Chapter 1 Introduction - Basic Nuclear Medicine Physics PDF

Chapter 1 Basic Nuclear Medicine Physics  Nuclear medicine Nuclear medicine is a multi-disciplinary specialty that develops and uses instrumentation and radiopharmaceuticals to study physiological process and non-invasively diagnose or treat disease.  Physics i...

Chapter 1 Basic Nuclear Medicine Physics  Nuclear medicine Nuclear medicine is a multi-disciplinary specialty that develops and uses instrumentation and radiopharmaceuticals to study physiological process and non-invasively diagnose or treat disease.  Physics in nuclear medicine The technology for producing radioactive tracers and for obtaining images of those tracer distributions is changing, however, the physics underlying nuclear medicine is not changing.  Atomstructure All matter is composed of atoms. An atom is the smallest unit into which an element can be broken down without losing its chemical identity. Atoms combine to form molecules and chemical compounds Suggested videos: Atoms: https://www.youtube.com/watch?v=M-1nzFZGaAM Molecules, compounds ,mixtures: https://www.youtube.com/watch?v=jBDr0mHyc5M  Atomstructure The atom consists of a very small, dense nucleus and electrons occupying the surrounding space and defining the size of the atom Electrons have negative charges and the nucleus has a positive charge  Atomstructure The electrons are orbiting the nucleus at high speeds In the electrically neutral atom, the number of orbital electrons is sufficient to balance exactly the number of positive charges, protons, in the nucleus.  Electrons shells Electrons are allowed discrete energy values and exist in energy “shells” and “subshells” The electron orbits can be depicted as the surfaces of spheres (called shells) The more significant characteristic of a shell is the energy that it signifies. The “closer” an electron is to the nucleus, the more tightly it is bound to the nucleus which mean more work (energy) is required to remove an inner-shell electron than an outer one  Quantumnumbers  The electrons in their shells are usually described by their quantum numbers, of which there are four types  The first is the principal quantum number (n), which identifies the energy shell.  Larger atoms have more shells  the maximum number of electrons associated with each energy shell is 2𝑛 , where n is the shell number. The first shell (the K shell) can contain a maximum of two electrons, the second shell (the L shell) can contain a maximum of 8 electrons, the third K, L, and M electron shells shell (the M shell) can contain a maximum of 18 electrons, and so on.  Other quantumnumbers The second (azimuthal), third (magnetic), and fourth (spin) quantum numbers refer to other physical properties of the electron. Each electron within an atom has a unique combination of the four quantum numbers  Binding energy  The energy that must be put into the atom to separate an electron is called the electron binding energy. It is usually expressed in electron volts (eV). From a few thousand electron volts (keV) for inner shell electrons to just a few eV for the less tightly bound outer-shell electron Atomic structure. The nucleus is surrounded by electron shells. The binding energy decreases as the distance from the nucleus increases (K > L > M).  Electron volt  The electron volt is a special unit defined as the energy required to move one electron against a potential difference of one volt. Conversely, it is also the amount of kinetic (motion) energy an electron acquires if it “falls” through a potential difference of one volt.  It is a very small unit on the everyday scale, at only 1.6 × 10−19 joules (J). The joule is the system International (SI) unit of energy.  Its a very convenient unit on the atomic scale.  Nucleus  Nucleus Like the atom itself, the atomic nucleus also has an inner structure.  The nucleus consists of two types of particles: protons, which carry a positive charge, and neutrons, which carry no charge.  The general term for protons and neutrons is nucleons.  Like electrons, nucleons have quantum The nucleus of an atom is composed properties, including spin. of protons and neutrons. The subatomic particles  Atoms Vs Ions Suggested videos: https://www.youtube.com/watch?v=fN8kH9Vvqo0 Atoms Vs Ions Atoms Vs Ions  Nomenclature (Atomic number and atomic mass) A nucleon is either a proton or a neutron The atomic number (Z) is the number of protons=the number of the orbital electrons in the electrical neutral atom and therefore the chemical elements to which the atom belongs The neutron number (N) is equal to A-Z  Standard atomic notation The notation used to summarize atomic and nuclear composition Chemical symbol for the element Elemental symbol  Annotated example of an atomic symbol Carbon Magnesium  Periodic table  All of the known elements, both natural and those made by humans, can be organized into the periodic table.  