Proceedings of the 1st Iberic Conference on Theoretical and Experimental Mechanics and Materials /

11th National Congress on Experimental Mechanics. Porto/Portugal 4-7 November 2018.

Ed. J.F. Silva Gomes. INEGI/FEUP (2018); ISBN: 978-989-20-8771-9; pp. 923-932.

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PAPER REF: 7398

NUMERICAL SIMULATION OF THE FLUID-STRUCTURE

INTERACTION TO EVALUATE THE STRESS IN ARTERIES

WITH ATHEROMAS

Concepción Paz1,2(*)

, Eduardo Suárez1,2, Christian Gil

1, Adrián Cabarcos

1

1School of Industrial Engineering, University of Vigo, Lagoas-Marcosende, 36310, Vigo, Spain

2Biofluids Research Group, Galicia Sur Heath Research Institute (IIS Galicia Sur). SERGAS-UVIGO, Spain

(*)Email: cpaz@uvigo.es

ABSTRACT

In this research, an analysis of the fluid-structure interaction in an artery damaged by

arteriosclerosis was carried out with Ansys software. The fluid side was solved with the

Fluent Inc. module, and the artery structural side with the Mechanical module. The aim of this

research is the study of the modification that arteriosclerosis causes on the blood flow, and the

effect of this modification on the walls of the arteries. Three different arteries were simulated:

a healthy artery and two damaged arteries with different degrees of arterial stenosis, 40% and

60%. The mechanical properties of the artery wall; and the rheological properties of the blood

flow, a non-Newtonian fluid, were modelled. The differences between the results of the three

simulations have been discussed in order to determine the effect of arteriosclerosis on the

artery and blood flow. The deposition of atheroma plaques increases the speed of blood flow,

which implies an increase in turbulence and shear stress. This increase supposes two main

risks: the damage that produces an increase of shear stress on the arteries; and this effort can

entail a detachment of parts of the atheroma plaque, which can produce the coagulation of the

blood inside the blood vessels and can cause a heart attack.

Keywords: CFD, FEM, arteriosclerosis, stenosis, hemodynamic, rheology, aorta.

INTRODUCTION

Several diseases, especially diseases related to the cardiac system, have increased since the

middle of the 20th century. One of the most important cardiac diseases is the arteriosclerosis,

characterized by narrowing of the arteries, increased thickness, and increased stiffness of the

walls of any artery. This obstruction is caused by plaque deposition, composed mostly of

cholesterol and fat. Depending on which arteries are affected, the arteriosclerosis can cause

several diseases such as ischemic heart disease, or chronic kidney disease. Some of the risk

factors are: smoke, hypertension, diabetes or cholesterol.

The hemodynamic studies, typically introducing catheters into arteries and veins from

different parts of the body, help in the understanding of the blood flow related diseases.

Computational methods, allow studying blood circulation without these procedures. The use

of Computational Fluid Dynamics (CFD) has been validated in several blood flow research:

Merih Cibis et al. [1], compare the shear stress obtained by magnetic resonance imaging

(MRI) and by CFD, obtaining very close results. Quaini et al. [2] validate a fluid-structure

interaction with a simulation of a passage of blood through an elastic opening. Khalil

Khanafer et al. [3], validate a CFD model simulating an aneurysm with previous in-vivo data,

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increasing the section of the artery. There are studies about the stress and deformation of the

artery and vein walls, and the interactions of both problems. Joao Janela et al. [4] focus on an

interesting part in development that is the Fluid Structure Interaction (FSI) coupling, in which

the structure interacts with the fluid, and vice versa. Philippe Reymond et al. [5] make a

comparison between the FSI coupling with an elastic wall, and a 3D model with a rigid wall.

Ashkan Javadzadegan et al. [6] study the effect of an aneurysm on the abdominal aorta using

FSI coupling.

