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Entry effects of droplet in a micro confinement


Entry effects of droplet in a micro confinement: Implications for deformation-based circulating tumor cell microfiltration
Zhifeng Zhang, Xiaolin Chen, and Jie Xu Citation: Biomicrofluidics 9, 024108 (2015); doi: 10.1063/1.4916645 View online: http://dx.doi.org/10.1063/1.4916645 View Table of Contents: http://scitation.aip.org/content/aip/journal/bmf/9/2?ver=pdfcov Published by the AIP Publishing Articles you may be interested in A microfluidic model for organ-specific extravasation of circulating tumor cells Biomicrofluidics 8, 024103 (2014); 10.1063/1.4868301 Label-free isolation of circulating tumor cells in microfluidic devices: Current research and perspectives Biomicrofluidics 7, 011810 (2013); 10.1063/1.4780062 Size-based hydrodynamic rare tumor cell separation in curved microfluidic channels Biomicrofluidics 7, 011802 (2013); 10.1063/1.4774311 Selective cell capture and analysis using shallow antibody-coated microchannels Biomicrofluidics 6, 044117 (2012); 10.1063/1.4771968 Novel platform for minimizing cell loss on separation process: Droplet-based magnetically activated cell separator Rev. Sci. Instrum. 78, 074301 (2007); 10.1063/1.2751414

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BIOMICROFLUIDICS 9, 024108 (2015)

Entry effects of droplet in a micro con?nement: Implications for deformation-based circulating tumor cell micro?ltration
Zhifeng Zhang,1 Xiaolin Chen,1 and Jie Xu2,a)
Department of Mechanical Engineering, Washington State University, Vancouver, Washington 98686, USA 2 Department of Mechanical and Industrial Engineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA (Received 26 January 2015; accepted 18 March 2015; published online 31 March 2015)
1

Deformation-based circulating tumor cell (CTC) microchips are a representative diagnostic device for early cancer detection. This type of device usually involves a process of CTC trapping in a con?ned microgeometry. Further understanding of the CTC ?ow regime, as well as the threshold passing-through pressure, is a key to the design of deformation-based CTC ?ltration devices. In the present numerical study, we investigate the transitional deformation and pressure signature from surface tension dominated ?ow to viscous shear stress dominated ?ow using a droplet model. Regarding whether CTC fully blocks the channel inlet, we observe two ?ow regimes: CTC squeezing and shearing regime. By studying the relation of CTC deformation at the exact critical pressure point for increasing inlet velocity, three different types of cell deformation are observed: (1) hemispherical front, (2) parabolic front, and (3) elongated CTC co-?owing with carrier media. Focusing on the circular channel, we observe a ?rst increasing and then decreasing critical pressure change with increasing ?ow rate. By pressure analysis, the concept of optimum velocity is proposed to explain the behavior of CTC ?ltration and design optimization of CTC ?lter. Similar behavior is also observed in channels with symmetrical cross sections like square and triangular but not in rectangular channels C 2015 AIP Publishing LLC. which only results in decreasing critical pressure. V [http://dx.doi.org/10.1063/1.4916645]

I. INTRODUCTION

Cancer is a major public health problem all over the world.1 Decades of efforts have been engaged in controlling cancer death rates with only marginal success. According to National Cancer Institute (NCI) database, cancers can be categorized into local stage, regional stage, and distant stage.2 Early detection at local stage has shown a high ?ve-year survival rate for many kinds of cancers such as cervical cancer,3 prostate cancer,4 and pancreatic cancer.5 Detecting cancers early when treatment is most successful offers substantial bene?ts to patients. There is growing evidence that circulating tumor cells (CTCs) present in the bloodstream of cancer patients are responsible for initiating cancer metastasis and can be used as a potential biomarker for early cancer detection.6–8 CTCs are tumor cells shed from primary tumors and can travel through circulation to distant metastatic sites or even self-seed into their tumors of origin.9 The detection of CTCs has important prognostic and therapeutic implications as they are found in the peripheral blood of early-stage cancer patients prior to the onset of clinical symptoms.8 A variety of technologies have been developed to separate CTCs from normal blood cells, based on the differences in cellular biochemical properties such as surface antigen expression or physical properties, such as size or deformability10 as well as differing acoustic,11,12 magnetic,13 optic,14 or dielectrophoretic characteristics.11,15–18 Among the various
a)

