The European Physical Journal B

, Volume 65, Issue 3, pp 435–442 | Cite as

Monitoring noise-resonant effects in cancer growth influenced by external fluctuations and periodic treatment

  • A. FiasconaroEmail author
  • A. Ochab-Marcinek
  • B. Spagnolo
  • E. Gudowska-Nowak


We investigate a mathematical model describing the growth of tumor in the presence of immune response of a host organism. The dynamics of tumor and immune cells populations is based on the generic Michaelis-Menten kinetics depicting interaction and competition between the tumor and the immune system. The appropriate phenomenological equation modeling cell-mediated immune surveillance against cancer is of the predator-prey form and exhibits bistability within a given choice of the immune response-related parameters. Under the influence of weak external fluctuations, the model may be analyzed in terms of a stochastic differential equation bearing the form of an overdamped Langevin-like dynamics in the external quasi-potential represented by a double well. We analyze properties of the system within the range of parameters for which the potential wells are of the same depth and when the additional perturbation, modeling a periodic treatment, is insufficient to overcome the barrier height and to cause cancer extinction. In this case the presence of a small amount of noise can positively enhance the treatment, driving the system to a state of tumor extinction. On the other hand, however, the same noise can give rise to return effects up to a stochastic resonance behavior. This observation provides a quantitative analysis of mechanisms responsible for optimization of periodic tumor therapy in the presence of spontaneous external noise. Studying the behavior of the extinction time as a function of the treatment frequency, we have also found the typical resonant activation effect: For a certain frequency of the treatment, there exists a minimum extinction time.


05.40.-a Fluctuation phenomena, random processes, noise, and Brownian motion 87.17.Aa Modeling, computer simulation of cell processes 87.15.A-Theory, modeling, and computer simulation 


