Steady Flow Through an Asymmetric Aneurysm Model
Group M1:
Laura Kaplan
Christina Croft
Danielle Antalffy
Abstract
The most precise length parameters in determining the Recrit values of an asymmetric cylindrical aneurysm were found to be the (d) parameter and the (h) parameter, with average Recrit values of 59.811 ± 13.55% and 1698.9 ± 13.48%. The (d) parameter represents the entrance and exit tube radii, while the (h) parameter represents the ratio of the upper radius to the lower radius of the aneurysm model multiplied by the path length to the center of the aneurysm. Several other determinations were made. First, it was determined, by performing t-tests, that the entrance and exit tube alignment produced statistically similar Recrit values for each aneurysm model. In addition, geometric similarity was proven to determine statistically similar Recrit values in the aneurysm models. It was also found that symmetric cylindrical aneurysm models produce Recrit values that are significantly different that an asymmetric cylindrical aneurysm of the same dimensions again by performing t-tests. Finally, it was determined that the Recrit values for a symmetric cylindrical aneurysm are significantly different from those obtained by the class for a circular aneurysm model. The Recrit values for the cylindrical aneurysm were found to be more precise than those obtained from class data.
Background
An aneurysm is the enlargement of a blood vessel caused by the weakening of the vessel wall. This weakening is usually due to arteriosclerosies (a thickening of arterial walls), an embolism (the lodging of a blood clot or object in an artery), syphilis, physical injury or congenital weakness of the vessel’s walls[i]. As the stresses on the walls of the aneurysm increase, the aneurysm is more likely to burst.
Aneurysms are more likely to occur in larger blood vessels; therefore, most aneurysms occur in the aorta, the largest blood vessel in the body. Three-fourths of the aneurysms occur in the abdominal region of the aorta, while most others occur in the thyroid region of the aorta[ii]. Rupture of abdominal aortic aneurysm is the 13th leading cause of the death in the United States[iii].
Abdominal aortic aneurysms have been known to come in different sizes and have different symmetry. Modeling has been done on these types of aneurysms in order to determine how the size or symmetry of the aneurysms influences the stress on the walls caused by flow through the aneurysms. For aneurysms that were symmetric and similar in shape it was determined that as the diameter of the aneurysm increases, the stresses on the aneurysms increase on the posterior surface (the surface of the aorta nearest to the spinal cord) and the inflection points (points where the surface changes from concave to convex)[iv]. As the aneurysm becomes more asymmetric, the stress on the vessel wall increases at the posterior surface and the inflection points as well. Therefore, it can be said that the size and positioning of the aneurysm influence the amount of stress on the vessel wall.
Furthermore, high blood pressure is also thought to weaken the aneurysm by causing turbulent flow in the blood vessels. This can be shown using Poiseuille's law (Equation 1), where Dp is the pressure, m is the kinematic viscosity, L is the length parameter, Q is the volumetric flow, and d is the radius of the vessel both before and after the aneurysm.
(1)
Equation 1 shows that the flow is proportional to the change in pressure of a tube. It is important to note that Poiseuille’s law is only valid for laminar fluid flow. Additionally, Q is also proportional to the fluid velocity, as seen in Equation 2, where U is the fluid velocity and d is the vessel radius.
(2)
Moreover, Equation 3 shows that the fluid velocity is proportional to the Reynolds number,
(3)
where U is the fluid velocity, l is the length parameter, Re is the Reynold’s number, and v is the fluid’s kinematic viscosity. Four parameters, as well as numerous combinations of these parameters, were used to calculate the Reynolds number. These parameters were substituted in place of l, and are: a, the top portion aneurysm interior radius; b, the bottom portion aneurysm interior radius; c, the length to the center of the aneurysm; and d, the upstream and downstream tube interior radius.
Apparatus and Materials
Cylindrical Plexiglas of diameters 1.8cm, 4.2cm and 6.5cm
Rubber Corks
Pipettes of diameter 1cm
Plastic Board
Epoxy Glue
Elevated Water flow tank
Syringe and tubing for dye injection
Evans blue water dye
Flow valve to regulate flow
Graduated Cylinder to collect output flow
Stopwatch to time flow collection
Manometer tubing
Tygon Tubing
Metal file
Sandpaper
Experimental Procedure
Various sizes of cylindrical asymmetric aneurysm models were used to determine the Reynolds number at the onset of turbulence. These aneurysms were made by first determining the desired length of the aneurysm and using a hacksaw to cut the Plexiglas to this desired length. The edges of the cylinder were then smoothed using the file and sandpaper. The pipettes were cut to about 2 inches long. For the asymmetrical aneurysms of 1.8 and 4.2 cm in diameter, rubber corks with holes of diameter 1cm were used to block the ends. For both the 6.2cm diameter aneurysm and the symmetric aneurysm, the ends were stopped by gluing the plastic boards to the ends and drilling holes 1 cm in diameter to the sides. Each of the asymmetrical aneurysms had holes in and out of the aneurysm placed proportionally in the same place. The pipettes were stuck into these holes.