The number appearing above each element’s abbreviation is referred to as the atomic number.  The elements in the periodic table are arranged in columns (called groups) and rows (called periods).  In general, elements within groups demonstrate similar properties. This is because elements in a group often have similar numbers of electrons in their outer shell; outer-shell electron configurations are more important in determining how an atom interacts with other elemental atoms.  All the elements have been assigned symbols or abbreviated chemical names, for example gold, Au; mercury, Hg; and helium, He. Some symbols are obvious abbreviations of the English name; others are derived from the original Latin name of the element. For example, Au is from aurum, the Latin word for gold Periodic table  Nomenclature same atomic number but different atomic mass different elements different elements same atomic mass  Atomand excitation Just as it takes energy to remove an electron from its atom, it takes energy to move an electron from an inner shell to an outer shell, which can also be thought of as the energy required to pull a negative electron away from the positively charged nucleus. Any vacancy in an inner shell creates an unstable condition, often referred to as an excited state. if the arrangement of the electrons in the shells is not in the stable state, they will undergo rearrangement in order to become stable, a process often referred to as de-excitation. Because the stable configuration of the shells always has less energy than any unstable configuration, the deexcitation releases energy as photons, often as X-rays. Different between excitation and ionization https://www.youtube.com/watch?v=KwOHJbE4Tro  Radiation  The term radiation refers to “energy in transit.”  In nuclear medicine, we are interested principally in the following two specific forms of radiation: 1. Particulate radiation, consisting of atomic or subatomic particles (electrons, protons, etc.) 2. Electromagnetic radiation, in which energy is traveling through space at the speed of light. electromagnetic radiation behaves as discrete “packets” of energy, called photons (also called quanta). Photons have no mass or electrical charge and also travel at the velocity of light. These characteristics distinguish them from the forms of particulate radiation.  Stability  Not all elements have stable nuclei, Some are unstable, even in their ground states  Most of the light and mid-weight elements are stable, those with atomic numbers (number of protons) around Z = 83. The exceptions are technetium (Z = 43) and promethium (Z = 61).  All elements with atomic numbers higher than 83, such as radium (Z = 88) and uranium (Z = 92), are inherently unstable because of their large size.  For those nuclei with a stable state, there is an optimal ratio of neutrons to protons. For the lighter elements, this ratio is approximately 1:1; for increasing atomic weights, the number of neutrons exceeds the number of protons. For heavy elements, it corresponds to N ≈ 1.5 Z, that is, approximately 50% more neutrons than protons  Stability  In general, there is a tendency toward instability in atomic systems composed of large numbers of identical particles confined in a small volume  This explains the instability of very heavy nuclei. It also explains why, for light elements, stability is favored by more or less equal numbers of neutrons and protons rather than grossly unequal numbers.  A moderate excess of neutrons is favored among heavier elements because neutrons provide only exchange forces (attraction), whereas protons provide both exchange forces and coulombic forces (repulsion). Exchange forces are effective over very short distances and thus affect only Neutrons can stabilize a “close neighbors” in the nucleus, whereas the repulsive nucleus, because they roughly coulombic forces are effective over much greater distances. act like bonds between protons. Thus an excess of neutrons is required in heavy nuclei to They will bind protons together overcome the long-range repulsive coulombic forces through what is called between a large number of protons. the strong force. The strong force is a very short range force which only dominates within very small distance, like Video: https://www.youtube.com/watch?v=F1ZwdVJbj7g the size of a nucleus. https://www.youtube.com/watch?v=mpDDQ4uEH6M  Line of stability  On a graph of neutron versus proton numbers the stable nuclides tend to be clustered about Isotopic line an imaginary line called the line of stability  The combinations of neutrons and protons that can coexist in a stable nuclear configuration all lie within the gray-shaded regions  More neutrons are needed to hold together larger numbers of protons.  