Tc

Sa

b

c

a

ARRA

KBCCMMC

1

tvaTphcivieabfi(fa

1d

Biochemical Engineering Journal 51 (2010) 110–116

Contents lists available at ScienceDirect

Biochemical Engineering Journal

journa l homepage: www.e lsev ier .com/ locate /be j

hree-dimensional CFD modelling of a continuous immunomagnetophoretic cellapture in BioMEMs

wati Mohantya,∗, Tobias Baierb, Friedhelm Schönfeldc

Institute of Minerals & Materials Technology (C.S.I.R.), Bhubaneswar 751013, IndiaTechnische Universität Darmstadt, Center of Smart Interfaces, Nanofluidics and Microfluidics, Petersenstr. 32, 64287 Darmstadt, GermanyFachhochschule Wiesbaden, Am Brückweg 26, 65428 Rüsselsheim, Germany

r t i c l e i n f o

rticle history:eceived 3 January 2010eceived in revised form 24 May 2010ccepted 6 June 2010

eywords:

a b s t r a c t

Separation of rare cells from blood stream using paramagnetic/superparamagnetic beads in microfluidicdevice has gained importance in recent years for early diagnosis of several critical diseases. However,the performance of immunomagnetophoretic cell sorters (ICS) crucially depends on their design andoperational conditions. Here, we present a three-dimensional CFD model based on the Navier–Stokesequations governing the fluid dynamics and continuum descriptions for the cell, bead and cell–beadcomplexes for a continuous ICS. The spatial-temporal evolution of the concentration fields are governedby convection–diffusion equations for non-magnetic cells and Nernst–Planck type equations for beads

ioMEMsell captureFDagnetophoresisagnetic beads

and cell–bead(s) complexes. The attachment rates between cells, cell–bead(s) complexes and beads arededuced from the collision probabilities which are derived by means of classical scattering theory. TheCFD model is used to investigate the performance of a generic continuous cell separation system. Sincethe cells are larger in diameter, more than one bead can get attached to the cells. Multiple beads binding to

ed inmanc

ontinuous sorting the cell has been considerwe investigate the perfor

. Introduction

Immunomagnetic separation of rare cells has gained impor-ance in bio-medical applications, primarily for early diagnosis ofarious types of serious diseases, for isolation of cells for geneticnd immunological studies as well as for regenerative medicine.he process involves mixing nano or micro-sized ferromagnetic,aramagnetic or superparamagnetic beads coated with antibodiesaving affinity for a specific type of antigens on the surface of theell, with a fluid sample containing the cells. Finally a magnetic fields applied to separate the beads and cell–bead(s) complexes. As theolume of sample to be handled is typically very small, design-ng and fabrication of such microdevices is difficult and continuousfforts are being made to improve upon these to make them suit-ble for use in lab-on-chip. A number of state-of-art reviews haveeen published [1–3], which discuss the application of magneticorce for manipulation of cells and magnetic beads in a microflu-

dic device. Several designs of micro immunomagnetic cell sortersICS) have been reported and research is on to improve upon theseor use in a continuous process. Continuous process has distinctdvantage over the batch process as it can be integrated into a lab-

∗ Corresponding author. Tel.: +91 674 2581635x235; fax: +91 674 2581637.E-mail address: swati.mohanty@gmail.com (S. Mohanty).

369-703X/$ – see front matter © 2010 Elsevier B.V. All rights reserved.oi:10.1016/j.bej.2010.06.002

this study, which has not been reported in literature till date. Exemplarily,e of Y-shaped geometry used for contacting of cells and beads.

© 2010 Elsevier B.V. All rights reserved.

on-chip system more easily, has high throughput and can be bettercontrolled.

Inokuchi et al. [4] propose a design for an on-chip separationof stem cells from peripheral blood. The mixing is first carried outin a laminated chaotic micro-mixer where the magnetic beads getattached to the target cells and then the cell–bead mixture anda buffer fluid are fed into a separator through 2 different inlets.The target cells captured by the magnetic beads migrate to thetop buffer layer due to the applied magnetic field generated bythe magnetic coil. Choi et al. [5] propose a glass microchip withmicro-channels and semi-encapsulated spiral electromagnet forefficient separation of target cells. Pekas et al. [6] designed a hybridmicromagnetic–microfluidic structure that exerts both repulsiveand attractive forces at microscale for better diversion of the tar-get particles. Xia et al. [7] have designed a micro device in whicha high gradient magnetic field concentrator is integrated into themicrofluidic channel. Target particles are efficiently pulled fromone fluid lamella to the other, flowing parallel to each other. Thetargeted particles are continuously drawn out as a separate streampreventing accumulation in the micro device and allowing contin-uous operation. A continuous cell sorter designed by Inglis et al.

