Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 229

interface between the single-phase flow domain (the surrounding fluid) and the
multiphase flow domain (the porous scaffold in which both scaffold and fluid
volume fractions must be considered). For example, Navier-Stokes equations may
be used to describe the surrounding fluid via

r u ¼ 0 ; o u

ot þ u r u ¼ 1 q r p þ m r 2 u

ð 3 Þ

where u is fluid velocity, p is fluid pressure, q is the fluid density and m is the
kinematic viscosity. The flow within the porous scaffold may be described by
Darcy's law

r u ¼ 0 ; u ¼ k

l r p ;

ð 4 Þ

where k is the permeability of the porous medium and l is the dynamic viscosity.
Alternatively, the Brinkman equations may be used:

r u ¼ 0 ; u ¼ k

l r p k r 2 u :

ð 5 Þ

In Eqs. ( 4 ) and ( 5 ) the Darcy flux u is the fluid velocity weighted by the scaffold
porosity.
Both the Darcy and Brinkman equations are examples of multiphase models
that may be obtained from Eqs. ( 1 ) and ( 2 ) via appropriate choices of constitutive
laws. Models of this type are reviewed in Sect. 3.2 . A key aspect of the macroscale
Darcy and Brinkman models is that the microscale properties are captured via a
parameter at the macroscale, for example, the scaffold permeability, without the
need for detailed knowledge of the pore geometry, information that is expensive to
obtain, and moreover, changes from one scaffold to another. Furthermore, if details
of the pore geometry are known, they can be incorporated into an expressions for
the permeability, for example via homogenisation techniques (Shipley et al. 2009 ).
An alternative to the above macroscale approach when considering flow and
nutrient transport problems is to consider a microscale one in which the fluid flow
within the interconnected pores is solved using Navier-Stokes equations. Such a
method, especially when complemented with imaging techniques such as l-CT
that provide detailed information about the geometry and the microstructure of the
porous scaffold, is a powerful tool for the full characterisation of the 3D flow fields
and stresses within dynamic culture systems. However, this can be computation-
ally intensive, and requires a simulation to be run for every scaffold architecture.
Examples of such studies are reviewed in Sect. 3.2 .
As an alternative to assuming that the cells occupy no volume, they may be
assumed to occupy volume, but that their interaction with the flow can be
neglected (so that the role of any culture medium in the system is simply to
provide a supply of nutrients to the cells). Such models, which are particularly
appropriate for static culture systems where the transport of nutrients and