High-throughput viscoelastic characterization of cells in hyperbolic microchannels

Lab on a Chip PAPER Cite this: Lab Chip,2024,24,2440 Received 10th December 2023, Accepted 2nd April 2024 DOI: 10.1039/d3lc01061a rsc.li/loc High-throughput viscoelastic characterization of cells in hyperbolic microchannels† Felix Reichel, abc Ruchi Goswami, ab Salvatore Girardo ab and Jochen Guck * abc Extensive research has demonstrated the potential of cell viscoelastic properties as intrinsic indicators of cell state, functionality, and disease. For this, several microfluidic techniques have been developed to measure cell viscoelasticity with high-throughput. However, current microchannel designs introduce complex stress distributions on cells, leading to inaccuracies in determining the stress–strain relationship and, consequently, the viscoelastic properties. Here, we introduce a novel approach using hyperbolic microchannels that enable precise measurements under a constant extensional stress and offer a straightforward stress–strain relationship, while operating at a measurement rate of up to 100 cells per second. We quantified the stresses acting in the channels using mechanical calibration particles made from polyacrylamide (PAAm) and found that the measurement buffer, a solution of methyl cellulose and phosphate buffered saline, shows strain-thickening following a power law up to 200 s −1 . By measuring oil droplets with varying viscosities, we successfully detected changes in the relaxation times of the droplets and our approach could be used to get the interfacial tension and viscosity of liquid–liquid droplet systems from the same measurement. We further applied this methodology to PAAm microgel beads, demonstrating the accurate recovery of Young's moduli and the near-ideal elastic behavior of the beads. To explore the influence of altered cell viscoelasticity, we treated HL60 human leukemia cells with latrunculin B and nocodazole, resulting in clear changes in cell stiffness while relaxation times were only minimally affected. In conclusion, our approach offers a streamlined and time-efficient solution for assessing the viscoelastic properties of large cell populations and other microscale soft particles. Introduction Cellular viscoelasticity has emerged as a central indicator of cell state and function, playing an increasingly significant role in disease characterization. 1–6 Established techniques such as micro-pipette aspiration, 1,7 atomic force microscopy, 8–10 or the optical stretcher, 11–14 while effective in revealing changes in cell deformability and relaxation times, operate on timescales of seconds with throughputs limited to a few hundred cells per hour max. This limitation makes them unsuitable for swift diagnostic applications. To address the need for faster measurements, microfluidic techniques have gained prominence. 3,5,15–17 Operating at timescales ranging from below 1 to 100 milliseconds, these platforms offer measurement rates of over 1000 cells per second. However, the quantification of stresses and cell strain in the microchannels remains challenging due to the complex three-dimensional stress distribution on the cell surface. 18,19 Viscoelastic analysis typically necessitates a onedimensional stress–strain relationship to derive mechanistic properties such as elastic moduli, relaxation times, or storage and loss moduli. 5,10,20,21 Moreover, in cell mechanics measurements, polymeric crowding agents are often employed to enhance the viscosity of the cell carrier solution. 5,16,22,23 This augmentation leads to non-Newtonian behavior, including shear-thinning, the occurrence of normal stress differences, or strain-thickening. 24–26 Notably, this behavior is frequently characterized solely in shear, neglecting the qualitatively different behavior in extension. 5,22,23 In this study, we address these challenges by introducing hyperbolic microchannels as a novel approach for accurate cell viscoelastic measurements under well-defined tensile stresses and strains. The flow pattern within the hyperbolic channel geometry creates a region of constant stress, simplifying the analysis of experiments. While hyperbolic 2440 |Lab Chip, 2024, 24, 2440–2453 This journal is © The Royal Society of Chemistry 2024 a Max Planck Institute for the Science of Light, Erlangen, Germany. E-mail: [email protected]g.de b Max-Planck-Zentrum für Physik und Medizin, Erlangen, Germany c Department of Physics, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Erlangen, Germany †Electronic supplementary information (ESI) available. See DOI: https://doi.org/ 10.1039/d3lc01061a Open Access Article. Published on 03 April 2024. Downloaded on 10/27/2025 3:30:27 PM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online View Journal | View Issue Lab Chip, 2024, 24, 2440–2453 | 2441This journal is © The Royal Society of Chemistry 2024 channels have been used before for cell mechanical measurements, the focus has predominantly been on measuring cell deformability, with little attention given to correlating stresses to strains within the hyperbolic region to quantify cell viscoelastic properties. 