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In this paper we present a simple deterministic model of a biological tissue growing within a porous scaffold. By neglecting the effects of nutrient limitations and intercellular pressure on cell growth, and by using Darcy's law to model the cells' movement through the scaffold, our model is formulated as a moving boundary problem. Due to the difficulty in solving the resulting system, we reformulate it as a linear complementarity problem using the Baiocchi transformation, and give both one-dimensional analytical solutions and two-dimensional numerical ones. We then focus on the behaviour of the moving boundary as the colony approaches confluence, using asymptotic analysis to derive the time of confluence and the shape of the moving boundary; we show in particular that the moving boundary evolves to an ellipse. We also show that pressures increase considerably in the tissue shortly before the scaffold is filled, and identify the potential problem for tissue engineers of a "slit" being left devoid of cells.

Original publication




Journal article


Mathematical Models and Methods in Applied Sciences


World Scientific Pub Co Pte Lt

Publication Date





1721 - 1750