Theoretical and experimental analysis of the sedimentation kinetics of concentrated red cell suspensions in a centrifugal field: determination of the aggregation and deformation of RBC by flux density and viscosity functions

The flow properties of blood are mostly determined using various viscometric approaches, and described in terms of a shear rate or shear stress dependent apparent viscosity. The interpretation of results are rather difficult, especially at low shear rates when particle sedimentation and migration within the viscometer gap are significant. By contrast, analysing the separation process in concentrated RBC suspensions in a centrifugal field also yields information about the viscosity function, including particle-particle interaction and deformation parameters. In this paper, the sedimentation process is approached by means of the theory of kinematic waves and theoretically described by solving the corresponding one-dimensional quasi-linear partial differential equation based on viscosity/flow function as a function of volume concentration. The sedimentation kinetics of rigid spherical RBC suspended in saline and normal RBC suspended in Dx-saline solutions were investigated by means of a separation analyser (LUMiFuge 114). The instrument detects the light transmission over the total length of the cell containing the suspension. During centrifugation the analyser automatically determines the position of the particle free fluid/suspension interface or the sediment by means of a special algorithm. The data obtained with sedimentation of rigid spherical RBC at different volume concentrations demonstrate that, in the case of suspensions rotated in containers of constant cross section, there is good agreement between the theory of kinematic waves developed by Anestis and Schneider (1983) and the results of the experiments. Such good agreement was obtained even though a restrictive one-dimensional model was used to obtain the theoretically derived sedimentation time course. In addition, we describe an algorithm enabling the experimental determination of the viscosity and related flux density function to be made for any suspension. Through this approach, we investigated in detail the rheological behavior of suspended rigid spheres at low Reynolds numbers ranging from 10(-6) to 10(-3). The method here introduced also enabled us to investigate RBC suspensions with respect to the deformability and interactions of the cells by means of the separation analysis. Normal, rigid as well as aggregating RBC exhibited marked differences in the sedimentation kinetics, which were quantified by means of the flux and viscosity functions based on the theory of kinematic waves.     

Transient lateral transport of platelet-sized particles in flowing blood suspensions.

Concentration profiles of 2.5 microns latex beads were measured to demonstrate lateral transport of platelet-sized objects in flows of blood suspensions; the flows had equivalent Poiseuille wall shear rates (WSRs) from 250 to 1220 s-1. Each experimental trial began with a steady flow of suspension without beads in a thin-walled capillary tube (219 microns ID; 10.2 microns SD). The tube entrance was then switched to a reservoir containing suspension of equal hematocrit, but with beads, for a short interval of flow at the same WSR. This process established a paraboloidal tongue of labeled suspension with a transient concentration gradient at its surface. The tube and contents were rapidly frozen to fix the suspended particles in flow-determined locations. Segments of frozen tube were collected at distances from the entrance corresponding to 13%, 39%, and 65% of the axial extent of the ideal paraboloidal tongue. Concentration profiles were estimated from distances measured on fluorescence microscope images of cross-cut tube segments. Experiments used tubes either 40 or 50 cm long, suspension hematocrits of 0, 15, or 40%, and bead concentrations in the range of 1.5-2.2 x 10(5)/mm3. Profiles for 0% hematocrit suspension, a dilute, single-component suspension, had features expected in normal diffusive mixing in a flow. Distinctly different profiles and more lateral transport occurred when the suspensions contained red cells; then, all profiles for 13% extent had regions of excess bead concentration near the wall. Suspension flows with 40% hematocrit exhibited the largest amount of lateral transport. A case is made that, to a first approximation, the rate of lateral transport grew linearly with WSR; however, statistical analysis showed that for 40% hematocrit, less lateral transport occurred when the WSR was 250 s-1 or 1220 s-1 than 560 s-1, thus indicating that the rate behavior is more complex.     

