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Mathematical Modeling of Electrocardiograms: A Numerical Study

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Abstract

This paper deals with the numerical simulation of electrocardiograms (ECG). Our aim is to devise a mathematical model, based on partial differential equations, which is able to provide realistic 12-lead ECGs. The main ingredients of this model are classical: the bidomain equations coupled to a phenomenological ionic model in the heart, and a generalized Laplace equation in the torso. The obtention of realistic ECGs relies on other important features—including heart–torso transmission conditions, anisotropy, cell heterogeneity and His bundle modeling—that are discussed in detail. The numerical implementation is based on state-of-the-art numerical methods: domain decomposition techniques and second order semi-implicit time marching schemes, offering a good compromise between accuracy, stability and efficiency. The numerical ECGs obtained with this approach show correct amplitudes, shapes and polarities, in all the 12 standard leads. The relevance of every modeling choice is carefully discussed and the numerical ECG sensitivity to the model parameters investigated.

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References

  1. Aehlert, B. ECGs Made Easy (3rd ed.). Mosby Jems, Elsevier, 2006.

  2. Antzelevitch, C. Cellular basis for the repolarization waves of the ECG. Ann. N. Y. Acad. Sci. 1080:268–281, 2006.

    Article  PubMed  Google Scholar 

  3. Barr, R. C., 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. 24(1):1–11, 1977.

    Article  CAS  PubMed  Google Scholar 

  4. Beeler, G., and H. Reuter. Reconstruction of the action potential of ventricular myocardial fibres. J. Physiol. (Lond.) 268:177–210, 1977.

    CAS  Google Scholar 

  5. Boulakia, M., M. A. Fernández, J.-F. Gerbeau, and N. Zemzemi. Towards the numerical simulation of electrocardiograms. In: Functional Imaging and Modeling of the Heart, Vol. 4466 of Lecture Notes in Computer Science, edited by F. B. Sachse and G. Seemann. Springer-Verlag, 2007, pp. 240–249.

  6. Boulakia, M., M. A. Fernández, J.-F. Gerbeau, and N. Zemzemi. Direct and inverse problems in electrocardiography. AIP Conf. Proc. 1048(1):113–117, 2008.

    Article  Google Scholar 

  7. Boulakia, M., M. A. Fernández, J.-F. Gerbeau, and N. Zemzemi. A coupled system of PDEs and ODEs arising in electrocardiograms modelling. Appl. Math. Res. Exp. 2008(abn002):28, 2008.

  8. Buist, M., and A. Pullan. Torso coupling techniques for the forward problem of electrocardiography. Ann. Biomed. Eng. 30(10):1299–1312, 2002.

    Article  PubMed  Google Scholar 

  9. Clements, J., J. Nenonen, P. K. J. Li, and B. M. Horacek. Activation dynamics in anisotropic cardiac tissue via decoupling. Ann. Biomed. Eng. 32(7):984–990, 2004.

    Article  PubMed  Google Scholar 

  10. Colli Franzone, P., and L. F. Pavarino. A parallel solver for reaction-diffusion systems in computational electrocardiology. Math. Models Methods Appl. Sci. 14(6):883–911, 2004.

    Article  Google Scholar 

  11. Colli Franzone, P., L. F. Pavarino, S. Scacchi, and B. Taccardi. Effects of anisotropy and transmural heterogeneity on the T-wave polarity of simulated electrograms. In: Functional Imaging and Modeling of the Heart, Vol. 5528 of Lecture Notes in Computer Science, edited by N. Ayache, H. Delingette, and M. Sermesant. Springer-Verlag, 2009, pp. 513–523.

  12. Colli Franzone, P., L. F. Pavarino, and B. Taccardi. Simulating patterns of excitation, repolarization and action potential duration with cardiac bidomain and monodomain models. Math. Biosci. 197(1):35–66, 2005.

    Article  CAS  PubMed  Google Scholar 

  13. Conrath, C. E., and T. Opthof. Ventricular repolarization: an overview of (patho)physiology, sympathetic effects and genetic aspects. Prog. Biophys. Mol. Biol. 92(3):269–307, 2006.

    Article  CAS  PubMed  Google Scholar 

  14. di Bernardo, D., and A. Murray. Modelling cardiac repolarisation for the study of the T wave: effect of repolarisation sequence. Chaos Solitons Fractals 13(8):1743–1748, 2002.

    Article  Google Scholar 

  15. Djabella, K., and M. Sorine. Differential model of the excitation-contraction coupling in a cardiac cell for multicycle simulations. In: EMBEC’05, Vol. 11. Prague, 2005, pp. 4185–4190.

