Bulletin of Mathematical Biology

, Volume 56, Issue 2, pp 295–321 | Cite as

Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis

  • Vladimir A. Kuznetsov
  • Iliya A. Makalkin
  • Mark A. Taylor
  • Alan S. Perelson
Article

Abstract

We present a mathematical model of the cytotoxic T lymphocyte response to the growth of an immunogenic tumor. The model exhibits a number of phenomena that are seenin vivo, including immunostimulation of tumor growth, “sneaking through” of the tumor, and formation of a tumor “dormant state”. The model is used to describe the kinetics of growth and regression of the B-lymphoma BCL1 in the spleen of mice. By comparing the model with experimental data, numerical estimates of parameters describing processes that cannot be measuredin vivo are derived. Local and global bifurcations are calculated for realistic values of the parameters. For a large set of parameters we predict that the course of tumor growth and its clinical manifestation have a recurrent profile with a 3- to 4-month cycle, similar to patterns seen in certain leukemias.

Keywords

Human Immunodeficiency Virus Effector Cell Phase Portrait Chimeric Mouse Dormant State 

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References

  1. Abrahms, S. I. and Z. Brahmi. 1988. Mechanism of K562-induced human natural killer cell inactivation using highly enriched effector cells isolated via a new single-step sheep erythrocyte rossette assay.Ann. Inst. Pasteur, Immunol. 139, 361–381.CrossRefGoogle Scholar
  2. Albert, A., M. Freedman and A. S. Perelson. 1980. Tumors and the immune system: The effects of a tumor growth modulator.Math. Biosciences 50, 25–58.MATHMathSciNetCrossRefGoogle Scholar
  3. Alsabti, A. 1978. Tumor dormancy: A review.Tumor Res. 13, 1–13.Google Scholar
  4. Beaumont, R. A. and R. S. Pierce. 1963.The Algebraic Foundations of Mathematics. Reading, MA: Addison-Wesley.MATHGoogle Scholar
  5. Brondz, B. D. 1987.T Lymphocytes and Their Receptors in Immunological Recognition (in Russian). Moscow: Nauka.Google Scholar
  6. Callewaert, D. M., P. Meyers, J. Hiernaux and G. Radcliff. 1988. Kinetics of cellular cytotoxicity mediated by cloned cytotoxic T lymphocytes.Immunobiol. 178, 203–214.Google Scholar
  7. Chen, L., Y. Suzuki, C.-M. Liu and E. F. Wheelock. 1990. Maintenance and cure of the L5178Y murine tumor dormant state by interleukin 2: Dependence of interleukin 2 on induced inteferon-g and on tumor necrosis factor for its antitumor effects.Cancer Res. 50, 1368–1374.Google Scholar
  8. Colmeraver, M. E., I. A. Loziol and V. H. Pilch. 1980. Enhancement of metastasis development by BCG immunotherapy.J. Surg. Oncology 15, 235–241.Google Scholar
  9. De Boer, R. J. and M. C. Boerlijst. 1993. Diversity and virulence thresholds in AIDS (submitted).Google Scholar
  10. De Boer, R. J. and P. Hogeweg. 1985. Tumor escape from immune elimination: Simplified precursor bound cytotoxicity models.J. theor. Biol. 113, 719–736.Google Scholar
  11. De Boer, R. J. and P. Hogeweg. 1986. Interactions between macrophages and T-lymphocytes: Tumor sneaking through intrinsic to helper T cell dynamics.J. theor. Biol. 120, 331–354.CrossRefGoogle Scholar
  12. Deichman, G. I. 1979. Current concepts on the immunological interaction between the tumor and the body. InTumor Growth as Problem of Development Biology, pp. 208–223. Moscow: Nauka.Google Scholar
