Numerical simulation of flow-diverting stent: comparison between branches in bifurcation brain aneurysm

The usage of flow-diverting stents in the treatment of intracranial aneurysms is widespread due to their high success and low complication rates. However, their use is still not officially recommended for bifurcation aneurysms, as there is a risk of generating ischemic complications due to the reduced blood flow to the jailed branch. Many works utilize computational fluid dynamics (CFD) to study how hemodynamic variables respond to flow diverter placement, but few seem to use it to verify flow variation between branches of bifurcation aneurysms and to aid in the choice of the best ramification for device placement. This investigation was performed in the present work, by comparing wall shear stress (WSS) and flowrates for a patient-specific scenario of a middle cerebral artery (MCA) aneurysm, considering device placement on each branch. A secondary objective was to follow a methodology that provides fast results, envisioning application to daily medical practice. The device was simplified as a homogeneous porous medium, and extreme porosity values were simulated for comparison. Results suggest that stent placement on either branch is both safe and effective, significantly reducing WSS and flow into the aneurysm while maintaining flow to the different ramifications within acceptable thresholds.


Introduction
Flow diverters are self-expanding, braided tubular devices that have increasingly been used in the treatment of intracranial aneurysms due to being a minimally invasive endovascular intervention, with low complication rates and high efficacy (Augsburger et al 2011;Shin et al 2020).
Unlike the predecessor endovascular intervention via microcoils, their healing mechanism does not target the aneurysm, but actually the parent artery that originated it.The device is positioned along the parent vessel so that its high metal coverage surface covers the aneurysm neck, decreasing blood flow into the aneurysm sac.This reduction leads to flow stasis inside the aneurysm, promoting thrombus formation and eventual endothelialization of the device, which is incorporated into the now reconstructed arterial wall (Saleme et al 2014;Shin et al 2020).It is important though that these devices have some degree of porosity, to guarantee that flow to perforators and other branches remain patent (Shin et al 2020).This degree of porosity will be especially important for bifurcation aneurysms, as the covered branch can suffer complications due to the reduction in blood flow (Saleme et al 2014;Shapiro et al 2018;Shin et al 2020).
Since they were first approved by the FDA (Food and drug administration) in 2011, treatment with flow diverter devices is only indicated for wide-necked intracranial aneurysms in the internal carotid artery area (Shin et al 2020) and their application to and beyond the Circle of Willis 1 3 locations, small aneurysms and bifurcations is still being debated (Saleme et al 2014;Shapiro et al 2018;Shin et al 2020).Nevertheless, their use has been pushed further and they have progressively been used to treat bifurcation aneurysms with a high success rate (Saleme et al 2014;Shapiro et al 2018).
Many biological mechanisms contribute to maintaining homeostasis and lowering the risk of ischemia in these cases: vessel wall remodeling and compensation via collateral support are cited in the literature (Saleme et al 2014;Shapiro et al 2018;Shin et al 2020).The first manifests as an increase in arterial diameter (vasodilatation), leading to decreased pressure and an increase in blood flow to the covered branch, and the latter represents an increase in blood flow in other arteries to maintain the metabolic needs of the tissues supplied by the covered branch.Depending on the extent of this collateral support, the covered branch can even eventually be occluded altogether (Shapiro et al 2018).Saleme et al. (2014) observed that this process seemed to safely contribute to complete aneurysm occlusion.
Even though these mechanisms are known to exist, they are difficult to predict and even identify due to the complexity of in vivo dynamics (Shapiro et al 2018), making it hard for physicians to rely on them when evaluating if a bifurcation aneurysm can be treated with flow diverters.The middle cerebral artery (MCA), for instance, is a region known for having low collateral support (Saleme et al 2014;Shapiro et al 2018) and is one of the most common sites for bifurcation aneurysms (Kumar et al 2007).Traditionally, this region is treated preferably via open surgery (microsurgical clipping) to assure the preservation of the bifurcation (Kumar et al 2007) and, when compared to treatment with flow diverters in other sites, studies have indicated ischemic complications to be significantly higher on this area (Shin et al 2020).Still, endovascular treatment is a more appealing option and, while they might not be the most common recommendation, successful cases with flow diverters have been reported in the literature (Saleme et al 2014).
