The application of the screen-model based approach for stents in cerebral aneurysms
3.4 Testing the continuum stent model under different con- con-ditionscon-ditions
3.4.4 Pulsatile flow: Womersley
A Womersley flow is a simple model of pulsatile flow in an artery. In order to test the applicability of our stent 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 pressure gradient[29,36]:
∂p
∂x =Real[Aeiωt] (3.10)
with an amplitudeAand a frequencyω. For 2D Poiseuille flowAis set as A= 8νumax
Ly2 (3.11)
Whereumax is the maximum value of the velocity in the middle of the channel.
Furthermore, parameterα is introduced as Womersley number and defined as α2= Ly2ω 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.4.9). The Womersley numberα is set to 1. A stent with porosity of
3.4 Testing the continuum stent model under different conditions 55 0.6 is adopted. All the other parameters are selected as before. Fig.3.17compares the norm of the velocities on the four measurement points respectively. The velocities of our stent 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 two time points which are marked by the dashed lines in Fig.3.17. The time points aret1=5.8909s,t2=8.8364s. In Fig.3.18, our stent model shows pretty similar velocity fields as for the fully resolved stent.
Fig. 3.16 Measurement points on the aneurysm wall
(a) Point A (b) Point B
(c) Point C (d) Point D
Fig. 3.17 Velocities at the measurement points
(a) fully resolved stent(t1) (b) our stent model(t1) (c) Raschi’s model(t1)
(d) fully resolved stent (t2) (e) our stent model (t2) (f) Raschi’s model (t2) Fig. 3.18 Velocity over the region of the aneurysm of Womersley flow. The units for the position and velocity aremmandm/s, respectively
Fig.3.19shows the WSSs on the four measurement points. Except for point D, our stent model are all larger than the fully resolved stent simulation, but smaller than Raschi’s model.
The wide lines of our stent model in Fig.3.17and Fig.3.19are caused by minor oscillation.
.
t1 fully resolved stent our stent 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 our stent 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 3.9 RMS and RMSD of velocities for the Womersley simulation
Tab.3.9compares the RMS and RMSD velocity of the simulated cases. The error of the two models on t1 are smaller than the error on t2 and it means that the error gets smaller with the increasing of the velocity. Our stent model has a smaller error compared with Raschi’s model. Both models underestimate the drag force for the stent for almost all the simulation cases, thus we would adjust the equations of our stent model in the future.
3.5 Conclusion 57
(a) Point A (b) Point B
(c) Point C (d) Point D
Fig. 3.19 WSS of the measurement points
3.5 Conclusion
We propose a new, extended validation of our stent model introduced in [40], which is designed to replace a flow diverter through a continuum approach in coarse-grain simulation. In this validation, the stent is placed in an artery with an aneurysm and two outlets. We study the performance of our stent model by testing different stent locations, porosities and inlet velocities.
A comparison between our stent model and Raschi’s model [55] is also carried out in this work.
We run simulations for two models to test both a shear flow and an inertial flow, and show that our model is able in both cases to reproduce a similar flow field and wall shear stresses as a fully resolved simulation. The results show that our methodology, which uses the theory of screen models to explicitly introduce a tangential force dependent on the flow deviation, leads to a more accurate flow prediction than Raschi’s model, based on an assumption of isotropy for the drag coefficients.
Section3.4.2explores how our stent model responds to a variation of the stent porosity.
Both our model and Raschi’s model perform better at higher porosityβ =0.8 than at the lower porosityβ =0.6. As before, we observe that our screen-model based approach is more accurate than Raschi’s model, qualitatively and quantitatively.
Although most stents used in practice are constructed with a homogeneous porosity, the porosity can become dependent on position as a result of the stent’s deployment in the artery.
We have therefore validated our model with a heterogeneous stent in Sec.3.4.3. In this case, the error exhibited by our model is larger than the one obtained with homogeneous porosities, but the model still produces results of an overall good quality.
In case of a pulsatile inlet flow, both Raschi’s model and ours perform more poorly than in the case of a constant velocity. However, the velocity profiles show that our screen-model based approach is able to represent the influence of a real stent with reasonable accuracy. The error decreases at larger flow velocities.
In summary, we have tested our novel macroscopic stent model with different stent place-ments in an artery, and different porosities and velocities. The test results highlight the validity of the model in a large range of common circumstances of aneurysm treatment with flow diverters. Our produces systematically convincing and reliable results and proves capable to represent a fully resolved stent at a coarser resolution. In the future, a 3D investigation of the stent model will be carried out. The gain of CPU time achieved in 3D is expected to be substantially more important than in 2D, and our model can lead to a numerical framework for patient-specific simulation of aneurysm treatments.
Acknowledgement
We acknowledge partial funding and access to high performance computing resources from the CADMOS center (http://www.cadmos.org) and partial funding from the European Union Hori-zon 2020 research and innovation program for the CompBioMed project (http://www.compbiomed.eu/) under grant agreement 675451.