Optimal control problems arising in cell-cycle-specific cancer chemotherapy

We explore mathematical properties of models of cancer chemotherapy including cell-cycle dependence. Using the mathematical methods of control theory, we demonstrate two assertions of interest for the biomedical community: 1 Periodic chemotherapy protocols are close to the optimum for a wide class of models and have additional favourable properties. 2 Two possible approaches, (a) to minimize the final count of malignant cells and the cumulative effect of the drug on normal cells, or (b) to maximize the final count of normal cells and the cumulative effect of the drug on malignant cells, lead to similar principles of optimization. From the mathematical viewpoint, the paper provides a catalogue of simplest mathematical models of cell-cycle dependent chemotherapy. They can be classified based on the number of compartments and types of drug action modelled. In all these models the optimal controls are complicated by the singular and periodic trajectories and multiple solutions. However, efficient numerical methods have been developed. In simpler cases, it is also possible to provide an exhaustive classification of solutions. We also discuss developments in estimation of cell cycle parameters and cell-cycle dependent drug action.