Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 244

3.3.1 Cell-Environment Interactions: Momentum Transfer

The above authors dramatically simplify the momentum transfer between phases
by employing the Darcy or Brinkman equations to relate the fluid flow to the fluid
pressure. A comprehensive multiphase formulation relevant to tissue growth
processes within a perfusion bioreactor system, which addresses in detail the
interactions between phases was presented by Lemon et al. ( 2006 ). The modelling
framework accommodates an arbitrary number of fluid phases (representing, for
instance, cells and culture medium), contained within a porous medium (such as an
artificial scaffold and/or ECM). The model differs from other fluid-based models
(such as Breward et al. ( 2002 ), Franks and King ( 2003 ) and references therein) by
the explicit consideration of a tissue's solid constituents (here, interpreted as a
porous scaffold and/or ECM), and the general nature of the interphase interaction
terms, which extends the range of tissue engineering applications that can be
addressed. In general, the formulation allows for deformation of the porous
medium; however, in Lemon et al. ( 2006 ), attention is restricted to a rigid porous
scaffold and as such, the stimulation provided by mechanical bioreactors (e.g. the
bioreactor system of El-Haj et al. 1990 ) may not be accommodated. The adoption
of fluid-like constitutive assumptions for the phases contained within the porous
scaffold, which is appropriate on the timescale of tissue growth, simplifies sig-
nificantly the modelling of tissue growth processes since mass transfer between
phases (representing tissue growth) does not generate stress, as is the case when
elastic constitutive modelling assumptions are made.
Referring to Eq. ( 2 )in Sect. 2.1 , the interphase force terms F ij are assumed to
comprise contributions from viscous drag, accommodated via terms proportional
to the difference in phase velocity, and active forces (such as those which arise
from interphase interactions), these being modelled via additional pressure con-
tributions (Lemon et al. 2006 ):

þ kh i h j ð u j u i Þ ; p ij ¼ p þ w ij ; j 6¼ i

F ij ¼ p ij h j r h i h i r h j ð 6 Þ

p i ¼ p þ / i þ X
j 6¼ i

h j w ij ð 7 Þ

where p ij denote the 'interphase pressures' which are assumed to comprise a
contact-independent pressure, p, common to all phases in the mixture, and an
additional term w ij which accounts for the effect of tractions between phases (e.g.
cell-ECM interactions). The pressure p i of each phase accommodates an addi-
tional internally-generated pressure / i (due to cell-cell interactions, for example).
The multiphase approach embodied by Eqs. ( 1 ), ( 2 ), ( 6 ) and ( 7 ), together with
suitable constitutive assumptions about r i and S i , have been employed by many
authors to investigate a range of tissue engineering problems. In each case, the
relevance to the biological system under consideration is ensured by restriction to
an appropriate numbers of phases, and by specification of r i ; S i , and the extra
pressures / i and w ij .