2.1. Flow model

Eight idealized aneurysm geometries were created (table 1). Consistent with a previous study (Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021; Chassagne et al. Reference Chassagne, Barbour, Chivukula, Machicoane, Kim, Levitt and Aliseda2021), the volume of the aneurysmal sac was kept constant. The sac aspect ratio, defined as $AR={a}/{b}$, was 0.4–1.6 ($AR=1$ corresponding to the spherical geometry in the previous study) (figure 1). The neck size $D_{neck}$ was 3–5 mm, matching physiological neck sizes (Khorasanizadeh et al. Reference Khorasanizadeh2022), and resulting in an opening angle of the cavity $\alpha$ equal to 73.7$^{\circ }$ and 102.7$^{\circ }$, respectively. Curvature of the parent vessel was also investigated. Each aneurysmal sac geometry was attached to parent vessels with two different curvatures: $\kappa = {1}/{r_C} = 0.0625$ and $0.22$ mm$^{-1}$ ($r_C = 16$ or $4.5$ mm). The angle between the straight sections of the parent vessel upstream and downstream of the aneurysm was kept constant. These eight idealized geometries correspond to a full factorial experimental design with three parameters (Tinsson Reference Tinsson2010), similar to Nair et al. (Reference Nair, Chong, Indahlastari, Ryan, Workman, Haithem Babiker, Yadollahi Farsani, Baccin and Frakes2016), who used the same approach for basilar tip aneurysms.

Table 1. Nomenclature for the different geometries and their corresponding geometrical parameters: curvature of the parent vessel $\kappa$, aspect ratio $AR$ and neck diameter $D_{neck}$.

Figure 1. (a) Illustration of the idealized aneurysm geometry and the different geometrical parameters. (b) Four idealized aneurysm geometries ($\kappa = 0.22$ mm$^{-1}$) with different aspect ratios ($AR$) and neck sizes ($D_{neck}$).

The lumen geometries were three-dimensionally printed to create a transparent silicone model (Sylgard 184, Dow Corning Corp., Auburn, MI, USA) via a methodology previously described (Chivukula et al. Reference Chivukula2019). We used a mixture of water, glycerine and NaCl (weight ratio 47.5:35.8:16.7) as working fluid, with a viscosity of 3.8 cP, matching the viscosity of blood, and index of refraction matching that of the silicone, to prevent optical distortion.

After the first round of experimental measurements, the models were treated with FDS by an experienced neurosurgeon. These FDS ($4\ {\rm mm}\times 20\ {\rm mm}$, Pipeline Embolization Devices, Medtronic) are tubular meshes, composed of braided platinum–tungsten and cobalt–chromium–nickel alloy wires ($\text {diameter} = 33\ \mathrm {\mu }{\rm m}$). The geometry of the FDS deployed in each model was imaged by synchrotron X-ray micro-tomography at ESRF, Grenoble, France, as previously described (Chivukula et al. Reference Chivukula2019).

2.2. Flow analysis

The models were studied with physiological unsteady waveforms in a flow loop with a pulsatile pump (Harvard Apparatus, Boston, MA, USA). The acceleration phase was set up to be shorter than the deceleration phase to match physiological waveforms (40 %/60 % of the period $T$). Two frequencies $f$ were selected for the periodic inflow, 0.8 and 1.6 Hz (50 and 100 beats per minute), corresponding to Womersley numbers $Wo = ({D_{PV}}/{2})\sqrt {{2{\rm \pi} f}/{\nu }} = 2.5$ and 3.2, respectively.

For each frequency, measurements were repeated for four different time-averaged flow rates $\bar {Q}_{PV} = 100 \unicode{x2013}400$ ml min$^{-1}$. These flow rates correspond to parent vessel Reynolds numbers $\overline {Re}_{PV}= 138$, 276, 414, 552, covering the full range of $Re$ in the human internal carotid artery (Ford et al. Reference Ford, Alperin, Lee, Holdsworth and Steinman2005). These experimental conditions result in Dean numbers $De=\overline {Re}_{PV} \sqrt {\kappa ({D}/{2})}$ from 50.9 to 384.0.

Two-dimensional, three-component velocity fields in the aneurysmal sac were obtained via stereoscopic PIV measurements in the plane of symmetry of the models, which were immersed in a tank filled with the working fluid. A full description of the optical set-up can be found in Chassagne et al. (Reference Chassagne, Barbour, Chivukula, Machicoane, Kim, Levitt and Aliseda2021). This approach provided 7653 image pairs for each experimental condition ($n=128$), achieving fully converged phase-averaged statistics.

2.3. Analysis of the velocity fields

The two-dimensional, three-component velocity fields $\boldsymbol {u} (x,y)$ obtained via stereo PIV measurements were eventually averaged onto one cycle. The Reynolds number in the aneurysmal sac, $Re_A$, was computed as the spatial average of the magnitude of the velocity fields $|\boldsymbol {u}|$. The circulation in the sac $\varGamma$ was computed as the spatial integral of the vorticity in the sac. The location of the centre of the vortex was estimated with the location of the minimum velocity magnitude in the cavity (Chassagne et al. Reference Chassagne, Barbour, Chivukula, Machicoane, Kim, Levitt and Aliseda2021).

