Clinical Hemorheology and Microcirculation 57 (2014) 159–173 DOI 10.3233/CH-141827 IOS Press
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Numerical analysis of 3D blood flow and common carotid artery hemodynamics in the carotid artery bifurcation with stenosis N. Antonovaa,∗ , X. Dongb , P. Toshevaa , E. Kaliviotisc and I. Velchevad a
Department of Biomechanics, Institute of Mechanics, Bulgarian Academy of Sciences, Bulgaria School of Civil Engineering, Tianjin University, Tianjin, China c Department of Engineering, University College London, London, UK d Department of Neurology, University Hospital of Neurology and Psychiatry “St. Naum”, Medical University, Sofia, Bulgaria b
Abstract. The results for blood flow in the carotid artery bifurcation on the basis of numerical simulation of Navier-Stokes equations are presented in this study. Four cases of carotid bifurcation are considered: common carotid artery (CCA) bifurcation without stenoses and cases with one, two and three stenoses are also presented. The results are obtained by performing numerical simulations considering one pulse wave period based on the finite volume discretization of Navier-Stokes equations. The structures of the flow around the bifurcation are obtained and the deformation of the pulse wave from common carotid artery (CCA) to the internal carotid artery (ICA) and external carotid artery (ECA) is traced. The axial velocity and wall shear stress (WSS) distribution and contours are presented considering the characteristic time points. The results of the WSS distribution around the bifurcation allow a prediction of the probable sites of stenosis growth. Keywords: 3D blood flow numerical analysis, carotid bifurcation, stenosis, wall shear stress, whole blood viscosity
1. Introduction An area of special interest is the carotid artery circulation, where stenoses, thromboses, atherosclerotic plaques and other lesions can cause cerebral disturbances. Possible relation between the carotid pathological lesions and the carotid blood flow give magnetic resonance imaging (MRI), computed tomography (CT) and Doppler ultrasound techniques, developed to measure velocity profiles and vessel morphology with high spatial and temporal resolution. However, complete description of the flow in the human vascular system is difficult because of the complex geometry of the carotid artery bifurcation and the unsteady flow. The use of imaging methods with mapping of wall shear stress (WSS) distribution in the carotid arteries in parallel with numerical simulation could help for detection of the vessel sites where atherosclerotic plaques in the separate individuals would develop. ∗ Corresponding author: N. Antonova, Department of Biomechanics, Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia Acad. G. Bonchev str., bl. 4, Bulgaria. Tel.: +35 929796413; Fax: +35 928707498; E-mail: [email protected].
1386-0291/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
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The WSS represents the frictional force that acts tangentially to the endothelial surface. It is accepted that the chronic exposure of the endothelial cells to high shear stress is atheroprotective, while its lower values are associated with thickening of the vessel wall and development of atherosclerotic plaques. In these cases the vascular endothelial cells are exposed to blood flow – induced shear stresses, which cause increase of their membrane fluidity and permeability and activation of the membrane-associated signal proteins – mechanochemical transduction [4, 5]. In laminar blood flow, where the fluid velocity profile is parabolic, WSS reflects the relationship of the blood flow velocity, the blood viscosity and the vessel radius. Significant changes in WSS occur in vivo due to different velocities at the sites of curvatures and bifurcations. The pulsatile blood flow, which varies with the cardiac cycle, also influences WSS, which assumes an oscillatory pattern [13]. All these phenomena justify the great interest in studying the hemodynamics around stenoses, bifurcations and other geometric irregularities as the most probable sites of generation of cerebral disturbances and pathological formations. The application of medical image reconstruction for modeling vessel to use in Computational Fluid Dynamics (CFD) has been of rapid development in recent decades. The typical process for performing numerical simulation of blood flow is based on medical imaging, image segmentation, 3D model reconstruction, grid generation and analysis of blood flow solving the Navier-Stokes equations. With the development of modern imaging technology, MRI, CT, etc., it is now possible to quantify arterial blood flow in subject-specific physiologic models and to evaluate the hemodynamics and WSS distribution [4, 5]. With the use of CFD simulations and MRI, the ability to evaluate the complex relationship between hemodynamics and the predictions of blood vessel damage may be possible. The aim of the study is to investigate the hemodynamics in the carotid artery bifurcation with stenoses by means of the 3D numerical analysis of blood flow based on the numerical solution of the Navier-Stokes and continuity equations.
2. Previous related studies 2.1. Common carotid artery hemodynamic factors in patients with cerebral infarctions and WSS The changes of the common carotid local hemodynamic factors like wall shear stress and tensile forces in 16 patients with chronic unilateral cerebral infarctions (CUCI), 58 patients with the main risk factors (RF) for cerebrovascular disease (CVD) like hypertension and hyperlipidemia and 25 healthy control subjects were investigated by Velcheva et al. [19]. The blood flow velocities (BFV), the internal diameters (D) and the vessel wall intima-media thickness (IMT) in the common carotid arteries (CCA) were recorded with color duplex sonography. Systolic (SBP) and diastolic (DBP) blood pressure we are measured and mean blood pressure (MBP) was calculated by the formula of Wiggers. Whole blood viscosity (WBV) at the shear rate of 94.5 s−1 was measured on the day of the Doppler ultrasound examination with a rotational viscometer Contraves LS 30. The ultrasound examination by color duplex scanning of the main neck arteries was performed by using a 5 MHz probe with the Versa plus - Siemens instrument. The vessel diameters of the common carotid artery (CCA) in the systole (Ds ) and the diastole (Dd ) and the IMT of the far vessel wall as the mean of 3 consecutive measurements were determined. The systolic (SBFV), mean (MBFV) and diastolic (DBFV) blood flow velocity were detected 1 to 2 cm proximal to the carotid bulb with sample volume in the center of the vessel at an angle of insonation of 45◦ . In the group with CUCI the velocity parameters on the infarction (IS) and non-infarction side (NS) were compared. The characteristics of the carotid plaques were estimated. The systolic (Ts ) and mean circumferential wall tension (Tm ) we
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are calculated by a formula according to the Laplace law [8]. The WSS, the circumferential wall tension (T) and the tensile stress (τ) we are calculated too. The SBP, WBV and IMT were significantly increased in the patients with CUCI and RF for CVD in comparison to controls. Lower systolic WSS and τ and higher T were established in the patients with CUCI. The IMT correlated with WSS and τ. The study confirms the complex influence of the changes in WBV and blood pressure for the development of carotid atherosclerosis. The findings in patients with cerebral infarctions indicated a higher degree of the vessel wall damage in these patients [19]. 2.2. Numerical studies of blood flow in the carotid artery bifurcation with stenosis A 3D blood flow analysis in an artery with a symmetric and asymmetric stenosis is done by Antonova et al. [1]. The blood flow is analysed in the straight section of the carotid artery with a symmetric and asymmetric stenosis, simultaneously solving the Navier-Stokes and mass transfer equation. Data of healthy 20 years old patients are obtained by averaging the dimensions of the carotid artery with length 8 cm and diameter 0.8 cm just before the bifurcation. The axial velocity component and strain rate are illustrated considering characteristic time points. For the symmetric stenoses two cases of a symmetric obstacle with a 3 cm length and maximal height 1 mm (about 11% of the arterial lumen) and 2 mm (about 44% of the arterial lumen) are used for simulations. In fact due to axial symmetry (at the maximal height) the stenosis reduces the arterial lumen from 8 mm to 6 mm and 4 mm. The strain rate (proportional to the wall shear) around the stenosis with maximum wall shear gradient is only part of the stenosis surface located on its top. Flow recirculation zones behind the stenosis for time characteristic values are obtained. The distribution of the strain rate (proportional to the wall shear) around the stenosis and the presence of recirculation zones during a sub-period of the considered period of time allow a prediction of the sites of stenosis growth.The results illustrating flow around a non-symmetric stenosis with different percentage of artery lumen reducing (16% and 25%) show well formed recirculation zone with considerably high intensity, located after the stenosis. The location of the area of high wall shear gradient is again on top of the obstacle, showing the site of intensive deposition. A 3D numerical analysis of blood flow based on the numerical solution of the Navier-Stokes and continuity equations in the carotid artery bifurcation with or without stenoses is simulated using the 3D source code is developed by Tosheva et al. [17]. Time dependent boundary conditions are used for modeling the development of the blood flow induced by a pulse wave (pressure wave) in the rigid tube. The fluid motion is modeled by solving numerically the system of the Navier-Stokes equations and the continuity equation. Blood is regarded as incompressible fluid with constant density and conservation of mass gives the incompressibility condition. The blood flow around carotid bifurcation (distribution of axial velocity and strain rate) and in the carotid bifurcation containing two and three asymmetrical stenoses placed on the wall of carotid artery is obtained [17]. The results are given at the time point of the maximal pulse wave velocity and are illustrated by the axial velocity and strain rate. The last one is the important factor in the deposition process on the vessel walls. The flow in the bifurcation is strongly unsteady and its visualization here is very difficult. The pattern of the velocity and the strain rate are presented in the character time point only. To investigate deeply the influence of the stenoses on the blood flow in the bifurcation and deposition processes around it, the stationary part of the flow has to be studied. A numerical study of stenosed carotid bifurcation models based on wall shear stress distribution is presented by Kelvin et al. [9]. Cardiovascular models based on medical image reconstruction of ten patientspecific carotid bifurcations in terms of their geometries and flow properties are used. The characteristics of stenoses at various sections of the anatomical structure are used as indicators for explaining the
