A discrete ordinates Boltzmann solver for application to inverse planning of photons and protons

Abstract

The aim of this work is to develop a discrete ordinates Boltzmann solver that can be used for calculation of absorbed dose from both photons and protons within an inverse planning optimiser, so as to perform accurate dose calculation throughout the whole of the inverse planning process. With photons, five transport sweeps were performed to obtain scattered photon fluence, and unscattered electron fluence was then obtained and used as a fixed source for solution of the electron transport equations. With protons, continuous slowing down was treated as a fixed source, and five transport sweeps were used to calculate scattered fluence. The total electron or proton fluence was multiplied by the stopping power ratio for the transport medium to obtain absorbed dose. The method was evaluated in homogeneous media and in a lung case where the planning target volume was surrounded by low-density lung material. Photon arc, proton passive scattering and proton arc treatments were considered. The results were compared to a clinically validated convolution dose calculation for photons, and with an analytical method for protons. In water-equivalent media, the discrete ordinates method agrees with the alternative algorithms to within 2%. Convergence is found to be sufficiently complete for water-, lung- and bone-equivalent materials after five iterations. The dose calculated by the relatively simple angular quadrature is seen to be very close to that calculated by a more comprehensive quadrature. For inhomogeneous lung plans, the method shows more heterogeneity of dose to the planning target volume than the comparative methods. The discrete ordinates Boltzmann solver provides a general framework for dose calculation with both photons and protons. The method is suitable for incorporation into an inverse planning optimiser, so that accurate dose calculation in a heterogeneous medium can be obtained throughout inverse planning, with the result that the final dose distribution is as predicted by the optimiser.