zbMATH — the first resource for mathematics

Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure in arteries. (English) Zbl 1175.76098
Summary: Flow and pressure waves emanate from the heart and travel through the major arteries where they are damped, dispersed and reflected due to changes in vessel caliber, tissue properties and branch points. As a consequence, solutions to the governing equations of blood flow in the large arteries are highly dependent on the outflow boundary conditions imposed to represent the vascular bed downstream of the modeled domain. The most common outflow boundary conditions for three-dimensional simulations of blood flow are prescribed constant pressure or traction and prescribed velocity profiles. However, in many simulations, the flow distribution and pressure field in the modeled domain are unknown and cannot be prescribed at the outflow boundaries. An alternative approach is to couple the solution at the outflow boundaries of the modeled domain with lumped parameter or one-dimensional models of the downstream domain. We previously described a new approach to prescribe outflow boundary conditions for simulations of blood flow based on the Dirichlet-to-Neumann and variational multiscale methods. This approach, termed the coupled multidomain method, was successfully applied to solve the non-linear one-dimensional equations of blood flow with a variety of models of the downstream domain. This paper describes the extension of this method to three-dimensional finite element modeling of blood flow and pressure in the major arteries. Outflow boundary conditions are derived for any downstream domain where an explicit relationship of pressure as a function of flow rate or velocities can be obtained at the coupling interface. We developed this method in the context of a stabilized, semi-discrete finite element method. Flow rate and pressure distributions are shown for different boundary conditions to illustrate the dramatic influence of alternative boundary conditions on these quantities.

76M10 Finite element methods applied to problems in fluid mechanics
76Z05 Physiological flows
92C35 Physiological flow
92-08 Computational methods for problems pertaining to biology
Full Text: DOI
[1] Taylor, C.A.; Draney, M.T., Experimental and computational methods in cardiovascular fluid mechanics, Ann. rev. fluid mech., 36, 197-231, (2004) · Zbl 1125.76414
[2] Kamiya, A.; Togawa, T., Adaptive regulation of wall shear stress to flow change in the canine carotid artery, Am. J. physiol., 239, 1, H14-H21, (1980)
[3] Wolinsky, H.; Glagov, S., Comparison of abdominal and thoracic aortic medial structure in mammals. deviation of man from the usual pattern, Circulation res., 25, 6, 677-686, (1969)
[4] Glagov, S.; Zarins, C.K.; Giddens, D.P.; Ku, D.N., Hemodynamics and atherosclerosis. insights and perspectives gained from studies of human arteries, Arch. pathol. lab. med., 112, 10, 1018-1031, (1988)
[5] Zarins, C.K.; Taylor, C.A., Hemodynamic factors in atherosclerosis. vascular surgery: A comprehensive review, (2002), W.S. Moore and W.B.S. Company, pp. 105-118
[6] Taylor, C.A.; Draney, M.T.; Ku, J.P.; Parker, D.; Steele, B.N.; Wang, K.; Zarins, C.K., Predictive medicine: computational techniques in therapeutic decision-making, Comput. aided surgery, 4, 5, 231-247, (1999)
[7] National Heart, Lung and Blood Institute report of the task force on research in pediatric cardiovascular disease, US Department of Health and Human Services, Public Health Service, National Institutes of Health, 2002.
[8] Stergiopulos, N.; Young, D.F.; Rogge, T.R., Computer simulation of arterial flow with applications to arterial and aortic stenoses, J. biomech., 25, 12, 1477-1488, (1992)
[9] Formaggia, L.; Nobile, F.; Quarteroni, A.; Veneziani, A., Multiscale modelling of the circulatory system: A preliminary analysis, Comput. visualization sci., 2, 2/3, 75-83, (1999) · Zbl 1067.76624
[10] Wan, J.; Steele, B.N.; Spicer, S.A.; Strohband, S.; Feijoo, G.R.; Hughes, T.J.R.; Taylor, C.A., A one-dimensional finite element method for simulation-based medical planning for cardiovascular disease, Comput. methods biomech. biomed. engrg., 5, 3, 195-206, (2002)
[11] Ruan, W.H.; Clark, M.E.; Zhao, M.D.; Curcio, A., A hyperbolic system of equations of blood flow in an arterial network, SIAM J. appl. math., 64, 2, 637-667, (2003) · Zbl 1126.76401
[12] Sherwin, S.J.; Formaggia, L.; Peiro, J.; Franke, V., Computational modelling of 1D blood flow with variable mechanical properties and its application to the simulation of wave propagation in the human arterial system, Int. J. numer. methods fluids, 43, 6-7, 673-700, (2003) · Zbl 1032.76729
[13] Womersley, J.R., Oscillatory motion of a viscous liquid in a thin-walled elastic tube—I: the linear approximation for long waves, The philos. mag., 7, 199-221, (1955) · Zbl 0064.43903
[14] Womersley, J.R., Oscillatory flow in arteries: the constrained elastic tube as a model of arterial flow and pulse transmission, Phys. med. biol., 2, 178-187, (1957)
[15] Taylor, M.G., Wave transmission through an assembly of randomly branching elastic tubes, Biophys. J., 6, 697-716, (1966)
[16] Olufsen, M.S., Structured tree outflow condition for blood flow in larger systemic arteries, Am. J. physiol., 276, H257-H268, (1999)
[17] B.N. Steele, C.A. Taylor, Simulation of blood flow in the abdominal aorta at rest and during exercise using a 1-D finite element method with impedance boundary conditions derived from a fractal tree, in: Proceedings of the 2003 ASME Summer Bioengineering Meeting, Key Biscayne, FL, 2003.
