Cardiovascular mathematics

International Congress of Mathematicians, ICM 2006

Volumn 1, Issue , 2006, Pages 479-512

  Quarteroni, Alfio ^a,b  

^a EPFL   (Switzerland)

^b POLITECNICO DI MILANO   (Italy)

Author keywords

Cardiovascular mathematics; Differential equations; Finite element method; Fluid dynamics; Mathematical modeling; Navier stokes equations; Numerical approximation

Indexed keywords

BLOOD FLOW; CIRCULATORY SYSTEMS; DIFFERENTIAL MODELS; MATHEMATICAL PROPERTIES; NUMERICAL APPROXIMATIONS; STOKES EQUATIONS;

DIFFERENTIAL EQUATIONS; FINITE ELEMENT METHOD; FLUID DYNAMICS; MATHEMATICAL MODELS; NAVIER STOKES EQUATIONS;

CARDIOVASCULAR SYSTEM;

EID: 84878078614     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: None     Document Type: Conference Paper

References (107)

1. Agoshkov, V., Quarteroni, A., Rozza, G., Shape design in aorto-coronaric bypass anastomoses using perturbation theory. SIAM J. Numer. Anal. 44 (2006), 367-384.
2. Ambrosio, L., Colli Franzone, P., Savaré, G., On the asymptotic behaviour of anisotropic energies arising in the cardiac bidomain model. Interfaces Free Bound. 2 (2000), 213-266.
3. Aris, R., Vectors, Tensors, and the Basic Equations of Fluid Mechanics. Prentice Hall, Englewood Cliffs, NJ, 1992.
4. Arada, N., Sequeira, A., Strong steady solutions for a generalized Oldroyd-B model with shear-dependent viscosity in a bounded domain. Math. Mod. Meth. Appl. Sci. 13 (9) (2003), 1303-1323.
5. Beirão da Veiga, H., On the existence of strong solutions to a coupled fluid-structure evolution problem. J. Math. Fluid Mechanics 6 (2004), 21-52.
6. Bellettini, G., Colli Franzone, P., Paolini, M., Convergence of front propagation for anisotropic bistable reaction-diffusion equations. Asymp. Anal. 15 (1997), 325-358.
7. Bestel, J., Clément, F., Sorine, M., A Biomechanical Model of Muscle Contraction. MICCAI (2001), 1159-1161.
8. Bratzler, R. L., Chisolm, G. M., Colton, C. K., Smith, K. A., Lees, R. S., The distribution of labeled low-density lipoproteins across the rabbit thoracic aorta in vivo. Atherosclerosis 28 (1977), 289-307.
9. Campeau, L., Enjalbert, M., Lesperance, J., Vaislic, C., Grondin, C. M., Bourassa, M. G., Atheroslerosis an dlate closure of aortocpronary saphenous vein grafts; sequential angiographic studies at 2weeks, 1 year, 5-7 year and 10-12 years after surgery. Circulation 68 (1983), 1-7.
10. Canic, S., Kim, E. H., Mathematical analysis of the quasilinear effects in a hyperbolic model of blood flow through compliant axi-symmetric vessels. Math. Models Methods Appl. Sci. 26 (14) (2003), 1161-1186.
11. Caro, C. G., Fitz-Gerald, J. M., Schroter, R. C., Atheroma and arterial wall shear stress. Observations, correlation and proposal of a shear dependent mass transfer mechanism for atherogenesis. Proc. Roy. Soc. B 177 (1971), 109-159.
12. Caro, C. G., Pedley, T. J., Schroter, R. C., Seed, W. A., The Mechanics of Circulation. Oxford University Press, 1978.
13. Causin, P., Gerbeau, J.-F., Nobile, F., Added-mass effect in the design of partitioned algorithms for fluid-structure problems. Comput. Methods Appl. Mech. Engrg. 194 (42-44) (2005), 4506-4527.
14. Cervera, M., Codina, R., Galindo, M., On the computational efficiency and implementation of block-iterative algorithms for nonlinear coupled problems. Engrg. Comput. 13 (6) (1996), 4-30.
15. Ciarlet, P.G., Introduction to Linear Shell Theory. Gauthiers-Villars, Paris 1998.
16. Colli Franzone, P., Pavarino, L., A parallel solver for reaction diffusion systems in computational electro-cardiology. Math. Models Methods Appl. Sci. 14 (6) (2004), 883-912.
17. Colli Franzone, P., Pavarino, L., Taccardi, B., Simulating patterns of excitation, repolarization and action potential duration with cardiac Bidomain and Monodomain models. Math. Biosci. 197 (2005), 35-66.
