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Mathematical Biosciences and Engineering

2011, Volume 8, Issue 2: 529-547. doi: 10.3934/mbe.2011.8.529
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Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer

  • 1.
    Ariel University Centerof of Samaria, Mathematics Department, Ariel
  • Received: 01 May 2010 Accepted: 29 June 2018 Published: 01 April 2011

  • MSC : Primary: 58F15, 58F17; Secondary: 53C35.

  • Understanding the dynamics of human hosts and tumors is of critical importance. A mathematical model was developed by Bunimovich-Mendrazitsky et al. ([10]), who explored the immune response in bladder cancer as an effect of BCG treatment. This treatment exploits the host's own immune system to boost a response that will enable the host to rid itself of the tumor. Although this model was extensively studied using numerical simulation, no analytical results on global tumor dynamics were originally presented. In this work, we analyze stability in a mathematical model for BCG treatment of bladder cancer based on the use of quasi-normal form and stability theory. These tools are employed in the critical cases, especially when analysis of the linearized system is insufficient. Our goal is to gain a deeper insight into the BCG treatment of bladder cancer, which is based on a mathematical model and biological considerations, and thereby to bring us one step closer to the design of a relevant clinical protocol.

    Citation: Svetlana Bunimovich-Mendrazitsky, Yakov Goltser. Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer[J]. Mathematical Biosciences and Engineering, 2011, 8(2): 529-547. doi: 10.3934/mbe.2011.8.529

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  • Understanding the dynamics of human hosts and tumors is of critical importance. A mathematical model was developed by Bunimovich-Mendrazitsky et al. ([10]), who explored the immune response in bladder cancer as an effect of BCG treatment. This treatment exploits the host's own immune system to boost a response that will enable the host to rid itself of the tumor. Although this model was extensively studied using numerical simulation, no analytical results on global tumor dynamics were originally presented. In this work, we analyze stability in a mathematical model for BCG treatment of bladder cancer based on the use of quasi-normal form and stability theory. These tools are employed in the critical cases, especially when analysis of the linearized system is insufficient. Our goal is to gain a deeper insight into the BCG treatment of bladder cancer, which is based on a mathematical model and biological considerations, and thereby to bring us one step closer to the design of a relevant clinical protocol.


  • This article has been cited by:

    1. K. E. Starkov, Svetlana Bunimovich-Mendrazitsky, Dynamical properties and tumor clearance conditions for a nine-dimensional model of bladder cancer immunotherapy, 2016, 13, 1551-0018, 1059, 10.3934/mbe.2016030
    2. Teddy Lazebnik, Niva Aaroni, Svetlana Bunimovich-Mendrazitsky, PDE based geometry model for BCG immunotherapy of bladder cancer, 2021, 200, 03032647, 104319, 10.1016/j.biosystems.2020.104319
    3. Cyrill A. Rentsch, Claire Biot, Joël R. Gsponer, Alexander Bachmann, Matthew L. Albert, Romulus Breban, David J. Klinke, BCG-Mediated Bladder Cancer Immunotherapy: Identifying Determinants of Treatment Response Using a Calibrated Mathematical Model, 2013, 8, 1932-6203, e56327, 10.1371/journal.pone.0056327
    4. Leonid Shaikhet, Svetlana Bunimovich-Mendrazitsky, Stability Analysis of Delayed Immune Response BCG Infection in Bladder Cancer Treatment Model by Stochastic Perturbations, 2018, 2018, 1748-670X, 1, 10.1155/2018/9653873
    5. Irina Volinsky, Salvo Danilo Lombardo, Paz Cheredman, Stability Analysis and Cauchy Matrix of a Mathematical Model of Hepatitis B Virus with Control on Immune System near Neighborhood of Equilibrium Free Point, 2021, 13, 2073-8994, 166, 10.3390/sym13020166
    6. Konstantin E. Starkov, Diana Gamboa, Localization of compact invariant sets and global stability in analysis of one tumor growth model, 2014, 37, 01704214, 2854, 10.1002/mma.3023
    7. Teddy Lazebnik, Cell-Level Spatio-Temporal Model for a Bacillus Calmette–Guérin-Based Immunotherapy Treatment Protocol of Superficial Bladder Cancer, 2022, 11, 2073-4409, 2372, 10.3390/cells11152372
    8. OPhir Nave, Moria Sigron, A mathematical model for cancer treatment based on combination of anti-angiogenic and immune cell therapies, 2022, 16, 25900374, 100330, 10.1016/j.rinam.2022.100330
    9. T. Lazebnik, S. Yanetz, S. Bunimovich-Mendrazitsky, 2021, Chapter 9, 978-981-16-6296-6, 119, 10.1007/978-981-16-6297-3_9
    10. Ophir Nave, A mathematical model for treatment using chemo-immunotherapy, 2022, 8, 24058440, e09288, 10.1016/j.heliyon.2022.e09288
    11. Amit Yaniv-Rosenfeld, Elizaveta Savchenko, Ariel Rosenfeld, Teddy Lazebnik, Scheduling BCG and IL-2 Injections for Bladder Cancer Immunotherapy Treatment, 2023, 11, 2227-7390, 1192, 10.3390/math11051192
    12. Irina Volinsky, Stability Analysis of a Mathematical Model of Hepatitis B Virus with Unbounded Memory Control on the Immune System in the Neighborhood of the Equilibrium Free Point, 2021, 13, 2073-8994, 1437, 10.3390/sym13081437
    13. Elizaveta Savchenko, Ariel Rosenfeld, Svetlana Bunimovich-Mendrazitsky, Mathematical modeling of BCG-based bladder cancer treatment using socio-demographics, 2023, 13, 2045-2322, 10.1038/s41598-023-45581-7
    14. Urszula Ledzewicz, Heinz Schättler, Analysis of a mathematical model for low-grade gliomas under chemotherapy as a dynamical system, 2025, 85, 14681218, 104344, 10.1016/j.nonrwa.2025.104344
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  • © 2011 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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