Forward problem of electrocardiography based on cardiac source vector orientations

To localize the unusual cardiac activities non-invasively, one has to build a prior forward model that relates the heart, torso, and detectors. This model has to be constructed to mathematically relate the geometrical and functional activities of the heart. Several methods are available to model the prior sources in the forward problem, which results in the lead field matrix generation. In the conventional technique, the lead field assumed the fixed prior sources, and the source vector orientations were presumed to be parallel to the detector plane with the unit strength in all directions. However, the anomalies cannot always be expected to occur in the same location and orientation, leading to misinterpretation and misdiagnosis. To overcome this, the work proposes a new forward model constructed using the VCG signals of the same subject. Furthermore, three transformation methods were used to extract VCG in constructing the time-varying lead fields to steer to the orientation of the source rather than just reconstructing its activities in the inverse problem. In addition, the unit VCG loop of the acute ischemia patient was extracted to observe the changes compared to the normal subject. The abnormality condition was achieved by delaying the depolarization time by 15ms. The results involving the unit vectors of VCG demonstrated the anisotropic nature of cardiac source orientations, providing information about the heart’s electrical activity.


Introduction
An action potential is a transmembrane potential (TMP) that occurs during the excitation of cardiac muscle cells [1].It consists of an activation (depolarization) and a recovery (repolarization) phase.To monitor and measure myocardial depolarization and repolarization, the electrodes are positioned on the surface of the body to measure the electrical activity of the heart.To examine the status of the heart's electrical activity non-invasively, electrocardiography (ECG) is a crucial medical diagnostic tool that aids in analyzing the variations in electric potentials on the body's surface.Before using invasive techniques or methods, cardiologists would prefer diagnostic procedures that reveal information about the structure and electrophysiology of the heart, such as ECG, magnetocardiography (MCG), computed tomography (CT), x-ray, cardiac magnetic resonance imaging (MRI), and, many others, to identify the location of the abnormality on the surface of the heart.
With acceptable accuracy, it is possible to practically examine the functionality of living organisms by constructing a model replicating the function.Investigating the activity of living tissue at the cellular and organic levels is made possible by bioelectromagnetism.As a result, it is possible to explore the function of bioelectric sources and conductors, which might then be quantitatively studied, by creating models that precisely depict the bioelectric behavior of the tissue they represent [2].The importance of non-invasively identifying cardiac problems has led to the development of several methods for heart modeling.Numerous techniques based on mathematical function methods [3] that are less reliant on the anatomical composition of the heart were suggested in the research on ECG signal simulation.Another group of researchers presented several complex mathematical models for simulating ischemia signals and real ECGs [4][5][6].
Boulakia et al [3] described the mathematical model for the ECG simulation numerically based on partial differential equations as a significant advancement in modeling ECGs.Bidomain equations with the Mitchell-Schaeer phenomenological ionic model, along with anisotropic conductivities and transmural action potential duration (APD) heterogeneity, were used in this article to illustrate the heart's electrical activity.Gidea et al [4] proposed another model for simulating the heart's electrical activity by integrating a quasi-periodic ECG signal with the Taylor-Chirikov standard map.Depending on deterministic chaos, these models replicated the heart's self-regulatory function and exhibited intrinsic non-linearity.Campos et al [5] used a three-dimensional thin-walled cardiac tissue preparation model to study the effect of 'superfusion-induced heterogeneities' on activation signals.The model considered the inner muscle fiber's electrophysiological variations for ischemia phases 1a, 1b, and myolysis.
Furthermore, a hardware membrane model that replicates the action potential of cardiac cells was constructed by Maeda et al [6] to study the synchronization in a diffusively coupled system.The acceleration phenomenon was achieved by a connected system created by two models with identical-period temporal waveforms but varied plateau durations.Fallahi et al [7] demonstrated cardiac action potential generation modeling by the standard electric model of cardiac cells developed by Maeda et al [6] and an ECG signalproducing mechanism in the heart.The presented model parameters were varied to simulate the cardiac ischemic signal.
