Numerical Determination of Wall Shear Stress in Red Blood Cell Membranes

Abstract

Red cells in microvessels deform extensively in order to flow, thus resulting in high observed apparent viscosities and high stress concentrations localized to the red cell membrane. Flow through microvessels is consequently aided by the high compliance in the red cell membrane. However, changes in the mechanical properties of Red Blood Cells (RBC) as a result of storage lesions, can increase stress concentrations in the membrane, impairing oxygen transport and resulting in increased vesiculation and likely cell lysis. The objective of this study is to determine the mechanical properties and thus wall shear stress experienced by a RBC membrane during capillary flow. The RBC membrane is modeled as a two dimensional Hookean Isotropic Solid. Videos of capillary flow of both fresh RBCs and RBCs stored for 2 weeks were taken in the rat cremaster muscle. The mechanical properties, including the elastic modulus, shear stress, and shear strain, of both conditions were compared for both groups. In order to determine strain, each RBC membrane will be parameterized by N (X,Y) points determined by the Moore Neighbor Tracing algorithm of videos of capillary flow obtained by intravital microscopy of the rat cremaster muscle, from which N deformation gradient tensors, [F], can be obtained numerically for each frame of the video. From these deformation gradient tensors, the updated Cauchy Strain Tensor will be determined as a function of position on the Cartesian plane relative to the centroid of the RBC. The strains determined from the updated Langrangian Green Strain are numerically fit to the Navier‐Lame Equations along the boundary of the RBC to experimentally fit the two Lame's Constants during capillary flow. These constants will then be used to determine the Cauchy Stress Tensor, using Hooke's Law, along the boundary of the RBC, allowing for the calculation of the Wall Shear Stress. The average Elastic Modulus of fresh cells was found to be 5.4 ± 0.4 × 106 N/m2, while of that of RBCs stored for two weeks was found to be statistically larger and was calculated to be 6.0 ± 1.0 × 106 N/m2. Additionally, stored RBCs experienced higher shear strains and wall shear strains over larger areas of the membrane as compared with stored cells. Additionally, stress concentrators in the membrane were observed at the points of contact with the capillary wall. However, for stored cells, the high stress acted on a larger segment of the membrane. This algorithm for determining wall shear stress experienced by RBC membranes in vivo during capillary flow was thus able to detect statistically significant differences in the both the mechanical properties and stress concentrations between fresh cells and stored cells.Support or Funding InformationThis work was supported by NIH grants from the Heart Lung and Blood Institute, P01‐HL110900, R56‐HL123015, and R01‐HL126945.