KAHIBARO
11.1. Radioactive Decay
Table of Contents
Radionuclides
In GATE, a radioactive source is built from one or more radionuclides, which are unstable nuclei that transform into more stable configurations by emitting radiation. For medical physics, radionuclides are not abstract. Each one is a specific isotope, such as $^{18}\text{F}$ or $^{99\text{m}}\text{Tc}$, with a defined nuclear mass, charge, half-life, and characteristic emission spectrum. When you choose a radionuclide for a simulation, you are implicitly defining which particles, which energies, and which time pattern of emission will appear in your simulation.
A radionuclide is described by its number of protons $Z$, number of neutrons $N$, and mass number $A = Z + N$. Isotopes of the same chemical element share $Z$ but have different $N$. In nuclear medicine and therapy you often meet notations like $^{18}\text{F}$, $^{11}\text{C}$, $^{68}\text{Ga}$, $^{99\text{m}}\text{Tc}$, or $^{131}\text{I}$. The superscript is the mass number $A$. The optional letter “m,” as in $^{99\text{m}}\text{Tc}$, means the nucleus is in a metastable excited state that can de-excite through internal transitions.
From the simulation point of view, the most important radionuclide property is its half-life. The half-life $T_{1/2}$ is the time after which, on average, half of the original nuclei have decayed. It sets the time scale of the problem: ultra short half-lives produce intense emission over a brief interval, while long half-lives produce low count rates spread over a long time. In GATE, half-life controls how the activity of a source changes during a time dependent simulation.
Another key property is the decay scheme. The decay scheme lists all possible ways the nucleus can decay, the type of radiation involved, and the probability of each branch. For example, a positron emitter has a decay branch in which a positron and a neutrino are emitted. A gamma emitter has one or more gamma lines at specific energies, each with a certain branching ratio. GATE uses these nuclear data to generate realistic decay products without you having to hard-code energies or probabilities.
For absolute beginners, it helps to think of a radionuclide in a GATE simulation as a black box with three time and emission related knobs: its initial activity (how many decays per second at the start), its half-life (how fast the activity drops with time), and its emission pattern (what particles and energies are released per decay). In later chapters, specific medical radionuclides and how to configure them in GATE will be discussed. Here the focus is on how their generic decay behavior is described.
Activity is the quantity that measures how many decays happen per unit time. If $N(t)$ is the number of undecayed nuclei at time $t$, the activity $A(t)$ is defined as
$$
A(t) = -\frac{\mathrm{d}N(t)}{\mathrm{d}t}.
$$
Activity is measured in becquerels (Bq), where $1\ \text{Bq} = 1$ decay per second. In practical simulations you often use kBq or MBq. In GATE you will frequently specify source strength by activity and use the radionuclide half-life to determine how that activity evolves during the run.
The time evolution of activity for a single radionuclide behaving as an isolated exponential decay is
$$
A(t) = A_0 \, e^{-\lambda t},
$$
where $A_0$ is the activity at $t = 0$ and $\lambda$ is the decay constant. The decay constant and half-life are linked by a simple relation that is essential in medical physics work.
For a simple exponential decay,
- $N(t) = N_0 e^{-\lambda t}$ and $A(t) = A_0 e^{-\lambda t}$.
- The relation between decay constant $\lambda$ and half-life $T_{1/2}$ is
$$
\lambda = \frac{\ln 2}{T_{1/2}}.
$$ - Activity is defined as $A(t) = -\dfrac{\mathrm{d}N(t)}{\mathrm{d}t}$ and is measured in Bq.
When you set up time dependent simulations in GATE, especially for dynamic imaging or therapy, these relations tell you how to translate between an initial activity, a half-life, and the expected activity at any later time point.
In practice, medical radionuclides may decay through multiple steps. A parent isotope may transform into a radioactive daughter, which decays again. This creates decay chains. GATE can follow such chains when the corresponding nuclear data are available. For beginners, it is often sufficient to think about a single nuclide that decays once. As you move toward more advanced internal dosimetry, multi step chains and the emissions from daughters also become important and are handled by the same underlying radioactive decay models.
In GATE, you will usually not type in decay constants or branching ratios by hand. Instead, you select the radionuclide by name and rely on Geant4’s radioactive decay database to supply half-lives and decay schemes. Understanding qualitatively what a radionuclide is and how its decay shapes the time and energy pattern of emitted radiation will help you choose suitable nuclides and interpret the output of your simulations.
Nuclear decay
Nuclear decay is the process by which an unstable nucleus transforms into a more stable configuration. For every decay an individual nucleus has a random decay time, but for a large population the overall behavior follows predictable statistics. Monte Carlo simulation, and thus GATE, samples these random decay times and decay channels one nucleus at a time to generate realistic histories of particle emissions.
