Modeling cell entry into a micro-channel

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Cell entry into a micro-channel has potential applications in cell sorting and cancer diagnostics. In this paper, we numerically model breast cancer cell entry into a constricted micro-channel. Our results indicate that the cell velocity decreases
  Biomech Model MechanobiolDOI 10.1007/s10237-010-0271-1 ORIGINAL PAPER Modeling cell entry into a micro-channel Fong Yew Leong  ·  Qingsen Li  ·  Chwee Teck Lim  · Keng-Hwee Chiam Received: 8 April 2010 / Accepted: 8 November 2010© Springer-Verlag 2010 Abstract  Cell entry into a micro-channel has potentialapplications in cell sorting and cancer diagnostics. In thispaper, we numerically model breast cancer cell entry into aconstricted micro-channel. Our results indicate that the cellvelocity decreases during entry and increases after entry, anobservation in agreement with experiments. We found thatthe cell entry time depend strongly on the cortical stiffnessand is minimum at some critical cortical elasticity. In addi-tion, we found that for the same entry time, a stiff nucleusis displaced toward the cell front, whereas a viscous nucleusis displaced toward the rear. In comparison, the nucleus isless sensitive to the viscosity of the cytoplasm. These obser-vations suggest that specific intra-cellular properties can bededucednon-invasivelyduringcellentry,throughtheinspec-tion of the nucleus using suitable illumination techniques,such as fluorescent labeling. Keywords  Cell entry  ·  Micro-channel  ·  Immersedboundary method · Cell sorting F. Y. Leong · K.-H. Chiam ( B )A*STAR Institute of High Performance Computing,1 Fusionopolis Way, #16-16 Connexis,Singapore 138632, Singaporee-mail: Li · C. T. LimDivision of Bioengineering, Department of MechanicalEngineering, National University of Singapore,Engineering Drive 1, Singapore 117576, SingaporeC. T. Lime-mail: 1 Introduction Cellular deformation in confined environment has attractedmuch attention, due to increasing applications of micro-fluidic devices in medical diagnostics (Suresh 2007). Cellsforced through micro-fluidic constrictions offer the potentialmeansofquantifyingcellmechanicalcharacteristicsin-vitro(Doerschuk et al. 1993; Yap and Kamm 2005). For instance, diseased cells such as cancer cells are known to have differ-entstiffnessandelasticitycomparedtotheirhealthycounter-parts(Gucketal.2005;LeeandLim2007).Suchdifferences couldbeusedtodistinguishbetweennormalandcancercells(Kamm 2002; Hou et al. 2009). Generally, continuum cell models can be classified intotwo broad categories, namely solid models and liquid dropmodels (Lim et al. 2006). For small cellular deforma-tions, a homogeneous visco-elastic solid model is applica-ble Schmid-Schönbein et al. (1981). But for large cellular deformations, a better model would be the Newtonian liquiddrop model, which describes the cell as a homogeneous liq-uiddropenclosedbyacortexwithconstant,isotropictension(Yeung and Evans 1989; Evans and Yeung 1989). To account for the effects of the nucleus on cell defor-mation (Dong et al. 1991), the compound drop model wasproposed (Hochmuth et al. 1993), which included an encap-sulated liquid drop as a model for the nucleus. The com-pound drop model effectively explains the observed rapidinitial response in micro-pipette aspiration and fast recoilon recovery (Tran-Son-Tay et al. 1998; Kan et al. 1998). Applications of the compound drop model include cellrecovery under extensional flows (Kan et al. 1999), micro- pipette aspiration and shear flow (Agresar et al. 1998),cell adhesion, and migration (N’Dri et al. 2003) as well as shear thinning and membrane elasticity (Marella andUdaykumar2004).Morerecentworksincludeshear-induced  1 3  F. Y. Leong et al. drop breakup and cell-surface adhesion (Khismatullin andTruskey 2005).Other related studies include the work of  Yap and Kamm(2005), who used multiple-particle-tracking micro-rheology(Tseng and Wirtz 2001) to probe the mechanical prop-erties of neutrophils within a micro-fabrication channelnon-invasively (McDonald et al. 2000). The experiment wassimulated(Zhouetal.2007)usingadiffuse-interfacemethod(Yue et al. 2004) and was extended to account for effects dueto the nucleus (Zhou et al. 2008a) and visco-elastic cyto-plasm (Zhou et al. 2008b). However, it remains unclear howthenucleusandcytoplasmcanindependentlyaffectcellentryin a confined channel.In the present study, we investigate the entry of breastcancerepithelialcellsintoamicro-channelthroughbothsim-ulation and experiment. Our objective is to relate observedentry behavior to mechanical properties so as to distinguishbetween benign (MCF-10A) and malignant cells (MCF-7) (Hou et al. 2009). This work could also be relevantto the study of cancer metastasis as malignant cells aresqueezed into micro-capillaries or undergo extra-vasationinto neighboring tissues. 