A 3D numerical study of the effect of channel height on leukocyte deformation and adhesion in parallel-plate flow chambers

Microvascular Research 68 (2004) 188 – 202 www.elsevier.com/locate/ymvre

A 3D numerical study of the effect of channel height on leukocyte deformation and adhesion in parallel-plate flow chambers Damir B. Khismatullin*, George A. Truskey Department of Biomedical Engineering, Duke University, Durham, NC 27708-0281, United States Received 6 May 2004 Available online 8 September 2004

Abstract The effect of channel height on leukocyte adhesion to a lower plate in a parallel-plate flow chamber is studied by direct numerical simulations in three dimensions. The numerical model takes into account deformability and viscoelasticity of the leukocyte, membrane ruffles (microvilli), and the presence of mechanically different regions inside the cell (nucleus and cytoplasm). Leukocyte adhesion is assumed to be mediated by interactions of adhesion molecules on the tips of microvilli with their counterparts on the lower plate. Results of this study indicate that an adherent leukocyte experiences much less drag than a rigid sphere due to its deformation and transient stress growth. While overall leukocyte deformation is modest at shear stresses encountered in the microcirculation, deformation in the contact region is significant. At fixed wall shear stress, the contact area of the cell membrane with the substrate increases with increasing the ratio of cell diameter to channel height, leading to greater adhesion. This suggests that in vitro flow chamber studies typically underestimate leukocyte adhesion that occurs in the microcirculation. D 2004 Elsevier Inc. All rights reserved. Keywords: Leukocyte–endothelial adhesion; Receptor–ligand interaction; Microvillus; Flow chamber; Microcirculation; Wall shear stress; Cell viscoelasticity; Cell deformation; 3D numerical modeling

Introduction Leukocyte adhesion to and transmigration across the vascular endothelium play a crucial role in several physiological and pathophysiological conditions such as acute and chronic inflammation (Ley, 1996, 2002), lymphocyte homing (Picker and Butcher, 1992; Wiedle et al., 2001), wound healing (Diegelmann and Evans, 2004; Gillitzer and Goebeler, 2001), and atherogenesis (Ross, 1999). Leukocyte–endothelial cell interactions are usually studied either in vivo by intravital imaging of the mesenteric vasculature of small animals (House and Lipowsky, 1987; * Corresponding author. Department of Biomedical Engineering, Duke University, 136 Hudson Hall, Box 90281, Durham, NC 27708-0281. Fax: +1 919 684 4488. E-mail addresses: [email protected] (D.B. Khismatullin)8 [email protected] (G.A. Truskey). 0026-2862/$ - see front matter D 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.mvr.2004.07.003

Ley, 1996) and by noninvasive imaging of the microcirculation in the human eye (Kirveskari et al., 2001) or in vitro in parallel-plate flow chambers (Lawrence and Springer, 1991, 1993). Both in vivo and in vitro experiments have revealed four steps of leukocyte trafficking that precede transendothelial migration: capture–rolling–slow rolling–firm adhesion. Firm adhesion of the leukocyte is mediated by h2 and a4 integrins which are present on the leukocyte membrane. Members of the selectin family of adhesion molecules are responsible for the first three steps: L-selectin located on the tips of leukocyte microvilli is critical for leukocyte capture and initiation of leukocyte rolling, P- and E-selectins expressed on the activated endothelium, and a4 integrins mediate rolling and slow rolling of the cell (Ley, 1996, 2002). Despite the qualitative agreement between in vitro and in vivo observations of the leukocyte adhesion cascade, leukocyte adhesion in flow chambers and in vivo is

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different. First, to mimic physiological conditions, parallelplate flow chamber experiments are performed at the volumetric flow rate at which the wall shear stress matches that observed in blood vessels in vivo. However, in all in vitro experiments, the flow chamber height exceeds 100 Am, which is considerably larger than the diameter of many postcapillary venules where neutrophil extravasation occurs. The theory of Goldman et al. (1967) for the drag force and torque acting on a solid sphere near a planar wall in a simple shear flow is used in many flow chamber studies. This theory is valid only if the upper plate of the chamber is infinitely far from the particle and the shear rate is constant everywhere. It cannot be used for deformable particles (Tempelman et al., 1994). Numerical calculations show that, for rigid cells in vessels with radii of the same order of magnitude as the cell diameter, adherent leukocytes experience a greater drag force than they would experience in an unbounded fluid (Chapman and Cokelet, 1998). For deformable cells, an increase in the drag force with a decrease in the channel height was shown numerically by Dong et al. (1999). Numerical corrections to the Goldman, Cox, and Brenner theory for confined channels were developed for rigid spheres (Pozrikidis, 2000; SugiharaSeki and Schmid-Scho¨nbein, 2003) and a two-dimensional (2D) deformable cell (Dong and Lei, 2000). Second, the difference in geometry between parallelplate flow chambers and blood vessels can have a significant effect on cell adhesion, especially if the vessel diameter is comparable with the cell diameter. Adhesion of the leukocyte to the cylindrical vessel wall is stronger than to the flat surface of the flow chamber because the region of the cell membrane, which is close enough to the substrate to form receptor–ligand bonds, increases in area with increasing the substrate curvature, that is, with bending the substrate toward the cell (Chapman and Cokelet, 1996). Third, in the microcirculation, adherent leukocytes interact with free-flowing erythrocytes. The erythrocyte– leukocyte interaction gives rise to an additional normal force on the leukocytes and can lead to increasing the contact time (Melder et al., 1995). This is because erythrocytes form a core region of the flow which pushes the leukocytes to the vessel wall. In this paper, we investigate numerically how changes in the channel height affect leukocyte adhesion to the lower plate in a parallel-plate flow chamber provided that the leukocyte is deformable and viscoelastic. As noted by Smith et al. (2002), bisolated neutrophils rolling in vitro cannot sustain interactions at the same wall shear stress or stresses as in microvessels in vivo, but the exact reason is not known.Q Our results indicate that the difference between the channel height and the vessel diameter is one possible reason why cell rolling in vitro is different from that in vivo. From the analysis, we describe an approach to relate in vitro adhesion results to estimate adhesion strength in vivo.

