M. Aoki, Y. Okamoto, T. Musha, and K. Harumi, Three-dimensional simulation of the ventricular depolarization and repolarization processes and body surface potentials: normal heart and bundle branch block, IEEE Trans. Biomed. Eng, vol.34, pp.454-462, 1987.

T. M. Austin, M. L. Trew, and A. J. Pullan, Solving the cardiac bidomain equations for discontinuous conductivities, IEEE Trans. Biomed. Eng, vol.53, pp.1265-1272, 2006.

S. Balay, S. Abhyankar, M. F. Adams, J. Brown, P. Brune et al., PETSc Web Page, 2017.

A. C. Barnard, I. M. Duck, M. S. Lynn, and W. P. Timlake, The application of electromagnetic theory to electrocardiology; II. Numerical solution of the integral equations, Biophys. J, vol.7, pp.463-490, 1967.

R. C. Barr, T. C. Pilkington, J. P. Boineau, and M. S. Spach, Determining surface potentials from current dipoles, with application to electrocardiography, IEEE Trans. Biomed. Eng, vol.13, pp.88-92, 1966.
DOI : 10.1109/tbme.1966.4502411

R. C. Barr, M. Ramsey, . Iii, and M. S. Spach, Relating epicardial to body surface potential distributions by means of transfer coefficients based on geometry measurements, IEEE Trans. Biomed. Eng, vol.24, pp.1-11, 1977.

E. Bartocci, E. M. Cherry, J. Glimm, R. Grosu, S. A. Smolka et al., Toward real-time simulation of cardiac dynamics, CMSB 2011: Proceedings of the 9th ACM International Conference on Computational Methods in Systems Biology, 2011.

J. D. Bayer, R. C. Blake, G. Plank, and N. A. Trayanova, A novel rule-based algorithm for assigning myocardial fiber orientation to computational heart models, Ann. Biomed. Eng, vol.40, pp.2243-2254, 2012.

R. Beyar and S. Sideman, A computer study of the left ventricular performance based on fiber structure, sarcomere dynamics, and transmural electrical propagation velocity, Circ. Res, vol.55, pp.358-375, 1984.

M. J. Bishop and G. Plank, Bidomain ECG simulations using an augmented monodomain model for the cardiac source, IEEE Trans. Biomed. Eng, vol.58, pp.2297-2307, 2011.

M. Boulakia, S. Cazeau, M. A. Fernández, J. Gerbeau, and N. Zemzemi, Mathematical modeling of electrocardiograms: a numerical study, Ann. Biomed. Eng, vol.38, pp.1071-1097, 2010.
URL : https://hal.archives-ouvertes.fr/inria-00400490

J. Chamorro-servent, R. Dubois, M. Potse, C. , and Y. , Improving the spatial solution of electrocardiographic imaging: a new regularization parameter choice technique for the Tikhonov method, Functional Imaging and Modeling of the Heart, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01564899

H. Cochet, R. Dubois, F. Sacher, N. Derval, M. Sermesant et al., Cardiac arrhythmias: multimodal assessment integrating body surface ECG mapping into cardiac imaging, Radiology, vol.271, pp.239-247, 2014.
DOI : 10.1148/radiol.13131331

URL : https://pubs.rsna.org/doi/pdf/10.1148/radiol.13131331

P. Colli-franzone, M. Pennacchio, G. , and L. , Accurate computation of electrograms in the left ventricular wall, Math. Mod. Methods Appl. Sci, vol.10, pp.507-538, 2000.

Y. Coudière, Y. Bourgault, and M. Rioux, Optimal monodomain approximations of the bidomain equations used in cardiac electrophysiology, Math. Models Methods Appl. Sci, vol.24, pp.1115-1140, 2014.

J. Duchateau, M. Potse, and R. Dubois, Spatially coherent activation maps for electrocardiographic imaging, IEEE Trans. Biomed. Eng, vol.64, pp.1149-1156, 2017.
DOI : 10.1109/tbme.2016.2593003

URL : https://hal.archives-ouvertes.fr/hal-01386890

R. Falgout, A. Baker, V. E. Henson, U. M. Yang, T. Kolev et al., , 2017.

P. Hypre-web,

H. L. Gelernter and J. C. Swihart, A mathematical-physical model of the genesis of the electrocardiogram, Biophys. J, vol.4, pp.285-301, 1964.