The line of stability ends at 209Bi (Z = 83, N = 126). All heavier nuclides are unstable  Nuclides that are not close to the line of stability are likely to be unstable. Unstable nuclides lying above the line of stability are said to be “proton deficient,” whereas those Isotonic line lying below the line are “neutron deficient.  Nuclides lying off the line of stability generally are radioactive.  Radioactivity If the nucleus is not in its stable state it will adjust itself until it is stable either by ejecting portions of its nucleus or by emitting energy in the form of photons (gamma ray). Radioactive decay is a process in which an unstable nucleus transforms into amore stable one by emitting particles, photons, or both, releasing energy in the process. Radioactivity decay is a process involving primarily the nucleus (Atomic electrons may become involved in some types of radioactive decay, but it is basically a nuclear process caused by nuclear instability) The elements which exhibit radioactivity are called radioactive elements, radioactive nuclides or radionuclides It is common terminology to call an unstable radioactive nucleus the parent and the more stable product nucleus the daughter. In many cases, the daughter also is radioactive and undergoes further radioactive decay  Radioactivity The radioactive decay is completely spontaneous there is no way to predict with certainty the exact moment at which an unstable nucleus will undergo its radioactive transformation into another, more stable nucleus  Each un-decayed nucleus has equal probability of decaying in the next second (statistical process) Radioactive decay results in the conversion of mass into energy. The total mass-energy conversion amount is called the transition energy, sometimes designated Q. Most of this energy is imparted as kinetic energy to emitted particles or converted to photons, with a small (usually insignificant) portion given as kinetic energy to the recoiling nucleus. Thus radioactive decay results not only in the transformation of one nuclear species into another but also in the transformation of mass into energy Suggested videos: https://www.youtube.com/watch?v=M0uw4ZNpqcI Radioactivity: Expect the unexpected - Steve Weatherall - Bing video  Radioactivity and chemistry What does a Radioactivity decay is a process involving primarily the nucleus mean?  Radioactive decay is a process involving primarily the nucleus, whereas chemical reactions involve primarily the outermost orbital electrons of the atom. Thus the fact that an atom has a radioactive nucleus does not affect its chemical behavior and, conversely, the chemical state of an atom does not affect its radioactive characteristics.  Independence of radioactive and chemical properties is of great significance in tracer studies with radioactivity—a radioactive tracer behaves in chemical and physiologic processes exactly the same as its stable, naturally occurring counterpart, and, further, the radioactive properties of the tracer do not change as it enters into chemical or physiologic processes  For example, an atom of the radionuclide 131I exhibits the same chemical behavior as an atom of 127I, the naturally occurring stable nuclide  Radioactivity However, There is a minor exceptions to these generalizations: The chemical behavior can be affected by differences in atomic mass. Because there are always mass differences between the radioactive and the stable members of an isotopic family (e.g., 131I is heavier than 127I), there may also be chemical differences. This is called the isotope effect. Note that this is a mass effect and has nothing to do with the fact that one of the isotopes is radioactive. Although the isotope effect is important in some experiments, it is, fortunately, of no practical consequence in nuclear medicine.  