22,27,28 Our study aims to fill this gap by simultaneously examining both cell deformability and associated stresses within hyperbolic microchannels. Additionally, our methodology extends to the determination of normal stresses in polymeric solutions at varying extension rates. To underscore the method's capability in measuring changes in relaxation times, we conducted experiments using silicone oil droplets of varying viscosities. Our results not only showcase the ability to detect changes in timescales but also highlight the method's utility in retrieving interfacial tension and droplet viscosity from a single experiment. Employing a Kelvin–Voigt model, we analyzed the stress– strain relationship to extract the Young's modulus and relaxation times of cells. Validation of our approach involved measurements on polyacrylamide (PAAm) microgel beads, 29 demonstrating the recovery of expected Young's moduli and indicating near-elastic behavior based on relaxation times. Finally, we used the hyperbolic microchannels to investigate changes in HL60 cell viscoelasticity under treatment with the actin polymerization inhibiting drug latrunculin B (LatB) and microtubule depolymerizing reagent nocodazole. Our findings reveal a concentration-dependent decrease in the cells' Young's modulus, while relaxation times show a slight increase. In conclusion, our results underscore the utility of hyperbolic microchannels for accurate measurements of cell viscoelastic properties in extension at up to 100 cells per second with potential applications in cell research and diagnostics. Theoretical background Stresses and deformations in extensional flow During flow through a converging channel in x-direction with a constant flow rate, Q, the fluid velocity, v, will increase as a Fig. 1 Microfluidic channel design and measurement principle. A) Full design of the microfluidic channel with magnified illustration of the hyperbolic region. The ellipse represents a deformed particle and the stresses in xand yare depicted. The shaded region indicates the ROI for a viscoelasticity measurement in the hyperbolic region. The inlet ROI shown in the full design indicates the region for inlet measurements to correct for optical distortions (see ESI†). B) Trajectory of an exemplary HL60 cell flowing through the hyperbolic region. The detected contours are shown in red. Scale bar represents 10 μm. At the contour of the right-most cell, the strain calculation is explained. C) Velocity trajectories as function of x for single 370 Pa PAAm beads (blue lines) at 0.02 μLs −1 flow rate. The orange line shows a polynomial fit to the data. The dashed, dark-gray line illustrates the theoretical velocity curve predicted by eqn (14). D) Extension rate of the fitted velocity curve in C as function of x. The shaded region highlights the zone of stable extension rate. Lab on a Chip Paper Open Access Article. Published on 03 April 2024. Downloaded on 10/27/2025 3:30:27 PM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online 2442 |Lab Chip, 2024, 24, 2440–2453 This journal is © The Royal Society of Chemistry 2024 function of x. The rate of this velocity increase is called the extension rate, : _ ε¼∂v ∂x(1) A planar, extensional flow field is defined with the following velocity components: v x =x,v y =−y,v z = 0 (2) at a constant extension rate. This results in a tensile stress component in x-direction, σ x , and compressive stress component in y-direction, σy(see Fig. 1A). The net stress acting on an object in this flow field is defined as: N 1 =σ x −σ y . (3) The measure N 1 is known in rheology as the first normal stress difference and is an intrinsic characteristic of a material's response to non-uniform flow conditions. 30,31 The stress in z-direction is zero: σ z = 0. For Newtonian liquids, there is a linear relationship between N 1 , the extension rate, and the shear viscosity, η. For planar extension, the case defined by eqn (2), the relation reads N 1 =4η. For nonNewtonian liquids, N 1 usually cannot be derived from ηand the behavior in extension has to be characterized independently. Generally, the relation between N 1 and the extension rate is described by the extensional viscosity η E :N 1 =η E and η E will often be dependent on . 25,26 For this stress definition, we can define a corresponding net strain as: ε=ε x −ε y . (4) Assuming that the particle is initially a sphere, it is expected to deform into an ellipsoidal shape, in first order approximation, when flowing in the extensional field. 32–35 We define the strain in xand yby the deviation of the contour from the initial sphere radius R 0 : εx¼a 2R0 −1;εy¼b 2R0 −1;(5) where aand bare the major and minor axes of the ellipse. Because there is no stress in z, the strain in z-direction is expected to be zero: ε z = 0. The resulting net strain can be computed from aand b, assuming the volume of the particle stays constant during the deformation: ε¼a−b ffiffiffiffiffi ab p:(6) The shapes and strain of an example HL60 cell flowing through the extensional field is shown in Fig. 1B. Deformation of liquid droplets in extensional flow The deformation of a liquid droplet flowing in an extensional field with interfacial tension, γ, droplet viscosity, η drop , and viscosity of the carrier solution, η 0 , is well described in the literature. 