On Existence of $L^1$-solutions for Coupled Boltzmann Transport Equation and Radiation Therapy Treatment Optimization

TL;DR

This paper proves the existence, uniqueness, and non-negativity of solutions for a coupled Boltzmann transport system modeling radiation therapy, and explores inverse treatment planning as an optimal control problem.

Contribution

It establishes the first rigorous mathematical results for coupled Boltzmann equations in radiation therapy modeling, including solution properties and inverse problem formulation.

Findings

Unique solutions exist under physically relevant assumptions.

Solutions are non-negative when data is non-negative.

Inverse problem is formulated as an optimal control problem for therapy planning.

Abstract

The paper considers a linear system of Boltzmann transport equations modelling the evolution of three species of particles, photons, electrons and positrons. The system is coupled because of the collision term (an integral operator). The model is intended especially for dose calculation (forward problem) in radiation therapy. It, however, does not apply to all relevant interactions in its present form. We show under physically relevant assumptions that the system has a unique solution in appropriate ($L^1$-based) spaces and that the solution is non-negative when the data (internal source and inflow boundary source) is non-negative. In order to be self-contained as much as is practically possible, many (basic) results and proofs have been reproduced in the paper. Existence, uniqueness and non-negativity of solutions for the related time-dependent coupled system are also proven. Moreover, we deal with inverse radiation treatment planning problem (inverse problem) as an optimal control problem both for external and internal therapy (in general $L^p$-spaces). Especially, in the case $p=2$ variational equations for an optimal control related to an appropriate differentiable convex object function are verified. Its solution can be used as an initial point for an actual (global) optimization.