Research Article A Numerical Simulation of Cell Separation ...1 $"+ ]$ 2 u + f. For the IB-LBM...

Research ArticleA Numerical Simulation of Cell Separation bySimplified Asymmetric Pinched Flow Fractionation

Jing-Tao Ma12 Yuan-Qing Xu13 and Xiao-Ying Tang13

1School of Life Science Beijing Institute of Technology Beijing 100081 China2School of Engineering and Information Technology University of New South Wales Canberra ACT 2600 Australia3Key Laboratory of Convergence Medical Engineering System and Healthcare TechnologyThe Ministry of Industry and Information Technology Beijing Institute of Technology Beijing 100081 China

Correspondence should be addressed to Yuan-Qing Xu bitxyqbiteducn

Received 15 April 2016 Accepted 11 July 2016

Academic Editor Yi Sui

Copyright copy 2016 Jing-Tao Ma et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

As a typical microfluidic cell sorting technique the size-dependent cell sorting has attracted much interest in recent years Inthis paper a size-dependent cell sorting scheme is presented based on a controllable asymmetric pinched flow by employing animmersed boundary-lattice Boltzmannmethod (IB-LBM)The geometry of channels consists of 2 upstream branches 1 transitionalchannel and 4 downstream branches (D-branches) Simulations are conducted by varying inlet flow ratio the cell size and the ratioof flux of outlet 4 to the total flux It is found that after being randomly released in one upstream branch the cells are aligned ina line close to one sidewall of the transitional channel due to the hydrodynamic forces of the asymmetric pinched flow Cells withdifferent sizes can be fed into different downstream D-branches just by regulating the flux of one D-branch A principle governingD-branch choice of a cell is obtained with which a series of numerical cases are performed to sort the cell mixture involving twothree or four classes of diameters Results show that for each case an adaptive regulating flux can be determined to sort the cellmixture effectively

1 Introduction

Sorting various categories of particles from the mixture toachieve pure sample is of great importance in biological andmedical engineering With the rapid development of micrototal analysis systems small sample volume high throughputsample processing high efficiency and precise particle frac-tionation are several representative requirements to guide thedesign of sorting scheme [1] And correspondingly a hostof particle sorting techniques have been developed in theseyears for example the fluorescence-activated cell sorting [2ndash4] magnetic-activated cell sorting [5ndash7] dielectrophoresissorting [8 9] and size-dependent sorting [10ndash12] The lastone has received a remarkable attention attributing to itspromising advantages of low cost high efficiency and beinglabel-free There are four typical size-dependent sortingmethods that are generally reported the deterministic lateraldisplacement [10 13] the pinched flow fractionation (PFF)[14ndash16] the cross-flow filtering [17] and the inertial focusingsorting [18] PFF is relatively simple because there is no extra

and specific microstructure needed in the channel and it hasbeen used to sort polymer beads [14] microparticles [19]and emulsion droplets [20] and for blood cells [21] in recentyears In these above researches an asymmetric pinched flowfractionation scheme (AsPFF) proposed experimentally firstby Takagi et al [19] is reported to perform a continuousseparation and collection for 15sim5 120583m particles it betteredthe traditional PFF remarkably while there are still someaspects that could be improved for example to perform ahydrodynamic analysis and further develop an active andcontrollable cell or particle sorter

In the present study a numerical AsPFF cell sorter modelis established with an immersed boundary-lattice Boltzmannmethod (IB-LBM) where the channel structure the flow themultiple sizes of cells and their interactions are consideredBased on the model cells with a prescribed size can bemanipulated to enter a desiredD-branch simply by regulatingthe flux of one D-branch (or the pressure of one outlet)The numerical results demonstrate that the numerical cellsorter is effective to perform an active and controllable cell

Hindawi Publishing CorporationComputational and Mathematical Methods in MedicineVolume 2016 Article ID 2564584 8 pageshttpdxdoiorg10115520162564584

2 Computational and Mathematical Methods in Medicine

sorting which suggests an improved scheme of AsPFF and isvaluable for guiding the experimental design of cell sorter onmicrofluidic chips

2 Models and Methods

21 Mathematical Models In the numerical model the fluidmotion is solved by LBM with D2Q9 lattice model Thediscrete lattice Boltzmann equation of a single relaxation timemodel is [26ndash28]

119892

119894(x + e

119894Δ119905 119905 + Δ119905) minus 119892

119894 (x 119905)

= minus

1

120591

[119892

119894 (x 119905) minus 119892eq

119894(x 119905)] + Δ119905119866119894

(1)

where 119892119894(x 119905) is the distribution function for particles of

velocity e119894at position x and time 119905Δ119905 is the time step 119892eq

119894(x 119905)

is the equilibrium distribution function 120591 is the nondimen-sional relaxation time and 119866

119894is the body force term In the

two-dimensional nine-speed (D2Q9)model [29] e119894are given

as follows

e0= (0 0)

e119894= (cos 120587 (119894 minus 1)

2

sin 120587 (119894 minus 1)2

)

ℎ

Δ119905

for 119894 = 1 to 4

e119894= (cos 120587 (119894 minus 92)

2

sin 120587 (119894 minus 92)2

)

radic2ℎ

Δ119905

for 119894 = 5 to 8

(2)

where ℎ is the lattice spacing In (1) 119892eq119894and 119866

119894are calculated

by [26 30]

119892

eq119894= 120596

119894120588[1 +

e119894sdot u119888

2

119904

+

uu (e119894e119894minus 119888

2

119904I)

2119888

4

119904

]

