E. Schuit July 2007 BMTE 08.15

Transcription

1 Internship Pulsatile flow in a stented and non-stented 2D cerebral aneurysm model E. Schuit July 2007 BMTE ir. G. Mulder prof. dr. ir. F.N. v.d. Vosse Eindhoven University of Technology Department of Biomedical Engineering

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3 Abstract In this study the influence of stent placement in a vessel with an aneurysm is investigated by examining hemodynamic parameters such as aneurysmal inflow, velocity field, wall shear stress and vortex strength in a two-dimensional computational fluid dynamics model in both a steady and a pulsatile flow case. These parameters are used to quantify the influence of the density of the wire and the thickness of the struts of the stent. The results show that the intra-aneurysmal flow velocity, wall shear stress, mean absolute velocity and vortex strength are reduced due to the stent insertion. Pulsatile inflow results in a wall shear stress with a pulsatile character. A pulsatile inflow may have a bigger influence on the intima than a steady flow. Future research might be concentrated on this pulsatile behavior in a 3-dimensional environment. 2

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5 Contents Page 1 Introduction 4 2 Methods Governing equations Geometry Characteristic numbers Boundary conditions Data analysis 14 3 Results Inflow Mean velocity Wall shear stress Vortex strength 27 4 Discussion 30 References 32 4

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7 1. Introduction Saccular aneurysms are focal dilatations of the arterial wall and may appear in any part of the human body s arterial system. A hypothesis is that their origin is mainly due to vascular malformations that grow during life 1. Cerebral aneurysms may eventually become symptomatic by rupture and hemorrhage into the subarachnoid space 1. The incidence of intracranial aneurysms is unknown but is estimated at 1-6% of the population and the risk of rupture is believed to be 1-2% per year. There is a 50-60% mortality rate associated with rupture 2. The rupture of a cerebral aneurysm leads in 80-90% of all cases to a subarachnoid hemorrhage (SAH), which has an incidence of about 10.5/100,000 persons per year 3. Main complications after a SAH are rebleeding and cerebral vasospasm 4, which may lead to a sudden surge of headache, nausea or neurological disorder 1,5. About half of the patients that suffered from SAH dies in the first 6 months and only 1/3 returns to their normal living situation due to severe neurologic deficits 6. The growth and rupture of saccular aneurysms are related to body systemic influences, such as blood pressure and pulsatility, as well as to local factors such as arterial wall properties and hemodynamic stress. The mechanism by which blood flow velocity influences aneurysm formation is thought to be through wall shear stresses and induced vibrations with secondary endothelial disruption 7. Taylor and Yamaguchi 8 showed that regions of maximum pressure moved in the pulsatile flow case compared to the steady flow case; local areas of maximum pressure were formed at the distal neck and then declined, putting an additional strain on the distal neck of the aneurysm. In a clinical study, Suzuki and Ohara 9 examined the origin, growth and rupture of the cerebral aneurysms and found that the walls of large aneurysms were usually thinner at the distal neck and the dome. So, they believe that the distal neck of the aneurysm and the dome are the most likely locations for rupture to take place. There are four methods of treatment, namely: clipping, coiling, stent placement and a combination of stent placement and coiling (Figure 1.1). Clipping has been the standard method of treatment for intracranial aneurysm occlusion. By clipping, the aneurysm is isolated from the intracranial circulation, so rebleeding is prevented. Disadvantages of clipping are high costs and the risk of surgical complications, Figure 1.1 Treatments of aneurysms: clipping (left), coiling (middle) and stent+coiling (right) 6

