The Effect of Blood Flow Rate
on Venous Collapse
During clinical processes such as blood donation and blood
pressure measurement, the pressure outside of a vessel may
become greater than the vessel’s internal pressure, causing
the vessel to collapse. In this experiment, the relationship
between these pressures and system parameters is analyzed to
determine the point at which collapse will occur. To simulate
collapse, a collapsible tubing model is created for experimentation, in which the diameter and type of material used as
tubing were varied. A dimensionless parameter for these
variables is created and verified in relation to Poiseuille’s
Law. According to the data and the dimensionless parameter,
higher flow rates require a greater pressure difference to
cause collapse and therefore the higher the flow rate, the
higher the pressure necessary for the tubing to eventually
collapse.
Betsy Hamme
Robert Jenkins
Rob Ledger
Jess LeMay
Group M4
Dept. Of Bioengineering
University of Pennsylvania
Philadelphia, PA
Introduction
Flow
has
collapse, or the cross sectional area may
medical
become very small. At this point, the lumen
Virtually all fluid-carrying
is reduced to two narrow channels separated
vessels in the body have flexible walls and
by a flat region of contact between opposite
can collapse when the transmural pressure,
walls. This twin-lobed shape is the lowest-
(pressure across the vessel wall) ptm, falls
energy
below a critical value. At large positive
flexible
values of ptm where the internal pressure is
constraints1.
numerous
applications.
in
collapsible
physiological
tubes
and
configuration
wall
except
for
a
close
uniformly
to
end
greater than the external pressure, collap-
For the case of steady-state flow, the
sible tubes are distended and stiff and
upstream-downstream pressure difference
behave as non-collapsible tubes.
(p) is related to flow rate and vessel
At
parameters through Poiseuille’s Law:
negative values the tubing may completely
1
Hamme, Jenkins, Ledger, LeMay
P 
8LQ
a 4
fluid, the compliance of the tubing, the flow
Equation 1
rate,
transmural
pressure
and
internal
pressure differences5. As of now, there are
where  is dynamic viscosity, L is the tubing
no simple or conclusive mathematical
length, Q is the flow rate, and a is the radius
equations modeling pulsatile flow.
Reynold’s number is a dimensionless
of the tubing. Large values of p tend to
variable that is defined as:
correspond with large positive values of ptm.
Oscillations occur in collapsible
Re 
U l

tubes when the external pressure exceeds the
Equation 2
internal pressure, which causes the vessel to
in which U is the mean fluid velocity, l is a
collapse. This difference in pressure occurs
length
when the flow rate exceeds a “critical” flow
diameter), and  is the kinematic viscosity
rate, at which point p becomes great
of the fluid. Collapse can occur in small
enough that the internal pressure is at the
vessels with low Reynolds number (Re) or
minimum needed to balance the given
in larger vessels with high Reynolds
external pressure and still maintain steady
numbers.
flow.
The tube begins collapse from the
accompanied by self-excited, flow-induced
downstream end, where pressure it at its
oscillations. There are several examples of
lowest. Collapse restricts flow, and pressure
physiological vessels that are prone to
builds up at the upstream end. The pressure
vascular collapse. Gravitational decrease in
in the tube eventually becomes great enough
internal pressure with height causes collapse
to overcome the effects of the external
in systemic veins above the heart. Systemic
pressure and flow is forced through the tube.
arteries can become compressed by a
With the increase in flow rate, the internal
pressure cuff, or within the chest during
pressure decreases and the process starts
cardiopulmonary resuscitation. The urethra
over.
and the ureter are both compressible tubes,
Higher external pressures require
parameter
(likely
the
tube’s
In some cases collapse is
lower flow rates before oscillations begin,
which
and likewise higher flow rates for given
micturition
external pressures cause greater oscillations.
respectively.
The rate of oscillation increases until the
biological occurrences of collapsible tubes,
vessel completely collapses. The period of
including intramyocardial coronary blood
oscillation depends on the viscosity of the
vessels during systole, pulmonary blood
2
may
be
or
compressed
peristaltic
There
are
during
pumping,
many other
Hamme, Jenkins, Ledger, LeMay
vessels in the upper levels of the lung, and
the systolic pressure over the diastolic
large intrathoracic airways during forced
pressure measured in mmHg4.
expiration or coughing2.