119866

119894= (1 minus

1

2120591

)120596

119894[

e119894minus u119888

2

119904

+

e119894sdot u119888

4

119904

119890

119894] sdot f

(3)

where 120596119894are the weights defined by 120596

0= 49 120596

119894= 19 for

119894 = 1 to 4 and 120596119894= 136 for 119894 = 5 to 8 u is the velocity of the

fluid 119888119904is the speed of sound defined by 119888

119904= ℎradic3Δ119905 and f is

the body force acting on the fluidThe relaxation time relatedto the kinematic viscosity of the fluid is in terms of

] = (120591 minus 05) 1198882119904Δ119905 (4)

Once the particle density distribution is known themacroscopical quantities including the fluid density velocityand pressure are then computed from

120588 = sum

119894

119892

119894

u =sum

119894e119894119892

119894+ 05fΔ119905120588

119901 = 120588119888

2

119904

(5)

Although the lattice Boltzmann method is original froma microscopic description of the fluid behavior the macro-scopic continuity (6) and momentum equations (7) can berecovered from it through the Chapman-Enskog multiscaleanalysis [31] Then the LBMmaybe can be viewed as a way ofsolving the macroscopic Navier-Stokes equations

120597120588

120597119905

+ nabla sdot (120588u) = 0 (6)

120597u120597119905

+ (u sdot nabla) u = minus1120588

nabla119901 + ]nabla2u + f (7)

For the IB-LBM frame the fluid motion is first solvedby LBM then the position of immersed boundary can beupdated within one-time step of Δ119905 through [32]

U (119904 119905) = intΩ

u (x 119905) 119863 (x minus X (119904 119905)) 119889x

120597X120597119905

= U (119904 119905) (8)

where X(119904 119905) is the position of the cell membrane 119904 at time119905 U(119904 119905) is the membrane velocity and u(x 119905) is the fluidvelocity119889x is the lattice side lengthΩ is the nearby area of themembrane defined by the Delta function119863(x minus X) [33ndash35]

119863 (x minus X) =119899

prod

119894=1

120575 (x119894minus X119894) (9)

where

120575 (x119894minus X119894) =

3 minus 2

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

+radic1 + 4

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

minus 4 (x119894minus X119894)

2

8

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

le 1

5 minus 2

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

minusradicminus7 + 12

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816


2

8

1 lt

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

le 2

0

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

gt 2

(10)

In (9) and (10) 119899 denotes the total dimension of themodel The fluid-structure-interaction is enforced by the fol-lowing equation [27 32 33 36]

f (x 119905) = intΓ

F (119904 119905) 119863 (x minus X (119904 119905)) 119889119904 (11)

where F(119904 119905) is Lagrangian force acting on the ambient fluidby the cell membrane In the present study the cell model isproposed as

F = F119897minus F119887+ F119904+ F119890 (12)

Computational and Mathematical Methods in Medicine 3

where F119897is the tensile force F

119887is the bending force F

119904is

the normal force on the membrane which controls the cellincompressibility and F

119890is the membrane-wall extrusion

acting on the cell The four force components are [33 37ndash39]

F119897=

120597

120597119904

[119870

119897(

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597X (119904 119905)120597119904

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus 1)

120597X (119904 119905)120597119904

] (13)

F119887= 119870

119887

120597

4X (119904 119905)120597119904

4

(14)

F119904= 119870

119904

119878 minus 119878

0

119878

0

n (15)

F119890=

119870

119890

X (119904 119905) minus X119908(

1003816

1003816

1003816

1003816

X (119904 119905) minus X1199081003816

1003816

1003816

1003816

)

3

1003816

1003816

1003816

1003816

X (119904 119905) minus X1199081003816

1003816

1003816

1003816

le 119903

119888

0

1003816

1003816

1003816

1003816

X (119904 119905) minus X1199081003816

1003816

1003816

1003816

gt 119903

119888

(16)

where 119870119897 119870119887 119870119904 and 119870

119890are the constant coefficients for the

corresponding force components In (15) 119878 is the evolvingcell area 119878

0is the reference cell area and n is unit normal

vector pointing to fluid In (16)X119908is the position of the vessel

wall and 119903119888is the cut-off distance of the effective scope in the

membrane-wall interaction

22 Physical Model and Simulation Setup The geometrymodel of for cell sorting is illustrated in Figure 1 whichconsists of 2 upstream branches (U-branches) 1 transi-tional channel and 4 downstream branches (D-branches)The U-branches and D-branches branches are labeled withthe numbers as well as the corresponding inlets and outletsThe two U-branches are perpendicular and symmetricalabout the center line of the transitional channel The transi-tional channel connects the U-branches and a circular bufferarea which assembles the entrances of the four D-branchesThe D-branches 1 and 4 are straight while 2 and 3 are foldedfor the convenience to conduct the boundary condition ofoutlets 1 and 4 are also symmetrical about the center lineof the transitional channel as well as 2 and 3 The entirelength 119909

0and width 119910

0of device are 458120583m and 400120583m

respectivelyThewidth of inlet 1 and inlet 2119908119894is 7071120583mThe

width of pinched segment 1199080is 30 120583mThe width of outlet 1

outlet 4 and unfolded part of outlets 2 and 3119908119887is 26120583mThe

width of folded part of outlets 2 and 3119908119890is 23 120583m119876 = Δ119901119877

is defined as [19] where119876 is the flux of a D-branch Δ119901 is thepressure difference between the buffer center and the outletand 119877 is the flow resistance produced by the microchannelIn order to allocate the flow averagely for all the D-branchesunder the same pressure boundary conditions 119877s in all D-branches should be equal A way to make 119877 be equal isdescribed as two steps First set the pressure of all outlet tobe the same Second change the length of the folded part ofD-branches 2 and 3 until the stable flows of all outlets areequal When sorting different size of cells set the pressureof outlets 1 2 and 3 to be the same while the pressure ofoutlet 4 is regulatable and the flows of D-branches can bereallocated by altering the outlet pressure To quantify the thecapacity of the reallocation of flow by regulating the flow ofoutlet 4 we define 120573 = 119876out4(sum