The above literature addresses numerous approaches for simulating ECG signals, whereas another group of researchers focused on computer simulations of activation sequences spreading throughout the surface of the heart.One such interactive simulation tool for modeling the 3D heart-torso model is ECGSIM.The simulation domain aspires to function as both a research and educational tool.It primarily investigates the relationship between the electric current sources of the heart and the corresponding ECG signals.The simulator provides information related to the potential fields on the body's surface and those on the surface of the heart [8].Huiskamp et al [9] employed the uniform double layer (UDL) and inhomogeneous volume conductor as a source and volume conductor, respectively, and the boundary integral method for computing ventricular activation sequences from body surface potentials.It was shown that the model determined the depolarization sequences on the heart's surface that closely resembled the information accomplished via invasive measurement.
Oosterom et al [10] described the numerical method for simulating the ECG signals on the thorax using an equivalent double layer (EDL) as a source model.The simulation study used a realistic thorax model to compute transfer factors between the cardiac current sources and the corresponding potentials on the surface of the body and heart.The authors suggested that the strength of the local source and timings of depolarization and repolarization on the ventricular surface were interactively changed in the simulation study.Concentrating on the physiological and clinical importance of activation and recovery sequences of the heart, Dam et al [11] described the inverse method using the EDL.The depolarization and recovery timings relied on the fastest route algorithm and electronic effects.Tysler et al [12] explained the robustness of the inverse solutions by two methods.A single and a set of five dipoles with the Gaussian noise included in input data were investigated to localize the sizes of various lesions, with the latter one providing better results for large ischemic lesions.Their work less concentrated on torso, heart geometry, and orientation uncertainties.
The information about the surface level of the heart plays a prominent role in the appropriate diagnosis of cardiac disorders.Solving the forward and inverse problems is key to obtaining this prominent information.Estimating cardiac activity at a particular location in the inverse problem avoids the need for invasive procedures.However, reconstructing the activity considering only location will not be efficient as the abnormality can occur in any portion of the heart.To appropriately diagnose the disease in any region of the heart, there is a need to focus on the orientation of the cardiac sources.
The authors in [9] worked on the computation of activation sequences of the ventricles with less priority on the orientation and position of the heart.In [13], the article focused on the magnitude and location of the electrophysiological sources, excluding the orientation.Tysler et al [12] employed multiple dipole and inhomogeneous torso models with the boundary element method (BEM) to solve the inverse problem but less focused on uncertainties of torso and heart geometries.Later, as an advancement, Lu et al [14] used the single equivalent moving current dipole (SEMCD) and piecewise homogeneous volume conductor model and the Nelder-Mead (NM) simplex algorithm to demonstrate the changing magnitude, orientation, and position.However, the study was limited to the single dipole.Hu et al [15] worked on a homogenous volume conductor using the modified FitzHugh-Nagumo (FHN) model called the Aliev-Panfilov model and Galerkin finite element method for solving the forward problem.The cardiac fiber orientation of a simple heart-torso model was not concentrated much in this article.
The above methods did not concentrate much on the orientations of the distributed source models, as the focus was mainly on stationary sources in the noninvasive procedures.The forward problem in the present work involves modeling the discretized heart and thorax to generate the transfer matrix based on vectorcardiography (VCG) signals to emphasize anisotropic orientations of the cardiac sources.

Method 2.1. Forward problem
In ECG, the forward problem comprises potential distribution calculation generated at the volume conductor's surface due to the presence of the chosen source model within it [16].Forward problems are generally well-posed as they have a unique solution that remains unaffected by minor variations in initial parameters [2].

