KAHIBARO
26.6. PET Detector Response
Table of Contents
Energy resolution
In PET, energy resolution describes how precisely a detector measures the energy of incoming photons around the 511 keV photopeak. Real detectors do not measure a single sharp energy value. Instead, they produce a spread of measured energies for identical 511 keV photons. This spread is often modeled as a Gaussian distribution centered at the true energy.
Energy resolution is usually specified as a full width at half maximum (FWHM) value at 511 keV, for example 12 % or 20 %. The FWHM relates to the standard deviation $\sigma$ of the Gaussian by:
$$
\text{FWHM} = 2.355 \, \sigma
$$
In PET simulations with GATE, you typically start from the deposited energy in a crystal that is obtained from hits or singles. To model realistic detector response, you then apply an energy blurring step using a digitizer module. The digitizer replaces the exact deposited energy $E$ by a blurred value $E^{\prime}$ sampled from a Gaussian with mean $E$ and a standard deviation that depends on the desired energy resolution.
A common way to describe energy resolution is:
$$
\frac{\Delta E}{E} = \frac{\text{FWHM}}{E}
$$
For a 511 keV photon and a detector with 12 % FWHM at 511 keV, the FWHM in keV is:
$$
\text{FWHM}_{511} = 0.12 \times 511 \,\text{keV}
$$
and the corresponding standard deviation for the blurring is:
$$
\sigma_{511} = \frac{\text{FWHM}_{511}}{2.355}
$$
In a simple PET simulation you often treat the fractional energy resolution as constant at 511 keV, although in reality it can vary with energy. When you configure PET digitization in GATE, you will usually specify the target FWHM at 511 keV, and GATE will internally apply an appropriate Gaussian blurring to the measured energies in singles.
Energy resolution directly influences your ability to separate true 511 keV events from scattered events. A better energy resolution means a narrower photopeak, a cleaner energy window around 511 keV, and reduced contamination from scattered photons. In contrast, a poor energy resolution produces a broad photopeak, forces you to use wider energy windows, and leads to higher scatter fraction.
Key relation for Gaussian energy blurring in PET:
$$
\text{FWHM} = 2.355 \, \sigma
\quad\text{and}\quad
\frac{\Delta E}{E} = \frac{\text{FWHM}}{E}
$$
These expressions link the detector energy resolution, given as FWHM, to the Gaussian standard deviation used to blur deposited energies in PET simulations.
The digitizer stage in GATE is where you introduce the energy resolution of the PET detector. The upstream physics and geometry define the true deposited energy. The downstream analysis, such as applying an energy window around 511 keV, operates on the blurred energy values and therefore reflects the energy resolution that you have chosen.
Timing resolution
Timing resolution characterizes how accurately a PET detector can measure the arrival time of photons. In coincidence PET, especially time of flight PET, timing information is used to decide whether two detection events belong to the same annihilation and to estimate the annihilation position along the line of response.
As for energy, real detectors exhibit timing uncertainty. If a photon arrives at time $t$, the detector reports a measured time $t^{\prime}$, which is spread around $t$ by a Gaussian distribution. The coincidence timing resolution is typically given as an FWHM of the time difference distribution between two detectors for back to back 511 keV photons.
The relation between FWHM and standard deviation is the same:
$$
\text{FWHM}_t = 2.355 \, \sigma_t
$$
In PET simulations, the timing resolution is used in two main ways. First, the digitizer applies time blurring to singles. It takes the true detection time from the particle transport and replaces it with a blurred value that reflects the detector timing resolution. Second, the coincidence sorter uses these blurred times to decide whether two singles fall within the coincidence time window and should be paired as a coincidence event.
Timing resolution is especially important in time of flight PET. If you can measure the difference in arrival times $\Delta t$ between two detectors, you can estimate where along the line of response the annihilation occurred. Neglecting other effects, the longitudinal position uncertainty $\Delta x$ from timing is:
$$
\Delta x = \frac{c \, \Delta t}{2}
$$
where $c$ is the speed of light. A better timing resolution reduces $\Delta t$ and therefore improves localization along the line of response.
Key timing relations in PET:
$$
\text{FWHM}_t = 2.355 \, \sigma_t
\quad\text{and}\quad
\Delta x = \frac{c \, \Delta t}{2}
$$
These link the detector timing resolution to the Gaussian blurring of detection times and to the spatial localization in time of flight PET.
In a basic PET simulation you may simply set a coincidence time window, for example a few nanoseconds, without modeling detailed time blurring. In a more realistic configuration you specify the detector timing resolution in the digitizer, which then smears detection times before coincidence sorting. This choice affects the rate of random coincidences, the precision of time of flight information, and the shape of timing spectra used in analysis.
Spatial resolution
Spatial resolution describes how accurately a PET system can localize the origin of detected photons in space. In PET, there are two main aspects to spatial resolution. One is the intrinsic resolution of the detector, which is related to the size and arrangement of the crystals and the precision of event positioning within each crystal. The other is the system resolution, which includes effects from positron range, photon non collinearity, and reconstruction.
In GATE simulations, PET detector spatial resolution is mainly controlled during geometry and digitization. The geometry defines where crystals are placed and how large they are. The digitizer can then apply spatial blurring on the measured interaction positions within each crystal to represent limitations in position decoding.
If you treat each crystal as a single detection channel, a simple assumption is that the measured position is at the crystal center. The effective intrinsic spatial resolution is then related to the crystal width. For a crystal of width $w$, a commonly used approximation is that the position uncertainty is uniform over the width, which corresponds to an equivalent Gaussian standard deviation:
$$
\sigma_x \approx \frac{w}{\sqrt{12}}
$$
You can combine such a geometrical contribution with additional optical and electronic effects by introducing explicit spatial blurring in the digitizer. In this case, the true interaction position is replaced by a blurred position sampled from a Gaussian centered at the true position, with a specified standard deviation in each direction.
In time of flight PET, the longitudinal spatial resolution along the line of response can also be related to timing resolution. As described earlier, a timing uncertainty $\Delta t$ translates into a spatial uncertainty $\Delta x = c \, \Delta t / 2$. This timing related component adds to the detector intrinsic spatial resolution.
Approximate intrinsic spatial resolution for a crystal of width $w$:
$$
\sigma_x \approx \frac{w}{\sqrt{12}}
$$
This relation is often used to link finite crystal size to an equivalent Gaussian spatial blurring in PET detector simulations.
In practice, when configuring PET detector response in GATE, you typically proceed in three steps. First, you define the crystal geometry, which determines the basic sampling grid of possible detection positions. Second, you enable spatial blurring in the digitizer if you want to model imperfect position decoding within or across crystals. Third, you analyze singles and coincidences using the blurred positions, which yields realistic spatial distributions of detected events that include the modeled resolution effects.
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