CFD analysis of non-Newtonian blood flow through human carotid artery bifurcation: Carotid sinus susceptible to atherosclerosis

Abstract

Fat accumulation on the arteries' walls has become a serious health concern. This study conducted an in-depth hemodynamic analysis to identify the regions of a human carotid artery most susceptible to fat accumulation or atherosclerosis, considering the following factors: blood velocity, secondary flow regions, pressure, and wall shear stress. The hemodynamic analysis used a generalized geometric model to analyze these factors at various locations within the carotid bifurcation over different time points. Results showed that the bifurcation region is comparatively more affected by the factors contributing to atherosclerosis. Especially, the sinus region experiences all the key factors: a larger secondary flow, higher blood pressure, lower mean WSS, and lower fluctuation in temporal WSS. The findings indicate that the bifurcation region, particularly the sinus is prone to fat accumulation. These results are consistent with clinical observations, emphasizing the bifurcation and sinus regions as key sites for plaque accumulation. Insights from this analysis will serve as a foundation for optimizing geometric models of bifurcated arteries to improve blood flow and, consequently, reduce the risk of atherosclerosis.

Nomenclature

WSS

Wall shear stress

CFD

Computational fluid dynamics

LES

Large eddy simulation

CCA

Common carotid artery

ICA

Internal carotid artery

ECA

External carotid artery

LOW

Left outer wall

LIW

Left inner wall

ROW

Right outer wall

RIW

Right inner wall

CC1, CC2, CC3, CC4

Specific lines on the bifurcation plane in the common carotid artery

SS1, SS3, SS6

Specific lines on the bifurcation plane in the sinus segment

EC1, EC2, EC3

Specific lines on the bifurcation plane in the external carotid artery

p1 to p4

Specific points along left common carotid artery walls/left outer wall

p5 to p7

Specific points along left sinus walls/left outer wall

p8 to p11

Specific points along right common carotid artery walls/right outer wall

p12 to p14

Specific points along right external carotid artery walls/right outer wall

p15 to p17

Specific points along right sinus walls/left inner wall

p18 to p20

Specific points along left external carotid artery walls/right inner wall

t2, t4, t5, t6

Specific time points in the cardiac cycle

1. Introduction

Blood removes waste products from our bodies. It also supplies oxygen and nutrients to vital organs and tissues. Blood circulates through a network of blood vessels. The major types of blood vessels are capillaries, veins, and arteries. Capillaries connect veins and arteries. Veins collect oxygen-poor blood and return it to the heart for re-oxygenation while arteries deliver oxygen-rich blood to tissues. A growing concern regarding arteries is the accumulation of fat on their walls, particularly in areas with low WSS, secondary flow, and recirculation zones [[1], [2], [3]]. High WSS, on the other hand, can cause mechanical damage to the arterial walls, activating immune cells and increasing inflammation, which also contributes to plaque formation. The pulsatile nature of blood flow causes periodic increases and decreases in shear stress. Plaques exposed to oscillatory WSS are more likely to rupture or erode [4]. Additionally, high blood pressure has been linked to the rapid development of atherosclerosis [5,6]. The Common Carotid Artery (CCA) is a major vessel that transports oxygen-rich blood to the brain. And, it is situated on both sides of the neck which can be easily palpated just beneath the jawline [7]. The common carotid artery is divided into two branches: the internal carotid artery (ICA) and the external carotid artery (ECA). The sinus, a bulge area right before the ICA, is crucial for regulating blood pressure. The carotid artery is particularly important as it supplies blood to the brain. Carotid artery disease, also known as carotid artery stenosis, is a condition where these arteries become narrowed or blocked due to the buildup of plaque. Plaque consists of fat, cholesterol, and other substances found in the blood, which disrupt and reduce blood flow to the brain. This condition can significantly increase the risk of stroke, which is why it is important to diagnose and treat carotid artery disease promptly. Carotid plaques are more common at the carotid bifurcation in patients with symptomatic stenosis due to temporary reverse blood flow [8]. According to Ref. [9], carotid disease causes 10%–20 % of ischemic strokes and is a common sign of localized atherosclerosis, often