Cells in 3D matrices under interstitial flow: Effects of extracellular matrix alignment on cell shear stress and drag forces

Interstitial flow is an important regulator of various cell behaviors both in vitro and in vivo, yet the forces that fluid flow imposes on cells embedded in a 3D extracellular matrix (ECM), and the effects of matrix architecture on those forces, are not well understood. Here, we demonstrate how fiber alignment can affect the shear and pressure forces on the cell and ECM. Using computational fluid dynamics simulations, we show that while the solutions of the Brinkman equation accurately estimate the average fluid shear stress and the drag forces on a cell within a 3D fibrous medium, the distribution of shear stress on the cellular surface as well as the peak shear stresses remain intimately related to the pericellular fiber architecture and cannot be estimated using bulk-averaged properties. We demonstrate that perpendicular fiber alignment of the ECM yields lower shear stress and pressure forces on the cells and higher stresses on the ECM, leading to decreased permeability, while parallel fiber alignment leads to higher stresses on cells and increased permeability, as compared to a cubic lattice arrangement. The Spielman–Goren permeability relationships for fibrous media agreed well with CFD simulations of flow with explicitly considered fibers. These results suggest that the experimentally observed active remodeling of ECM fibers by fibroblasts under interstitial flow to a perpendicular alignment could serve to decrease the shear and drag forces on the cell.     

A theoretical model study of the influence of fluid stresses on a cell adhering to a microchannel wall.

We predict the amplification of mechanical stress, force, and torque on an adherent cell due to flow within a narrow microchannel. We model this system as a semicircular bulge on a microchannel wall, with pressure-driven flow. This two-dimensional model is solved computationally by the boundary element method. Algebraic expressions are developed by using forms suggested by lubrication theory that can be used simply and accurately to predict the fluid stress, force, and torque based upon the fluid viscosity, muoffhannel height, H, cell size, R, and flow rate per unit width, Q2-d. This study shows that even for the smallest cells (gamma = R/H << 1), the stress, force, and torque can be significantly greater than that predicted based on flow in a cell-free system. Increased flow resistance and fluid stress amplification occur with bigger cells (gamma > 0.25), because of constraints by the channel wall. In these cases we find that the shear stress amplification is proportional to Q2-d(1-gamma)-2, and the force and torque are proportional to Q2-d(1-gamma2)-5/2. Finally, we predict the fluid mechanical influence on three-dimensional immersed objects. These algebraic expressions have an accuracy of approximately 10% for flow in channels and thus are useful for the analysis of cells in flow chambers. For cell adhesion in tubes, the approximations are accurate to approximately 25% when gamma > 0.5. These calculations may thus be used to simply predict fluid mechanical interactions with cells in these constrained settings. Furthermore, the modeling approach may be useful in understanding more complex systems that include cell deformability and cell-cell interactions.     

Parallel-plate Flow Chamber and Continuous Flow Circuit to Evaluate Endothelial Progenitor Cells under Laminar Flow Shear Stress

The overall goal of this method is to describe a technique to subject adherent cells to laminar flow conditions and evaluate their response to well quantifiable fluid shear stresses¹.Our flow chamber design and flow circuit (Fig. 1) contains a transparent viewing region that enables testing of cell adhesion and imaging of cell morphology immediately before flow (Fig. 11A, B), at various time points during flow (Fig. 11C), and after flow (Fig. 11D). These experiments are illustrated with human umbilical cord blood-derived endothelial progenitor cells (EPCs) and porcine EPCs^2,3.This method is also applicable to other adherent cell types, e.g. smooth muscle cells (SMCs) or fibroblasts.The chamber and all parts of the circuit are easily sterilized with steam autoclaving. In contrast to other chambers, e.g. microfluidic chambers, large numbers of cells (> 1 million depending on cell size) can be recovered after the flow experiment under sterile conditions for cell culture or other experiments, e.g. DNA or RNA extraction, or immunohistochemistry (Fig. 11E), or scanning electron microscopy⁵. The shear stress can be adjusted by varying the flow rate of the perfusate, the fluid viscosity, or the channel height and width. The latter can reduce fluid volume or cell needs while ensuring that one-dimensional flow is maintained. It is not necessary to measure chamber height between experiments, since the chamber height does not depend on the use of gaskets, which greatly increases the ease of multiple experiments. Furthermore, the circuit design easily enables the collection of perfusate samples for analysis and/or quantification of metabolites secreted by cells under fluid shear stress exposure, e.g. nitric oxide (Fig. 12)⁶.     

