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SCIENCE CHINA Physics, Mechanics & Astronomy

© Science China Press and Springer-Verlag Berlin Heidelberg 2010 phys.scichina.com www.springerlink.com

*Corresponding author (email: [email protected])

• Research Paper • May 2010 Vol.53 No.5: 944–953

doi: 10.1007/s11433-010-0190-2

Numerical simulations of the discontinuous progression of cerebral aneurysms based on fluid-structure interactions study

MA XiaoQi1, WANG YueShe1*, YU FangJun1 & WANG GuoXiang1,2

1 State Key Laboratory of Multiphase Flow in Power Engineering, Xi’an Jiaotong University, Xi’an 710049, China 2 Department of Mechanical Engineering, The University of Akron, Akron, OH 44325-3903, USA

Received December 2, 2009; accepted March 18, 2010

Investigations into the characteristics of hemodynamics will provide a better understanding of the pathology of cerebral aneu-rysms for clinicians. In this work, a steady state discontinuous-growth model of the cerebral aneurysms was proposed. With the assumption of the fluid-structure interaction between the wall of blood vessel and blood, a fluid-structure coupling numerical simulation for this model was built using software ANSYS and CFX. The simulation results showed that as the aneurysm volume increased, a blood flow vortex came forth, the vortex region became asymptotically larger, and eddy density became gradually stronger in it. After the emergence of the vortex region, the blood flow in the vicinity of the downstream in the an-eurysms volume turned into bifurcated flow, and the location of the flow bifurcated point was shifted with the aneurysm vol-ume growing while directions of the shear stress applied to two sides of the bifurcated point were opposite. The Von Mises stress distribution along the wall of aneurysm volume decreased in the prior period and increased gradually in the later period. The maximum stress was in the neck of the volume and the minimum was on the distal end in the whole process of growth. It was shown that as the aneurysm increased the maximum deformation location of the aneurysm, vertical to the streamline, was transferred from the distal end of the aneurysm to its neck, then back to its distal end of the aneurysm again.

cerebral aneurysm, fluid-structure interaction, wall shear stress, discontinuous progression

PACS: 83.80.Lz, 47.63.-b, 87.15.Aa

1 Introduction

Intracranial aneurysm, a kind of bump of cerebral aneu-rysm，results from an anomalous lesion of the arterial wall tissue, predominantly shaped as a capsule which is attached on an artery and is connected with the artery [1]. Since the 1950’s, many theoretical and experimental studies on the formation and rupture of cerebral aneurysm have been re-ported in the related literature. However, restricted with the conventional calculation method, the vessel wall was as-sumed to be rigid and the coupling interaction between the blood and the wall was ignored. Although previous studies have obtained some preliminary conclusions, there are still

many deficiencies. Fluid-structure interaction mechanics is a branch of en-

gineering mechanics related to fluid dynamics and solid mechanics, an interdisciplinary study of various behaviors of deformable solid under the influence of a flow field and the interaction between the fluid and solid.

As the fluid-structure interaction mechanics develops, it provides a new method to study the formation and rupture of the aneurysm. In order to analyze the interaction between the blood and the wall properly, and to avoid the assump-tions that aneurysm wall was set to be rigid, that the influ-ence of blood flow on the vessel wall was only considered, and that the deformed effect of vessel wall on blood flow was ignored, a novel simulation method of flow field in cerebral aneurysms will be proposed by the use of fluid-

MA XiaoQi, et al. Sci China Phys Mech Astron May (2010) Vol. 53 No. 5 945

structure interaction mechanics. So far as we know, only a few researchers have calcu-

lated the interaction between the blood and the vessel wall. Valencia [2] compared results of the elastic blood vessel model with results of the former rigid blood vessel model. The results showed much difference between the two mod-els, and the elastic vessel wall significantly affected the homodynamics results of the stenotic artery. Li [3] pre-dicted the homodynamics behavior of the different stenotic arteries by the fluid structure interaction method. Chat-ziprodromou [4] and Feng et al. [5] predicted the growth of aneurysm by the FSI method. Di Martino et al. [6] simu-lated an interaction of fluid and solid about the abdominal aortic aneurysm under the pulsant flow. He predicted the abdominal aortic aneurysm (AAA) rupture by the use of contrast between Von Mises stress and average shear strength of the AAA initially. Scotti [7] studied the effect of the asymmetric geometry and thickness of aneurysm on the vascular wall under the condition of laminar flow. Khalil et al. [8] calculated the aneurysm by the FSI method under the condition of turbulent flow.

