To what extent does a minimal atherosclerotic plaque alter the arterial wall shear stress distribution? A model study by an electrochemical method

An electrochemical surface shear stress measurement was applied to a model of very thin unilateral arterial stenosis (height of 1/8 of the model pipe diameter with very smooth surface). Three dimensional wall shear stress distribution was measured under steady flow field from a relatively low Reynolds number, Re = 270, to a high Reynolds number, Re = 1200. There was a characteristic high and low wall shear distribution pattern around the stenosis. There were also remarkable high shear stress areas on the opposite wall and both side walls of the stenosis. It was clearly shown that three dimensional structure of the flow field, hence, the wall shear stress distribution, is affected by a minimal change on the arterial wall.     

Wall shear stress estimates in coronary artery constrictions.

Wall shear stress estimates from laminar boundary layer theory were found to agree fairly well with the magnitude of shear stress levels along coronary artery constrictions obtained from solutions of the Navier Stokes equations for both steady and pulsatile flow. The relatively simple method can be used for in vivo estimates of wall shear stress in constrictions by using a vessel shape function determined from a coronary angiogram, along with a knowledge of the flow rate.     

Measured wall shear stress distribution pattern upstream and downstream of a unilateral stenosis by an electrochemical method

An electrochemical surface shear stress measurement was applied to a model of unilateral arterial stenosis. The unilateral stenosis model was made up of a removable stenosis plug, in an electrochemical shear stress measurement test section with 100 cathodes. Three dimensional wall shear stress distribution was measured under steady flow field. At a relatively low Reynolds number, Re = 270, there was a characteristic high and low wall shear distribution pattern downstream of the unilateral stenosis. There were also remarkable high shear stress areas on the opposite wall up- and downstream, and both side walls of the stenosis upstream. It was clearly shown that detailed three dimensional structure of the flow field must be studied in order to correlate it to pathological findings.     

Flow-induced Wall Shear Stress in Abdominal Aortic Aneurysms: Part II - Pulsatile Flow Hemodynamics

In continuing the investigation of AAA hemodynamics, unsteady flow-induced stresses are presented for pulsatile blood flow through the double-aneurysm model described in Part I. Physiologically realistic aortic blood flow is simulated under pulsatile conditions for the range of time-average Reynolds numbers 50 h Re m h 300. Hemodynamic disturbance is evaluated for a modified set of indicator functions which include wall pressure ( p w ), wall shear stress ( w ), Wall Shear Stress Gradient (WSSG), time-average wall shear stress ( w *), and time-average Wall Shear Stress Gradient WSSG *. At peak flow, the highest shear stress and WSSG levels are obtained at the distal end of both aneurysms, in a pattern similar to that of steady flow. The maximum values of wall shear stresses and wall shear stress gradients are evaluated as a function of the time-average Reynolds number resulting in a fourth order polynomial correlation. A comparison between numerical predictions for steady and pulsatile flow is presented, illustrating the importance of considering time-dependent flow for the evaluation of hemodynamic indicators.     

Flow-induced wall shear stress in abdominal aortic aneurysms: Part I--steady flow hemodynamics

Numerical predictions of blood flow patterns and hemodynamic stresses in Abdominal Aortic Aneurysms (AAAs) are performed in a two-aneurysm, axisymmetric, rigid wall model using the spectral element method. Homogeneous, Newtonian blood flow is simulated under steady conditions for the range of Reynolds numbers 10 < or =Re < or =2265. Flow hemodynamics are quantified by calculating the distributions of wall pressure (p(w)), wall shear stress (tau(w)), Wall Shear Stress Gradient (WSSG). A correlation between maximum values of hemodynamic stresses and Reynolds number is established, and the spatial distribution of WSSG is considered as a hemodynamic force that may cause damage to the arterial wall at an intermediate stage of AAA growth. The temporal distribution of hemodynamic stresses in pulsatile flow and their physical implications in AAA rupture are discussed in Part II of this paper.     

Experimental and numerical study of pulsatile flows through stenosis: wall shear stress analysis.

