An Investigation of Steady and Pulsatile Flow in Stenosed Vessels

Jeffrey Stamm6020992

Presented to: Prof. KademMech 691x

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Introduction

Stenosis in blood vessels is a symptom of various common forms of vascular disease and causes severe complications to the human body. A narrowing of a blood vessel due to plaque or cholesterol buildup influences the blood flow dynamics, which then play an important role in further development of the disease. Researchers have witnessed changes in wall shear stress and pressure distributions but also changes in blood flow as vortices, recirculation and even turbulence occur. Correlations between these properties and degree of stenosis are sought by numerous researchers as they are thought to be important in the development of the disease. Steady flow and pulsatile flow will both be studied and compared.

When stenosis is moderate to severe, doctors can detect the blood flow disturbances with the use of Doppler ultrasound or even a stethoscope due to noises caused by turbulence. However, mild stenosis which represents the first stages of disease is very hard to detect. With more understanding of the changes in flow and effects on surrounding tissue caused by stenosed blood vessels, it could be possible to discover new methods to detect disease in its early stages.

This paper will first go over some background information needed to familiarize the reader with terms pertaining to fluids mechanics and biology. Next, a large portion will be dedicated to investigating the results from many experiments and numerical models that aim to study hemodynamics in a stenosed vessel and find correlations between different flow properties. Steady flow and pulsatile flow will be dealt with separately and because so much data has been gathered, categories will be used to separate major findings. The major findings will then be analyzed to determine their biological effects.

Background

Stenosis

Stenosis is the abnormal narrowing of a vessel or organ. This report will mainly be referring to stenosed arteries, which is a disease commonly referred to as atherosclerosis. Atherosclerosis is a disorder which affects mainly the medium and large arteries. Substances such as fat and cholesterol build up to form a hard substance called plaque, which reduces the diameter of the artery in which blood is able to flow. Normally, the artery is said to have a degree of stenosis (percentage of which it is blocked) when referring to the level of plaque buildup.

Thrombosis

Thrombosis is a blood clot formation that obstructs flow in a blood vessel. When a vessel is injured or is subject to certain conditions (explained later), platelets (thrombocytes) accumulate to form a clot in order to prevent blood loss. This blood clot can be large enough to restrict blood flow.

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Pulsatile flow and flow through a pipe

In the human body, the flow of blood is pulsatile due to the pressure gradient being oscillatory in nature. It can be shown that (using Poiseuille flow theory) the resistance to flow in a pipe can be found by the following equation:

It says that resistance to flow is inversely proportional to the radius to the fourth power. For example if we narrow the vessel by a half, the resistance is 16 times more than it was before. However in the human body, the effect would not be as large as predicted due to it being coupled with other resistances [16]. But if we reduce the vessel to 25% of its original diameter, the resistance is 256 times its original value! As we will see, this is in fact not negligible and produces some very undesirable effects in blood flow.

Similar velocity waveforms to the one shown in Fig. 1 were used by various researchers when creating their numerical model. For experimental apparatuses, pulsatile pressure gradients were used as well.

Numerical and Experimental Methods

Since in vivo experiments are extremely difficult to perform and physically replicating the arterial system is quite complex, numerical models prove to be an excellent tool. Many researchers have studied the cases of steady and unsteady flow through stenosis. The steady flow problem is far easier to study and is thus used to determine how varying parameters affect the flow pattern, wall shear stress and pressure gradients. The unsteady flow model is then used as a better prediction to the flow behavior in the human body. Especially now since the technology is available, 3D flow simulations are also possible. The results in the next section were found using various methods in solving Navier-stokes equations and also experiments carried out using apparatuses able to produce steady or pulsatile flow. Pulsatile velocity wave forms were simplified to sine waves, cosine waves or a function of either.

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Results and Major Findings

Flow separation, recirculation and vortices

Recirculation and vortices are phenomena seen in basically every study that deals with flow through a stenotic channel. It is present in both steady and pulsatile flows however its behavior and occurrence is quite different in each flow type.

For steady flow, it is seen that recirculation only occurs after the stenosis. This recirculation zone varies in size depending on many factors such as Reynolds number, degree of stenosis and stricture length. If one has basic knowledge of fluid dynamics, it should be obvious that separation is promoted by increasing the Reynolds number of the flow (shown in Fig 2.). This is exactly what is found in numerous studies (numerical and experimental) [1,2,4,6,8].

