Inertial focusing of particles and cells in the microfluidic labyrinth device: Role of sharp turns

I. INTRODUCTION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTION <<II. EXPERIMENTAL METHODSIII. RESULTS AND DISCUSSI...IV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESPrevious sectionNext sectionInertial microfluidics has gained prominence in a variety of applications, such as sheath-less flow cytometry,1–71. A. A. S. Bhagat, S. S. Kuntaegowdanahalli, and I. Papautsky, “Enhanced particle filtration in straight microchannels using shear-modulated inertial migration,” Phys. Fluids 20(10), 101702 (2008). https://doi.org/10.1063/1.29988442. A. T. Ciftlik, M. Ettori, and M. A. Gijs, “High throughput-per-footprint inertial focusing,” Small 9(16), 2764–2773 (2013). https://doi.org/10.1002/smll.2012017703. D. R. Gossett, H. T. Tse, J. S. Dudani et al., “Inertial manipulation and transfer of microparticles across laminar fluid streams,” Small 8(17), 2757–2764 (2012). https://doi.org/10.1002/smll.2012005884. J. Hansson, J. M. Karlsson, T. Haraldsson, H. Brismar, W. van der Wijngaart, and A. Russom, “Inertial microfluidics in parallel channels for high-throughput applications,” Lab Chip 12(22), 4644–4650 (2012). https://doi.org/10.1039/c2lc40241f5. S. C. Hur, H. T. Tse, and D. Di Carlo, “Sheathless inertial cell ordering for extreme throughput flow cytometry,” Lab Chip 10(3), 274–280 (2010). https://doi.org/10.1039/B919495A6. A. J. Mach and D. Di Carlo, “Continuous scalable blood filtration device using inertial microfluidics,” Biotechnol. Bioeng. 107(2), 302–311 (2010). https://doi.org/10.1002/bit.228337. H. A. Nieuwstadt, R. Seda, D. S. Li, J. B. Fowlkes, and J. L. Bull, “Microfluidic particle sorting utilizing inertial lift force,” Biomed. Microdevices 13(1), 97–105 (2011). https://doi.org/10.1007/s10544-010-9474-6 fast particle transfer across fluid streams,33. D. R. Gossett, H. T. Tse, J. S. Dudani et al., “Inertial manipulation and transfer of microparticles across laminar fluid streams,” Small 8(17), 2757–2764 (2012). https://doi.org/10.1002/smll.201200588 filtration of biological matter,6,8,96. A. J. Mach and D. Di Carlo, “Continuous scalable blood filtration device using inertial microfluidics,” Biotechnol. Bioeng. 107(2), 302–311 (2010). https://doi.org/10.1002/bit.228338. K. Goda, A. Ayazi, D. R. Gossett et al., “High-throughput single-microparticle imaging flow analyzer,” Proc. Natl. Acad. Sci. U.S.A. 109(29), 11630–11635 (2012). https://doi.org/10.1073/pnas.12047181099. L. R. Huang, E. C. Cox, R. H. Austin, and J. C. Sturm, “Continuous particle separation through deterministic lateral displacement,” Science 304(5673), 987–990 (2004). https://doi.org/10.1126/science.1094567 and separating cancer cells from blood cells.1010. J. Zhou, A. Kulasinghe, A. Bogseth, K. O’Byrne, C. Punyadeera, and I. Papautsky, “Isolation of circulating tumor cells in non-small-cell-lung-cancer patients using a multi-flow microfluidic channel,” Microsyst. Nanoeng. 5(8), 1–12 (2019). https://doi.org/10.1038/s41378-019-0045-6,1111. J. Zhou, C. Tu, Y. Liang et al., “Isolation of cells from whole blood using shear-induced diffusion,” Sci. Rep. 8(1), 9411 (2018). https://doi.org/10.1038/s41598-018-27779-2 In one manifestation of inertial microfluidics, curved geometries are used, where, in addition to inertial forces, strong curvature forces driven by centrifugal effects are introduced.1212. A. A. Bhagat, S. S. Kuntaegowdanahalli, and I. Papautsky, “Continuous particle separation in spiral microchannels using dean