Dynamic Irregular Patterns and Invasive Wavefronts The Control of Tumour Growth by Cytotoxic T Lymphocytes

  • Anastasios MatzavinosEmail author
Part of the Modeling and Simulation in Science, Engineering and Technology book series (MSSET)


Hopf Bifurcation Travel Wave Solution Random Motility Stable Limit Cycle Numerical Continuation 
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  1. 1.
    A. Alsabti. Tumour dormant state.Tumour Res., 13(1):1–13, 1978.Google Scholar
  2. 2.
    D. Ambrosi, N. Bellomo, and L. Preziosi. Modelling tumor progression, heterogeneity, and immune competition.J. Theor. Medicine, 4:51–65, 2002.zbMATHCrossRefGoogle Scholar
  3. 3.
    E. de Angelis, M. Delitala, A. Marasco, and A. Romano. Bifurcation analysis for a mean field modelling of tumor and immune system competition.Math. Comp. Modelling, 37:1131–1142, 2003.zbMATHCrossRefGoogle Scholar
  4. 4.
    L. Arlotti, A. Gamba, and M. Lachowicz. A kinetic model of tumor/immune system cellular interactions.J. Theor. Medicine, 4:39–50, 2002.zbMATHCrossRefGoogle Scholar
  5. 5.
    P.B. Ashwin, M.V. Bartuccelli, T.J. Bridges, and S.A. Gourley. Travelling fronts for the KPP equation with spatio-temporal delay.Z. Angew. Math. Phys., 53:103–122, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    N. Bellomo, A. Bellouquid, and E. De Angelis. The modelling of the immune competition by generalized kinetic (Boltzmann) models: review and research perspectives.Math. Comp. Modelling, 37:65–86, 2003.zbMATHCrossRefGoogle Scholar
  7. 7.
    N. Bellomo, B. Firmani, and L. Guerri. Bifurcation analysis for a nonlinear system of integro-differential equations modelling tumor-immune cells competition.Appl. Math. Letters, 12:39–44, 1999.zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    N. Bellomo and L. Preziosi. Modelling and mathematical problems related to tumor evolution and its interaction with the immune system.Math. Comp. Modelling, 32:413–452, 2000.zbMATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    H. Berestycki, B. Larrouturou, and P.L. Lions. Multi-dimensional travelling wave solutions of a flame propagation model.Arch. Rational Mech. Anal., 111:33–49, 1990.zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    N. Breslow, C.W. Chan, G. Dhom, R.A. Drury, L.M. Franks, B. Gellei, Y.S. Lee, S. Lundberg, B. Sparke, N.H. Sternby, and H. Tulinius. Latent carcinoma of prostate at autopsy in seven areas.Intern. J. Cancer, 20(5):680–688, 1977.CrossRefGoogle Scholar
  11. 11.
    M. Chaplain and A. Matzavinos. Mathematical modelling of spatio-temporal phenomena in tumour immunology. In A. Friedman, editor,Tutorials in Mathematical Biosciences III: Cell Cycle, Proliferation, and Cancer, volume 1872 ofLecture Notes in Mathematics, pages 131–183. Springer, New York, 2006.Google Scholar
  12. 12.
    P.G. Coulie. Human tumor antigens recognized by T cells: new perspectives for anti-cancer vaccines?Molecular Medicine Today, 3:261–268, 1997.CrossRefGoogle Scholar
  13. 13.
    M.C. Cross and P.C. Hohenberg. Pattern formation outside equilibrium.Rev. Mod. Phys., 65:851–1112, 1993.CrossRefGoogle Scholar
  14. 14.
    R.A. Deweger, B. Wilbrink, R.M.P. Moberts, D. Mans, R. Oskam, and W. den Otten. Immune reactivity in SL2 lymphoma-bearing mice compared with SL2- immunized mice.Cancer Immun. Immunotherapy, 24:1191–1192, 1987.Google Scholar
  15. 15.
    A. Dhooge, W. Govaerts, and Yu. A. Kuznetsov. MATCONT: a MATLAB package for numerical bifurcation analysis of ODEs.ACM Trans. Math. Software, 29:141–164, 2003.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    A. Dhooge, W. Govaerts, and Yu. A. Kuznetsov. Numerical continuation of fold bifurcations of limit cycles in MATCONT. In P.M.A. Sloot et al., editors,Proceedings of the International Conference on Computational Science ICCS 2003, Melbourne, Australia and St Petersburg, Russia, 2-4 June 2003, Part I, volume 2657 ofLecture Notes in Computer Science, pages 701–710. Springer Verlag, Berlin, 2003.Google Scholar
