A comparative study on computational fluid dynamic, fluid-structure interaction and static structural analyses of cerebral aneurysm

ABSTRACT Computational Fluid Dynamics (CFD) and Fluid-Structure Interaction (FSI) are increasingly used to predict the hemodynamic and structural behaviors of cerebral aneurysms (CAs) whilst a less-pursued method is the static structural analysis (SSA) using the finite element method. In this paper, hemodynamic parameters including the flow velocity and wall shear stress predicted by CFD and FSI are compared whilst structural parameters including the wall displacement and wall stress predicted by SSA and FSI are compared for four patient-specific CA models under different systolic/diastolic pressures. The predicted distribution patterns of the same parameters for the same CA model under different pressures are similar. However, the percentage differences of the maximum hemodynamic parameters increase with increasing pressure. Conversely, the percentage differences of the maximum structural parameters decrease from a few to less than 1.5% when the systolic/diastolic pressure changes from 120/80 to 180/110 mmHg. The ratio of the computation times for CFD, SSA and FSI is typically 75:1:165. If the maximum wall stress under either ongoing or temporary hypertension is the most critical factor for immediate rupture and, thus, clinical treatment of CAs, SSA can provide a low cost and efficient predictions close to those of FSI for assessing the rupture risk.


Introduction
Cerebral aneurysm (CA), also known as intracranial aneurysm, is a localized dilation of the blood vessel. Previous autopsy studies reported that CA occurrence ranges from 0.2-9.9% (Vega et al., 2002;Wiebers et al., 2004). The rupture of CAs can lead to subarachnoid haemorrhage (SAH) in which blood enters the space surrounding the brain. About 50% of the SAH cases are fatal (Broderick et al., 1994;Kaminogo et al., 2003;Winn et al., 2002). In general, a clinician would formulate the treatment of a pre-rupture CA based on the following factors of the patient: age, overall health, medical history, signs and symptoms, tolerance for specific medications, the morphology of the CA, etc.
The underlying mechanism of the formation, progression and rupture of CAs has not been well-understood. However, some factors have been found to increase the rupture risk. For instance, hypertension and smoking are regarded to be strong risk factors (Brisman et al., 2006;Feigin et al., 2005;Juvela et al., 1993;Korja et al., 2014). Family history is another major risk factor. A study CONTACT Kam Yim Sze kysze@hku.hk based on 113 patients with unruptured CAs found that the rupture risk of patients with family history is 17 times higher than those without history (Brown et al., 2008). The aneurysm morphology is also believed to be a vital factor. A clinical study based on 5,720 Japanese patients reported that an independent daughter sac can increase the rupture risk (Investigators, 2012). Morphologic irregularity can result in a threefold rise in the rupture risk. In particular, aspect ratio (AR), defined as the ratio of the aneurysm height to the equivalent diameter ( = circumference/π ) of the aneurysm neck, is found to be an important factor of CA ruptures (Lindgren et al., 2016). The hemodynamics inside the aneurysm is believed to play a vital role in the formation and progression of CAs. For low wall shear stress (WSS, i.e. the fluid friction at the solid boundary), rupture risk is aggravated and long-term damage is inflicted due to abnormal alignment of endothelial cells (Irace et al., 2004;Sho et al., 2001;Shojima et al., 2004). On the other hand, the mechanical properties of the arterial wall play an impor-tant role in the prediction of aneurysm rupture (Couade et al., 2010;O'Rourke et al., 2002) and there are experimental measurements on the rupture stress of CAs. For instances, the aneurysm rupture stress in the range of 2-3 MPa was reported (Scott et al., 1972); the rupture stresses at the aneurysm neck and dome are 1.2 and 0.5 MPa, respectively, were reported (Steiger et al., 1989); rupture stress of CAs varies from 0.73-1.9 MPa due to material anisotropy was reported (MacDonald et al., 2000).
In the current study, CFD and FSI are conducted for four unruptured patient-specific CA models under systolic/diastolic pressure settings 120/80, 140/90, 160/100 and 180/110 mmHg. The predicted hemodynamic fluid velocity and WSS from the two methods are compared. The results show that the distribution patterns of the same parameters in the same CA models under the same pressure setting are similar. While the CFD predictions do not vary with the pressure, the FSI predictions decrease and the differences between CFD and FSI predictions increase with increasing pressure. SSA based on the finite element method is also attempted by prescribing the peak systole pressure on the vessel and aneurysm walls. The distribution patterns of the wall displacement and effective stress in the same CA models under the same systolic/diastolic pressure predicted by SSA and FSI are similar. While the FSI and SSA predictions increase with the increasing pressure, the percentage difference between the FSI and SSA predictions decreases with increasing pressure. Under all pressure settings, the percentage differences are within 3%.
A noteworthy observation is the extreme ratio of the computation times for CFD, SSA and FSI which is typically 75: 1: 165. If the wall stress under either ongoing or temporary hypertension is regarded as the most critical parameter for the immediate rupture risk and, thus, the clinical treatment of Cas, SSA can provide low cost and speedy predictions close to those of FSI for assessing the risk.