flows and differential migration,” Lab Chip 8(11), 1906–1914 (2008). https://doi.org/10.1039/b807107a These so-called Dean flows can manipulate differential migration of particles or cells, focusing them at different lateral positions depending on their sizes.1313. A. Russom, A. K. Gupta, S. Nagrath, D. Di Carlo, J. F. Edd, and M. Toner, “Differential inertial focusing of particles in curved low-aspect-ratio microchannels,” New J. Phys. 11, 075025 (2009). https://doi.org/10.1088/1367-2630/11/7/075025 Such inertial microfluidic systems incorporating channel curvature have been shown to be efficient at processing large volumes of suspensions in a relatively short time with good rates of particle separation efficacy and purity.1010. J. Zhou, A. Kulasinghe, A. Bogseth, K. O’Byrne, C. Punyadeera, and I. Papautsky, “Isolation of circulating tumor cells in non-small-cell-lung-cancer patients using a multi-flow microfluidic channel,” Microsyst. Nanoeng. 5(8), 1–12 (2019). https://doi.org/10.1038/s41378-019-0045-6,1111. J. Zhou, C. Tu, Y. Liang et al., “Isolation of cells from whole blood using shear-induced diffusion,” Sci. Rep. 8(1), 9411 (2018). https://doi.org/10.1038/s41598-018-27779-2In straight channels, particles with finite inertia get focused due to a balance of shear gradient lift force and the viscous Stokes drag.1414. D. Di Carlo, D. Irimia, R. G. Tompkins, and M. Toner, “Continuous inertial focusing, ordering, and separation of particles in microchannels,” Proc. Natl. Acad. Sci. U.S.A. 104(48), 18892–18897 (2007). https://doi.org/10.1073/pnas.0704958104 However, in curved channels, the fluid is pushed radially outward due to centrifugal forces, which set up a transverse pressure gradient. Thus, the fluid near the top and bottom walls moves inward, resulting in two symmetric, counter-rotating cross-sectional vortices or Dean vortices.1515. G. Vona, A. Sabile, M. Louha et al., “Isolation by size of epithelial tumor cells: A new method for the immunomorphological and molecular characterization of circulating tumor cells,” Am. J. Pathol. 156(1), 57–63 (2000). https://doi.org/10.1016/S0002-9440(10)64706-2,1616. L. Zabaglo, M. G. Ormerod, M. Parton, A. Ring, I. E. Smith, and M. Dowsett, “Cell filtration-laser scanning cytometry for the characterisation of circulating breast cancer cells,” Cytometry A 55(2), 102–108 (2003). https://doi.org/10.1002/cyto.a.10071 The onset of these vortices is determined by the Dean number1212. A. A. Bhagat, S. S. Kuntaegowdanahalli, and I. Papautsky, “Continuous particle separation in spiral microchannels using dean flows and differential migration,” Lab Chip 8(11), 1906–1914 (2008). https://doi.org/10.1039/b807107aDe=Refδ, where Ref is the fluid Reynolds number and δ is the curvature ratio. In addition to the aforementioned forces in straight channels, this vortex gives rise to an additional force called the Dean drag force acting on particles flowing in curved geometries.1212. A. A. Bhagat, S. S. Kuntaegowdanahalli, and I. Papautsky, “Continuous particle separation in spiral microchannels using dean flows and differential migration,” Lab Chip 8(11), 1906–1914 (2008). https://doi.org/10.1039/b807107a Here, Ref=ρfUfDh/μf is the fluid Reynolds number (ρf and μf denote fluid density and dynamic viscosity, respectively, Uf is the mean fluid velocity in the channel, and Dh is the hydraulic diameter) and the curvature ratio δ=Dh/2R, where R is the mean channel radius of curvature.The basic mechanism for particle separation in curved channels appears to be that the inertial