numerical simulation of lateral migration of red blood cells in poiseuille flows

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDSInt. J. Numer. Meth. FluidsPublished online in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/fld.2455

Numerical simulation of lateral migration of red blood cellsin Poiseuille flows

Lingling Shi, Tsorng-Whay Pan∗,† and Roland Glowinski

Department of Mathematics, University of Houston, Houston, TX 77204-3008, U.S.A.

SUMMARY

A spring model is applied to simulate the skeleton structure of the red blood cell (RBC) membrane andto study the RBC rheology in two-dimensional Poiseuille flows using an immersed boundary method.The lateral migration properties of the cells in Poiseuille flows have been investigated. The simulationresults show that the rate of migration toward the center of the channel depends on the swelling ratio andthe deformability of the cells. We have also combined the above methodology with a fictitious domainmethod to study the motion of RBCs in a two-dimensional micro-channel with a constriction with anapplication to blood plasma separation. Copyright � 2010 John Wiley & Sons, Ltd.

Received 30 April 2010; Revised 9 August 2010; Accepted 12 September 2010

KEY WORDS: red blood cells; lateral migration; elastic spring model; immersed boundary method;micro-channel; Poiseuille flow

1. INTRODUCTION

The microcirculation, which takes place in the microvessels of diameter smaller than 100�m, isessential to the human body since it is where exchanges of mass and energy take place. At themicrocirculatory level, the particulate nature of blood becomes significant. The rheological propertyof the red blood cells (RBCs) is a key factor of the blood flow characteristics in microvesselsdue to their large volume fraction (40–45%), so-called hematocrit (Hct), in the whole blood. TheRBC membrane is highly deformable so that an RBC can change its shape when an externalforce is acting on it and returns to the biconcave resting shape after the removal of the forces [1].In microvessels having an internal diameter close to the cell size, the RBCs exhibit well-knownparachute shapes under flow [2]. In a bigger microvessel, the RBCs tend to move to the center ofthe vessel so that there is a cell-free layer near the vessel wall. The non-uniform distribution ofHct within the cross-section of the vessel is the physical reason of the Fahraeus–Lindqvist effect[3] which is characterized by a decrease in the apparent blood viscosity in such microvessels.

Nowadays in silico mathematical modeling and numerical study of RBC rheology have attractedgrowing interest (see, e.g. [4, 5]) since it is difficult to deal with in vivo and in vitro experimentsto study microcirculation and RBC rheology due to the size limitation. For example, in [6] theparachute shape of RBCs in capillaries was investigated with different Hct and the apparent bloodviscosity in capillaries was studied by using the boundary-integral method with both Mooney–Rivlin and Skalak models plus bending resistance for the RBC membrane. In [7], an immersedboundary method was used to simulate 3D capsules and RBCs in shear flow with both neo-Hookean

∗Correspondence to: Tsorng-Whay Pan, Department of Mathematics, University of Houston, Houston, TX 77204-3008, U.S.A.

†E-mail: [email protected]

Copyright � 2010 John Wiley & Sons, Ltd.

23 October 20102012; 68:1 –140393 8

L. SHI, T.-W. PAN AND R. GLOWINSKI

and Skalak models for the membrane deformation. It was found that the bending resistance mustbe included in order to simulate complex shaped RBCs when they deform in shear flow. In [8], animmersed boundary method and a neo-Hookean model with and without bending resistance wereused to simulate the interaction of two deformable cells in a shear flow in two dimensions. It wasfound that aggregates made of deformable cells are easily dispersed by a shear flow, those madeof less deformable cells are not. In [9, 10], an immersed finite element method was presented forthe simulation of RBCs in three dimensions, the RBC membrane employing a Mooney–Rivlinmodel. The microscopic mechanism of RBC aggregation has been linked to the macroscopic bloodviscosity via direct numerical simulation and the relation between the effective viscosity of bloodflow and the diameter of capillaries has been derived. In [11], a semi-implicit particle methodcombined with a spring model was used to simulate a single file of RBCs between two parallelplates for various Hct in two dimensions. The parachute shape of RBCs in capillaries and flowresistance was investigated with different Hct . In [12], an immersed boundary method based onthe lattice-Boltzmann method has been developed with the Skalak model for the RBC membrane.Their simulations capture the RBC induced lateral platelet motion and the consequent developmentof a platelet concentration profile that includes an enhanced concentration within a few microns ofthe channel walls. In [13], a lattice-Boltzmann method has been coupled to a linear finite-elementanalysis to simulate suspensions of large numbers of RBCs and spherical fluid-filled capsules. In[14], a discrete model for the RBC membrane has been constructed by taking into account thevolume constraint of the RBC, the local area constraint on each triangle element from the meshfor the RBC membrane, the total area constraint of the RBC surface, the stretching force betweennodes on each edge of the surface triangle element, and the preferred angle between triangleelements sharing a common edge (the bending resistance). These constraints give different forcesacting on the nodes on the RBC surface. A lattice-Boltzmann method was combined with thisdiscrete model to simulate 200 densely packed RBCs in three-dimensional flows.

