Control of Tumor Growth Modeled by System of PDE, Numerical Analysis of Optimality Conditions

We analyze a mathematical model of a tumor that includes the interactions among tumor cells (live or dead), macrophages, endothelial cells, and the cytokines based on a system of partial differential equations which describes inhibition of tumor growth by granulocyte‐macrophage colony‐stimulating factor (GM‐CSF) treatment. The aim of the model is to predict the growth of the tumor volume under partial blocking of hypoxia‐inducible factor (HIF)‐1 or stabilization of HIF‐2, with injection of GM‐CSF. It was assumed that tumor is a spherical and the treatment by GM‐CSF is not optimized. In this paper, we do not assume that tumor is a spherical model and we develop a dual dynamic programming approach to formulate a verification condition of an approximate treatment for an injection of GM‐CSF. Next, we analyze numerically an ‐optimal treatment.