D. B. Geselowitz, On the theory of the electrocardiogram, Proc. IEEE, vol.77, pp.857-876, 1989.

D. B. Geselowitz, Description of cardiac sources in anisotropic cardiac muscle; application of the bidomain model, J. Electrocardiol, vol.25, pp.65-67, 1992.

S. Giffard-roisin, T. Jackson, L. Fovargue, J. Lee, H. Delingette et al., Non-invasive personalisation of a cardiac electrophysiology model from body surface potential mapping, IEEE Trans. Biomed. Eng, vol.64, pp.2206-2218, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01397393

K. Gima, R. , and Y. , Ionic current basis of electrocardiographic waveforms; A model study, Circ. Res, vol.90, pp.889-896, 2002.

F. Greensite and G. Huiskamp, An improved method for estimating epicardial potentials from the body surface, IEEE Trans. Biomed. Eng, vol.45, pp.98-104, 1998.

C. S. Henriquez, A brief history of tissue models for cardiac electrophysiology, IEEE Trans. Biomed. Eng, vol.61, pp.1457-1465, 2014.

V. E. Henson, M. Yang, and U. , BoomerAMG : a parallel algebraic multigrid solver and preconditioner, Appl. Numer. Math, vol.41, pp.155-177, 2002.

B. Hille, Ion Channels of Excitable Membranes, 2001.

M. G. Hoogendijk, M. Potse, A. C. Linnenbank, A. O. Verkerk, H. M. Den-ruijter et al., Mechanism of right precordial ST-segment elevation in structural heart disease: excitation failure by currentto-load mismatch, Heart Rhythm, vol.7, pp.238-248, 2010.

B. M. Horacek, Digital model for studies in magnetocardiography, IEEE Trans. Magn, vol.3, pp.440-444, 1973.

V. Jacquemet, An eikonal-diffusion solver and its application to the interpolation and the simulation of reentrant cardiac activations, Comput. Methods Programs Biomed, vol.108, pp.548-558, 2012.

V. Jacquemet, Modeling left and right atrial contributions to the ECG: a dipole-current source approach, Comput. Biol. Med, vol.65, pp.192-199, 2015.

V. Jacquemet, Equivalent dipole sources to estimate the influence of extracellular myocardial anisotropy in thin-walled cardiac forward models, Math. Biosci, vol.286, pp.31-38, 2017.

A. M. Janssen, D. Potyagaylo, O. Dössel, and T. F. Oostendorp, Assessment of the equivalent dipole layer source model in the reconstruction of cardiac activation times on the basis of BSPMs produced by an anisotropic model of the heart, Med. Biol. Eng. Comput, 2017.

M. Kania, Y. Coudière, H. Cochet, P. Jaïs, P. et al., Prediction of the exit site of ventricular tachycardia based on different ECG lead systems, 2017.
URL : https://hal.archives-ouvertes.fr/hal-01567961

D. U. Keller, F. M. Weber, G. Seemann, and O. Dössel, Ranking the influence of tissue conductivities on forward-calculated ECGs, IEEE Trans. Biomed. Eng, vol.57, pp.1568-1576, 2010.

D. U. Keller, D. L. Weiss, O. Dössel, and G. Seemann, Influence of I Ks heterogeneities on the genesis of the T-wave: a computational evaluation, IEEE Trans. Biomed. Eng, vol.59, pp.311-322, 2012.

E. Konukoglu, J. Relan, U. Cilingir, B. H. Menze, P. Chinchapatnam et al., Efficient probabilistic model personalization integrating uncertainty on data and parameters: application to eikonal-diffusion models in cardiac electrophysiology, Prog. Biophys. Mol. Biol, vol.107, pp.134-146, 2011.
URL : https://hal.archives-ouvertes.fr/inria-00616198

W. Krassowska and J. C. Neu, Effective boundary conditions for syncytial tissues, IEEE Trans. Biomed. Eng, vol.41, pp.143-150, 1994.