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Materials and methods The three-dimensional (3D) numerical simulation of leukocyte adhesion in a parallel-plate flow chamber (Fig. 1A) has been carried out by our own incompressible CFD code in which the volume-of-fluid (VOF) method (Gueyffier et al., 1999) is used for tracking leukocyte shapes over time. The Navier-Stokes equations are solved by Chorin’s projection method (Chorin, 1967) on a staggered markerand-cell (MAC) grid. The entire code is parallelized and run on the IBM p690 server of the National Center for Supercomputing Applications (University of Illinois at Urbana-Champaign). This code is the only available code which considers adhesion of deformable cells in 3D. The previous numerical simulations of receptor-mediated cell adhesion were restricted to 3D rigid (Chang and Hammer, 2000; Hammer and Apte, 1992; King and Hammer, 2001) and 2D deformable models (Dong et al., 1999; N’Dri et al., 2003). For details on the numerical algorithm including validation and comparison with experiments, the reader is referred to our previous paper (Khismatullin and Truskey, in press). We model a leukocyte as a compound viscoelastic drop (a viscoelastic nucleus covered by a thick layer of a viscoelastic cytoplasm) with a thin ruffled membrane that possesses a cortical tension (Fig. 1B). The viscoelasticity of the nucleus and cytoplasm is captured by the Giesekus model (Bird et al., 1987), which takes into account two distinct phases of a polymeric liquid: solvent phase and polymer phase. For a living cell, the solvent phase is the cell cytosol, and the polymer phase is the cell cytoskeleton. We have selected the Giesekus model because it gives good predictions about the stress growth in startup shear flow and about the steady shear-rate viscosity of polymeric liquids (Giesekus, 1982). The Giesekus constitutive equation is of the following form:
 BT 2 k1 þ ðudjÞT  ðjuÞT  TðjuÞ4 þ jT þ T Bt ¼ 2l p S;

S¼

1 ðju þ ju4Þ; 2

ð1Þ

where S is the rate-of-strain tensor; T is the extra stress tensor, which represents the polymer contribution to the stress, that is, the total stress tensor j ¼  pI þ 2l s S þ T;

ð2Þ

l s and l p are solvent and polymer viscosities, p is the pressure, k 1 is the relaxation time, and j is the Giesekus nonlinear parameter. In the code, the nucleus, cytoplasm, and extracellular fluid are characterized by different values of l s, l p, k 1, and j. In particular, the polymer viscosity of the nucleus l np is higher than the polymer p viscosity of the cytoplasm l cp. The solvent viscosities of the nucleus and cytoplasm are equal to the extracellular s fluid viscosity: l cp = l ns = l ext. It is evident from Eqs. (1)

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Fig. 1. Schematic drawing of (A) the flow domain and (B) the leukocyte model used in the numerical simulations. (C) Definition of the inclination angle H c, length L c, and breadth B c of the deformed leukocyte. The leukocyte is suspended in a low-viscosity liquid (extracellular fluid) between the fixed upper and lower plates (l ext b l c). The lower plate is covered by adhesion molecules. The parabolic velocity field is created by the pressure gradient. The x-axis coincides with the flow direction. The y- and z-axes are perpendicular to the flow. The leukocyte is modeled as a 3D viscoelastic drop which shell and core phases represent the leukocyte cytoplasm and nucleus. The leukocyte membrane is assumed to possess a cortical tension. The membrane ruffles (microvilli) are modeled as elastic rods with adhesion molecules on their tips. Bonds between adhesion molecules on the plate and their counterparts on the microvilli tips are Hookean springs.

and (2) that the cytoplasmic and nuclear viscosities of the cell are lcp ¼ lpcp þ lscp ; ln ¼ lpn þ lsn :

ð3Þ

The average viscosity of the cell l c is volume-weighted, that is, lc ¼ ð1  /Þlcp þ /ln ;

ð4Þ

where, / is the volume fraction of the nucleus. The membrane ruffles, or microvilli, are modeled as massless elastic rods that have adhesion molecules (receptors or ligands) on their tips. The adhesion molecules can interact with their counterparts distributed uniformly on the lower plate of a parallel-plate flow chamber if the separation distance between the microvillus and the plate does not exceed the unstressed bond length l b0. Receptor–ligand bonds behave as Hookean springs with spring constant j b. To

account for microvillus deformability, we assume that each receptor–ligand bond is in series with a bmicrovillus springQ characterized by spring constant j mv. This follows from the fact that the microvillus core is a bundle of parallel actin filaments that extends from the microvillus tip into the cell cytoplasm (Alberts et al., 2002). The filaments can be connected to the cytoplasmic tails of the adhesion molecules expressed on the microvillus tip. In our model, the bonds are aligned in the same direction as the filaments (normal to the membrane); that is, the microvillus and its bonds form a system of N b parallel springs (bmicrovillus-bond springsQ), where N b is the number of bonds. A microvillus-bond spring is characterized by spring constant j s = j bj mv / (j b + j mv). The spring-peeling kinetic model by Dembo et al. (1988), modified to include the effects of microvillus extension (Shao et al., 1998), receptor extraction (Shao and Hochmuth, 1999), and receptor shedding (HafeziMoghadam and Ley, 1999), is used to describe the

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receptor–ligand interaction. The modified model has the following form: dnb ¼ kf ðnl  nb Þðnr  nb Þ  kr nb ; dt " # jts ðls  lmv  lb0 Þ2 kf ¼ kf 0 exp  ; 2kB T

ð5Þ

ð6Þ

"

# ðcjs  jts Þðls  lmv  lb0 Þ2 k r ¼ k r0 exp ; 2kB T

ð7Þ

Fb ¼ Nb js ðls  lmv  lb0 ÞI:

