Dynamics of an oscillating microbubble in a blood-like Carreau fluid

A numerical model for cavitation in blood is developed based on the Keller-Miksis equation for spherical bubble dynamics with the Carreau model to represent the non-Newtonian behavior of blood. Three different pressure waveforms driving the bubble oscillations are considered: a single-cycle Gaussian waveform causing free growth and collapse, a sinusoidal waveform continuously driving the bubble, and a multi-cycle pulse relevant to contrast-enhanced ultrasound. Parameters in the Carreau model are fit to experimental measurements of blood viscosity. In the Carreau model, the relaxation time constant is 5-6 orders of magnitude larger than the Rayleigh collapse time. As a result, non-Newtonian effects do not significantly modify the bubble dynamics but do give rise to variations in the near-field stresses as non-Newtonian behavior is observed at distances 10-100 initial bubble radii away from the bubble wall. For sinusoidal forcing, a scaling relation is found for the maximum non-Newtonian length, as well as for the shear stress, which is 3 orders of magnitude larger than the maximum bubble radius.

A review on non-Newtonian fluid models for multi-layered blood rheology in constricted arteries

Haemodynamics is a branch of fluid mechanics which investigates the features of blood when it flows not only via blood vessels of smaller/larger diameter, but also under normal as well as abnormal flow states, such as in the presence of stenosis, aneurysm, and thrombosis. This review aims to discuss the rheological properties of blood, geometry of constrictions, dilations and the emergence of single-layered fluid to four-layered fluid models. To discuss further the influence of the aforesaid parameters on the physiologically important flow quantities, the mathematical formulation and solution methodology of the two-layered and four layered arterial blood flow problems studied by the authors (Afiqah and Sankar in ARPN J Eng Appl Sci 15:1129--1143, 2020, Comput Methods Programs Biomed 199:105907, 2021. 10.1016/j.cmpb.2020.105907) are recalled. It should be pointed out that the increasing resistive impedance to flow in three distinct states encompassing healthy, anaemic, and diabetic demonstrates that the greater the restriction in the artery, very few blood is carried to the pathetic organs, leading to subjects' death. It is also discovered that the pulsatile nature of blood movement produces a dynamic environment that poses a slew of intriguing and unstable fluid mechanical state. It is hoped that the intriguing results gathered from this literature survey and review conducted may help the medical practitioners to forecast blood behaviour mobility in stenotic arteries. Furthermore, the physiological information gathered from the available clinical data from the literature on patients diagnosed with diabetes and anaemia may be beneficial to doctors in deciding the therapeutic procedure for treating some particular cardiovascular disease.

The role of plasma in the yield stress of blood

BACKGROUND

Yielding and shear elasticity of blood are merely discussed within the context of hematocrit and erythrocyte aggregation. However, plasma might play a substantial role due its own viscoelasticity.

OBJECTIVE

If only erythrocyte aggregation and hematocrit would determine yielding, blood of different species with comparable values would present comparable yield stresses.

METHODS

rheometry (SAOS: amplitude and frequency sweep tests; flow curves) of hematocrit-matched samples at 37°C. Brillouin Light Scattering Spectroscopy at 38°C.

RESULTS

Yield stress for pig: 20mPa, rat: 18mPa, and human blood: 9mPa. Cow and sheep blood were not in quasi-stationary state supporting the role of erythrocyte aggregation for the development of elasticity and yielding. However, pig and human erythrocytes feature similar aggregability, but yield stress of porcine blood was double. Murine and ruminant erythrocytes both rarely aggregate, but their blood behavior was fundamentally different. Pig plasma was shear-thinning and murine plasma was platelet-enriched, supporting the role of plasma for triggering collective effects and gel-like properties.

CONCLUSIONS

Blood behavior near zero shear flow is not based solely on erythrocyte aggregation and hematocrit, but includes the hydrodynamic interaction with plasma. The shear stress required to break down elasticity is not the critical shear stress for dispersing erythrocyte aggregates, but the shear stress required to fracture the entire assembly of blood cells within their intimate embedding.

