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A Portrait of State-of-the-Art Research at the Technical University of Lisbon pp 65–87Cite as

An Overview of Some Mathematical Models of Blood Rheology

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Abstract

Experimental investigations over many years reveal that blood flow exhibits non-Newtonian characteristics such as shear-thinning, viscoelasticity and thixotropic behaviour. The complex rheology of blood is influenced by numerous factors including plasma viscosity, rate of shear, hematocrit, level of erythrocytes aggregation and deformability. Hemodynamic analysis of blood flow in vascular beds and prosthetic devices requires the rheological behaviour of blood to be characterized through appropriate constitutive equations relating the stress to deformation and rate of deformation.

The objective of this paper is to present a short overview of some macroscopic constitutive models that can mathematically characterize the rheology of blood and describe its known phenomenological properties. Some numerical simulations obtained in geometrically reconstructed real vessels will be also presented to illustrate the hemodynamic behaviour using Newtonian and non-Newtonian inelastic models under a given set of physiological flow conditions.

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References

  1. M. Anand, K. R. Rajagopal, A mathematical model to describe the change in the constitutive character of blood due to platelet activation, C.R. Mécanique, 330, 2002, pp. 557–562.

    Article  MATH  Google Scholar 

  2. M. Anand, K. Rajagopal and K. R. Rajagopal, A model incorporating some of the mechanical and biochemical factors underlying clot formation and dissolution in flowing blood, J. of Theoretical Medicine, 5, 2003, pp. 183–218.

    Article  MATH  MathSciNet  Google Scholar 

  3. M. Anand and K. R. Rajagopal, A shear-thinning viscoelastic fluid model for describing the flow of blood, Int. J. Cardiovascular Medicine and Science, 4, 2004, pp. 59–68.

    Google Scholar 

  4. N. Arada, and A. Sequeira, Strong Steady Solutions for a generalized Oldroyd-B Model with Shear-Dependent Viscosity in a Bounded Domain, Mathematical Models & Mehods in Applided Sciences, 13, no.9, 2003, pp. 1303–1323.

    Article  MATH  MathSciNet  Google Scholar 

  5. O. K. Baskurt and H. J. Meiselman, Blood rheology and hemodynamics, Seminars in Thrombosis and Hemostasis, 29, 2003, pp. 435–450.

    Article  Google Scholar 

  6. C. G. Caro, J. M. Fitz-Gerald and R. C. Schroter, Atheroma and arterial wall shear: observation, correlation and proposal of a shear dependent mass transfer mechanism of artherogenesis, Proc. Royal Soc. London, 177, 1971, pp. 109–159.

    Google Scholar 

  7. C. G. Caro, T. J. Pedley, R. C. Schroter and W. A. Seed, The Mechanics of the Circulation, Oxford University Press, Oxford, 1978.

    Google Scholar 

  8. P. J. Carreau, PhD Thesis, University of Wisconsin, Madison, 1968.

    Google Scholar 

  9. I. Chatziprodromoua, A. Tricolia, D. Poulikakosa and Y. Ventikos, Haemodynamics and wall remodelling of a growing cerebral aneurysm: A computational model, Journal of Biomechanics, accepted December 2005, in press.

    Google Scholar 

  10. S. Chien, S. Usami, R. J. Dellenback, M. I. Gregersen, Blood viscosity: Influence of erythrocyte deformation, Science, 157(3790), 1967, pp. 827–829.

    Article  Google Scholar 

  11. S. Chien, S. Usami, R. J. Dellenback, M. I. Gregersen, Blood viscosity: Influence of erythrocyte aggregation, Science, 157(3790), 1967, pp. 829–831.

    Article  Google Scholar 

  12. S. Chien, S. Usami, R. J. Dellenback, M. I. Gregersen, Shear-dependent deformation of erythrocytes in rheology of human blood, American Journal of Physiology, 219, 1970, pp. 136–142.

    Google Scholar 

  13. Y. I. Cho and K. R. Kensey, Effects of the non-Newtonian viscosity of blood on flows in a diseased arterial vessel. Part I: Steady flows, Biorheology, 28, 1991, pp. 241–262.

    Google Scholar 

  14. G. R, Cokelet, The rheology of human blood. In: Y. C. Fung and M. Anliker (Eds.), Biomechanics: its foundations and objectives, Ch. 4, Prentice Hall, 1972.

    Google Scholar 

  15. A. L. Copley, The rheology of blood. A survey, J. Colloid Sci., 7, 1952, pp. 323–333.

    Article  Google Scholar 

  16. A. L. Copley and G. V. F. Seaman, The meaning of the terms rheology, biorheology and hemorheology, Clinical Hemorheology, 1, 1981, pp. 117–119.

    Google Scholar 

  17. V. Cristini and G. S. Kassab, Computer modeling of red blood cell rheology in the microcirculation: a brief overview, Annals of Biomedical Engineering, 33, n.12, 2005, pp. 1724–1727.