4

119894=1119876out119894) where bigger 120573

means bigger flow through outlet 4 and smaller flow through1 2 and 3 In addition since the flow resistance 119877 in each D-branch is the same the flow 119876 is in proportion to Δ119875 thatis regulation of flow can be simply realized by regulating thepressure difference this means that 120573 also can be defined asΔ119875

4(sum

4

119894=1Δ119875

119894)

3 Results and Discussion

31 Validation The method and model are validated care-fully here by performing a simulation of flow past a stationarycircular cylinderThis simulation is carried out by employingIB-LBM model The computational domain is shown inFigure 2 The length 119871 and width 119867 of the computationaldomain are 1000 and 800 respectively The center point ofcylinder is located at 119909 = 301 and 119910 = 401 and the diameterof cylinder 119863 = 40 The cylinder is discretized into a seriesof points and the spacing between two adjacent points is06 The cylinder is handled by utilizing immersed boundarymethod (IB) and the feedback-force principle is adopted tocompute the force density on the cylinder which is describedas [22 40]

F (x119904 119905) = 120572

1int

119905

0

(u (x119904 119905) minus U (x

119904 119905)) 119889119905

+ 120572

2(u (x119904 119905) minus U (x

119904 119905))

(17)

where F(x119904 119905) denotes the interaction force between the fluid

and the immersed boundary (cylinder) 1205721and 120572

2are large

negative free constants u(x119904 119905) is the fluid velocity obtained

by interpolation at the IB and U(x119904 119905) is the velocity of the

cylinder expressed by U(x119904 119905) = 119889x

119904119889119905 Here U(x

119904 119905) equals

0 because cylinder is stationary In this case the ratio of lengthof the recirculation zone and cylinder diameter 119871

119908 the drag

force coefficient 119862119889(18) the lift force coefficient 119862

119897(19) and

the Strouhal number 119878119905are calculated at Reynolds numbers

40 and 100

119862

119889=

119865

119863

05120588119880

2

infin119863

(18)

119862

119897=

119865

119871

05120588119880

2

infin119863

(19)

The results are shown in Table 1 As shown in Table 1 thepresent results show close agreements with the general resultsreported by other literatures This means the IB-LBM modeladopted in present paper is accurate enough

32 Determination of the Inlet Flow Ratio 120572 In order toactualize the pinched flow to sort cells it is necessary toestablish an appropriate pinched segment in the transitionalchannel which is able to lead all cells to move along withthe lower sidewall of the transitional channelThere are threeaspects for establishing the pinched segment First the width119908

0of the transitional channel is better to set as 13sim15 times as

the largest diameter of the cells since it has been proved thata wider119908

0can reduce the fraction effect of pinched flow [14]

Second the length of the transitional channel is suggested


Inlet 1

Inlet 2

Outlet 4

Outlet 1

Outlet 2

Outlet 3

Qin1

Qin2

Qout1

Qout2

Qout3

Qout4

y0

x0

wb

wb

wb

wb

we

we

wp

wi

wi

w0

Figure 1 The basic schematic structure of the simulated device

L

H

InletOutlet

Solid wall

Solid wall

DH2

H203L

y

x u

Figure 2 The computational domain for flow past a stationary cir-cular cylinder

Table 1 Comparison of the flow characteristics

Literatures Re = 40 Re = 100119871

119908119862

119889119862

119889119862

119897119878

119905

Present 240 157 139 plusmn035 0160Reference [22] 259 158 139 plusmn035 0160Reference [23] 231 157 136 plusmn034 0163Reference [24] 235 166 138 plusmn034 0170Reference [25] 240 157 140 plusmn034 0162

to set as 2 times as 1199080 a too long transitional channel

may result in central tendency of the flexible cells whichis unfavourable to control the cells to move along with thelower sidewall Finally the inlet flow ratio 120572 = 119876in1 119876in2is also important to achieve the effective cell sorting To geta proper 120572 a set of numerical cases are performed by setting120572 = 18 16 14 12 1 2 4 6 8 and 10 where 20 cells with

Average positionCenter position

190

192

194

196

198

200

202

204

Cel

l cen

ter p

ositi

on (120583

m)

1 104 1 2 1 1 1 1 2 1 4 1 6 1 86 18 1120572

Figure 3 8120583m cell positions in pinched segment at different inletflow ratio

8 120583mdiameter (the smallest size) are initialized and randomlyplaced in the U-branch 2 to test the function of the pinchedflow The cell center positions at the end of the transitionalchannel are recorded and shown in Figure 3

As shown in Figure 3 the cell center positions whenleaving the pinched segment drop with the increase of 120572and finally they reach a relatively steady state when 120572 gt 6


Although 120572 = 8 and 120572 = 10 seem to be much better thismeans much higher shear stress which may do damage tothe cells Therefore 120572 = 6 is the choice for the present study

33 Effect of 120573 and Cell Size on D-Branch Choice In ourconsideration specific multiple classes of cells with differentsizes can be sorted if every class enters a D-branch In thissection the parameter 120573 and the cell size are regulated tomanipulate a specific-diameter cell to enter one D-branchand a series of numerical cases are performed to exhibit therelation of 120573 the cell size and the choice of D-branch To setup the numerical model 120573 is regulated from 01 to 09 withan increment of 01 Cells with the same initial diameter arereleased into U-branch 2 For each case of 120573 four sizes ofcell diameter are chosen as 8 120583m 16 120583m 20120583m and 24120583mto make clear which D-branch a specific diameter of cellsprefers to enter In order to eliminate the possible effectof the initial position of the cell to the D-branch choicein each case three randomly placed cells are released intothe U-branch and all the D-branch choices are taken intoaccount

