Multiscale modeling of red blood cell mechanics and blood flow in malaria

Red blood cells (RBCs) infected by a Plasmodium parasite in malaria may lose their membrane deformability with a relative membrane stiffening more than ten-fold in comparison with healthy RBCs leading to potential capillary occlusions. Moreover, infected RBCs are able to adhere to other healthy and parasitized cells and to the vascular endothelium resulting in a substantial disruption of normal blood circulation. In the present work, we simulate infected RBCs in malaria using a multiscale RBC model based on the dissipative particle dynamics method, coupling scales at the sub-cellular level with scales at the vessel size. Our objective is to conduct a full validation of the RBC model with a diverse set of experimental data, including temperature dependence, and to identify the limitations of this purely mechanistic model. The simulated elastic deformations of parasitized RBCs match those obtained in optical-tweezers experiments for different stages of intra-erythrocytic parasite development. The rheological properties of RBCs in malaria are compared with those obtained by optical magnetic twisting cytometry and by monitoring membrane fluctuations at room, physiological, and febrile temperatures. We also study the dynamics of infected RBCs in Poiseuille flow in comparison with healthy cells and present validated bulk viscosity predictions of malaria-infected blood for a wide range of parasitemia levels (percentage of infected RBCs with respect to the total number of cells in a unit volume).     

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.     

Nonequilibrium fluctuations of mechanically stretched single red blood cells detected by optical tweezers

We study the thermal and out-of-equilibrium mechanical dynamics of single, living human red blood cells (RBCs) by combining two-probe passive and active microrheology techniques. Both experiments were performed quasisimultaneously on the same cell using two identical polystyrene probes, biochemically attached to the cell membrane. We obtained compelling evidence of nonequilibrium fluctuations in the RBCs under physiological condition and without the influence of any external chemicals. The spectral distributions of metabolically driven forces and viscoelastic response were evaluated in the relaxed and stretched states, intended to simulate the varying natural environment of the cells during blood circulation. We found that the internally generated forces are more pronounced in the stretched state, suggesting a stress-dependent RBC activity.     

Dynamical clustering of red blood cells in capillary vessels

We have modeled the dynamics of a 3-D system consisting of red blood cells (RBCs), plasma and capillary walls using a discrete-particle approach. The blood cells and capillary walls are composed of a mesh of particles interacting with harmonic forces between nearest neighbors. We employ classical mechanics to mimic the elastic properties of RBCs with a biconcave disk composed of a mesh of spring-like particles. The fluid particle method allows for modeling the plasma as a particle ensemble, where each particle represents a collective unit of fluid, which is defined by its mass, moment of inertia, translational and angular momenta. Realistic behavior of blood cells is modeled by considering RBCs and plasma flowing through capillaries of various shapes. Three types of vessels are employed: a pipe with a choking point, a curved vessel and bifurcating capillaries. There is a strong tendency to produce RBC clusters in capillaries. The choking points and other irregularities in geometry influence both the flow and RBC shapes, considerably increasing the clotting effect. We also discuss other clotting factors coming from the physical properties of blood, such as the viscosity of the plasma and the elasticity of the RBCs. Modeling has been carried out with adequate resolution by using 1 to 10 million particles. Discrete particle simulations open a new pathway for modeling the dynamics of complex, viscoelastic fluids at the microscale, where both liquid and solid phases are treated with discrete particles. Figure A snapshot from fluid particle simulation of RBCs flowing along a curved capillary. The red color corresponds to the highest velocity. We can observe aggregation of RBCs at places with the most stagnant plasma flow.     

Red blood cell shape transitions and dynamics in time-dependent capillary flows

The dynamics of single red blood cells (RBCs) determine microvascular blood flow by adapting their shape to the flow conditions in the narrow vessels. In this study, we explore the dynamics and shape transitions of RBCs on the cellular scale under confined and unsteady flow conditions using a combination of microfluidic experiments and numerical simulations. Tracking RBCs in a comoving frame in time-dependent flows reveals that the mean transition time from the symmetric croissant to the off-centered, nonsymmetric slipper shape is significantly faster than the opposite shape transition, which exhibits pronounced cell rotations. Complementary simulations indicate that these dynamics depend on the orientation of the RBC membrane in the channel during the time-dependent flow. Moreover, we show how the tank-treading movement of slipper-shaped RBCs in combination with the narrow channel leads to oscillations of the cell's center of mass. The frequency of these oscillations depends on the cell velocity, the viscosity of the surrounding fluid, and the cytosol viscosity. These results provide a potential framework to identify and study pathological changes in RBC properties.     