The neck was defined as the curved line that completed the outline of the missing parent vessel in the images. The velocity at the neck, $\bar {u}_N$, was computed as follows:

(2.1)\begin{equation} \bar{u}_N=\frac{1}{2 L_{neck}} \int_{0}^{L_{neck}} | \boldsymbol{u}\boldsymbol{\cdot}\boldsymbol{n}| \,{\rm d}l, \end{equation}

where $L_{neck}$ is the neck line length. From this, the neck $Re$ can be computed as follows: $Re_{N} = {\bar {u}_N D_{neck}}/{\nu }$. If the velocity in some parts of the aneurysm was not captured properly by the measurement, because it was much lower than that of the main flow patterns, it was conservatively zeroed in the analysis.

2.4. Analysis of the effect of the different parameters on the haemodynamics

A statistical approach was used to assess the influence of the different parameters on the haemodynamics before and after treatment with FDS. The goal of this approach, whose detailed description and implementation can be found in previous publications (Tinsson Reference Tinsson2010; Frauziols et al. Reference Frauziols, Chassagne, Badel, Navarro, Molimard, Curt and Avril2016), is to quantify the fraction of the total observed changes that is induced by the variation of each isolated parameter, as well as their possible coupling. Based on previous work on intra-aneurysmal flow in spherical geometries with fixed neck areas (Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021; Chassagne et al. Reference Chassagne, Barbour, Chivukula, Machicoane, Kim, Levitt and Aliseda2021), which demonstrated the small influence of the Womersley number compared with the Dean number, it was decided to focus on three parameters and their interactions: $De$, $D_{neck}$, $AR$, $De*D_{neck}$, $De*AR$ and $D_{neck}*AR$.

These three parameters were first scaled between $-1$ and $1$. Then, a response function $Y$ was fitted to the values of the average intra-aneurysmal velocity $\langle \bar {u}\rangle$ before and after treatment separately:

(2.2)\begin{align} Y &= \beta_0 + \beta_1 * De + \beta_2 * D_{neck} + \beta_3 * AR + \beta_4 * De*D_{neck}\nonumber\\ &\quad + \beta_5 * De * AR + \beta_6 * D_{neck}*AR. \end{align}

An analysis of variance approach was then used to evaluate how well the response function fits the experimental data (i.e. $R^2 = 0.XX$ means that $XX\,\%$ of the experimental variance is replicated with this response function). Finally, the effect of each parameter $\beta _{1},\dots, \beta_{6}$ was normalized by the range of variation of the experimental data, quantifying the fraction of the total change induced by a variation of each parameter (i.e. the ‘influence’ of each parameter).

4.1. Flow topology

Stereo PIV measurements were repeated after the aneurysm models were treated with FDS, for the same flow conditions. First, implanting a FDS, i.e. a metal strut mesh with a cylindrical shape to match the parent artery that extends across the aneurysmal neck, results in a strong decrease in intra-aneurysmal velocity (about one order of magnitude) (figure 7). This strong decrease in velocity is in agreement with numerous previous studies (Cantón et al. Reference Cantón, Levy, Lasheras and Nelson2005; Bouillot et al. Reference Bouillot, Brina, Ouared, Lovblad, Farhat and Pereira2015; Clauser et al. Reference Clauser, Knieps, Büsen, Ding, Schmitz-Rode, Steinseifer, Arens and Cattaneo2018; Moriwaki et al. Reference Moriwaki, Tajikawa and Nakayama2020). The FDS also induced a strong modification of the intra-aneurysmal flow patterns, dependent on geometry. For all low-$AR$ aneurysms (K$-$/AR$-$/N$-$, K$+$/AR$-$/N$-$, K$-$/AR$-$/N$+$ and K$+$/AR$-$/N$+$), the flow topology at low and intermediate $Re_{PV}$ was identical (an example is presented in figure 7, top). When the velocity in the parent vessel is minimum, the flow in the aneurysmal sac is attached to the parent vessel (in blue in figure 7, top). It enters the sac at the proximal edge, expands in the aneurysm and exits at the distal edge. However, the vortex formed does not fully fill the cavity. As already observed for untreated flow, the velocities in the deep parts of the cavity are very low and were not properly captured by the stereo PIV measurements. When the flow in the vessel accelerates, separation starts to occur (in yellow in figure 7, top). The entrance of the flow in the sac is relocated to the centre of the neck and flow is towards the fundus of the cavity. When the flow in the sac is maximum (in red in figure 7, top), the flow is fully separated from the vessel wall: the flow enters the sac at the distal edge, expands in the sac and then reattaches to the flow in the parent vessel at the proximal edge. However, the flow topology is different from that before treatment: the vortex is not fully formed and stable, as it only exists for a short part of the cycle. Indeed, when the flow decelerates (in green in figure 7, top), the circulation in the sac is reversed again and the flow in the sac reattaches to the parent vessel. This cyclic change in the sense of circulation was observed for spherical aneurysms too, but only for low curvatures and $Re_{PV}$ (Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021), or for micro-spherical cavities (Shen et al. Reference Shen, Ai, Zhao, Yan and Liu2022). It is believed that this flow topology after treatment is more likely to lead to intracranial embolization, as it results in very low velocities twice per cycle (when the circulation changes sign). However, for the largest curvature of the parent vessel (K$+$/AR$-$/N$-$ and K$+$/AR$-$/N$+$) and high values of $Re_{PV}$, this inversion of circulation is not observed, as was the case for spherical aneurysms (Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021).