Mode of Decay The type of decay depends on which of the following rules for nuclear stability is violated 1. Excessive nuclear mass Alpha decay Fission 2. Unstable neutron–proton ratio Too many neutrons Beta decay Too many protons  Positron decay  Electron capture 3. Appropriate numbers of nucleons, but too much energy Isomeric transition  Gamma emission  Internal conversion  Excessive nuclear mass (Alpha decay)  Alpha decay Very large unstable atoms, atoms with high atomic mass, may split into nuclear fragments. The smallest stable nuclear fragment that is emitted is a particle consisting of two neutrons and two protons, equivalent to the nucleus of a helium atom. Because it was one of the first types of radiation discovered, the emission of a helium nucleus is called alpha radiation, and the emitted helium nucleus an alpha Videos: particle. https://www.youtube.com/watch?v=j5TJRtJxVfs https://www.youtube.com/watch?v=VeXpMijpazE Decay by α-particle emission results in a transmutation of elements, but it is not isobaric  Excessive nuclear mass (Alpha decay)  Alpha decay The α particle is emitted with kinetic energy usually between 4 and 8 MeV. Although quite energetic, α particles have very short ranges in solid materials, for example, approximately 0.03 mm in body tissues. Thus they present very difficult detection and measurement problems Illustration of series decay, starting from 238U and ending with stable 206Pb.  Excessive nuclear mass (Nuclear fission)  Nuclear fission Nuclear fission is the spontaneous fragmentation of a very heavy nucleus into two lighter nuclei. In the process a few (two or three) fission neutrons also are ejected. The distribution of nuclear mass between the two product nuclei varies from one decay to the next. Typically it is split in approximately a 60:40 ratio. The energy released is very large, often amounting to hundreds of MeV per nuclear fission, and is imparted primarily as kinetic energy to the recoiling nuclear fragments (fission fragments) and the ejected neutrons Video: https://www.youtube.com/watch?v=D91T-B-PVE0 Excessive nuclear mass (Alpha decay and Fission) Alpha decay and Fission Radionuclides that decay by α-particle emission or by nuclear fission are of relatively little importance for direct usage as tracers in nuclear medicine but are described here for the sake of completeness. Both of these decay modes occur primarily among very heavy elements that are of little interest as physiologic tracers. As well, they are highly energetic and tend to be associated with relatively large radiation doses Unstable neutron-proton ratio (toomany neutron) β− decay Nuclei with excess neutrons can achieve stability by a process that amounts to the conversion of a neutron into a proton and an electron. The proton remains in the nucleus, but the electron (beta “β−” particle) is emitted. Electron were given these names to contrast with the alpha particle before the physical nature of either was discovered. The beta particle generated in this decay will become a free electron until it finds a vacancy in an electron shell in another atom. Barium Caesium Unstable neutron-proton ratio (too many neutron) β− decay The conversion of a neutron to a proton involved more than the emission of a beta particle (electron). Beta emission satisfied the rule for conservation of charge in that the neutral neutron yielded one positive proton and one negative electron; however, it did not appear to satisfy the equally important rule for conservation of energy. Measurements showed that most of the emitted electrons simply did not have all the energy expected. To explain this apparent discrepancy, the emission of a second particle was postulated, and that particle was later identified experimentally. Called an antineutrino (ν)( the “neutrino” is for “small and neutral”), it carries the “missing” energy of the reaction. The neutrino is a “particle” having no mass or electrical charge. It undergoes virtually no interactions with matter and therefore is essentially undetectable. Its only practical consequence is that it carries away some of the energy released in the decay process. The energy released in β− decay is shared between the β− particle and the antineutrino. This sharing of energy is more or less random from one decay to the next Unstable neutron-proton ratio(toomany neutron)  β− decay  Decay by β− emission may be represented in standard nuclear notation as Schematically, the process is n p e energy The parent radionuclide (X) and daughter product (Y) represent different chemical elements because atomic number increases by one. Thus β− decay results in a transmutation of elements.  