# Dose Calculation Algorithms for External Radiation Therapy: An Overview for Practitioners
## 4 Dose Calculation Algorithms for Hadrontherapy  
Hadrontherapy utilizes relativistic charged particles, such as protons [45] and carbon ions [46], for targeted dose delivery. Although hadrontherapy uses considerably larger and more expensive accelerator facilities, its physical properties offer dosimetric advantages over other irradiation types. Three processes dominate the interaction of charged particles with matter in the therapeutic energy range [47]. Inelastic Coulomb interaction with electrons is responsible for most of the energy loss of the projectile, while leading to negligible direction changes due to the large mass difference of the projectile and the electrons. The energy loss is referred to as linear energy transfer (LET), in units of \(\mathrm{keV}/\mathrm{\SIUnitSymbolMicro m}\). The LET of individual ions is stochastic, but can be described for a particle beam using the Bethe equation [48]. In a wide energy range, the energy loss is inversely proportional to the square of the projectile's velocity \(v^{2}\), which leads to a sharp increase of the energy loss towards the end of its range, the so-called Bragg Peak [49]. Stochastic processes lead to a widening of this peak, a process known as range straggling. This effect is proportional to \(1/\sqrt{A}\), with \(A\) being the mass of the projectile. According to the Bethe equation, the range of the beam in a given material therefore depends on its initial energy, so that different points in the body can be reached by controlling this energy. In clinical practice, energies corresponding to approximately a 30-cm range in water are used: up to 220 MeV for protons, and 430 MeV/u for carbon ions. Several other ions have been explored in the past [50], and helium and oxygen are currently under pre-clinical investigation [51]. Elastic Coulomb interaction with target nuclei results in little energy loss, but scatters the projectile. The scattering behavior of a particle beam follows multiple Coulomb scattering theory. Furthermore, scattering is proportional to \(1/\sqrt{A}\). Inelastic, nuclear interactions lead to target fragmentation and, for particles heavier than protons to projectile fragmentation [49]. This causes a loss of primaries, so that with increasing projectile mass and initial energy, fewer primaries actually reach the Bragg Peak [52]. Heavy fragments of the target have a short range and a high LET, whereas light fragments have a larger range than the primaries, leading to the so-called fragment tail. Some target and projectile fragments, such as \({}^{11}\)C or \({}^{15}\)O, are short-lived positron emitters and enable beam detection via positron emission tomography [53]. A dose engine for hadrontherapy needs to take these principal effects into account--the energy loss and finite range, the multiple Coulomb scattering, as well as the nuclear fragmentation, which causes a complex mixed irradiation field, especially for heavier particles. Due to the dependence on particle mass, some effects that can be neglected for protons are highly significant for heavy ions and vice versa. In the following, the practical considerations for the two dominant dose engine types, analytic pencil beam algorithms as well as dedicated or general-purpose Monte Carlo algorithms, are presented. To irradiate a volumetric tumor, the beam has to be shaped laterally and longitudinally. Beam energies are stacked to form the'spread-out Bragg peak' (SOBP), whereas lateral conformity can be achieved using passive scattering and collimators or via active pencil beam scanning (PBS). PBS permits much more conformal delivery and uses a cleaner beam, as less material is in the beam path. Practically all new facilities are equipped with PBS; therefore, PBS is the main focus of this section. In modern pencil beam scanning, each field consists of several layers with a given primary energy and a corresponding penetration depth. Layer spacing in the beam direction is around 3 mm in water.