Biomedical Engineering Reference
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radius, because this is the farthest point that each capillary can supply. Note that the oxy-
gen concentration is not zero at the Krogh radius. Therefore, the second boundary condi-
tion is
r
5
r
K
5
dC
dr
0
Solving
Equation 7.12
using the two boundary conditions provides a means to obtain the
integration constants, which are
r
K
R
2
D
C
1
52
and
r
2
r
2
C
R
4
D
Substituting and simplifying the derived integration constants into
Equation 7.12
develops
a mathematical relationship for the change in concentration of a molecule with radial dis-
tance. This relationship is normally represented as
K
R
2
D
lnðr
C
Þ
2
C
2
5
C
P
1
r
K
R
4
C
P
D
r
2
2
r
C
r
K
1
C
ð
r
Þ
C
P
5
r
C
r
1
2
ln
ð
7
:
13
Þ
1
Example
Calculate the oxygen concentration at a distance of 10
μ
m and 20
μ
m from a capillary with a
radius of 4
μ
m. Assume that the Krogh radius is 35
μ
m, the diffusion coefficient for oxygen is
10
2
5
cm
2
/s, the plasma oxygen concentration is 4
10
2
8
mol/cm
3
, and that the oxygen reac-
2
3
3
10
2
8
mol/cm
3
s (see
Figure 7.3
). Also plot the oxygen concentration as a function
of radial distance. How does the plot change if the Krogh radius reduces to 25
tion rate is 5
3
μ
m?
FIGURE 7.3
Oxygen concentration as a
function of distance and the Krogh radius.
4.5
4
3.5
Krogh radius 25
µ
m
3
2.5
2
1.5
1
0.5
0
Krogh radius 35
µ
m
0
5
10
15
20
25
30
35
Distance (
µ
m)
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