Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 228

mixtures of interacting continua (Bowen 1976 ; Marle 1982 ; Passman and Nunziato
1984 ; Whitaker 2000 ; Kolev 2002 ; Ateshian 2007 ). Such approaches have been
used widely to model biological tissue mechanics (Mow et al. 1980 ; Lai et al.
1991 ) and, more recently, tissue growth and remodelling (Please and McElwain
1998 ; Please et al. 1999 ; Landman and Please 2001 ; Preziosi and Tosin 2009 ;
Ambrosi et al. 2010 ). Since the aim of this review is to highlight modelling
approaches and challenges, we do not provide a historical perspective of multi-
phase modelling approaches, nor do we present a comprehensive derivation of
relevant multiphase equations; it is markworthy, however, that although by defi-
nition such a multiphase continuum approach does not account for the precise
microscopic detail of (for example) cell-cell interactions, the averaging process
involved in deriving models of this type ensures that terms present in the model
equations arise from appropriate microscopic considerations (see, e.g. Drew 1983 ).
In Sect. 3 , we provide some example studies which highlight modelling issues in
tissue engineering and mixture theory approaches with which to investigate them.
We remark that in many cases, modelling tumour growth was the original focus of
these studies; the formulations are, nevertheless, relevant to tissue engineering
applications.
As indicated above, the basis of these multiphase models is a set of conser-
vation equations, governing mass and momentum transfer between the constituent
phases. For a system comprising n incompressible phases, if inertial effects and
body forces are negligible, then the equations governing the ith phase (with density
q i , volume fraction h i , velocity u i and stress tensor r i ) may be expressed:

¼ S i ;

o h i
ot þrð h i u i Þ

q i ð 1 Þ

Þþ X
j 6¼ i

r h i r i ð F ij ¼ 0 ; ð 2 Þ

where t denotes time, r is the spatial gradient operator, S i is the net rate of mass
transfer into the ith phase and F ij denotes the force acting on phase i as a result of
interactions with phase j. We note that in some cases, flow in bioreactors is
modelled by the full Navier-Stokes equation, in which case, inertial terms are
retained in Eq. ( 2 ).
Within the multiphase modelling context, assumptions can be made that sim-
plify the resulting systems of equations. For example, one approach is to assume
that the cells occupy no volume, and therefore have no effect on the fluid flow.
When determining the mechanical load that the flow exerts on the cells, it is
assumed that the shear stress exerted on the substrate will be that experienced by
the adherent cells; nutrient transport is incorporated by considering cells to be
sinks or sources of metabolites or waste products. In some cases, such flow and
transport problems require equations for the flow of nutrient-rich culture medium
surrounding a porous scaffold to be coupled to those describing the flow within
the scaffold. These equations are linked by specifying boundary conditions at the