Combination of direct methods and homotopy in numerical optimal control: application to the optimization of chemotherapy in cancer

TL;DR

This paper introduces a novel numerical approach combining direct methods and homotopy for solving complex, state-constrained optimal control problems in mathematical oncology, specifically optimizing chemotherapy to minimize tumor size and resistance.

Contribution

It proposes a new method that reformulates the problem to enable the use of Pontryagin Maximum Principle, improving reliability and efficiency over previous techniques.

Findings

The new approach effectively handles complex models with diffusion.

It outperforms traditional methods in solving the cancer chemotherapy optimization.

The method provides a reliable framework for similar high-dimensional control problems.

Abstract

We consider a state-constrained optimal control problem of a system of two non-local partial-differential equations, which is an extension of the one introduced in a previous work in mathematical oncology. The aim is to minimize the tumor size through chemotherapy while avoiding the emergence of resistance to the drugs. The numerical approach to solve the problem was the combination of direct methods and continuation on discretization parameters, which happen to be insufficient for the more complicated model, where diffusion is added to account for mutations. In the present paper, we propose an approach relying on changing the problem so that it can theoretically be solved thanks to a Pontryagin Maximum Principle in infinite dimension. This provides an excellent starting point for a much more reliable and efficient algorithm combining direct methods and continuations. The global idea is new and can be thought of as an alternative to other numerical optimal control techniques.