Biomechanics pp 108-205 | Cite as

Blood Flow in Arteries

  • Y. C. Fung


The larger systemic arteries, shown in Figure 3.1:1, conduct blood from the heart to the peripheral organs. Their dimensions are given in Table 3.1:1. In humans, the aorta originates in the left ventricle at the aortic valve, and almost immediately curves about 180°, branching off to the head and upper limbs. It then pursues a fairly straight course downward through the diaphragm to the abdomen and legs. The aortic arch is tapered, curved, and twisted (i.e., its centerline does not lie in a plane). Other arteries have constant diameter between branches, but every time a daughter branch forks off the main trunk the diameter of the trunk is reduced. Overall, the aorta may be described as tapered. In the dog, the change of area fits the exponential equation,
$$ A=A_{0}e^{(Bx/R_{o})}$$
where A is the area of the aorta, A 0 and R 0 are, respectively, the area and radius at the upstream site, x is the distance from that upstream site, and B is a “taper factor,” which has been found to lie between 0.02 and 0.05. Figure 3.1:2 shows a sketch of the dog aorta.


Reynolds Number Wall Shear Stress Pulse Wave Wave Speed Pulsatile Flow 
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  1. Anliker, M. (1972). Toward a nontraumatic study of the circulatory system. In Biomechanics: Its Foundations and Objectives (Y.C. Fung, N. Perrone, and M. Anliker, eds.), Prentice-Hall, Englewood Cliffs, NJ, pp. 337–379.Google Scholar
  2. Anliker, M., and Maxwell, J.A. (1966). The dispersion of waves in blood vessels. In Biomechanics (Y.C. Fung, ed.), American Society of Mechanical Engineers, New York, pp. 47–67.Google Scholar
  3. Anliker, M., and Raman, K.R. (1966). Korotkoff sounds at diastole—a phenomenon and dynamic instability of fluid-filled shells. Int. J. Solids Structures. 2: 467–492.CrossRefGoogle Scholar
  4. Anliker, M., Histand, M.B., and Ogden, E. (1968). Dispersion and attenuation of small artificial pressure waves in canine aorta. Circ. Res. 23: 539–551.PubMedCrossRefGoogle Scholar
  5. Aoki, T., and Ku, D.N. (1993). Collapse of diseased arteries with eccentric cross-section. J. Biomech. 26: 133–142.PubMedCrossRefGoogle Scholar
  6. Atabek, H.B. (1962). Development of flow in the inlet length of a circular tube starting from rest. Z. Angew. Math. Phys. 13: 417–430.CrossRefGoogle Scholar
  7. Atabek, H.B. (1980). Blood flow and pulse propagation in arteries. In Basic Hemodynamics and its Role in Disease Processes (DJ. Patel and R.N. Vaishnav, eds.), University Park Press, Baltimore, MD, pp. 253–361.Google Scholar
  8. Attinger, E.O. (ed.) (1964). Pulsatile Blood Flow. McGraw-Hill, New York.Google Scholar
  9. Benditt, E.P., and Benditt, J. M. (1973). Evidence for a monoclonal origin of human atherosclerotic plaques. Proc. Natl. Acad. Sci. USA, 70: 1753–1756.PubMedCrossRefGoogle Scholar
  10. Bergel, D.H. (ed.) (1972). Cardiovascular Fluid Dynamics, Vols. 1 & 2. Academic Press, New York.Google Scholar
  11. Bohr, D.F., Somlyo, A.P., and Sparks, H.V., Jr. (eds.) (1980). Handbook of Physiology, Sec. 2. The Cardiovascular System. Vol. 2, Vascular Smooth Muscles. American Physiological Society, Bethesda, MD.Google Scholar
