Proton loss model for therapeutic beam dose calculations.

Abstract

A transport algorithm called the proton loss (PL) model is developed for proton pencil beams of therapeutic energies. The PL model takes into account inelastic nuclear reactions, pathlength straggling, and energy-loss straggling and predicts the 3D dose distribution from a proton pencil beam. In proton beams, the multiple scattering and ionizational energy loss processes approach their diffusional limit where scattering and energy loss probability densities become Gaussian. Therefore we chose to derive the PL model from the Fermi-Eyges diffusional multiple scattering theory and the Gaussian theory of energy straggling. We first introduce a generalization of the Fermi-Eyges equation for proton pencil beams, labeled the proton loss (PL) transport equation. This new equation includes terms that model inelastic nuclear reactions as a depth-dependent absorption and pathlength straggling as a quasi-absorption. Then energy straggling is taken into account by using a weighted superposition of a discrete number of elementary pencil beams. These elementary pencil beams have different initial energies and lose energy according to the CSDA, thus they have different ranges of penetration. A final solution for the proton beam transport is obtained as a linear combination of elementary pencil beam solutions with weights defined by the Gaussian evolution of the proton energy spectrum with depth. A numerical comparison of the dose distribution predictions of the PL model with measurements and PTRAN Monte Carlo simulations indicates the model is both computational fast and accurate.