Microvascular Beds - Biofluidmechanics: An Introduction to Fluid Mechanics Macrocirculation and Microcirculation - page 192

which is developed from the continuity equation assuming that the fluid density within
each tube is the same. Again, this equation will hold for any experiment regardless of tube
cross-sectional area. To determine how the hematocrit is related to the tube velocities, mul-
tiple groups have conducted experiments where a known hematocrit is fed through a tube
of known diameter which branches into two daughter tubes with known diameters. The
fluid that is discharged from each of the daughter branches is collected and the hematocrit
in each collection tube is measured. The velocity of red blood cells, as well as the average
tube velocity can be quantified with optical means in these experiments (if the tubes are
transparent). A relationship has been developed for the quantity of red blood cells within
a particular cross section at an instant in time (tube hematocrit). The quantity of red blood
cells was found to be equal to the mean tube speed multiplied by the cross-sectional area
multiplied by the discharge hematocrit. This was also found to be equal to the mean red
blood cell speed multiplied by the cross-sectional area multiplied by the tube hematocrit.
If the cross-sectional areas are the same, then the discharge hematocrit is equal to

Tube Hematocrit Mean Red Blood Cell Speed
Mean Tube Flow Speed

Discharge Hematocrit ð 6 : 10 Þ

5

Different cross-sectional areas can be incorporated into Equation 6.10 as well.
Substituting Equation 6.10 into Equation 6.9

H inflow v inflow 5 H branch 1 v branch 1 1 H branch 2 v branch 2 ð 6 : 11 Þ

Again, assuming that the cross-sectional areas are the same, then Equation 6.9 can be sim-
plified to a relationship between the mean velocities within each tube. Using this to sim-
plify Equation 6.11 , we get

H branch 1 v branch 1 1 H branch 2 v branch 2

H inflow 5 ð 6 : 12 Þ

v branch 1 v branch 2

1

As a word of caution, this formulation is highly dependent on the relationship between
the cell diameter and the tube diameter. When the ratio of diameters is close to 1, then
Equations 6.10 through 6.12 are valid. As the ratio decreases, as occurs within the macro-
circulation, these equations are no longer valid because the cell velocity has little to no
effect on the tube velocity.
The hematocrit in capillaries is lower thank in the arterial hematocrit as shown in the
mathematical formulations above. This is because the red cells are fairly restricted to the
centerline (higher velocity flow) in the capillaries and thus within a particular time frame,
there will be a lower chance of “catching” a red blood cell as compared to “catching” the
slower plasma along the blood vessel wall (using still photography, for instance). With an
inflow hematocrit in the range of 40% to 50%, it has been shown that the capillary hemato-
crit can be as low as 10% but averages to around 20%. This reduction in hematocrit begins
to be seen in arterioles with a diameter in the range of 100 μ m (where the cell diameter/

tube diameter 0.1). This decreases steadily to the capillaries, where the hematocrit is

around 10% to 20%. The hematocrit in the venules rapidly increases to the inflow hemato-
crit level within vessels that have a diameter in the range of 50

D

μ m (hematocrit is typically