Wall shear stress-based model for adhesive dynamics of red blood cells in malaria

Red blood cells (RBCs) infected by the Plasmodium falciparum (Pf-RBCs) parasite lose their membrane deformability and they also exhibit enhanced cytoadherence to vascular endothelium and other healthy and infected RBCs. The combined effect may lead to severe disruptions of normal blood circulation due to capillary occlusions. Here we extend the adhesion model to investigate the adhesive dynamics of Pf-RBCs as a function of wall shear stress (WSS) and other parameters using a three-dimensional, multiscale RBC model. Several types of adhesive behavior are identified, including firm adhesion, flipping dynamics, and slow slipping along the wall. In particular, the flipping dynamics of Pf-RBCs observed in experiments appears to be due to the increased stiffness of infected cells and the presence of the solid parasite inside the RBC, which may cause an irregular adhesion behavior. Specifically, a transition from crawling dynamics to flipping behavior occurs at a Young's modulus approximately three times larger than that of healthy RBCs. The simulated dynamics of Pf-RBCs is in excellent quantitative agreement with available microfluidic experiments if the force exerted on the receptors and ligands by an existing bond is modeled as a nonlinear function of WSS.     

Viscoelastic properties of whole blood. Influence of fast sedimenting red blood cell aggregates

Red blood cell (RBC) aggregation is known to be of deciding influence on erythrocyte sedimentation-rate (ESR) and on whole blood viscoelastic properties. The rheological behaviour of blood collected from a control-group with normal ESR is compared to the viscoelastic behaviour of blood collected from two groups with high to very high ESR, whose individuals are suffering from chronical polyarthritis and Morbus Bechterew, respectively. The rheological properties are evaluated by means of an oscillating-flow capillary-rheometer where the viscous (eta') and elastic (eta") component of the complex viscosity (eta) is measured at a constant frequency of 2 Hz. Correcting for the varying hematocrit of the different blood samples according to an exponential equation, the viscoelastic data are found to be elevated in the groups with high ESR. For the viscous properties this is only due to the increase of the plasma viscosity. A correction for the plasma viscosity, however, shows that the viscous properties at low shear- rates (2s-1) are significantly reduced, whereas elastic properties in a range of medium shear-rates (10s-1 to 50s-1) are significantly increased (P less than 0.001, t-test of Student). This result is discussed to be due to the high packing density of the RBC in fast sedimenting aggregates. High packing density reduces the effective volume of the RBC but increases the stiffness of the aggregates.     

Computational Analysis of Dynamic Interaction of Two Red Blood Cells in a Capillary

The dynamic interaction of two red blood cells (RBCs) in a capillary is investigated computationally by the two-fluid model, including their deformable motion and interaction. For characterization of the deformation, the RBC membrane is treated as a curved two-dimensional shell with finite thickness by the shell model, and allowed to undergo the stretching strain and bending deformation. Moreover, a Morse potential is adopted to model the intercellular interaction for the aggregation behavior, which is characterized as the weak attraction at far distance and strong repulsion at near distance. For validation of the present technique, the dynamic interaction of two RBCs in static blood plasma is simulated firstly, where the RBCs aggregate slowly until a balanced configuration is achieved between the deformation and aggregation forces. The balanced configuration is in good agreement with the results reported previously. Three important effects on the dynamic behavior of RBCs are then analyzed, and they are the initial RBC shape, RBC deformability, and the intercellular interaction strength. It is found that the RBC is less deformed into a well-known parachute shape when the initial RBC shape is larger. Similarly, if the elastic shear modulus and bending stiffness of RBC membrane increase, the RBC resistance to deformation becomes higher, such that the RBC is less deformed. The simulation results also demonstrate that the RBC deformability strongly depends on the intercellular interaction strength. The RBCs deform more easily as the intercellular interaction strength increases.     

