Flow in tubes and arteries--a comparison

The cardiovascular circulatory system of the human body can be compared with a network of tubes. It consists of a pump and a system of branched vessels. The arteries transport the blood to the periphery in a manner similar to that of a water supply network. It is important to know what kind of forces act upon "fittings", bends and bifurcations. It is also essential to assess whether the flow is laminar or turbulent, attached or separated. The flow should be optimized in such a manner as to minimize the drop in pressure. This means that no additional pressure loss due to separation or turbulence should occur, since such losses increase the pump power requirements. The loss appears in heating and acoustic energy. The necessary understanding of blood flow in human vessels is also of great interest to physicians since it is believed that the local flow behavior of blood determines the formation of atherosclerotic plaques. As in tubing systems, deposits in blood vessels are found close to bends and bifurcations. These deposits lead to impaired cerebral circulation and to myocardial infarction. A partial review of recent research into the details of flow behavior (like separation, stagnation and reattachment points) in bends and bifurcations of arterial models is presented. Studies involving steady and pulsatile flow conditions in rigid and elastic models with Newtonian and non-Newtonian fluids are shown here. The most important differences between blood vessels and tubes are discussed. This modern biofluidmechanical approach of detailed flow examination is compared with the more classical hemodynamic approach considering only gross features such as pressure loss coefficients.     

Flow investigations in a model of a three-dimensional human artery with Newtonian and non-Newtonian fluids. Part I

Together with biochemical factors, fluid mechanical factors play a role in atherogenesis and the deposition of blood platelets at bends and bifurcations in human arteries. Hence, flow patterns were investigated in a simplified 3-dimensional model of a human renal artery bifurcation using Newtonian (aqueous glycerol) and non-Newtonian (aqueous solution of polyacrylamide) fluids. Studies were carried out in steady as well as pulsatile flow at inflow Reynolds numbers of 498 and 951 with flow rate ratios main tube V1: right branch V4: left branch V3 of 1: 0.25: 0.25 and 1: 0.18: 0.18 respectively. The velocity distribution proximal and distal to the bifurcations was measured using a laser-Doppler anemometer. In steady flow, zones of flow separation and reverse flow were observed distal to the bifurcations. In pulsatile flow using non-Newtonian fluids, there was a significant enlargement of these zones. Differences between the Newtonian and non-Newtonian fluids occurred especially distal to the bifurcations. Shear stresses along all measuring positions were computed from the velocity gradients.     

The effects of non-Newtonian viscoelasticity and wall elasticity on flow at a 90 degrees bifurcation

To study the fundamentals of hemodynamics in arteries, the flow parameters: pulsatility, elasticity and non-Newtonian viscoelasticity were considered in detail in a 90 degrees-T-bifurcation of a rigid and elastic model. The velocity distribution 2.5 mm behind the bifurcation in the straight tube was measured with a laser-Doppler-anemometer. The fluid used was an aqueous glycerine solution and a viscoelastic Separan mixture. Flow visualization studies were done with a sheet of laser light in the plane of the bifurcation. The velocity distribution was measured for both steady and pulsatile flows with a laser-Doppler-anemometer in a backward scattered way. From the velocity measurements the shear gradients were calculated. Substantial differences were found in the flow behavior of Newtonian and non-Newtonian fluids, especially behind the bifurcation in the main tube, where secondary flows and flow separation started. Also, differences due to the elastic and rigid wall could be seen. Very high shear gradients were found in the flow between main flow and the separation zone which can lead to a damage of the blood cells.     

