KAHIBARO
17.8 Time Blurring
Table of Contents
Detector timing resolution
Time blurring models the fact that real detectors do not measure the exact arrival time of a particle. Instead, each detected interaction time is spread around the true physical time with some uncertainty that is characterized by the detector timing resolution.
In GATE digitizers, you apply time blurring at the singles level. A typical workflow is that you first form singles from hits, sum the energy in each crystal during a short time, and then apply a timing model to each single. This blurred time is then used later in coincidence sorting, time-of-flight analysis, and any time-based cuts.
The timing resolution is usually specified as a Full Width at Half Maximum (FWHM) of a Gaussian distribution. Conceptually, for a true detection time $t_{\text{true}}$, the recorded time $t_{\text{meas}}$ is sampled from a normal distribution centered on $t_{\text{true}}$ with a standard deviation $\sigma$:
$$
t_{\text{meas}} = t_{\text{true}} + \Delta t,
\quad \Delta t \sim \mathcal{N}(0,\sigma^2).
$$
The connection between FWHM and $\sigma$ is very important when you configure time blurring:
For a Gaussian distribution:
$$
\text{FWHM} = 2 \sqrt{2 \ln 2}\,\sigma \approx 2.355\,\sigma
$$
Equivalently,
$$
\sigma = \frac{\text{FWHM}}{2.355}.
$$
GATE time blurring uses this relationship internally. You usually provide the time resolution as a FWHM (for example $300\,\text{ps}$), and the code converts it to $\sigma$ before sampling the random time offset. Always remember to include the time unit, for example 300 * gate.g4_units.ps, otherwise you will get an incorrect timing spread.
In medical imaging, the characteristic timing scales differ by modality. Conventional PET with non time of flight detectors might have a coincidence timing resolution of several nanoseconds. Time of flight PET aims for several hundred picoseconds or better. SPECT with gamma cameras is typically not used for coincidence timing, so very fine timing resolution is not as critical at the detector level. In all cases, the same mathematical blurring model applies, only the FWHM value changes.
When you configure time blurring in a digitizer chain, you are defining how the intrinsic detector timing resolution and the readout electronics timing resolution are represented in the simulation. For simple studies, you usually assume a single global timing resolution and apply one Gaussian blur to every single. More advanced simulations can use energy dependent or detector dependent timing, but these are built on the same Gaussian smearing idea.
Time blurring directly affects coincidence sorting and especially time of flight PET. A worse timing resolution gives a broader distribution of time differences between paired singles. In coincidence sorting, a broad timing distribution often requires a wider coincidence time window, which in turn increases the rate of random coincidences. In time of flight PET, the timing resolution limits how precisely you can localize the annihilation position along the line of response. The position uncertainty $\Delta x$ along the line of response is linked to the coincidence timing resolution $\Delta t_{\text{coinc}}$ by:
For time of flight along a line of response:
$$
\Delta x \approx \frac{c \, \Delta t_{\text{coinc}}}{2}
$$
where $c$ is the speed of light.
Here $\Delta t_{\text{coinc}}$ is the FWHM of the time difference distribution for two detectors. If each detector has a single detector timing resolution $\text{FWHM}_{\text{det}}$, and you assume they are independent and identical, then the coincidence timing resolution is worse by a factor of $\sqrt{2}$:
$$
\text{FWHM}_{\text{coinc}} \approx \sqrt{2}\,\text{FWHM}_{\text{det}}.
$$
This relationship is useful when you choose the FWHM time blurring value in the digitizer. Manufacturers might quote either a per detector or a coincidence timing resolution. Make sure you know which one you are mimicking in GATE, and convert correctly if needed.
In practice, implementing time blurring in a GATE digitizer involves three main choices. You choose a timing resolution value, usually as a FWHM in picoseconds or nanoseconds. You choose when in the digitizer chain the time blurring is applied, usually after energy summation, so that each single has one blurred time tag. Finally, you make sure the blurred times are written to the singles and coincidence outputs, because later analysis, for example coincidence time difference histograms, uses these blurred times.
If you find that your simulated coincidence timing spectrum is much narrower or much broader than you expect from a real scanner, the first thing to double check is the FWHM value, the FWHM to sigma conversion, and the units. A mistake of a factor of 10 in timing resolution can significantly change random rates, scatter characteristics, and time of flight localization in your simulations.
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