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A mathematical model of tumor growth. II. Effects of geometry and spatial nonuniformity on stability. (English) Zbl 0634.92002
Summary: A theoretical account of mitotic inhibition in one-, two-, and three- dimensional configurations is presented. Based on part I, ibid. 81, 229- 244 (1986; Zbl 0601.92007), the inhibitor production rate is taken to be nonuniform throughout the tissue, resulting in significant deviations from the prediction of uniform models. Geometry affects the stability of growth also. The analysis presented here represents a detailed study of the properties of highly nonuniform inhibition, from which information on intermediate inhibition models can be readily deduced. This information is used to compare such a model with experimental results in part III, see the following entry, Zbl 0634.92003.

92C50Medical applications of mathematical biology
92D25Population dynamics (general)
Full Text: DOI
[1] Abramowitz, M.; Stegun, I. A.: Handbook of mathematical functions. (1972) · Zbl 0543.33001
[2] Adam, J. A.: A simplified mathematical model of tumor growth. Math. biosci. 81, 229-244 (1986) · Zbl 0601.92007
[3] Adam, J. A.: A mathematical model of tumor growth. III. comparison with experiment. Math. biosci. 86, 213-227 (1987) · Zbl 0634.92003
[4] J.A. Adam, On complementary levels of description in applied mathematics: II. Mathematical models in cancer biology, Internat. Jnl. Math. Ed. Sci. Tech., to appear.
[5] Bullough, W. S.; Deol, J. U. R.: The pattern of tumor growth. Symp. soc. Exp. biol. 25, 255-275 (1971)
[6] Burton, A. C.: Rate of growth of solid tumors as a problem of diffusion. Growth 30, 157-176 (1966)
[7] Folkman, J.; Hochberg, M.: Self-regulation of growth in three dimensions. J. exp. Med. 138, 745-753 (1973)
[8] Glass, L.: Instability and mitotic patterns in tissue growth. J. dyn. Syst. meas. Control 95, 324-327 (1973)
[9] Greespan, H. P.: Models for the growth of a solid tumor by diffusion. Stud. appl. Math. 52, 317-340 (1972) · Zbl 0257.92001
[10] Post, J.; Hoffman, J.: Cell renewal patterns. New england J. Med. 279, 248-258 (1968)
[11] Shymko, R. M.; Glass, L.: Cellular and geometric control of tissue growth and mitotic instability. J. theoret. Biol. 63, 355-374 (1976)
[12] Stakgold, I.: Boundary-value problems in mathematical physics. (1967) · Zbl 0158.04801