Multi‐input Optimal Control Problems for Combined Tumor Anti‐angiogenic and Radiotherapy Treatments

A cell‐population‐based model for tumor growth under anti‐angiogenic treatment, with the tumor volume and its variable carrying capacity as variables, is combined with the linear‐quadratic model for damage done by radiation ionization. The resulting multi‐input system is analyzed as an optimal control problem with the objective of minimizing the tumor volume subject to isoperimetric constraints that limit the overall amounts of anti‐angiogenic agents, respectively, the damage done to healthy tissue by radiotherapy. For various model formulations, explicit expressions for singular controls are derived for both the dosage of the anti‐angiogenic therapeutic agent and the radiation dose schedule. Their role in the structure of optimal protocols is discussed.