Hydrodynamics in Porous Media with Applications to Tissue Engineering - Porous Media: Applications in Biological Systems and Biotechnology - page 83

3.3.2 Effects of Nutrient Transport
A key problem, often encountered in tissue engineering with large-size con-
structs, is the lack of spatial homogeneity in the distribution of cells and
extracellular matrix. Indeed, according to Martin et al . (2004), the supply
of oxygen and soluble nutrients becomes critically limiting for the in vitro
culture of three-dimensional tissues. The consequence of such a limitation is
exemplified by early studies showing that cellular spheroids larger than 1 mm
in diameter generally contain a hypoxic, necrotic center, surrounded by a rim
of viable cells (Sutherland et al . 1986). Similar observations were reported for
different cell types cultured on three-dimensional scaffolds under static con-
ditions. For example, glycosaminoglycan (GAG) deposition by chondrocytes
cultured on poly(glycolic acid) meshes was poor in the central part of the
constructs ( m from the outer surface) (Martin et al . 1999), and deposi-

tion of mineralized matrix by stromal osteoblasts cultured into poly(dl-lactic-
co -glycolic acid) foams reached a maximum penetration depth of 240

∼ 400 µ

µ m from

the top surface (Ishaug et al . 1997).
As the matrix density determines eventual mechanical functionality of
engineered tissues, such an inhomogeneous spatial distribution can result in
inadequate overall mechanical properties (Vunjak-Novakovic et al . 1999; Kelly
and Prendergast 2005; Sengers et al. 2007). Because engineered constructs
should be at least a few millimeters in size to serve as grafts for tissue replace-
ment, mass-transfer limitations represent one of the greatest challenges to be
addressed. The causes of these restrictions are diverse, implying supply of
nutrient and soluble biochemical factors, removal of waste products, and non-
homogeneity in consumption sites due to cell seeding or migration. The main
control parameter that determines whether solute gradient will occur is the
required surface density of the cells (corresponding to σ cel ) lining on the pore
wall associated with the cellular nutrient consumption rate V Max . Depending
on the tissue-engineering application, porous scaffold properties and cultiva-
tion processes can be different, as explained furthermore in Section 3.3, but the
overall transport restrictions have to be overcome by controlling the physical
and biochemical environments in “the heart of the bioreactor.” Thus it is nec-
essary to downscale the description of the phenomena to reach the length scale
a of the medium pore. Furthermore, the nondimensional parameter, named
Damkohler number Da , defined as the ratio between the consumption flux
of nutrient at the pore wall and the diffusion flux inside the pore, avers fun-
damental in the study of transport processes of nutrients such as oxygen or
glucose. This parameter can also be viewed as a good evaluation, at the pore
scale a , of the ratio between the consumption and diffusive rates of change in
nutrient concentration and is so expressed by

R V

= σ cel ×

V Max × a

Da =

c 0 a 2

(3.3)

D ×

D × c 0

In these expressions, c 0 stands for the reference concentration of the oxygen
or nutrient, whereas D is its diffusion coecient. As will be shown later on,