Wall Shear Stress Gradient Research Articles

Unsteady stenosis flow prediction: a comparative study of non-Newtonian models with operator splitting scheme

This paper presents a comparative study of non-Newtonian and Newtonian models of blood. A non-Newtonian incompressible 2-D Navier–Stokes (N-S) solver has been developed using Fasttalk language within the Fastflo environment. It is based on the method of operator splitting with artificial compressibility technique. The Power law and Casson models have been used as the constitutive equations for blood with a hematocrit of approximately 45%. These two non-Newtonian models and the Newtonian model are used to simulate unsteady flow through a hypothetical stenotic geometry over an aperiodic time interval of 1 s. Through comparison of the results of the three models, it was found that the wall shear stress (WSS) distribution over the time interval was comparable for both non-Newtonian models. The peak WSS for the Newtonian model had the lowest value. The peak wall shear stress gradient (WSSG) for the Power law was the highest, followed by the Casson and Newtonian models. Flow characteristics such as higher pressure drop across the stenosis, location and movement of vortex were similar in all three models. Non-Newtonian effects were most significant in the vicinity of the stenosis.

Rheological effects on pulsatile hemodynamics in a stenosed tube

Transient laminar axisymmetric flow through a tube with a smooth local area reduction of 75% is considered. Using an experimentally validated control volume method, the influence of three rheology models (i.e., Newtonian, power law and Quemada) are investigated. Three Womersley numbers ( Wo=r 0 ω/ν 1/2 :=4.0, 7.5 and 12.5) are compared for a sinusoidal input pulse which varies between 0 and 400 with a mean Reynolds number of 200. The primary application is pulsatile flow in axisymmetric stenosed artery segments. Results show that for the highest Womersley number considered here, a second co-rotating vortex is formed distal to the primary vortex. Also, the shear-thinning rheological models have a secondary effect on the flow field that primarily appears in terms of subtle changes to the hemodynamic wall parameters (i.e., the time-averaged wall shear stress, spatial wall shear stress gradient, and oscillatory shear index). The non-Newtonian models affect the entrainment of fluid-like particles in the post-stenotic region measurably. The particle residence time (PRT), defined as the ratio of transient to steady residence times, is found to be less than or equal to unity for the majority of fluid elements for all rheologies and Womersley numbers considered. The highest fraction of PRT < 1 is found at moderate Womersley numbers. Pathological fluid elements, where PRT > 10, is found to increase with increasing Wo and is decreased by the non-Newtonian formulations. This is due to a low-shear high-viscosity band that impedes the progress of fluid particles into the near wall region.

Hemodynamics analyses of arterial expansions with implications to thrombosis and restenosis

It is assumed that critical hemodynamic factors play an important role in the onset, localization and degree of post-operative complications, for example, thrombosis and restenosis. Of special interest are sudden expansion flows, which may occur in straight artery segments such as the common carotid after endarterectomy or end-to-end anastomoses. Sudden expansion geometries are possible origins of early post-operative emboli and significant myointimal hyperplasia resulting in early or late complications. Transient laminar axisymmetric and fully three-dimensional blood flows were simulated employing a validated finite volume code in conjunction with a Runge–Kutta particle tracking technique. Disturbed flow indicators, which may predict the onset of thrombosis and/or restenosis, were identified and employed to evaluate 90°-step and smooth expansion geometries. Smooth expansion geometries have weaker disturbed flow features than step expansion geometries. Specifically, the regions near the expansion wall and the reattachment point are susceptible to both atherosclerotic lesion and thrombi formations as indicated by non-uniform hemodynamic indicators such as near-zero wall shear stress and elevated wall shear stress gradients as well as blood particle accumulation and deposition. A new parameter, the wall shear stress angle deviation (WSSAD) has been introduced, which indicates areas of abnormal endothelial cell morphology and particle wall deposition. In turn, regions of low wall shear stress and high wall shear stress gradients are recognized as susceptible sites for arterial diseases. Thus, it is interesting to note that high WSSAD surface areas cover low wall shear stress, high wall shear stress gradient locations as well as high wall particle deposition. A gradual change in step expansion geometry provides better results in terms of WSSAD values and hence potentially reducing atherosclerosis as well as thrombi formation.

