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PAPER View Article OnlineView Journal | View Issue
2576 | Lab Chip, 2014, 14, 2576–2584 This journal is © The R
aMechanical Engineering, Department of ENCS, Washington State University,
Vancouver, 98686WA, USAb TeloVISION, LLC, 1281 Win Hentschel Blvd., West Lafayette, IN 47906, USA.
E-mail: [email protected]
Cite this: Lab Chip, 2014, 14, 2576
Received 7th March 2014,Accepted 4th May 2014
DOI: 10.1039/c4lc00301b
www.rsc.org/loc
The effects of 3D channel geometry on CTCpassing pressure – towards deformability-basedcancer cell separation
Zhifeng Zhang,a Jie Xu,a Bin Hongb and Xiaolin Chen*a
Various lab on a chip devices have been developed recently to detect and separate circulating tumour
cells (CTCs) for early stage cancer detection. Because CTCs are extremely rare in the blood, next
generation CTC microfilters aim at significant improvement in both efficiency and throughput. CTC
microfilters based on cell deformability seem to be a promising direction. In the present research, we
study a CTC passing event through a micro-filtering channel with various 3D geometries. The pressure
signatures for different types of cells passing through different channels are characterized numerically.
Specifically, five kinds of cross-sections, circular, square, triangular and two kinds of rectangular channels
with aspect ratios of 2 and 5, are studied in this work. The total pressures for cells passing through the
channels are calculated and reveal different behaviour from what is predicted by the static surface
tension model. Among all five cross-sections studied, the circular cross-section features the highest
critical pressure and thus is most suitable for high efficiency CTC separation. The square filtering channel
provides the second largest critical pressure, and the triangular cross-section provides the least critical
pressure among these three cross-sections. All these three cross-sections are better than the rectangular
channels with aspect ratios of 2 and 5. For the rectangular channel, a high aspect ratio channel may lead
to cell splitting at high speed, which will result in a periodic pressure signature. Our findings will provide
valuable information for the design of next generation CTC microfilters.
Introduction
Cancer remains a major global health problem.1 Besidesbeing locally invasive, cancer is known for its metastasizingcapability. The metastatic cancer cells in the circulatorysystem are called circulating tumor cells (CTCs). CTCs werefirst observed in the blood with metastasis in 1869. Withadvances in tumor biology and analytical technology, theunique seeding ability of CTCs was further demonstrated formany types of tumors such as breast carcinoma, colon carci-noma, and malignant melanoma.2 With multiple physico-chemical steps,3 detached tumor cells could traverse themicroenvironment and reach the blood vessels. After intra-vasation, CTCs could self-seed their tumors of origin or seedmetastases in remote organs.2 Cancers are most likely opera-ble and even curable if detected early, and therefore, earlycancer detection offers substantial benefit to patients. Con-ventional methods for early cancer detection mainly rely onmedical imaging techniques1 such as CT, X-ray, magneticresonance imaging (MRI), thermal texture maps (TTM), and
positron emission tomography (PET).4 However, these tech-niques are highly debatable in clinical practice for accuratecancer detection at early stage due to the uncertainty of thesetechniques by detecting abnormality instead of tumors aswell as their side effects.5
CTCs are shed by the solid tumor and found to be apotential indicator of early tumor formation due to theirpresence in the peripheral blood of cancer patients prior tothat of clinical symptoms.6 To detect CTCs, various methodshave been developed.6–8 Microfluidic devices, for instance,CTC microfilters,9,10 have provided an easy, fast and accuratedetection solution for non-invasive cancer diagnosis andclose-to-real-time monitoring of cancer progression. Variouscellular hallmarks can be employed for CTC detection in aCTC microfluidic device, and one category of label free CTCmicro-devices uses cellular physical characteristics, suchas size, deformability, acoustic properties11 and dielectro-phoretic features.12 Among all these physical characteristics,CTC microfilters based on cell deformability have the advan-tages of structural simplicity,13,14 stable performance and lowcost.15 For example, using channels, weirs,14 pillars,9 andmembranes,15 deformation-based CTC microfilters can auto-batch process blood purification or cancer cell capturing.16
Most existing studies on deformation-based CTC microfilters
oyal Society of Chemistry 2014
Fig. 1 The mechanical model for a cell passing through a filtering channel.
