Multiphysics Computational Modeling in Cartilage Tissue Engineering - Computational Modeling in Tissue Engineering - page 269

solutions of flow and mass transport problems. Since the mathematical models
representing such phenomena are based on partial differential equations, approx-
imation methods such as the finite element method or the finite difference (or finite
volume) methods may be equivalently applied. However, the forthcoming
description will be biased in the direction of the finite element method, which is a
highly flexible approach. The main challenges in modeling cellular constructs are
represented by the need to solve problems with spatially heterogeneous physical
parameters or to couple different mathematical models (viscous and inviscid flows,
free flows and porous media) within a genuine multiphysics framework, including:
Darcy's, Brinkman's or Stokes' models for perfusion flow in the scaffold porous
matrix with growing biomass, nutrient mass transport and delivery to growing
cells, cell growth and metabolism. The interfaces between different materials and
models have to be handled at the discrete level by a suitable treatment of the
transmission conditions.
As an example, for the approximation of mass transport through heterogeneous
media, which arises from the study of nutrient supply to cells, a correct quanti-
fication of the amount of nutrient reaching the target is achieved by properly
capturing the balance of mass fluxes at the interface between the fluid and the
growing tissue, as well as by accurately approximating the steep concentration
gradients that arise in the neighborhood of the fluid/solid interface. For the former
issue, an effective mathematical methodology is Domain Decomposition, for
which we refer to Quarteroni and Valli [ 33 ].
Such family of methods splits a coupled multiphysics system in single sub-
problems, which can be then solved by standard approximation techniques and
software packages. The global solution accounting for their interaction is then
recovered by means of iterative strategies. Domain decomposition techniques have
already been successfully applied to analyze fluid dynamics and mass transfer
through biological tissues [ 34 , 35 ], D'Angelo and Zunino [ 12 ]. Another important
issue in the simulation of heterogeneous problems is the development of robust
and conservative schemes. For the case of nutrient transport, a conservative
scheme would exactly preserve the local mass balance related to nutrient con-
centration. Robustness, in turn, refers to the stability of the numerical scheme,
which is the fundamental property that quantifies at what extent the approximate
solution is sensitive with respect to perturbations on the discrete problem data.
A numerical method is called robust when its stability properties are affected
neither by the magnitude of the problem coefficients nor by their variation. This is
achieved by means of Discontinuous Galerkin finite elements (DG). However,
a major drawback of DG schemes is their increased computational cost compared
to standard (conforming) finite element formulations, that in some cases makes
them unsuitable for large scale realistic problems.
The previous computational techniques allow for an accurate and efficient
approximation of the partial differential equations that describe the phenomena
governing artificial tissue growth. However, computational methods provide
quantitative results only when they are applied on the basis of realistic data
for the problem at hand, through a strong interaction between simulations and