Leukocyte relaxation properties

Study of the mechanical properties of leukocytes is useful to understand their passage through narrow capillaries and interaction with other cells. Leukocytes are known to be viscoelastic and their properties have been established by micropipette aspiration techniques. Here, the recovery of leukocytes to their normal spherical form is studied after prolonged deformation in a pipette which is large enough to permit complete entry of the leukocyte. The recovery history is characterized by the time history of the major diameter (d1) and minor diameter (d2). When the cell is removed from the pipette, it shows initially a small rapid recoil followed by a slower asymptotic recovery to the spherical shape. In the presence of cell activation and formation of pseudopods, the time history for recovery is prolonged compared with passive cell recovery. If a protopod pre-existed during the holding period, the recovery only begins when the protopod starts to retract.

Numerical study of asymmetric flows of red blood cells in capillaries

Experimental studies have shown that red blood cells in capillaries may flow in single-file or multifile arrangements. To model multifile rheological behavior, the asymmetric flows of rigid circular cylinders in a two-dimensional channel are studied by numerical analysis. The rigid circular cylinders are arranged off-center in a channel in a row or two rows with equal spacings. The motion of the suspending fluid is analyzed by the finite element method applied to the Stokes equation, and the motions of the particles are simultaneously determined under the zero force and zero moment conditions appropriate to neutorally buoyant particles. The velocity difference between the particles and the bulk flow is significantly affected by the arrangement of the particles. The particle velocity is reduced as the particles are moved away from the centerline of the channel. At a constant concentration of the particles, the relative apparent viscosity of an off-center arrangement is considerably higher than that of a single-file flow of the particles located on the centerline of the channel. The present results suggest that changes of the radial distribution of red cells flowing through narrow vessels may lead to alterations of the Fahraeus and Fahraeus-Lindqvist effects.

Transient relative motion of two cells in a channel flow

The hydrodynamic interaction of a red blood cell and a white blood cell in microvessels is studied, by use of a two-dimensional numerical model. The red blood cell, modeled as a small rigid circular cylinder, and the white blood cell, modeled as a larger rigid circular cylinder, are immersed in an incompressible Newtonian fluid in a two-dimensional channel. It is assumed that no external force or moment acts on the model cells, and the effect of inertia forces on the motion of the fluid and the cells is neglected. The velocity field of the suspending fluid and the instantaneous velocities of the two model cells are computed by the finite element method. Using the translational velocities of the model cells obtained, the trajectories of their relative motion are determined, for various initial positions. It is shown that the cells may or may not pass each other or separate, depending on the initial positions. The present results compare well to the experimental results.

Interaction of stress and growth in a fibrous tissue

The influence of stress on the growth and remodeling of a soft biological tissue is considered. For this purpose, the soft tissue is idealized as a fiber network. The stress-free lengths of the fibers composing the network are not fixed as in an inert elastic solid, but are assumed to evolve as a result of growth and stress adaptation. Similarly, the topology of the fiber network may also evolve under the application of stress. A set of constitutive equations are proposed which relate the tissue stress to the deformation of the tissue as well as to its growth and microstructure. It is shown that distinctly different growth patterns which may arise during initial growth or wound healing can be modeled by the proposed mathematical analysis.

Passive deformation analysis of human leukocytes

The following analysis presents an experimental and theoretical study of the passive viscoelastic behavior of human leukocytes. Individual neutrophils in EDTA were observed both during their partial aspiration into a small micropipette and after expulsion from a large micropipette where the cell had been totally aspirated and deformed into a sausage shape. To analyze the data, a passive model of leukocyte rheology has been developed consisting of a cortical shell containing a Maxwell fluid which describes the average properties of the cell cytoplasm. The cortical shell represents a crosslinked actin layer near the surface of the cell and is assumed to be under pre-stressed tension. This model can reproduce the results of experiments using micropipette for both short-time small deformation and slow recovery data after large deformation. In addition, a finite element scheme has been established for the same model which shows close agreement with the analytical solution.

