A two-phase model is presented to describe avascular tumour growth. Conservation of mass equations, including oxygen dependent cell growth and death terms, are coupled with equations of momentum conservation. The cellular phase behaves as a viscous liquid, while the viscosity of the extracellular water manifests itself as an interphase drag. It is assumed that the cells become mechanically stressed if they are too densely packed and that the tumour will try to increase its volume in order to relieve such stress. By contrast, the overlapping filopodia of sparsely populated cells create short-range attractive effects. Finally, oxygen is consumed by the cells as it diffuses through the tumour. The resulting system of equations are reduced to three, which describe the evolution of the tumour cell volume fraction, the cell speed and the oxygen tension. Numerical simulations indicate that the tumour either evolves to a travelling wave profile, in which it expands at a constant rate, or it settles to a steady state, in which the net rates of cell proliferation and death balance. The impact of varying key model parameters such as cellular viscosity, interphase drag, and cellular tension are discussed. For example, tumours consisting of well-differentiated ( i.e. viscous) cells are shown to grow more slowly than those consisting of poorly-differentiated (i.e. less viscous) cells. Analytical results for the case of oxygen-independent growth are also presented, and the effects of varying the key parameters determined (the results are in line with the numerical simulations of the full problem). The key results and their biological implications are then summarised and future model refinements discussed
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