KAHIBARO
25.1. Introduction to PET Simulation
Table of Contents
Positron emission
Positron emission tomography (PET) is based on radioactive isotopes that decay by emitting a positron, the antiparticle of the electron. In a PET simulation you do not model the full nuclear decay chain in every detail at the beginning of the course, but you need a basic understanding of what the positron source represents.
Many PET isotopes, such as $^{18}\text{F}$, $^{11}\text{C}$, or $^{15}\text{O}$, are proton rich. They transform toward stability through beta plus decay, usually written as
$$
p \rightarrow n + e^{+} + \nu_{e},
$$
inside the nucleus. At the macroscopic level you can think of this as the nucleus emitting a positron $e^{+}$ and a neutrino $\nu_{e}$. The neutrino escapes without interacting in the detector, while the positron is the particle of interest for PET.
In reality the positron is emitted with a continuous energy spectrum up to a characteristic end point that depends on the isotope. In a detailed PET simulation you might want to reproduce these beta spectra by using suitable energy distributions for the primary positrons. For a first, simplified study it is common to approximate the emission by a monoenergetic or average energy positron, or even to skip the beta spectrum and start directly from the annihilation photons. Later chapters in this example will show how to define practical sources in Geant4 that match the level of realism you need.
Once emitted, the positron travels through the surrounding medium, which is usually biological tissue or a tissue equivalent phantom. Along its path it loses energy through interactions such as ionization and excitation of atoms, while being deflected by multiple scattering. The typical range of a PET positron in tissue is on the order of a millimeter or less, depending on its initial energy and the isotope. This finite range causes a spatial blur between the true decay position and the place where the positron eventually annihilates. In PET this is one contribution to the intrinsic spatial resolution limit.
In a Geant4 PET simulation, the transport of the positron is handled by the electromagnetic physics processes in your chosen physics list. You define where the positron is created and with what energy and direction, and Geant4 takes care of simulating its slowing down until it eventually annihilates with an electron.
In PET, the decay position and the annihilation position are not the same. The positron travels a finite distance before annihilation, which affects spatial resolution and must be considered in realistic simulations.
Annihilation photons
When the positron has lost almost all of its kinetic energy, it encounters an electron in the surrounding material and the pair annihilates. In the simplest case, called two photon annihilation at rest, the positron and electron masses are fully converted into two gamma photons. Each photon has an energy of approximately $511\ \text{keV}$, which is the rest mass energy of the electron or positron, and the two gammas are emitted in nearly opposite directions.
The basic reaction can be written as
$$
e^{+} + e^{-} \rightarrow \gamma + \gamma,
$$
with each $\gamma$ carrying about $511\ \text{keV}$. Conservation of momentum requires that for annihilation at rest the two photons are emitted back to back. In practice there is always a small residual momentum from the positron and from the bound electron, so the angle between the two photons is not exactly $180^{\circ}$, but for most PET detector geometries it is very close to opposite.
In Geant4, once the positron slows down and comes close enough to an electron, an annihilation process is triggered. The physics list defines this process and controls how annihilation is modeled. You do not have to manually create the gamma photons; the process itself generates the secondary photons and assigns them the appropriate energies and directions. These photons then propagate through the surrounding geometry according to the gamma interaction processes that you have included in your physics list, such as photoelectric effect, Compton scattering, and pair production at higher energies.
For PET, the two annihilation photons are the signals that your detector ring measures. They leave the patient or phantom and may interact in the scintillator crystals of the PET scanner, depositing some or all of their energy. By recording where and when these interactions occur, and how much energy they deposit, you can attempt to reconstruct the line along which the original annihilation took place.
The initial source distribution in PET is usually a volume that mimics the tracer inside an organ or a phantom. Therefore, in your Geant4 simulation you will have a spatial distribution of annihilation events inside that volume, each generating a pair of 511 keV photons. The PET reconstruction, which is an analysis task outside Geant4, uses many such annihilation events to build an image of the activity distribution.
In PET, the main detectable signal is a pair of $511\ \text{keV}$ gamma photons from positron electron annihilation, emitted almost back to back. Modeling their creation and transport correctly is essential for a realistic PET simulation.
Coincidence detection
The key feature that distinguishes PET from many other gamma imaging techniques is coincidence detection. Instead of looking at single gamma interactions, a PET scanner looks for pairs of detector hits that are consistent with coming from the two annihilation photons of the same event.
Imagine a ring of detector crystals surrounding the patient. When an annihilation occurs inside the ring, the two photons travel in roughly opposite directions. If both photons escape the body without being absorbed, and if each deposits enough energy in a detector crystal, the electronics record two separate detector hits. If these hits occur within a short time window, they are considered a coincidence. The straight line connecting the centers of the two crystals that registered the hits is called a line of response, or LOR. The annihilation is assumed to lie somewhere along this line.
In a Geant4 simulation, the concept of coincidence is not built into the toolkit itself. Geant4 simulates individual events, where an event corresponds to one primary particle or decay, and it records all the interactions and energy depositions that result. Coincidence detection is instead implemented in your user code, typically by analyzing the hits and timing information after each event.
A typical coincidence logic in a PET simulation proceeds as follows. During an event, you collect all energy depositions in the detector crystals, often grouping them per crystal and summing the energy and possibly computing a time stamp. You then apply an energy window to reject interactions that do not look like 511 keV photons, for example by requiring that the deposited energy in a crystal lies within a certain range around 511 keV. Among the remaining detector hits, you search for pairs that represent valid coincidences, usually one hit in one crystal and another hit in a different crystal. In a full detector with many crystals, both hits will often lie in the same event, so your coincidence selection can focus on event by event analysis.
The coincidence time window defines how close in time two hits must be to be counted as a pair that came from the same annihilation. In realistic scanners this window is on the order of a few nanoseconds or less. In a Geant4 PET example, you use the time information associated with each hit, often derived from the earliest energy deposition in a crystal, to decide if two hits satisfy the coincidence condition. If the absolute difference between their times is less than your chosen coincidence window, you accept the pair as a coincidence and record the identifiers of the two crystals and the event information.
A PET coincidence event is defined by two detector hits that both pass an energy selection and occur within a short time window. These coincidences define lines of response, which are later used for image reconstruction.
Coincidence detection introduces additional effects that you may eventually want to model. True coincidences come from the two photons of the same annihilation, which is what you are trying to measure. Random coincidences occur when two unrelated gamma interactions happen to fall within the time window by chance, and scatter coincidences involve at least one photon that scattered before detection, which distorts the geometry of the line of response. While a beginner PET simulation might initially ignore these complications, Geant4 provides the detailed information needed to study them later, since you can inspect the full track and step history of each detected photon.
In summary, a PET simulation in Geant4 connects three key ideas. Positron emission produces a positron that travels and annihilates. Annihilation photons are generated as 511 keV gammas that propagate through the geometry and interact in your detectors. Coincidence detection is implemented in your analysis of detector hits, selecting pairs that correspond to potential annihilation events and forming lines of response that will eventually be used for image reconstruction.
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