Phospholipid asymmetry during erythrocyte deformation: maintenance of the unit membrane.

To assess the red blood cell (RBC) membrane's ability to maintain normal phospholipid orientation in the face of deforming stress, we examined RBC subjected to elliptical, tank-treading deformation. As determined by accessibility to phospholipase digestion and by labelling with fluorescamine, normal RBC are able to fully preserve their phospholipid asymmetry despite attaining over 96% of their maximal possible deformation. Phospholipid orientation is unchanged during deformation even for RBC that are ATP-depleted or vanadate-treated and for RBC that already have destabilized phospholipids due to treatment with t-butyl hydroperoxide. These data indicate that maintenance of phospholipid organization during marked deforming stress and tank-treading motion of the membrane is ascribable predominantly to the passive stabilizing effect of membrane proteins. This provides additional evidence for the concept of a unit membrane characterized by intimate associations between lipid and protein.     

Red blood cell mechanics and capillary blood rheology.

Blood contains a high vol fraction of erythrocytes (red blood cells), which strongly influence its flow properties. Much is known about the mechanical properties of red cells, providing a basis for understanding and predicting the rheological behavior of blood in terms of the behavior of individual red cells. This review describes quantitative theoretical models that relate red cell mechanics to flow properties of blood in capillaries. Red cells often flow in single file in capillaries, and rheological parameters can then be estimated by analyzing the motion and deformation of an individual red cell and the surrounding plasma in a capillary. The analysis may be simplified by using lubrication theory to approximate the plasma flow in the narrow gaps between the cells and the vessels walls. If red cell shapes are assumed to be axisymmetric, apparent viscosities are predicted that agree with determinations in glass capillaries. Red cells flowing in microvessels typically assume nonaxisymmetric shapes, with cyclic "tank-treading" motion of the membrane around the interior. Several analyses have been carried out that take these effects into account. These analyses indicate that nonaxisymmetry and tank-treading do not significantly influence the flow resistance in single-file or two-file flow.     

Red blood cell mechanics and capillary blood rheology

Blood contains a high vol fraction of erythrocytes (red blood cells), which strongly influence its flow properties. Much is known about the mechanical properties of red cells, providing a basis for understanding and predicting the rheological behavior of blood in terms of the behavior of individual red cells. This review describes quantitative theoretical models that relate red cell mechanics to flow properties of blood in capillaries. Red cells often flow in single file in capillaries, and rheological parameters can then be estimated by analyzing the motion and deformation of an individual red cell and the surrounding plasma in a capillary. The analysis may be simplified by using lubrication theory to approximate the plasma flow in the narrow gaps between the cells and the vessel walls. If red cell shapes are assumed to be axisymmetric, apparent viscosities are predicted that agree with determinations in glass capillaries. Red cells flowing in microvessels typically assume nonaxisymmetric shapes, with cyclic “tank-treading” motion of the membrane around the interior. Several analyses have been carried out that take these effects into account. These analyses indicate that nonaxisymmetry and tank-treading do not significantly influence the flow resistance in single-file or two-file flow.     

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.     

Viscoelastic behavior of erythrocyte membrane.

A nonlinear viscoelastic relation is developed to describe the viscoelastic properties of erythrocyte membrane. This constitutive equation is used in the analysis of the time-dependent aspiration of an erythrocyte membrane into a micropipette. Equations governing this motion are reduced to a nonlinear integral equation of the Volterra type. A numerical procedure based on a finite difference scheme is used to solve the integral equation and to match the experimental data. The data, aspiration length vs. time, is used to determine the relaxation function at each time step. The inverse problem of obtaining the time dependence of the aspiration length from a given relaxation function is also solved. Analytical results obtained are applied to the experimental data of Chien et al. 1978. Biophys. J. 24:463-487. A relaxation function similar to that of a four-parameter solid with a shear-thinning viscous term is proposed.     

