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Journal of Biomechanics 40 (2007) 2891–2897

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Shear modulation of intercellular contact area between twodeformable cells colliding under flow

Sameer Jadhava, Kit Yan Chanb, Konstantinos Konstantopoulosc, Charles D. Eggletonb,�

aDepartment of Chemical Engineering, Indian Institute of Technology Bombay, Mumbai 400 076, IndiabDepartment of Mechanical Engineering, University of Maryland Baltimore County, Baltimore, MD 21250, USA

cDepartment of Chemical and Biomolecular Engineering, The Johns Hopkins University, Baltimore, MD 21218, USA

Accepted 7 March 2007

Abstract

Shear rate has been shown to critically affect the kinetics and receptor specificity of cell–cell interactions. In this study, the collision

process between two modeled cells interacting in a linear shear flow is numerically investigated. The two identical biological or artificial

cells are modeled as deformable capsules composed of an elastic membrane. The cell deformation and trajectories are computed using the

immersed boundary method (IBM) for shear rates of 100–400 s�1. As the two cells collide under hydrodynamic shear, large local cell

deformations develop. The effective contact area between the two cells is modulated by the shear rate, and reaches a maximum value at

intermediate levels of shear. At relatively low shear rate, the contact area is an enclosed region. As the shear rate increases, dimples form

on the membrane surface, and the contact region becomes annular. The nonmonotonic increase of the contact area with the increase of

shear rate from computational results implies that there is a maximum effective receptor–ligand binding area for cell adhesion. This

finding suggests the existence of possible hydrodynamic mechanism that could be used to interpret the observed maximum leukocyte

aggregation in shear flow. The critical shear rate for maximum intercellular contact area is shown to vary with cell properties such as

radius and membrane elastic modulus.

r 2007 Elsevier Ltd. All rights reserved.

Keywords: Bulk flow; Cellular adhesion; Contact area; Simulation

1. Introduction

Polymorphonuclear leukocyte (PMN) recruitment tosites of inflammation/infection is orchestrated by thesequential involvement of distinct receptor–ligand pairs:the selectins, integrins and immunoglobulins. According tothis model, free-flowing PMNs first loosely attach (tether)and roll on activated endothelium via selectin–ligandinteractions, then stop, flatten and squeeze betweenendothelial cells into the afflicted tissues in an integrin/immunoglobulin-dependent manner (Simon and Green,2005). The paradigm of the coordinated action of aselectin-mediated tethering followed by integrin-supportedfirm adhesion has been extended to account for PMNhomotypic aggregation in cell suspensions stimulated by

e front matter r 2007 Elsevier Ltd. All rights reserved.

iomech.2007.03.017

ing author. Tel.: +1410 455 3334; fax: +1 410 455 1052.

ess: eggleton@umbc.edu (C.D. Eggleton).

bacterial peptides/chemokines typically found in bloodvessels proximate to the infected/inflamed tissue (Kuyperset al., 1990; Simon et al., 1998).Prior work has demonstrated that steady application of

a threshold level of shear rate is necessary to support PMNhomotypic aggregation in bulk suspensions (Goldsmithet al., 2001). The presence of the shear threshold pheno-menon, by which a reduction of shear rate below athreshold value diminishes the probability of cell adhesion,was also detected during the interaction of free-flowing andsurface-adherent PMNs (Kadash et al., 2004). Moreover,these studies revealed significant deformation duringcellular collision.Biological and artificial cell aggregation can currently be

predicted using mathematical models based on theSmoluchowski’s collision frequency, which assumeslinear trajectory of hard spheres (Smoluchowski, 1917).Given the evidence suggesting that cellular deformation

ARTICLE IN PRESSS. Jadhav et al. / Journal of Biomechanics 40 (2007) 2891–28972892

during shear-induced collisions affects the intercellularcontact area, and thus, the probability of receptor–ligandbond formation between the interacting cells (Goldsmithet al., 2001; Kadash et al., 2004), our attention focuses onthe development of mathematical models that incorporatecellular deformation.

