BE 310: Bioengineering Lab IV
Induced Oscillations
In a Collapsible Tube
May 12, 2000
Group M1
Grace Doe
Jeff Gross
Joe Smith
Abstract
The onset of Korotkov oscillations was studied in a continuous flow, collapsible tube system.
Both a pressure transducer and stethoscope sensor were employed to detect experimental signals.
With this setup, three distinct oscillations were observed: (a) low frequency high amplitude
signal, (b) high frequency small amplitude oscillation located within signal (a), and (c) small
amplitude high frequency oscillations. From the experimental data, a high correlation between
pressure and flow rate was observed. Specifically, the general trend line was determined to be y
= (1.210-9) x2 – (4.310-7) x + (4.510-5). Additionally, two dimensionless parameters,
(Pc*v*L) / (Pup*Q) and (Pcuff)/(Pup), were found to accurately modeled the experimental
results under all variable conditions.
Introduction
The objective of this experiment was to observe and describe the onset of induced oscillations in
a collapsible tube under steady state, continuous flow conditions. Korotkov oscillations,
commonly occurring during blood pressure measurement, can be best explained in this
situational context. Initially, a pressure cuff is placed on an arm and pumped to a certain
pressure. The pressure is then slowly released until oscillations occur in the walls of the human
arteries located in the arm. Blood flow in combination with an external pressure exerted on the
artery induces artery wall oscillations. The pressure range associated with these oscillations
corresponds to the blood pressure. These oscillations are called Korotkov oscillations and the
purpose of this experiment is to try to create a mathematical model for the onset of Korotkov
oscillations. The set-up used to simulate a human system is shown in figure 1 and uses several
important fluid dynamics principals that are present in the human circulatory system [2].
Application of the experimental study to the physiological system yields both similarities and
dissimilarities. Specifically, the change from rigid tubing to flexible tubing is not modeled in the
human system. The closest areas of comparison are located at the point in which the inferior
vena cava passes through and is attached to the diaphragm and the entry point into the thorax of
the jugular and brachial veins. As in the experiment, all oscillations are essentially derived from
outside or human intervention. However, unlike the physiological system, the tubular cross
sectional area, pusatile flow, and bending stiffness was not to be modeled in this experiment [1].
An application of Korotkov oscillations can be observed in “blood doping”. In this situation,
high altitude training, at which there is a lower partial pressure of oxygen, causes an increase in
red blood cells. Thus, this increase in whole blood density would affect the pressures at which
oscillations are approached.
Materials
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BioPac Pro 3.6
BioPac stethoscope
Graduated cylinder
Latex tube
Pressure cuff
Pressure transducer
Stop Watch
Sucrose
Water bottle
Water tank
Methods
Figure 1. The experimental set-up resembling a human arm under blood pressure measurement
The experiment setup is illustrated above in Figure 1. Note that clamps were used to keep the
latex tubing perfectly horizontal. Initially, the raised water tank was filled with 16 liters of water
to the desired height of 1.02 meters. Tubing coming from the water tank was attached to the
latex tubing of diameter 1.9 centimeters, which rested on a filled water bottle, simulating a
human arm. The pressure cuff held the tubing firmly against the water bottle. A pressure
transducer, placed in series with the pressure cuff, connected the pressure cuff to BioPac, which
recorded the pressure placed on the tubing during the trial. Immediately following the pressure
cuff, a stethoscope was placed against the tube, which recorded any vibrations of the tube during
trials. To record the pressure downstream of the tube, a pressure transducer was connected and
linked to BioPac for continuous recording. To control flow rate, a nozzle was placed at the tube
exit.
Trials were carried out at a steady upstream pressure, which was maintained by keeping the
water level constant in the tank. The pressure cuff was inflated until the desired pressure on the
latex tube was attained. For each experimental solution, the pressure cuff was initially set at 60
mmHg and subsequently increased in increments of 20 mmHg up to 200 mmHg. Two trials at
each unique pressure increment were conducted. Note that the range of pressures on the tube was
not chosen arbitrarily, 60 mmHg and 200 mmHg were the extremes that oscillation onset was
still possible. To induce the tube oscillations at a given pressure, the flow rate was varied until
the downstream pressure transducer and the stethoscope measured the desired oscillations. The
flow rate was then determined by measuring the amount of fluid exiting the system into a
graduated cylinder over a measured time interval. The oscillations from both the pressure
transducer and the stethoscope were recorded via the BioPac data acquisition program.
