title: Mathematical modelling of avascular-tumour growth.
abstract: A system of nonlinear partial differential equations is proposed as a model for the growth of an avascular-tumour spheroid. The model assumes a continuum of cells in two states, living or dead, and, depending on the concentration of a generic nutrient, the live cells may reproduce (expanding the tumour) or die (causing contraction). These volume changes resulting from cell birth and death generate a velocity field within the spheroid. Numerical solutions of the model reveal that after a period of time the variables settle to a constant profile propagating at a fixed speed. The travelling-wave limit is formulated and analytical solutions are found for a particular case. Numerical results for more general parameters compare well with these analytical solutions. Asymptotic techniques are applied to the physically relevant case of a small death rate, revealing two phases of growth retardation from the initial exponential growth, the first of which is due to nutrient-diffusion limitations and the second to contraction during necrosis. In this limit, maximal and "linear' phase growth speeds can be evaluated in terms of the model parameters.
authors.affiliation: Department of Theoretical Mechanics, University of Nottingham, UK.
authors.forename: J P
authors.initials: JP
authors.lastname: Ward
authors.forename: J R
authors.initials: JR
authors.lastname: King
mesh_terms: D000818:Animals; D016923:Cell Death; D002455:Cell Division; D006801:Humans; D007700:Kinetics; D008433:Mathematics; D008954:Models, Biological; D018874:Spheroids, Cellular / Q000473:pathology*