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The cardiovascular system: Mathematical modelling, numerical algorithms and clinical applications *

  • A. Quarteroni (a1) (a2), A. Manzoni (a1) and C. Vergara (a2)
Abstract

Mathematical and numerical modelling of the cardiovascular system is a research topic that has attracted remarkable interest from the mathematical community because of its intrinsic mathematical difficulty and the increasing impact of cardiovascular diseases worldwide. In this review article we will address the two principal components of the cardiovascular system: arterial circulation and heart function. We will systematically describe all aspects of the problem, ranging from data imaging acquisition, stating the basic physical principles, analysing the associated mathematical models that comprise PDE and ODE systems, proposing sound and efficient numerical methods for their approximation, and simulating both benchmark problems and clinically inspired problems. Mathematical modelling itself imposes tremendous challenges, due to the amazing complexity of the cardiocirculatory system, the multiscale nature of the physiological processes involved, and the need to devise computational methods that are stable, reliable and efficient. Critical issues involve filtering the data, identifying the parameters of mathematical models, devising optimal treatments and accounting for uncertainties. For this reason, we will devote the last part of the paper to control and inverse problems, including parameter estimation, uncertainty quantification and the development of reduced-order models that are of paramount importance when solving problems with high complexity, which would otherwise be out of reach.

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Abboud S., Berenfeld O. and Sadeh D. (1991), ‘Simulation of high-resolution QRS complex using a ventricular model with a fractal conduction system: Effects of ischemia on high-frequency QRS potentials’, Circ. Res. 68, 17511760.
Ahmed S. and Giddens D. (1984), ‘Pulsatile poststenotic flow studies with laser Doppler anemometry’, J. Biomech. 17, 695705.
Akkerman I., Bazilevs Y., Calo V., Hughes T. and Hulshoff S. (2008), ‘The role of continuity in residual-based variational multiscale modeling of turbulence’, Comput. Mech. 41, 371378.
Alauzet F., Fabrèges B., Fernández M. and Landajuela M. (2016), ‘Nitsche-XFEM for the coupling of an incompressible fluid with immersed thin-walled structures’, Comput. Methods Appl. Mech. Engrg 301, 300335.
Aliev R. and Panfilov A. (1996), ‘A simple two-variable model of cardiac excitation’, Chaos Solitons Fract. 7, 293301.
Álvarez A., Alonso-Atienza F., Rojo-Álvarez J., Garcia-Alberola A. and Moscoso M. (2012), ‘Shape reconstruction of cardiac ischemia from non-contact intracardiac recordings: A model study’, Math. Comput. Model. 55, 17701781.
Ambrosi D. and Pezzuto S. (2012), ‘Active stress vs. active strain in mechanobiology: Constitutive issues’, J. Elasticity 2, 199212.
Ambrosi D., Arioli G., Nobile F. and Quarteroni A. (2011), ‘Electromechanical coupling in cardiac dynamics: The active strain approach’, SIAM J. Appl. Math. 2, 605621.
Andreianov B., Bendahmane M., Quarteroni A. and Ruiz-Baier R. (2015), ‘Solvability analysis and numerical approximation of linearized cardiac electromechanics’, Math. Models Methods Appl. Sci. 25, 959993.
Antiga L., Peiró J. and Steinman D. (2009), From image data to computational domains. (Formaggia L., Quarteroni A. and Veneziani A., eds), Chapter 4 in Cardiovascular Mathematics , Springer, pp. 123175.
Antiga L., Piccinelli M., Botti L., Ene-Iordache B., Remuzzi A. and Steinman D. (2008), ‘An image-based modeling framework for patient-specific computational hemodynamics’, Med. Biol. Engrg Comput. 46, 10971112.
Antoulas A. (2005), Approximation of Large-Scale Dynamical Systems, SIAM.
Asch M., Bocquet M. and Nodet M. (2017), Data Assimilation: Methods, Algorithms, and Applications, SIAM.
Astorino M. and Grandmont C. (2009), ‘Convergence analysis of a projection semi-implicit coupling scheme for fluid–structure interaction problems’, Numer. Math. 45–46, 36033612.
Astorino M., Chouly F. and Fernández M. (2009), ‘Robin based semi-implicit coupling in fluid–structure interaction: Stability analysis and numerics’, SIAM J. Sci. Comput. 31, 40414065.
Astorino M., Gerbeau J.-F., Pantz O. and Traoré K. (2010), ‘Fluid–structure interaction and multi-body contact: Application to the aortic valves’, Comput. Methods Appl. Mech. Engrg 116, 721767.
Astorino M., Hamers J., Shadden C. and Gerbeau J.-F. (2012), ‘A robust and efficient valve model based on resistive immersed surfaces’, Int. J. Numer. Methods Biomed. Engrg 28, 937959.
Augustin C., Holzapfel G. and Steinbach O. (2014), ‘Classical and all-floating FETI methods for the simulation of arterial tissues’, Int. J. Numer. Methods Engrg 99, 290312.
Auricchio F., Ferrara A. and Morganti S. (2012), ‘Comparison and critical analysis of invariant-based models with respect to their ability in fitting human aortic valve data’, Ann. Solid Struct. Mech. 4, 114.
Auricchio F., Lefieux F., Reali A. and Veneziani A. (2016), ‘A locally anisotropic fluid–structure interaction remeshing strategy for thin structures with application to a hinged rigid leaflet’, Int. J. Numer. Methods Engrg 107, 155180.
Avolio A. (1980), ‘Multi-branched model of the human arterial system’, Med. Biol. Engrg Comput. 18, 709718.
Babuška I., Nobile F. and Tempone R. (2007), ‘A stochastic collocation method for elliptic partial differential equations with random input data’, SIAM J. Numer. Anal. 45, 10051034.
Babuška I., Tempone R. and Zouraris G. (2004), ‘Galerkin finite element approximations of stochastic elliptic partial differential equations’, SIAM J. Numer. Anal. 42, 800825.
Badia S., Nobile F. and Vergara C. (2008a), ‘Fluid–structure partitioned procedures based on Robin transmission conditions’, J. Comput. Phys. 227, 70277051.
Badia S., Nobile F. and Vergara C. (2009), ‘Robin–Robin preconditioned Krylov methods for fluid–structure interaction problems’, Comput. Methods Appl. Mech. Engrg 198, 27682784.
Badia S., Quaini A. and Quarteroni A. (2008b), ‘Modular vs. non-modular preconditioners for fluid–structure systems with large added-mass effect’, Comput. Methods Appl. Mech. Engrg 197, 42164232.
Badia S., Quaini A. and Quarteroni A. (2008c), ‘Splitting methods based on algebraic factorization for fluid–structure interaction’, SIAM J. Sci. Comput. 30, 17781805.
Bagci E., Vodovotz Y., Billiar T., Ermentrout B. and Bahar I. (2008), ‘Computational insights on the competing effects of nitric oxide in regulating apoptosis’, PLOS One 3, e2249.
Ball J. (1976), ‘Convexity conditions and existence theorems in nonlinear elasticity’, Arch. Rat. Mech. Anal. 63, 337403.
Ball J. (1977), Constitutive inequalities and existence theorems in nonlinear elastostatics. In Nonlinear Analysis and Mechanics (Knops R., ed.), Vol. I, Pitman, pp. 187241.
Ballarin F., Faggiano E., Ippolito S., Manzoni A., Quarteroni A., Rozza G. and Scrofani R. (2016), ‘Fast simulations of patient-specific haemodynamics of coronary artery bypass grafts based on a POD–Galerkin method and a vascular shape parametrization’, J. Comput. Phys. 315, 609628.
Ballarin F., Faggiano E., Manzoni A., Quarteroni A., Rozza G., Ippolito S., Antona C. and Scrofani R. (2017), ‘Numerical modeling of hemodynamics scenarios of patient-specific coronary artery bypass grafts’, Biomech. Model. Mechanobiol. doi:10.1007/s10237-017-0893-7
Balzani D., Brands D., Klawonn A., Rheinbach O. and Schroder J. (2010), ‘On the mechanical modeling of anisotropic biological soft tissue and iterative parallel solution strategies’, Arch. Appl. Mech. 80, 479488.
Balzani D., Neff P., Schroder J. and Holzapfel G. (2006), ‘A polyconvex framework for soft biological tissues: Adjustment to experimental data’, Int. J. Solids Struct. 43, 60526070.
Banks H. and Kunisch K. (1989), Estimation Techniques for Distributed Parameter Systems, Systems & Control: Foundations & Applications, Birkhäuser.
Banks J., Henshaw W. and Schwendeman D. (2014), ‘An analysis of a new stable partitioned algorithm for FSI problems, I: Incompressible flow and elastic solids’, J. Comput. Phys. 269, 108137.
Barbarotta L. (2014), A mathematical and numerical study of the left ventricular contraction based on the reconstruction of a patient specific geometry. MSc thesis, Mathematical Engineering, Politecnico di Milano.
Barker A. and Cai X. (2010a), ‘Scalable parallel methods for monolithic coupling in fluid–structure interaction with application to blood flow modeling’, J. Comput. Phys. 229, 642659.
Barker A. and Cai X. (2010b), ‘Two-level Newton and hybrid Schwarz preconditioners for fluid–structure interaction’, SIAM J. Sci. Comput. 32, 23952417.
Barnard A., Hunt W., Timlake W. and Varley E. (1966), ‘A theory of fluid flow in compliant tubes’, Biophys. J. 6, 717724.
Bayer J., Blake R., Plank G. and Trayanova N. (2012), ‘A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models’, Ann. Biomed. Engrg 40, 22432254.
Bazilevs Y., Calo V., Zhang Y. and Hughes T. (2006), ‘Isogeometric fluid–structure interaction analysis with applications to arterial blood flow’, Comput. Mech. 38, 310322.
Bazilevs Y., Gohean J., Hughes T., Moser R. and Zhang Y. (2009), ‘Patient-specific isogeometric fluid–structure interaction analysis of thoracic aortic blood flow due to implantation of the Jarvik 2000 left ventricular assist device’, Comput. Methods Appl. Mech. Engrg 198, 35343550.
Bazilevs Y., Takizawa K. and Tezduyar T. (2012), Computational Fluid–Structure Interaction: Methods and Applications, Wiley.
Beeler G. and Reuter H. (1977), ‘Reconstruction of the action potential of ventricular myocardial fibres’, J. Physiol. 268, 177210.
Beirão da Veiga H. (2004), ‘On the existence of strong solutions to a coupled fluid–structure evolution problem’, J. Math. Fluid Mech. 6, 2152.
Benner P., Gugercin S. and Willcox K. (2015), ‘A survey of model reduction methods for parametric dynamical systems’, SIAM Review 57, 483531.
Bentley J. and Friedman J. (1979), ‘Data structures for range searching’, ACM Comput. Surv. 11, 397409.
Benzi M., Golub G. and Liesen J. (2005), Numerical solution of saddle point problems. In Acta Numerica, Vol. 14, Cambridge University Press, pp. 1137.
Bertagna L. and Veneziani A. (2014), ‘A model reduction approach for the variational estimation of vascular compliance by solving an inverse fluid–structure interaction problem’, Inverse Problems 30, 055006.
Bertagna L., D’Elia M., Perego M. and Veneziani A. (2014), Data assimilation in cardiovascular fluid–structure interaction problems: An introduction. In Fluid–Structure Interaction and Biomedical Applications (Bodnár T., Galdi P. and Nečasová Š., eds), Springer, pp. 395481.
Bertoglio C., Moireau P. and Gerbeau J.-F. (2012), ‘Sequential parameter estimation for fluid–structure problems: Application to hemodynamics’, Int. J. Numer. Methods Biomed. Engrg 28, 434455.
Bertrand F., Tanguy P. and Thibault F. (1997), ‘A three-dimensional fictitious domain method for incompressible fluid flow problems’, Int. J. Numer. Methods Fluids 25, 719736.
Bevan R., Nithiarasu P., van Loon R., Sazonov I., Luckraz H. and Garnham A. (2010), ‘Application of a locally conservative Galerkin (LCG) method for modelling blood flow through a patient-specific carotid bifurcation’, Int. J. Numer. Methods Fluids 64, 12741295.
Biehler J., Gee M. and Wall W. (2015), ‘Towards efficient uncertainty quantification in complex and large-scale biomechanical problems based on a Bayesian multi-fidelity scheme’, Biomech. Model. Mechanobiol. 14, 489513.
Blanco P. and Feijóo R. (2013), ‘A dimensionally-heterogeneous closed-loop model for the cardiovascular system and its applications’, Med. Engrg Phys. 35, 652667.
Blanco P., Deparis S. and Malossi A. (2013), ‘On the continuity of mean total normal stress in geometrical multiscale cardiovascular problems’, J. Comput. Phys. 51, 136155.
Blanco P., Feijóo R. and Urquiza S. (2007), ‘A unified variational approach for coupling 3D–1D models and its blood flow applications’, Comput. Methods Appl. Mech. Engrg 196, 43914410.
Blanco P., Pivello M., Urquiza S. and Feijóo R. (2009), ‘On the potentialities of 3D–1D coupled models in hemodynamics simulations’, J. Biomech. 42, 919930.
Blanco P., Watanabe S. and Feijóo R. (2012), ‘Identification of vascular territory resistances in one-dimensional hemodynamics simulations’, J. Biomech. 45, 20662073.
Blanco P., Watanabe S., Passos M., Lemos P. and Feijóo R. et al. (2015), ‘An anatomically detailed arterial network model for one-dimensional computational hemodynamics’, IEEE Trans. Biomed. Engrg 62, 736753.
Blum J., Le dimet F.-X. and Navon I. (2009), Data assimilation for geophysical fluids. In Handbook of Numerical Analysis, (Temam R. and Tribbia J., eds), Vol. 14: Computational Methods for the Atmosphere and the Oceans , Elsevier, pp. 385441.
Bodnár T., Galdi G. and Nečasová Š (2014), Fluid–Structure Interaction and Biomedical Applications, Springer.
Boese J., Bock M., Schoenberg S. and Schad L. (2000), ‘Estimation of aortic compliance using magnetic resonance pulse wave velocity measurement’, Phys. Med. Biol. 45, 17031713.
Boffi D. and Gastaldi L. (2003), ‘A finite element approach for the immersed boundary method’, Comput. Struct. 81, 491501.
Boffi D., Brezzi F. and Fortin M. (2013), Mixed Finite Element Methods and Applications, Springer.
Boffi D., Gastaldi L. and Heltai L. (2007), ‘Numerical stability of the finite element immersed boundary method’, Math. Models Methods Appl. Sci. 17, 14791505.
Boffi D., Gastaldi L., Heltai L. and Peskin C. (2008), ‘On the hyper-elastic formulation of the immersed boundary method’, Comput. Methods Appl. Mech. Engrg 197, 22102231.
Boileau E., Nithiarasu P., Blanco P., Muller L., Fossan F., Hellevik L., Donders W., Huberts W., Willemet M. and Alastruey J. (2015), ‘A benchmark study of numerical schemes for one-dimensional arterial blood flow modelling’, Int. J. Numer. Methods Biomed. Engrg 31, e02732.
Boldak C., Rolland Y. and Toumoulin C. (2003), ‘An improved model-based vessel tracking algorithm with application to computed tomography angiography’, Biocybern. Biomed. 23, 4163.
Bonomi D., Vergara C., Faggiano E., Stevanella M., Conti C., Redaelli A., Puppini G., Faggian G., Formaggia L. and Luciani G. (2015), ‘Influence of the aortic valve leaflets on the fluid-dynamics in aorta in presence of a normally functioning bicuspid valve’, Biomech. Model. Mechanobiol. 6, 13491361.
Borazjani I., Ge L. and Sotiropoulos F. (2008), ‘Curvilinear immersed boundary method for simulating fluid structure interaction with complex 3D rigid bodies’, J. Comput. Phys. 227, 75877620.