found at the origin of the ICA. The post-mortem results from the study in Ref. [10] indicate fibrous and severe plaque were primarily found in the ICA and ECA branches [11]. Patients frequently develop atherosclerotic plaque in the anterior and posterior sections of CCA and ICA. In the United States and Western Europe, cardiovascular disease is still the leading cause of death. Atherosclerosis, which is the main cause of heart attacks and strokes, is responsible for most of these deaths [12]. While several factors can contribute to atherosclerosis, the geometry of the arteries is more likely to play the initial role in the development of atherosclerotic plaque [[13], [14], [15]]. Researchers conducted many numerical simulations to understand the behavior of blood flow, particularly in areas of recirculation and separated flow in curved or branching sections of blood vessels [[16], [17], [18]]. [19] Used a pressure correction FEA method to simulate blood flow in a 3D bifurcated carotid artery model under pulsatile flow conditions, and compared the velocity profile and WSS distribution with experimental results from Ref. [20] to better understand the relationship between atherogenesis and fluid dynamics in the carotid sinus. In these studies, blood was considered an incompressible Newtonian fluid. Blood is a complex non-Newtonian fluid with viscoelastic and shear-thinning characteristics, particularly at low shear rates. Red blood cells may aggregate at low shear rates, giving blood a non-Newtonian fluid characteristic. The non-Newtonian property of the blood gradually decreases when the aggregated red blood cells break apart, and its Newtonian property takes over. Taking this into account, several non-Newtonian models, including the Casson equation [21], power law [22], and the Carreau-Yasuda model [23], have been used in numerical studies of hemodynamics in blood circulation. Perktold et al., 1991 conducted a numerical study of blood flow at the carotid bifurcation and discovered that the flow behaviors at the bifurcation obtained using the Casson model of blood differed little from those obtained using the Newtonian model [24]. found that the non-Newtonian effect was significant near arterial stenosis in a comparative study. Therefore, a non-Newtonian viscosity model is required to more accurately estimate the blood flow behavior. In this study, the Carreau viscosity model was used to capture the non-Newtonian behavior of blood.

This study analyzes the hemodynamics to explain why clinical findings often show severe plaque in the bifurcation and origin of ICA compared to the CCA or ECA. Moreover, in most CFD analyses of bifurcated carotid arteries, the geometric models used are often patient-specific or average of small sample sizes, rather than general models. Since blood vessel geometry varies significantly among individuals, analyzing blood flow using a generalized model will provide a valuable starting point for optimizing geometric models to enhance blood flow and, consequently, reduce the risk of atherosclerosis. This work provides a comprehensive blood flow analysis, particularly, introducing zero axial velocity contours to offer a new perspective on blood hemodynamics. This study utilizes the bifurcated carotid artery model from Ref. [25] due to its large sample size and the thorough consideration given to sampling from different age groups.

2. Research method

A 3D model of the bifurcated carotid artery was constructed using SolidWorks 2017, as detailed in Section 2.2. Before this study, grid independence and validation tests were conducted to ensure the reliability of the methodology, detailed in Sections 2.3, 2.4. Blood flow was simulated in Ansys Fluent 15 to obtain blood velocity, pressure, and wall shear stress at specific locations and times, as detailed in Sections 2.1, 2.2, 2.6. The input velocity profile (representing the cardiac cycle) was divided into 86 equal-time segments, with the simulation covering six full cycles. Each time step required a residual in continuity of less than 10⁻⁶ for convergence. The Carreau model was employed to capture the non-Newtonian behavior of blood, as outlined in Section 2.5. Results were exported using Ansys CFD-Post for both qualitative and quantitative analyses. The study aimed to identify regions with larger secondary flow areas, higher blood pressure, a lower WSS value, and lower fluctuation in temporal WSS. The qualitative analysis included velocity and pressure contour plots (detailed in Sections 3.1, 3.2), while the quantitative analysis focused on velocity, pressure, and WSS distribution profiles (detailed in Sections 3.3, 3.4, 3.5).