ð8Þ

Here n r, n l, and n b are the surface densities of receptors (adhesion molecules on the microvillus tip), ligands (adhesion molecules on the plate), and bonds; k f and k r are the forward and reverse reaction rate coefficients; j ts is the transition state spring constant; l s is the length of the microvillus-bond spring, that is, the distance between the microvillus core and the plate; l mv is the unstressed length of the microvillus; k B = 1.38  1023 J/K is the Boltzmann constant; T is the temperature in Kelvin; and c is a correction factor, which is more than unity if receptor extraction/shedding occurs. The symbol Fb denotes the bond force that acts on the leukocyte due to the receptor– ligand bonds formed on the microvillus tip; l is the unit vector directed from the microvillus base to the plate. The total bond force is the sum of the bond forces for all the microvilli attached to the plate. If N b = n bS tip (S tip is the surface area of the microvillus tip) becomes less than unity, the link between the microvillus and the plate is considered to be broken. Flow chambers with parallel plate geometry are often used to study leukocyte adhesion to endothelium or a ligand-coated surface under laminar flow conditions (Alon et al., 1998; Lawrence and Springer, 1991; Lawrence et al., 1987). A parallel-plate flow chamber consists of two plastic or metal plates separated by a gasket (Lawrence et al., 1987). The gasket regulates the chamber height (100 to 500 Am). It has a hollow rectangular section needed to form the flow channel. The upper plate has inlet and outlet openings through which a physiological solution and leukocytes are perfused. A glass or plastic slide plate with the endothelial cell monolayer or ligand-coated substrate is assembled in a rectangular hole of the lower plate and mounted on the stage of an inverted phase-contrast microscope. The flow is generated by a syringe pump and is laminar inside the chamber due to a small gap between the plates. Because laminar flow occurs in postcapillary venules as well, the parallel-plate flow chamber allows for in vitro study of leukocyte trafficking in the venules provided physiological flow conditions are satisfied. As a rule, flow chamber assays simulate wall shear stresses that exist in postcapillary venules (Lawrence and Springer, 1991). For a given

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volumetric flow rate Q, the wall shear stress in the flow chamber s w is calculated from the solution of the Stokes equations for plane Poiseuille flow [cf. Eqs. (9)–(11) below]. In our simulations, the flow domain corresponds to a parallel-plate flow chamber. A leukocyte is suspended in a low-viscosity liquid (extracellular fluid) between the fixed upper and lower plates, separated by a distance h (Fig. 1A). Initially, the extracellular fluid is at rest and the leukocyte has a spherical shape. No-slip and periodic boundary conditions are applied at the plates and all other boundaries, respectively. The separation distance between the cell membrane and the lower plate is small enough to have a few microvilli attached to the plate. The flow is created by the negative pressure gradient G, which grows in magnitude linearly from zero to the fixed value G f after a few microseconds. Such a starting flow mimics detachment assays previously used to examine the strength of leukocyte or bead adhesion to the substrate and to determine the velocity and percent of rolling leukocytes (Alon et al., 1998; Lawrence and Springer, 1991, 1993; Tempelman et al., 1994). In the detachment assays, buffer flow is initiated after a static or low flow-rate incubation of leukocytes during which the cells settle on the lower plate and form receptor– ligand bonds. (In continuous flow assays, leukocytes are perfused through the chamber together with a physiological solution.) In the absence of the leukocytes and under steady-state conditions, the wall shear stress s w, wall shear rate c˙w, volumetric rate Q, and centerline velocity u max of the flow in the chamber depend on the pressure gradient in this fashion: sw ¼ lext c˙ w ¼

Gf h ; 2

Q ¼ whumean ¼

umax ¼

Gf h2 : 8lext

wh2 sw Gf wh3 ¼ ; 6lext 12lext

ð9Þ

ð10Þ

ð11Þ

Here, w is the plate width, l ext is the extracellular fluid viscosity, and u mean is the mean flow velocity. In the simulations, we keep the wall shear stress of the undisturbed flow constant but change the channel height h. We use Eq. (9) to calculate G f. The hydrodynamic force acting on the adherent cell is defined as Z F¼ ðndjÞdS; ð12Þ S

where S denotes integration along the cell surface, C is the total stress tensor determined in Eq. (2), and n is the outward normal vector to the cell surface. The hydrodynamic force has three components: the drag force F D (in

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the flow direction), lift force F L (perpendicular to the flow), and bshift forceQ F SH (perpendicular to the flow and parallel to the plates). The three-dimensional shapes of the cell are rendered with POV-Rayk (Persistence of Vision Raytracer Pty. Ltd.) from different views. This software is used to find the inclination angle H c, length L c, and breadth B c of the deformed cell (Fig. 1C). To calculate these parameters, we rotate the system of coordinates along y-axis, render the cell shape from a side view, and measure the horizontal and vertical lengths of the cell domain in the resulting pictures. The cell inclination angle is defined as the angle at which the horizontal length is maximal. The cell breadth is the vertical length of the cell domain in the picture where this maximum, representing the cell length, is reached. The cell deformation index D c is calculated from the formula (Taylor, 1934): Dc ¼ ðLc  Bc Þ=ðLc þ Bc Þ:

ð13Þ

The deformation index takes the value from zero to one. D c = 0 implies the spherical shape. D c = 1 suggests that the shape is a line of negligible thickness.

Results In this paper, we simulate numerically receptor-mediated adhesion of monocytic cells to the lower plate in a parallelplate flow chamber. The chamber height h ranges from 20 to

80 Am. The parameters whose values are the same for all runs are listed in Table 1. Because receptors are assumed to be located on microvilli only, the total number of cell 2 n r = 87462. This value corresponds to receptors N r = pr mv experimental data (Lawrence and Springer, 1991). We consider a higher value of the forward reaction rate than in experiments (Chesla et al., 1998) to decrease the computation time. In all the simulations, the Giesekus nonlinear parameter j is zero, that is, we do not consider the shear thinning behavior of the leukocyte. However, taking into account that the apparent viscosity of the cell decreases significantly with shear rate (Tsai et al., 1993), the average viscosity of the cell is assumed to be less than its zero shearrate value (Hochmuth et al., 1993). According to Eq. (4), the cell viscosity l c c 4.82 Pad s = 48.2 P. We ignore the effect of gravity on cell adhesion, that is, we assume that the densities of the nucleus, cytoplasm, and extracellular fluid are equal. Also, the surface tension at the nucleus– cytoplasm interface is supposed to be equal to the cortical tension. Two values of the wall shear stress are considered in the paper: (1) low WSS, s w = 0.8 Pa = 8 dyn/cm2, typical for blood flow in postcapillary venules of the mouse cremaster muscle (Smith et al., 2002), and (2) high WSS, s w = 4.0 Pa = 40 dyn/cm2, at which the effect of deformation is clearly seen. In the numerical code, the number of grid cells N x  N y  N z increases with height from 64  64  64 at h = 20 Am to 64  64  256 at h = 80 Am. (Further mesh refinement does not affect the leukocyte shapes.)