A D-branch choice for a rigid circular particle can bepredicted by the following experimental equations [19]

119908

0lowast (1 minus 120573)

119873

119861minus 1

(119873 minus 1) lt

119863

2

lt

119908


119873

119861minus 1

119873

(119873 = 1 2 3)

(20)

119863

2

gt

119908


119873

119861minus 1

(119873 minus 1)

(119873 = 4)

(21)

where 1199080is the width of pinched segment as marked in

Figure 1 120573 is the outflow ratio at outlet 4119873119861is the total num-

ber of outlets and 119863 is the particle diameter According tothe above two equations the particle will enter the119873th (119873 =1 2 3 4) D-branch if 119863 ranges in the scope which can bedescribed with (20) or (21) where (20) is for 119873 = 1 2 or3 and (21) is only for119873 = 4

The predicted and numerical results of the choice ofD-branch which is related to the cell diameter and 120573 areexhibited in Figure 4 In these results 11 numerical resultsout of 68 are found not to be consistent to the predictedresults which generally occur at the transition where the cellhas approximate probability to enter two neighbouring D-branches Amost possible reason to result in the 11 differencesis the predicted results are for rigid particles while cells areflexible

According to the results by regulating 120573 the 8 120583mand 16 120583m cells can be sent into any one of all four D-branches and some snapshots of the D-branch choice of16 120583m cell are displayed in Figure 5 By contrast the 20120583mand 24 120583m cells can select one of three D-branches labeled2 3 and 4 and the 20120583m cell snapshots are shown inFigure 6 The results indicate that by simply regulatingthe flux of one D-branch cells with the diameters ranging

Outlet 1 (simu)Outlet 2 (simu)Outlet 3 (simu)Outlet 4 (simu)

Outlet 1 (anal)Outlet 2 (anal)Outlet 3 (anal)Outlet 4 (anal)

8

16

20

24

01 02 03 04 0906 07 08 10 05120573

Cel

l dia

met

er (120583

m)

Figure 4 Comparison of simulation and predicted outflow posi-tion

from 8 to 24120583m can be manipulated to enter different D-branches which gives us an inspiration to sort cells withdifferent sizes if they enter different D-branches at a given120573

34 Size-Dependent Cell Sorting As discussed in Section 33cells with different diameters can be manipulated to choosea desired D-branch at a proper 120573 this gives us a potentialscheme for sorting cell mixture with different sizes if thecell-cell interaction is not present that is all cells in mixtureare discrete In this section a continuous size-dependent cellsorting is proposed based on the regulation of 120573 Accordingto Figure 4 it is clear which D-branch a certain cell will enterat a specific 120573 therefore two sizes of cells are sorted oncethey enter different D-branches For example at 120573 = 01the 8 120583m cell can be sorted from the 20 or 24 120583m cell since8 120583m will enter D-branch 1 while the latter two will enterD-branch 2 and the same result will happen if the 8 120583mcell is replaced by 16 120583m cell Some corresponding snapshotsare shown as in Figures 7(a) and 7(b) By this means at120573 = 04 it can be predicted that three sizes of cell can besorted they are 8 16 and 20120583mcells or 8 16 and 24120583mcellsTwo snapshots of the two cases are displayed as in Figures7(c) and 7(d) respectively Especially at 120573 = 06 the 8 16 20and 24120583m are predicted to enter four different D-branchesand the numerical experiment result validates this actually asexhibited in Figure 7(e)


000

3

000

6

000

9

001

2

001

5

001

8

002

1

002

4

002

7

003

0

003

3

000

0

U

(a)0

000

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3U

(b)

000

0

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3

U

(c)

000

00

006

000

90

012

001

50

018

002

10

024

002

70

030

003

3

000

3

U

(d)

Figure 5 The 16 120583m cell outflow positions at different outflow ratios (a) 01 (b) 03 (c) 07 and (d) 09

000

3

000

6

000

9

001

2

001

5

001

8

002

1

002

4

002

7

003

0

003

3

000

0

U

(a)

000

0

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3

U

(b)

000

0

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3

U

(c)

000

00

006

000

90

012

001

50

018

002

10

024

002

70

030

003

3

000

3

U

(d)


4 Summary and Conclusion

A size-dependent cell sorting model with an asymmet-ric pinched flow is investigated numerically by immersedboundary-lattice Boltzmann method In the model threeaspects are summarized as the following First the geometryof the channels is designed specially according to the effectivecell sorting where the size of the transitional channel forcontrolling the pinched segment is discussed in detail Sec-ond the parameters 120572 and 120573 are defined respectively for theflux ratio of the two inlets and the flux proportion of outlet4 in all outlets 120572 = 6 is considered as a proper value toprepare for the cell sorting based on which the regulationof 120573 can manipulate cells with different diameters to enter

different D-branches Finally four sizes of cells are takeninto account to exhibit the capacity of cell sorting and therelations of the regulation flux the cell size and the choice ofD-branch are analyzed systematically The simulation resultsindicate that cells with different diameters can be successfullysorted into different D-branches this evinces that the modelwe established is effective which can provide a directivereference for the design of microfluidic chip for sortingmultiple sizes of cells or particles

Competing Interests

The authors declare that they have no competing interests


003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6

U

(a)

003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6U

(b)

003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6

U

(c)

000

00

006

000

90

012

001

50

018

002

10

024

002

70

030

003

3

000

3

U

(d)

003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6

U

(e)

Figure 7 Separation of different-sized cells (a) separation of 8 120583m and 20 120583m cells at 120573 = 01 (b) separation of 16 120583m and 24 120583m cells at120573 = 02 (c) separation of 8 120583m 16 120583m and 20120583m cells at 120573 = 04 (d) separation of 8 120583m 16120583m and 24 120583m cells at 120573 = 05 and (e) separationof 8 120583m 16 120583m 20 120583m and 24 120583m cells at 120573 = 06

