Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T07:26:22.006Z Has data issue: false hasContentIssue false
  • English
  • Français

A wave propagation model of blood flow in large vessels using an approximate velocity profile function

Published online by Cambridge University Press:  21 May 2007

DAVID BESSEMS ,
MARCEL RUTTEN  and
FRANS VAN DE VOSSE
DAVID BESSEMS
Affiliation:
Department of Biomedical Engineering, Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands
MARCEL RUTTEN
Affiliation:
Department of Biomedical Engineering, Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands
FRANS VAN DE VOSSE
Affiliation:
Department of Biomedical Engineering, Eindhoven University of Technology, Postbus 513, 5600 MB Eindhoven, The Netherlands
Get access Rights & Permissions [Opens in a new window]

Abstract

Lumped-parameter models (zero-dimensional) and wave-propagation models (one-dimensional) for pressure and flow in large vessels, as well as fully three-dimensional fluid–structure interaction models for pressure and velocity, can contribute valuably to answering physiological and patho-physiological questions that arise in the diagnostics and treatment of cardiovascular diseases. Lumped-parameter models are of importance mainly for the modelling of the complete cardiovascular system but provide little detail on local pressure and flow wave phenomena. Fully three-dimensional fluid–structure interaction models consume a large amount of computer time and must be provided with suitable boundary conditions that are often not known. One-dimensional wave-propagation models in the frequency and time domain are well suited to obtaining clinically relevant information on local pressure and flow waves travelling through the arterial system. They can also be used to provide boundary conditions for fully three-dimensional models, provided that they are defined in, or transferred to, the time domain.

Most of the one-dimensional wave propagation models in the time domain described in the literature assume velocity profiles and therefore frictional forces to be in phase with the flow, whereas from exact solutions in the frequency domain a phase difference between the flow and the wall shear stress is known to exist. In this study an approximate velocity profile function more suitable for one-dimensional wave propagation is introduced and evaluated. It will be shown that this profile function provides first-order approximations for the wall shear stress and the nonlinear term in the momentum equation, as a function of local flow and pressure gradient in the time domain. The convective term as well as the approximate friction term are compared to their counterparts obtained from Womersley profiles and show good agreement in the complete range of the Womersley parameter α. In the limiting cases, for Womersley parameters α → 0 and α → ∞, they completely coincide. It is shown that in one-dimensional wave propagation, the friction term based on the newly introduced approximate profile function is important when considering pressure and flow wave propagation in intermediate-sized vessels.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 580 , 10 June 2007 , pp. 145 - 168
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

