zbMATH — the first resource for mathematics

Diffusion regulated growth characteristics of a spherical prevascular carcinoma. (English) Zbl 0712.92010
Summary: Recently a mathematical model of the prevascular phases of tumor growth by diffusion has been investigated by the authors [Math. Comput. Modelling 13, No.5, 23-38 (1990; Zbl 0706.92010)]. In this paper we examine in detail the results and implications of that mathematical model, particularly in the light of recent experimental work carried out on multicellular spheroids. The overall growth characteristics are determined in the present model by four parameters: Q, \(\gamma\), b, and \(\delta\), which depend on information about inhibitor production rates, oxygen consumption rates, volume loss and cell proliferation rates, and measures of the degree of non-uniformity of the various diffusion processes that take place.
The integro-differential growth equation is solved for the outer spheroid radius \(R_ 0(t)\) and three related inner radii subject to the solution of the governing time-independent diffusion equations (under conditions of diffusive equilibrium) and the appropriate boundary conditions. Hopefully, future experimental work will enable reasonable bounds to be placed on parameter values referred to in this model: meanwhile, specific experimentally-provided initial data can be used to predict subsequent growth characteristics of in vitro multicellular spheroids. This will be one objective of future studies.

92C50 Medical applications (general)
65L99 Numerical methods for ordinary differential equations
65R20 Numerical methods for integral equations
Full Text: DOI
[1] Adam, J. A. 1986. A simplified mathematical model of tumor growth.Math. Biosci. 81, 224–229. · Zbl 0601.92007
[2] Adam, J. A. 1987. A mathematical model of tumor growth. II. Effects of geometry and spatial nonuniformity on stability.Math. Biosci. 86, 183–211. · Zbl 0634.92002
[3] Adam, J. A. 1987. A mathematical model of tumor growth. III. Comparison with experiment.Math. Biosci. 86, 213–227. · Zbl 0634.92003
[4] Adam, J. A. 1989. Corrigendum. A mathematical model of tumor growth by diffusion.Math. Biosci. 94, 155. · Zbl 0719.92501
[5] Adam, J. A. and S. A. Maggelakis. 1989. Mathematical models of tumor growth. IV. Effects of a necrotic core.Math. Biosci.,97, 121–134. · Zbl 0682.92004
[6] Anderson, N. and A. M. Arthurs. 1980. Complementary variational principles for diffusion problems with Michaelis-Menten kinetics.Bull. math. Biol. 42, 131–135. · Zbl 0423.92040
[7] Arve, B. H. and A. I. Liapis. 1988. Oxygen tension in tumors predicted by a diffusion with absorption model involving a moving free boundary.Math. Comput. Modeling 10, 159–174. · Zbl 0644.92004
[8] Do, D. D. and P. F. Greenfield. 1981. A finite integral transform technique for solving the diffusion-reaction equations with Michaelis-Menten kinetics.Math. Biosci. 54, 31–47. · Zbl 0458.92008
[9] Folkman, J. 1974. Tumor angiogenesis.Adv. Cancer Res. 19, 331–338.
[10] Folkman J. and M. Hochberg. 1973. Self-regulation of growth in three dimensions.J. Exp. Med. 138, 745–753.
[11] Folkman, J. and M. Klagsbrun. 1987. Angiogenic factors.Science 235, 442–447.
[12] Franko, A. J. and H. I. Freedman. 1984. Model of diffusion on oxygen to spheroids grown in stationary medium–I. Complete spherical symmetry.Bull. math. Biol. 46, 205–217. · Zbl 0534.92009
[13] Franko, A. J. and R. M. Sutherland. 1979. Oxygen diffusion distance and the development of necrosis in multicell spheroids.Radiat. Res. 79, 439–453.
[14] Franko, A. J. and R. M. Sutherland. 1979. Radiation survival of cells from spheroids grown in different oxygen concentrations.Radiat. Res. 79, 454–467.
[15] Freyer, J. P. and R. M. Sutherland. 1983. Determination of diffusion constants for metabolites in multicell tumor spheroids. In:Oxygen Transport to Tissue–IV, pp., 463–475. New York: Plenum.
[16] Freyer, J. P., E. Tustanoff, A. J. Franko and R. M. Sutherland. 1984.In situ oxygen consumption rates of cells in V-79 multicellular spheroids during growth.J. Cell Physiol. 118, 53–61.
[17] Goldacre, R. J. and G. Sylven. 1962. On the access of blood-borne dyes to various tumor regions.Br. J. Cancer 16, 306.
[18] Greenspan, H. P. 1972. Models for the growth of a solid tumor by diffusion.Stud. appl. Math. 52, 317–340. · Zbl 0257.92001
[19] Greenspan, H. P. 1974. On the self-inhibited growth of cell cultures.Growth 38, 81–95.
[20] Grossman, U. 1984. Profiles of oxygen partial pressure and oxygen consumption inside multicellular spheroids.Recent Results Cancer Res. 95, 150–161.
