Dose Calculation in Proton Therapy using Finite Element Methods to solve the Fokker-Planck Transport Equation

Abstract

In proton therapy the development of dose calculation methods is of great importance
to optimize the dose. In this work a method is presented to solve the Fokker-Planck
transport equation using ray-tracing techniques and nite element methods.
The idea of the method is dividing the Fokker-Planck equation in two seperate equa-
tions, one without scattering and the other with scattering. The result of the unscat-
tered equation is then used to function as source for the second one. The equations
are discretized in all free variables: space, angle and energy. The spatial domain is
discretized such that each spatial element has its own angular and energy mesh. The
main solution technique is the discontinuous Galerkin (dG) method in combination
with ray-tracing techniques.
The results from the ray-tracing method for calculating the unscattered dose correspond
to known physical phenomena. The method for Gaussian sampling however shows a
great sensitivity to the number of sampling beams used and yields consistent results
only for a numer of sampling beams in the order of 103.
In future work the sampling needs to be done by using the quadrature method, which
results in a smaller error and uses only 4 sampling beams, resulting in a more favorable
computation time.