Methodologies
The four patient-specific models used in the current study are shown in Figure 1. The methods for obtaining the models are described in Section 2.1. CFD, SSI and FSI are briefly reviewed in Sections 2.2, 2.3 and 2.4, respectively. The settings for the solution procedures and the finite element meshes are discussed in Sections 2.5 and 2.6, respectively.

Patient-specific aneurysm models
The four patient-specific CA models used in a previous paper on assessing the effects of aspect ratio, wall thickness and hypertension on the CAs using FSI are adopted (Sun et al., 2019). The consent of the patients and the approval by the Institutional Review Board of the Hospital were obtained. The medical images of the models were obtained by computed tomography angiography (CTA) or magnetic resonance angiography (MRA). Based on the image data, three-dimensional models were constructed by selecting the solid-fluid boundary for every slice in the software Mimics 18.0 (Materialise, Leuven, Belgium). They were then exported in STL format and smoothed by SOLIDWORKS 2017 and its reverse engineering addin ScanTo3D. Since the resolution of the image was not sufficient to resolve the aneurysm wall, the wall thickness t w was assumed to be uniform at 0.3 mm which is about 10% of the inlet diameter (Riley et al., 1992). The morphology can be seen in Figure 1 and some elementary information of the patient-specific models are provided in Table 1. It can be seen that A and B are sidewall Cas whilst C and D are bifurcation Cas. In our previous work (Sun et al., 2019), the inlet sections of these models were artificially lengthened so as to secure fully developed flows before the aneurysms. While lengthened and un-lengthened inlet sections lead to different predictions at the inlet entrance, the difference in the aneurysm sag is of secondary effect. To reduce the computing time, the inlet sections are not lengthened here.

