KAHIBARO
35.1. Proton Physics
Table of Contents
Proton energy loss
In proton therapy simulations, one of the most important physical processes is the gradual loss of energy as protons travel through matter. In GATE, this behavior is handled by Geant4 electromagnetic and hadronic processes, but as a user you need a clear conceptual picture to configure and interpret simulations correctly.
When a proton enters a medium such as water or tissue, it interacts mainly with the electrons of the material through Coulomb forces. These interactions cause ionization and excitation of atoms and molecules. The average rate at which the proton loses energy per unit path length is called the stopping power, usually written as $-\mathrm{d}E/\mathrm{d}x$. For therapeutic proton energies, stopping power is dominated by electronic interactions, and this is what shapes the characteristic Bragg peak in depth dose curves.
At high energies, far from the end of the range, the stopping power is relatively modest and changes slowly with depth. The proton travels almost in a straight line, losing a small fraction of its energy in each millimeter of tissue. As the proton slows down, its stopping power increases. Near the end of its trajectory, the proton deposits a large amount of energy in a small distance, producing a sharp maximum in dose. After that depth, called the range, essentially no primary protons remain and the dose falls off rapidly.
In detailed transport codes, including Geant4 and GATE, stopping power is not taken from a single analytical formula for every case, but is computed using models and tabulated cross sections that approximate theories such as the Bethe formula, with corrections for low energy and for different materials. For a conceptual understanding, it is useful to recall the form of the electronic stopping power for a charged particle of charge $z$ and speed $v$ in a material of electron density $N_e$ and mean excitation energy $I$.
The Bethe formula for electronic stopping power in the continuous slowing down approximation can be written in one common form as
$$
- \left\langle \frac{\mathrm{d}E}{\mathrm{d}x} \right\rangle
= \frac{4 \pi N_A r_e^2 m_e c^2 z^2 Z}{A \beta^2}
\left[
\ln \left( \frac{2 m_e c^2 \beta^2 \gamma^2}{I} \right) - \beta^2
\right],
$$
where $N_A$ is Avogadro's number, $Z$ and $A$ are the atomic number and atomic mass of the absorber, $r_e$ is the classical electron radius, $m_e$ is the electron mass, $c$ is the speed of light, $\beta = v/c$, and $\gamma$ is the Lorentz factor. In practice, Monte Carlo codes use improved parameterizations and corrections over a wide range of energies, but the qualitative behavior is similar.
For therapy simulations, an especially useful quantity is the continuous slowing down approximation (CSDA) range $R_\mathrm{CSDA}$, which is the path length a proton with a given initial energy would travel if it lost energy continuously according to the average stopping power. It is given formally by
$$
R_\mathrm{CSDA}(E_0) = \int_{0}^{E_0} \frac{\mathrm{d}E}{-\mathrm{d}E/\mathrm{d}x},
$$
where $E_0$ is the initial proton energy. In water, ranges at therapeutic energies are often summarized in tables or parameterizations instead of computing this integral directly. For example, a proton of about 150 MeV has a range of roughly 15 cm in water, while a 70 MeV proton might travel around 4 cm. GATE uses Geant4 tables to model these ranges accurately for different materials.
The sharpness and position of the Bragg peak in depth dose curves come from this energy loss behavior. As the proton beam enters the medium, the dose gradually increases with depth because the number of protons is still large and each proton loses more energy as it slows down. Near the end of the range, protons are slow and deposit energy very efficiently, resulting in the peak. Beyond this point there is a steep falloff, which is a key advantage of proton therapy compared to photon therapy.
In reality, not all protons stop at exactly the same depth. There is always some statistical fluctuation in the energy lost in each interaction, a phenomenon called range straggling. As a result, the Bragg peak has a finite width and the falloff is not perfectly abrupt. Multiple Coulomb scattering with nuclei also causes lateral spreading of the proton beam. These effects are naturally handled by the Monte Carlo transport, but they are important when you interpret simulated dose distributions or compare them with measured data.
For treatment fields, you often do not want a single narrow Bragg peak. Instead, you need a region of uniform high dose that covers the tumor depth. This is achieved clinically by combining protons of different energies to create a spread out Bragg peak. In GATE, this is typically represented by defining a beam with either discrete energy layers or an energy spectrum. The underlying physics of proton energy loss remains the same, but the superposition of ranges and Bragg peaks leads to a flat dose plateau across the target volume.
