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

Tissue Engineering 833

The use of the dimensionless quantities facilitates com- TABLE VI Transport Parameters for a Few Metabolites
parison between systems with different geometries and Important for the Function of Tissue Engineered Constructs
Implanted in vivo
scales. For example, although two systems may be differ-
2
2
ent in several respects, they will behave similarly as far Metabolite D i (cm /s) a C i (mM) b Pe N i (µmol/cm /s)
as metabolite transport if they have similar P´eclet num-
Oxygen 2 × 10 −5 0.1 0.8 4 × 10 −5
bers. Furthermore, the P´eclet number gives an instant idea
Glucose 9 × 10 −6 5 1.8 120 × 10 −5
as to which major transport mechanisms operate in the
Insulin 1.5 × 10 −6 3 × 10 −8 11 5 × 10 −12
system under study. If Pe < 1, diffusional transport dom-
inates, while if Pe > 1, convective transport and electrical a D i are typical values measured at 37 C (the body temperature).
◦
migration are more important. b C i are typical values found in biological fluids in vivo (for oxygen,
We now consider the typical case of a tissue engineered arterial blood levels were used).
construct implanted in vivo. Although the precise values
of the transport parameters are not necessarily known, it is for oxygen, glucose, and a peptide hormone, insulin, in the
often a useful exercise to perform an order of magnitude tissue construct (Table VI). For the small metabolites oxy-
analysistodeterminetheapproximatecontributionofeach gen and glucose, P´eclet numbers are close to 1, which in-
term in the transport equation. For this purpose, we start dicates that diffusive and convestive transports have equal
withEq.(12)andproposethatareasonableestimateforthe contributions. Insulin, by virtue of its molecular size, has
concentration gradient ∂c i /∂x is c i /X. The flux equation a lower diffusivity, thus its transport is more dependent
becomes: on convection. One can notice that although oxygen dif-
fusivity is at least one order of magnitude greater that that
∂c i Pe c i Pe
N i = φD i − + c i ≈ φD i + c i of glucose, the transport of oxygen is almost two orders
∂x X X X
of magnitude slower. This is due to the fact that oxygen
φD i c i has a very low solubility in water and physiological fluids.
= (1 + Pe) (15)
X Thus, although oxygen diffuses rapidly, it is not possible
Most nutrient transport to the implant initially comes from to create large gradients to provide the necessary driving
thesurroundingcapillaries.Thesecapillariescontinuously force for its transport; for this reason, oxygen transport is
leak plasma into the tissue space because the pressure in- almost always the main factor limiting the size and cell
side capillaries is greater than in the tissue. Typical mea- density of tissue engineered constructs.
sured values for the capillary filtration coefficient and To obtain an estimate of the maximum thickness of a tis-
3
pressure gradient are 0.035 cm /min/mm Hg/100 g tis- sue engineered implant based on transport considerations,
sue and 27 mm Hg, respectively. Assuming capillaries are we must balance metabolite delivery with consumption by
distributed evenly 100 µm (or 0.01 cm) apart, on aver- the cells in the construct. For this purpose, we use the mass
3
3
age each capillary occupies (0.01 cm) = 10 −6 cm . Since balance or continuity equation, which is generally written
3
100 g tissue occupy a volume of about 100 cm , we can es- as:
timate the flow rate and velocity of fluid exiting capillaries ∂c i ∂N i
=− + (G i − R i ) (16)
and flowing through the implant: ∂t ∂x
i
cm 3 where G i and R i represent the generation and consump-
Q = 0.035 × 27 mmHg
3
min · mmHg · 100 cm tissue tion rates of metabolite i in the construct, respectively.
Substituting the expression for the flux N i yields:
−6
3
3
1 10 cm −8 cm
2

× × = 1.2 × 10 ∂c i ∂ c i Pe ∂c i
60s capillary capillary · s = D i +
∂t ∂x 2 X ∂x
The average surface area perfused by each capillary is

0.01 cm×0.01 cm, thus the velocity is + (G i − R i ) (17)
i
3
−8
1.2 × 10 cm /s −4
U = = 1.6 × 10 cm/s Let us now consider the specific case of oxygen trans-
−4
(100 × 10 cm) 2
port. Furthermore, to simplify the calculations, we assume
We next assume that the implant is 0.1 cm thick and highly that convective transport is negligible so that Pe = 0. This
porous (as would be the case for a collagen gel, for ex- would be a “worst-case scenario” where normal tissue per-
ample), so that φ = 1, with a mean pore size of >10 µm, fusion is disrupted due to the surgical trauma caused by
which is much larger than the molecular size of trans- the implantation procedure itself, or the implant is encap-
ported molecules, so that W i = 1. Using these values, we sulated into a membrane (i.e., to protect implanted cells
calculate the P´eclet number and corresponding flux rate from the host’s immune system) which does not allow