Accuracy of computational hemodynamics in complex arterial geometries reconstructed from magnetic resonance imaging

Reconstructing blood flow in data-poor regimes: a vasculature network kernel for Gaussian process regression

Blood flow reconstruction in the vasculature is important for many clinical applications. However, in clinical settings, the available data are often quite limited. For instance, transcranial Doppler ultrasound is a non-invasive clinical tool that is commonly used in clinical settings to measure blood velocity waveforms at several locations. This amount of data is grossly insufficient for training machine learning surrogate models, such as deep neural networks or Gaussian process regression. In this work, we propose a Gaussian process regression approach based on empirical kernels constructed by data generated from physics-based simulations-enabling near-real-time reconstruction of blood flow in data-poor regimes. We introduce a novel methodology to reconstruct the kernel within the vascular network. The proposed kernel encodes both spatiotemporal and vessel-to-vessel correlations, thus enabling blood flow reconstruction in vessels that lack direct measurements. We demonstrate that any prediction made with the proposed kernel satisfies the conservation of mass principle. The kernel is constructed by running stochastic one-dimensional blood flow simulations, where the stochasticity captures the epistemic uncertainties, such as lack of knowledge about boundary conditions and uncertainties in vasculature geometries. We demonstrate the performance of the model on three test cases, namely, a simple Y-shaped bifurcation, abdominal aorta and the circle of Willis in the brain.

Inverse problems in blood flow modeling: A review

Mathematical and computational modeling of the cardiovascular system is increasingly providing non-invasive alternatives to traditional invasive clinical procedures. Moreover, it has the potential for generating additional diagnostic markers. In blood flow computations, the personalization of spatially distributed (i.e., 3D) models is a key step which relies on the formulation and numerical solution of inverse problems using clinical data, typically medical images for measuring both anatomy and function of the vasculature. In the last years, the development and application of inverse methods has rapidly expanded most likely due to the increased availability of data in clinical centers and the growing interest of modelers and clinicians in collaborating. Therefore, this work aims to provide a wide and comparative overview of literature within the last decade. We review the current state of the art of inverse problems in blood flows, focusing on studies considering fully dimensional fluid and fluid-solid models. The relevant physical models and hemodynamic measurement techniques are introduced, followed by a survey of mathematical data assimilation approaches used to solve different kinds of inverse problems, namely state and parameter estimation. An exhaustive discussion of the literature of the last decade is presented, structured by types of problems, models and available data.

A Re-Engineered Software Interface and Workflow for the Open-Source SimVascular Cardiovascular Modeling Package

Patient-specific simulation plays an important role in cardiovascular disease research, diagnosis, surgical planning and medical device design, as well as education in cardiovascular biomechanics. simvascular is an open-source software package encompassing an entire cardiovascular modeling and simulation pipeline from image segmentation, three-dimensional (3D) solid modeling, and mesh generation, to patient-specific simulation and analysis. SimVascular is widely used for cardiovascular basic science and clinical research as well as education, following increased adoption by users and development of a GATEWAY web portal to facilitate educational access. Initial efforts of the project focused on replacing commercial packages with open-source alternatives and adding increased functionality for multiscale modeling, fluid-structure interaction (FSI), and solid modeling operations. In this paper, we introduce a major SimVascular (SV) release that includes a new graphical user interface (GUI) designed to improve user experience. Additional improvements include enhanced data/project management, interactive tools to facilitate user interaction, new boundary condition (BC) functionality, plug-in mechanism to increase modularity, a new 3D segmentation tool, and new computer-aided design (CAD)-based solid modeling capabilities. Here, we focus on major changes to the software platform and outline features added in this new release. We also briefly describe our recent experiences using SimVascular in the classroom for bioengineering education.

Reducing the impact of geometric errors in flow computations using velocity measurements

Numerical blood flow simulations are typically set up from anatomical medical images and calibrated using velocity measurements. However, the accuracy of the computational geometry itself is limited by the resolution of the anatomical image. We first show that applying standard no-slip boundary conditions on inaccurately extracted boundaries can cause large errors in the results, in particular the pressure gradient. In this work, we therefore propose to augment the flow model calibration by slip/transpiration boundary conditions, whose parameters are then estimated using velocity measurements. Numerical experiments show that this methodology can considerably improve the accuracy of the estimated pressure gradients and 3D velocity fields when the vessel geometry is uncertain.

COMPUTATIONAL STUDY OF THE INFLUENCE OF BIFURCATION ANGLE ON HAEMODYNAMICS AND OXYGEN TRANSPORT IN THE CAROTID BIFURCATION

In this study, blood flow associated with oxygen transport in the human carotid bifurcation was investigated numerically to assess the effects of bifurcation geometry on distribution and magnitude of the wall shear stress (WSS) and Sherwood number (Sh: dimensionless oxygen wall flux) at the favourable site of atherosclerotic lesion. Three-dimensional average models of the rigid-walled carotid bifurcation were constructed to perform simulations of steady blood flow under the wall boundary condition of a constant oxygen tension. The results demonstrated that changes in the bifurcation angle significantly altered the distribution of both the WSS and the Sh, even though the pattern of the axial flow was not very sensitive to the change in bifurcation angle. Flow with large inertia bifurcated at the flow divider and created a flow recirculation zone with low WSS and Sh on the outer wall of the internal carotid artery (ICA) sinus, where atherosclerotic lesions tend to develop. A wider bifurcation angle made the area of low Sh in the ICA sinus smaller, but the level of Sh along the outer wall of the ICA sinus extremely low. Another finding was that low Sh was associated with high WSS at the region distal to the ICA sinus. The Sh distribution did not readjust as fast as the WSS in this region, as reflected by the different rates of recovery of the WSS and Sh, thus uncoupling the transport process of oxygen transport from WSS.

Comparison between a generalized Newtonian model and a network-type multiscale model for hemodynamic behavior in the aortic arch: Validation with 4D MRI data for a case study

Blood is a complex fluid in which the presence of the various constituents leads to significant changes in its rheological properties. Thus, an appropriate non-Newtonian model is advisable; and we choose a Modified version of the rheological model of Phan-Thien and Tanner (MPTT). The different parameters of this model, derived from the rheology of polymers, allow characterization of the non-Newtonian nature of blood, taking into account the behavior of red blood cells in plasma. Using the MPTT model that we implemented in the open access software OpenFOAM, numerical simulations have been performed on blood flow in the thoracic aorta for a healthy patient. We started from a patient-specific model which was constructed from medical images. Exiting flow boundary conditions have been developped, based on a 3-element Windkessel model to approximate physiological conditions. The parameters of the Windkessel model were calibrated with in vivo measurements of flow rate and pressure. The influence of the selected viscosity of red blood cells on the flow and wall shear stress (WSS) was investigated. Results obtained from this model were compared to those of the Newtonian model, and to those of a generalized Newtonian model, as well as to in vivo dynamic data from 4D MRI during a cardiac cycle. Upon evaluating the results, the MPTT model shows better agreement with the MRI data during the systolic and diastolic phases than the Newtonian or generalized Newtonian model, which confirms our interest in using a complex viscoelastic model.