8 such as an associated vasospasm or brain edema. Because of these disadvantages new endovascular techniques have been investigated 10. In endovascular treatments the goal is to reduce the intra-aneurysmal blood flow so thrombus formation will be enhanced 6,10. Thrombus formation involves various factors and complex mechanisms, and the hemodynamics is one of the important factors that are involved in thrombus formation 10. Treatments that are suitable for thrombus formation are coil embolization, stent placement 11 and a combination of these two treatments. Coil embolization has become an established alternative for aneurysm treatment 5. Microcoils are inserted into the aneurysm using a microcatheter. As a result of the presence of microcoils in the aneurysm the intra-aneurysmal blood flow is reduced, which enhances a thrombus formation 10. Coiling cannot be used in wide-necked aneurysms because there are difficulties in obtaining complete filling of the aneurysm sac as well as the risk of coil protrusion into the parent artery 10. In very small aneurysms it is also not advisable to use coiling because the risk of perforation is high 6. Since both clipping and coiling are demanding in wide-necked intracranial aneurysms 1,6 alternative treatment options for patients with these kinds of aneurysms are desirable. Stent placement is such an alternative treatment. Stents are flexible cylindrical mesh tubes made of stainless steel or alloys, which are positioned at the aneurysm neck using a microcatheter. The insertion of a stent alters flow characteristics such as flow stasis, particle resident time, and wall shear stress, all of which are believed to affect thrombus formation inside the aneurysm sac 10. Furthermore, an amorphous layer of thrombus covers the stent wall, and this layer is progressively replaced by a smooth muscle layer of fibromuscular tissue. Endothelium grows into the stented part of the vessel wall, and neointimal formation is completed 10. The porosity of the stent seems to have a big influence on the way flow characteristics are changed. Alterations of vortex location, speed and reductions of aneurysmal flow and momentum exchange were found 12. Rhee et al. found similar results 10. The last method, stent placement in combination with coiling, can be used when stent placement alone is not sufficient. The coils can be placed in the aneurysm through the pores of the stent using a microcatheter, which does not have to be done simultaneously with the stent placement, but can also be performed at a later stage 6. Because stenting has the most promising prospective of the aforementioned methods, stent placement will be used as the treatment method in this study. Steiger et al. 7 found that the flow field of a steady flow was similar to the systolic field under pulsatile conditions. However, during the diastole, this flow field collapsed and flow reversal was found in some parts of the aneurysm. The most important difference between pulsatile and steady flow is that a pulsatile flow tends to lower the difference of the flow velocities between the parent vessel and the aneurysm. This leads to higher shear stress magnitude and a rapid change of the shear stress direction inside the aneurysm. Furthermore, Gonzalez et al. 13 had shown that the change in the direction of wall shear stress may be more important in rupture than the change in magnitude of wall shear stress. This is due to the fact that the intima is more susceptible to an oscillatory shear stress than unidirectional shear stress. Furthermore, the shear stress gradient is as important as the absolute stress level in determining intima response 1. 7

9 In a previous study of Groenendaal 14 the influence of stenting on possible thrombus formation was investigated by examining hemodynamic parameters. All simulations were performed using a steady flow. In this study the influence of stenting is investigated by examining the same hemodynamic parameters Groenendaal 14 investigated; the aneurysmal inflow, velocity field, wall shear stress and vortex strength. These parameters are investigated in a twodimensional computational fluid dynamics model under pulsatile flow conditions. The parameters are used to quantify the influence of the density of the wire and the thickness of the wire. 8

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11 2. Methods Computational simulations are performed to analyze the intra-aneurysmal flow. The finite element package SEPRAN 15 is used to derive approximate solutions of the Navier- Stokes equations. Firstly, the geometry of the artery, aneurysm and stent and the boundary conditions of the flow problem are defined. The intra-aneurysmal inflow, mean absolute velocity, wall shear stress and vortex strength are analyzed in order to examine the influence of stent placement. 2.1 Governing equations The Navier-Stokes equations can be divided into two equations, namely the balance of momentum and the balance of mass. Assuming the fluid follows a purely Newtonian law and the fluid is incompressible the general equations that have to be solved are 16,17,18 : r u r r r r 2r 1 r + u u ν u + p = 0 (2.1) t ρ r r u = 0 (2.2) with t the time, u r the velocity of the fluid, r the gradient, ν the kinematic viscosity, ρ the density of the fluid and p the pressure. It is common practice to rewrite the Navier-Stokes equations in their dimensionless form 16,18,19. The characteristic numbers, that follow from this dimensionless form, make it possible to compare different flow problems. After substituting the characteristic length scale L, time scale ω -1, and velocity V into (2.1) the non-dimensional Navier-Stokes equation reads 18 : ωl r u r r r ν r 2r r + ( u u ) u + p = 0 (2.3) V t VL from which two characteristic numbers can be distinguished; Strouhal and Reynolds number: ωl Sr = (2.4) V VL Re = (2.5) ν For a pulsatile flow case the Strouhal number has an important function. However, a more frequently used characteristic number instead of the Strouhal number is called the Womersley number 19. The Womersley number α can be calculated from both the Reynolds and Strouhal number: 2 α = Sr Re (2.6), so α = L ω (2.7) ν 10

