plane for the bite forces; (φ) is the angle of the total joint reaction force with the moment arm line (MAL)

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4 Mathematical Analysis of the Mandible

Another method is to separate each muscle force into components which are perpendicular and parallel to a line passing through the center of joint rotation and parallel to the occlusal plane. This line is known as the moment arm line (MAL) and is shown in Fig. 4.4. Only the force components that are perpendicular to the moment arm line generate a moment, and the length of the moment arm is determined by measuring the distance between the muscle force vector and the center of joint rotation (Fig. 4.4). Consequently, Eq. 4.11 can be further expanded: F(t) · sin θt · a + F(m) · sin θm · b + F(B) · sin θB · c = 0

(4.12)

Even though the magnitude of the forces exerted by the masseter and temporalis muscles cannot be measured directly, the magnitude of one muscle force can be expressed in terms of the other if the relative magnitudes of the muscle forces are estimated: F(m) = K · F(t)

(4.13)

where K is a proportionality constant between the forces exerted by the masseter and temporalis. The magnitudes of the forces exerted by the masseter and temporalis can now be determined by substituting the relative muscles forces into Eq. 4.12: [ F(m) =

F(B) · sin θB · c K · sin θt · a + sin θm · b

] (4.14)

Using Eq. 4.13, the value of F (t) can finally be calculated. Once the magnitude of each muscle force is calculated, the reaction force at the TMJ (F (TMJ) ) can be determined using the sum of forces: F(m) + F(t) − F(B) − F(T M J ) = 0

(4.15)

Each force is separated into horizontal components of force which are parallel to the MAL, and vertical components of force which are perpendicular to the MAL. Furthermore, the horizontal components of force produce translation and the vertical components of force generate rotation [40]: a. For the horizontal component of the reaction force at the TMJ (F (H.TMJ) ): F(H.TMJ) = F(m) · cos θm + F(t) · cos θt − F(B) · cos θB

(4.16)

b. For the vertical component of the reaction force at the TMJ (F (V.TMJ) ): F(V.TMJ) = F(m) · sin θm + F(t) · sin θt − F(B) · sin θB

(4.17)

References

39

c. Total joint reaction force (F (TMJ) ) and its direction φ: F(TMJ) =

/

2 2 F(H.TMJ) + F(V.TMJ)

(4.18)

F(V.TMJ) F(TMJ)

(4.19)

∅ = Ar c sin

4.6 Concluding Remarks Different approaches have been developed focusing on how to characterize the actions of the human mandible, all of which had to contend with the major obstacle that stress must be deduced by other means, and it is not something that can be directly measured. Free-body diagrams offer vectorial description of suggested external loads during mastication, and this in turn allows for a rough estimation of the nature of internal forces acting within the mandible. However, with this type of approach, any thorough examination of stress distribution might prove to be problematic. They are still useful when attempting to interpret results from theoretical and experimental studies as they might aid in the identification of how different external forces and moments interact to generate internal stresses.

References 1. Okeson J (1985) Fundamentals of occlusion and temporomandibular disorders. The CV Mosby Co., USA 2. Hildebrand GY (1931) Studies in the masticatory movements of the lower jaw. Walter De Gruyter, Berlin 3. Adams SH, Zander HA (1964) Functional tooth contacts in lateral and centric occlusion. J Am Dent Assoc 69:465 4. Glickman I, Pameijer JHN, Roeber FW et al (1969) Functional occlusion as revealed by miniaturized radio transmitters. Dent Clin North Am 13:667 5. Suit SR, Gibbs CH, Beng ST (1975) Study of gliding tooth contacts during mastication. J Periodontol 47:331 6. Brekhus PH, Armstrong WD, Simon WJ (1941) Stimulation of the muscles of mastication. J Dent Res 20:87 7. Garner LD, Kotwal NS (1973) Correlation study of incisive biting forces with age, sex, and anterior occlusion. J Dent Res 52:698 8. Worner HK, Anderson MN (1944) Biting force measurements in children. Aust Dent J 48:1 9. Worner HK (1939) Gnathodynamics: the measurements of biting force with a new design of gnathodynamometer. Dent J Aust 43:381 10. Howell AH, Manly RS (1948) An electronic strain gauge for measuring oral forces. J Dent Res 27:705 11. Brudevold F (1951) A basic study of the chewing forces of a denture wearer. J Am Dent Assoc 43:45 12. Amis AA, Dowson D, Wright V (1980) Elbow joint force predictions for some strenuous isometric actions. J Biomech 13:765–775

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13. Throckmorton GS, Throckmorton LS (1985) Quantitative calculations of temporomandibular joint reaction forces—I. the importance of the magnitude of the jaw muscle forces. J Biomech 18:445–452 14. Hylander WL (1979) An experimental analysis of temporomandibular joint reaction force in macaques. Am J Phys Anthropol 51:433–456 15. Hylander WL (1978) Incisal bite force direction in humans and the functional significance of mammalian mandibular translation. Am J Phys Anthropol 48:1–7 16. Hylander WL (1975) The human mandible: lever or link? Am J Phys Anthrop 43:227–242 17. Smith RJ (1978) Mandibular biomechanics and temporomandibular joint function in primates. Am J Phys Anthropol 49:341–349 18. Roberts D (1974) The etiology of the temporomandibular joint dysfunction syndrome. Am J Orthodont 66:498–515 19. Roberts D, Tattersall I (1974) Skull form and the mechanics of mandibular elevation in mammals. Am Mus Novitates 2356:1–9 20. Barbenel JC (1974) The mechanics of the temporomandibular joint: A theoretical and electromyographical study. J Oral Rehabil 1:19–27 21. Barbenel JC (1972) The biomechanics of the temporomandibular joint: a theoretical study. J Biomech 5:251–256 22. Gingerich PD (1971) Functional significance of mandibular translation in vertebrate jaw mechanics. Postilla 152:1–10 23. Roydhouse RH (1955) The temporomandibular joint: Upward force of the condyles on the cranium. J Am Dent Assoc 50:166–172 24. Craddock FW (1951) A review of Costen’s syndrome. Br Dent J 91:199–204 25. Robinson M (1946) The temporomandibular joint: theory of reflex controlled non-lever action of the mandible. J Am Dent Assoc 33:1260–1271 26. Gysi A (1921) Studies on the leverage problem of the mandible. Dent Digest 27;74–84, 184– 190, 203–208 27. Wilson GH (1920) The anatomy and physics of the temporomandibular joint. J Nat Dent Assoc 7:414–420 28. Pruim GJ, de Jongh HJ, ten Bosch JJ (1980) Forces acting on the mandible during bilateral static bite at different bite force levels. J Biomech 13:755–763 29. Schumacher GH (1961) Funktionelle Morphologie der Kinnmuskulatur VEB. Gustav Fischer, Jena 30. Ikai M, Fukunaga T (1968) Calculation of muscle strength per unit cross-sectional area of human muscle by means of ultrasonic measurement. Int Z Agnew Physiol 26:26–32 31. Hettinger T (1961) Physiology of strength. Charles C. Thomas, Illinois 32. Carlsoo S (1958) Motor units and action potentials in masticatory muscles. Acta Morph Neel Scand 2:13–19 33. Carlsoo S (1952) Nervous co-ordination and mechanical function of the mandibular elevators. an EMG study of the activity, and an anatomic analysis of the mechanics of the muscles. Acta Odont Scand 10:Suppl. 1–132 34. Morris CB (1948) The measurement of the strength of muscle relative to the cross-section. Res Q Am Assoc Hlth Phys Educ 19:295–303 35. Fick H (1904) Handbuch der Anatomie und Mechanik der Gelenke. Gustav Fisher, Jena 36. Mainland D, Hiltz JE (1934) Forces exerted on the human mandible by the muscles of mastication. J Dent Res 14:107–124 37. Pruim GJ, ten Bosch JJ, de Jongh HJ (1978) Jaw muscle EMG-activity and static loading of the mandible. J Biomech 11:389–395 38. Osborn JW, Baragar FA (1985) Predicted pattern of human muscle activity during clenching derived from a computer assisted model: symmetric vertical bite forces. J Biomech 18:599–612 39. Carlsoo S (1956) An electromyographic study of the activity, and an anatomic analysis of the mechanics of the lateral pterygoid muscle. Acta Anat 26:339–351 40. Bock WJ (1968) Mechanics of one- and two-joint muscles. Am Mus Novit 2319:1–45

Chapter 5

Understanding Bone Structures

Within the human body, bone is responsible for a number of functions including synthetic functions (production of blood cells), metabolic functions (fat storage, mineral storage and balance, and regulation of calcium and phosphate ions), and mechanical functions (under functional loadings). Numerous experimental investigations have been conducted to determine the elastic properties of human bones, and one of the key findings from these experiments is that the mechanical properties of bone will differ based on the orientation and position of the specimen; simply, bone is both anisotropic and heterogeneous in its mechanical properties. Given the fact that bone is an anisotropic and heterogeneous material, experimental results will be governed by the anatomical location of the specimen. A pivotal role is also played by the parameters or settings used during experimentations such as the duration of load and deformation rate in mechanical tests, or the vibration frequency in ultrasonic tests. The use of ultrasonic waves is well-established in determining the elastic properties of bone. A recent study has hypothesized that there is a clinical potential for ultrasounds to measure bone properties in different populations, i.e., between adult and juvenile groups [1]. The elastic properties are determined based on the velocity measurements of shear and longitudinal waves propagating in certain directions in the bone sample if the elastic anisotropy and density of bone are stated. Properties and features of bone will also change based on the type of bone as well as biological factors such as age, the extent of pathological degradation, and level of activity of the living bone. Variations in these features along with the preservation conditions of the specimen until it is used for experimentation all contribute to an irregular distribution of its mechanical properties [2, 3]. Bone occurs in two forms from a macroscopic point of view, and most bones in the human body have both types: the dense compact (cortical) bone forming an outer shell that surrounds a core of spongy cancellous bone. Both types consisted of onethird organic and two-thirds inorganic materials. The inorganic portions are mainly calcium phosphate and calcium carbonate, while the organic portions consist of cells. There are three types of cells that make up the organic component: osteoblasts, © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. H. Choi, Bone Remodeling and Osseointegration of Implants, Tissue Repair and Reconstruction, https://doi.org/10.1007/978-981-99-1425-8_5

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5 Understanding Bone Structures

which are generally thought of as the cells responsible for bone formation; osteoclasts, which form the matrix and are also responsible for bone demineralization; and the fibroblasts are responsible for the formation of collagenous fibers. Both the osteoblasts and osteoclasts are imprisoned within the “Haversian” and surrounding network, and it has been suggested that these cells may act as site-specific activators and regulators controlling the creation of more bone tissues [4, 5]. Observations from scanning electron microscopy studies revealed the sizes and distribution of the mineral and organic phases within the cortical bone that begins at the elemental level (less than 0.005 μm) and microstructural level (between 1 and 10 μm) will impact the mechanical characteristics of cortical bone [6]. A relationship exists between the mineral fraction and the mechanical properties of cortical bone [7–10]. Tiny changes in the mineral fraction will result in a significant change in the material properties. The mechanical behavior of cortical bone at the next level in the hierarchy (between 10 and 50 μm) can be ascertained from microhardness examination [11], which measure the physical effects of small-scale changes in the mineral content of the bone [12]. As mentioned earlier, nanoindentation has been used to investigate the properties of hard tissues such as bone since the 1990s [13–22]. Measuring the mechanical properties such as Young’s modulus and hardness at the surface of a material can be achieved through the use of this technique. The procedure for nanoindentation is much simpler particularly for small complex-shaped samples such as enamel, cementum, and dentine compared to other conventional mechanical examinations such as tensile testing [23]. Above all, nanoindentation allows the measurement of mechanical properties in a very small selected area within the specimen where the dimensions may be at a micrometer or even nanometer scale. This is particularly important when measuring local properties of non-homogeneous structures such as dental calcified tissues. Typically, the properties of mandibular cortical bone are deduced in three orthogonal directions relative to the surface of each sample: longitudinal (axial along the bone surface), radial (normal or perpendicular to the bone surface), and tangential (supero-inferior along the bone surface) (Fig. 5.1). A limitation of the loading experiments is that loading could not be carried out in the radial direction due to the overall thickness of the cortical bone in this direction. Hence, the yield stresses and elastic constants were estimated for the radial direction from the data obtained in the tangential direction (Table 2.1) [24, 25]. A number of studies have suggested that the cortical bone of the mandible is stiffer in the longitudinal than in the tangential and radial directions, indicating anisotropic behavior [24–28]. The average elastic moduli in the tangential and radial directions are approximately 40–70% of those along the longitudinal direction [21]. More importantly, the values of elastic moduli in the tangential and radial directions are roughly the same, and as a result, the cortical bone of the mandible can be considered transversely isotropic, with a greater Young’s modulus in the longitudinal direction and a lower Young’s modulus in all transverse directions. The strength of the mandible is also higher in the longitudinal direction than in transverse directions (Table 2.1). Similarly, the study by O’Mahony et al. also suggested that a model of transverse

5 Understanding Bone Structures

43

Fig. 5.1 Definition of directions

isotropy for the mandibular cancellous bone [29]. They observed Young’s modulus in the infero-superior direction was less than the bucco-lingual and mesio-distal directions, and the difference between the mesio-distal and bucco-lingual directions was negligible, indicating the symmetry axis is along the infero-superior (weakest) direction. Of vital significance is the constitutive properties of cancellous or trabecular bone as it is this bone that is in direct contact with the implant or prosthesis. Given that cancellous bone is a cellular material consisted of a connected network of rods or plates where a network of rods produces open cells and a network or plates creates closed cells, its mechanical behavior is typical of a cellular material. The stress– strain curve of such materials has three distinct regimes of behavior. The compressive behavior in the first regime is linear elastic as the cell walls bend or compress axially, and eventually, the cells start to collapse at high enough loads via elastic buckling, which also leads to brittle fracture or plastic yielding of the cell walls. This second phase of collapse progresses at a roughly constant load until the cell walls meet and touch. Once this happens, the resistance to load increases, giving rise to a final increasingly steep portion of the stress–strain curve (Fig. 5.2) [30]. The direction of the applied loads will govern the symmetry of the structure in cancellous bone. For instance, the trabeculae often develop a columnar structure with cylindrical symmetry in bone where the loading is primarily uniaxial such as the vertebrae. However, if the stress pattern in the cancellous bone is complex, then the structure of the network of trabeculae is also complex and highly asymmetric [31, 32]. It has been proven both theoretically and experimentally that Young’s modulus of cancellous bone is strongly dependent on the bone’s structure density or sometimes referred to as apparent density [33–38]. Cancellous bone has a structure with distinctive textural anisotropy in many areas within the skeleton and subsequently cannot be adequately predicted by a single scalar measure such as structural density.

