Objective - Penn Engineering



Measurement of Viscous Fluid Flow through a Branched Tube Network: Modeling the Left Coronary Artery in Humans

Group Number: M4

Date Submitted: May 9, 2001

Group Members:

Askwin K. Desikan

Matthew H. Fink

Alan M. Lee

Abstract

A branched tube network modeling the left coronary artery and its descending branches in humans was constructed and tested for situations of arterial occlusion and subsequent coronary artery bypass graft. Pressure drops and flow rates via timed collection and thermodilution were measured and resistances for segments of the network were calculated. It was found that the resistance of a first generation branch simulating the left anterior descending artery increased by factors of 8.92 and 14.06 for timed collection and thermodilution measurement in a 50% occluded segment, and by an infinite amount for both measurement techniques in a 100% occluded segment, corresponding with a no-flow situation. These factors represent a significant increase in resistance.

Resistance was found to increase to a large degree only in the segment that was occluded. This is illustrated by confidence intervals that are factors of 3.2 to 9.6 times greater for timed collection and 2.0 to 3.2 times greater for thermodilution than those of the un-occluded segments.

It was also found that, when a bypass graft was in place, the combined resistance increased over the un-occluded resistance by a factor of 3.05 for thermodilution in a 50% occluded segment with bypass, and by a factor of 3.21 and 10.46 for timed and thermodilution measurement in a 100% occluded segment with bypass. It was originally hypothesized that bypasses in these two situations would result in combined resistances 94.12% and 100% of the original un-occluded resistance for the 50% and 100% occlusion, respectively.

Errors in this experiment were attributed to difficulties encountered when using thermodilution flow measurement, changes in flow patterns associated with modifications in the system layout, and the overall sensitivity of the apparatus to external influences. It was determined that the higher than anticipated combined resistances were due mainly to the developing flow present in a majority of the system. This developing flow contributed to pressures higher than those for Poiseuille flow, and consequently to the higher resistances.

Objective

The primary objective of this experiment was to measure pressure drops and flow rates in different segments of a branched tube network modeling the coronary arterial system under steady flow. These measurements were to demonstrate the similarities and differences in flow and resistance inherent to arterial occlusion and coronary artery bypass grafts in the human circulatory system. In order to accomplish this several specific aims had to be met, including:

• Designing a branched tube network modeling the initial generations of the human coronary arteries.

• Measuring the flow rate - using both the timed collection and thermodilution methods - and pressure drop in each of the seven tube segments of our model in order to calculate the resistances in each segment.

• Measuring these same components while employing a 50% and 100% blockage of one tube segment both with and without a bypass in position.

• Comparing the resistances of these four systems to those of the unblocked system to demonstrate the similarities and differences in flow and resistances between the systems.

• Comparing these resistances to those predicted by the Poiseuille Law and Ohm's Law.

Hypotheses

• The resistances in a completely unblocked segment will be significantly less than those in either a half blocked or completely blocked segment of tubing.

• When an occlusion is in place, the resistance will increase in the segment that is occluded, while the remaining the same in subsequent generations of the system as well as in segments parallel to the occluded segment.

• Imposing a half-occlusion on a segment of the tube network will increase the resistance in that segment, only, by a factor of 16.

• Employing a bypass graft of a diameter equal to that of the occluded segment will decrease the combined resistance of the half-occluded segment and bypass to 94.12% that of the un-occluded segment alone.

• Imposing a full-occlusion on a segment of the network will result in an infinite resistance in that segment, due to the fact that no flow can pass.

• Employing a bypass graft of a diameter equal to that of the occluded segment will decrease the combined resistance of the full-occluded segment and bypass to 100% that of the un-occluded segment alone.

