A computer simulation of the non-Newtonian blood flow at the aortic bifurcation

Toward a Mesoscopic Modeling Approach of Magnetohydrodynamic Blood Flow in Pathological Vessels: A Comprehensive Review

The investigation of magnetohydrodynamic (MHD) blood flow within configurations that are pertinent to the human anatomy holds significant importance in the realm of scientific inquiry because of its practical implications within the medical field. This article presents an exhaustive appraisal of the diverse applications of magnetohydrodynamics and their computational modeling in biological contexts. These applications are classified into two categories: simple flow and pulsatile flow. An alternative approach of traditional CFD methods called Lattice Boltzmann Method (LBM), a mesoscopic method based on kinetic theory, is introduced to solve complex problems, such as hemodynamics. The results show that the flow velocity reduces considerably by increasing the magnetic field intensity, and the flow separation area is minimized by the increase of magnetic field strength. The LBM with BGK collision model has shown good results in terms of precision. Finally, this literature review has revealed a number of potential avenues for further research. Suggestions for future works are proposed accordingly.

Modeling and analysis of biomagnetic blood Carreau fluid flow through a stenosis artery with magnetic heat transfer: A transient study

We present a numerical investigation of tapered arteries that addresses the transient simulation of non-Newtonian bio-magnetic fluid dynamics (BFD) of blood through a stenosis artery in the presence of a transverse magnetic field. The current model is consistent with ferro-hydrodynamic (FHD) and magneto-hydrodynamic (MHD) principles. In the present work, blood in small arteries is analyzed using the Carreau-Yasuda model. The arterial wall is assumed to be fixed with cosine geometry for the stenosis. A parametric study was conducted to reveal the effects of the stenosis intensity and the Hartman number on a wide range of flow parameters, such as the flow velocity, temperature, and wall shear stress. Current findings are in a good agreement with recent findings in previous research studies. The results show that wall temperature control can keep the blood in its ideal blood temperature range (below 40°C) and that a severe pressure drop occurs for blockages of more than 60 percent. Additionally, with an increase in the Ha number, a velocity drop in the blood vessel is experienced.

Recirculation zone length in renal artery is affected by flow spirality and renal-to-aorta flow ratio

Haemodynamic perturbations such as flow recirculation zones play a key role in progression and development of renal artery stenosis, which typically originate at the aorta-renal bifurcation. The spiral nature of aortic blood flow, division of aortic blood flow in renal artery as well as the exercise conditions have been shown to alter the haemodynamics in both positive and negative ways. This study focuses on the combinative effects of spiral component of blood flow, renal-to-aorta flow ratio and the exercise conditions on the size and distribution of recirculation zones in renal branches using computational fluid dynamics technique. Our findings show that the recirculation length was longest when the renal-to-aorta flow ratio was smallest. Spiral flow and exercise conditions were found to be effective in reducing the recirculation length in particular in small renal-to-aorta flow ratios. These results support the hypothesis that in renal arteries with small flow ratios where a stenosis is already developed an artificially induced spiral flow within the aorta may decelerate the progression of stenosis and thereby help preserve kidney function.

Numerical Investigation of Oxygenated and Deoxygenated Blood Flow through a Tapered Stenosed Arteries in Magnetic Field

Current paper is focused on transient modeling of blood flow through a tapered stenosed arteries surrounded a by solenoid under the presence of heat transfer. The oxygenated and deoxygenated blood are considered here by the Newtonian and Non-Newtonian fluid (power law and Carreau-Yasuda) models. The governing equations of bio magnetic fluid flow for an incompressible, laminar, homogeneous, non-Newtonian are solved by finite volume method with SIMPLE algorithm for structured grid. Both magnetization and electric current source terms are well thought-out in momentum and energy equations. The effects of fluid viscosity model, Hartmann number, and magnetic number on wall shear stress, shearing stress at the stenosis throat and maximum temperature of the system are investigated and are optimized. The current study results are in agreement with some of the existing findings in the literature and are useful in thermal and mechanical design of spatially varying magnets to control the drug delivery and biomagnetic fluid flows through tapered arteries.

