Biomedical Engineering Reference
In-Depth Information
where
k
is the kinetic association constant,
x
is a characteristic length, and
D
is the diffusion
coefficient. The oxygen-hemoglobin kinetic association constant within red blood cells is
equal to 128 s
2
1
and the diffusion coefficient for oxygen within red blood cells is on the
order of 6
10
2
6
cm
2
/s. If we take the characteristic length of the red blood cell to be
3
1.5
μ
m, which is the average thickness of the red blood cell, then the Thiele modulus is
s
128 s
2
1
2
ð
1
:
5
μ
m
Þ
φ
5
0
:
6928
5
6
E
10
2
6
cm
2
=
s
Combining the five diffusion events for the oxygenation of red blood cells, we can
see that only the diffusion of oxygen across the respiratory boundary, the convection/
diffusion of oxygen within plasma, and the association of oxygen with hemoglobin can
limit the time it takes for red blood cells to become oxygenated. The remaining two events
occur so rapidly that they will have little to no effect on the overall time required for red
blood cell oxygenation under normal conditions. The first two diffusion events that we
will consider are described by permeability coefficients through a particular medium. To
combine these coefficients, we will consider that the barriers to diffusion are in series and
that they act to resist the motion. Permeability coefficients are equivalent to conductances
in circuit theory, and therefore, the combined oxygen permeability is
1
P
barrier
1
plasma
5
1
P
barrier
1
1
K
plasma
ð
7
:
8
Þ
x
plasma
Sh
plasma
D
plasma
1
P
barrier
1
plasma
5
x
barrier
p
c
2
barrier
D
barrier
1
x
plasma
1
P
barrier
1
plasma
5
x
barrier
p
c
2
barrier
D
barrier
1
½
Sh
0
:
25
1
0
:
84
ð
Hct
2
0
:
25
ÞD
plasma
1
P
barrier
1
plasma
5
2
:
2
μ
m
0
:
875
μ
m
Þ
1
5cm
2
5cm
2
1
ð
2
:
3
E
2
=
s
½
1
:
3
1
0
:
84
ð
0
:
4
2
0
:
25
Þð
2
E
2
=
s
Þ
P
barrier
1
plasma
5
792
μ
m
=
s
assuming that the hematocrit is 40%. To add the resistance of oxygen diffusion through
the red blood cell into this combined permeability, a mass transfer Biot number and an
effectiveness factor is needed. The Biot number relates the mass transfer of the molecules
to the diffusion of the molecules with the following relationship:
Px
D
Bi
5
ð
7
:
9
Þ
For our case the Biot number will equal
Bi
5
ð
μ
=sÞð
:
μ
Þ
792
m
1
5
m
1
:
98
5
6
E
2
6cm
2
=s
for the mass transfer/diffusion of oxygen within the red blood cell (using the red blood
cell characteristics). To combine all of the resistances and diffusion along with the kinetic
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