Background

This experiment was designed to study the effects of a Coronary Artery Bypass Graft (CABG) on a model of the primary generations of the left coronary artery as they relate to relative pressures, flows, and resistances. The coronary arteries are the first two branches off of the aorta in the human circulatory system. The left coronary artery supplies mainly the anterior and lateral portions of the left ventricle, whereas the right coronary artery supplies most of the right ventricle as well as the posterior part of the left ventricle.[i] Ischemic heart disease is usually caused by arteriosclerosis, which occurs when fatty deposits become calcified on the walls of the coronary arteries, specifically in the first few centimeters of these arteries. This causes a narrowing or occlusion of the arteries and prevents blood and oxygen from reaching the heart muscle. Eventually a blockage can lead to myocardial infarction, resulting in damage or death of the heart muscle tissue.

A CABG is performed to restore normal blood flow to the heart muscle. The saphenous vein from the leg or the internal thoracic artery may be used to create a bypass of the blocked section of coronary artery. These new grafts allow blood to flow freely around the blockage to the heart muscle[ii].

Figure 1: Diagram of a Double Coronary Artery Bypass Graft

In this experiment, our main goal was to explore the effects of arteriosclerosis and the effect of a coronary artery bypass graft on the left coronary artery using a branching tube network model, exploring the fluid dynamical effect of each situation.

Materials

The two-generation coronary arterial system model was made from 10mm, 8mm and 6mm Tygon tubing. In decreasing diameter, the tubing models the left coronary artery, the left circumflex and left anterior descending branches of the left coronary artery, and one additional branch of the arterial system. A Y-joint splits each branch with the two outlets modified to fit the smaller diameter tubing. A T-joint is placed at the end of the first and second generation branches and at the beginning and end of the primary tube, where monometers will be attached to measure pressure drops.

10 ml pipettes are used as monometers. The monometers across which the drop is to be measured are connected at the tops by Tygon tubing to create a differential monometer. This prevents the pressure from causing water to leak from the tops of the tubes. The occlusion clamp is placed on the first-generation tube before the monometer.

The length of tubing between each monometer was set as 1 foot, or about 305 mm. This allows enough length between the injectate hole on the catheter and the thermistor so that the entire temperature change can be recorded. The flow rate through the network is gravity driven by an elevated reservoir, with the flow rate controlled by a needle valve. The tank was constantly re-filled with water to provide a constant pressure head and flow rate.

The following diagram illustrates the design of our branched tube network model.

Figure 2: Diagram of Branched Tube Network Constructed for Experiment

Methods

This experiment was carried out in six steps performed over 4 weeks allotted for lab work. These steps were 1) construction of the model, 2) testing a free flowing model, 3) applying a 50% occlusion, 4) applying a 50% occlusion with a graft, 5) applying a 100% occlusion, and 6) applying a 100% occlusion with a graft. Step 1 was carried out in the first week, step 2 in the second week, steps 3 and 4 in the third week, and steps 5 and 6 in the fourth week.

In the first week, various materials were collected to construct a working model. The materials necessary were Tygon tubing, the reservoir tank, various sized T-joints, Y-joints, 10 ml pipettes, and clamps. The materials were sized, cut, and connected to complete our full model. The model was then tested for leaks and structural stability, as well as for the function of the manometers and catheter insertion techniques.

The following week, trials were done with no occlusions present in the system. Measurements were taken for the four, second-generation tubes by timed collection then combined to determine the flow rates in each segment. The thermodilution technique was used to measure the same flow rates in each segment. The pressure drop was also measured across each segment for three trials by measuring the height difference between the monometer tubes being tested. From these values we determined the resistance in each segment of the network.

In the third week, a clamp was applied to one branch (segment 7) of the first generation tube to reduce the area of by half creating a 50% occlusion. This was accomplished by tightening the clamp until the height of the tube was reduced by ½. Again we measured the flows via thermodilution and timed collection, as well as the pressure drops at each segment, to calculate the individual resistance. A bypass graft was then applied with a diameter equal to that of the original un-occluded tube. Again pressure and flow were measured to calculate resistance.

In the final week, the procedure was identical to that in week three, except that a full occlusion was formed by completely closing the clamp on the first generation branch. Pressure and flow were measured, and resistances were calculated in an identical manner to the previous week.