Analysis of non-Newtonian effects on Low-Density Lipoprotein accumulation in an artery

In this work, non-Newtonian effects on Low-Density Lipoprotein (LDL) transport across an artery are analyzed with a multi-layer model. Four rheological models (Carreau, Carreau-Yasuda, power-law and Newtonian) are used for the blood flow through the lumen. For the non-Newtonian cases, the arterial wall is modeled with a generalized momentum equation. Convection-diffusion equation is used for the LDL transport through the lumen, while Staverman-Kedem-Katchalsky, combined with porous media equations, are used for the LDL transport through the wall. Results are presented in terms of filtration velocity, Wall Shear Stresses (WSS) and concentration profiles. It is shown that non-Newtonian effects on mass transport are negligible for a healthy intramural pressure value. Non-Newtonian effects increase slightly with intramural pressure, but Newtonian assumption can still be considered reliable. Effects of arterial size are also analyzed, showing that Newtonian assumption can be considered valid for both medium and large arteries, in predicting LDL deposition. Finally, non-Newtonian effects are also analyzed for an aorta-common iliac bifurcation, showing that Newtonian assumption is valid for mass transport at low Reynolds numbers. At a high Reynolds number, it has been shown that a non-Newtonian fluid model can have more impact due to the presence of flow recirculation.

Investigation of blood flow in the external carotid artery and its branches with a new 0D peripheral model

BACKGROUND

Patient-specific modelling in clinical studies requires a realistic simulation to be performed within a reasonable computational time. The aim of this study was to develop simple but realistic outflow boundary conditions for patient-specific blood flow simulation which can be used to clarify the distribution of the anticancer agent in intra-arterial chemotherapy for oral cancer.

METHODS

In this study, the boundary conditions are expressed as a zero dimension (0D) resistance model of the peripheral vessel network based on the fractal characteristics of branching arteries combined with knowledge of the circulatory system and the energy minimization principle. This resistance model was applied to four patient-specific blood flow simulations at the region where the common carotid artery bifurcates into the internal and external carotid arteries.

RESULTS

Results of these simulations with the proposed boundary conditions were compared with the results of ultrasound measurements for the same patients. The pressure was found to be within the physiological range. The difference in velocity in the superficial temporal artery results in an error of 5.21 ± 0.78 % between the numerical results and the measurement data.

CONCLUSIONS

The proposed outflow boundary conditions, therefore, constitute a simple resistance-based model and can be used for performing accurate simulations with commercial fluid dynamics software.

MULTIPHASE FLOW OF BLOOD THROUGH ARTERIES WITH A BRANCH CAPILLARY: A THEORETICAL STUDY

In this paper, we present a theoretical analysis of the problem of hematocrit reduction (due to plasma skimming) in a capillary that emerges from an artery making an angle α with the parent artery. The analysis bears the potential to explore a variety of information regarding some phenomenological aspects of this important physiological problem. The flow is considered to consist of three distinct phases, viz., the peripheral plasma layer, the cell-depleted middle layer, and the core region which usually has a high concentration of erythrocytes. This study deals with both steady and pulsatile flow of blood, which is treated as a non-Newtonian fluid of Herschel–Bulkley type. A computational procedure is developed for a quantitative measure of the velocity profile, the volumetric flow rate, and the hematocrit of blood in a specific situation. The procedure also gives us an opportunity to examine the nature of variation of these important hemodynamic factors; this observation holds true irrespective of whether the flow of blood is steady or pulsatile. The study reveals that the velocity of blood in the parent artery reduces when the fluid index/yield stress increases. It is further revealed that the volumetric flow rate of blood in the capillary also decreases with an increase in the value of the fluid index/yield stress of blood.

