1. Introduction
An electrocardiogram (ECG) is an essential tool for the early detection of heart disease [
1]. A 12-lead ECG displays the records of signals expanded to 12 patterns based on the potentials measured at standardized electrode positions: six electrodes are located on the patient’s chest, and one electrode is attached to each arm and left leg [
1]. The ECG waveforms record a combination of electrical activity from various cardiac cells; a typical waveform consists of three phases: P-wave, QRS-complex, and T-wave. Much attention has been paid to the classification of diseases and localization of cardiac sources with ECG waveform (e.g., [
2,
3,
4,
5]). By estimating the time-varying source of the electrical activity from the potential changes, several types of heart disease can be noninvasively identified. However, most previous studies are based on signal processing, and thus additional approach based on physics would be helpful, especially for localization.
The localization of the cardiac source based on a 12-lead ECG is a well-known ill-posed problem [
6]. This is because only nine observation points are considered, whereas the unknown or potential source locations/directions are substantial. A unique solution to the source localization problem cannot be exactly determined. Moreover, it is difficult to model the observed potential in a straightforward manner because of the inhomogeneity of the human body, which is composed of tissues with different electrical conductivity values. Therefore, the development of a cardiac source localization technique using a 12-lead ECG is an essential research topic for several clinical applications.
Previous studies have been conducted to solve the inverse problem of electrocardiography [
7,
8,
9]. In these studies, a forward problem analysis was conducted using geometric conductor models representing a human torso or realistic torso model. However, misplacement of the electrode positions causes substantial errors in clinical ECG signals [
10]. Moreover, the position of the limb electrodes directly influences all leads in terms of the shape and amplitude of the ECG waveform [
11]. Thus, to reproduce a realistic 12-lead ECG, it is essential to analyze the electric potential over a whole-body model [
12,
13].
Substantial computational cost is required to analyze the cardiac phenomena in a whole-body model, which commonly consists of a relatively large number of elements (e.g., voxel models applying finite difference methods and tetrahedral models using finite element methods). One of the limitations to processing whole-body models is the computationally expensive forward problems that need to be resolved to solve the main inverse problem [
14].
To solve a single forward problem, the volume conductor model must be solved through an iterative process [
15,
16]. Therefore, a fast-forward problem solver can significantly contribute to accelerating the entire process. Owing to its notable success in solving finite-difference problems, a geometric multi-grid method [
16] was used in our previous study [
17].
A common cardiac source localization technique is based on the lead field matrix (LFM) algorithm [
18,
19,
20,
21]. An LFM is a projection matrix that defines the ratio between the cardiac current density (electric dipoles) and measured potential at the electrodes located on the body surface. The cardiac current density is then estimated through a linear inverse filter or a reconstruction algorithm based on the LFM. However, in principle, the computations required to construct an LFM associated with the test dipoles are extremely expensive.
The localization algorithm is also an important factor in the problem. The most common solution to this ill-posed problem is the minimum norm estimation [
22,
23]. However, this method contains non-physiological characteristics and estimates overly smeared sources [
24,
25,
26]. Therefore, numerous studies have considered the inclusion of regularization terms [
19,
20,
27,
28]. These methods require a computational memory proportional to the square number of voxels (O(N2)). Conversely, sparse signal processing solutions have been applied to avoid estimating an overly smeared source in a bioelectromagnetic inverse problem [
25,
29,
30]. Our focus here is the sparse-based ECG localization algorithm applied toward more accurate source estimation. In [
31], numerous sparse reconstruction methods were proposed. An orthogonal matching pursuit (OMP), which is a simple, sparse reconstruction algorithm, requires a computational time and memory space proportional to the number of modeled voxels (i.e., O(N) only).
In addition to current source localization, a more detailed diagnostic based on the current direction equivalent to a vectorcardiogram is proposed [
1,
32]. However, a detailed analysis of this has not been demonstrated in the aforementioned studies. Hence, we propose to estimate the current location and direction simultaneously, which could be useful to improve the estimation accuracy. This information could then be useful for time-series analysis to compensate for the localization error (LE).
In the present study, we propose an ECG source localization method that combines the scalar-potential finite-difference (SPFD) method as a forward problem analysis and an OMP for solving the source localization (inverse problem). The localization performance of the proposed method is demonstrated in a whole-body human model with a spatial resolution of 2.0 × 2.0 × 2.0 mm. Moreover, we evaluate the localization performance variation corresponding to a homogeneous model, which is a model obtained by converting the anatomical human body model into a uniform homogenous tissue. We then compare the effect of the model inhomogeneity, cardiac volume, and cardiac orientation on the localization accuracy with that of the previously mentioned human body model. The novel feature of our formulation is that it estimates not only the current source location but also the current direction. For a time-domain ECG waveform, the method is augmented with a Kalman filter to improve the accuracy of the localization.
4. Discussion
In this study, we proposed a new method of ECG source localization that considers a realistically shaped and inhomogeneous human whole-body model based on an OMP. One feature of our method is that it estimates the source location and direction simultaneously. The estimated result was processed in the time domain using a Kalman filter to improve the localization performance.