Results

Before meaningful data could be collected the validity of our apparatus must be verified. First we calibrated the Swan-Ganz Catheter. The thermistor in the catheter was calibrated against a laboratory thermometer by placing the catheter tip in water of known temperatures. These temperatures ranged from 00C (ice-water) to 400C, encompassing the entire range of temperatures that were achieved in our experiment. The voltage was recorded using the acquire.vi program on LabView. A plot of voltage vs. temperature was constructed and found to be linear. Regression analysis determined the coefficients of the relationship and their 95% confidence intervals. The equation relating temperature to voltage was

[pic]

with a correlation of R2 = 0.9959. The graph of the calibration curve is presented below.

Figure 3: Calibration Curve of Swan-Ganz Catheter

This temperature calibration curve, along with the thermodilution.vi program and the equation for volumetric flow, [pic], were used to calculate flow rate via thermodilution.

In order to measure pressure differences between the ends of each tube segment differential monometers were utilized. The pressure difference was measured in mm of H2O and converted to SI Pascals by the formula [pic]. Fluid dynamic theory states that for viscous fluid flow through a horizontal tube the drag effects of the tube wall on the fluid impose head losses, which cause a negative pressure gradient to exist along the path of flow. This means that we should see lower pressures as our tube network progresses. Indeed, this was the situation observed, proving that our results are consistent with fluid dynamic theory. Our results were also proven reliable as repeated pressure measurements on a segment indicated a constant pressure drop between trials within confidence limits. The chart below shows the mean pressure drops in each segment of the system with their confidence intervals over three trials for the situation of 100% occlusion with graft. All values fell within the confidence limits indicated.

Table 1: Mean Pressure Drops with Confidence Intervals for 100% Occlusion with Graft

Measurements of flow in our experiment were obtained by two methods; the timed collection method and the thermodilution method. Timed collection was performed by collecting the outflow from the four open ends of our tube network. The volume collected was divided by the time period of collection to get a volumetric flow rate for each of the four end tubes. Because our system exhibited steady flow, these values were then summed to get the flow rates in the parent tubes. Three trials of this method were performed for each situation.

This method of measuring flow rates cannot be used in all situations, though, since when the bypass is active there are two parallel flows feeding into one tube. In this case thermodilution is used to determine the flow rates in all tubes. By measuring the flow in the downstream tube and the occluded tube, the flow in the parallel graft tube can be determined. Again, three trials using thermodilution were performed for each situation, as with the timed collection method. The table below indicates both the timed and thermodilution flow rates obtained for the situation of 50% occlusion without a bypass.

Table 2: Timed and Thermodilution Flow Rates with Confidence Intervals for 50% Occlusion

As is seen for the 50% occlusion situation, most confidence intervals fell within 5% of the mean flow rate for any given tube segment, flow situation, and method, with only a few exceptions. These small confidence intervals indicate that the data are consistent and fall within a relatively small spread. The data obtained for all trials fell within the confidence limits of their respective means, indicating no significant difference between trials.

When we compare the timed collection and thermodilution methods of measuring flow rate, we unfortunately find gross disparities. This is indicated best by the following representative graph for 50% occlusion without bypass.

Figure 4: Timed Collection and Thermodilution Flow Data for 50% Occlusion without Bypass

From this graph we see that the discrepancies in the thermodilution flows range from 46% – 157% of the corresponding values using timed collection for this situation. Similar discrepancies exist for the other four situations examined in this experiment. This particular situation was illustrated because both the timed collection and thermodilution methods are valid for determining flows in all segments of the network, since there is no bypass in use.

A disturbing fact is revealed in this representation by the yellow bars labeled “Thermo*”. Theoretically we should be able to recover the flows in the parent tubes (5, 6, and 7) from the thermodilution measurements in the end tubes (1, 2, 3, and 4). However, when we add these lower thermodilution rates to acquire the higher ones, we obtain the yellow bars as opposed to the experimentally measured red bars. The actual thermodilution rates are 60% – 70% lower than predicted by the summing of lower tube flows.