Assuming that the vessel wall is linear elastic, the pre-sent study aims to develop a discontinuous growth model for cerebral aneurysm. By the use of this model, various behaviors at the different stages of aneurysm growth in a steady flow state will be simulated. A comparison of the flow field in the aneurysm, distribution of shear stress and distribution of Von Mises stress at the vessel wall will be made. It will help investigations into the mechanism of hemodynamics in the aneurysm growth, which in turn will provide the theoretical guidance for further studies about the formation and rupture of aneurysm.

2 Discontinuous-growth model of the cerebral aneurysms

In order to investigate the flow field of the cerebral aneu-rysms in different stages of the progression, eight typical stages in the progression of the internal cerebral aneurysms (ICA) are chosen as samples of this discontinuous growth model, and their corresponding geometric structures are shown in Figure 1. For the fluid-structure interactions be-tween the wall of a blood vessel and blood, a fluid-structure coupling numerical simulation for each geometric structure is made by using Software ANSYS and CFX. Such infor-mation as flow field profile in an artery and the intensity and distribution of shear stress at the blood vessel will be obtained, and comparisons will be made. Those will be sig-nificant for studying the mechanism of hemodynamics in the growth of cerebral aneurysms.

2.1 Physical models of discontinuous growth

In the present study a sample of the ICA is chosen. The de-

tails of the sampling of actual physiological parameters are shown in Table 1 [9,10], where D, L and T denote the geo-metrical parameters of the model, U and P denote boundary condition. Initiated from a model of a healthy vessel wall (M1 in Figure 1 and Table 2), the calculation of discon-tinuous growth of aneurysm will be made.

Figure 1 illustrates the geometrical profiles of 8 discon-tinuous progression models. The ratio H/N of the width of the neck of aneurysm N and the height of aneurysm H serves as a parameter to characterize the volume and profile of the aneurysm. By adjusting the value of the height of aneurysm H, the influence of geometrical factors of aneu-rysm on the flow field will be considered during the com-parison of various stages of progressions.

Table 1 Physiological parameters of ICA

Item Symbol Value

Inner diameter of ICA (mm) Diameter (D) 4.4

Inlet velocity of ICA (m s−1) Velocity (U) 0.36

Outlet pressure of ICA (Pa) Pressure (P) 12500

Density of blood (kg m−3) Density (ρ) 1056

Length of ICA (mm) Length (L) 30

Thickness of ICA (mm) Thickness of arterial wall (T) 0.3

Viscosity of blood (Pa s) Dynamic viscosity (μ) 0.0035

Figure 1 Sections of models in the X=0 plane.

Table 2 The geometry parameters of models

Geometry parameters of models (mm) Model Width of aneurysm

neck N Height of aneurysm

volume H H/N

M1 3 0 0 M2 3 0.35 0.117

M3 3 0.54 0.19

M4 3 0.78 0.26

M5 3 1.05 0.35

M6 3 1.2 0.4

M7 3 1.5 0.5

M8 3 3 1

946 MA XiaoQi, et al. Sci China Phys Mech Astron May (2010) Vol. 53 No. 5

2.2 Blood flow control equations

Perktold et al. [11] found that the Newtonian fluids instead of non-Newtonian fluids in the numerical calculation caused the error within 2% when the size of a blood vessel is larger than 0.5 mm under the actual physiological condition. So, in this study the blood is considered as Newtonian fluid. The viscosity of blood is supposed to be constant. The wall thickness of arterial aneurysm is identical. The effect of wall thickness of arterial aneurysm on hemodynamics is ignored. The fundamental conservation equations control-ling flow include incompressible fluid continuity equations and Navier-Stokes equations. Because heat transfer effect is not accounted for, energy conservation equations are omit-ted from the conservation equations. Meanwhile, due to small sizes, the gravity effect is neglected from the mo-mentum conservation equations. Therefore, the blood flow equations in an artery are shown below.