2 Theory and numerical method We use the immersed boundary method (Peskin 1977) toanalyze the entry of a cell in a model micro-channel. Theimmersed boundary method is well known (Peskin 2002),therefore only a brief outline will be provided here. Detailson a more specific mixed Eulerian–Lagrangian numericalscheme are available elsewhere (Udaykumar et al. 1997;Shyy et al. 2001).2.1 Model geometryFigure 1 shows a schematic sketch of the experimental setupusingamicro-channelwithasquarecross-section(150 × 10 × 10 µ m) and a 45 ◦ tapered entrance. The length and width of the taper section are  L 1  and  W  1  and those of the channel sec-tion are  L 2  and  W  2 , respectively. The computational domainis segregated into three regions, namely, the extra-cellularfluid (f), the cytoplasm (c), and the nucleus (n). The corticaland nuclear diameters of the cell are  d  c  and  d  n , respectively.Flowinthemicro-channelisdrivenbyadifferentialpressure   p  established between both ends of the setup.A two-dimensional model is used to obtain insights atmodest computational costs. Using an axisymmetric testmodel (see Appendix), we found the cell entry behavior tobe similar to one in a two-dimensional model and thereforeinsensitive to depth. Fig. 1  Schematic of a micro-channel with a tapered entrance and thecomputational domain 2.2 Governing equationsAssuming that the fluid is Newtonian and incompressible,the momentum and continuity equations within each distinctregion (i = f, c, n) are ρ i  ∂ u i ∂ t  + u i ∇· u i  =−∇   p i  + µ i ∇  2 u i  +  f   j  (1) ∇· u i  = 0 (2)where  u i  is the fluid velocity,  ρ i  is the fluid density,  µ i  isthe fluid dynamic viscosity and  p i  is the fluid pressure.  f   j  isthe force contributed by the interfacial forces  F   j  at adjacentinterface j:  f   j  =    F   j  · δ(  x  i  −  x   j ( s )) ds  (3)where δ  isthedeltafunctiontodistributetheinterfacialforcespatially,  x  i  isthepositionvectorsofthefluidgridpoints,and  x   j  isthepositionvectorsofthemarkerpointsalonginterface j whose arc length is s. Material properties such as viscosityare distributed spatially using the Heaviside function. Inter-facial markers are advected by the fluid using the followingkinematic relation: ∂  x   j ∂ t  =    u i  · δ(  x  i  −  x   j ( s )) dx  i  (4)The above system of Eqs. (1–4) are discretized on 2-D staggered grid and solved using the projection methodon a finite difference scheme. The convection terms aresolved explicitly on Adams–Bashforth scheme, and thediffusion terms are solved implicitly on Crank–Nicholsonscheme. Both schemes are second-order accurate (N’Driet al. 2003). To ensure adequate resolution for tracking of markers, the horizontal and vertical fluid grid spacing arerestricted to  |    x  ,  y  |≤  0 . 05 W  2 . In addition, the initialspacing between the markers is specified at 0.8 of fluid  1 3  Modeling cell entry into a micro-channel grid spacing. This minimizes the material loss due toinaccuracyoftheimmersedboundarymethodalongtheinter-face (Marella and Udaykumar 2004). The error is verified tobe no more than 1% of the srcinal cell volume.The following boundary conditions are implemented:positive inlet pressure  (  p in  >  0 )  relative to null outlet pres-sure  (  p out  =  0 ) . No-slip condition is imposed on all walls ( u wall  =  0 ) . Near the tapered walls, the velocity is approx-imated as  u  f   −  u w  =  0 , where  u  f   is the adjacent fluidnode and  u w  is the adjacent wall node. For other geometrieswhose tapered slope is off-diagonal, the cut-cell method isrecommended (Agresar et al. 1998; Udaykumar et al. 1997). 