12 Using the Womersley number it is possible to calculate the entrance length of a flow situation. The entrance length is defined as the length needed for the boundary layer to reach its final thickness 19. The inlet length is in the order of magnitude: L Re Le = O α 2 (2.8) 2.2 Geometry The same geometry is used as in the study of Groenendaal 14. The geometry of the aneurysm is based on average values of size for a middle cerebral aneurysm investigated by Parlea et al. 20. The diameter of the artery is 3.6 mm and the size of the neck is 3.4 mm (Figure 2.2a). The aneurysm is modeled as a circle, instead of using average values for height and diameter. This is done because of the average values for height 5.6 mm and diameter 6.1 mm 20, which makes it reasonable to approximate the aneurysm as a circle (Figure 2.2a). The mesh is shown in Figure 2.2b. Figure 2.2 The geometry (left) and a close-up of the mesh (right) The width of the wire of the stent is 3.6 mm 21, which is slightly bigger than the width of the neck of the aneurysm, so the stent is also covering a small part of the parent vessel (Figure 2.3). The stent is placed at the entrance of the aneurysm in such way that the stent is not positioned in the neck but only in the parent vessel. To simplify meshing of the stent, its wires are assumed to be square in cross section (Figure 2.3). The height of the wire is equal to mm in all simulations and the width is varied throughout the different simulations. Six different settings are used to perform the simulations. First, a simulation without stent is performed, which serves as a comparison for the simulations with stent. The other five settings involve a stent (Table 2.1). The width B and the spacing A are varied, resulting in 3 different values for the density B/(A+B). 11

13 Setting Wire thickness [mm] Spacing [mm] Density [%] Table 2.1 Settings Figure 2.3 Stent 2.3 Characteristic numbers Groenendaal 14 used a velocity of 97.2 mm/s, a diameter of the tube of 3.6 mm and a viscosity of 3.5 mm 2 /s 22. These values result in a Reynolds number of 100. Using the same Reynolds number as for the situation above ensures the same flow characteristics are used in the simulation. A healthy person has a heart rate of approximately 75 beats/min 23. Combined with the tube diameter and viscosity mentioned above this would result in a Womersley number of 5.4. The same viscosity is used as determined above, so the angular frequency can be determined: 2 ω = α ν = 0.291, (2.9) The inlet length can be determined using the values for L (section 2.2), α and Re using Equation 2.8. The inlet length is in the order of 12 mm, which is small enough for the boundary layer to reach its final thickness. 12

14 2.4 Boundary conditions There is a no slip condition on the rigid walls and stress free outflow has been assumed. The velocity profile at the entrance of the tube in the stationary situation is a Poiseuille profile: 2 ( ) v( y) = 4 y + y (2.10) With the values calculated in 2.3 it is possible to set up a velocity profile for the pulsatile situation. Again, a Poiseuille profile is used, now multiplied with a time dependent function: 2 v( y, t) = 4 y + y a cos bt + d, (2.11) ( )( ) in which a is the amplitude, b is the angular frequency (Eq. 2.9) and d is the equilibrium state (Figure 2.4). Figure 2.4 The graph of the time dependent function that is multiplied with the Poiseuille profile As can be seen in Figure 2.4, the velocity is normalized with the mean velocity, which means the equilibrium state d is set to one. With an amplitude a of one the velocity would vary between zero and two, giving a peak Reynolds number of 200. Filling in these values and the angular frequency stated in section 2.3 results in: 2 v( y, t) = 4 y + y cos 0.291t + 1 (2.12) ( )( ) 13

