The deformation of spherical vesicles with permeable, constant-area membranes: application to the red blood cell

The deformation of an initially spherical vesicle of radius a with a permeable membrane under extensive forces applied at its poles is calculated as a function of the in-plane shear modulus, H, and the out-of-plane bending modulus, B, using an axisymmetric theory that is valid for large deformations. Suitably nondimensionalized, the results depend upon a single nondimensional parameter, C identical with a(2)H/B. For small deformations, the calculated force-polar strain curves are linear and, under these conditions, the slope of the curve determines only C, not the values of H and B separately. Independent determination of H and B from experimental measurements require deformations that are large enough to produce nonlinear behavior. Simple approximations for large and small C are given, which are applied to experimental measurements on red blood cell ghosts that have been made permeable by treatment with saponin.     

Dielectro-deformations of erythrocyte: analysis of the ellipsoidal shear model

Using the theoretical analysis within the framework of the proposed ellipsoidal shear electromechanical model of erythrocyte, the main mechanisms and relationships have been established and studied for the deformations of erythrocytes caused by a spatially homogeneous high-frequency electric field. The main types of the stress-strain curves characteristic of stationary and dynamic deformations caused by the rectangular-pulse and harmonic modulations of the field amplitude have been calculated. The relationship has been established between the parameters of essentially nonlinear stress-strain curves and mechanical, electric, and geometric parameters of erythrocyte. The impossibility of unlimited elongation of erythrocyte by the field, due to the conservation of the cell volume and surface area, has been shown, and the dependence of the maximum possible elongation of the cell on its volume has been calculated. It has been shown that the relaxation time of dynamic deformations of erythrocyte in the presence of an electric field considerably differs from that characteristic of the membrane material and sharply decreases with the increase of the initial elongation of the cell.     

Effect of long-term anemia and retransfusion on central circulation during exercise

The bulk modulus and the shear modulus describe the capacity of material to resist a change in volume and a change of shape, respectively. The values of these elastic coefficients for air-filled lung parenchyma suggest that there is a qualitative difference between the mechanisms by which the parenchyma resists expansion and shear deformation; the bulk modulus changes roughly exponentially with the transpulmonary pressure, whereas the shear modulus is nearly a constant fraction of the transpulmonary pressure for a wide range of volumes. The bulk modulus is approximately 6.5 times as large as the shear modulus. In recent microstructural modeling of lung parenchyma, these mechanisms have been pictured as being similar to the mechanisms by which an open cell liquid foam resists deformations. In this paper, we report values for the bulk moduli and the shear moduli of normal air-filled rabbit lungs and of air-filled lungs in which alveolar surface tension is maintained constant at 16 dyn/cm. Elevating surface tension above normal physiological values causes the bulk modulus to decrease and the shear modulus to increase. Furthermore, the bulk modulus is found to be sensitive to a dependence of surface tension on surface area, but the shear modulus is not. These results agree qualitatively with the predictions of the model, but there are quantitative differences between the data and the model.     

An elastic network model based on the structure of the red blood cell membrane skeleton.

A finite element network model has been developed to predict the macroscopic elastic shear modulus and the area expansion modulus of the red blood cell (RBC) membrane skeleton on the basis of its microstructure. The topological organization of connections between spectrin molecules is represented by the edges of a random Delaunay triangulation, and the elasticity of an individual spectrin molecule is represented by the spring constant, K, for a linear spring element. The model network is subjected to deformations by prescribing nodal displacements on the boundary. The positions of internal nodes are computed by the finite element program. The average response of the network is used to compute the shear modulus (mu) and area expansion modulus (kappa) for the corresponding effective continuum. For networks with a moderate degree of randomness, this model predicts mu/K = 0.45 and kappa/K = 0.90 in small deformations. These results are consistent with previous computational models and experimental estimates of the ratio mu/kappa. This model also predicts that the elastic moduli vary by 20% or more in networks with varying degrees of randomness. In large deformations, mu increases as a cubic function of the extension ratio lambda 1, with mu/K = 0.62 when lambda 1 = 1.5.     

Effect of wheat germ agglutinin on the viscoelastic properties of erythrocyte membrane

The elasticity and viscosity of the human erythrocyte membrane were measured as a function of the concentration of wheat germ agglutinin (WGA) in a suspending solution containing 1 mg/ml albumin, approximately 5 X 10(5) cells/ml and between 0.0 and 0.2 microgram/ml WGA. Membrane elasticity was characterized by the elastic shear modulus, which provided a measure of the resistance of the membrane to constant-area elastic deformations that occurred in the membrane plane. The elastic shear modulus was determined by aspirating a portion of the membrane into a micropipette and measuring the extension of the membrane into the pipette as a function of the suction pressure. The results indicated no significant change in shear modulus for concentrations of WGA between 0.0 and 0.2 microgram/ml. Membrane viscosity was characterized by the coefficient of surface viscosity, which, in effect, was a measure of the membrane's resistance to rates of deformation. This coefficient was determined from the time required for an erythrocyte to recover its undeformed shape after it had been elongated by the application of an equal and opposite force applied at diametrically opposite points on the erythrocyte rim. The value for the coefficient of surface viscosity was found to increase by a factor of almost three when the WGA concentration was increased from 0.0 to 0.2 microgram/ml. These results indicated that, in the presence of albumin, WGA can increase membrane dissipation (viscosity) without altering the structural rigidity (elasticity) of the membrane.     

