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.

Time-dependent behavior of interstitial fluid pressure in solid tumors: implications for drug delivery

Elevated interstitial fluid pressure (IFP) may constitute a significant physiological barrier to drug delivery in solid tumors. Strategies for overcoming this barrier have not been developed to date. To identify and characterize various mechanisms regulating IFP and to develop strategies for overcoming the IFP barrier, we modeled the tumor as a poroelastic solid. We used this model to simulate the effect of changes in microvascular pressure and tumor blood flow (TBF) on IFP. To test model predictions, the effects of changes in arterial pressure and TBF on IFP were measured using a tissue-isolated tumor preparation. IFP in the center of an isolated tumor was predicted to follow variation of the arterial pressure with a time delay of the order of magnitude of 10 s, and this delay was found to be 11 +/- 6 s experimentally. Following a cessation of TBF, the time constant of the drop in IFP was predicted to be of the order of 1000 s and was found to be 1500 +/- 900 s experimentally. The former time scale is characteristic of transcapillary fluid exchange, and the latter of percolation of fluid through the interstitial matrix. Relying on the good agreement between theoretical predictions and experimental data, we estimated the effect of blood pressure modulation on macromolecular uptake in solid tumors. Our results show that no appreciable increase of macromolecular uptake should occur either by an acute or by a chronic increase of blood pressure. On the other hand, higher uptake would result from periodic modulation of blood pressure. Therefore, the effectiveness of a vasoconstrictor such as angiotensin II to increase macromolecular delivery should be significantly enhanced by periodic rather than bolus or continuous administration of the vasoactive agent.

A stochastic model of leukocyte rolling

Selectin-mediated leukocyte rolling under flow is an important process in leukocyte recruitment during inflammation. The rolling motion of individual cells has been observed to fluctuate randomly both in vivo and in vitro. This paper presents a stochastic model of the micromechanics of cell rolling and provides an analytical method of treating experimental data. For a homogeneous cell population, the velocity distribution is obtained in an analytical form, which is in good agreement with experimentally determined velocity histograms obtained previously. For a heterogeneous cell population, the model provides a simple, analytical method of separating the contributions of temporal fluctuations and population heterogeneity to the variance of measured rolling velocities. The model also links the mean and variance of rolling velocities to the molecular events underlying the observed cellular motion, allowing characterization of the distribution and release rate of the clusters of molecular bonds that tether the cell to substratum. Applying the model to the analysis of data obtained for neutrophils rolling on an E-selectin-coated surface at a wall shear stress of 1.2 dyn/cm2 yields estimations of the average distance between bond clusters (approximately micron) and the average time duration of a bond cluster resisting the applied fluid force (approximately 0.5 s).

Modeling of molecular mechanisms of cell adhesion

Cellular adhesion is a process of great importance in biology. We present a simple model of the adhesion process in which the molecular mechanisms involve a receptor, a ligand, and the cytoskeleton of the cell. Based on the energetic consideration of the process, we propose a molecular interpretation of the existing experimental data. The model suggests that the interaction of the receptor and (or) receptor-ligand with the cytoskeleton can have important influence on the formation and strength of the adhesion complex as well as on the subsequent interaction with different ligands. When conformational changes take place during the adhesion process, the characterization of the adhesion bonds based on chemical kinetics alone seems to be incomplete and must be supplemented by parameters, describing the functionality of the complex, i.e., change of the affinity for different ligands, as in the signal transduction, or the strength of the bond, as in the adhesion process.

Membrane model of endothelial cells and leukocytes. A proposal for the origin of a cortical stress

Previous models of the erythrocyte membrane have been based on the assumption that the resting curvature of the membrane is either flat or has a small curvature relative to the overall cell dimension. In contrast, several recent experimental observations, both in leukocytes and in endothelial cells, suggest that local regions of the membrane may have high membrane curvature in the resting state. The resting curvature may be of the order of plasmalemmal vesicles in endothelial cells or surface membrane folds on leukocytes. A tension is required to unfold the membrane with strain energy which depends largely on mean curvature. It is proposed that the tendency of endothelial or leukocyte membranes to wrinkle in the unstressed state may provide a restoring force, i.e. a cortical tension.