  16. Ebrard, G., M. A. Fernández, J.-F. Gerbeau, F. Rossi, and N. Zemzemi. From intracardiac electrograms to electrocardiograms. models and metamodels. In: Functional Imaging and Modeling of the Heart, Vol. 5528 of Lecture Notes in Computer Science, edited by N. Ayache, H. Delingette, and M. Sermesant. Springer-Verlag, 2009, pp. 524–533.

  17. Ethier, M., and Y. Bourgault. Semi-implicit time-discretization schemes for the bidomain model. SIAM J. Numer. Anal. 46:2443, 2008.

    Article  Google Scholar 

  18. Fenton, F., and A. Karma. Vortex dynamics in three-dimensional continuous myocardium with fiber rotation: filament instability and fibrillation. Chaos 8(1):20–47, 1998.

    Article  PubMed  Google Scholar 

  19. Fitzhugh, R. Impulses and physiological states in theoretical models of nerve membrane. Biophys. J. 1:445–465, 1961.

    Article  CAS  PubMed  Google Scholar 

  20. Franz, M. R., K. Bargheer, W. Rafflenbeul, A. Haverich, and P. R. Lichtlen. Monophasic action potential mapping in human subjects with normal electrocardiograms: direct evidence for the genesis of the T wave. Circulation 75(2):379–386, 1987.

    CAS  PubMed  Google Scholar 

  21. Frey, P. Yams: a fully automatic adaptive isotropic surface remeshing procedure. Technical report 0252, Inria, Rocquencourt, France, November 2001.

  22. George, P. L., F. Hecht, and E. Saltel. Fully automatic mesh generator for 3d domains of any shape. Impact Comput. Sci. Eng. 2:187–218, 1990.

    Article  Google Scholar 

  23. Gerardo-Giorda, L., L. Mirabella, F. Nobile, M. Perego, and A. Veneziani. A model-based block-triangular preconditioner for the bidomain system in electrocardiology. J. Comput. Phys. 228(10):3625–3639, 2009.

    Article  Google Scholar 

  24. Goldberger, A. L. Clinical Electrocardiography: A Simplified Approach (7th ed.). Mosby–Elsevier, 2006.

  25. Green, L. S., B. Taccardi, P. R. Ershler, and R. L. Lux. Epicardial potential mapping. effects of conducting media on isopotential and isochrone distributions. Circulation 84(6):2513–2521, 1991.

    CAS  PubMed  Google Scholar 

  26. Gulrajani, R. M. Models of the electrical activity of the heart and computer simulation of the electrocardiogram. Crit. Rev. Biomed. Eng. 16(1):1–6, 1988.

    CAS  PubMed  Google Scholar 

  27. Higuchi, T., and Y. Nakaya. T wave polarity related to the repolarization process of epicardial and endocardial ventricular surfaces. Am. Heart J. 108(2):290–295, 1984.

    Article  CAS  PubMed  Google Scholar 

  28. Huiskamp, G. Simulation of depolarization in a membrane-equations-based model of the anisotropic ventricle. EEE Trans. Biomed. Eng., 5045(7):847–855, 1998.

    Article  Google Scholar 

  29. Irons, B., and R. C. Tuck. A version of the aitken accelerator for computer implementation. Int. J. Numer. Methods Eng., 1:275–277, 1969.

    Article  Google Scholar 

  30. Keller, D. U. J., G. Seemann, D. L. Weiss, D. Farina, J. Zehelein, and O. Dössel. Computer based modeling of the congenital long-qt 2 syndrome in the visible man torso: from genes to ECG. In: Proceedings of the 29th Annual International Conference of the IEEE EMBS, 2007, pp. 1410–1413.

  31. Krassowska, W., and J. C. Neu. Effective boundary conditions for syncitial tissues. IEEE Trans. Biomed. Eng. 41(2):143–150, 1994.

    Article  CAS  PubMed  Google Scholar 

  32. Lines, G. T., M. L. Buist, P. Grottum, A. J. Pullan, J. Sundnes, and A. Tveito. Mathematical models and numerical methods for the forward problem in cardiac electrophysiology. Comput. Vis. Sci. 5(4):215–239, 2003.

    Article  Google Scholar 

  33. Luo, C., and Y. Rudy. A dynamic model of the cardiac ventricular action potential. I. Simulations of ionic currents and concentration changes. Circ. Res. 74(6):1071–1096, 1994.

    CAS  PubMed  Google Scholar 

  34. Luo, C. H., and Y. Rudy. A model of the ventricular cardiac action potential. depolarisation, repolarisation,and their interaction. Circ. Res. 68(6):1501–1526, 1991.

    CAS  PubMed  Google Scholar 

  35. Malmivuo, J., and R. Plonsey. Bioelectromagnetism. Principles and Applications of Bioelectric and Biomagnetic Fields. New York: Oxford University Press, 1995.

