Red blood cell shape transitions and dynamics in time-dependent capillary flows

Abstract

The dynamics of single red blood cells (RBCs) determine microvascular blood flow by adapting their shape to the flow conditions in the narrow vessels. In this study, we explore the dynamics and shape transitions of RBCs on the cellular scale under confined and unsteady flow conditions using a combination of microfluidic experiments and numerical simulations. Tracking RBCs in a comoving frame in time-dependent flows reveals that the mean transition time from the symmetric croissant to the off-centered, nonsymmetric slipper shape is significantly faster than the opposite shape transition, which exhibits pronounced cell rotations. Complementary simulations indicate that these dynamics depend on the orientation of the RBC membrane in the channel during the time-dependent flow. Moreover, we show how the tank-treading movement of slipper-shaped RBCs in combination with the narrow channel leads to oscillations of the cell's center of mass. The frequency of these oscillations depends on the cell velocity, the viscosity of the surrounding fluid, and the cytosol viscosity. These results provide a potential framework to identify and study pathological changes in RBC properties.

Figure 1

(a) Schematic of the experimental setup. (b) Tracking of an RBC during an increasing pressure drop ramp from 100 mbar to 1000 mbar. While the cell travels through the channel, as shown in the snapshot, the stage speed is adjusted in a feedback loop to keep the cell in the field of view. The velocity of the cell (green line) is calculated for each frame based on the stage velocity (black circles) and the position of the cell in the field of view (magenta coordinates).

Figure 2

Classification of RBC shapes in microfluidic channels under constant flow conditions. (a) Fraction of RBC shapes as a function of the applied pressure drop (bottom axis) and the mean cell velocity (top axis). (b) Representative examples of a croissant (top) and a slipper (bottom) for experiments with p = 100 mbar and p = 600 mbar, corresponding to cell velocities of v ≈ 1 mm s^–1 and v ≈ 5.8 mm s^–1, respectively. The simulation snapshots are obtained at similar velocities. The flow is in x direction, and the scale bars represent a length of 5 μm.

Figure 3

Dynamics of single RBCs in time-dependent flows at t_ramp = 0.5 s for (a) an experiment and (b) a simulation. The left and right columns in (a) and (b) correspond to upward and downward ramps, respectively. In (a), the top panels show the applied pressure drop signal over time, ramping from $p_{\text{low}}$= 100 mbar to $p_{\text{high}}$ = 600 mbar for the upward ramp, and vice versa for the downward situation. The middle panels in (a) show the cell velocity as green line. The gray dashed-dotted lines correspond to analytical solutions of the maximum and mean velocity inside the channel (78). The corresponding simulations in (b) show the cells transitioning from a velocity $v_{\text{low}}$≈ 1 mm s^–1 and $v_{\text{high}}$≈ 5.8 mm s^–1. Vertical dotted lines indicate the start and end of the ramps. The bottom panels in (a) and (b) show the y coordinate of the cell's center of mass. The inset images show the cells at the start and end points of the shape transitions, highlighted by the magenta markers. The horizontal, dashed black lines correspond to the channel center axis. The scale bars represent a length of 5 μm. Corresponding videos are provided in the Supporting Material for the upward experiment and simulation (Videos S1 and S2) and for the downward experiment and simulation (Videos S3 and S4).

Figure 4

Effect of the ramping parameters on the shape transition. Experimental data are shown as boxplots with superimposed individual gray data points. Outliers are marked using “+” symbols. Simulation results are plotted as red symbols. Cells that show pronounced rotations during the shape transitions are highlighted as stars. (a) Transition time $\text{Δ}t$ for an upward ramp as a function of the total pressure difference at a constant ramp duration of t_ramp = 0.5 s. For $\text{Δ}p=500\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$ and $\text{Δ}p=650\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, the velocities in the simulations are $\text{Δ}v\approx 4.7\mathrm{m}\mathrm{m}{\mathrm{s}}^{-1}$ and $\text{Δ}v\approx 6.2\mathrm{m}\mathrm{m}{\mathrm{s}}^{-1}$, respectively. (b) Transition time for different ramping duration for an upward ramp with $\text{Δ}p=500\mathrm{m}\mathrm{b}\mathrm{a}\mathrm{r}$, corresponding to $\text{Δ}v\approx 4.7\mathrm{m}\mathrm{m}{\mathrm{s}}^{-1}$ in the numerical simulation. In (c), all individual measurements for the upward and downward ramp directions are combined. The inset shows the experimentally observed transition times normalized by the mean transition time $\overline{\text{Δ}t}$. (d) Transition length $\text{Δ}x$ as a function of the ramping direction, combining all individual measurements, similar to (c). The individual measurements of $\text{Δ}t$ and $\text{Δ}x$ are shown Figs. S5 and S6 in the Supporting Material, respectively.

Figure 5

Simulated dependency of the downward transition process on the RBC's dimple position for a downward ramp from $v_{\text{high}}$≈ 7.2 mm s^–1 to $v_{\text{low}}$≈ 1 mm s^–1 within t_ramp = 1 s. (a) Simulation snapshots of the RBC at rest (left) and at the two starting positions (middle and right) of the downward ramp at $t=0$. (b) The y coordinate of the cell's center of mass, corresponding to the two scenarios marked by the black and red boxes in (a).

Figure 6

Dynamics of slipper-shaped RBCs. Panel (a) shows the y position of the cell's center of mass for a simulation (top) during a velocity ramp between $v_{\text{low}}$≈ 1 mm s^–1 and $v_{\text{high}}$≈ 5.8 mm s^–1, and for an experiment (bottom) during an upward pressure ramp from $p_{\text{low}}$= 100 mbar to $p_{\text{high}}$ = 600 mbar within t_ramp = 0.5 s. (b) Oscillations of the slipper-shaped cells over one oscillation period T, corresponding to the time windows marked by the rectangular boxes in (a). Red and black lines represent sinusoidal fits to the simulation and experimental results, respectively. (c) Representative image sequences of the slipper-shaped cells for the simulation (top) and experiment (bottom), corresponding to the points in time marked by large symbols in (b). Dashed black lines indicate the channel centerline, and the black dots represent the cell's center of mass. The scale bars represent a length of 5 μm. The frequency and amplitude of the fits as a function of the applied pressure drop or cell velocity are shown in (d) and (e), respectively. Numerical results are plotted for different viscosity ratios λ as red symbols and experiments are shown as black boxplots with gray dots. Outliers are marked using “+” symbols. Panel (f) shows the oscillation frequency as a function of the cell velocity and for different outer viscosities ${\eta }_{\text{o}}$. Error bars correspond to averaging over different measurements at a set pressure drop. Black full symbols correspond to the experimental results, and red open symbols show the simulation results. Simulations are performed with a constant inner viscosity of ${\eta }_{\text{i}}$= 12 mPa s.

Figure 7

The motion of a wrinkle on the RBC surface, marked with cyan circles in the experiment (a) and a simulation (c), at v ≈ 7 mm s^–1. The time between each consecutive image in (a) and (c) is 10 ms and 7.6 ms, respectively. The black dotted line indicates the channel's centerline and dark gray dots represent the cell's center of mass. The white scale bars correspond to a length of 5 μm. The wrinkles in the membrane of the slipper-shaped cell periodically move along the RBC surface. The frequency of this movement is plotted in (b) and (d) as cyan symbols for the experiments and simulations, respectively, together with the frequency of oscillation of the cell's center of mass. Error bars in (b) correspond to averaging over different measurements at a set pressure drop. For the simulation in (c), the positions of the two RBC membrane dimples are highlighted with blue and green markers. The oscillation frequency of the green dimple is plotted in (d).