Motion of nonaxisymmetric red blood cells in cylindrical capillaries

We analyze theoretically the single-file flow of asymmetric red blood cells along cylindrical capillaries. Red cells in narrow capillaries are typically nonaxisymmetric, with the cell membrane moving continuously around the cell. In our analysis, cell shape and streamlines of membrane motion are prescribed. Lubrication theory is used to compute velocities and pressures in the fluid surrounding the cell. Conditions of zero lift, zero torque, zero drag, and energy conservation in the cell are imposed. Predicted tank-treading frequency, cell inclination and transverse displacement are small. Cell asymmetry and tank-treading are found to have little effect on the apparent viscosity of blood in capillaries with diameters up to 7 microns.     

Dynamic finite element implementation of nonlinear,anisotropic hyperelastic biological membranes

We present a novel method for the implementation of hyperelastic finite strain, non-linear strain-energy functions for biological membranes in an explicit finite element environment. The technique is implemented in LS-DYNA but may also be implemented in any suitable non-linear explicit code. The constitutive equations are implemented on the foundation of a co-rotational uniformly reduced Hughes-Liu shell. This shell is based on an updated-Lagrangian formulation suitable for relating Cauchy stress to the rate-of-deformation, i.e. hypo-elasticity. To accommodate finite deformation hyper-elastic formulations, a co-rotational deformation gradient is assembled over time, resulting in a formulation suitable for pseudo-hyperelastic constitutive equations that are standard assumptions in biomechanics. Our method was validated by comparison with (1) an analytic solution to a spherically-symmetric dynamic membrane inflation problem, incorporating a Mooney-Rivlin hyperelastic equation and (2) with previously published finite element solutions to a non-linear transversely isotropic inflation problem. Finally, we implemented a transversely isotropic strain-energy function for mitral valve tissue. The method is simple and accurate and is believed to be generally useful for anyone who wishes to model biologic membranes with an experimentally driven strain-energy function.     

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.     

Extensional recovery of an intact erythrocyte from a tank-treading motion

Normal human erythrocytes suspended in shear flow are stretched into quasi ellipsoidal forms while their membranes rotate smoothly (tank-treading). Following abrupt cessation of shear the cells recover their discoidal shapes approximately exponentially, in the manner of a Kelvin-Voigt (K-V) solid. To test the hypothesis that the recovery process is membrane-controlled, the effects of initial deformation, cytoplasmic viscosity and membrane surface-to-volume ratio were studied. It was concluded that the membrane dynamics dominates the transient shape recovery, and that the characteristic recovery time is dependent on the initial deformation. Hence, the usual simplified analysis based on retraction of a plane sheet of K-V material with constant moduli appears to be an inadequate treatment of transient whole cell recovery.     

Effect of the surface charge on deformation dynamics in lipid membranes]

A theoretical analysis of equations of motion describing the dynamics of a charged spherical membrane in a viscous liquid medium was performed. It was shown that the curvature of the membrane substantially depends on the static surface charge. In addition, it was found that a local variation in surface charge density induces a deformation of the membrane and conversely.     

Tank-treading of erythrocytes in strong shear flows via a nonstiff cytoskeleton-based continuum computational modeling

We develop a computationally efficient cytoskeleton-based continuum erythrocyte algorithm. The cytoskeleton is modeled as a two-dimensional elastic solid with comparable shearing and area-dilatation resistance that follows a material law (Skalak, R., A. Tozeren, R. P. Zarda, and S. Chien. 1973. Strain energy function of red blood cell membranes. Biophys. J. 13:245-264). Our modeling enforces the global area-incompressibility of the spectrin skeleton (being enclosed beneath the lipid bilayer in the erythrocyte membrane) via a nonstiff, and thus efficient, adaptive prestress procedure which accounts for the (locally) isotropic stress imposed by the lipid bilayer on the cytoskeleton. In addition, we investigate the dynamics of healthy human erythrocytes in strong shear flows with capillary number Ca = O(1) and small-to-moderate viscosity ratios 0.001 ≤ λ ≤ 1.5. These conditions correspond to a wide range of surrounding medium viscosities (4-600 mPa s) and shear flow rates (0.02-440 s⁻¹), and match those used in ektacytometry systems. Our computational results on the cell deformability and tank-treading frequency are compared with ektacytometry findings. The tank-treading period is shown to be inversely proportional to the shear rate and to increase linearly with the ratio of the cytoplasm viscosity to that of the suspending medium. Our modeling also predicts that the cytoskeleton undergoes measurable local area dilatation and compression during the tank-treading of the cells.     

