Direct Estimation of Equivalent Bioelectric Sources Based on Huygens' Principle

Abstract

An estimation of the electric sources in the heart was conducted using a novel method, based on Huygens' Principle, aiming at a direct estimation of equivalent bioelectric sources over the heart's surface in real time. The main scope of this work was to establish a new, fast approach to the solution of the inverse electrocardiography problem. The study was based on recorded electrocardiograms (ECGs). Based on Huygens' Principle, measurements obtained from the surfaceof a patient's thorax were interpolated over the surface of the employed volume conductor model and considered as secondary Huygens' sources. These sources, being non-zero only over the surface under study, were employed to determine the weighting factors of the eigenfunctions' expansion, describing the generated voltage distribution over the whole conductor volume. With the availability of the potential distribution stemming from measurements, the electromagnetics reciprocity theorem is applied once again to yield the equivalent sources over the pericardium. The methodology is self-validated, since the surface potentials calculated from these equivalent sources are in very good agreement with ECG measurements. The ultimate aim of this effort is to create a tool providing the equivalent epicardial voltage or current sources in real time, i.e., during the ECG measurements with multiple electrodes.

Keywords: Huygens’ Principle; electroencephalography (ECG); finite element method; reciprocity theorem.

Figure 1

(a) Vertical view of the human torso, where the basic organs are presented alongside the horizontal sections of the anatomic atlas [45]. The numbering on the left corresponds to the original sections, whereas on the right is the employed numbering for the model. (b) Side view of the spine, where the reference axis and the model’s sections are also illustrated. (c) Discretized thorax model for FEM simulation, where the cardiac elements and electrodes position are also illustrated, the red dots are the electrodes (78, 114, and 141) in the front (thorax), and the blue dot is the electrode 9 in the back.

Figure 2

Illustration of (a) the actual model, (b) Love’s equivalent internal model, and (c) the equivalent problem of the thorax and epicardium currents.

Figure 3

The employed thorax model, where (a) the different recorded sites and (b) the anticipated surface potential distribution are presented.

Figure 4

The pyramidal interpolation function as denoted upon the utilized node configuration. The voltage is measured on purple nodes and is additionally sought and interpolated on black nodes.

Figure 5

Comparison between pyramid interpolation (red line) and the original measurements (black crosses) for the surface nodes at 130 ms.

Figure 6

Distribution of eigenvalues for (a) the whole range and (b) the first 200 eigenvalues with the different energy percentages considered.

Figure 7

Four indicative low–order eigenvectors of thoraxes at the $20\mathrm{th}$ cross–section, including a slice of heart, as depicted in Figure 1.

Figure 8

The epicardium potential distribution resulting from the inverse direct method for $t=8$ ms (a), $t=30$ ms (b), $t=130$ ms (c), $t=170$ ms (d), $t=200$ ms (e), $t=420$ ms (f), and $t=740$ ms (g) compared to a physiological PQRST electrocardiograph.

Figure 9

Original ECG measurements versus generated potential on electrodes (a) 9, (b) 78, (c) 114, and (d) 141 after the evaluation of the resulted equivalent source.