Among the methodologies and models available for simulating the motion of the RBCs in flows,we have chosen to combine the immersed boundary method with spring models since one of ourmain goals is to investigate the interaction of blood cells and rigid particles in microvessels (seeReferences [15, 16] for cardiovascular and oncological applications). For simulating hundreds ofrigid particles freely moving in Newtonian fluid in three dimensions, we have already developedvery efficient methodologies, called distributed Lagrange multiplier/fictitious domain (DLM/FD)methods (see, e.g. [17, 18]). Actually, the DLM/FD methods are closely related to the immersedboundary methods since they both use uniform grids on simple-shaped computational domainsand the Lagrange multipliers play a role similar to the force acting on the elastic membraneimmersed in the fluid for the immersed boundary methods. For modeling the RBC membrane,the general organization of the RBC membrane has been well characterized. It has been shownthat the human RBC is an inflated closed membrane filled with a viscous fluid, called cytoplasm.The RBC membrane has a phospholipid bilayer with the attached glycocalyx at the plasmatic faceof the bilayer and a network of spectrins, called the cytoskeleton, fastened to the bilayer at itscytoplasmic face [1, 19]. The cytoskeleton is an elastic network of spectrin which has a triangularstructure (and most of these triangles form hexagons) in the network (e.g. see [20]). This particulargeometry, as well as the intrinsic elastic properties of the spectrin, allows the RBC to be highlydeformable and elastic. Owing to its special structure, the RBC membrane has strong resistanceto changes in area/volume and shear deformation [19]. Therefore, it is of significance to takeinto consideration the structure of the RBC membrane skeleton in the study of RBC rheology.Several spring models [11, 14, 19, 21–24] and spectrin-based models [25, 26] have been developedto illustrate the structure of the RBC membrane skeleton and to describe the deformability of theRBCs. In this paper, the mechanical properties of the RBC membrane are predicted by a recentlyproposed elastic spring model in [11]. In [27], we have developed an immersed boundary methodcombined with the elastic spring model of [11] in which the fluid motion is computed using anoperator splitting technique and finite element method with a fixed regular triangular mesh sothat fast solvers can be used for simulating the fluid flow. The elastic spring model is validatedby comparing with previous experimental data [28], theoretical results (Keller and Skalak model

Copyright � 2010 John Wiley & Sons, Ltd.DOI: 10.1002/fld

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Int. J. Numer. Meth. Fluids 2012; 68:1393–1408

NUMERICAL SIMULATION OF LATERAL MIGRATION OF RED BLOOD CELLS

[29]), and simulation results [30, 31] for the inclination angles and tank-treading frequencies of asingle RBC in shear flows.

In this paper, we have tested the capability of the methodology developed in [27] by applyingit to simulate the motion of many RBCs in two-dimensional Poiseuille flows and validated themethodology by comparing the size of the cell-free layer next to the walls with the experimentaldata in [32]. Then the methodology has been combined with a DLM/FD method. We have appliedthe resulting methodology to simulate the motion of several RBCs in a two-dimensional micro-channel with a constriction, which has many practical applications (see, e.g. [33, 34]). The structureof this paper is as follows: we discuss the elastic spring model and numerical methods in Section 2.In Section 3, we first study the lateral migration of few cells of different shapes in Poiseuilleflows. Then we present the results of simulations involving many RBCs in Poiseuille flows andvalidate them by comparing the size of the cell-free layer next to the walls. Finally, the resultsof the simulation of the motion of RBCs in a micro-channel with a constriction are shown. Theconclusions are summarized in Section 4.

