Analysis of lumped parameter models for blood flow simulations and their relation with 1D models
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 4, p. 613-632

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.

Classification:  35L50,  35M20,  47H10,  65L05,  76Z05
Keywords: multiscale modelling, parabolic equations, hyperbolic systems, lumped parameters models, blood flow modelling
     author = {Mili\v si\'c, Vuk and Quarteroni, Alfio},
     title = {Analysis of lumped parameter models for blood flow simulations and their relation with 1D models},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {38},
     number = {4},
     year = {2004},
     pages = {613-632},
     doi = {10.1051/m2an:2004036},
     zbl = {1079.76053},
     mrnumber = {2087726},
     language = {en},
     url = {}
Milišić, Vuk; Quarteroni, Alfio. Analysis of lumped parameter models for blood flow simulations and their relation with 1D models. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 38 (2004) no. 4, pp. 613-632. doi : 10.1051/m2an:2004036.

[1] A.P. Avolio, Multibranched model of the human arterial system. Med. Biol. Eng. Comput. 18 (1980) 709-119.

[2] B.S. Brook, S.A.E.G. Falle and T.J. Pedley, 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. | Zbl 0971.76052

[3] S. Čanić and E.H. Kim, 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. | Zbl 1141.76484

[4] S. Čanić and A. Mikelić, 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. | Zbl 1088.76077

[5] 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). | Zbl 1081.35073

[6] L. De Pater and J.W. Van Den Berg, An electrical analogue of the entire human circulatory system. Med. Electron. Biol. Engng. 2 (1964) 161-166.

[7] C.A. Desoer and E.S. Kuh, Basic Circuit Theory. McGraw-Hill (1969).

[8] L. Formaggia, J.F. Gerbeau, F. Nobile and A. Quarteroni, 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. | Zbl 1007.74035

[9] 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.

[10] E. Godlewski and P.-A. Raviart, Hyperbolic systems of conservation laws. Math. Appl., 3/4. Ellipses, Paris (1991). | MR 1304494 | Zbl 0768.35059

[11] W.P. Mason, Electromechanical Transducers and Wave Filters (1942).

[12] F. Migliavacca, G. Pennati, G. Dubini, R. Fumero, R. Pietrabissa, G. Urcelay, E.L. Bove, T.Y. Hsia and M.R. De Leval, Modeling of the norwood circulation: effects of shunt size, vascular resistances, and heart rate. Am. J. Physiol. Heart Circ. Physiol. 280 (2001) H2076-H2086.

[13] V. Milišić and A. Quarteroni, Coupling between linear parabolic and hyperbolic systems of equations for blood flow simulations, in preparation.

[14] A. Quarteroni, R. Sacco and F. Saleri, Numerical Mathematics, 37 Texts Appl. Math. Springer-Verlag, New York (2000). | MR 1751750 | Zbl 0957.65001

[15] 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.

[16] P. Segers, F. Dubois, D. De Wachter and P. Verdonck, Role and relevancy of a cardiovascular simulator. J. Cardiovasc. Eng. 3 (1998) 48-56.

[17] S.J. Sherwin, V. Franke, J. Peiro and K. Parker, One-dimensional modelling of a vascular network in space-time variables. J. Engng. Math. 47 (2003) 217-250. | Zbl pre02068972

[18] 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). | Zbl 1023.76061

[19] J.A. Spaan, J.D. Breuls and N.P. Laird, Diastolic-systolic coronary flow differences are caused by intramyocardial pump action in the anesthetized dog. Circ. Res. 49 (1981) 584-593.

[20] N. Stergiopulos, D.F. Young and T.R. Rogge, Computer simulation of arterial flow with applications to arterial and aortic stenoses. J. Biomech. 25 (1992) 1477-1488.

[21] 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). | MR 1005330 | Zbl 0681.65064

[22] N. Westerhof, F. Bosman, C.J. De Vries and A. Noordergraaf, Analog studies of the human systemic arterial tree. J. Biomechanics 2 (1969) 121-143.

[23] F. White, Viscous Fluid Flow. McGraw-Hill (1986). | Zbl 0356.76003