Bioengineering 310 Final Project
A Study of Steady Flow Through a Plastic Saccular Aneurysm Model
M3
Renee Deehan
Stefanie Ostfeld
Emily Rothman
Abstract
In this experiment, two plastic, saccular aneurysm models were designed, built,
and tested. These models more accurately represent aneurysm physiology in the body
than those used in Bioengineering Laboratory IV, experiment 4. These models depict
aneurysms at different stages of growth in the same vessel. Critical flow rates at the
onset of turbulence were determined for two aneurysm models using dye streamlines.
Furthermore, critical Reynolds numbers were determined from the critical flow rates for
aneurysm models with 6.0 cm and 10.0 cm diameters. The average Reynold’s numbers
were found to be 2675  436 and 2813  358, respectively.
Background
An aneurysm is a bulge or enlargement of some point in the wall of an artery,
vein, or the microcirculation, resulting from disease of the vessel wall. Disease or injury
can weaken a vessel or cause thinning of its walls, which tend to balloon outward from
the pressure of the circulating blood, forming a sac. In a typical aneurysm, the two
innermost layers of the arterial wall, the tunica intima and tunica media, have ruptured
and a blood-filled bulge or sac is formed by the vessel's outermost layer, the tunica
adventitia. In a false aneurysm, all three layers have ruptured, and the arterial blood is
held in the vicinity only by the surrounding tissues.
Flow throughout the arteries can be described as laminar, exerting equal pressure
on the sides of the venous or arterial walls. When the artery is weakened, the pressure
exerted on the wall causes it to balloon outward. Blood flowing through the artery exerts
pressure on the enlarged portion and causes the aneurysm to increase in diameter. When
flow through the sac becomes turbulent this too causes an increase in pressure on the
already ballooned walls and the vessel may burst, causing serious hemorrhaging and
other medical problems.
Aneurysms may form as a result of arteriosclerosis (thickening of arterial walls),
embolism (a blood clot or foreign object that travels through the bloodstream and
eventually becomes lodged in an artery), syphilis, physical injury, or congenital weakness
of the artery walls. A small aneurysm can exist for many years without causing any
symptoms. A popliteal artery aneurysm is easily detected by the affected person because
it causes a noticeable, pulsating bulge behind the knee. An aneurysm in this location may
lead to a blood clot and a resultant cutoff of circulation to the lower leg (with danger of
gangrene) unless circulation is restored by surgery. Since aneurysms tend to enlarge over
time and blood vessel walls tend to weaken with age, there is risk that an aneurysm will
eventually burst, or rupture, an event marked by serious, even massive, internal bleeding.
The rupture of an aortic aneurysm causes severe pain and results in immediate collapse.
The cerebral hemorrhage that accompanies a ruptured aneurysm in the brain is one of the
chief causes of strokes.
The principal artery prone to aneurysms, however, is the aorta, particularly as it
descends through the chest and abdomen. The symptoms of an aortic aneurysm vary
with the size and location of the defect. The three most common causes of an aneurysm
in the aorta are arteriosclerosis, syphilis, and cystic medionecrosis. Arteriosclerotic
aneurysms are the most prevalent and tend to occur more often in males over the age of
fifty. All aneurysms in the abdominal region of the aorta that have a fusiform shape,
which is spindle-like, a cylindroid shape, which is an extended fusiform, or a saccular, or
spherical, shape are arteriosclerotic.1 Most aneurysms in the thoracic region can be
classified as caused by syphilis. They can be saccular, fusifom, or cylindroid in shape.
These luetic aneurysms are prevalent in a younger age group than those associated with
arteriosclerosis. Again, they are more common among males. This type of aneurysm is
associated with many clinical symptoms such as respiratory problems or difficulty in
swallowing due to compression on the lungs or esophagus.2 If an aortic aneurysm
presses against the windpipe and the bronchi, it may interfere with breathing and lead to
coughing. The dissecting aneurysm is the least common aortic aneurysm. It results from
idiopathic cystic medial necrosis’s weakening of the aorta. Rather than be characterized
by a marked dilation of the aorta like arteriosclerotic and syphilitic aneurysms, they are
associated with long hemorrhagic cleavage, or dissections, of the laminar planes of the
aortic media.3 Dissecting aneurysms are mainly found in males between the ages of forty
and sixty.
