KAHIBARO
19.6. Coincidence Timing
Table of Contents
Time difference
In PET, each coincidence event is built from two singles that are close in time. The time difference is simply the detection time of one single minus the detection time of the other. If the two detection times are $t_1$ and $t_2$, the time difference is usually defined as
$$
\Delta t = t_2 - t_1.
$$
GATE stores a time for every single event after digitization. When coincidences are sorted, the coincidence module or actor reads the times of the two singles and calculates this $\Delta t$. The sign of $\Delta t$ depends on which detector is considered first, so in many analyses you use $|\Delta t|$ rather than $\Delta t$ itself.
In non time of flight PET, the absolute value of $\Delta t$ is mainly used to decide if two singles belong to the same annihilation event. In time of flight PET, the value of $\Delta t$ also carries position information along the line of response, but that is addressed in the dedicated TOF chapter.
The time difference is affected by several factors: the true physical emission times of the photons, their different path lengths through the phantom and the scanner, and the detector timing response, including any timing blurring you have configured in the digitizer. In GATE, if you add a timing blurring component, each single time is shifted by a random delay that follows a chosen distribution, often Gaussian. This directly broadens the distribution of $\Delta t$.
In analysis, you often look at a histogram of $\Delta t$ values for all coincidences. For a stationary source and a realistic timing model this histogram has a narrow peak around zero. The width of this peak is related to the coincidence timing resolution that you have configured in your simulation.
The coincidence time difference is
$$\Delta t = t_2 - t_1.$$
A narrow distribution of $\Delta t$ around zero is essential for accurate coincidence selection and for realistic time of flight behavior.
Coincidence timing window
The coincidence timing window is the maximum allowed time difference between two singles for them to be accepted as a coincidence. In other words, two singles are considered coincident if
$$
|\Delta t| \le \Delta t_{\text{window}},
$$
where $\Delta t_{\text{window}}$ is the timing window half width that you choose.
In GATE, the coincidence sorter in the digitizer chain uses this rule. You configure the timing window as a parameter of the coincidence module, usually in units like ns. During the simulation, the sorter examines the ordered list of singles. When it finds two singles that come from different detectors and whose times satisfy this condition, it groups them into a coincidence event and writes them to the coincidence output.
The choice of $\Delta t_{\text{window}}$ has a strong impact on coincidence data quality. If the window is too narrow, many true coincidences are lost because detector timing blurring and different photon paths make their $|\Delta t|$ slightly larger than the ideal value. This reduces sensitivity and increases statistical noise. If the window is too wide, many unrelated singles that happen to occur close in time are grouped into coincidences, which increases random coincidences and degrades image quality.
For a given detector timing resolution, there is a practical range of useful timing windows. In a basic non time of flight PET simulation, the window is often chosen to be several times larger than the single detector timing resolution so that most true coincidences are accepted, while still rejecting a large fraction of randoms.
In time of flight PET, the same timing window also limits which events can be considered for TOF reconstruction. The detailed relation between $\Delta t_{\text{window}}$ and spatial localization along the line of response is described in the TOF PET chapter, but the basic coincidence selection rule is the same.
When you configure and analyze coincidences in GATE, it is useful to check the distribution of $\Delta t$ after the simulation. If the chosen timing window barely covers the central peak of that distribution, you are discarding many valid events. If the window is much wider than the central peak, you are allowing many unnecessary randoms. Adjusting $\Delta t_{\text{window}}$ is a simple but important part of realistic PET coincidence modeling.
Coincidence selection rule:
$$|\Delta t| \le \Delta t_{\text{window}}.$$
A smaller window reduces randoms but can lose true events. A larger window keeps more trues but increases random coincidences.
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