High-Frequency Fluctuations in Post-stenotic Patient Specific Carotid Stenosis Fluid Dynamics: A Computational Fluid Dynamics Strategy Study
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Abstract
Purpose
Screening of asymptomatic carotid stenoses is performed by auscultation of the carotid bruit, but the sensitivity is poor. Instead, it has been suggested to detect carotid bruit as neck’s skin vibrations. We here take a first step towards a computational fluid dynamics proof-of-concept study, and investigate the robustness of our numerical approach for capturing high-frequent fluctuations in the post-stenotic flow. The aim was to find an ideal solution strategy from a pragmatic point of view, balancing accuracy with computational cost comparing an under-resolved direct numerical simulation (DNS) approach vs. three common large eddy simulation (LES) models (static/dynamic Smagorinsky and Sigma).
Method
We found a reference solution by performing a spatial and temporal refinement study of a stenosed carotid bifurcation with constant flow rate. The reference solution \(\left( {\Delta x = 1.92 \times 10^{ - 4} \;{\text{m}},\; \Delta t = 5 \times 10^{ - 5} \;{\text{s}}} \right)\) was compared against LES for both a constant and pulsatile flow.
Results
Only the Sigma and Dynamic Smagorinsky models were able to replicate the flow field of the reference solution for a pulsatile simulation, however the computational cost of the Sigma model was lower. However, none of the sub-grid scale models were able to replicate the high-frequent flow in the peak-systolic constant flow rate simulations, which had a higher mean Reynolds number.
Conclusions
The Sigma model was the best combination between accuracy and cost for simulating the pulsatile post-stenotic flow field, whereas for the constant flow rate, the under-resolved DNS approach was better. These results can be used as a reference for future studies investigating high-frequent flow features.
Keywords
CFD Finite elements Carotid stenosis Oasis LESIntroduction
Carotid stenosis is a progressive and local buildup of plaque in the carotid bifurcation, leading to a local narrowing of the lumen. The major risk consists of plaque rupture with subsequent debris and thrombi being transported downstream where they can cause a blockage leading to a stroke and consequent neurologic deficits.35 Asymptomatic carotid artery stenoses (ACAS), which affects 1.6% of the population,8 are rarely detected unless diagnosed with another associated cardiovascular disease.42
A characteristic feature of stenoses is that the downstream flow is turbulent-like, with high-frequent pressure fluctuations.3 These fluctuations can traverse the soft neck tissue as mechanical waves, and present as a bruit or skin vibration. The stenosis-induced turbulent-like flow is therefore a strong marker for inferring the presence of a stenosis.
The current clinical practice for ACAS screening is auscultation of the carotid bruit.32 Auscultation is only applied if the physician suspects presence of a stenosis, i.e., if correlated risk factors are present,42 and is operator-dependent, with low sensitivity23 due to the presence of background noise.37 Carotid auscultation is hence not sufficient to infer the presence of a stenosis, whose diagnosis hence requires confirmation by techniques which are usually not available to a general practitioner, i.e., ultrasound or tonometry.
To overcome the abovementioned challenges, the CARDIS project proposes to instead infer the presence of stenosis by measuring skin vibrations using a newly developed multi-beam laser Doppler vibrometry device,21 with increased temporal resolution, with 10 μs at a sample rate of 100 kHz, and reduced noise level, with less than 1 μm for 5 s time measurement in the 1–1000 Hz range.22 The device has already proven suitable for measuring physiological signals from skin movements, such as pulse wave velocity and heart rate.7, 38 The new device could allow rapid and consistent non-contact screening for ACAS, and thus detection prior to a traumatic event. As part of this project, we combine in vitro experiments and computational fluid dynamics (CFD) flow simulations to show a ‘theoretical’ proof of concept of the device before clinical testing. However, the efficacy of CFD relies on the robustness of the numerical methods and their ability to reproduce experimental results, and has proven challenging, especially for transitional flows.4 In particular, the use of numerical schemes, such as first order UPWIND schemes, well-known to be dissipative,34 is common in the biomedical literature.46 The choice of the correct numerical methodology is hence crucial for reliable simulations.
We have previously used an under-resolved DNS approach with rigid walls for two biomedical benchmarks4, 16 and biomedical applications such as aneurysms15, 44, 45 and vascular junctions.28 The first aim of this study was to find an adequate under-resolved DNS solution from a spatial and temporal refinement study with respect to the smallest scales, and from a pragmatic biomedical engineering point of view balancing the computational cost with accuracy.