Acknowledgments

Thiswork is supported by theNational Natural Science Foun-dation of China (no 81301291) and the Beijing Higher Educa-tion Young Elite Teacher Project (no YETP1208)

References

[1] J Autebert B Coudert F-C Bidard et al ldquoMicrofluidic aninnovative tool for efficient cell sortingrdquo Methods vol 57 no3 pp 297ndash307 2012

[2] S H Cho C H Chen F S Tsai J M Godin and Y-HLo ldquoHuman mammalian cell sorting using a highly integratedmicro-fabricated fluorescence-activated cell sorter (120583FACS)rdquoLab on a Chip vol 10 no 12 pp 1567ndash1573 2010

[3] V E Gallardo and M Behra ldquoFluorescent activated cell sorting(FACS) combined with gene expression microarrays for tran-scription enrichment profiling of zebrafish lateral line cellsrdquoMethods vol 62 no 3 pp 226ndash231 2013

[4] I Van Brussel R Ammi M Rombouts et al ldquoFluorescent acti-vated cell sorting an effective approach to study dendritic cellsubsets in human atherosclerotic plaquesrdquo Journal of Immuno-logical Methods vol 417 pp 76ndash85 2015

[5] H Lee A M Purdon and R M Westervelt ldquoManipulationof biological cells using a microelectromagnet matrixrdquo AppliedPhysics Letters vol 85 no 6 pp 1063ndash1065 2004

[6] K Schriebl G Satianegara A Hwang et al ldquoSelective removalof undifferentiated human embryonic stem cells usingmagneticactivated cell sorting followed by a cytotoxic antibodyrdquo TissueEngineering Part A vol 18 no 9-10 pp 899ndash909 2012

[7] G Welzel D Seitz and S Schuster ldquoMagnetic-activated cellsorting (MACS) can be used as a large-scale method forestablishing zebrafish neuronal cell culturesrdquo Scientific Reportsvol 5 article 7959 2015

[8] A Valero T Braschler N Demierre and P Renaud ldquoA minia-turized continuous dielectrophoretic cell sorter and its applica-tionsrdquo Biomicrofluidics vol 4 no 2 Article ID 022807 2010

[9] H J Song J M Rosano Y Wang et al ldquoContinuous-flowsorting of stem cells and differentiation products based on die-lectrophoresisrdquo Lab on a Chip vol 15 no 5 pp 1320ndash1328 2015

[10] J V Green M Radisic and S K Murthy ldquoDeterministiclateral displacement as a means to enrich large cells for tissueengineeringrdquoAnalytical Chemistry vol 81 no 21 pp 9178ndash91822009

[11] J Sun C LiuM Li et al ldquoSize-based hydrodynamic rare tumorcell separation in curved microfluidic channelsrdquo Biomicroflu-idics vol 7 no 1 Article ID 011802 2013

[12] S Song M S Kim J Lee and S Choi ldquoA continuous-flowmicrofluidic syringe filter for size-based cell sortingrdquo Lab on aChip vol 15 no 5 pp 1250ndash1254 2015

[13] J McGrath M Jimenez and H Bridle ldquoDeterministic lateraldisplacement for particle separation a reviewrdquo Lab on a Chip-Miniaturisation for Chemistry and Biology vol 14 no 21 pp4139ndash4158 2014

[14] M Yamada M Nakashima and M Seki ldquoPinched flow frac-tionation continuous size separation of particles utilizing alaminar flow profile in a pinched microchannelrdquo AnalyticalChemistry vol 76 no 18 pp 5465ndash5471 2004

[15] A L Vig and A Kristensen ldquoSeparation enhancement inpinched flow fractionationrdquo Applied Physics Letters vol 93 no20 Article ID 203507 2008

[16] C Cupelli T Borchardt T Steiner N Paust R Zengerle andM Santer ldquoLeukocyte enrichment based on amodified pinchedflow fractionation approachrdquo Microfluidics and Nanofluidicsvol 14 no 3-4 pp 551ndash563 2013

[17] Q Wei Y-Q Xu F-B Tian T-X Gao X-Y Tang and W-HZu ldquoIB-LBM simulation on blood cell sorting with a micro-fence structurerdquo Bio-Medical Materials and Engineering vol 24no 1 pp 475ndash481 2014

[18] XWang and I Papautsky ldquoSize-basedmicrofluidic multimodalmicroparticle sorterrdquo Lab on a Chip vol 15 no 5 pp 1350ndash13592015

[19] J Takagi M Yamada M Yasuda and M Seki ldquoContinuousparticle separation in a microchannel having asymmetrically


arranged multiple branchesrdquo Lab on a Chip vol 5 no 7 pp778ndash784 2005

[20] H Maenaka M Yamada M Yasuda andM Seki ldquoContinuousand size-dependent sorting of emulsion droplets using hydro-dynamics in pinched microchannelsrdquo Langmuir vol 24 no 8pp 4405ndash4410 2008

[21] J T Ma Y Q Xu F B Tian and X Y Tang ldquoIB-LBM studyon cell sorting by pinched flow fractionationrdquo Bio-MedicalMaterials and Engineering vol 24 no 6 pp 2547ndash2554 2014

[22] D V Le B C Khoo and K M Lim ldquoAn implicit-forcingimmersed boundary method for simulating viscous flows inirregular domainsrdquo Computer Methods in Applied Mechanicsand Engineering vol 197 no 25ndash28 pp 2119ndash2130 2008

[23] JWu and C Shu ldquoImplicit velocity correction-based immersedboundary-lattice Boltzmannmethod and its applicationsrdquo Jour-nal of Computational Physics vol 228 no 6 pp 1963ndash19792009

