Basics of Medical Physics [1 ed.] 9788024638843, 9788024638102

Citation preview

Basics of Medical Physics Daniel Jirák František Vítek

Reviewed by: Ing. Milan Hájek, DrSc., Institute for Clinical and Experimental Medicine, Prague doc. RNDr. Otakar Jelínek, CSc., Institute of Biophysics and Informatics, First Faculty of Medicine, Charles University, Prague Published by Charles University Karolinum Press Prague 2017 Edited by Alena Jirsova Setting and layout by Studio Lacerta (www.sazba.cz) First English edition © Charles University, 2017 © Daniel Jirák, František Vítek, 2017 ISBN 978-80-246-3810-2 ISBN 978-80-246-3884-3 (pdf)

Charles University Karolinum Press 2018 www.karolinum.cz [email protected]

CONTENTS

1. STRUCTURE OF MATTER. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Particles and force interactions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Quantum effects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Quantum numbers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.1 Spectrum of the hydrogen atom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Electron structure of heavy atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Excitation and ionisation of atoms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Binding energy of electrons in an atom. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Principle of mass spectroscopy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Atomic nuclei . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Binding energy of a nucleus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Magnetic properties of nuclei. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Forces acting between atoms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.1 Ionic bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9.2 Covalent bonds. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Physical basis of nuclear magnetic resonance tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 9  9 11 13 15 18 20 20 22 23 24 26 27 28 29 29 30 32

2. MOLECULAR BIOPHYSICS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Molecular Bonds and Forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Phases of matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 Gaseous phase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Liquid phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4 Plasma. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Change of phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.1 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Gibbs law of phases. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Classification of dispersion system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Analytical dispersions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Colloidal dispersions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Water . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Transport phenomena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Basic laws of fluids. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Law of Laplace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Viscosity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.4 The Hagen-Poiseuille law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.5 Stokes law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.6 Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 41 43 44 49 49 50 50 50 51 52 53 54 60 63 63 64 66 68 69 70

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2.7 Colligative properties of solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Raoult laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.2 Osmotic pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Phase border phenomena. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Surface tension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.2 Adsorption. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 71  72  73  75  75  76

3. THERMODYNAMICS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Thermodynamic system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Work and heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Heat transport . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Functions of state. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Internal energy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.2 Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.3 Entropy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.4 Free energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5.5 Free enthalpy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Chemical potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Reaction heat. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Thermodynamics of biological system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Transformation and accumulation of energy in biological system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Measurement of temperature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.1 Liquid thermometers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.2 Medical thermometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.3 Calorimetric thermometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.4 Thermocouple. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.5 Electrical resistance thermometer. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.6 Thermistor. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.7 Thermography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10.8 A bimetallic strip. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Calorimetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Thermal losses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 The laws of thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 77  77  79  81  82  83  84  85  87  87  88  88  89  91  92  92  93  93  93  93  94  94  94  94  95  96

4. BIOPHYSICS OF ELECTRIC PHENOMENA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Coulomb law and permittivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Electric potential, potentials of phase boundary-lines. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Donnan equilibrium. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Electric phenomena in alive organism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Resting membrane potential of nerve cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Action potential of nerve fibre . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Action potential in heart cell. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.4 Electrocardiogram (ECG). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.5 Heart’s electrical sequence and interpretation of electrocardiogram . . . . . . . . . . . . . . . . . . . . . 4.2.6 Electroencephalograph (EEG) and Electromyograph (EMG). . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Electric field, electric current and voltage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 Conduction of electric current in organism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Effect of electric current on organism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Conductometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Oscilloscope . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

 98  98  98 100 103 103 104 106 110 111 113 115 115 115 118 120 122

5.  ACOUSTICS AND PHYSICAL PRINCIPLES OF HEARING. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.1.1 Basic quantities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

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5.1.2 The Doppler effect. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 Weber-Fechner’s law. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.4 Complex tones. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 The principles of hearing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Ultrasound. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Piezoelectric effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Ultrasound imaging. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.3 Effect of ultrasound on tissue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Shock waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

127 129 131 133 135 135 136 141 141

6. OPTICS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Propagation of light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Ray optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Dispersion of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Light scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Rayleigh scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Raman scattering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Absorption of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6 Polarisation of light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.6.1 Polarimetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.7 Quantum optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8 Wave optics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.1 Interference of light. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.8.2 Diffraction of light . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9 Lenses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.9.1 Compound microscope. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.10 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11 Optics of the human eye . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.1 Eye defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.11.2 Biophysics of vision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

143 143 145 149 149 150 150 151 152 154 155 155 155 157 158 160 162 164 166 167

7.  X-RAY PHYSICS AND MEDICAL APPLICATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 General features of X-rays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.1 Production of braking radiation (bremsstrahlung) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.2 Production of characteristic X-rays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.3 The attenuation of X-radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1.4 X-ray contrast . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Use of X-rays for diagnostic purposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.1 X-ray imaging methods. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.2 Computed tomography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Risks of X-ray radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3 Therapeutic application of X-rays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

169 169 170 172 172 174 177 179 180 184 185

8.  RADIOACTIVITY AND IONISING RADIATION. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Natural and artificial radioactivity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.1 Basic law of radioactive decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Radioactive equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.3 Radioactive series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1.4 Types of radioactive decay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Ionising radiation and its sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.1 Positively charged particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.2 Linear accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3 Circular accelerators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.4 Negatively charged particles – electrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.5 Neutrons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

187 187 188 190 192 193 196 196 196 197 198 199

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8.3

8.4

8.5 8.6 8.7

8.2.6 Radionuclide sources of neutrons. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.7 γ-radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2.8 Cosmic rays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interaction of radiation with matter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.1 Interaction of α-particles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.2 Interaction of β-radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.3 Interaction of γ-radiation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.3.4 Neutron interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Detection of ionising radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.1 Ionisation chambers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 Geiger-Müller counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Scintillation counter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 Semiconductor-based detector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.5 Integral and selective detection of γ-radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic quantities in radiation dosimetry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5.1 Personal dosimeters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of nuclear medicine in therapy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Use of nuclear medicine in diagnostics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.1 Radionuclides . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Scintigraphy (planar gamma radiography). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.3 Single photon emission computerised tomography. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.4 Positron emission tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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1.

STRUCTURE OF MATTER

1.1 PARTICLES AND FORCE INTERACTIONS There are two forms of matter: particles and fields. Under physical conditions, particle forms of matter exist in four states: solid, liquid, gas and plasma. Force interactions are characteristic of individual field types, of which there are four: gravitational and electromagnetic fields (existing as part of the environment) and strong and weak nuclear fields (existing at the atomic level). Individual forms of matter can mutually transform, e.g. the formation of an electromagnetic wave due to the annihilation of particles and antiparticles. The creation of an electron-positron pair during the absorption of γ-radiation is an example of the opposite transformation of a field into particles. The corpuscular form of matter consists of two groups of fundamental particles: leptons and quarks. Leptons do not interact with strong nuclear forces. Both groups consist of three generations. The first generation of leptons contains an electron and an electron neutrino, the second contains a muon and a muon neutrino and the third contains particle τ and its neutrino (See Table 1.1). Table 1.1 Fundamental particles Quarks

Leptons electron

electron neutrino

Flavour

Charge

u (up)

+2/3

e

νe

d (down)

−1/3

muon

muon neutrino

c (charm)

+2/3

μ

νm

s (strange)

−1/3

tau

tau neutrino

t (top)

+2/3

τ

νt

b (bottom)

−1/3

The three generations of quarks differ according to a property called flavour. Each generation has two flavour quarks: u quarks (up) and d quarks (down) in the first, c quarks (charm) and s quarks (strange) in the second and t quarks (top) and b quarks (bottom) in the third. As 9

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well as flavour, each quark is characterised by a non-integer electric charge that equals +2/3 of the elementary charge for the first quark of each pair and −1/3 for the second quark of the corresponding generation (Table 1.1). Quarks also differ according to another property known as colour. Each quark possesses a red, green or blue colour. All fundamental particles, leptons and quarks are also distinguished by a spin quantum number equalling ±1/2. Each particle has its own antiparticle. When charged, the antiparticle possesses an opposite electric charge. In the case of flavour and colour, these properties are denoted by the prefix anti-, i.e. the flavours (quarks) antiu and antid and the colours antired, antigreen and antiblue. Although the antiparticle has the same mass as the particle and the same value of spin (integer or half-value), it has opposite rotation (clockwise or counter-clockwise) and opposite magnetic moment (see Table 1.2). If a particle and antiparticle are in the appropriate quantum states, then they can annihilate each other and produce other particles. Table 1.2 Selected basic characteristics of antiparticles Same mass Identical value of spin (integer, non-integer) but opposite rotation (clockwise, counter-clockwise) Opposite magnetic moment (positive, negative) – if half-value Opposite charge – if not without charge Opposite colour (anticolour)

Quarks also form composite particles called hadrons. Hadrons must possess an integer electric charge and have a colour combination that is colourless or white. These conditions are achieved in two different ways. Hadrons of the first group are composed of two quarks called mesons (a quark and an antiquark). Mesons have an integer value of spin. A typical example is the pion particle, π. It is formed by a u quark and an antiu in the case of meson π0, by a u quark and an antid in the case of meson π+ and by a d quark and an antiu in the case of meson π−. Baryons are another group of hadrons. Baryons are composed of three quarks of different colours (red, green and blue). Baryons have half-value spin. For example, a proton consists of two u quarks and one d quark, while a neutron consists of two d quarks and one u quark. Elementary particles (fundamental hadrons and quanta of fields) consist of two large groups according to spin value. The first are fermions, which are characterised by a half-value spin quantum number. Their behaviour can be explained using Fermi-Dirac statistics. They also function in accordance with Pauli’s exclusion principle, i.e. no two fermions of identical energies can exist in one system. The particles of the second group are called bosons, which possess an integer spin quantum number value. Their behaviour can be explained using Einstein-Bose statistics. In the case of bosons, the number of particles at the same energy level is not limited. Table 1.3 Hadrons Particles composed of quarks – have an integer value of electric charge – are white (colourless) Mesons Baryons

– 2 quarks: quark + antiquark (integer spin) – 3 quarks: (half-value spin)

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As previously mentioned, there are four types of force interactions: strong, electromagnetic, weak and gravitational. The force interactions of all field types contain non-contact and exchange characters, i.e. they occur due to the exchange of the quanta of these fields. Basic Bose particles represent the excitations of these fields. Thus, the photon corresponds to the electromagnetic field, gluons (of three different colours) to strong nuclear force, the particles W± and Z0 to weak interaction and the hypothetical graviton to gravitational interaction. The ranges of the gravitational field (source is mass) and electromagnetic field (source is electric charge) are not limited, whereas the range of the strong interaction (source is colour) is approximately 10−15 m and the range of the weak interaction (enabling the change of flavour) is approximately 10−18 m. These last two are called saturated fields. At distances corresponding to the sizes of the respective atomic nuclei, i.e. approximately 10−15 m, the relative ratios of the strong, electromagnetic, weak and gravitational interactions are 1 : 10−3 : 10−15 : 10−40. These ratios indicate that gravitation is negligible in particle physics but very important for macroscopic objects. Gravitational interaction occurs only in particles with mass; it cannot be absorbed, transformed or shielded against and it always attracts and never repels. The electromagnetic force acts between electrically charged particles and can be cancelled out; it can also attract and repel. The weak interaction is responsible for changing one quark to another. The strong interaction binds protons and neutrons together to form the nucleus of an atom. Of the particles with a mass other than zero, only electrons and protons are stable; other particles are unstable. For example, a free neutron decays after approximately 103 s due to β-decay into a proton, electron and electron antineutrino, n → pe–v´e. This decay corresponds to the transmutation of a d quark into a u quark. Of the particles with non-zero mass, muon μ– possesses the longest life span (2.10−6 s). Most hadrons decay immediately after formation since they exist no longer than 10−12 s.

1.2 ENERGY Energy is a scalar physical quantity and represents the ability to work. The law of conservation of energy states that the total amount of energy in an isolated system remains constant over time. This means that energy cannot be created or destroyed and that it is capable of being transformed from one form to another or transferred from one place to another. For example, the result of the annihilation of an electron and a positron (with a weight equal to the energy equivalent of 0.51 MeV) is two light quanta (photons) of the same energy. Total energy E of a particle (or system of particles) in a force field is given by the sum of its resting energy E0, kinetic energy Ek and potential energy Ep E = E0 + Ek + Ep(1.1) where E0 is the energy related to the particle mass according to Einstein’s relationship E0 = m0 c 2,

(1.2)

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where m0 is the mass at rest and c is the velocity of light in a vacuum (the highest velocity of propagation of energy). A photon, which possesses zero mass and a rest energy of zero, does not exist at rest and moves throughout a system of coordinates at velocity c. The mass m of a particle moving with relativistic velocity v (almost at the velocity of light in a vacuum) increases according to the relation m=

m0 v2 1− 2 c

,(1.3)

where v is the velocity related to the observer. Particles with non-zero mass m0 > 0, energy E, velocity of movement v and momentum p = mv are related by the equation E 2 = m02 c 4 + p 2 c 2(1.4) Kinetic energy Ek is defined by the following equation = Ek

mv 2 p2 = (1.5) 2 2m

Kinetic energy is energy due to motion. It is also independent of direction and can only possess positive or zero values, Ek ≥ 0. Generally, potential energy may be positive or negative according to the zero level chosen. In central fields of Newton-type forces where the force of interaction is inversely proportional to the squared distance, (e.g. Coulomb’s law, Newton’s law of gravitation), the zero level is set where there is no interaction, i.e. at “infinity”; therefore, potential energy Ep is negative (Ep < 0). It must be negative because positive force is required to remove the particle (body, electric charge), which is attracted to the distance. Here, the force of interaction is negligible and energy equals zero. In the mechanics of a mass point, the equation for potential energy is as follows: Ep = mgh, where m is mass, g is gravity acceleration and h is height. In this case, if the zero energy level is defined at the surface of the earth and h > 0, then the potential energy will be positive and will equal the product, mgh. Energy is expressed in joules. However, in atomic physics and radiation physics, this unit is not suitable. In these cases, energy is mostly expressed in electronvolts (eV). 1 eV is the energy obtained by an electron accelerated by the potential difference of 1 volt. Since 1 J = 1 C.1 V and charge 1 C equals a total charge of approximately 6×1018 electrons, 1 eV = 1.6×10−19  J. The relation of 1 eV to 1 J is the same as that for the charge of 1 electron to 1 C. In the physics of elementary particles, the unit eV is also applied as a unit of mass according to equation (1.2). Based on this relationship, the rest mass of electron me = 0.51 MeV/c2. The c2 unit is usually omitted. Thus, similarly, the mass of proton mp = 938.28 MeV and the mass of neutron mn = 939.57 MeV.

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1.3 QUANTUM EFFECTS The laws of classical physics are not sufficient to describe the physical processes that occur at the atomic level. Within the region of very small distances, there are processes that exist separate to the macro-level and that have other physical quantities. One of them is an effect related to Planck’s constant h or Dirac’s constant, ħ = 1.05×10−34 J.s. These constants are related by ħ = h/2π. Planck’s constant in circular motion with radius r represents the smallest amount of radiated energy. One of the effects of quantum properties is the discrete value of angular momentum. Angular momentum L is defined as a vector product (cross-product) of position vector r and the vector of momentum, p = mv. L = [r × p]

(1.6)

In general, the cross-product of two vectors is the vector perpendicular to the plane determined by the vectors multiplied. Its magnitude equals the product of their magnitudes multiplied by the sine of their angle. In the case of regular circular motion, the directions of momentum of vector p and position vector r change at every moment, whereas the magnitude and direction of the cross-product remain constant (see Fig. 1.1). The two vectors r and v are perpendicular to each other. Therefore, angular momentum L = rmv, since sin (π/2) = 1. z L p r y x

Figure 1.1: Orbital angular momentum L of a particle with momentum p at circular motion with the radius r.

According to quantum mechanics, the angular momentum of the orbital motion of a particle can possess only certain discrete values, which are multiples of Dirac’s constant. Similarly, the projections of angular momentum of an atom in the direction of the coordinate axes can only possess well-defined values (see later). As well as orbital angular momentum, elementary particles possess their own angular momentum, spin and magnetic moment due to rotation. Particles with half-value spin are called Fermi particles (fermions) and those with integer-value spin are called Bose particles (bosons). For example, the spin of an electron or of a nucleon equals ½ and the spin of a photon equals 1 (in multiples of ħ). The spin value determines the behaviour of the particle. Thus, Fermi particles with identical spin cannot exist at the same energy level. This explains why all of the electrons in heavy atoms occupy the higher energy levels more distanced from the nucleus instead of the lowest energy levels. On the other hand, Bose particles tend to occupy the same energy state. 13

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Elementary particles and their system (atom, molecule) possess both corpuscular and wave properties – a finding originally resulting from experimentation into the properties of light. Interference or diffraction of light demonstrates that light is represented by waves, while the photoelectric effect demonstrates that light is a flux of quanta of energy in the form of photons. Energy E of a photon (J) is related to frequency f (s−1) of the wave and to its wavelength λ (m) by the equation E = hf = hc/λ, 

(1.7)

where c is the velocity of light in a vacuum and h = 6.63×10−34 J.s = 4.13×10−15 eV.s is Planck’s constant. Therefore, Planck’s constant reflects the sizes of energy quanta in quantum mechanics. The motion of each particle with mass m, momentum p and energy E is related to wavelengths λ of the de Broglie wave given by the equation

λ=

h = p

h 2mE

(1.8)

and to frequency f given by the equation f =

E (1.9) h

Equation (1.8) reveals that the wavelengths of elementary particles are very short. In the case of a wavelength of an electron in an electron microscope accelerated by a voltage of l kV, its energy (expressed in eV) will be 1 keV = 103eV×1.6×10−19 J/eV = 1.6×10−16 J. Using equation (1.8) we get

λ=

6.63 × 10−34 J.s (2 × 9.11× 10

−31

kg ).(1.6 × 10

−16

)J

= 3.88 × 10−11 m = 0.039 nm

Therefore, the wavelength of this electron is four orders shorter than that of visible light. That is why the resolving power of an electron microscope is more accurate than that of an optical microscope (see later in Chapter 6). The corpuscular-wave dualism of a subatomic particle has consequences. For example, it is not possible to simultaneously estimate position vector r of a particle (or its coordinates) or its momentum p with an arbitrary accuracy. Heisenberg’s uncertainty principle holds for the uncertainty of position vector Δr and momentum Δp ∆r.∆p ≥ (1.10) Thus, the smaller region of motion results in a higher uncertainty of momentum. A similar relation also holds for the simultaneous determination of an energy level and its duration. If the uncertainty of an energy level is ΔE and the time interval in which the measurement is performed is Δt, then 14

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∆E.∆t ≥ (1.11) Therefore, if the given energy state lasts for a long time, its energy may be established with a high degree of certainty. For example, the mean time between atom excitation and photon emission during de-excitation is approximately 10−8 s. With respect to the above equation, the uncertainty of estimating the energy level is ∆E =

 1.05 × 10−34 Js = ≅ 1.1× 10−26 J ≅ 7 × 10−6 eV ∆t 10−8 s

Quantum effects can be quantitatively described by quantum mechanics.

1.3.1 Quantum numbers From a quantum-mechanical perspective, the motion of an electron in the force field of the atomic nucleus is not represented by a certain trajectory but by a “cloud”. Its form and distance from the nucleus is determined by other parameters such as orbital angular momentum, magnetic moment and spin. The region of space in which the electron moves is called the orbital. The electron state is described by the wave function, which involves a number of dimensionless parameters that equal the number of degrees of freedom. With regard to the rotation of the electron, the number of degrees of freedom is 4. Therefore, the state of the electron can be completely expressed by four quantum numbers. These numbers are natural integers (with the exception of spin) that determine the geometry and symmetry of the electron cloud. No two electrons in the same atom can have the same four quantum numbers. Principal quantum number n determines the total electron energy. In accordance with the quantum theory of the hydrogen atom, an electron may exist at various energy levels En given by the equation E0 = −

1 me 4 .( 2 ),(1.12) 2 2 8ε 0 h n

where m = 9.11×10−31 kg is the rest mass of the electron, ε0 = 8.854×10−12 F.m−1 is the permittivity of the vacuum and e = 1.6×10−19 C is the electron charge. The principal quantum number is a natural number, which can possess values of n = 1, 2, 3 and so on. Moreover, its value estimates the shell in which the electron appears. The shells K, L, M, N, O, P and Q correspond to the values n = 1, 2, 3, 4, 5, 6 and 7, respectively. As n increases, the orbital becomes larger and the electron spends more time farther from the nucleus. The electron also moves at a higher potential energy and is, therefore, less tightly bound to the nucleus. n also expresses the maximal number of electrons in the shells according to the relation 2n2 (for more detail, see paragraph 1.5). Orbital quantum number ℓ of an electron in a shell expressed by n may possess the values ℓ = 0, 1, 2, … (n − 1) and determines the form and symmetry of the electron cloud, which is in turn determined by angular momentum L. The magnitude of L is given by equation L =  ( + 1) (1.13) 15

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The orbital quantum number is used for notating states in spectroscopy. Values of ℓ are denoted by letters so that the values of ℓ = 0, 1, 2, 3, 4 and 5 correspond to the letters s, p, d, f, g and h, respectively. According to equation (1.13), the value of orbital angular momentum corresponding to state s equals zero and the value corresponding to state p is 2., etc. The notation of states is a combination of the principal quantum number and letter. Thus, a state with n = 2 and ℓ = 0 is 2s, a state with n = 4 and ℓ = 2 is 4d, etc. If an arbitrary direction (axis z) is chosen in the space, then the values of projection of the angular momentum in this direction, Lz, are discrete values given by the product ml  where ml is the magnetic quantum number. Magnetic quantum number ml can possess the values ml = ±0, ±1, ±2, ... , ±ℓ for a given ℓ, which determines the spatial position of the orbital. The magnitudes of the orbital angular momentum can only have discrete values given by equation (1.13). Moreover, the direction of angular momentum is not arbitrary and is limited with respect to the orientation of the external magnetic field. The magnetic quantum number estimates the direction of vector L by determining its component in the direction of the external magnetic→field. → Orbital magnetic moment µorb is related to angular momentum L (see Fig. 1.1) and given by the equation →

µ orb = −(

e → ) L ,(1.14) 2me

where me is the mass of the electron. The unit used to express the orbital magnetic moment of the electron is the Bohr magneton, uB = eħ/2me = 9.28×10−24 A.m2 (or J.T−1, since J/T = J/(Wb.m−2) = J.m2/Wb = J.m2/(V.s) = = C.V.m2/(V.s) = A.m2). For a given ℓ, the number of possible orientations of the orbital angular momentum in the external magnetic field equals 2ℓ+1 since the values of m may vary within a range from −ℓ through 0 to +ℓ. Thus, the magnetic quantum number estimates the  magnitude of the projection of mechanical angular momentum L and of magnetic moment →  ∝ in a certain direction. For the z-component of orbital magnetic moment µorb , z , it holds that µorb , z = ml .µ B . Orbital magnetic dipole moments are multiples of uB. Spin quantum number s is a value that describes the angular momentum of an electron. An electron possesses its own, internal angular momentum, which does not depend on its orbital angular momentum. However, it also possesses its own magnetic moment, which is related to its internal angular momentum. The magnitude of angular momentum S due to the spin of the electron is given for any electron, bound or free, by S =  s ( s + 1) where s = ½ is the spin quantum number of the electron. In an external magnetic field, the vector of the spin angular momentum can have two orientations. Component Sz of the spin angular momentum of an electron along the external magnetic field in direction z is determined by spin magnetic 1 1 quantum number ms, which has two values, ± , and thus the value of this component is ± . 2 2 Similar to orbital angular momentum, spin dipole magnetic momentum μs is also related to e spin angular momentum S by µ s = − S , where e is the charge and m is the mass of the elecm tron. Spin dipole moments of electrons (and of other elementary particles) are multiples of uB. 16

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Thus, the state of an electron in an atom is completely determined by a set of 4 quantum numbers: n, ℓ, m and s. Electron configurations of atoms with more electrons are governed by the Pauli exclusion principle, which states that any two electrons in an atom cannot exist in an identical quantum state. There is a different set of quantum numbers for each electron in a given atom. An electron can transit from one state to another due to the absorption or emission of energy. Transitions between quantised states occur as a result of two photon processes: emission and excitation. Absorption of energy is connected with the transition of electrons from lower to higher energy levels. A downward transition involves the emission of energy (photons). All of these processes require that the photon energy given by the Planck relationship defined in equation 1.7 is equal to the energy separation of the participating pair of quantum energy states (see Fig. 1.2.). E2 ∆E = hf E1

E2 Figure 1.2: Energy separation between energy states. ∆E = hf

During electron transitions from one state to another due to the absorption or emission of energy, only these transitions occur.E By means of this process, the principal quantum number 1 can vary arbitrarily, whereas the orbital quantum number varies only by ±1. These transitions are called allowed while other transitions are called forbidden. Thus, of the 3×2 = 6 possible n = 3, l = 2 3d transitions from shell M (n = 3, ℓ = 0, 1, 2) to shell L (n = 2, ℓ = 0, 1), only those from 3d to n = 3, l = 1 3p 2p, from 3p to 2s and from 3s to 2p are allowed (see Fig. 1.3). n = 3, l = 0 3s n = 3, l = 2 n = 3, l = 1 n = 3, l = 0

n = 2, l = 1 n = 2, l = 0 n = 2, l = 1 n = 2, l = 0

3d 3p 3s

possible

possible

allowed

allowed

2p 2s

2p 2s

Figure 1.3: Transitions from orbital n = 3 to orbital n = 2.

E (eV)

+10 E (eV)

0

+10

17

−100 basics_medical_physics.indd 17

−10 −20

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1.4 HYDROGEN ATOM The simplest system composed of nucleons and electrons is the hydrogen atom. In this system, one electron moves in the central electric field of one proton. The distance from the nucleus, at which the electron appears with the highest probability, can be estimated from the relationship of uncertainty (see equation 1.10). If the electron moves at distance r from the nucleus then the uncertainty only equals r. From equation (1.10), the uncertainty of momentum p is  ∆p = (1.15) r The total energy of the electron in the field of the atom is given by the sum of its kinetic and potential energies. According to equation (1.5) via equation (1.15), kinetic energy Ek is = Ek

2 p2 = ,(1.16) 2me 2me r 2

where me is the mass of the electron. Potential energy Ep of an electron with charge −e in the force field of a proton with charge +e at distance r is given by Ep = −

1 e2 . (1.17) 4πε 0 r

where ε0 is the permittivity of the vacuum. The potential energy of an electron in the field of the nucleus is negative. It reaches the highest (zero) value at “infinite” distance from the nucleus, where the force action of charges of the electron and nucleus is negligible. Thus, total energy E of an electron in the field of one proton is E = Ek + E p =

2 1 e2 − . (1.18) 2me r 2 4πε 0 r

Values of total electron energy can be calculated and plotted as a function of distance r from the nucleus (see Fig. 1.4). This curve of total energy manifests in a minimum value for a certain distance, r0. It holds generally in physics that each system is stable at the minimum value of its energy. Therefore, the highest probability of the appearance of an electron is only at this distance. An electron in its stable state does not emit energy. Distance r0 calculated from equation (1.18) using dE/dr = 0 (the extreme of the function can be calculated on the condition that its first derivative equals zero) is given by r0 =

4πε 0  2 (1.19) me e 2

If numerical values are substituted for electron mass me = 9.1×10−31 kg, permittivity of vacuum ε0 = 8.8×10−12 F.m−1, electron charge e = 1.6×10−19 C and ħ = 1.05×10−34 Js, then r0 = 5.29×10−11 m. This distance is called the Bohr radius. After substituting r0 back into 18

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n = 2, l = 0

possible

allowed

2s

E (eV) +10 0 −10 −20 r0 0.05

0.1

0.15

r (nm)

Figure 1.4: Total electron energy E as a function of its distance r0 from the nucleus.

equation (1.18) for the total electron energy, the energy of the hydrogen atom in its basic state is givenEvby the equation me e 4 E3 1 E=− . (1.20) E2 32π 2ε 2  2 0

Equation (1.20) corresponds to the solution provided by Schrödinger’s equation for an electron in Ethe field of a proton where n = 1. A state n = 1 is the basic state, while states n = 2, 3, ... 1 are excited states to which an electron may transit after the absorption of energy. If numerical values are substituted for the quantities in equation (1.12), subsequently n = 1, 2, 3, ... up to infinity according to the following energy values: E1 = −13.6 eV, E2 = −3.38 eV, E3 = −1.5 eV, a) b) c) etc. up to E° = 0, respectively. It can be demonstrated that for each energy level, value En corresponds to the most probable distance rn from the nucleus, given by the following equation rn = n 2 r0(1.21) The most probable distance from the nucleus increases with n2. Based on the wave theory of matter, the wavelength of an electron depends on its momentum as defined by equation (1.8). It can be shown that it equals the value of path 2πr0 = 3.3×10−10 m. In simplistic terms, an electron can move around the nucleus for an infinitely long period without emitting energy if its path is an integer multiple of the de Broglie wavelength, i.e. if nλ = 2π rn,(1.22) where n is the principal quantum number. During transitions from higher to lower energy levels, the hydrogen atom emits photons that possess a discrete (line) spectrum. 19

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1.4.1 Spectrum of the hydrogen atom An electron at a higher energy level than its ground state is not stable. A fast transition to either a lower or ground state occurs with the simultaneous emission of a photon. If the state changes from energy Ek to energy En, k > n, then according to equation (1.12) a quantum of radiation is emitted with the following energy E = E k − En =

me e 4 1 1 .( 2 − 2 )(1.23) 2 2 2 k 32π ε 0  n

The frequency or wavelength of this radiation is given by equation (1.7) when the energy value is substituted into the above equation. Since there are discrete values of electron energies, only certain energies (frequencies, wavelengths) may be emitted by the atom. Therefore, a line spectrum of radiation is observed. The set of spectral lines observed during transitions from all higher levels to a certain energy level (corresponding to a given n) is called a series (see Fig. 1.5). The spectral emission lines of the hydrogen atom correspond to a transition to the basic energy level (with n = 1). They can be observed in the ultraviolet region of light, forming the Lyman series. The Balmer series corresponds to a transition to a level of n = 2 and only this series can be observed within the region of visible light. The series corresponding to n = 3 (Paschen series) and to higher values of n are within the region of infrared light. Thus, the highest energy emitted due to a transition from n equal to infinity to a basic state of n = 1 will correspond to equation (1.23) E=

me e 4 0.91× 10−30 × (1.6 × 10−19 ) 4 = = 2.16 × 10−18 J = 13.6 eV 32π2ε 02  2 32 × (3.14) 2 × (8.86 × 10−12 ) 2 × (1.05 × 10−34 ) 2

It corresponds to the wavelength

λ=

hc 6.6 × 10−34 × 3 × 108 = = 9.2 × 10−8 m = 92 nm E 2.16 × 10−18

Analogously, the highest energy in the Balmer series (n = 2) is En = 2 =

me e 4 1 . 2 = 0.54 × 10−18 J = 3.38 eV 2 2 2 32π ε 0  2

Its wavelength is 368 nm, which reaches the region of visible light.

1.5 ELECTRON STRUCTURE OF HEAVY ATOMS The electron structure of atoms with multiple electrons is mainly determined by two rules: 1. The system of particles is stable at a minimum total energy. 2. Only one electron exists in each individual quantum state of the atom. 20

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E (eV) –0 –1 N M –2 –3 L –4 –5 –6 –7 –8 –9 –10 –11 –12 –13 K –14 Lyman series

0

200

n=∞ n=4 n=3 n=2 Balmer series

n=1

400

600

λ (nm)

Figure 1.5: Series of spectral lines for the hydrogen atom.

Like the hydrogen atom, the state of each electron in a heavy atom is determined by four quantum numbers. Each electron moves in the central force field of the nucleus with charge Ze (where Z is the atomic number) and is shielded by the presence of other electrons at a lesser distance from the nucleus. All electrons with an identical principal quantum number are at an approximately equal distance from the nucleus. Therefore, they interact with the same field intensity and possess approximately the same energies. According to this arrangement, the same shell – denoted by letters K, L, M, etc. – is occupied. Since the (2ℓ + 1) values of the magnetic quantum number correspond to each orbital quantum number ℓ and given that the spin number has two possible values, the highest number of electrons present in each shell is given by n −1

2∑ (2 + 1) = 2n 2 (1.24)  =0

However, the electron energy also depends on the orbital quantum number, which rises with increasing ℓ. Its dependence on ℓ also increases in tandem with the increasing number of electrons in the atom. The shell (given by n) or subshell (given by ℓ), which is completely occupied by electrons, is closed. Closed subshell s (ℓ = 0) contains two electrons, closed subshell p (ℓ = 1) six electrons and subshell d (ℓ = 2) ten electrons, etc. The total orbital and spin angular momentum of the electrons in a closed subshell equals zero and the distribution of their effective charge is completely symmetrical. 21

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n = 2, l = 1 n = 2, l = 0

possible

allowed

2p 2s

The periodicity of the physical and chemical properties of the elements corresponds to the order in which electron shells are filled. In the case of heavy atoms, higher shells are filled before the lower shells are completely filled (with respect to the potential number of electrons given by the value 2n2). This scenario occurs when a minimum total energy is needed for the system to achieve stability. The order in which shells are filled in heavy atoms is as follows: 1s, 2s, 2p, 3s, 3p, 4s,E (eV) 3d, 4p, 5s, 4d, 5p, 6s, 4f, 5d, 6p, 7s, 6d. For example, since the energy of state 3d exceeds that of state 4s, level 4s is filled beforehand. However, states s and p in +10 the previous shell must be completely filled before filling the states in shells with higher n. In addition to the Pauli principle, Hund’s rule plays an important role in filling shells with electrons. Where possible, 0 electrons in shells generally remain unpaired, i.e. they possess parallel spins. This results from the mutual repulsion of electrons. Electrons with parallel spin are more separated in −10 space compared with paired electrons and thus this arrangement allows for lower energy and higher stability. −20 r0

1.6 EXCITATION AND IONISATION OF ATOMS 0.05

0.1

0.15

r (nm)

The quantum state of a bound electron with a minimum energy is called the ground state. States with higher energies are called excited states. An excited state is reached through the absorption of energy. An electron can absorb only energy that corresponds to the difference between the ground level and one of the excited energy levels (see Fig. 1.6). Ev E3 E2

E1

a)

b)

c)

Figure 1.6: (a) Excitation, (b) emission of fluorescence radiation, (c) ionisation.

One of the ways an electron gains the energy necessary for the transition to a higher energy level is through the absorption of a photon. Absorption and quantum transitions occur provided the bound electron can absorb a photon whose energy hf equals the energy difference between the initial state and the final state (higher energy). If the energy absorbed exceeds the given potential, then the electron is emitted as a free electron. This process is called (positive) ionisation. In general, ionisation is the physical process of converting an atom or molecule to an ion by adding or removing charged particles, such as electrons or other ions. An ionised atom is not stable and tends to return to a ground state of minimum energy. However, if the electron does not absorb enough energy to leave the atom, then the electron briefly enters an excited state until the energy absorbed is radiated out. The electron does not exist in an excited 22

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state for a long time and eventually transits (de-excites) spontaneously to a lower energy state. It can emit a photon whose energy equals the energy difference between the initial state and the final state (lower energy). Spontaneous de-excitation (fluorescence) occurs over a short period of time (10−5–10−7 s). Excitation energy is emitted in the form of one or more photons. During de-excitation, the electron can reach such an energy level that the transition to the ground state cannot take place (see section 1.3.1). In this case, the electron remains in a metastable state for a substantially longer time and emits radiation later. This scenario is known as phosphorescence. De-excitation is followed by the emission of radiation. Radiation transitions of electrons from higher to lower energy levels result in luminescence.

1.6.1 Binding energy of electrons in an atom The binding energy of a particle in a system generally equals the work that must be achieved in order to remove the particle from the system. Therefore, the binding energy of an electron equals the energy that must be supplied to remove the electron from the action of the electrostatic forces of the nucleus, i.e. to remove it to a place of zero potential energy. Total energy E of an electron in the field of a nucleus is negative (see equation 1.12) and its highest value is zero (for n → ∞ and also simultaneously r → ∞; see equation 1.18) at infinite distance from the nucleus. Therefore, binding energy Eb in this system is determined under the condition Eb + E = 0 and hence Eb = − E (1.25) The binding energy of an electron in the field of a nucleus is positive and numerically equals its total energy given by equation (1.12). For heavy atoms, other factors apply to the equation since the total energy is also a function of atomic number Z. As a result, the binding energies of an electron in these atoms are Z2 times higher for the same n. Thus, the binding energy of an electron in the K-shell of the hydrogen atom is −(−13.6 eV) = 13.6 eV, while in the uranium atom (Z = 92) in the same shell its order of magnitude is 105 eV (922 times higher). The binding energy is also called the ionisation potential. Electrons in heavy atoms have various ionisation potential values since these electrons possess different total energies. Naturally, valence electrons have the lowest ionisation potential values. If an electron is of a higher energy than its binding energy due to the absorption of a quantum of radiation at energy hf, some part of this energy must be consumed for the work required to remove this electron from the system. The remaining part manifests as the kinetic energy of the removed electron. Thus, the law of conservation of energy directly relates to Einstein’s equation for the photoelectric effect: 1 hf = Eb + mv 2(1.26) 2 Electrons are emitted from matter (metals and non-metallic solids, liquids and gases) as a consequence of their absorbing energy from electromagnetic radiation (such as visible or 23

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ultraviolet light). In this way, a current can be induced in a circuit simply by shining a light on a metal plate. One photon gives all of its energy to one electron. Therefore, the rate at which the electron is ejected is dependent on radiation intensity. The emission of photo-electrons only occurs if the frequency of incident radiation exceeds the threshold frequency. The emission of photoelectrons starts immediately once the surface becomes irradiated. From equation (1.26), it is evident that increasing the frequency of the incident radiation has the effect of increasing the kinetic energy of all emitted electrons. The photoelectric effect is used in many areas in medicine. For example, scintillation counters contain a material that produces flashes of light when struck by radiation. These counters then count and measure the number and intensity of the flashes. They are used in nuclear tracer analysis to identify particular isotopes and by computed tomography (CT) scanners to detect x-rays. A positively charged ion is formed by the ionisation of an atom, since the positive charge of the nucleus prevails. The ionisation of the atom also increases the total energy of the system since the presence of the electron in the system decreases its total energy; the sign of the total energy of the electron is negative. Therefore, an ionised atom is not stable and it tends to return to a ground state of minimum energy through the emission of fluorescence radiation. Changes occurring in the electron envelope of the nucleus after energy absorption depend on the amount of energy absorbed. If the absorbed energy is in the order of eV, the excitation or ionisation of slightly bound electrons occurs. The bonds of electrons in the inner shells of heavy atoms are in the order of tens or hundreds of keV. Therefore, the excitation or ionisation of electrons from these shells may result in the emission of ultraviolet light or x-rays. In general, IR-light, visible or UV-light and x-rays are emitted by excited atoms depending on the difference between the excited and basic energy levels.

1.7 PRINCIPLE OF MASS SPECTROSCOPY Mass spectrometry is an analytical technique for determining particle mass, the elemental composition of samples/molecules and the chemical structures of molecules. In order to measure the characteristics of individual molecules, a mass spectrometer converts them to ions so that they can be moved about and manipulated by external electric and magnetic fields. The method is based on the premise that the ions of various isotopes in a given element have different values of specific charge q/m (the ratio of electric charge q to mass m), which means that their trajectories in a magnetic field are different. Mass spectroscopy involves the following steps: a) A sample is transformed into a gas state and then ionised (usually using a beam of electrons). b) A longitudinal electric field is applied to accelerate these ions. c) The beam of accelerated ions is decomposed in a magnetic field according to various values of specific charge. d) The intensities of the separated ion beams are detected and evaluated. For example, ions with mass m and charge q are obtained from a source of ions and accelerated in an electric field with potential difference U; the kinetic energy of the accelerated ions is then 24

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1 2 = mv qU (1.27) 2

= E

These accelerated ions enter a magnetic field with induction B perpendicular to their direction. (see Fig. 1.7 – perpendicular to the plane of the figure). Thus, the ions are affected by magnetic force Fmag (Lorentz force), the magnitude of which is given by the cross-product of the vectors v and B. Since the angle of these vectors is π/2, the value of the sine equals 1 and Fmag = qvB(1.28) Due to this force, the path of the ions in the magnetic field is a circle with radius r, which can be calculated from the equilibrium of the centrifugal force, Fcentr = mv2/r, and magnetic force, Fmag, i.e. mv 2 = qvB (1.29) r From (1.34) it follows that r=

mv (1.30) qB

Calculating velocity v from equation (1.32) and substituting it into equation (1.35), r=

2U B

where A =

2U B

−

1

m q 2 = A   (1.31) q m

The value of constant A depends only on the accelerating voltage and magnetic induction. Thus, its value is the same for different isotopes. The radius of a circle in relation to a certain isotope is a function of its specific charge; various isotopes yield different paths (see Fig. 1.7). If ion detectors are located along a straight line, the impacts of the different ions are likely to occur at different places, m1, m2, etc., at the same accelerating voltage applied. Measuring the relative amount of isotopes then yields the isotopic composition of the sample. The quadrupole mass analyser is a modern, widely used device. It applies an oscillating electric field formed by four electrodes to filter ions. This quadrupole filter consists of an ion trap that accumulates ions over a certain period of time before releasing them into a detector according to the ratio q/m. Another type of analyser measures the time of flight (time-of-flight method, TOF). As ions of the same charge have identical kinetic energies, they are identified according to their respective masses. Thus, ions with lower mass arrive at the detector before heavy ions. The analyser works in pulse mode by measuring the time required to reach the detector according to the ratio q/m. 25

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magnetic field

Figure 1.7: Principle of mass spectroscopy. The ions are accelerated by voltage U so that they all have the same kinetic energy. The ions are then deflected by a magnetic field B according to their masses. The lighter they are, the more they are deflected. This allows ions of various masses (m1, m2, m3) to be separated.

1.8 ATOMIC NUCLEI The nuclei of atoms are very dense and consist of nucleons, i.e. protons and neutrons. The main features of a nucleus are: atomic number Z, which determines the number of protons present in the nucleus; mass number A, which determines the total number of nucleons; and neutron number N, which determines the number of neutrons. Thus, A = Z + N. The total electric charge of a nucleus is Z.1.6×10−19 C. The mass of an atom is mostly contained in its nucleus, since the mass of a nucleon is approximately 2.103 times higher than the mass of an electron. The mass of an atom is usually expressed in mass units. 1 mass unit (m.u.) is represented by 1/12 mass of carbon isotope 12C. Numerically, 1 mass unit = 1.66×10−27 kg. Its energy equivalent is 931 MeV. Atomic mass is estimated using a mass spectrometer. Isotopes are types of nuclei with identical charge but different mass (identical Z, different A). There are approximately 280 stable isotopes in nature. Together with those produced artificially, their number exceeds 1100. Isobars are atoms whose nuclei contain an identical number of nucleons but a different number of protons (different Z, identical A). Isomers are atoms with the same number of protons and neutrons but with different energies of nuclei. Therefore, an isomer is not stable and after some time its excess energy is emitted in the form of a quantum of radiation. The radius of a proton is 1.23×10−15 m. Radius RA of the nucleus of a heavy atom can be calculated by RU = 1.23 × 10−15. A1/ 3 (m) (1.32)

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For example, for the nucleus of the 238U isotope, RU = 1.23×10−15×2381/3 = 1.23×10−15×6.2 ≅ ≅ 7.6×10−15 m. From hydrogen to uranium, the radii of nuclei increase only by a factor of 6.2. This can result in extremely high values for nuclear mass density (ρ ≅ 1014 kg/m3). Nuclear forces are the result of a strong interaction but the range of their action does not exceed the radius of the nucleus. Forces of strong interaction do not depend on nucleon charge. Their range is 10−15 m and at this distance these forces are the strongest known in nature.

1.8.1 Binding energy of a nucleus The binding energy of a nucleus determines its stability. It can be estimated from the total mass defect of the nucleus. For example, with a mass of proton mp and a mass of neutron mn, the total mass of a nucleus consisting of protons and neutrons is given by the sum Zmp + Nmn. However, measurements can yield a nucleus mass (mnucleus) lower than the calculated value. The difference between the calculated and experimentally measured values is the mass defect Δm. Δm = (Z.mp + N.mn) – mnucleus 

(1.33)

Thus, a certain part of the rest energy of nucleons, represented by their rest mass, is converted into the binding energy, which keeps the system together. The disintegration of a nucleus into individual nucleons requires energy given by ΔE = Δm.c2(1.34) A greater mass defect results in a higher binding energy. The binding energy of nucleus ΔE is related to all of its nucleons. For example, the mass defect of helium 4He is 0.030 mu. This value corresponds to the binding energy value of 28 MeV. The mass defect of uranium 235U is 1.908 mu (mu is the unified atomic mass unit or Dalton and is defined as a one twelfth of the mass of an unbound neutral atom of carbon-12 in its nuclear and electronic ground state and at rest) and yields ΔE = 1741 MeV. Binding energies related to one nucleon, ΔE/A, are different for various nuclei. For light and heavy nuclei, they amount to 7–7.5 MeV, while for nuclei in the middle of the periodic system of elements they are approximately 8.5 MeV (see Fig. 1.8). Note that the binding energies of a chemical system (molecules) are in the order of several eV, while the above energies are in the order of MeV. This means that the nuclei of elements of middle mass are the most stable. In contrast to electromagnetic forces where one charge acts on all other charges, nuclear forces are saturated, i.e. one nucleon interacts with only one nucleon or with a very low number of other nucleons present in the nucleus. At low distances, strong interactions are much stronger than electromagnetic interactions. The electric charge of a nucleus, Ze, forms an electrostatic force field around it with a potential U(r), which is a function of distance r from the nucleus. Therefore, due to electromagnetic interaction, a potential barrier exists for a positively charged particle (proton, deuteron, α-particle) entering the nucleus (see Fig. 1.9). 27

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E/A (MeV)

mass number Figure 1.8: Binding energy per nucleon as a function of mass number A.

Figure 1.9: Potential barrier of an atomic nucleus. The maximum potential barrier is at the surface of the nucleus (distance R), while increasing distance r is weaker.

1.8.2 Magnetic properties of nuclei Like electrons, protons and neutrons also possess their own angular momentum (spin) and conjugated spin dipole magnetic moment. For protons, these vectors are of the same direction, while for neutrons they are anti-parallel. The value of spin is a half-integer (in multiples of ħ). Therefore, nuclei that contain an odd number of protons and neutrons or an odd number of nucleons possess a resulting spin (given as the vector spin sum of the individual nucleons) different to zero. These nuclei also possess a non-zero magnetic moment since protons as well as neutrons are formed by electrically charged quarks. Examples of these nuclei consist of the following nuclides: 1H1, 2D1, 7Li3, 13C6, 14N7, 19F9, 23Na11, 127I53, etc. There are more than one hundred stable atoms with magnetic moment and non-zero spin. All nuclei composed of even numbers of protons or neutrons, e.g. 12C6, 16O8, 32S16, 40Ga20 and so on, possess zero spin. 28

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Nuclear spin number I and the Dirac constant (e.g. Iħ) yield the spin of the nucleus. Nuclei with even mass numbers have integer values of spin (in ħ) while nuclei with odd mass numbers have half-integer spin. The effects of nuclear magnetic resonance are observed for nuclei with I > 0, (see later). Analogous to the Bohr magneton (eħ/2me), which is the unit for the magnetic moment of an electron, the magnitude of the magnetic moment of nucleus μj is expressed as the nuclear magneton (nm) defined by 1 nm = eħ/2mp, where mp is the mass of the proton. As is evident, the numerical value of the nuclear magneton is lower than that of the Bohr magneton by a factor of me /mp, which is approximately 1/1836. The value of 1 nm = 5.05×10−27 J.T−1 (or A.m2). The magnetic moment of a proton is 2.8 nm and the magnetic moment of a neutron is 1.9 nm. Thus, the magnetic moment of a proton is 1.41×10−26 J.T−1, which is approximately 658 times lower than that of an electron. That is why the magnetic phenomena of nuclei are so weak and why sophisticated devices are utilised to observe their effects. The magnetic properties of nuclei serve as the basis for the most advanced imaging method, nuclear magnetic resonance imaging (see section 1.10).

1.9 FORCES ACTING BETWEEN ATOMS Electrons can be rearranged in external shells to yield higher stability, whereby a chemical bond between atoms is formed creating valence electrons. Chemical bonds mostly do not affect internal electrons. Bonds between neutral atoms cannot be explained within the framework of classical physics. However, forces that create a covalent bond can be quantitatively calculated using quantum mechanics. Molecules are stable because the energy of the combined system of atoms is lower than that of the system of separated atoms. Thus, if an interaction between atoms decreases their total energy, these atoms can create a molecule. If two atoms come near to each other, a covalent/ion bond is formed or no bond is formed. If electron shells overlap, they form one physical system. But according to the Pauli principle, two electrons cannot exist in the same quantum state. Therefore, if interacting electrons are forced to occupy higher energy states than those in separated atoms, the system has higher energy and is unstable. In such a case, the mutual overlap of electrons results in Born repulsive forces. These forces possess a very small range of action and drop with distance r proportional to r−13.

1.9.1 Ionic bonds Coulomb attractive forces create ionic bonds. They have a larger range than covalent bonds and the distance between the bound atomic nuclei is larger than the sum of the radii of the bound atoms. The charge is shifted from one atom to another (see Fig. 1.10). Ionic bonds are spherically symmetrical and have no preferred direction. The bond is not saturated, which means that the number of mutually attracted ions is not limited. Through the interaction of ions of opposite charge, a stable spatial configuration is formed in which ions periodically 29

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occupy regular places in space. This is why substances with ionic bonds mostly form crystals. Ionic bonds are typically compounds of metals and halogens, e.g. Na+ and Cl−.

electron transition from one to another atom

positive ion

negative ion

Figure 1.10: Ion bond. Electrons are donated by one atom to another, resulting in positive and negative ions which attract each other.

