Optimal control of tumor-immune system interaction with treatment

This paper concerns the optimal control of a mathematical model of a growing tumor and its interaction with the immune system. This model consists of four populations - tumor cells, dendritic cells (as an innate immune system), cytotoxic T cells, and helper T cells (as a specific immune system) - in the form of a system of ordinary differential equations. Some tumors present dendritic cell and such cells have a potential role in regulating the immune system. In this model, we assume that dendritic cells can activate cytotoxic T cells and, in turn, can clear out tumor cells. Furthermore, by adding controls as a treatment to the model, we minimize both the tumor cell population and the cost of treatment. We do this by applying the optimal control for this problem. First, Pontryagin's Principle is used to characterize the optimal control. Then, the optimal system is solved numerically using the Forward-Backward Runge- Kutta method. Finally, the effect of each treatment is investigated. The numerical results show that these controls are effective in reducing the number of tumor cells.