KAHIBARO
11.6. Time-Dependent Activity
Table of Contents
Radioactive decay
Activity in GATE is always linked to physical time, so once you introduce radioactivity the number of emitted particles is no longer constant. Instead, it follows the radioactive decay law. For a single radionuclide with initial activity $A_0$ at time $t = 0$, the activity at time $t$ is given by
$$
A(t) = A_0 \, e^{-\lambda t}
$$
where $\lambda$ is the decay constant. The decay constant is related to the physical half-life $T_{1/2}$ of the radionuclide by
$$
\lambda = \frac{\ln 2}{T_{1/2}}.
$$
In a time-dependent simulation, GATE uses these relations internally to change the effective emission rate during the run. When you specify a radionuclide and a half-life for a source, GATE can compute how many decays occur in each time interval and how the activity decreases.
For a radionuclide with half-life $T_{1/2}$, the activity as a function of time is
$$
A(t) = A_0 \, 2^{-t / T_{1/2}} = A_0 \, e^{-(\ln 2)\, t / T_{1/2}}.
$$
Always use consistent time units for $t$ and $T_{1/2}$, for example both in seconds.
In practice, you rarely interact directly with $A(t)$ inside your Python script, because GATE handles the decay law internally once you define the radionuclide and the simulation time structure. However, it is important to understand what happens physically. If you run a simulation over a long time interval compared with the half-life, the activity will drop significantly and later parts of the acquisition will contribute fewer events.
When you configure time-dependent activity for a source in GATE, you typically specify:
- The radionuclide, which fixes the half-life.
- The initial activity, which sets $A_0$ at the chosen reference time.
- The acquisition duration, which defines for how long the source is active.
- Optional time intervals or frames if you want to divide the acquisition into segments.
GATE will then generate decays at a rate that follows the radioactive decay law. If you later compare simulated count rates to analytical expectations or measurements, you should always include the exponential decay of activity when predicting how many decays occur during a given time interval.
You can also work with effective decay when biological processes are involved, for example for internal dosimetry, but detailed kinetics and compartment models are covered in other chapters. Here, it is enough to understand that any time-dependent activity curve, whether purely physical or effective, will be mapped to a time-varying source strength in the simulation.
Dynamic simulations
Time-dependent activity becomes especially relevant when you divide the simulation into multiple time frames. This is called a dynamic simulation. Instead of one long acquisition with a single output, you have a set of shorter intervals, each with its own start and end time. For each interval, the activity is different, so the expected number of decays and detected events changes over time.
In GATE, dynamic simulations are typically organized as a series of time windows. Each window has a start time $t_{\text{start}}$ and an end time $t_{\text{end}}$, and the source activity within that window follows the decay law. The total number of decays in one frame can be written as
$$
N_{\text{decays}} = \int_{t_{\text{start}}}^{t_{\text{end}}} A(t) \, dt.
$$
If $A(t)$ is purely exponential with constant $A_0$ at $t = 0$, you can evaluate this integral analytically. For most GATE users, it is more important to understand the qualitative behavior. Early frames, close to the start of the simulation, have higher count rates because the activity is higher. Later frames suffer from reduced statistics unless you increase the acquisition time to compensate.
Dynamic simulations allow you to model:
Temporal changes in source strength, as in radionuclide decay during long imaging acquisitions.
Time-varying activity distributions, for example when tracer uptake or clearance changes the spatial pattern of activity. The exact definition of these changing distributions is handled elsewhere, but they always rely on assigning a specific activity value as a function of time.
Multi-frame acquisitions, such as dynamic PET studies, where you reconstruct a sequence of images that show how the tracer distribution evolves.
GATE supports this by providing a simulation time structure and by allowing you to define the activity and, when relevant, the geometry or source distribution as functions of time. The key idea is that your simulation is no longer static. You consider both where and when particles are emitted.
In a dynamic simulation, every source parameter that depends on time, such as activity, must be consistent with the defined time frames. Make sure that:
- Time units in the simulation match the units used for activity curves.
- Frame start and end times are ordered correctly and cover the intended acquisition.
- The radionuclide half-life is compatible with the total simulation duration.
From a practical point of view, dynamic simulations affect how you interpret the results. Each output frame, file, or segment corresponds to a specific time period with its own mean activity. When you analyze event rates, reconstructed images, or dose in each interval, you should relate them back to the corresponding activity level and time window. This is essential when comparing with clinical dynamic studies or when validating that the simulated count rates and dose follow the expected time dependence from radioactive decay.
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