Comparison of blood viscosity models in different degrees of carotid artery stenosis

Data source

This study included patients who underwent CTA of the head and neck due to symptoms suggestive of cerebrovascular insufficiency at Xianning Central Hospital between January 1, 2022 and May 31, 2023 and processing of patient images was performed in accordance with institutional ethics committee guidelines. Written informed consent was obtained from all participants prior to their inclusion in the study, allowing the use of their clinical imaging data for research purposes under institutional review board (IRB) protocol (No. 202401003). The data were accessed for research purposes from March 12, 2024, to May 10, 2024.

Study population

This study included 214 patients diagnosed with carotid atherosclerosis. We identified cases with different stenosis degrees in the C1 segment of the internal carotid artery. Inclusion Criteria: Patients diagnosed with carotid atherosclerosis based on clinical evaluation and CTA imaging. Degree of stenosis between 1% and 90%, as defined by the NASCET criteria (Kumar et al., 2020). Patients who provided informed consent for participation in the study. To ensure that the study population reflected symptomatic carotid artery disease, patients were included only if they had a confirmed history of transient ischemic attack (TIA), ischemic stroke, or central retinal artery occlusion (CRAO). Exclusion criteria: (1) Patients presenting with dizziness alone without a documented ischemic event were excluded to prevent confounding; (2) patients with severe internal carotid artery occlusion or aneurysms; (3) patients who have undergone any prior interventional procedures (e.g., balloon dilation or stent placement); (4) patients with other cardiovascular or neurological conditions that may interfere with the study results. Finally, 33 narrow carotid arteries were selected: 16 females and 17 males, and 11 cases of mild stenosis, 11 cases of moderate stenosis, and 11 cases of severe stenosis were included. The 3D models used in this study are available at MorphoSource (refer to the Data Availability section). The degree of stenosis was classified according to the NASCET criteria as follows: Mild stenosis: 1% to 29% narrowing. Moderate stenosis: 30% to 69% narrowing. Severe stenosis: 70% to 99% narrowing. Cases of complete occlusion (100% luminal narrowing) were excluded from this study. CFD models were constructed based on CTA images of each patient to simulate both the Newtonian (N group) and non-Newtonian (NN group) models. The differences in hemodynamic parameters between the two models were analyzed and compared.

Data processing and carotid artery modeling

Mimics is used to reconstruct the imported CTA images by threshold segmentation. Smooth the 3D model with Geomagic Studio, clear the wrong data, and enter it into FLUENT for meshing and simulation. Blood flow was assumed to be laminar. Physiological velocity waveforms were applied to the inlet of the common carotid artery (CCA) (Albadawi et al., 2021). Due to the difficulty in measuring the outlet flow rates of the internal carotid artery (ICA) and external carotid artery (ECA), we set the boundary conditions to a ternary Windkessel model with a proximal resistor R₁, a distal resistor R₂, and a capacitor to avoid errors that might be caused by assumptions on blood pressure or flow rate (Liu et al., 2021; Jung et al., 2022). A single pulsation period is considered to be 0.8 s. Newtonian’s model assumes a constant viscosity, a density of 1,060 kg/m³, and a dynamic viscosity coefficient of 0.0035 Pa s. The Carreau model of a non-Newtonian fluid satisfies η₀ = 0.25 Pa s, η_∞ = 0.0035 Pa s, n = 0.25, λ = 25, a density of 1,060 kg/m³. The inlet flow rate is constant. η₀ and η_∞ are the zero shear rate viscosity and infinite shear rate viscosity, n is the power index, λ is the time constant.

Grid and time step independence tests

Figure 2A is a time-independent experiment involving four different time intervals: 0.1s, 0.01s, 0.001s, and 0.0001s. The results show that there is very little change beyond the time interval of 0.001s, so 0.001s is chosen for all numerical simulations. In addition, the grid independence test is shown in Fig. 2B. Using the same vascular model, WSS maxima with mesh numbers of 84,110, 164,822, 372,615, 540,280, 761,138, 1,031,332, and 1,429,980 are compared. When the number of grids exceeds 1,031,332, the effect on the maximum WSS is extremely weak. Therefore, about one million grids are eventually selected for meshing. The mesh division of one blood vessel in this study is shown in Fig. 2C.

Verification

Moreover, in this study, OSI and WSS were utilized to compare the findings with those of research (Razavi, Shirani & Sadeghi, 2011), to validate and confirm the accuracy and consistency of the numerical simulation assessment. Additionally, as illustrated in Fig. 3, the comparison of the presently projected WSS and OSI outcomes with those derived by research (Razavi, Shirani & Sadeghi, 2011) reveals substantial concurrence. Nonetheless, a slight variance in the values is noted, which could be attributed to a variance in the estimation approach employed in this paper.