15 2.5 Data analysis Intra-aneurysmal flow can be determined by investigating parameters like inflow into the aneurysm, mean velocity, wall shear stress and vortex strength inside the aneurysm. The inflow into the aneurysm is the amount of fluid flowing in negative y-direction (Figure 2.2) into the aneurysm. It is assumed that this flow measure decreases by placement of a stent. Then flow is calculated by the following formula: φ = u l, (2.13) e y, e e with φ as the flow into the aneurysm, u y, e the average velocity in negative y-direction of the e th element, and l e the length of the e th element. A way to quantify the stagnant intra-aneurysmal flow is to calculate the mean absolute velocity u inside the aneurysm 24. The mean absolute velocity is calculated as follows: ui i u =, (2.14) N u = u r, (2.15) i i where u r i is the velocity in node i and u i the velocity magnitude in node i. This is also considered a measure for the energy inside the aneurysm. Another parameter that is widely used in numerical analysis of intra-aneurysmal flow is the wall shear stress 1,7,10. The wall shear stress is related to arterial wall remodeling and is believed to affect thrombus formation. It may be an important indicator of aneurysm rupture and is included in this research for those reasons. The wall shear stress is calculated according to the following formulas: t r w = t r t r, (2.16) n t r = σ n r, (2.17) where t r w is the wall shear stress vector, t r n the normal component of the stress vector, t r the stress vector, σ the Cauchy stress tensor and n r the normal vector. To calculate the normal between two nodes of the aneurysm, the boundary between the two nodes is assumed to be a straight line. This assumption can be made because the nodes are close together. The wall shear stress is calculated at five different locations (Figure 2.5). These locations are chosen so not only the magnitude change of the wall shear stress because of the stent can be seen, but also the change of the distribution of the wall shear stress can be investigated. 14

16 Figure 2.5 Locations at which the wall shear stress is calculated It is also possible to quantify the flow inside the aneurysm by calculating the vortex strength. The vortex strength is calculated using 25 : Γ = u nda = ω da z, (2.18) A with Γ as the vortex strength, d l as the element length, da as the element surface and ω z the z-component of the rotation vector. In the two-dimensional form the normal is in z-direction and therefore the equation can be reduced to ω da z (Eq 2.18). The integral is approximated with a sum over all elements. The advantage of the vortex strength is that the force inside the aneurysm generated by the flow is characterized with one single number, on the condition that the vortex structure does not change. In order to characterize the flow inside the aneurysm, the vortex strength of the whole aneurysm is calculated. The neck is not taken into account as shown in Figure 2.6, because there are a lot of small vortices in the upper part of the neck because of the stent. Taking the stent into account would have caused a loss of information about the vortex strength in the aneurysm. Figure 2.6 The black line indicating from which point the vortex strength is calculated 15

17 3. Results The different hemodynamic parameters mentioned in Section 2.5 are analyzed in this Chapter. First, aneurysmal inflow (p13) is discussed followed by mean absolute velocity (p18), wall shear stress (p20) and vortex strength (p24). The results of the steady flow simulation are discussed first with the non-stented situation followed by the stented situation. Afterwards, the results of the pulsatile flow simulation are discussed in similar way and compared to the results of the steady flow simulation. 3.1 Inflow Steady flow In the non-stented situation the inflow is situated at the distal part of the aneurysm neck and outflow is seen at the proximal part of the neck (Figure 3.1a and 3.1b). Figure 3.1 Velocity field in the artery and the aneurysm (a,c) and the velocity in y-direction in the aneurysm neck (b,d) in the non-stented (a,b) and 25% stented (c,d) situation 16