2.1. Numerical method and Governing equations

A pressure-based solver was selected to solve the incompressible flow equations due to the nature of the flow. The velocity formulation was set to absolute, meaning velocities were calculated in an inertial frame of reference. The simulation was run under transient conditions to capture the flow's time-dependent behavior. To resolve larger turbulent structures, the Large Eddy Simulation (LES) model was used, with subgrid-scale stresses modeled by the Smagorinsky-Lilly model, where the Smagorinsky constant $C_s$ was set to 0.1. Pressure and velocity coupling were handled by the SIMPLE (Semi-Implicit Method for Pressure-Linked Equations) algorithm, which ensures mass conservation at each time step. For spatial discretization, the Least Squares cell-based method was applied to compute gradients, a second-order scheme was used for pressure calculation, and bounded central differencing was chosen for momentum discretization. The time integration was carried out using a bounded second-order implicit scheme.

The filtered continuity and momentum equations for non-Newtonian and incompressible fluid flow can be expressed as follows [26]:

Filtered Continuity Equation:

$$\frac{\partial {\bar{u}}_j}{{\partial x}_j}=0$$ (1)

Here, ${\bar{u}}_j=\left(\bar{u},\bar{v},\bar{w}\right)$ represents the filtered velocity components along the corresponding coordinate axes, $x_j=\left(x,y,z\right)$, respectively.

Filtered momentum equation:

$$\frac{\partial \rho {\bar{u}}_i}{\partial t}+\frac{\partial \rho {\bar{u}}_i{\bar{u}}_j}{{\partial x}_j}=-\frac{\partial \bar{p}}{{\partial x}_i}+\frac{\partial }{{\partial x}_j}\left[\mu \left(\left|\dot{\gamma }\right|\right)\left(\frac{\partial {\bar{u}}_i}{{\partial x}_j}+\frac{\partial {\bar{u}}_j}{{\partial x}_i}\right)\right]-\frac{{\partial \tau }_{ij}}{{\partial x}_j}$$ (2)

Here, fluid density, $\rho =1050\text{kg}/m^3$, $\bar{p}$ denotes the filtered pressure, $t$ represents time. The blood viscosity, $\mu \left(\left|\dot{\gamma }\right|\right)$ depends on the shear rate, defined as ${\dot{\gamma }}_{ij}=\frac{1}{2}\left(\frac{\partial {\bar{u}}_i}{{\partial x}_j}+\frac{\partial {\bar{u}}_j}{{\partial x}_i}\right)$. The magnitude of the shear rate is defined as $\left|\dot{\gamma }\right|=\sqrt{2{\dot{\gamma }}_{ij}{\dot{\gamma }}_{ji}}$ [27]. When blood is considered a Newtonian fluid, its viscosity approaches a constant value, represented as ${\mu }_{\infty }$ = 0.0035 kg/m-s. However, for a non-Newtonian model, the constitutive equations for the apparent viscosity of blood are discussed in Section 2.5. The subgrid-scale stress terms, ${\tau }_{ij}$, are modeled using the Smagorinsky model [28],

$${\tau }_{ij}-\frac{1}{3}{\partial }_{ij}{\tau }_{kk}=-2{\nu }_{sgs}{S}_{ij}=-2{\left(C_s\Delta \right)}^2\left|S\right|S_{ij}$$ (3)

Here, $C_s$ is the Smagorinsky constant, set to 0.1. $\Delta =\sqrt[3]{\Delta x\Delta y\Delta z}$ is the filtered width and $\left|S\right|=\sqrt{2S_{ij}{S}_{ij}}$ represents the magnitude of the large-scale strain rate tensor S.

