Physics and Radiobiology of Nuclear Medicine PDF

Physics and Radiobiology of Nuclear Medicine Gopal B. Saha, Ph.D. Physics and Radiobiology of Nuclear Medicine Fourth Edition 13 Gopal B. Saha, Ph.D. Emeritus Staff Cleveland Clinic Cleveland, OH 44195 USA ISBN 978-1-4614-4011-6     ISBN 978-1-4614-4012-3 (eBook) DOI 10.1007/978-1-4614-4012-3...

Physics and Radiobiology of Nuclear Medicine Gopal B. Saha, Ph.D. Physics and Radiobiology of Nuclear Medicine Fourth Edition 13 Gopal B. Saha, Ph.D. Emeritus Staff Cleveland Clinic Cleveland, OH 44195 USA ISBN 978-1-4614-4011-6     ISBN 978-1-4614-4012-3 (eBook) DOI 10.1007/978-1-4614-4012-3 Springer New York Dordrecht Heidelberg London Library of Congress Control Number: 2012948009 © Springer Science+Business Media New York 1993, 2001, 2006, 2013 All rights reserved. This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden. The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights. While the advice and information in this book are believed to be true and accurate at the date of going to press, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)  To All my benefactors Preface The fourth edition of this book has been prompted by a need to provide up-to-date information in keeping up with perpetual growth and improvement in instrumen- tation and techniques employed in nuclear medicine since its last edition in 2006. Like the past editions, the book is intended for the radiology and nuclear medi- cine residents to prepare for the American Board of Nuclear Medicine, American Board of Radiology and American Board of Science in Nuclear Medicine exami- nations, all of which require strong physics background. Also, it will serve as a textbook on nuclear medicine physics and instrumentation for nuclear medicine technologists taking the Nuclear Medicine Technology Certification Board ex- amination. The organization of the book has been kept the same as in the previous edi- tions consisting of 16 chapters—starting with basics in initial chapters and then progressing into more developed techniques in later chapters. Chapters 1–4, 6, 8 and 11 have no major changes. An expanded description of a cyclotron and two examples of calculation of yield of activity produced are added to Chapter 5 along with the updating of Table 5.1. Chapter 7 includes additional information on the ion chamber survey meter, proportional counter, and G-M counter. The section on solid state digital cameras has been revised in Chapter 9. In Chapter 10, sev- eral changes have been made, namely, an expanded collimator section, a detailed description of “contrast”, and a more detailed quality control section. Changes in Chapter 12 include: detailed iterative reconstruction method, description of CT scanners in SPECT/CT, detailed description of attenuation correction in SPECT/ CT, and expanded sections on partial volume effects and quality control of CT scanners. Included in Chapter 13 are: upgraded information on detectors and PET scanners; a new section on PET/MR including the attenuation correction method and its quality control tests; the time of flight technique; several scatter correction techniques; and a section on accreditation of nuclear medicine and PET facilities. Doses in Table 14.4 have been updated, and new tables with revised pediatric dosages have been included in Chapter 14. For consistency, the section, “Sources of Radiation exposure in USA” has been removed from Chapter 16 and added to Chapter 15 with revised data, while the sections “Dirty Bombs”, “Verification Card for Radioactive Patients” and “Radiation Phobia” have been moved from vii viii Preface Chapter 15 and added to Chapter 16. “European Regulations Governing Radia- tions” in Chapter 16 has been revised. A number of figures and tables have been added for better explanation of the text contents throughout the book. It has been a great pleasure to work with, for which I am ever grateful to, Andrew Moyer, Senior Editor of Clinical Medicine at Springer, for his sincere support during the publication of several of my books by Springer including this one. I would like to thank Ian Hayes, Editorial Assistant, and Joseph Quatela, Senior Production Editor of Springer for their support in the production of this book. Gopal B. Saha, Ph.D. Contents Preface vii Chapter 1 Structure of Matter    1 Matter and Energy    1 Radiation    2 The Atom    3 Electronic Structure of the Atom    3 Structure of the Nucleus    6 Nuclear Binding Energy    7 Nuclear Nomenclature    8 Chart of the Nuclides    8 Questions 10 Suggested Readings 10 Chapter 2 Radioactive Decay 11 Spontaneous Fission 11 Isomeric Transition 12 Gamma (γ)-Ray Emission 12 Internal Conversion 12 Alpha (α)-Decay 14 Beta (β−)-Decay 15 Positron (β+)-Decay 17 Electron Capture 18 Questions 20 Suggested Readings 20 Chapter 3 Kinetics of Radioactive Decay 21 Radioactive Decay Equations 21 General Equation 21 Half-Life 22 Mean Life 24 Effective Half-Life 25 ix x Contents Units of Radioactivity 25 Specific Activity 26 Calculation 27 Successive Decay Equations 30 General Equation 30 Transient Equilibrium 30 Secular Equilibrium 31 Questions 32 Suggested Readings 34 Chapter 4 Statistics of Radiation Counting 35 Error, Accuracy, and Precision 35 Mean and Standard Deviation 36 Gaussian Distribution 36 Standard Deviation of Count Rates 38 Propagation of Errors 38 Chi-Square Test 41 Minimum Detectable Activity 43 Evaluation of Diagnostic Tests 43 Questions 45 Suggested Readings 45 Chapter 5 Production of Radionuclides 47 Cyclotron-Produced Radionuclides 47 Reactor-Produced Radionuclides 50 Fission or (n, f) Reaction 51 Neutron Capture or (n,γ) Reaction 52 Target and Its Processing 52 Equation for Production of Radionuclides 52 Radionuclide Generators 57 99 Mo–99mTcGenerator 59 Cyclotron production of 99mTc 60 Questions 60 References and Suggested Readings 61 Chapter 6 Interaction of Radiation with Matter 63 Interaction of Charged Particles with Matter 63 Specific Ionization 64 Linear Energy Transfer 65 Range 65 Bremsstrahlung 67 Annihilation 67 Interaction of γ-Radiations with Matter 68 Mechanism of Interaction of γ-Radiations 68 Attenuation of γ-Radiations 72 Interaction of Neutrons with Matter 75  Contents xi Questions    76 Suggested Readings    77 Chapter 7 Gas-Filled Detectors    79 Principles of Gas-Filled Detectors    79 Ionization Chambers    81 Ion Chamber Survey Meter    82 Dose Calibrator    83 Pocket Dosimeter    86 Proportional Counters    87 Geiger–Müller Counters    87 Questions    90 Suggested Readings    90 Chapter 8 Scintillation and Semiconductor Detectors    91 Scintillation Detectors    91 Solid Scintillation Detectors    92 Solid-State Detectors    94 Solid Scintillation Counters    95 NaI(Tl) Detector    95 Photomultiplier Tube    95 Preamplifier    97 Linear Amplifier    97 Pulse-Height Analyzer    97 Display or Storage    98 Gamma-Ray Spectrometry    98 Photopeak    98 Compton Valley, Edge, and Plateau    99 Characteristic X-Ray Peak 100 Backscatter Peak 100 Iodine Escape Peak 100 Annihilation Peak 101 Coincidence Peak 101 Liquid Scintillation Counters 102 Characteristics of Counting Systems 104 Energy Resolution 104 Detection Efficiency 106 Dead Time 108 Gamma Well Counters 110 Calibration of Well Counters 110 Counting in Well Counters 111 Effects of Sample Volume 112 Thyroid Probe 113 Thyroid Uptake Measurement 114 Questions 114 Suggested Readings 116 xii Contents Chapter 9 Gamma Cameras 117 Gamma Cameras 117 Principles of Operation 117 Detector 119 Collimator 120 Photomultiplier Tube 121 X-, Y-Positioning Circuit 121 Pulse-Height Analyzer 123 Display and Storage 123 Digital Cameras 124 Solid State Digital Cameras 124 Questions 125 Suggested Readings 126 Chapter 10 Performance Parameters of Gamma Cameras 127 Spatial Resolution 127 Intrinsic Resolution 127 Collimator Resolution 128 Scatter Resolution 132 Evaluation of Spatial Resolution 132 Bar Phantom 132 Line-Spread Function 134 Modulation Transfer Function 135 Sensitivity 137 Collimator Efficiency 137 Uniformity 139 Pulse-Height Variation 139 Nonlinearity 139 Edge Packing 140 Gamma Camera Tuning 141 Effects of High Counting Rates 141 Contrast 142 Quality Control Tests for Gamma Cameras 145 Daily Checks 146 Weekly Checks 148 Monthly Checks 148 Annual, Semiannual, or As-Needed Checks 149 Questions 149 References and Suggested Readings 151 Chapter 11 Digital Computers in Nuclear Medicine 153 Basics of a Computer 153 Central Processing Unit 154 Computer Memory 155 External Storage Devices 155  Contents xiii Input/Output Devices 155 Operation of a Computer 156 Digitization of Analog Data 156 Digital-to-Analog Conversion 157 Digital Images 157 Application of Computers in Nuclear Medicine 158 Digital Data Acquisition 158 Static Study 159 Dynamic Study 160 Gated Study 160 Reconstruction of Images 162 Fusion and Subtraction of Images 162 Display 162 Software and DICOM 163 PACS 164 Questions 166 Suggested Readings 166 Chapter 12 Single Photon Emission Computed Tomography 167 Tomographic Imaging 167 Single Photon Emission Computed Tomography 167 Data Acquisition 169 Image Reconstruction 170 SPECT/CT 184 Factors Affecting SPECT 187 Performance of SPECT Cameras 196 Spatial Resolution 196 Sensitivity 197 Other Parameters 198 Quality Control Tests for SPECT