Nowadays there are many computational studies of the cardiac system. Many of them

combine in-vitro and in-silico results, or simply collect CFD data that would be impossible, or

very difficult to measure, in an in-vivo study. Most of these studies focus on the fluid or the

structural part, very few cover both parts of the problem. In this work, the CFD software,

Ansys Fluent 18.0, has been used to study the fluid flow. The structural calculation of the

walls was solved by Finite Element Method (FEM), Mechanical, coupling with the fluid

dynamic problem. The obtained blood flow velocities and pressure behaviours, in a healthy

artery and in arteries with different degrees of arteriosclerosis, have been compared in this

research.

METHODOLOGY

In this study, three different geometries were used: a healthy artery, and two arteries with two

different degrees of arteriosclerosis, with 40% and 60% of stenosis. The dimensions of the

studied arteries correspond to a descending aorta artery and have been obtained from the

article [7], in which measurements are established for the aorta artery depending on gender,

age, etc. The geometries have been created with Rhinoceros, and have been imported into the

Ansys workbench. The healthy artery was simplified as a smooth cylinder, with an inner

diameter of 25mm, a length of 150mm, and wall thickness of 2mm, as shown Figure 1. The

stenosis produced by the atheroma deposit was modelled using a half-ellipsoid centred in the

middle of the geometry; the main dimensions of the artery were maintained constant.

Fig. 1 - Dimensions of the three studied arteries.

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The three geometries were imported into the DesignModeler module of Ansys Workbench,

where the separation between the structural and the fluid sides was made. Two different

meshes have been created for each of the arteries, one for the structural side and the other for

the fluid side. The meshes were created maintaining a balance between the mesh size and the

computational resources. For the creation of the fluid meshes, the fine meshing, and high

smoothing options have been chosen, to obtain the best possible quality. In the zone around

the atheroma, a smaller size mesh was created using size function, to better capture of the

pressure variations. A maximum mesh size of 1 mm was fixed to the superficial elements, and

the grow rate was set to 1.2. Using the inflation tool, 20 prismatic cell layers with a 1.2 grow

rate were created. After the meshes were created, the quality and mesh statistics were

checked, parameters summarized in Table 1.

Table 1 - Mesh statistics of the meshes used for the fluid side.

Healthy artery 40% stenosis 60% stenosis

Cell number 50000 88427 96013

Skewness Max. 0.04 0.75 0.76

Skewness Average 0.04 0.20 0.22

Orthogonal Quality Max. 0.999 0.998 0.993

Orthogonal Quality Average 0.998 0.86 0.85

in Figure 2 details of the mesh aspect are shown in the three arteries, and an example of the

boundary layer aspect, which is similar in the three cases, is shown in the upper right area.

Fig. 2 - Details of the meshes used in the fluid side.

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Analogously to the creation of the mesh of the fluid side, a mesh has been created for each

structural side. Although a very refined mesh is not necessary, it was created with good

quality. All the mesh parameters were maintained, only the sizing groups were modified, and

the inflation tool, which was not used.

BOUNDARY CONDITIONS AND SOLVER

The blood flow is a pulsating flow, so a specific User Define Function (UDF) reproducing the

pulsating flow conditions was created. There are different approaches parameterizing the

blood flow evolution curve, some of them are based in Womersley number [8], in which they

propose a sinusoidal function for the volumetric flow. In this study the blood flow curve is

based on the mass flow curve by Inga Voges et al. [9]. This curve was obtained with the help

of the equations developed by Gupta et al. [10]. They characterizes coughing, respiratory

flow, through the use of a series of equations based on the maximum peak flow rate (CPFR),

the time of this maximum peak (PVT), and the total expired volume (CEV). These parameters

and equations have been modified to achieve an adaptation to the simplified blood flow curve.

