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Optimal Control for a Mathematical Model of Glioma Treatment with Oncolytic Therapy and TNF-\(\alpha \) Inhibitors

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Abstract

A mathematical model for combination therapy of glioma with oncolytic therapy and TNF-\(\alpha \) inhibitors is analyzed as an optimal control problem. In the objective, a weighted average between the tumor volume and the total amount of viruses given is minimized. It is shown that optimal controls representing the virus administration are generically of the bang-bang type, i.e., the virus should be applied at maximal allowed dose with possible rest periods. On the other hand, optimal controls representing the dosage of TNF-\(\alpha \) inhibitors follow a continuous regimen of concatenations between pieces that lie on the boundary and in the interior of the control set.

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Acknowledgements

U. Ledzewicz’s research was partially supported by the National Science Foundation under collaborative research Grants No. DMS 1311733. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

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Authors and Affiliations

  1. Institute of Mathematics, Lodz University of Technology, 90-924, Lodz, Poland

    Elzbieta Ratajczyk & Urszula Ledzewicz

  2. Department of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL, 62026-1653, USA

    Urszula Ledzewicz

  3. Department of Electrical and Systems Engineering, Washington University, St. Louis, MO, 63130-4899, USA

    Heinz Schättler

Authors
  1. Elzbieta Ratajczyk
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  2. Urszula Ledzewicz
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  3. Heinz Schättler
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Correspondence to Urszula Ledzewicz.

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Alberto d’Onofrio.

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Ratajczyk, E., Ledzewicz, U. & Schättler, H. Optimal Control for a Mathematical Model of Glioma Treatment with Oncolytic Therapy and TNF-\(\alpha \) Inhibitors. J Optim Theory Appl 176, 456–477 (2018). https://doi.org/10.1007/s10957-018-1218-4

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  • DOI: https://doi.org/10.1007/s10957-018-1218-4

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