Computational haemodynamics analysis and comparison study of arterio-venous grafts

Haemodialysis arterio-venous graft failure is related to the development of stenotic lesions most commonly located near the venous anastomosis, especially in the toe region. 'Disturbed flow' interaction with the vessel wall surface, characterized by haemodynamic parameters based on the local wall shear stress or the radial pressure gradient, have been widely recognized as the trigger mechanism of a cascade of abnormal biological events leading to occlusive developments. Assuming incompressible laminar flow and rigid, in-plane vessel walls, validated haemodynamics are numerically simulated for a constant-diameter end-to-side base case, the VenafloTM graft, and an improved graft-end configuration. The geometric design of the new graft-end was based on the reduction of three time- and area-averaged haemodynamic parameters, i.e. the wall shear stress gradient, wall shear stress angle gradient, and radial pressuregradient. Considering the critical toeregion, the VenafloTM graft has demonstrated measurable improvements over the base case configuration in predictive computer simulations as well as in clinical trials. The performance improvement should be further enhanced with the modifications illustrated by the new design.

Computer simulation and geometric design of endarterectomized carotid artery bifurcations.

The main goal of this computational study is to establish surgical guidelines for optimal geometries of carotid endarterectomy reconstructions that may measurably reduce postoperative complications, that is, thrombosis, stroke, and/or restenosis. The underlying hypotheses are that nonuniform hemodynamics, or "disturbed flows," are linked to arterial diseases and consequently that minimization of "disturbed flow" indicators leads to geometric bifurcation designs that lower postoperative complication rates. Considering transient 3-D laminar blood flow in partially occluded, in-plane, rigid-wall carotid artery bifurcations, the results presented include time-averaged indicators of "disturbed flow", such as the wall shear stress, spatial wall shear stress gradient, and wall shear stress angle deviation. In addition, trajectories and deposition patterns of critical blood particles (i.e., monocytes) are shown and evaluated. Within given physiological constraints, the vessel geometry was then changed in order to reduce the magnitudes of key indicators associated with thrombosis (i.e., blood clot formation) or restenosis (e.g., renewed atherosclerosis and/or hyperplasia). The quantitative results and knowledge base generated will be crucial for future clinical trials.

Local Measurement of Loss Using Heated Thin-Film Sensors

A calibration equation is derived linking the nondimensional entropy generation rate per unit area with the nondimensional aerodynamic wall shear stress and free-stream pressure gradient. It is proposed that the latter quantities, which can be measured from surface sensors, be used to measure the profile entropy generation rate. It is shown that the equation is accurate for a wide range of well-defined laminar profiles. To measure the dimensional entropy generation rate per unit area requires measurement of the thickness of the boundary layer. A general profile equation is given and used to show the range of accuracy of a further simplification to the calibration. For flows with low free-stream pressure gradients, the entropy generation rate is very simply related to the wall shear stress, if both are expressed without units. An array of heated thin film sensors is calibrated for the measurement of wall shear stress, thus demonstrating the feasibility of using them to measure profile entropy generation rate.

Relation between non-uniform hemodynamics and sites of altered permeability and lesion growth at the rabbit aorto-celiac junction

Using the rabbit’s aorto-celiac junction as a representative atherosclerotic model, the hemodynamics of a bifurcating blood vessel are numerically simulated and three hemodynamic parameters are compared. The wall shear stress (WSS), the oscillatory shear index (OSI), and the spatial wall shear stress gradient (WSSG) are considered in this study. Locally enhanced wall permeabilities and intimal macrophages are generally considered to be involved in atherogenesis, and here the primary concern is with the hemodynamic influence on these early stages of the disease process. In comparing the segmental averages of the indicator functions and previously published intimal white blood cell densities, only the WSSG shows a statistically significant correlation. All three indicators have selective strengths in determining sites of early lesion growth around the aorto-celiac flow divider. At the proximal end of the flow divider on the lateral side of the orifice, there are elevated values of the OSI as well as WSSG and low WSS values. Regions of elevated wall permeabilities compare with the regions of elevated WSSG along the lateral and distal portions of the flow divider. Largely dependent upon the present input pulse with reverse flow, the OSI indicates relatively high values throughout the flow domain, however, it is important when utilized in conjunction with low WSS regions. This study presents a rationale for further quantitative correlative studies in the rabbit model based on additional histological data sets.