[8] consists of a magnetic strip integrated to the micro-channel sothat the captured cells flow in the direction of the magnetic striprather than the direction of the main fluid flow. Rong et al. [9] havedesigned micromachined magnetic tips for microfluidic device forseparation of biological cells using magnetic beads.

ngineering Journal 51 (2010) 110–116 111

wagfapuni[mNccKhoibsimobfsiFstoibnbfliastaaocrs

fimfdznbwicipadcud

therefore be written as:

S. Mohanty et al. / Biochemical E

Mathematical modelling helps in optimal design of any deviceithout much experimentation, thus saving time, raw material

nd allowing simulating at conditions which may be quite dan-erous to carry out experimentally. Several mathematical modelsor microseparators have been reported in literature. In general,ll the models assume that there is no interaction between thearticles and no body force on the fluid. Pekas et al. [6] havesed the equation of motion, taking into consideration the mag-etic and viscous drag forces, to predict the particle trajectory

n a hybrid repulsion–attraction microseparator. Smistrup et al.10] have simulated a microfluidic channel with planar spiral

icro-electromagnets to predict the flow profile of the fluid usingavier–Stokes equation without the inertial terms and the parti-le trajectory using the Newton’s equation of motion taking intoonsideration the viscous drag force and the force due to gravity.inetic modelling of interaction between cells and magnetic beadsas been reported by Deponte et al. [11] with the assumption thatnly one bead gets attached to each cell. Kim et al. [12] have exper-mentally studied a continuous separation of T lymphocytes fromiological suspensions and computed the binding probabilities. Aample stream containing target cells and a buffer stream contain-ng magnetic beads flow side by side in a single channel. A first

agnet pulls the magnetic beads into the sample stream and a sec-nd magnet further downstream pulls the beads–cell complexesack into the buffer stream such that the target cells are separatedrom the original sample stream. Mikkelsen and Bruus [13] havetudied the motion of paramagnetic beads in a microfluidic devicen the presence of a magnetic field using continuum approximation.urlani et al. [14] have developed a model for a batch bioseparatorimilar to the design suggested by Choi et al. [5] to track the particlerajectory in the presence of a magnetic field. We [15] have devel-ped a two-dimensional model for immunomagnetic cell capturen a flow-through microfluidic device, considering more than oneead binding to a cell. The two-way coupling between the mag-etic beads and the fluid as well as the magnetic interaction haseen studied by Mikkelsen et al. [16], considering two beads forow through a microfluidic device. The magnetic interaction was

nsignificant compared to hydrodynamic interaction. Zolgharni etl. [17] have developed a two-dimensional steady state model totudy the extent of mixing as well as the particle–particle collisionhat could predict the probability of a cell being tagged. Modak etl. [18] have developed an Eulerian–Lagrangian model for a straightnd T-shaped microfluidic channel with line dipole for separationf biological cells. The model takes into consideration two-wayoupling of the fluid–particle momentum interaction. They haveeported that for large loadings of the particles the interaction wasignificant.

However, there is a lack of a suitable three-dimensional modelor predicting the cell capture and the flow of different speciesn continuous micromagnetic sorters, particularly when possibly

ore than one bead gets attached to each cell. Necessity waselt to extend the model to three-dimension so as to study theistribution of the cell, bead and cell–bead complexes in the-direction. In this paper we present a three-dimensional hydrody-amic and magnetophoretic model which explicitly accounts forinding kinetics for the formation of cell–bead(s) complexes andhich can easily be integrated into a computational fluid dynam-

cs (CFD) code. We apply the model for a specific application ofontinuous magnetic cell sorting considering a generic microflu-dic geometry used for continuous cell sorting. The model allowsredicting the concentration profiles of the unbound cells, beadsnd cell–bead(s) complexes. In this way, the model facilitates toesign a device that can efficiently separate the target cells from

omplex mixtures. Moreover, the simulation methods could besed to deduce details of the binding kinetics from experimentalata.