Mathematical analysis of mural thrombogenesis. Concentration profiles of platelet-activating agents and effects of viscous shear flow.

The concentration profiles of adenosine diphosphate (ADP), thromboxane A2 (TxA2), thrombin, and von Willebrand factor (vWF) released extracellularly from the platelet granules or produced metabolically on the platelet membrane during thrombus growth, were estimated using finite element simulation of blood flow over model thrombi of various shapes and dimensions. The wall fluxes of these platelet-activating agents were estimated for each model thrombus at three different wall shear rates (100 s-1, 800 s-1, and 1,500 s-1), employing experimental data on thrombus growth rates and sizes. For that purpose, whole human blood was perfused in a parallel-plate flow chamber coated with type l fibrillar human collagen, and the kinetic data collected and analyzed by an EPl-fluorescence video microscopy system and a digital image processor. It was found that thrombin concentrations were large enough to cause irreversible platelet aggregation. Although heparin significantly accelerated thrombin inhibition by antithrombin lll, the remaining thrombin levels were still significantly above the minimum threshold required for irreversible platelet aggregation. While ADP concentrations were large enough to cause irreversible platelet aggregation at low shear rates and for small aggregate sizes, TxA2 concentrations were only sufficient to induce platelet shape change over the entire range of wall shear rates and thrombi dimensions studied. Our results also indicated that the local concentration of vWF multimers released from the platelet alpha-granules could be sufficient to modulate platelet aggregation at low and intermediate wall shear rates (less than 1,000 s-1). The sizes of standing vortices formed adjacent to a growing aggregate and the embolizing stresses and the torque, acting at the aggregate surface, were also estimated in this simulation. It was found that standing vortices developed on both sides of the thrombus even at low wall shear rates. Their sizes increased with thrombus size and wall shear rate, and were largely dependent upon thrombus geometry. The experimental observation that platelet aggregation occurred predominantly in the spaces between adjacent thrombi, confirmed the numerical prediction that those standing vortices are regions of reduced fluid velocities and high concentrations of platelet-activating substances, capable of trapping and stimulating platelets for aggregation. The average shear stress and normal stress, as well as the torque, acting to detach the thrombus, increased with increasing wall shear rate. Both stresses were found to be nearly independent of thrombus size and only weekly dependent upon thrombus geometry. Although both stresses had similar values at low wall shear rates, the average shear stress became the predominant embolizing stress at high wall shear rates.     

Tortuosity triggers platelet activation and thrombus formation in microvessels

Tortuous blood vessels are often seen in humans in association with thrombosis, atherosclerosis, hypertension, and aging. Vessel tortuosity can cause high fluid shear stress, likely promoting thrombosis. However, the underlying physical mechanisms and microscale processes are poorly understood. Accordingly, the objectives of this study were to develop and use a new computational approach to determine the effects of venule tortuosity and fluid velocity on thrombus initiation. The transport, collision, shear-induced activation, and receptor-ligand adhesion of individual platelets in thrombus formation were simulated using discrete element method. The shear-induced activation model assumed that a platelet became activated if it experienced a shear stress above a relative critical shear stress or if it contacted an activated platelet. Venules of various levels of tortuosity were simulated for a mean flow velocity of 0.10?cm s(-1), and a tortuous arteriole was simulated for a mean velocity of 0.47?cm s(-1). Our results showed that thrombus was initiated at inner walls in curved regions due to platelet activation in agreement with experimental studies. Increased venule tortuosity modified fluid flow to hasten thrombus initiation. Compared to the same sized venule, flow in the arteriole generated a higher amount of mural thrombi and platelet activation rate. The results suggest that the extent of tortuosity is an important factor in thrombus initiation in microvessels.     