lift force FL stabilizes particle position (i.e., particle focusing), while the Dean drag force FD aids in lateral migration due to the cross-sectional circulation (i.e., particle separation).12,1712. A. A. Bhagat, S. S. Kuntaegowdanahalli, and I. Papautsky, “Continuous particle separation in spiral microchannels using dean flows and differential migration,” Lab Chip 8(11), 1906–1914 (2008). https://doi.org/10.1039/b807107a17. H. Amini, W. Lee, and D. Di Carlo, “Inertial microfluidic physics,” Lab Chip 14(15), 2739 (2014). https://doi.org/10.1039/c4lc00128a Both these forces are dependent on particle diameter dp, the fluid Reynolds number Ref, and channel curvature ratio δ requiring optimization of channel geometries and flow conditions to achieve high fidelity in focusing and particle separation.1212. A. A. Bhagat, S. S. Kuntaegowdanahalli, and I. Papautsky, “Continuous particle separation in spiral microchannels using dean flows and differential migration,” Lab Chip 8(11), 1906–1914 (2008). https://doi.org/10.1039/b807107a Experimentally, two geometrical parameters are available to optimize particle focusing and separation in curved channels. The first is the focusing length Lf, which is defined as the channel length required to focus particles. Since Lf depends strongly on particle diameter dp, i.e., Lf∝dp−3, larger particles can be focused at short channel lengths.1212. A. A. Bhagat, S. S. Kuntaegowdanahalli, and I. Papautsky, “Continuous particle separation in spiral microchannels using dean flows and differential migration,” Lab Chip 8(11), 1906–1914 (2008). https://doi.org/10.1039/b807107a Smaller particles, in contrast, require much longer channel lengths to focus due to the reduced inertial lift forces FL acting on them since FL∝dp4.12,1412. A. A. Bhagat, S. S. Kuntaegowdanahalli, and I. Papautsky, “Continuous particle separation in spiral microchannels using dean flows and differential migration,” Lab Chip 8(11), 1906–1914 (2008). https://doi.org/10.1039/b807107a14. D. Di Carlo, D. Irimia, R. G. Tompkins, and M. Toner, “Continuous inertial focusing, ordering, and separation of particles in microchannels,” Proc. Natl. Acad. Sci. U.S.A. 104(48), 18892–18897 (2007). https://doi.org/10.1073/pnas.0704958104 The second is the curvature ratio, where tighter curvatures lead to higher Dean drag forces. Since FD∝dp, Dean drag forces are weaker for smaller particles, necessitating tighter curvatures to focus them.An important application where size-based separation in curved geometries has been effective is the separation of cancer cells from blood cells in the context of isolating circulating tumor cells from patient blood.18–2118. H. W. Hou, M. E. Warkiani, B. L. Khoo et al., “Isolation and retrieval of circulating tumor cells using centrifugal forces,” Sci. Rep. 3, 1259 (2013). https://doi.org/10.1038/srep0125919. M. E. Warkiani, G. Guan, K. B. Luan et al., “Slanted spiral microfluidics for the ultra-fast, label-free isolation of circulating tumor cells,” Lab Chip 14(1), 128–137 (2014). https://doi.org/10.1039/C3LC50617G20. M. E. Warkiani, B. L. Khoo, D. S. Tan et al., “An ultra-high-throughput spiral microfluidic biochip for the enrichment of circulating tumor cells,” Analyst 139(13), 3245–3255 (2014). https://doi.org/10.1039/C4AN00355A21. M. E. Warkiani, B. L. Khoo, L. Wu et al., “Ultra-fast, label-free isolation of circulating tumor cells from blood using spiral microfluidics,” Nat. Protoc. 