KAHIBARO
29.2. Detector Timing Resolution
Table of Contents
Coincidence timing resolution
Detector timing resolution describes how precisely a PET system can measure the detection time of each 511 keV photon. In time of flight PET this timing precision directly determines how well you can localize the annihilation point along the line of response. In GATE you do not usually implement detailed electronics, you instead model timing uncertainties as a statistical blurring of the true detection times.
Coincidence timing resolution, often written as CTR, is usually specified for the whole coincidence pair, not for individual detectors. Experimentally, CTR is measured as the full width at half maximum of the distribution of time differences between two detectors that are irradiated by back to back photons from a very small source at the center of the field of view. Typical values in modern clinical TOF PET systems are between 200 ps and 600 ps.
In many publications the timing resolution of a single detector is quoted as $\sigma_{\text{det}}$ and the coincidence timing resolution as $\sigma_{\text{coin}}$. If both detectors have the same timing performance and their errors are independent, then the coincidence resolution is related to the single detector resolution by
$$
\sigma_{\text{coin}} = \sqrt{ \sigma_{\text{det,1}}^2 + \sigma_{\text{det,2}}^2 } = \sqrt{2}\,\sigma_{\text{det}} .
$$
The corresponding FWHM values are related by $\text{FWHM} = 2.355\,\sigma$.
In practice, manufacturers and experimental studies report CTR as a single FWHM number for the coincidence system. When you configure a GATE digitizer with timing blurring, you normally feed in the single event timing resolution, meaning the spread on each measured detection time, not directly the coincidence FWHM. If you only know the coincidence FWHM from measurements, you convert to the equivalent single event $\sigma$ using the relations above.
The link between timing resolution and spatial localization along the LOR is central for TOF PET. A time difference uncertainty of $\Delta t$ corresponds to a position uncertainty of
$$
\Delta x = \frac{c \,\Delta t}{2} ,
$$
where $c$ is the speed of light in vacuum. The factor of one half appears because the extra distance traveled by one photon compared to the other is split symmetrically around the annihilation point. In practice you plug in the timing resolution (for example the coincidence FWHM) to estimate the effective TOF kernel width along the LOR.
Key timing relations:
- Relation between standard deviation and FWHM for Gaussian timing resolution:
$$
\text{FWHM} = 2.355\,\sigma .
$$ - Coincidence timing resolution from identical detectors:
$$
\sigma_{\text{coin}} = \sqrt{2}\,\sigma_{\text{det}} .
$$ - Spatial uncertainty along the LOR from timing uncertainty:
$$
\Delta x = \frac{c \,\Delta t}{2} .
$$
When you implement timing resolution in a GATE digitizer, the blurring step will take the true interaction time, then add a random value drawn from a Gaussian distribution with zero mean and a user provided standard deviation. This standard deviation typically represents the single detector timing resolution in seconds. Every recorded single will therefore have a slightly different measured time than the true simulation time. When coincidences are built later in the chain, GATE uses these blurred times to compute the time difference, and the resulting distribution of $\Delta t$ values will reproduce the coincidence timing resolution implied by your choice of per detector $\sigma$.
For example, assume you want to simulate a PET scanner with a measured CTR of 400 ps FWHM. First convert to $\sigma_{\text{coin}}$:
$$
\sigma_{\text{coin}} = \frac{400\ \text{ps}}{2.355} \approx 170\ \text{ps} .
$$
Then deduce the single detector resolution:
$$
\sigma_{\text{det}} = \frac{\sigma_{\text{coin}}}{\sqrt{2}} \approx \frac{170\ \text{ps}}{1.414} \approx 120\ \text{ps} .
$$
You then use $\sigma_{\text{det}} \approx 120\ \text{ps}$ as the Gaussian standard deviation in the timing blurring of singles. As a check, if you later reconstruct the time difference between two such blurred times for many coincident events, the FWHM of that distribution will be close to 400 ps.
The table below summarizes the different quantities you may encounter and how they relate.
| Quantity | Symbol | Typical units | Relation |
|---|---|---|---|
| Single detector timing SD | $\sigma_{\text{det}}$ | ps | Input for timing blurring in GATE |
| Coincidence timing SD | $\sigma_{\text{coin}}$ | ps | $\sigma_{\text{coin}} = \sqrt{2}\,\sigma_{\text{det}}$ |
| Single detector FWHM | $\text{FWHM}_{\text{det}}$ | ps | $\text{FWHM} = 2.355\,\sigma$ |
| Coincidence FWHM (reported CTR) | $\text{FWHM}_{\text{coin}}$ | ps | $\text{FWHM}_{\text{coin}} = 2.355\,\sigma_{\text{coin}}$ |
| TOF position uncertainty (FWHM) | $\Delta x_{\text{FWHM}}$ | mm or cm | $\Delta x_{\text{FWHM}} = c\,\text{FWHM}_{\text{coin}} / 2$ |
In real detector systems the timing resolution can depend on energy, detector position, or angle of incidence. Basic simulations often start with a constant Gaussian timing resolution for every single event, which is sufficient to study the main impact of TOF information on image quality or to compare non TOF and TOF scenarios. More advanced simulations can vary the timing resolution for different detectors or energy ranges by applying conditional timing blurring or using different digitizer chains for different detector regions.
For GATE based TOF studies it is important to keep the distinction clear between timing resolution and coincidence time window. The timing resolution describes the intrinsic uncertainty on each time measurement, while the coincidence window is the time interval within which two singles are considered a coincidence. The coincidence window must be much larger than the intrinsic timing resolution so that true pairs are not lost, but it is typically only a few nanoseconds even when the CTR is a few hundred picoseconds.
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