### Prof. Dr. Heinz Schättler (Washington University, St. Louis, USA): Geometric Methods in Optimal Control Theory Case Study of a Biomedical Problem

##### Wednesday, 01.07.2009 16:15 im Raum M4

In the talk, the use of geometric methods in optimal control theory will be illustrated in the construction of an optimal synthesis of controlled trajectories for a mathematical model of tumor anti-angiogenesis. This is a novel cancer treatment approach that aims at preventing the development of the blood vessel network a tumor needs for growth. A low-dimensional mathematical model originally introduced by a group of researchers from Harvard School of Medicine will be analyzed as an optimal control problem with the objective of minimizing the size of the tumor with a constraint on the total amount of anti-angiogenic agents to be given. The dynamics of the system describes the growth of the primary tumor volume and its vascularization under the effects of control functions representing the dosage of these anti-angiogenic agents. The focus of the talk will be on the mathematical methods used in deriving a theoretical solution to this problem in terms of a complete synthesis of optimal controls and trajectories. Using tools of geometric control theory that are based on Lie bracket computations and high-order necessary conditions for optimality, analytic formulas for an optimal singular arc can be computed. The associated singular arc becomes the center piece for the synthesis. A geometric analysis of trajectories near the singular arc (including saturation effects) reveals the structure of optimal controls as specific concatenations of bang-bang controls (representing therapies of full dose with rest periods) and singular controls (therapies with specific time-varying partial doses). The optimal synthesis gives a benchmark value to which other protocols can be compared and thus allows to judge the quality of simple, heuristically chosen, protocols. It also provides the foundation for solutions of medically more realistic formulations that, for example, include a pharmacokinetic model for the anti-angiogenic agents or consider combination therapies.

Angelegt am 22.04.2009 von Frank Wübbeling

Geändert am 15.06.2009 von Frank Wübbeling

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Angewandte Mathematik Münster

Kolloquium der angewandten Mathematik