Most of the previous theoretical/computational studiesinvolving deformable cells were limited to single cells inshear flow. Barthes-Biesel and Rallison (1981); Barthes-Biesel and Sgaier (1985) studied the motion of an elasticcapsule in a linear shear flow under the small deformationregime using perturbation analysis, and obtained thedeformation and orientation of the capsule in the shearfield. Deformation was found to increase with an increasein the capillary number. Large deformation of red bloodcell ghosts was simulated by Eggleton and Popel (1998)using the immersed boundary method (IBM) that repro-duced the tank treading behavior observed experimentallyin shear flow (Fischer et al., 1978). Lac and Barthes-Biesel(2005) computed elastic capsule deformation in simpleshear flow and hyperbolic flow using the boundary elementmethod, and showed that steady shapes were obtained onlywithin stable capillary number ranges. Outside of the stablecapillary number ranges, the capsules either go throughcontinuous elongation or a membrane buckling instabilitydevelops. Numerical simulations by Pozrikidis (2001) usingboundary element method showed that membrane bendingstiffness significantly affected capsule deformation in shearflow. Recently, Bagchi et al. (2005) simulated the aggrega-tion of erythrocytes in two dimensions and showed thatrheological cell properties effect the aggregate dynamics.

Although there have been a handful of numerical studieson multiple particle deformation (Loewenberg and Hinch,1997; Breyiannis and Pozrikidis, 2000; Zinchenko andDavis, 2002), the particles considered in these studies wereeither liquid droplets or two-dimensional capsules. Thispaper investigates the effects of fluid shear on theintercellular contact area of two identical cells modeledas three-dimensional (3-D) elastic capsules. The celldeformation and trajectories are calculated using IBM.The cell contact area and the contact duration obtainedfrom the simulations are relevant to liposome or polymer-some interactions as well as homotypic leukocyte aggrega-tion in a linear shear field.

2. Problem statement

The off-center binary collision of two cells in anunbounded linear shear flow is simulated using the IBM(Peskin and McQueen, 1989). The cells are modeled as 3-Delastic capsules containing a Newtonian liquid, and thefluid flow is governed by the continuity and Navier–Stokesmomentum balance equations (Eggleton and Popel, 1998;Jadhav et al., 2005):

r � uðxÞ ¼ 0, (1)

rquðxÞ

qtþ ruðxÞ � ruðxÞ ¼ �rpðxÞ þ mr2uðxÞ þ F ðxÞ, (2)

where r is the fluid density, m is the internal and externalviscosity, u(x) is the velocity vector at position xðx1;x2;x3Þ,p is the pressure, and F(x) is the total force exerted by theelastic membrane onto the fluid. In the IBM, thecomputational domain is comprised of an EulerianCartesian fluid grid xðx1; x2; x3Þ and a Langragian trian-gular finite element grid X ðX 1;X 2;X 3Þ that tracks cellmotion and deformation (Eggleton and Popel, 1998;Jadhav et al., 2005).The force F(x) exerted on the fluid by the membrane is

calculated at each time step by the interpolation of theforces obtained on the surface grid node F(X) as follows:

F ðxÞ ¼X

F ðX Þ �DhðX � xÞ for jX � xjp2h, (3)

where h is the uniform fluid grid spacing. Hence, therestoring force of the membrane located at the surface gridnode X is distributed onto the fluid grid node x accordingto the 3-D discrete delta function,

DhðX � xÞ ¼ dhðX 1 � x1ÞdhðX 2 � x2ÞdhðX 3 � x3Þ (4)

and

dhðzÞ ¼14h

1þ cos p�z2h

� �� �for jzjp2h;

0 for jzj42h:

((5)

Similarly, the velocity at the membrane node X iscalculated from the velocity at the fluid grid nodes x usingthe delta function as

uðX Þ ¼X

uðxÞ �DhðX � xÞ for jX � xjp2h. (6)

This also implies velocity continuity condition on themembrane surface since its velocity matches that of thefluid. Periodic boundary conditions on the velocity andpressure are imposed, and the fast Fourier Transformmethod is used to solve the flow equations (Peskin andMcQueen, 1989; Jadhav et al., 2005).The elastic membrane is assumed to have an initial

spherical stress-free shape. The cell membrane mechanics isdescribed by the neo-Hookean membrane model, whichassumes that the membrane material is incompressible andisotropic. The strain energy density, W, of a neo-Hookeanmembrane is given by (Green and Adkins, 1970)

