Mathematical modelling for dose deposition in photon-therapy

Radiotherapy treatments consists in irradiating the patient with beams of energetic particles (typically photons) targeting the tumor. Such particles are transported through the medium and deposit energy in the medium. This deposited energy is the so called dose, responsible for the biological effect of the radiations. The present work aim to develop numerical methods for dose computation and optimization that are competitive in terms of computational cost and accuracy compared to reference method. The motion of particles is first studied through a system of linear transport equations at the kinetic level. However, solving directly such systems is numerically too costly for medical application. Instead, the moment method is used with a special focus on the Mn models. Those moment equations are non-linear and valid under a condition called realizability. Standard numerical schemes for moment equations are constrained by stability conditions which happen to be very restrictive when the medium contains low density regions. Inconditionally stable numerical schemes adapted to moment equations (preserving the realizability property) are developed. Those schemes are shown to be competitive in terms of computational costs compared to reference approaches. Finally they are applied to in an optimization procedure aiming to maximize the dose in the tumor and to minimize the dose in healthy tissues. (author)