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5 Understanding Bone Structures

Fig. 5.2 Compressive behavior of a stress–strain curve for cancellous bone. According to the study by Hayes and Carter [30], Young’s modulus and compressive strength increase as the relative density increases ((1) apparent density = 962 gm/cc; (2) apparent density = 697 gm/cc; and (c) apparent density = 559 gm/cc) (Modified and adapted from [30])

The elastic constants of cancellous bone in such instances depend upon its textural anisotropy in addition to its structural density. Even though the elastic moduli and yield strength ascertained in the studies by Arendts and Sigolotto displayed regional variation (up to a factor of one and a half times between minimum and maximum values), their values did not indicate systematic differences between particular locations of the mandible such as lower margin vs. upper margin or ramus vs. corpus [24, 25]. Conversely, a later study by Dechow et al. revealed the stiffest region of the mandibular corpus is at the lower border inferior to the canine, suggesting that this region has the highest resistance to torsion and is the region in which torsional stresses are believed to change during function [39]. Furthermore, additional data from two other studies indicated that [40, 41]: I.

An increase in the longitudinal Young’s modulus going from the molar region to the symphysis, II. The lingual cortex is stiffer than the buccal cortex in the symphysis and premolar regions, and III. The cortical bone is stiffer at the lower border of the corpus than at the alveolus. In comparison with the molar region in the body of the mandible, a greater longitudinal Young’s modulus at the symphysis may compensate for the anticipated higher torsional loadings experienced in that area. This hypothesis was reinforced by the findings on apparent density and mineralization density, revealing that the areas of the highest density are located at the symphysis [42, 43].

5.1 Bone Healing Process The process of bone healing is well-coordinated and complex. Once a fracture occurs, bone begins healing indirectly through the formation of callus or directly through

5.1 Bone Healing Process

45

Fig. 5.3 Stages of bone healing process. a Hematoma formation, b Fibrocartilaginous callus formation, c Bony callus formation, d Bone remodeling. Modified and adapted with permission from [46] (http://creativecommons.org/licenses/by/4.0/)

bone union. As healing progresses, cartilaginous callus or soft callus is created through the activities of endothelial and skeletal cells that bridge the gap between the bone fragments. Soft callus then develops into hard callus (Fig. 5.3) [44, 45]. Intramembranous and endochondral ossification are two typical mechanisms of bone formation [47, 48]. Cartilage is formed during endochondral ossification when mesenchymal stem cells (MSCs) are differentiated into chondrocytes and will eventually be replaced by bone. On the other hand, bone tissue in intramembranous ossification is produced directly by osteoblasts created through the differentiation of MSCs [44, 48, 49]. Bone healing can be categorized into both primary bone healing and secondary bone healing. If the bony fragments are firmly united under compression from implantation, then primary bone healing will take place and the two fragments are joined and healed directly through the activities of osteoclasts and osteoblasts. More importantly, there is no formation of callus [44, 45, 50]. The most common category of bone healing is secondary bone healing and this occurs if there is a small quantity of motion in the fracture site. Formation of soft callus is the result of interfragmentary motion, leading to secondary bone formation through both intramembranous and endochondral ossification [45, 51]. Osteoblasts, osteocytes, osteoclasts, and osteoprogenitor bone cells (osteogenic and develop into osteoblast cells) are the four most important cells in this bone healing process, which together are recognized as the basic multicellular unit (BMU) involved in bone regeneration [52, 53]. The osteoblasts are responsible for the production of bone matrix constituents and are found in clusters along the bone surface, “coating” the layer of bone matrix they are creating [54]. The origin of the osteoblast cells can be traced back to multipotent MSCs, which has the capacity to separate into osteoblasts as well as into myoblasts, adipocytes, fibroblasts, and chondrocytes [54, 55]. Conversely, osteoclast cells are responsible for bone resorption and they are derived from hematopoietic cells of

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5 Understanding Bone Structures

the mononuclear lineage and are giant multinucleated cells with a diameter of up to 100 nm [56]. The term basic multicellular unit is frequently used to describe the close collaborative efforts of osteoblasts and osteoclasts in the remodeling process. Bone formation and bone resorption are balanced in a homeostatic equilibrium. On a cellular level, the simultaneous biological and mechanical actions play a critical role in the delicate equilibrium between bone formation, growth, and resorption. Moreover, the combined efforts of osteoblasts and osteoclasts also played a part in the bone remodeling of defects such as microfractures [57, 58]. The clinical evaluation of fracture healing involves a number of stages, and the complete process can be assessed using an assortment of testing methods such as histology, mechanical tests, radiographic inspections, and computerized tomography (CT) scans [59]. The mechanical properties of bone undergoing the healing process can be evaluated using testing methods such as torsion, tensile compression, or flexure [60, 61]. Torsion and four-point-bending tests are the most ideal for assessing the mechanical function of a healing bone and in particular for long bones. Normally, torsion test is a more suitable approach as the same amount of torque is applied across the entire cross-sectional area of the callus. Moreover, the test provides a more realistic simulation of orthopedic surgical procedures. On the other hand, a non-uniform bending moment might be generated throughout the callus by fourpoint bending [62]. Precautions must also be taken if three-point bending tests are used to evaluate the mechanical properties of a healing bone, especially during the early stages of the healing process. This is due to concerns arising from the site where the force is applied might be situated at the original fracture spot, and at these early stages of healing, the new bone tissue is mainly composed of cartilage, calcified cartilage, or less mature bone tissue depending on how far the healing has progressed [62]. Histological analysis as well as radiographic inspections can be used if necessary to assess biological repairs. In 2001, it was suggested that acoustic emission could be utilized as a non-destructive method for monitoring fracture healing [63]. In their study, acoustic emission was used to monitor the yield strength of healing callus during external fixation in 35 patients [64]. The most common radiographic investigations of fracture healing involve the bridging of fracture site by callus, obliteration of the fracture line, and continuity. At the moment, 3-D CT scan reconstructed images or microtomography can be used in research projects to determine the volume of callus [65]. It has been suggested that there is a necessity to quantitatively and non-invasively estimate the quality and strength of fracture callus during bone healing in an effort to determine the effectiveness of certain treatments and that bone strength could be predicted using a patient/specimen-specific finite element (FE) model created using CT scans to compensate for the inability to measure quantitatively the bone strength in the fracture healing process in clinical settings. Moreover, the prediction of callus strength using the FE approach would enable an objective treatment plan [66, 67]. The study by Shefelbine et al. suggested that the mechanical properties of fracture callus could be estimated using micro-CT scan data and FE modeling [67]. Later,

5.1 Bone Healing Process

47

using New Zealand white rabbits where a 10 mm long bone defect was created in the central femur, a relationship was developed by Suzuki et al. that converts bone density measured by quantitative CT scans to Young’s modulus. A specimenspecific FE model was generated based on CT data after the femur was removed after 3, 4, and 5 weeks, and mechanical testing was carried out at the same time. Based on their findings, they suggested that the bone strength could be quantitatively predicted during the bone healing process by finite element modeling (FEM) using appropriate equations that convert data from CT scans to material properties [66]. Such CT scan-material properties relationships will be discussed in details in the patient-specific modeling chapter. Compared to CT scans, bone histology is an invasive approach used in the clinical environment to investigate the bone structure during fracture healing. Tissue heterogeneity within callus is not fully captured in a majority of healing studies that involve animal trials as well as compared to single longitudinal sections of a healing fracture. Consequently, transverse sections at the fracture line level can offer the most accurate measurement of cross-sectional area in addition to an estimation of tissue heterogeneity [68]. Concerning callus cross-section, bone tissue, and fibrous cartilage can be represented as the percentage of the total cross-sectional callus area [68]. Bone metabolism can be examined by staining both osteoblasts and osteoclasts, and their activities can then be quantified. Furthermore, bone formation-related factors can be assessed using fluorochromes and these can be utilized to assess and determine bone remodeling and stress shielding particularly during the middle and final stages of fracture healing [69]. In animal trials, calcein can be injected before the animals such as rats or mice are sacrificed typically at 2, 4, 6, and 16 weeks after femoral osteotomy. Linear calcein labeling indicated lamellar bone formation during callus remodeling. Fracture healing is a regenerative bone healing process where bone is repaired without the formation of any scar tissue. The healing sequence begins with a cycle of inflammation, cell migration, proliferation, and differentiation of progenitors into mature osteoblasts and osteoclasts under certain biomechanical factors and remodeling [46]. Immune cells infiltrate into the hematoma during the bone healing process. This triggers the release of cytokines and thus causes inflammation [44]. It has long been considered that the immune system is vital in the healing of bone fractures given that inflammation goes before bone regeneration. Clinicians have reported that there is a delay in bone healing in patients treated with immunosuppressants, and the occurrence of non-union is more frequent in certain immune-deficient patients. It has been well-proven that bone healing begins with an inflammatory reaction that set off the regenerative healing process and ultimately results in the reconstitution of bone. During this initial period of bone healing, an unbalanced immune reaction has been postulated to disturb the healing cascade in a way that delays the bone healing process. The immune cell composition and expression pattern of angiogenic factors can be examined using a sheep bone osteotomy model and compared to a mechanically induced delayed/impaired bone healing group [70]. It has been suggested that significantly high T cell percentages were present in the bone hematoma in the delayed healing and the bone marrow adjacent to the osteotomy

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5 Understanding Bone Structures

Fig. 5.4 Bone healing and remodeling sequence. Modified and adapted with permission from [71] (http://creativecommons.org/licenses/by/4.0/)

gap in comparison to the normal healing bone. Moreover, this was reflected in the higher cytotoxic T cell percentage discovered under delayed bone healing conditions signifying longer pro-inflammatory process. It has also been reported that the highly activated periosteum adjourning the osteotomy gap displayed lower expression of hematopoietic stem cell markers and angiogenic factors such as vascular endothelial growth factor and heme oxygenase. This indicates that a deferred revascularization of the injured area as a result of ongoing pro-inflammatory processes in the delayed healing in bone fractures [70]. Observations from a number of studies also revealed that there are unfavorable immune cells and factors contributing in the initial healing phase (Fig. 5.4) [71]. Observations from the study by Könnecke and co-workers in which the healing of femoral fractures in mice was analyzed using confocal microscopy revealed that immune cells are discovered only in the endosteal region near the fractured bone segments and edges during the formation of soft callus [72]. It was also discovered that T and B cells were absent in the avascular cartilage filling the fracture callus. In areas of newly formed woven bone, T and B cells reappear in the callus upon subsequent hypertrophy of the cartilage and the region was revascularized. The presence of large quantities of T and B cells recorded during the bone growth and remodeling process correlates with the increasing numbers of osteoblasts and osteoclasts in the vicinity of newly formed bone. Reaction of the regional lymphoid tissue, lymphatics, and lymph nodes takes place after tissue damage via mechanical injury and inflammation. Clinical studies have shown that closed fractures of a lower limb triggered reaction of the local lymphoid tissue. Lymphatics were dilated draining the site of the fracture and enlargement

5.2 Fracture Healing and Numerical Simulation

49

of inguinal lymph nodes. Furthermore, these changes continued even after clinical healing of the fracture. On the other hand, the draining lymphatics became obliterated and the lymph nodes disappeared in the long-lasting non-healing fractures [73–75]. Changes in iliac and popliteal lymph nodes draining lymph in rats were examined from the region of tibial fracture and adjacent soft tissue injury [75]. Observations displayed a reaction of local lymph nodes to internal wounds took place through mobilization of cells from the blood circulation accompanied by activation of cellular subsets. The authors of the study also stated that the absence of major changes in the frequency of lymph node cell subpopulation indicated that lymph nodes are constitutively prepared for influx of antigens from damaged tissues and only react with increases in cell number and cell activation. The fracture gap tissue was assumed to be the leading source of signals to the lymph nodes, releasing cellular and humoral regulatory factors [74]. Later, the existence of a functional axis was hypothesized between the local lymphatic (immune) system and injuries to bone and surrounding soft tissue, suggesting that fast healing is controlled by influx into the wound of lymph node regulatory cells while prolonged healing resulted in gradual exhaustion of the regional lymph node functional elements and reciprocally impairment in sending regulatory cells to the fracture gap [73]. In the context of bone fracture healing, B cells are recognized to increase at the injury site and in the peripheral blood [72, 76]. It has been suggested that a portion of B cells suppresses excessive and/or prolonged inflammation. The functions of other B cells subsets in bone regeneration remain largely unclear and further investigations are essential.