Fig. 1. Schematic diagram of the modeled microfluidic device for cell capture.

2. Materials and methods

In the present study we investigate immunomagnetic tagging ofcells in a Y-channel which is probably one of the most often usedmicrofluidic geometries. The Y-shaped micro-channel under studyhas a length of 1.12 cm, width (end-to-end of the arms) of 0.16 cmand a depth of 0.01 cm. Two streams containing beads and the tar-get cells are fed into the reaction channel from 2 separate inlets asshown in Fig. 1. An external magnet, which is placed at a distancefrom the channel pulls the magnetic beads into the sample streamwhere cells and beads collide and immunological tagging of thetarget cells takes place. Due to the antibody–antigen reaction, thebeads get attached to the cells. As the cells are typically much largerthan the beads, more than one bead can bind to a single cell. How-ever, the bead can bind to the cell only if it comes in contact with thefree surface of the cell. The collisions can either be due to directionalmovement caused by external force fields like gravitational or mag-netic fields or the collisions can occur through non-directionalmovement, viz. diffusion. More details about cell–bead binding aregiven in our earlier paper [15].

3. Mathematical model

A three-dimensional model was developed based on the fol-lowing assumptions: the cells and the beads have been treatedas continuum. The flow of both streams is laminar. The collisionbetween a bead and a cell results in binding of the bead to the cellwith a certain probability, which is assumed to be constant. Thesedimentation of the cells and the beads is negligible. The externalmagnetic field is created by a magnetic dipole. The fluid is a New-tonian fluid and the properties are same as that of water. The fluidflow is not affected by the motion of the beads, cell-bead(s) com-plexes or cells, but the fluid has an influence on the motion of thecells, beads and cell–bead(s) complex, i.e., a one-way coupling hasbeen considered. Note, basically all the assumptions are made inorder to focus on the essential aspects of simulating immunomag-netic cell capture. The developed model can straightforwardly beexpanded to account for more complex binding kinetics, sedimen-tation, arbitrary magnetic fields and two-way coupling betweenparticle and fluid motion.

Because of the small time constant associated with movement ofmicro particles in water, acceleration phases can safely be neglected[19]. Thus, it is assumed that the cells have the same velocity asthe fluids and beads as well as cell–bead(s) complex have a veloc-ity equal to that of fluid plus an additional velocity contributiondue to the magnetic force. The Navier–Stokes equations for incom-pressible fluid is used to model the fluid phase neglecting the bodyforce. The unsteady state continuum model for the fluid phase can

∂�u∂t

= −(�u · ∇)�u − 1�

∇P + �∇2 �u. (1)

1 nginee

T

∇wo

vc[sisioaoiidtuieGcebb

R

wt

r

Afltnd

B

wrm

m

wFim

H

e

F

wi

fif

12 S. Mohanty et al. / Biochemical E

he continuity equation is given as:

· �u = 0, (2)

here �u is the velocity vector of the fluid, � the kinematic viscosityf the fluid, P the pressure and � is the density of the fluid.

For binary collision of rigid spheres, the rate of collision per unitolume depends on the concentration of the particles colliding,haracteristic radii of the particles as well as the relative velocity20]. When the Reynolds number is small, the external forces on thephere balance the hydrodynamic forces and the relative velocitys a function of the sum of the external forces [21]. In the presenttudy, therefore, the rate of binding depends on the relative veloc-ty between the bead and the cell–bead(s) complex, concentrationf the beads and the cell–bead(s) complex. The velocity of beadsnd cell–bead(s) complex depends on the magnetic field, velocityf the fluid as well as the drag force whereas the velocity of the cells influenced by the velocity of the fluid. The difference in veloc-ty of bead and a cell-one bead complex would be basically due torag force as the surface area for a cell-one bead would be greaterhan that of a bead. Since the magnetic force depends on the vol-me of the magnetic material, the magnetic force would be equal

n both the cases where there is only one bead. Due to the differ-nce in velocity there is a possibility of binding of more than one.enerally, the binding efficiency can be expected to be less than theollision rate, i.e., not every collision necessarily leads to a bindingvent. Assuming a binding efficiency of p, the rate of binding rateetween beads and cells and beads and cell–bead(s) complexes cane written as:

n = p�CBCCnB(rB + rCnB)2|wB − wCnB|, (3)

here n = 0, N − 1, wB denotes the velocity of the beads and wCnBhe velocity of the cells with n beads bound.

