KAHIBARO
17.3. Energy Summation
Table of Contents
Combining energy deposits
In a real detector a single gamma or charged particle can interact several times inside one crystal or one detector channel. Each interaction deposits part of the total energy and may happen at slightly different positions and times. The electronics of the detector cannot resolve every microscopic interaction. Instead, it collects all the light or charge generated within a short time and within one readout element, and produces one measured signal. Energy summation in GATE digitizers is the process that mimics this behavior.
Before digitization you typically have a hit collection. Each hit contains information such as the volume, energy deposited $E_{\text{dep}}$, position, time, and event or track identifiers. If you passed this directly to analysis, one physical detection could appear as many entries, which is not what an energy spectrum or a PET or SPECT system measures. The digitizer therefore groups hits according to rules that represent the detector readout and then sums their energies.
The typical grouping rules to form a single detector signal are: all hits that belong to the same detector channel, occur within a specified coincidence-in-crystal or integration time window, and occur in the same event are combined. Within each group, the digitizer computes the total deposited energy as
$$
E_{\text{sum}} = \sum_{i=1}^{N} E_{\text{dep},i},
$$
where $N$ is the number of hits in the group. The group then becomes one single, often called a pulse or event, with energy $E_{\text{sum}}$. The position and time associated with this single are usually derived from the individual hits, for example the energy weighted average position or the earliest time.
Energy summation is essential whenever one physical detection produces multiple hits in the same detector channel. Always ensure that your digitizer configuration groups hits by detector ID and within an appropriate time window before applying energy thresholds or windows.
If you do not perform energy summation correctly, your simulation can underestimate the number of detected events above threshold or distort the shape of the energy spectrum. For example, a 511 keV gamma that undergoes Compton scatter followed by photoelectric absorption in one crystal should contribute a single pulse near 511 keV. Without summation, the same interaction would appear as two lower energy deposits and could fall below your photopeak window.
Energy summation also allows you to approximate effects of optical transport and charge collection without explicitly simulating every optical photon. The summed energy is treated as proportional to the light or charge that the detector electronics see. Later steps in the digitizer chain can then apply blurring, nonlinearity, or thresholds to the summed value rather than to many small deposits.
Crystal response
In GATE you usually treat the scintillation crystal or semiconductor sensor as the basic readout element in a gamma camera or PET scanner. Energy summation is the link between the microscopic energy deposits in the crystal volume and the macroscopic crystal response that the electronics measure. The goal is to transform a cluster of hits in a given crystal into one realistic detector signal.
The first part of this response is the relationship between deposited energy and produced signal. For many detectors you assume a linear response: the signal amplitude is proportional to $E_{\text{sum}}$. In simulation this corresponds to using the summed deposited energy directly as the measured energy, followed by energy blurring that accounts for finite energy resolution. For example, after summation, the measured energy $E_{\text{meas}}$ might be modeled as
$$
E_{\text{meas}} = \mathcal{N}\bigl(E_{\text{sum}}, \sigma_E^2\bigr),
$$
where $\mathcal{N}$ denotes a Gaussian distribution and $\sigma_E$ is determined by the detector energy resolution at that energy. Energy summation must be done before this step, because the resolution applies to the entire pulse, not to individual microscopic interactions.
Always perform energy summation in the crystal before modeling energy resolution. The electronics measure one pulse per crystal, not separate contributions from each interaction, so the blurring must be applied to the summed energy.
Another aspect of crystal response is how you assign a single position and time to the summed signal. The digitizer often sets the signal time to the earliest hit time in the group or to an energy weighted combination of hit times. For the position, a common choice is the energy weighted centroid of the hits inside the crystal. This ensures that the final singles data resemble what a position sensitive crystal readout would reconstruct, while still relying on summed energy as the main observable.
Energy summation is particularly important in thick crystals or high energy applications where multiple scatters within a single crystal are common. In PET and SPECT simulations, proper summation within each crystal volume allows the rest of the digitizer chain to apply thresholds and energy windows at the level of physically meaningful crystal events. In semiconductor detectors, where charge sharing between neighboring pixels can occur, you may use energy summation separately within each pixel, and optionally introduce additional logic to merge signals across pixels if your electronics behave that way. In all these cases, energy summation defines how the idealized crystal response is constructed from the raw Monte Carlo hits.
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