Cameras 198 Daily Tests 198 Weekly Tests 198 Quality Control Tests for CT Scanners 199 Questions 200 References and Suggested Readings 201 Chapter 13 Positron Emission Tomography 203 PET Radiopharmaceuticals 203 Detectors in PET Scanners 204 PM Tubes and Pulse-Height Analyzers 205 PET Scanners 205 Block Detectors 205 Coincidence Timing Window 207 PET/CT Scanners 209 PET/MR Scanners 210 xiv Contents Principles of MR Imaging 212 MR scanner 216 Commercial PET/MR Scanners 217 Mobile PET or PET/CT 218 Micro-PET 219 Dual- and Triple-Head Gamma Cameras 220 Data Acquisition 221 Time of Flight Method 223 Two-Dimensional Versus Three-Dimensional Data Acquisition 224 Image Reconstruction 225 Factors Affecting PET 226 Normalization 227 Photon Attenuation Correction 227 Random Coincidences 230 Scatter Coincidences 230 Dead Time 232 Radial Elongation 232 Performance of PET Scanners 232 Spatial Resolution 232 Sensitivity 235 Noise Equivalent Count Rate 235 Quality Control Tests for PET Scanners 236 Daily Tests 236 Weekly Tests 236 Quality Control Tests for MR Scanners 237 Accreditation of Nuclear Medicine and PET Facilities 238 Questions 240 References and Suggested Readings 241 Chapter 14 Internal Radiation Dosimetry 243 Radiation Units 243 Dose Calculation 246 Radiation Dose Rate 246 Cumulative Radiation Dose 247 Radiation Dose in SI Units 251 Effective Dose Equivalent and Effective Dose 252 Pediatric Dosages 256 Questions 260 References and Suggested Readings 261 Chapter 15 Radiation Biology 263 The Cell 263 Effects of Radiation 266 DNA Molecule 266 Chromosome 269  Contents xv Direct and Indirect Actions of Radiation 270 Radiosensitivity of Cells 272 Cell Survival Curves 274 Factors Affecting Radiosensitivity 276 Dose Rate 277 Linear Energy Transfer 277 Chemicals 278 Stage of Cell Cycle 281 Apoptosis 281 Classification of Radiation Damage 281 Sources of Radiation Exposure in the United States 283 Stochastic and Deterministic Effects 284 Acute Effects of Total Body Irradiation 284 Hemopoietic Syndrome 285 Gastrointestinal Syndrome 285 Cerebrovascular Syndrome 286 Long-Term Effects of Radiation 286 Somatic Effects 286 Genetic Effects 293 Risk Versus Benefit in Diagnostic Radiology and Nuclear Medicine 295 Risk to Pregnant Women 296 Questions 297 References and Suggested Readings 299 Chapter 16 Radiation Regulations and Protection 301 License 301 Radiation Protection 303 Definition of Terms 303 Caution Signs and Labels 304 Occupational Dose Limits 305 ALARA Program 305 Principles of Radiation Protection 306 Personnel Monitoring 309 Dos and Don’ts in Radiation Protection Practice 310 Bioassay 311 Receiving and Monitoring of Radioactive Packages 311 Radioactive Waste Disposal 312 Radioactive Spill 313 Recordkeeping 314 Medical Uses of Radioactive Materials 314 Applications, Amendments, and Notifications 314 Authority and Responsibilities of the Licensee 315 Supervision 315 Mobile Nuclear Medicine Service 315 Written Directives 316 xvi Contents Measurement of Dosages 316 Calibration, Transmission, and Reference Sources 317 Requirement for Possession of Sealed Sources 317 Labeling of Vials and Syringes 317 Surveys of Ambient Radiation Exposure Rate 317 Calibration of Survey Instruments 318 Training and Experience Requirements for Medical Uses of By-Product Materials 318 Report and Notification of a Medical Event 319 Report and Notification of a Dose to an Embryo/Fetus or a Nursing Child 320 Release of Patients Administered with Radiopharmaceuticals 320 Recordkeeping 323 Dirty Bombs 323 Types of Accidental Radiation Exposure 323 Protective Measures in Case of Explosion of a Dirty Bomb 325 Verification Card for Radioactive Patients 326 Radiation Phobia 327 Transportation of Radioactive Materials 328 European Regulations Governing Radiation 330 Questions 332 References and Suggested Readings 333 Appendix A Units and Constants 335 Appendix B Terms Used in Text 337 Appendix C Answers to Questions 343 Index 345 1 Structure of Matter Matter and Energy The existence of the universe is explained by two entities: matter and energy. These two entities are interchangeable and exist in different forms to make up all things visible or invisible in the universe. Whereas matter has a definite size, shape, and form, energy has different forms but no size and shape. Matter is characterized by its quantity, called the mass, and is composed of the smallest unit, the atom. In atomic physics, the unit of mass is the atomic mass unit (amu), which is equal to 1.66 × 10−27 kg. Energy is the capacity to do work and can exist in several forms: kinetic en- ergy (which is due to the motion of matter); potential energy (which is due to the position and configuration of matter); thermal energy (which is due to the motion of atoms or molecules in matter); electrical energy (which is due to the flow of electrons across an electric potential); chemical energy (which is due to chemical reaction); and radiation (energy in motion). Energy can change from one form to another. Of all these forms, radiation is of great importance in nuclear medicine and, therefore, will be discussed in detail. Mass and energy are interchangeable, and one is created at the expense of the other. This is predicted by the Einstein’s mass–energy relationship:  E = mc2 (1.1) where E is energy in ergs, m is the mass in grams, and c is the velocity of light in a vacuum given as 3 × 1010 cm/s. This relationship states that everything around us can be classified as matter or energy. G. B. Saha, Physics and Radiobiology of Nuclear Medicine, 1 DOI 10.1007/978-1-4614-4012-3_1, © Springer Science+Business Media New York 2013 2 1. Structure of Matter Radiation Radiation is a form of energy in motion through space. It is emitted by one object and absorbed or scattered by another. Radiations are of two types: 1. Particulate radiations: Examples of these radiations are energetic electrons, protons, neutrons, α-particles, and so forth. They have mass and charge, except neutrons, which are neutral particles. The velocity of their motion depends on their kinetic energy. The particulate radiations originate from radioactive decay, cosmic rays, nuclear reactions, and so forth. 2. Electromagnetic radiations: These radiations are a form of energy in motion that does not have mass and charge and can propagate as either waves or dis- crete packets of energy, called the photons or quanta. These radiations travel with the velocity of light. Various examples of electromagnetic radiations include radio waves, visible light, heat waves, γ-radiations, and so forth, and they differ from each other in wavelength and hence in energy. Note that the sound waves are not electromagnetic radiations. The energy E of an electromagnetic radiation is given by hc  E = hv = (1.2)  where h is the Planck constant given as 6.625 × 10−27 erg · s/cycle, ν is the frequen- cy in hertz (Hz), defined as 1 cycle per second, λ is the wavelength in centimeters, and c is the velocity of light in vacuum, which is equal to nearly 3 × 1010 cm/s. The energy of an electromagnetic radiation is given in electron volts (eV), which is defined as the energy acquired by an electron when accelerated through a potential difference of 1 V. Using 1 eV = 1.602 × 10−12 erg, Eq. (1.2) becomes  1.24 × 10−4 (1.3) E(eV) =  where λ is given in centimeters. Table 1.1 lists the different electromagnetic radia- tions along with their frequencies and wavelengths. Table 1.1.   Characteristics of different electromagnetic radiations. Type Energy (eV) Frequency (Hz) Wavelength (cm) Radio, TV 10 –10 −10 −6 10 –10 4 8 102–106 Microwave 10−6–10−2 108–1012 10−2–102 Infrared 10−2–1 1012–1014 10−4–10−2 Visible 1–2 1014–1015 10−5–10−4 Ultraviolet 2–100 1015–1016 10−6–0−5 x-Rays and γ-rays 100–107 1016–1021 10−11–10−6  The Atom 3 Table 1.2. Characteristics of electrons and nucleons. Particle Charge Mass (amu)a Mass (kg) Mass (MeV)b Electron −1 0.000549 0.9108 × 10−30 0.511 Proton +1 1.00728 1.6721 × 10−27 938.78 Neutron 0 1.00867 1.6744 × 10−27 939.07 a amu = 1 atomic mass unit = 1.66 × 10−27 kg = 1/12 of the mass of 12C. b 1 atomic mass unit = 931 MeV. The Atom For the purpose of this book, the atom can be considered as the smallest unit in the composition of matter. The atom is composed of a nucleus at the center and one or more electrons orbiting around the nucleus. The nucleus consists of protons and neutrons, collectively called nucleons. The protons are positively charged particles with a mass of 1.00728 amu, and the neutrons are electrically neutral particles with a mass of 1.00867 amu. The electrons are negatively charged par- ticles with a mass of 0.000549 amu. The protons and neutrons are about 1836 times heavier than the electrons but the neutron is heavier than the proton by one electron mass (i.e., by 0.511 MeV). The number of electrons is equal to the num- ber of protons, thus resulting in a neutral atom of an element. The characteristics of these particles are given in Table 1.2. The size of the atom is about 10−8 cm (called the angstrom, Å), whereas the nucleus has the size of 10−13 cm (termed the Fermi, F). The density of the nucleus is of the order of 1014 g/cm3. The elec- tronic arrangement determines the chemical properties of an element, whereas the nuclear structure dictates the