Viscoelasticity of packed erythrocyte suspensions subjected to low amplitude oscillatory deformation.

Concentrated adult erythrocyte suspensions were subjected to low amplitude oscillatory shear in a Weissenberg rheogoniometer equipped with a cone-and-plate assembly. The dynamic viscoelastic properties of the suspension were measured over a broad range of frequency by a numerical solution that accounted for fluid inertia. Variation of shear amplitude and cell volume percent, and comparison of buffered saline, plasma, and dextran as suspending media showed that the cellular elements had undergone small bending and shearing deformations. Studies of normal adult erythrocytes, hypotonically swollen cells, temperature-altered cells, and erythrocyte ghosts suggested that the method was evaluating membrane material properties. The normal membrane was found to exhibit a shear rate dependent elastic modulus that increased by more than a factor of 20 over a frequency range from 0.0076 Hz to 60 Hz. The membrane viscosity showed a substantial drop with frequency indicative of a frequency thinning phenomenon. At high frequency of deformation the viscous response of normal erythrocytes was no longer indicative of a membrane property due to the dominant influence of the internal hemoglobin solution. The studies generally supported the ability of the method to quantify relative membrane material properties and detect changes in membrane structure.     

Incremental elastic modulus for ventricles in diastole

Utilizing the formulation of so-called 'small deformations superimposed on a large initial deformation' the incremental pressure modulus of a ventricle in diastole is studied and the explicit expression of it is obtained as a function of intraventricular pressure. In the analysis the ventricular wall material is assumed to be homogeneous, incompressible, isotropic and the stress-strain relation is exponential. The numerical results for a dog left ventricle indicate that above a critical value of inner pressure the incremental pressure modulus increases with increasing intra-ventricular pressure. Furthermore, the relationship between the stiffness and pressure is seen to be curvilinear (particularly for low pressure level), but for large values of inner pressure the behavior of the curve may be approximated by a set of straight lines.     

Quantification of the mechanical properties of noncontracting canine myocardium under simultaneous biaxial loading

Detailed understanding of cardiac mechanics depends upon accurate and complete characterization of the three-dimensional properties of both normal and diseased myocardial tissue. This, however, can only be obtained by performing multiaxial tests on cardiac tissue. In this study we subjected thin sheets of passive canine left ventricular myocardium to various combinations of simultaneous biaxial stretching. During each stretch the ratio of the orthogonal strains was kept constant and the corresponding stresses remained proportional. We fitted the biaxial stress-strain data both with exponential strain-energy functions with quadratic powers of strains as well as with an alternative function with nonintegral powers of strains. We used our recently developed nonparametric method to assess the reliability of the coefficients for each of these functions. The quadratic strain-energy functions resulted in wide intra- and interspecimen variability in the coefficients. Moreover, both their absolute and relative values demonstrated marked load history dependence such that interpretation of the direction of anisotropy was difficult. Fitting the data with the alternative nonintegral strain-energy function seemed to alleviate these problems. This alternative strain-energy function may provide more self-consistent results than the more commonly used quadratic strain-energy functions.     

On a nonlinear theory for muscle shells: Part I--Theoretical development

This paper presents a theory for studies of the large-strain behavior of biological shells composed of layers of incompressible, orthotropic tissue, possibly muscle, of arbitrary orientation. The intrinsic equations of the laminated-shell theory, expressed in lines-of-curvature coordinates, account for large membrane [O(1)] and moderately large bending and transverse shear strains [O(0.3)], nonlinear material properties, and transverse normal stress and strain. An expansion is derived for a general two-dimensional strain-energy density function, which includes residual stress and muscle activation through a shifting zero-stress configuration. Strain-displacement relations are given for the special case of axisymmetric deformation of shells of revolution with torsion.     

A tethered adhesive particle model of two-dimensional elasticity and its application to the erythrocyte membrane.