Directly calculating the smallest scales of the turbulent-like flow requires large computational resources, and is therefore not routinely performed in the biomedical literature. However, we also know that the smallest scales have little energy, and only contribute to dissipation. We can therefore model these scales, for instance by means of large eddy simulations (LES). Applying LES can allow for the use of a coarser grid, since we can model the scales we do not capture, referred to as the sub-grid scales (SGS). LES simulations depend on the properties of the SGS model, for instance its ability to mimic the near-wall behavior. The second aim of this study was therefore to assess whether three commonly used SGS models (Smagorinsky, Sigma and Dynamic Smagorinsky) are able to replicate our reference solution for both constant—at peak systole—and pulsatile flow rates on a coarser mesh, and thus reducing computational cost without significant loss of resolution of the high-frequent flow features. The study was performed in an anatomically correct model geometry, retrieved from a patient with significant carotid stenosis, subjected to physiologically relevant boundary conditions.
Methods
Computed tomography angiography images of a common carotid bifurcation with severe stenosis (76% narrowing computed by means of the NASCET method36) in the internal carotid artery (ICA) were obtained from a 75 years old male patient, who gave informed consent for the use and further processing of the images.
Mesh characteristics.
Mesh name | Number elements (–) | Average cell length \((\Delta x\)) (m) | Number of boundary layers (–) |
---|---|---|---|
200K | \(2 \times 10^{5}\) | \(9.13 \times 10^{ - 4}\) | 1 |
2M | \(2 \times 10^{6}\) | \(4.63 \times 10^{ - 4}\) | 4 |
6M | \(6 \times 10^{6}\) | \(3.04 \times 10^{ - 4}\) | 4 |
22M | \(22 \times 10^{6}\) | \(1.92 \times 10^{ - 4}\) | 4 |
50M | \(50 \times 10^{6}\) | \(1.44 \times 10^{ - 4}\) | 4 |
We assumed rigid walls, prescribed a no-slip boundary condition along the vessel walls, and enforced a flow split of 43.8% in the ICA/CCA, based on the model presented in Groen et al.10 by prescribing a Womersley profile on the ECA outlet and zero pressure at the ICA outlet.
To cheaply washout initial transients associated with the artificial initial conditions we computed the flow at the 11M mesh for 2 physical seconds, equivalent of one and a half flow-throughs with model length of 0.2 m and peak inlet velocity of 0.15 m/s. We projected the last time step of the solution onto each mesh as an initial condition.
Simulations were performed using the open-source verified27 and validated4 finite element CFD solver Oasis,27 where special care was taken to ensure a kinetic energy-preserving and minimally-dissipative numerical solution of the Navier–Stokes equation. We used linear Lagrange elements (\({\mathbb{P}}_{1} - {\mathbb{P}}_{1}\)) for both velocity and pressure.
The spatial refinement study was simulated with \(\Delta t = 5 \times 10^{ - 5}\) s, while the temporal refinement study was performed on the least computationally expensive mesh which gave adequate results with varying time step: \(\Delta t = 1 \times 10^{ - 4} , \;5 \times 10^{ - 5} , \;1 \times 10^{ - 5}\) and \(5 \times 10^{ - 6}\) seconds. The pair of \(\Delta t\) and mesh size that provided the best tradeoff between computational cost, and accuracy of resolving high-frequent flow features was used as the reference solution for comparison with the LES simulations.
The SGS tensor \(\tau_{ij}^{SGS}\) depends on the strain tensor of the resolved scaled \(S_{ij}\) and on the subgrid-scale viscosity \(\nu_{SGS}\), which is a function of the cut-off length scale \(\Delta\) and of two model-specific parameters \(D_{m}\) and \(C_{m} .\)\(D_{m}\) is the model specific differential operator related to the resolved velocity field \(\varvec{u}\) and it sets the properties of the model, for instance, the near wall behavior. \(C_{m}\) is the model constant and it sets the amount of energy drained from the resolved scales.
We applied three different SGS models: the static Smagorinsky model39 with \(C_{m} = 0.168\), the Sigma model29 with \(C_{m} = 1.5\), and the dynamic Smagorinsky model, for which the model-specific parameters were updated every time step following Meneveau et al.26 The LES simulations were performed on the mesh with one spatial refinement level lower, i.e., on a coarser mesh, relative to the reference solution, but with the same \(\Delta t\) to ensure a fair comparison.
Number of cores and workload per core for each simulation.