Bordas R., Gillow K., Gavaghan D., Rodríguez B. and Kay D. (2012), ‘A bidomain model of the ventricular specialized conduction system of the heart’, SIAM J. Appl. Math. 72, 16181643.
Borzì A. and Schulz V. (2011), Computational Optimization of Systems Governed by Partial Differential Equations, SIAM.
Boulakia M., Fernández M., Gerbeau J.-F. and Zemzemi N. (2008), ‘Direct and inverse problems in electrocardiography’, AIP Conference Proceedings 1048, 113117.
Boulakia M., Schenone E. and Gerbeau J.-F. (2012), ‘Reduced-order modeling for cardiac electrophysiology. application to parameter identification’, Int. J. Numer. Meth. Biomed. Engrg 28, 727744.
Bourgault Y., Coudière Y. and Pierre C. (2006), ‘Existence and uniqueness of the solution for the bidomain model used in cardiac electrophysiology’, Nonlinear Analysis: Real World Applications 10, 458482.
Bourgault Y., Ethier M. and Leblanc V. (2003), ‘Simulation of electrophysiological waves with an unstructured finite element method’, ESAIM Math. Model. Numer. Anal. 37, 649661.
Brault A., Dumas L. and Lucor D. (2016), ‘Uncertainty quantification of inflow boundary condition and proximal arterial stiffness-coupled effect on pulse wave propagation in a vascular network’, Int. J. Numer. Methods Biomed. Engrg. doi:10.1002/cnm.2859
Bueno-Orovio A., Cherry E. and Fenton F. (2008), ‘Minimal model for human ventricular action potentials in tissue’, J. Theoret. Biol. 3, 544560.
Burman E. and Fernández M. (2009), ‘Stabilization of explicit coupling in fluid–structure interaction involving fluid incompressibility’, Comput. Methods Appl. Mech. Engrg 198, 766784.
Burman E. and Fernández M. (2014), ‘An unfitted Nitsche method for incompressible fluid–structure interaction using overlapping meshes’, Comput. Methods Appl. Mech. Engrg 279, 497514.
Burman E., Claus S., Hansbo P., Larson M. and Massing A. (2015), ‘CutFEM: Discretizing geometry and partial differential equations’, Int. J. Numer. Methods Engrg 104, 472501.
Campbell I., Ries J., Dhawan S., Quyyumi A., Taylor W. and Oshinski J. (2012), ‘Effect of inlet velocity profiles on patient-specific computational fluid dynamics simulations of the carotid bifurcation’, J. Biomech. Engrg 134, 051001.
Carew T., Vaishnav R. and Patel D. (1968), ‘Compressibility of the arterial wall’, Circ. Res. 23, 6168.
Carr J., Fright W. and Beatson R. (1997), ‘Surface interpolation with radial basis functions for medical imaging’, IEEE Trans. Med. Imaging 16, 96107.
Causin P., Gerbeau J.-F. and Nobile F. (2005), ‘Added-mass effect in the design of partitioned algorithms for fluid–structure problems’, Comput. Methods Appl. Mech. Engrg 194, 45064527.
Celik I., Ghia U., Roache P., Freitas C., Coleman H. and Raad P. (2008), ‘Procedure for estimation and reporting of uncertainty due to discretization in CFD applications’, J. Fluids Engrg: Trans. ASME 130, 078001.
Chabiniok R., Moireau P., Lesault P.-F., Rahmouni A., Deux J.-F. and Chapelle D. (2012), ‘Estimation of tissue contractility from cardiac cine-MRI using a biomechanical heart model’, Biomech. Model. Mechanobiol. 11, 609630.
Chabiniok R., Wang V., Hadjicharalambous M., Asner L., Lee J., Sermesant M., Kuhl E., Young A., Moireau P., Nash M., Chapelle D. and Nordsletten D. (2016), ‘Multiphysics and multiscale modelling, data–model fusion and integration of organ physiology in the clinic: Ventricular cardiac mechanics’, Interface Focus 6, 20150083.
Chapelle D., Fragu M., Mallet V. and Moireau P. (2013a), ‘Fundamental principles of data assimilation underlying the Verdandi library: Applications to biophysical model personalization within euHeart’, Med. Biol. Engrg Comput. 51, 12211233.
Chapelle D., Gariah A., Moireau P. and Sainte-Marie J. (2013b), ‘A Galerkin strategy with proper orthogonal decomposition for parameter-dependent problems: Analysis, assessments and applications to parameter estimation’, ESAIM Math. Model. Numer. Anal. 47, 18211843.
Charonko J., Kumar R., Stewart K., Little W. and Vlachos P. (2013), ‘Vortices formed on the mitral valve tips aid normal left ventricular filling’, Ann. Biomed. Engrg 41, 10491061.
Chavent G. (2010), Nonlinear Least Squares for Inverse Problems: Theoretical Foundations and Step-by-Step Guide for Applications, Scientific Computation, Springer.
Chen J., Lu X.-Y. and Wang W. (2006), ‘Non-Newtonian effects of blood flow on hemodynamics in distal vascular graft anastomoses’, J. Biomech. 39, 19831995.
Chen P. and Schwab C. (2015), ‘Sparse-grid, reduced-basis Bayesian inversion’, Comput. Methods Appl. Mech. Engrg 297, 84115.
Chen P., Quarteroni A. and Rozza G. (2013), ‘Simulation-based uncertainty quantification of human arterial network hemodynamics’, Int. J. Numer. Methods Biomed. Engrg 29, 698721.
Cheng L., Bodley J. and Pullan A. (2003), ‘Comparison of potential- and activation-based formulations for the inverse problem of electrocardiology’, IEEE Trans. Biomed. Engrg 50, 1122.
Cheng Y., Oertel H. and Schenkel T. (2005), ‘Fluid–structure coupled CFD simulation of the left ventricular flow during filling phase’, Ann. Biomed. Engrg 5, 567576.
Cherubini C., Filippi S., Nardinocchi P. and Teresi L. (2008), ‘An electromechanical model of cardiac tissue: Constitutive issues and electrophysiological effects’, Prog. Biophys. Molec. Biol. 2–3, 562573.
Cheung S., Wong K. K. L., Yeoh G. H., Yang W., Tu J., Beare R. and Phan T. (2010), ‘Experimental and numerical study on the hemodynamics of stenosed carotid bifurcation’, Australas. Phys. Engrg Sci. Med. 33, 319328.
Chinchapatnam P., Rhode K., Ginks M., Rinaldi C., Lambiase P., Razavi R., Arridge S. and Sermesant M. (2008), ‘Model-based imaging of cardiac apparent conductivity and local conduction velocity for diagnosis and planning of therapy’, IEEE Trans. Med. Imaging 27, 16311642.
Ching J., Beck J. and Porter K. (2006), ‘Bayesian state and parameter estimation of uncertain dynamical systems’, Probab. Engrg Mech. 21, 8196.
Chnafa C., Mendez S. and Nicoud F. (2014), ‘Image-based large-eddy simulation in a realistic left heart’, Comput. Fluids 94, 173187.
Choi Y., Constantino J., Vedula V., Trayanova N. and Mittal R. (2015), ‘A new MRI-based model of heart function with coupled hemodynamics and application to normal and diseased canine left ventricles’, Front. Bioeng. Biotechnol. 3, 140.
Chorin A. (1968), ‘Numerical solution of the Navier–Stokes equations’, Math. Comp. 22, 745762.
Chung J. and Hulbert G. (1993), ‘A time integration algorithm for structural dynamics with improved numerical dissipation: The generalized- $\unicode[STIX]{x1D6FC}$ method’, Trans. ASME J. Appl. Mech. 60, 371375.
Ciarlet P. (1988), Mathematical Elasticity. Vol. 1: Three Dimensional Elasticity , Elsevier Science.
Ciarlet P. and Necas J. (1985), ‘Unilateral problems in nonlinear, three-dimensional elasticity’, Arch. Rat. Mech. Anal. 87, 319338.
Clayton R. and Panfilov A. (2008), ‘A guide to modelling cardiac electrical activity in anatomically detailed ventricles’, Prog. Biophys. Molec. Biol. 96, 1943.
Clayton R., Bernus O., Cherry E., Dierckx H., Fenton F., Mirabella L., Panfilov A., Sachse F., Seemann G. and Zhang H. (2011), ‘Models of cardiac tissue electrophysiology: Progress, challenges and open questions’, Prog. Biophys. Molec. Biol. 104, 2248.
Cocosco C., Netsch T., Sénǵas J., Bystrov D., Niessen W. and Viergever M. (2004), Automatic cardiac region-of-interest computation in cine 3D structural MRI. In Computer Assisted Radiology and Surgery: Proceedings of the 18th International Congress and Exhibition, Vol. 1268 of International Congress Series, Elsevier, pp. 11261131.
Codina R. and Badia S. (2006), ‘On some pressure segregation methods of fractional-step type for the finite element approximation of incompressible flow problems’, Comput. Methods Appl. Mech. Engrg 195, 29002918.
Cohen A. and Devore R. (2015), Approximation of high-dimensional parametric PDEs. In Acta Numerica, Vol. 24, Cambridge University Press, pp. 1159.
Colciago C., Deparis S. and Quarteroni A. (2014), ‘Comparisons between reduced order models and full 3D models for fluid–structure interaction problems in haemodynamics’, J. Comput. Appl. Math. 2754, 120138.
Colli Franzone P. and Guerri L. (1993), ‘Spreading excitation in 3-D models of the anisotropic cardiac tissue I: Validation of the Eikonal model’, Math. Biosci. 113, 145209.
Colli Franzone P. and Pavarino L. (2004), ‘A parallel solver for reaction-diffusion systems in computational electrocardiology’, Math. Models Methods Appl. Sci. 14, 883911.
Colli Franzone P. and Savaré G. (2002), Degenerate evolution systems modeling the cardiac electric field at micro- and macroscopic level. In Evolution Equations, Semigroups and Functional Analysis: In Memory of Brunello Terreni (Lorenzi A. and Ruf B., eds), Birkhäuser, pp. 4978.
Colli Franzone P., Guerri L. and Rovida S. (1990), ‘Wavefront propagation in an activation model of the anisotropic cardiac tissue: Asymptotic analysis and numerical simulations’, J. Math. Biol. 28, 121176.
Colli Franzone P., Guerri L., Viganotti C. and Taccardi B. (1985), ‘Finite element approximation of regularized solution of the inverse potential problem of electrocardiography and applications to experimental data’, Calcolo 12, 91186.
Colli Franzone P., Pavarino L. and Scacchi S. (2014), Mathematical Cardiac Electrophysiology, Springer.
Colli Franzone P., Pavarino L. and Scacchi S. (2016), ‘Bioelectrical effects of mechanical feedbacks in a strongly coupled cardiac electro-mechanical model’, Math. Models Methods Appl. Sci. 26, 2757.
Colli Franzone P., Pavarino L. and Taccardi B. (2005), ‘Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models’, Math. Biosci. 197, 3566.
Colli Franzone P., Taccardi B. and Viganotti C. (1978), ‘An approach to inverse calculation of epicardial potentials from body surface maps’, Adv. Cardiol. 21, 5054.
Corrado C., Gerbeau J.-F. and Moireau P. (2015), ‘Identification of weakly coupled multiphysics problems: Application to the inverse problem of electrocardiography’, J. Comput. Phys. 283, 271298.
Costa K., Holmes J. and Mcculloch A. (2001), ‘Modelling cardiac mechanical properties in three dimensions’, Phil. Trans. Royal Soc. A 359, 12331250.
Crosetto P., Deparis S., Fourestey G. and Quarteroni A. (2011), ‘Parallel algorithms for fluid–structure interaction problems in haemodynamics’, SIAM J. Sci. Comput. 33, 15981622.
Cui T., Marzouk Y. and Willcox K. (2015), ‘Data-driven model reduction for the Bayesian solution of inverse problems’, Int. J. Numer. Methods Engrg 102, 966990.
Dacorogna B. (2000), Direct Methods in the Calculus of Variations, Vol. 78 of Applied Mathematical Sciences, Springer.
D’Angelo C. (2007), Multiscale modelling of metabolism and transport phenomena in living tissues. PhD thesis, Ecole Polytechnique Fédérale de Lausanne.
Degroote J. (2011), ‘On the similarity between Dirichlet–Neumann with interface artificial compressibility and Robin–Neumann schemes for the solution of fluid–structure interaction problems’, J. Comput. Phys. 230, 63996403.
Degroote J. and Vierendeels J. (2011), ‘Multi-solver algorithms for the partitioned simulation of fluid–structure interaction’, Comput. Methods Appl. Mech. Engrg 25–28, 21952210.
Degroote J., Swillens A., Bruggeman P., Haelterman, Segers P. and Vierendeels J. (2010), ‘Simulation of fluid–structure interaction with the interface artificial compressibility method’, Int. J. Numer. Methods Biomed. Engrg 26, 276289.
D’Elia M. and Veneziani A. (2013), ‘Uncertainty quantification for data assimilation in a steady incompressible Navier–Stokes problem’, ESAIM Math. Model. Numer. Anal. 47, 10371057.
D’Elia M., Perego M. and Veneziani A. (2012), ‘A variational data assimilation procedure for the incompressible Navier–Stokes equations in hemodynamics’, J. Sci. Comput. 52, 340359.
Delingette H., Billet F., Wong K., Sermesant M., Rhode K., Ginks M., Rinaldi C., Razavi R. and Ayache N. (2012), ‘Personalization of cardiac motion and contractility from images using variational data assimilation’, IEEE Trans. Biomed. Engrg 59, 2024.
Deparis S. (2004), Numerical analysis of axisymmetric flows and methods for fluid–structure interaction arising in blood flow simulation. PhD thesis, Ecole Polytechnique Fédérale de Lausanne.
Deparis S., Discacciati M., Fourestey G. and Quarteroni A. (2006), ‘Fluid–structure algorithms based on Steklov–Poincaré operators’, Comput. Methods Appl. Mech. Engrg 195, 57975812.
Deparis S., Forti D., Grandperrin G. and Quarteroni A. (2016), ‘FaCSI: A block parallel preconditioner for fluid–structure interaction in hemodynamics’, J. Comput. Phys. 700718.
Deparis S., Grandperrin G. and Quarteroni A. (2014), ‘Parallel preconditioners for the unsteady Navier–Stokes equations and applications to hemodynamics simulations’, Comput. Fluids 92, 253273.
Dick J., Kuo F. and Sloan I. (2013), High-dimensional integration: The quasi-Monte Carlo way. Acta Numerica, Vol. 22, Cambridge University Press, pp. 133288.
Difrancesco D. and Noble D. (1985), ‘A model of cardiac electrical activity incorporating ionic pumps and concentration changes’, Phil. Trans. Royal Soc. B 307, 353398.
Dihlmann M. and Haasdonk B. (2016), ‘A reduced basis Kalman filter for parametrized partial differential equations’, ESAIM Control Optim. Calc. Var. 22, 625669.
Do H., Owida A. A., Yang W. and Morsi Y. (2011), ‘Numerical simulation of the haemodynamics in end-to-side anastomoses’, Int. J. Numer. Methods Fluids 67, 638650.
Dohrmann C. and Widlund O. (2009), ‘An overlapping Schwarz algorithm for almost incompressible elasticity’, SIAM J. Numer. Anal. 4, 88118823.
Dokos S., Smaill B., Young A. and Legrice I. (2002), ‘Shear properties of passive ventricular myocardium’, Amer. J. Physiol. 283, H2650H2659.
Donders W., Huberts W., van de Vosse F. and Delhaas T. (2015), ‘Personalization of models with many model parameters: An efficient sensitivity analysis approach’, Int. J. Numer. Methods Biomed. Engrg 31, e02727.
Donea J. (1982), ‘An arbitrary Lagrangian–Eulerian finite element method for transient dynamic fluid–structure interaction’, Comput. Methods Appl. Mech. Engrg 33, 689723.
Dur O., Coskun S., Coskun K., Frakes D., Kara L. and Pekkan K. (2011), ‘Computer-aided patient-specific coronary artery graft design improvements using CFD coupled shape optimizer’, Cardiovasc. Engr. Tech. 113.