A novel two-layer, coupled finite element approach for modeling the nonlinear elastic and viscoelastic behavior of human erythrocytes

A novel finite element approach is presented to simulate the mechanical behavior of human red blood cells (RBC, erythrocytes). As the RBC membrane comprises a phospholipid bilayer with an intervening protein network, we propose to model the membrane with two distinct layers. The fairly complex characteristics of the very thin lipid bilayer are represented by special incompressible solid shell elements and an anisotropic viscoelastic constitutive model. Properties of the protein network are modeled with an isotropic hyperelastic third-order material. The elastic behavior of the model is validated with existing optical tweezers studies with quasi-static deformations. Employing material parameters consistent with literature, simulation results are in excellent agreement with experimental data. Available models in literature neglect either the surface area conservation of the RBC membrane or realistic loading conditions of the optical tweezers experiments. The importance of these modeling assumptions, that are both included in this study, are discussed and their influence quantified. For the simulation of the dynamic motion of RBC, the model is extended to incorporate the cytoplasm. This is realized with a monolithic fully coupled fluid-structure interaction simulation, where the fluid is described by the incompressible Navier–Stokes equations in an arbitrary Lagrangian Eulerian framework. It is shown that both membrane viscosity and cytoplasm viscosity have significant influence on simulation results. Characteristic recovery times and energy dissipation for varying strain rates in dynamic laser trap experiments are calculated for the first time and are found to be comparable with experimental data.     

A dynamic and stationary rheological study of erythrocytes incubated in a glucose medium

A higher than normal glucose concentration in a suspending medium may produce non-enzymatic glycosylation of erythrocyte proteins. This process can modify the viscoelastic properties of erythrocytes. In this paper, we studied the possible relationship between glucose concentration in a suspending medium and erythrocyte rheological parameters. Human venous blood was obtained from the antecubital veins of 10 healthy volunteers. Blood samples were anticoagulated with EDTA and centrifuged. Red blood cells (RBCs) were washed and subsequently divided in aliquots, which were incubated in vitro with glucose solutions of different concentrations. Dynamic and stationary viscoelastic parameters of RBCs were determined by laser diffractometry in an Erythrodeformeter. Aggregate shape parameter (ASP) of the RBCs was determined by digital image processing. Significant changes were observed both in ASP and in rheological parameters when the glucose concentration in the medium was increased, demonstrating that a glucose concentration as low as 1% induces alterations in the mechanical properties of RBCs.     

Rheology of concentrated suspensions of deformable elastic particles such as human erythrocytes

New models for the viscosity of concentrated suspensions of deformable elastic particles are developed using the differential effective medium approach (DEMA). The models are capable of describing the rheological behavior of un-aggregated suspensions of human red blood cells (RBCs). With the increase in shear rate, a shear-thinning behavior is predicted similar to that observed in the case of un-aggregated suspensions of RBCs. A decrease in relative viscosity and an enhancement of shear-thinning behavior is predicted when either the particle rigidity (elastic modulus) is decreased or the continuous medium viscosity is increased. These predictions are similar to those observed in suspensions of human RBCs. The proposed models are evaluated using experimental data on normal and hardened human RBC suspensions in protein-free saline.     

Computational fluid dynamic simulation of aggregation of deformable cells in a shear flow

We present computational fluid dynamic (CFD) simulation of aggregation of two deformable cells in a shearflow. This work is motivated by an attempt to develop computational models of aggregation of red blood cells (RBCs). Aggregation of RBCs is a major determinant of blood viscosity in microcirculation under physiological and pathological conditions. Deformability of the RBCs plays a major role in determining their aggregability. Deformability depends on the viscosity of the cytoplasmic fluid and on the rigidity of the cell membrane, in a macroscopic sense. This paper presents a computational study of RBC aggregation that takes into account the rheology of the cells as well as cell-cell adhesion kinetics. The simulation technique considered here is two dimensional and based on the front tracking/immersed boundary method for multiple fluids. Results presented here are on the dynamic events of aggregate formation between two cells, and its subsequent motion, rolling, deformation, and breakage. We show that the rheological properties of the cells have significant effects on the dynamics of the aggregate. A stable aggregate is formed at higher cytoplasmic viscosity and membrane rigidity. We also show that the bonds formed between the cells change in a cyclic manner as the aggregate rolls in a shearflow. The cyclic behavior is related to the rolling orientation of the aggregate. The frequency and amplitude of oscillation in the number of bonds also depend on the rheological properties.     