1.9.2 Covalent bonds As demonstrated in Fig. 1.11, one or more pairs of electrons are simultaneously shared with both bound atoms. This is a consequence of a quantum-mechanical effect called the exchange effect. A significant characteristic of the covalent bond is its dependence on electron spin. This type of bond is very strong when the directions of the spins are anti-parallel. A further important characteristic of this bond is its saturation effect since it requires only a couple of electrons. In the covalent bond, only electrons of external, incompletely occupied shells can participate. The electric charge is not distributed symmetrically in the molecule formed and thus the resulting molecule possesses the electrical properties of a dipole. The covalent bond is the strongest of all bond types. For example, the energy of the bond between two atoms of hydrogen (H–H) is 4.3 eV. This means that energy of only 4.3 eV is released during the formation of a hydrogen molecule. An energy of 6.02×1023×4.3×1.6×10−19 = = 417 kJ is released when forming 1 mole H2. The atoms bound by this type of bond lie close together. For example, the distance between protons in a hydrogen molecule is 0.074 nm, whereas the radius of the internal path of an electron in the hydrogen atom (Bohr radius) is ≅ 0.054 nm. 30

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sharing of electrons

molecule

Figure 1.11: Covalent bond. Two atoms share a pair of electrons to complete the outer shell.

At a certain distance between atoms, a molecule is formed whereby the repulsive and attractive forces are equilibrated; this distance corresponds to the minimum potential energy of the system (see Fig. 1.12). It is impossible to remove atoms at larger distances or to reach them without externally increasing the energy of the system. Of course, the binding energy equals the amount of energy released during the formation of the molecule. Bonds are also discussed in section 2.1.

r1

Figure 1.12: Potential energy E of the system as a function of distance r. W – binding energy, r0 – distance of minimal potential energy. magnetic field

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1.10 PHYSICAL BASIS OF NUCLEAR MAGNETIC RESONANCE TOMOGRAPHY Nucleons possess their own half-integer angular momentum spin. As previously mentioned in section 1.8.2, the magnetic moment of a proton and a neutron is 2.8 nm and 1.9 nm, resp. The magnetic properties of atomic nuclei are the physical basis for the nuclear magnetic resonance effects applied in imaging techniques. Most atomic nuclei possess their own magnetic moment. This principle is the basis for determining the presence and relative amounts of suitable nuclei in samples of matter and for diagnostic imaging computer techniques. All chemical elements have at least one isotope of non-zero spin number in the nucleus and, therefore, they also have magnetic moment. Nuclear magnetic resonance (NMR) is a method based on the distribution and behaviour of magnetic moments of specific isotopes in a magnetic field. As the method is non-invasive and provides information about biochemical processes in living tissue, it has become the most important non-invasive radiological technique. In clinical practice, the term magnetic resonance (MR) imaging is used. Since MR imaging uses low energies, it does not induce ionising radiation damage. The principle of the MR method is based on the absorption of radiofrequencies by nuclei placed in a strong static magnetic field, B0. In general, all isotopes with a non-zero magnetic moment can be used. However, for routine clinical applications, only hydrogen nuclei are used because their MR sensitivity is greater than all other nuclei. Hydrogen is present in all organic compounds and occurs in water (the basic material of biological tissue) in very high concentrations. The distribution of water molecules reflects the structural composition of tissue. As changes in the water properties of tissue also closely reflect pathologic processes (Fig.1.13), this relationship is an important factor in the high success rate of MR imaging in medical applications.

Figure 1.13 MR image of a rat brain (coronal plane). Here, the distinct hyperintense lesion in the brain is a lesion caused by stroke.

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Figure 1.14 The magnetic moments of protons in non-zero magnetic field B0. The sum of all magnetic moments is zero in the case of a zero B0 because they are aligned in a random fashion. In the case of a non-zero B0, macroscopic magnetisation vector M is observed as the sum of all magnetic moments because the moments have an anti-parallel alignment with B0 and all the transversal components are suppressed by the precession.

MR is perhaps most elegantly explained by describing the interaction between radiofrequency radiation and atomic magnetic moments. When nuclei are inserted into a strong external magnetic field, B0, they align themselves with it and start to precess around an axis parallel to B0 (see Fig. 1.14). Transversal components are erased by this precession because of the random phase of the precession of each proton. Macroscopic magnetisation vector M is thus obtained. It rotates around an axis parallel to external magnetic field B0. The frequency of precession ω depends on the magnetic field intensity of external field B0 and on the type of nucleus, which is expressed by gyromagnetic constant γ. This dependence is expressed by the Larmor formula (1.34), the basic equation used in MR

ω = γ ⋅ B0 (1.34) When calculating the frequency value of the rotating magnetic field for 1H protons localised in an external magnetic field with a magnetic induction of 1 T, since the magnetic moment of 1H proton is 1.41×l0−26 J.T−1, it follows from eq. (1.34) and (1.31) that f =

γ B µ B 1.41× 10−26 (J.T −1 ) × 1 (T) = = = 42.6 MHz. The frequency value for the protons 2π π 3.14 × 1.05 × 10−34 (J.s)

is 42.6 MHz. At B0 = 1.5 T, which is the most widely used magnetic field in medical imaging, ω is 63.6 MHz. Since other nuclei have different values of gyromagnetic ratios γ, the resonance effects occur at different frequencies. For B0 = 1 T and 13C nuclei, the frequency is 33

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10.7 MHz, for 19F the frequency is 40.1 MHz and for 31P nuclei the frequency is 17.2 MHz. These examples demonstrate that the values of resonance frequencies are very different for various nuclei. The gyromagnetic ratio is defined as the ratio of the magnetic moment to its own angular momentum and spin, and is given by

γ =

µ (1.35) /2

Resonance frequencies also depend on the chemical state of the compound. If the nucleus of an atom is located in an external magnetic field with induction B0, a weaker field B (calculated as (1 − σ) B0 where σ is the shielding constant depending on the electron density in the vicinity of the nucleus) influences its magnetic moment due to the presence of electrically charged electrons in its envelope. As its exact value depends on the type of electron envelope, it will be different for various chemical compounds of the same element. This effect is the foundation for determining the composition of samples. For example, three peaks can be observed in an NMR signal of ethyl alcohol samples for protons located in the CH3CH2 and OH groups. The interaction between the magnetic moment of an element and magnetic field B0 occurs after the element is inserted into a static external field. This results in the splitting of energy levels. In the case of hydrogen, for any given nucleus there are two energy states. Protons occupy lower energy levels (corresponding to the parallel alignment with B0) and higher energy levels (corresponding to the anti-parallel alignment with B0). The number of protons at each energy level is calculated by Boltzmann distribution. The distance between adjacent energy levels ∆E is described by formula (1.35)

∆E = γ ⋅

h ⋅ B0(1.35) 2π

where h is the Planck constant. Although the difference in the occupancy of protons at lower and higher energy levels is very small (only 7 protons per one million protons at 1 T), it is sufficient for manipulating vector M given that 1 cm3 equates to approximately 1023 hydrogen nuclei. This difference linearly increases with increasing magnetic field (Fig. 1.15), which means that the sensitivity of MR methods used must be greater for higher magnetic fields. Vector M is affected by the external high frequency magnetic field B1 (perpendicular to static field B0), the frequency of which corresponds to the Larmor frequency. In this way, the system of nuclei starts to absorb the energy of electromagnetic field B1. This is is called the nuclear magnetic resonance phenomenon. From a quantum perspective, this represents a transition of the element between individual adjacent stationary energy levels; the probability of passing between levels is the same in both directions (two energy levels in the case of hydrogen 1H). At a given constant temperature, the population of low energy levels is greater (according to Boltzmann distribution) in a steady state. Consequently, the effect of magnetic resonance manifests in energy absorption, because the transition to a higher energy state is dominant. Electromagnetic field B1 is produced by a short radiofrequency (RF) impulse (in the range of 10 MHz – 1 GHz), which is emitted by transmission coils (Fig. 1.16). Depending on the pulse intensity and the duration, resulting magnetic moment M can be flipped in any orientation, most often onto a plane that is either perpendicular or parallel to external magnetic field 34

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Figure 1.15: Splitting of energy levels in hydrogen protons 1H. The distance between adjacent energy levels ∆E is dependent on magnetic field strength.

Figure 1.16: Flipping of magnetisation M onto a perpendicular plane (ϕ = 90°) in the direction of external static magnetic field B0 by applying electromagnetic field B1 transmitted from the transmission coils.

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B0 (at a 90° or 180° pulse, respectively). Immediately after the RF pulse is applied, all excited nuclei are at the same phase and start to return to equilibrium. The return is called the relaxation process, during which the nuclei release absorbed energy in the form of electromagnetic radiation detected in the receiver coils. This is based on Faraday’s law of electromagnetic induction because nuclear magnetisation M precesses (according to Larmor precession) during its return. The induced alternating electromotive force with angular frequency ω is called the MR signal or the free induction decay (FID) signal (Fig. 1.17). The FID signal contains a signal from each compound containing hydrogen nuclei (water, ice, protein, tissue, etc.) and its amplitude is proportional to the number of nuclei that contribute to its formation. A

B

amplitude 100%

x–y plane

50% 0% Mxy

FID t

50% 100%

receiver coil Figure 1.17: FID signal and FID detection. (A) The FID signal is detected in the receiving coils due to the non-zero transversal component of magnetisation M. (B) – The maximum amplitude of the FID is detected immediately after applying the RF pulse, after which the amplitude decreases due to relaxation processes and local magnetic field inhomogeneities.

The speed of relaxation is generally in the range of several milliseconds to a few seconds. As nuclei are parts of molecules, their relaxation processes depend on various factors such as chemical bonds, different types of molecular motions (rotation, vibration and translation), molecule size, temperature, etc. Relaxation plays a key role in MR imaging because it affects the contrast between tissues. Two independent relaxation processes of exponential character can be generated: spin-lattice relaxation and spin-spin relaxation. Spin-lattice relaxation, also called T1 or longitudinal relaxation, is characterised by the recovery of magnetisation vector M in the direction of external magnetic field B0. This is due to the energy transfer from excited atoms in the surrounding areas (lattice) in the form of RF radiation and heat. Magnetisation vector M reaches equilibrium in an exponential manner. This is represented in the expectation probability of a collision between two atoms wherein energy is exchanged as discrete quanta. Longitudinal relaxation is characterised by time T1, which is defined as the time that an excited system needs to recover to 63% of equilibrium (in direction B0) after an RF pulse (see Fig. 1.18). The dependence is described by formula (1.36) 36

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M = M 0 ⋅ (1 − e

−t T1

)(1.36)

where M0 is the amplitude of the magnetisation vector at equilibrium. For example, relaxation times T1 of liver tissue are 576 ms at 1.5 T and 812 ms at 3T. M 100% tissue A tissue B 63%

short T1

long T1

TR

Figure 1.18: Longitudinal relaxation after a 90° RF pulse. This type of relaxation is characterised by time T1. Tissue A has a shorter longitudinal relaxation time than tissue B. In practice, a system is fully relaxed if time TR is approximately 5×T1. A satisfactory T1 tissue contrast cannot be obtained in this case (dotted line). The dashed line shows a greater T1 tissue contrast. MT

100% Whereas longitudinal relaxation causes a loss of energy from the system, spin-spin relaxation (transversal or T2 relaxation) occurs due to a mutual swapping of energy between protons within the excited system. T2 relaxation is also characterised by exponential decay, which is represented by time constant T2 (see Fig. 1.19) and described as follows

M xy = M xy 0 ⋅ e

−t T2

(1.37) tissue B

where Mxy0 is the37% amplitude of the magnetisation vector on the transversal plane immediately after the RF pulse is switched off. Time T2 reflects the speed of loss of the measurable macroscopic magnetisation in the tissue A plane perpendicular to external magnetic field B0. It also indicates the magnetic inhomogeneities inside the excited sample. These inhomogeneities affect the speed of phase decay, i.e. T2 expresses how quickly lose Tphase coherence. For example, relaxation times short Tprotons long TE 2 2 T2 of liver tissue are 46 ms at 1.5 T and 42 ms at 3T. Magnetic field differences in the sample arise from two distinct sources: (1) static magnetic field inhomogeneities throughout the volume of the sample (which can be minimised by shimming the static magnetic field) and 37

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M 100% tissue A tissue B 63%

(2) local magnetic fields (arising from intermolecular interactions in the sample). In contrast to T1, there is no energy change in the system of excited atoms in the case of T2 relaxation. Relaxation time T2 is shorter than T1 and is defined as the time in which macroscopic magnetisation in the plane perpendicular to magnetic field B0 decays to 37% of its original value (see Fig. 1.19). The time of relaxation, which comprises the external field inhomogeneities (in addition to spin-spin relaxation), is specified as the effective T2* (always lower than T2). Equation (1.38) shows the T2 − T2* relationship short T

long T

TR 1 1 1 1 = + γ ⋅ ∆B(1.38) T2 * T2

where ∆B is the mean deviation from external static magnetic field B0. MT

100%

tissue B 37% tissue A short T2

long T2

TE

Figure 1.19: Spin-spin relaxation. This type of relaxation is characterised by time T2, which corresponds to the time in which T2 transversal magnetisation Mxy decreases to 37% of the original value. As shown, T2 of tissue B is longer than tissue A. T2 tissue contrasts depend mainly on time (TE). For a shorter TE, a stronger MR signal is achieved from the tissue.

MR images are characterised by signal intensity and contrast, which are affected by T1 and T2 relaxation as well as proton density. This means that for the same object from the same area, different MR contrasts are produced according to the dominant influence (weighting), i.e. proton density or T1 and T2 (T2*) relaxation. The level of weighting depends on various combinations and on the order of RF pulses and gradients that create the imaging sequences. The MR signal is detected in the receiver coils (due to transverse magnetisation only). Its initial magnitude is equal to the number of excited nuclei in the sample. The radiofrequency signal is converted into digital form and stored in k-space in a computer. The final MR image is reconstructed from k-space using mathematical operations (Fourier transformation). Short TR and TE (which are the basic MR sequence parameters) produce T1-weighted images, long 38

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TR and TE produce T2-weighted images and long TR and short TE produce proton density-weighted images. Spatial information is obtained by different spatial-encoding and gradient techniques. Gradients are small magnetic fields that are generated by gradient coils located within the bore of the magnet. The currents in these coils induce a gradient magnetic field around them, which either add to, or subtract from, static magnetic field B0. Their magnetic induction is a linear function of spatial coordinates, so in this way they unambiguously determine the location from which a signal originates. Thus, Larmor frequency depends on spatial positioning, which enables the exact reconstruction of an MR image. The principle of spatial encoding is shown in Fig. 1.20. The simultaneous application of RF pulses and synchronised changes in the gradients lead to signal acquisition from different places in space. This organisation of pulses and gradients is known as a pulse sequence. Many parameters characterise the pulse sequence. The most important, which influence image contrast, are repetition time (TR) and echo time (TE). TR controls T1 relaxation and is defined as the time interval between two excitation RF pulses (usually a 90° pulse). TE is the time from the application of an excitation pulse to the maximum signal induced in the receiver coil. TE controls the decay of transverse magnetisation and determines the level of T2 relaxation.

B

B0

B0

B B < B0 z

S(ω)

A

Larmor frequency ω = |γ|B0 ω0

B > B0

B >> B0 z ω = |γ|B0[f(x,y,z)]

S(ω) ω < ω0

ω

B0 + ∆B

B0

B

ω > ω0 ω0

ω0 + ∆ω

ω >> ω0 ω

Figure 1.20: Principle of spatial encoding. A – No gradient is used and, therefore, the nuclei in both tubes are of the same Larmor frequency. B – A gradient is used and, therefore, the frequency is dependent on the positions of the tubes. The gradient causes a higher magnetic field at the location of the larger tube. Consequently, the nuclei in the larger tube are of higher Larmor frequency compared to the nuclei in the smaller tube.

From a technical point of view, imaging devices that use NMR signals are complicated and expensive. An MR scanner consists of the following main parts: a magnet, a shim system, an RF transmitter, transmitter and receiver coils, a gradient system, and computer and work stations for systems control and data evaluation (console) (see Fig. 1.21). The magnet provides external magnetic field B0 with a field strength in the range of 0.2–21 T. The magnetic field strength is usually 1.5 T or 3 T in clinical practice. There are three different types of magnet: permanent, resistive and superconductive. The most common MR magnets are based on superconductivity. These magnets have very reliable field homogeneity and do not require the constant application of power. However, they need to be cooled to the temperature of 39

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liquid He. In experimental systems for in vivo measurements of animals, the diameter usually ranges from 10 to 40 cm, while in clinical set-ups the diameter can range up to 1 metre. An important part of the magnet is the shim coil system, which is used to further improve field homogeneity within the central part of the magnet. The gradient system is positioned in the vicinity of the shim coils in the magnet. The current in the gradient coil induces a gradient magnetic field, which slightly changes the strengths of static magnetic field B0 (in a spatially predictable way) and the Larmor frequency (see Fig. 1.20). An MR scanner uses three gradient fields, each designed to vary the field along one of the three primary axes of the machine. The RF transmitter generates an RF pulse of appropriate frequency ω, amplitude, shape and duration (in ms). The transmitter coils represents an antenna, which emits magnetic field B1 into the measured volume, usually in the form of RF pulses. Receiver coils detect the MR signal released by the relaxation processes. This signal is then demodulated and digitised, after which the data is stored in k-space. The same coil commonly serves as a transmitter as well as a receiver. In general, the coils are divided into two main types: volume and surface coils. In volume coils, the object is positioned inside the coil and ideally produces a homogenous signal from the whole volume of the coil. In contrast, a surface coil is positioned very closely to the measured area and, therefore, a higher signal-to-noise ratio (SNR) can be achieved. Nevertheless, the signal intensity strongly depends on the geometry of the coil. Problems may be encountered in patients with cardiostimulators or with implanted metals due to the application of a strong magnetic field.

Figure 1.21: MR scanner. 1 – Patient table, 2 – magnet, 3 – gradient coils, 4 – radio frequency coil.

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2.

MOLECULAR BIOPHYSICS

2.1 MOLECULAR BONDS AND FORCES A molecule is formed when two or more atoms are bound by interactions that are transmitted primarily by valence electrons. For each bond type the outermost electron shell layer absorbs a minimum energy which transforms the system and attains a stable state. Heteropolar (or ionic compounds) bonds are formed when one or more electrons are transferred from one atom to another (e.g. when NaCl is formed, Na+ and Cl– are formed, energy is released, and ions attract each other by electric forces). For more details please consult paragraph 1.9.1 and Fig. 1.10. Homopolar (or covalent) bonds can be explained on the basis of quantum mechanics: electrons are not removed from the atoms, and the outermost electrons of two atoms are shared between them and both the atoms are united into one system. In pure covalent bonds the centres of positive and negative charges coincide (for more details please consult paragraph 1.9.2 and Fig. 1.11. However, there are a considerable number of covalent compounds in which the shared electrons are found with greater probability in one part of the molecule. In this case, the centres of charges are separated and the molecule behaves as electric dipole. For example, the atoms in both H2 or N2 molecules are linked by pure covalent bonds whereas the H2O molecule is a dipole (see Fig. 2.1). This is because of the geometry of the H2O molecule and the large difference in electronegativity between the hydrogen atom and oxygen atom. The result of this pattern of unequal electron association is a charge separation in the molecule, where one part of the oxygen molecule has a partial negative charge and the hydrogens have a partial positive charge. This type of molecule is not an ion because there are no excess protons or electrons, but there is a simple charge separation in this electrically neutral molecule. Similar to the water molecule, amino acid molecules, lipids, and proteins possess dipole properties. Metallic bonds may be found in metals, and are characterized by valence electrons that are removed and shared by all the other atoms. Thus, the lattice points of metallic crystals are occupied by positive metal ions; their valence electrons move more or less freely in the lattice. The bond energy is defined as the energy required to remove the partner particles from each other and separate them by an infinite distance. The bond energy is usually given for 1 mole. In ionic, covalent and metallic bonds the bond energies are in the range of 100–400 kJ/mol, which corresponds to 1–5 eV per bond. The total bond energy of a molecule is the sum of all bond energies present. 41

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Water molecule

–

+

Oxygen

+ Hydrogens

Figure 2.1: Polar covalent bond in a water molecule. The electrons shared by the atoms spend a greater amount of time closer to the nucleus of one type of atom (e.g. Oxygen) than to the nucleus of the other (e.g. Hydrogen), which creates a permanent dipole. The small grey circle represents a proton, and the small dark circle, an electron.

Dipole-dipole and ion-dipole bonds are called van der Waals bonds. These bonds cover all types of attractive forces among electrical neutral molecules. Van der Waals bonds are weaker – (0.001–0.1 eV, or 0.08–8 kJ/mol), and their bond length than covalent, ionic or metallic bonds range is 200–400 pm. Hydrogen bonds are also the result of interacting + dipoles. When a molecule having polar covalent bonds is charged, it forms an electrostatic (charge, as in positive attracted to negaThe Hydrogen atom tive) interaction with a substance of opposite charge (see Fig. 2.2 below). – can form bonds with two other atoms in some compounds containing F, O, and N atoms, or + the hydrogen atoms are bound to two oxygen atoms. The FH, OH and NH radicals. In water, energy of hydrogen bonds is about 10–45 kJ/mol. Hydrogen bonds are of importance in the structure of alcohol, carboxylic acids, amines, fats, carbohydrates, nucleic acids and proteins. Hydrogen bond + than that of a van der Waals + bond. Table 1 The binding energy is usually several times higher summarizes the basic characteristics of selected chemical bonds. All forces that act upon molecules and ions are of electrostatic nature, characterised by the attractive or repulsive behaviour of electric charges. The magnitude of attractive force between the particles equals the gradient of the corresponding potential energy. Therefore, for forces whose action does not depend on direction, it is possible to write F = −dU/dr, where U is potential energy and r is the distance between the particles. There are various mechanisms that exert attractive forces that vary based on the distance between molecules (these forces decrease proportional to the square of the distance between molecules). Coulomb’s law describes quantitatively the interaction among electric charges, where like charges repel, and unlike charges attract. The potential energy of this interaction is inversely proportional to the distance between charges. Potential energy decreases proportionally to r−1 and the force decreases proportionally to r−2. Many molecules are electric dipoles. The changes of their mutual orientation appear during their thermal motion that may result in attractive or repulsive action. The attractive orientation prevails. Keesom’s forces are responsible for the interaction of permanent dipoles. Their attractive force decreases proportionally to r−7. The ions, due to their charge, can induce a dipole in the molecules of compounds. Thus, an attractive action may appear among ions and induced dipoles, with a force that decreases proportionally to r−5. Debye forces are responsible for the interaction among permanent and induced dipoles. These forces decrease proportionally to r−7. 42

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– + – + Hydrogen bond

+

+

Figure 2.2: Hydrogen bond. The small grey circle – proton. the small dark circle – electron.

London forces represent the third type of forces acting among electrically neutral particles (atoms or molecules). These forces are observed in molecules of single-atom gases. Electron clouds exist around the nuclei of these atoms. The average distribution of electric charge in this envelope has spherical symmetry, however, fluctuation in this symmetry may appear at a given moment and thus the atom becomes a weak dipole which may induce further dipoles in other molecules. The force of this type of interaction decreases proportionally to r−7. The common name for all these attractive forces among neutral molecules is van der Waals forces. These forces are responsible for the appearance of surface tension of liquids and play an important role in the formation of phase boundaries in systems of a high complexity (e.g. living organisms). Table 1. The length and strength of chemical bonds. The strongest bond is the covalent bond. In biological systems, covalent bonds are not normally broken unless by enzymic catalysis. Bond type

Length of bond [nm]

Strength (kcal/mole) In vacuum

In water

Covalent

0.15

90

90

Ionic

0.25

80

3

Hydrogen

0.30

4

1

Van der Waals

0.35

0.1

0.1

2.2 PHASES OF MATTER Matter is a general term for the substance of which all physical objects consist, and is described by physical quantities (mainly pressure and temperature) according to its state. It may appear in various phases: gaseous, liquid, solid or plasma. However, there are states corresponding to the transitions between these phases as well. The molecules of fluids and solids are in continuous, but irregular motion (thermal motion). In solids, the thermal motion 43

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of the atomic constituents is mainly restricted to vibration and rotation around their equilibrium positions.

2.2.1 Gaseous phase The gaseous phase can be formed from atoms (i.e. noble gases such as neon), elemental molecules (i.e. one type of atom such as oxygen), or compounds of molecules (which includes various types of atoms, i.e. carbon dioxide). It is characterized by vast separation of the individual gas particles at very low density (around 1%) and viscosity, therefore gases are very compressible. The interactions of gas particles in the presence of electric and gravitational fields are considered negligible. All gases are very diffusible and transparent, and most are colorless: major exceptions are fluorine F2, chlorine Cl2, bromine Br2 or iodine I2. A very important gas property is its high mobility. Gases flow readily from one space to another, occupying all available space, and assume the shape of their container unless prevented from doing so by a solid or liquid barrier or a force. Two or more gases form homogeneous mixtures (solutions) in all proportions; a typical example is air – a mixture of gases. The model of an ideal gas is based on the following assumptions: i) The gas consists of a very large number of identical molecules moving with random velocities; ii) All kinetic energy of molecules is only energy of translation; iii) The molecules do not interact except during brief elastic collisions with each other and the wall of the container; iv) The average distance between the molecules is much greater than their diameter. 2.2.1.1 The ideal gas law The product of absolute pressure p of an ideal gas and of its volume V is directly proportional to the product of the Kelvin temperature T and the number of moles n of the gas. The constant of proportionality R = 8.31 J.mol−1.K−1 (in SI units) is the universal gas constant. pV = nRT (2.1) The Avogadro’s constant NA = 6.02×1023mol−1 is the number of molecules in 1 mole. If we denote the total number of particles N, then the number of particles in n moles is N = nNA. The Boltzmann’s constant k = R/NA. By using this constant, the ideal gas law can also be written in the form: pV = NkT (2.2) Other well-known laws follow from the above-mentioned state equation of a gas. For isothermal processes (T = constant), we get Boyle’s law: pV = constant. For isobaric processes (p = constant), one gets Charles’ law: V/T = constant. Finally, for V = constant, p/T = constant. Charles’ law helps us to understand the definition of absolute zero. This law states that the volume of a gas is directly proportional to its thermodynamic temperature, provided that the amount of gas and the pressure remain constant. The result is that an increase in the temperature of a gas will cause the gas to expand. On the contrary, if we cool a gas sufficiently, its volume will become zero at −273 °C. Hence −273 °C is referred 44

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to as the absolute zero temperature: it is impossible to go any lower (i.e. impossible to get negative volume). Dalton’s law of partial pressure states that the pressure of a gas is proportional to the amount of gas, and in a mixture of two or more gases, the total pressure is the sum of the partial pressures of all the components. Each gas in the mixture creates its own pressure and behaves as though the other gases were not present. This law plays a significant role in medicine and other areas of respiratory science (i.e. anaesthesia, clinical therapy). For example, different proportions of gases have different therapeutic effects, so it is important to know the partial pressures of each gas, in a gas line or gas tank (etc). In addition, oxygen tents used for therapeutic purposes make use of the principles of Dalton’s law (where an increase in the concentration of oxygen, making air’s mixture more oxygen-rich, results in a higher gas pressure, and an increased amount of oxygen delivered to a patient with diminished lung capacity. 2.2.1.2 The Maxwell-Boltzmann distribution The distribution of velocities of the random motion of molecules is given by a function called the Maxwell-Boltzmann distribution. An example of the distribution curve is plotted in Fig. 2.3. The form of this curve plotted as a function of velocity v depends on temperature. The higher the temperature, the broader the form of the curve, and the position of its peak is shifted to higher velocities. While the lower limit of speed is zero, there is no limit for the highest speed. The asymmetry of distribution means that the average velocity differs from the most probable velocity. This is due to the fact that the lower limit velocity is zero, and there is no limit for the maximum speed (actually the maximum velocity equals to the speed of the light). The position of the peak determines the most probable velocity, vmp. According to the Maxwell-Boltzmann distribution, the most probable velocity may be expressed as: vmp =

2kT (2.3) m

where m is mass of molecule, k = R/NA = (8.31 J.mol−1.K−1)/(6.02×1023 mol−1) = 1.38×10–23 J.K−1 is the Boltzmann’s constant and T is absolute temperature. The average velocity, vav is defined as a sum of the speeds of all of the particles divided by the number of particles, and is given by vav =

8kT (2.4) π.m

The root-mean-square velocity, vrms = v 2 , can be calculated as the square root of the mean of squared velocities. The Maxwell-Boltzmann distribution yields for its value vrms =

3kT (2.5) m

For illustration, the values of vmp,, vav, and vrms for O2 molecules at 0 °C are 377, 425, and 462 m.s−1, respectively. The velocities of translation motion of molecules in a gas are relatively high, and they increase with increasing temperature (see Fig. 2.4). The root mean square 45

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dN/N (%) 2,0

1,0 1 2 3

0 400 600 800 1000

v (m/s)

Figure 2.3: Distribution curve of oxygen molecules at 0 °C. 1 – Most probable velocity, 2 – Average velocity, 3 – Root mean square velocity. probability of occurence

100 °K 200 °K 400 °K

velocity Figure 2.4: Maxwell-Boltzmann distribution of the velocities of translation motion of molecules in a gas at temperature.

velocity of O2 molecules in the air at T = 400 K (27 °C) is 483 m.s−1, and that of N2 molecules is 517 m.s−1. Therefore, the exchange of gases during respiration owes thanks to fast diffusion processes in the lungs. In diffusion, the translation motion of molecules is in a straight-line zigzag pattern due to their mutual collisions. The average distance travelled by a molecule between two collisions is called the mean free path. The collisions among molecules results in the mutual transfer of energy and momentum. The relationship between the macroscopic variables of the system (pressure, temperature, or volume) and microscopic variables can be solved, and is known as a kinetic theory of gases, which describes the interactions that lead to macroscopic relationships like the 46

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ideal gas law. This theory postulates that molecules can collide with each other and with the walls of the container. Collisions with the walls account for the pressure of the gas. When collisions occur, the molecules do not lose kinetic energy; that is, the collisions are said to be perfectly elastic: the total kinetic energy of all the molecules remains constant. The molecules exert no attractive or repulsive forces on one another except during the process of collision; between collisions, they move in straight lines. When ρ = Nm/V is the density, the pressure (macroscopic variable) of gas is related to the root mean square velocity by equation p = (1 / 3) ρ vrms 2(2.6) Similarly, the macroscopic variable temperature is related to the average kinetic energy of a molecule by equation = Ekin , av

1 2 3 = mvrms kT (2.7) 2 2

The mean kinetic energy of the molecule is directly proportional to the absolute temperature. It is necessary to stress that temperature is a state variable related to the root-mean-square velocity. Thus, to speak about the temperature of one molecule is meaningless. Equation 2.7 above also contributes to comprehending the physical meaning of the Boltzmann constant k; it is related to the change in mean kinetic energy corresponding to an increase of gas temperature by 1 K (or °C). 2.2.1.3 Theorem of the equipartition of energy Besides translation, further kinds of motion can be found in a system containing molecules composed of many atoms. Each type of motion contributes to the number of degrees of freedom. The number of degrees of freedom, i, depends on the number of atoms in the molecule. We have i = 3 for a single-atom gas, i = 5 for molecules composed of two atoms, and i = 6 for molecules composed of 3 or more atoms. The number of degrees of freedom for a molecule of the ideal gas equals 3 because three coordinates uniquely determine its position in space. The relation (2.7) can be generalized with respect to the other types of motion by the theorem of the equipartition of energy. This theorem quantitatively relates the temperature of the system with the mean energy of individual degrees of freedom assuming that the same average energy appertains to each degree of freedom. The energy of other types of motion, vibration and rotation (see Fig. 2.5) are increased due to thermal energy supplied to molecules composed of many atoms. Thus, according to Maxwell’s theorem of the equipartition of energy, each degree of freedom has an average energy (1/2)kT, and the total mean energy Eav of a molecule can be expressed as 1 Eav = i kT (2.8) 2 47

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Furthermore, since i = 3 for a single-atom gas and k = R/NA, the total mean kinetic energy U of 1 mole of a rare gas is U = NA×(3/2)(R/NA)T = 3/2 RT. At a given temperature, each atom at equilibrium has the same mean kinetic energy of translation motion. Since the kinetic energy depends on mass, the velocity of more heavy atoms (xenon) is smaller than that of light atoms (helium).

A

B

Figure 2.5: Vibration (a) and rotation motion (b) of biatomic molecule.

Spectral lines of excited vibration and rotation states are seen in spectra of systems containing molecules composed of many atoms. Since the changes of these states correspond with small energy differences, these lines can be observed in the infrared spectrum. 2.2.1.4 Real gases At normal conditions such as standard temperature and pressure, most real gases behave qualitatively like an ideal gas. Hydrogen, oxygen, nitrogen, helium, or neon deviate from the ideal gas law less than 0.1 percent at room temperature and atmospheric pressure. Other gases, such as carbon dioxide or ammonia, have stronger intermolecular forces and consequently greater deviation from ideality. The equation of state for an ideal gas (pV = nRT) holds for real gases at low densities and high temperatures (which does not take into account the attractive forces among the molecules or the volume of molecules). The behaviour of real gases can be well predicted by using van der Waals equation ( p + a / V 2 )(V − b) = nRT (2.9) Two empirical constants, a and b are introduced. While a is related to the correction of attractive forces and depends on the properties of the given gas, b represents the effect of own volume of the molecules. Units must be chosen so that the unit of a/V 2 corresponds to pressure and b is expressed in the unit of volume. When a molecule is near the wall, the attractions between it and its neighbours are unbalanced, tending to pull it away from the wall. The molecule produces slightly less impact than it would if there were no intermolecular forces. This means that all collisions with the walls are softer, and that the pressure is less than would be predicted by the ideal gas law.

48

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2.2.2 Liquid phase The essential liquid of life is water (details are found in part 2.5). Contrary to the gas phase, liquids exhibit interaction among molecules that cannot be ignored. The number of molecules present in a unit of volume of a liquid is much higher than that of the gas phase. Liquids are much less compressible and volume changes with temperature are much lower as compared with those in gases. Surface tension (see part 2.8) appears as a result of the presence of van der Waals forces. Common liquids behave as an isotropic medium, i.e. their properties do not depend on direction. However, in anisotropic liquids, mesomorphous states with special arrangement of molecules called liquid crystals or crystalline liquids can be also encountered. These structures are also found in living organisms. The most typical example of the mesomorphous state in the human body is the arrangement of molecules in some regions of striated muscles that may be confirmed by birefringence effects. These states combine some properties of liquids but with a conservation of some structure. The mesomorphous state can be observed in substances consisting of molecules with non-spherical symmetry (rod-like or thread-like molecules). The refraction index of liquid crystals depends on temperature. Some crystalline liquids (e.g. cholesterol derivatives) are applied in thermography which depict temperature changes on the surface of the body. Amorphous solids (glasses) are structurally liquids. Amorphous solids are super cooled and strongly viscous liquids without fluidity.

2.2.3 Solids Solids are characterised by their crystalline structure and resistance to changes of shape or volume. In the solid phase, ions, atoms or molecules form a lattice with a special spatial arrangement, which give the solid its form (see Fig. 2.6). Hydrogen bonds play an important role in crystals of ice. The components of a lattice oscillate around an equilibrium position and the amplitude of the vibrations is a function of temperature. A supply of thermal energy increases the amplitude of vibration motion, and the lattice may collapse at high temperatures. Due to the interaction between components, the electrons of a complex system are not strictly confined to individual levels, though they cannot move completely freely either. In this case, energy bands are formed instead of sharp energy levels.

Na+ Cl−

A

B

Figure 2.6: Crystal lattice of NaCl (a) and graphite (b).

49

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2.2.4 Plasma The plasma state is formed from a gas by partial ionization of gas molecules. This may be caused by a high temperature, and/or exposure to ionizing radiation etc. Plasma is formed together with neutral and electrically charged particles. The mobility of the particles in plasma is similar to that of particles in the gaseous state, nevertheless the mutual action of force of the individual particles is not negligible.

2.3 CHANGE OF PHASES When thermal energy is changed (supplied or removed), the phase may change. The physical quantities describing the state of a system (i.e. pressure and temperature) define the conditions of the change. The change of phases is described in the following figure 2.7: n tio za ni on  o i i De izat  Ion

 Deposition Sublimation 

Gas

Solid

Plasma

 Co Vapo nd ens rizati atio on n

Liquid zing ree  F lting  Me

Figure 2.7: The change of phases.

Any change in phase of matter is accompanied by structural changes. Molecules with relatively high kinetic energies overcome the attractive forces and break away from the liquid (in the case of evaporation) or solid (in the case of sublimation). Condensation or deposition involves the transformation in the opposite direction. Melting, sublimation and evaporation are energy requiring structural changes. Freezing, deposition and condensation are accompanied by heat liberation.

2.3.1 Phase diagram The behaviour of a gas can be understood by means of the phase diagram (see Fig. 2.8). A plot of pressure as a function of temperature yields areas corresponding to solid (III), liquid (I) and vapour phases (II) separated by the lines of sublimation (a), fusion (b) and evaporation (c). At the phase transition point the two phases of a substance are equally likely to exist. 50

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pressure

b

c I water

III ice

II vapour

a

temperature Figure 2.8: Equilibrium p-T diagram of single component system.

The equilibrium water-vapour phase is represented by the line of evaporation that also yields the boiling points at the given pressure. This line ends at critical temperature. A vapour cannot be converted into liquid at temperature higher than its critical temperature. A liquid can be converted into vapour phase at a temperature below the normal boiling point. Similarly, the line of fusion represents the melting points. In the case of water, the increase in pressure results in a decrease of melting point temperature. At the given temperature, ice can be made to melt by the application of pressure. The line of sublimation gives the pressure and temperature of direct conversion of solid into the vapour phase. The three phase equilibrium lines intersect at the triple point. At this pressure and temperature, the three phases can coexist in equilibrium. For water, this equilibrium appears at pressure 610.6 Pa and temperature 273.16 K (0.01 °C). A

B

2.3.2 Gibbs law of phases As a dispersion system we understand a system containing at least two phases or two portions (chemical unique). One of them, the dispersive portion, is dispersed within the other, the dispersing medium. The dispersive portion is not continuous and it is dispersed in a continuous dispersing medium. If there is a boundary between the particles of the dispersive portion and dispersing medium then the system is heterogeneous (e.g. if oil is added to water and the system is shaken, the oil droplets are distributed but undissolved). If the refraction index of the both phases is not identical, heterogeneity with respect to transmission of light appears. On the other hand, if the system contains two portions and one phase only, then the compound representing dispersive portion is dispersed within the dispersing medium in the form of particles so small that the boundary between them cannot be observed. Such system is homogeneous, e.g. sugar in water. A single-phase system is optically homogeneous. Single-phase system may contain more than two portions. For example, if n gases are mixed, a single-phase system of n components is formed. Gibbs Law relates the number of components (c), phases (p) and degrees of freedom of the system (d): 51

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p + d = c + 2(2.10) The number of degrees of freedom of a heterogeneous system is the number of independent variables that define the equilibrium state (i.e. pressure, temperature, concentration). These independent variables can be independently changed without changing the count of present phases (e.g., a single-portion system containing a liquid and its vapour can be at equilibrium at temperatures from the solidification point up to the critical temperature). If we choose a temperature within this range, the equilibrium pressure is uniquely defined. To keep two phases in equilibrium, only one variable may be varied, and the system has one degree of freedom. At the triple point, no variable can be changed if three phases are to be in equilibrium (i.e. p = 3), and this situation appears only at the unique pressure and temperature in a single-compound system. This three-phase system has no degree of freedom. In this case, the system of water vapor, liquid water, and solid ice has zero degrees of freedom because the three phases of vapor, liquid, and solid coexist in one component, water. According to the Gibbs law, the degrees of freedom of a heterogeneous system are uniquely determined by the number of coexisting phases together with the number of independent portions that create the system. Besides the choice of the pressure and temperature, the degree of freedom is enlarged by the choice of concentration ratio in a system containing more than one portion.

2.4 CLASSIFICATION OF DISPERSION SYSTEM Various criteria may be applied for classification of a dispersion system. The size of particles of dispersion portion are commonly used. The reciprocal value of particle diameter is called the dispersion degree, which has the units m−1. Avery fine dispersion possesses a high degree of dispersion. A dispersion system may also be classified according to the phases of dispersive medium and the dispersive portion. This classification is seen in the following table: Table 2: Examples of dispersion systems. Dispersive Dispersive medium portion Gaseous Gaseous

Liquid

Solid

Coarse dispersions

Colloidal dispersions

Analytical dispersions

–

–

Mixture of gases

Liquid

Rain, fog

Aerosols

Vapours of liquids

Solid

Dust, smoke

Aerosols

Vapours of solids

Gaseous

Bubbles, foams

Foams

Solution of gasses in liquids Solutions of solids in liquids

Liquid

Emulsions

Lyosols

Solid

Suspensions

Lyosols

Gaseous

Rigid foams, bubbles of gas in solids

Rigid foams Gas dissolved in a solid

Liquid

Bubbles closed in solids Rigid foams Crystalline water

Solid

Rigid mixtures

Rigid sols

Rigid solutions, doped crystals

52

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A monodispersed system contains dispersed particles of the same size, whereas a polydispersed system contains particles of various sizes. The following classification may be applied according to the particle size: 1) Analytical dispersion (up to 1 nm). The dispersed portion cannot be estimated by physical methods, only chemically; 2) Colloidal dispersion (1–1000 nm); 3) Coarse dispersion (1 μm and greater).

2.4.1 Analytical dispersions All analytical dispersions are homogeneous. A dispersion portion is dispersed as ions, molecules or atoms in a dispersion medium. Dalton’s law (also mentioned in paragraph 2.2.1.1) estimates the total pressure observed in a mixture of gases. Each gas contributes by its partial pressure pi to the total resulting pressure p in a mixture of gases. Therefore, p = p1 + p2 + ....(2.11) The partial pressure is that exerted by the gas if it were distributed alone in the same volume and at the same temperature. Similarly, Amagad’s law holds for volumes. It says that the total volume of a mixture of gases is the sum of partial volumes of its components. The partial volume is the volume of the gas portion if it were distributed alone at the same pressure and temperature. The vapours of a liquid are dispersed in the form of molecules. A limited amount of water vapours can be dispersed in a gas at a given pressure and temperature. Water vapours in the air are of the highest biologically very important. The quantity of water vapours in the air, or absolute humidity(φ) is expressed in kg.m−3. The maximum humidity (φmax) corresponds to the saturation of air by water vapours. The relative humidity, φrel, is expressed as the ratio of absolute humidity to that at saturation state, i.e. φrel = φ / φmax. Multiplication of this ratio by a factor of 100 yields the relative humidity expressed in per cent. If a gas is in contact with a liquid, it dissolves within the liquid until the equilibrium state is achieved. At equilibrium, the same number of gas molecules enter and simultaneously leave the liquid phase equilibrium any one time. Henry’s law states that the amount of a gas dissolved in a liquid is directly proportional to the partial pressure of the gas above the liquid. If a mixture of gases is dissolved, the partial pressure of each of its components has to be considered. If m grams of gas is dissolved in the volume Vliq (litre) of a liquid at temperature T (K) and pressure p (Pa), then m = kp (2.12) Vliq When the concentration, c, of gas present in the liquid phase is expressed in mole/litre, then c = α ∗ p (2.13) 53

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where the coefficient α* is called Bunsen’s absorption coefficient. If Vliq denotes the volume of liquid in which 1 mole of gas is dissolved and Vgas denotes the volume of one mole of gas, then Vliq Vgas

= α (2.14)

The constant α is called Ostwald’s absorption coefficient. All the above relations hold at constant temperature. The solubility of a gas in liquids decreases with increasing temperature. Various gases possess different values of absorption coefficients. Thus, portions of gases dissolve in liquids at different ratios than in the gaseous phase e.g., α*O2 = 0.031 for oxygen dissolved in water at 20 °C while α*N2 = 0.0164 for nitrogen at the same temperature. Therefore, the ratio of oxygen to nitrogen concentrations in water is higher than that in air. The law of Henry does not hold for gases that react chemically with a solvent. This law is of great importance in the physiology of breathing. When air is inhaled at a pressure higher than atmospheric pressure, a greater amount of nitrogen is dissolved in the blood. A fast return to normal pressure may result in the formation of nitrogen bubbles in the capillaries and a nitrogen embolism may result. An excess of nitrogen can be exhaled out of body during the slow return to the normal atmospheric pressure of air. Three situations are observed when mixing together liquids. A homogeneous analytical dispersion is formed when two liquids can be mixed at any concentration ratio (e.g. alcohol in water). Limited mixing is found in the case of mixing ether and water. Two layers are found after shaking this mixture: the upper represents the solution of water in ether, and the lower that of ether in water. No analytical dispersion can be formed when mixing two liquids that are not mixing-able. Solutions of solids are ionic (e.g. NaCl in water) or molecular solutions (e.g. glucose in water). The solubility of a compound depends mainly on the polarity of a solvent. The substances able to dissociate into ions and organic compounds with polar groups (e.g. −SO3H, −COOH, −NH2, etc.) are more easily dissolved in polar solvents. The most common solvent is water. When an amount of a solid substance is added into the given volume of water, the number of particles released into the solvent from the solid phase is greater than the number of those absorbed, and the concentration of the solution increases. A saturated solution is that in which a dynamic equilibrium is achieved, i.e. the number of particles released equals the number entering. In most cases the solubility of solids increases with increasing temperature. Solidification of the solid phase should appear upon cooling of the saturated solution, and occur with certainty if solidification centres are present in the solution. If solidification centres are not present, an over saturated solution is obtained. The amount of compound contained in a volume or mass unit of solution is called concentration.

2.4.2 Colloidal dispersions Colloidal solutions are biologically very significant. There are two types of colloidal solutions: lyophilic and lyophobic, named according to their behaviour with respect to the solvent. Lyophilic compounds (liquid loving colloids) keep easily in water, manifest high hydration, are easily dissolved in water, and colloidal particles of this type also remain stable in water. The process of preparing a colloidal solution is easy; it can be prepared directly by mixing the 54

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colloid with a liquid. Additional stabilizers are not required during preparation. A lyophilic colloidal solution is relatively stable as strong forces of interaction exist between colloidal particles and liquids. Examples of lyophilic colloidal solutions are solutions of organic substances like gelatine, gum, starch and proteins. Lyophobic compounds (liquid hating colloids) are not able to keep water or dissolve spontaneously in water and thus the preparation of a colloidal solution is more complicated. Lyophobic compounds are less stable. An example of lyophilic colloidal solution are solutions of inorganic substances like Arsenic (As2S3) and Iron (Fe(OH)3). Macromolecules or micelles form colloidal particles. Macromolecules are molecular polymers formed by smaller molecules chemically bound in a larger system (e.g., macromolecules of proteins are formed by chemically bound amino acids). Micelles are clusters of particles that lack a chemical bond. Colloidal particles move in solution as individual particles. Molecules of solvent continuously contact these particles which results in their zigzag motion (Brown motion) provided their size is smaller than 4 μm (since strokes from various directions cannot be equilibrated). A colloidal particle with density ρ, moving in a dispersive medium with density ρ0 and viscosity coefficient η, is exposed to the gravitational field force with gravity acceleration g. The particle falls down if ρ > ρ0 (sedimentation) or moves up if ρ < ρ0. The force of gravity Fg = mg affects a spherical particle with radius r, volume V and mass m = Vρ. Simultaneously, the lifting force Fl = Vρ0 g operates in the opposite direction due to Archimedes law. The resulting force is given by the difference Fg − Fl = V(ρ − ρ0 )g and, if ρ > ρ0, the particle falls down. The force Fr, due to the resistance of the medium, operates against this motion. This force is described by Stokes Law (see part 2.6.3 and 2.6.5) and it is defined by Fr = 6πη rv ,(2.15) where v is the velocity and η is the viscosity coefficient. The volume V of a spherical particle is V = (4/3) πr3. It follows from the force balance Fg − Fb = Fr that the sedimentation rate v of a spherical particle is given by v=

2( ρ − ρ0 ) gr 2 (2.16) 9η

The thermal motion of molecules of liquid disturbs the sedimentation process, but after a certain time interval an equilibrium state, called sedimentation equilibrium, is achieved. An equilibrium concentration is established along the direction of the force of gravity. The value of the sedimentation rate is negligible in analytical solutions but it is very large in the case of coarse dispersions. The sedimentation rate also depends on temperature. Since, the earth’s gravitational field is weak the sedimentation rate is very small and very long time would be required for a colloidal solution to achieve sedimentation equilibrium. The use of a high rotation velocity centrifuge strongly accelerates the sedimentation process due to large value of centrifugal acceleration (up to 600,000 g in the case of an ultra-centrifuge). A very important property of colloidal particles is their permeability (different rate of passage for various molecules) or non-permeability across membranes. This property can be used for their separation from the analytical portion of solutions (dialysis or electrodialysis) or even from the dispersing medium (ultra-filtration). 55

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Scattering of light on colloidal particles takes place when a beam of light passes through a cuvette containing a colloidal dispersion (the Tyndall phenomenon). The Tyndall effect is caused by the reflection of light by very small particles in suspension in a transparent medium. It can be observed from the dust in the air when sunlight peers through a window, or beams down through holes in clouds. Since the intensity of scattered light depends on the particle size, the measurement of the intensity of scattered light can be applied for estimating the concentration in a monodispersed colloidal system. Therefore, in solution the beam of light is transmitted, and in colloidal solutions the beam of light is scattered onto surfaces of the suspended particles. This is the basis nephelometry, a technique used for example in immunology to determine the levels of several blood plasma proteins. As a source of light is usually used as a laser. The permeability of colloidal particles is important in dialysis. In general, dialysis separates suspended colloidal particles from dissolved ions or molecules that are (crystalloids) by means of their unequal rates of diffusion through the pores of semipermeable membranes. Separation by dialysis is a slow process, its speed depends on the differences in particle size and diffusion rates between the colloidal and crystalloidal constituents, which can be accelerated by heating or, if the crystalloids are charged, by applying an electric field. Dialysis in medicine is a process for removing waste and excess water from the blood. It is used primarily as an artificial replacement for lost kidney function in people with renal failure. Dialysis works on the principles of the diffusion of solutes and ultrafiltration of fluid across a semi-permeable membrane. Diffusion is a property of substances in water; substances in water tend to move from an area of high concentration to an area of low concentration. The semi-permeable membrane is a thin layer of material that contains various sized pores and plays a key role in diffusion. Smaller solutes and fluids pass through the membrane, but the membrane blocks the passage of larger substances (for example, red blood cells, large proteins). We distinguish two types of dialysis: hemodialysis and peritoneal dialysis. Hemodialysis utilizes counter current flow, where the dialysate is flowing in the opposite direction to blood flow in the extracorporeal circuit. Counter-current flow maintains the concentration gradient across the membrane at a maximum and increases the efficiency of the dialysis. Fluid removal (ultrafiltration) is achieved by altering the hydrostatic pressure of the dialysate compartment, causing free water and some dissolved solutes to move across the membrane along a created pressure gradient. A schematic of hemodialysis is shown in figure 2.9. The blood is filtrated in a dialyser. A semi-permeable membrane contains thousands of small ‘capillary’ like tubes and fitted into a cartridge (see Fig. 2.10). Cellulose and polysulfone are usually used as membrane. Second type of dialysis is peritoneal dialysis. Principle of peritoneal dialysis is shown in figure 2.11. It uses the patient’s peritoneum in the abdomen as a membrane across which fluids and dissolved substances (electrolytes, urea, glucose, albumin and other small molecules) are exchanged from the blood. Peritoneal dialysis is cheaper than hemodialysis, allows greater patient mobility, and produces fewer swings in symptoms due to its continuous nature. However, compared to hemodialysis, it is less efficient at removing wastes from the body. Another disadvantage is the presence of the tube which creates a risk of peritonitis due to the potential to introduce bacteria to the abdomen.