In order for our model to accurately represent the flow characteristics seen in the coronary arterial system we must first match the Reynolds Numbers in both situations and second ensure that the dimensions of our model are scaled a constant amount from the dimensions of the actual arterial branched network. Numerous sources place the Reynolds Number in the left coronary artery between 150 and 500[iii],[iv],[v]. Utilizing the flow data from our experiment, the dimensions of the tube network, and the equation for the Reynolds Number with flow in a horizontal tube, [pic], where U is the velocity of the fluid, D is the diameter of the tube, and ( = 8.926 x 10-7 m2/s is the kinematic viscosity of water, we find the Reynolds Numbers in each tube segment for each flow situation. The tables below show the mean Reynolds Numbers for each segment and flow situation for both timed collection and thermodilution flow measurements.

Table 3: Timed Collection Reynolds Numbers

Table 4: Thermodilution Reynolds Numbers

Locations where the Reynolds Number is N/A indicate that flow in this segment under these conditions was zero, as is to be expected in cases with a 100% occlusion, for example. Values are N/A for the 50% with bypass for timed collection because this method cannot be used in this situation, as was previously stated. When the graft is in use parallel flows make timed measurements impossible. In the case of the 100% occlusion with graft, though, we know that flow in the occluded segment is zero, so timed collection is still valid.

Utilizing our measured pressure drops and both the flow rates determined by timed collection and by thermodilution, we were able to calculate the resistances in each tube segment. This was accomplished using the analogue of Ohm's Law for fluid flow, [pic], where (P is the pressure drop across a segment and Q is the volumetric flow rate through that segment. Because no changes were made to the system in any segment except the occluded segment, we should expect to see no change in the resistances in these segments. Increases in flow rates should be accompanied by increases in pressure to give the same resistance. When we calculate the resistances all the resistances for our experiment, we obtain the following tables.

Table 5: Resistances for Timed Collection Flow Measurements

Table 6: Resistances for Thermodilution Flow Measurements

Values of N/A indicate areas where no flow or pressure drops occurred. This is to be expected when there is no flow as in the 100% occlusion case. The 50% occlusion with no bypass for timed collection was N/A because no timed collection measurements were taken for this situation for reasons previously stated. Infinite resistances occurred when a pressure drop was present, but flow was not. This was the case only in the 100% occluded segment; segment 7.

Because the resistances should be equal for all cases for segments 1 thru 6, we found the means and confidence intervals for each segment and represented those confidence intervals as percentages of the means. These percentages are presented in the table below.

Table 7: Confidence Intervals as Percentages of the Means for Each Tube Segment

In calculating confidence intervals for segment seven, it is important to note that infinite resistance cannot be mathematically incorporated into the calculation. In this case, a value of 1 x 10100 Pa-s/m3 was substituted for infinite resistance. It was judged that this value was sufficiently large to accurately depict infinite resistance when compared to the typical range of resistance in this experiment (around 107 Pa-s/m3).

We can see from this table that there is a wide spread of confidence intervals. More importantly we see that the confidence intervals for segment seven are much greater than those of the other six segments, indicating that the resistances in this segment varied the most between trials. Four values were incorporated for segments 1, 2, 5, and 7 for timed collection, and for segments 3, 4, and 6 for thermodilution. Segments 1, 2, 5, and 7 for thermodilution incorporated five values of resistance. The reason for the larger confidence intervals on segments 3, 4, and 6 for timed collection can be attributed to the fact that only three values of resistance were available, due to the lack of flow when completely occluded.

When the confidence intervals using only three values are disregarded, we see that the %CI's of segment 7 are between 3.2 and 9.6 times greater than other segments for timed collection and between 2.0 and 3.4 times greater for thermodilution. This indicates that the resistances in segment 7 varied much more significantly than those in other, non-occluded segments of the tube network.

When we consider the fact that resistances measured with timed collection and thermodilution should show no difference for a given segment, 1 thru 6, we find that this is not the case. This difference is again accounted for by the discrepancies in flow rate measurement by the two methods. We can see the significant differences that exist between the two sets by performing a two-sample equal variance t-test for each segment. The results are presented below. (t-crit > t-stat ( statistically significant difference between sets).

