Pulsatile flow of non-Newtonian fluid in distensible models of human arteries

In addition to biochemical factors, hydromechanical influences are responsible for atherogenesis and deposits of blood platelets at bends and bifurcations of human arteries. Hence the flow patterns were simulated in a true-to-scale three-dimensional bifurcation of a human renal artery model and of an arterial femoralis with Newtonian and non-Newtonian blood like fluid. Investigations were made with steady and pulsatile flow. The velocity profiles (at physiological Re-numbers) were measured after the bifurcations with a laser-Doppler-anemometer. In previous works Newtonian fluids were used to investigate the flow in bends and bifurcations of rigid and elastic simplified models. In this paper, emphasis is placed on the difference between rigid and elastic models and also Newtonian and non Newtonian flow behavior. Differences between Newtonian and non Newtonian fluids may especially be expected to occur after branches where the flow has local strong convective elements such as in reverse zones and flow separation points.     

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.     

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.     

Low Reynolds number equi-bifurcation flow in a two-dimensional channel

The fluid flow in some physiological vessels such as the blood flow in blood vessels and the air flow through bronchi and bronchioles in the lungs undergoes a large number of bifurcations. The understanding of the bifurcation flow is of importance for a better comprehension of its effect in the blood and the air circulatory systems of the living body. The Reynolds number of flow in large blood vessels and bronchi is high and fluid inertia plays a dominant role in the bifurcation flow in such vessels. In small caliber blood vessels such as arterioles and capillaries, and bronchioles, the Reynolds number of flow is quite low and the effect of fluid inertia is negligible compared to the pressure and shear forces. In order to have a quantitative understanding of the bifurcation flow at low Reynolds numbers, the low Reynolds number equi-bifurcation flow in a two-dimensional channel at zero bifurcation angle is studied based on the Stokes approximation. The solution of the problem is posed as an infinite series, where the truncated version is used in numerical calculations. The results of this analysis is discussed in connection with the bifurcation flow of blood in small caliber blood vessels and that of the air in bronchioles in the lung.     

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.     

Vortex generation in pulsatile flow through arterial bifurcation models including the human carotid artery

Visualization experiments were performed to elucidate the complicated flow pattern in pulsatile flow through arterial bifurcations. Human common carotid arteries, which were made transparent, and glass-models simulating Y- and T-shaped bifurcations were used. Pulsatile flow with wave forms similar to those of arterial flow was generated with a piston pump, elastic tube, airchamber, and valves controlling the outflow resistance. Helically recirculating flow with a pattern similar to that of the horseshoe vortex produced around wall-based protuberances in circular tubes was observed in pulsatile flow through all the bifurcations used in the present study. This flow type, which we shall refer to as the horseshoe vortex, has also been demonstrated to occur at the human common carotid bifurcation in steady flow with Reynolds numbers above 100. Time-varying flows also produced the horseshoe vortex mostly during the decelerating phase. Fluid particles of dye solution approaching the bifurcation apex diverged, divided into two directions perpendicularly, and then showed helical motion representing the horseshoe vortex formation. While this helical flow was produced, the stagnation points appeared on the wall upstream of the apex. Their position was dependent upon the flow distribution ratio between the branches in the individual arteries. The region affected by the horseshoe vortex was smaller during pulsatile flow than during steady flow. Lowering the Reynolds number together with the Womersley number weakened the intensity of helical flow. A separation bubble, resulting from the divergence or wall roughness, was observed at the outer or inner wall of the branch vessels and made the flow more complicated.     

One-dimensional model for propagation of a pressure wave in a model of the human arterial network: comparison of theoretical and experimental results

Pulse wave evaluation is an effective method for arteriosclerosis screening. In a previous study, we verified that pulse waveforms change markedly due to arterial stiffness. However, a pulse wave consists of two components, the incident wave and multireflected waves. Clarification of the complicated propagation of these waves is necessary to gain an understanding of the nature of pulse waves in vivo. In this study, we built a one-dimensional theoretical model of a pressure wave propagating in a flexible tube. To evaluate the applicability of the model, we compared theoretical estimations with measured data obtained from basic tube models and a simple arterial model. We constructed different viscoelastic tube set-ups: two straight tubes; one tube connected to two tubes of different elasticity; a single bifurcation tube; and a simple arterial network with four bifurcations. Soft polyurethane tubes were used and the configuration was based on a realistic human arterial network. The tensile modulus of the material was similar to the elasticity of arteries. A pulsatile flow with ejection time 0.3 s was applied using a controlled pump. Inner pressure waves and flow velocity were then measured using a pressure sensor and an ultrasonic diagnostic system. We formulated a 1D model derived from the Navier-Stokes equations and a continuity equation to characterize pressure propagation in flexible tubes. The theoretical model includes nonlinearity and attenuation terms due to the tube wall, and flow viscosity derived from a steady Hagen-Poiseuille profile. Under the same configuration as for experiments, the governing equations were computed using the MacCormack scheme. The theoretical pressure waves for each case showed a good fit to the experimental waves. The square sum of residuals (difference between theoretical and experimental wave-forms) for each case was <10.0%. A possible explanation for the increase in the square sum of residuals is the approximation error for flow viscosity. However, the comparatively small values prove the validity of the approach and indicate the usefulness of the model for understanding pressure propagation in the human arterial network.     

