The application of the screen model for stents in cerebral aneurysms

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Article

Reference

The application of the screen-model based approach for stents in cerebral aneurysms

LI, Sha, LATT, Jonas, CHOPARD, Bastien

Abstract

Numerical modeling of the blood flow through flow diverters in intracranial aneurysms is a crucial tool to personalize and optimize endovascular stenting procedures in the treatment of aneurysms. However, large computational meshes are needed to resolve the difference of scale between the struts of a flow diverter and the blood vessel leads, leading to a high computational cost. This article presents a screen-model based approach, a heuristic continuum method which can effectively reduce the computation time by avoiding to fully resolve the stent. We study the performance of the screen-model based stent model under various conditions, such as different stent placements inside an artery, varying homogeneous porosities, and pulsatile inlet velocities. The results show that the stent model can effectively prevent the blood from flowing into the aneurysm as a fully resolved stent does, but at a coarser resolution. The application range of the stent model is tested, and the validity of the model is assessed.

LI, Sha, LATT, Jonas, CHOPARD, Bastien. The application of the screen-model based approach for stents in cerebral aneurysms. Computers & Fluids, 2018, vol. 172, p. 651-660

DOI : 10.1016/j.compfluid.2018.02.007

Available at:

http://archive-ouverte.unige.ch/unige:104548

Disclaimer: layout of this document may differ from the published version.

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The application of the screen model for stents in cerebral aneurysms

Sha Li, Jonas Latt, Bastien Chopard

Department of Computer Science, University of Geneva, Carouge, 1227, Switzerland

Abstract

Modeling the blood flow through flow diverters in intracranial aneurysms is a crucial tool to personalize and optimize endovascular stenting procedures in the treatment of aneurysms. However, the large difference of the scale between the struts of flow diverter and the aneurysm leads to a massive number of meshes and thus an extremely expensive computational cost. Instead of fully resolving the stent, the screen model, as a heuristic continuum method, can effectively reduce the computation time. In this paper, we studied the performance of the screen model on various conditions, such as different stent implanted locations, homogeneous porosities and pulsatile inlet velocities. The results show that the screen model can effectively prevent the blood from flowing into the aneurysm as a fully resolved stent does, but at a coarser resolution. Through the work of this article, we tested the application range of the screen model and validated its effectiveness.

Keywords:

aenurysm, stent, screen model, CFD, lattice Boltzmann 1. Introduction

Flow diverters or stents have been widely adopted for the treatment of intracranial aneurysms which is a dilation of an artery that has a risk of rupture [1]. Numerical simulation provides a crucial way to acknowledge the complex nature of the aneuris- mal flow and thus to optimize the design of the flow diverters [2, 3]. The struts which compose stents are rather dense and fine. The diameter of strut is about 30 ∼ 50 µm [1], even can differ from the diameter of the artery by more than two orders of magnitude, so the computation cost is prohibitively expensive.

Corresponding author

Email address: sha.li@unige.ch(Sha Li)

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To solve this problem, Augsburger et al. [1] proposed an alternative strategy, which models the device as a porous medium. An addition of the momentum source term to the standard fluid flow equations is taken as a representation of the fully resolved stent under coarser mesh. Afterwards, Raschiet al.[2] developed the porous medium model to a more general application. The parameters of Augsburger’s model only could be used on the specific stent that he employed in the experiment. While in Raschi’s model, the parameters are adjusted according to the geometries of stents, and can be applied to any stent designs. In this framework, Li et al. [6], inspired by the porous medium model, presented a screen model according to the methodol- ogy of the hydrodynamics of screen. The screen model provides formulas for drag coefficients for both perpendicular and inclined screen, and also for the tangential force which causes the flow deviation. Besides, this model manages to compute the body force from the local and instantaneous velocity. Compared with Raschi’s model which assumes that the drag coefficients are the same for all directions, the screen model presents a more accurate way to compute the tangential force.