5.2 Fracture Healing and Numerical Simulation Maintaining the integrity of the skeleton has been accepted as the principle behind the bone remodeling sequence. The bone remodeling sequence consisted of three consecutive phases: resorption, reversal, and formation. The resorption phase begins with the creation of multinucleated osteoclasts as a result of partially differentiated mononuclear pre-osteoclasts migrating to the bone surface. The commencement of the reversal phase is marked by the appearance of mononuclear cells on the bone surface in addition to the initiation of osteoclastic resorption. Signals for the differentiation and migration of osteoblasts and the creation of surfaces for new osteoblasts to begin bone formation are provided by mononuclear cells. The formation phase is the final stage and it occurs when the new bone deposited by osteoblasts completely replaces the resorbed bone. Flattened lining cells over the entire surface once the formation phase are completed. A prolonged resting period will take place until the beginning of a new remodeling sequence [54]. In determining the ideal mechanical-based treatment or reconstruction after an accident or illness, the use of numerical models to simulate the process of fracture healing may prove to be advantageous. Numerical simulation using finite element analysis (FEA) to predict the fracture healing process has been previously reported

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with the prediction validated to a certain extent utilizing the experimental works of Claes et al. on sheep models [77, 78]. A FE model of a fracture callus was used by Lacroix and Prendergast to simulate the time course of tissue differentiation during fracture healing. Their simulation began with granulation tissue (postinflammation phase) and ended with bone resorption. The biomechanical regulatory model assumed that the combination of fluid flow acting within the tissue and shear strain governed tissue differentiation. Low fluid flows and shear strain stimulated the formation of cartilage and ossification at even lower levels. On the contrary, high fluid flows and shear strain stimulated the formation of fibrous connective tissue based on the assumption that at high levels, the precursor cells are deformed. The appearance and disappearance of various tissues found in a callus were similar to histological observation when their mechano-regulatory scheme was examined by simulating healing in fractures with different loading magnitudes and gap sizes [79]. Later, the influence of mechanical stimuli on cellular processes (proliferation, migration, and differentiation) that occurred during fracture healing was attempted in the study by Gómez-Benito and co-workers. The model, on the basis of these three processes, then simulated the evolution of geometry, distributions of cell types, and elastic properties inside a healing fracture. The three processes were implemented in a FE code as a combination of three coupled analysis stages (a biphasic, a diffusion, and a thermoelastic step). The callus geometry, tissue differentiation patterns, and fracture stiffness predicted by the model were similar to experimental observations when the healing patterns of fractures with different mechanical stimuli and different gap sizes were analyzed [80]. Ghiasi et al. theorized that the initial phase of healing has a contributory mechanobiological effect on the overall bone healing process, resulting in the creation of an initial callus with an ideal range of material properties and geometry to achieve the most efficient healing time [81]. Using the FE model by Lacroix and Prendergast [79], Ghiasi and co-workers simulated the bone healing process in models with different diffusion co-efficients of MSC migration, granulation tissue Young’s moduli, callus geometries, and interfragmentary gap sizes, as these parameters modulate the outcome of bone healing during its initial phase, which involves inflammatory stage, hematoma evolution to form granulation tissue and initial callus development during the first few days post-fracture. Their results suggested that how the speed of MSC migration, stiffer granulation tissue, thicker callus, and smaller interfragmentary gap improved healing to some degree but a state of saturation was reached after a certain threshold. A number of studies have been carried out to model and simulate the course of healing under various loading conditions and mechanical stimulations. In 2008, Prendergast and co-workers proposed the development of a tissue differential model for fracture healing that could be applied in complicated 3-D geometries and loading condition as well as being capable of capturing the influence of the mechanical environment on the number of cellular and tissue processes. Poroelastic FEA was used in their study so that the effect of the frequency of dynamic loading could be modeled. According to the authors, their model predicted that asymmetric loading generated an asymmetric distribution of tissues in the callus but only for high bending moments. In

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51

addition, the frequency of loading was also predicted to have an effect [82]. The same group later postulated that computational simulations of fracture healing in the human tibia under realistic muscle loading and with an external fixator applied could predict healing similar to that found in vivo [83]. The simulation was accomplished using a discrete lattice modeling approach combined with a mechano-regulation algorithm to describe the cellular processes involved in the healing process—namely proliferation, migration, apoptosis, and differentiation of cells. The main phases of fracture healing were predicted by the simulation, including the bone resorption phase. Bone healing was simulated beyond the reparative phase by modeling the transition of woven bone into lamellar bone. In another study, a mechano-regulatory bone healing model was attempted that combined directly cellular mechanisms to mechanical stimulation during bone healing. This was based on the belief that the cells serve as transducers during tissue regeneration. The evolutions of MSCs, fibroblasts, chondrocytes, and osteoblasts as well as the production of extracellular matrices of fibrous tissue, bone, and cartilage were calculated. Furthermore, the cells in the model within the matrix proliferate, differentiate, migrate, and produced extracellular matrix, all at cell-phenotype specific rate, according to the mechanical stimulation they experience. The potential of their model was evaluated using a 2-D FE model of a long bone osteotomy [84]. The authors claimed that their model was able to accurately predict several aspects of bone healing, including cell and tissue distributions during normal fracture healing. Furthermore, it was able to predict experimentally established alterations due to excessive mechanical stimulation, periosteal stripping, and impaired effects of cartilage remodeling. Experimental investigations have demonstrated that one of the primary mechanisms for the course of bone healing is interfragmentary movement even though there are also other variables that will have an influence on fracture healing [85]. The effectiveness of computational models to address the complex relations between tissue formation and mechanical environment was first proposed by García-Aznar et al. [86] and later by Comiskey et al. [87] where they examined the formation of callus in two cases of bone fracture healing. Compressive strain field experienced by the immature callus tissues due to the interfragmentary movement was calculated using plane strain models of the oblique fractures. The external formation of calluses was determined using an optimization algorithm that iteratively excludes tissue that encounters low strains from a large domain. A combined failure-repair mechanistic computation model was also suggested to describe the healing process of a bone fracture. The bone fracture gap was created in the model using interface elements to connect the two fracture ends and to simulate the discontinuity in the displacement field between fragments [85]. As mentioned above, predicting the key phases of fracture healing utilizing mathematical models and in particular FE simulations have been reported in a number of studies [82–84, 88, 89]. The study by Grivas et al. attempted to address the limitations of FE simulation in the prediction of cell proliferation during bone healing [88]. The authors suggested that the application of a meshless approach where no background cells were required for the numerical solution of the integrals would

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address the shortfall in which the solution to diffusion equations in most studies is achieved using the FE approach that nonetheless requires global remeshing in situations with moving or newly created surfaces or material phases. In their study, a meshless local boundary integral equation method was used to derive predictions of cell proliferation during bone healing in a 2-D fractured model. It must be remembered that these structural properties of a fracture callus depend collectively on the individual tissues, including cartilage, calcified cartilage, and woven bone, and the spatial distribution of these tissues, as well as the overall geometry of the callus. Despite the aforementioned studies are focused on simulating numerically the fracture healing process in long bones such as the femur, the same concept can be employed in an oral environment. Above all, investigations have been carried out to assess the possibility of bone remodeling and formation in addition to the risk of further bone fracture induced by fixation plates used to treat mandibular fractures [90– 97]. It has been well-proven that the absence of excessive mobility or immobilization is crucial for the unification of the fracture segments. Furthermore, this stability plays a key role in the healing of both hard and soft tissues in the fractured or injured area. This indicates that the fracture site must be stabilized to aid in the physiological process toward normal bony healing using mechanical means. The application of plate fixation should produce undisturbed primary fracture healing, and it has been stated that the required fracture mobility essential to favor primary bone healing should be no greater than 0.15 mm between the fracture surfaces [78, 95, 98–100]. Similarly, it has been hypothesized that the differentiation of callus tissue is governed by the amount of biomechanical strain along existing calcified surfaces in the fracture callus [101]. The study by Claes and Heigele predicted intramembranous bone formation would occur for strains smaller than 5%, while endochondral ossification for strains less than 15%. They also suggested that all other conditions would result in fibrous cartilage or connective tissue [101]. Based on the observations of their fracture experiments using metallic fixation devices, Orassi et al. postulated that bone healing would take place at lower mechanical strains for fractures located around the mandibular body and symphysis, and at higher strains for fractures around the mandibular angle and condyles where the muscle forces are applied [90]. Their findings were consistent with the observations made by Kimsal et al. in which they assess the ideal fixation candidate as a less invasive fixation approach to fractures of the mandibular angle using principal strains in the callus at the fracture site [97].

5.3 Bone Remodeling Around Dental Implants The interface between the implant and bone tissue during mandibular movement is of great concern to the osseointegration process as the mechanical environment of the mandible may be altered by dental implants. This may also result in the remodeling and adaptation of the surrounding cortical and cancellous bone tissues. As a result, it is imperative that the effect of bone remodeling and its influence on the longevity

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of implants and prostheses be considered in order to improve its performance and reliability [57, 102]. During functional movements such as chewing, forces on the prosthesis will be transferred to the supporting implants and this will lead to stresses being generated within the bone tissue surrounding the implant. During the first year of function, bone remodeling occurs in response to occlusal forces as well as establishing the normal dimensions of the peri-implant soft tissues. Bone remodeling is categorized as either constructive or destructive, and the changes in the internal stress state will determine the type of remodeling that will take place in the bone tissue surrounding the implant. Bone resorption and stress shielding will occur if no load is transmitted from the implant to the supporting bone tissues. On the other hand, abnormally high stress concentration can result in implant failure. For these reasons, it is crucial to consider the effect of bone remodeling on the performance of dental implants and prostheses in order to improve its efficiency. The mechanostat hypothesis for bones proposed by Frost focuses on how it adapts their strength to the mechanical loads on them as well as how strain magnitudes might affect the biological machinery of load-bearing bones [103, 104]. The hypothesis further describes four microstrain zones and the role they play in mechanical adaptation of bone tissue. If bone strain is less than a certain threshold, such as between 50 and 100 microstrain (equivalent to around 1 to 2 MPa), then disuse-mode remodeling occurs and bone strength is reduced. Bone strain between 50 and 1500 microstrain represents no net gain or loss in bone strength, and bone in this strain environment remains in a steady state (lazy zone), and according to Frost, 1000 to 1500 microstrain is equivalent to around 20 MPa. On the other hand, bone strain greater than 3000 microstrain (equivalent to around 60 MPa) could lead to physiological overload and bone resorption will take place. Strains between 1500 and 3000 microstrain can be considered to be mildly overloaded and repeated bone strain result in microscopic fatigue damage in bone. Typically, this microdamage can be detected and repaired by bone if the strain is below 3000 microstrains and this new or repaired bone is usually in the form of woven bone [103, 104], which is weaker than lamellar bone [105]. It has been suggested that this state of bone could be similar to an overloaded endosteal implant attempting to alter the strain environment at the bone-implant interface back to the steady-state zone [106]. Bone fractures in young adults occur if the strain exceeds 25,000 microstrain or 120 MPa. An important note also is the stress values mainly apply to cortical lamellar bone in healthy young adult mammals [103, 104]. It has been hypothesized that the mechanostat theory could also be applied to the field of dental implantology and in particular, implants placed in the alveolar jaw, but there were also concerns on the applicability of the strain thresholds identified by Frost to the alveolar bone [107]. Achieving osseointegration will involve a number of factors such as the design, geometry, and material composition of the implant, sufficient bone quality at the implantation site, interaction at the bone-implant interface, surgical technique, and the absence of overheating during preparation at the site of implantation.

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Fig. 5.5 Simple illustration displaying the steps involved in bone remodeling calculation using the FE approach. Modified from Isaksson [89]

The process of bone remodeling around dental implants has been simulated in a number of studies using a variety of models (Fig. 5.5). Mathematical algorithms such as mechano-regulation mechanisms and strain energy density algorithms have been integrated into numerical models to study bone formation and osseointegration of dental implants. In addition, time-dependent technique for imitating bone modeling and remodeling in response to the daily loading history has also been adapted to demonstrate bone remodeling induced by dental implants and fixed partial dentures.

5.3.1 Strain Energy Density The 1970s saw the beginning of the development of the first mathematical theory for bone remodeling with the theory of adaptive elasticity [108, 109]. A study was conducted afterward utilizing a similar theory on the correlation between orthopedic implants and long-term adaptive bone remodeling in both 2-D and 3-D FE models [109]. The shape or bone density adaptations to different functional requirements were governed by the strain energy density as a feedback control variable. Strain energy density in their study was expressed in the general form of:  Strain energy density = 1 2εi j : σi j

(5.1)

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where σij and εij are the components of the local stress and strain tensors and the strain energy density discusses the rate of variation in bone density (ρ) using the following relation when it is applied to bone remodeling: ∂ρ = C(S E Da − S E Dh ) ∂t

(5.2)

where C is the rate of adaptation. The homeostatic strain energy density (SEDh ) is referred to as the local actual strain energy density, and the difference between the actual strain energy density (SEDa ) and a site-specific homeostatic strain energy density was suggested to be the driving force for adaptive activity. As previously suggested by Carter, bone is added above and removed below certain thresholds of tissue loading [110]. A “lazy zone” or a “dead zone” exists within these thresholds in which no net change in bone mass occurs [108, 109, 111]. Adaptive activity is initiated if: S E Da > (1 + s)S E Dh

(5.3)

S E Da < (1 − s)S E Dh

(5.4)

Or

where s is the threshold level, and the rate of variation in bone density in the “lazy zone” can be described as: ⎧ ⎨ C[S E Da − S E Dh (1 + s)] ∂ρ = (5.5) 0 ⎩ ∂t C[S E Da − S E Dh (1 − s)] If S E Da > S E Dh (1 + s) (1 − s)S E Dh ≤ S E Da ≤ (1 + s)S E Dh S E Da < S E Dh (1 − s) Recently, a FE analysis was carried out to determine the strain energy density in a 2-D mandibular model under normal chewing and biting forces. A load consisting of a vertical component of 100 N and a lateral component of 10 N was applied to the three teeth contained in the model (canine, lateral incisor, and central incisor). Included in the 2-D model were enamel, dentin, pulp, cementum, periodontal ligament, gingiva, cortical and trabecular bone. The effects of model parameters (boundary conditions, adjacent teeth, initial bone density, reference value for the mechanical stimulus, and width of the equilibrium zone) on the resulting trabecular bone density were investigated [112].

