KAHIBARO
29.5. Analyzing TOF Events
Table of Contents
Timing distributions
In a time of flight PET simulation, each coincidence event carries a detection time for each photon. GATE stores these times in the singles and coincidence outputs, typically as absolute times $t_1$ and $t_2$ in seconds or nanoseconds, depending on your configuration. To analyze TOF, the first quantity you extract is the time difference
$$
\Delta t = t_1 - t_2 .
$$
The sign convention is arbitrary, but you must use it consistently when interpreting the results.
For a static point source at the center of the scanner, and in the absence of detector effects, the true value of $\Delta t$ is zero for every annihilation, because the two 511 keV photons travel the same distance and arrive at the detectors simultaneously. In a realistic GATE simulation with a TOF digitizer, you will observe a spread of $\Delta t$ values around zero. This spread comes from the coincidence timing resolution that you configured, which is usually modeled as a Gaussian blurring applied to the detection times.
If you fill a histogram of $\Delta t$ for many coincidences, you obtain the timing distribution. For a simple centered point source and symmetric detectors, this distribution is approximately Gaussian:
$$
P(\Delta t) \approx \frac{1}{\sqrt{2\pi}\,\sigma_t} \exp\!\left(-\frac{\Delta t^2}{2\sigma_t^2}\right),
$$
where $\sigma_t$ is the standard deviation of the time difference. If the digitizer uses a per photon timing resolution $\sigma_{\text{det}}$, then the coincidence time resolution is
$$
\sigma_t \approx \sqrt{2}\,\sigma_{\text{det}},
$$
because the time difference combines two independent blurred measurements.
From a fitted Gaussian timing distribution $P(\Delta t)$ you can extract:
- The mean time difference $\mu_{\Delta t}$, which ideally should be close to zero for a symmetric system and a centered source.
- The standard deviation $\sigma_t$, which characterizes the coincidence timing resolution.
- The full width at half maximum (FWHM) of the timing distribution,
$$
\text{FWHM}_t = 2 \sqrt{2 \ln 2}\,\sigma_t \approx 2.355\,\sigma_t.
$$
This quantity is commonly quoted as the TOF timing resolution of the PET system.
In a GATE study, you typically compare the measured $\text{FWHM}_t$ from the simulation to the target timing performance of your scanner design. If you have defined a Gaussian time blurring with a specified FWHM in the digitizer, the resulting coincidence timing FWHM should agree with that value within statistical uncertainty, provided that you impose a sufficiently wide coincidence window that does not clip the Gaussian tails.
You can also inspect timing distributions under more complex conditions. For example, if you move the source along the scanner axis, or place it off center along the line connecting two detectors, the peak of the timing distribution will shift away from zero, because one photon travels a longer path than the other. For a fixed detector pair, the expected shift in time is
$$
\mu_{\Delta t} = \frac{\Delta L}{c},
$$
where $\Delta L$ is the difference in path length between the two photons and $c$ is the speed of light. In the coincidence data, plotting $\Delta t$ as a function of known source position provides a consistency check that your TOF modeling is correct and that the sign convention used when computing $\Delta t$ is understood.
In practice you will perform these analyses outside GATE using ROOT or Python. You read the coincidence tree, compute $\Delta t$ event by event, and fill histograms, optionally separated by detector pair, crystal ring, or energy selection. By applying energy windows around the 511 keV photopeak, you can study how scattering affects the timing distribution. Scattered coincidences usually have broader timing distributions or shifted means, because the scattered photons follow longer, more complex paths.
Finally, timing distributions are also useful to check the effect of the coincidence time window that you configured. If you choose a coincidence window that is much larger than the timing FWHM, you will accept almost all true coincidences, but also more random coincidences. If the window is too narrow, you will truncate the timing distribution and lose part of the Gaussian tails, which reduces sensitivity and can bias the apparent timing performance if you fit only the remaining central part. Examining the full $\Delta t$ histogram and how it is truncated by the coincidence window is an essential part of TOF PET performance analysis.