  12. Boussinesq, J. (1891). Maniere dont les vitesses, se distrib. depui l’entree—Moindre longueur d’un tube circulaire, pour qu’un regime uniforme s’y établisse. Comptes Rendus, 113: 9, 49.Google Scholar
  13. Caro, C.G., Fitzgerald, J.M., and Schroter, R.C. (1969). Arterial wall shear and distribution of early atheroma in man. Nature 223: 1159–1161.PubMedCrossRefGoogle Scholar
  14. Caro, C.G., Fitzgerald, J.M., and Schroter, R.C. (1971). Atheroma and arterial wall shear. Observation, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis. Proc. Roy. Soc. Lond. B. 177: 109–159.CrossRefGoogle Scholar
  15. Caro, C.G., Pedley, T.J., and Seed, W.A. (1974). Mechanics of the circulation. In Cadiovascular Physiology (A.C. Guyton, ed.), Chapter 1, Medical and Technical Publishers, London.Google Scholar
  16. Caro, C.G., Pedley, T.J., Schroter, R.C., and Seed, W.A. (1978). The Mechanics of the Circulation. Oxford University Press, Oxford.Google Scholar
  17. Dai, K., Xue, H., Dou, R., and Fung, Y.C. (1985). On the detection of messages carried in arterial pulse waves. J. Biomech. Eng. 107: 268–273.PubMedCrossRefGoogle Scholar
  18. DeBakey, M.E., Lawrie, G.M., and Glaeser, D.H. (1985). Patterns of atherosclerosis and their surgical significance. Ann. Surg. 201: 115–131, Lippincott Co., Philadelphia, PA.PubMedCrossRefGoogle Scholar
  19. Deshpande, M.D., and Giddens, D.P. (1980). Turbulence measurements in a constricted tube. J. Fluid Mech. 97: 65–90.CrossRefGoogle Scholar
  20. Euler, L. (1775). Principia pro motu sanguins per arterias determinado. Opera posthuma mathematica et physica anno 1844 detecta, ediderunt P.H. Fuss et N. Fuss. Petropoli, Apud Eggers et socios, Vol. 2, pp. 814–823.Google Scholar
  21. Friedman, M.H., and Deters, O.J. (1987). Correlation among shear rate measures in vascular flows. J. Biomech. Eng. 109: 25–26.PubMedCrossRefGoogle Scholar
  22. Friedman, M.H., Deters, O.J., Mark, F. F., Bargeron, C.B., and Hutchins, G.M. (1983). Arterial geometry affects hemodynamics. A potential risk factor for atherosclerosis. Atherosclerosis 46: 225–231.PubMedCrossRefGoogle Scholar
  23. Fry, D.L. (1968). Acute vascalar endothelial changes associated with increased blood velocity gradients. Circ. Res. 22: 165–197.PubMedCrossRefGoogle Scholar
  24. Fry, D.L. (1973). Responses of the arterial wall to certain factors. In Atherosclerosis: Initiating Factors. Ciba Foundation Symp. Elsevier, Amsterdam, p. 93.Google Scholar
  25. Fry, D.L. (1977). Aortic Evans blue dye accumulation: Its measurement and interpretation. Am. J. Physiol. 232: H204–H222.PubMedGoogle Scholar
  26. Fry, D.L., Griggs, Jr., D.M., and Greenfield Jr., J.C. (1964). In vivo studies of pulsatile blood flow: the relationship of the pressure gradient to the blood velocity. In Pulsatile Blood Flow (E.O. Attinger, ed.), McGraw-Hill, New York, Chap. 5, pp. 101–114.Google Scholar
  27. Fung, Y.C. (1990). Biomechanics: Motion, Flow, Stress, and Growth. Springer-Verlag, New York.Google Scholar
  28. Fung, Y.C. (1993a). A First Course in Continuum Mechanics. 3rd ed. Prentice-Hall, Englewood Cliffs, N.J.Google Scholar
  29. Fung, Y.C. (1993b). Biomechanics: Mechanical Properties of Living Tissues. 2nd ed. Springer-Verlag, New York.Google Scholar
