Narrow Results

A Boltzmann transport model for dose calculation in radiation therapy is considered. We formulate an optimal control problem for the desired dose. We prove existence and uniqueness of a minimizer. Based on this model, we derive optimality conditions. The P_N discretization in angle of the full model is considered. We show that the P_N approximation of the optimality system is in fact the optimality system of the P_N approximation, provided that, instead of the usually used Marshak boundary conditions, Mark's boundary conditions are used. Numerical results in one and two dimensions are presented.

In this paper we study a problem in radiotherapy treatment planning. This problem is formulated as an optimization problem of a functional of the radiative flux. It is constrained by the condition that the radiative flux, which depends on position, energy and direction of the particles, is governed by a Boltzmann integro-differential equation. We show the existence, uniqueness and regularity of solutions to this constrained optimization problem in an appropriate function space. The main new difficulty is the treatment of the energy loss term. Furthermore, we characterize optimal controls by deriving first-order optimality conditions.