Number of cells | Number of cores | Workload | Workload/6M workload | |
---|---|---|---|---|
200K | \(174.37 \times 10^{3}\) | 16 | \(1.09 \times 10^{4}\) | 0.06 |
2M | \(1.84 \times 10^{6}\) | 16 | \(1.15 \times 10^{5}\) | 0.62 |
6M | \(5.97 \times 10^{6}\) | 32 | \(1.87 \times 10^{5}\) | 1.00 |
22M | \(22.43 \times 10^{6}\) | 96 | \(2.34 \times 10^{5}\) | 1.25 |
50M | \(51.14 \times 10^{6}\) | 128 | \(4.00 \times 10^{5}\) | 2.14 |
List of the simulations performed for the three studies reported in this paper, with details on the mesh size, time step size, inlet type, end time and LES model.
Study | Mesh size | Time step size \((\Delta t\)) [s] | Inlet type | LES model |
---|---|---|---|---|
Spatial refinement | 200K | \(5 \times 10^{ - 5}\) | Constant | None |
2M | \(5 \times 10^{ - 5}\) | Constant | None | |
6M | \(5 \times 10^{ - 5}\) | Constant | None | |
22M | \(5 \times 10^{ - 5}\) | Constant | None | |
50M | \(5 \times 10^{ - 5}\) | Constant | None | |
Temporal refinement | 22M | \(\bf{1 \times 10^{ - 4}}\) | Constant | None |
22M | \(\bf{5 \times 10^{ - 5}}\) | Constant | None | |
22M | \(\bf{1 \times 10^{ - 5}}\) | Constant | None | |
22M | \(\bf{5 \times 10^{ - 6}}\) | Constant | None | |
LES | 6M | \(5 \times 10^{ - 5}\) | Constant | Smagorinsky |
6M | \(5 \times 10^{ - 5}\) | Constant | Dynamic Smagorinsky | |
6M | \(5 \times 10^{ - 5}\) | Constant | Sigma | |
6M | \(5 \times 10^{ - 5}\) | Pulsatile | Smagorinsky | |
6M | \(5 \times 10^{ - 5}\) | Pulsatile | Dynamic Smagorinsky | |
6M | \(5 \times 10^{ - 5}\) | Pulsatile | Sigma | |
Reference solution | 6M | \(5 \times 10^{ - 5}\) | Pulsatile | None |
22M | \(5 \times 10^{ - 5}\) | Pulsatile | None |
Reynolds decomposition was used for all constant flow rate simulations to separate the instantaneous velocity, \(\varvec{u}(\varvec{x}, t)\), from the time averaged, \(\bar{\varvec{u}}(\varvec{x})\), and fluctuating, \(\varvec{u^{\prime}}(\varvec{x}, t)\), components, i.e., \(\varvec{u} = \bar{\varvec{u}} + \varvec{u^{\prime}}.\) Taking the fluctuating velocity magnitude signal, \(|\varvec{u^{\prime}}|\), as input we computed the power spectral density (PSD) using Welch’s method12 with 16 segments and a Hanning windowing function with 50% overlap. The turbulent kinetic energy (\(tke\)) was calculated as \(tke = \frac{1}{2}\mathop \sum \limits_{i = 1}^{3} \overline{{\varvec{u^{\prime}}({x}_{{i}} , t)^{2} }}\) where \(\varvec{u^{\prime}}(x_{i = 1:3} , t)\) are the components of the fluctuating velocity.
Results
General Flow Features
Sensitivity Analysis
Spatial Refinement Study
To further support these results, we also show the \(tke\) for all available meshes in slices A–D (Fig. 12 in Appendix A) and the PSD of the fluctuating velocity, \(\varvec{u^{\prime}}(\varvec{x},t)\), in five additional points along the ICA.
Based on these observations, the 22M simulation was considered to be the best tradeoff between computational cost and accuracy. The temporal refinement study was hence performed on the 22M mesh.
Temporal Refinement Study
The temporal refinement simulations were evaluated similarly to the spatial refinement study. In all lines, the \(tke\) with \(\Delta t\) equal to \(5 \times 10^{ - 5}\), \(1 \times 10^{ - 5}\), and \(5 \times 10^{ - 6}\) (Fig. 6b) were close to indistinguishable. In contrast, the \(1 \times 10^{ - 4}\) simulation differed slightly in the lines on slices A, B, and C. The contours for the available time steps are shown as well in the appendix A, with consistent results. Furthermore, the PSD of the fluctuating velocity compoent shown in Fig. 7b confirms that the impact of temporal refinement is small, however the plots in Figs. 17, 18, 19, 20, 21, and 22 of Appendix B show a slight difference between \(\Delta t = 1 \times 10^{ - 4}\) and the rest.
