Biomedical Engineering Reference
In-Depth Information
10 red cells, whereas the second plug may consist of 20 red cells. Within that one plug
however, the density, viscosity, velocity, and all other parameters do not change. The fluid
plugs can have different volumes with different compositions, but again, it is uniform
across the volume (i.e., perfectly mixed). To simplify the problem, it is easiest to also
assume that the flow is steady and that there is no wave propagation throughout the fluid.
As discussed before, this is a relatively good assumption within the microcirculation.
The first governing equation for this type of problem is the Conservation of Mass
throughout the individual plugs. For red blood cells, this is simply the number of red
blood cells that enter the region of interest must exit the region of interest. Because under
normal conditions, red blood cells cannot pass through the endothelial cell wall and are
not produced within the capillary; this is the exact solution (although this assumes that a
red blood cell cannot leave one plug and join another). For the plasma portion of the fluid,
as well as any compound dissolved within the plasma, the Conservation of Mass in its full
form will be (we will use the example of oxygen)
ð
O 2
Þ accumlated 5 ð
O 2
Þ entering 2 ð
O 2
Þ exiting 1 ð
O 2
Þ generated 2 ð
O 2
Þ consumed
ð
6
:
14
Þ
Clearly, the amount of oxygen generated and consumed would be dependent on the
kinetics of particular molecular reactions within the plasma components. Summing all of
these components together, this can be formulated as
r 2 u
r 2 R O 2 Zdx
Þ
where r is the radius of the blood vessel, C is the concentration of a particular species at
location x (the entrance of the vessel, for our example we are referring to oxygen) or x
π
ð
C O 2 ð
x
Þ 2
C O 2 ð
x
dx
ÞÞ 1 π
0
ð
6
:
15
1
5
1
dx (the exit of the vessel), u is the fluid velocity, R is the kinetic reaction rate of the specific
species (this would be positive if generating species, or negative if consuming species;
again, we are referring to oxygen), and Z is a stoichiometric coefficient. Simplifying
Equation 6.15 into a differential form we obtain
u dC O 2
dx
5
ZR O 2
ð
6
:
16
Þ
In typical kinetic reactions, the kinetic reaction rate is dependent on the concentration of
the particular species present. As an example, if the kinetic reaction rate can be modeled
as a first-order kinetic reaction, then
R O 2 5
kC O 2
dC O 2
dx
kZC O 2
u
5
dC O 2
C O 2 5
kZ
u dx
kZx
u
ln
ð
C O 2 Þ 5
C
1
C inlet e kZx
C O 2 ð
x
Þ 5
u
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