2.2. Geometric model construction

The 3D model of the artery was designed using SolidWorks 2017. The dimensions of the model were taken from Bharadvaj et al. (1982). The length of CCA is 130 mm. Fig. 1 shows the bifurcation plane of the 3D model, which includes four artery walls: left outer wall (LOW), left inner wall (LIW), right inner wall (RIW), and right outer wall (ROW); ten lines: CC1, CC2, CC3, CC4, SS1, SS3, SS6, EC1, EC2, and EC3; and a total of twenty points (P1 to P20).

Fig. 1.

Bifurcation plane of the 3D model arteries.

2.3. Grid independence test

A grid independence test was conducted for the geometric model using a uniform steady flow with a velocity of 0.333 m/s and an 80:20 flow split between the ICA and ECA. Three different mesh densities were tested, ranging from 729,035 to 1,042,713 tetrahedral cells. The axial velocity 'w' along the ECA axis was very similar across the three meshes (see Fig. 2), indicating that the number of cells had reached a saturation point. Based on this result, the mesh with 848,201 computational grid cells was selected.

Fig. 2.

Axial velocity ‘w’ in the bifurcation plane along the ECA axis for the geometric model.

2.4. Numerical validation

We conducted a computer simulation replicating the experiment by Bharadvaj et al. (1982) with a Reynolds number of 400 and a 70:30 outflow ratio. The simulated axial velocity profile along the CC1 line was compared to the corresponding profile from Bharadvaj et al. (1982) (Fig. 3(a)). Our simulation results closely matched those of the original experiment. The mesh used for this simulation is shown in Fig. 3(b).

Fig. 3.

(a) Comparison between Simulated and Experimental Non-dimensional Axial Velocity along the CC1 line. (b) Mesh Nodes and Edges at CC1 Cross Section.

2.5. Fluid property

The Carreau model was chosen to represent blood as a non-Newtonian fluid, with a density of 1050 kg/m³. The non-Newtonian viscosity model, as proposed by Carreau [29], was utilized for this characterization.

$$\mu \left(\left|\dot{\gamma }\right|\right)={\mu }_{\infty }+\left({\mu }_0-{\mu }_{\infty }\right){\left[1+{\left(\lambda \dot{\gamma }\right)}^2\right]}^{\left(n-1\right)/2}$$ (4)

Here, infinite shear viscosity, ${\mu }_{\infty }$ = 0.0035 kg/m-s, zero shear viscosity, ${\mu }_0$ = 0.056 kg/m-s, power-law index, n = 0.3568, and time constant, λ = 3.313 s [30].

2.6. Boundary conditions

A cardiac cycle of 0.857 s was assigned as the boundary condition at the CCA inlet [31]. This cycle includes the following key velocity points: minimum at 0.27 s, maximum at 0.33 s, local minimum at 0.58 s, and local maximum at 0.68 s. The simulation results were extracted for the sixth cardiac cycle at the minimum (t2 = 4.57 s), maximum (t4 = 4.63 s), local minimum (t5 = 4.88 s), and local maximum (t6 = 4.98 s) velocity points. An outflow boundary condition with a 70:30 ratio was applied at the ICA and ECA outlets. The boundary wall, set as a rigid wall, has a no-slip boundary condition.

3. Results and discussion

3.1. Velocity distribution qualitative analysis

Velocity distribution plays a crucial role in the progression of atherosclerosis. In areas of low velocity, blood flow stagnates, allowing plaque to build up. Conversely, high-velocity areas can damage arterial walls, causing inflammation and promoting further plaque formation.

3.1.1. Axial velocity (w) contours on the bifurcation plane

(See Fig. 4) Blood at a higher velocity from the CCA inlet approaches the area of bifurcation, where it splits into two separate flows with a sharp velocity gradient (as shown in Fig. 4(b)). As the cross-sectional area at the sinus gradually increases up to SS3, the flow velocity decreases and the outer wall along the sinus experiences a significant adverse pressure gradient, leading to the development of a secondary flow region near the outer sinus wall. The cross-sectional area starts to decrease from SS3 to SS6, leading to an increase in flow velocity. After SS6, there is a noticeable increase in velocity, and the peak velocity has been observed here. And, near the outer wall of the ECA's beginning, a relatively smaller secondary flow zone has developed. Due to this secondary flow region, the higher velocity region shifted towards the ECA inner wall.