Figure 7. Velocity fields $\boldsymbol {u}$ after treatment with FDS for four geometries and for $\overline {Re}_{PV} = 414$ at four time points of the cycle: A (yellow), the flow in the parent vessel accelerates; B (red), the flow rate in the parent vessel is maximum; C (green), the flow in the parent vessel decelerates; D (blue), the flow rate in the parent vessel is minimum. The background colour represents the out-of-plane velocity and the grey scale of the arrows represents the magnitude of the in-plane velocity.

For the high aspect ratio, the flow topology varies with parent vessel curvature, $\kappa$, as well as with neck size, $D_{neck}$. For low curvature (K$-$/AR$+$/N$-$ and K$-$/AR$+$/N$+$), independently of the neck size, the flow in the sac remains attached, as shown in figure 7 (second row). The flow enters the sac at the proximal edge, expands in the sac and exits at the distal edge. The velocity magnitude increases with parent vessel flow, but never reaches high enough inertia for flow separation. For high parent vessel curvature, the flow topology is influenced by the size of the neck, as before treatment. For the smaller neck size (figure 7, third row), the treatment with FDS did not result in a change in flow topology. Throughout the cycle, a counter-rotating vortex exists, moving deeper in the cavity as parent vessel flow increases. When flow decelerates, the vortex loses strength and moves towards the neck. This flow topology, presented here for $\overline {Re}_{PV} = 414$ and observed for all values of $\overline {Re}_{PV}$ in this geometry (K$+$/AR$+$/N$-$), was also observed for spherical aneurysms at high $De$ (Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021). For this geometry and those with low curvature, circulation does not reverse during the cycle. Finally, for the case with the high curvature and large neck (K$+$/AR$+$/N$-$) (figure 7, bottom), the flow enters the cavity at the centre of the neck aiming towards the fundus and recirculates on both distal and proximal walls. The velocity in the sac increases when the flow in the parent vessel accelerates. During deceleration, two clear recirculation areas can be observed above the proximal and distal edges, breaking down at the end of deceleration.

Unlike before treatment, the flow topology greatly varies between the different geometrical and haemodynamic configurations. These observations differ from those of Moriwaki et al. (Reference Moriwaki, Tajikawa and Nakayama2020), where for all geometries the flow remained attached after treatment with FDS. This difference can be due to the fact that they performed the experiments with steady inflow and that the aneurysms were on straight vessels. Another experimental study (Bouillot et al. Reference Bouillot, Brina, Ouared, Lovblad, Farhat and Pereira2015), with pulsatile inflow but also straight vessels, observed a change in circulation only for the lowest porosity stent ($\phi \approx 70\,\%$). On the other hand, Cantón et al. (Reference Cantón, Levy, Lasheras and Nelson2005) characterized the intra-aneurysmal velocities after the deployment of one to up to three high-porosity (non-FDS) stents. Even with three stents, where the reduction of velocity was dramatic, flow separation always occurred. For this patient-specific case, the aneurysm was located on a parent vessel with high curvature, which confirms our observations.

4.2. Inertial and unsteady effects

The flow patterns can be categorized into four topologies (figure 8, top): (1) no separation between the flow in the sac and the flow in the parent vessel; (2) two opposing vortices (inflow located at the centre of the neck); (3) separation of the flow in the sac for part of the cycle (inversion of circulation during the cycle); and (4) counter-rotating vortex. Velocity profiles at the neck of a representative example of each flow topology are presented in figure 8.

Figure 8. Velocity profile at the neck for K$-$/AR$+$/N$+$ (a), K$-$/AR$-$/N$+$ (b), K$+$/AR$+$/N$+$ (c) and K$+$/AR$+$/N$-$ (d) during the acceleration phase of the flow in the parent vessel (line becoming darker as the velocity in the parent vessel increases) for $\overline {Re}_{PV} = 414$ (i.e. $Q_{PV}=300$ ml min$^{-1}$).

