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Measurement of Pressure Drop across a Stenosis

Null Hypothesis

This experiment models the pressure drop across a stenosis in an artery. You will test two null hypotheses:

1. The pressure drop across the stenosis is independent of the stenosis length. (Alternative hypothesis: Pressure drop increases with stenosis length).
2. The pressure drop across the stenosis is independent of the flow rate. (Alternative hypothesis: Pressure drop increases with flow rate).

Theory for Pressure Drops Across a Stenosis

Young (1979) summarized the results of a number of studies on arterial stenoses with an equation for pressure loss () that is recast here in terms of flow rate () instead of Reynolds number:

Eq. (1)

where is dynamic viscosity, is fluid density, D_v is diameter of the vein, D_s is stenosis diameter at its narrowest point, and is the turbulent loss coefficient (1.52). The expression in square brackets is called the laminar loss coefficient, and the expression within in angled brackets is the effective stenosis length. Physically, the first term represents losses caused by separation and turbulence, where as the second term represents viscous (or laminar) losses. As a consequence, the dynamic viscosity appears only in the second term.

For the laboratory you will be doing, the following values or ranges of parameters are appropriate:

for 50% diameter reduction

will range from 2 ml/s to 20 ml/s

will be 1 to 2 cm

Pressure Transducers

The pressure transducers you will be using are known as “strain gauge” type transducers. The pressure on the sensing element changes the resistance of a resistor according to the equation . When a wire is stretched, its length, , increases slightly and its cross-sectional area, , decreases. The small change in resistance can be detected by the use of a Wheatstone bridge setup. The bridge is shown in Figure 1.

Initially the bridge is balanced so that all of the resistances are the same. As one resistor is strained, its resistance changes and the bridge becomes out of balance. If the positive supply voltage (at the top of the bridge) is equal in magnitude to the negative supply voltage, the two output voltages at balance are zero, which is convenient since it is then not necessary to subtract the DC offset from the output. However, in our case we will use a positive supply voltage at the top of the bridge and ground the bottom. If we use a 9 V supply, we expect that both V_out and -V_out wil be +4.5 V. These two outputs can be input to a differential amplifier so that the DC offset does not appear in the output.

Before You Come to the Laboratory

You should have some understanding of the behavior of pressure as a function of stenosis diameter. Use Excel to plot pressure drop as a function of flow rate for a few typical cases. Use the typical parameter values given above. Plot three curves on the same plot, one for a 25% diameter reduction, one for a 50% diameter reduction, and one for a 66% diameter reduction. An example Excel file that sets up the equation has been generated. The file uses the following techniques to make the equation easier to create and easier to verify. You should use these techniques whenever you set up equations in Excel:

1. Use named variables. Not only does these improve the readability of your equation, but they also make it easy to change parameters quickly (e.g., if you go from a 50% stenosis to a 75% stenosis, or if you change the vessel diameter).
2. Break up equations into smaller pieces. The syntax becomes easier, and it is easier to debug the pieces individually than to debug a giant complicated equation. In this case, I have broken up the equation into the following pieces,

and by defining , , , and separately I have eliminated the need to use a large number of confusing parentheses in setting up the equation. Furthermore, I can now easily look at the pieces of this equation individually.

3. Make plots not only of the final results (i.e. pressure loss), but also of the individual terms.
4. Display the results in units that you have some feel for, if possible. For example, since you know that 120 mm Hg is a typical blood pressure, it is helpful to relate the pressure losses across a stenosis to this physiological number.

Set up the formulas in Excel to do the following:

1. Translate flow rate from the volume of fluid collected and the time over which it was collected.
2. Calculate theoretical pressure drop for a given measured flow rate value.
3. Calculate the pressure drop, given the transducer voltage measurement and the transducer calibration constant.

If this is set up properly, you can obtain the expected pressure drop immediately whenever you take a flow rate measurement. This pressure can then be immediately compared to the pressure drop you obtained from the pressure transducer to see if there is an obvious error in either your procedure or your calculations.

Experimental Setup

You will need the following materials for this experiment:

You will need the following:

1. Graduated cylinder to measure volume.
2. Stop watch to measure time of volume collection.
3. Tape measure to measure height of the transducer.
4. Two large white buckets (fluid reservoirs)
5. A length of ¼ inch inner diameter tubing
6. Smaller pieces of tubing to make connections to the pressure transducer
7. A pressure transducer that is connected to a signal conditioner
8. A digital multimeter
9. Stopcocks to control connections to the pressure transducer.
10. Plastic glue.
11. Three stenoses: 50% 1 cm long, 50% 2 cm long, 66% 1 cm long.

The experimental setup is as shown in Figure 2.

You will need to set up this flow configuration with the materials provided. The pressure transducer has two fluid inlets to it, but it is not a differential transducer and only senses the pressure that is applied to its sensing element. One of the ports of the transducer is to be connected to your experiment, and the other is to be used to “bleed” air from the transducer. Once all air has been removed from the transducer and the tubing, shut off the extra port. Connect the transducer to one of the pressure taps at a time. If you wish, you can physically connect the transducer to both of the pressure taps and use the stopcocks to control which of the taps is open to the transducer. However, if you do this, make sure that one tap is open to the transducer and the other is closed to the transducer when you make your measurement.

Preliminaries

You will need two pressure taps, one upstream and one downstream of the stenosis.

Insert the first stenosis in the tubing. The easiest way to do this is to cut the tubing at the location where you wish to place the stenosis. The stenosis is then inserted halfway into one side and halfway into the other so that the stenosis itself holds the two pieces of tubing together (Figure 3).

Connect the pressure transducer to the upstream pressure tap. Make sure that there are no air bubbles in the line.