Durrer D., van Dam R., Freud G., Janse M., Meijler F. and Arzbaecher R. (1970), ‘Total excitation of the isolated human heart’, Circulation 41, 899912.
Eck V., Donders W., Sturdy J., Feinberg J., Delhaas T., Hellevik L. and Huberts W. (2016), ‘A guide to uncertainty quantification and sensitivity analysis for cardiovascular applications’, Int. J. Numer. Methods Biomed. Engrg 32, e02755.
Einstein D., Kunzelman K., Reinhall P., Nicosia M. and Cochran R. (2005), ‘The effects of cellular contraction on aortic valve leaflet flexural stiffness’, J. Heart Valve Disease 14, 376385.
Eitel C., Hindricks G., Dagres N., Sommer P. and Piorkowski C. (2010), ‘EnSite Velocity™ cardiac mapping system: A new platform for 3D mapping of cardiac arrhythmias’, Expert Rev. Med. Devices 7, 185192.
Elman H. and Silvester D. (1996), ‘Fast nonsymmetric iterations and preconditioning for Navier–Stokes equations’, SIAM J. Sci. Comput. 17, 3346.
Elman H., Silvester D. and Wathen A. (2005), Finite Elements and Fast Iterative Solvers, Oxford Science Publications.
Enden G. and Popel A. (1992), ‘A numerical study of the shape of the surface separating flow into branches in microvascular bifurcations’, J. Biomech. Engrg 114, 398405.
Eriksson T., Prassl A., Plank G. and Holzapfel G. (2013), ‘Modeling the dispersion in electromechanically coupled myocardium’, Int. J. Numer. Methods Biomed. Engrg 29, 12671284.
Ernst O., Sprungk B. and Starkloff H.-J. (2014), Bayesian inverse problems and Kalman filters. In Extraction of Quantifiable Information from Complex Systems, Vol. 102 of Lecture Notes in Computational Science and Engineering, Springer, pp. 133159.
Ervin V. and Lee H. (2007), ‘Numerical approximation of a quasi-Newtonian Stokes flow problem with defective boundary conditions’, SIAM J. Numer. Anal. 45, 21202140.
Ethier C., Steinman D., Zhang X., Karpik S. and Ojha M. (1998), ‘Flow waveform effects on end-to-side anastomotic flow patterns’, J. Biomech. 31, 609617.
Ethier M. and Bourgault Y. (2008), ‘Semi-implicit time-discretization schemes for the bidomain model’, SIAM J. Numer. Anal. 5, 24432468.
Euler L. (1775), Principia pro motu sanguinis per arterias determinando. In Opera posthuma mathematica et physica anno 1844 detecta, editerunt P. H. Fuss et N. Fuss, Petropoli, apud Eggers et socios, Vol. 1, pp. 814823.
Evensen G. (1994), ‘Sequential data assimilation with a nonlinear quasi-geostrophic model using Monte Carlo methods to forecast error statistics’, J. Geophys. Res. 9, 1014310162.
Evensen G. (2003), ‘The ensemble Kalman filter: Theoretical formulation and practical implementation’, Ocean Dynamics 53, 343367.
Evensen G. (2009), ‘The ensemble Kalman filter for combined state and parameter estimation’, IEEE Control Syst. Mag. 29, 83104.
Faggiano E., Antiga A., Puppini G., Quarteroni A., Luciani G. and Vergara C. (2013), ‘Helical flows and asymmetry of blood jet in dilated ascending aorta with normally functioning bicuspid valve’, Biomech. Model. Mechanobiol. 4, 801813.
Faggiano E., Lorenzi T. and Quarteroni A. (2014), ‘Metal artefact reduction in computed tomography images by a fourth-order total variation flow’, Comput. Methods Biomech. Biomed. Engrg: Imaging & Visualization 3–4, 202213.
Farhat C. and Roux F. (1991), ‘A method of finite element tearing and interconnecting and its parallel solution algorithm’, Int. J. Numer. Methods Engrg 32, 12051227.
Fasano A., Santos R. and Sequeira A. (2012), Blood coagulation: A puzzle for biologists, a maze for mathematicians. (Ambrosi D., Quarteroni A. and Rozza G., eds), Chapter 3 in Modeling of Physiological Flows , Springer, pp. 4175.
Fedele M., Faggiano E., Barbarotta L., Cremonesi F., Formaggia L. and Perotto S. (2015), Semi-automatic three-dimensional vessel segmentation using a connected component localization of the region-scalable fitting energy. In 2015 9th International Symposium on Image and Signal Processing and Analysis (ISPA), IEEE, pp. 7277.
Fedele M., Faggiano E., Dede’ L. and Quarteroni A. (2016), A patient-specific aortic valve model based on moving resistive immersed implicit surfaces. MOX-Report 23-2016, Department of Mathematics, Politecnico di Milano.
Fenton F. and Karma A. (1998), ‘Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: Filament instability and fibrillation’, Chaos 8, 2047.
Fernández M. (2013), ‘Incremental displacement-correction schemes for incompressible fluid–structure interaction: Stability and convergence analysis’, Numer. Math. 123, 2165.
Fernández M. and Moubachir M. (2005), ‘A Newton method using exact Jacobians for solving fluid–structure coupling’, Comput. Struct. 83, 127142.
Fernández M. and Zemzemi N. (2010), ‘Decoupled time-marching schemes in computational cardiac electrophysiology and ECG numerical simulation’, Math. Biosci. 226, 5875.
Fernández M., Gerbeau J.-F. and Grandmont C. (2007), ‘A projection semi-implicit scheme for the coupling of an elastic structure with an incompressible fluid’, Int. J. Numer. Methods Engrg 69, 794821.
Fernández M., Milisic V. and Quarteroni A. (2005), ‘Analysis of a geometrical multiscale blood flow model based on the coupling of ODEs and hyperbolic PDEs’, Multiscale Model. Simul. 4, 215236.
Figueroa C., Vignon-Clementel I., Jansen K., Hughes T. and Taylor C. (2006), ‘A coupled momentum method for modeling blood flow in three-dimensional deformable arteries’, Comput. Methods Appl. Mech. Engrg 195, 56855706.
Fin L. and Grebe R. (2003), ‘Three dimensional modeling of the cerebrospinal fluid dynamics and brain interactions in the aqueduct of sylvius’, Comput. Methods Biomech. Biomed. Engrg 3, 163170.
Fischer P., Loth F., Lee S., Lee S., Smith D. and Bassiouny H. (2007), ‘Simulation of high-Reynolds number vascular flows’, Comput. Methods Appl. Mech. Engrg 196, 30493060.
Fishman G. (1996), Monte Carlo: Concepts, Algorithms, and Applications, Springer Series in Operations Research and Financial Engineering, Springer.
Fitzhugh R. (1961), ‘Impulses and physiological states in theoretical models of nerve membrane’, Biophys. J. 1, 445466.
Formaggia L. and Vergara C. (2012), ‘Prescription of general defective boundary conditions in fluid-dynamics’, Milan J. Math. 80, 333350.
Formaggia L., Gerbeau J.-F., Nobile F. and Quarteroni A. (2001), ‘On the coupling of 3D and 1D Navier–Stokes equations for flow problems in compliant vessels’, Comput. Methods Appl. Mech. Engrg 191, 561582.
Formaggia L., Gerbeau J.-F., Nobile F. and Quarteroni A. (2002), ‘Numerical treatment of defective boundary conditions for the Navier–Stokes equation’, SIAM J. Numer. Anal. 40, 376401.
Formaggia L., Lamponi D. and Quarteroni A. (2003), ‘One-dimensional models for blood flow in arteries’, J. Engrg Math. 47, 251276.
Formaggia L., Moura A. and Nobile F. (2007), ‘On the stability of the coupling of 3D and 1D fluid–structure interaction models for blood flow simulations’, M2AN Math. Model. Numer. Anal. 41, 743769.
Formaggia L., Nobile F., Quarteroni A. and Veneziani A. (1999), ‘Multiscale modelling of the circulatory system: A preliminary analysis’, Comput. Vis. Sci. 2, 7583.
Formaggia L., Quarteroni A. and Veneziani A. (2009a), Cardiovascular Mathematics: Modeling and Simulation of the Circulatory System, Springer.
Formaggia L., Quarteroni A. and Vergara C. (2013), ‘On the physical consistency between three-dimensional and one-dimensional models in haemodynamics’, J. Comput. Phys. 244, 97112.
Formaggia L., Veneziani A. and Vergara C. (2008), ‘A new approach to numerical solution of defective boundary value problems in incompressible fluid dynamics’, SIAM J. Numer. Anal. 46, 27692794.
Formaggia L., Veneziani A. and Vergara C. (2009b), ‘Flow rate boundary problems for an incompressible fluid in deformable domains: Formulations and solution methods’, Comput. Methods Appl. Mech. Engrg 199, 677688.
Fornefett M., Rohr K. and Stiehl H. (2001), ‘Radial basis functions with compact support for elastic registration of medical images’, Image Vision Comput. 19, 8796.
Forster C., Wall W. and Ramm E. (2007), ‘Artificial added mass instabilities in sequential staggered coupling of nonlinear structures and incompressible viscous flow’, Comput. Methods Appl. Mech. Engrg 196, 12781293.
Forti D. and Dede’ L. (2015), ‘Semi-implicit BDF time discretization of the Navier–Stokes equations with VMS-LES modeling in a high performance computing framework’, Comput. Fluids 117, 168182.
Frangi A., Niessen W., Hoogeveen R., van Walsum T. and Viergever M. (1999), ‘Model-based quantitation of 3-D magnetic resonance angiographic images’, IEEE Trans. Med. Imaging 18, 946956.
Frazier D., Krassowska W., Chen P., Wolf P., Danieley N., Smith W. and Ideker R. (1988), ‘Transmural activations and stimulus potentials in three-dimensional anisotropic canine myocardium’, Circ. Res. 63, 135146.
Fritz T., Wieners C., Seemann G., Steen H. and Dossel O. (2014), ‘Simulation of the contraction of the ventricles in a human heart model including atria and pericardium’, Biomech. Model. Mechanobiol. 13, 627641.
Fung Y. (1993), Biomechanics: Mechanical Properties of Living Tissues, Springer.
Fung Y. C., Fronek K. and Patitucci P. (1979), ‘Pseudoelasticity of arteries and the choice of its mathematical expression’, Amer. J. Physiol. 237, H620H631.
Galvin K. and Lee H. (2013), ‘Analysis and approximation of the cross model for quasi-Newtonian flows with defective boundary conditions’, Appl. Math. Comput. 222, 244254.
Galvin K., Lee H. and Rebholz L. (2012), ‘Approximation of viscoelastic flows with defective boundary conditions’, J. Non-Newton. Fluid Mech. 169/170, 104113.
Gee M., Kuttler U. and Wall W. (2011), ‘Truly monolithic algebraic multigrid for fluid–structure interaction’, Int. J. Numer. Methods Engrg 85, 9871016.
van der Geest R., Jansen E., Buller V. and Reiber J. (1994), Automated detection of left ventricular epi- and endocardial contours in short-axis MR images. In Computers in Cardiology 1994, IEEE, pp. 3336.
Geiger B. (1993), Three-dimensional modeling of human organs and its application to diagnosis and surgical planning. Technical Report RR-2105, INRIA.
Geneser S., Kirby R. and Macleod R. (2008), ‘Application of stochastic finite element methods to study the sensitivity of ECG forward modeling to organ conductivity’, IEEE Trans. Biomed. Engrg 55, 3140.
Gerardo-Giorda L., Mirabella L., Nobile F., Perego M. and Veneziani A. (2009), ‘A model-based block-triangular preconditioner for the bidomain system in electrocardiology’, J. Comput. Phys. 228, 36253639.
Gerardo-Giorda L., Nobile F. and Vergara C. (2010), ‘Analysis and optimization of Robin–Robin partitioned procedures in fluid–structure interaction problems’, SIAM J. Numer. Anal. 48, 20912116.
Gerbeau J.-F., Lombardi D. and Schenone E. (2015), ‘Reduced order model in cardiac electrophysiology with approximated lax pairs’, Adv. Comput. Math. 41, 11031130.
Gervasio P., Saleri F. and Veneziani A. (2006), ‘Algebraic fractional-step schemes with spectral methods for the incompressible Navier–Stokes equations’, J. Comput. Phys. 214, 347365.
Ghanem R. and Spanos P. (2003), Stochastic Finite Elements: A Spectral Approach, revised edition, Dover, Reprint of the Springer 1991 edition.
Gigante G. and Vergara C. (2015), ‘Analysis and optimization of the generalized Schwarz method for elliptic problems with application to fluid–structure interaction’, Numer. Math. 131, 369404.
Giles M. (2015), Multilevel Monte Carlo methods. In Acta Numerica, Vol. 24, Cambridge University Press, pp. 259328.
Giordana S., Sherwin S., Peiró J., Doorly D., Crane J., Lee K., Cheshire N. and Caro C. (2005), ‘Local and global geometric influence on steady flow in distal anastomoses of peripheral bypass grafts’, J. Biomech. Engrg 127, 10871098.
Girault V. and Raviart P. (1986), Finite Element Methods for Navier–Stokes Equations, Springer.
Glagov S., Zarins C., Giddens D. and Ku D. (1988), ‘Hemodynamics and atherosclerosis: Insights and perspectives gained from studies of human arteries’, Arch. Pathol. Lab. Med. 112, 10181031.
a priori Glowinski R., Pan T. and Periaux J. (1997), ‘A Lagrange multiplier/fictitious domain method for the numerical simulation of incompressible viscous flow around moving rigid bodies (I): Case where the rigid body motions are known’, CR Acad. Sci. I Math. 324, 361369.
Gnyaneshwar R., Kumar R. and Balakrishnan K. (2002), ‘Dynamic analysis of the aortic valve using a finite element model’, Ann. Thorac. Surg. 73, 11221129.
Goktepe S. and Kuhl E. (2010), ‘Electromechanics of the heart: A unified approach to the strongly coupled excitation-contraction problem’, Comput. Mech. 45, 227243.
Goshtasby A. and Turner D. (1995), ‘Segmentation of cardiac cine MR images for extraction of right and left ventricular chambers’, IEEE Trans. Med. Imaging 14, 5664.
Grandmont C. (1998), Analyse mathématique et numérique de quelques problèmes d’interaction fluide-structure. PhD thesis, Laboratoire d’Analyse Numérique de Paris VI.
Griffith B., Hornung R., Mcqueen D. and Peskin C. (2007), ‘An adaptive, formally second order accurate version of the immersed boundary method’, J. Comput. Phys. 223, 10419.
Griffith B., Luo X., Mcqueen D. and Peskin C. (2009), ‘Simulating the fluid dynamics of natural and prosthetic heart valves using the immersed boundary method’, Int. J. Appl. Mech. 1, 137176.
Grinberg L., Cheever E., Anor T., Madsen J. and Karniadakis G. (2010), ‘Modeling blood flow circulation in intracranial arterial networks: A comparative 3D/1D simulation study’, Ann. Biomed. Engrg 39, 297309.
Grinberg L., Yakhot A. and Karniadakis G. (2009), ‘Analyzing transient turbulence in a stenosed carotid artery by proper orthogonal decomposition’, Ann. Biomed. Engrg 37, 22002217.
Guccione J., Mcculloch A. and Waldman L. (1991), ‘Passive material properties of intact ventricular myocardium determined from a cylindrical model’, J. Biomech. Engrg 113, 4255.
Guerciotti B., Vergara C., Azzimonti L., Forzenigo L., Buora A., Biondetti P. and Domanin M. (2015), ‘Computational study of the fluid-dynamics in carotids before and after endarterectomy’, J. Biomech. 195, 20882099.
Guerciotti B., Vergara C., Ippolito S., Quarteroni A., Antona C. and Scrofani R. (2017), ‘Computational study of the risk of restenosis in coronary bypasses’, Biomech. Model. Mechanobiol. 16, 313332.