The measure of systolic and diastolic
Materials/Apparatus
blood pressure relies on the fact that arteries
1. Tygon tubing
are collapsible tubes and that Korotkoff
2. Penrose tubing, 16 mm internal diameter
sounds are created during oscillation. Study
3. Latex tubing, 2mm internal diameter
of these sounds has determined that they are
4. Tank with supports for holding water
generated by self-excited tube oscillations
5. Pressure transducer
and are specifically due to the increasing
6. Graduated cylinder to collect water
steepness of the wave front during the
7. Stop watch
propagation of a pressure wave3. When a
8. Flow cutoff valve
blood pressure cuff on the arm is highly
9. LabView (flow.vi and pressure.vi)
pressurized, the external pressure around the
10. Pressurized Rubbermaid® chamber
artery causes the artery to collapse. Nothing
11. Pipettes to act as manometers
is heard at this time since there is no flow.
12. Sphygmomanometer
The cuff pressure is slowly released, and
reaches a specific pressure at which blood
Procedure
can be forced through the artery during
systole. During diastole, the artery is forced
The experimental setup is shown in Figure
closed again, causing a Korotkoff sound.
1. Water was used as the fluid medium. A
The first cuff pressure at which these sounds
thin-walled undistended tube was mounted
are heard is called the systolic pressure,
between
which
pressurizable chamber (Figure 2).
is
measured
using
a
rigid
couplings
We
significantly decreased, it becomes low
Rubbermaid® container measuring (7.5in x
enough that there are no fluctuations in the
4.5in x 2.25in). The inlet end view can be
artery wall and blood flow becomes silent
seen in Figure 3.
and laminar, as it is normally. The lowest
identical except that there is no inlet for air.
pressure. Blood pressure readings consist of
3
using
a
constructed
sounds are heard is called the diastolic
chamber
in
sphygmomanometer. As the cuff pressure is
cuff pressure at which these Korotkoff
the
brass
a
The outlet end view is
Hamme, Jenkins, Ledger, LeMay
Figure 1: Experimental Set-up
Figure 2: Pressurizable chamber
Figure 3: End view of chamber
4
Hamme, Jenkins, Ledger, LeMay
Thus, the chamber could be pressurized.
experiment, an initial flow rate was set and
The pressure within the chamber provided
external
external pressure on the tubing (pe) and was
atmospheric
controlled by an ideally airtight system
collapsed and flow ceased; measurements of
consisting of a sphygmomanometer pump
p and Q were recorded for each new pe.
and
pressure
This was repeated for several initial flow
The transducer and Labview
rates. The maximum initial flow rate for
tubing
transducer.
connecting to
a
were used to measure pe. The tubing in the
pressure
was
pressure
increased
until
the
from
tubing
any trial was 40 ml/sec.
chamber consisted of both 2mm (internal
diameter) latex tubing and 16mm Penrose
Results and Discussion
tubing with lengths of 14cm and 9cm,
respectively.
An
upstream
At low external pressures (.002 - .01
reservoir
atm gage), the flow through the latex rubber
provided the mean upstream pressure while
collapsible tubing was found to obey
an upstream valve controlled flow rate (Q)
within the tubing.
Poiseuille’s Law. Four flow studies fell into
Upstream (p1) and
this range, with outside gage pressures at
downstream (p2) pressures within the tubing
.0026, .0045, .0069, and .011 atmospheres.
were measured using manometers mounted
The internal pressure drop for these flow
at the inlet and outlet ends of the chamber.
situations varied linearly with the flow rate,
Two different experiments were
with R-square values ranging from .96 at
performed for both types of tubing. In the
.011 atm to .998 were plotted against the
first, external pressure was held constant at a
theoretical Poiseuille flow, given the same
pressure and the flow rate was varied. The
viscosity,
internal change in pressure, p = p1 - p2, was
tube
at
.0069
atm.
The
pressure/flow relationships length, and tube
measured for each flow rate. The flow rates
diameter (see Figure 4).
were increased until large oscillations of the
From this data, it was determined that the
tubing occurred and flow became pulsatile.
collapsible nature of the tubing does not
A new pe was selected and the experiment
affect flow until the outside-inside pressure
was repeated. The external pressures ranged
difference is great enough to cause it to
between 262 Pa and 1,114 Pa. In the second
5
Hamme, Jenkins, Ledger, LeMay
and 1,113 Pa latex tubing trials.