Source model
The distributed source model of 256 cardiac sources is considered as a source model in the forward study.The heart is located within the thorax at a position (0.0364, 0.0351, −0.0022), with conductivity, s throughout the torso volume.Figure 1 illustrates a visualization of the heart positioned within the torso with eight electrodes (V1-V6, Lead I, and Lead II) attached to its surface in SCIRun as per locations in table 1.
The cardiac currents generate an electric field within the torso, and potentials distributed on the body's surface are determined by employing finite element method (FEM).The computation of FEM was performed on the discretized torso (volume conductor) model of 300 nodes and 596 tetrahedral meshes.Figure 2 shows the visualization of the calculated potentials distributed on the surface of the torso geometries in the SCIRun environment using the ViewScene module.The generated transfer matrix based on the activation times, signifies the spatial relationship between the elementary double-layer sources around 256 nodes evenly distributed over the myocardial surface S h to 8 ECG electrodes on the torso.The computation of the lead field focused on a piecewise homogeneous with a multi-compartmental volume conductor model.This model mainly specifies the various inhomogeneity interfaces, like the lungs and the blood-filled cavities.These compartments had conductivity values of 0.2 S/m for lungs and 3.0 S/m for blood filled cavities [17][18][19], [20].These conductivity values were considered to be relative to the overall conductivity of the body.Considering this, the conductivity value of 0.2 S m / was used in the present study for the forward problem computation.The transfer matrices were computed using the finite element method (FEM) of dimension 8 256. 2.1.2.
Conventional lead field After modeling the heart and volume conductors, the next step is generating the lead field, which provides the spatial sensitivity between sources and electrodes.The prior locations on the heart's surface are assigned with the unit strength, and the vectors are chosen to be unit normal vectors of source locations.The forward problem is expressed as: Here, dipolar current sources are located on the ventricular surface with the directions pointed along the unit vector n ˆas S n d dS H . , = represents the Heaviside step function corresponding to the cardiac sources at r ¢ locations activated at the time r ( ) ¢ t [18].r, r A( ) ¢ includes the transfer matrix that relates the geometries of the heart and torso as a volume conductor.In this article, for a node at any location on the myocardial surface S h has been assigned with the transmembrane potential (TMP) of a ventricular muscle cell, as given in figure 3.