Normalization of Blood Viscosity According to the Hematocrit and the Shear Rate

The rheological properties of blood depend highly on the properties of its red blood cells: concentration, membrane elasticity, and aggregation. These properties affect the viscosity of blood as well as its shear thinning behavior. Using an experimental analysis of the interface advancement of blood in a microchannel, we determine the viscosity of different samples of blood. In this work, we present two methods that successfully normalize the viscosity of blood for a single and for different donors, first according to the concentration of erythrocytes and second according to the shear rate. The proposed methodology is able to predict the health conditions of the blood samples by introducing a non-dimensional coefficient that accounts for the response to shear rate of the different donors blood samples. By means of these normalization methods, we were able to determine the differences between the red blood cells of the samples and define a range where healthy blood samples can be described by a single behavior.

Microfluidics Approach to the Mechanical Properties of Red Blood Cell Membrane and Their Effect on Blood Rheology

In this article, we describe the general features of red blood cell membranes and their effect on blood flow and blood rheology. We first present a basic description of membranes and move forward to red blood cell membranes’ characteristics and modeling. We later review the specific properties of red blood cells, presenting recent numerical and experimental microfluidics studies that elucidate the effect of the elastic properties of the red blood cell membrane on blood flow and hemorheology. Finally, we describe specific hemorheological pathologies directly related to the mechanical properties of red blood cells and their effect on microcirculation, reviewing microfluidic applications for the diagnosis and treatment of these diseases.

Recent advances in blood rheology: a review

Due to the potential impact on the diagnosis and treatment of various cardiovascular diseases, work on the rheology of blood has significantly expanded in the last decade, both experimentally and theoretically. Experimentally, blood has been confirmed to demonstrate a variety of non-Newtonian rheological characteristics, including pseudoplasticity, viscoelasticity, and thixotropy. New rheological experiments and the development of more controlled experimental protocols on more extensive, broadly physiologically characterized, human blood samples demonstrate the sensitivity of aspects of hemorheology to several physiological factors. For example, at high shear rates the red blood cells elastically deform, imparting viscoelasticity, while at low shear rates, they form "rouleaux" structures that impart additional, thixotropic behavior. In addition to the advances in experimental methods and validated data sets, significant advances have also been made in both microscopic simulations and macroscopic, continuum, modeling, as well as novel, multiscale approaches. We outline and evaluate the most promising of these recent developments. Although we primarily focus on human blood rheology, we also discuss recent observations on variations observed across some animal species that provide some indication on evolutionary effects.

Continuum Scale Non Newtonian Particle Transport Model for Hæmorheology

We present a continuum scale particle transport model for red blood cells following collision arguments, in a diffusive flux formulation. The model is implemented in FOAM, in a framework suitable for haemodynamics simulations and adapted to multi-scaling. Specifically, the framework we present is able to ingest transport coefficient models to be derived, prospectively, from complimentary but independent meso-scale simulations. For present purposes, we consider modern semi-mechanistic rheology models, which we implement and test as proxies for such data. The model is verified against a known analytical solution and shows excellent agreement for high quality meshes and good agreement for typical meshes as used in vascular flow simulations. Simulation results for different size and time scales show that migration of red blood cells does occur on physiologically relevany timescales on small vessels below 1 mm and that the haematocrit concentration modulates the non-Newtonian viscosity. This model forms part of a multi-scale approach to haemorheology and model parameters will be derived from meso-scale simulations using multi-component Lattice Boltzmann methods. The code, haemoFoam, is made available for interested researchers.

A comparative study of blood rheology across species

Recent advances in hemorheology are extended to study blood rheology across species, which has important clinical implications particularly in intravenous drug scaleup as drugs undergoing clinical trials are first tested in animals. Some of the first hemorheological measurements from seven different species under both steady and transient shear conditions are presented and modeled using a rheological model developed and validated on human blood rheology fit to 20 different donors. Despite similar physiological properties across the blood samples from different species, significant differences are observed, particularly at low shear rates. Blood from species that form rouleaux exhibit a yield-like behavior and enhanced viscoelasticity at low shear rates, while blood from species without rouleaux exhibit nearly Newtonian behavior at similar shear rates. Viscoelasticity due to blood cell deformation is evident for all species at high shear rates. Novel, unidirectional large amplitude oscillatory shear measurements differentiate species. Using the newly acquired data in combination with previous literature data, a new allometric scaling relation is suggested for the low-shear blood viscosity for various mammalian evolutionary orders. Using an established model for arterial branching across species, it is conjectured that the observed hemorheological scaling across species is driven by maintaining a constant wall shear stress in arterial vessels.

