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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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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

Correspondence to Miguel A. Fernández.

Additional information

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 abc, 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
figure29

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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Keywords

  • 12-Lead electrocardiogram
  • Mathematical modeling
  • Numerical simulation
  • Bidomain equation
  • Ionic model
  • Heart–torso coupling
  • Monodomain equation
  • Sensitivity analysis