Author to whom correspondence should be addressed. Electronic mail: jiexu@uic.edu. 9, 024108-1
C 2015 AIP Publishing LLC V

1932-1058/2015/9(2)/024108/14/$30.00

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extant methods for CTC detection and enumeration, deformation-based CTC ?ltration offers the advantages of structural simplicity,19,20 stable performance, and low cost.21 For example, deformation-based CTC micro?lters, such as ori?ce,22 weirs,20 ratchet,23 pillars,24 or membranes21-based devices, can batch-process samples of whole blood without elaborate chemical labeling25 and provide a compelling label-free26 strategy for blood puri?cation and cancer cell capturing. Most of the existing studies on deformation-based CTC micro?lters are experimental work focusing on device design and manufacturing.15,27–29 In order to achieve high capture ef?ciency, high isolation purity, and high system throughput,27,30 CTC micro?lter designs need to be optimized for performance. Numerical modeling can play a crucial role in increasing our ability to understand the complex cell/?ow/channel interactions in the cell entrained pressuredriven ?ow passing through micro?uidic channels and may deliver important insights into optimum design of future lab-on-a-chip devices. Previous numerical studies have employed a liquid-droplet model for cells and reported ?ndings on entry channel pressure in?uences for semi-static device design,16 with relevant pressure and deformation correlation,31,32 as well as the effect of 3D channel geometry on cell deformation.16 The pressure signature33 and possible deformation evolution under the in?uence of both surface tension of the cancer cell and viscous shear stresses acting on the cell at different device operating conditions remain elusive. Transitional operating conditions between surface tension-dominant and viscous shear stress-dominant ?ow behaviors are also characterized. Considerations important to the device design include achieving maximum ?ltration capability, high throughput, and high ef?ciency. Understanding of the interplay between the effects of interfacial, viscous, and inertial forces, as well as the balance between the viscous shear stress and surface tension,34 is essential to develop a micro?lter capable of ef?cient, speci?c, and reliable capture of CTCs. In this paper, we study the deformation of CTC, the pressure signature of CTC passing through the micro?lter, and cross-section in?uence over a wide range of ?ow rates. The pressure signature of a typical CTC deformation process is investigated for four types of 3-D geometries with uniform lengthwise cross-sectional areas: circular, square, triangular, and rectangular. The numerical study is based on Continuum Surface Force (CSF) and Volume of Fluid (VOF) methods.35 The dynamic variance of CTC passing-through pressure is studied and compared for different channel geometries. These data analyses provide useful insight not only for high ef?ciency CTC micro?lter development but also aids in understanding capillary blockage,36 cancer metastasis,37 and drug delivery processes.38
II. PROBLEM DESCRIPTION

CTCs are generally stiffer than normal blood cells39 although malignant CTCs with increased metastatic potential tend to have increased deformability.40,41 Isolation of CTCs using deformation is achieved as the stiffer CTCs are quickly trapped in narrow channels when a ?uid sample is injected into a micro?uidic system. In the present study, we consider ?ltering channels with four distinct cross sectional shapes, namely, circular, square, rectangular, and triangular. Effective CTC capturing has been reported to have a critical diameter in the range of 5–12 lm for circular micro?lters.20,42 A circular channel with a diameter of 5 lm is adopted as the baseline case in this study to target high ?ltering performance. The key dimensions of noncircular channels are determined based on arriving at the same pressure drop in the viscous channel ?ows as the circular baseline calculation. Fig. 1 illustrates schematically the micro?ltration setup. Carrier ?ow and CTC proceed from the left to the right along the microchannel. The device is working on the characteristic dimensions on the order of 10 lm. We model a cell, which is an aqueous core surrounded by a thin lipid bilayer membrane, as a liquid droplet. In the droplet model, the cell’s ability to resist deformation is described by the surface energy (interfacial tension) of the droplet. CTCs derived from different cancer types are investigated. Table I gives the cancer cell types and their corresponding interfacial tension coef?cients extracted from single cell deformability experiments reported in the literature. Note

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FIG. 1. A cell passing through a ?ltering channel.