Merging computational fluid dynamics and 4D Flow MRI using proper orthogonal decomposition and ridge regression

Time resolved phase-contrast magnetic resonance imaging 4D-PCMR (also called 4D Flow MRI) data while capable of non-invasively measuring blood velocities, can be affected by acquisition noise, flow artifacts, and resolution limits. In this paper, we present a novel method for merging 4D Flow MRI with computational fluid dynamics (CFD) to address these limitations and to reconstruct de-noised, divergence-free high-resolution flow-fields. Proper orthogonal decomposition (POD) is used to construct the orthonormal basis of the local sampling of the space of all possible solutions to the flow equations both at the low-resolution level of the 4D Flow MRI grid and the high-level resolution of the CFD mesh. Low-resolution, de-noised flow is obtained by projecting in vivo 4D Flow MRI data onto the low-resolution basis vectors. Ridge regression is then used to reconstruct high-resolution de-noised divergence-free solution. The effects of 4D Flow MRI grid resolution, and noise levels on the resulting velocity fields are further investigated. A numerical phantom of the flow through a cerebral aneurysm was used to compare the results obtained using the POD method with those obtained with the state-of-the-art de-noising methods. At the 4D Flow MRI grid resolution, the POD method was shown to preserve the small flow structures better than the other methods, while eliminating noise. Furthermore, the method was shown to successfully reconstruct details at the CFD mesh resolution not discernible at the 4D Flow MRI grid resolution. This method will improve the accuracy of the clinically relevant flow-derived parameters, such as pressure gradients and wall shear stresses, computed from in vivo 4D Flow MRI data.

Uncertainty Quantification in a Patient-Specific One-Dimensional Arterial Network Model: EnKF-Based Inflow Estimator

Successful clinical use of patient-specific models for cardiovascular dynamics depends on the reliability of the model output in the presence of input uncertainties. For 1D fluid dynamics models of arterial networks, input uncertainties associated with the model output are related to the specification of vessel and network geometry, parameters within the fluid and wall equations, and parameters used to specify inlet and outlet boundary conditions. This study investigates how uncertainty in the flow profile applied at the inlet boundary of a 1D model affects area and pressure predictions at the center of a single vessel. More specifically, this study develops an iterative scheme based on the ensemble Kalman filter (EnKF) to estimate the temporal inflow profile from a prior distribution of curves. The EnKF-based inflow estimator provides a measure of uncertainty in the size and shape of the estimated inflow, which is propagated through the model to determine the corresponding uncertainty in model predictions of area and pressure. Model predictions are compared to ex vivo area and blood pressure measurements in the ascending aorta, the carotid artery, and the femoral artery of a healthy male Merino sheep. Results discuss dynamics obtained using a linear and a nonlinear viscoelastic wall model.

Minimizing the blood velocity differences between phase-contrast magnetic resonance imaging and computational fluid dynamics simulation in cerebral arteries and aneurysms

The integration of phase-contrast magnetic resonance images (PC-MRI) and computational fluid dynamics (CFD) is a way to obtain detailed information of patient-specific hemodynamics. This study proposes a novel strategy for imposing a pressure condition on the outlet boundary (called the outlet pressure) in CFD to minimize velocity differences between the PC-MRI measurement and the CFD simulation, and to investigate the effects of outlet pressure on the numerical solution. The investigation involved ten patient-specific aneurysms reconstructed from a digital subtraction angiography image, specifically on aneurysms located at the bifurcation region. To evaluate the effects of imposing the outlet pressure, three different approaches were used, namely: a pressure-fixed (P-fixed) approach; a flow rate control (Q-control) approach; and a velocity-field-optimized (V-optimized) approach. Numerical investigations show that the highest reduction in velocity difference always occurs in the V-optimized approach, where the mean of velocity difference (normalized by inlet velocity) is 19.3%. Additionally, the highest velocity differences appear near to the wall and vessel bifurcation for 60% of the patients, resulting in differences in wall shear stress. These findings provide a new methodology for PC-MRI integrated CFD simulation and are useful for understanding the evaluation of velocity difference between the PC-MRI and CFD.

Incorporating an immersed boundary method to study thermal effects of vascular systems during tissue cryo-freezing

In this paper, the three-dimensional thermal effects of a clinically-extracted vascular tissue undergoing cryo-freezing are numerically investigated. Based on the measured experimental temperature field, the numerical results of the Pennes bioheat model combined with the boundary condition-enforced immersed boundary method (IBM) agreed well with experimental data with a maximum temperature discrepancy of 2.9°C. For simulating the temperature profile of a tumor sited in a dominantly vascularized tissue, our model is able to capture with ease the thermal effects at specified junctions of the blood vessels. The vascular complexity and the ice-ball shape irregularity which cannot be easily quantified via clinical experiments are also analyzed and compared for both two-dimensional and three-dimensional settings with different vessel configurations and developments. For the three-dimensional numerical simulations, a n-furcated liver vessels model from a three-dimensional segmented volume using hole-making and subdivision methods is applied. A specific study revealed that the structure and complexity of the vascular network can markedly affect the tissue's freezing configuration with increasing ice-ball irregularity for greater blood vessel complexity.

SimVascular: An Open Source Pipeline for Cardiovascular Simulation

Patient-specific cardiovascular simulation has become a paradigm in cardiovascular research and is emerging as a powerful tool in basic, translational and clinical research. In this paper we discuss the recent development of a fully open-source SimVascular software package, which provides a complete pipeline from medical image data segmentation to patient-specific blood flow simulation and analysis. This package serves as a research tool for cardiovascular modeling and simulation, and has contributed to numerous advances in personalized medicine, surgical planning and medical device design. The SimVascular software has recently been refactored and expanded to enhance functionality, usability, efficiency and accuracy of image-based patient-specific modeling tools. Moreover, SimVascular previously required several licensed components that hindered new user adoption and code management and our recent developments have replaced these commercial components to create a fully open source pipeline. These developments foster advances in cardiovascular modeling research, increased collaboration, standardization of methods, and a growing developer community.

