The Effects of a Constriction in a Branched Artery

The Effects of a Constriction in a Branched Artery A Fluid Dynamic Model Study

BE 310 May 3, 1999

Group M3 Taso Argeros Jessica Barag Eric Brahin Alexander Taich Abstract This experiment was conducted to determine whether the ratio of the Reynolds numbers in the branches changes with the change in the upstream Reynolds number. Specifically, we determined the flow rate in each branch while varying the percent constriction as well as the upstream flow rate. We found that the ratio of the Reynolds numbers in the branches remained constant as the upstream Reynolds number varies. Our

1/2 results therefore verified the derived relationship (Re1/Re2=(1+KL) )

Background Cardiac ischemia is an insufficient flow of blood through the coronary arteries into the heart. This insufficiency is usually caused by atherosclerotic plaque, which builds up in the coronary arteries (see Figure 1), gradually diminishing the flow of blood through them. In addition to causing angina, such occlusion of the arteries is thought to increase the risk of heart attack. Coronary heart disease caused 476,124 deaths in 1996 and is the single leading cause of death in America today. This year an estimated 1,100,000 Americans will have a new or recurrent coronary attack, and about one-third of them will die.1 Therefore study in coronary and aortic flow is pertinent to today’s society.

Figure 1: Drawing of an artery blocked with plaque

The effect on coronary blood flow with a coronary artery stenosis has been studied in mechanical models, experimental animals, and human subjects, but to our knowledge there has been only one limited study of the fluid dynamic effects of a stenosis in the coronary artery using proper modeling techniques.2 In that study, a single vessel with a stenosis was used, instead of a branched vessel system. In this study, a branched artery model was considered. Virtually all arteries and veins branch off to create a vascular system that can cover a large area. Examples are the coronary arteries, the aorta, and the capillaries.

2 Stenosis in one artery of a branched system will affect the flow characteristics in other nearby arteries, which can be predicted by the conservation laws. In a bifurcated system with the branches being the same diameter, stenosis in artery one will increase the flow of artery two, therefore also increasing the Reynolds Number. What is unclear is how the fraction of flow between the two arteries change as the degree of stenosis increases. The purpose of this experiment is to study the flow through a model of a single branched arterial system when applying various degrees of constriction on one of the upstream branches. Specifically the aim is to examine how the fraction of flow between the two branches changes as the degree of constriction increases.

Mathematical Model The general assumptions that went into the development of the mathematical model are as follow: 1. Incompressible Flow. The fluids used in this experiment, (water and sucrose solutions) do not vary in density from point to point in the apparatus. 2. Steady State There was no accumulation of fluid inside the apparatus; the flow rate in was equal to the flow rate out over time. 3. Fully Developed Flow The points in the apparatus were the velocity of the fluid was measured were at least 30 diameters away from the point were the fluid entered the apparatus.3 This ensures that the average velocity of the fluid is not a function of position.

The above assumptions satisfy Bernoulli’s equation.

2 P/g=-( V /2g+ z-htotal) (Equation1)

3 Where P is the difference in pressure of the two points of interest,  is the density of the fluid, g is the gravitational acceleration, V is the velocity of the fluid, z is the

difference in height of the two points of interest, and htotal are the losses. Since the Bernoulli equation can be applied to a streamline, two streamlines of interest can be drawn in a diagram of the apparatus (see Figure 2).

Figure 2: Diagram of two streamlines. Streamline 1 goes though branch with constriction; Streamline 2 goes through branch without constriction.

The Bernoulli equation can be applied to each of the two streamlines with z=0. However the P/g term is equal in both streamlines. This is shown in Figure 2, since P1=P2=atmospheric and P is the same for both streamlines. In addition, the densities and accelerations of gravity are equal. Therefore, the right side of Equation 1 is equal for both streamlines (Equation 2).

2 2 ( V2 /2g-htotal 2) = ( V1 /2g-htotal 1) (Equation 2)

Where V2 and V1 are the changes in velocities of streamline 2 and streamline 1

respectively and htotal 1 and htotal 2 are the head losses of the unconstricted and

constricted tubes respectively. The losses in the unconstricted tube htotal 1 are only equal to the losses due to the tubing.

2 htotal 1 = f*l/D*V1 /2g (Equation 3)

4 Where f is the friction factor, l is the length of the tubing and D is the diameter of the tube.

The losses in the constricted tube htotal 2 are equal to the losses due to the tubing as well as the losses due to the constriction. In order to determine the loss due to the constriction we approximated the constriction as a gate valve given by the following equation.