18 In the stented situation the locations of inflow and outflow are reversed (Figure 3.1b versus 3.1d). Furthermore, the flow velocities inside the aneurysm in the stented situation are lower than in the non-stented situation, indicated by the velocity scales in both Figure 3.1a and 3.1c. Furthermore, the location of the vortex centre is changed from approximately y = -0.3 in Figure 3.3 to approximately y = 0 in Figure 3.3. In x-direction there is no change in vortex centre location. The aforementioned changes are observed in all stented simulations. The placement of a stent inside the vessel results in a reduction of the inflow into the aneurysm in all cases. A comparison between the total inflow in the different simulations can be seen in Figure ,5 2 Inflow [mm^3/s] 1,5 1 0, Simulation Figure 3.2 Inflow for the different simulations From the results in Figure 3.2 it is possible to calculate the factor by which the inflow is decreased by comparing the values of the stented situation to the inflow in the nonstented simulation (Table 3.1). Simulation Factor of decrease Table 3.1 The factor of decrease of inflow for the different simulations It can be seen from both Figure 3.2 and Table 3.1 that increased density leads to a decrease of inflow. Furthermore, it is clear from simulation 3, 4 and 5 (See Table 2.1 for explanation of the different simulations) that increased spacing of the stent leads to an increased inflow. A stent with a density of 25% has the least effect on the inflow, while a stent with a 75% density leads to a decrease of the inflow into the aneurysm that is 34 times bigger than in the non-stented simulation. 17

19 18 Figure 3.3 Velocity field at different moments of the puls. Upper left t 0.1 s, upper right t 0.2 s, etc. till t 0.7 s (left)

20 Pulsatile flow In the non-stented situation the inflow is located at the proximal part of the aneurysm neck and the outflow at the distal part for low velocities (Figure 3.3a and 3.3g). However, when the velocity increases with time the location of inflow changes to the distal part of the aneurysm neck (Figure 3.3b-f), as is also seen in the steady flow case (Figure 3.1a). Furthermore, a change in vortex centre location can be distinguished. At low velocities the centre is located in the middle of the aneurysm in the x-direction and at approximately y = -0.5 (Figure 3.3a). When the velocities are higher, like at t = 0.4, the centre is located more to the distal part of the aneurysm and closer to the parent vessel (Figure 3.3d). The change of inflow over time can be seen in figure 3.4. It can be seen that there are two minima present at time point t = 0.13 s and t = 0.67 s, which correspond to the change in inflow location that can be seen in Figure 3.3 (between a and b, and c and d) ,03 0,06 0,1 0,13 0,16 0,19 0,22 0,26 0,29 0,32 0,35 0,38 0,42 0,45 0,48 0,51 0,54 0,58 0,61 Inflow [mm^3/s] 0,64 0,67 0,7 0,74 0,77 Time [s] Figure 3.4 The change of inflow over one complete pulse in the non-stented situation In the stented situation the inflow is, just like in the steady flow situation, located at the proximal part of the aneurysm. The outflow is located at the distal part of the aneurysm. Over time, there is no difference of in- and outflow region like observed in the nonstented situation. The fluid can not enter the aneurysm at low velocities and starts entering the aneurysm at higher velocities at the distal part of the aneurysm neck. Furthermore, the profile of the inflow over time does not show the two minima as observed in the non-stented situation. Furthermore, the location of the centre of the vortex does not seem to change and the value of inflow is in all stented situations lower than in the non-stented situation. A comparison of the inflow in all different simulations can be seen in figure 3.5. All values shown are the values of inflow at t = 0.2 s, because the magnitude of the steady flow and pulsatile flow were similar at this time (See Figure 2.4), and at t = 0.4 s, when the velocity in the pulsatile situation reaches its maximum. 19

21 7 6 Inflow [mm^3/s] Steady Pulsatile [t=0.2 s] Pulsatile [t=0.4 s] Simulation Figure 3.5 Inflow for the different simulations Figure 3.5 shows that the trend of all three situations is similar. Furthermore are the values of inflow in the steady and pulsatile situation at t = 0.2 s of the same order of magnitude. The non-stented situation has the highest inflow value followed by simulation 2, 5, 4, 3 and 6, respectively. From the results in Figure 3.5 it is possible to calculate the factor by which the inflow is decreased by comparing the values of the stented situation to the inflow in the nonstented simulation (Table 3.2). Simulation Steady Pulsatile t = 0.2 s Pulsatile t = 0.4 s Table 3.2 The factor of decrease of inflow for the different simulations As can be seen from Table 3.2 both the steady and pulsatile situation at t = 0.2 s show a factor of decrease that is in the same order of magnitude for the different simulations. At t = 0.4 s the decrease factor is slightly higher compared to the factor of decrease at t = 0.2 s. 20