With the advances in computational power, more and more medical research has benefited and gained insights from computational fluid dynamic (CFD) numerical simulations, both to assist diagnostics and study treatment outcome.Aneurysms are no exception, with numerous works analyzing hemodynamics to study their growth, rupture, and treatment (He et al 2009;Augsburger et al 2011;Mut et al 2011;Raschi 2014;Munarriz et al 2016;Sforza et al 2016;Li et al 2018;Yadollahi-Farsani et al 2019).
Although computational power has increased, it still faces limitations, and large simulation times can still be a constraint, especially if these simulations are to be used in day-to-day clinical practice (Raschi 2014).To mitigate this, many simplifications are adopted in CFD simulations, a common one being treating the flow diverter stent as a porous medium.This consideration significantly reduces simulation times, as it eliminates meshing complexities associated with the smaller scale of the device wires in relation to the aneurysm, allowing to reduce the total number of mesh elements (Augsburger et al 2011;Raschi 2014;Li et al 2018;Yadollahi-Farsani et al 2019).Augsburguer et al. (2011) validated this approach by comparing results of porous medium simulations against simulations considering the actual stent geometry.Porous medium coefficients were taken from the second-order polynomial pressure drop curve fit, resultant of multiple numerical simulations of a flat screen in a long pipe with various velocities.Although results displayed some differences, especially for the bifurcation case (referred to as "inertiadriven" by the authors), qualitative differences were small, and even quantitative differences were not so significant, considering the errors introduced due to other simplifications usually adopted on this type of blood-flow simulations.The porous medium approach reduced simulation times significantly, while still allowing to reproduce faithful flow and wall shear stress (WSS) patterns.
Based on these results and using the same process to obtain porous medium parameters, Li et al. ( 2018) simulated flow diverter devices from different manufacturers as homogeneous porous mediums to obtain calibrated flow diverter coefficients according to device properties.This work showed that even for the same aneurysm geometry and device porosity, stent properties such as thickness significantly alter results and should be taken into consideration when simulating.
A well-known limitation of these studies is the assumption of a homogeneous porous medium.It is known that porosity varies significantly along device curvature and its value also depends on artery and stent sizes and deployed configuration (Raschi 2014;Shapiro et al 2014;Li et al 2018), making the actual values hard to predict.Yadollahi-Farsani et al. (2019) simulated fully resolved geometries as well as homogeneous and heterogeneous porous medium geometries and compared steady-state results.The authors found that the heterogeneous approach leads to smaller differences from the fully resolved solution without impairment to computational time.Even though relative error differences between the two approaches are impressive (less than 10% error for heterogeneous against a variation between 15 and 64% for homogeneous for aneurysmal kinetic energy per unit volume), because biological flowrates are very small, absolute differences are lower than 0.1 kg m −1 s −2 .Furthermore, heterogeneous porosity coefficients were calculated by an algorithm developed by the authors to perform a mapping between the untreated mesh and the geometry of the stent deployed virtually, which requires specialized software that might not always be available.
The most practical and simple approach to estimate device porosity, without the need to run extra CFD simulations or perform a virtual deployment, is by calculating it through the use of correlations.Raschi et al. (2014) proposed a model based on empirical correlations for infinite flat screens that depends solely on stent properties, such as the braiding angle, wire thickness, diameter, and number of wires.While practical, this still assumes a homogeneous porous medium, and result comparisons with fully resolved simulations showed good agreement with more uniform scenarios, but larger errors for curved geometries.
Although many works study the resulting hemodynamics after placement of a flow diverter device and many others mention physicians' concerns regarding flow reduction in bifurcation aneurysms, there seems to be a lack of studies that compare resulting hemodynamics for flow diverters placed on different branches of a bifurcation in terms of both efficacy and ischemic risk.
The objective of this work is to evaluate through numerical simulations whether a patient-specific bifurcation aneurysm case located in the MCA is eligible for treatment with a flow diverter device and if there is a preferred branch to place the stent to avoid ischemic complications.Furthermore, the objective is to do so by utilizing a simple but effective framework, capable of generating results fast and that can be used in day-to-day clinical practice.To accomplish so, CFD simulations were performed to compare results between the untreated case and placement of the flow diverter device on each branch of the bifurcation.Simulations utilized the homogeneous porous medium approximation for simplification.To reduce uncertainties related to this assumption, simulations were run considering a range of porosity values.