In the case of no separation, the flow enters and exits the sac at the proximal and distal edges, respectively. As the velocity in the parent vessel accelerates (going from light grey to dark grey in figure 8a), the velocity at the neck increases and the peak of the velocity profile shifts towards the centre of the neck. However, despite high inflow velocities, separation never occurs for theses geometries. As described in Sobey (Reference Sobey1980), two pressure gradients influence the flow at the proximal edge: the first is due to the expansion at the cavity, leading to the deceleration of the flow, and the second, opposed to the first, is caused by acceleration along the cardiac cycle. The expansion-based gradient, which depends on $Q^2(t)$, is positive for an expanding section, whereas the unsteady gradient, which depends on $\dot {Q}(t)$, is negative when parent vessel flow accelerates. Thus, separation will occur when the magnitude of the pressure gradient induced by the acceleration of the flow becomes lower than the magnitude of the expansion-based pressure gradient. The magnitude of the gradient resulting for the flow in the parent vessel increases for the first part of the acceleration phase and then decreases. However, as it decreases in the second part of the acceleration phase (i.e. systole), the flow keeps increasing and the expansion-based gradient keeps increasing until becoming dominant, resulting in separation of the flow. In these geometries with low parent vessel curvature and high aspect ratio (K$-$/AR$+$/N$-$ and K$-$/AR$+$/N$+$), the magnitude of the adverse pressure gradient never becomes large enough to result in separation. However, for the same curvature of the parent vessel but a lower aspect ratio (K$-$/AR$-$/N$+$), flow separation is observed for part of the cycle suggesting the impact of the sac geometry on the expansion-based pressure gradient. Figure 8(b) shows the velocity profiles at the neck as the velocity in the sac increases for this aneurysm geometry. At first, the flow in the sac is attached, and velocities are low. As velocity increases, the inflow at the proximal edge decreases until becoming negative when flow separates. After separation, the inflow velocity, located at the distal edge, increases, as does the outflow velocity at the proximal edge. This difference of flow patterns for two models with the same curvature and neck size demonstrates the importance of the sac geometry.

As the parent vessel curvature increases, the velocities at the neck become higher (figure 8c,d). Increased curvature tends to shift the velocity profile in the parent vessel towards the entrance of the sac, increasing the inflow in the aneurysmal sac. Figure 8(d) represents the velocity profiles in a case of a counter-rotating vortex maintained throughout the cycle (similar to the pre-treatment flow pattern). As the velocity in the parent vessel accelerates, the velocity at the neck increases but does not shift. Finally, for the extreme geometry with the high curvature, large aspect ratio and neck (K$+$/AR$+$/N$+$), the flow enters the sac at the centre of the neck, as shown in figure 8(c). This flow topology is very different from the others, as was before treatment. A study by Ouared et al. (Reference Ouared, Larrabide, Brina, Bouillot, Erceg, Yilmaz, Lovblad and Mendes Pereira2016) on 12 patients (9 fully occluded) suggested that the shift of the inflow zone from proximal to the middle of the neck facilitates aneurysm occlusion. Chong et al. (Reference Chong, Zhang, Qian, Lai, Parker and Mitchell2014) obtained similar conclusions from CFD simulations of four occluded and four non-completely occluded aneurysms and were able to define flow patterns more favourable for occlusion: reduction of inflow speed and central diversion of the inflow. These results suggest that the flow topology observed for K$+$/AR$+$/N$+$, where the inflow is coming through the centre of the neck, is highly favourable for aneurysm occlusion. Another study concluded that higher flow rates in the sac and larger inflow areas seem to delay the occlusion of the aneurysm (Su et al. Reference Su, Reymond, Brina, Bouillot, Machi, Delattre, Jin, Lövblad and Vargas2020). Similarly, using CFD in patient-specific geometries, Mut et al. (Reference Mut, Raschi, Scrivano, Bleise, Chudyk, Ceratto, Lylyk and Cebral2015) observed that the ‘fast occlusion’ group had lower post-treatment mean velocities and inflow rates than the ‘slow occlusion’ group. These observations support the fact that a counter-rotating vortex after treatment with FDS is detrimental for the complete occlusion of the sac.

The analysis of the inflow at the neck highlighted the impact of geometrical parameters, mostly the sac aspect ratio and the parent vessel curvature, on flow topology in the sac. However, as previously described by Barbour et al. (Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021), this topology is also influenced by inertial ($Re$ and $De$) and unsteady ($Wo$) effects. Figure 9(a) shows the circulation in the sac throughout the cycle, for a geometry in which flow separates at some point of the cardiac cycle (K$-$/AR$-$/N$+$). The circulation was normalized by its amplitude over the cycle and plotted for different $Re$ and $Wo$. As $Re_{PV}$ increases (from light pink to dark pink in figure 9a) while keeping the sac geometry constant, separation occurs earlier in the cycle, consistent with our previous results (Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021). This can be explained by the fact that the expansion-based pressure gradient is more sensitive to an increase in $Q(t)$ than the pressure gradient induced by the acceleration in the parent vessel, as they depend on $Q^2(t)$ and $\dot {Q}(t)$, respectively. A comparison of $\varGamma (t)$ for the same $Re$ and two values of $Wo$ suggests that the separation tends to occur slightly later in the cycle for the highest value of $Wo$. This observation confirms the work by Sobey (Reference Sobey1980), in which the separation time was found to increase with Strouhal number (the Womersley number in this study) while keeping $Re$ constant. It is also in agreement with our previous observations for spherical aneurysms (Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021).