2.3 Interfacial constitutive modelThe compound drop model accounts for two interfaces,namelythecortexandthenuclearmembrane.Thecommonlyassumed model for the cortex is that of constant, isotropictension of a liquid drop (Yeung and Evans 1989; Evans and Yeung 1989). Here, we include elastic contributions, so thatthe net interfacial force can be expressed as the sum of atension term and an elastic term (Agresar et al. 1998). F  m , c  = σ  c · κ · ˆ n + η c · λ ·ˆ t   (5) F  m , n  = σ  n · κ · ˆ n + η n · λ ·ˆ t   (6)where σ   istheinterfacialtensionand η  istheelasticmodulusfor surface area dilation. The subscripts  c  and  n  refer to thecortex and the nucleus, respectively.  κ  is the interfacial cur-vature,  λ  is the interfacial strain,  ˆ n  and  ˆ t   are the normal andtangential unit vectors, respectively. The vector componentsare evaluated through parameterization of the arc length fol-lowed by cubic-spline interpolation of the marker points. Asa simplification, we assume that both cytoskeletal and mem-brane properties do not vary in time. Experimental cell entrytimesaretypicallyontheorderof0.1s,atimescalethatmaybe too fast for active processes to remodel the membrane.However,theremaybeotherfactorsthatcausethemembraneproperties to change over the time scale of cell entry. Onesuch scenario could be the slipping of adhesion bonds thatresults in changes to the membrane’s visco-elasticity. How-ever, as we will discuss in Sect. 3.2, the channel surfaces aretreatedwithbovineserumalbumintoavoidadhesion.There-fore, we have neglected transient elastic membrane effectsand assumed that our membrane properties are constant.The cell surface has many folds and wrinkles whichprovide between 80 and 100% excess area (Schmid-Schönbein et al. 1980). Based on known stress–strainresponse, non-linear stiffening only occurs at strains above40%(MarellaandUdaykumar2004).However,suchstiffen- ing was not found to be significant in typical micro-pipettestudies. We assume that the stress–strain response of thecortex is linear before this limit is reached.In addition, we consider two other known numericalissues. Firstly, the interfacial markers can become eitherclustered or dilated leading to numerical instability (Marellaand Udaykumar 2004). Incidentally, this problem is avoideddue to our inclusion of interfacial elasticity. Secondly, amarker could cross a wall boundary, leading to a violation of no-penetration condition. However, it is assumed that a thinlubricating fluid layer exists between the cell and the wall,and so the cell should not penetrate the wall. To implementthis, virtual compressive springs are introduced between thecell and the wall. This approach is shown to be reasonableby (Dong and Skalak 1992). The equilibrium length of thespring is such that it is between one or two grid spaces toensure that (1) the dimension of the gap is relatively smallcompared to the width of the channel and (2) there is at leastone node to resolve the fluid velocity in that layer. The stiff-ness issufficiently large to ensure that the gap width ismain-tained but not so large as to lead to numerical instability. Wefound that, within such limits, the exact magnitude of lengthand stiffness of the springs does not affect the overall rhe-ology significantly. This agrees with the observations madeby (Marella and Udaykumar 2004), who also found that thevirtual spring model does not affect the cell and fluid flowintoamodelmicro-pipette.Thismethodobviatestheneedtoenforce an artificial free-slip boundary condition on the wall(Drury and Dembo 2001).2.4 Dimensionless parametersThe characteristic scales for length  l , pressure  p , velocity  u ,and time  t   are  L  ≡  H  2 ,  P  ≡    p ·  L L 2 ,  U   ≡  L · P 12 µ  f  ,  T   ≡  LU  ,  (7)where  µ  f   is the viscosity of the suspending fluid. Based onthese characteristic scales, we obtain the following dimen-sionless parameters, ˜ l  ≡  l L ,  ˜  p ≡  pP ,  ˜ u  ≡  uU  ,  ˜ t   ≡  t T  ,  (8) Ca − 1 c  ≡  σ  c µ  f   · U  , α c  ≡  d  c  L , β c  ≡  µ c µ  f  , γ  c  ≡  η c ·  L σ  c ,  (9) Ca − 1 n  ≡  σ  n µ  f   · U  , α n  ≡  d  n  L , β n  ≡  µ n µ  f  , γ  n  ≡  η n ·  L σ  n ,  (10)where Ca − 1 isthereciprocalofthecapillarynumber, α  istheinterfacialgeometricratio, β  istheinterfacialviscosityratio,and  γ   is the ratio between interfacial elasticity and tension.Subscripts  c  and  n  refer to the cortex and nuclear membrane,respectively. For the present study, the Reynolds numberis sufficiently small  (  Re  ≪  1 )  that we do not consider itsinfluence on cell entry explicitly.  