Active motion of polymorphonuclear leukocytes in response to chemoattractant in a micropipette

A novel experimental method of producing and observing the active motion of polymorphonuclear leukocytes (PMNs) using a micropipette technique has been recently developed (Usami et al., 1992). The present paper develops a quantitative theory for the chemoattractant gradients and cell locomotion observed in these experiments. In previous experimental methods (e.g., the Boyden chamber, the Zygmond chamber and the Dunn chamber) for study chemotaxis of leukocytes, fibroblasts, and PMNs, the exact nature of the concentration gradient of the chemoattractant is unknown. The cells may themselves modify the local gradient of the chemoattractant. In experiments using the micropipette, an internal source of chemoattractant provides well-defined boundary and initial conditions which allow the computation of the chemoattractant concentration gradient during the active locomotion of the PMNs. Since the cell completely fills the pipette lumen, convection is limited to the motion of the cells themselves. In coordinates moving with cell, it is assumed that diffusion is the only mechanism of mass transport of the chemoattractant (fMLP). Computations of the fMLP concentration during locomotion of the cell were carried out for a range of rates of fMLP binding by the receptors expressed on the front face of the cell membrane. The results show that the front face of the cell is subjected to increasing fMLP concentration during the cell motion. The sequence of events involve receptor binding of fMLP, signal transduction, polymerization of the cell cytoskeleton at the membrane of the front face, spatially dependent adhesion to the pipette wall, and localized contraction of the cytoskeleton. This sequence of events leads to the steady locomotion of the leukocytes in the micropipette. The computation of the distribution of the fMLP concentration during cell locomotion with constant velocity in micropipette experiments shows that the cell is exposed to increasing concentration of fMLP. This suggests that chemotaxis maybe induced by temporal gradient of an attractant.

Development of a device for measuring adherence of skin grafts to the wound surface

Adherence of a biological graft to the wound surface is the most important factor influencing the ultimate success of graft viability. A machine has been developed to test the adherence of biological graft materials to a substrate such as a wound surface. The peeling mode, which yields reproducible quantitative measurements of adherence, is a standard method for testing adhesives. The device is designed to continuously measure the force required to peel the graft from the substrate at a constant rate. This force is a function of the energy of adhesion per unit area of adhered surface. This device has been used to measure the peeling force of (2 x 2 cm) skin grafts which are applied to full-thickness wounds on mice. Results of tests on adherence of autografts on mice show that the peeling force increases significantly with time over the first 9 days of healing. Thus, this device is useful in quantitative comparison of various skin grafting techniques and artificial grafts.

Design and construction of a linear shear stress flow chamber

A new parallel plate flow chamber that has a linear variation of shear stress, starting from a predetermined maximum value at the entrance and falling to zero at the exit, has been designed and tested. This is in contrast to the usual rectangular channel plan which produces a constant shear stress over the entire length. The new design is based on the theory of Hele-Shaw flow between parallel plates. To verify the efficacy of the flow channel, the effect of fluid shear stress on platelet adhesion to a fibrinogen-coated glass surface was tested. The percentage of attached platelets after 5 min of shear stress is shown to be a function of shear stress. With this new flow chamber, cell-cell interactions can be studied efficiently over a wide range of shear stress using a single run at constant discharge.

Locomotion forces generated by a polymorphonuclear leukocyte

There have been very few studies which have measured the physical forces generated by cells during active movements. A special micropipette system has been designed to make it possible to observe cell motion within the pipette and to apply a pressure to counter the chemotactic migration of the cell. This provides a direct measure of the locomotion force generated by the cell. The average velocity of forward motion is 0.33 microns/s in the absence of counter-pressure. The application of a positive counter-pressure (C-P) causes a decrease in the velocity of the forward motion of the cell. At 17 cm H2O of C-P, the cell velocity drops to zero and even moves backward with a higher C-P. The results show that the decrement of velocity is linearly related to the magnitude of the C-P with a complete stoppage at a pressure of 17 cm H2O which corresponds to a force of 0.003 dyn. The maximum work rate of the cell is approximately 2.5 x 10(-8) erg/s.

