zbMATH — the first resource for mathematics

Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Growth of necrotic tumors in the presence and absence of inhibitors. (English) Zbl 0856.92010
Summary: A mathematical model is presented for the growth of a multicellular spheroid that comprises a central core of necrotic cells surrounded by an outer annulus of proliferating cells. The model distinguishes two mechanisms for cell loss: apoptosis and necrosis. Cells loss due to apoptosis is defined to be programmed cell death, occurring, for example, when a cell exceeds its natural lifespan, whereas cell death due to necrosis is induced by changes in the cell’s microenvironment, occurring, for example, in nutrient-depleted regions. Mathematically, the problem involves tracking two free boundaries, one for the outer tumor radius, the other for the inner necrotic radius. Numerical simulations of the model are presented in an inhibitor-free setting and an inhibitor-present setting for various parameter values. The effects of nutrients and inhibitors on the existence and stability of the time-independent solutions of the model are studied using a combination of numerical and asymptotic techniques.

92C50Medical applications of mathematical biology
35R35Free boundary problems for PDE
35K57Reaction-diffusion equations
65C20Models (numerical methods)
65Z05Applications of numerical analysis to physics
Full Text: DOI
[1] Greenspan, H. P.: Models for the growth of a solid tumour by diffusion. Stud. appl. Math. 52, 317-340 (1972) · Zbl 0257.92001
[2] Mcelwain, D. L. S.; Ponzo, P. J.: A model for the growth of a solid tumor with non-uniform oxygen consumption. Math. biosci. 35, 267-279 (1977) · Zbl 0364.92020
[3] Adam, J. A.: A mathematical model of tumour growth. II. effects of geometry and spatial uniformity on stability. Math. biosci. 86, 183-211 (1987) · Zbl 0634.92002
[4] Adam, J. A.: A mathematical model of tumor growth. III. comparison with experiment. Math. biosci. 86, 213-227 (1987) · Zbl 0634.92003
[5] Adam, J. A.; Maggelakis, S. A.: Diffusion regulated growth characteristics of a spherical prevascular carcinoma. Bull. math. Biol. 52, 549-582 (1990) · Zbl 0712.92010
[6] Maggelakis, S. A.; Adam, J. A.: Mathematical model of a prevascular growth of a spherical carcinoma. Math. comput. Modelling 13, 23-38 (1990) · Zbl 0706.92010
[7] Chaplain, M. A. J.; Britton, N. F.: On the concentration profile of a growth inhibitory factor in multicell spheroids. Math. biosci. 115, 223-245 (1993) · Zbl 0771.92009
[8] Marusic, M.; Bajzer, Z.; Freyer, J. P.; Vuk-Pavlovic, S.: Analysis of growth of multicellular tumour spheroids by mathematical models. Cell prolif. 27, 73-94 (1994)
[9] Tubiana, M.: The kinetics of tumour cell proliferation and radiotherapy. Br. J. Radiol. 44, 325-347 (1971)
[10] Sutherland, R. M.; Durand, R. E.: Radiation response of multicell spheroids--an in vitro tumour model. Int. J. Radiat. biol. 23, 235-246 (1973)
[11] Sutherland, R. M.; Durand, R. E.: Growth and cellular characteristics of multicell spheroids. Recent results cancer res. 95, 24-49 (1984)
[12] Freyer, J. P.; Sutherland, R. M.: Regulation of growth saturation and development of necrosis in EMT6/ro multicellular spheroids by the glucose and oxygen supply. Cancer res. 46, 3504-3512 (1986)
[13] Freyer, J. P.; Sutherland, R. M.: Proliferative and clonogenic heterogeneity of cells from EMT6/ro multicellular spheroids induced by the glucose and oxygen supply. Cancer res. 46, 3513-3520 (1986)
[14] Sutherland, R. M.: Cell and environment interactions in tumor microregions: the multicell spheroid model. Science 240, 177-184 (1988)
[15] Adam, G.; Steiner, U.; Maier, H.; Ulrich, S.: Analysis of cellular interactions in density-dependent inhibition of 3T3 cell proliferation. Biophys. struct. Mech. 9, 75-82 (1982)
[16] Iversen, O. H.: Epidermal chalones and squamous cell carcinomas. Virchows arch. B cell pathol. 27, 229-235 (1978)
[17] Iversen, O. H.: The chalones. Tissue growth factors, 491-550 (1981)
[18] Iversen, O. H.: What’s new in endogenous growth stimulators and inhibitors (chalones). Pathol. res. Pract. 180, 77-80 (1985)
[19] Byrne, H. M.; Chaplain, M. A. J.: Growth of non-necrotic tumours in the presence and absence of inhibitors. Math. biosci. 130, 151-181 (1995) · Zbl 0836.92011
[20] Kerr, J. F. R.: Shrinkage necrosis; a distinct mode of cellular death. J. pathol. 105, 13-20 (1971)
[21] Kerr, J. F. R.; Wyllie, A. H.; Currie, A. R.: Apoptosis: a basic biological phenomenon with wide ranging implications in tissue kinetics. Br. J. Cancer 26, 239-257 (1972)
[22] Darzynkiewicz; Bruno, S.; Del Bino, G.; Gorczyca, W.; Hotz, M. A.; Lassota, P.; Traganos, F.: Features of apoptotic cells measured by flow cytometry. Cytometry 13, 795-808 (1992)
[23] Mcelwain, D. L. S.; Morris, L. E.: Apoptosis as a volume loss mechanism in mathematical models of solid tumour growth. Math. biosci. 39, 147-157 (1978)
[24] Durand, R. E.: Cell cycle kinetics in an in vitro tumour model. Cell tissue kinet. 9, 403-412 (1976)
[25] Durand, R. E.: Multicell spheroids as a model for cell kinetic studies. Cell tissue kinet. 23, 141-159 (1990)
[26] Folkman, J.; Hochberg, M.: Self-regulation of growth in three dimensions. J. exp. Med. 138, 745-753 (1973)
[27] Protter, M. H.; Weinberger, H. F.: Maximum principles in differential equations. (1984) · Zbl 0549.35002
[28] Byrne, H. M.; Chaplain, M. A. J.: The role of cell-cell adhesion in the growth & development of carcinomas. (1996) · Zbl 0883.92014