2. MODELS AND METHODS

Let �\� be a bounded region filled with blood plasma which is incompressible, Newtonian, andcontains RBCs. Assume that � is a rectangular domain and � is the blockade (see Figure 1). Forsome T>0, the governing equations for the fluid–cell system are

�

[�u�t

+u ·∇u]=−∇p+��u+f in �\�, t ∈ (0,T ), (1)

∇ ·u=0 in �\�, t ∈ (0,T ). (2)

Equations (1) and (2) are completed by the following boundary and initial conditions:

u=0 on the top and bottom of �\� and is periodic in the x1 direction, (3)

u(0)=u0. (4)

In (1)–(3), u and p are the fluid velocity and pressure, respectively, � is the fluid density, and �is the fluid viscosity, which is assumed to be constant for the entire fluid (but in reality � is not aconstant since the viscosity of the cytoplasm is about five times that of the blood plasma viscosity).When considering Poiseuille flow, we have �=∅. We also assume that the flow is periodic in thex1 direction with period L , L being the common length of the channel � (see Figure 1). In (1),f is a body force which is the sum of fp and fB, where fp is a constant force pointing in the x1direction obtained with a fixed pressure drop and fB accounts for the force acting on the fluid/cellinterface.

2.1. Elastic spring model for the RBC membrane

In this article, a two-dimensional elastic spring model used in [11] is considered to describe thedeformable behavior of the RBCs. Based on this model, the RBC membrane can be viewed as

ωΩ

X1

X2

ωL

H

Figure 1. An example of computational domain with a constriction region �.


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θl

Figure 2. The elastic spring model of the RBC membrane.

membrane particles connecting with the neighboring membrane particles by springs. Elastic energystores in the spring due to the change in the length l of the spring with respect to its referencelength l0 and the change in angle � between two neighboring springs (see Figure 2). The totalelastic energy of the RBC membrane, E= El+Eb, is the sum of the total elastic energy forstretch/compression and the total energy for bending which, in particular, are

El = kl2

N∑i=1

(li −l0l0

)2

(5)

and

Eb= kb2

N∑i=1

tan2(�i/2). (6)

In Equations (5) and (6), N is the total number of spring elements and kl and kb are springconstants for changes in length and bending angle, respectively.

Remark 2.1In the process of creating the shape of RBCs described in [11], the membrane particles of the RBCare uniformly distributed on a circle of radius R0=2.8�m, initially. The circle is discretized intoN =76 membrane particles so that 76 springs are formed by connecting the neighboring particles.The shape change is stimulated by reducing the total area of the circle through a penalty function

�s = ks2

(s−sese

)2

, (7)

where s and se are the time-dependent area and the targeted area of the RBC, respectively, andthe total energy is modified as E+�s . Based on the principle of virtual work the force acting onthe i th membrane particle is now

Fi =−�(E+�s )

�ri, (8)

where ri is the position of the i th membrane particle. When the area is reduced, each RBCmembrane particle moves on the basis of the following equation of motion:

mri +�ri =Fi . (9)

Here, () denotes the time differentiation, m and � represent the membrane particle mass and themembrane viscosity of the RBC, respectively. The position ri of the i th membrane particle isobtained by discretizing (9) via a second-order finite difference method.

For the shapes of the RBCs used in Section 3, we have chosen for kb, kl , and ks the valuesused in [11], that is: the bending constant in (6) is kb =5×10−10Nm, the spring constant in (5) iskl =5×10−8Nm, and the penalty coefficient in (7) is ks =10−5Nm. The penalty function definedby (7) constrains the area only. In general, the spring constant has to be larger than the bending


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constant to have the conservation of the circumference and of the local arclength (based on ourcomputational results). The final shape is obtained by minimization of the total elastic energy (see[27]). The area of the final shape differs by less than 0.001% from the given targeted area se;similarly the length of the perimeter of the final shape differs by less than 0.005% circumferenceof the initial circle. In this article, the swelling ratio of an RBC is defined by s∗ =4�se/(2�R0)2.