As indicated in
Figure 5, the LE was less than 10 mm for more than 90% of the source positions in the homogeneous model. Moreover, it was confirmed that the DE was approximately 10° or less. For a homogeneous model, the propagation of the potential only follows the geometry of the volume conductor, and thus, the influence of the tissue inhomogeneity on the estimation accuracy is excluded. As indicated in
Figure 6b, the LE varied for different places because the tissue surrounding the cardiac tissue differs depending on the position of the estimation target. In particular, the high conductivity of the blood could result in a change in the potential propagation and consequently influence the estimation accuracy if not modeled accurately. As indicated in
Figure 7, the LE was stable and more accurate when the SNR was 20 dB or greater in the proposed method for the anatomical model. This suggests that a denoizing technique is required for an SNR less than 20 dB (see [
48,
49] for a reduction in the Gaussian noise). However, these LEs are for single isolated time points of each dipole. Therefore, when multiple time points are used, as in ECG signals, the Kalman filter can improve the estimation, as demonstrated in
Section 3.4.
Figure 10 indicates that the proposed method is sensitive to simple rotations of the cardiac tissue, which depends on the cardiac model used for constructing the LFM. Cardiac tissue is a moving object whose exact static orientation is difficult to recognize. Thus, the anatomical models are provided with static phase representation of the cardiac tissue except for the 4D extended cardiac-torso model [
50]. An imaging technique that avoids spatial artifacts such as the position and angle of the cardiac tissue is necessary for personalizing the proposed method [
51,
52]. We applied the proposed method to multiple source localizations during cardiac activity, as displayed in
Figure 12. Mislocalization can easily occur if the time series is computed as indicated by the green vector in
Figure 12b. However, the physical laws, i.e., the current continuity, must be satisfied. As indicated in
Table 5, the application of a Kalman filter significantly improved the estimation results, where the state-space model uses estimated directions as input. Thus, improvement of the localization is supported by the proposed method for an estimation of the current source location and direction simultaneously. Note that the estimation of the current direction is stable, unlike the location, which is influenced by the model inhomogeneity.
We compared our localization accuracy with that of a previous study [
53,
54].
Table 6 summarizes the localization performance of the proposed method, and that reported in previous studies. Note that the majority of recent papers [
8,
20,
53] have considered a greater number of electrodes (e.g., 100) to improve the localization accuracy. The minimum LE for an SNR of greater than 30 dB was approximately 5.0 and 12.6 mm in the homogeneous and inhomogeneous models, respectively. The latter value is somewhat greater than 10.1 mm, as reported by Svehlikova et al. [
53].
The position of test dipoles in [
53] was selected just from the voxels (points) used for developing transfer matrices, whereas it was selected randomly from all voxels in the heart volume in our study. In general, the inverse problem for unknown points has a greater localization error than for known points. Moreover, the input source in [
53] is 30 ms of signal for one estimation point, which may not be suitable for time-series estimation.
The minimum LE in [
54] was 4.4 mm. A straightforward comparison is infeasible because the edge length of the tetrahedral torso model in [
54] is 6.7 ± 1.5 mm, whereas it is 2 mm in our study. In general, models constructed with a lower resolution provide a greater estimation accuracy, although the definition of exact location inside a single voxel or tetrahedral is arbitrary. An LE less than the tetrahedral dimension could result in zero errors (e.g., [
55] for localization of an electroencephalogram source). Moreover, as described in [
54], the skin, fat, and muscle are homogenized as one tissue. In our previous study [
56] on EEG (electroencephalography), we demonstrated that tissue inhomogeneity may cause the localization error, which is attributable to the abrupt change of electrical conductivity). Thus the inhomogeneity in this study and that in [
54] are different.
The maximum error in our estimation was 54 mm, which is comparable to 60 mm, as reported in [
54]. The worst estimation was observed in the left ventricle summit region and at the endocardial apex of the right ventricle in [
54], whereas it was near the surface of the central part of the heart in our study. As indicated in
Figure 6b, the estimation error in the corresponding region would be less than 20 mm. In our study, this difference can be attributed to the higher inhomogeneity of the conductivity. In general, our computational results provide a comparable or somewhat better accuracy than the previous study. This computational approach is thus applicable not only for accurate localization but also for the design of wearable ECG sensing systems. The limitation of this study is that, as is similar to other computational studies, personalized human body models are needed to apply measured results. Thus, further study on the morphing human body models would be needed [
57,
58].
5. Conclusions
A localization method for determining cardiac sources was proposed by combining an electrical analysis of a realistic human body as the forward problem and a sparse reconstruction method as the inverse problem. For a 12-lead electrocardiogram system, a sensitivity analysis of the localization to the cardiac volume, tilted angle, and model inhomogeneity was conducted. Once an LFM was constructed, the estimation of the source location was virtually instantaneous.
For a noise-free condition, the average LE for an isolated time point was 12.6 mm, which is comparable to or somewhat superior to that reported in a previous study. Time-series source localization with Kalman filtering for the estimated location in terms of estimated current direction and location demonstrated that source mislocalization could be compensated, suggesting the effectiveness of the proposed method. For the ECG R-wave, the mean distance error was reduced to less than 7.3 mm using the proposed method. This highly accurate estimation was achieved because the proposed approach uses an estimation of the current direction, which is less sensitive to different error sources. Considering the physical properties of the human body with Kalman filtering enables highly accurate estimation of the cardiac electric signal source location and direction. Our proposal is applicable to the electrode configuration in wearable sensing systems where non-conventional locations would be more essential.