Computational Modeling of Mass Transport and Its Relation to Cell Behavior in Tissue Engineering Constructs - Computational Modeling in Tissue Engineering - page 84

where C i is the concentration of the solute of interest; C i is the mass flux; R i is a
reaction term which accounts for consumption, production, degradation, or binding
of solute i to the matrix; and t is time. The mass flux due to molecular diffusion is
proportional to the gradient in solute concentration (C i =-D ij r C i ), while con-
vective transport is driven by the velocity field v (C i = C i v)[ 6 ]. Substitution of
both terms into the previous equation gives (in the case of an incompressible
medium),

oC i x ; y ; z ; t ð Þ

Þ 2 C i x ; y ; z ; t

¼ D ij x ; y ; z ; t ð ð Þ vx ; y ; z ; t ð Þ C i x ; y ; z ; t ð Þ

ot

þ R i x ; y ; z ; t ð Þ

where v is the solute velocity vector; D ij is the diffusion coefficient of the solute in
solvent j; and r 2 is the Laplacian operator. This general equation applies for most
biomaterial setups used in tissue engineering and its constitutive transport parame-
ters can be determined either from experiments or from theoretical formulations.

2.1 Diffusion

Experimental quantification of solute diffusion rates through a carrier have been
performed in well-controlled release kinetics experiments and by fitting analytical
solutions to Fick's diffusion law [ 16 , 25 ]. Also well established are fluorescence
techniques to measure dispersal of fluorescently labeled target molecules, such as
Fluorescence Recovery After Photobleaching (FRAP) [ 11 ], photoactivation [ 99 ],
photoconversion [ 47 ] or photoswitching [ 3 ] of these fluorescent molecules. Major
advantage of the latter methods is that they are less time-consuming as compared
to release kinetics [ 11 ] and also have the ability to record local differences in
solute diffusivity, which have been shown to result from structural matrix heter-
ogeneities [ 118 ].
Alternatively, solute diffusion rates for a specific carrier matrix can be esti-
mated from existing literature values. Reported values are obtained either for
diffusion in free solution or for a given solute carrier combination. Based on
microscale structural differences, several authors have formulated relationships
which try to explain discrepancies in diffusion coefficients between often seem-
ingly equal matrices [ 55 , 57 , 62 , 95 ]. These models take into account the steric
interactions of solutes diffusing through the matrix. The interplay between struc-
tural matrix features and effective solute diffusivity can also be described in terms
of the hydrodynamic obstructions of diffusing solute. Correlations have resulted
from this approach using techniques of volume averaging, which requires the use
of periodic structural models [ 128 , 129 ], the effective-medium approximation,
which does not impose any restrictions on the structural model but is at the cost of
a reduced validity and reliability of the predictions [ 18 , 103 ] or by using a random
walk approach [ 118 ].