Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 236

cell seeding results in competition for nutrients which, when combined with
limitations in nutrient transport, will reduce the overall cell growth rate.
3.1.3 The Cells' Mechanical Environment
While nutrient availability undoubtedly plays a crucial role in regulating cell
growth within tissue constructs, mechanical effects are also important. In Wilson
et al. ( 2007 ), Wilson and coworkers develop a simple deterministic model of tissue
growth in which the cells are initially seeded around the periphery of a porous
scaffold. They assume that nutrient is freely available, and use Darcy's law to
model cell movement towards the centre of the scaffold. As in Galban and Locke
( 1997 ), a moving boundary problem is introduced to delineate regions of the
scaffold that have been colonised by cells from regions which are devoid of cells.
The Baiocchi transformation is used to transform the model to a linear comple-
mentarity problem for which one-dimensional analytical solutions and two-
dimensional numerical ones are presented. Attention focusses on the behaviour of
the moving boundary as the cells reach the centre of the scaffold and the colony
approaches confluence: asymptotic techniques are used to derive approximate
expressions for the time to confluence and to show that, near closure, the moving
boundary evolves to an ellipse and that the ratio of its semi-major and semi-minor
axes is identical to that of the two-dimensional, rectangular scaffold. The pressure
within the scaffold is found to increase considerably shortly before it is filled,
highlighting the potential problem for tissue engineers of a ''slit'' persisting within
the tissue construct and compromising its mechanical integrity (see Fig. 3 ).
3.2 Modelling Dynamic Tissue Culture: Cell Volume Neglected
As discussed in Sect. 1.1.3 , since static culture systems rely on diffusive transport
of solutes, it is not possible to engineer constructs of a size suitable for implan-
tation. Dynamic culture systems exploit the flow of culture medium to enhance
nutrient and waste product transport by advection. Furthermore, such systems can
provide mechanical load to mechanosensitive tissues, e.g. via the application of
fluid shear stress to the cells. We review here models in which the cells occupy no
volume, and have no affect on the fluid flow. We start by describing computational
approaches and then consider approaches in which simplifying assumptions are
used to derive reduced models that may be solved using analytical or simpler
numerical techniques.