The average radius of a cell with n beads bound is calculated as:

CnB = (r3C + nr3

B)1/3

. (4)

s stated above all particles are assumed to be convected with theuid flow and possibly have an additional velocity component dueo the external magnetic field. For simplicity we assume a perma-ent magnet and model the induced magnetic field (�B), using aipole approximation in cylindrical co-ordinates [21]:

� = �0m

4�r3(2 cos � r + sin � �), (5)

here m is the magnetic moment, �0 the permeability constant,the distance from the magnet, and r and � are unit vectors. Theagnetic moment is given by

= BiV

�0, (6)

here Bi is the intensity of the intrinsic or remanent magnetic field.or a Nd–Fe–B permanent magnet, Bi is given as 1.4 T [13]. Assum-ng that there is no other magnetic material around, the external

agnetic field, �H can be written as:

� =�B

�0. (7)

For a paramagnetic bead, the force acting on the bead due to thexternal magnetic field can be written as [19]:

� = 2��0r3B

+ 3( �H · ∇) �H, (8)

here is the magnetic susceptibility of the bead material and rBs the radius of the bead.

The velocity of beads or cell–bead(s) complex due to magneticeld, vmnB, can be obtained by equating the drag force with the

orce due to the magnetic field gradient [22]. The drag force exists

ring Journal 51 (2010) 110–116

due to the difference in the velocity of fluid and the magnetic par-ticle. Neglecting velocity gradients in the fluid and hydrodynamicparticle–particle interactions we use the Stokes formula for a par-ticle of radius r′ moving with a velocity, v, in a stationary fluid:

Fdrag = 6�r′v. (9)

Particles are accelerated if exposed to a magnetic field but typ-ically reach a constant velocity within microseconds. In this casethe force due to the magnetic field gradient is balanced by the dragforce on the beads or cell–bead(s) complexes. Thus from Eqs. (8)and (9) we get

vmnB = n�0r3B

3rCnB

(

+ 3

)∇ �H2. (10)

Within the continuum approximation cell transport is governedby the time dependent convection–diffusion equation:

∂CC

∂t= DC∇2CC − (�u · ∇)CC − R0. (11)

The reaction terms R0 accounts for the loss in the cell concen-tration since cells and beads ‘react’ to cell–bead complexes, cf. Eq.(3).

For the bead concentration the transport equation has to beaugmented by the flux resulting from the external magnetic field:

∂CB

∂t= DB∇2CB − (�u · ∇)CB − bB∇ · (CB �F) −

N−1∑n=0

Rn. (12)

Moreover the reaction term accounts for the loss in bead con-centration due to all possible reactions with cells and cell–bead(s)complexes.

The cell–bead(s) complex concentrations obey the same equa-tion, except that the ‘reactive’ loss is restricted to the bindingreaction CnB + bead → C(n + 1)B and additional new species are cre-ated via the reaction C(n − 1)B + bead → CnB:

∂CCnB

∂t= DCnB(∇2CCnB) − ∇ · (CCnB �u) − bCnB∇ · (CCnB

�F) + Rn−1 − Rn,

(13)

where Rn is defined in Eq. (3) and the reaction cascade is terminatedby setting, RN = 0. Generally, CC, CB and CCnB, denote the concentra-tions of the cells, beads and cell with n bead(s) per unit volume,respectively. DC, DB, and DCnB, are their respective diffusivities. Nis maximum number of beads bound to a cell and n is the numberof beads bound to the cell. The parameter bi, denotes the particlemobility and is defined as

bi = 16�ri

, (14)

where i stands for cell, bead or cell–bead(s) complex.The following boundary conditions were applied to solve the

model equations:

• At the inlet: CB = CBi; CC = CCi, CCnB = 0, |�u| = u, velocity normal tothe inlet cross-section.

• At the outlet: �CB =�CC =�CCnB = 0, P = 1 atm (abs).• At the wall: no slip.