The mass conservation equation is expressed as

0,u∇⋅ = (1)

conservation of momentum reads

( ) .g

uu u u p T

tρ ∂⎛ ⎞+ − ⋅∇ = −∇ +∇⋅⎜ ⎟∂⎝ ⎠

(2)

The viscous stress component T reads

2 ,T Dη γ•⎛ ⎞= ⎜ ⎟

⎝ ⎠ (3)

( )1,

2TD u u= ∇ +∇ (4)

where u is velocity of fluid, ug is the velocity of the grid, p is the pressure, ρ is density, T is stress tensor, γ is shear

rate and η is viscosity.

2.3 Vessel wall (solid) constitution equations

The tissue of the vessel wall is considered linear elastic in-stead of the traditional rigid wall, which makes the blood flow close to the real state. The actual material of the vessel wall is viscoelastic, which significantly affects the attenua-tion of the arterial wave. Especially for shorter aortic the influence is less obvious, because the vessel wall is mainly composed of the elastin and collagen fibers. The tensile modulus of the former is small as from 3×105 N/m2 to 6×105 N/m2. Its tensile strength is low, its hysteresis loop is very small, and its stress relaxation is extremely unnoticeable. So, its mechanical properties approximate to the ideal elastic body whereas the tensile modulus of the latter is larger, ap-proximately the dimension of 109 N/m2，and its tensile strength is higher. When the radial expansion of the vessel wall increases and the strain is not significant, most of the collagen fibers are loose and curly. All the stresses are born

by the elastic fiber. When the radial expansion of the blood vessels increases the stress, the collagen fibers are straight-ened while its born stress becomes large and the blood ves-sel stiffness increases [12]. The aim of this paper is mainly to study the steady blood flow. Ignoring the effects of vessel wall creeping and stress relaxation may overestimate the risk of aneurysm rupture relative to the actual situation. Nevertheless, this assumption will not affect our under-standing of the internal flow field of aneurysm and instead this simplification will help grasp the predominant factors and get what we want. According to refs. [5–13], the value of the young’s modulus of Elasticity is set to 1.2 MPa and the Poisson ratio is set as 0.4 for a normal physiological blood vessel while one of the young’s modulus of Elasticity is set to 0.3 MPa and the Poisson ratio 0.4 for a diseased blood vessel. The aneurysm wall is set as uniform and iso-tropic, which is consistent with Hooke’s Law, the thin-walled elastic model. The solid model control equation is expressed as:

s s s ,σ ρ∇⋅ = a (5)

where sσ is the stress tensor of the vessel wall; sρ is the

density of the vessel wall; sa is the acceleration of the

vessel wall. At the interface of solid and fluid, the following equations should be met.

s f s s f f s f, , d d n n u uσ σ= ⋅ = ⋅ =

where d is the displacement, n is the boundary normal vec-tor, and the subscripts s and f denote solid and fluid, respec-tively.

2.4 Calculation process in the software of ANSYS and CFX

Regarding the FSI numerical simulation, the solid model control equations are solved by ANSYS while the fluid flow control equations are solved by CFX. The convergence re-sults can be obtained through the real-time exchange of data (as shown in Figure 2).

2.5 Model validation

The study predicts the swelling behavior of a typical inter-nal carotid artery (ICA) model in physiological conditions by fluid-structure interaction method. Geometric and boundary parameters of the model are denoted in Table 3. The inlet and outlet of the model are set to be fixed. Com-paring the results of this model with that of Chatziprodro-mou et al. [13] will yield the following result.

It is obvious that the blood vessel becomes expanded as shown in Figure 3. The results show that the diameter of blood vessel expands from 6.5 mm to 7.8 mm, which is ex-tremely close to Chatziprodromou et al.’s results (from 6.5 mm to 8 mm).