W h ¼Eh

6ðl21 þ l22 þ l�21 l�22 � 3Þ, (7)

where l1 and l2 are the principal stretching ratio, E is theYoung’s modulus and h is the membrane thickness.The finite element implementation of the force F(x)

calculation of the membrane is based on the modeldeveloped by Charrier et al. (1989) and Shrivastava andTang (1993). The membrane is discretized into triangularfinite elements to obtain the forces acting at the discretenodes of the membrane surface, which are then distributedonto the fluid grid as described above. Given thedisplacement of the three nodes of an element, its state of

ARTICLE IN PRESSS. Jadhav et al. / Journal of Biomechanics 40 (2007) 2891–2897 2893

strain (l1, l2) is obtained. The material properties of theelement determine the forces that are required to maintainthe element in a given state of strain/stress. The principle ofvirtual work is used to calculate the forces at the threenodes of an element. The resultant force F(X) on amembrane node X is simply the sum of the forces exertedby the triangular elements attached to that node. An equaland opposite force acts on the fluid in the manner describedby the IBM. Further details of the numerical implementa-tion and validation of the model can be found in (Eggletonand Popel, 1998).

The fluid domain is a cube with a side that is eight timesthe cell radius, a, and the uniform grid used in thesimulations has 2563 nodes, and grid spacing of a/32. Thefinite element grid of each cell has 20,480 triangularelements. A time step of 10�7 s was used to ensurenumerical stability. The computations require upwards of2GB of memory and run times of 2–4 weeks depending onthe CPUs employed and other hardware characteristics.Our simulations are conducted on an IBM SP3 and SUNFire Server.

3. Results and discussion

In our simulations, biological or artificial cells with aradius, a, of 3.75 mm, equivalent to that of a PMN, aremodeled as elastic capsules whose membrane elasticity, Eh,varies from 0.03 to 3 dyn/cm (Jadhav et al., 2005), and aresuspended in medium with fluid viscosity, m, of 0.8 cP. Themembrane stiffness values of 0.3–1.2 dyn/cm have beenshown to match the extent of PMN deformation previouslyobserved in vivo (Damiano et al., 1996; Smith et al., 2002),while higher values are used to observe the effects ofmembrane stiffening. The physiological shear rate, G, thatcells or liposomes/polymersomes can experience in thevenular circulation varies from 100 to 400 s�1. Theseparameter values give a dimensionless capillary number(Ca ¼ mGa/Eh) of 0.0001–0.04. The capillary numbercompares the relative importance of the hydrodynamicviscous forces causing cell deformation to the modeledcell’s elastic tension forces resisting deformation. Thesimulated results are presented first in terms of this

Fig. 1. Cell deformation and trajectory shift between two cells in shear flow.

interaction between cells (modeled as elastic capsules) in a linear sh

Dx1 ¼ 2:4a;Dx2 ¼ 0:24a;Dx3 ¼ 0. The fluid grid used in the computations has

elements (not all shown in the figure above). The parameters in this simulatio

time evolution of cell deformation during collisional contact and subsequent s

dimensionless capillary number so the behavior becomesindependent of the specific values of the cell properties. Thedimensionless results can be applied to a wide variety ofbiological or artificial cells by specifying particular valuesof membrane elasticity and radius, from which the shearrate can be determined for any given capillary number, asdiscussed below.The x2 component of velocity of the far-field shear flow

can be written as: u ¼ Gx2, and the initial offset betweenthe centers of the cells is: Dx1 ¼ 2:4a;Dx2 ¼ 0:24a;Dx3 ¼ 0:At this initial offset, the deformation of the individual cellsreaches that of a single isolated cell at the given shear rate,before hydrodynamic interactions are observed. Therelatively close initial proximity of the cells greatly reducesthe computational time required. A greater offset in the x1

