Procedure - Penn Engineering



Modeling the Effects of Atherosclerosis on the Circulatory System

BE 310 Final Project

May 4, 1999

T3

Lytal Kaufman

Minhthe Luu

Dan Pincus

Zachary Shinar

Abstract

This experiment achieves the following goals: to express the increase in circulatory system power expended by the body as a function of percent of arterial blockage and to assess the utility of bypass surgery’s rerouting blood flow around the obstruction in the artery. A physical model of fluid flow through an obstruction was created and tested using parameters that are scaled back to the circulatory system through the use of dimensional analysis. By matching the Reynolds number of model and prototype and using these values for velocity and diameter in the dimensionless parameter [pic], we were able to generate a scaling criteria. Experimental results showed increases of pressure on the orders of 600% for the occlusion extremes

(62% occlusion vs. 0%). The physiologic conclusion from these experimental results is that percent occlusion is exponentially related to the power increase of the system with vessel diameter of 6mm. Applying this knowledge to medicinal purposes, the advantages of the use of bypass surgery seems to be most beneficial when the occlusion is 70% or greater.

Background

The human circulatory system is a closed loop consisting of 3 primary elements: the heart, the blood, and the pathways. Movement of the blood through the pathways is done not only by the behavior of the heart but by the compliance of the vessels and the activity of the skeletal muscles. On average, 5.2 Lblood/min is circulated through the systemic and pulmonary tracts.

Fluid, such as blood, flow through a tube can be described by the following equation:

[pic] Equation 1

for a pressure difference, P, tube radius a, dynamic viscosity, (, and length of tube, L.

The work done by a system moving the fluid through the circulatory system may be computed from the equation of conservation of energy:

[pic] Equation 2

Canceling terms based on the following assumptions about the blood flow in the body: steady state, fully developed flow, Newtonian fluid, no heat exchange, non-compliant tubes and no gravitational effects and the following relationship for losses:

[pic] Equation 3

the relationship for work remains:

[pic] Equation 4

Flow through a given pathway is dependent on the cross-sectional area, the number of pathways with which it is in parallel, and the pressure drop across a length of the pathway. By electrical analog, we may consider blood flow rate as current, cross sectional area and length as resistance and pressure drop as voltage. As in a circuit, we may consider the properties of a circuit in parallel and in series:

Series Circuit Equation: Rtotal = R1 + R2 Equation 5

Parallel Circuit Equation: Rtotal = (R1-1 + R2-1)-1 Equation 6

A manometer allows measurement of pressure, which can be expressed as the following:

P = (gh Equation 7

Blood flow is variable and is regulated by the autonomic nervous system. Physical exercise or emotional stress may increase blood flow rate by increasing heart rate. Atherosclerosis is a disease of the arteries where plaque builds up and inhibits the blood from flowing freely through the arteries. Such an obstruction may be modeled and the increased work can be evaluated using the above relationships.

Modeling any biological system requires a scaling factor if the exact parameters as used in the prototype and model are not geometrically, kinematically, and dynamically identical. The Buckingham pi method for non-dimensionalizing units to develop a scaling factor for the model here used is the following. The dimensions of the parameters used are:

Pressure: Mass/ (Length*Time2)

Length of blockage: Length

Diameter of tube: Length

Flow Rate: Length3/Time

Viscosity: Mass/(Length*Time)

Employing the Buckingham pi technique, the non-dimensional parameter [pic]

Procedure

The setup of the experiment is shown in Figure 2.

Figure 2: Apparatus used in the modeling the circulatory system

A standpipe was used to flow water through the apparatus. Using T-tubes, manometers were attached perpendicular to the direction of flow. Blockages of different sizes were inserted in the tubing and the pressure drop between the manometers was measured. This was done for many different flow rates. The blockages were created using tubes with smaller diameters (varying from 4.7mm to 12.5mm in diameter), each 8mm in length. The flow rates were determined using a stopwatch and a graduated cylinder.

The same apparatus was used to model the bypass, but instead of one tube being used between the manometers, two tubes were used. Each of the two tubes were identical in length to the original tube. The tubes were attached using Y-tubes. Blockages were placed in one of the two tubes and measurements of pressure drop and flow rate, were determined. The other tube was unobstructed.