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venous pressure monitor

air trap and air detector clean blood

saline solution fresh dialysate dialyser

patient

used dialysate inflow pressure monitor

removed blood for blood pump cleaning heparin pump arterial (to prevent clotting) pressure monitor

Figure 2.9: Schematic of hemodialysis. The patient’s blood is sucked by a pump into the dialysis unit and cleaned by the artificial semi-permeable membrane. After filtration of toxic substances, the blood is pumped back through a central venous catheter or vascular short circuit to the patient. After connecting the patient to the dialysis unit, blood thinners (i.e heparin) are administered to prevent clotting. In chronic renal failure, dialysis is performed 3 times per week for 4–5 hours.

Blood inlet Header Tube sheet Solution outlet Fibers Jacket Solution inlet

Blood outlet

Figure 2.10: Schematic of dialyzer. Fibers serve as a membrane.

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saline solution

switch connector

used dialysate

Figure 2.11: The principle of peritoneal dialysis, which consists of the following steps: hookup, infusion, diffusion and finally drainage. The blood is filtrated by the patient‘s peritoneum in the abdomen which serves as a membrane.

2.4.2.1 Electric double-layer of colloidal particles The existence of a double-layer of charged particles on the surface of a colloidal particle represents the most important property of colloidal particles. This property distinguishes them from all other dispersions. This structure appears on the surface of an object when it is placed into a liquid. In a simplified model, two parallel layers of charge surround the object. In the first layer, the surface charge (either positive or negative) is comprised of ions which are adsorbed directly onto the object due to a host of chemical interactions. The second layer is composed of ions attracted to the surface charge via the coulomb force, which electrically screen the first layer. This second layer is loosely associated with the object, because it is made of free ions which move in the fluid under the influence of electric attraction and thermal motion rather than being firmly anchored. It is thus called the diffuse layer. An electric double-layer can be arranged by electrolytical dissociation or by an adsorption of ions (see Fig. 2.12). Formation of an electric double-layer by adsorption of ions will be explained by using an example of the formation of micelles of AgI. The compound AgI is not soluble in water. It can be prepared by mixing KI solution with AgNO3 solution. KI and AgNO3 dissociate in water as follows: AgNO3 ↔ Ag+ + NO3− KI ↔ K+ + I − 58

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micela AgI

Figure 2.12: Electric double-layer colloidal particle. When the particles are placed in the liquid, two parallel layers are formed around the particle. The first layer determines the surface charge (here is an example of a negative particle, therefore, the surface charge is positive), forming the ions adsorbed by chemical interactions directly with the particle. The second layer consists of ions attracted by Coulomb force. This second layer is loosely coupled to the particle because it is formed by ions which move in the liquid due to electrical attraction and thermal movement. This is a so-called diffusion layer.

They exist in the solution in the form of ions. Let us assume an excess of AgNO3 with respect to KI in the solution. After mixing, the reactions Ag+ + I − ↔ AgI occur and AgI would coagulate and fall to the bottom. However, only some opalescence appears which demonstrates formation of colloidal particles, micelles of AgI. These micelles are formed by unification of several molecules of AgI and possess a crystalline structure. The micelle tends to bind the ions from which is formed, in our case silver ions, which are in excess in the solution. Ag+ ions attract ions with the opposite sign, i.e., the NO3− ions, which are also in excess, and adsorption forces bind only some of them (see circle in Fig. 2.12). Electrostatic forces attract the others; the magnitude of electrostatic forces decreases with squared distance. During the motion of the colloidal particle, only the portion of ions bound by adsorption forces is fixed to the particle. In this portion there are more positively charged ions and therefore the micelle is positively charged. This positive charge inhibits further unification of micelles into larger particles due to repulsive electrostatic forces and thus the colloidal state is stabilised. Naturally, if more KI would be added, the Ag+ ions would be removed from the surface of colloidal particles by formation of other AgI molecules, the colloidal particles would lose their charge and coagulate, forming larger and larger particles, and finally fall to the bottom. Thanks to the coating by electric double layer, the colloidal particles possess an electrokinetic potential ξ, sometimes called zeta potential. The significance of zeta potential is that its value can be related to the stability of colloidal dispersions (stable = high value). Let’s imagine a colloidal particle surface with one layer positively charged and with the second layer negatively charged and dispersed due to diffusion (see Fig. 2.13). The surface possesses 59

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a certain positive potential E with respect to its surroundings. This potential E decreases as a function of distance from the as depicted in Fig. 2.13.

A

B

x

Figure 2.13: Electrokinetic potential. A – ions adsorption, B – dependence of electrokinetic potential on the distance from the colloidal particle.

The electrokinetic potential ξ is the potential difference between the boundary of the adjacent liquid film and remaining liquid phase, in another words the potential difference between the dispersion medium and the stationary layer of fluid attached to the dispersed particle. The magnitude of this potential plays a decisive role in the motion of colloidal particles in an electric field. The transport velocity towards the oppositely charged electrode at the given intensity of electric field depends on their charge, size, and form, which enables the separation of colloidal particles by electrophoresis. If solution containing particles with electrokinetic potential are exposed to a direct electric field (two electrodes – a cathode and anode connected with external source of voltage are immerged) then the particles move with different velocities towards the oppositely charged electrodes. Such arrangement is called free electrophoresis. It enables separation of individual parts from the solution, and can be followed by refractometry. To achieve more perfect separation, free electrophoresis may be replaced by an electrophoresis on carrier. Filtration paper, agar gel, cellulose, starch and other materials can be applied as carriers. According to the material, the carrier behaves as a molecular sieve; the transit of large molecules has a lower probability than that of smaller ones. Thus, for a given voltage, the path travelled by each particle will be different during the given time interval. The principle of electrophoresis is presented in figure 2.14, with the real output below in figure 2.15.

2.5 WATER The most important liquid is water, the H2O molecule. It is the most abundant compound covering approximately 70% of the earth’s surface. Water is essential to humans and other lifeforms; all known forms of life depend on water. The human body consist of up to 80% 60

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1

3

2

4

Figure 2.14: Principle of electrophoresis: In the porous material (e.g. a polyacrylamide gel) are small wells. Samples are placed into wells (1, 2). Then, a DC electric field is applied (3). A negatively charged electric field is applied on the side plate, where wells with the samples are. The electric field causes the particles to move in the porous material towards the anode with a speed that depends on the size of electric field, as well as the particle size and shape (4). Thus, the smaller and lighter particles in the porous plate travel a greater distance from their initial position, which can be separated from each other for varying particle sizes.

Figure 2.15: DNA Gel Electrophoresis. Gel contains agarose and ethidium bromide for UV visualization of nucleic acids. Individual DNA samples are applied in admixture with “loading buffer”. On the left is a control marker containing DNA samples of defined molecular weight. Signal intensity corresponds to DNA amount. Provided by Mgr. I. Leontovyc from Department of Experimental Medicine, Institute for Clinical and Experimental Medicine, Prague, Czech Republic.

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of the water. The H2O molecule is a dipole because its shared electrons are asymmetrically. Two hydrogen atoms are bound by two covalent bonds with one oxygen atom of oxygen. The shared electrons are not located symmetrically. The atoms are not linked in a straight line but they form a triangle. The angle between hydrogen atoms and oxygen atom is about 105° (see Fig. 2.16). A negative charge of nonbinding electrons prevails at the site of oxygen. A hydrogen atom of one molecule can bind to the oxygen atom of another molecule via hydrogen bond. A water molecule is shown in figure 2.16. One O–H distance is about 100 pm (covalent bond) and the other is 180 pm (the weaker hydrogen bond). The water molecules are interconnected by hydrogen bonds in both the liquid and the solid phase. Every oxygen atom occupies the centre of a nearly regular tetrahedron, and the adjacent oxygen atoms occupy the vertices of the tetrahedron. The hydrogen atoms between the oxygen atoms are situated such that four hydrogen atoms are linked to one oxygen atom by two covalent bonds and two hydrogen bonds. In water, the molecules are more closely packed than in ice. Contrary to other liquids, whose density decreases with increasing temperature, the highest density of water is observed at about 4 °C. An increase of temperature (at 4 °C) is accompanied by an increase of the mean kinetic energy of the molecules, together with the progressive splitting of hydrogen bonds. This later process requires a considerable amount of energy, which explains large values of the latent heat of water. The latent heat of fusion at 0 °C is 335 kJ/kg; the latent heat of vaporisation at 37 °C is 2.25 MJ/kg. The energy of one hydrogen bond is about 0.2 eV, which means that a single water molecule is bound with the energy of about 0.4 eV (about 40 kJ/mol). The hydrogen bonds result in a relatively high melting point (0 °C) and boiling point (100 °C) of water at normal pressure.

8− 8+

8+

hydrogen bond

8− 8− 8+

8+ 8−

Figure 2.16: A molecule of water.

A water molecule is most common solvent because it possesses a relatively large dipole moment. That is why water is known as the polar solvent and has effective solvent power. Aggregated water molecules form the hydrate sheath when ions are no longer surrounded 62

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by their partners, but by water molecules (e.g., the radius of Na ion in the lattice is 0.095 nm while its effective radius in water solution is 0.24 nm). An interaction between water and the surface of some hydrophilic substances (through hydrogen bonds or by van der Waals forces) results in bound water, which is biologically significant. Under normal conditions, a considerable proportion of cellular water content is bound water. As an example, one lipid molecule in a phospholipid membrane binds about 11 water molecules. Due to the large dipole moment of water molecules, the relative permittivity of water is very large (≈ 80). Water is the product of many biochemical reactions in cells. Digestion of 100 g of fats or glycides results in the formation of 107 or 5 g of water, respectively. Water is a substantial component of all body liquids and organs. The content of water in blood, the skeleton, and in muscles is 79%, 22%, and 76%, respectively. Water represents the dispersion medium for macromolecules, molecules, and ions in cells and enables their interactions. Water in a body is partially free, and partially bound to hydrophilic colloids. The water in this hydrate layer is a little compressed, and has a higher density and lower tension than that of saturated vapours. For example, 1 g of albumin bounds about 1.3 g of water. Besides the aforementioned functions (dispersion medium, solvent), water is fundamental to photosynthesis and respiration, and is vitally important for thermoregulation, as well as for transport processes.

2.6 TRANSPORT PHENOMENA Transport phenomena are related to the motion of molecules and to the processes of interactions of molecules. These processes result in the transport of physical quantities. From this point of view, the viscosity is considered as the transport of momentum, conduction of heat as transport of energy, and finally diffusion as transport of molecules. The necessary condition for the transport of momentum, energy, or even molecules is the presence of the appertaining gradient of flow velocity, temperature, and concentration, respectively.

2.6.1 Basic laws of fluids When a force of the magnitude F is exerted perpendicular to a plane with surface S, then the scalar quantity P defined by the relation P=

F (2.17) S

is called pressure. The S.I. unit of pressure is pascal (Pa); 1 Pa = 1 N.m−2. The normal pressure is equivalent to 1 atmosphere = 1 atm = 1.013×105 Pa = 760 torr = 760 mm Hg. Pascal’s law states that the magnitude of pressure does not depend on direction and is identical in all points of a horizontal plane. A liquid with density ρ = m/V (kg.m−3) exerts hydrostatic pressure at the depth h under a free surface given by p = h ρ g ,(2.18) 63

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where g (m.s−2) is gravity acceleration. Thus, the hydrostatic pressure of the Hg column with a height of 1 mm (1 torr) is l×10−3.13.6×103×9.8 = 133.3 Pa. Since the static pressure in a fluid depends only on depth, Pascal stated the following principle: An external pressure applied to a fluid in an enclosed container is undiminished and transmitted to all the parts of the fluid and the walls of the container. This principle has many practical applications. Since pressure is force/area, a small force applied to a piston with a small area results in a larger force acting on a larger area. If a certain volume of an ideal liquid (frictionless, incompressible) enters one end of the tube per unit time, the same volume must leave the other end. The mathematical form of this principle is called the equation of continuity. The flow rate Q is the volume of fluid flowing past a point in a tube per unit time, Q = ∆V / ∆t(2.19) The S.I. unit of flow rate is m3.s−1. Over the time interval Δt, the fluid moves a distance Δx = v Δt and the volume leaving the tube is ΔV = AΔx = AvΔt, where A is the cross-sectional area of the tube and v is the velocity of displacement (m.s−1). Therefore, Q = A.v; the flow rate equals the cross-sectional area of the tube times the velocity of the fluid. The equation of continuity may be expressed as: A1v1 = A2 v2(2.20) The product of fluid cross-sectional area and fluid velocity is constant. A decrease in the cross-sectional area results in increased flow velocity. Bernoulli’s equation states that the work done on a flowing fluid is equal to the change of its mechanical energy, expressed as: p + (1 / 2) ρ v 2 + h ρ g = const (2.21) Applying Bernoulli’s principle, the sum of pressure and total mechanical energy of liquid per unit volume is constant everywhere in a flow tube. The term (1/2)ρv2 represents kinetic energy, and the term hρg is the potential energy of the fluid per unit volume. It follows from Bernoulli’s equation that the pressure at the same depth at two different places in a fluid at rest is the same.

2.6.2 Law of Laplace Laplace’s law describes the relationship between the pressure difference ΔP across the surface of a closed curved membrane and the wall tension T (N.m−1) (see Fig. 2.17). According to Pascal’s principle, the pressure inside the object is equivalent throughout (for example in a balloon at equilibrium), however, there are great differences in wall tension on different parts of the object. In general, the larger the vessel radius, the larger the wall tension required to withstand a given internal fluid pressure.

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Figure 2.17: The diagram of action of pressure and of wall tension in elastic tube.

The shape is characterized for each point on the membrane by two principal radii of curvature, R1 and R2. The surface is generally a curved surface of concave or convex form. Therefore, the direction of pressure (defined by the force perpendicular to the area) is different from that in the case of a plane surface. The pressure difference and wall tension are related by  1 1  ∆P = T  +  (2.22)  R1 R2  For a cylindrical form of the membrane, one of the radii is infinitely large and thus ΔPcylinder = T/R(2.22a) For a sphere R1 = R2 = R and thus ΔPsphere = 2T/R(2.22b) This relation between pressure and wall tension in a curved elastic surface is important in physiology. Laplace’s Law may be applied in Alveoli of lung to explain their role in exhalation, the difficulty of the baby’s first breath, and the unfortunate effects of emphysema. The net pressure required for lung inflation is dictated by the surface tension and radii of the tiny balloon-like alveoli. During inhalation, the radii of the alveoli increase from about 0.05 mm to 0.1 mm. Laplace’s law also helps us understand the circulatory system. The wall tension required to keep the wall from rupturing is proportional to the radius of the vessel. Larger arteries must have stronger walls: an artery of twice the radius is required to be able to withstand twice the wall tension. Arteries are reinforced by fibrous bands to strengthen them against the risks of an aneurysm. Tiny blood vessels called capillaries rely on their small size; they resist relatively high blood pressure thanks to their small radius. Equation 2.22a enables to one to calculate the total tension in the walls of different categories of blood vessels from the 65

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physiologic mean values of internal pressure and their histologically determined radii. Some of the parameters of the human circulatory system are summarized in table 2 below. Table 2: Some parameters of the circulatory system Type of vessel Aorta

Pressure (kPa)

Radius

Tension in wall (N.m−1)

Thickness

13.3

1.3 cm

170

2 mm

Total cross section (cm2) 4.5

Arterioles

8

0.15–0.06 mm

1.2–0.5

20 μm

400

Capillaries

4

4 μm

1.6×10−2

1 μm

4500

Venules

2.6

10 m

2.6×10−2

2 μm

4000

Veins

2

200 μm or more

0.4

0.5 mm

Vena cava

1.3

1.6 cm

21

1.5 mm

40 1.8

Despite a thickness of only 1 μm, the walls of the capillaries are able to resist relatively high blood pressure thanks to their small radii (see eq’n 2.22a). Similarily, an increase in the size of the heart increases the load of the heart (see eq’n 2.22b).

2.6.3 Viscosity In real fluids, the internal friction has to be considered to describe the flow. Let us consider laminar flow in a horizontal rigid cylindrical tube of radius R. The distribution of velocity vectors is represented by a paraboloid, situated along the axis of the tube with the top at the centre of the tube (see Fig. 2.18). A gradient of velocity, ∆v/∆r (where both ∆v → 0 and ∆r → 0), exists along the radius of the tube. When the cylindrical area of contact of liquid layers in the presence of a velocity gradient is denoted S, then the force of internal friction F results in tangental tension σ = F/S. The unit of tangental tension is Pa. The tangental tension is proportional to the vector of velocity gradient,

σ =η

∆v (2.23) ∆r

The proportionality coefficient η is called the dynamic viscosity. Its unit in the SI system of units is Pa.s. The kinematic viscosity ηk is defined as dynamic viscosity divided by the density,

ηk =

η (2.24) ρ

The liquids whose tangental tension is directly proportional to the velocity gradient are called Newtonian liquids. Single component liquids and analytical solutions belong to this group. Colloidal solutions, suspensions and emulsions are usually the non-Newtonian. In general, fluids that have smaller values of viscosity are near to ideal fluids, because they flow more readily with only weak viscous forces impeding their movement. The ideal fluid has zero viscosity. 66

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Figure 2.18: Field of velocity vectors

The unit of kinematic viscosity is m2.s−1. A motion of molecules depends on the temperature; therefore the viscosity is a function of temperature. The effect of the temperature is implicitly involved in the gradient of velocity. The viscosity decreases with increasing temperature in liquids, and increases in gases. The viscosity of an “easily flowing” fluid is small, whereas viscous fluids have high viscosity. Blood is a suspension of red blood cells in blood plasma. The viscosity of blood increases with increasing values of the hematocrit (the percentage of blood volume occupied by the red blood cells). The viscosity of blood is about 3 mPa.s at 20 °C. The viscosity of blood plasma is of about 1.5 mPas, and that of water at 18 °C is 1.1 mPa.s. The relative viscosity of blood as related to water is of about 3. Generally, the viscosity of a suspension ηs depends on the volume concentration c of particles by

η s = η (1 + kc)(2.24a) where η is the viscosity of the medium and k is a constant that characterizes physical properties of the particle. The effective viscosity ηeff of a suspension of particles of diameter d flowing through a tube of radius R is given by

ηeff =

η∞  d 1 +   R

2

(2.24b)

where η∞ is the viscosity value for a tube with an infinite radius. The effective value of suspension viscosity decreases as a consequence of the accumulation of particles in the axial part of the tube. This effect is partially compensated by higher values of velocity gradient near to the wall, because the gradient decreases toward the centre of the tube. The highest velocity vmax is observed in the centre of the tube and it is given by vmax =

∆PR 2 (2.25) 4η L 67

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where ΔP is the pressure drop (difference between the pressures at the both ends of tube) along the horizontal tube and L is the length of the tube. The velocity decreases with increasing distance from the centre towards the tube walls, and at the distance r is  r2 v = vmax 1 − 2  R

 (2.26) 

The velocity equals zero just at the wall of the tube, where r = R. It can be shown that the average velocity vav equals half the maximum velocity, vav = (1/2)vmax. = ΔPR2/(8ηL). The pressure drop along a horizontal tube is proportional to the viscous forces and to the length of the tube, since the work done against viscous forces is proportional to the displacement. The average velocity vav and the volumetric flow Q = πR2vav of the fluid are proportional to the pressure gradient along the tube, ΔP/L.

2.6.4 The Hagen-Poiseuille law Let’s consider a laminar flow in a rigid tube of radius R and length L, with pressure difference at the ends of tube ∆P. The volumetric flow Q = ΔV/Δt is given by equation Q=

πR 4 ∆P (2.27) 8η L

Equation (2.27) is known as Hagen-Poiseuille’s law. It demonstrates that the volumetric flow is extremely dependent on the radius. An increase of the radius by 19% results in twofold increased volumetric flow. Thus, there is a huge effect of vasodilatation and vasoconstriction on volumetric flow. System of small vessels can constrict flow to one part of the body while enhancing the flow to another to meet changing demands for oxygen and nutrients. For example, an emergency requirement for a five-fold increase in blood volume flowrate would need a five-fold increase in blood pressure (from 120 mmHg to 600 mmHg). But the sufficient blood volume flowrate with constant blood pressure (120 mmHg) could be simply established by vasodilatation: according to Hagen-Poiseuille’s law just by increasing the radius about ratio 1.5 (1.54 = 5.04). The flow resistance Rf is defined as the ratio of the pressure drop to the volumetric flow, R f = ∆p / Q (2.28) For laminar flow, we get by using the equation (2.27) Rf =

8η L (2.29) πR 4

It follows from the definition that the S.I. unit of flow resistance is Pa.s.m−3. It is seen that the flow resistance strongly depends of the radius of the tube. The contribution of arteries and arteriols to the total blood flow resistance in man is 66%, and of capillaries and the venous part of the circulatory system to the total blood flow resistance in man is 27, and 7%, respectively 68

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2.6.5 Stokes law The internal frictional force F for a sphere of radius r moving in a medium with viscosity η at the velocity v is given by F = 6πη rv(2.30) This relation is called Stokes law. This relation can be applied for measurement of viscosity. In the case of motion at constant velocity, the driving and frictional forces differ from each other only in sign, and the velocity is proportional to the driving force. The coefficient of the proportionality is called mobility u. The value of u gives the mean velocity of a colloid particle or macromolecule that moves in some medium by unit driving force. In the case of spherical particles, u=

1 (2.31) 6πη r

The streamline flow is that at which the streamlines do not intersect. If streamlines swirl and mix, the flow is said to be turbulent. The value of the dimensionless parameter called the Reynolds number, Re, can be used to predict whether the flow will be laminar or turbulent. Consider a fluid of viscosity η and density ρ. If it is flowing in a tube of radius R and has an average velocity vav, then the Reynolds number is defined by Re =

2 ρ vav R (2.32) η

It has been found experimentally that if Re < 2000, the flow is laminar, and if Re > 3000, the flow is turbulent. In a turbulent flow, some energy is dissipated as sound. Thus, the noise associated with a turbulent flow in the arteries facilitates blood pressure measurement. 2.6.5.1 Measurement of viscosity The measurement of viscosity of solutions is important for determining the molar weight, especially in the case of macromolecular substances of higher viscosity. The viscosity can be estimated on the basis of equation (2.27) by measurement of the time t required for the flow through a pipe of a given volume of liquid at the pressure difference ΔP = P1 − P2, where P1 and P2 are the pressures at the ends of the pipe (see Fig. 2.19). Because the pressure difference is given by the weight of liquid, and the volume and tube parameters are known, the viscosity can be easily calculated. However, relative measurement is often done so that the time ts is measured for the given volume of a standard liquid of known viscosity ηs at the same conditions (barometric pressure, temperature). It follows that

η tρ = (2.33) η s ts ρ s 69

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where ρ and ρs are the densities of measured and standard liquid, respectively. The ratio of times for standard and measured liquid equals the ratio of their kinematic viscosities for the given volume and pipe. The Ostwald viscosimeter is seen in Fig. 2.19. The liquid measured is soaked from the wide pipe into the narrow one. Then the given volume flows back and the corresponding time is measured. Body viscosimeters utilize Stokes law (see eq. (2.30). A spherical body is allowed to fall down in the given liquid and the time of its fall (or of the rise of gas bubble) is measured again.

R

v1

I II

v2

Fig: 2.19: Ostwald viscosimeter

x1

x2

0

2.6.6 Diffusion In spite of the random motion of molecules, the net transport of molecules of a dissolved substance can be observed. This transport occurs from the area A of higher concentration towards the area of lower concentration, against the vector of concentration gradient ∆c/∆x, where ∆c = c1 − c2 and ∆x = x1 − x2 (see Fig. 2.20).

C1 C2

A

x2

x1

Figure 2.20: The scheme for diffusion. The arrows show direction of concentration gradient.

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The density of diffusion flux, n/Aτ, where n is the number of moles, A is the area through that diffusion takes place, and τ is time, is expressed in mol.m−2.s−1. It is proportional to the concentration gradient. Thus, n ∆c = − D (2.34) Aτ ∆x where D (m2.s−1) is the diffusion coefficient. From equation 2.34, is clear that diffusion is a passive transport which is dependent on time. The above equation is called the 1st law of Fick. The negative sign appears in the above equation due to the fact that the direction of the flux is opposite to the direction of the vector of concentration gradient. The statistics of the random motion of a particle shows that the mean distance travelled by a particle in a certain direction due to its zigzag motion increases as the square root of the number of steps. Since the number of steps is proportional to time, the mean squared displacement xrms2 into one direction is related to the diffusion coefficient by xrms 2 = 2 Dτ (2.35) This equation can be applied to calculate the time required for displacement of the diffusing particle to a certain distance, from the place of its origin, due to diffusion. The value of D depends on the nature of the diffusing atom or molecule and the choice of the solvent or medium. E.g., the value of D for haemoglobin in water at 20 °C is 6.9×10−11 m2.s−1. The diffusion coefficient is a function of temperature. For spherical particle of the radius r moving in a medium with viscosity η and at the absolute temperature T, the diffusion coefficient is given by D=

kT (2.36) 6πη r

where k is the Boltzmann constant. Thus, an increase in temperature accelerates diffusion. Diffusion is the primary mechanism in the body in absorbing and distributing the substances required by living cells. The release of the by-products of cellular function, such as carbon dioxide, also proceeds by diffusion.

2.7 COLLIGATIVE PROPERTIES OF SOLUTIONS There are four properties of solutions that depend only on the number of particles of a solute dissolved in the solution. They are vapor pressure lowering, boiling point elevation, freezing point depression, and osmotic pressure These properties are independent of the size or form of particles, their chemical behaviour, and finally state (molecules, ions, or colloids). These properties can be described with the help of physical quantities. Generally, if any of these properties is denoted Φ, then Φ = kCm (2.37) 71

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where k is the proportionality constant and Cm is the molar concentration expressed in the R 1 number of moles per unit volume ofvthe solution. All of these properties can be applied to I determine the molar mass of solute particles. In fact, molar concentration can be calculated II v 1 x2 or kg/m3 as Cm = cg /M, where on the basis of mass concentration cg2, and expressed in xg/litre M is the molar mass. Therefore, 0

Φ=k

cg M

(2.38)

If Φ is measured and the constant of proportionality is known, then M can be determined based on the known value of mass concentration.

2.7.1 Raoult laws

A

C1 C2

If a substance is dissolved in a solvent, the partial pressure of vapours of the solution decreases with respect to that of the pure solvent (see Fig. 2.21). The decrease Δp = p0 − p, x1 x2 where p0 is the pressure above pure solvent and p the pressure above solution, is proportional to the number of particles dissolved (n2), and is given by n2 ∆p = (2.39) p0 n1 + n2

pressure

where n1 is the number of particles of the solvent. The decrease of pressure above solution with respect to that above pure solvent is called 1st law of Raoult.

p0 p temperature

Figure 2.21: Pressure of saturated vapours above solution (p) and pure solvent (p0).

Similarly, the presence of dissolved particles in a solution increases its boiling point at the given pressure with respect to that in pure solvent. The shift of the boiling point towards higher temperature is given by ∆Tb. p = K e Cm,(2.40) 72

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where ΔTb.p. = Tb.p.solution − Tb.p.solvent is the difference between the boiling point of solution and the boiling point of the pure solvent. The constant of proportionality Ke is called the ebullioscopy constant and its values are tabled for various solutes and solvents. Analogously, the freezing point of the solution at the given pressure is shifted towards lower temperatures with respect to its value for the pure solvent, ∆T f . p = − K c Cm (2.41) where ΔTf.p. = Tfp.solution − Tf.p.solvent is the difference between the freezing point of solution and the freezing point of the pure solvent. The constant of proportionality Kc is called the cryoscopy constant and its values are tabled for various solutes and solvents. The shifts of the boiling and freezing points are called 2nd and 3rd law of Raoult The fourth colligative property, the osmotic pressure, is of great importance for cells and living organisms.

2.7.2 Osmotic pressure If a solution is separated from a medium by a semi permeable membrane, the molecules of solvent can pass freely through the membrane while the dissolved compound molecules are restrained. At the beginning of the process, the concentration of solvent molecules in a medium is higher than that in the solution, and net transport of solvent molecules into solution occurs from the area of higher concentration to the area of lower concentration. After a certain time, equilibrium is achieved when the number of solvent molecules that pass the membrane in both directions is the same. This process is called osmosis. More simply, osmosis can be defined as a diffusion of water across a cell membrane. The schematic is seen in Fig. 2.22. The net transport of solvent molecules stops if the hydrostatic pressure, given by the product of hρg (ρ – density of the solution, g – gravity acceleration), equals the osmotic pressure of the solution. The osmotic pressure is the extra pressure that must be applied to stop the flow of solvent molecules into solution.

Figure 2.22: Demonstration of osmotic pressure. The osmotic pressure corresponds to the hydrostatic pressure given by height h. A – tube (filled with a solution and ended with semi-permeable membrane) submersed into container B which is filled with pure solvent, for example water. In this set up, the water will begin to flow from container B to tube A through the membrane to equilibrate the concentration of solution on both sides of membrane.

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It is necessary to stress that only the molecules that do not pass the given membrane can contribute to the osmotic pressure. For example, some simple inorganic ions can pass through semipermeable membranes and thus, their dilution is the result of simple diffusion into the solvent only. Van’t Hoff laws describe quantitatively osmotic pressure: (i) At constant pressure, the osmotic pressure is directly proportional to the number of particles in the solution, i.e. to its molar concentration Posm = kCm(2.42)

by

Of course, this property holds for all colligative properties. (ii) At the given concentration, the change of osmotic pressure with temperature is given Posm = P0 (1 + γ T ) (2.43)

where T is absolute temperature and γ =1/273.15 is the volume expansion coefficient of gases. The dependence of osmotic pressure on concentration and temperature can be described by Posm = iRTCm(2.44) where R is the universal gas constant and i denotes the degree of dissociation (splitting of molecules or complexes into smaller molecules or ions). (iii) At identical osmotic pressure and temperature, the same volumes of different solutions contain the same number of molecules dissolved. The equation (2.44) is similar to that of the ideal gas. The molar concentration Cm must be expressed in mol.m−3. Since the osmotic pressure depends on the number of particles in the solution and not on their size, analytical solutions usually demonstrate higher osmotic pressures as compared to colloidal solutions. Colloidal particles are relatively large and their concentration is usually lower when compared with that of molecules or ions in analytical solutions. In completely dissociated ion solutions, each ion individually contributes to the osmotic pressure. Thus, completely dissociated NaCl has two times the osmotic pressure compared with glucose at identical concentrations. Osmosis is very important in understanding biological processes. Tissues are composed of cells containing complex solutions. The fluid surrounding the cells are also complex solutions but with different composition. At equilibrium, the total osmotic pressures due to impermeable molecules or ions must be the same outside and inside the cell. If not, water will move into or out of the cells (from a solution of lower (hypotonic) into solution of higher (hypertonic) osmotic pressure). Solutions of the same osmotic pressure are called isotonic solutions. If a red blood cell is placed into pure water, the water molecules enter the cell, which raises the internal pressure since the cell cannot expand appreciably. Equilibrium occurs at about 800 kPa, but the cell membrane usually ruptures before this pressure is achieved (haemolysis). Osmosis is important for water distribution in blood and tissues in the capillaries. In a capillary vessel, blood pressure and osmotic pressure have the opposite action. According to Starling’s hypothesis (see Fig.2.23), in the arterial part of capillary tube, the blood pressure is higher than the osmotic pressure, and thus a net transport of water outside the capillary 74

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vessel occurs. In the venous part of tube, the osmotic pressure inside the tube is higher than the blood pressure and thus water intake prevails. A spontaneous osmotic dilution is related to the liberation of work, the increase of concentration with a supply of work. The organ that controls the osmotic pressure in the body is the kidney. Urine is often hypertonic as compared with blood, and due its excretion, body liquids maintain a constant osmotic pressure.

Capillary

Blood flow

capillary wall

larger hydrostatic pressure = 35 mm

smaller osmotic pressure = 25 mm

net flow out of capillary into tissues = 10 mm

smaller hydrostatic pressure = 15 mm

larger osmotic pressure = 25 mm

net flow into capillary = 10 mm

Figure 2.23: Starling’s hypothesis. Left: arterial part. Right: venous part.

2.8 PHASE BORDER PHENOMENA Due to the action of forces among the molecules, changes of concentrations appear along the phase boundary between solid and liquid or gas phases together with the formation of surface tension of liquid. Both concentration changes as well as surface tension depend on temperature.

2.8.1 Surface tension Molecules in a liquid interact their action among them by attractive forces. In a liquid, molecules are surrounded by like molecules, thus these attractive forces are at equilibrium. The situation changes along the border between the phases, i.e. at the place of the contact of a liquid with solid or gas phase or with a non-mixable liquid. The result of these forces at the surface that have direction towards the centre of liquid is the tendency of liquid to have what is known as the minimum surface. That is why a drop of a liquid flowing in the other non-mixable liquid has spherical shape; a sphere has the minimum surface for the given volume. The Surface tension at the boundary of a liquid with gas is a force acting perpendicularly to the unit of length of the liquid surface. It is a tangental force to the surface with unit N.m−1 and dimension kg.s−2. Besides surface tension, another quantity is defined as the capillary constant, i.e. as the density of surface energy. It represents the work that must be done to increase the surface of the liquid by 1 m2. Its unit is J.m−2, which is the same as N.m−1. Therefore, the 75

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capillary constant numerically equals the surface tension. In addition, the surface tension does not depend on the area of the liquid surface and decreases with increasing temperature. In solutions, the surface tension is affected by the presence of a dissolved substance. For example, the presence of a surface-active substance in solution decreases the surface tension. Surface tension plays a role on all phase boundaries of the human body. It is of great importance in numerous processes, e.g. respiration, since the walls of alveoli in the lungs are wet.

2.8.2 Adsorption A solution demonstrates the same concentration in all its parts at equilibrium, however we can observe an increased concentration of the given atom, molecules or ions dissolved in the solution along the boundaries. This phenomenon is called adsorption. It is caused by decreased surface tension due to adsorption of a substance on the surface (phase boundary). Liquids have the tendency to decrease their surface tension and the adsorption of the surface-active substances enables this phenomenon. Against this tendency are the requirements to reach the equilibrium concentration and to decrease the concentration gradient due to diffusion. Therefore, adsorption equilibrium is established along the phase boundary. This equilibrium state is described by Gibb’s adsorption equation that yields the surface concentration of a substance Г (mol.m−2) by Γ=−

c dσ . (2.44) RT dc

where c (mol.m−3) is molar concentration, R (J.K−1.mol−1) is the universal gas constant, T (K) is temperature, and dσ/dc (J.m.mol−1) is the change of surface tension with respect to the concentration. If surface-active substances that decrease the surface tension are considered, then dσ/dc < 0 and the resultant surface concentration in equation 2.44 is positive.

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3.

THERMODYNAMICS

3.1 THERMODYNAMIC SYSTEM Thermodynamics involves processes of energy transformation. Living systems possess a high degree of spatial-structural arrangement and temporal coordination of processes, and are coupled with chemical reactions. In nature, there are various kinds of energies that may be converted or transformed, however, the law of conservation of energy always holds true. According to this law, the amount of energy in a closed and isolated system remains constant whenever any physical or chemical processes occur in the system. This law is generally valid. However, a living organism is not isolated from its environment, and its functions are maintained by the uptake of energy from its environment and by the simultaneous transfer of thermal energy away to the surroundings. Thermodynamics is the method suitable for quantitative description of energy conversion. In addition, the concepts and laws of thermodynamics can be used to quantitatively describe chemical reactions, transport processes and the equilibrium state. Surroundings Q Boundary

System

W

n

Figure3.1: A Thermodynamic system and its surroundings.

A thermodynamic system may be described as a part of nature, separated from its surroundings by real or imaginary boundaries, containing a great number of interacting particles, and being composed of a great number of subsystems. Heat (Q), work (W) or a substance, described by the number of moles (n), may penetrate the boundaries (Fig 3.1). According to the penetrability, the system is isolated if Q = 0, W = 0, and Σni = 0 (no exchange of heat, work and substance occurs), closed if Q ≠ 0, W = 0, and Σni = 0 (only heat is exchanged), and open if Q ≠ 0, W ≠ 0, and Σni ≠ 0. A living system is open. 77

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There are two types of system parameters that may be encountered in any system: global (extensive) and local (intensive) parameters. Global parameters describe the system as a whole and possess additive properties, i.e. the total value of the parameter for the whole system is the sum of the values of its individual parts (e.g. mass, total charge, total number of particles within the system, etc.). Local parameters are those that depend not only on time but also on spatial coordinates (e.g. temperature, pressure, chemical potential, etc.). The variables and functions of state describe the state of a system, which is determined by its composition (number of moles n), pressure P, temperature T, volume V. The variables are related among themselves by the state equations (e.g. pV = nRT, where R = 8.314 J.K−1.mol−1 is the universal molar gas constant). The value of a state variable depends only on the state of the system and does not depend on path (its history). All physical or chemical processes that occur in a system are related to changes of variables and functions of state. The functions of state (enthalpy, internal energy, entropy, etc.) are functions of state variables, and their value does not depend on path (the history of the process). Thanks to its time-evolution, the state achieved in an isolated system is the equilibrium state. Equilibrium corresponds to the most probable arrangement of the system. In the same way, a spontaneous evolution of an isolated thermodynamic system points toward the most probable arrangement of this system, which interestingly has been demonstrated experimentally, but cannot be derived from more general laws. The rate by which a system deviates from its equilibrium and returns to an equilibrium state is determined by the relaxation time. The deviation of I from an equilibrium state at time τ is given by I = I0 e−τ/T, where I0 is the initial deviation at time τ = 0, e = 2.71 is the base of natural logarithms, and T is the relaxation time. Since 1/e = 1/2.71 = 0.37, the relaxation time is the time at which the deviation from equilibrium achieves 37% of its initial value. Relaxation times may vary to a large extent (e.g. the duration of some diffusion process may last for 10−8 s or for hours). A spontaneous evolution of a non-equilibrated thermodynamic system results in an equilibrium state when the state functions approach their extreme values (maximum or minimum). However, an open system may evolve into a stationary state since the parameters of the system do not change and are independent of time. Each equilibrium state is stationary, but each stationary state is not at equilibrium. A system evolves toward a stationary state thanks to the exchange of matter (chemical compounds) or energy with the surrounding of the system. Thermodynamic processes are related to the evolution of the thermodynamic system(s) in time and are described by thermodynamic functions. The courses of these functions fulfil the laws of conservation (of energy, mass, momentum, electric charge, etc.) and of the evolution in the equations of state. Reversible and irreversible processes can be encountered. Irreversible processes are the result of non-reversible finite changes that occur; therefore, the system is not at equilibrium throughout the process. In the case of a reversible process, the change in trajectory can be achieved by an infinitesimally small change in some of the state variables. The process is cyclic, without loss or dissipation of energy. Equilibrium is related to reversible processes. Let us consider a hot gas situated in a cylinder closed by a piston. The volume of the gas is a function of its temperature, and follows the state equation of an ideal gas (pV = nRT). If the initial temperature (T1) is higher than the temperature of the environment (T2), T1 > T2, then the volume of gas decreases during cooling and its volume at state 2 will be V2 < V1. During this process of cooling, the gas passes through a series of equilibrium states corresponding to an actual temperature. If heat is supplied to the system, the direction of the process can be reversed. This is why equilibrium states are related to reversible processes. Gradients of 78

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concentration, temperature, etc. which cause the flow of a substance, heat, electric charge, etc. are called thermodynamic forces, while the physical quantities of other types, (e.g. the flow of a substance, intensity of electric current, the flow of heat, etc.) are called thermodynamic flows. The physical laws of these phenomena demonstrate direct proportionality between thermodynamic flows and forces (i.e. Fick’s 1st law of diffusion, the law of heat conduction, etc.).

3.2 WORK AND HEAT When work is done, the transfer of energy from one body to another is related to the change of external state, i.e. as the body or of its part change location. Mechanical work is a measure of the transfer of mechanical energy from one body to another. However, the transfer of heat is not related to either the change of external state, or with displacement. Therefore, work is the quantity of total energy transferred from one system to another, not counting energy that’s transferred by heat. The quantity of heat ΔQ is the amount of the thermal motion of molecules transferred from one body to another by heat exchange. Heat can be defined as the energy transferred between two bodies as a consequence of a temperature difference between them. If a quantity of heat ΔQ produces a change in temperature ΔT in a body, its heat capacity is defined as ΔQ/ΔT and its SI unit is J/K. Note that the older unit of heat was 1 calorie = 4.18 J. The quantity of heat ΔQ required to produce a change in temperature ΔT is proportional to the mass of the sample m, and to ΔT (for small ΔT). It also depends on the substance. These facts result in the equation ∆Q = mc∆T ,(3.1) The quantity c is called the specific heat capacity (or specific heat) of the substance. Its unit is J.kg−1.K−1. The specific heat of liquid water at 15 °C is 4.186 kJ.kg−1.K−1 and that of ice (−15 °C) is 2 kJ.kg−1.K−1. That of human body (37 °C, average) is 3.5 kJ.kg−1.K−1. Specific heat is a property of the given substance. If we work with the number of moles n instead of mass (expressed in kg), then equation (3.1) changes into ΔQ = n.C.ΔT, where C is the molar specific heat (J.K−1mol−1). Since the number of moles n = m/M, where M is the molar mass, it holds that C = Mc(3.2) The specific heat of a substance generally varies with temperature. It can be shown that the specific heat of a gas kept at constant pressure, cP is greater than the specific heat of a gas kept at constant volume, cV. Let us consider 1 mole of gas at pressure p, temperature T and volume V. The state equation of this gas is pV = RT. Let’s increase its temperature by 1 K while keeping its volume constant. The amount of heat required will be just Cv:its pressure increased during heating at constant volume. Now let’s arrange its pressure to the initial value by increasing its volume by ΔV. Further heat is required and the gas does work equal to p.ΔV. Therefore, Cp = Cv + pΔV. Taking the state equation at the new temperature we have p.(V + ΔV ) = R.(T + 1). By dividing this equation by the equation for initial conditions, we get 79

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V + ∆V T + 1 = , V T and ∆V =

V R = . T p

Substituting for ΔV into the equation for specific heats, one can solve for molar specific heats resulting in the equation CP − CV = R. Their difference equals just the value of the universal gas constant R. The specific heat capacities of water vapours (100  °C) are cp = 2020 J.kg−1.K−1 and cV = 1520 J.kg−1.K−1. The ratio cP / cV = CP / CV is known as Poisson’s constant. Since the change of volume with temperature is small in both solids and liquids, the difference is generally small for solids and liquids and cP is usually a measured value. Heat plays an important role as matter changes from one phase into another. A solid can melt or fuse into a liquid if heat is added, while the liquid can freeze into a solid if heat is removed. Similarly, a liquid can evaporate into a gas if heat is supplied, while the gas can condense into a liquid if heat is taken away. If a liquid evaporates quickly forming vapour bubbles within the liquid, then we are dealing with boiling. Sometimes, a solid can change directly into gas phase if heat is provided, which is called sublimation. The temperature remains constant when a substance changes its phase, for example from solid to liquid or from liquid to gas. Let’s consider a sample of mass m that changes its phase. The heat exchanged with its surroundings is related to the latent heat L (J/kg) by ∆Q = mL (3.3) The latent heat can be defined as the heat released or absorbed by a chemical substance or a thermodynamic system during a process that occurs without a change in temperature, thus only the phase changes (i.e. melting ice). Compared to other substances, water possesses a relatively high latent heat of fusion and evaporation. For water at normal pressure (1.01×105 Pa), the specific latent heat of fusion at melting temperature 0 °C is 334 kJ/kg, and the specific latent heat of evaporation at boiling temperature 100 °C is 2.26 MJ/kg. For comparison, ethyl alcohol has melting point −114.4 °C and boiling point 78.3 °C, and its value of the latent heat of evaporation is 8.55×105 J/kg. In thermodynamics, we are concerned with the work done by a system on its surroundings or on the system by its surroundings. In a quasistatic process, the thermodynamic variables (P, V, T, n, etc.) change infinitely slowly. Thus the system is always arbitrarily close to an equilibrium state is which the whole system is characterized by single values of the macroscopic variables. Let us consider a cylinder filled with a gas closed by a piston. The system is represented by the gas, and the surroundings by the piston. When the gas quasistatically expands, the piston moves and the work is done by the system. If the piston rises by dx, the work dW done by the force of the gas is dW = F.dx = (PA)dx, where A is the cross-sectional area of the piston. Since the change of the volume of the gas dV = A.dx, the work can be written as dW = P.dV. Thus, 80

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if the system transits quasistatically from an initial equilibrium state i into final equilibrium state f, the total work W done by the system is Vf

W=

∫ PdV (3.4)

Vi

If Vf > Vi then the work done is positive. If the volume decreases, the work done is negative and can be considered as work done on the gas by the surroundings. If we plot a PV diagram, this work is represented by the area below the curve that describes P as a function of V. Therefore the work done depends not only on the initial and final state, but also on the thermodynamic path between the states. It is necessary to know how the pressure changes with the volume. In an isobaric process the expansion occurs at constant pressure and thus the work done is W = P.(Vi − Vi). In an isothermal process the temperature is kept constant. For the special case of an ideal gas, we can apply the state equation pV = nRT and thus P = nRT/V, thus by using the equation (3.4) we get W = nRT ln(Vf / Vi). If a membrane replaces the piston and this membrane is punctured, the gas expands without doing any work, which is the case of free expansion into vacuum. Due to the fact that both work and heat depend on the thermodynamic path, both work and heat are not state variables (however, later it will be shown that their sum is the state variable).

3.4 HEAT TRANSPORT There are three processes involved in heat transport: conduction, convection and radiation. Conduction is based on energy transfer among atoms in contact due to a temperature gradient. Heat flows from regions of greater heat to regions of lessor heat until eventually all areas are at the same temperature. The rate of heat transfer, dQ/dt, is proportional to the cross-sectional area A and to the temperature gradient dT/dx along the chosen direction x. Therefore dQ  dT = −κ A  dt  dx

 (3.5) 

Since dT/dx is negative, the negative sign must be applied in order for the rate of heat transfer to be positive. The constant κ is called thermal conductivity, with units of W.m−1.K−1. The values for the thermal conductivity of water and air are 0.6 and 0.025 W.m−1.K−1, respectively. Convection is the process in which heat is carried out from place to place by the bulk movement of a fluid. Therefore, this transfer of heat is accompanied by transport of mass. Free convection occurs because the density of fluid varies with its temperature. The hot expanded fluid has lower density. Radiation involves the transfer of heat without the action of an intervening medium. The energy is transferred by means of electromagnetic waves. The most important source of radiation for living organisms is the heat from the sun. Microscopically, the radiation comes about because the oscillating ions and electrons in a warm solid are accelerating electric charges. Since the colour black is associated with nearly complete absorption, the term “perfect black body” is used when referring to an object that absorbs all the electromagnetic waves falling upon it. A good absorber is a good emitter, and poor absorber is a poor emitter. 81

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It has been found experimentally that the power irradiated from a body with area A at absolute temperature T equals dQ = eσ AT 4 dt

(3.6)

where the quantity e is called emissivity and depends on the nature of the surface (e = 1 for a true black body, which are not found naturally; e ≈ 0.95 for a mat black surface) and σ = 5.67×10−8 W.m−2.K−4. This equation is called the Stefan-Boltzmann law, which states that all bodies which are not at absolute zero temperature radiate, and at room temperature the radiation is in the infrared spectrum. If the temperature of the emitting body is T1, and the temperature of its surroundings T2, then the net rate at which heat is irradiated by the body is dQ = eσ A(T14 − T24 )(3.7) dt Note that a good heat emitter is also a good heat absorber. The wavelength distribution of thermal radiation from a black body at any temperature has essentially the same shape as the distribution at any other temperature. When the temperature of a blackbody radiator increases, the overall radiated energy increases and the peak of the radiation curve moves to shorter wavelengths. This describes Wien’s displacement law

λmax T = const (3.8) where λmax is wavelength and T temperature.

3.5 FUNCTIONS OF STATE Functions of state such as internal energy, enthalpy, entropy, free enthalpy, etc. describe the state of a thermodynamic system by state variables. Each process is described at the initial and final state. For each process, during which the system is described at the initial state i is described by the state variables: pressure pi, volume Vi, temperature Ti, and the number of moles of chemical substances (Σn)i. For each process, the initial state variables are transferred into the final state f, determined by pf , Vf , Tf , (Σn)f , it holds that the resulting change of any state function F(p, V, T, (Σn)) is given by f

∆F = ∫ dF = Ff − Fi(3.9) i

The change of a function of state does not depend on integration (or reaction) path. Thus, the resulting change of a function of state is only given by the difference between the values at the final and initial states and it does not depend on the history of the process. 82

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3.5.1 Internal energy It was already mentioned that heat and work are not state variables since their values depend of integration (reaction) path. However, their linear combination defines the state function called internal energy U (unit J). ∆U = U f − U i = Q − W (3.10) Q is positive when the system gains heat and negative when it loses heat. W is positive when work is done by the system and negative when work is done on the system. The supply of heat into the system and/or the work done on the system increases the internal energy of the thermodynamic system. Equation (3.10) is a mathematical expression of the first law of thermodynamics, and corresponds to the law of conservation of energy. Since ΔU = Uf  − Ui, from equation (3.9) we can solve for W = Q − ΔU, and the system can produce work if heat is supplied to the system or by the decrease of its internal energy. The internal energy of a system changes when work is done on the system (or by it), and when it exchanges heat with the environment. The kinetic and potential energies, associated with the random motion of particles in the system, form a part of internal energy called thermal energy. For an isolated system, for which there is no heat exchange and no work done, it follows from the first law ΔU = 0 J or U = constant. The internal energy of an isolated system remains constant. In a cyclic process, the system transits from the state 1 to the state 2 and the work W12 is done by the system; then it returns back to state 1 and the work W21 is done on the system. From the first law ΔU = 0 J and thus Q = W. The net work done by the system in each cycle equals the net heat input into the system. In a constant-volume process, the volume does not change, and thus W = 0. From the first law it can be seen that ΔU = Q. All the heat entering the system contributes to the increase in internal energy. In an adiabatic process, the system does not exchange heat with the surroundings, and thus Q = 0 J. During the adiabatic process, the system performs work through motion or mechanical work, and does not exchange heat with its surroundings. Within an adiabatic process, a temperature change will occur only due to the work that it performs, but not due to heat loss with its environment. Therefore, the internal energy increases during adiabatic compression and the temperature of a gas increases while the internal energy decreases and temperature decreases during adiabatic expansion. In an adiabatic free expansion, the gas is allowed to expand adiabatically without doing any work. Thus W = 0 J and also Q = 0 J, and therefore from the first law ΔU = 0 J. In an adiabatic free expansion the internal energy of any gas does not change. For an adiabatic quasistatic process of an ideal gas (which proceeds very slowly), no heat exchange occurs, and the work is done by the internal energy of gas is: pV γ = const.(3.11) where γ = Cp /Cv is the Poisson’s constant. We meet with adiabatic processes in nature each day. In the Earth’s atmosphere, air masses will undergo adiabatic expansion and cool down, or they will experience adiabatic 83

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compression, and heat up. Warm air at the surface is less dense than cold air and will rise. As warm air rises, it expands and cools, and at a certain temperature, the relative humidity in the rising air mass reaches 100% and water will condense as clouds.