Studies of fluids simulating blood-like rheological properties and applications in models of arterial branches

We studied several non-Newtonian fluids to determine how closely they simulate the flow behavior of human blood. The viscous and viscoelastic properties of these fluids were compared with human blood samples in steady flow and transient flow Couette viscometers and in an oscillatory tube flow viscoelasticity analyzer. We examined: 1) A polyacrylamide suspension (Separan AP30 and AP45) to which we added 4% isopropanol and 0.01% magnesium chloride. 2) A suspension of 2% Dextran with 16% by weight biconcave disc-shaped particles simulating red blood cells. 3) 40% ghost cells prepared according to Dodge in Tri (hydroxymethyl) aminomethane. These ghost cells were used to simulate the two-phase flow behavior of blood. 4) A suspension of 5% Dextran (70,000) with 12% polystyrene particles (diameter of 1 micron) and 10 mMol calcium chloride. All these fluids closely approximate the flow behavior of blood and can be used in a variety of different experimental situations. To measure velocity distribution using a laser-Doppler-anemometer, we used fluids #1 and #3 in a rigid T-junction simulating the first septal branch of the left descending coronary artery. The measurements were done in steady and pulsatile flow experiments at different flow rate ratios. The fluids showed large differences in velocity profiles compared to Newtonian fluids.     

Effect of hematocrit on wall shear rate in oscillatory flow: do the elastic properties of blood play a role?

Porcine blood was used to examine the relationship between hematocrit levels and wall shear rate patterns in straight and curved artery models under fixed oscillatory flow conditions characteristic of larger arteries. It is demonstrated that porcine blood models both the viscous and elastic components of the 2 Hz complex viscosity of human blood quite accurately over a broad range of shear rates (1-1000 s-1) and hematocrits (20%-80%). For a fixed oscillatory flow waveform (Poiseuille peak shear rate = 168 s-1; mean shear rate 84 s-1), increases in hematocrit produced a decrease in the peak wall shear rate in both the straight and curved artery models and a corresponding decrease in wall shear rate reversal on the inside wall of the curved artery model. The same trends were also observed for oscillatory flows of aqueous glycerin solutions of increasing viscosity in the range of viscosity of the blood samples tested. Aqueous glycerin solutions produced wall shear rate waveforms of the same magnitude and shape as the porcine blood. This indicates that variations in the shear rate, and therefore the shear stress, were caused primarily by changes in the viscous and not the elastic properties of blood. The results suggest that simple Newtonian fluids may be sufficient for in vitro determination of the first order effects to be expected of human blood flow in large vessels having complex geometries and shear rates in or above the range of the present study.     

An introduction to biofluid mechanics--basic models and applications

Cardiovascular disease is the primary cause of morbidity and mortality in the western world. Complex hemodynamics play a critical role in the development of atherosclerosis and the processes of aging, as well as many other disease processes. Biofluid mechanics play a major role in the cardiovascular system and it is important to understand the forces and movement of blood cells and whole blood as well as the interaction between blood cells and the vessel wall. Fundamental fluid mechanical, which are important for the understanding of the blood flow in the cardiovascular circulatory system of the human body aspects are presented. Measurement techniques for model studies such as LDA, ultrasound, and MRI studies will be discussed. Viscosity and flow behavior changes specifically the creation of vortices and flow disturbances can be used to show how medication can influence flow behavior. Experiments have shown that hemodynamics may have a strong influence on the creation of aneurysms and varicose veins. Other factors such as vessel wall structure are also important. In preliminary studies, it has been demonstrated that geometry and elasticity of vessel walls help determine flow behavior. High velocity fluctuations indicate flow disturbances that should be avoided. Health care practitioners must understand fluid dynamic factors such as flow rate ratio, pressure and velocity gradients, and flow behavior, velocity distribution, shear stress on the wall and on blood cells. These mechanical factors are largely responsible for the deposit of blood cells and lipids, a leading cause of atherosclerosis. The interaction between blood cells and of the cells with the vessel, leads to the formation of plaques and agglomerations. These deposits are found predominantly at arterial bends and bifurcations where blood flow is disturbed, where a secondary flow is created, and where flow separation regions are found. Experiments on hemodynamic effects in elastic silicon rubber models of the cardiovascular system with flow wire, stents, or patches for vessel surgery will be discussed. These studies can be important in improving diagnostics and therapeutic applications.     