Fig. 4.

Axial Velocity (w) contours on the bifurcation plane.

3.1.2. Zero axial velocity (w) contours on the bifurcation plane

(See Fig. 5) The blue lines on the bifurcation plane indicate zero axial velocity. The regions enclosed by these lines experience secondary flow and flow recirculation. At the sinus segment, flow separation starts slightly after CC3 at all the time points. Visual examination reveals that the extent of the secondary flow region at the sinus is smallest at t4 and largest at t2. Secondary flow is also observed in the ECA, except at t5. Compared to the sinus, the ECA and CCA regions are less affected by secondary flow.

Fig. 5.

Zero Axial Velocity (w) contours on the bifurcation plane.

3.1.3. Zero axial velocity (w) contours on the perpendicular planes (at CC3 and CC4) of the bifurcation plane

(See Fig. 6) Up to the cross-sectional area CC3, no secondary flow has been observed. However, at CC4, a secondary flow region forms near the LOW and the ROW, while the top and bottom boundary regions remain unaffected. From visual inspection, the CC4 cross-section is significantly affected by secondary flow at time points t4 and t6. Additionally, the area near the LOW is more affected by secondary flow at the CC4 cross-section.

Fig. 6.

Zero axial velocity (w) contours on the perpendicular planes (at CC3 and CC4) of the bifurcation plane.

3.1.4. Zero axial velocity (w) contours on the perpendicular planes (at SS1 and SS3) of the bifurcation plane

(See Fig. 7) Secondary flow occupies a significant area on the perpendicular planes at SS1 and SS3, but none is observed after SS6. In each instance, a secondary flow zone is present near the outer wall of the sinus. At times t4 and t6, this zone extends from the LOW to the top and bottom walls. Notably, secondary flow is consistently absent near the inner wall of the sinus.

Fig. 7.

Zero axial velocity (w) contours on the perpendicular planes (at SS1 and SS3) of the bifurcation plane.

3.1.5. Zero axial velocity (w) contours on the perpendicular planes (at EC1 and EC2) of the bifurcation plane

(See Fig. 8) Compared to the sinus segment, the ECA has experienced significantly less disruption from secondary flow. Secondary flow regions were observed near the ROW at EC1 only at the t2 and t4 time points. No secondary flow zones formed at or beyond EC2 at any time point.

Fig. 8.

Zero axial velocity (w) contours on the perpendicular planes (at EC1 and EC2) of the bifurcation plane.

3.1.6. Axial velocity (w) contours on the perpendicular planes (at CC4) of the bifurcation plane

(See Fig. 9) Since no secondary flow zone forms up to CC3, the analysis of velocity contours up to CC3 has been excluded from this discussion. At CC4, low-velocity regions developed near both the LOW and ROW, with the zone near the LOW being comparatively larger. Additionally, the top and bottom walls exhibit significantly higher velocity zones compared to the LOW and ROW at CC4. The highest velocity zone is located near the center of the CC4 cross-section.

Fig. 9.

Axial velocity (w) contours on the perpendicular planes (at CC4) of the bifurcation plane.

3.1.7. Axial velocity (w) contours on the perpendicular planes (at SS1 and SS3) of the bifurcation plane

(See Fig. 10) The lower-velocity region near the outer wall of the sinus extends toward both the top and bottom walls. The maximum velocity shifts towards the inner wall as the secondary flow region occupies space near the outer wall, the fluid has a smaller area to flow through.

Fig. 10.

Axial velocity (w) contours on the perpendicular planes (at SS1 and SS3) of the bifurcation plane.

3.1.8. Axial velocity (w) contours on the perpendicular planes (at EC1 and EC2) of the bifurcation plane

(See Fig. 11) Velocity profiles at the ECA shift towards the RIW, resulting in a lower velocity area near the ROW and a higher velocity region near the RIW.

