General Aspects of Modeling Tumor Growth and Immune Response

  • John A. Adam
Part of the Modeling and Simulation in Science, Engineering, & Technology book series (MSSET)


This chapter opens with a summary of the philosophy and methodology of mathematical modeling (Sections 2.1 and 2.2). In Section 2.3 a survey of deterministic diffusion models of spheroid growth is provided. Section 2.4 addresses a particular type of model, based on a predator-prey description of the immune response to cancer illustrating the modeling pro- cess in some detail. This is followed in Section 2.5 by a spatially-dependent approach to the immune response in a one-dimensional system. Section 2.6 is more speculative in nature: the suggestion is made that the concepts of “tunneling” (as used in quantum mechanics) and “catastrophe” may be applicable to both the development of cancer and the effectiveness of the immune response. Two appendices follow on (i) catastrophe theory and (ii) certain mathematical properties of solutions to diffusion equations.


Modeling Tumor General Aspect Necrotic Core Catastrophe Theory Tumor Surface 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [ADa]
    Adam J.A., A simplified mathematical model of tumor growth, Math. Biosci., 81 (1986), 229–244.MATHCrossRefGoogle Scholar
  2. [ADb]
    Adam J.A., A mathematical model of tumor growth: II. Effects of geometry and spatial non-uniformity on stability, Math. Biosci., 86 (1987), 183–211.MATHCrossRefGoogle Scholar
  3. [ADc]
    Adam J.A., A mathematical model of tumor growth: III. Comparison with experiment, Math. Biosci., 86 (1987), 213–227.MATHCrossRefGoogle Scholar
  4. [ADd]
    Adam J.A. and Maggelakis S.A., Diffusion regulated growth characteristics of a prevascular carcinoma, Bull. Math. Biol., 52 (1990), 549–582.MATHGoogle Scholar
  5. [ADe]
    Adam J.A., On complementary levels of description in applied mathematics. II. Mathematical models in cancer biology, Int. Jnl. Math. Ed. Sci. Tech., 19 (1988), 519–535.MathSciNetCrossRefGoogle Scholar
  6. [ADf]
    Adam J.A. and Noren R., Equilibrium model of a vascularized spherical carcinoma with central necrosis: some properties of the solution, J. Math. Biol, 31 (1993), 735–745.MathSciNetMATHCrossRefGoogle Scholar
  7. [ADg]
    Adam J.A., The effects of vascularization on lymphocyte-tumor cell dynamics: qualitative features, Math. Comp. Modelling-Special issue on modeling and simulation problems on tumor/immune system dynamics, Bellomo N. ed., 23 (1996), 1–10.Google Scholar
  8. [ADh]
    Adam J.A., The dynamics of growth-factor-modified immune response to cancer growth: one dimensional models, Math. Comp. Modelling, 17 (1993), 83–106.MathSciNetMATHCrossRefGoogle Scholar
  9. [ADi]
    Adam J.A. and Panetta J.C., A simple mathematical model and alternative paradigm for certain chemotherapeutic regimes, Math. Comp. Modelling, 22 (1995), 49–60.MathSciNetMATHCrossRefGoogle Scholar
  10. [ADj]
    Adam J.A., Solution uniqueness and stability criteria for a model of growth factor production, Appl. Math. Lett., 5 (1992), 89–92.MathSciNetMATHCrossRefGoogle Scholar
  11. [ADk]
    Adam J. A., Asymptotic solutions and spectral theory of linear wave equations, Phys. Repts., 86 (1982), 217–316.MathSciNetCrossRefGoogle Scholar
  12. [ADI]
    Adam J.A., Critical layer singularities and complex eigenvalues in some differential equations of mathematical physics, Phys. Repts., 142 (1986), 263–356.MathSciNetCrossRefGoogle Scholar
  13. [ADm]
