Proton loss model for therapeutic beam dose calculations

A transport algorithm called the proton loss (PL) model is developed for pr oton pencil beams of therapeutic energies. The PL model takes into account inelastic nuclear reactions, pathlength straggling, and energy-loss straggl ing and predicts the 3D dose distribution from a proton pencil beam. In pro ton beams, the multiple scattering and ionizational energy loss processes a pproach their diffusional limit where scattering and energy loss probabilit y densities become Gaussian. Therefore we chose to derive the PL model from the Fermi-Eyges diffusional multiple scattering theory and the Gaussian th eory of energy straggling. We first introduce a generalization of the Fermi -Eyges equation for proton pencil beams, labeled the proton loss (PL) trans port equation. This new equation includes terms that model inelastic nuclea r reactions as a depth-dependent absorption and pathlength straggling as a quasi-absorption. Then energy straggling is taken into account by using a w eighted superposition of a discrete number of elementary pencil beams. Thes e elementary pencil beams have different initial energies and lose energy a ccording to the CSDA, thus they have different ranges of penetration. A fin al solution for the proton beam transport is obtained as a linear combinati on 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 fa st and accurate. (C) 2000 American Association of Physicists in Medicine. [ S0094-2405(00)01109-3].