[18] Vignon, I.; Taylor, C.A., Outflow boundary conditions for one-dimensional finite element modeling of blood flow and pressure waves in arteries, Wave motion, 39, 4, 361-374, (2004) · Zbl 1163.74453
[19] Cebral, J.R.; Castro, M.; Soto, O.; Lohner, R.; Yim, P.J.; Alperin, N., Finite element modeling of the circle of willis from magnetic resonance data, Proceedings of SPIE—the international society for optical engineering, 5031, 11-21, (2003)
[20] Oshima, M.; Torii, R.; Kobayashi, T.; Taniguchi, N.; Takagi, K., Finite element simulation of blood flow in the cerebral artery, Comput. methods appl. mech. engrg., 191, 6-7, 661-671, (2001) · Zbl 0999.76081
[21] Perktold, K.; Rappitsch, G., Computer simulation of local blood flow and vessel mechanics in a compliant carotid artery bifurcation model, J. biomech., 28, 7, 845-856, (1995)
[22] Stroud, J.; Berger, S.; Saloner, D., Numerical analysis of flow through a severely stenotic carotid artery bifurcation, J. biomech. engrg., 124, 1, 9-20, (2002)
[23] Stuhne, G.R.; Steinman, D.A., Finite-element modeling of the hemodynamics of stented aneurysms, J. biomech. engrg., 126, 3, 382-387, (2004)
[24] Taylor, C.A.; Hughes, T.J.R.; Zarins, C.K., Finite element modeling of blood flow in arteries, Comput. methods appl. mech. engrg., 158, 155-196, (1998) · Zbl 0953.76058
[25] Pennati, G.; Migliavacca, F.; Dubini, G.; Pietrabissa, R.; de Leval, M., A mathematical model of circulation in the presence of the bidirectional cavopulmonary anastomosis in children with a univentricular heart, Med. eng. phys., 19, 3, 223-234, (1997)
[26] Migliavacca, F.; de Leval, M.R.; Dubini, G.; Pietrabissa, R.; Fumero, R., Computational fluid dynamic simulations of cavopulmonary connections with an extracardiac lateral conduit, Med. engrg. phys., 21, 3, 187-193, (1999)
[27] Migliavacca, F.; Kilner, P.J.; Pennati, G.; Dubini, G.; Pietrabissa, R.; Fumero, R.; de Leval, M.R., Computational fluid dynamic and magnetic resonance analyses of flow distribution between the lungs after total cavopulmonary connection, IEEE trans. biomed. engrg., 46, 4, 393-399, (1999)
[28] Shim, E.B.; Kamm, R.D.; Heldt, T.; Mark, R.G., Numerical analysis of blood flow through a stenosed artery using a coupled multiscale simulation method, Comput. cardiol., 219-222, (2000)
[29] Quarteroni, A.; Ragni, S.; Veneziani, A., Coupling between lumped and distributed models for blood flow problems, Comput. visualization sci., 4, 2, 111-124, (2001) · Zbl 1097.76615
[30] Guadagni, G.; Bove, E.L.; Migliavacca, F.; Dubini, G., Effects of pulmonary afterload on the hemodynamics after the hemi-fontan procedure, Med. engrg. phys., 23, 5, 293-298, (2001)
[31] Lagana, K.; Dubini, G.; Migliavacca, F.; Pietrabissa, R.; Pennati, G.; Veneziani, A.; Quarteroni, A., Multiscale modelling as a tool to prescribe realistic boundary conditions for the study of surgical procedures, Biorheology, 39, 3-4, 359-364, (2002)
[32] Quarteroni, A.; Veneziani, A., Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations, Multiscale model. simul., 1, 2, 173-195, (2003) · Zbl 1060.35003
[33] Lagana, K.; Balossino, R.; Migliavacca, F.; Pennati, G.; Bove, E.L.; de Leval, M.R.; Dubini, G., Multiscale modeling of the cardiovascular system: application to the study of pulmonary and coronary perfusions in the univentricular circulation, J. biomech., 38, 5, 1129-1141, (2005)
