Multiphysics Computational Modeling in Cartilage Tissue Engineering - Computational Modeling in Tissue Engineering - page 266

These approaches are aimed at accounting for the fact that the environment in a
tissue-engineered construct is strongly heterogeneous. Multiphase models allow an
explicit consideration of these interactions. Each constituent is considered as a
distinct phase within the multiphase system with corresponding constitutive laws
and interactions with neighbouring phases, the inherent complexity of this
approach, compared to that of solving the complete problem of Eqs. ( 1 - 3 ) can be
conveniently reduced by an averaging process, yielding a single equation which
holds uniformly in the material, upon characterization of effective parameters such
as nutrient diffusivity, biochemical reaction rates and construct permeability.
Derivation of multiphase models applied to a wide range of problems in
computational biology has been given extensive treatment by many authors,
including, for example, Marle [ 24 ], Whitaker [ 48 ] and Byrne et al. [ 6 ]. In the series
of papers by Galban and Locke [ 15 ], a two phase (fluid and biomass) model for
cell growth and nutrient diffusion in a polymer scaffold with no perfusion is
presented. A single, averaged reaction-diffusion equation for the nutrient con-
centration in the two phase system is derived using the volume-averaging method
of Whitaker [ 48 ], Wood et al. [ 50 ] and the effective diffusion coefficient and
reaction rate are calculated as a function of the local cell volume fraction which
evolves according to a cell population balance equation. In the paper by Chung
et al. [ 9 ] a two-phase (fluid and biomass) model analogous to the one of Galban
and Locke [ 15 ] is proposed, with the inclusion of a self-consistent computation of
the fluid-dynamic field via an averaged Stokes-Brinkman model stemming upon
volume averaging of Stokes equations [ 20 ]. To reflect the fact that cell growth into
the scaffold reduces the effective pore size, Chung et al. [ 9 ] propose to include the
dependency on the cell volume fraction via a Carman-Kozeny type relation for the
permeability.
In our group, an homogenized approach in the line of the above cited literature
has been recently investigated in Sacco et al. [ 44 ]. The main novel contribution of
this work is the systematic inclusion of experimental data, in particular the
dependence of the biomass growth rate on the local fluid-dynamical shear stress,
and the inclusion of the effect of the solid scaffold fraction in the characterization
of the effective parameters arising from the homogenization procedure (hydraulic
permeability and nutrient diffusivity). The set of results presented in Fig. 5 are
obtained adopting the same model as in Sacco et al. [ 44 ] but investigating the use
of a modified biomass growth model based on two contributions: a promotion and
an inhibition term. This latter term is calibrated with respect to the inlet velocity
modulus, that is used as an indicator of the typical fluid-dynamical shear stress in
the device. In this analysis, we track the spatial and temporal evolution of the
nutrient concentration and fluid-dynamical variables as well as the volume fraction
occupied by the growing biomass which modifies the porosity of the scaffold
matrix, thus altering the fluid flow. Computations, corresponding to different time
levels, refer to the same geometry discussed in Sacco et al. [ 56 ] enforcing a
Poiseuille flow profile at the inlet. Simulations show that biomass growth is
enhanced close to the inlet section of the scaffold, due to a larger local availability
of nutrient. In turn, this results into a higher occlusion of the porous matrix in this