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A branched one-dimensional model of vessel networks

Published online by Cambridge University Press:  12 February 2009

JOSE-MARIA FULLANA  and
STÉPHANE ZALESKI
JOSE-MARIA FULLANA
Affiliation:
Service de Biophysique, Laboratoires Innothera, F-94110 Arcueil, France
STÉPHANE ZALESKI*
Affiliation:
UPMC Univ Paris 06, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
*
†Email address for correspondence: stephane.zaleski@upmc.fr
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Abstract

We introduce a model representing the venous network of the leg. The network consists of a coupled system of elastic tubes. The flow through each elastic tube is assumed to be unsteady, incompressible and one-dimensional. The network topology, as well as the lengths and diameters of the tubes, is based on literature data. As in the human leg the network is composed of two sub-networks, deep and superficial, which are connected by transverse segments. We introduce a new model for confluences or branching points, as well as models of the valvular system and of the muscular activity. We perform a numerical study of the transmission and reflection of waves at a confluence. Our model valvular system imposes a privileged direction of the flow towards the heart. Muscular activity is modelled using a modification of the tube law of the vessel and through an inflow of blood when muscle contraction pushes blood from the microcirculation to the veins. The model is capable of simulating several motions such as walking, dorsal flexion and tiptoe. Numerical tests show the physical relevance of the model, and in particular demonstrate that when the system is excited at the foot level, a two-frequency response appears. These frequencies are closely related to the characteristic lengths of the typical segments of the deep and of the superficial networks. We find good qualitative agreement between experimental and numerical flow rates, using clinical data corresponding to a single ‘tiptoe’ motion. We make numerical predictions of the internal venous pressure at the foot level in a valvular-incontinent system which agree with clinical observations.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 621 , 25 February 2009 , pp. 183 - 204
Copyright
Copyright © Cambridge University Press 2009