11(1), 134–148 (2016). https://doi.org/10.1038/nprot.2016.003 The diameter of tumor cells, white blood cells (WBCs), and red blood cells (RBCs) lies in the range of ∼10–20, ∼7–12, and ∼8 μm, respectively.1818. H. W. Hou, M. E. Warkiani, B. L. Khoo et al., “Isolation and retrieval of circulating tumor cells using centrifugal forces,” Sci. Rep. 3, 1259 (2013). https://doi.org/10.1038/srep01259 Given these differences in cell size, spiral geometries where Dean flow can be invoked have been developed for separating cancer cells.18–2118. H. W. Hou, M. E. Warkiani, B. L. Khoo et al., “Isolation and retrieval of circulating tumor cells using centrifugal forces,” Sci. Rep. 3, 1259 (2013). https://doi.org/10.1038/srep0125919. M. E. Warkiani, G. Guan, K. B. Luan et al., “Slanted spiral microfluidics for the ultra-fast, label-free isolation of circulating tumor cells,” Lab Chip 14(1), 128–137 (2014). https://doi.org/10.1039/C3LC50617G20. M. E. Warkiani, B. L. Khoo, D. S. Tan et al., “An ultra-high-throughput spiral microfluidic biochip for the enrichment of circulating tumor cells,” Analyst 139(13), 3245–3255 (2014). https://doi.org/10.1039/C4AN00355A21. M. E. Warkiani, B. L. Khoo, L. Wu et al., “Ultra-fast, label-free isolation of circulating tumor cells from blood using spiral microfluidics,” Nat. Protoc. 11(1), 134–148 (2016). https://doi.org/10.1038/nprot.2016.003 However, a potential drawback of the spiral geometries is that smaller cells or particles might be difficult to focus due to weaker Dean forces (FD∝dp). To compensate for this, it is necessary to use tighter curvatures.Recently, a microfluidic labyrinth device was proposed, which isolates cancer cells from blood cells,2222. E. Lin, L. Rivera-Baez, S. Fouladdel et al., “High-throughput microfluidic labyrinth for the label-free isolation of circulating tumor cells,” Cell Syst. 5(3), 295–304.e4 (2017). https://doi.org/10.1016/j.cels.2017.08.012 with the aim of improving the focus of smaller-sized blood cells. In addition to spiral sections, the labyrinth device contains multiple sharp turns. The turns were proposed to help in two ways: (i) allow longer channel lengths to be packaged into a small device footprint, thereby increasing the opportunity to focus smaller particles and (ii) augment the local Dean forces due to their tight curvatures, which helps push smaller particles toward their equilibrium positions, focusing them. Using zigzag serpentine geometries, a few other studies23–2523. T. T. Jin, S. Yan, J. Zhang, D. Yuan, X. F. Huang, and W. H. Li, “A label-free and high-throughput separation of neuron and glial cells using an inertial microfluidic platform,” Biomicrofluidics 10(3), 3952–3960 (2016). 24. J. Zhang, S. Yan, R. Sluyter, W. H. Li, G. Alici, and N. T. Nguyen, “Inertial particle separation by differential equilibrium positions in a symmetrical serpentine micro-channel,” Sci. Rep. 4, 032002 (2014).25. J. Zhang, W. H. Li, M. Li, G. Alici, and N. T. Nguyen, “Particle inertial focusing and its mechanism in a serpentine microchannel,” Microfluid. Nanofluid. 17(2), 305–316 (2014). https://doi.org/10.1007/s10404-013-1306-6 have also suggested that turns can enhance the focusing of smaller particles and cells. Despite the proposed hypothesis that turns can aid in focusing of smaller particles and, therefore, provide efficient separation, currently quantitative data characterizing how particle size and fluid Reynolds number influence the focusing dynamics is lacking. Moreover, the turns present are U-shaped, which leads to quick and successive changes in the flow direction that can complicate focusing dynamics, which is yet to be quantified.