The aim of the present article is to test the applicability of the screen model in situations not explored in [6]. An artery with two outlets in 2-dimension (2D) is employed in the numerical experiments. Different stent implantation locations and porosities are simulated in this paper. Given that the blood flow is pulsatile, Womersley flow is tested as the inlet velocity. An comparison to Raschi’s model is carried out.

2. Force model

2.1. Raschi’s model

Raschi’s model [2] utilizes a porous medium equation (Forchheimer’s law) to recreate the pressure drop of blood flow through a flow diverter:

0 = δp δxi − µ

κui−Cρui|ui|, (1)

with p the pressure in the porous medium, κ the permeability coefficient, which is assumed to be the same for each direction i, and C the Forchheimer coefficient.

Idelchik’s equation [5] for the drag coefficient of screen is adopted by Raschi’s model:





∆P = 12Kρu2 K = Re22

h + 1.3(1−β) + (β1 −1)2 Reh = ρudµh

(2)

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where Reh is the Reynolds number based on hydraulic diameter dh which is defined as four times the ratio of the connected void volume to the wetted surface area Aw.

dh = 4Vvoid

Aw (3)

Sinceκfrom Eq. 1 is assumed to be the same for all directions andCis a constant, K in Eq. 2 should also be the same for each direction.

2.2. The screen model

Distinct from the isotropic assumption of Raschi’s model, the screen force model [6]

deals with the normal force (the direction which is normal to the surface of the stent) and the tangential force in different ways. The normal force, namely the pressure drop across the stent, is computed through drag coefficient which depends on the Reynolds number, porosity and incident angle:









∆P = 12knρu2

kn =k0(Rend) or kn =k0(Rend)1+secθ

2

k0 =

12.25(1−β) + 41.321−β

β

2

1 Red +

1.41(1−β) + 1.111−β

β

2 ,

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wherek0 andknare the drag coefficients of the stent: k0 refers to the stent whose surface is normal to the flow direction, while kn refers to the stent whose normal direction deviates from the flow at an arbitrary angelθ, andβ is the porosity of the stent.

(Red= U dν

Rend = Uνn (5)

whered is the diameter of the strut of the stent. U is the speed of fluid, while Un is the speed of the norm of the normal component of the velocity. The superscript n of k and Red in Eq. 4 and Eq. 5 denotes the normal component.

The tangential force caused by the flow on the stent is calculated by the deflection coefficient B which represents the deviation of the flow velocity. According to the experiments done by Schubauer et al. [7], B is a function of the drag coefficient kn. The main limitation of this methodology is that the forces are computed from a uniform and constant upstream flow in front of the screen; condition that can not be met in a real situation with the stent deployed in aneurysm for the blood flow orientation is always affected by the artery wall. Therefore, a reduction coefficient

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r, defined as the ratio between the local velocity and the upstream velocity in the tangential direction, is introduced to overcome this shortcoming:

r =v/v1 (6)

Then the tangential stress τ can be computed as









τ =Buv/r B = 1 + 3.4kn −h

kn 3.4

3

+ 0.797i1/3

r = 1 +aRend−h

(aRend)32 + 1i23 ,

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where

a= 7.68β2−8.842β+ 2.734

For a detailed description of the screen model, one could refer to [6].

3. Validation

3.1. geometry

We applied the screen force model introduced in Sec. 2 on an artificial artery with aneurysm in 2D, as shown in Fig. 1. The artery includes one inlet and two outlets.

A stent is implanted into the artery in a way which could cause the blood to flow toward the right outlet.

Figure 1: The artery with two outlets.

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The diameter of the artery isD= 3.6mm. The diameter of struts which compose the stent isd= 40µm. The porosity of the stent is 0.6. The density and viscosity of the fluid areρ= 1000kg/m3 andµ= 3.3×10−3P a·s. An analytic velocity profile of a Poiseuille flow is applied as the inlet boundary condition. The outlets implement pressure boundary condition. The Reynolds number is Re = ρumaxD/µ= 400, and Red =ρumaxd/µ = 4.44, whereumax is the maximum velocity at the inlet boundary.

Thus if we choose the average velocity as the characteristic velocity, the Reynolds number would be equal to half of that based on maximum velocity, which is 200 and 2.22 for Red.