stability and radioactive transformation of the atom. Electronic Structure of the Atom Several theories have been put forward to describe the electronic structure of the atom, among which the theory of Niels Bohr, proposed in 1913, is the most plausible one and still holds today. The Bohr’s atomic theory states that electrons rotate around the nucleus in discrete energy shells that are stationary and arranged in increasing order of energy. These shells are designated as the K shell, L shell, M shell, N shell, and so forth. When an electron jumps from the upper shell to the lower shell, the difference in energy between the two shells appears as electro- magnetic radiations or photons. When an electron is raised from the lower shell to the upper shell, the energy difference is absorbed and must be supplied for the process to occur. The detailed description of the Bohr’s atomic structure is provided by the quantum theory in physics. According to this theory, each shell is designated by a quantum number n, called the principal quantum number, and denoted by inte- gers, for example, 1 for the K shell, 2 for the L shell, 3 for the M shell, 4 for the N shell, and 5 for the O shell. Each energy shell is subdivided into subshells or orbit- 4 1. Structure of Matter als, which are designated as s, p, d, f, and so on. For a principal quantum number n, there are n orbitals in a given shell. These orbitals are assigned the azimuthal quantum numbers, l, which represent the electron’s angular momentum and can assume numerical values of l = 0, 1, 2… n−1. Thus for the s orbital, l = 0; the p orbital, l = 1; the d orbital, l = 2; the f orbital, l = 3; and so forth. According to this description, the K shell has one orbital, designated as 1s, the L shell has two orbit- als, designated as 2s and 2p, and so forth. The orientation of the electron’s mag- netic moment in a magnetic field is described by the magnetic quantum number, m. The values of m can be m = −l, − ( l − 1),…, 0,… ( l − 1), l. Each electron rotates about its own axis clockwise or anticlockwise, and the spin quantum number, s ( s = − 1/2 or + 1/2) is assigned to each electron to specify this rotation. The electron configuration of the atoms of different elements is governed by the following rules: 1. No two electrons can have the same values for all four quantum numbers in a given atom. 2. The orbital of the lowest energy will be filled in first, followed by the next higher energy orbital. The relative energies of the orbitals are 1s < 2s < 2p < 3s < 3p < 4s < 3d < 4p < 5s < 4d < 5p < 6s < 4f < 5d < 6p < 7s. This order of energy is valid for lighter elements and is somewhat different in heavier elements. 3. There can be a maximum of 2(2l + 1) electrons in each orbital. 4. For given values of n and l, each of the available orbitals is first singly occu- pied such that no electron pairing occurs. Only when all orbitals are singly occupied does electron pairing take place. 5. Each energy shell contains a maximum of 2n2 electrons. The hydrogen atom has one proton in the nucleus and one electron in the orbit. Its electronic structure is represented as 1s1. The helium atom has two electrons, which are accommodated in the 1s orbital, and thus has the structure of 1s2. Now let us consider the structure of 168 O, which has eight electrons. The first two elec- trons will fill the 1s orbital. The next two electrons will go to the 2s orbital. There are three p orbitals, designated as px, py, pz, which will be occupied by three electrons individually. The eighth electron will occupy the px orbital pairing with the electron already in it. Thus, the electronic configuration of 168 O is given by 1s22s22p4. The electron configurations in different orbitals and shells are illustrated in Table 1.3, and the structure of 28Ni is shown in Fig. 1.1. The electronic structure of the atom characterizes the chemical properties of elements. The outermost shell in the most stable and chemically inert elements such as neon, argon, krypton, and xenon has the electronic structure of ns2np6. Helium, although a noble gas, has the 1s2 configuration. Elements having electronic configurations different from that of the noble gases either lose or gain electrons to achieve the structure ns2np6 of the nearest noble gas atom. The electrons in these shells are called the valence electrons and are primarily responsible for the chemical bond formation.  The Atom 5 Table 1.3.   Electron configurations in different energy shells. Principal quantum number Orbital No. of electrons = 2(2l  + 1) 2n2 Principal shell ( n) ( l) in each orbital K 1 s(0) 2 2 L 2 s(0) 2 p(1) 6 8 M 3 s(0) 2 p(1) 6 d(2) 10 18 N 4 s(0) 2 p(1) 6 d(2) 10 f(3) 14 32 O 5 s(0) 2 p(1) 6 d(2) 10 f(3) 14 g(4) 18 50 Electrons in different shells are held by binding energy in different shells of the atom. The binding energy of an electron is defined as the energy that is required to be supplied to remove it completely from a shell. The binding energy of the electron is the greatest in the K shell and decreases with higher shells such as L, M, and so on. The binding energy also increases with increasing atomic number of the elements. Thus, the K-shell binding energy (21.05 keV) of technetium, with atomic number 43, is higher than the K-shell binding energy (1.08 keV) of sodium, with atomic number 11. The K-shell binding energy of electrons in sev- eral elements are: carbon, 0.28 keV, gallium, 10.37 keV, technetium, 21.05 keV; indium, 27.93 keV; iodine, 33.16 keV; lead, 88.00 keV. When an electron is removed completely from an atom, the process is called ionization. The atom is said to be ionized and becomes an ion. On the other hand, when the electron is raised from a lower energy shell to an upper energy shell, the process is called excitation. Both ionization and excitation processes require a supply of energy from outside the atom such as heating, applying an electric field, and so forth. In the excited atoms, electrons jump from the upper energy shell to Fig. 1.1. The electronic configuration of 28Ni. The K shell has 2 electrons, the L shell has 8 electrons, and the M shell has 18 electrons. 6 1. Structure of Matter the lower energy shell to achieve stability. The difference in energy appears as electromagnetic radiations or photons. Thus, if the binding energy of K-shell elec- trons in, say, bromine is 13.5 keV and the L-shell binding energy is 1.8 keV, the transition of electrons from the L shell to the K shell will occur with the emission of 11.7 keV (13.5 − 1.8 = 11.7 keV) photons. As we shall see later, these radia- tions are called the characteristic x-rays of the product atom. Structure of the Nucleus As already stated, the nucleus of an atom is composed of protons and neutrons. The number of protons is called the atomic number of the element and denoted by Z. The number of neutrons is denoted by N, and the sum of the protons and neutrons, Z + N, is called the mass number, denoted by A. The symbolic represen- tation of an element, X, is given by A Z XN. For example, sodium has 11 protons and 12 neutrons with a total of 23 nucleons. Thus, it is represented as 23 11 Na12. However, the atomic number Z of an element is known, and N can be calculated as A−Z; therefore, it suffices to simply write 23Na (or Na-23). To explain the various physical observations related to the nucleus of an atom, two models for the nuclear structure have been proposed: the liquid drop model and the shell model. The liquid drop model was introduced by Niels Bohr and assumes a spherical nucleus composed of closely packed nucleons. This model explains various phenomena, such as nuclear density, energetics of particle emis- sion in nuclear reactions, and fission of heavy nuclei. In the shell models, both protons and neutrons are arranged in discrete energy shells in a manner similar to the electron shells of the atom in the Bohr atomic theory. Similar to the electronic configuration of the noble gas atoms, nuclei with 2, 8, 20, 28, 50, 82, or 126 protons or neutrons are found to be very stable. These nucleon numbers are called the magic numbers. It is observed that atomic nuclei containing an odd number of protons or neu- trons are normally less stable than those with an even number of protons or neu- trons. Thus, nuclei with even numbers of protons and neutrons are more stable, whereas those with odd numbers of protons and neutrons are less stable. For example, 12C with six protons and six neutrons is more stable than 13C containing six protons and seven neutrons. There are 280 naturally-occurring stable nuclides of which 166 are even N–even Z, 57 are even–odd, 53 are odd–even, and only 4 are odd–odd. The stability of these elements is dictated by the configuration of protons and neutrons in the nucleus. The ratio of the number of neutrons to the number of pro- tons ( N/Z) is an approximate indicator of the stability of a nucleus. The N/Z ratio is 1 in low-Z elements such as 126 C, 147 N, and 168 O, but it increases with increasing atomic number of elements. For example, it is 1.40 for 127 208 53 I and 1.54 for 82 Pb. The plot of the atomic number versus the neutron number of all nuclides is shown in Fig. 1.2. All stable nuclear species fall on or around what is called the line of stability. The