1. Introduction
Electrocardiography (ECG) is currently the most widely used method of sensing electrical activity in the human heart. Vectorcardiography (VCG) is an investigation method that was previously compared to ECG several times and was evaluated as a useful investigation method [
1,
2,
3]. It achieved a higher sensitivity for the detection of hypertrophy as well as ischaemic heart disease [
4,
5,
6,
7]. Three leads scanned in three mutually orthogonal axes (vertical, transversal, and sagittal) are used for VCG measurements, but this is not commonly used in clinical practice due to the more complex interpretation of the recordings. Mathematical transformations can be made between ECG and VCG leads. Previously, ECG transformations from VCG leads were used, but later only 12-lead ECG spread and VCG leads were derived from ECG leads instead of measurements. These transformations are based on transformation coefficients that have been defined by various authors [
8,
9,
10]. For example, the authors used thoracic models to obtain the coefficients, or, based on the similarity of one of the ECG leads, they determined the orthogonal lead. Transformation methods have also been developed based on a regression approach for multiple recordings of simultaneously measured ECG and VCG.
VCG is a diagnostic and less specific method than the standard 12-lead ECG measurement [
1]. This method can be measured using several different lead systems. In clinical practice, Frank’s orthogonal lead system is the most widely used. It is measured using three orthogonal leads described by Frank, which measure cardiac activity with the same sensitivity. The problem of needing to connect multiple electrodes to measure both ECG and VCG was solved by synthesis from the currently recorded ECG leads. Compared to ECG, VCG has been recognized as a very useful investigative clinical method for cardiac activity sensing [
1,
3]. In addition, it shows the balance of cardiac cycle forces at any moment and contains more information in comparison to ECG that may have diagnostic significance, such as the display of temporal or phase relationships between complexes. The importance of VCG is, for example, in the detection and localization of acute myocardial infarction, right ventricular hypertrophy, and Tawar arm blockade. Despite all the benefits of VCG, ECG is increasingly being used.
VCG recognizes three planes, sagittal, transversal, and frontal. The frontal one is located between the XY leads, the transverse one between the XZ leads, and the sagittal one between the YZ leads. The electrical activity of the heart is described by three loops that represent the individual phases, and their contours, rotation, or the direction of the cardiac axis are taken into account. The first loop corresponds to the P wave, the second one, which is also the largest loop, corresponds to the QRS complex, and the third one corresponds to the T wave. The loops can be displayed in a 1-D image as three scalar recordings (
Figure 1), in a 2-D image using three planes (
Figure 2), or in one 3-D image (
Figure 3). The axes are represented as time and voltage dependence in the 1-D image. In the case of 2-D and 3-D images, the dependence between the individual leads in mV is displayed. In the analysis, the greatest attention was paid to the QRS complex, which is oval in shape and faces in the same direction as the cardiac axis of the heart [
2].
The Frank’s orthogonal lead system is most widely used for its simplicity and is almost orthogonal. This is a bipolar circuit. It consists of seven electrodes whose position is marked with capital letters: I, E, C, A, M, F, and H.Each electrode has its specific location, see
Figure 4. The E electrode is located on the front, in the middle of the chest, and the M electrode is located opposite, in the rear, on the back. The I electrode is located in the right central axillary line and the A electrode is in the left central axillary line. The position of the C electrode is obtained by halving the distance between the A electrode and E electrode. The F Electrode is positioned on the left leg and the H electrode on the neck. Furthermore, one electrode attached to the right leg is used as a ground electrode. On the X-axis, the left side of the body is positive with respect to the right side of the body, on the Y-axis, the left foot is a positive electrode with respect to the head, and on the Z-axis, the back of the chest is positive with respect to the front [
2]. The individual electrodes are routed to a resistor network. Malmivuo et al., 1995 [
11] explained the correct values of each resistors in their book. The signals on the leads are derived using mathematical Equations (
1)–(
3), where P represents the potentials on the individual electrode clips and I, E, C, A, M, F, H represent individual electrodes [
8]. Macfarlane et al., 2010 [
12] explained calculations of Equations (
1)–(
3), which are based on the resistor network in
Figure 4. In addition to Frank’s orthogonal leads, other leads, such as McFee–Parungao leads [
13] and Schmitt–Simonson [
14] leads called Svec-III, are used for sensing VCG [
15,
16,
17].
