Bulletin of Mathematical Biology

, Volume 73, Issue 1, pp 2–32 | Cite as

Interactions Between the Immune System and Cancer: A Brief Review of Non-spatial Mathematical Models

  • Raluca EftimieEmail author
  • Jonathan L. Bramson
  • David J. D. Earn
Review Article


We briefly review spatially homogeneous mechanistic mathematical models describing the interactions between a malignant tumor and the immune system. We begin with the simplest (single equation) models for tumor growth and proceed to consider greater immunological detail (and correspondingly more equations) in steps. This approach allows us to clarify the necessity for expanding the complexity of models in order to capture the biological mechanisms we wish to understand. We conclude by discussing some unsolved problems in the mathematical modeling of cancer-immune system interactions.


Cancer Immunology Tumor-immune system interaction Ordinary differential equations (ODEs) 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adam, J., Bellomo, N., 1997. A Survey of Models for Tumor-Immune System Dynamics. Birkhäuser, Basel. zbMATHGoogle Scholar
  2. Alarcon, T., Byrne, H., Maini, P., 2003. A cellular automaton model for tumor growth in inhomogeneous environment. J. Theor. Biol. 225, 257–274. MathSciNetCrossRefGoogle Scholar
  3. Araujo, R., McElwain, D., 2004. A history of the study of solid tumor growth: the contribution of mathematical modeling. Bull. Math. Biol. 66, 1039–1091. MathSciNetCrossRefGoogle Scholar
  4. Arciero, J., Jackson, T., Kirschner, D., 2004. A mathematical model of tumor-immune evasion and siRNA treatment. Discrete Contin. Dyn. Syst., Ser. B 4(1), 39–58. zbMATHMathSciNetGoogle Scholar
  5. Bajzer, Z., Maruĉić, M., Vuk-Pavlović, S., 1996. Conceptual frameworks for mathematical modeling of tumor growth dynamics. Math. Comput. Model. 23(6), 31–46. zbMATHCrossRefGoogle Scholar
  6. Banerjee, S., Sarkar, R., 2008. Delay-induced model for tumor-immune interaction and control of malignant tumor growth. BioSystems 91, 268–288. CrossRefGoogle Scholar
  7. Bellomo, N., Delitala, M., 2008. From the mathematical kinetic, and stochastic game theory to modeling mutations, onset, progression and immune competition of cancer cells. Phys. Life Rev. 5, 183–206. CrossRefGoogle Scholar
  8. Bellomo, N., Preziosi, L., 2000. Modelling and mathematical problems related to tumor evolution and its interactions with the immune system. Math. Comput. Model. 32, 413–452. zbMATHMathSciNetCrossRefGoogle Scholar
  9. Bellomo, N., Li, N., Maini, P., 2008. On the foundations of cancer modeling: selected topics, speculations, and perspectives. Math. Mod. Methods Appl. Sci. 18(4), 593–646. zbMATHMathSciNetCrossRefGoogle Scholar
  10. Bennett, S., Carbone, F., Karamalis, F., Miller, J., Heath, W.R., 1997. Induction of a CD8+ cytotoxic T lymphocyte response by cross-priming requires cognate CD4+ T cell help. J. Exp. Med. 186(1), 65–70. CrossRefGoogle Scholar
  11. Berner, V., Liu, H., Zhou, Q., Alderson, K.L., Sun, K., Weiss, J.M., Back, T.C., Longo, D.L., Blazar, B.R., Wiltrout, R.H., Welniak, L.A., Redelman, D., Murphy, W.J., 2007. IFN-γ mediates CD4+ T-cell loss and impairs secondary antitumor responses after successful initial immunotherapy. Nat. Med. 13, 354–360. CrossRefGoogle Scholar
  12. Boon, T., van der Bruggen, P., 1996. Human tumor antigens recognized by T lymphocytes. J. Exp. Med. 183, 725–729. CrossRefGoogle Scholar