that a more deformable cell possesses a lower interfacial tension coef?cient. As expected, surface tension of white blood cells (WBCs) is substantially lower than those of the stiffer cancer cells listed in Table I. The droplet model adopted here neglects the cell internal contents such as nucleus and micro?laments and neither does it account for heterogeneity of the cell nor does it allow the description of active responses exhibited by the living cell, such as rigidity sensing
TABLE I. Cell types and their interfacial tension coef?cients. Cell type Neutrophils (WBC) Macrophage Myeloid cell Brain cancer Cell line … … HL60 U-87 Mg LN-229 U-118MG Esthesioneuroblastoma Salmonella enteritidis Cervical cancer ENB … … Interfacial tension coefficient (mN/m) 0.027 0.14 0.15 7 10 17 20 40 50 47 48 41 References 44 45 45 46

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or cell pulsation. However, it readily allows insights into the ?ltration effects of different cancer cell types under various operating conditions for cell capturing. Indeed, previous studies also show that the droplet model is suitable for simulating cells especially when deformation is large.43 The size distribution of CTCs (12–25 lm in diameter) has shown to overlap with that of normal WBCs (10–18 lm in diameter44). In our model, CTCs and WBCs are both considered to be 16 lm in diameter, to rule out effects resulting from the size differences in cells. The erythrocyte (7–8 lm in diameter44) is not considered in the study as they are highly deformable and can pass freely through any ?ltering channel within the design range.
III. MODELING OF TWO-PHASE FLOW WITH SURFACE TENSION

The problem under consideration consists of a dispersed phase (the cell) entrained in a continuous phase (the ambient ?ow) passing through a micro?uidic contraction.49,50 The interface that separates the two phases can be solved using VOF,16,51 level-set,43 front-tracking,52 or phase-?eld method. Among these methods, the VOF method, which has superior mass conservation property, is adopted in our model to track the interface between the cell and its surrounding ?uid. The VOF method uses a simple and economical way to track free surface.53 It de?nes and stores only a function a whose value is 1 at any point occupied by CTC and 0 otherwise. The value between 0 and 1 indicates the membrane of CTC. Transient simulation of a cell passing through a ?ltering channel is performed using the commercial software ANSYS Fluent with an explicit time stepping scheme. In?ation layers are employed to properly resolve the ?ow within the boundary layer. The 3-D mesh is in?ated along the walls of the non-circular channels. The circular channel is modeled using a 2-D axisymmetric model as shown in Fig. 2, with in?ated mesh distributed along the symmetry line as well as the channel walls. The total number of elements used for the 3-D simulation is approximately 200 000, with nearly 5000 elements patched to the cell (Fig. 3). Both CTCs and white blood cells are assumed to be incompressible. The interface is tracked by solving the transport equation for the dispersed phase volume fraction a. Assuming no mass transfer between phases, the continuity equation can be written as ? *? @a ? r ? aV ? 0: @t (1)

Besides the interface tracking, simulation needs to take into account surface tension effects. Our surface tension model is based on the CSF method. With this model, the addition of surface tension to the VOF calculation results in a source term in the conventional incompressible Navier-Stokes momentum equation35 h  i     * ** * *T * @ qV ? r ? qV V ? ?rp ? r ? l rV ? rV ? F; @t (2)

FIG. 2. Mesh and dimensions for circular channel. The CTC locates at 15 lm away from the ?lter inlet. Area patched with the volume fraction of 1 indicates the CTC.

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FIG. 3. Mesh and dimensions for the non-circular micro?lter (rectangular).

where V is the velocity vector. q and l are volume-fraction-averaged density and viscosity, * respectively. F is the source term representing the surface tension force localized at the interface. The surface tension force can be transformed into a volume force in the region near the interface and expressed as35 F?r
*

*

2qjra ; ?q1 ? q2 ?

(3)

where r is the interfacial tension coef?cient between the dispersed phase and the continuous phase, j is the mean curvature of the interface in the control volume, and q1 and q2 are densities of the continuous phase and the dispersed phase, respectively. The density is interpolated from q ? a q1 ? ?1 ? a? q2 . For the incompressible two-phase ?ow problem, a constant volume ?ow rate is used as the inlet boundary condition. No-slip boundary conditions are prescribed on the channel walls. The walls are assumed to be non-wetting with respect to the dispersed phase with a 180 contact angle.
IV. RESULTS AND DISCUSSION