direction would require the majority of the simulation timeto be dedicated to convecting the cells towards each withminimal hydrodynamic interaction (Loewenberg andHinch, 1997). The simulated trajectory profile as the two-modeled cells interact with each other under the influenceof a linear shear flow is shown in Fig. 1 at a capillarynumber of 0.04. The initially stress-free spherical cellsdeform into oblate spheroid tank-treading cells withnegligible hydrodynamic interaction under the influenceof the imposed shear flow. As the cells approach eachother, the cell contact regions flatten and a thin lubricationgap develops (Fig. 1B). Note that the contact time betweencapsules scales as 1=G; while lubrication theory predictsthat the time for film drainage scales as a2=ðGhoÞ; where hois the film thickness. The ratio of drainage time to contacttime scales as a2/ho51. Thus, there is insufficient time forfilm drainage to occur. Then, the cells start to rotate andspin-off each other with respect to the center point betweenthe cells (Fig. 1C and D). Lastly, the cells separate andcontinue on their own path as oblate tank-treadingspheroids (Fig. 1E). Moreover, there is a permanent shiftin the trajectories of the modeled cells after this cell–cellinteraction. For example, for L-selectin-mediated recep-tor–ligand bond formation between PMNs, the bondlength is �70 nm, the length of microvillus on themembrane surface is �350 nm. Thus, a minimum separa-tion distance between the two cells of about 800 nm or less

The immersed boundary method was used to simulate the hydrodynamic

ear field. The initial offset between the centers of the cells is:

2563 nodes, and the finite element grid of each cell has 20,480 triangular

n are G ¼ 400 s�1, Eh ¼ 0.03 dyn/cm, Ca ¼ 0.04. Panels (A)–(E) show the

eparation.

ARTICLE IN PRESSS. Jadhav et al. / Journal of Biomechanics 40 (2007) 2891–28972894

is defined to be the contact area. This intercellular contactarea represents the available area for cell adhesion; thelarger the contact area the greater the possibility for theinvolvement of multiple receptor–ligand bonds and thussuccessful adhesion (Gourier et al., 2004; Jadhav et al.,2005; Lin et al., 2006). The time evolution of the contactarea with contact time is illustrated in Fig. 2a. The contactarea first increases, reaches a maximum, and then decreasesback to zero when the cells separate at all capillarynumbers examined here. Interestingly, Fig. 2b and c

0

0.04

0.08

0.12

0.16

0.0 2.0

Contact

Co

nta

ct

are

a (

a/(

4*p

i*r2

)

0

1

2

3

4

5

6

0 0.01 0.02 0.03 0.04 0.05

Ca

Co

nta

ct

Du

rati

on

(G

*t)

Fig. 2. Dimensionless intercellular contact area and contact duration as

(a), nondimensionalized with surface area of the undeformed cell and the interc

as a function of the capillary number. The two cells were assumed to be in conta

account the radius of PMNs as 3.75mm, receptor–ligand bond length of appro

length is 350 nm). The evolution of the contact area as a function of contact d

indicate that the maximum dimensionless contact areaand the dimensionless contact duration occur at anintermediate capillary number. The existence of themaximum intercellular contact area can be explained byexamining the detailed cell shape evolution during thebinary-collision process. As the modeled cells makecontact, a thin lubrication fluid layer appears and theshape, size and thickness of this gap are influenced by thestrength of the shear flow and the membrane deformability(i.e. capillary number). Fig. 3 depicts the cell deformation

4.0 6.0

duration (G*t)

0.0006

0.004

0.01

0.02

0.04

0.00

0.02

0.04

0.06

0.08

0.10

0.12

0.14

0.16

0 0.01 0.02 0.03 0.04 0.05

Ca

Co

nta

ct

Are

a (

a/(

4*p

i*r2

)

a function of the capillary number. The intercellular contact area

ellular contact duration, (b) nondimensionalised with shear rate is plotted

ct when the distance between the surfaces was less than 800nm taking into

ximately 70 nm and surface roughness due to PMN microvilli (microvillus

uration (c) is plotted for capillary numbers ranging from 0.0006 to 0.04.

ARTICLE IN PRESS

A D

B

C

A

B

C

E

D

E

Fig. 3. Profiles of interacting cells and shape of intercellular contact area.

Dimple formation in the contact region observed in the plane of shear

passing through the centers of interacting cells (a) and the shape of the

intercellular contact area (b) shown for different capillary numbers: (A)

0.0006, (B) 0.004, (C) 0.01, (D) 0.02, (E) 0.04.

