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BCG and IL − 2 model for bladder cancer treatment with fast and slow dynamics based on SPVF method—stability analysis

1 Department of Mathematics, Jerusalem College of Technology (JCT)
2 Department of Mathematics, Ben-Gurion University, Azrieli College of Engineering
3 Department of Bioinformatics, Jerusalem College of Technology (JCT)
4 Department of Computer Science, Jerusalem College of Technology (JCT)
5 Department of Mathematics, Ariel University

In this study, we apply the method of singularly perturbed vector field (SPVF) and its application to the problem of bladder cancer treatment that takes into account the combination of Bacillus CalmetteGurin vaccine (BCG) and interleukin (IL)-2 immunotherapy (IL − 2). The model is presented with a hidden hierarchy of time scale of the dynamical variables of the system. By applying the SPVF, we transform the model to SPS (Singular Perturbed System) form with explicit hierarchy, i.e., slow and fast sub-systems. The decomposition of the model to fast and slow subsystems, first of all, reduces significantly the time computer calculations as well as the long and complex algebraic expressions when investigating the full model. In addition, this decomposition allows us to explore only the fast subsystem without losing important biological/ mathematical information of the original system.The main results of the paper were that we obtained explicit expressions of the equilibrium points of the model and investigated the stability of these points.
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Keywords mathematical modeling; therapy schedule; impulse differential equations; dirac deltafunction; gamma distribution function; BCG and IL-2 combined therapy

Citation: OPhir Nave, Shlomo Hareli, Miriam Elbaz, Itzhak Hayim Iluz, Svetlana Bunimovich-Mendrazitsky. BCG and IL − 2 model for bladder cancer treatment with fast and slow dynamics based on SPVF method—stability analysis. Mathematical Biosciences and Engineering, 2019, 16(5): 5346-5379. doi: 10.3934/mbe.2019267


