Optimal control of therapies related to an oxytaxis glioblastoma model

TL;DR

This paper develops an optimal control framework for a glioblastoma growth model influenced by oxygen, incorporating chemoattraction, and proposes a numerical scheme to optimize therapies with demonstrated effectiveness.

Contribution

It introduces a novel optimal control approach for a Keller-Segel type glioblastoma model with oxygen influence, including existence, uniqueness, and numerical approximation of optimal therapies.

Findings

Existence of a global optimal therapy solution

Derivation of first-order optimality conditions

Numerical experiments showing scheme effectiveness

Abstract

We propose and analyze an optimal control problem associated with a Keller-Segel type parabolic system with chemoattraction, modeling the glioblastoma growth in a bi-dimensional bounded domain, influenced by the presence of oxygen where the controls are two different (chemotherapy and antiangiogenic) therapies. The model considers the random diffusion of tumor cells and oxygen, the movement of cells towards the oxygen gradient (oxytaxis), and reaction terms describing the interaction between cells and oxygen. We establish a mathematical framework to analyze the existence and uniqueness of weak-strong solution of the model and subsequently we analyze an optimal control problem considering a cost functional that minimizes both the tumor growth and the oxygen concentration. We prove the existence of a global optimal solution and derive necessary first-order optimality conditions. Finally, we propose a methodology for approximating the optimal therapies. We use the gradient of the reduced cost functional through the adjoint scheme, and minimize the cost functional implementing the Adam gradient optimization method. Some numerical experiments are provided to demonstrate the effectiveness of the proposed scheme.