Flow-Induced Transitions of Red Blood Cell Shapes under Shear

A recent study of red blood cells (RBCs) in shear flow [Lanotte et al., Proc. Natl. Acad. Sci. U.S.A. 113, 13289 (2016)] has demonstrated that RBCs first tumble, then roll, transit to a rolling and tumbling stomatocyte, and finally attain polylobed shapes with increasing shear rate, when the viscosity contrast between cytosol and blood plasma is large enough. Using two different simulation techniques, we construct a state diagram of RBC shapes and dynamics in shear flow as a function of shear rate and viscosity contrast, which is also supported by microfluidic experiments. Furthermore, we illustrate the importance of RBC shear elasticity for its dynamics in flow and show that two different kinds of membrane buckling trigger the transition between subsequent RBC states.

Figure 1

RBC shapes observed in microfluidic experiments ($\lambda \approx 8$) and SDPD simulations ($\lambda \approx 5$) at various dimensionless shear rates ${\dot{\gamma }}^{*}=\dot{\gamma }\tau$ ($\tau \approx 1.2\times {10}^{-3}\mathrm{s}$). The shapes are rolling discocyte, rolling stomatocyte, TB stomatocyte, trilobe, and multilobe, observed at ${\dot{\gamma }}^{*}=0.012$, 0.18, 0.3, 0.9, and 2.15 in experiments and at ${\dot{\gamma }}^{*}=0.014$, 0.18, 0.34, 0.93, and 3.3 in simulations, respectively. Two views, vorticity and flow-gradient directions, are shown by the arrows with unequal and equal lengths, respectively. See also movies S1–S4.

Figure 2

Shapes and dynamics of RBCs in shear flow as a function of ${\dot{\gamma }}^{*}$ and $\lambda$. Different areas, representing rolling discocyte, rolling stomatocyte, TB stomatocyte, TT, and multilobes, are based on simulation results, where dashed lines serve as a guide to the eyes. Two sets of simulations are denoted by triangles (SDPD) and circles (YALES2BIO). The colors indicate RBC shape or dynamics. All simulation data are for Föppl-von Kármán number $\alpha =\mu D^2/\kappa =680$. The two sets of circles at $\lambda \approx 5.3$ and $\lambda \approx 8.3$ correspond to simulations at $\lambda =5$ and $\lambda =8$, respectively, and are just shifted up in the diagram for visual clarity. The square symbols ($\lambda \approx 8$) correspond to experiments from Ref. [22], the plus symbols ($\lambda \le 1$) to data from Ref. [10], and the crosses ($\lambda <1$) to data from Ref. [38]. Diamond symbols ($\lambda \approx 1$ and $\lambda \approx 5$—shifted down to 4.7) represent most probable states from our microfluidic experiments, since no unique state, but a distribution of different states is obtained for fixed flow conditions, see distributions in Fig. S1. All experimental shear rates are normalized by $\tau$ based on average RBC properties (i.e., $D=6.5\times {10}^{-6}\mathrm{m}$ and $\mu =4.8\times {10}^{-6}\mathrm{N}/\mathrm{m}$).

Figure 3

Transition between different shapes and dynamics at $\lambda =5$. (a) Change in RBC total energy, including shear and bending elasticity, from YALES2BIO simulations as a function of shear rate. (b) Change in shear elasticity and bending energies of a RBC compressed between two plates in SDPD simulations. The cell buckles from a discocyte to a stomatocyte shape, when the distance between plates becomes approximately $6.3\mu \mathrm{m}$ (see movie S5).

Figure 4

Trilobe formation. (a) Inclination angle (square and plus symbols, left axis) and aspect ratio (maximum to minimum size, red bullets, right axis) of a RBC as a function of $\lambda$ at ${\dot{\gamma }}^{*}=1.36$ from YALES2BIO simulations. The square and bullet symbols correspond to simulation measurements, while the plus symbols are obtained from the Keller-Skalak theory for fluid vesicles [41] using cell dimensions measured in simulations. (b) Time evolution ($t^{*}=\dot{\gamma }t$) of shapes at $\lambda =3.5$ (starting from a simulation with $\lambda =3.1$), when TT becomes unstable and a trilobe is formed (see movie S6). (c) Membrane’s elastic tension in the direction of the smallest (top) and largest (bottom) local strains.