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The same strain energy density algorithm was applied in a number of studies where bone remodeling was induced by the use of dental implants and was chosen as the mechanical stimulus initially by Mellah et al. in 2004 to examine the stresses at the bone-implant interface before and after osseointegration [113]. In another study, Li and co-workers suggested an alternative mathematical model that describes changes in bone density as a function of the mechanical stimulus [114]. Their proposed algorithm includes an additional quadratic term based on the work carried out by Weinans and co-workers [115] and can simulate both overload and underload resorptions that frequently occur in dental implant treatments using the strain energy density as the stimulus for bone remodeling. Their FE results demonstrated that bone resorption occurred at the neck of the implant as a result of occlusal overload but then resorption ceased after some time prior to reaching the coarse threads. The authors also noticed from their simulation that the density of the bone deeper into the mandible increased slightly due to the additional mechanical stimuli provided by the occlusal load, a phenomenon observable in some clinical situation. This was similar to a number of studies aimed at gaining an insight into the effects of mandibular bone remodeling induced by dental implantation particularly changes in bone density [116] and stiffness over a period of 48 months [117]. The result of Lin et al., which were qualitatively validated by comparing with the X-ray images from clinical follow-up, displayed signs of visible bone apposition in the peri-implant cancellous region. The study by Nu¸tu attempted to correlate the effect of initial bone quality and density distribution as an essential factor for long-term survival of dental implants [118]. Two strain energy density-based bone remodeling theories were applied (one of which accounted for overload resorption). It was also applied in a number of hypothetical and comparative studies. In 2010, a modified remodeling algorithm based on existing theories including strain energy density was used to examine the influence of various degrees of bone-implant contact (between 25 and 100%) on bone remodeling following dental implantation through a 2-D FE model [119]. Later, the effect of cusp inclination and occlusal loading on mandibular bone remodeling was simulated [120]. Strain energy density obtained from 2-D plane strain FEA was used as the mechanical stimulus to drive cortical and cancellous bone remodeling in a bucco-lingual mandibular section. It was applied to study the relationship between stress shielding and low-stiffness implants [121]. Marginal bone loss around low-stiffness implants was determined by applying the calculated strain energy density values to the “lazy zone” range as suggested by Huiskes et al. [109]. Recently, the influence of subcrestal implant placement depths (between 0 and 3 mm) on cortical and cancellous bone remodeling over the first 12 months after final prosthesis delivery was investigated [122]. The morphology of bone around dental implants with the consideration of both short-term and long-term bone healing was also predicted using a hybrid two-step algorithm in which the short-term stage was simulated using a tissue differentiation model [123]. The distribution of strain, fluid velocity, and stem cell diffusion in the short-term or initial model was first calculated by FEM. Granulation tissues were then differentiated into various tissue phenotypes based on the mechano-regulation algorithm. The authors also claimed that their hybrid model was able of reproducing a

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number of features discovered in experiments such as high-strength bone connective tissue bands, stress shielding effect, and marginal bone loss. The remodeling of both maxillary and mandibular bone also plays a vital role in the long-term application and survival of implant-supported fixed partial dentures and overdentures. Using an adaptive algorithm that included an overloading bone resorption process, bone remodeling simulations were carried out on two-unit fixed partial dentures with and without cantilever configuration incorporated into a patientspecific 3-D FE model of a maxillary bone with two absent central incisors [124]. Rungsiyakull et al. explored the biomechanics and associated bone remodeling responses between implant-implant-supported and tooth-implant-supported fixed partial dentures. The strain energy density induced by occlusal loading was used as a mechanical stimulus for driving the bone remodeling [125]. Later, the stress distribution in the mandible and changes in bone density was attempted by Li and co-workers to examine the effect of implant number on the remodeling of the peri-implant bone and posterior mandible under occlusal force induced by implant-supported overdentures [126]. It has also been suggested that the combination of FEA containing mechanobiological stimuli such as strain energy density and medical imaging such as positron emission tomography (PET) could offer a new approach to clinically monitor and examine bone remodeling driven by fixed partial dentures [127].

5.3.2 Mechanobiological Approach As discussed previously, the regeneration of skeletal tissue is achieved through a cascade of biological process that may involve differentiation of pluripotential tissue, endochondral ossification, and bone remodeling. The local tissue mechanical loading history has been shown to influence all these processes in a significant manner [128]. Mechanobiology is the study on how mechanical loads regulate biological processes and responses of cells within tissue [129–131]. Gaining deeper insight into the mechanobiological behavior of bone tissue is made possible through the combined use of FEM and mathematical models. Determining the quantitative rules that govern the effects of mechanical loading on tissue differentiation, growth, adaption, and maintenance has been attempted using computational mechanobiology [89, 132]. According to Isaksson, the modeling is based on the principle that local mechanical variables stimulate cell expression to regulate tissue density, structure, or composition. Moreover, modeling considerations include force application at the boundary, transmission of force through the tissue matrix, mechanosensation and transduction by cells, and transformation of extracellular matrix characteristics [89]. Physical forces play a key role in the differentiation of the cell populations which arise over the healing period during bone regeneration, and a number of hypothesis for the mechano-regulation of tissue differentiation have been developed by comparing differentiation patterns during tissue repair to predict the mechanical environment within the mesenchymal tissue [131, 133]. Experimental evidence by Caplan has

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demonstrated the capacity of MSCs to differentiate into bone and cartilage as well as other musculoskeletal tissues [134, 135]. The theory concerning the role of mechanical stimuli and the regulation of tissue differentiation within a fracture callus was first suggested in a fracture callus study by Pauwels, and it was based on the understanding that stress and deformation of the MSCs are the result of physical factors and that the cell differentiation pathway could be governed by these stimuli [89, 131, 136]. The work of Pauwels hypothesized the formation of fibrous and bone tissues was the consequence of MSCs being stimulated into fibroblasts due to shear strain. Similarly, cartilaginous tissue is created as a result of hydrostatic compression stimulating MSCs into chondrocytes. Furthermore, Pauwel also hypothesized that a stable mechanical environment is needed for primary bone formation and the necessity for soft tissues to generate this low strain environment in order for endochondral bone formation to proceed [89, 131, 136]. The histological pattern of fracture healing has been theorized to be regulated at least to a certain degree by the local mechanical strains in the interfragmentary region. The term “interfragmentary strain” was used to describe the strain generated in the fracture gap when the fracture fragments dislocate relative to each other as soon as a load is applied to the fracture bone [100, 137–140]. The interfragmentary strain is calculated by dividing the magnitude of the gap movement in the longitudinal or axial direction by the size of the gap. The notion behind the interfragmentary strain theory is that tissues located within the fracture gap must be able to sustain the interfragmentary strain without experiencing rupture. Tissue differentiation within the fracture gap is governed by the scale of the interfragmentary strain. Bone tissue has the lowest tolerance level to this strain, while on the other hand granulation tissue has the highest tolerance. Cartilage has an intermediate tolerance level. As stated by the interfragmentary strain theory, granulation tissue is the only tissue that can be formed if the intrafragmentary strain is high. Once this tissue is formed, interfragmentary strain will be reduced as the tissues in the fracture gap stiffens (formation of callus). An environment more ideal for chondrogenesis and where chondrocytes can maintain the intrafragmentary strain and proliferate is created by this reduction in the movement of the fracture fragments [100, 131, 137–140]. A number of studies have reported the use of FE modeling to predict changes in mechanical properties during fracture healing. The study by Cheal et al. [138] predicted the complex tissue strains using a non-linear 3-D FE model of the interfragmentary tissue at the initial stage of healing. Later, Gardner and co-workers [141] developed 2-D FE models of a diaphyseal tibial fracture. Stress and strain distributions in the callus and bone were calculated from peak interfragmentary displacements measured during natural walking activity and were correlated with the subsequently observed pattern of tissue differentiation and maturation of the callus. Their study also confirmed the growth and stiffening of the external callus progressively reduced the interfragmentary gap strain. Lacroix et al. [142] created an algorithm to predict the time course of intramembranous and endochondral ossification and this algorithm assumes that there are precursor cells in the undifferentiated tissue and that these cells differentiate into either fibroblasts, chondrocytes, or osteoblasts, based on a combination of biophysical stimuli derived from strain in the collagenous matrix and flow of the interstitial fluid. It was tested

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in an investigation of the fracture healing of a long bone using an axisymmetric FE model. They suggested that the origin of the precursor cells will have a fundamental effect on the healing pattern and on the rate of reduction of the interfragmentary strain. FEA was also used by Cleas and co-workers to support the theory that correlates the local tissue formation in a fracture gap to the local stress and strain and that the magnitudes of hydrostatic pressure and strain along existing calcified surfaces in the fracture callus govern the differentiation of the callus tissue, which was initially suggested by Carter and co-workers in 1987 [128, 143, 144]. The theory takes into consideration the intermittent or repeated mechanical forces which represent the loading history on the chondro-osseous skeleton and the local tissue stress history plays a vital part in determining the biology of connective tissue [144]. The results of their 2-D FE analysis suggested that intermittent hydrostatic (dilatational) stresses could play a significant role in affecting revascularization and tissue differentiation as well as regulating the morphological patterns of initial fracture healing [143]. Furthermore, according to Carter et al., cyclic motion and the associated shear stresses cause cell proliferation and the production of a large callus during the early stages of fracture healing. Fibrocartilage development is predicted to take place under tensile strain with a superimposed hydrostatic compressive stress. The production of fibrous tissue occurs when tissue experienced high tensile strain. Direct intramembranous bone formation is allowed in areas of low stress and strain. Intramembranous ossification could be encouraged by low to moderate degrees of tensile strain and hydrostatic tensile stress. Hydrostatic compressive stress could spur chondrogenesis, and it can be promoted by poor vascularity in an otherwise osteogenic environment [128]. Claes and co-workers hypothesized that the scales of hydrostatic pressure and strain along existing calcified surfaces in the fracture callus will control the differentiation of the callus tissue. In their study, the local stresses and strains in the callus as determined from a FE model were compared to the histological observations using an animal fracture model, and this permits the attribution of certain mechanical scenarios to the kind of tissue differentiation. The formation of intramembranous bone will occur for hydrostatic pressures smaller than 0.15 MPa and strains smaller than approximately 5%. If the hydrostatic pressure was more than 0.15 MPa and strains smaller than 15%, this situation would stimulate endochondral ossification. Moreover, all other conditions appeared to result in fibrous cartilage or connective tissue [77, 101]. In addition to mechanical strain, microdamage as the stimulus driving cellular responses to predict bone remodeling has also been examined, and certain aspects of bone remodeling have been successfully predicted using damage-based models [145]. The study by McNamara and Prendergast [145] examined the theory that bone remodeling could be governed by signals due to both microdamage and strain. This was based on experimental data implicating both stimuli in bone cell regulation. Four mechano-regulation algorithms were examined (strain, damage, combination of strain and damage, and either strain or damage) with damage-adaptive remodeling prioritized when damage is above a critical level. Each algorithm is applied with both osteocyte cell sensors (internal) and bone lining cell sensors (surface) and used to predict a bone multicellular unit (BMU) remodeling on the surface of a

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bone trabecula. Utilizing this approach, the effect of implant geometry (length and diameter) on its long-term stability was examined [146]. The temporal evolution of porosity and microstructural damage of the peri-implant bone were the variables investigated in their study. FE simulations were carried out in the study by Wang et al. [147] using a model that combined both adaptive bone remodeling and microdamage-based mechano-sensory mechanisms to predict the evolution of the trabecular architecture around dental implants. Their study was based on the value of damage accumulation (ω) determined using Miner’s rule as suggested in the study by McNamara and Prendergast [145]: 1 Nf

ω˙ =

(5.6)

where Nf is defined as the number of cycles to failure of the material at a given stress level. An empirical equation can be utilized to determine the value of Nf for bone according to a study by Carter et al. [148]: log N f = H log σ i + J · T + K · ρ i + M

(5.7)

where σ is the stress in MPa, T is the temperature (°C), ρ is the density of the material in g/cm3 . H, J, K, and M are empirical constants. The damage accumulates can then be calculated using the following equation: t

ω = ∫ ωdt ˙

(5.8)

0

where ω is the state of local damage bone tissue, and t is the duration of the calculation. Once the damage accumulation (ω) is below the critical damage threshold, strainadaptive remodeling was considered to be the determining mechanism in the remodeling process. If the damage accumulation value exceeds the damage threshold on the other hand, then damage-induced remodeling will take over. The combined algorithms of adaptive and microdamage remodeling will alter the local bone density values. Using the equation derived by Currey [8], the value of Young’s modulus is updated based on the change in the bone density value: E i = Cρin

(5.9)

According to the authors, the suggested algorithms were shown to be effective in simulating the remodeling process of trabecular architecture around the dental implant systems and the predicted orientational and ladder-like architecture around the implants were similar to those observed in animal experiments and clinical studies. The theory behind poroelasticity centers on the interaction of deformation and fluid flow in a fluid-saturated porous medium, and it has been applied to the study

