The effect of blood viscoelasticity on pulsatile flow in stationary and axially moving tubes

An analytical solution for pulsatile flow of a generalized Maxwell fluid in straight rigid tubes, with and without axial vessel motion, has been used to calculate the effect of blood viscoelasticity on velocity profiles and shear stress in flows representative of those in the large arteries. Measured bulk flow rate Q waveforms were used as starting points in the calculations for the aorta and femoral arteries, from which axial pressure gradient ▿P waves were derived that would reproduce the starting Q waves for viscoelastic flow. The ▿P waves were then used to calculate velocity profiles for both viscoelastic and purely viscous flow. For the coronary artery, published ▿P and axial vessel acceleration waveforms were used in a similar procedure to determine the separate and combined influences of viscoelasticity and vessel motion.Differences in local velocities, comparing viscous flow to viscoelastic flow, were in all cases less than about 2% of the peak local velocity. Differences in peak wall shear stress were less than about 3%.In the coronary artery, wall shear stress differences between viscous and viscoelastic flow were small, regardless of whether axial vessel motion was included. The shape of the wall shear stress waveform and its difference, however, changed dramatically between the stationary and moving vessel cases. The peaks in wall shear stress difference corresponded with large temporal gradients in the combined driving force for the flow.     

Frequency dependence of dynamic curvature effects on flow through coronary arteries

The flow through a curved tube model of a coronary artery was investigated computationally to determine the importance of time-varying curvature on flow patterns that have been associated with the development of atherosclerosis. The entry to the tube was fixed while the radius of curvature varied sinusoidally in time at a frequency of 1 or 5 Hz. Angiographic data from other studies suggest that the radius of curvature waveform contains significant spectral content up to 6 Hz. The overall flow patterns were similar to those observed in stationary curved tubes; velocity profile skewed toward the outer wall, secondary flow patterns, etc. The effects of time-varying curvature on the changes in wall shear rate were expressed by normalizing the wall shear rate amplitude with the shear rate calculated at the static mean radius of curvature. It was found that the wall shear rate varied as much as 94 percent of the mean wall shear rate at the mid wall of curvature for a mean curvature ratio of 0.08 and a 50 percent change in radius of curvature. The effects of 5 Hz deformation were not well predicted by a quasi-static approach. The maximum values of the normalized inner wall shear rate amplitude were found to scale well with a dimensionless parameter equivalent to the product of the mean curvature ratio (delta), normalized change in radius of curvature (epsilon), and a Womersley parameter (alpha). This parameter was less successful at predicting the amplitudes elsewhere in the tube, thus additional studies are necessary. The mean wall shear rate was well predicted with a static geometry. These results indicate that dynamic curvature plays an important role in determining the inner wall shear rates in coronary arteries that are subjected to deformation levels of epsilon delta alpha > 0.05. The effects were not always predictable with a quasi-static approach. These results provide guidelines for constructing more realistic models of coronary artery flow for atherogenesis research.     

Blood flow regulation in the cerebral microvasculature with an arcadal network: a numerical simulation

Blood flow regulation in the cerebral microvasculature with an arcadal network was investigated using a numerical simulation. A mathematical model for blood flow in the arcadal network, based on in vivo data of cat cerebral microvasculature and flow velocity was developed. The network model consists of 45 vessel segments and 25 branching points. To simulate microvascular response to blood flow, non-reactive (solid), cerebral arteriole-like, or skeletal muscle arteriole-like responses to wall shear stress were taken into account. Numerical calculation was carried out in the flow condition where the inlet (arterial) pressure was changed from 60 to 120 mmHg. Flow-rate in each efferent vessel and the mean flow-rate over all efferent vessels were evaluated for assessment of blood supply to the local area of cerebral tissue. The simulation demonstrated the wall shear stress-induced vasodilation in the arcadal network worked to maintain the blood flow at a constant level with pressure variable in a wide range. It is suggested that an individual microvessel (segment) should join in the regulatory process of flow, interacting with other microvessels (cooperative regulation).     