Effect of cytosol viscosity on the flow behavior of red blood cell suspensions in microvessels

OBJECTIVE

The flow behavior of blood is strongly affected by red blood cell (RBC) properties, such as the viscosity ratio C between cytosol and suspending medium, which can significantly be altered in several pathologies (e.g. sickle-cell disease, malaria). The main objective of this study is to understand the effect of C on macroscopic blood flow properties such as flow resistance in microvessels, and to link it to the deformation and dynamics of single RBCs.

METHODS

We employ mesoscopic hydrodynamic simulations to investigate flow properties of RBC suspensions with different cytosol viscosities for various flow conditions in cylindrical microchannels.

RESULTS

Starting from a dispersed cell configuration which approximates RBC dispersion at vessel bifurcations in the microvasculature, we find that the flow convergence and development of RBC-free layer (RBC-FL) depend only weakly on C, and require a convergence length in the range of 25D-50D, where D is channel diameter. In vessels with D ≤ 20 μ m , the final resistance of developed flow is nearly the same for C = 5 and C = 1, while for D = 40 μ m , the flow resistance for C = 5 is about 10% larger than for C = 1. The similarities and differences in flow resistance can be explained by viscosity-dependent RBC-FL thicknesses, which are associated with the viscosity-dependent dynamics of single RBCs.

CONCLUSIONS

The weak effect on the flow resistance and RBC-FL explains why RBCs can contain a high concentration of hemoglobin for efficient oxygen delivery, without a pronounced increase in the flow resistance. Furthermore, our results suggest that significant alterations in microvascular flow in various pathologies are likely not due to mere changes in cytosolic viscosity.

Non-identifiability of parameters for a class of shear-thinning rheological models, with implications for haematological fluid dynamics

Choosing a suitable model and determining its associated parameters from fitting to experimental data is fundamental for many problems in biomechanics. Models of shear-thinning complex fluids, dating from the work of Bird, Carreau, Cross and Yasuda, have been applied in highly-cited computational studies of hemodynamics for several decades. In this manuscript we revisit these models, first to highlight a degree of uncertainty in the naming conventions in the literature, but more importantly to address the problem of inferring model parameters by fitting to rheology experiments. By refitting published data, and also by simulation, we find large, flat regions in likelihood surfaces that yield families of parameter sets which fit the data equally well. Despite having almost indistinguishable fits to experimental data these varying parameter sets can predict very different flow profiles, and as such these parameters cannot be used to draw conclusions about physical properties of the fluids, such as zero-shear viscosity or relaxation time of the fluid, or indeed flow behaviours. We verify that these features are not a consequence of the experimental data sets through simulations; by sampling points from the rheological models and adding a small amount of noise we create a synthetic data set which reveals that the problem of parameter identifiability is intrinsic to these models.

Front microrheology of the non-Newtonian behaviour of blood: scaling theory of erythrocyte aggregation by aging

We introduce a new framework to study the non-Newtonian behaviour of fluids at the microscale based on the analysis of front advancement. We apply this methodology to study the non-linear rheology of blood in microchannels. We carry out experiments in which the non-linear viscosity of blood samples is quantified at different haematocrits and ages. Under these conditions, blood exhibits a power-law dependence on the shear rate. In order to analyse our experimental data, we put forward a scaling theory which allows us to define an adhesion scaling number. This theory yields a scaling behaviour of the viscosity expressed as a function of the adhesion capillary number. By applying this scaling theory to samples of different ages, we are able to quantify how the characteristic adhesion energy varies as time progresses. This connection between microscopic and mesoscopic properties allows us to estimate quantitatively the change in the cell-cell adhesion energies as the sample ages.

The Impact of Blood Rheology on Drug Transport in Stented Arteries: Steady Simulations

BACKGROUND AND METHODS

It is important to ensure that blood flow is modelled accurately in numerical studies of arteries featuring drug-eluting stents due to the significant proportion of drug transport from the stent into the arterial wall which is flow-mediated. Modelling blood is complicated, however, by variations in blood rheological behaviour between individuals, blood's complex near-wall behaviour, and the large number of rheological models which have been proposed. In this study, a series of steady-state computational fluid dynamics analyses were performed in which the traditional Newtonian model was compared against a range of non-Newtonian models. The impact of these rheological models was elucidated through comparisons of haemodynamic flow details and drug transport behaviour at various blood flow rates.

