KAHIBARO
29.4. Timing Blurring
Table of Contents
Realistic detector timing
In an ideal PET simulation, every detected gamma would be assigned its true interaction time. In reality, PET detectors measure time with limited precision, so the measured detection time is slightly shifted from the true time by a random error. This effect is called timing blurring.
In GATE, you typically start from unblurred singles, where each interaction in a detector crystal has a true time given by the particle transport. The digitizer then adds timing blurring to simulate the finite timing resolution of the detectors. This is essential for any realistic time-of-flight PET (TOF PET) simulation, because the TOF information comes from differences in these measured, blurred times.
Timing blurring is usually modeled as a Gaussian smearing of the true time. If $t_{\text{true}}$ is the physical interaction time of the gamma in the crystal, the digitizer samples a time error $\Delta t$ from a normal distribution with standard deviation $\sigma_t$ and defines the measured time as
$$
t_{\text{meas}} = t_{\text{true}} + \Delta t,
\quad \Delta t \sim \mathcal{N}(0,\sigma_t^2).
$$
The parameter $\sigma_t$ is related to the detector coincidence timing resolution, which is commonly specified as a full width at half maximum (FWHM). For a Gaussian distribution, the conversion is
$$
\text{FWHM} = 2\sqrt{2\ln 2}\,\sigma \approx 2.355\,\sigma.
$$
For a Gaussian timing blurring:
$$
\sigma_t = \frac{\text{FWHM}}{2.355}, \quad
t_{\text{meas}} = t_{\text{true}} + \Delta t, \quad \Delta t \sim \mathcal{N}(0,\sigma_t^2).
$$
Always convert the timing resolution FWHM to $\sigma_t$ before configuring timing blurring.
When you configure timing blurring in a GATE PET digitizer, you typically specify the time resolution in units of time, for example in picoseconds or nanoseconds, using the unit system discussed earlier in the course. It is important to keep units consistent. If the digitizer expects a standard deviation, you must give $\sigma_t$, not the FWHM. Some high-level interfaces may allow you to specify the FWHM and perform the conversion internally, but you should always verify the convention in the documentation.
Timing blurring affects TOF PET analysis through the coincidence time difference. In a coincidence event, two detectors register interaction times $t_{1,\text{meas}}$ and $t_{2,\text{meas}}$. The coincidence time difference is
$$
\Delta t_{\text{meas}} = t_{1,\text{meas}} - t_{2,\text{meas}}.
$$
This time difference is used to estimate the annihilation position along the line of response, so any timing blur directly broadens the spatial localization. The relationship between the timing difference and the spatial uncertainty along the line is
$$
\sigma_z = \frac{c\,\sigma_{\Delta t}}{2},
$$
where $c$ is the speed of light and $\sigma_{\Delta t}$ is the standard deviation of the measured time difference. Better timing resolution, that is smaller $\sigma_t$ and smaller $\sigma_{\Delta t}$, leads to a narrower distribution of reconstructed positions and improved TOF PET image quality.
Timing resolution directly limits TOF spatial resolution along the LOR:
$$
\sigma_z = \frac{c\,\sigma_{\Delta t}}{2}.
$$
Smaller timing blur yields better TOF localization.
In practice, detector timing resolution may depend on energy, crystal, or detector channel. For introductory simulations, it is common to use a single global timing resolution for all detectors, for example a coincidence timing FWHM of a few hundred picoseconds. As you move toward realistic scanner modeling, you can refine the timing blurring configuration to match experimental measurements for a specific PET system, including different resolutions for different detector modules or energy dependent timing jitter if supported by your digitizer setup.
When you validate a TOF PET simulation, it is useful to check the distribution of measured time differences between detector pairs and compare the measured FWHM from the simulation output to the target timing resolution that you configured. If the simulated timing spectrum is too narrow or too broad, this often indicates that the timing blurring parameter was misinterpreted as FWHM or as $\sigma_t$, or that units such as nanoseconds and picoseconds were mixed.
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