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focus on experimental tests based on a static surface tensionmodel. In order to achieve high capture efficiency, high isola-tion purity and high system throughput,17 the CTC micro-filters need to be optimized for the best geometry, whichcannot be done if the complex cell/flow/channel interactionsare not fully understood. Numerical simulation can be apowerful tool for studying the cell behaviour inside a filteringchannel. For example, previous models have explored theentry channel pressure influence17–19 and entrance time.18
However, to the best of our knowledge, the performance of3D channel geometries on the deformation of CTC remainselusive in the literature.
Various models are developed using either the micro/nanostructural approach or the continuum approach. Theformer was developed to investigate the contribution ofthe cell membrane and cytoskeleton to the mechanicalbehaviours of the suspended cells.20–23 The latter treats thecell as comprising materials with certain continuum materialproperties.24 Although providing less insight into the detailedinteractions at the subcellular level, the continuum approachis easier and more straightforward to use in modelling thecell/fluid/channel interaction if the effective mechanicalresponse at the cell level is all that is needed as in the case ofthe filtering process. The continuum approach can be catego-rized into liquid or solid models.19,20 By considering the cellas a liquid droplet, the liquid model uses the governing equa-tions consisting of mass and momentum balance for boththe cell and the surrounding flow and is generally consideredto perform better than solid models for large deformationproblems.7,18
In the present research, we explore key parameters suchas pressure behaviour and velocity influence under various3D geometries of a CTC filtering channel. A cell fluid modelis employed for proper treatment of the large deformationof cells.7,18 The deformability of CTC and inlet pressureare numerically simulated by the volume of fluid (VOF)algorithm.25 We first study the behaviour of CTC and whiteblood cells when they are pushed through a micrometre sizedfiltering channel. These channels feature uniform lengthwisecross-sections with various cross-sectional shapes. The pres-sure signatures of cells passing these filtering channels aredescribed under a constant flow rate. Surface tension termsobtained in the present paper are compared with the quasi-static Young–Laplace surface tension model for differentgeometries. The discrepancies between our results and theYoung–Laplace model are interpreted via a detailed celldeformation analysis. These data and analyses will provideuseful insight into the future development of high efficiencyCTC microfilters.
Model description
When the blood flows through a filtering channel in a micro-fluidic chip, the more deformable normal blood cells cansimply flow or squeeze through but the stiffer CTCs will beblocked. The critical channel diameter is reported to be
This journal is © The Royal Society of Chemistry 2014
within 5–12 μm for effective filtering of CTCs according todifferent cell sorting methods.14,26 In this study, we use acircular channel with a diameter of 5 μm as a baseline geometry.Fig. 1 shows an illustration of the circular channel design.
Both CTCs and normal blood cells vary significantly insize and deformability.27–29 One consistent finding has beenobserved, that is, CTCs are typically larger and stiffer thannormal blood cells although malignant CTCs with increasedmetastatic potential tend to have increased deformability.30,31
Considering the extremely heterogeneous CTC population, adeformability-based microfilter is capable of targeting thelargest possible subset of CTC population compared with theconventional antibody approach which targets only certainsubpopulations of CTCs. For convenience, typical cell proper-ties that fall within the reported range have been used in ourmodel. For example, the surface tension of a CTC13,32,33 anda white blood cell29 is considered to be 50 × 10−3 and 30 ×10−6 N m−1, respectively. The cell deformation is describedusing the fluid model of a droplet bounded by surfacetension. Note that the red blood cells are neglected in ourmodel, since they are small and highly deformable to passany filtering channel within the normal design range.
Theoretical background
The total inlet pressure Pt during a cell passing event througha channel is mainly used to overcome two types of resistance:1) viscous resistance and 2) resistance caused by surface tension,which acts to preserve the integrity of the cell surface:
Pt = Pvis + Psur (1)
Pressure drop due to viscosity (Pvis)
For pressure driven flow, the viscous resistance is causedby fluid viscosity which can be summarized by the Hagen–Poiseuille law:34
Pmajor = Rhyd × QV (2)
where Pmajor is the viscous pressure drop of the channel, Rhyd
is the hydraulic resistance, and QV is the volume flow rate.
