Identification of Some Unknown Parameters in an Aggressive–Invasive Cancer Model Using Adjoint Approach

Abstract

This study deals with the identification of unknown parameters in an important class of continuous models of aggressive–invasive cancers. This model is obtained as a system of non-linear reaction–diffusion equations from the evolution of cancer cells based on population dynamics. The proposed identification problem is formulated as an optimal control problem with PDEs’ constraint in which an objective functional is defined with respect to the patient and experimental data and an adjoint problem is derived to develop an iterative procedure. A gradient-based iteration method is established to solve this optimal control problem. In each iteration, two systems of nonlinear initial and boundary-value problems (namely, direct and adjoint problems) should be solved. To this end, a nonstandard finite-difference approach is proposed. The robustness of the numerical approach is examined by a test problem without and with noisy data. The numerical results are in good agreement with the results in the literature.