Biomedical Engineering Reference
In-Depth Information
L
is impermeable. Substituting these boundary
conditions into (10.54),
A
and
B
can be determined.
as the boundary condition at
x
=
10.4.2 Case 2: The Krogh Tissue Cylinder
Consider the case of drug eluting stent, where drug is radially diffusing in a tube in
one-dimension (Figure 10.11). Since the drug has to be dispersed along the thick-
ness of the blood vessel, one has to determine the concentration profile of the drug
across the thickness of the artery. Consider the length of the stent to be
L
and radial
thickness
r
at
r
radius in the flow system. Performing a material balance on the
differential element from
r
to
r
Δ
+
Δ
r
yields
∂
Ci
(
)
(
)
(10.56)
2
π
rL r
Δ=
N r
−
N
r
+Δ
r
2
π
L
+
R
2
π
rL r
Δ
ir
ir
+Δ
r
i
∂
t
Dividing both sides of the equation by the volume element 2
π
Lr
Δ
r
, taking the
limit as
Δ
r
→
0, and using the definition of the derivative results in
∂
CrN
R
t
1
∂
i
=
ir
+
i
∂
r
∂
r
Assuming that the solution is dilute and using Fick's first law, flux can be rep-
resented as
C
ND
t
∂
=
i
ir
AB
∂
Substituting the above expression and assuming
D
AB
to be constant with re-
spect to radius
∂
C
1
∂
r C
∂
∂
C
D
∂
∂
C
⎛
⎞
⎛
⎞
i
=
D
i
+
R
or
i
=
AB
r
i
+
R
(10.57)
⎜
⎟
⎜
⎟
AB
i
i
⎝
⎠
⎝
⎠
∂
t
r
∂
r
∂
t
∂
t
r
∂
r
∂
t
Δ
r
R
r
L
Figure 10.11
One-dimensional radial diffusion.
