Effects of shear rate and suspending viscosity on deformation and frequency of red blood cells tank-treading in shear flows

The tank-treading rotation of red blood cells (RBCs) in shear flows has been studied extensively with experimental, analytical, and numerical methods. Even for this relatively simple system, complicated motion and deformation behaviors have been observed, and some of the underlying mechanisms are still not well understood. In this study, we attempt to advance our knowledge of the relationship among cell motion, deformation, and flow situations with a numerical model. Our simulation results agree well with experimental data, and confirm the experimental finding of the decrease in frequency/shear-rate ratio with shear rate and the increase of frequency with suspending viscosity. Moreover, based on the detailed information from our simulations, we are able to interpret the frequency dependency on shear rate and suspending viscosity using a simple two-fluid shear model. The information obtained in this study thus is useful for understanding experimental observations of RBCs in shear and other flow situations; the good agreement to experimental measurements also shows the potential usefulness of our model for providing reliable results for microscopic blood flows.     

Motion and deformation of a red blood cell in a shear flow: a two-dimensional simulation of the wall effect

The motion and deformation of a single red blood cell in a simple shear flow between two parallel walls is studied theoretically. A two-dimensional deformable microcapsule is adopted as a model for the cell, which has a thin moving membrane, like a tank-tread, around the interior and is deformed into an elliptical shape with a constant area. Applying the finite element method to the Stokes equations, the tank-tread motion and deformation is determined in a stationary motion, under fluid dynamic interaction between the cell and the walls. It is shown that the motion and deformation of the microcapsule crucially depends on the channel width between the two walls. As the width decreases, the microcapsule is more elongated and the frequency of tank-tread motion decreases at a constant shear rate. In addition, the angle of inclination decreases at the low range of the viscosity ratio of internal to external fluids and increases at the high range. The results obtained are compared with experimental observations and applied to the behavior of cells under mutual interaction.     

Sticky chain model for shear response of red blood cells

A sticky chain model has been proposed to describe the unfolding of spectrin network under applied mechanical loads. With the model, the response of a red blood cell (RBC) under static and cyclic shear loading has been predicted, which agrees qualitatively with relevant experimental results.     

Wall shear stress in backward-facing step flow of a red blood cell suspension.

An experimental investigation of the wall shear stress distribution downstream of a backward-facing step is carried out. The wall shear stress distribution was determined by measuring the deformation of a gel layer, attached to the wall downstream of the step. Speckle pattern interferometry was applied to measure the deformation of the gel layer. The measured deformation, combined with the properties of the gel layer, served as an input for a finite element solid mechanics computation to determine the stress distribution in the gel layer. The wall shear stress, required to generate the measured deformation of the gel layer, was determined from these computations. A Newtonian buffer solution and a non-Newtonian red blood cell suspension were used as measuring fluids. The deformation of the gel layer was determined for a Newtonian buffer solution to evaluate the method and to obtain the properties of the gel layer. Subsequently, the wall shear stress distribution for the non-Newtonian red blood cell suspension was determined for three different flow rates. The inelastic non-Newtonian Carreau-Yasuda model served as constitutive model for the red blood cell suspension. Using this model, the velocity and wall shear stress distribution were computed by means of a finite element fluid mechanics computation. From the comparison between the numerical and the experimental results, it can be concluded that wall shear stresses, induced by the red blood cell suspension, can be modeled accurately by employing a Carreau-Yasuda model.     

The dynamics of shear disaggregation of red blood cells in a flow channel

Red blood cell (RBC) rouleaux were formed in a flow channel in the presence of 2 g/dl dextran (molecular weight 76,000). The partial separation of RBC rouleau doublets adhering to the floor of the flow channel in response to small oscillatory shear stresses was observed experimentally. Theoretical analyses on displacement and drag force were performed to determine whether the motion of the cell involves membrane rotation (i.e., rolling) or sliding. From the experimental data and the results of theoretical analyses, it is concluded that, under the conditions of the experiments, the RBCs in a doublet separate from each other by rolling, rather than sliding of the sheared cell.     