3.5.2 Enthalpy If an extrenal pressure is applied on the the system and increases its volume by dV, the work dW = pdV is done. Many chemical reactions occur at constant pressure, in which case: equation (3.10) may be rewritten as: dQ = dU + pdV (3.12) Therefore, a new function of state called enthalpy (H) is introduced, and defined by H = U + pV (3.13) The equation of enthalpy enables the measurement of the total energy of a thermodynamic system. This energy creates a system and ensures the work needed to make room for that system. If the process changes the volume in a chemical reaction which produces a gaseous product, then work must be done to produce the change in volume. Thus, the product pV can be interpreted as the work. The change in enthalpy (ΔH = H2 − H1) is related to the change of state 1 into state 2 at constant pressure, and represents the quantity of heat released or absorbed by the system. If ΔH > 0 J, then the system absorbs energy from the surroundings in the form of heat and is called endothermic reaction. If ΔH < 0 J, then the process or reaction releases energy from the system (i.e. when liquid water freezes in the form of ice) and is called an exothermic process. In the course of a spontaneous chemical exothermal reaction at constant pressure, energy is released in the form of heat, and the enthalpy of the system decreases to and it achieves its minimum final equilibrium state. An example, i.e when glucose is formed during photosynthesis from carbon dioxide and water according to equation: 6CO 2 + 6H 2 O → C6 H12 O6 + 6O 2, Equation (above) reveals that the molar enthalpy change ΔH = Hf − Hi = 2.81 MJ/mol. When ΔH > 0 J, the reaction is isobaric and the energy is consumed in the given direction (endothermic reaction). In the reverse direction, the same quantity of energy can be released across many intermediate states. The molar specific heat Cp mentioned above is related to the molar enthalpy by  ∂H Cp =   ∂T

 (3.14a) 

i.e. as the partial change of enthalpy which corresponds to the change of temperature at a constant pressure when otherwise the composition of the system remains constant. Similarly, the molar specific heat CV is related to the molar internal energy (i.e. internal energy related to one mole) by 84

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 ∂U  CV =  (3.14b)  ∂T  i.e. as the partial change of internal energy which corresponds to the change of temperature at a constant volume when the composition of the system remains constant. The Specific heat of a gas kept at constant pressure, Cp is greater than the specific heat of a gas kept at constant volume, CV; the relation between specific heats is Cp = Cv + pΔV.

3.5.3 Entropy The first law of thermodynamics confers the ability toto transform heat into work and into an increase of internal energy, and is described by equation (3.10). However, the first law does not define the conditions of the conversion. Experiments have demonstrated that it is not possible to obtain work through repeated cyclic processes, since the heat Q1 released from a reservoir (at temperature T1) is greater that the heat released (Q2)) from another reservoir at lower temperature T2 < < T1. What follows is the well-known relationship for the efficiency η of any thermal engine operating on the basis of reversible cycles.

η=

Q1 − Q2 T1 − T2 Q T = = 1 − 2 = 1 − 2 (3.15) Q1 T1 Q1 T1

For this equation we can see that the efficiency of any thermal engine is less than 1, i.e. less than 100%. The portion of heat, which was transferred into the reservoir at lower temperature can be converted into work only by using another reservoir an even lower temperature, and thus its “quality” or ability to be converted into work is decreased in a process known as the degradation of energy. The thermodynamic function that describes a degradation of energy is called entropy. The nature of all matter and energy tends to proceed from order to disorder in an isolated system and entropy is a measure of disorder. For any two isolated systems, nature tends toward maximum entropy (Fig. 3.2). entropy


0 J) and a system at equilibrium (ΔG = 0 J). Gibbs energy (referred to as ∆G) is also the chemical potential that is minimized when a system reaches equilibrium at constant pressure and temperature.

3.6 CHEMICAL POTENTIAL Chemical potential is the energy which a substance can produce to alter a system. When chemical reactions are going on in a system, its composition changes and thus its state changes as well. Therefore, the number of moles of the systems components also defines its state. The change in composition are related to the changes of energy. Each kind of energy can be expressed as a product of intensive and extensive factors. In the case of chemical energy, the extensive factor is represented by the increase in number of moles of the given substance i, Δni,. The intensive factor is represented by its chemical potential, μi. Thus, ΔE = μi.Δni. Most reactions go on at constant pressure and temperature. At constant pressure and temperature, the chemical potential can be defined by  ∂G  µi =   (3.23)  ∂ni T , p... i.e. as the partial change of free enthalpy which corresponds to the small change in the number of moles of the given substance at a constant pressure and temperature where` otherwise the composition of the system remains constant. Chemical potential is a measure of the affinity of a given substance. The reactions that take place in a system and their rates depend on chemical potential and the amount of substance in the system.

3.7 REACTION HEAT In chemical reactions, the liberation or absorption of heat (exothermic or endothermic processes) accompany the changes in the mutual positions of the atoms. The reaction heat is a measure of the overall change in potential energies of the combining atoms. When a reaction goes on at constant pressure, the amount of heat released or consumed is equal to the change of enthalpy. According to Hess’s Law holds: The reaction heat does not depend on the reaction path (i.e. on intermediate states), but only on the final and initial state of the system. Practical applications of this law are demonstrated in the following example, where carbon dioxide is formed by the subsequent reactions 1) C + ½ O2→ CO and 2) CO + ½ O2 → CO2, or directly C + O2 → CO2. If the change of enthalpy of the first reaction is denoted ΔHI, 88

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the second reaction ΔH2, and for the direct reaction ΔH, it holds ΔH = ΔHI + ΔH2. Thus, the okolí enthalpy of individual steps can be calculated if the reaction is decomposed into several steps. The reaction heat is the heat that’s released into the system and thus its value is negative for exothermic reactions sand positive for endothermic reactions. termodynamický systém

Q

W

T, p, V, n1, n2, …

3.8 THERMODYNAMICS OF BIOLOGICAL SYSTEM Biological thermodynamic systems contain complicated structures and high degree of arrangement. By the arrangement we understand i) the arrangement of cells, ii) the arrange∑ni ment of cells from polymers, and iii) the arrangement of protein structures and of nucleic acids. Calculation reveals a low effect of biological arrangement on entropy changes. The entropy of human organisms composed of about 1013 cells differs from the entropy of a set of 1013 single cell organisms (ΔS = 10−9 J.K−1). If the internal arrangement of a cell contains about 108 components of protein, nucleic acids, and phospholipides, than the own arrangement of proteins fromT aminoacids and the arrangement of nucleic acids, then the unique arrangement of human body corresponds to the decrease of entropy of about (ΔS = 1300 J.K−1. ireverzibilní Simple physical or chemical processes can easily compensate such a decrease. The change procesy of entropy by 1300 J.K−1 corresponds to evaporation of 170 cm3 of water or to the oxidation reverzibilní of 900 g of glucose. A living body is an openproces thermodynamic system. Processes that act against the thermodynamic equilibrium occur due to the continuous uptake of nutrients. An organism thrives and continue to live as a result of these changes in state. However, no process or even an evolutionary leap represents a contradiction to the laws ofS thermodynamics. It is only after S0 death when all possible thermodynamic processes that are going towards an equilibrium state occur spontaneously. In non-equilibrium systems (i.e. in open systems where exchange of matter and energy with their surroundings occur), the increase of entropy can be separated into two components as seen in Fig. 3.3 surroundings deS

diS system

Figure 3.3: An open thermodynamic system. Its entropy changes due to internal entropy production diS and entropy exchange deS with its surroundings.

The diS corresponds to the production of entropy due to internal processes that are going in the system, and deS corresponds to the exchange of entropy with surroundings. He total change of entropy is then Synthesis of sugar

Sun

hf

Green plants

Animal cells O2

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biomacromolecules

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dS = di S + d e S (3.24) Similar to an isolated system, it holds that diS ≥ 0, where the sign = holds for the equilibrium. A living system with low entropy may be enabled by the inflow of matter or energy into the system from its surroundings. If a living organism gathers sufficiently high free energy (or quanta of light at photosynthesis in plants) from the surroundings, then part of this energy is used for its conservation and the other part is consumed to increase its arrangement and remove waste products of its metabolism. Therefore, processes can take place for which

d e S < −di S ≤ 0 (3.25) The states with low entropy can be maintained for a long time under the assumption that the system can achieve a stationary state when dS = 0. Thus,

d e S = −di S ≤ 0 ,(3.26) A negative flux of entropy is required from the surroundings into the system to maintain a highly arranged state compared with its surrounding. If −deS > −diS, then the entropy of the living system wll decrease according to:

dS = di S + d e S < 0.

(3.27)

This explains why biological processes do not contradict the 2nd law of thermodynamics and are similarily governed by this law according to all processes in nature. Identical physical laws hold for inorganic and living matter. It follows that the temporal change of entropy, dS/dt, can be written in the form

dS = P( S ) + J ( S )(3.28) dt where P(S) denotes the evolution of entropy due to irreversible processes in the system during over time and J(S) denotes the flux of entropy across the system boundaries over the period. The total production of entropy can be written in the form P( S ) =

di S = σ ( S )dV (3.29) dt V∫

where σ(S) denotes the density of entropy production, i.e., the amount of entropy produced in the unit volume over time. The calculation of entropy production is the integral of the production density over the entire volume of the system. An important property of living systems is the formation of non-equilibrium stationary states. Under assumption of local equilibrium, a living system develops towards a stationary state so that its production of entropy decreases over time and reaches the minimum just at the stationary state. Therefore, the evolution of a living system may be expressed as: 90

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proces

S

S0

dσ =≤ 0(3.30) dt The sign of equality holds for the stationary state corresponding to a constant (and minimum) density of entropy production, its derivative over time equals zero. surroundings deS

diS

3.9 TRANSFORMATION AND ACCUMULATION OF ENERGY system IN BIOLOGICAL SYSTEM The primary source of energy for all organisms living on the Earth (i.e. for plants and bacteria that use the photosynthesis) is light or the energy that is released from chemical reactions during the processing of nutrients. The total radiation energy of the Sun is (0.4–1)×1025 J/year, of which approximately 0.5% is consumed in the biosphere for photosynthesis (6CO2 + 6H2O → C6H12O6 + 6O2, ΔG = 2.686 MJ/mol) and the rest is irradiated back into space. The schema is seen in Fig. 3.4 shows the flu of free energy in nature. Synthesis of biomacromolecules

sugar Sun

hf

Green plants

Animal cells O2

CO2

H2O

Useful work Heat

CO2

H2O

Figure 3.4: Flux of energy in biosphere.

With the help of light and further with the help of animal cells, Macromolecules containing large amounts of free energy a formed from the combination of light and proteins (i.e. CO2 and H2O). Chemical processes in cells proceed through a number of intermediate steps, and for individual processes, electron transfer is the primary source of yielding energy. The average energy content per unit of mass of food is as follows: carbohydrate 17.2, protein 17.6, and fat 38.9 MJ/kg. Phosphates, mainly ATP (adenosinetriphosphate) produced by various phosphorylation processes play an important role in living organisms. These molecules act as the universal energy storage reservoir in plant and animal cells. During the reversible enzymatic splitting of ATP, energy is released again. The reaction (ATP)4– + H2O → (ADP)3– + (HPO4)2– + H+ yields ΔG = −31 kJ/mol at pH = 8 and this free enthalpy obtained is used in coupled endothermic reactions. ATP also plays a role in the mechanical work of muscles and in processes of active transport. Other energy-dense phosphate rich compounds exist in an organism. In humans, 91

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the consumption of ATP is about 70 kg/day, thus ATP must be produced continuously to meet the energy requirements of the body. Cellular chemical reactions are the studied in detail in the biochemistry.

3.10 MEASUREMENT OF TEMPERATURE Temperature is an objective measure of the thermal state of matter and its value denotes the individual temperature states. The temperature scale is defined by the zero point and the appertaining unit. The unit of temperature is 1 kelvin (K) degree, and is one of the base units of the S.I. system. The Kelvin temperature is related to Celsius temperature (expressed in °C) by T (K ) = 273.15 + tC ( 0 C)(3.31) The Fahrenheit scale used in the USA has a smaller unit and different zero point. The relationship between Fahrenheit and Celsius temperatures is given by 9 TF = tC + 32(3.32) 5 Temperature can be measured by a number of methods. Each method is based on various changes of thermometric properties of materials with temperature (expansion, change of electric resistance, the appearance of electromotoric voltage in thermocouples, emission of light, etc.). For example, the thermometric property of the mercury thermometer is the length of the mercury column, and the thermometric property of thermocouple is the electric voltage difference between the two junctions. The lower limit of temperature is called absolute zero, and is designated as 0 K on the Kelvin temperature scale.

3.10.1 Liquid thermometers Mercury thermometers are widely used in laboratories, and their thermometric property is volumetric expansion. When a liquid at initial volume V0 is heated, its temperature increases by ΔT, and its volume changes by ΔV = β V0ΔT, where β is the coefficient of volume expansion (for mercury, β = 1.8×10−4 K−1). Similarly, a thick-walled capillary vessel is enlarged at its lower end into thin-walled cylindrical vessel when it is filled with mercury at the level of the capillary. Above the mercury surface there is a vacuum, and the scale is depicted on the glass of the capillary. A mercury thermometer has a temperature range from −39 to +250 °C. Thermometers filled by other liquids (toluene, ethyl alcohol) are used for measuring temperatures below −39 °C.

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3.10.2 Medical thermometer A medical thermometer is used to measure body temperature. It is a mercury thermometer designed to measure the maximum temperature. The temperature scale ranges from 35–42 °C.

3.10.3 Calorimetric thermometer A calorimetric thermometer is used to measure small temperature differences within a limited scale range. Its capillary is enlarged at the lower end, which results in a shorter thermometer length, and a custom scale around both base points is maintained.

3.10.4 Thermocouple The thermocouple consists of thin wires of different metals (often copper and constantan), welded together at the ends at two different junctions (see Fig. 3.5). One of the junctions (the so called “hot” junction) is placed in thermal contact with the object whose temperature is being measured. The other junction is kept at constant temperature. The thermocouple generates a voltage that depends on the difference in temperature and is measured by a voltmeter. The temperature of the hot junction can be determined from this voltage by using calibration tables.

Metal A T1

T2

U

Metal B

Figure 3.5: A thermocouple, which generates a voltage that depends on the difference in temperature (metals responded differently to the temperature difference, creating a current loop).

3.10.5 Electrical resistance thermometer The electrical resistance of a metallic wire changes with temperature. Since platinum possesses excellent electro-mechanical properties in the temperature range from −270 to +700 °C, platinum wire is may be placed in thermal contact with another substance and the electric resistance is measured. The voltage produced by a single thermocouple is low. By connecting several thermocouples in series into a thermobattery the measurement sensitivity can be increased. 93

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3.10.6 Thermistor The density of free electrons in a semiconductor is lower than in a metal. Semiconductors possess higher electric resistance as compared to metals. The density of electrons in a semiconductor strongly increases with increasing temperature and thus, its electrical resistance decreases exponentially. The resistance is measured by suitable methods. The temperature measurement accuracy of a thermistor is very high, on the order of mK, but is not often it is not completely exploited in medical measurements.

3.10.7 Thermography Temperature can also be evaluated on the basis of the radiation emitted by an object. At low to moderate temperatures, the predominant type of emissions is infrared radiation. The intensity of radiation increases significantly with increasing temperature. In medicine, radiation from the surface of the body is measured by a thermo camera connected to a colour monitor that displays the different infrared intensities as different colours, and a thermogram is produced. This method of temperature investigation is called thermography, and is used in oncology to investigate the presence of malignant tissue which is often associated with elevated body surface temperature above the malignant tissue.

3.10.8 A bimetallic strip A bimetallic strip can be used as a temperature-sensitive switch or thermometer. Two metals of different expansion coefficients are welded together as a bimetallic strip, and as the temperature increases, the strip bends toward the side with the lower expansion coefficient. The strip may be made into a coil, and is often used temperature control in thermal devices where a constant temperature is required (thermostats).

3.11 CALORIMETRY Calorimetry is the method for measuring thermal energy. Calorimeters are devices that quantitatively measure the heat required or evolved during a chemical process, the most common being the mixing calorimeter (see Fig. 3.6). The simplest calorimeter is the Dewar’s vessel with coupled walls and a vacuum between the walls which is filled with water. Heat is supplied into the vessel and consumed for heating the walls, thermometer, and the mixing device. The water value K of the calorimeter is equal to the amount of heat required to heat 1 Kg of water by 1 °C which is consumed by the device. This amount of energy is then added to the mass of water present in the calorimeter. The heat supplied, Q (J), is then calculated from the calorimetric equation Q = ( M + K )c∆T (3.33) 94

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where M (kg) denotes the mass of heated water, K (kg) the water value of calorimeter, c – specific heat of water, and ΔT the temperature difference before and after heating. Calorimetry is used for measuring the energy requirements of organism as well as evaluating the energy content in nutrients. During the combustion of nutrients, heat is released in the presence of oxygen. For proteins and sugars, the amount of heat released is 17 MJ/kg, and for fats is 38 MJ/kg. In addition, energy turnover can be measured by direct or indirect calorimetry. In direct calorimetry, the subject is situated in an isolated space, and the heat formed is measured by the temperature of the volume of water circulating in the calorimeter. Simultaneously, the quantity of oxygen consumption, and carbon dioxide/nitrogen released are estimated. In adults, the total energy turnover is about 11 MJ/day. Energy turnover is not proportional to body mass but to the body surface and its value is about 5 MJ/m2/day. For the most part, body organs participate in metabolic processes (i.g. energy exchange in the kidneys is about 20–25 times higher per unit mass than in the rest of the body). Most of energy supplied is in the form of chemical energy and other energy sources may be neglected. Motorized stirrer Electrical leads for igniting sample Thermometer

Insulated container O2 inlet Bomb (reaction chamber) Fine wire in contact with sample Cup holding sample Water

Figure 3.6: The mixing calorimeter. Both constant volume and pressure instruments use the fact that the heat evolved from the reaction changes the temperature of a working substance (usually a water bath) with a known heat capacity.

3.12 THERMAL LOSSES The rate of chemical reactions going on in a body are affected by temperature. In warm blooded animals, body temperature is kept constant (at about 37 °C in humans). The conditions for a constant body temperature are that the amount of heat produced in the body (by the work of muscles and by chemical reactions) must be equal to that transferred away to the surroundings. Thermal equilibrium is maintained mainly by the regulation of thermal losses. Four physical processes take part in thermal losses: radiation, flow, conduction of heat, and evaporation of water. Each body at a unique temperature emits energy in the form of electromagnetic waves. The wavelength of that electromagnetic radiation is in the ultraviolet spectrum at high temperatures, in the visible spectrum at lower temperature, and finally in the infrared region at 95

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temperatures corresponding to those of the surroundings. The Stefan-Boltzmann law states that the energy irradiated by a black body over time is directly proportional to the fourth power of its absolute temperature. Since the body simultaneously absorbs the irradiation emitted by its surroundings, the net quantity of heat lost by this physical process is proportional to the difference of the square of squares of the temperatures of the body and to its surroundings. This process represents about 40–60% of the total thermal losses in animals. The quantity of thermal energy irradiated depends on a supply of blood to the skin, its surface, and is decreased by clothing. The wavelengths of the radiation of skin are 3–30 μm, with the maximum at about 9 μm. It is difficult to distinguish thermal losses due to the heat flow and conduction. The circulation of blood is importance in processes of heat distribution within a body. The amount of heat lost by these processes and their contribution to the total losses is of 15–30%. At surface body temperature, the heat of evaporation of water is 2.4 MJ/, which is why this process contributes significantly to thermal losses through the skin and mucous membranes of a body and represents about 20–25% of total heat losses. Evaporation, occurs during respiration and sweating, and is very important for the maintenance of thermal equilibrium at temperatures over 31 °C. Sweat production increases with increased activity, and results in a higher body temperature and increased temperature of the surroundings. Relative humidity is very important in biological systems. The relative humidity equals 100% when the partial pressure of water vapour equals the equilibrium vapour pressure at the given temperature. In this situation, the vapour is said to be saturated because it is present in the maximum amount. If the relative humidity is less than 100%, the water vapour is said to be unsaturated. When air containing a given amount of vapour is cooled, the equilibrium vapor pressure is reached when the partial pressure of the vapour equals the equilibrium vapour pressure at a temperature known as the dew point. Sweating stops when air is saturated by water vapours, and air-motion replaces the saturated air layer with an unsaturated layer which accelerates the transport of water molecules from the body surface. Each person has a unique metabolic rate, and thus the thermal comfort of a body depends on a number of factors, including: the temperature of the air, relative humidity, air-motion, and clothing worn. Thermal losses are minimized when environmental temperatures are below 19 °C. The equilibrium between heat production and losses occurs at temperatures between 19–31 °C. Finally, radiation and convection are insufficient to ensure thermal losses at temperatures over 31 °C, and the process of evaporation becomes very important for maintaining thermal equilibrium.

3.13 THE LAWS OF THERMODYNAMICS The zeroth law of thermodynamics: When two individual systems are in thermal equilibrium with a third system, they are all in thermal equilibrium with each other. No flow of heat within a system can occur at equilibrium. The first law of thermodynamics: The internal energy of a system changes from an initial value Ui to a final value Uf  due to the heat Q, and work W: ΔU = Uf − Ui = Q − W 96

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The heat Q is positive when the system gains heat and negative when it loses heat. The work W is positive when work is done by the system and negative when work is done on the system. The second law of thermodynamics: Heat flows spontaneously from a substance at higher temperature to a substance at lower temperature and does not flow in the reverse direction. Carnot principle (An alternative statement of the second law of thermodynamics): No irreversible engine operating between two reservoirs at constant temperatures can have a greater efficiency than a reversible engine operating between the same temperatures. Furthermore, all reversible engines operating between the same temperatures have the same efficiency. The second law of thermodynamics stated in terms of entropy: The total entropy of the universe does not change when a reversible process occurs (ΔS = 0), and increases when an irreversible process occurs (ΔS > 0). The third law of thermodynamics: It is not possible to reduce the temperature of any system to absolute zero in a finite number of steps.

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4.

BIOPHYSICS  ELECTRIC PHENOMENA OF

Bioelectric phenomena include cell membrane potentials which are of particular importance in nerve cells where they produce action potentials, include heart electrical phenomena such as the natural pacemaker which triggers the heart electrical sequence, and include other bioelectric measurements.

4.1 INTRODUCTION Electric biosignals are generated by nerve cells and muscle cells. Its source is the membrane potential, which under certain conditions may be excited to generate an action potential. In single cell measurements, the action potential itself is the biomedical signal. The electric field propagates through the biologic medium, and thus the potential may be acquired at relatively convenient locations on the surface. Electrical biosignals are usually taken to be electric currents produced by the sum of electrical potential differences across a specialized tissue, organ or cell system. Electric charge is present in all bodies of our surroundings and also in our body. It is one of the basic physical properties of elementary particles and it belongs to the basic physical quantities. The charge is positive or negative. The quantum of charge is the charge of one electron or of one proton, i.e. 1.6×10−19 (C). The unit of electric charge is 1 coulomb (C). Since the number of particle present in 1 mole is given by Avogadro’s constant NA, total charge of 1 mole of univalent ions is so called Faraday’s constant F = e.NA= 1.6×10−19(C)×6.02×1023(mol−1) = = 96.484 kC.mol−1. In any system the law of conservation of electric charge holds: During any process, the net electric charge of an isolated system remains constant (is conserved).

4.1.1 Coulomb law and permittivity Coulomb law quantitatively describes the interaction among electric charges. Attractive or repulsive force F acting between two charges, q0 and q is directly proportional to their product and inversely proportional to the squared distance r between them. 98

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F=

1 q0 q .  4πε r 2

(4.1)

The constant ε is called permittivity and its value for vacuum is ε0 = 8.85×10−12 F.m−1. Permitivity is a measure of how an electric field affects, and is affected by, a dielectric medium. Dielectric polarisation is observed in insulators and thus the force between the charges is decreased as compared with that in vacuum. Therefore, permittivity value in insulators is higher than ε0. The relative permittivity εrel = ε/ε0 is usually tabled. Relative permittivity is the ratio of the capacitance of a capacitor using that material as a dielectric, compared to a capacitor that has a vacuum as its dielectric. For illustration, some values are presented in the following table. Tab. 4.1 Relative permittivity in some media Medium

ε/ε0

Medium

ε/ε0

Vacuum

1

Cell membrane

8

Air

1.0006

Water

78.5

High value of relative permittivity of water enables very good solubility of salts in water. In water, ions may be distanced since water molecules decrease the attractive forces between negatively and positively charged ions. From the energy point of view, hydration of ions is accompanied by a decrease of free enthalpy (Gibbs energy). Various ions are hydrated in various extents. The measure of hydration is given by the number of water molecules that have lost their translation degrees of freedom due to the interaction with the given ion. The positively charged ions possess a higher hydration number since positive charge manifests a higher polarisation effect on electron shells in water molecule as compared with negative charge. The ions of smaller size possess a higher hydration number than larger ones of the same charge. Therefore, the order of relative size of hydrated K+ and Na+ ions in water solution is reversed in comparison with the order of their radii in crystalline lattice. That is why the cell membranes are more permeable for K+ than for Na+ ions. This fact plays its role at the transport of ions across the membrane and at the formation of membrane potential. Tab. 4.2 Radius of some ions (nm) Radius of ion in crystalline lattice

Effective radius in water solution

Na+

0.095

0.24

K

0.133

0.17

+

The electric field that exists at a point is the electrostatic force F experienced by a small test charge q0 placed at this point divided by the charge itself. It is a vector. The intensity of electric field E is the force per unit charge. Therefore, its unit is N/C. It can be also expressed by using the unit of potential, volt. Since 1 J = 1C×1V, then N/C = N.m/C.m = J/(C.m) = V/m. The intensity of electric field is given by E=

1 q F = . (4.2) q0 4πε 0 r 2 99

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Direction of electric field intensity is shown with field map lines. In general, it directed from positive charge to negative.

4.1.2 Electric potential, potentials of phase boundary-lines Tab. 4.3 Review of symbols applied for various potentials and their explanation Symbol and quantity

Explanation

V Electric potential

Physical quantity expressed in volts. Zero potential is theoretically considered in an infinite distance from conductor, practically in such distance when the action of electrostatic forces can be neglected. In practice, potential of Earth is considered as zero potential.

φ Internal (Galvani) potential

φ = ψ + χ; internal potential of the given phase (e.g. solid phase) in the medium of other phase (e.g. liquid phase)

Ψ External (Volt) potential

The work required for the transport of unite charge from infinity to the distance of about 10−6 cm from the surface of the given phase (to win electrostatic forces)

χ Surface potential difference

The work required for the transport of unite charge across the surface

µ i = Electrochemical potential

µ i = µi + zi Fφ ; The work required for transport of 1 mole of the i-th component (ion or electron) inside the given phase, defined as sum of chemical and electrostatic components (F = 96 484 C/mol – ­Faraday’s constant, zi – number of elementary charges of the i-th ion).

μi Chemical potential of the i-th component

µi = µi0 + RT .ln ai ; Partial molar Gibbs energy, i.e. the work corresponding to the change of the i-th component by 1 mole (R – universal gas constant, T – absolute temperature, ai – activity of the i-th component).

E Electrode potential

Potential on electrode surface. Potential of standard hydrogen electrode is usually considered as zero potential.

The electric potential V at a given point is the electric potential energy EPE of a small test charge situated at this point divided by the charge itself. Its unit is volt (V), and it holds volt = joule/coulomb. Potential of a point charge is given by V=

1 q EPE = . (4.3) 4πε 0 r q0

Let us consider electric field around charged conductor. Let us assume that the charge q of the same sign is transported from infinity to the place of the conductor. The work W must be done to win the repulsive force F. This work equals the potential energy of the charge q. Potential energy of the unite charge depends on the position of the charge in the field. It is called the potential at the given place. 100

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Let us consider a point charge q located in a medium of permittivity ε. Potential around this charge decreases with increasing distance r according to the equation (4.3). Potential at any place equals the work, which must be done to transport unit charge from infinity to that place. However, it is true for one medium. If a conductor, represented by solid phase, is immersed in a liquid phase then the potential of this conductor grows along the direction from “infinity” to its surface. The inverse proportionality between the potential and distance is conserved to the distance of 10−5–10−6 cm from the solid phase (see Fig. 4.1). Then it remains roughly constant. This constant value is called external or Volt’s potential ψ. It may be defined as the work required for the transport of the unite charge from infinity to the distance of 10−6 cm from the surface of the given charged phase (in our case the solid one) placed inside liquid phase. An electric double-layer is usually formed at the surface of solid phase due to orientation of molecules, which are usually dipoles. For the transport of the unite charge through this double-layer, the work, called surface potential difference χ, must be done. Then the internal (Galvani) potential φ is defined as the sum

ϕ = ψ + χ (4.4) The quantity ψ is given by the existence of the charge of solid phase, the value χ by the existence of electric double-layer. Potential cannot be measured, however, potential difference between two places of the same phase can be measured as voltage. V

χ – surface

φ – internal

ψ – external

r

solid phase ~10–6 cm surrounding medium

Figure 4.1: Potentials between two phases.

Therefore, the values of ψ can be determined as potential difference between the potential distanced roughly 10−6 cm from the surface of solid phase and the place for which r → ∞, which is considered as zero potential. The values of φ and χ cannot be measured since the transport of unit charge inside the solid phase results in chemical changes accompanied by release or consumption of work, which can be expressed by using chemical potential μ. Why these chemical changes appear? Up to now the transport of “charge” has been considered. However, the transport of electric charge is represented by the transport of electrically charged particles, i.e., ions or electrons. Chemical potential cannot be measured as well but 101

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the electrochemical potential can be determined as the work required for the transport of unit charge inside the given phase considering chemical and electric components. For 1 mole of the i-th component (ion or electron) that takes part in the transport of charge and chemical change considered, it can be written

µ i = µi + zi Fϕ (4.5) where F is Faraday constant, the meaning of which follows from Faraday law of electrolysis m It = (4.6) M zF where m is the mass of the element formed at the electrode by the current I during time t, z is the number of elementary charges (irrespective the sign) transported by its ions and M is the molar mass of the given element. The value of F = 96 484 C/mol. The value φ is multiplied by ziF in order to ensure that the second component in the equation (4.5) describes the work related to 1 mole of particles transporting the charge since the quantities µ i = and μi are related to 1 mole as well. Chemical potential μi of the given component represents the change of free enthalpy that takes place if 1 mole of the i-th component is added or removed at constant pressure and temperature while the amounts of other components are not changed. Therefore, it is also called as partial molar free enthalpy and written as (∂G / ∂ni )T , p , n that denotes the partial derivative of the function G with respect to the variable ni (number of moles of the i-th component) while other variables (pressure, temperature, amounts of other components nj are kept constant. For the gas phase, it holds

µi = µi0 + RT .ln Pi (4.7) where μi0 is standard chemical potential of the component i, i.e., chemical potential at standard barometric pressure 1.01×105 Pa, and Pi is partial pressure of the i-th component. Instead of partial pressure, the activity ai, is used for estimation of the amount of the i-th component in liquid phase. Therefore,

µ = µi0 + RT .ln ai (4.8) where μi0 is the chemical potential of the i-th component at ai = l. In diluted solutions, the activity may be replaced by concentration ci (mol/dm3), so that

µi = µi0 + RT .ln ci (4.9) where μi0 is chemical potential of the i-th component at its concentration 1 mol/dm3. The ratio of the activity of the given component to its concentration may be expressed by activity coefficient γ; ai = γi.ci, where γi = ai/ci equals 1 for ideal solutions. Non-ideal behaviour, expressed by the difference from 1, is determined by interactions (dissociation, association, and hydration) among the particles of dissolved substance and the molecules of solvent. 102

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4.1.3 Donnan equilibrium When there is an ion on one side of a membrane that cannot diffuse through the membrane, the distribution of other ions to which the membrane is permeable is affected in a predictable way. For example, the negative charge of non-diffusible anions hinders diffusion of the diffusible cations and favours diffusion of the diffusible anions. Let us suppose that ions K+, Cl–, are present both in a compartments X and Y, separated by a semi permeable membrane. Moreover, compartment X contains negatively charged proteins Prot–, which cannot pass the membrane. Ions K+ and Cl– can freely pass and thus Cl– tends to diffuse down its concentration gradient from Y to X and some K+ moves with the negatively charged Cl–. Therefore,  K +X  >  K Y+  and Cl−X  < ClY−  where by [] is denoted the concentration of the given ion. Furthermore,  K +X  + Cl−X  +  Prot −X  >  K Y+  + ClY−  i.e., there are more osmotically active particles at the side X than at Y. Donnan and Gibbs showed that in the presence of a nondiffusible ion, the diffusible ions distribute themselves so that, at equilibrium, the product of the concentration of the diffusible ions on one side equals that on the other side. Thus, in this case,  K +X  .Cl−X  =  K Y+  .ClY−  Electrochemical neutrality would require that the sum of the anions on each side of the membrane equal the sum of the cations on that side,  K +X  = Cl−X  +  Prot −X  and  K Y+  = ClY− . There is slight excess of cations at side Y together with slight excess of anions at side X at equilibrium and thus there is a difference in electrical potentials between X and Y. However, it should be noted that the difference between the number of anions and the number of cations on either side of the membrane is extremely small relative to the total number of anions and cations present.

4.2 ELECTRIC PHENOMENA IN ALIVE ORGANISM In the next part, the facts mentioned above will be applied for evaluation of the situation in living cells, especially for explanation of the existence of a resting membrane potential of cells of excitable tissues (nerve and muscle tissues). Every living cell maintains an electric potential difference across its plasma membrane. Usually, inside is negative compared to outside, by 40 to 90 mV. The cell membrane acts as capacitor and as highly variable resistor. A living cell functions as a battery, i.e. uses chemical energy (ATP) to recharge the membrane 103

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capacitor. Signalling (information transmission) in the neural system is based on electric excitations traveling along the membrane. Muscle activities or heart rhythm are also controlled by electric excitations. All living cells are bounded by membranes. Membranes main role is to protect the cell from its surroundings. They composed primarily of phospholipids and proteins and are typically described as phospholipid bi-layer. The cell membrane is selectively permeable to ions and organic molecules and controls the movement of substances in and out of cells, thus facilitating the transport of materials needed for survival. The movement of substances across the membrane can be either “passive”, occurring without the input of cellular energy, or “active”, requiring the cell to expend energy in transporting it. Cell membrane permeability is affected by lipid solubility of the diffusing substance, size and shape of the diffusing substance, temperature and membrane thickness. Generally, the more lipid soluble a substance is the greater the permeability to that substance. Lipid solubility has the strongest influence on permeability as most substances in the body are hydrophilic and don’t easily cross the lipid bilayer. In case of size and shape of diffusing substance, the larger and more irregular in shape the molecule the lower the membrane permeability is. In the same way membrane thickness affects permeability, the thinner the membrane the more permeable it is to various molecules. On the contrary, temperature factor is rarely important because of the constancy of temperature in the human body. However, the higher the temperature maintains the greater permeability.

4.2.1 Resting membrane potential of nerve cell Let us start for the simplest case, assuming that the cell membrane is permeable for one kind of ion (i-th ion). At equilibrium, the electrochemical potentials are equal on both sides of the membrane, so that

µ iin = µ iex(4.10) where the upper index in denotes the interior, and ex the external side of the cell. The electrochemical potentials can be expressed according to equation (4.5) and thus

µiin + zi Fϕ in = µiex + zi Fϕ ex (4.11) Chemical potentials can be described by using the equation (4.8) and we get RT .ln aiin + zi Fϕ in = RT .ln aiex + zi Fϕ ex (4.12) The values μi0 inside and outside the cell are identical, since it is standard chemical potential of the same substance and thus they disappeared from the equation. Equation (4.12) may be rearranged and thus zi F (ϕiin − ϕiex ) = RT .ln (

aiex )(4.13) aiin

104

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The difference ϕiin − ϕiex may be denoted as membrane potential difference Δφmem. This potential difference can be measured experimentally by using suitable electrodes. It can also be estimated from the above equation and thus we get resting membrane potential ∆ϕ mem =

a ex RT .ln iin (4.14) zi F ai

As the name suggests, the resting membrane potential is the potential at resting state, In the resting cell there are more anions within the cell compared to outside the cell. Hence the potential is negative (about −70 mV). This resting membrane potential is sometimes called Donnan potential. The equation (4.14), that describes this potential difference, is analogous to the Nernst equation for electrode potential. This analogy follows from the fact that the definition of the chemical potential (which is a part of the electrochemical potential) is always given by equation (4.8). The potential difference between the internal and external membrane surface is determined by concentration difference of the given ion inside and outside the cell. Concentration gradients provide the potential energy to drive the formation of the membrane potential. Voltage is established when the membrane has permeability to one or more ions. The most important the ion transporter Na+/K+-ATPase pumps sodium ions from the inside to the outside, and potassium ions from the outside to the inside of the cell. Other mechanisms which establish the resting state, are for example diffusion, osmosis. These mechanisms establish two concentration gradients: a gradient for sodium (higher concentration outside the cell), and a gradient for potassium (higher concentration inside the cell). Table 4.4 shows concentration on Na and K ions of the nerve cell at rest state. Tab. 4.4 Concentration of Na and K ions. Ion

Outside of cells

K+ (mmol/L) Na (mmol/L) +

Inside cells

4

155

145

12

The Sodium-Potassium Pump Na+/K+-ATPase is an active transport mechanism driven by breakdown of Adenosine triphosphate (ATP). The sodium-potassium pump moves Na+/K+ ions, both against their concentration gradients. The carrier is an ATP-ase and that it pumps three sodium ions out of the cell for every two potassium ions pumped in. At rest, the membrane of nerve cell is significantly more permeable for potassium ions as compared to all other ions present in the intracellular and extra cellular fluid. For example, the permeability for K+ ions is about 40 times higher than that for Na+ ions. Therefore, in the first approximation, we can neglect the contribution of other ions to the value of the rest membrane potential as calculated by using the equation (4.14) for K+ ions. Thanks to the active transport, the concentration of potassium ions inside the cell is about 20 fold higher that outside the cell. If the activities in (4.14) are replaced by concentrations, we get for the voltage U mem between interior and external side of membrane U mem =

c ex RT .ln iin  F ci

(4.15) 105

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since z = 1 for univalent K+ ions. After substitution of 1/20 for concentration ratio and numerical values of other constants we get membrane potential −77 mV at temperature 310 K. Internal side of the membrane is negative with respect to the external side. The calculated value is near to the value measured experimentally and thus the rest membrane potential is affected mainly by potassium ion concentrations. The potential difference can be measured by using glass microelectrodes, which can be inserted into nerve cell. Drawing out glass capillary pipets makes these electrodes so thin that the diameter of the tip is of 1 μm and filled by a solution of electrolyte, usually potassium chloride. Besides potassium ions, Na+ and Cl- ions also contribute to the rest membrane potential value. In this case, the value of rest membrane potential can be calculated by using Goldmann equation U mem =

ex ex P + .c ex+ + PNa + .cNa + + P − .c − RT Cl Cl .ln K Kin (4.16) in in zF PK + .cK + + PNa + .cNa + + P − .c − Cl Cl

where cK+, cNa+ and cCl– are the concentrations of K+, Na+, and Cl- ions (mol.m−3) and PK+, PNa+ a PCl– are permeability coefficients of the membrane for the given ions. For example, ion Clconcentrations for typical neuron outside is 125 mol.m−3 and inside is 9 mol.m−3. The rest membrane potential may be also expressed by using membrane conductivity for individual ions, GK+, GNa+ and GCl– and by potential differences for these ions, UK+, UNa+ and UCl– by so called Hodgkin-Horowitz equation in the form U mem =

GK + .U K + + GNa + .U Na + + GCl− .U Cl− GK + + GNa + + GCl−

(4.17)

or by using transport numbers for individual ions Ti = Gi / ∑ Gi in the form i

U mem = TKU K + TNaU Na + TClU Cl (4.18)

4.2.2 Action potential of nerve fibre The action potential sequence is essential for neural communication. The simplest action in response to thought requires many such action potentials for its communication and performance. For the nerve cell, equilibrium (the rest state) is disturbed by the arrival of a suitable stimulus. The dynamic changes in the membrane potential in response to the stimulus is called an action potential. In response to the appropriate stimulus (voltage), the cell membrane of a nerve cell goes through a sequence of depolarization from its rest state followed by repolarization to that rest state. Within this action, it reverses polarity for a brief period before reestablishing the rest potential. Using two electrodes and some external source of voltage we can stimulate the axon. One electrode is inserted inside and the other is located on the surface of axon. As it was already demonstrated, at the presence of resting membrane potential, the interior of the cell is negative. If the external electrode is positive, then the positive charge on the surface is 106

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increased and we speak about hyper polarization. If, on the contrary, the internal electrode will be positive, depolarisation appears. If the absolute value of the rest membrane potential is decreased due to depolarisation bellow certain threshold value (so-called threshold potential), a response will be observed in the form of so called action potential (see Fig. 4.2). The difference between the threshold potential and the rest membrane potential is usually about 5–15 mV, i.e., if the value of the rest membrane potential is of −70 mV, the threshold potential may be of −60 mV. The production of action potential may be explained as follows. During the stimulation, ion channels in the membrane are open and thus the permeability is changed for several ions. Only a small increase is observed for potassium ions, while the permeability for sodium ions increases roughly by a factor of 600. This effect results in a substantially faster flux of sodium ions from the extra cellular medium inside the cell. The negative charge inside is equilibrated and the potential increases from initially negative value of the rest membrane potential to zero and even to positive values. Permeability changes of channels are related with conformation changes of their constituent proteins. The current of order pA flows in one single channel, which corresponds to the transport of 107 ions/s. Due to the transport of ions, transpolarization appears and interior of the cell becomes positive. Simultaneously the spike potential appears and its value is near to the value calculated by using equation (4.14) for sodium ions. During further phase, the permeability for sodium ions returns to its low initial value and membrane potential returns to equilibrium values for potassium ions. +30 mV

0

K+ Na+ Na+ K+

Na+ Na+

Sodium gates close

Potassium gates open

Depolarization Na

+

Na

+

Na+ K+

–55 mV –70 mV –90 mV

K+

Gate treshold

Active sodium and potassium pumps

Repolarization Rest potential

Stimulus

Na+ Na+ K+

Na+ Na+ K+ Hyperpolarization

Figure 4.2: The course of the action potential sequence of nerve cell. A stimulus is received by the dendrites of a nerve cell. This causes the Na+ channels to open. It change the interior potential from −70 mV up to −55 mV. Having reached the action threshold, more Na+ channels open. The Na+ influx drives the interior of the cell membrane up to about +30 mV – this process is called depolarization. Then the Na+ channels close and the K+ channels open. Since the K+ channels are much slower to open, the depolarization has time to be completed. Reaching the peak (maximal amplitude), the membrane begins to repolarize back toward its rest potential and even hyperpolarization can be achieved. After that, the Na+/K+ pump eventually brings the membrane back to its resting state.

107

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It is possible to determine minimal intensity of stimulating current (threshold intensity) that will just produce action potential. This threshold varies with the experimental conditions and the type of axon, but once it is reached, a full-fledged action potential is produced. Further increase in the intensity of stimulus produce no increment or other change in the action potential as long as other experimental conditions remain constant. The action potential fails to occur if the stimulus is subthreshold in magnitude, and it occurs with constant amplitude and form regardless of the strength of the stimulus if the stimulus is at or above threshold intensity. The action potential is therefore “all or none” in character, and said to obey the “all or none law”. Electric stimulation of nerve can result in involuntary muscle construction with or without pain. It depends on the strength and duration of electrical stimulus. It can be described by rheobase and chronaxie. Classically, the magnitude of the current just sufficient to excite a given nerve or muscle is called the rheobase and the time for which it must be applied the utilization time. Another common quantity is the chronaxie, represented by duration of response time interval if the current of twice rheobase is applied to produce a response. The chronaxie of nerve is short, usually < 1 ms; denervated muscle has longer chronaxie (>> 1 ms). Characterization of rheobase and chronaxie is in Fig. 4.3. Slowly rising currents sometimes fail to fire the nerve because the nerve in some way adapts to the applied stimulus. This process is called accommodation. stimulus strength

2× rheobase rheobase

chronaxie

stimulus duration

Figure 4.3: The rheobase and chronaxie definition.

The amount of ions passed through the membrane during action potential is very small – about 3.10−12 to 4.10−12 mol/cm2 of membrane surface. Therefore the process of production of action potential and further renewal of the rest membrane potential has very low energy requirements. Also the time interval during which the ion channels are open is very short (less than 1 ms). Therefore, the maximum value of action potential only approaches near to the value of rest membrane potential for sodium ions – there is not time enough at disposal to reach the value. 108

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For a certain time interval (10–15 ms) after action potential it is not possible to evoke new action potential by a threshold stimulus while it may be produced by a super–threshold stimulus. This time interval is called the relative refractory period. However, very short time interval exists (about 1 ms) during which action potential cannot be produce by any stimulus. This time interval is called absolute refractory period. Thus, to the time course of potential changes which represent action potential, corresponds a certain excitability cycle. The period just before the spike potential corresponds to a little increased excitability and the period of spike potential (1–2 ms in the case of nerve fibre or 200 or more ms in the case of heart muscle fibres) corresponds to the absolute refractory period. If we summarize the course of action potential of nerve cell, it consist of these steps: 1. A stimulus is received by the dendrites of a nerve cell. This causes the Na+ channels to open. It changes the interior potential from −70 mV up to −55 mV; 2. Having reached the action ­threshold, more Na+ channels open. The Na+ influx drives the interior of the cell membrane up to about +30 mV. Depolarization begins; 3. The Na+ channels close and the K+ channels open. Since the K+ channels are much slower to open, the depolarization has time to be completed; 4. The membrane begins to repolarize back toward its rest potential; 5. Hyperpolarization occurs, it is important in the transmission of information. It prevents the neuron from receiving another stimulus during this time (at least raises the threshold for any new stimulus); 6. After hyperpolarization, the Na+/K+ pump eventually brings the membrane back to its resting state. The function of a nerve fibre is not only the production of action potential but also its conduction. The propagation of action potential along the axon is based on ion fluxes. Ion fluxes flow across the membrane at the place of action potential appearance and therefore the polarity of the membrane is changed. Due to the tendency to equilibrate electric charge, longitudinal ion fluxes take place along the fibre that induces a decrease of the resting membrane potential in the neighbour section and thus action potential is produced in this section. One-way propagation takes place due to the existence of refractory period. Thus, action potential propagates along the nerve fibre. The speed of signal propagation along the nerve (several m/s up to more than 100 m/s) depends on axon type. Conduction in myelinated axons depends upon a similar pattern of circular current flow. Thanks to the fact that myelin is a relatively effective insulator, depolarisation in myelinated axons jumps from one node of Ranvier to the next. This jumping of depolarisation from node to node is called saltatory conduction. It is a rapid process, and myelinated axons conduct up to 50 times faster than the fastest unmyelinated fibres. Principle of conduction of action potential is shown in Fig. 4.4. If the axon under one of the external electrodes is damaged, the damaged area becomes negative relative to the healthy portion at rest. The potential difference between the intact nerve fibers or muscle fibers and the injured area of the same fibers is called the demarcation potential. Its magnitude is variable, depending upon the extent of the disruption of the membrane. Various values of the resting membrane potential as well as well as different forms of action potential are observed in the cells of bone muscle, smooth muscle and heart muscle. Especially, action potential of heart muscle fibre possesses a wide plateau. Action potentials produced during the activity of various organs yield the basis for various electro diagnostic methods. 109

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Status of Na+ channels:

Closed and inactivated

Closed

Open

Axon

Membrane potential (mV)

Time: 0 50

Passive spread

Refractory period

0

Depolarization

–50 0

1

Closed

2 3 4 Distance along axon (mm) Closed and inactivated

Open

5

6

Closed

Membrane potential (mV)

Time: 1 ms 50 Hyperpolarization

0 –50 0

1

Closed

2 3 4 Distance along axon (mm) Closed and inactivated

5

Open

6

Closed

Membrane potential (mV)

Time: 2 ms 50

Resting potential

0 –50 0

1

2 3 4 Distance along axon (mm)

5

6

Figure 4.4: Conduction of action potential. The membrane depolarization spreads passively in both direction along the axon but the Na channels at one side are still inactivated and cannot be reopened for a few milliseconds. Only one side is polarized (rest membrane potential is completely established) and to that direction is action potential conducted.

4.2.3 Action potential in heart cell The heart is a muscular organ in humans and other animals, which pumps blood through the blood vessels of the circulatory system with a rhythm (in healthy adults around 72 beats per min) determined by a group of pace-making cells in the sinoatrial node. These generate a current that causes contraction of the heart, traveling through the atrioventricular node and along the conduction system of the heart. Action potential in heart cell (Fig. 4.5) occurs within systole (contraction that drives blood out of the heart). The rest is the period of time when the heart refills with blood after systole (−80 – −90mV). 110

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mV 20 0

U

−20 −40 −60 −80 −100 0

150

300

ms

Figure 4.5: Action potential of heart cell. It has faster depolarization compared to nerve cells.