A hybrid one-dimensional/Womersley model of pulsatile blood flow in the entire coronary arterial tree

Using a frequency-domain Womersley-type model, we previously simulated pulsatile blood flow throughout the coronary arterial tree. Although this model represents a good approximation for the smaller vessels, it does not take into account the nonlinear convective energy losses in larger vessels. Here, using Womersley's theory, we present a hybrid model that considers the nonlinear effects for the larger epicardial arteries while simulating the distal vessels (down to the 1st capillary segments) with the use of Womersley's Theory. The main trunk and primary branches were discretized and modeled with one-dimensional Navier-Stokes equations, while the smaller-diameter vessels were treated as Womersley-type vessels. Energy losses associated with vessel bifurcations were incorporated in the present analysis. The formulation enables prediction of impedance and pressure and pulsatile flow distribution throughout the entire coronary arterial tree down to the first capillary segments in the arrested, vasodilated state. We found that the nonlinear convective term is negligible and the loss of energy at a bifurcation is small in the larger epicardial vessels of an arrested heart. Furthermore, we found that the flow waves along the trunk or at the primary branches tend to scale (normalized with respect to their mean values) to a single curve, except for a small phase angle difference. Finally, the model predictions for the inlet pressure and flow waves are in excellent agreement with previously published experimental results. This hybrid one-dimensional/Womersley model is an efficient approach that captures the essence of the hemodynamics of a complex large-scale vascular network. The present model has numerous applications to understanding the dynamics of coronary circulation.     

Confocal imaging of flows in artificial venular bifurcations

We describe a new experimental methodology for visualizing three-dimensional structures in microscopic tubes under flow conditions. Through the use of microfabrication techniques, artificial venular bifurcations are constructed from glass tubes with semicircular cross sections (radius = 50 mu). Aqueous fluorescent solutions are infused into the tubes at flow rates of about 1 microliter/min, a value comparable to blood flow in the microcirculation. The flow is imaged using a combination of confocal microscopy and three-dimensional image reconstruction software techniques. The quantitative accuracy of the experimental method is evaluated by measuring the "separation surface," a formation resulting from converging flows at a bifurcation. Details of the fabrication process, fluidics, confocal microscopy, image reconstructions, optical effects, and computations are described. We show the first three-dimensional visualization of a microscopic flow structure using confocal microscopy, and within certain limitations, quantitative agreement between the measured and computed positions of the separation surface.     

Reversible and irreversible components of central-airway flow resistance

The flow energy loss (head loss) through a cast of canine central airways is found to be nearly independent of flow direction. By contrast, head loss in geometrically-simpler branching sections at comparable flow conditions is highly irreversible, with inspiratory loss being greater by nearly two units of dynamic pressure (2 X 1/2 rho V2). In these branching sections head loss appears to be independent of important geometric parameters such as the branch length/diameter ratio and the exit/inlet flow-area ratio. An analysis of these observations suggests that kinetic energy factors, not shear stresses, account for most of the energy dissipated in central airways and in simple bifurcating sections. Inspiratory loss in bifurcations is greatly increased by the onset of flow separation: irreversibility is minimal in central airways, where separation either is absent or else is much less pronounced.     

Influence of bifurcations on forced oscillations in an airway model.

Forced oscillations is a technique to determine respiratory input impedance from small amplitude sinusoidal pressure excursions introduced at the airway opening. Models used to predict respiratory input impedance typically ignore the direct effect of bifurcations on the flow, and treat airway branches as individual straight tubes placed appropriately in parallel and series. The flow within the individual tubes is assumed equivalent to that which would occur in infinitely long tubes. In this study we examined the influence of bifurcations on impedance for conditions of the forced oscillatory technique. We measured input impedance using forced oscillations in straight tubes and in an anatomically-relevant, four generation physical model of a human airway network. The input impedance measured experimentally compared well to that obtained theoretically using model predictions. The predictive scheme was based on appropriate parallel and series combinations of theoretically computed individual tube impedances, which were computed from solutions to oscillatory flow of a compressible gas in an infinitely long rigid tube. The agreement between experimental measurements and predictions indicates that bifurcations play a relatively minor direct role on the flow impedance for conditions of the forced oscillations technique. These results are explained in terms of the small tidal volumes used, whereby the axial distance traveled by a fluid particle during an oscillation cycle is appreciably smaller than branch segment lengths. Accordingly, only a small fraction of fluid particles travel through the bifurcation region, and the remainder experience an environment approaching flow in an infinite straight tube. The relevance of the study to the prediction of impedances in the human lung during forced oscillations is discussed.     