Computation of hemodynamics in the left coronary artery with variable angulations

The purpose of this study was to investigate the hemodynamic effect of variations in the angulations of the left coronary artery, based on simulated and realistic coronary artery models. Twelve models consisting of four realistic and eight simulated coronary artery geometries were generated with the inclusion of left main stem, left anterior descending and left circumflex branches. The simulated models included various coronary artery angulations, namely, 15°, 30°, 45°, 60°, 75°, 90°, 105° and 120°. The realistic coronary angulations were based on selected patient's data with angles ranging from narrow angles of 58° and 73° to wide angles of 110° and 120°. Computational fluid dynamics analysis was performed to simulate realistic physiological conditions that reflect the in vivo cardiac hemodynamics. The wall shear stress, wall shear stress gradient, velocity flow patterns and wall pressure were measured in simulated and realistic models during the cardiac cycle. Our results showed that a disturbed flow pattern was observed in models with wider angulations, and wall pressure was found to reduce when the flow changed from the left main stem to the bifurcated regions, based on simulated and realistic models. A low wall shear stress gradient was demonstrated at left bifurcations with wide angles. There is a direct correlation between coronary angulations and subsequent hemodynamic changes, based on realistic and simulated models. Further studies based on patients with different severities of coronary artery disease are required to verify our results.

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.

Non-Newtonian effects of blood flow on hemodynamics in distal vascular graft anastomoses

Non-Newtonian fluid flow in a stenosed coronary bypass is investigated numerically using the Carreau-Yasuda model for the shear thinning behavior of the blood. End-to-side coronary bypass anastomosis is considered in a simplified model geometry where the host coronary artery has a 75% severity stenosis. Different locations of the bypass graft to the stenosis and different flow rates in the graft and in the host artery are studied. Particular attention is given to the non-Newtonian effect of the blood on the primary and secondary flow patterns in the host coronary artery and the wall shear stress (WSS) distribution there. Interaction between the jet flow from the stenosed artery and the flow from the graft is simulated by solving the three-dimensional Navier-Stokes equation coupled with the non-Newtonian constitutive model. Results for the non-Newtonian flow, the Newtonian flow and the rescaled Newtonian flow are presented. Significant differences in axial velocity profiles, secondary flow streamlines and WSS between the non-Newtonian and Newtonian fluid flows are revealed. However, reasonable agreement between the non-Newtonian and the rescaled Newtonian flows is found. Results from this study support the view that the residual flow in a partially occluded coronary artery interacts with flow in the bypass graft and may have significant hemodynamic effects in the host vessel downstream of the graft. Non-Newtonian property of the blood alters the flow pattern and WSS distribution and is an important factor to be considered in simulating hemodynamic effects of blood flow in arterial bypass grafts.

Numerical investigation of the non-Newtonian pulsatile blood flow in a bifurcation model with a non-planar branch

The pulsatile flow of non-Newtonian fluid in a bifurcation model with a non-planar daughter branch is investigated numerically by using the Carreau-Yasuda model to take into account the shear thinning behavior of the analog blood fluid. The objective of this study is to deal with the influence of the non-Newtonian property of fluid and of out-of-plane curvature in the non-planar daughter vessel on wall shear stress (WSS), oscillatory shear index (OSI), and flow phenomena during the pulse cycle. The non-Newtonian property in the daughter vessels induces a flattened axial velocity profile due to its shear thinning behavior. The non-planarity deflects flow from the inner wall of the vessel to the outer wall and changes the distribution of WSS along the vessel, in particular in systole phase. Downstream of the bifurcation, the velocity profiles are shifted toward the flow divider, and low WSS and high shear stress temporal oscillations characterized by OSI occur on the outer wall region of the daughter vessels close to the bifurcation. Secondary motions become stronger with the addition of the out-of-plane curvature induced by the bending of the vessel, and the secondary flow patterns swirl along the non-planar daughter vessel. A significant difference between the non-Newtonian and the Newtonian pulsatile flow is revealed during the pulse cycle; however, reasonable agreement between the non-Newtonian and the rescaled Newtonian flow is found. Calculated results for the pulsatile flow support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.