Proposed VCG based lead field
The orientations of cardiac sources parallel to the detector plane cannot always effectively diagnose the disorders present in the other region of the heart.The proposed methodology focuses on the fibre orientations based on the depolarization times, whereas the conventional method assumed the orientations to be normal to the heart surface.The nodes of the heart are assigned with the transmembrane potentials, and the orientations are assigned with the unit vectors of the VCG to represent each cardiac node with varying orientations [19].
VCG is a diagnostic method that provides information about the electrical activity of the heart, representing the vector's spatial orientation and magnitude at every moment [21].The VCG can be extracted from the standard 12-lead ECG system in clinical applications, employing the VCG transformation matrices [22][23][24].Kors regression transformation method (KRT), Inverse Dower transformation (IDT), Q Least Square Value Transformation (QLSV), P Least Square Value Transformation (PLSV), and Kors quasi-orthogonal transformation are some of the methods used to extract VCG from the measured ECG signals based on transformation matrices [22,23,25,26].These transformation matrices can be obtained based on a mathematical model of the torso (e.g., inverse Dower) or by regression approach for directly measured ECG and VCG (e.g., the Kors transformation) [27].VCG signals with the 3-orthogonal leads than the 12 lead ECG have proven to be useful methods for diagnosing various cardiac disorders, such as myocardial infarction, ischemia, ventricular hypertrophy, myocardial scars, long QT syndrome [22,23,[25][26][27][28].
The current work focuses on employing the unit vectors of VCG extracted using Kors Regression Transformation (KRT), Inverse Dower Transformation (IDT), and Linear Regression Based Transformation (LRBTQLSV) [26,27] methods with an emphasis on QRS complex.Here, an abnormal condition of unusual R-wave amplitudes in acute ischemia is indicated to highlight the significance of the unit vector VCG-based lead field in the non-invasive detection of the mentioned cardiac disorder.A brief explanation of the transformations used in this paper is as follows: Kors Regression transformation (KRT): Kors Regression transformation (regression-based approach) is one of the most widely used VCG extraction approaches to obtain the leads X and Y more accurately than other methods [26].This approach relies on mathematical regression to calculate the coefficients of the matrix.The reconstruction coefficients in [26] were obtained by minimizing the Mean Squared Error (MSE) between the measured and derived VCG [27].The studies [23] and [26] described the Kors regression-based transformation as suitable for representing the QRS-T angle, as the values obtained from using this method did not differ much from the values of Frank's lead system.
Inverse Dower Transformation (IDT): Another transformation method, whose matrix coefficients relied on the torso model (model-based approach) [29] described by Frank [30], was introduced by Dower et al [31] for extracting 12 lead ECGs from 3 lead VCG.Later, Edenbrandt et al [32] derived a pseudoinverse matrix to the matrix derived by Dower for the reverse transformation from 8 lead ECG to VCG [27].The pseudoinverse transformation coefficients for the procedure were extracted from [26,27].The procedure of deriving the transformation matrix uses the Moore-Penrose pseudoinverse.Compared to KRT, evaluating the QRS-T angle and atrial fibrillation cases using IDT in [31] and [27] did not produce the expected results.
Linear Regression-Based Transformations (LRBTQLSV): Many transformations have been derived to improve the accuracy of only some parts of the cardiac cycle to achieve the same accuracy as the regression approach [22,25].Guillem et al presented their transformation matrix derived using the regression method, which is optimized for the Q wave.The matrix was called QLSV (Q Least Square Value) [27,29].A regression model was found for each patient using the least squares method.The resulting transformation matrix was given by the mean value of the transformation matrices for all patients [27].In the Least-square value (LSV), with an emphasis on the QRS complex, a matrix for transformation (QLSV) was obtained by focusing on the interval from the beginning to the end of the QRS complex [26].The LRBTQLSV provides better results in detecting Z leads [26,27].
The regression transformation proved statistically the most accurate for X and Y leads compared to other transformation techniques.For Z lead, there was no statistically significant difference between the PLSV and QLSV approaches and the Kors regression transformation [25].Regarding physiological records, the Z lead was obtained most accurately by the QLSV approach and the X and Y lead by the Kors regression method.The most reliable technique for pathological recordings for all leads was Kors regression [23].The IDT and the KRT are the most accurate approaches; they lag slightly behind the accuracy of the P and T waves.
The present study solved the forward problem by allocating the prior source locations with the unit VCG based oriented vectors.The block diagram in figure 4 illustrates the procedure for deriving the VCG and solving the suggested forward problem.The transformation matrices 'T's derived coefficients depend on the described torso model or the regression methods that rely on measured data from the patient [26,27].
The work uses transformation matrix T [25] to generate dynamic lead fields based on unit vectors of VCG.The VCG vectors are formed by projecting T on the eight-lead ECG.