RESULTS

Recirculation lengths were found to reduce by as much as 24% with the inclusion of a non-Newtonian rheological model. Another model possessing the viscosity and density of blood plasma was also implemented to account for near-wall red blood cell losses and yielded recirculation length increases of up to 59%. However, the deviation from the average drug concentration in the tissue obtained with the Newtonian model was observed to be less than 5% in all cases except one. Despite the small sensitivity to the effects of viscosity variations, the spatial distribution of drug matter in the tissue was found to be significantly affected by rheological model selection.

CONCLUSIONS/SIGNIFICANCE

These results may be used to guide blood rheological model selection in future numerical studies. The clinical significance of these results is that they convey that the magnitude of drug uptake in stent-based drug delivery is relatively insensitive to individual variations in blood rheology. Furthermore, the finding that flow separation regions formed downstream of the stent struts diminish drug uptake may be of interest to device designers.

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.

SIMULATION OF CASSON FLUID FLOW AND HEAT TRANSPORT IN DIFFERENTLY SHAPED STENOSES

The present investigation deals with a mathematical model representing the response of heat transfer to blood streaming through the arteries under stenotic condition. The flowing blood is represented as the suspension of all erythrocytes assumed to be Casson fluid and the arterial wall is considered to be rigid having differently shaped stenoses in its lumen arising from various types of abnormal growth or plaque formation. The governing equations of motion accompanied by the appropriate choice of the boundary conditions are solved numerically by Marker and Cell (MAC) method. The necessary checking for numerical stability has been incorporated into the algorithm for better precision of the results computed. The quantitative analysis carried out finally includes the respective profiles of the flow-field and the temperature along with their individual distributions over the entire arterial segment as well. The key factors like the pressure drop, wall shear stress, flow separation, Nusselt number and streamlines are examined for qualitative insight into the blood flow and heat transport phenomena through arterial stenosis. In conformity with other several existing findings the present simulation predicts that the pressure drop and Nusselt number diminishes with increasing yield stress values, and significant enhancement in values of Nusselt number is observed with increasing severity of the stenosis. However, the effect of the shapes of the stenoses on flow separation cannot be ruled out from the present investigation.

Computational biorheology of human blood flow in health and disease

Hematologic disorders arising from infectious diseases, hereditary factors and environmental influences can lead to, and can be influenced by, significant changes in the shape, mechanical and physical properties of red blood cells (RBCs), and the biorheology of blood flow. Hence, modeling of hematologic disorders should take into account the multiphase nature of blood flow, especially in arterioles and capillaries. We present here an overview of a general computational framework based on dissipative particle dynamics (DPD) which has broad applicability in cell biophysics with implications for diagnostics, therapeutics and drug efficacy assessments for a wide variety of human diseases. This computational approach, validated by independent experimental results, is capable of modeling the biorheology of whole blood and its individual components during blood flow so as to investigate cell mechanistic processes in health and disease. DPD is a Lagrangian method that can be derived from systematic coarse-graining of molecular dynamics but can scale efficiently up to arterioles and can also be used to model RBCs down to the spectrin level. We start from experimental measurements of a single RBC to extract the relevant biophysical parameters, using single-cell measurements involving such methods as optical tweezers, atomic force microscopy and micropipette aspiration, and cell-population experiments involving microfluidic devices. We then use these validated RBC models to predict the biorheological behavior of whole blood in healthy or pathological states, and compare the simulations with experimental results involving apparent viscosity and other relevant parameters. While the approach discussed here is sufficiently general to address a broad spectrum of hematologic disorders including certain types of cancer, this paper specifically deals with results obtained using this computational framework for blood flow in malaria and sickle cell anemia.

A Porous Media Model for Blood Flow within Reticulated Foam

A porous media model is developed for non-Newtonian blood flow through reticulated foam at Reynolds numbers ranging from 10-8 to 10. This empirical model effectively divides the pressure gradient versus flow speed curve into three regimes, in which either the non-Newtonian viscous forces, the Newtonian viscous forces, or the inertial fluid forces are most prevalent. When compared to simulation data of blood flow through two reticulated foam geometries, the model adequately captures the pressure gradient within all three regimes, especially that within the Newtonian regime where blood transitions from a power-law to a constant viscosity fluid.