  17. 17.
    S.R. Dunbar. Travelling wave solutions of diffusive Lotka-Volterra equations.J. Math. Biology, 17:11–32, 1983.zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    G.B. Ermentrout.Simulating, Analyzing, and Animating Dynamical Systems: A Guide to XPPAUT for Researchers and Students, volume 14 ofSoftware, Environments, and Tools. SIAM, Philadelphia, 2002.Google Scholar
  19. 19.
    G.B. Ermentrout, X. Chen, and Z. Chen. Transition fronts and localized structures in bistable reaction-diffusion systems.Physica D, 108:147–167, 1997.zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    N. Fenichel. Geometric singular perturbation theory for ordinary differential equations.J. Diff. Eqns., 31:53–98, 1979.zbMATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    G. Forni, G. Parmiani, A. Guarini, and R. Foa. Gene transfer in tumour therapy.Annals Oncol., 5:789–794, 1994.Google Scholar
  22. 22.
    U. Fory’s. Marchuk’s model of immune system dynamics with application to tumour growth.J. Theor. Medicine, 4:85–93, 2002.CrossRefGoogle Scholar
  23. 23.
    J.A.A. Ghiso. Inhibition of FAK signaling activated by urokinase receptor induces dormancy in human carcinoma cells in vivo.Oncogene, 21(16):2513–2524, 2002.CrossRefGoogle Scholar
  24. 24.
    J. Guckenheimer and P. Holmes.Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields, volume 42 ofApplied Mathematical Sciences. Springer, New York, 1983. Corrected seventh printing, 2002.Google Scholar
  25. 25.
    L. Holmberg and M. Baum. Work on your theories!Nat. Med., 2(8):844–846, 1996.CrossRefGoogle Scholar
  26. 26.
    C.G. Ioannides and T.L. Whiteside. T-cell recognition of human tumours– implications for molecular immunotherapy of cancer.Clin. Immunol. Immunopath., 66:91–106, 1993.CrossRefGoogle Scholar
  27. 27.
    J. Jaaskelainen, A. Maenpaa, M. Patarroyo, C.G. Gahmberg, K. Somersalo, J. Tarkkanen, M. Kallio, and T. Timonen. Migration of recombinant IL-2- activated T-cells and natural killer cells in the intercellular space of human H-2 glioma spheroids in vitro—a study on adhesion molecules involved.J. Immunol., 149:260–268, 1992.Google Scholar
  28. 28.
    Y. Kawakami, M.I. Nishimura, N.P. Restifo, S.L. Topalian, B.H. O’Neil, J. Shilyansky, J.R. Yannelli, and S.A. Rosenberg. T-cell recognition of humanmelanoma antigens.J. Immunotherapy, 14:88–93, 1993.CrossRefGoogle Scholar
  29. 29.
    C.E. Kelly, R.D. Leek, H.M. Byrne, S.M. Cox, A.L. Harris, and C.E. Lewis. Modelling macrophage infiltration into avascular tumours.J. Theor. Medicine, 4:21–38, 2002.zbMATHCrossRefGoogle Scholar
  30. 30.
    R. Kobayashi, T. Ohta, and Y. Hayase. Self-organized pulse generator.Physica D, 84:162–170, 1995.CrossRefGoogle Scholar
  31. 31.
    V.A. Kuznetsov and G.D. Knott. Modeling tumor regrowth and immunotherapy.Math. Comp. Modelling, 33:1275–1287, 2001.zbMATHCrossRefMathSciNetGoogle Scholar
  32. 32.
    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., 56:295–321, 1994.zbMATHGoogle Scholar
  33. 33.
    D. Loeffler and S. Ratner. In vivo localization of lymphocytes labeled with low concentrations of HOECHST-33342.J. Immunol. Meth., 119:95–101, 1989.CrossRefGoogle Scholar
  34. 34.
    E.M. Lord and G. Burkhardt. Assessment of in situ host immunity to syngeneic tumours utilizing the multicellular spheroid model.Cell. Immunol., 85:340–350, 1984.CrossRefGoogle Scholar
  35. 35.
    A. Matzavinos and M.A.J. Chaplain. Travelling-wave analysis of a model of the immune response to cancer.C.R. Biologies, 327:995–1008, 2004.CrossRefGoogle Scholar
  36. 36.