4. Results and discussions

The model equations (1) and (11)–(13) were solved using the

commercial CFD software Fluent 6.2.16. User Defined Functions(UDFs) were written for calculating the loss of beads as well as lossand formation of cell–bead(s) complexes and linked with the Flu-ent solver. The three-dimensional geometry was meshed by 14,784hexahedral cells with 19,840 nodes. An adaptive time step with a

S. Mohanty et al. / Biochemical Engineering Journal 51 (2010) 110–116 113

Table 1Parameter values used in the present study.

rC 3.75 �m Radius of the cellrB 2.25 �m Radius of the beadCB 4 × 1014 m−3 Concentration of the beads at the inletCC 3 × 1013 m−3 Concentration of the cells at the inletp 0.1 Probability of a bead being attached to a cell or cell–bead complexes� 1000 kg/m3 Density of the fluid 1 Magnetic susceptibilityDi 1 × 10−9 m2/s Diffusion co-efficient of the cell, beads, cell–bead(s)N 4 Maximum number of beads attached to a cell�0 12.57 × 10−7 H/m Magnetic constant 1 mPa s Dynamic viscosity of the fluid�u 10−3 m/s Velocity normal to the inlet cross-section

IntenPositPositVolu

mofutBaIfifac

liidromsrovbwsl5fibob1bcf

ttttmnpwp(ci

Bi 1.4 Txm 4 × 10−3 mym 5 × 10−3 mV 1.8 × 10−8 m3

inimum time step size of 1 × 10−5 s and a maximum time stepf 0.01 s were used for the simulation. The convergence criterionor the residuals was set to 10−4 for all the species. The parameterssed in the present study are listed in Table 1. The magnetic suscep-ibility of paramagnetic beads has been taken from Mikkelsen andruus [13]. This is also in the range reported by Fonnum et al. [23],fter transforming the data given by them into dimensionless form.n order to ensure numerical stability relatively large diffusion coef-cients has been used. The values are much larger than those which

or instance result from the Stokes–Einstein equation. Consideringn average convective velocity of 0.001 m/s, and the length of thehannel is 0.01 m, the average residence time is 10 s. The diffusion

ength scale can be approximated as√

10 × 10−13 = 10−6 m, whichs much smaller than the convective length scale, clearly indicat-ng that convection dominates over diffusion. The Peclet numberefined as hu/D, is of the order of 107, which is very large. At thisegime, numerical simulations become extremely difficult becausef convergence problem. For convergence, it is necessary to haveesh Peclet number less than unity [13]. The characteristic mesh

ize is of the order of 10−5 m. Taking the velocity in the criticalegion as 10% of the inlet velocity, we get mesh Peclet number of therder of 104, which leads to unstable solution. In order to get a con-erged solution, therefore the diffusion coefficient was increasedy an order of 104, so that Peclet number was approximately unity,ithout affecting the numerical solution significantly, as has been

hown earlier. Mikkelsen and Bruss [13] also observed similar prob-em and had increased the diffusion coefficient artificially to about0 times than that predicted by Stokes–Einstein equation. Thus, thenal conclusions drawn from the model results are not dominatedy the large diffusion coefficients used. This was also observed inur earlier studies [15]. The maximum number of beads that canind to a single cell, based on the radius of the bead and the cell, is0. However, assuming that there is free space between the beadsound to a single cell, in the present study, 40% of the surface of theell is covered with the beads has been assumed, i.e., maximum ofour beads are attached to a single cell.

The simulations were carried out with different magnet posi-ions. The final position of the magnet was fixed keeping in mindhat the minimum number of beads and cell–bead(s) complex sticko the walls, as it cannot be completely avoided with the geome-ry chosen for simulation. If the magnet position is too far, proper

ixing of the cells and the beads will not be possible whereas tooear will result in accumulation of beads and cell–bead(s) com-lex near the wall. The model was simulated till a steady state

as reached, which was approximately 15 s. The fluid velocityrofile was obtained by solving Eq. (1). The transport equations11)–(13) were solved to predict the concentration profile of theell, bead and cell–bead(s) complexes. The cell-one-bead complexs first formed, followed by cell-two-bead, cell-three-bead and cell-

sity of the remanent magnetic field for Nd–Fe–B magnetion of the magnet in the x-directionion of the magnet in the y-directionme of the Nd–Fe–B magnet

