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

834 Tissue Engineering

convective transport into the implant. Furthermore, there
is no generation of oxygen by the tissue, so G i = 0. The
consumption of metabolites by cells often follows first-
order kinetics at low concentrations (i.e., the consumption
rate is proportional to the concentration of metabolite)
and progressively becomes zero order as the concentra-
tion increases. At one point, the cells are “saturated” and
cannot take up more. This behavior is often described by
Michaelis–Menten kinetics:
V max c i
R i = (18)
K M + c i
where V max is the maximal uptake rate and K M is the
metabolite concentration at which the uptake rate is
half-maximal. Substituting this expression into the
continuity equation and taking into account other
assumptions described above yield:
2
∂c i ∂ c i V max c i
= D i − (19)
∂t ∂x 2 K M + c i
If we next assume steady state, we obtain: FIGURE 12 Correlation for the maximum thickness of a tissue
2 construct to avoid nutrient depletion.
∂ c i V max c i
0 = D i 2 − (20)
∂x K M + c i
tificial liver device possessing 10% of the detoxification
One can integrate this expression with the following
andproteinsynthesiscapacityofthenormalhumanliver(a
boundary conditions:
rough estimate of the minimum processing and secretory
x = 0, c i = C 0 (21) capacities that can meet a human body’s demands) would
10
x = X, ∂c i /∂x = 0 (22) contain a total of 10 adult hepatocytes. Thus, to keep the
total bioreactor volume within reasonable limits (1 L or
to yield the concentration profile throughout the system. 7
less), 10 cells/mL or more are required. For comparison,
However, rather than looking at the whole concentration 8
the normal human liver contains approximately 10 hep-
profile, we really only want to know if at any point within
atocytes/mL. Three main types of bioreactor design have
the system there will be a significant depletion of metabo-
been considered in tissue engineering: (1) suspension cul-
lite. As a rule of thumb to estimate when a cell is starving,
ture methods using microcarriers, (2) cells immobilized
the condition c i = K M is often used. Figure 12 shows the in hollow-fiber systems, and (3) suspension culture in ro-
results for the maximum thickness X of a construct with- tating wall vessel bioreactors.
out c i going below K M anywhere in the construct. Using
this chart to estimate the thickness of an implantable con- a. Microcarrier-based systems. Microcarriers are
struct containing liver parenchymal cells (hepatocytes), one of the first methods used for supporting large-scale
we find that a construct containing even a relatively low mammalian cell culture. Microcarriers are small beads
7
3
density of cells (10 cells/cm ) cannot have a thickness
exceeding about 500 µm (Table VII). At tissue-level cell TABLE VII Maximum Thickness (X) of Tissue
8
3
densities of 10 cells/cm , that thickness can be as low as Constructs Estimated by Order of Magnitude
Transport Analysis a
100 µm, which is consistent with the in vivo density of
capillary vessels. 10 cells/cm 3 10 cells/cm 3
8
7
Geometry X (µm) X (µm)
3. Bioreactor Technologies
Slab 300 95
For certain applications it is necessary to maintain a large Cylinder 430 135
number of cells that transform an input of reactants into Sphere 520 165
an output of products. This is the case for the bioartifi-
a These estimates are based on the assumptions
cial liver or pancreas and more recently for the production
that that surface of construct is exposed to arterial
of blood cells from hematopoietic tissue. These systems 3
oxygen levels (C 0 = 100 nmol/cm ), hypoxic damage
require maintenance of the function of a large number of
occurs when c i = K M , and for hepatocytes V MAX =
3
6
cells in a small volume. For example, a hypothetical bioar- 0.4 nmol/10 cells/s, K M = 0.6 nmol/cm .