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Fluid-structure interaction of an aortic heart valve prosthesis driven by an animated anatomic left ventricle. (English) Zbl 1377.76045

Summary: We develop a novel large-scale kinematic model for animating the left ventricle (LV) wall and use this model to drive the fluid-structure interaction (FSI) between the ensuing blood flow and a mechanical heart valve prosthesis implanted in the aortic position of an anatomic LV/aorta configuration. The kinematic model is of lumped type and employs a cell-based, FitzHugh-Nagumo framework to simulate the motion of the LV wall in response to an excitation wavefront propagating along the heart wall. The emerging large-scale LV wall motion exhibits complex contractile mechanisms that include contraction (twist) and expansion (untwist). The kinematic model is shown to yield global LV motion parameters that are well within the physiologic range throughout the cardiac cycle. The FSI between the leaflets of the mechanical heart valve and the blood flow driven by the dynamic LV wall motion and mitral inflow is simulated using the curvilinear immersed boundary (CURVIB) method [L. Ge and F. Sotiropoulos, J. Comput. Phys. 225, No. 2, 1782–1809 (2007; Zbl 1213.76134); with I. Borazjani, ibid. 227, No. 16, 7587–7620 (2008; Zbl 1213.76129)] implemented in conjunction with a domain decomposition approach. The computed results show that the simulated flow patterns are in good qualitative agreement with in vivo observations. The simulations also reveal complex kinematics of the valve leaflets, thus, underscoring the need for patient-specific simulations of heart valve prosthesis and other cardiac devices.

MSC:

76Z05 Physiological flows
74F10 Fluid-solid interactions (including aero- and hydro-elasticity, porosity, etc.)
92C30 Physiology (general)
92C35 Physiological flow
Full Text: DOI

References:

[1] Ge, L.; Sotiropoulos, F., A numerical method for solving the 3d unsteady incompressible Navier-Stokes equations in curvilinear domains with complex immersed boundaries, Journal of Computational Physics, 225, 2, 1782 (2007) · Zbl 1213.76134
[2] Borazjani, I.; Ge, L.; Sotiropoulos, F., Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3d rigid bodies, Journal of Computational Physics, 227, 16, 7587-7620 (2008) · Zbl 1213.76129
[3] Hong, G.-R.; Pedrizzetti, G.; Tonti, G.; Li, P.; Wei, Z.; Kim, J. K.; Baweja, A.; Liu, S.; Chung, N.; Houle, H.; Narula, J.; Vannan, M. A., Characterization and quantification of vortex flow in the human left ventricle by contrast echocardiography using vector particle image velocimetry, Journal of the American College of Cardiology Imgaging, 1, 6, 705-717 (2008)
[4] Lee, J.; Niederer, S.; Nordsletten, D.; Le Grice, I.; Smail, B.; Kay, D.; Smith, N., Coupling contraction, excitation, ventricular and coronary blood flow across scale and physics in the heart, Philosophical Transactions of the Royal Society A, 367, 1896, 2311-2331 (2009) · Zbl 1185.92034
[5] Faludi, R.; Szulik, M.; D’hooge, J.; Herijgers, P.; Rademakers, F.; Pedrizzetti, G.; Voigt, J.-U., Left ventricular flow patterns in healthy subjects and patients with prosthetic mitral valves: an in vivo study using echocardiographic particle image velocimetry, Journal of Thoracic and Cardiovascular Surgery, 139, 6, 1501-1510 (2010)
[6] Markl, M.; Kilner, P.; Ebbers, T., Comprehensive 4d velocity mapping of the heart and great vessels by cardiovascular magnetic resonance, Journal of Cardiovascular Magnetic Resonance, 13, 1, 7 (2011)
[7] Kvitting, J.-P. E.; Dyverfeldt, P.; Sigfridsson, A.; Franzn, S.; Wigstrm, L.; Bolger, A. F.; Ebbers, T., In vitro assessment of flow patterns and turbulence intensity in prosthetic heart valves using generalized phase-contrast MRI, Journal of Magnetic Resonance Imaging, 31, 5, 1075-1080 (2010)