  30. Fung, Y.C., and Liu, S.Q. (1993). Elementary mechanics of the endothelium of blood vessels. J. Biomech. Eng. 115: 1–12.PubMedCrossRefGoogle Scholar
  31. Fung, Y.C., Fronek, K., and Patitucci, P. (1979). On pseudo-elasticity of arteries and the choice of its mathematical expression. Am. J. Physiol. 237: H620–H631.PubMedGoogle Scholar
  32. Giddens, D.P., Zarins, C.K., and Glagov, S. (1990). Response of arteries to near-wall fluid dynamic behavior. Appl. Mech. Rev. 43: S98–S102.CrossRefGoogle Scholar
  33. Hamilton, W.F., and Dow, P. (1939). An experimental study of the standing waves in the pulse propagated through the aorta. Am. J. Physiol. 125: 48–59.Google Scholar
  34. Jacobs, R.B. (1953). On the propagation of a disturbance through a viscous liquid flowing in a distensible tube of appreciable mass. Bull. Math. Biophys. 5: 395–409.CrossRefGoogle Scholar
  35. Jones, E., Anliker, M., and Chang, I.D. (1971). Effects of viscosity and constraints on the dispersion and dissipation of waves in large blood vessels. I & II. Biophys. J. 11: 1085–1120, 1121-1134.PubMedCrossRefGoogle Scholar
  36. Joukowsky, N.W. (1900). Ueber den hydraulischen Stoss in Wasserheizungsrohren. Memoires de l’Academie Imperiale des Science de St. Petersburg, 8 series, Vol. 9, No. 5.Google Scholar
  37. Kamiya, A., and Togawa, T. (1972). Optimal branching of the vascular tree (minimum volume theory). Bull. Math. Biophys. 34: 431–438.PubMedCrossRefGoogle Scholar
  38. Kamiya, A., and Togawa, T. (1980). Adaptive regulation of wall shear stress to flow change in the canine carotid artery. Am. J. Physiol. 239: H14–H21.PubMedGoogle Scholar
  39. Kamiya, A., Bukhari, R., and Togawa, T. (1984). Adaptive regulation of wall shear stress optimizing vascular tree function. Bull. Math. Biol. 46: 127–137.PubMedGoogle Scholar
  40. Kassab, G.S., and Fung, Y.C. (1995). The pattern of coronary arteriolar bifurcations and the uniform shear hypothesis. Ann. Biomed. Eng., 23: 13–20.PubMedCrossRefGoogle Scholar
  41. Kassab, G.S., Rider, C.A., Tang, N.J., and Fung, Y.C. (1993). Morphometry of the pig coronary arterial trees. Am. J. Physiol., 266: H350–H365.Google Scholar
  42. King, A.L. (1947). Waves in elastic tubes: velocity of the pulse wave in large arteries. J. Appl. Phys. 18: 595–600.CrossRefGoogle Scholar
  43. Klip, W. (1958). Difficulties in the measurement of pulse wave velocity. Am. Heart J. 56: 806–813.PubMedCrossRefGoogle Scholar
  44. Klip, W. (1962). Velocity and Damping of the Pulse Wave. Martinus Nijhoff, The Hague.Google Scholar
  45. Klip, W. (1967). Formulas for phase velocity and damping of longitudinal waves in thick-walled viscoelastic tubes. J. Appl. Phys. 38: 3745–3755.CrossRefGoogle Scholar
  46. Korteweg, D.J. (1878). Ueber die Fortpflanzungesgeschwindigkeit des Schalles in elastischen Rohren. Ann. Physik. Chemie 5: 525–542.CrossRefGoogle Scholar
  47. Ku, D.N., Giddens, D.P., Zarins, C.K., and Glagov, S. (1985). Pulsatile flow and atherosclerosis in the human carotid bifurcation. Arteriosclerosis 5: 293–302.PubMedCrossRefGoogle Scholar
  48. Lamb, H. (1897-1898). On the velocity of sound in a tube, as affected by the elasticity of the walls. Phil. Soc. Manchester Memoirs Proc., lit. A, 42: 1–16.Google Scholar
  49. Lambert, J.W. (1958). On the nonlinearities of fluid flow in nonrigid tubes. J. Franklin Inst. 266: 83–102.CrossRefGoogle Scholar