The simplest mathematical description of decay assumes that each nucleus has a constant probability per unit time to decay, independent of its age. This leads to the exponential decay law. If you start with $N_0$ identical unstable nuclei at time $t = 0$, the expected number that remain undecayed at time $t$ is
$$
N(t) = N_0 e^{-\lambda t},
$$
where $\lambda$ is the decay constant. The decay constant is a measure of how quickly the nuclei disappear. The half-life is related to $\lambda$ by the relation in the previous section. In GATE, the radioactive decay models use this basic law to sample when each individual decay happens.
For a large sample, the number of decays that occur between times $t$ and $t + \Delta t$ is
$$
\Delta N \approx N(t)\,\lambda\,\Delta t,
$$
as long as $\Delta t$ is small. This is equivalent to saying that the probability a given nucleus decays in a short time interval $\Delta t$ is approximately $\lambda \Delta t$. GATE uses this probability concept in a Monte Carlo sense. It draws random numbers to decide if a particular nucleus decays during a given time step and, if it does, which decay channel is taken.
Each nuclear decay emits one or more particles. Depending on the nuclide and the decay mode, these may include photons, electrons, positrons, alpha particles, or heavier fragments. In medical physics, the modes of most interest are beta minus decay, beta plus decay, and gamma emission. Although GATE relies on Geant4 to implement the detailed physics of each decay mode, you should understand qualitatively what these terms mean, because they directly determine what your detectors or dose scoring actors will see.
In beta minus decay, a neutron in the nucleus transforms into a proton, and the nucleus emits an electron and an anti neutrino. The atomic number $Z$ increases by one, the mass number $A$ stays the same, and the emitted beta electron has a continuous energy spectrum up to some endpoint energy. In a GATE simulation this electron will then undergo electromagnetic interactions in surrounding materials and deposit energy, contributing to dose.
In beta plus decay, or positron emission, a proton in the nucleus transforms into a neutron, and the nucleus emits a positron and a neutrino. The atomic number $Z$ decreases by one. The emitted positron again has a continuous energy spectrum. It slows down in tissue, travels a certain distance (the positron range), and eventually annihilates with an electron, producing two 511 keV gamma photons emitted nearly back to back. Positron emission, positron transport, and annihilation are central to PET simulations, which are covered later, but the underlying decay is handled by Geant4’s radioactive decay module when you select a positron emitting radionuclide in GATE.
Gamma emission occurs when a nucleus in an excited state drops to a lower energy state and emits a photon. These photons have discrete energies that correspond to differences between nuclear energy levels. Many medical gamma emitters have simple schemes where one or a few gamma lines dominate. When you simulate such a radionuclide with GATE, you will see clear peaks in the detected energy spectrum at those characteristic energies, provided your detector and digitizer are configured correctly.
Often a single nuclide has multiple possible decay branches. For example, it may emit different gamma rays from different excited levels, or it may have both beta decay and electron capture branches. Each branch is assigned a branching ratio, which is the probability that a decay follows that particular path. The sum of all branching ratios is 1. GATE samples these branches stochastically, so that the relative frequencies of emitted particles match the nuclear data on average.
Key features of nuclear decay used in simulations:
- Exponential law: $N(t) = N_0 e^{-\lambda t}$ and $A(t) = \lambda N(t)$.
- Decay is random for each nucleus, but ensemble behavior is predictable.
- Decay modes (beta minus, beta plus, gamma, etc.) have fixed branching ratios that sum to 1.
- The energy spectrum and particle types from each decay are determined by the decay scheme of the radionuclide.
Besides single step decays, some radionuclides belong to decay chains. A parent nucleus decays to a daughter that is itself radioactive, and so on, until a stable product is reached. The populations of parent and daughter nuclides then evolve according to a set of coupled exponential equations, not a single one. Although the detailed mathematics of decay chains is beyond what you need at this stage, you should be aware that GATE can simulate chained decays when configured with appropriate radioactive decay data. This is important for internal dosimetry and for some therapeutic isotopes that have significant daughter emissions.
From the GATE user’s perspective, the practical implication of nuclear decay is how it controls the timing of emissions and how many particles are produced during a given acquisition. For a static acquisition with a very short duration compared to the half-life, activity can be treated as nearly constant. For long acquisitions or dynamic studies, the decrease of activity over time must be taken into account. GATE allows you to define simulation intervals and acquisition durations that relate directly to the decay law, and the underlying models will adjust the rate of emitted decays automatically based on the radionuclide half-life you have chosen.
Finally, nuclear decay in GATE is tightly coupled to the rest of the simulation workflow. The decay products become primary particles in the transport step, interact with materials according to the physics list, and are recorded by actors or digitizers. Understanding nuclear decay qualitatively helps you connect a radionuclide definition at the source to patterns you later observe in energy spectra, time distributions, and dose maps.
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