Solving the Inverse Problem of Electrocardiography on the Endocardium Using a Single Layer Source.

Alexander Kalinin,Danila Potyagaylo,Vitaly Kalinin

doi:10.3389/fphys.2019.00058

Abstract

The inverse problem of electrocardiography consists in reconstructing cardiac electrical activity from given body surface electrocardiographic measurements. Despite tremendous progress in the field over the last decades, the solution of this problem in terms of electrical potentials on both epi- and the endocardial heart surfaces with acceptable accuracy remains challenging. This paper presents a novel numerical approach aimed at improving the solution quality on the endocardium. Our method exploits the solution representation in the form of electrical single layer densities on the myocardial surface. We demonstrate that this representation brings twofold benefits: first, the inverse problem can be solved for the physiologically meaningful single layer densities. Secondly, a conventional transfer matrix for electrical potentials can be split into two parts, one of which turned out to posess regularizing properties leading to improved endocardial reconstructions. The method was tested in-silico for ventricular pacings utilizing realistic CT-based heart and torso geometries. The proposed approach provided more accurate solution on the ventricular endocardium compared to the conventional potential-based solutions with Tikhonov regularization of the 0th, 1st, and 2nd orders. Furthermore, we show a uniform spatio-temporal behavior of the single layer densities over the heart surface, which could be conveniently employed in the regularization procedure.

Highlights

Non-invasive electrocardiographic imaging (ECGI) is a novel imaging modality which is based on numerical reconstruction of cardiac electrical activity using the so-called body surface potential maps (BSPM) and patient-specific heart and torso geometries (Ramanathan et al, 2004)
Though we considered this method in the context of the endo-epicardial potential inverse problem, this approach can be used for the reconstruction of electrical potentials on the epicardial surface only, in contrast to the equivalent double layer (EDL)
We proposed a novel statement of the inverse problem of ECG which is based on a representation of the electrical potential on the cardiac surface as equivalent single layer (ESL) on the same surface

Summary

Introduction

Non-invasive electrocardiographic imaging (ECGI) is a novel imaging modality which is based on numerical reconstruction of cardiac electrical activity using the so-called body surface potential maps (BSPM) and patient-specific heart and torso geometries (Ramanathan et al, 2004). A mathematical problem underlaying ECGI is known as the inverse problem of ECG This problem can be formulated in several ways with respect to the unknown physical values that are sought after. Inverse ECG Problem in Terms of ESL by definition neglect electrophysiological processes ongoing on the inner heart walls (Franzone et al, 1978; Rudy and MessingerRapport, 1988). This is a Cauchy problem for the Laplace equation, a classical example of ill-posed problem: even small amount of noise in the Cauchy boundary data can lead to arbitrary high errors in the solution (Kubo, 1994; Takeuchi and Yamamoto, 2008). Special regularizing algorithms must be used in order to obtain a stable solution approximation

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