Numerical investigation of the non-Newtonian blood flow in a bifurcation model with a non-planar branch

The non-Newtonian fluid flow in a bifurcation model with a non-planar daughter branch is investigated by using finite element method to solve the three-dimensional Navier-Stokes equations coupled with a non-Newtonian constitutive model, in which the shear thinning behavior of the blood fluid is incorporated by the Carreau-Yasuda model. The objective of this study is to investigate the influence of the non-Newtonian property of fluid as well as of curvature and out-of-plane geometry in the non-planar daughter vessel on wall shear stress (WSS) and flow phenomena. In the non-planar daughter vessel, the flows are typified by the skewing of the velocity profile towards the outer wall, creating a relatively low WSS at the inner wall. In the downstream of the bifurcation, the velocity profiles are shifted towards the flow divider. The low WSS is found at the inner walls of the curvature and the lateral walls of the bifurcation. Secondary flow patterns that swirl fluid from the inner wall of curvature to the outer wall in the middle of the vessel are also well documented for the curved and bifurcating vessels. The numerical results for the non-Newtonian fluid and the Newtonian fluid with original Reynolds number and the corresponding rescaled Reynolds number are presented. Significant difference between the non-Newtonian flow and the Newtonian flow is revealed; however, reasonable agreement between the non-Newtonian flow and the rescaled Newtonian flow is found. Results of this study support the view that the non-planarity of blood vessels and the non-Newtonian properties of blood are an important factor in hemodynamics and may play a significant role in vascular biology and pathophysiology.

Reconstruction of carotid bifurcation hemodynamics and wall thickness using computational fluid dynamics and MRI

A thorough understanding of the relationship between local hemodynamics and plaque progression has been hindered by an inability to prospectively monitor these factors in vivo in humans. In this study a novel approach for noninvasively reconstructing artery wall thickness and local hemodynamics at the human carotid bifurcation is presented. Three-dimensional (3D) models of the lumen and wall boundaries, from which wall thickness can be measured, were reconstructed from black-blood magnetic resonance imaging (MRI). Along with time-varying inlet/outlet flow rates measured via phase contrast (PC) MRI, the lumen boundary was used as input for computational fluid dynamic (CFD) simulation of the subject-specific flow patterns and wall shear stresses (WSSs). Results from a 59-year-old subject with early, asymptomatic carotid artery disease show good agreement between simulated and measured velocities, and demonstrate a correspondence between wall thickening and low and oscillating shear at the carotid bulb. High shear at the distal internal carotid artery (ICA) was also colocalized with higher WSS; however, a quantitative general relationship between WSS and wall thickness was not found. Similar results were obtained from a 23-year-old normal subject. These findings represent the first direct comparison of hemodynamic variables and wall thickness at the carotid bifurcation of human subjects. The noninvasive nature of this image-based modeling approach makes it ideal for carrying out future prospective studies of hemodynamics and plaque development or progression in otherwise healthy subjects.

Experimental analysis of flow calculations based on HRCT imaging of individual bifurcations

Flow simulations in airways and arteries allow the non-invasive study of transport and deposition processes in these vessel systems. Individual vessel geometries as input for such simulations are highly desirable. Computed tomography (CT) permits the acquisition of binary data to reconstruct such geometries. To prove the suitability of this reconstruction method, we compared measured with simulated velocities within model bifurcations. Particle image velocimetry was applied to measure flow velocities. Numerical simulations of these velocities were carried out by using the CT data of the same models as input to flow calculations (CFD). Within the resolution limits good agreement between measured and simulated velocities was found. For the smallest bifurcation (tube diameter: 2 mm) the agreement was less, indicating a methodical limitation by the actual resolution of the CT-scan technique. The study showed that a combination of CT and CFD can be considered as an appropriate step towards realistic simulations of particle transportation and deposition in individual geometries of the respiratory or cardiovascular systems.

The influence of the non-Newtonian properties of blood on the flow in large arteries: unsteady flow in a 90 degrees curved tube

A numerical and experimental investigation of unsteady entry flow in a 90 degrees curved tube is presented to study the impact of the non-Newtonian properties of blood on the velocity distribution. The time-dependent flow rate for the Newtonian and the non-Newtonian blood analog fluid were identical. For the numerical computation, a Carreau-Yasuda model was employed to accommodate the shear thinning behavior of the Xanthan gum solution. The viscoelastic properties were not taken into account. The experimental results indicate that significant differences between the Newtonian and non-Newtonian fluid are present. The numerical results for both the Newtonian and the non-Newtonian fluid agree well with the experimental results. Since viscoelasticity was not included in the numerical code, shear thinning behavior of the blood analog fluid seems to be the dominant non-Newtonian property, even under unsteady flow conditions. Finally, a comparison between the non-Newtonian fluid model and a Newtonian fluid at a rescaled Reynolds number is presented. The rescaled Reynolds number, based on a characteristic rather than the high-shear rate viscosity of the Xanthan gum solution, was about three times as low as the original Reynolds number. Comparison reveals that the character of flow of the non-Newtonian fluid is simulated quite well by using the appropriate Reynolds number.