Mass number A does not change because the total number of nucleons in the nucleus does not change. This is therefore an isobaric decay mode, that is, the parent and daughter are isobars Unstable neutron-proton ratio(toomany neutron)  β− decay Radioactive decay processes often are represented by a decay scheme diagram. A diagram for 14C, a radionuclide that decays solely by β− emission. The line representing 14C (the parent) is drawn above and to the left of the line representing 14N (the daughter). Decay is “to the right” because atomic number increases by one (reading Z values from left to right). The vertical distance between the lines is proportional to the total amount of energy released, that is, the transition energy for the decay process (Q = 0.156 MeV for 14C) There is no physical difference between a beta particle and an electron; the term beta particle is applied to an electron that is emitted from a NOTE radioactive nucleus. The symbol β without a minus or plus sign attached always refers to a beta-minus particle, or electron. Unstable neutron-proton ratio (toomany neutron) Beta particles and nuclear medicine Beta particles present special detection and measurement problems for nuclear medicine applications. These arise from the fact that they can penetrate only relatively small thicknesses of solid materials. For example, the thickness is at most only a few millimeters in soft tissues. Therefore it is difficult to detect β− particles originating from inside the body with a detector that is located outside the body. For this reason, radionuclides emitting only β− particles rarely are used when measurement in vivo is required. Unstable neutron-proton ratio (toomany proton)  an unstable nucleus with too many protons can undergo a decay that has the effect of converting a proton into a neutron. There are two ways this can occur: 1. Positron decay (β+ decay ) 2. Electron capture  In general, proton-rich nuclei decay by a combination of these two processes.  Among the radioactive nuclides, one finds that β+ decay occurs more frequently among lighter elements, whereas EC is more frequent among heavier elements, because in heavy elements orbital electrons tend to be closer to the nucleus and are more easily captured.  Unstable neutron-proton ratio (toomany proton) Positron decay (β+ decay )  In radioactive decay by positron emission, a proton in the nucleus is transformed into a neutron and a positively charged electron.  The positively charged electron is also referred to as a positive beta particle, positron, or antielectron.  In many ways, positron decay is the mirror image of beta decay: positive electron instead of negative electron, neutrino instead of antineutrino. Schematically, the process is:  Unstable neutron-proton ratio(toomany proton) Positron decay (β+ decay ) Remember we will talk about A positron is the antiparticle of an ordinary this again when electron. After ejection from the nucleus, it loses we talk about PET resolution its kinetic energy in collisions with atoms of the surrounding matter and comes to rest, usually within a few millimeters of the site of its origin in body tissues. More accurately, the positron and an electron momentarily form an “atom” called positronium, which has the positron as its “nucleus” and a lifetime of approximately 10 sec. The positron then combines with the negative electron in an annihilation reaction, in which their masses are converted into energy The mass-energy equivalent of each particle is Schematic representation of annihilation reaction between a positron (β+ ) and an 0.511 MeV. This energy appears in the form of ordinary electron. A pair of 0.511 MeV two 0.511 MeV annihilation. annihilation photons are emitted “back- a total equivalent to 1.02MeV to-back” at 180 degrees to each other Unstable neutron-proton ratio (too many proton) Electron capture (EC) Through a process that competes with positron decay, a nucleus can combine with one of its inner orbital electrons to achieve the net effect of converting one of the protons in the nucleus into a neutron. An outer shell electron in the daughter atom, then fills the vacancy in the inner shell left by the captured electron. The energy lost by the “fall” of the outer-shell electron to the inner shell is emitted as an characteristic x rays or Auger electrons Usually, the electron is captured from orbits that are closest to the nucleus, that is, the K and L shells. The notation EC(K) is used to indicate capture of a K-shell electron, EC(L) an L-shell electron Unstable neutron-proton ratio (too many proton) Electron capture (EC) Unstable neutron-proton ratio (too many proton) Electron capture (EC) The Auger effect is a physical phenomenon in which the filling of