Figure 9. (a) Normalized time-averaged circulation in the sac across the cardiac cycle period for K$-$/AR$-$/N$+$ and for three values of $\overline{Re}_{PV}$, as well as for $Wo = 2.5$ and $Wo = 3.2$. (b) Time-averaged circulation in the sac as a function of $De$. The marker shape corresponds to the sac $AR$, triangle for AR$-$ and circle for AR $+$, and the colour corresponds to the flow topology in the sac.

Figure 9(b) presents the time-averaged vorticity $\langle \varGamma \rangle$ as a function of $De$ for all geometries. The flow topology (as defined in figure 8) is illustrated by colour. First, it is important to notice that, as before treatment, the circulation in the sac is linear with $De$, but varies with flow topology. For K$+$/AR$+$/N$+$ (figure 8, (2)), for which two opposing vortices are observed in the aneurysmal sac, the circulation remains very low, even for high $De$. For low values of $De$, circulation in the sac remains very low for all flow topologies except for the K$-$/AR$+$/N$+$ geometry (figure 8, (1)). For this geometry, even if the flow in the sac remains attached, the circulation strongly decreases with increasing $De$. Despite very high absolute value of the circulation, the flow never separates. This result shows once more that, in addition to the strong impact of parent vessel curvature, the sac aspect ratio affects the flow topology. Moreover, higher velocities entering the sac are not the only factor impacting flow separation in the sac.

The two other geometries exhibiting high value of circulation are K$+$/AR$+$/N$-$ and K$+$/AR$-$/N$+$ (figure 8, (3) and (4)). For these two geometries and at low $De$, the flow in the sac only separates for part of the cycle (figure 8, (3)), but when $De$ increases, the vortex is maintained throughout the cycle (figure 8, (4)) resulting in higher time-averaged circulation.

5.1. Effect of the haemodynamic and geometric parameters

The analysis of the different haemodynamics metrics highlights the complex and interdependent role of the different geometrical and haemodynamic parameters. The study of haemodynamics metrics in spherical aneurysm (Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021; Chassagne et al. Reference Chassagne, Barbour, Chivukula, Machicoane, Kim, Levitt and Aliseda2021) demonstrated the relatively small influence of the Womersley number compared with the Dean number. Thus, $Wo$ was not included in the following statistical analysis and only three parameters were considered: the Dean number $De$ (from 0 to 368), the size of the neck $D_{neck}$ (3 or 5 mm) and the sac aspect ratio $AR$ (0.4, 1.0 or 1.6). Focusing on the effect of these parameters and their interactions ($De*D_{neck}$, $De*AR$ and $D_{neck}*AR$), a model was fitted on the values of the time-averaged velocity magnitude $\langle \bar {u}\rangle$ in the aneurysm before and after treatment. Figure 10 presents the part of the variation of $\langle \bar {u} \rangle$ (before and after treatment) induced by the variation of each parameters of the model (within the range previously defined), i.e. $De$, $D_{neck}$, $AR$ and their interactions. About 3/4 of the total variance of $\langle \bar {u} \rangle$ can be explained by these six parameters, both before and after treatment ($R^2 = 0.72$ and $R^2 = 0.75$, respectively). For both conditions, 1/3 of the total variation of $\langle \bar {u}\rangle$ is a consequence of variation of $De$ alone. An increase in this parameter leads to an increase of flow in the sac, due to both the increased velocity in the vessel and the shift in velocity profile towards the neck induced by the curvature of the parent vessel. The Dean number was highlighted as the most impactful parameter in our previous studies (Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021; Chassagne et al. Reference Chassagne, Barbour, Chivukula, Machicoane, Kim, Levitt and Aliseda2021). A numerical study by Xu et al. (Reference Xu, Wu, Yu, Lv, Wang, Karmonik, Liu and Huang2015), performed on idealized spherical aneurysms with varying curvatures of the parent vessel, also demonstrated that before and after treatment, inflow velocity and flow rate in the sac increased with increasing curvature of the parent vessel. Based on CFD simulations of 47 patients treated with FDS, Chen et al. (Reference Chen, Zhang, Tian, Li, Zhang, Zhang, Liu and Yang2019) found a correlation between regions with inflow jets after treatment and regions that are not occluded post-treatment. These results tend to suggest that an increased $De$ would be detrimental for the occlusion of the aneurysmal sac after treatment with FDS.

Figure 10. Percentage of the variation of $\langle \bar {u}\rangle$ induced by the three parameters $De$, $D_{neck}$ and $AR$ and their interactions before (in black) and after (in grey) treatment with FDS.