Guermond J. and Quartapelle L. (1998), ‘On the approximation of the unsteady Navier–Stokes equations by finite element projection methods’, Numer. Math. 80, 207238.
Guermond J. and Shen J. (2003), ‘Velocity-correction projection methods for incompressible flows’, SIAM J. Numer. Anal. 41, 112134.
Guermond J. and Shen J. (2006), ‘An overview of projection methods for incompressible flows’, Comput. Methods Appl. Mech. Engrg 195, 60116045.
Guermond J., Minev P. and Shen J. (2005), ‘Error analysis of pressure-correction schemes for the time-dependent Stokes equations with open boundary conditions’, SIAM J. Numer. Anal. 43, 239258.
Guermond J., Minev P. and Shen J. (2006), ‘An overview of projection methods for incompressible flows’, Comput. Methods Appl. Mech. Engrg 195, 60116045.
Guidoboni G., Glowinski R., Cavallini N. and Canic S. (2009), ‘Stable loosely-coupled-type algorithm for fluid–structure interaction in blood flow’, J. Comput. Phys. 228, 69166937.
Gultekin O., Sommer G. and Holzapfel G. (2016), ‘An orthotropic viscoelastic model for the passive myocardium: Continuum basis and numerical treatment’, Comput. Methods Biomech. Biomed. Engrg 19, 16471664.
Gunzburger M. (2003), Perspectives in Flow Control and Optimization, Advances in Design and Control, SIAM.
Gunzburger M., Webster C. and Zhang G. (2014), Stochastic finite element methods for partial differential equations with random input data. In Acta Numerica, Vol. 23, Cambridge University Press, pp. 521650.
Gupta A., von Kurowski L., Singh A., Geiger D., Liang C., Chiu M., Adler L., Haacke M. and Wilson D. (1993), Cardiac MR image segmentation using deformable models. In Computers in Cardiology, IEEE, pp. 747750.
Gurev V., Pathmanathan P., Fattebert J., Wen H., Magerlein J., Gray R., Richards D. and Rice J. (2015), ‘A high-resolution computational model of the deforming human heart’, Biomech. Model. Mechanobiol. 14, 132140.
Haggerty C., Mirabella L., Restrepo M., de Zélicourt D., Rossignac J., Sotiropoulos F., Spray T., Kanter K., Fogel M. and Yoganathan A. (2013), Patient-specific surgery planning for the fontan procedure. In Computer Models in Biomechanics: From Nano to Macro (Holzapfel G. and Kuhl E., eds), Springer, pp. 217228.
Hansbo A. and Hansbo P. (2002), ‘An unfitted finite element method, based on Nitsche’s method, for elliptic interface problems’, Comput. Methods Appl. Mech. Engrg 191, 55375552.
Hansbo A., Hansbo P. and Larson M. (2003), ‘A finite element method on composite grids based on Nitsche’s method’, ESAIM: Math. Model. Numer. Anal. 37, 495514.
Hariton I., de Botton G., Gasser T. and Holzapfel G. (2006), ‘Stress-driven collagen fiber remodeling in arterial walls’, Biomech. Model. Mechanobiol. 3, 163175.
de Hart J., Baaijens F., Peters G. and Schreurs P. (2003), ‘A computational fluid–structure interaction analysis of a fiber-reinforced stentless aortic valve’, J. Biomech. 36, 699712.
de Hart J., Peters G., Schreurs P. and Baaijens F. (2000), ‘A two-dimensional fluid–structure interaction model of the aortic value’, J. Biomech. 33, 10791088.
Haruguchi H. and Teraoka S. (2003), ‘Intimal hyperplasia and hemodynamic factors in arterial bypass and arteriovenous grafts: A review’, J. Artif. Organs 6, 227235.
He X., Ku D. and Moore J. Jr (1993), ‘Simple calculation of the velocity profiles for pulsatile flow in a blood vessel using Mathematica’, Ann. Biomed. Engrg 21, 4549.
Heil M. (2004), ‘An efficient solver for the fully coupled solution of large-displacement fluid–structure interaction problems’, Comput. Methods Appl. Mech. Engrg 193, 123.
Hesthaven J., Rozza G. and Stamm B. (2016), Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs in Mathematics, Springer.
Heywood J., Rannacher R. and Turek S. (1996), ‘Artificial boundaries and flux and pressure conditions for the incompressible Navier–Stokes equations’, Int. J. Numer. Methods Fluids 22, 325352.
Hilgemann D. and Noble D. (1987), ‘Excitation–contraction coupling and extracellular calcium transients in rabbit atrium: Reconstruction of basic cellular mechanisms’, Proc. Royal Soc. London B 230, 163205.
Hillen B., Hoogstraten H. and Post L. (1986), ‘A wave propagation model of blood flow in large vessels using an approximate velocity profile function’, J. Biomech. 19, 187194.
Hintermüller M., Kunisch K., Spasov Y. and Volkwein S. (2004), ‘Dynamical systems-based optimal control of incompressible fluids’, Int. J. Numer. Methods Fluids 46, 345359.
Hinze M., Pinnau R., Ulbrich M. and Ulbrich S. (2009), Optimization with PDE Constraints, Vol. 23 of Mathematical Modelling: Theory and Applications, Springer.
Hirt C., Amsden A. and Cook J. (1974), ‘An arbitrary Lagrangian Eulerian computing method for all flow speeds’, J. Comput. Phys. 69, 277324.
Hodgkin A. and Huxley A. (1952), ‘A quantitative description of membrane current and its application to conduction and excitation in nerve’, J. Physiol. 117, 500544.
Holzapfel G. (2000), Nonlinear Solid Mechanics: A Continuum Approach for Engineering, Wiley.
Holzapfel G. and Gasser T. (2001), ‘A viscoelastic model for fiber-reinforced composites at finite strains: Continuum basis, computational aspects, and applications’, Comput. Methods Appl. Mech. Engrg 190, 43794403.
Holzapfel G. and Ogden R. (2009), ‘Constitutive modelling of passive myocardium: A structurally based framework for material characterization’, Philos. Trans. Royal Soc. A 367, 34453475.
Holzapfel G. and Ogden R. (2010), ‘Constitutive modelling of arteries’, Proc. Royal Soc. Lond. A: Math. Phys. Engrg Sci. 466, 15511596.
Holzapfel G., Gasser T. and Ogden R. (2000), ‘A new constitutive framework for arterial wall mechanics and a comparative study of material models’, J. Elast. 61, 148.
Hopf E. (1951), ‘Über die Anfangswertaufgabe für die hydrodynamischen Grundgliechungen’, Math. Nachrichten 4, 213231.
Hoteit I., Pham D. and Blum J. (2002), ‘A simplified reduced order Kalman filtering and application to altimetric data assimilation in Tropical Pacific’, J. Marine Systems 36, 101127.
Houtemaker P. and Zhang F. (2016), ‘Review of the ensemble Kalman filter for atmospheric data assimilation’, Monthly Weather Rev. 122, 44894532.
Hsu M., Kamensky D., Bazilevs Y., Sacks M. and Hughes T. (2014), ‘Fluid–structure interaction analysis of bioprosthetic heart valves: Significance of arterial wall deformation’, Comput. Mech. 54, 10551071.
Huberts W., de Jonge C., van der Linden W., Inda M., Passera K., Tordoir J., van de Vosse F. and Bosboom E. (2013a), ‘A sensitivity analysis of a personalized pulse wave propagation model for arteriovenous fistula surgery, B: Identification of possible generic model parameters’, Med. Engrg Phys. 35, 827837.
Huberts W., de Jonge C., van der Linden W., Inda M., Tordoir J., van de Vosse F. and Bosboom E. (2013b), ‘A sensitivity analysis of a personalized pulse wave propagation model for arteriovenous fistula surgery, A: Identification of most influential model parameters’, Med. Engrg Phys. 35, 810826.
Hughes T. (1974), A study of the one-dimensional theory of arterial pulse propagation. PhD thesis, University of California, Berkeley.
Hughes T. (1995), ‘Multiscale phenomena: Green’s function, the Dirichlet-to-Neumann formulation, subgrid scale models, bubbles and the origins of stabilized formulations’, Comput. Methods Appl. Mech. Engrg 127, 387401.
Hughes T. and Lubliner J. (1973), ‘On the one-dimensional theory of blood flow in the larger vessels’, Math. Biosci. 18, 161170.
Hughes T., Mazzei L. and Jansen K. (2000), ‘Large eddy simulation and the variational multiscale method’, Comput. Vis. Sci. 3, 4759.
Huiskamp G. (1998), ‘Simulation of depolarization in a membrane- equations-based model of the anisotropic ventricle’, IEEE Trans. Biomed. Engrg 45, 847855.
Humpherys J., Redd P. and West J. (2012), ‘A fresh look at the Kalman filter’, SIAM Review 54, 801823.
Humphrey J. and Yin F. (1987), ‘On constitutive relations and finite deformations of passive cardiac tissue, I: A pseudostrain-energy function’, J. Biomech. Engrg 109, 298304.
Hunter P., Nash M. and Sands G. (1997), Computational electromechanics of the heart. In Computational Biology of the Heart (Panfilov A. and Holden A., eds), Wiley-Blackwell, pp. 345407.
Iaizzo P. (2009), Handbook of Cardiac Anatomy, Physiology, and Devices, Springer.
Ide K., Courtier P., Ghil M. and Lorenc A. (1997), ‘Unified notation for data assimilation: Operational, sequential and variational’, J. Met. Soc. Japan 75, 181189.
Iglesias M., Law K. and Stuart A. (2013), ‘Ensemble Kalman methods for inverse problems’, Inverse Problems 29, 045001.
Ijiri T., Ashihara T., Yamaguchi T., Takayama K., Igarashi T., Shimada T., Namba T., Haraguchi R. and Nakazawa K. (2008), ‘A procedural method for modeling the Purkinje fibers of the heart’, J. Physiol. Sci. 58, 90100.
and Isaksen J., Bazilevs Y., Kvamsdal T., Zhang Y., Kaspersen J., Waterloo K., Romner B. and Ingebrigtsen T. (2008), ‘Determination of wall tension in cerebral artery aneurysms by numerical simulation’, Stroke 39, 31723178.
Johnstone R., Chang E., Bardenet R., de Boer T., Gavaghan D., Pathmanathan P., Clayton R. and Mirams G. (2016), ‘Uncertainty and variability in models of the cardiac action potential: Can we build trustworthy models?J. Molec. Cell. Cardiol. 96, 4962.
Julier S. and Uhlmann J. (2004), ‘Unscented filtering and nonlinear estimation’, Proc. IEEE 92, 401422.
Julier S., Uhlmann J. and Durrant-Whyte H. (1995), A new approach for filtering nonlinear systems. In Proceedings of the 1995 American Control Conference, Vol. 3, pp. 16281632.
Julier S., Uhlmann J. and Durrant-Whyte H. (2000), ‘A new method for the nonlinear transformation of means and covariance in filters and estimators’, IEEE Trans. Automat. Control 45, 477482.
Juntunen M. and Stenberg R. (2009), ‘Nitsche’s method for general boundary conditions’, Math. Comp. 78, 13531374.
Kaipio J. and Somersalo E. (2005), Statistical and Computational Inverse Problems, Vol. 160 of Applied Mathematical Sciences, Springer.
Kalman R. (1960), ‘A new approach to linear filtering and prediction problems’, ASME. J. Basic Engrg 82, 3545.
Karniadakis G., Israeli M. and Orszag S. (1991), ‘High order splitting methods for the incompressible Navier–Stokes equations’, J. Comput. Phys. 59, 414443.
Keener J. (1991), ‘An eikonal-curvature equation for action potential propagation in myocardium’, J. Math. Biol. 29, 629651.
Keener J. and Bogar K. (1998), ‘A numerical method for the solution of the bidomain equations in cardiac tissue’, Chaos 8, 234241.
Kefayati S., Holdsworth D. and Poepping T. (2014), ‘Turbulence intensity measurements using particle image velocimetry in diseased carotid artery models: Effect of stenosis severity, plaque eccentricity, and ulceration’, J. Biomech. 47, 253263.
Keldermann R., Nash M. and Panfilov A. (2009), ‘Modeling cardiac mechano-electrical feedback using reaction-diffusion-mechanics systems’, Physica D: Nonlinear Phenomena 238, 10001007.
Kelley C. (1999), Iterative Methods for Optimization, SIAM.
Kelly D., Law K. and Stuart A. (2014), ‘Well-posedness and accuracy of the ensemble Kalman filter in discrete and continuous time’, Nonlinearity 27, 2579.
Kennedy M. and O’Hagan A. (2000), ‘Predicting the output from a complex computer code when fast approximations are available’, Biometrika 87, 113.
Kerckoffs B., Faris O., Boveenderd P., Prinzen F., Smits K. and Arts T. (2003), ‘Timing of depolarization and contraction in the paced canine left ventricle’, J. Cardiovasc. Electrophysiol. 14, S188S195.
Keynton R., Evancho M., Sims R., Rodway N., Gobin A. and Rittgers S. (2001), ‘Intimal hyperplasia and wall shear in arterial bypass graft distal anastomoses: An in vivo model study’, J. Biomech. Engrg 123, 464473.
Khalafvand S., Zhong L. and Ng E. (2014), ‘Three-dimensional CFD/MRI modeling reveals that ventricular surgical restoration improves ventricular function by modifying intraventricular blood flow’, Int. J. Numer. Methods Biomed. Engrg 30, 10441056.
Kim H., Lu J., Sacks M. and Chandran K. (2008), ‘Dynamic simulation of bioprosthetic heart valves using a stress resultant shell model’, J. Biomech. 36, 262275.
Kim H., Vignon-Clementel I., Figueroa C., Ladisa J., Jansen K., Feinstein J. and Taylor C. (2009), ‘On coupling a lumped parameter heart model and a three-dimensional finite element aorta model’, Ann. Biomed. Engrg 37, 21532169.
Kleinstreuer C. (2006), Biofluid Dynamics: Principles and Selected Applications, CRC Press.
Kohl P. and Sachs F. (2001), ‘Mechanoelectric feedback in cardiac cells’, Philos. Trans. Royal Soc. A 359, 11731185.
Kohl P., Hunter P. and Noble D. (1999), ‘Stretch-induced changes in heart rate and rhythm: Clinical observations, experiments and mathematical models’, Prog. Biophys. Molec. Biol. 71, 91138.
Konukoglu E., Relan J., Cilingir U., Menze B., Chinchapatnam P., Jadidi A., Cochet H., Hocini M., Delingette H., Jaïs P., Haïssaguerre M., Ayache N. and Sermesant M. (2011), ‘Efficient probabilistic model personalization integrating uncertainty on data and parameters: Application to eikonal-diffusion models in cardiac electrophysiology’, Prog. Biophys. Molec. Biol. 107, 134146.
Korakianitis T. and Shi Y. (2006a), ‘A concentrated parameter model for the human cardiovascular system including heart valve dynamics and atrioventricular interaction’, Med. Engrg Phys. 28, 613628.
Korakianitis T. and Shi Y. (2006b), ‘Numerical simulation of cardiovascular dynamics with healthy and diseased heart valves’, J. Biomech. 39, 19641982.
Koutsourelakis P. (2009), ‘Accurate uncertainty quantification using inaccurate models’, SIAM J. Sci. Comput. 31, 32743300.
Krishnamurthy A., Villongco C., Chuang J., Frank L., Nigam V., Belezzuoli E., Stark P., Krummen D., Narayan S., Omens J., McCulloch A. and Kerckhoffs R. (2013), ‘Patient-specific models of cardiac biomechanics’, J. Comput. Phys. 244, 421.
Kufahl R. and Clark M. (1985), ‘A circle of Willis simulation using distensible vessels and pulsatile flow’, J. Biomech. Engrg 107, 112122.
Kuhl E. and Holzapfel G. (2006), ‘A continuum model for remodeling in living structures’, J. Mater. Sci. 21, 88118823.