[8] Pierrakos, O.; Vlachos, P. P.; Telionis, D. P., Time-resolved DPIV analysis of vortex dynamics in a left ventricular model through bileaflet mechanical and porcine heart valve prostheses, Journal of Biomechanical Engineering, 126, 6, 714-726 (2004)
[9] Cenedese, A.; Del Prete, Z.; Miozzi, M.; Querzoli, G., A laboratory investigation of the flow in the left ventricle of a human heart with prosthetic, tilting-disk valves, Experiments in Fluids, 39, 322-335 (2005)
[10] Pierrakos, O.; Vlachos, P. P., The effect of vortex formation on left ventricular filling and mitral valve efficiency, Journal of Biomechanical Engineering, 128, 4, 527-539 (2006)
[11] Querzoli, G.; Fortini, S.; Cenedese, A., Effect of the prosthetic mitral valve on vortex dynamics and turbulence of the left ventricular flow, Physics of Fluids, 22, 4, 041901 (2010) · Zbl 1188.76125
[12] Hart, J. D.; Peters, G.; Schreurs, P.; Baaijens, F., A two-dimensional fluid-structure interaction model of the aortic value, Journal of Biomechanics, 33, 9, 1079-1088 (2000)
[13] Hart, J. D.; Baaijens, F. P.T.; Peters, G. W.M.; Schreurs, P. J.G., A computational fluid-structure interaction analysis of a fiber-reinforced stentless aortic valve, Journal of Biomechanics, 36, 5, 699-712 (2003), cardiovascular Biomechanics
[14] van Loon, R.; Anderson, P. D.; de Hart, J.; Baaijens, F. P.T., A combined fictitious domain/adaptive meshing method for fluid-structure interaction in heart valves, International Journal for Numerical Methods in Fluids, 46, 5, 533-544 (2004) · Zbl 1060.76582
[15] Dasi, L. P.; Ge, L.; Simon, H. A.; Sotiropoulos, F.; Yoganathan, A. P., Vorticity dynamics of a bileaflet mechanical heart valve in an axisymmetric aorta, Physics of Fluids, 19, 6, 067105 (2007) · Zbl 1182.76181
[16] De Tullio, M. D.; Cristallo, A.; Balaras, E.; Verzicco, R., Direct numerical simulation of the pulsatile flow through an aortic bileaflet mechanical heart valve, Journal of Fluid Mechanics, 622, 259-290 (2009) · Zbl 1165.76388
[18] Borazjani, I.; Ge, L.; Sotiropoulos, F., High-resolution fluid structure interaction simulations of flow through a bi-leaflet mechanical heart valve in an anatomic aorta, Annals of Biomedical Engineering, 38, 326-344 (2010)
[19] Yoganathan, A.; Chandran, K.; Sotiropoulos, F., Flow in prosthetic heart valves: state-of-the-art and future directions, Annals of Biomedical Engineering, 33, 1689-1694 (2005)
[20] Ge, L.; Sotiropoulos, F., Direction and magnitude of blood flow shear stresses on the leaflets of aortic valves: Is there a link with valve calcification?, Journal of Biomechanical Engineering, 132, 1, 014505 (2010)
[21] Hunter, P. J.; Pullan, A. J.; Smaill, B. H., Modeling total heart function, Annual Review of Biomedical Engineering, 5, 1, 147-177 (2003)
[22] Nordsletten, D.; Niederer, S.; Nash, M.; Hunter, P.; Smith, N., Coupling multi-physics models to cardiac mechanics, Progress in Biophysics and Molecular Biology, 104, 1-3, 77-88 (2011), cardiac Physiome project: Mathematical and Modelling Foundations
[23] Dominguez, G.; Fozzard, H. A., Effect of stretch on conduction velocity and cable properties of cardiac purkinje fibers, American Journal of Physiology - Cell Physiology, 237, 3, C119-C124 (1979)
[24] Sengupta, P. P.; Khandheria, B. K.; Korinek, J.; Wang, J.; Jahangir, A.; Seward, J. B.; Belohlavek, M., Apex-to-base dispersion in regional timing of left ventricular shortening and lengthening, Journal of the American College of Cardiology, 47, 1, 163-172 (2006)
[25] Nash, M.; Hunter, P., Computational mechanics of the heart, Journal of Elasticity, 61, 113-141 (2000) · Zbl 1071.74659