22 3.2 Mean velocity Steady flow The mean absolute velocities in the aneurysm for different simulations are shown in Figure ,2 1 Mean velocity [mm/s] 0,8 0,6 0,4 0,2 O(10-2 ) O(10-1 ) O(10-1 ) O(10-3 ) Simulation Figure 3.6 The mean velocity for the different simulations This graph follows the same trend as was shown in Figure 3.5, with the values of mean velocity of the stented simulations considerably lower than in the non-stented simulation. Of the stented simulations, simulation 2 with the 25% density has the highest value of mean velocity. Furthermore, the 50% density stent simulations 3,4 and 5 show increasing mean velocity with increasing spacing. The 75% stent has the lowest mean velocity of all the simulations. Just like for the inflow a factor of decrease can be calculated for the different simulations for the mean velocities (Table 3.3). Simulation Factor of decrease Table 3.3 The factor of decrease for the different simulations for the mean velocity 21

23 Pulsatile flow The mean velocities in the aneurysm for different simulations are shown in Figure 3.7. The values were calculated at t = 0.2 s and t = 0.4 s for the same reasons as mentioned in the inflow section. 2,5 Mean absolute velocity [mm/s] 2 1,5 1 0,5 O(10-2 ) O(10-1 ) O(10-1 ) O(10-3 ) Steady Pulsatile [t=0.2 s] Pulsatile [t=0.4 s] Simulation Figure 3.7 Mean velocity for the different simulations Again, the trend of the tree situations is the same and the order of magnitude for the mean velocity in the steady situation and the pulsatile situation at t = 0.2 s are similar. The factor of decrease for the different simulations for the mean velocities is shown in Table 3.4. Simulation Steady Pulsatile t = 0.2 s Pulsatile t = 0.4 s Table 3.4 The factor of decrease for the different simulations for the mean velocity As expected from Figure 3.7 the factors of decrease in the pulsatile situation at t = 0.2 s are of the same order of magnitude than the ones in the steady flow simulations. In contrast to the inflow the factor of decrease of the mean absolute velocity in the pulsatile situation at t = 0.4 s is lower than the decrease factor of the steady situation. 22

24 3.3 Wall shear stress Steady flow The stress in the non-stented situation is higher at the distal part of the aneurysm compared to the proximal part of the aneurysm (Figure 3.8a). This can also be seen in simulation 2 with a 25% density stent and simulation 5 with a 50% density stent with a spacing of mm and a wire thickness of mm (Figure 3.8b,e), which both have the highest spacing of mm. In the other stented situations, which have a medium (0.216 mm) or low spacing (0.108 mm), there is no clear distinction possible between the proximal and distal part of the aneurysm, indicating that the stent evens out the values of the wall shear stress (Figure 3.8a,c,d,f). Furthermore, it can be seen that the values of the stress in the stented situation are considerably lower than in the non-stented situation. The wall shear stress is the highest in the non-stented situation and the lowest in the 75% stent situation, which was to be expected. Pulsatile flow The wall shear stress is higher at the distal part than at the proximal part of the aneurysm in the same simulations as seen in the steady situation (Figure 3.8). Both in the stented and non-stented simulations the distributions are the same as in the steady flow case. 23

25 No stent 140 Wall shear stress [10^-3 Pa] Steady Pulsatile [t=0.2 s] Pulsatile [t=0.4 s] Location (a) Stent 25% 7 Wall shear stress [10^-3 Pa] Steady Pulsatile [t=0.2 s] Pulsatile [t=0.4 s] Location (b) 24