    Google Scholar 

  36. Mitchell, C. C., and D. G. Schaeffer. A two-current model for the dynamics of cardiac membrane. Bull. Math. Biol. 65:767–793, 2003.

    Article  CAS  PubMed  Google Scholar 

  37. Neu, J. C., and W. Krassowska. Homogenization of syncytial tissues. Crit. Rev. Biomed. Eng. 21(2):137–199, 1993.

    CAS  PubMed  Google Scholar 

  38. Noble, D., A. Varghese, P. Kohl, and P. Noble. Improved guinea-pig ventricular cell model incorporating a diadic space, ikr and iks, and length- and tension-dependent processes. Can. J. Cardiol. 14(1):123–134, 1998.

    CAS  PubMed  Google Scholar 

  39. Pennacchio, M., G. Savaré, and P. Colli Franzone. Multiscale modeling for the bioelectric activity of the heart. SIAM J. Math. Anal. 37(4):1333–1370, 2005.

    Article  Google Scholar 

  40. Potse, M., G. Baroudi, P. A. Lanfranchi, and A. Vinet. Generation of the t wave in the electrocardiogram: lessons to be learned from long-QT syndromes. In: Canadian Cardiovascular Congress, 2007.

  41. Potse, M., B. Dubé, and M. Gulrajani. ECG simulations with realistic human membrane, heart, and torso models. In: Proceedings of the 25th Annual Intemational Conference of the IEEE EMBS, 2003, pp. 70–73.

  42. Potse, M., 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. 53(12):2425–2435, 2006.

    Article  PubMed  Google Scholar 

  43. Potse, M., B. Dubé, and A. Vinet. Cardiac anisotropy in boundary-element models for the electrocardiogram. Med. Biol. Eng. Comput. doi:10.1007/s11517-009-0472-x.

  44. Pullan, A. J., M. L. Buist, and L. K. Cheng. Mathematically modelling the electrical activity of the heart: from cell to body surface and back again. Hackensack, NJ: World Scientific Publishing Co. Pte. Ltd., 2005.

    Google Scholar 

  45. Quarteroni, A., R. Sacco, and F. Saleri. Numerical Mathematics, Vol. 37 of Texts in Applied Mathematics (2nd ed.). Berlin: Springer-Verlag, 2007.

  46. Quarteroni, A., and A. Valli. Domain decomposition methods for partial differential equations. In: Numerical Mathematics and Scientific Computation. New York: The Clarendon Press, Oxford University Press, Oxford Science Publications, 1999.

  47. Sachse, F. B. Computational Cardiology: Modeling of Anatomy, Electrophysiology, and Mechanics. Springer-Verlag, 2004.

  48. Scacchi, S., L. F. Pavarino, and I. Milano. Multilevel Schwarz and Multigrid preconditioners for the Bidomain system. Lect. Notes Comput. Sci. Eng. 60:631, 2008.

    Article  Google Scholar 

  49. Sermesant, M., Ph. Moireau, O. Camara, J. Sainte-Marie, R. Andriantsimiavona, R. Cimrman, D. L. Hill, D. Chapelle, and R. Razavi. Cardiac function estimation from mri using a heart model and data assimilation: advances and difficulties. Med. Image Anal. 10(4):642–656, 2006.

    Article  CAS  PubMed  Google Scholar 

  50. Shahidi, A. V., P. Savard, and R. Nadeau. Forward and inverse problems of electrocardiography: modeling and recovery of epicardial potentials in humans. IEEE Trans. Biomed. Eng. 41(3):249–256, 1994.

    Article  CAS  PubMed  Google Scholar 

  51. Sundnes, J., G. T. Lines, X. Cai, B. F. Nielsen, K.-A. Mardal, and A. Tveito. Computing the Electrical Activity in the Heart. Springer-Verlag, 2006

  52. Sundnes, J., 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. Methods Biomech. Biomed. Eng. 5(6):397–409, 2002.

    Article  CAS  Google Scholar 

  53. Toselli, A., and O. Widlund. Domain Decomposition Methods—Algorithms and Theory, Vol. 34 of Springer Series in Computational Mathematics. Berlin: Springer-Verlag, 2005.

  54. Trudel, M.-C., B. Dubé, M. Potse, R. M. Gulrajani, and L. J. Leon. Simulation of qrst integral maps with a membrane-based computer heart model employing parallel processing. IEEE Trans. Biomed. Eng. 51(8):1319–1329, 2004.

    Article  PubMed  Google Scholar 

  55. Tung, L. A Bi-Domain Model for Describing Ischemic Myocardial D–C potentials. Ph.D. thesis, MIT, 1978.

  56. van Capelle, F. H., and D. Durrer. Computer simulation of arrhythmias in a network of coupled excitable elements. Circ. Res. 47:453–466, 1980.