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of bone tissue for many decades [149]. It is evident that the tissue itself is a material composed of both fluid and solid constituents and since the cells that undergo differentiation are contained in the fluid until such time as they become precursors for tissue-forming cells, Prendergast and Huiskes hypothesized that a model describing the tissue behavior derived from the fundamental biphasic nature of tissue would be more accurate than a model based on the elastic approach despite the convenience [150]. Based on the observations from their study in which an axisymmetric 2-D FE model of the gap tissue and surrounding cancellous bone was developed to investigate the biomechanical behavior of the environment surrounding the implant, the prediction of osteogenic index was more suitable when the tissue was modeled as a biphasic material. The study by Prendergast et al. [151] further highlighted the necessity for biphasic analysis in which bone tissue is modeled as a biphasic material composed of both solid and fluid constituents. Using a biphasic FE model, the interfacial tissue formation adjacent to a micromotion device implanted into the condyles of dogs was investigated. Their results showed as tissue differentiation progressed, subtle but systematic mechanical changes took place on cells in the interfacial tissue. In particular, the implant changed to being governed by the maximum-available load (force-control) rather than being regulated by the maximum-allowable displacement (motion-control) as the forces opposing motion increase. This resulted in a reduction in the velocity of the fluid phase relative to the solid phase and a decrease in interstitial fluid pressure accompanied by a reduction in peri-prosthetic tissue strains [151]. This approach was later used in another study to examine bone regeneration within the callus of an osteotomized mandible. The stimulus regulating tissue differentiation within the callus was hypothesized to be a function of the strain and fluid flow computed by the poroelastic FE model. This model was then used to investigate tissue differentiation during a 15-day latency period, defined as the time between the day of the osteotomy and the day when the first distraction is given to the device [152]. Later, a mechano-regulatory tissue differentiation model that takes into consideration the stimuli through the solid and the fluid components of the healing tissue as well as the diffusion of pluripotent stem cells into the healing callus was used in a study to examine the effect of immediate loading on the peri-implant bone healing. The influence of micromotion, size of the healing callus, and thread design on the length of the bone-to-implant contact and bone volume formed in the healing callus were examined [153]. Their work was further expanded in a later study in which the long-term osseointegration and bone remodeling around dental implants were examined by simulating bone healing followed by bone remodeling process. The assumption that wound healing and remodeling around oral implants would be the same as in long bone. Development of soft tissue was observed both in the coronal region due to high fluid velocity, and on the vertical sides of the healing gap due to high shear stress. Furthermore, tissue between the implant threads differentiated into bone during the healing phase with small implant micromotion but resorbed during remodeling. Conversely, higher percentage of the healing region differentiated into soft tissue with large implant micromotion resulting in smaller volume of bone tissue available for remodeling. However, the remaining bone region developed

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higher-density bone tissue [154]. The work Chou and Müftü [153] was later used by Li et al. [155] to validate their results when the authors attempted to examine the influence of implant geometry on biophysical stimuli and healing patterns. The mechanobiological approach was also used by Vanegas-Acosta et al. [156] in an attempt to describe the process of osseointegration at the bone-implant interface in relations to the implant surface and mechanical and biological factors. In their study, an injury area of 1 μm measured from the implant surface was defined for equations related to the concentration of osteogenesis and osteogenic chemical compound which resembles the contact area between the tissues in formation and the implant surface. Their study also assumed that new osteoid formation in the injury area is dependent on the displacement of fibrin matrix, and subsequently, the mechanical properties of tissue are influenced by surface roughness.

5.3.3 Stanford Theory Developed originally by Carter [144] and further extended by Beauprè et al. [157], the so-called Stanford Theory is a time-dependent technique for imitating bone modeling and remodeling in response to the daily loading history. In their approach, the daily tissue level stress stimulus (ψb ) was defined by the following equation, where ni is the number of cycles of load type i, and m is the stress component: ψb =

 

 m1 n i σim

(5.10)

d

The continuum level effective stress (σi ) can be defined as: σi =

√

2EU

(5.11)

where U is the continuum strain energy density and E is the continuum average Young’s modulus. In addition, the authors devised an equation that can be applied to both internal and external remodeling, and this can be applied directly to estimate the rate of bone apposition or resorption on external surfaces. The bone apposition or resorption rate (˙r) can be expressed as: ⎧ 

⎨ (ψb − ψb·as )c + c · w r˙ μm day = 0 ⎩ (ψb − ψb·as )c − c · w

(5.12)

5.4 Concluding Remarks

63

If [(ψb − ψb·as ) < −w] [−w ≤ (ψb − ψb·as )] ≤ w [(ψb − ψb·as ) > w] where c is the empirical rate constant, w is half the width of the central, normal activity region, and ψb.as is the average daily tissue level attractor state stress stimulus. This algorithm was first adopted by Eser et al. to predict time-dependent biomechanics and remodeling of bone around immediately loaded dental implants with various macrogeometric designs and later in patients with reduced bone width using a 3-D FE model [158, 159]. Sotto-Maior et al. [160] simulated bone remodeling around single implants of different lengths (7, 10, and 13 mm) and to validate the theoretical prediction with the clinical findings of crestal bone loss. Occlusal loading cycles of 200 N were applied to the implants to simulate daily mastication, and bone remodeling was examined via changes in the strain energy density of bone after 3, 6, and 12 months of function.

5.4 Concluding Remarks The finite element method can be used to calculate local stress–strain distributions in geometrically complex structures. The predictive accuracy of the finite element model is influenced by the geometric detail of the object to be modeled, the material properties, and the applied boundary conditions. Finite element analysis has become widely used in all biomechanical fields, especially for assessing stresses and strains in dental implants and the surrounding bone structures as well as in normal bone. Bone remodeling around various options for tooth restoration can be examined under functional movements and any possible problems such as stress shielding eliminated by proper implant design and surgical protocol prior to implantation. Furthermore, the use of numerical models to simulate the process of fracture healing may prove to be advantageous when deciding the optimal mechanical-based treatment for fracture fixation after a trauma. Nevertheless, considerations need to be taken to account for the variations between FEA simulation and the natural healing process of the human body. For example, in addition to being governed by mechanical factors, bone healing is also regulated by chemical, genetic, and biological factors. These factors need to be addressed in order to improve the predictive accuracy of the bone fracture healing process.

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Chapter 6

Patient-Specific Modeling

The usefulness of finite element analysis (FEA) from a clinical point of view as a biomechanical foundation in the construction of a predictive tool is ideal for determining the influence of a specific implant design on the magnitude of bone remodeling and osseointegration and avoiding any potential issues such as stress shielding and bone resorption based on the anatomical conditions of individual patients such as bone health and muscle force. The application of computerized modeling technique will have countless applications in both craniomaxillofacial surgery and dentistry, not only at a research level but also at a clinical level. For instance, it is a common practice to generate a computer model based on data from CT scans and micro-CT or other suitable imaging techniques such as magnetic resonance imaging (MRI) of individual patients [1]. 2-D or 3-D finite element (FE) meshes are created by importing the image data into specialized modeling software where this information is converted into solid models (Fig. 6.1). This model can then be utilized in the following applications: • Traumatology: Trauma and dental surgeons can determine through accurate patient-specific 3-D modeling the best form and engineering requirements of hardware such as bone fixation devices used in the treatment of injuries such as mandibular fractures. This approach enables both a reduction and fixation of fractures while minimizing the number and size of the osteosynthesis plates used at the same time. Above all, bone remodeling and the healing process of bone fractures have been simulated using the combination of FEA and patient-specific modeling. Computational models and simulations of fracture healing process may prove to be useful in defining the optimal mechanical-based treatments after a trauma or accident. • Temporomandibular joint surgery: Intense scrutiny has been placed on temporomandibular joint prostheses in the past as a consequence of breakdowns primarily due to material failure. It is anticipated that patient requiring temporomandibular joint replacement can benefit from improvements in the design and © The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. H. Choi, Bone Remodeling and Osseointegration of Implants, Tissue Repair and Reconstruction, https://doi.org/10.1007/978-981-99-1425-8_6

71

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6 Patient-Specific Modeling

Fig. 6.1 Illustration showing the stages involved in the creation of a patient-specific FE model

material used in these prostheses based on computational simulation. Furthermore, computational modeling technology can also play a part in the in vitro examination of these prostheses by fine-tuning its design so that damage due to numerous materials failures in the past can be avoided. • Distraction osteogenesis: The success of distraction osteogenesis of the mandible depends on the accurate planning of the distraction vector required to produce the final position of the distracted segment. Currently, this planning process can be refined through the combined use of FEA and patient-specific computational modeling, and in particular, the beginning of the widespread utilization of intraoral multidirectional distraction devices. • Dental implants: Determining the correct implant placement location as well as the number, size, and configuration of implants needed is now possible in an effort to address the patient’s restorative and functional needs. In such instance, the dental clinician or dentist is able to estimate the functional loads that an implantsupported restoration may develop based on the patient’s own bone properties and muscle forces. Accordingly, any essential surgeries can be planned beforehand by the clinician in an effort to avoid harmful overloading on the restoration that could jeopardize the success of the treatment plan. In doing so, the likelihood of over-treatment can also be prevented. Moreover, it is possible to simulate and predict bone remodeling and osseointegration induced by dental implants using mathematical algorithms.

6.1 Mechanical Properties of Bone Tissue from CT Scans

73

• Restorative dentistry: In a similar fashion to dental implant restorations, different restorative tooth options can be investigated under simulated functional loading and designed to resist these loads prior to manufacturing. • Pathology resection and reconstruction: Reconstructions after the resection of various pathologies involving the mandible can be conducted using patientspecific models so that form and function are returned to the premorbid state as closely as possible. • Aesthetic facial surgery: Patient-specific computational modeling can also be utilized to investigate various aspects of facial bone osteotomies such as postoperative functional change and stability of the orthognathic movements.

6.1 Mechanical Properties of Bone Tissue from CT Scans As discussed earlier, assigning appropriate physical and mechanical properties of bone tissues to a FEA model is of paramount importance, as it will strongly influence the distributions of stresses, strains, and deformations within a structure. A suitable approximation of the physical and mechanical properties of bone tissue can be acquired using CT scans. Subsequently, a more relevant physiological model on a subject-specific basis can be achieved using this approach.

6.1.1 Hounsfield Units to Density The raw CT values can be converted into Hounsfield units (HU) according to the study by Rho et al. [2] by relating the bone values to air where the value of HU is − 1000 [3] and to water where the value of HU is zero using the following equation: HU = 1000

C T − C TH2 O C TH2 O − C Tair

(6.1)

In recent years, the introduction of cone-beam CT (CBCT) has permitted dental clinicians to examine craniofacial structures in 3-D at a relatively high spatial resolution with less exposure time and radiation dose than the conventional clinical multidetector CT [4, 5]. Consequently, this resulted in the widespread application of CBCT in dentistry for 3-D imaging and in the diagnostics of dental complications [5, 6]. It has also been suggested that the 3-D image generated by CBCT allows for detailed morphological examination of bone [5]. According to the review by Pauwels et al. [4], it is unclear if the gray values derived from CBCT units are similar to the HU values obtained from CT scanners. In 2010, Mah et al. investigated the relationship between gray levels in dental CBCT and HU in CBCT scanners [7]. The authors demonstrated that a linear relationship between the attenuation coefficients and the

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6 Patient-Specific Modeling

gray levels of each material tested. Furthermore, by utilizing linear attenuation coefficients, HU can be calculated from the gray levels in dental CBCT scanners using a standard equation: HU = 1000

μmaterial − μ H2 O μ H2 O

(6.2)

where μmaterial and μ H2 O are the linear attenuation coefficients for the material and water, respectively. Bone density is defined as mean value expressed in HU in each pixel. The apparent density of bone (ρ app ) can be determined using a linear equation from HU according to Rho et al. [2]: ρapp = a + b · HU

(6.3)

A linear calibration derived from two reference points in one of the CT scan slices is used to determine the values of a and b [8]. Non-bone condition should represent the first reference point where the density should be equal to 0 g/cm3 . The second reference point can be cortical bone with an average density of 1.768 g/cm3 taken from 17 mandibular samples [9] or between 1.85 and 2.0 g/cm3 obtained from both edentulous and dentate human mandibular cortical bones according to the studies by Dechow and co-workers [10, 11]. The CT gray values (HU) of both of the reference points are then substituted into Eq. 6.3 together with the data on apparent density. The apparent density can be determined at any point in bone using linear interpolation of the raw CT gray values once the values of a and b are calculated. According to the study by Keyak et al. [12], correlations between various measures of density are needed in order to compare numerous Young’s modulus-density relations that have been published over the years. The relationships between different density measurements such as ash density (ρash ), apparent dry density (ρdry ), quantitative CT equivalent mineral density (ρQCT ), apparent wet density (ρapp ), as well as the connection between density and bone tissue volume can be expressed mathematically using the equations shown in Table 6.1. Bone tissue ash mass, which is used to determine the ash density (ρash ), can be obtained based on the methods described by Galante et al. [13] and later by Keyak et al. [12], and the process essentially involves weighing the tissue after defatting and firing in a furnace at an elevated temperature over a period of time. Apparent dry density (ρdry ) can be determined by weighing tissue samples in air once it has been defatted and dried at moderate temperatures and dividing it by the total specimen volume. Mineral equivalent density (ρQCT ) can be converted to ash density (ρash ) using either dipotassium hydrogen phosphate (K2 HPO4 ) phantoms [12] or hydroxyapatite phantoms [14] and the average HU value of all voxels within a region of interest of the known calibration phantom sample rods [15]. As discussed in the reviews by Helgason et al. [17] and later by Poelert et al. [18], the biggest challenge is determining which Young’s modulus-density relationship is

6.1 Mechanical Properties of Bone Tissue from CT Scans

75

Table 6.1 Mathematical conversions between the various measures of density using apparent density (ρapp ) as input Density (g/cm3 )

Conversion

Reference

ρash

0.55 · ρapp

Keyak et al. [12]

ρdry

0.92 · ρapp

Keyak et al. [12]

Bone volume/total volume (BV/TV)

ρapp 1.8

Gibson [16]

ρQCT [hydroxyapatite phantoms]

(0.55ρapp −0.0789) 0.877

Schileo et al. [14]

ρQCT [K2 HPO4 phantoms]

(0.55ρapp −0.0389)

Keyak et al. [12]

1.06

the most appropriate. The vast majority of these Young’s modulus-density relationships are expressed either as a linear form (Eq. 6.4) or as an exponential function (Eq. 6.5): E =a+b·ρ

(6.4)

E = Cρ d

(6.5)

where a, b, c, and d are constants derived by statistical techniques based on mechanical testing results. It should be mentioned that there are substantial differences between the proposed relationships. The lack of standardized testing approach (such as how strain is measured), sample preparation, and the angle between the main trabecular orientation and loading direction may to a certain degree contribute to these variations observed between different studies. The review by Helgason et al. [17] also commented on the accuracy of utilizing a single Young’s modulus-density relationship over the entire range of densities or multiple relationships are needed for cortical and cancellous bone. Extensive investigations have been carried out since the late 1970s to study the mathematical correlations between Young’s modulus of human cortical and cancellous bone and the apparent density values, and these numerical relationships are considerably different from each other [17]. A greater number of Young’s modulus-density relationships can be found in the review by Helgason et al. [17].

