Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 234

further growth. This behaviour was evident in their numerical results which showed
the cell volume fraction increasing towards a fixed value at long times, this value
depending on the system parameters and, for example, decreasing as the scaffold
thickness was increased. In this respect, the one-dimensional model yielded results
which were in good agreement with the experimental observations reported in Freed
et al. ( 1993 , 1994 ). However, the predicted changes were not large enough to match
those observed in the experimental data, possibly because the cell volume fraction
was assumed to be uniform throughout the scaffold. Therefore, in Galban and
Locke ( 1999b ), the two-phase model was generalised by extending it to two spatial
dimensions and allowing the cell and fluid volume fractions, and, hence, the
effective nutrient diffusion coefficient, to vary with position and time. For typical
simulations, spatial variation in the cell volume fraction and the effective diffusion
coefficient became more pronounced over time, with the cell density being maximal
on the periphery of the scaffold, where nutrient levels were highest, and declining
towards the centre, where nutrient levels were lowest (the diffusion coefficient
exhibited the opposite behaviour, with transport inhibited where the cell volume
fraction was high). The agreement between the simulations obtained from this model
and Freed et al.'s experimental data was much better than that obtained for the
earlier, simpler models: the model was able to capture the dynamic trends that had
been observed in the spatial distribution of the cell volume fraction and nutrient
concentration when the scaffold thickness was varied.
The predictions generated from Galban and Locke's work, particularly the
results in Galban and Locke ( 1999b ), highlight some of the problems associated
with culturing cells in scaffolds under static conditions. For example, as the
thickness of the desired tissue construct (and, hence, the scaffold) increases,
limitations associated with transporting nutrient to cells at the centre of the scaf-
fold can lead to the formation of nutrient-deprived regions which are characterised
by cell quiescence and even necrotic cell death. As we explain in Sects. 3.2 and
3.3 , perfusion bioreactors, in which nutrient-rich culture medium is driven through
the scaffold, offer scope for increasing nutrient supply to the central regions of the
scaffolds and, thereby, improving the integrity of the engineered tissue.
3.1.2 Reaction and Diffusion of Nutrients
Galban and Locke's work has inspired the development of a number of similar
mathematical models. For example, Lewis et al. ( 2005 ) develop a simple, one-
dimensional model to investigate interactions between the evolving nutrient pro-
files and cell distributions within cartilaginous tissue constructs. The nutrient
profile is modelled with a reaction-diffusion equation in which the diffusion
coefficient is assumed to be constant and the local rate of nutrient consumption
taken to be proportional to both the nutrient concentration and the cell density. The
cells are assumed to be immobile and to proliferate at a rate proportional to that at
which they consume nutrients. By comparing their simulations with experimental
data for the spatio-temporal evolution of the oxygen and cell distributions, they