Biomedical Engineering Reference
In-Depth Information
If we assume that the area under the Gaussian tail from
−∞
to0is
negligible, we have approximately:
∞
A
t
=
A
at
+
A
bt
≈
f
(
s
)
ds
(8.106)
−∞
Therefore, we obtain the following formula for the reflection coecient,
σ,
of a given solute:
erf
s
x
−
+
erf
s
σ
=
1
2
s
h
√
2
h
√
2
(8.107)
Although, equation (8.107) expresses the dependence of
σ
on the solute dim
en
-
sions,
s
x
, it also involves other unknown distribution parameters (
h
and
s
).
We can find the latter using experimental data for the membrane (the reflec-
tion coecients and molecular dimensions of various solutes). We plot the
reflection coecient,
σ,
as a function of
s
x
. Next we find a function of the
form (8.107) that fits these experimental data (the plot
σ = f(s
x
))
.
Data for
human erythrocytes collected from available literature is given in Table 8.4,
and the fitting was done using a simulated annealing algorithm to minimize
the
χ
2
error. In our computations we had an additional constra
i
nt that the
majority of the pores ought to be permeab
le
to water, that is,
s
s
w
>
3
h
.
Using data shown in Table 8.4 we obtained
s
= 0.137 [nm
2
] and
h
= 0.024.
When transport is described using the KK equation where the volume
flux is defined as the rate of volume flow, ∆
V/
∆
t,
per unit surface area of the
membrane, then assuming ∆Π = 0, we get
−
∆
V
A
tm
∆
t
J
v
=
L
p
∆
P
=
(8.108)
where
A
tm
is the membrane surface area. For porous membranes, an alterna-
tive definition of the volume flux can be considered: the volume flow per unit
TABLE 8.4
Transport Data for Human Erythrocytes and Selected Solutes
Solute
Water Formamide Urea
Acetamide Propionamide
Molecular
radius
r
[nm]
∗
0.15
0.207
0.211
0.23
0.231
Molecular
cross-section
area
s
x
[nm
2
]
∗
0.07
0.134
0.134
0.166
0.168
L
p
[m
3
/(Ns)]
†
10
−
12
10
−
12
1
10
−
12
1
10
−
12
1
.
27
×
1
.
27
×
.
27
×
.
27
×
σ
†
0.58
0.53
0.80
0.55
Source:
∗
Levitt 1974, Goldstein and Solomon 1961;
†
Sha'afi and Gary-Bobo 1973.
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