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Mathematical Biosciences and Engineering

2005, Volume 2, Issue 3: 657-670. doi: 10.3934/mbe.2005.2.657
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Analysis and Optimization of Drug Resistant an Phase-Specific Cancer

  • 1.
    Institute of Automatic Control, Silesian University of Technology, Akademicka 16, 44-101 Gliwice
  • Received: 01 January 2005 Accepted: 29 June 2018 Published: 01 August 2005

  • MSC : 92D25, 93C23, 92C50.

  • This paper presents analysis and biomedical implications of a certain class of bilinear systems that can be applied in modeling of cancer chemotherapy. It combines models that so far have been studied separately, taking into account both the phenomenon of gene amplification and drug specificity in chemotherapy in their different aspects. The methodology of analysis of such models, based on system decomposition, is discussed. The mathematical description is given by an infinite dimensional state equation with a system matrix, the form of which allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the second one is infinite-dimensional and tridiagonal. Then the optimal control problem is defined in l1 space. To derive necessary conditions for optimal control, the model description is transformed into an integrodifferential one.

    Citation: Andrzej Swierniak, Jaroslaw Smieja. Analysis and Optimization of Drug Resistant an Phase-Specific Cancer[J]. Mathematical Biosciences and Engineering, 2005, 2(3): 657-670. doi: 10.3934/mbe.2005.2.657

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  • This paper presents analysis and biomedical implications of a certain class of bilinear systems that can be applied in modeling of cancer chemotherapy. It combines models that so far have been studied separately, taking into account both the phenomenon of gene amplification and drug specificity in chemotherapy in their different aspects. The methodology of analysis of such models, based on system decomposition, is discussed. The mathematical description is given by an infinite dimensional state equation with a system matrix, the form of which allows decomposing the model into two interacting subsystems. While the first one, of a finite dimension, can have any form, the second one is infinite-dimensional and tridiagonal. Then the optimal control problem is defined in l1 space. To derive necessary conditions for optimal control, the model description is transformed into an integrodifferential one.


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    1. Hode Sbeity, Rafic Younes, Suat Topsu, Imad Mougharbel, 2011, Comparative study of the optimization theory for cancer treatment, 978-1-4244-9352-4, 927, 10.1109/BMEI.2011.6098370
    2. Andrzej Świerniak, Marek Kimmel, Jaroslaw Smieja, Krzysztof Puszynski, Krzysztof Psiuk-Maksymowicz, 2016, Chapter 2, 978-3-319-28093-6, 9, 10.1007/978-3-319-28095-0_2
    3. Mst. Shanta Khatun, Md. Haider Ali Biswas, Modeling the effect of adoptive T cell therapy for the treatment of leukemia, 2020, 2, 2577-7408, 10.1002/cmm4.1069
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    5. Andrzej Swierniak, 2012, Three levels of objects of control in anticancer therapy, 978-1-4673-2124-2, 421, 10.1109/MMAR.2012.6347852
    6. Orit Lavi, Michael M. Gottesman, Doron Levy, The dynamics of drug resistance: A mathematical perspective, 2012, 15, 13687646, 90, 10.1016/j.drup.2012.01.003
    7. Chen-Hsiang Yeang, Robert A. Beckman, Long range personalized cancer treatment strategies incorporating evolutionary dynamics, 2016, 11, 1745-6150, 10.1186/s13062-016-0153-2
    8. Mst. Shanta Khatun, Md. Haider Ali Biswas, Mathematical analysis and optimal control applied to the treatment of leukemia, 2020, 64, 1598-5865, 331, 10.1007/s12190-020-01357-0
    9. Jessica J. Cunningham, Joel S. Brown, Robert A. Gatenby, Kateřina Staňková, Optimal control to develop therapeutic strategies for metastatic castrate resistant prostate cancer, 2018, 459, 00225193, 67, 10.1016/j.jtbi.2018.09.022
    10. Tim Cardilin, Torbjörn Lundh, Mats Jirstrand, Optimization of additive chemotherapy combinations for an in vitro cell cycle model with constant drug exposures, 2021, 00255564, 108595, 10.1016/j.mbs.2021.108595
    11. Marzena Dolbniak, Jaroslaw Smieja, Andrzej Swierniak, 2018, Structural Sensitivity of Control Models Arising in Combined Chemo-Radiotherapy, 978-1-5386-4325-9, 339, 10.1109/MMAR.2018.8486088
    12. Andrzej Świerniak, 2007, Control problems arising in cancer chemotherapy under evolving drug resistance - finite versus infinite dimensional models, 978-3-9524173-8-6, 3103, 10.23919/ECC.2007.7068245
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  • © 2005 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)
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