In the literature, the problem of modeling a ?uid body passing through a channel con?nement has mainly focused on either the in?uence of surface tension or the viscous shear stress on the cell or droplet.54 Brie?y, droplet motion at low velocity can be described by surface-tension-dominated ?ow, while high-speed motion can be described as shear-stress-dominated ?ow. The dimensionless numbers that can be used to describe the relations of important forces include Ca and We numbers as de?ned in Eq. (4). For low ?ow rate cases, the threshold pressure of CTC passing through a microchannel is determined by the Young-Laplace equation assuming quasi-static conditions. Examples of slow motion application of the Young-Laplace equation include micropipette aspiration55 and ratchet CTC chips.19,23 For high ?ow rate cases, research has been focused on the friction- and viscosity-induced non-linear droplet transport pressure.
A. CTC deformation behavior at different operational conditions

Circulating tumor cells can be categorized into three types by their stiffness: rigid, medium, and compliant, according to the list of interfacial tension coef?cients in Table I. Together with extremely compliant WBC, the ?ow regimes of four types of cells are illustrated in Fig. 4. Three characteristic stages are selected to represent different ?ow regimes.

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FIG. 4. Phase chart of CTC deformation behavior. (H1) CTCs (rigid) at Re ? 1.8, We ? 1.3 ? 10?2, ?ow rate 7 nl/s; (H2) CTCs (rigid) at Re ? 36, We ? 5.2, ?ow rate 140 nl/s; (M1) CTCs (medium) at Re ? 0.18, We ? 6.48 ? 10?4, ?ow rate 0.7 nl/s; (M2) CTCs (medium) at Re ? 3.6, We ? 6.48 ? 10?2, ?ow rate 14 nl/s; (S1) CTCs (compliant) at Re ? 0.18, We ? 4.3 ? 10?2, ?ow rate 0.7 nl/s; (S2) CTCs (compliant) at Re ? 1.8, We ? 4.3, ?ow rate 7 nl/s; (W1) WBC at Re ? 0.18, We ? 2.4 ? 10?1; and (W2) WBC at Re ? 1.8, We ? 24, ?ow rate 7 nl/s.

A ?ow regime chart based on dimensionless numbers is often used in multiphase ?ow study.56 Here, we choose the x-axis of the ?ow regime as the We number, and the Y-axis is selected as the Ca number, which makes the diagonal lines tilted 45 the Re number. These dimensionless numbers are de?ned below: We ? qV 2 d r Ca ? lV r Re ? qVd : l (4)

Here, d is the diameter of con?nement channel, q is the density of cell, V is the velocity of carrier ?ow, r is the interfacial tension coef?cient of CTC, and l is the viscosity of carrier ?ow. A series of CTC deformation simulations was conducted as listed in Fig. 4. The W, S, M, and H represent white blood cells, CTCs (compliant), CTCs (medium), and CTCs (rigid), respectively. Three stages were plotted and illustrated. For purpose of pressure analysis, we are most interested in the ?rst stage, where the maximum threshold pressure is reached. As can be seen for CTCs (rigid), we illustrated the cell deformation at four different ?ow rates. (H1) deformation is a representative of CTCs (rigid) passing through the micro?lter at an extremely low velocity. Since the viscous shear stress in?uence is negligible, the deformation of the cell front at critical pressure follows a spherical shape and the cell is squeezed into the con?nement channel. (H2) deformation shows an elongated cell front, and the cell is detached from the wall of the microchannel, resulting in co-?ow of the carrier ?uid inside the ?ltering channel. In this case, the cell is sheared into the con?nement channel. Similar phenomena can be observed for other classes of cell stiffness. Deformation at (S1) and (M1) represents that cells being squeezed through the channel without co-?ow of carrier ?uid. Design at this stage should focus

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on CTC experiencing “friction”40 with the wall of channel.57 With increasing velocity, for deformation at (W2), (S2), and (M2), the carrier ?uid begins to co-?ow with CTC resulting in shear dominated ?ow. Fig. 5 further illustrates the detailed ?ow streamlines at the stage when threshold pressure is reached at three different ?ow rates for CTCs (rigid). We found the behavior of CTCs (compliant) predictable by studying WBC and CTCs (medium). Deformation behavior of CTCs (compliant) is more like WBC. If the velocity is small enough, the CTC deformation follows spherical deformation when passing through the channel as in Fig. 5(a). The cell is passing with wall friction51 at a low Re number. This is the so called “squeezing regime,” and the process of CTC passing is similar to the micropipette aspiration that has been described previously.55,58 With increasing Re, shear force begins to dominate, and the front end of CTC will follow a parabolic shape as in Fig. 5(b). With even higher Re, a complete cell-wall detachment will happen, a situation that helps high speed cell passing, and the CTC front end becomes extremely thin and elongated.
B. Maximum passing pressure at different operational conditions

