The circadian timing system is composed of molecular clocks, which drive 24-h changes in xenobiotic metabolism and detoxification, cell cycle events, DNA repair, apoptosis, and angiogenesis. The cellular circadian clocks are coordinated by endogenous physiological rhythms, so that they tick in synchrony in the host tissues that can be damaged by anticancer agents. As a result, circadian timing can modify 2- to 10-fold the tolerability of anticancer medications in experimental models and in cancer patients. Improved efficacy is also seen when drugs are given near their respective times of best tolerability, due to (a) inherently poor circadian entrainment of tumors and (b) persistent circadian entrainment of healthy tissues. Conversely, host clocks are disrupted whenever anticancer drugs are administered at their most toxic time. On the other hand, circadian disruption accelerates experimental and clinical cancer processes. Gender, circadian physiology, clock genes, and cell cycle critically affect outcome on cancer chronotherapeutics. Mathematical and systems biology approaches currently develop and integrate theoretical, experimental, and technological tools in order to further optimize and personalize the circadian administration of cancer treatments.

Resistance to chemotherapies, particularly to anticancer treatments, is an increasing medical concern. Among the many mechanisms at work in cancers, one of the most important is the selection of tumor cells expressing resistance genes or phenotypes. Motivated by the theory of mutation-selection in adaptive evolution, we propose a model based on a continuous variable that represents the expression level of a resistance gene (or genes, yielding a phenotype) influencing in healthy and tumor cells birth/death rates, effects of chemotherapies (both cytotoxic and cytostatic) and mutations. We extend previous work by demonstrating how qualitatively different actions of chemotherapeutic and cytostatic treatments may induce different levels of resistance.The mathematical interest of our study is in the formalism of constrained Hamilton-Jacobi equations in the framework of viscosity solutions. We derive the long-term temporal dynamics of the fittest traits in the regime of small mutations. In the context of adaptive cancer management, we also analyse whether an optimal drug level is better than the maximal tolerated dose.