[26] Noble, D., Modeling the heart – from genes to cells to the whole organ, Science, 295, 5560, 1678-1682 (2002)
[27] Kovacs, S. J.; Mcqueen, D. M.; Peskin, C. S., Modelling cardiac fluid dynamics and diastolic function, Philosophical Transactions of the Royal Society of London Series A: Mathematical, Physical and Engineering Sciences, 359, 1783, 1299-1314 (2001) · Zbl 0994.92016
[28] Krittian, S.; Janoske, U.; Oertel, H.; Bhlke, T., Partitioned fluid-solid coupling for cardiovascular blood flow, Annals of Biomedical Engineering, 38, 1426-1441 (2010)
[29] Arts, T.; Reneman, R. S.; Veenstra, P. C., A model of the mechanics of the left ventricle, Annals of Biomedical Engineering, 7, 299-318 (1979)
[30] Beyar, R.; Sideman, S., The dynamic twisting of the left ventricle: a computer study, Annals of Biomedical Engineering, 14, 547-562 (1986)
[31] Yellin, E. L.; Meisner, J. S., Physiology of diastolic function and pressure-flow relations, Cardiology Clinics, 18, 3, 411-433 (2000)
[32] Thomas, J.; Weyman, A., Numerical modeling of ventricular filling, Annals of Biomedical Engineering, 20, 19-39 (1992)
[33] Kim, H.; Vignon-Clementel, I.; Figueroa, C.; LaDisa, J.; Jansen, K.; Feinstein, J.; Taylor, C., On coupling a lumped parameter heart model and a three-dimensional finite element aorta model, Annals of Biomedical Engineering, 37, 2153-2169 (2009)
[34] Vierendeels, J. A.; Riemslagh, K.; Dick, E.; Verdonck, P. R., Computer simulation of intraventricular flow and pressure gradients during diastole, Journal of Biomechanical Engineering, 122, 6, 667-674 (2000)
[35] Baccani, B.; Domenichini, F.; Pedrizzetti, G.; Tonti, G., Fluid dynamics of the left ventricular filling in dilated cardiomyopathy, Journal of Biomechanics, 35, 5, 665-671 (2002)
[36] Baccani, B.; Domenichini, F.; Pedrizzetti, G., Model and influence of mitral valve opening during the left ventricular filling, Journal of Biomechanics, 36, 3, 355-361 (2003)
[37] Domenichini, F.; Pedrizzetti, G.; Baccani, B., Three-dimensional filling flow into a model left ventricle, Journal of Fluid Mechanics, 539, 179-198 (2005) · Zbl 1075.76065
[38] Schenkel, T.; Malve, M.; Reik, M.; Markl, M.; Jung, B.; Oertel, H., MRI-based CFD analysis of flow in a human left ventricle: methodology and application to a healthy heart, Annals of Biomedical Engineering, 37, 503-515 (2009)
[39] Saber, N. R.; Wood, N. B.; Gosman, A. D.; Merrifield, R. D.; Yang, G.-Z.; Charrier, C. L.; Gatehouse, P. D.; Firmin, D. N., Progress towards patient-specific computational flow modeling of the left heart via combination of magnetic resonance imaging with computational fluid dynamics, Annals of Biomedical Engineering, 31, 42-52 (2003)
[40] Long, Q.; Merrifield, R.; Xu, X. Y.; Kilner, P.; Firmin, D. N.; Yang, G. Z., Subject-specific computational simulation of left ventricular flow based on magnetic resonance imaging, Proceedings of the Institution of Mechanical Engineers, Part H: Journal of Engineering in Medicine, 222, 475-485 (2008)
[41] Saber, N.; Gosman, A.; Wood, N.; Kilner, P.; Charrier, C.; Firmin, D., Computational flow modeling of the left ventricle based on in vivo MRI data: initial experience, Annals of Biomedical Engineering, 29, 275-283 (2001)
[42] Lemmon, J. D.; Yoganathan, A. P., Three-dimensional computational model of left heart diastolic function with fluid-structure interaction, Journal of Biomechanical Engineering, 122, 2, 109-117 (2000)
[43] McQueen, D. M.; Peskin, C. S., A three-dimensional computer model of the human heart for studying cardiac fluid dynamics, SIGGRAPH Computer Graphics, 34, 56-60 (2000)
[44] Cheng, Y.; Oertel, H.; Schenkel, T., Fluid-structure coupled CFD simulation of the left ventricular flow during filling phase, Annals of Biomedical Engineering, 33, 567-576 (2005)