From a simulation point of view, all of these aspects are governed by the physics models and transport parameters you select. The correct modeling of stopping power is embedded in the electromagnetic and hadronic physics lists that you choose for proton therapy. As you will see in other sections of this course, production cuts and step limits can also influence how precisely the Bragg peak shape and range are represented.
For proton therapy simulations, accurate proton energy loss is essential for predicting the Bragg peak position and the dose falloff beyond the target. Always validate simulated ranges and depth dose curves against reference data for your materials.
Nuclear interactions
In addition to continuous energy loss through ionization and excitation, protons can undergo discrete nuclear interactions with the atoms of the medium. These interactions are less frequent than electronic interactions, but they have a large impact on dose distributions and on secondary radiation fields in proton therapy. GATE models these processes using Geant4 hadronic physics models, which you choose through the physics list configuration.
When a therapeutic proton interacts with a nucleus, several things can happen. In inelastic nuclear collisions, the proton may be absorbed or scatter with significant energy transfer, and the target nucleus can be left in an excited state. The nucleus may then de-excite by emitting gamma rays, nucleons, or light fragments such as deuterons, tritons, or alpha particles. These secondary particles transport energy away from the primary beam direction and can contribute to dose in regions outside the Bragg peak.
Elastic proton nucleus scattering is another important process. In an elastic event, the proton changes direction without changing the internal state of the nucleus. For proton therapy, such scattering contributes to multiple Coulomb scattering and broadening of the beam, especially at depth. Inelastic interactions, by contrast, can remove protons from the primary beam, reducing the number that reach the nominal range and slightly modifying the distal falloff of the depth dose curve.
The combination of electronic stopping and nuclear interactions leads to three key features in dose and fluence distributions. First, the primary proton fluence decreases gradually with depth not only because of energy loss but also due to nuclear attenuation. Second, there is a low level dose beyond the nominal range, called the dose tail, caused mainly by secondary particles, especially neutrons and long range nuclear fragments. Third, out of field dose in lateral directions is increased by large angle scattering and secondary emission.
In a Monte Carlo simulation, each nuclear interaction is treated stochastically. The probability per unit path length for a given interaction is determined by the nuclear cross section, which depends on proton energy and target material. In GATE, you do not normally supply these cross sections yourself. Instead, they are taken from physics data libraries and parameterizations within Geant4. By selecting a suitable hadronic physics list for proton therapy, such as those that include models optimized for medical applications, you ensure that these interactions are modeled reasonably well in the energy range of interest.
Among the nuclear secondaries, neutrons are particularly important. They are produced in inelastic proton nucleus reactions and can travel long distances before interacting. Although neutron dose inside the target volume is usually small compared to proton dose, neutrons can contribute to out of field dose in normal tissues and to whole body effective dose. In GATE, you can record neutron fluence and energy deposition using appropriate actors to study secondary neutron fields if needed.
Another consequence of nuclear interactions is the production of positron emitting isotopes within the patient or phantom. When certain nuclei are activated by the proton beam, they may produce isotopes that decay via positron emission, such as $^{11}\mathrm{C}$ or $^{15}\mathrm{O}$. These positron emitters can be imaged with PET to provide information about the proton range in vivo. While detailed activation studies require careful physics choices and may not be needed for basic dose simulations, they illustrate how proton nuclear interactions can be exploited for treatment verification.
From a depth dose perspective, nuclear interactions cause a gradual loss of primary fluence that modifies the shape of the Bragg peak compared to a simple CSDA model. The energy carried by secondaries leads to a modest increase in entrance dose and a small but non zero dose beyond the peak. If your application is highly sensitive to distal dose or out of field dose, you must ensure that your chosen physics configuration models these nuclear processes adequately and that your simulation statistics are sufficient to reduce the uncertainty in low dose regions.
In GATE, you will typically control nuclear interactions through the choice of hadronic physics lists and through production cuts that define the tracking of secondaries. Activating or deactivating certain hadronic processes, or changing cut values, can significantly alter predicted secondary particle spectra and low level dose. For example, using a physics list that explicitly models low energy neutron transport may be essential if you want to quantify neutron dose or activation, but may not be necessary for a simple Bragg peak range study.
Nuclear interactions of protons create secondary particles and modify the distal and out of field dose. For realistic proton therapy simulations, you must include appropriate hadronic physics and validate the distal dose tail and secondary spectra against reference data when they are clinically relevant.
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