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References

REFERENCES

Alimi, Y. S., Barthelemy, P. & Juhan, C. 1994 Venous pump of the calf: a stduy of venous and muscular pressures. J. Vascu. Surg. 20 (5), 728–735.Google Scholar
Anliker, M., Rockwell, L. & Ogden, E. 1971 Nonlinear analysis of flow pulses and shock waves in arteries. ZAMP 22, 217–246.Google Scholar
Avolio, A. 1980 Multi-branched model of the human arterial system. Med. Biol. Engng Comp. 18, 709–718.Google Scholar
Bassez, S., Chauveau, M. & Flaud, P. 2001 Modeling of the deformation of flexible tubes using a single law. application to veins of the lower limb. J. Biomech. Engng 123, 58–65.Google Scholar
Bernhard, S., Mohlenkamp, S., Erbel, R. & Tilgner, A. 2005 Oscillatory flow in tube with time-dependent wall deformation and its application to myocardial bridges. In ESAIM: Proceeding (ed. Cancés, E. & Gerbeau, J-F.), vol. 14, pp. 25–40.CrossRefGoogle Scholar
Bertram, C. D. & Pedley, T. J. 1982 A mathematical model of unsteady collapsible tube behaviour. J. Biomech. 15 (1), 39–50.Google Scholar
Brook, Bindi S. 1997 The effect of gravity on the haemodynamics of the giraffe jugular vein. PhD thesis, University of Leeds.Google Scholar
Cancelli, C. & Pedley, T. J. 1985 A separated flow model for collapsible tube oscillations. J. Fluid Mech. 157, 375–404.CrossRefGoogle Scholar
Comolet, R. 1984 Biomecanique Circulatoire. Masson.Google Scholar
Cros, F. 2003 Confluences, remplissage et vidange: deux aspects singuliers du système veineux jambier. PhD thesis, Université de Paris 7.Google Scholar
Dai, G., Gertler, J. P. & Kamm, R. D. 1999 The effects of external compression on venous blood flow and tissue deformation in the lower leg. J. Biomech. Engng 121 (6), 557–564.Google Scholar
Elad, D., Kamm, R. D. & Shapiro, A. H. 1987 Choking phenomena in a lung like model. ASME J. Biomech. Engng 109, 1–9.Google Scholar
Elad, D., Katz, D., Kimmel, E. & Einav, S. 1991 Numerical schemes for unsteady fluid flow through collapsible tubes. J. Biomed. Engng 13, 10–18.Google Scholar
Falson, O. B., Haond, C., Chaslon, M., Moenner, M., Naili, S., Perrault, R., Finet, M. & Ribreau, C. 1998 Endothelial cells in culture: an in vitro model of the wall shear stress gradient effect. J. Biomech. 31 (1001), 173–173.Google Scholar
Fernandez, M. A., Milisic, V. & Quarteroni, A. 2005 Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs. Multiscale Model Simul. 4, 215–236.Google Scholar
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 Applied Mech. Engng 191, 561–582.CrossRefGoogle Scholar
Formaggia, L., Nobile, F., Quarteroni, A. & Veneziani, A. 1999 Multiscale modeling of the circulatory system: A preliminary analysis. Comput. Visual. Science 2, 75–84.Google Scholar
Fullana, J. M., Cros, F., Flaud, P. & Cros, F. 2003 Filling a collapsible tube. J. Fluid Mech. 494, 285–296.Google Scholar
Fung, Y. C. 1996 Biomechanics:Circulation. Springer.Google Scholar
Griffiths, D. J. 1971 Hydrodynamics of male micturation. I. theory of steady flow through elastic walled tubes. Med. Biol. Engng 9 (6), 581–588.CrossRefGoogle Scholar
Guyton, A. C. & Jones, C. E. 1973 Central venous pressure: physiological significance and clinical implications. Am. Heart J. 86 (4), 431–437.Google Scholar
Heil, M. & Pedley, T. J. 1996 Large post-buckling deformations of cylindrical shells conveying viscous flow. J. Fluids Struct. 10, 565–599.Google Scholar
Jan, D. L., Kamm, R. D. & Shapiro, A. H. 1983 Filling of partially collapsed compliant tube. J. Biomech. Engng 105, 12–18.Google Scholar
Kamm, R. D. 1982 Bioengineering studies of periodic external compression as prophylaxis against deep vein thrombosis. Part I: numerical studies. J. Biomech. Engng 104 (2), 87–95.Google Scholar
Kamm, R. D. & Pedley, T. J. 1989 Flow in collapsible tube: a brief review. J. Biomech. Engng 111, 177–179.Google Scholar
Kamm, R. D. & Shapiro, A. H. 1979 Unsteady flow in a collapsible tube subjected to external pressure of body forces. J. Fluid Mech. 95, 1–78.Google Scholar
Kimmel, E., Kamm, R. D. & Shapiro, A. H. 1988 Numerical solutions for steady and unsteady flow in a model of the pulmonary airways. J. Biomech. Engng 110 (4), 292–299.Google Scholar
