Annals of Biomedical Engineering

, Volume 34, Issue 4, pp 575–592 | Cite as

Blood Flow in Compliant Arteries: An Effective Viscoelastic Reduced Model, Numerics, and Experimental Validation

  • Sunčica ČanićEmail author
  • Craig J. Hartley
  • Doreen Rosenstrauch
  • Josip Tambača
  • Giovanna Guidoboni
  • Andro Mikelić


The focus of this work is on modeling blood flow in medium-to-large systemic arteries assuming cylindrical geometry, axially symmetric flow, and viscoelasticity of arterial walls. The aim was to develop a reduced model that would capture certain physical phenomena that have been neglected in the derivation of the standard axially symmetric one-dimensional models, while at the same time keeping the numerical simulations fast and simple, utilizing one-dimensional algorithms. The viscous Navier–Stokes equations were used to describe the flow and the linearly viscoelastic membrane equations to model the mechanical properties of arterial walls. Using asymptotic and homogenization theory, a novel closed, “one-and-a-half dimensional” model was obtained. In contrast with the standard one-dimensional model, the new model captures: (1) the viscous dissipation of the fluid, (2) the viscoelastic nature of the blood flow – vessel wall interaction, (3) the hysteresis loop in the viscoelastic arterial walls dynamics, and (4) two-dimensional flow effects to the leading-order accuracy. A numerical solver based on the 1D-Finite Element Method was developed and the numerical simulations were compared with the ultrasound imaging and Doppler flow loop measurements. Less than 3% of difference in the velocity and less than 1% of difference in the maximum diameter was detected, showing excellent agreement between the model and the experiment.


Blood flow modeling Viscoelasticity of arterial walls Fluidstructure interaction 



Research of Čanić, Hartley, Rosenstrauch, Tambača and Mikelić is supported in part by the joint National Science Foundation and the National Institutes of Health grant DMS-0443826. In addition, research of Čanić is supported in part by the National Science Foundation under grants DMS-0245513, DMS-0337355, and the research of Hartley is supported in part by the National Institutes of Health under Grant HL22512. The experimental design and the in vitro work at Dr. Rosenstrauch's laboratory at the Texas Heart Institute was partially supported by a grant from the Roderick Duncan McDonald Foundation at the St. Luke's Episcopal Hospital. Kent Elastomer Inc. donation of latex tubing is also acknowledged. The authors would like to thank undergraduate student Joy Chavez for the help with data collection and flow loop experiments. Chavez's research was supported by the NSF under grant DMS-0337355.


