Biomechanics and Modeling in Mechanobiology

, Volume 18, Issue 2, pp 347–359 | Cite as

Red blood cell simulation using a coupled shell–fluid analysis purely based on the SPH method

  • Meisam SoleimaniEmail author
  • Shahab Sahraee
  • Peter Wriggers
Original Paper


In this paper, a novel 3D numerical method has been developed to simulate red blood cells (RBCs) based on the interaction between a shell-like solid structure and a fluid. RBC is assumed to be a thin shell encapsulating an internal fluid (cytoplasm) which is submerged in an external fluid (blood plasma). The approach is entirely based on the smoothed particle hydrodynamics (SPH) method for both fluid and the shell structure. Both cytoplasm and plasma are taken to be incompressible Newtonian fluid. As the kinematic assumptions for the shell, Reissner–Mindlin theory has been introduced into the formulation. Adopting a total Lagrangian (TL) formulation for the shell in the realm of small strains and finite deflection, the presented computational tool is capable of handling large displacements and rotations. As an application, the deformation of a single RBC while passing a stenosed capillary has been modeled. If the rheological behavior of the RBC changes, for example, due to some infection, it is reflected in its deformability when it passes through the microvessels. It can severely affect its proper function which is providing the oxygen and nutrient to the living cells. Hence, such numerical tools are useful in understanding and predicting the mechanical behavior of RBCs. Furthermore, the numerical simulation of stretching an RBC in the optical tweezers system is presented and the results are verified. To the best of authors’ knowledge, a computational tool purely based on the SPH method in the framework of shell–fluid interaction for RBCs simulation is not available in the literature.


Shell Red blood cell Smoothed particle hydrodynamics Fluid–solid interaction 



The authors sincerely acknowledge the financial support of this research by the state of Lower Saxony, Germany, within the program ”wissenschaftsallianz.”