The
Volume Flow (m^3/s)
0.00006
critical point was defined as the internal
Poiseuille Flow
0.00005
0.00004
264 Pa
0.00003
460 Pa
0.00002
696 Pa
0.00001
1113 Pa
pressure difference that corresponded to the
first oscillations in flow.
Of course, a
minor amount of error was induced because
the flow itself oscillated with the diameter
of the tube, but this was negligible by
0
0
500
1000
1500
2000
design (the critical point was chosen at the
2500
very smallest amount of oscillation).
Pressure Drop (Pa)
Figure 5 shows the “flutter points” in
Fig. 4: Internal pressure/volume flow relationships
for low outside pressures
Essentially, at these pressure
0.00006
differences (both transmural and internal),
0.00005
Volume Flow (m^3/s)
flutter.
relationship to the rest of the flow trials.
flow functions as it does in rigid tubes.
However, as the internal pressure
difference increases (i.e. as the flow velocity
increases),
the
transmural
pressure
difference increases to a critical point. That
Poiseuille Flow
0.00004
460 Pa
0.00003
696 Pa
0.00002
0.00001
1113 Pa
0
0
is, as the pressure downstream continues to
500
1000
1500
2000
2500
Pressure Drop (Pa)
drop, the pressure across the membrane
Fig. 5: Internal Pressure/flow relationship including
point of deviation from Poiseuille flow
increases (because the external pressure is
held constant).
At the critical point, the
The critical point of oscillation is clearly
outside pressure overcomes the elastic force
shown as the last point on each of the data
of the tube, and the tubing begins to neck.
series. The sharp decrease in volume flow
Water builds up behind the neck, which re-
makes sense, since less fluid is allowed
opens the tube. This process causes high-
through the tube due to the fluctuations in
frequency oscillations of the diameter of the
diameter.
tube, resulting in the commonly known
“bronx-cheer effect.”
As the transmural pressure
increases, the degree of the effect of the
At the critical
oscillations also increases (i.e. larger effect
pressures, the flow through a collapsible
on volume flow). This effect results from
tube no longer obeys Poiseuille’s law. This
the higher transmural pressure collapsing the
phenomenon was observed in the 460, 696,
tube to a greater degree. Note also that as
6
Hamme, Jenkins, Ledger, LeMay
the transmural pressure increases, the tube
relationships were linear, the actual flow
collapses at a smaller internal pressure
rate was three orders of magnitude lower
difference.
Collapse occurs because the
than the expected flow rate (10-5 vs. 10-2
increased transmural pressure allows the
m3/s). The flow rates and internal pressure
rubber to collapse at greater internal
differences were within the same order of
pressures.
magnitude as those in the latex trials. That
Unfortunately,
the
latex-rubber
is, the radius of the tubing increased by eight
tubing could not be collapsed. This failure
times, but the flow through the tube did not.
was primarily due to the high elastic
Because the pressure drop remained the
modulus and small diameter of the tubing, in
same, and the radius increased, Poiseuille’s
conjunction with the limitations of the
law predicts a greatly increased flow rate for
pressure chamber. The latex tubing required
the Penrose trials, ranging from .021 to .274
too large a transmural pressure difference
m3/s.
for the pressure chamber to handle – while
As shown in Figure 6, these
predictions were false.
0.00005
Volume Flow (m^3/s)
the critical transmural pressure difference
would only have been around .5 atm, the
pressure upon the lid of the Rubbermaid
container was too great to hold such
pressures.
In an attempt to overcome these
460 Pa
0.00004
0.00003
745 Pa
0.00002
965 Pa
0.00001
0
0
difficulties, a series of experiments was
500
1000
1500
2000
Pressure Drop (Pa)
completed using Penrose rubber tubing,
experiments
Fig. 6: Pressure/flow relationship for Penrose
tubing, showing a 10^3 drop in flow rate from
Poiseuille
previously used on the latex rubber (internal
The flow through the Penrose tubing did not
pressure vs. volume flow at constant
act in accordance with Poiseuille’s law
transmural pressure) were carried out using
because of errors in the experimental set-up.
the Penrose tubing. However, the data for
Primarily, the Penrose tubing was wider
the Penrose rubber trials did not comply
than the upstream and the downstream
with
same
Tygon tubing. In fact, the Penrose tubing
transmural pressure differences as with the
was around 3 times as wide as the Tygon
latex). While the internal pressure vs. flow
tubing (Penrose ~ 8mm and Tygon ~
which had a greater diameter and a smaller
elastic
modulus.