Computational fluid dynamics (CFD)
The governing equations of the incompressible fluid flow are the Cauchy momentum equation and the continuity equation. Under the Eulerian description in which the computational mesh is fixed in space, they can be expressed as where ∇ = [∂/∂x, ∂/∂y, ∂/∂z] T is the gradient operator, t is the time, ρ f is the fluid density, u f is the fluid velocity vector, σ f is the fluid stress tensor and the body force is ignored. The fluid stress can be decomposed as σ f = τ -I 3 p in which τ is the shear stress tensor and p is the pressure. The shear stress can be related to the shear strain rate (Morrison, 2001), i.e.
where µ is the viscosity coefficient. For Newtonian flow, µ is a constant. WSS can be obtained by projecting τ onto the normal-tangential coordinate system defined at the solid boundary.
Previous studies indicate that hemodynamic parameters of CAs are sensitive to the pulsatile feature of the blood flow (Caro et al., 2012;Elad & Einav, 2003;Ku, 1997). On the other hand, the velocity profile across the inlet section has little effect on WSS in regions away from the inlet and a uniform inlet velocity profile can be used for simplicity (Campbell et al., 2012). In the current study, the time-varying inlet blood volume flow rate in Figure 2 is adopted (Ku et al., 1985). Different outlet boundary conditions may lead to large variations of WSS (Grinberg & Karniadakis, 2008;Ramalho et al., 2012). In computational hemodynamics of CAs, the outlet boundary is usually prescribed with a pulsatile pressure waveform with systolic/diastolic pressure equal to 120/80 mmHg (Tang et al., 2015;Torii et al., 2009;Valencia et al., 2008).
The pulsatile pressure waveform of normal range 120/80 mmHg in Figure 2 is employed at the outlet(s) in the current study (2007;Valencia & Torres, 2017, 2013bTorii et al., 2006;Lee et al., 2013a). It can be seen that the peak systolic flow and peak systolic pressure occur at around 0.07 and 0.32 s, respectively. The pressure at any time t under the hypertension condition p hypertension (t) is obtained from that of the normal range p normal (t) based on the linear scaling, i.e.
in which d is the vessel's internal diameter. As the maximum Reynolds number among all current models is 638, which is based on the systolic velocity 0.331 m/s and the 7.1 mm inlet diameter of Patient A, the flow can be considered as laminar.

Structural Analysis and static Structural Analysis (SSA)
The governing equation for solid materials is the momentum equation. Under the Lagrangian description in which the computation mesh follows the material deformation, the equation reads in which gravity is ignored; ρ s , u s and σ s are the density, velocity and Cauchy (also, known as true) stress tensor of the solid material, respectively. Comparing the two momentum equations (1) and (6), the only difference is the advective derivative term u·∇u in (1). SSA of the vessel and aneurysm walls can be considered by the static finite element method in which u s vanishes. On the boundary conditions, the structural nodes at the inlet and outlet sections of the blood vessels are restrained from moving whilst the peak systole pressure is prescribed to the inner surfaces of the vessel and aneurysm walls. The arterial and aneurysmal walls are assumed to be an incompressible Mooney-Rivlin hyperelastic material with the following strain energy function.
where c 1 = 0.429, c 2 = −0.119, c 3 = 0.585, c 4 = 0.579 and c 5 = 0.564 MPa (Valencia et al., 2015); I 1 and I 2 are the first and second invariants of the Cauchy-Green deformation tensor, respectively. To give ideas on how the nonlinear stress-strain relation varies, the true stress σ 1 is plotted against the engineering strain which is comparable to 0.602 N/mm ( = 7 MPa×0.086 mm) and 0.621 N/mm ( = 1.7742 MPa ×0.35 mm) used in (Valencia et al., 2009) and (Valencia et al., 2013), respectively, which also present CFD, computational structural dynamics (CSD) and FSI predictions for CAs under normal blood pressure and linear elastic material assumption.

Fluid-Structural interaction (FSI)
As the fluid boundary often moves in FSI, the fluid motion cannot be conveniently described by a fixed Eulerian mesh. Meanwhile, a Lagrangian mesh following the fluid particles would become highly distorted and entangle quickly. In this light, fluids in FSI is commonly considered by the Arbitrary-Lagrangian-Eulerian (ALE) description under which the momentum balance equation becomes (Bazilevs et al., 2010): in which the only undefined term u g is the velocity of the computational grid or mesh. Comparing the momentum balance equations in Equation (1), Equation (6) and Equation (8), it can be noted that the ALE description turns into the Eulerian and the Lagrangian descriptions when the mesh is static, i.e. u g = 0, and the mesh follows the fluid particles, i.e. u g = u f , respectively.
In the present FSI studies, the vessel and aneurysm walls are considered by the dynamic finite element method. The boundary conditions at the inlet and outlet of the CA are the same as used in CFD and SSA. On the fluid-structure interface, i.e. the inner surface of the vessel and aneurysm walls, the no-slip condition was assumed, i.e.
where d f and d s are fluid and solid displacements, respectively. Furthermore, the tractions of the fluid and solid domains at the interface are in equilibrium, i.e.
where n f and n s ( = −n f ) denote the unit outward normal vectors of the fluid and solid at the interface.