Based on the results from the spatial and temporal refinement study the 22M mesh (\(\Delta x = 1.92^{ - 4}\) m) and time step of \(\Delta t = 5 \times 10^{ - 5}\) seconds offered the best tradeoff between computational cost and accuracy, and is now referred to as the reference solution.
Large Eddy Simulations
Constant Flow Simulations
CPU hours of the reference solution (22M-None) and of the most relevant SGS models compared to the 6M-None for constant flow rate.
Mesh | SGS model | CPU h/6M-None CPU h | s/time step |
---|---|---|---|
22M | None | 4.60 | 2.43 |
6M | None | 1.00 | 1.58 |
6M | Sigma | 1.77 | 2.80 |
6M | Dynamic Smagorinsky | 2.66 | 4.20 |
Pulsatile Flow Simulations
In the middle row of Fig. 11 we show \(tke\) of the pulsatile simulations, where \(\varvec{u}^{\varvec{'}} \left( {\varvec{x},t} \right)\) was computed with a constant \(\bar{u}\), in slice C for the reference solution, the 6M with Sigma model, and 6M with Dynamic Smagorinsky model, left to right respectively. Visually, there was a large similarity between the \(tke\) fields.
The bottom row of Fig. 11 shows that the \(tke\) computed by using high-pass filtered fluctuating velocity components, \(\varvec{u'}(\varvec{x},t\)), now referred to as \(\widetilde{{tke^{\prime}}}\), is highly comparable. It is hence clear that the LES simulations harbor the same turbulent kinetic energy as the reference solution. The power spectral densities in Appendix B Figs. 18, 19, 20, 21 and 22, showing no difference in between the LES models and the reference solution, also backed up these results. Of note are also the large differences between the two measures of \(tke\) in the pulsatile simulation. The bottom row represents the regions with turbulent kinetic energy with a higher frequency fluctuation than those introduced from the flow waveform at the inlet. In the context of this study, we emphasize these more as the high frequency content of the simulated flow is our quantity of interest. For completeness, the \(\nu_{SGS}\), the \(tke\) and the \(\widetilde{{tke^{\prime}}}\) in slices A–D are shown in Appendix A, Figs. 14b, 15b and 16, respectively.
CPU hours of the reference solution (22M-None) and of the most relevant SGS models compared to the 6M-None for pulsatile flow rate.
Mesh | SGS model | CPU h/6M-None CPU h | s/time step |
---|---|---|---|
22M | None | 4.18 | 3.65 |
6M | None | 1.00 | 2.62 |
6M | Sigma | 1.49 | 3.90 |
6M | Dynamic Smagorinsky | 2.53 | 6.64 |
Since the computational results of the two SGS models were comparable to the reference solution, the reduction in computational cost favored the Sigma model.
Discussion
Space and cycle-averaged velocity compared and extrapolated with Richardson’s extrapolation method for different grid sizes.
Number of elements | \(\bar{u}\) (m/s) | A (m^{2}) | f (m^{3}/3) | % Error |
---|---|---|---|---|
6M | \(2.2265 \times 10^{ - 1}\) | \(1.8289 \times 10^{ - 4}\) | \(4.1426 \times 10^{ - 4}\) | 8.19 |
22M | \(2.3295 \times 10^{ - 1}\) | \(1.7165 \times 10^{ - 4}\) | \(3.9989 \times 10^{ - 4}\) | 4.44 |
50M | \(2.3636 \times 10^{ - 1}\) | \(1.6691 \times 10^{ - 4}\) | \(3.9453 \times 10^{ - 4}\) | 3.04 |
Richardson extrapolation | – | – | \(3.8290 \times 10^{ - 4}\) | 0.00 |
Focusing firstly on the spatial and temporal refinement studies, we found a grid spacing of \(\Delta x = 1.92 \times 10^{ - 4} \; {\text{m}}\) and time step size of \(\Delta t = 5 \times 10^{ - 5} \;{\text{s}}\) to be the best tradeoff between computational cost and accuracy from a pragmatic biomedical engineering point of view.