    A. Matzavinos, M.A.J. Chaplain, and V.A. Kuznetsov. Mathematical modelling of the spatio-temporal response of cytotoxic T-lymphocytes to a solid tumour.Mathematical Medicine and Biology: A Journal of the IMA, 21:1–34, 2004.zbMATHCrossRefGoogle Scholar
  37. 37.
    A. Matzavinos-Toumasis.Mathematical Modelling of the Spatio-temporal Response of Cytotoxic T-lymphocytes to a Solid Tumour. PhD thesis, University of Dundee, Scotland, 2004.Google Scholar
  38. 38.
    J.H. Merkin and M.A. Sadiq. The propagation of travelling waves in an open cubic autocatalytic chemical system.IMA J. Appl. Math., 57:273–309, 1996.zbMATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    M.R. Owen and J.A. Sherratt. Pattern formation and spatio-temporal irregularity in a model for macrophage-tumour interactions.J. Theor. Biol., 189:63–80, 1997.CrossRefGoogle Scholar
  40. 40.
    M.R. Owen and J.A. Sherratt. Modelling the macrophage invasion of tumours: effects on growth and composition.IMA J. Math. Appl. Med. Biol., 15:165–185, 1998.zbMATHCrossRefGoogle Scholar
  41. 41.
    M.R. Owen and J.A. Sherratt. Mathematical modelling of macrophage dynamics in tumours.Math. Models Meth. Appl. Sci., 9:513–539, 1999.zbMATHCrossRefGoogle Scholar
  42. 42.
    V. Schirrmacher. T-cell immunity in the induction and maintenance of a tumour dormant state.Seminars in Cancer Biology, 11:285–295, 2001.CrossRefGoogle Scholar
  43. 43.
    J.A. Sherratt, B.T. Eagan, and M.A. Lewis. Oscillations and chaos behind predator-prey invasion: mathematical artifact or ecological reality?Phil. Trans. R. Soc. Lond. B, 352:21–38, 1997.CrossRefGoogle Scholar
  44. 44.
    J.A. Sherratt, M.A. Lewis, and A.C. Fowler. Ecological chaos in the wake of invasion.Proc. Natl. Acad. Sci. USA, 92:2524–2528, 1995.zbMATHCrossRefGoogle Scholar
  45. 45.
    J.A. Sherratt, A.J. Perumpanani, and M.R. Owen. Pattern formation in cancer. In M.A.J. Chaplain, G.D. Singh, and J.C. McLachlan, editors,On Growth and Form: Spatio-temporal Pattern Formation in Biology. John Wiley – Sons Ltd., Chicester, 1999.Google Scholar
  46. 46.
    H. Siu, E.S. Vitetta, R.D. May, and J.W. Uhr. Tumour dormancy. regression of BCL tumour and induction of a dormant tumour state in mice chimeric at the major histocompatibility complex.J. Immunol., 137:1376–1382, 1986.Google Scholar
  47. 47.
    Z. Szyma’nska. Analysis of immunotherapy models in the context of cancer dynamics.Appl. Math. Comp. Sci., 13:407–418, 2003.MathSciNetGoogle Scholar
  48. 48.
    T. Udagawa, A. Fernandez, E.G. Achilles, J. Folkman, and R.J. D’Amato. Persistence of microscopic human cancers in mice: alterations in the angiogenic balance accompanies loss of tumor dormancy.FASEB J., 16(11):1361–1370, 2002.CrossRefGoogle Scholar
  49. 49.
    J.W. Uhr and R. Marches. Dormancy in a model of murine B cell lymphoma.Seminars in Cancer Biology, 11:277–283, 2001.CrossRefGoogle Scholar
  50. 50.
    A.I. Volpert, V.A. Volpert, and V.A. Volpert.Traveling Wave Solutions of Parabolic Systems, volume 140 ofTranslations of Mathematical Monographs. American Mathematical Society, Providence, RI, 2000.Google Scholar
  51. 51.
    S.D. Webb, J.A. Sherratt, and R.G. Fish. Cells behaving badly: a theoretical model for the Fas/FasL system in tumour immunology.Mathematical Biosciences, 179:113–129, 2002.zbMATHCrossRefMathSciNetGoogle Scholar
  52. 52.
    K.M. Wilson and E.M. Lord. Specific (EMT6) and non-specific (WEHI-164) cytolytic activity by host cells infiltrating tumour spheroids.Brit. J. Cancer, 55:141–146, 1987.Google Scholar
  53. 53.
    E. Yefenof. Cancer dormancy: from observation to investigation and onto clinical intervention.Seminars in Cancer Biology, 11:269–270, 2001.CrossRefGoogle Scholar

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© Birkhäuser Boston 2008

Authors and Affiliations

  1. 1.Department of MathematicsOhio State UniversityColumbusUSA

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