Final Experimental Model - Penn Engineering



Arterial Resistance as a Function of Compliance

BE 310 Final Project

Group T1

Mike Dolan

Sigmund Kulessa

David Thakker

Bryan Wells

BACKGROUND

Biological Relevance

Basic physiological changes occur in the circulatory system as a result of increased age. The large arteries gradually dilate and stiffen. These changes come about because of progressive degeneration of the vascular wall, a process quite independent of the disease, atherosclerosis. Collagen and ground substance increase, the elastic fibers of the media become fragmented, and the collagen/elastin ratio rises. As a result, the compliance of the aorta decreases by more then a factor of 2 between the ages of 20 and 60 years of age; the same change occurs in other large systemic and pulmonary arteries. The diameter of the ascending aorta increases (from 1.0 to 1.8 cm), but wall thickness enlarges proportionately more than vessel diameter [1]. It is evident that the characteristic impedance of the aorta is affected more by the decrease in compliance and thickening of the vessel walls than by vessel dilation. Thus, peripheral resistance rises with age. Cardiac output also declines with age, about 7% per decade. The declining output is apparently due to the greater load imposed by the increase in peripheral resistance and not to the depressed myocardial contractility. The decreased cardiac output and increased sluggishness of the blood, which have been shown to be effects of increased vascular resistance, put older patients at a higher risk for stroke. For this reason, we have decided to study the relationship between the compliance of a vessel and its peripheral resistance. To do this we have employed the Windkessel model of arterial flow, where the parameter that defines arterial compliance is known as the modulus of volumetric elasticity (E’).

The expected trend for arteries in the body with age is that an increase in modulus of volumetric elasticity correlates to an increase in peripheral resistance (Rs). Younger people have a lower E’ and have more compliant arteries, in contrast, elderly people have arteries that are more rigid with a higher E’. Increasing the resistance to flow in the arteries, can be fatal such is the case with arteriosclerosis [2]. Arteries throughout the body may be affected by hardening, which causes symptoms because hardened arteries cannot carry enough blood to the body. Narrowing or hardening of the arteries that feed the heart (the coronary arteries) can lead to a heart attack. Narrowing and hardening of the arteries in the legs can cause pain and difficulty walking. Arteries that lead to the brain may cause a stroke if they become narrow and hard. Understanding the degree to which peripheral resistance is affected by the hardening of the arteries therefore becomes an important point in the pathology of arterial diseases.

Windkessel Model

The Windkessel theory models the arteries as a system of interconnected tubes with fluid storage capacity. The length of artery is roughly approximated to operate as a reservoir, which receives blood in an intermittent fashion and issues blood in a different time dependent fashion [3]. The property of the system enabling it to store blood is defined by its modulus of volume elasticity E’ as E’ = dp/dVo.

Figure 1: Model of the Windkessel concept.

The following mathematical model (see Equation 1), relating flow rate, pressure, resistance, and the elastic properties of the artery define the Windkessel concept.

Equation 1: Windkessel equation

When the assumption is made that inflow during systole is constant and equal to Qo, pressure during systole equals:

Equation 2: Pressure during systole

Equation 2 is for 0 < t < ts, where ts is the duration of systole and po is the pressure at the onset of systole. During diastole, when Q=0

Equation 3: Pressure during diastole

Equation 3 is for ts < t < T, where T is the duration of the cycle and pT is the arterial pressure at the end of diastole. The total peripheral resistance can be calculated if the stroke volume Vs and the pressure trace from 0 to T are known.

Equation 4: Peripheral resistance

MATERIALS

• C-FLEX L/S 18 E-06424-18 (5/16” inner diameter) (55 cm in length)

• TYGON R-3603 (5/16” inner diameter) (55 cm in length)

• Long balloon

• T-connector (1/4” inner diameter)

• Tube clamp

• 50-cc syringe

• Pressure transducer

• Water tank with flow cutoff valve

• Threaded tube connector (used at the end of tube as a downstream resistor)

APPARATUS

Figure 2: Experimental apparatus modeling the Windkessel concept

Legend

1) Water tank used to fill syringe and experimental artery with water.

2) Clamp used to block H2O flow to the syringe and tube after they have been filled.

3) 50 cc syringe with expanded opening to enable a 50mL injection of water into the artery within ~ 1 second.

4) Pressure transducer

5) Output from pressure transducer sent to A/D converter into LabVIEW.

6) Experimental artery (silicon tube, tygon tube or balloon), 55 cm in length

7) Resistive valve placed at the end of the artery to provide peripheral resistance.

8) Wood block used to keep the artery horizontally aligned with the pressure transducer.

9) Collection bucket used to collect water ejected from the artery after injection.

10) Plastic T-connector

11) Brass T-connector

12) Water supply tubing

The water supply tubing (12) was connected into a plastic T-connector (10) that joined the 50cc syringe (3) to the brass T-connector. The 50cc syringe was joined to the plastic T-connector with a small piece of rigid tubing that fit securely around both the opening of the syringe and the connector. The brass T-connector (11) was screwed into the pressure transducer (4) and joined the plastic transducer to the experimental tubing (6) (artery). The plastic T-connector was threaded so it screwed into the brass T-connector. The experimental tubing was connected to the brass connector via a threaded tube connector. Using the apparatus shown above, ten trials were conducted for each type of tubing. The tank (1) was used to fill the tubing and the syringe with water taking care not to allow any air bubbles into the setup. The water supply from the tank was then turned off and the tube was clamped (2) to prevent hydrostatic pressure from interfering with the trial. LabVIEW was configured to take 200 points per second with a total of 2000 points. The program was set to ‘Run’ and the bolus injection was sent through the tubing and collected in the bucket at the other end. The same procedure was repeated for remainder of the trials.