Remember an inner-shell vacancy of an atom is accompanied by the emission of an electron from the same atom. When a core electron is removed, leaving a vacancy, an electron from a higher energy level may fall into the vacancy, resulting in a release of energy. Although most often this energy is released in the form of an emitted photon, the energy can also be transferred to another electron, which is ejected from the atom; this second ejected electron is called an Auger electron. Unstable neutron-proton ratio (toomany proton) Electron capture The neutrino is emitted from the nucleus and carries away some of the transition energy. Appropriate number of nucleons, but too much energy Isomeric transition  If the nucleus has a more favorable physical configuration of nucleons but usually contains an excess of energy.  The nucleus is said to be in an excited state when the energy of the nucleus is greater than its resting level.  If the excited state is stable enough (has a half-life longer than 10 seconds) then the nuclide is referred to as an isomer or in metastable or isomeric state, and the excess energy is shed by an isomeric transition.  This may occur by either or both of two competing reactions: gamma emission and internal conversion. Most isomeric transitions occur as a combination of these two reactions. Appropriate number of nucleons, but too much energy Gamma emission In this process, excess nuclear energy is emitted as a gamma ray. The name “gamma” was given to this radiation, before its physical nature was understood, because it was the third (alpha, beta, gamma) type of radiation discovered. A gamma ray is a photon (energy) emitted by an excited nucleus. Despite its unique name, it cannot be distinguished from photons of the same energy from a different source, for example X-rays.  Gamma ray and X-ray X-rays and gamma rays have the same basic properties BUT X-ray Gamma ray X-rays are emitted gamma rays originate from processes inside the nucleus outside the nucleus Emitted by excited nucleus itself after radioactive decay less energy higher energy Have strong ionizing ability  Appropriate number of nucleons, but too much energy  Internal conversion The excited nucleus can transfer its excess energy to an orbital electron (generally an inner-shell electron), causing the electron to be ejected from the atom. This can only occur if the excess energy is greater than the binding energy of the electron. This electron is called a conversion electron The resulting inner-orbital vacancy is rapidly filled with an outer-shell electron The energy released as a result of the “fall” of an outer-shell electron to an inner shell is emitted as an X-ray or as a free electron, an Auger electron  Decay notation Decay schematics Notice that a pathway ending to the left, as in electron capture or positron emission, corresponds to a decrease in atomic number. On the other hand, a line ending to the right, as in beta emission, corresponds to an increase in atomic number.  Decay mode and the line of stability The type of radioactive decay that occurs Isotopic line usually is such as to move the nucleus closer to this line. A radionuclide that is proton deficient (above the line) usually decays β− emission, because this transforms a neutron into a proton, moving the nucleus closer to the line of stability. A neutron-deficient radionuclide (below the line) usually decays by EC or β+ emission, because these modes transform a proton into a neutron. Heavy nuclides frequently decay by α Isotonic line emission or by fission, because these are modes that reduce mass number.  Decay mode and the line of stability It also is worth noting that β− , β+ , and Isotopic line EC decay all can transform an “odd-odd” nucleus into an “even-even” nucleus. even-even nuclei are relatively stable because of pairing of alike particles within the nucleus. There are in fact a few odd-odd nuclides lying on or near the line of stability that can decay either by β− emission or by EC and β+ emission. An example is 40K (89% β− , 11% EC or β+ ). In this example, the instability created by odd numbers of protons and neutrons is sufficient to Isotonic line cause decay in both directions away from the line of stability; however, this is the exception rather than the rule.  Stability and line of stability Note Stability Strictly speaking, stability is a relative term. We call a nuclide stable when its half-life is so long as to be practically immeasurable—say, greater than 108 years. The isotope of potassium 40K, for example, which makes up about 1% of the potassium found in nature, is considered stable but actually has a half-life of 109 years.  