A delay differential equation model for tumor growth

Abstract.We present a competition model of tumor growth that includes the immune system response and a cycle-phase-specific drug. The model considers three populations: Immune system, population of tumor cells during interphase and population of tumor during mitosis. Delay differential equations are used to model the system to take into account the phases of the cell cycle. We analyze the stability of the system and prove a theorem based on the argument principle to determine the stability of a fixed point and show that the stability may depend on the delay. We show theoretically and through numerical simulations that periodic solutions may arise through Hopf Bifurcations.

[1]  G. Steel.  Cell loss as a factor in the growth rate of human tumours. , 1967, European journal of cancer.

[2]  Ami Radunskaya,et al.  A mathematical tumor model with immune resistance and drug therapy: an optimal control approach , 2001 .

[3]  T. Jacks,et al.  The cell cycle and cancer. , 1997, Proceedings of the National Academy of Sciences of the United States of America.

[4]  J. Cerottini,et al.  Studies on the mechanism of T cell-mediated lysis at the single effector cell level. I. Kinetic analysis of lethal hits and target cell lysis in multicellular conjugates. , 1979, Journal of Immunology.

[5]  Joseph M. Mahaffy,et al.  A test for stability of linear differential delay equations , 1982 .

[6]  J. Mahaffy.  Periodic solutions for certain protein synthesis models , 1980 .

[7]  Z. Agur,et al.  The growth law of primary breast cancer as inferred from mammography screening trials data. , 1998, British Journal of Cancer.

[8]  Jack K. Hale,et al.  Introduction to Functional Differential Equations , 1993, Applied Mathematical Sciences.

[9]  Richard Bellman,et al.  Differential-Difference Equations , 1967 .

[10]  John A. Adam,et al.  A mathematical model of cycle-specific chemotherapy , 1995 .

[11]  Martin M. Eisen,et al.  Mathematical Models in Cell Biology and Cancer Chemotherapy , 1979 .

[12]  H. Knolle,et al.  Cell Kinetic Modelling and the Chemotherapy of Cancer , 1988 .

[13]  M. Tubiana,et al.  Comparison of cell proliferation kinetics in human and experimental tumors: response to irradiation. , 1976, Cancer treatment reports.

[14]  R. Thomlinson,et al.  Measurement and management of carcinoma of the breast. , 1982, Clinical radiology.

[15]  S Gallivan,et al.  A mathematical model of the development of drug resistance to cancer chemotherapy. , 1987, European journal of cancer & clinical oncology.

[16]  D. Kirschner,et al.  Modeling immunotherapy of the tumor – immune interaction , 1998, Journal of mathematical biology.

[17]  Fathalla A. Rihan,et al.  Modelling and analysis of time-lags in some basic patterns of cell proliferation , 1998, Journal of mathematical biology.

[18]  P. García-Peñarrubia,et al.  Kinetic analysis of effector cell recycling and effector-target binding capacity in a model of cell-mediated cytotoxicity. , 1989, Journal of immunology.

[19]  R. Hromas,et al.  Regulation of human natural killer cell migration and proliferation by the exodus subfamily of CC chemokines. , 2000, Cellular immunology.

[20]  A. Perelson,et al.  Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. , 1994, Bulletin of mathematical biology.

[21]  E K Afenya,et al.  Some perspectives on modeling leukemia. , 1998, Mathematical biosciences.

[22]  J. Panetta,et al.  A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. , 1996, Bulletin of mathematical biology.

[23]  J. Merrill,et al.  Interferon‐Induced NK Augmentation in Humans , 1981, Scandinavian journal of immunology.

[24]  L. Norton.  A Gompertzian model of human breast cancer growth. , 1988, Cancer research.

[25]  John Carl Panetta,et al.  A mathematical model of periodically pulsed chemotherapy: Tumor recurrence and metastasis in a competitive environment , 1996 .

[26]  K. Cooke,et al.  On zeroes of some transcendental equations , 1986 .

[27]  G. Hedlund,et al.  Proliferation and differentiation of alloselective NK cells after alloimmunization-evidence for an adaptive NK response. , 1999, Cellular immunology.

[28]  Renato Baserga.  The Cell cycle and cancer , 1971 .

[29]  W. Stigelman,et al.  Goodman and Gilman's the Pharmacological Basis of Therapeutics , 1986 .

[30]  L. Norton,et al.  Predicting the course of Gompertzian growth , 1976, Nature.