four-bead. Hence there is a slight shift towards the right, for thefirst appearance of cell–bead(s) complex, as the number of beadsincreases. Fig. 2 shows the concentration profile of the beads, cellsand cell–bead(s) complexes for a steady state condition. The beadsare deflected towards the cell stream and then move along the cellstream. As the beads bind to the cells, the concentration of the beadsdecreases towards the outlet (Fig. 2a). The cell concentration alsodecreases with length (Fig. 2b). It can be seen that as the cells andbeads move from the inlet to the outlet, the concentration of thecell–bead(s) complexes increases. The beads and the cell–bead(s)complexes also move faster than the cells on the left side of themagnet, due to additional force acting on the magnetic beads. Oncrossing the magnet, the force acting on the beads is in the oppo-site direction to that of the flow, hence the velocity decreases, butonce it has moved away from the influence of the magnet, the flowis dominated by convective and diffusive forces. The concentra-tion profile of the cell-one-bead complex is shown in Fig. 2c. Thecell–bead complex is formed when cell and bead come in contact.The cell–bead complex is drawn towards the magnet resulting inmaximum concentration near the magnet. As it moves away fromthe magnet, because of the magnetic pull in the opposite directionto that of flow, the velocity decreases. The cell–bead complex alsobinds with beads to form cell-two-bead complex. This results indrop in concentration of the cell-one-bead complex. After travellingcertain distance, when the cell-one-bead complexes are out of theinfluence of the magnetic field, the velocity increases and the con-centration also increases but due to again attachment of the beadsto the cell-one-bead, the concentration decreases. The behaviour ofthe cell-two-beads is similar to cell-one-bead complex. The max-imum concentration is near the magnet and once the complex isaway from the influence of the magnet the concentration increasesand then decreases due to formation of cell-three-bead complex(Fig. 2d). The concentration profile of cell-three-bead complex isshown in Fig. 2e. The maximum concentration is near the magnetbut over a small section only. This is because the concentration ofcell-two-bead is low and hence rate of binding is low. After cross-ing the magnet and the influence of the magnet, the concentrationagain increases since the concentration of the cell-two-bead com-plex also increases. Again there is a drop in concentration due toformation of cell-three-bead complex. The formation of the cell-four-bead is the lowest but since there is no loss of cell-four-beadcomplex with attachment with more beads, the concentration ishigher than cell-three-bead complex (Fig. 2f). There is a local max-ima near the magnet but the global maximum concentration is near

the outlet as more and more cell-four-bead complexes are formedtowards the outlet.

Cells with more number of beads experience stronger mag-netic force, and hence it can be seen that cells with three beadsare drawn more towards the magnet than cells with one bead.

1 nginee

Adttflccawcvzrm

Fm

14 S. Mohanty et al. / Biochemical E

s the distance from the magnet increases, the magnetic forcesecreases, and the net velocity increases. Since it is assumedhat the maximum number of beads attached to the cell is four,here is no loss of the cell-four-bead complex. So as they moveorward the concentration increases with a maximum at the out-et. For all other cell–bead(s) complexes, the net increase in theoncentration depends on the rate of formation and loss of theell–bead(s) complexes. Since 10% binding efficiency has beenssumed, the concentration of cell–bead(s) complexes decreasesith number of beads attached to a cell. Fig. 3 shows the con-

entration of the species at the outlet along the y-direction forarious z-location. Since the magnet is placed mid-way in the-direction, the concentration of the species is almost symmet-ic along the z-direction. The maximum concentration is at theid-point in the z-direction. The concentration of the bead at

ig. 2. Concentration profile of (a) bead, (b) cell, (c) cell-one-bead, (d) cell-two-bead, (e) cagnet are (0.004, 0.005, 0.0).