Mechanics of blood flow

The historical development of the mechanics of blood flow can be traced from ancient times, to Leonardo da Vinci and Leonhard Euler and up to the present times with increasing biological knowledge and mathematical analysis. In the last two decades, quantitative and numerical methods have steadily given more complete and precise understanding. In the arterial system wave propagation computations based on nonlinear one-dimensional modeling have given the best representation of pulse wave propagation. In the veins, the theory of unsteady flow in collapsible tubes has recently been extensively developed. In the last decade, progress has been made in describing the blood flow at junctions, through stenoses, in bends and in capillary blood vessels. The rheological behavior of individual red blood cells has been explored. A working model consists of an elastic membrane filled with viscous fluid. This model forms a basis for understanding the viscous and viscoelastic behavior of blood.     

Improved reduced-order modelling of cerebrovascular flow distribution by accounting for arterial bifurcation pressure drops

Reduced-order modelling offers the possibility to study global flow features in cardiovascular networks. In order to validate these models, previous studies have been conducted in which they compared 3D computational fluid dynamics simulations with reduced-order simulations. Discrepancies have been reported between the two methods. The loss of energy at the bifurcations is usually neglected and has been pointed out as a possible explanation for these discrepancies. We present distributed lumped models of cerebrovasculatures created automatically from 70 cerebrovascular networks segmented from 3D angiograms. The outflow rate repartitions predicted with and without modelling the energy loss at the bifurcations are compared against 3D simulations. When neglecting the energy loss at the bifurcations, the flow rates though the anterior cerebral arteries are overestimated by 4.7±6.8% (error relative to the inlet flow rate, mean ± standard deviation), impacting the remaining volume of flow going to the other vessels. When the energy loss is modelled, this error is dropping to 0.1±3.2%. Overall, over the total of 337 outlet vessels, when the energy losses at the bifurcations are not modelled the 95% of agreement is in the range of ±13.5% and is down to ±6.5% when the energy losses are considered. With minimal input and computational resources, the presented method can estimate the outflow rates reliably. This study constitutes the largest validation of a reduced-order flow model against 3D simulations. The impact of the energy loss at the bifurcations is here demonstrated for cerebrovasculatures but can be applied to other physiological networks.     

Effect of nonaxisymmetric hematocrit distribution on non-Newtonian blood flow in small tubes

Hematocrit distribution and red blood cell aggregation are the major determinants of blood flow in narrow tubes at low flow rates. It has been observed experimentally that in microcirculation the hematocrit distribution is not uniform. This nonuniformity may result from plasma skimming and cell screening effects and also from red cell sedimentation. The goal of the present study is to understand the effect of nonaxisymmetric hematocrit distribution on the flow of human and cat blood in small blood vessels of the microcirculation. Blood vessels are modeled as circular cylindrical tubes. Human blood is described by Quemada's rheological model, in which local viscosity is a function of both the local hematocrit and a structural parameter that is related to the size of red blood cell aggregates. Cat blood is described by Casson's model. Eccentric hematocrit distribution is considered such that the axis of the cylindrical core region of red cell suspension is parallel to the axis of the blood vessel but not coincident. The problem is solved numerically by using finite element method. The calculations predict nonaxisymmetric distribution of velocity and shear stress in the blood vessel and the increase of apparent viscosity with increasing eccentricity of the core.     

Effects of flow geometry on blood viscoelasticity

The viscoelastic properties of blood are dominated by microstructures formed by red cells. The microstructures are of several types such as irregular aggregates, rouleaux, and layers of aligned cells. The dynamic deformability of the red cells, aggregation tendency, cell concentration, size of confining vessel and rate of flow are determining factors in the microstructure. Viscoelastic properties, viscosity and elasticity, relate to energy loss and storage in flowing blood while relaxation time and Weissenberg number play a role in assessing the importance of the elasticity relative to the viscosity. These effects are shown herein for flow in a large straight cylindrical tube, a small tube, and a porous medium. These cases approximate the geometries of the arterial system: large vessels, small vessels and vessels with many branches and bifurcations. In each case the viscosity, elasticity, relaxation time and Weissenberg number for normal human blood as well as blood with enhanced cell aggregation tendency and diminished cell deformability are given. In the smaller spaces of the microtubes and porous media, the diminished viscosity shows the possible influence of the F?hraeus-Lindqvist effect and at high shear rates, the viscoelasticity of blood shows dilatancy. This is true for normal, aggregation enhanced and hardened cells.     