The numerical simulations are launched with the help of the open-scource fluid solver Palabos 1 based on lattice Boltzmann method (see for instance Chen and Doolen [8]).

3.2. Resolution study

3.2.1. Resolution for the fully resolved stent

In order to choose a proper resolution for the simulation of the fully resolved stent, three resolutions are tested. The nodes for the diameter of the artery and the diameter of the strut are shown in Tab. 1. The diameter of the artery is the characteristic length for the resolution. From the table, we see that the diameter of the strut is 2, which means that the radius is just 1 node, so the resolution 180 is the minimum resolution for the fully resolved stent simulation.

Nodes Resolution 1 Resolution 2 Resolution 3

artery 540 360 180

strut 6 4 2

Table 1: Nodes for the diameters of artery and strut of three different resolutions

The velocity fields of the whole artery of the three resolutions are shown in Fig. 2.

There is no visible difference between the three simulations. Given the fact that the whole domain is too large for the observation of the velocity in aneurysm, we draw figures of the local velocity field over the region of the aneurysm in Fig. 3. Through the figure, we can distinguish that at the connection corner between the aneurysm and artery, the velocity of resolution 180 is smaller than the velocities of resolution 360 and 540.

For a quantitative comparison of the results, the root mean square (RMS) values of the velocities inside the aneurysm and the root mean square deviation (RMSD) of

1http://www.palabos.org

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(a) resolution 540 (b) resolution 360 (c) resolution 180

Figure 2: Velocity for the fully resolved simulation at different resolutions (β= 0.6). The units for the position and velocity aremmandm/s, respectively.

(a) resolution 540 (b) resolution 360 (c) resolution 180

Figure 3: Velocity over the region of the aneurysm for the fully resolved simulation at different resolutions (β = 0.6). The units for the position and velocity aremmand m/s, respectively.

the velocity of resolution 360 and 180 are computed and shown in Tab. 2. The RMS of the velocity of resolution 360 is similar to the velocity of resolution 540, while the RMS of resolution 180 shows a large difference, so does its RMSD. Therefore, resolution 180 is too coarse for the simulation of the fully resolved stent. Taking into account the computation cost, 360 is chosen as the option for the simulation of the fully resolved stent.

Resolution 540 360 180

RMS (m/s) 5.6548×10−3 5.4894×10−3 3.2019×10−3 RMSD (m/s) 1.7318×10−4 2.4177×10−3

Table 2: RMS and RMSD values of the velocity field for the three resolutions of the fully resolved stent simulation.

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3.2.2. Resolution for the force model

After choosing the proper resolution for the simulation of the fully resolved stent, we continue to test the resolutions for the force model. Since the force model does not need to have a real stent geometry, its resolution could be less than the resolution of the fully resolved stent. Therefore, the resolutions we chose to test are 1/2, 1/4 and 1/8 of the selected resolution (360) of the fully resolved stent simulation. They are equal to 180, 90 and 45 respectively. Fig. 4 shows the local velocity in the aneurysm obtained from the simulation at the three resolutions. The velocity from the simulation at three resolutions are similar to each other. Table 3 shows the RMS values of the velocity inside the aneurysm for each resolution case. Resolution 45 can reproduce the velocity at the resolution 180. The results indicate that we can reduce the resolution for the force model to 1/8 of the fully resolved stent without losing accuracy in velocity. However, compared with resolution 180 and 90, the quality of the figure of resolution 45 is coarse, especially at the dome part and the nodes where the forces are applied. As a result of balancing between accuracy and the computation cost, we choose 90 as the resolution for the simulation of the force models in the following part.

(a) resolution 180 (b) resolution 90 (c) resolution 45

Figure 4: Velocity obtained from the simulation using the screen model at different resolutions.

The units for the position and velocity aremmandm/s, respectively.

Resolution 180 90 45

RMS (m/s) 8.0484×10−3 8.0101×10−3 8.0386×10−3

Table 3: RMS of the velocity field from the simulation at the three resolutions using the screen model.

