Annals of Biomedical Engineering

, Volume 33, Issue 12, pp 1724–1727 | Cite as

Computer Modeling of Red Blood Cell Rheology in the Microcirculation: A Brief Overview

  • Vittorio CristiniEmail author
  • Ghassan S. Kassab


One of the major functions of the cardiovascular system is to deliver blood to the microcirculation where exchange of mass and energy can take place. In the present article, we will provide an overview of the state-of-the-art computational methods for modeling of red blood cell (RBC) rheology and dynamics in the microcirculation. While significant progress has been made in simulation of single-file motion of deformable RBCs in capillaries and of diluted sheared suspensions of RBCs in infinite domains, detailed understanding of the mechanics of blood flow in intermediate diameter microvessels (8–1000μm) has presented formidable challenges. The difficulties are largely due to modeling the motion of multiple, interacting, highly deformable particles. The current computational tools consist mainly of three-dimensional (3D) boundary-integral methods for single RBC dynamics and deformation; and for rheology of large systems of droplets at large volume fractions using periodic boundary conditions and novel adaptive computational meshes. Further advances will result from combination of these tools to produce new algorithms capable of describing the motion and deformation of large systems of RBCs in microvessels at physiologically relevant volume fractions.


Red blood cell Microcirculation 3D boundary-integrals 3D mesh adaptivity 


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  1. 1.
    Aidun, C. K., and E.-J. Ding. Direct numerical simulation of red blood cell flow and scaling relation for the aggregate size distribution. In: International Biofluid Mechanics Conference, California Institute of Technology, December 12–14, 2003.Google Scholar
  2. 2.
    Barthes-Biesel, D., and H. Sgaier. Role of membrane viscosity in the orientation and deformation of a spherical capsule suspended in shear flow. J. Fluid Mech. 160:119–135, 1985.MathSciNetGoogle Scholar
  3. 3.
    Bugliarello, G., and J. Sevilla. Velocity distribution and other characteristics of steady and pulsatile blood flow in fine glass tubes. Biorheology 7:85–107, 1970.Google Scholar
  4. 4.
    Cokelet, G. R., and H. L. Goldsmith. Decreased hydrodynamic resistance in the two-phase flow of blood through small vertical tubes at low flow rates. Circ. Res. 68:1–17, 1991.Google Scholar
  5. 5.
    Coulliette, C., and C. Pozrikidis. Motion of an array of drops through a cylindrical tube. J. Fluid Mech. 358:1–28, 1998.CrossRefMathSciNetGoogle Scholar
  6. 6.
    Cristini, V. Adaptive multiscale numerical simulation of blood rheology and red cell dynamics in the microcirculation. In: International Biofluid Mechanics Conference, California Institute of Technology, December 12–14, 2003.Google Scholar
  7. 7.
    Cristini, V., J. Blawzdziewicz, and M. Loewenberg. An adaptive mesh algorithm for evolving surfaces: Simulations of drop breakup and coalescence. J. Comp. Phys. 168:445–463, 2001.Google Scholar
  8. 8.
    Cristini, V., J. Blawzdziewicz, and M. Loewenberg. Drop breakup in three-dimensional viscous flows. Phys. Fluids 10:1781–1783, 1998.CrossRefGoogle Scholar
  9. 9.
    Cristini, V., J. Blawzdziewicz, M. Loewenberg, and L. R. Collins. Breakup in stochastic Stokes flows: Sub-Kolmogorov drops in isotropic turbulence. J. Fluid Mech. 492:231, 2003.CrossRefMathSciNetGoogle Scholar
  10. 10.
    Cristini, V., and J. Lowengrub. Three-dimensional crystal growth-II: Nonlinear simulation and suppression of the Mullins–Sekerka instability. J. Cryst. Growth 266:552, 2004.CrossRefGoogle Scholar
  11. 11.
    Cristini, V., S. Guido, A. Alfani, J. Blawzdziewicz, and M. Loewenberg. Drop breakup and fragment size distribution in shear flow. J. Rheol. 47:1283, 2003.CrossRefGoogle Scholar
  12. 12.
    Dintenfass, L. Inversion of the Fahraeus–Lindquist phenomena in blood through capillaries of diminishing radius. Nature 215:1099–1100, 1967.Google Scholar
  13. 13.
    Eggleton, C. D., and A. S. Popel. Large deformation of red blood cell ghosts in a simple shear flow. Phys. Fluids 10:1834–1845, 1998.CrossRefGoogle Scholar
  14. 14.
    Evans, E. A., and R. Skalak. Mechanics and Thermodynamics of Biomembranes. CRC Press, Boca Raton, 1980.Google Scholar
  15. 15.