Table 1 Values of parameters used in the numerical simulations Parameter

Notation

Value

Reference

Cell radius Volume fraction of the nucleus Cortical tension Cytoplasmic viscosity Cytoplasmic relaxation time Nuclear viscosity Nuclear relaxation time Extracellular fluid viscosity Extracellular fluid density Number of microvilli Number of initially attached microvilli Microvillus radius Unstressed microvillus length Microvillus spring constant Unstressed bond length Bond spring constant Transition state spring constant Correction factor Temperature Forward reaction rate Reverse reaction rate Surface density of receptors Surface density of ligands Length of the computational box Width of the computational box Time step

R u Tc l cp k 1cp ln k 1n l ext q ext N mv N a0 r mv l mv j mv l b0 jb j ts c T k f0 k r0 nr nl l w Dt

6.5 Am 0.2 30 AN/m 3.53 Pad s 0.176 s 10.00 Pad s 0.200 s 0.001 Pad s 103 kg/m3 928 3 0.10 Am 0.30 Am 210 pN/Am 0.01 Am 5300 pN/Am 100 pN/Am 1 310 K 100 Am2/(sd molec) 10 s1 3  103 molec/Am2 1.5  103 molec/Am2 50 Am 50 Am 107 s

Erl et al. (1995) Schmid-Scho¨nbein (1986) Zhelev et al. (1994) Tsai et al. (1993) Schmid-Scho¨nbein (1986) Tsai et al. (1993)

Lawrence and Springer (1991) Bell (1978) Shao et al. (1998) Park et al. (2002) Springer (1990) Fritz et al. (1998) Dembo et al. (1988)

Ramachandran et al. (1999) Lawrence and Springer (1991) Lawrence and Springer (1991)

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Fig. 2 shows snapshots of the monocyte shape for 8 dyn/cm2 at different instants, according to the numerical simulation. The monocyte is attached to the lower plate in the flow chamber of height 30 Am. Shear flow induces monocyte deformation. In response to the force exerted on the cell membrane due to receptor–ligand interaction, the monocyte is elongated. Stretching of the monocyte membrane is significant in the contact zone. Similar behavior, but with different degree of deformation, is observed for the cell in the flow chamber of smaller or larger height. As illustrated in Fig. 3, elongation of the cell body is more pronounced at a higher wall shear stress. The cell deforms to a teardrop shape typical for rolling leukocytes in vivo (Damiano et al., 1996) due to its viscoelasticity (Khismatullin and Truskey, in press). When viewed from the top (Fig. 2), no deformation is apparent, and the cell moves with constant velocity along the plate. Because in vitro cell adhesion assays provide only a top view of the cells, they are unable to capture cell

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Fig. 3. Computed shape of an adherent monocyte (side view) under low WSS (s w = 0.8 Pa = 8 dyn/cm2) and high WSS (s w = 4.0 Pa = 40 dyn/cm2) at the instant when the monocyte undergoes maximal deformation. The remaining parameters are given in Table 1. The cell deforms to a teardrop shape if the shear stresses are sufficiently high.

Fig. 2. Side and top views of a monocyte adherent to the lower plate in a parallel-plate flow chamber of height 30 Am. The wall shear stress s w = 0.8 Pa = 8 dyn/cm2. The remaining parameters are given in Table 1. The cell deforms in response to fluid shear stresses and receptor–ligand interaction. At the contact zone, stretching of the cell membrane is clearly seen. The stretched portion of the cell membrane retracts after cell detachment. No deformation is observed from a top view.

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deformation, especially if the wall shear stress is low. Fig. 2 indicates that the cell can be attached to the plate, but its deformation may be interpreted as motion along the plate when viewed from above. The time dependence of the drag and lift forces experienced by the adherent leukocyte in chambers of different height is illustrated in Fig. 4. The third component of the hydrodynamic force, normal to the flow but parallel to the plates, is nearly zero at all times. The drag force changes with time because the cell deforms and its microvilli are extended. Fig. 4A shows that the peak drag force increases as the chamber height decreases. The peak drag force is 1.5 times higher at h = 20 Am than at h = 80 Am. The drag force depends nonlinearly on the wall shear stress: a fivefold increase in the wall shear stress results in an about 3.3-fold increase in the drag force at h = 20 Am (Figs. 4A, B). This is in line with numerical results of Dong et al. (1999) for a 2D deformable cell. An increase in the drag force with the wall shear stress becomes less pronounced for a larger height of the chamber. The lift force is less affected by changes in the height, as compared to the drag force. It is interesting that the peak lift force decreases with h at low WSS (Fig. 4C) but increases with h at high WSS (Fig. 4D). Also, the lift force is comparable in magnitude with the drag force; that is, it

should be taken into account in the analysis of bond dynamics. Fig. 5 illustrates how changes in the chamber height and wall shear stress affect the cell deformation index D c, defined in Eq. (13), and cell inclination angle to flow direction. Note that the wall shear stress is kept constant when the chamber height is altered. The deformation index gives an indication of the overall deformation of the cell body (mostly, shear deformation of the cell). If the cell deforms at the contact region only, the deformation index will be close to zero. In the case of low WSS, D c reaches its peak value just before detachment of the initially attached microvilli. The maximum deformation index falls with a decrease in the chamber height (Fig. 5A). Note that even a small change in the deformability index means a significant change in cell deformation [see Eq. (13)]. In relation to the microcirculation, this result suggests that leukocytes are less deformable in small venules than in large venules provided that the wall shear stress is held constant. It is important to note that shear deformation of the leukocyte is less at h = 80 Am than at h = 50 Am for the first 2 ms (Fig. 5A). Also, the maximum deformation index at h = 50 Am is slightly more than that at h = 80 Am if the wall shear stress is high (Fig. 5B). This is solely the effect of the transient stress growth after flow startup. In the case of Poiseuille flow, the time needed to establish the steady-state flow is proportional to

Fig. 4. (A, B) Drag and (C, D) lift forces on an adherent monocyte as functions of time at different heights of the chamber. A and C are the case of 8 dyn/cm2. B and D are the case of 40 dyn/cm2. The drag and lift forces change with time due to cell deformation. The drag force decreases with height both at low and high wall shear stresses. The lift force is less sensitive to changes in the chamber height.

