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Mathematical Medicine and Biology 1997 14(1):39-69; doi:10.1093/imammb/14.1.39
© 1997 by Institute of Mathematics and its Applications
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Mathematical modelling of avascular-tumour growth

J. P. WARD and J. R. KING

Department of Theoretical Mechanics, University of Nottingham Nottingham, NG7 2RD, UK

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.

Keywords: tumour growth; avascular; mathematical modelling; numerical solution; asymptotic analysis


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