Materials and methods
Geometries and meshes for three scenarios were generated based on the images of a real patient: the untreated case, Configuration 1 (flow diverter placed on the left-side branch), and Configuration 2 (flow diverter placed on the right-side branch).
Steady-state simulations were initially performed for the untreated case, to compare two possible outlet boundary conditions.
Once the outlet boundary condition of the model was defined, transient simulations comparing hemodynamic variables for the three scenarios were performed.Then, Configuration 2 was selected as a scenario to be further evaluated in the porosity analysis simulations, which were also conducted in steady state, but considering extreme values for stent porosity.

Geometry and mesh
The case studied consisted of a small saccular aneurysm (around 5.5 mm) located in a bifurcation in the MCA.A set of 373 images acquired via digital angiography with contrast injection was used for the generation of the patient-specific geometry.The usage of these images was approved by both the patient and the hospital's Ethical Committee.
The open-source software 3DSlicer-http:// www.slicer.org (Fedorov et al 2012;Kikinis et al 2014) was used for volume reconstruction, and the resulting geometry was treated on Meshlab (Cignoni et al 2008) for manual removal of overlaid vessels, closing of surface holes and surface smoothing.Finally, some simplifications and scale adjustments were performed on ANSYS® Spaceclaim® 2019 R2.To reduce the dependency on the boundary condition, while keeping geometry size reasonable for simulations, the simulated portion of the artery was truncated to contain 5 ramifications on the outlets.The final geometry utilized for the untreated case is illustrated in Fig. 1a.
That same geometry served as a basis to create the ones for the two stented configurations: stent placed on the leftside bifurcation, occluding the right-side branch (Configuration 1), and stent placed on the right-side bifurcation, occluding the left-side branch (Configuration 2).
Using ANSYS ® Spaceclaim ® , a porous medium with a thickness of 300 µm was inserted into the original untreated geometry, to represent the flow-diverting stent.Instead of simulating a tube-like geometry for the flow diverter, patches were inserted just covering the aneurysm neck area (Raschi 2014).Figure 1b-e illustrates the final stent geometry for Configurations 1 and 2.
One limitation of the porous medium approach is that it does not account for inappropriate stent apposition to the wall (Raschi 2014).Therefore, before preparing the geometry for the stented scenario, an assessment was made of whether the nominal diameters commercially available for a Pipeline Embolization Device (PED; Medtronic, Dublin, Ireland) would fit the studied case and could be properly positioned against the vessel wall.This was checked by calculating the maximum cross-sectional area along the stent curvature, and assuming the device is able to deform freely and conform to the vessel wall (Shapiro et al 2014).Based on this simple assumption, nominal diameters of 2.75 and 3.25 mm were considered for Configurations 1 and 2, respectively.Although vessel diameters for inlet and outlets are in the vicinity of 1.5 mm, a larger stent diameter is required due to the enlargement of the artery on the aneurysm neck area.
Tetrahedral meshes with structured local refinement near the wall were created on ANSYS ® Meshing ™ 2019 R2 for the 3 scenarios (untreated case, Configuration 1, and Configuration 2).For the stented geometries, element size was selected to guarantee at least 4 layers of element across the porous medium (Raschi 2014).All 3 meshes had around 2 million elements each.
The Grid Convergence Index (GCI) method (Celik et al 2008) was used to assess mesh independence for the untreated case, as this is the scenario with the most complex flow pattern.While the GCIs are very low for the flowrates at the five outlets (below 0.2%), the GCI for average WSS is relatively high (19%) and results suggested the occurrence of oscillatory convergence for this variable.
Nevertheless, mesh quality metrics were found to be within acceptable ranges (average skewness of 0.21 and average orthogonal quality of 0.79) and absolute differences for WSS were inferior to 4 Pa, a considerably low WSS value.Finally, even though there is a relatively high variation in average WSS value, a comparison of results between the different meshes showed that qualitative results were very similar, and areas of high and low WSS remained the same.
Real patient geometries are quite irregular and lead to a challenging meshing process.Even though a more refined grid could be more appropriate from a numerical point of view, higher mesh levels would be impractical from a daily medical practice point of view, which is the main objective of the present study.Conclusions expected to be taken from the proposed methodology are not expected to change if mesh is further refined and therefore, grids with about 2 million elements were considered to return reliable results for the present work.