32,33,36 Here, we use the derivations from Cabral and Hudson to compute γand η drop from the droplet deformation and extension rate. 36 The droplet deformation is defined by the Taylor deformation: 32 D¼a−b aþb:(7) The interfacial tension at the steady-state deformation, D ∞ , can be computed with: γ¼5 2þ3αη 0 _ εR0 D∞ ;(8) with the viscosity ratio =η drop /η 0 , and the effective viscosity αη 0 with: α¼2þ3ðÞ 19þ16 40 þ1ðÞ :(9) In the absence of flow or at uniform flow velocity, a droplet with deformation D 0 at time t= 0 will relax towards equilibrium following an exponential: 36 D(t)=D 0 e −t/τ , (10) with the droplet relaxation time, τ. The relaxation time is related to the other system parameters via: 33,36 γ¼aR0 τmax η0;ηdrop  :(11) τand D ∞ can be derived from the time evolution of the droplet deformation at constant extension rate. We assume that R 0 and η 0 are known and that η 0 <η drop .γand η drop can be derived by equating eqn (8) and (11) (see ESI†text). Kelvin–Voigt model analysis We characterized the viscoelasticity of microgel beads and cells using the Kelvin–Voigt material model: σ¼Eεþη∂ε ∂t:(12) Here, σis a one-dimensional stress, Ean apparent elastic modulus and ηan apparent bulk viscosity. For a general stress function in time, σ(t), and an initial strain, ε 0 ,att=0, the general solution of the model is: εtðÞ¼e−t=τε0þ1 ηðt 0 σvðÞev=τdv  ;(13) with the relaxation time τ=η/E. Materials and methods Microfluidic chip design The entire chip geometry is shown in Fig. 1A. Flow is introduced via FEP-tubing into the sheath and sample inlet. The sample particles are only delivered to the channel through the sample inlet and the sheath only contains the Lab on a ChipPaper Open Access Article. Published on 03 April 2024. Downloaded on 10/27/2025 3:30:27 PM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online Lab Chip, 2024, 24, 2440–2453 | 2443This journal is © The Royal Society of Chemistry 2024 carrier solution. At the point where sheath and sample flow meet, the sample particles get focused into the channel center before the measurement region (shown as zoomed-in inset). The dotted structures after the inlets show filter pillars that prevent large particles or particle aggregates from clogging or contaminating the measurement region. After sheath focusing, the channel continues straight for 2.4 mm. This was introduced to give particles enough time (several seconds depending on flow rate) to relax from the deformation that is induced by the sheath focusing before reaching the measurement region. The channel height of all chips used for this study was 30 μm. This height was chosen to ensure cells with an approximate diameter of 15 μm will be focused at the flow centerline. For higher channels, cells would take up a lateral equilibrium position off the centerline. 37,38 The measurement region, also called hyperbolic region, was designed to ensure a linear increase of the centerline velocity along the x-axis. The centerline velocity, u 0 , was computed with the following equation: 39 u0¼3 2 Q Hw−0:63HðÞ ;(14) with flow rate Q, channel height, H, and channel width, w, where w=w(x). Eqn (14) is a good approximation for u 0 when w≫H. To ensure a constant extension rate ,w(x) must be constructed such that ∂u 0 /∂x== const. This leads to the following equation for the channel width in the hyperbolic region: wxðÞ¼0:63H−2H 3Qxþ1 0:63H−wc  −1 ;(15) with the contraction width, w c (see Fig. 1A and ESI†text). A channel geometry is fully defined by the upper contraction width, w u , lower contraction width, w c , and contraction length, L c The measures relate to the flow parameters with the following relation (see ESI†text): 2H_ ε 3Q¼1 Lc 1 0:63H−wu −1 0:63H−wc  :(16) The resulting velocity as function of xfor PAAm microgel beads with a Young's modulus of 370 Pa at a flow rate of 0.02 μLs −1 are shown in Fig. 1C. The blue lines depict velocity trajectories of single beads and the orange line shows a 12thorder polynomial fit to the data. We chose the 12th order to avoid underfitting of the data which has strong effects on the resulting extension rates. The velocity curve predicted by eqn (14) is illustrated by a dashed gray line. It can be seen that the measured velocities deviate from the predicted curve especially when getting closer to x=L c . Eqn (14) describes the fluid velocity while the data was measured from the bead velocity. It is expected that a particle with a finite dimension will move slower through a channel than the liquid itself. 40 Furthermore, eqn (14) is an approximation for rectangular channels and the deviation from the true velocity depends on the channels' aspect ratio. 39 The aspect ratio of the cross section changes along x, causing an additional deviation from the predicted values. The extension rates for the sample was computed from the fitted velocity curve and is shown in Fig. 1D. After the hyperbolic contraction, the channel continues straight for about 2 mm to observe the relaxation from the previous stress. Before the waste outlet, an expansion region mirroring the hyperbolic contraction was added to observe objects in expansion. We provide the mask for our chip design in the ESI.