Different shapes of pulsatile flows through a model of stenosis are experimentally and numerically modeled to validate both methods and to determine the wall shear stress temporal evolution downstream from the stenosis. Two-dimensional velocity measurements are performed in a 75% severity stenosis using a pulsed Doppler ultrasonic velocimeter. Finite element package is employed for the transient numerical simulations. Polynomial method, based on the experimental velocity values, is proposed to determine the wall shear stress temporal evolution. There is a good agreement between the numerical and experimental results. The wall shear stress temporal analysis shows oscillating wall shear stress values during the cycle with high wall shear stress values at the throat of about 120 dyn/cm2, and low values downstream from the stenosis of about - 2.5 dyn/cm2. The key result of the study is that the presence of the stenosis leads the artery to work in a direction which is opposite to the direction of a healthy artery.     

Steady flow and wall compression in stenotic arteries: a three-dimensional thick-wall model with fluid-wall interactions.

Severe stenosis may cause critical flow and wall mechanical conditions related to artery fatigue, artery compression, and plaque rupture, which leads directly to heart attack and stroke. The exact mechanism involved is not well understood. In this paper a nonlinear three-dimensional thick-wall model with fluid-wall interactions is introduced to simulate blood flow in carotid arteries with stenosis and to quantify physiological conditions under which wall compression or even collapse may occur. The mechanical properties of the tube wall were selected to match a thick-wall stenosis model made of PVA hydrogel. The experimentally measured nonlinear stress-strain relationship is implemented in the computational model using an incremental linear elasticity approach. The Navier-Stokes equations are used for the fluid model. An incremental boundary iteration method is used to handle the fluid-wall interactions. Our results indicate that severe stenosis causes considerable compressive stress in the tube wall and critical flow conditions such as negative pressure, high shear stress, and flow separation which may be related to artery compression, plaque cap rupture, platelet activation, and thrombus formation. The stress distribution has a very localized pattern and both maximum tensile stress (five times higher than normal average stress) and maximum compressive stress occur inside the stenotic section. Wall deformation, flow rates, and true severities of the stenosis under different pressure conditions are calculated and compared with experimental measurements and reasonable agreement is found.     

Blood flow in abdominal aortic aneurysms: pulsatile flow hemodynamics

Numerical predictions of blood flow patterns and hemodynamic stresses in Abdominal Aortic Aneurysms (AAAs) are performed in a two-aneurysm, axisymmetric, rigid wall model using the spectral element method. Physiologically realistic aortic blood flow is simulated under pulsatile conditions for the range of time-averaged Reynolds numbers 50< or =Re(m)< or =300, corresponding to a range of peak Reynolds numbers 262.5< or =Re(peak) < or = 1575. The vortex dynamics induced by pulsatile flow in AAAs is characterized by a sequence of five different flow phases in one period of the flow cycle. Hemodynamic disturbance is evaluated for a modified set of indicator functions, which include wall pressure (p(w)), wall shear stress (tau(w)), and Wall Shear Stress Gradient (WSSG). At peak flow, the highest shear stress and WSSG levels are obtained downstream of both aneurysms, in a pattern similar to that of steady flow. Maximum values of wall shear stresses and wall shear stress gradients obtained at peak flow are evaluated as a function of the time-average Reynolds number resulting in a fourth order polynomial correlation. A comparison between predictions for steady and pulsatile flow is presented, illustrating the importance of considering time-dependent flow for the evaluation of hemodynamic indicators.     

Flow-induced wall shear stress in abdominal aortic aneurysms: Part II--pulsatile flow hemodynamics

In continuing the investigation of AAA hemodynamics, unsteady flow-induced stresses are presented for pulsatile blood flow through the double-aneurysm model described in Part I. Physiologically realistic aortic blood flow is simulated under pulsatile conditions for the range of time-average Reynolds numbers 50< or =Re(m) < or =300. Hemodynamic disturbance is evaluated for a modified set of indicator functions which include wall pressure (p(w)), wall shear stress (tau(w)), Wall Shear Stress Gradient (WSSG), time-average wall shear stress (tau(w)*), and time-average Wall Shear Stress Gradient WSSG*. At peak flow, the highest shear stress and WSSG levels are obtained at the distal end of both aneurysms, in a pattern similar to that of steady flow. The maximum values of wall shear stresses and wall shear stress gradients are evaluated as a function of the time-average Reynolds number resulting in a fourth order polynomial correlation. A comparison between numerical predictions for steady and pulsatile flow is presented, illustrating the importance of considering time-dependent flow for the evaluation of hemodynamic indicators.     