Simulations and assumptions
All CFD simulations in this work were performed with the finite-volume solver OpenFoam v1912 (Weller et al 1998).
Rigid vessel walls with no-slip boundary condition were assumed, a simplification widely adopted in the literature (Rayz et al 2008;Augsburger et al 2011;Li et al 2018) and considered reasonable for aneurysmal vessels, due to the smaller amount of elastin (Rayz et al 2008;Jones 2014).
Blood was assumed as an incompressible, Newtonian fluid with density and viscosity values equal to 1000 kg.m −3 and 0.004 Pa.s., respectively (He et al 2009;Mut et al 2011;Raschi 2014;Sforza et al 2016).Although blood is known to be a non-Newtonian, pseudoplastic fluid, containing proteins, salts, and cells suspended in the plasma (Kundu and Cohen 2008;Jones 2014), many studies show that for medium and large arteries, the non-Newtonian effects are negligible and blood can be considered a Newtonian fluid without impairing the results (Kundu and Cohen 2008;Rayz et al 2008;Mut et al 2011).Important non-Newtonian effects such as the Fahraeus-Lindqvist are observed only for narrow vessels (diameter between 15 and 500 µm), in which red blood cells migrate away from the vessel walls toward the center, decreasing blood viscosity and therefore facilitating flow through capillaries (Kundu and Cohen 2008).This effect could apply if the stent's real geometry, which contains microchannels, had been considered.As the stent has been modeled as a porous medium and, for the studied geometry, the smallest vessel diameter is 1400 µm, it is reasonable to assume the blood as a Newtonian fluid.
The simpleFoam (steady state) and pimpleFoam (transient) incompressible solvers were selected for solving the governing Navier-Stokes equations based on the SIMPLE (semi-implicit method for pressure-linked equations) and PISO (pressure-implicit split-operator) algorithms for all simulations (Untreated case, Configuration 1 and Configuration 2).Gaussian linear integration was selected for all variable's discretization schemes, except for velocity gradient, which used a Point Cells Least Squares scheme, and the Laplacian terms, which used the Gauss harmonic scheme.In the porous medium region, OpenFoam's DarcyForchheimer A velocity profile obtained from the literature for the MCA was utilized as the inlet boundary condition for the transient simulations, and a variation of OpenFoam's total-Pressure boundary condition was used in the outlets.