Kunisch K. and Rund A. (2015), ‘Time optimal control of the monodomain model in cardiac electrophysiology’, IMA J. Appl. Math. 80, 16641683.
Kunisch K. and Vexler B. (2007), ‘Optimal vortex reduction for instationary flows based on translation invariant cost functionals’, SIAM J. Control Optim. 46, 13681397.
Kunzelman K. and Cochran R. (1990), ‘Mechanical properties of basal and marginal mitral valve chordae tendineae’, ASAIO Trans. 36, M405.
Kunzelman K., Cochran R., Chuong C., Ring W., Verrier and Eberhart R. (1993), ‘Finite element analysis of the mitral valve’, J. Heart Valve Disease 2, 326340.
Kuttler U. and Wall W. (2008), ‘Fixed-point fluid–structure interaction solvers with dynamic relaxation’, Comput. Mech. 43, 6172.
Kuttler U., Gee M., Forster C., Comerford A. and Wall W. (2010), ‘Coupling strategies for biomedical fluid–structure interaction problems’, Int. J. Numer. Methods Biomed. Engrg 26, 305321.
Laadhari A. and Quarteroni A. (2016), ‘Numerical modeling of heart valves using resistive Eulerian surfaces’, Int. J. Numer. Methods Biomed. Engrg 32, e02743.
Lal R., Mohammadi B. and Nicoud F. (2016), ‘Data assimilation for identification of cardiovascular network characteristics’, Int. J. Numer. Methods Biomed. Engrg. doi:10.1002/cnm.2824
Lancellotti R., Vergara C., Valdettaro L., Bose S. and Quarteroni A. (2015), Large eddy simulations for blood fluid-dynamics in real stenotic carotids. MOX-Report 63-2015, Department of Mathematics, Politecnico di Milano.
Land S., Niederer S., Aronsen J., Espe E., Zhang L., Louch W., Sjaastad I., Sejersted O. and Smith N. (2012), ‘An analysis of deformation-dependent electromechanical coupling in the mouse heart’, J. Physiol. 590, 45534569.
Lassila T., Manzoni A., Quarteroni A. and Rozza G. (2013a), ‘Boundary control and shape optimization for the robust design of bypass anastomoses under uncertainty’, ESAIM Math. Model. Numer. Anal. 47, 11071131.
Lassila T., Manzoni A., Quarteroni A. and Rozza G. (2013b), ‘A reduced computational and geometrical framework for inverse problems in haemodynamics’, Int. J. Numer. Methods Biomed. Engrg 29, 741776.
Lassila T., Quarteroni A. and Rozza G. (2012), ‘A reduced basis model with parametric coupling for fluid–structure interaction problems’, SIAM J. Sci. Comput. 34, A1187A1213.
Law K., Stuart A. and Zygalakis K. (2015), Data Assimilation: A Mathematical Introduction, Vol. 62 of Texts in Applied Mathematics, Springer.
Le T. and Sotiropoulos F. (2013), ‘Fluid–structure interaction of an aortic heart valve prosthesis driven by an animated anatomic left ventricle’, J. Comput. Phys. 224, 4162.
Le Dimet F.-X. and Talagrand O. (1986), ‘Variational algorithms for analysis and assimilation of meteorological observations: Theoretical aspects’, Tellus A 38, 97110.
Le Maître O. and Knio O. (2010), Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics, Springer.
Lee H. (2011), ‘Optimal control for quasi-Newtonian flows with defective boundary conditions’, Comput. Methods Appl. Mech. Engrg 200, 24982506.
Lee J., Cookson A., Roy I., Kerfoot E., Asner L., Vigueras G., Sochi T., Deparis S., Michler C., Smith N. and Nordsletten D. (2016), ‘Multiphysics computational modeling in CHeart’, SIAM J. Sci. Comput. 38, C150C178.
Lee S., Lee S., Fischer P., Bassiouny H. and Loth F. (2008), ‘Direct numerical simulation of transitional flow in a stenosed carotid bifurcation’, J. Biomech. 41, 25512561.
Legato M. (1973), ‘Ultrastructure of the atrial, ventricular, and Purkinje cell, with special reference to the genesis of arrhythmias’, Circulation 47, 178189.
Leguy C., Bosboom A., Belloum A., Hoeks A. and van de Vosse F. (2011), ‘Global sensitivity analysis of a wave propagation model for arm arteries’, Med. Engrg Phys. 33, 10081016.
Lei M., Archie J. and Kleinstreuer C. (1997), ‘Computational design of a bypass graft that minimizes wall shear stress gradients in the region of the distal anastomosis’, J. Vasc. Surg. 25, 637646.
Leiva J., Blanco P. and Buscaglia G. (2011), ‘Partitioned analysis for dimensionally-heterogeneous hydraulic networks’, Multiscale Model. Simul. 9, 872903.
Leray J. (1934), ‘Sur le mouvement d’un liquide visqueux emplissant l’espace’, Acta Math. 63, 193248.
Lesage D., Angelini E., Bloch I. and Funka-Lea G. (2009), ‘A review of 3D vessel lumen segmentation techniques: Models, features and extraction schemes’, Med. Image Anal. 13, 819845.
Leuprecht A., Perktold K., Prosi M., Berk T., Trubel W. and Schima H. (2002), ‘Numerical study of hemodynamics and wall mechanics in distal end-to-side anastomoses of bypass grafts’, J. Biomech. 2, 225236.
Leveque R. (1992), Numerical Methods for Conservation Laws, Vol. 132 of Lectures in Mathematics, Birkhäuser.
Li C. and Vuik C. (2004), ‘Eigenvalue analysis of the simple preconditioning for incompressible flow’, Numer. Linear Algebra Appl. 11, 511523.
Li D. and Robertson A. (2013), ‘A structural multi-mechanism damage model for cerebral arterial tissue’, J. Biomech. Engrg 131, 101013.
Liang F. and Liu H. (2006), ‘Simulation of hemodynamic responses to the valsalva maneuver: An integrative computational model of the cardiovascular system and the autonomic nervous system’, J. Physiol. Sci. 1, 4565.
Lions J. and Prodi G. (1959), ‘Un théorème d’existence et d’unicité dans les équations de Navier–Stokes en dimension 2’, CR Acad. Sci. Paris 248, 35193521.
Liu Y., Charles C., Gracia M., Gregersen H. and Kassab G. S. (2007), ‘Surrounding tissues affect the passive mechanics of the vessel wall: Theory and experiment’, Amer. J. Physiol.: Heart Circ. Physiol. 293, H3290H3300.
Lombardi D. (2014), ‘Inverse problems in 1D hemodynamics on systemic networks: A sequential approach’, Int. J. Numer. Methods Biomed. Engrg 30, 160179.
Lorenzo-Valdés M., Sanchez-Ortiz G., Elkington A., Mohiaddin R. and Rueckert D. (2004), ‘Segmentation of 4D cardiac MR images using a probabilistic atlas and the EM algorithm’, Med. Image Anal. 8, 255265.
Loth F., Fischer P. and Bassiouny H. (2008), ‘Blood flow in end-to-side anastomoses’, Annu. Rev. Fluid. Mech. 40, 367393.
Luo C. and Rudy Y. (1991), ‘A model of the ventricular cardiac action potential: Depolarization, repolarization, and their interaction’, Circ. Res. 68, 15011526.
Luo C. and Rudy Y. (1994a), ‘A dynamic model of the cardiac ventricular action potential, I: Simulations of ionic currents and concentration changes’, Circ. Res. 74, 10711096.
Luo C. and Rudy Y. (1994b), ‘A dynamic model of the cardiac ventricular action potential, II: Afterdepolarizations, triggered activity, and potentiation’, Circ. Res. 74, 10971113.
Lykaser M. and Nielsen B. (2006), ‘Towards a level set framework for infarction modeling: An inverse problem’, Int. J. Numer. Anal. Model. 3, 377394.
Maclachlan M., Nielsen B., Lysaker M. and Tveito A. (2006), ‘Computing the size and location of myocardial ischemia using measurements of ST-segment shift’, IEEE Trans. Biomed. Engrg 53, 10241031.
Maday Y. (2009), Analysis of coupled models for fluid–structure interaction of internal flows. (Quarteroni A., Formaggia L. and Veneziani A., eds), Chapter 8 in Cardiovascular Mathematics , Springer, pp. 279306.
Mahmoud A., El-Barkouky A., Farag H., Graham J. and Farag A. (2013), A non-invasive method for measuring blood flow rate in superficial veins from a single thermal image. In 2013 IEEE Conference on Computer Vision and Pattern Recognition Workshops (CVPRW), IEEE, pp. 354359.
Malossi A. (2012), Partitioned solution of geometrical multiscale problems for the cardiovascular system: Models, algorithms, and applications. PhD thesis, Ecole Polytechnique Fédérale de Lausanne. Thesis 5453.
Malossi A., Blanco P. and Deparis S. (2012), ‘A two-level time step technique for the partitioned solution of one-dimensional arterial networks’, Comput. Methods Appl. Mech. Engrg 237–240, 212226.
Malossi A., Blanco P., Crosetto P., Deparis S. and Quarteroni A. (2013), ‘Implicit coupling of one-dimensional and three-dimensional blood flow models with compliant vessels’, Multiscale Model. Simul. 11, 474506.
Mansi T., Voigt I., Georgescu B., Zheng X., Mengue E., Hackl M., Ionasec R., Noack T., Seeburger J. and Comaniciu D. (2012), ‘An integrated framework for finite-element modeling of mitral valve biomechanics from medical images: Application to MitralClip intervention planning’, Med. Image Anal. 16, 13301346.
Manzoni A. (2012), Reduced models for optimal control, shape optimization and inverse problems in haemodynamics. PhD thesis, Ecole Polytechnique Fédérale de Lausanne.
Manzoni A., Lassila T., Quarteroni A. and Rozza G. (2014), A reduced-order strategy for solving inverse Bayesian shape identification problems in physiological flows. In Modeling, Simulation and Optimization of Complex Processes, HPSC 2012: Proceedings of the Fifth International Conference on High Performance Scientific Computing (Bock G., Hoang P., Rannacher R. and Schlöder P., eds), Springer, pp. 145155.
Manzoni A., Pagani S. and Lassila T. (2016), ‘Accurate solution of Bayesian inverse uncertainty quantification problems combining reduced basis methods and reduction error models’, SIAM/ASA J. Uncert. Quant. 4, 380412.
Manzoni A., Quarteroni A. and Rozza G. (2012a), ‘Model reduction techniques for fast blood flow simulation in parametrized geometries’, Int. J. Numer. Methods Biomed. Engrg 28, 604625.
Manzoni A., Quarteroni A. and Rozza G. (2012b), ‘Shape optimization of cardiovascular geometries by reduced basis methods and free-form deformation techniques’, Int. J. Numer. Methods Fluids 70, 646670.
Marchesseau S., Delingette H., Sermesant M. and Ayache N. (2012), ‘Fast parameter calibration of a cardiac electromechanical model from medical images based on the unscented transform’, Biomech. Model. Mechanobiol. 12, 815831.
Marchesseau S., Delingette H., Sermesant M., Cabrera-Lozoya R., Tobon-Gomez C., Moireau P., Figueras i Ventura R., Lekadir K., Hernandez A., Garreau M., Donal E., Leclercq C., Duckett S., Rhode K., Rinaldi C., Frangi A., Razavi R., Chapelle D. and Ayache N. (2013), ‘Personalization of a cardiac electromechanical model using reduced order unscented Kalman filtering from regional volumes’, Med. Image Anal. 17, 816829.
Mardal K., Nielsen B., Cai X. and Tveito A. (2007), ‘Semi-implicit time-discretization schemes for the bidomain modelan order optimal solver for the discretized bidomain equations’, Numer. Linear Algebra Appl. 14, 8398.
Margaris K. and Black R. (2012), ‘Modelling the lymphatic system: Challenges and opportunities’, J. Royal Soc. Interface 69, 601612.
Marom G. (2015), ‘Numerical methods for fluid–structure interaction models of aortic valves’, Arch. Comput. Methods Engrg 22, 595620.
Marsden A. (2014), ‘Optimization in cardiovascular modeling’, Annu. Rev. Fluid Mech. 46, 519546.
Marsden A., Feinstein J. and Taylor C. (2008), ‘A computational framework for derivative-free optimization of cardiovascular geometries’, Comput. Methods Appl. Mech. Engrg 197, 18901905.
Martin V., Clément F., Decoene A. and Gerbeau J.-F. (2005), ‘Parameter identification for a one-dimensional blood flow model’, ESAIM Proc. 14, 174200.
Matthys K., Alastruey J., Peiró J., Khir A., Segers P., Verdonck P., Parker K. and Sherwin S. (2007), ‘Pulse wave propagation in a model human arterial network: Assessment of 1-D numerical simulations against in vitro measurements’, J. Biomech. 40, 34763486.
Maury B. (2013), The Respiratory System in Equations, Springer.
May-Newman K. and Yin F. (1998), ‘A constitutive law for mitral valve tissue’, J. Biomech. Engrg 120, 3847.
May-Newman K., Lam C. and Yin F. (2009), ‘A hyperelastic constitutive law for aortic valve tissue’, J. Biomech. Engrg 131, 081009.
Merryman W., Huang H., Schoen F. and Sacks M. (2006), ‘The effects of cellular contraction on aortic valve leaflet flexural stiffness’, J. Biomech. 39, 8896.
Migliavacca F. and Dubini G. (2005), ‘Computational modeling of vascular anastomoses’, Biomech. Model. Mechanobiol. 3, 235250.
Migliavacca F., Balossino R., Pennati G., Dubini G., Hsia T., de Leval M. and Bove E. (2006), ‘Multiscale modelling in biofluidynamics: Application to reconstructive paediatric cardiac surgery’, J. Biomech. 39, 10101020.
Mittal R. and Iaccarino G. (2005), ‘Immersed boundary methods’, Annu. Rev. Fluid Mech. 37, 239261.
Mittal R., Seo J., Vedula V., Choi Y., Liu H., Huang H., Jain S., Younes L., Abraham T. and George R. (2016), ‘Computational modeling of cardiac hemodynamics: Current status and future outlook’, J. Comput. Phys. 305, 10651082.
Moireau P. and Chapelle D. (2011), ‘Reduced-order unscented Kalman filtering with application to parameter identification in large-dimensional systems’, ESAIM Control Optim. Calc. Var. 17, 380405.
Moireau P., Bertoglio C., Xiao N., Figueroa C., Taylor C., Chapelle D. and Gerbeau J.-F. (2013), ‘Sequential identification of boundary support parameters in a fluid–structure vascular model using patient image data’, Biomech. Model. Mechanobiol. 12, 475496.
Tallec Moireau P., Chapelle D. and Le P. (2008), ‘Joint state and parameter estimation for distributed mechanical systems’, Comput. Methods Appl. Mech. Engrg 197, 659677.
Tallec Moireau P., Chapelle D. and Le P. (2009), ‘Filtering for distributed mechanical systems using position measurements: perspectives in medical imaging’, Inverse Problems 25, 035010.
Moireau P., Xiao N., Astorino M., Figueroa C., Chapelle D., Taylor C. and Gerbeau J.-F. (2012), ‘External tissue support and fluid–structure simulation in blood flows’, Biomech. Model. Mechanobiol. 11, 118.
Moore J., Steinman D. and Ethier C. (1997), ‘Computational blood flow modelling: Errors associated with reconstructing finite element models from magnetic resonance images’, J. Biomech. 31, 179184.
Moradkhani H., Sorooshian S., Gupta H. and Houser P. (2005), ‘Dual state-parameter estimation of hydrological models using ensemble Kalman filter’, Adv. Water Resour. 28, 135147.
Morbiducci U., Ponzini R., Rizzo G., Cadioli M., Esposito A., De Cobelli F., Del Maschio A., Montevecchi F. and Redaelli A. (2009), ‘ In vivo quantification of helical blood flow in human aorta by time-resolved three-dimensional cine phase contrast magnetic resonance imaging’, Ann. Biomed. Engrg 37, 516531.
