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A mathematical model is adapted to the description of the growth of an avascular tumour spheroid embedded in a deformable medium (gel). Attention is focused on the influence that the mechanical properties of the gel have on the tumour's growth dynamics. Following the work of Landman and Please (2000), the tumour is treated as a two phase material, whereby the tumour cells and the extracellular fluid form distinct phases. Its growth is modelled by combining mass conservation and force balances for each phase with the usual concepts of diffusion limited growth in response to an externally supplied nutrient. The mechanical properties of the gel are characterised by a strain energy function. The stress induced in the gel by the tumour's expansion is incorporated into the force balance equations, thereby linking it to the tumour's growth. Numerical simulations of the model equations show that as the stiffness of the gel increases, the tumour's growth rate and equilibrium size decrease and the time at which necrosis is initiated is delayed (if it occurs at all). Similar results are obtained when the initial size of the tumour is reduced whilst the mechanical properties of the surrounding gel are held fixed. Such results, which are in good qualitative agreement with available experimental data, suggest that mechanical interactions between a tumour and the tissue or medium in which it is located can significantly influence its growth dynamics.


Journal article


J Math Biol

Publication Date





191 - 220


Animals, Cell Division, Computer Simulation, Humans, Models, Biological, Necrosis, Neoplasms, Spheroids, Cellular