Abstract
A novel finite element approach is presented to simulate the mechanical behavior of human red blood cells (RBC, erythrocytes). As the RBC membrane comprises a phospholipid bilayer with an intervening protein network, we propose to model the membrane with two distinct layers. The fairly complex characteristics of the very thin lipid bilayer are represented by special incompressible solid shell elements and an anisotropic viscoelastic constitutive model. Properties of the protein network are modeled with an isotropic hyperelastic third-order material. The elastic behavior of the model is validated with existing optical tweezers studies with quasi-static deformations. Employing material parameters consistent with literature, simulation results are in excellent agreement with experimental data. Available models in literature neglect either the surface area conservation of the RBC membrane or realistic loading conditions of the optical tweezers experiments. The importance of these modeling assumptions, that are both included in this study, are discussed and their influence quantified. For the simulation of the dynamic motion of RBC, the model is extended to incorporate the cytoplasm. This is realized with a monolithic fully coupled fluid-structure interaction simulation, where the fluid is described by the incompressible Navier–Stokes equations in an arbitrary Lagrangian Eulerian framework. It is shown that both membrane viscosity and cytoplasm viscosity have significant influence on simulation results. Characteristic recovery times and energy dissipation for varying strain rates in dynamic laser trap experiments are calculated for the first time and are found to be comparable with experimental data.
Similar content being viewed by others
References
Boey SK, Boal DH, Discher DE (1998) Simulations of the erythrocyte cytoskeleton at large deformation. I. Microscopic models. Biophys J 75: 1573–1583
Bornemann PB, Wall WA (2009) An incompressible solid-shell element for finite deformations in statics, internal report
Canham PB (1970) The minimum energy of bending as a possible explanation of the biconcave shape of the human red blood cell. J Theor Biol 26: 61–81
Chee CY, Lee HP, Lu C (2008) Using 3d fluid-structure interaction model to analyse the biomechanical properties of erythrocyte. Phys Lett A 372: 1357–1362
Chien S, Sung KL, Skalak R, Usami S, Tozeren A (1978) Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane. Biophys J 24: 463–487
Dao M, Lim CT, Suresh S (2003) Mechanics of the human red blood cell deformed by optical tweezers. J Mech Phys Solids 51: 2259–2280
Dao M, Li J, Suresh S (2006) Molecularly based analysis of deformation of spectrin network and human erythrocyte. Mat Sci Eng C 26: 1232–1244
Deuling HJ, Helfrich W (1976) Red blood cell shapes as explained on the basis of curvature elasticity. Biophys J 16: 861–868
Discher DE, Mohandas N, Evans EA (1994) Molecular maps of red cell deformation: hidden elasticity and in situ connectivity. Science 266(5187): 1032–1035
Discher DE, Mohandas N (1996) Kinematics of red cell aspiration by fluorescence-imaged microdeformation. Biophys J 71: 1680–1694
Discher DE, Boal DH, Boey SK (1998) Simulations of the erythrocyte cytoskeleton at large deformation. II. Micropipette aspiration. Biophys J 75: 1584–1597
Dohrmann CR, Bochev PB (2004) A stabilized finite element method for the Stokes problem based on polynomial pressure projections. Int J Numer Meth Fluid 46: 183–201
Eggleton CD, Popel AS (1998) Large deformation of red blood cell ghosts in a simple shear flow. Phys Fluids 10: 1834–1845, AIP
Evans EA, Fung Y-C (1972) Improved measurements of the erythrocyte geometry. Microvasc Res 4: 335–347
Feng F, Klug WS (2006) Finite element modeling of lipid bilayer membranes. J Comput Phys 220: 394–408
Fischer TM (2004) Shape memory of human red blood cells. Biophys J 86: 3304–3313
Förster C, Wall WA, Ramm E (2006) On the geometric conservation law in transient flow calculations on deforming domains. Int J Numer Meth Fluid 50: 1369–1379
Förster C, Wall WA, Ramm E (2009) Stabilized finite element formulation for incompressible flow on distorted meshes. Int J Numer Meth Fluid 60: 1103–1126
Gee M, Küttler U, Wall WA (2010) Truly monolithic algebraic multigrid for fluid-structure interaction. Int J Numer Meth Eng (accepted)
Gompper G (2004) Fluid vesicles with viscous membranes in shear flow. Phys Rev Lett 93:258102, American Physical Society