Dependence of adhesive behavior of neutrophils on local fluid dynamics in a region with recirculating flow

We have recently described patterns of adhesion of different types of leukocytes downstream of a backward facing step. Here the predicted fluid dynamics in channels incorporating backward facing steps are described, and related to the measured velocities of flowing cells, patterns of attachment and characteristics of rolling adhesion for neutrophils perfused over P-selectin. Deeper (upstream depth 300 microm, downstream depth 600 microm, maximum wall shear stress approximately 0.1 Pa) and shallower (upstream depth 260 microm, downstream depth 450 microm, maximum wall shear stress approximately 0.3 Pa) channels were compared. Computational fluid dynamics (CFD) predicted the presence of vortices downstream of the steps, distances to reattachment of flow, local wall shear stresses and components of velocity parallel and perpendicular to the wall. Measurements of velocities of perfused neutrophils agreed well with predictions, and suggested that adhesion to P-selectin should be possible in the regions of recirculating flow, but not downstream in re-established flow in the high shear channel. When channels were coated with a P-selectin-Fc chimaera, neutrophils were captured from flow and immobilised. Capture showed local maxima around the reattachment points, but was absent elsewhere in the high shear chamber. In the low shear chamber there was depression of adhesion just beyond the reattachment point because of expansion of flow and depletion of neutrophils near the wall. Inside the recirculation zones, adhesion decreased approaching the step because of an increasing, vertically upward velocity component. When channels were coated with P-selectin, neutrophils rolled in all regions, but lifted off the surface as they rolled backwards into low shear regions near the step. Rolling velocity in the recirculation zone was independent of shear stress, possibly because of the effects of vertical lift. We conclude that while local wall shear stress influences adhesive behavior, delivery of cells to the wall and their behavior after capture also depend on components of flow perpendicular to the wall.     

Shear stress,vascular remodeling and neointimal formation

The role of shear stress in atherosclerosis has been well documented. However, its role in restenosis was underexposed. In this paper a novel in vivo measuring technique and several of its applications related to restenosis will be described. The technique consists of a combination of 3D reconstruction of blood vessels and computational fluid dynamics (CFD). The 3D imaging techniques use either of 3D intravascular ultrasound (IVUS) as a stand-alone technique or a fusion of biplane angiography and IVUS (ANGUS). CFD is applied in order to relate local shear stress distribution to the morphology of the vessel wall. In the applications of these techniques it will be demonstrated that shear stress plays a role in the prediction of neointimal formation in in-stent restenosis and in vascular remodeling after balloon angioplasty. Attempts to locally increase shear stress by a newly developed flow divider indicate that shear stress reduce in-stent neointimal formation by 50%.     

Numerical modelling of simulated blood flow in idealized composite arterial coronary grafts: transient flow

In composite arterial coronary grafts (CACGs), transport phenomena and geometry may considerably alter blood flow dynamics. CACGs aim at revascularizing pathological arteries according to the human anatomy. However, the exact mechanisms causing the failure of coronary bypass grafting are not yet well elucidated. In the present study, computational fluid dynamics (CFD) techniques are applied for the simulation of multi-branched CACGs under physiologically realistic inflow waveforms. The numerical solution is obtained by a finite-volume method formulated in non-orthogonal, curvilinear coordinates and a multi-grid approach. The geometrical models, consisting of idealized and rigid vessels, include the typical T- and a rather new pi-graft configuration. The stenotic effect is also investigated by comparing computational results for three different degrees of area constriction, namely 25%, 50% and 75%, as well as the case without stenosis. Different grafting distances and various inflow rate ratios are imposed, to give an insight into haemodynamical alterations of CACGs and to study the process of restenosis. The results focus on the interaction between the grafts and coronary flows in terms of spatial and temporal variations of velocity and wall shear stress (WSS) distribution. Prominent variations among the different geometries, concerning the velocity profiles and secondary flow motion, are shown. Moreover, the residual flow emerging from different degrees of area constriction shows that low and oscillating shear stresses may arise for even moderate stenotic fields.     

