Mathematical Models of Tumor Growth: From Empirical Description to Biological Mechanism

  • John A. Adam
Part of the Advances in Experimental Medicine and Biology book series (AEMB, volume 537)


Cancer is a complex phenomenon (Weinberg, 1998), and in order to attempt to model any of its different facets (such as avascular spheroid growth, angiogenesis and vascularization, invasion, or metastasis) in a reasonable mathematical manner, many simplifications are necessary. If the simplifications are reasonable, the model may be of considerable use, not only as a receptacle for what is already known but also for its predictive capabilities.


Necrotic Core Catastrophe Theory Multicellular Spheroid Critical Size Defect Cusp Catastrophe 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Adam, J.A., 1986, A simplified mathematical model of tumor growth, Math. Biosci. 81:229–244.zbMATHCrossRefGoogle Scholar
  2. Adam, J.A., 1987, A mathematical model of tumor growth: II. Effects of geometry and spatial non-uniformity on stability, Math. Biosci. 86:183–211.zbMATHCrossRefGoogle Scholar
  3. Adam, J.A., 1987, A mathematical model of tumor growth: III. Comparison with experiment, Math. Biosci. 86:213–227.zbMATHCrossRefGoogle Scholar
  4. Adam, J. A., 1988, On complementary levels of description in applied mathematics. II. Mathematical models in cancer biology, Int. J. Math. Ed. Sci. Tech. 19:519–535.MathSciNetCrossRefGoogle Scholar
  5. Adam, J.A., 1996, Mathematical models of prevascular spheroid development and catastrophe-theoretic description of rapid metastatic growth/tumor remission, Invas. Metast. 16:247–267.Google Scholar
  6. Adam, J.A., 1997, General aspects of modeling tumor growth and immune response, in: A Survey of Models for Tumor-Immune System Dynamics, J.A. Adam and N. Bellomo, eds, Birkauser, Boston.Google Scholar
  7. Adam, J.A., 1999, A simplified model of wound healing (with particular reference to the critical size defect), Math. Comput. Model. 30:23–32.MathSciNetCrossRefGoogle Scholar
  8. Adam, J.A., and Maggelakis, S., 1989, A mathematical model of tumor growth. IV. Effects of a necrotic core, Math. Biosci. 97:121–136.zbMATHCrossRefGoogle Scholar
  9. Adam, J.A., and Maggelakis, S.A., 1990, Diffusion regulated growth characteristics of a prevascular carcinoma, Bull. Math. Biol. 52:549–582.zbMATHGoogle Scholar
  10. Adam, J.A., and Noren, R., 1993, Equilibrium model of a vascularized spherical carcinoma with central necrosis: some properties of the solution, J. Math. Biol. 31:735–745.MathSciNetzbMATHCrossRefGoogle Scholar
  11. Arnold, J.A., and Adam, J.A., 1999, A simplified model of wound healing. II: The critical size defect in two dimensions, Math. Comput. Model. 30:47–60.MathSciNetzbMATHCrossRefGoogle Scholar
  12. Burton, A.C., 1966, Rate of growth of solid tumors as a problem of diffusion, Growth 30:159–176.Google Scholar
  13. Byrne, H.M., and Chaplain, M.A.J., 1995, Growth of non-necrotic tumors in the presence and absence of inhibitors, Math. Biosci. 130:151–181.zbMATHCrossRefGoogle Scholar
  14. 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, H.G. Othmer, P.K. Maini, and J.D. Murray, eds, Plenum Press, New York.Google Scholar
  15. Chaplain, M.A.J., and Sleeman, B.D., 1992, 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. 111:169–215.MathSciNetzbMATHCrossRefGoogle Scholar