D. Krause, M. Potse, T. Dickopf, R. Krause, A. Auricchio et al., Hybrid parallelization of a large-scale heart model, Facing the Multicore-Challenge II, vol.7174, pp.120-132, 2012.

N. Kudryashova, V. Tsvelaya, K. Angaldze, P. , and A. , Virtual cardiac monolayers for electrical wave propagation, Sci. Rep, vol.7, p.7887, 2017.

L. J. Leon and B. M. Horácek, Computer model of excitation and recovery in the anisotropic myocardium. I. Rectangular and cubic arrays of excitable elements, J. Electrocardiol, vol.24, pp.1-15, 1991.

G. T. Lines, P. Grøttum, and A. Tveito, Modeling the electrical activity of the heart; A bidomain model of the ventricles embedded in a torso, Comput. Vis. Sci, vol.5, pp.195-213, 2003.

Z. Liu, C. Liu, and B. He, Noninvasive reconstruction of threedimensional ventricular activation sequence from the inverse solution of distributed equivalent current density, IEEE Trans. Biomed. Eng, vol.25, pp.1307-1318, 2006.

M. Lorange and R. M. Gulrajani, A computer heart model incorporating anisotropic propagation: I. Model construction and simulation of normal activation, J. Electrocardiol, vol.26, pp.245-261, 1993.

M. C. Maclachlan, J. Sundnes, and G. T. Lines, Simulation of ST segment changes during subendocardial ischemia using a realistic 3-D cardiac geometry, IEEE Trans. Biomed. Eng, vol.52, pp.799-807, 2005.

G. E. Mailloux and R. M. Gulrajani, Theoretical evaluation of the McFee and Frank vectorcardiographic lead systems using a numerical inhomogeneous torso model, IEEE Trans. Biomed. Eng, vol.29, pp.322-332, 1982.

R. Mcfee and F. D. Johnston, Electrocardiographic leads; I. introduction, Circulation, vol.8, pp.554-568, 1953.

R. Mcfee and F. D. Johnston, Electrocardiographic leads, Circulation, vol.9, pp.868-880, 1954.

V. M. Meijborg, M. Potse, C. E. Conrath, C. N. Belterman, J. M. De-bakker et al., Reduced sodium current in the lateral ventricular wall induces inferolateral J-waves, Front. Physiol, vol.7, p.365, 2016.
URL : https://hal.archives-ouvertes.fr/hal-01386905

A. Mena, J. M. Ferrero, and J. F. Matas, GPU accelerated solver for nonlinear reaction-diffusion systems. Application to the electrophysiology problem, Comput. Phys. Commun, vol.196, pp.280-289, 2015.

W. T. Miller, G. Iii, and D. B. , Simulation studies of the electrocardiogram; I. The normal heart, Circ. Res, vol.43, pp.301-315, 1978.

M. Munteanu, L. F. Pavarino, and S. Scacchi, A scalable Newton-KrylovSchwarz method for the bidomain reaction-diffusion system, SIAM J. Sci. Comput, vol.31, pp.3861-3883, 2009.

A. Neic, F. O. Campos, A. J. Prassl, S. A. Niederer, M. J. Bishop et al., Efficient computation of electrograms and ECGs in human whole heart simulations using a reaction-eikonal model, J. Comput. Phys, vol.346, pp.191-211, 2017.

A. Neic, M. Liebmann, E. Hoetzl, L. Mitchell, E. J. Vigmond et al., Accelerating cardiac bidomain simulations using graphics processing units, IEEE Trans. Biomed. Eng, vol.59, pp.2281-2290, 2012.

U. C. Nguyên, M. Potse, F. Regoli, M. L. Caputo, G. Conte et al., An in-silico analysis of the effect of heart position and orientation on the ECG morphology and vectorcardiogram parameters in patients with heart failure and intraventricular conduction defects, J. Electrocardiol, vol.48, pp.617-625, 2015.

S. A. Niederer, E. Kerfoot, A. P. Benson, M. O. Bernabeu, O. Bernus et al., Verification of cardiac tissue electrophysiology simulators using an N-version benchmark, Philos. Trans. A Math. Phys. Eng. Sci, vol.369, pp.4331-4351, 2011.