The Impact of Cardiac Motion on Aortic Valve Flow Used in Computational Simulations of the Thoracic Aorta

Advancements in image-based computational modeling are producing increasingly more realistic representations of vasculature and hemodynamics, but so far have not compensated for cardiac motion when imposing inflow boundary conditions. The effect of cardiac motion on aortic flow is important when assessing sequelae in this region including coarctation of the aorta (CoA) or regurgitant fraction. The objective of this investigation was to develop a method to assess and correct for the influence of cardiac motion on blood flow measurements through the aortic valve (AoV) and to determine its impact on patient-specific local hemodynamics quantified by computational fluid dynamics (CFD). A motion-compensated inflow waveform was imposed into the CFD model of a patient with repaired CoA that accounted for the distance traveled by the basal plane during the cardiac cycle. Time-averaged wall shear stress (TAWSS) and turbulent kinetic energy (TKE) values were compared with CFD results of the same patient using the original waveform. Cardiac motion resulted in underestimation of flow during systole and overestimation during diastole. Influences of inflow waveforms on TAWSS were greatest along the outer wall of the ascending aorta (AscAo) (∼30 dyn/cm2). Differences in TAWSS were more pronounced than those from the model creation or mesh dependence aspects of CFD. TKE was slightly higher for the motion-compensated waveform throughout the aortic arch. These results suggest that accounting for cardiac motion when quantifying blood flow through the AoV can lead to different conclusions for hemodynamic indices, which may be important if these results are ultimately used to predict patient outcomes.

SAPHENOUS VEIN GRAFT PATENCY AFTER GEOMETRY REMODELING

Tortuous saphenous vein graft (SVG) hemodynamics was investigated using computational fluid dynamics (CFD) techniques. Computed tomography (CT) technology is used for noninvasive bypass graft assessment seven days after surgery. CT investigation shows two regions with severe shape remodeling, one is an angle type contortion and the other one is a sharp curvature with tortuous area reduction. The numerical analysis carefully examines the effect of an SVG geometry remodeling through flow separation, particle deposition, and wall shear stress (WSS). During the cardiac cycle, overall pressure drop increases from 2.6[Formula: see text]mmHg to 4.4[Formula: see text]mmHg. In the accelerating part of the systolic phase, particles released in the inlet section move downstream toward the first narrowed part (elbow type contortion) with a helical motion. WSS range along the cardiac cycle varies from 2[Formula: see text]Pa to 42[Formula: see text]Pa, enough to damage the endothelial cells. Vessel torsion induced helical flow can reduce the flow disturbance and separation. Additionally, in the distal end of the graft, the high particle concentrations can promote the inflammatory processes in the vessels.

Inter-operator Reliability of Magnetic Resonance Image-Based Computational Fluid Dynamics Prediction of Cerebrospinal Fluid Motion in the Cervical Spine

For the first time, inter-operator dependence of MRI based computational fluid dynamics (CFD) modeling of cerebrospinal fluid (CSF) in the cervical spinal subarachnoid space (SSS) is evaluated. In vivo MRI flow measurements and anatomy MRI images were obtained at the cervico-medullary junction of a healthy subject and a Chiari I malformation patient. 3D anatomies of the SSS were reconstructed by manual segmentation by four independent operators for both cases. CFD results were compared at nine axial locations along the SSS in terms of hydrodynamic and geometric parameters. Intraclass correlation (ICC) assessed the inter-operator agreement for each parameter over the axial locations and coefficient of variance (CV) compared the percentage of variance for each parameter between the operators. Greater operator dependence was found for the patient (0.19 < ICC < 0.99) near the craniovertebral junction compared to the healthy subject (ICC > 0.78). For the healthy subject, hydraulic diameter and Womersley number had the least variance (CV = ~2%). For the patient, peak diastolic velocity and Reynolds number had the smallest variance (CV = ~3%). These results show a high degree of inter-operator reliability for MRI-based CFD simulations of CSF flow in the cervical spine for healthy subjects and a lower degree of reliability for patients with Type I Chiari malformation.

Distinctive flow pattern of wall shear stress and oscillatory shear index: similarity and dissimilarity in ruptured and unruptured cerebral aneurysm blebs

Object

The difference in the hemodynamics of wall shear stress (WSS) and oscillatory shear index (OSI) between ruptured and unruptured aneurysms is not well understood. The authors investigated the hemodynamic similarities and dissimilarities in ruptured and thin-walled unruptured aneurysm blebs.

Methods

Magnetic resonance imaging–based fluid dynamics analysis was used to calculate WSS and OSI, and hemodynamic and intraoperative findings were compared. The authors also compared ruptured and unruptured thin-walled blebs for the magnitude of WSS and OSI.

Results

Intraoperatively, 13 ruptured and 139 thin-walled unruptured aneurysm blebs were identified. Twelve of the ruptured (92.3%) and 124 of the unruptured blebs (89.2%) manifested low WSS and high OSI. The degree of WSS was significantly lower in ruptured (0.49 ± 0.12 Pa) than in unruptured (0.64 ± 0.15 Pa; p < 0.01) blebs.

Conclusions

Ruptured and unruptured blebs shared a distinctive pattern of low WSS and high OSI. The degree of WSS at the rupture site was significantly lower than in the unruptured thin-walled blebs.

Including aortic valve morphology in computational fluid dynamics simulations: initial findings and application to aortic coarctation

Computational fluid dynamics (CFD) simulations quantifying thoracic aortic flow patterns have not included disturbances from the aortic valve (AoV). 80% of patients with aortic coarctation (CoA) have a bicuspid aortic valve (BAV) which may cause adverse flow patterns contributing to morbidity. Our objectives were to develop a method to account for the AoV in CFD simulations, and quantify its impact on local hemodynamics. The method developed facilitates segmentation of the AoV, spatiotemporal interpolation of segments, and anatomic positioning of segments at the CFD model inlet. The AoV was included in CFD model examples of a normal (tricuspid AoV) and a post-surgical CoA patient (BAV). Velocity, turbulent kinetic energy (TKE), time-averaged wall shear stress (TAWSS), and oscillatory shear index (OSI) results were compared to equivalent simulations using a plug inlet profile. The plug inlet greatly underestimated TKE for both examples. TAWSS differences extended throughout the thoracic aorta for the CoA BAV, but were limited to the arch for the normal example. OSI differences existed mainly in the ascending aorta for both cases. The impact of AoV can now be included with CFD simulations to identify regions of deleterious hemodynamics thereby advancing simulations of the thoracic aorta one step closer to reality.