The Reynolds number was determined from the various length parameters, l, of the aneurysm. The measured parameters are: (a), the top portion aneurysm interior radius; (b), the bottom portion aneurysm interior radius; (c), the length to the center of the aneurysm; (d), the upstream and downstream tube interior radius. Following the measurement of each of these parameters, the model was arranged as shown in Figure 2.
Figure One Diagram depicting the asymmetrical placement of the tubes flowing in and out of the aneurysm.
Figure Two Diagram of experimental setup
It was necessary to make certain that the tube in the upstream was straight and had minimal elevation. In addition, air bubbles were removed throughout the model.
The next step in the experiment was the slow injection of the dye and gradual opening of the flow clamp. The flow was measured using the graduated cylinder and stopwatch for each amount that the flow clamp was opened. The flow clamp was opened gradually so that the exact point of the onset of turbulence could be determined. The critical value of the flow was determined when the flow changed from laminar to turbulent at the center of the aneurysm, which was the point at which the streamline flow breaks up as shown in Figure 2 above. This procedure was repeated several times to obtain an average value of the critical flow rate at which the flow becomes turbulent. The entire procedure was repeated for flow through the aneurysms of varying dimensions. Furthermore, this process was also repeated with the entrance and exit tubes both aligned and misaligned (Figure 3).
Figure 3 Diagram depicting the two different entrance and exit tube alignments tested in the experiment
Finally, the results gained from the above process were compared to those of a symmetric aneurysm that was built as in the above procedure; however, the entrance and exit tubes were placed directly in the center of the blocking boards to produce symmetry. In this case, there were only three length parameters because the upper and lower radii in the symmetrical aneurysm model were identical.
Results
The data collected in this experiment was used to calculate the critical Reynolds numbers of water flow through various cylindrical asymmetric aneurysm models. The values for viscosity used in the calculation of the Reynolds number were values of kinematic viscosity. These values were used so that the Reynolds number found would be dimensionless. The kinematic viscosity value used for water was 1.795 x 10-6 m2/sec.
Furthermore, the values for the Reynolds number were be determined based on four different measured parameters. Different combinations of these parameters were calculated and used to measure the Reynolds number (Table 1). All of these combinations yielded a number with units of meters.
Different Length Parametersa= upper ane. Radius / f = (a/e)*c / k = volume (total ane) / area (lower radius) / p = area (total)/ circumference (inner and outer)
b= lower ane. Radius / g = (b/e)*c / l = volume (total ane) / area (upper portion) / q = length from center to tube
c= length to center of ane. / h= (a/b)*c / m = volume (total ane) /area (lower portion) / r = (Area (total) - Area (q))/c
d= inner and outer tube radius. / i = (a/b)*d / n = volume (total ane) /area (inner and outer) / s = (Area (total) / Area (q)) x c
e = total radius of ane. / j = volume (total ane) /area (radius upper) / o = area (inner and outer) /circumference (total ane)
Table 1 The different length parameters used to determine the Reynolds number of the aneurysm models.
The first test was to determine the significance of the tubing alignment in calculating the Reynolds number. T-tests were performed on each of the calculated Reynolds numbers to determine the significance of the positioning of the tubes, aligned or misaligned, in calculating the Reynolds number. It was determined that there was no significant difference between the Reynolds numbers obtained for the tube alignments except for the aneurysm of length 9.1 cm and diameter 4.2 cm. Given that all other aneurysms were found to be statistically similar regardless of the tube alignment, the difference in the aneurysm of length 9.1cm was determined to be due to experimental error.
Comparison of Aligned and Misaligned AneurysmsLength (cm) / Diameter (cm) / t Stat / t Critical one-tail / t Critical two-tail
4.8 / 4.2 / 1.333 / 1.860 / 2.306
14.4 / 4.2 / 0.3634 / 1.740 / 2.110
14.2 / 6.5 / 0.5898 / 1.740 / 2.110
14.2 / 1.8 / 0.5045 / 1.740 / 2.110
9.1 / 4.2 / 2.388 / 1.771 / 2.160
Table 2 T-test comparing the Reynolds number of aligned and misaligned aneurysm of all different diameters and lengths.