VCG
T ECG . 2 ´is a matrix with rows corresponding to the VCG leads (X, Y, and Z).Further the unit vector of the derived VCG is computed to assign the cardiac sources with varying orientations.To implement the proposed methodology, a few assumptions must be considered in the forward problem.
On the surface of the heart, S h the activity of the ventricles comprising of the entire ventricular tissue can be expressed as a uniform double layer (UDL).The source amplitudes assigned to any position on S h are proportional to the TMP h f of the cardiac cells at r¢ discrete locations on the boundary of the heart.The TMPs are formulated and shown as Heaviside step functions.
The cardiac source's maximum amplitudes are assumed to be identical at any position on the S , h for the uniform double layer source.In the UDL model, the strengths of dipoles were assumed to be oriented along the unit normal.In the proposed work, the strengths of dipoles were slightly changed to move/ vary in the direction of unit vectors derived from the VCG loops.The orientations of these unit vectors were linked with the activation times of the discrete nodes.The slightly modified forward problem is expressed as, with t corresponding to the activation times, electrode locations (r') and sources (r) as In the present study, the abnormality condition was evaluated for the increased R-wave amplitude of acute ischemia cases.In the normal subject, the amplitude of the R-wave increases gradually from V1 to V5 followed by the decrease in amplitude from V5-V6.Here, the activation times of the chosen source (node index 32) and its surrounding region from the posterior region of the heart were delayed by a duration of 15 ms resulting in the increased R-wave amplitude [17] in the ECG electrodes as shown in figure 5, particularly in V6, which is unusal.The hexagonal patch of around 2 cm of the central node 32 from the posterior region is considered as abnormal.Figure 5 shows the normal and abnormal transmembrane potentials (TMP) of the node index 32.The activation time deferred by 15 ms increases R-wave amplitude, resulting in the corresponding changes in the ECG detectors.Further, the cardiac sources are assigned with unit VCG KRT vector based orientations to analyze the efficiency of the generated transfer matrix in detecting the cardiac disease non-invasively.
Further, the derived VCG and unit VCG loops from KRT, IDT and LRBTQLSVare shown in figure 6.The unit VCGs computed indicate the QRS-T loops, with the KRT achieving better results in the R-peak detection.In addition the phase angle between the static and the unit VCG KRT vector orientations for the normal and diseased case is given by where u VCG ˆcorresponds to unit VCG KRT vector orientations assigned to the distributed source models for normal and abnormal subjects extracted using the KRT approach.The n, ˆrepresenting the static unit normal vectors are compared with the aforementioned time varying vectors.