  13. Deichman, G. I., T. E. Klyuchareva, L. M. Kashkina and V. A. Matveyeva. 1979. Reproducibility and relation to specific and nonspecific antitumor resistance of the “sneaking through” phenomenon.Int. J. Cancer 23, 571–584.Google Scholar
  14. DeLisi, C. and A. Rescigno. 1977. Immune surveillance and neoplasia—1. A minimal mathematical model.Bull. math. Biol. 39, 201–221.MATHMathSciNetCrossRefGoogle Scholar
  15. Dozmorov, I. M. and V. A. Kuznetsov. 1988. The role of cellular ratios in the maintenance of organism immune homeostasis. InProblems and Perspectives of Modern Immunology: Methodological Analysis (in Russian), R. V. Petrov and V. P. Lozovoy (Eds), pp. 43–66. Novosibirsk: Nauka.Google Scholar
  16. Emanuel, N. M. 1981. Chemical and biological kinetics.Russian Chem. Rev. 50, 901–947.CrossRefGoogle Scholar
  17. Fidler, I. J. 1973.In vitro studies of cellular-mediated immunostimulation of tumor growth.J. Natl Cancer Inst. 50, 1307–1312.Google Scholar
  18. Fishelson, Z. and G. Berke. 1981. Tumor cell destruction by cytotoxic T lymphocytes: The basis of reduced antitumor cell activity in syngeneic hosts.J. Immunol. 125, 2048–2052.Google Scholar
  19. Gatenby, P. A., A. Basten and P. Creswick. 1981. “Sneaking through”: A T-cell-dependent phenomenon.Br. J. Cancer 44, 753–756.Google Scholar
  20. Gray D. and T. Leanderson. 1990. Expansion, selection and maintenance of memory B-cell clones.Current Topics Microbiol. Immunol. 159, 1–17.Google Scholar
  21. Greenberg, P. D. 1991. Adoptive T cell therapy of tumors: Mechanisms operative in the recognition and climination of tumor cells.Adv. Immunol. 49, 281–355.CrossRefGoogle Scholar
  22. Grossman, Z. and G. Berke. 1980. Tumor escape from immune elimination.J. theor. Biol. 83, 267–296.CrossRefGoogle Scholar
  23. Hellström, K. E. and I. Hellström. 1969. Cellular immunity against tumor antigens.Adv. Cancer Res. 12, 167–223.Google Scholar
  24. Herberman, R. B. 1974. Cell-mediated immunity to tumor cells.Adv. Cancer Res. 19, 207–263.Google Scholar
  25. Hiernaux, J. R., R. Lefever, C. Uyttenhove and T. Boon. 1986. Tumor dormancy as a result of simple competition between tumor cells and cytolytic effector cells. InParadoxes in Immunology, G. W. Hoffman, J. G. Levy and G. T. Nepom (Eds), pp. 95–109. Florida, CRC Press.Google Scholar
  26. Hooke, R. and T. A. Jeeves. Direct search solution of numerical and statistical problems.J. Assoc. Comput. Machin. 8, 212–229.Google Scholar
  27. Jeejeebhoy, H. F. 1977. Stimulation of tumor growth by the immune response.Int. J. Cancer 13, 665–678.Google Scholar
  28. Krikorian, J. G., C. S. Portlock, D. P. Cooney and S. A. Rosenberg. 1980. Spontaneous regression of non-Hodgkin's lymphoma: A report of nine cases.Cancer 46, 2093–2099.CrossRefGoogle Scholar
  29. Krolick, K. A., P. C. Isakson, I. W. Uhr and E. S. Vitetta. 1979. BCL1, a murine model for chronic lymphocytic leukemia: Use of the surface immunoglobulin idiotype for the detection and treatment of tumor.J. Immunol. Rev. 48, 81–106.CrossRefGoogle Scholar
  30. Kukain, R. A., L. I. Nagayeva, V. P. Lozha, S. Ya Laganovsky, S. V. Chapenko, O. I. Bratsslavskaya, V. P. Ose and G. V. Kudeleva. 1982.Bovine Leukemia Virus (in Russian). Riga: Zinatne.Google Scholar
  31. Kuznetsov, V. A. 1979. The dynamics of cellular immunological antitumor reactions. I. Synthesis of a multi-level model. InMathematical Methods of Systems Theory (in Russian), Vol. 1, pp. 57–71.Google Scholar
