Optimal control of a mathematical model of CAR-T cell therapy for glioblastoma

CAR-T cell immunotherapy involves the genetic reprogramming of T-lymphocytes, enabling them to recognize and attack cancer cells, thereby triggering an anti-tumor immune response. While this treatment has been approved for hematological cancers, tackling solid tumors presents new challenges. These include the heterogeneity of antigen expression within solid tumors – encompassing antigen-positive non-tumoral cells–the presence of immune-inhibitory molecules, and the difficulty of CAR-T cell trafficking within the tumor microenvironment. In this article, we investigate an optimal control problem for a mathematical model based on differential equations that describes the interactions between tumor cells and CAR-T cells. Specifically, this model focuses on the dynamics of glioblastoma, the most aggressive brain tumor, and its response to CAR-T cell treatment. We formulate an optimal control problem and analyze it using the Pontryagin Maximum Principle and the Legendre-Clebsch condition. Subsequently, we solve the problem numerically. Additionally, we conduct a sensitivity analysis to identify the model parameters with the greatest influence. Finally, we demonstrate how numerical artifacts may arise in such problems, depending on the software used.