As mentioned before, geometrically similar aneurysms should have the same Reynolds number. Since two of the aneurysms tested were geometrically similar, the best length parameter was determined from the precision in the Reynolds number calculations of these aneurysms. The values for each Reynolds number was calculated for both aneurysms and then averaged together. The length parameter that yielded the most precise Reynolds number was considered to be the true length parameter to be used in Equation 3 (Table 3). Furthermore, to show that these geometrically similar aneurysms yielded the same critical Reynolds numbers, t-test were preformed on the Reynolds numbers derived from the (d) and the (h) parameters (Table 4).
Flow Rates (cm3/sec) / U Critical (m/sec) / Re(h) / Re(d)Average / 0.02147 / 1698.9 / 59.811
Standard Deviation / 0.002908 / 228.94 / 8.1019
Percent Deviation / 13.45% / 13.48% / 13.55%
Table 3 The two best Reynolds Numbers for the geometrically similar aneurysms
Comparison of Proportional/Geometrically Similar AneurysmsLength (cm) / Diameter / Re(d) / %Stdev / t Stat / t Critical one-tail / t Critical two-tail
9.1 / 4.2 / 60.35 / 15.965%
14.4 / 6.5 / 59.38 / 11.570% / 0.3385 / 1.706 / 2.056
Length (cm) / Diameter / Re(h) / %Stdev / t Stat / t Critical one-tail / t Critical two-tail
9.1 / 4.2 / 1689.74 / 15.965%
14.4 / 6.5 / 1706.23 / 11.570% / 0.2045 / 1.703 / 2.052
Table 4 Comparison of the Reynolds numbers from the (d) and the (h) parameters for geometrically similar aneurysms.
Since the length parameters h and d yielded the most precise Reynolds numbers, these values for the Reynolds number were utilized for the remainder of the data calculations. The effect of varying diameters and lengths on the Reynolds number was also tested. This was done by comparing the different values for the Reynolds number for aneurysms of the same length and different diameters (Tables 5 and 6). Aneurysms with the same diameter but varying lengths were also compared (Tables 7 and 8).
Significance of Diameter on the Critical Re(d)Diameter (cm) / Re(d) / %St.dev / t Stat / t Critical one-tail / t Critical two-tail
4.2 / 88.734 / 8.132%
6.5 / 59.381 / 11.57% / 12.9975 / 1.699 / 2.045
1.8 / 96.342 / 7.154%
6.5 / 59.381 / 11.57% / 16.9858 / 1.687 / 2.024
1.8 / 96.342 / 7.154%
4.2 / 88.734 / 8.132% / 3.3636 / 1.687 / 2.026
Table 5 The effect of diameter on the critical Reynolds number using the length parameter (d) for aneurysms with a length of 14.4 cm.
Significance of Diameter on the Critical Re(h)Diameter (cm) / Re(h) / %St.dev / t Stat / t Critical one-tail / t Critical two-tail
4.2 / 4030.52 / 8.132%
6.5 / 1706.23 / 11.57% / 26.6570 / 1.699 / 2.045
1.8 / 4104.18 / 7.154%
6.5 / 1706.23 / 11.57% / 30.3110 / 1.692 / 2.035
1.8 / 4104.18 / 7.154%
4.2 / 4030.52 / 8.132% / 38.3605 / 1.703 / 2.052
Table 6 The effect of diameter on the critical Reynolds number using the length parameter (h) for aneurysms with a length of 14.4 cm.
Significance of Length of Aneurysm on the Critical Re(d)Length (cm) / Re(d) / %St.dev / t Stat / t Critical one-tail / t Critical two-tail
4.8 / 84.72 / 9.436%
9.1 / 60.35 / 15.965% / 8.4231 / 1.699 / 2.045
4.8 / 84.72 / 9.436%
14.4 / 88.73 / 8.132% / 0.4827 / 1.701 / 2.048
9.1 / 60.35 / 15.965%
14.4 / 88.73 / 8.132% / 9.7126 / 1.703 / 2.052
Table 7 The effect of length on the critical Reynolds number using the length parameter (d) for aneurysms with a diameter of 4.2 cm.
Significance of Length of Aneurysm on the Critical Re(h)Length (cm) / Re(h) / %St.dev / t Stat / t Critical one-tail / t Critical two-tail
4.8 / 1305.65 / 9.436%
9.1 / 1689.74 / 15.965% / 5.1509 / 1.721 / 2.080
4.8 / 1305.65 / 9.436%
14.4 / 4030.52 / 8.132% / 33.3741 / 1.711 / 2.064
9.1 / 1689.74 / 15.965%
14.4 / 4030.52 / 8.132% / 23.1744 / 1.692 / 2.035
Table 8 The effect of length on the critical Reynolds number using the length parameter (h) for aneurysms with a diameter of 4.2.