Proteoglycan synthesis in the endothelium of embryonal vertebrate hearts: significance for the development of cardiovascular function

The endothelium of the embryonic heart is able to synthetize proteoglycans (PG) as it is the myocardium. In the extracellular matrix, PG form highly polymeric visco-elastic networks, which besides others act as shock absorber. That is apparently of evidence for the modulation of embryonic heart actions. Because during the embryonic period the large arteries are simple endothelial tubes without having an elastic-muscular wall. That means the typical "windkessel" function such as dumping of pulse waves or a continues pressure distribution is not existent. The embryonic vessels are perfused like rigid tubes. The continuous rhythmic flow pattern in the endothelial tubes, necessary for perfusion of the different organs, is apparently compensated by a high initial pressure level initiated by the heart. It is concluded that the continuity of the pressure profile is caused by intracardial PG. The endothelial synthesis of the PG of the heart decreases with increasing development of the muscular wall of the vessels and disappears completely post partum.     

Geometrical optimisation of a biochip microchannel fluidic separator

This article reports on the geometric optimisation of a T-shaped biochip microchannel fluidic separator aiming to maximise the separation efficiency of plasma from blood through the improvement of the unbalanced separation performance among different channel bifurcations. For this purpose, an algebraic analysis is firstly implemented to identify the key parameters affecting fluid separation. A numerical optimisation is then carried out to search the key parameters for improved separation performance of the biochip. Three parameters, the interval length between bifurcations, the main channel length from the outlet to the bifurcation region and the side channel geometry, are identified as the key characteristic sizes and defined as optimisation variables. A balanced flow rate ratio between the main and side channels, which is an indication of separation effectiveness, is defined as the objective. It is found that the degradation of the separation performance is caused by the unbalanced channel resistance ratio between the main and side channel routes from bifurcations to outlets. The effects of the three key parameters can be summarised as follows: (a) shortening the interval length between bifurcations moderately reduces the differences in the flow rate ratios; (b) extending the length of the main channel from the main outlet is effective for achieving a uniformity of flow rate ratio but ineffective in changing the velocity difference of the side channels and (c) decreasing the lengths of side channels from upstream to downstream is effective for both obtaining a uniform flow rate ratio and reducing the differences in the flow velocities between the side branch channels. An optimisation process combining the three parameters is suggested as this integration approach leads to fast convergent process and also offers flexible design options for satisfying different requirements.     

Modeling the effect of blood vessel bifurcation ratio on occlusive thrombus formation

Vascular geometry is a major determinant of the hemodynamics that promote or prevent unnecessary vessel occlusion from thrombus formation. Bifurcations in the vascular geometry are repeating structures that introduce flow separation between parent and daughter vessels. We modelled the blood flow and shear rate in a bifurcation during thrombus formation and show that blood vessel bifurcation ratios determine the maximum shear rate on the surface of a growing thrombus. We built an analytical model that may aid in predicting microvascular bifurcation ratios that are prone to occlusive thrombus formation. We also observed that bifurcation ratios that adhere to Murray’s law of bifurcations may be protected from occlusive thrombus formation. These results may be useful in the rational design of diagnostic microfluidic devices and microfluidic blood oxygenators.     

Geometrical optimisation of a biochip microchannel fluidic separator

This article reports on the geometric optimisation of a T-shaped biochip microchannel fluidic separator aiming to maximise the separation efficiency of plasma from blood through the improvement of the unbalanced separation performance among different channel bifurcations. For this purpose, an algebraic analysis is firstly implemented to identify the key parameters affecting fluid separation. A numerical optimisation is then carried out to search the key parameters for improved separation performance of the biochip. Three parameters, the interval length between bifurcations, the main channel length from the outlet to the bifurcation region and the side channel geometry, are identified as the key characteristic sizes and defined as optimisation variables. A balanced flow rate ratio between the main and side channels, which is an indication of separation effectiveness, is defined as the objective. It is found that the degradation of the separation performance is caused by the unbalanced channel resistance ratio between the main and side channel routes from bifurcations to outlets. The effects of the three key parameters can be summarised as follows: (a) shortening the interval length between bifurcations moderately reduces the differences in the flow rate ratios; (b) extending the length of the main channel from the main outlet is effective for achieving a uniformity of flow rate ratio but ineffective in changing the velocity difference of the side channels and (c) decreasing the lengths of side channels from upstream to downstream is effective for both obtaining a uniform flow rate ratio and reducing the differences in the flow velocities between the side branch channels. An optimisation process combining the three parameters is suggested as this integration approach leads to fast convergent process and also offers flexible design options for satisfying different requirements.     

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.