Before treatment, almost half of the variation can be traced to the geometry of the sac ($D_{neck}$, $AR$ and $D_{neck}*AR$) and the interaction between inertial and geometrical parameters ($De*D_{neck}$, $De*AR$) was less influential. However, after treatment, the influence of purely geometrical parameters decreases and is compensated by an increase in the effects of $De*D_{neck}$ and $De*AR$. Overall, the analysis of variance for the velocity in the sac before and after treatment suggests that the influence of $De$ alone or coupled with geometrical parameters increases after treatment. However, it is important to note that this approach is built from $n=112$ measurements and did not consider the case K$+$/AR$+$/N$+$, as the flow topology is very different from the others. Indeed, this approach is based on the assumption of a linear relationship between the variation of a parameter and the variation of the value of interest (i.e. the time-averaged velocity magnitude), an assumption that would fail for K$+$/AR$+$/N$+$. Thus, the conclusions here are valid only if, before treatment, a single counter-rotating vortex is observed in the sac.

The second approach to characterize the effect of the treatment is to compare the haemodynamics before and after treatment. Chen et al. (Reference Chen, Zhang, Tian, Li, Zhang, Zhang, Liu and Yang2019) concluded that the reduction of velocity in the sac was the only factor affecting treatment outcome. Similarly, Paliwal et al. (Reference Paliwal, Damiano, Davies, Siddiqui and Meng2017) observed a slightly stronger reduction in velocity for successful treatments. Pereira et al. (Reference Pereira2013) used the MAFA ratio (i.e. the decrease in velocity with respect to the decrease in flow rate in the parent vessel) to predict the outcomes of the treatment with FDS for 21 patients. A numerical study has determined that a reduction in the inflow volume of 91.5 % leads to complete occlusion (Chong et al. Reference Chong, Zhang, Qian, Lai, Parker and Mitchell2014) while another study set the threshold at a reduction of 30 % for total occlusion (Ouared et al. Reference Ouared, Larrabide, Brina, Bouillot, Erceg, Yilmaz, Lovblad and Mendes Pereira2016). There is for now no consensus threshold, but it is accepted that a larger reduction is more likely to result in complete occlusion. Another metric to characterize the effect of the treatment with FDS on haemodynamics could be the time-averaged circulation in the sac, $\langle \varGamma \rangle$. Figure 11(a) presents the magnitude of the circulation after treatment as a function of its magnitude before treatment: this metric can be used as an estimation of the stagnation of the platelets in the sac and, thus, as a rough estimate of the thrombogenic potential. The colour of the markers represents the flow topology after treatment, as previously described in figure 8, and the shape of the marker corresponds to the sac $AR$ (triangles for $AR=0.4$, stars for $AR=1$ and disks for $AR = 1.6$). Notice that, independent of the sac $AR$, the reduction in circulation is very high ($90\,\%$) when the circulation before treatment is low, which is in agreement with the fact that strong inflow could be detrimental to thrombus formation (Chen et al. Reference Chen, Zhang, Tian, Li, Zhang, Zhang, Liu and Yang2019; Su et al. Reference Su, Reymond, Brina, Bouillot, Machi, Delattre, Jin, Lövblad and Vargas2020).

Figure 11. (a) Ratio between the magnitude of the time-averaged circulation $\langle |\varGamma |\rangle$ after and before FDS treatment, as a function of $De$, for the three aspect ratios. (b) Effect of the three parameters $De$, $D_{neck}$ and $AR$ and their interactions on the reduction of the magnitude of the time-averaged circulation ($R^2 = 0.55$).

Treatment with FDS is also very efficient at reducing circulation for the case K$+$/AR$+$/N$+$, for which no stable vortex was observed pre-treatment. Similarly, for high aspect ratio and low curvature of the parent vessel, the treatment strongly reduces circulation. For this geometry, the flow after treatment remains attached to the parent vessel throughout the entire cycle. Finally, the treatment effect is less strong if circulation is high pre-treatment and, after treatment, there is separation of the flow in the sac for part or the entire cycle. These cases mostly correspond to aneurysms for which the curvature of the parent vessel is high. Comparing different values of $AR$, it seems that treatment is successful at reducing circulation by 90 % for any curvature and neck size if $AR$ is low, but it is less efficient for higher $AR$.

The same analysis of variance approach, described earlier in this section, was implemented to assess the influence of each parameter and their interactions on the reduction of circulation in the sac induced by FDS treatment (figure 11a). However, a linear model with interactions is less adequate for the description of the variation of the reduction in circulation induced by the treatment ($R^2 = 0.55$). All the parameters have a negative effect, which means that increasing any of these parameters leads to a lower reduction of the circulation (i.e. less effective treatment). This analysis highlights the detrimental effect of $De$, alone or combined with the neck size and aspect ratio. The two geometrical parameters ($D_{neck}$, $AR$ and $D_{neck}*AR$), alone (and even more so combined together), have a smaller influence on the effect of the treatment. Increasing the neck size and/or the aspect ratio results in a small decrease in the reduction of the circulation in the sac. More interestingly, the interaction between these geometrical parameters and $De$ has a stronger effect on treatment.