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For different cross-sections, the hydraulic resistance Rhyd isdifferent.
Due to constriction from the inlet of the filtering channeland the expansion from the exit of the filtering channel, theminor pressure drop Pminor exits and can be calculated byusing the following equation:
P K V K Vminor C E
2 2
2 2(3)
where V is the flow velocity in the filtering channel, KC is theconstriction coefficient and KE is the expansion coefficient.KC is selected to be 0.5 and KE = 1 assuming the filteringchannel is much smaller than the outside chambers.
Pressure drop due to surface tension (Psur)
The surface tension–geometry relation can be demonstratedusing the cortical shell–liquid core model. The model uses aquasi-static Young–Laplace relation, which predicts a criticalpressure needed for a cell to be squeezed into a filteringchannel and is given by the following equation:
PR Rsurchannel cell
2 1 1
(4)
Here, σ is the surface tension coefficient, Rchannel is theradius of a circular filtering channel, and Rcell is the radius ofthe main cell body. This model is successfully used in micro-pipette aspiration experiments to evaluate the surface tensionof cells.35 The model is based on a near-static process andhas been widely used for CTC microfilter design. Based onthe Young–Laplace analysis, the critical pressure for a non-circular channel is given by34
2578 | Lab Chip, 2014, 14, 2576–2584
Table 1 Designed geometries and their critical pressures for a CTC
Cross-sectional shapesof the filtering channel
Hydraulicresistance Rhyd
Key dimensioconstant Pvis
Circular 8 14
La
a = 2.50
Square28 4 1
4. La
a = 4.57
Triangular 3203
14La
a = 7.30
Rectangular
tV• 0
β = b/a = 2, aβ = b/a = 5, a
PCA Rsurpw
P cell
cos 2
(5)
Cpw is the wet perimeter of the cross-section; AP is the areaof the cross-section, and θ is the contact angle betweenthe cell and fluid media (180° for our model). In thepresent study, we will compare our numerical results to thisYoung–Laplace relation and show its shortcomings fordynamic flow conditions and non-circular geometries.
A summary of the pressure calculations of a CTC micro-filter with various channel shapes is tabulated in Table 1.Five different cross-sections are studied, circular, square,triangular and two rectangular channels with aspect ratios of2 and 5. For meaningful comparison of these shapes, wefixed Pvis at a constant value using the circular channel as abaseline and calculated the dimensions of all other shapes.
Numerical method
The transient simulation of a cell passing through a filteringchannel is performed using the commercial software ANSYSFluent. The volume of fluid (VOF) method is employed totrack the interface between the cell and its surroundingmedium. An explicit time stepping scheme is used with theCourant number set within 0.5–5 to ensure that the schemeis both stable and convergent. The cell–water interface isreconstructed using the Geo-reconstruct scheme available inFluent. The channel walls are set as no-slip and stationary.
Inflation layers are used to achieve accurate resolution ofthe boundary layer. The 3-D mesh is inflated along the wallregions of the narrow channel where the cell is passingthrough. For the circular channel, a 2-D axisymmetric modelis employed, and the mesh is inflated along the symmetryline as well as the walls of the narrow channel. The total
This journal is © The Royal Society of Chemistry 2014
ns based on a(μm)
Critical pressureexpression Psur
Critical Psurvalue (kPa)
cos
2 2a Rcell
27.5
cos
4 2a Rcell
31.26
cos
123
2a Rcell
34.95
= 3.41 and b = 6.82 cos
2 1 1 2
a b Rcell
31.49= 2.55, b = 12.75 34.55
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number of elements used for the 3-D simulation is around200 000, with nearly 5000 elements patched for the cell. BothCTCs and white blood cells are assumed incompressible.
The VOF method36 is based on the fact that the twophases form an impenetrable interface, which can be easilyunderstood as the membrane of a cell. The method isachieved through the volume fraction of a primary phase (α),which varies between 0 and 1. An example of how theprimary-phase volume fraction varies near the interface isillustrated schematically in Fig. 2.
