Optimal and suboptimal protocols for a mathematical model for       tumor anti-angiogenesis in combination with chemotherapy

Urszula Ledzewicz; Helmut Maurer; Heinz Schättler

doi:10.3934/mbe.2011.8.307

Article Contents

2011, Volume 8, Issue 2: 307-323. Doi: 10.3934/mbe.2011.8.307

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Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy

1.
Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653
2.
Institut für Numerische und Angewandte Mathematik, Universität Münster, D-48149 Münster
3.
Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899

Received: February 28, 2010

Revised: August 31, 2010

Published: March 2011

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Abstract

We consider the problem of minimizing the tumor volume with a priori given amounts of anti-angiogenic and cytotoxic agents. For one underlying mathematical model, optimal and suboptimal solutions are given for four versions of this problem: the case when only anti-angiogenic agents are administered, combination treatment with a cytotoxic agent, and when a standard linear pharmacokinetic equation for the anti-angiogenic agent is added to each of these models. It is shown that the solutions to the more complex models naturally build upon the simplified versions. This gives credence to a modeling approach that starts with the analysis of simplified models and then adds increasingly more complex and medically relevant features. Furthermore, for each of the problem formulations considered here, there exist excellent simple piecewise constant controls with a small number of switchings that virtually replicate the optimal values for the objective.

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Mathematics Subject Classification: Primary: 92C50; Secondary: 49K15, 49M99.

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