Particle depositions and related hemodynamic parameters in the multiple stenosed right coronary artery

BACKGROUND

Blood flow analysis of the human right coronary artery (RCA) has been carried out to investigate the effects of serial stenosis on coronary hemodynamics. A 3-D model of a serial stenosed RCA was reconstructed based on multislice computerized tomography images.

METHODS

A velocity waveform in the proximal RCA and a pressure waveform in the distal RCA of a patient with a severe stenosis were acquired with a catheter delivered wire probe and applied as boundary conditions. The numerical analysis examines closely the effect of a multiple serial stenosis on the hemodynamic characteristics such as flow separation, wall shear stress (WSS) and particle depositions.

RESULTS AND CONCLUSIONS

Energy loss associated with such flow expansion after each constriction will be large and consequently the pressure drop will be higher. Overall pressure drop increased from 1700 Pa (12.75 mmHg) at the end diastole to 11000 Pa (82.5 mmHg) at the peak systole. At the peak systole the WSS values reached 110 Pa in the stenosis with 28% diameter reduction and 210 Pa in the stenosis with 54% diameter reduction, which is high enough to damage the endothelial cells. However at the end of one cardiac cycle a percent of 1.4% (15 from 1063 particles release at the inlet section) remain inside the stenosed RCA.

Fluid Structure Interaction With Contact Surface Methodology for Evaluation of Endovascular Carotid Implants for Drug-Resistant Hypertension Treatment

Drug-resistant hypertensive patients may be treated by mechanical stimulation of stretch-sensitive baroreceptors located in the sinus of carotid arteries. To evaluate the efficacy of endovascular devices to stretch the carotid sinus such that the induced strain might trigger baroreceptors to increase action potential firing rate and thereby reduce systemic blood pressure, numerical simulations were conducted of devices deployed in subject-specific carotid models. Two models were chosen—a typical physiologic carotid and a diminutive atypical physiologic model representing a clinically worst case scenario—to evaluate the effects of device deployment in normal and extreme cases, respectively. Based on the anatomical dimensions of the carotids, two different device sizes were chosen out of five total device sizes available. A fluid structure interaction (FSI) simulation methodology with contact surface between the device and the arterial wall was implemented for resolving the stresses and strains induced by device deployment. Results indicate that device deployment in the carotid sinus of the physiologic model induces an increase of 2.5% and 7.5% in circumferential and longitudinal wall stretch, respectively, and a maximum of 54% increase in von Mises arterial stress at the sinus wall baroreceptor region. The second device, deployed in the diminutive carotid model, induces an increase of 6% in both circumferential and longitudinal stretch and a 50% maximum increase in von Mises stress at the sinus wall baroreceptor region. Device deployment has a minimal effect on blood-flow patterns, indicating that it does not adversely affect carotid bifurcation hemodynamics in the physiologic model. In the smaller carotid model, deployment of the device lowers wall shear stress at sinus by 16% while accelerating flow entering the external carotid artery branch. Our FSI simulations of carotid arteries with deployed device show that the device induces localized increase in wall stretch at the sinus, suggesting that this will activate baroreceptors and subsequently may control hypertension in drug-resistant hypertensive patients, with no consequential deleterious effects on the carotid sinus hemodynamics.

Modeling n-furcated liver vessels from a 3-D segmented volume using hole-making and subdivision methods

It is difficult to build an accurate and smooth liver vessel model due to the tiny size, noise, and n-furcations of vessels. To overcome these problems, we propose an n-furcation vessel tree modeling method. In this method, given a segmented volume and a point indicating the root of the vessels, centerlines and cross-sectional contours of the vessels are extracted and organized as a tree first. Then, the tree is broken up into separate branches in descending order of length, and polygonal meshes of all the branches are separately constructed from the cross-sectional contours. Finally, all the meshes are combined sequentially using our hole-making approach. Holes are made on a coarse mesh, and a final fine mesh is generated using a subdivision method. The hole-making approach with the subdivision method provides good efficiency in mesh construction as well as great flexibilities in mesh editing. Experiments show that our method can automatically construct smooth mesh models for n-furcated vessels with mean absolute error of 0.92 voxel and mean relative error of 0.17. It is promising to be used in diagnosis, analysis, and surgery simulation of liver diseases, and is able to model tubular structures with tree topology.

Evaluation of aortic geometries created by magnetic resonance imaging data in healthy volunteers

INTRODUCTION

The development of atherosclerotic plaques has been associated with the patterns of wall shear stress (WSS). However, much is still uncertain with the methods used to calculate WSS. Correct vessel geometries are mandatory to get reliable estimations, and the purpose of this study was to evaluate an in vivo method for creating aortic 3D geometry in human based on data from magnetic resonance imaging (MRI) with ultrasound as reference.

METHODS

The aortas of ten healthy men, 23·4 ± 1·6 years of age, were examined with a 1·5 T MRI system using a 3D gadolinium-enhanced gradient-echo sequence. Three-dimensional geometries were created using manual segmentation of images. Lumen diameters (LD) were measured in the abdominal aorta (AA) and the thoracic aorta (TA) with non-invasive B-mode ultrasound as a reference.

RESULTS

The anteroposterior diameter of the AA was 13·6 ± 1·1 mm for the MRI and 13·8 ± 1·3 mm for the ultrasound (NS). Intraobserver variability (CV) for MRI and ultrasound was <0·92% and <0·40%, respectively. Interobserver variability for MRI and ultrasound was 0·96% and 0·56%, respectively. The diameter of the TA was 19·2 ± 1·4 mm for the MRI, and the intraobserver variability (CV) was <0·78% and interobserver variability (CV) was 0·92%.

CONCLUSION

  Specific arterial geometries can be constructed with a high degree of accuracy using MRI. This indicates that the MRI geometries may be used to create realistic and correct geometries in the calculation of WSS in the aorta of human.

Endothelial shear stress from large-scale blood flow simulations

We discuss the optimal evaluation of endothelial shear stress for real-life case studies based on anatomic data acquisition. The fluid dynamic simulations require smoothing of the geometric dataset to avoid major artefacts in the flow patterns, especially in the proximity of bifurcations. A systematic series of simulations at different corrugation levels shows that, below a smoothing length of about 0.5 mm, the numerical data are insensitive to further smoothing.