1 3  F. Y. Leong et al. 3 Results 3.1 Base case modelWe introduce a base case simulation, whose model param-eters are  Ca − 1 c  =  Ca − 1 n  =  18 ,α c  =  2 ,α n  =  1 . 5 ,β c  = β n  =  1, and  γ  c  =  γ  n  =  10 − 4 . The capillary number isbasedonthemeancorticalstiffness(0.01dyn/cm)referencedfrom the range of literature values (0.005–0.035dyn/cm)(Evans and Yeung 1989; Yeung and Evans 1989; Needham and Hochmuth 1992). The diameters of the cell and nucleusare based on mean experimental values of 20 and 15 µ m,respectively, with standard deviations of up to 5 µ m (Fig. 4).Viscosity is initialized as unity for subsequent time-scaling(see 3.2). Elasticity tends to be small compared to stiffness,andtheselectedvalueof10 − 4 correspondsto0 . 001dyn / cm 2 indimensionalunits.Thisissubsequentlyshowntobeasuit-able choice, as we found that an optimal entry time existswithin a range from 10 − 5 to 10 − 3 .The cell is initially stationary, and its center is positionedat a normalized distance of 1.5 from the channel entrance.Figure 2 shows snapshots of cell entry, and Fig. 3 shows the dependence of cell leading and trailing edge displacement(normalized by  L ) and velocity (normalized by  α c  ·  L ) ontime  ˜ t  . The cell entry time  τ   is defined as the time intervalbetween the leading edge and the trailing edge crossing theentrance of the micro-channel.After a brief initial transient period ( ˜ t   <  0 . 01), the fluidflow is nearly steady (Fig. 2a). The cell velocity remainssteady at  ˜ t   ∼  1 . 5 before decreasing after  ˜ t   ∼  1 . 5 (Fig. 3b).Midway through the entrance ( ˜ t   ∼  3 . 9), the cytoplasmicvelocity is observed to uniform (Fig. 2c). Furthermore, thepressure gradient is confined to the cell interior, which sug-gests a plugged flow behavior (Fig. 2d). Meanwhile, thecell velocity increases until a maximum is reached at  ˜ t   ∼ 6 (Fig. 3b). Beyond the entry region, cell deformation isminimal and is of no further interest. Fig. 2  Snapshots of cell entry showing velocity ( ˜ u ) and pressure ( ˜  p ) fields, using base case parameters:  Ca − 1 c  = Ca − 1 n  = 18,  α c  = 2,  α n  = 1 . 5, β c  = β n  = 1 and  γ  c  = γ  n  = 10 − 4  1 3  Modeling cell entry into a micro-channel Fig. 3  Cell displacement ( a ) and velocity ( b ) time-plots using base case parameters:  Ca − 1 c  = Ca − 1 n  = 18,  α c  = 2,  α n  = 1 . 5,  β c  = β n  = 1, and γ  c  = γ  n  = 10 − 4 . Horizontal dotted lines at ˜ l  = 0 . 6 and 2.4 denote the distances of the channel entrance from the initial leading and trailing edges,respectively Fig. 4  Snapshots of a transfected MCF-10A breast epithelial cancercell entering a micro-channel. The fluorescent-labeled nucleus is illu-minated in ( a – c ), whereas the cell membrane outline is distinct underbrightfield ( d – f  ). The pressure gradient of the micro-channel is main-tained at constant    p = 3 . 33Pa / µ m (scale bar=10 µ m) 3.2 Experimental validationCell entry can be visualized and recorded experimentallyusing a high-speed video camera, as shown in Fig. 4. AMCF-10A cell has been pre-transfected, so that the nucleusis fluorescent. The nucleus outline is distinct under fluores-cence (Fig. 4a-c), whereas the cell outline is distinct underbrightfield (Fig. 4d-f). This allows us to infer the mechanicalproperties of the nucleus and will be further elaborated inSect. 4. To avoid cell adhesion, we use bovine serum albu-min (BSA) to prepare the surfaces prior to each experiment.Previously, Zhou et al. (2007) compared their numerical model with experiments (Yap and Kamm 2005) using an  a priori  specification of a viscosity ratio for a real cell ( β c  = 220), followed by an extrapolation through an assumedpowerlawfittedatlowviscosityratios.However,itisunclearhow cell entry depends on the viscosity of the nucleus.Furthermore, cell viscosities are highly variable even for thesame cell type (Yeung and Evans 1989; Evans and Yeung 1989;Hochmuthetal.1993;Tran-Son-Tayetal.1998;Need- ham and Hochmuth 1990; Valberg and Albertini 1985; Tsai et al. 1993).  1 3
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