6.1.2 Cortical Bone A power-law relationship was suggested by Carter and co-workers that could be utilized to calculate Young’s modulus (E) of both cortical and cancellous bone based on the apparent density values [19–21]: B E = Aρapp

(6.6)

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6 Patient-Specific Modeling

where A and B are constants derived experimentally. Equation 6.6 can be rearranged to give: log E = log A + B log ρapp

(6.7)

Subsequently, a linear relationship exists between the apparent density and Young’s modulus if log E is plotted against log ρapp on a graph. The gradient or the slope of the line will be the value of B and log A is the intercept with the yaxis. Carter and co-workers [19–21] concluded that axial Young’s modulus in GPa is calculated using the apparent density to the third power and the strain rate š to the power of 0.06. As a result, Eq. 6.6 can be rewritten as: E = 3.79 · ε

0.06

3 · ρapp

(6.8)

where š is in inverse seconds and ρapp in g/cm3 . Carter and co-workers [19–21] further concluded that the power-law relationship has been shown to be valid for all bone tissues within the skeleton. This also enables meaningful predictions and comparisons of strength and stiffness of bone based on in vivo density measurements. It has been suggested that the stiffness of compact bone was non-linearly and highly dependent on its apparent density, bone volume fraction, and porosity [22]. Young’s modulus increases both as a power of increasing apparent density and bone tissue volume but decreases as a power of increasing porosity. Subsequently, the power-law relation between Young’s modulus (in GPa) and apparent density (in g/cm3 ) was proposed: 7.4 E = 0.09ρapp

(6.9)

Destructive mechanical testing on 496 cubic specimens of human cortical and trabecular bones taken from the lumbar vertebrae and femoral metaphyses and diaphysis was carried out in the study by Keller [23] along the superior-inferior axis. The results showed the apparent dry density ranged from 0.05 to 1.89 g/cm3 and the apparent ash density ranged between 0.03 and 1.22 g/cm3 . Based on these findings, the following power-law relationship between Young’s modulus (in GPa) and apparent ash density (in g/cm3 ) was proposed: 2.75±0.04 E = 10.5ρash

(6.10)

Similarly, Jacobs [24] also suggested the relationship between Young’s modulus (in GPa) and apparent density (in g/cm3 ): 2.5 E = 2.014ρapp (if ρapp ≤ 1.2

g ) cm3

(6.11)

3.2 E = 1.763ρapp (if ρapp > 1.2

g ) cm3

(6.12)

6.1 Mechanical Properties of Bone Tissue from CT Scans

77

Later, a linear relationship between apparent density (in g/cm3 ) and Young’s modulus of cortical bone (in GPa) was hypothesized by Lotz et al. [25] when they investigated the structural behavior of specimens harvested from the metaphyseal shell of the cervical and intertrochanteric regions of human proximal femora. All specimens were oriented in either the local longitudinal or transverse directions. E = −13.43 + 14.261ρapp

(6.13)

Snyder and Schneider [26] examined the possibility of estimating the mechanical properties of cortical bone using a CT system typically employed in the clinical setting. Samples from the mid-diaphyses of adult human tibiae were tested in threepoint bending to failure and the mechanical properties as well as the density and ash fraction determined. Hence, the following equation was proposed: 2.39 E = 3.891ρapp

(6.14)

where E is in GPa and apparent density is in g/cm3 . Similarly, Rho et al. [2] also developed bone-specific relationships to predict Young’s modulus from density and CT numbers in human bone. Seventy-two cortical bone specimens were obtained from the mandibles of eight fresh un-embalmed human cadavers: three transverse sections at 40, 60, and 80% of the total bone length and three annular samples originated from the buccal, lingual, and inferior aspects of the mandibular corpus. According to their study, the increase in percentage in bone length signifies the positions from the chin to the ramus. Three linear relationships were derived between Young’s modulus (in GPa) and apparent density for the three orthogonal directions: E 1 = 6.382 + 0.255E 3

(6.15)

E 1 = −13.05 + 13ρ

(6.16)

E 3 = −23.93 + 24ρ

(6.17)

where 1 is the radial direction and 3 is the superior-inferior direction and apparent density in g/cm3 .

6.1.3 Cancellous Bone Rice et al. [27] presented a statistical analysis of the pooled data from several experiments centered on the dependence of Young’s modulus of cancellous bone and

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6 Patient-Specific Modeling

apparent density. Their findings revealed that Young’s modulus is proportional to the square of apparent density of the tissue. In the longitudinal direction: 2 E compr ession = 0.06 + 0.90ρapp

(6.18)

2 E tension = 0.06 + 1.65ρapp

(6.19)

In the transverse direction: 2 E compr ession = 0.06 − 0.15ρapp

(6.20)

2 E tension = 0.06 + 0.60ρapp

(6.21)

A regression was suggested by Hodgskinson and Currey [28] that can be applied to describe the relationship between the apparent density (in g/cm3 ) and mean Young’s modulus (in MPa) of human cancellous bone: 1.96 E mean = 0.003715ρapp

(6.22)

O’Mahony et al. [29] also determined Young’s modulus values in three orthogonal directions for cancellous bone taken from an edentulous mandible, and the relationship between apparent wet density and Young’s modulus (in MPa) was determined. 2.15 E Mesio−distal = 2349ρwet

(6.23)

2.12 E Bucco−lingual = 1274ρwet

(6.24)

1 E I n f er o−superior = 194ρwet

(6.25)

The correlation between mineral density measured via QCT (g/cm3 ) and Young’s modulus of human vertebral trabecular bone (in MPa) were predicted using both the linear and power-law models [30]: E = −34.7 + 3230ρ QC T

(6.26)

1.05 E = 2980ρ QC T

(6.27)

In addition, linear and power-law relations were also derived based on the apparent wet density and used as a comparison:

6.2 Muscle Forces and other Boundary Conditions

79

E = −97.1 + 2130ρwet

(6.28)

1.34 E = 2580ρwet

(6.29)

Later, Morgan and co-workers [31] proposed power-law regressions between Young’s modulus (in MPa) and apparent density (in g/cm3 ) of human trabecular bone samples from various anatomic sites: 1.72 E = 7540ρapp

(6.30)

1.94 E = 10550ρapp

(6.31)

The reported range:

The mean regression reported when the data from all anatomic sites were pooled: 1.83 E = 8920ρapp

(6.32)

6.2 Muscle Forces and other Boundary Conditions It is vital to determine appropriate magnitudes of muscle forces to be applied to a patient-specific mathematical model if the intention is to examine the in vivo biomechanical behavior of bone, particularly the mandible given the fact that muscular forces cannot be measured directly under non-invasive conditions [32–35]. The following steps are used to estimate the force magnitudes of masticatory muscles during various functional movements of the mandible such as clenching, opening, and protrusion [32–34]: I. II.

III.

IV. V.

All the forces are assumed to have an equal magnitude on both sides of the mandible and to be symmetrical with respect to the mid-line. Vectors can be used to represent the forces exerted by contracting muscles. The direction of these vectors can be defined by connecting lines between the insertions of the muscles and the centroids of their origins. The maximum force that a muscle can exert can be determined using the value of maximum muscle tension, which relates directly to its physiological crosssectional area. The lines of masticatory muscle actions and their moment arms are calculated. The moments produced by the reaction forces applied at the condylar region must be equal to the moments produced by the sum of the bite and applied muscle forces.

80 Table 6.2 Muscle actions during mastication [34]

Table 6.3 Magnitudes of bite, muscle, and joint reaction forces (N) acting on the mandible during simulated clenching (one side only)

6 Patient-Specific Modeling Muscles of Mastication

Actions

Digastrics (openers)

Depression of the mandible

Masseter

Elevation of the mandible

Temporalis

Elevation of the mandible

Medial pterygoid

Elevation and protrusion of the mandible

Lateral pterygoid

Protrusion of the mandible

Force magnitude (N) Masseter

340.0

Temporalis (anterior + posterior)

528.6

Lateral pterygoid

378.0

Medial pterygoid

191.4

Openers

155.9

Bite force

403.7

Joint reaction force

471.9

As previously mentioned, muscles provide the energy to move the mandible and permit functioning of the masticatory system. Despite being previously ignored in the biomechanical analysis of the muscles of mastication, the digastrics also play a vital role in mandibular depression (Table 6.2). In the works of Choi et al. [34, 36, 37], all muscles were assumed to be active during clenching. The reaction forces were assumed to be acting at the center of the condyles (Table 6.3).

6.3 Concluding Remarks The creation of a patient-specific 3-D model using appropriate imaging technologies is already a common practice. The developed model can also be coupled with computer-aided designs and computer-assisted manufacturing to produce dental structures and restorations with greater accuracy and relative ease. Furthermore, patient-specific implants, tissue engineering scaffolds, and fracture fixation devices for bone tissue repair and reconstruction can be manufactured using 3-D, and more recently, 4-D and 5-D printing technologies. The methodology of 3-D printing begins with the construction of a 3-D model based on 2-D images from CT or MRI. The printer then utilizes this model to print a 3-D representation by layering thin crosssections on top of one another. 3-D printing technology has developed in a significant manner, and it has found applications in other clinical settings such as the production of customized patient models, surgical templates, and cutting guides. More importantly, outstanding precision and accuracy are needed for 3-D printing

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in order to generate models or representations of true anatomic structures in addition to providing clinical practicality in dentistry as well as in craniomaxillofacial, orthopedic, and spinal surgeries.

References 1. Tanaka E, Rodrigo DP, Tanaka M et al (2001) Stress analysis in the TMJ during jaw opening by use of a three-dimensional finite element model based on magnetic resonance images. Int J Oral Maxillofac Surg 30:421–430 2. Rho JY, Hobatho MC, Ashman RB (1995) Relations of mechanical properties to density and CT numbers in human bone. Med Eng Phys 17:347–355 3. Hvid I, Bentzen SM, Linde F et al (1989) X-ray quantitative computed tomography: the relations to physical properties of proximal tibial trabecular bone specimens. J Biomech 22:837–844 4. Pauwels R, Jacobs R, Singer SR et al (2015) CBCT-based bone quality assessment: are Hounsfield units applicable? Dentomaxillofac Radiol 44:20140238. https://doi.org/10.1259/ dmfr.20140238 5. Kim DG (2014) Can dental cone beam computed tomography assess bone mineral density? J Bone Metab 21:117–126 6. Scarfe WC, Farman AG, Sukovic P (2006) Clinical applications of cone-beam computed tomography in dental practice. J Can Dent Assoc 72:75–80 7. Mah P, Reeves TE, McDavid WD (2010) Deriving Hounsfield units using grey levels in cone beam computed tomography. Dentomaxillofac Radiol 39:323–335 8. Khan SN, Warkhedkar RM, Shyam AK (2014) Analysis of Hounsfield unit of human bones for strength evaluation. Procedia Mater Sci 6:512–519 9. Dechow PC, Nail GA, Schwartz-Dabney CL et al (1993) Elastic properties of human supraorbital and mandibular bone. Am J Phys Anthropol 90:291–306 10. Schwartz-Dabney CL, Dechow PC (2002) Edentulation alters material properties of cortical bone in the human mandible. J Dent Res 81:613–617 11. Schwartz-Dabney CL, Dechow C (1997) Variations in cortical material properties from throughout the human mandible. J Dent Res 76:249 12. Keyak JH, Lee IY, Skinner HB (1994) Correlations between orthogonal mechanical properties and density of trabecular bone: use of different densitometric measures. J Biomed Mater Res 28:1329–1336 13. Galante J, Rostoker W, Ray RD (1970) Physical properties of trabecular bone. Calcif Tissue Res 5:236–246 14. Schileo E, Dall’ara E, Taddei F et al (2008) An accurate estimation of bone density improves the accuracy of subject-specific finite element models. J Biomech 41:2483–2491 15. Knowles NK, Reeves JM, Ferreira LM (2016) Quantitative Computed Tomography (QCT) derived Bone Mineral Density (BMD) in finite element studies: a review of the literature. J Exp Orthop 3:36. https://doi.org/10.1186/s40634-016-0072-2 16. Gibson LJ (1985) The mechanical behavior of cancellous bone. J Biomech 18:317–328 17. Helgason B, Perilli E, Schileo E et al (2008) Mathematical relationships between bone density and mechanical properties: a literature review. Clin Biomech 23:135–146 18. Poelert S, Valstar E, Weinans H et al (2013) Patient-specific finite element modeling of bones. Proc Inst Mech Eng H 227:464–478 19. Hernandez CJ, Beaupré GS, Keller TS et al (2001) The influence of bone volume fraction and ash fraction on bone strength and modulus. Bone 29:74–78 20. Carter DR, Hayes WC (1977) The compressive behavior of bone as a two-phase porous material. J Bone Jt Surg 49:954–962 21. Carter DR, Hayes WC (1976) Bone compressive strength: the influence of density and strain rate. Science 194:1174–1176