nuclear species on the left side of the line have fewer neutrons and more protons; that is, they are proton-rich. On the other hand, those on the right  The Atom 7 100 90 80 70 N = 60 ATOMIC NUMBERZ Z 50 40 30 20 10 10 20 30 40 50 60 70 80 90 100 110 120 130 NEUTRON NUMBER N Fig. 1.2. The plot of atomic number ( Z) versus the number of neutrons ( N) for allnuclides. The proton-rich nuclides fall on the left ( dotted) and the neutron-rich nuclides fall on the right ( cross-hatched) of the line of stability, indicated by the dark-shaded area. The solid line represents nuclides with Z = N. side of the line have fewer protons and more neutrons; that is, they are neutron- rich. The nuclides away from the line of stability are unstable and disintegrate to achieve stability. Nuclear Binding Energy According to the classical electrostatic theory, the nucleus of an atom cannot exist as a single entity, because of the electrostatic repulsive force among the protons in the nucleus. The stability of the nucleus is explained by the existence of a strong binding force called the nuclear force, which overcomes the repulsive force of the protons. The nuclear force is effective equally among all nucleons and exists only in the nucleus, having no influence outside the nucleus. The short range of the nuclear force leads to a very small size (~10−13 cm) and very high density (~1014 g/cm3) of the nucleus. The mass M of a nucleus is always less than the combined masses of the nucle- ons A in the nucleus. The difference in mass ( M−A) is termed the mass defect, which has been used as binding energy for all nucleons in the nucleus. The aver- age binding energy of a nucleon is equal to the total binding energy (calculated from the mass defect) divided by the number of nucleons. It is of the order of 6–9 MeV, although the binding energy of an individual nucleon has a definite 8 1. Structure of Matter value, depending on the shell it occupies. The binding energy of a nucleon must be supplied to completely remove it from the nucleus. Note that whereas the bind- ing energy of the nucleons is in the megaelectron volt (MeV) range, the electron binding energy in the atomic orbital is of the order of kiloelectron volts (keV), a factor of 1000 lower. Nuclear Nomenclature A nuclide is an atomic species with a definite number of protons and neutrons arranged in a definite order in the nucleus. Radionuclides are those nuclides that are unstable and thus decay by emission of particles or electromagnetic radiations or by spontaneous fission. Isotopes are the nuclides having the same atomic number Z but different mass number A. Isotopes exhibit the same chemical properties. Examples of carbon isotopes are 116 C, 126 C, and 136 C. Isotones are the nuclides having the same number of neutrons N but different numbers of protons. Examples of isotones are: 134 55 Cs, 54 Xe, and 53 I, each 133 132 having 79 neutrons. Isobars are the nuclides with the same number of nucleons, that is, the same mass number A, but a different combination of protons and neutrons. For example: 82 Y, 82Sr, 82Rb, and 82Kr are all isobars having the mass number 82. Isomers are the nuclides with the same number of protons and neutrons, but having different energy states and spins. 99Tc and 99mTc are isomers of the same nuclide. Individual nuclides can exist in different energy states above the ground state due to excitation. These excited states are called the isomeric states, which can have a lifetime varying from picoseconds to years. When the isomeric states are long-lived, they are referred to as metastable states. These states are denoted by “m” as in 99mTc. Chart of the Nuclides Nearly 3700 nuclides, both stable and unstable, are arranged in the form of a chart, called the chart of the nuclides, a section of which is presented in Fig. 1.3. Each square in the chart represents a specific nuclide, containing various infor- mation such as the half-life, type and energy of radiations, and so forth of the nuclide, and neutron capture cross section of the stable nuclide. The nuclides are arranged in increasing neutron number N horizontally and in increasing proton number Z vertically. Each horizontal group of squares contains all isotopes of the same element, whereas the vertical group contains all isotones with the same number of neutrons. For isomers, the square is subdivided into sections represent- ing each isomer. Fig. 1.3. A section of the chart of nuclides. (Courtesy of Knolls Atomic Power Laboratory, Schenectady, New York, operated by the General Electric Company for Naval Reactors, the U.S. Department of Energy.) Chart of the Nuclides 9 10 1. Structure of Matter Questions 1. If a mass of matter ( m) is converted to electromagnetic radiation, what should be the energy of this radiation? 2. Describe the Bohr’s atomic theory in terms of the electronic configuration of the atom. 3. What is the difference between the orbital electron binding energy and the nuclear binding energy of an atom? 4. Define the mass defect and mass number of an atom. What does the mass defect account for? 5. Write the electronic configuration of 99mTc and 131I. 6. How many electrons can the 3d orbital contain? 7. The electron binding energy of the K shell in an atom is higher than that of the L shell. True or false? 8. What is the difference between ionization and excitation of an atom? 9. What is a metastable state of a nuclide? How is it designated? Suggested Readings Evans RD. The Atomic Nucleus. Malabar, FL: Kreiger; 1982. Friedlander G, Kennedy TW, Miller JM. Nuclear and Radiochemistry. 3rd ed. New York: Wiley; 1981. Turner JE. Atoms, Radiation, and Radiation Protection. 2nd ed. New York: Wiley; 1995. 2 Radioactive Decay In 1896, Henri Becquerel first discovered natural radioactivity in potassium ura- nyl sulfate. Artificial radioactivity was not produced until 1934, when I. Curie and F. Joliot made boron, aluminum, and magnesium radioactive by bombarding them with α-particles from polonium. This introduction of artificial radioactivity prompted the invention of cyclotrons and reactors in which many radionuclides are now produced. So far, more than 3400 radionuclides have been artificially produced and characterized in terms of their physical properties. Radionuclides are unstable and decay by emission of particle or γ-radiation to achieve stable configuration of protons and neutrons in the nucleus. As already mentioned, the stability of a nuclide in most cases is determined by the N/Z ratio of the nucleus. Thus, as will be seen later, whether a nuclide will decay by a par- ticular particle emission or γ-ray emission is determined by the N/Z and/or exci- tation energy of the nucleus. Radionuclides can decay by one or more of the six modes: spontaneous fission, isomeric transition (IT), alpha (α) decay, beta (β−) decay, positron (β+) decay, and electron capture (EC) decay. In all decay modes, energy, charge, and mass are conserved. Different decay modes of radionuclides are described later in detail. Spontaneous Fission Fission is a process in which a heavy nucleus breaks into two fragments accompa- nied by the emission of two or three neutrons. The neutrons carry a mean energy of 1.5 MeV and the process releases about 200 MeV energy that appears mostly as heat. Spontaneous fission occurs in heavy nuclei, but its probability is low and in- creases with mass number of the nuclei. The half-life for spontaneous fission is 2 × 1017 years for 235U and only 55 days for 254Cf. As an alternative to the sponta- neous fission, the heavy nuclei can decay by α-particle or γ-ray emission. G. B. Saha, Physics and Radiobiology of Nuclear Medicine, 11 DOI 10.1007/978-1-4614-4012-3_2, © Springer Science+Business Media New York 2013 12 2. Radioactive Decay Isomeric Transition As previously mentioned, a nucleus can exist in different energy or excited states above the ground state, which is considered as the state involving the arrangement of protons and neutrons with the least amount of energy. These excited states are called the isomeric states and have lifetimes of fractions of picoseconds to many years. When isomeric states are long-lived, they are referred to as metastable states and denoted by “m” as in 99mTc. An excited nucleus decays to a lower energy state by giving off its energy, and such transitions are called isomeric tran- sitions (ITs). Several isomeric transitions may occur from intermediate excited states prior to reaching the ground state. As will be seen later, a parent radionu- clide may decay to an upper isomeric state of the product nucleus by α-particle or β-particle emission, in which case the isomeric state returns to the ground state by one or more isomeric transitions. A typical isomeric transition of 99mTc is il- lustrated in Fig. 2.1. Isomeric transitions can occur in two ways: gamma (γ)-ray emission and internal conversion. Gamma (γ)-Ray Emission The common mode of an isomeric transition from an upper energy state of a nucleus to a lower energy state is by emission of an electromagnetic radiation, called the γ-ray. The energy of the γ-ray emitted is the difference between the two isomeric states. For example, a decay of a 525-keV isomeric state to a 210-keV isomeric state will result in the emission of a 315-keV γ-ray. Internal Conversion An alternative to the γ-ray emission is the internal conversion process. The ex- cited nucleus transfers the excitation energy to an orbital electron—preferably 99m Tc (6.02 h) 43 142 keV 140 keV 99 Tc (2.12 x 105 yr) 43 Fig. 2.1. Isometric transition of   99mTc. Ten percent of the decay follows internal conversion.  