Mixed Exciton-Charge-Transfer States in Photosystem II: Stark Spectroscopy on Site-Directed Mutants

Sickle erythrocytes exhibit abnormal morphology and membrane mechanics under deoxygenated conditions due to the polymerization of hemoglobin S. We employed dissipative particle dynamics to extend a validated multiscale model of red blood cells (RBCs) to represent different sickle cell morphologies based on a simulated annealing procedure and experimental observations. We quantified cell distortion using asphericity and elliptical shape factors, and the results were consistent with a medical image analysis. We then studied the rheology and dynamics of sickle RBC suspensions under constant shear and in a tube. In shear flow, the transition from shear-thinning to shear-independent flow revealed a profound effect of cell membrane stiffening during deoxygenation, with granular RBC shapes leading to the greatest viscosity. In tube flow, the increase of flow resistance by granular RBCs was also greater than the resistance of blood flow with sickle-shape RBCs. However, no occlusion was observed in a straight tube under any conditions unless an adhesive dynamics model was explicitly incorporated into simulations that partially trapped sickle RBCs, which led to full occlusion in some cases.     

Interaction forces between red cells agglutinated by antibody. I. Theoretical.

A general method of calculating forces, torques, and translational and rotational velocities of rigid, neutrally buoyant spheres suspended in viscous liquids undergoing a uniform shear flow has been given by Arp and Mason (1977). The method is based on the matrix formulation of hydrodynamic resistances in creeping flow by Brenner and O'Neill (1972). We describe the solution of the Brenner-O'Neill force-torque vector equation in terms of the particle and external flow field coordinates and derive expressions for the normal force acting along, and the shear force acting perpendicular to, the axis of the doublet of spheres, the latter explicitly given for the first time. The equations consist of a term comprising force and torque coefficients obtained from the matrices of the hydrodynamic resistances (functions of the distance h between sphere surfaces which have been computed), and terms comprising the orientation of the doublet axis relative to the coordinates of the external flow field and the shear stress (which can be experimentally determined). We have applied the theory to a system of doublets of sphered, hardened human red cells of group A or B antigenic type cross-linked by the corresponding antibody at a fixed interparticle distance. Working from studies of the breakup of doublets of red cells in an accelerating Poiseuille flow, given in the succeeding paper, we are able to compute the hydrodynamic force required to separate the two spheres. Previous work has shown that the theory can be applied to doublets in a variable shear, Poiseuille flow, provided the ratio of particle to tube diameter is small. In calculating the force-torque coefficients it was assumed that the cells are crosslinked by antibody with h = 20 nm.     

Development of wall surface tangent DPIV measurement techniques for arterial branch models.

Experimental techniques for measuring unsteady flow in a glass arterial bifurcation model have been developed to aid in quantifying three-dimensional wall shear fluctuations associated with arterial disease. The unique feature of the current technique is the use of a "curved" laser sheet, which was everywhere tangent to the inner wall of a daughter tube in an arterial bifurcation model. Surface tangent velocity vector field measurements were made to demonstrate the potential of this technique. Ensemble-averaged data showing weak secondary flows as well as statistical distributions of flow angles are presented. Measurements of this type may be used to estimate mean and instantaneous wall shear magnitude and direction, data that are necessary for understanding the importance of circumferential motions on arterial disease.     

On the in vivo material functions of mammalian blood

The present paper explores the implications of employing time-averaged true continuum fields to investigate blood rheology in "steady viscometric" flows. This approach is in contrast with the spatially-averaged interpretation of field variables which is generally employed. On the basis of four plausible constitutive assumptions it is then possible to deduce the qualitative in vivo behavior of all three of the material functions of whole mammalian blood from inspection of the corresponding velocity profile. Quantitative results, and the evaluation of the material constants for specific constitutive models, can be obtained through curve-fitting procedures, as is illustrated. The development reconfirms, and puts on a formal basis, the earlier conclusion of Bugliarello, et al, that whole blood can have a dilatant response at low rates of shear. In addition, the normal stress forces are shown to have off-axis extrema in tube flow, and to be large enough to influence particle migration across streamlines. The existing data on particle migration in whole blood is reviewed, and shown to be in accord with these results.     

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.     

An interpretation of partial plug flow of concentrated suspensions.

The partial plug flow of a concentrated suspension of rigid particles in a circular tube has been previously studied experimentally. It has been shown that a central core may exist in which the mean velocities of the particles and the suspending fluid are equal and constant within a cylindrical core of the flow. This behavior has been attributed to hydrodynamic interaction of the particles within the core. In the present analysis this interaction is interpreted in terms of passing vs. non-passing motions of adjacent particles. A hypothesis of a critical parameter alpha c involving the shear stress and the pressure gradient is explored and a new form of the relation of core diameter to particle size and concentration is developed based on alpha c.     

