Parameter identification for nonlocal phase field models for tumor growth via optimal control and asymptotic analysis

TL;DR

This paper develops a method for identifying parameters in nonlocal tumor growth models using optimal control theory, asymptotic analysis, and relaxation techniques to handle complex PDE systems.

Contribution

It introduces a novel approach combining optimal control and asymptotic analysis for parameter identification in nonlocal PDE tumor models, including relaxed and original systems.

Findings

Derived first-order optimality conditions for relaxed models.

Proved convergence of adjoint systems as relaxation parameters vanish.

Established a framework for solving parameter identification in complex tumor growth PDEs.

Abstract

We introduce the problem of parameter identification for a coupled nonlocal Cahn-Hilliard-reaction-diffusion PDE system stemming from a recently introduced tumor growth model. The inverse problem of identifying relevant parameters is studied here by relying on techniques from optimal control theory of PDE systems. The parameters to be identified play the role of controls, and a suitable cost functional of tracking-type is introduced to account for the discrepancy between some a priori knowledge of the parameters and the controls themselves. The analysis is carried out for several classes of models, each one depending on a specific relaxation (of parabolic or viscous type) performed on the original system. First-order necessary optimality conditions are obtained on the fully relaxed system, in both the two and three-dimensional case. Then, the optimal control problem on the non-relaxed models is tackled by means of asymptotic arguments, by showing convergence of the respective adjoint systems and the minimization problems as each one of the relaxing coefficients vanishes. This allows obtaining the desired necessary optimality conditions, hence to solve the parameter identification problem, for the original PDE system in case of physically relevant double-well potentials.