    Adam J.A., A linear scattering problem in magnetohydrodynamics: transmission resonances in a magnetic slab, Astrophys. Sp. Sci., 133 (1987), 317–337.MATHCrossRefGoogle Scholar
  14. [ADn]
    Adam J.A., Non-radial stellar oscillations: a perspective from potential scattering. I. Theoretical foundations, Astrophys. Sp. Sci., 220 (1994), 179–233.MathSciNetMATHCrossRefGoogle Scholar
  15. [ADo]
    Adam J.A. and Maggelakis S.A., A mathematical model of tumor growth. IV. Effects of a necrotic core, Math. Biosci., 97 (1989), 121–136.MATHCrossRefGoogle Scholar
  16. [BAa]
    Baym G., Lectures on Quantum Mechanics, Benjamin (1989).Google Scholar
  17. [BDa]
    Bender E.A., An Introduction to Mathematical Modeling, Wiley (1978).Google Scholar
  18. [BEa]
    Bell G.I., Predator-prey equations simulating an immune response, Math. Biosci., 16 (1973), 291–314.MATHCrossRefGoogle Scholar
  19. [BEb]
    Bell G.I., Some models for the interaction between cells of the immune system, in Systems Theory in Immunology, Brini C., Doria G., Koch G., and Strom R. eds., Lecture Notes in Biomath-ematics, Vol. 32, Springer-Verlag (1979), 66–74.Google Scholar
  20. [BJa]
    Bajzer Z. and Vuk-Pavlovic S., Quantitative aspects of autocrine regulation in tumors, Crit. Rev. Oncog., 2 (1990), 53–73.Google Scholar
  21. [BJb]
    Bajzer Z., Marusic M., and Vuk-Pavlovic S., Conceptual frameworks for mathematical modeling of tumor growth dynamics, Math. Comp. Modelling-Special Issue on Modelling and Simulation Problems on Tumor-Immune System Dynamics, Bellomo N. ed., 23 (1996), 31–46.Google Scholar
  22. [BLa]
    Bellomo N. and Forni G., Dynamics of tumor interaction with the host immune system, Math. Comp. Modelling, 20 (1994), 107–122.MATHCrossRefGoogle Scholar
  23. [BNa]
    Burton A.C., Rate of growth of solid tumors as a problem of diffusion, Growth, 30 (1966), 159–176.Google Scholar
  24. [BRa]
    Britton N.F. and Chaplain M.A.J., A qualitative analysis of some models of tissue growth, Math. Biosci., 113 (1993), 77–89.MATHCrossRefGoogle Scholar
  25. [BUa]
    Bullough W.W., Mitotic and functional homeostasis: a speculative review, Cancer Res., 25 (1965), 1683–1727.Google Scholar
  26. [BUb]
    Bullough W.W. and Deol J.U.R., The pattern of tumor growth, Symp. Soc. Exp. Biol., 25 (1971), 225–275.Google Scholar
  27. [BYa]
    Byrne H.M. and Chaplain M.A.J., Growth of non-necrotic tumors in the presence and absence of inhibitors, Math. Biosci., 130 (1995), 151–181.MATHCrossRefGoogle Scholar
  28. [CAa]
    Casciari J.J., Sotiochos S.V., and Sutherland R.M., Mathematical modelling of microenvironment and growth in EMT6/Ro multicellular tumor spheroids, Cell Prolif., 25 (1992), 1–22.CrossRefGoogle Scholar
  29. [CHa]
    Chaplain M.A.J. and Britton N.F., On the concentration profile of a growth inhibitory factor in multicell spheroids, Math. Biosci., 115 (1993), 233–243.MATHCrossRefGoogle Scholar
  30. [CHb]
    Chaplain M.A.J., Benson D.L., and Maini P.K., Nonlinear diffusion of a growth inhibitory factor in multicell spheroids, Math. Biosci., 121 (1994), 1–13.MATHCrossRefGoogle Scholar
  31. [CHc]
    Chaplain M.A.J., The development of a spatial pattern in a model for cancer growth, in Experimental and Theoretical Advances in Biological Pattern Formation, Othmer H.G., Maini P.K., and Murray J.D. eds., Plenum Press (1993), 45–60.Google Scholar
  32. [CHd]
    Chaplain M.A.J. and Sleeman B.D., A mathematical model for the growth and classification of a solid tumor: a new approach via nonlinear elasticity theory using strain-energy functions, Math. Biosci., 111 (1992), 169–215.MathSciNetMATHCrossRefGoogle Scholar