Three-layered Couette flow of polar fluid with non-zero particle spin boundary condition at the interfaces with applications to blood flow

In this paper, Couette flow of blood is modelled as a three-layered flow. The model basically consists of a core (red-cell suspension) and plasma (a Newtonian fluid) in the top (near the moving plate) and bottom (near the stationary plate) layers. Flow is assumed to be steady and laminar and fluids are incompressible. A spin boundary condition at the interfaces is used by introducing two parameters. Analytic expressions for velocity, total angular velocity and effective viscosity have been obtained and their variations with spin parameters S and s, layer thickness, coupling number N and characteristic length ratio L are computed and shown graphically. One of the important observations of the analysis is the permissible values of the coupling number N is between 0 and 1/square root2 (in the existing literature, the range of N is 0 to 1). The present model includes Couette flow of one and three-layered Newtonian fluids and one-layered polar fluid models as its special cases. Applications of the proposed model to blood flow have been briefly discussed.     

The influence of the non-Newtonian properties of blood on the flow in large arteries: steady flow in a carotid bifurcation model.

Laser Doppler anemometry experiments and finite element simulations of steady flow in a three dimensional model of the carotid bifurcation were performed to investigate the influence of non-Newtonian properties of blood on the velocity distribution. The axial velocity distribution was measured for two fluids: a non-Newtonian blood analog fluid and a Newtonian reference fluid. Striking differences between the measured flow fields were found. The axial velocity field of the non-Newtonian fluid was flattened, had lower velocity gradients at the divider wall, and higher velocity gradients at the non-divider wall. The flow separation, as found with the Newtonian fluid, was absent. In the computations, the shear thinning behavior of the analog blood fluid was incorporated through the Carreau-Yasuda model. The viscoelastic properties of the fluid were not included. A comparison between the experimental and numerical results showed good agreement, both for the Newtonian and the non-Newtonian fluid. Since only shear thinning was included, this seems to be the dominant non-Newtonian property of the blood analog fluid under steady flow conditions.     

Hemodynamic Analysis in an Idealized Artery Tree: Differences in Wall Shear Stress between Newtonian and Non-Newtonian Blood Models

Development of many conditions and disorders, such as atherosclerosis and stroke, are dependent upon hemodynamic forces. To accurately predict and prevent these conditions and disorders hemodynamic forces must be properly mapped. Here we compare a shear-rate dependent fluid (SDF) constitutive model, based on the works by Yasuda et al in 1981, against a Newtonian model of blood. We verify our stabilized finite element numerical method with the benchmark lid-driven cavity flow problem. Numerical simulations show that the Newtonian model gives similar velocity profiles in the 2-dimensional cavity given different height and width dimensions, given the same Reynolds number. Conversely, the SDF model gave dissimilar velocity profiles, differing from the Newtonian velocity profiles by up to 25% in velocity magnitudes. This difference can affect estimation in platelet distribution within blood vessels or magnetic nanoparticle delivery. Wall shear stress (WSS) is an important quantity involved in vascular remodeling through integrin and adhesion molecule mechanotransduction. The SDF model gave a 7.3-fold greater WSS than the Newtonian model at the top of the 3-dimensional cavity. The SDF model gave a 37.7-fold greater WSS than the Newtonian model at artery walls located immediately after bifurcations in the idealized femoral artery tree. The pressure drop across arteries reveals arterial sections highly resistive to flow which correlates with stenosis formation. Numerical simulations give the pressure drop across the idealized femoral artery tree with the SDF model which is approximately 2.3-fold higher than with the Newtonian model. In atherosclerotic lesion models, the SDF model gives over 1 Pa higher WSS than the Newtonian model, a difference correlated with over twice as many adherent monocytes to endothelial cells from the Newtonian model compared to the SDF model.     