S. A. Niederer, L. Mitchell, N. Smith, and G. Plank, Simulating human cardiac electrophysiology on clinical time-scales, Front. Physiol, vol.2, p.14, 2011.

B. F. Nielsen, T. S. Ruud, G. T. Lines, and A. Tveito, Optimal monodomain approximations of the bidomain equations, Appl. Math. Comput, vol.184, pp.276-290, 2007.

D. Noble, R. , and Y. , Models of cardiac ventricular action potentials: iterative interaction between experiment and simulation, Philos. Trans. R. Soc. A, vol.359, pp.1127-1142, 2001.

P. Oosterhoff, V. Meijborg, P. M. Van-dam, P. F. Van-dessel, C. N. Belterman et al., Experimental validation of noninvasive epicardial and endocardial activation imaging, Circ. Arrhythm. Electrophysiol, vol.9, p.4104, 2016.

D. Ottino and S. Scacchi, BPX preconditioners for the bidomain model of electrocardiology, J. Comput. Appl. Math, vol.285, pp.151-168, 2015.

E. Passini, O. J. Britton, H. R. Lu, J. Rohrbacher, A. N. Hermans et al., Human in silico drug trials demonstrate higher accuracy than animal models in predicting clinical pro-arrhythmic cardiotoxicity, Front. Physiol, vol.8, p.668, 2017.
DOI : 10.3389/fphys.2017.00668

URL : https://www.frontiersin.org/articles/10.3389/fphys.2017.00668/pdf

S. Pezzuto, P. Potse, M. Prinzen, F. W. Auricchio, A. Krause et al., Evaluation of a rapid fully anisotropic model for ECG simulation, Front. Physiol, vol.8, p.265, 2017.

T. C. Pilkington, M. N. Morrow, and P. C. Stanley, A comparison of finite element and integral equation formulations for the calculation of electrocardiographic potentials-II, IEEE Trans. Biomed. Eng, vol.34, pp.258-260, 1987.

M. Potse, B. Dubé, J. Richer, A. Vinet, and R. M. Gulrajani, A comparison of monodomain and bidomain reaction-diffusion models for action potential propagation in the human heart, IEEE Trans. Biomed. Eng, vol.53, pp.2425-2435, 2006.

M. Potse and N. H. Kuijpers, Simulation of fractionated electrograms at low spatial resolution in large-scale heart models, Computing in Cardiology, vol.37, pp.849-852, 2010.

M. Potse, A. Vinet, T. Opthof, C. , and R. , Validation of a simple model for the morphology of the T wave in unipolar electrograms, Am, 2009.

, J. Physiol. Heart Circ. Physiol, vol.297

A. J. Pullan and C. P. Bradley, A coupled cubic Hermite finite element/boundary element procedure for electrocardiographic problems, Comput. Mech, vol.18, pp.356-368, 1996.

C. Ramanathan, R. Ghanem, P. Jia, K. Ryu, R. et al., Noninvasive electrocardiographic imaging for cardiac electrophysiology and arrhythmia, Nat. Med, vol.10, pp.422-428, 2004.

D. F. Richards, J. N. Glosli, E. W. Draeger, A. A. Mirin, B. Chan et al., Towards real-time simulation of cardiac electrophysiology in a human heart at high resolution, Comput. Meth. Biomech. Biomed. Eng, vol.16, pp.802-805, 2013.

S. Rush and H. Larsen, A practical algorithm for solving dynamic membrane equations, IEEE Trans. Biomed. Eng, vol.25, pp.389-392, 1978.

H. I. Saleheen and K. T. Ng, New finite difference formulations for general inhomogeneous anisotropic bioelectric problems, IEEE Trans. Biomed. Eng, vol.44, pp.800-809, 1997.

G. Shou, L. Xia, M. Jiang, Q. Wei, F. Liu et al., Truncated total least squares: a new regularization method for the solution of ECG inverse problems, IEEE Trans. Biomed. Eng, vol.55, pp.1327-1355, 2008.

M. Stenroos and J. Haueisen, Boundary element computations in the forward and inverse problems of electrocardiography: comparison of collocation and Galerkin weightings, IEEE Trans. Biomed. Eng, vol.55, pp.2124-2133, 2008.

