Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 233

Cells
Void space
Fig. 2 In Galban and Locke ( 1997 ), moving boundary problems were developed to study the
influence of nutrient transport and consumption on cell growth. The moving boundaries separate
regions devoid of cells from those containing cells. Two seeding protocols were considered: in
one case, cells were seeded on the scaffold periphery (where nutrient levels are highest), and in
the second case the cells were seeded on a plane passing through the centre of the scaffold. The
direction of cell growth is indicated by the arrows in the figure
boundary separating regions containing cells from those devoid of cells was
determined by assuming that cell growth is localised at the moving boundary
where it occurs at a rate which is proportional to the local nutrient concentration.
This phenomenological model was used to compare two different seeding proto-
cols: in one case, cells were seeded on the periphery of the scaffold, where nutrient
levels are highest, and in the second case the cells were seeded in a plane passing
through the centre of the scaffold. The direction of cell growth in each case is
indicated in Fig. 2 .
Numerical simulations revealed that the second case, with cells initially
migrating outwards from the centre of the scaffold and in the direction of increasing
nutrient levels, gave results which were in better agreement with Freed et al.'s
experimental data than those obtained for the case in which the cell migrate towards
the centre of the scaffold and into regions with lower nutrient levels. Good agree-
ment was, however, only achieved for scaffolds of intermediate thickness (i.e.
neither excessively thick nor overly thin). This deficiency motivated Galban and
Locke to consider more detailed, multiphase models, in which the scaffold was
viewed as a two-phase mixture of (effectively) cells and extracellular fluid, and cell
growth was no longer localised on a moving boundary (Galban and Locke 1999a , b ).
In Galban and Locke ( 1999a ), a one-dimensional geometry was employed, and the
nutrient concentration assumed to vary parallel to the axis of the cylindrical scaffold
only. The cell and fluid volume fractions were taken to be spatially uniform and to
evolve over time, the cell volume fraction increasing at a rate which depends on the
nutrient distribution within the scaffold. As in Galban and Locke ( 1997 ), the nutrient
distribution was governed by a quasi-steady, reaction-diffusion equation. However
the diffusion coefficient in Galban and Locke ( 1999a ) was no longer taken to be a
constant: it was assumed to decrease as the cell volume fraction increased. In this
way, cell growth was found to be self-limiting: as cells proliferate, their volume
fraction increases but this hinders nutrient transport through the scaffold and, hence,
slows the rate of cell growth until, eventually, nutrient levels are too low to support