Rheology of leukocytes

The rheological properties of human leukocytes (WBCs) have been studied by micropipette aspiration and filtration tests. A small aspiration pressure applied via a micropipette (diameter approximately equal to 3 micron) causes the WBC to undergo a rapid elastic deformation followed by a slow creep. The data can be analyzed with a viscoelastic model: an elastic element K1 in parallel with a Maxwell element (elastic element K2 in series with viscous element mu). Neutrophils and B lymphocytes are similar in K1, K2, and mu, but these values are higher for T lymphocytes. Treatment of neutrophils with colchicine decreases K2 and mu without changing K1, whereas cytochalasin B decreases all three coefficients; these results indicate the importance of cytoskeletal microtubules and microfilaments in WBC rheology. In the presence of Ca2+, WBCs form protopods which have increased viscoelastic coefficients. Inhibition of protopod formation with pentoxifylline is associated with an increase in WBC deformability. The ruffled surface of the apparently round WBC provides an area about twice that needed to enclose a smooth sphere of the same volume; this geometric feature plays an important role in whole WBC deformability tested through 4-5 micron filter pores or micropipettes. Because of its larger volume and higher cellular viscosity, each WBC is equivalent to approximately 700 erythrocytes in its tendency to block 5-micron channels. The rheology of WBCs has significant implications in their functional behavior, including flow through the microcirculation and interaction with endothelial cells.

Morphological analysis and computer-aided, three dimensional reconstruction of chondrocytic columns in rabbit growth plates

Proximal tibial growth plates of New Zealand white rabbits were serially sectioned in parasagittal and horizontal planes for three dimensional, light microscopic analysis of the chondrocytic columns. A total of 431 columns was analysed. Of these, 258 columns extended through the full height of the growth plate. The remaining columns were considerably shorter, being located either predominantly in the epiphyseal half of the growth plate (100) or in the metaphyseal half of the growth plate (73). The epiphyseal and metaphyseal columns were found in clusters in the plate. Some columns in all three groups had interruptions along their length, while others had duplications. Computer-aided, three dimensional graphic reconstructions were prepared of a selected group of columns. The reconstructions illustrated the variability in the morphology and the dimensions of the neighbouring chondrocytic columns. The observations suggest that chondrocytic columns in rabbit growth plates are replaced regularly and that the small cell zone may play an important role as the cellular source for column renewal.

Studies on orthocephalization: growth behavior of the rat skull in the period 13-49 days as described by the finite element method

Rat cranial skeletal growth was studied, using a cross-sectional data set, for the period 13-49 days by the application of the concepts of continuum mechanics and the numerical techniques of the finite element method (FEM). In contrast to the methods of conventional craniometry (CM) and roentgenographic cephalometry (RCM) the FEM permits fine scale, reference frame invariant descriptions and analysis of growth behavior. This advantage was demonstrated by a numerical example of the use of FEM. The skull was discretized into a number of two-dimensional, triangular elements, whose enclosed areas corresponded closely to both specific skeletal structures and to related functional matrices. Since it was assumed presently that the growth behavior of all of the points enclosed within a given element was similar, the application of the functional matrix hypothesis permitted an integrated description of the growth of the skeletal structure and functional matrix related to each element. The principal locus of rotation of the facial skull, relative to the cranial base, is the inferior frontoethmoidal articulation, a motion that includes a rigid body rotation. Other active and passive skeletal and visceral growth events associated with orthocephalization were located and described. Finally it was shown that the morphogenetically important growth behavior of other portions of the rat head were not directly involved in orthocephalization.

Theory of filtration of mixed blood suspensions

A theory is developed for the flow of suspensions of blood cells through filters in which the properties of the cells are defined by statistical distributions. It is shown that conditions are generally transient, and computational procedures are developed to compute the pressure drop and the fraction of the pores of the filter containing cells of various types as a function of time. The computations show a large influence of very small concentrations of stiff cells which gradually collect in the filter and effectively plug the filter during the time of a typical test. It is also shown that the mean value of the resistance offered by a cell population with a limited distribution of resistances is more important than dispersion of resistances about the mean in determining the observable pressure curve. Experimental data are presented demonstrating that the drug pentoxifylline reduces the stiffness of leukocytes.

Finite element method modeling of craniofacial growth

The application of the concepts of continuum mechanics and of the numerical techniques of the finite element method permits the development of a new and potentially clinically useful method of describing craniofacial skeletal growth. This new method differs from those associated with customary roentgenographic cephalometry in that its descriptions and analyses are invariant; that is, they are independent of any method of registration and superimposition. Such invariance avoids the principal geometric constraint explicit in all analytical methods associated with conventional roentgenographic cephalometry. The conceptual and mathematical bases of the finite element method (FEM) are presented and illustrated by the numerical and graphic descriptions of the two-dimensional growth of the rat skull, for which two sets of longitudinal growth data are used. In practice, the FEM permits analysis of the skull at a scale significantly finer than previously possible, by considering cranial structure as consisting of a relatively large number of contiguous finite elements. For each such element, independently, it is then possible to describe and depict both the magnitude and the direction of temporal size and shape changes occurring in that element relative to itself at some initial time. It is emphasized that such descriptions are completely independent of any local reference frame.

