zbMATH — the first resource for mathematics

Examples
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.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
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)
Mathematical modelling of the effects of mitotic inhibitors on avascular tumour growth. (English) Zbl 0981.92012

Summary: We build on our previous mathematical model [see IMA J. Math. Appl. Med. Biol. 16, No. 2, 171-211 (1999; Zbl 0943.92019); ibid. 14, No. 1, 39-69 (1997; Zbl 0866.92011)] to study the effects of high molecular mass mitotic inhibitors released at cell death. The model assumes a continuum of living cells which, depending on the concentration of a generic nutrient, generate movement (described by a velocity field) due to the changes in volumes caused by cell birth and death. The necrotic material is assumed to consist of two diffusible materials: 1) basic cellular material which is used by living cells as raw material for mitosis; 2) a generic non-utilizable material which may inhibit mitosis.

Numerical solutions of the resulting system of partial differential equations show all the main features of tumour growth and heterogeneity. Material 2) is found to act in an inhibitive fashion in two ways: i) directly, by reducing the mitotic rate and ii) indirectly, by occupying space, thereby reducing the availability of the basic cellular material. For large time the solutions to the model tend either to a steady-state, reflecting growth saturation, or to a traveling wave, indicating continual linear growth. The steady-state and traveling wave limits of the model are derived and studied, the regions of existence of these two types of long-time solution being explored in parameter space using numerical methods.

MSC:
92C50Medical applications of mathematical biology
35Q80Appl. of PDE in areas other than physics (MSC2000)
65N99Numerical methods for BVP of PDE