Flush-mounted hot film anemometer accuracy in pulsatile flow

The accuracy of a flush-mounted hot film anemometer probe for wall shear stress measurements in physiological pulsatile flows was evaluated in fully developed pulsatile flow in a rigid straight tube. Measured wall shear stress waveform based on steady flow anemometer probe calibrations were compared to theoretical wall shear stress waveforms based on well-established theory and measured flow rate waveforms. The measured and theoretical waveforms were in close agreement during systole (average deviation of 14 percent at peak systole). As expected, agreement was poor during diastole because of flow reversal and diminished frequency response at low shear rate.     

Influence of erythrocyte aggregation on radial migration of platelet-sized spherical particles in shear flow

Blood platelets when activated are involved in the mechanisms of hemostasis and thrombosis, and their migration toward injured vascular endothelium necessitates interaction with red blood cells (RBCs). Rheology co-factors such as a high hematocrit and a high shear rate are known to promote platelet mass transport toward the vessel wall. Hemodynamic conditions promoting RBC aggregation may also favor platelet migration, particularly in the venous system at low shear rates. The aim of this study was to confirm experimentally the impact of RBC aggregation on platelet-sized micro particle migration in a Couette flow apparatus. Biotin coated micro particles were mixed with saline or blood with different aggregation tendencies, at two shear rates of 2 and 10 s⁻¹ and three hematocrits ranging from 20 to 60%. Streptavidin membranes were respectively positioned on the Couette static and rotating cylinders upon which the number of adhered fluorescent particles was quantified. The platelet-sized particle adhesion on both walls was progressively enhanced by increasing the hematocrit (p < 0.001), reducing the shear rate (p < 0.001), and rising the aggregation of RBCs (p < 0.001). Particle count was minimum on the stationary cylinder when suspended in saline at 2 s⁻¹ (57 ± 33), and maximum on the rotating cylinder at 60% hematocrit, 2 s⁻¹ and the maximum dextran-induced RBC aggregation (2840 ± 152). This fundamental study is confirming recent hypotheses on the role of RBC aggregation on venous thrombosis, and may guide molecular imaging protocols requiring injecting active labeled micro particles in the venous flow system to probe human diseases.     

Flow around cells adhered to a microvessel wall II: comparison to flow around adherent cells in channel flow

The flow around adherent cells in a parallel-plate channel and that in a circular cylindrical tube are numerically analyzed, and their effects on the adherent cells are compared. The cells are modeled as rigid spherical particles and they are assumed to be attached to a wall of a 2D channel uniformly in a square array, or a wall of a circular tube regularly in a line along the tube axis. It is found that, when the size ratios of the particle-to-channel height and the particle-to-tube diameter are smaller than approximately 0.2, the distributions of the shear stress and the pressure exerted on the surface of an adherent particle as well as the drag force and torque acting on it compare favorably in the 2D channel flow and tube flow. As the size ratios increase from 0.2, the differences between the 2D channel and the tube increase drastically, especially when separation distances between neighboring particles are large.     

Three-dimensional focusing of red blood cells in microchannel flows for bio-sensing applications

Three-dimensional (3D) focusing of particles in microchannels has been a long-standing issue in the design of biochemical/biomedical microdevices. Current microdevices for 3D cell or bioparticle focusing involve complex channel geometries in view of their fabrication because they require multiple layers and/or sheath flows. This paper proposes a simple method for 3D focusing of red blood cells (RBCs) in a single circular microcapillary, without any sheath flows, which is inspired from the fluid dynamics phenomenon in that a spherical particle lagging behind a Poiseuille flow migrates toward and along the channel axis. More explicitly, electrophoresis of RBCs superimposed on the pressure-driven flow is utilized to generate an RBC migration mode analogous to this phenomenon. A particle-tracking scheme with a sub-pixel resolution is implemented to spatially position red blood cells flowing through the channel, so that a probability density function (PDF) is constructed to evaluate the tightness of the cell focusing. Above a specific strength of the electric field, approximately 90% of the sheep RBCs laden in the flow are tightly focused within a beam diameter that is three times the cell dimension. Particle shape effect on the focusing is discussed by making comparisons between the RBCs and the spherical particles. The lateral migration velocity, predicted by an existing theoretical model, is in good agreement with the present experimental data. It is noteworthy that 3D focusing of non-spherical particles, such as RBCs, has been achieved in a circular microchannel, which is a significant improvement over previous focusing methodologies.     