  33. [CHe]
    Chaplain M.A.J. and Sleeman B.D., Modelling the growth of solid tumors and incorporating a method for their classification using nonlinear elasticity theory, J. Math. Biol., 31 (1993), 431–473.MathSciNetMATHCrossRefGoogle Scholar
  34. [CHf]
    Chaplain M.A.J., Avascular growth, angiogenesis, and vascular growth in solid tumors: the mathematical modelling of the stages of tumor development, Math. Comp. Modelling-Special issue on modeling and simulation problems on tumor/immune system dynamics, Bellomo N. ed., 23 (1996), 47–88.Google Scholar
  35. [CRa]
    Craig I.J.D. and Brown J.C., Inverse Problems in Astronomy, Adam Hilger (1986).Google Scholar
  36. [DEa]
    Deakin A., Model for the growth of a solid in-vitro tumor, Growth, 39 (1975), 159–165.Google Scholar
  37. [DEb]
    Deakin M.A.B., Applied catastrophe theory in the social and biological sciences, Bull. Math. Biology, 42 (1980), 647–679.MathSciNetMATHGoogle Scholar
  38. [DLa]
    Delisi C. and Rescigno A., Immune surveillance and neoplasia-I. A minimal mathematical model, Bull. Math. Biol., 39 (1977), 201–221.MathSciNetMATHGoogle Scholar
  39. [EDa]
    Edelstein-Keshet L., Mathematical Models in Biology, Random House (1988).Google Scholar
  40. [EKa]
    Ekeland I., Mathematics and the Unexpected, University of Chicago Press (1988), Chapter 3.Google Scholar
  41. [FIa]
    Fife P.C., Mathematical aspects of reacting and diffusing systems, in Lecture Notes in Biomathematics, 28 (1979), Springer-Verlag.Google Scholar
  42. [FKa]
    Franko A.J. and Sutherland R.M., Oxygen diffusion distance and the development of necrosis in multicell spheroids, Radiat. Res., 79 (1979), 439–453.CrossRefGoogle Scholar
  43. [FOa]
    Folkman J., Tumor angiogenesis, Adv. Cancer Res., 19 (1974), 331–358.CrossRefGoogle Scholar
  44. [FOb]
    Folkman J. and Greenspan H.P., Influence of geometry on control of cell growth, Biochim. Biophys. Acta, 417 (1975), 211–236.Google Scholar
  45. [FOc]
    Folkman F. and Klagsbrun M., Angiogenic factors, Science, 235 (1987), 442–447.CrossRefGoogle Scholar
  46. [FOd]
    Folkman J. and Hochberg M., Self-regulation of growth in three dimensions, J. Exp. Med., 138 (1973), 745–753.CrossRefGoogle Scholar
  47. [FRa]
    Freyer J.P., Regulation of growth saturation and development of necrosis in EMT6/Ro multicellular spheroids by the glucose and oxygen supply, Cancer Res., 46 (1986), 3504–3512.Google Scholar
  48. [FRb]
    Freyer J.P., Role of necrosis in regulating the growth saturation of multicellular spheroids, Cancer Res., 48 (1988), 2432–2439.Google Scholar
  49. [GDa]
    Goodwin B.C. and Trainor L.E.H., A field description of the cleavage process in embryogenesis, J. Theor. Biol., 85 (1980), 757–770.CrossRefGoogle Scholar
  50. [GFa]
    Griffel D.H., Applied Functional Analysis, Ellis Harwood (1981).Google Scholar
  51. [GHa]
    Ghosh Roy D.N., Methods of Inverse Problems in Physics, CRC Press (1991).Google Scholar
  52. [GLa]
    Glass L., Instability and mitotic patterns in tissue growth, J. Dyn. Syst. Meas. Control, 95 (1973), 324–327.CrossRefGoogle Scholar
  53. [GOa]
    Goldacre R.J. and Sylven B., On the access of blood-borne dyes to various tumor regions, Br. J. Cancer, 16 (1962), 306–322.CrossRefGoogle Scholar
  54. [GRa]
    Greenspan, H.P., Models for the growth of a solid tumor by diffusion, Stud. Appl. Math., 51 (1972), 317–340.MATHGoogle Scholar
  55. [GRb]