Pulsatile flow in a constricted channel.

A nonuniform channel is used as a simple model of a constricted arterial vessel. Flow patterns have been calculated for pulsatile flow with both sinusoidal and nonsinusoidal flow rates for a range of Reynolds number, Re, and Strouhal number, St. The results show that even for relatively low frequency flows a strong vortex wave will be generated with a complex wall shear stress distribution and peak values much greater than those found in steady or unsteady parallel flow. The vortex wave increases in strength with increasing Re and St, with its total length and wavelength independent of Re but inversely proportional to St. The form of the imposed flow rate is found to have an important effect on the flow and the shear stress distribution.     

Low Reynolds number turbulence modeling of blood flow in arterial stenoses.

Moderate and severe arterial stenoses can produce highly disturbed flow regions with transitional and or turbulent flow characteristics. Neither laminar flow modeling nor standard two-equation models such as the kappa-epsilon turbulence ones are suitable for this kind of blood flow. In order to analyze the transitional or turbulent flow distal to an arterial stenosis, authors of this study have used the Wilcox low-Re turbulence model. Flow simulations were carried out on stenoses with 50, 75 and 86% reductions in cross-sectional area over a range of physiologically relevant Reynolds numbers. The results obtained with this low-Re turbulence model were compared with experimental measurements and with the results obtained by the standard kappa-epsilon model in terms of velocity profile, vortex length, wall shear stress, wall static pressure, and turbulence intensity. The comparisons show that results predicted by the low-Re model are in good agreement with the experimental measurements. This model accurately predicts the critical Reynolds number at which blood flow becomes transitional or turbulent distal an arterial stenosis. Most interestingly, over the Re range of laminar flow, the vortex length calculated with the low-Re model also closely matches the vortex length predicted by laminar flow modeling. In conclusion, the study strongly suggests that the proposed model is suitable for blood flow studies in certain areas of the arterial tree where both laminar and transitional/turbulent flows coexist.     

Effects of elastic property of the wall on flow characteristics through arterial stenoses

Hemodynamic characteristics of blood flow through arterial stenoses are numerically investigated in this work. The blood is assumed as a Newtonian fluid and the pulsatile nature of flow is modeled by using measured values of the flowrate and pressure for the canine femoral artery. An isotropic elastic and incompressible material is assumed for the wall at each axial section, but a non-uniform distribution of the shear modulus in axial direction is used to model the high stiffness of the wall at the stenosis location. Full Navier equations for a thick wall are used as the governing equations for the wall displacements. A continuous grid extending over the flow field and the wall is considered and governing equations are transformed for use in the computational domain. Discretized forms of the transformed wall and flow equations, which are coupled through the boundary conditions at their interface, are obtained by control volume method and simultaneously solved using the well-known SIMPLER algorithm. To study the effects of wall deformability, solutions are obtained for both rigid and elastic walls. The results indicate that deformability of the wall causes an increase in the time average of pressure drop, but a decrease in the maximum wall shear stress. Displacement and stress distributions in the wall are presented.     

Large deforming buoyant embolus passing through a stenotic common carotid artery: a computational simulation

Arterial embolism is responsible for the death of lots of people who suffers from heart diseases. The major risk of embolism in upper limbs is that the ruptured particles are brought into the brain, thus stimulating neurological symptoms or causing the stroke. We presented a computational model using fluid-structure interactions (FSI) to investigate the physical motion of a blood clot inside the human common carotid artery. We simulated transportation of a buoyant embolus in an unsteady flow within a finite length tube having stenosis. Effects of stenosis severity and embolus size on arterial hemodynamics were investigated. To fulfill realistic nonlinear property of a blood clot, a rubber/foam model was used. The arbitrary Lagrangian-Eulerian formulation (ALE) and adaptive mesh method were used inside fluid domain to capture the large structural interfacial movements. The problem was solved by simultaneous solution of the fluid and the structure equations. Stress distribution and deformation of the clot were analyzed and hence, the regions of the embolus prone to lysis were localized. The maximum magnitude of arterial wall shear stress during embolism occurred at a short distance proximal to the throat of the stenosis. Through embolism, arterial maximum wall shear stress is more sensitive to stenosis severity than the embolus size whereas role of embolus size is more significant than the effect of stenosis severity on spatial and temporal gradients of wall shear stress downstream of the stenosis and on probability of clot lysis due to clot stresses while passing through the stenosis.     