Figure 2 Calculation flow chart.

Table 3 Geometric and boundary parameters of models used

Item Nomenclature Value

Diameter of vessel (mm) Diameter (D) 6.5

The velocity of inlet (m s−1) Velocity (U) 0.1

The pressure of outlet (Pa) Pressure (P) 12500

Density of blood (kg m−3) Density (ρ) 1056

Length of vessel (mm) Length (L) 65

Thickness of vessel (mm) Thickness of arterial wall 0.75

Viscosity of blood (Pa s) Dynamic viscosity (μ) 0.0035

Figure 3 The expanded vessel.

Figure 4 shows the distribution of Von Mises stress along with the vessel wall. The Von Mises stress has changed greatly in the inlet and outlet of the vessel wall and the stress of the internal wall is higher than that of the outer wall. The immediate cause for this profile is that deforma-tion of the vessel tissue at the both ends and at the interior wall is larger than that of other parts. That case is consistent with the actual situation. This case has verified the validity to solve the model via FSI method.

3 Numerical results

3.1 Deformation of the vessel wall

After the fluid-structure interaction calculation, a marked deformation occurred in the maternal vessel of aneurysm

Figure 4 Distribution of Von Mises stress of the vascular wall.

and tumor. From the perspective of the discontinuous pro-gression, the last geometric distortion of aneurysm is the initial step for its further growth. By combining FSI calcu-lation of the viscoelastic properties of vessel wall, results of subsequent geometric distortion of the wall can be obtained from the given geometric figure, which accords to the actual physiological progression of aneurysm. So it is useful to predict the growth and rupture of the aneurysm by this method.

As shown in Figure 5, when the initial deformation of the aneurysm is not significant (H/N=0–0.26, denoted as M1–M4 at Figure 5), the maximum deformation location of the aneurysm, vertical to the streamline (named the Y axis), is at the distal end of aneurysm and its value increases gradually. When the deformation of the aneurysm is sig-nificant (H/N=0.35–0.5, denoted as M5–M7 in Figure 5), the maximum deformation location of the aneurysm in the Y-axis is transferred from the distal end of the aneurysm to its neck. When the deformation becomes larger (H/N=1), the maximum deformation location is back to the distal end of the aneurysm again.

Deformation of aneurysm in the Z axis will change the width of the neck of the aneurysm. For the model with a slight initial deformation (denoted as M1–M2 in Figure 5), when the width of the neck of the aneurysm decreases, the vessel wall around the aneurysm deforms a little toward the aneurysm. When the deformation of the aneurysm becomes significant (denoted as M3–M8 in Figure 5), the numerical results show that the width of the necks increases. Its de-formation range is the magnitude of 10−5 m, which can be neglected relative to the one in the Y axis of 10−4 m.

3.2 Flow field in an aneurysm

In this study, comparisons of flow field profile in 8 various geometrical models in plane X=0 are made. As shown in Figure 6, there is no vortex in M1–M3, in which the flow pattern is a stratified flow, and at whose distal end the flow


Figure 5 The deformation of flow field of M1–M8 along the Y axis.

field looks like a thicker boundary layer. From Model M4, a counterclockwise vortex emerges in it. Blood stream flows into the aneurysm from the downstream of the neck of the

tumor, and flows back to the maternal vessel along with the internal wall of the aneurysm. Because of the neck of the selected aneurysm model is relatively wide in this study, the


Figure 6 The distribution of flow field in the X=0 plane.

intensity of the reflux is weak, and the vortex appears evi-dently as a flat large eddy. As the aneurysm depth increases, the vortex becomes clearer and vortex velocity becomes

larger. Meanwhile, the wall shear stress is intensified. However, in any event the velocity of blood flow in the an-eurysm is less than the velocity of the mainstream.


For Models M4–M8, when blood stream flows through the neck of the aneurysm, one stream of blood flows inward into the volume of the tumor to present a counterclockwise vortex while another stream flows downstream along the maternal vessel as the main flow. The location of the bifur-cated point for two streams of blood flow is defined as the stagnation point. Compared to Model M4–M8, it is found that the stagnation point is unstable. The stagnation point was located in the downstream slightly above the neck of aneurysm in M4–M6 while the stagnation point was located

in the downstream neck of aneurysm in M7–M8. Two typi-cal flow bifurcated patterns were shown in Figure 7.