0.00001

0.0001

0.001

0.01

0.1

1

1 10 100 1000 10000 100000

shear rate (s-1)

shear rate (s-1)

co

nta

ct

du

rati

on

(s)

co

nta

ct

are

a (

µm2)

0.03 dyn/cm

0.3 dyn/cm

3.0 dyn/cm

0

10

20

30

1 10 100 1000 10000 100000

0.03 dyn/cm

0.3 dyn/cm

3.0 dyn/cm

Fig. 4. Intercellular contact area and contact duration as a function of the

membrane stiffness. The intercellular contact area (a) and the intercellular

contact duration (b) are plotted as a function of the shear rate. The two

cells were assumed to be in contact when the distance between the surfaces

was less than 800 nm taking into account the radius of PMNs as 3.75mm,

receptor–ligand bond length of approximately 70 nm and surface rough-

ness due to PMN microvilli (microvillus length is 350 nm). The values of

the membrane elastic modulus are 0.03, 0.3 , and 3 dyn/cm.

S. Jadhav et al. / Journal of Biomechanics 40 (2007) 2891–2897 2895

and contact area shape during cell contact. The lubricationlayer widens and thickens with increasing capillary number(Fig. 3a). Moreover, dimple formation is more pronouncedat large capillary numbers, in which the edge of thelubrication film is significantly thinner than the middle ofthe film. The increase in size of the lubrication region leadsto a bigger contact area circumference, while the formationof dimples leads to the transition from solid to annularshape of the contact area, as shown in Fig. 3b. Therefore,the net cell contact area (the shaded area in Fig. 3b) showsa maximum value as capillary number increases. It isnoteworthy that dimple formation has been simulated inliquid droplets (Zinchenko and Davis, 2002), and observedin experiments (Horn et al., 2006; Zdravkov et al., 2006).Thus, it may represent a characteristic of deformableparticles.

Since the capillary number depends on the fluid shearrate G, the cell radius a, and the membrane elasticity Eh, wewished to investigate how specific modulations of theseparameters would affect the intercellular contact area andduration. To this end, we first chose to vary the cellmembrane elastic modulus from 0.03 to 3 dyn/cm. Oursimulations indicate that stiffening the cell membraneincreases monotonically the shear rate at which maximalintercellular contact area and contact duration are detected(Fig. 4) and note that shear rates beyond 2000 and10,000 s�1 are considered supraphysiologic and hemolytic,respectively. Similarly, our analysis predicts that increasingcell size decreases the shear rate at which maximal contactparameters occur at a given membrane elastic modulus(not shown). This analysis can be used to predict themodulation of the homotypic intercellular contact area

between biological or artificial cells of different membranecharacteristics and varying cell sizes in a linear shear field.Our key observation of the maximum intercellular

contact area as a function of shear rate predicts ahydrodynamic mechanism that would contribute to max-imal homotypic cell aggregation. This finding may providea basis for interpreting experimental observations showingthe existence of an intermediate shear rate at whichmaximal homotypic PMN aggregation occurs (Taylor etal., 1996; Jadhav et al., 2001; Jadhav and Konstantopou-los, 2002; Simon and Goldsmith, 2002; Kadash et al.,2004). However, recent experimental observations alsosuggest that selectins exhibit a ‘‘catch-slip’’ bond transition,where increasing tensile forces initially prolong, and subse-quently diminish bond lifetimes (Marshall et al., 2003;

ARTICLE IN PRESSS. Jadhav et al. / Journal of Biomechanics 40 (2007) 2891–28972896

Yago et al., 2004). Thus, a kinetic and hydrodynamicmechanism may both contribute to the observed shearthreshold phenomenon in which the number of tetheredPMNs first increases and then decreases with monotoni-cally increasing shear (Finger et al., 1996; Lawerence et al.,1997; Hammer, 2005).

Taken altogether, the simulation results presented in thispaper provide qualitative evidence of a hydrodynamicmechanism contributing to the maximum homotypicaggregation observed between biological or artificial cellsat intermediate shear rates in vitro. Moreover, it can guidethe design of artificial cells for in vivo-targeted drugdelivery applications by prescribing the combination ofmembrane properties and vesicle size that lead tomaximum contact area with biological cells.

Acknowledgments

This work was supported by the National Institute ofHealth Grant RO1 AI063366 and NSF CAREER AwardBES0093524. We thank the Center for Imaging Science atJohns Hopkins University and the National Center forSupercomputing Applications for computational re-sources.

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