Deformation-based CTC detection uses pressure-driven ?ow for cell capturing. For static pressure analysis,16 the total inlet pressure, Pt, during a cell passing through a channel is mainly used to overcome two types of resistance: (1) viscous resistance Pvis and (2) resistance caused by cell deformation Pcell (due to surface tension), Pt ? Pvis ? Pcell : The viscous pressure involves major and minor losses,59 (5)

FIG. 5. Streamline and CTC deformation behavior for ?ow rate of 7 nl/s, 77 nl/s, and 140 nl/s for cervical cancer, respectively. (a) CTC deformation at low ?ow rate (7 nl/s), the squeezing regime dominated by Young-Laplace equation. (b) CTC deformation at optimum ?ow rate (77 nl/s). If a higher ?ow rate is used, the cell may detach from the ?lter and allow carrier ?uid co-?ow through the ?ltration channel. (c) CTC deformation at high ?ow rate (140 nl/s). Carrier ?uid exhibits co-?ows while the CTC passes through the channel.

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Biomicrofluidics 9, 024108 (2015)

Pvis ?

8gL qV 2 : V ? ? K ? K ? C E 2 R2 C

(6)

The ?rst term on the right is the viscous pressure drop along the channel, g is the viscosity coef?cient, and L is the length of micro?lter channel. Besides, KC and KE are the constriction and expansion coef?cient, respectively. V is the average ?ow velocity in the ?ltering channel. When a cell is passing through the ?lter, the pressure will increase ?rst as the cell is getting squeezed and then will decrease as the cell is exiting the ?lter. Here, we de?ne the maximum increase in pressure in the ?lter as critical pressure. Further simulation observation collects the CTC deformation and ?ltration critical pressure together at a wide range of ?ow velocity in Fig. 6. As seen in Fig. 6, with increasing of ?ow rate, there is a maximum critical pressure for both cells with interfacial tension coef?cients of 10 mN/m and 50 mN/m. Similar behavior is observed for cells with different interfacial tension coef?cients. Illustrated in Fig. 6, we can understand that the critical pressure is primarily dominated by the CTC surface tension, which implies that critical pressure increase is mainly caused by the increase in leading-edge curvature (e.g., from spherical to parabolic-like shape). With increasing ?ow speed, the cell will start to detach from the channel wall, and the critical pressure reaches a maximum value—we de?ne this ?ow velocity as optimum velocity (Vcr), which is dependent on cell type and surface tension. At optimum velocity, the dynamic pressure in the carrier ?uid is just strong enough to overcome the surface tension force generated by the CTC front. After the velocity is higher than the optimum velocity, the carrier ?uid pressure will overcome the surface tension of CTC. Consequently, the carrier ?uid will co-?ow with the CTC. With a further increase in ?ow rate, the gap between the CTC and the micro?lter wall becomes larger, CTC passthrough resistance, as well as the ?ltering capabilities of the device decrease accordingly.
C. Cell specific optimum operational conditions

Motivated by these observations, we wish to ?nd the maximum ?ltering ability of a micro?lter for certain types of CTC for earlier and more speci?c CTC detection. According to Fig. 6,

FIG. 6. Critical pressure with the in?uence of channel ?ow rate. The deformations are (a) critical pressure at low ?ow rate, (b) critical pressure at optimum velocity, and (c) critical pressure at high ?ow rate with CTC co-?ow with the carrier ?uid. Insets (a), (b), and (c) show a schematic plot of CTC deformation on each characteristic stage.