A finite element study of mechanical stimuli in scaffolds for bone tissue engineering

Mechanical stimuli are one of the factors that affect cell proliferation and differentiation in the process of bone tissue regeneration. Knowledge on the specific deformation sensed by cells at a microscopic level when mechanical loads are applied is still missing in the development of biomaterials for bone tissue engineering. The objective of this study was to analyze the behavior of the mechanical stimuli within some calcium phosphate-based scaffolds in terms of stress and strain distributions in the solid material phase and fluid velocity, fluid pressure and fluid shear stress distributions in the pores filled of fluid, by means of micro computed tomographed (CT)-based finite element (FE) models. Two samples of porous materials, one of calcium phosphate-based cement and another of biodegradable glass, were used. Compressive loads equivalent to 0.5% of compression applied to the solid material phase and interstitial fluid flows with inlet velocities of 1, 10 and 100 microm/s applied to the interconnected pores were simulated, changing also the inlet side and the viscosity of the medium. Similar strain distributions for both materials were found, with compressive and tensile strain maximal values of 1.6% and 0.6%, respectively. Mean values were consistent with the applied deformation. When 10 microm/s of inlet fluid velocity and 1.45 Pas viscosity, maximal values of fluid velocity were 12.76 mm/s for CaP cement and 14.87 mm/s for glass. Mean values were consistent with the inlet ones applied, and mean values of shear stress were around 5 x 10(-5)Pa. Variations on inlet fluid velocity and fluid viscosity produce proportional and independent changes in fluid velocity, fluid shear stress and fluid pressure. This study has shown how mechanical loads and fluid flow applied on the scaffolds cause different levels of mechanical stimuli within the samples according to the morphology of the materials.     

Effects of extracellular fiber architecture on cell membrane shear stress in a 3D fibrous matrix

Interstitial fluid flow has been shown to affect the organization and behavior of cells in 3D environments in vivo and in vitro, yet the forces driving such responses are not clear. Due to the complex architecture of the extracellular matrix (ECM) and the difficulty of measuring fluid flow near cells embedded in it, the levels of shear stress experienced by cells in this environment are typically estimated using bulk-averaged matrix parameters such as hydraulic permeability. While this is useful for estimating average stresses, it cannot yield insight into how local matrix fiber architecture-which is cell-controlled in the immediate pericellular environment-affects the local stresses imposed on the cell surface. To address this, we used computational fluid dynamics to study flow through an idealized mesh constructed of a cubic lattice of fibers simulating a typical in vitro collagen gel. We found that, in such high porosity matrices, the fibers strongly affect the flow fields near the cell, with peak shear stresses up to five times higher than those predicted by the Brinkman equation. We also found that minor remodeling of the fibers near the cell surface had major effects on the shear stress profile on the cell. These findings demonstrate the importance of fiber architecture to the fluid forces on a cell embedded in a 3D matrix, and also show how small modifications in the local ECM can lead to large changes in the mechanical environment of the cell.     

Microscopic investigation of erythrocyte deformation dynamics

The understanding of erythrocyte deformation under conditions of high shear stress and short exposure time is central to the study of hemorheology and hemolysis within prosthetic blood contacting devices. A combined computational and experimental microscopic study was conducted to investigate the erythrocyte deformation and its relation to transient stress fields. A microfluidic channel system with small channels fabricated using polydimethylsiloxane on the order of 100 mum was designed to generate transient stress fields through which the erythrocytes were forced to flow. The shear stress fields were analyzed by three-dimensional computational fluid dynamics. Microscopic images of deforming erythrocytes were experimentally recorded to obtain the changes in cell morphology over a wide range of fluid dynamic stresses. The erythrocyte elongation index (EI) increased from 0 to 0.54 with increasing shear stress up to 123 Pa. In this shear stress range, erythrocytes behaved like fluid droplets, and deformed and flowed following the surrounding fluid. Cells exposed to shear stress beyond 123 Pa (up to 5170 Pa) did not exhibit additional elongation beyond EI=0.54. Two-stage deformation of erythrocytes in response to shear stress was observed: an initial linear elongation with increasing shear stress and a plateau beyond a critical shear stress.     

A model for strain amplification in the actin cytoskeleton of osteocytes due to fluid drag on pericellular matrix.

