A simple model to understand the effect of membrane shear elasticity and stress-free shape on the motion of red blood cells in shear flow

Abstract

An analytical model was proposed by Keller and Skalak in 1982 to understand the motion of red blood cells in shear flow. The cell was described as a fluid ellipsoid of fixed shape. This model was extended in 2007 to introduce shear elasticity of the red blood cell membrane. Here, this model is further extended to take into account that the cell discoid shape physiologically observed is not a stress-free shape. The model shows that spheroid stress-free shapes allow us to fit the experimental data with the values of shear elasticity typical to that found with micropipette and optical tweezer experiments. In the range of moderate shear rates (for which RBCs keep their discoid shape) this model enables us to quantitatively determine (i) an effective cell viscosity, which combines membrane and hemoglobin viscosities and (ii) an effective shear modulus of the membrane that combines the shear modulus and the stress-free shape. This model can also be used to determine RBC mechanical parameters not only in the tanktreading regime when cells are suspended in medium of high viscosity but also in the tumbling regime characteristic of cells suspended in media of low viscosity. In this regime, a transition is predicted between a rigid-like tumbling motion and a fluid-like tumbling motion above a critical shear rate, which is directly related to the mechanical parameters of the cell.

Red cells’ dynamic morphologies govern blood shear thinning under microcirculatory flow conditions

“…Under physiological conditions, membrane tank treading is forbidden due to the relatively high internal viscosity as predicted by viscous ellipsoidal models of the RBC (26). However, during each tumbling period the membrane elements oscillate around a given position and this local oscillatory strain seems to destabilize RBCs from tumbling toward rolling (27) instead of tank treading. Although not fully settled, this phenomenon is well captured by our simulations and is described as a stable motion in several recent numerical simulations of capsules (28) and RBCs (29).…”

In-plane elasticity controls the full dynamics of red blood cells in shear flow

“…2, inset). The simulations highlight the importance of the membrane stress-free shape [12,13,32], and thus its in-plane elasticity, as a key parameter controlling the 3-D dynamics of RBCs. However, the roles of inertia or shape deformations, inherently present both in simulations and in experiments, remain unclear.…”

“…This reference shape is first deflated until an equilibrium biconcave shape is reached, which is then subjected to shear. The higher v ref , the lower the elastic energy barrier for membrane circulation [13,20,32]. In all simulations, G s = 2.5 µN.m −1 [31,33], and the bending modulus is κ b = 3.0 × 10 −19 J [16].…”

Optical Tweezers in Studies of Red Blood Cells

“…Particularly, regarding the RBCs ontogeny, RBCs undergo different cellular morphologies during the different stages of erythropoiesis. The membrane flexibility of the stress-free spheroidal shaped premature RBCs and their motion in shear flow can be analyzed by OTs experiments [132]. The optical forces can be applied directly to several points on the RBC membrane [133,134] or through several silica beads bound to the RBC membrane to reduce possible heating of the trapped cells [104,105].…”