ring Journal 51 (2010) 110–116

the outlet increases in the y-direction with a maxima aroundy = 0.00015 m (Fig. 3a). The concentration of the cells also increaseswith a maxima at y = 0.0001 (Fig. 3b). Since concentration of celland bead is more in the positive direction of y, rate of attach-ment is higher resulting in loss of cells as well as beads. Hence,beyond a certain value of y, there is a drop in the concentra-tion of the cells and beads. The concentration of the cell–bead(s)complexes depend on the net rate of formation and loss of thecomplexes. Hence we find that there is a steady increase in theconcentration in the positive y-direction but as the concentration

of the beads starts decreasing the rate of attachment decreases andhence the concentration in the y-direction decreases (Fig. 3c–f).Unlike, the 2-D model which assumes that there is no variationalong the z-direction and therefore predicts a flat concentrationprofile, in the present study we have been able to predict the con-

ell-three-bead and (f) cell-four-bead at steady state (m−3). The co-ordinates of the

S. Mohanty et al. / Biochemical Engineering Journal 51 (2010) 110–116 115

cell, (c

ca

5

fithtoabidl

Fig. 3. Concentration profile at the outlet along the y-direction: (a) bead, (b)

entration profile along the z-direction taking the wall effect intoccount.

. Conclusions

A three-dimensional fluid dynamic model has been developedor a continuous immunomagnetophoretic cell sorters (ICS) tak-ng into consideration binding kinetic and the attachment of morehan one bead per cell. Attachment of up to four beads per cellas been considered. Although magnetic field is assumed to bewo-dimensional, the diffusion is three-dimensional. One of thebjectives of the paper is to show that ICS can be simulated by

ugmenting standard CFD approaches and the present model cane implemented in virtually any CFD solver which allows defin-

ng user specified convection–diffusion equations. The work is aemonstration that the approach can be implemented in CFD simu-

ation and can be easily extended for a 3-D B-field. It was found that

) cell-one-bead, (d) cell-two-bead, (e) cell-three-bead and (f) cell-four-bead.

the position of the magnet was very crucial for the flow of beads andcell–bead(s) complexes. The position of the magnet was optimizedby seeing that the magnetic beads were drawn to the cell stream butdid not accumulate near the wall but flowed out towards the out-let. The number of beads can be extended to more number of beadsattached to a cell, however, the probability will decrease as the freesurface available on the cell would decrease. A simple geometryhas been chosen initially to simulate and study the flow of the dif-ferent species in the device. It was noticed that due to magneticforce the beads and cell–bead(s) complexes move towards the walland their flow is hindered by the wall. The present model can beused to study the flow behaviour of different species in an ICS and

design a device so that efficient separation of unbound cells andbound cells can be obtained. This would significantly reduce thenumber of experiments to be carried out to obtain the best design.Since the beads and cell–bead(s) complexes get deflected towardsthe magnet, one possibility is to bifurcate the micro-channel into

1 nginee

twrbbtcrt

A

StwEa

R

[

[

[

[

[

[

[

[

[

[

[

[Sci. Technol. 30 (7–9) (1995) 1169–1187.

[22] http://www.britannica.com/EBchecked/topic/357334/magnetism (accessed22 August 2009).

[23] G. Fonnum, C. Johansson, A. Molteberg, S. Mørup, E. Aksnes, Characterisation of

16 S. Mohanty et al. / Biochemical E

wo after crossing the magnet, one which is curved and the otherhich is straight. The straight channel could be the outlet for the

esidual cells whereas the curved channel could be the outlet for theeads and the cell–bead(s) complexes. The other possibility woulde to use a second magnet in the opposite side of the first magnet,owards the outlet which could pull back the beads and cell–bead(s)omplexes. Again, the outlet can be bifurcated to separate out theesidual cells and the bead and cell–bead(s) complexes. By simula-ion, the optimum position of the magnet(s) can be obtained.

cknowledgements

This work was supported by DFG-Forschergruppe FOR 516/1.t.wati Mohanty acknowledges Alexander von Humboldt Founda-ion, Bonn, Germany, for the research fellowship. Part of this workas presented in the World Congress in Computer Science and

ngineering 2008 held at San Francisco, USA. The authors gratefullycknowledge organizers for permission to reuse the material.

eferences

[1] M.A.M. Gijs, Magnetic bead handling on-chip: new opportunities for analyticalapplications, Microfluid. Nanofluid. 1 (2004) 22–40.

[2] N. Pamme, Magnetism and microfluidics, Lab Chip 6 (2006) 24–38.[3] J. Nilsson, M. Evander, B. Hammarström, T. Laurell, Review of cell and

particle trapping in microfluidic systems, Anal. Chim. Acta 649 (2) (2009)141–157.

