Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 242

may be used to describe flow in the lumen and ECS. Analysis of the width of the
boundary layer region where slip effects are important, together with the sensi-
tivity of the retentate and permeate equations to the slip parameter, showed that
slip was not significant for these membranes. The model was then further reduced
by assuming no-slip conditions at the membrane surface, and comparison of the
theoretical predictions for the retentate and permeate with the experimental data
enabled the membrane permeability to be determined. The validated and param-
eterized model was then used to determine the operating conditions that enable the
lumen inlet flowrate and pressure at the lumen outlet to be controlled to provide a
specific permeate to lumen inlet flowrate ratio. In Shipley et al. ( 2011 ) the com-
plementary nutrient transport problem was studied: although the modelling was for
a generic nutrient, the parameters were specialised for oxygen. Here fluid flow in
the lumen was modelled by Poiseuille's law, while membrane and ECS flow were
neglected. Nutrient concentration was governed by an advection-diffusion equa-
tion in the lumen, a diffusion equation in the membrane, and an advection-reac-
tion-diffusion equation in the ECS, with Michaelis-Menten kinetics for the
reaction term. These equations were coupled via specification of continuity of
concentration and flux at the lumen-membrane and membrane-ECS boundaries.
Analytical progress was made in the limit in which the ECS oxygen concentration
is much larger than the half-maximal concentration of oxygen, so that the
Michaelis-Menten reaction term can be approximated by a constant. These
solutions were complemented by computations of the full system of equations,
using COMSOL. The study enabled operating conditions to be defined to enable
the user to determine the medium flow rate, lumen length and ECS depth that
ensures a minimum oxygen concentration throughout the HFMB. In Shipley and
Waters ( 2011 ), this work was extended to consider flow throughout the HFMB and
the transport of lactate, in addition to oxygen. The study indicated that oxygen (as
opposed to lactate) is the limiting solute with respect to bioreactor design, and that
opening the ECS port to promote radial flow through the bioreactor provides
significant benefit, enabling a greater volume of cells to be cultured within the
desired nutrient environment.
All the models described above consider timescales appropriate to transport
processes rather than cell proliferation, differentiation etc, and neglect neo-tissue
formation, so that, for example, the effect of cell proliferation on the scaffold
permeability and hence the resulting fluid flow is not considered. In Sect. 3.3 we
consider models which account for the evolution of cell volume fraction, and the
full coupling between the cell dynamics and their environment.