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difference in flow properties such as wall shear stress. The geometry of the carotid vessels is extracted from MRI imaging of six patients. The segmentation process includes thresholding and region growing, followed by 3D anatomical reconstruction to obtain a very coarse solid model. During thresholding, a range of gray scale values are selected such that the region to be selected is of best contrast within this range. After the regions of interest were selected and extracted, the voxels are grouped together to form a 3D geometry. Then the reconstruction of carotid bifurcation anatomy into a Computer Aided Design (CAD) model is performed based on the segmentation information. The fluid is assumed to have a density (ρ) of 1.176 g/cm3 , viscosity (η) of 4.0 mPa·s and molar mass of 25 kg/kmol. The flow is assumed to be Newtonian and the vessel walls are rigid. The time period T of one cardiac cycle is 0.92 s. The k- turbulence model is applied in the simulation. To ensure the convergence of each time step, the convergence criterion for the relative residual of all dependent variables was set to 1 × 10– 4 . For time discretization, the second order backward Euler transient scheme is used. Steady and unsteady flow patterns are analysed in rigid models of the carotid artery bifurcation by van Steenhoven et al. [18]. Both measurements and calculations were carried out enabling an experimental validation of the numerical method, based on Galerkin’s finite element approximation of the Navier-Stokes and continuity equation. Using patient-specific carotid artery bifurcation data from ultrasound imaging Sousa et al. [14] realized a semi-automatic generation of a structured and conformal hexahedral mesh of a nearly planar carotid bifurcation. The geometry, velocity at the inlet (common carotid artery) and at the oulet (internal carotid artery) obtained from ultrasound measurements are imported into the finite element model software to simulate the flow dynamics. The three-dimensional, unsteady, incompressible Navier-Stokes equations are solved with the assumptions of rigid vessel walls and constant viscosity. A good agreement between ultrasound imaging and computational simulated results is achieved. One example of grid reconstruction and blood flow simulation for a patient with internal carotid artery aneurysms is presented by Xu Bai-nan et al. [20]. The method accurately duplicates the geometry to provide computer simulations of the blood flow. Initial images are obtained by using CT angiography in DICOM format. The image is processed by using MIMICS software, and the 3D fluid model (blood flow) and 3D solid model (wall) are generated. The subsequent output is exported to the Ansys workbench software to generate the volumetric mesh for further hemodynamic study. The fluid model is defined and simulated in CFX software, while the solid model is calculated in ANSYS software. The force data calculated firstly in the CFX software are transferred to ANSYS software, and after receiving the force data, total mesh displacement data are calculated in Ansys software. Then the mesh displacement data are transferred back to CFX software. The results of simulation could be visualized in CFX-post. The wall shear stress and wall total pressure are in good agreement with the clinical data. 2.3. Comparison of wall shear stress Individual variation in the anatomy of arteries in human is considerable. The degree of blockage is from 9% to 60% and all the bifurcations exhibit some degree of geometrical non-planarity [11]. It has been indicated by previous studies that wall shear stress promotes luminal thinning and plaque rupture and low WSS regions are more prone to athersclerosis. The growth of lesion will lead to stenosis at the artery wall, and further narrowing downstream of the blood vessel will occur. Therefore, this is one of the key parameters that is used for examination of diseased carotid bifurcations. Topological maps of the predicted maximum wall shear stress at peak pressure for each of the ten case studies are given. In all of the models, the maximum wall shear stress (WSSmax ) is shown to have peak values on the inner
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walls of ICA and ECA near the bifurcation points, and significantly elevated values spiralling around the larger angular branch (either the ICA or ECA) from the inner walls along the superior orientation. The maximum WSSmax value appeared at the stenosed section due to vessel tapering. 2.4. Three-dimensional numerical analysis of pulsatile flow and wall shear stress in the carotid artery bifurcation To analyse the pulsatile flow field and the mechanical stresses in a three-dimensional carotid artery bifurcation model, computer simulation is applied by Perktold et al. [12]. Numerical results are presented for axial and secondary flow velocity and wall shear stresses with special emphasis on the fluid dynamics in the carotid sinus. Flow and stress patterns in human carotid artery bifurcation models, which differ in the bifurcation angle, are analysed numerically under physiologically relevant flow conditions. The governing Navier-Stokes equations describing pulsatile, three-dimensional flow of an incompressible non-Newtonian fluid are approximated using a pressure correction finite element method. The blood non-Newtonian behaviour is modelled by Casson’s relation, based on measured dynamic viscosity at supposed rigid blood vessel wall. The results show that the complex flow in the sinus is affected by the angle variation. The magnitude of reversed flow, the extension of the recirculation zone in the outer sinus region and the duration of flow separation during the pulse cycle as well as the resulting wall shear stress are clearly different in the small angle and in the large angle bifurcation. The haemodynamic phenomena, which are important in atherogenesis, are more pronounced in the large angle bifurcation. To simulate the transition zone flow around carotid bifurcation with and without stenoses the wall turbulence model is used by Perktold et al. [12]. Results for different reduction of arterial lumen are obtained. To investigate the effect of the distensible artery wall on the local flow field and to determine the mechanical stresses in the artery wall, a numerical model for the blood flow in the human carotid artery bifurcation has been developed. The wall displacement and stress analysis use geometrically non-linear shell theory where incrementally linearly elastic wall behavior is assumed. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier-Stokes equations for non-Newtonian inelastic fluids. In an iteratively coupled approach the equations of the fluid motion and the transient shell equations are numerically solved using the finite element method. The results show the occurring characteristics in carotid artery bifurcation flow, such as strongly skewed axial velocity in the carotid sinus with high velocity gradients at the internal divider wall and with flow separation at the outer common-internal carotid wall and at the bifurcation side wall. Flow separation results in locally low oscillating wall shear stress. Further strong secondary motion in the sinus is found. The comparison of the results for a rigid and a distensible wall model demonstrates quantitative influence of the vessel wall motion. With respect to the quantities of main interest, it can be seen, that flow separation and recirculation slightly decrease in the sinus and somewhat increase in the bifurcation side region, and the wall shear stress magnitude decreases by 25% in the distensible model. The global structure of the flow and stress patterns remains unchanged. The deformation analysis shows that the tangential displacements are generally lower by one order of magnitude than the normal directed displacements. The maximum deformation is about 16% of the vessel radius and occurs at the side wall region of the intersection of the two branches. The analysis of the maximum principal stresses at the inner vessel surface shows a complicated stress field with locally high gradients and indicates a stress concentration factor of 6.3 in the apex region. The effect of different plaques of trapezoidal, elliptical and triangular shape, imposing 30% reduction of arterial lumen on blood flow through the carotid artery leading to plaque growth and rupture have been
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analysed [7]. A carotid artery model based on statistical analysis called tuning fork model is constructed. Comparison of the results of various CFD simulations show that trapezoidal shape has greater effect on blood flow producing highest flow velocities and wall shear stresses, and creates lowest wall shear stress region downstream of the plaque which may increase the fatty deposits in that area. In 3D transient analysis it is observed that variations of wall shear stress followed velocity pulse variations and maximum WSS corresponded to maximum velocity. So a steady analysis was performed at maximum velocity of pulse on fine meshes to save computational time. Previous studies show that one of the main factors involved in plaque buildup is low wall shear stress and on the other hand plaque rupture can be due to high shear stress as it causes high flow changes in the vessel. The maximum value of wall shear stress was found on trapezoidal plaque while maximum average wall shear stress was found on elliptical plaque because it causes a continuous change in velocity across its surface. So plaques similar in shape to the trapezoidal plaque may result in more downstream fatty deposit on artery wall. 2.5. Numerical analysis of flow through a severely stenotic carotid artery bifurcation A model based on an endarterectomy specimen of the plaque in a carotid bifurcation was examined by Stround et al. [15]. The work modeled the flow through the severely stenotic up to 70% occluded carotid bifurcation as the flow through a rigid-walled vessel and the blood is assumed to have a constant dynamic viscosity of 3.5 mPa.s and mass density 1060 kg/m3 . The flow conditions include steady flow at Reynolds numbers of 300, 600 and 900 as well as unsteady, pulsatile flow. Both dynamic pressure and wall shear stress are very high, with shear values up to 70 N/m2 , proximal to the stenosis throat in the internal carotid artery, and both vary significantly through the flow cycle. The wall shear stress gradient is also strong along the throat. Vortex shedding is observed downstream of the most severe occlusion.