26 Stent 50% Thickness: low Spacing: low Wall shear stress [10^-3 Pa] 1 0,9 0,8 0,7 0,6 0,5 0,4 0,3 0,2 0,1 Steady Pulsatile [t=0.2 s] Pulsatile [t=0.4 s] Location (c) Stent 50% Thickness: med Spacing: med 3 Wall shear stress [10^-3 Pa] 2,5 2 1,5 1 0,5 Steady Pulsatile [t=0.2 s] Pulsatile [t=0.4 s] Location (d) 25

27 Stent 50% Thickness: high Spacing: high Wall shear stress [10^-3 Pa] 4,5 4 3,5 3 2,5 2 1,5 1 0,5 Steady Pulsatile [t=0.2 s] Pulsatile [t=0.4 s] Location (e) Stent 75% Wall shear stress [10^-3 Pa] 0,5 0,45 0,4 0,35 0,3 0,25 0,2 0,15 0,1 0,05 Steady Pulsatile [t=0.2 s] Pulsatile [t=0.4 s] Location (f) Figure 3.8 Wall shear stress over the aneurysm wall in the different simulations 26

28 3.4 Vortex strength Steady flow The centre of the vortex is located at approximately y = -0.3 in the non-stented situation and at y = 0 in the stented simulations (Figure 3.1a and 3.1c, respectively). The vortex strength for the different simulations is shown in Figure Vortex strength [1/s] O(10-1 ) O(1) O(1) O(10-1 ) Simulation Figure 3.9 The vortex strength for the different simulations Figure 3.9 shows the same trend as Figure 3.2 and 3.6 from section 3.1 and 3.2, respectively. The vortex strength is the strongest in the non-stented situation. From the stented simulations, simulation 2 with the 25% density stent has the highest vortex strength. Of the three 50% stents, the vortex strength increases with increased spacing. The vortex strength is the lowest for the stent with the smallest spacing and the thickest wire, simulation 6. The factors with which the vector strength is decreased are shown in Table 3.5. Simulation Factor of decrease Table 3.5 The factor of decrease for the different simulations for the vortex strength 27

29 Pulsatile flow The vortex strength increases with increasing velocity. In the non-stented situation there is seen a movement of the centre of the vortex over the cycle from the middle of the aneurysm towards the distal part of the aneurysm, while no shift occurs in all of the stented situations. The vortex strengths can be seen in Figure Vortex strength [1/s] O(10-1 ) O(1) O(1) O(10-1 ) Simulation Steady Pulsatile [t=0.2 s] Pulsatile [t=0.4 s] Figure 3.10 Vortex strength of the different simulations The trends of both Figure 3.9 and 3.10 are the same, with the same order of magnitude for the vortex strengths for the steady and pulsatile simulation at t = 0.2 s. Using Figure 3.10 the decrease factor of the vortex strength of the different stents is calculated. The results are shown in Table 3.6. Simulation Steady Pulsatile t = 0.2 s Pulsatile t = 0.4 s Table 3.6 The factor of decrease for the different simulations for the vortex strength As expected from Figure 3.10 also the factors of decrease in the pulsatile simulation at t = 0.2 s are of the same order of magnitude than the ones in the steady flow simulations. 28