  1. 1.
    Armentano, R. L., J. G. Barra, J. Levenson, A. Simon, and R. H. Pichel. Arterial wall mechanics in conscious dogs: Assessment of viscous, iner-tial, and elastic moduli to characterize aortic wall behavior. Circ. Res. 76:468–478, 1995.Google Scholar
  2. 2.
    Armentano, R. L., J. L. Megnien, A. Simon, F. Bellenfant, J. G. Barra, and J. Levenson. Effects of hypertension on viscoelasticity of carotid and femoral arteries in humans. Hypertension 26:48–54, 1995.Google Scholar
  3. 3.
    Barnard, A. C. L, W. A. Hunt, W. P. Timlake, and E. Varley. A theory of fluid flow in compliant tubes. Biophys. J. 6:717–724, 1966.Google Scholar
  4. 4.
    Bauer R. D., R. Busse, A. Shabert, Y. Summa, and E. Wetterer. Separate determination of the pulsatile elastic and viscous forces developed in the arterial wall in vivo. Pflugers Arch. 380:221–226, 1979.Google Scholar
  5. 5.
    Biot, M. A. Theory of propagation of elastic waves in a fluid-saturated porous solid. I. Lower frequency range, and II. Higher frequency range. J. Acoust. Soc. Am. 28(2):168–178, 179–191, 1956.Google Scholar
  6. 6.
    Čanić, S., J. Tambača, G. Guidoboni, A. Mikelić, C. J. Hartley, and D. Rosenstrauch. Modeling viscoelastic behavior of arterial walls and their interaction with pulsatile blood flow. Submitted.Google Scholar
  7. 7.
    Čanić, S., and E. H. Kim. Mathematical analysis of the quasilinear effects in a hyperbolic model of blood flow through compliant axisym-metric vessels. Math. Methods Appl. Sci. 26(14):1161–1186, 2003.Google Scholar
  8. 8.
    Čanić, S., and A. Mikelić. Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries. SIAM J. Appl. Dyn. Sys. 2(3):431–463, 2003.Google Scholar
  9. 9.
    Čanić, S., A. Mikelić, D. Lamponi, and J. Tambača. Self-consistent effective equations modeling blood flow in medium-to-large compliant arteries. SIAM J. Multisc. Anal. Simul. 3(3):559–596, 2005.Google Scholar
  10. 10.
    Čanić, S., A. Mikelić, and J. Tambača. A two-dimensional effective model describing fluid–structure interaction in blood flow: Analysis, simulation and experimental validation. Comptes Rendus Mech. Acad. Sci. Paris 333:867–883, 2005.Google Scholar
  11. 11.
    Čanić, S., J. Tambača, A. Mikelić, C. J. Hartley, D. Mirković, and D. Rosenstrauch. Blood flow through axially symmetric sections of compliant vessels: New effective closed models. In: Proceedings of the 26th Annual International Conference. IEEE Eng. Med. Bio. Soc., 2004, 10–13 pp.Google Scholar
  12. 12.
    Chmielewski, C. Master of Science Thesis, Department of Mathematics, North Carolina State University, 2003.Google Scholar
  13. 13.
    Eringen, A. Cemal. Mechanics of continua. New York: Wiley, 1967, 365 pp.Google Scholar
  14. 14.
    Formaggia, L., D. Lamponi, and A. Quarteroni. One-dimensional models for blood flow in arteries. J. Eng. Math. 47:251–276, 2003.Google Scholar
  15. 15.
    Formaggia, L., F. Nobile, and A. Quarteroni. A one-dimensional model for blood flow: Application to vascular prosthesis. In: Mathematical Modeling and Numerical Simulation in Continuum Mechanics, edited by Babuska, Miyoshi, and Ciarlet), Lect. Notes Comput. Sci. Eng. 19:137–153, 2002.Google Scholar
  16. 16.
    Haidekker, M. A., C. R. White, and J. A. Frangos. Analysis of temporal shear stress gradients during the onset phase of flow over a backward-facing step. J. Biomech. Eng. 123:455–463, 2001.Google Scholar
  17. 17.
    Hartley, C. J. Ultrasonic blood flow and velocimetry. In: McDonald's Blood Flow in Arteries, Theoretical, Experimental and Clinical Principles, 4th edn. Ch. 7, edited by W. W. Nichols and M. F. O'Rourke. London: Arnold, 1998, pp. 154–169.Google Scholar
  18. 18.
    Hartley, C. J. G. Taffet, A. Reddy, M. Entman, and L. Michael. Noninvasive cardiovascular phenotyping in mice. ILAR J. 43:147–158, 2002.Google Scholar
  19. 19.
    Nichols, W. W., and M. F. O'Rourke. McDonald's Blood Flow in Arteries: Theoretical, Experimental and Clinical Principles, 4th edn. New York: Arnold and Oxford University Press, 2000.Google Scholar
  20. 20.
    Olufsen, M. S., C. S. Peskin, W. Y. Kim, E.M. Pedersen, A. Nadim, and J. Larsen. Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Eng. 28:1281–1299, 2000.Google Scholar
  21. 21.
    Pontrelli, G. Modeling the fluid–wall interaction in a blood vessel. Prog. Biomed. Res. 6(4):330–338, 2001.Google Scholar
  22. 22.
    Scott-Burden, T., J. P. Bosley, D. Rosenstrauch, K. Henderson, F. Clubb, H. Eichstaedt, K. Eya, I. Gregoric, T. Myers, B. Radovancevic, and O. H. Frazier. Use of autologous auricular chondrocytes for lining artificial surfaces: A feasibility study. Ann. Thor. Surg. 73(5):1528–1533, 2002.Google Scholar
  23. 23.
    Smith, N. P., A. J. Pullan, and P. J. Hunter. An anatomically based model of transient coronary blood flow in the heart. SIAM J. Appl. Math. 62(3):990–1018, 2002.Google Scholar
  24. 24.
    Tambača, J., S. Čanić, and A. Mikelić. Effective model of the fluid flow through elastic tube with variable radius. Grazer Math. Ber., ISSN1016 7692 Bericht Nr. 3:1–22, 2005.Google Scholar

Copyright information

© Biomedical Engineering Society 2006

Authors and Affiliations

  • Sunčica Čanić
    • 1
    Email author
  • Craig J. Hartley
    • 2
  • Doreen Rosenstrauch
    • 3
  • Josip Tambača
    • 4
  • Giovanna Guidoboni
    • 1
  • Andro Mikelić
    • 5
  1. 1.Department of MathematicsUniversity of HoustonHoustonUSA
  2. 2.Sections of Cardiovascular Sciences, Department of MedicineBaylor College of MedicineHoustonUSA
  3. 3.Texas Heart Institute and the University of Texas Health Science Center at HoustonHoustonUSA
  4. 4.Department of MathematicsUniversity of ZagrebZagrebCroatia
  5. 5.Department of MathematicsUniversité Claude Bernard Lyon 1Villeurbanne CedexFrance

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