Table 8: Two-Sample t-test with Equal Variance Performed on Sets of Resistances

In order to see the effects of applying the bypass graft to the system we calculated the combined resistances of the first generation segments under each situation. Depending on the situation, these resistances included the occluded segment, the bypass graft, and the succeeding length of tube that was fed by the previous two. These resistances are listed below.

Table 9: Combined Resistances for the Occlusion and Bypass Segments

The combined resistances display an increase for 50% occlusion by a factor of 4.55 and 8.07 for timed collection and thermodilution, respectively. They also show that resistance increased by a factor of 3.05 for the 50% occlusion with bypass for thermodilution, an infinite increase in resistance for the 100% occlusion, and increases by factors of 3.21 and 10.46 for 100% occlusion with bypass for timed and thermodilution, respectively. It was originally hypothesized that a 50% occlusion would increase resistance by a factor of 16, a 50% occlusion with bypass would decrease resistance by a factor of 0.94, a 100% occlusion would increase resistance infinitely, and a 100% occlusion would return the same resistance as initially.

Analysis

Timed collection is considered to be the gold standard of measuring flow rates since it is the simplest and most accurate method available; therefore the discrepancies (as seen in Table 2) are believed to be with the thermodilution method. One possible explanation for these disagreements is that when the catheter was inserted into the system to measure flow it obstructed a portion of the tube and blocked the path of some water. With a diameter of 2 mm the catheter accounted for a 3.14 mm2 reduction in the cross sectional area, while the tubes themselves have diameters of 6, 8, and 10 mm with cross sectional areas of about 28, 50, and 79 mm2, respectively. This resulted in a reduction in area of 11%, 6%, and 4% in each of those diameter tubes.

This reduction in area increased the resistance in the tubes and reduced the flow rate. In most cases the flow rate obtained by thermodilution was less than that obtained by timed collection. These results agree with the obstruction hypothesis. Unfortunately, because of the randomness of the constantly changing flows and flow patterns in each tube segment, the inconsistencies between timed collection and thermodilution cannot be quantitatively accounted for.

As seen in Tables 3 and 4, most of the Reynolds Numbers in this experiment fall within the indicated range of 150 – 500 for the coronary arterial systemiii,iv,v. While no data was available to us on the specific dimensions of the coronary artery, attempts were made nonetheless to approximate the arrangement of artery sizes present in the human heart. "Y-tubes" were used to create the branches in the network, approximating to the best ability the branching seen in the coronary arteries, and tubing used in subsequent generations was of a diameter smaller than that of the parent generation. Specifically an inner diameter of 10 mm was used for the primary artery (left coronary artery), 8 mm for the first generation (left anterior descending and left circumflex branches), and 6 mm for the second generation (succeeding arteries of decreasing diameter). Thus with geometries approximated and Reynolds Numbers matching for both the model and the coronary arterial system, our model represented an accurate depiction of the natural branched tube network.

The resistance of the un-occluded tube was found to be significantly less than in the 50% or 100% occluded tube, as predicted by our hypotheses. The resistance of the free flowing tube 7 in timed flow trials was 11.26 x 106 Pa-s/m3. The 50% and 100% occluded were 100 x 106 Pa-s/m3 and infinite resistance, respectively. This shows an increase in resistance by a factor of 8.92 between the 50% and unblocked systems. Furthermore the thermodilution flow rate data also yields similar results. The un-occluded tube had resistance of 6.75 x 106 Pa-s/m3 while the 50% occluded resistance was 94.68 x 106 Pa-s/m3, increasing by a factor of 14.06. Infinite resistance was observed for the 100% occlusion with thermodilution, also. It was originally hypothesized that the resistance in this case should increase by a factor of 16. The infinite resistance of the 100% occluded segments, however, is in agreement with the hypothesis.