18. Curry, F. R. E., Mechanics and thermodynamics of transcapillary exchange. In Handbook of Physiology, ed. by E. M. Renkin, American Physiological Society, 1984.
19. Dáger, R., Zuazua, E., Controllability of tree-shaped networks of vibrating strings. C. R. Acad. Sci. Paris 332 (12) (2001), 1087-1092.
20. D'Angelo, C., Papelier, Y., Mathematical modelling of the cardiovascular system and skeletal muscle interaction during exercice. In CEMRACS 2004-mathematics and applications to biology and medicine (ed. by Eric Cancès and J-F. Gerbeau), ESAIM Proceedings 14, EDP Sciences, Les Ulis, 2005 72-88.
21. D'Angelo, C., Milisic, V., Reduced model for coupling of axisymmetric Navier-Stokes equations with a reaction-diffusion model for concentration. EPFL-IACS report 02.2006, submitted.
22. Deen, W. M., Hindered transport of large molecules in liquid-filled pores. AIChE Journal 33 (9) (1987), 1409-1425.
23. Deparis, S., Fernandez, M., Formaggia, L., Acceleration of a fixed point algorithm for fluid-structure interaction using transpiration conditions. Math. Model. Numer. Anal. 37 (4) (2003), 601-616.
24. Deparis, S., Discacciati, M., Fourestey, G., Quarteroni, A., Fluid-structure algorithms based on Steklov-Poincaré operators. Comput. Methods Appl. Mech. Engrg. 195 (2006), 5797-5812.
25. Deparis, S., Numerical Analysis of Axisymmetric Flows and Methods for Fluid-Structure Interaction Arising in Blood Flow Simulation. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, 2004.
26. Deparis, S., Gerbeau, J.-F., Vasseur, X., A dynamic preconditioner for Newton-Krylov algorithm. Application to fluid structure interaction. INRIA, Rapport de Recherche, No. 5277, 2004.
27. Desjardins, B., Esteban, M. J., Existence of a weak solutions for a model of fluid-rigid structure interaction. Arch. Ration. Mech. Anal. 146 (1999), 59-71.
28. Deuflhard, P., Hochmuth, R., Multiscale Analysis of Thermoregulation in the Human Microvascular System. Math. Methods Appl. Sci. 27 (2004), 971-989.
29. Fernandez, M., Moubachir, M., A Newton method using exact jacobians for solving fluidstructure coupling. Computers & Structures 83 (2-3) (2005), 127-142.
30. Fernandez, M., Gerbeau, J.-F., Grandmont, C., A projection semin-implicit scheme for the coupling of an eleastic structure with an incompressible fluid. INRIA, Rapport de Recherche, No. 5700, 2005.
31. Fitzgibbon, G. M., Kafka, H. P., Keon, W. J., Coronary bypass fate: long-term angiographic study. J. Amer. Coll. Cardiol. 17 (5) (1991), 1557-1565.
32. 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 (2001), 561-582.
33. Formaggia, L., Nobile, F., Quarteroni, A., Veneziani, A., Multiscale Modelling of the Circulatory System: a Preliminary Analysis. Comput. Vis. Sci. 2 (1-2) (1999), 75-83.
34. Formaggia, L., Gerbeau, J.-F., Nobile, F., Quarteroni, A., Numerical treatment of defective boundary conditions for the Navier-Stokes equations. SIAM J. Numer. Anal. 40 (1) (2002), 376-401.
35. Formaggia, L., Lamponi, D., Quarteroni, A., One dimensional models for blood flow in arteries. J. Engrg. Math. 47 (2003), 251-276.
36. Formaggia, L., Quarteroni, A., Veneziani, A., The circulatory system: from case studies to mathematical modeling. In Complex Systems in Biomedicine (ed. by A. Quarteroni, L. Formaggia, A. Veneziani), Springer Italia, Milano 2006, 243-287.
37. Formaggia, L., Veneziani, A., Reduced and Multiscale Models for the Human Cardiovascular System. Von Karman Institute Lecture Notes, Seventh Lecture Series on Biological Fluid Dynamics, 2003.
38. Fung, Y. C., Biomechanics: Mechanical Properties of Living Tissues. Springer-Verlag, NewYork 1993.
39. Fung, Y. C., Biodynamics: Circulation. Springer-Verlag, NewYork 1997.
40. Galdi, G. P., Sequeira, A., Further Existence Results for Classical Solutions of the Equation of a Second Grade Fluid. Arch. Ration. Mech. Anal. 128 (1994), 297-312.
