Resolution study

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 struts are shown in Tab.3.1. We use the diameter of the artery as a reference to define the resolution

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of the simulation. From the table, we can see that the coarsest acceptable mesh for a fully resolved simulation has a resolution of 180 cells across the diameter of the artery, in which case a cell strut has a diameter of just 2 cells.

Nodes Resolution 1 Resolution 2 Resolution 3

artery 540 360 180

strut 6 4 2

Table 3.1 Number of lattice nodes along the diameters of the artery and stent struts at three different resolutions.

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

Fig. 3.2 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

Fig. 3.3 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 aremmandm/s, respectively.

The velocity fields of the whole artery at the three resolutions are shown in Fig.3.2. There is no visible difference between the three simulations. For a better representation of the flow inside the aneurysm, Fig.3.3provides a representation of this region, with a rescaled velocity color map. It can be seen that at the connection corner between the aneurysm and artery, the velocity at a resolution 180 is smaller than the velocities at a resolution 360 and 540.

3.3 Validation 45 For a quantitative comparison of these results, the root mean square (RMS) values of the velocities inside the aneurysm and the root mean square deviation (RMSD) of the velocity at resolution 360 and 180 are computed and

The definitions of RMS and RMSD of the velocity are

 WhereNis the total number of mesh nodes contained in the aneurysm and refers to the number of nodes at coarser resolution in the formula of RMSD.u^{re f}_i is the reference velocity which refers to the velocity at resolution 540 for the RMSD computation of resolution 360 and 180.

The results are shown in Tab.3.2. Both the RMS and RMSD of the velocity are similar at resolution 360 and 560, while the results at resolution 180 show a large difference. For this reason, resolution 360 is selected to serve as a reference solution in this article, as it provides a best tradeoff between accuracy and numerical efficiency.

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

Resolution for the stent model

After choosing the proper resolution for the fully resolved reference simulation, we proceed to measure the impact of mesh resolution on the quality of our screen-model based stent model.

Since the purpose of our model is to reduce computational cost, the model is applied to coarser grids. The resolutions we chose to test are 1/2, 1/4 and 1/8 of the reference resolution (360), which means, resolutions of 180, 90 and 45 nodes along the diameter of the artery respectively.

Fig. 3.4 shows the local velocity in the aneurysm obtained by the simulation at the three resolutions. Qualitatively, it is concluded that the velocity is very similar at all three resolutions, and Tab.3.3shows a quantitative match of the RMS values of the velocity inside the aneurysm between all three cases. This shows that the model would allow for a coarsing factor of 8 and still produce quantitatively acceptable results. Nevertheless, resolution 45 is so coarse that it leaves the user with little data to post-process and visualize the results. As a tradeoff between accuracy and computational cost, we therefore choose a resolution of 90 to validate our model in the following.

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

Fig. 3.4 Velocity obtained from the simulation using our stent 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 3.3 RMS of the velocity field from the simulation at the three resolutions using our stent model.

Validation of the stent model

In order to investigate the performance of our screen-model based stent model [40], four cases are simulated: the aneurysm without stent, the aneurysm with a fully resolved stent, our model described in [40] and the force model of Raschi [55]. Both Raschi’s model and ours are executed at the same resolution.

Fig.3.5shows the velocities inside the aneurysm in all four cases. Compared with the case without stent, the fully resolved stent and the two models yield a reduction of the flow inside the aneurysm. Our model shows a good consistency with the fully resolved stent simulation, while Raschi’s model over-predicts the velocity, as compared to the fully resolved simulation.

As a further validation, we measure the wall shear stress (WSS) along the aneurysm wall.

Figure3.6adepicts the WSS on the aneurysm wall for all four cases. Figure3.6billustrates the definition of the angleθ which labels the x-axis in Fig.3.6a. In the fully resolved simulation with stent, as well as in the coarse-grained simulations with both models, the presence of a stent appears to substantially reduce the WSS. Both our model and Raschi’s model however slightly over-predict the WSS.

For a quantitative comparison, we again compute the RMS and RMSD of the velocity inside the aneurysm in all four cases, along with a value of the WSS which is averaged over the aneurysm wall. The results are summarised in Tab.3.4. In the fully resolved simulations, it can be seen from the velocity RMS that the introduction of a stent introduces a flow reduction

3.3 Validation 47

(a) without stent (b) fully resolved stent (c) our stent model (d) Raschi’s model Fig. 3.5 Velocity over the region of the aneurysm of stent 1 (β =0.6). The units for the position and velocity aremmandm/s, respectively.

(a) Wss on the wall of aneurysm

(b) Definition ofθ for the WSS measurement.

Fig. 3.6 WSS on the wall of aneurysm of stent 1 (β =0.6)

of approximately 1/18 inside the aneurysm. Our stent model predicts a velocity reduction of approximately 1/12, while Raschi’s model predicts a reduction of 1/7. This leads to the

conclusion that our model has an error of 47.29% for the RMS velocity, while the error of Raschi’s model is 191.68%. Both models over-predict the RMS velocity, which means that they propose drag forces that are weaker than the ones produced by a real stent. Still, our model yields a significant improvement over Raschi’s model. Furthermore, Tab.3.4leads to similar conclusions concerning the average WSS.

without stent fully resolved stent our stent 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 3.4 Comparison of the results obtained from the four simulation cases (β =0.6).

In our simulations based on the Palabos library, and executed on a parallel computer, the full simulation time could be reduced by a factor 21.9 by applying our model rather than running a fully resolved simulation, and by a factor 24.7 by using Raschi’s model.

3.4 Testing the continuum stent model under different