Fig. 11.

Axial velocity (w) contours on the perpendicular planes (at EC1 and EC2) of the bifurcation plane.

3.2. Pressure distribution qualitative analysis

Higher blood pressure may accelerate the progression of atherosclerosis. Elevated blood pressure can damage the arterial walls, creating areas where fat can accumulate. Over time, these deposits build up, leading to the development of atherosclerosis.

3.2.1. Pressure contours on the bifurcation plane

(See Fig. 12) The highest-pressure region has developed at the apex, which is most prominent at t4. The peak pressure is slightly shifted from the apex towards the inner wall of the ECA. Overall, a higher pressure region forms in both the ECA and the sinus region.

Fig. 12.

Pressure contours on the bifurcation plane.

3.2.2. Pressure contours on the perpendicular planes (at CC4) of the bifurcation plane

(See Fig. 13) Since no flow disturbance forms until CC3, the analysis of pressure contours prior to CC3 has been excluded from this discussion. In the CC4 section, relatively higher-pressure regions are observed at the top and bottom walls near the ROW. Additionally, a relatively lower pressure region has developed adjacent to the LOW.

Fig. 13.

Pressure contours on the perpendicular planes (at CC4) of the bifurcation plane.

3.2.3. Pressure contours on the perpendicular planes (at SS1 and SS3) of the bifurcation plane

(See Fig. 14) A high-pressure region has developed near the inner wall of the sinus, with a peak near the intersection of the inner wall and the bifurcation plane. The pressure gradually decreases as we approach the outer wall.

Fig. 14.

Pressure contours on the perpendicular planes (at SS1 and SS3) of the bifurcation plane.

3.2.4. Pressure contours on the perpendicular planes (at EC1 and EC2) of the bifurcation plane

(See Fig. 15) A higher-pressure zone forms near the inner wall of the EC1 section. In EC2, the maximum pressure is slightly lower compared to EC1. Also, there isn't a clear distinction between higher and lower pressure zones on the EC2 plane.

Fig. 15.

Pressure contours on the perpendicular planes (at EC1 and EC2) of the bifurcation plane.

3.3. Velocity distribution quantitative analysis

Areas prone to fat accumulation can be pinpointed by studying velocity profiles in various locations. Atherosclerotic plaque is more likely to develop in regions experiencing secondary flows and lower velocities.

3.3.1. Axial velocity (w) along specific lines on the bifurcation plane

(See Fig. 16) Up to the CC3 cross-section, the risk of plaque formation is low as no flow stagnation, backflow or secondary flow has been observed. The CC4 line experiences greater backflow near the LOW compared to the LIW. As blood approaches towards the ICA, backflow occupies more area. Backflow has been observed in the outer sinus region, while the inner sinus remains unaffected. At SS1, backflow occupies 29.15 %, 26.62 %, 25.66 %, and 23.13 % of the SS1 line at time points t2, t4, t5, and t6, respectively. At SS3, backflow occupies 33.37 %, 3.03 %, 34.71 %, and 26.74 % of the SS3 line at time points t2, t4, t5, and t6, respectively. Additionally, backflow occupies approximately 33.92 % and 19.28 % of the EC1 line at t2 and t4, respectively. No backflow is observed in the EC2 line at any time point. Velocity profiles at SS1 and SS3 shifted slightly towards the right as secondary flow occupied a portion of the region along the left sinus wall. Conversely, velocity profiles in EC1 and EC2 shifted towards the right inner wall. In all cases, maximum velocities are reached at t4. Locations other than SS1 and SS3 are comparatively less affected by the secondary flow region.

Fig. 16.

Axial velocity (w) on the bifurcation plane.

3.3.2. Axial velocity (w) along specific perpendicular lines of the bifurcation plane

(See Fig. 17) In this section, the lines perpendicular to and passing through the center of CC4, SS1, SS3, EC1, and EC2 are the locations of interest. The highest velocities have consistently been observed at time point t4 across all the locations. No flow reversal or secondary flow regions were formed near the top or bottom walls of the artery for any line. The velocity profiles at SS3 and EC2 show an exception: at both lines, the peaks in velocities occurred near the top and bottom walls with a dip in the middle. However, at the t4 time point, this dip in velocity profiles decreases.

