RUI: Collaborative Research: Optimal Control of Mathematical Models for Cancer Treatments

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In this project mathematical models for various cancer treatments will be investigated with the methods of modern optimal control theory. A class of models based on chemotherapy with single or multiple killing agents will be formulated and analyzed under evolving drug resistance. In the case of a single drug this leads to bilinear structures, but for the multiple drug case unconventional formulations arise and new tools need to be applied. Aside from problems involving drug resistance, attention also will be directed to models representing treatments with angiogenic inhibitors (which slow tumor growth by depriving it from recruiting new blood vessels). The dynamics in these models is fully nonlinear and its analysis within the framework of geometric methods of optimal control is expected to answer some open questions about this novel therapy. Drug resistance and other side effects related to chemo-, radio- or antiangiogenic therapy naturally lead to the need for combination treatments. Mathematically such treatments provide a challenge not only in the formulation of the complex dynamics, but also in the choice of proper objectives which would capture the various and multi-fold aspects reflecting the medical criteria by which to judge these therapies. All of these efforts will serve the overall goal of finding the optimal control, which could aid in designing improved therapy protocols.Conventional cancer treatments aim at directly killing tumor cells, be it by means of drugs in chemotherapy or by means of radiation in radiation treatments. However, there exist many limiting factors and probably the single most important, and what has been called "certainly the most frustrating one," is drug resistance. As of today no medical solution exists for this obstacle to developing effective cancer treatments. In the project tools of modern optimal control theory will be employed to analyze a variety of mathematical models representing cancer treatments with the focus on the important medical issue of drug resistance. New models that capture these phenomena will be analyzed in an effort to better reflect the underlying biological situation and goals of the treatment. The research conducted will provide mathematical insights into an understanding of these important biomedical models aimed at designing more effective therapy protocols which may be difficult, or at least very expensive, if not impossible, to test in a laboratory setting. It will also contribute to optimal control theory by developing and employing new techniques aimed at significant applications. Due to its applied and interdisciplinary character the project is of interest to students and consequently will contain a substantial educational component.