Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 239

concentration are required, then a more detailed modelling approach, where the
scaffold geometry is reconstructed using l-CT images is more appropriate.
3.2.2 Asymptotic Model Simplification
As outlined in Sect. 2.1.1 , asymptotic methods can be used to simplify the gov-
erning equations, leading to reduced models that provide insights into the key
underlying mechanisms involved in tissue growth within bioreactors. This
approach is exemplified in Cummings and Waters ( 2006 ) (for complementary
computational approaches see e.g. Lappa 2003 ) where a model for a rotating
bioreactor is developed, in which a cylindrical vessel of circular cross-section
rotates about its longitudinal axis. The bioreactor is filled with nutrient-rich culture
medium and contains a growing tissue construct, which is modelled as a cylin-
drical solid object of circular cross-section. The axial length of the bioreactor is
small relative to its radius. This fact, together with the observation that the reduced
Reynolds number for the system is small, leads to a simplified model, in which the
dependent variables are averaged across the axial direction, reducing the problem
to two space dimensions. The rotation of the system introduces Coriolis and
centrifugal force terms in the governing equations. The resulting fluid-dynamical
problem is similar to that for flow around a moving circular obstacle (the tissue
construct) within a rotating Hele-Shaw cell (see, e.g. Schwartz 1989 ; Waters and
Cummings 2005 ). Additionally, the Peclet number is typically large in these
biological applications, which simplifies the nutrient transport problem to one in
which the nutrient concentration is constant along the two-dimensional fluid flow
streamlines, with diffusion taking place perpendicular to the streamlines.
The authors exploit the fact that fluid flow, nutrient transport, and construct
growth occur on very different time-scales, and decompose the problem into four
distinct stages. Firstly, the fluid flow around the tissue construct is determined,
assuming its location within the bioreactor is known. Then the position of the
construct is determined by considering the balance of forces acting on the con-
struct. Two classes of periodic motions are found, distinguished by whether or not
the centre of the construct orbits the bioreactor centre (a special subcase of the
latter being where the tissue construct remains stationary in the laboratory frame of
reference). The fluid flow and construct trajectory model were validated in
Cummings et al. ( 2009 ) by comparing the theoretical results with an experimental
study of the trajectory of a large solid cylindrical disc suspended within a fluid-
filled rotating vessel. All three flow regimes described above were observed
experimentally, and good agreement between experiment and theory was found.
Having successfully solved the fluid flow problem, the nutrient distribution
around the construct was then found, assuming a fresh supply of nutrient is
maintained at the bioreactor's circular boundary and that at the tissue construct
boundary nutrient is taken up at a rate proportional to how much is available in
the surrounding culture medium. Using the nutrient concentration solution, and