[34] Formaggia, L.; Gerbeau, J.F.; Nobile, F.; Quarteroni, A., On the coupling of 3D and 1D Navier-Stokes equations for flow problems in compliant vessels, Comput. methods appl. mech. engrg., 191, 561-582, (2001) · Zbl 1007.74035
[35] Formaggia, L.; Gerbeau, J.; Nobile, F.; Quarteroni, A., Numerical treatment of defective boundary conditions for the Navier-Stokes equations, SIAM J. numer. anal., 40, 1, 376-401, (2002) · Zbl 1020.35070
[36] Givoli, D.; Keller, J.B., A finite element method for large domains, Comput. methods appl. mech. engrg., 76, 1, 41-66, (1989) · Zbl 0687.73065
[37] Hughes, T.J.R., Multiscale phenomena: green’s functions, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized methods, Comput. methods appl. mech. engrg., 127, 387-401, (1995) · Zbl 0866.76044
[38] Nichols, W.W.; O’Rourke, M.F., Mcdonald’s blood flow in arteries: theoretical, experimental and clinical principles, (1998), Oxford University Press New York
[39] Gresho, P.M.; Sani, R.L., Incompressible flow and the finite element method, (2000), Wiley · Zbl 0988.76005
[40] Brooks, A.N.; Hughes, T.J.R., Streamline upwind Petrov-Galerkin formulation for convection dominated flows with particular emphasis on the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 32, 199-259, (1982) · Zbl 0497.76041
[41] Franca, L.P.; Frey, S.L., Stabilized finite element methods II. the incompressible Navier-Stokes equations, Comput. methods appl. mech. engrg., 99, 2-3, 209-233, (1992) · Zbl 0765.76048
[42] Whiting, C.H.; Jansen, K.E., A stabilized finite element method for the incompressible Navier-Stokes equations using a hierarchical basis, Int. J. numer. methods fluids, 35, 93-116, (2001) · Zbl 0990.76048
[43] Jansen, K.E.; Whiting, C.H.; Hulbert, G.M., Generalized-alpha method for integrating the filtered Navier-Stokes equations with a stabilized finite element method, Comput. methods appl. mech. engrg., 190, 3-4, 305-319, (2000) · Zbl 0973.76048
[44] J.R. Cebral, R. Lohner, O. Soto, P.J. Yim, On the modelling of carotid blood flow from magnetic resonance images, in: Proceedings of the 2001 Summer Bioengineering Meeting, 2001.
[45] Holdsworth, D.W.; Norley, C.J.D.; Frayne, R.; Steinman, D.A.; Rutt, B.K., Characterization of common carotid artery blood-flow waveforms in normal subjects, Physiol. meas., 20, 219-240, (1999)
[46] Zhao, S.Z.; Xu, X.Y.; Hughes, A.D.; Thom, S.A.; Stanton, A.V.; Ariff, B.; Long, Q., Blood flow and vessel mechanics in a physiologically realistic model of a human carotid arterial bifurcation, J. biomech., 33, 8, 975-984, (2000)
[47] B.T. Tang, C.P. Cheng, P.S. Tsao and C.A. Taylor, Subject-specific finite element modeling of three-dimensional pulsatile flow in the human abdominal aorta: comparison of resting and exercise conditions, in: Proceedings of the 2003 Summer Bioengineering Meeting, Key Biscayne, FL, 2003.
[48] Ganong, W.F.M., Review of medical physiology, (1995), Appleton & Lange Englewood Cliffs
[49] Guyton, A.C., Physiology of the human body, (1984), Saunders College Publishing San Francisco
[50] Taylor, C.A.; Hughes, T.J.R.; Zarins, C.K., Finite element modeling of three-dimensional pulsatile flow in the abdominal aorta: relevance to atherosclerosis, Ann. biomed. engrg., 26, 6, 1-14, (1998)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.