Modelling the transport of ionizing radiation using the finite element method

Radiation therapy treatment planning is based on the calculation of the absorbed dose in the patient domain. For exact dose calculations, the solution of three coupled Boltzmann transport equations (BTEs) is needed to cover the transport of photons, electrons and positrons. In many situations, however, two coupled systems for photons and electrons are enough. The use of numerical methods in finding the exact solution of the unknown particle fluxes is necessary. In the stationary case, the BTE has six variables, three spatial, two directional and one energy variable. In this paper, we describe an approach in which the finite element method (FEM) is used to solve the six-dimensional problem. For the coupled photon-electron system, the variational formulation and the existence and uniqueness of the solution are derived. We simulate the solution of two coupled BTEs describing the travelling of photons and electrons in two spatial dimensions. The results are compared to Monte Carlo calculations with good agreement.