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.
We study here the three-dimensional motion of an elastic structure immersed in an incompressible viscous fluid. The structure and the fluid are contained in a fixed bounded connected set Ω. We show the existence of a weak solution for regularized elastic deformations as long as elastic deformations are not too important (in order to avoid interpenetration and preserve orientation on the structure) and no collisions between the structure and the boundary occur. As the structure moves freely in the fluid, it seems natural (and it corresponds to many physical applications) to consider that its rigid motion (translation and rotation) may be large.The existence result presented here has been announced in [4]. Some improvements have been provided on the model: the model considered in [4] is a simplified model where the structure motion is modelled by decoupled and linear equations for the translation, the rotation and the purely elastic displacement. In what follows, we consider on the structure a model which represents the motion of a structure with large rigid displacements and small elastic perturbations. This model, introduced by [15] for a structure alone, leads to coupled and nonlinear equations for the translation, the rotation and the elastic displacement.
A reduced-order model based on proper orthogonal decomposition (POD) is proposed for the bidomain equations of cardiac electrophysiology. Its accuracy is assessed through electrocardiograms in various configurations, including myocardium infarctions and long-time simulations. We show in particular that a restitution curve can efficiently be approximated by this approach. The reduced-order model is then used in an inverse problem solved by an evolutionary algorithm. Some attempts are presented to identify ionic parameters and infarction locations from synthetic electrocardiograms.
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