3. Methods 3.1. Navier-Stokes equations The blood flows in the arteries are treated as incompressible viscous flow. Fluid motion is modeled by solving numerically the system of equations of the continuity equation and the equations of motion using the finite volume method. The Navier-Stokes equations for incompressible viscous flows read: ∂t u + u · ∇ u = −∇p + µ∇ 2 u + f
(1)
∇ ·u=0
(2)
where u is the vector of fluid velocities, p is the pressure normalized with the fluid density and f is a body force term. A fluid solver was developed using unstructured mesh. It is a three-dimensional fluid solver with second order accuracy in both time and space and is based on a finite volume formulation. For time-stepping, a Crank-Nicolson scheme was used for the diffusion term and the convection term is discretised using an explicit two-step Adams-Bashforth scheme. To create the complicated shape of the computation area a CAD system is applied with tetrahedral elements.
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Fig. 1. Entrance boundary condition scheme.
3.2. Rheological model for non-Newtonian fluid The blood flow is well known for its complex rheological behaviour. The non-Newtonain features, such as shear thinning and viscoelasticity of blood, have been widely observed [3, 16], and are closely related to the deformation and the aggregation of red blood cells in the blood. To describe the rheological property of the blood correctly, Carreau-Yasuda model [2] is used to compute the non-Newtonian viscosity: n−1 η − η∞ ˙ a a = 1 + (λγ) η0 − η
(3)
where γ˙ is a scalar quantity of the rate of deformation tensor, namely γ˙ =
2tr D2
(4)
∇ u+(∇ u)T ] where D = [ . The other parameters in Eq. (3) are employed from the experimental data of Gijsen 2 et al. [6] with the value set as: η∞ = 2.2 × 10−3 Pa s; η0 = 2.2 × 10−2 Pa s; λ = 0.110 s; a = 0.644; n = 0.392; ρ = 1410 kg m−3 . Numerical simulations by Perktold et al. [11] show that incorporation of the shear thinning behavior of blood did not alter the flow characteristics significantly. The experimental results of Liepsch et al. [10] however, indicated that the viscoelasticity of the fluid could be of importance for the region of flow reversal. The experiments are carried out for a Newtonian fluid and a shear-thinning Carreau fluid. Their comparison between Newtonian and non-Newtonian fluid shows considerable difference between the sizes of the recirculation zone formed after the step. Differences between the distribution of the wall shear in both cases are observed, too.
3.3. Entrance effects and computational configurations Blood flow in the carotid artery bifurcation with or without stenoses is simulated using the finite volume method. Time dependent boundary conditions are used for modeling the development of the blood flow induced by a pulse wave (pressure wave) (see Figs. 1 and 2). It is also assumed that the tube is rigid. The computational area of the considered problem is a rigid cylindrical tube with bifurcation. Four cases of the tube with bifurcations are considered: without stenosis and with one, two and three stenoses. Non-slip boundary conditions for velocities are used (U = 0, V = 0 and W = 0) over the rigid wall of the tube. The continuity boundary conditions are applied for velocities at the end of the bifurcations. A modified boundary condition representing the pulse wave is used at the tube entrance. It is shown schematically in Fig. 2.
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Fig. 2. Pulse wave.
Fig. 3. Surface mesh for the common carotid bifurcation without stenosis and with one, two and three stenoses.
A fully developed Suasion profile for z-velocity, with a Reynolds number pulsing like that shown in Fig. 2, has the following form: ⎛
Uz =
⎞ 2
Re .ν ⎜ d 2
z ⎟ ⎝1 − 2 ⎠
(5)
d 2
where d is diameter of the tube, Re is the Reynolds number, ν is kinematic viscosity, z is the axial coordinate, Uz is the axial velocity component at the entrance cross section and Umax is the maximal axial velocity component at the entrance cross section.
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Tetrahedral elements are adopted for its flexibility for complex geometries. The average mesh size is around 0.05 cm and the total amount of elements is around 250, 000 for each case. See Fig. 3 for the surface mesh for all of the 4 simulation cases. The time step for the simulation is dt = 0.005 s, yielding a CFL number of around 0.1. 4. Results Blood flow in carotid artery bifurcation is studied on the basis of Navier-Stokes equations. The results are found by performing numerical simulations considering one pulse wave period and using a finite volume method. The structures of the flow around the bifurcation are obtained. The axial velocity distribution in the CCA without stenoses is illustrated considering characteristic time points. Using the numerical simulation the deformation of the pulse wave from CCA to the ICA and ECA has been traced. The axial velocity and WSS distribution around the bifurcation allow a prediction of the probable sites of stenosis growth. Four cases are considered in the present work: blood flow in the carotid bifurcation without stenoses (Figs. 4–6) and blood flow in the carotid bifurcation containing one (Fig. 7), two (Fig. 8) and three stenoses (Fig. 9). The changes of the axial velocity in six characteristic time points of the pulse wave in the CCA bifurcation without stenoses are shown on Fig. 4 (T = 0 s, 0.1 s and 0.2 s) and on Fig. 5 (T = 0.3 s, 0.4 s and 0.5 s). The axial velocity of the flow around the carotid bifurcation without stenoses is increasing at T = 0.1 s in comparison with the initial moment T = 0 s, especially in the external carotid artery (ECA) – left branch of the bifurcation. The distribution of the axial velocity at another three characteristic time points T = 0.3 s, 0.4 s and 0.5 s, in the CCA bifurcation without stenoses, shown on Fig. 5 reveal increasing axial velocity and flow disturbances at T = 0.4 s. The greatest change in the velocity distribution Z
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Fig. 8. Velocity magnitude contour and wall shear stress in the carotid bifurcation of the carotid artery with 2 stenoses. Left: velocity magnitude contour unit: m/s Right: Wall shear stress distribution unit: Pa.
and increased non-uniformity and the velocity of flow in the area of the carotid bifurcation was observed in the specific time point of the maximum speed of the pulse wave during systole (T = 0.1 s) – Fig. 4, and in diastole (T = 0.4 s) – Fig. 5. The results show that the blood flow in the carotid bifurcation is unsteady and the velocity changes with time and spatial location in the vessel. The results are given at the time
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Fig. 9. Velocity magnitude contour and wall shear stress in the carotid bifurcation of the carotid artery with 3 stenoses. Left: velocity magnitude contour unit: m/s Right: Wall shear stress distribution unit: Pa.