41. Galdi, G. P., Rajagopal, K. R., Slow motion of a body in a fluid of second grade. Internat. J. Engrg. Sci. 35 (1997), 33-54.
42. Gerbeau, J.-F., Vidrascu, M., Frey, P., Fluid-structure interaction in blood flows on geometries coming from medical imaging. Computers & Structures 83 (2005), 155-165.
43. Gerbeau, J.-F, Vidrascu, M., A quasi-Newton algorithm based on a reduced model for fluid-structure interaction problems in blood flows. Math. Model. Numer. Anal. 37 (4) (2003), 663-680.
44. Grandmont, C., Existence for a three-dimensional steady state fluid-structure interaction problem. J. Math. Fluid Mech. 4 (1) (2002), 669-694.
45. Grandmont, C., Maday, Y., Existence for an unsteady fluid-structure interaction problem. Math. Model. Numer. Anal. 34 (3) (2000), 609-636.
46. Heil, M., An efficient solver for the fully coupled solution of large displacement fluidstructure interaction problems. Comput. Methods Appl. Mech. Engrg. 193 (1-2) (2004), 1-23.
47. Hochmuth, R., Deuflhard, P., Multiscale Analysis for the Bio-Heat-Transfer Equation- The Nonisolated Case. Math. Models Methods Appl. Sci. 14 (11) (2004), 1621-1634.
48. Holzapfel, G. A., Gasser, T. C., Ogden, R.W., A new constitutive framework for arterial wall mechanics and a comparative study of material models. J. Elasticity 61 (2000), 1-48.
49. Hoppensteadt, F., Peskin, C., Modeling and Simulation in Medicine and the Life Sciences. Second edition, Texts Appl. Math. 10, Springer-Verlag, NewYork 2002.
50. Katchalsky, A., Curran, P. F., Nonequilibrium Thermodynamics in Biophysics. Harvard University Press, 1981.
51. Kedem, O., Katchalsky, A., Thermodynamic analysis of the permeability of biological membranes to non-electrolytes. Biochimica et Biophysica Acta (1958), 229-246.
52. Keener, J., Sneyd, J., Mathematical Physiology. Springer-Verlag, NewYork 1998.
53. Lagnese, J., Leugering, G., Schmidt, E. J. P. G., Modeling, analysis and control of dynamic elastic multi-link structures. Systems Control Found. Appl., Birkhäuser, Boston, MA, 1994.
54. Lamponi, D., One dimensional and Multiscale Models for Blood Flow Circulation. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, 2004.
55. Lions, J. L., Zuazua, E., Approximate controllability of a hydroelastic coupled system. ESAIM Control Optim. Calc. Var. 1 (1995), 1-15.
56. Lowe, D. O., Clinical Blood Rheology. Vol. I, II, CRC Press, Boca Raton, Florida, 1988.
57. Luo, X. Y., Kuang, Z. B., A study of the constitutive equation of blood. J. Biomech. 25 (1992), 929-934.
58. Mallabiabarrena, I., Experimental set-up and numerical simulations of intravenous gas transfer devices. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, 2003.
59. Matthies, H., Steindorf, J., Partitioned but strongly coupled iteration schemes for nonlinear fluid-structure interaction. Computers & Structures 80 (2002), 1991-1999.
60. Matthies, H., Steindorf, J., Partitioned strong coupling algorithms for fluid-structure interaction. Computers & Structures 81 (2003), 805-812.
61. Matthies, H., Steindorf, J., Numerical efficiency of different partitioned methods for fluidstructure interaction. Z. Angew. Math. Mech. 2 (80) (2000), 557-558.
62. Michler, C., Van Brummelen, E. H., De Borst, R., An interface Newton-Krylov solver for fluid-structure interaction. Int. J. Numer. Methods Fluids 47 (10-11) (2005), 1189-1195.
63. Mok, D. P., Wall, W. A., Partitioned analysis schemes for the transient interaction of incompressible flows and nonlinear flexible structures. In Trends in computational structural mechanics (ed. by K. Schweizerhof, W. Wall, K. Bletzinger), International Center for Numerical Methods in Engineering (CIMNE), Barcelona 2001.
64. Mok, D. P., Wall, W. A., Ramm, E., Accelerated iterative substructuring schemes for instationary fluid-structure interaction. In Computational Fluid and Solid Mechanics (ed. by K. Bathe), Elsevier, Amsterdam 2001, 1325-1328.
