Optimal control for cancer chemotherapy under tumor heterogeneity with michealis-menten pharmacodynamics

Abstract

We analyze mathematical models for cancer chemotherapy under tumor heterogeneity as optimal control problems. Tumor heterogeneity is incorporated into the model through a potentially large number of states which represent sub-populations of tumor cells with different chemotherapeutic sensitivities. In this paper, a Michaelis-Menten type functional form is used to model the pharmacodynamic effects of the drug concentrations. In the objective, a weighted average of the tumor volume and the total amounts of drugs administered (taken as an indirect measurement for the side effects) is minimized. Mathematically, incorporating a Michaelis-Menten form creates a nonlinear structure in the controls with partial convexity properties in the Hamiltonian function for the optimal control problem. As a result, optimal controls are continuous and this fact can be utilized to set up an efficient numerical computation of extremals. Second order Jacobi type necessary and sufficient conditions for optimality are formulated that allow to verify the strong local optimality of numerically computed extremal controlled trajectories. Examples which illustrate the cases of both strong locally optimal and non-optimal extremals are given.