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Mathematical Biosciences and Engineering

2015, Volume 12, Issue 6: 1257-1275. doi: 10.3934/mbe.2015.12.1257
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Dynamics and control of a mathematical model for metronomic chemotherapy

  • 1.
    Dept. of Mathematics and Statistics, Southern Illinois University, Edwardsville, Il 62025
  • 2.
    Dept. of Electrical and Systems Engineering, Washington University, St. Louis, Mo 63130
  • Received: 01 November 2014 Accepted: 29 June 2018 Published: 01 August 2015

  • MSC : Primary: 92C50; Secondary: 37N25.

  • A $3$-compartment model for metronomic chemotherapy that takes intoaccount cancerous cells, the tumor vasculature and tumorimmune-system interactions is considered as an optimal controlproblem. Metronomic chemo-therapy is the regular, almost continuousadministration of chemotherapeutic agents at low dose, possibly withsmall interruptions to increase the efficacy of the drugs. Thereexists medical evidence that such administrations of specificcytotoxic agents (e.g., cyclophosphamide) have both antiangiogenicand immune stimulatory effects. A mathematical model for angiogenicsignaling formulated by Hahnfeldt et al. is combined with theclassical equations for tumor immune system interactions byStepanova to form a minimally parameterized model to capture theseeffects of low dose chemotherapy. The model exhibits bistablebehavior with the existence of both benign and malignant locallyasymptotically stable equilibrium points. In this paper, thetransfer of states from the malignant into the benign regions isused as a motivation for the construction of an objective functionalthat induces this process and the analysis of the correspondingoptimal control problem is initiated.

    Citation: Urszula Ledzewicz, Behrooz Amini, Heinz Schättler. Dynamics and control of a mathematical model for metronomic chemotherapy[J]. Mathematical Biosciences and Engineering, 2015, 12(6): 1257-1275. doi: 10.3934/mbe.2015.12.1257

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  • A $3$-compartment model for metronomic chemotherapy that takes intoaccount cancerous cells, the tumor vasculature and tumorimmune-system interactions is considered as an optimal controlproblem. Metronomic chemo-therapy is the regular, almost continuousadministration of chemotherapeutic agents at low dose, possibly withsmall interruptions to increase the efficacy of the drugs. Thereexists medical evidence that such administrations of specificcytotoxic agents (e.g., cyclophosphamide) have both antiangiogenicand immune stimulatory effects. A mathematical model for angiogenicsignaling formulated by Hahnfeldt et al. is combined with theclassical equations for tumor immune system interactions byStepanova to form a minimally parameterized model to capture theseeffects of low dose chemotherapy. The model exhibits bistablebehavior with the existence of both benign and malignant locallyasymptotically stable equilibrium points. In this paper, thetransfer of states from the malignant into the benign regions isused as a motivation for the construction of an objective functionalthat induces this process and the analysis of the correspondingoptimal control problem is initiated.


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