
Mathematical Biosciences and Engineering, 2019, 16(5): 53465379. doi: 10.3934/mbe.2019267
Research article
Export file:
Format
 RIS(for EndNote,Reference Manager,ProCite)
 BibTex
 Text
Content
 Citation Only
 Citation and Abstract
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, BenGurion 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
Received: , Accepted: , Published:
References
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 steadystate approximation: perturbation and accelerated convergence methods, J. Chem. Phys., 93 (1990), 1072–1081.
6. M. R. Roussel and S. J. Fraser, Accurate steadystate 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), 44118.
14. U. Maas and S. B. Pope, Implementation of simplified chemical kinetics based on intrinsic lowdimensional manifolds (PDF), Symposium (International) on Combustion, TwentyFourth Symposium on Combustion, (1992), 103–112.
15. U. Maas and S. B. Pope, Simplifying chemical kinetics: Intrinsic lowdimensional manifolds in composition space, Combust. Flame, 88 (1992), 239–264.
16. H. Bongers, J. A. Van Oijen and L. P. H. De Goey, Intrinsic lowdimensional 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 Manifold 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. AlTameemi, M. Chaplain and A. d'Onofrio, Evasion of tumours from the control of the immune system: consequences of brief encounters, Biol. Direct, 7 (2012), 1–22.
22. I. Kareva, F. Berezovskaya and C. CastilloChavez, Myeloid cells in tumourimmune interactions, J. Biol. Dyn., 4 (2010), 315–327.
23. L. Derbel, Analysis of a new model for tumorimmune system competition including long time scale effects, Math. Model Methods Appl. Sci., 14 (2004), 1657–1681.
24. K. E. Starkov and S. BunimovichMendrazitsky, Dynamical properties and tumor clearance conditions for a ninedimensional model of bladder cancer immunotherapy, Math. Biosci. Eng., 13 (2016), 1059–1075.
25. S. BunimovichMendrazitsky, S. Halachmi and N. Kronik, Improving Bacillus CalmetteGuerin (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 nonmuscle invasive bladder 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 CalmetteGuerin immunotherapyincreasing dose as a means of improving therapy?, Trans. Cancer Res., 6 (2017), 168–173.
29. A. Kiselyov, S. BunimovichMendrazitsky and V. Startsev, Treatment of nonmuscle invasive bladder cancer with Bacillus CalmetteGuerin (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 antigenspecific 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 Tcell immunotherapy for melanoma through a mathematically motivated strategy: efficacy in numbers?, J. Immunother., 35 (2012), 116–124.
© 2019 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution Licese (http://creativecommons.org/licenses/by/4.0)