American Institute of Mathematical Sciences

Advanced
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

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.  Google Scholar

[2]

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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.  Google Scholar

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A. D. Bruno, "Local Methods in Nonlinear Diff. Equations," Springer Verlag, New York, 1989.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," New York: Springer, 1982.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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Y. M. Goltser, Some bifurcation problems of stability, Nonlinear Analysis, TMA, 30 (1997), 1461-1467. doi: 10.1016/S0362-546X(97)00044-8.  Google Scholar

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J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Applied Mathematical Sciences, New York: Springer, 1983.  Google Scholar

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G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," World Scientific, Singapore, 1992.  Google Scholar

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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.  Google Scholar

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[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. Google Scholar

[24]

A. M. Liapunov, "Probléme géné ral de la stabilité du Mouvement," Princeton University Press, Ann. of Math. Stud, 1947. Google Scholar

[25]

I. G. Malkin, "Theory of Stability of Motion," Translated by Atomic Energy Commission, 1952, 92-94. Google Scholar

[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.  Google Scholar

[27]

V. A Pliss, The reduction principle in the theory of the stability motion, Izv.Akad. Nauk SSSR, Ser. Mat. 28 (1964), 1297-1324.  Google Scholar

[28]

H. Poincaré, Oeuvres, Paris, 1928. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

V. I. Arnold, "Geometrical Methods in the Theory of Ordinary Differential Equations," Springer Verlag, New York, 1982. Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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. Google Scholar

[6]

Y. N. Bibikov, "Local Theory of Nonlinear Analytic Ordinary Differential Equations," Lecture Notes in Mathematics, Springer Verlag, New York, 702 (1979).  Google Scholar

[7]

G. D. Birkhoff, "Dynamical Systems," New York, 1927. Google Scholar

[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.  Google Scholar

[9]

A. D. Bruno, "Local Methods in Nonlinear Diff. Equations," Springer Verlag, New York, 1989.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

S. N. Chow and J. K. Hale, "Methods of Bifurcation Theory," New York: Springer, 1982.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

Y. M. Goltser, Some bifurcation problems of stability, Nonlinear Analysis, TMA, 30 (1997), 1461-1467. doi: 10.1016/S0362-546X(97)00044-8.  Google Scholar

[19]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields," Applied Mathematical Sciences, New York: Springer, 1983.  Google Scholar

[20]

G. Iooss and M. Adelmeyer, "Topics in Bifurcation Theory and Applications," World Scientific, Singapore, 1992.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[24]

A. M. Liapunov, "Probléme géné ral de la stabilité du Mouvement," Princeton University Press, Ann. of Math. Stud, 1947. Google Scholar

[25]

I. G. Malkin, "Theory of Stability of Motion," Translated by Atomic Energy Commission, 1952, 92-94. Google Scholar

[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.  Google Scholar

[27]

V. A Pliss, The reduction principle in the theory of the stability motion, Izv.Akad. Nauk SSSR, Ser. Mat. 28 (1964), 1297-1324.  Google Scholar

[28]

H. Poincaré, Oeuvres, Paris, 1928. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

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