Deformation of leukocytes on a hematological blood film

Human leukocytes in a blood film exhibit a significantly larger diameter than in the circulation. This is due to the fact that white cells are highly deformed during preparation of a blood film. Instead of having the usual spherical shape, the cells are compressed to "pancake" forms with a thickness of about 1 micron. Hematological investigation is usually performed on these compressed cells, but in the circulation they are not observed. The deformation of the cells on a blood film is due to compression by the glass edge used to spread the blood. After deformation leukocytes do not have enough time to recover since the blood film usually dries in a shorter period than is needed for cell recovery. The shape and size of the leukocyte on the blood film is not only determined by cell volume but also by the cell membrane area. This is shown for each kind of leukocyte by independent prediction of the pancake dimensions from previous measurements of cell volume and membrane area. Leukocytes which are strongly compressed during blood film preparation may exhibit mechanical damage with rupture of membranes.

Aggregation and disaggregation of red blood cells

The aggregation of red blood cells may be analyzed as an interaction of an adhesive surface energy and the elastic stored energy which results from deformation of the cell. The adhesive surface energy is the work required to separate a unit adhered area and is the resultant of adhesive forces due to the bridging molecules that bind the cells together and the electrostatic repulsion due to surface charge. The elastic strain energy in the case of the red blood is associated with the membrane elasticity only since the interior of the cell is liquid. The membrane elasticity is due both to bending stiffness and shear. The area expansion is small and may be neglected. These assumptions allow realistic computation of red cell shapes in rouleaux. The disaggregation of rouleaux requires an external force which must overcome the adhesive energy and also supply additional elastic energy of deformation. Depending on the geometry, the initial effect of elastic energy may tend to aid disaggregation. In a shear flow, the stresses on a suspended rouleau alternately tend to compress and to disaggregate the cells if they are free to rotate. This introduces a time dependence so that viscous effects due to the viscosity of the cell membrane, the cell cytoplasm and the external fluid may play a role in determining whether disaggregation proceeds to completion or not.

Biomechanics at the cellular level. The ALZA distinguished lecture

The mechanical behavior of individual cells presents a variety of problems of interest in many different biologic phenomena. The rheology of single red blood cells is well developed, and passive properties of leukocytes and endothelial cells are currently being explored. Dynamic aspects of single-cell mechanics, including growth, cell division, active motion, contractile mechanisms, phagocytosis, and locomotion, offer many challenging aspects to be analyzed. Transduction mechanisms of neurosensory cells and mechanical stresses and damage of neural structures are relatively unexplored.

Hydrodynamic and mechanical aspects of endothelial permeability

The passive transport of water through the endothelial cell layer junctions is considered from the standpoint of hydrodynamic theories based on ultrastructural information. The local geometry of tight junctions based on molecular level forces and elastic membrane properties has been modeled and leads to estimates of the hydraulic resistance of the clefts. It is shown that the large resistance measured experimentally can be accounted for in this model. The transport of large macromolecules via vesicles which diffuse across the endothelial cell has been developed, but recent experimental data do not appear to support this mechanism as a primary pathway. Fused vesicles forming an open channel appear to be rare. Leaky junctions, such as around dying endothelial cells or produced by cytoskeletal changes within the cells, may be important in control of endothelial permeability. Another kind of model is a fiber matrix model of the endocapillary layer, extending into the intercellular clefts which can also account for the molecular seiving properties of the endothelial cell layer but may produce a large resistance to water flux.

An allometric network model of craniofacial growth

This study of cranial skeletal growth kinematics details the conceptual principles underlying the development of an allometric network model of such growth. This model is tested by the analysis of longitudinal rat and cross-sectional human growth data and by comparison of this model with a previously described allometric centered model. It is shown that the network model is superior to the centered model in three ways: (1) The allometric network model permits growth prediction when allometric constants are known; (2) the network model has significantly smaller errors than the centered model; and (3) the network model is capable of displaying growth kinematics of both the neural and facial skulls while in time there are marked transformations, such as relative rotations of two sets of cranial anatomic points.