The electrical potentials that are generated in the body have their origin in membrane potentials where differences in the concentrations of positive and negative ions give a localized separation of charges (polarization). Changes in voltage occur when some event triggers a depolarization of a membrane (also upon the repolarization). The depolarization and repolarization of the sinoatrial node and the other elements of the heart’s system produce a strong pattern of voltage change.

4.2.4 Electrocardiogram (ECG) Voltage change can be measured with electrodes on the skin. Voltage measurements on the skin of the limb and chest wall are called an electrocardiogram (ECG). The ECG is no other than the graphical representation of heart electrical activity and it is a major diagnostic tool for the assessment of the health of the heart. Voltage measurement is done by electrodes. The stretch between two limbs (arm or leg) or between limb and ground electrodes is called a lead. We distinguish two types of leads: standard (bipolar) leads which are between two limbs and unipolar leads which are between ground and limb. Bipolar leads I, II, III are called Einthoven leads (named so after the inventor of the Electrocardiogram) (Fig. 4.6). The three bipolar leads form what is called the Einthoven’s Triangle. These leads maintain a mathematical proportion explained by the Einthoven’s Law, which says: II = I + III. This law is of great value when interpreting an electrocardiogram because it allows us to determine whether the limb electrodes are correctly placed; if the position of any electrode is altered, this law would not hold, thus allowing us to realise the ECG is not correctly done. 111

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Unipolar leads are called Wilson and Goldberger leads. Wilson leads (or Wilson central terminals) are formed by connecting a 5 kΩ resistance to each limb electrode and interconI ΦR free wires;Lead necting the the central terminal ΦL (CT) is the R common point (ground) (Fig L 4.6). R

L

Lead I

ΦR

ΦL

p

R

5k

R

5k

5k

p

IR

α

F ΦF

5 kΩ

Lead III VIII = ΦF – ΦL

Lead II VII = ΦF – ΦR

L

CT Ω 5k IL

Ω 5 kΩ

Lead II VII = ΦF – ΦR

Ω IL

Ω

IR

L

α

Lead III VIII = ΦF – ΦL

IF

F

CT

IF

F

F ΦF

Figure 4.6: Einthoven leads I, II, III (left) and Wilson leads (right) connection.

Signals can be augmented by omitting that resistance from the Wilson central terminal, which is connected to the measurement electrode (Fig. 4.7). This connection is called Goldberger augmented leads and 50% larger signal than the signal with the Wilson central terminal chosen as reference could be obtained.

R

L

R

L

5 kΩ 5 kΩ

5 kΩ

5 kΩ

5 kΩ

5 kΩ

R R

L

R

L

L

R

L 5 kΩ

aVF 5 kΩ 5 kΩ

F

aVF F

F

F

F

F

aVL

Figure 4.7: Goldberger augmented leads (aVL, aVF, aVR). aVL

5 kΩ

5 kΩ

5 kΩ

aVR aVR

In clinical practice, for a standard electrocardiogram 10 electrodes are used, divided in two groups: four limb electrodes (one on the right leg is a ground) and six precordial (chest) electrodes. The chest leads record the ECG in the transverse or horizontal plane, and are called 112

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V1, V2, V3, V4, V5 and V6. Their location is shown in Fig. 4.8. The 12 electrocardiogram leads (I, II, II, aVL, aVF, aVR, V1–V6) are obtained from all data provided by them and offer more complex information about heart function.

Figure 4.8: Location of chest electrodes. The unipolar chest leads record the ECG in the transverse or horizontal plane, and are called V1, V2, V3, V4, V5 and V6.

4.2.5 Heart’s electrical sequence and interpretation of electrocardiogram The heart has four chambers; the upper two chambers are called the atria, and the lower two chambers are called the ventricles. The atria are thin-walled, low-pressure pumps that receive blood from the venous circulation. Located in the top right atrium are a group of cells that act as the primary heart’s natural pacemaker of the heart. It is called the sinoatrial (SA) node. Through a complex change of ionic concentration across the cell membranes (the current source), an extracellular potential field is established which then excites neighbouring cells, and a cell-to-cell propagation of electrical events occurs. The ECG signals are typically in the range of ±2 mV and require a recording bandwidth of 0.05 to 150 Hz. We describe more in details the spreading od ECG signal. ECG impulse is initiated by the SA node. It continues via its neurons to the right atrium, left atrium, and atrioventricular (AV) node simultaneously. Since the right atrium is closer to the SA node, it depolarizes first, resulting in pumping action by the right atrium before the left atrium. At the AV node, the impulse is delayed to allow for the ventricles to fill up with blood, then the impulse continues to the Bundle of His and the Purkinje fibres. This triggers the contraction of the ventricles to send blood either to the lungs or out to the body. 113

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A

C

B

ECG is recording of summation of all electrical potential vectors from multiple myocardial fibres, not only from cardiac conduction system. Every myocardial fibre can generate electrical potential. These electrical potentials are conducted by body fluids (which are very good conductors) to the body surface, where they are recorded by an electrocardiograph. It is important to know that leads should not be analysed separately but as a whole complex data, as each lead is a different point of view of the same electrical stimulus. The basic aspects of ECG interpretation include examinations of amplitudes and intervals of ECG waveforms. However, there is wide range of normal variability in the 12 lead ECG, characteristics are not absolute. Therefore, there is required ECG reading experience to discover all normal variants. From ECG, we can easily determine heart rate (between 60–90 beats per minute in healthy adults). Mostly is shown signal obtained by the II lead. Its basic curve is in Fig. 4.9. P-wave reflects depolarization of the atria in response to SA node triggering. It is common to see notched or biphasic P-waves of right and left atrial activation. P-wave duration should be shorter than 0.12 s and amplitude should be lower than < 2.5 mV s in healthy adults. PR interval shows the delay of AV node to allow filling of ventricles. A PR interval should be between 0.12–0.20 s in healthy adult person. PR interval shorter than 120 ms suggests that the electrical impulse is bypassing the AV node. PR interval significantly longer than 200 ms diagnoses first degree atrioventricular block. QRS complex is typically in range 0.06–0.1 s, it represents the simultaneous activation of the right and left ventricles, although most of the QRS waveform is derived from the larger left ventricular musculature. If the QRS complex is wide (longer than 120 ms) it suggests disruption of the heart’s conduction system. Metabolic issues such as severe hyperkalemia can also widen the QRS complex. An unusually high amplitude of QRS complex may represent left ventricular hypertrophy while a very low-amplitude QRS complex may represent a pericardial effusion or infiltrative myocardial disease. The ST segment connects the QRS complex and the T wave; it represents the period when the ventricles are depolarized. It is usually isoelectric, but may be depressed or elevated with myocardial infarction or ischemia. The T wave represents the repolarization of the ventricles. It is generally upright in all leads except aVR and lead V1, its duration is around 160 ms. U [mV]

1.0 0.5 0

t [s] Figure 4.9: Normal ECG signal obtained from the II. lead.

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4.2.6 Electroencephalograph (EEG) and Electromyograph (EMG) We can also assess electrical activity of the brain and muscles. Electroencephalograph (EEG) can read brain activity data from the entire head simultaneously. It determines the relative strengths and positions of electrical activity in different brain regions and records complex patterns of neural activity occurring within fractions of a second after a stimulus has been administered. Voltage fluctuations resulting from ionic current flows within the neurons of the brain. Brain waves are made up of many frequencies (Alpha wave (f = 8–13 Hz), ­Theta wave (f = 4–8 Hz), Delta wave (f ≤ 4 Hz), Beta wave (f = 13–30 Hz), …) which can be determined. EEG can help diagnose conditions such as epilepsy, brain tumors, brain injury, cerebral palsy, stroke, liver or kidney disease (metabolic conditions), or even brain death. It can also help physicians find the cause of problems such as headaches, weakness, blackouts or dizziness. Unfortunately, this simple non-invasive method has less spatial resolution than functional magnetic resonance imaging or positron emission tomography. Electromyograph (EMG) records the electrical activity produced by skeletal muscles when they are electrically or neurologically activated. Electrode is inserted into the muscle or placed on the skin. The electrical activity is recorded on electromyogram or displayed on an oscilloscope and heard through a loudspeaker. Typically, EMG is used as a diagnostics tool for identifying neuromuscular diseases, or as a research tool for studying kinesiology, and disorders of motor control.

4.3 ELECTRIC FIELD, ELECTRIC CURRENT AND VOLTAGE An electric field is defined as a vector field that associates to each point in space the Coulomb force that would be experienced per unit of electric charge. Electric current I is the rate of charge flow past a given point in an electric circuit. The SI unit is ampere A. Voltage is a potential energy per unit charge, its unit is the volt V. The difference in voltage is equal to the work, which would have to be done, per unit charge, against the electric field. The principle of conservation of electric charge implies two electric (Kirchhoff’s) laws: the current law and voltage law. The current law states that the sum of the currents into any junction is equal to the sum of the currents out. Voltage law states that the net voltage drop around any closed loop path must be zero.

4.3.1 Conduction of electric current in organism Ohm’s law holds if electric current flows through a conductor with resistance R (unit is ohm Ω), U = RI. In another words, the electric current, which flows through conductor is directly proportional to the applied voltage. If alternating voltage is applied to a conductor that has also its capacity C and induction L, then instead the resistance the impedance, Z, has to be considered. The impedance consists of three components: resistance R, capacitive reactance, RC = 1/ωC, and the inductive reactance, RL= ωL. The impedance can be measured and also Ohm’s law holds, U = ZI. 115

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The relation between impedance Z and its components is given by Z = R 2 + (ω L −

1 2 ) (4.19) ωC

In tissues, the inductive reactance may be neglected and thus Z = R 2 + RC2 (4.20) Total resistance of a conductor of the length l and cross-section A is given by R=ρ

l (4.21) A

where ρ denotes the resistivity. The resistivity of conductors is of order 10−8 Ω.m while its very high values may be found in insulators, e.g. wood (maple) 3.1010 or teflon (PTFE) 1013 Ω.m. Resistance of conductors is also dependent on temperature. Increase of temperature increases the resistance. In case of semiconductors, the dependence on temperature is very strong and negative; increase of temperature decreases the resistance exponentially. The reason is that density of electrons in a semiconductor strongly increases with increasing temperature. Resistance of the human body is affected by many factors as are temperature, path etc. Every person has different electrical resistance, men tend to have lower resistance than woman. Total resistivity for electric current flowing through body is sum of resistance of the skin where current is entering into body, internal resistance and resistance of the skin where current is leaving the body. The majority of the body’s resistance is the skin, dry skin as in the range between 1000–100000 Ω. Internal resistance of the human body is 250–1100 Ω. The greatest resistance have bone and fat, nerves and muscle have the lowest resistance. Pure ohmic resistance R plays an important role in liquids. There is sufficiently high number of free ions for transport of electric charge in extra cellular as well as in intracellular medium. The property of a solution to conduce electric current is described by specific conductivity κ, which is inverse value of resistivity ρ:

κ=

1 (4.22) ρ

The unit of specific conductivity is S.m−1. In order to relate the dependence of conductivity on ion concentration, the molar conductivity can be used. The molar conductivity of the i-th ion, Λi, is given by: Λi =

κ (4.23) ci

where ci (mol.m−3) is the molar concentration of the i-th ion. In diluted solution, where ions move independently, the molar conductivity has additive property, i.e., Λ = ∑ Λ i (4.24) i

For most of ions in biological systems, the molar conductivity does not differ substantially in order. Significantly higher is molar conductivity of OH− ions (near to 2×10−2 S.m2.mol−1) and H+ ions (near to 3.5×10−2 S.m2.mol−1). The molar conductivity increases by about 2% at temperature increase by 1 °C. 116

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Properties of biological membranes in relation to the conduction of electric current are quite different from liquid intracellular or extra cellular media. Biological membranes possess a high capacity – about 1 μF.cm−2. Their resistance is by 8 orders higher than that of protoplasmatic medium. That is why their conductivity for direct current is very low and their capacitive reactance, RC, plays an important role in conduction of alternating current. The capacity C of a capacitor formed by two plates of the area S located at the distance l is given by C=

εS (4.25) l

If this relation is applied into the definition formula of the capacitive reactance then we get RC =

l (4.26) ωε S

For the angular frequency it holds ω = 2πf, therefore RC =

l (4.27) 2π f ε S

Thus, the capacitive reactance is inversely proportional to both frequency and permittivity. Frequency dependence of the capacitive reactance is a complex problem. Permittivity is frequency dependent in the media as water, water solutes of electrolytes, and hence also in biological tissues and liquids (see Fig. 4.10). This complicated behaviour of the capacitive reactance results from the fact that different mechanisms play their role at various frequency ranges (motion of opposite charges through the membrane, charging and discharging of membranes, rotation of polar molecules, etc). Since RC and κ are inversely proportional, we can assume that frequency dependent conductivity may be added to frequency independent component of conductivity and thus total conductivity of tissue κ = κ0 + κC. The frequency dependent component of conductivity κC prevails over the component κ0 corresponding to ohmic conductivity at high frequencies. Membrane serves as both an insulator and diffusion barrier to the movement of ions. Conduction of electric current through membrane can be simplified by the scheme given in Fig. 4.11. C represents the capacity of biological membranes, it is determined by the properties of the lipid bilayer. The voltage is determined by the concentrations of the ion on each side of the membrane (the same for each ionic pathway). R represents the resistance of membrane. Resistance of plasma membrane is highly variable, it depends on number and state of ion channels. Typical value 1 MΩ/mm2. Extracellular and intracellular liquids can be also represented by resistances. For the circuit, in which the resistor R is parallel to the capacitor with the capacity C and capacitive reactance RC, it holds for total impedance Z 1 1 1 = 2 + 2 (4.28) 2 Z R RC and thus Z=

R.RC R 2 + RC2

(4.29)

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ε/ε0

S.m−1

κ

εrel

ε/ε0 κ

εrel

102

S.m−1 102 1

1 10−2 κ0

10−2 κ0

Figure 4.10: Relative permitivity εrel and specific conductivity κ of biological tissues as a function of frequency, κ0 conductivity independent component. Ue-i

R

Extracellular

Intracellular

Ue-i

R

Extracellular

C

Intracellular

Figure 4.11: Simplified electrical model of cell membrane. C

4.3.2 Effect of electric current on organism There are three types of effects of electric current on living system. We may distinguish thermal, electrolytic and irritation effects. The manifestation of these effects depends mainly on type of electric current. Their review is seen in Table 4.5. Water medium in organism (intracellular liquids, extra cellular liquids and body liquids) contains ions and therefore it behaves as electrolyte. That is why direct current must have electrolytic effects. The accumulation of alkaline compounds below the cathode and acid compounds below the anode is the effect of electrolysis. It results in changed stimulation of nerves. These changes may be observed at current densities 0.2–0.3 mA/cm2. Current densities around 0.5 mA/cm2 can already result in tissue damage. Stimulating effects of direct current can be observed only at switching off, or -in, or at sudden change of the intensity. 118

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Tab. 4.5 Effects of electric current on alive system Type of current

Effects Electrolytic

Stimulation

Thermal

Direct current

Strong

At sudden change of intensity

–

Alternating low-frequency

Weak

Strong

–

Alternating high-frequency

–

–

Strong

Tissue excitability can be expressed by the quantities called rheobase and chronaxie already mentioned. If rectangular pulses stimulate a muscle, then a certain minimum value of intensity exists and the tissue does not respond to the intensity below this threshold value. This minimum intensity is called rheobase. Chronaxie is the time of pulse duration required to evoke excitation at two-fold rheobase. Low-frequency alternating current (50–500 Hz) has only very weak electrolytic effects since the polarity changes in time. Weak electrolytic effect can be observed only at low frequency, when the products of electrolytic effects have time enough to remove from the electrode before the change of its polarity. However, very distinct stimulating effects are observed in the case of low-frequency alternating current. The stimulation increases with frequency up to about 100 Hz and then decreases, so that above 100 Hz the threshold value of its intensity is proportional to the second root of frequency. If the current passes the heart, its activity is disturbed and it may lead to lethal effects. Its thermal effects may be neglected with respect to the excitation effects. High-frequency alternating current has no electrolytic effects; the excitation effects decrease with increasing frequency and vanish at frequencies above 100 kHz. This type of current has mainly thermal effects. Therefore, it is applied for heating the tissues by diathermy. It is obvious that all types of current have thermal effects. However, speaking about alive system, thermal effect of direct current or low frequency alternating current would be observed only at intensities that cannot be applied. The current density 50–200 μA/cm2 is below stimulus. The following review yields the relation of ac and dc intensities (mA) and the effect evoked. AC

DC

Weak shock sensation

1

5–6

Beginning of danger, “can’t let go”

15

70–80

Respiratory spasm, heavy pain

25

90–100

Ventricular fibrillation, danger of death

80

300

Cardiac paralysis, clinical death

100

500

Micro shock and fibrillation may be observed even at 10 μA when applied by catheter. Electric injury symptoms range from skin burns, damage to internal organs and other soft tissues to cardiac arrhythmias and respiratory arrest. In addition to burn injuries, AC can freeze the patient’s hand to the current source, while DC can throw the patient, causing injury. Important factors, how injury is severe, are the pathway of current determining the 119

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specific tissue damage and the duration of exposure. The longer exposure increases injury severity. Body resistance also affect the burn severity. Although skin burn severity does not predict the degree of internal damage, internal damage is more severe if the skin has low resistance. The use of electric energy can be used also as a medical treatment. It is used mostly in rehabilitation to speed wound healing. In electrotherapy is lifesaving defibrillator heart machine. A defibrillator delivers an electric shock to restore a person’s heart rhythm. Huge application of electric energy is in electrosurgery. Electrosurgery is the application of a high-frequency electrical current to biological tissue as a means to cut, coagulate, desiccate, or fulgurate tissue. Typical example of set up in electrosurgery is shown in Fig. 4.12.

Active electrode

Electrical generator

Ground electrode

Figure 4.12: Electrosurgery. Active electrode serves as surgery tool.

4.3.3 Conductometry If one chemical compound, which is an electrolyte, is present in a solution then the electric conductivity is proportional to the concentration of this compound. Therefore, the concentration can be estimated by measuring the specific conductivity. The total resistance R of a conductor is given by equation (4.21) being dependent on the length l and cross section q of the conductor and on its resistivity ρ that is characteristic for the material of the conductor. The specific conductivity κ is inversely proportional to ρ (see equation (4.22)). By using the equation (4.21) we get

κ=

l (4.30) qR

Since the resistivity depends on the composition of conductor material, the specific conductivity depends on its composition as well. Besides solid conductors, liquid conductors exist and the above relations also hold for them. If some pure chemical compound is dissolved in a solvent, then the specific conductivity will depend on the concentration of this compound. Thus, by measuring the specific conductivity, the concentration of the compound can be estimated. 120

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RA UA

A V

The measurement of specific conductivity can be realized by using conducting vessel containing two platinum electrodes. The Ux vesselRx(N) containing the measured solution is located into the Wheatstone bridge circuit as an unknown resistance (see Fig. 4.13). To calculate the specific conductivity from the measured value of R according to equation (4.30), the unknown values of l and q are required. Their ratio l/q can be determined experimentally as follows. The vessel is filled by a standard solution of a known specific conductivity κs, (e.g. by 0.1 N KCl solution). Then it holds

κs =

l (4.31) qRs

Thus R1

R3

l = κ s Rs (4.32) q +

G Value l/q is −called the “capacity” C of the conducting vessel. Once C being estimated by using standard solution, the value of the resistance Rx of unknown solution can be measured and its specific conductivity κx is then given by R

κx =

R

2 x C (4.33) Rx

Alternating current, preferably high-frequency ac is applied for conductivity measurements of solutions to avoid polarization of electrodes or even electrolytic changes. Therefore, the impedance instead of resistance is measured.

V

N

R G

Rx

Ry

~ Figure 4.13: Scheme of Wheatstone bridge for conductivity measurement. R – known resistance, Rx, Ry – resistances of the resistors for balancing of the bridge. Bridge is balanced if RxR = RyN; in this case galvanometer G detects zero current.

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4.4 OSCILLOSCOPE The electric quantities as is voltage that changes periodically in time may be visualized by using oscilloscope. An oscilloscope is a device that displays a graph of an electrical signal. The vertical Y axis represents amplitude (voltage, current), the horizontal X axis represents time. The main units of oscilloscope are power supply unit, cathode ray tube, vertical amplifier, horizontal amplifier, time base generator, trigger circuit, delay line (Fig. 4.14). Input signal

Vertical amplifier

Delay line

To cathode ray tube

Electron gun

Luminous spot Screen

HV supply LV supply

Electron beam Vertical deflection plates

To all circuits

Trigger circuit

Time base generator

Horizontal deflection plates

Horizontal amplifier

Figure 4.14: Block diagram of oscilloscope.

The cathode ray tube (electron gun) is used for that purpose. Its function is based on the motion of electric charge in electric field. A particle of mass m and charge q in an electric field of the intensity E experiences a force F = qE and, according the second law of mechanics, its acceleration is a = qE/m. Electrons are emitted from a thin, heated filament, accelerated by anode (with hole) that they pass through, and finally collimated by a negatively charged cylinder into a narrow beam. Then the collimated beam passes between two pairs of deflection plates and finally strikes the screen coated by a phosphorescence material (e.g. ZnS). As each electron strikes the screen, a tiny flash of light is emitted. The plates that due to their charge deflect the beam in the horizontal plane serve for the “time basis” of the oscilloscope. The time basis may be created by a periodical, linearly increasing voltage that rapidly returns back to zero at the end of the period (saw-like course). The voltage under investigation is led to the second pair of plates that deflect the beam in the vertical plane. If no voltage is put to this second pair, we see a horizontal straight-line abscissa on the screen. If alternating current of the frequency identical to the frequency of the time basis is led to the second pair, we can see one cycle of that current on the screen. If the frequency of the time basis would be one half of that of the alternating current, we would see two waves of the current. Many of physiological processes possess periodical character and thus they may be studied by using an oscilloscope, e.g. electric heart activity or breathing (after transformation of mechanical into electric phenomena), etc. 122

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5.

ACOUSTICS AND PHYSICAL PRINCIPLES OF HEARING

5.1 INTRODUCTION Acoustics is the science that deals with the study of all mechanical waves in gases, liquids, and solids including vibration, sound, ultrasound and infrasound.Sound waves are mechanical waves produced by a vibration source of suitable frequency, which pass through different media at various velocities. No transport of medium particles occurs when a sound wave passes through the medium, but the medium’s particles oscillate around their equilibrium positions (see Fig. 5.1). Wave of longitudinal undulation Distance from source

Motion of particles

Density of particles

Pressure wave

Figure 5.1: Scheme of sound propagation and pressure changes in a medium.

In gas and liquid media, these oscillations are longitudinal, and in solid phases, both longitudinal and transversal sound waves can occur. The particles do not move from point A to point B with the wave; they simply oscillate back and forth parallel to the direction of longitudinal wave propagation, or up and down perpendicular to the direction of transverse wave 123

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propagation about their individual equilibrium positions as the wave passes by. The wave is seen as the motion of the compressed region (called a pressure wave), which moves from point A to point B. The simplest example of longitudinal wave is radiation from a monopole stationary source. In this case, the source radiates sound uniformly from the source in all directions in the shape of a sphere undergoing sinusoidal (alternating) radial contraction and expansion.

5.1.1 Basic quantities The velocity of sound, c, in a gas medium is related to its pressure p and density ρ by

χp (5.1) ρ

c=

where χ = Cp /CV is the Poisson’s constant (see part 3.2). The propagation of sound velocity depends on the temperature, which is not apparent in the above equation. However, since the density equals total mass/volume, and the mass m is given by the product of the number of moles, n, and the molar mass M, then by using the state equation of gas (see equation (2.1) in the part 2), and after rearrangement equation 5.1 the velocity of sound can be solved in terms of the temperature T:

χ RT (5.2) M

c=

where R is the universal gas constant. In liquid media (and therefore in soft tissue), the velocity of sound propagation is given by

κ (5.3) ρ

c=

where κ denotes the volume modulus of elasticity and ρ is the density. The elasticity modulus is defined as the ratio of pressure changes to the relative volume changes, dp/(dV/V). The velocity of the sound propagation in the air at 20 °C is 344 m.s−1; in water at 13 °C is equal to 1441 m.s−1, in soft tissue 1400–1600 m.s−1, in bones is 2800–4800 m.s−1, and in glass 6000 m.s−1. The wavelength of an acoustic wave. The wavelength λ is related to the velocity of propagation c and frequency f by

λ=

c (5.4) f

The Frequency f is the same during the transition through different media. Acoustic amplitude. The acoustic amplitude, a, is the amplitude of the vibration motion of medium’s particles, and thus, it varies between zero and its maximum value amax. It can be described by 124

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a = amax sin (2πft )(5.5) In water, amax reaches values on the same order as the size of the hydrogen atom. The Acoustic velocity v is the velocity of the vibrating motion of the medium’s particles, and can be obtained by taking the first derivative of equation (5). The resultant equation is: v=

da π = amax .2πf .cos (2πft ) = vmax .cos (2πft ) = vmax . sin (2πft + )(5.6) 2 dt

where amax.2πf is considered to be the maximum acoustic velocity vmax.The acoustic velocity also varies between zero and the maximum acoustic velocity. For this reason, the effective acoustic velocity is defined (analogously to effective voltage or intensity of alternating electric current) by = vef

vmax = 0, 7vmax (5.7) 2

Acoustic pressure. Oscillations of medium particles result in periodical density changes and thus, they induce periodically changing acoustic pressure p. Acoustic pressure is in phase with acoustic velocity. Therefore, with respect to equation (5.6), it holds π p = pmax .sin (2πft + )(5.8) 2

Presssure

The effective acoustic pressure is defined analogously to the effective acoustic velocity, i.e. pef = 0.7 pmax. Acoustic pressure is superposed on the barometric pressure (see Fig. 5.2).

infrasound

sound

acoustik pressure

air pressure

time Figure 5.2: Acoustic and barometric pressure.

The effective acoustic pressure is related to the effective acoustic velocity, the density ρ of the medium and the velocity of sound propagation c by pef = vef ρ c (5.9) 125

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Acoustic impedance, z, of a plane acoustic wave passing through a medium is defined by the ratio of the effective acoustic pressure to the effective acoustic velocity, i.e. by z=

pef vef

= ρ c(5.10)

Acoustic impedance indicates how much sound pressure is generated by a given air vibration at that frequency. When a wave encounters a medium of higher acoustic impedance, there is no phase change upon reflection. When a wave encounters a medium of lower acoustic impedance, there is phase reversal in reflected wave. According to the above equation, the acoustic impedance of air zair = 1.29 kg.m−3×340 m.s−1 = = 0.44 kPa.s.m−1 (or 440 Pa.s.m−1). Soft tissue with a high content of water possesses a density of about 1000 kg.m−3, and the velocity of sound propagation in water is about 1500 m.s−1. Therefore, acoustic impedance of soft tissue is about 1.5×106 Pa.s.m−1. Intensity of sound, I, (i.e. the amount of energy passing through the area of 1 m2 perpendicular to the direction of sound wave propagation within 1 s) is expressed in W.m−2 and it can also be defined by I = vef pef =

pef2

ρc

(5.11)

In a given medium, the intensity of sound is directly proportional to the squared effective acoustic pressure. Intensity level. Due to very large range of intensities detected by human ears, relative units bell (B), or decibel (dB) were introduced for the intensity level L. The sound intensity level expressed in bells or decibels are defined by I I = L log = (B) 10.log (dB)(5.12) I0 I0 E.g., if the intensity I of a sound increases 100 fold with respect to its basic value I0, i.e., I = 100 I0, then the intensity level increases by 10 log (100 I0 /I0) = 10 log 100 = 20 dB. The intensity level can be expressed by using acoustic pressure as well. However, since with respect to the equation (5.11) I/I0 = (p/p0)2, then p p = L 2= log (B) 20 log (dB)(5.13) p0 p0 By using the above relations, the acoustic pressure, the amplitude of oscillation motion or the velocity of oscillation motion of molecules of the medium through which a sound wave propagates can be evaluated. Let us consider sound with frequency 1 kHz and intensity level 50 dB propagating in air. The intensity corresponding to that intensity level can be calculated by using equation (5.12) 50 = 10.log

10

−12

I W.m −2

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Dividing both sides by the factor 10 and considering that 5 = log 105, the above equation may be rewritten into the form log 105 = log

I 10−12 W.m −2

Thus, the arguments of the function logarithm must be equal on both sides and therefore I = 105.10−12 = 10−7 W.m−2. It follows from equations (5.10) and (5.11) that the intensity, effective acoustic pressure, and acoustic impedance are related by pef = I .z . After substitution of numerical values we get pef = (10−7 W.m −2 ).(0.44 × 103 Pa.s.m −1 = 0.44.10−4 N 2 .m −4 = 0.66×10−2 Pa. Since pmax = 2 . pef , the maximum acoustic pressure of this sound will be pmax = = 1.41×0.66×10−2 Pa = 0.93×10−2 Pa. By substituting numerical values of intensity and effective acoustic pressure into equation (5.11) vef =I/pef, we solve for the effective acoustic velocity vef =10−7 W.m−2/0.66 ×10−2 Pa = = 1.51×10−5 m.s−1. Similarly, substitution of the intensity and effective acoustic velocity, the maximum acoustic velocity (vmax = 2 .vef ) becomes vmax = 1.41×1.51×10−5 m.s−1= = 2.14×10−5 m.s−1. Finally, since vmax = 2πf.amax, for the maximum acoustic amplitude given by amax = vmax/2πf, we get amax = 2.14×10−5 m.s−1/(2.π.1000 s−1) = 0.34×10−8 m = 3.4 nm. These results demonstrate that the maximum acoustic amplitude is shorter than the wavelength of visible light. If a similar calculation is carried out for propagation of the same sound intensity and frequency in water that possesses the acoustic impedance of 1.48 MPa.s.m−1, then analogously, we get pef = 0.38 Pa, pmax = 0.53 Pa, vef = 2.63×10−7 m.s−1 a vmax = 3.7×10−7 m.s−1. Finally, the maximum amplitude in water at this frequency will be amax = 3.7×10−7 m.s−1/(2.π.1000 s−1) = = 5.9×10−11 m, i.e. of the same order as that corresponding to the size of hydrogen atom.

5.1.2 The Doppler effect The Doppler effect is the change of frequency or wavelength observed by an observer (detected by detectors) which is caused by the relative motion of the observer or source with respect to the medium. If the observer is at rest and the source of sound moves with velocity vsource, then the

λ = λ0 ±

vsource (5.14) f0

where λ0 = c/f0 is the wavelength when the source is at rest, its frequency is f0 and the waves propagate with the velocity c. The sign + or − is used if the source moves from or to the observer, respectively. When both the source and the observer move, the frequency observed is given by f = f0

c ± vdetector (5.15) c  vsource 127

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where vdetector is the velocity of the detector motion with respect to the medium towards the source, and vsource is the velocity of source with respect to the medium towards the detector. The upper sign is applied if the source and detector approach each other and the lower sign at the opposite situation. Sound waves are produced by both stationary and moving sound sources with velocity v, and are shown in Fig 5.3 below. If the velocity v of the sound source is even higher than velocity of sound propagation c, then the sound source breaks through the sound speed barrier. Since the source is moving faster than the sound waves it creates, it actually leads the advancing wavefront. In that case, the sound source will pass by a stationary observer before the observer actually hears the sound it creates.

A

B

C

Figure 5.3: Sound waves produced by stationary and moving sound source. (a) Stationary sound source. Sound waves are produced at a constant frequency f0, and the wavefronts propagate symmetrically away from the source at a constant speed c, which is the speed of sound in the medium. The distance between wavefronts is the wavelength. All observers will hear the same frequency, which will be equal to the actual frequency of the source. (b) Moving sound source where v < c. The wavefronts are produced with the same frequency f0. However, since the source is moving (with velocity v) from left to the right, the center of each new wavefront is now slightly displaced to the right. As a result, the wavefronts begin to bunch up on the right side (in front of) and spread further apart on the left side (behind) of the source. An observer in front of the source will hear a higher frequency f´ > f0, and an observer behind the source will hear a lower frequency f´ < f0. (c) Moving sound source v = c. The wavefronts in front of the source are now all bunched up at the same point. As a result, an observer in front of the source will detect nothing until the source arrives. The pressure front will be quite intense (a shock wave), due to all the wavefronts adding together, and will not be perceived as a pitch but as a “thump” of sound as the pressure wall passes by.

The Doppler effect is used in medicine to evaluate the blood flow velocity in blood vessels. The principle is that an ultrasonic pulse probe detects the reflected sound from moving blood. The frequency of the reflected sound is different, and the beat frequency Δf (mostly in acoustic range) between the direct (frequency ft transmitted by transducer probe) and reflected sounds (frequency fr detected by receiving probe) can be amplified and used in earphones to hear the pulse sound. Because the speed of blood depends on vessel diameter (see the equation of continuity in section 2.6.1), we can easily detect the area where the change of blood velocity occurs which can be utilized to diagnose steatosis and other blood flow anomolies. The ultrasound probe is used in neonatology to examine the heart function of a premature infant. The sound produced by the pulse of a premature infant is extremenly 128

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.

faint, and may be very difficult to detect with a stethoscope. A sensitive Doppler pulse probe can be used to its advantage because it detects the movement of the blood through an artery. The ultrasonic echo from the moving blood can be mixed with the source frequency to produce a beat frequency. As the blood surges with the pumping action of the heart, the beat frequency signal changes in frequency and amplitude. The principle of Doppler’s pulse detection is shown in Fig. 5.4. Ultrasonic source

Receiving transducer

Figure 5.4: Doppler’s pulse detection. The transmitting transducer transmits sound waves with frequenΔf blood = fvysílaná fpřijímaná cy ft. These waves are reflected from moving (red−blood cells) and detected in the receiving transducer with a different frequency fr. The difference (beat) frequency Δf = | fr − ft| can be easily detected in earphones. The surges in blood speed with the pumping action of the heart can cause detectable changes in the beat frequency. The increase in blood speed caused by a constriction or any obstruction in a vessel can be detected as a change in beat frequency.

5.1.3 Weber-Fechner’s law The threshold intensity isthe lowest detectable intensity detected by human’s healthy ears at frequency of 1 kHz is about 10−12 W.m−2. This value is used for I0 when calculating the intensity level in dB from intensity I, expressed in W.m−2 (see section 5.1.1). The frequency of 1 kHz is selected as a reference frequency and by definition its threshold intensity corresponds to the intensity level at 0 dB. The physiological quantity loudness, is described as the intensity of sound in the frequency range between 16 Hz to 16 kHz that reaches the sensory organ or ear, and results in hearing and perception of sound. However, sound perception is subjective and, moreover, the human ear’s has various sensitivities to various frequencies and amplitudes, thus loudness is not proportional to the intensity. The dependence of loudness on the intensity of its stimulus is described by Weber-Ferchner’s law: The change of loudness ΔL is proportional to the relative change of the stimulus ΔI/I. ΔL = k . ΔI/I(5.16) 129

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Integration results in the equation formally similar to equation (5.12). For this reason, a linear increase of intensity results in a logarithmic increase of loudness. The unit phone (Ph) was introduced for quantitative description of loudness, which, thanks to the frequency-dependence of the human ear, is a function of the sound frequency. The highest sensitivity of the human ear is observed at frequencies of 1–5 kHz. This sensitivity is related to the resonance of the auditory canal. To hear at the same loudness, a sound with a higher or lower frequency than that laying within the above mentioned region, a higher intensity is required. The loudness level expressed in phones corresponds numerically to the intensity level in dB only at the reference frequency, i.e. at 1 kHz. The field of hearing (see Fig. 5.5) is a region of sound intensities and frequencies inducing the effect of hearing. The curve of the lowest intensity level expressed in dB and plotted as a function of frequency represents the threshold of hearing. The whole curve corresponds to 0 Ph at each frequency. Hearing involves the perception of pitch, loudness and timbre. Pitch is the frequency of sound (concert A = 440 Hz), loudness is related to the sound intensity and timbre expresses the sound quality. Timbre allows the ear to distinguish sounds which have the same pitch and loudness. Timbre is then a general term for the distinguishable characteristics of a tone. Timbre is mainly determined by the harmonic content of a sound and the dynamic characteristics of the sound. level of sound intensity (dB)

region of speaking

region of music treshold of pain

treshold of hearing

frequency (Hz) Figure 5.5: Field of hearing.

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Intensity levels of about 130 dB represent the threshold of pain, which practically does not depend on the frequency. In the field between these two curves, the levels of the same loudness may be plotted which correspond to the identical values of loudness in phones at different frequencies (see Fig. 5.6). Each curve corresponds to the identical number of phones (units of loudness) given by the value of intensity level at the reference frequency, i.e. 1 kHz. The loudness of normal speech is about 40–60 Ph, the loudness of the street about 60–90 Ph, and that of starting jet engine between 120–130 Ph. intensity level (dB)

phones

frequency (Hz)

Figure 5.6: Curves of the same loudness.

5.1.4 Complex tones Every complex periodical function of time can be mathematically decomposed into an amplitude infinite sum of simple functions by the Fourier series: F (t ) = a1 sin(2π ft + Φ1 ) + a2 sin(4π ft + Φ 2 ) + a3 sin(6π ft + Φ 3 ) + ...(5.17) Each complex tone can be expressed as a sum of simple tones of certain amplitudes and phases, i.e. every complex sound has its spectrum of amplitudes and phases, and frequencies of its components (higher harmonics) are given by multiplication of a certain basic frequency by a factor 1, 2, 3, etc. The example is seen in Fig. 5.7 and 5.8.

f

2f

3f

4f

5f

6f

7f

frequency

131

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frequency (Hz)

amplitude

f

2f

3f

4f

5f

6f

7f

frequency

phase +π

frequency −π

Figure 5.7: Line spectrum of amplitudes and phases of a complex tone. x

a)

x

T

b)

T

t

t

Complex tone Basic 1st harmonic 2nd harmonic

Figure 5.8: (A) Tone composed of the basic and two harmonics. (B) the same at identical amplitudes and different phases

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5.2 THE PRINCIPLES OF HEARING Hearing is one of the sensorial functions with a remarkably wide dynamic range. It occurs when audible sound (mechanical waves of frequency 16 Hz–16 kHz) enter the ear structure via the pinna and auditory canal, vibrating the tympanic membrane which transfer energy through the ossicles to the inner ear that sends electrical impulses through the auditory nerve to the auditory region of the brain. The human ear is very sensitive, and the ear’s structure and organization allow detection of pressure variations of less than one billionth of atmospheric pressure. There is a very efficient amplification of the sound signal by the outer and middle ear structures and at the same time there is an effective mechanism that reduces the ear’s response to very loud sounds. The difference between threshold of hearing and threshold of pain is 1013×I0 – the widest range of stimuli of any of the senses. During the perception of a sound stimulus coming into the outer ear from the surrounding medium, sound energy is transformed into electric potentials in the receptor cells of the inner ear and conducted by the acoustic nerve to the brain. The tympanic membrane separates the middle ear from the outer ear as seen in Fig. 5.9: The middle ear is separated from the inner ear by the oval and round windows, which are covered by thin membranes. The middle ear contains three small bones called middle ear ossicles: these are the malleus (hammer), the incus (anvil) and the stapes (stirrup), whose base is attached to the oval window. The sound vibrations arriving from the air are transmitted by the middle ear with minimal energy loss to the inner ear fluid (endolymph and perilymph) in the coiled tube-like organ, the cochlea. An amplification of the pressure occurs in the middle ear in which both the tympanic membrane and the ossicles participate. The surface area of the tympanic membrane is about 55 mm2, and is attached to the the handle of the hammer, whereas the the stapes footplate has a surface area of only 3.2 mm2. As sound propagates through the middle ear, the force is distributed first on the tympanic membrane, and subsequently on the smaller surface area of the stapes footplate, which corresponds to a 17-fold pressure amplification. Another reason for the pressure increase is the lever system formed by the ossicles, since the ratio of lever arms is 1.3 : 1 and thus the pressure on the stirrup footplate is about 20-times of that on the tympanic membrane. These favourable circumstances of energy transmission in the middle ear result in the more sensitive hearing of air-conducted sound than that conducted through the skull bones. Pressure changes reach the cochlear fluid mainly through the tympanic membrane-ossicles system, but also through the tympanic cavity by air-conduction, but the role of the last can be neglected. However, the situation will be quite different if the tympanic membrane and the auditory ossicles are missing. Total dysfunction of the middle ear leads to a hearing loss of 40–60 phones. The cochlea serves as the body’s microphone (converting sound pressure impulses from the outer ear into electrical impulses which are passed on to the brain via the auditory nerve). It is a coiled tube with two and a half turns and a membranous channel narrowed at its end. The cochlea is divided into three parts by its partly osseous and partly fibrous walls. The elastic fibrous separating wall is the basilar membrane (more important for hearing), and the thinner wall is the Reissner’s membrane.

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4

Figure. 5.9: Schema of hearing organ. A - External ear, B – middle ear, C – inner ear. 1 – tympanum, 2 – oscicles, 3 – oval window, 4 – round window, 5 – scala vestibuli, 6 – scala tympani, 7 – Eustach tube

The width of basilar membrane increases from 0.04 to 0.5 mm from the oval window towards the apex of the cochlea. On the basilar membrane, there is the organ of Corti, which contains the endings of the auditory nerve fibres. These are connected with elongated cells covered with hairs on top, the haired cells, which are further covered with the tectorial membrane. The vibrations arriving from the middle ear to the cochlear fluid are transmitted from here to the tectorial membrane and Reissner’s membrane. In both model and cadaver experiments, it has been shown that the movement of the stapes, through the mediation of the cochlear fluid, induced travelling waves in the basilar membrane, with a frequency equal to the sound frequency. The shape of these waves is affected not only by the frequency, but also by the elasticity of the membrane, friction, etc. The overall result is that the amplitude of the travelling wave varies along the membrane, even at fixed intensity. At low frequencies the maximum amplitude is formed close to the apex of the cochlea, while at sufficiently high frequencies lies close to the oval window. The hearing process relies on deformation of the structures of the basilar membrane caused by membrane vibration. The sound analysis relates basically to the frequency and intensity, i.e. it is connected with the formation of a pitch level and sound intensity sensation. The basilar membrane participates in the first step of the analysis. The shear forces acting on the receptor cells induce the receptor potential characteristic of the hearing process. This is the microphone potential (cochlear potential). Practically, the frequency of the microphone potential is equal to the frequency of the sound stimulus. A microphone potential is produced in every haired cell located on the vibrating part of the basilar membrane. The amplitude distribution of the potential changes follows the amplitude distribution of the travelling waves produced on the basilar membrane (the maximum cochlear potential is found at the maximum vibrating amplitude). The intensity of the sound stimulus is manifested partly via the amplitude of the produced microphone potential and partly via the size of the area where the microphone potential is 134

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actually produced. The microphone potentials generate the action potentials in the auditory nerve endings of the haired cells. The properties of these action potentials are related to the sound stimulus analysis in the following way: For a sound stimulus at a given frequency, the microphone potential has a characteristic amplitude distribution; accordingly, in nerve fibres in the environment of the haired cells, the microphone potentials of different amplitudes generate an action potential series whose frequency changes from fibre to fibre. The frequency distribution of the action potential series and the localization of the maximum frequency characterize the frequency of the sound stimulus. Thus, the number of active auditory nerve fibres is characteristic for the generation range of the microphone potential, while the magnitude of the microphone potential is expressed by the frequency of the action potential series. Thus, the following conclusions may be expressed: For basilar membrane: a) The frequency of a strongly damped travelling wave agrees with that of the stimulus; b) The amplitude maximum position depends upon the frequency; c) The magnitude of the amplitude increases with the intensity of the stimulus; d) The size of the vibrating region increases with the intensity. For haired cells: a) The frequency of the microphone potential agrees with that of the stimulus; b) The position of the maximum amplitude of the microphone potential depends upon the frequency; c) The amplitude of the microphone potential increases with the intensity of the stimulus; d) The region, where the microphone potential appears, increases with the intensity of the stimulus. For an auditory nerve: a) The frequency distribution and the position of the maximum frequency are characteristics of the stimulus frequency; b) The frequency of the action potential increases with the intensity of the stimulus; c) The number of active fibres increases with the intensity of the stimulus.

5.3 ULTRASOUND Humans can not detect ultrasound waves (mechanical waves with frequencies higher than 20 kHz) but auditory organs of some animals detect them. Mechanical, magnetic or piezoelectric generators may produce these waves. Piezoelectric generators are very important in medicine, which in this case of an ultrasound wave are generated when an electric field is applied to an array of piezoelectric crystals located on the transducer surface. In medicine, we use ultrasound for diagnostics as well as for therapy.

5.3.1 Piezoelectric effect The piezoelectric effect is an accumulation of charge in the walls of certain solid materials in response to applied mechanical stress, notably crystals, certain ceramics, and biological matter such as bone, and DNA. On the other hand, alternating voltage may result in its deformation. The schematic of the direct piezoelectric effect is seen in Fig. 5.10.

135

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pressure

A

B

draw

C

Figure 5.10: Piezoelectric effect. (A) A piezoelectric material is not compressed and its electrical charged is balanced. (B) and (C) the piezoelectric material is under stress and therefore electrical charge is accumulated on its wall.

Piezoelectric materials are generally crystals which acquire a charge when compressed, twisted or distorted. This provides a convenient transducer effect between electrical and mechanical oscillations. Ultrasonic transducers in medical diagnostic applications are usually used crystals from lead zirconate titanate (PbZrTi) – PZT crystals. In piezoelectric generators, mechanical waves are produced by suitable materials vibrating due to an action of high-frequency alternated electrical field in a liquid medium (oil). These generators can produce intensities up to 10 W.m−2. PZT crystals will generate measurable piezoelectricity when their static structure is deformed by about 0.1% of the original dimension. Conversely, those same crystals will change about 0.1% of their static dimension when an external electric field is applied to the material.

5.3.2 Ultrasound imaging Ultrasound imaging (also known as diagnostic sonography or ultrasonography) is a non-invasive radiologic diagnostic method based on reflection of ultrasound waves from different tissues. Anatomic structures are identified based on their location and echogenicity. The frequency spectrum commonly used in ultrasound imaging is 2.5–60 MHz. Each piezoelectric crystal produces an ultrasound wave. As ultrasound waves travel through tissue, they undergo attenuation and are also deflected by scattering, refraction, and reflection; it is the reflection of the waves that forms the basis of the ultrasonographic images. The summation of all waves generated by the piezoelectric crystals forms the ultrasound beam. Ultrasound pulses must be spaced with enough time between pulses to permit the sound to reach the target of interest and return to the transducer before the next pulse is generated. Higher frequency allows higher resolution but lower penetration of waves through tissue. The principle of ultrasound imaging is shown in Fig. 5.11. The pulse wave emitted from the transducer is transmitted into the body (electric-mechanical energy conversion), reflected off the tissue interface and returned to the transducer (mechanical-electric energy conversion). 136

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Electric Impulse

Probe

Crystal Skin Sound beam

Returning beam organ

organ

Figure 5.11: Principle of ultrasound imaging.

For ultrasound image creation, there are two main parameters: the amplitude of the signal and the timing of the returning signal. The amplitude of the signal indicates how much energy is reflected and the timing of returning signal is related to the distance (depth) of the target from the emitting probe. Energy reflection depends on the difference in sound transmission characteristics of the tissue, or the acoustic impedance, which is a measure of the resistance to propagation of sound waves from one medium to another. Acoustic impedance of the tissue determines the amount of reflection (and transmission) of mechanical waves. At the boundary between two tissue structures with different impedances, an ultrasound pulse is partially reflected. The greater the difference in acoustic impedance, the greater the strength of the reflected ultrasound signal. Different structures will reflect different amounts of the emitted energy, and thus the reflected signal from different depths will have different amplitudes. The time before a new pulse is sent out, is dependent on the maximum A C desired depth that is desired to image. As the velocityBof sound in tissue is fairly constant, the time between the emission of a pulse and the reception of a reflected signal is dependent on distance; i.e. the depth of the reflecting structure. The velocity of an ultrasound wave in a medium is the same as that of audible sound. However, it possesses much shorter wavelength with respect to its high frequency. Since the wavelength equals the velocity of propagation/frequency, and the velocity in soft tissue is about 1500 m.s−1, its wavelength at 1 MHz is 1.5 mm and at 10 MHz is 0.15 mm. Let us determine the value of effective acoustic pressure in water (soft tissue) for the intensity of 1.5 W.cm−2 =1.5×104 W.m−2 at frequency 1 MHz. By using the value of the acoustic impedance of water 1.48 MPa.s.m−1 and equation (5.10) and (5.11), it follows that pef = I .z , and the effective acoustic pressure pef = (1.5 × 104 W.m −2 ).(1.48 × 106 Pa.s.m −1 ) = 2.22 × 1010 N 2 .m −4 = 1.45×105 Pa. 137

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Since pmax = 2 . pef , the value of maximum acoustic pressure is pmax = 1.41×1.45×105 Pa = = 2.04×105 Pa. Because it holds also that vef= I/pef, then by substituting numerical values of the intensity and of the effective acoustic pressure we get for the effective acoustic velocity vef = 1.5×104 W.m−2/1.45×105 Pa = 0.1 m.s−1 and for the maximum acoustic velocity given by vmax = 2 .vef we get the value vmax = 1.41×0.1 m.s−1 = 0.14 m.s−1. Finally, since vmax = 2πf.amax and thus amax = vmax/2πf, the maximum amplitude amax = 0.14 m.s−1/(2.π.106 s−1) = 2.22×10−8 m. These calculations demonstrate that substantial pressure changes take place along very short distances. At high intensity, these changes may result in damage of cell membranes or other pathological changes at cellular level. At the boundary of tissue with various acoustic impedances, the reflection and refraction of sound waves are observed. A similar law holds for light waves in optical media: sin θ1 c1 = (5.18) sin θ 2 c2 where θ1 is the angle of incidence, θ2 is the angle of reflection and c1 and c2 are the velocities of sound in the corresponding media. If we denote z1 and z2 the acoustic impedances of these media, then the ratio R of reflected to the incident intensity (at perpendicular incidence to the boundary) is given by 2

z −z  R =  1 2  (5.19)  z1 + z2  The magnitude of this ratio is shown in the following table Tab. 5.1 The values of R The value z2 is R=

>> z1

10 z1

1.1 z1

z1

0.9 z1

0.1 z1

z2 then R = 1, while z1 = z2 yields R = 0. About 50% of incident wave energy is reflected at the boundary between soft tissue and lung and about 30% at the boundary soft tissue and bone. The difference in impedance between air and tissue is the greatest; therefore, a coupling gel is necessary to remove all of the air from between the transducer and the skin to allow ultrasound energy to enter the body. Due to the energy loss, the sound waves are damped in each medium to various extents; the attenuation of intensity in some medium is described by the attenuation coefficient α expressed in dB/cm. Since the absorption of ultrasound is frequency-dependent, the attenuation coefficient is a function of frequency and its value is roughly proportional to the frequency f. Therefore, the quantity α /f is nearly constant for biological tissues: For soft tissue, at frequencies ranging from 0.1 to 10 MHz, it is from 0.2 to 1 dB.cm−1.MHz−1, whereas for lungs containing a higher amount of gas it is from 20 to 50 dB.cm−1.MHz−1. The absorption of ultrasound in gas media is much higher than that in liquid ones. The layer of air and water that decreases the intensity of ultrasound at various frequencies to 50% of its initial value is shown in the following table. 138

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organ

organ

Tab. 5.2 Thickness of layers of air and water that absorbs half of the incident intensity at various frequencies Frequency

10 kHz

100 kHz

0,5 MHz

1 MHz

Air

220 m

220 cm

4.8 cm

2.2 cm

Water

400 km

4 km

100 m

40 m

The transducers emit the ultrasound beam. We distinguish basically 3 types of probes according to the shape of the beam: sector probe, convex probe and linear probe (Fig. 5.12). A

B

C

Figure 5.12: Ultrasound transducers. They consist of many piezoelectric crystals, each of them produces an ultrasound wave. The summation of all waves generated by the piezoelectric crystals forms the ultrasound beam. According to the shape of the beam we distinguish a sector probe (A), convex probe (B) and linear probe (C).