Tank treading of optically trapped red blood cells in shear flow

Tank-treading (TT) motion is established in optically trapped, live red blood cells (RBCs) held in shear flow and is systematically investigated under varying shear rates, temperature (affecting membrane viscosity), osmolarity (resulting in changes in RBC shape and cytoplasmic viscosity), and viscosity of the suspending medium. TT frequency is measured as a function of membrane and cytoplasmic viscosity, the former being four times more effective in altering TT frequency. TT frequency increases as membrane viscosity decreases, by as much as 10% over temperature changes of only 4°C at a shear rate of ∼43 s⁻¹. A threshold shear rate (1.5 ± 0.3 s⁻¹) is observed below which the TT frequency drops to zero. TT motion is also observed in shape-engineered (spherical) RBCs and those with cholesterol-depleted membranes. The TT threshold is less evident in both cases but the TT frequency increases in the latter cells. Our findings indicate that TT motion is pervasive even in folded and deformed erythrocytes, conditions that occur when such erythrocytes flow through narrow capillaries.     

The bulk rheology of close-packed red blood cells in shear flow

A theoretical analysis is made of the dynamical behavior and bulk rheology of close-packed red blood cell suspensions subjected to simple shear flow. The model for the polyhedral cell shapes and tank-treading membrane motion developed in the companion paper (1) is used. The flow in the thin lubricating plasma layers between cells is analyzed taking into account the mechanical properties of the membrane at the corner regions of sharp membrane curvature. This leads to predictions for the apparent viscosity as a function of hematocrit and shear rate. Good agreement with experimental results is obtained at moderate and high shear rates (above 20 s-1). At lower shear rates, a rapid rise in apparent viscosity has been found experimentally, and the mechanisms leading to this behavior are examined.     

Local deformation of human red blood cells in high frequency electric field

A method of local and general deformation of single erythrocytes by external forces in high-frequency electric field is described. The method allows the avoidance of any mechanical contact of the cell with electrodes. Under the action of the forces applied human erythrocytes change their shape and produce various membrane structures: long filopodia-like processes, retraction fibers and lamella-like structures. These structures are never formed by erythrocytes under normal conditions, but are typical for fibroblasts, macrophages and epithelium cells. By the method developed the elastic properties of spicules on the membranes of echinocytes were also studied. Deformation of echinocyte in high-frequency electric field leads to the smoothing out of spicules. However, after the electric field is turned off, echinocyte restores its initial forms including the number and localization of all initial spicules on the cell surface.     

Membrane stress and internal pressure in a red blood cell freely suspended in a shear flow.

Presented is an algorithm for the approximate calculation of the membrane stress distribution and the internal pressure of a steadily tank-treading red cell. The algorithm is based on an idealized ellipsoidal model of the tank-treading cell (Keller, S.R., and R. Skalak, 1982, J. Fluid Mech., 120:27-47) joined with experimental observations of projected length, width, and tank-treading frequency. The results are inexact because the membrane shape and velocity are assumed a priori, rather than being determined via appropriate material constitutive relations for the membrane; these results are, nevertheless, believed to be approximately correct, and show that internal pressure builds up slowly as cell elongation increases, rising more rapidly as the deformed cell approaches the limiting geometry of a prolate ellipsoid. The maximum shear stress resultant in the membrane was found to be below but approaching the yield point range at the highest shear rate applied.     

Deformation and fragmentation of human red blood cells in turbulent shear flow.

By means of glutaraldehyde fixation, human erythrocytes are "frozen" while suspended in turbulent shear flow. As the shearing is increased in steps from 100 to 2,500 dyn/cm2, the deformed cells evolve gradually toward a smooth ellipsoidal shape. At stresses above 2,500 dyn/cm2, approximately, fragmentation of the cells occurs with a concomitant increase in free hemoglobin content of the suspending medium. The photographic evidence suggests that the cells rupture in tension in the bulk flow.     

Tank-treading dynamics of red blood cells in shear flow: On the membrane viscosity rheology

In this article, extensive three-dimensional simulations are conducted for tank-treading (TT) red blood cells (RBCs) in shear flow with different cell viscous properties and flow conditions. Apart from recent numerical studies on TT RBCs, this research considers the uncertainty in cytoplasm viscosity, covers a more complete range of shear flow situations of available experiments, and examines the TT behaviors in more details. Key TT characteristics, including the rotation frequency, deformation index, and inclination angle, are compared with available experimental results of similar shear flow conditions. Fairly good simulation-experiment agreements for these parameters can be obtained by adjusting the membrane viscosity values; however, different rheological relationships between the membrane viscosity and the flow shear rate are noted for these comparisons: shear thinning from the TT frequency, Newtonian from the inclination angle, and shear thickening from the cell deformation. Previous studies claimed a shear-thinning membrane viscosity model based on the TT frequency results; however, such a conclusion seems premature from our results and more carefully designed and better controlled investigations are required for the RBC membrane rheology. In addition, our simulation results reveal complicate RBC TT features and such information could be helpful for a better understanding of in vivo and in vitro RBC dynamics.     