A new model of two-dimensional elasticity with application to the erythrocyte membrane is proposed. The system consists of a planar array of self-adhesive particles attached to nearest neighbors with flexible tethers. Stretching from the equilibrium dimension is resisted because force is required to dissociate the particle clusters and to decrease the distribution entropy. Release of the external force is accompanied by a contraction as thermal diffusion randomizes the particles and allows interparticle attachments to form again. Analysis of membrane thermodynamics and mechanics under the two-state particle assumption results in a shear softening stress-strain relation. The shear modulus is found proportional to the square root of the surface density of particles, the interparticle adhesive energy, and is inversely proportional to the tether length. Applied to the erythrocyte membrane under the assumption that band 3 tetramer represents the particle and spectrin the tether, the shear modulus predicted corresponds to the measured value when the interparticle adhesive energy is approximately 4.0-5.9 kT, where kT is the Boltzmann constant multiplied by the temperature. This model suggests a mechanism wherein erythrocyte membrane deformability depends on integral protein homomultimeric interactions and can be modulated from the external surface.     

Temperature dependence of the viscoelastic recovery of red cell membrane.

The time-dependent recovery of an elongated red cell is studied as a function of temperature. Before release, the elongated cell is in static equilibrium where external forces are balanced by surface elastic force resultants. Upon release, the cell recovers its initial shape with a time-dependent exponential behavior characteristic of a viscoelastic solid material undergoing large ("finite") deformation. The recovery process is characterized by a time constant, tc, that decreases from approximately 0.27 s at 6 degrees C to 0.06 s at 37 degrees C. From this measurement of the time constant and an independent measurement of the shear modulus of surface elasticity for red cell membrane, the value for the membrane surface viscosity as a function of temperature can be calculated.     

Shear modulation of intercellular contact area between two deformable cells colliding under flow

Shear rate has been shown to critically affect the kinetics and receptor specificity of cell-cell interactions. In this study, the collision process between two modeled cells interacting in a linear shear flow is numerically investigated. The two identical biological or artificial cells are modeled as deformable capsules composed of an elastic membrane. The cell deformation and trajectories are computed using the immersed boundary method (IBM) for shear rates of 100-400s(-1). As the two cells collide under hydrodynamic shear, large local cell deformations develop. The effective contact area between the two cells is modulated by the shear rate, and reaches a maximum value at intermediate levels of shear. At relatively low shear rate, the contact area is an enclosed region. As the shear rate increases, dimples form on the membrane surface, and the contact region becomes annular. The nonmonotonic increase of the contact area with the increase of shear rate from computational results implies that there is a maximum effective receptor-ligand binding area for cell adhesion. This finding suggests the existence of possible hydrodynamic mechanism that could be used to interpret the observed maximum leukocyte aggregation in shear flow. The critical shear rate for maximum intercellular contact area is shown to vary with cell properties such as radius and membrane elastic modulus.     

Axisymmetric optical-trap measurement of red blood cell membrane elasticity

The elastic properties of the cell membrane play a crucial role in determining the equilibrium shape of the cell, as well as its response to the external forces it experiences in its physiological environment. Red blood cells are a favored system for studying membrane properties because of their simple structure: a lipid bilayer coupled to a membrane cytoskeleton and no cytoplasmic cytoskeleton. An optical trap is used to stretch a red blood cell, fixed to a glass surface, along its symmetry axis by pulling on a micron-sized latex bead that is bound at the center of the exposed cell dimple. The system, at equilibrium, shows Hookean behavior with a spring constant of 1.5×10(-6)?N/m over a 1-2 μm range of extension. This choice of simple experimental geometry preserves the axial symmetry of the native cell throughout the stretch, probes membrane deformations in the small-extension regime, and facilitates theoretical analysis. The axisymmetry makes the experiment amenable to simulation using a simple model that makes no a priori assumption on the relative importance of shear and bending in membrane deformations. We use an iterative relaxation algorithm to solve for the geometrical configuration of the membrane at mechanical equilibrium for a range of applied forces. We obtain estimates for the out-of-plane membrane bending modulus B≈1×10(-19)?Nm and an upper limit to the in-plane shear modulus H<2×10(-6)?N/m. The partial agreement of these results with other published values may serve to highlight the dependence of the cell's resistance to deformation on the scale and geometry of the deformation.     

Measurement of the Elastic Modulus for Red Cell Membrane Using a Fluid Mechanical Technique

Red cells which adhere to a surface in a parallel plate flow channel are stretched when acted on by a fluid shear stress. Three types of stretching are studied: whole cell stretching, the stretching of a red cell evagination, and tether (long, thin membrane process) stretching. In addition, the stretching of a large scale model cell attached to a surface is studied in a Couette flow channel. The results indicate that the uniaxial stretching of red cell membrane can be described by a linear stress-strain relationship. Simple theories developed from free body diagrams permit the calculation of a value for the modulus of elasticity of cell membrane in each of the three experiments. In all cases the value for the modulus is on the order of 10⁴ dyn/cm² for an assumed membrane thickness of 0.01 μm. It was also observed that red cell tethers steadily increase in length when the fluid shear stress is greater than approximately 1.5 dyn/cm² and tether lengths in excess of 200 μm have been achieved. Tethers appear to possess both fluid and elastic properties.     