Results and discussion
The comparative analysis was performed to observe the changes in cardiac source orientations obtained using different transformation methods with the static unit vectors oriented normal to the heart surface.The vectors were chosen as orthonormal unit vectors of source locations, as shown in figure 7(a) in the static lead field.Apart from identifying only the abnormality region, the orientations were allocated to the cardiac sources based on the unit VCG vectors derived from IDT, KRT, and LRBTQLSV methods for the mentioned T [25] in the dynamic lead field.
The limitation of a static lead field with the orientations normal to the surface of the heart as given in figure 7(a) was overcome by including the unit VCG vectors in the transfer matrix [33].The transfer matrix was computed using the SCIRun module 'BuildMa-trixOfSurfaceNormals.Figures 7(b)-(d A cardiac disease for the increased R-peak amplitude was estimated for the time-varying lead field based on unit KRT vector orientations.This variation in the individual source models wasachieved by loading the dipoles with VCG orientations extracted using KRT method into the SCIRun network along with the spatial geometries of thorax, heart and eight ECG electrodes [19,34].The 'GetColumnOrRowFromMatrix' module of the SCIRun network collects the varying orientations that were to be assigned to the distributed source models. The source model has 256 distributed source models assigned with the static and VCG based unit vectors.In figure 8(a), the source models with orientations perpendicular to the myocardial surface of the heart localize the cardiac anomalies present in that orientation.With an intention to focus on cardiac diseases occurring in all the orientations, the generated lead field with the changing orientations was assigned to the cardiac sources.accordingly based on the coefficients of transfer matrices.
The comparison of unit vectors oriented perpendicular to the source locations with the assigned unit VCG IDT, unit VCG KRT, and unit VCG LRBTQLSV vectors in figures 9(b)-(d) at time t = 30 ms, 80 ms, and 150 ms, respectively, were performed to analyse the varying source orientations and to understand how they differ from the conventional ones thereby helping in identifying the abnormal cardiac region effectively.The changing orientations can help in diagnosing acute myocardial infarction (AMI) accompanied by the left anterior fascicular block (LAFB) and inferior infarction extended to the left ventricle (LV) anterior wall with greater sensitivity [29].Figure 9(a) depicts VCG loops demonstrating the heart's electrical activity.The arrow pointing to the loop region indicates the occurrence of R-peak, where the maximum electrical activity of the ventricles is observed for each transformation applied.
The proposed lead field was evaluated for its efficiency in the non-invasive localization of cardiac disorders occurring in any region of the myocardial surface.A hexagonal patch with 7 nodes from the posterior region of the myocardial surface was assigned with the time varying unit VCG KRT vectors.Table 2 tabulates the node index, with their normal and abnormal activation times.The abnormal activation of the nodes with the delayed initiation was considered till the 2 cm surrounding the central node 32 of the patch.The orientations with unit VCG vectors of the selected cardiac sources derived from the KRT transformation method were compared with the static unit vectors for the normal and abnormal TMPs at time instants 30 ms, 80 ms and 150 ms respectively.These results demonstrate the orientation changes, indicating the possibility of cardiac abnormality identifications in all regions of the heart.
The phase angle q was calculated between the static and time varying KRT based orientations for normal and abnormal cases.The results tabulated in table 2 showed an angular variation for the KRT extraction approach for each node at three different time instants.
The phase angle between the static and time varying orientations for normal and abnormal cases at time instants t = 30 ms, 80 ms and 150 ms is tabulated in table 2. For the variation in the activation time by 15 ms, the generated lead field was able to provide the The authors in [19,[21][22][23][24][25][26][27][28][29][30][31][32][33][34][35][36][37][38] worked on the VCG, another great medical technique for the non-contact detection of heart problems.The KRT method is the most common and accurate approach to obtain X and Y leads.It is a method with higher accuracy and is used in QRS-T angle detection, which cannot be detected using the IDT.The LRBTQLSV is another method that provides better results in detecting Z leads [25,26].Depending on the techniques employed in obtaining VCG, the changing orientations of the cardiac sources are achieved, as depicted in figures 8(b)-(d).Additionally, the potentials distributed on the body's surface vary accordingly for each method, as the coefficients of the transfer matrices [23] depend on the different parameters in each approach.The VCG loops shown in figure 9(a) indicate the R-wave, with the KRT method achieving greater efficiency than the IDT and LRBTQLSV.
The R-wave, the largest component of the QRS complex, signifies the heart's electrical activity propagating through the major portion of ventricular walls.The variation in R-wave amplitude corresponds to various cardiac disorders.Acute ischemia, an unusual cardiac condition resulting from the deferred initiation and reduced rate of rise in the action potential upstroke, results in enlarged R-peak amplitude [36].Irregular condition in this paper was achieved by delaying the normal depolarization time by 15 ms to represent the late initiation.The FEM was performed to obtain the ECGs of normal and abnormal subjects.The VCG and unit VCG loops were derived from the measured ECGs to identify the abnormality that could occur in any heart region.Figure 11 illustrates the unit    VCG loops of normal and acute ischemia cases for the KRT approach.The R-peak of the subject with normal depolarization times was present at the location (0.0159831, −0.0811992, 0.0591414).The patient with the depolarization time delayed by 15 ms had the R-peak located at (0.0159962, − 0.0812791, 0.0591126).The results illustrated the deviation in the R-wave for the suggested anomaly.

Conclusion
The present study proposed an innovative approach for constructing lead fields in the forward problem to analyze the cardiac source's electrical activity at the surface level of the heart.This method utilized a novel technique to generate lead field matrices that relied on VCG-based unit vector orientations assigned to the distributed source models to localize the cardiac abnormality non-invasively.The unit VCG vectors extracted using IDT, KRT, and LRBTQLSV transformation approaches were incorporated into the transfer matrices to set the cardiac sources in all orientations to assist clinicians or cardiologists in easily diagnosing heart disorders that go unnoticeable in ECG.The suggested methodology demonstrated the varying orientations of the distributed source models for all three VCG extraction approaches at time instants 30 ms, 80 ms and 150 ms.The unit vector VCG loops from different transformations clearly depicted the QRS-T loops.The visual respresentation of the cardiac sources activating at the different time instants with the larger arrow indicating the R-peak of the cardiac cycle was achieved in the present work.An abnormality cardiac condition of increased R-wave amplitude was evaluated for the time varying transfer matrix with the orientations based on unit KRT vectors.For the chosen nodes from the posterior patch with the abnormality spread to a distance of 2 cm, the developed transfer matrix was able to non-invasively detect the R-wave amplitude increase in electrode V6, without necessitating the need for additional electrodes.Additionally, the phase angle was computed between the static and unit VCG KRT vector orientations for the normal and abnormal subject.The suggested methodology effectively illustrated the deviation, making it capable of non-invasive locaization of cardiac disorders in any region on the myocardial surface.The difference between the maximum potential distributions of the proposed forward solutions at time t = 35 ms was compared with the static to analyze the deviation.Similarly, the deviation in potentials was computed by taking the difference between the minimum values.The extracted unit VCG loops were analyzed for the unusual cardiac condition, with increased R-wave amplitude in the abnormal loops.In future work, the generated lead fields will be used to estimate the cardiac activity in the inverse problem.