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3.2.3. Validation of the force models

In order to investigate the performance of the screen model [6], four cases are simulated: the anuerysm without stent, the aneurysm with a fully resolved stent, the screen force model [6] and the force model of Raschi [2]. Raschi’s model has the same resolution as the screen model.

Figure 5 shows the velocities inside the aneurysm of the four cases. Compared with the case without stent, the fully resolved stent and the two force models reduced the velocity inside the aneurysm. The screen model shows a good consistency with the fully resolved stent simulation, while the velocity of Raschi’s model is higher than the fully resolved stent simulaiton.

(a) without stent (b) fully resolved stent

(c) screen model (d) Raschi’s model

Figure 5: Velocity over the region of the aneurysm of stent 1 (β= 0.6). The units for the position and velocity aremmandm/s, respectively.

The wall shear stress(WSS) is measured at the points located at the internal side of the aneurysm wall. Figure 6a illustrates WSS on the wall of aneurysm of the four cases and Fig. 6b shows the defination of θ used in Fig. 6a. The WSS of the aneurysm without stent is much higher than the fully resolved stent and the two force models. The WSS obtained from both the screen model and Raschi’s model are a little higher than the fully resolved stent.

For a quantitative resolution, we computed the RMS of the velocity inside the aneurysm for the four cases. We also computed the RMSD of the screen model and the Raschi’s model compared with the fully resolved stent simulation and their errors. These results are summarised in Tab. 4 together with the WSS averaged over all the measured points. The fully resolved stent shows a significant resistance by reducing the RMS velocity to about 1/18 of the velocity from the simulation without stent. The reduction from the screen model and Raschi’s model is about 1/12 and 1/7, respectively. Compared with the RMS velocity of the fully resolved simulation, the error of the force model is 47.29%, and Raschi’s model, 191.68%. Both force

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(a) Wss on the wall of aneurysm (b) Definition ofθfor the WSS measurement.

Figure 6: WSS on the wall of aneurysm of stent 1 (β= 0.6)

models have higher RMS velocity, which means that the drag forces computed by the equations of the models are not big enough to represent the real stent. However, although having some error compared to the fully resolved stent, the screen model poses an improvement over with Raschi’s model. The values of WSS show similar results as the velocity as compared between the simulated cases.

without stent fully resolved stent screen model Raschi’s model RMS(m/s) 9.9371×10−2 5.4894×10−3 8.0101×10−3 1.5923×10−2

RMSD(m/s) 2.5960×10−3 1.0522×10−2

Error(%) 47.29 191.68

WSS(Pa) 0.5574 0.0417 0.0576 0.1024

Table 4: Comparison of the results obtained from the four simulation cases (β = 0.6).

The run time of the fully resolved simulation is 21.9 times of the run time of the screen model, and 24.7 times of the Raschi’s model. Therefore, both the force models significantly reduced the computational cost.

4. Test the screen model with various conditions

4.1. Different implanted location

In order to test the performance of the screen model, we implant the stent in another location which would cause the blood to flow preferentially towards the left

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outlet as shown in Fig. 7. The stent employed in Sec. 3 is denoted as “stent 1”, and the stent employed here is denoted as “stent 2”. The corner between the vertical channel and the left outlet channel is rounded in Fig. 7 to avoid pressure oscillation.

Figure 7: Two outlets: stent 2

The velocity fields of the whole artery are shown in Fig. 8. The blood flows out of the artery through the right outlet in the simulation without stent. Stent 2 changed the flow direction as seen in Fig. 8b, and the two force models also succeed in presenting this phenomenon as shwon in Figs. 8c, 8d. Since the orientation of stent 1 is consistent with the original flow as in the simulation without stent, it could be taken as being implemented in a shear flow. While the stent 2 deflects the flow direction, thus it could be taken as being implemented in an inertia flow.

Furthermore, comparing the velocity fields from the simulations, we can distin- guish that the velocity of Raschi’s model is higher than that of the fully resolved stent in the right channel of the artery, while the screen model is slight higher than the fully resolved stent simulation. The local velocity inside the aneurysm is also drawn in Fig. 9. The figure shows that the velocity of the screen model is slightly higher than the velocity of the fully resolved stent, while Raschi’s model shows an apparent difference.

