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Mathematical Model of Pulsed Immunotherapy for Superficial Bladder Cancer

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Abstract

We present a theoretical study of superficial bladder cancer growth and its treatment via pulsed immunotherapy with Bacillus Calmette–Guérin (BCG), an attenuated strain of Mycobacterium bovis. BCG pulsed immunotherapy is a clinically established procedure for the treatment of superficial bladder cancer. In this paper, periodic BCG instillations are modeled using impulsive differential equations, which are studied using a combination of analytical and numerical techniques. In this way, we determine critical threshold values of the BCG instillation dose and rate of pulsing for successful treatment. We also identify treatment regimes in which tumor destruction occurs, but undesirable side effects are maintained at low levels by the immune system.

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References

  • Alexandroff, A.B., Jackson, A.M., O’Donnell, M.A., James, K., 1999. BCG immunotherapy of bladder cancer: 20 years on. Lancet 353, 1689.

    Article  Google Scholar 

  • Archuleta, J., Mullens, P., Primm, T.P., 2002. The relationship of temperature to desiccation and starvation tolerance of the Mycobacterium avium complex. Arch. Microbiol. 178, 311–14.

    Article  Google Scholar 

  • Bainov, D., 1993. Impulsive Differential Equations. Longman, Harlow.

    MATH  Google Scholar 

  • Bevers, R.F.M., Kurth, K.H., Schamhart, D.H.J., 2004. Role of urothelial cells in BCG immunotherapy for superficial bladder cancer. Br. J. Cancer 91, 607–12.

    Google Scholar 

  • Bohle, A., Brandau, S., 2003. Immune mechanisms in bacillus Calmette–Guérin immunotherapy for superficial bladder cancer. J. Urol. 170, 964969.

    Article  Google Scholar 

  • Bunimovich-Mendrazitsky, S., Shochat, E., Stone, L., 2007. Mathematical model of BCG immunotherapy in superficial bladder cancer. Bull. Math. Biol. Online: http://dx.doi.org/10.1007/s11538-007-9195-z

  • Cheng, C.W., Ng, M.T., Chan, S.Y., Sun, W.H., 2004. Low dose BCG as adjuvant therapy for superficial bladder cancer and literature review. ANz J. Surg. 74(7), 569–72.

    Article  Google Scholar 

  • Chopin, D., Gattegno, B., 2002. Superficial bladder tumors. Eur. Urol. 42, 533–1.

    Article  Google Scholar 

  • De Boer, E.C., Bevers, R.F., Kurth, K.H., Schamhart, D.H., 1996. Double fluorescent flow cytometric assessment of bacterial internalization and binding by epithelial cells. Cytometry 25, 381–87.

    Article  Google Scholar 

  • De Pillis, L.G., Radunskaya, A.E., Wiseman, C.L., 2005. A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res. 65(17), 7950–958.

    Google Scholar 

  • De Pillis, L.G., Gu, W., Radunskaya, A.E., 2006. Mixed immunotherapy and chemotherapy of tumors: modeling, applications and biological interpretations. J. Teor. Biol. 238, 841–62.

    Article  Google Scholar 

  • Durek, C., Brandau, S., Ulmer, A.J., Flad, H.D., Jocham, D., Bohle, A., 1999. Bacillus Calmette–Guérin (BCG) and 3D tumors: an in vitro model for the study of adhesion and invasion. J. Urol. 162, 600–05.

    Article  Google Scholar 

  • Glendinning, P., 1994. Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations. Cambridge University Press, Cambridge.

    MATH  Google Scholar 

  • Guckenheimer, J., Holmes, P., 1983. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields. Springer, New York.

    MATH  Google Scholar 

  • Hartman, P., 1964. Ordinary Differential Equations. Wiley, New York.

    MATH  Google Scholar 

  • Iooss, G., Joseph, D., 1980. Elementary Stability and Bifurcation Theory. Springer, New York.

    MATH  Google Scholar 

  • Kim, J.C., Steinberg, G.D., 2001. The limits of bacillus Calmette–Guerin for carcinoma in situ of the bladder. J. Urol. 165(3), 745–56.

    Google Scholar 

  • Kirschner, D., Panetta, J., 1998. Modelling immunotherapy of the tumor-immune interaction. J. Math. Biol. 37(3), 235–52.

    Article  MATH  Google Scholar 

  • Kuznetsov, V.A., Makalkin, I.A., Taylor, M.A., Perelson, A.S., 1994. Nonlinear dynamics of immunogenic tumours: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 56, 295–21.

    MATH  Google Scholar 

  • Lakmeche, A., Arino, O., 2001. Nonlinear mathematical model of pulsed therapy of heterogeneous tumors. Nonlinear Anal. R. World Appl. 2, 455–65.