Numerical Study on Three-Dimensional Unsteady Blood Flow through an Artery with a Small Side Branch.

Three-dimensional unsteady blood flow through an artery with a small side branch was numerically analyzed using the finite-element method. Our main focus was on the model in which the angle between the trunk and the small side branch was larger than 90 deg. This arterial geometry is generally observed in a real arterial system. For this study, the arterial wall was assumed to be rigid and the non-Newtonian effect of blood was not taken into consideration. The calculation results are summarized as follows. 1) The branch angle does not have a strong effect on the flow rate through the side branch. 2) In the case of a large branch angle, the still-standing stagnation point appears near the outer wall of the small side branch. Moreover, this still-standing stagnation point has an effect of increasing the values of wall shear stress gradient (WSSG).

Computational study of the effect of geometric and flow parameters on the steady flow field at the rabbit aorto-celiac bifurcation

Arterial hemodynamic forces may play a role in the localization of early atherosclerotic lesions. We have been developing numerical techniques based on overset or “Chimera” type formulations to solve the Navier-Stokes equations in complex geometries simulating arterial bifurcations. This paper presents three-dimensional steady flow computations in a model of the rabbit aorto-celiac bifurcation. The computational methods were validated by comparing the numerical results to previously-obtained flow visualization data. Once validated, the numerical algorithms were used to investigate the sensitivity of the computed flow field and resulting wall shear stress distribution to various geometric and hemodynamic parameters. The results demonstrated that a decrease in the extent of aortic taper downstream of the celiac artery induced looping fluid motion along the lateral walls of the aorta and shifted the peak wall shear stress from downstream of the celiac artery to upstream. Increasing the flow Reynolds number led to a sharp increase in spatial gradients of wall shear stress. The flow field was highly sensitive to the flow division ratio, i.e., the fraction of total flow rate that enters the celiac artery, with larger values of this ratio leading to the occurrence of flow separation along the dorsal wall of the aorta. Finally, skewness of the inlet velocity profile had a profound impact on the wall shear stress distribution near the celiac artery. While not physiological due to the assumption of steady flow, these results provide valuable insight into the fluid physics at geometries simulating arterial bifurcations.

Flow waveform effects on end-to-side anastomotic flow patterns

Purpose: Restenosis due to distal anastomotic intimal hyperplasia, a leading cause of arterial bypass graft failure, is thought to be promoted by hemodynamic effects, specifically ‘abnormal’ wall shear stress patterns. The purpose of this study was to quantify the effects of flow waveform on peri-anastomotic flow and wall shear stress patterns. Methods: Blood flow and wall shear stress patterns were numerically computed in a representative three-dimensional anastomosis using femoral, iliac and coronary flow waveforms suitable for humans at rest. Numerical results were validated against experimental data. Results: Peri-anastomotic wall shear stress patterns were influenced by a complex interplay between secondary flow effects and unsteadiness. Peripheral flow waveforms (iliac, femoral) produced large temporal and spatial wall shear stress gradients on the host artery bed. In comparison, the coronary flow waveform produced normalized bed wall shear stress gradients that were a factor of 2–3 less than for the peripheral waveforms, even though average bed wall shear stress magnitudes were similar for the two waveforms. Conclusions: If anastomotic intimal hyperplasia is promoted by large spatial and/or temporal gradients of wall shear stress, as has been proposed, this study predicts that there will be markedly less intimal hyperplasia on the host artery bed of coronary bypass grafts than for peripheral bypass grafts. This information, in conjunction with a comparative histopathologic study of intimal hyperplasia distribution, could help determine specific wall shear stress factors promoting intimal hyperplasia.

