The later part of the atrial Ta wave is not observed in sinus rhythm subjects. The results of the present study are valid only for resting, supine male subjects in sinus rhythm and in patients with AV block. This is in agreement with previous studies done on the Ta wave. In the present study, the MLL system was used to record the Ta wave in sinus rhythm subjects. The lead system needs to be evaluated in patients, perhaps with atrial infarction or even pericarditis, to see what changes might be found in the P-Ta segment.
Since atrial repolarization waves are often hidden within the QRS complex, they remain difficult to analyze. Better delineation of atrial Ta waves may help to decode the mysteries surrounding atrial repolarization. Repolarization abnormalities are the major electrophysiological substrate for arrhythmias. Analyzing the Ta waves helps in understanding the mechanism of the onset of atrial arrhythmias. The Ta wave measurement will be especially important in decoding the mechanism for the triggers of atrial fibrillation, which is a very common clinical problem.
Precise measurement of Ta waves will also help to measure the P-Ta interval, which is the atrial equivalent of the QT interval. Analyzing the P-Ta interval dynamics will shed further insight into the mechanism of atrial tachycardia. Further studies are warranted to determine the effect of various anti-arrhythmic drugs on the Ta wave duration.
The authors would like to thank Professor Peter Macfarlane of the University of Glasgow for reviewing the draft manuscript and for suggesting the improvements to the text. Conflict of interest: None declared. Peer-review: Externally peer-reviewed. Authorship contributions: Concept - S. National Center for Biotechnology Information , U. Journal List Anatol J Cardiol v. Anatol J Cardiol. Published online Dec Author information Article notes Copyright and License information Disclaimer. Address for Correspondence: Dr.
Accepted Jul 4. This article has been cited by other articles in PMC. Abstract Objective: In the present study, a modified limb lead MLL system was used to record the Ta wave in sinus rhythm and with AV block in male patients. Introduction Human atrial depolarization is represented by the P wave and it is well observed and recorded by the standard lead ECG in sinus rhythm subjects. Table 1 Basic statistics of the age of the subjects studied. Open in a separate window.
Modified limb electrode placement The modified limb electrode placement 22 of the MLL system is briefly described as follows Fig. Figure 1. Results The lead ECG recorded on male subjects in sinus rhythm revealed no trace of cardiac disorders. Figure 2. Figure 3. Figure 4. Figure 5. SD - standard deviation; all values are in microvolts. Mean S. P wave duration I 87 5.
SD - standard deviation; all values are in milliseconds. Discussion Placement of the modified limb electrodes The modified limb electrode placement 22 that produces the MLL system is designed according to a the movement of atrial depolarization and repolarization wavefronts spreading inferiorly from the SA node and b the direction of the mean electrical vector of atrial depolarization, as well as that of ven tricular depolarization.
Study limitations No female subjects were involved in this study. Clinical implications Since atrial repolarization waves are often hidden within the QRS complex, they remain difficult to analyze.
Acknowledgments The authors would like to thank Professor Peter Macfarlane of the University of Glasgow for reviewing the draft manuscript and for suggesting the improvements to the text. Footnotes Conflict of interest: None declared. References 1. Briggs KL. A digital approach to cardiac cycle. Clinical observations on the T wave of the auricle appearing in the human electrocardiogram.
J Clin Invest. A study of electrical activity in the auricles. Am Heart J. Atrial repolarization-its importance in clinical electrocardiography. Atrial T Ta loop in patients with AV block: a trial to differentiate normal and abnormal groups. Atrial T Ta wave and atrial gradient in patients with AV block. Detailed ECG analysis of atrial repolarization in humans. In [ 13 ], the problem was initially addressed via computer simulation. Without involving inverse problems, [ 13 ] facilitated a forward model, which mapped current dipoles onto atrial mid-myocardium to surface ECGs under a set of predetermined parameters to understand the contributions of right and left atria activities to the observed P waves during sinus rhythm and atrial fibrillation.
In clinical situations, Ta waves are not only hidden, but P waves are also tangled with the adjacent QRS complex. In [ 7 ] and [ 8 ], statistical and signal processing methods were, respectively used to single out P waves from the rest of the QRS part. Surface signals are complex combinations of current stimuli from millions of cardiomyocytes; thus, the signal separation task must be performed at the level of myocardium cells, and solving an ill-posed inverse problem is inevitable.
Pioneer source models, such as those in [ 14 , 15 , 16 ], have been integrated into and advanced for contemporary computational electrocardiology models that establish models for various processes ranging from cellular bioelectrical activities to body surface potential distribution [ 17 , 18 , 19 ].
The forward problems involve mapping from inter- to intra-cellular currents onto body surface potential distributions [ 20 , 21 ]. Given the nature of complexity in the biological field, the large degree of freedom poses a challenge in evaluating inverse problems. When dealing with the inverse problem, most regulation methods can only condition numerical difficulty from the mathematical point of view; however, the problem of multiple solutions must be addressed by restoring possible missing constraints.