Flush mounted hot film anemometer measurement of wall shear stress distal to a tri-leaflet valve for Newtonian and non-Newtonian blood analog fluids

Wall shear stress has been measured by flush-mounted hot film anemometry distal to an Ionescu-Shiley tri-leaflet valve under pulsatile flow conditions. Both Newtonian (aqueous glycerol) and non-Newtonian (aqueous polyacrylamide) blood analog fluids were investigated. Significant differences in the axial distribution of wall shear stress between the two fluids are apparent in flows having nearly identical Reynolds numbers. The Newtonian fluid exhibits a (peak) wall shear rate which is maximized near the valve seat (30 mm) and then decays to a fully developed flow value (by 106 mm). In contrast, the shear rate of the non-Newtonian fluid at 30 mm is less than half that of the Newtonian fluid and at 106 mm is more than twice that of the Newtonian fluid. It is suggested that non-Newtonian rheology influences valve flow patterns either through alterations in valve opening associated with low shear separation zones behind valve leaflets, or because of variations in the rate of jet spreading. More detailed studies are required to clarify the mechanisms. The Newtonian wall shear stresses for this valve are low. The highest value observed anywhere in the aortic chamber was 2.85 N/m2 at a peak Reynolds number of 3694.     

Blood's critical Taylor number and its flow behavior at supercritical Taylor numbers

When the inner cylinder of a fluid-filled Couette viscometer is rotated rapidly, a vortical flow pattern develops when a dimensionless value referred to as the critical Taylor number (Tc) is reached. We have determined its magnitude in our viscometer for three Newtonian fluids and for blood at 37 degrees C, using the inflection point of torque/RPM vs. RPM (sudden rise in apparent viscosity). Its position was identified by least squares line fitting. Because blood was studied, the viscosity used in Tc calculation was the apparent bob shear stress/shear rate ratio at the inflection marking vortical flow onset. For glycerol-water mixtures Tc was 41.8 +/- 0.3 (N = 11), for propylene glycol 42.0 +/- 0.2 (N = 14), for silicone oil 41.8 +/- 0.2 (N = 11). For healthy blood Tc was 40.7 +/- 0.9 (N = 140). This evidence against blood's increased resistance to flow instability was accompanied by a slower rate of rise in torque both above and below Tc compared to the three Newtonian fluids. Newtonian fluids and blood both developed wavy vortical flow at a rotation rate moderately higher than Tc. Blood resisted this unstable flow behavior more than the Newtonian fluids but it also experienced a slower rate of rise in torque with increasing rotation rate above the critical Taylor number. Shear-thinning is the simplest explanation for blood's mildly altered Taylor vortex behavior; blood's resistance to flow instability is otherwise not found to be sufficient to affect its flow stability in man.     

Development of a general method for designing microvascular networks using distribution of wall shear stress

In the present study, theoretical formulations for calculation of optimal bifurcation angle and relationship between the diameters of mother and daughter vessels using the power law model for non-Newtonian fluids are developed. The method is based on the distribution of wall shear stress in the mother and daughter vessels. Also, the effect of distribution of wall shear stress on the minimization of energy loss and flow resistance is considered. It is shown that constant wall shear stress in the mother and daughter vessels provides the minimum flow resistance and energy loss of biological flows. Moreover, the effects of different wall shear stresses in the mother and daughter branches, different lengths of daughter branches in the asymmetric bifurcations and non-Newtonian effect of biological fluid flows on the bifurcation angle and the relationship between the diameters of mother and daughter branches are considered. Using numerical simulations for non-Newtonian models such as power law and Carreau models, the effects of optimal bifurcation angle on the pressure drop and flow resistance of blood flow in the symmetric bifurcation are investigated. Numerical simulations show that optimal bifurcation angle decreases the pressure drop and flow resistance especially for bifurcations at large Reynolds number.     

