### Prof. Dr. Urszula Lewdzewicz (Southern Illinois University at Edwardsville, USA): Analysis of Mathematical Models for Tumor Anti-Angiogenesis and Combination Therapy

##### Wednesday, 15.07.2009 16:15 im Raum M4

A class of mathematical models for tumor anti-angiogenesis that are based on medical research originally done at Harvard School of Medicine and the National Cancer Institute (NIH) will be considered as optimal control problems with the objective of minimizing the size of the tumor with a constraint on the total amount of agents to be administered. Optimal controls consist of concatenations of bang-bang controls that correspond to full dose therapies with rest periods and singular controls which give rise to specific, state-dependent, partial dose therapies. Because of the feedback form of the singular portion, these optimal solutions are not medically realizable, but they provide the benchmark to which other, simple and medically realizable protocols can be compared. Piecewise constant suboptimal protocols with a small number of pieces will be discussed that come within 1% of the optimal values. In the second part of the talk an extension of the model will be considered that incorporates the pharmacokinetics (PK) of the anti-angiogenic agents. It will be analyzed how such an extension effects the structure of optimal solutions. For the standard linear model for PK, the optimality status of the singular control is preserved, but its order increases from 1 to 2. From a theoretical point of view, this significantly complicates the structure of optimal solutions since concatenations with the singular control can now only be achieved by means of so-called chattering controls that have infinitely many switchings on a finite interval. From a practical point of view, however, again simple suboptimal protocols can be constructed and numerical examples of such approximations will be given. The talk will conclude with some recent results for models when anti-angiogenic treatments are combined with chemotherapy. The model for such a combination therapy also includes a cytotoxic agent that introduces a second control into the system. Due to the multi-control aspect, even with simplified dynamical equations, this becomes a mathematically challenging problem. However, the optimal solution for the angio-monotherapy problem discussed earlier, plays an important part in the solution for the combination therapy problem and both analytical and numerical results will be presented.

Angelegt am 25.05.2009 von Frank Wübbeling

Geändert am 15.06.2009 von Frank Wübbeling

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Kolloquium der angewandten Mathematik