Biomedical Engineering Reference
In-Depth Information
radial direction). The permeability of oxygen through plasma is typically modeled through
the use of a mass transfer coefficient, which is related to the Sherwood number (Sh), the dif-
fusion coefficient, and a characteristic length. The Sherwood number is a dimensionless
parameter that is the ratio of the convective mass transport to the diffusive mass transport:
Kx
D
Sh
ð
7
:
5
Þ
5
In Equation 7.5 , K is the mass transfer coefficient, x is the characteristic length, and D is
the diffusion coefficient. In plasma, the diffusion coefficient for oxygen is on the order of
2
10 2 5 cm 2 /s. The plasma layer (characteristic length) ranges from 0.35
m,
depending on the red blood cell orientation and capillary size. The mass transfer coeffi-
cient is based on many physiological parameters, such as blood viscosity, blood speed,
and dissolved species within the blood, and can only be calculated from the other para-
meters and not measured. The Sherwood number has been found to be dependent on the
blood hematocrit and can be represented mathematically as
3
μ
m to 1.4
μ
Sh
Sh 0 : 25
0
:
84
ð
Hct
0
:
25
Þ
ð
7
:
6
Þ
5
1
2
In Equation 7.6 , Hct is the hematocrit which is represented as the decimal equivalent of the
percent hematocrit. Sh 0.25 is the Sherwood number at a hematocrit of 0.25 (25%), which has
been experimentally found to be equal to 1.3. Knowing the hematocrit and the characteris-
tic plasma thickness, one can easily calculate the mass transfer coefficient that describes the
particular flow conditions and then relate this to the other diffusion/permeability steps.
This event must also be considered during normal red blood cell oxygenation.
The fourth diffusion event during red blood cell oxygenation is the diffusion of oxygen
across the red blood cell wall. The thickness of the red blood cell membrane is on the
order of 10 nm, which is similar for most human cells. The diffusion coefficient for oxygen
across the red blood cell membrane has been found to be similar to the diffusion coeffi-
cient for oxygen within plasma. Using Equation 7.4 , it can be seen that the permeability of
the red blood cell wall will be approximately 100 times greater than the permeability of
oxygen across the respiratory boundary (due to thickness differences). This assumes that
the partition coefficient is close to 1 for both cases (this is actually a good assumption).
Therefore, this diffusion occurs significantly faster than the second diffusion event during
blood oxygenation and does not need to be considered in the overall red blood cell oxy-
genation model.
The fifth diffusion event that occurs for the oxygenation of blood is the diffusion of oxy-
gen within the red blood cell until the oxygen comes into contact with and reacts with
hemoglobin. Similar to the convection and diffusion that oxygen experiences while mov-
ing within the plasma, once in the red blood cell, oxygen molecules can either react imme-
diately with nearby hemoglobin or diffuse through the red blood cell to interact with a
different hemoglobin molecule. The Thiele modulus is a dimensionless parameter that
relates the kinetic reaction (in this case, oxygen-hemoglobin binding) to the molecular dif-
fusion through the particular medium. The Thiele modulus is defined as
r
kx 2
D
φ 5
ð
7
:
7
Þ
Search WWH ::




Custom Search