Isomeric Transition 13 Fig. 2.2. Internal conversion process. The excitation energy of the nucleus is transferred to a K-shell electron, which is then ejected with kinetic energy equal to Eγ–EB, and the K-shell vacancy is filled by an electron from the L shell. The energy difference between the L shell and K shell appears as the characteristic K x-ray. Alternatively, the characteristic K x-ray may transfer its energy to an L-shell electron, called the Auger electron, which is then ejected. the K-shell electron—of its own atom, which is then ejected from the shell, provided the excitation energy is greater than the binding energy of the elec- tron (Fig. 2.2). The ejected electron is called the conversion electron and carries the kinetic energy equal to Eγ – EB, where Eγ is the excitation energy and EB is the binding energy of the electron. Even though the K-shell electrons are more likely to be ejected because of the proximity to the nucleus, the electrons from the L shell, M shell, and so forth also may be ejected by the internal conversion process. The ratio of the number of conversion electrons ( Ne) to the number of observed γ-radiations ( Nγ) is referred to as the conversion coefficient, given as α = Ne/Nγ. The conversion coefficients are subscripted as αK, αL, αM… depending on which shell the electron is ejected from. The total conversion coefficient αT is then given by αT = αK + αL + αM + · · · Problem 2.1 If the total conversion coefficient (αT) is 0.11 for the 140-keV γ-rays of 99m Tc, calculate the percentage of 140-keV γ-radiations available for imaging. Answer Ne αT = = 0.11 Nγ Ne = 0.11Nγ 14 2. Radioactive Decay Total number of disintegrations = Ne + Nγ = 0.11Nγ + Nγ = 1.11Nγ Thus, the percentage of γ-radiations Nγ = × 100 1.11Nγ 1 = × 100 1.11 = 90 % An internal conversion process leaves an atom with a vacancy in one of its shells, which is filled by an electron from the next higher shell. Such situations may also occur in nuclides decaying by electron capture (see later). When an L electron fills in a K-shell vacancy, the energy difference between the K shell and the L shell appears as a characteristic K x-ray. Alternatively, this transition energy may be transferred to an orbital electron, which is emitted with a kinetic energy equal to the characteristic x-ray energy minus its binding energy. These electrons are called Auger electrons, and the process is termed the Auger process, analogous to internal conversion. The Auger electrons are monoenergetic. Be- cause the characteristic x-ray energy (energy difference between the two shells) is always less than the binding energy of the K-shell electron, the latter cannot undergo the Auger process and cannot be emitted as an Auger electron. The vacancy in the shell resulting from an Auger process is filled by the tran- sition of an electron from the next upper shell, followed by emission of similar characteristic x-rays and/or Auger electrons. The fraction of vacancies in a given shell that are filled by emitting characteristic x-ray emissions is called the fluo- rescence yield, and the fraction that is filled by the Auger processes is the Auger yield. The Auger process increases with the increasing atomic number of the atom. Alpha (α)-Decay The α-decay occurs mostly in heavy nuclides such as uranium, radon, plutonium, and so forth. Beryllium-8 is the only lightest nuclide that decays by breaking up into two α-particles. The α-particles are basically helium ions with two protons and two neutrons in the nucleus and two electrons removed from the orbital of the helium atom. After α-decay, the atomic number of the nucleus is reduced by 2 and the mass number by 4. 222 86 Rn → 218 84 Po + α  Beta (β−)-Decay 15 The α-particles from a given radionuclide all have discrete energies correspond- ing to the decay of the initial nuclide to a particular energy level of the prod- uct (including, of course, its ground state). The energy of the α-particles is, as a rule, equal to the energy difference between the two levels and ranges from 1 to 10 MeV. The high-energy α-particles normally originate from the short-lived heavy radionuclides and vice versa. The range of the α-particles is very short in matter and is approximately 0.03 mm in body tissue. The α-particles can be stopped by a piece of paper, a few centimeters of air, and gloves. Beta (β−)-Decay When a radionuclide is neutron rich—that is, the N/Z ratio is greater than that of the nearest stable nuclide—it decays by the emission of a β−-particle (note that it is an electron1) and an antineutrino, v̄. In the β−-decay process, a neutron is converted to a proton, thus raising the atomic number Z of the product by 1. Thus: n → p + β − + v̄ The difference in rest masses between the parent nuclide and the daughter nuclide plus β−-particle appears as the kinetic energy, which is called the transition or decay energy, denoted by Emax. The β−-particles carry Emax or part of it, exhibiting a spectrum of energy as shown in Fig. 2.3. The average energy of the β−-particles is about one-third of Emax. This observation indicates that β−-particles often carry only a part of the transition energy, and energy is not apparently conserved in β−-decay. To satisfy the law of energy conservation, a particle called the antineu- trino, v̄ , with no charge and a negligible mass has been postulated, which carries the remainder of Emax in each β−-decay. The existence of antineutrinos has been proven experimentally. In β−-decay, the parent nuclide may decay to the ground state or an excited state of the daughter nuclide and also, if energetically permitted, may emit several β−-particles. The excited states then decay to the ground state by γ-ray emission or internal conversion (Fig. 2.4). The decay process of a radionuclide is normally represented by what is called the decay scheme. Typical decay schemes of 131I and 99Mo are shown in Figs. 2.4 and 2.5, respectively. The β−-decay is shown by a left-to-right arrow from the par- ent nuclide to the daughter nuclide, whereas the isomeric transition is displayed by a vertical arrow between the two states. (Note: The β+-decay is shown by a two-step right-to-left arrow between the two states, the electron capture decay by a right-to-left arrow, and the α-decay by a down arrow). Although it is often said that 131I emits 364-keV γ-rays, it should be understood that the 364-keV γ-ray 1 The difference between a β−-particle and an electron is that a β−-particle originates from the nucleus, and an electron originates from the extranuclear electron orbitals. 16 2. Radioactive Decay 70 RELATIVE NUMBER OF 60 β–-PARTICLES (%) 40 20 Eβ = 0.695 E max = 1.7 MeV 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 ENERGY OF β–-PARTICLE (MeV) Fig. 2.3. A typical energy spectrum of the β−-particles of 32P. 131 3 I (8 da days) s) 53 723 kkeV V 1.6% 637 6 9% 6.9% 90.4% 364 80 131 Xe (stable) 54 Fig. 2.4. Decay scheme of 131I. Eighty-one percent of the total 131I radionuclides decay by 364-keV γ-ray emission. The 8.0-day half-life of 131I is shown in parentheses. belongs to 131Xe as an isomeric state. This is true for all β−-, β+-, or electron cap- ture decays that are followed by γ-ray emission. Some examples of β−-decay follow: 99 − 42 Mo → 99m 43 Tc + β + v̄ 131 − 53 I → 131 54 Xe + β + v̄ 67 − 29 Cu → 67 30 Zn + β + v̄ 90 − 38 Sr → 90 39Y + β + v̄  Positron (β+)-Decay 17 Fig. 2.5. Decay scheme of 99Mo. Approximately 87 % of the total 99Mo ultimately decays to 99mTc, and the remaining 13 % decays to 99Tc. A 2-keV transition occurs from the 142-keV level to the 140-keV level. All the 2-keV γ-rays are internally converted. (The energy levels are not shown in scale.) It should be noted that in β−-decay, the atomic number of the daughter nuclide is increased by 1 and the mass number remains the same. Positron (β+)-Decay When a radionuclide is proton rich—that is, the N/Z ratio is low relative to that of the nearest stable nuclide—it can decay by positron (β+) emission accompanied by the emission of a neutrino ( ν), which is an opposite entity of the antineutrino. In β+-decay, essentially a proton is converted to a neutron plus a positron, thus, decreasing the atomic number Z of the daughter nuclide by 1. Thus, p → n + β+ + ν Positron emission takes place when the parent nuclide has a minimum of mass-energy equivalent of 1.022 MeV more than the daughter nuclide. The re- quirement of 1.022 MeV for β+-decay arises from the fact that one electron mass has to be added to a proton to produce a neutron and one positron is created. Since each electron or positron mass is equal to 0.511 MeV, one electron and one positron are equal to 1.022 MeV, which is required as a minimum for β+-decay. Energy in excess of 1.022 MeV (Emax–1.022) is shared as kinetic energy between 18 2. Radioactive Decay Fig. 2.6. Decay scheme of 68Ga. The positrons are annihilated in medium to give rise to two 511-keV γ-rays emitted in