  13. Bose, T., Trimper, S., 2009. Stochastic model for tumor growth with immunization. Phys. Rev. E 79, 5. MathSciNetGoogle Scholar
  14. Bunimovich-Mendrazitsky, S., Shochat, E., Stone, L., 2007. Mathematical model of BCG immunotherapy in superficial bladder cancer. Bull. Math. Biol. 69, 1847–1870. zbMATHMathSciNetCrossRefGoogle Scholar
  15. Burger, P., Vogel, F., Green, S., Strike, T., 1985. Glioblastoma multiforme and anaplastic astrocytoma, pathologic criteria and prognosis implications. Cancer 56, 1106–1111. CrossRefGoogle Scholar
  16. Burnet, F., 1957. Cancer: a biological approach. Br. Med. J. 1, 779–786. CrossRefGoogle Scholar
  17. Burnet, F., 1967. Immunological aspects of malignant disease. Lancet 1, 1171–1174. CrossRefGoogle Scholar
  18. Byrne, H., Cox, S., Kelly, C., 2004. Macrophage-tumor interactions: in vivo dynamics. Discrete Contin. Dyn. Syst., Ser. B 4(1), 81–98. zbMATHMathSciNetGoogle Scholar
  19. Byrne, H., Alarcon, T., Owen, M., Webb, S., Maini, P., 2006. Modeling aspects of cancer dynamics: a review. Philos. Trans. R. Soc. A 364, 1563–1578. MathSciNetCrossRefGoogle Scholar
  20. Cappuccio, A., Elishmereni, M., Agur, Z., 2006. Cancer immunotherapy by Interleukin-21: potential treatment strategies evaluated in a mathematical model. Cancer Res. 66(14), 7293–7300. CrossRefGoogle Scholar
  21. Castiglione, F., Piccoli, B., 2007. Cancer immunotherapy, mathematical modeling and optimal control. J. Theor. Biol. 247, 723–732. MathSciNetCrossRefGoogle Scholar
  22. Castro, M., Klamt, F., Grieneisen, V., Grivicich, I., Moreira, J., 2003. Gompertzian growth pattern correlated with phenotypic organization of colon carcinoma, malignant glioma and non-small cell lung carcinoma cell lines. Cell Proliferation 36, 65–73. CrossRefGoogle Scholar
  23. Chaplain, M., 2008. Modelling aspects of cancer growth: insight from mathematical and numerical analysis and computational simulation. In: Multiscale Problems in the Life Sciences. Lecture Notes in Mathematics, vol. 1940, pp. 147–200. Springer, Berlin CrossRefGoogle Scholar
  24. Chatterjee, A., Mayawala, K., Edwards, J., Vlachos, D., 2005. Time accelerated Monte Carlo simulations of biological networks using the binomial τ-leap method. Bioinformatics 21(9), 2136–2137. CrossRefGoogle Scholar
  25. Chignola, R., Foroni, R., 2005. Estimating the growth kinetics of experimental tumors from as few as two determinations of tumor size: implications for clinical oncology. IEEE Trans. Biomed. Eng. 52(5), 808–815. CrossRefGoogle Scholar
  26. de Boer, R., Hogeweg, P., Dullens, H., de Weger, R., den Otter, W., 1985. Macrophage T lymphocyte interactions in the anti-tumor immune response: a mathematical model. J. Immunol. 134(4), 2748–2758. Google Scholar
  27. de Pillis, L., Radunskaya, A., 2001. A mathematical tumor model with immune resistance and drug therapy: an optimal control approach. Comput. Math. Methods Med. 3(2), 78–100. Google Scholar
  28. de Pillis, L., Radunskaya, A., 2003a. The dynamics of an optimally controlled tumor model: a case study. Math. Comput. Model. 37, 1221–1244. zbMATHCrossRefGoogle Scholar
  29. de Pillis, L., Radunskaya, A., 2003b. A mathematical model of immune response to tumor invasion. In: Computational Fluid and Solid Mechanics. Proceedings of the Second M.I.T. Conference on Computational Fluid Dynamics and Solid Mechanics, pp. 1661–1668. Google Scholar
  30. de Pillis, L., Radunskaya, A., Wiseman, C., 2005. A validated mathematical model of cell-mediated immune response to tumor growth. Cancer Res. 65(17), 7950–7958. Google Scholar