In [ 22 ], this difficulty was solved by reducing the inverse problem in limited mapping from epicardial to body surfaces.
In [ 23 ], the inverse problem was addressed by proposing a special equipment that could collect thousands of potentials on the body surface instead of the standard lead ECG. Activation time sequencing or imaging can be evaluated for primitive diagnosis without retrieving detailed cellular activities [ 24 , 25 , 26 , 27 , 28 ]. However, electric current constraints from the ionic behavior of individual cells, such as in [ 17 ], impose additional computational challenge given the millions of myocytes.
Moreover, additional constraints introduce other unknown parameters, and the degree of freedom remains high. Therefore, advanced statistical methods should be applied to obtain reliable solutions. Existing studies continuously contribute to addressing the problem of physiological models, and specific body surface electrical data from patients are always being corrupted by noise and the incorrect construction of organ geometries.
In [ 29 ], spatial covariance in a volume conductor was facilitated for maximum a posteriori MAP equation. In [ 30 ], temporal and spatial covariances were estimated under certain mathematical assumptions based on structures that were inherent in the space—time correlation matrix.
In [ 31 ], the facilitation of multiple information sources to improve the efficiency of Bayesian MAP formulation was suggested. In [ 32 ], TMPs were constrained using a diffusion—reaction model from cellular activation dynamics, which limited the inverse problem in both spatial and temporal dimensions. This work suggested relying on a statistical method to address both model and data errors in terms of prior knowledge on cell current dynamics and evidence for surface potential data.
In [ 33 ], the progress of statistical identification from the perspective of systems biology was reviewed. As mentioned earlier, the extraction of P waves should be conducted at the electric current level in myocardial sources. The model for the cardiac computational system comprises two parts according to the component guideline in [ 34 ].
The first part involves mapping between body surface potentials and intra-cellular TMPs. Evaluating TMPs is considered a difficult inverse problem given a potential map of a body surface [ 35 , 36 ]. The second part aims to constrain the inverse problem, in which the constraint describes changes in TMPs in terms of electrical propagation between myocardia. Most electrophysiological models are diffusion—reaction systems [ 17 , 36 , 37 , 38 ]. We first consider the forward problem from equivalent current—dipole sources to body surface potentials.
The sources of bioelectric currents across cell membranes excite the movement of cardiomyocytes and induce potential fields, which can be detected via surface electrodes. To model equivalent current density, the entire myocardium is divided into grid meshes.
Following the suggestion in [ 39 ], boundary element methods are applied. By tessellating and vectorizing all cardiac and thorax surfaces, a discrete matrix Eq. The geometric coordinates are segmented and discretized via magnetic resonance imaging MRI or computed tomography for a specific patient. Given numerical sensitivity and unavoidable movement, the forward model may suffer from geometric errors and should be incorporated as a part of modeling [ 9 , 41 ]. In [ 42 ], geometric errors were suggested to be overcome by using Bayesian MAP estimation or Kalman filtering with Gaussian geometric errors.
In the present study, we do not rely on the accuracy of geometry and conductivity. We estimate the parameters along with the process of estimating TMPs [ 43 , 44 ]. Bayesian estimation in error covariance enables performance analysis to statistically characterize solutions. Phenomenological models focus at the macroscopic level and ranges from 2-variable equations [ 14 , 37 ] to the complicated variable Luo—Rudy model [ 45 ].
Resolution is not a concern in extracting P waves. Electrical propagation is captured using the reaction—diffusion system [ 37 ] with the same setting as that in [ 46 , 47 ]. Considering the balance between precision and computation, a simple system is sufficient to constrain the ill-posed inverse problem.
Therefore, we adopt the system from [ 37 ] as follows:. By converting the equation into finite element meshes [ 47 ], the reaction—diffusion system can then be used as an effective constraint in solving the inverse problem. Our problem contains a large number of uncertainties, and thus, advanced Bayesian statistics can be a viable approach [ 44 ].
When 1 and 2 are combined, we obtain the data model as follows 3 :. To deal with a large number of parameters, the guideline in [ 46 ] and [ 47 ] indicates that the complicated joint distribution in data model 3 can be formulated as a hierarchical model and factorized into a series of conditional distributions. Therefore, the joint posterior distribution can be written in a hierarchical form as follows:.
Following the suggestion in [ 47 ], a Monte Carlo Markov chain MCMC slice sampler [ 48 ] is applied in the Bayesian computation model because of the high dimension in our complex problem. A full Bayesian analysis of this problem is achieved by sampling the joint posterior distribution 13 using an MCMC technique called slice sampling [ 49 ]. Another potential solution for reducing the constraining effects of prior knowledge is the simultaneous estimation of the TMP dynamics and electrophysiological properties of the myocardium.