Dye was injected into the apparatus in order to determine whether our experiment actually had laminar flow. This is important since this was an assumption of our derived equations.

Throughout this paper, blockage and occlusion are used interchangeably and refer to the blocked off area of the tube. The term orifice refers to the open area of the smaller tube, the blockage. Percent occlusion is defined according to equation 8.

[pic][pic] Equation 8

where Aocclusion is the area of the occlusion, Rartery is the radius of the original unobstructed tube and Rorifice is the radius of the orifice, as defined above. These terms are schematically displayed in Figure 3.

Figure 3: Enlarged schematic of the orifice

Results

Bypass versus non-bypass procedure was modeled in our experiment. For non-bypass, the pressure change versus flow rate for six different blockage sizes are summarized in Figure 4. Our data suggests that for small blockages there is no statistical difference between the pressure drop and flow. However, as the blockage is increased, a sharp rise in pressure is observed for a small increase in flow rate. On the other hand, for a bypass technique, shown in figure 5, the pressure drop versus flow rate collapses onto a single curve. Both results do not take into account a branching system.

Figure 4: There is no statistical difference between pressure drop and flow rate for small blockages.

Figure 5: Data for bypass modeling collapses onto a single curve for different blockages.

By making the appropriate derivation for scaling the experimental results to the circulatory system (derived below) and then taking this pressure increase and correlating it to a power increase, we were able to give significant data. Power increase as a function of diameter size for non-bypass and bypass methods are shown in Figure 6 and Figure 7 respectively. Applications of our experiment have been generalized for different size arteries and percent occlusions. The corresponding power increase can therefore be approximated based upon the data obtained from our lab procedure.

Figure 6: Greatest power increase occurs for small arteries and largest occlusion size as expected.

Figure 7: Our data suggests bypass surgery drastically reduces the power increase of the circulatory system.

Power increase of the circulatory system as a function of percent blockage for a particular vessel diameter size using non-bypass and bypass procedure are graphed for comparison in Figure 8. Both curves are described by an exponential function. Our data for bypass surgery, in large vessels of 12.5mm, shows to be most effective for occlusions greater than 70%. However, extrapolating our data for smaller vessels, it is expected that bypass surgery will be effective for smaller percent blockages.

Figure 8: Advantages to using bypass surgery, according to our data, occur for blockages of 70% or greater.

Discussion

Analysis

The aim of this experiment was to develop a relationship between the percent blockage of an artery and the increase in power necessary to keep a given blood flow. Data from Figure 4 suggests that pressure drop across a blockage is greatly increased as the blockage increases. This is evident by the steeper slope of the lines when graphing the flow versus change in pressure for each blockage size. However, there seems to be no statistical significance on the system when the blockages are relatively small.

In modeling the effects of plaque build up in large vessels on the system, our model has served to differentiate the advantages of bypass versus non-bypass techniques. Figures 6 and 7 show the contrast in the increase of power versus thrombus size for the bypass and non-bypass system. The increase of the blockage effects the non-bypass system more greatly than the bypass system. The results from both techniques are shown in Figure 8, which compares both techniques and highlights situations where one may be more beneficial. Power increase on the heart as a function of percent thrombus size was found experimentally to be defined by an exponential function as expected. When comparing the bypass versus non-bypass technique, our data suggests that use of a bypass greatly reduces the power output of the heart when percent occlusion is greater than 50%. However, this applies directly for only arteries of a diameter of 12.5 mm. We expect for smaller vessels the exponential portion of the power graph will occur at a lower percent occlusion.

Laminar/Turbulent Flow

Laminar flow was investigated by injecting a dye stream into our model to determine the point of transition to turbulent flow. This was especially crucial for our model given the derived formula holds for laminar flow only. The critical flow rate was found mathematically, given the properties of the fluid and the subsequent Reynolds numbers. We found the transition to turbulent flow to be at 4.3 liters per minute. Experimentally we found that a similar flow rate produced a similar transition for laminar to turbulent flow.

Error Analysis

Experimental error was mainly due to human factors. The flow rate was taken with a stopwatch and graduated cylinder. This is expected to account for approximately 0.1 seconds in variance. The manometer readings added to the cumulative error by fluctuating under high flow rates. We approximated this error by stating a 0.2 cm error at low flow up to 0.6 cm error under high flow rates. In addition, the radius of the blockage was measured using calipers lending to slight inaccuracies. These errors were offset by “zeroing” the manometers to ensure the calipers were at the same height. The cumulative error is calculated by seeing the subsequent increase in pressure caused by a given error in time and adding the error of the reading the manometers. This gives a cumulative error of approximately 300 Pa.