Mozaffarian D., Benjamin E., Go A., Arnett D., Blaha M., Cushman M., Das S., de Ferranti S., Després J.-P., Fullerton H., Howard V., Huffman M., Isasi C., Jiménez M., Judd S., Kissela B., Lichtman J., Lisabeth L., Liu S., Mackey R., Magid D., McGuire D., Mohler E., Moy C., Muntner P., Mussolino M., Nasir K., Neumar R., Nichol G., Palaniappan L., Pandey D., Reeves M., Rodriguez C., Rosamond W., Sorlie P., Stein J., Towfighi A., Turan T., Virani S., Woo D., Yeh R. and Turner M. (2015), ‘Heart disease and stroke statistics: 2016 update’, Circulation 133, 447454.
Muller J., Sahni O., Lia X., Jansen K., Shephard M. and Taylor C. (2005), ‘Anisotropic adaptive finite element method for modelling blood flow’, Comput. Methods Biomech. Biomed. Engrg 8, 295305.
Muller L. and Toro E. (2013), ‘Well-balanced high-order solver for blood flow in networks of vessels with variable properties’, Int. J. Numer Methods Biomed. Engrg 29, 13881411.
Muller L. and Toro E. (2014), ‘A global multiscale mathematical model for the human circulation with emphasis on the venous system’, Int. J. Numer. Methods Biomed. Engrg 30, 681725.
Munteanu M., Pavarino L. and Scacchi S. (2009), ‘A scalable Newton–Krylov–Schwarz method for the bidomain reaction-diffusion system’, SIAM J. Sci. Comput. 31, 38613883.
Murillo M. and Cai X. (2004), ‘A fully implicit parallel algorithm for simulating the non-linear electrical activity of the heart’, Numer. Linear Algebra Appl. 2/3, 261277.
Muszkiewicz A., Britton O., Gemmell P., Passini E., Sánchez C., Zhou X., Carusi A., Quinn T., Burrage K., Bueno-Orovio A. and Rodriguez B. (2016), ‘Variability in cardiac electrophysiology: Using experimentally-calibrated populations of models to move beyond the single virtual physiological human paradigm’, Prog. Biophys. Molec. Biol. 120, 115127.
Nagaiah C., Kunisch K. and Plank G. (2011), ‘Numerical solution for optimal control of the reaction-diffusion equations in cardiac electrophysiology’, Comput. Optim. Appl. 49, 149178.
Nagaiah C., Kunisch K. and Plank G. (2013a), ‘On boundary stimulation and optimal boundary control of the bidomain equations’, Math. Biosci. 245, 206215.
Nagaiah C., Kunisch K. and Plank G. (2013b), ‘Optimal control approach to termination of re-entry waves in cardiac electrophysiology’, J. Math. Biol. 67, 130.
Nagaiah C., Kunisch K. and Plank G. (2016), ‘PDE constrained optimization of electrical defibrillation in a 3D ventricular slice geometry’, Int. J. Numer. Methods Biomed. Engrg 32, e02742.
Nagler A., Bertoglio C., Gee M. and Wall W. (2013), Personalization of cardiac fiber orientations from image data using the unscented Kalman filter. In Functional Imaging and Modeling of the Heart, (Ourselin S., Rueckert D. and Smith N., eds), Vol. 7945 of Lecture Notes in Computer Science, Springer, pp. 132140.
Nagler A., Bertoglio C., Stoeck C., Kozerke S. and Wall W. (2015), Cardiac fibers estimation from arbitrarily spaced diffusion weighted MRI. In Functional Imaging and Modeling of the Heart: FIMH 2015, (van Assen H. et al. , ed.), Vol. 9126 of Lecture Notes in Computer Science, Springer.
Nash M. and Panfilov A. (2004), ‘Electromechanical model of excitable tissue to study reentrant cardiac arrhythmias’, Prog. Biophys. Molec. Biol. 2/3, 501522.
Negri F. (2016), Efficient reduction techniques for the simulation and optimization of parametrized systems: Analysis and applications. PhD thesis, Ecole Polytechnique Fédérale de Lausanne.
Nestola M., Faggiano E., Vergara C., Lancellotti R., Ippolito S., Antona C., Quarteroni A. and Scrofani R. (2017), ‘Computational comparison of aortic root stresses in presence of stentless and stented aortic valve bio-prostheses’, Comput. Methods Biomech. Biomed. Engrg 20, 171181.
Newmark N. (1959), ‘A method of computation for structural dynamics’, J. Engrg Mech. 85, 6794.
Nichols N. (2010), Mathematical concepts of data assimilation. In Data Assimilation: Making Sense of Observations (Lahoz W., Khattatov B. and Menard R., eds), Springer, pp. 1339.
Nichols W. & O’Rourke M. (Eds) (2005), McDonald’s Blood Flow in Arteries, Hodder Arnold.
Niederer S. and Smith N. (2007), ‘A mathematical model of the slow force response to stretch in rat ventricular myocytes’, Biophys. J. 92, 40304044.
Niederer S. and Smith N. (2008), ‘An improved numerical method for strong coupling of excitation and contraction models in the heart’, Prog. Biophys. Molec. Biol. 96, 90111.
Niederer S., Hunter P. and Smith N. (2006), ‘A quantitative analysis of cardiac myocyte relaxation: A simulation study’, Biophysical Journal 90, 16971722.
Niederer S., Kerfoot E., Benson A., Bernabeu M., Bernus O., Bradley C., Cherry E., Clayton R., Fenton F. and Garny A. et al. (2011), ‘Verification of cardiac tissue electrophysiology simulators using an $N$ -version benchmark’, Phil. Trans. R. Soc. A 369, 43314351.
Nielsen B., Cai X. and Lykaser M. (2007a), ‘On the possibility for computing the transmembrane potential in the heart with a one shot method: An inverse problem’, Math. Biosciences 210, 523553.
Nielsen B., Lykaser M. and Tveito A. (2007b), ‘On the use of the resting potential and level set methods for identifying ischemic heart disease: An inverse problem’, J. Comput. Phys. 220, 772790.
Nitsche J. (1970/71), ‘Uber ein Variationsprinzip zur lösung von Dirichlet-Problemen bei verwendung von Teilräumen, die keinen Randbedingungen unterworfen sind’, Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg 36, 915.
Nobile F. (2001), Numerical approximation of fluid–structure interaction problems with application to haemodynamics. PhD thesis, Ecole Polytechnique Fédérale de Lausanne. Thesis 2458.
Nobile F. and Vergara C. (2008), ‘An effective fluid–structure interaction formulation for vascular dynamics by generalized Robin conditions’, SIAM J. Sci. Comput. 30, 731763.
Nobile F. and Vergara C. (2012), ‘Partitioned algorithms for fluid–structure interaction problems in haemodynamics’, Milan J. Math. 80, 443467.
Nobile F., Pozzoli M. and Vergara C. (2013), ‘Time accurate partitioned algorithms for the solution of fluid–structure interaction problems in haemodynamics’, Comput. Fluids 86, 470482.
Nobile F., Pozzoli M. and Vergara C. (2014), ‘Inexact accurate partitioned algorithms for fluid–structure interaction problems with finite elasticity in haemodynamics’, J. Comput. Phys. 273, 598617.
Nobile F., Tempone R. and Webster C. (2008), ‘A sparse grid stochastic collocation method for partial differential equations with random input data’, SIAM J. Numer. Anal. 46, 23092345.
Nocedal J. (1992), Theory of algorithms for unconstrained optimization. In Acta Numerica, Vol. 1, Cambridge University Press, pp. 199242.
Nordsletten D., Mccormick M., Kilner P., Hunter P., Kayand D. and Smith N. (2011a), ‘Fluid–solid coupling for the investigation of diastolic and systolic human left ventricular function’, Int. J. Numer. Methods Biomed. Engrg 27, 10171039.
Nordsletten D., Niederer S., Nash M., Hunter P. and Smith N. (2011b), ‘Coupling multi-physics models to cardiac mechanics’, Prog. Biophys. Molec. Biol. 104, 7788.
Oberai A., Gokhale N. and Feijóo G. (2003), ‘Solution of inverse problems in elasticity imaging using the adjoint method’, Inverse Problems 19, 297313.
O’Donnell T., Jolly M. and Gupta A. (1998), ‘A cooperative framework for segmentation using 2D active contours and 3D hybrid models as applied to branching cylindrical structures’, Proc. IEEE Int. Conf. Computer Vision 454459.
Olufsen M., Peskin C., Kim W., Pedersen E., Nadim A. and Larsen J. (2000), ‘Numerical simulation and experimental validation of blood flow in arteries with structured-tree outflow conditions’, Ann. Biomed. Engrg 28, 12811299.
Orszag S., Israeli M. and Deville M. (1986), ‘Boundary conditions for incompressible flows’, J. Sci. Comput. 1, 75111.
Osnes H. and Sundnes J. (2012), ‘Uncertainty analysis of ventricular mechanics using the probabilistic collocation method’, IEEE Trans. Biomed. Engrg 59, 21712179.
Owida A., Do H. and Morsi Y. (2012), ‘Numerical analysis of coronary artery bypass grafts: An over view’, Comput. Methods Programs Biomed. 108, 689705.
Padala M., Sacks M., Liou S., Balachandran K., He Z. and Yoganathan A. (2010), ‘Mechanics of the mitral valve strut chordae insertion region’, J. Biomech. Engrg 132, 081004.
Pagani S. (2016), Reduced order models for inverse problems and uncertainty quantification in cardiac electrophysiology. PhD thesis, Mathematical Models and Methods in Engineering, Department of Mathematics, Politecnico di Milano.
Pagani S., Manzoni A. and Quarteroni A. (2016), A reduced basis ensemble Kalman filter for state/parameter identification in large-scale nonlinear dynamical systems. MOX-Report 24-2016, Department of Mathematics, Politecnico di Milano.
Palamara S., Vergara C., Catanzariti D., Faggiano E., Centonze M., Pangrazzi C., Maines M. and Quarteroni A. (2014), ‘Computational generation of the Purkinje network driven by clinical measurements: The case of pathological propagations’, Int. J. Numer. Meth. Biomed. Engrg 30, 15581577.
Palamara S., Vergara C., Faggiano E. and Nobile F. (2015), ‘An effective algorithm for the generation of patient-specific Purkinje networks in computational electrocardiology’, J. Comput. Phys. 283, 495517.
Panfilov A. (1999), ‘Three-dimensional organization of electrical turbulence in the heart’, Phys. Rev. E 59, R6251R6254.
Pant S., Fabrèges B., Gerbeau J.-F. and Vignon-Clementel I. (2014), ‘A methodological paradigm for patient-specific multi-scale CFD simulations: From clinical measurements to parameter estimates for individual analysis’, Int. J. Numer. Meth. Biomed. Engrg 30, 16141648.
Papadakis G. (2009), ‘Coupling 3D and 1D fluid–structure-interaction models for wave propagation in flexible vessels using a finite volume pressure-correction scheme’, Comm. Numer. Methods Engrg 25, 533551.
Patankar S. and Spalding D. (1972), ‘A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows’, Int. J. Heat Mass Transfer 15, 17871806.
Pathmanathan P., Chapman S., Gavaghan D. and Whiteley J. (2010), ‘Cardiac electromechanics: The effect of contraction model on the mathematical problem and accuracy of the numerical scheme’, Quart. J. Mech. Appl. Math. 63, 375399.
Pathmanathan P., Mirams G., Southern J. and Whiteley J. (2011), ‘The significant effect of the choice of ionic current integration method in cardiac electro-physiological simulations’, Int. J. Numer. Methods Biomed. Engrg 27, 17511770.
Pavarino L. and Scacchi S. (2008), ‘Multilevel additive Schwarz preconditioners for the bidomain reaction-diffusion system’, SIAM J. Sci. Comput 31, 420443.
Pavarino L., Scacchi S. and Zampini S. (2015), ‘Newton–Krylov–BDDC solvers for nonlinear cardiac mechanics’, Comput. Methods Appl. Mech. Engrg 295, 562580.
Pedley T. (1980), The Fluid Mechanics of Large Blood Vessels, Cambridge Monographs on Mechanics and Applied Mathematics, Cambridge University Press.
Peiró J., Veneziani A., Quarteroni A., Formaggia L. and Veneziani A. (2009), Reduced models of the cardiovascular system. Chapter 10 in Cardiovascular Mathematics , Springer, pp. 347394.
Peiró J., Formaggia L., Gazzola M., Radaelli A. and Rigamonti V. (2007), ‘Shape reconstruction from medical images and quality mesh generation via implicit surfaces’, Int. J. Numer. Methods Fluids 53, 13391360.
Pennacchio M. and Simoncini V. (2002), ‘Efficient algebraic solution of reaction-diffusion systems for the cardiac excitation process’, J. Comput. Appl. Math. 145, 4970.
Perdikaris P. and Karniadakis G. (2015), ‘Model inversion via multi-fidelity Bayesian optimization: A new paradigm for parameter estimation in haemodynamics, and beyond’, J. Royal Soc. Interface 13, 20151107.
Perego M., Veneziani A. and Vergara C. (2011), ‘A variational approach for estimating the compliance of the cardiovascular tissue: An inverse fluid–structure interaction problem’, SIAM J. Sci. Comput. 33, 11811211.
Perktold K. and Hilbert D. (1986), ‘Numerical simulation of pulsatile flow in a carotid bifurcation model’, J. Biomed. Engrg 8, 193199.
Perktold K., Thurner E. and Kenner T. (1994), ‘Flow and stress characteristics in rigid walled and compliant carotid artery bifurcation models’, Med. Biol. Engrg Comput. 32, 1926.
Perona P. and Malik J. (1990), ‘Scale-space and edge detection using anisotropic diffusion’, IEEE Trans. Pattern Anal. Mach. Intel. 12, 629639.
Peskin C. (1972), ‘Flow patterns around heart valves: A numerical method’, J. Comput. Phys. 10, 252271.
Peskin C. (1989), ‘Fiber architecture of the left ventricular wall: An asymptotic analysis’, Comm. Pure Appl. Math. 42, 11261131.
Peskin C. (2002), The immersed boundary method. In Acta Numerica, Vol. 11, Cambridge University Press, pp. 479517.
Petitjean C. and Dacher J.-N. (2011), ‘A review of segmentation methods in short axis cardiac MR images’, Med. Image Anal. 15, 169184.
Pezzuto S. (2013), Mechanics of the heart: Constitutive issues and numerical experiments. PhD thesis, Department of Mathematics, Politecnico di Milano.
Pham D. (2001), ‘Stochastic methods for sequential data assimilation in strongly nonlinear systems’, Monthly Weather Rev. 129, 11941207.
Pham D., Xu C. and Prince J. (2000), ‘Current methods in medical image segmentation’, Annu. Rev. Biomed. Engrg 2, 315337.
Piccinelli M., Vergara C., Antiga L., Forzenigo L., Biondetti P. and Domanin M. (2013), ‘Impact of hemodynamics on lumen boundary displacements in abdominal aortic aneurysms by means of dynamic computed tomography and computational fluid dynamics’, Biomech. Model. Mechanobiol 12, 12631276.
Pierre C. (2012), ‘Preconditioning the bidomain model with almost linear complexity’, J. Comput. Phys. 231, 8297.
Piperno S. and Farhat C. (2001), ‘Partitioned prodecures for the transient solution of coupled aeroelastic problems, II: Energy transfer analysis and three-dimensional applications’, Comput. Methods Appl. Mech. Engrg 190, 31473170.
Plank G., Liebmann M., dos Santos R., Vigmond E. and Haase G. (2007), ‘Algebraic multigrid preconditioner for the cardiac bidomain model’, IEEE Trans. Biomed. Engrg 54, 585596.
Ponzini R., Vergara C., Redaelli A. and Veneziani A. (2006), ‘Reliable CFD-based estimation of flow rate in haemodynamics measures’, Ultrasound Med. Biol. 32, 15451555.
