Optimal Combined Radio- and Anti-Angiogenic Cancer Therapy

A mathematical model for combination of radio- and anti-angiogenic therapy is considered as optimal control problem with the objective of minimizing the tumor volume subject to isoperimetric constraints that limit the total radiation dose and the overall amount of anti-angiogenic agents to be given. The dynamics combines a model for tumor development under angiogenic inhibitors with the linear-quadratic model for the damage done by radiation ionization. The system has been investigated analytically as an optimal control problem and explicit expressions for possible singular controls were derived before. In this paper, for varying total radiation doses, examples of numerically computed optimal controls are given that verify and confirm these analytical structures: optimal schedules for the anti-angiogenic agents typically start with a brief full-dose segment, and then use up all inhibitors along a time-varying singular control while optimal radiotherapy schedules intensify the dosing and, after a brief period when the control is singular, end with a maximum dose segment. Singular controls occur for both the anti-angiogenic and radiotherapy dose rates. A discussion of the difficulties in proving the strong local optimality of corresponding trajectories is included.