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Figure 2: Streamline variation with Re [1]

It was also discovered that the recirculation zone is most prominent for high degrees of stenosis (75% and above, Fig. 3) [1,2]. At a high Reynolds number of 500 and a 50% stenosis, there was found to be minimal flow separation. However at lower Reynolds numbers (100 and 200) but higher degree of stenosis (75%), separation and recirculation were very noticeable. [1,2]. The result that degree of stenosis is more of an influence on flow separation than Reynolds number is also shown by experiments done by Donald F. Young as separation was observed at higher Reynolds numbers for models with a

stenosis of 56%, compared to others that had one of 89%. Interestingly, he (and others) also noted that models with an axisymmetric

stenosis promoted separation at lower Reynolds number values than models with nonsymmetrical stenosis shapes [4, 5]. Other researchers used a 25% degree of stenosis in flows of up to Re = 1000 and observed little change in flow patterns [3]. For a 75% value, they obtained recirculation zones as much as 5 diameters downstream of the stenosis.

For pulsatile flow it’s a bit more complicated as the size of the recirculation zone is dependent on time due to the velocity being so as well. Curiously, a recirculation zone is also present upstream of the stenosis during a certain time in each cycle [1,9,12].

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A study consisting of a mild stenosis with pulsatile flow of mean Reynolds number of 100 was conducted in order to determine the main effects of the unsteady flow since this flow and stenosis condition result in no separation or turbulence in steady state [1]. The flow begins with an instantaneous velocity equal to the mean velocity. When T = 0.25 the flow reaches its peak velocity and there is still no separation present. However during deceleration, a vortex forms and continues to grow. At T = 0.5 the flow is again equal to the mean velocity however we see that the recirculation zone has increased in size and separation is occurring upstream of the stenosis. As the flow continues to decelerate, flow upstream starts to recirculate and each vortex becomes larger and moves away from the stenosis. At T= 0.75 the flow reaches minimum velocity and most of the artery is filled upstream and downstream with vortices. The flow starts to accelerate again and the vortices eventually disappear as the cycle repeats.

The flow behavior seen in the experiment described above was also found in another which used a finite element analysis approach to solving the Navier-Stokes equations. They used a stenosis of 70% and a peak Reynolds number of 337. As velocity increased, separation eventually occurred and recirculation followed (this was expected due to studying steady flow). Interestingly, the recirculation zone increased in size even during deceleration. Another vortex forms upstream when the flow near the wall reverses. When the flow at the entrance reverses direction at T = 0.6, the vortices detached from the stenosis and fill most of the artery. The vortices eventually dissipate when flow is returned in the normal direction. Finally, deceleration during the diastole causes a small vortex to appear but then disappear slightly afterward. The cycle then repeats. Interestingly, if we observe flow patterns at T = 0.2 and 0.5 where the velocities are the same, recirculation is more present at 0.5 (deceleration). Deceleration after the second peak also causes a small vortex for a short time. It would appear that deceleration of flow affects flow more than initially predicted. [1].

Wall shear stress

Wall shear stress caused by fluid flow is another important area of study due to the fact that low shear stress and high shear stress can pose certain problems in the vascular system (this will be

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discussed later). Many researchers aim to determine what flow and stenosis properties would promote such changes in shear stresses to occur. The general result for steady flow is that there is a large shear stress value just upstream of the stenosis and it drops to low values afterwards (possibly negative due to recirculation) then returns to slightly below its initial value (due to irrecoverable viscous loses) [1,2,3,5,9,13].

As always, studying steady flow is desirable to determine flow and stenosis properties that change the shear stress in the vessel in order to pinpoint the actual influence of flow unsteadiness. Increasing the Reynolds number resulted in the increase of the large sharp peak of shear stress just prior to the stenosis location. Negative shear stress was also observed in certain cases and was present due to the reverse flow in recirculation zones. Even for low Re values, shear stress was high [1,2,3]. Some example graphs shown below show results from two different papers [1,2]. Wall vorticity is proportional shear stress so the shape of the second graph holds true for shear stress as well. We see very similar shapes if not identical from both researchers. These results were also found by others [3,5,9,13].

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Increasing the degree of stenosis also caused an increase in shear stress [1]. We can see from figures 7 and 8 that a mild and moderate stenosis only result in a minimum change in wall shear stress. However at 75% degree of stenosis the stress value jumps by a large amount. It was also shown that at a lower Reynolds number, a stenosis of 89% results in more wall shear stress than a stenosis of 75% at a higher Reynolds number [2].

Results for pulsatile flow are somewhat similar although we can predict that the shear will dependent on the velocity during the cycle. Naturally, the max velocity occurs at the peak of the systole. The following graph shows the variation between degree of stenosis and wall shear stress. Notice the curves are identical to the study on steady flow.

