×

zbMATH — the first resource for mathematics

Nonlinear dynamics of immunogenic tumors: Parameter estimation and global bifurcation analysis. (English) Zbl 0789.92019
Summary: 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 seen in 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 \(\text{BCL}_ 1\) in the spleen of mice. By comparing the model with experimental data, numerical estimates of parameters describing processes that cannot be measured in 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.

MSC:
92C50 Medical applications (general)
PDF BibTeX XML Cite
Full Text: DOI
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.
[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. · Zbl 0439.92004
[3] Alsabti, A. 1978. Tumor dormancy: A review.Tumor Res. 13, 1–13.
[4] Beaumont, R. A. and R. S. Pierce. 1963.The Algebraic Foundations of Mathematics. Reading, MA: Addison-Wesley. · Zbl 0121.25902
[5] Brondz, B. D. 1987.T Lymphocytes and Their Receptors in Immunological Recognition (in Russian). Moscow: Nauka.
[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.
[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.
[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.
[9] De Boer, R. J. and M. C. Boerlijst. 1993. Diversity and virulence thresholds in AIDS (submitted). · Zbl 0786.92020
[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.
[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.
[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.
[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.
[14] DeLisi, C. and A. Rescigno. 1977. Immune surveillance and neoplasia–1. A minimal mathematical model.Bull. math. Biol. 39, 201–221. · Zbl 0357.92028
[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.
[16] Emanuel, N. M. 1981. Chemical and biological kinetics.Russian Chem. Rev. 50, 901–947.
[17] Fidler, I. J. 1973.In vitro studies of cellular-mediated immunostimulation of tumor growth.J. Natl Cancer Inst. 50, 1307–1312.
[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.
[19] Gatenby, P. A., A. Basten and P. Creswick. 1981. ”Sneaking through”: A T-cell-dependent phenomenon.Br. J. Cancer 44, 753–756.
[20] Gray D. and T. Leanderson. 1990. Expansion, selection and maintenance of memory B-cell clones.Current Topics Microbiol. Immunol. 159, 1–17.
[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.
[22] Grossman, Z. and G. Berke. 1980. Tumor escape from immune elimination.J. theor. Biol. 83, 267–296.
[23] Hellström, K. E. and I. Hellström. 1969. Cellular immunity against tumor antigens.Adv. Cancer Res. 12, 167–223.
[24] Herberman, R. B. 1974. Cell-mediated immunity to tumor cells.Adv. Cancer Res. 19, 207–263.
[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.
[26] Hooke, R. and T. A. Jeeves. Direct search solution of numerical and statistical problems.J. Assoc. Comput. Machin. 8, 212–229. · Zbl 0111.12501
[27] Jeejeebhoy, H. F. 1977. Stimulation of tumor growth by the immune response.Int. J. Cancer 13, 665–678.
[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.
[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.
[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.
[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.
[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.
[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.
[34] Kuznetsov, V. A. 1984. Analysis of population dynamics of cells that exhibit natural resistance to tumors.Soviet Immunol. (Immunologiya) 3, 58–68.
[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.
[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.
[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.
[38] Kuznetsov, V. A. 1992.Dynamics of Immune Processes During Tumor Growth (in Russian). Moscow: Nauka.
[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.
[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.
[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.
[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.
[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.
[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. · Zbl 0747.92017
[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.
[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.
[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.
[48] Mathe, G. and P. Rejzenstein. 1986. Managing minimal residual malignant disease.Oncology 43, 137–142.
[49] Menta, B. C. and M. B. Agarwal. 1980. Cyclic oscillations in leukocyte count in chronic myeloid leukemia.Acta. Haematol. 63, 68–70.
[50] Merrill, S. J. 1982. Foundations of the use of enzyme kinetic analogy in cell-mediated cytotoxicity.Math. Biosci. 62, 219–236. · Zbl 0497.92008
[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. · Zbl 0598.92009
[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.
[53] Nelson, D. S. and M. Nelson. 1987. Evasion of host defenses by tumors.Immunol. Cell. Biol. 65, 287–304.
[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.
[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.
[56] Perelson, A. S. and C. A. Macken. 1984. Kinetics of cell-mediated cytotoxicity: Stochastic and deterministic multistage models.Math. Biosci. 170, 161–194. · Zbl 0571.92012
[57] Prehn, R. T. 1972. The immune reaction as a stimulator of tumor growth.Science 4031, 170–171.
[58] Prehn, R. T. 1983. Review/commentary. The dose-response curve in tumor immunity.Int. J. Immunopharm. 5, 255–257.
[59] Prigogine, I. and R. Lefever. 1980. Stability problems in cancer growth and nucleation.Comp. Biochem. Physiol. 67, 389–393.
[60] Rescigno, A. and C. DeLisi. Immune surveillance and neoplasia. II. A two-stage mathematical model.Bull. math. Biol. 39, 487–497. · Zbl 0376.92017
[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.
[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.
[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.
[64] Slavin, S. and S. Strober. 1978. Spontaneous murine B-cell leukemia.Nature 272, 624–626.
[65] Stewart, T. H. M. and E. F. Wheelock. 1992.Cellular Immune Mechanisms and Tumor Dormancy. Boca Raton, FL: CRC.
[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.
[67] Swan, G. W. 1977.Some Current Mathematical Topics in Cancer Research. Ann Arbor, MI: University Microfilms International.
[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.
[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.
[70] Thorn, R. M. and C. S. Henney. 1976. Kinetic analysis of target cell destruction by effector T cell.J. Immunol. 117, 2213–2219.
[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.
[72] Umiel, T. and N. Trainin. 1974. Immunological enhancement of tumor growth by syngeneic thymus-derived lymphocytes.Transplant 18, 244–250.
[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.
[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.
[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.
[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.
[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.
[78] Wheelock, E. F. and M. K. Robinson. 1983. Biology of disease. Endogenous control of the neoplastic process.Lab. Investigation 48, 120–139.
[79] Wheelock, E. F., K. J. Weinhold and J. Levich. 1981. The tumor dormant state.Adv. Cancer Res. 34, 107–135.
[80] Wiggins, S. 1990.Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York, NY: Springer. · Zbl 0701.58001
[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.
[82] Yermakova, A., P. Valko and S. Vajda. 1982. Direct intergral method via spline approximation for estimating rate constant.Appl. Catalysis 2, 139–154.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.