Fig. 10. Comparison of peak wall shear stress for different cases.
points when the pulse wave reaches the maximum value and are illustrated by the axial velocity and WSS distribution. The WSS has an important role in the deposition process of the plaques on the vessel walls. The results from the numerical simulation for the case without stenoses and the velocity magnitude contour in the CCA bifurcation to the ICA and ECA are shown on Fig. 6 (Left panel). The flow is unsteady and the velocity magnitude changes between 0.8 m/s at the entrance of the CCA to 0.1 m/s at the entrance of the ICA after bifurcation to the maximum value of 0.8 m/s at the outlet of the ECA. It is seen from Fig. 6 (Left panel) that with the exception of the entrance of the CCA, where the maximum
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velocity magnitude is on the axis of the artery (0.8 m/s), in the remainder of the artery to the bifurcation the axial velocity is almost constant (0.4–0.5 m/s). The results show that the flow becomes unsteady at the bifurcation and after this region. In contrast to the ECA (left branch of the bifurcation), where the flow is accelerated to the maximum in the end of the corresponding section, at the beginning of the internal carotid artery (ICA) the flow is significantly slowed down, the velocity magnitude is only 0.2 m/s to about 0.1 m/s on the inside of the artery. This low velocity region in the initial inner part of ICA (0.1–0.2 m/s) creates a corresponding region of low WSS seen on Fig. 6 (Right panel) where the distribution of WSS has the following characteristics: relatively high WSS (3–3.5 Pa) to the bifurcation and low WSS between 0.5–1 Pa at the beginning of the ICA (right branch). This area of low shear stress is the site where the conditions of the flow are predisposing for plaque deposition and narrowing of the artery. In the case of the CCA with one stenosis before the bifurcation, the velocity magnitude contour, shown on Fig. 7 (Left panel) is the following: the area, where the velocity magnitude is only 0.2 m/s to about 0.1 m/s on the inside of the artery is increasing. This low velocity region after the bifurcation at the beginning of the inner part of ICA (0.1–0.2 m/s) results in the increase of the corresponding area of low WSS seen on Fig. 7 (Right panel). Noticeably smaller is the area with the lowest wall shear stress on the inner side at the entrance of the internal carotid artery (0.5–1 Pa), but this portion is displayed on the outside of the ICA, shown on Fig. 7 (Right). The numerical results of the velocity distribution in the case of CCA bifurcation with two stenoses indicate acceleration of the flow in the narrowed part of the artery, shown on Fig. 8 (Left panel), where the velocity magnitude varies between 0.6 m/s – 0.7 m/s. An increase of the area of lower velocity on the inside and outside at the entrance of the internal carotid artery (ICA) after the bifurcation in comparison with the cases with one and without stenoses is observed on Fig. 8 (Left panel). A substantial change in the distribution of the WSS after the bifurcation, and a significant increase of the WSS at the beginning of the left branch (ECA) is also observed. An increase in the WSS above the bifurcation is found too. Similar characteristics show the trends of the velocity magnitude and the velocity contour for the case with three stenoses, shown on Fig. 9 (Left panel) and the WSS distribution on Fig. 9 (Right panel). The relationship between the peak wall shear stress and time for one pulse wave, shown on Fig. 10 has 4 inflection points at which the curvature changes its sign. The curve changes from being concave downwards (negative curvature) to concave upwards (positive curvature) at T = 0.12 s to a maximum at T = 0.2 s and vice versa. The second maximum as at T = 0.45 s, which is also the characteristic time. The dependence of the peak WSS on time reflects the changes due to the velocity of the pulse wave. Comparison of the peak wall shear stress for the four different cases (without stenoses and with 1, 2 and 3 stenoses correspondingly) shown on Fig. 10 reveals the peak WSS maximum value of about 6.7 Pa at the characteristic point of T = 0.2 s for the cases with two and three stenoses. This is the characteristic point where the pulse wave has a maximal velocity. In the case of a bifurcation without or with one stenosis, the maximum WSS is about 5.6 Pa, as it is achieved with a delay of 0.02 seconds compared to the other two cases. The differences in the maximal WSS value (1.8 Pa) for the cases with 1, 2 and 3 stenoses in comparison to the case without stenosis (1.5 Pa) are observed at the characteristic point of the pulse wave at T = 0.45 s. Again it is achieved with a delay of 0.02 seconds compared to the other three cases (Fig. 10).
5. Discussion and conclusion Blood flow in carotid artery bifurcation is studied on the basis of Navier-Stokes equations performing numerical simulations by a finite volume method and considering one wave period. Four different cases of
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the common carotid bifurcation were examined: without stenoses, with one, two and three stenoses. Based on geometry reconstruction a mesh generation is done. The case studies are based on different anatomies presented by the one, two or three stenoses of common carotid bifurcation vessel. The physiological geometry can be imported into a CFD solver. The numerical results of the blood flow in the common carotid bifurcation give detailed picture of the axial velocity and WSS distribution. The flow in the bifurcation is strongly unsteady and its visualisation is very difficult. For the case of carotid bifurcation without stenoses results for the axial velocity distribution are presented in the character time points T = 0 s, 0.1 s, 0.2 s, 0.3 s, 0.4 s, 0.5 s. The results show the influence of the stenoses on blood flow in the bifurcation and deposition processes around it. The recirculation zone behind the stenosis is the area of low WSS. This is the most probable area for thrombus formation. The results show that the blood flow in the carotid bifurcation is unsteady and the flow disturbances depend on the time and type of the stenoses. The pattern of the velocity and the WSS are obtained and comparison of the peak WSS is done for the four considered cases – without stenoses, with 1, 2 and 3 stenoses. The recirculation zone behind the bifurcation and stenosis is the area with low shear stress. The peak WSS is increasing and the maximum is being achieved earlier with increase of the number of stenoses. The results of computational simulations may supplement MRI, CT, Doppler and other in vivo diagnostic techniques to provide an accurate picture of the hemodynamics in particular vessels, which may help demonstrate the risks of embolism or plaque rupture posed by particular plaque deposits. The use of imaging investigation with mapping of WSS distribution in the carotid arteries in parallel with numerical simulation could help for detection of the vessel sites where atherosclerotic plaques in the separate individuals would develop. The examination of the hemodynamic profile in healthy subjects and patients in parallel with numerical analysis of blood flow and common carotid artery hemodynamics could have a prognostic value for development of carotid atherosclerosis. As such, this technique can be applied non-invasively at arterial sites where vascular anatomy typically exhibits substantial inter-individual variability. The results play an important role in understanding the formation, growth, rupture and prognosis of the damaged vessel wall and may be a practical tool for planning treatment and follow-up of patients after neurosurgical or endovascular interventions with 3D angiography. The results also present the potential of using numerical simulation to provide existing clinical prerequisites for diagnosis and treatment. The numerical results from the changes in the hemodynamic profile could also guide the therapeutic plan in the examined patients. Acknowledgments The Ministry of Education, Youth and Science of Bulgaria supported the study - funded by the Operational Programme “Human Resources Development” within the Project No. BG051PO001-3.3-05/0001 regimen Science-Business” under Grant No. D-803/2012 is gratefully acknowledged. References [1] N.M. Antonova, P.E. Tosheva and E.T. Toshev, A 3D flow analysis in an artery with symmetric and asymmetric stenosis, J Series on Biomechanics 24(1) (2009), 15–23. [2] R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymer Liquids, Vol. 1, 2nd edition. Wiley, NewYork, 1987.
