Optimal Solutions of Chronotherapy Models

Optimal Solutions of Chronotherapy Models

Mathematics

Xingyu (Steve) Wu

2023

It is hypothesised that both cancer cell growth and chemotherapy drug tolerance can be dependent on the circadian rhythm, causing differences in treatment
efficiency at different times of the day. Chronotherapy aims to exploit this dependency to provide an optimal treatment plan for patients. This plan seeks to maximise the elimination of cancer cells while minimising drug toxicity resulting from chemotherapy. This paper aims to extend previous works and provide analyses of optimal solutions for various cancer treatment models, including scenarios in which growth is partially coupled with the circadian rhythm, cases where drugs target growth, and models that account for delayed drug effects. Using Pontryagin’s Minimum Principle (PMP), we found the switching function of each model that decides the optimal time period to apply or stop the drugs. In the first two models, we noted the aperiodic switching between full and no drugs in optimal solutions and explored the nature of the switching zones. While the optimal solution for the last case is complex, it is proven to be bang-bang and can be reduced to a simpler growth model using the Pseudo-Steady State Hypothesis (PSSH) for sufficiently fast drug absorption speeds. Furthermore, we presented the backward integration method to analyse the nature of optimal solutions. The exploration of optimal solutions provides a better understanding of optimal treatment plans for
chronotherapy models and offers insights into potential future analyses of more complex and clinically applicable models.