Anliker, M., Rockwell, L. & Ogden, E. 1971 Nonlinear analysis of flow pulses and shock waves in arteries. Z. Angew. Math. Phys. 22, 217–246.CrossRefGoogle Scholar
Brands, P. J., Hoeks, A. P. G., Rutten, M. C. M. & Reneman, R. S. 1996 A non-invasive method to estimate arterial impedance by means of the assessment of the local diameter change and the local centre-line blood flow velocity using ultrasound. Ultrasound Med. Biol 7, 895–905.CrossRefGoogle Scholar
Canuto, C., Hussaini, M. Y., Quarteroni, A. & Zang, T. A. 1988 Spectral Methods in Fluid Dynamics. Springer.CrossRefGoogle Scholar
Cox, R. H. 1968 Wave propagation through a newtonian fluid contained within a thick-walled viscoelastic tube. Biophys. J. 8, 691–709.CrossRefGoogle Scholar
Cox, R. H. 1970 Wave propagation through a newtonian fluid contained within a thick-walled viscoelastic tube: the influence of wall compressibility. J. Biomech. 3, 317–335.CrossRefGoogle ScholarPubMed
Formaggia, L., Gerbeau, J. F., Nobile, F. & Quarteroni, A. 2001 On the coupling of 3d and 1d navier-stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Engng 191, 561–582.CrossRefGoogle Scholar
Fung, Y. C. 1993 Biomechanics: Mechanical Properties of Living Tissues. Springer.CrossRefGoogle Scholar
Gijsen, F. J. H., van, de Vosse, F. N. & Janssen, J. D. 1999 a Influence of the non-newtonian properties of blood on the flow in large arteries: Steady flow in a carotid bifurcation model. J. Biomech. 32, 601–608.CrossRefGoogle Scholar
Gijsen, F. J. H., van, de Vosse, F. N. & Janssen, J. D. 1999 b Influence of the non-newtonian properties of blood on the flow in large arteries: Unsteady flow in a 90-degree curved tube. J. Biomech. 32, 705–713.CrossRefGoogle Scholar
Hughes, T. J. R. & Lubliner, J. 1973 On the one-dimensional theory of blood flow in the large vessels. Math. Bioscie. 18, 161–170.CrossRefGoogle Scholar
Lagrée, P.-Y. 2000 An inverse technique to deduce the elasticity of a large artery. Euro. Phys. J. AP 9, 153–163.CrossRefGoogle Scholar
Olufsen, M. S. 1999 Structured tree outflow condition for blood flow in larger systemic arteries. Am. J. Physiol. 276, H257–H268.Google ScholarPubMed
Olufsen, M. S. & Peskin, C. S. 2000 Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Engng 28, 1281–1299.CrossRefGoogle ScholarPubMed
de, Pater, L. & van, den Berg, J. W. 1964 An electrical analogue of the entire human circulatory system. Med. Electron. Biol. Engng 2, 161–166.Google Scholar
Pedley, T. J. 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.CrossRefGoogle Scholar
Porenta, G., Young, D. F. & Rogge, T. R. 1986 A finite element model of blood flow in arteries including taper, branches and obstructions. J. Bio. Engng 108, 161–167.CrossRefGoogle ScholarPubMed
Schlichting, H. 1960 Boundary Layer Theory. McGraw-Hill.Google Scholar
Sherwin, S. J., Formaggia, L., Peiró, J. & Franke, V. 2003 Computational modelling of 1d blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system. Intl J. Num. Meth. Fluids 43, 673–700.CrossRefGoogle Scholar
Stergiopulos, N., Westerhof, B. E. & Westerhof, N. 1999 Total arterial inertance as the fourth element of the windkessel model. Am. J. Physiol. 276, H81–H88.Google ScholarPubMed
Stergiopulos, N., Young, D. F. & Rogge, T. R. 1992 Computer simulation of arterial flow with applications to arterial and aortic stenoses. J. Biomech. 25, 1477–1488.CrossRefGoogle ScholarPubMed
Taylor, C. A., Hughes, T. J. R. & Zarins, C. K. 1998 Finite element modeling of blood flow in arteries. Comput. Meth. Appl. Mech. Engng 158, 155–196.CrossRefGoogle Scholar
Tsou, F. K., Chou, P. C., Frankel, S. N. & Hahn, A. W. 1971 An integral method for the analysis of blood flow. Bull. Math. Biophys. 33, 117–128.CrossRefGoogle ScholarPubMed
van, de Vosse, F. N., de, Hart, J., van, Oijen, C. H. G. A., Bessems, D., Gunther, T. W. M., Segal, A., Wolters, B. J. B. M., Stijnen, J. M. A. & Baaijens, F. P. T. 2003 Finite-element-based computational methods for cardiovascular fluid-structure interaction. J. Engng Maths 47, 335–368.Google Scholar
Wan, J., Steele, B., Spicer, S. A., Strohband, S., Feijoo, C. R., Hughes, T. J. R. & Taylor, C. A. 2002 A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease. Computer Meth. Biomech. Biomed. Engng, pp. 195–206.Google Scholar
Wang, J. J. & Parker, H. H. 2004 Wave propagation in a model of the arterial circulation. J. Biomech. 37, 457–470.CrossRefGoogle Scholar
Westerhof, N., Bosman, F., de, Vries, Co. J. & Noordergraaf, A. 1969 Analog studies of the human systemic arterial tree. J. Biomech. 2, 121–143.CrossRefGoogle ScholarPubMed
Womersley, J. R. 1957 An elastic tube theory of pulse transmission and oscillatory flow in mammalian arteries. Tech. Rep. WADC-TR-56-614.Google Scholar
Young, D. F. & Tsai, F. Y. 1973 Flow characteristics in models of arterial stenoses-ii. unsteady flow. J. Biomech. 6, 547–559.CrossRefGoogle ScholarPubMed
Zagzoule, M., Khalid-Naciri, J. & Mauss, J. 1991 Unsteady wall shear stress in a distensible tube. J. Biomech. 24, 435–439.CrossRefGoogle Scholar