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.     

A membrane bending model of outer hair cell electromotility

We propose a new mechanism for outer hair cell electromotility based on electrically induced localized changes in the curvature of the plasma membrane (flexoelectricity). Electromechanical coupling in the cell's lateral wall is modeled in terms of linear constitutive equations for a flexoelectric membrane and then extended to nonlinear coupling based on the Langevin function. The Langevin function, which describes the fraction of dipoles aligned with an applied electric field, is shown to be capable of predicting the electromotility voltage displacement function. We calculate the electrical and mechanical contributions to the force balance and show that the model is consistent with experimentally measured values for electromechanical properties. The model rationalizes several experimental observations associated with outer hair cell electromotility and provides for constant surface area of the plasma membrane. The model accounts for the isometric force generated by the cell and explains the observation that the disruption of spectrin by diamide reduces force generation in the cell. We discuss the relation of this mechanism to other proposed models of outer hair cell electromotility. Our analysis suggests that rotation of membrane dipoles and the accompanying mechanical deformation may be the molecular mechanism of electromotility.     

Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion.

Measurements of the dimensions and membrane rotational frequency of individual erythrocytes steadily tank-treading in a rheoscope are used to deduce the surface shear viscosity of the membrane. The method is based on an integral energy principle which says that the power supplied to the tank-treading cell by the suspending fluid is equal to the rate at which energy is dissipated by viscous action in the membrane and cytoplasm. The integrals involved are formulated with the aid of an idealized mathematical model of the tank-treading red blood cell (RBC) (Keller and Skalak, 1982, J. Fluid Mech., 120:24-27) and evaluated numerically. The outcome is a surface-averaged value of membrane viscosity which is representative of a finite interval of membrane shear rate. The numerical values computed show a clear shear-thinning characteristic as well as a significant augmentation of viscosity with cell age and tend toward agreement with those determined for the rapid phase of shape recovery in micropipettes (Chien, S., K.-L. P. Sung, R. Skalak, S. Usami, and A. Tozeren, 1978, Biophys. J., 24:463-487). The computations also indicate that the rate of energy dissipation in the membrane is always substantially greater than that in the cytoplasm.     

An Eulerian/XFEM formulation for the large deformation of cortical cell membrane

Most animal cells are surrounded by a thin layer of actin meshwork below their membrane, commonly known as the actin cortex (or cortical membrane). An increasing number of studies have highlighted the role of this structure in many cell functions including contraction and locomotion, but modelling has been limited by the fact that the membrane thickness (about 1?μm) is usually much smaller than the typical size of a cell (10-100?μm). To overcome theoretical and numerical issues resulting from this observation, we introduce in this paper a continuum formulation, based on surface elasticity, that views the cortex as an infinitely thin membrane that can resists tangential deformation. To accurately model the large deformations of cells, we introduced equilibrium equations and constitutive relations within the Eulerian viewpoint such that all quantities (stress, rate of deformation) lie in the current configuration. A solution procedure is then introduced based on a coupled extended finite element approach that enables a continuum solution to the boundary value problem in which discontinuities in both strain and displacement (due to cortical elasticity) are easily handled. We validate the approach by studying the effect of cortical elasticity on the deformation of a cell adhering on a stiff substrate and undergoing internal contraction. Results show very good prediction of the proposed method when compared with experimental observations and analytical solutions for simple cases. In particular, the model can be used to study how cell properties such as stiffness and contraction of both cytoskeleton and cortical membrane lead to variations in cell's surface curvature. These numerical results show that the proposed method can be used to gain critical insights into how the cortical membrane affects cell deformation and how it may be used as a means to determine a cell's mechanical properties by measuring curvatures of its membrane.     