3.3 Distribution of shear stress

Figure 8 shows the distribution of wall shear stress of 8 models. It is shown that the location of the maximum stress occurred at the inflow side near the neck of aneurysm. The value of wall shear stress is about 3 MPa–5 MPa in the downstream of aneurysm while the normal stress is about

Figure 7 The flow field of M6 and M7 in the neck of aneurysm.

Figure 8 Distribution of wall shear stress in 8 models.


1.5 MPa–2.8 MPa in the healthy vessel wall. So the location of downstream neck of aneurysm is prone to rupture. The wall shear stress is close to 0 MPa, due to lack of shear stress at the distal end of aneurysm, which made vascular endothelial cells deform and degrade.

The wall shear stress of the downstream aneurysm wall of Models M5–M8 is depicted in Figure 9. Angle θ, which is between the connected line of one point at the aneurysm wall and the center point of the aneurysm neck and the plane through the neck serves as the coordination of the referenced point. The value of θ varies in the range of 0°– 90°. The positive direction of shear stress is clockwise along the boundary.

As shown in Figure 10, when θ varies from 0° to 90°, the value of wall shear stress decreases from the initial positive value in M5 and M6. When θ is about 23°, the value of the wall shear stress is 0. As θ increases continuously, the value of the wall shear stress becomes negative, and the direction of the stress is counter-clockwise. The value of the wall shear stress is always negative for all cases in M7 and M8. The direction of the stress is counter-clockwise. This phe-nomenon can be explained properly by the internal flow field of the aneurysm.

As shown in Figure 10, the location of Von Mises stress of the position corresponding to 0 in M5–M6 corresponds to

Figure 9 Wall shear stress point coordinates of the downstream wall of aneurysm.

Figure 10 The wall shear stress of downstream aneurysm wall in M5– M8.

the position of stagnation point in Figure 7. The flow direc-tions of two sides of the stagnation point are opposite, which results in opposite directions of shear stress. The po-sition of the stagnation point is in the neck of aneurysm in M7 and M8. The whole flow field in the aneurysm is eddy zone. The direction of the wall shear stress applied to the wall is consistent, named as counter-clockwise. So, the valve of wall shear stress of M7–M8 is negative in Figure 10.

3.4 Von Mises stress distribution of the vessel wall

For 8 models from different discontinuous growth progres-sions, the distribution of Von Mises stress along the volume of the aneurysm is predicted, shown in Figure 11. In order to understand the mechanism of aneurysm growth, the dis-tribution and variation of shear stress along the outline of aneurysm (curve abcd shown in Figure 11) are depicted.

From Figure 12, it is obviously found that the distribution of Von Mises stress at the inlet of curve ab is the same as the one at the outlet of curve cd, which shows that emer-gence and evolution of the aneurysm have little effect on the healthy vessel wall. In addition, the value of the Von Mises stress along the healthy vessel wall is about 80 kPa, while the Von Mises stress decreases monotonously to minimum and then increases monotonously in the initial inlet and the end of outlet along the mainstream. This profile results from the assumption that two ends of the vessel are set to be fixed during the numerical simulation. This boundary condition is reasonable for large deformation cases. For the aneurysm volume of curve bc of Models M1–M4 (as shown in Figure 12(a)), the Von Mises stress decreases gradually to a certain value below 80 kPa, which shows the Von Mises stress of aneurysm wall decreases in the initial progression of aneu-rysm. Its minimum valve reaches about 20 kPa, and the Von Mises stress of aneurysm wall is below the Von Mises stress of the healthy vessel wall. Nevertheless, the yield stress of aneurysm wall has been significantly reduced at this situa-tion. So the aneurysm wall is thought in danger of rupture. It is noticed that the Von Mises stress at the vicinity of points b and c occurs to maximum values, which indi-cates that stress concentration exists in the neck of aneu-rysm, and stress concentration becomes strong as the aneu-rysm volume increases. Such results are directly related to the ideal model used in this paper. Actually, the deforma-tion of the real aneurysm is slightly flat and uniform and the stress concentration is weakened consequently. The Von

Figure 11 Sketch of the outline of the aneurysm volume.