Ponzini R., Vergara C., Rizzo G., Veneziani A., Roghi A., Vanzulli A., Parodi O. and Redaelli A. (2010), ‘Womersley number-based estimates of blood flow rate in Doppler analysis: In vivo validation by means of phase contrast magnetic resonance imaging’, IEEE Trans. Biomed. Engrg 57, 18071815.
Porpora A., Zunino P., Vergara C. and Piccinelli M. (2012), ‘Numerical treatment of boundary conditions to replace lateral branches in haemodynamics’, Int. J. Numer. Methods Biomed. Engrg 28, 11651183.
Potse M., Dubé B., Richer J., Vinet A. and Gulrajani R. (2006), ‘A comparison of monodomain and bidomain reaction-diffusion models for action potential propagation in the human heart’, IEEE Trans. Biomed. Engrg 53, 24252435.
Pravdin S., Berdyshev V., Panfilov A., Katsnelson L., Solovyova O. and Markhasin V. (1989), ‘Mathematical model of the anatomy and fibre orientation field of the left ventricle of the heart’, Biomed. Engrg OnLine 12, 54.
Prodi G. (1962), ‘Teoremi di tipo locale per il sistema di Navier–Stokes e stabilitá delle soluzioni stazionarie’, Rendiconti del Seminario Matematico della Universitá di Padova 32, 374397.
Prot V., Skallerud B. and Holzapfel G. (2007), ‘Transversely isotropic membrane shells with application to mitral valve mechanics. constitutive modelling and finite element implementation’, Int. J. Numer. Methods Engrg 71, 9871008.
Pullan A., Cheng L., Nash M., Bradley C. and Paterson D. (2001), ‘Noninvasive electrical imaging of the heart: Theory and model development’, Ann. Biomed. Engrg 29, 817836.
Pullan A., Cheng L., Nash M., Ghodrati A., Macleod R. and Brooks D. (2010), The inverse problem of electrocardiography. In Comprehensive Electrocardiology (Macfarlane P. et al. , ed.), Springer, pp. 299344.
Puwal S. and Roth B. (2007), ‘Forward Euler stability of the bidomain model of cardiac tissue’, IEEE Trans. Biomed. Engrg 5, 951953.
Qu Z. and Garfinkel A. (1998), ‘An advanced algorithm for solving partial differential equation in cardiac conduction’, IEEE Trans. Biomed. Engrg 46, 11661168.
Quarteroni A. (Ed.) (2015), Modeling the Heart and the Circulatory System, Springer.
, Quarteroni A. and Formaggia L. (2004), Mathematical modelling and numerical simulation of the cardiovascular system. In Handbook of Numerical Analysis, Vol. 12: Computational Models for the Human Body , Elsevier, pp. 3127.
Quarteroni A. and Rozza G. (2003), ‘Optimal control and shape optimization of aorto-coronaric bypass anastomoses’, Math. Models Methods Appl. Sci. 13, 18011823.
Quarteroni A. & Rozza G. (Eds) (2014), Reduced Order Methods for Modeling and Computational Reduction, Vol. 9 of Modeling, Simulation and Applications, Springer.
Quarteroni A. and Valli A. (1994), Numerical Approximation of Partial Differential Equations, Springer.
Quarteroni A. and Veneziani A. (1997), Modeling and simulation of blood flow problems. In Computational Science for the 21st Century (Bristeau M.-O. et al. , ed.), Wiley.
Quarteroni A. and Veneziani A. (2003), ‘Analysis of a geometrical multiscale model based on the coupling of ODE and PDE for blood flow simulations’, Multiscale Model. Simul. 1, 173195.
Quarteroni A., Lassila T., Rossi S. and Ruiz-Baier R. (2017), ‘Integrated heart-coupling multiscale and multiphysics models for the simulation of the cardiac function’, Comput. Methods Appl. Mech. Engrg 314, 345407.
Quarteroni A., Manzoni A. and Negri F. (2016), Reduced Basis Methods for Partial Differential Equations. An Introduction, Vol. 92 of Unitext, Springer.
Quarteroni A., Ragni S. and Veneziani A. (2001), ‘Coupling between lumped and distributed models for blood flow problems’, Comput. Vis. Sci. 4, 111124.
Quarteroni A., Sacco R. and Saleri F. (2000a), Numerical Mathematics, Springer.
Quarteroni A., Saleri F. and Veneziani A. (1999), ‘Analysis of the Yosida method for the incompressible Navier–Stokes equations’, J. Math. Pures Appl. 78, 473503.
Quarteroni A., Saleri F. and Veneziani A. (2000b), ‘Factorization methods for the numerical approximation of Navier–Stokes equations’, Comput. Methods Appl. Mech. Engrg 188, 505526.
Quarteroni A., Tuveri M. and Veneziani A. (2000c), ‘Computational vascular fluid dynamics: Problems, models and methods’, Comput. Vis. Sci. 2, 163197.
Quarteroni A., Veneziani A. and Vergara C. (2016c), ‘Geometric multiscale modeling of the cardiovascular system, between theory and practice’, Comput. Methods Appl. Mech. Engrg 302, 193252.
Querzoli G., Fortini S. and Cenedese A. (2010), ‘Effect of the prosthetic mitral valve on vortex dynamics and turbulence of the left ventricular flow’, Phys. Fluids 22, 041901.
Raghavan M. and Vorp D. (2000), ‘Towards a biomechanical tool to evaluate rupture potential of abdominal aortic aneurysm: Identification of a finite strain constitutive model and evaluation of its applicability’, J. Biomech. 33, 475482.
Rannacher R. (1992), On Chorin’s Projection Method for Incompressible Navier–Stokes Equations, Vol. 1530 of Lecture Notes in Mathematics, Springer, pp. 167183.
Raya S. and Udupa J. (1990), ‘Shape-based interpolation of multidimensional objects’, IEEE Trans. Med. Imaging 9, 3242.
Rayz V., Berger S. and Saloner D. (2007), ‘Transitional flows in arterial fluid dynamics’, Comput. Methods Appl. Mech. Engrg 196, 30433048.
Rees T., Dollar H. and Wathen A. (2010), ‘Optimal solvers for PDE-constrained optimization’, SIAM J. Sci. Comput. 32, 271298.
Relan J., Chinchapatnam P., Sermesant M., Rhode K., Ginks M., Delingette H., Rinaldi C., Razavi R. and Ayache N. (2011), ‘Coupled personalization of cardiac electrophysiology models for prediction of ischaemic ventricular tachycardia’, Interface Focus 1, 396407.
Reymond P., Merenda F., Perren F., Rufenacht D. and Stergiopulos N. (2009), ‘Validation of a one-dimensional model of the systemic arterial tree’, Amer. J. Physiol.: Heart Circ. Physiol. 297, H208H222.
Robert C. and Casella G. (2004), Monte Carlo Statistical Methods, second edition, Springer.
Robertson A., Sequeira A. and Owens R. (2009), Rheological models for blood. (Formaggia L., Quarteroni A. and Veneziani A., eds), Chapter 6 in Cardiovascular Mathematics , Springer, pp. 211241.
Robertson D., Yuan J., Wang G. and Vannier M. (1997), ‘Total hip prosthesis metal-artifact suppression using iterative deblurring reconstruction’, J. Comput. Assist. Tomogr. 21, 293298.
Rogers J. and Mcculloch A. (1994), ‘A collocation-Galerkin finite element model of cardiac action potential propagation’, IEEE Trans. Biomed. Engrg 41, 743757.
Rossi S. (2014), Anisotropic modeling of cardiac mechanical activation. PhD thesis, Ecole Polytechnique Fédérale de Lausanne.
Rossi S., Lassila T., Ruiz-Baier R., Sequeira A. and Quarteroni A. (2014), ‘Thermodynamically consistent orthotropic activation model capturing ventricular systolic wall thickening in cardiac electromechanics’, Europ. J. Mech. A: Solids 48, 129142.
Roth B. (1991), ‘Action potential propagation in a thick strand of cardiac muscle’, Circ. Res. 68, 162173.
Roth B. (1997), ‘Electrical conductivity values used with the bidomain model of cardiac tissue’, IEEE Trans. Biomed. Engrg 44, 326328.
Rousseau O. (2010), Geometrical modeling of the heart. PhD thesis, Université d’Ottawa.
Rudy Y. and Messinger-Rapport B. (1988), ‘The inverse problem in electrocardiography: Solutions in terms of epicardial potentials’, Crit. Rev. Biomed. Engrg 16, 215268.
Rudy Y. and Silva J. (2006), ‘Computational biology in the study of cardiac ion channels and cell electrophysiology’, Quart. Rev. Biophys. 39, 57116.
Saltelli A., Ratto M., Andres T., Campolongo F., Cariboni J., Gatelli D., Salsana M. and Tarantola S. (2008), Global Sensitivity Analysis: The Primer, Wiley.
Sankaran S. and Marsden A. (2010), ‘The impact of uncertainty on shape optimization of idealized bypass graft models in unsteady flow’, Phys. Fluids 22, 121902.
Sankaran S. and Marsden A. (2011), ‘A stochastic collocation method for uncertainty quantification and propagation in cardiovascular simulations’, J. Biomech. Engrg 133, 031001.
Sankaran S., Grady L. and Taylor C. (2015), ‘Impact of geometric uncertainty on hemodynamic simulations using machine learning’, Comput. Methods Appl. Mech. Engrg 297, 167190.
Sankaran S., Kim H., Choi G. and Taylor C. (2016), ‘Uncertainty quantification in coronary blood flow simulations: Impact of geometry, boundary conditions and blood viscosity’, J. Biomech. 49, 25402547.
dos Santos R., Plank G., Bauer S. and Vigmond E. (2005), Preconditioning techniques for the bidomain equations. In Domain Decomposition Methods in Science and Engineering, (Kornhuber R. et al. , ed.), Vol. 40 of Lecture Notes in Computational Science and Engineering, Springer, pp. 571580.
Särkkä S. (2013), Bayesian Filtering and Smoothing, Cambridge University Press.
Savader S., Lund G. and Osterman F. (1997), ‘Volumetric evaluation of blood flow in normal renal arteries with a Doppler flow wire: A feasibility study’, J. Vasc. Intervent. Radiol. 8, 209214.
Sazonov I., Yeo S., Bevan R., Xie X., van Loon R. and Nithiarasu P. (2011), ‘Modelling pipeline for subject-specific arterial blood flow: A review’, Int. J. Numer. Methods Biomed. Engrg 27, 18681910.
Schiavazzi D., Arbia G., Baker C., Hlavacek A., Hsia T., Marsden A., Vignon-Clementel I. and the Modeling Of Congenital Hearts Alliance (MOCHA) investigators (2016), ‘Uncertainty quantification in virtual surgery hemodynamics predictions for single ventricle palliation’, Int. J. Numer. Methods Biomed. Engrg 32, 02737.
Schilling R., Peters N. and Davies D. (1998), ‘Simultaneous endocardial mapping in the human left ventricle using a noncontact catheter comparison of contact and reconstructed electrograms during sinus rhythm’, Circulation 98, 887898.
Sebastian R., Zimmerman V., Romero D. and Frangi A. (2011), ‘Construction of a computational anatomical model of the perpheral cardiac conduction system’, IEEE Trans. Biomed. Engrg 58, 90100.
Seo J., Vedula V., Abraham T., Lardo A., Dawoud F., Luo H. and Mittal R. (2014), ‘Effect of the mitral valve on diastolic flow patterns’, Phys. Fluids 26, 121901.
Sermesant M., Chabiniok R., Chinchapatnam P., Mansi T., Billet F., Moireau P., Peyrat J., Wong K., Relan J., Rhode K., Ginks M., Lambiase P., Delingette H., Sorine M., Rinaldi C., Chapelle D., Razavi R. and Ayache N. (2012), ‘Patient-specific electromechanical models of the heart for the prediction of pacing acute effects in CRT: A preliminary clinical validation’, Med. Image Anal. 16, 201215.
Sermesant M., Moireau P., Camara O., Sainte-Marie J., Andriantsimiavona R., Cimrman R., Hill D., Chapelle D. and Razavi R. (2006), ‘Cardiac function estimation from MRI using a heart model and data assimilation: Advances and difficulties’, Med. Image Anal. 10, 642656.
Sethian J. (1999), Level Set Methods and Fast Marching Methods, Cambridge University Press.
Sherwin S., Formaggia L., Peiró J. and Franke V. (2003a), ‘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, 673700.
Sherwin S., Franke V., Peiró J. and Parker K. (2003b), ‘One-dimensional modelling of a vascular network in space-time variables’, J. Engrg Math. 47, 217259.
Simon D. (2006), Optimal State Estimation: Kalman, H-Infinity, and Nonlinear Approaches, Wiley.
Smith N., Nickerson D., Crampin E. and Hunter P. (2004), Multiscale computational modelling of the heart. Acta Numerica, Vol. 13, Cambridge University Press, pp. 371431.
Spilker R. and Taylor C. (2010), ‘Tuning multidomain hemodynamic simulations to match physiological measurements’, Ann. Biomed. Engrg 38, 26352648.
Stankovičová T., Bito V., Heinzel F., Mubagwa K. and Sipido K. (2003), ‘Isolation and morphology of single Purkinje cells from the porcine heart’, Gen. Physiol. Biophys. 22, 329340.
Steele B., Wan J., Ku J., Hughes T. and Taylor C. (2003), ‘ In vivo validation of a one-dimensional finite-element method for predicting blood flow in cardiovascular bypass grafts’, IEEE Trans. Biomed. Engrg 50, 649656.
Steinman D., Thomas J., Ladak H., Milner J., Rutt B. and Spence J. (2001), ‘Reconstruction of carotid bifurcation hemodynamics and wall thickness using computational fluid dynamics and MRI’, Magnet. Reson. Med. 47, 149159.
Stergiopulos N., Westerhof B. and Westerhof N. (1999), ‘Total arterial inertance as the fourth element of the windkessel model’, Amer. J. Physiol.: Heart Circ. Physiol. 276, H81H88.
Stergiopulos N., Westerhof B., Meister J. and Westerhof N. (1996), The four-element windkessel model. In Bridging Disciplines for Biomedicine: Proceedings of the 18th Annual International Conference of the IEEE, Vol. 4, pp. 17151716.
Stijnen J., de Hart J., Bovendeerd P. and van de Vosse F. (2004), ‘Evaluation of a fictitious domain method for predicting dynamic response of mechanical heart valves’, Int. J. Numer. Methods Fluids 19, 835850.
Stroud J., Berger S. and Saloner D. (2002), ‘Numerical analysis of flow through a severely stenotic carotid artery bifurcation’, J. Biomech. Engrg 124, 920.
Stuart A. (2010), Inverse problems: A Bayesian perspective. Acta Numerica, Vol. 19, Cambridge University Press, pp. 451559.
Sullivan T. (2015), Introduction to Uncertainty Quantification, Vol. 63 of Texts in Applied Mathematics, Springer.
Sun W., Starly B., Nam J. and Darling A. (2005), ‘Bio-CAD modeling and its applications in computer-aided tissue engineering’, Comput. Aided Design 11, 10971114.
Sundaram G., Balakrishnan K. and Kumar R. (2015), ‘Aortic valve dynamics using a fluid structure interaction model: The physiology of opening and closing’, J. Biomech. 48, 17371744.
Sundnes J., Lines G. and Tveito A. (2005), ‘An operator splitting method for solving the bidomain equations coupled to a volume conductor model for the torso’, Math. Biosci. 2, 233248.
Sundnes J., Wall S., Osnes H., Thorvaldsen T. and Mcculloch A. (2014), ‘Improved discretisation and linearisation of active tension in strongly coupled cardiac electro-mechanics simulations’, Comput. Methods Biomech. Biomed. Engrg 6, 604615.
Swim E. and Seshaiyer P. (2006), ‘A nonconforming finite element method for fluid–structure interaction problems’, Comput. Methods Appl. Mech. Engrg 195, 20882099.
