Conceptual frameworks for mathematical modeling of tumor growth dynamics

The paper discusses the applicability of some empirical and functional mathematical models of tumor growth to multicell tumor spheroids which are a three-dimensional experimental paradigm of prevascular tumors. A new mathematical model based on receptor-mediated regulation of growth is developed and tested and two additional functional models (autostimulation, competing populations) are analyzed. It is demonstrated that these models, defined by two autonomous differential equations, can describe spontaneous tumor regression. The concept of tumor magnification factor, a quantifier of positive feedback mechanisms, is elaborated.

[1]  Douglas S. Riggs,et al.  Control theory and physiological feedback mechanisms , 1970 .

[2]  Lee A. Segel,et al.  Modeling Dynamic Phenomena in Molecular and Cellular Biology , 1984 .

[3]  Z Bajzer,et al.  Modeling autostimulation of growth in multicellular tumor spheroids. , 1991, International journal of bio-medical computing.

[4]  C P Calderón,et al.  Modeling tumor growth. , 1991, Mathematical biosciences.

[5]  M Gyllenberg,et al.  Quiescence as an explanation of Gompertzian tumor growth. , 1989, Growth, development, and aging : GDA.

[6]  P. Durbin,et al.  Construction of a growth curve for mammary tumors of the rat. , 1967, Cancer research.

[7]  L. Révész,et al.  Analysis of the growth of tumor cell populations , 1974 .

[8]  Z Bajzer,et al.  Analysis of growth of multicellular tumour spheroids by mathematical models , 1994, Cell proliferation.

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

[10]  Partial purification of a protein growth inhibitor from multicellular spheroids. , 1988, Biochemical and biophysical research communications.

[11]  Z Bajzer,et al.  Growth self-incitement in murine melanoma B16: a phenomenological model. , 1984, Science.

[12]  J A Adam,et al.  Diffusion regulated growth characteristics of a spherical prevascular carcinoma. , 1990, Bulletin of mathematical biology.

[13]  J. Freyer.  Role of necrosis in regulating the growth saturation of multicellular spheroids. , 1988, Cancer research.

[14]  S. Piantadosi,et al.  A model of growth with first-order birth and death rates. , 1985, Computers and biomedical research, an international journal.

[15]  G. W. Swan.  Role of optimal control theory in cancer chemotherapy. , 1990, Mathematical biosciences.

[16]  J. Leith,et al.  Tumor micro-ecology and competitive interactions. , 1987, Journal of theoretical biology.

[17]  Max Kurtz.  Handbook of Applied Mathematics for Engineers and Scientists , 1991 .

[18]  M. Sporn,et al.  Autocrine secretion and malignant transformation of cells. , 1980, The New England journal of medicine.

[19]  Željko Bajzer,et al.  Mathematical modeling of cellular interaction dynamics in multicellular tumor spheroids , 1995 .

[20]  James P. Freyer,et al.  Tumor growthin vivo and as multicellular spheroids compared by mathematical models , 1994, Bulletin of mathematical biology.

[21]  M A Savageau.  Allometric morphogenesis of complex systems: Derivation of the basic equations from first principles. , 1979, Proceedings of the National Academy of Sciences of the United States of America.

[22]  Miljenko Marušić,et al.  Modeling hormone-dependent cell-cell interactions in tumor growth: A simulation study , 1991 .

[23]  Miljenko Marušić,et al.  Generalized two-parameter equation of growth , 1993 .

[24]  Edward J. Beltrami,et al.  Mathematics for Dynamic Modeling , 1987 .

[25]  M E Fisher,et al.  A mathematical model of cancer chemotherapy with an optimal selection of parameters. , 1990, Mathematical biosciences.

[26]  R. Makany.  A Theoretical Basis for Gompertz'S Curve , 1991 .

[27]  E. Bradley,et al.  A theory of growth , 1976 .

[28]  A S Glicksman,et al.  Growth in solid heterogeneous human colon adenocarcinomas: comparison of simple logistical models , 1987, Cell and tissue kinetics.

[29]  J. Leith,et al.  Growth factors and growth control of heterogeneous cell populations. , 1993, Bulletin of mathematical biology.

[30]  L. Bertalanffy.  Quantitative Laws in Metabolism and Growth , 1957 .

[31]  M. Richardson,et al.  Fundamentals of mathematics , 1960 .

[32]  Z Bajzer,et al.  Quantitative aspects of autocrine regulation in tumors. , 1990, Critical reviews in oncogenesis.

[33]  C. Frenzen,et al.  A cellk kinetics justification for Gompertz' equation , 1986 .

[34]  Raoul Kopelman,et al.  Fractal Reaction Kinetics , 1988, Science.

[35]  P. G. Doucet.  On the static analysis of nonlinear feedback loops , 1986 .

[36]  M. Chaplain,et al.  Modelling the growth of solid tumours and incorporating a method for their classification using nonlinear elasticity theory , 1993, Journal of mathematical biology.

[37]  M. Nowak,et al.  Oncogenes, anti-oncogenes and the immune response to cancer : a mathematical model , 1992, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[38]  J. Leith,et al.  Autocrine and paracrine growth factors in tumor growth: a mathematical model. , 1991, Bulletin of mathematical biology.

[39]  Anita B. Roberts,et al.  Autocrine growth factors and cancer , 1985, Nature.

[40]  A. Perelson.  Immune Network Theory , 1989, Immunological reviews.

[41]  R. Sutherland.  Cell and environment interactions in tumor microregions: the multicell spheroid model. , 1988, Science.

[42]  M A Woodbury,et al.  A new model for tumor growth analysis based on a postulated inhibitory substance. , 1980, Computers and biomedical research, an international journal.

[43]  H. H. Lloyd,et al.  Kinetic parameters and growth curves for experimental tumor systems. , 1970, Cancer chemotherapy reports.

[44]  W. M. Gray,et al.  Mitotic autoregulation, growth control and neoplasia. , 1973, Journal of theoretical biology.