  32. Kuznetsov, V. A. 1981. A model for cytotoxic cellular immune process and its experimental application (in Russian). InApplied Problems in the Theory of Dynamic Systems, Gorky, Vol. 4, pp. 14–43. Manuscript submitted to the All-Union Institute of Science and Technology Information, 25 December 1981, No. 5851.Google Scholar
  33. Kuznetsov, V. A. 1983. Bifurcations in a model of the two-level reactivity of an immune system to antigens of a developing neoplasm. InDynamics of Biological Populations, Gorky (in Russian), pp. 52–64. Gor'ki State University.Google Scholar
  34. Kuznetsov, V. A. 1984. Analysis of population dynamics of cells that exhibit natural resistance to tumors.Soviet Immunol. (Immunologiya) 3, 58–68.Google Scholar
  35. Kuznetsov, V. A. 1987. Mathematical modelling of the processes of dormant tumors formation and immunostimulation of their growth (in Russian).Cybernetics 4, 96–102.MATHGoogle Scholar
  36. Kuznetsov, V. A. 1988. Nonlinear effects of the dynamics of antitumor cellular immune system (preprint; in Russian). Moscow: Institute of Chemical Physics, Academy of Sciences, USSR.Google Scholar
  37. Kuznetsov, V. A. 1991. A mathematical model for the interaction between cytotoxic lymphocytes and tumour cells. Analysis of the growth, stabilization and regression of the B cell lymphoma in mice chimeric with respect to the major histocompatibility complex.Biomed. Sci. 2, 465–476.Google Scholar
  38. Kuznetsov, V. A. 1992.Dynamics of Immune Processes During Tumor Growth (in Russian). Moscow: Nauka.Google Scholar
  39. Kuznetsov, V. A., A. V. Inshina and Z. G. Kadagidze. 1988. Computer-aided determination of the number of active natural killers, their avidity and the rate of recycling in a lytic cycle.Soviet Immunology (Immunologiya) 5, 25–30.Google Scholar
  40. Kuznetsov, V. A. and M. V. Volkenshtein 1978. Mathematical model of cellular immune response to tumor growth (in Russian). InThe Reports at the Third All-Union Conference on Biology and Medical Cybernetics (Sukhumi), pp. 58–61. Moscow: USSR Academy of Science.Google Scholar
  41. Kuznetsov, V. A. and M. V. Volkenshtein. 1979. Dynamics of cellular immunological antitumor reactions. II. Qualitative analysis of the model (in Russian). InMathematical Methods of Systems Theory, pp. 72–100. Frunze: Kirghiz State University.Google Scholar
  42. Kuznetsov, V. A., V. P. Zhivoglyadov and L. A. Stepanova. 1993. Kinetic approach and estimation of parameters of cellular interaction between the immunity system and a tumor.Archiv. Immunol. Ther. Exp. 41, 21–32.Google Scholar
  43. Lefever, R. and T. Erneaux. 1984. On the growth of cellular tissues under constant and fluctuating environmental conditions. InNonlinear Electrodynamics in Biological Systems, P. Adley and A. F. Lowrence (Eds), pp. 287–305. New York and London: Plenum Press.Google Scholar
  44. Lefever, R., J. Hiernaux, J. Urbain and P. Meyers. 1992. On the kinetics and optimal specificity of cytotoxic reactions mediated by T-lymphocyte clones.Bull. math. Biol. 54, 839–873.MATHCrossRefGoogle Scholar
  45. Liu, Ch.-M., Y. Suzuki, L. Chen, T. Okayasu, C. E. Calkins and E. F. Wheelock. 1990. Maintenance and cure of the L5178 murine tumor dormant state by interleukin-2:In vivo andin vitro effects.Cancer Res. 50, 1361–1367.Google Scholar