In this study, we use the labyrinth device as a prototypical system that contains both spiral arcs and turns to investigate how particle size and fluid Reynolds number influence the focusing dynamics at all the corners present in the device. Additionally, we simulate the fluid flow in the U-turn region to understand how variations in focusing dynamics can be explained by the cross-sectional flow fields. Next, we develop a quantitative measure of separability S to track the separation between focused streams of size-differing particles as they approach and exit the sharp turn. A comparison of S before and after the turn allows us to determine whether the tight-curvature turns help improve particle separation or not and if so, under what flow conditions. Finally, we characterize the separability in the downstream expansion section where particle streams are collected, to report on the overall performance of the labyrinth device for size-based separation of particles and cells.

II. EXPERIMENTAL METHODS

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. EXPERIMENTAL METHODS <<III. RESULTS AND DISCUSSI...IV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESPrevious sectionNext section

A. Labyrinth device design

The microfluidic labyrinth device has a width of 500 μm and a total length of 637 mm. It consists of 11 spiral loops and 56 corners, with the curvature ratio in the loops varying from 5.29×10−3 to 3.7×10−2. The channel height H is uniformly 97 um. Multiple sharp turns throughout the device induce an abrupt increase in geometrical curvature. The fabricated devices were received from the Sunitha Nagrath laboratory at the University of Michigan.

B. Sample preparation

In this study, three types of samples were used: MCF-7 breast cancer cells, WBCs, and fluorescent polystyrene microspheres. MCF-7 breast cancer cells were cultured and stained with a live fluorescent dye (CellTracker™ Red CMTPX dye, Invitrogen™, Carlsbad, USA). WBCs were isolated from human whole blood and also stained with a live cell marker (10 μm CMTPX, Invitrogen™, Carlsbad, USA). Detailed methods for MCF-7 cell culture and staining of cancer cells and WBCs are provided in our previous works.26–2826. S. M. Ahmmed, S. S. Bithi, A. A. Pore et al., “Multi-sample deformability cytometry of cancer cells,” APL Bioeng. 2(3), 032002 (2018). https://doi.org/10.1063/1.502099227. A. Gangadhar, H. Sari-Sarraf, and S. A. Vanapalli, “Staining-free, in-flow enumeration of tumor cells in blood using digital holographic microscopy and deep learning,” bioRxiv (2022).28. G. Moallem, A. A. Pore, A. Gangadhar, H. Sari-Sarraf, and S. A. Vanapalli, “Detection of live breast cancer cells in brightfield microscopy images containing white blood cells by image analysis and deep learning,” bioRxiv (2021). Tagged WBCs and MCF-7 cells with a mean size of 10 and 21 μm, respectively, were diluted with 1X phosphate buffer saline to make up working concentrations of 100 000 and 500 000 cells/ml. Gentle pipetting prior to the experiment ensured that the cells did not aggregate or settle.

Fluorescent polystyrene beads of four different sizes were selected with sizes comparable to the cell types used in the study. 7.32 μm TRITC (Bangs Laboratories Inc., Fishers, USA), 12 μm FITC, 15 μm TRITC (Thermo Fisher, Carlsbad, USA), and 20 μm FITC (Polysciences Inc., Warrington, USA) particles were suspended in de-ionized water to make up a final concentration of 1×106 beads/ml. 0.1% (V/V) Tween 20 (Sigma-Aldrich, St. Louis, USA) was added to minimize aggregation of microspheres. Particle suspensions were vortexed prior to the experiment to ensure adequate dispersion of the microspheres in suspension.

C. Flow conditions

Samples were processed through the devices using a syringe pump (PHD 2000, Harvard Apparatus, Holliston, USA). Prior to the experiment, devices were flowed with 1% Pluronic® F-127 (Sigma-Aldrich, St. Louis, USA) solution, diluted in 1× PBS at 100 μl/min for 10 min followed by incubation at room temperature for 30 min to minimize unspecific cell adhesion to microchannel walls.2222. E. Lin, L. Rivera-Baez, S. Fouladdel et al., “High-throughput microfluidic labyrinth for the label-free isolation of circulating tumor cells,” Cell Syst. 5(3), 295–304.e4 (2017). https://doi.org/10.1016/j.cels.2017.08.012 In order to experimentally vary the fluid Reynolds number Ref, four different flow rates were tested: 1.5, 2.5, 3.5, and 5 ml/min. Table I shows the experimental parameter space explored in this study.Table icon

TABLE I. Range of dimensionless parameters used in the study.