Leukocyte deformability: finite element modeling of large viscoelastic deformation

An axisymmetric deformation of a viscoelastic sphere bounded by a prestressed elastic thin shell in response to external pressure is studied by a finite element method. The research is motivated by the need for understanding the passive behavior of human leukocytes (white blood cells) and interpreting extensive experimental data in terms of the mechanical properties. The cell at rest is modeled as a sphere consisting of a cortical prestressed shell with incompressible Maxwell fluid interior. A large-strain deformation theory is developed based on the proposed model. General non-linear, large strain constitutive relations for the cortical shell are derived by neglecting the bending stiffness. A representation of the constitutive equations in the form of an integral of strain history for the incompressible Maxwell interior is used in the formulation of numerical scheme. A finite element program is developed, in which a sliding boundary condition is imposed on all contact surfaces. The mathematical model developed is applied to evaluate experimental data of pipette tests and observations of blood flow.

Red cell membrane elasticity as determined by flow channel technique

The elasticity of red cell membrane was determined in a rectangular flow channel under controlled shear flow. The relation between shear stress and cell extension ratio (lambda) has been analyzed with the use of Evans' two-dimensional model. The deformed cell shapes observed experimentally agreed well with the model with lambda up to 1.4. The best correlation was found at lambda = 1.2. The analysis suggests a nonlinear extensional membrane modulus in the low stress range encountered in the flow channel. In terms of an appropriate strain parameter, the elastic modulus is shown to rise toward the level encountered in micropipette aspiration experiments. The implications of the present findings in modeling of cell mechanics and in cell hemolysis are discussed.

An interpretation of partial plug flow of concentrated suspensions

The partial plug flow of a concentrated suspension of rigid particles in a circular tube has been previously studied experimentally. It has been shown that a central core may exist in which the mean velocities of the particles and the suspending fluid are equal and constant within a cylindrical core of the flow. This behavior has been attributed to hydrodynamic interaction of the particles within the core. In the present analysis this interaction is interpreted in terms of passing vs. non-passing motions of adjacent particles. A hypothesis of a critical parameter alpha c involving the shear stress and the pressure gradient is explored and a new form of the relation of core diameter to particle size and concentration is developed based on alpha c.

Cytoplasmic rheology of passive neutrophils

The rheological properties of leukocytes are important to their effectiveness in the microcirculation. Previous studies based on in vitro data from micropipette experiments suggest that a Maxwell fluid bounded by a cortical shell with persistent tension is a realistic model for non-activated neutrophils in both the rapid and slow deformation phases. However, various viscoelastic coefficients have been obtained depending on the degree of cell deformation. In the present paper it is demonstrated that the cytoplasmic apparent viscosity and elasticity vary continuously, depending on the degree of deformation. These apparent variations are due to the inhomogeneous nature of the neutrophil internal structure. It is shown that the nucleus is much stiffer than the cytoplasm. The composite structure of the cell results in the deformation-dependent properties.

Folding of red blood cells in capillaries and narrow pores

The geometric features of red blood cells in narrow channels in vivo and in vitro were studied by electron microscopy. In rabbit myocardial capillaries about half of the red cells were folded. In polycarbonate filters with pore diameters of 2.2-4.5 microns approximately one third of the trapped red blood cells were folded. The frequency of folding did not depend on the applied pressure, which ranged from 0.1 to 8.0 cm H2O. The folding of the red blood cells in filter pores was used to estimate the bending stiffness of the membrane. An analysis based on the large deformation theory of bending of an elastic sheet was developed. Using pressures of 0.2 and 1.0 cm H2O, the bending stiffness of human red cell membranes was estimated to be approximately 2.4 - 11.6 x 10(-12) dyn-cm, which is in good agreement with other methods. A limiting radius of curvature of about 85 nm was found at higher pressures.