[45] Watanabe, H.; Sugiura, S.; Kafuku, H.; Hisada, T., Multiphysics simulation of left ventricular filling dynamics using fluid-structure interaction finite element method, Biophysical Journal, 87, 3, 2074-2085 (2004)
[46] Nakamura, M.; Wada, S.; Yamaguchi, T., Computational analysis of blood flow in an integrated model of the left ventricle and the aorta, Journal of Biomechanical Engineering, 128, 6, 837-843 (2006)
[47] Vigmond, E. J.; Clements, C.; McQueen, D. M.; Peskin, C. S., Effect of bundle branch block on cardiac output: a whole heart simulation study, Progress in Biophysics and Molecular Biology, 97, 2-3, 520-542 (2008), life and Mechanosensitivity
[48] Tang, D.; Yang, C.; Geva, T.; del Nido, P. J., Image-based patient-specific ventricle models with fluid-structure interaction for cardiac function assessment and surgical design optimization, Progress in Pediatric Cardiology, 30, 1-2, 51-62 (2010), proceedings of the 1st International Conference on Computational Simulation in Congenital Heart Disease
[49] Peskin, C. S., Flow patterns around heart valves: a numerical method, Journal of Computational Physics, 10, 2, 252-271 (1972) · Zbl 0244.92002
[50] S Peskin, C., Numerical analysis of blood flow in the heart, Journal of Computational Physics, 25, 3, 220-252 (1977) · Zbl 0403.76100
[51] Peskin, C. S.; McQueen, D. M., A three-dimensional computational method for blood flow in the heart I. Immersed elastic fibers in a viscous incompressible fluid, Journal of Computational Physics, 81, 2, 372-405 (1989) · Zbl 0668.76159
[52] McQueen, D. M.; Peskin, C. S., A three-dimensional computational method for blood flow in the heart. II. Contractile fibers, Journal of Computational Physics, 82, 2, 289-297 (1989) · Zbl 0701.76130
[53] Sotiropoulos, F.; Borazjani, I., A review of state-of-the-art numerical methods for simulating flow through mechanical heart valves, Medical and Biological Engineering and Computing, 47, 245-256 (2009)
[54] Cheng, R.; Lai, Y. G.; Chandran, K. B., Three-dimensional fluid-structure interaction simulation of bileaflet mechanical heart valve flow dynamics, Annals of Biomedical Engineering, 32, 1471-1483 (2004)
[55] Ge, L.; Leo, H.-L.; Sotiropoulos, F.; Yoganathan, A. P., Flow in a mechanical bileaflet heart valve at laminar and near-peak systole flow rates: CFD simulations and experiments, Journal of Biomechanical Engineering, 127, 5, 782-797 (2005)
[56] Borazjani, I.; Ge, L.; Sotiropoulos, F., High-resolution fluidstructure interaction simulations of flow through a bi-leaflet mechanical heart valve in an anatomic aorta, Annals of Biomedical Engineering, 38, 326-344 (2010)
[57] Gilmanov, A.; Sotiropoulos, F., A hybrid Cartesian/immersed boundary method for simulating flows with 3d, geometrically complex, moving bodies, Journal of Computational Physics, 207, 2, 457 (2005) · Zbl 1213.76135
[58] Rodriguez, L.; Thomas, J.; Monterroso, V.; Weyman, A.; Harrigan, P.; Mueller, L.; Levine, R., Validation of the proximal flow convergence method, Calculation of orifice area in patients with mitral stenosis, Circulation, 88, 3, 1157-1165 (1993)
[59] Henry, W.; Griffith, J.; Michaelis, L.; McIntosh, C.; Morrow, A.; Epstein, S., Measurement of mitral orifice area in patients with mitral valve disease by real-time, two-dimensional echocardiography, Circulation, 51, 5, 827-831 (1975)
[60] Le, T.; Bukkapatnam, S.; Sangasoongsong, A.; Komanduri, R., Towards virtual instruments for cardiovascular healthcare: real-time modeling of cardiovascular dynamics using ecg signals, IEEE Conference on Automation Science and Engineering (CASE), 2010, 903-910 (2010)
