Abstract
•A new variable-order model with modified parameters for cancer treatment.•Formulation of the optimal control cancer treatment problem.•Derivation of the necessary optimality conditions.•Two numerical methods for the simulations of the control problem.•Comparative studies.
This article presents a fractional-order mathematical model of the biological phenomena that occur in cancer therapy. The formulation generalizes the one proposed by Soto-Ortiza and Finley that consists of eighteen integer order differential equations and intends to serve as a platform for cancer treatment design. The fractional model is used to test the hypothesis that a combination of anti-Vascular Endothelial Growth Factor (VEGF) treatment with immunotherapy, involving injections of unlicensed dendritic cells (DC), can lead to the tumor eradication by concealing the suppressor VEGF. The new approach adopts derivatives defined in the Caputo sense and implements an optimal control strategy. Two control variables, one for immunotherapy and another for anti-angiogenic therapy, are considered for reducing the number of the cancer cells. Moreover, two numerical methods are introduced and their stability analysis is studied. Numerical simulations illustrate the proposed concepts.