Numerical solutions
In FSI, the fluid traction, formed by pressure and WSS, causes the wall deformation which, in turn, alters the fluid flow. When the structural deformation is small and can barely affect the fluid flow, the flow can be solved by CFD in which the structure is rigid and the structure deformation can then be solved by applying the fluid traction predicted by CFD. This uncoupled analysis is often termed the one-way coupling in FSI. When the structural deformation is large enough to alter the fluid flow considerably, the two-way coupling should be opted for. Since two-way coupling is commonly used in the FSI analysis of CAs, it will be used in the present study. All the CFD, FSI and SSA simulations were conducted using the software ADINA 9.4.1 installed in a workstation with an Intel Xeon CPU (E5-1960 v4 @ 3.20 GHz with 8 cores) and 128 GB RAM. In the simulations, the fluid domains were modeled by Flow-Condition-Based-Interpolation 4-node tetrahedral elements whilst solid domains were modeled mainly by 4-node 3D shell elements with a small portion of 3-node 3D shell elements. The linearized matrix equation was solved by the direct computing method. The implicit Euler scheme with 1 ms time step was used for all CFD and FSI analyses. With a larger time step size, the computing time generally reduces but the risk of divergence in the iterative solution procedure increases. The present time step size was chosen by balancing the computing time and the iterative convergence. By default, the convergence criteria including force tolerance < 0.01 and displacement tolerance < 0.01 were adopted.

Mesh convergence study
Mesh convergence studies for the four models had been previously conducted to ensure the prediction yielded by a mesh would be practically the same as that by a finer mesh (Sun et al., 2019). In this regard, the maximum wall effective stress (σ e ) predicted by the structural elements had been taken to be the indicators for the mesh convergence. The effective stress σ e is a positive scalar commonly used as material failure criteria. It is given as in which σ ij s are the Cauchy stress components in the aneurysm wall. Finally, the meshes generated by the 0.3 mm maximum mesh edge length setting had been adopted for the models of Patients A, B and C whilst the mesh generated by the 0.20 mm setting had been adopted for the model of Patient D. These meshes, generated by ADINA, would also be used here and their information are summarized in Table 2.

Results and discussions
In Section 3.1, hemodynamic parameters in the fluid domain at peak systolic flow predicted by CFD and FSI are compared. In Section 3.2, the wall parameters at peak systolic pressure predicted by SSA and FSI are compared with those predicted by SSA. The predictions in CFD and FSI become periodic after two cardiac cycles and predictions in the third cycle were extracted. It should be remarked that the CFD and FSI predictions for the four patient-specific models under the systolic/diastolic pressure settings 120/80 and 160/100 mmHg had been previously reported (Sun et al., 2019).

Comparison between CFD and FSI
The fluid velocity and WSS predicted by CFD and FSI at peak systolic flow under different systolic/diastolic pressure settings are compared. As changes in the distribution patterns of the hemodynamic and structural parameters from 120/80 and 180/110 mmHg are progressive, only those at the two extremes systolic/diastolic pressure settings are shown.

Fluid velocity at peak systolic flow
The fluid velocities predicted by CFD and FSI at peak systolic flow are shown in Figures 4-7 for the models of Patient A to D. The distribution patterns of the velocity field in both types of analysis and different systole/diastole pressures are highly similar. For all models,     blood flows enter the aneurysms at the distal necks and leave at the proximal neck. A vortex is observed inside each of the aneurysms. It can also be seen that the maximum flow velocities occurred inside the parent vessels but not the aneurysms. Table 3 lists the maximum fluid velocities predicted by CFD and FSI. As the volume influx does not vary with the systole/diastole pressure and the vessel/aneurysm walls are taken to be rigid in CFD, the fluid velocity predicted by CFD do not vary with the systole/diastole pressure. On the other hand, the fluid velocity predicted by FSI drops successively with the systole/diastole pressure because the vessel/aneurysm walls undergo temporal elastic dilatation under the consideration of FSI. For the same volume flow rate, the larger the vessel and aneurysm, the smaller would be the flow velocity. For the same reason, the difference of the maximum flow speed increases with increasing dilation of the vessel/ aneurysm walls or, equivalently, systolic/diastolic pressure. In the four patient-specific models, the percentage difference increases successively from 3.55 ∼ 4.06% to 4.62 ∼ 5.44% when the systole/diastole pressure changes from 120/80 to 180/110 mmHg.