Hemodynamics of human carotid artery bifurcations: Computational studies with models reconstructed from magnetic resonance imaging of normal subjects

Purpose: The precise role played by hemodynamics, particularly wall shear stress, in the development and progression of vascular disease remains unclear, in large part because of a lack of in vivo studies with humans. Although technical challenges remain for noninvasively imaging wall shear stresses in humans, vascular anatomy can be imaged with sufficiently high resolution to allow reconstruction of three-dimensional models for computational hemodynamic studies. In this paper we present an entirely noninvasive magnetic resonance imaging (MRI) protocol that provides carotid bifurcation geometry and flow rates from which the in vivo hemodynamics can be computed. Maps of average, oscillatory, and gradients of wall shear stress are presented for two normal human subjects, and their data are compared with those computed for an idealized carotid bifurcation model. Methods: An MRI protocol was developed to acquire all necessary image data in scan times suitable for patient studies. Three-dimensional models of the carotid bifurcation lumen were reconstructed from serial black blood MR images of two normal volunteers. Common and internal carotid artery flow rate waveforms were determined from MRI phase-contrast velocity imaging in the same subjects and were used to impose fully developed velocity boundary conditions for the computational model. Subject-specific time-resolved velocities and wall shear stresses were then computed with a finite element–based Navier-Stokes equation solver. Results: Models reconstructed from in vivo MRI of two subjects showed obvious differences in branch angle, bulb size and extent, and three-dimensional curvature. Maps of a variety of wall shear stress indices showed obvious qualitative differences in patterns between the in vivo models and between the in vivo models and the idealized model. Secondary, helical flow patterns, induced primarily by the asymmetric and curved in vivo geometries, were found to play a key role in determining the resulting wall shear stress patterns. The use of in vivo flow rate waveforms was found to play a minor but noticeable role in some of the wall shear stress behavior observed. Conclusions: Conventional “averaged” carotid bifurcation models mask interesting hemodynamic features observed in realistic models derived from noninvasive imaging of normal human subjects. Observation of intersubject variations in the in vivo wall shear stress patterns supports the notion that more conclusive evidence regarding the role of hemodynamics in vascular disease may be derived from such individual studies. The techniques presented here, when combined with subject-specific MRI measurements of carotid artery plaque thickness and composition, provide the tools necessary for entirely noninvasive, prospective, in vivo human studies of hemodynamics and the relationship of hemodynamics to vascular disease.(J Vasc Surg 1998;27:143-56.)

Shear stress gradient over endothelial cells in a curved microchannel system

Our purpose was to test a scale model of the microcirculation by measuring the shear forces to which endothelial cells were exposed, and comparing this to computer simulations. In vitro experiments were performed to measure the 2-dimensional projected velocity profile along endothelial cell lined microchannels (D-shaped, 10–30 μm radius, n = 15), or in microchannels without endothelial cells ( n = 18). Microchannels were perfused with fluorescently labeled microspheres (0.5 μm dia., < 1%) suspended in cell culture media. The velocity of individual microspheres was obtained off-line (videorecording), using an interactive software program; velocity was determined as the distance traveled in one video field (1/60 s). Mass balance was verified in the microchannels by comparing the microsphere velocities to the perfusion pump rate. In confluent endothelial cell lined microchannels, a velocity profile was obtained as microspheres passed an endothelial cell nucleus (identified by fluorescent dye), and again, for a paired region 100 μm away without nuclei (cytoplasm region). The velocity profile was significantly shifted and sharpened by the endothelial cell nucleus, as anticipated. Over the nucleus, data are consistent with a normal sized nucleus extending into the lumen, further confirming that this scale model can be used to determine the wall shear stress to which endothelial cells are exposed. Using the experimental bulk phase fluid parameters as boundary conditions, we used computational fluid dynamics (CFD) to predict the expected wall shear stress gradient along an endothelial cell lined D-shaped tube. The wall shear stress gradient over the nucleus was 2-fold greater in the radial versus axial directions, and was sensitive to lateral versus midline positioned nuclei.

Endothelial cells in culture: An in vitro model of the wall shear stress gradient effect

Endothelial cells in culture: An in vitro model of the wall shear stress gradient effect

Numerical Study on the Unsteady Flows through the Bifurcation of the Porcine Renal Artery.