Spatial localization
The main motivation for TOF PET is to localize the annihilation point along the line of response by exploiting the difference in arrival times. For each coincidence between detectors 1 and 2, with measured detection times $t_1$ and $t_2$, you define the position offset along the line of response, relative to the midpoint between the two detectors, as
$$
\Delta x = \frac{c}{2}\,(t_1 - t_2) = \frac{c}{2}\,\Delta t,
$$
where $c$ is the speed of light in the medium, usually approximated by the speed of light in vacuum for PET reconstruction purposes.
The basic TOF localization formula is
$$
\Delta x = \frac{c}{2}\,\Delta t,
$$
where $\Delta x$ is the displacement of the annihilation point along the line connecting the two detectors, and $\Delta t$ is the measured time difference between the two detected photons.
In a GATE simulation, every coincidence has known detector positions. Let $\mathbf{r}_1$ and $\mathbf{r}_2$ be the 3D positions of the two detecting crystals. The line of response is the line through these points, and its midpoint is
$$
\mathbf{r}_{\text{mid}} = \frac{\mathbf{r}_1 + \mathbf{r}_2}{2}.
$$
If you define the unit vector from detector 2 to detector 1 as
$$
\hat{\mathbf{u}} = \frac{\mathbf{r}_1 - \mathbf{r}_2}{\|\mathbf{r}_1 - \mathbf{r}_2\|},
$$
then the TOF estimated annihilation position is
$$
\mathbf{r}_{\text{TOF}} = \mathbf{r}_{\text{mid}} + \Delta x \,\hat{\mathbf{u}}.
$$
With these expressions you can reconstruct a cloud of estimated annihilation points directly from the coincidence list, without performing full image reconstruction. This is very useful for understanding how timing resolution affects spatial localization.
If you simulate a point source displaced along a known direction, for example along the line between two opposite detectors, then compute $\mathbf{r}_{\text{TOF}}$ for many events, you will see that the reconstructed positions form a one dimensional distribution centered near the true source position, with a width that reflects the timing resolution. The standard deviation of the localization along the LOR, $\sigma_x$, is related to the timing standard deviation by
$$
\sigma_x = \frac{c}{2}\,\sigma_t,
$$
and the FWHM of the spatial kernel is
$$
\text{FWHM}_x = 2.355\,\sigma_x = 2.355\,\frac{c}{2}\,\sigma_t.
$$
This quantity is the TOF spatial resolution along the line of response.
You can verify this relationship in GATE by first measuring $\sigma_t$ from the timing distribution, then computing $\mathbf{r}_{\text{TOF}}$ for many events and fitting the spatial distribution along the line to obtain $\sigma_x$. Agreement between the two is a strong indication that your digitizer timing model and TOF analysis are consistent.
Spatial localization analysis also helps you see the benefit of TOF for extended sources. For a uniform cylindrical phantom, traditional non TOF coincidences provide no preferred position along each line of response. With TOF, each event carries a Gaussian localization kernel centered near the true annihilation point. If you histogram the TOF estimated positions in 3D, you will recover an image of the phantom that appears blurred along the LOR with a width corresponding to $\text{FWHM}_x$, even before any reconstruction algorithm is applied. Comparing this direct TOF localization cloud with the true phantom geometry that you defined in GATE provides intuitive insight into how TOF information reduces uncertainty along each line and improves image signal to noise ratio.
Finally, analyzing spatial localization in simulation allows you to study how various effects degrade TOF performance. You can change the timing resolution in the digitizer and observe how the spatial kernel broadens. You can introduce detector depth of interaction, scattering in the patient, or imperfect calibration, and then examine how the cloud of TOF estimated positions shifts or becomes asymmetric. Because GATE records ground truth information such as the true annihilation position and event ID, you can correlate each $\mathbf{r}_{\text{TOF}}$ with its corresponding true position, compute residuals, and derive quantitative metrics such as bias and variance of the TOF localization along the line of response.
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