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[3] S. Chien, S. Usami, R. Dellenback and M. Gregersen. Shear dependent deformation of erythrocytes in rheology of human Blood, American Journal of Physiology 219 (1970), 136–142. [4] S. Chien, S. Li and Y.J. Shyy, Effects of mechanical forces on signal transduction and gene expression in endothelial cells, Hypertension 31 (1998), 162–169. [5] J.J. Chiu, S. Usami and S. Chien, Vascular endothelial responses to altered shear stress: Pathologic implications for atherosclerosis, Ann Med 41 (2009), 19–28. [6] F.J.H. Gijsen, F.N. van de Vosse and J.D. Janssen, The infuence of the non-Newtonian properties of blood on the flow in large arteries: Steady flow in a carotid bifurcation model, J Biomechanics 32(6), (1999), 601–608. [7] A.U. Jabbar, U.A. Rana, P. Khalid and Umar H.K. Niazi, Three-dimensional numerical analysis of pulsatile blood flow around different plaque shapes in human carotid, artery, International Journal of Bioscience, Biochemistry and Bioinformatics, 2(5) (2012), 305-308. [8] Y. Jiang, K. Kohara and K. Hiwada, Association between risk factors for atherosclerosis and mechanical forces in carotid artery, Stroke 31 (2000), 2319–2324. [9] K.L. Kelvin, Wong, J. Dong and Tu. Jiyuan, Numerical Study of Stenosed Carotid Bifurcation Models Based on Wall Shear Stress Distribution, 2nd Int Conf Biomedical Engineering and Technology, IPCBEE 34 (2012), 40–44. [10] D. Liepsch and St. Moravec, Pulsatile flow of non-Newtonian fluid in distensible models of human arteries, Biorheology 21 (1984), 571–586. [11] K. Perktold, R.O. Peter, M. Resch and G. Langs, Pulsatile non-newtonian blood flow in three-dimensional carotid bifurcation models: A numerical study of flow phenomena under different bifurcation angles, J Biomed Engineering 13(6) (1991), 507–515. [12] K. Perktold and G. Rappitsch, Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model, J Biomechanics 28(7) (1995), 845–856. [13] M.A. Shaaban and J. Duerinckx, Wall shear stress and early atherosclerosis, AJR 174 (2000), 1657–1665. [14] L.C. Sousa, C.F. Castro, C.C. Antonio, R. Chaves, R. Santos, P. Castro and E. Asevedo Blood flow study in a stenosed carotid bifurcation, in: J.M. Natal Jorge, J.M.R.S. Tavares, J. Belinda, M.P.L. Parente and P.A.L.S. Martins (Eds.), 5th Portuguese Congress on Biomechanics, Espinho, Portugal, 2013, 177–180. [15] J.S. Stroud, S.A. Berger and D. Saloner, Numerical analysis of flow through a severely stenotic carotid artery bifurcation, transactions of ASME, J Biomechanical Engineering 9 2002 (124), 9–20. [16] Thurston, Rheological parameters for the viscosity, viscoelasticity and thixotropy of blood, Biorheology 16 (1979), 149–162. [17] P.E. Tosheva, N.M. Antonova and E.T. Toshev, A 3D blood flow analysis in carotid artery bifurcation, Series on Biomechanics 25(1-2) (2010), 157–160. [18] A.A. van Steenhoven, F.N. van de Vosse, C.M. Rindt, J.D. Janssenu and S. Reneman Experimental and Numerical Analysis of Carotid Artery Blood Flow, in: D.W. Liepsch (ed.), Blood Flow in Large Arteries: Applicationst to Atherogenesias and Clinical Medicine. Monogr Atheroscler, Basel, Karger, 15 (1990), 250–260. [19] I. Velcheva, N. Antonova, P. Damianov and P. Dimitrov, Common carotid artery hemodynamic factors in patients with cerebral infarctions, Clin Hemorheol Microcirc 45(2-4) (2010), 233–238. [20] X. Bai-nan, W. Fu-yu, L. Lei and Z. Xiao-jun, Numerical Simulation of Blood Flow in Internal Carotid Artery Aneurysms Based on CTA, Bioinformatics and Biomedical Engineering (iCBBE), 2010 4th Int. Conference, 2010, 1–4.
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Numerical analysis of 3D blood flow and common carotid artery hemodynamics in the carotid artery bifurcation with stenosis N. Antonovaa,∗ , X. Dongb , P. Toshevaa , E. Kaliviotisc and I. Velchevad a
Department of Biomechanics, Institute of Mechanics, Bulgarian Academy of Sciences, Bulgaria School of Civil Engineering, Tianjin University, Tianjin, China c Department of Engineering, University College London, London, UK d Department of Neurology, University Hospital of Neurology and Psychiatry “St. Naum”, Medical University, Sofia, Bulgaria b
Abstract. The results for blood flow in the carotid artery bifurcation on the basis of numerical simulation of Navier-Stokes equations are presented in this study. Four cases of carotid bifurcation are considered: common carotid artery (CCA) bifurcation without stenoses and cases with one, two and three stenoses are also presented. The results are obtained by performing numerical simulations considering one pulse wave period based on the finite volume discretization of Navier-Stokes equations. The structures of the flow around the bifurcation are obtained and the deformation of the pulse wave from common carotid artery (CCA) to the internal carotid artery (ICA) and external carotid artery (ECA) is traced. The axial velocity and wall shear stress (WSS) distribution and contours are presented considering the characteristic time points. The results of the WSS distribution around the bifurcation allow a prediction of the probable sites of stenosis growth. Keywords: 3D blood flow numerical analysis, carotid bifurcation, stenosis, wall shear stress, whole blood viscosity
1. Introduction An area of special interest is the carotid artery circulation, where stenoses, thromboses, atherosclerotic plaques and other lesions can cause cerebral disturbances. Possible relation between the carotid pathological lesions and the carotid blood flow give magnetic resonance imaging (MRI), computed tomography (CT) and Doppler ultrasound techniques, developed to measure velocity profiles and vessel morphology with high spatial and temporal resolution. However, complete description of the flow in the human vascular system is difficult because of the complex geometry of the carotid artery bifurcation and the unsteady flow. The use of imaging methods with mapping of wall shear stress (WSS) distribution in the carotid arteries in parallel with numerical simulation could help for detection of the vessel sites where atherosclerotic plaques in the separate individuals would develop. ∗ Corresponding author: N. Antonova, Department of Biomechanics, Institute of Mechanics, Bulgarian Academy of Sciences, 1113 Sofia Acad. G. Bonchev str., bl. 4, Bulgaria. Tel.: +35 929796413; Fax: +35 928707498; E-mail: [email protected].
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The WSS represents the frictional force that acts tangentially to the endothelial surface. It is accepted that the chronic exposure of the endothelial cells to high shear stress is atheroprotective, while its lower values are associated with thickening of the vessel wall and development of atherosclerotic plaques. In these cases the vascular endothelial cells are exposed to blood flow – induced shear stresses, which cause increase of their membrane fluidity and permeability and activation of the membrane-associated signal proteins – mechanochemical transduction [4, 5]. In laminar blood flow, where the fluid velocity profile is parabolic, WSS reflects the relationship of the blood flow velocity, the blood viscosity and the vessel radius. Significant changes in WSS occur in vivo due to different velocities at the sites of curvatures and bifurcations. The pulsatile blood flow, which varies with the cardiac cycle, also influences WSS, which assumes an oscillatory pattern [13]. All these phenomena justify the great interest in studying the hemodynamics around stenoses, bifurcations and other geometric irregularities as the most probable sites of generation of cerebral disturbances and pathological formations. The application of medical image reconstruction for modeling vessel to use in Computational Fluid Dynamics (CFD) has been of rapid development in recent decades. The typical process for performing numerical simulation of blood flow is based on medical imaging, image segmentation, 3D model reconstruction, grid generation and analysis of blood flow solving the Navier-Stokes equations. With the development of modern imaging technology, MRI, CT, etc., it is now possible to quantify arterial blood flow in subject-specific physiologic models and to evaluate the hemodynamics and WSS distribution [4, 5]. With the use of CFD simulations and MRI, the ability to evaluate the complex relationship between hemodynamics and the predictions of blood vessel damage may be possible. The aim of the study is to investigate the hemodynamics in the carotid artery bifurcation with stenoses by means of the 3D numerical analysis of blood flow based on the numerical solution of the Navier-Stokes and continuity equations.