2 hconstriction=KL*V /2g (Equation 4)

Where KL is the loss coefficient. The loss coefficient depends on the amount of constriction. Because the literature values of the loss coefficient were given for fully open, ½ closed, ¼ closed and ¾ closed only we graphed the percent constriction versus the loss coefficient and interpolated to determine the loss coefficients. (Figure 3).4

Figure 3: Interpolation of literature values of loss coefficient (K) versus constriction fraction

K vs. constriction fraction

18 16 y = 0.0945e6.512x 2 14 R = 0.949 K

n 12 e i c i

f 10 f e

s 6 o L 4 2 0 0 0.2 0.4 0.6 0.8 Constriction Fraction

Therefore the total losses of the constricted tube are:

2 2 htotal 2 = f*l/D*V1 /2g + KL*V /2g (Equation 5)

However, the difference in the head losses in each branch due to tubing can be taken as negligible (see discussion). Equation 2 therefore becomes:

5 2 2 2 2 ( V2 /2g) = ( V1 /2g-hconstriction) = ( V2 /2g- KL*V /2g) (Equation 6)

Finally rearranging Equation 6 we get:

1/2 V1/V2=(1+KL) (Equation 7)

The Equation of Figure 3 can be substituted in Equation 7 for KL in addition, V1/V2 can be converted into Re2/Re1 giving:

6.512*X 1/2 Re2/Re1 =1/(1+e ) (Equation 8)

Methods and Apparatus The following is a list of the components used for our model: 1. Raised water tank with a flow valve 2. Water 3. 5% and 10% Sucrose Solutions 4. Tubing 5. Clamp 6. Scoops 7. Y connectors 8. Graduated cylinders 9. Buckets 10. Stopwatch

In the present study, a branched vessel system was used (see Figure 3). The system used dimensional analysis to study the effects of a constriction in one branch of our model and to compare the one branch to the unconstricted branch. Plastic tubing of with an inside and outside diameters of 0.0129 m and 0.0175 m, respectively, was used to construct the system. Three sections of tubing were cut to a length of over 30 diameters, or 0.387 meter. The purpose of this was to ensure a fully developed velocity profile formed before reaching the fork and the constriction. To assemble the model, a tube (tube 1) was attached to a flow valve on the water tank. The other end of the tube was connected to a Y connector. The Y connector has angles between the elements of 60 and 120. Tube 1 is connected to the element of the connector that is 120 from the other elements. Tubes 2 and 3 are then connected to the remaining elements of the connector. To make sure the model does not change throughout experimentation, duct tape was used to secure the connector to the middle of

6 the lab table and it was used to secure tubes 2 and 3 to the table. A schematic diagram is shown in Figure 4. Figure 4: Our Model

The most difficult and somewhat trivial part of the lab was making sure that the in the unconstricted state, equal flow exited from tubes 2 and 3. Equal flow was achieved by taping scoops to the ends of the tubes so the curvature of the previously coiled tubes was eliminated. To simulate constrictions, an adjustable screw clamp was used. Each constriction was marked on the clamp to ensure uniformity between lab periods. We observed that as the constrictions made by the clamp were increased, the constriction in the tube approached the shape of a well-defined rectangle. Therefore, we quantified the constriction using two parameters, length and width. A table of the constrictions is below.

Table 1: Constriction Sizes Constriction % Constriction Area (m2) base line 0 1.30E-04 1 5.3 8.18E-05 2 36 4.70E-05 3 62.7 6.90E-06

Besides using water, 5% and 10% sucrose solutions were prepared and utilized. The actual experiment consisted of varying the flow rate for a given constriction. For each trial, a volume of water from each tube was collected in graduated cylinders while being timed. A variety of trials were conducted by varying the flow rate, constriction size, and solution composition. By measuring the flow rate, Reynolds numbers for each condition were then calculated and analysis was performed.

7 Results We chose to represent our results in dimensionless format. Consistent with the goal of the experiment to measure the effect of an increase in blood flow on fraction flow in the branching artery we compared the fraction of flow in left and right tubes to the Reynolds number in the tube upstream from bifurcation. These data are presented in Figure 5 below for all constriction sizes. Figure 5: Fraction of flow through constricted vs. open branches compared to Re in the tube upstream from the bifurcation.

Flow fraction in branches vs. Total Flow

1.2 n e

p 1.0 o

r 0.8 t s n o

n 0.4 o i t c

a 0.2 r F 0.0 0 1000 2000 3000 4000 5000 6000 7000 Re in tube upstream from bifurcation

Baseline Constriction 1 Constriction 2 Constriction 3

The mathematical model developed for our project was tested by comparing the relationship between fraction of flow in the two branches vs. the constriction fraction. Constriction fraction was defined as the fraction of the cross sectional area of the constricted tube versus the cross sectional area of the open tube.

Figure 6: Comparison of flow vs. constriction fractions in the model.

