FSI Simulation of Common Carotid under Normal and High Blood Pressures

Abstract

There has been a tremendous progress in most of the computational domains in recent times. One of the key areas is analyzing the complex interaction of blood flow through elastic artery. Such study will aid in predicting the progression and behaviour of diseases like atherosclerosis and hypertension. In the present investigation, a simple straight common carotid artery having an eccentric blockage is analyzed for normal as well as high blood pressure condition. The clinical data has been obtained from ultrasound measurements on a volunteer patient and a model is generated, ANSYS, a commercially available Finite Element Analysis software. The ANSYS Fluid-Structure Interaction (FSI) solver is used to perform the coupled field transient analysis for three pulse cycles and the obtained results demonstrate the pulsatile flow. The simulation shows that the influence of hypertension has high deformation than normal case, severely altering the flow dynamics in downstream of the stenosis. The results obtained agree very well with available clinical observation of normal blood pressure condition.

1. Introduction

The simulation of blood flow has helped in investigating the complex interaction of flow dynamics within cardiovascular system. The altered haemodynamics as observed in development of cardiovascular disease such as atherosclerosis (stenosis) and aneurysms further influences the development of the disease and arterial deformity and changes the regional blood rheology. The study of physiologically realistic pulsatile flow through stenosis has profound implications for the diagnosis and treatment of vascular disease. Although the actual geometry of a stenotic artery may be varied and complex, it has been frequently simplified as symmetrical or asymmetrical constriction in a cylindrical tube [1]. The flow through constricted tubes is generally characterized by a high velocity jet generated from the narrowest section and flow separation distal to the stenosis. The characteristics of the prestenotic flow is usually laminar; however the flow in poststenotic is extremely turbulent depending upon the flow conditions and severity of the stenosis. The accurate simulation of realistic physiological phenomenon is possible only through FSI and using CFD alone will not be beneficial to demonstrate the interaction of blood flow through elastic artery.

There are numerous FSI studies conducted in the past assuming both the simple and complex geometry models generated from in vivo data. Initially, the flow dynamics in poststenotic region was demonstrated on a simple straight tube stenosed model [2]. In this study, 25%, 50%, and 75% axisymetrical and symmetrical models were analyzed. A highly complex flow is observed in the downstream side of stenosis leading to flow separation as observed from simulated results. Later, the flow interaction with elastic structure was adopting using interactive approach. This study was more realistic as experimental data was used as boundary condition forcomputation [3]. Similar study was conducted on stenotic arteries using both thick and thin wall model. The simulated results revealed that severity of stenosis and asymmetry in thick-wall models changed not only the wall geometry, but also the stiffness of the tube wall which in turn affected the wall deformation [4].

Since the arterial wall has a complex structure, it cannot be treated as a homogenous material. Considering this fact, FSI simulation of blood flow in carotid was conducted on a nonlinear thick-wall model [5]. The obtained results were more significant compared to analysis performed considering linear model. The asymmetric stenosis was studied to quantify the effects of stenosis severity, eccentricity, and pressure conditions on blood flow and artery compression. It is observed that, particularly in poststenotic region, the velocity field is highly dependent on the flow waveform along with the uniform distribution of flow recirculation along the circumference of the walls. Also the strong flow reversal and recirculation were found to be uniformly distributed around the circumference of the walls in the downstream of stenosis. From the available literature, it is observed that most of the studies have been performed at normal blood pressure. But the important aspect of flow dynamics in the downstream of stenosis under the influence of high blood pressure is yet to be studied as it can prove to be one of the major risk factors [6]. The flow resistance in arteries increases abruptly due to the influence of high-grade stenosis. This forces the body to raise the blood pressure to maintain the necessary blood supply. The constriction and high blood pressure causes high flow velocity, high wall shear stress, and low pressure at the throat of the stenosis, while in the distal side of stenosis, there will be low wall shear stress, flow separation, and wall compression.

In the present study, a patient specific case is investigated having eccentric stenosis using normal blood pressure. The results are compared with results of the influence of high blood pressure, so as to prognose the crucial factors. The results indicate the significant changes with hypertension. Considerable changes in the wall deformation, wall shear stress, flow behaviour in the downstream of the stenosis and flow separation zone are observed.

2. Methodology

The blood flow in carotid artery is assumed to be laminar, incompressible and governed by the Navier-Stokes equations of incompressible flows. The fluid domain is solved using modified momentum equation adopting moving velocity along with continuity equation as given in equation below [7, 8]:

\begin{matrix} \frac{\partial}{\partial t} \int_{Ω}^{} ρ \partial Ω + \int_{S}^{} ρ (υ - υ_{b}) \cdot n \partial S = \int_{S}^{} (τ_{i j} i_{j} - P i_{i}) \cdot n \partial S + \int_{Ω}^{} b_{i} \partial Ω, \end{matrix}

where ρ is the density, τ is the stress tensor, ν is the velocity vector, νb is the grid velocity, P is the pressure, and b_i is the body force at time t.

