KAHIBARO
28.3. Coincidence Time Window
Table of Contents
Selecting coincidence events
In PET, a coincidence event is accepted only if the two detected photons arrive within a short time interval called the coincidence time window. This window is applied to the detection times of singles after digitization, during the coincidence sorting stage. Understanding and choosing this window correctly is essential for realistic PET simulations and for controlling the balance between sensitivity and the number of random coincidences.
Definition of the coincidence time window
Each detected gamma that survives the digitizer becomes a single, with at least an energy and a detection time $t$. To decide if two singles belong to the same annihilation event, their detection times are compared.
Let $t_1$ and $t_2$ be the detection times of two singles in different detector elements. The absolute time difference is:
$$
\Delta t = |t_1 - t_2|.
$$
A coincidence is accepted if $\Delta t$ is smaller than a predefined time window, usually denoted $T_\text{coin}$.
A pair of singles is accepted as a coincidence if and only if
$$
\Delta t = |t_1 - t_2| \le T_\text{coin}.
$$
In a GATE PET digitizer, $T_\text{coin}$ is a configurable parameter of the coincidence sorter. You do not change time at the source or in the physics list to adjust coincidences, you change the digitizer coincidence window.
Role of the coincidence window in PET
The coincidence time window serves as a temporal filter that tries to separate photons from the same annihilation event from unrelated photons. True and scattered coincidences from the same annihilation have very small $\Delta t$ values, limited mainly by the detector timing resolution and, in TOF PET, by the difference in flight path lengths of the two photons. Unrelated photons that occur far apart in time have large $\Delta t$ and are rejected.
The choice of $T_\text{coin}$ affects three key categories of coincidence events that you will analyze later:
- True coincidences, where both detected photons originate from the same annihilation and have not scattered.
- Scattered coincidences, where at least one photon scattered before detection but both photons still come from the same annihilation.
- Random coincidences, where two unrelated singles, from different annihilations, fall by chance within the window.
A good coincidence window should keep almost all true coincidences while limiting the number of random coincidences.
Relationship with detector timing resolution
The achievable coincidence time window is constrained by detector timing performance. Each detector has a timing resolution, often expressed as a full width at half maximum (FWHM) value, for example $500$ ps or $300$ ps. This resolution represents the statistical spread of measured detection times around the true interaction time.
If the single-detector timing resolution is $\sigma_t$ (standard deviation), the coincidence timing resolution for a pair of detectors is approximately
$$
\sigma_\text{coin} = \sqrt{\sigma_{t,1}^2 + \sigma_{t,2}^2},
$$
which simplifies to $\sigma_\text{coin} \approx \sqrt{2}\,\sigma_t$ for identical detectors. In practice, the FWHM is often used rather than $\sigma$:
$$
\text{FWHM} \approx 2.355\,\sigma.
$$
The time window $T_\text{coin}$ is usually chosen larger than the FWHM of the coincidence timing distribution to keep most annihilation events. For a non TOF PET scanner, typical $T_\text{coin}$ values are on the order of a few to several nanoseconds, depending on timing performance.
As a rule of thumb, select a coincidence window $T_\text{coin}$ that is a few times larger than the coincidence timing resolution FWHM, for example
$$
T_\text{coin} \approx 2 \text{ to } 4 \times \text{FWHM}.
$$
This keeps most true events while limiting randoms.
In GATE, the detector timing resolution is modeled in the digitizer with a time blurring step. The blurred detection times are then used when applying the coincidence window. If you reduce the blurring, the simulated timing resolution improves and you can choose a smaller $T_\text{coin}$.
Impact on true and random coincidence rates
The coincidence window has a strong influence on the rates of different coincidence types, especially random coincidences. For a given activity and scanner geometry, the rate of true coincidences is almost independent of $T_\text{coin}$ once the window is wide enough to include the full timing distribution of true pairs. Making the window even wider will not increase true coincidences significantly.
Random coincidences, in contrast, grow approximately linearly with the coincidence time window, because a larger window increases the chance that two unrelated singles fall within it.
If $R_1$ and $R_2$ are the singles rates in two detector channels, a simplified expression for the random rate $R_\text{rand}$ between that pair is:
$$
R_\text{rand} \propto R_1 R_2 T_\text{coin}.
$$
Random coincidence rate increases approximately in proportion to the coincidence time window $T_\text{coin}$, while true coincidence rate saturates once the window is wide enough. Increasing $T_\text{coin}$ improves sensitivity only slightly but increases randoms strongly.
In a full PET ring, there are many detector pairs, so total randoms can become large if $T_\text{coin}$ is too wide, especially at high activity levels.
Practical selection in simulations
When you configure the PET digitizer in GATE, you will typically set:
- A timing resolution (time blurring parameter) that models your detector.
- A coincidence time window $T_\text{coin}$ in the coincidence sorter.
To choose $T_\text{coin}$:
- Look at the detector timing resolution that you want to simulate. If you know the experimental coincidence timing resolution FWHM, base your choice on that.
- Ensure that $T_\text{coin}$ is comfortably larger than the FWHM so that more than 95 percent of true coincidences are accepted. For instance, if the coincidence FWHM is $500$ ps, a window of $2$ to $4$ ns is conservative for a non TOF PET system.
- Check the effect on randoms by analyzing the simulated coincidence output. In later sections you will learn how to estimate the fraction of random, scattered, and true coincidences from the simulation.
In time of flight PET, the coincidence time window is often narrower, since timing resolution is much better. The principles are the same, but the chosen $T_\text{coin}$ can be closer to the FWHM of the timing distribution to keep a reasonable balance between true and random coincidences.
Implementation aspects in GATE
In a typical GATE PET simulation, coincidence sorting works on the list of singles that pass energy and time blurring. The coincidence module groups singles that occur within $T_\text{coin}$, usually in pairs, and writes them as coincidence events to an output file. You do not need to manually compare times in your Python analysis code for basic coincidence selection, although you can reapply or tighten time cuts in post processing if needed.
If you change $T_\text{coin}$ between runs while keeping source activity and other parameters fixed, you can directly study its impact on total coincidence counts, random fraction, and image quality. This is a useful way to validate that your simulation behaves as expected and to understand the trade offs involved in selecting coincidence events.
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