    Article  MATH  MathSciNet  Google Scholar 

  • Lakshmikantham, V., Bainov, D.D., Simeonov, P.S., 1989. Theory of Impulsive Differential Equations. World Publishers, Singapore.

    MATH  Google Scholar 

  • Lämmle, M., Beer, A., Settles, M., Hanning, C., Schwaibold, H., Drews, C., 2002. Reliability of MR imaging-based virtual cystoscopy in the diagnosis of cancer of the urinary bladder. Am. J. Roentgenol. 178, 1483–488.

    Google Scholar 

  • Mackey, M., Glass, L., 1977. Oscillation and chaos in physiological control systems. Science 197, 287–89.

    Article  Google Scholar 

  • Meyer, J.P., Persad, R., Gillatt, D.A., 2002. Use of bacille Calmette–Guérin in superficial bladder cancer. Postgrad. Med. J. 78, 449–54.

    Article  Google Scholar 

  • Murray, J.D., 1993. Mathematical Biology, 2nd edn. Springer, London.

    MATH  Google Scholar 

  • Nseyo, U.O., Lamm, D.L., 1997. Immunotherapy of Bladder Cancer. Sem. Surg. Oncol. 13, 342–49.

    Article  Google Scholar 

  • Ojea, A., Nogueira, J.L., Solsona, E., et al., 2007. A multicentre, randomized prospective trial comparing three intravesical adjuvant therapies for intermediate-risk superficial bladder cancer: low-dose bacillus Calmette–Guérin (27 mg) versus very low-dose bacillus Calmette–Guérin (13.5 mg) versus mitomycin C. Eur. Urol. 52, 1398–06.

    Article  Google Scholar 

  • Panetta, J., 1996. A mathematical model of periodically pulse chemotherapy: tumor recurrence and metastasis in a competition environment. Bull. Math. Biol. 58, 425–47.

    Article  MATH  Google Scholar 

  • Patard, J.J., Saint, F., Velotti, F., Abbou, C.C., Chopin, D.K., 1998. Immune response following intravesical bacillus Calmette–Guérin instillations in superficial bladder cancer: a review. Urol. Res. 26(3), 155–59.

    Article  Google Scholar 

  • Schenk-Braat, E.A.M., Bangma, C.H., 2005. Immunotherapy for superficial bladder cancer. Cancer Immunol. Immunother. 54(5), 414–23.

    Article  Google Scholar 

  • Shulgin, B., Stone, L., Agur, Z., 1998. Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60, 1–6.

    Article  Google Scholar 

  • Stone, L., Shulgin, B., Agur, Z., 2000. Theoretical examination of the pulse vaccination policy in the SIR epidemic model. Math. Comput. Mod. 31(4–), 207–15.

    Article  MATH  MathSciNet  Google Scholar 

  • Schenkman, E., Lamm, D.L., 2004. Superficial bladder cancer therapy. Sci. World 4, 387–99.

    Google Scholar 

  • Shochat, E., Hart, D., Agur, Z., 1999. Using computer simulations for evaluating the efficacy of breast cancer chemotherapy protocols. Math. Models Methods Appl. Sci. 9(4), 599–15.

    Article  MATH  Google Scholar 

  • Swanson, K.R., Bridge, C., Murray, J.D., Alvord, E.C., 2003. Virtual and real brain tumors:using mathematical modeling to quantify glioma growth and invasion. J. Neurol. Sci. 216, 1–0.

    Article  Google Scholar 

  • Zlotta, A.R., van Vooren, J.P., Huygen, K., et al., 2000. What is the optimal regimen for BCG intravesical therapy? Eur. Urol. 37, 470.

    Article  Google Scholar 

  • Zwillinger, D., 1989. Handbook of Differential Equations. Academic Press, New York.

    MATH  Google Scholar 

  • Wigginton, J., Kirschner, D., 2001. A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis. J. Immunol. 166, 1951–967.

    Google Scholar 

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Authors and Affiliations

  1. Biomathematics Unit, Department of Zoology, Faculty of Life Science, Tel-Aviv University, Tel-Aviv, 69978, Israel

    Svetlana Bunimovich-Mendrazitsky & Lewi Stone

  2. Centre for Mathematical Medicine and Biology, School of Mathematical Sciences, Nottingham, NG7 2RD, UK

    Helen Byrne

Authors
  1. Svetlana Bunimovich-Mendrazitsky
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  2. Helen Byrne
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  3. Lewi Stone
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Correspondence to Svetlana Bunimovich-Mendrazitsky.

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Bunimovich-Mendrazitsky, S., Byrne, H. & Stone, L. Mathematical Model of Pulsed Immunotherapy for Superficial Bladder Cancer. Bull. Math. Biol. 70, 2055–2076 (2008). https://doi.org/10.1007/s11538-008-9344-z

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  • DOI: https://doi.org/10.1007/s11538-008-9344-z

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