KAHIBARO
12.5. Positron Physics
Table of Contents
Energy loss
Positrons, written as $e^+$, are the antimatter partners of electrons. In GATE, their transport is handled by the electromagnetic physics models provided by Geant4. From the point of view of energy loss, a positron in matter behaves in many ways like an electron, but with some important differences that are relevant for medical imaging and dosimetry.
When a positron enters a material, its initial kinetic energy is gradually reduced through many small interactions. The dominant process in the energy range used in nuclear medicine is ionization and excitation of atoms in the medium. In each interaction the positron transfers a part of its kinetic energy to the atomic electrons, which leads to a progressive slowing down. This continuous energy loss is usually described by the stopping power, often written as $- \mathrm{d}E / \mathrm{d}x$, which expresses how much energy is lost per unit path length. In a homogeneous medium, such as water, the stopping power depends mainly on the positron energy and the material properties.
In GATE, you do not usually configure energy loss explicitly for positrons, because it is already embedded in the selected electromagnetic physics list. However, you should remember that choosing a physics list appropriate for low energy electromagnetic transport, and setting production cuts that are small enough, will control how accurately positron slowing down is described. If cuts are too large, very short segments of the track and low energy secondaries can be artificially suppressed, which may affect the apparent path and the distribution of energy deposition.
A characteristic feature of positron transport, compared with electrons, is the so‑called positron range. Before annihilation, the positron travels a finite distance from its emission point, following a tortuous path due to multiple scattering. The path length is generally longer than the straight line distance between the starting point and the annihilation point, but in imaging applications we are usually interested in the displacement between emission and annihilation. This displacement depends strongly on the initial positron energy and on the material.
For a given radionuclide, the emitted positron energies are distributed over a beta spectrum. Higher energy positrons travel further before stopping, so isotopes with higher maximum energy have a larger average positron range. For example, in water, the mean distance between emission and annihilation for F‑18 positrons is of the order of a fraction of a millimeter, while for higher energy emitters such as Rb‑82 or Ga‑68 the mean range can be several millimeters. This range limits the intrinsic spatial resolution in PET, even if the detector and reconstruction were perfect.
In GATE simulations, the apparent positron range is influenced not only by physics processes but also by your transport parameters. The step length that Geant4 uses to propagate the particle, the multiple scattering model, and the production cuts for secondary electrons and photons all affect how detailed the track is. Very coarse steps or overly large cuts can shorten or otherwise bias the simulated range. For accurate PET resolution studies or internal dosimetry, it is therefore important to use physics settings that are suitable for low energy charged particle transport and to verify that your step and range settings are consistent with experimental data or reference calculations.
Multiple scattering also plays an important role. As the positron loses energy, its direction becomes more and more randomized due to elastic scattering on nuclei and electrons. The combination of gradual energy loss and angular deflections creates a convoluted trajectory. In dense, high atomic number materials, such as detector crystals, the positron range is shorter than in soft tissue because the stopping power is larger. This is relevant when you simulate positron emitters inside or near detector materials, for example in calibration setups where a source is embedded in a solid object.
Because positrons carry positive charge, their interaction cross sections for some processes differ slightly from those of electrons. For instance, close to the end of their track, positrons can form positronium, a bound state with an electron, which changes the details of the final annihilation. The detailed modeling of positronium formation and decay is handled internally by the physics models, and you usually control it indirectly through the choice of electromagnetic physics options rather than by configuring separate processes in GATE.
For dosimetric applications, most of the absorbed dose from a positron emitter in soft tissue comes from the local energy deposition of the slowing positrons and the subsequent annihilation photons. The continuous slowing down approximation range of a positron provides an estimate of how locally this energy is deposited. In practice, if you use a fine spatial grid in a dose actor, you can resolve the spatial distribution of energy deposition from positrons, especially near interfaces between materials with different densities or compositions where ranges change.
To summarize, positron energy loss in GATE is governed by electromagnetic physics lists that simulate ionization, excitation, bremsstrahlung, and multiple scattering. The main observable consequence for medical physics is the positron range, which introduces a blur between the physical origin of the decay and the point where annihilation photons are produced. When you design simulations that depend on spatial accuracy, such as PET resolution studies or small‑scale internal dosimetry, you should pay particular attention to physics list selection, transport cuts, and step parameters that influence how faithfully positron energy loss and range are reproduced.
Positron range is a key limiting factor for PET spatial resolution, and its accurate simulation in GATE depends critically on the chosen electromagnetic physics models and on sufficiently small production cuts and step limits.
Annihilation
After a positron has lost most of its kinetic energy, it eventually encounters an electron in the surrounding material. The interaction of a positron and an electron leads to annihilation, where their rest mass energy is converted into photons. In medical physics simulations, this process is particularly important for PET, because annihilation photons are the signals detected by the scanner.