Transducer Calibration

Calibrate the pressure transducer using the weight of water as a standard. The pressure at the transducer is rgh, where r is the density of water, g is the acceleration of gravity, and h is the height of the fluid surface above the pressure transducer. For example, if the transducer height changes by 5 cm, the pressure changes by

.

With the experiment set up as in Figure 1, fill the upstream reservoir to about ½ its height and stop up the end of the tubing so that there is no flow in the system. Make sure that there is no air in the tubing. Now you can change the pressure to which the transducer is exposed by changing the height of the transducer relative to the surface of the water in the upper reservoir. This height difference is the in . Take several readings of voltage from the signal conditioner as a function of the height of the reservoir surface above the transducer. Take measurements for heights that are both above and below the level of the pressure tap up to ±12”. Read the voltage for each transducer position and obtain about 10 total data points. Pressure transducer calibration data should be tabulated in a table similar to Table 1. If you generate this table in Excel, the linear fit to find the slope of the calibration will be simple. You should calculate the linear fit of the transducer before you move on to the next stage of the experiment so that you can find and check any incorrect values in the calibration data.

Before Experiment		After Experiment
Height Difference (in)	Voltage Out (mV)	Height Difference (in)	Voltage Out (mV)

To obtain the calibration factor that converts volts to pressure, perform a linear least squares fit of pressure as a function of voltage (V). The slope of the derived function () allows voltage to be converted to pressure. We will not be concerned with the y-intercept, only the slope because all calculations will be done in terms of a voltage difference (voltage upstream – voltage downstream). Thus, to obtain the pressure drop you will use .

You will need to know the true diameter of the tubing used in this experiment. This may not be the same as what is written on the tubing. Fill a length of the tubing with water, and use the volume of water required to fill the tubing to calculate back to tubing diameter (, hence ).

Flow Rate

You do not have a valve to adjust the flow rate. However, you can adjust the flow rate by changing the vertical distance from the fluid level in the upper reservoir to the outlet of the tubing. You can also adjust the flow rate by inserting the second stenosis into the end of the tubing. Calculate the flow rate that is required to obtain a cross-sectional average velocity () of 50 cm/sec (). Set up the system so that the surface of the upper reservoir is 2 to 4 feet above the outlet of the tubing into the lower reservoir, and use a graduated cylinder to measure the flow rate. If the flow rate is significantly different from the target of 50 cm/s (say, by a factor of 2), then readjust the height difference and re-measure. This time, measure the flow rate as precisely as possible. You can reasonably expect to obtain a flow rate that varies by no more than 5% from measurement to measurement.

The flow rate will change somewhat as a result of changes in the height of the fluid in the reservoir. Be sure to maintain this height constant as much as possible. Also, determine how much a given change in height changes the flow rate. To do this, take repeated flow rate measurements as the fluid reservoir height decreases.

Pressure

Record the voltage on the digital multimeter both when flow is turned on and when flow is turned off. You can insert a plug (made from the same material used to make the stenosis) to turn off the flow. When you remove the plug, the flow rate should return to its original value, assuming that you have made no other changes in the configuration (such as changing the vertical position of the end of the tubing).

Now connect the pressure transducer to the downstream pressure tap. Again, measure pressure with the flow on and the flow off. The pressure with the flow off should be the same as the no-flow measurement upstream, but may not be as a result of (1) transducer drift or (2) a significant change in the height of fluid in the reservoir. You should doublecheck the height of the fluid in the reservoir before doing the downstream measurement and add water if the height has changed significantly. The drift should not be a large problem as long as the flow-on and flow-off measurements are taken close enough to one another in time. Repeat these measurements five times.

Reduce the flow rate by a factor of 2, and repeat the set of 5 measurements again. In the end you should have enough data to fill Table 2.

		For Flow Rate		For Pressure Drops
Conditions		Volume (ml)	Time (sec)	Voltage Up-stream No Flow	Voltage Up-stream With Flow	Voltage Down-stream No Flow	Voltage Down-stream With Flow
50% Short Stenosis	Flow Rate 1
	Flow Rate 2
50% Long Stenosis	Flow Rate 1
	Flow Rate 2

Recalibration

To ensure that your calibration has not changed, perform a post-experiment calibration, again using the weight of water as your standard.

Data analysis

1. Plot the measurements of output voltage as a function of pressure for the transducer calibration. Perform a least square fit of pressure as a function of voltage. The slope is the transducer sensitivity, in volts/(dynes/cm²). Show the raw data for the calibration on a plot as individual points without connecting lines. Show the least square fit model as a solid line. If the pre-experiment and post-experiment calibrations are significantly different, perform least squares fits for these data sets separately.

2. For each data set value of (Volume Collected, Time of Collection), translate to flow rate (Q).

3. For each data value of (Change in voltage upstream, Change in voltage downstream), translate to pressure drop.

4. Plot Pressure as a function of Q (use symbols for these data).

5. Plot Young’s equation for the same flow rate range (use a solid line for these data).

6. Discuss reasons for any differences (theory vs experiment).

7. Perform Student’s T-tests to test the two hypotheses. The first T-test will compare data taken with different flow rates but the same stenosis. The second T-test will compare the two stenoses of different lengths. Provide a p-value in both cases. Are the results significant.

8. As part of your discussion, determine whether the overall pressure drop in the system () is accounted for by the pressure drop across the stenosis and the Poiseuille flow pressure drop in the tube. Do you expect the measured flow rate to be higher or lower than what would be predicted by the combination of Young’s equation and Poiseuille’s equation?

References

Young DF: Fluid mechanics of arterial stenosis. J Biomech Eng 101: 157-175, 1979

Biomedical Engineering Senior Laboratory (BIEN 435)

Louisiana Tech University

Steven A. Jones

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