2. Previous related studies 2.1. Common carotid artery hemodynamic factors in patients with cerebral infarctions and WSS The changes of the common carotid local hemodynamic factors like wall shear stress and tensile forces in 16 patients with chronic unilateral cerebral infarctions (CUCI), 58 patients with the main risk factors (RF) for cerebrovascular disease (CVD) like hypertension and hyperlipidemia and 25 healthy control subjects were investigated by Velcheva et al. [19]. The blood flow velocities (BFV), the internal diameters (D) and the vessel wall intima-media thickness (IMT) in the common carotid arteries (CCA) were recorded with color duplex sonography. Systolic (SBP) and diastolic (DBP) blood pressure we are measured and mean blood pressure (MBP) was calculated by the formula of Wiggers. Whole blood viscosity (WBV) at the shear rate of 94.5 s−1 was measured on the day of the Doppler ultrasound examination with a rotational viscometer Contraves LS 30. The ultrasound examination by color duplex scanning of the main neck arteries was performed by using a 5 MHz probe with the Versa plus - Siemens instrument. The vessel diameters of the common carotid artery (CCA) in the systole (Ds ) and the diastole (Dd ) and the IMT of the far vessel wall as the mean of 3 consecutive measurements were determined. The systolic (SBFV), mean (MBFV) and diastolic (DBFV) blood flow velocity were detected 1 to 2 cm proximal to the carotid bulb with sample volume in the center of the vessel at an angle of insonation of 45◦ . In the group with CUCI the velocity parameters on the infarction (IS) and non-infarction side (NS) were compared. The characteristics of the carotid plaques were estimated. The systolic (Ts ) and mean circumferential wall tension (Tm ) we
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are calculated by a formula according to the Laplace law [8]. The WSS, the circumferential wall tension (T) and the tensile stress (τ) we are calculated too. The SBP, WBV and IMT were significantly increased in the patients with CUCI and RF for CVD in comparison to controls. Lower systolic WSS and τ and higher T were established in the patients with CUCI. The IMT correlated with WSS and τ. The study confirms the complex influence of the changes in WBV and blood pressure for the development of carotid atherosclerosis. The findings in patients with cerebral infarctions indicated a higher degree of the vessel wall damage in these patients [19]. 2.2. Numerical studies of blood flow in the carotid artery bifurcation with stenosis A 3D blood flow analysis in an artery with a symmetric and asymmetric stenosis is done by Antonova et al. [1]. The blood flow is analysed in the straight section of the carotid artery with a symmetric and asymmetric stenosis, simultaneously solving the Navier-Stokes and mass transfer equation. Data of healthy 20 years old patients are obtained by averaging the dimensions of the carotid artery with length 8 cm and diameter 0.8 cm just before the bifurcation. The axial velocity component and strain rate are illustrated considering characteristic time points. For the symmetric stenoses two cases of a symmetric obstacle with a 3 cm length and maximal height 1 mm (about 11% of the arterial lumen) and 2 mm (about 44% of the arterial lumen) are used for simulations. In fact due to axial symmetry (at the maximal height) the stenosis reduces the arterial lumen from 8 mm to 6 mm and 4 mm. The strain rate (proportional to the wall shear) around the stenosis with maximum wall shear gradient is only part of the stenosis surface located on its top. Flow recirculation zones behind the stenosis for time characteristic values are obtained. The distribution of the strain rate (proportional to the wall shear) around the stenosis and the presence of recirculation zones during a sub-period of the considered period of time allow a prediction of the sites of stenosis growth.The results illustrating flow around a non-symmetric stenosis with different percentage of artery lumen reducing (16% and 25%) show well formed recirculation zone with considerably high intensity, located after the stenosis. The location of the area of high wall shear gradient is again on top of the obstacle, showing the site of intensive deposition. A 3D numerical analysis of blood flow based on the numerical solution of the Navier-Stokes and continuity equations in the carotid artery bifurcation with or without stenoses is simulated using the 3D source code is developed by Tosheva et al. [17]. Time dependent boundary conditions are used for modeling the development of the blood flow induced by a pulse wave (pressure wave) in the rigid tube. The fluid motion is modeled by solving numerically the system of the Navier-Stokes equations and the continuity equation. Blood is regarded as incompressible fluid with constant density and conservation of mass gives the incompressibility condition. The blood flow around carotid bifurcation (distribution of axial velocity and strain rate) and in the carotid bifurcation containing two and three asymmetrical stenoses placed on the wall of carotid artery is obtained [17]. The results are given at the time point of the maximal pulse wave velocity and are illustrated by the axial velocity and strain rate. The last one is the important factor in the deposition process on the vessel walls. The flow in the bifurcation is strongly unsteady and its visualization here is very difficult. The pattern of the velocity and the strain rate are presented in the character time point only. To investigate deeply the influence of the stenoses on the blood flow in the bifurcation and deposition processes around it, the stationary part of the flow has to be studied. A numerical study of stenosed carotid bifurcation models based on wall shear stress distribution is presented by Kelvin et al. [9]. Cardiovascular models based on medical image reconstruction of ten patientspecific carotid bifurcations in terms of their geometries and flow properties are used. The characteristics of stenoses at various sections of the anatomical structure are used as indicators for explaining the
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difference in flow properties such as wall shear stress. The geometry of the carotid vessels is extracted from MRI imaging of six patients. The segmentation process includes thresholding and region growing, followed by 3D anatomical reconstruction to obtain a very coarse solid model. During thresholding, a range of gray scale values are selected such that the region to be selected is of best contrast within this range. After the regions of interest were selected and extracted, the voxels are grouped together to form a 3D geometry. Then the reconstruction of carotid bifurcation anatomy into a Computer Aided Design (CAD) model is performed based on the segmentation information. The fluid is assumed to have a density (ρ) of 1.176 g/cm3 , viscosity (η) of 4.0 mPa·s and molar mass of 25 kg/kmol. The flow is assumed to be Newtonian and the vessel walls are rigid. The time period T of one cardiac cycle is 0.92 s. The k- turbulence model is applied in the simulation. To ensure the convergence of each time step, the convergence criterion for the relative residual of all dependent variables was set to 1 × 10– 4 . For time discretization, the second order backward Euler transient scheme is used. Steady and unsteady flow patterns are analysed in rigid models of the carotid artery bifurcation by van Steenhoven et al. [18]. Both measurements and calculations were carried out enabling an experimental validation of the numerical method, based on Galerkin’s finite element approximation of the Navier-Stokes and continuity equation. Using patient-specific carotid artery bifurcation data from ultrasound imaging Sousa et al. [14] realized a semi-automatic generation of a structured and conformal hexahedral mesh of a nearly planar carotid bifurcation. The geometry, velocity at the inlet (common carotid artery) and at the oulet (internal carotid artery) obtained from ultrasound measurements are imported into the finite element model software to simulate the flow dynamics. The three-dimensional, unsteady, incompressible Navier-Stokes equations are solved with the assumptions of rigid vessel walls and constant viscosity. A good agreement between ultrasound imaging and computational simulated results is achieved. One example of grid reconstruction and blood flow simulation for a patient with internal carotid artery aneurysms is presented by Xu Bai-nan et al. [20]. The method accurately duplicates the geometry to provide computer simulations of the blood flow. Initial images are obtained by using CT angiography in DICOM format. The image is processed by using MIMICS software, and the 3D fluid model (blood flow) and 3D solid model (wall) are generated. The subsequent output is exported to the Ansys workbench software to generate the volumetric mesh for further hemodynamic study. The fluid model is defined and simulated in CFX software, while the solid model is calculated in ANSYS software. The force data calculated firstly in the CFX software are transferred to ANSYS software, and after receiving the force data, total mesh displacement data are calculated in Ansys software. Then the mesh displacement data are transferred back to CFX software. The results of simulation could be visualized in CFX-post. The wall shear stress and wall total pressure are in good agreement with the clinical data. 2.3. Comparison of wall shear stress Individual variation in the anatomy of arteries in human is considerable. The degree of blockage is from 9% to 60% and all the bifurcations exhibit some degree of geometrical non-planarity [11]. It has been indicated by previous studies that wall shear stress promotes luminal thinning and plaque rupture and low WSS regions are more prone to athersclerosis. The growth of lesion will lead to stenosis at the artery wall, and further narrowing downstream of the blood vessel will occur. Therefore, this is one of the key parameters that is used for examination of diseased carotid bifurcations. Topological maps of the predicted maximum wall shear stress at peak pressure for each of the ten case studies are given. In all of the models, the maximum wall shear stress (WSSmax ) is shown to have peak values on the inner