Four solutions were used to determine the effect of viscosity on the onset of tube oscillation.
The four solutions consisted of water, viscosity=1.1 cP, 5% sucrose, viscosity=1.144 cP, 10%
sucrose, viscosity= 1.333 cP, and 34% sucrose, viscosity= 4.0 cP. The 34% sucrose was used
because its viscosity was close to the viscosity of human whole blood.
Results
Flow rates were calculated for every trial within a given solution subset and plotted against the
pressure exerted by the pressure cuff. Figures A1 to A4 located in the appendix show the
individual graphs of each solution (H2O, 5%, 10%, and 34% sucrose). These graphs illustrate
that a decreasing flow rate was required to produce the onset of oscillatory flow. Since two trials
were performed at every pressure on the tube, the flow rates from each trial were averaged, and
the calculated values are located below in Table 1.
Average Flow Rates (m3/s)
Pressure
Water
5%
10%
34%
60 mmHg
2.56E-05
2.86E-05
1.58E-05
2.32E-05
80 mmHg
1.77E-05
2.16E-05
1.94E-05
1.80E-05
100 mmHg
1.18E-05
1.49E-05
1.37E-05
1.03E-05
120 mmHg
1.00E-05
1.22E-05
1.13E-05
7.74E-06
140 mmHg
7.19E-06
9.54E-06
9.84E-06
6.16E-06
160 mmHg
7.56E-06
8.43E-06
7.83E-06
5.54E-06
180 mmHg
5.77E-06
7.68E-06
7.03E-06
4.41E-06
200mmHg
6.84E-06
7.82E-06
6.46E-06
4.40E-06
Table 1. Average flow rates of the four solutions.
To find a general correlation between pressure exerted on the tube by the pressure cuff and the
flow rate, the averaged flow rates from the four solutions were plotted together on a graph versus
the exerted pressure on the tube, Figure 2. A trend line was added to the four solutions, which
yielded a very close 2nd order polynomial fit. Figure 2 has a trend line that represents all four
solutions. Equation 1 gives the equation of the trend line.
0.000035
Flow Rate (m^3/s)
0.00003
0.000025
5%
0.00002
10%
34%
0.000015
Water
0.00001
Trend line
0.000005
0
50
100
150
200
Pressure Cuff (mmHg)
y  (1.2 109 ) x 2  (4.3 107 ) x  (4.5 105 )
Equation 1
Figure 2. Average flow rates versus the pressure exerted on the tube via the pressure cuff.
The flow rates appeared to decrease as the viscosity increased, as observed in Figure 2. Note
that the averaged water flow rate was expected to be higher than that of the averaged 5% sucrose.
This discrepancy in Figure 2 is explained by the larger variation in the water trials (Figure A1) in
comparison to the sucrose solution trials. Additionally, more trials with a larger range of
viscosities would have yielded a better resolution in the trend line.
A leveling off of flow rates appears to occur at 180 mmHg and above. The rapid increase of
flow rates around 60 mmHg is the likely explanation of why oscillations were unattainable at
lower pressures. To achieve the flow rates that were needed, an increased upstream pressure
would be required. However, since this was to remain constant for all trials an increased flow
rate was unachievable.
The oscillation signals recorded from the sensors were analyzed. The pressure transducer
observed many different types of fluid oscillations. Figure 3 shows graphs of the most
commonly observed oscillations. However, note that the oscillations picked up by the
stethoscope did not appear to correlate to the oscillations observed by the pressure transducer.
Much of the signals recorded by the stethoscope were attributed to noise and not the actual wall
oscillations. Therefore, only the oscillations observed by the down-stream pressure transducer
were analyzed.