We use various modes for diagnostic ultrasound imaging which provides different information. The basic are the Doppler mode, A-mode, B-mode, C-mode and M-mode. Doppler mode (Color Doppler, Continuous Doppler, Pulsed wave Doppler etc.) makes use of the Doppler effect in measuring and visualizing blood flow (see section 5.1.2). Adding colors to the image demonstrates the direction and rate of blood flow more clearly. A-mode (Amplitude mode) is the simplest type, actually the basis for other modes. The principle is that a single transducer scans a line through the body with the echoes (reflected sound waves) plotted on screen as a function of depth (Fig. 5.13). The created image is a one-dimensional presentation of a reflected sound wave in which echo amplitude is displayed along the vertical axis and echo delay (depth) along the horizontal axis. B-mode is the most common type of ultrasound imaging. It is a two-dimensional diagnostic ultrasound presentation of echo producing interfaces. Transducers simultaneously scans a plane through the body that can be viewed as a two-dimensional image on screen (Fig. 5.13). Amplitude of the echo intensity is represented by the brightness of the spot in the monitor. The position of the echo is determined by the angular position of the transducer and the transit time of the acoustical pulse and its echo. Fig. 5.14 shows example of B-mode ultrasound images. 139

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transducer signal intensity

1

1

2

2

3

3

1 position / time 2 3

beam

1 2 3

position

Figure 5.13: Creation of image in ultrasound. Up: object; Middle: A-mode; Bellow: B-mode. (transducer, signal intensity, position/time, beams). Position of amplitudes (A-mode) or pixels (B-mode) corresponds to time when the reflected echo is detected. Amplitude (A-mode) or pixel intensity (B-mode) correspond to amount of reflected sound energy. In this example a linear probe is used. In case of acoustic wave 3, all energy is reflected from high dense tissue (square). Behind this structure (usually bony structures such as ribs) is anechoic region called Acoustic shadows. Acoustic shadows provide no ultrasound information, therefore no structure (circle behind the square) is detected in that region.

a)

b)

c)

Figure 5.14: B-mode ultrasound images. (a) Healthy rat kidney, (b) Transplanted liver graft in child – VP marks hepatic portal vein, (c) Fetus at 13th week of gestation. These images show typical applications of diagnostic imaging – in Obstetrics and Gynecology and imaging of soft tissue (kidney, liver, heart, tendons, muscles, etc.).

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In C-mode, the image is formed in a plane normal to a B-mode image. A gate that selects data from a specific depth from an A-mode line is used; then the transducer is moved in the 2D plane to sample the entire region at this fixed depth. M-mode stands for motion. Reflected energy is shown as areas of brightness traced from left to right on screen with time on the x-axis. It is often used as an adjunct to B-mode. In this mode ultrasound pulses are emitted in quick succession – each time, either an A-mode or B-mode image is taken. Over time, this is analogous to recording a video in ultrasound. As the organ boundaries that produce reflections move relatively to the probe, this can be used to determine the velocity of specific organ structures. The ultrasound image can be impaired by artifacts. The most important are acoustic shadow and reverberation echoes. An acoustic shadow is an area through which sound waves fail to propagate, due to topographical obstructions (ribs etc.). Reverberation occurs when sound encounters two highly reflective layers. The sound is bounced back and forth between the two layers before traveling back. The probe will detect a prolonged traveling time and assume a longer traveling distance and display additional ‘reverberated’ images in a deeper tissue layer. Ultrasound imaging is very cheap, real-time, radiation-free, which extends the possibilities for diagnosis and treatment. The choice of transducer probes depends on frequency (compromising between spatial resolution and depth of scan) and footprint (determining acoustic window size). A key disadvantage of ultrasonography is that it is operator-dependent, thereby limiting retrospective analysis of previously acquired images.

5.3.3 Effect of ultrasound on tissue The effects of ultrasound are: mechanical (cavitations), thermal (increase of temperature within regions where ultrasound is absorbed), physically-chemical (coagulation), chemical and electrochemical (decomposition of some high molecular compounds, polymerisation) and finally biological which is the combination of the previously mentioned effects (structural changes, changes of membrane permeability and of conductivity of nerves, altered pH values in tissue, analgesic or spasmolytic effects, an increased metabolic exchange, etc.). The biological effects of ultrasound depend mainly on its intensity. The intensities up to 1.5 W.m−2 exert predominantly reversible bio-positive effects whereas those over 3 W.m−2 may result in irreversible morphological changes (i.e. rupture of cell membrane, decomposition of cell nuclei, thermal coagulation of proteins, etc.).

5.4 SHOCK WAVES Shock waves are mechanical waves, which may result from a sudden release of mechanical, electrical, chemical, or nuclear energy in a limited space. They are defined as a sharp discontinuity through which a sudden change in pressure, density, temperature, entropy, and velocity occurs. Shock waves may propagate in a manner different from that of ordinary acoustic waves. They are produced, in general, in a medium when the source moves with a velocity higher than the velocity of sound in the given medium. These waves have the form 141

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of a cone with the source located at its top. Shock waves share properties with conventional ultrasound; however they differ significantly. Ultrasound is characterized by alternate compressions. It consists of a sinusoidal wave or modulated pulse train, having a defined frequenTlak cy. Shock waves are high-energy waves, consisting of a single high-pressure peak with a steep onset and a gradual decline into a pressure trough (Fig. 5.15). At certain circumstances, these waves are applied to destruction of stones in kidneys or in bile bladder (lithotripsy) as is p+ shown if Fig 5.16. Tlak t+

p–

Čas

p–

Čas

t

–

p+

t+

t

–

Figure 5.15: Shock wave. p+ reaches up to 100 MPa p− about 10 MPa. The rise in positive pressure wave is within 100 ns.

water filled cushion

F2

ellipsoidal reflector

F1

wave. Principal of Extracorporeal Figure 5.16: Application of the shockwater shock wave lithotripsy treatF2 ment. For that, an electrohydraulic extracorporeal lithotripter is used; F1 and F2 demonstrate focal filled cushion points. ellipsoidal reflector

F1

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6.

OPTICS

Optics is a branch of physics that deals with the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Light is electromagnetic radiation and contains electric and magnetic field components, which oscillate in phases that are both perpendicular to each other and to the direction of energy propagation. Electromagnetic waves span an immense range of frequencies – from very long wavelength radio waves with frequencies at around 10 Hz to extremely high energy γ rays from space with frequencies at around 1023 Hz (Fig. 6.1). Visible light is electromagnetic radiation that is visible to the human eye. It is also responsible for our sense of sight at a wavelength range of approximately 380 to 760 nm. An approximate range of wavelengths is associated with each colour: violet (400–450 nm), blue (450–520 nm), green (520–560 nm), yellow (560–600 nm), orange (600–625 nm) and red (625–700 nm). As electrons undergo transitions between energy levels in an atom, light is produced at well-defined wavelengths. Light covering a continuous range of wavelengths is produced by the random accelerations of electrons in hot bodies. Our sense of sight and the processes of biosynthesis in plants have evolved within the range of the wavelengths of sunlight that our atmosphere does not absorb, which is between 300–1100 nm. The ultraviolet (UV) region extends from 400 to approximately 10 nm. It plays a role in the production of vitamin D in the skin and leads to tanning. In large doses, it can kill bacteria and can induce cancer in humans. Glass absorbs UV radiation and can provide some protection against sunrays. If the ozone in our atmosphere did not absorb UV radiation below 300 nm, there would be a large number of cell mutations. The infrared (IR) region starts at 700 nm and extends to approximately 1 mm. IR radiation is associated with the vibration and rotation of molecules, which we perceive as heat. Because of its ability to scan minute temperature variations in the human body, IR radiation is used in the early detection of tumours, which are warmer than surrounding tissue. Snakes and “night vision” instruments can detect IR radiation emitted by the warm bodies of animals.

6.1 PROPAGATION OF LIGHT Light consists of transverse electromagnetic waves. The propagation speed of transverse electromagnetic waves in a vacuum does not depend on its wavelength or frequency and is given by 143

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wavelength λ [m]

frequency [Hz]

10−15 10

−14

1023 γ-rays

1022

10

1021

10

1010

−13 −12

10

−11

10

−10

X-rays

10

−9

10

−8

1019 1018 1017

ultraviolet light

1016

10−7

1015

10−6

1014

10−5

infrared light

1013

−4

10

1012

10

−3

1011

10

1010

−1

10

109

1

108

−2

10 102

radiowaves

107 106

103

105

10

104

10

103

10

102

4 5 6

visible light

Figure 6.1: Spectrum of electromagnetic waves.

c=

1 (6.1) µ0ε 0

144

lní hustota vyzařování [Wm]

where μ0 is the magnetic permeability constant of the vacuum, μ0 = 4π.10−7 H.m−1 and ε0 is the permittivity constant of the vacuum, ε0 = 8.85.10−12 F.m−1. If these values are inserted into the above equation, the approximate result obtained is c = 3.108 m.s−1. The intensity vectors of electric field E and magnetic field B are in-phase and, in the case of a plane electromagnetic wave, perpendicular to each other and to the direction of propagation. The magnitudes of the fields are related by

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E = cB (6.2) The propagation of electromagnetic waves does not require a medium. However, when the wave travels through a substance composed of atoms and molecules, the fields do interact with charges in the atoms. The strength of the interaction is related to permittivity ε and permeability μ of the substance. Therefore, the speed of light in a medium is reduced from c to the value 1/ µε . Light transports energy and momentum. Energy flow is perpendicular to both vectors E and B. The instantaneous power that crosses a unit area normal to the direction of propagation is the intensity. Its value is determined by the magnitude of Poynting vector S, defined as the vector product of vectors E and B, i.e. S=

[ E × B] (6.3) µ0

In an electromagnetic wave, the magnitudes of E and B fluctuate over time and can be demonstrated according to the harmonic function of time, e.g. as E = E0sin(ωt) and B = B0sin(ωt), respectively, where ω = 2πf. Similarly, the magnitude of S that follows from eq. (6.3) as the vector product of E and B fluctuates as well. Since the average of sin2(ωt) over one period equals ½, then using eq. (6.2) average intensity Sav is given by S av =

E0 B0 E2 = 0 (6.4) 2 µ0 2 µ0 c

The intensity of light is measured in W.m−2. It is the average power incident per unit area normal to the direction of propagation. In an electromagnetic wave, the energy density of the electric field (uE = ε0E2/2) equals that of the magnetic field (uB = B2/(2μ0). Therefore, the portion of energy transported by the electric field equals that transported by the magnetic field. An electromagnetic wave transports linear momentum and exerts radiation pressure (force/area). Radiation pressure equals energy density (N.m−2 = J.m−3). In the case of the perpendicular incidence of light on a perfectly reflecting surface, surface pressure is doubled due to the transport of linear momentum alone.

6.2 RAY OPTICS In a homogeneous medium, the propagation of light is characterised by a straight line. A ray is equivalent to a very narrow beam of light and indicates the path along which the energy of the wave travels. Rays are drawn perpendicular to the wave front. Geometrical optics pertains to the behaviour of straight-line rays at the interfaces between two media using simple geometrical constructions. There are many optical effects caused by refraction, which is the bending of rays at the boundary between two media. The directions of the incident ray and the refracted ray are specified by angle of incidence θ1 and angle of refraction θ2, both measured in the context of the normal to the boundary. It holds that 145

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sin θ1 v1 = (6.5) sin θ 2 v2 According to this equation, θ1 > θ2 if speed of propagation v1 > v2. Therefore, the ray bends toward the normal when it enters a medium in which the wave velocity is lower. The above equation is usually expressed in terms of the refractive index of each medium. The refractive index, n, of a medium is defined as the ratio of the speed of light in vacuum c to speed v in the medium, c n = (6.6) v Table 1 shows refractive indices of some common substances. Thus, the equation (6.5) can be rewritten by using equation (6.6) in the form of Snell’s law: n1 sin θ1 = n2 sin θ 2 (6.7) When n2 > n1, it follows that θ2< θ1, i.e. on entering a medium with a higher refractive index the ray bends toward the normal (Fig. 6.2). Table 6.1 Refractive indices n for various media. n

Material Vacuum

1

Air

1.00028

Water

1.33333

Ice

1.31

Glass

1.60

Diamond

2.417

Fast Medium

Slow Medium

n2

Smaller index of refreaction

n1 φ1

φ2 The amount of bending depends on the change in index of refraction

Normal line

Figure 6.2: Snell’s law relates the indices of refraction n of two media to the directions of propagation in terms of angles to the normal. An example of bending toward the normal is shown (n1 < n2).

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The frequency of the wave, which is determined by the source, is the same on both sides of the boundary. If the wavelength in the vacuum is λ0, then v = f.λn and c = f.λ0. Therefore, using equation (6.6)

λn =

λ0 (6.8) n

Equation 6.8 means that the refractive index is either a function of the frequency or the wavelength (Fig. 6.3). Since n > 1 for each medium, the wavelength in the medium is shorter than the medium in the vacuum.

f1

f2

f

Figure 6.3: Refractive index n as a function of frequency f or wavelength λ.

A ray approaching the boundary from the medium of a higher refractive index is refracted away from the normal (see Fig. 6.2). However, at a given critical angle of incidence, αc, the refracted ray emerges parallel to the boundary. For any angle of incidence greater than αc, the light is completely reflected back into the medium of the higher refractive index. This is called total internal reflection (Fig. 6.4). By setting θ1 = αc and θ2 = 90° it follows from equation (6.7) that n1 sin α c = n2 (6.9) For example, when the medium with a lower refractive index is air, then n2 = 1. When the medium with a higher refractive index is glass (n = 1.5), then the value for the critical angle of incidence is αc = 42°. 147

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n2

low index material

αc

n2

n

low index 1 high index material material

Light source

αc n

1 Figure 6.4: Total internal reflection. Light striking a medium with a lower refractive index (n1 > n2) can index be totally reflected. Although the ray normal to the surface is nothigh bent, part of the normal ray is reflected. material On the other side, light incident at any angle higher then critical angle αc is completely reflected. Light striking the surface between 0° and αc is partly reflected, the amount of which depends on the angle.

Light source

Total internal refraction has several applications in medicine. One of them is endoscopy, which enables internal organs to be viewed. Glass and plastics can be made into thin fibres between 10 μm and 50 μm in thickness. When a fibre is coated with a material of lower refractive index (n2) than glass (n1), then a ray entering one end of a fibre at angle α, given that the condition sin α < n12 − n22 holds, will undergo total internal reflection and travel down the fibre without much loss through the sides. Optical fibres have nowadays replaced wires for land-based communications. In endoscopy (Fig. 6.5), light from a bright lamp is directed into an endoscope tube. The light bounces along the walls of the fibre-optic endoscope tube into the body cavity of the patient. nLight 1 reflected off the body part travels back up a second fibre-optic tube, bouncing off the glass walls of the endoscope as it travels. The light then shines up into the physician’s eyepiece (camera). n2

n2 < n1

n1 n2 n2 < n1

Figure 6.5: The principle of endoscopy. An endoscope uses total internal reflection, whereby n1 > n2.

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6.3 DISPERSION OF LIGHT Nearly white light from the sun consists of a continuous distribution of wavelengths. The wavelengths (spectral colours) of white light can be separated using a dispersive medium such as a prism or diffraction grating. While the speed of light of any wavelength in a vacuum is constant, it is a function of the wavelength in any medium and, therefore, the refractive index is a function of the wavelength as well. The rate of change of the refractive index with respect to the wavelength is called dispersion. Normal dispersion corresponds to the condition dn/dλ < 0. A particular small range of wavelengths is perceived as a single colour. Red, which has the longest wavelength, has a lower refractive index than violet, which has the shortest wavelength. When a beam of white light, which is a mixture of all visible wavelengths, is incident at an angle to a glass surface, it is dispersed into a spectrum of colours. If the glass consists of two parallel surfaces, the rays that emerge from the second surface will be parallel to the incident ray. The nonparallel sides of a triangular prism serve to increase the angular separation between the colours. Each colour has its own angle of deviation relative to the original ray. This effect is applied in prism spectroscopes (see Fig. 6.6). A diverging beam of white light is emitted from a source. It passes through a narrow aperture (S) to the collimating lens (K), which ensures the rays are parallel. The rays are refracted by prism P, where they are decomposed before passing the objective lens and finally forming the spectrum on indicator I. The indicator may be an eyepiece (aided by the eye) in the case of spectroscopy, a photographic plate in the case of spectrography or a photomultiplier in the case of quantometry.

O K

I P

s source

Figure 6.6: Spectroscope scheme.

6.4 LIGHT SCATTERING Light scattering is a phenomenon in which light changes its direction of propagation. It is caused by the interaction of light with matter. As a result of this interaction, a vibration of electrons in the matter causes polarisation, which in turn generates secondary light waves. In optically homogeneous media these secondary waves cancel each other; whereas in optically inhomogeneous media, which are always > 0 K, they do not. There are three types of scattering: elastic scattering, dynamic scattering (quasielastic) and inelastic scattering. Linear

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Circular

Elliptical

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In the case of elastic scattering, the wavelength of scattered light is the same as the wavelength of incident light. The energy (and therefore the wavelength) of the incident photon is conserved; only its direction changes. This means that the scattered photons have the same energy as the incident photons. In this case, the scattering intensity is proportional to the fourth power of the reciprocal wavelength of the incident photon. An example of elastic scattering is Rayleigh scattering where the wavelength of incident light λ >> the size of the molecule (uncharged). Mie scattering occurs when the wavelength of incident light λ ≤ the size of the molecule. In the case of dynamic scattering a change of frequencies is observed, corresponding to the relative movement of molecules. Inelastic scattering is associated with a change in the energy of a molecule due to a transition to another (usually higher) energy level. The frequency of the photon shifts in favour of the colours red or blue. A red shift is observed when part of the energy of the photon is transferred to the interacting matter, where it adds to its internal energy. A blue shift is observed when the internal energy of the matter is transferred to the photon.

6.4.1 Rayleigh scattering When light passes through diluted gas, elastic scattering of light is observed due to its interaction with very small particles, e.g. molecules. This type of scattering may occur when wavelength λ >> a (where a is the size of the molecule). The intensity of the electric field of the incident electromagnetic wave induces an oscillating magnetic moment, which emits an electromagnetic wave of the same frequency and wavelength. This effect is called elastic scattering. Although the intensity of light scattered in all directions is very low, it can be used to obtain information on scattering objects. Given a ratio of the intensity of scattered light Is to incident intensity I0, it holds that Is M2 = k 4 (6.10) I0 λ where k is a constant, M is the molar mass and λ is the wavelength of incident light. The value of k depends on both the concentration of particles and the measurement angle. Therefore, using monochromatic light, the molar mass can be estimated if the concentration is known. The above equation demonstrates the strong dependence of scattered intensity on wavelength. In the case of natural illumination, for example, the blue component is scattered 16 times more strongly than red light and at twice the wavelength. The shorter the wavelength, the higher the intensity of scattered light. This explains the bluish colour of the sky and the reddish colour of sunsets (where there is minimal scattering).

6.4.2 Raman scattering Like Rayleigh scattering, Raman scattering depends upon the polarisability of molecules. For polarisable molecules, the incident photon energy can excite vibrational modes in molecules. Therefore, spectral lines of shorter or longer wavelengths may appear in a spectrum 150

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of scattered light compared to incident light. Incident photons (of energy hf0) interact with the scattering molecules, which changes their vibrational and rotational energy. Two types of energy change exist: The vibrational and rotational energy of the molecules increase compared with the original state (in this case the energy of the scattered photons will be smaller than incident energy hf0) or the excited molecules pass into a vibrational or rotational state of lower energy (where the energy of scattered photons is larger than hf0). Both energy changes are influenced by the vibrational and rotational energy changes of the molecule. As a result, new lines appear on both sides of the hf0 line in the spectrum of scattered light. The probability of Raman scattering is very small and thus the intensity of the spectral lines is very weak. They cannot be detected by the naked eye and can only be viewed with a sensitive photodetector or using photographic plates after prolonged exposure. Laser radiation can be used to increase light intensity, while photomultipliers can be used to detect radiation.

6.5 ABSORPTION OF LIGHT When light passes through a medium, some part of its energy is absorbed. When incident intensity I0 passes through a medium of thickness d, then the intensity which has passed, I, is given by I = I 0 .e −α d (6.11) where α is the coefficient of absorption. Its value for visible light in air is around 10−3 m−1, for glass l m−1 and for metals 106 m−1. It follows from equation (6.9) that the intensity decreases to 1/e = 1/2.7 = 0.36, i.e. to 36% of its initial value when passing through 103 m of air, 1 m of glass or 10−6 m of metal. The absorption coefficient is a function of the wavelength. Therefore, absorption is a selective process. For example, water has a great capacity to absorb infrared light; normal glass is transparent in visible light but absorbs infrared and completely absorbs ultraviolet light. When light passes through a solution of a concentration cm (mol.m−3), then the absorption coefficient is proportional to the concentration, α = ε´cm. Thus, after substitution and rearrangement according to equation (6.9) ln

I0 = ε ′cm d (6.12) I

or using the decade logarithm log

I0 = ε cm d (6.13) I

where ε (mol−1.m2) is the molar extinction coefficient. Its value depends on the molecule types of the dissolved substance and solvent. It is also a function of the wavelength of the light applied. The quantity defined by the relation 151

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E = ε cm d = log

I0 I

O

(6.14)

K

I

is called extinction (absorbance) where equation (6.14) corresponds to the Beer-Lambert law. This quantity is applied in absorption photometry for concentration measurements. P s

6.6 POLARISATION OF LIGHT source In a beam of natural light, the orientation of vectors E and B is distributed uniformly in all planes that intersect in a straight line corresponding to the direction of propagation of the wave. The light is generally unpolarised, all planes of propagation being equally probable. When an oscillation of vectors E and B occurs in two perpendicular planes only, the light is linearly polarised. Polarisation pertains to the orientation of a wave’s electric field at a point in space over one period of the oscillation. The basic types of polarisation are linear, circular and elliptical (Fig. 6.7).

Linear

Circular

Elliptical

Figure 6.7: Polarisation of light. Only the electric field is shown (the transverse electric field wave is accompanied by a magnetic field).

Linear polarisation is an electromagnetic field propagating in one plane only. This occurs as a result of reflection, refraction (in dielectrics) or birefringence. Polarisation by reflection is based on the premise that reflection coefficients are different for waves parallel and perpendicular to the plane of incidence. At a certain angle of incidence α, the reflected ray is linearly polarised. At this angle of incidence, the reflected and refracted rays are perpendicular, i.e. α + β = 90°, where β is the angle of refraction. Since α + β = 90°, then sin β = cos α and according to Snell’s law (equation 6.7) tgα =

n2 n1

(6.15)

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Angle α is called Brewster’s angle. At this angle of incidence, the reflected rays are linearly polarised (Fig. 6.8). For air, n1 = l; for glass, n2 = 1.5. Therefore, tg α = 1.5 and α = 57°. II Brewster angle

T

α

α

90°

β II

Brewster angle

α

TT

II

α

90°angle and β is the angle of refraction. Figure 6.8: Polarisation by reflection. α is Brewster’s

β

Light can also be polarised by birefringence, which occursIIwhen a ray of normal light enters a crystal of Iceland spar (calcite). The ordinary rays (O) obey Snell’s law and are absorbed in the walls, whereas the extraordinary rays (E) do not and pass the prism (see Fig. 6.9). Liquids and amorphous solids (such as glass) are isotropic because their properties do not depend on direction. In particular, the Polarization speed ofbylight is the same in all directions. Birefrinbirefringence or double-refraction gence occurs in optically anisotropic crystals. The arrangement of atoms in an anisotropic crystal is such that the speed of light in the given direction Calcite (CaCO3) depends on its state of polarisation. Along an optical axis, the O and E rays are of the sameE speed and consequently of the same refractive index. The index for ray E depends on its direction relative to the optical axis, O 68° reaching a maximum normal to the axis. When an unpolarised beam passes through a calcite =along 1.4864 perpendicular directions. nE crystal, rays O and E are linearly polarised T

nbalsam = 1.526 = 1.6584 nO

Polarization by birefringence or double-refraction

Calcite (CaCO3) E 68°

O nE = 1.4864 nbalsam = 1.526 = 1.6584 nO

Figure 6.9: Polarisation by birefringence. A Nicol prism (shown above) consists of two prisms of calcite cemented with Canada balsam. The ordinary ray (the propagation of light along the optical axis is independent of polarisation, while its electric field is perpendicular at every point to the optical axis) reflects fully off the prism boundary, leaving only the extraordinary ray (the wave with the electric field parallel to the optical axis).

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6.6.1 Polarimetry Polarimetry is an instrumental analytical method that uses the rotation of polarised light by some substances to measure their concentration in a solution. When linearly polarised light is produced by one crystal (polariser) and enters a second (analyser), then only the component of the electric field along its polarisation axis is transmitted (see Fig. 6.10). Light passes when the orientation of both is identical, but cannot pass when the polarisation axis of the analyser turns by 90°. Since the intensity is proportional to the square of the amplitude, the intensity of the transmitted light is given by I = I 0 cos 2 α ,(6.16) where I0 denotes the intensity transmitted at α = 0, i.e. when the polarisation axis is parallel. The equation (6.16) is called Malus’s law. Polaroid glasses have a vertical transmission axis, which means they absorb the horizontal components of reflections from horizontal surfaces (roads, water) and reduce glare. An optically active substance turns the plane of linearly polarised light. In a solution, the rotation angle is directly proportional to the concentration of the optically active substance. Using a polarimeter (see Fig. 6.10), this principle is applied to determine the concentration as follows: A beam of light from a light source first passes through a polarising filter and then through a cuvette containing the analysed sample. It is then analysed in the objective and eyepiece, where its intensity is evaluated. When the cuvette contains only pure water, then the maximum intensity of light (which has passed through the analyser) is observed at a parallel polarisation axis by the polariser or analyser. If the cuvette contains a solution of an optically active substance, the observer must turn the polarisation axis of the analyser to attain the maximum intensity of light. This can be observed in the eyepiece by the angle corresponding to the change in the plane of polarisation caused by the optically active substance. If a beam of linearly polarised light passes through a cuvette of length l containing a solution of an optically active substance at concentration c (g/ml), then the angle of rotation of the plane of polarisation, α, is given by α = [α].l.c, where [α] is the specific rotation of the substance (tabled for each substance at a cuvette length of 10 cm). For instance, its value for glucose is 52.8°ml.g−1.dm−1 in the case of yellow sodium light at a temperature of 20 °C.

Light source

Polarized light Rotated polarized light

Unpolarized light Polarizer (Nicol prism)

α

Degree of rotation measured

Sample tube Analyzer (Nicol prism)

Figure 6.10: Polarised light/polariser/analyser/unpolarised light.

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6.7 QUANTUM OPTICS Atoms and molecules emit radiation energy in the form of energy quanta and photons according to Planck’s equation: E = hf (6.17) where h = 6,626.10−34 J.s (Planck’s constant) and c = 3.108 m.s−1 is the speed of electromagnetic waves in the vacuum. Since λ = cT = c/f, the relation of energy E and wavelength λ is given by E=

by

hc (6.18) λ

The momentum of the photon relative to a wave with frequency f and energy E is described

p=

hf h = (6.19) c λ

The momentum of the photon plays an important role in the inelastic scattering processes of photons on electrons (see part 8.3.3.2). As demonstrated previously, the energy of visible light is of the order eV, the energies of X-rays are of the order 104–105 eV and the energies of γ-rays are of the order MeV or even higher.

6.8 WAVE OPTICS Direct evidence of the wave characteristics of light is the effect characteristic for any type of wave, i.e. interference and diffraction.

6.8.1 Interference of light Interference is observed only in coherent waves. Coherent waves have the same frequency (wavelength) and differ from each other only by a constant phase shift, which does not change over time. Path difference Δδ must be distinguished from phase difference Δφ, which are related by ∆φ =

2π ∆δ (6.20) λ

In general, constructive interference (maximum) occurs if the path difference is an integer multiple of the whole wavelength 155

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Δδ = kλ, k = 1, 2, ….

(6.21)

Destructive interference (minimum) occurs if the path difference is an odd number of half the wavelength

λ ∆δ = (2k + 1). (6.22) 2 If the wavelength in the vacuum is λ0 and the speed is c, then in a medium of refractive index n the wavelength is λ = λ0 /n and the phase speed is v = c/n. Therefore, when one of the waves passes path d1 in a medium of refractive index n1 and the second wave passes path d2 in a medium of refractive index n2, then their phase difference Δφ is given by  n2 d 2 −light d d  n1d1  Light source ∆ϕ = 2π  2 − 1  = 2π Polarized  (6.23) λ0  λ2 λ1    Rotated Degree of rotation polarized light

α

measured

Thus, it is equal to the difference of the optical paths multiplied by 2π/λ0. The difference Unpolarized light is sufficient for a maximum or minimum value. The optical path is the in the optical paths Sample tube product of the refractive index and the geometrical path. Polarizer prism)a thin layer of a medium of refractive index n between two parWhen light passes(Nicol through Analyzer allel planes, the interference of the refracted (Nicol and prism) reflected waves takes place (see Fig. 6.11). Assuming the refractive index of air equals 1, the difference in the optical paths is given by

λ ∆δ = 2d n 2 − sin 2 α + (6.24) 2 where d is the thickness of the layer, α is the angle of incidence and λ is the wavelength of the light applied. When light encounters a medium of a higher refractive index, the reflected wave undergoes a phase change equal to π. Therefore, λ/2 must be added to the right part of the equation since, with respect to equation (6.17), phase change π corresponds to the path change of λ/2.

α

d

Figure 6.11: Interference of light on a thin layer.

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The interference of the reflected and refracted waves will increase the intensity if Δδ = 2k.λ /2 = k.λ, while the resulting intensity will be zero if Δδ = (2k + 1).λ/2. If thickness d and refractive index n are known, the relation (6.21) can be applied for determining the wavelength. These principles are applicable when using an interferometer. When white light passes through a thin layer, a maximum and minimum appear for each colour (due to the dependence of the refractive index on the wavelength), as observed in soap bubbles or oil spots on the surface of water.

6.8.2 Diffraction of light Diffraction of light is the effect of deviation from the direction of propagation. As waves bend around small obstacles, they spread out past small openings (Fig. 6.12). This can be observed in an object of a size comparable to the wavelength of light. Diffraction phenomena S are caused by the interference of light. Diffraction can be studied using slits. When the separation between slits is d, then the path difference between the two outgoing rays at angle θ is Δδ = d.sin θ (see Fig. 6.13). A maximum appears when d.sin θ = kλ and a minimum when d.sin θ = (2k + 1).λ /2, k = 0, 1, 2, ... Integer k is referred to as the order fringe. The diffraction spectrum is observed when white light shines through an optical grating. It differs from the refraction spectrum because the colour red exhibits the maximum deviation. α

S

α

Figure 6.12: Deviation of light by a slit.

Figure 6.13: Diffraction of light. Diffraction is less in the wider slit (left). The narrower aperture in the slit causes greater diffraction (right).

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6.9 LENSES A lens is an optical device of perfect or approximate axial symmetry that transmits and refracts light by converging or diverging light beams. In a converging lens, the central section is thicker than the rim (Fig. 6.14). Such a lens causes rays parallel to the principal axis to focus on a real focal point. In a diverging lens, the central section is thinner than the rim (Fig. 6.14). Such a lens causes a parallel beam to diverge from a virtual focal point. Both lenses have two focal points at an equal distance from the lens. The distance of the focal point from the lens axis is called the focal distance, which depends on the refractive index and the radius of curvature of the lens. The thickness of a thin lens is much less than its diameter. Direction of Light

Converging Lens

F

Diverging Lens

F

Figure 6.14: Converging and diverging lenses. F – focal point.

Lenses can be susceptible to what is known as chromatic aberration. Different colours have different focal points because focal length depends on refraction and the index of refraction. Each colour has its own angle of deviation relative to the original ray, while the index of refraction decreases as the wavelength increases. Blue light travels more slowly in material than red light, resulting in spherical aberration (even a monochromatic parallel beam is not brought to a unique focal point). Chromatic aberration can be corrected with a second (diverging) lens made of glass with different dispersion. Using paraxial rays only decreases the effect of spherical aberration. Ray diagrams are useful for locating images generated by a lens. Any two of the following principal rays are suitable for locating an imageLens produced by a lens (Fig. 6.15): (1) a ray passing through the centre of a lens (not deviating); (2) a ray directed parallel to the axis (passing Object F − focal point through a focal point); (3) a ray directed toward, or away from, a focal point (emerging parallel to the axis). When the object distance is denoted a, the image distance b and the focal distance f, then F the position of an image can be calculated using the thin lens formula: 158

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Image

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1 1 1 + = (6.25) a b f Sign convention: a is positive (real) on the left and negative (virtual) on the right; b is positive on the right and negative on the left; f is positive for a converging lens and negative for a diverging lens (light travels from left to right). Lens Object

F − focal point

F

Image

Figure 6.15: Construction of an image in a converging lens.

Instead of focal distance, optical power D may be used to describe a lens. The optical power is defined as the reciprocal of the focal length in metres, D=

1 (6.26) f

The unit of optical power is 1 dioptre. A converging lens has a positive power while a diverging lens has a negative power. Transverse or linear magnification m is defined as the ratio of image height y1 to object height y0, m=

y1 b = − (6.27) y0 a

The negative sign indicates that the image is inverted if both values a and b have the same sign. In order to see detail on an object, it must be brought as close as possible to the eye. The closest distance one can comfortably bring an object to the eye is called the least distance of distinct vision and is taken to be 25 cm for a normal eye (10 cm for children and greater for elderly people). Thus, 25 cm is used as a convenient reference value. The maximum angle that an object can be subtended is reached when at the least distance of distinct vision. This position corresponds to the maximum resolution since the retinal image is at its largest size. For such an angle, it holds that tan α ≅ α =

y0 (6.28) 0.25 159

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where y0 is in metres (for small angles the function tangent can be approximated by its argument). At this approximation, the angle subtended by the image is β = y1 /b = y0 /a. The angular magnification, M, of a single lens, called a simple magnifier, is defined as the ratio of the angle subtended by the image produced by the lens to the angle subtended when the object itself is at 25 cm, M =

β (6.29) α

On inserting the expressions for α and β, it becomes M =

0.25 (6.30) a

where a is expressed in metres. It is easier for the eye to view an image in infinity, because then the normal eye is relaxed. The object is then at the focal point, i.e. a = f and equation (6.26) becomes M∞ =

0.25 (6.31) f

The shorter the focal distance, the higher the magnification. The highest magnification achieved using a simple magnifier is approximately 40. Using a converging lens, the image emerges as real or virtual, small or enlarged, erected or reversed, according to the position of the object with respect to the focal point. Using a diverging lens, only a virtual, small, erected image is formed. An optical system composed of several lenses may be created. The lenses are located along the optical axis. For an optical system composed of two lenses with optical powers D1 and D2, the total optical power of the system is given by D = D1 + D2 − D1 D2 d (6.32) where d is their distance. Converging and diverging lenses can be combined. A composed optical system converges when the resulting optical power of system D > 0 and diverges when D < 0.

6.9.1 Compound microscope In a compound microscope the first lens is the objective and the second is the eyepiece (see Fig. 6.16). The function of the objective is to place an enlarged image of the object at a point closer to the eyepiece than its focal length fE. The eyepiece then acts as a simple magnifier. The focal length of the objective is of the order of 5 mm. This allows the instrument to be placed close to the object under study. The focal length of the eyepiece is approximately 15 mm. The distance between the focal points of the objective and the eyepiece is called the optical tube length, Δ. The distance between the lenses is d = Δ + fO + fE. If an object is 160

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located just beyond the focal point of the objective (a > fO) then the real, enlarged, inverted image is produced by the objective (b > Δ + fO). This image acts as a real object for the eyepiece. Since it is located between the focal point of the eyepiece and the eyepiece itself, the final virtual image is enlarged and inverted with respect to the object. eyepiece

objective Fob

F’ob

Fe

F’e

image by eyepiece

image by objective

object

Figure 6.16: Diagram of a compound microscope.

The basic parameters of an objective are focal distance fO, magnification MO and numerical aperture A. The magnification of an objective is given by MO = −

fO + ∆ ∆ ≅ − (6.33) fO fO

The negative is inserted because the image is inverted. Numerical aperture A is given by A = n sin ε (6.34) where ε is half of the aperture angle at which the objective lens is seen from the object point on the optical axis and n is the refractive index of the medium between the object and the objective. Since the magnification of the eyepiece is given by equation (6.31), total magnification M, calculated as the product of the magnifications of the objective and the eyepiece, is given by M = MOM E = −

0.25.∆ (6.35) fO f E

The image is a result of interference phenomena. The shortest distance between two points that can be resolved, e, is given by

λ e = 0, 61 (6.36) A 161

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The resolving power of a microscope is then 1/e. The shorter the wavelength, the higher the resolving power of the microscope. Resolving power increases along with the increasing value of the numerical aperture. The resolving power limits the magnification of the microscope. A short straight line of length d0, which is seen from the least distance of distinct vision δ at angle u = d0 /δ, is seen by a microscope at angle u´ = d0Z/δ. The angle must be u´ > 1´ for resolution. It is illogical to increase this angle by increasing the magnification since, at the given wavelength and numerical aperture, no more detail can be resolved due to the diffraction phenomena. In this context, the numerical aperture is an important component. When there is air (n ≅ 1) between the object and objective (dry objective), the highest theoretical value of the numerical aperture equals 1, practically 0.9. The value of the numerical aperture can be increased using the immersion objective, i.e. by filling the space between the cover glass and the objective lens with a liquid of a higher refractive index. The liquid is dropped onto the cover slide and the frontal lens of the objective is immersed in the droplet. For example, the refractive index of cedar oil is 1.51. Since the maximum value of ε is approximately 70° (sin 70° ~ 0.95), the smallest distance still resolved at visible light (λ = 550 nm) is approximately 600 nm for a dry objective and 400 nm using immersion.

6.10 LASER Laser (an acronym for Light Amplification by Stimulated Emission of Radiation) is a source of highly coherent light. Its function is based on the stimulated emission of radiation. Atomic and molecular systems can only exist in discrete energy states (see Chapter 6.1–6.7). The transition from higher state En to lower energy state Em is followed by the emission of a radiation quantum with frequency fnm, which is given by f nm =

En − Em (6.37) h

An atom in a basic state of energy, Em, has the lowest energy and therefore can only absorb electromagnetic radiation. An atom in a higher energy state, En, can transit spontaneously to a lower energy state or as a result of interacting with an external electromagnetic field. Spontaneous transitions are mutually independent. Therefore, spontaneous emission is not coherent. Transitions between energy levels due to an external electromagnetic field are called induced transitions, whereby the energy of the external field must equal the energy difference of the transition levels. In this case, the induced radiation possesses the same frequency, direction and polarisation as the radiation that induced the emission. Energy levels are distributed according to Boltzmann’s law under normal conditions in thermodynamic equilibrium, where Ni = N0 e

−

Ei − E0 kT

(6.38)

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Ni denotes the number of atoms with energy Ei, N0 is the number of atoms in basic energy state E0, k = 1.38.10−23 J.K−1 is Boltzmann’s constant and T is the absolute temperature. Only a small portion of atoms exists in an excited state under normal conditions, i.e. Ni < N0 at T > 0. That is why the quantum system absorbs electromagnetic radiation under normal conditions. The state that corresponds to Ni > N0 (the number of atoms in an excited state greater than the number of atoms in a ground state) is known as inversion. It is also called a state with negative temperature (see equation (6.38), where the exponent of Euler’s number e must be positive if the ratio Ni / N0 > 1). The intensity of an electromagnetic wave passing a medium in a state of inversion increases on account of the energy of the excited atoms. The laser principle refers to the supply of external energy needed for an induced transition to a higher energy level. Electrons are pumped into excited states optically, electrically, chemically, etc. Electrons can be elevated to excited states by energy absorption, but no significant collection of electrons can be accumulated by absorption alone since both spontaneous emission (10−8 s) and stimulated emission bring them back down from the excited energy state. Significantly, most of the excited ions rapidly (10−8 s) transit from an excited state – not to the ground energy level – but to a metastable level. In a metastable state, spontaneous emission is significantly delayed, resulting in the population inversion Ni > N0. In this case, there is a higher probability of interaction with another photon by stimulated emission. As a consequence of that interaction, emission light of the same phase and frequency is produced. To achieve sufficient intensity of the laser beam, the light constrains to a path. In stimulated emission, electrons in the metastable energy level transit to a lower energy level after interaction with an incident photon. This is followed by the emission of a photon of the same energy as that of the incoming photon. The emitted photons have a definite time and phase relation to each other, while the light has a high degree of coherence. The light from a typical laser emerges in an extremely thin beam with very little divergence. The optical cavity of the laser has parallel front and back mirrors, which constrain the final laser beam to a path perpendicular to the mirrors. Light passes back and forth between the mirrors many times in order to gain intensity through stimulated emission of more photons at the same wavelength. The function of the laser can be explained using the example of a ruby laser (see Fig. 6.17). The ruby crystal, doped with Cr3+ ions possessing a wide band of energy (3) over a ground state (1), easily absorbs energy in a wide range. Most of the excited ions rapidly (within 10−8 s) transit to a metastable level (2) and remain at this level for a relatively long time (10−3 s). In this way, many of the excited Cr ions can accumulate at this level, resulting in inversion. When light is of energy 2.6×10−19 J (wavelength 694.3 nm), which corresponds to the energy difference between the levels (1) and (2) passing through the crystal, all excited atoms transit into the ground state simultaneously. This process results in a pulse of coherent light of that wavelength. The intensity of the pulse is inversely proportional to its duration. The power of the emitted light band is approximately 107 W and the intensity of the emitted light is approximately 1013 W.m−2. For comparison, the intensity of sunlight in the high atmosphere is approximately 1.39 kW.m−2. With lasers in pulse operation, a very large power density (intensity) can be obtained on a small surface, resulting in the evaporation of all substances.

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3 S32 2 A13

W31 W21 1

Figure 6.17: Principle of ruby laser. It demonstrates energy levels of quantum transitions. A13 - pumping energy, S32 – radiationless spontaneous transition, W13 – spontaneous emission, W21 – induced emission

Laser light is coherent, monochromatic and collimated. Coherence is secure because different parts of the laser beam are related to each other in-phase. These phase relationships are maintained over a long enough time to allow the interference effects to be seen or recorded photographically. This coherence property is what makes holograms possible. A laser consists of essentially one wavelength (monochromatic light), which is initiated by stimulated emission from one set of atomic energy levels. A laser is collimated because light must pass between the mirrors many times in order to amplify and to be as perpendicular as possible to the mirrors. These beams are very narrow and do not extend to a large degree. The unique properties of the laser require special safety precautions. Even low-power lasers with only a few milliwatts of output power can be hazardous to human eyesight, causing permanent damage in seconds. This happens when a laser beam hits the eye directly or after reflecting from a shiny surface. In medical practice, a highly collimated laser beam can be focused on a microscopic dot of an extremely high energy density. Laser light is used for cutting and cauterising instruments, for photocoagulation of the retina to halt retinal haemorrhaging and for tacking down retinal tears. It can be used after cataract surgery if the supportive membrane surrounding the implanted lens becomes milky. In dentistry, it is used to remove decay within a tooth, to prepare surrounding enamel for fillings, in whitening procedures and to “cure” or harden fillings. It is also used to destroy small-sized tissues, coagulate tissues (e.g. retinal detachment) and heal ulcers.

6.11 OPTICS OF THE HUMAN EYE The eye is the peripheral organ of vision, accounting for approximately 90% of all sensory input. Its physical structure (see Fig. 6.18) enables rays of light from external objects to focus upon the retina. There, nerve impulses are generated and transmitted by fibres of the optic nerve to the visual area in the cortex of the brain. Light enters the lens through the pupil. The size of the pupil is controlled by the iris, which itself changes with the intensity of light that enters the eye. Light is then focused on the retina, which contains sensitive rods and cones. Information is carried by the optic nerve to the 164

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Cross section of human eye Ciliary muscle

Retina

Ciliary fibers

Cornea

Vitreous humor

Fovea centralis

Blind spot

Aqueous humor Iris

Crystalline lens

Optic nerve

Figure 6.18: Cross-section of the human eye/aqueous humour/vitreous humour/ciliary fibres.

brain. The eye is able to focus objects at different distances by varying the focal length of the crystalline lens. This process is called accommodation. It is accomplished by the contraction or relaxation of the ciliary muscles. This tension affects the radii of curvature of the two surfaces of the lens. Because the eye is composed of various media with different indices of refraction, it is difficult to accurately trace the entire path of light. However, the course of light rays through the eye may be followed with sufficient accuracy by means of simplification (see Fig. 6.19). The reduced eye is an idealised model of the optics of the human eye. All refraction is assumed to occur at a single interface between air and the contents of the eye, here assumed to be homogeneous and to have the same index of refraction as water, 1.333. The geometrical axis of the eye is a straight line passing through the centres of the cornea and crystalline lens. The interface corresponding to the surface of the cornea has a radius of 5.5 mm and its centre of curvature is the optical centre or H nodal point N of the system. Nodal point N is the point through which light rays travel undeviated. The retina lies 17 mm posterior to the nodal point and 22.5 mm from the cornea. This is also the principal focal distance of the system, so that distant objects are focused on the retina of the reduced eye at rest. The anterior principal focus, i.e. the F1point at which rays parallel within N the eye converge on F2 emerging light, lies 17 mm in front of the cornea. The anterior and posterior focal distances are different because light travels in air outside the eye and in denser media inside the eye. If the interior focal distance is divided by the index of refraction of the reduced eye, the result equals the anterior focal distance (22.5/1.333 = 17). The image on the retina is inverted and smaller than the object. However, all objects (regardless of size) located at various distances from the eye and seen under the same angle of vision yield identical images on the retina (Fig. 6.20). Two different points can be distinguished by the eye when seen at a vision angle of 1’ (which mm of 0.00035.5 17 to the size of photosensitive corresponds to the angular 17 distance rad) with respect elements (with a cone diameter of approximately 0.005 mm). The average optical power of the eye is + 60 D (± 3.5 D) but the power of the lens is only 20 D. The total optical power of the eye cannot be estimated as a sum of the individual components. 165

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urychlující elektroda

fotokatoda

dynody

anoda výstupní

foton

signál The far point (punctum remotum) is the farthest distance at which an object can be focused by the unaided eye. The closest distance that an object can be brought to the eye and focused without difficulty is the near point (punctum proximum). For a normal eye, the far point is at infinity and the near point is 25 cm. The distance of the near point increases withR age. This condition is called age-related long-sight or presbyopia.

zaostřovací elektroda

R

R

R

R

R

R

R

R

R

C

C

R

H napětí dělič

vyzářený fotoelektron

VN ≈ 1300 V N

F1

17 mm

5.5

C

F2

17

Figure 6.19: The reduced eye. The interface corresponding to the surface of the cornea has a radius of 5.5 mm, while its centre of curvature is the optical centre or nodal point N of the system. The anterior principal focus point F1 lies 17 mm in front of the cornea. The retina lies 17 mm posterior to the nodal point and 22.5 mm from the cornea. This is also the principal focal distance of the system in which the posterior focus point F2 lies.

Objects of different sizes

N α

The same image

Figure 6.20: Image generated in the human eye. N – nodal point, α – angle of vision

6.11.1 Eye defects Emmetropia is the refractive state of the eye in which parallel rays focus on the sensitive layer of the retina without accommodation or in which the far point is infinitely distant. Any deviation from the condition of emmetropia is called ametropia. When a person is farsighted, the eyeball is too short compared to the focal length of the eye. Objects at infinity are focused behind the retina. This eye defect is called hyperopia. In the eye at rest, light reaches the retina before it comes into focus, where a diffusion circle represents each point source of light. The near point for an eye of this type can extend beyond the normal distance of 25 cm. Another kind of farsightedness is presbyopia. As a person gets older, the eye muscles become weaker and the lens hardens. The near point extends beyond 25 cm, which makes nearby objects difficult to focus on and renders the optical power of the eye too weak. A converging lens helps both of these conditions. 166

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When a person is nearsighted, the eyeball is too long compared to the focal length of the relaxed eye. Objects at infinity are focused in front of the retina and thus the optical power of the eye is compromised. This eye defect is called myopia and may be compensated for by a diverging lens. Eye defects can be corrected by either wearing glasses positioned approximately 10 mm in front of the eye or using contact lenses. The principle of correction lies in positioning the object focal point of the corrective lens at the far point of the eye. In such a case, the eye has distinct vision of the object infinitely distanced without accommodation. Myopia is corrected using a diverging lens, and hyperopia using a converging lens (Fig. 6.21). Presbyopia is also corrected using a converging lens.

Figure 6.21: Correction of eye defects. Left: myopia correction. Right: hyperopia correction.

