Continuum Modelling of In Vitro Tissue Engineering: A Review - Computational Modeling in Tissue Engineering - page 243

3.3 Modelling Dynamic Tissue Culture: Multiphase Phase
Flow Models
As discussed in Sect. 3.2 , a simple study of perfusion within porous tissue con-
structs was presented by Whittaker et al. ( 2009 ); here, the scaffold geometry is
fixed, and the effects of cell proliferation on scaffold porosity, and hence fluid flow
and nutrient transport fields, are neglected. Multiphase models which consider
these features include Coletti et al. ( 2006 ) and Causin and Sacco ( 2011 ). In these
studies, the dynamics of a cell seeded porous scaffold incubated in a perfusion
bioreactor were considered. The scaffold was modelled as a rigid porous material,
saturated with a viscous fluid. To account for cell proliferation, the porosity of the
rigid scaffold is defined to be the sum of a time-invariant component (the porosity
of the decellularised porous scaffold) and a time-dependent component (due to the
proliferation of the cells); that is, the cell phase is assumed to be immobile, and
with identical material properties to the scaffold phase. By appealing to the sep-
aration of timescales between flow and cell proliferation, the momentum balance
equation for a viscous culture medium is simplified by assuming that fluid flow
may be modelled by the Brinkman equations. Occlusion of scaffold pores due to
cell proliferation and its influence on the flow field is accommodated via a cell
density-dependent Darcy permeability of Carman-Kozeny type. The influence of
this flow field on the distribution of a passive nutrient is modelled by reaction-
diffusion-advection equations; nutrient consumption by the cells is modelled by
Michaelis-Menton kinetics. Cell proliferation in response to this nutrient field is
described by a Contois equation.
A more comprehensive study, investigating such ideas is presented by Chung
et al. ( 2007 ). Here, cell movement is accommodated via linear diffusion, the fluid
flow and nutrient fields being modelled in a similar manner to that described
above. The cell density and nutrient profiles were calculated by numerical simu-
lation, to demonstrate that perfusion of cell cultures can lead to enhanced prolif-
eration and a more spatially-uniform cell distribution. Shakeel et al. ( 2011 ) extend
this approach by employing nonlinear diffusion to represent cell movement, and by
accommodating the influence of the mechanical environment on cell proliferation.
Such a change in cell phenotype may be accommodated by suitable specification
of the mass transfer rates S i ; Shakeel et al. ( 2011 ) capture the dependence of cell
proliferation and nutrient consumption on fluid shear stress by adapting functional
forms proposed by O'Dea and coworkers (reviewed below) to describe enhanced
proliferation and nutrient consumption when cells are exposed to physiologically-
relevant levels of fluid shear stress. By employing the commercial finite element
software COMSOL Multiphysics, the influence on eventual tissue construct
composition of different scaffold porosity distributions, and cell seeding protocols
was investigated. The authors conclude that an effective means to ensure nutrient
delivery to large tissue constructs in such bioreactors is through the use of high-
porosity channels which span the construct.