Numerical Solution of an Optimal Control Problem in Cancer Treatment: Combined Radio and Anti-Angiogenic Therapy

Abstract

Several therapies and also combined therapies for cancer treatment exist mathematical models of which have partly been also optimized by means of optimal control methods. Here we focus on an optimal control problem describing a combined radiotherapy and an antiangiogenic treatment. The underlying model is taken from literature where, however, despite rigorous analytical investigations a complete numerical solution is missing. In order to fill this gap, a direct solution approach has been developed for which both state and control functions of the optimal control problem are discretized. The resulting nonlinear programming problem is solved via the modelling language AMPL and the interior point solution method IPOPT. In an a posteriori step we try to check the main necessary conditions of optimal control theory in order to verify the candidate optimality of the continuous problem via the approximate discrete optimal solution. The most interested feature of this model from the viewpoint of optimal control theory is the fact that singular control subarcs exist for a two-dimensional control vector. However, the arising numerical difficulties caused by chattering controls prevented us so far from a complete candidate optimal solution.