Numerical study of the impact of non-Newtonian blood behavior on flow over a two-dimensional backward facing step

Endothelial cell (EC) responsiveness to shear stress is essential for vasoregulation and plays a role in atherogenesis. Although blood is a non-Newtonian fluid, EC flow studies in vitro are typically performed using Newtonian fluids. The goal of the present study was to determine the impact of non-Newtonian behavior on the flow field within a model flow chamber capable of producing flow disturbance and whose dimensions permit Reynolds and Womersley numbers comparable to those present in vivo. We performed two-dimensional computational fluid dynamic simulations of steady and pulsatile laminar flow of Newtonian and non-Newtonian fluids over a backward facing step. In the non-Newtonian simulations, the fluid was modeled as a shear-thinning Carreau fluid. Steady flow results demonstrate that for Re in the range 50-400, the flow recirculation zone downstream of the step is 22-63% larger for the Newtonian fluid than for the non-Newtonian fluid, while spatial gradients of shear stress are larger for the non-Newtonian fluid. In pulsatile flow, the temporal gradients of shear stress within the flow recirculation zone are significantly larger for the Newtonian fluid than for the non-Newtonian fluid. These findings raise the possibility that in regions of flow disturbance, EC mechanotransduction pathways stimulated by Newtonian and non-Newtonian fluids may be different.     

Effects of vessel compliance on flow pattern in porcine epicardial right coronary arterial tree

The compliance of the vessel wall affects hemodynamic parameters which may alter the permeability of the vessel wall. Based on experimental measurements, the present study established a finite element (FE) model in the proximal elastic vessel segments of epicardial right coronary arterial (RCA) tree obtained from computed tomography. The motion of elastic vessel wall was measured by an impedance catheter and the inlet boundary condition was measured by an ultrasound flow probe. The Galerkin FE method was used to solve the Navier–Stokes and Continuity equations, where the convective term in the Navier–Stokes equation was changed in the arbitrary Lagrangian–Eulerian (ALE) framework to incorporate the motion due to vessel compliance. Various hemodynamic parameters (e.g., wall shear stress—WSS, WSS spatial gradient—WSSG, oscillatory shear index—OSI) were analyzed in the model. The motion due to vessel compliance affects the time-averaged WSSG more strongly than WSS at bifurcations. The decrease of WSSG at flow divider in elastic bifurcations, as compared to rigid bifurcations, implies that the vessel compliance decreases the permeability of vessel wall and may be atheroprotective. The model can be used to predict coronary flow pattern in subject-specific anatomy as determined by noninvasive imaging.     

Newtonian and non-Newtonian blood flow in coiled cerebral aneurysms

Endovascular coiling aims to isolate the aneurysm from blood circulation by altering hemodynamics inside the aneurysm and triggering blood coagulation. Computational fluid dynamics (CFD) techniques have the potential to predict the post-operative hemodynamics and to investigate the complex interaction between blood flow and coils. The purpose of this work is to study the influence of blood viscosity on hemodynamics in coiled aneurysms. Three image-based aneurysm models were used. Each case was virtually coiled with a packing density of around 30%. CFD simulations were performed in coiled and untreated aneurysm geometries using a Newtonian and a Non-Newtonian fluid models. Newtonian fluid slightly overestimates the intra-aneurysmal velocity inside the aneurysm before and after coiling. There were numerical differences between fluid models on velocity magnitudes in coiled simulations. Moreover, the non-Newtonian fluid model produces high viscosity (>0.007$>0.007$ [Pa s]) at aneurysm fundus after coiling. Nonetheless, these local differences and high-viscous regions were not sufficient to alter the main flow patterns and velocity magnitudes before and after coiling. To evaluate the influence of coiling on intra-aneurysmal hemodynamics, the assumption of a Newtonian fluid can be used.     

Endothelial nitric oxide synthase and calcium production in arterial geometries: an integrated fluid mechanics/cell model

