Deformation of a single red blood cell in a microvessel

Red blood cells (RBCs) are the most common type of cells in human blood and they exhibit different types of motions and deformed shapes in capillary flows. The behaviour of the RBCs should be studied in order to explain the RBC motion and deformation mechanism. This article presents a numerical simulation method for RBC deformation in microvessels. A two dimensional spring network model is used to represent the RBC membrane, where the elastic stretch/compression energy and the bending energy are considered with the constraint of constant RBC surface area. The forces acting on the RBC membrane are obtained from the principle of virtual work. The whole fluid domain is discretized into a finite number of particles using smoothed particle hydrodynamics concepts and the motions of all the particles are solved using Navier--Stokes equations. Minimum energy concepts are used to simulate the deformed shape of the RBC model. To verify the model, the motion of a single RBC is simulated in a Poiseuille flow and the characteristic parachute shape of the RBC is observed. Further simulations reveal that the RBC shows a tank treading motion when it flows in a linear shear flow. References D. A. Fedosov, B. Caswell, and G. E. Karniadakis. A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys. J. , 98(10):2215–2225, 2010. doi:10.1016/j.bpj. 2010.02. 002 T. M. Fischer, M. Stohr-Lissen, and H. Schmid-Schonbein. The red cell as a fluid droplet: tank tread-like motion of the human erythrocyte membrane in shear flow. Science , 202(4370):894–896, 1978. doi:10.1126/science.715448 R. A. Frcitas. Exploratory design in medical nanotechnology: a mechanical artificial red cell. Artif. Cell. Blood. Sub. , 26(4):411–430, 1998. doi:10.3109/10731199809117682 H. N. P. Gallage, Y. T. Gu, S. C. Saha, W. Senadeera, and A. Oloyede. Numerical simulation of red blood cells' deformation using SPH method. In Y. T. Gu and S. C. Saha, editors, 4th International Conference on Computational Methods (ICCM 2012) , Crowne Plaza, Gold Coast, QLD, November 2012. H. N. P. Gallage, Y. T. Gu, S. C. Saha, W. Senadeera, and A. Oloyede. Numerical simulation of red blood cells' motion : a review. In Y. T. Gu and S. C. Saha, editors, 4th International Conference on Computational Methods (ICCM 2012) , Crowne Plaza, Gold Coast, QLD, November 2012. Y. T. Gu. Meshfree methods and their comparisons. Int. J. Comput. Meth. , 2(04):477–515, 2005. doi:10.1142/S0219876205000673 D. V. Le, J. White, J. Peraire, K. M. Lim, and B. C. Khoo. An implicit immersed boundary method for three-dimensional fluid–membrane interactions. J. Comput. Phys. , 228(22):8427–8445, 2009. doi:10.1016/j.jcp. 2009.08. 018 G. R. Liu and Y. T. Gu. An introduction to meshfree methods and their programming . Springer, 2005. G. R. Liu and M. B. Liu. Smoothed particle hydrodynamics: a meshfree particle method . World Scientific, 2003. doi:10.1142/5340 T. W. Pan and T. Wang. Dynamical simulation of red blood cell rheology in microvessels. Int. J. Numer. Anal. Mod. , 6:455–473, 2009. L. Shi, T. W. Pan, and R. Glowinski. Deformation of a single red blood cell in bounded Poiseuille flows. Phys. Rev. E , 85(1):016307, 2012. doi:10.1103/PhysRevE.85. 016307 C. Sun and L. L. Munn. Particulate nature of blood determines macroscopic rheology: a 2-D lattice Boltzmann analysis. Biophys. J. , 88(3):1635–1645, 2005. doi:10.1529/biophysj.104. 051151 K. I. Tsubota, S. Wada, and T. Yamaguchi. Particle method for computer simulation of red blood cell motion in blood flow. Comput. Meth. Prog. Bio. , 83(2):139–146, 2006. doi:10.1016/j.cmpb. 2006.06. 005 K. I. Tsubota, S. Wada, and T. Yamaguchi. Simulation study on effects of hematocrit on blood flow properties using particle method. J. Biomech. Sci. Eng. , 1(1):159–170, 2006. doi:10.1299/jbse.1. 159 A. Vadapalli, D. Goldman, and A. S. Popel. Calculations of oxygen transport by red blood cells and hemoglobin solutions in capillaries. Artif. Cell. Blood. Sub. , 30(3):157–188, 2002. doi:10.1081/BIO-120004338