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31 4. Discussion Various endovascular techniques have been introduced for the treatment of aneurysms in the last decade. Stenting has been used to treat wide necked aneurysms. Even though characteristic hemodynamic parameters cannot fully explain the complicated embolization process, they can provide a favorable environment for aneurysm embolization. Therefore, hemodynamic changes caused by the insertion of a stent in both a steady and a pulsatile flow case are examined. This is done using a two-dimensional computational fluid dynamics model to clarify the effects of stent porosity in a saccular aneurysm. Hemodynamic parameters The non-stented simulation shows in both flow cases an inflow motion along the distal side of the aneurysm neck and a vortex formation inside the aneurysm. Furthermore, the vortex moves towards the distal part of the aneurysm when the velocity increases. Figure 3.4 shows two minima of the inflow over one pulse. An explanation can be that these minima occur because of the change of inflow location. At both time points at which the minima occur the flow is almost horizontal in the neck of the aneurysm (Figure 3.3b and 3.3f), while before and after these time points the flow tends to be in a more vertical direction (Figure 3.3c,d and e). This will lead to a higher inflow value for the latter compared to the former. The stented models show an outflow at the distal wall, and inflow is observed along the pores of the stent at the proximal part of the aneurysm neck. The intra-aneurysmal flow is significantly reduced in the aneurysm models that have a stent. The decreased inflow into the aneurysm and the slow vortex motion causes the aneurysmal flow to decline and the particle resident time to increase, which facilitates thrombus formation 10,12. Stent porosity is an important parameter, but other stent characteristics, such as wire size and shape of the opening may also be important for stent characterization 10. Considering the effects of porosity, less porous stents reduce intraaneurysmal flow further than stents that have a higher porosity. However, less porous stents are rarely used because of their reduced flexibility. Therefore, the stent porosity should be chosen that results in intra-aneurysmal flow reduction, but also has a stent flexibility that makes clinical use easy. The values of wall shear stress over most of the aneurysmal pouch are smaller than the wall shear stress at regions close to the proximal and distal part of the aneurysm neck. This is seen in both flow cases. Furthermore, placement of a stent lowers the wall shear stress significantly in both the steady and pulsatile flow case. Together with the aforementioned changes this suggest a more favorable environment for thrombus formation 26,27,28. The wall shear stress has a pulsatile character, due to the pulsatile character of the flow. According Gonzalez et al 13 the intima may be much more susceptible to oscillatory shear stress, which is the case in the pulsatile flow case and therefore also seen in this study, than unidirectional shear stress as seen in the steady flow case. When a stent is inserted the vortex did not disappear completely, but the vortex strength is significantly lower and the direction of the vortex changes in both the steady and pulsatile flow case compared to the non-stented situation. Furthermore, the change of 30

32 location of the vortex centre seen in the non-stented situation did not occur after stent placement, due to the strongly diminished driving flow at the location of the stent. The intra-aneurysmal flow is quantified by calculating the mean velocity and vortex strength inside the aneurysm. This velocity is reduced in the models with stent compared to the model without stent and decreases further when the density is increased. According to Groenendaal 14 the velocity is also decreased when the stent was meshed using more elements. Improvements Although blood is a non-newtonian fluid, in this research blood is modeled as a Newtonian fluid. This can be done, because the influence of low shear rates on the viscosity of blood is small for tubes with a diameter larger than 0.5 mm. If this is the case the difference between simulations with Newtonian and non-newtonian flow is about 1-2% 29. It was also investigated that there are only small differences in the basic flow characteristics and flow phenomena remained virtually unchanged 30. However, the pores of the stents are considerably smaller than 0.5 mm, so this might have an influence on the outcome of this study and is something to consider in future studies. Simulations are performed with rigid walls. Löw et al. 31 investigated the difference between rigid and distensible saccular aneurysms 32,33. The basic flow patterns are the same in both models. Furthermore, cerebral arteries are considered to be less distensible than large systemic arteries 34,35, hence the role of the compliance in augmenting flow patterns is reduced. The values used in this study for the Reynolds number and also the Womersley number are not physiological when investigating a cerebral artery. Therefore simulations should also be performed using a Reynolds number of 200 and a Womersley number of 2.7, which are more physiological numbers 10,19. Furthermore, the time dependent function by which the Poiseuille flow is multiplied is not a physiological one, so this might be improved in future studies. The in- and outflow regions seen in this study are strongly influenced by the geometry of the parent vessel. If the aneurysm is, for example, located at the outer side of a curved vessel, it could be that the inflow is located at the distal part of the aneurysm neck and outflow at the proximal part of the aneurysm neck. Therefore, simulations with different vessel geometries are advisable for future studies. Conclusion The results show that the intra-aneurysmal flow, the wall shear stress, the mean velocity and the vortex strength are reduced due to the stent insertion. However, the effects of low flow rates and low wall shear stress on an in vivo aneurysm embolization should be studied further because the biological responses of the arterial wall, neointimal formation on the stent wall and an excessive thrombus formation are important for the success of intravascular stenting

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