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

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.

Keywords: Tikhonov regularization; endocardial surface; inverse ECG problem; single layer potential; transfer matrix.

Figure 1

Physical model underlying the inverse ECG problem.

Figure 2

Schematic geometric relationships of the inverse potential problem in the internal statement. Ω is the passive volume conductor domain, Ω_M is the myocardial domain, Γ₀ is the body surface, Γ₁ is the myocardial surface (endo- and epicardial surface), $\overrightarrow{\text{n}}$ its unit normal vector directed inward, P_i, i = 1..N₀+N₁ are collocation points used in direct boundary element method, N₀ is the number of collocation points on the Γ₀, N₁ is the number of collocation points on Γ₁.

Figure 3

SVD plot of the considered matrices, (A) is ill-conditioned matrices, (B) is well-conditioned matrices.

Figure 4

Schematic geometric relationships of the inverse potential problem in the external statement. Ω_M is the myocardial domain, Γ₀ is the body surface, Γ₁ is the myocardial surface (endo- and epicardial surface), $\overrightarrow{\text{n}}$ is unit normal vector, P_i, i = 1..N₀+N₁ are collocation points used in direct boundary element method, N₀ is the number of collocation points on the Γ₀, N₁ is the number of collocation points on Γ₁.

Figure 5

Distribution of the ESL density (function w₁) on the heart surface for the fixed time moments of cardiocycle. Cardiac excitation was initiated in the apical area. LV, left ventricle; RV, right ventricle; RVOT, right ventricle outflow tract.

Figure 6

Example of ESL and transmembrane action potential signals (simulation data). Cardiac excitation was initiated in the apical area. (A) is locations of the points where the signal was computed, (B) is transmembrane action potentials (left panel) and the ESL signals (right panel), (C) is merged transmembrane action potentials and ESL signals. Notation mC/m² is the millicoulomb per square meter, the unit for an electrical charge density.

Figure 7

Given (red curves) and inverse reconstructed electrograms (blue curves) in different point of epicardial and endocardial surface for the simulation of the pacing from the RVOT. (A) Is the reconstruction with the conventional EP transfer matrix A and Tikhonov regularization of 2^nd order, (B) is the reconstruction with the ESL transfer matrix G₀₁ and Tikhonov regularization of 0 order.