Biomedical Engineering Reference
In-Depth Information
Therefore, the volumetric flow rate can be approximated as
5 πΔ
PR 4
Q
ð
5
:
6
Þ
8
μ
L
where
P / L is the pressure gradient across a tube of length L and radius R . The fluid
within the tube must have a constant viscosity of
Δ
and a uniform cross-sectional area over
the entire length, L . As we have seen, the Hagen-Poiseuille solution is derived from integrat-
ing the velocity of the fluid (solved from the Navier-Stokes equations) with respect to blood
vessel area. In other words, this is a weighted average for each section of fluid that has the
same velocity. The lamina of fluid against the wall would be characterized by the circumfer-
ence of the tube (or total arc length of an irregular shaped container) with a velocity of zero.
The lamina in the center of the tube would have a differential circular area with the maxi-
mum velocity of the fluid. Using the generalized integral relationship, one can derive a for-
mula for volumetric flow rate for any shaped container under pressure driven flow.
What is interesting to note about Equation 5.6 is that small changes in the vessel radius
cause very large changes in the flow rate. A doubling of the blood vessel radius would be
associated with a 16-fold increase in the volumetric flow rate throughout the tube, assum-
ing everything else remains the same. This is not a trivial discussion, because radius dou-
bling is quite possible when the blood vessel goes from a maximally constricted state to a
maximally dilated state. It is common for the resistance blood vessels (arterioles) to experi-
ence changes in vessel diameter on the order of 4-fold, which theoretically would increase
the volumetric flow rate 256 times. Also what one can infer from this is that the blood ves-
sel radius does not have to change significantly to increase (or decrease) flow to a region
that has non-normal (low or high) tissue oxygenation. This is important for energy usage
(or consumption) for the smooth muscle cells. If flow rate did not change rapidly with
small changes in vessel diameter, then blood vessels would need to dilate or constrict sig-
nificantly to counterbalance the tissue oxygenation level changes. For instance, to accom-
modate a large oxygen depletion, the vessel radius would need to increase significantly.
This would have the effect of depleting the smooth muscle cells of energy, and they would
be unable to respond to changes in the tissue needs in the near future. Instead, the biologi-
cal system is designed so that minimal energy is required for vast blood flow changes.
μ
Example
Calculate the volumetric flow rate within an arteriole with a length of 100
μ
m and a radius of
35
m. The pressure difference across the arteriole is 10 mmHg. Also calculate the change in
diameter needed to reduce the volumetric flow rate by 5% and to increase the volumetric flow
rate by 10%.
μ
Solution
Using the Hagen-Poiseuille's formulation to calculate the volumetric flow rate, we get
4
PR 4
5 πΔ
5 πð
10 mmHg
Þð
35
μ
m
Þ
Q
0
:
135 mL
=
min
5
8
μ
L
8
ð
3
:
5cP
Þð
100
μ
m
Þ
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