Multiscale modeling of blood flow: from single cells to blood rheology

Mesoscale simulations of blood flow, where the red blood cells are described as deformable closed shells with a membrane characterized by bending rigidity and stretching elasticity, have made much progress in recent years to predict the flow behavior of blood cells and other components in various flows. To numerically investigate blood flow and blood-related processes in complex geometries, a highly efficient simulation technique for the plasma and solutes is essential. In this review, we focus on the behavior of single and several cells in shear and microcapillary flows, the shear-thinning behavior of blood and its relation to the blood cell structure and interactions, margination of white blood cells and platelets, and modeling hematologic diseases and disorders. Comparisons of the simulation predictions with existing experimental results are made whenever possible, and generally very satisfactory agreement is obtained.

Fluid dynamics of heart development

The morphology, muscle mechanics, fluid dynamics, conduction properties, and molecular biology of the developing embryonic heart have received much attention in recent years due to the importance of both fluid and elastic forces in shaping the heart as well as the striking relationship between the heart's evolution and development. Although few studies have directly addressed the connection between fluid dynamics and heart development, a number of studies suggest that fluids may play a key role in morphogenic signaling. For example, fluid shear stress may trigger biochemical cascades within the endothelial cells of the developing heart that regulate chamber and valve morphogenesis. Myocardial activity generates forces on the intracardiac blood, creating pressure gradients across the cardiac wall. These pressures may also serve as epigenetic signals. In this article, the fluid dynamics of the early stages of heart development is reviewed. The relevant work in cardiac morphology, muscle mechanics, regulatory networks, and electrophysiology is also reviewed in the context of intracardial fluid dynamics.

CONSTITUTIVE EQUATION OF BLOOD: A MESOMECHANICAL MODEL

The general author's mesomechanical concept is applied to constitutive modeling of rheological behavior of human blood. The thermomechanical properties of blood are very complicated and their mathematical modeling — although studied for many years — deserves still attention, as phenomenological descriptions do not seem to achieve the due generality and accuracy. Our mesomechanical approach is rooted within the mesoscale description of the substructure of erythrocyte aggregates and their changes in the course of different time-dependent loading paths. Good agreement of the outcomes of our model with experimental findings received under complicated conditions corroborates the usefulness of this approach.

Predicting human blood viscosity in silico

The viscosity of blood has long been used as an indicator in the understanding and treatment of disease, and the advent of modern viscometers allows its measurement with ever-improving clinical convenience. However, these advances have not been matched by theoretical developments that can yield a quantitative understanding of blood's microrheology and its possible connection to relevant biomolecules (e.g., fibrinogen). Using coarse-grained molecular dynamics and two different red blood cell models, we accurately predict the dependence of blood viscosity on shear rate and hematocrit. We explicitly represent cell-cell interactions and identify the types and sizes of reversible rouleaux structures that yield a tremendous increase of blood viscosity at low shear rates. We also present the first quantitative estimates of the magnitude of adhesive forces between red cells. In addition, our simulations support the hypothesis, previously deduced from experiments, of yield stress as an indicator of cell aggregation. This non-Newtonian behavior is analyzed and related to the suspension's microstructure, deformation, and dynamics of single red blood cells. The most complex cell dynamics occurs in the intermediate shear rate regime, where individual cells experience severe deformation and transient folded conformations. The generality of these cell models together with single-cell measurements points to the future prediction of blood-viscosity anomalies and the corresponding microstructures associated with various diseases (e.g., malaria, AIDS, and diabetes mellitus). The models can easily be adapted to tune the properties of a much wider class of complex fluids including capsule and vesicle suspensions.

Movement of a spherical cell in capillaries using a boundary element method

This study aims to investigate the translation and rotation of a spherical particle in capillaries and overcomes limitations in previous studies by using a boundary element method. The capillary, a straight cylindrical tube, is filled with a Newtonian viscous fluid. A spherical particle is arbitrarily positioned in the capillary either co-centrically or eccentrically and is free to translate and rotate. Flow in the capillary is first assumed to be caused solely by the movement of the sphere under the gravity. When a steady state is reached, the net force and torque on the sphere are zero. The translating velocity and rotation of the particle are calculated from equilibrium equations. For a co-centric sphere, our result agrees to Bohlin's analytical solution (Bohlin, 1960) and the difference is less than 1%. For an eccentrically positioned sphere in the tube, there are no analytical solutions unless the eccentricity is infinitesimal. Results by boundary element method (BEM) give an improved estimations on the velocity and rotation of the sphere than earlier results by a boundary singularity method (BSM), particularly when the clearance between the tube and the sphere becomes small. Movement of a spherical particle in a capillary driven by a pressure gradient is further investigated, which has closer relevance to movement of blood cells in capillaries. The current study using BEM enables investigation on cell movement in close proximities of the capillary wall.

