Pflügers Archiv

, Volume 380, Issue 3, pp 221–226 | Cite as

Separate determination of the pulsatile elastic and viscous forces developed in the arterial wall in vivo

  • R. D. Bauer
  • R. Busse
  • A. Schabert
  • Y. Summa
  • E. Wetterer
Heart, Circulation, Respiration and Blood; Environmental and Exercise Physiology

Abstract

The viscoelastic behaviour of arteries in vivo is analyzed by separate representation of the purely elastic and the purely viscous properties, using natural pressure and diameter pulses of various dog arteries recorded under steady-state conditions. The circumferential wall stress (σ) and the radius (r) of the mean wall layer are calculated as functions of time and the hysteresis of the σ-r diagram is represented. The stress is regarded as the sum of an elastic stress (σel) which is a function ofr, and a viscous stress (σvis) which is a function ofdr/dt. Thus σel=σ−σvis. Since the σel-r diagram must be free from hysteresis, the disappearance of the loop is the criterion that indicates that σel has been found.

σvis is formulated as a second degree polynomial ofdr/dt whose coefficients are determined using that criterion.

The σel-r curve is always nonlinear and the elastic modulus increases with increasing radius. The σvis-dr/dt curve, too, is nonlinear. Its slope decreases with increasingdr/dt. The same applies to the wall viscosity (pseudoplastic behaviour). The nonlinear properties can be represented adequately by processing the experimental data in the time domain. Problems inherent in investigations based on the frequency domain, as reported in the literature, are pointed out.

Key words

Arterial elasticity Arterial wall viscosity Arterial stress-strain relationship Models of the arterial wall 

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References

  1. 1.
    Apter, J. T.: Correlation of viscoelastic properties of large arteries with microscopic structure. I, III. Circ. Res.19, 104–121 (1966)Google Scholar
  2. 2.
    Bauer, R. D., Pasch, Th.: The quasistatic and dynamic circumferential elastic modulus of the rat tail artery studied at various wall stresses and tones of the vascular smooth muscle. Pflügers Arch.330, 335–346 (1971)Google Scholar
  3. 3.
    Bauer, R. D., Busse, R., Schabert, A., Summa, Y., Wetterer, E.: The determination of the nonlinear elastic behaviour of viscoelastic solids demonstrated on arteries in vivo. Pflügers Arch.368, R7 (1977)Google Scholar
  4. 4.
    Bergel, D. H.: The dynamic elastic properties of the arterial wall. J. Physiol. (Lond.)156, 458–469 (1961)Google Scholar
  5. 5.
    Bergel, D. H., Schultz, D. L.: Arterial elasticity and fluid dynamics. Progr. Biophys. Mol. Biol.22, 1–36 (1971)Google Scholar
  6. 6.
    Cox, R. H.: A model for the dynamic mechanical properties of arteries. J. Biomech.5, 135–152 (1972)Google Scholar
  7. 7.
    Cox, R. H.: Pressure dependence of the mechanical properties of arteries in vivo. Am. J. Physiol.229, 1371–1375 (1975)Google Scholar
  8. 8.
    Flügge, W.: Viscoelasticity, 2nd ed. Berlin, Heidelberg, New York: Springer 1975Google Scholar
  9. 9.
    Fung, Y. C.: Comparison of different models of the heart muscle. J. Biomech.4, 289–295 (1971)Google Scholar
  10. 10.
    Fung, Y. C.: Stress-strain-history relations of soft tissues in simple elongation. In: Biomechanics — Its Foundations and Objectives (Y. C. Fung, N. Perrone, and M. Anliker, eds.) pp. 181–208. New Jersey: Prentice-Hall 1972Google Scholar
  11. 11.
    Goedhard, W. J. A., Knoop, A. A.: A model of the arterial wall. J. Biomech.6, 281–288 (1973)Google Scholar
  12. 12.
    Gow, B. S.: The influence of vascular smooth muscle on the viscoclastic properties of blood vessels. In: Cardiovascular fluid dynamics, Vol. II (D. H. Bergel, ed.), pp. 65–110. London, New York: Academic Press 1972Google Scholar
  13. 13.
    Gow, B. S., Taylor, M. G.: Measurement of viscoelastic properties of arteries in the living dog. Circ. Res.23, 111–122 (1968)Google Scholar
  14. 14.
    Hardung, V.: Vergleichende Messungen der dynamischen Elastizität und Viskosität von Blutgefäßen, Kautschuk und synthetischen Elastomeren. Helv. Physiol. Pharmacol. Acta11, 194–211 (1953)Google Scholar
  15. 15.
    Hardung, V.: Dynamische Elastizität und innere Reibung muskulärer Blutgefäße bei verschiedener durch Dehnung und tonische Kontraktion hervorgerufener Wandspannung. Arch. Kreisl.-Forsch.61, 83–100 (1970)Google Scholar
  16. 16.
    Learoyd, B. M., Taylor, M. G.: Alterations with age in the viscoelastic properties of human arterial walls. Circ. Res.18, 278–292 (1966)Google Scholar
  17. 17.
    McDonald, D. A., Taylor, M. G.: The hydrodynamics of the arterial circulation. Progr. Biophys. Biophys. Chem.9, 105–173 (1959)Google Scholar
  18. 18.
    Ranke, O. F.: Die Dämpfung der Pulswelle und die innere Reibung der Arterienwand. Z. Biol.95, 179–204 (1934)Google Scholar
  19. 19.
    Summa, Y.: Determination of the tangential elastic modulus of human arteries in vivo. In: The Arterial system-dynamics, control theory and regulation (R. D. Bauer and R. Busse, eds.), pp. 95–100. Berlin, Heidelberg, New York: Springer 1978Google Scholar
  20. 20.
    Taylor, M. G.: Hemodynamics. Annu. Rev. Physiol.35, 87–116 (1973)Google Scholar
  21. 21.
    Wesseling, K. H., Weber, H., de Wit, B.: Estimated five component viscoelastic model parameters for human arterial walls. J. Biomech.6, 13–24 (1973)Google Scholar
  22. 22.
    Westerhof, N., Noordergraaf, A.: Arterial viscoelasticity: a generalized model. J. Biomech.3, 357–379 (1970)Google Scholar
  23. 23.
    Wetterer, E., Kenner, Th.: Grundlagen der Dynamik des Arterienpulses. Berlin, Heidelberg, New York: Springer 1968Google Scholar
  24. 24.
    Wetterer, E., Bauer, R. D., Busse, R.: New ways of determining the propagation coefficient and the visco-elastic behaviour of arteries in situ. In: The Arterial system-dynamics, control theory and regulation (R. D. Bauer and R. Busse, eds.), pp. 35–47. Berlin, Heidelberg, New York: Springer 1978Google Scholar
  25. 25.
    Wetterer, E., Busse, R., Bauer, R. D., Schabert, A., Summa, Y.: Photoelectric device for contact-free recording of the diameters of exposed arteries in situ. Pflügers Arch.368, 149–152 (1977)Google Scholar

Copyright information

© Springer-Verlag 1979

Authors and Affiliations

  • R. D. Bauer
    • 1
  • R. Busse
    • 1
  • A. Schabert
    • 1
  • Y. Summa
    • 1
  • E. Wetterer
    • 1
  1. 1.Institut für Physiologie und Kardiologie der Universität Erlangen-NürnbergErlangenGermany

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