The graph and color coded diagram below (fig. 9) show the stresses over the entire flow cycle. As predicted, the maximum shear stress is at the peak velocity (systole). It also confirms previous findings where the largest recirculations are during the deceleration of flow (notice negative shear stress) [13].

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Another study did the same analysis and obtained similar results. The extremely large peak of shear stress is at the maximum velocity during the systole (fig. 10). [1] Minimum shear stress occurs near the recirculation zone as there are areas of stagnation when the flow changes direction (transition from positive to negative wall shear stress).

Pressure changes

Pressure changes and gradients are very important due to the fact that blood flow is governed by pressure differences in the cardiovascular system. The presence of a stenosis produces more

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resistance to the flow and as the flow speeds increases therefore the pressure falls due to Bernoulli’s law. Pressure drops are highly undesirable as they reduce blood flow and thus oxygen to the body. For steady flow, it was found that increasing flow Reynolds number through a stenosis produces increasingly large pressure drops as seen in the figures below [1,2,3,4,5,7]. At low Reynolds numbers, the slight pressure drop is most likely due to viscous effects [4].

The pressure gradient over the stenosis is also very much affected by the degree of the stenosis [1,2,3,4,5,7,13]. Figure 13 shows normalized pressure drops over a stenosis with increasing Reynolds number for different stenosis models; a comparison to Poiseuille flow is also made. The models with the highest degree of stenosis (89%) produce the highest pressure drop while the other two (56% and 75%) show lower pressure drops not far off from Poiseuille flow.

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Since the flow velocity in pulsatile flow is time dependent, it is safe to say that so is the pressure gradient. This is clearly shown in Figure 16 below [15]. A pressure graph showing pressures along the center axis is also shown for different times during one cycle. The pressure drops can be seen in figure 15 for different times during a cycle.

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Turbulence

Turbulence is a phenomenon present in the body due to certain flow and stenosis conditions. A problem with many academic papers is that they neglect to include turbulence into their numerical models. Turbulence is mainly seen in experiments that consist of apparatuses that are able to pump flows steadily or in a pulsatile fashion through different models of constricted tubes. Turbulence is a very important area of study due to the fact that it is a big cause of energy loss [3,4,16,17].

It is mainly of interest to know when the transition to turbulence takes place, namely, at what Reynolds numbers and at what degrees of stenosis. This could lead to knowing exactly where in the body (assuming Re is known) turbulence is likely to occur for a specific degree of stenosis.

The image to the right (fig. 18) (taken with laser Doppler velocimetry) shows variation of Reynolds number (500, 750, 1000) for a 50% degree of stenosis. For Re of 500, we can see separation but then reattachment. The flow is lightly disturbed. As Reynolds number is increased to 750, flow is a bit more disturbed but never reaches turbulence. While at Re = 1000 the flow does appear to be random but further investigation revealed that it did not transition to turbulent. It was only at Re = 2000 that turbulence occurred [3]. The same experiment was repeated for a 75% stenosis and very turbulent flows were discovered when the Reynolds number was 1000. Compared to laminar flow it was also found that turbulence delays flow reattachment but also has shorter recirculation zones. Also, as previously seen, velocity increases when diameter starts to decrease. But when the flow transitions to turbulence, the velocity experiences a large drop.

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This large velocity drop is seen below (fig 19, Re = 2000) for the 50% stenosis and an even larger velocity drop for the 75% stenosis while 25% hardly experiences any. This low velocity results in low shear stress for turbulent regions downstream [3].

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Another study discovered that once Reynolds number is high enough to cause turbulence, pressure drop is no longer dependent on the Reynolds number. It was also interesting that they conducted tests on numerous pipes with different degrees of stenosis and obtained critical Reynolds numbers for when flow would turn turbulent. The probe position shows inches downsteam of the stenosis. We clearly see a correlation between percent stenosis and the Reynolds number required for turbulent flow. [4]

Summary of ResultsThe large amount of data collected from numerous papers agreed upon the following

correlations between flow and stenosis properties.

An increase in Reynolds number or velocity promotes separation and continued increase will result in eventual recirculation zones and possible turbulence upstream of the stenosis. Specifically for pulsatile flow, the deceleration portion of the cycle consists of larger recirculation zones not seen in steady flow models and experiments. Increased flow also causes higher shear stresses at the throat of the vessel and pressure drops across it. So for pulsatile flow, the pressures and stresses will be maximum when maximum velocity is reached.