In the experimental set-up three tubes of varying compliance were used as part of our arterial model. The most compliant tube consisted of a stretched balloon, the tube of intermediate compliance was a silicon tube while the least compliant tube was made of tygon plastic. The tubes were selected in order to provide the broadest range of compliance.

RESULTS and DISCUSSION

Experimental Data

Using the pressure transducer, pressure vs. time traces were collected for three different types of tubing of varying compliance. In each trial, the same procedure was followed: a bolus injection of 50 ml of water was made into the tube while the pressure transducer recorded the pressure data. Ten sets of pressure vs. time data were collected for each of the three types of tubing. Using equations 3 and 4 above, values for Rs and E’ were calculated for each type of tubing. First, a value for Rs was determined by integrating over the duration of the pressure trace and dividing by the stroke volume (see equation 4). An extrapolation of the exponential region of the pressure trace (i.e. diastole region) was made. The resulting exponential fit was compared to equation 3 and values for E’ were computed. Figures 3, 4 and 5 show representative pressure traces for each of the three tubes.

Figure 3: Pressure vs. Time trace for soft silicon tubing

Figure 4: Pressure vs. Time trace for rigid tygon tubing

Figure 5: Pressure vs. Time trace for the balloon

For the soft and rigid tubing shown in Figures 3 and 4, there is a noticeable oscillation that occurs during diastole. The oscillations are attributed to the standing wave phenomenon, a characteristic that is ignored in the Windkessel model. In order to compute the exponential decay during diastole (using equation 3), the peaks of the damped sinusoid were fit to an exponential regression.

During experimentation, the only set of trials where a noticeable bulge was seen (characteristic of the Windkessel concept, Figure 1) were the trials using the balloon tubing. Specifically, a radial expansion of the balloon could be seen immediately following the bolus injection. The bulge acted like a capacitor and discharged water in a time dependent fashion as can be seen by the regression analysis of Figure 5. The graph shows that the peak pressure was reached at the end of injection (i.e. end of systole), followed by a sharp dip at which time the pressure remaining in the tube was due to the radial bulge. Finally, after the bulge discharged the liquid, the pressure inside the tube returned to atmospheric pressure.

The diastole region of the pressure traces obtained from the hard and soft tube trials can be seen to follow the Windkessel plot; where the envelope of the decaying sinusoid of the experimental data is taken to reflect the exponential decay displayed in the theoretical model. As can be seen in Figure 5, the diastolic regions for the balloon trials differed significantly from the hard and soft tube data trends. The balloon diastoles demonstrated a slight exponential decay and a nominal rise in pressure immediately followed by an abrupt pressure drop.

The slight exponential decay in the balloon trials was attributed to the beginning of the diastolic portion of the graph (as indicated by the pink exponential fit on Figure 5). This diastolic region reflects the emptying of water from the balloon due to the elastic properties and response of the balloon. The beginning of diastole was identified on the pressure trace through comparison of time measurements (using a stopwatch) with the time scale of the pressure trace itself. Once the diastolic portion of the trace was identified, the curve was fit with an exponential regression as demanded by the Windkessel model. Even though the diastolic pressure traces inherent in the balloon trials did not conform to the typical trace of the Windkessel, the fitted exponential was nonetheless used to define the relationship between peripheral resistance and the modulus of volumetric elasticity (equation 3). Again, it must be stressed that the use of the exponential fit on this particular portion of the trace is justified based on the results of the time measurements taken and compared to the time domain of the LabVIEW pressure trace. This comparison led the experimenters to conclude that the diastolic region of the graph must correspond to the above mentioned region and therefore, must be described by an exponential regression.

A summary of all major results of the project are displayed in the table 1 and in graphical format (see figure 6).

Table 1: The following table displays the means and the standard deviations for the balloon, soft tube, and hard tube.

Figure 6: The following figure displays graphically the data found in Table 1.

Overall, the individual trials for the balloon, soft tube, and hard tube are consistent within their respective sets of data. The data are precise since the standard deviations are small (averaging 7.2%) as compared to the means. Ten trials were performed for each type of vessel. In Figure 6, the trend that resistance increases with the modulus of volumetric elasticity is shown. The Windkessel model predicts that peripheral resistance would decrease as compliance was increased.

In order to justify the existence of the observed trend the data points had to be shown to be statistically distinguishable. A statistical analysis involving the 95% confidence limits of all the averaged points was carried out on the data. The results of this statistical analysis are displayed in tables 2 and 3 below. All of the final mean values of E’ and Rs were shown to be statistically different, with one exception. The peripheral resistance values between the hard and soft tubes are statistically the same. This result indicates that the material properties between the tygon and silicon tubing are too similar in order for their calculated resistances to be distinguishable (outside of the given range of uncertainty). As future consideration the tubing used should be chosen such that material properties vary significantly in order to fully validate the hypothesis.

Table 2: Statistical analysis of the Rs values between tubes

Table 3: Statistical analysis of the E’ values between tubes

Further Experimentation

A potential variation on the lab would include doing multiple, periodic pulses. A second variation may involve putting tubes (of varying compliance) in series. This would allow for additional types of analysis, specifically, incorporating electrical analogs into the analysis for the tubes in series. If more time were allotted, one could place several tubes together using y-connections as a model for an arterial tree. Lastly, a model could be developed that would account for the standing wave phenomena observed in the data.

REFERENCES

1. Wilson, Robert Francis M.D., Critical Care Manual F. A. Davis Company, Philadelphia., 1992.

2. United Care-Cepts, Inc., Hardening of the Arteries

3. Noordergraaf, Abraham., Circulatory System Dynamics Academic Press, New York., 1978

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