† Device fabrication Microfluidic chips were produced using standard soft lithography methods with polydimethylsiloxane (PDMS). To produce the master mold, a 30 μm-thick layer of AZ®15nXT (450 CPS) photoresist (MicroChemicals GmbH, Germany) was spin-coated onto a 4-inch silicon wafer (2000 rpm, 5000 rpm s −1 , for 2.8 s) using a spin coater (WS-650Mz-23NPP, Laurell, USA) and soft-baked at 110 °C for 3 minutes. The chip design, depicted on a chromium photomask, was transferred on the coated substrate via UV exposure at 600 mJ cm −2 (MA6 Gen4, Suess MicroTec GmbH, Germany), followed by a postbaking at 120 °C for 1 minute and development in a 1 : 3 dilute solution of AZ®400 K developer (MicroChemicals GmbH, Germany) in deionized water for 150 seconds. The final master mold received a hydrophobic coating through vapor deposition of 1H,1H,2H,2H-perfluorooctyltrichlorosilane (Sigma-Aldrich 448931, CAS 78560-45-9) in a vacuum desiccator for 1 day, facilitating the easy peel-off of replicas during replica molding. PDMS (base to curing agent ratio of 10 :1 w/w; VWR 634165S, SYLGARD 184) was poured over the master, degassed, and cured at 78 °C for 1 hour and 15 minutes. Following the cutting of the chips and the creation of 1.5 mm inlet and outlet holes, the PDMS replica was bonded to a glass cover slip (40 ×24 mm 2 , thickness 2, Hecht, Germany) through an air plasma treatment (Plasma system Atto, Diener Electronic, Germany) at 75 W for 2 min, 20 sccm air flow and 0.4 mbar pressure. Production of carrier media Measurements on oil droplets were performed by dispersing the concentrated droplet solution into a mixture of 83.8 v/v% glycerol (Sigma-Aldrich G5516, CAS 56-81-5) with water. This concentration of glycerol corresponds to a dynamic viscosity of 100 mPa s at 25 °C. 41,42 Before measurements, cells and beads were suspended into a 0.6 w/w% solution of methyl cellulose (MC) (4000 cPs, Alfa Aesar 036718.22, CAS 9004-67-5) dissolved in phosphate buffered saline (PBS) (Gibco Dulbecco 14190144) following the protocol of Büyükurgancıet al. 16,24 The osmolality of PBS was adjusted to values between 280–290 mOsm kg −1 before the addition of MC (VAPRO Vapor Pressure Osmometer 5600, Wescor, USA). To prepare MC–PBS solutions for 1 L of PBS (1010 g), 6.00 g MC powder were utilized, resulting in a 0.594 w/w% MC concentration in the final solution. Following the Lab on a Chip Paper Open Access Article. Published on 03 April 2024. Downloaded on 10/27/2025 3:30:27 PM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online 2444 |Lab Chip,2024,24,2440–2453 This journal is © The Royal Society of Chemistry 2024 addition of the powder, the mixtures were maintained at 4 °C under continuous rotation for two days to allow for the dissolution of MC. This process involved placing a rotating mixer (RS-TR05, Phoenix Instrument) in a high-performance pharmacy refrigerator (TSX Series, Thermo Fisher Scientific), with the bottles containing the MC–PBS solution rotating inside. Subsequently, the pH value was adjusted to 7.4 (Orion Star A211 benchtop pH meter, Thermo Scientific) by the addition of NaOH. In the next step, the solutions were sterilefiltered using membrane filters (Merck Millipore Steritop-GP polyethersulfone (PES) membrane, 0.22 μm pore size). The buffer solutions were then stored at 4 °C. Latrunculin B and nocodazole treatments. A stock solution of latrunculin B (Sigma-Aldrich L5288, CAS 7634394-7) was prepared by dissolving the powder in DMSO at a concentration of 10 mM. This stock solution was then further diluted in DMSO to 10 000 times the desired concentration, ensuring an equivalent DMSO concentration in all treatments (0.01 v/v%). Subsequently, LatB was diluted by a factor of 10 000 in a 0.6 w% MC–PBS buffer, resulting in final LatB concentrations of 0.1, 1, 5, 10, 25, 50, 100, and 250 nM. The same approach was used for nocodazole (Sigma-Aldrich SML1665, CAS 31430-18-9) to get a final concentration of 1 μM. Production of PAAm-microgel beads PAAm beads were produced according to the protocol introduced by Girardo et al. 29 Briefly, a PDMS-based flowfocusing microfluidic chip was used to produce polyacrylamide pre-gel droplets in fluorinated oil (3 M™ Novec™7500, Iolitec Ionic Liquids Technologies GmbH, Germany). The fluorinated oil contained ammonium Krytox® surfactant (2.4 w/v%) as the emulsion stabilizer and 0.4 v/v% N,N,N′,N′-tetramethylethylenediamine (TEMED) (SigmaAldrich T9281, CAS 110-18-9) as the catalyst. The pre-gel mixture contained acrylamide (40 w/v%) (Sigma-Aldrich A8887) as the monomer, bis-acrylamide (2 w/v%) (SigmaAldrich 146072) as the crosslinker, and ammonium persulphate (0.05 w/v%) (Cytiva GE17-1311-01) as the free radical initiator, diluted in 10 mM Tris buffer (pH = 7.48). The total monomer concentration, %c T , in the pre-gel mixture together with the droplet diameter was fine-tuned to obtain beads with precise diameter and elasticity within the desired range. After in-drop polymerization at 65 °C, the beads were washed and resuspended in 1 ×PBS (pH = 7.4, Gibco), and ultimately stored at 4 °C. A list of the PAAm beads used in this study is reported with both their total monomer concentration (%c T ), mean diameter, and Young's modulus in Table S1.