[21] Hiltman, P. and P. Lory. 1983. On oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics.Bull. math. Biol. 45, 661–664. · Zbl 0512.92008
[22] Jain, R. K. and J. Wei. 1977. Dynamics of drug transport in solid tumors: distributed parameter model.J. Bioengng 1, 313–330.
[23] King, W. E., D. S. Schultz and R. A. Gatenby. 1986. Multi-region models for describing oxygen tension profiles in human tumors.Chem. Engng Commun. 47, 73–91.
[24] King, W. E., D. S. Schultz and R. A. Gatenby. 1988. An analysis of systematic tumor oxygenation using multi-region models.Chem. Engng Commun. 64, 137–153.
[25] Laird, A. K. 1975. Dynamics of tumor growth. Comparisons of growth rates and extrapolation of growth curve to one cell.Br. J. Cancer 19, 278.
[26] Landry, J., J. P. Freyer and R. M. Sutherland. 1982. A model for the growth of multicellular spheroids.Cell Tissue Kinet. 15, 585–594.
[27] Liapis, A. I., G. G. Lipscomb and O. K. Crosser. 1982. A model of oxygen diffusion in absorbing tissue.Math. Modeling 3, 83–92. · Zbl 0516.92004
[28] Lin, S. H. 1976. Oxygen diffusion in a spherical cell with nonlinear oxygen uptake kinetics.J. theor. Biol. 60, 449–457.
[29] Lin, S. H. 1979. Nonlinear diffusion in biological systems.Bull. math. Biol. 41, 151–162. · Zbl 0394.92006
[30] Maggelakis, S. A. and J. A. Adam. 1989. Mathematical model of prevascular growth of a spherical carcinoma.Math. Comput. Modeling, in press. · Zbl 0706.92010
[31] Maggelakis, S. A. and J. A. Adam. 1989. Note on a class of nonlinear time independent diffusion equations.Appl. Math. Lett. 2, 141–145. · Zbl 0708.34006
[32] McElwain, D. L. S. 1978. A re-examination of oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics.J. theor. Biol. 71, 255–263.
[33] McElwain, D. L. S. 1981. A comment on Lin’s paper on nonlinear diffusion in biological systems.Bull. math. Biol. 43, 117–120. · Zbl 0445.92003
[34] McElwain, D. L. S. and P. J. Ponzo. 1977. A model for the growth of solid tumor with nonuniform oxygen consumption.Math. Biosci. 35, 267–279. · Zbl 0364.92020
[35] McElwain, D. L. S., R. Callcott and L. E. Morris. 1979. A model of vascular compression in solid tumors.J. theor. Biol. 78, 405–415.
[36] Mueller-Klieser, W. F. 1984. Method for the determination of oxygen consumption rates and diffusion coefficients in multicellular spheroids.Biophys. J. 46, 343–348.
[37] Mueller-Klieser, W. F. and R. M. Sutherland. 1982. Oxygen tensions in multicell spheroids of two cell lines.Br. J. Cancer 45, 256–263.
[38] Mueller-Klieser, W. F. and R. M. Sutherland. 1984. Oxygen consumption and oxygen diffusion properties of multicellular spheroids from two different cell lines.Adv. exp. Med. Biol., in press.
[39] Mueller-Klieser, W. F., J. P. Freyer and R. M. Sutherland. 1983. Evidence for a major role of glucose in controlling development of necrosis in EMT6/R0 multicell tumor spheroids. In:Oxygen Transport to Tissue–IV, pp. 487–495. New York: Plenum.
[40] Schultz, D. S. and W. E. King. 1987. On the analysis of oxygen diffusion in biological systems.Math. Biosci. 83, 179–190. · Zbl 0613.92011
[41] Shymko, R. M. and L. Glass. 1976. Cellular and geometric control of tissue growth and mitotic instability.J. theor. Biol. 63, 355–374.
[42] Sutherland, R. M. 1988. Cell and environment interactions in tumor microregions: the multicell spheroid model.Science 240, 177–184.
[43] Sutherland, R. M. and R. E. Durand. 1976. Radiation response of multicellular spheroids–anin vitro tumor model.Curr. Top. Radiat. res. 11, 87–139.
[44] Swan, G. W. 1981. Optimization of human cancer radiotherapy.Lecture Notes in Biomathematics, Vol. 42. Berlin: Springer. · Zbl 0464.92002
[45] Tannock, I. 1976. Oxygen distribution in tumours: influence on cell proliferation and implications for tumour therapy.Adv. exp. Med. Biol. 75, 597–603.
[46] Tannock, I. F. 1968. The relation between cell proliferation and the vascular system in a transplanted mouse mammary tumor.Br. J. Cancer 22, 258–273.
[47] Thews, G. and P. Vaupel. 1976. Oxygen supply conditions in tumor tissuein vivo.Adv. exp. Med. Biol. 75, 537–546.
[48] Tosaka, N. and S. Miyaka. 1982. Analysis of a nonlinear diffusion problem with Michaelis-Menten kinetics by an integral equation method.Bull. math. Biol. 44, 841–849. · Zbl 0498.92004
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.