Numerical modeling of simulated blood flow in idealized composite arterial coronary grafts: steady state simulations

This paper presents a comparative study of simulated blood flow in different configurations of simplified composite arterial coronary grafts (CACGs). Even though the composite arterial grafting is increasingly used in cardiac surgery, it is still questionable whether or not the blood flow in such grafts can adequately meet the demands of the native myocardial circulation. A computational fluid dynamics (CFD) model was developed to conduct computer-based studies of simulated blood flow in four different geometric configurations of CACGs, corresponding to routinely used networks in cardiac surgery coronary grafts (T, Y, Pi and sequential). The flow was assumed three-dimensional, laminar and steady and the fluid as Newtonian, while the vessel walls were considered as inelastic and impermeable. It was concluded that local haemodynamics, practically described by velocity, pressure drop, wall shear stress (WSS) and flow rates, may be strongly influenced by the local geometry, especially at the anastomotic sites. The computations were made at mean flow rates of 37.5, 75 and 150ml/min. The side-branch outflow rates, computed for each bypass graft, showed noticeable differences. The results, which were found both qualitatively and quantitatively consistent with other studies, indicate that the Pi-graft exhibits significantly less uniform distribution of outflow rates than the other geometric configurations. Moreover, prominent variations in WSS and velocity distribution among the assessed CACGs were predicted, showing remarkable flow interactions among the arterial branches. The lowest shear stress regions were found on the lateral walls of bifurcations, which are predominantly susceptible to the occurrence of coronary artery disease (CAD). In contrast, the highest WSS were observed at the turn of the arterial branches.     

Breaking symmetry in non-planar bifurcations: distribution of flow and wall shear stress

Non-planarity in blood vessels is known to influence arterial flows and wall shear stress. To gain insight, computational fluid dynamics (CFD) has been used to investigate effects of curvature and out-of-plane geometry on the distribution of fluid flows and wall shear stresses in a hypothetical non-planar bifurcation. Three-dimensional Navier-Stokes equations for a steady state Newtonian fluid were solved numerically using a finite element method. Non-planarity in one of the two daughter vessels is found to deflect flow from the inner wall of the vessel to the outer wall and to cause changes in the distribution of wall shear stresses. Results from this study agree to experimental observations and CFD simulations in the literature, and support the view that non-planarity in blood vessels is a factor with important haemodynamic significance and may play a key role in vascular biology and pathophysiology.     

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.     

Circumferential vascular deformation after stent implantation alters wall shear stress evaluated with time-dependent 3D computational fluid dynamics models.

The success of vascular stents in the restoration of blood flow is limited by restenosis. Recent data generated from computational fluid dynamics (CFD) models suggest that stent geometry may cause local alterations in wall shear stress (WSS) that have been associated with neointimal hyperplasia and subsequent restenosis. However, previous CFD studies have ignored histological evidence of vascular straightening between circumferential stent struts. We tested the hypothesis that consideration of stent-induced vascular deformation may more accurately predict alterations in indexes of WSS that may subsequently account for histological findings after stenting. We further tested the hypothesis that the severity of these alterations in WSS varies with the degree of vascular deformation after implantation. Steady-state and time-dependent simulations of three-dimensional CFD arteries based on canine coronary artery measurements of diameter and blood flow were conducted, and WSS and WSS gradients were calculated. Circumferential straightening introduced areas of high WSS between stent struts that were absent in stented vessels of circular cross section. The area of vessel exposed to low WSS was dependent on the degree of circumferential vascular deformation and axial location within the stent. Stents with four vs. eight struts increased the intrastrut area of low WSS in vessels, regardless of cross-sectional geometry. Elevated WSS gradients were also observed between struts in vessels with polygonal cross sections. The results obtained using three-dimensional CFD models suggest that changes in vascular geometry after stent implantation are important determinants of WSS distributions that may be associated with subsequent neointimal hyperplasia.     

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.     

Effect of inlet velocity profiles on patient-specific computational fluid dynamics simulations of the carotid bifurcation

