Global Stability and Thermal Optimal Control Strategies for Hyperthermia Treatment of Malignant Tumors

Malignant tumor (cancer) is the leading cause of death globally and the annual cost of managing cancer is trillions of dollars. Although, there are established therapies including radiotherapy, chemotherapy and phototherapy for malignant tumors, the hypoxic environment of tumors and poor perfusion act as barriers to these therapies. Hyperthermia takes advantage of oxygen deficiency and irregular perfusion in the tumor environment to destroy malignant cells. Despite successes recorded with hyperthermia, there are concerns with the post-treatment condition of patients as well as the required thermal dose to prevent harm. The investigation of the dynamics of tumor-induced immune suppression with hyperthermia treatment using mathematical analysis and optimal control theory is potentially valuable in the development of hyperthermia treatment. The role of novel tumor-derived cytokines in counterattacking immune cells is considered in this study as a mechanism accounting for the aggressiveness of malignant tumors. Since biological processes are not instantaneous, a discrete time delay is used to model biological processes involved in tumor inhibitory mechanisms by secretion, the elaboration of suppressive cells, and effector cell differentiation to produce suppressive cells. Analytical results obtained using Lyapunov’s function indicate the conditions required for global stability of the tumor-present steady-state. A thermal optimal control strategy is pursued based on optimal control theory, and the best strategy to avoid adverse outcomes is obtained. We validate the analytical results numerically and demonstrate the impact of both inadequate and excessive heat on the dynamics of interactive cell functioning.