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Encyclopedia of Physical Science and Technology EN016J-783 August 1, 2001 10:58

832 Tissue Engineering

FIGURE 11 Transport of metabolites into a tissue construct implanted next to a blood vessel. U is the velocity of
fluid extravasated into the tissue; see text for additional explanations.
engineered blood vessels. A potential design of such tissue construct of thickness X implanted in vivo, assum-
vascular grafts would have endothelial cells on the inside ing that this construct is avascular but surrounded with
surface of the graft aligned with the direction of blood blood vessels from the hosts’s tissue (Fig. 11). The fun-
flow, as is observed in vivo in arteries. On the other hand, damental equation describing the flux N of a particular
the appropriate direction of smooth muscle cells within species i is given by Fick’s law of diffusion, to which
the blood vessel wall would be along the circumference terms to account for the electrical migration of species
of the vessel in order to be able to perform their natural and convection are added:
function of modulating the diameter of the vessel.
∂c i z i
N i = φ −D i + µ i c i E + W i c i U (10)
∂x |z i |
2. Metabolite Transport
To facilitate the comparison between systems of different
In normal tissues, the circulatory system brings in nutri- geometries and scales, we next rewrite the previous equa-
ents and removes waste products. Typically, no cell in vivo tion using dimensionless quantities. The P´eclet number
is farther than about 100 µm, or even sometimes less, from is a dimensionless number defined as the ratio of nondif-
a blood vessel. Thus, transport by diffusion does not have fusive transport (convection and electrical migration) to
to occur over distances beyond 100 µm. Engineered tissue transport by diffusion in a particular system:
constructs are mostly devoid of any vascular system, and
W i U + φ z i µ i E
although there may be culturing methods allowing trans- |z i |
Pe = X (11)
port by convection throughout the cell mass, as described φD i
later in the bioreactor section, after implantation the en-
Substituting into the flux equation yields:
gineered tissue is no longer perfused and the situation
remains so until vascularization by angiogenesis occurs ∂c i Pe
from the surrounding host’s tissues. Vascularization in situ N i = φD i − + c i (12)
∂x X
can be accelerated by implanting tissues that release an-
We can further simplify this equation by defining the fol-
giogenic factors, such as fibroblast-derived growth factor
lowing dimensionless variables:
and vascular endothelial growth factor. Although tissue
constructs may include vascular endothelial cells, it is not N i c i x
∗ ∗ ∗
yet possible to create three-dimensional vascular networks N = , c = , x = (13)
i
i
φD i X X
in vitro that are patent.
which yields a new and simplified flux equation containing
Transport through tissues can be modeled using the ba-
only the P´eclet number as the unknown or “adjustable”
sic transport equations used in nonliving systems and can
parameter, the value of which depends on the system under
be useful to predict the concentration profiles of metabo-
consideration:
lites throughout engineered tissues. In the following pre-
∗
sentation, we apply these equations with the goal of pro- ∗ ∂c i ∗
N = − + Pe c i (14)
i
viding design criteria for tissue constructs. We consider a ∂x ∗