[24] Z Wang J Fan and K Cen ldquoImmersed boundary method forthe simulation of 2D viscous flow based on vorticity-velocityformulationsrdquo Journal of Computational Physics vol 228 no 5pp 1504ndash1520 2009

[25] S K Kang and Y A Hassan ldquoA comparative study of direct-forcing immersed boundary-lattice Boltzmannmethods for sta-tionary complex boundariesrdquo International Journal for Numer-ical Methods in Fluids vol 66 no 9 pp 1132ndash1158 2011

[26] ZGuoCG Zheng andBC Shi ldquoDiscrete lattice effects on theforcing term in the lattice Boltzmann methodrdquo Physical ReviewE vol 65 no 4 Article ID 046308 6 pages 2002

[27] F-B Tian H Luo L Zhu J C Liao and X-Y Lu ldquoAnefficient immersed boundary-lattice Boltzmann method forthe hydrodynamic interaction of elastic filamentsrdquo Journalof Computational Physics vol 230 no 19 pp 7266ndash7283 2011

[28] H-B Deng Y-Q Xu D-D Chen H Dai J Wu and F-BTian ldquoOn numerical modeling of animal swimming and flightrdquoComputational Mechanics vol 52 no 6 pp 1221ndash1242 2013

[29] Y H Qian D Humieres and P Lallemand ldquoLattice BGKmodels for NavierStokes equationrdquo Europhysics Letters vol 17article 479 1992

[30] D-K Sun D Jiang N Xiang K Chen and Z-H Ni ldquoAnimmersed boundary-lattice boltzmann simulation of particlehydrodynamic focusing in a straight microchannelrdquo ChinesePhysics Letters vol 30 no 7 Article ID 074702 2013

[31] S Chapman and T G Cowling The Mathematical Theory ofNon-Uniform Gases An Account of the Kinetic Theory of Vis-cosity Thermal Conduction and Diffusion in Gases CambridgeUniversity Press 1991

[32] C S Peskin ldquoThe immersed boundary methodrdquo Acta Numer-ica vol 11 pp 479ndash517 2002

[33] Y-Q Xu F-B Tian and Y-L Deng ldquoAn efficient red blood cellmodel in the frame of IB-LBMand its applicationrdquo InternationalJournal of Biomathematics vol 6 no 1 Article ID 1250061 2013

[34] Y Q Xu X Y Tang F B Tian Y H Peng Y Xu and Y J ZengldquoIB-LBM simulation of the haemocyte dynamics in a stenoticcapillaryrdquo Computer Methods in Biomechanics and BiomedicalEngineering vol 17 no 9 pp 978ndash985 2014

[35] X Yang X Zhang Z Li andG-WHe ldquoA smoothing techniquefor discrete delta functions with application to immersedboundary method in moving boundary simulationsrdquo Journal ofComputational Physics vol 228 no 20 pp 7821ndash7836 2009

[36] Y Cheng and H Zhang ldquoImmersed boundary method andlattice Boltzmann method coupled FSI simulation of mitralleaflet flowrdquoComputers amp Fluids vol 39 no 5 pp 871ndash881 2010

[37] Y Q Xu F B Tian H J Li and Y L Deng ldquoRed bloodcell partitioning and blood flux redistribution in microvascularbifurcationrdquo Theoretical and Applied Mechanics Letters vol 2no 2 Article ID 024001 2012

[38] F-B Tian ldquoRole of mass on the stability of flagflags in uniformflowrdquo Applied Physics Letters vol 103 no 3 Article ID 0341012013

[39] F-B Tian H X Luo L D Zhu and X-Y Lu ldquoCouplingmodes of three filaments in side-by-side arrangementrdquo Physicsof Fluids vol 23 no 11 Article ID 111903 2011

[40] W-X Huang and H J Sung ldquoAn immersed boundary methodfor fluid-flexible structure interactionrdquo Computer Methods inApplied Mechanics and Engineering vol 198 no 33ndash36 pp2650ndash2661 2009

Submit your manuscripts athttpwwwhindawicom

Stem CellsInternational

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014


MEDIATORSINFLAMMATION

of


Behavioural Neurology

EndocrinologyInternational Journal of



Disease Markers


BioMed Research International

OncologyJournal of



Oxidative Medicine and Cellular Longevity


PPAR Research

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Immunology ResearchHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

ObesityJournal of



Computational and Mathematical Methods in Medicine

OphthalmologyJournal of


Diabetes ResearchJournal of



Research and TreatmentAIDS


Gastroenterology Research and Practice


Parkinsonrsquos Disease

Evidence-Based Complementary and Alternative Medicine

Volume 2014Hindawi Publishing Corporationhttpwwwhindawicom


sorting which suggests an improved scheme of AsPFF and isvaluable for guiding the experimental design of cell sorter onmicrofluidic chips

2 Models and Methods

21 Mathematical Models In the numerical model the fluidmotion is solved by LBM with D2Q9 lattice model Thediscrete lattice Boltzmann equation of a single relaxation timemodel is [26ndash28]

119892

119894(x + e

119894Δ119905 119905 + Δ119905) minus 119892

119894 (x 119905)

= minus

1

120591

[119892

119894 (x 119905) minus 119892eq

119894(x 119905)] + Δ119905119866119894

(1)

where 119892119894(x 119905) is the distribution function for particles of

velocity e119894at position x and time 119905Δ119905 is the time step 119892eq

119894(x 119905)

is the equilibrium distribution function 120591 is the nondimen-sional relaxation time and 119866

119894is the body force term In the

two-dimensional nine-speed (D2Q9)model [29] e119894are given

as follows

e0= (0 0)

e119894= (cos 120587 (119894 minus 1)

2

sin 120587 (119894 minus 1)2

)