†The Young's modulus of the beads was measured with real-time deformability cytometry (RT-DC, see ESI†text). 16,19,24 Production of silicone oil droplets The same chip design and setup employed in the PAAm bead production was also used for generating silicone oil droplets. This process involved a PDMS-based chip with a channel height of 14 μm and channel width of 15 μm at the cross junction. Before starting the droplet production, the microfluidic chips underwent a treatment by an air plasma (Plasma system Atto, Diener Electronic, Germany) at 75 W for 4 min, 20 sccm air flow and 0.4 mbar pressure. This treatment rendered the surface of the PDMS channel walls hydrophilic, a crucial step to enable on-chip production of an oil-in-water microemulsion. 43 Two distinct types of droplets were generated using two silicone oils with kinematic viscosities of 500 and 1000 cSt (Sigma-Aldrich 378380 & 378399, CAS 63148-62-9) as the dispersed phase. The continuous phase was a aqueous solution of 2 v/v% poly(ethylene glycol)monooleate (SigmaAldrich 460176, CAS 9004-96-0) used as a emulsifier. Droplets with a kinematic viscosity of 500 cSt were produced under a set pressure of 480 mbar for the continuous phase and 310 mbar for the dispersed phase, resulting in droplets with an average diameter of 16 μmata production rate of 13.5 Hz. In the case of 1000 cSt droplets, both the continuous and dispersed phases were pressurized at 400 mbar, resulting in droplets with an average diameter of 15.5 μm at a rate of 27 Hz. Droplets were collected inside a 15 mL Falcon tube. Cell culture The HL60/S4 cell subline (ATCC Cat# CRL-3306, RRID:CVCL_II77) was cultured in RPMI 1640 medium with 2 mM L-glutamine (Thermo Fisher #A1049101) with 1% penicillin and streptomycin (Gibco) and 10% heat-inactivated fetal bovine serum (Sigma Aldrich F4135, lot no. 13C519). Cells were grown at 37 °C, with 5% CO 2 , at densities between 10 5 – 10 6 cells per mL with subculturing every 48–72 hours. Cells used for experiments originated all from the same frozen batch after thawing and were measured in 14–26 passages after initial seeding. Experimental protocol Measurements were performed using an AcCellerator device with temperature control (Zellmechanik Dresden GmbH, Germany). This device offers high-speed imaging with online contour analysis through the software ShapeIn 2 (Zellmechanik Dresden). All measurements were performed at a chamber temperature of 25 ± 0.5 °C. For a measurement, the microfluidic chip was placed on the stage of an inverted microscope (Axio Observer Z1, Zeiss, Germany) and flow was introduced using syringe pumps (neMESyS 290 N, Cetoni GmbH, Germany) connected to the chip with FEP tubing (0.0625″OD, 0.03″ID; no. 1520XL, Postnova Analytics). The sheath to sample flow rate ratio was 1.5 : 1 for oil droplet experiments and 3 : 1 for all other measurements. Imaging was done with a pulsed, high power, blue LED (Zellmechanik Dresden) that was synchronized to a CMOS camera (EoSens CL, MC1362, Mikrotron GmbH, Germany). Lab on a ChipPaper Open Access Article. Published on 03 April 2024. Downloaded on 10/27/2025 3:30:27 PM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online Lab Chip, 2024, 24, 2440–2453 | 2445This journal is © The Royal Society of Chemistry 2024 Fig. 2 Stress characterization of 0.6 w% MC–PBS in the hyperbolic region. A) ROIs for the different stress measurements. The red region in ROI hyper represents the analysis window for the strain data shown in B. B) Strain and velocities of 300 Pa PAAm beads in ROI hyper at different flow rates. Data points represent the median values of all beads in the measurement at the respective x-position. Error bars in all panels indicate standard error of the mean (SEM) C) median strain of PAAm beads with varying stiffness as function of extension rate in the strain analysis region. D) Median strain and apparent shear stress scaled by bead size for different beads in ROI inlet. The line in the scaled stress plot shows a linear fit to the data with 2-σerror band (eqn (23)). E) Median strain and apparent shear stress scaled by bead size for different beads in ROI channel. The line in the scaled stress plot shows a power law fit to the data with 2-σerror band (eqn (24)). F) First normal stress difference N 1 () for different beads measured in ROI hyper. The line shows a linear fit to the data with 2-σerror band (eqn (27)). G) Shear stress, first normal stress difference, and total stress, calculated from the median extension rate curves with eqn (25), (27), and (28), for 832 Pa PAAm beads flowing through the hyperbolic region (see Fig. 1A) at a flow rate of 0.04 μLs −1 . Lab on a Chip Paper Open Access Article. Published on 03 April 2024. Downloaded on 10/27/2025 3:30:27 PM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online 2446 |Lab Chip, 2024, 24, 2440–2453 This journal is © The Royal Society of Chemistry 2024 Experiments for the viscoelasticity of PAAm beads, oil droplets and HL60 cells were