Periodic optimization of a class of bilinear systems with application to control of cell proliferation and cancer therapy

The authors are concerned with the optimization of a class of bilinear systems under an exponential stability constraint by periodic control functions. Such problems arise in many practical applications, a typical one being the design of optimal strategies for cancer chemotherapy by a suitable control of the proliferation kinetics of cell populations. To provide a focus to the approach used for optimization and the utility of periodic controls, this particular application is discussed in detail. An integration of a pharmacokinetic model with a multicompartmental model for cell proliferation is made. This is done in order to obtain a mathematical formulation of the problem of designing treatment strategies as a parameter optimization problem of determining the optimal dose and the optimal period for minimizing the total quantity of drug administered (and hence the host toxicity) under a specified rate of cure constraint. A procedure for solving this problem is developed, and an illustration is made by designing strategies for administering the phase-specific drug ara-c on L1210 leukemia.