On the structure of optimal controls for a mathematical model of tumor anti-angiogenic therapy with linear pharmacokinetics

A mathematical model for tumor anti-angiogenesis that was formulated and biologically validated by Hahnfeldt et al. is considered as an optimal control problem. In earlier research, the optimal scheduling of anti-angiogenic agents has been analyzed under the simplifying assumption that dosage and concentration were identified. In this case, there exists an optimal singular arc of order 1 that forms the centerpiece of a synthesis of optimal controlled trajectories. Here we consider the same model with standard pharmacokinetic equations added that define the concentration as the state of a first-order linear system driven by the dosage. The singular arc and its optimality status are preserved under this modelling extension and an explicit feedback formula that defines the optimal singular control in the simplified model now becomes the singular concentration for the extended system. Optimal controls track this concentration of inhibitors along the singular arc. However, the order of the singular arc increases from 1 to 2 and the overall concatenation structure in the synthesis of optimal trajectories changes. Now optimal transitions to and from the singular arc can only occur through chattering arcs.