Wall shear stress
WSS is the fluid friction acting on the vessel and aneurysmal walls. Following the definition in Equation (3), WSS is equal to the viscosity coefficient times the gradient of the tangential fluid velocity normal to and at the solid boundary. For low WSS, rupture risk is aggravated and long-term damage is inflicted due to abnormal alignment of endothelial cells (Irace et al., 2004;Sho et al., 2001;Shojima et al., 2004). In the CFD and FSI predictions, WSS on the aneurysm wall is lower than that on the vessel wall as depicted in Figure 8, suggesting that the aneurysmal wall would be more vulnerable to long-term damage than the vessel wall. For the same model, both CFD and FSI give similar WSS distribution patterns. WSSs at the inlets are not shown because the prescribed fluid velocity profile is spatially uniform over the inlet cross-sections, leading to large non-physical WSS. Table 4 shows the WSS at control points taken in proximal necks, distal necks and domes of the patient-specific models, see Figure 1, at the peak systole flow. Same as the maximum fluid velocity listed in Table 3, WSSs predicted by CFD do not vary with the systole/diastole pressure. On the other hand, those predicted by FSI drops as the pressure increases due to the dilation of the vessel and the aneurysm. In the four patient-specific models, the percentage difference between the CFD and FSI predictions increases from 2.9 ∼ 13.7% to 12.7 ∼ 53.6% when the systole/diastole pressure changes from 120/80 to 180/110 mmHg. The difference is more obvious at the dome than the neck due to the dilation of the CA sag. Noteworthily, the peak WSSs predicted by CFD in two out of the three considered CAs in (Valencia et al., 2009(Valencia et al., , 2013, see the brief description given at the end of Section 2.3, under normal blood pressure are smaller than those predicted by FSI. Here, the WSSs predicted by CFD are unanimously larger than those by FSI in all the four CAs.

Computing time
The computing times over three cardiac cycles for CFD and FSI are given in Table 5. It is well-expected that FSI is more computing-time consuming than CFD. The ratio of the computing times consumed by CFD and FSI ranges from 43.5% to 47.2% for different CA models. In particular, the computing time consumed by CFD  is practically unaffected by the systole/diastole pressure setting.

Comparison between SSA and FSI
In the previous section, it is noted that the WSS due to the fluid motion is in the order of 10 Pa, see Table 4, which is small compared with the blood pressure which is in the order of 10 kPa ( ∼ 75 mmHg). Moreover, the instantaneous pressure inside the aneurysm sag is very close to uniform, see Figure 9. It is also noted that the maximum wall displacement and wall stress always occur at or very close to peak systolic pressure at which the wall motion is close to stationary. These observations lead to the hypothesis that the maximum wall displacement and wall stress can be predicted by SSA of the solid domain with the systolic pressure applied over the inner surfaces of the vessel and aneurysm walls. The benefits of using SSA are significant including shorter computing time as well as lower hardware and software requirements. This section examines the above hypothesis by using the four patient-specific models under the four systolic/ diastolic pressure settings. Same as the hemodynamic parameters, the distribution patterns of the structural parameters from 120/80 and 180/110 mmHg change gradually. Thus, only the distribution patterns at the two extremes systolic/diastolic pressure settings are shown.