Effects of inherited membrane abnormalities on the viscoelastic properties of erythrocyte membrane.

Several workers have identified molecular abnormalities associated with inherited blood disorders. The present work examines how these alterations in molecular structure affect the viscoelastic properties of the red blood cell membrane. Changes in the membrane shear modulus, the membrane viscosity, and the apparent membrane bending stiffness were observed in cells of eight patients having a variety of disorders: Two had reductions in the number of high-affinity ankyrin binding sites, two had abnormalities associated with the protein band 4.1, and six were known to be deficient in spectrin. The data suggest that the membrane shear modulus is proportional to the density of spectrin on the membrane and support the view that spectrin is primarily responsible for membrane shear elasticity. Although membranes having abnormalities associated with the function of ankyrin or band 4.1 exhibited reduced elasticity, the degree of mechanical dysfunction was quantitatively inconsistent with the extent of the molecular abnormality. This indicates that these skeletal components do not play a primary role in determining membrane shear elasticity. The membrane viscosity was reduced in seven of the eight patients studied. The reduction in viscosity was usually greater than the reduction in shear modulus, but the degree of reduction in viscosity was variable and did not correlate well with the degree of molecular abnormality.     

Is the surface area of the red cell membrane skeleton locally conserved?

The incompressibility of the lipid bilayer keeps the total surface area of the red cell membrane constant. Local conservation of membrane surface area requires that each surface element of the membrane skeleton keeps its area when its aspect ratio is changed. A change in area would require a flow of lipids past the intrinsic proteins to which the skeleton is anchored. in fast red cell deformations, there is no time for such a flow. Consequently, the bilayer provides for local area conservation. In quasistatic deformations, the extent of local change in surface area is the smaller the larger the isotropic modulus of the skeleton in relation to the shear modulus. Estimates indicate: (a) the velocity of relative flow between lipid and intrinsic proteins is proportional to the gradient in normal tension within the skeleton and inversely proportional to the viscosity of the bilayer; (b) lateral diffusion of lipids is much slower than this flow; (c) membrane tanktreading at frequencies prevailing in vivo as well as the release of a membrane tongue from a micropipette are fast deformations; and (d) the slow phase in micropipette aspiration may be dominated by a local change in skeleton surface.     

Spectrin folding versus unfolding reactions and RBC membrane stiffness

Spectrin (Sp), a key component of the erythrocyte membrane, is routinely stretched to near its fully folded contour length during cell deformations. Such dynamic loading may induce domain unfolding as suggested by recent experiments. Herein we develop a model to describe the folding/unfolding of spectrin during equilibrium or nonequilibrium extensions. In both cases, our model indicates that there exists a critical extension beyond which unfolding occurs. We further deploy this model, together with a three-dimensional model of the junctional complex in the erythrocyte membrane, to explore the effect of Sp unfolding on the membrane's mechanical properties, and on the thermal fluctuation of membrane-attached beads. At large deformations our results show a distinctive strain-induced unstiffening behavior, manifested in the slow decrease of the shear modulus, and accompanied by an increase in bead fluctuation. Bead fluctuation is also found to be influenced by mode switching, a phenomenon predicted by our three-dimensional model. The amount of stiffness reduction, however, is modest compared with that reported in experiments. A possible explanation for the discrepancy is the occurrence of spectrin head-to-head disassociation which is also included within our modeling framework and used to analyze bead motion as observed via experiment.     

The mechanical behaviour of brain tissue: large strain response and constitutive modelling

The non-linear mechanical behaviour of porcine brain tissue in large shear deformations is determined. An improved method for rotational shear experiments is used, producing an approximately homogeneous strain field and leading to an enhanced accuracy. Results from oscillatory shear experiments with a strain amplitude of 0.01 and frequencies ranging from 0.04 to 16 Hz are given. The immediate loss of structural integrity, due to large deformations, influencing the mechanical behaviour of brain tissue, at the time scale of loading, is investigated. No significant immediate mechanical damage is observed for these shear deformations up to strains of 0.45. Moreover, the material behaviour during complex loading histories (loading-unloading) is investigated. Stress relaxation experiments for strains up to 0.2 and constant strain rate experiments for shear rates from 0.01 to 1 s(-1) and strains up to 0.15 are presented. A new differential viscoelastic model is used to describe the mechanical response of brain tissue. The model is formulated in terms of a large strain viscoelastic framework and considers non-linear viscous deformations in combination with non-linear elastic behaviour. This constitutive model is readily applicable in three-dimensional head models in order to predict the mechanical response of the intra-cranial contents due to an impact.     

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.