Statistical analysis

All statistical analyses were performed using IBM SPSS Statistics 27.0.1 (SPSS Inc., Chicago, IL, USA), Since the data did not follow a normal distribution, we used the Wilcoxon signed-rank test for paired comparisons between Newtonian and Non-Newtonian hemodynamic parameters (Table 2). The significance level was set at p < 0.05.

Comparison of hemodynamic parameters

According to the Wilcoxon signed rank test analysis, the difference between WSS_N and WSS_NN was statistically significant under transient conditions (p = 0.001); the difference between TAWSS_N and TAWSS_NN was statistically significant (p < 0.001). The difference between OSI_N and OSI_NN was statistically significant (p < 0.001). The difference between RRT_N and RRT_NN was statistically significant (p < 0.001).

Wall shear stress

All the results data come from the 33 models mentioned above. As shown in Fig. 4A, when evaluating WSS using static analysis, there are significant computational differences compared to transients. For WSS, in mild stenosis, the differences between the Newtonian (WSS_N) and non-Newtonian (WSS_NN) models ranged from 4% to 16% (Fig. 4B). In moderate stenosis, the differences ranged from 0% to 25% (Fig. 4C), while in severe stenosis, the differences ranged from 0% to 22% (Fig. 4D). The trend of increasing WSS with higher degrees of stenosis was consistent across both models. Notably, there is a 0% difference when S is 38%, and the difference ranges from 0% to 6% when S is between 72% and 84%.

Time-averaged wall shear stress

As shown in Fig. 5A, both TAWSS_N and TAWSS_NN exhibit an increasing trend as S increases. For TAWSS, there was no significant difference between the two models in mild stenosis, with differences ranging from 0% to 8% (Fig. 5B). In moderate stenosis, the differences ranged from 0% to 28% (Fig. 5C), and in severe stenosis, the difference ranged from 0% to 14% (Fig. 5D). Overall, TAWSS_N were consistently higher than those TAWSS_NN, especially in moderate to severe stenosis. Notably, no difference is observed when S is 12%, 18%, 26%, 54%, or 76%.

Oscillatory shear index

As shown in Fig. 6A, both OSI_N and OSI_NN exhibit an increasing trend as stenosis progresses, with a similar overall pattern; however, OSI_N is consistently higher than OSI_NN. In mild stenosis cases, OSI differences range from 0% to 24% (Fig. 6B). In moderate stenosis, OSI differences range from 2% to 52% (Fig. 6C), while in severe stenosis, the differences range from 2% to 24% (Fig. 6D). Notably, there is no difference in OSI when S is 16%, 18%, or 26%.

Relative residence time

As shown in Fig. 7A, both RRT_N and RRT_NN exhibit an increasing trend as S increases. While the trends are similar, notable differences are still observed. Among them, RRT_N is larger than RRT_NN. RRT differences range from 0% to 22% in mild stenosis cases (Fig. 7B), 2% to 20% in moderate stenosis (Fig. 7C), and 0% to 69% in severe stenosis (Fig. 7D). Notably, no difference is observed when S is 18% or 84%.

Analysis of the influence of the Newtonian model and the non-Newtonian model on blood flow parameters

In traditional hemodynamic studies, the relationship between hemodynamic parameters and S is typically established by comparing parameters under the assumption of constant blood viscosity (Cho & Kensey, 1991; Malota et al., 2018). A recent study analyzed the WSS in both Newtonian and non-Newtonian models (Albadawi et al., 2021) and reported that WSS_N was consistently higher than WSS_NN. In this study, 33 cases with varying S were selected to investigate the influence of WSS, TAWSS, OSI, and RRT are influenced by changes in viscosity effects. The Newtonian model tends to simplify complex physiological processes, which can result in overestimation of hemodynamic parameters. Therefore, the potential limitations of the Newtonian model should be carefully considered when applying it to hemodynamic analysis.

Analysis of the effects of Newtonian and non-Newtonian models on WSS

High WSS (>2.5 Pa) has been linked to the formation of vulnerable plaques (Albadawi et al., 2021), while low WSS (<0.5 Pa) can lead to intimal thickening (Gharleghi, Sowmya & Beier, 2022). As shown in Fig. 8, this study found that regions with WSS below 2.5 Pa are predominant, while areas with high WSS tend to occur in the stenotic regions. These findings are consistent with previous studies (Zhang et al., 2021). Figure 8 shows that the distribution of WSS_N and WSS_NN is roughly similar when S ranges from 20% to 66%. Thus, the Newtonian model can be used to reduce computational time in this range. However, when S reaches 82% to 84%, the differences become pronounced, and the non-Newtonian model should be used to improve result accuracy. Our results indicate that for WSS, differences between Newtonian and non-Newtonian models were minimal in mild stenosis but became significant in moderate to severe cases. This observation aligns with previous studies, such as studies (Al-Azawy, Kadhim & Hameed, 2020; Kumar et al., 2020), which also reported that non-Newtonian effects become more pronounced as shear rates decrease in stenotic regions. Specifically, study (Liu et al., 2021) demonstrated that Newtonian models tend to overestimate WSS values in highly stenotic arteries, a trend also observed in our study.