  50. Lanczos, C. (1952). Introduction. In Tables of Chebyshev Polynomials. National Bureau of Standards, Appl Math, Ser. 9, U.S. Govt. Printing Office, Washington, D.C., pp.7–9.Google Scholar
  51. Landowne, M. (1958). Characteristics of impact and pulse wave propagation in brachial and radial arteries. J. Appl. Physiol. 12: 91–97.PubMedGoogle Scholar
  52. Lei, M., Kleinstreuer, G, and Truskey, G.A. (1995). Numerical investigation and prediction of atherogenic sites in branching arteries. J. Biomech. Eng. 17: 350–357.CrossRefGoogle Scholar
  53. Lew, H.S., and Fung, Y.C. (1970). Entry flow into blood vessels at arbitrary Reynolds number. J. Biomech. 3: 23–38.PubMedCrossRefGoogle Scholar
  54. Liebow, A.A. (1963). Situations which lead to changes in vascular patterns. In: Handbook of Physiology, Section 2: Circulation, Vol. 2. Washington, D.G: American Physiological Society, pp. 1251–1276.Google Scholar
  55. Lighthill, M.J. (1978). Waves in Fluids. Cambridge University Press, London, UK.Google Scholar
  56. Ling, S.C., and Atabek, H.B. (1972). A nonlinear analysis of pulsatile flow in arteries. J. Fluid. Mech. 55: 493–511.CrossRefGoogle Scholar
  57. Liu, S.Q., Yen, M., and Fung, Y.C. (1994). On measuring the third dimension of cultured endothelial cells in shear flow. Proc. Natl. Acad. Sci. U.S.A. 91: 8782–8786.PubMedCrossRefGoogle Scholar
  58. Maxwell, J.A., and Anliker, M. (1968). The dissipation and dispersion of small waves in arteries and veins with viscoelastic wall properties. Biophys. J. 8: 920–950.PubMedCrossRefGoogle Scholar
  59. McCord, B.N., and Ku, D.N. (1993). Mechanical rupture of the atherosclerotic plaque fibrous cap. Bioengineering Conf. ASME, BED Vol. 24, pp. 324–326.Google Scholar
  60. McDonald, D.A. (1960, 1974). Blood Flow in Arteries. 1st ed., 2nd ed. Williams & Wilkins, Baltimore, MD.Google Scholar
  61. McGill, H.C. Jr., Geer, J.C., and Holman, R.L. (1957). Sites of vascular vulnerability in dogs demonstrated by Evans Blue. AMA Arch. Pathol. 64: 303–311.PubMedGoogle Scholar
  62. Morgan, G.W., and Ferrante, W.R. (1955). Wave propagation in elastic tubes filled with streaming liquid. J. Acoust. Soc. Amer. 27: 715–725.CrossRefGoogle Scholar
  63. Morgan, G.W, and Kiely, J.P. (1954). Wave propagation in a viscous liquid contained in a flexible tube. J. Acoust. Soc. Amer. 26: 323–328.CrossRefGoogle Scholar
  64. Motomiya, M., and Karino, T. (1984). Flow patterns in the human carotid artery bifurcation. Stroke 15: 50–56.PubMedCrossRefGoogle Scholar
  65. Murray, C.D. (1926). The physiological principle of minimum work. I. The vascular system and the cost of blood volume. Proc. Natl. Acad. Sci. U.S.A. 12: 207-214; J. Gen. Physiol. 9: 835–841.Google Scholar
  66. Nerem, R.M., Seed, W.A., and Wood, N.B. (1972). An experimental study of the velocity distribution and transition to turbulence in the aorta. J. Fluid Mech. 52: 137–160.CrossRefGoogle Scholar
  67. Oka, S. (1974). Rheology-Biorheology. Syokabo, Tokyo. (In Japanese).Google Scholar
  68. Patel, D.J., and Vaishnav, R.N. (eds.) (1980). Basic Hemodynamics and its Role in Disease Process. University Park Press, Baltimore, MD.Google Scholar