TEM immunogold staining of C3 from plasma onto titanium oxides

Immunogold staining in conjunction with TEM was used to observe C3 adsorption from plasma in relation to the underlying titanium structure of thermal, anodic, and electropolished oxides. Heat treatments and oxide thickness were found to have no significant effect on the adsorption behavior of C3, while surface oxide type possibly has. Surface concentration of C3 was found to be time- and plasma concentration-dependent. Evidence is given for the possible involvement of C3 in protein exchange, i.e., the Vroman effect. Diluted plasma resulted in a random distribution of gold colloids, whereas clustering occurred with undiluted plasma. Although C3 concentrations present on grain boundaries followed the same trend as that found on the surface, C3 was found to have a higher grain boundary than bulk concentration for 0.1% plasma.

Two-dimensional analysis of two-file flow of red cells along capillaries

The effect of a tank-treading motion of the red cell membrane on a staggered "zipper"-type red cell flow in capillaries is examined, using a two-dimensional theoretical model. An approximately triangular cell with a thin flexible membrane enclosing a viscous fluid is adopted as a model of the red cell. The motion of model red cells arranged periodically along a two-dimensional channel in an idealized zipper-type arrangement is analyzed numerically by a finite element method applied to the Stokes equations for the flow both inside and outside the model cells. It is shown that, if the viscosity ratio of the internal fluid to the suspending fluid is lower than a critical value, there exists a stable zipper-type arrangement of cells. In that arrangement, the cells remain stationary relative to each other with the membrane tank-treading. In contrast, inhibiting tank-treading by increasing the viscosity ratio above the critical value induces a cyclic oscillatory motion of red cells. The critical viscosity ratio increases if the channel is narrowed or if the spacing between cells is reduced. The present results suggest that the membrane tank-treading tends to stabilize zipper-type arrangements of red cells in capillaries at high hematocrit.

Passive deformations and active motions of leukocytes

The purpose of this paper is to review the development of continuum mechanics models of single leukocytes in both passive deformations and active motions and to indicate some future directions. Models of passive deformations describe the overall rheological behavior of single leukocytes under externally applied forces and predict the average mechanical properties from experimental data. Various "apparent" viscoelastic coefficients are obtained depending on the models assumed and the types of test used. Models of spontaneous motions postulate active driving mechanisms which must be derived internally from the cell itself and probably have different bases for different kind of motions. For pseudopod protrusion on leukocytes, energy transduction from chemical potential to mechanical work associated with actin polymerization at the tip of the projection is assumed to supply the motive power. For pseudopod retraction, active contraction due to actin-myosin interaction is assumed to be the driving force. The feasibility of the hypotheses are tested via numerical examples and comparison of the theoretical results with experimental measurements.

Rheological aspects of red blood cell aggregation

The rheological aspects of red blood cell aggregation include molecular phenomena, cell viscoelasticity, and bulk flow rheology. At the molecular level, rates at which bonds are formed and broken, the chemical energy liberation from bond formation, the elasticity of the cross-bridges and lateral mobility of cross-linking molecules must all be considered for a complete description of bond formation and distribution. Lateral migration of binding molecules occurs due to diffusion in the surface of the membrane but may also be influenced by the stresses in the membrane during separation of adhering cells. In red blood cell disaggregation, fluorescent probes have shown concentration of ligands in the region of contact close to the line of separation. The chemical potential decrement that occurs when a bond is formed provides the energy source that may deform red blood cells in the process of aggregation. The degree of aggregation and the extent of cell deformation depends on the viscoelastic properties of the cell as well as the dynamics of bond formation and repulsive potential of surface charges present, which is governed by an equation representing a balance of these energies. In flowing blood, the hydrodynamic forces applied by the plasma and surrounding cells must be added to the bond forces and elastic response of the cell. Under sufficiently strong aggregation, plug flow or large aggregates may result. At high shear rates, aggregation may be prevented due to the small contact time and high shear stresses so that no effects of aggregation may be observed. At intermediate shear stresses, transitory contact, adhesion and disaggregation may occur between neighboring cells. Such phenomena have not been analyzed in detail, but simplified models suggest that plug-like flow can occur due to hydrodynamic cell-cell interaction even when cells are not aggregated.