Figure 12 The distribution of the Von Mises stress along the curve abcd in 8 models.

Mises stress of the aneurysm wall increases from M5 to M8 in Figure 12(b), different from Figure 12(a). Its value is still below 80 kPa. As the volume of the aneurysm changes greatly, the mechanical properties of vessel tissue gradually degenerate. The rupture of aneurysm occurs easily. When the aneurysm continues to increase till the Von Mises stress reaches the limit of the aneurysm wall tissue, the aneurysm will rupture at any time once affected by other external dis-turbances.

Figure 13 shows the distribution of Mises stress contours for 8 models. It is obvious that the stress at the internal ves-sel wall is larger than the one at the outer vessel wall, which is caused by the non-uniform deformation of the maternal vessel. Although the Von Mises stress of the distal end of

the aneurysm relative to the healthy wall is less, the distal end of the aneurysm is likely to be broken in the clinical medicine. So, the Von Mises stress can’t be used alone as a criterion for judging the rupture of aneurysm. Thus, in the study of abdominal aneurysm, Raghavan et al. [14] found the variation of yield stress of vessel tissue is a very impor-tant factor which can affect the rupture of aneurysm. They also pointed out that the Yong’s modulus of Elasticity reached 4.66 MPa, which is three times larger than that of the normal vessel wall, while the yield stress is half of the normal vessel wall. Although a large Young’s modulus will reduce the wall stress, the yield stress will be smaller. The above combined effect leads to the final rupture. Since tis-sue and mechanical properties of the vessel wall are com-

Figure 13 The distribution of Von Mises stress of the section of aneurysm wall in 8 models along the streamline.


plicated, understanding of the mechanism of cerebral aneu-rysm is limited, and the data about the yield stress of aneu-rysm are scarce at the present time. Trapped by the current level of research, the author assumes the value, that is, the Von Mises stress of the aneurysm wall subtracts the yield stress, should be a criterion for judging the aneurysm rup-ture, and the greatest value region along the aneurysm is the most dangerous region, which corresponds to the distal end of the aneurysm.

4 Results and discussion

The discontinuous progression of the aneurysm is simulated in this study. Compared to the flow field in the volume of aneurysm, the distribution of wall shear stress and distribu-tion of Von Mises stress along the blood vessel in different growth stages, and the effect mechanism of hemodynamic factors on the aneurysm growth are obtained, which aids understanding of the growth and rupture of the aneurysm. The conclusions are presented below.

(1) In the growth of the aneurysm, a blood flow vortex emergences in the volume of the aneurysm, where the vor-tex velocity is below one of the mainstream. As the aneu-rysm increases, the vortex velocity slightly increases and the profile of the vortex becomes clearer.

(2) In the initial growth of aneurysm (M1–M4), the loca-tion of maximum deformation of aneurysm, vertical to the streamline, is at the distal end of the aneurysm. When the deformation of the initial models is significant(M5–M7), the location of the maximum shifts from the distal end of the aneurysm to its neck. When the deformation is more sig-nificant (M8), the maximum deformation location is back to its distal end of the aneurysm again.

(3) In the M4–M6, the stagnation point is located in the downstream aneurysm wall, while the stagnation point is located in the neck of aneurysm in a big spherical crown model (M7–M8). Two different locations cause many dif-ferences in the distribution of the wall shear stress, which affects the growth of aneurysm.

(4) The Von Mises stress of aneurysm wall decreases gradually from M1 to M4, and increases gradually from M5 to M8. In the whole process of aneurysm growth, the maximum Von Mises stress is in the neck of the aneurysm and the minimum Von Mises stress is in the distal end of the aneurysm. Meanwhile, the mechanical properties of aneu-rysm tissue have changed greatly during its growth, which leads to the easy aneurysm rupture. As the aneurysm vol-ume increases, the Von Mises stress of the aneurysm wall will reach the limit of the aneurysm tissue, and the aneu-rysm will rupture at any time. These data point to a new way for diagnosing and predicting aneurysm.