In a study of the effects of blood flow on aneurysms, it is important to generate
aneurysm models to analyze turbulent flow and characterize the flow mathematically
using the Reynolds number. The critical Reynolds number is used to determine the onset
of turbulent flow in this experiment. In order to determine the values for the critical
Reynolds numbers, the following equation is used:
Re crit 
U crit  l

Equation 1
where Ucrit is the critical mean velocity through the upstream and downstream tubes, l is a
length parameter (the diameter of the aneurysm in this experiment), and  is the fluid
kinematic viscosity.4 The critical mean velocity Ucrit is determined by the following
equation:
Ucrit = Qcrit/a2
Equation 2
where Qcrit is the critical flow rate and a is the diameter of the tube.
1
Addendum to BE 310 Bioengineering Laboratory IV Manual, experiment 4, p. 615.
Ibid. p. 617.
3
Ibid. p. 617.
4
BE 310 Bioengineering Laboratory IV Manual
2
Apparatus and Materials
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Plastic, saccular aneurysm models
Water spicket
Syringe and Tygon tubing for dye injection
Evans Blue dye
Flow valve to regulate flow
Graduated cylinder to collect flow
Buckets to collect flow
Stop watch to time flow collection
Pocket knife
Hacksaw
File
Al’s Super Epoxy
Ringstands
3 cc syringes
Procedure
1. Different sized clear, plastic spheres were purchased. Each sphere was comprised of
two halves.
2. Two circles on opposite sides of each sphere were whittled using the pocket knife.
They had the same diameter as the syringes.
3. The ends of each syringe were hacksawed off and filed. They were placed in the
whittled holes with the long end sticking out of the sphere. Epoxy was used to keep
them in place to allow the fluid to flow from one end to the other. Once the syringes
were perfectly aligned and epoxyed, the two halves of the sphere were epoxyed and
left to dry.
4. Tygon tubing was attached from the spicket in the wall to two flow stabilizers and to
the aneurysm model. Tubing was connected to the other end of the aneurysm to
allow the flow to empty into a bucket. The tubing extended along the entirety of the
counter to allow fully formed flow prior to the entrance of the plastic model.
5. In the tubing a centimeter in front of the syringe entrance to the aneurysm a hole was
punctured with a needle to allow tubing (approximately 1 mm in diameter) to be
inserted into the tubing. A flow solenoid valve and a 15cc syringe were attached to
the tubing in order to inject the dye into the apparatus.
6. Air bubbles were removed from all tubing and the aneurysm model. Evans Blue dye
was slowly injected into the tubing where it traveled through the aneurysm model.
The critical flow rate, where the dye streamline changes from laminar to turbulent,
was recorded. This is where flow becomes unsteady and breaks up on the
downstream side of the aneurysm. Fifteen trials were taken using water with the
small diameter (6 cm) aneurysm and the large diameter (10 cm) aneurysm.
Results
Table 1: Early stage saccular aneurysm model. (6.0 cm diameter)
Entrance diameter to aneurysm: 0.87 cm
Q (ml/sec)
10.66667
7.666667
15
10.33333
9.666667
10.66667
10
11.66667
10.66667
10.33333
9
9.666667
10.33333
13.33333
10
Re
2691.483
1934.503
3784.898
2607.374
2439.156
2691.483
2523.265
2943.809
2691.483
2607.374
2270.939
2439.156
2607.374
3364.353
2523.265
The average critical flow rate was 10.6  1.73 cm3/sec and the average Reynold’s number
at the onset of turbulence was 2675  436.
Table 2: Late stage saccular aneurysm model. (10.0 cm diameter)
Entrance diameter to aneurysm: 0.87 cm
Q (ml/sec)
6
8
6.333333
6.333333
5
6.666667
7
7.5
6.5
7
7.166667
8.333333
5.833333
6.166667
6.5
Re
2523.265
3364.353
2663.446
2663.446
2102.721
2803.628
2943.809
3154.081
2733.537
2943.809
3013.9
3504.535
2453.174
2593.356
2733.537
The average critical flow rate was 6.89  0.852 cm3/sec and the average Reynold’s
number at the onset of turbulence was 2813  358.
Figure 1:
Streamline patterns for laminar flow through a spherical plastic aneurysm. The Reynolds
number is 700 and the diameter of the sphere is 6.0 cm. The diameter of the upstream
and downstream tubes is 0.87 cm. The direction of fluid flow is from right to left.
Figure 2:
Streamline patterns for turbulent flow through a spherical plastic aneurysm. The
Reynolds number is 2850 and the diameter of the sphere is 6.0 cm. The diameter of the
upstream and downstream tubes is 0.87 cm. The direction of fluid flow is from right to
left. Trapped circulating flow can be seen in the bulb outside the core.
