A 3-Compartment Model for Chemotherapy of Heterogeneous Tumor Populations
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A 3-Compartment Model for Chemotherapy of Heterogeneous Tumor Populations Urszula Ledzewicz · Kenneth Bratton · Heinz Schättler
Received: 5 October 2013 / Accepted: 7 April 2014 © Springer Science+Business Media Dordrecht 2014
Abstract We consider a mathematical model for cancer chemotherapy with a single agent that distinguishes three levels of sensitivities calling the subpopulations ‘sensitive’, ‘partially sensitive’ and ‘resistant’. We analyze the dynamic properties of the system under what could be considered metronomic (continuous, low-dose, constant) chemotherapy and, more generally, also consider the optimal control problem of minimizing the tumor burden over a prescribed therapy interval. Interestingly, when several levels of chemotherapeutic sensitivities are taken into account in the model, lower time-varying dose rates as they are given by singular controls become a treatment option. This is only the case once a significant residuum of resistant cells has been created in a simpler 2-compartment model that only considers sensitive and resistant cells. For heterogeneous tumor populations, a more modulated approach that varies the dose rates of the drugs may be more beneficial than the classical maximum tolerated dose approach pursued in medical practice. Keywords Optimal control · Singular control · Metronomic chemotherapy
This material is based upon work supported by the National Science Foundation under collaborative research Grants Nos. DMS 1311729/1311733. 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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U. Ledzewicz ( ) · K. Bratton Dept. of Mathematics and Statistics, Southern Illinois University Edwardsville, Edwardsville, IL 62026-1653, USA e-mail: [email protected] K. Bratton e-mail: [email protected] H. Schättler Dept. of Electrical and Systems Engineering, Washington University, St. Louis, MO 63130-4899, USA e-mail: [email protected]
U. Ledzewicz et al.
1 Introduction: Chemotherapy for Heterogeneous Tumor Populations The prevailing paradigm in cancer chemotherapy is to give as much of the drug as possible (MTD-maximum tolerated dose) immediately. The reason is that cancer is a widely symptomless disease which, once finally detected, often is in an advanced stage where immediate action is required. Then the aim simply is to be as toxic as possible to the cancerous cells. If the tumor consists of a homogeneous agglomeration of chemotherapeutically sensitive cells, simple mathematical models confirm such a strategy as optimal (e.g., see [5, 16, 18, 27–29]). However, malignant cancer cell populations are genetically unstable and coupled with fast proliferation rates, this leads to a great variety in the structure of the cells within one tumor— the number of genetic errors present within one cancer cell can lie in the thousands [19]. Consequently, tumors often consist of heterogeneous agglomerations of subpopulations that show widely varying sensitivities t
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