Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 235

found that their model represents a good description of cartilaginous tissue growth
for the first two weeks after seeding. At later times, mechanisms not included in
the model, such as contact inhibition of cell proliferation and reduced nutrient
transport, are likely to become significant and may explain the poor agreement
with the data after two weeks in culture.
Dunn et al. ( 2006 ) extended Lewis et al.'s model to account for contact inhibition
of growth, using a logistic growth law to describe cell growth, the growth rate being
proportional to the nutrient concentration, and by considering a two-dimensional
cylindrical geometry. Their model was found to give good agreement with data from
experiments in which pre-osteoblasts were initially seeded uniformly throughout a
porous scaffold, these results being qualitatively similar to those reported by Freed
et al. for chondrocytes. Dunn et al.'s model was, however, unable to reproduce
experiments in which the scaffolds initially consisted of a series of thinner scaffolds
that alternated between cell-seeded and unseeded: when the initial conditions were
altered to mimic the layered structure, the experimentally observed peak in cell
density at the interface between seeded and unseeded regions was not evident in the
numerical simulations. Two possible explanations for this discrepancy were pro-
posed. First, convective flow, which is not included in the mathematical model, may
enhance nutrient transport at the boundaries between the seeded and unseeded
regions. Alternatively, cell migration may not be negligible: cells may migrate away
from regions of low oxygen and accumulate at the interface between the seeded and
unseeded regions, where oxygen levels are higher.
More recently, Jeong et al. ( 2011 ) have focussed on developing an efficient
numerical method for solving an extension of Lewis et al.'s model for the growth
of cartilaginous tissue constructs in two-dimensional cylindrical geometry, in
which the cells are allowed to move by random motion. Jeong and coworkers
propose an operator-splitting algorithm to solve the resulting pair of coupled
reaction-diffusion equations. Their approach involves alternating finite-difference
approximations and analytical solutions in order to update the cell and nutrient
profiles on successive time-steps. We remark that the model that Jeong et al. solve
is similar in form to an earlier model by Obradovic et al. ( 2000 ) that was used to
investigate how the local oxygen concentration within a cartilaginous construct
influences the rate at which cells seeded within the scaffold produce glycosami-
noglycan, an important component of cartilage.
The work of Chung et al. ( 2006 ) places the phenomenological reaction-diffu-
sion models developed in Obradovic et al. ( 2000 ) and Jeong et al. ( 2011 )ona
stronger theoretical foundation. Following Galban and Locke ( 1999a , b ), Chung
and coworkers use a volume-averaging approach to derive an extension to Galban
and Locke's two-phase model that accounts for a small degree of random cell
movement. Their model simulations suggest that cell motility leads to more uni-
form cell distributions within the scaffold and higher overall rates of cell growth
than when cell movement is neglected. Chung et al. also use their model to
demonstrate that uniform cell seeding is likely to be a better strategy for tissue
engineering (in static culture) than non-uniform seeding, arguing that concentrated