Shear rate and viscosity corrections for a Casson fluid in cylindrical (Couette) geometries

Shear rate and viscosity correction factors for a Casson fluid in cylindrical (Couette) geometries--type I, concentric cylinder with a narrow gap, and type II, cylinder in a semi-infinite medium--are developed in this work. The analytical correction factors compare well with published data corrected by general methods. Corrected viscosities were also compared to viscosity-shear rate data on chocolate in an absolute viscometer (cone-and-plate), which requires no corrections. This comparison showed excellent agreement of the corrected data in the higher shear rate range, but not very good agreement in the lower shear rate range. Reasons for this are discussed. In addition, a Reiner-Riwlin type formulation was developed for determination of model constants directly from rotational viscometry data without the need to correct for non-Newtonian shear rate, but this method is less accurate than existing slope-intercept methods for determining model constants. The analytical correction procedure developed here is expected to be useful wherever the Casson model is used to describe the rheology of a fluid, particularly one exhibiting yield-stress pseudoplastic behavior.

Determination of in vivo velocity and endothelial shear stress patterns with phasic flow in human coronary arteries: a methodology to predict progression of coronary atherosclerosis

BACKGROUND

Although the coronary arteries are equally exposed to systemic risk factors, coronary atherosclerosis is focal and eccentric, and each lesion evolves in an independent manner. Variations in shear stress elicit markedly different humoral, metabolic, and structural responses in endothelial cells. Areas of low shear stress promote atherosclerosis, whereas areas of high shear stress prevent atherosclerosis. Characterization of the shear stresses affecting coronary arteries in humans in vivo may permit prediction of progression of coronary disease, prediction of which plaques might become vulnerable to rupture, and prediction of sites of restenosis after percutaneous coronary intervention.

METHODS

To determine endothelial shear stress, the 3-dimensional anatomy of a segment of the right coronary artery was determined immediately after directional atherectomy by use of a combination of intracoronary ultrasound and biplane coronary angiography. The geometry of the segment was represented in curvilinear coordinates and a computational fluid dynamics technique was used to investigate the detailed phasic velocity profile and shear stress distribution. The results were analyzed with several conventional indicators and one novel indicator of disturbed flow.

RESULTS

Our methodology identified areas of minor flow reversals, significant swirling, and large variations of local velocity and shear stress--temporally, axially, and cirumferentially--within the artery, even in the absence of significant luminal obstruction.

CONCLUSIONS

We have described a system that permits, for the first time, the in vivo determination of pulsatile local velocity patterns and endothelial shear stress in the human coronary arteries. The flow phenomena exhibit characteristics consistent with the focal nature of atherogenesis and restenosis.

Rheology of blood by NMR

Pipe flow of blood in tubes of 1 and 7 mm inner diameter, respectively, was investigated employing two-dimensional NMR velocity imaging and PFG propagator measurements at different Reynolds numbers between 10 and 3500. The results are compared to flow of a water/glycerol mixture of matching viscosity under identical conditions. The transition from laminar to turbulent flow is observed by both a flattening of the velocity profile and a change of the propagator shape. For blood flow this transition is found to be shifted toward higher Reynolds numbers as compared to the transition of the water/glycerol mixture. This observation is in agreement with predictions from hydraulic measurements and is a consequence of the non-Newtonian flow characteristics of blood as a suspension of erythrocytes and plasma. Likewise, a deviation from the laminar flow condition is observed for blood at low Reynolds numbers between 10 and 100. This phenomenon is unknown for Newtonian liquids and is explained by the onset of a geometrical arrangement of the erythrocytes, the so-called rouleaux effect.