Simulation of gravitational field variation on fluid-filled biological membranes

The knowledge of the behavior of biological organs in a gravitational field is important to understand the functioning of the human body in the aerospace environment. The disturbances in biological transport processes in microgravity have indicated adverse effects on humans engaged in space operations. The relationship between the deformations in the biological organs and the transport phenomena that take place in them has been long established and widely reported in biological sciences and engineering literature. A number of soft tissue organs such as brain, lungs, heart, kidney, bladder, stomach, and the circulatory system can be modeled as fluid-filled membranes. In this investigation, a mathematical model of a fluid-filled biological membrane is developed, and its deformation and spatial configuration in a variable gravitational field are calculated. The variation in the gravitational field in the range 1g to zero-g is simulated by partial submergence of the fluid-filled membrane which, by virtue of buoyancy, gains an effective density as if it is in a different gravitational field. The equations of motion are derived using the theory of large elastic deformations and numerically solved in conjunction with a constitutive equation suitably selected for the biological membrane.     

Continuum rheology of muscle contraction and its application to cardiac contractility.

A set of constitutive equations is proposed to describe the mechanics of contraction of skeletal and heart muscle. Fiber tension is assumed to depend on the degree of chemical activation, the stretch ratio, and the rate of stretching of the fibers. The time rate of change of activation is governed by a differential equation. The proposed constitutive equations are used to model the time courses of isotonic and isometric twitches during contraction and relaxation phases of the muscle response to stimulation. Various contractility indices of the left ventricle are considered next by using the proposed constitutive equations. The present analysis introduces a new interpretation of the index of contractility (dP/dt)/P used in cardiac literature. It is shown that this index may not be related at all to the maximum speed of shortening and that it may be dependent on both preload and afterload. The development of pressure during isovolumetric contraction of the left ventricle is shown to be governed by a differential equation describing the time rate of change of tension during isometric contraction of myocardium fibers.     

Visualization study of motion and deformation of red blood cells in a microchannel with straight,divergent and convergent sections

The size of red blood cells (RBC) is on the same order as the diameter of microvascular vessels. Therefore, blood should be regarded as a two-phase flow system of RBCs suspended in plasma rather than a continuous medium of microcirculation. It is of great physiological and pathological significance to investigate the effects of deformation and aggregation of RBCs on microcirculation. In this study, a visualization experiment was conducted to study the microcirculatory behavior of RBCs in suspension. Motion and deformation of RBCs in a microfluidic chip with straight, divergent, and convergent microchannel sections have been captured by microscope and high-speed camera. Meanwhile, deformation and movement of RBCs were investigated under different viscosity, hematocrit, and flow rate in this system. For low velocity and viscosity, RBCs behaved in their normal biconcave disc shape and their motion was found as a flipping motion: they not only deformed their shapes along the flow direction, but also rolled and rotated themselves. RBCs were also found to aggregate, forming rouleaux at very low flow rate and viscosity. However, for high velocity and viscosity, RBCs deformed obviously under the shear stress. They elongated along the flow direction and performed a tank-treading motion.     

Erythrocyte deformation in simulated weightless human and rabbits

Effect and mechanism of simulated weightlessness (SWL) in humans and rabbits erythrocyte deformation were studied. Erythrocyte deformation and membrane fluidity in humans and rabbits, and erythrocyte morphology and hemorreology indices in control and HDT rabbits were measured. The results were a decrease in erythrocyte deformation and membrane fluidity in humans and rabbits during SWL, a significant increase in abnormal erythrocyte, blood viscosity, hematocrit, fibrinogen, and red blood cell aggregation index in HDT rabbits. These results show that the changes in erythrocyte shape, increase of erythrocyte internal viscosity and changes in erythrocyte visco-elasticity may be causing the decrease of erythrocyte deformation in SWL humans and rabbits.