A numerical study of magnetic resonance images of pulsatile flow in a two dimensional carotid bifurcation: a numerical study of MR images

A numerical method to simulate magnetic resonance angiographic images is proposed. The new method greatly simplifies the calculation of the average phase in a voxel, the bottleneck of previous simulations, and reduces the computation time by more than a factor of 5. Both the Navier-Stokes and the Bloch equations are solved on the same mesh to obtain the distributions of the modulus and phase of the magnetization. The data in the frequency domain are reordered according to the gating strategy to generate the final images. Pulsatile flow through a 2D normal carotid bifurcation is considered as a test case. Images for magnetic resonance angiography with an uncompensated gradient waveform, a velocity-compensated gradient waveform and an uncompensated short-TE gradient waveform are compared. Systolic gating images are shown to have degraded image quality. Images acquired with diastolic-gating have little variation in magnetization strength throughout the pulsatile cycle and provide a better representation of the vessel lumen.

Association between secondary flow in models of the aorto-celiac junction and subendothelial macrophages in the normal rabbit

In order to examine the association between arterial fluid dynamics and the distribution of subendothelial macrophages in the normal rabbit aorta, steady and pulsatile particle flow visualization was performed in a geometrically realistic model of the rabbit aorto-celiac junction region. Over a range of aorto-celiac steady flow ratios, particle pathlines along the upstream lateral aortic walls curved to enter the celiac orifice, while two asymmetric regions of reversing spiral secondary flow originated along the downstream lateral portions of the orifice flow divider. These regions increased in size as either the Reynolds number or flow into the celiac artery increased. In pulsatile flow studies, particles along the lateral aortic walls near the celiac orifice began to spiral into the branch during peak systole. During systolic deceleration, the size of this spiral flow region increased as particles reversed direction to enter the celiac orifice. This contrasted with flow patterns directly upstream and downstream of the orifice, which remained unidirectional throughout this period even along the distal lip of the orifice. The highest frequency of subendothelial white blood cells in the normal rabbit aorta was associated with regions where secondary flow patterns occurred, and where the orientation of endothelial cell nuclei deviated from the major direction of aortic flow. Secondary flow patterns may aid the accumulation of monocytes and macrophages about the lateral regions of the celiac artery flow divider by transporting monocytes to the walls, allowing them time to attach to the endothelial cells, or by stimulating the endothelial cells to express leukocyte adhesion molecules. These same regions are associated with increased endothelial permeability to low density lipoprotein and, under hypercholesterolemic conditions, lesion origination.

Simulation of pressure drop and energy dissipation for blood flow in a human fetal bifurcation

The pressure drop from the umbilical vein to the heart plays a vital part in human fetal circulation. The bulk of the pressure drop is believed to take place at the inlet of the ductus venosus, a short narrow branch of the umbilical vein. In this study a generalized Bernoulli formulation was deduced to estimate this pressure drop. The model contains an energy dissipation term and flow-scaled velocities and pressures. The flow-scaled variables are related to their corresponding spatial mean velocities and pressures by certain shape factors. Further, based on physiological measurements, we established a simplified, rigid-walled, three-dimensional computational model of the umbilical vein and ductus venosus bifurcation for stationary flow conditions. Simulations were carried out for Reynolds numbers and umbilical vein curvature ratios in their respective physiological ranges. The shape factors in the Bernoulli formulation were then estimated for our computational models. They showed no significant Reynolds number or curvature ratio dependency. Further, the energy dissipation in our models was estimated to constitute 24 to 31 percent of the pressure drop, depending on the Reynolds number and the curvature ratio. The energy dissipation should therefore be taken into account in pressure drop estimates.

