Optimal controlled trajectories for a mathematical model of anti-angiogenic therapy in cancer

Anti-angiogenic therapy is a novel treatment approach in cancer therapy that aims at preventing a tumor from developing a network of blood vessels and capillaries that it needs for its supply of nutrients to further its growth. In this paper, a mathematical model for anti-angiogenic treatment that is based on a biologically validated model by Hahnfeldt, Panigrahy, Folkman and Hlatky is considered. Using geometric methods from optimal control theory, in [20] a full solution was given for the problem of scheduling an a priori given amount of anti-angiogenic agents when dosage and effectiveness of the agent are identified. The anchor piece of the optimal synthesis is an order 1 singular arc whose control saturates. In this paper the structure of this optimal synthesis near the saturation point is developed in detail.