  69. Pedley, T.J. (1980). The Fluid Mechanics of Large Blood Vessels. Cambridge University Press, London.CrossRefGoogle Scholar
  70. Poiseuille, J.L. (1841). Recherches expérimentales sur le mouvement des liquides dans les tubes de très petits diamètres. Compte Rendus, Académie des Sciences. Paris.Google Scholar
  71. Prandtl, L. (1904). Über Flüssigkeitsbewegung bei sehr Kleiner Reibung. Proc. 3rd Intern. Math. Congress. Heidelberg.Google Scholar
  72. Rodbard, S. (1975). Vascular caliber. Cardiology 60: 4–49.PubMedCrossRefGoogle Scholar
  73. Rokitansky, C. von (1852). A Manual of Pathological Anatomy. Translated by G.E. Day. Vol. 4. London, The Sydenham Society.Google Scholar
  74. Rosen, R. (1967). Optimality Principles in Biology. Butterworth, London.Google Scholar
  75. Ross, R. (1988). The pathogenesis of atherosclerosis. In Heart Disease (E. Braunwald, ed.) Saunders, Philadelphia, Chapter 35, pp. 1135–1152.Google Scholar
  76. Ross, R., and Glomset, J. (1973). Atherosclerosis and the arterial smooth muscle cell. Science 180: 1332.PubMedCrossRefGoogle Scholar
  77. Rubinow, S.I., and Keller, J.B. (1972). Flow of a viscous fluid through an elastic tube with applications to blood flow. J. Theo. Biol. 35(2): 299–313.CrossRefGoogle Scholar
  78. Schiller, L. (1922). Die Entwicklung der laminaren Geschwindigkeitsverteilung und ihre Bedeutung für Zahigkeitsmessungen. Z. Angew. Math. Mech. 2: 96–106.CrossRefGoogle Scholar
  79. Schlichting, H. (1968). Boundary Layer Theory, 6th ed. McGraw-Hill, New York.Google Scholar
  80. Schwartz, C.J., Valente, A.J., Sprague, E.A., Kelley, J.L., and Nerem, R.M. (1991). The pathogenesis of atherosclerosis: An overview. Clin. Cardiol. 14: 1–16.CrossRefGoogle Scholar
  81. Skalak, R. (1966). Wave propagation in blood flow. In Biomechanics Symposium (Y.C. Fung, ed.). American Society of Mechanical Engineers, New York, pp. 20–40.Google Scholar
  82. Skalak, R. (1972). Synthesis of a complete circulation. In Cardiovascular Fluid Dynamics (D.H. Bergel, ed.), Vol. 2. Academic Press, New York, pp. 341–376.Google Scholar
  83. Sramek, B.B., Valenta, X, and Klimes, F. (eds.) (1995). Biomechanics of the Cardiovascular System, Czech Technical Univ. Press Prague.Google Scholar
  84. Szegö, G. (1939). Orthogonal Polynomials, 4th ed. American Math. Soc. Colloquium Vol. 23.Google Scholar
  85. Taber, L.A. (1995). Biomechanics of growth, remodeling, and morphogenesis. Appl. Mech. Rev. 48: 487–545.CrossRefGoogle Scholar
  86. Targ, S.M. (1951). Basic Problems of the Theory of Laminar Flows. Moscow (In Russian).Google Scholar
  87. Taylor, M.G. (1959). An experimental determination of the propagation of fluid oscillations in a tube with a viscoelastic wall. Phys. Med. Biol. 4: 63–82.PubMedCrossRefGoogle Scholar
  88. Taylor, M.G. (1966a). Use of random excitation and spectral analysis in the study of frequency-dependent parameters of the cardiovascular system. Circ. Res. 18: 585–595.PubMedCrossRefGoogle Scholar
  89. Taylor, M.G. (1966b). Input impedance of an assembly of randomly branching elastic tubes. Biophys. J. 6: 29-51, 6: 697–716.CrossRefGoogle Scholar
  90. Thoma, R. (1893). Untersuchungen über die Histogenese und Histomechanik des Gefassy sternes. Stuttgart: Enke.Google Scholar