2.2. Immersed boundary method

The immersed boundary method developed by Peskin, e.g. [35–37], is employed in this studybecause of its distinguishing features in dealing with the problem of fluid flow interacting with aflexible fluid/structure interface. Over the years, it has demonstrated its CFD capability includingblood flow simulations. Based on the method, the boundary of the deformable structure isdiscretized spatially into a set of boundary nodes. The force located at the immersed boundary noderi = (ri,1,ri,2) affects the nearby fluid mesh nodes x= (x1, x2) through a 2D discrete �-functionDh(x−ri ):

fB(x)=∑

Fi Dh(x−ri ) for |x−ri |�2h, (10)

where h is the uniform finite element mesh size and

Dh(x−ri )=�h(x1−ri,1)�h(x2−ri,2), (11)

the 1D discrete �-functions being defined by

�h(z)=

⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1

8h(3−2|z|/h+

√1+4|z|/h−4(|z|/h)2), |z|�h,

1

8h(5−2|z|/h−

√−7+12|z|/h−4(|z|/h)2), h�|z|�2h,

0, otherwise.

(12)

The velocity of the immersed boundary node ri is also affected by the surrounding fluid and istherefore enforced by summing the velocities at the nearby fluid mesh nodes x weighted by thesame discrete �-function:

U(ri )=∑

h2u(x)Dh(x−ri ) for |x−ri |�2h. (13)

After each time step, the position of the immersed boundary node is updated by

rn+1i =rni +�tU(rni ). (14)

2.3. A fictitious domain formulation of the model problem and the operator splitting technique

The DLM/FD formulation for the flow problem (1)–(4) in a channel of constriction reads asfollows:

For a.e. t>0, find u(t)∈W0,P , p(t)∈ L20, k∈� such that

⎧⎪⎪⎪⎨⎪⎪⎪⎩

�∫

�

[�u�t

+(u ·∇)u]·vdx+�

∫�

∇u :∇vdx−∫�p∇·vdx

=∫�f·vdx+〈k,v〉 ∀v∈W0,P,

(15)

∫�q∇·u(t)dx=0 ∀q∈ L2(�), (16)

〈l,u(t)〉=0 ∀l∈�, (17)

u(x,0)=u0(x) (18)


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with

W0,P = {v|v∈ (H1(�))2, v=0 on the top and bottom of � and

v is periodic in the x1 direction},

L20=

{q|q∈ L2(�),

∫�q dx=0

},

�= (H1(�))2.

In (15)–(18), k is a Lagrange multiplier associated with relation (17) and 〈·, ·〉 is an inner producton � (see [38] for further information). We also use, if necessary, the notation (t) for the functionx→(x, t).

Remark 2.2Although we have applied the fictitious domain methodology in Section 3 to simulate the motionof the RBCs in a micro-channel with a simple shape of �, this approach will be used to investigatethe separation of cells and plasma in a more complicated micro-channel as in [33] and the motionof the RBCs in a curved channel in the near future.

2.4. Space approximation and time discretization

Concerning the finite element-based space approximation of {u, p} in problem (15)–(18), we usethe P1-iso-P2 and P1 finite element approximation (e.g. see [17] (Chapter 5)). Suppose that arectangular computational domain �⊂ R2 is chosen with length L , h is a space discretization step,Th is a finite element triangulation of � for velocity, and T2h is a twice coarser triangulation forpressure. Letting P1 be the space of polynomials in two variables of degree �1, we introduce thefinite dimensional spaces:

W0h = {vh|vh ∈C0(�)2,vh|T ∈ P1×P1 ∀T ∈Th ,vh =0 on the top and bottom

of � and is periodic in the x1 direction with period L},

L2h = {qh |qh ∈C0(�),qh |T ∈ P1 ∀T ∈T2h qh is periodic in the x1

direction with period L},

L2h,0=

{qh |qh ∈ L2

h,

∫�qh dx=0

}.

A finite dimensional space approximating � is defined as follows: let {xi}Mi=1 be a set of pointsfrom � which cover � (uniformly, for example); we define then

�h ={lh |lh =

M∑i=1li�(x−xi ), li ∈R2, ∀i=1, . . . ,N

},

where �(·) is the Dirac measure at x=0. Then we shall use 〈·, ·〉 defined by

〈lh ,vh〉=M∑i=1li ·vh(xi ) ∀lh ∈�h , vh ∈W0h

A typical choice of points for defining �h is to take the grid points of the velocity mesh internalto the region � and whose distance to the boundary of � is greater than, e.g. h/2, and to completewith selected points from the boundary of �, but we exclude those points which are also on theboundary of �.