The parameters and equations used are the following:

����������� = ���� × ���� (1)

� = ������

(2)

�� =�������exp −�

� "Г(%�)���

(3)

�' =�'(� − 1.2)�+��exp,−(� − 1.2)

' -Г(%')'�+

(4)

�. = ���� < 1.2 (5)

�. = ��+�'�� ≥ 1.2

where the constant values fitted to this research:

a1 = 1.68; b1 = 3.338; c1 = 0.428; a2 = -0.0133; b2 = 6.8603; c2 = 0.30715 (6)

Due to the model imposed at the mass flow inlet condition, realistic pressure and velocity

evolutions were obtained, shown in Figures 3 and 4 respectively. In addition, the direction of

this flow was selected to normal to the surface (normal to boundary). In the output surface, a

pressure-outlet condition was used, with a constant value of outlet pressure, 13332 Pa was

chosen from [11]. All the walls are no-slip walls, so it was not necessary to modify any

parameter.

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Fig. 3 - Pressure evolution. Fig. 4 - Velocity evolution.

The blood, is a shear thinning no-Newtonian fluid, and was modelled with the Carreau-

Yasuda model [12], following the equation 7.

2(34) = 25 + (26−25) ∙ (1 + (834 )9):��9 (7)

The constants of the model were obtained experimentally [4]. The initial viscosity µ0, the final

viscosity µ∞, the deformation velocity γ, and the other parameters and properties used are

summarized in Table 2.

Table 2 - Blood properties.

Density 1060 kg/m3

Viscosity

Model Carreau

Time Constant, λ 3.313 s

Power-Law Index, n 0.3568

Zero Shear Viscosity, µ0 0.056 Poise

Infinite Shear Viscosity, µ∞ 0.0345 Poise

The solver used was a 3d, double precision, pressure based, transient, and with gravity

activated in the general conditions field. The SIMPLE scheme was selected. The simulations

started with first order, then were changed to Second Order for all the fluid variables. The

convergence of the simulation was controlled with monitors of wall shear stress, inlet and

outlet pressures, and mass flow. The time step was fixed to 4·10-5

s, and 12500 time steps

were calculated, with a maximum of 20 iterations per time step. The fluid side in Fluent was

coupled with the structural side of the artery in the Ansys Static Structural module, with the

workbench platform. In the coupling was indicated that the solution of the blood flow,

specifically the pressure on the walls, was an input parameter in the structural side. The load

and transfer of the information data was automatically done during the simulation process.

Once the coupling was done, standard parameters of the structural solver were defined.

RESULTS AND DISCUSSION

The analysis of the velocity evolution along the entire domain is of interest. For this purpose,

during the simulation the velocity contours in the middle plane of the domain were extracted.

In Figure 5, which is a compilation of several figures, the evolution of blood flow throughout

the domain is observed. In the healthy artery, the speed is uniform and the speed distribution

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is radial due to the non-slip condition. The pulse of speed is appreciable in the three cases,

during the first instants the speed increases and then decreases, although the decrease is

slower. In the cases with arteriosclerosis, the blood flow is completely different due to the

presence of the stenosis. The effect of the non-slip condition on the walls is still appreciated

but the velocity varies without following a clear pattern. In areas before the stenosis, the flow

behaviour is similar to the healthy artery. In areas near atheroma, the flow begins to adapt due

to the change of section, and the speed increases due to the smaller section, and recover the

previous values downstream. The inertia of the flow, combined with the zones of low and

high pressures, produces a series of small vortices in which the flow recirculates and acts in a

random manner. The more severe stenosis, the more severe is this effect, therefore the

potential damage to the walls of the artery increases.

Fig. 5 - Evolution of velocity contours for the three arteries.

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The shear stress is directly proportional to the velocity. Therefore, in the arteries affected by

arteriosclerosis, the shear stress should be greater than in a healthy artery, this is verified in

the Figure 6. This figure shows the distribution of shear stress at different times of the

simulation. In the healthy artery, in the first instants of the simulation, the greatest shear stress

is at the entrance and exit, and the central zone has a lower and constant shear stress

throughout all the area. It is also possible to observe the increase and decrease of the shear

stress, following the imposed velocity, observing the temporal evolution.

Fig. 6 - Evolution of wall shear stress contours for the three arteries.

Fig. 7 - Evolution of wall shear stress for the three

arteries.

Fig. 8 - Evolution of Von Misses stress for the three

arteries.