There are many different transformation methods of 3-lead VCG to 8-lead ECG, or 3-lead ECG to 8-lead VCG. Kors et al., 1990 [
18] compared the Kors quasi-orthogonal transform to the inverse Dower transform, the Kors regression transform, and the Frank’s orthogonal lead system. A total of 90 recordings were used. The regression method achieved the best results in this study, while the quasi-orthogonal method achieved the worst results. Other quasi-orthogonal transforms were described in articles [
19,
20,
21,
22]. The authors use different coefficients in equations to perform ECG to VCG transformations. Burger et al., 1952 [
23] introduced first linear transformation method. They showed that it would be possible to transform ECG to VCG. Levkov, 1987 [
3] introduced five different transformation methods in his work. One transformation method is based on one-dipole model and four transformation methods are derived by regression. His methods provide high accuracy, particularly th the methods based on regression. Edenbrandt et al., 1994 [
4] tested the inverse Dower transformation. Authors compared this method with the directly measured Frank’s leads, the quasi-orthogonal method described by Bjerl, and the quasi-orthogonal method implemented by Marquette Electronics Inc. control systems. The comparison was performed on 80 people with different types of heart attacks. The inverse Dower transformation achieved the best QRS complex image with respect to other methods and its recording was the most similar to the Frank’s. Guillem et al., 2006 [
9] tested linear regression-based transformations for deriving the P wave (PLSV) and QRS complex (QLSV). They compared these methods with the inverse Dower transformation methods during the QRS complex and P wave detection on 247 recordings. In the P wave transformation, the PLSV method achieved the best results and the QLSV method achieved the best results for QRS complex detection. Both methods achieved better results than the inverse Dower method. Acar and Koymen, 1999 [
24] optimized the singular value decomposition (SVD) transformation to derive VCG indications. They used data from 23 patients. Their approach does not perform transformation of three VCG leads but provides three non-correlated orthogonal leads. Dawson et al., 2009 [
25] used the Dower transform and statistical method called affine transform to derive 8-lead ECG from 3-lead VCG and their inverse forms to derive 3-lead VCG from 8-lead ECG. They concluded that for subjects with myocardial infarction and healthy control subjects, affine transform provides better accuracy than Dower transform. Maheshwari et al., 2016 [
26] showed that principal component analysis (PCA) could also be used for deriving the VCG from ECG. They compared PCA with the inverse Dower transform and the Kors regression transform. Their results support the theoretical basis behind the PCA-based reconstruction methodology and show better performance then the compared transforms. Vozda et al., 2014 [
27] used transformation methods based on artificial neural networks. They showed that these transformation methods offer even higher accuracy then other reviewed transforms.
In this paper, Kors quasi-orthogonal transform, inverse Dower transform, and Kors regression transform were chosen, because they are the most commonly used transformation methods. A very large number of transformation methods and matrices are optimizing the transformations of the QRS complex. Methods based on linear regression transformations are focused not only on the QRS complex, but also on optimizing the transformations of the P wave. That is the main reason why this transform was chosen.
4. Results
Before conducting individual transformations, signal pre-processing was performed using a frequency selective filter with a finite impulse response (FIR filter). Based on recommendations [
38,
39,
40], filter cut-off frequencies should be selected from 0.05 to 100 Hz. Since we only analyzed the recordings of healthy volunteers, the limit values were set from 0.2 to 100 Hz with a filter order of 500, which is in line with the recommendation stated in the articles [
38,
39,
40].
Subsequently, R-oscillations were detected using the Pan–Tompkins detector. The individual positions of the R-oscillations determine the beginning of the heart rate as ms and the end of the heart rate as ms. Thus, all heart rates of the individual leads were cut out. The resulting transformations could be influenced by extrasystoles, so they were removed using the library function in MATLAB. Once the extrasystoles were removed, the individual heart rates were averaged, so that the average heart rate for each ECG and VCG lead was created to perform the transformation and the transformation accuracy check.