opposite directions. the β+particle and ν. This results in an energy spectrum of β+ particles similar to the β−-particles. The parent nuclide may decay by one or more ground states of the daughter nuclide, followed by γ-ray emission or internal conversion. Some examples of β+-decay follow: 18 9F → 188 O + β + + v 68 + 31 Ga → 68 30 Zn + β + v 13 7N → 136 C + β + + v 15 + 8O → 15 7N+β +v The energetic β+-particle loses energy while passing through matter. The range of positrons is short in matter. When it loses almost all of its energy, it combines with an atomic electron of the medium and is annihilated, giving rise to two photons of 511 keV emitted in opposite directions. These photons are called annihilation radiations. The decay scheme of 68Ga is presented in Fig. 2.6. Note that the β+-decay is represented by a two-step right-to-left arrow. Electron Capture Decay by electron capture (EC) is an alternative to the β+-decay for proton-rich radionuclides with N/Z lower than that of the stable nuclide. In EC decay, an elec- tron from an extranuclear shell, particularly the K shell because of its proximity,  Electron Capture 19 111 49 In (2 81 days) (2.81 > 99% EC 173 keV 245 keV 111 48 Cd (stable) Fig. 2.7. Decay scheme of 111In illustrating the electron capture process. The abundances of 171 and 245-keV γ-rays are 90 and 94 %, respectively. is captured by a proton in the nucleus, forming a neutron accompanied by the emission of a neutrino for conservation of energy. Thus, p + e− → n + ν In this process, the atomic number of the daughter nuclide is lowered by 1. The EC process occurs usually in nuclides having mass-energy equivalent less than 1.022 MeV. In nuclides having energy greater than 1.022 MeV, both EC and β+- decay can occur, although the probability of β+-decay increases with higher en- ergy. The decay scheme of 111In is shown in Fig. 2.7. The EC decay is indicated by a right-to-left arrow. Some examples of EC decay follow: 111 49 In + e− → 111 48 Cd + v 67 31 Ga + e− → 67 30 Zn + v 125 53 I + e− → 125 52 Te + v 57 27 Co + e− → 57 26 Fe + v 123 53 I + e− → 123 52 Te + v In EC decay, analogous to the situation in internal conversion, a vacancy is cre- ated in the shell from which the electron is captured. It is filled in by the transition of an electron from the next upper shell, in which case the difference in energy between the two shells appears as a characteristic x-ray of the daughter nuclide. Also, as described earlier, instead of characteristic x-ray emission, the Auger pro- cess can occur, whereby an Auger electron is emitted. 20 2. Radioactive Decay Questions 1. What are the primary criteria for β+ and β−-decay? 2. If the mass-energy difference between the proton-rich parent nuclide and the daughter nuclide is 1.2 MeV, could the parent radionuclide decay by β+ decay and/or electron capture? If the energy difference is 0.8 MeV, what should be the mode of decay? 3. If the total conversion coefficient (αT) of 195-keV γ-rays of a radionuclide is 0.23, calculate the percentage of 195-keV photons available for imaging. 4. Can a K-shell electron be emitted as an Auger electron? Explain. 5. Explain how characteristic x-rays and Auger electrons are emitted. 6. Why is an antineutrino emitted in β −-decay? 7. A K-shell electron is ejected by the internal conversion of a 155-keV γ-ray photon. If the binding energy of the K-shell electron is 25 keV, what is the kinetic energy of the electron? 8. What is the average energy of the β−-particles emitted from a radionuclide? 9. Explain the production of annihilation radiations. Suggested Readings Evans RD. The Atomic Nucleus. Malabar, FL: Kreiger; 1982. Friedlander G, Kennedy JW, Miller JM. Nuclear and Radiochemistry. 3rd ed. New York: Wiley; 1981. Turner JE. Atoms, Radiation, and Radiation Protection. 2nd ed. New York: Wiley; 1995. 3 Kinetics of Radioactive Decay Radioactive Decay Equations General Equation As mentioned in Chapter 2, radionuclides decay by spontaneous fission, α-, β−-, and β+-particle emissions, electron capture, or isomeric transition. The radioac- tive decay is a random process, and it is not possible to tell which atom from a group of atoms disintegrates at a specific time. Thus, one can only talk about the average number of radionuclides disintegrating during a period of time. This gives the disintegration rate of a particular radionuclide. The disintegration rate of a radionuclide, that is, the number of disintegrations per unit time, given as −dN/dt, is proportional to the total number of radioactive atoms present at that time. Mathematically, −dN  = N (3.1) dt where N is the number of radioactive atoms present, and λ is referred to as the decay constant of the radionuclide. As can be seen from Eq. (3.1), it is a small fraction of the radioactive atoms that decays in a very short period of time. The unit of λ is (time)−1. Thus, if λ is 0.2 s−1 for a radionuclide, then 20 % of the radio- active atoms present will disappear per second. The disintegration rate −dN/dt is referred to as the radioactivity or simply the activity of the radionuclide and denoted by A. It should be understood from Eq. (3.1) that the same amount of radioactivity means the same disintegration rate for any radionuclide, but the total number of atoms present and the decay constants differ for different radionuclides. For example, a radioactive sample A containing 106 atoms and with λ = 0.01 min−1 would give the same disintegration rate (10,000 disintegrations per minute) as that by a radioactive sample B contain- ing 2 × 106 atoms and with a decay constant 0.005 min−1. G. B. Saha, Physics and Radiobiology of Nuclear Medicine, 21 DOI 10.1007/978-1-4614-4012-3_3, © Springer Science+Business Media New York 2013 22 3. Kinetics of Radioactive Decay Now from the preceding discussion, the following equation can be written:  A = N (3.2) From a knowledge of the decay constant and radioactivity of a radionuclide, one can calculate the total number of atoms or mass of the radionuclides present (us- ing Avogadro’s number 1 g · atom = 6.02 × 1023 atoms). Because Eq. (3.1) is a first-order differential equation, the solution of this equation by integration leads to  Nt = N0 e−t (3.3) where N0 and Nt are the number of radioactive atoms at t = 0 and time t, respec- tively. Equation (3.3) is an exponential equation indicating that the radioactivity decays exponentially. By multiplying both sides of Eq. (3.3) by λ, one obtains  At = A0 e−t (3.4) The factor e−λt is called the decay factor. The decay factor becomes e+λt if the ac- tivity at time t before t = 0 is to be determined. The plot of activity versus time on a linear graph gives an exponential curve, as shown in Fig. 3.1. However, if the activity is plotted against time on semilogarithmic paper, a straight line results, as shown in Fig. 3.2. Half-Life Every radionuclide is characterized by a half-life, which is defined as the time required to reduce its initial activity to one half. It is usually denoted by t1/2 and is unique for a radionuclide. It is related to the decay constant λ of a radionuclide by 0.693  = (3.5) t1/2 Ao Acvity Ao 2 Ao 4 Fig. 3.1. Plot of radioactivity versus- 1 2 3 4 5 6 7 time on a linear graph indicating an Time (half-lives) exponential curve.  Radioactive Decay Equations 23 Fig. 3.2. Plot of radioactivity against 100 time on a semilogarithmic graph 50 indi­cating a straight line. The half- life of the radionuclide can be 20 determined from the slope of the Activity line, which is given as the decay 10 constant λ. Alternatively, an activ- 5 ity and half its value and their cor- responding times are read from the plot. The difference in the two time 2 readings gives the half-life. 1 2 3 4 5 6 Time (half-lives) From the definition of half-life, it is understood that A0 is reduced to A0/2 in one half-life; to A0/4, that is, to A0/22 in two half-lives; to A0/8, that is, to A0/23 in three half-lives; and so forth. In n half-lives of decay, it is reduced to A0/2n. Thus, the radioactivity At at time t can be calculated from the initial radioactivity A0 by A0 A0  At = = (t/t ) = A0 (0.5)t/t1/2 (3.6) 2 n 2 1/2 where t is the time of decay. Here, t/t1/2 can be an integer or a fraction depending on t and t1/2. For example, a radioactive sample with t1/2 = 3.2 days decaying at a rate of 10,000 disintegrations per minute would give, after seven days of decay, 10,000/2(7/3.2) = 10,000/22.2 = 10,000/4.59 = 2178 disintegrations per minute. It should be noted that ten half-lives of decay reduce the radioactivity by a fac- tor of about 1000(210 = 1024), or to 0.1 % of the initial activity. The half-life of a radionuclide is determined by measuring the radioactivity at different time intervals and plotting them on semilogarithmic paper, as shown in Fig. 3.2. An initial activity and half its value are read from the line, and the corre- sponding times are noted. The difference in time between the two readings gives the half-life of the radionuclide. For a very long-lived radionuclide, the half-life is determined by Eq. (3.2) from a knowledge of its activity and the number of atoms present. The number of atoms N can be calculated from the weight W of the radionuclide with atomic weight A and Avogadro’s number 6.02 × 1023 atoms per g · atom as follows: W  