  31. de Vladar, H., González, J., 2004. Dynamic response of cancer under the influence of immunological activity and therapy. J. Theor. Biol. 227, 335–348. CrossRefGoogle Scholar
  32. di Carlo, E., de Totero, D., Piazza, T., Fabbi, M., Ferrini, S., 2007. Role of IL-21 in immune-regulation and tumor immunotherapy. Cancer Immunol. Immunother. 56, 1323–1334. CrossRefGoogle Scholar
  33. Diefenbach, A., Jensen, E., Jamieson, A., Raulet, D., 2001. Rael and H60 ligands of the NKG2D receptor stimulate tumor immunity. Nature 413, 165–171. CrossRefGoogle Scholar
  34. d’Onofrio, A., 2005. A general framework for modeling tumor-immune system competition and immunotherapy: mathematical analysis and biomedical inferences. Physica D 208, 220–235. zbMATHMathSciNetGoogle Scholar
  35. d’Onofrio, A., 2008. Metamodeling tumor-immune system interaction, tumor evasion and immunotherapy. Math. Comput. Model. 47, 614–637. zbMATHMathSciNetCrossRefGoogle Scholar
  36. Dranoff, G., 2004. Cytokines in cancer pathogenesis and cancer therapy. Nat. Rev. Cancer 4, 11–22. CrossRefGoogle Scholar
  37. Dullens, H., Tol, M.V.D., de Weger, R., Otter, W.D., 1986. A survey of some formal models in tumor immunology. Cancer Immunol. Immunother. 23, 159–164. Google Scholar
  38. Fidler, I., 1985. Macrophages and metastasis: a biological approach to cancer therapy: presidential address. Cancer Res. 45, 4714–4726. Google Scholar
  39. Forys, U., Waniewski, J., Zhivkov, P., 2006. Anti-tumor immunity and tumor anti-immunity in a mathematical model of tumor immunotherapy. J. Biol. Syst. 14(1), 13–30. zbMATHCrossRefGoogle Scholar
  40. Gajewski, T., 2007. Failure at the effector phase: immune barriers at the level of melanoma tumor microenvironment. Clin. Cancer Res. 13(18), 5256–5261. CrossRefGoogle Scholar
  41. Galach, M., 2003. Dynamics of the tumor-immune system competition: the effect of time delay. Int. J. Appl. Math. Comput. Sci. 13(3), 395–406. zbMATHMathSciNetGoogle Scholar
  42. Gerlee, P., Anderson, A., 2009. Evolution of cell motility in an individual-based model of tumor growth. J. Theor. Biol. 259(1), 67–83. CrossRefGoogle Scholar
  43. Gillespie, D., 1976. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22(4), 403–434. MathSciNetCrossRefGoogle Scholar
  44. Goldstein, B., Faeder, J., Hlavacek, W., 2004. Mathematical and computational models of immune-receptor signaling. Nat. Rev. Immunol. 4(6), 445–456. CrossRefGoogle Scholar
  45. Gompertz, B., 1825. On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philos. Trans. R. Soc. Lond. 115, 513–583. CrossRefGoogle Scholar
  46. Greenblatt, M., Bennett, W., Hollstein, M., Harris, C., 1994. Mutations in the p53 tumor suppressor gene: clues to cancer etiology and molecular pathogenesis. Cancer Res. 54, 4855–4878. Google Scholar
  47. Hamilton, D., Bretscher, P., 2008. The commonality in the regulation of the immune response to most tumors: the prevalence of immune class deviation as a tumor escape mechanism and its significance for vaccination and immunotherapy. Cancer Ther. 6, 745–754. Google Scholar
  48. Hart, D., Shochat, E., Agur, Z., 1998. The growth law of primary breast cancer as inferred from mammography screening trials data. Br. J. Cancer 78, 382–387. CrossRefGoogle Scholar
  49. Horny, H.-P., Horst, H.-A., 1986. Lymphoreticular infiltrates in invasive ductal breast cancer. Virchows Arch. 409, 275–286. CrossRefGoogle Scholar
