Approximation methods and reduced models for optimal radiation treatment

Radiation therapy becomes an important and common tool for cancer treatment with increasing the number of cancer deceases. Most common tools in clinics and medical institutions are based on statistical and experimental methods, e.g. a Pencil Beam, a Collapsed Cone or a Monte Carlo method. Nevertheless, in the present work weconsider deterministic models for radiation therapy which are based on transport theory of the particles. The Boltzmann transport equation is used to describe evolution of charged particles in tissue. The radiation transport problem is approximated in terms of moments of the transfer equation. We introduce two different methods to solve spherical harmonics moments systems. We study properties of the numerical methods and show that the presented numerical method preserves asymptotic limit while the transport theory transmits into diffusion theory. The process of radiotherapy is an inverse problem. The main aim of the radiation therapy is to find position and dose of external beam such that particles will be distributed in a domain in desired way: much more coupled in cancer region and almost no particles in other regions. Since deterministic methods have explicit mathematical structures it allows us to formulate the radiation treatment process as optimal control problem constrained by the Boltzmann equation. We show existence, uniqueness and regularity of the optimal control and present the first-order optimality conditions. Geometrical or physical parameters of the problem may change during the treatment process which requires quick and efficient re-computation of the treatment plan. We propose low-dimensional reduced basis formulation which allows us to solve optimal control problem efficiently. We derive a posteriori error estimators for reduced basis solution with respect to the solution of high dimensional system