  16. Chaplain, M.A.J., and Sleeman, B.D., 1993, Modelling the growth of solid tumors and incorporating a method for their classification using nonlinear elasticity theory, J. Math. Biol. 31:431–473.MathSciNetzbMATHCrossRefGoogle Scholar
  17. Chaplain, M.A.J., and Anderson, A.R.A., 1996, Mathematical modelling, simulation and prediction of tumour-induced angiogenesis, Invas. Metast. 16:222–234.Google Scholar
  18. Edelstein-Keshet, L., 1988, Mathematical Models in Biology, Random House, New York.zbMATHGoogle Scholar
  19. Freyer, J.P., 1988, The role of necrosis in regulating the growth saturation of multicellular spheroids, Cancer Res. 48:2432–2439.Google Scholar
  20. Foulds, L., 1954, The experimental study of tumor progression: a review, Cancer Res. 14:327–339.Google Scholar
  21. Glass, L., 1973, Instability and mitotic patterns in tissue growth, J. Dyn. Syst. Meas. Contr. 95:324–327.CrossRefGoogle Scholar
  22. Greller, L.D., Tobin, F.L., and Poste, G., 1996, Tumor heterogeneity and progression: conceptual foundations for modeling, Invas. Metast. 16:177–208.Google Scholar
  23. Greenspan, H.P., 1972, Models for the growth of a solid tumor by diffusion, Stud Appl. Math. 51:317–340.zbMATHGoogle Scholar
  24. Greenspan, H.P., 1974, On the self-inhibited growth of cell cultures, Growth 38:81–95.Google Scholar
  25. Greenspan, H.P., 1976, On the growth and stability of cell cultures and solid tumors, J. Theor. Biol. 56:229–242.MathSciNetCrossRefGoogle Scholar
  26. Jones, D.S., and Sleeman, B.D., 1983, Differential Equations and Mathematical Biology, George Allen and Unwin, London.zbMATHCrossRefGoogle Scholar
  27. Laird, A.K., 1965, Dynamics of tumor growth. Comparison of growth rates and extrapolation of growth curve to one cell, Brit. J. Cancer 19:278–291.CrossRefGoogle Scholar
  28. Maggelakis, S., and Adam, J.A., 1990, Mathematical model for prevascular growth of a spherical carcinoma, Math. Comp. Model. 13:23–38.zbMATHCrossRefGoogle Scholar
  29. Marusic, M., Bajzer, Z., Vuk-Pavlovic, S., and Freyer, J.P., 1994, Tumor growth in-vivo and as multicellular spheroids compared by mathematical models, Bull. Math. Biol, 56:617–631.zbMATHGoogle Scholar
  30. Panetta, J.C., Chaplain, M.A.J., and Adam, J.A., 1998, The mathematical modelling of cancer: a review, in: Mathematical Models in Medical and Health Science, M.A. Horn, MA, G. Simonett, and G.F. Webb, eds, Vanderbilt University Press, Nashville.Google Scholar
  31. Swan, G.W., 1992, The diffusion of inhibitor in a spherical tumor, Math. Biosci. 108:75–79.CrossRefGoogle Scholar
  32. Thorn, R., 1989, Structural Stability and Morphogenesis: An Outline of a General Theory of Models, Addison- Wesley, New York.Google Scholar
  33. Vaidya, V.G., and Alexandro, F.J., 1982, Evaluation of some mathematical models for tumor growth, Int. J. Bio-Med. Comp. 13:19–35.MathSciNetCrossRefGoogle Scholar
  34. Weinberg, R.A., 1998, One Renegade Cell, Basic Books, New York.Google Scholar
  35. Wheldon, T.E., 1988, Mathematical Models in Cancer Research, Adam Hilger, Bristol.zbMATHGoogle Scholar
  36. Williams, T., and Bjerknes, R., 1972, Stochastic model for an abnormal clone spread through epithelial basal layer, Nature 236:19–21.CrossRefGoogle Scholar
  37. Zeeman, E.C., 1973, Applications of Catastrophe Theory, Tokyo University Press,Tokyo.Google Scholar

Copyright information

© Springer Science+Business Media New York 2003

Authors and Affiliations

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

Personalised recommendations