imaged through a 20×objective (Plan-Apochromat, ×20/0.8; no. 420650-9902, Zeiss). To measure the stresses acting in the channel based on PAAm bead deformation (see Fig. 2), we used a 40×objective (ECPlan-Neofluar, ×40/0.75; no. 420360-9900, Zeiss). Tubings for sheath and sample were filled with the respective carrier solution at 0.1 μLs −1 before connecting to the chip and then chips were filled through the sheath inlet using the syringe pump until liquid was pushed out of the sample inlet. Sample suspensions were loaded into the tubing by placing the tubing inside the suspension and setting a negative flow rate of −2μLs −1 for the sample. The filled chip was then connected to the filled sample tubing to avoid the introduction of air bubbles into the chip. The samples were pushed into the chip at a flow rate of 0.1 μLs −1 before setting the measurement flow rate. Oil droplets. Oil droplets accumulated at the top of the emulsion inside the storage container. 1 μL of this concentrated droplet emulsion was pipetted into 50 μL of the glycerol–water mixture and mixed by up-down pipetting before loading into the sample tubing. The silicone oil droplet samples were measured at a flow rate of 0.04 μLs −1 . PAAm beads. PAAm beads accumulated at the bottom of the storage Eppendorf tube. 1–2μL of this concentrated bead suspension was pipetted into 98–99 μL of 0.6 w% MC–PBS and mixed by up-down pipetting and vortexing before loading into the sample tubing. HL60 cells. HL60 cells were counted on the measurement day and then a volume of the cell culture medium was centrifuged at 200 ×gfor 2 minutes to reach a final concentration of 1–2×10 6 cells per mL in the cell carrier solution. After centrifugation, the supernatant was taken off and the cell pellet was re-suspended in 100 μL of 0.6 w% MC–PBS, containing the respective amount of LatB or nocodazole, and mixed by up-down pipetting. The samples were then stored for an incubation time of 30 minutes at 25 °C inside the heating chamber of the AcCellerator before loading into the sample tubing. Data acquisition Data was recorded using the program ShapeIn 2 (Zellmechanik Dresden). The program analyses the images in real-time by finding a contour based on background subtraction and thresholding. 16 A number of contour and brightness features are extracted from the image and saved to an hdf5 file. For a viscoelasticity experiment in the hyperbolic region, objects were recorded in a 680 ×54 μm region of interest (ROI, see Fig. 1A), imaged through the 20×objective. The border of the ROI was placed 50 μm after the end of the hyperbolic region (x=L c +50μm). The frame rate was adjusted according to the flow rate to capture about 50 events per object flowing through the ROI. PAAm beads and silicone oil droplets showed optical distortions dependent on the position in the ROI. To correct for these effects, we measured each sample 500 μm before the hyperbolic region with an ROI of 680 ×102 μm at a flow rate of 0.01 μLs −1 (see inlet ROI in Fig. 1A). In this region, no extensional field is present and at this flow rate, the influence of shear stresses is negligible. The measured deviations in the deformation can be attributed to optical distortions in the setup. The correction curves are shown in Fig. S4–S6.† Stress measurements. For the stress measurements on PAAm beads inside the hyperbolic region, we used an ROI of 230 ×30 μm, and for measurements in the inlet or channel 102 ×34 μm (see Fig. 2A), imaged through the 40×objective. We used a lager ROI for measurements in the hyperbolic region to measure multiple events per bead, to determine bead velocities and derive extension rates. ROI hyper in Fig. 2A shows the window used for the strain analysis (see Results section). Data analysis The full analysis pipeline is documented at https://gitlab. gwdg.de/cell-viscoelasticity-in-hyperbolic-channels. Object tracking. The data from the hdf5 files was tracked to get trajectories for every single object using a custommade python package called dctrack. Velocities and extension rates were computed from the tracked object trajectories. Velocities were computed for single objects from the displacement change between frames. The velocities were then assigned to the second frame and no velocity data is recorded for the start of the ROI. The combined velocity data of the sample was then fitted to a 12th order polynomial to get v(x). Event times were calculated from the velocity curve v(x) in reference to a point x 0 with: txðÞ¼ðx x0 dχ vχðÞ :(17) Strain computation. Object contours only extend a few pixels in xand yand the computation of the strain from the raw contour bounding box size is very susceptible to noise in the contour detection. To have a more robust measure, the strain was calculated from the second moments of area of the raw contour over xand ydefined by: 44 Ix¼ððA y2dxdy;Iy¼ððA x2dxdy;(18) with the contour area, A. The ratio of I¼ffiffiffiffiffiffiffiffiffiffi Iy=Ix pis readily available as a feature in the measurement hdf5 files and known as the inertia ratio. For a perfect ellipse or rectangle, I is equal to the aspect ratio of the contour. The strain definition in eqn (6) can then be rewritten using the inertia ratio: Lab on a ChipPaper Open Access Article. Published on 03 April 2024. Downloaded on 10/27/2025 3:30:27 PM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online Lab Chip,2024,24,2440–2453 | 2447This journal is © The Royal Society of Chemistry 2024 ε¼I−1 ffiffiI p:(19) Data binning. Data curves as function of xor time as shown in Fig. 1–5 were made by creating equal sized bins in xand computing the median from all datapoints inside the respective x-bin. Datapoints represent the median value of all objects at the respective x-position in the measurement. Results and discussion Characterizing stresses in the hyperbolic region The first normal stress differences of 0.6 w% MC–PBS in extensional flow were not reported before. We measured the stresses acting in the hyperbolic region using PAAm microgel beads. The Young's modulus, E, of the beads was measured using RT-DC (see ESI†text and Fig. S1). The stress was then deducted from the strain measured in the hyperbolic region: σ=Eε. (20) To find a relation of N 1 and the extension rate , beads were measured at different flow rates and the ROI was placed 225 μm after the start of the hyperbolic region (see “ROI hyper” in Fig. 2A). The ROI was chosen to ensure beads were measured at a constant extension rate and had enough time to comply to the acting stress. The relaxation time of the beads was reported before at around 0.1 ms, 29 allowing them to fully deform during the several milliseconds they spend in the ROI. Fig. 2B shows the median strain and velocity curves of beads with E= 300 Pa. Medians were computed from all beads at a respective x-position for all flow rates and each curve represents 400–2400 beads (more beads at higher flow rates). We report the number of beads included at every flow condition in Tables S2 and S3.† As discussed below, the velocity gradient between top and bottom wall causes shear stresses on the beads. The influence of these shear stresses on bead deformation increases with velocity and during flow through the hyperbolic region. To keep the influence of shear stresses on the bead deformation as small as possible when getting closer to x=L c , the analysis region was restricted to 250 <x <325 μm. It can be seen that the strain is constant over xin the chosen analysis region for all flow rates, indicating the stress in the chosen ROI can be considered constant. The velocity curves were analyzed from the full ROI width of 230 μm. In Fig. 2B we show the median velocities for 250 μm<x. The extension rates were computed from the slope of a linear fit to the velocity data. The resulting median strains and extension rates for three bead types of different stiffness are shown in Fig. 2C. The further analysis was restricted to small strains with ε<0.125 to stay within the linear elastic regime. For larger strains we observed deviations from linearity for the beads with E= 300 Pa (see Fig. S2E†). The strain values were corrected for influences of optical distortions from the instrument (see Fig. S2A and B†). Typical object diameters for our experiments were in the range of 10–20 μm. At a channel height of 30 μm, the influence of shear stresses on the deformation cannot be neglected. To quantify this influence, we measured the deformation of the beads 500 μm before entering and after leaving the hyperbolic region (see “ROI inlet”and “ROI channel”in Fig. 2A). The resulting strains are shown in Fig. 2D and E and S2C and D.†The apparent shear stress, σ s , can then be calculated similarly to eqn (20). The shear stress on an object inside the channel is sizedependent because shear stresses increase closer to the channel walls. We found empirically that scaling the apparent stresses quadratically by size resulted in the datapoints collapsing on one line (see Fig. 2D and E). We performed the scaling with the confinement λdefined by: λ¼object diameter channel height :(21) The scaled stress σ λ is defined by: σλ¼σs λ2:(22) The data for σ λ (Q) at the inlet was best described by a linear function while it showed shear-thinning power law behavior inside the channel (Fig. 2D and E). Fits to the data resulted in the following functions for the shear influence: σs;inlet Q;λðÞ¼138 ±5ðÞ·Q Q0 ·λ2Pa (23) σs;channel Q;λðÞ¼1283 ±67ðÞ·Q Q0  0:88±0:02 ·λ2Pa;(24) with Q 0 =1μLs −1 . We assumed that the power law exponent presented in eqn (24) can be used to describe the relation of shear stresses and object velocity inside the hyperbolic region. Since the velocity increases approximately linear as a function of x, the apparent shear stress at any point xin the contraction can then be calculated with: σsxðÞ¼σs;channel −σs;inlet  x Lc  0:88 þσs;inlet:(25) Considering the influence of the shear stress, the first normal stress difference can then be computed from the data in Fig. 2C, eqn (20) and correcting for σ s : N 1 =Eε−σ s . (26) The resulting values for N 1 as a function of the extension rate are shown in Fig. 2F. It can be seen that the datapoints from different bead types all collapse on one line and follow a power law with: Lab on a Chip Paper Open Access Article. Published on 03 April 2024. Downloaded on 10/27/2025 3:30:27 PM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online 2448 |Lab Chip, 2024, 24, 2440–2453 This journal is © The Royal Society of Chemistry 2024 N1 _ εðÞ¼ 0:16 ±0:03ðÞ· _ ε _ ε0  1:22±0:04 Pa;(27) with  0 =1s −1 . A power law index greater than 1 describes strain-thickening behavior and is commonly observed for polymer solutions. 