Asymmetric flows of spherical particles in a cylindrical tube

To study the rheological behavior of blood cells in various flow patterns through narrow vessels, we analyzed numerically the motion of blood cells arranged in one row or two rows in tube flow, at low Reynolds numbers. The particles are assumed to be identical rigid spheres placed periodically along the vessel axis at off-axis positions with equal spacings. The flow field of the suspending fluid in a circular cylindrical tube is analyzed by a finite element method applied to the Stokes equations, and the motion of each particle is simultaneously determined by a force-free and torque-free condition. In both cases of single- and two-file arrangements of the particles, their longitudinal and angular velocities are largely affected by the radial position and the axial spacing between neighboring particles. The apparent viscosity of the asymmetric flows is higher than that of the symmetric flow where particles are located on the tube centerline, and this is more pronounced when particles are placed farther from the tube centerline and when the axial distance between neighboring particles is reduced.     

Conductometric study of shear-dependent processes in red cell suspensions. II. Transient cross-stream hematocrit distribution

A novel experimental approach based on electrical properties of red blood cell (RBC) suspensions was applied to study the effects of the size and morphology of RBC aggregates on the transient cross-stream hematocrit distribution in suspensions flowing through a square cross-section flow channel. The information about the effective size of RBC aggregates and their morphology is extracted from the capacitance (C) and conductance (G) recorded during RBC aggregation, whereas a slower process of particle migration is manifested by delayed long-term changes in the conductance. Migration-induced changes in the conductance measured at low shear rates (< or =3.1 s(-1)) for suspensions of RBCs in a strongly aggregating medium reveal an increase to a maximum followed by a decrease to the stationary level. The ascending branch of G(t) curves reflects the aggregate migration in the direction of decreasing shear rate. A further RBC aggregation in the region of lower shear stresses leads to the formation of RBC networks and results in the transformation of the rheological behavior of suspensions from the thinning to the thickening. It is suggested that the descending branches of the G(t) curves recorded at low shear rates reflect an adjustment of the Hct distribution to a new state caused by a partial dispersion of RBC networks. For suspensions of non-aggregating RBCs it is found that depending on whether the shear rate is higher or lower compared with the prior value, individual RBCs migrate either toward the centerline of the flow or in the opposite direction.     

Numerical study on flows of red blood cells with liposome-encapsulated hemoglobin at microvascular bifurcation

Flow analysis at microvascular bifurcation after partial replacement of red blood cell (RBC) with liposome-encapsulated hemoglobin (LEH) was performed using the lattice Boltzmann method. A two-dimensional symmetric Y bifurcation model with a parent vessel diameter of 20 mum and daughter branch diameters of 20 microm was considered, and the distributions of the RBC, LEH, and oxygen fluxes were calculated. When only RBCs flow into the daughter branches with unevenly distributed flows, plasma separation occurred and the RBC flow to the lower-flow branch was disproportionately decreased. On the other hand, when half of RBC are replaced by LEH, the biasing of RBC flow was enhanced whereas LEH flowed favorably into the lower-flow branch, because many LEH within the parent vessel are suspended in the plasma layer, where no RBCs exist. Consequently, the branched oxygen fluxes became nearly proportional to flows. These results indicate that LEH facilitates oxygen supply to branches that are inaccessible to RBCs.     

Developing steady laminar flow through uniform straight tubes with varying wall cross curvature

Numerical calculations are used to determine not only the wall shear stress but also the entry length in a laminar steady flow of an incompressible Newtonian fluid. The fluid is conveyed through rigid straight tubes with axially uniform cross sections, which mimic collapsed vessels. For each tube configuration, the "Navier-Stokes" equations are solved using the finite element method. The numerical tests are performed with the same value of the volume flow-rate whatever the tube configuration for three "Reynolds numbers". The wall shear stress is computed and determined along the axis of the tube, then the entry length is estimated by introducing two indexes by using: (i) the axial fluid velocity, and (ii) the wall shear stress. The results are analysed in order to exhibit the mechanical environment of cultured endothelial cells in the flow chamber for which the test conditions will be well-defined. For example, in a tube configuration where the opposite walls are in contact for which the inner perimeter and the area of the cross section are respectively given by 45 mm and 37.02 mm(2), the computed entry lengths with the criteria defined by (i) and (ii) are equals to about 118 and 126 mm, respectively for R(e0) = 500.     