    Greenspan H.P., On the self-inhibited growth of cell cultures, Growth, 38 (1974), 81–95.Google Scholar
  56. [GRc]
    Greenspan H.P., On the growth and stability of cell cultures and solid tumors, J. Theor. Biol, 56 (1976), 229–242.MathSciNetCrossRefGoogle Scholar
  57. [GSa]
    Goustin A.S., Loef E.B., Shipley G.D., and Moses H.L., Growth factors and cancer, Cancer Res., 46 (1986), 1015–1018.Google Scholar
  58. [HEa]
    Henkart P.A., Mechanism of lymphocyte-mediated cytotoxicity, Ann. Rev. Immunol., 3 (1985), 31–58.CrossRefGoogle Scholar
  59. [JAa]
    Jain R.K. and Wei J., Dynamics of drug transport in solid tumors: distributed parameter model, J. Bioeng., 1 (1977), 313–330.Google Scholar
  60. [JAb]
    Jain R.K., Barriers to drug delivery in solid tumors, Sci. Amer., 271 (1994), 58–65.CrossRefGoogle Scholar
  61. [JAc]
    Jain R.K., Comment made by him during a lecture at the International Center for Mathematical Sciences, February 1995.Google Scholar
  62. [JOa]
    Jones D.S. and Sleeman B., Differential Equations and Mathematical Biology, George Allen and Unwin (1983).Google Scholar
  63. [KEa]
    Kendall D.G., A form of wave propagation associated with the equation of heat conduction, Proc. Camb. Phil. Soc, 44 (1948), 591–594.MathSciNetMATHCrossRefGoogle Scholar
  64. [KIa]
    King W.E., Schultz D.S., and Gatenby R.A., Multi-region models for describing oxygen tension profiles in human tumors, Chem. Eng. Commun., 47 (1986), 73–91.CrossRefGoogle Scholar
  65. [LAa]
    Laird A.K., Dynamics of tumor growth. Comparison of growth rates and extrapolation of growth curve to one cell, Br. J. Cancer, 19 (1965), 278–291.CrossRefGoogle Scholar
  66. [LEa]
    Lefever R. and Garay R.P., A mathematical model of the immune surveillance against cancer, in Theoretical Immunology, Bell G.I., Perelson A.S., and Pimbley G. eds., Marcel Dekker (1978), 481–518.Google Scholar
  67. [LEb]
    Lefever R. and Garay R.P., Local description of immune tumor rejection, in Biomathematics and Cell Kinetics, Vallerron A.J. and MacDonald P.D.M. eds., Elsevier (1978), 333–344.Google Scholar
  68. [LEc]
    Lefever R. and Erneux T., On the growth of cellular tissues under constant and fluctuating environmental conditions, in Nonlinear Electrodynamics in Biological Systems, Ross W. and Lawrence A. eds., Plenum Press (1984), 287–305.Google Scholar
  69. [LEd]
    Lefever R. and Horsthemke W., Bistability in fluctuating environments. Implications in tumor immunology, Bull. Math. Biol., 41 (1979), 469–490.MATHGoogle Scholar
  70. [LEe]
    Lefever R., Hiernaux J., Urbain J., and Meyers P., On the kinetics and optimal specificity of cytotoxic reactions mediated by T-lymphocyte clones, Bull. Math. Biol., 54 (1992), 839–873.MATHGoogle Scholar
  71. [LVa]
    Levins R., Evolution in Changing Environments, Princeton University Press (1968), 7.Google Scholar
  72. [MAa]
    Maggelakis S., Type αand type ßtransforming growing factors as regulators of cancer cellular growth: a mathematical model, Math. Comp. Modelling, 18 (1993), 9–16.MATHCrossRefGoogle Scholar
  73. [MAb]
    Maggelakis S. and Adam J.A., Mathematical model for prevascu-lar growth of a spherical carcinoma, Math. Comp. Modelling, 13 (1990), 23–38.MATHCrossRefGoogle Scholar
  74. [MCa]
    McElwain D.L.S. and Ponzo P.J., A model for the growth of solid tumor with non-uniform oxygen consumption, Math. Biosci., 35 (1977), 267–279.MATHCrossRefGoogle Scholar
