×

zbMATH — the first resource for mathematics

Mathematical model of prevascular growth of a spherical carcinoma. (English) Zbl 0706.92010
Summary: A mathematical model of prevascular tumor growth by diffusion, which is an extension of previous models, has been constructed. The effects of nonuniform nutrient consumption and nonuniform inhibitor production on the growth rate of a spherically symmetric carcinoma are examined in this model, and the complete growth history of the tissue is followed analytically. A four-layer structure in the dormant steady state is predicted, and the evolution of a multicellular spheroid from an initial form is determined by an integro-differential (growth) equation. A parameter is used to measure the degree of nonuniformity of the inhibitor production rate, and growth patterns are presented for several values of this parameter. As this nonuniformity parameter is varied from zero to unity, there is a considerable relative increase in the asymptotic steady-state outer tumor radius, and also in the corresponding inner necrotic core size. A detailed understanding of the mathematical model presented here can form the basis for a further level of description in deterministic models of tumor growth.

MSC:
92C50 Medical applications (general)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Adam, J.A., A simplified mathematical model of tumor growth, Mathl biosci., 81, 224-229, (1986) · Zbl 0601.92007
[2] Adam, J.A., A mathematical model of tumor growth. II. effects of geometry and spatial nonuniformity on stability, Mathl biosci., 86, 183-211, (1987) · Zbl 0634.92002
[3] Greenspan, H.P., Models for the growth of a solid tumor by diffusion, Stud. appl. math., 52, 317-340, (1972) · Zbl 0257.92001
[4] Greenspan, H.P., On the self inhibited growth of cell cultures, Growth, 38, 81, (1974)
[5] Deakin, A.S., Model for the growth of a solid in vitro tumor, Growth, 39, 159, (1975)
[6] McElwain, D.L.S.; Ponzo, P.J., A model for the growth of a solid tumor with nonuniform oxygen consumption, Mathl biosci., 35, 267-279, (1977) · Zbl 0364.92020
[7] Arve, B.H.; Liapis, A.I., Oxygen tension in tumors predicted by a diffusion with absorption model involving a moving free boundary, Mathl comput. modelling, 10, 159-174, (1988) · Zbl 0644.92004
[8] Liapis, A.I.; Lipscomb, G.G.; Crosser, O.K., A model of oxygen diffusiuon in absorbing tissue, Mathl modelling, 3, 83-92, (1982) · Zbl 0516.92004
[9] Maggelakis, S.A.; Adam, J.A., Note on a diffusion model of tissue growth, Appl. math. lett., 3, 27-31, (1990) · Zbl 0706.92017
[10] Old, L.J., Tumor necrosis factor, Scient. am., 59-75, (1988), May
[11] Folkman, J.; Klagsbrun, M., Angiogenic factors, Science, 235, 442-447, (1987)
[12] King, W.E.; Schultz, D.S.; Gatenby, R.A., Multi-region models for describing oxygen tension profiles in human tumors, Chem. engng commun., 47, 73-91, (1986)
[13] King, W.E.; Schultz, D.S.; Gatenby, R.A., An analysis of systemic tumor oxygenation using multi-region models, Chem. engng commun., 64, 137-153, (1988)
[14] Schultz, D.S.; King, W.E., On the analysis of oxygen diffusion and reaction of biological systems, Mathl biosci., 83, 179-190, (1987) · Zbl 0613.92011
[15] Mueller-Klieser, W.F.; Sutherland, R.M., Oxygen tensions in multicell spheroids of two cell lines, Br. J. cancer, 45, 256-263, (1982)
[16] J.A. Adam and S.A. Maggelakis, Diffusion regulated growth characteristics of a spherical carcinoma. Bull. math. Biol. (in press) · Zbl 0712.92010
[17] Sutherland, R.M., Growth of multicell spheroids in tissue culture as a model of nodular carcinomas, J. natn. cancer inst., 46, 113-120, (1971)
[18] Adam, J.A., A mathematical model of tumor growth. III. comparison with experiment, Mathl biosci., 86, 213-227, (1987) · Zbl 0634.92003
[19] Sutherland, R.M., Cell and environment interactions in tumor microregions: the multicell spheroid model, Science, 240, 177-184, (1988)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.