  50. Kammertoens, T., Schüler, T., Blankenstein, T., 2005. Immunotherapy: target the stroma to hit the tumor. Trends Mol. Med. 11(5), 225–231. CrossRefGoogle Scholar
  51. Khar, A., 1997. Mechanisms involved in natural killer cell mediated target cell death leading to spontaneous tumor regression. J. Biosci. 22, 23–31. CrossRefGoogle Scholar
  52. Kim-Schulze, S., Taback, B., Kaufman, H., 2007. Cytokine therapy for cancer. Surg. Oncol. Clin. N. Am. 16(4), 793–818. CrossRefGoogle Scholar
  53. Kirk, D., 2004. Optimal Control Theory: An Introduction. Dover, New York. Google Scholar
  54. Kirschner, D., Panetta, J., 1998. Modeling immunotherapy of the tumor-immune interaction. J. Math. Biol. 37, 235–252. zbMATHCrossRefGoogle Scholar
  55. Knudson, A., 1971. Mutations and cancer: statistical study of retinoblastoma. Proc. Natl. Acad. Sci. 68(4), 820–823. CrossRefGoogle Scholar
  56. Kronik, N., Kogan, Y., Vainstein, V., Agur, Z., 2008. Improving alloreactive CTL immunotherapy for malignant gliomas using a simulation model of their interactive dynamics. Cancer Immunol. Immunother. 57, 425–439. CrossRefGoogle Scholar
  57. Kruse, C., Cepeda, L., Owens, B., Johnson, S., Stears, J., Lillehei, K., 1997. Treatment of recurrent glioma with intracavity alloreactive cytotoxic T lymphocytes and Interleukin-2. Cancer Immunol. Immunother. 45, 77–87. CrossRefGoogle Scholar
  58. Kurtz, T., 1971. Limit theorems for sequences of jump Markov processes approximating ordinary differential processes. J. Appl. Probab. 8(2), 344–356. zbMATHMathSciNetCrossRefGoogle Scholar
  59. Kuznetsov, Y., 1994. Elements of Applied Bifurcation Theory. Springer, London. Google Scholar
  60. Kuznetsov, V., Makalkin, I., Taylor, M., Perelson, A., 1994. Nonlinear dynamics of immunogenic tumors: parameter estimation and global bifurcation analysis. Bull. Math. Biol. 2(56), 295–321. Google Scholar
  61. Laird, A., 1964. Dynamics of tumor growth. Br. J. Cancer 18, 490–502. CrossRefGoogle Scholar
  62. Lala, P., Patt, H., 1966. Cytokinetic analysis of tumor growth. Proc. Natl. Acad. Sci. 56, 1735–1742. CrossRefGoogle Scholar
  63. Ledzewicz, U., Munden, J., Schättler, H., 2009. Scheduling of angiogenic inhibitors for Gompertzian and logistic tumor growth models. Discrete Contin. Dyn. Syst. Ser. B 12(2), 415–438. zbMATHMathSciNetCrossRefGoogle Scholar
  64. Leek, R., Harris, A., 2002. Tumor-associated macrophages in breast cancer. J. Mammary Gland Biol. Neoplasi 7(2), 177–189. CrossRefGoogle Scholar
  65. Lejeune, O., Chaplain, M., Akili, I.E., 2008. Oscillations and bistability in the dynamics of cytotoxic reactions mediated by the response of immune cells to solid tumors. Math. Comput. Model. 47, 649–662. zbMATHCrossRefGoogle Scholar
  66. Lin, A., 2004. A model of tumor and lymphocyte interactions. Discrete Contin. Dyn. Syst. Ser. B 4(1), 241–266. zbMATHMathSciNetCrossRefGoogle Scholar
  67. Lollini, P., Motta, S., Pappalardo, F., 2006a. Discovery of cancer vaccination protocols with a genetic algorithm driving and agent based simulator. BMC Bioinform. 7, 352. CrossRefGoogle Scholar
  68. Lollini, P., Motta, S., Pappalardo, F., 2006b. Modeling tumor immunology. Math. Mod. Methods Appl. Sci. 16(7S), 1091–1124. zbMATHMathSciNetCrossRefGoogle Scholar
  69. Malet, D., de Pillis, L., 2006. A cellular automata model of tumor-immune system interactions. J. Theor. Biol. 239, 334–350. CrossRefGoogle Scholar