Wall displacement
The wall displacement distribution patterns predicted by SSA and FSI at peak systolic pressure are shown in  Figure 10. In each of the models, the patterns predicted by SSA and FSI for the same systole/diastole pressure are visually indistinguishable. Table 6 further shows that maximum displacement predicted by FSI at peak systolic pressure is slightly larger than that by SSA probably due to the ignored impingement force which causes the directional change of the fluid velocity. In the four patient-specific models, the percentage difference in the predicted maximum wall displacements decreases from 0.26 ∼ 1.22% to 0.11 ∼ 0.37% when the systole/diastole pressure changes from 120/80 to 180/110 mmHg.

Wall effective stress
Predictions of solid material failures are often based on the stress state. Among the material failure criteria, the effective stress in Equation (12) is commonly used for ductile materials. In recent years, it has also been adopted as a criterion for predicting the aneurysm rupture risk   (Volokh, 2008) and immediate rupture may occur when the effective stress of the wall exceeds the strength of the aneurysm wall (Isaksen et al., 2008).
This subsection compares the wall effective stresses predicted by SSA and FSI. Figure 11 shows that effective stress distribution patterns predicted by SSA and FSI are visually indistinguishable in each of the models at the same systole/diastole pressure. Table 7 lists the predicted maximum wall effective stresses. Same as the maximum wall displacement, the SSA prediction is consistently smaller than the FSI prediction probably due to the ignored impingement force which causes the directional change of the fluid velocity. In the four patient-specific models, the percentage difference in the predicted maximum wall effective stress decreases from 2.18 ∼ 2.58% to 1.19 ∼ 1.30% when the systole/diastole pressure changes from 120/80 to 180/110 mmHg.
The dome of the CA is the common rupture side due to wall thinning as the CA grows. Though the employed imaging equipment cannot resolve the wall thickness, it is still of reference value to compare the predicted wall effective stresses at the dome listed in Table 8. They are around one-eighth of the maximum effective stresses reported in Table 7. In the four patient-specific models, the percentage difference in the predicted wall effective stress at the dome decreases from 1.50 ∼ 1.97% to 0.49 ∼ 0.95% when the systole/diastole pressure changes from 120/80 to 180/110 mmHg.
Besides the effective stress, the first principal stress is another commonly-used scalar stress measure for predicting material failures. Same as the effective stress, the distribution patterns, maximum valves and the values at the dome of the first principal stresses predicted by SSA and FSI are close. The percentage difference between the SSA and FSI predictions also diminishes as the pressure increases. To keep this paper in reasonable length, the related results are not presented explicitly.
Effective stresses predicted by CSD and FSI under normal blood pressure are compared in (Valencia et al., 2009(Valencia et al., , 2013, see the brief description given at the end of Section 2.3. In (Valencia et al., 2009), the peak effective stresses predicted by CSD and FSI at the dome of aneurysm 1 are ∼ 1130 and ∼ 610 kPa, respectively, whilst the peak effective stresses predicted by CSD and FSI at the dome of aneurysm 2 are ∼ 245 and ∼ 470 kPa, respectively. In (Valencia et al., 2013), the peak maximum effective stresses predicted by CSD and FSI are ∼ 425 and ∼ 460 kPa, respectively whilst the maximum effective stresses at peak systole predicted by CSD and FSI are ∼ 428 and ∼ 461 kPa, respectively. The large differences in the above predictions are markedly different from the present results in which the differences of the maximum SSA and FSI predictions as well as the differences at the aneurysm domes in all the four CA models are unanimously less than 3%.

Computing time
Since FSI deals with the temporal analysis of the solid domain, the fluid domain and their coupling over three cardiac cycles whilst the SSA only deals with a one-off static analysis of the solid domain, it can be expected that the computing time consumed by SSA would be significantly lower than that by FSI. Table 9 gives the computing times consumed in the two analysis methods. In all CA models, the computing times consumed by SSA are only 0.39 ∼ 0.74% of those by FSI.

