The Application of Power Series Expansion to Optimal Control of an ImmuneOncology Nonlinear Dynamic Problem

This paper is concerned with the eradication of tumor cells in the human body by defining an optimal protocol using a polynomial approximation technique for the injection of chemotherapy drugs. The dynamics of the system are described based on immune-oncology. Variation of host, tumor, and immune cells’ populations are studied in the model during the injection of the chemotherapeutic drugs. The objective is the minimization of cancerous cells' average population by minimum drug injection to avoid the destructive side-effects of these chemotherapeutic substances. It should be done by stabilizing the population of host and immune cells around a free-tumor desirable health condition. This optimization problem by considering the nonlinear model of the system makes applying nonlinear optimal control inevitable. Solving Hamilton-Jacobi-Bellman (HJB) nonlinear partial differential equation (PDE) for the system is put into our perspective to cope with this problem. Since the dynamics of the system are not polynomial, it comprises fractional terms, this PDE cannot be solved straightforwardly. We take advantage of the power series expansion technique to approximate the solution of the PDE with satisfactory accuracy. Finally, a series of simulations are carried out to prove the capability of the controller in terms of robustness and sensitivity, increasing convergence rate for the elimination of cancerous cells, and enlargement of the domain of attraction.