Device rationale and fabrication. (a) Sketch representing the passage of a cell in a double constriction device. (i) Spherical cell blocked at the entrance of a rectangular constriction. Red arrows illustrate flow leaks around the cell in the channel corners. (ii) The cell is squeezed in constriction C1. The deformed cell fills the cross-section of the channel and plugs the passage of the fluid around the cell. (iii) Cell deformation during the penetration into constriction C2, L is the projection length of the cell in C2. (iv) End of the stage of cell entrance in constriction C2. (b) Design of the microfluidic device showing the inputs E 1 and E 2 for medium and cell suspensions and the output EO . The junction Ji divides the flow between the analysis circuit (red dashed box), and a bypass (black dashed box). Ji is used to select and isolate cells of interest. The dark circle localizes the portion of the device zoomed in (c). The scale bar corresponds to 500 μm. (c) Design of the double constriction device, showing the entrance and exit zones (grey), the first constriction C1 of cross-section W × H 1 = 6 × 11 μm (green) and the second constriction C2 of cross-section W × H 2 = 6 × 8.5 μm (red). The entrance channel leading to C1 has a radius of curvature Rcu of 150 μm. All channels except C2 have a targeted height of 17 μm. Theblack disk has a diameter of 14 μm like the cells used in the experiments. The scale bar corresponds to 100 μm. (d)SEMmicrograph of the resin mould of double constriction zone (the scale bar corresponds to 20 μm).
Cell force vs. deformation experiment in a double constriction. Series of micrographs showing THP-1 cells of diameter Di = 17 μm (a) flowing freely with the flow in the entrance zone of C1, (b) squeezed during entry in C1, (c) stopped in C1 at the entrance of C2, (d) and (e) transferring from C1 to C2, and (f) after complete entry in C2. The green and red bars in (a) indicate the position along the axis of the transition between constrictions C1 and C2, the black arrow in (a) shows the direction of the fluid flow in the whole experiment, and the yellow arrows point to the back and front edges of the cell. The scale bar corresponds to 20 μm (enhanced online). [URL: http://dx.doi.org/10.1063/1.4802272.1]doi: 10.1063/1.4802272.1.
Reduction of flow leaks around a deformed cell in a double constriction. Micrographs showing the fluid flow, materialized by fluorescent nanoparticles, when a cell has just arrived at the entrance of (a) the constriction of a simple constriction device 5 of size 4 × 16 μm, and (b) the constriction C2 of a double constriction device. The flow is stopped by the cell in (b) but not (a). White dashed lines underline the edges of the cell. The green and red bars in (b) indicate the position along the axis of the transition between constrictions C1 and C2. The scale bar corresponds to 10 μm.
Measurement with a double construction device. (a) Comparison of experimental (dots) and computed (solid line) of the maximum fluid velocities Vmax at two positions in the device, before the bifurcation Ji (red data) and in the analysis circuit (black data) at positions indicated by, respectively, red and black arrows in Fig. 1(b) . (b) Cell projection length in C2, L, versus time, t, of a THP-1 cell of diameter 14 μm at an applied pressure ΔPext = 400 Pa. Stages I, II, and III correspond, respectively, to the entry of the cell front in C2, the transfer of the cell body from constriction C1 to C2, and the final entry of the rear of the cell. The slope in stage II yields the velocity of the projection length, VL . (c) Velocity of the projection length in stage II, VL , against the external applied pressure ΔPext . (d) Cumulative fraction of cells vs. VL at ΔPext = 400 Pa of normal THP-1 cells (dots) and fit (black line) with a log-normal distribution in semi-log representation.
Effect of the size of the cell and of the contact zone on the cell flow. (a) RICM imaging of the cell-wall interface of different THP-1 cells at the end of constriction C2 moving at a steady-state velocity showing: (i) a flat homogeneous contact zone without measurable gutters along the corners, (ii) a contact zone with gutters of width 1.5 μm along the corner, and (iii) an inhomogeneous contact zone. The scale bar corresponds to 10 μm. (b) Cell velocity in C2, VC 2, measured 500 μm away from the entrance of C2, normalized by the mean fluid velocity in C2 in the absence of cells, VMF at the same applied pressure versus the initial diameter of the cell, Di , for different cells and applied pressures. (c) Cell-to-wall distance in the contact zone, h, estimated from RICM images of cells traveling in C2, versus the cell diameter, Di , for different cells and applied pressures. (d) Distance h versus cell velocity in C2, VC 2, for cells of diameters limited to a range of 13–15 μm. Plots (b)–(d) correspond to pooled data at different ΔPext in the range of 300-2000 Pa (enhanced online). [URL: http://dx.doi.org/10.1063/1.4802272.2]doi: 10.1063/1.4802272.2.
Pressure and friction force. (a) Sketch showing the distribution of pressure drop in the analysis circuit with a single cell in the channel. (b) Sketch of the contact zone around a cell squeezed in a channel. Inset is a zoom on the cross-section in the plane (xy) of the gutter. The width of the gutter, G, which is also the radius of curvature of the cell along the corner, can be directly measured in RICM pictures on the interval between the edges of the channel and the contact zone of the cell. (c) Comparison of the section of a bubble or a liquid droplet (yellow) obstructing a channel of side 2R with the case of a living cell (grey). Section plane is (yz) in (i) and (xy) (ii) and (iii). The tip of droplet or bubble has a spherical cap shape with a radius close to R leading to rounded sections with radii R/2 in corners. From our observations, corners of cells are much sharper. (d) ΔPcell * (red circles) and ΔPpiston * (black circles) correspond to ΔPcell and ΔPpiston calculated with Eqs. (8) and (5) and normalized by ΔPac . Data are reported against the diameter of the cells, Di , and correspond to a range of ΔPac between 100 and 1000 Pa. (e) Thickness of lubrication film, hv , estimated from the ratio of Ffriction and Fpressure as calculated using Eqs. (3) and (10) . The red dots correspond to data where ΔPpiston < 0 and ΔPcell > 0, which supports further the validity of Eq. (8) .
Constant viscous dissipation in the vicinity of the step. Sketch representing the different stages of the cell entry from constrictions C1 to C2, (a) stage I, (b) and (c) stage II, and (d) stage III. During stage II, the viscous dissipation due to cell deformation is constant and localized in the vicinity of step (red shadow).
Measurement of the cell loss modulus with a double constriction setup. (a) Apparent cell viscosity, ηapparent measured for different cells versus applied pressure. The grey zone corresponds to the range of pressure where friction is abnormally high. (b) Velocity of a cell in C2, VC 2, versus applied pressure ΔPext for 3 different cells of diameter Di = 15.5 μm (•) and 13.5 μm (▲,○). The solid line is a guide to the eye for the linear behavior at high pressure. The black arrows indicate the threshold of pressure required for moving the cell. The grey area underlines the range of pressures, where VC 2 is abnormally lower than the linear dependence.
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