The wall shear stress produced with stent 2 is shown in Fig. 10. The WSSs obtained from the screen model is close to the WSS of the fully resolved stent simu- lation, but Raschi’s model is distinctively higher.

Table 5 summarizes the quantitative results. The error of the RMS velocity of the screen model is 25.63%, compared to 129.9% obtained from Raschi’s model.

The screen model is more accurate than Raschi’s model to approximate the results obtained from the fully resolved stent.

The run time of the fully resolved simulation is 24.2 times of both the run times of the screen model and Raschi’s model.

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(a) without stent (b) fully resolved stent

(c) screen model (d) Raschi’s model

Figure 8: Velocity field with stent 2 (β = 0.6). The units for the position and velocity aremmand m/s, respectively.

(a) without stent (b) fully resolved stent

(c) screen model (d) Raschi’s model

Figure 9: Velocity over the region of the aneurysm with stent 2 (β = 0.6). The units for the position and velocity aremmandm/s, respectively

Figure 10: WSS on the wall of aneurysm with stent 2 (β= 0.6).

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without stent fully resolved stent screen model Raschi’s model RMS(m/s) 9.9790×10−2 5.5142×10−3 6.7231×10−3 1.2555×10−2

RMSD(m/s) 1.4134×10−3 7.1629×10−2

Error(%) 25.63 129.90

WSS(Pa) 0.5747 0.0609 0.0866 0.1392

Table 5: Quantitative comparison for stent 2 withβ= 0.6

4.2. Different porosity

To validate the performance of the screen model when dealing with different porosities, both stents with a same porosity of 0.8 are tested. The diameter of the strut is kept the same, and the porosity is changed by enlarging the distance between the struts. Figure 11 and Fig. 12 compare the local velocities inside the aneurysm for the case with stent 1 and 2, respectively. For the case with stent 1, the velocity at the aneurysm dome with the fully resolved stent is smaller than with the two force models. Except this area, the velocity in all the other parts are similar to each other.

For the case with stent 2, the two force models are consistent with the fully resolved stent.

(a) without stent (b) fully resolved stent

(c) screen model (d) Raschi’s model

Figure 11: Velocity over the region of the aneurysm with stent 1 (β = 0.8). The units for the position and velocity aremmandm/s, respectively.

The WSSs of the four simulation cases are compared in Fig. 13a and Fig. 13b.

For stent 1, the two force models are close to that of the fully resolved stent. For stent 2, the results from the screen model seems lower than that of the Raschi’s model, but higher than that of the fully resolved stent.

A quantitative comparison is also done for the two stents of porosity 0.8. Table 6 and Table 7 illustrates that the RMS of the velocities of the fully resolved stent simulation for both stents are higher than the velocities of the stents of porosity 0.6.

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(a) without stent (b) fully resolved stent

(c) screen model (d) Raschi’s model

Figure 12: Velocity over the region of the aneurysm with stent 2 (β = 0.8). The units for the position and velocity aremmandm/s, respectively.

(a) Wss on the wall of aneurysm of stent 1 (b) Wss on the wall of aneurysm of stent 2 Figure 13: WSS on the wall of aneurysm (β= 0.8)

The errors of porosity 0.8 of the force models are smaller than the errors of porosity 0.6. Still, the screen model shows a better accuracy than Raschi’s model.

For stent 1 at porosity of 0.8, the simulation time of the screen model is 19.7% of the fully resolved simulation, while the ratio of Raschi’s model is 20.5%. For stent 2, the percentages ares 20.2% and 18.9%, corresponding to the screen model and Raschi’s model respectively.

