Optimal multitherapy strategy in mathematical model of dynamics of the number of nonuniform tumor cells

Mathematical model of dynamics of the number of tumor cells is considered. A tumor is assumed to consist of two cell types, each being under the influence of a specific chemotherapeutic agent that is capable of destroying cells of this specific type. The laws of growth of the number of all types of cells are considered to be given by logistic equations. The measure of the influence of each chemotherapeutic agent on the tumor is defined by a therapy function. Two types of therapy functions are used: a monotonically increasing function and a nonmonotonic function with a threshold value. In the first case, the higher the concentration of the agent, the stronger its influence on the tumor. In the second case, there exists a certain threshold value of the chemotherapeutic agent concentration: once it is exceeded, therapy efficiency decreases. The variant, when the total amount of each agent has an integral limit, is also studied. Necessary optimality conditions are formulated using the Pontryagin’s maximum principle. They are used as a base for making important conclusions about the character of the optimal therapy strategy. We find numerically solutions to the optimal control problems, when the control is aimed at minimizing the total number of tumor cells for the cases of monotonic and threshold therapy functions, as well as with account of the integral constraints for the amounts of chemotherapeutic agents.