6.11.2 Biophysics of vision The eye contains two systems of receptors – cones and rods. Cones specifically function in daylight when surroundings are brightly illuminated. In these conditions, the pupil constricts and visual acuity is at its best. The fovea centralis contains 4–7 million cones. These photoreceptor cells allow colours to be perceived. There are at least three types of colour-sensitive cones: blue, green and red. They facilitate finer perception of detail and rapid changes in images because their response times to stimuli are faster than those of rods. Rods function for twilight and night vision. Numbering 75–150 million in total, rods are sensitive enough to respond to a single photon of light. As they respond more slowly to light than cones do, the stimuli they receive increase to approximately 100 ms. By a chemical process in the retina, the eye becomes considerably more sensitive to light (dark adaptation), resulting in the dilation of the pupil and the admittance of more photons into the eye. The retina also consists of the fovea centralis for colour and detail vision and the periphery for light and dark vision. The fovea centralis contains only cones. Rods prevail in the periphery of the retina. Before being detected by photoreceptors in the retina, light must pass a few layers of cells (see Fig. 6.22). The retinal layer nearest the choroid is made up of pigmented cells that probably store and produce photochemicals as visual purple. The layer next to the pigment cells contains two types of neurons. The axons of rod-bearing and cone-bearing neurons end upon the dendrites of ganglion cells. The axons of ganglion cells form the optic nerve. Due to this structure and arrangement, only about 10% of the light intensity that enters the eye is used for stimulating photoreceptors. In spite of this complicated structure, the adapted eye has a very high sensitivity for detecting light. 167

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pigment epithelium rods cones

Direction of light

outer limiting membrane Müller cells horizontal cells bipolar cells amacrine cells

ganglion cells nerve fiber layer inner limiting membrane

Figure 6.22: Arrangement of the retina.

Direction of light

The eye is sensitive to a narrow band of wavelengths that make up the visible spectrum (400–750 nm). The wavelengths within this range are not equally effective in stimulating the retina. Moreover, the total intensity of light influences sensitivity. There are basically three pigment epithelium types of colour-sensitive cones in the retina of the human eye, corresponding approximately rods cones to red, green and blue sensitive detectors.Light The “green” (population: 32% sensitive cones) and Rhodopsin Retinene protein outer limiting membrane “red” cones (64%) are mostly packed into the fovea centralis. The+“blue” cones (2%) have Müller cells Darkoutside the fovea. The shapes of the curves are the highest sensitivity and are mostly found cells obtained by horizontal measuring absorption by the cones. The maximum sensitivity in daylight for cells photopic visionbipolar (illumination >102 cd.m−2) is considered 550 nm (green-yellow colour), while in darkness itamacrine shiftscells toward shorter wavelengths (maximum at 505 nm) for scotopic vision (illumination < 10−3 cd.m−2). ganglion The change thatcells takes place in rods and cones, which translates physical energy and light into nerve impulses, consists of a photochemical step. Retinal rods contain a red pigment that nerve fiber layer is bleached by light called visual purple or rhodopsin. Rhodopsin, which has a high molecular mass (270inner 000), ismembrane a conjugated protein, i.e. a protein molecule united to a pigment group limiting (retinene). Rhodopsin is only stable when not exposed to light. After exposure to light, it dissociates into protein and retinene. In darkness, it is reconstituted with the help of vitamin A as demonstrated in Fig. 6.23. The rate of synthesis depends on the intensity of light, which plays an important role in the adaptation processes of the eye.

Light Rhodopsin

Dark

Retinene + protein

Figure 6.23: Rhodopsin: dissociated in light and reconstituted in darkness.

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7.

X-RAY PHYSICS AND  MEDICAL APPLICATION

In 1895 Wilhelm Conrad Rӧntgen discovered X-rays, a significant scientific advancement that would benefit a variety of fields in industry and in medicine. Rӧntgen’s discovery bridged medicine towards the modern era and became an important diagnostic tool in medicine, allowing doctors to see inside the human body for the first time without surgery. In 1897, X-rays were first used on a military battlefield, during the Balkan War, to find bullets and broken bones inside soldiers. Nowadays in medicine, X-rays are used both for diagnostic and therapeutic purposes. The diagnostic application is based on the fact that various tissues absorb X-radiation to various extents. Therapeutic application (treatment of malignant tumours) is based on various sensitivities of cells (young and actively dividing cells are the most sensitive ones).

7.1 GENERAL FEATURES OF X-RAYS X-rays are electromagnetic waves produced whenever high-energy electrons arestopped after striking a target. Their wavelengths lay within the region of 5–120 pm that corresponds to photon energy of about 0.1–200 keV. Due to high photon energy, they possess both a high ionizing capacity as well as penetrating ability through materials. The source of X-radiation is an evacuated X-ray tube containing heated cathode and cooled anode. The cathode, emits electrons into the vacuum and an anode collects the electrons, thus establishing a flow of electrical current, known as the beam, through the tube. A high voltage power source is applied between the cathode and anode to accelerate the electrons. The X-ray spectrum depends on the anode material and the accelerating voltage. The generation of X-ray photons is a result of two phenomena. First, the high-energy electrons are suddenly decelerated by the fields of atomic nuclei. This electromagnetic interaction of accelerated electrons with fields of atomic nuclei forms radiation with a continuous spectrum called bremsstrahlung radiation, or braking radiation. The second type of X-ray radiation is characteristic radiation. In this case, the electrons make transitions between lower atomic energy levels in heavy elements due to their former ionisation. X-rays produced in this way have definite energies just like other line spectra from atomic electrons. The primary effects produced by X-rays in atoms are ionisation and excitation. The remaining effects that of practical importance are: 169

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i) The excitation of luminescence: luminescence of certain materials (e.g. zinc sulphide doped with silver or copper) may be observed in response to X-ray irradiation; ii) The photographic effect: a photographic plate is darkened similarly as in the case of visible light; iii) Ionising effect: electrical conductivity of some materials is increased; iv) Chemical effect: production of hydrogen peroxide in water; v) Biological effects: production of morphological and functional changes in cell. Beside these, the formation of secondary X-radiation (X-ray scattering) that always accompanies propagation of X-rays in certain medium’s is noteworthy.

7.1.1 Production of braking radiation (bremsstrahlung) Electrons that escape from a heated X-ray cathode tube and are accelerated by the accelerating voltage U (volt) strike the anode with energy E = eU (see Fig. 7.1). The kinetic energy of decelerating electrons is partially transformed into electromagnetic waves of energy in the field of atomic nuclei (with positive electric charge). The highest possible photon energy is then Emax = eU = hf max (7.1) where h is the Planck constant and fmax is the frequency of radiation with the highest energy. Since λ = c/f, the maximum frequency fmax corresponds to the shortest wavelength λmin, given by

λmin =

hc 1234.6 (nm) (7.2) = eU U

where U is in the unit volts. The shortest wavelength (if all electron energy is converted to radiation energy) depends on voltage. CB W A C

TF OF

Figure 7.1: Scheme of X-ray tube. C – cathode, A – anode, W – window, OF – optical focus, TF – thermal focus, CB – direction of central beam.

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This interaction occurs at various distances from the atomic nucleus. Therefore, its strength is different and electromagnetic waves of various wavelengths are produced resulting in a continuous X-ray spectrum (see Fig. 7.2). The peak of this spectrum is shifted to the left (shorter wavelengths) with increasing accelerating voltage. At a constant voltage, the most frequently occurring λ´ = 1.3 λmin. The intensity of anode current does not influence the shape of this spectrum practically. The emitted power P of bremsstrahlung is proportional to the squared voltage on the tube, intensity I of the electron current, and atomic number Z of the target (anode) material P = kU 2 IZ (7.3) If voltage is given in V, current in A, and the power in W, the value of the proportionality factor k is about 10−9. On the other hand, it should be pointed out that only about 1% of energy transported by electrons into anode is transformed into electromagnetic waves while 99% is transformed into thermal energy (therefore cooling of anode by water, air, oil, is required or rotating anode is applied). Since the invested electric power is given by product UI, the efficiency η of the X-ray tube as a radiation source may be calculated by η = kUZ(7.4) From this relationship, it is evident that the higher the atomic number of anode material, the higher the efficiency of bremsstrahlung production. However, due to the production of heat, a high melting point of anode material is also required. 50 kV

relative intensity of radiation

12 10 40 kV 8 35 kV

6 4

30 kV 25 kV

2

20 kV 0.03

0.05

0.07

0.09 λ [nm]

Figure 7.2: Energy spectrum of X-rays generated at various voltage.

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From the above observations, the following two conclusions may be derived: a) Accelerating voltage (kV) may control penetration ability of X-radiation. The higher this voltage, the higher X-ray energy and thus, higher penetration ability (short wave length) and more homogeneous X-radiation is produced at its higher total intensity. b) The power of X-rays produced in an X-ray tube is directly proportional to the intensity of the anode current (mA), i.e. to the heating of cathode and to the squared voltage. If it is desired to change the intensity without varying the energy of radiation, only the current should be changed (by variation of the cathode filament heating).

7.1.2 Production of characteristic X-rays If the energy of accelerated electrons that strike the anode in X-ray tube is higher than the binding energy of electrons in the electron shell of the nucleus (in the case of U about 70 keV), ionisation of inner electron shells may be expected. The subsequent deexcitation and electron transition result in production of the line X-ray spectrum since electron energy levels are well defined. Thus K-, L-, and other line series are well defined for the given atom. The increasing atomic number Z of the target material results in the shift of these line spectra to shorter wavelengths. Both types of spectra are produced simultaneously in an X-ray tube at high accelerating voltages and superposition of both continuous and line spectrum occurs (Fig 7.3). Kα

I (λ) (relative)

3

Characteristic radiation

2 Kβ 1

Braking radiation

V = 35 kV 0

0

0.2

0.4 λM

0.6

0.8

1.0

1.2 λ, A

Figure 7.3: Braking and characteristic radiation. Both continuous and line spectrum are simultaneously produced.

7.1.3 The attenuation of X-radiation The X-ray image reflects absorption of X-ray radiation in matter (tissue). Absorption is characterized by average linear attenuation coefficient between X-ray source and the detector. Attenuation is given by absorption of X-ray radiation in the matter (tissue) – their mutual 172

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interaction. In case of bigger atoms, there is higher energy level difference between orbitals and therefore there is higher probability of absorption of X-ray photons. Because soft tissue mostly consists of small atoms, it shows lower absorption of X-ray. If monochromatic, parallel X-ray beam propagates in some medium, its intensity I decreases according to the law similar to that for some other photon radiation, i.e. I = I 0 e− µ x = I 0 e

−

0.693 x D

(7.5)

where μ is the linear attenuation coefficient, D is the half-thickness and x is the path passed in an absorbing medium. Since μ depends on the density ρ of absorbing material, the mass-attenuation coefficient μm is frequently used and it is defined by

µm =

µ (7.6) ρ

Thus, while μ is given in m−1, μm is given in m2.kg−1since ρ is given in kg.m−3. Two following processes cause the attenuation of the intensity of the X-ray beam: the photoelectric effect and the Compton scattering. 7.1.3.1 The photoelectric effect The photoelectric effect (photoeffect) consists of an interaction between a photon with energy hf with one of the electrons bound to the atom by transferring its whole energy to this electron, which is discussed in details in chapter 1, section 1.6.1. Thus, this electron could be ejected from the atom and the following energy balance holds: 1 hf = Eb + mv 2(7.7) 2 where Eb denotes the binding energy (work necessary to raise the electron from some inner shell to the atomic surface (of order magnitude of 5–100 keV for K-shells) and the second term denotes its original kinetic energy after its release from the atom. After leaving the atom, this electron induces ionisation and excitation until its excess energy is lost. Therefore, X-rays do not cause excitation and ionisation directly; rather the high-energy electrons produced by X-rays do this. This process is of course accompanied by the emission of characteristic X-radiation of absorbing material. Probability of this type of interaction depends strongly on the atomic number Z of target nucleus and on the incident photon energy. Therefore, the linear attenuation coefficient due to the photoeffect, τ, is given by

τ = k1 ρ E −3 Z 4 = k2 ρλ 3 Z 4(7.8) where k1 and k2 are proportionality constants, E – photon energy, λ – wavelength. If a complex material is considered, the effective atomic number Zef must be used instead of Z. 173

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It can be calculated from material composition by using the following formula: Z ef =

3

p1 Z14 + ... + pn Z n4 (7.9) p1 Z1 + ... + pn Z n

where pi is the relative amount of an element of the atomic number Zi. Soft tissue (containing mostly H, C, N, and O atoms) possesses Zef value of about 7.6 whereas Zef of bones (containing relatively high amounts of Ca and P atoms) equals of about 13.8. Thus, the attenuation in bones due to the photoeffect is much higher than that in soft tissue. 7.1.3.2 Compton scattering. During this interaction, only a part of the total photon energy hf is transferred to a free electron and scattered photon moves in a changed direction with lower energy hf´ < hf. The photon energy decrease does not depend on energy but only on the scattering angle θ that ranges between 0 and π. The original wavelength λ changes to λ´ > λ. The increase of the wavelength of scattered photon Δλ = λ´ – λ is given by ∆λ =

h (1 − cos θ )(7.10) me c

where me is mass of electron and c is the velocity of electromagnetic waves in vacuum. The expression h/mec is called the Compton wavelength of electron. This type of interaction does not depend on the atomic number of absorber and the probability of its occurrence depends on incident photon energy (the highest attenuation coefficient σ belong to energy 0.5–5 MeV). In spite of the fact that X-ray photons possess lower energy, this interaction plays an important role in the quality of X-ray contrast. Total linear (mass) attenuation coefficient is then a sum of attenuation coefficients for the two above-mentioned effects, μ = τ + σ. Compton scattering is also discussed in details in chapter 8, section 8.3.3.2.

7.1.4 X-ray contrast The intensity of X-ray beam passing through a patient decreases to various extents as a function of attenuation due to the above-mentioned effects. The contrast Cr of traditional X-ray image (shadow image) resulting from different X-ray absorption in various tissue (of various density and effective atomic number) is defined by Cr = ln

I2 (7.11) I1

where I1 and I2 are X-ray intensities coming to two neighbouring areas of the screen or photographic plate whose contrast is evaluated (or the corresponding brightness). The contrast between bone and soft tissue decreases with increasing accelerating voltage installed on the X-ray tube, since the values of linear attenuation coefficients of a bone or 174

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2

[kV] soft tissue decrease with increasing incident photon energy. In order to distinguish tissue with similar absorbing properties, the contrast materials (positive or negative) have to be applied to the patient. Calculating the contrast of areas corresponding to the intensities I1 and I2 passing through an object with linear absorption coefficient μ and containing a region with a different attenuation coefficient μ0 (see Fig. 7.4) by using the above equation for X-ray beam attenuation, we get

I1 = I 0 e

−[ µ x ]

I2 = I0e

(7.12)

−[ µ ( x − ∆x ) + µ0 ∆x ]

(7.13)

and the contrast will be given, according to equation (7.11), by Cr = ln(

I e − µ x e µΛx e − µ0 ∆x I2 ) = ln 0 = ln e − ( µ0 − µ ) ∆x = −( µ0 − µ )∆xx (7.14) I1 I 0 e− µ x X-ray beam

x

Δx

μ0

μ

Screen I2 Figure 7.4: Scheme for evaluation of X-ray contrast.

I1

I0

The following conclusions can be derived: a) The contrast is negative, i.e. the presence of more absorbing material with respect to the surrounding tissue (μ0 > μ) results in lower X-ray beam intensity on the screen. b) The contrast does not depend on the thickness of irradiated object. c) For empty space with the thickness Δx (i.e. μ0 = 0), the resulting contrast will be positive, i.e. + μΔx, as it is seen from equation (7.14). d) If some material with a different attenuation coefficient, μ0, will be present in the original absorber, the contrast will be proportional to the absolute value of the difference of linear absorption coefficients, |μ0 − μ|. 175

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It is seen in Fig. 7.5 that the value of the mass absorption coefficient (μ/ρ) for bones is higher than that for soft tissue. For this reason, the contrast in bones is higher than that in soft tissue (at the same thickness difference). Since the values of absorption coefficients decrease with increasing voltage, the contrast decreases with increasing voltage both in bones and soft tissue. Because the difference between absorption coefficients decreases with increasing photon energy, the contrast between bone and soft tissue decreases with increasing generating voltage applied. [µ/ρ] 1

2

[kV]

Figure 7.5: Mass absorption coefficient as a function of energy. 1 – bone, 2 – soft tissue.

If there is no essential difference between the investigated tissue and its surroundings, an artificial contrast may be applied by using some contrast substance. For instance, stomach can be seen if a barium sulphate suspension is introduced. Liquid contrast substances, e.g. iodine-containing solutions may be used to obtain contrast images of the kidney, the gall bladder, the blood vessel, etc. Materials that absorb more strongly than their surroundings are called positive contrast materials and the weaker absorbers are called negative contrast materials. Fig. 7.6 shows various contrast of X-ray image of fingers. We can observe very good contrast between soft tissue and bone and very good contrast between the air and soft tissue. Unfortunately, the absorption between muscles, nerves and tendons is similar; therefore, we observe very low or none contrast between these tissues. X-ray beam

x

Δx

μ0

μ

Screen I2

I1

I0

Figure 7.6: X-ray image of fingers.

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7.2 USE OF X-RAYS FOR DIAGNOSTIC PURPOSES X-ray imaging is one of the most important imaging modalities in medicine. It is simple and cheap. X-ray machine consists of evacuated x-ray tube (containing heated cathode and cooled anode), filters and shielding system (Fig. 7.7). Only about 1% of energy transported by electrons into anode is transformed into electromagnetic waves, while 99% is transformed into thermal energy. Therefore cooling of anode by water, air, oil, is required or rotating anode is applied. Lead Shielding Cathode

Oil Bath

Electron Beam Filter

X-ray Radiation

Vacuum

Motor Wolfram Anode

Figure 7.7: X-ray machine.

X-rays emitted from the optical focus of an X-ray tube pass through a certain part of the body and X-ray image is observed on the screen of the X-ray apparatus, thanks to luminescence properties of X-rays in crystals of suitable materials in detectors. The traditional X-ray image is a simple shadow image. When observing an X-ray image on the screen or photographic plate or film, we have a summation image of the object. We can distinguish the depth of a structure observed in patient’s body only by moving the patient. In conventional X-ray imaging, we can scan patient in different projections and by that to get at least some information about depth of the visualized structure. Projection is the direction of X-ray beam to the position of the body in space. Fig 7.8 shows the basic radiographic projections are Antero-Posterior (AP), Postero-Anterior (PA) and lateral (oblique). AP projection is directed from the front toward the back. PA projection is directed from back towards the front and lateral projection is from the side. 177

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AP

PA

Lateral

Figure 7.8: The basic projections in X-ray imaging. Antero-Posterior (AP), Postero-Anterior (PA) and lateral (oblique).

Each x-ray image is a little fuzzy. The reason of blurred image is the generation of penumbra PU, because the focus point O is never a point, in practice just a small area (Fig. 7.9). Penumbra is smaller if the area of optical focus point is smaller. O

S

I

PU

Figure 7.9: X-ray image. PU – penumbra, O – optical focus point, S – scanned structure (object), I – ideal image of structure.

From Fig. 7.9, it is evident that image quality depends on area of optical focus point. Other factors which can reduce image quality are movement artefacts. They can be suppressed by shortening of exposition and by fixation of the patient. Other three factors controlling unsharpness are source size, source to object distance, and object to detector distance. Total unsharpness is given by the biggest one. 178

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7.2.1 X-ray imaging methods In conventional X-ray imaging, there are various imaging methods. The most important ones are fluoroscopy (sciascopy), skiagraphy, angiography, digital subtraction angiography and mammography. Skiagraphy does static X-ray images reflecting absorption in the tissue. Sciascopy image usually is not so quality compared to skiagraphy image, but it allows to observe target structures in the real time and visualize dynamic processes as is for example the insertion of the catheter; lower dose of radiation is used for these purposes. Angiography is mostly used for visualization of the blood vessels and organs of the body, with particular interest in the arteries, veins, and the heart chambers. It is done by injecting a radio-opaque contrast agent. Digital subtraction angiography (DSA) is modification of angiography, it provides better image contrast. DSA consists of three steps: 1) The mask image is taken before the injection, then only the background is imaged; 2) After injection with a contrast agent a radiogram will show enhanced absorption due to the applied contrast agent, but there will also be a large background of absorption; 3) These two images acquired before and after contrast agent application are subtracted mathematically, point by point, the difference between images shows only the difference (blood vessels containing the contrast agent) in absorption. Bones and other strong X-ray absorbers are removed by subtraction because their contrast is not changed. Mammography is the radiographic method for examining the human breast. The main purpose is the detection of breast cancer, typically through detection of characteristic masses and/or microcalcifications. It is done by special X-ray machine called mammogram. Mammogram utilizes low energy, mostly 20–35 keV to maximize radiographic contrast. They are lower energy but still produce ionizing radiation. As stated above, conventional X-ray images are summation images which might visualize inner structures inaccurately. Computed tomography (CT) acquires sectional images which provide superior anatomical information compared to conventional X-ray images (Fig. 7.10). It is tomographic imaging – the most advanced non-invasive radiological diagnostic method based on X-ray; different internal structures appear in different grayscale based on the attenuation of the incident beam of X-rays. CT is unsubstitutable especially in acute medicine because it allows rapid acquisition of large amounts of information that may result in a significant effect of the treatment.

Figure 7.10: CT imaging. Section image of alligator (left) and 3D reconstruction of head’s alligator from a set of section images (right). Density on CT images represents absorption property of tissue.

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7.2.2 Computed tomography Principle of tomography imaging is the image reconstruction based on image projections acquired under various angles (Fig. 7.11). The simplest reconstruction algorithm is back projection (Fig. 7.12). It was used in the first CT systems, but quality of reconstructed images was bad. Axial section of the scanned subject

Source of radiation

Bundle of beams

Rotation

Array of detectors

Figure 7.11: Principle of CT. Sectional image is reconstructed from the set of projections acquired under various angles. Usually X-ray tube (source of radiation) rotates around fixed object, X-ray tube and detectors are coupled. X-ray source

z

y

a11

7 6 A

a12

4

3

a21

a22

5

1 x

z

9 y

4

B

x

imaging Figure 7.12: Back projection reconstruction algorithm. This algorithm consists of two steps: (a) acquisiand control tion of profiles (of absorptions) and (b) back projection devicesof profiles under appropriate angle into empty matrix for image reconstruction. In this figure, two profiles are acquired under two angles shifted 90°. source Then these absorption profiles are projected in empty matrix and their intersection represents reconstructed image. reference detectors

computer

additional devices

180 detectors basics_medical_physics.indd 180

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of the scanned subject

Source of radiation

Bundle of beams

Other reconstruction methods are algebraic (iterative) reconstruction and analytical reconstruction. The principle of algebraic method can be explained by using the following example. Let us consider an area (see Fig. 7.13) separated into four squares. Let us denote the square in the first row andRotation first column a11, in the first row and second column a12, etc. Let Array of detectors us assume that each square is characterized by a certain value of a parameter, e.g. its value for a11 equals 4, for a12 equals 3, etc. It is impossible to measure the value of the parameter in each pixel, but it is possible to measure the sum of the parameters for different directions. Thus, we get for the rows a11 + a12 = 7, and a21 + a22 = 6, and for the columns a11 + a12 = 9, and a12 + a22 = 4. We have in this case four linear equations for four unknowns that can be solved by some suitable method. X-ray source

7 6

a11

a12

4

3

a21

a22

5

1

9

4

Figure 7.13: Principle of algebraic reconstruction

The same principle is applied to computed tomography. The area of a certain cross section of the patient’s body is separated into small pixels (order of tens or hundreds micrometres). The intensity of narrow beam of X-rays that has passed through the body in various directions is measured by a set of photomultipliers. Thus, the value of absorption coefficient of each individual pixel can be obtained by solving thousands of equations with thousands of imaging unknowns. It represents no problem for fast computer. and control intensity I0 by The emerging intensity I is related to the incident devices I = I0e

− ( µi 1∆x + ...+source µij ∆x + ...+ µin ∆x )

(7.15)

reference where Δx represents the width of the matrix element and computer μi are attenuationadditional coefficients of detectors devices individual elements. The resulting image is calculated from a matrix of measured intensities by a computer and displayed. Block scheme of CT is shown in Fig. 7.14. Much higher contrast as compared with other X-ray techniques is achieved. Calculated image represents some advantages (the area of interest may be chosen to observe more details, storage of image in computer memory).detectors Devices for whole-body measurement or for the examination of only memory a part of body (e.g. head) have been developed. The radiation load for patient (absorbed dose of radiation) is comparable to that in classical X-ray investigation.

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zdroj záření pole detektorů

imaging and control devices source reference detectors

computer

additional devices 360° X-ray source

detectors

memory 40°

X-ray fan beam

Detector array

Rotary mechanism

Figure 7.14: Block scheme of CT and typical arrangement of clinical CT scanner.

In the present, most analytical reconstruction algorithms are used in commercial clinical CT scanners, especially filtered back projection. Filtered back projection is modified back projection reconstruction method. Image quality depends heavily on convolution filter (Fig. 7.15). In reconstruction algorithm, firstly multiplication of profiles by weighted function (filtered projection) and after that back projection from filtered projections into empty image reconstruction matrix are performed.

a)

b)

c)

Figure 7.15: Effect of convolution filter on image quality reconstruction. (a) original object (abdomen), (b) reconstruction from 32 projections with no filtering and (c) reconstruction from 32 projections convoluted with Cosine filter.

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Other important parameters affecting reconstruction quality are number of projections (i.e. number of different angles) and number of rays used in one projection. More projections and more number of rays enable to achieve better quality of image reconstruction. Effect of number of projection on image reconstruction is shown in Fig. 7.16.

a)

b)

c)

d)

e)

f)

Figure 7.16: Effect of number of projection on image reconstruction. (a) original object, (b) image reconstruction from one projection, (c) from two projections, (d) from four projections, (e) from eight projections and (f) from sixty four projections.

Image quality also depends on detectors. In CT we use semiconductor detectors (they have good signal to noise ratio, high efficiency, very fast recovery (in ms) after incidence and registration of x-ray radiation quantum. Older CT systems used scintillation crystals (Bismuth germanium oxide, Thallium doped Sodium Iodide) with photomultipliers or photosensitive semiconductors. These detectors have high efficiency (> 99%), unwanted longer flashes, low sensitivity to vibrations. Principle of their detection is that incident photon induces secondary emission of electrons, subsequently electrons return to basic state, which accompanied with light flashes (scintillation); flashes are registered by photodiode (1 foton ~ 100 fA). The third type of detectors are ionization chambers with noble gases (xenon). They are based on the change of ionization intensity by incident photons. They are cheap and have lower efficiency (50–60%). In modern CT, spiral CT scanner, table moves in a continuous linear motion through the tunnel while the X-ray emitter and detector rotate continuously over 360 degrees, and data can be acquired within seconds in a single sweep over the entire volume of interest. Intensity of CT images are expressed in Hounsfield numbers. Hounsfield number is a standardized unit 183

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for displaying reconstructed X-ray computed tomography CT values. The system of units represents a line transformation from the original linear attenuation coefficient measurements into one where water is assigned a value of zero and air is assigned a value of −1000. Range of Hounsfield numbers H: –1000 – 3000 and can be calculated as H=

µ − µ water ∗1000(7.15) µ water

where is a linear attenuation coefficient. Linear attenuation coefficient for water is used as reference. Therefore, according to equation 7.15, H = 0 for water.

7.2.3 Risks of X-ray radiation Since the application of X-rays always represents a risk to the patient, the lowest possible intensity has to be used and the eyes of physician have to be adapted to darkness when observing the screen. Electronic image intensifier may be used to increase the brightness of the image. The increase of brightness is achieved by acceleration of electrons induced by photoeffect of X-ray photons passing through the body. Scintigraphy technique results in lower radiation doses absorbed by patient’s body as compared with those occurring during examinations based on observing the screen. X-ray films have photographic emulsion on both sides, which increases their sensitivity. Density of the blackening describes the blackening. When the intensity of incident light is denoted as I0 and the intensity of passed light as I, then the density of blackening is defined by D = log

I0 (7.16) I

For example, if 1/10 of intensity of the light passed then the density equals l. The difference of the densities between two usually neighbouring areas is called radiographic contrast. Since the curve of density plotted against the logarithm of exposure possesses a linear character with a slope of 2–5, the contrast observed on the film is higher than that observed on the screen. Therefore, the image localised on photographic plate or film enables higher resolution as compared to that of screen. And more details can be distinguished. In general, all X-ray procedures pose a potential high risk of radiation-induced cancer to the patient. Normal efficient dose (can be different in various types of X-ray machines) in mSv is 0.02 for thorax X-ray examination, 5.3 for thorax CT, 1.5 for head CT. It was several times stated that the highest source of radiation pollution in population in developed countries is CT examination. For example, in Czech Republic (population 10 000 000), around 500 000 CT examinations and around 10 000 000 all X-ray examinations are performed each year. Radiosensitivity of tissue is different. The most sensitive tissue to X-ray radiations is lymphatic tissue, bone marrow, gonads and young dividing cells. We distinguish two types of effects on the body: deterministic and stochastic. Deterministic effect may only affect the dose exceeds a certain value. It causes for example acute radiation sickness (during nuclear disasters) or local effects on the skin (in radiotherapy). Stochastic effect is cumulative; it is caused by long-term accumulation of the effects of exposure to radiation. It cannot be completely prevented. It arises later after exposure. 184

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7.3 THERAPEUTIC APPLICATION OF X-RAYS Thanks to biological effects of X-rays, they are also used in therapy of malignant tumours. X-ray therapy uses high energy to destroy cancer cells or keep them from reproducing. It is a local treatment, the radiation acts only on the part of the body that is exposed to the radiation (different from chemotherapy in which drugs circulate throughout the whole body). We distinguish two main types of radiation therapy: 1) External radiation therapy where a beam of radiation is directed from outside the body at the cancer; 2) Internal radiation therapy (brachytherapy or implant therapy) where a source of radiation is surgically placed inside the body near the cancer. It is beneficial for patient because X-ray therapy is painless. The low-energy photons have to be cut off out of the X-ray beam by filters since the damage of superficial tissue could appear together with the irradiation of tumour located at a certain depth below the body surface. The filters (Zn, Cu, Al plates or composed filters) are located in front of the output window of X-ray tube. The filters also attenuate the high-energy photons, but to a lesser extent then the low-energy ones. As a consequence, the radiation irradiating the body is more homogeneous. The narrower energy ranges of photons in the beam, the higher its quality. The half-value layer (HVL) that reduces the initial intensity by 50% estimates the quality of X-ray beam. The HVL cuts off more the longer wavelengths from the continuous spectrum of the bremsstrahlung produced by the X-ray tube since photons with lower energy are more easily absorbed. Therefore, a further HVL must possess a higher thickness to absorb 50% of incident intensity since shorter wavelengths prevail in the filtered spectrum as compared with the spectrum originally produced. The quality factor h = HVL1/HVL2 should be about 1.5. For example, for 124 keV photons, i.e. wavelength of 10 pm, the half-value layer of water is 4.55 cm, that of the air is 38 m. The soft tissue of the body absorbs practically in the same way as water. The biological effect of radiation depends mainly on the absorbed dose of radiation or exposure. Absorbed dose is the energy absorbed per unit of mass. Its unit is grey (Gy); 1 Gy = 1 J/kg. The exposure is expressed in C/kg. (See also Chapter 8). We distinguish various types of dose (exposure). The air dose (exposure) is the dose (exposure) measured in the air at certain distance from the focus of X-ray tube. Surface dose (exposure) is measured at the surface of patient’s body. It is higher than the air dose measured at that place due to the contribution of radiation scattered back by patient’s body. Depth dose is what is absorbed at a certain depth below the surface. Geometrical irradiation conditions play an important role in X-ray therapy. Since the intensity of radiation emitted by a point source of radiation decreases proportionally to the squared distance from the source, the depth dose Dd observed at a depth d is related to the surface dose Ds as follows: Dd FS 2 = (7.17) Ds ( FS + d ) 2 where FS is the distance between the focus of X-ray tube and body surface. The tissue of irradiated object localised near to the source is more exposed as compared with those more distant, however, the ratio of corresponding doses approaches the values near to 1 at increasing FS distance, as shown from equation (7.17). 185

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The depth dose is frequently expressed as percentage of the surface dose, i.e. Dd (%) =

100 Dd (7.18) Ds

With respect to this relation, high-energy photons from a source sufficiently distant from the irradiated subject must be used to avoid the damage of superficial body layers (above the tumour) during the therapy of tumours. The use of low-energy photons (low acceleration voltage) with a source near to the skin is recommended for the therapy of superficial lesions. Accurate determination of absorbed dose of radiation cannot be done since it depends on many factors that cannot be measured. The value of absorbed dose is affected by different attenuation coefficients of different tissue and by their spatial arrangement. Therefore, measurements on phantoms are usually made before the therapy. Phantom is a model of irradiated object filled with a material that absorbs similarly to the tissue (water, rice, etc.). Dose meter is placed into this material and the dose is measured at different places at the given irradiation conditions. The values measured are plotted in graphs. Continuous lines connect the places of the same dose. In this way, the net of isodose curves is obtained which yields the information about the dose obtained by tissue. Nowadays, the 60Co or 137Cs γ-ray therapy often replaces the X-ray therapy, thanks to high-energy γ-photons emitted from these radioisotopes (suitable half-life and photon energy). Physicians should be protected when using X-ray techniques (especially haemopoietic tissue or gonads) in a suitable way (shielding, distance, and short time of exposure) and the absorbed dose of radiation of person working with ionising radiation has to be measured. The radiation hazard has to be checked by film dosimeters. The maximum permissible dose is of 1 mGy/week. For more details see Chapter 8.

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8.

RADIOACTIVITY AND  IONISING RADIATION

Radioactivity is particles that are emitted from nuclei as a result of nuclear instability. Nuclear instability is a consequence of the conflict between the two strongest forces in nature (strong interaction and electromagnetic interaction), both of which act on the nucleus. Atoms that contain unstable nuclei are radioactive. Radioactive nuclei spontaneously decay by emitting a particle and/or quantum of electromagnetic radiation. The most common types of radiation are alpha-, beta- and gamma-radiation. Radiation, which is the product of radioactivity, can be analysed by an electric or a magnetic field. Different types of radioactivity lead to different decay paths, which transmute nuclei into other stable or radioactive chemical elements. Radioactive dating consists of examining the amount of decay products. Since radioactivity is related to the nucleus, the decay rate can be influenced by either chemical or physical processes.

8.1 NATURAL AND ARTIFICIAL RADIOACTIVITY Radioactive nuclei occurring in nature are natural, while others produced artificially in atomic reactors or accelerators are artificial. Natural radionuclides are divided into the following two groups: a) Light natural radionuclides with atomic numbers Z ≤ 75, e.g. 14C, 40K, 115In, 139La, etc. These radionuclides do not form decay series and upon disintegration result in stable 14 nuclei. Potassium 40 19 K and carbon 6 C are the most important elements in this group and are of interest in biomedical disciplines. b) Heavy natural radionuclides are members of three decay series (see later). The radionuclide that decays and forms another radionuclide is the parent radionuclide and its decay product is the daughter radionuclide. Elements with atomic numbers Z ≥ 93 (transuranium elements) are also produced by the bombardment of heavy nuclides in accelerators. All of these elements are short-lived radionuclides.

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8.1.1 Basic law of radioactive decay In the case of a sample containing radioactive nuclei of number N where the number at time t = 0 is N0, since radioactive decay is a stochastic process the number of nuclei, dN, decayed over an infinitesimally short time interval dt is proportional to N. Denoting the constant of proportionality λ and given that N decreases with increasing time, then dN = −λ N (8.1) dt By solving this differential equation with the initial condition N = N0 for t = 0, then N = N 0 e − λt,(8.2) where N is the number of radioactive nuclei at time t and number e is Euler’s number (the basis of natural logarithms, e = 2,71..). Constant λ is called the disintegration constant or decay constant and represents the relative rate of decay as shown in equation (8.1) after it is rearranged: dN λ = − dt (8.3) N It follows from equation (8.2) that the number A of radioactive nuclei exponentially decreasN0 the ratio of the number of nuclei es over time (see Fig. 8.1A). Decay constant λ represents per time interval transformed to the total number of those present, but as yet untransformed, nuclei. Taking the logarithm of the equation (8.2), then N0

2 ln N = ln N 0 − λ t (8.4) N0

which represents the equation of a straight line; 4the values of the natural logarithm of N decrease linearly over time, the slope being −λ. Therefore, using the semi-logarithmic co-ordinates, N can be easily plotted as a function of time (see Fig. 8.1B). T

A

2T

t

B

N0

N0 N0 2 N0 4

N0 2 N0 4

T

2T

t

T

2T

t

B Figure 8.1: Activity as a function of time. A – y-axis linear, B – y-axis logarithmic.

188

N0 N0 2 N0 4

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As activity A is defined as the number of atoms that decay in 1 s, it is also an estimation of the decay rate. Equation (8.1) also shows that the product of the disintegration constant and the number of radioactive atoms, i.e. λN, equates to the decay rate. Thus, it follows from equation (8.2) that the activity also decreases exponentially over time. In fact, the multiplication of both sides of this equation by factor λ results in an equation analogous to equation (8.2) provided A0 = λN0 (activity at time t = 0). The unit of activity is 1 becquerel (Bq). A radioactive sample possesses an activity of 1 Bq when the number of atoms that decay in 1 s equals 1. Disintegration constant λ is a characteristic parameter of a radionuclide. As decay constant λ is different for each particular radionuclide, each has its own decay speed. Its dimension is time−1. Two radioisotopes of the same decay constant value have yet to be identified. An important parameter for radioactive nuclei is radioactive half-life. The radioactive halflife for a given radioisotope is a measure of the tendency of the nucleus to disintegrate and, as such, is based purely upon that probability, i.e. during a given time half the radioactive nuclei in any sample will undergo radioactive decay. After two half-lives, it will be one-fourth the original sample and after three half-lives one eighth the original sample, which is expressed by exponential dependence (see Fig. 8.1A). The half-life is independent of temperature, pressure, physical state (solid, liquid, gas) and any other outside influence. The half-life can only be altered if it comes into direct nuclear interaction with an exterior particle possessing high energy (typically a collision in an accelerator). There are three types of half-life in a medical context: physical half-life, biological halflife and effective half-life. Physical half-life Tf  is defined as the time during which only one half of the initial number of radioactive nuclei, present in the sample at zero time, disintegrates. The dimension of this constant is time and its value is usually expressed in suitable time units (second, hour, day, year). If Tf  is substituted for t and N0/2 for N in equation (8.2), the following relationship between the physical half-life and the disintegration constant after rearrangement is obtained Tf =

ln 2 0.693 = (8.5) λ λ

Physical half-life values are given in various radionuclide tables. Straight lines in a semi-logarithmic scale can easily plot decay curves. The biological half-life Tb of a given substance is defined as the time required for the biological elimination of only one half of the initial amount administered to the organism, assuming there is uniform distribution throughout the organism and there are no further administrations. The biological half-life has no relation to radioactivity. Analogously, as mentioned above, the relationship may be expressed between the biological excretion rate, λb, and the biological half-life Tb =

0.693 (8.6) λb

Effective half-life Tef is primarily of importance in medical applications. If a radionuclide is administered to an organism, then both radioactive decay and biological excretion govern its relative disappearance rate. Therefore, the relative disappearance rate, λef, is a sum of the excretion rate and the decay, i.e. 189

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λef = λ + λb(8.7) If the effective half-life is defined analogously as above, then the following relationship between the effective, biological and physical half-lives is obtained 1 1 1 = + (8.8) Tef T f Tb If a radionuclide is deposited within the organism and elimination does not occur, the value of its biological half-life will be consequently very high; thus, its inverse value is negligible. In this case, the effective half-life practically equals the physical half-life. The effective halflife is used for calculating absorbed doses of radiation as a result of administering radionuclides for diagnostic or therapeutic purposes. The mean lifetime of radionuclide τ is defined as the inverse value of the decay constant. It can be expressed as a mean time of the existence of the radionuclide until its disintegration. With respect to equation (8.5), it is given by

τ=

Tf 1 = ≅ 1.44T f (8.9) λ 0.693

8.1.2 Radioactive equilibrium Radioactive equilibrium occurs in a decay series of radionuclides, where identical numbers of the nuclei of the parent and daughter radionuclides decay per unit time. If daughter radionuclide B with half-life T2 and decay constant λ2 is formed from parent radionuclide A (decaying with half-life T1 and decay constant λ1), then schematically A → B → C. NA0 is the number of parent radionuclides at time t = 0. NB0 = NC0 = 0 at time t = 0. The rate of change of the number of parent nuclei is given by dN1 = −λ1 N11(8.10) dt while the rate of change in the number of daughter nuclei is dN 2 = λ1 N1 − λ2 N 2,(8.11) dt since the number of daughter nuclei increases by the decay rate of the parent nuclei and decreases by the decay rate of the daughter nuclei, λ2N2. This scenario can also be explained without exactly solving the above differential equations as follows: If T1 T2, as in the previous case. However, the half-life of the daughter radionuclide is shorter than that of the parent. Therefore, the number of daughter radionuclides increases up to a certain maximum, but its activity later decreases proportionally to the activity of the 190

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parent radionuclide (see Fig. 8.2B). Starting from a certain time tt, the ratio of the activities of both radionuclides will remain constant and transitional equilibrium achieved. Thus, the following equation holds

λ1 N1 = k (8.12) λ2 N 2

activity

A

λ1N1

λ2N2

t

t0 B

activity

λ1N1 λ2N2

t0

t

λ1N1 activity

C

λ2N2

t

Figure 8.2: Course of the activity of parent (λ1N1) and daughter (λ2N2) radionuclide A – no radionuclide, B – transition equilibrium, C – radioactive equilibrium.

If T1 >> T2 and if the half-life of the parent radionuclide is of a sufficient duration that activity remains unchanged during the measurements, the activity of the daughter radionuclide will increase and finally match the activity of the parent radionuclide. The number of daughter nuclei formed per unit time will equal the number of decaying nuclei (see Fig. 8.2C). 191

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Radionuclides A and B then achieve a state of permanent radioactive equilibrium. In this case, the ratio of the number of parent and daughter radionuclides equals the ratio of their half-lives: N1 = N2

N A M1 A1 M 1 A2 T1 = = (8.13) NAM 2 M 2 A1 T2 A2

Since the number of atoms in 1 gram-atom is constant and equals Avogadro’s constant NA, the ratio of the mass of parent radionuclide M1 to that of daughter radionuclide M2 equals the ratio of their mass numbers times the ratio of their half-lives, i.e. M 1 T1 A1 = (8.14) M 2 T2 A2 where A1 and A2 are the mass numbers of the parent and daughter radionuclides, respectively. 222 For example, for radium 226 88 Ra, which decays with a half-life of 1622 years into radon 86 Rn with a half-life of 3.8 days, the equilibrium amount of radon corresponding to 1 g of Ra is 6.4×10−6g.

8.1.3 Radioactive series The decay of natural radionuclides with high atomic numbers results in the formation of decay series. Three natural decay series start with the elements of a very long physical half-life. 9 1. The parent element of the uranium-radium series is uranium 238 92 U (Tf ≅ 4.56×10 y). 206 A isotope of lead, 82 Pb, is the last stable element of this series. Mass numbers of this series are given by the general formula A = 4n + 2, e.g. for radium 226 88 Ra, 226 = 4×56 + 2. This series is he most important, since it involves two radionuclides that have applica222 tions in therapeutic medicine: radium 226 88 Ra and radon 86 Rn. 235 235 8 2. Actinouranium 92 U or 92 AcU (Tf ≅ 8.5×10 y) is the parent element of the actinium series. Its last element is a stable isotope of lead, 207 82 Pb. Its general formula is A = 4n + 3. 3. Thorium 23290Th (Tf ≅ 1.39×1010 y) is the parent element of the thorium series, the last element of which is a stable isotope of lead, 208 82 Pb. The general formula for its mass numbers is A = 4n. Another decay series with the general formula A = 4n + 1 is the neptunium series. This series involves plutonium 241 94 Pu, americium and neptunium and its last member is a stable isotope of bismuth, 209 Bi. The neptunium series is so-called because of all its members nep83 tunium 237 Np possesses the longest physical half-life at 2.2×106 years. 93 It should be noted that some artificial radionuclides also form decay series, especially some isobaric fragments of uranium fission, e.g. 89 36

Kr (Tf = 2.6 min) → → 89 39 Y (stable)

89 37

Rb (Tf = 15.4 min) →

89 38

Sr (Tf = 53 days) →

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8.1.4 Types of radioactive decay The decay of some radioactive nuclei is followed by the emission of a particle. The type of particle depends on the number of protons and neutrons in the decaying nucleus. The following physical quantities are always conserved during this process: electric charge, the number of nucleons, momentum and energy. They are defined according to the following basic radioactivity laws: The law of conservation of electric charge. The algebraic sum of the charges of the nucleus and of the emitted particles remains constant. Thus, if a negatively charged electron is emitted, the nucleus produces one positive charge. The law of conservation of nucleon number. The number of nucleons of parent nucleus X before transmutation equals the sum of the nucleons of daughter nucleus X´ and the emitted nucleons. The law of conservation of momentum. During the transmutation of nucleus X into X´, as described by the scheme X → X´ + the emitted particle, the sum of moments of the daughter nucleus and of the particle emitted equals zero. Therefore, the daughter nucleus and the particle emitted move in opposite directions. The law of conservation of energy. During the transmutation of a parent nucleus according to the scheme: X → X´+ particle + quantum with energy hf, it holds that M 0 c 2 = M 0´ c 2 + m p , 0 c 2 + E´+ E p , k + hf (8.15) where index 0 denotes the rest mass, E´ the kinetic energy of the daughter nucleus and Ep,k the kinetic energy of the particle emitted. Transmutation energy Q is given by the relationship Q = ( M 0 − M ´−m p , 0 )c 2(8.16) 8.1.4.1 α-Decay The decay of heavy radionuclides very frequently results in the emission of an α-particle. This particle is composed of two protons and two neutrons, which are bound tightly so that they behave as one particle, e.g. the nucleus of helium 42 He. The charge of the α-particle is positive due to the presence of the two protons, equalling two elementary charges. Due to the emission of the α-particle, the nucleus changes according to the scheme A Z

X → 42 He + ZA −− 42 X´(8.17)

The nucleus resulting from this type of decay is shifted to the left in the periodic system of elements by two positions. Monoenergetic α-particles are also emitted. If the energy of the α-particle emitted is lower than the transmutation energy, a daughter nucleus is formed in an excited state. It immediately transits into a basic energy state by the emission of one or more quanta of gamma-radiation (see Fig. 8.3). The kinetic energy of the α-particles is of the order MeV, which corresponds to their velocities (approximately 104 km/s). Experiments demonstrate that short-lived nuclei emit 193

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high-energy α-particles and vice versa. This type of decay cannot be explained by classical physics, only by tunnel-effect and quantum physics. 212

0

Bi

0.1 E = 5.6 MeV

0.2 0.3 0.4

E = 5.7

0.5

E = 5.9 5.5 5.6

E = 6.2

5.7 5.8 5.9 6.0 6.1 6.2

208

Ti

Figure 8.3: Diagram of bismuth disintegration into thalium.

8.1.4.2 β-decay β-decay is an isobaric transmutation of the nucleus, i.e. the total number of nucleons is conserved. Emitted high-energy particles 2(e.g. electrons) have a greater range of penetration than α-particles, but a much lesser range than gamma rays. There are three types of β-decay: electron emission, positron emission and electron capture. a) Electron emission. If an electron is emitted by the nucleus, the last electron changes according to the scheme A Z

 X → Z +1A X´ + −10 e + 00υe

(8.18)

A negatively charged electron and its antineutrino (which shares the momentum and enerenergie gy of the decay) are emitted by the parent nucleus. The atomic number of the daughter nucleus [MeV] exceeds that of the parent by 1, i.e. 3 1

 T → 23 He + −10 e + 00υe

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In this process, the neutron-to-proton conversion occurs in the nucleus accompanied by the simultaneous emission of an electron and its antineutrino, i.e. 1 0

 n → 11 p + −10 e + 00υe

The electrons emitted possess a continuous energy spectrum. The mean energy of one emitted electron equals approximately one third of its maximum energy. b) Positron emission. Positron β+-decay corresponds to the transmutation of the nucleus according to the following scheme A Z

X → Z −1A X´ + 10 e + 00υe ,(8.19)

64 e.g. 64 29 Cu → 28 Ni + positron + electron + neutrino. A proton is converted into a neutron in the nucleus with a simultaneous emission of a positron and its neutrino according to the scheme 1 1

p → 01 n + 10 e + 00υe

The daughter nucleus is shifted to the left by one unit in the periodical system of elements to offset the position of the parent nucleus. Similar to the electron energy spectrum, the positron energy spectrum is continuous. This type of decay can be observed only with artificially produced radionuclides of excess energy. c) Electron capture (mostly from shell K). The scheme of this transmutation is as follows A Z

X + −10 e → Z −1A X´ + 00υe (8.20)

Since one energy level is empty in the electron cloud surrounding the nucleus after the capture of the electron, this transmutation must be followed by the emission of characteristic electromagnetic radiation from the electron envelope. A proton is changed into a neutron in the nucleus according to the scheme 1 1

p + −10 e → 01 n + 00υe

Excited daughter nuclei are mostly formed during all three types of β-decay. In this case, immediate transition into the basic energy state occurs followed by simultaneous emission of a quantum of gamma-radiation. 8.1.4.3 Nuclear isomers 234 During the β-decay of thorium 234 90 Th, palladium 91 Pa is formed in 99.65% of cases, which 234 decays at a half-life of 1.22 minutes into uranium 92 U. However, in 0.35% of cases, metastable 234m 234 91 Pa is formed, which again decays the other half-life (6.7 hours) into uranium 92 U. A meta­ stable state is usually denoted by the letter m behind the nucleon number of the radionuclide. Nuclear isomerism also occurs in artificially produced radionuclides.

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8.1.4.4 Spontaneous fission Spontaneous fission can be observed during the disintegration of certain heavy radionuclides (e.g. 252Cf), but it is of lower importance in medical physics.

8.2 IONISING RADIATION AND ITS SOURCES Ionising radiation is radiation that has enough quantum energy to eject an elementary particle, such as an electron. Ionising radiation can produce a number of physiological effects, such as those associated with the risk of mutation or cancer, which non-ionising radiation cannot directly produce at any intensity. The threshold for radiation risk applies to the ionisation of human tissue. Since the ionisation energy of a hydrogen atom is 13.6 eV, the level around that value is considered a threshold. Since the energies associated with nuclear radiation are many orders of magnitude (MeV) above this threshold, all nuclear radiation is deemed ionising radiation. Ionising radiation is generated by nuclear reactions, nuclear decay, very high temperatures or the acceleration of charged particles in electromagnetic fields. There are two types of ionising sources: natural and artificial. An example of natural sources are stars including the sun, lightning and supernova explosions. Artificial sources include nuclear reactors, particle accelerators and x-ray tubes that are often used in medical fields.