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walls of ICA and ECA near the bifurcation points, and significantly elevated values spiralling around the larger angular branch (either the ICA or ECA) from the inner walls along the superior orientation. The maximum WSSmax value appeared at the stenosed section due to vessel tapering. 2.4. Three-dimensional numerical analysis of pulsatile flow and wall shear stress in the carotid artery bifurcation To analyse the pulsatile flow field and the mechanical stresses in a three-dimensional carotid artery bifurcation model, computer simulation is applied by Perktold et al. [12]. Numerical results are presented for axial and secondary flow velocity and wall shear stresses with special emphasis on the fluid dynamics in the carotid sinus. Flow and stress patterns in human carotid artery bifurcation models, which differ in the bifurcation angle, are analysed numerically under physiologically relevant flow conditions. The governing Navier-Stokes equations describing pulsatile, three-dimensional flow of an incompressible non-Newtonian fluid are approximated using a pressure correction finite element method. The blood non-Newtonian behaviour is modelled by Casson’s relation, based on measured dynamic viscosity at supposed rigid blood vessel wall. The results show that the complex flow in the sinus is affected by the angle variation. The magnitude of reversed flow, the extension of the recirculation zone in the outer sinus region and the duration of flow separation during the pulse cycle as well as the resulting wall shear stress are clearly different in the small angle and in the large angle bifurcation. The haemodynamic phenomena, which are important in atherogenesis, are more pronounced in the large angle bifurcation. To simulate the transition zone flow around carotid bifurcation with and without stenoses the wall turbulence model is used by Perktold et al. [12]. Results for different reduction of arterial lumen are obtained. To investigate the effect of the distensible artery wall on the local flow field and to determine the mechanical stresses in the artery wall, a numerical model for the blood flow in the human carotid artery bifurcation has been developed. The wall displacement and stress analysis use geometrically non-linear shell theory where incrementally linearly elastic wall behavior is assumed. The flow analysis applies the time-dependent, three-dimensional, incompressible Navier-Stokes equations for non-Newtonian inelastic fluids. In an iteratively coupled approach the equations of the fluid motion and the transient shell equations are numerically solved using the finite element method. The results show the occurring characteristics in carotid artery bifurcation flow, such as strongly skewed axial velocity in the carotid sinus with high velocity gradients at the internal divider wall and with flow separation at the outer common-internal carotid wall and at the bifurcation side wall. Flow separation results in locally low oscillating wall shear stress. Further strong secondary motion in the sinus is found. The comparison of the results for a rigid and a distensible wall model demonstrates quantitative influence of the vessel wall motion. With respect to the quantities of main interest, it can be seen, that flow separation and recirculation slightly decrease in the sinus and somewhat increase in the bifurcation side region, and the wall shear stress magnitude decreases by 25% in the distensible model. The global structure of the flow and stress patterns remains unchanged. The deformation analysis shows that the tangential displacements are generally lower by one order of magnitude than the normal directed displacements. The maximum deformation is about 16% of the vessel radius and occurs at the side wall region of the intersection of the two branches. The analysis of the maximum principal stresses at the inner vessel surface shows a complicated stress field with locally high gradients and indicates a stress concentration factor of 6.3 in the apex region. The effect of different plaques of trapezoidal, elliptical and triangular shape, imposing 30% reduction of arterial lumen on blood flow through the carotid artery leading to plaque growth and rupture have been
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analysed [7]. A carotid artery model based on statistical analysis called tuning fork model is constructed. Comparison of the results of various CFD simulations show that trapezoidal shape has greater effect on blood flow producing highest flow velocities and wall shear stresses, and creates lowest wall shear stress region downstream of the plaque which may increase the fatty deposits in that area. In 3D transient analysis it is observed that variations of wall shear stress followed velocity pulse variations and maximum WSS corresponded to maximum velocity. So a steady analysis was performed at maximum velocity of pulse on fine meshes to save computational time. Previous studies show that one of the main factors involved in plaque buildup is low wall shear stress and on the other hand plaque rupture can be due to high shear stress as it causes high flow changes in the vessel. The maximum value of wall shear stress was found on trapezoidal plaque while maximum average wall shear stress was found on elliptical plaque because it causes a continuous change in velocity across its surface. So plaques similar in shape to the trapezoidal plaque may result in more downstream fatty deposit on artery wall. 2.5. Numerical analysis of flow through a severely stenotic carotid artery bifurcation A model based on an endarterectomy specimen of the plaque in a carotid bifurcation was examined by Stround et al. [15]. The work modeled the flow through the severely stenotic up to 70% occluded carotid bifurcation as the flow through a rigid-walled vessel and the blood is assumed to have a constant dynamic viscosity of 3.5 mPa.s and mass density 1060 kg/m3 . The flow conditions include steady flow at Reynolds numbers of 300, 600 and 900 as well as unsteady, pulsatile flow. Both dynamic pressure and wall shear stress are very high, with shear values up to 70 N/m2 , proximal to the stenosis throat in the internal carotid artery, and both vary significantly through the flow cycle. The wall shear stress gradient is also strong along the throat. Vortex shedding is observed downstream of the most severe occlusion.
3. Methods 3.1. Navier-Stokes equations The blood flows in the arteries are treated as incompressible viscous flow. Fluid motion is modeled by solving numerically the system of equations of the continuity equation and the equations of motion using the finite volume method. The Navier-Stokes equations for incompressible viscous flows read: ∂t u + u · ∇ u = −∇p + µ∇ 2 u + f
(1)
∇ ·u=0
(2)
where u is the vector of fluid velocities, p is the pressure normalized with the fluid density and f is a body force term. A fluid solver was developed using unstructured mesh. It is a three-dimensional fluid solver with second order accuracy in both time and space and is based on a finite volume formulation. For time-stepping, a Crank-Nicolson scheme was used for the diffusion term and the convection term is discretised using an explicit two-step Adams-Bashforth scheme. To create the complicated shape of the computation area a CAD system is applied with tetrahedral elements.
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Fig. 1. Entrance boundary condition scheme.
3.2. Rheological model for non-Newtonian fluid The blood flow is well known for its complex rheological behaviour. The non-Newtonain features, such as shear thinning and viscoelasticity of blood, have been widely observed [3, 16], and are closely related to the deformation and the aggregation of red blood cells in the blood. To describe the rheological property of the blood correctly, Carreau-Yasuda model [2] is used to compute the non-Newtonian viscosity: n−1 η − η∞ ˙ a a = 1 + (λγ) η0 − η
(3)
where γ˙ is a scalar quantity of the rate of deformation tensor, namely γ˙ =
2tr D2
(4)
∇ u+(∇ u)T ] where D = [ . The other parameters in Eq. (3) are employed from the experimental data of Gijsen 2 et al. [6] with the value set as: η∞ = 2.2 × 10−3 Pa s; η0 = 2.2 × 10−2 Pa s; λ = 0.110 s; a = 0.644; n = 0.392; ρ = 1410 kg m−3 . Numerical simulations by Perktold et al. [11] show that incorporation of the shear thinning behavior of blood did not alter the flow characteristics significantly. The experimental results of Liepsch et al. [10] however, indicated that the viscoelasticity of the fluid could be of importance for the region of flow reversal. The experiments are carried out for a Newtonian fluid and a shear-thinning Carreau fluid. Their comparison between Newtonian and non-Newtonian fluid shows considerable difference between the sizes of the recirculation zone formed after the step. Differences between the distribution of the wall shear in both cases are observed, too.
3.3. Entrance effects and computational configurations Blood flow in the carotid artery bifurcation with or without stenoses is simulated using the finite volume method. Time dependent boundary conditions are used for modeling the development of the blood flow induced by a pulse wave (pressure wave) (see Figs. 1 and 2). It is also assumed that the tube is rigid. The computational area of the considered problem is a rigid cylindrical tube with bifurcation. Four cases of the tube with bifurcations are considered: without stenosis and with one, two and three stenoses. Non-slip boundary conditions for velocities are used (U = 0, V = 0 and W = 0) over the rigid wall of the tube. The continuity boundary conditions are applied for velocities at the end of the bifurcations. A modified boundary condition representing the pulse wave is used at the tube entrance. It is shown schematically in Fig. 2.
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Fig. 2. Pulse wave.
Fig. 3. Surface mesh for the common carotid bifurcation without stenosis and with one, two and three stenoses.
A fully developed Suasion profile for z-velocity, with a Reynolds number pulsing like that shown in Fig. 2, has the following form: ⎛
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Tetrahedral elements are adopted for its flexibility for complex geometries. The average mesh size is around 0.05 cm and the total amount of elements is around 250, 000 for each case. See Fig. 3 for the surface mesh for all of the 4 simulation cases. The time step for the simulation is dt = 0.005 s, yielding a CFL number of around 0.1. 4. Results Blood flow in carotid artery bifurcation is studied on the basis of Navier-Stokes equations. The results are found by performing numerical simulations considering one pulse wave period and using a finite volume method. The structures of the flow around the bifurcation are obtained. The axial velocity distribution in the CCA without stenoses is illustrated considering characteristic time points. Using the numerical simulation the deformation of the pulse wave from CCA to the ICA and ECA has been traced. The axial velocity and WSS distribution around the bifurcation allow a prediction of the probable sites of stenosis growth. Four cases are considered in the present work: blood flow in the carotid bifurcation without stenoses (Figs. 4–6) and blood flow in the carotid bifurcation containing one (Fig. 7), two (Fig. 8) and three stenoses (Fig. 9). The changes of the axial velocity in six characteristic time points of the pulse wave in the CCA bifurcation without stenoses are shown on Fig. 4 (T = 0 s, 0.1 s and 0.2 s) and on Fig. 5 (T = 0.3 s, 0.4 s and 0.5 s). The axial velocity of the flow around the carotid bifurcation without stenoses is increasing at T = 0.1 s in comparison with the initial moment T = 0 s, especially in the external carotid artery (ECA) – left branch of the bifurcation. The distribution of the axial velocity at another three characteristic time points T = 0.3 s, 0.4 s and 0.5 s, in the CCA bifurcation without stenoses, shown on Fig. 5 reveal increasing axial velocity and flow disturbances at T = 0.4 s. The greatest change in the velocity distribution Z
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Fig. 8. Velocity magnitude contour and wall shear stress in the carotid bifurcation of the carotid artery with 2 stenoses. Left: velocity magnitude contour unit: m/s Right: Wall shear stress distribution unit: Pa.