Dimensionless Analysis

The goal of this project is to relate the data back to the circulatory system. This is only possible through the use of dimensionless parameters and scaling factors. The three requirements as far as similarity is concerned are geometrical, kinematic, and dynamic similarity. Geometrical similarity refers to every segment of the model being spatially similar to the prototype. This quality was achieved to the best of the capabilities of the materials at our disposal. Tubing resembling the shape of true arteries was used. Because the arteries were assumed to be non-compliant, this parameter was also similar. The second quality, kinematic similarity, means all the velocities were similar over given points on the model. This is achieved through the use of dimensionless parameters. The first dimensionless parameter that we wanted satisfied for our data was the Reynolds number. This would ensure that similar degrees of laminar versus turbulent flow were achieved for our setup. This involved equating

Equation 9

where U is velocity of the fluid, D is the diameter, and ( is the kinematic viscosity. Using the Buckingham Pi method for solving for dimensionless parameters, we were able to derive a second dimensionless parameter:

[pic] Equation 10

Using this parameter we can set up an equality between the model and the prototype where.

[pic] Equation 11

By solving this equation for pressure of the prototype (circulatory system), we were able to find the pressure drop across a thrombus in the circulatory system given parameters we found experimentally.

From this pressure drop across an artery we were able to derive a relationship for the amount of power increase that the circulatory system needs to generate as a result of the occlusion.

[pic] Equation 12

This relationship comes from the conservation of energy (derived above) where D is the diameter of the thrombus, U is the velocity of the blood through the occlusion, and P is the change in pressure.

Significance of Results to Circulatory System and Branching Analog

The graph of pressure increase as a function of the artery size shows that as artery diameter decreases the power needed to be generated increases exponentially. This trend is consistent with predictions that percent occlusion as smaller diameters has a much greater impact than at larger diameters. This assumes a non-branching system and therefore can only be used to model small segments of the arterial system. Using the electrical analog mentioned we can predict how branching will affect the system.

Taking an arterial system where the diameter is 1/8 the size of the aorta and branching has occurred twice (resulting in four arterioles), we can predict the total increase in pressure and the subsequent increase in power. This can be done by taking the resistance found from the graph of diameter versus pressure for 0% occlusion and using it as the resistance for three of the branches. The fourth branch consists of different size occlusions. The resulting resistance is equal to the inverse of the sum of all the inverses of individual resistances.

[pic]Equation 13

where N is the number of artery branching that has occurred. In this example N= 4. Using this equation, we can map the pressure increase in a branched system of four arteries. Using this analogy, we can loosely map the entire circulatory system using the experimentally determined resistance values. These values though must go through the same process of dimensionless analysis and scaling that the power values went through.

So for the example of a four-branched system with an occlusion of 50%, we can use the experimentally derived data of the pressure drop on the circulatory system of a 50% occlusion to find the corresponding resistance. Then using equation 13 we can relate the equivalent resistance and find the new pressure drop. This can then be related to power using equation 12.

Using this derivation and trends from our data, the expected graph combining arterial branching with thrombus power increase results in:

This graph shows a maximum power value where positive

contributor of decreased diameter is offset by increasing branching of the system. This shows that there is a diameter where occlusion will be more destructive to the body’s circulatory system than others.

Conclusion

The goal of any biomedical research is to improve treatment techniques so health care is more effective and efficient. The results of this study allow doctors to more easily and correctly assess whether coronary bypass surgery is necessary and beneficial to the patient. By knowing the extent of the blockage and the vessel size at risk, doctors will be able to predict the necessary increase power output of the circulatory system and the advantageous decrease in power by opting for bypass surgery. As a result, a more educated decision can be made.

References





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Water inlet from standpipe

Manometers

occlusion

Pressure Drop

Flow Rate

occlusion

Orifice

Diameter

Artery Diameter

orifice

[pic]

[pic]

[pic]

[pic]

[pic]

[pic]

Figure 9: Power output is for small arteries and least branching.

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