J. Sundnes, G. T. Lines, K. A. Mardal, and A. Tveito, Multigrid block preconditioning for a coupled system of partial differential equations modeling the electrical activity in the heart, Comput. Meth. Biomech. Biomed. Eng, vol.5, pp.397-409, 2002.

K. H. Ten-tusscher and A. V. Panfilov, Alternans and spiral breakup in a human ventricular tissue model, Am. J. Physiol. Heart Circ. Physiol, vol.291, 2006.

J. Tilt, J. S. Taggart, L. Tufts, D. L. Bacon, P. Durant et al., , 2013.

M. Trudel, B. Dubé, M. Potse, R. M. Gulrajani, L. et al., Simulation of propagation in a membrane-based computer heart model with parallel processing, IEEE Trans. Biomed. Eng, vol.51, pp.1319-1329, 2004.

L. ;. Tung, . Mit, and M. A. Cambridge, A Bi-Domain Model for Describing Ischemic Myocardial D-C Potentials, 1978.

M. Tysler, P. Kneppo, M. Turzová, J. ?vehlíková, S. Karas et al., Noninvasive assessment of local myocardium repolarization changes using high resolution surface ECG mapping, Physiol. Res, vol.56, pp.133-141, 2007.

H. A. Van-der-vorst, Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems, SIAM J. Sci. Stat. Comput, vol.13, pp.631-644, 1992.

A. Van-oosterom, Genesis of the T wave as based on an equivalent surface source model, J. Electrocardiol, vol.34, pp.217-227, 2001.

A. Van-oosterom, J. , and V. , Genesis of the P wave: atrial signals as generated by the equivalent double layer source model, Europace, vol.7, pp.21-29, 2005.

N. Vandersickel, T. P. De-boer, M. Vos, and A. V. Panfilov, Perpetuation of torsade de pointes in heterogeneous hearts: competing foci or re-entry?, J. Physiol, vol.594, pp.6865-6878, 2016.

M. Vázquez, R. Arís, G. Houzeaux, R. Aubry, P. Villar et al., A massively parallel computational electrophysiology model of the heart, Int. J. Numer. Methods Biomed. Eng, vol.27, pp.1911-1929, 2011.

E. J. Vigmond, F. Aguel, and N. A. Trayanova, Computational techniques for solving the bidomain equations in three dimensions, IEEE Trans. Biomed. Eng, vol.49, pp.1260-1269, 2002.

D. Wang, R. M. Kirby, R. S. Macleod, J. , and C. R. , Inverse electrocardiographic source localization of ischemia: an optimization framework and finite element solution, J. Comput. Phys, vol.250, pp.403-424, 2013.

L. Wang, F. Dawoud, S. Yeung, P. Shi, K. C. Wong et al., Transmural imaging of ventricular action potentials and postinfarction scars in swine hearts, IEEE Trans. Med. Imaging, vol.32, pp.731-747, 2013.

R. Weber-dos-santos, G. Plank, S. Bauer, and E. J. Vigmond, Parallel multigrid preconditioner for the cardiac bidomain model, IEEE Trans. Biomed. Eng, vol.51, 1960.

D. Western, B. Hanson, and P. Taggart, Measurement bias in activationrecovery intervals from unipolar electrograms, Am. J. Physiol. Heart Circ. Physiol, vol.308, pp.331-338, 2015.

N. Zemzemi, M. O. Bernabeu, J. Saiz, J. Cooper, P. Pathmanathan et al., Computational assessment of drug-induced effects on the electrocardiogram: from ion channel to body surface potentials, Br. J. Pharmacol, vol.168, pp.718-733, 2013.
URL : https://hal.archives-ouvertes.fr/hal-00764102

N. Zemzemi, C. Dobrzynski, L. Bear, M. Potse, C. Dallet et al., Effect of the torso conductivity heterogeneities on the ECGI inverse problem solution, Computing in Cardiology, vol.42, pp.233-236, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01241750

, Conflict of Interest Statement: The author declares that the research was

©. Copyright, This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner are credited and that the original publication in this journal is cited, 2018.