Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 241

scaffold. There are therefore two flow domains—the fibres and the space occupied
by the scaffold around them—and the flows are coupled via continuity conditions
at the interfaces. The scaffold is modelled as a uniform isotropic porous medium,
with a point source and point sink modelling the inlet and outlet pipes respectively,
and line sources representing flow from the fibres. A separate fluid dynamical
problem is solved within the fibres to compute the strength of the source terms in
the line sources representing flux of fluid from the fibres into the scaffold. Again,
geometric features are exploited to simplify the governing equations: the slender
geometry of the fibres allows lubrication theory to be used to simplify the full
Navier-Stokes equations. A Poisson problem is obtained for flow in the scaffold,
which can be solved numerically, and is a much easier to solve than direct
numerical simulation of the full Navier-Stokes equations within the complex
scaffold geometry. Having solved for the fluid flow, the authors were then able to
determine the distributions of shear stress, nutrients and waste products, and asses
the implications for cell proliferation.
3.2.3 Bioreactor Design
In a series of papers (Shipley et al. 2010 , 2011 ; Shipley and Waters 2011 ), Shipley
and co-workers consider flow and nutrient transport problems in hollow fibre
membrane bioreactors (HFMB). HFMBs are designed to enhance nutrient delivery
to, and metabolite removal from, cells by using fluid flow to provide advective
transport in addition to diffusion. A single hollow fibre consists of a central lumen,
surrounded by a porous fibre wall or membrane, which separates the lumen from
the extra-capillary space (ECS). Cells are seeded in a single layer on the fibre
walls, or throughout a matrix surrounding the fibre. Fluid is driven through the
fibre lumen under the action of an applied pressure gradient. The porous fibre wall
allows the passage of nutrients, metabolites and growth factors both to and from
the cells, and acts as a membrane to protect the cells from the direct effect of fluid
shear due to flow through the lumen, thus enabling relatively high flow rates to be
used without cell damage. A bundle of such fibres is then housed within a bio-
reactor: in addition to flow through the fibre lumen, the bioreactor has entry and
exit ports for ECS flow. By developing a series of theoretical models, Shipley
et al. were able to specify a set of operating conditions which the end user can use
to prescribe the bioreactor geometry (e.g. fibre length and ECS depth) and oper-
ating parameters (e.g. pressures, flow rates, nutrient inlet concentrations, cell-
seeding density) to obtain the optimum cell culture environment for the tissue
under consideration.
In Shipley et al. ( 2010 ) theoretical predictions for the fluid retentate (lumen
outlet flowrate) and permeate (ECS outlet flowrate) are derived, and compared
against experimental data to determine the membrane permeability and slip. Fluid
transport in the lumen and ECS is described using Navier-Stokes equations, and
flow through the porous fibre walls using Darcy equations. The model is simplified
by exploiting the slender geometry of bioreactor system, so that lubrication theory