Shapiro et al. (Reference Shapiro, Becske and Nelson2017), for example, concluded that lower dome-to-neck ratios were associated with treatment failure. In the present study, the highest dome-to-neck ratio would correspond to a high $AR$ and a small neck. For the high curvature of the parent vessel, this model shows the lowest reduction of the velocity and no change in the flow patterns compared with pre-treatment. Other studies (Cebral et al. Reference Cebral, Mut, Raschi, Hodis, Ding, Erickson, Kadirvel and Kallmes2014; Chung et al. Reference Chung, Mut, Kadirvel, Lingineni, Kallmes and Cebral2015) observed that aneurysms partially occluded after treatment had significantly larger necks than those fully occluded. However, Sarrami-Foroushani et al. (Reference Sarrami-Foroushani, Lassila, MacRaild, Asquith, Roes, Byrne and Frangi2021) observed no significant difference in velocity reduction in patients with large neck (>10 mm) or large $AR$ compared with small neck and $AR$. These observations are consistent with the results of our statistical analysis, where purely geometrical parameters do not seem to have a strong impact. Finally, as for other studies, we identified $De$, via an increase in the curvature of the parent vessel and $Re$, as the most influential parameter. For instance, Meng et al. (Reference Meng, Wang, Kim, Ecker and Hopkins2006) characterized stagnation in stented aneurysms with different parent vessel curvatures and observed lower stasis in aneurysms on higher-curvature parent vessels. Inflow in the sac was found to be larger for a parent vessel with increasing curvature (Liou et al. Reference Liou, Li and Wang2008; Augsburger et al. Reference Augsburger, Farhat, Reymond, Fonck, Kulcsar, Stergiopulos and Rüfenacht2009; Barbour et al. Reference Barbour, Chassagne, Chivukula, Machicoane, Kim, Levitt and Aliseda2021). Also, it is important to note that for our idealized aneurysms, the sac is always located on the outer convexity of the vessel. This location of the aneurysm with respect to the parent vessel has been identified as less favourable for successful embolization of the sac after FDS treatment (Sunohara et al. Reference Sunohara2021). A study comparing the clearance of aneurysmal sac on the outer convexity versus the concave side of the parent vessel also observed significantly faster clearance for the aneurysm on the outer convexity (Epshtein & Korin Reference Epshtein and Korin2018). As previously explained, in a curved vessel, the velocity profile is shifted towards the outer part of the vessel, i.e. towards the aneurysmal sac if located on the outer convexity. Thus, the flow is impinging on the stent with a larger angle (almost orthogonal to the stent surface), while for a straight vessel the flow is not directly impinging on the stent. These topologies are sometimes referred to as ‘shear-driven’ or ‘inertia-driven’ flow, respectively. In their study, Augsburger et al. (Reference Augsburger, Farhat, Reymond, Fonck, Kulcsar, Stergiopulos and Rüfenacht2009) observed that flow reduction was smaller for ‘inertia-driven’ flow. Indeed, a flow in the parent vessel directed towards the sac leads to stronger inflow into the sac, but also is less affected by the placement of the stent (as the pressure gradient is more orthogonal to the stent surface), which is why $De$ is such an influential parameter. This suggests that stent porosity should be lower for aneurysms exhibiting strong inflow in the sac before treatment (Lieber, Stancampiano & Wakhloo Reference Lieber, Stancampiano and Wakhloo1997). An approach to a solution could be to deploy more than one FDS in the vessel, which has been shown to result in higher occlusion rate (Waqas et al. Reference Waqas, Vakharia, Gong, Rai, Wack, Fayyaz, Snyder, Davies, Siddiqui and Levy2020).