  91. Thubrikar, M.J., and Robicsec, F. (1995). Pressure-induced arterial wall stress and atherosclerosis. Ann. Thorac. Surg. 59: 1594–1603.PubMedCrossRefGoogle Scholar
  92. Thubrikar, M.J., Baker, J.W., and Nolan, S.P. (1988). Inhibition of atherosolerosis associated with reduction of arterial intramural stress in rabbits. Arteriosclerosis 8: 410–420.PubMedCrossRefGoogle Scholar
  93. Thubrikar, M.J., Roskelley, S.K., and Eppink, R.T. (1990). Study of stress concentration in the walls of the bovine coronary arterial branch. J. Biomech. 23: 15–26.PubMedCrossRefGoogle Scholar
  94. Valenta, J. (ed.) (1993). Biomechanics, Academia, Prague, Elsevier, Amsterdam.Google Scholar
  95. Van der Werff, T.J. (1973). Periodic method of characteristics. J. Comp. Phys. 11: 296–305.CrossRefGoogle Scholar
  96. Virchow, R. (1856). Phlogose und thrombose im gefassystem. Gessamette Abhandlungen zur Wissenschaftlichen Medicin. Frankfurt-am-Main, Meidinger Sohn u. Co. p. 458.Google Scholar
  97. Werlé, H. (1974). Le Tunnel Hydrodynamique au Service de la Recherche Aérospatiale. Publication No. 156, ONERA, France. Paris.Google Scholar
  98. Wetterer, E., and Kenner, T. (1968). Grundlagen der Dynamik des Arterienpulses. Springer-Verlag, Berlin.Google Scholar
  99. Witzig, K. (1914). Über erzwungene Wellenbewegungen zaher, inkompressibler Flüssigkeiten in elastischen Rohren. Inaugural Dissertation, Universitat Bern, K.J. Wyss, Bern.Google Scholar
  100. Womersley, J.R. (1955a). Method for the calculation of velocity, rate of flow, and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127: 553–563.PubMedGoogle Scholar
  101. Womersley, J.R. (1955b). Oscillatory motion of a viscous liquid in a thin-walled elastic tube-I: The linear approximation for long waves. Phil. Mag. 46(Ser. 7): 199–221.Google Scholar
  102. Xue, H., and Fung, Y.C. (1989). Persistence of asymmetry in nonaxisymmetric entry flow in a circular cylindrical tube and its relevance to arterial pulse wave diagnosis. J. Biomech. Eng. 111: 37–41.PubMedCrossRefGoogle Scholar
  103. Yao, L.S., and Berger, S.A. (1975). Entry flow in a curved pipe. J. Fluid Mech. 67: 177–196.CrossRefGoogle Scholar
  104. Yih, C.S. (1977). Fluid Mechancs. West River Press, Ann Arbor, MI.Google Scholar
  105. Young, T. (1808). Hydraulic investigations, subservient to an intended Croonian lecture on the motion of the blood. Phil. Trans. Roy. Soc. London 98: 164–186.CrossRefGoogle Scholar
  106. Young, T. (1809). On the functions of the heart and arteries. Phil. Trans. Roy. Soc. London 99: 1–31.Google Scholar
  107. Zamir, M. (1976). The role of shear forces in arterial branching. J. Gene. Biolo. 67: 213–222.Google Scholar
  108. Zamir, M. (1977). Shear forces and blood vessel radii in the cardiovascular system. J. Gen. Physiol. 69: 449–461.PubMedCrossRefGoogle Scholar
  109. Zarins, C.K., Zatina, M.A., Giddens, D.P., Ku, D.N., and Glagov, S. (1987). Shear stress regulation of artery lumen diameter in experimental atherogenesis. J. Vasc. Surg. 5: 413–420.PubMedGoogle Scholar

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© Springer Science+Business Media New York 1997

Authors and Affiliations

  • Y. C. Fung
    • 1
  1. 1.Department of BioengineeringUniversity of California, San DiegoLa JollaUSA

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