26,31 It is noticeable that the extensional viscosity of the solution (see Fig. S3†) is much higher than the zero-shear viscosity, which is reported at 30 mPa s, highlighting the need for distinct measurements in extension to accurately determine the stresses exerted by MC-solutions, or polymer solutions in general. Finally, the total stress on an object at any position xinside the hyperbolic region can be calculated from the extension rate (x), the flow rate, and the confinement: σ total (x,,Q,λ)=N 1 ()+σ s (x,Q,λ). (28) The stress curves for beads with an apparent Young's modulus of 832 Pa and a radius of 6.6 μm at a flow rate of 0.04 μLs −1 are shown in Fig. 2G. It can be seen that the total stress will stay stable after about 150 μm. Mechanics of silicone oil droplets To verify that our approach is sensitive to detect different relaxation times, we measured silicone oil droplets with different viscosities. The lower viscosity droplets had a dynamic viscosity of 485 mPa s and the higher viscosity sample 970 mPa s, according to the manufacturer. Droplets were analyzed by the Taylor deformation as introduced in eqn (7). The full deformation and extension rate curves as function of xare shown in Fig. S4.†A stable extension rate was achieved after x 0 = 110 μm. The droplet deformation was analyzed in the range 110 μm<x<390 μm. The upper limit for xwas chosen to avoid strong effects of the shear stress influencing the deformation. The resulting Taylor deformations as a function of time, where t(x 0 ) = 0, are shown in Fig. 3. According to eqn (10), the data follows an exponential and will equilibrate at D ∞ .Att= 0, the data shows a predeformation D 0 caused by stresses acting on the droplets before reaching x 0 . With these, the deformation curve will have the form: D T (t)=D ∞ (1 −e −t/τ )+D 0 e −t/τ . (29) Eqn (29) was fitted to the data to get D ∞ and the relaxation time, τ. We found τ(485 mPa s) = 18 ± 4 ms and τ(970 mPa s) = 52 ± 25 ms, highlighting that we are sensitive to different time scales of droplet deformation. Error margins show standard errors of the fit. τ(970 mPa s) has a large error margin of almost 50%. This large uncertainty can be explained because the estimated relaxation time is much larger than the observation time of about 12 ms, which leads to inaccuracies when fitting the exponential. Utilizing the analysis in eqn (8)–(11), we determined the interfacial tensions and viscosities of the droplet systems. For this, the extension rate was taken as a constant average, calculated from all datapoints in the analysis region. For the 485 mPa s droplets, we got an interfacial tension of γ= 1.0 ± 0.1 mN m −1 and the viscosity η drop = 552 ± 73 mPa s, which resulted in the expected viscosity of the silicone oil, within error margins. For 970 mPa s droplets we got γ= 0.6 ± 0.2 mN m −1 and η drop = 711 ± 212 mPa s. The deviation of the resulting viscosity from the expected value can be explained by the large uncertainty of the relaxation time. Both emulsified droplet systems showed a small interfacial tension of around 1 mN m −1 . Similar values for such interfaces have been reported before. 45 Viscoelasticity of PAAm-microgel beads The strain curves of PAAm-microgel beads with Young's moduli of 379, 669, and 832 Pa were measured at flow rates from 0.04–0.10 μLs −1 and analyzed for their viscoelastic behavior with the Kelvin–Voigt model explained in eqn (13). Object times were calculated in reference to x 0 =−40 μm. The analysis was restricted to values for −40 μm<x<490 μm. Strain curves were corrected for optical distortions as shown in Fig. S5A.†An example stress and strain curve as function of time with Kelvin–Voigt fit for 669 Pa beads at 0.04 μLs −1 is shown in Fig. 4A. The data for all conditions are shown in Fig. S5B.† Fig. 4B–D shows the Young's moduli, relaxation times and bulk viscosities resulting from the fit. The error bars indicate standard errors of the fit. Young's moduli showed a slight increase with flow rate for all bead types and within error margins, the expected values for all conditions could be recaptured, showing that our approach is able to accurately measure object stiffness. Relaxation times were longer for stiffer beads and showed no clear trend with flow rate. The bulk viscosities followed the same trend as the relaxation times. Fig. 3 Taylor deformation vs. time for silicone oil droplets at the stable extension rate. The solid lines illustrate fit curves of eqn (29) to the datapoints. Shaded regions show the 2-σerror band of the fit data. The number of droplets Nper condition was: N 485 = 90, N 970 = 421. Lab on a ChipPaper Open Access Article. Published on 03 April 2024. Downloaded on 10/27/2025 3:30:27 PM. This article is licensed under a Creative Commons Attribution 3.0 Unported Licence. View Article Online