Outlet boundary condition selection
Most works in the literature use a fixed-pressure boundary condition (He et al 2009;Raschi 2014;Li et al 2018) for the outlets, and some even specify flow (Sforza et al 2016), in the lack of real physiological measurements.Since the objective of the simulations of this work was to evaluate flowrate distribution between the different ramifications after stent placement, an adaptation of OpenFoam's totalPressure condition was suggested, so as not to specify flowrate nor pressure.With the placement of the flow diverter device, an additional head loss is introduced in the system, and this effect might be masked if these 2 variables are fixed.
Opposed to the more commonly used fixed-pressure condition, in which static pressure is defined, in the totalPressure condition, total pressure, also known as "stagnation pressure" is specified.Static and stagnation pressure are related through the following relation: where p stands for pressure, is the fluid's density, and v is the velocity vector.
This allows static pressure to vary freely on each outlet, according to pulsatile fluid velocity.Steady-state simulations were run for the untreated geometry, in which total pressure was specified as zero for all outlets, and results were compared to the fixed-pressure condition for validation.A small adjustment to the original totalPressure boundary condition was required so it could be used in the outlets of the geometry, as the original condition available in OpenFoam assumes a positive flow (inflow).This newly adapted version was named the "stagnation pressure" condition, and its code can be found in Appendix B.
After evaluating the results, the "stagnation pressure" condition was selected for the other simulations.

Transient simulations
For the transient simulations, since the patient-specific velocity profile was not available, 4 cardiac cycles were computed considering the waveform proposed by He et al. (2009) for an aneurysm located at the MCA bifurcation (see Fig. 2).Steady-state simulations considered the initial  2016) for a bifurcation aneurysm in the MCA with a similar inflow velocity value and geometry is 351, and Reynolds numbers lower than 258 are expected for the whole cardiac cycle, especially within the aneurysm when the stent is placed, laminar flow was considered for all simulations in this work.
Timestep was allowed to vary so that the Courant number had a maximum value of 20 and the Euler discretization scheme was selected.Timestep independence tests were performed by comparing results with lower thresholds for Courant.Residual independence tests were also performed to check whether simulations could be considered as converged, and a threshold of 10 -4 was found to be acceptable.
For the stented scenarios, the porous medium was simulated as homogeneous, with Darcy and Forchheimer coefficients calculated using the equations presented below, according to the procedure presented in Raschi et al.(2014).
(2) where is blood's viscosity, is blood's density, ΔL is the porous medium thickness, d h is the hydraulic diameter of a stent cell (analogous to a pore diameter), and is the stent porosity.The hydraulic diameter and the porosity can be calculated using the stent's characteristics as inputs, namely the braiding angle, thickness of wire, nominal diameter, and number of strands.The equations used to calculate d h and are presented in Appendix C.This correlation was chosen due to its simplicity, since its parameters are all related to the device geometry and because it takes porous medium thickness into consideration.The wire thickness was set as 30 µm and the number of strands was considered as 48, based on PED characteristics reported in the literature (Shapiro et al 2014).Stent nominal diameters were 2.75 mm for Configuration 1 and 3.25 mm for Configuration 2, taken from the maximum cross-sectional area assessment mentioned previously.This choice took into consideration PED's ability to expand up to 0.25 mm beyond their nominal diameter (Shapiro et al 2014).Since the maximum area occurs on the curvature located on the aneurysm neck, the nominal diameter was selected as the one up to 0.25 mm below the maximum diameter required, based on the assumption that the device would be unconstrained and allowed to fully open on this part of the artery.Also, based on this assumption, the braiding angle was assumed as 120°, as this is the angle usually found for an unconstrained device on a straight vessel segment according to Shapiro et al.(2014).This represents a porosity of 74%, which is within the limits usually reported in the literature (Shapiro et al 2014(Shapiro et al , 2018;;Shin et al 2020).
While a homogeneous porous medium can be a fairly good assumption for a few cases, it is known that porosity varies between a certain range of values for devices located in curvatures due to the elongation and contraction of device cells (Raschi 2014).Furthermore, transition zones of porosity will exist when there is a mismatch between distal and landing zone diameters (Shapiro et al 2014).To evaluate this effect without having to resort to a heterogeneous porous media approach, steady-state simulations were performed for Configuration 2 considering extreme braiding angles: 90° and 150° representing, respectively, porosities of 81 and 53%.Particularly, the braiding angle of 90° corresponds to the configuration of minimum metal coverage possible for a flow diverter, when the device's strands assume a square configuration (Shapiro et al 2014).Table 1 displays all Darcy and Forchheimer coefficients used in the simulations, considering the above-mentioned input parameters and the correlation from Raschi et al. (2014)