Discussion
There is a high degree of clinical and physiological importance in studying steady
flow through a saccular aneurysm. Disease and injury can weaken a blood vessel causing
its walls to thin. When the walls of a vessel weaken and lose their elasticity they tend to
balloon outward from the pressure of the circulating blood, forming a sac. In terms of
fluid dynamics, there is data that turbulent flow through a vessel can cause the elastin to
degenerate at a higher rate5. The atrophy of elastin can increase the size of the aneurysm.
This atrophy is due to the increased pressure on the walls of the vessel from turbulent
flow. For this reason it is important to generate aneurysm models to analyze turbulent
flow and characterize the flow mathematically using the Reynolds number. In particular,
we looked at the critical Reynolds number to determine the onset of turbulent flow
through the aneurysms.
For further discussion, the term core flow refers to the flow through the aneurysm
that exits one vessel and enters the opposite vessel, or the path outlined by the laminar
flow line in figure 1. In an aneurysm model it is possible to retain laminar flow through
the core at Reynolds numbers that are below the critical Reynolds number. In our model
it is also possible to observe trapped or recirculating flow outside of the core flow. It has
also been shown that there exists a relationship between the flow rate, size of the core
volume, and the volume of the recirculating fluid. When the flow rate is lowered, the
core volume increases and the volume of recirculating fluid decreases.6 This is an
important result clinically. Recirculating fluid places an added pressure on the walls of
the vessel and encourages turbulent flow because it decreases the core volume (which is
non-turbulent) and we know that turbulent flow can lead to increased aneurysm size.
From a clinical perspective, it has been shown that statistically, abdominal aneurysms
that are greater than 6 cm in diameter are much more likely to rupture.7 This is assuming
a blood kinematic viscosity of 0.027 cm2/sec, an aortic diameter of 2 cm, a peak flow rate
of 5.0 L/min, and a Reynolds number calculated to be about 2900.8
It is important to understand that our experiment is based on a mathematical
exploration of fluid flow through aneurysms. We are able to manipulate our models in
ways that are impossible in vivo. For example, we can vary the flow rate through the
aneurysm and inject dye to visibly ascertain turbulent or laminar flow. However, there
are several limitations of an in vitro model. Instead of examining pulsatile flow that
exists in vessels we looked at steady flow. Steady flow is a good approximation of the
microcirculation but it does not replicate flow in the aorta. Water was used as a
substitute for blood, so there is obviously a difference in viscosity. A fluid that better
Fry, D. L., “Acute vascular endothelial changes associated with increased blood velocity gradients,”
Circ. Res. Vol. 22, 1968, pp. 165-197.
Roach, M. R., “Changes in arterial distensibility as a cause of poststenotic dilatation,” Am. J. Cardiol.,
Vol. 12, 1963, pp. 802-815.
6
Scherer, op. cit., pp. 696.
7
Crane, C., “Arteriosclerotic aneurysm of the abdominal aorta,” New Engl. J.Med.,, Vol. 253, 1955,
pp.954-961.
8
Scherer, op. cit., pp. 699.
5
approximates the viscosity of blood could not be used in this experiment because it would
require a much higher flow rate than can be achieved in this laboratory to incite
turbulence. Probably the most significant difference is that water contains no clotting
mechanisms so it is impossible to replicate the clotting mechanisms that often occur
where the blood recirculates in an aneurysm.
A saccular aneurysm better approximates an aneurysm in vivo instead of the
idealized fusiform model. There are three variables to be considered in the idealized
fusiform model: the vessel diameter (a), the aneurysm diameter (b), and the width of the
aneurysm (c). The idealized fusiform model had different values for b and c, creating a
flatter aneurysm than what might have been seen in the body. To replicate a saccular
aneurysm the model built had equal values of b and c since the aneurysm was a sphere.
This in vitro model represents two stages of a growing aneurysm. In this
experiment it was determined that in a larger, later stage aneurysm the onset of
turbulence will occur at a lower flow rate. Conversely, a higher flow rate is needed to
evoke the onset of turbulence in an early stage (smaller) aneurysm. Therefore, a more
advanced aneurysm requires a lower flow rate to achieve turbulence, and this turbulent
flow exerts a higher pressure on the vessel walls. This higher pressure will cause faster
growth of the aneursym. The critical flow rate for the smaller aneurysm model used in
this study was 10.6  1.73 cm3/sec, while the critical flow rate for the larger aneurysm
was only 6.89  0.852 cm3/sec. These results show that an aneurysm will begin to grow
faster over time because lower and lower flow rates will be required to cause turbulence.