  75. [MCb] McElwain D.L.S. and Morris L.E., Apoptosis as a volume loss mechanism in mathematical models of solid tumor growth, Math. Biosci., 39 (1978), 147–157.CrossRefGoogle Scholar
  76. [MCc] McElwain D.L.S., Callcott R., and Morris L.E., A model of vascular compression in solid tumors, J. Theor. Biol., 78 (1979), 405–415.CrossRefGoogle Scholar
  77. [MEa] Messiah A., Quantum Mechanics, North-Holland (1961).Google Scholar
  78. [MKa] Müller-Klieser W.F. and Sutherland R.M., Influence of convection in the growth medium on oxygen tensions in multicell tumor spheroids, Cancer Res., 42 (1982), 237–242.Google Scholar
  79. [MKb] Müller-Klieser W.F. and Sutherland R.M., Oxygen tensions in multicell spheroids of two cell lines at different stages of growth, Br. J. Cancer, 45 (1982), 256–264.CrossRefGoogle Scholar
  80. [MKc] Müller-Klieser W.F. and Sutherland R.M., Frequency distribution histograms of oxygen tensions in multicell spheroids, in Oxygen Transport to Tissue, Bicker H.I. and Briley D.F. eds., Plenum Press (1983), Vol. IV, 497–508.Google Scholar
  81. [MRa] Marusic M., Bajzer Z., Freyer J.P., and Vuk-Pavlovic S., Modeling autostimulation of growth in multicellular tumor spheroids, Int. J. Biomed. Comput., 29 (1991), 149–158.CrossRefGoogle Scholar
  82. [MRb] Marusic M. and Bajzer Z., Generalized two-parameter equation of growth, J. Math. Anal. Applics., 179 (1993), 446–462.MathSciNetMATHCrossRefGoogle Scholar
  83. [MRc] Marusic M., Bajzer Z., Freyer J.P., and Vuk-Pavlovic S., Analysis of growth of multicellular tumor spheroids by mathematical models, Cell Prolif., 27 (1994), 73–94.CrossRefGoogle Scholar
  84. [MRd] Marusic M., Bajzer Z., Vuk-Pavlovic S., and Freyer J.P., Tumor growth in-vivo and as multicellular spheroids compared by mathematical models, Bull. Math. Biol., 56 (1994), 617–631.MATHGoogle Scholar
  85. [MUa] Murray J.D., Mathematical Biology, Springer-Verlag (1989).Google Scholar
  86. [MXa] Marx J.L., How cancer cells spread in the body, Science, 244 (1989), 47–48.Google Scholar
  87. [OLa] Old L.J., Tumor necrosis factor, Sci. Amer., 258 (1988), 59–75.CrossRefGoogle Scholar
  88. [PEa] Perelson A.S. and Kauffman S.A. eds., Molecular Evolution on Rugged. Landscapes: Proteins, RNA and the Immune System, Addison-Wesley (1991).Google Scholar
  89. [PGa] Prigogine I. and Lefever R., Stability problems in cancer growth and nucleation, Comp. Biochem. Physiol, 67B (1980), 389–393.Google Scholar
  90. [PIa] Pimbley G.H., Periodic solutions of predator-prey equations simulating an immune response, I, Math. Biosci., 20 (1974), 27–51.MathSciNetMATHCrossRefGoogle Scholar
  91. [PIb] Pimbley G.H., Periodic solutions of predator-prey equations simulating an immune response, II, Math. Biosci., 21 (1974), 251–277.MathSciNetMATHCrossRefGoogle Scholar
  92. [PRa] Prehn R.T., Stimulatory effects of immune reactions upon the growth of transplanted tumors, Cancer Res., 55 (1994), 908–914.Google Scholar
  93. [QIa] Qi A.-S., Multiple solutions of model describing cancerous growth, Bull. Math. Biol, 50 (1988), 1–17.MathSciNetMATHGoogle Scholar
  94. [REa] Rescigno A. and Le Lisi C., Immune surveillance and neoplasia-II. A two-stage mathematical model, Bull. Math. Biol., 39 (1977), 487–497.MATHGoogle Scholar
  95. [ROa] Roberts A.B., Anzano M.A., Wakefield L.M., Roche N.S., Roche D.F., Stern D.F., and Sporn M.B., Type beta-transforming growth factor: a bifunctional regulator of cellular growth, Proc. Nat. Acad. Sci., (1985), 119–121.Google Scholar
  96. [SCa] Schiff L.I., Quantum Mechanics, McGraw-Hill (1968).Google Scholar