Effect of lanthanum on red blood cell deformability

Prior reports describing the effects of lanthanum (La(3+)) on red blood cells (RBC) have focused on the effects of this lanthanide on cell fusion or on membrane characteristics (e.g., ion movement across membrane, membrane protein aggregation); the present study explores its rheological and biophysical effects. Normal human RBC were exposed to La(3+) levels up to 200 microM then tested for: (1) cellular deformability using a laser-based ektacytometer and an optical-based rheoscope; (2) membrane viscoelastic behavior via micropipettes; (3) surface charge via micro electrophoresis. La(3+) concentrations of 12.5 to 200 microM caused dose-dependent decreases of deformability that were greatest at low stresses: these rheological changes were completely reversible upon removing La(3+) from the media either by washing with La(3+)-free buffer or by suspending La(3+)-exposed cells in La(3+)-free media (i.e., viscous dextran solution). Both membrane shear elastic modulus and membrane surface viscosity were increased by 25-30% at 100 or 200 microM. As expected, La(3+) decreased RBC electrophoretic mobility (EPM), with EPM inversely but not linearly associated with deformability; changes of EPM were also completely reversible. These results thus indicate novel aspects of RBC cellular and membrane rheological behavior yet raise questions regarding specific mechanisms responsible for La(3+)-induced alterations.     

Curling and Local Shape Changes of Red Blood Cell Membranes Driven by Cytoskeletal Reorganization

Human red blood cells (RBCs) lack the actin-myosin-microtubule cytoskeleton that is responsible for shape changes in other cells. Nevertheless, they can display highly dynamic local deformations in response to external perturbations, such as those that occur during the process of apical alignment preceding merozoite invasion in malaria. Moreover, after lysis in divalent cation-free media, the isolated membranes of ruptured ghosts show spontaneous inside-out curling motions at the free edges of the lytic hole, leading to inside-out vesiculation. The molecular mechanisms that drive these rapid shape changes are unknown. Here, we propose a molecular model in which the spectrin filaments of the RBC cortical cytoskeleton control the sign and dynamics of membrane curvature depending on whether the ends of the filaments are free or anchored to the bilayer. Computer simulations of the model reveal that curling, as experimentally observed, can be obtained either by an overall excess of weakly-bound filaments throughout the cell, or by the flux of such filaments toward the curling edges. Divalent cations have been shown to arrest the curling process, and Ca²⁺ ions have also been implicated in local membrane deformations during merozoite invasion. These effects can be replicated in our model by attributing the divalent cation effects to increased filament-membrane binding. This process converts the curl-inducing loose filaments into fully bound filaments that arrest curling. The same basic mechanism can be shown to account for Ca²⁺-induced local and dynamic membrane deformations in intact RBCs. The implications of these results in terms of RBC membrane dynamics under physiological, pathological, and experimental conditions is discussed.     

Effects of r—HuEPO on the biophysical characteristics of erythrocyte membrane in patients with anemia of chronic renal failure

Using electron spin resonance (ESR) spin labeling technique,we have studied the conformation of sulfhydryl groups(-SH) binding sites in membrane proteins and mem brane fluidity of red blood cells(RBCs) from two groups of patients with anemia of chronic renal failure(ACRF).One of the groups is composed of patients who were untreated with recombinant human erythropoietin(r-HuEPO),and the other is composed of patients who were treated with r-HuEPO.The results indicated:1)the conformation of SH group binding site in RBC membrane proteins from former group was different from those of healty people.2)the fluidity in the region near the surface of RBC membrane from former group was lower than those of healthy people.3)However,the above biophysical properties of RBC membrane from later group were normal.We concluded that RBC membrane in patients with ACRF was abnormal,and the treatment of r-HuEPO may promote the production of normal RBCs,thus ameliorate the biophysical properties of RBCs from the patients with ACRF.     