ℎ

Δ119905

for 119894 = 1 to 4

e119894= (cos 120587 (119894 minus 92)

2

sin 120587 (119894 minus 92)2

)

radic2ℎ

Δ119905

for 119894 = 5 to 8

(2)

where ℎ is the lattice spacing In (1) 119892eq119894and 119866

119894are calculated

by [26 30]

119892

eq119894= 120596

119894120588[1 +

e119894sdot u119888

2

119904

+

uu (e119894e119894minus 119888

2

119904I)

2119888

4

119904

]

119866

119894= (1 minus

1

2120591

)120596

119894[

e119894minus u119888

2

119904

+

e119894sdot u119888

4

119904

119890

119894] sdot f

(3)

where 120596119894are the weights defined by 120596

0= 49 120596

119894= 19 for

119894 = 1 to 4 and 120596119894= 136 for 119894 = 5 to 8 u is the velocity of the

fluid 119888119904is the speed of sound defined by 119888

119904= ℎradic3Δ119905 and f is

the body force acting on the fluidThe relaxation time relatedto the kinematic viscosity of the fluid is in terms of

] = (120591 minus 05) 1198882119904Δ119905 (4)

Once the particle density distribution is known themacroscopical quantities including the fluid density velocityand pressure are then computed from

120588 = sum

119894

119892

119894

u =sum

119894e119894119892

119894+ 05fΔ119905120588

119901 = 120588119888

2

119904

(5)

Although the lattice Boltzmann method is original froma microscopic description of the fluid behavior the macro-scopic continuity (6) and momentum equations (7) can berecovered from it through the Chapman-Enskog multiscaleanalysis [31] Then the LBMmaybe can be viewed as a way ofsolving the macroscopic Navier-Stokes equations

120597120588

120597119905

+ nabla sdot (120588u) = 0 (6)

120597u120597119905

+ (u sdot nabla) u = minus1120588

nabla119901 + ]nabla2u + f (7)

For the IB-LBM frame the fluid motion is first solvedby LBM then the position of immersed boundary can beupdated within one-time step of Δ119905 through [32]

U (119904 119905) = intΩ

u (x 119905) 119863 (x minus X (119904 119905)) 119889x

120597X120597119905

= U (119904 119905) (8)

where X(119904 119905) is the position of the cell membrane 119904 at time119905 U(119904 119905) is the membrane velocity and u(x 119905) is the fluidvelocity119889x is the lattice side lengthΩ is the nearby area of themembrane defined by the Delta function119863(x minus X) [33ndash35]

119863 (x minus X) =119899

prod

119894=1

120575 (x119894minus X119894) (9)

where

120575 (x119894minus X119894) =

3 minus 2

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

+radic1 + 4

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816


2

8

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

le 1

5 minus 2

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

minusradicminus7 + 12

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816


2

8

1 lt

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

le 2

0

1003816

1003816

1003816

1003816

x119894minus X119894

1003816

1003816

1003816

1003816

gt 2

(10)

In (9) and (10) 119899 denotes the total dimension of themodel The fluid-structure-interaction is enforced by the fol-lowing equation [27 32 33 36]

f (x 119905) = intΓ

F (119904 119905) 119863 (x minus X (119904 119905)) 119889119904 (11)

where F(119904 119905) is Lagrangian force acting on the ambient fluidby the cell membrane In the present study the cell model isproposed as

F = F119897minus F119887+ F119904+ F119890 (12)




119904is




F119897=

120597

120597119904

[119870

119897(

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597X (119904 119905)120597119904

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus 1)

120597X (119904 119905)120597119904

] (13)

F119887= 119870

119887

120597

4X (119904 119905)120597119904

4

(14)

F119904= 119870

119904

119878 minus 119878

0

119878

0

n (15)

F119890=

119870

119890

X (119904 119905) minus X119908(

1003816

1003816

1003816

1003816

X (119904 119905) minus X1199081003816

1003816

1003816

1003816

)

3

1003816

1003816

1003816

1003816

X (119904 119905) minus X1199081003816

1003816

1003816

1003816

le 119903

119888

0

1003816

1003816

1003816

1003816

X (119904 119905) minus X1199081003816

1003816

1003816

1003816

gt 119903

119888

(16)

where 119870119897 119870119887 119870119904 and 119870








0and width 119910







4

119894=1119876out119894) where bigger 120573


4(sum

4

119894=1Δ119875

119894)



F (x119904 119905) = 120572

1int

119905

0

(u (x119904 119905) minus U (x

119904 119905)) 119889119905

+ 120572

2(u (x119904 119905) minus U (x

119904 119905))

(17)



2are large




119904119889119905 Here U(x

119904 119905) equals


119908 the drag


119897(19) and


40 and 100

119862

119889=

119865

119863

05120588119880

2

infin119863

(18)

119862

119897=

119865

119871

05120588119880

2

infin119863

(19)








Inlet 1

Inlet 2

Outlet 4

Outlet 1

Outlet 2

Outlet 3

Qin1

Qin2

Qout1

Qout2

Qout3

Qout4

y0

x0

wb

wb

wb

wb

we

we

wp

wi

wi

w0


L

H

InletOutlet

Solid wall

Solid wall

DH2

H203L

y

x u




119908119862

119889119862

119889119862

119897119878

119905





190

192

194

196

198

200

202

204

Cel

l cen

ter p

ositi

on (120583

m)

1 104 1 2 1 1 1 1 2 1 4 1 6 1 86 18 1120572








119908


119873

119861minus 1

(119873 minus 1) lt

119863

2

lt

119908


119873

119861minus 1

119873

(119873 = 1 2 3)

(20)

119863

2

gt

119908


119873

119861minus 1

(119873 minus 1)

(119873 = 4)

(21)