Fig. 17.

Axial Velocity (w) on the perpendicular Planes of the Bifurcation Plane.

3.4. Pressure distribution quantitative analysis

Maintaining a healthy blood pressure level is crucial in preventing the progression of atherosclerosis and protecting the cardiovascular system. High blood pressure or hypertension, increases the stress on the arterial walls, causing them to become damaged over time. This damage promotes plaque formation and adhesion to the arterial walls, leading to the buildup of fatty deposits.

3.4.1. Pressure along specific lines on the bifurcation plane

(See Fig. 18) The minimum and maximum pressure profiles on the lines CC4, SS1, SS3, EC1, and EC2, were observed at t2 and t4, respectively. The peak pressures developed on the lines CC4, SS1, SS3, EC1, and EC2 are 145, 546, 305, 626, and 217 Pa, respectively. On the CC4 line, a higher-pressure region has developed near the ROW. In the case of SS1 and SS3, the area near the LIW experiences higher pressure than the area near the LOW. As blood approaches SS3 from SS1 the maximum pressure decreases to 305 Pa, while the minimum pressure rises to 193.26 Pa. On the other hand, the EC1 segment experiences higher pressure near the RIW than the area near the ROW. As blood approaches the EC2 line, the pressure becomes more uniformly distributed than in the case of EC1, with the maximum pressure decreasing to 217 Pa.

Fig. 18.

Pressure on the bifurcation plane.

3.4.2. Pressure along specific perpendicular lines of the bifurcation plane

(See Fig. 19) In this section, the lines perpendicular to and passing through the center of CC4, SS1, SS3, EC1, and EC2 are the locations of interest. The highest and lowest pressure profiles were observed at t4 and t2, respectively, for all the locations. Along the CC4 perpendicular line, two distinct high-pressure regions were identified near the top and bottom walls at t4, with lower pressure observed immediately adjacent to these walls. On the SS1 and SS3 perpendicular lines, higher pressure is formed just adjacent to both the top and bottom walls. The top and bottom walls at SS3 perpendicular line experience the highest pressure, exceeding 250 Pa, compared to CC4, SS1, EC1, and EC2. On the EC1 perpendicular line, the pressure profile showed a minimum value at the top and bottom artery walls, which gradually increased to a maximum in the middle. On the other hand, on the EC2 perpendicular line, the pressure distribution was nearly uniform, with no clearly defined high or low-pressure regions.

Fig. 19.

Pressure on the perpendicular Planes of the Bifurcation Plane.

3.5. Wall shear stress (WSS) quantitative analysis

Areas experiencing low WSS are prone to lipoprotein buildup, which can oxidize and become trapped in arterial walls. Over time, this process can lead to plaque formation, obstructing blood flow and increasing the risk of heart attack and stroke. Conversely, high wall shear stress can mechanically damage arterial walls, triggering immune cell activation and inflammation, which also promotes plaque formation. Maintaining a balanced wall shear stress is crucial for healthy blood vessels. Disruption of this balance can contribute to the initiation and progression of atherosclerosis.

3.5.1. WSS distribution at Specific Times

(See Fig. 20) The highest and lowest WSS profiles were observed at the t4 and t2 time points, respectively, for all locations of interest. The LIW starts with a higher WSS at P17. As blood flows forward, the WSS decreases, reaching a minimum at P16. After this point, the WSS increases again, peaking at P15. Along the LOW, a relatively low WSS value was observed up to P6, followed by a gradual increase, reaching a maximum at P7. The RIW starts with a higher WSS at P18. As blood flows forward, the WSS initially decreases, reaching a minimum near P19. After P19, the WSS gradually increases. Finally, along the ROW, a relatively low WSS was observed up to P13, after which it began to increase.