The simulation of the discontinuous progression of

aneurysm is made under the steady flow of blood, corre-sponding to the average effect of blood flow. To some ex-tent the rupture can also depend on the long-term effect of blood flow on the aneurysm wall. However, the actual ve-locity and pressure of blood are not consistent all the time. The pulse factors have an enormous effect on the flow field of the aneurysm [15,16]. For further studies, the authors will consider the pulse factor that affects the growth of the an-eurysm, and the model used in this study will be improved to approximate to the actual application in clinical medi-cine.

This work was supported by the National Natural Science Foundation of China (Grant No. 50976088).

1 Skirgaudas M, Awad I, Kim J, et al. Expression of angiogenesis fac-tors and selected vascular wall matrix proteins in intracranial saccular aneurysms. Neurosurgery, 1996, 39: 537–547

2 Valencia A, Villanueva M. Unsteady flow and mass transfer in mod-els of stenotic arteries considering fluid-structure interaction. Int Commun Heat Mass Transfer, 2006, 33: 966–975

3 Li M X, Beech B J, John L, et al. Numerical analysis of pulsatile blood flow and vessel wall mechanics in different degrees of stenoses. J Biomech, 2007, 40: 3715–3724

4 Chatziprodromou I, Tricoli A, Poulikakos D, et al. Haemodynamics and wall remodelling of a growing cerebral aneurysm: A computa-tional model. J Biomech, 2007, 40: 412–426

5 Feng Y, Wada S, Tsubota K, et al. A model-based numerical analysis in the early development of intracranial aneurysms. Conf Proc IEEE Eng Med Biol Soc, 2005, 1: 607– 610

6 Di Martino E S, Guadagni G, Fumero A, et al. Fluid-structure inter-action within realistic three-dimensional models of the aneurysmatic aorta as a guidance to assess the risk of rupture of the aneurysm. Med Eng Phys, 2001, 23: 647–655

7 Scotti C M, Shkolnik A D, Muluk S C, et al. Fluid-structure interac-tion in abdominal aortic aneurysms: Effects of asymmetry and wall thickness. BioMed Eng OnLine, 2005, 4: 1–22

8 Khalil M, Hanafer K, Josephbull L, et al. Fluid-structure interaction of turbulent pulsatile flow within a flexible wall axisymmetric aortic aneurysm model. J Mech B, 2009, 28: 88–102

9 Pedley T J, Luo X. The Fluid Mechanics of Large Blood Vessels (in Chinese). Xi’an: Shaanxi Normal University Press, 1994. 14–15

10 Liu Z, Li X. Hemodynamic Principles and Methods(in Chinese). Shanghai: Fudan University Press, 1997. 16–17

11 Perktold K, Peter R, Resch M, et al. Pulsatile non-newtonian blood flow simulation through a bifurcation with an aneurysm. Biorheology, 1989, 26: 1011–1030

12 Tian X, Bi P. Biomechanics Foundation (in Chinese). Beijing: Sci-ence Press, 2007. 94

13 Chatziprodromou I, Tricoli A, Poulikakos D, et al. Haemodynamics and wall remodelling of a growing cerebral anerysm: A computa-tional model. J Biomech, 2007, 40: 412–426

14 Raghavan M, Vorp D. Wall stress distribution on three-dimensionally reconstructed models of human abdominal aortic aneurysm. J Vasc Surgery, 2000, 31: 760–769

15 Zhang X, Yao Z, Zhang Y, et al. Experimental and computational studies on the flow fields in aortic aneurysms associated with de-ployment of AAA stent-grafts. Acta Mech Sin, 2007, 23: 495–501

16 Zhang X W, Zhang Y, He F. Fluid-structure interaction analysis of dissecting aneurysm and stent-graft model(in Chinese). Mech Q, 2007, 28: 592–598

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