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.     

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.     

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.     

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.     

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.     

On the energy dissipation in a tank-treading human red blood cell.

The energy dissipation in the membrane (ED mem) and in the cytoplasm (ED cyt) of tank-treading human red blood cells is estimated. The tank-tread motion of the membrane occurs when the cells in a sheared suspension assume a steady-state of orientation (Fischer et al., 1978, Science [Wash. D. C.], 202:894). The kinematic data used are from red cells suspended either in a dextran-saline solution at a low hematocrit, or in plasma at a hematocrit of 45%. The viscosities of the cytoplasm and the membrane are taken from the literature. The cell in dextran was subjected to seven different shear rates. Both ED mem and ED cyt showed a strong increase with shear rate. Their ratio, however, was always of the order of 1. From this value and the value which was given by Hochmuth et al. (1979, Biophys. J., 26:101) for a shape recovery of a red cell, it is concluded that the range of ED mem/ED cyt for all possible geometries is 1-100.     

Flow conditions of red cells and plasma in microvascular bifurcations

The flow conditions of red cells and plasma in microvascular ramifications were investigated in a biological model of the frog's retrolingual membrane. Upon the controllable reduction of blood flow from the arterioles into the microvascular bed, with an appropriate decrease of red cell: plasma ratio in the blood, a tendency of the red cells to be drawn along the parent main capillaries without entering the branching capillaries was in evidence. These latter thus transformed into the plasmatic capillaries deprived of red cells. The factors being responsible for this process were found to be as follows: (a) the diameter of branching capillaries, (b) the angles of off-shoots, (c) the degree of slow-down of blood flow velocity in the branches, and (d) the reduction of red cell: plasma ratio in parent vessels. The direct relationship was found between these factors and the transformations of the off-shoots into the plasmatic capillaries.     

Intraerythrocytic parasites and red cell deformability: Plasmodium berghei and Babesia microti

In the studies reported here, we examined the effects of two intraerythrocytic parasites (Plasmodium berghei and Babesia microti) on the deformability of their host red cells. Red cell deformability was assessed by three criteria: 1) the prevalence of tank-treading (the tank-tread-like movement of the red cell membrane around its cytoplasmic contents), 2) elongation under fluid shear stress (the steady-state length: width ratio), and 3) the time required for the red cell to reduce its steady-state elongation by 63.2% after the abrupt release of the shear stress (the characteristic shape-recovery time). Trophozoite-stage parasites of both species reduced the prevalence of tank-treading. Ring- and trophozoite-stage parasites of both species reduced steady-state elongation, and ring-stage P. berghei prolonged the shape-recovery time. These results suggest that altered red cell deformability is a common feature of infection with intraerythrocytic parasites.     

Characterization of erythrocyte membrane tension for hemolysis prediction in complex flows

Hemolysis is a persistent issue with blood-contacting devices. Many experimental and theoretical contributions over the last few decades have increased insight into the mechanisms of hemolysis in both laminar and turbulent flows, with the ultimate goal of developing a comprehensive, mechanistic hemolysis model. Many models assume that hemolysis scales with a resultant, scalar stress representing all components of the fluid stress tensor. This study critically evaluates this scalar stress hypothesis by calculating the response of the red blood cell membrane to different types of fluid stress (laminar shear and extension, and three turbulent shear and extension cases), each with the same scalar stress. It was found that even though the scalar stress is the same for all cases, membrane tension varied by up to three orders of magnitude. In addition, extensional flow causes constant tension, while tank-treading in shear flow causes periodic tension, with tank-treading frequency varying by three orders of magnitude among the cases. For turbulent flow, tension also depends on eddy size. It is concluded, therefore, that scalar stress alone is inadequate for scaling hemolysis. Fundamental investigations are needed to establish a new index of the fluid stress tensor that provides reliable hemolysis prediction across the wide range of complex flows that occur in cardiovascular devices.     

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.     

Red cell distribution and the recruitment of pulmonary diffusing capacity.

The distribution of red blood cells in alveolar capillaries is typically nonuniform, as shown by intravital microscopy and in alveolar tissue fixed in situ. To determine the effects of red cell distribution on pulmonary diffusive gas transport, we computed the uptake of CO across a two-dimensional geometric capillary model containing a variable number of red blood cells. Red blood cells are spaced uniformly, randomly, or clustered without overlap within the capillary. Total CO diffusing capacity (DLCO) and membrane diffusing capacity (DmCO) are calculated by a finite-element method. Results show that distribution of red blood cells at a fixed hematocrit greatly affects capillary CO uptake. At any given average capillary red cell density, the uniform distribution of red blood cells yields the highest DmCO and DLCO, whereas the clustered distribution yields the lowest values. Random nonuniform distribution of red blood cells within a single capillary segment reduces diffusive CO uptake by up to 30%. Nonuniform distribution of red blood cells among separate capillary segments can reduce diffusive CO uptake by >50%. This analysis demonstrates that pulmonary microvascular recruitment for gas exchange does not depend solely on the number of patent capillaries or the hematocrit; simple redistribution of red blood cells within capillaries can potentially account for 50% of the observed physiological recruitment of DLCO from rest to exercise.     