3a
3b
Pressure (mmHg)
4
3
2
1
0
80
90
100
110
Tim e (sec)
3c
3d
Figure 3. Commonly observed oscillations in all trials. Figures 3a and 3b show high frequency small
amplitude oscillations within lower frequency high amplitude oscillations. Figures 3c and 3d are
representative of the smaller amplitude high frequency oscillations without the high amplitude low
frequency oscillations.
Figures 3a and 3b represent a little over a half of the water and 5% sucrose trials. Low amplitude
high frequency oscillations are observed within high amplitude slow oscillation. The small
amplitude high frequency oscillations become more apparent when the tube collapsed, which is
seen at the lowest downstream pressures. Figures 3c and 3d are characteristic of the majority of
graphs. The graphs are small amplitude high frequency with no low frequency high amplitude
oscillations. Another trend was seen in the lower viscosity solutions, the downstream pressure at
oscillation onset was always subzeroed. Only the highly viscous solution, 34% sucrose, showed
downstream pressures above zero and the oscillations were all small amplitude high frequency.
The three types of oscillations observed in the experiment were consistent with the three types of
oscillations observed by Bertram et al. However, the onset of the three types of oscillation in
this experiment occurred at random, meaning that whether the oscillation was fast, slow or mixed
did not follow any pattern. Bertram et al demonstrated that the three types of oscillations should
be producible at any given cuff pressure by changing the flow rate, Figure A5. This experiment
was unable to duplicate the onset of all three oscillations by changing the flow rate. This
experiment only produced one of the three types of oscillations at a given pressure and flow rate
and any deviation from this flow rate would halt all oscillations.
In order to describe the relationship between the onset of the Korotkov oscillations and the
parameters involved, dimensionless parameters reveal that the critical flow of the fluid was
directly related to the force (pressure) pushing down from the pressure, and hence the pressure
within the cuff (used in plot). As stated before, two dimensionless variables were calculated:
(Pcuff *v* L/Pupstream *Q) and (Pcuff/Pup) where v is the kinematic viscosity of the fluid, L is the
length of the tube in the collapsed region, and Q is the flow rate. The Reynolds number was
calculated as (U*D/v) where U is the velocity of the fluid, D is the diameter of the tube, and v is
the kinematic viscosity of the fluid. Figure 4 below is the plot for all trials except for 34%
sucrose solution, which followed the same trend but was found to be an outlier. The equation of
the relationship is:
y = .0074*x + 2x10-17
Equation 2
Figure 4. Plot of Reynolds number times the Dimensionless parameter vs. the
pressure within the pressure cuff.
Discussion
As stated earlier, there were several fluid dynamics principals involved in this experiment that
enable the set-up to simulate a human arm during a blood pressure measurement. The principals
involved are: maintained upstream pressure (raised water tank), critical flow rate to cause
oscillation, increased upstream potential, and increased downstream resistance.
The occurrence of Korotkov oscillations can be explained in simple terms of these few fluid
dynamics principals. The pressure transducer was pumped to a certain pressure between 60 and
200 mmHg in intervals of 20mmHg for each trial. The valves were opened for constant flow
through the collapsible tube. As the flow upstream before the collapsible tube stayed constant,
the flow downstream was decreased with a downstream valve. This caused a fluid build up and
hence a pressure build up in the tube at the point where the pressure cuff reduced the cross
sectional area. When this pressure build up reached a numerical value close to the upstream
pressure (about 73 mmHg due to gH), the potential energy due to pressure build up turned into
mechanical energy in the form of oscillations in the tube wall. These oscillations begin as a
“push” back against the actual pressure pushing down by the pressure cuff (refer to Figure 5).
As the pressure cuff reduces in size, it’s pressure increases until it can reduce the cross sectional
area of the tube once again, and hence an oscillatory motion. This initial local oscillation
continues down the length of the tube and repeats.
Figure 5. Example of the Korotkov oscillation in the collapsible tube as it begins. The pressure cuff and
the water bottle caused the indentions in the side before oscillation.