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Fig. 5. (A, B) Deformation index and (C, D) inclination angle of an adherent monocyte as functions of time at different heights of the chamber. A and C are the case of 8 dyn/cm2. B and D are the case of 40 dyn/cm2. A fivefold increase in the wall shear stress enhances the overall deformation of the cell by a factor of about 2.5. When the monocyte has diameter comparable with the chamber height, its overall deformation is small. In the case of low WSS, the maximum deformation occurs at the same time as the minimum drag force (Fig. 4A). The cell inclination angle to flow direction decreases with the wall shear stress.

the square of the channel height (Batchelor, 1967). Hence, it takes 2.56 as long to establish steady flow in a chamber of height 80 Am than in a chamber of height 50 Am. The simulations show that the flow establishment time is about 2.5 ms at h = 80 Am. The cell inclination angle is inversely related to the cell deformation index. The more the cell deforms, the less the cell is inclined with respect to flow direction. However, the difference in the inclination angle between chambers of 20 and 80 Am is within 58 (Figs. 5C and D). The wall shear stress has a greater effect on both the leukocyte deformation index and inclination angle. A fivefold increase in the wall shear stress leads to about 2.5-fold rise in leukocyte deformation (Fig. 5B) and to a fall in the average inclination angle from about 508 to 348 (Fig. 5D). Fig. 6 demonstrates that, for a fixed wall shear stress, the decreased deformation of the leukocyte in a chamber of smaller height is a consequence of decreasing the shear stresses acting on the cell surface at the top of the cell. The shear rate inside the cell, that is, the rate of shear deformation of the cell, is clearly less at h = 20 Am than at h = 80 Am (Figs. 6A and C). Also, the linear shear flow profile is observed inside the cell. This suggests that the horizontal flow velocity at a position of one cell diameter above the substrate multiplied by the

fluid viscosity can be used as a matching parameter for in vitro and in vivo studies. Note that the velocity field across the cell membrane is continuous. Extracellular fluid motion is, therefore, retarded by a more viscous cell. Nevertheless, the higher the fluid viscosity and the higher the unperturbed fluid velocity at a position of one cell diameter above the substrate, the greater the cell deformation is expected. Fig. 7 illustrates the effect of chamber height on adhesion strength of the leukocyte. As seen in Figs. 7A and B, both cell and microvilli contact times are rate-of-shear deformation-dependent. The less the shear deformation of the cell, the longer the cell and its microvilli attach to the substrate. Therefore, the cell contact time and average microvilli contact time fall as the chamber height increases from 20 to 60 Am. The difference in the cell contact time between h = 20 and 60 Am is about 1.3 ms. It can be detected in experiments. A further increase in height results in a rise of the contact times. This rise is due to slow stress growth after flow startup in chambers of large height, as explained above. Under continuous flow conditions, the left tail of the curve in Fig. 7A will straighten; that is, the cell contact time will be saturated at a sufficiently large height (more than 60 Am) of the chamber. This is, however, not the case for detachment assays.

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attach initially, other microvilli attach later to the plate). After detachment, microvilli can again interact with the substrate. They form short-lived bonds until the distance between the microvilli tip and the substrate exceeds the bond length. Because the microvilli make contact with the plate several times during cell adhesion, the number of microvilli contacts is always more than the number of microvilli involved in adhesion. At low WSS, cell adhesion in the flow chamber of smaller height is characterized by the increased number of microvilli contacts. This can happen due to a decrease in the lift force with height at low WSS (Fig. 4C) and an increase in contact area. The fact that the lift force affects cell adhesion is evident from dashed line in Fig. 7C: at high WSS, microvilli interact more often with the plate as the chamber height increases from 50 to 80 Am because of a decrease in the peak lift force (Fig. 4D). However, an increase in contact area has a greater effect on cell adhesion than a decrease in the lift force. For example, the number of microvilli contacts is always more at high WSS than at low WSS (Fig. 7C), in spite of twofold increase in the lift force (compare Figs. 4C and D). The cumulative number of attached microvilli is constant when the chamber height decreases from 80 to 30 Am, at least at low WSS. However, it grows with a further decrease in height (Fig. 7D). This happens due to the increased torque acting on the cell. The torque is large if the drag force is large and deformation is small (Olivier and Truskey, 1993), just what is observed for h = 20 Am (Figs. 4A and 5A). The simulations show that microvilli attach to the plate with a higher rate in a chamber of smaller height (not shown here). This means that the rate of the contact region growth increases with decreasing height. We also observed that an increase in the wall shear stress magnifies the rate of microvilli attachment despite the significant deformation of the cell. Fig. 6 clearly demonstrates that a decrease in channel height (or vessel diameter) increases the size of the contact region (adhesion zone). Hence, the deformation occurring in the adhesion zone is very sensitive to the ratio of cell diameter to channel height and is not correlated with the overall deformation of the cell. For a higher diameter-toheight ratio, the cell looks less deformable (Figs. 5 and 6) but undergoes the increased deformation in the adhesion zone (Figs. 6 and 7). Fig. 6. Computed monocyte shape and velocity field at the midplane of the cell ( y = 25 Am) at t = 2 ms. The wall shear stress is equal to 8 dyn/cm2. (A) h = 80 Am. (B) h = 50 Am. (C) h = 20 Am. The remaining parameters are given in Table 1. The shear field inside the monocyte is linear and of less magnitude at h = 20 Am than at h = 50 or 80 Am: shear deformation of the monocyte decreases with decreasing the chamber height.