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22. Schaffler MB, Burr DB (1988) Stiffness of compact bone: effects of porosity and density. J Biomech 21:13–16 23. Keller TS (1994) Predicting the compressive mechanical behavior of bone. J Biomech 27:1159– 1168 24. Jacobs C (1994) Numerical simulation of bone adaptation to mechanical loading. Dissertation, Stanford University 25. Lotz JC, Gerhart TN, Hayes WC (1991) Mechanical properties of metaphyseal bone in the proximal femur. J Biomech 24:317–329 26. Snyder SM, Schneider E (1991) Estimation of mechanical properties of cortical bone by computed tomography. J Orthop Res 9:422–431 27. Rice JC, Cowin SC, Bowman JA (1988) On the dependence of the elasticity and strength of cancellous bone on apparent density. J Biomech 21:155–168 28. Hodgskinson R, Currey JD (1990) The effect of variation in structure on the Young’s modulus of cancellous bone: a comparison of human and non-human material. Proc Inst Mech Eng H 204:115–121 29. O’Mahony AM, Williams JL, Katz JO et al (2000) Anisotropic elastic properties of cancellous bone from a human edentulous mandible. Clin Oral Impl Res 11:415–421 30. Kopperdahl DL, Morgan EF, Keaveny TM (2002) Quantitative computed tomography estimates of the mechanical properties of human vertebral trabecular bone. J Orthop Res 20:801–805 31. Morgan EF, Bayraktar HH, Keaveny TM (2003) Trabecular bone modulus-density relationships depend on anatomic site. J Biomech 36:897–904 32. Choi AH, Ben-Nissan B (2018) Anatomy, modeling and biomaterial fabrication for dental and maxillofacial applications. Bentham Science Publishers, United Arab Emirates 33. Choi AH, Conway RC, Taraschi V et al (2015) Biomechanics and functional distortion of the human mandible. J Investig Clin Dent 6:241–251 34. Choi AH, Ben-Nissan B, Conway RC (2005) Three-dimensional modelling and finite element analysis of the human mandible during clenching. Aust Dent J 50:42–48 35. Koseki M, Inou N, Maki K (2005) Estimation of masticatory forces for patient-specific analysis of the human mandible. In: Ursino M, Brebbia CA, Pontrelli G et al (eds) Modelling in medicine and biology VI. WIT Transactions on Biomedicine and Health, vol 8. WIT Press, Boston, pp 491–500 36. Choi AH, Matinlinna J, Ben-Nissan B (2013) Effects of micromovement on the changes in stress distribution of partially stabilized zirconia (PS-ZrO2 ) dental implants and bridge during clenching: a three-dimensional finite element analysis. Acta Odontol Scand 71:72–81 37. Choi AH, Matinlinna JP, Ben-Nissan B (2012) Finite element stress analysis of Ti-6Al-4V and partially stabilized zirconia dental implant during clenching. Acta Odontol Scand 70:353–361

Chapter 7

Artificial Intelligence, Machine Learning, and Neural Network

As a field of computer science, artificial intelligence or more commonly known as AI is conceived with the intention of executing numerous specific tasks that normally require human intelligence. Fuzzy logic, machine learning, and neural network fall under the umbrella of AI where algorithms are trained to carry out tasks by studying patterns from data instead of explicit programming [1]. It has been suggested that the clinical implications of AI are its ability to predict the survivability of dental implants as well as offering the opportunity to optimize its design [2]. AI in conjunctions with numerical simulations could also reduce the computational cost associated with predicting new bone formation within synthetic bone tissue scaffolds used to treat large bone defects.

7.1 Fuzzy Logic Considered as a subcategory of AI, fuzzy logic is a technique of reasoning which mimic the human behavior of reasoning. As a data handling approach that allows ambiguity or imprecise descriptions, it acknowledges that in the real world most things would fall somewhere in between with varying shades of gray [3]. A definite output which can be partially true or false similar to a human “yes” or “no” decisions is generated instead of absolutely true or false [3, 4]. Introduced in 1965 by Zadeh [5], a fuzzy set is a class of objects with a continuum of grades of membership. Such a set is characterized by a membership function that assigns to each object a grade of membership ranging between zero and one. The fuzzy “if–then” rules are a more general concept that plays a vital role in the fuzzy sets approach to examine imprecise description [6]. Generated from a set of given training patterns, the classification behavior of the fuzzy “if–then” rule-based classifier can be easily interpretable by human users as the rules are fundamentally expressed in linguistic forms [7, 8]. The Mamdani fuzzy rule-based system (implemented as an interpreter

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. H. Choi, Bone Remodeling and Osseointegration of Implants, Tissue Repair and Reconstruction, https://doi.org/10.1007/978-981-99-1425-8_7

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for a set of rules expressed as fuzzy conditional statements) and the Sugeno (Takagi– Sugeno-Kang) fuzzy rule-based system (has a fuzzy input and its consequence is a linear input–output relation) are the two main class of fuzzy systems [9, 10]. The structure of a fuzzy system can be broken down into three blocks. Fuzzification is the first block that converts the given numbered-input to a linguistic variable utilizing the membership functions stored within the knowledge base. The inference engine is the heart of the fuzzy system capable of human decision-making by operating based on the input data and the fuzzy rules provided. The final block translates the fuzzy output into a single value (categorical or numerical) [7, 11]. Fuzzy logic mechanobiological modeling is another approach that can be used to simulate bone healing, in which the mechanical stimulus calculations can be carried out using finite element analysis (FEA) and the fuzzy rules simulate the mechanobiological regulations [12]. The combination of FEA and a set of fuzzy rules based on medical knowledge of the fracture healing process to describe tissue transformation and blood supply such as intramembranous ossification, chondrogenesis, endochondral ossification, and tissue destruction were attempted in a number of studies [13, 14]. The complex interactions of mechanical stability, revascularization, and tissue differentiation in secondary fracture healing were examined by Simon and coworkers through the development of a dynamic model that includes blood perfusion as a spatiotemporal state variable to simulate the revascularization process. A 2-D axisymmetrical finite element (FE) model was based on a standardized callus geometry of an ovine metatarsus with a transverse osteotomy and loaded axially to represent the amplitude of the major metatarsal loading during normal walking in sheep. Each callus element had its own material properties, which were updated at each time step based on the current tissue concentrations, and the properties of the pure tissue. Dilatational strain (hydrostatic strain) and distortional strain were the mechanical stimuli in their study. A Mamdani-type fuzzy logic controller was used to simulate the biological processes that consisted of eight linguistic “if–then” rules which described processes of angiogenesis, intramembranous ossification, chondrogenesis, cartilage calcification, endochondral ossification, and tissue destruction, all of which depended on local strain state and local blood perfusion. Membership functions, correlating quantitative values such as strain, concentrations, and perfusion, to linguistic values (such as low, medium, and high), were defined for each of the fuzzy inputs and the fuzzy output variables. Using the numerical model, the study simulated different healing situations corresponding to previously published experimental fracture healing cases in sheep and the calculated course of the interfragmentary movement were compared with weekly measured axial movements [13, 15]. Steiner and co-workers [16] further calibrated the fracture healing algorithm developed by Simon et al. [15] to be applicable to a greater range of different mechanical conditions, and in particular, predicting fracture healing under axial compressive and torsional rotational isolated loading. In addition, the authors suggested that the model should also be capable of estimating the healing processes under nonaxially symmetric loading such as translational shear or bending. The local mechanical stimuli (distortional and dilatational strains) were determined in an iterative

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loop and utilized as input for a fuzzy logic controller along with the current tissue composition and blood supply. Twenty linguistic fuzzy logic rules control how the tissue composition and vascularization for each FE within the healing region changes depending on local mechanical and biological stimuli. The rules were in part based on a previously published mechano-regulatory model and represent intramembranous ossification, chondrogenesis, endochondral ossification, revascularization, and tissue destruction. Rules concerning the chondrogenesis process were modified for an improved representation of the tissue differentiation. Later, a study adopted from the work of Simon et al. [15] was attempted using 3-D biphasic FEA instead of 2-D model [17]. Biphasic FE model was used to obtain shear strain and fluid flow in their study, which was then used in their calculations to determine the biophysical stimulus. According to the authors, it has been previously suggested that fluid shear strain is the mechanical signals that are sensed by bone cells and activities of cells regulated in a process known as mechanotransduction. Hence, in their model, they were able to include this process and examine the way cells sense the mechanical signals from tissue level and regulate its response. The fuzzy controller consisted of fourteen linguistic “if–then” rules and was compared to Simon et al. [15], and more rules were used to describe the processes of angiogenesis, chondrogenesis, and endochondral ossification. Furthermore, no rule was used to describe the cartilage calcification, which occurred in the presence of higher mechanical stimuli and independently from perfusion. The quantitative values of biophysical stimulus, tissue concentrations, and perfusion were translated to linguistic values through the membership functions. Hydrostatic and distortional strains determined from the principal strains within the healing region were selected as the mechanical stimuli in an effort to gain an understanding into the spatiotemporal healing phenomena for plated bone regulated by both FEA and fuzzy logic control [18]. 3-D FE models of femur with a 20 mm fracture gap in the subtrochanteric region were constructed using a manufacturer supplied CAD model of the left femur. The callus region representing the initial fracture hematoma was constructed to capture the tissue composition resulting from biological tissue characterization. Each callus element contains its own characteristic material property which was updated after each iteration based on current tissue concentrations. The callus region was assumed to be full of connective tissues during the initial stages. The fuzzy controller consisted of nineteen linguistic “if–then” rules that consider the process of angiogenesis, intramembranous ossification chondrogenesis, cartilage calcification, endochondral ossification, and bone and cartilage destruction. Mamdani-type controller was used to link the input variables (mechanical stimuli, perfusion, cartilage, and bone concentration) to predict the changes in the output variables (perfusion, bone and cartilage concentration). A fuzzy logic mechano-regulation fracture healing simulation was attempted and applied to 3-D models representing idealized tibia diaphysis fractures based on AO/OTA classification of tibial fractures with intramedullary nail fixation and multiaxial loading conditions [19]. Hydrostatic and distortional strains as mechanical stimuli determined from FEA using 3-D idealized CAD models of a cortical shaft with a trabecular bone cylinder were used as inputs for the fuzzy logic controller

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composed of twenty-four “if–then” rules governing tissue differentiation during the healing process (intramembranous ossification, endochondral ossification, bone and cartilage remodeling, cartilage calcification and formation, and tissue destruction).

7.2 Machine Learning and Neural Network Machine learning centers on the learning aspect of AI through the development of algorithms based on a wide array of data inputs commonly referred to as covariates. Machine learning models can account for complex, non-linear relationships due to their flexibility [20]. Unsupervised, semisupervised, supervised, and reinforcement learning are the four commonly used learning methods in machine learning, and they are useful for solving different tasks [21]. Furthermore, prognostic models derived from machine learning can examine and study a vast amount of data and have shown the capacity to recognize new relationships and associations believed to be undetectable through human observations [20]. In a recent clinical study, two different supervised learning methods (a decision tree model and a support vector machine) were used in an effort to predict the most significant factors in dental implant prognosis with both methods yielding similar results [22]. Deep learning is a category of machine learning that allows computers to understand the world in terms of a hierarchy of concepts and learn from experience. The hierarchy of concepts permits the computer to learn complex concepts by building them out of simpler ones. Furthermore, it is unnecessary for a human computer operator to specify correctly all the knowledge that the computer requires due to the fact that the computer collects knowledge from experience [23]. Deep learning can generate models with exceptional performance because of increasing computer power, and these models are multilayer artificial neural networks [1]. Inspired by the biological neural networks that make up the human and animal brain, artificial neural networks (ANNs) are computational analytical tools and are the most popular AI approach in medicine [3, 24]. The first and simplest type of artificial neural network conceived was the feedforward neural network, in which there are no loops or cycles in the network, and as the name suggests, the information can only move in the forward direction. ANNs consist of layers of neurons, normally an input layer (where the initial data are inputted for processing), one or two hidden layers (where the computation is conducted), and a final layer of output neurons (where the results are generated), and all these layers are fully interconnected. Hence, in a feedforward neural network, information moves from the input layer through to the output layer, with the hidden layer in between [25]. “Neurons” are used in the creation of ANNs, which are networks of highly interconnected computer processors that are able of conducting parallel computations for knowledge representation and data processing. Links are used to connect the neurons, and each of these connections is associated with a numeric number known as weight. In addition, there are also bias and activation functions. In essence, neurons take the input value and multiply it by the weight vector. Next, the bias is added to this weighted input. Finally, this

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summation is further processed by the activation function and an output is produced [26]. Extensively utilized in neural networks, the backpropagation algorithm is a learning algorithm in which the weights in the connections are repeatedly attuned so that the neural network will produce the desired output. In other words, the variations between the actual output and the desired output are minimized [27]. The capacity of ANN to correctly recognize and categorize patterns has led to their application in medicine, and in particular solving clinical problems due to their capability to gain knowledge from historical examples, deal with inexact information, and evaluate non-linear data [3]. The ability to learn from examples is a vital feature of ANNs, and hence, the application of ANNs can be categorized into a training phase, a test phase, and an application phase [28]. The repeated amendments of weights enable the network to learn [3, 29]. ANNs are rapidly discovering new fields of application within clinical biomechanics and in particular biomechanical modeling, where the sequence of input and output variables follow common biomechanical ideas about movement control without having deterministic relationships of these variables on hand like electromyogram and force relationship [28]. Deep convolution neural network, which is a type of deep learning, has produced notable results in prediction and diagnosis in pathology and radiology. In essence, it is used to process data that contain a grid pattern such as images and is designed to learn spatial hierarchies of features automatically and adaptively [30]. Constructed from three types of layers, extractions of features are carried out by the first two layers of the neural network (convolution and pooling layers). These extracted features are normally transformed into 1-D arrays of vectors or numbers and then fed as inputs into the third layer of the network known as a fully connected layer. Learnable weights are applied to the inputs, and probabilities are predicted for each class in a classification task [30]. Convolution neural network has also been investigated in dentistry and oral and maxillofacial surgery due to their ability in object recognition and in handling complex images such as cone-beam CT (CBCT) images for applications such as tooth numbering and root morphology [31, 32], dental and maxillofacial radiological examination [33], diagnosis and detection of cysts and lesions [34, 35], automated segmentation of the mandibular canals for dental implant planning [36–40], and bony changes of the temporomandibular joint condyle head [41]. Despite the fact that CBCT is capable of providing high-quality 3-D images, a study by Minnema et al. [42] suggested that there are still uncertainties as to which convolution neural network training strategy is the most ideal for 3-D medical image segmentation. Different training strategies (2-D, 2.5D, majority voting, randomly oriented 2-D cross-sections, and 3-D patches) were compared using both simulated and experimental CBCT images with the intention of providing the most ideal training strategy for CBCT image segmentation. It should be mentioned that another well-known model of deep learning is the recurrent neural network where the connections between nodes can generate a cycle as in the case of the human brain. This allows a network to recycle its limited computational resources over time and carry out deeper sequence of non-linear transformations. Recurrent neural networks are more ideal in tasks which require temporal patterns to be generated or recognized such as linguistic tasks (speech recognition

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and text translation) and time series prediction such as drug pharmacokinetics and pharmacodynamics [43, 44]. It has been postulated that the ability for a machine learning model to study a function without prior knowledge of the problem and able to map inputs and outputs will create an opportunity in which finite element method (FEM) can be employed to generate data that machine learning models can use to calculate a function that maps inputs such as mechanical properties and outputs such as stresses offline. The machine learning models can then make predictions for complex biomechanical behaviors in real time once the mapping functions are fully trained and evaluated [45]. This approach could potentially be useful in easing computational costs associated with the use of FEM in applications such as design optimization where testing every possible design scheme is impractical. This is particularly true when it comes to dental implants in which the stress developed at the bone-implant interface is governed by a number of design variables such as materials used and the implant geometry [46, 47]. Four research directions have emerged regarding the integration of deep learning into FEM. These include multiscale simulations, constitutive modeling, element formulations, and surrogate modeling. Surrogate models are most useful if the FE model needs to be solved repeatedly because of minor alterations made in the analysis such as changing boundary conditions. One FE model is substituted by each surrogate model at the model level. The removal of iterative methods speeds up the computational process by several orders of magnitude. However, the use of surrogate models does have its drawbacks. Surrogate models need to be trained specifically for each problem, and ultimately, this limits their flexibility. In addition, the resources needed to generate computational data used in the training also need to be considered [48]. Below are some of the recent examples where FEA is combined with artificial neural network in applications centered on bone tissue repair and remodeling.