[61] Nash, M. P.; Panfilov, A. V., Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias, Progress in Biophysics and Molecular Biology, 85, 2-3, 501-522 (2004), modelling Cellular and Tissue Function
[62] Chadwick, R., Mechanics of the left ventricle, Biophysical Journal, 39, 3, 279-288 (1982)
[63] Panfilov, A.; Hogeweg, P., Spiral breakup in a modified FitzHugh-Nagumo model, Physics Letters A, 176, 5, 295-299 (1993)
[64] Aliev, R. R.; Panfilov, A. V., A simple two-variable model of cardiac excitation, Chaos, Solitons and Fractals, 7, 3, 293-301 (1996)
[65] FitzHugh, R., Impulses and physiological states in theoretical models of nerve membrane, Biophysical Journal, 1, 6, 445-466 (1961)
[66] Rogers, J.; McCulloch, A., A collocation-Galerkin finite element model of cardiac action potential propagation, IEEE Transactions on Biomedical Engineering, 41, 8, 743-757 (1994)
[67] Kanai, H., Propagation of spontaneously actuated pulsive vibration in human heart wall and in vivo viscoelasticity estimation, IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control, 52, 11, 1931-1942 (2005)
[68] Pernot, M.; Fujikura, K.; Fung-Kee-Fung, S. D.; Konofagou, E. E., Ecg-gated, mechanical and electromechanical wave imaging of cardiovascular tissues in vivo, Ultrasound in Medicine and Biology, 33, 7, 1075-1085 (2007)
[69] Sengupta, P. P.; Tondato, F.; Khandheria, B. K.; Belohlavek, M.; Jahangir, A., Electromechanical activation sequence in normal heart, Heart Failure Clinics, 4, 3, 303-314 (2008)
[70] Beyar, R.; Sideman, S., A computer study of the left ventricular performance based on fiber structure, sarcomere dynamics, and transmural electrical propagation velocity, Circulation Research, 55, 3, 358-375 (1984)
[71] Miller, W.; Geselowitz, D., Simulation studies of the electrocardiogram. I. The normal heart, Circulation Research, 43, 2, 301-315 (1978)
[72] Jung, B.; Markl, M.; Foll, D.; Hennig, J., Investigating myocardial motion by MRI using tissue phase mapping, European Journal Cardio-Thoracic Surgery, 29, Suppl. 1, S150-157 (2006)
[73] Yoganathan, A. P.; He, Z.; Casey Jones, S., Fluid mechanics of heart valves, Annual Review of Biomedical Engineering, 6, 1, 331-362 (2004)
[74] Kuttler, U.; Wall, W., Fixed-point fluidstructure interaction solvers with dynamic relaxation, Computational Mechanics, 43, 61-72 (2008) · Zbl 1236.74284
[76] Gilmanov, A.; Sotiropoulos, F.; Balaras, E., A general reconstruction algorithm for simulating flows with complex 3d immersed boundaries on Cartesian grids, Journal of Computational Physics, 191, 2, 660-669 (2003) · Zbl 1134.76406
[77] Le, T. B.; Borazjani, I.; Kang, S.; Sotiropoulos, F., On the structure of vortex rings from inclined nozzles, Journal of Fluid Mechanics, 686, 1-33 (2011)
[78] Webster, D. R.; Longmire, E. K., Vortex rings from cylinders with inclined exits, Physics of Fluids, 10, 2, 400-416 (1998)
[79] Troolin, D. R.; Longmire, E. K., Volumetric velocity measurements of vortex rings from inclined exits, Experiments in Fluids, 48, 409-420 (2010)
[80] Kheradvar, A.; Gharib, M., On mitral valve dynamics and its connection to early diastolic flow, Annals of Biomedical Engineering, 37, 1-13 (2009)
[81] Fujimoto, S.; Mohiaddin, R. H.; Parker, K. H.; Gibson, D. G., Magnetic resonance velocity mapping of normal human transmitral velocity profiles, Heart and Vessels, 10, 236-240 (1995)
[82] Port, S.; Cobb, F. R.; Coleman, R. E.; Jones, R. H., Effect of age on the response of the left ventricular ejection fraction to exercise, New England Journal of Medicine, 303, 20, 1133-1137 (1980)