Figs. 7C and D show how the number of microvilli contacts and the number of microvilli involved in cell adhesion depend on the chamber height under low and high shear-stress conditions. Here, the number of microvilli involved in adhesion counts all individual microvilli that attach to the plate during cell adhesion (some microvilli

Discussion Several factors are responsible for greater adhesion of leukocytes to activated endothelium in postcapillary venules relative to in vitro flow channels at the same wall shear stress. Collisions of red blood cells with leukocytes increase the frequency of leukocyte contacts with endothelium resulting in a greater frequency of adhesion and lower rolling velocity (Melder et al., 1995). Several previous studies have examined the stresses acting on static rigid cells in a confined channel (Brooks and Tozeren, 1996; Pozriki-

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Fig. 7. Dependence of (A) monocyte contact time and (B) average microvillus contact time on the flow chamber height in the case of 8 dyn/cm2. (C) Number of contacts of microvilli with the plate and (D) number of microvilli involved in monocyte adhesion as functions of the chamber height at 8 and 40 dyn/cm2. The cell contact time is characterized by the parabolic dependence on the chamber height with a minimum at about four cell diameters. The number of microvilli contacts increases with decreasing the chamber height and with increasing the wall shear stress.

dis, 2000) or cylinder (Sugihara-Seki and Schmid-Scho¨nbein, 2003). The current study represents the first effort to consider the time-dependent forces acting during adhesion on a spherical cell in a rectangular channel. Like these previous studies, we have observed that the net force acting on the cells is much less than that predicted by the results obtained by Goldman et al. (1967) for a sphere near a planar surface and exposed to a shear flow. In addition, we identified an additional mechanism responsible for enhanced adhesion arising from transient stress growth and cell deformation in a channel with dimensions comparable to the red blood cell dimensions. While overall cell deformation is modest at shear stresses encountered in the microcirculation, there is increased deformation in the contact region as the ratio of cell diameter to channel height increases. Drag force The drag and lift forces depend on time due to the leukocyte deformability, microvillus extension, and transi-

ent stress growth. Shear deformation and drag force-induced translational motion of the cell lead to stretching the attached microvilli and receptor–ligand bonds which exert a pulling force on the cell membrane. Because the drag and lift forces are initially zero and grow during flow establishment, the pulling force increases with time and leads to the increased deformation of the cell near the contact region. When the overall deformation of the cell becomes sufficiently large (and microvilli are stretched significantly), the drag force begins to decrease. Microvilli extension and shear deformation of the cell are the reasons why the peak drag force is reached earlier in the simulations than expected from the transient stress growth (see below). The drag force goes through a minimum at which (a) cell deformation is maximal (compare Figs. 4A and 5A) and (b) the initially attached microvilli reach the critical lengths and begin to detach from the lower plate (compare Fig. 4A and dashed line in Fig. 7B). After detachment of these microvilli, the stretched portion of the cell body membrane retracts (Fig. 2). However, the cell does not detach at this instant. There are other microvilli attached to the cell after flow initiation.

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Also, receptors on the detached microvilli can form transient (short-lived) bonds with ligands on the plate until the distance between the microvilli tips and the plate is more than the bond length. The cell detaches from the plate when the drag force goes through another maximum (compare Figs. 4A and 7A). After detachment, the cell translates with a higher velocity. Because the drag force acting on a translating cell is proportional to the relative velocity of the cell, it drops after cell detachment. Later, it should reach a small positive value due to the quadratic flow contribution (Nadim and Stone, 1991). For a solid cell in simple shear flow, the drag force is proportional to the wall shear rate c˙w and the distance between the cell center and the lower plate z c V R + l b0 + l mv (Goldman et al., 1967): FDGCB ¼ 6pkD Rzc lext c˙ w :

ð14Þ

Here k D is the correction factor due to the wall effects. However, the channel flow is nonlinear (parabolic), and the theory by Goldman et al. (1967), hereafter called GCB theory, is not applicable to this problem. Based on the boundary-integral numerical simulations, Pozrikidis (2000) found that, for small diameter-to-height ratios, the drag force on a rigid static sphere in the channel flow was approximately equal to   FD ¼ FDGCB  11:64p R3 =h lext c˙ w : ð15Þ Eq. (15) shows that the drag force is less than predicted by the GCB theory, and the difference between F DGCB and F D increases with a decrease in height. Our simulations also show that the drag force on the leukocyte is less than F DGCB (Fig. 4). The GCB theory gives k D c 1.65 for R = 6.5 Am and z c = 6.81 Am. In this case, F DGCB is about 1101 pN at the wall shear rate c˙w = 800 s1 (low WSS) and about 5505 pN at c˙w = 4000 s1 (high WSS). According to Eq. (15), F D is less than F DGCB by a factor of about 1.17 at h = 50 Am and by a factor of about 1.57 at h = 20 Am. As seen in Fig. 4, the computed drag force of a deformable cell is much less than predicted by Eq. (15). This happens due to overall (shear) deformation of the cell and transient stress growth after flow startup. (Overall cell deformation is not small if the channel height is large as evident from Figs. 5 and 6). Brooks and Tozeren (1996) and Dong et al. (1999) showed numerically that cell elongation in flow direction resulted in a decrease in the drag force and torque. As a result, the drag force and torque on a deformable cell increase disproportionately less than the wall shear stress (Dong et al., 1999). According to Dong et al., the drag force on an adherent neutrophil is less than the value predicted by the GCB theory by a factor of more than two if the wall shear rate is 8 dyn/cm2 in a chamber of height 30 Am. The drag force further decreases with increasing the channel height (Fig. 7 in that paper). Our calculations fully support these results. First, the peak drag force is much smaller than that predicted by the GCB theory

both at low and high WSS (Figs. 4A, B). Second, the GCB theory predicts that a fivefold increase in the wall shear stress results in a fivefold increase in the drag force. The numerical calculations show that the peak drag force rises by a factor of about 3.3 at h = 20 Am (Figs. 4A, B). The peak drag force is less than that calculated by Dong et al. (1999) because we simulate the starting channel flow (see discussion below), while Dong et al. assume the fixed velocity profile far from the cell. Eq. (15) predicts that the drag force decreases with decreasing channel height if the wall shear rate is kept constant. The numerical simulations for deformable cells show opposite behavior. There are several reasons why the drag force on an adherent leukocyte is more in chambers of smaller height (and hence in smaller vessels): ˙ (a) increased pressure drop across the leukocyte for fixed c w, (b) fast flow establishment, and (c) decreased deformation of the leukocyte. Adherent leukocytes disturb fluid flow. When the channel height (or vessel diameter) is large as compared to the leukocyte diameter, the flow is weakly affected by the leukocyte. The flow through narrower chambers (or smaller vessels) is more disturbed. It is occluded by the leukocyte. Flow occlusion results in an increase in the pressure drop across the cell according to conservation of the volumetric flow rate and Bernoulli’s law. Because the region of the disturbed flow increases in size with decreasing the distance between the cell surface and the vessel wall, the adherent leukocyte significantly increases flow resistance (pressure drop divided by volumetric flow rate) when their size is comparable with vessel diameter (Chapman and Cokelet, 1997; Lipowsky et al., 1980). As evident from Eqs. (2) and (12), the hydrodynamic force experienced by the adherent cell depends on both the pressure drop across the cell (that is, normal stresses) and the shear stresses. Hence, the drag force on the cell should grow with decreasing channel height (or vessel diameter) due to the increased pressure drop across the cell. We consider the starting channel flow to mimic detachment assays. As shown by Batchelor (1967), the time needed to establish the steady-state pipe flow is proportional to a 2/v, where a is the pipe diameter and v is the fluid kinematic viscosity. The same is expected for the channel flow, that is, the flow establishment time (Truskey et al., 2004) ttransient f