  70. Mansury, Y., Diesboeck, T., 2003. The impact of “search precision” in an agent-based tumor model. J. Theor. Biol. 224, 325–337. CrossRefGoogle Scholar
  71. Mansury, Y., Diggory, M., Deisboeck, T., 2006. Evolutionary game theory in an agent-based brain tumor model: exploring the ‘Genotype-Phenotype’ link. J. Theor. Biol. 238, 146–156. MathSciNetCrossRefGoogle Scholar
  72. Mantovani, A., Sozzani, S., Locati, M., Allavena, P., Sica, A., 2002. Macrophage polarization: tumor-associated macrophages as a paradigm for polarized M2 mononuclear phagocytes. TRENDS Immunol. 23(11), 549–555. CrossRefGoogle Scholar
  73. Martins, M., Ferreira, S.C. Jr., Vilela, M., 2007. Multiscale models for the growth of avascular tumors. Phys. Life Rev. 4, 128–156. CrossRefGoogle Scholar
  74. Matkowski, B., 1970. Nonlinear dynamics stability. SIAM J. Appl. Math. 18, 872–883. MathSciNetCrossRefGoogle Scholar
  75. Mattes, J., Hulett, M., Xie, W., Hogan, S., Rothenberg, M., Foster, P., Parish, C., 2003. Immunotherapy of cytotoxic T cell-resistant tumor by T helper 2 cells: an eotaxin and STAT6-dependent process. J. Exp. Med. 197(3), 387–393. CrossRefGoogle Scholar
  76. Matzavinos, A., Chaplain, M., Kuznetsov, V., 2004. Mathematical modeling of the spatiotemporal response of cytotoxic T-lymphocytes to a solid tumour. Math. Med. Biol. 21(1), 1–34. zbMATHCrossRefGoogle Scholar
  77. Menta, B., Agarwal, M., 1980. Cyclic oscillations in leukocyte count in chronic myeloid leukemia. Acta Haematol. 63, 68–70. CrossRefGoogle Scholar
  78. Merola, A., Cosentino, C., Amato, F., 2008. An insight into tumor dormancy equilibrium via the analysis of its domain of attraction. Biomed. Sign. Process. Control 3, 212–219. CrossRefGoogle Scholar
  79. Michelson, S., Leith, J., 1993. Growth factors and growth control of heterogeneous populations. Bull. Math. Biol. 55, 993–1011. zbMATHGoogle Scholar
  80. Michelson, S., Miller, B., Glicksman, A., Leith, J., 1987. Tumor micro-ecology and competitive interactions. J. Theor. Biol. 128, 233–246. MathSciNetCrossRefGoogle Scholar
  81. Moore, H., Li, N., 2004. A mathematical model for chronic myelogenous leukemia (CML) and T cell interaction. J. Theor. Biol. 227, 513–523. MathSciNetCrossRefGoogle Scholar
  82. Mueller, M., Fusenig, N., 2004. Friends or foes—bipolar effects of the tumor stroma in cancer. Nat. Rev. 4, 839–849. Google Scholar
  83. Nagy, J., 2005. The ecology and evolutionary biology of cancer: a review of mathematical models of necrosis and tumor cells diversity. Math. Biosci. Eng. 2(2), 381–418. MathSciNetGoogle Scholar
  84. Naito, Y., Saito, K., Shiiba, K., Ohuchi, A., Saigenji, K., Nagura, H., Ohtani, H., 1998. CD8+ T cells infiltrated within cancer cell nests as a prognostic factor in human colorectal cells. Cancer Res. 58, 3491–3494. Google Scholar
  85. Nani, F., Freedman, H., 2000. A mathematical model of cancer treatment by immunotherapy. Math. Biosci. 163, 159–199. zbMATHMathSciNetCrossRefGoogle Scholar
  86. Norton, L., 1988. A Gompertzian model of human breast cancer growth. Cancer Res. 48, 7067–7071. Google Scholar
  87. Nowak, M., Komarova, N., Sengupta, A., Jallepalli, P., Shih, I., Vogelstein, B., Lengauer, C., 2002. The role of chromosomal instability in tumor initiation. Proc. Natl. Acad. Sci. 99(25), 16226–16231. CrossRefGoogle Scholar
  88. Orucevic, A., Lala, P., 1998. Role of nitric oxide in IL-2 therapy-induced capillary leak syndrome. Cancer Metastasis Rev. 17(1), 127–142. CrossRefGoogle Scholar