[4] H. Inokuchi, Y. Suzuki, N. Kasagi, N. Shikazono, K. Furukawa, T. Ushida, Micromagnetic separator for stem cell sorting system, in: Proceedings of the 22ndSensor Symposium, Tokyo, October 20–21, 2005, pp. 125–128.

[5] J.-W. Choi, T.M. Liakopoulos, C.H. Ahn, An on-chip magnetic bead separatorusing spiral electromagnets with semi-encapsulated permalloy, Biosens. Bio-electron. 16 (2001) 409–416.

[6] N. Pekas, M. Granger, M. Tondra, A. Popple, M.D. Porter, Magnetic particlediverter in an integrated microfluidic format, J. Magn. Magn. Mater. 293 (2005)584–588.

[7] N. Xia, T.P. Hunt, B.T. Mayers, E. Alsberg, G.M. Whitesides, R.M. Westervelt,D.E. Ingber, Combined microfluidic-micromagnetic separation of living cells incontinuous flow, Biomed. Microdev. 8 (2006) 299–308.

ring Journal 51 (2010) 110–116

[8] D.W. Inglis, R. Riehn, R.H. Austin, J.C. Sturm, Continuous microfluidic immuno-magnetic cell separation, Appl. Phys. Lett. 85 (21) (2004) 5093–5095.

[9] R. Rong, J.-W. Choi, C.H. Ahn, An on-chip magnetic bead separator for biocellsorting, J. Micromech. Microeng. 16 (2006) 2783–2790.

10] K. Smistrup, O. Hansen, H. Bruus, M.F. Hansen, Magnetic separation inmicrofluidic systems using microfabricated electromagnets—experiments andsimulations, J. Magn. Magn. Mater. 293 (2005) 597–604.

11] S. Deponte, J. Steingroewer, C. Löser, E. Boschke, T. Bley, Biomagnetic sepa-ration of Escherichia coli by use of anion-exchange beads: measurement andmodelling of the kinetics of cell–bead interaction, Anal. Bioanal. Chem. 379(2004) 419–426.

12] J. Kim, U. Steinfeld, H.-H. Lee, H. Seidel, Development of a novel micro immune-magnetophoresis cell sorter, in: The 6th Annual IEEE Conference on Sensors,Atlanta, 2007, pp. 1081–1084.

13] C.I. Mikkelsen, H. Bruus, Microfluidic capturing-dynamics of paramagneticbead suspensions, Lab Chip 5 (2005) 1293–1297.

14] E.P. Furlani, Y. Sahoo, K.C. Ng, J.C. Wortman, T.E. Monk, A model for predictingmagnetic particle capture in a microfluidic bioseparator, Biomed. Microdev. 9(2007) 451–463.

15] T. Baier, S. Mohanty, K.S. Drese, F. Rampf, J. Kim, F. Schönfeld, Modellingimmunomagnetic cell capture in CFD, Microfluid. Nanofluid. 7 (2009) 205–216.

16] C.I. Mikkelsen, M.F. Hansen, H. Bruus, Theoretical comparison of magnetic andhydrodynamic interactions between magnetically tagged particles in microflu-idic systems, J. Magn. Magn. Mater. 293 (2005) 578–583.

17] M. Zolgharni, S.M. Azimi, M.R. Bahmanyar, W. Balachandran, A numericaldesign study of chaotic mixing of magnetic particles in a microfluidic bio-separator, Microfluid. Nanofluid. 3 (2007) 677–687.

18] N. Modak, A. Datta, R. Ganguly, Cell separation in a microfluidic channel usingmagnetic microspheres, Microfluid. Nanofluid. 6 (2009) 647–660.

19] C.I. Mikkelsen, Magnetic separation and hydrodynamic interactions inmicrofluidic systems, Ph.D. Dissertation, Technical University of Denmark,2005.

20] X. Zhang, R.H. Davis, The rate of collisions due to Brownian or gravitationalmotion of small drops, J. Fluid Mech. 230 (1991) 479–504.

21] X. Zhang, O.A. Basaran, Electric field enhanced coalescence of liquid drops, Sep.

Dynabeads (R) by magnetization measurements and Mossbauer spectroscopy,J. Magn. Magn. Mater. 293 (2005) 41–47.