The occurrence of a sub-zero downstream pressure recorded by the pressure transducer can be
explained again in terms of fluid dynamics principles. During the one cycle of the oscillatory
process stated above, fluid builds up and therefore pressure builds up within the collapsible tube
allowing the downstream fluid to flow out into the bucket. As this downstream fluid (near the
downstream valve) flows out of the system and most of the fluid within the collapsible tube has
not passed the throat, the downstream pressure just after the throat becomes sub-atmospheric. As
a result, the pressures recorded by the transducer were sub-zero. However, with the 34% sucrose
solution, the viscosity is much higher and therefore the fluid has more momentum in flowing
through the tube. Because of this, the potential energy build up of the fluid allowed it to flow
more constantly through the collapsed throat of the tube and not allow a sub-atmospheric
pressure region after the throat.
This agrees with the general trend that was found from the data. As the viscosity and density of
the fluid increased, more momentum enables pressure build up in the collapsible tube at lower
velocities and yielding lower flow rates than fluids with lower viscosity.
Several adjustments could have been made to the methods of this experiment that would have
made it much easier to analyze. First, the pressure cuff pressure was assumed to be the pressure
shown on the pressure gauge. However, this is obviously not the pressure (force) pushing down
on the tube, rather, it is the pressure of air within the cuff that is shown on the pressure gauge.
Some method of correlation between the cuff’s internal pressure given from the gauge and the
pressure pushed down on the walls of the tube could have been made. In addition, changing the
apparatus, such as using different heights of water and running the water through the reservoir
could allow us to calculate the actual pressure down on the tube.
Another source of error is the loss of upstream pressure during the actual trials due to conducting
the trial. While conducting the trial, the water level in the tank decreased slightly and therefore
the upstream pressure was not exactly constant during the trial.
Perhaps a method of adding water at a rate at which it is lost would be more accurate. Also, the
time interval at which the flow rate was recorded was at the end of the trial when oscillation was
obtained. However, the time of oscillation was much larger than the time period for the sample
graphs, hence the sample graphs may not represent exactly when the flow rate was measured.
The method of data acquisition may have been improved by keeping one variable constant
between the flow rate and the downstream pressure. If there was a way to keep the flow rate
constant while the downstream “back pressure” (caused by opening and closing of valve)
changed, the relationship between the two could have been found.
Conclusions
General trends related to critical oscillation factors were determined. It was observed that by
increasing pressure cuff pressure, the critical flow rate for oscillation decreased. Additionally,
increasing viscosity and density resulted in decreased critical flow rate for oscillations. A notable
2nd degree polynomial relationship between pressure cuff pressure and flow rate was found.
Additionally a linear relationship was found between the dimensionless variables (Pc*v*L) /
(Pup*Q) and (Pcuff)/(Pup). Unfortunately, upon application of FFT analysis, no correlation
between the frequency of oscillations and all other experimental parameters involved in the
experiment was determined.
References
[1] Bertram, C, K. Butcher, and C. Raymond. Oscillations in aCollapsed-Tube Analog of the
Brachial Artery Under a Sphygmomanometer Cuff. Journal of Biomechanical
Engineering, Vol 111 pp. 185-191, 1989.
[2] Drzewiecki, Gary and James J. Pilla. Noninvasive Measurement of the Human Brachial
Artery Pressure Area Relation in Collapse and Hypertension. Annals of Biomedical
Engineering, Vol 26, pp. 965-974, 1998.
[3] Middleman, Stanley. Introduction to Fluid Mechanics. John Wiley and Sons, Inc.
United States of America, 1992.
Appendix
Figure A1. Flow rate vs. pressure due to
pressure cuff for water
Figure A2. Flow rate vs. pressure
pressure cuff for 5% sucrose solution
34%
0.000025
Flow Rate (m^3/s)
0.00002
0.000015
Trial 1
Trial 2
0.00001
0.000005
0
0
100
200
300
Pressure Cuff (m m Hg)
Figure A3. Flow rate vs. pressure due to
pressure cuff for 10% sucrose solution
Figure A4. Flow rate vs. pressure due to
pressure cuff for 34% sucrose solution
Pressure vs. flow rate illustrating location of
observed oscillations
Figure A5.
Figure A5. Oscillation regions defined by Pressure vs. flow rate