A stochastic collocation method for uncertainty quantification and propagation in cardiovascular simulations

Simulations of blood flow in both healthy and diseased vascular models can be used to compute a range of hemodynamic parameters including velocities, time varying wall shear stress, pressure drops, and energy losses. The confidence in the data output from cardiovascular simulations depends directly on our level of certainty in simulation input parameters. In this work, we develop a general set of tools to evaluate the sensitivity of output parameters to input uncertainties in cardiovascular simulations. Uncertainties can arise from boundary conditions, geometrical parameters, or clinical data. These uncertainties result in a range of possible outputs which are quantified using probability density functions (PDFs). The objective is to systemically model the input uncertainties and quantify the confidence in the output of hemodynamic simulations. Input uncertainties are quantified and mapped to the stochastic space using the stochastic collocation technique. We develop an adaptive collocation algorithm for Gauss-Lobatto-Chebyshev grid points that significantly reduces computational cost. This analysis is performed on two idealized problems--an abdominal aortic aneurysm and a carotid artery bifurcation, and one patient specific problem--a Fontan procedure for congenital heart defects. In each case, relevant hemodynamic features are extracted and their uncertainty is quantified. Uncertainty quantification of the hemodynamic simulations is done using (a) stochastic space representations, (b) PDFs, and (c) the confidence intervals for a specified level of confidence in each problem.

Effect of posture change on the geometric features of the healthy carotid bifurcation

Segmented cross-sectional MRI images were used to construct 3-D virtual models of the carotid bifurcation in ten healthy volunteers. Geometric features, such as bifurcation angle, internal carotid artery (ICA) angle, planarity angle, asymmetry angle, tortuosity, curvature, bifurcation area ratio, ICA/common carotid artery (CCA), external carotid artery (ECA)/CCA, and ECA/ICA diameter ratios, were calculated for both carotids in two head postures: 1) the supine neutral position; and 2) the prone sleeping position with head rotation to the right ( ∼ 80°). The results obtained have shown that head rotation causes 1) significant variations in bifurcation angle [32% mean increase for the right carotid (RC) and 21% mean decrease for the left carotid (LC)] and internal carotid artery angle (97% mean increase for the RC, 43% mean decrease for the LC); 2) a slight increase in planarity and asymmetry angles for both RC and LC; 3) minor and variable curvature changes for the CCA and for the branches; 4) slight tortuosity changes for the braches but not for the CCA; and 5) unsubstantial alterations in area and diameter ratios (percentage changes %). The significant geometric changes observed in most subjects with head posture may also cause significant changes in bifurcation hemodynamics and warrant future investigation of the hemodynamic parameters related to the development of atherosclerotic disease such as low oscillating wall shear stress and particle residence times.

Comparison of hemodynamics of intracranial aneurysms between MR fluid dynamics using 3D cine phase-contrast MRI and MR-based computational fluid dynamics

INTRODUCTION

Hemodynamics is thought to play a very important role in the initiation, growth, and rupture of intracranial aneurysms. The purpose of our study was to compare hemodynamics of intracranial aneurysms of MR fluid dynamics (MRFD) using 3D cine PC MR imaging (4D-Flow) at 1.5 T and MR-based computational fluid dynamics (CFD).

METHODS

4D-Flow was performed for five intracranial aneurysms by a 1.5 T MR scanner. 3D TOF MR angiography was performed for geometric information. The blood flow in the aneurysms was modeled using CFD simulation based on the finite element method. We used MR angiographic data as the vascular models and MR flow information as boundary conditions in CFD. 3D velocity vector fields, 3D streamlines, shearing velocity maps, wall shear stress (WSS) distribution maps and oscillatory shear index (OSI) distribution maps were obtained by MRFD and CFD and were compared.

RESULTS

There was a moderate to high degree of correlation in 3D velocity vector fields and a low to moderate degree of correlation in WSS of aneurysms between MRFD and CFD using regression analysis. The patterns of 3D streamlines were similar between MRFD and CFD. The small and rotating shearing velocities and higher OSI were observed at the top of the spiral flow in the aneurysms. The pattern and location of shearing velocity in MRFD and CFD were similar. The location of high oscillatory shear index obtained by MRFD was near to that obtained by CFD.

CONCLUSION

MRFD and CFD of intracranial aneurysms correlated fairly well.

Study of reproducibility of human arterial plaque reconstruction and its effects on stress analysis based on multispectral in vivo magnetic resonance imaging

PURPOSE

To quantify the uncertainties of carotid plaque morphology reconstruction based on patient-specific multispectral in vivo magnetic resonance imaging (MRI) and their impacts on the plaque stress analysis.

MATERIALS AND METHODS

In this study, three independent investigators were invited to reconstruct the carotid bifurcation with plaque based on MR images from two subjects to study the geometry reconstruction reproducibility. Finite element stress analyses were performed on the carotid bifurcations, as well as the models with artificially modified plaque geometries to mimic the image segmentation uncertainties, to study the impacts of the uncertainties to the stress prediction.

RESULTS

Plaque reconstruction reproducibility was generally high in the study. The uncertainties among interobservers are around one or the subpixel level. It also shows that the predicted stress is relatively less sensitive to the arterial wall segmentation uncertainties, and more affected by the accuracy of lipid region definition. For a model with lipid core region artificially increased by adding one pixel on the lipid region boundary, it will significantly increase the maximum Von Mises Stress in fibrous cap (>100%) compared with the baseline model for all subjects.

CONCLUSION

The current in vivo MRI in the carotid plaque could provide useful and reliable information for plaque morphology. The accuracy of stress analysis based on plaque geometry is subject to MRI quality. The improved resolution/quality in plaque imaging with newly developed MRI protocols would generate more realistic stress predictions.

Vertebrobasilar junction fenestration with dumbbell-shaped aneurysms formation: computational fluid dynamics analysis

BACKGROUND

We report 8 rare cases of paired ANs involving fenestrated vertebrobasilar junction and demonstrate the flow patterns of the paired ANs by qualitative CFD analysis in 5 cases.

METHODS

Two-dimensional and 3-dimensional angiographic features of 8 cases were reviewed. Nine patient-specific geometries of CFD models in 5 cases were created for flow analysis.