A model is presented that provides a resolution to a fundamental paradox in bone physiology, namely, that the strains applied to whole bone (i.e., tissue level strains) are much smaller (0.04-0.3 percent) than the strains (1-10 percent) that are necessary to cause bone signaling in deformed cell cultures (Rubin and Lanyon, J. Bone Joint Surg. 66A (1984) 397-410; Fritton et al., J. Biomech. 33 (2000) 317-325). The effect of fluid drag forces on the pericellular matrix (PM), its coupling to the intracellular actin cytoskeleton (IAC) and the strain amplification that results from this coupling are examined for the first time. The model leads to two predictions, which could fundamentally change existing views. First, for the loading range 1-20MPa and frequency range 1-20Hz, it is, indeed, possible to produce cellular level strains in bone that are up to 100 fold greater than normal tissue level strains (0.04-0.3 percent). Thus, the strain in the cell process membrane due to the loading can be of the same order as the in vitro strains measured in cell culture studies where intracellular biochemical responses are observed for cells on stretched elastic substrates. Second, it demonstrates that in any cellular system, where cells are subject to fluid flow and tethered to more rigid supporting structures, the tensile forces on the cell due to the drag forces on the tethering fibers may be many times greater than the fluid shear force on the cell membrane.     

Model for the alignment of actin filaments in endothelial cells subjected to fluid shear stress

Cultured vascular endothelial cells undergo significant morphological changes when subjected to sustained fluid shear stress. The cells elongate and align in the direction of applied flow. Accompanying this shape change is a reorganization at the intracellular level. The cytoskeletal actin filaments reorient in the direction of the cells' long axis. How this external stimulus is transmitted to the endothelial cytoskeleton still remains unclear. In this article. we present a theoretical model accounting for the cytoskeletal reorganization under the influence of fluid shear stress. We develop a system of integro-partial-differential equations describing the dynamics of actin filaments, the actin-binding proteins, and the drift of transmembrane proteins due to the fluid shear forces applied on the plasma membrane. Numerical simulations of the equations show that under certain conditions, initially randomly oriented cytoskeletal actin filaments reorient in structures parallel to the externally applied fluid shear forces. Thus, the model suggests a mechanism by which shear forces acting on the cell membrane can be transmitted to the entire cytoskeleton via molecular interactions alone.     

Shear-induced resistance to neutrophil activation via the formyl peptide receptor

The application of fluid shear stress on leukocytes is critical for physiological functions including initial adhesion to the endothelium, the formation of pseudopods, and migration into tissues. The formyl peptide receptor (FPR) on neutrophils, which binds to formyl-methionyl-leucyl-phenylalanine (fMLP) and plays a role in neutrophil chemotaxis, has been implicated as a fluid shear stress sensor that controls pseudopod formation. The role of shear forces on earlier indicators of neutrophil activation, such as L-selectin shedding and α(M)β(2) integrin activation, remains unclear. Here, human neutrophils exposed to uniform shear stress (0.1-4.0 dyn/cm(2)) in a cone-and-plate viscometer for 1-120 min showed a significant reduction in both α(M)β(2) integrin activation and L-selectin shedding after stimulation with 0.5 nM of fMLP. Neutrophil resistance to activation was directly linked to fluid shear stress, as the response increased in a shear stress force- and time-dependent manner. Significant shear-induced loss of FPR surface expression on neutrophils was observed, and high-resolution confocal microscopy revealed FPR internalized within neutrophils. These results suggest that physiological shear forces alter neutrophil activation via FPR by reducing L-selectin shedding and α(M)β(2) integrin activation in the presence of soluble ligand.     

Osteoblasts respond to pulsatile fluid flow with short-term increases in PGE(2) but no change in mineralization.

Although there is no consensus as to the precise nature of the mechanostimulatory signals imparted to the bone cells during remodeling, it has been postulated that deformation-induced fluid flow plays a role in the mechanotransduction pathway. In vitro, osteoblasts respond to fluid shear stress with an increase in PGE(2) production; however, the long-term effects of fluid shear stress on cell proliferation and differentiation have not been examined. The goal of this study was to apply continuous pulsatile fluid shear stresses to osteoblasts and determine whether the initial production of PGE(2) is associated with long-term biochemical changes. The acute response of bone cells to a pulsatile fluid shear stress (0.6 +/- 0.5 Pa, 3.0 Hz) was characterized by a transient fourfold increase in PGE(2) production. After 7 days of static culture (0 dyn/cm(2)) or low (0.06 +/- 0.05 Pa, 0.3 Hz) or high (0.6 +/- 0.5 Pa, 3.0 Hz) levels of pulsatile fluid shear stress, the bone cells responded with an 83% average increase in cell number, but no statistical difference (P > 0.53) between the groups was observed. Alkaline phosphatase activity per cell decreased in the static cultures but not in the low- or high-flow groups. Mineralization was also unaffected by the different levels of applied shear stress. Our results indicate that short-term changes in PGE(2) levels caused by pulsatile fluid flow are not associated with long-term changes in proliferation or mineralization of bone cells.     