Three-dimensional steady and unsteady flows through the bifurcation of the porcine renal artery were analyzed numerically using the finite-element method. As the bifurcation geometry plays a primary role in determining the hemodynamical variables, the present bifurcation geometry was precisely developed in accordance with the measured data. In the present calculations, the arterial wall was assumed to be rigid and the non-Newtonian effect of blood was not taken into consideration. The calculated results showed that the present difurcation geometry had an effect of suppressing the flow separation. In addition, it was shown that the present bifurcation geometry flattened the spatial distributions of wall shear stress, which has important implications for the concept of Wall Shear Stress Gradient.

Hemodynamic simulations and computer-aided designs of graft-artery junctions.

Severe occlusion of graft-artery junctions due to restenosis, e.g., excessive tissue overgrowth and renewed plaque formation, may occur within a few months or years after bypass surgery. Our hypothesis is that nonuniform hemodynamics, represented by large sustained wall shear stress gradients, trigger abnormal biological processes leading to rapid restenosis and hence early graft failure. In turn, this problem may be significantly mitigated by designing graft-artery bypass configurations for which the wall shear stress gradient (WSSG) is approximately zero and hence nearly uniform hemodynamics are achieved. Focusing on the distal end of several femoral artery bypass junctions, a validated finite volume code has been used to compute the transient three-dimensional velocity vector fields and its first and second surface derivatives in order to test the idea. Specifically, it is shown that the Taylor patch, which generates higher patency rates than standard end-to-side anastomoses, exhibits lower WSSG levels than standard configurations, and that further geometric design improvements reduce the WSSG in magnitude and local extent even more.

Microcinematographic studies of flow patterns in the excised rabbit aorta and its major branches

Arterial fluid mechanics may play a role as a localizing factor for early atherosclerosis. Flow patterns in natural rabbit aortas rendered transparent were studied using a microcinematographic visualization technique. The aortic arch exhibited a single cell of clockwise-rotating helical secondary flow along the ventral and inner walls. Flow separation occurred proximal to the two arch branches with flow reversal proximal to the brachiocephalic artery. Sinusoidal flow rendered the helical motion more pronounced in systole, while the reverse flow zone periodically expanded and contracted. Steady flow in the abdominal aorta revealed streamlines which follow slow looping trajectories lateral to ostia before tracing helical paths into the branches. Flow separation was present along the dorsal wall of the aorta opposite the superior mesenteric artery. With the exception of the left renal artery, steady flow wall shear stresses were higher distal to ostia than proximal. Spatial gradients of wall shear stress were larger around branches than elsewhere. Similar to observed flow patterns, sites of enhanced macromolecular permeability, as observed previously in the normal rabbit aorta, follow a clockwise helical pattern in the arch and exhibit a distribution around ostia that correlates to some degree with regions of elevated shear stress gradients.

Computational design of a bypass graft that minimizes wall shear stress gradients in the region of the distal anastomosis.

Recent experimental and theoretic studies show that large wall shear stress gradients characterize disturbed flow patterns associated with the location of myointimal hyperplasia, atheroma, or both. Graft-to-artery anastomoses that minimize wall shear stress gradients may reduce the degree of myointimal development and the propensity for thrombosis. This study analyzes the distribution of distal anastomotic wall shear stress gradients for conventional geometries and for the optimized geometry assuming idealized merging of the graft with the artery. A validated computational fluid dynamics program was used to solve the transient three-dimensional partial differential equations and auxiliary equations that describe laminar incompressible blood flow. Time-averaged wall shear stresses and wall shear stress gradients were calculated for three distal graft-artery anastomoses: a standard end-to-side, a Taylor patch, and an optimized geometry. The latter was obtained iteratively by minimizing the local wall shear stress gradients and was analyzed under resting and exercise inflow waveforms. Both the standard and Taylor patch anastomoses have relatively high wall shear stress gradients in the regions of the toe and heel. For all flow inputs studied nonuniform hemodynamics in the optimized graft design are largely eliminated, and the time-averaged wall shear stress gradients are greatly reduced throughout the anastomotic zone. At resting flow the Taylor patch produces slightly lower wall shear stress gradients in the anastomotic region than the standard end-to-side anastomosis. The optimized design reduces wall shear stress gradients to almost one half of that of the standard and Taylor patch geometries. At exercise flow wall shear stress gradients almost triple in the standard anastomosis and increase approximately 30% in the Taylor patch. In contrast, the geometrically optimized design is basically independent of the type of flow input waveform in terms of time-averaged wall shear stress gradients and disturbed flow patterns. This study demonstrates that it is possible to design a terminal graft geometry for an end-to-side anastomosis that significantly reduces wall shear stress gradients. If the wall shear stress gradient is confirmed to be a major hemodynamic determinant of intimal hyperplasia and restenosis, these results may point to the design of optimal bypass graft geometries.