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The degree of stenosis was found to be the most important factor as it is the most influential on the flow. It gives the same general results as increased Reynolds number (flow velocity) however the consequences are vastly more pronounced as large falls in pressure, large shear stress and recirculation zones were seen for degrees of stenosis 70% and above even for low Re. It also largely decreased the required Reynolds number for turbulence to occur downstream.

Discussion

Now that correlations between stenosis and changes in flow properties have been establish, it is necessary to evaluate these changes and determine how they affect the human body and the development of the disease.

High and low wall shear stress

It was found that both high and low wall shear stress caused by high velocity at the throat and low velocity in zones of recirculation or turbulence affect the arterial wall and arterial structure.

High cyclic stress is very important and is the main cause of plaque fatigue and ruptures which, in turn, can break off the arterial wall and possibly cause blockage downstream.

High shear stress near the stenosis throat can also cause platelet activation and thus induce thrombosis [11,18,21,22]. An in vivo experiment on dogs showed a direct correlation with high shear stress promoting platelet accumulation. This high shear stress at the stenosis resulted in platelets accumulating and forming a thrombosis, which further constricted the artery. A vessel with a 70% degree of stenosis was found to promote thrombosis to complete blockage of the lumen [18]. The platlet deposition rate was shown to increase with shear rate by three separate researchers [11].

Low shear stress on the other hand, was found to lead to a condition called hypercholesterolemia. It was observed that cholesterol started to accumulate and plaques started to form in low-shear stress areas [23]. From the results previously discussed, this could be where velocity transitions from positive to negative. It also promotes red blood cells to clump up into larger particles (cell adhesion) [3,11]. Coagulation is also affected by how long blood is exposed to low shear stress. Thrombosis and blood clotting are induced due to the flow stimulating homeostasis [11]. The thickness increase for low shear stresses can be shown in figure 23.

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From this information is it reasonable to assume that locations of low shear stress, mostly found in areas of low velocity, turbulence and recirculation, promote thrombosis, blood clotting, red blood cell adhesion and further cholesterol buildup. High stress mainly promotes thrombosis, endothelial cell damage and plaque rupture depending on its value.

Pressure gradient and pressure changes

A pressure drop across a stenosis can be viewed as an energy loss and a sign of the heart being overworked. To maintain the same amount of blood flow as a stenosis forms in a vessel, the heart must provide a higher pressure to drive the flow. This puts strain on the heart and could result in heart failure. This high pressure then can also lead to the further development of disease since it may cause damage to the endothelial lining. This damage could very well lead to thrombosis. Another possibility is for the high pressure to cause an aneurysm which is an outpouch or bulk in the vessel. If this pouch bursts it could be fatal due to internal bleeding. Low pressure on the other hand can be much lower than the outside pressure acting on the arterial wall. Artery collapse could very well occur during the peak of the systole near the stenosis where velocity is maximum and pressure is minimum.

Turbulence and recirculation

Turbulence and recirculation are both causes of energy loss as flow velocity can be negative (opposite of desired flow) and have velocity components not in the direction of where the flow is supposed to go. These are examples of kinetic energy loss which in turn reduces flow rate. However, the body compensates by providing more pressure to sustain a desired flow rate. The same problem with high arterial pressure is explained above. Recirculation zones consist of low flow velocities which result in low shear stresses thus also furthering the development of the disease.

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Conclusion

Understanding how atherosclerosis and thrombosis affect the flow in the arterial system is crucial to discovering new ways on how to detect it early and evaluate possible threats. With the correlations found above and some knowledge on basic flow characteristics in specific arteries, detection of any sort of change of normal flow should provide clues that disease is present. For example, figure in a review by David Ku on the following page shows different sections of the arterial system with they’re corresponding pressure and velocity ranges [11]. However, the lack of proper safe techniques to measure pressure and other flow parameters in vivo makes this a very difficult task. For example, if one knows the Reynolds number in an artery of question and obtains an approximate value for the degree of stenosis (using laser Doppler velocimetry), many conclusions can be hypothesized and the severity of the threat can be assessed.

One very important fact that was discovered nearly by all the literature investigated is that a stenosis above 70% is very dangerous as all negative effects brought upon by a blockage increase quite quickly. Another is that the disease actually promotes its own development by causing flow conditions which in turn, cause further cholesterol buildup, thrombosis and cell adhesion.

While approximate correlations were indeed discovered, the methods and experiments used were not accurate models of flow in the human body. Although some of the experiments were performed on animals which backed up studies done numerically, no numerical data could be compared to actually values in the human body; no experiments were performed that even aimed to accomplish this.

While this knowledge is useful in understanding the effects on an artery when it becomes sclerosed, it cannot yet be fully applied due to limited technology.

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