Cardiovascular Engineering

, Volume 7, Issue 2, pp 51–73 | Cite as

A One-dimensional Model of Blood Flow in Arteries with Friction and Convection Based on the Womersley Velocity Profile

  • Karim AzerEmail author
  • Charles S. Peskin
Original Paper


In this paper, we present a one-dimensional model for blood flow in arteries, without assuming an a priori shape for the velocity profile across an artery (Azer, Ph.D. thesis, Courant Institute, New York University, 2006). We combine the one-dimensional equations for conservation of mass and momentum with the Womersley model for the velocity profile in an iterative way. The pressure gradient of the one-dimensional model drives the Womersley equations, and the velocity profiles calculated then feed back into both the friction and nonlinear parts of the one-dimensional model. Besides enabling us to evaluate the friction correctly and also to use the velocity profile to correct the nonlinear terms, having the velocity profile available as output should be useful in a variety of applications. We present flow simulations using both structured trees and pure resistance models for the small arteries, and compare the resulting flow and pressure waves under various friction models. Moreover, we show how to couple the one-dimensional equations with the Taylor diffusion limit (Azer, Int J Heat Mass Transfer 2005;48:2735–40; Taylor, Proc R Soc Lond Ser A 1953;219:186–203) of the convection-diffusion equations to drive the concentration of a solute along an artery in time.


One-dimensional blood flow Womersley MRI Shear stress Velocity profile Structured tree Hypertension Compliance Taylor diffusion 



The authors would like to thank Mette Olufsen for helpful discussion on structured trees, and for making available the MRI data of a healthy male subject which was recorded by E.M Pedersen and Y. Kim at Skejby University Hospital in Denmark. This research was supported by the Applied Computer Science Department at Merck & Co., Inc. We would especially like to thank Jeffrey Saltzman, Robert Nachbar and Christopher Tong for their interest in this work and for helpful discussions.


  1. Azer K. Taylor diffusion in time-dependent flow. Int J Heat Mass Transfer 2005;48:2735–40CrossRefGoogle Scholar
  2. Azer K. A one-dimensional model of blood flow in arteries with friction, convection and unsteady Taylor diffusion based on the Womersley velocity profile. Ph.D. thesis, Courant Institute, New York University; 2006Google Scholar
  3. Calfisch R, Majda G, Peskin CS, Strumolo G. Distortion of the arterial pulse. Math BioSci 1980; 229–60Google Scholar
  4. Caro C, Pedley T, Schroter R, Seed W. The mechanics of the circulation. Oxford University Press; 1978Google Scholar
  5. Chorin AJ, Marsden JE. A mathematical introduction to fluid mechanics, 3rd ed. New York: Springer; 1998Google Scholar
  6. Coleman T, Harris J, Guyton A. Whole-body circulatory autoregulation and hypertension. Circ Res 1971;XXVIII and XXIX, 76–87Google Scholar
  7. Courant R, Friedrichs K. Supersonic flow and shock waves. Springer, Applied Mathematical Sciences; 1999Google Scholar
  8. Dong S, Karniadakis G, Karonis N. Cross-site computations on the teragrid. Comput Sci Eng 2005;7(5):14–23CrossRefGoogle Scholar
  9. Feinberg AW, Lax H. Studies of the arterial pulse wave. Circulation 1958;XVIII:1125–30Google Scholar
  10. Formaggia L, Gerbau J-F, Nobile F, Quarteroni A. On the coupling of 3d and 1d Navier-Stokes equations for flow problems in compliant vessels. Comp Methods Appl Mech Eng 2001;191(6–7):561–82CrossRefGoogle Scholar
  11. Franklin S. Systolic blood pressure. Am J Hypertens 2004;17:49S–54SPubMedCrossRefGoogle Scholar
  12. Griffith T, Klassen P, Franklin S. Systolic hypertension: an overview. Am Heart J 2005;149(5):769–75PubMedCrossRefGoogle Scholar
  13. Guyton A, Hall J. Textbook of medical physiology. W. B. Saunders Company; 2000Google Scholar
  14. Hoppensteadt FC, Peskin CS. Modeling and simulation in medicine and the life sciences. 2nd ed. Springer Verlag; 2002Google Scholar
  15. Iberall A. Anatomy and steady flow characteristics of the arterial system with an introduction to its pulsatile characterisitcs. Math BioSci 1967;197:375–85CrossRefGoogle Scholar
  16. Lax H, Feinberg A, Cohen B. Studies of the arterial pulse wave; the normal pulse wave and its modification in the presence of human arteriosclerosis. J Chronic Dis 1956;3(6):618–31Google Scholar
  17. Mendelsohn M. In hypertension, the kidney is not always the heart of the matter. J Clin Invest 2005;115(4):840–4PubMedCrossRefGoogle Scholar
  18. Nichols MW, O’Rourke MF. McDonald’s blood flow in arteries, theoretical, experimental and clinical principles. 4th ed. Edward Arnold; 1998Google Scholar
  19. Nichols W. Clinical measurement of arterial stiffness obtained from noninvasive pressure waveforms. Am J Hypertens 2005;18:3S–10SCrossRefGoogle Scholar
  20. Olufsen MS, Peskin CS, Kim WK, Pedersen EM, Nadim A, Larsen J. Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann Biomed Eng 2000;28:1281–99PubMedCrossRefGoogle Scholar
  21. Ottesen J, Olufsen M, Larsen J. Applied mathematical models in human physiology. SIAM Monographs on Mathematical Modeling and Computation; 2004Google Scholar
  22. Pedley TJ. The fluid mechanics of large blood vessels. 3rd ed. Cambridge: Cambridge University Press; 1980Google Scholar
  23. Richtmeyer, Morton. Difference methods for initial-value problems. 2nd ed. John Wiley and Sons; 1967Google Scholar
  24. Saltzman J. An unsplit 3d upwind method for hyperbolic conservation laws. J Comput Phys 1994;115(1):153–68CrossRefGoogle Scholar
  25. Steele B, Taylor C. Simulation of blood flow in the abdominal aorta at rest and during exercise using a 1-d finite element method with impedance boundary conditions derived from a fractal tree. In: Summer Bioengineering Conference. Summer Bioengineering Conference; 2003Google Scholar
  26. Stergiopulos N, Young DF, Rogge TR. Computer simulation of arterial flow with applications to arterial and aortic stenoses. J Biomech 1992;25:1477–88PubMedCrossRefGoogle Scholar
  27. Taylor G. Dispersion of soluble matter in solvent flowing slowly through a tube. Proc R Soc Lond Ser A 1953;219:186–203CrossRefGoogle Scholar
  28. Zamir M. On fractal properties of arterial trees. J Theor Biol 1999;197(4):517–26Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Courant Institute of Mathematical SciencesNew York UniversityNew YorkNYUSA
  2. 2.Applied Computer Science and Mathematics DepartmentRahwayNJUSA

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