Limitations
There are limitations in the current study. The present work investigates the effect of hypertension by increasing the systolic/diastolic pressure alone without concurrent changes of mechanical properties of the walls, cardiac output and blood viscosity (or non-Newtonian flow effects). In terms of the clinical perspective, testing and analysis with brains of human cadavers show that arterial wall actually breaks at very high pressure, perhaps several times the largest systolic pressure observed (Ciszek et al., 2013). Moreover, even from a purely computational perspective, dynamics and rupture risk may depend on the precise location of the blood vessel in the brain (Sarrami-Foroushani et al., 2015). All these issues provide a glimpse of the complexity of the problems under investigation. Further intensive efforts would definitely yield more fruitful results.

Conclusions
In this paper, four patient-specific cerebral aneurysms (CA) models are analyzed by Computational Fluid Dynamics (CFD), Static Structural Analysis (SSA) and Fluid-Structure Interaction (FSI) under normal and hypertension systole/diastole pressure settings. The hemodynamics parameters predicted by CFD and FSI at peak systolic flow are first compared. The fluid velocity and wall shear stress (WSS) distribution patterns predicted by both methods are qualitatively similar. While the CFD predictions do not change with the systole/diastole pressure, the FSI predictions decrease with increasing systole/diastole pressure due to the dilation of the CA. The percentage difference between the maximum fluid velocity predictions increases from 3.6 ∼ 4.1% to 4.6 ∼ 5.4% and the percentage difference of WSS at the neck and dome increases from 2.9 ∼ 13.7% to 12.7 ∼ 53.6% when the pressure changes from 120/80 to 180/110 mmHg.
Driven by observations that the maximum stress always occurs at or very close to the peak systolic pressure and the fluid pressure is close to uniform and far larger than the WSS, it is hypothesized that the predicted maximum wall displacement and stress predicted by FSI may be close to that by SSA prescribed with the peak systolic pressure. The hypothesis is then tested. The distribution patterns of the wall displacement and stress predicted by SSA and FSI at the peak systolic pressure can hardly be distinguished. The SSA predictions are consistently and marginally smaller than the FSI predictions probably due to the ignored impingement reaction force for driving the directional change of the fluid velocity over the solid boundary in SSA. The percentage difference of the maximum displacements decreases monotonically from 0.3 ∼ 1.2% to 0.1 ∼ 0.4% and the percentage difference of the maximum effective stresses decreases monotonically from 2.2 ∼ 2.6% to 1.2 ∼ 1.3% when the systolic/diastolic pressure increases from 120/80 to 180/110 mmHg. The smallness of the differences is in favor of the hypothesis.
A noteworthy observation is the extreme ratio of the computation times consumed by CFD, SSA and FSI typically at 75: 1: 165. If the wall stress under either ongoing or temporary hypertension is regarded as the most critical parameter for the immediate rupture risk and, thus, the clinical treatment of CAs, SSA can provide low cost and speedy predictions close to those of FSI for assessing the risk. The closeness of the FSI and SSA predictions is probably due to the small physical size of CAs. Noteworthily, CFD, SSA and FSI predictions for three patient-specific abdominal aortic aneurysm (AAA) models with wall stiffness Et w = 2.7 MPa × 1.5 mm = 4.05 N/mm under normal blood pressures are compared in (Leung et al., 2006). Leung et al. (2006) concluded that the flow-induced pressure variations are too small to cause a noticeable difference in the wall stress. Though AAAs are much larger than Cas, the conclusion is in line with ours probably because the flow structures in the AAAs are much simpler than those in CAs. For instance, vortexes are localized to the aneurysm walls. Finally, it should be remarked that SSA cannot provide any hemodynamic parameters which are believed to play a vital role in the formation and progression of CAs.

Disclosure statement
No potential conflict of interest was reported by the author(s).

Funding
This work was supported by Innovation and Technology Commission of the Hong Kong Special Administrative Region Government [Grant Number ITS/150/15] .