A Nonspecialist's Introduction to some Biomedical Applications of Optimal Control

Heinz Schättler, Department of Electrical and Systems Engineering, Washington University, St. Louis, Missouri

I will give a self-contained, non-specialist introduction to some issues centered around chemotherapy treatments of cancer that can be formulated mathematically as optimization and, more generally, optimal control problems. (No prior knowledge of optimal control theory will be assumed in the talk.) They include such fundamental questions as the temporal scheduling of a drug over a therapy interval or the appropriate sequencing of drugs in case of multi-drug therapies. Although the precise molecular mechanisms of the drugs? actions can be forbiddingly complex (and are not necessarily always understood), the general picture and treatment approaches derived from it are much clearer and can easily be explained. In spite of tremendous progress in the medical understanding of the disease(s) - cancer is caused by ?mistakes? in the cell cycle and consequently a vast number of widely differing phenomena are summarized by this term - a ?cure? remains elusive. The most significant obstacle to the success of chemotherapy is drug resistance which, unfortunately, severely limits many treatments in the long run. I will conclude the talk with one novel approach to cancer treatment that is currently pursued also in clinical trials in the hope to overcome this problem. For one of the models in the medical literature, using Lie bracket computations and other tools of optimal control theory in combination with elementary geometric considerations, I?ll construct a complete synthesis of optimal controls for the problem of minimizing the tumor volume at the end of therapy given a constraint on the total amount of inhibitors.