The interface is tracked by solving the transport equationfor the primary-phase volume fraction. Assuming no masstransfer between the two phases, the continuity equation canbe written as
tV 0 (6)
Here V is the velocity vector. According to eqn (6), the sub-stantial derivative of the primary-phase volume fraction iszero so that the interface-tracking is achieved through thevelocity field at the interface.
The following momentum equation is solved for the wholecomputational domain and the velocity field is sharedbetween the two phases:
tV VV p V V g F
T (7)
where g is the gravitational acceleration vector and F is thesource term. In the two-phase flow, surface tension forcecontributes to the source term. ρ and μ are volume-fraction-averaged density and viscosity, respectively. The materialproperties are determined by considering the volume fractionof each phase in a control volume. For the two-phase flow,for example, the volume-fraction-averaged density can begiven as follows:
ρ = αρ2 + (1 − α)ρ1 (8)
Here ρ2 is the density of the cell, while ρ1 is the density ofwater. Using the divergence theorem, the surface tensionforce, i.e., the source term added to the momentum equation,is given by
This journal is © The Royal Society of Chemistry 2014
Fig. 2 Values of the volume fraction for each of the phases and theinterface (α = 1 in cell, α = 0 in water, 0 < α < 1, at interface).
F vol
2
1 2(9)
where σ is the surface tension coefficient between the celland water, and κ is the mean curvature of the interface in thecontrol volume.
The curvature of the surface near the wall is adjusted byimposing the so-called dynamic boundary conditions andconsidering the effect of static contact angle as follows:
ni = nw cos(θst) + nt sin(θst) (10)
where nw is the unit vector normal to the wall, nt is a vectoron the wall and normal to the contact line, and θst is thestatic contact angle. In the present study, the static contactangle is selected as 180° for phase interaction.
Results and discussionPressure analysis
When a cell is passing through a filtering channel at a con-stant speed, the pressure in the channel will first increaseand then decrease. The maximum pressure occurs when thecell is entering the filtering channel and the minimum pres-sure occurs when the cell is exiting the channel. This maxi-mum pressure is therefore the critical pressure to ensure asuccessful passing event of a cell. In practical CTC micro-filters, the flow is often driven by pressure. Therefore, for suc-cessful CTC separation, the operating pressure of the deviceneeds to be kept below the critical pressure of a CTC andabove the critical pressure of normal blood cells. As men-tioned before, this inlet pressure is used for balancing twotypes of resistance and can be decomposed into Pvis and Psur.In this section, we first verify this concept by studying theinfluence of the flow velocity on the inlet pressure Pt for a cir-cular channel. As can be seen from Fig. 3, the inlet pressurePt for a CTC to pass through a circular channel increases withincreasing flow velocity. At the same time, Pvis also increasesat roughly the same speed with increasing velocity. Simulation
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Fig. 3 The effect of flow rate on pressure components when a cancercell passes through a circular channel.
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results match well with theoretical Pvis calculated from eqn (2)and (3). It is therefore evident that the increase in the inletpressure at a high flow rate is mainly due to the viscous resis-tance in the channel, while Psur is relatively constant. Similarresults can be obtained for other geometry cross-sections. Forcomparison purposes, we fixed the viscous pressure for all thefollowing studies at 7.9 kPa by fixing the flow rate at 7 nL s−1
(unless otherwise noted).We next investigate the influence of the filter hole size
on the critical pressure of CTC passing a circular channel.As can be seen from Fig. 4, the simulated critical pressureof CTC passing through a circular channel decreases withincreasing microfilter radius. According to the modelling ofthe filtering process, a CTC fails to pass through the filterwhen the inlet pressure is less than the corresponding criticalpressure for a given filter size. The shaded area in Fig. 4 pro-vides conditions that the CTC cannot pass through the filter.
Fig. 5 plots the pressure profiles of cells passing throughdifferent filtering channels. As can be seen from Fig. 5, at thesame viscous pressure Pvis, the filtering channel with a
2580 | Lab Chip, 2014, 14, 2576–2584
Fig. 4 The microfilter size effect on the critical pressure.