In the simplest and most common case, a thermalized positron annihilates almost at rest with an electron also essentially at rest. Conservation of energy and momentum then requires the creation of two photons, each with an energy very close to $511 \,\text{keV}$, emitted in nearly opposite directions. In vacuum, they would be exactly back‑to‑back, separated by an angle of $180^\circ$. In matter, the small residual motion of the positron and electron at the moment of annihilation introduces a slight non‑collinearity, so the angle between the two photons deviates by a small amount from $180^\circ$. This non‑collinearity contributes an additional blurring to PET images, especially for scanners with large diameters.
In GATE, annihilation is modeled as part of the electromagnetic physics for positrons. You do not need to explicitly request annihilation, it is automatically included once you enable standard electromagnetic processes for $e^+$. The annihilation photons are created as new gamma particles in the simulation, with energies and directions sampled according to the underlying physics model. These photons then undergo further interactions such as Compton scattering or photoelectric absorption, depending on the materials and the physics list you have selected.
Although two‑photon annihilation dominates in most PET relevant conditions, there are other channels. In particular, when the positron forms positronium with an electron, the bound state can annihilate into two or three photons. Para‑positronium decays predominantly into two photons, while ortho‑positronium decays mostly into three photons with a continuous energy distribution. In condensed matter, collisions often shorten the lifetime of positronium and modify the branching ratios, so the effective contribution of three‑photon annihilation is small but not strictly zero. The detailed treatment of positronium is model dependent, and for many PET applications the two‑photon approximation is sufficient. However, when you study fine effects in time or energy spectra, these small components can in principle play a role.
From an imaging perspective, annihilation defines the location of the PET signal. The pair of 511 keV photons defines what is called a line of response, which is a straight line connecting the two detection points in the scanner. In the ideal case, this line would pass exactly through the annihilation point. In practice, non‑collinearity and detector effects such as finite spatial resolution and timing uncertainties cause the reconstructed line to deviate from the true one. GATE provides the tools to simulate the full chain from annihilation through photon transport, interaction, and detector response, so that you can study how annihilation physics contributes to image blurring and quantification errors.
The energy of the annihilation photons is fixed by the electron and positron rest mass and is approximately $511 \,\text{keV}$. In the simulation, this value may appear slightly shifted or broadened in the detected energy spectra because of detector energy resolution and photon interactions before detection. For example, if a photon undergoes Compton scattering in the patient or in the detector, the measured energy will be lower than 511 keV. In PET simulations, you typically apply an energy window around the 511 keV photopeak at the singles or coincidence level in the digitizer chain, in order to reject scattered photons and improve image quality.
Timing is another important aspect of annihilation. The time at which the positron annihilates is later than the time of radioactive decay, due to the short but finite delay associated with the positron slowing down and thermalization. This delay is usually extremely small compared with the time resolution of clinical detectors, but in principle it is part of the full time history of the event. The annihilation time plus the photon flight times to the detectors determines the detection times recorded in the simulation. In time of flight PET, differences in detection times are used to estimate the position of the annihilation along the line of response. GATE can simulate the annihilation time, photon propagation times, and detector timing resolution, which together determine the time distributions you observe in the coincidence data.
For dose calculations, annihilation photons contribute to the dose delivered at some distance from the site of the decay. Because 511 keV photons have a relatively long mean free path in soft tissue, part of their energy can be deposited far from where the positron emitter is located. In internal dosimetry, the combination of local energy deposition from positrons and nonlocal deposition from annihilation photons must be accounted for in order to obtain accurate organ doses and voxel dose maps.
When you configure physics in GATE for applications that involve positron emitters, you should verify that annihilation is properly handled by the selected physics list. Standard electromagnetic lists already include $e^+$ annihilation at rest and in flight. Annihilation in flight occurs when the positron still has significant kinetic energy at the time of annihilation, which can slightly modify the angular and energy distributions of the photons. For most medical physics scenarios, annihilation in flight plays a minor role compared with annihilation at rest, but it is naturally included in the detailed transport models.
Finally, annihilation provides a clear signature in energy spectra. In simulated and measured PET detector data, you will see a prominent photopeak at 511 keV, corresponding to events where at least one annihilation photon deposits its full energy in the detector crystal. Below this peak, the Compton continuum arises from photons that undergo one or more Compton scatters and escape. By comparing simulated and experimental annihilation spectra, you can validate your physics configuration and detector modeling.
Annihilation of positrons with electrons produces photons of approximately $511 \,\text{keV}$, emitted almost back to back. These annihilation photons are the fundamental signal in PET, and accurate modeling of their production and subsequent transport in GATE is essential for realistic PET imaging and dosimetry simulations.
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