    Article  Google Scholar 

  18. M. M. Cross, Rheology of non-Newtonian fluids: a new flow equation for pseudo-plastic systems, J. Colloid Sci., 20, 1965, pp. 417–437.

    Article  Google Scholar 

  19. R. Fåhraeus, Die Strömungsverhältnisse und die Verteilung der Blutzellen im Gefässsystem, Klin. Wschr., 7, 1928, pp. 100–106.

    Article  Google Scholar 

  20. R. Fåhraeus and T. Lindqvist, The viscosity of blood in narrow capillary tubes, Am. J. Physiol. 96. 1931, pp. 562–568.

    Google Scholar 

  21. A. L. Fogelson, Continuum models of platelet aggregation: formulation and mechanical properties, SIAM J. Appl. Math., 52, 1992, 1089–1110.

    Article  MATH  MathSciNet  Google Scholar 

  22. P. J. Frey, Génération et adaptation de maillages de surfaces à partir de données anatomiques discrètes, Rapport de Recherche, 4764, INRIA, 2003.

    Google Scholar 

  23. V. Girault and P.-A. Raviart, Finite Element Methods for the navier-Stokes Equations, Springer-Verlag Berlin, Heidelberg, New York, Tokyo, 1986.

    MATH  Google Scholar 

  24. P. M. Gresho and R. L. Sani, Incompressible Flow and the Finite Element Method, Vol.2, Jphn Wiley and Sons, Chichester, 2000.

    MATH  Google Scholar 

  25. C. R. Huang, N. Siskovic, R. W. Robertson, W. Fabisiak, E. H. Smith-Berg and A. L. Copley, Quantitative characterization of thixotropy of whole human blood, Biorheology, 12, 1975, pp. 279–282.

    Google Scholar 

  26. R. Keunings, A survey of computational rheology, in: Proceedings of the XIIIth International Congress on Rheology (D.M. Binding et al. ed.), British Soc. Rheol., 1, 2000, pp. 7–14.

    Google Scholar 

  27. A. Kuharsky, Mathematical modeling of blood coagulation, PhD Thesis, Univ. of Utah, 1998.

    Google Scholar 

  28. A. Kuharsky, A. L. Fogelson, Surface-mediated control of blood coagulation: the role of binding site densities and platelet deposition, Biophys. J., 80(3), 2001, pp. 1050–1074.

    Google Scholar 

  29. D. O. Lowe, Clinical Blood Rheology, Vol. I, II, CRC Press, Boca Raton, Florida, 1998.

    Google Scholar 

  30. D. E. McMillan, J. Strigberger and N. G. Utterback, Rapidly recovered transient flow resistance: a newly discovered property of blood, American Journal of Physiology, 253, pp. 919–926.

    Google Scholar 

  31. R. G. Owens and T. N. Phillips, Computational Rheology, Imperial College Press/World Scientific, London, UK, 2002.

    MATH  Google Scholar 

  32. R. G. Owens, A new microstructure-based constitutive model for human blood, J. Non-Newtonian Fluid Mech., 2006, to appear.

    Google Scholar 

  33. M. J. Perko, Duplex Ultrasound for Assessment of Superior Mesenteric Artery Blood Flow, European Journal of Vascular and Endovascular Surgery, 21, 2001, pp. 106–117.

    Article  Google Scholar 

  34. K. Perktold and M. Prosi, Computational models of arterial flow and mass transport, in:Cardiovascular Fluid Mechanics (G. Pedrizzetti, K. Perktold, Eds.), CISM Courses and Lectures n.446, Springer-Verlag, pp. 73–136, 2003.

    Google Scholar 

  35. W. M. Phillips, S. Deutsch, Towards a constitutive equation for blood, Biorheology, 12(6), 1975, pp. 383–389.

    MathSciNet  Google Scholar 

  36. A. S. Popel and P. C. Johnson, Microcirculation and hemorheology, Annu. Rev. Fluid Mech., 37, 2005, pp. 43–69.

    Article  MathSciNet  Google Scholar 

  37. A. R. Pries and T. W. Secomb, Rheology of the microcirculation, Clinical Hemorheology and Microcirculation, 29, 2003, pp. 143–148.

    Google Scholar 

  38. D. A. Quemada, A non-linear Maxwell model of biofluids-Application to normal blood, Biorheology, 30(3–4), 1993, pp. 253–265.

    Google Scholar 

  39. A. Quarteroni and A. Valli, Numerical Approximation of Partial Differential Equations, Springer-Verlag, Heidelberg, 1994.

    MATH  Google Scholar 

  40. K. R. Rajagopal, A. Srinivasa, A thermodynamic framework for rate type fluid models, J. of Non-Newtonian Fluid Mech., 88, 2000, pp. 207–228.

