Semi-empirical formulation of multiple scattering for Gaussian beam model of heavy charged particles stopping in tissue-like matter

Dose calculation for radiotherapy with protons and heavier ions deals with a large volume of path integrals involving a scattering power of body tissue. This work provides a simple model for such demanding applications. There is an approximate linearity between RMS end-point displacement and range of incident particles in water, empirically found in measurements and detailed calculations. This fact was translated into a simple linear formula, from which the scattering power that is only inversely proportional to residual range was derived. The simplicity enabled analytical formulation for ions stopping in water, which was designed to be equivalent with the extended Highland model and agreed with measurements within 2% or 0.02 cm in RMS displacement. The simplicity will also improve the efficiency of numerical path integrals in the presence of heterogeneity.Comment: 6 pages, 3 figures, submitted to Physics in Medicine and Biolog

Computational modeling of beam-customization devices for heavy-charged-particle radiotherapy

A model for beam customization with collimators and a range-compensating filter based on the phase-space theory for beam transport is presented for dose distribution calculation in treatment planning of radiotherapy with protons and heavier ions. Independent handling of pencil beams in conventional pencil-beam algorithms causes unphysical collimator-height dependence in the middle of large fields, which is resolved by the framework comprised of generation, transport, collimation, regeneration, range-compensation, and edge-sharpening processes with a matrix of pencil beams. The model was verified to be consistent with measurement and analytic estimation at a submillimeter level in penumbra of individual collimators with a combinational-collimated carbon-ion beam. The model computation is fast, accurate, and readily applicable to pencil-beam algorithms in treatment planning with capability of combinational collimation to make best use of the beam-customization devices.Comment: 16 pages, 5 figure

Theoretical methods for the calculation of Bragg curves and 3D distributions of proton beams

The well-known Bragg-Kleeman rule RCSDA = A dot E0p has become a pioneer work in radiation physics of charged particles and is still a useful tool to estimate the range RCSDA of approximately monoenergetic protons with initial energy E0 in a homogeneous medium. The rule is based on the continuous-slowing-down-approximation (CSDA). It results from a generalized (nonrelativistic) Langevin equation and a modification of the phenomenological friction term. The complete integration of this equation provides information about the residual energy E(z) and dE(z)/dz at each position z (0 <= z <= RCSDA). A relativistic extension of the generalized Langevin equation yields the formula RCSDA = A dot (E0 +E02/2M dot c2)p. The initial energy of therapeutic protons satisfies E0 << 2M dot c2 (M dot c2 = 938.276 MeV), which enables us to consider the relativistic contributions as correction terms. Besides this phenomenological starting-point, a complete integration of the Bethe-Bloch equation (BBE) is developed, which also provides the determination of RCSDA, E(z) and dE(z)/dz and uses only those parameters given by the BBE itself (i.e., without further empirical parameters like modification of friction). The results obtained in the context of the aforementioned methods are compared with Monte-Carlo calculations (GEANT4); this Monte-Carlo code is also used with regard to further topics such as lateral scatter, nuclear interactions, and buildup effects. In the framework of the CSDA, the energy transfer from protons to environmental atomic electrons does not account for local fluctuations.Comment: 97 pages review pape