It is unknown why the resistances observed in the 50% occlusion situation are less than those predicted by Poiseuille’s Law. We may speculate that perhaps the method of obtaining a ½ occlusion was not as accurate as we would have liked. Because the occlusion was obtained by pinching the tube to half its original height, it is possible that less than half the cross sectional area of the tube was blocked, allowing more flow to pass for the given pressure drop resulting in lower than anticipated resistances. However, as will be shown later, resistances were greater than anticipated for the 50% occlusion with bypass, so this theory is not in agreement with all the data. Further experimentation may be necessary to resolve this discrepancy.

Resistance increased in blocked first generation tubes, but remained relatively constant in unblocked first & second-generation tubes as predicted by our hypothesis. As shown by Table 7, the confidence interval for segment 7 (the first generation tube used as the site of occlusion) is 183.74 % for the timed flow data, which is 3.2 to 9.6 times greater as compared to the confidence intervals of the other tube segments. This demonstrates that the resistance in this tube changed to a significantly larger degree than in any of the other tube segments. The thermodilution flow rate data yielded similar results as the confidence interval of tube seven 170.02% approximately 2.0 to 3.4 times greater than the unblocked first and second generation tube segments. Both thermodilution and timed flow data illustrate that resistance is relatively stable beyond the point of occlusion, as hypothesized.

Based on Poiseuille’s Law and Ohm’s Law applied to resistors in parallel and series combinations, the hypothesized values for the combined resistances of the two bypass systems were calculated. The hypothesized values of the resistances for these systems were significantly less than the values obtained. For the 50% occluded-bypass system, the total resistance increased by a factor of 3.05 over the unblocked system, using thermodilution flow values shown in Table 9. This figure represents a 224.47% error over the hypothesized value of resistance, which was 94.12% that of the unblocked system. A similar discrepancy was seen in the 100% occluded-bypass system, where the total resistance was seen to increase by a factor of 3.21 for the timed collection values and 10.46 for the thermodilution values. These values were substantially different from what was hypothesized, as no change was expected in this system, and represented a 221% and 946% error, respectively.

Primarily, this data shows that Poiseuille’s Law does not hold for this situation. Although laminar flow was exhibited in the branched tube network, as shown by the corresponding Reynolds Numbers, the flow was not fully developed. Using the entrance length equation [pic], the 305 mm lengths of the individual tube segments proved sufficient for fully developed flow for Reynolds number values smaller than 1690 in 6mm ID tubes, 1270 in 8mm ID tubes, and 1016 in 10mm ID tubes. However, since the catheter was only inserted about 125 mm, and the flow was generally not fully developed at this length, Poiseuille’s Law does not accurately depict the pressure drops and, accordingly, the resistances.

In these tubes, flow was not fully developed for the major portion of the length of a segment. In developing flows, pressure drops along a horizontal tube are generally greater that those predicted by Poiseuille’s Law. This corresponds to the results obtained for the systems with bypasses in place. Resistances were greater than those predicted by Poiseuille assumptions by several hundred percent.

Due to the difficulty in obtaining reliable flow measurements with the thermodilution technique, we present some alternate methods of measuring flow that may yield better results in a model such as this. These would include the dye-injection technique or a closed system where the flow rate is controlled, measured, and re-circulated with an electrical pumping system. The latter system would also eliminate the error associated with changes in the height of water in the reservoir, which was a constant problem and may have contributed to error in all trials.

In performing this experiment, mistakes were made that may have contributed to the error seen in the results. For the first two weeks the flow rate was modeled after the coronary arteries at approximately 400 ml per min. However, due the size of the network, it was difficult to keep all of the 2nd generation tubes at a constant flow rate. In free flow trials the segments 1 and 2 (first two second generation tubes) had visibly stronger flow rates than segments 3 and 4 (last two second generation tubes). Due to the length of the catheter, segments 3 and 4 had to be moved to enable insertion. These tubes segments were twisted 90( clockwise, thus changing the network geometry. The flow rate for segment 3 and 4 measured with timed collection were 90 and 101 ml/min before and 108ml/min and 122 ml/min after the change, respectively. This shows that a change in network geometry can greatly affect the flow pattern. Undoubtedly this played a factor in the four weeks of testing, with the network changing its geometry from week to week, if not from trial to trial.