Exponential decay radio active decay is characterized by the disappearance of a constant fraction of the activity present in the sample during a given time interval. If a sample containing N radioactive atoms of a certain radionuclide, the average decay rate, ΔN/Δt, for that sample is given by: ∆𝑵⁄∆𝒕 =- 𝝀 𝑵 where λ is the decay constant for the radionuclide. The decay constant has a characteristic value for each radionuclide. It is the fraction of the atoms in a sample of that radionuclide undergoing radioactive decay per unit of time This equation is valid only as an estimate of the average rate of decay for a radioactive sample. From one moment to the next, the actual decay rate may differ from that predicted by this equation. This is called statistical fluctuations in decay rate  Exponential decay Activity of a sample (𝐀 = ∆𝑵⁄∆𝒕) is the average decay rate and it’s a measure of how radioactive the sample is. The SI unit of activity is the bequerel (Bq) is the number of disintegration per second (dps) or disntergration per minute (dpm) The –Ve sign A sample has an activity of 1 Bq if it is decaying at a rate 1 dps indicates that N is decreasing A (Bq)= −∆𝑵⁄∆𝒕 =𝝀 𝑵 with time The absolute value is used to indicate that activity is a “positive” quantity. The traditional unit for activity is the curie (Ci) which is defined as 3.7 × 10 dps (2.22 × 1012 dpm) The amount of activity used for nuclear medicine studies typically are in the MBq and GBq range  Exponential decay 𝝀 is the decay constant for the radionuclide Decay constant has a characteristic value for each radionuclide Decay constant: is the fraction of the atoms in a sample of that radionuclide undergoing radioactive decay per unit of time. The unit of decay constant is 𝑡𝑖𝑚𝑒 Thus decay constant 0.01 sec-1 means that, on the average , 1% of the atoms undergo radioactive decay each second Example: Tc-99m has λ= 0.1151 hr-1, i.e., 11.5% decay/hr Mo-99 has λ = 0.252 day-1, i.e., 25.2% decay/day  Exponential decay  With the passage of time the number N of radioactive atoms in a Decay of a radioactive sample during successive 1- sec increments of time, starting with 1000 atoms, for λ = 0.1 sec−1. Both the number of atoms remaining and activity (decay rate) decrease with time. The values shown are approximations Exponential decay Exponential decay  Physical half life It is not possible to predict when an individual nuclide atom will decay, just as in preparing popcorn, where one cannot determine when any particular corn will open. However, the average behavior of a large number of the popcorn is predictable. From experience with microwave popcorn, one knows that half of the corn will pop within 2min and most of the bag will be done in 4min. In a like manner, the average behavior of a radioactive sample containing billions of atoms is predictable. The time it takes for half of these atoms to decay is called the half-life or, in scientific notation, T1/2 (pronounced “T one-half ”). the time it takes for half of the remaining atoms to decay is also T1/2. This process continues until the number of nuclide atoms eventually comes so close to zero that we can consider the process complete.  Physical half life Physical Half-life (𝑻𝟏/𝟐 ) of a radionuclide is the time required for it to decay to 50% of its initial activity level. Time required for the amount of the radionuclides to reduce to half =(physical) half life Usually, tables or charts of radionuclides list the half-life of the radionuclide rather than its decay constant The half-life and decay constant of a radionuclide are related as 𝑇 ⁄ = ln ⁄ λ = ln  Exponential decay Decay curve radioactive(parent) atom Activity A(t) =Number of time Time required for the amount of the radionuclides to reduce to half =(physical) half life Exponential decay  Exponential decay  With the passage of time the number N of radioactive atoms in a sample decrease, therefore the activity A of a sample also decrease  N= the number of radioactive atoms present at any time t. Decay equations An exact mathematical expression for N(t) can be derived using methods of calculus. N(t) = N e λt Note that because activity A A(t) = A e λt is proportional to the number of atoms N λ = 0.693/𝑇 One can detect the presence of a radioactive sample not by the radioactive atom present but by the radiation emitted by these atoms when they disintegrate.  