Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer

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2011, 8(2): 529-547. doi: 10.3934/mbe.2011.8.529

Use of quasi-normal form to examine stability of tumor-free equilibrium in a mathematical model of bcg treatment of bladder cancer

Svetlana Bunimovich-Mendrazitsky ^1, and Yakov Goltser ^1,

1.	Ariel University Centerof of Samaria, Mathematics Department, Ariel, Israel, Israel

Received May 2010 Revised October 2010 Published April 2011

Abstract
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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.

Keywords: nonlinear dynamics., Normal form, critical case, immune response, zero eigenvalue.

Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.

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. Mathematical Biosciences & Engineering, 2011, 8 (2) : 529-547. doi: 10.3934/mbe.2011.8.529

References:

[1]	A. Algaba, E. Friere and E. Gamero, Characterizing and computing normal forms using Lie transforms: A survey, Dyn. Continuous, Discrete Impulsive Systems, 8 (2001), 449-476.
[2]	V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations," Springer Verlag, New York, 1982.
[3]	J. Archuleta, P. Mullens and T. P. Primm, The relationship of temperature to desiccation and starvation tolerance of the Mycobacterium avium complex, Arch. Microbiol., 178 (2002), 311-314. doi: 10.1007/s00203-002-0455-x.
[4]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, in "Dynamical Systems III, Encyclopaedia Mathematical Science," 2nd ed., Springer-Verlag, New York, 1993.
[5]	R. F. M. Bevers, K. H. Kurth and D. J. H. Schamhart, Role of urothelial cells in BCG immuno-therapy for superficial bladder cancer, Brit. J. Cancer, 91 (2004), 607-612.
[6]	Y. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations," Lecture Notes in Mathematics, Springer Verlag, New York, 702 (1979).
[7]	G. D. Birkhoff, "Dynamical Systems," New York, 1927.
[8]	A. Bohle and S. Brandau, Immune mechanisms in bacillus CalmetteGuerin immunotherapy for superficial bladder cancer, J. Urol., 170 (2003), 96496. doi: 10.1097/01.ju.0000073852.24341.4a.
[9]	A. D. Bruno, "Local Methods in Nonlinear Diff. Equations," Springer Verlag, New York, 1989.
[10]	S. Bunimovich-Mendrazitsky, E. Shochat and L. Stone, Mathematical Model of BCG Immunotherapy in Superficial Bladder Cancer, Bull. Math. Biol., 69 (2007), 1847-1870. doi: 10.1007/s11538-007-9195-z.
[11]	S. Bunimovich-Mendrazitsky, H. M. Byrne and L. Stone, Mathematical Model of Pulsed Immunotherapy for Superficial Bladder Cancer, Bull. Math. Biol., 70 (2008), 2055-276. doi: 10.1007/s11538-008-9344-z.
[12]	C. W. Cheng, M. T. Ng, S. Y. Chan and W. H. Sun, Low dose BCG as adjuvant therapy for superficial bladder cancer and literature review, Anz Journal of Surgery, 74 (2004), 569-572. doi: 10.1111/j.1445-2197.2004.02941.x.
[13]	S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," New York: Springer, 1982.