Investigation of red blood cell mechanical properties using AFM indentation and coarse-grained particle method

Background

Red blood cells (RBCs) deform significantly and repeatedly when passing through narrow capillaries and delivering dioxygen throughout the body. Deformability of RBCs is a key characteristic, largely governed by the mechanical properties of the cell membrane. This study investigated RBC mechanical properties using atomic force microscopy (AFM) with the aim to develop a coarse-grained particle method model to study for the first time RBC indentation in both 2D and 3D. This new model has the potential to be applied to further investigate the local deformability of RBCs, with accurate control over adhesion, probe geometry and position of applied force.

Results

The model considers the linear stretch capacity of the cytoskeleton, bending resistance and areal incompressibility of the bilayer, and volumetric incompressibility of the internal fluid. The model’s performance was validated against force–deformation experiments performed on RBCs under spherical AFM indentation. The model was then used to investigate the mechanisms which absorbed energy through the indentation stroke, and the impact of varying stiffness coefficients on the measured deformability. This study found the membrane’s bending stiffness was most influential in controlling RBC physical behaviour for indentations of up to 200 nm.

Conclusions

As the bilayer provides bending resistance, this infers that structural changes within the bilayer are responsible for the deformability changes experienced by deteriorating RBCs. The numerical model presented here established a foundation for future investigations into changes within the membrane that cause differences in stiffness between healthy and deteriorating RBCs, which have already been measured experimentally with AFM.

     

Theoretical model and experimental study of red blood cell (RBC) deformation in microchannels

The motion and deformation of red blood cells (RBCs) flowing in a microchannel were studied using a theoretical model and a novel automated rheoscope. The theoretical model was developed to predict the cells deformation under shear as a function of the cells geometry and mechanical properties. Fluid dynamics and membrane mechanics are incorporated, calculating the traction and deformation in an iterative manner. The model was utilized to evaluate the effect of different biophysical parameters, such as: inner cell viscosity, membrane shear modulus and surface to volume ratio on deformation measurements. The experimental system enables the measurement of individual RBCs velocity and their deformation at defined planes within the microchannel. Good agreement was observed between the simulation results, the rheoscope measurements and published ektacytometry results. The theoretical model results imply that such deformability measuring techniques are weakly influenced by changes in the inner viscosity of the cell or the ambient fluid viscosity. However, these measurements are highly sensitive to RBC shear modulus. The shear modulus, estimated by the model and the rheoscope measurements, falls between the values obtained by micropipette aspiration and laser trapping. The study demonstrates the integration of a theoretical model with a microfabricated device in order to achieve a better understanding of RBC mechanics and their measurement using microfluidic shear assays. The system and the model have the potential of serving as quantitative clinical tools for diagnosing deformability disorders in RBCs.     

A Model of Piezo1-Based Regulation of Red Blood Cell Volume

A red blood cell (RBC) performs its function of adequately carrying respiratory gases in blood by its volume being ～60% of that of a sphere with the same membrane area. For this purpose, human and most other vertebrate RBCs regulate their content of potassium (K⁺) and sodium (Na⁺) ions. The focus considered here is on K⁺ efflux through calcium-ion (Ca²⁺)-activated Gárdos channels. These channels open under conditions that allow Ca²⁺ to enter RBCs through Piezo1 mechanosensitive cation-permeable channels. It is postulated that the fraction of open Piezo1 channels depends on the RBC shape as a result of the curvature-dependent Piezo1-bilayer membrane interaction. The consequences of this postulate are studied by introducing a simple model of RBC osmotic behavior supplemented by the dependence of RBC membrane K⁺ permeability on the reduced volume (i.e., the ratio of cell volume to its maximal possible volume) of RBC discoid shapes. It is assumed that because of its intrinsic curvature and strong interaction with the surrounding membrane, Piezo1 tends to concentrate in the dimple regions of these shapes, and the fraction of open Piezo1 channels depends on the membrane curvature in that region. It is shown that the properties of the described model can provide the basis for the formation of the negative feedback loop that interrelates cell volume and its content of potassium ions. The model predicts the relation, valid for each cell in an RBC population, between RBC volume and membrane area, thus explaining the large value of the measured membrane area versus the volume correlation coefficient. The mechanism proposed here for RBC volume regulation is in accord with the loss of this correlation in RBCs of Piezo1 knockout mice.     

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.