RESULTS

All 8 cases had 2 ANs, one large and the other small, projecting to the opposite sides at the proximal end of fenestrated vertebrobasilar junction. The different angiographic findings between right VA and left VA suggested the different hemodynamic characteristics of the respective VAs. Computational fluid dynamics analysis also demonstrated that the inflows of these paired ANs were different between right VA and left VA. Flow simulations by CFD were consistent with angiographic findings.

CONCLUSION

Intrinsic wall defects at fenestrated vertebrobasilar junction and specific hemodynamic stresses from 2 inflows may contribute to the formation of a pair of dumbbell-shaped ANs.

Recent advances in the application of computational mechanics to the diagnosis and treatment of cardiovascular disease

During the last 30 years, research into the pathogenesis and progression of cardiovascular disease has had to employ a multidisciplinary approach involving a wide range of subject areas, from molecular and cell biology to computational mechanics and experimental solid and fluid mechanics. In general, research was driven by the need to provide answers to questions of critical importance for disease management. Ongoing improvements in the spatial resolution of medical imaging equipment coupled to an exponential growth in the capacity, flexibility and speed of computational techniques have provided a valuable opportunity for numerical simulations and complex experimental techniques to make a contribution to improving the diagnosis and clinical management of many forms of cardiovascular disease. This paper contains a review of recent progress in the numerical simulation of cardiovascular mechanics, focusing on three particular areas: patient-specific modeling and the optimization of surgery in pediatric cardiology, evaluating the risk of rupture in aortic aneurysms, and noninvasive characterization of intraventricular flow in the management of heart failure.

Choice of in vivo versus idealized velocity boundary conditions influences physiologically relevant flow patterns in a subject-specific simulation of flow in the human carotid bifurcation

Accurate fluid mechanics models are important tools for predicting the flow field in the carotid artery bifurcation and for understanding the relationship between hemodynamics and the initiation and progression of atherosclerosis. Clinical imaging modalities can be used to obtain geometry and blood flow data for developing subject-specific human carotid artery bifurcation models. We developed subject-specific computational fluid dynamics models of the human carotid bifurcation from magnetic resonance (MR) geometry data and phase contrast MR velocity data measured in vivo. Two simulations were conducted with identical geometry, flow rates, and fluid parameters: (1) Simulation 1 used in vivo measured velocity distributions as time-varying boundary conditions and (2) Simulation 2 used idealized fully-developed velocity profiles as boundary conditions. The position and extent of negative axial velocity regions (NAVRs) vary between the two simulations at any given point in time, and these regions vary temporally within each simulation. The combination of inlet velocity boundary conditions, geometry, and flow waveforms influences NAVRs. In particular, the combination of flow division and the location of the velocity peak with respect to individual carotid geometry landmarks (bifurcation apex position and the departure angle of the internal carotid) influences the size and location of these reversed flow zones. Average axial wall shear stress (WSS) distributions are qualitatively similar for the two simulations; however, instantaneous WSS values vary with the choice of velocity boundary conditions. By developing subject-specific simulations from in vivo measured geometry and flow data and varying the velocity boundary conditions in otherwise identical models, we isolated the effects of measured versus idealized velocity distributions on blood flow patterns. Choice of velocity distributions at boundary conditions is shown to influence pathophysiologically relevant flow patterns in the human carotid bifurcation. Although mean WSS distributions are qualitatively similar for measured and idealized inlet boundary conditions, instantaneous NAVRs differ and warrant imposing in vivo velocity boundary conditions in computational simulations. A simulation based on in vivo measured velocity distributions is preferred for modeling hemodynamics in subject-specific carotid artery bifurcation models when studying atherosclerosis initiation and development.

Patient-specific modeling of cardiovascular mechanics

Advances in numerical methods and three-dimensional imaging techniques have enabled the quantification of cardiovascular mechanics in subject-specific anatomic and physiologic models. Patient-specific models are being used to guide cell culture and animal experiments and test hypotheses related to the role of biomechanical factors in vascular diseases. Furthermore, biomechanical models based on noninvasive medical imaging could provide invaluable data on the in vivo service environment where cardiovascular devices are employed and on the effect of the devices on physiologic function. Finally, patient-specific modeling has enabled an entirely new application of cardiovascular mechanics, namely predicting outcomes of alternate therapeutic interventions for individual patients. We review methods to create anatomic and physiologic models, obtain properties, assign boundary conditions, and solve the equations governing blood flow and vessel wall dynamics. Applications of patient-specific models of cardiovascular mechanics are presented, followed by a discussion of the challenges and opportunities that lie ahead.

Wall shear stress gradient analysis within an idealized stenosis using non-Newtonian flow

OBJECTIVE

The endothelium is functionally regulated by the magnitude and spatiotemporal gradients of wall shear stress (WSS). Although flow separation and reversal occur beyond high-grade stenoses, little is known of the WSS pattern within clinically relevant mild to moderate stenoses.

METHODS

An axisymmetric geometry with 25, 50, and 75% stenosis criteria (quantified in accordance with the North American Symptomatic Carotid Endarterectomy Trial) was used to generate a high-resolution, hybrid, tetrahedral-hexahedral computational mesh with boundary-layer enrichment to improve near-wall shear stress gradient (WSSG) computation. Time-dependent computational fluid dynamic analysis was performed using a non-Newtonian Carreau-Yasuda model of blood to yield the shear-dependent viscosity.

RESULTS

Transition to secondary flow patterns was demonstrated in stenoses of 25, 50, and 75%. A focal region with near-wall flow reversal and retrograde WSS was identified within the stenosis itself and was found to migrate cyclically during the cardiac pulse. A zone of zero WSS and divergent WSSG that shifts in toward the throat with increasing stenotic severity was identified. Focal zones of high WSSG with converging and/or diverging direction were uncovered within the stenosis itself, as were expected changes in the distal poststenotic region. These zones of divergent WSSG shift over a substantial length of the stenosis during the course of the cardiac cycle.

CONCLUSION

Luminal WSS demonstrates dynamic direction reversal and high spatial gradients within the distal stenosis throat of even clinically moderate lesions. These findings shed light on the complex vessel wall hemodynamics within clinical stenoses and reveal a mechanical microenvironment that is conducive to perpetual endothelial functional dysregulation and stenosis progression.

Computational fluid dynamics modeling of intracranial aneurysms: qualitative comparison with cerebral angiography

RATIONALE AND OBJECTIVE

The purpose of this study is to determine whether computational fluid dynamics modeling can correctly predict the location of the major intra-aneurysmal flow structures that can be identified by conventional angiography.