65. Mouro, J., Interactions Fluide Structure en Grands Déplacements. Resolution Numerique et Application aux Composants Hydrauliques Automobiles. Ph.D. thesis, École Polytechnique, Paris 1996.
66. Murea, C. M., Vazquez, C., Sensitivity and approximation of coupled fluid-structure equations by virtual control method. Appl. Math. Optim. 52 (2) (2005), 357-371.
67. Murea, C. M., Optimal control approach for the fluid-structure interaction problems. In Elliptic and parabolic problems (ed. by J. Bemelmans et al.), World Scientific Publishing Co., River Edge, NJ, 2002, 442-450.
68. Nazarov, S., Sequeira, A., Videman, J. H., Steady flows of Jeffrey-Hamel type from the half-plane into an infinite channel. Linearization on an anti-symmetric solution. J. Math. Pures Appl. 80 (12) (2001), 1069-1098.
69. Nichols, W. W., O'Rourke, M. F., Mc Donald's Blood Flow in Arteries. Third edition, Edward Arnold Ltd., London 1990.
70. Nobile, F., Numerical Approximation of Fluid-Structure Interaction Problems with Application to Haemodynamics. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, 2001.
71. Novotny, A., Sequeira, A., Videman, J. H., Steady motions of viscoelastic fluids in 3-D exterior domains - existence, uniqueness and asymptotic behaviour. Arch. Ration. Mech. Anal. 149 (1999), 49-67.
72. Osses, A., Puel, J. P., Approximate controllability for a linear model of fluid structure interaction. ESAIM Control Optim. Calc. Var. 4 (1999), 497-513.
73. Peskin, C., The immersed boundary method. Acta Numerica 11 (2002), 479-517.
74. Prosi, M., Zunino, P., Perktold, K., Quarteroni, A., Mathematical and numerical models for transfer of low density lipoproteins through the arterial walls: a new methodology for the model set up with applications to the study of disturbed lumenal flow. J. Biomech. 38 (2005), 903-917.
75. Quarteroni, A., Formaggia, L., Mathematical Modelling and Numerical Simulation of the Cardiovascular System. In Modelling of Living Systems (ed. by P. G. Ciarlet and J. L. Lions), Handbook of Numerical Analysis Series 12, Elsevier, Amsterdam 2004, 3-127.
76. Quarteroni, A., Rozza, G., Optimal control and shape optimization in aorto-coronaric bypass anastomoses. Math. Models Methods Appl. Sci. 13 (12) (2003), 1801-1823.
77. Quarteroni, A., Valli, A., Numerical Approximation of Partial Differential Equations. Springer Ser. Comput. Math. 23, Springer-Verlag, Berlin 1994.
78. Quarteroni, A., Valli, A., Domain Decomposition Methods for Partial Differential Equations. Oxford University Press, NewYork 1999.
79. Quarteroni, A., Veneziani, A., Zunino, P., Mathematical and numerical modelling of solute dynamics in blood flowand arterialwalls. SIAM J.Numer.Anal. 39 (5) (2002), 1488-1511.
80. Quarteroni, A., Veneziani, A., Zunino, P., A Domain Decomposition Method for Advection-Diffusion Processes with Application to Blood Solutes. SIAM J. Sci. Comput. 23 (6) (2002), 1959-1980.
81. Quarteroni, A., Veneziani, A., Analysis of a geometrical multiscale model based on the coupling of ODEs and PDEs for blood flow simulations. SIAM Mult. Models Sim. 1 (2) (2003), 173-195.
82. Raback, P., Ruokolainen, J., Lyly, M., Järvinen. Fluid-structure interaction boundary conditions by artificial compressibility. In ECCOMAS Computational Fluid Dynamics Conference, Swansea, 2001.
83. Rajagopal, K. R., Bhatnagar, R. K., Flow of an Oldroyd-B fluid due to a stretching sheet in the presence of a free stream velocity. Internat. J. Non-Linear Mech. 30 (3) (1995), 391-405.
84. Robertson, A. M., Sequeira, A., A director theory approach for modeling blood flow in te arterial system: an alternative to classical 1d models. Math. Models Methods Appl. Sci. 15 (6) (2005), 871-906.