Exponential decay The decay factor This factor e λt , the fraction of radioactive atom remaining after a time t , is called the decay factor (DF) The decay factor is an exponential function of time t (it is a curve gradually approaching zero)  Exponential decay Table of decay factor: It is essential that an individual working with radionuclides know how to determine decay factors. The simplest and most straightforward approach is to use tables of decay factors, which are available from vendors of radiopharmaceuticals, instrument manufacturers, and so forth. Tables of decay factors cover only limited periods; however, they can be extended by employing principles based on the properties of exponential functions specifically 𝒆𝒂 𝒃 = 𝒆𝒂 x 𝒆𝒃. For example, suppose that the desired time t does not appear in the table but that it can be expressed as a sum of times, t = t1 + t2 + · · ·, that do appear in the table.  Average lifetime The actual lifetimes of individual radioactive atoms in a sample range anywhere from “very short” to “very long.” Some atoms decay almost immediately, whereas a few do not decay for a relatively long time. The average lifetime τ of the atoms in a sample has a value that is characteristic of the nuclide and is related to the decay constant λ by The average lifetime for the atoms of a radionuclide is therefore longer than its half-life, by a factor 1/ln 2 (≈1.44). The concept of average lifetime is of importance in radiation dosimetry calculations Example What is the decay factor for 99mTc after 16 hours? Express 16 hours as 6 hours + 10 hours. Then, from the table DF(16 hr) = DF(10 hr) × DF(6 hr) = 0.315 × 0.5 = 0.1575. Other combinations of times totaling 16 hours provide the same result. Example What if you do not have the table ?? What is the decay factor for 99mTc after 16 hours? N(t) = N e λt DF (𝑡) = e λt DF (16) = e[−1.848] =0.15755 λ = 0.693/𝑇 [(. / )t] DF (16) = e DF (16) = e[(. / ) (16)] Example What if you do not have the table ?? Example  Example N(t) = N e λt N(t) = N e λt λ = 0.693/𝑇 λ = 0.693/𝑇 [(. / )t] [(. / )t] N(t) = N e N(t) = N e N(60 min) = 1000 e[(. / ) 60] N(40 min) = 1000 e[(. / ) 40] N(60 min) = 1000 e[. ] = 125.06 N(40 min) = 1000 e[. ] = 250 atoms atoms Example Think differently  Example N(t) = N e λt t= 197.36 min λ = 0.693/𝑇 t= 3 hr and 17 min 1 = 1000 e[(. /20 )t] 1/1000 = e[(.035 ) t] 1 ln = −0.035 𝑡 1000 Example Example A vial containing 99mTc is labeled “75 kBq/mL at 8 am.” What volume should be withdrawn at 4 pm on the same day to prepare an injection of 50 kBq for a patient? Answer From the table ; the DF for 99mTc after 8 hours is found to be 0.397. Therefore the concentration of activity in the vial after 8 hours will be 0.397 × 75 kBq/ mL = 29.8 kBq/mL. The volume required for 50 kBq is 50 kBq divided by 29.8 kBq/mL = 1.68 mL. Example A vial containing 99mTc is labeled “50 kBq at 3 pm.” What is the activity at 8 am on the same day? Answer The decay time is t = −7 hours. From Table 4-1, DF(7 hr) = 0.445. Thus DF(−7 hr) = 1/0.445 = 2.247. The activity at 8 am is therefore 2.247 × 50 kBq = 112.4 kBq.  The specific activity A radioactive sample may contain stable isotopes of the element represented by the radionuclide of interest. For example, a given 131I sample may also contain the stable isotope 127I. When stable isotopes of the radionuclide of interest are present in the sample, they are called carrier, and the sample is said to be with carrier. A sample that does not contain stable isotopes of the element represented by the radionuclide is called carrier-free. Radionuclides may be produced carrier-free or with carrier, depending on the production method The ratio of radioisotope activity to total mass of the element present is called the specific activity of the sample. Specific activity has units of becquerels per gram The highest possible specific activity of a radionuclide is its carrier-free specific activity (CFSA) Decay of a mixed radionuclide sample When a sample contains a mixture of unrelated species, the total activity A is just the sum of the individual activities of the various species:  Decay Vs Counts A related term that is frequently confused with decay is the count, which refers to the registration of a single decay by a detector such as a Geiger counter. Most of the detectors used in nuclear medicine detect only a fraction of the decays, principally because the radiation from many of the decays is directed away from the detector. The count rate refers to the number of decays actually counted in a given time, usually counts per minute. The count rate will be proportional to the decay rate.  Decay Vs Counts Decay or counts Decay Vs Counts Decay Vs Counts The End