Confinement ratio, λ0.04–0.13Fluid Reynolds number, Ref83, 139, 194, 278Particle Reynolds number, Rep0.16–4.41Dean number, De6–53

D. Fluorescent streakline imaging and streak width measurement

Imaging was performed using an Olympus IX81 microscope (Massachusetts, USA). After 1 min of flow stabilization, fluorescence streak images were recorded using a Hamamatsu digital camera (ImagEM X2 EM-CCD, New Jersey, USA). At each flow rate tested, an automated stage (Thorlabs, New Jersey, USA) was used to capture images at all 56 corners and expansion region of the device at multiple time points using the Slidebook 6.1 software (3i Intelligent Imaging Innovations Inc., Denver, USA). Images were taken at ×4 objective magnification with an exposure time of 100 ms and a frame rate of 10 fps. ImageJ software (NIH, Bethesda, USA) was used to obtain a single, mean intensity image from a stack of images at every corner.

Streak width was obtained using ImageJ software (NIH, Bethesda, USA). At the corners, this was measured as the width of the fluorescence streak (in micrometers) along the line connecting the inner and outer corner vertices. Previous attempts to standardize our measurements involved taking a line scan connecting the two corner vertices and calculating the full width at half-maximum (FWHM) of the Gaussian-fit intensity profile.2929. D. R. Gossett and D. Di Carlo, “Particle focusing mechanisms in curving confined flows,” Anal. Chem. 81(20), 8459–8465 (2009). https://doi.org/10.1021/ac901306y However, it was discovered that the fit was below par in many cases. We attribute this to the strongly asymmetric nature of the corner geometry. Moreover, tight curvatures at the turns strongly impact the particle streaks and, in many cases, they were found to deviate significantly from the classical Gaussian intensity profile. Subsequently, we decided to measure the streak width using the above-mentioned method.

E. Computational fluid dynamics simulations

To visualize flow near the corner, 3D CFD fluid simulations were performed using ANSYS Fluent (v. 19.3). The geometry was imported from AutoCAD (v. 2019, Autodesk). We selected a region containing corners 55 and 56 of the labyrinth device. A steady-state viscous laminar model was chosen with water as the working fluid. No slip condition was applied at the microchannel walls. At the inlet, a velocity boundary condition was imposed to match the experimental flow rates while an outflow condition was imposed at the outlet. For discretization, we chose a mesh resolution of 20 μm in the streamwise plane and 10 μm for the cross section. To solve for the flow field, the SIMPLEC algorithm was selected for pressure–velocity coupling along with least squares-based gradient and second order scheme for pressure and momentum. The maximum number of iterations was set to 1000 and convergence tolerance was 10−6. For computing the 2D cross-sectional flow streamlines, the number of spatial points was set to 300.

III. RESULTS AND DISCUSSION

Section:

ChooseTop of pageABSTRACTI. INTRODUCTIONII. EXPERIMENTAL METHODSIII. RESULTS AND DISCUSSI... <<IV. CONCLUSIONSUPPLEMENTARY MATERIALREFERENCESPrevious sectionNext section