Results
For all simulations, aneurysm WSS, velocity streamlines, outlet flowrates, and pressure were analyzed and compared.Results from the third and fourth cycles coincided, so only results for the last cycle will be displayed.All images were generated in Paraview 5.6 (Ahrens et al 2005), the software used for post-processing results, and all charts were generated in Excel.

Boundary condition comparison
Simulation results matched between the fixed-pressure and the stagnation pressure boundary conditions, except for pressure results.As one can see in Figs. 3 and 4, WSS and the split of flowrate through the outlets are the same for both conditions.Both numerical simulations also took similar times to converge.
Pressure profiles displayed only subtle differences, as can be seen in Fig. 5.The pressure did vary between the outlets for the Pstag condition, but this variation was very small to be considered significant (less than 1 Pa), as can be seen in Fig. 6.
Based on these observations, it was considered that for the untreated case, the use of the stagnation pressure boundary condition is equivalent to the constant pressure boundary condition, largely used in the literature, displaying coherent results.
The stagnation pressure condition, therefore, was selected as a valid and preferred setup for the stented scenarios, since it allows pressure at the outlets to vary according to flow velocity, leading to a more realistic pressure profile.

Hemodynamic alterations due to flow diverter
The transient simulations for the untreated scenario indicated an impinging jet flow entering the aneurysm and causing large WSS values on a portion of the aneurysm dome.This jet is then dispersed, generating vortices and recirculation zones.This behavior was more accentuated during systole (t = 3.4 s).
After stent placement, both maximum and average WSS on the aneurysm wall were reduced significantly.Table 2 compares the maximum and average WSS during systole for all three scenarios.Furthermore, WSS profile plots in Fig. 7 show that once the stent is inserted, the maximum WSS no longer occurs on the aneurysm dome, but actually on the artery's wall.As expected, flow into the aneurysm reduced after stent placement, but it did not cease completely.Figure 8 displays still some recirculation of fluid inside the aneurysm, but at a much lower flowrate and velocity and with a less complex pattern.Most importantly, the impinging jet flow into the aneurysm dome that lead to the high WSS area depicted in Fig. 7a ceased to exist.
Analysis of transient results indicated that the stent does not provide an equal effect of flowrate reduction for the entire cardiac cycle, and during systole, proportionally more blood still enters the aneurysm (Table 3).For Configuration 1, the average flowrate reduction is 54% and for Configuration 2 the average flow reduction is 46%.
To investigate the risk of ischemia, mass flowrate on each of the geometry outlets was also studied.Table 4 summarizes the percentage reduction, compared to the untreated case.Positive values represent a decrease in flow, and negative values an increase.
Opposed to the initial expectation, when the stent is placed on the left branch for Configuration 1, an increase in blood flow is observed in the outlets located on the right branch (R1, R2, and R3), instead of in the outlets This can be explained due to the high porosity of the flow diverter stent (70-74%), which allows a large amount of blood to still enter the aneurysm.This blood recirculates inside it and then leaves through the branch that does not contain the stent, following the least resistance path, as can be seen in Fig. 8.
Even though the insertion of the stent leads to flowrate variations, it is important to notice that they represent a very small change in actual blood volume.The maximum reduction observed is 6% at outlet L1, during and shortly after systole (t = 3.4 and 3.6 s) for Configuration 1, which represents a variation of only 0.035 g.s −1 .On average, when the entire cycle is considered, this reduction is only 4%.

