Multiphysics Computational Modeling in Cartilage Tissue Engineering - Computational Modeling in Tissue Engineering - page 270

Fig. 6 (left) Fluorescence image of the engineered tissue growing on the fiber scaffold
represented in Fig. 1 b (scaffold fibers are visualized with blue, biomass with green). (middle) The
outcome of image segmentation approach targeting the biomass. The contours of the biomass film
are highlighted in red.(right) Numerical simulation of a diffusion problem where the biomass
region features lower diffusivity than the surrounding fluid. Owing to the extended finite element
technique, the computational mesh does not fit the complex contours of the fluid/biomass
interface, although the computational accuracy is not significantly compromised
experiments. On the one hand, problem specific parameters such as nutrient dif-
fusivity and consumption rates as well as fluid viscosities and hydraulic resistances
of biomass should be provided. On the other hand, equations must be solved on
realistic geometrical configurations, to allow for model validation based on direct
comparison between simulations and observations.
Micro-CT and fluorescence microscopy images at the cell scale should be the
starting point to reconstruct the geometrical models for numerical simulations.
More precisely, the geometry of the scaffold matrix and the biomass available
from images, has to be translated into quantitative terms in order to be exploited
for the numerical approximation of fluid perfusion and transport processes.
To address this task, several approaches are available. In principle, one can attempt
to set up a parametric mathematical description of the domain boundaries or
interfaces. However, the highly irregular shape of tissue-engineered constructs
makes this task hardly achievable in the present context (Fig. 6 ). Alternatively,
recent advances on numerical approximation schemes for PDEs have shown a
promising way to override this difficulty. In particular, non-standard finite element
schemes (such as the Extended Finite Element method, XFEM) have the ability to
use Cartesian, non-boundary-fitted meshes to solve problems in complicated
domains. Such approach, generally called the fictitious domain method [ 18 ],
consists in embedding the computational domain into a fictitious volume with
simple geometry, in such a way that the material interfaces do no longer need to be
geometrically approximated. Recent developments in this direction Hansbo and
Hansbo [ 19 ], Burman and Hansbo [ 4 ], Burman and Zunino [ 5 ] give rise to robust
and efficient finite element schemes that are capable to approximate partial dif-
ferential equations with interfaces separating between highly heterogeneous
material properties even when the coefficient discontinuities are not exactly
captured by the computational mesh.