Virotherapy, using herpes simplex virus, represents a promising therapy of glioma. But the innate immune response, which includes TNF-α produced by macrophages, reduces the effectiveness of the treatment. Hence treatment with TNF-α inhibitor may increase the effectiveness of the virotherapy. In the present paper we develop a mathematical model that includes continuous infusion of the virus in combination with TNF-α inhibitor. We study the efficacy of the treatment under different combinations of the two drugs for different scenarios of the burst size of newly formed virus emerging from dying infected cancer cells. The model may serve as a first step toward developing an optimal strategy for the treatment of glioma by the combination of TNF-α inhibitor and oncolytic virus injection.

Abstract. We consider an optimal control problem for a general mathematical model of drug treatment with a single agent. The control represents the concentration of the agent and its effect (pharmacodynamics) is modelled by a Hill function (i.e., Michaelis-Menten type kinetics). The aim is to minimize a cost functional consisting of a weighted average related to the state of the system (both at the end and during a fixed therapy horizon) and to the total amount of drugs given. The latter is an indirect measure for the side effects of treatment. It is shown that optimal controls are continuous functions of time that change between full or no dose segments with connecting pieces that take values in the interior of the control set. Sufficient conditions for the strong local optimality of an extremal controlled trajectory in terms of the existence of a solution to a piecewise defined Riccati differential equation are given.

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