    Google Scholar 

  57. Vigmond, E. J., and C. Clements. Construction of a computer model to investigate sawtooth effects in the purkinje system. IEEE Trans. Biomed. Eng. 54(3):389–399, 2007.

    Article  PubMed  Google Scholar 

  58. Vigmond, E. J., R. Weber dos Santos, A. J. Prassl, M. Deo, and G. Plank. Solvers for the cardiac bidomain equations. Prog. Biophys. Mol. Biol. 96(1–3):3–18, 2008.

    Article  CAS  PubMed  Google Scholar 

  59. Yan, G.-X., and C. Antzelevitch. Cellular basis for the normal T wave and the electrocardiographic manifestations of the long-QT syndrome. Circulation 98:1928–1936, 1998.

    CAS  PubMed  Google Scholar 

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Acknowledgments

This work was partially supported by INRIA through its large scope initiative CardioSense3D. The authors wish to thank Elsie Phé (INRIA) for her work on the anatomical models and meshes, and Michel Sorine (INRIA) for valuable discussions regarding, in particular, the heart–torso transmission conditions.

Author information

Authors and Affiliations

  1. Laboratoire Jacques-Louis Lions, Université Pierre et Marie Curie-Paris 6, UMR 7598, 75005, Paris, France

    Muriel Boulakia

  2. Département de Rythmologie-Stimulation, Hôpital Saint-Joseph, 185, rue Raymond Losserand, 75014, Paris, France

    Serge Cazeau

  3. INRIA, CRI Paris-Rocquencourt, Rocquencourt BP 105, 78153, Le Chesnay Cedex, France

    Miguel A. Fernández, Jean-Frédéric Gerbeau & Nejib Zemzemi

  4. Laboratoire de mathématiques d’Orsay, Université Paris 11, Bâtiment 425, 91405, Orsay Cedex, France

    Nejib Zemzemi

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  1. Muriel Boulakia
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Correspondence to Miguel A. Fernández.

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Associate Editor Kenneth R. Lutchen oversaw the review of this article.

Appendix: External Stimulus

Appendix: External Stimulus

In order to initiate the spread of excitation within the myocardium, we apply a given volume current density to a thin subendocardial layer of the ventricles during a small period of time t act. In the left ventricle, this thin layer (1.6 mm) of external activation is given by

$$ S \,{\mathop{=}\limits^{\rm def}}\, \{ (x,y,z) \in {\Upomega_{\rm H}} / c_1 \le a x^2 +b y^2 +c z^2 \le c_2 \} , $$

where a, b, c, c 1 and c 2 are given constants, with c 1 < c 2, see Fig. 29. The source current I app, involved in (2.7), is then parametrized as follows:

$$ {I_{\rm app}} (x,y,z,t) = I_0(x,y,z) \chi_S(x,y,z) \chi_{[0,t_{\rm act}]}(t) \psi(x,z,t), $$

where

$$ I_0(x,y,z)\,{\mathop{=}\limits^{\rm def}}\, i_{\rm app} \left[\frac{c_2}{c_2 -c_1} -\frac{1}{c_2-c_1}\left(a x^2 +b y^2 +c z^2\right)\right], $$

with i app the amplitude of the external applied stimulus,

$$ \begin{aligned} \chi_S (x,y,z)& \,{\mathop{=}\limits^{\rm def}}\, \left\{ \begin{array}{ll} 1& \hbox{if }(x,y,z)\in S, \\ 0& \hbox{if } (x,y,z)\notin S, \\ \end{array} \right.\\ \chi_{[0,t_{\rm act}]}(t) & \,{\mathop{=}\limits^{\rm def}}\, \left\{ \begin{array}{ll} 1&\hbox{if } t\in [0,t_{\rm act}],\\ 0& \hbox{if } t \notin[0, t_{\rm act}],\\ \end{array} \right.\\ \psi(x,z,t) & \,{\mathop{=}\limits^{\rm def}}\, \left\{ \begin{array}{ll} 1&\hbox{if }\hbox{atan}\left( \frac{x-x_0}{z-z_0}\right)\le \alpha(t),\\ 0& \hbox{if }\hbox{atan} \left(\frac{x-x_0}{z-z_0}\right) > \alpha(t), \end{array} \right. \end{aligned} $$

the activated angle \( \alpha(t) \,{\mathop{=}\limits^{\rm def}}\, \frac{t \pi}{2 t_{\rm act}}\) and t act = 10ms. The activation current in the right ventricle is built in a similar fashion.

Figure 29
figure 29

Geometrical description of the external stimulus (plane cut y = 0)

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Boulakia, M., Cazeau, S., Fernández, M.A. et al. Mathematical Modeling of Electrocardiograms: A Numerical Study. Ann Biomed Eng 38, 1071–1097 (2010). https://doi.org/10.1007/s10439-009-9873-0

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