Relative to others, Lancellotti et al.18 reported using \(\Delta t = 6.25 \times 10^{ - 4}\) and an “effective” mesh size of \(\Delta x = 6.5 \times 10^{ - 5}\) as a reference solution, i.e., a time step one order of magnitude larger, and a cell size almost three times smaller. Even with a different working fluid, their Reynolds number was only 25% smaller at the stenosis. Lee et al., which was the first to performed true DNS in a patient-specific geometry, is not directly comparable with respect to mesh size since they used spectral finite elements, but they reported a time step of \(\Delta t = 1 \times 10^{ - 5}\), although argued from a numerical stability point of view.19
Furthermore, we found that simulations on a coarser mesh with SGS models were not able to capture the high-frequency flow features for a constant flow rate equivalent to the peak systolic flow rate (\(Re = 1380\)). If pulsatile flow was applied (\(Re_{average} = 980, Re_{\hbox{max} } = 1380\)) the SGS models were able to replicate the flow features of the reference solution. The SGS models are thus applicable for investigating the turbulent-like flow in multiple patients or configurations, and ultimately, with fluid structure interaction (FSI), how the flow fluctuations can present as skin vibrations on the neck of affected patients. The discrepancies between constant and pulsatile flow simulations can be attributed to the differences in flow rate, which for the constant flow rate simulation, was higher than the average flow rate for the pulsatile flow rate simulations, although with the same peak flow.
The SGS viscosity of the Sigma model was overall higher than the one resulting from the application of the Dynamic Smagorinsky model, which is the opposite of what Baya Toda et al.2 found in their study of an internal combustion (IC) engine. In their IC chamber, the viscosity of the Dynamic Smagorinsky model increased when the flow jet impinged a wall, as a consequence of the increase of the strain-rate tensor. On the contrary, the Sigma model did not produce an increased SGS viscosity in that region, as its \(D_{m}\) is not affected by the magnitude of the strain-rate tensor. The different dissipation provided by the two model was, therefore, a direct consequence of the physics of an impinging jet, which are not comparable to the physics of a free jet such as the one considered in this manuscript. We therefore recommend a careful evaluation of the choice of SGS model with respect to the physical problem, keeping in mind that there is a sensitivity which can affect the results.
Although the mean flow was converged, the question remains if the flow was well resolved at the smallest scales. We previously discussed this in Mancini et al.,24 where we computed the Kolmogorov length scale (μ).17 We computed the ratio of the local cell length, \(\Delta x\), and the Kolmogorov length scale, μ, in each cell, and reported the temporal and spatial global maximum. The ratio on the two finest meshes were below 10, typically sufficient to capture > 95% of the dissipation.31 However, considering the temporal averaged ratio in the post-stenotic region, we obtained a mean/max of 0.347/1.11, and 0.463/1.755 for the 22M and 50M simulation, respectively, showing that the flow is well-resolved. We also obtained equal results from computing \(l^{ + }\), a surrogate measure for the Kolmogorov length scale compared to the local cell length,43 with \(l_{\hbox{max} }^{ + } = 6.2\) located along the wall in the stenosis. Performing a true DNS simulation would require building a mesh with cell lengths of roughly seven times smaller than the finest mesh, yielding mesh consisting of roughly 21 billion cells and thus requiring an enormous amount of computational resources while, from a pragmatic biomedical engineering point of view, having no added value. That being said, using an adaptive mesh strategy would yield a DNS simulation with a smaller mesh, however our local refinement approach is simple to adapt, and the local refinement is consistent between spatial refinement levels. It is also noteworthy that the homogeneous isotropic assumption of the Kolmogorov hypothesis was not met in our simulations. The Kolmogorov hypothesis therefore underestimates the smallest scales, and the simulation might therefore be even more well resolved than the Kolmogorov length scale indicates.
Based on our numerical investigations, we have a high level of confidence in our numerical results, but how do the results compare to flow in vivo? Firstly, we assumed a Newtonian fluid with properties mimicking water. With realistic flow rates, the Reynolds number was hence 3.3 times larger compared to in vivo blood flow, since blood has a higher kinematic viscosity compared to water. As a result, the intensity of the post-stenotic turbulent-like flow is higher in our numerical experiments than what can be expected in vivo. The spatial and temporal resolutions of this study still hold for physiological realistic situations, in terms of being well-resolved. On the other hand, the use of water also allows for valuable comparison with in vitro experiments, in which water is often used instead of glycerin-based blood-mimicking fluids. Moreover, the relative effect of assuming a non-Newtonian fluid has shown to be small in the carotid bifurcation since, due to the high share rates in the carotid bifurcation, the non-Newtonian models work in a regime where the viscosity is close to a constant value.20 Furthermore, compared to other uncertainties like modeling choices and segmentation, the assumption of a Newtonian fluid is marginal. Moreover, from a physical point of view, blood is more complex than just being a non-Newtonian fluid, it is also multiphase flow. The presence of small particles, to mimic red blood cells, has been found to dampen flow instabilities,1 but should not phenotypically change the flow.
