KAHIBARO
29.3. Time Difference
Table of Contents
Detection time difference
In time of flight PET you detect two 511 keV photons in coincidence. Each detection is timestamped by the PET electronics. Let the detection times of the two photons be $t_1$ and $t_2$, typically measured in picoseconds or nanoseconds relative to a common clock.
The central TOF quantity is the detection time difference
$$
\Delta t = t_1 - t_2 .
$$
By convention you are free to define which detector is “1” and which is “2”, but you must use the same convention consistently in your analysis. In GATE, each coincidence event will usually contain at least the following timing information for the two contributing singles: their individual times and the coincidence time difference derived from them.
The measured $\Delta t$ is not exact. It is blurred by the detector timing resolution, electronics jitter, and sometimes additional effects like light transport in long crystals. Even if the annihilation happens exactly at the center of the line of response, where you would expect $t_1 \approx t_2$ and so $\Delta t \approx 0$, the measured time difference will be distributed around zero with a spread characterized by the coincidence timing resolution, often quoted as a full width at half maximum.
In a simulation, you can distinguish three conceptually separate times:
- The true physical times $t_{1,\mathrm{true}}$ and $t_{2,\mathrm{true}}$ when each photon actually interacts in the crystal.
- The “ideal” detector times before timing blurring is applied, which may include delays such as light transport within the crystal or electronics response but no random noise.
- The measured times $t_{1,\mathrm{meas}}$ and $t_{2,\mathrm{meas}}$ after GATE has applied timing resolution effects.
In TOF PET analysis, you always work with the measured time difference
$$
\Delta t_{\mathrm{meas}} = t_{1,\mathrm{meas}} - t_{2,\mathrm{meas}} ,
$$
because that is what the real scanner provides. By comparing the distribution of $\Delta t_{\mathrm{meas}}$ for many events, you can evaluate the system timing resolution and verify that your simulation settings are realistic.
For each coincidence event, the sign of $\Delta t$ tells you which detector detected its photon first, and the magnitude $|\Delta t|$ tells you how much earlier that happened. This is the key information that allows TOF PET to localize the annihilation point along the line of response.
The fundamental TOF timing quantity is the detection time difference
$$
\Delta t = t_1 - t_2 ,
$$
where $t_1$ and $t_2$ are the measured detection times of the two photons in a coincidence event.
TOF position estimate
The detection time difference can be converted into a position estimate along the line connecting the two detectors, the line of response. Assume two detectors separated by a distance $L$. A positron annihilates at some point along this line. The resulting two photons travel in opposite directions with speed very close to the speed of light in vacuum, $c$.
If the annihilation occurs exactly in the middle between the detectors, the photons travel equal distances and arrive at the same time, so ideally $\Delta t = 0$. If the annihilation occurs closer to detector 1, that photon travels a shorter distance and arrives earlier at detector 1, so $t_1 < t_2$ and $\Delta t$ becomes negative with our $t_1 - t_2$ convention.
Let $x$ be the displacement of the annihilation point from the midpoint of the line of response, positive toward detector 1. Then the distances traveled by the photons are
$$
d_1 = \frac{L}{2} - x, \qquad d_2 = \frac{L}{2} + x ,
$$
and their ideal arrival times are
$$
t_1 = \frac{d_1}{c}, \qquad t_2 = \frac{d_2}{c} .
$$
The ideal time difference is therefore
$$
\Delta t = t_1 - t_2
= \frac{d_1 - d_2}{c}
= \frac{(\tfrac{L}{2} - x) - (\tfrac{L}{2} + x)}{c}
= \frac{-2x}{c} .
$$
Solving for $x$ gives the position estimate along the line of response in terms of the measured time difference:
$$
x = -\frac{c}{2} \, \Delta t .
$$
The minus sign comes from the convention used for $\Delta t$. If you prefer, you can define $\Delta t$ as $t_2 - t_1$ instead, which flips the sign in the formula. The physically important quantity is the magnitude and consistency of your definition, not the sign itself.
In practice, you replace the ideal $\Delta t$ with the measured value $\Delta t_{\mathrm{meas}}$, which already includes timing blurring. The TOF-based position estimate for an event then becomes
$$
x_{\mathrm{TOF}} = -\frac{c}{2} \, \Delta t_{\mathrm{meas}} .
$$
This $x_{\mathrm{TOF}}$ tells you, relative to the midpoint of the detector pair, where along the line the event is more likely to have happened. Because the timing measurement has finite resolution, you cannot know the exact annihilation point, but you can say that it lies within a region around $x_{\mathrm{TOF}}$ whose width is related to the timing resolution.
If the coincidence timing resolution is characterized by a full width at half maximum $\Delta t_{\mathrm{FWHM}}$, you can define an approximate TOF spatial resolution along the line of response as
$$
\Delta x_{\mathrm{FWHM}} \approx \frac{c}{2} \, \Delta t_{\mathrm{FWHM}} .
$$
Better timing resolution (smaller $\Delta t_{\mathrm{FWHM}}$) directly improves the localization of the annihilation point along the line of response.
In a GATE simulation of TOF PET, you use this relationship in analysis, not inside the digitizer itself. The digitizer generates singles and coincidences with realistic timing, and then in your analysis code you convert the stored time differences into TOF position estimates using the formula.
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