Constitutive equations of erythrocyte membrane incorporating evolving preferred configuration

The erythrocyte membrane is modeled as a two-dimensional viscoelastic continuum that evolves under the application of stress. The present analysis of the erythrocyte membrane is motivated by the recent development of knowledge about its molecular structure. The constitutive equations proposed in the present analysis explain in a consistent manner the data on both the deformation and recovery phases of the micropipette experiment. The rheological equations of the present study are applied in a later section to the analysis of a plane membrane deformation that is quantitatively similar to the tank-treading motion of the erythrocytes in a shear field. The computations yield useful information on how the membrane viscosity becomes a more dominant feature in tank-treading motion. The material constants appearing in the proposed constitutive equations may be useful indications of the biochemical state of the membrane in health and disease.

Continuum mechanical model of leukocytes during protopod formation

A new continuum mechanical theory for protopod extension in leukocytes is developed. Protopod formation is an active process which is the basis for amoeboid displacement on substrates. Leukocytes may form protopods both when adhering to a substrate and when freely suspended in plasma. Therefore the required energy is derived from the cell itself. Protopods are depleted of granules and other organelles, they have a fine fibrillar ultrastructure, and they are covered by a cell membrane. They grow at about 5 micron/min until they reach a length of 4-5 micron. A period of protopod retraction follows during which granules re-enter via the protopod base by Brownian motion. Micropipette experiments have indicated that the protoplasm in the leukocyte has viscoelastic properties, whereas the protopod is stiffer and shows elastic behavior. We propose a continuum theory based on the polymerization of the actin matrix in the cell which results in gelation with a preferred orientation. It is triggered by influx of Ca++ across local regions of the cell membrane and the polymerization occurs along an interface at the base of the polymerized protopod. As cytoplasm passes through the interface it is subject both to a volumetric strain due to exclusion of granules and a shear strain due to alignment of actin molecules. The polymerization provides an active force leading to projection of the protopod and cell deformation. The base of the protopod rests on the unpolymerized cytoplasm along the interface. As the external plasma medium and the cell membrane, if it is not stretched taut, offer little resistance, the projection of the protopod proceeds outward with simultaneous unfolding of the membrane.(ABSTRACT TRUNCATED AT 250 WORDS)

Energy balance in red cell interactions

Experiments were performed to elucidate the balance of energies involved in the formation of red blood cell (RBC) aggregates and in their disaggregation. In order to achieve a mean stable rouleau formation, the aggregating energy provided by macromolecular binding to the cell membrane must overcome the disaggregation energy of electrostatic repulsion between RBC surfaces and the effects of mechanical shear stress. In a quiescent suspension the net aggregation energy is largely stored in the membrane as a change in strain energy. The alterations in strain energy cause the curvature of the end cells in rouleaux of normal RBCs in Dx 80 to change from concave to convex and back again to concave as [Dx 80] was increased from 1 to 4 to 6 g/dl; computation of net aggregation energy per unit area (gamma) from changes in membrane strain energy yielded values on the order of 10(3) ergs/cm2. The end cells of neuraminidase-treated RBCs remained convex with [Dx 80] above 2 g/dl, and gamma is probably on the order of 10(2) ergs/cm2. The variations in gamma with [Dx 80] and RBC surface charge are similar to variations in reflectometric aggregation index without shear ( RAI0 ), indicating that RAI0 reflects gamma. The difference in gamma between normal and neuraminidase-treated RBCs represents the electrostatic repulsive energy, the magnitude of which varied inversely with dextran molecular size and directly with [Dx]. Moderate shearing in the reflectometer enhanced RBC aggregation by promoting cell-cell encounter, but high shear stresses cause RBC disaggregation. The energy required to disaggregate a unit interacting area of normal RBCs in Dx 80 in a flow channel is on the order of 10(4) ergs/cm2, which is much lower than gamma. These results suggest that the release of the stored membrane strain energy during disaggregation aids in the separation process. The results show that the understanding of RBC aggregation requires the considerations of surface charge, properties of aggregating agents, and the rheology of the cell membrane.

Determination of erythrocytes transit times through a 5 mu "nuclepore" filter

Filtration experiments through 5 or 3 mu Nuclepore membranes are often performed in order to assess the so-called erythrocyte deformability. The relation between this parameter and the RBC filterability is not straightforward. A simple theoretical treatment relating filtration index (as determined by the initial flow rate method) to the average RBC flow resistance through an average pore is presented. In order to deduce the average RBC transit time through the membrane from the initial flow rate data, the suspension hematocrit change after filtration has been determined. The calculated average transit time is comparable to experimental values, as determined by KIESEWETTER et al. with the single pore technique.