Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost

Urszula Ledzewicz; Mozhdeh Sadat Faraji Mosalman; Heinz Schättler

doi:10.3934/dcdsb.2013.18.1031

Article Contents

2013, Volume 18, Issue 4: 1031-1051. Doi: 10.3934/dcdsb.2013.18.1031 open access

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Optimal controls for a mathematical model of tumor-immune interactions under targeted chemotherapy with immune boost

1.
Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, Illinois, 62026-1653
2.
Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Missouri, 63130-4899

Received: November 30, 2011

Revised: March 31, 2012

Published: May 2013

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Abstract

An optimal control problem for combination of cancer chemotherapy with immunotherapy in form of a boost to the immune system is considered as a multi-input optimal control problem. The objective to be minimized is chosen as a weighted average of (i) the number of cancer cells at the terminal time, (ii) a measure for the immunocompetent cell densities at the terminal point (included as a negative term), the overall amounts of (iii) cytotoxic agents and (iv) immune boost given as a measure for the side effects of treatment and (v) a small penalty on the free terminal time that limits the overall therapy horizon. This last term is essential in obtaining a mathematically well-posed problem formulation. Both analytical and numerical results about the structures of optimal controls will be presented that give some insights into the structure of optimal protocols, i.e., the dose rates and sequencing of drugs in these combination treatments.

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Mathematics Subject Classification: Primary: 49K15, 92B05; Secondary: 93C95.

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