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Biomicrofluidics 9, 024108 (2015)

very compliant CTCs with a low interfacial tension coef?cient follow the carrier ?ow well when entering the micro?lter channel, and accordingly, the radius of curvature of compliant CTCs at the ?lter is dif?cult to estimate. Rigid CTCs maintain a spherical front well under pressure, and membrane curvature is thus easier to estimate. Assuming that a narrow lubrication ?lm exists before the cell fully blocks the channel, we can estimate the optimum velocity by equating the surface tension pressure 2r?1=rf ? 1=RC ? to pressure drop in the lubrication ?lm, which includes the pressure drop due to kinetic energy increase, i.e., 2 =2 and the viscous loss in the liquid ?lm between the cell and the loss in dynamic pressure qVA channel wall. VA is ?ow velocity at the CTC front. Assuming the thickness of the lubrication ?lm is e, and the viscous shear stress in the channel can be estimated to be as s ? lVL=, with an area along the ?ow direction Af ? 2pRC L and an area perpendicular to the ?ow direction A ? 2pRC e. Therefore, the pressure drop caused by the lubrication ?lm is estimated by lQL2 =2pRC e2 . By combining the relation of the static pressure, lubrication ?lm pressure drop, and the dynamic pressure of the carrier ?ow, we can obtain the following equation for optimum velocity: ? ? q?VA ?2 lQL2 ? ? 2r 1=rf ? 1=RCTC : 2 2 2pRC e (7)

At a velocity just high enough to cause CTC yield to carrier ?uid, the pressure is the highest critical pressure and ?ow rate Q in the ?lm is zero. Assuming fully developed ?ow in the con?nement channel, the velocity at the front of CTC VA along the centerline of the channel is twice the average velocity in the con?nement channel. cr in the con?nement From Eq. (7), we can see the optimum velocity (average velocity V channel) for CTC and can be described by the following equation: ? s???????????????????????????????   r 1 1 cr ? ? : V q RC RCTC (8)

When the velocity is higher than optimum velocity, we predict the existence of co-?ow of CTC and carrier ?uid and a decreasing of CTC ?ltration pressure. Theoretical relation in Fig. 7 is plotted from Eq. (8).

FIG. 7. Surface tension-optimum velocity relation. The velocity is obtained by repeated determination of two velocities. V? < Vcr < V?, V? represents the velocity at which a cell is squeezed through the channel, and V? represents the velocity at which a cell is sheared through the channel. The criterion for detachment is selected as lubrication ?lm less than one mesh thickness near the wall of ?lter.

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Zhang, Chen, and Xu

Biomicrofluidics 9, 024108 (2015)

In Fig. 8, we plotted the phase chart again to indicate typical operational parameter of a deformation based CTC micro?lter. The optimum velocity should be a vertical line in the phase chart. Indeed, for the CTC of the parameter of 16 lm, micro?lter channel of the diameter of 5 lm, the critical We number can be calculated by Eqs. (4) and (8), We ? 2 ? ?1 ? RC =RCTC ?. For our case Wec ? 1.375, which indicates that the transition happens when inertial force is comparable to surface tension force. However, our model becomes less accurate for extremely soft cells (white blood cell) when searching for the optimum velocity. This is probably due to too small a value for surface tension and errors involved in the detachment criterion (as explained in caption of Fig. 7). There are several physical and engineering constrains for CTC lab-on-a-chip device design. (1) The total pressure should be smaller than 200 kPa for typical micro?uidic bonding strength, which limits the operating ?ow rate. (2) Acceptable time duration (ideally < 1 h) for processing a standard sized blood sample (8 ml).60 (3) Interfacial tension coef?cient value of different cells. These constrains give the shaded area on the ?ow regime chart in Fig. 8, where the vertical dashed line indicates optimum ?ow velocity to obtain the highest maximum passing pressure. Further derivation yields the following expression among dimensionless numbers, which helps to de?ne the boundaries of (b) and (e) in Fig. 8, ! 1 1 1 l2 : (9) lg?Ca? ? lg?We? ? lg?r? ? lg qd 2 2 2 Equation (10) indicates that contours of constant r, i.e., a speci?c cell type, are tilted lines with a slope of 1=2, which helps to explain the data distribution in Fig. 4. We can get the corresponding boundary under the con?nement of the following Ca, We, Re limit due to the con?nement above. Re spans the range of Re  (10?2, 102), suggesting the time duration of processing time. The lower limit re?ects slow processing tolerance and the high Re re?ects a high pressure limit by limitation (1).

FIG. 8. CTC detection operation chart. CTCs with a surface tension coef?cient higher than 7 mN/m show an optimum velocity near We ? 1.375. However, the compliant CTCs have smaller optimum velocity values than predicted.