  46. Look, A. T., T. J. Schriber, J. F. Nawrocki and W. H. Murphy. 1981. Computer simulation of the cellular immune response to malignant lymphoid cells: Logic of approach, model design and laboratory verification.Immunol. 43, 677–690.Google Scholar
  47. Macken, C. A. and A. S. Perelson. 1984. A multistage model for the action of cytotoxic T lymphocytes in multicellular conjugates.J. Immunol. 132, 1614–1624.Google Scholar
  48. Mathe, G. and P. Rejzenstein. 1986. Managing minimal residual malignant disease.Oncology 43, 137–142.CrossRefGoogle Scholar
  49. Menta, B. C. and M. B. Agarwal. 1980. Cyclic oscillations in leukocyte count in chronic myeloid leukemia.Acta. Haematol. 63, 68–70.Google Scholar
  50. Merrill, S. J. 1982. Foundations of the use of enzyme kinetic analogy in cell-mediated cytotoxicity.Math. Biosci. 62, 219–236.MATHMathSciNetCrossRefGoogle Scholar
  51. Merrill, S. J. and S. Sathananthan. 1986. Approximate Michaelis-Menthen kinetics displayed in a stochastic model of cell-mediated cytotoxicity.Math. Biosci. 80, 223–238.MATHMathSciNetCrossRefGoogle Scholar
  52. Mohler, R. R. and K. S. Lee. 1989. Dynamic analysis and control of cancer. InInt. Conf. IEEE Engng Med. Biol. Seattle, pp. 1–2.Google Scholar
  53. Nelson, D. S. and M. Nelson. 1987. Evasion of host defenses by tumors.Immunol. Cell. Biol. 65, 287–304.Google Scholar
  54. Old, L. J., E. A. Boyse, D. A. Clarke and F. A. Carswell. 1962. Antigenic properties of chemically induced tumors.Ann. N. Y. Acad. Sci. 101, 80–106.Google Scholar
  55. Perelson, A. S. and G. I. Bell. 1982. Delivery of lethal hits by cytotoxic T lymphocytes in multicellular conjugates occurs sequentially but at random.J. Immunol. 129, 2796–2801.Google Scholar
  56. Perelson, A. S. and C. A. Macken. 1984. Kinetics of cell-mediated cytotoxicity: Stochastic and deterministic multistage models.Math. Biosci. 170, 161–194.MathSciNetCrossRefGoogle Scholar
  57. Prehn, R. T. 1972. The immune reaction as a stimulator of tumor growth.Science 4031, 170–171.Google Scholar
  58. Prehn, R. T. 1983. Review/commentary. The dose-response curve in tumor immunity.Int. J. Immunopharm. 5, 255–257.CrossRefGoogle Scholar
  59. Prigogine, I. and R. Lefever. 1980. Stability problems in cancer growth and nucleation.Comp. Biochem. Physiol. 67, 389–393.CrossRefGoogle Scholar
  60. Rescigno, A. and C. DeLisi. Immune surveillance and neoplasia. II. A two-stage mathematical model.Bull. math. Biol. 39, 487–497.Google Scholar
  61. Reynolds, C. W., R. H. Wiltrout, S. Reichardi and R. B. Herberman. 1985. Measurements of thein vivo turnover rates of rat peripheral blood and spleen large granular lymphocytes.Natural Immun. Cell Growth Regul. 9, 272.Google Scholar
  62. Sampson, D., T. G. Peter, S. D. Lewis, J. Metzig and B. E. Murtz. 1977. Dose dependence of immunopotentiation and tumor regression induced by levamisole.Cancer Res. 37, 3526–3528.Google Scholar
  63. Siu, H., E. S. Vitetta, R. D. May and I. W. Uhr. 1986. Tumor dormancy. I. Regression of BCL1 tumor and induction of a dormant tumor state in mice chimeric at the major histocompatibility complex.J. Immunol. 137, 1376–1382.Google Scholar
  64. Slavin, S. and S. Strober. 1978. Spontaneous murine B-cell leukemia.Nature 272, 624–626.CrossRefGoogle Scholar
  65. Stewart, T. H. M. and E. F. Wheelock. 1992.Cellular Immune Mechanisms and Tumor Dormancy. Boca Raton, FL: CRC.Google Scholar
  66. Strober, S., E. S. Gronowicz, M. R. Knapp and S. Slavin. 1979. Immunobiology of a spontaneous murine B cell Leukemia (BCL).Immunol. Rev.,48, 169–195.CrossRefGoogle Scholar
