# Direct Estimation of Equivalent Bioelectric Sources Based on Huygens’ Principle

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Volume Conductor Problem

#### 2.2. Direct Inverse Problem Solution

- Interpolate the discrete electrode voltages to generate equivalent surface potential source distribution.
- Consider an eigenfunction expansion for the volume potential distribution and estimate its weighing factors by equating it to the source distribution (of step 1) by exploiting the eigenfunction orthogonality.
- Assume a source distribution over the epicardium (or inside the brain) as an expansion of the FEM basis functions. Its weighing factors are estimated by equating to the volume potential and exploiting the FEM basis function’s orthogonality.
- The resulting internal sources are validated by comparing their generated potential to the original ECG or EEG measurements.

**g**and a surface

**S**that encloses it, there is a certain source

**h**distributed over the surface

**S**that gives the same field outside (or inside) this surface. Thus, two source distributions that produce the same field within a region are said to be equivalent within that region. Consequently, when we are interested in the field of a specific region, prior knowledge of the actual sources is not needed as long as the equivalent sources can be sought.

**g**corresponds to the equivalent heart (or brain) activity sought,

**S**corresponds to the thorax or scalp surface, and the auxiliary Huygens sources (

**h**) can be extracted from the recorded surface potentials through interpolation (Figure 2c). As mentioned above, the surface distributed source

**w**can produce the same field either inside or outside the enclosing surface

**S**. We consider the case of electrostatic sources and electric fields, i.e., $\overrightarrow{{M}_{1}}=0$ and $\partial /\partial t=0$, when the external field is considered null. The field equivalence principle has been exemplified for various different cases by Branko Popovich in his classic work [48]. Therein, the present case of the static or quasi-static problem is also analyzed. This distributed source can be either a current or a potential source. The problem formulation for both cases follows.

#### 2.2.1. Step 1: Surface Source Distribution

**${p}_{k}$**, the potential for a specific node can be calculated as

^{th}time sample (130 ms), where the Q waveform is presented in this specific case. The x-axis displays the surface node numbering based on the thorax’s surface meshing of the FEM. The above procedure yields the potentials at all the time samples. However, the basic Huygens’ Principle asks for a continuous surface source distribution (${V}_{d}$). This was obtained by exploiting the FEM interpolation-shape functions as implemented in [8], estimating every quadrilateral surface element surface potential (${V}_{d}^{e}$):

#### 2.2.2. Step 2a: Volume Potential Eigenfunction Expansion—Current Sources

#### 2.2.3. Step 2b: Volume Potential Eigenfunction Expansion—Voltage Sources

#### 2.2.4. Step 3: Estimate the Internal Equivalent Sources

- Acquisition of a data set corresponding to measurements recorded from the surface of the thorax.
- Interpolation of the acquired recordings throughout the surface of the model, i.e., the thorax.
- Calculation of the weighting factor ${w}_{m}$, exploiting Equation (14) or (18), depending on the selected source type.
- Adaptation of the problem to describe the heart’s surface (Huygens’ Principle) and selection of the appropriate shape functions for them to comprise an orthogonal basis.
- Exploitation of the orthogonality over the heart and integration over the epicardium.
- Extraction of the epicardium potentials using Equation (24).

#### 2.3. Numerical Implementation

- ${f}_{1}\to \left[{V}_{n}\right]$, which is an $\left[n\right]$ array containing the m eigenvectors for each of the n nodes, and
- ${f}_{2}\to \left[{V}_{d}\right]$, which is an $\left[n\right]$ array consisting of the n nodes of the model from which only the ones corresponding to the surface of the thorax are non-zero and the different time instances t.

## 3. Numerical Results

## 4. Discussion

## 5. Conclusions

## 6. Future Extensions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**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.

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## Share and Cite

**MDPI and ACS Style**

Theodosiadou, G.; Arnaoutoglou, D.G.; Nannis, I.; Katsimentes, S.; Sirakoulis, G.C.; Kyriacou, G.A.
Direct Estimation of Equivalent Bioelectric Sources Based on Huygens’ Principle. *Bioengineering* **2023**, *10*, 1063.
https://doi.org/10.3390/bioengineering10091063

**AMA Style**

Theodosiadou G, Arnaoutoglou DG, Nannis I, Katsimentes S, Sirakoulis GC, Kyriacou GA.
Direct Estimation of Equivalent Bioelectric Sources Based on Huygens’ Principle. *Bioengineering*. 2023; 10(9):1063.
https://doi.org/10.3390/bioengineering10091063

**Chicago/Turabian Style**

Theodosiadou, Georgia, Dimitrios G. Arnaoutoglou, Ioannis Nannis, Sotirios Katsimentes, Georgios Ch. Sirakoulis, and George A. Kyriacou.
2023. "Direct Estimation of Equivalent Bioelectric Sources Based on Huygens’ Principle" *Bioengineering* 10, no. 9: 1063.
https://doi.org/10.3390/bioengineering10091063