Patient-specific computational fluid dynamics (CFD) is a powerful tool for researching the role of blood flow in disease processes. Modern clinical imaging technology such as MRI and CT can provide high resolution information about vessel geometry, but in many situations, patient-specific inlet velocity information is not available. In these situations, a simplified velocity profile must be selected. We studied how idealized inlet velocity profiles (blunt, parabolic, and Womersley flow) affect patient-specific CFD results when compared to simulations employing a "reference standard" of the patient's own measured velocity profile in the carotid bifurcation. To place the magnitude of these effects in context, we also investigated the effect of geometry and the use of subject-specific flow waveform on the CFD results. We quantified these differences by examining the pointwise percent error of the mean wall shear stress (WSS) and the oscillatory shear index (OSI) and by computing the intra-class correlation coefficient (ICC) between axial profiles of the mean WSS and OSI in the internal carotid artery bulb. The parabolic inlet velocity profile produced the most similar mean WSS and OSI to simulations employing the real patient-specific inlet velocity profile. However, anatomic variation in vessel geometry and the use of a nonpatient-specific flow waveform both affected the WSS and OSI results more than did the choice of inlet velocity profile. Although careful selection of boundary conditions is essential for all CFD analysis, accurate patient-specific geometry reconstruction and measurement of vessel flow rate waveform are more important than the choice of velocity profile. A parabolic velocity profile provided results most similar to the patient-specific velocity profile.     

Experimental determination of wall shear rate in canine carotid arteries perfused in vitro

The mathematical model of Hung (Tsai and Hung, 1984) is employed to determine the wall shear rate acting on canine carotid arteries perfused in vitro. Model equations for pulsatile flow in a deformable vessel are coupled with experimental data of dynamic pressure drop, flow rate, vessel radius and radial wall motion. Derived quantities, e.g. velocity profiles and wall shear, are obtained for vessels exposed to 'normotensive' hemodynamics, 'hypertension' simulations and perfusions in which the compliance of the vessel wall is deliberately altered. Our results indicate that wall shear varies markedly as a function of the hemodynamic environment. The effects of vessel radius vs flow rate on the development of wall shear are also demonstrated. It is found that convective processes correlate with the magnitude of wall shear in the 'hypertension' simulations. The present findings and complementary published data may explain, at least in part, the variations in vessel wall transport and endothelial cell biology we observe as a function of the hemodynamic environment. For example we have documented that the exposure of canine carotids to 'hypertensive' (vs 'normotensive') hemodynamics is associated with an increased flux of lipoproteins (LDL) into the intima and luminal media. Alternations in wall compliance, on the other hand, profoundly influence endothelial shape, orientation and cytoskeletal array.     

MRI and CFD studies of pulsatile flow in healthy and stenosed carotid bifurcation models

Pulsatile flow was studied in physiologically realistic models of a normal and a moderately stenosed (30% diameter reduction) human carotid bifurcation. Time-resolved velocity measurements were made using magnetic resonance imaging, from which wall shear stress (WSS) vectors were calculated. Velocity measurements in the inflow and outflow regions were also used as boundary conditions for a computational fluid dynamics (CFD) model. Experimental flow patterns and derived WSS vectors were compared qualitatively with the corresponding CFD predictions. In the stenosed phantom, flow in the bulb region of the "internal carotid artery" was concentrated along the outer wall, with a region of low and recirculating flow near the inner wall. In the normal phantom, the converse was found, with a low flow region near the outer wall of the bulb. Time-averaged WSS and oscillatory shear index were also markedly different for the two phantoms.     

Flow in t-bifurcations: Effect of the sharpness of the flow divider

The local geometry of a bifurcation has been hypothesized to be a potential geometrical risk factor for the development of atherosclerosis. While flow division and branch area ratios clearly affect the flow field, the importance of the flow divider shape is not as clear. A fast spectral element computational fluid mechanics (CFD) solver was used to simulate flow through 90 ° T-bifurcations with three different flow divider shapes. Other factors, such as flow partition, area ratio, and bifurcation angle, were kept constant. A Reynolds number range of 15 to 350 was studied to bracket experimental results in the literature. The variation in the sharpness of the corners had a dramatic effect on both the flow field and wall shear stress distribution in the side branch, but little effect on the flow in the main tube. The magnitude of reverse velocities and wall shear stress in the side branch increased linearly over a physiological range of Reynolds number and corner shape. This paper verifies the accuracy and usefulness of spectral element CFD in studying three-dimensional hemodynamics.     