7.3 Predicting the Osseointegration Process An inverse procedure was first attempted by Deng et al. [49] to identify Young’s modulus of interfacial tissue around a dental implant using neural network and FEA. 3-D FEA model with known interfacial properties was used to obtain displacement responses that were then used to train the neural network model. In their study, perfect bonding was assumed between the dental implant and bone interface; hence, it cannot describe precisely the property and structure of interfacial bone during the bone healing process where various degrees of osseointegration and patterns can take place. Later, the group developed a rapid inverse analysis approach based on the reduced-basis method (to compute the displacement response of the bone-implant structure subjected to a harmonic loading) and a trained neural network to identify “unknown” Young’s modulus of the interfacial tissue between a dental implant and the surrounding bones. A self-curing resin was used to simulate the changes in stiffness of the interfacial tissue during the osseointegration process and the authors

7.4 Design Optimization

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did not mention at what stage during the curing process the harmonic loading was applied to the simulated aluminum implant [50]. Although the authors claimed their proposed approach was able to predict Young’s modulus of the resin with an error percentage of 2.5%, no other experimental testing was carried out to further validate their inverse analysis approach such as using a different resin. Additionally, perfect bonding between the bone and implant was assumed in their study and the ability of this study to identify Young’s modulus of interfacial bone tissue during the different stages of osseointegration is also unclear. The abovementioned studies demonstrated the feasibility of combining FEA with machine learning to predict the outcomes of implants mathematically. In spite of that, they failed to represent realistic clinical scenarios or appropriate bone properties. Bone remodeling was also not considered in their analysis as perfect bonding between the implant and bone tissue was assumed (100% osseointegration). Furthermore, evaluations of the simulation outcomes should be carried out against in vivo examinations or clinical trials in order to determine the validity of the training of the neural network. It has been postulated that the thickness of soft tissue at the bone-implant interface can be assessed utilizing a method based on the analysis of its ultrasonic response by means of a simulation-based convolution neural network. Osseointegration phenomena were determined based on the estimation of the soft tissue thickness present at the bone-implant interface that was obtained by analyzing the interaction of the ultrasonic waves with the bone-implant interface. The implant surface roughness was modeled by a sinusoidal function. A number of assumptions were made in their study including the absence of cancellous bone tissues and the simplification in the modeling of the bone-implant interface. In addition, their work only demonstrates the feasibility of such approach to retrieve quantitative information on the bone-implant interface and was not supported by any experimental studies or clinical trials given the fact that the practical applications of ultrasound techniques in determining the stability of a dental implant are hindered by the constant changing variables associated with the osseointegration process [51].

7.4 Design Optimization As mentioned previously, the clinical success of an implant is largely determined by the manner in which the mechanical stresses are transferred from implant to the surrounding bone without generating a force of a magnitude that would jeopardize the longevity of the implant and prostheses. Carefully planned functional occlusal loading will result in the preservation of osseointegration and possible increased bone-to-implant contact. In contrast, insufficient loading may lead to bone destruction/resorption, while excessive loading may lead to bone loss and/or component failure. Hence, it is of paramount importance to discover a means of examining, reducing, and optimizing the stress at the bone-implant interface to reduce the rate of failure of dental implants [47].

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It has been postulated that the use of machine learning can be utilized to reduce the computational cost associated with the application of FEM in determining the influence of implant design variables such diameter, length, and percentage porosity on the stresses at the bone-implant interface [52, 53]. A study hypothesized that the osseointegration process of a dental implant could be improved if it was designed or selected in such a manner that it possesses specific dimensions and physical features such as porosity ideal for the patient’s bone conditions and a microstrain value at the bone-implant interface falls within the range that favors bone growth [53]. An approach based on 3-D FEA of the human mandible and surrogate models using artificial neural network was employed to identify the best combination of porosity and geometry to reach as close as possible to the desired stress (350 MPa for titanium alloy) and microstrain in the bone-implant interface (2500 microstrain) for five different bone conditions (very strong, strong, normal, weak, and very weak bone). In their model, all materials were assumed linearly elastic and Young’s modulus was determined based on CT scan data. The interface between bone and implant was modeled as fully bonded, simulating the ideal osseointegration scenario and without any relative motion at the interface. An average biting force of 120 N was applied on the dental implant. Later, support vector regression as surrogate models was introduced as a replacement for FEM in a study aimed at reducing the stress magnitude at the bone-implant interface [52]. A FEA model based on the first molar region of an edentulous mandible was generated using CT scan data. Both cortical and cancellous bones were simulated as anisotropic materials and the mechanical properties were selected from previously published data. The contacts between cortical bone, cancellous bone, and the implant were modeled as fully bonded. An occlusal force of 150 N was applied vertically to the upper surface of abutment, which according to the authors, was not suitable to represent the real chewing cycle. It should be noted that both studies focused on the reducing the stress at boneimplant interface as well as attempting to enhance the osseointegration process through various optimization approaches and algorithms, and no comparisons were made between their results and predictions with clinical observations of dental implantations; hence, its translation to the clinical environment will need to be further investigated. Mandibular fractures are common facial lesions normally treated with titanium plates and screws. However, the need for its subsequent removal has led to the use of absorbable plate and screw systems as an alternative approach in fracture fixation. Yet, issues such as screw breakage due to high pretension in the screws caused by incorrect application of insertion torque will need to be addressed. A theoretical model was attempted to investigate the possibility of using 2-D FE simulation to train an artificial neural network and its inverse to identify the optimal screw pretension in absorbable plate and screw systems, which must satisfy a desired maximum von Mises strain value [54]. In their study, the neural network was used as a surrogated method to predict the optimal value of maximum von Mises strain by considering the screw pretension as the design factor. Simplified bone segments were constructed and used as training for the network. The model assumes that the cortical bone, trabecular

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bone, absorbable screws, and plates are perfectly bonded to each other. Bone tissues were considered linearly elastic isotropic material, and the thicknesses of the cortical bone and trabecular bone were arbitrary set as 2 mm and 13 mm, respectively. An optimal screw pretension was determined but no validation was carried out in their study against published data.

7.5 Bone Healing and Remodeling A multiscale approach was utilized by Hambli and co-workers [55] in an effort to simulate the bone remodeling process in which 2-D FE was used to calculate changes only at the macro level of a human proximal femur and a trained 3-D neural network serving as a numerical substitute for the FE code necessary for the mesoscale (trabecular architecture) prediction. The application of this multiscale hierarchical hybrid model based on FEA and neural network enabled a reduction in computational time. According to the authors, macroscale bone tissue transfers information to the trabecular network in the form of macroscopic variables and boundary conditions based on the results of the 2-D FE macroanalysis. This information was then used by the trained neural network to predict the responses of the trabecular architecture. Finally, the trabecular network transfers information back to the macroscale bone tissue in the form of averaged updated material properties. The effective properties of trabecular bone were modeled as elastic isotropic behavior coupled with damage incorporating strain and microdamage stimulus. The same approach was later used in another study to investigate the elastic properties of different bone tissue levels. 3-D FE models for each nanoscopic structure of bone ultrastructure (mineralized collagen microfibrils, fibrils, and fibers) were proposed. The models were used to perform parametric studies to examine the influence of geometrical and mechanical properties of the elementary constituents such as cross-links and hydroxyapatite crystals on the equivalent properties. These results were then used as training phase for the neural network. The results were compared and validated using other studies found in the literature, and the authors claimed that a good agreement was observed between the simulation outcomes and literature values [56]. Bone adaptation models are frequently solved in the forward direction where bone response to a given set of loads is calculated through bone tissue adaptation model. The study by Zadpoor et al. [57] proposed an inverse approach in which an artificial neural network was used to predict the load that resulted in tissue adaptation from a given density distribution of trabecular bone. Strain energy density was used in the tissue adaptation model. 2-D FEA was used to calculate the resulting stress and strains once the loads were applied, which were then used to determine the remodeling stimulus signal and the density. The mechanical properties of the bone tissue were then updated according to the stimulus signal. Linearly varying line load and a combination of four constant line loads applied on the edges of the geometry were the loads considered in their analysis. The anisotropic behavior of trabecular bone was a vital issue the authors stated that it should be implemented if this approach

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was to be used in a clinical setting. However, despite the originality of such approach, its practicality in the clinical environment could be confined to applications where it is important to ascertain the magnitude of physiological load a biomaterial will experience if used as a bone substitute or tissue engineering scaffold during bone healing and regeneration process. It has been suggested that machine learning has considerable potential in addressing the limitation of the current FE procedures in predicting bone healing and regeneration around dental implants or in bone tissue engineering scaffolds at both macroscopic and microscopic levels where existing numerical models are computationally demanding [58, 59]. The study by Hsu et al. [58] proposed an alternative to FE simulations during the design process of dental implants with a workflow consisting of a series of machine learning algorithms (deep neural networks). Mechano-regulatory method was used to generate data of bone healing around dental implants, and these data were subsequently used to train the deep learning models. The authors stated that properties such as principal strains and stem cell concentrations of the surrounding bones could be accurately predicted using less time with the neural network than calculations based on FEM. The authors also claimed that this level of accuracy was achieved through the development of a tailor-made deep learning network which permitted the prediction of results 35 days after implant surgery using only the input data from the first day of implantation. Moreover, the design guidelines predicted by the network were similar to industry know-hows without prior knowledge of dental clinics. In another study, a machine learning-based method to replace the homogenizationbased micro-FE analyses in the conventional multilevel FE framework was proposed by Wu et al. [59] in an attempt to estimate efficiently and effectively bone remodeling inside tissue scaffolds using less computational resources than the conventional approach. The proposed approach was applied to simulate scaffold-based bone formation in a large segmental defect of an in vivo sheep tibia. Comparisons of the bone ingrowth results were carried out between those obtained from the conventional multilevel FE model and the predictions made by the proposed machine learningbased procedure to examine its efficiency and credibility. A bone remodeling algorithm based on Wolff’s law that considers the strain energy density evaluated at the microscope level as a mechanical stimulus after sufficient MSC attachment, and proliferation was adopted to predict bone regeneration outcome at the microstructural level in the scaffold. Satisfactory accuracy was observed when the authors compared the machine learning-based time-dependent prediction of bone ingrowth with conventional multilevel FE model.

7.6 Concluding Remarks The clinical translation and application of bone healing and remodeling simulations in areas such as determining the treatment strategies after a fracture or assessing the suitability of a particular implant design based on the bone quality of the patient is

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an objective worth pursuing. Currently, FEA is the primary choice when it comes to simulating and analyzing bone mechanics and structures. However, its dependence on the use of mesh and the accuracy relies on the mesh quality is time and computationally intensive, thus reducing its clinical practicality and impractical for scenarios where real-time responses are needed such as minor design changes due to the anatomical condition of the patient. Studies that utilized AI as a design optimization tool have all shown the potential applicability of such approach in improving design variables to reduce the stresses at the bone-implant interface. Fuzzy logic and ANN in conjunction with numerical simulations could also reduce the computational cost associated with predicting osseointegration and new bone formation. This reduction is achieved through the utilization of FEA to generate datasets that the AI algorithm can learn from and ultimately makes predictions for complex bone healing and remodeling process once the algorithm is properly trained and tested. Furthermore, this approach can be used to carry out inverse calculations to predict the load that resulted in tissue adaptation from a given density distribution of bone tissue. This could potentially be beneficial in determining the magnitude of physiological load a biomaterial will experience if used as a bone substitute or tissue engineering scaffold during the healing and regeneration process. Nevertheless, providing an appropriate learning scheme is vital, as AI (fuzzy logic, machine learning, or ANN) is heavily reliant on the datasets used in the training phase.

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7 Artificial Intelligence, Machine Learning, and Neural Network

58. Hsu CW, Yang AC, Kung PC et al (2021) Engineer design process assisted by explainable deep learning network. Sci Rep 11:22525. https://doi.org/10.1038/s41598-021-01937-5 59. Wu C, Entezari A, Zheng K et al (2021) A machine learning-based multiscale model to predict bone formation in scaffolds. Nat Comput Sci 1:532–541