Under static conditions, the distinction between the Newtonian and non-Newtonian models is critical across all cases. The Newtonian model tends to overestimate hemodynamic parameters, reducing the accuracy of predictive results. This phenomenon may be attributed to the reduced blood flow rate and shear rate under static conditions, which enhance the non-Newtonian characteristics of blood, particularly its higher viscosity. Since the Newtonian model fails to capture these changes, it results in an overestimation of hemodynamic parameters (Skiadopoulos, Neofytou & Housiadas, 2017).

Under transient conditions, the difference between WSS_N and WSS_NN is minimal (0–6%) in cases of severe stenosis (S < 86%), but becomes more pronounced in mild to moderate stenosis. This phenomenon can be attributed to the rapid blood flow and elevated shear rates in severe stenosis. At this stage, blood exhibits Newtonian-like fluid behavior, which minimizes the variance in WSS values between the Newtonian and non-Newtonian models (Stamou, Radulovic & Buick, 2023). The Newtonian model provides a reasonable approximation of blood flow at high shear rates, while in mild to moderate stenosis, where blood flow is slower and shear rates are lower, the non-Newtonian model more accurately reflects the shear-dependent viscosity of blood.

In this study, WSS exhibited an increasing trend as S progressed. This phenomenon can be explained by the fact that as arteries narrow, blood is forced through a reduced cross-sectional area, which increases the velocity of blood flow. According to the continuity equation and Bernoulli’s principle in fluid dynamics, an increase in blood flow velocity is directly associated with a rise in shear rate, ultimately leading to higher WSS. Previous studies (Ahmadpour et al., 2021; Md Kabir, Md Alam & Md Uddin, 2021) have demonstrated that as S increases, fluid dynamics adjust to maintain a constant flow rate, resulting in a significant rise in WSS. These findings further confirm the correlation between S and both WSS_N and WSS_NN.

Analysis of the effects of Newtonian and non-Newtonian models on TAWSS

Elevated TAWSS (>2.5 Pa) can stimulate platelet activation and promote thrombus formation (Albadawi et al., 2021). Low TAWSS (<0.4 Pa) is associated with a higher risk of atherosclerosis and plaque progression (Buradi & Mahalingam, 2020). As illustrated in Fig. 9, TAWSS increases with the progression of S in the non-Newtonian model, which is consistent with findings in existing literature (Azar et al., 2019) and observations from this study. This trend can be attributed to the substantial increase in blood flow velocity at the stenotic site as stenosis progresses, resulting in an elevated velocity gradient and consequently leading to a rise in TAWSS, as predicted by fundamental hydrodynamic principles. Figure 9 reveals that while the overall distributions of TAWSS_N and TAWSS_NN are similar, local variations are evident. Thus, selecting the Newtonian model for observing the TAWSS distribution can help reduce computational resources.

For TAWSS, our study found that differences between the models were negligible in mild stenosis but increased progressively in moderate and severe cases. This is consistent with findings by Ahmadpour et al. (2021), who demonstrated that the non-Newtonian model more accurately captures local hemodynamic disturbances in high-degree stenosis. However, unlike previous studies, our work quantifies the percentage differences in various stenosis degrees, providing additional guidance for model selection in CFD simulations. As S increases, Fig. 5A shows an increasing divergence between TAWSS_N and TAWSS_NN. As reported in the literature (Li, 2008), the presence of strong blood flow in stenotic segments, coupled with large variations in blood velocity and shear rate, reduces the accuracy of the Newtonian model, introducing biases in TAWSS calculations. The analysis reveals that TAWSS in mild stenosis is higher than in moderate stenosis, indicating that luminal narrowing is not the sole determinant of cerebral ischemia risk. This may be influenced by the characteristics and morphology of intravascular plaques.

