Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T00:10:31.243Z Has data issue: false hasContentIssue false
  • English
  • Français

Analysis of lumped parameter models for blood flow simulations and their relation with 1D models

Published online by Cambridge University Press:  15 August 2004

Vuk Milišić  and
Alfio Quarteroni
Vuk Milišić
Affiliation:
Chair of Modelling and Scientific Computing, IACS, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. vuk.milisic@epfl.ch.
Alfio Quarteroni
Affiliation:
Chair of Modelling and Scientific Computing, IACS, École Polytechnique Fédérale de Lausanne, 1015 Lausanne, Switzerland. vuk.milisic@epfl.ch. MOX, Dipartimento di Matematica, Politecnico di Milano, 20133 Milano, Italy. alfio.quarteroni@epfl.ch.
Get access

Abstract

This paper provides new results of consistence and convergence of the lumped parameters (ODE models) toward one-dimensional (hyperbolic or parabolic) models for blood flow. Indeed, lumped parameter models (exploiting the electric circuit analogy for the circulatory system) are shown to discretize continuous 1D models at first order in space. We derive the complete set of equations useful for the blood flow networks, new schemes for electric circuit analogy, the stability criteria that guarantee the convergence, and the energy estimates of the limit 1D equations.

Keywords

Multiscale modellingparabolic equationshyperbolic systemslumped parameters modelsblood flow modelling.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 38 , Issue 4 , July 2004 , pp. 613 - 632
Copyright
© EDP Sciences, SMAI, 2004

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

Avolio, A.P., Multibranched model of the human arterial system. Med. Biol. Eng. Comput. 18 (1980) 709–119. CrossRef
Brook, B.S., Falle, S.A.E.G. and Pedley, T.J., Numerical solutions for unsteady gravity-driven flows in collapsible tubes: evolution and roll-wave instability of a steady state. J. Fluid Mech. 396 (1999) 223–256. CrossRef
Čanić, S. and Kim, E.H., Mathematical analysis of quasilinear effects in a hyperbolic model of blood flow through compliant axi-symmetric vessels. Math. Meth. Appl. Sci. 26 (2003) 1161–1186. CrossRef
Čanić, S. and Mikelić, A., Effective equations modeling the flow of a viscous incompressible fluid through a long elastic tube arising in the study of blood flow through small arteries. SIAM J. Appl. Dyn. Sys. 2 (2003) 431–463. CrossRef
A. Čanić, D. Lamponi, S. Mikelić and J. Tambaca, Self-consistent effective equations modeling blood flow in medium-to-large compliant arteries. SIAM MMS (2004) (to appear).
de Pater, L. and van den Berg, J.W., An electrical analogue of the entire human circulatory system. Med. Electron. Biol. Engng. 2 (1964) 161–166. CrossRef
C.A. Desoer and E.S. Kuh, Basic Circuit Theory. McGraw-Hill (1969).
Formaggia, L., Gerbeau, J.F., Nobile, F. and Quarteroni, A., On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels. Comput. Methods Appl. Mech. Engrg. 191 (2001) 561–582. CrossRef
L. Formaggia and A. Veneziani, Reduced and multiscale models for the human cardiovascular system. Technical report, PoliMI, Milan (June 2003). Collection of two lecture notes given at the VKI Lecture Series 2003–07, Brussels 2003.
E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Math. Appl., 3/4. Ellipses, Paris (1991).
W.P. Mason, Electromechanical Transducers and Wave Filters (1942).
Migliavacca, F., Pennati, G., Dubini, G., Fumero, R., Pietrabissa, R., Urcelay, G., Bove, E.L., Hsia, T.Y. and de Leval, M.R., Modeling of the norwood circulation: effects of shunt size, vascular resistances, and heart rate. Am. J. Physiol. Heart Circ. Physiol. 280 (2001) H2076–H2086.
V. Milišić and A. Quarteroni, Coupling between linear parabolic and hyperbolic systems of equations for blood flow simulations, in preparation.
A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, 37 Texts Appl. Math. Springer-Verlag, New York (2000).
V.C. Rideout and D.E. Dick, Difference-differential equations for fluid flow in distensible tubes. IEEE Trans. Biomed. Eng. BME-14 (1967) 171–177.
Segers, P., Dubois, F., De Wachter, D. and Verdonck, P., Role and relevancy of a cardiovascular simulator. J. Cardiovasc. Eng. 3 (1998) 48–56.
Sherwin, S.J., Franke, V., Peiro, J. and Parker, K., One-dimensional modelling of a vascular network in space-time variables. J. Engng. Math. 47 (2003) 217–250. CrossRef
N.P. Smith, A.J. Pullan and P.J. Hunter, An anatomically based model of transient coronary blood flow in the heart. SIAM J. Appl. Math. 62 (2001/02) 990–1018 (electronic).
Spaan, J.A., Breuls, J.D. and Laird, N.P., Diastolic-systolic coronary flow differences are caused by intramyocardial pump action in the anesthetized dog. Circ. Res. 49 (1981) 584–593. CrossRef
Stergiopulos, N., Young, D.F. and Rogge, T.R., Computer simulation of arterial flow with applications to arterial and aortic stenoses. J. Biomech. 25 (1992) 1477–1488. CrossRef
J.C. Strikwerda, Finite difference schemes and partial differential equations. The Wadsworth & Brooks/Cole Mathematics Series. Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA (1989).
Westerhof, N., Bosman, F., De Vries, C.J. and Noordergraaf, A., Analog studies of the human systemic arterial tree. J. Biomechanics 2 (1969) 121–143. CrossRef
F. White, Viscous Fluid Flow. McGraw-Hill (1986).