The effect of pharmacokinetics on optimal protocols for a mathematical model of tumor anti-angiogenic therapy

A mathematical model for the scheduling of angiogenic inhibitors that includes a pharmacokinetic equation is considered as an optimal control problem. When dosage and concentration of the inhibitor are identified, there exists an optimal singular arc of order 1 that forms the core of a synthesis of optimal controls. Under the standard pharmacokinetic linear model for the concentration this singular arc is preserved and still optimal, but its order increases to 2. This prevents concatenations of the singular arc with the constant bang controls and now the transitions to and from the singular arc are through chattering arcs. Optimal controls have the property that the associated concentration of inhibitors tracks the optimal singular arc for the reduced model without pharmacokinetic equations.