Heterogeneous response of microvascular endothelial cells to shear stress

We investigated changes in calcium concentration in cultured bovine aortic endothelial cells (BAECs) and rat adrenomedulary endothelial cells (RAMECs, microvascular) in response to different levels of shear stress. In BAECs, the onset of shear stress elicited a transient increase in intracellular calcium concentration that was spatially uniform, synchronous, and dose dependent. In contrast, the response of RAMECs was heterogeneous in time and space. Shear stress induced calcium waves that originated from one or several cells and propagated to neighboring cells. The number and size of the responding groups of cells did not depend on the magnitude of shear stress or the magnitude of the calcium change in the responding cells. The initiation and the propagation of calcium waves in RAMECs were significantly suppressed under conditions in which either purinergic receptors were blocked by suramin or extracellular ATP was degraded by apyrase. Exogenously applied ATP produced similarly heterogeneous responses. The number of responding cells was dependent on ATP concentration, but the magnitude of the calcium change was not. Our data suggest that shear stress stimulates RAMECs to release ATP, causing the increase in intracellular calcium concentration via purinergic receptors in cells that are heterogeneously sensitive to ATP. The propagation of the calcium signal is also mediated by ATP, and the spatial pattern suggests a locally elevated ATP concentration in the vicinity of the initially responding cells.     

Effects of shear rate and suspending medium viscosity on elongation of red cells tank-treading in shear flow

Elongation measurements of red cells subjected to simple shear flow are usually performed using a single suspending medium (viscosity η(0) ) and varying the mean shear rate (y). Such data are often plotted versus the shear stress (tau = no(y) suggesting that the elongation scales with τ. In this work, normal blood samples were tested in a rheoscope varying both η(0) and(y.). The ranges of (y.) were chosen to restrict the elongation of the red cells to low values where the behavior is dominated by their intrinsic properties. It was found that the elongation scales with [formula: see text] with s decreasing from two at η(0) = 20 mPas to unity at η(0) = 70 mPas. Above η(0) = 70 mPas, the elongation is therefore essentially determined by the membrane elasticity alone. A side observation was a large variation of the elongation both intraindividually and interindividually.     

Viscoelastic deformation response of red blood cells under conditions of oscillating centrifugal field

The red cell deformation under the conditions of oscillating centrifugal fields was studied. Experiments were carried out with a modified Cell-Elastometer operating in oscillating mode (0.02 to 0.30 Hz). Gravitational acceleration was sinusoidally modulated between 620 g and 2250 g. At low frequencies (below 0.08 Hz), native red cells followed the applied stress without delay. At 0.09 Hz and up, the cellular deformation was still periodical and included an additional perturbation due to intracellular movements. This perturbation was analysed and quantified. The influence of alterations on the erythrocyte membrane by diamide was analysed to verify the sensitivity of this method. On increasing the membrane stiffness with low concentrations of diamide, the response to oscillatory centrifugal stress was impaired characteristically in terms of amplitude deformation. Based on tangential and centrifugal accelerations, a physical model was developed that describes the basic observable changes on varying the oscillation frequency. From the data it can be concluded that viscoelastic properties of red cells can be analysed and quantified using oscillatory centrifugal accelerations. The described method can become a valid tool to differentiate between membrane alterations or intracellular viscous modifications.     

Dynamic adhesion of umbilical cord blood endothelial progenitor cells under laminar shear stress

Late outgrowth endothelial progenitor cells (EPCs) represent a promising cell source for rapid reendothelialization of damaged vasculature after expansion ex vivo and injection into the bloodstream. We characterized the dynamic adhesion of umbilical-cord-blood-derived EPCs (CB-EPCs) to surfaces coated with fibronectin. CB-EPC solution density affected the number of adherent cells and larger cells preferentially adhered at lower cell densities. The number of adherent cells varied with shear stress, with the maximum number of adherent cells and the shear stress at maximum adhesion depending upon fluid viscosity. CB-EPCs underwent limited rolling, transiently tethering for short distances before firm arrest. Immediately before arrest, the instantaneous velocity decreased independent of shear stress. A dimensional analysis indicated that adhesion was a function of the net force on the cells, the ratio of cell diffusion to sliding speed, and molecular diffusivity. Adhesion was not limited by the settling rate and was highly specific to α₅β₁ integrin. Total internal reflection fluorescence microscopy showed that CB-EPCs produced multiple contacts of α₅β₁ with the surface and the contact area grew during the first 20 min of attachment. These results demonstrate that CB-EPC adhesion from blood can occur under physiological levels of shear stress.     