MATERIALS AND METHODS

Patient-specific models of three cerebral aneurysms were constructed from three-dimensional rotational angiography images and computational fluid dynamic simulations performed. Using these velocity fields, contrast transport was simulated and visualizations constructed to provide a "virtual" angiogram. These models were then compared to images from high frame rate conventional angiography to compare flow structures.

RESULTS

Computational fluid dynamics simulations showed three distinct flow types ranging from simple to complex. Virtual angiographic images showed good agreement with images from conventional angiography for all three aneurysms with analogous size and orientation of the inflow jet, regions of impaction, and flow type. Large intra-aneurysmal vortices and regions of outflow also corresponded between the images.

CONCLUSIONS

Patient-specific image-based computational models of cerebral aneurysms can realistically reproduce the major intra-aneurysmal flow structures observed with conventional angiography. The agreement between computational models and angiographic structures is less for slower zones of recirculation later in the cardiac cycle.

The integration of medical imaging and computational fluid dynamics for measuring wall shear stress in carotid arteries

The link between atherosclerosis and wall shear stress (WSS) has lead to considerable interest in the in vivo estimation of WSS. Both magnetic resonance imaging (MRI) and three-dimensional ultrasound (3DUS) are capable of providing the anatomical and flow data required for subject-specific computational fluid dynamics (CFD) simulations. This study compares, for the first time, predicted 3D flow patterns based on black blood MRI and 3DUS. Velocity fields in the carotid arteries of nine subjects have been reconstructed, and the haemodynamic wall parameters WSS, oscillatory shear index (OSI), WSS gradients (WSSG) and angle gradients (WSSAG) were computed and compared. There was a good qualitative agreement between results derived from MRI and 3DUS, embodied by a strong linear correlation between the patched representations of the haemodynamic wall parameters. The root-mean-square error between haemodynamic wall parameters was comparable to the range of the expected variability of each imaging technique (WSS: 0.411 N/m; OSI: 0.048; temporal WSSG: 2.29 N/(s.m/sup 2/); spatial WSSG: 150 N/m/sup 3/; WSSAG: 87.6 rad/m). In conclusion, MRI and 3DUS are comparable techniques for combining with CFD in the carotid artery. The relatively high cost of MRI favour 3DUS to MRI for future haemodynamic studies of superficial arteries.

The influence of vessel wall elasticity and peripheral resistance on the carotid artery flow wave form: A CFD model compared to in vivo ultrasound measurements

The Doppler flow wave form and its derived measures such as the pulsatility index provide clinically important tools for the investigation of arterial disease. The typical shape of Doppler flow wave forms is physiologically known to be largely determined by both peripheral resistance and elastic properties of the arterial wall. In the present study we systematically investigate the influence of both vessel wall elasticity and peripheral resistance on the flow wave form obtained from a CFD-simulation of blood flow in the carotid bifurcation. Numerical results are compared to in vivo ultrasound measurements. The in vivo measurement provides a realistic geometry, local elasticities and an input flow wave form for the numerical experiment. Numerical and experimental results are compared at three different sites in the carotid branches. Peripheral resistance has a profoundly decreasing effect on velocities in the external carotid artery. If elasticity is taken into account, the computed peak systolic velocities are considerably lower and a more realistic smoothing of the flow wave form is found. Together, the results indicate that only if both vessel wall elasticity and positive peripheral resistance are taken into account, experimentally obtained Doppler flow wave forms can be reproduced numerically.

Validation of CFD simulations of cerebral aneurysms with implication of geometric variations

BACKGROUND

Computational fluid dynamics (CFD) simulations using medical-image-based anatomical vascular geometry are now gaining clinical relevance. This study aimed at validating the CFD methodology for studying cerebral aneurysms by using particle image velocimetry (PIV) measurements, with a focus on the effects of small geometric variations in aneurysm models on the flow dynamics obtained with CFD.

METHOD OF APPROACH

An experimental phantom was fabricated out of silicone elastomer to best mimic a spherical aneurysm model. PIV measurements were obtained from the phantom and compared with the CFD results from an ideal spherical aneurysm model (S1). These measurements were also compared with CFD results, based on the geometry reconstructed from three-dimensional images of the experimental phantom. We further performed CFD analysis on two geometric variations, S2 and S3, of the phantom to investigate the effects of small geometric variations on the aneurysmal flow field. Results. We found poor agreement between the CFD results from the ideal spherical aneurysm model and the PIV measurements from the phantom, including inconsistent secondary flow patterns. The CFD results based on the actual phantom geometry, however, matched well with the PIV measurements. CFD of models S2 and S3 produced qualitatively similar flow fields to that of the phantom but quantitatively significant changes in key hemodynamic parameters such as vorticity, positive circulation, and wall shear stress.

CONCLUSION

CFD simulation results can closely match experimental measurements as long as both are performed on the same model geometry. Small geometric variations on the aneurysm model can significantly alter the flow-field and key hemodynamic parameters. Since medical images are subjected to geometric uncertainties, image-based patient-specific CFD results must be carefully scrutinized before providing clinical feedback.

Computational fluid dynamics: hemodynamic changes in abdominal aortic aneurysm after stent-graft implantation

The aim of this study was to demonstrate quantitatively and qualitatively the hemodynamic changes in abdominal aortic aneurysms (AAA) after stent-graft placement based on multidetector CT angiography (MDCT-A) datasets using the possibilities of computational fluid dynamics (CFD). Eleven patients with AAA and one patient with left-side common iliac aneurysm undergoing MDCT-A before and after stent-graft implantation were included. Based on the CT datasets, three-dimensional grid-based models of AAA were built. The minimal size of tetrahedrons was determined for grid-independence simulation. The CFD program was validated by comparing the calculated flow with an experimentally generated flow in an identical, anatomically correct silicon model of an AAA. Based on the results, pulsatile flow was simulated. A laminar, incompressible flow-based inlet condition, zero traction-force outlet boundary, and a no-slip wall boundary condition was applied. The measured flow volume and visualized flow pattern, wall pressure, and wall shear stress before and after stent-graft implantation were compared. The experimentally and numerically generated streamlines are highly congruent. After stenting, the simulation shows a reduction of wall pressure and wall shear stress and a more equal flow through both external iliac arteries after stenting. The postimplantation flow pattern is characterized by a reduction of turbulences. New areas of high pressure and shear stress appear at the stent bifurcation and docking area. CFD is a versatile and noninvasive tool to demonstrate changes of flow rate and flow pattern caused by stent-graft implantation. The desired effect and possible complications of a stent-graft implantation can be visualized. CFD is a highly promising technique and improves our understanding of the local structural and fluid dynamic conditions for abdominal aortic stent placement.