Stability of particle motions in a narrow channel flow

Single rows or two rows of identical circular cylinders spaced regularly in a narrow channel flow have been shown to be capable of steady flow provided the cylinders are located at equal lateral positions and with equal spacings in the flow direction. The stability of such steady flows of circular cylinders is studied for periodic perturbations of the particle positions, assuming that every other cylinder is equally perturbed in lateral position and spacing along the channel. This results in two rows which are not symmetrically placed. The suspending fluid is assumed to be an incompressible Newtonian fluid. It is assumed that no external forces or moments act on the cylinders and the effects of inertia forces on the motion of the fluid and the cylinders are negligible. The velocity field of the suspending fluid and the instantaneous velocities of the cylinders are computed by the finite element method. The translational velocities of the cylinders are obtained for a large number of particle positions, from which the trajectories of their relative motion are determined for various initial positions near the regular single-file and two-file arrangements. It is shown that when the initial arrangements of the cylinders are slightly perturbed from the regular (alternating) two-file flows, the trajectories of their relative motions become small closed loops around the regular two-file arrangements. In contrast, such small closed trajectories are not obtained when they start from the arrangements near the regular single-file flows or regular (symmetric) double-file flows, suggesting that these flows are unstable under the conditions examined.     

Shear‐induced diffusion of red blood cells measured with dynamic light scattering‐optical coherence tomography

Quantitative measurements of intravascular microscopic dynamics, such as absolute blood flow velocity, shear stress and the diffusion coefficient of red blood cells (RBCs), are fundamental in understanding the blood flow behavior within the microcirculation, and for understanding why diffuse correlation spectroscopy (DCS) measurements of blood flow are dominantly sensitive to the diffusive motion of RBCs. Dynamic light scattering‐optical coherence tomography (DLS‐OCT) takes the advantages of using DLS to measure particle flow and diffusion within an OCT resolution‐constrained three‐dimensional volume, enabling the simultaneous measurements of absolute RBC velocity and diffusion coefficient with high spatial resolution. In this work, we applied DLS‐OCT to measure both RBC velocity and the shear‐induced diffusion coefficient within penetrating venules of the somatosensory cortex of anesthetized mice. Blood flow laminar profile measurements indicate a blunted laminar flow profile and the degree of blunting decreases with increasing vessel diameter. The measured shear‐induced diffusion coefficient was proportional to the flow shear rate with a magnitude of ~0.1 to 0.5 × 10^?6 mm². These results provide important experimental support for the recent theoretical explanation for why DCS is dominantly sensitive to RBC diffusive motion.      

Finite element analysis of the lift on a slightly deformable and freely rotating and translating cylinder in two-dimensional channel flow

Motivated by the lateral migration phenomena of fresh and glutaraldehyde-fixed red blood cells in a field flow fractionation (FFF) separation system, we studied the transverse hydrodynamic lift on a slightly flexible cylinder in a two-dimensional channel flow. The finite element method was used to analyze the flow field with the cylinder at different transverse locations in the channel. The shape of the cylinder was determined by the pressure on the surface of the cylinder from the flow field solution and by the internal elastic stress. The cylinder deformation and the flow field were solved simultaneously. The transverse lift exerted on the cylinder was then calculated. The axial and angular speed of the cylinder were iterated such that the drag and torque on the cylinder were nulled to represent a freely translating and rotating state. The results showed that the transverse lift on a deformable cylinder increased greatly and the equilibrium position moved closer to the center of the channel compared to a rigid cylinder. Also, with the same elastic modulus but a higher flow rate, a larger deformation and higher equilibrium location were found. The maximum deformation of the cylinder occurred when the cylinder was closest to the wall where a larger shear rate existed. The numerical results and experimental studies are discussed.