Poiseuille’s
The
Law
(at
the
7
Hamme, Jenkins, Ledger, LeMay
2.5mm).
Furthermore, the fluid flowed
through several centimeters of Tygon tubing
0.00005
downstream manometer.
Volume Flow (m^3/s)
after the upstream manometer and before the
This change in
radius created most of the errors in
predicting flow through the Penrose tube.
For example, while the radius of the Penrose
was wide enough to allow a 10-2 m3/s
1/3 Highest Vel.
0.00004
0.00003
2/3 Highest Vel.
0.00002
Highest Vel.
0.00001
0
volume flow for a 1000 Pa internal pressure
0
5000
10000 15000 20000 25000 30000
Atmospheric Pressure (Pa)
difference, the radius of the Tygon forced
the volume flow down to about 10-5 m3/s.
Fig. 7: Figure 7 shows the relationship between
Furthermore, because the Penrose was so
volume flow and atmospheric pressure for Penrose
much weaker than the latex tubing, the
diameter began to oscillate at very low
In Figure 7, the needle valve at 1/3, 2/3, and
transmural and internal pressure differences.
completely open positions corresponded to
In fact, it was nearly impossible to
increasing volume flow at near atmospheric
distinguish
pressure. The flow curves then converge as
between
the
steady
and
the atmospheric pressure is increased – this
oscillating flow in the Penrose tubing trials.
trials
corresponds to the beginning and increase of
conducted on the Penrose tubing involved
oscillations of the tube diameter. Finally,
opening the needle valve to a given
the atmospheric pressure is increased until
diameter, and increasing the atmospheric
flow is completely stopped. The pressure
pressure until the tube completely collapsed.
needed to completely collapse the tube
Essentially, the pressure behind the tubing
increased directly with the size of the valve
was kept at a constant value (by keeping the
aperture. This result is fairly easy to explain
needle valve set and the water level in the
–the height of the water in the tank was kept
tank constant) and the atmospheric pressure
constant, and the size of the valve aperture
was increased from zero gage until the flow
directly
through the tube completely stopped. Three
upstream of the tube. An opening of the
trials were completed, with the needle valve
aperture from 1/3 to 2/3 of it’s maximum
at 1/3, 2/3, and completely open. The data is
value creates a greater pressure behind the
The
final
experimental
presented in Figure 7.
8
corresponded
to
the
pressure
Hamme, Jenkins, Ledger, LeMay
tube, which in turn increases the necessary
behavior of flow at greater transmural
transmural pressure for collapse.
pressures would be beneficial. For both the
It is useful to relate all of the system
latex and Penrose, it would be very
variables to each other and enable the
interesting to investigate the flow behavior
system to be a prototype for a biological
at atmospheric pressures closer to the point
system.
two
of collapse. One of the major problems with
created
this study, however, is that once the
which together contain all of the necessary
diameter begins to oscillate, it is nearly
variables. The variables were created using
impossible to determine the internal pressure
the Buckingham Pi method.
difference or flow rate, because pressure and
In
order
dimensionless
1 
to
parameters
L
Q
2 
do
this,
were
Q 2   Pext
D 3 (P) 2
flow rate oscillate as well.
For the latex rubber trials, the
A relationship between these two parameters
experimenters would like to have applied
can be derived by plotting one against the
pressures great enough to observe complete
other. The determined relationship is shown
collapse. The total pressures applied were
in Figure 8.
actually fairly small, but the surface area of
the lid of the pressure chamber was too large
to remain sealed at higher pressures. With a
revised
experimental
set-up,
greater
pressures could be applied, and collapse
could be observed.
For the Penrose tubing trials, the
experimenters would like to use tubing with
a smaller diameter. If tubes were used that
had diameters comparable in size to the
Tygon tubing, the deviations from Poisuille
Fig. 8: This graph shows the relationship between the
two undimensional variables.
flow would disappear. Then more thorough,
meaningful
Further Investigation
This
experiment
studies
of
pressure/flow
relationships could be completed.
would
benefit
greatly from a minor amount of further
investigation. Primarily, more study of the
9
Hamme, Jenkins, Ledger, LeMay
References
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Downing, J.Micah, “One-Dimensional
Steady Inviscid Flow through a
Stenotic Collapsible Tube.” Journal of
Biomechanical Engineering. 112: 444449, November 1990.
9. Jensen, O.E. “Chaotic Oscillations in a
Simple Collapsible-Tube Model.”
Journal of Biomechanical
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