Optimising the tumour elimination payoff in cancer therapy

A new payoff function is proposed for cancer treatment optimisation, the tumour elimination payoff (TEP), that incorporates the increase in lifespan if tumour elimination is achieved. The TEP is discounted by drug toxicity and by potential risks, such as metastasis and mutation. An approximation is used for the probability of tumour elimination, $e^{-\alpha N\left(T\right)}$, giving a terminal payoff with an exponential dependence on the final tumour size $N\left(T\right)$. The optimal solutions for this payoff for simple tumour growth models, (logistic and Gompertz growth), are determined. Using Pontryagin's maximum principle it is proved that bang–bang optimal solutions exist with a single switch; specifically delayed treatment and treat-and-stop solutions at maximum tolerated dose (MTD) exist. There is also a singular arc with constant tumour size. Solutions either have a high probability, respectively, low probability of tumour elimination; these correspond to a post-treatment high probability of cure, and a high probability of relapse, respectively. Optimising over the time horizon $T$ results in solutions that are either MTD throughout or no treatment, that is, treatment is either beneficial or detrimental. For the logistic growth model, the treatment benefit phase diagram is derived with respect to the patient's expected increase in lifespan and tumour size.