Blood yield stress in systemic sclerosis

Blood is a weak percolating physical gel at low shear rates, in which clusters of aggregates can be reversibly disaggregated or formed again. This phenomenon is of potential importance in the microvascular pathophysiology of ischemic and vasospastic disorders such as systemic sclerosis. The aim of this work was to determine blood yield stress using low-shear-rate rheometry with a homemade roughened Couette device in 10 patients with systemic sclerosis compared with 10 healthy controls. Biochemical plasmatic parameters were assessed independently. Results showed a significantly increased stress (+56%, P < 0.05 at 60% hematocrit) for scleroderma patients. The best biochemical predictor for yield stress was the ratio of albumin to globulins; 69% of its variance was explained by plasmatic factors (albumin, fibrinogen, and globulins) in scleroderma patients and 23.4% in healthy controls. Additional microscopic observations showed different microstructures. These results support the hypothesis of an abnormal red blood cell organization process in scleroderma patients that could be partly responsible for the severity of ischemic complications of the disease.

Fluid-structure interaction problems in bio-fluid mechanics: a numerical study of the motion of an isolated particle freely suspended in channel flow

In this paper a problem belonging to the moving boundary class is tackled with a 2-D application of computational fluid dynamics techniques. The motion of an isolated rigid particle freely suspended in an incompressible Newtonian fluid in a narrow channel is studied numerically at a low Reynolds number, yet different from zero. The actual problem consists of two coupled problems: the motion of the viscous fluid and that of the rigid particle suspended and convected with the fluid. The full Navier-Stokes equations (i.e. both transient and convective terms are included) are solved in the fluid domain by means of the finite element method, while the motion of the particle is determined on the basis of a rigid act of motion. Results from simulations corresponding to differential initial positions of the particle are shown in this paper: they allow one to study the rotational motions of the particle as well as its displacements. The goal of the paper is to analyse the lateral displacement behaviour of the particle, already observed in experimental studies in microcirculation. In particular, lateral migrations are supposed to be due to inertial forces acting in the fluid around the moving particle combined with the proximity of the resting wall (wall effect). Preliminary results are in fairly good agreement with those available in the literature.

A non-Newtonian fluid model for blood flow through arteries under stenotic conditions

This paper presents an analytical study on the behaviour of blood flow through an arterial segment having a mild stenosis. The artery has been treated as a thin-walled initially stressed orthotropic non-linear viscoelastic cylindrical tube filled with a non-Newtonian fluid representing blood. The analysis is restricted to propagation of small-amplitude harmonic waves, generated due to blood flow whose wave length is large compared to the radius of the arterial segment. For the equations of motion of the arterial wall consideration is made of a pair of appropriate equations derived by using suitable constitutive relations and the principle of superimposition of a small additional deformation on a state of known finite deformation. It has been shown through numerical computations of the resulting analytical expressions that the resistance to flow and the wall shear increase as the size of the stenosis increases. A quantitative analysis is also made for the frequency variation of the flow rate at different locations of the artery, as well as of the phase velocities and transmission per wavelength.

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

A two-dimensional numerical model was developed to determine the effect of the non-Newtonian behavior of blood on a pulsatile flow at the aortic bifurcation. The blood rheology was described by a weak-form Casson equation. The successive-over-relaxation (SOR) method was used to solve both the vorticity and Poisson equations numerically. It was disclosed that the non-Newtonian property of blood did not drastically change the flow patterns, but caused an appreciable increase in the shear stresses and a slightly higher resistance to both flow separations and the phase shifts between flow layers.

A study on the constitutive equation of blood

This paper proposes and studies a new three-parameter constitutive equation for whole human blood, tau = tau y+eta 2 gamma 1/2 + eta 1 gamma. The model aims at a proper description of the shear thinning behavior of blood at both low and high shear rates. While empirically based, it relies on continuum constitutive theories. The model has been verified by fitting the experimental data available in the literature using the weighted least squares. Results show that the proposed model fits the experimental data with nearly constant parameters in a wide shear range, and with average deviations epsilon less than 6.24%. Formulae to calculate the velocity profile and flow rate of the proposed model in a straight tube flow were deduced. Compared to Casson's and Newtonian models, it is concluded that the proposed model is more effective in describing the shear thinning behavior of blood within a wide shear range.