N= × 6.02 × 1023 (3.7) A When two or more radionuclides are present in a sample, the measured count of such a sample comprises counts of all individual radionuclides. A semiloga- rithmic plot of the activity of a two-component sample versus time is shown in 24 3. Kinetics of Radioactive Decay 50 20 Acvity (log scale) a+b 10 5 b(t1/2= 5.8 hr) 2 a(t1/2 = 27 hr) 10 20 30 40 50 60 70 Time (hours) Fig. 3.3. A composite radioactive decay curve for a sample containing two radionuclides of different half-lives. The long-lived component ( a) has a half-life of 27 h and the short- lived component ( b) has a half-life of 5.8 h. Fig. 3.3. The half-life of each of the two radionuclides can be determined by what is called the peeling or stripping method. In this method, first, the tail part (sec- ond component) of the curve is extrapolated as a straight line up to the ordinate, and its half-life can be determined as mentioned previously (e.g., 27 h). Second, the activity values on this line are subtracted from those on the composite line to obtain the activity values for the first component. A straight line is drawn through these points, and the half-life of the first component is determined (e.g., 5.8 h). The stripping method can be applied to more than two components in the similar manner. Mean Life Another relevant quantity of a radionuclide is its mean life, which is the average lifetime of a group of radionuclides. It is denoted by τ and is related to the decay constant λ and half-life t1/2 as follows: 1  τ= (3.8)  t1/2  τ= = 1.44 t1/2 (3.9) 0.693 In one mean life, the activity of a radionuclide is reduced to 37 % of its initial value.  Radioactive Decay Equations 25 Effective Half-Life As already mentioned, a radionuclide decays exponentially with a definite half- life, which is called the physical half-life, denoted by Tp (or t1/2). The physical half-life of a radionuclide is independent of its physicochemical conditions. Analogous to physical decay, radiopharmaceuticals administered to humans dis- appear exponentially from the biological system through fecal excretion, urinary excretion, perspiration, or other routes. Thus, after in vivo administration every radiopharmaceutical has a biological half-life ( Tb), which is defined as the time needed for half of the radiopharmaceutical to disappear from the biologic system. It is related to decay constant λb by λb = 0.693/Tb. Obviously, in any biologic system, the loss of a radiopharmaceutical is due to both the physical decay of the radionuclide and the biologic elimination of the radiopharmaceutical. The net or effective rate (λe) of loss of radioactivity is then related to λp and λb by  e = p + b (3.10) Because λ = 0.693/t1/2, it follows that 1 1 1  = + (3.11) Te Tp Tb or, Tp × Tb  Te = (3.12) Tp + Tb The effective half-life, Te, is always less than the shorter of Tp or Tb. For a very long Tp and a short Tb, Te is almost equal to Tb. Similarly, for a very long Tb and short Tp, Te is almost equal to Tp. Units of Radioactivity The unit of radioactivity is a curie. It is defined as 1curie (Ci) = 3.7 × 1010 disintegrations per second (dps) = 2.22 × 1012 disintegrations per minute (dpm) 1 millicurie (mCi) = 3.7 × 107 dps = 2.22 × 109 dpm 26 3. Kinetics of Radioactive Decay 1 microcurie (µCi) = 3.7 × 104 dps = 2.22 × 106 dpm The System Internationale (SI) unit for radioactivity is the becquerel (Bq), which is defined as 1 dps. Thus, 1 becquerel (Bq) = 1 dps = 2.7 × 10−11 Ci 1 kilobecquerel (kBq) = 103 dps = 2.7 × 10−8 Ci 1 megabecquerel (MBq) = 106 dps = 2.7 × 10−5 Ci 1 gigabecquerel (GBq) = 109 dps = 2.7 × 10−2 Ci 1 terabecquerel (TBq) = 1012 dps = 27 Ci Similarly, 1 Ci = 3.7 × 1010 Bq = 37 GBq 1 mCi = 3.7 × 107 Bq = 37 MBq 1µCi = 3.7 × 104 Bq = 37 kBq Specific Activity The presence of “cold,” or nonradioactive, atoms in a radioactive sample always induces competition between them in their chemical reactions or localization in a body organ, thereby compromising the concentration of the radioactive atoms in the organs. Thus, each radionuclide or radioactive sample is characterized by specific activity, which is defined as the radioactivity per unit mass of a radionu- clide or a radioactive sample. For example, suppose that a 200-mg 123I-labeled monoclonal antibody sample contains 350-mCi (12.95-GBq) 123I radioactivity. Its specific activity would be 350/200 = 1.75 mCi/mg or 64.75 MBq/mg. Sometimes, it is confused with concentration, which is defined as the radioactivity per unit volume of a sample. If a 10-ml radioactive sample contains 50 mCi (1.85 GBq), it will have a concentration of 50/10 = 5 mCi/ml or 185 MBq/ml. Specific activity is at times expressed as radioactivity per mole of a labeled compound, for example, mCi/mole (MBq/mole) or mCi/μmole (MBq/μmole) for 3 H-, 14C-, and 35S-labeled compounds. The specific activity of a carrier-free (see Chapter 5) radionuclide sample is related to its half-life and mass number A: the shorter the half-life and the lower the A, the higher the specific activity. The specific activity of a carrier-free  Calculation 27 radionuclide with mass number A and half-life t1/2 in hours can be calculated as follows: Suppose 1 mg of a carrier-free radionuclide is present in the sample. 1 × 10−3 6.02 × 1020 Number of atoms in the sample = × 6.02 × 1023 = A A 0.693 Decay constant  = sec−1 t1/2 × 60 × 60 Thus, disintegration rate D = λN 0.693 × 6.02 × 1020 = t1/2 × A × 60 × 60 1.1589 × 1017 = dps A × t1/2  1.1589 × 1017 Thus, specific activity (mCi/mg) = A × t1/2 × 3.7 × 107 3.13 × 109 = (3.13) A × t1/2 where A is the mass number of the radionuclide, and t1/2 is the half-life of the radionuclide in hours. From Eq. (3.13), specific activities of carrier-free 99mTc and 131I can be cal- culated as 5.27 × 106 mCi/mg (1.95 × 105 GBq/mg) and 1.25 × 105 mCi/mg (4.6 × 103 GBq/mg), respectively. Calculation Some examples related to the calculation of radioactivity and its decay follow: Problem 3.1 Calculate the total number of atoms and total mass of Tl present in 201 10 mCi (370 MBq) of 201Tl ( t1/2 = 3.04d). Answer For 201Tl, 0.693 λ= = 2.638 × 10−6 sec−1 3.04 × 24 × 60 × 60 A = 10 × 3.7 × 107 = 3.7 × 108 dps 28 3. Kinetics of Radioactive Decay Using Eq. (3.2), A 3.7 × 108 N= = = 1.40 × 1014 atoms λ 2.638 × 10−6 Because 1 g · atom 201Tl = 201 g 201Tl = 6.02 × 1023 atoms of 201Tl (Avoga- dro’s number), 1.40 × 1014 × 201 Mass of 201 Tl in 10 mCi(370 MBq) = 6.02 × 1023 = 46.7 × 10−9 g = 46.7 ng Therefore, 10 mCi of 201Tl contains 1.4 × 1014 atoms and 46.7 ng. Problem 3.2 At 10:00 a.m., the 99mTc radioactivity was measured as 150 mCi (5.55 GBq) on Wednesday. What was the activity at 6 a.m. and 3 p.m. on the same day ( t1/2of 99mTc = 6 h)? Answer Time from 6 a.m. to 10 a.m. is 4 h: 0.693  for 99m Tc = = 0.1155 h−1 6 At = 150 mCi (5.55 GBq) A0 =? Using Eq. (3.4) 150 = A0 e+0.1155×4 A0 = 150 × e0.462 = 150 × 1.5872 = 238.1mCi (8.81 GBq)at 6 a.m. Time from 10 a.m. to 3 p.m. is 5 h: A0 = 150 mCi At =?  Calculation 29 Using Eq. (3.4) At = 150 × e−0.1155×5 = 150 × e−0.5775 = 150 × 0.5613 = 84.2 mCi (3.1 GBq) at 3 p.m. Problem 3.3 If a radionuclide decays at a rate of 30 %/h, what is its half-life? Answer  = 0.3 h−1 0.693 = t1/2 0.693 0.693 t1/2 = = h = 2.31 h  0.3 Problem 3.4 If 11 % of 99mTc-labeled diisopropyliminodiacetic acid (DISIDA) is elimi- nated via renal excretion, 35 % by fecal excretion, and 3.5 % by perspi- ration in 5 h from the human body, what is the effective half-life of the radiopharmaceutical ( Tp = 6 h for 99mTc)? Answer Total biological elimination = 11 % + 35 % + 3.5 % = 49.5 % in 5 h Therefore , Tb ≈ 5 h Tp = 6 h Tb × Tp 5×6 30 Te = = = = 2.7 h Tb + Tp 5+6 11 30 3. Kinetics of Radioactive Decay Successive Decay Equations General Equation In the preceding section, we derived equations for the activity of any radionuclide that is decaying. Here we shall derive equations for the activity of a radionuclide that is growing from another radionuclide and at the same time is itself decaying. If a parent radionuclide p decays to a daughter radionuclide d, which in turn decays to another radionuclide (i.e., p → d →), then the rate of growth of d be- comes dNd  = p Np − d Nd (3.14) dt By integration, Eq. (3.14) becomes d (Ap )0 −p t  (Ad )t = d Nd = (e − e − d t ) (3.15) d − p Equation (3.15) gives the activity of the daughter nuclide d at time t as a result of growth from the parent nuclide p and also due to the decay of the daughter itself. Transient Equilibrium If λd > λp, that is, ( t1/2)d < ( t1/2)p, then e−λd t in Eq. (3.15) is negligible compared to e−λp t when t is sufficiently long. Then Eq. (3.15) becomes d (Ap )0 −p t (Ad )t = e d − p  d (Ap )t = (3.16) d − p (t1/2 )p (Ap )t  = (3.17) (t1/2 )p − (t1/2 )d This relationship is called the transient equilibrium. This equilibrium holds good when ( t1/2)p and ( t1/2)d differ by a factor of about 10–50. The semilogarithmic plot of this equilibrium equation is shown in Fig. 3.4. The daughter nuclide initially builds up as a result of the decay of the parent nuclide, reaches a maximum, and then achieves the transient equilibrium decaying with an apparent half-life of the parent nuclide. In equilibrium, the ratio of the daughter to parent activity is con- stant. It can be seen from Eq. (3.17) that the daughter activity is always greater than the parent activity, because ( t1/2)p/(( t1/2)p − ( t1/2)d) is always greater than 1. The time to reach maximum daughter activity is given by the formula:  Successive Decay Equations 31 Parent 100 Daughter RADIOACTIVITY 10 1 0 4 8 12 16 Time (hours) Fig. 3.4. Plot of activity versus time on a semilogarithmic graph illustrating the transient equilibrium. Note that the daughter activity reaches a maximum, then transient equilib- rium, and follows an apparent half-life of the parent. The daughter activity is higher than the parent activity at equilibrium. 