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.

The assignment of velocity profiles in finite element simulations of pulsatile flow in arteries

In this paper we present a new method for the assignment of pulsatile velocity profiles as input boundary conditions in finite element models of arteries. The method is based on the implementation of the analytical solution for developed pulsatile flow in a rigid straight tube. The analytical solution provides the fluid dynamics of the region upstream from the fluid domain to be investigated by means of the finite element approach. In standard fluid dynamics finite element applications, the inlet developed velocity profiles are achieved assuming velocity boundary conditions to be easily implementable-such as flat or parabolic velocity profiles-applied to a straight tube of appropriate length. The tube is attached to the inflow section of the original fluid domain so that the flow can develop fully. The comparison between the analytical solution and the traditional numerical approach indicates that the analytical solution has some advantages over the numerical one. Moreover, the results suggest that subroutine employment allows a consistent reduction in solving time especially for complex fluid dynamic model, and significantly decreases the storage and memory requirements for computations.

Experimental investigation of pulsatile flows in tubes

Based on cam-piston-valve arrangement, a mechanical pulsatile flow generator is designed to investigate sinusoidal flow and other types of pulsatile flow in straight rigid tube. Measurement reveals the relation between pressure gradient and flow rate. Numerical simulation using the k-epsilon turbulence model are carried out to compare the pulsatile flow produced by the generator with a sinusoidal flow and a physiological flow in a rigid tube. The results show that the pulsatile flow generated has similar dynamic properties to the physiological flow. Hence, the present setup can be used for in-vitro investigation of biofluid phenomena.

Hemodynamics of an artery with mild stenosis

In this study the hemodynamics in the early stages of the atherosclerotic process--when a neointimal hyperplasia or an intimal fibrocellular hypertrophy takes place--is theoretically investigated. A local, slight increase in the wall thickness of a canine femoral artery is simulated using an original two-dimensional mathematical model of arterial hemodynamics and the effects induced on the velocity field by the simulated mild stenosis--only 2% of area reduction--are analysed. The model incorporates: fluid non-linear inertial forces, viscoelastic wall motion, anatomical taper, unsteady flow, pressure propagation and reflections on both the proximal and distal vessel ends. Two different physiological pulsatile flows are considered: a basal flow condition and a light vasodilation state inducing in the vessel segment a limited increase in mean flow (50%). The distribution along the vessel during the cardiac cycle of both the velocity profile and wall shear stress, are shown. The shape of velocity distributions is strongly perturbed by the stenosis and disturbances are clearly evident whatever instant of the cardiac cycle is considered. After vasodilatation, during the phase of systolic deceleration, a vortex circulation appears in the post-stenotic region. The vortex persists for the whole diastolic phase, causing a very strong stress at the arterial wall: wall shear stress in the distal part of the simulated mild stenosis is at least five times the basal value. The reported results provide a coherent explanation of the critical role that hemodynamic factors may play in the early stages of atherogenic process.

Errors in the estimation of arterial wall shear rates that result from curve fitting of velocity profiles

An analysis was performed to determine the error that results from the estimation of the wall shear rates based on linear and quadratic curve-fittings of the measured velocity profiles. For steady, fully developed flow in a straight vessel, the error for the linear method is linearly related to the distance between the probe and the wall, dr1, and the error for the quadratic method is zero. With pulsatile flow, especially a physiological pulsatile flow in a large artery, the thickness of the velocity boundary layer, delta is small, and the error in the estimation of wall shear based on curve fitting is much higher than that with steady flow. In addition, there is a phase lag between the actual shear rate and the measured one. In oscillatory flow, the error increases with the distance ratio dr1/delta and, for a quadratic method, also with the distance ratio dr2/dr1, where dr2 is the distance of the second probe from the wall. The quadratic method has a distinct advantage in accuracy over the linear method when dr1/delta << 1, i.e. when the first velocity point is well within the boundary layer. The use of this analysis in arterial flow involves many simplifications, including Newtonian fluid, rigid walls, and the linear summation of the harmonic components, and can provide more qualitative than quantitative guidance.