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Then we apply the Lie’s scheme [17, 39] to Equations (15)–(18) with the backward Euler methodin time for some subproblems and obtain the following fractional step subproblems (some of thesubscripts h have been dropped):

u0=u0 is given; for n�0, un being known, we compute the approximate solution via thefollowing fractional steps:

1. Solve

�∫

�

un+1/3−un

t·vdx−

∫�pn+1/3(∇ ·v)dx=0 ∀v∈W0h,

∫�q∇·un+1/3 dx=0 ∀q∈ L2

h,

un+1/3∈W0h , pn+1/3∈ L2h,0.

(19)

2. Update the position of the membrane by (13) and (14) and then compute the force fB on thefluid/cell interface by (8) and (10).

3. Solve ∫�

�u(t)�t

·vdx+∫

�(un+1/3 ·∇)u(t)·vdx=0 on (tn, tn+1) ∀v∈W0h,

u(tn)=un+1/3,

u(t)∈W0h on (tn, tn+1),

(20)

and set un+2/3=u(tn+1).4. Finally solve

�∫

�

un+1−un+2/3

t·vdx+�

∫�

∇un+1 :∇vdx=∫�f·vdx+〈k,v〉 ∀v∈W0h,

〈l,un+1〉=0 ∀l∈�h,

un+1∈W0h , k∈�h .

(21)

The degenerated quasi-Stokes problem (19) is solved by a conjugate gradient method (see, e.g.[17]). System (20) is an advection-type subproblem. It is solved by a wave-like equation method,which is described in detail in [40, 41]. Problem (21) is a discrete elliptic system which is aclassical problem if �=∅. When there is a constriction in the channel, we have applied a conjugategradient method (see [38]) to solve problem (21).

3. NUMERICAL RESULTS AND DISCUSSIONS

In [27], we have validated the elastic spring model by comparing the numerical results with theprevious experimental data [28], theoretical results (Keller and Skalak model [29]), and simulationresults [30, 31] for the inclination angles and tank-treading frequencies of a single RBC in shearflows. In this paper, the lateral migration properties of the cells in Poiseuille flows have beeninvestigated. The simulation results show that the lateral migration toward the center of the channeldepends on the swelling ratio, the deformability of the cells, and the height of the channel. Thenwe have tested the capability of the methodology developed in [27] by applying it to simulate themotion of many RBCs in Poiseuille flows and validated it by comparing the size of the cell-freelayer next to the walls with the one obtained in [32]. We have also combined the methodology witha DLM/FD method to simulate the motion of several RBCs in a micro-channel with a constriction.

The initial shapes of cells used in the flow simulations in this article are obtained via theprocedure described in [27] and Remark 2.1. We have chosen the parameter values equal to thoseused in [11], that is: the bending constant in (6) is kb=5×10−10Nm, the spring constant in (5)


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is kl =5×10−8Nm, and the penalty coefficient in (7) is ks =10−5Nm. The cells are suspended inblood plasma which has a density �=1.00g/cm3 and a dynamical viscosity �=0.012g/(cms).The viscosity ratio which describes the viscosity contrast of the fluid inside and outside the RBCmembrane is fixed at 1.0. The computational domain is a two-dimensional horizontal channel.In the simulation, a constant pressure drop is prescribed as a body force. In addition, periodicconditions are imposed at the left and right boundary of the domain. The Reynolds number isdefined by Re=�UH/�, where U is the averaged velocity in the channel and H is the height ofthe channel.

