Biomedical Engineering Reference
In-Depth Information
done using experimental transient
SC
diffusion flux data in combination with
the analytic solution to the one-dimensional transient flux through the
SC
barrier. The following equations depicting solute flux are developed by Kush-
ner et al. (2007a), which incorporates the porosity and tortuosity for single
tortuosity models:
Q
(
t
)=(
εK
b
C
2
τL
)
D
b
exp
∞
1)
n
n
2
D
b
n
2
π
2
t
(
τL
)
2
1
6
−
2
π
2
(
−
−
(
τL
)
2
−
(9.5)
n
=1
and for the two-tortuosity model:
Q
(
t
)=(
εK
b
C
2
τ
volume
L
)
D
b
τ
flux
τ
volume
L
2
−
(9.6)
exp
−
∞
1)
n
n
2
D
b
n
2
π
2
t
τ
flux
τ
volume
L
2
1
6
−
2
π
2
(
−
×
n
=1
The study finds that the two-tortuosity model most closely predicts
D
b
and
K
b
in a comparison between the various models listed in Table 9.1, and com-
pares well with experimental results when predicting
D
b
and
K
b
of hydropho-
bic permeants.
It should be noted that the models listed in Table 9.1 can produce varying
results. For example, using the following input values suggested for normal
SC
by Kushner et al. (2007a):
N
= 15,
h
=1
µ
m,
g
= 0.1
µ
m,
w
=3,
d
=40
10
−
4
(Michaels et al.
µ
m
,
the values of
ε
/
τ
of the studies are as follows: 2
.
60
×
10
−
4
10
−
4
1975), 2.93
×
(Cussler et al. 1988), 2.78
×
(Lange-Lieckfeldt and
10
−
4
(Johnson et al. 1997), and
ε
/
τ
volume
of the (Kushner
et al. 2007a) two-tortuosity model has a value of 3.37
Lee 1992), 3.92
×
10
−
4
.
×
9.5.2 Models Based on Lipid Microstructure: Free Volume
Diffusion
For very small hydrophobic solutes (
MW <
400 Da), it is believed that dif-
fusion transport takes place within the lipid bilayers where the fluctuations
of the bilayer lipids provide transport pathways through “free volume pock-
ets.” These fluctuations produce free pockets with radii estimated at 4 A that
exist for 1.6
µ
s, while the time for the molecule to “jump” through the free
volume is on the order of ns (Mitragotri 2003). The steady state free volume
permeability is defined as
1
τ
∗
D
b
K
b
L
P
=
(9.7)
The representation here of the tortuosity,
τ
∗
, includes contributions from
the diffusion path length within the intercellular spaces as well as contributions
from the area fraction occupied by the lipid bilayers (a value of
τ
∗
=3
.
6 cm)
is used in this study as referenced from Johnson et al. (1997).
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