Observations on induced carcinosis peritonei of the guinea pig

A diethylnitrosamine-induced hepatocellular carcinoma cell suspension (line-10), injected intraperitoneally in Sewall Wright strain-2 guinea pigs, causes ascites with implantation of malignant cells on the peritoneal surface. At these sites, swelling of the mesothelial cells and simultaneous proliferation of underlying fibroblasts and capillaries are seen. When there are about four layers of malignant cells and the mesothelial lining is disrupted, papillary projections of fibroblasts with capillaries, covered by malignant cells develop. These begin to behave as a "tissue". In these areas basement membrane destruction and lymphatic and blood vessel infiltration are demonstrable. These developments have been investigated by light microscopy, histochemistry, transmission- and scanning electron microscopy.     

Velocity distribution on the membrane of a tank-treading red blood cell

The kinematics of an area-conserving tank-treading disk-shaped red blood cell membrane is studied using the stream function method suggested by Secomb and Skalak (Q. Jl Mech. appl. Math. 35, Pt 2, 233–247, 1982). Two simple area-conserving velocity fields are superimposed to satisfy the continuity condition at the curved edges of the disk. A differential equation for the trajectory of any material point of the membrane is derived. The requirement of synchrony of the cycle for all membrane points leads to an integral equation which determines a magnitude function. An approximate solution is made possible by assuming small trajectory deflections.     

ELECTRON MICROSCOPY OF THE AVIAN RENAL GLOMERULUS

Electron microscopy of sections of chicken glomeruli shows them to possess a large central cell mass, occupying the hilum and the centre of the glomerulus, and continuous with the adventitia of the afferent and efferent arterioles. The glomerular capillaries form a much simpler system than in mammals and are spread over the surface of the central cell mass. Between the capillaries the mass is limited externally by the major component of the glomerular capillary basement membrane, which continues over the surface of the mass from one capillary to the next. Projections of the central cell mass characteristically form the support for glomerular capillaries, and smaller knobs of the central mass may project actually into the lumen of the capillaries, but always carry a layer of endothelial cytoplasm before them. They are never in direct contact with blood. The basement membrane of the glomerular capillary loop has a central dense layer and two lateral less dense layers as in mammals. The central dense layer is continuous with similar appearing dense material in the intercellular spaces of the adventitiae of the arterioles, and also with that of the central cell mass. The two less dense layers can also be traced into direct continuity with the less dense regions of this intercellular substance. The endothelial cytoplasm is spread as a thin sheet over the inner surface of the capillary basement membrane, and shows scattered "pores" resembling those described in mammals. Epithelial cells with interlacing pedicels are at least as prominent as those in mammals. Bowman's capsular membrane also possesses three layers similar to but less wide than those of the capillary basement membrane, and all three layers can be traced into continuity with the dark and light regions of the intercellular material of the adventitial cells of the arterioles, and beyond them with that of the central cell mass. At the hilum Bowman's capsular membrane also fuses with the capillary basement membrane.     

Correlation between local cell membrane displacements and filterability of human red blood cells.

Local mechanical fluctuations of the cell membrane of human erythrocytes were shown to involve MgATP- and Mg(2+)-driven fast membrane displacements. We propose that these local bending deformations of the cell membrane are important for cell passage through capillaries. In order to verify this hypothesis, we examined cell membrane fluctuations and filterability of erythrocytes over a wide range of medium osmolalities (180-675 mosmol/kg H2O). The results indicate the existence of a positive correlation between the amplitude of local cell membrane displacements and cell filterability. We suggest that the occurrence of metabolically driven membrane displacements on the side surface of the red blood cell diminishes its bending stiffness and enables it to fold more efficiently upon entrance into blood capillaries. Thus, local cell membrane displacements seem to play an important role in microcirculation.     

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.     

A fluid mechanical analysis of the velocity,adhesion, and destruction of cancer cells in capillaries during metastasis

Metastasis, a multistep process by which cancer disseminates through the body, mainly by intravascular routes, constitutes a major problem in cancer. When cancer cells are injected directly into the veins of animals, they are apparently arrested in the vascular bed of the first organ encountered and gradually released over the next 24 h. These interactions with the microvasculature are often associated in some manner with the death of many cancer cells, and are thought to contribute to the inefficiency of the metastatic process. We have made a theoretical analysis of cancer cells deformed into capillaries with respect to their intravascular velocity, adhesion to the vascular endothelium and intravascular destruction, in terms of the dynamics of the thin liquid film separating the surfaces of the blood vessels and cancer cells. Our calculations, which are based on previously reported experimental observations, indicate that the transit of cancer cells through the microvasculature is discontinuous, being interrupted by adhesions between the two. In addition, in some cases cell membrane rupture (and cell death) will occur when the critical membrane tension of the cancer cells is exceeded by the sum of their initial equilibrium membrane tension and the increased tension in the cancer cell membranes caused by friction generated as they move over the intraluminal surfaces of the capillaries. Our calculations on membrane rupture are consistent with previously unexplained observations by Sato and Suzuki relating cancer cell deformability to death on transpulmonary passage, and constitute a novel mechanism for “metastatic inefficiency” in terms of intravascular cancer cell death.     

Fission and refusion of red blood cells using a micropipette technique

In micropipette experiments with small capillaries and moderate high pressure difference (approximately 1000 Pa) cell fragmentation (fission) of human red blood cells without hemolysis was observed by TV-system for a large number of fresh red blood cells of different donors. After separation, the fragment moves away from the residual cell. In seven cases this process was evaluated quantitatively and was shown that the rate of the fragment was constant in time. Two mechanisms for this phenomenon are discussed. In particular cases a spontaneous re-fusion with the residual cell body in the capillary can be observed. In our opinion probably protein-depleted membrane surfaces arise and membrane fusion is possible simply by mechanical contact without additional electric fields and/or fusion agents.     

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.