Flow imaging and computing: large artery hemodynamics

The objective of our session at the International Bio-Fluid Mechanics Symposium and Workshop was at the International Bio-Fluid Mechanics Symposium and Workshop to review the state-of-the-art in, and identify future directions for, imaging and computational modeling of blood flow in the large arteries and the microcirculation. Naturally, talks in other sessions of the workshop overlapped this broad topic, and so here we summarize progress within the last decade in terms of the technical development and application of flow imaging and computing, rather than the knowledge derived from specific studies. We then briefly discuss ways in these tools may be extended, and their application broadened, in the next decade. Furthermore, owing to the conceptual division between the hemodynamics of large arteries, and those within the microcirculation, we review these regimes separately: The former here by Steinman and Taylor; and the latter in a separate paper by Cristini.

Inlet Conditions for Image-Based CFD Models of the Carotid Bifurcation: Is it Reasonable to Assume Fully Developed Flow?

Background: Computational fluid dynamics tools are useful for their ability to model patient specific data relevant to the genesis and progression of atherosclerosis, but unavailable to measurement tools. The sensitivity of the physiologically relevant parameters of wall shear stress (WSS) and the oscillatory shear index (OSI) to secondary flow in the inlet velocity profiles was investigated in three realistic models of the carotid bifurcation. Method of Approach: Secondary flow profiles were generated using sufficiently long entrance lengths, to which curvature and helical pitch were added. The differences observed were contextualized with respect to effect of the uncertainty of the models’ geometry on the same parameters. Results: The effects of secondary velocities in the inlet profile on WSS and OSI break down within a few diameters of the inlet. Overall, the effect of secondary inlet flow on these models was on average more than 3.5 times smaller than the effect of geometric variability, with 13% and 48% WSS variability induced by inlet secondary flow and geometric differences, respectively. Conclusions: The degree of variation is demonstrated to be within the range of the other computational assumptions, and we conclude that given a sufficient entrance length of realistic geometry, simplification to fully developed axial (i.e., Womersley) flow may be made without penalty. Thus, given a choice between measuring three components of inlet velocity or a greater geometric extent, we recommend effort be given to more accurate and detailed geometric reconstructions, as being of primary influence on physiologically significant indicators.

Blood Flow in Major Blood Vessels—Modeling and Experiments

Although primarily motivated by an interest in atherosclerosis, modeling of arterial blood flow is also important to an understanding of congenital effects and to improvements in therapeutics. A variety of methods are available to estimate the flow field in living arteries, each with its own advantages and limitations. Tradeoffs must be made among the realism of the technique, spatial resolution, geometric fidelity, and the reliability of assumed wall mechanical properties. Once the velocity field is obtained, each differentiation, to obtain wall shear or its spatial or temporal derivatives, adds additional uncertainty into the results, demanding cautious interpretation. A distinction is made between "macro" and "micro" levels of flow structure detail: macro level structure is relatively coarse and more descriptive of the flow field, pressure, and shear distribution than the cellular response; the micro approach tries to relate a more local hemodynamic description to vascular pathology. The applications of each, and the interactions between them, are described. Issues related to these approaches, including the use of clinical data, animal experimentation, the role of cell and organ culture, and in vivo flow measurement, are briefly discussed. The summary closes with a list of recommendations for future developments in this area.

Regional material property alterations in porcine femoral arteries with atheroma development

We have developed a novel methodology that permits assessment of regional vascular mechanical property alterations in the presence of atheroma in vivo employing a Yucatan miniswine model with induced lesions. Femoral arteries were imaged with intravascular ultrasound. Image data were segmented and, following three-dimensional reconstruction, underwent finite element and sensitivity analysis with optimization to identify regions with altered vascular mechanical properties. All regions were compared to histological analysis. In 12 animals with 8 weeks of endothelial cell denudation and high cholesterol diet (induced atherosclerosis), the elastic modulus initially decreased with early lesion development and then increased with increasing fibrosis-(elastic modulus-all values x10(4)Pa-mean+/-SEM) histologically normal (non-denuded control segment) elements 9.73+/-0.01, fatty elements 9.53+/-0.01, fibrofatty elements 9.41+/-0.03, and fibrous elements 9.68+/-0.02 (all p<0.001 vs. normal elements). Wall thickness, however, increased with atheroma formation. These data demonstrate decreasing vascular material properties with early lesions, followed by an increase as lesions progress. This methodology permits determination of areas with early atheroma development, follow atheroma progression, and potentially evaluate interventions aimed at decreasing atheroma load and normalizing vascular material properties.

Generating an ex vivo vascular model

Realistic ex vivo anthropometric vascular environments are required for endovascular device optimization and for preclinical evaluation of interventional procedures. The objective of this research is to build an anthropomorphic model of the human carotid artery. The combination of magnetic resonance angiography image processing and computer-aided design and manufacturing techniques allowed fabrication of multicomponent morphologically precise casts of the carotid artery. The lost core technique was used to produce a hollow vessel prototype incorporating polyvinyl alcohol cryogel (PVA-C) as a tissue-mimicking vessel wall material. PVA-C was mechanically characterized by uniaxial tensile testing after different numbers of freeze/thaw cycles. The novel model construction approach outlined in this study accounts for the morphologic complexities of the human vasculature, and proved successful for the production of realistic compliant ex vivo arterial model.

A Computational Study of Flow in a Compliant Carotid Bifurcation–Stress Phase Angle Correlation with Shear Stress

The present study presents a three-dimensional, unsteady supercomputer simulation of the coupled fluid-solid interaction problem associated with flow through a compliant model of the bifurcation of the common carotid artery into the internal and external carotid arteries. The fluid wall shear stress (WSS) and solid circumferential stress/strain (CS) are computed and analyzed for the first time using the complex ratio of CS to WSS (CS/WSS). This analysis reveals a large negative phase angle between CS and WSS (stress phase angle--SPA) on the outer wall of the carotid sinus where atherosclerotic plaques are localized. This finding is consistent with other measurements and computations of the SPA in coronary arteries and the aortic bifurcation that show large negative SPA correlating with sites of plaque location and in vitro studies of endothelial cells showing that large negative SPA induces pro-atherogenic gene expression and metabolite release profiles.