A. Dynamics of particle focusing at the turns in the labyrinth device

The geometry of the microfluidic labyrinth device is complex compared to standard curved or spiral geometries. Two important differences are (i) the existence of multiple U-shaped turns in the labyrinth compared to curved geometries [Fig. 1(b)], where the turns might induce strong circulatory forces affecting particle focusing (ii) unlike spiral geometries, where, typically, the curvature ratio δ monotonically decreases from inlet to outlet due to increasing channel radius of curvature, the curvature ratio is non-monotonic in the labyrinth [Fig. 1(c)]. Both U-shaped turns and non-monotonic curvature ratio may lead to variation in focusing as particles traverse through the labyrinth. Thus, we programmed the microscope stage to image particle focusing on the corner locations [Fig. 1(d)], enabling us to understand the role that sharp turns play on determining the particle focusing behavior.To characterize particle focusing at the turns of the labyrinth device, we varied particle diameter dp and the fluid Reynolds number Ref and quantified the dispersity in particle focusing. Polystyrene microspheres of four different sizes: 7, 12, 15, and 20 μm as well as MCF-7 breast cancer cells, WBCs and RBCs were tested at four different flow rates: 1.5,2.5,3.5and5ml/min corresponding to Ref=83,139,194,278, respectively. From the acquired fluorescence streak images [Fig. 1(d)], the streak width ws was measured at every corner [Fig. 1(e)]. To assess the experimental variability in the streak width measurement, we performed three independent trials with the 7 and 12 μm particles at Ref=139 and evaluated the coefficient of variation (CoV) at each of the 56 corners. We found the mean CoV from the 56 corners to be within 9% and 12%, respectively, for 7 and 12 μm particles at Ref=139 (Fig. S1 in the supplementary material), suggesting good reproducibility in streak width measurements.Figure 2 shows the streak widths at all the labyrinth corners for all the conditions investigated. In all cases, even though the general trend is that the streak width declines with corner number, we observe that the streak width data show intermittent fluctuations. Stated differently, even though particles appear to focus at specific locations evident by the lower streak width values, they tend to de-focus at other locations as they traverse through the multiple sharp turns in the labyrinth device. Qualitatively, the degree of these intermittent fluctuations is high for smaller particles and at higher Reynolds numbers. Interestingly, the MCF-7 and WBC cells show the least fluctuations at the lower Reynolds numbers. We note that WBCs at Ref=278 and the RBCs could not focus, so, therefore, streak width data are not shown.

To quantify the degree of fluctuations in the streak width data due to turns, we develop a measure of focusing dispersity FDt. We chose a region of interest (ROI) between corners 30 and 56 in the labyrinth device, away from the high streak width region. We defined the floor of the shaded ROI as the lowest streak width value wmin among all the corners in the labyrinth. The roof is calculated as: (μ+σ), where μ and σ denote the mean and standard deviation of all the streak widths in the ROI. From this, we quantify focusing dispersity as: FDt=hs/dp, where hs (=μ+σ−wmin) is the height of the shaded ROI. The ideal case of tight focusing arises when hs=dp, i.e., FDt=1, which would indicate that the fluctuations in ws are of the same order as particle size dp.

Figure 3 shows the focusing dispersity for all the conditions investigated. For particles, we observe that for any given Ref, FDt decreases with an increase in the particle size. For the 12, 15, and 20 μm particles as well as MCF-7 cells, the focusing dispersity increases at higher Ref=278, suggesting greater variability in focusing. Interestingly, WBCs, despite their smaller size (dp=10μm), displayed lower dispersity (FDt=3–6) compared to 15 μm particles (FDt=8–14). Likewise, the MCF-7 cells (dp=21μm) have slightly better focusing dispersity (FDt=1–4) than 20 μm rigid particles (FDt=2–7). Unlike particles, for both cell types, increasing fluid inertia has a less pronounced effect on the measured FDt.To explain the effect of particle size, we note that for larger particles, stronger inertial lift forces are able to overcome the cross-sectional mixing forces induced by curvature, resulting in tight streaks without much fluctuation as indicated by the lower values of focusing dispersity.14,3014. D. Di Carlo, D. Irimia, R. G. Tompkins, and M. Toner, “Continuous inertial focusing, ordering, and separation of particles in microchannels,” Proc. Natl. Acad. Sci. U.S.A. 104(48), 18892–18897 (2007). https://doi.org/10.1073/pnas.070495810430. B. Ho and L. Leal, “Inertial migration of rigid spheres in two-dimensional unidirectional flows,” J. Fluid Mech. 65(2), 365–400 (1974). https://doi.org/10.1017/S0022112074001431 For small particles, the “mixing” forces dominate, causing higher streak dispersion. At high Ref=278, the increased strength of curvature forces disturbs the particle streaks even further causing significant variability in focusing. In the case of cells, it seems that tight focusing can be achieved that is less sensitive to Ref, probably due to additional lift forces arising from cell deformability.3131. S. C. Hur, N. K. Henderson-MacLennan, E. R. McCabe, and D. Di Carlo, “Deformability-based cell classification and enrichment using inertial microfluidics,” Lab Chip 11(5), 912–920 (2011). https://doi.org/10.1039/c0lc00595a

