Biomedical Engineering Reference
In-Depth Information
C
A
C
D
C
B
C
R
L
FIGURE 5.4
Illustration of diffusion process through membrane-based system (left) and coated microsphere (right).
In this situation, Fick's first law inside the membrane can be written as
ds
dt
D
dC
dx
= −
(5.1)
where
s
is the mass per unit area.
Due to mass conservation, the amount of drug transported from region A to region B
results in that the concentration in region A is reduced by
s
V
A
, whereas the concentration
A
s
V
A
B
in region B of the membrane is increased by
. Therefore, at any instance, the concen-
Therefore, at any instance, the concen-
V
V
tration difference in the membrane is
V
− − .
V
A
and
V
B
denote the volumes of
regions A and B, respectively, and
A
denotes the effective area of the membrane. Therefore,
the equation becomes
B
A
C
C
A
B
B
C
V
V
C
−
B
A
−
C
A
B
B
V
A
d
d
C
t
B
B
= −
D
(5.2)
L
Under an infinite source condition, there is a constant source outside the system (mem-
brane) to maintain
C
A
at a constant value at all times; and initial condition
C
B
(0) = 0,
Equation 5.2 becomes
C
V
V
ln
C
−
C
−
B
A
A
B
B
AD
V L
t
ln
C
−
=
−
A
(5.3)
V
V
V
V
B
B
B
1
+
1
+
A
A
Since A and B are interfacial regions, it can be inferred that
V
A
=
V
B
as long as surface area
of the membrane at both sides are equal; therefore, the above equation can be simplified
to be
C
−
2
C
AD
V L
t
ln
A
B
= −
2
(5.4)
C
A
B