[14]	S. N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcations of Planar Vector Fields," Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511665639.
[15]	Y. M. Goltser, On the strong stability of resonance systems with parametrical perturbations, Applied Mathematics and Mechanics (PMM), Acad. Sc. USSR, 41 (1977), 251-261.
[16]	Y. M. Goltser, Bifurcation and stability of neutral systems in the neighborhood of third order resonance, Applied Mathematics and Mechanics (PMM), Acad. Sc. USSR, 43 (1979), 429-440.
[17]	Y. M. Goltser, On the extent of proximity of neutral systems to internal resonance, Applied Mathematics and Mechanics (PMM), Acad. Sc. USSR, 50 (1986), 945-954.
[18]	Y. M. Goltser, Some bifurcation problems of stability, Nonlinear Analysis, TMA, 30 (1997), 1461-1467. doi: 10.1016/S0362-546X(97)00044-8.
[19]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Applied Mathematical Sciences, New York: Springer, 1983.
[20]	G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," World Scientific, Singapore, 1992.
[21]	A. Jemal, T. Murray, E. Ward, A. Samuels, R. C. Tiwari, A. Ghafoor, E. J. Feuer and M. J. Thun, Cancer Statistics, CA Cancer. J. Clin., 55 (2005), 10-30. doi: 10.3322/canjclin.55.1.10.
[22]	F. A. Kelley, The stable, center stable, center, center unstable, and unstable manifolds, J. Diff. Eqns, 3 (1967), 546-570. doi: 10.1016/0022-0396(67)90016-2.
[23]	V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumours: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321.
[24]	A. M. Liapunov, "Probléme géné ral de la stabilité du Mouvement," Princeton University Press, Ann. of Math. Stud, 1947.
[25]	I. G. Malkin, "Theory of Stability of Motion," Translated by Atomic Energy Commission, 1952, 92-94.
[26]	J. J Patard, F. Saint, F. Velotti, C. C. Abbou and D. K. Chopin, Immune response following intravesical bacillus Calmette-Guerin instillations in superficial bladder cancer: A review, Urol. Res., 26 (1998), 155-159. doi: 10.1007/s002400050039.
[27]	V. A Pliss, The reduction principle in the theory of the stability motion, Izv.Akad. Nauk SSSR, Ser. Mat. 28 (1964), 1297-1324.
[28]	H. Poincaré, Oeuvres, Paris, 1928.
[29]	E. Shochat, D. Hart and Z. Agur, Using computer simulations for evaluating the efficacy of breast cancer chemotherapy protocols, Math. Models&Methods in Applied Sciences, 9 (1999), 599-615. doi: 10.1142/S0218202599000312.
[30]	K. R. Swanson, C. Bridge, J. D. Murray and E. C. Alvord, Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion, J. Neurol. Sci., 216 (2003), 1-10. doi: 10.1016/j.jns.2003.06.001.
[31]	J. Wigginton and D. Kirschner, A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J. Immunol., 166 (2001), 1951-1967.
[32]	Q. S. Zhang, A. Y. T. Leung and J. E. Cooper, Computation of normal forms for higher dimensional semi-simple systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, 8 (2001), 559-574.