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  1. 1.
    S.A. Rosenberg, P. Spiess, R. Lafreniere, Science 233, 4770 1318 (1986)CrossRefADSGoogle Scholar
  2. 2.
    R.O. Dillman, Future Drugs 5, 6 1041 (2005)Google Scholar
  3. 3.
    R.M. Thorn, C.S. Henney, J. Immunol. 117, 6 2213 (1976)Google Scholar
  4. 4.
    P.M. Moy, E.C. Holmes, S.H. Golub, Cancer Research, 45, 1 57 (1985)Google Scholar
  5. 5.
    D. Kirschner, J.C. Panetta, J. Math. Biol. 37, 235 (1998)zbMATHCrossRefGoogle Scholar
  6. 6.
    A. Matzavinos, M.A.J. Chaplain, V.A. Kuznetsov, Math. Med. Biol. 21, 1 (2004); D. Wodarz, N.L. Komarova, Computational Biology Of Cancer: Lecture Notes And Mathematical Modeling (World Scientific, 2005), p. 185zbMATHCrossRefGoogle Scholar
  7. 7.
    R.P. Garay, R. Lefever, J. Theor. Biol. 73, 417 (1978)CrossRefMathSciNetGoogle Scholar
  8. 8.
    A. LeFever, S. Micha, Scand. J. Immunol. 29, 417 (1989)CrossRefGoogle Scholar
  9. 9.
    A. Ochab-Marcinek, E. Gudowska-Nowak, Physica A, 343, 557 (2004)ADSGoogle Scholar
  10. 10.
    A. Fiasconaro, B. Spagnolo, A. Ochab-Marcinek, E. Gudowska-Nowak, Phys. Rev. E 74, 041904 (2006)CrossRefADSGoogle Scholar
  11. 11.
    A. Ochab-Marcinek, A. Fiasconaro, E. Gudowska-Nowak, B. Spagnolo, Acta Physica Polonica B 37 1651 (2006)ADSGoogle Scholar
  12. 12.
    B. Spagnolo et al., Acta Phys. Pol. B 38, 1925 (2007)ADSGoogle Scholar
  13. 13.
    R. Lefever, R. Garay, Local description of immune tumor rejection, Dev. Cell Biol., edited by A.J. Valleron, P.D.M. Macdonald (Elsevier, Amsterdam, 1978), Vol. 2Google Scholar
  14. 14.
    R. Lefever, W. Horsthemke, Bull. of Math. Biol 41, 469 (1979)zbMATHGoogle Scholar
  15. 15.
    I. Prigogine, R. Lefever, Comp. Biochem. Physiol. 67B, 389 (1980)Google Scholar
  16. 16.
    W. Horsthemke, R. Lefever, Noise-Induced Transitions (Springer-Verlag, Berlin, 1984)zbMATHGoogle Scholar
  17. 17.
    W. Ebeling, B. Röder, L. Schimansky-Geier, Studia Biophys. 113, 1–2 151 (1986)Google Scholar
  18. 18.
    E. Gudowska-Nowak, Acta Phys. Pol. A. 64, 341 (1983)Google Scholar
  19. 19.
    E. Gudowska-Nowak, Acta Phys. Pol. A. 65, 573 (1984)Google Scholar
  20. 20.
    L. Gammaitoni, P. Hänggi, P. Jung, F. Marchesoni, Rev. Mod. Phys. 70, 234 (1998); P. Hänggi, Chem. Phys. Chem. 3, 285 (2002)CrossRefADSGoogle Scholar
  21. 21.
    B. McNamara, K. Wiesenfeld, Phys. Rev. A 39, 4854 (1989)CrossRefADSGoogle Scholar
  22. 22.
    L.K. Andersen, M.C. Mackey, J. Theor. Biol. 209, 113 (2001)CrossRefGoogle Scholar
  23. 23.
    F. Michor, M.A. Nowak, Y. Iwasa, J. Theor. Biol. 240, 521 (2006)CrossRefMathSciNetGoogle Scholar
  24. 24.
    M. Molski, J. Konarski, Phys. Rev. E. 68, 021916 (2003)CrossRefADSGoogle Scholar
  25. 25.
    C.R. Doering, J.C. Gadoua, Phys. Rev. Lett. 69, 2318 (1992); M. Bier, R.D. Astumian, Phys. Rev. Lett. 71, 1649 (1993); P. Reimann, Phys. Rev. Lett. 74, 4576 (1995); M. Marchi et al., Phys. Rev. E 54, 3479 (1996); J. Iwaniszewski, Phys. Rev. E 54, 3173 (1996); M. Bogu~n’a, J.M. Porra, J. Masoliver, K. Lindenberg, Phys. Rev. E 57, 3990 (1998); R.N. Mantegna, B. Spagnolo, Phys. Rev. Lett. 84, 3025 (2000); P. Pechukas, P. Hänggi, Phys. Rev. Lett. 73, 2772 (1994)CrossRefADSGoogle Scholar
  26. 26.
    N. Agudov, B. Spagnolo, Phys. Rev. E 64, (2001) 035105(R)CrossRefADSGoogle Scholar
  27. 27.
    C. Schmitt, B. Dybiec, P. Hänggi, C. Bechinger, Europhys. Lett. 74, 937 (2006)CrossRefADSMathSciNetGoogle Scholar
  28. 28.
    B. Dybiec, E. Gudowska-Nowak, Acta Phys. Pol. B. 38, 1759 (2007)ADSGoogle Scholar
  29. 29.
    A.G. Papatsorisa, C. Deliveliotisb, A. Giannopoulosb, C. Dimopoulosb, Urol. Int. 72, 284 (2004)CrossRefGoogle Scholar
  30. 30.
    M.D. Hiroki Shirato et al., Int. J. Radiation Oncology Biol. Phys. 56, 240 (2003)Google Scholar

Copyright information

© Springer 2008

Authors and Affiliations

  • A. Fiasconaro
    • 1
    • 2
    • 3
    Email author
  • A. Ochab-Marcinek
    • 2
    • 4
  • B. Spagnolo
    • 3
  • E. Gudowska-Nowak
    • 1
    • 2
  1. 1.Mark Kac Complex Systems Research CenterJagellonian UniversityKrak’owPoland
  2. 2.Marian Smoluchowski Institute of PhysicsJagellonian UniversityKrak’owPoland
  3. 3.Dipartimento di Fisica e Tecnologie Relative and CNISM, Group of Interdisciplinary PhysicsUniversità di PalermoPalermoItaly
  4. 4.Institut für PhysikUniversität Augsburg, Universitätsstraβe 1AugsburgGermany

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