  67. Swan, G. W. 1977.Some Current Mathematical Topics in Cancer Research. Ann Arbor, MI: University Microfilms International.Google Scholar
  68. Tanaka, K., T. Yoshioka, C. Bieberich and G. Jay. 1988. Role of the major histocompatibility complex class I antigens in tumor growth and metastasis.Ann. Rev. Immunol. 6, 359–380.CrossRefGoogle Scholar
  69. Thoma, J. A., G. J. Thoma and W. Clark. 1978. The efficiency and linearity of the radiochromium release assay for cell-mediated cytotoxicity.Cell Immunol 40, 404–418.CrossRefGoogle Scholar
  70. Thorn, R. M. and C. S. Henney. 1976. Kinetic analysis of target cell destruction by effector T cell.J. Immunol. 117, 2213–2219.Google Scholar
  71. Thorn, R. M. and C. S. Henney. 1977. Kinetic analysis of target cell destruction by effector cells. II. Changes in killer cell avidity as a function of time and dose.J. Immunol. 119, 1973–1978.Google Scholar
  72. Umiel, T. and N. Trainin. 1974. Immunological enhancement of tumor growth by syngeneic thymus-derived lymphocytes.Transplant 18, 244–250.Google Scholar
  73. Uhr, J. W., T. Tucker, R. D. May, H. Siu and E. S. Vitetta. 1991. Cancer dormancy: Studies of the murine BCL1 lymphoma.Cancer Res. (Suppl.) 51, 5045s-5053s.Google Scholar
  74. Uyttenhove, C., J. Maryanski and T. Boon. 1983. Escape of mouse mastocytoma P815 after nearly complete rejection is due to antigen-loss variants rather than immunosuppression.J. Expl Med. 157, 1040–1052.CrossRefGoogle Scholar
  75. Weinhold, K. J., L. T. Goldstein and E. F. Wheelock. 1979a. The tumor dormant state. Quantitation of L5178Y cells and host immune response during the establishment.J. Expl Med. 149, 732–744.CrossRefGoogle Scholar
  76. Weinhold, K. J., D. A. Miller and E. F. Wheelock. 1979b. The tumor dormant state. Comparison of L5178Y cells used to establish dormancy with those that emerge after its termination.J. Expl Med. 149, 745–747.CrossRefGoogle Scholar
  77. Weiss, L., S. Morecki, E. S. Vitetta and S. Slavin. 1983. Suppression and elimination of BCL1 leukemia by allogeneic bone marrow transplantation.J. Immunol. 130, 2452–2455.Google Scholar
  78. Wheelock, E. F. and M. K. Robinson. 1983. Biology of disease. Endogenous control of the neoplastic process.Lab. Investigation 48, 120–139.Google Scholar
  79. Wheelock, E. F., K. J. Weinhold and J. Levich. 1981. The tumor dormant state.Adv. Cancer Res. 34, 107–135.CrossRefGoogle Scholar
  80. Wiggins, S. 1990.Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York, NY: Springer.MATHGoogle Scholar
  81. Yefenof, E., L. J. Picker, R. H. Scheuermann, T. F. Tucker, E. S. Vitetta and J. W. Uhr. 1993. Cancer dormancy: Isolation and characterization of dormant lymphoma cells.Proc. Natl Acad. Sci. USA 90, 1829–1833.CrossRefGoogle Scholar
  82. Yermakova, A., P. Valko and S. Vajda. 1982. Direct intergral method via spline approximation for estimating rate constant.Appl. Catalysis 2, 139–154.CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 1994

Authors and Affiliations

  • Vladimir A. Kuznetsov
    • 1
  • Iliya A. Makalkin
    • 1
  • Mark A. Taylor
    • 2
  • Alan S. Perelson
    • 2
  1. 1.Laboratory of Mathematical Immunobiophysics, Institute of Chemical PhysicsRussian Academy of SciencesMoscowRussia
  2. 2.Theoretical DivisionLos Alamos National LaboratoryLos AlamosU.S.A.

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