1.44 × (t1/2 )p × (t1/2 )d × ln((t1/2 )p /(t1/2 )d )  t max = (3.18) ((t1/2 )p − (t1/2 )d ) A typical example of transient equilibrium is 99 Mo ( t1/2 = 66 h) decaying to 99mTc ( t1/2 = 6 h). Because 87 % of 99 Mo decays to 99mTc, and the remaining 13 % to the ground state, Eqs. (3.15), (3.16), and (3.17) must be multiplied by a factor of 0.87. Therefore, in the time–activity plot, the 99mTc daughter activity will be lower than the 99 Mo parent activity (Fig. 3.5). Also, the 99mTc activity reaches maximum in about 23 h, i.e., 4( t1/2)d (Eq. (3.18)). Secular Equilibrium When λd  λp, that is, when the parent half-life is much longer than that of the daughter nuclide, in Eq. (3.16), we can neglect λp compared to λd. Then Eq. (3.16) becomes 32 3. Kinetics of Radioactive Decay 99 100 Mo (66 hrs) 50 99m Tc (6 hrs) Activity 10 5 2 1 2 3 4 5 6 Time (days) Fig. 3.5. Plot of logarithm of 99 Mo and 99mTc activities versus time showing transient equilibrium. The activity of the daughter 99mTc is less than that of the parent 99 Mo, because only 87 % of 99Mo decays to 99mTc. If 100 % of the parent were to decay to the daughter, then the daughter activity would be higher than the parent activity after reaching equilib- rium, as recognized from Eq. (3.17), and Fig. 3.4.  (Ad )t = (Ap )t (3.19) Equation (3.19) represents the secular equilibrium. This equilibrium holds when the half-life of the parent is much longer than that of the daughter nuclide by more than a factor of 100 or so. In secular equilibrium, both parent and daughter activities are equal, and both decay with the half-life of the parent nuclide. A semilogarithmic plot of activity versus time representing secular equilibrium is shown in Fig. 3.6. Typical examples of secular equilibrium are 113Sn ( t1/2 = 117 days) decaying to 113mIn ( t1/2 = 100 min), and 68Ge ( t1/2 = 280 days) decaying to 68 Ga ( t1/2 = 68 min). Questions 1. Calculate (a) the total number of atoms and (b) the total mass of 131I present in a 30-mCi (1.11-GBq) 131I sample ( t1/2 = 8.0 days). 2. Calculate (a) the disintegration rate per minute and (b) the activity in curies and becquerels present in 1 mg of 201Tl ( t1/2 = 73 h). 3. A radiopharmaceutical has a biologic half-life of 10 h in humans and a physi- cal half-life of 23 h. What is the effective half-life of the radiopharmaceutical? 4. If the radioactivity of 111In ( t1/2 = 2.8 days) is 200 mCi (7.4 GBq) on Monday noon, what is the activity (a) at 10:00 a.m. the Friday before and (b) at 1:00 p.m. the Wednesday after?  Questions 33 Parent 100 Daughter Radioactivity 10 1 0 4 8 12 16 Time (hours) Fig. 3.6. Plot of activity versus time illustrating secular equilibrium. In equilibrium, the daughter activity becomes equal to that of the parent. 5. How long will it take for a 10-mCi (370-MBq) sample of 123I ( t1/2 = 13.2 h) and a 50-mCi (1.85-MBq) sample of 99mTc ( t1/2 = 6 h) to possess the same activity? 6. What is the time interval during which 111In ( t1/2 = 2.8 days) decays to 37 % of the original activity? 7. For antibody labeling, 5 mCi of 111InCl3 is required. What amount of 111InCl3 should be shipped if transportation takes one day? 8. What are the specific conditions of transient equilibrium and secular equilibrium? 9. How long will it take for the decay of 7/8 of an 18F ( t1/2 = 110 min) sample? 10. What fraction of the original activity of a radionuclide has decayed in a period equal to the mean life of the radionuclide? 11. A radioactive sample initially gives 8564 cpm and 2 h later gives 3000 cpm. Calculate the half-life of the radionuclide. 12. If N atoms of a sample decay in one half-life, how many atoms would decay in the next half-life? 34 3. Kinetics of Radioactive Decay 13. The 99 Mo ( t1/2 = 66 h) and 99mTc ( t1/2 = 6 h) are in transient equilibrium in a Moly generator. If 600 mCi (22.2 GBq) of 99 Mo is present in the generator, what would be the activity of 99mTc 132 h later, assuming that 87 % of 99Mo decays to 99mTc? 14. A radionuclide decays with a half-life of 10 days to a radionuclide whose half-life is 1.5 h. Approximately how long will it take for the daughter activ- ity to reach a maximum? Suggested Readings Cherry SR, Sorensen JA, Phelps ME. Physics in Nuclear Medicine. 3rd ed. Philadelphia: W. B Saunders; 2003. Friedlander G, Kennedy JW, Miller JM. Nuclear Chemistry and Radiochemistry. 3rd ed. New York: Wiley; 1981. Saha GB. Fundamentals of Nuclear Pharmacy. 6th ed. New York: Springer; 2010. 4 Statistics of Radiation Counting As mentioned in previous chapters, radioactive decay is a random process, and therefore one can expect fluctuations in the measurement of radioactivity. The detailed discussion of the statistical treatment of radioactive measurements is be- yond the scope of this book. Only the salient points of statistics related to radia- tion counting are given here. Error, Accuracy, and Precision In the measurement of any quantity, an error in or deviation from the true value is likely to occur. Errors can be two types: systematic and random. Systematic er- rors appear as constant deviations and arise from malfunctioning instruments and inappropriate experimental conditions. These errors can be eliminated by rectify- ing the incorrect situations. Random errors are variable deviations and arise from the fluctuations in experimental conditions such as high-voltage fluctuations or statistical fluctuations in a quantity such as radioactive decay. The accuracy of a measurement of a quantity indicates how closely it agrees with the “true” value. The precision of a series of measurements describes the reproducibility of the measurement, although the measurements may differ from the “average” or “mean” value. The closer the measurement is to the average value, the higher the precision, whereas the closer the measurement is to the true value, the more accurate the measurement. Remember that a series of measure- ments may be quite precise, but their average value may be far from the true value (i.e., less accurate). Precision can be improved by eliminating the random errors, whereas better accuracy is obtained by removing both the random and systematic errors. G. B. Saha, Physics and Radiobiology of Nuclear Medicine, 35 DOI 10.1007/978-1-4614-4012-3_4, © Springer Science+Business Media New York 2013 36 4. Statistics of Radiation Counting Mean and Standard Deviation When a series of measurements is made on a radioactive sample, the most likely value of these measurements is the average, or mean value, which is obtained by adding the values of all measurements divided by the number of measurements. The mean value is denoted by n. The standard deviation of a series of measurements indicates the deviation from the mean value and is a measure of the precision of the measurements. Ra- dioactive decay follows the Poisson distribution law, from which one can show that if a radioactive sample gives an average count of n, then its standard devia- tion σ is given by √  σ = n (4.1) The mean of measurements is then expressed as n±σ Gaussian Distribution If a series of measurements are made repeatedly on a radioactive sample giving a mean count n, then the distribution of counts would normally follow a Poisson distribution. If the number of measurements is large, the distribution can be ap- proximated by a Gaussian distribution, illustrated in Fig. 4.1. It can be seen that Number Of Measurements 68% 95% 99% -3σ -2σ -1σ n 1σ 2σ 3σ Counts (n) Fig. 4.1. A Gaussian distribution of radioactive measurements. Note the 68 % confidence level at ± 1σ, 95 % confidence level at ± 2σ, and 99 % confidence level at ± 3σ.  Mean and Standard Deviation 37 68 % of all measurements fall within one standard deviation on either side of the mean, that is, within the range n ± σ ; 95 % of all measurements fall within the range n ± 2σ ; and 99 % fall within the range n ± 3σ. Also the Gaussian curve shows that half of the measurements are below the mean value, and the other half are above it. The standard deviations in radioactive measurements indicate the statistical fluctuations of radioactive decay. For practical reasons, only single counts are ob- tained on radioactive samples instead of multiple repeat counts to determine the mean value. In this situation, if a single count n of a radioactive √ sample is large, then n can be estimated as close to n; that is, n = n and σ = n. It can then be said that there is a 68 % chance that the true value of the count falls within n ± σ or that the count n falls within one standard deviation of the true value (Fig. 4.1). This is called the 68 % confidence level. That is, one is 68 % confident that the count n is within one standard deviation of the true value. Similarly, 9