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

Urszula Ledzewicz; Helmut Maurer; Heinz Schättler; Urszula Ledzewicz; Helmut Maurer; Heinz Schättler

doi:10.3934/mbe.2011.8.307

Mathematical Biosciences and Engineering

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: 01 March 2010 Accepted: 29 June 2018 Published: 01 April 2011
MSC : Primary: 92C50; Secondary: 49K15, 49M99.

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.

Keywords:

Citation: Urszula Ledzewicz, Helmut Maurer, Heinz Schättler. Optimal and suboptimal protocols for a mathematical model for tumor anti-angiogenesis in combination with chemotherapy[J]. Mathematical Biosciences and Engineering, 2011, 8(2): 307-323. doi: 10.3934/mbe.2011.8.307

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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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