Abstract
We reveal that under moderate shear stress (_ 0:1 Pa) red blood cells present an oscillation of their inclination (swinging) superimposed to the long-observed steady tank treading (TT) motion. A model based on a fluid ellipsoid surrounded by a viscoelastic membrane initially unstrained (shape memory) predicts all observed features of the motion: an increase of both swinging amplitude and period (1=2 the TT period) upon decreasing _ , a _-triggered transition toward a narrow _ range intermittent regime of successive swinging and tumbling, and a pure tumbling at low _ values. A human red blood cell (RBC) is a biconcave disk-shaped membrane encapsulating a Newtonian solution of hemoglobin. The membrane is composed by a fluid incom-pressible lipid bilayer underlined by a thin elastic cytoske-leton [1]. This complex structure determines the RBC behavior in shear flow, which greatly influences flow and mass transport in the microcirculation in both health and disease [2]. However, this behavior is not well understood yet and important questions remain open. First, the state of deformation of the elastic membrane at rest is still debated. Does RBC present shape memory (membrane elements of the rim and the dimples nonmechanically equivalent) as recently suggested [3]? Second, flowing RBCs were observed [4-7] only when the cells were suspended in plasma and present an unsteady tumbling (T) solidlike motion [4] or when they are subjected to a high shear stress and exhibit a droplike tank treading (TT) motion characterized by a steady orientation and membrane rotation about the internal fluid. The RBC regime of motion at smaller shear-stress and close to the T-TT transition has not been studied, although it is of crucial importance. Indeed, the simplest models, which treat RBCs like fluid ellipsoids [7-9] retrieve T and TT motions but do not capture the shear-rate dependency of the transition, thus raising the question of the role of the elasticity of the membrane on the cell behavior. Here, by using a recent method of cell imaging parallel to the shear plane [10], we explore the RBC movement close to the T-TT transition. In the TT regime, we reveal that RBCs present an oscillation of their inclination with a period equal to half the TT period that we name swinging (S). We characterize the shear-stress dependence of this oscillation down to the T-TT transition. We show that the transition to pure T is preceded by a narrow critical shear-stress regime where the RBC exhibits an intermittent S-T behavior. Finally, we propose a model, assuming an elastic nonspherical RBC membrane, which captures the main features of the observed behavior. Materials and methods.-Direct measurements of the orientation of the cells with respect to the flow direction (angle) and cell shape (lengths of the long and small axis of the cell cross section, a 1 and a 2 , respectively) are provided from side-view microscopic imaging [10] [Fig. 1(a)]. We varied the wall shear rate _ (in the range 0-5 s ÿ1) and the outer viscosity 0 by suspending RBCs in various solutions of dextran (concentration 6%, 7.5%, or 9% w=w and viscosity 22, 31, and 47 mPa s, respectively). Correspondingly, the wall shear stress, 0 _ , varies in a range from 0 to 0.25 Pa. We observed (i) the motion of flowing RBCs at a fixed 0 _ , (ii) the motion of individual cells at various 0 _ (for 8 RBCs), and (iii) the T-TT transition of 35 cells by increasing and/or decreasing 0 _. FIG. 1. Units i 0 m mPa s, Pa and _ s ÿ1. (a) Schematic drawing of a TT ellipsoid in a shear flow. (b) RBC swinging (_ 1:33, 0 47). Time sequence of 2 s. (c) Rotation of a bead (diameter 1 m) stuck on the membrane of a RBC with (_ 6, 0 47). Time sequence of 1 s. (d) The transition from S to T induced by decreasing _ is associated with a transient localized deformation (0 47, _ 2:66). Time sequence of 1 s.