The test was performed on the recordings from the PTB database and the MSE and R result vector in relation to the reference Frank’s leads was obtained for each method. Since the mean value of the results was different from the median of the results, it was determined that the data did not come from a normal distribution. Statistical testing of data normality was performed at work [
27]. Therefore, the Mann–Whitney nonparametric median test was selected for statistical analysis. If the resulting
p-value was less than the significance level of
, the zero hypothesis would be rejected in favor of the alternative hypothesis. This would mean that the difference between the medians compared was statistically significant. The determination of the zero and alternative hypotheses can be seen in Equations (
10) and (
11), where the zero hypothesis indicates that the median results were statistically identical, and the alternative hypothesis indicates that the difference between the median results was statistically significant. The test was bilateral, so the medians for Kors regression transformation and the method compared must be determined.
4.1. Visual Evaluation
The testing focused on the regression method described by Kors because, according to studies, this transformation has achieved the best results in comparison to other transformations. Statistical analysis was performed for recordings measured on healthy volunteers that were received from the PTB database. For comparison, box plots are created, see
Figure 5 and
Figure 6. In the figures and tables, the result of the Kors quasi-orthogonal transformation will be referred to as
Quasi, the result of inverse Dower transformation will be referred to as
Dower, the result of Kors regression transformation will be referred to as
Regress, the result of the PLSV method will be referred to as
PLSV, and the result of the QLSV method will be referred to as
QLSV. It can be concluded from the figure that Kors regression transformation was the most accurate because the median MSE was lower than the other methods for X and Y leads, see
Figure 5. Similarly, it can be seen that the median R was higher than the other methods for the X and Y leads, see
Figure 6. In the box plots, the median is represented by the horizontal line and the mean by the circle. Red crosses represent outliers, which are commonly used in statistics to represent samples which are removed from evaluation. Statistical verification that this method is better must be proved by a statistical test.
4.2. Statistical Analysis of MSE Results
Figure 5 indicates that the MSE medians of Kors regression transformation for X and Y leads are lower than the medians of the other transformation methods and, for Z lead, it could not be assessed. According to the statistical test, all
p-values for the X and Y leads were less than the significance level of
, see
Table 5, and, thus, it was impossible to reject the zero hypothesis. This means that Kors regression transformation has been recognized as more accurate for X and Y leads than all other transformation methods, based on the medians of the MSE results.
For Z lead, the p-values between Kors regression transformation and Kors quasi-orthogonal transformation, and between Kors regression transformation and inverse Dower transformation were less than the significance level of , therefore, the zero hypothesis could not be rejected here. Thus, Kors regression transformation was recognized as more accurate for Z lead than Kors quasi-orthogonal transformation and inverse Dower transformation, based on the median of the MSE results. For Z lead, the p-values between Kors regression transformation and the PLSV method, and Kors regression transformation and the QLSV method were greater than the significance level of , therefore, the zero hypothesis was rejected in favor of the alternative hypothesis. This indicates that, for Z lead, it cannot be asserted that Kors regression transformation would achieve a more accurate transformation than the PLSV and QLSV methods, hence, the difference between the methods was not statistically significant.
4.3. Statistical Analysis of R Results
Figure 6 indicates that the R medians of Kors regression transformation for X and Y leads were higher than the medians of the other transformation methods and, for Z lead, it could not be assessed with certainty. According to the statistical test, all
p-values for the X and Y leads were higher than the significance level of
, see
Table 6, and, thus, it was impossible to reject the zero hypothesis. This means that Kors regression transformation has been recognized as more accurate for X and Y leads than all other transformation methods, based on the medians of the R results. For Z lead, the
p-values between Kors regression transformation and Kors quasi-orthogonal transformation, and between Kors regression transformation and inverse Dower transformation were less than the significance level of
, therefore, it was impossible to reject the zero hypothesis here, either. Thus, Kors regression transformation was recognized as more accurate for Z lead than Kors quasi-orthogonal transformation and inverse Dower transformation, based on the medians of the R results. For Z lead, the
p-valuess between Kors regression transformation and the PLSV method, and Kors regression transformation and the QLSV method were greater than the significance level of
, therefore, the zero hypothesis was rejected in favor of the alternative hypothesis. This indicates that, for Z lead, it cannot be asserted that Kors regression transformation would achieve a more accurate transformation than the PLSV and QLSV methods based on the medians of the R results, hence, the difference between the methods was not statistically significant.
