Analysis of optimal controls for a mathematical model of tumor anti-angiogenesis

Urszula Ledzewicz Department of Mathematics and Statistics, Southern Illinois University, Edwardsville, USA

Abstract: In the talk a new development in cancer treatment, anti-angiogenesis, will be discussed. The importance of this type of novel treatment is that by targeting the cells of the vascularization of the tumor rather then the tumor itself they are "resistant to drug resistance". In the talk a mathematical model for cancer treatments using angiogenic inhibitors (by Ergun et al.) will be presented and analyzed with the tools of geometric optimal control theory. The two-dimensional state represents the volume of tumor and endothelial cells and the nonlinear dynamics illustrates how control functions representing angiogenic inhibitors effect their growths. The goal of the therapy is to minimize the size of the tumor, the therapy interval is taken to be free while we are looking for the best therapy administering a fixed total amount of drug. With the use of Lie bracket theoretical computations, a full synthesis of optimal controls will be constructed. It turns out that the optimal strategies for this model are concatenations of bang-bang controls (representing therapies of full doses with rest periods) and singular controls (therapies of varying partial doses). A biological interpretation of the results will be given.

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