Multiphysics Computational Modeling in Cartilage Tissue Engineering - Computational Modeling in Tissue Engineering - page 268

area, giving rise to an increase in the local state of shear stress (represented in the
model by the Darcy stress).
6 Multiscale Modeling
In the attempt of including in the mathematical description of the tissue-engineered
construct a wider spectrum of phenomena, while keeping the computational
complexity at an acceptable level, multiscale homogenization techniques have been
applied in a certain number of recent studies. A first example in this direction is
represented by the work of Cioffi et al. [ 11 ], where a Macroscale simplified model
provides appropriate boundary conditions for a Microscale model. The former model
is set in an axi-symmetric geometry and steady Navier-Stokes equations are solved
coupled with a reaction-diffusion equation for the nutrient with a constant volumetric
consumption rate. This model provides the velocity and nutrient concentration
profiles at the inlet of the Microscopic model formulated on a small (sample) portion
of the bioreactor which consists of the union of a few unit cells. In this Microscale
computational domain the same equations as above are solved, but on the real
geometry extracted from micro-CT images.
In Raimondi et al. [ 43 ], two complementary multiscale models are presented.
The first approach follows a concept similar to the one described above. Namely, a
2D model of the whole scaffold seeded with a cell monolayer is considered,
coupled with a 3D model of a functional sub-unit of such construct. The main
novelty of this approach consists in the use of a moving boundary formulation
originally proposed in Galban and Locke [ 15 ] and based on a phenomenological
relation for the time evolution of the biomass-fluid interface, which consistently
updates the geometry. To handle this time-evolving domains, an Arbitrary
Lagrangian-Eulerian (ALE) formulation is adopted and periodic remeshing is
applied to adapt the computational mesh to large deformations of the computa-
tional domain. In the second approach, a lumped discrete approximation to the
problem of mass transfer, nutrient uptake and biomass growth at the Macroscopic
scale of the scaffold is devised. In this description, the scaffold is represented by a
simplified geometry characterized by piecewise constant biophysical parameters,
whose values are extracted from the Microscale model. In turn, the Macroscale
model supplies the nutrient concentration boundary condition to a discrete set of
Microscale problems, whose solution is the nutrient distribution and biomass
evolution at the pore-size level, to be compared with real-time microscopic data.
7 Scientific Computing Techniques in Multiphysics Modeling
The aforementioned models of the engineered tissue growth feature specific dif-
ficulties with respect to their numerical approximation, which determine signifi-
cant criteria to select suitable numerical schemes to come up with computational