3.1. Migration of cells in Poiseuille flows

3.1.1. One cell migration. We present first the results on the simulation of the migration of a singlecell in a Poiseuille flow through a narrow channel. The fluid domain is a 20�m×10�m rectangle.The pressure drop is set as a constant for this study so that the Reynolds number of the Poiseuilleflow without cells is 0.5556. The initial velocity is zero everywhere. The grid resolution for thecomputational domain is 80 grid points per 10�m. The initial shapes of the cells we consider arebiconcave and circular with the swelling ratio s∗ =0.55, 0.7, and 1, respectively (see Figure 3).The initial position of the mass center of the single cell is located at (5, 3). Because of the periodicboundary condition at the inflow and outflow boundaries, this configuration corresponds to the

0

2.5

5

7.5

10

t = 0.00ms t = 0.05ms t = 0.10ms t = 0.20ms t = 0.30ms t = 0.50ms t = 0.80ms

(a)

Y (

μm)

0

2.5

5

7.5

10


Y (

μ m)

0

2.5

5

7.5

10


Y (

μ m)

0

2.5

5

7.5

10t = 0.90ms t = 1.00ms t = 1.25ms t = 1.50ms t = 2.00ms t = 3.00ms t = 7.50ms

Y (

μm)

0

2.5

5

7.5

10


Y (

μm)

0

2.5

5

7.5

10


Y (

μm)

(b)

(c)

Figure 3. The lateral migration of a single cell in a Poiseuille flow: (a)–(c) show the position and shapeof a single cell with s∗ =0.55, 0.7, and 1.0, respectively.


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0 1 2 3 4 5 6 72.5

3

3.5

4

4.5

5

5.5

Time(ms)

Y(μ

m)

s* = 0.55

s*= 0.7

s*=1.0

Figure 4. The histories of the height of the mass center of a single cell in a narrow channel.

0 100 200 300 400 500 600 700 8005

10

15

20

25

30

Time(ms)

Y(μ

m)

s* = 0.55

s*= 0.7

s*=1.0

Figure 5. The histories of the height of the mass center of a single cell in a larger channel.

Hct=6.77, 8.62, and 12.32% for s∗ =0.55, 0.7, and 1, respectively. The Reynolds numbers basedon the averaged velocity of the channel flow for the cell with s∗ =0.55, 0.7, and 1 are 0.5176,0.5023, and 0.5028, respectively. In Figure 3, the well-known parachute shape of RBCs has beenobserved for the cases of s∗ =0.55 and 0.7, and a slightly bullet-like shape has been observedfor the case of s∗ =1. Three different cells that migrate to the center of the channel are shown inFigure 4. The cell with s∗ =1 moves slowest toward the center of the channel, which is qualitativelyin good agreement with the experimental results of vesicles in [42]. But for the behavior of the twoother cell migration rates, we believe that the wall effect plays some role here since the channelis very narrow. In order to find out more about the wall effect on the cell migration, we havesimulated the migration of a single cell in a larger channel of the size of 100�m×50�m. Thegrid resolution for the computational domain is 32 grid points per 10�m. All other parameters arekept the same. As shown in Figure 5, the cell with smaller swelling ratio moves faster toward thecenter of the channel, which is similar to the experimental results in [42]. This confirms that thewall effect plays a significant role on the cell migration when the channel is narrow. The furtherstudy of the effects of the wall and the swelling ratio s∗ will be reported in the near future.

3.1.2. Four cell migration. To study the effect of the deformability on the migration of the cell,we have considered first the case of four cells with s∗ =1 in a Poiseuille flow so that we cancompare their motions with the computational results of four neutrally buoyant disks of the samearea obtained by the DLM/FD method described in [43]. Then we have compared the results withthose with s∗ =0.7 to study the effect of the swelling ratio on the lateral migration.

In the simulations, we have kept the same values for kl and ks and considered four differentvalues of the bending constant, namely 1000kb, 100kb, 10kb, and kb. The fluid domain is a100�m×50�m rectangle. The pressure drop is set as a constant for this study so that the Reynoldsnumber of the Poiseuille flow without cells is 2.778. The initial velocity is zero everywhere. Thegrid resolution for the computational domain is 32 grid points per 10�m. The initial shapes of


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0 25 50 75 100 125 150 175 2000

10

20

30

40

50

Y(μ

m)

Time(ms)0 20 40 60 80 100

0

10

20

30

40

50

X(μm)

Y(μ

m)

0 25 50 75 100 125 150 175 2000

10

20

30

40

50

Y(μ

m)

Time(ms)0 20 40 60 80 100

0

10

20

30

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X(μm)

Y( μ

m)

0 25 50 75 100 125 150 175 2000

10

20

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Y( μ

m)

Time(ms)0 20 40 60 80 100

0

10

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X(μm)

