On a Class of Systems Describing Tumor Anti-Angiogenesis under Gompertzian Growth

Abstract: Anti-angiogenic therapy is a novel treatment approach for cancer that aims at preventing a tumor from developing its own blood supply system that it needs for growth. In the course of the treatment the lining of the newly developing blood vessels formed by so-called endothelial cells is targeted with anti-angiogenic inhibitors which halt their growth. In this paper a general class of models for anti-angiogenic treatment is analyzed as a 3-dimensional optimal control problem with tumor cells, endothelial cells, and anti-angiogenic inhibitors as variables. It is assumed that the growth of the tumor is Gompertzian with a variable carrying capacity given by the endothelial cells, but the dynamics of endothelial cells is kept general. For this class of models, the optimality properties of singular controls are determined in great generality using Lie-algebraic computations.