The artery wall is assumed to be elastic, isotropic, incompressible, and homogeneous. The transient dynamic structural solution is given by the following equation [4]. The stiffness matrix is updated in each time step. The Newmark method is used for updating of displacement at each time interval and then stiffness matrix is solved using direct solver in particular sparse solver, for every time step:

\begin{matrix} [M] {\overset{ʨ}{U}} + [C] {\dot{U}} + [K] {U} = {F^{a}}, \end{matrix}

(2)

where M is the structural mass matrix, C is the structural damping matrix, K is structural stiffness matrix, F_a is the applied load vector, and $\ddot{U}, \dot{U}$ , and U represent acceleration, velocity, and displacement vector.

2.1. FSI Algorithm

The FSI analysis is performed using sequentially coupled FSI solver in ANSYS [9, 10]. This solver uses ANSYS FLOTRAN elements for fluid domain and ANSYS structural for solid domain. The FSI algorithm is shown in form of flow chart in Figure 1. In the beginning of the FSI analysis, first fluid domain has to be modeled using appropriate element then required boundary conditions are applied followed by solid domain.

Figure 1:

FSI algorithm.

The FSI algorithm solves the fluid and solid domains independently. After fluid domain is solved, the surface load to the solid domain is transferred through fluid-structure interface and solid domain gets solved. After the convergence is achieved, the fluid mesh has to be morphed using Arbitrary Lagrangian-Eulerian (ALE) formulation. The algorithm continues to loop through the solid and fluid domain until convergence is reached for that time step. Later the loop continues to solve for the time period specified.

3. Modeling and Analysis

In the patient, the right side common carotid is normal and left side has 66% eccentric stenosis. The model of stenosed common carotid is generated in ANSYS-11.0 as shown in Figure 2. The model is generated based on the details taken at different sections depending upon the convenience using M-mode scan for velocity and B-mode scan for diastolic diameter [3, 9]. The generated stenosed model has a diastolic diameter of 6.5 mm, total length of 100 mm, throat of stenosis is at a distance of 40 mm from inlet, and artery wall has a thickness of 0.7 mm.

Figure 2:

Stenosed model.

Both the fluid and the structural parts are modelled with 8 nodded brick element and the grid distribution across the cross-section of stenosis is shown in the Figure 3. Initially, a static FSI analysis was carried to perform grid independency test. A cross-section at throat of stenosis was considered to calculate the average pressure and maximum wall shear stress and the observed variation is shown in the Figure 4. Finally, the fluid and structural models with 50000 and 12000 hexahedral elements were considered for further transient FSI simulation.

Figure 3:

Grid distribution across stenosis.

Figure 4:

Grid independency results.

In this study, the inlet boundary condition is set by a velocity pulse at the inlet for a period of 0.8 sec as shown in Figure 5 and outlet boundary condition is set by a pulse pressure, applied at the outlet as shown in Figure 6. Each pulse cycle is discretized into 100 time steps to simulate the flow behaviour more accurately. The range of pulse pressure is different as the simulation is carried out for both normal pressure and high pressure having80–125 mm Hg and 100–170 mm Hg, respectively, as shown in Figure 6 [6]. The inlet and outlet of solid model are constrained by specifying zero displacement in all the directions and the rest of the nodes are left free to undergo displacement in any direction [10, 11].

Figure 5:

Input velocity BC.

Figure 6:

Output pressure BC.

The blood is considered to be Newtonian in the present study as the focus is on large arteries where velocity and shear rate will be high. The density and dynamic viscosity of the blood are considered to be 1050 kg/m³ and 0.004 N-sec/m², respectively. The artery is assumed to be linearly elastic material, having density of 1120 kg/m³, Poisson's ratio of 0.40, and elastic modulus is 0.9 Mpa [10, 11]. Since the patient is an aged person, the arterial wall is stiffened and elasticity is less when compared with that of healthy young person [12, 13]. A generalized ANSYS Parametric Development Language (APDL) written in ANSYS generates the geometric model solves and performs the postprocessing of the results. The convergence criteria of fluid flow and across the fluid-surface interface are set at 10⁻⁴ and 10⁻³, respectively.