Y(μ

m)

0 25 50 75 100 125 150 175 2000

10

20

30

40

50

Y( μ

m)

Time(ms)0 20 40 60 80 100

0

10

20

30

40

50

X(μm)

Y( μ

m)

Figure 6. Histories of the height of the mass centers (left) and the shape and the position of four cells (right)at t=200ms in a Poiseuille flow with different bending constants kb=5×10−10Nm, 10kb=5×10−9Nm,

100kb =5×10−8Nm, and 1000kb =5×10−7Nm (from top to bottom).

the cells we consider are circular with the swelling ratio s∗ =1. The initial positions of the masscenters of the four cells are at (25, 10), (25, 40), (65, 10), and (65, 40). The histories of the heightof the mass centers of the cells in the channel with different values of bending constant kb areshown in Figure 6. The Reynolds numbers based on the averaged velocity of the channel flowwith four cells are 2.710, 2.705, 2.685, and 2.673 for the bending constants kb, 10kb, 100kb, and1000kb, respectively.

The cells with s∗ =1 move slowly toward the central region of the bigger channel. In Figure 6,the averaged distances of the mass centers of the four cells to the central line of the channel forthe last millisecond are 9.22, 9.38, 11.10, and 12.23�m for the bending constants kb, 10kb, 100kb,and 1000kb, respectively. Thus the migration of four cells with s∗ =1 in a Poiseuille flow dependson the strength of the bending property of the cells. The shape and the position of the four cells


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0 25 50 75 100 125 150 175 2000

10

20

30

40

50

Time(ms)

Y(μ

m)

0 20 40 60 80 1000

10

20

30

40

50

X(μm)

Y(μ

m)

Figure 7. Left figure: histories of the height of the mass centers of four neutrally buoyant disks. Rightfigure: position of neutrally buoyant disks at t=200ms.

0 25 50 75 100 125 150 175 2000

10

20

30

40

50

Y(μ

m)

Time(ms)0 20 40 60 80 100

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X(μm)

Y(μ

m)

Figure 8. Left figure: histories of the height of the mass centers of four cells with s∗ =0.7. Right figure:position of four cells at t=200ms.

at t=200ms are shown in Figure 6. For the cases of the bending constant kb, 10kb, and 100kb,the four cells are not circular at t=200ms. When the bending constant is 1000kb, the four almostcircular cells behave like four neutrally buoyant disks of radius 2.8�m moving in a Poiseuille flow.Via the DLM/FD method described in [43], the averaged distance of the mass centers of the fourneutrally buoyant disks to the central line is 12.73�m, which is in good agreement with the distancefor the case of 1000kb. The results of four neutrally buoyant disks obtained by the DLM/FDmethod are shown in Figure 7. For the case of the four cells with s∗ =0.7 shown in Figure 8,these cells move much faster toward the central line of the channel. The Reynolds number basedon the averaged velocity and the channel height is Re=2.760. There is a big difference betweenthe results of s∗ =1 and those of s∗ =0.7. The cells with s∗ =1 behave like these rigid disks inthe larger channel since the wall effect is weak and the flow inertia takes over the control of thecell lateral migration. For the cells with s∗ =0.7, the effect of the flow inertia must be weaker sothat they easily move to the central region of the channel. Hence, the interplay between the walleffect, deformation, and inertia forces has strong influence on the cell lateral migration rate.

We have noticed that both four cells with s∗ =1 and the four neutrally buoyant disks becomestaggered as shown in Figures 6 and 7. One reason for having these kinds of patterns is that theinitial positions of the four cells and the four neutrally buoyant disks are not stable since particlesdo not move side by side in Newtonian fluids, the force on each particle makes them to moveaway from each other (see, e.g. [44]).

3.1.3. Many cell migration. To apply the methodology to many-cell cases, we have consideredthe cases of 50 and 100 RBCs with s∗ =0.7 in a Poiseuille flow. We have used the same valuesof kb, kl , and ks for the previous one-cell cases. The fluid domain is a 100�m×50�m rectangle.Hence, the Hct of the 50-cell case (resp., 100 cell case) is 17.24% (resp., 34.48%). The pressuredrop is set as a constant for this study so that the Reynolds number based on the averaged velocityof the Poiseuille flow without cells is 2.778. The initial velocity is zero everywhere. The gridresolution for the computational domain is 32 grid points per 10�m. The positions and shapes


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Figure 9. Shapes and positions of 50 cells at t=0, 50, 100, and 150ms.

