Biomedical Engineering Reference
In-Depth Information
is possible to describe the two hindering factors discussed above theoretically. Ferry [10]
addressed the steric hindrance issue by only allowing molecules without striking the edge
to enter the pore. Assuming the pore is cylindrical with inner diameter
r
, and the solvent
molecule is so small that the system could be treated as a continuum, solutes of radius
a
could pass through the pore only if
r
-
a
> 0. Hence, the effective area of opening
A
eff
(the
area inside the dashed line) is given by
2
a
r
A
eff
=
A
1
−
(5.14)
0
where
A
0
is the total cross-sectional area of the pore. With a given geometry of the pore,
the effective opening is very sensitive to the shape of the solute. Although we limit our dis-
shape of the solute. Although we limit our dis-
cussion on spherical solutes, even proteins classifi ed as globular proteins are not necessar-
the shape of the solute. Although we limit our dis-
, even proteins classiied as globular proteins are not necessar-
ily sphere, bovine serum albumin (BSA) is an ellipsoid of axial ratio 3.4 [11]. Fifty percent
reduction of the effective diffusion coefficient of BSA through a noncharged membrane
was reported [12] as compared with the value predicted by taking BSA as a sphere; there-
fore, caution has to be taken when a hydrodynamic radius is used to predict the effective
diffusion coefficient in a release medium.
The second factor, friction of the solute moving inside the pore, was correlated to the
effective surface area by [13]
s, even proteins classifi ed as globular proteins are not necessar-
3
5
A
A
a
r
a
r
a
r
eff
0
= −
1 2 104
.
+
2 09
.
−
0 95
.
(5.15)
The total effect due to steric hindrance and frictional resistance is given by the Renken
equation [14]
2
3
5
A
A
a
r
a
r
a
r
a
r
eff
0
=
1
−
1 2 104
−
.
+
2 09
.
−
0 95
.
(5.16)
After we obtained the total effective area of pore opening, we need to relate it to the total
surface area (
A
) to formulate the relationship between the effective diffusion coefficient
(
D
eff
) and the diffusion coefficient in a pure solvent (
D
). To do so, the volume fraction of the
void space, also known as porosity (
ε
) is introduced, such that
A
0
-
A
ε
(5.17)
Another factor that needs to be considered is the distance traveled by the solute. In pore
models, solute molecules are only transported through pores. However, the pore length
and the distance that the solute travels are not necessarily equal to each other. To take this
factor into account, tortuosity, which is defined as the ratio of the path length traveled by
the solute
d
to the end distance
L
is introduced (Figure 5.6).
Since only the total surface area, the porosity, and the membrane thickness can be
measured, the diffusion coefficient calculated from the Fick's first law is only an effective
value (
D
eff
)
