KAHIBARO
26.1. PET Physics
Table of Contents
Positron emission
In positron emission tomography, the imaging signal begins with a radioactive decay that produces a positron. Certain proton‑rich radionuclides, such as F‑18 and C‑11, transform a proton in the nucleus into a neutron. To conserve charge and lepton number, this process emits a positron and a neutrino. The positron is the antimatter partner of the electron and has the same mass but positive electric charge.
For GATE simulations, you typically do not model the full nuclear transition in detail. Instead, you use radionuclides whose decay schemes are already known in Geant4 or you define a positron source directly with a particular energy spectrum. The positron leaves the decay site with kinetic energy that is characteristic of the radionuclide. This energy is usually a continuous distribution up to a maximum, not a single monoenergetic value.
Once emitted, the positron travels through matter, such as water, tissue, or a phantom, and interacts through electromagnetic processes. It loses energy mainly by ionizing and exciting atoms and molecules along its path. This random path has a finite length, often called the positron range, and depends strongly on the initial energy and on the material. Higher energy positrons travel further before they slow down. In PET, this range introduces a small but real spatial uncertainty between the original decay position and the eventual annihilation point. In high‑resolution PET simulations, you must include realistic positron transport to capture this effect.
When the positron energy has decreased to almost zero, it is effectively stopped in the medium. At that point, it can form a short‑lived bound state with an electron called positronium, or it can annihilate directly with an electron. GATE uses electromagnetic physics models from Geant4 to handle these transport and stopping processes automatically, as long as you choose an appropriate physics list that includes positron interactions.
Positron annihilation
After the positron has slowed down and found an electron, the particle and antiparticle annihilate. In this process, their rest mass energy is converted into photons. In most cases relevant to PET, the annihilation produces two photons. Conservation of energy and momentum strongly constrains the energy and directions of these photons.
At rest, each particle has rest mass energy $m_e c^2$, where $m_e$ is the electron mass and $c$ is the speed of light. Since there are two particles, the total rest mass energy available is $2 m_e c^2$. If the positron and electron annihilate essentially at rest, this energy appears as two photons that share it equally. Their directions are approximately opposite so that the total momentum remains zero in the annihilation frame.
In reality, there is usually a small amount of residual momentum from the positron and electron system even at annihilation, and there can be bound‑state effects when positronium is involved. This leads to a slight non‑collinearity of the two photons. Instead of being exactly 180 degrees apart, there is a small angular deviation. For most PET scanner geometries, this effect is very small compared with the size of the detector ring, but for very high spatial resolution simulations it can become relevant. Geant4 physics can model this non‑collinearity if you select an appropriate electromagnetic physics configuration.
Another subtlety is that positron annihilation can sometimes produce three photons. This occurs mainly in one of the two spin states of positronium. Three‑photon annihilation is much less probable than two‑photon annihilation and contributes only a small fraction of events in clinical PET. Many practical simulations ignore it, but more detailed physics configurations can include it if needed for specific research studies.
In GATE, when you simulate a positron emitter and track positrons with the electromagnetic physics models, the annihilation process and its resulting photons are handled automatically. You do not need to create separate gamma sources for annihilation. Instead, you configure your PET source as a radionuclide or as positrons, and the annihilation emerges naturally from the transport physics.
511 keV photons
The two dominant photons produced by positron annihilation each have an energy of 511 keV when the annihilation occurs at rest. This value comes directly from the electron rest mass. The rest mass energy of an electron is
$$
m_e c^2 \approx 0.511 \text{ MeV} = 511 \text{ keV}.
$$
Because a positron has the same mass as an electron, the total rest mass energy in the electron–positron pair is
$$
2 m_e c^2 \approx 1.022 \text{ MeV}.
$$
If this energy is converted to two photons of equal energy, each photon has
$$
E_\gamma = \frac{2 m_e c^2}{2} = m_e c^2 \approx 511 \text{ keV}.
$$
In PET, the fundamental annihilation photons each have an energy of about $511 \text{ keV}$, given by the electron rest mass $m_e c^2 \approx 0.511 \text{ MeV}$, and the pair together carries approximately $1.022 \text{ MeV}$.
These 511 keV photons travel in nearly opposite directions and can be detected by the PET scanner. The detection system is designed so that when two 511 keV photons are recorded within a short time window, they are considered a coincidence and associated with the same annihilation event. This is the basis of PET imaging, and later chapters on PET detector response and coincidences explain this in more detail.
As the 511 keV photons propagate through tissue and detector materials, they interact via processes such as the photoelectric effect and Compton scattering. The relative importance of these processes depends on the photon energy and the atomic composition of the material. At 511 keV in soft tissue, Compton scattering dominates, which leads to multiple scatter events before absorption or escape. In PET simulations, this scattering is crucial because it generates scattered coincidences and degrades image contrast.
GATE uses Geant4 electromagnetic physics models to handle 511 keV photon transport, interactions, and energy deposition in detectors. When configuring a PET simulation, you do not typically specify 511 keV photons directly as the source. Instead, you select an appropriate positron emitter or positron source, ensure that annihilation and gamma physics are enabled, and let the simulation generate, track, and interact the 511 keV photons according to the underlying physics.
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