  1. Adami S, Hu XY, Adams NA (2012) A generalized wall boundary condition for smoothed particle hydrodynamics. J Comput Phys 231:7057–7075MathSciNetCrossRefGoogle Scholar
  2. Antoci C, Gallati M, Sibilla S (2007) Numerical simulation of fluidstructure interaction by SPH. Comput Struct 85:879–890CrossRefGoogle Scholar
  3. Aristodemo F, Federico I, Veltri P, Panizzo A (2010) Two-phase SPH modeling of advective diffusion processes. Environ Fluid Mech 10:451–470CrossRefGoogle Scholar
  4. Ay C, Lien CC, Wu MC (2014) Study on the Youngs modulus of red blood cells using atomic force microscope. Appl Mech Mater 627:197–201CrossRefGoogle Scholar
  5. Belytschko T, Guo Y, Kam Liu W, Ping Xiao S (2000) A unified stability analysis of meshless particle methods. Int J Numer Methods Eng 48:13591400MathSciNetCrossRefzbMATHGoogle Scholar
  6. Belytschko T, Liu WK, Moran B, Elkhodary K (2014) Nonlinear finite elements for continua and structures, Second edn. Wiley, New YorkzbMATHGoogle Scholar
  7. Betsch P, Menzel A, Stein E (1998) On the parametrization of finite rotations in computational mechanics: A classification of concepts with application to smooth shells. Comput Methods Appl Mech Eng 155(3):273–305MathSciNetCrossRefzbMATHGoogle Scholar
  8. Cleary PW, Monaghan JJ (1999) Conduction modeling using smoothed particle hydrodynamics. J Comput Phys 148:227–264MathSciNetCrossRefzbMATHGoogle Scholar
  9. Cordasco D, Bagchi P (2013) Orbital drift of capsules and red blood cells in shear flow. Phys Fluids 25:091902CrossRefGoogle Scholar
  10. Dupire J, Socol M, Viallat A (2012) Full dynamics of a red blood cell in shear flow. PNAS 109(51):20808–20813CrossRefGoogle Scholar
  11. Fedosov D, Caswell B, Karniadakis G (2010) A multiscale red blood cell model with accurate mechanics, rheology, and dynamics. Biophys J 98(10):2215–25CrossRefGoogle Scholar
  12. Fedosov DA, Noguchi H, Gompper G (2014) Multiscale modeling of blood flow: from single cells to blood rheology. Biomech Model Mechanobiol. 13(2):239–58CrossRefGoogle Scholar
  13. Freund JB (2014) Numerical simulation of flowing blood cells. Annu Rev Fluid Mech 46:67–95MathSciNetCrossRefzbMATHGoogle Scholar
  14. Gingold RA, Monaghan JJ (1977) Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon Notices R Astron Soc 181:375CrossRefzbMATHGoogle Scholar
  15. Gomez-Gesteira M, Rogers BD, Dalrymple RA, Crespo AJ (2010) State-of-the-art of classical SPH for free-surface flows. J Hydraul Res 48:6–27CrossRefGoogle Scholar
  16. Gray JP, Monaghan JJ, Swift RP (2001) SPH elastic dynamics. Comput Methods Appl Mech Eng 190:6641–6662CrossRefzbMATHGoogle Scholar
  17. Hochmuth RM, Mohandas N, Blackshear PL (1997) Measurement of the elastic modulus for red cell membrane using a fluid mechanical technique. Biophys J. 13(8):747–762CrossRefGoogle Scholar
  18. Ju M, Ye SS, Namgung B, Cho S, Low HT, Leo HL, Kim S (2015) A review of numerical methods for red blood cell flow simulation. Comput Methods Biomech Biomed Eng 18(2):130–140CrossRefGoogle Scholar
  19. Keller SR, Skalak R (1982) Motion of a tank-treading ellipsoidal particle in a shear flow. J Fluid Mech 120:27–47CrossRefzbMATHGoogle Scholar
  20. Kristof P, Benes B, Krivanek J, Stava O (2009) Hydraulic erosion using smoothed particle hydrodynamics. Comput Graphics Forums 28:219–228CrossRefGoogle Scholar
  21. Krüger T, Gross M, Raabe D, Varnik F (2013) Crossover from tumbling to tank-treading-like motion in dense simulated suspensions of red blood cells. Soft Matters 9:9008–9015CrossRefGoogle Scholar
  22. Lanotte L, Mauer J, Mendez S, Fedosov DA, Fromental J-M, Claveria V, Nicoud F, Gompper G, Abkarian M (2016) Red cells dynamic morphologies govern blood shear thinning under microcirculatory flow conditions. PNAS 13(47):13289–13294CrossRefGoogle Scholar
  23. Li S, Liu WK (2002) Mesh-free and particle methods and their applications. Appl. Mech. 55(1):1–34MathSciNetCrossRefGoogle Scholar
  24. Libersky LD, Petschek AG, Carney TC, Hipp JR, Allahdadi FA (1993) High strain Lagrangian hydrodynamics: a three-dimensional SPH code for dynamic material response. J Comput Phys 109(1):67–75CrossRefzbMATHGoogle Scholar
  25. Lin J, Naceur H, Coutellier D, Laksimi A (2014) Efficient meshless SPH method for the numerical modeling of thick shell structures undergoing large deformations. Int J Nonlinear Mech 65:1–13CrossRefGoogle Scholar
  26. Lucy LB (1977) Numerical approach to the testing of the fission hypothesis. Astron J 82:1013CrossRefGoogle Scholar
  27. Maurel B, Combescure A (2008) An SPH shell formulation for plasticity and fracture analysis in explicit dynamics. Numer Method Eng 76(7):949–9715MathSciNetCrossRefzbMATHGoogle Scholar
  28. Mills J, Qie L, Dao M, Lim C, Suresh S (2004) Nonlinear elastic and viscoelastic deformation of the human red blood cell with optical tweezers. Mech Chem Biosyst 1(3):169–180Google Scholar