Influence of microflow on hepatic sinusoid blood flow and red blood cell deformation

Hepatic sinusoids present complex anatomical structures such as the endothelial sieve pores and the Disse space, which govern the microscopic blood flow in the sinusoids and are associated with structural variations in liver fibrosis and cirrhosis. However, the contributions of the permeability of endothelial and collagen layers and the roughness of hepatocyte microvilli to the features of this microflow remain largely unknown. Here, an immersed boundary method coupled with a lattice Boltzmann method was adopted in an in vitro hepatic sinusoidal model, and flow field and erythrocyte deformation analyses were conducted by introducing three new source terms including permeability of the endothelial layer, resistance of hepatocyte microvilli and collagen layers, and deformation of red blood cells (RBCs). Numerical calculations indicated that alterations in endothelial permeability could significantly affect the flow velocity and flow rate distributions in hepatic sinusoids. Interestingly, a biphasic regulating pattern of shear stress occurred simultaneously on the surface of hepatocytes and the lower side of endothelium, i.e., the shear stress increased with increased thickness of hepatocyte microvilli and collagen layer when the endothelial permeability was high but decreased with the increase of the thickness at low endothelial permeability. Additionally, this specified microflow manipulates typical RBC deformation inside the sinusoid, yielding one-third of the variation of deformable index with varied endothelial permeability. These simulations not only are consistent with experimental measurements using in vitro liver sinusoidal chip but also elaborate the contributions of endothelial and collagen layer permeability and wall roughness. Thus, our results provide a basis for further characterizing this microflow and understanding its effects on cellular migration and deformation in the hepatic sinusoids.     

Research of smooth muscle cells response to fluid flow shear stress by hyaluronic acid micro-pattern on a titanium surface

The morphology of vascular smooth muscle cells (SMCs) in the normal physiological state depends on cytoskeletal distribution and topology beneath, and presents vertical to the direction of blood flow shear stress (FFSS) although SMCs physiologically are not directly exposed to the shear conditions of blood flow. However, this condition is relevant for arteriosclerotic plaques and the sites of a vascular stent, and little of this condition in vitro has been studied and reported till now. It is unclear what will happen to SMC morphology, phenotype and function when the direction of the blood flow changed. In this paper, the distribution of SMCs in a specific area on Ti surface was regulated by micro-strips of hyaluronic acid (HA). Cell morphology depended on the distribution of the cytoskeleton extending along the micrographic direction. Simulated vascular FFSS was perpendicular or parallel to the direction of the cytoskeleton distribution. Based on investigating the morphology, apoptotic number, phenotypes and functional factors of SMCs, it was obtained that SMCs of vertical groups showed more apoptosis, expressed more contractile types and secreted less TGF-β1 factor compared with SMCs of parallel groups, the number of ECs cultured by medium from SMCs of parallel groups was larger than vertical groups. This study could help to understand the effect of direction change of FFSS on patterned SMC morphology, phenotype and function.     

Laser tweezers as a biophotonic tool to investigate the efficacy of living sickle red blood cells in response to optical deformation

A laser tweezer technique based on single and/or dual-laser beams is proposed as a biophotonic tool to trap single cells and investigate their biophysical and biomechanical characteristics. Optical deformability and changes in size and cellular morphology of living and nonliving cells can be measured using the proposed technique. Representative results of red blood cell (RBC) optical deformability of 20 homozygous patients with sickle cell disease, including follow-up patients after treating with hydroxyurea (HU) for at least 3 months and 20 healthy control groups, are presented and compared. Shape recovery of deformed RBCs and relaxation time are recorded for each RBC. Results showed that healthy blood and patients treated with HU demonstrate significantly higher optical deformability and degree of optical elongation with morphological change of RBCs than untreated patients. Moreover, the healthy control group and patients treated with HU exhibited faster relaxation time for RBCs than untreated patients. A trapping power that reaches 180 mW caused no observable photo-damage at a wavelength 1064 nm.Supplementary InformationThe online version contains supplementary material available at 10.1007/s12551-021-00790-0.