Elastic behavior of a red blood cell with the membrane’s nonuniform natural state: equilibrium shape, motion transition under shear flow, and elongation during tank-treading motion

Abstract

Direct numerical simulations of the mechanics of a single red blood cell (RBC) were performed by considering the nonuniform natural state of the elastic membrane. A RBC was modeled as an incompressible viscous fluid encapsulated by an elastic membrane. The in-plane shear and area dilatation deformations of the membrane were modeled by Skalak constitutive equation, while out-of-plane bending deformation was formulated by the spring model. The natural state of the membrane with respect to in-plane shear deformation was modeled as a sphere ([Formula: see text]), biconcave disk shape ([Formula: see text]) and their intermediate shapes ([Formula: see text]) with the nonuniformity parameter [Formula: see text], while the natural state with respect to out-of-plane bending deformation was modeled as a flat plane. According to the numerical simulations, at an experimentally measured in-plane shear modulus of [Formula: see text] and an out-of-plane bending rigidity of [Formula: see text] of the cell membrane, the following results were obtained. (i) The RBC shape at equilibrium was biconcave discoid for [Formula: see text] and cupped otherwise; (ii) the experimentally measured fluid shear stress at the transition between tumbling and tank-treading motions under shear flow was reproduced for [Formula: see text]; (iii) the elongation deformation of the RBC during tank-treading motion from the simulation was consistent with that from in vitro experiments, irrespective of the [Formula: see text] value. Based on our RBC modeling, the three phenomena (i), (ii), and (iii) were mechanically consistent for [Formula: see text]. The condition [Formula: see text] precludes a biconcave discoid shape at equilibrium (i); however, it gives appropriate fluid shear stress at the motion transition under shear flow (ii), suggesting that a combined effect of [Formula: see text] and the natural state with respect to out-of-plane bending deformation is necessary for understanding details of the RBC mechanics at equilibrium. Our numerical results demonstrate that moderate nonuniformity in a membrane's natural state with respect to in-plane shear deformation plays a key role in RBC mechanics.