8

16

20

24

01 02 03 04 0906 07 08 10 05120573

Cel

l dia

met

er (120583

m)





000

3

000

6

000

9

001

2

001

5

001

8

002

1

002

4

002

7

003

0

003

3

000

0

U

(a)0

000

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3U

(b)

000

0

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3

U

(c)

000

00

006

000

90

012

001

50

018

002

10

024

002

70

030

003

3

000

3

U

(d)


000

3

000

6

000

9

001

2

001

5

001

8

002

1

002

4

002

7

003

0

003

3

000

0

U

(a)

000

0

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3

U

(b)

000

0

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3

U

(c)

000

00

006

000

90

012

001

50

018

002

10

024

002

70

030

003

3

000

3

U

(d)





Competing Interests



003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6

U

(a)

003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6U

(b)

003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6

U

(c)

000

00

006

000

90

012

001

50

018

002

10

024

002

70

030

003

3

000

3

U

(d)

003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6

U

(e)


Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of



















119904is




F119897=

120597

120597119904

[119870

119897(

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

120597X (119904 119905)120597119904

1003816

1003816

1003816

1003816

1003816

1003816

1003816

1003816

minus 1)

120597X (119904 119905)120597119904

] (13)

F119887= 119870

119887

120597

4X (119904 119905)120597119904

4

(14)

F119904= 119870

119904

119878 minus 119878

0

119878

0

n (15)

F119890=

119870

119890

X (119904 119905) minus X119908(

1003816

1003816

1003816

1003816

X (119904 119905) minus X1199081003816

1003816

1003816

1003816

)

3

1003816

1003816

1003816

1003816

X (119904 119905) minus X1199081003816

1003816

1003816

1003816

le 119903

119888

0

1003816

1003816

1003816

1003816

X (119904 119905) minus X1199081003816

1003816

1003816

1003816

gt 119903

119888

(16)

where 119870119897 119870119887 119870119904 and 119870








0and width 119910







4

119894=1119876out119894) where bigger 120573


4(sum

4

119894=1Δ119875

119894)



F (x119904 119905) = 120572

1int

119905

0

(u (x119904 119905) minus U (x

119904 119905)) 119889119905

+ 120572

2(u (x119904 119905) minus U (x

119904 119905))

(17)



2are large




119904119889119905 Here U(x

119904 119905) equals


119908 the drag


119897(19) and


40 and 100

119862

119889=

119865

119863

05120588119880

2

infin119863

(18)

119862

119897=

119865

119871

05120588119880

2

infin119863

(19)








Inlet 1

Inlet 2

Outlet 4

Outlet 1

Outlet 2

Outlet 3

Qin1

Qin2

Qout1

Qout2

Qout3

Qout4

y0

x0

wb

wb

wb

wb

we

we

wp

wi

wi

w0


L

H

InletOutlet

Solid wall

Solid wall

DH2

H203L

y

x u




119908119862

119889119862

119889119862

119897119878

119905





190

192

194

196

198

200

202

204

Cel

l cen

ter p

ositi

on (120583

m)

1 104 1 2 1 1 1 1 2 1 4 1 6 1 86 18 1120572








119908


119873

119861minus 1

(119873 minus 1) lt

119863

2

lt

119908


119873

119861minus 1

119873

(119873 = 1 2 3)

(20)

119863

2

gt

119908


119873

119861minus 1

(119873 minus 1)

(119873 = 4)

(21)








8

16

20

24

01 02 03 04 0906 07 08 10 05120573

Cel

l dia

met

er (120583

m)





000

3

000

6

000

9

001

2

001

5

001

8

002

1

002

4

002

7

003

0

003

3

000

0

U

(a)0

000

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3U

(b)

000

0

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3

U

(c)

000

00

006

000

90

012

001

50

018

002

10

024

002

70

030

003

3

000

3

U

(d)


000

3

000

6

000

9

001

2

001

5

001

8

002

1

002

4

002

7

003

0

003

3

000

0

U

(a)

000

0

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3

U

(b)

000

0

003

0

000

90

012

001

50

018

002

10

024

002

7

000

6

003

3

000

3

U

(c)

000

00

006

000

90

012

001

50

018

002

10

024

002

70

030

003

3

000

3

U

(d)





Competing Interests



003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6

U

(a)

003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6U

(b)

003

0

000

0

000

90

012

001

50

018

002

10

024

002

7

000

3

003

3

000

6

U

(c)

000

00

006

000

90

012

001

50

018

002

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(e)


Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















Inlet 1

Inlet 2

Outlet 4

Outlet 1

Outlet 2

Outlet 3

Qin1

Qin2

Qout1

Qout2

Qout3

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y0

x0

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wb

wb

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we

we

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L

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InletOutlet

Solid wall

Solid wall

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119908119862

119889119862

119889119862

119897119878

119905





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1 104 1 2 1 1 1 1 2 1 4 1 6 1 86 18 1120572








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119863

2

lt

119908


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119861minus 1

119873

(119873 = 1 2 3)

(20)

119863

2

gt

119908


119873

119861minus 1

(119873 minus 1)

(119873 = 4)

(21)








8

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01 02 03 04 0906 07 08 10 05120573

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Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of




















119908


119873

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(119873 minus 1) lt

119863

2

lt

119908


119873

119861minus 1

119873

(119873 = 1 2 3)

(20)

119863

2

gt

119908


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119861minus 1

(119873 minus 1)

(119873 = 4)

(21)








8

16

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24

01 02 03 04 0906 07 08 10 05120573

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Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















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Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of

















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Acknowledgments


References
















































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of












































of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of





















of






Disease Markers



OncologyJournal of





PPAR Research



Journal of

ObesityJournal of
















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Research Article A Numerical Simulation of Cell Separation ...1 $"+ ]$ 2 u + f. For the IB-LBM...

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