Furthermore the network was very sensitive to external influences. The slightest motion or pressure on the system caused the open ended second generation tubes to change their flow rates or completely stop. This directly affected the pressure measurements from the differential monometers. In order to measure pressure, movement of the tubes was often necessary, affecting flow rate and possibly causing some of the error associated with pressure results. Small errors in pressure measurements dramatically affected the results. Even a 1 mm H2O difference in pressure produced large confidence intervals, as seen specifically in segment 4 of the 100% occlusion with bypass trial, as seen in Table 1.

Conclusions

In constructing our branched tube network model to study the effect of occlusions and bypasses, several hypotheses were made. It was found that the total resistance for the free flowing tube was significantly less than both the 50% and 100% occluded tubes without bypasses. The resistance of the free flowing tube in the timed flow trials was 11.26 x 106 Pa-s/m3, and 100.49 x 106 Pa-s/m3 and infinity for the 50% and 100% occluded tubes without bypasses, respectively. Furthermore, the thermodilution flow rate data also yielded similar results as the free flowing system had resistance of 6.75 x 106 Pa-s/m3, while the 50% occluded resistance was 94.68 x 106 Pa-s/m3 and the 100% occluded resistance was infinite.

As hypothesized, resistance change was much greater in the occluded tube segment than in any other segment of the branched tube network model. For the timed flow data, a 183.74 % confidence interval was seen for the occluded tube, which was 3.2 to 9.6 times greater compared to the confidence intervals of the other tube segments. The thermodilution flow rate data yielded similar results as the confidence interval of tube seven 170.02% approximately 2.0 to 3.4 times greater than the unblocked first and second generation tube segments. Both thermodilution and timed flow data illustrated that resistance was relatively stable beyond the point of occlusion.

Employing the bypass graft in the occluded systems was shown to effectively reduce resistances in occluded sections of the coronary arterial model; however, the total combined resistances of these systems were much higher than hypothesized. The total combined resistance obtained for the 50% occluded system with a bypass was 3.05 times that of the un-occluded segment, which represented a 224.47% error over the hypothesized value (using thermodilution data). Similarly, the total resistance of the 100% occluded system w/bypass incurred a 221% and a 946% error for timed flow and thermodilution data, respectively.

The results indicated that Poiseuille’s Law does not hold in this experiment. Entrance length restrictions and system design resulted in non-Poiseuille pressure distributions and flow patterns. The lack of fully developed flow resulted in higher pressures, and consequently higher resistances, than were anticipated based on Poiseuille assumptions. Also, thermodilution was found to be difficult to use and yielded results that did not correlate with flows using timed collection. Employing a more accurate flow rate method is recommended. The error associated in this experiment can also be attributed to the sensitivity of the network.

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[i] Guyton, Arthur C., M.D, Hall, John E., Ph.D., Textbook of Medical Physiology, 10th Ed., Ch. 21: "Muscle Blood Flow and Cardiac Output During Exercise; the Coronary Circulation and Ischemic Heart Disease", W.B Saunders Co., Philadelphia, 2000.

[ii] Sundt, T.M., M.D., Coronary Artery Bypass Grafting, The Society of Thoracic Surgeons, 2000.

[iii] Liepsch, D. PhD., An Introduction to Biofluid Mechanics – Basic Models and Applications, 12th Conference of the European Society of Biomechanics, Dublin, 2000, pp. 262 – 265.

[iv] Bertolotti, C. & V. Delpano, Numerical Study of 3-D Flow in a Stenosed Coronary Bypass, Laboratoire de Biomécanique Cardiovasculaire, Marseille, France.

[v] Wada, Shigeo, & Takeshi Karino, Relationship Between Wall Shear Stress and Concentration of Lipoproteins Calculated for a Multiple Bend of the Human Right Coronary Artery, Research Institute for Electronic Science, Sapporo, Japan.

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