85. Rozza, G., Shape Design by Optimal FlowControl and Reduced Basis Techniques: Applications to Bypass Configurations in Haemodynamics. Ph.D. thesis, École Polytéchnique Fédérale de Lausanne, 2005.
86. Serrin, J., Mathematical principles of classical fluid mechanics. In Handbuch der Physik (ed. by S. Flügge and C. Truesdell), Vol. VIII/1, Springer-Verlag, Berlin 1959, 125-263.
87. Smith, N. P., Nickerson, D. P., Crampin, E. J., Hunter, P. J., Multiscale computational modelling of the heart. Acta Numer. 13 (2004), 371-431.
88. Tallec, P. L., Mouro, J., Fluid structure interaction with large structural displacements. Comput. Methods Appl. Mech. Engrg. 190 (2001), 3039-3067.
89. Jo, H., Dull, R. O., Hollis, T. M., Tarbell, J. M., Endothelial albumin permeability is shear dependent, time dependent and reversible. Amer. J. Physiol. 260 (1991), H1992-H1996.
90. Wang D. M., Tarbell, J. M., Modelling interstitial flow through arterial media. ASME J. Biomech. Eng. 117 (1995), 358-363.
91. Tarbell, J. M., Lever, M. J., Caro, C. G., The effect of varying albumin concentration and hydrostatic pressure on hydraulic conductivity of the Rabbit common carotid artery. Microvasc. Res. 35 (1988), 204-220.
92. 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. Computer Aided Surgery 4 (5) (1999), 231-247.
93. Tezduyar, T., Finite element methods for fluid dynamics with moving boundaries and interfaces. Arch. Comput. Methods Engrg. 8 (2001), 83-130.
94. Thurston, G. B., Viscoelstic properties of blood and blood analogs. Avances in Hemodynamics and hemorheology 1 (1996), 1-30.
95. Truskey, G. A., Roberts, W. L., Herrmann, R. A., Malinauskas, R. A., Measurement of endothelial permeability to I-low density lipoproteins in rabbit arteries by use of en face preparations. Circ. Res. 7 (4) (1992), 883-897.
96. Varty, K., Allen, K. E., Bell, P. R. F., London, N. J. M., Infra-inguinal vein graft stenosis. Br. J. Surg. 80 (1993), 825-833.
97. Veneziani, A., Vergara, C., Flow rate defective boundary conditions in haemodynamics simulations. Int. J. Numer. Methods Fluids 47 (2005), 803-816.
98. Veneziani, A., Vergara, C., An approximate method for solving incompressible Navier- Stokes problems with flow rate conditions. MOX report 70, Department of Mathematics, Politecnico di Milano, 2005.
99. Wada, S., Karino, T., Computational study on LDL transfer from flowing blood to arterial walls. In Clinical Application of Computational Mechanics to the Cardiovascular System (ed. by T.Yamaguchi), Springer-Verlag, 2000.
100. Wang, J. J., Parker, K. H., Wave propagation in a model of the arterial circulation. J. Biomech. 37 (2004), 457-470.
101. Womersley J. R., Method for the calculation of velocity, rate of flow and viscous drag in arteries when the pressure gradient is known. J. Physiol. 127 (1955), 553-563.
102. Yeleswarapu, K. K, Evaluation of Continuum Models for Characterizing the Constitutive Behavior of Blood. Ph.D. thesis, Department of Mechanical Engineering, University of Pittsburgh, 1996.
103. Zhang, X., Zuazua, E., Polynomial decay and control of a 1-d hyperbolic-parabolic coupled system. J. Differential Equations 204 (2) (2004), 380-438.
104. Zhang, X., Zuazua, E., Long time behavior of a coupled heat-wave system arising in fluid-structure interaction. Arch. Ration. Mech. Anal. 184 (2007), 49-120.
105. Zunino, P., Mathematical and numerical modeling of mass transfer in the vascular system. Ph.D. thesis, École Polytechnique Fédérale de Lausanne, 2002.
106. Zunino, P., Multidimensional pharmacokinetic models applied to the design of drug eluting stents. Cardiov Eng. 4 (2) (2004), 181-191.
107. Zunino, P., Mastalli, D., Quarteroni, A., VanBiesen, W., Vecten, D., Pacitti, A., Lameire, N., Neftel, F., Wauters, J. P., Development of a new mathematical approach to optimize peritoneal dialysis. EPFL-IACS report 03.2006.