  29. Monaghan JJ (1992) Smoothed particle hydrodynamics. Annu Rev Astron Astrophys 3:543–574CrossRefGoogle Scholar
  30. Monaghan JJ (1994) Simulating free surface flows with SPH. J Comput Phys 110(2):399–406MathSciNetCrossRefzbMATHGoogle Scholar
  31. Monaghan JJ (2000) SPH without a Tensile Instability. J Comput Phys 159:290–311CrossRefzbMATHGoogle Scholar
  32. Monaghan JJ (2005) Smoothed particle hydrodynamics. Rep Progress Phys 68:1703–1759MathSciNetCrossRefzbMATHGoogle Scholar
  33. Monaghan JJ, Gingold RA (1983) Shock simulation by the particle method SPH. J Comput Phys 52(2):374–389CrossRefzbMATHGoogle Scholar
  34. Nayanajith H, Gallage P, Saha SC, Sauret E, Flower R, Senadeera W, YuanTong G (2016) SPH-DEM approach to numerically simulate the deformation of three-dimensional RBCs in non-uniform capillaries. BioMed Eng Online 15(2):350–370Google Scholar
  35. Owen B, Bojdo N, Jivkov A, Keavney B, Revell A (2018) Structural modelling of the cardiovascular system. Biomech Model Mechanobiol.
  36. Pozrikidis C (2001) Effect of membrane bending stiffness on the deformation of capsules in simple shear flow. J Fluid Mech 440:269–291CrossRefzbMATHGoogle Scholar
  37. Reddy JN (2006) Theory and analysis of elastic plates and shells, Second edn. CRC Press, Boca RatonGoogle Scholar
  38. Skalak R, Tozeren A, Zarda RP, Chien S (1973) Strain energy function of red blood cell membranes. Biophys J 13(3):245–264CrossRefGoogle Scholar
  39. Soleimani M, Wriggers P (2016) Numerical simulation and experimental validation of biofilm in a multi-physics framework using an SPH based method. Comput Mech 58(4):619–633MathSciNetCrossRefGoogle Scholar
  40. Suzuki Y, Tateishi N, Soutani M, Maeda N (1996) Deformation of erythrocytes in microvessels and glass capillaries: effects of erythrocyte deformability. Microcirculation 3(1):49–57CrossRefGoogle Scholar
  41. Tartakovsky AM, Meakin P, Scheibe TD (2007) Simulations of reactive transport and precipitation with smoothed particle hydrodynamics. J Comput Phys 222:654–672MathSciNetCrossRefzbMATHGoogle Scholar
  42. Tenghu W, Feng JJ (2013) Simulation of malaria-infected red blood cells in microfluidic channels: passage and blockage. Biomicrofluidics 7:044115CrossRefGoogle Scholar
  43. Tomaiuolo G (2014) Biomechanical properties of red blood cells in health and disease towards microfluidics. Biomicrofluidics 8(5):051501CrossRefGoogle Scholar
  44. Tran-Son-Tay R, Sutera SP, Zahalak GI, Rao PR (1987) Membrane stress and internal pressure in a red blood cell freely suspended in a shear flow. Biophys J 51(6):915–924CrossRefGoogle Scholar
  45. Vahidkhah K, Fatouraee N (2012) Numerical simulation of red blood cell behaviour in a stenosed arteroile using the immersed boundary-lattice Boltzmann method. Int J Numer Method Biomed Eng 28:239–256CrossRefzbMATHGoogle Scholar
  46. Vahidkhah K, Balogh P, Bagchi P (2016) Flow of red blood cells in stenosed microvessels. Sci Rep 6:281–94CrossRefGoogle Scholar
  47. Valizadeh A, Monaghan JJ (2015) A study of solid wall models for weakly compressible SPH. J Comput Phys 300:5–19MathSciNetCrossRefzbMATHGoogle Scholar
  48. Vignjevic R, Campbell J, Liberskyb L (2000) A treatment of zero-energy modes in the smoothed particle hydrodynamics method. Comput Methods Appl Mech Eng 184(1):67–85MathSciNetCrossRefzbMATHGoogle Scholar
  49. Vignjevic R, Campbell J (2009) Review of development of the smooth particle hydrodynamics (SPH) method. In: Predictive modeling of dynamic processes, pp 367–396Google Scholar
  50. Wanner GW (1973) Modelling the mechanical behavior of red blood cells. Biorheology 10(2):229–38CrossRefGoogle Scholar
  51. Wriggers P (2008) Non-linear finite element method. Springer, Heidelberg, pp 142–148Google Scholar
  52. Wriggers P, Simo JC (1990) A general procedure for the direct computation of turning and bifurcation points. Int J Numer Methods Eng l 30:155176zbMATHGoogle Scholar
  53. Yazdani A, Baghchi P (2012) Three dimensional numerical simulation of vesicle dynamics using a front tracking method. Phys Rev E 85:056308CrossRefGoogle Scholar
  54. Zarda PR, Chien S, Skalak R (1997) Elastic deformations of red blood cells. J Biomech 10:211–221CrossRefGoogle Scholar
  55. Zhong-can O-Y, Helfrich W (1989) Bending energy of vesicle membranes: general expressions for the first, second, and third variation of the shape energy and applications to spheres and cylinders. Phys. Rev. A 39:5280–5288CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Meisam Soleimani
    • 1
    Email author
  • Shahab Sahraee
    • 1
  • Peter Wriggers
    • 1
  1. 1.Institute of Continuum MechanicsLeibniz Universität HannoverHannoverGermany

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