Figure 1 .
Figure 1.The visualization of the heart and torso with electrodes in SCIRun.

Figure 2 .
Figure 2. The computation of FEM on the discretized torso model.

Figure 3 .
Figure 3.The transmembrane potential of the node index 32 from the posterior region of the myocardial surface.

Figure 4 .
Figure 4. Procedure depicting the extraction of VCG and construction of the proposed forward problem.

Figure 5 .
Figure 5. Node index 32 selected from the posterior region of the myocardial surface in ECGSIM with the normal (depolarization time of 56 ms) and abnormal (depolarization time of 71 ms) TMPs resulting in the increasedR-wave amplitude in ECG electrodes.
) shows the visualization of the varying orientations of the distributed source models at 30 ms, 80 ms, and 150 ms, respectively.The time instants 30 ms, 80 ms, and 150 ms indicate the R-peak, ST and T waves respectively.The arrows of varied sizes represent each unit vectors of the cardiac nodes, signifying their depolarization times.The tiny arrows in figure 7(b) indicate the ventricular nodes which are at the verge of getting activated, while the thicker arrows in the same figure represent the occurrence of R-peak of the cardiac cycle.
Figures 8(b)-(d) depict the simulation result of the varying orientations of the 256 sources at the time t = 35 ms for unit VCG IDT, unit VCG KRT, and unit VCG LRBTQLSV based oriented vectors, respectively.Here, the orientations vary

Figure 7 .
Figure 7. (a) Distributed source models with orientations normal to the heart's surface.The varying unit VCG vector orientations based on KRT assigned to cardiac sources at (b) 30 ms, (c) 80 ms and (d)150 ms respectively, with the inset of normal QRS-T waves.

Figure 8 .
Figure 8.(a) Static unit vectors oriented perpendicular to the source locations.Cardiac sources assigned with (b) unit VCG IDT, (c) unit VCG KRT, and (d) unit VCG LRBTQLSV based oriented vectors at the time, t = 35 ms.

Figure 9 .
Figure 9.An example illustrating the VCG loops assigned to 6 nodes showing deviations from the normal oriented vectors.(a) VCG loops extracted using IDT, KRT, and LRBTQLSV transformation methods.Comparison of unit vectors oriented perpendicular to the source locations with the assigned (b) unit VCG IDT, (c) unit VCG KRT, and (d) unit VCG LRBTQLSV at time t = 30 ms, 80 ms, and 150 ms.

Figure 10 .
Figure 10.The forward solution of ECG to construct static and time-varying lead fields.Body surface potential distributions observed at t = 35 ms for (a) static, (b) unit VCG IDT, (c) unit VCG KRT and (d) unit VCG LRBTQLSV based orientations assigned on sources.

Figure 11 .
Figure 11.The simulated unit VCG loops of normal subject and acute ischemia patient.

Table 2 .
Comparison between the static and unit VCG KRT based oriented vectors of the selected nodes on the myocardium for normal and increased R-wave amplitude case at 30 ms , 80 ms and 150 ms respectively.