Flow shear stress regulates endothelial barrier function and expression of angiogenic factors in a 3D microfluidic tumor vascular model

Endothelial cells lining blood vessels are exposed to various hemodynamic forces associated with blood flow. These include fluid shear, the tangential force derived from the friction of blood flowing across the luminal cell surface, tensile stress due to deformation of the vessel wall by transvascular flow, and normal stress caused by the hydrodynamic pressure differential across the vessel wall. While it is well known that these fluid forces induce changes in endothelial morphology, cytoskeletal remodeling, and altered gene expression, the effect of flow on endothelial organization within the context of the tumor microenvironment is largely unknown. Using a previously established microfluidic tumor vascular model, the objective of this study was to investigate the effect of normal (4 dyn/cm²), low (1 dyn/cm²), and high (10 dyn/cm²) microvascular wall shear stress (WSS) on tumor-endothelial paracrine signaling associated with angiogenesis. It is hypothesized that high WSS will alter the endothelial phenotype such that vascular permeability and tumor-expressed angiogenic factors are reduced. Results demonstrate that endothelial permeability decreases as a function of increasing WSS, while co-culture with tumor cells increases permeability relative to mono-cultures. This response is likely due to shear stress-mediated endothelial cell alignment and tumor-VEGF-induced permeability. In addition, gene expression analysis revealed that high WSS (10 dyn/cm²) significantly down-regulates tumor-expressed MMP9, HIF1, VEGFA, ANG1, and ANG2, all of which are important factors implicated in tumor angiogenesis. This result was not observed in tumor mono-cultures or static conditioned media experiments, suggesting a flow-mediated paracrine signaling mechanism exists with surrounding tumor cells that elicits a change in expression of angiogenic factors. Findings from this work have significant implications regarding low blood velocities commonly seen in the tumor vasculature, suggesting high shear stress-regulation of angiogenic activity is lacking in many vessels, thereby driving tumor angiogenesis.     

Shear stress gradient over endothelial cells in a curved microchannel system.

Our purpose was to test a scale model of the microcirculation by measuring the shear forces to which endothelial cells were exposed, and comparing this to computer simulations. In vitro experiments were performed to measure the 2-dimensional projected velocity profile along endothelial cell lined microchannels (D-shaped, 10-30 microns radius, n = 15), or in microchannels without endothelial cells (n = 18). Microchannels were perfused with fluorescently labeled microspheres (0.5 micron dia., < 1%) suspended in cell culture media. The velocity of individual microspheres was obtained off-line (videorecording), using an interactive software program; velocity was determined as the distance traveled in one video field (1/60 s). Mass balance was verified in the microchannels by comparing the microsphere velocities to the perfusion pump rate. In confluent endothelial cell lined microchannels, a velocity profile was obtained as microspheres passed an endothelial cell nucleus (identified by fluorescent dye), and again, for a paired region 100 microns away without nuclei (cytoplasm region). The velocity profile was significantly shifted and sharpened by the endothelial cell nucleus, as anticipated. Over the nucleus, data are consistent with a normal sized nucleus extending into the lumen, further confirming that this scale model can be used to determine the wall shear stress to which endothelial cells are exposed. Using the experimental bulk phase fluid parameters as boundary conditions, we used computational fluid dynamics (CFD) to predict the expected wall shear stress gradient along an endothelial cell lined D-shaped tube. The wall shear stress gradient over the nucleus was 2-fold greater in the radial versus axial directions, and was sensitive to lateral versus midline positioned nuclei.     

The effect of shear stress on protein conformation: Physical forces operating on biochemical systems: The case of von Willebrand factor

Macromolecules and cells exposed to blood flow in the circulatory tree experience hydrodynamic forces that affect their structure and function. After introducing the general theory of the effects of shear forces on protein conformation, selected examples are presented in this review for biological macromolecules sensitive to shear stress. In particular, the biochemical effects of shear stress in controlling the von Willebrand Factor (VWF) conformation are extensively described. This protein, together with blood platelets, is the main actor of the early steps of primary haemostasis. Under the effect of shear forces > 30 dyn/cm², VWF unfolding occurs and the protein exhibits an extended chain conformation oriented in the general direction of the shear stress field. The stretched VWF conformation favors also a process of self aggregation, responsible for the formation of a spider web network, particularly efficient in the trapping process of flowing platelets. Thus, the effect of shear stress on conformational changes in VWF shows a close structure-function relationship in VWF for platelet adhesion and thrombus formation in arterial circulation, where high shear stress is present. The investigation of biophysical effects of shear forces on VWF conformation contributes to unraveling the molecular interaction mechanisms involved in arterial thrombosis.     

