Analysis of a cell-cycle specific model for cancer chemotherapy

摘要

A class of mathematical models for cancer chemotherapy which has been described in the literature takes the form of an optimal control problem with dynamics given by a bilinear system. In this paper we analyze a three-dimensional model in which the cellcycle is broken into three compartments. The cytostatic agent used as control to kill the cancer cells is active in a compartment which combines the second growth phase and mitosis where cell-division occurs. A blocking agent is used as a second control to slow down the transit of cells during synthesis, but does no kill cells. The cumulative effect of the killing agent is used to model the negative effect of the treatment on healthy cells. It is shown that singular controls are not optimal. This eliminates treatments where during some time only a portion of the full drug dose is administered. Consequently only treatments which alternate between a full and no dose, i.e., so-called bang-band controls, can be optimal for this model. Both necessary and sufficient conditions for optimality of treatment schedules of this type are given.