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of cells at different times are shown in Figures 9 and 10. We have observed that the cells movetoward the center of the channel, an effect attributed to their deformability. Similar two-dimensionalresults have been observed numerically in [12, 45]. There are cell-free layers next to the walls asshown in Figures 9 and 10. In [32], the cell-free layer has been estimated to be roughly 100/Hctin cylindrical tubes with diameters between 40 and 83�m. In [12], the numerical results of thesize of the cell-free layer are in agreement with the above estimation reported in [32]. Thus forHct=17.24%, the estimation of the cell-free layer is 5.8�m (2.9�m for the Hct=34.48% case).We have computed the size of the cell-free layer by averaging the gap size between the wall andthe closest cell to the wall for the last millisecond of the simulation. Our results show that the sizes


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of cell-free layer are 6.24 and 2.94�m for the Hct=17.24 and 34.48%, respectively, which are inagreement with the estimation in [32]. The Reynolds numbers based on the averaged velocity andthe channel height for the 50-cell case and 100-cell case are Re=2.346 and 1.677, respectively.

3.2. Motion of cells in a channel with constriction

In [33], micro-channels with constriction have been studied for their application to blood plasmaseparation, which is based on the fact that rapid variation of the geometry and the deformabilityof the cells can dramatically modify the cell spatial distribution in the channel. In this section, wehave shown the computational results of a similar test problem obtained by the combination of aDLM/FD method with the spring model and immersed boundary method.

The computational domain is a 120�m×30�m rectangle. The region of constriction shown inFigure 11 has a width of 40�m and a height of 7.5�m hence the height of the narrow channel is15�m. We have considered the cases of 12 RBCs with s∗ =0.481 and used for kb, kl , and ks thevalues considered in the previous one-cell cases. The Hct is 4.74%. The initial position of the 12cells is shown in Figure 11. The pressure drop is set as a constant for this study so that the Reynoldsnumber based on the averaged velocity of the Poiseuille flow without cells and constriction is1.667. The initial velocity is zero everywhere. The grid resolution for the computational domainis 32 grid points per 10�m. In Figure 11, we have shown the position and the shape of 12 cellsat different times. The snapshots of the velocity field with 12 cells are shown in Figure 12. Thecells do recover the biconcave shape once they pass the neighborhood of the constriction in thechannel, which shows the shape memory ability of the membrane described by the spring model.

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Figure 12. Velocity field and shapes and positions of 12 cells at t=60 and 100ms.


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Figure 13. Histories of the mass centers of 12 cells.

The histories of the mass centers of 12 cells are shown in Figure 13. We can see that the cellsmove quickly toward the central region of the channel due to the presence of the constriction.The averaged flow velocity in the wider (resp., narrow) portion of the channel is U =1.70cm/s(resp., U =3.33cm/s). The Reynolds number for the wider (resp., narrow) portion of the channelis Re=0.4250 (resp., Re=0.4163).

4. CONCLUSIONS

In summary, in this article a numerical model has been tested for simulating the motion of manyRBCs in Poiseuille flows. We have obtained that the RBCs in a narrow micro-channel deformthemselves into parachute shapes and the lateral migration of cells depends on their swellingratio, bending property, and the height of the channel. For many-cell cases, our results show thatthe size of cell-free layer is in agreement with the estimation in [32]. When cells are moving ina channel with constriction, they move toward the central region of the channel quickly due tothe presence of the blockades. The numerical results in this article are quantitatively/qualitativelysimilar to experimental observations and other investigators’ findings thus show the poten-tial of this numerical algorithm for future studies of blood flow in microcirculation andmicro-channels.

ACKNOWLEDGEMENTS

The authors acknowledge the support of NSF (grants ECS-9527123, CTS-9873236, DMS-9973318, CCR-9902035, DMS-0209066, DMS-0443826, DMS-0914788). We acknowledge the helpful comments ofJames Feng, Ming-Chih Lai, and Sheldon X. Wang.

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numerical simulation of lateral migration of red blood cells in poiseuille flows

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