4.3. Heterogeneity

For a real stent implanted in an artery, the bending of the stent always causes the nonuniform distribution of the struts, which we call heterogeneity. Heterogeneity

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without stent fully resolved stent screen model Raschi’s model RMS (m/s) 9.9371×10−2 2.7050×10−2 3.0648×10−2 3.6993×10−2

RMSD (m/s) 5.9275×10−3 1.4081×10−2

Error (%) 21.91 52.06

WSS (Pa) 0.5574 0.1475 0.1619 0.1849

Table 6: Results from the simulation using different models for stent 1 withβ= 0.8.

without stent fully resolved stent screen model Raschi’s model RMS(m/s) 9.9790×10−2 2.2217×10−2 2.2752×10−2 3.2236×10−2

RMSD(m/s) 2.5818×10−3 1.1945×10−2

Error(%) 11.62 53.77

WSS(Pa) 0.5747 0.1798 0.2111 0.2724

Table 7: Results from the simulation using different models for stent 2 withβ= 0.8.

makes the local porosity of the stent vary with the position. In order to simulate this phenomenon, and test the application range of the screen model, we carry out a simple numerical experiment on stent 1 by dividing it into two parts according to its radian. The porosity of the upper part of the stent is set to 0.6, and the lower part 0.8.

(a) without stent (b) fully resolved stent

(c) screen model (d) Raschi’s model

Figure 14: Velocity over the region of the aneurysm with the heterogeneous stent. The units for the position and velocity aremmandm/s, respectively

Fig.14 presents the local velocity inside the aneurysm. The velocity of the fully resolved stent shows clearly that the struts on the upper part of the stent are dis- tributed more densely, while the struts on the lower part are coarser. An evident stronger stream of flow occurs at the junction of the two parts. The screen model successfully simulates this phenomenon at the junction and presents a high similarity

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with the velocity field of the fully resolved stent. Fig. 15 shows that the WSS of the two force models have a good consistency with the fully resolved stent simulation.

Figure 15: Wss on aneurysm wall with an heterogeneous porosity

Tab. 8 compares the fully resolved stent simulation with the force model simula- tion quantitatively. The RMS velocity of the screen model is higher than the velocity of the fully resolved stent by 61.44%. The error of the Raschi’s model is 244.84%.

Both force models show larger errors than homogeneous porosities.

without stent fully resolved stent screen model Raschi’s model RMS(m/s) 9.9371×10−2 4.5774×10−3 7.3070×10−3 1.5699×10−2

RMSD(m/s) 2.8121×10−3 1.1207×10−2

Error(%) 61.44 244.84

WSS(Pa) 0.5574 0.0748 0.0863 0.1153

Table 8: RMS velocities in aneurysm with heterogeneous stents

The run time of the fully resolved simulation is 18.0 times of the run time of the screen model, and 19.0 times of the Raschi’s model.

4.4. Pulsatile flow: Womersley

A Womersley flow is a simple model of pulsatile flow in an artery. In order to test the applicability of the screen model under pulsatile flow, a two-dimensional Womersley flow is employed as the inlet velocity. The geometric configuration of the Womersley flow is the same as the Poiseuille flow, but is driven by a periodic

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pressure gradient[4, 9]:

∂p

∂x = Real[Aeiωt] (8)

A is the maximum value of the velocity in the middle of the channel and is set to A=−8/Re. ω is the frequency of variation of the pressure gradient.

Furthermore, parameter α is introduced as Womersley number and defined as α2 = Ly2ω

4ν (9)

The analytical Womersley profile is ux = Real[A

iωeiωt(1− cosh(√

2(y− 12)(α+ iα)) cosh(

2

2 (α+ iα)) )] (10)

whereyis the dimensionless coordinate characterized by the diameter of the channel.

Velocity and WSS were measured in four points located on the internal surface of the aneurysm wall (see Fig. 16). The Womersley number α is set to 1. A stent with porosity of 0.6 is adopted. All the other parameters are selected as before. Fig. 17 compares the norm of the velocities on the four measurement points respectively. The velocities of the screen model on the measurement points exhibit some differences from the fully resolved stent simulation. They are similar to the fully resolved case at point B and C, but higher at point A and point D. We record the velocity field in the aneurysm at three time points which are marked by the dashed lines in Fig. 17.

The time points are t1 = 5.8909s, t2 = 8.8364s. In Fig. 18, the screen model shows pretty similar velocity fields as for the fully resolved stent.