Lighthill, J. M. 1989 Pulse Propagation Theory, chap. 12, pp. 227–252. SIAM.Google Scholar
MacCormack, R. W. 1969 The effect of viscosity in hypervelocity impact cratering. AIAA Paper No. 69-454.Google Scholar
Marzo, A., Luo, X. Y. & Bertram, C. D. 2005 Three-dimensional flow through a thick-walled collapsible tube. J. Fluids Struct. 20, 817–835.Google Scholar
Maton, B., Thiney, G., Dang, S., Tra, S., Bassez, S., Wicart, P. & Ouchene, A. 2006 a Human muscle fatigue and elastic compressive stockings. Eur. J. Appl. Physiol. 97 (4), 432–442.Google Scholar
Maton, B., Thiney, G., Ouchène, A., Flaud, P. & Barthelemy, P. 2006 b Intramuscular pressure and surface emg in voluntary ankle dorsal flexion: Influence of elastic compressive stockings. J. Electromyogr. Kinesiol. 16 (3), 291–302.Google Scholar
Matsuzaki, Y. & Matsumoto, T. 1989 Flow in a two-dimensional collapsible channel with rigid inlet and outlet. ASME J. Biomech. Engng 111, 180–184.CrossRefGoogle Scholar
McClurken, M. E., Kececioglu, I., Kamm, R. D. & Shapiro, A. H. 1981 Steady, supercritical flow in collapsible tubes. Part 2. theoretical studies. J. Fluid. Mech. 109, 391–415.Google Scholar
Nicolaides, A. N. & Zukowski, A. J. 1986 The value of dynamic venous pressure measurements. World J. Surg. 10 (6), 919–924.Google Scholar
Olufsen, M. S. 1999 Structured tree outflow condition for blood flow in larger systemic arteries. Am. J. Physiol. 276 (1 Pt 2), H257–H268.Google ScholarPubMed
Olufsen, M. S. 2000 A one-dimensional fluid dynamic model of the systemic arteries. Stud. Health Technol. Inform. 71, 79–97.Google Scholar
Olufsen, M. S., Peskin, C. S., Kim, W. Y., Pedersen, E. M., Nadim, A. & Larsen, J. 2000 a Simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Engng 28 (11), 1281–1299.Google Scholar
Olufsen, M. S., Peskin, C. S., Kim, W. Y., Pedersen, E. M., Nadim, A. & Larsen, J. 2000 b Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions. Ann. Biomed. Engng 28 (11), 1281–1299.Google Scholar
Ozawa, E. T., Bottom, K. E., Xiao, X. & Kamm, R. D. 2001 Numerical simulation of enhanced external counterpulsation. Ann. Biomed. Engng 29 (4), 284–97.Google Scholar
Peyret, R. & Taylor, T. D. 1983 Computational Methods for Fluid Flow. Springer.Google Scholar
Quarteroni, A. & Formaggia, L. 2003 Mathematical Modelling and Numerical Simulation of the Cardiovascular System. In Handbook of Numerical Analysis Series, Vol. XII (special volume): Computational Models for the Human Body, (ed. Ciarlet, P. G. & guest editor, Ayache, N.). Elsevier.Google Scholar
Ribreau, C., Naili, S. & Langlet, A. 1994 Head losses in smooth pipes obtained from collapsed tubes. J. Fluids Struct. 8, 183–200.Google Scholar
Rosar, M. E. & Peskin, C. S. 2001 Fluid flow in collapsible elastic tubes: a three-dimensional numerical model. N.Y. J. Math. 7, 281–302.Google Scholar
Shapiro, A. H. 1977 Steady flow in collapsible tubes. ASME J. Biomech. Engng 99, 126–147.Google Scholar
Sherwin, S. J., Franke, V., Peiro, J. & Parker, K. 2003 One-dimensional modelling of vascular network in space-time variables. J. Engng Math. 47, 217–250.Google Scholar
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.Google Scholar
Stettler, J. C., Niederer, P. & Anliker, M. 1981 Theoretical analysis of arterial hemodynamics including the influence of bifurcations. Part I: mathematical models and prediction of normal pulse patterns. Ann. Biomed. Engng 9 (2), 145–164.Google Scholar
Wan, Jing, Steele, , Brooke, A., Sean, Strohband, Sven, G. R. Feijóo, Hughes, Thomas J. R. & Taylor, Charles A. 2002 A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease. Comput. Methods Biomech. Biomed. Engng 5 (3), 195–206.Google Scholar
Wang, J. J. & Parker, K. H. 2004 Wave propagation in a model of the arterial circulation. J. Biomech. 37 (4), 457–470.Google Scholar
Westerhof, N., Bosman, F., Vries, C. & Noordergraaf, A. 1969 Analog studies of the human systemic arterial tree. J. Biomech. 2, 121–143.Google Scholar
Wild, R., Pedley, T. J. & Riley, D. S. 1977 Viscous flow in collapsible tubes of slowly-varying elliptical cross-section. J. Fluid Mech. 81, 273–294.CrossRefGoogle Scholar
Young, D. F. & Tsai, F. Y. 1975 Flow characteristics in models of arterial stenosis. I. steady flow. J. Biomech. 6, 395–410.Google Scholar