Fig. 5 Pressure signatures of CTC and white blood cells when theyare passing through various 3D cross-sections at a constant flow rate.The y-axis is the inlet pressure measured Pt in the simulation. Whiteblood cells produce a negligible pressure drop which is magnified inthe inset.
circular cross-section requires the highest inlet pressure Ptfor a CTC to pass through. The square channel produces thesecond largest pressure followed by the triangular filteringchannel. All these three geometries are better than the rect-angular filtering channels with β = 2 and 5. Since the higherinlet pressure Pt means the easier it is to separate a CTC fromnormal blood cells, we can rank the channel performanceaccording to the predicted critical pressure: circular > square >triangular > rectangular (β = 2) > rectangular (β = 5). Here,we introduce the concept of roundness of geometry to assistour comparison:
Roundness P
AP
4
2
AC
(11)
where AP is the cross-sectional area and CAP is the perimeterof the cross-section. Understandably, the roundness of a cir-cular shape is 1, and the more deviation from the circularshape, the smaller the roundness value will be. The calcu-lated roundness values of our studied cross-sectional shapesare listed in Fig. 6 and plotted with the critical pressures forCTC filtering obtained from simulations. As can be seen fromFig. 6, with the decrease in roundness, the pressure neededto separate the cancer cell will also decrease. In Fig. 6, wealso plotted the critical pressures calculated from the classi-cal quasi-static Young–Laplace model (see the last column inTable 1), which indicates a reverse trend, i.e., the criticalpressure increases with increasing roundness. Therefore, it isprecarious to apply the classical model for the design of CTCmicrofilters, since no dynamic effects are considered in themodel. For example, the main assumption for the classicalmodel is that the cell can fully occupy the channel cross-sectional area, while our simulation shows that the cells didnot fill up the entire channel cross section if there are sharpcorners, which may induce a leakage flow through the channelfrom inlet to outlet. This issue is discussed in greater detailin the next section.
This journal is © The Royal Society of Chemistry 2014
Fig. 6 The relation between the critical pressures and the channelcross-section roundness – comparison between theory and simulation.
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The effects of cell deformation
As can be seen in Fig. 7, the CTC deforms significantly whenentering a filtering channel due to the channel constriction.
Fig. 8 compares the deformation of a CTC and a whiteblood cell when they are inside a circular filtering channel.We can see that the CTC can fully fill the channel while thewhite blood cell shares the channel with the surroundingfluid. Another observation is that the front and rear parts ofthe cancel cell that are outside the channel is nearly sphericalin shape, while the white blood cell shows a more stream-lined shape.
We can analyse the deformation using capillary number37
(Ca). Ca number has been used to study biological cells,18
which is defined as Ca = μV/σ. It is the ratio of viscous forceand surface tension force. The Ca number for a white bloodcell in the studied case is approximately 1, which means thatthe surface tension force is comparable to the viscous force.38
At the same velocity, Ca number for a CTC is about 6 × 10−3,meaning that surface tension force governs the cancer cell
This journal is © The Royal Society of Chemistry 2014
Fig. 7 Typical deformation of a CTC when entering a circular filteringchannel. The channel is made invisible to aid the cell visualization.
Fig. 8 Deformation of a cancer cell and a white blood cell when theyare passing through a filtering channel. a) Cancer cell profile whenpassing through the channel. b) White blood cell profile when passingthrough the channel.
deformation, which can explain why the cancer cell main-tains a spherical shape easily and can fill the entire channel.
The whole process of pressure change during cell passingthrough a circular filtering channel is demonstrated in Fig. 9.In stage a, the cancer cell has not reached the filteringchannel yet, and the pressure is nearly constant. The celldeformation is negligible at stage a.
After initial deformation, the cell goes through very quickdeformation to the critical pressure at stage b. The front partdeforms into a near half spherical convex. With continuouspushing from fluid flow, this convex front will be finallysqueezed into the channel. Between stage b and stage c, thepressure is maintained at a high level, when the cancer cellgoes through huge deformation to pass through the channelfrom a 16 μm sphere to a 5 μm cylinder-like shape. Afterstage c, the cell begins to come out of the filtering channelon the other side. As soon as the front part of the cell movesout of the channel, its radius of curvature starts to increase.Meanwhile, the back part of the cell keeps shrinking toproduce smaller and smaller radius of curvature. Therefore,the net surface tension resistance force of the cell starts todecrease. At point d, both ends of the cell are equal in sizeand the net surface tension force reduces to zero, and inturn, the inlet pressure shown in the figure reaches the pureviscous pressure level. After this tipping point, the net surfaceforce changes its direction and actually pulls the cell out ofthe channel, which causes the decrease in the fluid pressure.
