Asymptotic dynamics and optimal treatment for a model of tumour resistance to chemotherapy

Failure in cancer treatment often stems from drug resistance, which can manifest as either intrinsic (pre-existing) or acquired (induced by drugs). Despite extensive efforts, overcoming this resistance remains a challenging task due to the intricate and highly individualized biological mechanisms involved. This paper introduces an innovative extension of an already well-established mathematical model to account for tumour resistance development against chemotherapy. This study examines the existence and local stability of model solutions, as well as exploring the model asymptotic dynamics. Additionally, a numerical analysis of the optimal control problem is conducted using an objective functional. The numerical simulations demonstrate that a constant anti-angiogenic treatment leads to a concatenation of bang-bang and singular intervals in chemotherapy control, resembling a combined protocol comprising maximal tolerated dose and metronomic protocols. This observation lends support to the hypothesis that mean-dose chemotherapy protocols may help circumvent acquired drug resistance. Lastly, a sensitivity analysis is undertaken to scrutinize the dependence of model parameters on the outcomes of the previously examined therapeutic protocols.