5.2. Effect of FDS porosity

The predominant effect of the Dean number in the post-treatment haemodynamics can be explained by the fact that curvature of the parent vessel does not only impact the flow but also the FDS geometry. Flow-diverting stents are metal cylindrical meshes that are meant to be slightly larger (uncompressed) than the parent vessel, in order to apply a radial force on the artery to maintain the location of deployment. To quantify how FDS porosity was impacted by the geometry of the parent vessel and the aneurysmal sac, synchrotron X-ray micro-tomography (ESRF, Grenoble, France) of flow phantoms treated with FDS was performed to obtain high-resolution geometries of the deployed stents. Two examples of reconstruction of the stent geometry are shown in figure 12(a). Porosity of the FDS at the neck was then computed from these reconstructions. It was found to increase with parent vessel curvature: $\bar {\varPhi }_{K-} = 0.65 \ [0.59\unicode{x2013}0.72]$ and $\bar {\varPhi }_{K+} = 0.70 \ [0.66 \unicode{x2013} 0.74]$, for low and high curvature of the parent vessel, respectively, which is consistent with previous characterizations (Makoyeva et al. Reference Makoyeva, Bing, Darsaut, Salazkin and Raymond2013). This increase of porosity is another reason why the Dean number plays such an important role in the haemodynamics after treatment. An increase in $De$ results in an increased inflow in the sac, but also in a reduction of the stent porosity, which reduces the effect of the treatment. However, curvature is not the only parameter influencing the stent porosity. As the stent unconstrained diameter is slightly larger than the artery diameter, it expands at the neck. Consequently, a larger neck area results in a larger stent expansion at the neck. This is illustrated in figure 12(b), which presents the distribution of the measured stent porosity at the neck for three different geometries: K$-$/AR$-$/N$+$, K$+$/AR$-$/N$+$ and K$+$/AR$-$/N$-$. However, the variation of porosity induced by the change of neck size is smaller than that induced by the change of curvature of the parent vessel. These results demonstrate that the stent porosity is influenced by the geometry of the aneurysm. However, the FDS porosity can also be influenced by the way the FDS is deployed in the vessel (Roloff & Berg Reference Roloff and Berg2022).

Figure 12. (a) Stent geometries for two aneurysm models reconstructed from synchrotron X-ray micro-tomography images. (b) Distribution of the porosity at the neck for three different models, highlighting the variation of porosity with neck size $D_{neck}$ and curvature $\kappa$ of the parent vessel.

The fact that the treatment is not efficient enough at reducing the flow in the sac is a combination of two factors: first, before treatment the inflow in the sac was high (i.e. a strong inflow jet) and, second, that the stent porosity is not low enough to counter this strong inflow (which tends to impinge orthogonally on the stent). This is why the curvature of the parent vessel plays such an important role as it leads to both detrimental effects. Many previous experimental studies investigated the effect of the stent on haemodynamics for straight vessels, which is the more favourable case for flow reduction as there is no jet entering the cavity in this geometrical configuration. Bouillot et al. (Reference Bouillot, Brina, Ouared, Lovblad, Farhat and Pereira2014) observed, in an aneurysm attached to a straight vessel, that low porosities of around 70 % led to averaged velocities below 20 % of the pre-treatment velocities. They also observed that porosities higher than 90 % only reduced the velocity to 60 to 30 % of its pre-treatment value, which is likely insufficient for successful occlusion of the aneurysm. Another numerical study, of patient-specific geometries, has concluded that a porosity lower than 70 % was more favourable for thrombus formation (Zhang et al. Reference Zhang, Wang, Kao, Flórez-Valencia and Courbebaisse2019). A study of rabbits concluded that a metal coverage of 35 %, which corresponds to the highest value of metal coverage measured in the present study, was enough to induce full occlusion in 95 % of aneurysms (Wang et al. Reference Wang, Huang, Hong, Li, Fang and Liu2012). Similar flow reduction was observed by Yu et al. (Reference Yu, Matsumoto, Shida, Kim and Ohta2012) for PIV measurements in a straight parent vessel with steady inflow. They observed that the velocity reduction was already very high ($\approx 94\,\%$) for high porosity ($\phi =80\,\%$), and that it reached 98 % for $\phi = 64\,\%$. Another numerical study (Ouared et al. Reference Ouared, Larrabide, Brina, Bouillot, Erceg, Yilmaz, Lovblad and Mendes Pereira2016) suggested that a mean porosity of 83 %, resulting in a reduction of intra-aneurysmal velocity by 70 %, was sufficient to achieve stable occlusion of the sac. More generally, a retrospective study of patient data concluded that higher metal coverage (i.e. lower porosity) was correlated with better treatment outcomes (Kole et al. Reference Kole, Miller, Cannarsa, Wessell, Jones, Le, Jindal, Aldrich, Simard and Gandhi2019). Even if the porosity measured in our study was on the low range, it was not always low enough to strongly reduce the inflow into the sac. Indeed, a porosity of around 70 % does not seem to be low enough to counter the inflow jet into the sac, for geometries where the inflow pre-treatment is already high. Finally, an experimental and numerical study of the flow in aneurysms on straight vessels (Bouillot et al. Reference Bouillot, Brina, Ouared, Yilmaz, Lovblad, Farhat and Mendes Pereira2016), treated with stents with different porosities, showed that the reduction of the velocity in the sac is not linear with stent porosity. Measurements performed in a similar geometry (Roszelle et al. Reference Roszelle, Babiker, Hafner, Gonzalez, Albuquerque and Frakes2013) compared the performance of a single FDS ($1 -\phi \approx 30\,\%$) with the combination of three very high-porosity stents ($1-\phi \approx 10\,\%$ for each stent). They observed that for low flow rates in the parent vessel, the FDS (low porosity) was as efficient as the combination of three very high-porosity stents at reducing velocities in the sac. However, it underperformed for high values of flow rates. These results demonstrate that porosity is not the only parameter that plays a role in the efficacy of the treatment, and that permeability of the mesh should also be characterized.