Adjoint Method for Beam Angle Selection in Radiotherapy

Abstract

In this thesis, a new methodology called the adjoint method is developed to select an optimal set of beam angles and beam weights in external beam radiotherapy. The Linear Boltzmann Transport Equation is specifically adjusted for proton transport and forms the Fokker-Planck approximation. The Fokker-Planck approximation models proton transport. Only the continuous slowing down operator, the straggling operator and a source are considered in the approximation. Numerical analysis is performed to solve the Fokker-Planck approximation. The discretization of the energy domain is done with the Discontinuous Galerkin method, and ray-tracing of beams in the spatial domain is done with the Crank-Nicolson method. A minimum least squares objective function is introduced to find the optimal selection of beam angles. The gradient descent method is applied to find a local minimum of the objective function. The gradient is calculated by performing adjoint transport of protons. Adjoint transport is based on an adjoint source, which is changing every iteration based on the dose deposition. The dose deposition is computed using forward transport of protons and uses the newest optimization variables. This way, several iterations can be performed.