Aperiodic Optimal Chronotherapy in Simple Compartment Tumour Growth Models Under Circadian Drug Toxicity Conditions

Cancer cells typically divide with weaker synchronisation with the circadian clock than normal cells, with the degree of decoupling increasing with tumour maturity. Chronotherapy exploits this loss of synchronisation, using drugs with circadian-clock-dependent activity and timed infusion to balance the competing demands of reducing toxicity toward normal cells that display physiological circadian rhythms and of efficacy against the tumour. We analysed optimal chronotherapy for one-compartment nonlinear tumour growth models that were no longer synchronised with the circadian clock, minimising a cost function with a periodically driven running cost accounting for the circadian drug tolerability of normal cells. Using Pontryagin’s Minimum Principle (PMP), we show, for drugs that either increase the cell death rate or kill dividing cells, that optimal solutions are aperiodic bang–bang solutions with two switches per day, with the duration of the daily drug administration increasing as treatment progresses; for large tumours, optimal therapy can in fact switch mid treatment from aperiodic to continuous treatment. We illustrate this with tumours grown under logistic and Gompertz dynamics conditions; for logistic growth, we categorise the different types of solutions. Singular solutions can be applicable for some nonlinear tumour growth models if the per capita growth rate is convex. Direct comparison of the optimal aperiodic solution with the optimal periodic solution shows the former presents reduced toxicity whilst retaining similar efficacy against the tumour. We only found periodic solutions with a daily period in one-compartment exponential growth models, whilst models incorporating nonlinear growth had generic aperiodic solutions, and linear multi-compartments appeared to have long-period (weeks) periodic solutions. Our results suggest that chronotherapy-based optimal solutions under a harmonic running cost are not typically periodic infusion schedules with a 24 h period.