Secondly, we modeled the walls as rigid, but they are naturally compliant. Modeling the stenosis with compliant models might dampen some of the unstable flow features. However, determining the material properties of the stenosed region are therefore challenging, since it consists of both plaque and lipid pools. The former is stiffer than a healthy vessel wall, while the latter is more compliant. Therefore, anticipating how this might affect the results are challenging. Moreover, the soft tissue embedding the carotid arteries would further dampen the instabilities that would present on the skin surface, generally with a lower amplitude relative to the fluctuations induced by the turbulent-like post-stenotic flow.5
In total, the simulation results cannot directly be translated into an in vivo situation, but is an important first step towards a trustworthy patient-specific flow simulation for investigating the high-frequent flow fluctuations. Future efforts will be directed towards validating the flow with in vitro experiments, and using an FSI approach, i.e., incorporating the effect of compliant walls, which is well-known to better mimic physiological conditions.40
To ease comparison with other solvers, increase reproducibility, and promote openness in science, we also provide an additional repository25 with our problem file for the Oasis solver, the used meshes, and averaged results from the reference solution with constant flow rate. The results from this study can therefore easily be compared to other solvers, and potentially ease the amount of work needed to show that a given solution strategy is adequate for investigating turbulent-like flow features for a post-stenotic flow.
In our line of research we are interested in the high-frequent flow features, however a plurality of studies using CFD to investigate flow in the carotid bifurcation are interested in (time-averaged) wall share stress (WSS), or other WSS-derived quantities.4, 34, 46 We have not investigated whether a coarser mesh would be sufficient for investigating WSS, and furthermore, how LES models would affect the results. For comparison, Valen-Sendstad et al. found that time-averaged wall shear stress (WSS) was relatively insensitive to the applied computational solution strategy, whereas the OSI, more sensitive to flow fluctuations, changed significantly.46 We therefore caution readers about extrapolating our results to studies investigating WSS-derived quantities.
Lancellotti et al.18 investigated the applicability of LES models in pulsatile simulations of a stenosed patient-specific carotid bifurcation. They, like us, found a static Sigma model to perform best. Although we agree with most of their conclusions, there are some caveats of the study, e.g., they used a streamline upwind/Petrov–Galerkin pressure stabilized Petrov–Galerkin formulation for the reference solution, known to add numerical diffusion,41 which would not be equal between the two meshes. Despite these limitations, we can conclude that both Lancellotti et al.18 and the current study could be used as points of reference for LES modeling and spatial and temporal refinement when investigating flow in the carotid bifurcation. In particular, Lancellotti et al. for WSS and the current study for the high frequent flow features.
We can thus state that future studies investigating the high-frequent features of post-stenotic flow can use an equivalent spatial and temporal resolution reported here, with a Sigma SGS model. Of note is that if a higher Reynolds number is applied, like in the constant flow rate simulations presented here, the turbulence models were not applicable at reported spatial resolution. More specifically, we will use these results to investigate the possibility to diagnose carotid stenosis based on the amplitude of the unstable flow’s frequency content by measuring the neck skin vibrations.
Conclusions
The numerical methodology applied in this study allowed us to properly resolve the flow field of a stenosed patient-specific carotid bifurcation and hence detect instabilities induced by the stenosis. When compared to the reference solution, only Sigma and Dynamic Smagorinsky were able to replicate the averaged mean flow features from the constant flow rate simulation, and the turbulent flow features in the pulsatile flow rate simulations. The computational cost was lower for the Sigma model, and therefore the best choice balancing accuracy with computational cost for studying high frequency flow instabilities. However, for higher Reynolds numbers, similar to the constant flow rate simulation, the LES models were not sufficient. Future efforts on this subject should be conducted with the abovementioned SGS model, while taking advantage of a robust high-order numerical solver such as the one used in this study.
Notes
Acknowledgments
The study was partially funded by the H2020 European funded Early stage CARdio Vascular Disease Detection with Integrated Silicon Photonics (CARDIS) 644798 project and by the Flemish Fund for Scientific Research (FWO) G086917 N project.
Conflict of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
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