Endothelial cell morphometry near branch junctions of rabbit aortae

Endothelial cell morphometric data were gathered from corrosion casts of the aorta and its branches of six New Zealand white rabbits weighing 2-3 kg. The endothelial cell outlines were ditigized to provide cell orientation index (COI) from the equation COI = 2Ms/2M1-1, where 2Ms and 2M1 are the second moments of area about short and long axes of the cell. The COI varies from zero to infinity. The former occurs when cells are symmetrical (e.g., circular) since 2Ms = 2M1 and the latter occurs for a straight line. Large values of COI were found distal to intercostals, lumbars, and the aortorenal junction, and probably reflect relatively stable high shear regions. Values close to zero occur on the flow divider where the shear stress approaches zero. Other values were less predictable and we concluded that while the long axis of the cell appears to indicate flow direction, the shape of the cell does not appear to be a reliable indicator of either tensile forces or shear stress acting in it, but probably reflects a combination of the two.     

Coronary stent implantation changes 3-D vessel geometry and 3-D shear stress distribution

Mechanisms of in-stent restenosis are not fully understood. Shear stress is known to play a role in plaque and thrombus formation and is sensitive to changes in regional vessel geometry. Hence, we evaluated the regional changes in 3-D geometry and shear stress induced by stent placement in coronary arteries of pigs.Methods. 3-D reconstruction was performed, applying a combined angiographic and IVUS technique (ANGUS), from seven Wallstents (diameter 3.5 (n=3) and 5mm (n=4)), which were implanted in seven coronary arteries of five pigs. This 3-D geometry was used to calculate locally the curvature, while the shear stress distribution was obtained by computational fluid dynamics. Local changes in shear stress were obtained at the entrance and exit of the stent for baseline (0. 65+/-0.22 ml/s) and hyperemic flow (2.60+/-0.86 ml/s) conditions. Results. After stent implantation, the curvature increased by 121% at the entrance and by 100% at the exit of the stent, resulting in local changes in shear stress. In general, at the entrance of the stent local maxima in shear stress were generated, while at the exit both local maxima and minima in shear stress were observed (p<0.05). Additionally, the shear stress at the entrance and exit of the stent were correlated with the local curvature (r: 0.30-0.84).Conclusion. Stent implantation changes 3-D vessel geometry in such a way that regions with decreased and increased shear stress occur close to the stent edges. These changes might be related to the asymmetric patterns of in-stent restenosis.     

The fluid shear stress distribution on the membrane of leukocytes in the microcirculation

Recent in-vivo and in-vitro evidence indicates that fluid shear stress on the membrane of leukocytes has a powerful control over several aspects of their cell function. This evidence raises a question about the magnitude of the fluid shear stress on leukocytes in the circulation. The flow of plasma on the surface of a leukocyte at a very low Reynolds number is governed by the Stokes equation for the motion of a Newtonian fluid. We numerically estimated the distribution of fluid shear stress on a leukocyte membrane in a microvessel for the cases when the leukocyte is freely suspended, as well as rolling along or attached to a microvessel wall. The results indicate that the fluid shear stress distribution on the leukocyte membrane is nonuniform with a sharp increase when the leukocyte makes membrane attachment to the microvessel wall. In a microvessel (10 microns diameter), the fluid shear stress on the membrane of a freely suspended leukocyte (8 microns diameter) is estimated to be several times larger than the wall shear stress exerted by the undisturbed Poiseuille flow, and increases on an adherent leukocyte up to ten times. High temporal stress gradients are present in freely suspended leukocytes in shear flow due to cell rotation, which are proportional to the local shear rate. In comparison, the temporal stress gradients are reduced on the membrane of leukocytes that are rolling or firmly adhered to the endothelium. High temporal gradients of shear stress are also present on the endothelial wall. At a plasma viscosity of 1 cPoise, the peak shear stresses for suspended and adherent leukocytes are of the order of 10 dyn/cm2 and 100 dyn/cm2, respectively.