  97. [SHa] Shymko R.M. and Glass L., Cellular and geometric control of tissue growth and mitotic instability, J. Theor. Biol., 63 (1976), 355–374.CrossRefGoogle Scholar
  98. [SPa] Sporn M.B. and Todaro G.J., Autocrine secretion and malignant transformation of cells, New Engl. J. Med., 303 (1980), 878–880.CrossRefGoogle Scholar
  99. [SPb] Sporn M.B. and Roberts A.B., Autocrine growth factors and cancer, Nature, 313 (1985), 745–747.CrossRefGoogle Scholar
  100. [SRa] Sherratt J.A. and Nowak M.A., Oncogenes, antioncogenes and the immune response to cancer, Proc. Roy. Soc. Lond., B 248 (1992), 261–271.CrossRefGoogle Scholar
  101. [SRb] Sherratt J.A., Cellular growth control and traveling waves of cancer, SIAM J. Appl. Math., 53 (1993), 1713–1730.MathSciNetMATHCrossRefGoogle Scholar
  102. [SUa] Sutherland R.M., McCredie J.A., and Inch W.R., Growth of multicellular spheroids in tissue culture as a model of nodular carcinomas, J. Natl. Cancer Inst., 46 (1971), 113–120.Google Scholar
  103. [SUb] Sutherland R.M. and Durand R.E., Radiation response of multicellular spheroids: an in-vitro tumor model, Curr. Top. Rad. Res., 11 (1976), 87–139.Google Scholar
  104. [SUc] Sutherland R.M. and Durand R.E., Growth and cellular characteristics of multicell spheroids, Rec. Res. Cane. Res., 95 (1985), 24–49.CrossRefGoogle Scholar
  105. [SUd] Sutherland R.M., Cell and environment interactions in tumor mi-croregions: the multicell spheroid model, Science, 240 (1988), 177–184.CrossRefGoogle Scholar
  106. [SWa] Swan G.W., Some Current Mathematical Topics in Cancer Research, University Microfilms International (1977).Google Scholar
  107. [SWb] Swan G.W., The diffusion of inhibitor in a spherical tumor, Math. Biosci., 108 (1992), 75–79.CrossRefGoogle Scholar
  108. [TAa] Tannock I.F., The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumor, Br. J. Cancer, 22 (1968), 258–273.CrossRefGoogle Scholar
  109. [THa] Thorn R., Structural Stability and Morphogenesis: An Outline of a General Theory of Models, Addison-Wesley (1989).Google Scholar
  110. [VAa] Vaupel P., Hypoxia in neoplastic tissue, Microvasc. Res., 13 (1977), 399–408.CrossRefGoogle Scholar
  111. [VAb] Vaupel P., Oxygen supply to malignant tumors, in Tumor Blood Circulation. Angiogenesis, Morphology and Blood Flow of Experimental and Human Tumors, Peterson H.I. ed., CRC Press (1980), 144–168.Google Scholar
  112. [VAc] Vaupel P., Namz R., Múller-Klieser W.F., and Grunewald W.A., Intracapillary Hb02 saturation in malignant tumors during nor-moxia and hyperoxia, Microvasc. Res., 17 (1979), 181–191.CrossRefGoogle Scholar
  113. [VYa] Vaidya V.G. and Alexandra F.J., Evaluation of some mathematical models for tumor growth, Int. J. Bio-Med. Comp., B (1982), 19–35.CrossRefGoogle Scholar
  114. [WHa] Wheldon T.E., Mathematical Models in Cancer Research, Adam Hilger (1988).Google Scholar
  115. [WOa] Woodcock A.E.R., Cellular differentiation and catastrophe theory, Ann. N.Y. Acad. Sci., 231 (1974), 60–76.MATHCrossRefGoogle Scholar
  116. [WOb] Woodcock A.E.R., Catastrophe theory and cellular determination, transdetermination and differentiation, Bull. Math. Biol., 41 (1979), 101–117.MathSciNetGoogle Scholar
  117. [ZEa] Zeeman E.C., Applications of Catastrophe Theory, Tokyo University Press (1973).Google Scholar

Copyright information

© Springer Science+Business Media New York 1997

Authors and Affiliations

  • John A. Adam
    • 1
  1. 1.Department of Mathematics and StatisticsOld Dominion UniversityNorfolkUSA

Personalised recommendations