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Zhang, Chen, and Xu

Biomicrofluidics 9, 024108 (2015)

D. Effects of 3D geometry on maximum passing pressure

Previous results show that due to the shear ?ow, the Young-Laplace equation does not accurately describe a non-circular channel. Channel geometry was found to be in?uential on static Laplace pressure prediction for the dynamic process.16 The in?uence of channel geometry on pressure31 and deformation32 in a low speed dynamic process is investigated for circular, square, triangular, and rectangular16 channel cross-sections. Circular,61 square,51,62,63 and rectangular cross-section channels have been extensively investigated, but triangular channel has not been well-studied due to dif?culties in manufacturing. Based on the Young-Laplace analysis, the maximum passing pressure for a non-circular channel is given by59 Psur   CPw 2 : ?r ? AP Rcell (10)

Cpw is the wet perimeter of cross-section; AP is the area of cross-section. Theoretical calculation of critical surface tension can be obtained from the previous reference. The values of passing pressure for a cell entering a microchannel of particular channel cross-section are plotted in Fig. 9. The x-axis is the normalized time for ease of comparison. As seen in Fig. 9, optimum velocity is observed to in?uence the ?ltration pressure of CTC entering a micro?lter channels. For these three types of channels, maximum passing pressure increases ?rst with increasing velocity and then decreases after the optimum velocity. However, such behavior is not observed in a rectangular channel with an aspect ratio of 2. For this rectangular cross-section, the pressure continues to decrease with increasing inlet velocity; no optimum velocity is observed as in the other cross-sections. This is due to a lubrication ?lm64,65 of carrier ?uid between the cell and wall. It is ?lled with the carrier ?uid and con?ned by the cell and wall. As seen in Fig. 10, due to the anisotropic elongation of the cell body, the cell is more easily compressed from the wider side of the channel. Thus, the lubrication ?lm becomes larger and pressure in the channel drops. Consequently, the cell may pass more easily through the micro?lter. Note that the negative ?ltration pressure occurs when the net surface tension force changes its direction. Initially, this net force is resisting the ?ow when the cell is entering the ?lter. Afterwards, this force changes direction and helps to pull the remaining cell out of the ?lter into

FIG. 9. Optimum velocity-?ow rate relation for four microchannel cross-sections: circular, square, triangular, and rectangular for a CTC with interfacial tension coef?cient of 50 mN/m.

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024108-12

Zhang, Chen, and Xu

Biomicrofluidics 9, 024108 (2015)

FIG. 10. Dynamic pressure signature for of a CTC passing through a rectangular microchannel with aspect ratio of 2. The x-axis is the normalized time: 0 stands for the starting point of cell entering and 1 stands for the cell recovery to sphere. The y-axis is the ?ltration pressure, which is the difference between inlet pressure and the viscous pressure drop.

the exiting chamber. Rather than roundness, a microchannel can be characterized by the order of rotational symmetry—the number of equal sides a regular polygon has. For a regular polygon, channels with an order of rotational symmetry greater than 2 have an advantage in CTC ?ltering.
V. CONCLUSION

In this study, the transition condition of soft matter CTC is conducted numerically using the droplet model. We have demonstrated the deformation evolution of CTC in various microchannel con?nements and under different ?ow conditions. A phase chart for CTC deformation is obtained. The velocity variation line for a speci?c cell is found to be linear in We-Ca log-log coordinates with slope of 1=2. Also, two different ?ow regimes of CTC entering a con?ned microchannel are observed: (1) cell squeezing with no carrier ?uid lubrication ?lm and (2) cell shearing with co-?ow of carrier ?uid. Correspondingly, there are mainly three types of CTC pro?le at critical pressure (1) hemisphere CTC front with CTC fully blocks channel inlet, (2) parabolic pro?le at higher speed with also fully blocked inlet, and (3) extreme elongation of CTC with a surrounding carrier ?ow. Moreover, the maximum passing pressure is also observed to vary with the increase of inlet velocity. For circular channel and square channel, the maximum passing pressure increases ?rst then decreases after optimum velocity. However, for a rectangular channel, the maximum passing pressure decreases continuously with the increase of inlet velocity. No optimum velocity is observed. Finally, the optimum velocity is discussed and expressed with numerical veri?cation.
ACKNOWLEDGMENTS

We thank Adam Popma for proofreading the manuscript.
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