Gov NS, Safran SA (2005) Red blood cell membrane fluctuations and shape controlled by ATP-induced cytoskeletal defects. Biophys J 88(3): 1859–1874
Hansen JC, Skalak R, Chien S, Hoger A (1996) An elastic network model based on the structure of the red blood cell membrane skeleton. Biophys J 70: 146–166
Hartmann D (2010) A multiscale model for red blood cell mechanics. Biomech Model Mechanobiol 9: 1–17
Heinrich V, Svetina S, Zeks B (1993) Nonaxisymmetric vesicle shapes in a generalized bilayer-couple model and the transition between oblate and prolate axisymmetric shapes. Phys Rev E 48: 3112
Heinrich V, Ritchie K, Mohandas N, Evans EA (2001) Elastic thickness compressibility of the red cell membrane. Biophys J 81: 1452–1463
Helfrich W (1973) Elastic properties of lipid bilayers: theory and possible experiments. Z Naturforsch 28C: 693–703
Henon S, Lenormand G, Richert A, Gallet F (1999) A new determination of the shear modulus of the human erythrocyte membrane using optical tweezers. Biophys J 76: 1145–1151
Hochmuth RM (1993) Measuring the mechanical properties of individual human blood cells. J Biomech Eng 115: 515–519, ASME
Hochmuth RM, Worthy PR, Evans EA (1979) Red cell extensional recovery and the determination of membrane viscosity. Biophys J 26: 101–114
Holzapfel G (2000) Nonlinear solid mechanics. A continuum approach for engineering. Wiley, Chichester, UK
Holzapfel GA, Gasser TC (2001) A viscoelastic model for fiber-reinforced composites at finite strains: continuum basis, computational aspects and applications. Comput Meth Appl Mech Eng 190: 4379–4403
Khairy K, Foo JJ, Howard J (2008) Shapes of red blood cells: comparison of 3d confocal images with the bilayer-couple model. Cell Mol Bioeng 1: 173–181
Küttler U, Förster C, Wall WA (2006) A solution for the incompressibility dilemma in partitioned fluid-structure interaction with pure dirichlet fluid domains. Comput Mech 38: 417–429
Küttler U, Gee M, Förster Ch, Comerford A, Wall WA (2010) Coupling strategies for biomedical fluid-structure interaction problems. Int J Numer Meth Biomed Eng 26: 305–321
Le DV, White J, Peraire J, Lim KM, Khoo BC (2009) An implicit immersed boundary method for three-dimensional fluid-membrane interactions. J Comput Phys 228: 8427–8445
Li J, Dao M, Lim CT, Suresh S (2005) Spectrin-level modeling of the cytoskeleton and optical tweezers stretching of the erythrocyte. Biophys J 88: 3707–3719
Lim CT, Dao M, Suresh S, Sow CH, Chew KT (2004) Large deformation of living cells using laser traps. Acta Mater 52: 1837–1845
McClain BL, Finkelstein IJ, Fayer MD (2004) Vibrational echo experiments on red blood cells: comparison of the dynamics of cytoplasmic and aqueous hemoglobin. Chem Phys Lett 392: 324–329
Mills JP, Qie L, Dao M, Lim CT, Suresh S (2004) Nonlinear elastic and viscoelastic deformation of the human red blood cell with optical tweezers. Mech Chem Biosyst 1: 169–180
Noguchi H, Gompper G (2005) Dynamics of fluid vesicles in shear flow: effect of membrane viscosity and thermal fluctuations. Phys Rev E 72: 011901–011914, APS
Pozrikidis C (2003) Numerical simulation of the flow-induced deformation of red blood cells. Ann Biomed Eng 31: 1194–1205
Puig-de-Morales-Marinkovic M, Turner KT, Butler JP, Fredberg JJ, Suresh S (2007) Viscoelasticity of the human red blood cell. Am J Physiol Cell Physiol 293: C597–605
Svetina S, Zeks B (1989) Membrane bending energy and shape determination of phospholipid vesicles and red blood cells. Eur Biophys J 17: 101–111
Tran-Son-Tay R, Sutera SP, Rao PR (1984) Determination of red blood cell membrane viscosity from rheoscopic observations of tank-treading motion. Biophys J 46: 65–72
Vu-Quoc L, Tan XG (2003) Optimal solid shells for non-linear analyses of multilayer composites. I. Statics. Comput Meth Appl Mech Eng 192: 975–1016
Yeoh OH (1990) Characterization of elastic properties of carbon-black-filled rubber vulcanizates. Rubber Chem Tech 63: 792–805
Yoon Y-Z, Kotar J, Yoon G, Cicuta P (2008) The nonlinear mechanical response of the red blood cell. Phys Biol 5: 036007
Zhou H, Pozrikidis C (1995) Deformation of liquid capsules with incompressible interfaces in simple shear flow. J Fluid Mech 283: 175–200
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Klöppel, T., Wall, W.A. A novel two-layer, coupled finite element approach for modeling the nonlinear elastic and viscoelastic behavior of human erythrocytes. Biomech Model Mechanobiol 10, 445–459 (2011). https://doi.org/10.1007/s10237-010-0246-2
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10237-010-0246-2