Porosity analysis
Configuration 2 was selected for the porosity analyses because it was the one that lead to the smallest reductions in WSS and flow in the aneurysm, and therefore possibly is the less efficient alternative.
Figure 9 illustrates both the velocity streamlines and WSS profiles on the aneurysm, obtained as a result of steadystate simulations for the three porosity scenarios considering Configuration 2.
As expected, higher porosities yielded higher velocities and flowrates into the aneurysm, as well as higher values of WSS.For the highest porosity scenario (braiding angle of 90°), a jet of blood can still be seen entering the aneurysm, but it does not reach the aneurysm dome.This jet reduces for the intermediate scenario (braiding angle of 120°) and ceases to exist for the highest metal coverage case (150°), where blood inflow appears more uniform and depicts a less complex circulation pattern, as can be seen in Fig. 9a-c.
However, regardless of the blood inflow pattern into the aneurysm, high WSS values on the dome no longer occur: for all three porosities, the highest values of WSS are located on the artery wall, as can be seen in Fig. 9d-f.
Regarding changes in the flowrate distribution, for the lowest porosity scenario (53%) there is a maximum flow reduction of 4% at the L2 outlet.In fact, for this scenario,  as porosity is lower, flowrate to the aneurysm is reduced in such a way that the unexpected effect of increasing flow to the jailed branch no longer happens, as can be seen in Table 5, in which negative values represent an increase in flow.

Discussion
A patient-specific aneurysm located in the MCA bifurcation was investigated through CFD numerical simulations.For this patient-specific geometry, on the untreated scenario, one can observe that a small area of the aneurysm dome is under larger values of WSS throughout the cardiac cycle, undergoing the highest stress during systole, due to an inflow jet impinging on the wall.This situation has been shown to be a cause of probable aneurysm rupture for small aneurysms (Mut et al 2011;Munarriz et al 2016;Sforza et al 2016) and is a case in which treatment is recommended.
Simulation results indicate that, for the studied case, placement of the stent on either bifurcation leads to an efficient treatment outcome; the area under large WSS values disappears and tension on the wall is reduced by at least 60% during systole.Furthermore, flow to the aneurysm is reduced by at least 42% and displays a less complex pattern, forming an environment that favors aneurysm occlusion (Raschi 2014;Munarriz et al 2016;Li et al 2018;Shin et al 2020).
One major concern regarding the treatment of bifurcation aneurysms with flow-diverting techniques is the risk of ischemic complications or thrombosis of the covered side branches (Saleme et al 2014).Therefore, this work analyzed not only treatment efficacy regarding the reduction of flow and WSS in the aneurysm, but also flowrate alteration to the bifurcations after stent placement.Once again, stent placement on either bifurcation does not greatly alter flow distribution between outlets, with a maximum average reduction of 4% in outlet L1 for Configuration 1. Available literature establishes a threshold of 60% as acceptable (Augsburger et al 2011;Shapiro et al 2018).
This shows that the flow diverter stent has a much bigger effect on reducing WSS than diverting flowrate.
It is important to highlight, however, that all these results are directly linked to the porosity values assigned to the stent during the simulation, which is one of the biggest uncertainties of this work, since uniform porosity was assumed and calculated through a correlation.Results of the porosity analysis indicate that even for the highest porosity scenario (81%, 90° braiding angle) WSS in the aneurysm dome is still significantly reduced, and on a lower porosity scenario (53%, 150° braiding angle) flowrate variation on the different outlets is still maintained well below the acceptable threshold.
While it is certain that porosity is not uniform along the stent, it can be assumed that the exact actual values for WSS and flowrate will be near the range of results encountered for the porosity analyses.As those indicate a good treatment outcome, treatment with a flow diverter can be considered an effective and safe approach.