8.2.1 Positively charged particles Positively charge particles lead to primary ionisation and include α-particles, protons, deuterons and ions of light elements. α-particles are released from the nuclei of heavy natural radionuclides, while protons are produced by the ionisation of hydrogen. A deuteron is the nucleus of deuterium 21 D (also known as heavy hydrogen) and is produced as a result of nuclear reactions. A deuteron is composed of one proton and one neutron. Their binding energy is relatively low at 2.2 MeV. However, a particle generated due to the decay of radionuclides or nuclear reactions possesses a relatively low energy. Therefore, accelerators are used to increase this energy for practical purposes. An accelerator is a device used to accelerate electrically charged particles of a high kinetic energy. These accelerated, high-energy particles are used to bombard target nuclei in order to induce nuclear reactions or high-energy bremsstrahlung. Accelerators such as the cyclotron are used to produce short-lived radionuclides and are important for medical applications. The betatron is an electron accelerator that generates high-energy bremsstrahlung for the therapy of malignant tumours. Linear and circular accelerators are used according to the path shape of the accelerated particles.

8.2.2 Linear accelerators Electrostatic or high-frequency linear accelerators are used according to the source of electric power applied. The Van de Graaff generator is an electrostatic generator. The klystron 196

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energie [rel. j.]

generator is used to generate high-frequency voltage. Particles are accelerated as they pass through a straight accelerating tube or high-frequency linear accelerator. High-frequency linear accelerator. A high-frequency generator is a source of voltage (20–40 kV) for the acceleration of particles emitted from a source. Ions are accelerated when passing through gaps between metal cylinders of alternating polarity that are connected to a power supply. The charge of the first cylinder must be of the opposite sign to the accelerated ion. The cylinders are of stepwise-increasing length, since the velocity of the accelerated ions increases when passing through each cylinder (see Fig. 8.4). The acceleration process occurs in a high vacuum, with the target located at the end of the last cylinder.

V1

S

V2

V3

V4

V5

~ Figure 8.4: Diagram of a high-frequency linear accelerator. S – source of ions, Vi – cylinders

8.2.3 Circular accelerators A cyclotron is used to accelerate heavy particles such as protons, deuterons, α-particles and ions of various elements. It consists of an evacuated cylindrical vessel separated into two parts (duants), which are located in a magnetic field oriented along its axis (see Fig. 8.5). These two parts are connected to a source of high-frequency (HF) alternating voltage, which creates an alternating electric field in the space between them. The source of the particles is located in the centre of the split between the duants. The particle emitted from the source is attracted by an oppositely charged duant and is simultaneously accelerated with entering the magnetic field in such a way that its path lies perpendicular to the vector of the magnetic field intensity. Particles move inside hollow accelerating electrodes (also called dees) until inertia. The path is curved to the circular form by a magnetic field oriented perpendicular to the plane of the particle trajectories. The magnetic field is used only for guiding particle beams and not for acceleration. Dees act as a Faraday cage and therefore the particles inside the electrodes do not affect the electrical pathway, only the magnetic field. Acceleration occurs only in the gap between the dees. These are powered by a high-frequency alternating current. The electric field between the dees always acts in such a direction as to increase the velocity of the particles (Ek ≈ 50 MeV). The polarity of the duants changes at the moment the particle leaves the appertaining duant, which is attracted by the second duant. The radius of the path increases due to its higher velocity (faster particles move along a circle of greater radius) and energy. The particles finally leave the accelerator in a continuous flux. Cyclotrons are mostly located in a conglomeration of diagnostic centres and used to produce short-lived radionuclides for diagnostic purposes. 197

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Accelerated particles Trajectory of the particles Source of the particles

B

B

B

B

B

B

HF duants

Duants

High-frequency (HF) alternating voltage

͢ Figure 8.5: Diagram of a cyclotron. B – the vector of the magnetic field intensity

8.2.4 Negatively charged particles – electrons Pure or mixed β-emitters, thermal sources and particle accelerators produce electrons. Heated wires of heavy metals, which are relatively weak thermal sources, are used in many branches of physics. The most intensive generator of electrons is the betatron (see Fig. 8.6).

target

Figure 8.6: Diagram of a betatron.

Acceleration takes place in an evacuated glass ring, which is the secondary coil of the transformer. This ring, which receives pulsed jets of electrons, is situated at the poles of the electromagnet. 198

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Periodically changing magnetic induction, B = Bmsin(ωt), results in a variable magnetic flux, Φ = 2πr2Bmsin(ωt), which induces an electromotive force proportional to the variation of the magnetic flux. This force accelerates the electrons during the first part of a process in which magnetic induction increases. Therefore, the betatron is a pulse accelerator of electrons moving at relativistic velocities. The following relationship holds for their energy E = 3.108 r0 Bm (8.21) where energy E is expressed in eV with the radius of path r0 expressed in metres and the highest value of magnetic induction Bm in tesla. The accelerated electrons finally hit their target, producing high-energy bremsstrahlung.

8.2.5 Neutrons The velocity range of a neutron varies greatly. Due to the absence of an electric charge, a neutron cannot be accelerated. A fast neutron possesses an energy greater than 0.5 MeV, while a slow neutron has a lower energy. All neutrons present in an atomic reactor are called reactor neutrons. Their energy spectra vary greatly from 10−3 eV to 15 MeV. Neutrons that have not slowed down are called fission neutrons. A thermal neutron reaches thermal equilibrium through a medium (moderator) and acts according to the Maxwell distribution of velocity. Resonance or epithermal neutrons are moderated neutrons that have not reached thermal equilibrium.

8.2.6 Radionuclide sources of neutrons Radionuclide sources of neutrons come from the release of a neutron from a target nucleus due to the impact of an α-particle or γ-quantum or from the spontaneous fission of a certain transuranium element. Beryllium is frequently used as the target element. Natural or artificially produced radioisotopes can be used as sources of α-particles or γ-radiation, respectively. The total neutron flux from a source containing 10 MBq is in the order of 104–107 neutron.s−1 for various radioactive emitters. Higher neutron yields, 1010–1011 neutron.s−1, are obtained from neutron generators based on reactions (d,n). Using a suitable accelerator, charged particles can be accelerated. The highest neutron fluxes are produced by nuclear reactors.

8.2.7 γ-radiation γ-radiation is electromagnetic radiation in the form of photons emitted by the nuclei of radioactive elements. These photons possess a line energy spectrum, since only a portion of energy is released (corresponding to the transitions between the quantum energy states of the nuclei). A γ-ray photon is in fact identical to an x-ray, since both generate electromagnetic rays. The terms x-ray and γ-ray in fact denote origin rather than implying different kinds of radiation. Nevertheless, most γ-rays possess higher energy than x-rays and therefore penetrate 199

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to a greater degree. This type of radiation frequently involves the emission of alpha and beta radiation. The cobalt gun is used as a source of γ-radiation in radiotherapy. Radioactive cobalt 60Co emits γ-radiation of an energy higher than 1 MeV. A special container (bomb) is filled with radionuclide 60Co at an activity of approximately 1011–1012 Bq. γ-radiation intensity decreases slowly over time. Since the half-life of this radionuclide is approximately 5 years, the device can be used for several years. It also has therapeutic applications as it uses ionising radiation. Besides cobalt, 137Cs (which has an energy of 0.66 MeV and a half-life of 26.6 years) is also used as a source of γ-radiation.

8.2.8 Cosmic rays Cosmic rays contain ionising radiation and strike the earth from the solar system. They consist of primary and secondary components. Primary components contain radiation that has not interacted with the atmosphere (fast protons, α-particles and approximately 1% of light nuclei). The average energy of protons is approximately 10 GeV. However, energy values of 1018 eV can also be detected. Secondary components can either be “soft” or “hard”. Soft secondary components (electrons, positrons, photons, protons and light nuclei) can be shielded by approximately 10 cm of a Pb layer. Hard secondary components consist of fast mesons, high-energy secondary protons and neutrons and almost all elementary particles. The intensity of cosmic radiation depends on the altitude above sea level and geographical width. Its intensity rises up to an altitude of 20 km, decreases up to an altitude of 50 km and remains all but constant at even higher altitudes. Two Van Allen bands with cosmic rays of very high intensity (dose rate of up to 1 Gy/hr) have been detected by satellites and are highly dangerous to humans. The effect of the magnetic field of the earth results in a variation of intensity in accordance with geographical width. The lowest values are found at the equator, while values at the poles are approximately 25% higher. Cosmic radiation is of importance in nuclear medicine, particularly with regard to its use in detection devices (see later). Suitable shielding can decrease its effect.

8.3 INTERACTION OF RADIATION WITH MATTER Nuclear radiation loses energy when passing through an absorber. This energy loss depends on the type of radiation and the absorber. The processes involved include ionisation, excitation, scattering, bremsstrahlung formation (braking radiation) and nuclear reactions. Ionisation is the most important process of energy loss by charged particles. The kinetic energy of a particle decreases as it passes through an absorber due to ionisation and even thermal motion energy values. For example, energy of 34 eV is required to form one ion pair in the air. The distance passed by the particle in the absorber is called the range. Linear energy transfer (LET) or −(dE/dx) is the energy loss of a charged particle due to ionisation along the unit path. LET is higher at lower energies and vice versa. This means that the number of ion pairs formed along the path of the particle is not uniformly distributed, 200

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becoming higher at the end of the path. LET is a function of energy, particle charge and mass. It is usually expressed in keV/μm. Primary ionisation is the total number of ion pairs formed by the ionising particle. However, some electrons released through primary ionisation are of an energy that enables further ionisation of atoms in the absorbing medium. Ionisation caused by secondarily released electrons (δ-electrons) is called secondary ionisation. Total ionisation is the sum of these primary and secondary processes. If the energy absorbed by the electron envelope of an atom is not sufficient for electron release and subsequent ion pair formation, excitation occurs. De-excitation results in the emission of photons in the x-ray, ultraviolet, visible, or infrared region of electromagnetic radiation. Excitation may also result in the decomposition of a large molecule into two or more molecules.

8.3.1 Interaction of α-particles Due to relatively large mass and electric charge (two elementary charges), ionisation losses of energy are high and several thousands of ion pairs can be formed during the absorption of one α-particle. Energy losses as a result of ionisation and excitation are approximately 50 : 50%. Due to their short range of absorption and inability to penetrate the outer layers of the skin, α-particles are not by and large dangerous to life. However, if the source is ingested or inhaled they become extremely dangerous. The range of α-particles is very small; at an energy of 10 MeV it is approximately 10 cm in the air and several μm in soft tissue or water. Therefore, the negative biological effect of incorporated α-emitters is considerable since all the energy emitted is absorbed in a very small volume of tissue. When isotope-emitting α-particles are ingested, they can be more dangerous than their half-life or decay rate might suggest. Ingested α-emitter radioisotopes are on average approximately 20 times more dangerous than the equivalent activity of β-emitting or γ-emitting radioisotopes. α-radiation can be used in cancer therapy by directing the damaging α-emitting radionuclides inside the body in small amounts towards the tumour. These elements damage the tumour and stop its growth, while their low penetration depth prevents radiation damage of the surrounding healthy tissue. This type of therapy is called unsealed source radiotherapy.

8.3.2 Interaction of β-radiation The processes of ionisation and excitation result in the highest energy losses of electrons during passage through an absorber. In comparison with α-particles, the specific linear ionisation of electrons is lower due to their smaller mass and charge. Therefore, they also have greater range. In addition to the ionisation and excitation processes, bremsstrahlung (braking radiation) can be produced due to the interaction of electrons with matter. Bremsstrahlung is electromagnetic radiation produced when electrons are stopped in the electrostatic field of an atomic nucleus as the result of Coulomb interaction (see Fig. 8.7). This mechanism also occurs during the production of x-rays in an x-ray tube. Bremsstrahlung intensity is directly proportional to the atomic number of the absorber and to the energy of the electron. 201

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e– photon

nucleus e–

Figure 8.7: Diagram of bremsstrahlung generation.

Energy losses due to bremsstrahlung are relatively low but become more significant at higher energies. The ratio of energy loss due to radiation, −(dE/dx)rad, to the energy loss due to ionisation, −(dE/dx)ion, is given by the following empirical formula (dE / dx) radabsorber EZ = (8.22) thickness (dE / dx)ion 800 [mm]

water or

where E is the energy of β-radiation in MeV Z is the atomic number of the absorber. softand tissues umaplex In the case of lead10(Z = 82), radiation losses exceed ionisation losses only at energies higher than 10 MeV. On the other hand, radiation losses amount to only 1% at an energy of 0.1 MeV. Bremsstrahlung is characterised by a continuous energy spectrum. 8 Intensity I of a beam of monoenergetic electrons decreases during passage through an aluminium absorber according to the relationship 6

I = I 0 e − µ d (8.23) 4

where μ is the linear absorption coefficient (given in reciprocal length units), d is the thickness of the absorber and I0 is the intensity of the incident radiation. copper Instead of the linear2absorption coefficient, the mass attenuation coefficient, μ/ρ, can be used (ρ is the absorber density). Equation (8.23) gives I = I0e

−

µR ρ

0.5 1.5 2.5 3.5 (8.24) energy [MeV]

where R is the mass of the absorber per unit area (kg.m ). The maximum range of β-particles depends on their energy. The thicknesses of the various materials that can completely absorb β-particles of a given energy are shown in Fig. 8.8. Half-thickness D1/2 is of such an absorption thickness that it reduces the intensity of incident radiation by 50%. It follows from equation (8.24) that the relationship of the linear attenuation coefficient and the half-thickness is given by −2

D1/ 2 =

ln 2 0.693 =  µ µ

(8.25)

Half-layer R1/2 is of such a mass per unit area of the absorber (kg.m−2) at density ρ that it reduces the intensity of incident radiation by 50%. Analogous to equation (8.24), it holds that 202

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photon

nucleus e–

R1/ 2 =

0.693ρ (8.26) µ

The range of β-radiation is a function of its energy. In soft tissue, the range is in an order of mm for most β-emitters. The radiation hazard from betas is greatest when they are ingested. absorber thickness [mm]

water or soft tissues

umaplex

10 8

aluminium 6 4 copper

2

0.5

1.5

2.5

3.5 energy [MeV]

Figure 8.8: Absorption of β-radiation in various materials.

8.3.3 Interaction of γ-radiation The interaction of γ-radiation in an absorber is expressed by an exponential relationship similar to that given by equation (8.23). However, attenuation coefficient μ is the sum of three components corresponding to the absorption by photoeffect τ, Compton scattering σ and the formation of electron-positron pairs κ, so that

µ = τ + σ + κ (8.27) Analogously, the mass attenuation coefficient is also the sum of the corresponding coefficients for these absorption mechanisms:

µ τ σ κ = + + (8.28) ρ ρ ρ ρ The half-layer and half-thickness are defined by equations (8.25) and (8.26) in the case of γ-radiation. 203

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8.3.3.1 Photoelectric effect The photoelectric effect is discussed in detail in chapter 1, section 1.6.1 and in chapter 7, section 7.1.3.1. In this interaction, the total energy of the photon is transferred to one of the electrons in the envelope. A part of this energy is consumed and released from the atom (ionisation potential W = binding energy Eb), while the rest is converted into the kinetic energy of the electron (see Fig. 8.9). The energy of the photon (hf) is used for ionisation and the rest is transformed into the kinetic energy of the secondary electron. Thus, the energy balance of this effect is given by the relationship

hf = W +

mv 2 (8.29) 2

The photon vanishes during this interaction, while the photoelectron released interacts with atoms or molecules along its path by ionisation. The probability of the absorption by the photoelectric effect is directly proportional to the fourth power of the atomic number of the absorber (Z 4) and inversely proportional to the third power of γ-quantum energy (E 3). That is why this type of absorption is of greater probability at low energies and in heavy absorbers. Thus, absorption in bones (the average Zbone = 13.6) is approximately 16-times higher than that in soft tissue (Zsoft tissue = 7.6) of the same photon energy. The probability of this type of absorption in biological tissue is very low at energies over 1 MeV. secondary electron

primary photon

φ

Figure 8.9: Diagram of the photoelectric effect.

8.3.3.2 Compton scattering Compton scattering is also discussed in chapter 7, section 7.1.3.2. In this interaction only a part of the photon energy is transferred to a free electron, while the scattered photon moves in a changed direction with lower energy. This type of interaction does not depend on the atomic number of the absorber and the probability of its occurrence depends on the incident photon energy (the highest attenuation coefficients are at energies of 0.5–5 MeV). Therefore, Compton scattering occurs at higher photon energies. In this type of interaction, a photon of energy hf interacts with a free electron of the absorber. Because it possesses momentum, a part of its energy is transferred to the electron (see Fig. 8.10). The laws of conservation of energy and momentum hold. Therefore, the electron moves with the energy obtained in a direction deflected from that of the primary photon by the angle φ, while the scattered, 204

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secondary photon moves with the energy hf´ < hf in another direction deflected by the angle ϑ. This process is repeated until the energy loss of the photon results in its final absorption by the photoelectric effect. secondary electron E1 = hf1 primary photon

φ θ E2 = hf2

secondary photon

Figure 8.10: Diagram of Compton scattering.

The range of the deflection angle of the photon is 0 < ϑ < 180°. The energy of the scattered photon depends on the scattering angle. Initial wavelength λ changes due to scattering to λ´, where λ´ > λ. The value of the difference Δλ = λ´− λ does not depend on the energy of the incident photon but on the scattering angle only. The shift in the wavelength increases with the scattering angle and is given by the following equation e−

λ′ − λ =

h (1 − cos ϑ ) (8.30) me c E1 = 0.51 MeV

E > 1.02 MeV

where h is the Planck constant, c is the velocity of light in the vacuum and me is the mass of the electron. The energy balance of this interaction is given by e+

m v2 hf = hf ´+ e (8.31) 2 e− Although the decrease in photon energy is independent of its initial value and only on the scattering angle, the relative loss of photon energy depends on its initial energy. The relative energy loss increases with increasing energy. It follows equation (8.30) that the highE2 =from 0.51 MeV est decrease may be expected for backscattering, i.e. for ϑ = 180°. For the scattering angle, cos180° = −1 and according to equation (8.30) ∆λ =

2 × 6.6 × 10−34 J.s = 4.8 × 10−12 m = 4.8 pm (9.11× 10−31 kg )(3.0 × 108 m.s −1 )

8.3.3.3 Electron-positron pairs The third absorption mechanism is the formation of electron-positron pairs where energy is converted into mass. A high-energy photon splits into an electron and its antiparticle, the 205

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positron. The probability of this interaction increases with the energy of the incident photon. At high energies generated by γ-radiation (x-ray energy generated by an x-ray machine is not sufficient) in the vicinity of the atomic nucleus or other particles (necessary for the acceptance of the momentum of the photon), photon energy is completely converted to form an electron and positron with relativistic kinetic energy, Ek,e and Ek,p. The photon then disappears and the electron-positron pair is formed (see Fig. 8.11). A process known as annihilation involves the weight conversion of the electron-positron pair to ansecondary electromagnetic wave, whereby a subatomic particle collides with its respective antiparticleelectron to produce other particles. The energy balance of the electron-positron pair interaction is given by E1 = hf1

φ

hf = Ek , e + Ek , p + 2me c 2 (8.32) primary photon

θ

It follows from this equation that the energy of the incident photon must be higher than E2 =electron hf2 that which corresponds to the sum of the rest masses of the and positron, i.e. higher secondary than 2×0.51 MeV = 1.02 MeV. This is the condition for the creation of the electron-positron photon pair. The particles formed lose their kinetic energy through the processes of ionisation and excitation. When the positron energy finally corresponds to the energy of the thermal motion of the absorber particles, annihilation of the electron occurs with a simultaneous emission of two quanta of γ-radiation at energies of 0.51 MeV moving in opposite directions. The life span of a positron is approximately 10−7 s. As mentioned above, the formation of the electron-positron pair takes place only at high energies and is more probable in absorbers of high atomic numbers. e−

E1 = 0.51 MeV

E > 1.02 MeV

e+ e−

E2 = 0.51 MeV

Figure 8.11: Diagram showing electron-positron generation followed by positron annihilation.

The specific linear ionisation of γ-radiation is low and thus its range is large. The penetrability of γ-radiation is very high. Therefore, absorbers of high atomic numbers (lead, steel) must be used to protect against its dangerous effects. The course of the mass absorption coefficient, μ/ρ, of air is plotted as a function of the photon energy at a log-log scale (shown in Fig. 8.12). Its value is given by the sum of the coefficients of the interactions. Compton scattering plays a dominant role at photon energies from 0.1 to 3 MeV. 206

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μ/ρ [cm2/g] 0.1

μ/ρ

0.01

0.01

0.1

1

10

E [MeV]

100

Figure 8.12: Mass absorption coefficient (μ/ρ) for air as a sum of of the mass absorption coefficient of photoeffect (τ/ρ), Compton scattering (κ/ρ).

8.3.4 Neutron interactions Neutrons cannot interact by ionisation directly because they lack an electric charge. However, indirect ionisation can occur due to elastic collisions with hydrogen nuclei (protons), causing ionisation effects. This mechanism plays an important role in living systems that contain a large amount of water. Of the possible neutron interactions, interactions with the atomic nuclei of absorbers become significant under forces of strong interaction. When a neutron approaches a target nucleus, it can be captured by nuclear forces to form a compound nucleus in an excited state, (X + n)*. The energy obtained by the nucleus is released in various ways over a very short period of time (10−16–10−12 s). It can be released in the form of a particle/group of particles or by emission of a quantum of electromagnetic radiation. This process is called a nuclear reaction and occurs as follows: decay of the composed nucleus

X + n ⇒ target nucleus

(X + n)* ⇒ X´ + x composed new particle or quantum nucleus nucleus emitted by the nucleus

A nuclear reaction in short notation is written X (n,x)X´. The excitation energy supplied, E*, which excites the nucleus (assuming the mass of the neutron is less than that of nucleus M ), is given by E* = Ekin + Eb , n (8.33) 207

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where Ekin is the kinetic energy of the neutron and Eb.n is its binding energy. The binding energy Eb.n transfers to the nucleus with the incident neutron, while the emission of particle (particles) x is accompanied by an energy decrease in the nucleus and the binding energy of particle (particles) Eb.x. If the difference Q = Eb , n − Eb , x(8.34) is positive, an exothermic reaction takes place, which may occur without any supply of energy. However, if Q < 0, i.e. if the binding energy of particle (particles) x is higher than that of the neutron, then an endothermic reaction takes place. This can only occur if the incident neutron possesses sufficiently high kinetic energy to overcome the endothermic nature of reaction Q. The minimum neutron energy required for the endothermic reaction is called the energy threshold of the reaction. Naturally, exothermic reactions are non-threshold reactions and can occur at any energy of the incident neutron. De-excitation of the compound nucleus occurs upon the emission of a particle identical to the one absorbed, i.e. the neutron. This (n,n´) reaction is called elastic scattering, whereby the sum of kinetic energies of the nucleus and neutron before and after the reaction is conserved. The resulting nucleus is identical to the target nucleus in (n,n) reactions. Inelastic scattering occurs when the nucleus remains in an excited state after the emission of the neutron and is denoted (n,n´). In this case, the atomic and mass numbers of the target and resulting nuclei are also identical. Various radionuclides are produced in reactions of type (n,x), which differ from the target nucleus in atomic and/or mass numbers. The highest yields possess reactions of type (n,γ) where de-excitation occurs by the emission of a quantum of γ-radiation. Reactions of type (n,γ) involve a process known as radiation capture. The compound nucleus can also be split to form two nuclei. This process is called a fission reaction and is denoted by (n,f  ). The number of elements that can be split by neutrons is limited. Different types of interactions have various probabilities and depend on the target nucleus, the energy of the incident neutron and the type of reaction. The probability is given by the value of its effective cross-section. If a sample containing N target nuclei is exposed to a neutron flux with a flux density φ (number of neutrons.m−2.s−1), the number of interactions Xi of a given type in 1 s is given by the relationship X i = σ i N ϕ (8.35) The proportionality constant σi represents the cross-section of the reaction. The cross-section is a function of the neutron energy; its dimension is in m2. Cross-section values for respective reactions are given in various tables.

8.4 DETECTION OF IONISING RADIATION The detection of various types of ionising radiation is based on the interaction of radiation with the sensitive part of a detector. Detectors convert radiation energy into other forms of 208

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energy, which can be registered by devices and techniques used in various branches of physics. The ratio of the number of registered particles or photons to the number that pass through a detector is called the detection efficiency. Various types of detectors are used according to the effects of radiation interaction. They include ionisation, excitation, chemical, thermal, photographic, scintillation and secondary emission detectors. An electric pulse appears at the detector output after the absorption of a particle or photon. Detectors are the input parts of measurement devices. The electric pulses produced are further amplified, formed into an appropriate shape, measured and registered individually by a counter (measurement of activity). Alternatively, the mean count rate is evaluated (measurement of the mean intensity of radiation). There are two types of detector connections: coincidence and non-coincidence connections. Coincidence connections only register impulses generated simultaneously in two or more detectors. This connection is used in the diagnostic imaging method, positron emission tomography (PET). Non-coincidence connections are used only for registering impulses that occur separately in two detectors.

8.4.1 Ionisation chambers An ionisation chamber is a capacitor with two electrodes filled with gas (mostly air). Gas molecules are ionised by charged particles entering the chamber. This produces ion pairs (positive ion and electron) that are attracted to the oppositely charged electrodes of the capacitor (see Fig. 8.13). A very weak ionisation current passes through the chamber. The ionisation currents used in chambers that can be registered have values of approximately 10−5–10−11 A. Photons and quanta of x-rays and γ-rays can be detected by the ionisation chamber, indirectly filling ionised gas. The photoelectric effect, Compton scattering and the formation of electron-positron pairs all involve the release of ionising particles. Anode

Ionising radiation

Cathode

Figure 8.13: Diagram of ionization chamber.

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Ions may disappear in various ways: through electrodes, chamber walls or by recombi-­ nation. Ions with charge e move in the electric field of the chamber since the force at F = eE (where E is the intensity of the electric field due to the presence of a voltage difference between the electrodes) attracts them to oppositely charged electrodes. The higher the operating voltage, the higher the accelerating force and the higher the intensity of the electric current flowing through the chamber. Two regions can be observed along the current curve plotted as a function of the applied voltage (see Fig. 8.14). The first part, where the intensity of the electric current is proportional to the applied voltage applied, is called the region of Ohm’s law. The second, flat part, where the intensity does not depend on the voltage, is called the region of saturated current (or saturated ionisation). The second part starts at a certain voltage value, where the acceleration force is so high that the ions have no time to recombine. For detecting activities over 10 MBq, a galvanometer or electrometer is used to measure the total electric current. Ionisation chambers can also operate in a pulsed mode. Ionisation chambers also come in the form of pocket personal dosimeters for evaluating exposure to ionising radiation and estimating absorbed doses of radiation or in the form of exposure rate meters used in x-ray dosimetry.

U Figure 8.14: Current intensity as a function of the voltage applied.

8.4.2 Geiger-Müller counter The Geiger-Müller counter or Geiger-Müller tube (GMT) is mostly used for detecting β-radiation; scintillation detectors are used for detecting γ-radiation. The detection efficiency for γ-radiation is only approximately 1%. The measuring part of the GMT consists of a tube and fibre surrounded by gas with wires under high voltage. Transiting particles strike the gas atoms, generating ions and electrons. Electrons incident on the anode are then registered as pulses. A metallic cylinder represents the cathode and an axially mounted tungsten wire represents the anode (see Fig. 8.15). The tube is filled with a mixture of inert gas, mostly argon (90%) and polyatomic gas (10%), e.g. ethanol vapours. End-window tubes are used to detect β-radiation. 210

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A relatively high voltage is loaded between the cathode and anode (in an order of 102–103 V). When the electrically charged particle enters the gas filling of the tube it collides with gas atoms, resulting in ionisation. Due to the relatively high voltage difference between the electrodes, the ions formed are accelerated into high kinetic energies, which can in turn lead to further ionisation of argon atoms. Thus, an electron avalanche can occur as a consequence of the formation of only one ion pair. This avalanche of electrons results in a pulse (which is further registered) in the electric current between the electrodes. The count rate of the pulses plotted as a function of the operating voltage reveals a plateau, along which the count rate is essentially independent of the operating voltage (see Fig. 8.16).

Cathode ica w M do n wi

Anode Atom

Ionising radiation

Electron Ion

Counter

R 500 V

Figure 8.15: Diagram of a Geiger-Müller tube.

Figure 8.16: Count-rate of GMT as function of voltage U.

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8.4.3 Scintillation counter The scintillation counter converts absorbed energy into a corresponding quantity of light. It consists of the following: (1) a scintillator for transforming radiation energy into luminescence radiation, (2) a photomultiplier for detecting flashes (scintillation) and (3) electronic parts. Thallium-activated sodium iodide crystal (NaI(Tl)) is frequently used as a scintillator. Plastic, liquid or other solid scintillators can also be used. When a particle or photon of γ-radiation passes through the scintillator, the excitation of some atoms occurs due to the absorption of radiation in the scintillator. A photon of visible or ultraviolet light is emitted during de-excitation. The luminescence photon impacts with the photocathode of the photomultiplier. The resulting emission of electron occurs due to the photoelectric effect. The voltage pulse induced by the particle or photon is directly proportional to its energy. The photomultiplier contains approximately 8–14 dynodes, electrodes of subsequently increasing voltage (see Fig. 8.17). The electrons reach the first dynode of the lowest voltage. After this impact, the secondary electrons are ejected and attracted to the second dynode of a higher and more positive voltage and so on. Consequently, multiple electrons are formed due to the impact of the first. The total amplification of the photomultiplier is approximately 105–107. This avalanche of electrons ends at the last dynode, where the anode and the pulse of voltage is formed at the resistance connected to the anode. The pulses are amplified and then counted. Radiation

Scintillator

Photocathode Dynods

Anode

Figure 8.17: Diagram of a scintillation

212

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The scintillator and photomultiplier are usually placed in a container (to avoid light entering from outside) along with a pre-amplifier and scintillation head. In in vivo applications, the scintillation head detects γ-radiation from only one direction as a result of shielding by a steel or lead collimator (see Fig. 8.18). An advantageous property of the scintillation detector is that amplitude U of the voltage pulse, which results from the absorption of a particle or photon in the scintillator, is proportional to energy E, i.e. U = kEγ (8.36) Therefore, the scintillation facilitates radiation spectrometry and enables various incorporated γ-emitters to be distinguished, i.e. the double tracer technique. Crystal NaI(Tl), most frequently used in medical devices, has an emission spectrum of fluorescence radiation with a maximum wave length of 410 nm emitted within a time interval of approximately 2.5×10−7 s. This type of crystal must be protected against humidity. The detection efficiency of a scintillation detector is approximately 30–50% for γ-radiation. Liquid scintillators are used for detecting soft β-emitters mostly dissolved inside the scintillator. 2 1 3

3

Figure 8.18. Diagram of a collimator 1 – Crystal, 2 – Photomultiplier, 3 – Shielding, Dotted lines – rays of radiation.

8.4.4 Semiconductor-based detector

Countrate

A semiconductor-based detector is a device that uses a semiconductor to detect traversing charged particles or absorbed photons. Semiconductors are either crystalline (usually silicon or germanium) or amorphous solids. They have electrical conductivity opposite to temperatures of metal and although they offer higher electrical resistance than typically resistant materials their resistance is still much lower than insulators. The conducting properties of semiconductors can be altered to pass an electric current more easily in one direction than the other and to show variable resistance sensitivity to light or heat. This is achieved by the controlled introduction of impurities (called doping) into the crystal structure, which lowers its resistance but also permits the creation of semiconductor junctions between differently doped regions in the crystal. The behaviour of charge carriers – including electrons, ions and electron holes at these junctions – is the basis of diodes, transistors and all modern electronics. 213 ES basics_medical_physics.indd 213

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The energy required for the production of electron-hole pairs is very low compared to the energy required for the production of paired ions in gas detectors. Consequently, in semiconductor detectors, the statistical variation of the pulse amplitude is smaller and the energy resolution is higher. Moreover, the density of a semiconductor detector is very high, and charged particles of high energy can emit energy in a semiconductor of relatively small dimensions 2 compared with gaseous detectors.

1

8.4.5 Integral and selective detection of γ-radiation 3

3

Scintillation detectors with NaI(Tl) crystals are used in nuclear medicine for in vivo measurements as well as in laboratories for in vitro experiments. Because the amplitude spectrum is of a continuous character due to the presence of Compton scattering and the use of an amplitude discriminator, two modes of detection can be performed, i.e. selective or integral detection. All pulses of amplitude higher than a chosen level are detected in the integral detection; the total number of pulses is proportional to the area under the curve of the number of pulses plotted as a function of energy (see Fig. 8.19). Selective detection (photopeak detection) occurs when only pulses with an amplitude higher than the lower discrimination level and lower than the higher discrimination level are registered. Selective detection is advantageous for enhancing the spatial resolution of the scintillation head and decreasing the detection background caused by cosmic rays and scattered radiation. The energy resolving power, R, of a scintillation head for spectrometric measurements is defined by the width of the photopeak, ΔE, measured at half its height and expressed as a percentage of the mean energy, Emean, as follows ∆E .100(%)(8.37) Emean

Countrate

R=

ES

E/keV

Figure 8.19: Amplitude spectrum of pulses.

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8.5 BASIC QUANTITIES IN RADIATION DOSIMETRY Emission is the basic quantity for describing a radiation source, defined as the number of particles (quanta) emitted by the source in a unit of time. Its dimension is s−1. If the source is a radionuclide, the emission is related to activity A, which estimates the number of decays occurring within the given amount of radionuclide in one second. The unit of activity is 1 becquerel (Bq), which corresponds to the disintegration of one nucleus in 1 s. Its dimension is s−1. The formerly applied unit was Ci (curie). 1 Ci = 3.7×1010 Bq. A certificate containing all data about activity and time (to which the latter is related) is supplied together with the radioactive preparation upon delivery. Based on the decay curve, it is easy to estimate the activity of the pharmaceutical preparation at the time of administering to the patient. The activity decreases exponentially over time and thus a larger volume of the preparation must be injected to ensure the count-rate required for detection purposes. This is frequently a limiting factor in the application of short-lived radiopharmaceuticals. Various radioisotopes possess various masses at the same activity. The mass M of a radionuclide can be calculated by M = 2.073 × 10−15 A.(MBq )T f (8.38) where M is expressed in μg, A is the nucleon number, MBq is the activity expressed in MBq and Tf is the physical half-life in days. The characteristic property of an x- or γ-radiation field is the exposure. This quantity is defined on the basis of ionisation effects and estimates the ionising power of radiation in the air. Exposure X is defined as the ratio of electric charge ΔQ of ions created by the complete absorption of corpuscular particles (electrons, positrons) formed due to the interaction of x- or γ-rays in a volume of air with mass Δm, i.e. X = ΔQ/Δm. The unit of exposure is C.kg−1 with a dimension of kg−1.s.A. The unit formerly applied was 1 roentgen = 2.58×10−4 C.kg−1. Exposure rate dX/dt is defined as exposure ΔX formed in time interval Δt, i.e. dX/dt = ΔX/Δt. The exposure rate unit is A.kg−1 with a dimension of m2.s−2.A. Absorbed dose D is the ratio of the mean energy of ionising radiation ΔE, absorbed by the volume element to its mass Δm, D = ΔE/Δm. The unit is gray (Gy) = J.kg−1. 1 Gy is defined as the absorption of one joule of radiation energy per kilogram of matter or tissue. Its dimension is m2.s−2. The unit formerly applied was rad = 10−2 Gy. Dose rate dD/dt is the mean increase of the dose, where ΔD is within the time interval of Δt and dD/dt = ΔD/Δt. Its unit is 1 W.kg−1. Dose equivalent H is applied to evaluate the radiation effect on living organisms. This quantity represents a modified dose, H = D.Q.N, which correlates with the magnitude or probability of the biological effect of various types of ionising radiation. Q is the quality factor of the type of radiation (Q = 1 for x- or γ-radiation, Q = 2 for fast neutrons and protons, Q = 20 for α-radiation) and N represents the other factors that determine the conditions of irradiation. Its unit is the sievert (S) and its dimension is identical to that of the absorbed dose, i.e. m2.s−2. The sievert represents the equivalent biological effect of the deposit of a joule of radiation energy in a kilogram of human tissue. The equivalence to the absorbed dose is denoted by parameter Q. Energy of 34 eV = 34×1.6×10−19 J is required to produce one ion pair in the air. Therefore, exposure 1 C.kg−1 in the air corresponds to a dose of [1/1.6×10−19]×34×1.6×10−19 = 34 Gy. 215

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In tissues rich in water, this exposure corresponds to a dose of 36 Gy. For other tissue compositions (e.g. the effective atomic number of bone, Zef.bone = 13.7 is much higher than that of skeletal muscle, Zef.muscle = 7.6), the number of secondary electrons released due to the photoelectric effect is higher and thus the absorbed dose must also be higher (see Fig. 8.20). 150

Gy (C.kg–1)

100

50

0.01

bone muscle air fat 0.1

1

10

100

E (MeV)

Figure 8.20: Ratio Gy/(C/kg) for various tissues.

8.5.1 Personal dosimeters Technicians working with radionuclides or ionising radiation must check the dose absorbed by their bodies during operation. Various dosimeters are used based on ionising effects (ionisation chambers), effects on photographic emulsion (film-dosimeters) or excitation effects (thermoluminescence dosimeters). A film dosimeter is similar to the x-ray film used in dentistry. The emulsion is sensitive to ionising radiation and the dose is evaluated based on measuring its blackening after the development of the film. Since the blackening is a function of the radiation energy, metal filters are situated in the film-badges to evaluate the energy to which the emulsion is exposed. An evaluation at an exposure less than 5 μC.kg−1 cannot be accurate. Film-badges are evaluated centrally. The thermoluminescence dosimeter is based on the excitation effect of radiation in suitable materials, e.g. the crystal form of powder-like lithium fluoride (LiF) or in the form of small rods of 1 mm thickness and a cross-section of 3 mm2. The electrons remain in excited states in the crystal lattice after exposure to radiation. After heating to a temperature above 100 °C, de-excitation occurs followed by the emission of visible light. Its intensity can be measured by a photomultiplier, which is proportional to the dose absorbed by the crystal. As this type of dosimeter can be exposed to room temperature and humidity, it can be used over longer time intervals (up to 3 months). The pocket dosimeter is a small ionisation chamber in the form of a fountain pen or small ovoid filled with air and loaded with an electric charge. The ionisation effects create ion pairs in the gas filling, thus decreasing the charge. The dose or dose-rate can be measured but the disadvantage of this dosimeter is its sensitivity to humidity. 216

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8.6 USE OF NUCLEAR MEDICINE IN THERAPY Nuclear medicine offers therapeutic procedures that use small amounts of radioactive material to treat cancer. There are two basic therapeutic procedures: (1) irradiation and (2) stereotactic radiotherapy or radiosurgery. Irradiation is based on the selective uptake of radio­pharmaceuticalspecific tissue and on the premise that rapidly dividing cells are particularly sensitive to damage by radiation. For this reason, some cancerous growths can be controlled or eliminated by irradiating the area containing the growth. Charged particles emitted from radioactive nuclei (usually β-electrons) ionise atoms of the treated tissue, leading to the damage and eventual destruction of DNA in the cell nuclei. In radiation therapy, the amount of radiation applied varies depending on the type and stage of cancer. For curative cases, the typical dose for a solid tumour ranges from 60 to 80 Gy. Lymphomas are treated with a dose ranging from 20 to 40 Gy. Preventive doses are typically approximately 45–60 Gy, while small fractions (1.8–2 Gy) are applied for breast, head and neck cancers. In diagnostics, the average radiation dose is significantly lower. For example, in the case of an abdominal x-ray it is 0.7 mGy, while in the case of an abdominal CT scan it is 8 mGy. Whole-body acute exposure to 5 Gy or more of high-energy radiation usually leads to death within 14 days. This dose represents 375 joules for a 75 kg adult. Stereotactic radiotherapy and radiosurgery consist of applications of extreme energy doses in one or a few fractions. Therefore, extreme requirements are needed to ensure the geometric accuracy of the irradiation. Such an example is the Leksell Gamma Knife. The common source of radiation in the Leksell Gamma Knife is 60Co with two energies (1.17 MeV and 1.33 MeV) of emitted gamma photons. The half-life of 60Co is 5.3 years. A high number of multiple 60Co beams directed in one focus point are used. This geometry leads to a high dose to the focus point and a very low dose to the surrounding tissue. There are more than 200 sources of radiation. Radiosurgical treatment is performed in one single fraction. A Leksell Gamma Knife stereotactic frame, which is attached to the head of the patient, is used for precise imaging and targeting. The whole procedure typically involves only one single fraction of radiation and is carried out over the course of one treatment day.

8.7 USE OF NUCLEAR MEDICINE IN DIAGNOSTICS Diagnostic techniques in nuclear medicine consist of imaging methods. They have a unique ability to visualise metabolic changes even at the molecular level, thus enabling information to be obtained on the biological activity of tissues or the function of the organ under investigation. In comparison with radiodiagnostic methods, nuclear imaging methods commonly involve the electromagnetic radiation flux of photons. Radiodiagnostic methods use x-ray radiation with a continuous spectrum and spectral lines. The source of radiation (x-ray tube) lies outside the body of the patient. In the case of nuclear imaging methods (scintigraphy – gamma camera, single photon emission computerised tomography, positron emission tomography), the source of the gamma photons (with spectral lines) lies inside the body of the patient. These tracers are generally short-lived isotopes linked to chemical compounds that can analyse specific physiological processes. A stable radionuclide-labelled carrier is administered to the organ/tissue under investigation (by injection, inhalation or orally) and selectively absorbed by the target organ (accumulated in various pathological 217

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and healthy tissues). The chemical composition of the radiopharmaceuticals determines the pharmacokinetics and the inclusion of certain metabolic processes. The primary parameter is volume activity [Bqcm−3], which indicates the number of unstable nuclei per unit volume. The disadvantage of these imaging methods is that they are expensive because of the high price of the cyclotrons used to produce radionuclides with short half-lives. Their spatial resolution is also lower in comparison with CT and MR.

8.7.1 Radionuclides Radiopharmaceuticals consist of radionuclide and bound agents specific to particular tissues, e.g. water, glucose or tissue receptors. The radionuclide itself should ideally originate outside the patient without any interaction (i.e. without ionising the energy transfer). The burden of radiation imitation is that the radionuclide must be presented in the body only for examination and then decomposed. With regard to detection efficiency, the energy used is usually in the range of 50–600 keV. Higher detection efficiency is achieved for lower energies. The most commonly used radionuclide in nuclear medicine is 99mTc, which produces pure γ-rays with an energy of 140 keV and a half-life of 6 hours. In clinical practice, especially in oncology, 18F-FDG fluorodeoxyglucose (half-life of 2 hours) and 18F-FLT fludeoxythymidine (half-life of 2 hours) are mostly used in positron emission tomography examinations. In the case of 18F-FDG, the higher uptake of radionuclides is expressed by the affinity for cells with increased metabolism (increased need for sugar/glucose), thus bypassing respective transport proteins before phosphorylation. 18F-FDG, unlike true glucose, is not further metabolised in the cell and thus accumulates in tumour cells that are metabolically very active. 18F-FDG lacks the 2′ hydroxyl group (present in normal glucose) needed for further glycolysis (metabolises glucose by splitting). 18F-FLT mainly accumulates in proliferating cells and promotes the activity of the enzyme thymidine kinase, which characterises the intensity of cell division. It achieves a better contrast in proliferating tumour lesions than 18F-FDG.

8.7.2 Scintigraphy (planar gamma radiography) The gamma camera is the basic imaging detector used in nuclear medicine. A detection head scintillation camera consists of a scintillation crystal, a light guide, a set of photomultipliers and a collimator. Through the interaction of the photon in the scintillator with the photocathode of the photomultipliers, a photoelectron is formed. It first strikes a dynode (i.e. electrode), leading to the emission of more secondary electrons. Consequently, on each dynode the number of electrons that strike the anode photomultipliers increases greatly. These electrons create a photomultiplier output current or voltage pulse. The brightness of scintillation is proportional to the photon energy. The sum of the amplitude of all pulses provides information about the energy transmitted by one photon to the gamma radiation detector. The most basic, cost-effective and simple imaging method is planar gamma radiography (also called scintigraphy), which produces a 2D-image projection of the radionuclide distribution. Spatial resolution is poor, resulting in inaccuracies with localisation flashes in the scintillation crystal system that uses the photomultiplier tubes (i.e. internal spatial resolution of the detector head) and with the spatial resolution of the collimator. The spatial resolution of the 218

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detector heads disimproves with increasing distance from the face of the parallel-holed collimator. Other types of collimators use converging holes, diverging holes and pinholes. Image quality is affected by scattered radiation in the tissue (disimproving with increasing distance from the detector), by quantum noise due to radioactive decay (followed by the emission of photons subjected to random fluctuations) and by the linear attenuation coefficient (which depends on radiation energy and the atomic number and density of the tissue). Scintigraphy images reflect the absorption reduction in the number of pulses in the structures located at greater depths, compared to structures on the surface. The principle of gamma camera scanning is shown in Fig. 8.21. The image of a skeleton generated by planar gamma radiography is shown in Fig. 8.22. A summation image (similar to a conventional x-ray image) is generated as a result of the radiation emitted from different depths. This is collated as information on superposition and summation with regard to the distribution of radiopharmaceuticals from all depth layers of tissues and organs in one image.

Figure 8.21: Gamma camera imaging. The gamma camera only detects partially emitting photons. The 2D images only provide information on the distribution of radiopharmaceuticals because the signal is superposed from all depth layers.

Figure 8.22: Whole-body planar scintigraphy of a skeleton. Accumulated radionuclides are visualised as dark areas in the lesion (the shoulders and joints of the left hand) and in the urinary bladder through which they are excreted.

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8.7.3 Single photon emission computerised tomography Single photon emission computerised tomography (SPECT) is an imaging method that produces 3D distribution of radionuclides, which are determined from emission projections using reconstructive absorption collimation. The principle of SPECT is that the camera rotates around the object and under a number of different angles to acquire planar (scintigraphy) projection data (Fig. 8.23). Because it employs the same type of data acquisition used in CT imaging, the same reconstruction algorithms are also used (iteration methods, Fourier transform and filtered back-projection). Because it fundamentally affects the signal-to-noise ratio, the most important part of SPECT scanners is the collimation system (Fig. 8.24). Absorption collimator efficiency only registers up to 10%, while performance decreases rapidly with increasing energy. The quality of SPECT images is affected mainly by the properties of the collimator, inhomogeneity detectors (different number of registered pulses), acquisition and detection geometry (the distance of displayed scenes from the face of the collimator), absorption of radiation and the mechanical instability of the rotation axis (camera detectors can weigh in the hundreds of kilograms). Most of the detectors used in current SPECT systems are based on a single or multiple NaI(TI) scintillation detectors. SPECT images are shown in figure 8.25. Rotating Nal(TI) Detector Module with Pb/W Collimator in Front Direction of Rotation Signals to electronic Emitted γ-rays Scintillation site for γ-ray parallel to collimator holes Absorbed in collimator

Patient

tracer

Tc

99m

Radioisotope decay by γ emission

tracer

Tc + γ

99

Figure 8.23: Direction of rotation/gamma(sign)-emission/Scintillation site for gamma(sign)-rays parallel to the collimator holes.

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Detectors array

L

h Photon is detected Photon is absorbed in collimator Source of radiation

Figure 8.24: Principle of collimator

Figure 8.25: SPECT imaging. Data obtained from various angles allows images to be reconstructed for any plane (left) as well as 3D visualisation (right).

8.7.4 Positron emission tomography Positron emission tomography (PET) is an imaging method that produces 3D images of radionuclide distribution (introduced in a stable state). The transformation of proton to neutron is accompanied by the emission of β+ positron radiation with a continuous spectrum. After random transit at a few millimetres at most, a β+ positron interacts with a β– electron. This is called annihilation (see section 8.3.3.3). As a consequence, two γ-rays (photons) with an energy of 511 keV travelling in opposite directions are simultaneously released from the site of annihilation (Fig. 8.26). These two photons are detected by the PET detector system 221

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and correspond to multiple rings of the scintillation crystals. Image reconstruction is key to detecting a pair of photons in coincidence (propagating in opposite directions). Coincidence detection is used for electronic collimation of γ-radiation and the subsequent reconstruction of tomographic images. By collecting a statistically significant number of radioactive events, mathematical algorithms reconstruct a three-dimensional image depicting the distribution of the positron-emitting molecules. The site of emission (and thus the monitored radiopharmaceutical) lies at the junction sites of these two detection events (Fig. 8.27). 18

F

18

O

2 hours

p

E = mc2 = 511 keV

n β+ ≈ 2 mm

e– ~ 180 deg

Figure 8.26: Principle of PET. Two photons are simultaneously emitted as a consequence of annihilation due to the interaction of a β+ positron with a β− electron. During its short life-time, the positron (10−7 s) travels in a random direction, the distance of which also depends on the type of radionuclide.

∆t < 10 ns: record

Figure 8.27: Principle of PET image reconstruction. Only detectors that detect photons at the same time (in practice the time difference can be less than 10 ns) are suitable for image reconstruction. The lesion lies on a straight line between these two detectors.

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The main limitation with the resolution of PET images is the random trajectory of the positron before annihilation. Each isotope has a different maximal range; 18F has 2.6 mm, 68 Ga has 9 mm and 82Rb has 16.5 mm. Therefore, PET images are often overlaid with CT or MR images (Fig 8.28). Another important factor that affects PET images is that the number of detected photons is smaller than the number of emitted positrons. Compton scattering and random coincidence can impair the reconstruction of PET images, because they can activate incorrect detectors (see Fig. 8.29).

PET

CT

CT-PET

Figure 8.28: PET imaging. PET image (left), CT image (middle) and fusion of CT and PET images (right) of a brain with glioblastoma. The PET image was obtained 15 min after 18F-FLT application. The combination of high-quality imaging of anatomical structures and functional (biological activity) imaging provides more detailed information about the patient.

A

B

Figure 8.29: False detection in PET. Compton scattering (left) and random coincidence (right) activate incorrect detectors. The thick line represents the real trajectory of the photons, while the dotted line shows the resulting line connecting the activated detectors. This line does not intersect the lesions (circles).

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There are key differences between PET and SPECT. As they operate with different types of radionuclides, two γ-photons (travelling in the opposite direction) are emitted from the transformation site of the radionuclide in PET instead of only one in SPECT. The advantage with PET is that instead of an absorption collimator, an electronic collimator is used. Moreover, as the PET detector system uses ring geometry, more photons contributing to the reconstruction of the image can be detected. The image quality of PET is also better than SPECT.

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