qext h2 : 4lext

ð16Þ

This is confirmed by the numerical simulations: the velocity field far from the cell does not change after about 2.5 ms at h = 80 Am. In the case of h = 20 Am, flow establishment is reached within 0.2 ms. The more the flow establishment time, the less shear-induced cell deformation and more cell contact time, as compared to the steady-state case. However, this is not to say that the adherent cells do not deform and

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their microvilli do not detach from the substrate during the transient process. According to Eq. (11), the steady-state centerline velocity increases linearly with increasing height if the wall shear stress is kept constant. In the detachment assays (flow chamber heights are usually more than 100 Am), the transient stresses should, therefore, be of sufficient magnitude to induce cell deformation and extension of microvillus-bond complexes before flow establishment. Because cell deformation decreases the drag force on the cell, the peak drag force should be observed earlier than flow reaches the steady state (Fig. 4A). It is less than it could be after flow establishment. Also, leukocyte deformation and its adhesive interaction with the plate for a few milliseconds after flow startup will affect the strength of cell adhesion. If the cell is able to form a sufficient number of bonds for this time, it will attach firmly to the plate. If the number of bonds is small, the cell will detach from or roll along the plate. Cell deformation The drag force defines a translational velocity of the cell. It is not a measure of cell deformation. On the contrary, the drag force depends on cell shape and hence on cell deformation. The less the cell deforms, the more drag it experiences. As shown in Figs. 5 and 6, leukocytes are less deformable in chambers of smaller height and hence in smaller vessels. Here, we talk about shear deformation of the cell of diameter more than the channel height (or vessel diameter). When the vessel diameter becomes less than the cell diameter, the cell undergoes significant deformation during aspiration. However, this deformation is extensional in character. Fig. 6 shows that the shear field inside the cell is linear even if the fluid shear field is quadratic. Hence, shear deformation of the cell can be characterized by the horizontal component of the fluid velocity, taken at the top of the cell, multiplied by the fluid viscosity. This product can be called bdeforming shear stress.Q Because the flow velocity depends on z-coordinate as z U f ð zÞ ¼ 1  ð17Þ z˙c w ; h the deforming shear stress, under continuous flow conditions, can be roughly approximated as ztop  ztop  sdef ¼ lext c˙ def ¼ lext c˙ w 1  ¼ sw 1  ; ð18Þ h h where c˙ def is the rate of shear deformation of the cell. Eq. (18) can be obtained by taking the sum of the wall shear stress and fluid shear stress at the top of the cell and dividing the result by two. In reality, a more viscous fluid inside the cell retards extracellular fluid motion, as seen in Fig. 6A. Therefore, c˙ def should be calculated as c˙ def ¼

utop  ubottom ; ztop  zbottom

ð19Þ

199

where u top and u bottom are the fluid velocities at the top and bottom edges of the cell body (without microvilli), z bottom is the distance from the bottom edge of the cell body to the plate. [For simplicity, we assumed that z bottom = 0 and used Eq. (17) to calculate u top and u bottom when derived Eq. (18).] Note that for a solid cell, u top and u bottom are equal to the translational velocity of the cell, due to the no-slip condition. As seen from Eq. (18), s def is always less than s w, and the difference between these two stresses grows with decreasing h. Therefore, shear deformation of the leukocyte should be diminished with a decrease in channel height (or vessel diameter) if the steady-state wall shear stress is kept constant. This leads to decreasing the drag force with increasing height. Of course, s def rises due to the increased pressure drop. However, this effect seems to be smaller than the second term in the right-hand side of Eq. (18) because (a) the fluid motion is retarded by the cell, as discussed above, and (b) the volumetric flow rate is low in a flow chamber of small height: it is proportional to h 2 at fixed s w [see Eq. (10)]. The distance z top can be approximated as ztop ¼ 2Rc þ lmv ¼ lb0 :

ð20Þ

Using the values of parameters from Table 1, we have z top = 13.31 Am. If the wall shear stress s w = 8 dyn/cm2, Eq. (18) gives that, under continuous flow conditions, the shear stress acting on the cell top s def c 6.67 dyn/cm2 and 2.68 dyn/cm2 at h = 80 Am and 20 Am, respectively. Upon ignoring the increased pressure gradient and fluid retardation, shear deformation of the cell should be 2.5 times larger at height of 80 Am than at height of 20 Am. All these results can be applied to microvessels. Under the assumption that the shear field inside the cell is linear, the Hagen-Poiseuille solution results in the formula for s def similar to Eq. (18): 

rtop sdef ¼ sw 1  ; ð21Þ 2Rv where r top = z top and R v is the vessel radius. Unless the cell diameter is much less than the vessel diameter (realized only in large vessels), deformation and adhesion of the cell are defined by the deforming shear stress s def and not by the wall shear stress s w. Therefore, for in vitro study of the leukocyte-adhesion cascade, flow chamber assays should simulate the deforming shear stress (or rate) acting on leukocytes in postcapillary venules; that is, the deforming shear stress in the flow chamber [Eq. (18)] should be equal to the deforming shear stress in the venule of interest [Eq. (21)]. For a monocyte of diameter 6.5 Am attached to a vessel of diameter 30 Am by microvilli of length 0.3 Am, Eq. (21) gives that the deforming shear stress acting on this cell is about 4.45 dyn/cm2 if the wall shear stress s w = 8 dyn/ cm2. From Eq. (18) it follows that the deforming shear stress acting on the cell in the chamber of height 80 Am is about 1.5 times larger than s def in the vessel if the flow chamber assays are performed at the same wall shear stress. Hence, to