    Fahraeus, R., and T. Lindqvist. The viscosity of blood in narrow capillary tubes. Am. J. Physiol. 96:562–568, 1931.Google Scholar
  16. 16.
    Fung, Y. C. Stochastic flow in capillary blood vessels. Microvasc. Res. 5:34–49, 1973.CrossRefGoogle Scholar
  17. 17.
    Goldsmith, H. L. Red cell motions and wall interactions in tube flow. Fed. Proc. 30:1578–1590, 1971.Google Scholar
  18. 18.
    Keller, S. R., and R. Skalak. Motion of a tank-treading ellipsoidal particle in a shear flow. J. Fluid Mech. 120:27–47, 1982.Google Scholar
  19. 19.
    Kennedy, M., C. Pozrikidis, and R. Skalak. Motion and deformation of liquid drops, and the rheology of dilute emulsions in shear flow. Comput. Fluids 23:251–278, 1994.CrossRefGoogle Scholar
  20. 20.
    Li, X., J. Lowengrub, Q. Nie, V. Cristini, and P. Leo. Microstructure evolution in three-dimensional inhomogeneous elastic media. Met. Mater. Trans. A-Physica 34:1421, 2003.Google Scholar
  21. 21.
    Loewenberg, M., and E. J. Hinch. Numerical simulation of a concentrated emulsion in shear flow. J. Fluid Mech. 321:395–419, 1996.Google Scholar
  22. 22.
    Patel, P. D., E. S. G. Shaqfeh, J. E. Butler, V. Cristini, J. Blawzdziewicz, and M. Loewenberg. Drop breakup in the flow through fixed fiber beds: An experimental and computational investigation. Phys. Fluids 15:1146–1157, 2003.CrossRefGoogle Scholar
  23. 23.
    Pozrikidis, C. Numerical simulation of the flow-induced deformation of red blood cells. Ann. Biomed. Eng. 31:1194, 2003.Google Scholar
  24. 24.
    Pozrikidis, C. ed. Modeling and Simulation of Capsules and Biological Cells. CRC Mathematical Biology and Medicine Series, Chapman& Hall, Boca Raton, 2003.Google Scholar
  25. 25.
    Pozrikidis, C. Dynamical simulation of the flow of suspensions: Wall-bounded and pressure-driven channel flow. Ind. Eng. Chem. Res. 41:6312–6322, 2002.CrossRefGoogle Scholar
  26. 26.
    Pozrikidis, C. Effect of bending stiffness on the deformation of liquid capsules in simple shear flow. J. Fluid Mech. 440:269–291, 2001.CrossRefzbMATHGoogle Scholar
  27. 27.
    Pries, A. R., D. Neuhaus, and P. Gaehtgens. Blood viscosity in tube flow: dependence on diameter and hematocrit. Am. J. Physiol. 263:H1770–H1778, 1992.Google Scholar
  28. 28.
    Pries, A. R., K. Ley, and P. Gaehtgens. Generalization of the Fahraeus principle for microvessel networks. Am. J. Physiol. 251:H1324–H1332, 1986.Google Scholar
  29. 29.
    Reinke, W., P. Gaehtgens, and P. C. Johnson. Blood viscosity in small tubes: Effect of shear rate, aggregation, and sedimentation. Am. J. Physiol. 253:H540–H547, 1987.Google Scholar
  30. 30.
    Schmid-Schonbein, G. W., R. Skalak, S. Usami, and S. Chien. Cell distribution in capillary networks. Microvasc. Res. 19:18–44, 1980.Google Scholar
  31. 31.
    Secomb, T. W. Mechanics of red blood cells and blood flow in narrow tubes. In: C. Pozrikidis, ed. Modeling and Simulation of Capsules and Biological Cells. CRC Mathematical Biology and Medicine Series, Chapman& Hall, Boca Raton, 2003, pp. 163–190.Google Scholar
  32. 32.
    Secomb, T. W., S. Chien, K. M. Jan, and R. Skalak. The bulk rheology of close-packed red blood cells in shear flow. Biorheol. 20:295–309, 1983.Google Scholar
  33. 33.
    Secomb, T. W., T. M. Fischer, and R. Skalak. The motion of close-packed red blood cells in shear flow. Biorheol. 20:283–294, 1983.Google Scholar
  34. 34.
    Yen, R. T., and Y. C. Fung. Inversion of Fahraeus effect and effect of mainstream flow on capillary hematocrit. J. Appl. Physiol. 42:578–586, 1977.Google Scholar
  35. 35.
    Zheng X., S. Wise, and V. Cristini. Nonlinear simulation of tumor necrosis, neo-vascularization and tissue invasion via an adaptive finite-element/level-set method. Bull. Math. Biol. 67:211–259, 2005.CrossRefGoogle Scholar
  36. 36.
    Zhou H., and C. Pozrikidis. Deformation of liquid capsules with incompressible interfaces in simple shear flow. J. Fluid Mech. 283:175–200, 1995.Google Scholar
  37. 37.
    Zinchenko A. Z., and R. H. Davis. Shear flow of highly concentrated emulsions of deformable drops by numerical simulations. J. Fluid Mech. 455:21–62, 2002.CrossRefGoogle Scholar

Copyright information

© Biomedical Engineering Society 2005

Authors and Affiliations

  1. 1.Department of Biomedical EngineeringUniversity of CaliforniaIrvine
  2. 2.Department of MathematicsUCI
  3. 3.Department of Biomedical EngineeringUCIUSA

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