Hemorheological aspects of the pathophysiology of cerebral ischemia

In summary, these considerations show that the hemodynamic role of blood rheology in cerebrovascular insufficiency or stroke is still not clear. However, the arguments and data presented allow some conclusions that are of practical and theoretical value in the process leading to further clarification: Changes of blood viscosity as determined by coaxial viscometry cannot be used to predict quantitatively the changes of cerebral blood flow occurring. It is likely that characteristic rheological phenomena such as red cell aggregation and deformability, plasma viscosity, and protein composition are more important for the rheological aspects of microcirculatory supply function than is reflected by their contribution to macroscopic viscosity of the blood. Further studies of blood rheology in stroke should therefore certainly include quantification of those rheological properties that are more closely related to the in vivo flow behavior of blood and thus the magnitude and distribution of flow in the cerebral microcirculation.

Effect of shear rate variation on apparent viscosity of human blood in tubes of 29 to 94 microns diameter

In order to test the hypothesis that the increase of vascular resistance observed in vivo at low flow rates is due in part to blood rheological properties, the apparent viscosity of human blood was measured in small tubes in a range of shear rates. Pressure-flow relationships were obtained in vertical glass tubes (29 to 94 microns i.d.) perfused with blood at hematocrits between 0.13 and 0.65. Viscosity of blood and plasma was calculated using Poiseuille's law. With the exception of data obtained in the largest tube at a hematocrit of 0.6, relative blood viscosity was found to be independent of shear rate in the range between 1 and 120 s-1. Microscopic observation revealed pronounced red cell aggregation at low shear rates. Velocity profiles obtained by the use of fluorescence-labelled red cells showed increased blunting with decreasing shear rate. The Fahraeus-Lindqvist effect was evident in a reduction of viscosity with tube size at a given feed hematocrit. The observed constancy of apparent blood viscosity with decreasing shear is attributed to the opposing effects of a cell-depleted marginal layer and red cell aggregation or deformation in the cell core. The findings indicate that the increase of vascular resistance at low arterial pressure cannot be explained by shear-dependent changes of apparent blood viscosity observed in macroviscometers.

Microrheology and light transmission of blood. IV. The kinetics of artificial red cell aggregation induced by Dextran

Employing both microscopic and photometric methods the rheology of pathological red cell aggregation was studied in model experiments. Suspensions of washed human red blood cells in dextran solutions containing rising concentrations of dextrans (M.W. 40000, 70000, 110000, 250000, 500000) were used. At low concentrations (less than 500 mg-%) of high molecular weight dextrans (greater than 70000) red cell suspensions formed aggregates similar to the ones found in normal human blood. At higher concentrations, the aggregates were similar to those observed in pathological human blood. The aggregates were studied under the condition of stasis, slow flow and at shear rate of their hydrodynamic dispersion. Besides, the flow behavior of the dispersed cells at high shear rates was studied. We found: 1. In all samples the rate of spontaneous aggregate re-formation in stasis (following hydrodynamic desaggregation) rose with rising dextran concentration up to 5.0 g-%. 2. The shear resistance of the aggregates, as measured by the shear stress necessary to keep them dispersed, rose up to concentrations of 2.5g-%, but fell at higher concentrations. 3. Only with dextran of a molecular weight above 110000 coarse agglomerates could be produced at high concentrations. Loose elastic meshes were rapidly produced at high concentrations of Dx 70. 4. When subjected to steady state low shear (m sec-1) only the agglomerates, but not the meshes rapidly grew in size. Most of the aggregation kinetics recorded by photometry and microscopy evaded detection by viscometry.

The microcirculation in shock

Some aspects of blood fluidity in hypoperfusion with particular reference to the important role of the red cell are discussed. Changes in different organ beds and the internal fluidity of the red cell, which is of obvious importance in capillary flow are not considered.

It is not certain to what extent the phenomena discussed are critical factors in morbidity and mortality in shock because of paucity of objective data but there are good reasons for assuming them to be of importance and for directing treatment towards their correction.

The properties of Dextran 40 applicable to the therapy of disordered blood fluidity are described.

Rheological comparison of hemoglobin solutions and erythrocyte suspensions

Hemoglobin solutions prepared from hemolyzed human erythrocyte packs have Newtonian flow properties. Diluted solutions are also Newtonian. All solutions have a viscosity lower than the apparent viscosity of erythrocyte suspensions of equal oxygen-carrying capacity. The presence of cell debris in hemoglobin solutions causes non-Newtonian (pseudoplastic or rheopectic) flow behavior.