Flow of couple stress fluid through stenotic blood vessels

The effects of an axially symmetric mild stenosis on the flow of blood, when blood is represented by a couple stress fluid model, have been studied. It is found that, for a fixed stenosis size, the resistance to flow and wall shear stress increase as the couple stress parameter eta decreases from unity. A comparison of the results with those of the Newtonian case shows that the magnitude of resistance to flow and wall shear under a given set of conditions, is greater in the case of the couple stress fluid model. It is seen that even in the case of a mild stenosis (19% area reduction), resistance to flow and wall shear values are increased over those for no stenosis by 60% and 62%, respectively, when compared with the case of a Newtonian fluid.     

Comparison of the hemodynamics in 6mm and 4-7 mm hemodialysis grafts by means of CFD

The aim of our study is to investigate with computational fluid dynamics (CFD) whether different arterial anastomotic geometries result in a different hemodynamics at the arterial (AA) and venous anastomosis (VA) of hemodialysis vascular access grafts. We have studied a 6mm graft (CD) and a 4-7 mm graft (TG). A validated three-dimensional CFD model is developed to simulate flow in the two graft types. Only the arterial anastomosis (AA) geometry differs. The boundary conditions applied are a periodic velocity signal at the arterial inlet and a periodic pressure wave at the venous outlet. Flow rate is set to 1,000 ml/min. The time dependent Navier-Stokes equations are solved. Wall shear stress (WSS), wall shear stress gradient (WSSG) and pressure gradient (PG) are calculated. Anastomotic flow is asymmetric although the anastomosis geometry is symmetric. The hemodynamic parameters, WSS, WSSG and PG, values at the suture line of the arterial anastomosis of the TG are at least twice as much as in the CD. Comparing the parameters at the two AA indicate that little flow rate increase introduces the risk of hemolysis in the TG whereas the CD is completely free of hemolysis. The hemodynamic parameter values at the venous anastomosis of the CD are 24 till 35% higher compared to the values of the TG. WSS values (> 3 Pa) in the VA are in the critical range for stenosis development in both graft geometries. The zones where the parameters reach extreme values correspond to the locations where intimal hyperplasia formation is reported in literature. In all anastomoses, the hemodynamic parameter levels are in the range where leucocytes and platelets get activated. Our simulations confirm clinical results where TG did not show a better outcome when compared to the CD.     

Haemodynamic assessment of human coronary arteries is affected by degree of freedom of artery movement

Abnormal haemodynamic parameters are associated with atheroma plaque progression and instability in coronary arteries. Flow recirculation, shear stress and pressure gradient are understood to be important pathogenic mediators in coronary disease. The effect of freedom of coronary artery movement on these parameters is still unknown. Fluid–structure interaction (FSI) simulations were carried out in 25 coronary artery models derived from authentic human coronaries in order to investigate the effect of degree of freedom of movement of the coronary arteries on flow recirculation, wall shear stress (WSS) and wall pressure gradient (WPG). Each FSI model had distinctive supports placed upon it. The quantitative and qualitative differences in flow recirculation, maximum wall shear stress (MWSS), areas of low wall shear stress (ALWSS) and maximum wall pressure gradient (MWPG) for each model were determined. The results showed that greater freedom of movement was associated with lower MWSS, smaller ALWSS, smaller flow recirculation zones and lower MWPG. With increasing percentage diameter stenosis (%DS), the effect of degree of freedom on flow recirculation and WSS diminished. Freedom of movement is an important variable to be considered for computational modelling of human coronary arteries, especially in the setting of mild to moderate stenosis.