Analysis of the effects of Newtonian and Non-newtonian models on OSI

A high OSI (>0.3) indicates frequent changes in flow direction, which is associated with a higher likelihood of neointimal hyperplasia. In contrast, a low OSI (<0.3) suggests more stable flow patterns (Yao et al., 2019). Regions with high OSI are prone to atherosclerotic lesions. Albadawi et al. (2021) has suggested that OSI values rise with increasing S. As shown in Fig. 6A, we observed a significant increase in OSI under the non-Newtonian model, primarily in moderate to severe stenosis cases, which supports previous research by Rahman et al. (2018) and Buradi & Mahalingam (2020). However, in severe stenosis, these differences became less pronounced, likely due to the dominance of turbulent flow. This suggests that while non-Newtonian models provide better accuracy in moderate stenosis, their impact may be reduced in severe cases where recirculation zones and disturbed flow patterns override viscosity effects. In Fig. 10, the overall distribution of OSI_N and OSI_NN appears similar, but notable differences exist in the bifurcation and stenosis regions. Therefore, the Newtonian model can be selected to observe OSI distribution, reducing computational effort.

In cases of moderate to severe stenosis, increased blood flow velocity leads to a more complex and unstable flow pattern, resulting in frequent changes in shear stress direction and a significant increase in OSI (Carvalho et al., 2021). However, in cases of very severe stenosis (86%–90%), OSI decreases. In very severe stenosis, despite stronger turbulence, the flow becomes more linear. The elevated blood flow velocity in the narrowed region reduces the fluctuations in shear stress direction, leading to a decrease in OSI (Singh & Singh, 2024). OSI_N is typically higher than OSI_NN because in the non-Newtonian model, the low OSI results from the consideration of blood viscosity as a function of shear rate. In contrast, the Newtonian model, which does not account for this viscosity variation, changes direction more frequently, leading to a higher OSI (Carvalho et al., 2021). In cases of severe stenosis, the non-Newtonian OSI values range from 0.204 to 0.435. An OSI value exceeding 0.3 is generally considered indicative of substantial alterations in blood flow shear stress direction, suggesting potential endothelial dysfunction and arterial instability.

Analysis of the effects of Newtonian and Non-newtonian models on RRT

Elevated RRT (>8 Pa⁻¹) is widely recognized as a marker for areas more susceptible to atherosclerosis and plaque progression. Regions with elevated RRT are more susceptible to the development of atherosclerotic lesions. As illustrated in Fig. 11, both models exhibit prolonged RRT in regions with moderate to severe stenosis, indicating that blood flow near the vessel wall becomes relatively stagnant as S increases. This phenomenon occurs because blood exhibits shear thinning (decreased viscosity) in regions of high shear rate (stenotic zone) and shear thickening (increased viscosity) in regions of low shear rate (post-stenotic dilatation zone). This thickening effect slows blood flow and increases the residence time of fluid in these regions (Lagache et al., 2021). Figure 11 shows that while the overall distribution of RRT_N and RRT_NN is similar, there are local differences. Therefore, selecting the Newtonian model to observe RRT distribution can significantly reduce computational time.

This study found that RRT_N was generally higher than RRT_NN. This phenomenon may be explained by the Newtonian model overestimating viscosity changes in these regions, leading to an overestimation of RRT. In contrast, the non-Newtonian model more accurately captures the high viscosity of blood at low shear rates, resulting in a shorter RRT (Mehri, Mavriplis & Fenech, 2018; Trenti et al., 2022). Previous studies (Azar et al., 2019; Hou et al., 2023) have suggested a consistent negative correlation between RRT and S, with the strongest correlation observed in vessels with minimal stenosis. However, this study did not identify a correlation between RRT and S, which may be attributed to the limited sample size. Further studies are required to determine the accuracy of using RRT to assess carotid stenosis. RRT exhibited the largest discrepancies between the two models, particularly in severe stenosis. Unlike Tu et al. (2023), who reported less variation in RRT between Newtonian and non-Newtonian models, our study found that non-Newtonian effects significantly influence RRT in severe stenotic regions. This discrepancy could be attributed to differences in boundary conditions, as our study incorporated patient-specific inlet velocity profiles rather than generalized assumptions.

Our results suggest that while Newtonian models are computationally efficient, they may introduce errors in moderate to severe stenosis cases, particularly when predicting OSI and RRT. Non-Newtonian models offer greater physiological accuracy in these cases but require higher computational resources. These findings align with Bernabeu et al. (2013); Wong et al. (2020), who emphasized the trade-off between accuracy and computational efficiency when choosing viscosity models for CFD-based hemodynamic analysis.

Limitation and recommendation

This study has some limitations. This study’s findings are limited by the small sample size, highlighting the need for a larger cohort to further validate the results and their correlation with clinical outcomes. In the models used, a rigid vascular wall was assumed; however, the outcomes may vary in cases involving deformable vascular walls. Future research should incorporate arterial wall elasticity, include broader hemodynamics indicators, and validate findings with a larger sample size to obtain more reliable estimates of hemodynamic parameters.