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After certain instant, the effect of reflux can be observed (instant 0.18 s), that causes the shear

force gradient to be inverted. In the stenotic arteries, the shear stress contours follow a similar

shape than the velocities contours. The central part of the artery accumulates the maximum

shear stress, due to the decrease in section which implies an increase in speed. In areas

downstream of the stenosis, the turbulence and the effect of the vortices are clearly observed

on the shear stress contours. Analogous to what happened with the velocity contours, the

artery with a more severe stenosis (60%) has a similar shear stress distribution than less

severe (40%), but greater turbulence, and a remarkable increase in the shear stress are

observed, due to a greater narrowness that produces a greater increase of blood speed.

For easier discussion, all the results shown in Figure 6, are presented summarized in the curve

evolution of Figure 7. At the beginning of the curve, the shear stress is proportional to the

velocity, as mentioned above, after that the behaviour changes. When the velocity is

maximum (± 0.21 s), this point does not coincide with the point of inversion of the pressure (±

0.125 s) due to the inertia of the flow. The inversion of the pressure produces a reflux. The

pressure is reversed but the reflux cannot be achieved until the inertia of the flow is overcome

and the flow is accelerated in the opposite direction. The arteries with stenosis have a similar

behaviour, but a big increase in the scale of the shear stress was obtained. The vortices and

the recirculation of the flow generate irregularities in the more severe stenosis case.

Moreover, the effect of flow behaviour on the arteries walls, with and without stenosis, has

been studied. The deformation suffered by the artery, when a pulse is imposed on the entrance

and moves throughout the domain, was studied. The presence of atheroma plaques produces a

change in the mechanical properties of the artery wall, which was considered in the

simulations. The results obtained from the three different arteries were compared, so that the

effect of arteriosclerosis on the stress and deformation of the walls of the artery was observed.

The evolution of the maximum Von-Mises equivalent stress that appears on the walls,

following the same pulse than in the flow analysis, are presented in Figure 8. The Von-Mises

stress curves follow the same profile than the pressure evolution.

In the Figure 9, the contours of the deformation of the three arteries studied are compared at

the same time. The deformation increases as the degree of stenosis increases. The reduction of

free flow section causes an increase in pressure and consequently a greater dilation. In the

contours of the arteries with stenosis, a cut has been made in the middle plane to be able to

observe the interior and as the distribution obtained in the atheroma plaque, this cut has no

meaning in the healthy artery.

The deformation contours shown in Figure 9, have been increased six times in order to the

deformation can be appreciated correctly, and the maximum values were summarized in

Figure 10 for easier comparison under the same scale. These curves have a clear relation with

the stress curves, a similar evolution, greater stress implies a greater deformation, both

dependent on the imposed pressure.

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Fig. 9 - Deformation contours at t=0.044s, for the three arteries.

Fig. 10 - Evolution of the maximum deformation for the three artery cases.

CONCLUSIONS

The results obtained have allowed to evaluate the effect of atherosclerosis on the walls of the

artery and on the blood flow. Arteriosclerosis causes a reduction of the free transversal

section of the blood flow due to the deposition of the atheroma plaque. This reduction induces

an increase of blood velocity and consequently an increase in turbulence and shear stress.

This supposes two main risks: the damage that produces an increase of shear stress on the

walls of the arteries; and the increase of the shear effort can entail a detachment of parts of the

atheroma plate, which can produce the coagulation of the blood inside the blood vessels and

can cause a heart attack.

In cases of very severe stenosis, the flow changes to turbulent. The increase in speed and

turbulence, increases the shear stress on the walls. A large increase in the shear stress may

cause a detachment of the atheroma plaque or even rupture of the artery wall. The reduction

of the free section of the blood flow causes an overpressure that increases the deformation of

the arteries and the stress of the walls of the arteries.

The arteriosclerosis produces an increase in stiffness of the arteries walls that causes an

increase in the stress that appear in the arteries walls, and as consequence a reduction in the

deformation.

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