Constitutive Equations of Erythrocyte

The erythrocyte membrane is modeled as a two-dimensional viscoelastic continuum that evolves under the application of stress. The present analysis of the erythrocyte membrane is motivated by the recent development of knowledge about its molecular structure. The constitutive equations proposed in the present analysis explain in a consistent manner the data on both the deformation and recovery phases of the micropipette experiment. The rheological equations of the present study are applied in a later section to the analysis of a plane membrane deformation that is quantitatively similar to the tank-treading motion of the erythrocytes in a shear fi'eld. The computations yield useful information on how the membrane viscosity becomes a more dominant feature in tank-treading motion. The material constants appearing in the proposed constitutive equations may be useful indications of the biochemical state of the membrane in health and disease. INTRODUCTION The erythrocyte membrane is modeled as a two-dimensional viscoelastic continuum that evolves under the application of stress. The present analysis of the erythrocyte membrane is motivated by the recent development of knowledge about its molecular structure and by the complex behavior it exhibits in dynamic micropipette testing and in tank-treading during shear flow. It is well established that the erythrocyte membrane deforms at constant surface area under physiological circumstances (1, 2). The constant area condition is considered to be a reflection of the structure of the lipid bilayer of the membrane. The membrane elasticity exhibited in experiments is primarily due to the network of protein macromolecules embedded in the lipid bilayer. The molecular organization of the entangled protein molecules evolves continuously during a prolonged deformation. The preferred configuration of the membrane from which the elastic strains are measured may not coincide with the initial stress-free configuration. The present model takes into account the evolution of the preferred configuration. The viscoelastic properties of the erythrocyte membrane were modeled previously by Evans and Hochmuth (3) by using a two-dimensional Kelvin model. Analyzing experimental data with the viscoelastic model of Evans and Hochmuth is simple and useful for obtaining an estimate of Dr. Tozeren's present address is National Institutes of Health, Division of Research Services, Biomedical Engineering and Instrumentation Branch, Bethesda, MD 20205. BIOPHYS. J.© Biophysical Society * 0008-3495/84/03/541/09 Volume 45 March 1984 541-549 the membrane viscosity during rapid deformation. However, the model needs to be further developed to explain the complicated membrane behavior observed in the rapidtransient micropipette experiments of Chien et al. (4). For example, the Kelvin model predicts essentially exponential aspiration length vs. time curves with a single time parameter for both the loading and the recovery phases of the experiment. Chien et al. (4) showed that the deformation of a cell into a micropipette following a step load consists of two phases. The initial rapid phase of deformation has a shear thinning behavior and was modeled recently by Tozeren et al. (5). The rapid phase is followed by a slow deformation phase with high membrane viscosity. The experimental data of Chien et al. (4) also indicate that the time constant of the recovery phase increases with increasing duration of the creep phase for durations of aspiration "up to 20 s. The constitutive equations proposed in the present analysis explain in a consistent manner the data on both the deformation and recovery phases of the micropipette experiment. The rheological equations of the present study are applied in a later section to the analysis of a plane membrane deformation that is quantitatively similar to the tank-treading motion of the erythrocytes in a shear field. The computations yield useful information on how the membrane viscosity becomes a more dominant feature in tank-treading motion. The present constitutive equations have a form similar to that proposed by Evans and Hochmuth (3). However, the membrane viscosity is made to depend on the rate of strain as suggested by Tozeren et al. (5), and the elastic strain tensor is measured from the