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


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.

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M. Anliker , L. Rockwell & E. Ogden 1971 Nonlinear analysis of flow pulses and shock waves in arteries. Z. Angew. Math. Phys. 22, 217246.

C. Canuto , M. Y. Hussaini , A. Quarteroni & T. A. Zang 1988 Spectral Methods in Fluid Dynamics. Springer.

R. H. Cox 1968 Wave propagation through a newtonian fluid contained within a thick-walled viscoelastic tube. Biophys. J. 8, 691709.

R. H. Cox 1970 Wave propagation through a newtonian fluid contained within a thick-walled viscoelastic tube: the influence of wall compressibility. J. Biomech. 3, 317335.

L. Formaggia , J. F. Gerbeau , F. Nobile & A. Quarteroni 2001 On the coupling of 3d and 1d navier-stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Engng 191, 561582.

F. J. H. Gijsen , de Vosse, F. N. van & J. D. Janssen 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, 601608.

F. J. H. Gijsen , de Vosse, F. N. van & J. D. Janssen 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, 705713.

T. J. R. Hughes & J. Lubliner 1973 On the one-dimensional theory of blood flow in the large vessels. Math. Bioscie. 18, 161170.

P.-Y. Lagrée 2000 An inverse technique to deduce the elasticity of a large artery. Euro. Phys. J. AP 9, 153163.

M. S. Olufsen & C. S. Peskin 2000 Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Engng 28, 12811299.

T. J. Pedley 1980 The Fluid Mechanics of Large Blood Vessels. Cambridge University Press.

G. Porenta , D. F. Young & T. R. Rogge 1986 A finite element model of blood flow in arteries including taper, branches and obstructions. J. Bio. Engng 108, 161167.

S. J. Sherwin , L. Formaggia , J. Peiró & V. Franke 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, 673700.

N. Stergiopulos , D. F. Young & T. R. Rogge 1992 Computer simulation of arterial flow with applications to arterial and aortic stenoses. J. Biomech. 25, 14771488.

C. A. Taylor , T. J. R. Hughes & C. K. Zarins 1998 Finite element modeling of blood flow in arteries. Comput. Meth. Appl. Mech. Engng 158, 155196.

F. K. Tsou , P. C. Chou , S. N. Frankel & A. W. Hahn 1971 An integral method for the analysis of blood flow. Bull. Math. Biophys. 33, 117128.

J. J. Wang & H. H. Parker 2004 Wave propagation in a model of the arterial circulation. J. Biomech. 37, 457470.

N. Westerhof , F. Bosman , Vries, Co. J. de & A. Noordergraaf 1969 Analog studies of the human systemic arterial tree. J. Biomech. 2, 121143.

D. F. Young & F. Y. Tsai 1973 Flow characteristics in models of arterial stenoses-ii. unsteady flow. J. Biomech. 6, 547559.

M. Zagzoule , J. Khalid-Naciri & J. Mauss 1991 Unsteady wall shear stress in a distensible tube. J. Biomech. 24, 435439.

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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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