    Article  MATH  Google Scholar 

  41. G. Rappitsch, K. Perktold and E. Pernkopf, Numerical modelling of shear-dependent mass transfer in large arteries, Int. J. Numer. Meth. Fluids, 25, 1997, pp. 847–857.

    Article  MATH  Google Scholar 

  42. M. Renardy, Existence of slow steady flows of viscoelastic fluids with a differential constitutive equation, Z. Angew. Math. Mech., 65, 1985, pp. 449–451.

    Article  MATH  MathSciNet  Google Scholar 

  43. M. Renardy, Mathematical Analysis of Viscoelastic Flows, CBMS 73, SIAM, Philadelphia, 2000.

    MATH  Google Scholar 

  44. G. W. Scott-Blair, An equation for the flow of blood, plasma and serum through glass capillaries, Nature, 183, 1959, pp. 613–614.

    Article  Google Scholar 

  45. R. Tabrizchi and M. K. Pugsley, Methods of blood flow measurement in the arterial circulatory system, Journal of Pharmacological and Toxicological Methods, 44(2), 2000, pp. 375–384.

    Article  Google Scholar 

  46. C. A. Taylor, M. T. Draney, J. P. Ku, D. Parker, B. N. Steele, K. Wang, C. K. Zarins, Predictive medicine: Computational techniques in therapeutic decision-making, Computed Aided Surgery, 4(5), 1999, pp.231–247.

    Article  Google Scholar 

  47. R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis, North Holland, Amsterdam, 1984.

    MATH  Google Scholar 

  48. G. B. Thurston, Viscoelasticity of human blood, Biophys. J., 12, 1972, 1205–1217.

    Article  Google Scholar 

  49. G. B. Thurston, Rheological parameters for the viscosity, viscoelasticity and thixotropy of blood, Biorheology, 16, 1979, pp. 149–162.

    Google Scholar 

  50. G. B. Thurston, Light transmission through blood in oscillatory flow, Biorheology, 27, 1990, pp. 685–700.

    Google Scholar 

  51. G. B. Thurston, Non-Newtonian viscosity of human blood: flow-induced changes in microstructure, Biorheology, 31(2), 1994, pp. 179–192.

    MathSciNet  Google Scholar 

  52. R. Unterhinninghofen, J. Albers, W. Hosch, C. Vahl and R. Dillmann, Flow quantification from time-resolved MRI vector fields, International Congress Series, 1281, 2005, pp. 126–130.

    Article  Google Scholar 

  53. G. Vlastos, D. Lerche, B. Koch, The superimposition of steady and oscillatory shear and its effect on the viscoelasticity of human blood and a blood-like model fluid, Biorheology, 34(1), 1997, pp. 19–36.

    Article  Google Scholar 

  54. F. J. Walburn, D. J. Schneck, A constitutive equation for whole human blood, Biorheology, 13, 1976, pp. 201–210.

    Google Scholar 

  55. N. T. Wang, A. L. Fogelson, Computational methods for continuum models of platelet aggregation, J. Comput. Phys., 151, 1999, pp. 649–675.

    Article  MATH  MathSciNet  Google Scholar 

  56. B. J. B. M. Wolters, M. C. M. Rutten, G.W. H. Schurink, U. Kose, J. de Hart and F. N. van de Vosse, A patient-specific computational model of fluid-structure interaction in abdominal aortic aneurysms, Medical Engineering & Physics, 27, 2005, pp. 871–883.

    Article  Google Scholar 

  57. K. K. Yelesvarapu, M. V. Kameneva, K. R. Rajagopal, J. F. Antaki, The flow of blood in tubes: theory and experiment,Mech. Res. Comm, 25(3), 1998, pp. 257–262.

    Article  Google Scholar 

  58. J.-B. Zhang and Z.-B. Kuang, Study on blood constitutive parameters in different blood constitutive equations, J. Biomechanics, 33, 2000, pp. 355–360.

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Dept. Matemática, IST and CEMAT/IST, Universidade Técnica de Lisboa, Av. Rovisco Pais, 1, 1049-001, Lisboa, Portugal

    Adélia Sequeira

  2. Dept. Matemática, ISEG and CEMAT/IST, Universidade Técnica de Lisboa, Rua do Quelhas, 6, 1200, Lisboa, Portugal

    João Janela

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  1. Adélia Sequeira
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  2. João Janela
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Editors and Affiliations

  1. Technical University of Lisbon, Lisbon, Portugal

    Manuel Seabra Pereira

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Sequeira, A., Janela, J. (2007). An Overview of Some Mathematical Models of Blood Rheology. In: Pereira, M.S. (eds) A Portrait of State-of-the-Art Research at the Technical University of Lisbon. Springer, Dordrecht. https://doi.org/10.1007/978-1-4020-5690-1_4

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  • DOI: https://doi.org/10.1007/978-1-4020-5690-1_4

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