Blood flow patterns in the proximal human coronary arteries: relationship to atherosclerotic plaque occurrence

Atherosclerotic plaques in human coronary arteries are focal manifestations of systemic disease, and biomechanical factors have been hypothesized to contribute to plaque genesis and localization. We developed a computational fluid dynamics (CFD) model of the ascending aorta and proximal sections of the right and left coronary arteries of a normal human subject using computed tomography (CT) and magnetic resonance imaging (MRI) and determined the pulsatile flow field. Results demonstrate that flow patterns in the ascending aorta contribute to a pro-atherosclerotic flow environment, specifically through localization of low and oscillatory wall shear stress in the neighborhood of coronary orifices. Furthermore, these patterns differ in their spatial distribution between right and left coronary arteries. Entrance effects of aortic flow diminish within two vessel diameters. We examined relationships between spatial distributions of wall shear stress and reports of plaque occurrence in the literature. Results indicate low wall shear stress is co-located with increased incidence of lesions, and higher wall shear stresses are associated with lesion-resistant areas. This investigation does not consider plaque progression or advanced lesions, inasmuch as the CFD model was developed from a normal individual and the clinical data used for comparisons were obtained from autopsy specimens of subjects who died from non-cardiovascular causes. The data reported are consistent with the hypothesis that low wall shear stress is associated with the localization of atherosclerotic lesions, and the results demonstrate the importance of aortic flow on flow patterns in the proximal segments of the coronary arteries.     

Compound Ex Vivo and In Silico Method for Hemodynamic Analysis of Stented Arteries

Hemodynamic factors such as low wall shear stress have been shown to influence endothelial healing and atherogenesis in stent-free vessels. However, in stented vessels, a reliable quantitative analysis of such relations has not been possible due to the lack of a suitable method for the accurate acquisition of blood flow. The objective of this work was to develop a method for the precise reconstruction of hemodynamics and quantification of wall shear stress in stented vessels. We have developed such a method that can be applied to vessels stented in or ex vivo and processed ex vivo. Here we stented the coronary arteries of ex vivo porcine hearts, performed vascular corrosion casting, acquired the vessel geometry using micro-computed tomography and reconstructed blood flow and shear stress using computational fluid dynamics. The method yields accurate local flow information through anatomic fidelity, capturing in detail the stent geometry, arterial tissue prolapse, radial and axial arterial deformation as well as strut malapposition. This novel compound method may serve as a unique tool for spatially resolved analysis of the relationship between hemodynamic factors and vascular biology. It can further be employed to optimize stent design and stenting strategies.     

Physiological flow analysis in significant human coronary artery stenoses

To evaluate the local hemodynamics in flow limiting coronary lesions, computational hemodynamics was applied to a group of patients previously reported by Wilson et al. (1988) with representative pre-angioplasty stenosis geometry (minimal lesion size d(m)=0.95 mm; 68% mean diameter stenosis) and with measured values of coronary flow reserve (CFR) in the abnormal range (2.3+/-0.1). The computations were at mean flow rates (Q) of 50, 75 and 100 ml/min (the limit of our converged calculations). Computed mean pressure drops Deltap were approximately 9 mmHg for basal flow (50 ml/min), approximately 27 mmHg for elevated flow (100 ml/min) and increased to an extrapolated value of approximately 34 mmHg for hyperemic flow (115 ml/min), which led to a distal mean coronary pressure p(rh) of approximately 55 mmHg, a level known to cause ischemia in the subendocardium (Brown et al., 1984), and consistent with the occurrence of angina in the patients. Relatively high levels of wall shear stress were computed in the narrow throat region and ranged from about 600 to 1500 dyn/cm(2), with periodic (phase shifted) peak systolic values of about 3500 dyn/cm(2). In the distal vessel, the interaction between the separated shear layer wave, convected downstream by the core flow, and the wall shear layer flow, led to the formation of vortical flow cells along the distal vessel wall during the systolic phase where Reynolds numbers Re(e)(t) were higher. During the phasic vortical mode observed at both basal and elevated mean flow rates, wide variations in distal wall shear stress occurred, distal transmural pressures were depressed below throat levels, and pressure recovery was larger farther along the distal vessel. Along the constriction (convergent) and throat segments of the lesion the pulsatile flow field was principally quasi-steady before flow separation occurred. The flow regimes were complex in the narrow mean flow Reynolds number range Re(e)=100-230 and a frequency parameter of alphae=2.25. The shear layer flow disturbances diminished in strength due to viscous damping along the distal vessel at these relatively low values of Re(e), typical of flow through diseased epicardial coronary vessels. The distal hyperemic flow field was likely to be in an early stage of turbulent flow development during the peak systolic phase.