Optimal Control and Tumour Elimination by Maximisation of Patient Life Expectancy

We propose a life-expectancy pay-off function (LEP) for determining optimal cancer treatment within a control theory framework. The LEP averages life expectancy over all future outcomes, outcomes that are determined by key events during therapy such as tumour elimination (cure) and patient death (including treatment related mortality). We analyse this optimisation problem for tumours treated with chemotherapy using tumour growth models based on ordinary differential equations. To incorporate tumour elimination we draw on branching processes to compute the probability distribution of tumour population extinction. To demonstrate the approach, we apply the LEP framework to simplified one-compartment models of tumour growth that include three possible outcomes: cure, relapse, or death during treatment. Using Pontryagin’s maximum principle (PMP) we show that the best treatment strategies fall into three categories: (i) continuous treatment at the maximum tolerated dose (MTD), (ii) no treatment, or (iii) treat-and-stop therapy, where the drug is given at the MTD and then halted before the treatment (time) horizon. Optimal treatment strategies are independent of the time horizon unless the time horizon is too short to accommodate the most effective (treat-and-stop) therapy. For sufficiently long horizons, the optimal solution is either no treatment (when treatment yields no benefit) or treat-and-stop. Patients, thus, split into an untreatable class and a treatable class, with patient demographics, tumour size, tumour response, and drug toxicity determining whether a patient benefits from treatment. The LEP is in principle parametrisable from data, requiring estimation of the rates of each event and the associated life expectancy under that event. This makes the approach suitable for personalising cancer therapy based on tumour characteristics and patient-specific risk profiles.