  • 1. V. M. Gol'dshtein and V. A. Sobolev, Singularity theory and some problems of functional analysis, Amer. Math. Soc., 1 (1992), 73–92.
  • 2. V. I. Babushok and V. M. Gol'dshtein, Structure of the thermal explosion limit, Combust. Flame, 72 (1988), 221–226.
  • 3. A. C. McIntosh, V. M. Gol'dshtein, I. Goldfarb, et al., Thermal explosion in a combustible gas containing fuel droplets, Combust. Th. Mod., 2 (1998), 153–165.
  • 4. I. Goldfarb, V. M. Gol'dshtein, D. Katz, et al., Radiation effect on thermal explosion in a gas containing evaporating fuel droplets, Int. J. Ther. Sci., 46 (2007), 358–370.
  • 5. M. R. Roussel and S. J. Fraser, Geometry of the steady-state approximation: perturbation and accelerated convergence methods, J. Chem. Phys., 93 (1990), 1072–1081.
  • 6. M. R. Roussel and S. J. Fraser, Accurate steady-state approximations: implications for kinetics experiments and mechanism, J. Chem. Phys., 95 (1991), 8762–8770.
  • 7. M. R. Roussel and S. J. Fraser, Global analysis of enzyme inhibition kinetics, textitJ. Chem. Phys., 97 (1993), 8316–8327.
  • 8. M. R. Roussel and S. J. Fraser, Invariant manifold methods for metabolic model reduction, Chaos,196 (2001), 196–206.
  • 9. A. Zagaris, H. G. Kaper and T. J. Kaper, Analysis of the computational singular perturbation reduction method for chemical kinetics, J. Non. Sci., 14 (2004), 59–91.
  • 10. A. Zagaris, H. G. Kaper and T. J. Kaper, Fast and slow dynamics for the computational singular perturbation method, Soc. Indust. App. Math., 2 (2004), 613–638.
  • 11. N. Berglunda and B. Gentzd, Geometric singular perturbation theory for stochastic differential equations, J. Diff. Eq., 191 (2003), 1–54.
  • 12. N. Fenichel, Geometric singular perturbation theory for ordinary differential equations, J. Diff. Eq., 31 (1979), 53–98.
  • 13. C. K. Jones, Geometric singular perturbation theory, 1609 of the series, Lec. Notes Math., Dyn. Syst. (2006), 44-118.
  • 14. U. Maas and S. B. Pope, Implementation of simplified chemical kinetics based on intrinsic low-dimensional manifolds (PDF), Symposium (International) on Combustion, Twenty-Fourth Sym-posium on Combustion, (1992), 103–112.
  • 15. U. Maas and S. B. Pope, Simplifying chemical kinetics: Intrinsic low-dimensional manifolds in composition space, Combust. Flame, 88 (1992), 239–264.
  • 16. H. Bongers, J. A. Van Oijen and L. P. H. De Goey, Intrinsic low-dimensional manifold method extended with diffusion, Proc. Combust. Inst., 291 (2002), 1371–1378.
  • 17. S. T. Alison, L. Whitehouse and L. Richard, The Estimation of Intrinsic Low Dimensional Man-ifold Dimension in Atmospheric Chemical Reaction Systems, Air Poll. Modell Simul., (2002), 245–263.
  • 18. G. K. Hans and J. K. Tasso, Asymptotic analysis of two reduction methods for systems of chemical reactions, Phys. D, 165 (2002), 66–93.
  • 19. V. Bykov, I. Goldfarb and V. Gol'dshtein, Singularly perturbed vector fields, J. Phys. Conf. Ser., 55 (2006), 28–44.
  • 20. O. Nave, Singularly perturbed vector field method (SPVF) applied to combustion of monodisperse fuel spray, Diff. Eqs. Dyn. Syst., 27 (2018), 1–18.
  • 21. M. Al-Tameemi, M. Chaplain and A. d'Onofrio, Evasion of tumours from the control of the im-mune system: consequences of brief encounters, Biol. Direct, 7 (2012), 1–22.
  • 22. I. Kareva, F. Berezovskaya and C. Castillo-Chavez, Myeloid cells in tumour-immune interactions, J. Biol. Dyn., 4 (2010), 315–327.
  • 23. L. Derbel, Analysis of a new model for tumor-immune system competition including long time scale effects, Math. Model Methods Appl. Sci., 14 (2004), 1657–1681.
  • 24. K. E. Starkov and S. Bunimovich-Mendrazitsky, Dynamical properties and tumor clearance con-ditions for a nine-dimensional model of bladder cancer immunotherapy, Math. Biosci. Eng., 13 (2016), 1059–1075.
  • 25. S. Bunimovich-Mendrazitsky, S. Halachmi and N. Kronik, Improving Bacillus Calmette-Guerin (BCG) immunotherapy for bladder cancer by adding interleukin 2 (IL−2): A mathematical model, Math. Med. Biol., 33 (2015), 1–30.
  • 26. S. Brandau and H. Suttmann, Thirty years of BCG immunotherapy for non-muscle invasive blad-der cancer: a success story with room for improvement, Biomed. Pharmacother., 61 (2007), 299–305.
  • 27. A. N. Tikhonov, Systems of differential equations containing small parameters multiplying the derivatives. Mat. Sborn., 31 (1952), 575–586.
  • 28. R. Mahendran, Bacillus Calmette-Guerin immunotherapy-increasing dose as a means of improv-ing therapy?, Trans. Cancer Res., 6 (2017), 168–173.
  • 29. A. Kiselyov, S. Bunimovich-Mendrazitsky and V. Startsev, Treatment of non-muscle invasive blad-der cancer with Bacillus Calmette-Guerin (BCG): biological markers and simulation studies, BBA Clin., 10 (2015), 27–34.
  • 30. C. Pettenati and M. A.Ingersoll, Mechanisms of BCG immunotherapy and its outlook for bladder cancer, Nat. Rev. Urol., 15 (2018), 615–625.
  • 31. A. H. Kitamur and T. Tsukamoto, Immunotherapy for urothelial carcinoma, Current status and perspectives, Cancers, 29 (2011), 3055–3072.
  • 32. C. Yee, J. A. Thompson, D. Byrd, et al., Adoptive T cell therapy using antigen-specific CD8+ T cell clones for the treatment of patients with metastatic melanoma: in vivo persistence, migration, and antitumor effect of transferred T cells, Proc. Natl. Acad. Sci. U S A, 10 (2002), 16168–16173.
  • 33. N. Kronik, Y. Kogan, P. G. Schlegel, et al., Improving T-cell immunotherapy for melanoma through a mathematically motivated strategy: efficacy in numbers?, J. Immunother., 35 (2012), 116–124.


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