5. Discussion
In
Figure 7,
Figure 8,
Figure 9,
Figure 10,
Figure 11,
Figure 12,
Figure 13 and
Figure 14, you can see 2-D and 3-D comparisons of tested tranformations. For visual comparison, we chose four random recordings out of the whole tested data set. 2-D figures were selected, as they are ideal for comparisons to the reference Frank’s orthogonal lead system. According to the MSE and R parameters, Kors regression transformation provided the best results, which also corresponds to the mentioned figures.
Kors regression transformation has advantages over Frank’s orthogonal lead system. It is much easier to identify myocardial infarction [
41,
42,
43,
44,
45,
46], hypertrophy [
47,
48,
49], and ischaemic heart disease [
50,
51] from the VCG compared to the ECG. However it is important to properly estimate the VCG from the ECG. Transformations need a lower number of required leads in comparison to Frank’s orthogonal lead system combined with the conventional ECG system. To better diagnose heart diseases, doctors often require both signals at once. In the case of a successful extraction of the VCG from the ECG, it is possible to identify multiple symptoms, such as the QRS-T angle, QRS-T spatial angle, heart axis slope, and QRS loop vector size [
52]. A high accuracy of Kors regression transformation could lead to a complete deprecation of Frank’s orthogonal lead system, which also provides higher comfort for patients.
Apart from the presented solutions, a number of different VCG transformations were tested by other teams, with varying results. Other teams carried out experiments based on affine transform, PCA, or transformation methods based on artificial neural networks that offer even higher accuracy [
25,
26,
27]. These transformations will be a basis for our future research.
VCG systems based on transformations are much cheaper to conventional Frank’s orthogonal lead system, since it does not need a dedicated hardware platform. Newly graduated doctors could potentially choose this system for their clinical practice, since it has significantly lower initial costs.
6. Conclusions
The accuracy of the ECG transformation to the VCG is very important for possibly determining symptoms of hypertrophy, ischaemic heart disease, etc. A total of five different transformation methods, which were tested on PTB database recordings, were selected for the testing. The results of the transformations were compared with the original Frank’s leads measured on the basis of the MSE and R. From the results, the medians of the individual results were determined and statistical analysis was performed with emphasis on Kors regression transformation in relation to other transformation methods. The results were determined separately for each lead. A nonparametric statistical test was used to calculate the p-values by means of which the significance of the difference of values between the transformations was determined.
During testing, Kors regression transformation was statistically the most accurate for X and Y leads compared to other transformation methods. The smallest accuracy of transformation was provided by Kors quasi-orthogonal transformation. The difference between Kors regression transformation and the PLSV and QLSV methods was not statistically significant for Z lead. Overall, the results showed that Kors regression transformation was the most accurate of the methods tested and could be significant for transforming the ECG into the VCG for automatic computer analysis of ECG recordings based on VCG symptoms. Evaluating the accuracy of transformation methods for patients suffering from some heart disease and comparing the effects of transformations on pathological recordings could be the subject of the next work. Furthermore, newer types of transformation methods could be added. In the future, ECG transformations into VCG could be used in clinical practice, which could facilitate diagnosing heart diseases.
2-D/3-D graphs (
Figure 7,
Figure 8,
Figure 9 and
Figure 10) were reconstructed after processing tested transformations via statistical analysis. 2-D/3-D graphs consisting of VCG loops are very important in clinical practice, as they are provide more information than the ECG and therefore help doctors identify heart diseases. Comparisons of visual signals correlates with statistical analysis. Kors regression transformation tend to be the most similar to Frank’s orthogonal lead system, so its safe to say that it could be used for morphological analysis of 2-D/3-D graphs (for example pathological phenomena).