The two stages of a growing aneurysm are represented in a small model with a
diameter of 6.0 cm and a large model with a diameter of 10.0 cm. Both models had a
vessel diameter of 0.87 cm. As previously stated, the average diameter of the aorta is 2
cm. By calculating the aneurysm diameter to vessel diameter ratio, the model’s
aneurysm diameters of 6.0 cm and 10.0 cm correspond to in vivo aortic saccular
aneurysm diameters of 13.79 cm and 22.99 cm respectively. Aneurysms in the aorta can
vary in size from 1 cm up to 20 cm but frequently fall within the 5 to 10 cm range. Our
model better resembles a saccular syphilitic aneurysm. Saccular syphilitic aneurysms are
mostly found in the thoracic aorta and achieve diameters between 15 to 20 cm. The
reason the models did not have a larger vessel size, and subsequently aneurysm sizes that
more closely resemble those found in the body, was that it is impossible to achieve the
high flow rates necessary to cause turbulent flow in the laboratory. However, the large
aneurysm diameter to vessel diameter ratio replicates some of the larger aneurysms that
are found in the human body.
The onset of turbulent flow occurs at a Reynolds number of approximately 2900.9
In this experiment, the point at which laminar flow begins to become turbulent was
gauged visually. The critical flow rates were measured when the dye streamlines began
to oscillate and break up inside the aneurysm model. To maintain consistency from trial
to trial the same team member determined when the dye streamlines began to break up
and another member was responsible for measuring the corresponding flow rate.
The Reynolds number was calculated utilizing the measured flow rate and
Equation 1 in the background. An accurate calculation of the onset of turbulent flow in
an aneurysm must take all of the parameters affecting flow into account. Obviously the
Scherer. Peter W., “Flow in Axisymmetrical Glass Model Aneurysms”, J. Biomechanics, 1973, Vol. 6,
pp. 699.
9
kinematic viscosity of the fluid and the flow rate through the aneurysm are factors; in
addition, it is imperative that the dimensions of the upstream vessel entering the
aneurysm and the aneurysm itself are considered variables in the mathematical equation.
Both of these dimensions are integral components of the calculation because it would not
be logical to only focus in on one parameter. If the aneurysm diameter were ignored, the
same Reynolds number and flow conditions would be concluded for all shapes and sizes
of aneurysms. An aneurysm in its early stages of development and one that has been
growing for some time would have drastically different flow conditions, but a calculation
in this manner would yield identical Reynolds numbers. Likewise, ignoring the vessel
size would produce the same results for aneurysms of the same diameter in different sized
vessels. The dimensionless parameter of the ratio of aneurysm diameter to vessel
diameter is necessary to determine the flow conditions. In an aneurysm of another shape,
such as the idealized fusiform aneurysm used in lab 2, a third parameter becomes
important. This model also has a width, which differs from the other dimensions, that
affects the properties of flow.
In this experiment, the equation used to determine the Reynolds number
(Equation 1 in the background) included the diameter of the vessel and the aneurysm.
This equation was determined to be the best approximation, to date, of flow through a
saccular aneurysm by Peter Scherer in a similar study of critical flow.10 The results of
this study verify Scherer’s equation. The visual determination of the onset of flow
yielded a flow rate that was plugged into Scherer’s equation to calculate the Reynolds
number. It was decided that the onset of turbulence occurs at Reynolds numbers
approximating 2900, and it can be seen that the Reynolds numbers resulting from
Scherer’s equation support this hypothesis. The results for the 6cm diameter aneurysm
show a Reynolds number of 2675  436 and 2813  358 for the 10cm aneurysm. A
Reynolds number of 2900 is included within the standard deviation for both aneurysm
models, thus verifying that Scherer’s equation can be used as a valid method for the
determination of critical Reynolds numbers.
10
Ibid, pp. 696.
References
Addendum to BE 310 Bioengineering Laboratory IV Manual, experiment 4, pp. 614-620.
BE 310 Bioengineering Laboratory IV Manual, experiment 4.
Crane, C., “Arteriosclerotic aneurysm of the abdominal aorta,” New Engl. J.Med.,, Vol.
253, 1955, pp.954-961.
Fry, D. L., “Acute vascular endothelial changes associated with increased blood velocity
gradients,” Circ. Res. Vol. 22, 1968, pp. 165-197.
Roach, M. R., “Changes in arterial distensibility as a cause of poststenotic dilatation,”
Am. J. Cardiol., Vol. 12, 1963, pp. 802-815.
Scherer. Peter W., “Flow in Axisymmetrical Glass Model Aneurysms”, J.
Biomechanics, 1973, Vol. 6, pp. 695-700.