Figure 16: Measurement points on the aneurysm wall

Fig. 19 shows the WSSs on the four measurement points. Except for point D, the screen model are all larger than the fully resolved stent simulation, but smaller than Raschi’s model. But both the velocity in Fig. 17 and WSS in Fig. 19 show that

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(a) Point A (b) Point B

(c) Point C (d) Point D

Figure 17: Velocities at the measurement points

the screen model is not as stable as the fully resolved stent simulation and Raschi’s model.

Tab. 9 compares the RMS and RMSD velocity of the simulated cases. The error of the two force models on t1 are smaller than the error on t2 and it means that the error gets smaller with the increasing of the velocity. The screen model has a smaller error compared with Raschi’s model. We also find that both force models underestimate the drag force for the stent for almost all the simulation cases, so the equation of the model should be adjusted a little in the future.

5. Conclusion

We validate the screen model of [6] which is designed to replace a flow diverter through a continuum approach in coarse-grain simulation in the following situation:

the stent is placed in an artery with an aneurysm and two outlets. We study the performance of the screen model by testing different stent locations, porosities and

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(a) fully resolved stent(t1) (b) screen model(t1) (c) Raschi’s model(t1)

(d) fully resolved stent (t2) (e) screen model (t2) (f) Raschi’s model (t2)

Figure 18: Velocity over the region of the aneurysm of Womersley flow. The units for the position and velocity aremmandm/s, respectively

inlet velocities. A comparison between the screen model and Raschi’s model [2] is also carried out in this work.

In the comparison with Raschi’s model in Sec. 4.1, the screen model provides results similar to the fully resolved simulation in velocity and WSS for stent 1 in shear flow and stent 2 in inertia flow. It proves that the methodology of the tangential force based on the flow deviation in the screen model is more accurate than the isotropy assumption for the drag coefficients in Raschi’s model.

A different porosity is tested in Sec. 4.2. The two force models perform better at higher porosity β = 0.8 than at β = 0.6 in Sec. 4.1. Moreover, it is observed that the screen model is again more accurate than Raschi’s model, qualitatively and quantitatively.

Given that the local porosity of the flow diverter may vary with the position while the flow diverter is deployed into the artery, the screen model is validated with a heterogeneous stent in Sec. 4.3. A good consistency between the screen model and fully resolved stent is shown for the velocity field and WSS, which demonstrates that the screen model has the ability to handle the heterogeneous stent with a varying local porosity.

For the pulsatile inlet flow, the two force models perform not as well as in the constant velocity cases, but the velocity profiles figures show that the screen model

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(a) Point A (b) Point B

(c) Point C (d) Point D

Figure 19: WSS of the measurement points

still can represent the influence of real stent approximately. The error is smaller when the velocity is larger.

In summary, we have applied the screen model with different stent locations, porosities and velocities, thus further tested its applicability with the most common circumstances for flow diverter in aneurysm of a human artery. The screen model always gives convincing and reliable results as a proper representation of a fully resolved stent at a coarser-grained resolution. In the future, a 3D investigation of the force model will be carried out in order to make the patient-specific simulation possible. In the 3D case, the gain in CPU caused by the screen model, is even more important than it is in 2D.

Acknowledgement

We acknowledge partial funding and access to high performance computing re- sources from the CADMOS center (http://www.cadmos.org) and partial funding from the European Union Horizon 2020 research and innovation programme for

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t1 fully resolved stent screen model Raschi’s model RMS (m/s) 3.9446×10−3 5.5890×10−3 1.0251×10−2

RMSD (m/s) 1.6948×10−3 6.3553×10−3

Error (%) 30.32 161.11

t2 fully resolved stent screen model Raschi’s model RMS (m/s) 1.1599×10−3 2.0635×10−3 3.9020×10−3

RMSD (m/s) 9.8346×10−4 2.9031×10−3

Error (%) 84.79 250.30

Table 9: RMS and RMSD of velocities for the Womersley simulation

the CompBioMed project (http://www.compbiomed.eu/) under grant agreement 675451.

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