8 Experimental vs. Calculated Results

n 1.2 e p o

r 0.8 t s n

o 0.6 c

0.4 n o i t

c 0.2 a r F 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 Constriction Fraction Experimental Mathematical Model

Discussion The mathematical model developed for this experiment led us to predict that the fraction of flow of constricted versus open branch would not be affected by an increase in the flow velocity upstream of bifurcation. We found that this fraction depends only on the constriction fraction and not fluid velocity. (Equation 8) The experimental data confirms this hypothesis. Figure 5 shows how the flow fraction varies with the upstream Re. For a given constriction size we found that the above relationship to be flat, that is the flow fraction does not vary with the upstream flow increase. The experimental and theoretical results of this experiment lead us to predict that an increase in blood flow (e.g. due to exercise) does not affect the fraction of blood delivered to tissues through constricted versus unconstricted vessels. This finding indicates that while cardiac disease is caused by an insufficient blood supply to the cardiac muscle, this shortage is not exaggerated by fractional decrease in blood flow through the constricted vessel. It is important to establish the extent of relevance of our model to the physiological conditions. Our initial design called for Reynolds numbers in the tubing upstream from the bifurcation to match those of the blood stream in the coronary arteries, about 200-500. The actual Reynolds numbers for flows used in our experiment ranged from 2500-5000 (see Figure 5). The materials limitations prevented us from being able to achieve the desired flow. We used the smallest diameter tubing possible that would

9 enable us to accurately quantify the constrictions. Smaller tubing was tried but we were faced with problems in identifying the cross-sectional area of the constriction. Fluid velocity was minimized to the fullest extent possible; lowering it any further introduced air bubbles into the tubing of the model. The only component left to vary was the viscosity, but we were unable to come up with a sufficiently viscous fluid to use with the material limitations of the laboratory. It is important to note, however, that despite the fact that our experimental flow Re were much higher than those in the coronary arteries, our model can be used to approximate flow in the aorta and its branches. We also assume that the conclusions derived would still be valid in the lower Re number range. Part of the assumptions that were made in modeling of the coronary arteries was that the fluid was homogeneous and Newtonian. Blood is known not to be either. In fact, blood flowing in arteries separates so that the red blood cells are closer to the center of the flow and plasma borders the vessel walls. Despite the fact that Bernoulli’s equation cannot give accurate results when applied to such a fluid, the main trend derived from our data will not be significantly affected. In addition to the fluid properties, the fluid dynamics between our model and the coronary arteries must be compared. In the development of our mathematical and experimental model we assumed the flow to be fully developed and steady. In fact, blood flow in arteries is neither steady nor is it fully developed.5 Thus our results are inherently incorrect, but we again hold that the general trend of flow fraction not being affected by the increase in flow velocity should still hold. Complex equipment and much more thorough analysis should be used to model pulsatile flow. Along the same lines, it can be noted that the geometry of a real arterial bifurcation is never as simple as that of our model. For example, the bifurcation axis may not be symmetrical and the vessels may not be straight. The plane of bifurcation may also not be orthogonal to the gravitational vector. A more rigorous three-dimensional set- up would be needed to model this kind of situation. In creation of the mathematical model describing our experimental set-up, we made a few notable assumptions that bear commentary. As described above, we approximated our constriction as a gate valve. Although we believe the geometry of the constriction to be similar to the gate valve, it is not identical to it, and, therefore,

10 introduces certain error in model predictions. Also, we took the difference in head losses due to tubing in each branch to be negligible. The head loss due to tubing is given by Equation 3. Using the maximum and minimum velocities of our data set, we determined the difference in head losses, due to tubing, between the left and the right branch to be at most 0.005 meters. For other types of flow, the head loss differences may not be negligible. Our results would not apply in any such cases. The difference in head losses in the two branches depends on difference in velocities. In cases of mild or moderate stenosis, velocity difference is not great so that our head loss assumption, and thus the results, hold. Some error in our results can be attributed to the experimental procedures. For example, flow rates were determined manually and thus reaction times of two people affect the results. In addition, an attempt was made to keep the pressure gradient constant by keeping the water in the tank at a constant level, however, some deviation in the water level is inherent in the procedure adding to the total error. These errors are of human nature and thus cannot be accurately quantified without making gross assumptions. In addition we estimate these errors to be the greatest contribution to the overall precision of our set-up, therefore rendering the uncertainty intrinsic to the equipment used of little consequence.

References 1. American Heart Association Web page at: http://www.americanheart.org/catalog/Scientific_catpage70.html 2. DiGiovanni PR, Einav S: Hemodynamics of coronary artery saphenous vein bypass using laser Doppler anemometry. In Proceedings of the Second International Workshop on Laser Velocimetry, Purdue University, March 27-29, 1974, vol 1, edited by Thompson HD, Stevenson WH. West Lafayette, Indiana, Enigineering Experiment Station, Purdue University, Bull No 144. 3. BE Lab Manual, Lab 3: Measurement of the Pressure-Flow Relationship for Steady Flow in a Straight, Horizontal Tube – Additional reading (Authors: Hughes and Brighton) 4. Munson, Young, and Okiiski, Fundamentals of Fluid Mechanics; John Wiley and Sons, Inc.: New York, 1990, p.505. 5. Mark, Bargeron, Deters, and Friedman, “Nonquasi-Steady Character Flow of Pulsatile Flow in Human Coronary Arteries,” Transactions of the ASME, Vol. 107, February 1985, pp 24-27.