It is well known that atherosclerosis occurs at very specific locations throughout the human vasculature, such as arterial bifurcations and bends, all of which are subjected to low wall shear stress. A key player in the pathology of atherosclerosis is the endothelium, controlling the passage of material to and from the artery wall. Endothelial dysfunction refers to the condition where the normal regulation of processes by the endothelium is diminished. In this paper, the blood flow and transport of the low diffusion coefficient species adenosine triphosphate (ATP) are investigated in a variety of arterial geometries: a bifurcation with varying inner angle, and an artery bend. A mathematical model of endothelial calcium and endothelial nitric oxide synthase cellular dynamics is used to investigate spatial variations in the physiology of the endothelium. This model allows assessment of regions of the artery wall deficient in nitric oxide (NO). The models here aim to determine whether 3D flow fields are important in determining ATP concentration and endothelial function. For ATP transport, the effects of a coronary and carotid wave form on mass transport is investigated for low Womersley number. For the carotid, the Womersley number is then increased to determine whether this is an important factor. The results show that regions of low wall shear stress correspond with regions of impaired endothetial nitric oxide synthase signaling, therefore reduced availability of NO. However, experimental work is required to determine if this level is significant. The results also suggest that bifurcation angle is an important factor and acute angle bifurcations are more susceptible to disease than large angle bifurcations. It has been evidenced that complex 3D flow fields play an important role in determining signaling within endothelial cells. Furthermore, the distribution of ATP in blood is highly dependent on secondary flow features. The models here use ATP concentration simulated under steady conditions. This has been evidenced to reproduce essential features of time-averaged ATP concentration over a cardiac cycle for small Womersley numbers. However, when the Womersley number is increased, some differences are observed. Transient variations are overall insignificant, suggesting that spatial variation is more important than temporal. It has been determined that acute angle bifurcations are potentially more susceptible to atherogenesis and steady-state ATP transport reproduces essential features of time-averaged pulsatile transport for small Womersley number. Larger Womersley numbers appear to be an important factor in time-dependent mass transfer.     

Comparative study of Newtonian and non-Newtonian simulations of drug transport in a model drug-eluting stent

To elucidate the difference between Newtonian and shear thinning non-Newtonian assumptions of blood in the analysis of DES drug delivery, we numerically simulated the local flow pattern and the concentration distribution of the drug at the lumen-tissue interface for a structurally simplified DES deployed in a curved segment of an artery under pulsatile blood flow conditions. The numerical results showed that when compared with the Newtonian model, the Carreau (shear thinning) model could lead to some differences in the luminal surface drug concentration in certain areas along the outer wall of the curved vessel. In most areas of the vessel, however, there were no significant differences between the 2 models. Particularly, no significant difference between the two models was found in terms of the area-averaged luminal surface drug concentration. Therefore, we believe that the shear thinning property of blood may play little roles in DES drug delivery. Nevertheless, before we draw the conclusion that Newtonian assumption of blood can be used to replace its non-Newtonian one for the numerical simulation of drug transport in the DES implanted coronary artery, other more complex mechanical properties of blood such as its thixotropic behavior should be tested.     

Spatial and temporal variation of secondary flow during oscillatory flow in model human central airways

Axial and secondary velocity profiles were measured in a model human central airway to clarify the oscillatory flow structure during high-frequency oscillation. We used a rigid model of human airways consisting of asymmetrical bifurcations up to third generation. Velocities in each branch of the bifurcations were measured by two-color laser-Doppler velocimeter. The secondary velocity magnitudes and the deflection of axial velocity were dependent not only on the branching angle and curvature ratio of each bifurcation, but also strongly depended on the shape of the path generated by the cascade of branches. Secondary flow velocities were higher in the left bronchus than in the right bronchus. This spatial variation of secondary flow was well correlated with differing gas transport rates between the left and right main bronchus.     

Non-Newtonian blood flow in human right coronary arteries: transient simulations

This study looks at pulsatile blood flow through four different right coronary arteries, which have been reconstructed from biplane angiograms. A non-Newtonian blood model (the Generalised Power Law), as well as the usual Newtonian model of blood viscosity, is used to study the wall shear stress in each of these arteries over the entire cardiac cycle. The difference between Newtonian and non-Newtonian blood models is also studied over the whole cardiac cycle using the recently generalised global non-Newtonian importance factor. In addition, the flow is studied by considering paths of massless particles introduced into the flow field. The study shows that, when studying the wall shear stress distribution for transient blood flow in arteries, the use of a Newtonian blood model is a reasonably good approximation. However, to study the flow within the artery in greater detail, a non-Newtonian model is more appropriate.