show all references

References:

[1]	A. Algaba, E. Friere and E. Gamero, Characterizing and computing normal forms using Lie transforms: A survey, Dyn. Continuous, Discrete Impulsive Systems, 8 (2001), 449-476.
[2]	V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations," Springer Verlag, New York, 1982.
[3]	J. Archuleta, P. Mullens and T. P. Primm, The relationship of temperature to desiccation and starvation tolerance of the Mycobacterium avium complex, Arch. Microbiol., 178 (2002), 311-314. doi: 10.1007/s00203-002-0455-x.
[4]	V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, in "Dynamical Systems III, Encyclopaedia Mathematical Science," 2nd ed., Springer-Verlag, New York, 1993.
[5]	R. F. M. Bevers, K. H. Kurth and D. J. H. Schamhart, Role of urothelial cells in BCG immuno-therapy for superficial bladder cancer, Brit. J. Cancer, 91 (2004), 607-612.
[6]	Y. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations," Lecture Notes in Mathematics, Springer Verlag, New York, 702 (1979).
[7]	G. D. Birkhoff, "Dynamical Systems," New York, 1927.
[8]	A. Bohle and S. Brandau, Immune mechanisms in bacillus CalmetteGuerin immunotherapy for superficial bladder cancer, J. Urol., 170 (2003), 96496. doi: 10.1097/01.ju.0000073852.24341.4a.
[9]	A. D. Bruno, "Local Methods in Nonlinear Diff. Equations," Springer Verlag, New York, 1989.
[10]	S. Bunimovich-Mendrazitsky, E. Shochat and L. Stone, Mathematical Model of BCG Immunotherapy in Superficial Bladder Cancer, Bull. Math. Biol., 69 (2007), 1847-1870. doi: 10.1007/s11538-007-9195-z.
[11]	S. Bunimovich-Mendrazitsky, H. M. Byrne and L. Stone, Mathematical Model of Pulsed Immunotherapy for Superficial Bladder Cancer, Bull. Math. Biol., 70 (2008), 2055-276. doi: 10.1007/s11538-008-9344-z.
[12]	C. W. Cheng, M. T. Ng, S. Y. Chan and W. H. Sun, Low dose BCG as adjuvant therapy for superficial bladder cancer and literature review, Anz Journal of Surgery, 74 (2004), 569-572. doi: 10.1111/j.1445-2197.2004.02941.x.
[13]	S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," New York: Springer, 1982.
[14]	S. N. Chow, C. Li and D. Wang, "Normal Forms and Bifurcations of Planar Vector Fields," Cambridge University Press, Cambridge, 1994. doi: 10.1017/CBO9780511665639.
[15]	Y. M. Goltser, On the strong stability of resonance systems with parametrical perturbations, Applied Mathematics and Mechanics (PMM), Acad. Sc. USSR, 41 (1977), 251-261.
[16]	Y. M. Goltser, Bifurcation and stability of neutral systems in the neighborhood of third order resonance, Applied Mathematics and Mechanics (PMM), Acad. Sc. USSR, 43 (1979), 429-440.
[17]	Y. M. Goltser, On the extent of proximity of neutral systems to internal resonance, Applied Mathematics and Mechanics (PMM), Acad. Sc. USSR, 50 (1986), 945-954.
[18]	Y. M. Goltser, Some bifurcation problems of stability, Nonlinear Analysis, TMA, 30 (1997), 1461-1467. doi: 10.1016/S0362-546X(97)00044-8.
[19]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Applied Mathematical Sciences, New York: Springer, 1983.
[20]	G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," World Scientific, Singapore, 1992.
[21]	A. Jemal, T. Murray, E. Ward, A. Samuels, R. C. Tiwari, A. Ghafoor, E. J. Feuer and M. J. Thun, Cancer Statistics, CA Cancer. J. Clin., 55 (2005), 10-30. doi: 10.3322/canjclin.55.1.10.
[22]	F. A. Kelley, The stable, center stable, center, center unstable, and unstable manifolds, J. Diff. Eqns, 3 (1967), 546-570. doi: 10.1016/0022-0396(67)90016-2.
[23]	V. A. Kuznetsov, I. A. Makalkin, M. A. Taylor and A. S. Perelson, Nonlinear dynamics of immunogenic tumours: Parameter estimation and global bifurcation analysis, Bull. Math. Biol., 56 (1994), 295-321.
[24]	A. M. Liapunov, "Probléme géné ral de la stabilité du Mouvement," Princeton University Press, Ann. of Math. Stud, 1947.
[25]	I. G. Malkin, "Theory of Stability of Motion," Translated by Atomic Energy Commission, 1952, 92-94.
[26]	J. J Patard, F. Saint, F. Velotti, C. C. Abbou and D. K. Chopin, Immune response following intravesical bacillus Calmette-Guerin instillations in superficial bladder cancer: A review, Urol. Res., 26 (1998), 155-159. doi: 10.1007/s002400050039.
[27]	V. A Pliss, The reduction principle in the theory of the stability motion, Izv.Akad. Nauk SSSR, Ser. Mat. 28 (1964), 1297-1324.
[28]	H. Poincaré, Oeuvres, Paris, 1928.
[29]	E. Shochat, D. Hart and Z. Agur, Using computer simulations for evaluating the efficacy of breast cancer chemotherapy protocols, Math. Models&Methods in Applied Sciences, 9 (1999), 599-615. doi: 10.1142/S0218202599000312.
[30]	K. R. Swanson, C. Bridge, J. D. Murray and E. C. Alvord, Virtual and real brain tumors: Using mathematical modeling to quantify glioma growth and invasion, J. Neurol. Sci., 216 (2003), 1-10. doi: 10.1016/j.jns.2003.06.001.
[31]	J. Wigginton and D. Kirschner, A model to predict cell-mediated immune regulatory mechanisms during human infection with Mycobacterium tuberculosis, J. Immunol., 166 (2001), 1951-1967.
[32]	Q. S. Zhang, A. Y. T. Leung and J. E. Cooper, Computation of normal forms for higher dimensional semi-simple systems, Dynamics of Continuous, Discrete and Impulsive Systems, Series A: Mathematical Analysis, 8 (2001), 559-574.

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