Conclusions
The objective of this work was to analyze a real bifurcation aneurysm case with CFD and evaluate the possibility of treatment using a flow diverter device, using a methodology that can be easily applied to day-to-day clinical practice.Therefore, the framework proposed needed to be not only able to predict treatment outcome with sufficient accuracy, but also generate results quickly.
This imposes some limitations regarding simulation and geometry complexity, which is why a homogeneous porous medium approach was assumed, along with other simplifications (rigid walls, Newtonian fluid, non-specific, and uniform velocity profile).
The meshing process can also pose a limitation, as it can be expected to be challenging for every simulation involving real-patient scenarios.A compromise between simulation time and computational cost must always be assessed to make such methodology viable.A framework for generating more accurate and smaller meshes for real-patient geometries could be of great contribution to this field of study.However, even by adopting more complex simulation techniques (such as heterogeneous porous medium, for instance), one might question how assertive can the results be, given all the sources of variability that exist in stent deployment procedures, such as folding, "push and pull" techniques (Raschi 2014) and "shape memory" effects (Shapiro et al 2014), as well as other in vivo dynamics, such as deposition of solid components (proteins, cells, etc.) in the stent cells, possibly altering porosity (Augsburger et al 2011) and mechanisms biologic systems adopt to return to homeostasis (Shapiro et al 2018).Given all these unpredictable factors, the strategies adopted in this work might be the simplest, best-guess solution that can still provide valuable insights for clinical decision-making.Further work could consider the comparison of the present approach of using a homogeneous porous medium with a range of porosities with heterogeneous porous medium simulations and/or the real stent geometry.
For the studied case, results suggest that treatment with a flow diverter device is both safe and effective, regardless of the selected bifurcation for device placement.Even though placement on the left-side bifurcation appears to lead to higher WSS reduction, the difference might not be significant enough to be conclusive, considering all the simplifications assumed.
Many other factors beyond the scope of the present simulations still need to be considered before a final decision can be reached, such as collateral support availability, vasodilatation effects, and, most importantly, pre-, and post-treatment procedures (Shin et al 2020), which is why physician's experience is key on making the final call.The goal of CFD simulations is to aid in such decisions, providing qualitative and quantitative backing.

3
Finally, porosity β is calculated as the relation between the free area and the total area of a stent cell:

Fig. 1
Fig. 1 Simulated geometry for untreated and stented cases; a untreated scenario, where L1 and L2 are the left-side outlets and R1, R2, and R3 are the right-side outlets; b, c frontal view of the porous medium used to represent the stent (Configurations 1 and 2, respectively); d, e lateral view of the stented geometry (Configurations 1 and 2, respectively).Volumes in blue represent the theoretical flow diverter configuration.Volumes in red represent the actual porous medium utilized to represent the flow diverter.Images used courtesy of ANSYS, Inc cycle and were used as an initial converged state for transient simulations.Even though blood flow is usually laminar(Kundu and Cohen 2008;Rayz et al 2008;Jones 2014;Li et al 2018), some studies have shown that transition to turbulent flow can occur in specific scenarios.The maximum Reynolds number in the present case was calculated by considering the peak velocity provided in Fig.2and the parent artery diameter, leading to a Reynolds number of 258.As the Reynolds transition number reported byJain et al. (

Fig. 2
Fig. 2 Velocity profile, digitized from He et al. (2009), used as the inlet boundary condition for the transient simulations

Fig. 3 Fig. 4
Fig. 3 Comparison of WSS in the Aneurysm for the two assessed boundary conditions: a fixed pressure at the outlets (Pconst) and b fixed stagnation pressure at the outlets (Pstag)

Fig. 5 Fig. 6
Fig. 5 Comparison of pressure drop profile for the two assessed boundary conditions: a fixed pressure at the outlets (Pconst) and b fixed stagnation pressure at the outlets (Pstag)

Table 1
Darcy and Forchheimer coefficients used on the porous medium of the stented simulations

Table 2
Comparison of aneurysmal WSS between untreated and stent-treated scenarios during systole

Table 3
Mass flowrate reduction inside the aneurysm after stent placement