Tagliabue A., Dede’ L. and Quarteroni A. (2015), Fluid dynamics of an idealized left ventricle: The extended Nitsche’s method for the treatment of heart valves as mixed time varying boundary conditions. MOX-Report 61-2015, Department of Mathematics, Politecnico di Milano.
Talbot H., Cotin S., Razavi R., Rinaldi C. and Delingette H. (2015), Personalization of cardiac electrophysiology model using the unscented Kalman filtering. In Computer Assisted Radiology and Surgery: CARS 2015.
Tarantola A. (2004), Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM.
Taylor C. and Figueroa C. (2009), ‘Patient-specific modeling of cardiovascular mechanics’, Annu. Rev. Biomed. Engrg 11, 109134.
Taylor C., Hughes T. and Zarins C. (1996), ‘Finite element analysis of pulsatile flow in the abdominal aorta under resting and exercise conditions’, Amer. Soc. Mech. Engineers, Bioengineering Division 33, 8182.
Taylor C., Hughes T. and Zarins C. (1998), ‘Finite element modeling of blood flow in arteries’, Comput. Methods Appl. Mech. Engrg 158, 155196.
Temam R. (1969), ‘Sur l’approximation de la solution des équations de Navier–Stokes par la méthode des pas fractionaires (I)’, Arch. Rat. Mech. Anal. 32, 135153.
Tezduyar T., Sathe S., Cragin T., Nanna B., Conklin B., Pausewang J. and Schwaab M. (2007), ‘Modelling of fluid–structure interactions with the space–time finite elements: Arterial fluid mechanics’, Int. J. Numer. Methods Fluids 54, 901922.
Thompson J., Soni B. & Weatherill N. (Eds) (1999), Handbook of Grid Generation, CRC Press.
Timmermans L., Minev P. and van de Vosse F. (1996), ‘An approximate projection scheme for incompressible flow using spectral elements’, Int. J. Numer. Methods Fluids 22, 673688.
Tomlinson K., Hunter P. and Pullan A. (2002), ‘A finite element method for an eikonal equation model of myocardial excitation wavefront propagation’, SIAM J. Appl. Math. 1, 324350.
Toro E. (2016), ‘Brain venous haemodynamics, neurological diseases and mathematical modelling: A review’, Appl. Math. Comput. 272, 542579.
Trayanova N. (2006), ‘Defibrillation of the heart: Insights into mechanisms from modelling studies’, Exp. Physiol. 91, 323337.
Trayanova N., Li W., Eason J. and Kohl P. (2004), ‘Effect of stretch-activated channels on defibrillation efficacy’, Heart Rhythm 1, 6777.
Trenhago P., Fernandes L., Müller L., Blanco P. and Feijóo R. (2016), ‘An integrated mathematical model of the cardiovascular and respiratory systems’, Int. J. Numer. Methods Biomed. Engrg 32, e02736.
Tröltzsch F. (2010), Optimal Control of Partial Differential Equations: Theory, Methods and Applications, Vol. 112 of Graduate Studies in Mathematics, AMS.
Tu C. and Peskin C. (1992), ‘Stability and instability in the computation of flows with moving immersed boundaries: A comparison of three methods’, SIAM J. Sci. Statist. Comput. 6, 13611376.
Turek S. (1999), Efficient Solvers for Incompressible Flow Problems, Vol. 6 of Lecture Notes in Computational Science and Engineering, Springer.
ten Tusscher K. and Panfilov A. (2006), ‘Cell model for efficient simulation of wave propagation in human ventricular tissue under normal and pathological conditions’, Phys. Med. Biol. 51, 61416156.
Unser M. (1999), ‘Splines: A perfect fit for signal and image processing’, IEEE Trans. Signal Process. Mag. 16, 2238.
Usyk T., Legrice I. and Mcculloch A. (2002), ‘Computational model of three-dimensional cardiac electromechanics’, Comput. Vis. Sci. 4, 249257.
Veneziani A. (1998a), Boundary conditions for blood flow problems. In Proceedings of ENUMATH (Rannacher R. et al. , ed.), World Scientific.
Veneziani A. (1998b), Mathematical and numerical modeling of blood flow problems. PhD thesis, University of Milan.
Veneziani A. (2003), ‘Block factorized preconditioners for high-order accurate in time approximation of the Navier–Stokes equations’, Numer. Methods Partial Differ. Equations 19, 487510.
Veneziani A. and Vergara C. (2005), ‘Flow rate defective boundary conditions in haemodynamics simulations’, Int. J. Numer. Methods Fluids 47, 803816.
Veneziani A. and Vergara C. (2007), ‘An approximate method for solving incompressible Navier–Stokes problems with flow rate conditions’, Comput. Methods Appl. Mech. Engrg 196, 16851700.
Veneziani A. and Vergara C. (2013), ‘Inverse problems in cardiovascular mathematics: Toward patient-specific data assimilation and optimization’, Int. J. Numer. Methods Biomed. Engrg 29, 723725.
Veneziani A. and Villa U. (2013), ‘ALADINS: An ALgebraic splitting time ADaptive solver for the Incompressible Navier–Stokes equations’, J. Comput. Phys. 238, 359375.
Vergara C. (2011), ‘Nitsche’s method for defective boundary value problems in incompressibile fluid-dynamics’, J. Sci. Comput. 46, 100123.
Vergara C., Lange M., Palamara S., Lassila T., Frangi A. and Quarteroni A. (2016), ‘A coupled 3D–1D numerical monodomain solver for cardiac electrical activation in the myocardium with detailed Purkinje network’, J. Comput. Phys. 308, 218238.
Vergara C., Palamara S., Catanzariti D., Nobile F., Faggiano E., Pangrazzi C., Centonze M., Maines M., Quarteroni A. and Vergara G. (2014), ‘Patient-specific generation of the Purkinje network driven by clinical measurements of a normal propagation’, Med. Biol. Engrg Comput. 52, 813826.
Vergara C., Ponzini R., Veneziani A., Redaelli A., Neglia D. and Parodi O. (2010), ‘Womersley number-based estimation of flow rate with Doppler ultrasound: Sensitivity analysis and first clinical application’, Comput. Methods Programs Biomed. 98, 151160.
Vergara C., Viscardi F., Antiga L. and Luciani G. (2012), ‘Influence of bicuspid valve geometry on ascending aortic fluid-dynamics: A parametric study’, Artificial Organs 36, 368378.
Vierendeels J. A., Riemslagh K., Dick E. and Verdonck P. (2000), ‘Computer simulation of intraventricular flow and pressure gradients during diastole’, J. Biomech. Engrg 6, 667674.
Vigmond E., Aguel F. and Trayanova N. (2002), ‘Computational techniques for solving the bidomain equations in three dimensions’, IEEE Trans. Biomed. Engrg 49, 12601269.
Vigmond E. and Clements C. (2007), ‘Construction of a computer model to investigate sawtooth effects in the Purkinje system’, IEEE Trans. Biomed. Engrg 54, 389399.
Vigmond E., dos Santos R., Prassl A., Deo M. and Plank G. (2008), ‘Solvers for the cardiac bidomain equations’, Prog. Biophys. Molec. Biol. 96, 318.
Vignon-Clementel I., Figueroa C., Jansen K. and Taylor C. (2006), ‘Outflow boundary conditions for three-dimensional finite element modeling of blood flow and pressure waves in arteries’, Comput. Methods Appl. Mech. Engrg 195, 37763996.
Virag N., Jacquemet V., Henriquez C., Zozor S., Blanc O., Vesin J.-M., Pruvot E. and Kappenberger L. (2002), ‘Study of atrial arrhythmias in a computer model based on magnetic resonance images of human atria’, Chaos 12, 754763.
Viscardi F., Vergara C., Antiga L., Merelli S., Veneziani A., Puppini G., Faggian G., Mazzucco A. and Luciani G. (2010), ‘Comparative finite element model analysis of ascending aortic flow in bicuspid and tricuspid aortic valve’, Artificial Organs 34, 11141120.
Voss H. U., Timmer J. and Kurths J. (2004), ‘Nonlinear dynamical system identification from uncertain and indirect measurements’, Int. J. Bifurc. Chaos 14, 19051933.
Voss J. (2013), An Introduction to Statistical Computing: A Simulation-Based Approach, Wiley.
Vossoughi J., Vaishnav R. and Patel D. (1980), Compressibility of the myocardial tissue. In Advances in Bioengineering 1980: Papers Presented at the Winter Annual Meeting of the American Society of Mechanical Engineers (Mow V. C., ed.), ASME, pp. 4548.
Votta E., Le T., Stevanella M., Fusinic L., Caiani E., Redaelli A. and Sotiropoulos F. (2013), ‘Toward patient-specific simulations of cardiac valves: State-of-the-art and future directions’, J. Biomech. 46, 217228.
Votta E., Maisano F., Bolling S., Alfieri O., Montevecchi F. and Redaelli A. (2007), ‘The geoform disease-specific annuloplasty system: A finite element study’, Ann. Thorac. Surg. 84, 92101.
Waiter G., Mckiddie F., Redpath T., Semple S. and Trent R. (1999), ‘Determination of normal regional left ventricular function from cine-MR images using a semi-automated edge detection method’, Magnet. Reson. Imaging 17, 99107.
Wall W., Wiechert L., Comerford A. and Rausch S. (2010), ‘Towards a comprehensive computational model for the respiratory system’, Int. J. Numer. Methods Biomed. Engrg 26, 807827.
Wallman M., Smith N. and Rodriguez B. (2014), ‘Computational methods to reduce uncertainty in the estimation of cardiac conduction properties from electroanatomical recordings’, Med. Image Anal. 18, 228240.
Wang D., Kirby R., Macleod R. and Johnson C. (2013), ‘Inverse electrocardiographic source localization of ischemia: An optimization framework and finite element solution’, J. Comput. Phys. 250, 403424.
Wang K., Dutton R. and Taylor C. (1999), ‘Improving geometric model construction for blood flow modeling’, IEEE Engrg Med. Biol. Mag. 18, 3339.
Wang L., Wong K. L., Zhang H., Liu H. and Shi P. (2011), ‘Noninvasive computational imaging of cardiac electrophysiology for 3-D infarct’, IEEE Trans. Biomed. Engrg 58, 10331043.
Wang L., Zhang H., Wong K. and Shi P. (2009), A reduced-rank square root filtering framework for noninvasive functional imaging of volumetric cardiac electrical activity. In 2009 IEEE International Conference on Acoustics, Speech and Signal Processing, pp. 533536.
Watanabe H., Hisada T., Sugiura S., Okada J. and Fukunari H. (2002), ‘Computer simulation of blood flow, left ventricular wall motion and their interrelationship by fluid–structure interaction finite element method’, JSME Int. J. C: Mechanical Systems, Machine Elements and Manufacturing 45, 10031012.
Watson D. (1981), ‘Computing the $n$ -dimensional Delaunay tessellation with application to Voronoi polytopes’, Comput. J. 24, 167172.
Weatherill N. and Hassan O. (1994), ‘Efficient three-dimensional Delaunay triangulation with automatic point creation and imposed boundary constraints’, Int. J. Numer. Methods Engrg 37, 20052039.
Weinberg E. and Kaazempur-Mofrad M. (2006), ‘A large-strain finite element formulation for biological tissues with application to mitral valve leaflet tissue mechanics’, J. Biomech. 39, 15571561.
Wenk J., Ge L., Zhang Z., Soleimani M., Potter D., Wallace A., Tseng E., Ratcliffe M. and Guccione J. (2013), ‘Mechanics of the mitral valve strut chordae insertion region’, Comput. Methods Biomech. Biomed. Engrg 16, 807818.
Westerhof N., Lankhaar J. and Westerhof B. (2009), ‘The arterial windkessel’, Med. Biol. Engrg Comput. 47, 131141.
Wong J. and Kuhl E. (2014), ‘Generating fibre orientation maps in human heart models using Poisson interpolation’, Comput. Methods Biomech. Biomed. Engrg 17, 12171226.
Wong J., Goktepe S. and Kuhl E. (2013), ‘Computational modeling of chemo-electro-mechanical coupling: A novel implicit monolithic finite element approach’, Int. J. Numer. Methods Biomed. Engrg 29, 11041133.
Xi J., Lamata P., Lee J., Moireau P., Chapelle D. and Smith N. (2011), ‘Myocardial transversely isotropic material parameter estimation from in-silico measurements based on a reduced-order unscented Kalman filter’, J. Mech. Behav. Biomed. Materials 4, 10901102.
Xie F., Qu Z., Yang J., Baher A., Weiss J. and Garfinkel A. (2004), ‘A simulation study of the effects of cardiac anatomy in ventricular fibrillation’, J. Clin. Invest. 113, 686693.
Xiong F. and Chong C. (2008), ‘A parametric numerical investigation on haemodynamics in distal coronary anastomoses’, Med. Engrg Phys. 30, 311320.
Xiu D. and Hesthaven J. (2005), ‘High-order collocation methods for differential equations with random inputs’, SIAM J. Sci. Comput. 27, 11181139.
Xiu D. and Karniadakis G. (2002a), ‘Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos’, Comput. Methods Appl. Mech. Engrg 191, 49274948.
Xiu D. and Karniadakis G. (2002b), ‘The Wiener–Askey polynomial chaos for stochastic differential equations’, SIAM J. Sci. Comput. 24, 619644.
Xiu D. and Sherwin S. (2007), ‘Parametric uncertainty analysis of pulse wave propagation in a model of a human arterial network’, J. Comput. Phys. 226, 13851407.
Yamashita Y. (1982), ‘Theoretical studies on the inverse problem in electrocardiography and the uniqueness of the solution’, IEEE Trans. Biomed. Engrg BME‐29, 719725.
Yanagihara K., Noma A. and Irisawa H. (1980), ‘Reconstruction of sino-atrial node pacemaker potential based on the voltage clamp experiments’, Japan. J. Physiol 30, 841857.
Yang H. and Veneziani A. (2015), ‘Estimation of cardiac conductivities in ventricular tissue by a variational approach’, Inverse Problems 31, 115001.
Yim P., Cebral J., Mullick R., Marcos H. and Choyke P. (2001), ‘Vessel surface reconstruction with a tubular deformable model’, IEEE Trans. Med. Imaging 20, 14111421.
Yin M., Luo X., Wang T. and Watton P. (2009), ‘Effects of flow vortex on a chorded mitral valve in the left ventricle’, Int. J. Numer. Methods Biomed. Engrg 26, 381404.
Younis H., Kaazempur-Mofrad M., Chan R., Isasi A., Hinton D., Chau A., Kim L. and Kamm R. (2004), ‘Hemodynamics and wall mechanics in human carotid bifurcation and its consequences for atherogenesis: Investigation of inter-individual variation’, Biomech. Model. Mechanobiol. 3, 1732.
Yu Y., Baek H. and Karniadakis G. (2013), ‘Generalized fictitious methods for fluid–structure interactions: Analysis and simulations’, J. Comput. Phys. 245, 317346.
Zhao S., Xu X., Hughes A., Thom S., Stanton A., Ariff B. and Long Q. (2000), ‘Blood flow and vessel mechanics in a physiologically realistic model of a human carotid arterial bifurcation’, J. Biomech. 33, 975984.
Zhu F. and Tian J. (2003), ‘Modified fast marching and level set method for medical image segmentation’, J. X-Ray Sci. Technol. 11, 193204.
Zienkiewicz O. and Taylor R. (2005), The Finite Element Method for Solid and Structural Mechanics, sixth edition, Butterworth-Heinemann.
Zonca S., Formaggia L. and Vergara C. (2016), An unfitted formulation for the interaction of an incompressible fluid with a thick structure via an XFEM/DG approach. MOX-Report 35-2016, Department of Mathematics, Politecnico di Milano.
Zunino P. (2009), ‘Numerical approximation of incompressible flows with net flux defective boundary conditions by means of penalty techniques’, Comput. Methods Appl. Mech. Engrg 198, 30263038.