[83] Remme, E. W.; Lyseggen, E.; Helle-Valle, T.; Opdahl, A.; Pettersen, E.; Vartdal, T.; Ragnarsson, A.; Ljosland, M.; Ihlen, H.; Edvardsen, T.; Smiseth, O. A., Mechanisms of preejection and postejection velocity spikes in left ventricular myocardium: interaction between wall deformation and valve events, Circulation, 118, 4, 373-380 (2008)
[84] Buckberg, G. D.; Mahajan, A.; Jung, B.; Markl, M.; Hennig, J.; Ballester-Rodes, M., MRI myocardial motion and fiber tracking: a confirmation of knowledge from different imaging modalities, European Journal Cardio-Thoracic Surgery, 29, 1, 165-177 (2006)
[85] Gatehouse, P.; Keegan, J.; Crowe, L.; Masood, S.; Mohiaddin, R.; Kreitner, K.-F.; Firmin, D., Applications of phase-contrast flow and velocity imaging in cardiovascular MRI, European Radiology, 15, 2172-2184 (2005)
[86] Thomas, J.; Weyman, A. E., Echocardiographic Doppler evaluation of left ventricular diastolic function. physics and physiology, Circulation, 84, 977-990 (1991)
[87] Oh, J. K.; Appleton, C. P.; Hatle, L. K.; Nishimura, R. A.; Seward, J. B.; Tajik, A., The noninvasive assessment of left ventricular diastolic function with two-dimensional and Doppler echocardiography, Journal of the American Society of Echocardiography, 10, 3, 246-270 (1997)
[88] Kim, W. Y.; Walker, P. G.; Pedersen, E. M.; Poulsen, J. K.; Oyre, S.; Houlind, K.; Yoganathan, A. P., Left ventricular blood flow patterns in normal subjects: a quantitative analysis by three-dimensional magnetic resonance velocity mapping, Journal of the American College of Cardiology, 26, 1, 224-238 (1995)
[89] Sabbah, H.; Khaja, F.; Brymer, J.; McFarland, T.; Albert, D.; Snyder, J.; Goldstein, S.; Stein, P., Noninvasive evaluation of left ventricular performance based on peak aortic blood acceleration measured with a continuous-wave Doppler velocity meter, Circulation, 74, 2, 323-329 (1986)
[90] Pollak, S. J.; McMillan, S. A.; Knopff, W. D.; Wharff, R.; Yoganathan, A. P.; Felner, J. M., Cardiac evaluation of women distance runners by echocardiographic color Doppler flow mapping, Journal of the American College of Cardiology, 11, 1, 89-93 (1988)
[91] Sengupta, P. P.; Khandheria, B. K.; Korinek, J.; Wang, J.; Belohlavek, M., Biphasic tissue Doppler waveforms during isovolumic phases are associated with asynchronous deformation of subendocardial and subepicardial layers, Journal of Applied Physiology, 99, 3, 1104-1111 (2005)
[92] Dasi, L. P.; Simon, H. A.; Sucosky, P.; Yoganathan, A. P., Fluid mechanics of artificial heart valves, Clinical and Experimental Pharmacology and Physiology, 36, 2, 225-237 (2009)
[93] Long, Q.; Merrifield, R.; Yang, G. Z.; Xu, X. Y.; Kilner, P. J.; Firmin, D. N., The influence of inflow boundary conditions on intra left ventricle flow predictions, Journal of Biomechanical Engineering, 125, 6, 922-927 (2003)
[94] Takatsuji, H.; Mikami, T.; Urasawa, K.; Teranishi, J.-I.; Onozuka, H.; Takagi, C.; Makita, Y.; Matsuo, H.; Kusuoka, H.; Kitabatake, A., A new approach for evaluation of left ventricular diastolic function: spatial and temporal analysis of left ventricular filling flow propagation by color m-mode Doppler echocardiography, Journal of the American College of Cardiology, 27, 2, 365-371 (1996)
[96] Le, T. B.; Sotiropoulos, F., On the three-dimensional vortical structure of early diastolic flow in a patient-specific left ventricle, European Journal of Mechanics - B/Fluids, 35, 20-24 (2012) · Zbl 1446.76198
[98] Borazjani, I.; Sotiropoulos, F., The effect of implantation orientation of a bileaflet mechanical heart valve on kinematics and hemodynamics in an anatomic aorta, Journal of Biomechanical Engineering, 132, 11, 111005 (2010)
[99] Grinberg, L.; Karniadakis, G., Outflow boundary conditions for arterial networks with multiple outlets, Annals of Biomedical Engineering, 36, 1496-1514 (2008)
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