Abbreviations: 3D: Three-dimensional; 3DR: Three-dimensional Reconstruction; 3D-QCA: Three-dimensional quantitative coronary angiography; ALWSS: Areas of low wall shear stress; CAD: Coronary artery disease; CFD: Computational fluid dynamics; %DS: Diameter stenosis percentage; EPCS: End point of counter-rotating streamlines; FSI: Fluid–structure interaction; IVUS: Intravascular ultrasound; LAD: Left anterior descending; MWSS: Maximum wall shear stress; SST: Shear stress transport; TAWSS: Time-averaged wall shear stress; WSS: wall shear stress; WPG: Wall pressure gradient; MWPG: Maximum wall pressure gradient; FFR: Fractional flow reserve; iFR: Instantaneous wave-free ratio     

Computational modeling of coupled blood-wall mass transport of LDL: effects of local wall shear stress

The work herein represents a novel approach for the modeling of low-density lipoprotein (LDL) transport from the artery lumen into the arterial wall, taking into account the effects of local wall shear stress (WSS) on the endothelial cell layer and its pathways of volume and solute flux. We have simulated LDL transport in an axisymmetric representation of a stenosed coronary artery, where the endothelium is represented by a three-pore model that takes into account the contributions of the vesicular pathway, normal junctions, and leaky junctions also employing the local WSS to yield the overall volume and solute flux. The fraction of leaky junctions is calculated as a function of the local WSS based on published experimental data and is used in conjunction with the pore theory to determine the transport properties of this pathway. We have found elevated levels of solute flux at low shear stress regions because of the presence of a larger number of leaky junctions compared with high shear stress regions. Accordingly, we were able to observe high LDL concentrations in the arterial wall in these low shear stress regions despite increased filtration velocity, indicating that the increase in filtration velocity is not sufficient for the convective removal of LDL.     

Mathematical modelling of flow through an irregular arterial stenosis.

A mathematical model of flow through an irregular arterial stenosis is developed. The model is two-dimensional and axi-symmetric with the stenosis outline obtained from a three-dimensional casting of a mildly stenosed artery. Agreement between modelled and experimental pressure drops (obtained from an axi-symmetric machined stenosis with the same profile) is excellent. Results are also obtained for a smooth stenosis model, similar to that used for most mathematical modelling studies. This model overestimates the pressure drop across the stenosis, as well as the wall shear stress and separation Reynolds number. Also, the smooth model predicts one instead of three recirculation zones present in the irregular model. The original stenosis is modified to increase the severity from 48 and 87% areal occlusion, while maintaining the same general shape. This has the effect of increasing the pressure drop by an order of magnitude and decreasing the number of recirculation zones to one, with a lower separation Reynolds number.     

Wall shear stress distribution along the cusp of a tri-leaflet prosthetic valve

High levels of wall shear stress on the surface of valvular cusps can cause mechanical damage to the blood cells and the cusp surfaces. The shear stresses are also responsible for mechanical failure of prosthetic heart valves. Quatitative measurements of wall shear stress in the vicinity of the leaflets are thus essential for diagnosis of suspected complications and provide important information for the design and fabrication of bioprosthetic heart valves. For this purpose we measured the velocity distribution along the inside wall of the cusps of a tri-leaflet heart valve with a two colour laser Doppler anemometer system. The wall shear stresses on the cusp surface were computed and found to range from 80 to 120 N/m² during the ejection phase. Wall shear stresses of up to 180 N/m² were measured in loci of cusp flexure and the accelerated boundary layer. The results of this study show a correlation between the high shear stress loci and the clinically (animal) observed regions of cusp calcification.     

Radial artery wall shear stress evaluation in patients with arteriovenous fistula for hemodialysis access

It has been extensively documented that changes in blood flow induce vascular remodeling and this phenomenon seems to be correlated to the shear forces imposed on the vessel wall by motion of blood. Wall shear stress, the tractive force that acts on the endothelium, has been shown to influence endothelial cell function. To study changes in wall shear stress that develop on the vessel wall upon changes of blood flow, we set up a technique that allows estimation of shear stress in the radial artery of patients on chronic hemodialysis therapy. The technique is based on color-flow Doppler examination of the radial artery before and after surgical creation of radiocephalic fistula for hemodialysis. Calculation of time function wall shear stress and blood flow rate in the radial artery is performed on the basis of arterial diameter, center-line velocity waveform and blood viscosity, using a numerical method developed according to Womersley's theory for pulsatile flow in tubes. The results presented confirm that the model developed is suitable for calculation of the wall shear stress that develops in the radial artery of patients before and after surgical creation of an arteriovenous fistula for hemodialysis. This methodology was developed for characterization of wall shear stress in the radial artery but may be well applied to other vessels that can be examined by echo-Doppler technique.