After the cell moves out entirely from the filteringchannel, we notice that the rear part of the cell may gothrough a rebound process forming a slight concave at stage g.This should be due to fluid inertia. The concave-shapeddeformation is more obvious at higher velocity.
The streamlines inside the microfilter induced by a cellpassing through different stages are shown in Fig. 10. Due tothe blockage of the cell, the flow has to increase its speed in
Lab Chip, 2014, 14, 2576–2584 | 2581
Fig. 9 Pressure–deformation relation of a cancer cell. The stages aregiven as follows: a) initial state, b) maximum pressure begins, c) maximumpressure finishes, d) pressure balanced and is equal to the backgroundpressure, e) minimum pressure begins, f) minimum pressure stagefinishes, and g) rear part bouncing.
Fig. 10 Streamline of CTC passing microfilter at different stages.
Fig. 11 Cell deformation for a 3D filtering channel at the criticalpressure point, a) triangular case, b) square case, c) rectangular casewith β = 2, and d) rectangular case with β = 5.
Fig. 12 Pressure oscillation due to cell split for a high aspect ratiochannel (β = 5). Inlet velocity is selected as 0.05 (m s−1).
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the small gap between the cell and the wall. A pressure differ-ence is generated across the cell which drags the cell into thefilter. The cell follows the underlying fluid streamlines in themicrofilter and recovers its spherical shape by the surfacetension effect after passing through the microfilter.
We also compared the deformation of cells in several non-circular channels as shown in Fig. 11. One major differenceis that the cell could not fully fill the channel as it does in thecircular channel case. Therefore, the use of the Young–Laplaceequation in Table 1 will definitely involve errors.
As can be seen at the entrance of a triangular filteringchannel, the centre part is occupied by the cancer cell, andthe corners near the three vertexes are filled by the surroundingfluids. The same situation happens for all other 3D geometries.We can also see that if the aspect ratio of the rectangularchannel increases, the fluids will occupy more space insidethe filtering channel. Another interesting observation for alarge aspect ratio channel is that cells can split into smallerparts as illustrated in Fig. 12. Cell splitting is a nonlinearphenomenon39 for a droplet model. It has a strong correlationwith channel geometry and flow velocity. Detached daughtercells are reported in a similar droplet dispenser device. For atheoretical study, the splitting phenomenon was reported tohave a relation with Ca number and shear stress.40,41
In the present study, cell splitting for circular, triangular,square and rectangular channels with β = 2 is not observed
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under the studied parameters. However, for the higher aspectratio rectangular channel with β = 5, the CTC splits at theoutlet. The ruptured daughter cells will merge and keep split-ting and merging until they both pass through the hole. Thecell merging results from the lack of cell membrane in ourdroplet model. The cell split is due to the surface-tension-driven Rayleigh–Plateau instability42 with the main source ofperturbation coming from the anisotropic elongation of thecell body at the exit of the rectangular channel.43 Loweringthe fluid speed will prevent the cell from splitting, whichagrees with a previous numerical study.39
Conclusions
A CTC passing event through a 3D micro-filtering channel isstudied in this work. The pressure signatures for differenttypes of cells passing through different channels are charac-terized numerically. The effect of 3D channel geometry onthe total pressure for cells passing through the channels isexamined. Among the five cross-sections studied, the circularcross-section features the highest critical pressure and thusis most suitable for high efficiency CTC separation. As thecross-section deviates from a round shape, the pressureneeded to separate the cancer cell decreases with decreasingroundness of the cross-section. Between the two equilateralpolygon cross-sections, the square filtering channel providesa larger critical pressure than that of the triangular channel.The rectangular channels provide the least critical pressuresamong all the different cross-sections studied. The highaspect ratio rectangular channel may lead to cell splitting ata high speed, which will result in a periodic pressure signa-ture. Our findings will aid in the design of next generationCTC microfilters by offering insight into the role that 3Dchannel geometry plays in deformation-based CTC separation.
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