and increased non-uniformity and the velocity of flow in the area of the carotid bifurcation was observed in the specific time point of the maximum speed of the pulse wave during systole (T = 0.1 s) – Fig. 4, and in diastole (T = 0.4 s) – Fig. 5. The results show that the blood flow in the carotid bifurcation is unsteady and the velocity changes with time and spatial location in the vessel. The results are given at the time
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Fig. 10. Comparison of peak wall shear stress for different cases.
points when the pulse wave reaches the maximum value and are illustrated by the axial velocity and WSS distribution. The WSS has an important role in the deposition process of the plaques on the vessel walls. The results from the numerical simulation for the case without stenoses and the velocity magnitude contour in the CCA bifurcation to the ICA and ECA are shown on Fig. 6 (Left panel). The flow is unsteady and the velocity magnitude changes between 0.8 m/s at the entrance of the CCA to 0.1 m/s at the entrance of the ICA after bifurcation to the maximum value of 0.8 m/s at the outlet of the ECA. It is seen from Fig. 6 (Left panel) that with the exception of the entrance of the CCA, where the maximum
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velocity magnitude is on the axis of the artery (0.8 m/s), in the remainder of the artery to the bifurcation the axial velocity is almost constant (0.4–0.5 m/s). The results show that the flow becomes unsteady at the bifurcation and after this region. In contrast to the ECA (left branch of the bifurcation), where the flow is accelerated to the maximum in the end of the corresponding section, at the beginning of the internal carotid artery (ICA) the flow is significantly slowed down, the velocity magnitude is only 0.2 m/s to about 0.1 m/s on the inside of the artery. This low velocity region in the initial inner part of ICA (0.1–0.2 m/s) creates a corresponding region of low WSS seen on Fig. 6 (Right panel) where the distribution of WSS has the following characteristics: relatively high WSS (3–3.5 Pa) to the bifurcation and low WSS between 0.5–1 Pa at the beginning of the ICA (right branch). This area of low shear stress is the site where the conditions of the flow are predisposing for plaque deposition and narrowing of the artery. In the case of the CCA with one stenosis before the bifurcation, the velocity magnitude contour, shown on Fig. 7 (Left panel) is the following: the area, where the velocity magnitude is only 0.2 m/s to about 0.1 m/s on the inside of the artery is increasing. This low velocity region after the bifurcation at the beginning of the inner part of ICA (0.1–0.2 m/s) results in the increase of the corresponding area of low WSS seen on Fig. 7 (Right panel). Noticeably smaller is the area with the lowest wall shear stress on the inner side at the entrance of the internal carotid artery (0.5–1 Pa), but this portion is displayed on the outside of the ICA, shown on Fig. 7 (Right). The numerical results of the velocity distribution in the case of CCA bifurcation with two stenoses indicate acceleration of the flow in the narrowed part of the artery, shown on Fig. 8 (Left panel), where the velocity magnitude varies between 0.6 m/s – 0.7 m/s. An increase of the area of lower velocity on the inside and outside at the entrance of the internal carotid artery (ICA) after the bifurcation in comparison with the cases with one and without stenoses is observed on Fig. 8 (Left panel). A substantial change in the distribution of the WSS after the bifurcation, and a significant increase of the WSS at the beginning of the left branch (ECA) is also observed. An increase in the WSS above the bifurcation is found too. Similar characteristics show the trends of the velocity magnitude and the velocity contour for the case with three stenoses, shown on Fig. 9 (Left panel) and the WSS distribution on Fig. 9 (Right panel). The relationship between the peak wall shear stress and time for one pulse wave, shown on Fig. 10 has 4 inflection points at which the curvature changes its sign. The curve changes from being concave downwards (negative curvature) to concave upwards (positive curvature) at T = 0.12 s to a maximum at T = 0.2 s and vice versa. The second maximum as at T = 0.45 s, which is also the characteristic time. The dependence of the peak WSS on time reflects the changes due to the velocity of the pulse wave. Comparison of the peak wall shear stress for the four different cases (without stenoses and with 1, 2 and 3 stenoses correspondingly) shown on Fig. 10 reveals the peak WSS maximum value of about 6.7 Pa at the characteristic point of T = 0.2 s for the cases with two and three stenoses. This is the characteristic point where the pulse wave has a maximal velocity. In the case of a bifurcation without or with one stenosis, the maximum WSS is about 5.6 Pa, as it is achieved with a delay of 0.02 seconds compared to the other two cases. The differences in the maximal WSS value (1.8 Pa) for the cases with 1, 2 and 3 stenoses in comparison to the case without stenosis (1.5 Pa) are observed at the characteristic point of the pulse wave at T = 0.45 s. Again it is achieved with a delay of 0.02 seconds compared to the other three cases (Fig. 10).
5. Discussion and conclusion Blood flow in carotid artery bifurcation is studied on the basis of Navier-Stokes equations performing numerical simulations by a finite volume method and considering one wave period. Four different cases of
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the common carotid bifurcation were examined: without stenoses, with one, two and three stenoses. Based on geometry reconstruction a mesh generation is done. The case studies are based on different anatomies presented by the one, two or three stenoses of common carotid bifurcation vessel. The physiological geometry can be imported into a CFD solver. The numerical results of the blood flow in the common carotid bifurcation give detailed picture of the axial velocity and WSS distribution. The flow in the bifurcation is strongly unsteady and its visualisation is very difficult. For the case of carotid bifurcation without stenoses results for the axial velocity distribution are presented in the character time points T = 0 s, 0.1 s, 0.2 s, 0.3 s, 0.4 s, 0.5 s. The results show the influence of the stenoses on blood flow in the bifurcation and deposition processes around it. The recirculation zone behind the stenosis is the area of low WSS. This is the most probable area for thrombus formation. The results show that the blood flow in the carotid bifurcation is unsteady and the flow disturbances depend on the time and type of the stenoses. The pattern of the velocity and the WSS are obtained and comparison of the peak WSS is done for the four considered cases – without stenoses, with 1, 2 and 3 stenoses. The recirculation zone behind the bifurcation and stenosis is the area with low shear stress. The peak WSS is increasing and the maximum is being achieved earlier with increase of the number of stenoses. The results of computational simulations may supplement MRI, CT, Doppler and other in vivo diagnostic techniques to provide an accurate picture of the hemodynamics in particular vessels, which may help demonstrate the risks of embolism or plaque rupture posed by particular plaque deposits. The use of imaging investigation with mapping of WSS distribution in the carotid arteries in parallel with numerical simulation could help for detection of the vessel sites where atherosclerotic plaques in the separate individuals would develop. The examination of the hemodynamic profile in healthy subjects and patients in parallel with numerical analysis of blood flow and common carotid artery hemodynamics could have a prognostic value for development of carotid atherosclerosis. As such, this technique can be applied non-invasively at arterial sites where vascular anatomy typically exhibits substantial inter-individual variability. The results play an important role in understanding the formation, growth, rupture and prognosis of the damaged vessel wall and may be a practical tool for planning treatment and follow-up of patients after neurosurgical or endovascular interventions with 3D angiography. The results also present the potential of using numerical simulation to provide existing clinical prerequisites for diagnosis and treatment. The numerical results from the changes in the hemodynamic profile could also guide the therapeutic plan in the examined patients. Acknowledgments The Ministry of Education, Youth and Science of Bulgaria supported the study - funded by the Operational Programme “Human Resources Development” within the Project No. BG051PO001-3.3-05/0001 regimen Science-Business” under Grant No. D-803/2012 is gratefully acknowledged. References [1] N.M. Antonova, P.E. Tosheva and E.T. Toshev, A 3D flow analysis in an artery with symmetric and asymmetric stenosis, J Series on Biomechanics 24(1) (2009), 15–23. [2] R.B. Bird, R.C. Armstrong and O. Hassager, Dynamics of Polymer Liquids, Vol. 1, 2nd edition. Wiley, NewYork, 1987.
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