  89. Owen, M., Sherratt, J., 1998. Modeling the macrophage invasion of tumors: effects on growth and composition. Math. Med. Biol. 15, 165–185. zbMATHCrossRefGoogle Scholar
  90. Owen, M., Byrne, H., Lewis, C., 2004. Mathematical modeling of the use of macrophages as vehicles for drug delivery to hypoxic tumor sites. J. Theor. Biol. 226, 377–391. MathSciNetCrossRefGoogle Scholar
  91. Page, K., Uhr, J., 2005. Mathematical models of cancer dormancy. Leuk. Lymphoma 46, 313–327. CrossRefGoogle Scholar
  92. Panetta, J., 1998. A mathematical model of drug resistance: heterogeneous tumors. Math. Biosci. 147, 41–61. zbMATHCrossRefGoogle Scholar
  93. Pappalardo, F., Lollini, P.-L., Castiglione, F., Motta, S., 2005. Modelling and simulation of cancer immunoprevention vaccine. Bioinformatics 21, 2891. CrossRefGoogle Scholar
  94. Pappalardo, F., Motta, S., Lollini, P.-L., Mastriani, E., 2006. Analysis of vaccine’s schedule using models. Cell. Immunol. 244, 137–140. CrossRefGoogle Scholar
  95. Parish, C., 2003. Cancer immunotherapy: the past, the present and the future. Immunol. Cell Biol. 81, 106–113. CrossRefGoogle Scholar
  96. Parmiani, G., Rivoltini, L., Andreola, G., Carrabba, M., 2000. Cytokines in cancer therapy. Immunol. Lett. 74(1), 41–44. CrossRefGoogle Scholar
  97. Perez-Diez, A., Joncker, N., Choi, K., Chan, W., Anderson, C., Lantz, O., Matzinger, P., 2007. CD4 cells can be more efficient at tumor rejection than CD8 cells. Blood 109, 5346–5354. CrossRefGoogle Scholar
  98. Perko, L., 2001. Differential Equations and Dynamical Systems. Springer, Berlin. zbMATHGoogle Scholar
  99. Quesnel, B., 2008. Dormant tumor cells as therapeutic target? Cancer Lett. 267, 10–17. CrossRefGoogle Scholar
  100. Roose, T., Chapman, S., Maini, P., 2007. Mathematical models of avascular tumor growth. SIAM Rev. 49(2), 179–208. zbMATHMathSciNetCrossRefGoogle Scholar
  101. Rosenberg, S., 1991. Immunotherapy and gene therapy of cancer. Cancer Res. 51, 5074s–5079s. Google Scholar
  102. Rosenberg, S., Yang, J., Restifo, N., 2004. Cancer immunotherapy: moving beyond current vaccines. Nat. Med. 10, 909–915. CrossRefGoogle Scholar
  103. Sachs, R., Hlatky, L., Hahnfeldt, P., 2001. Simple ODE models of tumor growth and anti-angiogenic or radiation treatment. Math. Comput. Model. 33, 1297–1305. zbMATHMathSciNetCrossRefGoogle Scholar
  104. Samad, H.E., Khammash, M., Petzold, L., Gillespie, D., 2005. Stochastic modeling of gene regulatory networks. Int. J. Robust Nonlinear Control 15, 691–711. zbMATHCrossRefGoogle Scholar
  105. Sarkar, R., Banerjee, S., 2005. Cancer self remission and tumor stability—a stochastic approach. Math. Biosci. 196, 65–81. zbMATHMathSciNetCrossRefGoogle Scholar
  106. Skipper, H., Schabel, F.H. Jr., 1982. Quantitative and cytokinetic studies in experimental tumor systems. In: Holland, J., Frey E. III (Eds.), Cancer Medicine, 2nd edn., pp. 636–648. Google Scholar
  107. Smyth, M., Godfrey, D., Trapani, J., 2001. A fresh look at tumor immunosurveillance and immunotherapy. Nat. Immunol. 2, 293–299. CrossRefGoogle Scholar
  108. Sotolongo-Costa, O., Molina, L.M., Perez, D.R., Antoraz, J., Reyes, M.C., 2003. Behavior of tumors under nonstationary therapy. Physica D 178, 242–253. zbMATHMathSciNetGoogle Scholar
  109. Spratt, J., Meyer, J., Spratt, J., 1996. Rates of growth of human neoplasms: part ii. J. Surg. Oncol. 61, 68–83. CrossRefGoogle Scholar