Fig. 20.

Wall shear stress distribution profile at specific times.

3.5.2. Temporal wall shear stress distributions

(See Fig. 21) The pulsatile nature of blood flow causes periodic increases and decreases in shear stress. Plaques exposed to oscillatory WSS are more likely to rupture or erode. All artery walls experience significant variations in shear stress over the cardiac cycle, with peaks occurring around t4 seconds for all points (P1-P20). The maximum and minimum WSS along the LOW occur at P7 and P4, respectively. Along the ROW, the maximum and minimum WSS are at P14 and P11. Along the LIW, the maximum and minimum WSS are at P15 and P16. Along the RIW, the maximum and minimum WSS are at P18 and P19.

Fig. 21.

Temporal WSS distributions at the sixth cardiac cycle.

3.5.3. Mean WSS and RMSE Values of Temporal WSS at Location Points p1 to p20

(See Fig. 22) Locations with high mean WSS are likely subject to higher forces, which can contribute to mechanical stress on the vessel walls, potentially reducing plaque formation. In addition, points with high RMSE values indicate greater variability in shear stress over time and are more likely to experience destabilizing forces, leading to plaque rupture. Conversely, locations with low mean WSS and low RMSE are more suitable for plaque formation. Points along the right sinus wall (p15, p16, and p17) have very high mean WSS values and tend to have higher RMSE values. In contrast, points along the left sinus wall (p5 and p6) have much lower mean WSS and RMSE values. Points along the right EC (p12, p13, p14) experience lower mean WSS and RMSE values compared to those along the left EC (p18, p19, p20). These observations conclude that the left sinus wall is a more suitable spot for plaque formation considering the factors of mean WSS and oscillatory WSS.

Fig. 22.

Mean WSS and RMSE Values of Temporal WSS at Location Points p1 to p20.

4. Conclusions

Atherosclerosis or plaque accumulation on artery walls is mainly advanced by, a larger secondary flow area, higher blood pressure, a lower WSS value, and lower fluctuation in temporal WSS. This study identified the areas of the bifurcated carotid artery affected by these conditions. A) Secondary flow was observed between the CC3 and SS6 cross-sections, with flow separation starting slightly after CC3 and extending into the sinus region. The ECA segment of the bifurcation plane also experiences a secondary flow region, though it is less severe than in the sinus. B) The highest pressure is observed at the bifurcation point, with higher pressure near the bifurcation along the left ECA wall and within the sinus region. C) Points along the left sinus wall (p5 and p6) and the right EC wall (p12 and p13) experience comparatively lower mean WSS and lower fluctuation in temporal WSS value. These findings show that the bifurcation region, particularly the sinus is significantly affected by the factors that contribute to atherosclerosis. However, this study has some limitations, such as treating the artery wall as rigid and using a constant outflow ratio as the outlet boundary conditions. A more precise outcome could be achieved by addressing these issues in future research.

CRediT authorship contribution statement

Md. Syamul Bashar: Writing – review & editing, Writing – original draft, Visualization, Validation, Supervision, Software, Resources, Project administration, Methodology, Investigation, Formal analysis, Data curation, Conceptualization. Rifat Hossain: Writing – original draft, Validation, Software, Resources, Data curation. Md. Habibur Rahman: Writing – original draft, Software. Utsha Roy: Writing – original draft, Software. Md. Shafiqul Islam: Writing – review & editing, Supervision, Project administration, Conceptualization.

Ethical approval

This declaration is “not applicable”.

Availability of data and materials

The velocity, wall shear stress (WSS), and pressure data were calculated using the Ansys Fluent 15 software. These data can be replicated by following the instructions outlined in the Research Methodology section of the study.

Declaration of generative AI and AI-assisted technologies in the writing process

During the preparation of this work, the author(s) used ChatGPT to improve language and readability. After using this tool/service, the author(s) reviewed and edited the content as needed and take(s) full responsibility for the content of the publication.

Funding

This research was not funded by any financial organizations.

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.