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

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

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.

Stability of particle motions in a narrow channel flow

Single rows or two rows of identical circular cylinders spaced regularly in a narrow channel flow have been shown to be capable of steady flow provided the cylinders are located at equal lateral positions and with equal spacings in the flow direction. The stability of such steady flows of circular cylinders is studied for periodic perturbations of the particle positions, assuming that every other cylinder is equally perturbed in lateral position and spacing along the channel. This results in two rows which are not symmetrically placed. The suspending fluid is assumed to be an incompressible Newtonian fluid. It is assumed that no external forces or moments act on the cylinders and the effects of inertia forces on the motion of the fluid and the cylinders are negligible. The velocity field of the suspending fluid and the instantaneous velocities of the cylinders are computed by the finite element method. The translational velocities of the cylinders are obtained for a large number of particle positions, from which the trajectories of their relative motion are determined for various initial positions near the regular single-file and two-file arrangements. It is shown that when the initial arrangements of the cylinders are slightly perturbed from the regular (alternating) two-file flows, the trajectories of their relative motions become small closed loops around the regular two-file arrangements. In contrast, such small closed trajectories are not obtained when they start from the arrangements near the regular single-file flows or regular (symmetric) double-file flows, suggesting that these flows are unstable under the conditions examined.

One-dimensional steady continuum model of retraction of pseudopod in leukocytes

A one-dimensional steady state continuum mechanics model of retraction of pseudopod in leukocytes is developed. The retracting pseudopod is assumed to move bodily toward the main cell body, the bulk motion of which can be represented by cytoplasmic flow within a typical stream tube through the leukocyte. The stream tube is approximated by a frictionless tube with prescribed geometry. The passive rheological properties of cytoplasm in the main cell body and in the pseudopod are modeled, respectively, by Maxwell fluid and Hookean solid. The two regions are assumed to be separated by a sharp interface at which actin gel solates and thereby changes its rheological properties as it flows from the pseudopod to the main cell body. The driving mechanism responsible for the active retraction motion is hypothesized to be a spontaneous deformation of the actin gel, analogous but not necessarily equal to the well known actin-myosin interaction. This results in an active contractile stress being developed in the pseudopod as well as in the cell cortex. The transverse traction pulls against the inclined wall of the stream tube and is transduced into an axial stress gradient, which in turn drives the flow. The tension on the tube wall is picked up by the prestressed cortical shell. Governing equations and boundary conditions are derived. A solution is obtained. Sample data are computed. Comparison of the theory with experiments shows that the model is compatible to the observations.

A continuum model of protrusion of pseudopod in leukocytes

The morphology of human leukocytes, the biochemistry of actin polymerization, and the theory of continuum mechanics are used to model the pseudopod protrusion process of leukocytes. In the proposed model, the pseudopod is considered as a porous solid of F-actin network, the pores of which are full of aqueous solution. G-actin is considered as a "solute" transported by convection and diffusion in the fluid phase. The pseudopod grows as actin filaments elongate at their barbed ends at the tip of the pseudopod. The driving force of extension is hypothesized as being provided by the actin polymerization. It is assumed that elongation of actin filaments, powered by chemical energy liberated from the polymerization reaction, does mechanical work against opposing pressure on the membrane. This also gives rise to a pressure drop in the fluid phase at the tip of the pseudopod, which is formulated by an equation relating the work done by actin polymerization to the local state of pressure. The pressure gradient along the pseudopod drives the fluid filtration through the porous pseudopod according to Darcy's Law, which in turn brings more actin monomers to the growing tip. The main cell body serves as a reservoir of G-actin. A modified first-order equation is used to describe the kinetics of polymerization. The rate of pseudopod growth is modulated by regulatory proteins. A one-dimensional moving boundary problem based on the proposed mechanism has been constructed and approximate solutions have been obtained. Comparison of the solutions with experimental data shows that the model is compatible with available observations. The model is also applicable to growth of other cellular systems such as elongation of acrosomal process in sperm cells.