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simulate the deforming shear stress realized in the vessel, the wall shear stress in the flow chamber should be about 5.3 dyn/cm2, that is, 1.5 times smaller than that in the vessel. Note that the difference in the deforming shear stress between the flow chamber and the vessel will further increase with increasing the chamber height or decreasing the vessel diameter if the wall shear stresses in vitro and in vivo are kept equal. Contact time Cell adhesion in vivo or in vitro is a dynamic process. Cell microvilli at the leading edge of the adhesion zone attach to the substrate and form new bonds. They increase contact area of the cell. Microvilli near the trailing edge lose their bonds and eventually detach from the substrate. They are responsible for a decrease in contact area. The speed with which microvilli attach, that is, the contact length increase per unit time, is the bonding velocity of the cell u bond. We define contact length as the length of the adhesion zone in flow direction. In addition, cell contact area widens by attachment of peripheral microvilli, especially in microvessels where the substrate curvature is much higher than in arteries and flow chambers. The microvilli detachment speed, that is, the contact length decrease per unit time, is the peeling velocity of the cell u peel. Again, there is an additional decrease in contact area due to detachment of peripheral microvilli. If the bonding velocity is significantly higher than the peeling velocity for a few seconds after the first contact, firm adhesion of the cell to the substrate will occur. Otherwise, the cell will either roll along the substrate (if the velocities do not differ too much from each other) or detach from the substrate. Our numerical simulations show that, for fixed wall shear rates in channels of different heights, the microvillus contact time and the cell contact time depend on the rate of shear deformation of the cell. The effect of the drag-induced translational cell motion is much less than the effect of shear deformation. The less the shear deformation of the cell, the longer the cell and its microvilli attach to the substrate. For example, the dashed line in Fig. 7B shows that the microvilli attached initially to the plate separate from the plate faster in a chamber of larger height, where the cell is more deformed. (A slight increase in contact time at large heights is the consequence of slow flow establishment, as discussed above.) These results indicate that the peeling velocity of the adherent cell is an increasing function of c˙ def. Based on Eq. (19), the peeling velocity can be evaluated as   upeel ¼ u0peel þ kp utop  ubottom :

ð22Þ

Here k p is a nondimensional coefficient defined by adhesion parameters (on and off rates, spring constants), u0peel is the peeling velocity of a nondeformable cell. The latter depends on the translational and rotational velocities of the cell, microvilli extension (Shao et al., 1998), and adhesion

parameters. Imagine that someone pulls the top edge of a viscoelastic body, which is adherent to a horizontal plate, in horizontal direction. As a result of such pulling, the body deforms and exerts, due to its viscoelasticity, the pulling force on adhesive bonds. The stronger the top edge is pulled, the stronger the pulling force experienced by bonds and the less the contact time. The same situation is for a leukocyte in shear flow. Flow-induced shear stresses at the top edge of the leukocyte (and not in the center of mass of the cell as in the case of the drag force) induce shear deformation of the cell, which in its turn exerts the pulling force on microvilli and their bonds. The higher the flow velocity at the top of the cell, the higher the peeling velocity of the cell will be. Because of the decreased shear deformation, the peeling velocity is less in small vessels than in large vessels provided the wall shear rate is kept fixed. On the other hand, the cell in a chamber of smaller height (or in a smaller vessel) has more torque due to the increased drag force and decreased deformation. The more the torque acting on the cell, the faster the microvilli located at the leading edge of the contact region attach to the substrate. Hence, the rate of contact area growth increases with decreasing channel height (Figs. 7C and D). Our simulations show that microvilli attach to the plate with a higher rate in a chamber of smaller height. These results indicate that the bonding velocity of the cell u bond increases, and hence the difference between the peeling and bonding velocities decreases, with decreasing channel height. Consequently, if the wall shear stress is kept constant, adherent cells in narrower chambers (or smaller vessels) form a larger contact area; that is, they are characterized by stronger adhesion and greater deformation in the contact region. Certainly, a rise in the wall shear stress not only leads to a greater shear deformation of the cell, but it also magnifies the torque acting on the cell, that is, the bonding velocity of the cell. Although the microvillus contact time is about three times smaller at s w = 40 dyn/cm2 (between 0.9 and 1.1 ms) than at 8 dyn/cm2, the number of attached microvilli increases with the wall shear stress (Fig. 7D). The cell forms rapidly large contact area at high wall shear stresses. Although the peeling velocity is high under high shear-stress conditions, more time is required to detach all the microvilli in a larger adhesion zone and hence to initiate the next phase of microvilli attachment at the leading edge of the adhesion zone. (Attached microvilli exert countertorque on the cell body; that is, they slow down clockwise rotation of the cell. After microvilli detachment, cell rotation accelerates.) This can explain why the rolling velocities of neutrophils are saturated at high wall shear stresses (Chen and Springer, 1999; Rinker et al., 2001; Smith et al., 2002). In summary, the results of this paper indicate that leukocyte–endothelial cell interactions are stronger in smaller vessels even if the wall shear stress is kept constant. Apart from the increased curvature of microvessels and erythro-

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cyte–leukocyte interactions (Melder et al., 1995), we identified a new feature that affects cell adhesion in the microcirculation: increased contact area with increasing the cell-to-vessel diameter ratio. Because the flow chamber height (gap thickness) is much more than the diameter of postcapillary venules, the cells in the flow chambers bcannot sustain adhesive interactions at the same wall shear stress or stresses as in microvessels in vivoQ (Smith et al., 2002). Flow chamber studies (with small aspect ratio) underestimate adhesion strength occurring in microvessel if they are performed at the volumetric flow rate at which the wall shear stress matches that observed in microvessels. The deforming shear rate or deforming shear stress [cf. Eqs. (18) and (19), and (21)] can be used as an alternative matching parameter for in vitro experiments. This parameter can predict the peeling velocity of the cell in the microcirculation. Nevertheless, good matching between in vitro and in vivo experiments seems to be possible only if the flow chamber height is comparable with the diameter of blood vessel of interest. Flow chambers of very small height can be produced, for example, by microfabrication techniques (Voldman et al., 1999). Also, the increased overall (shear) deformation of the leukocyte does not mean the increased leukocyte deformation in the adhesion zone, that is, increased contact area. The leukocyte is less deformed in smaller vessels but has greater contact area.

Acknowledgments This work is supported by NIH Grant HL-57446 and NCSA Grant BCS040003N, and utilized the NCSA IBM p690. The authors wish to thank Klaus Ley for useful discussions.

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