4. Results and Discussion

The simulated results obtained for normal blood pressure (NBP) and high blood pressure (HBP) conditions are compared to observe the variations in the flow behaviour. The velocity contours, wall displacement, and wall shear obtained at specified instants of pulse cycle are considered for observation. The computed results like numerical values of deformation, wall shear stress, and velocity patterns are normalized to facilitate the comparison. The velocity, WSS, and wall deformation are found to be maximum during peak systole due to high flow rate.

The wall deformation patterns are shown in Figures 7 and 8 during early systole and peak systole for NBP and HBP conditions, respectively. The maximum wall deformation is within 10% and 16% of diastolic diameter for NBP and HBP conditions, respectively. The wall deformation is uniform in upstream as well as in downstream of the model apart from the throat region during the entire pulse cycle [11]. There is significant difference in wall deformation due to HBP condition, which is maximum during peak systole when compared with NBP case. The nonuniform structural deformation is observed to be severe in poststenotic region than in prestenotic region. Due to the instant increase in the pressure, the wall collapsibility is observed after peak systole and further increases during later part of the pulse cycle because of low pressure gradient.

Figure 7:

Wall deformation at early systole in NBP.

Figure 8:

Wall deformation at peak systole in HBP.

The velocity contours during peak systole for NBP and HBP conditions are shown in Figures 9 and 10, respectively. The characteristic of stenotic flow shows a sudden increase in velocity and drop in pressure at the throat during the entire pulse cycle [14]. The flow jet covers a longer distance in HBP compared to NBP due to large pressure gradient. The flow is more predominant even during later part of pulse cycle in HBP compared to NBP. However, there is no significant difference in the flow rate for both the NBP and HBP cases. Figure 11 shows the velocity contour during early diastole in HBP and velocity profile during late diastole in NBP is shown in Figure 12. Due to the eccentric stenosis, flow separation in downstream of stenosis is intense in case of HBP than in NBP leading to severe eddy formation.

Figure 9:

Velocity contour at peak systole in NBP.

Figure 10:

Velocity contour at peak systole in HBP.

Figure 11:

Velocity contour at early diastole in HBP.

Figure 12:

Velocity contour at late diastole in NBP.

During the later part of the pulse cycle, the flow in downstream side stabilizes within shorter distance in NBP compared to HBP. Due to flow deceleration, the back flow is observed from early diastole in NBP, but in case of HBP this effect is reduced instantly because of large pressure gradient.

The wall deformation is compared during NBP and HBP as shown in Figure 13. The maximum displacement is found to be in HBP condition. The pattern of the displacement is almost similar to the given input pulse cycle, which shows the pulsatile behaviour. The maximum difference is seen during peak systole and in early part of diastole than in any other part of the cycle. Due the high pressure, the time taken for flow stabilization is high in HBP compared to NBP. Also due to wall collapsible behaviour during late diastole, the difference is less and almost the same between NBP and HBP [6, 14]. Due to this, the effect on flow dynamics in downstream is significant causing high turbulence. Figure 14 shows the comparison of WSS during NBP and HBP conditions and there is a minor difference because of the same flow rate in both cases. The maximum WSS is at the throat region compared to upstream and downstream side of the model. There is a sudden increase in WSS during peak systole in both the HBP and NBP cases. The variation of maximum WSS is similar to pulse cycle, but the gradual reduction lags by few milliseconds as compared to normal input velocity cycle due to increase in velocity in the throat region. The difference in pressure gradient is less in early part of diastole and because of this flow WSS is constant during this phase of the cycle. The results obtained in NBP agree well with the clinically observed results and with available literature. From the obtained results, it is observed that, including the effect of hypertension can simulate the risk or prognosis of arterial disease and it cannot be neglected because it has profound influence on altering the flow dynamics, WSS distribution, stress levels, and wall deformation.

Figure 13:

Wall deformation comparison.

Figure 14:

Wall shear stress comparison.

5. Conclusions

The effect of hypertension is simulated for an eccentric stenosed model in the present study. The variations in velocity distribution, WSS, wall deformation, pressure are compared for NBP and HBP conditions. The poststenotic deformation and WSS is found to be maximum during peak systole and increased velocity at the throat region. In both the NBP and HBP cases, the wall collapsibility is observed from the early diastole which influences the intense flow separation for later part of pulse cycle. The flow is highly turbulent in downstream especially in the downstream of the throat region. The results demonstrate that HBP causes the significant changes in the wall deformation, stress distribution and flow especially in downstream during early diastole part of the pulse cycle than in NBP condition. The variation of wall deformation in the poststenotic region and the velocity contours obtained under the NBP conditions agree well with the clinically observed results.

Footnotes

Nomenclature

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