Effects of Pharmacokinetics and DNA Repair on the Structure of Optimal Controls in a Simple Model of Radio-Chemotherapy

Abstract

In this study optimal control framework is applied to two models of radiochemotherapy. A base, simple model adopts a classical log-kill hypothesis for chemotherapy and a linear-quadratic response for radiotherapy. The model is represented by the first order nonlinear differential equations with two control variables which can be transformed into a linear differential equations with a nonlinear control action. The multi-input optimal control problem formulated for this model is solved explicitly. It is shown that optimal treatment should apply radiotherapy first, while the onset of chemotherapy should be delayed. The second model takes into account the effects of drug metabolism (pharmacokinetics) and DNA repair of adjacent strand breaks. By obtaining full analytical results for chemotherapy control and partial results for radiotherapy it is shown that an introduction of these additional processes alters the structure of optimal control, which sometimes leads to opposite conclusions. A numerical example is provided which suggests that in practice the differences may not be as critical as the purely theoretical results would suggest.