Pulsatile Wall Shear Stress Gradient and Aortoseptal Angle: Implications for Subaortic Stenosis † 147

A better understanding of the etiology of discrete subaortic stenosis (SAS) would be useful in identifying patients at risk and in surgical decision making since the onset, progression, and recurrence of SAS are difficult to predict. Previous studies have shown that congenital defects and morphologic abnormalities associated with SAS (such as steepened aortoseptal angle [AoSA]) cause significant, localized elevations in peak wall shear stress and the wall shear stress gradient (WSSG). High WSSG has been associated with cellular changes in the endothelium. Since the magnitude of wall shear stress varies thoughout the cardiac cycle, we addressed the hypothesis that these morphologic abnormalities have similar effects on the time-averaged shear stress and WSSG. Methods: A finite element model of the left ventricular outflow tract was implemented on the Cray C90 at the Pittsburgh Supercomputing Center for typical pulsatile flow conditions. AoSA was varied between 120 and 150 degrees, maximum aortic velocities from 0.5 to 1 m/s, and aortic diameter from 1 to 1.5 cm. Shear stress and WSSG were directly calculated from the velocity field at discrete points in time. The time-averaged wall shear stress and WSSG at the location of the peak was determined. Results: At an aortic velocity of 1 m/s, the peak and time-averaged shear stress increased with steeper AoSA by 52% and 49.5% respectively. WSSG was more sensitive to changes in AoSA with the peak WSSG increasing by 217%, compared to the time-averaged WSSG which increased by 225%. Conclusions: Geometric variables had similar effects on both the peak and mean shear stress and WSSG. Although the progression and recurrence of SAS may be dependent on time-varying exposure to shear stress, this exposure may be quantified using either a peak or a mean value.

Geometric design improvements for femoral graft-artery junctions mitigating restenosis

The present study is based on the hypothesis that nonuniform hemodynamics, represented by large time-averaged wall shear stress gradients, trigger abnormal biological processes leading to rapid restenosis, i.e. excessive tissue overgrowth and renewed plaque formation, and hence early graft failure. It implies that this problem may be significantly mitigated by finding graft-artery bypass configurations for which the wall shear stress gradient is approximately zero and hence nearly uniform hemodynamics is achieved. These fluid flow and geometric design considerations are applied to four different end-to-side anastomoses for the distal end of a femoral artery bypass with an appropriate test input pulse and a typical 20–80 flow division. A validated finite-volume code has been used to compute the transient three-dimensional velocity vector fields, wall shear stress distributions and surface contours of the wall shear stress gradients. It is shown that large anastomotic flow areas, small continuously changing bifurcation angles, and smooth junction wall curvatures reduce local time-averaged wall shear stress gradients significantly and hence should mitigate restenosis.

Geometric design improvements for femoral graft-artery junctions mitigating restenosis

The present study is based on the hypothesis that nonuniform hemodynamics, represented by large time-averaged wall shear stress gradients, trigger abnormal biological processes leading to rapid restenosis, i.e. excessive tissue overgrowth and renewed plaque formation, and hence early graft failure. It implies that this problem may be significantly mitigated by finding graft-artery bypass configurations for which the wall shear stress gradient is approximately zero and hence nearly uniform hemodynamics is achieved. These fluid flow and geometric design considerations are applied to four different end-to-side anastomoses for the distal end of a femoral artery bypass with an appropriate test input pulse and a typical 20–80 flow division. A validated finite-volume code has been used to compute the transient three-dimensional velocity vector fields, wall shear stress distributions and surface contours of the wall shear stress gradients. It is shown that large anastomotic flow areas, small continuously changing bifurcation angles, and smooth junction wall curvatures reduce local time-averaged wall shear stress gradients significantly and hence should mitigate restenosis.