B. Cross-sectional flow visualization in a U-turn using CFD simulations

From Sec. , we observed that in the labyrinth device as particles are advected downstream, traversing the numerous corners, their streak width measured at the corners fluctuates in a non-monotonic fashion. These fluctuations in streak width could arise due to changes in flow field as the fluid moves from a spiral section into a U-shaped turn and exits into a spiral section. We also evaluated whether the streak width increases or decreases after the entry and exit in the U-turn and did not find any systematic trend suggesting that the turns may not always be beneficial for particle focusing (Fig. S2 in the supplementary material). In this section, we performed three-dimensional CFD simulations to characterize the flow kinematics at various locations in a U-shaped turn. The observed cross-sectional flow fields at various fluid Reynolds numbers help explain the fluctuations in streak width.To study the fluid transition behavior in the U-shaped turn, we chose a representative corner pair 55–56, as shown in Fig. 4(a). We divided the entire geometry into three regions: (1) entry, containing the spiral section prior to corner 55, (2) straight section between corners 55 and 56, and (3) exit consisting of the spiral segment downstream of corner 56. A total of eight cross-sectional cuts are taken spanning these regions as shown in Fig. 4(a). A qualitative view of how the fluid motion changes in the U-shaped turn is shown in Fig. 4(b). The fluid is moving clockwise as it enters the spiral arc and is expected to be pushed toward wall 2. The fluid then changes direction as it traverses through the horizontal section and exits the spiral arc in a counterclockwise direction with the fluid pushed toward wall 1. This change in direction may lead to a cross-sectional flow, the strength of which can depend on the fluid Reynolds number.Figure 4(c) shows the 2D cross-sectional fluid streamlines computed at different downstream distances measured from the corner points, spanning the three regions of interest described above at different Ref. Two features of the flow are evident—a high velocity zone (shown in red) and a vortical flow. We observe that the high velocity zone moves from wall 2 to wall 1 as it enters from the spiral arc into the horizontal section. The vortical flow is most prominent at the higher Ref. We observe the vortices at the entrance of the horizontal section, which appears to vanish mid-way and re-appear at the exit of the spiral arc and disappear downstream.It is now evident that the presence of sharp turns in the labyrinth device leads to the generation of Dean vortices. Given that the labyrinth has numerous such turns, it is expected that the local hydrodynamics in the turn plays an important role in dictating the overall focusing performance of the device. Rather than the Dean vortices aiding in particle focusing at the U-turns, we suggest that these vortices are responsible for causing the streak dispersions discussed in Sec. . It is important to note that Dean vortices are induced by spiral channels also. Conducting CFD fluid simulations, we found that, unlike the sharp U-turn which induces abrupt changes in curvature, the structure of the vortex does not qualitatively change along a spiral arc of constant radius (Fig. S3 in the supplementary material). Therefore, particles traversing spiral sections are not subjected to any fluctuations in the cross-sectional flow field and, thus, remain focused after achieving initial equilibrium.

The ensuing vortex-driven circulatory forces act to disrupt the existing equilibrium positions of particles. Larger particles are associated with stronger inertial lift forces, which help them withstand these destabilizing circulatory forces much more than their smaller counterparts (FL∝dp4). Therefore, their focused streaks are disturbed to a muc

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