  110. Stepanova, N., 1980. Course of the immune reaction during the development of a malignant tumor. Biophysics 24, 917–923. Google Scholar
  111. Stuart, J., 1960. On the nonlinear mechanism of wave disturbances in stable and unsable parallel flows. Part I.. J. Fluid Mech. 9, 353–370. zbMATHMathSciNetCrossRefGoogle Scholar
  112. Swan, G., 1985. Optimal control applications in the chemotherapy of multiple myeloma. IMA J. Math. Appl. Med. Biol. 2(3), 139–160. zbMATHMathSciNetCrossRefGoogle Scholar
  113. Szymanska, Z., 2003. Analysis of immunotherapy models in the context of cancer dynamics. Int. J. Appl. Math. Comput. Sci. 13, 407–418. zbMATHMathSciNetGoogle Scholar
  114. Takayanagi, T., Ohuchi, A., 2001. A mathematical analysis of the interactions between immunogenic tumor cells and cytotoxic T lymphocytes. Microbiol. Immunol. 45(10), 709–715. Google Scholar
  115. Teng, M., Swann, J., Koebel, C., Schreiber, R., Smyth, M., 2008. Immune-mediated dormancy: an equilibrium with cancer. J. Leukoc. Biol. 84, 988–993. CrossRefGoogle Scholar
  116. Thorne, B., Bailey, A., Peirce, S., 2007. Combining experiments with multi-cell agent-based modeling to study biological tissue patterning. Brief. Bioinform. 8(4), 245–257. CrossRefGoogle Scholar
  117. Villasana, M., Radunskaya, A., 2003. A delay differential equation model for tumor growth. J. Math. Biol. 47, 270–294. zbMATHMathSciNetCrossRefGoogle Scholar
  118. von Bertalanffy, L., 1957. Quantitative laws in metabolism and growth. Q. Rev. Biol. 32, 217–231. CrossRefGoogle Scholar
  119. von Stebut, E., Metz, M., Milon, G., Knop, J., Maurer, M., 2003. Early macrophage influx to sites of cutaneous granuloma formation is dependent on MIP-1α/β released from neutrophils recruited by mast cell-derived TNF-α. Blood 101, 210–215. CrossRefGoogle Scholar
  120. Waldhauer, I., Steinle, A., 2008. NK cells and cancer immunosurveillance. Oncogene 27, 5932–5943. CrossRefGoogle Scholar
  121. Wang, S., Hinow, P., Bryce, N., Weaver, A., Estrada, L., Artega, C., Webb, G., 2009. A mathematical model quantifies proliferation and motility effects of TGF-β on cancer cells. Comput. Math. Methods Med. 10(1), 71–83. MathSciNetCrossRefGoogle Scholar
  122. Webb, S., Owen, M., Byrne, H.M., Murdoch, C., Lewis, C., 2007. Macrophage-based anti-cancer therapy: modeling different modes of tumour targeting. Bull. Math. Biol. 69(5), 1747–1776. zbMATHMathSciNetCrossRefGoogle Scholar
  123. Weedon-Fekjaer, H., Lindqvist, B., Vatten, L., Aalen, O., Tretli, S., 2008. Breast cancer tumor growth estimated through mammography screening data. Breast Cancer Res. 10, R41. CrossRefGoogle Scholar
  124. Weldon, T., 1988. Mathematical Models in Cancer Research. Hilger, Bristol. Google Scholar
  125. Zhang, S., Bernard, D., Khan, W., Kaplan, M., Bramson, J., Wan, Y., 2009. CD4+ T-cell-mediated anti-tumor immunity can be uncoupled from autoimmunity via the STAT4/STAT6 signaling axis. Eur. J. Immunol. 39, 1252–1259. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2010

Authors and Affiliations

  • Raluca Eftimie
    • 1
    Email author
  • Jonathan L. Bramson
    • 2
    • 3
  • David J. D. Earn
    • 1
    • 3
  1. 1.Department of Mathematics and StatisticMcMaster UniversityHamiltonCanada
  2. 2.Centre for Gene Therapeutics, Department of Pathology and Molecular MedicineMcMaster UniversityHamiltonCanada
  3. 3.Michael G. DeGroote Institute for Infectious Disease ResearchMcMaster UniversityHamiltonCanada

Personalised recommendations