This method has the advantage that the constraining models can be modified according to the collected data of patients with filtering of unknown parameters. To conduct the following experiments, 3D geometric models of a complete heart and torso are necessary.
Cardiac geometric data were adopted from the ECGSim data set, which described a healthy normal young male using complete atria and ventricles Fig. Given that a 3D imaging will not be constructed on the epicardial surface, the requirement for grid size is low. Resolution is further reduced to prevent the introduction of excessive numerical difficulties from the source of the standard lead ECG. The geometry of a torso was adopted from the PhysioNet data archive, which also originated from the body surface mapping data of Dalhousie University [ 51 , 52 , 53 ].
Although accuracy is not a concern, mapping between surface nodes to the electrode positions of standard leads should be specified. Given the well-prepared recording and documentation in the data set, the detailed mapping from the surface nodes to the 15 standard leads was elaborated. The signals were preprocessed to eliminate electromagnetic interference, baseline wandering e.
This study develops a model that retrieves hidden atrial repolarization waves by solving an inverse problem from surface ECG to cardiac TMPs Fig. The modeling approach can only be maintained at a coarse level because the source data are limited by the number of channels in the standard lead ECG. By contrast, cardiac electrical signals can be estimated by being modeled as a stochastic process with unknown excitation parameters and continuous acquisition of signals.
In the solving process, several issues are encountered and need to discuss further. The experiment presents good results. As shown in Fig. The figure reflects the correct excitation sequence starting from the atrium to the end of the apex. When we multiply the entire TMPs to the transfer matrix, the forward problem restores the original ECG, as shown in the third panel.
The figure exhibits good approximation of the original ECG second panel , except for several ripples near the end of the cycle. This result is considered good because the resolution is under 14 nodes on the body surface and 20 nodes in the myocardium.
The bottom panel shows the extracted atrial electric activities. Each line in the graph corresponds to one of the 14 nodes that constitute the standard lead ECG.
The error caused by spatial digitization may overwhelm other sources of error terms because of the resolution limitation in mesh grids. The errors caused by modeling, geometric, and measuring uncertainties are integrated into a reaction—diffusion system, which is a stochastic state-space system with a powerful capability to estimate true values among noise background. The two-variable reaction—diffusion system 2 is linearized for convenient computation experiments. All the parameters are embedded into each individual matrix element, which will be estimated in the subsequent MCMC step.
Although sophisticated and accurate models for cardiac electrophysiological dynamics can be used, the estimation results may not improve in our case. One of the error sources originates from the inhomogeneity and anisotropy of conductivity among different torso tissues.
To obtain an acceptable imaging of the heart surface, the anisotropy of the intracellular conductivity tensor along the fiber structure of myocardia must be considered.
In this study, we adopt a real-time estimation approach and leave all the parameters to be estimated during the online training step. As mentioned earlier, given the lack of spatial resolution, we do not expect the complete accuracy of the inverse and forward simulations. Although an ordinary Moore—Penrose pseudo-inverse can obtain a perfectly low root-mean-square error after performing backward and forward projections, the cardiac potential maps obtained from this pseudo-inverse are meaningless.
Therefore, the computation objective is to reconstruct a TMP distribution that closely mimics a real TMP distribution [ 17 ]. We determine that the sequence and heterogeneity of TMPs throughout the myocardium significantly influence the final waveform of surface ECG. The selection of prior distributions and initial conditions can influence the final results of the inverse solution when the Bayesian approach is used.
During our MCMC estimation step, we perform our experiments by assuming that the noise and error terms are Gaussian distributed. Accordingly, we also select Gaussian as their conjugate prior for assigning prior distribution. The Gaussian assumption is reasonable for the coarse computation level.
If further investigation is conducted, then non-Gaussian or empirical distribution can be used to estimate unknown parameters under the hierarchical Bayesian estimation framework. By contrast, initial conditions are considerably sensitive to the algorithm. If parameters or matrices are available in published studies, then we will use their well-known values as initial amounts.
This occurs because the last cells to depolarize are located in the subepicardial region of the ventricles and these cells have shorter action potentials than found in the subendocardial regions of the ventricular wall. So, although the depolarization of the subepicardial cells occurs after the subendocardial cells, the subepicardial cells undergo phase 3 repolarization before the subendocardial cells.
Therefore, repolarization waves generally are oriented opposite of depolarization waves green versus red arrows in figure , and repolarization waves moving away from a postive recording electrode produce a positive voltage.
The T wave is longer in duration than the QRS complex that represents depolarization. The longer duration occurs because conduction of the repolarization wave is slower than the wave of depolarization. The reason for this is that the repolarization wave does not utilize the high-velocity bundle branch and purkinje system, and therefore primarily relies on cell-to-cell conduction.
Sometimes a small positive U wave may be seen following the T wave not shown in figure at top of page. This wave represents the last remnants of ventricular repolarization. Inverted T waves or prominent U waves indicates underlying pathology or conditions affecting repolarization. The QT interval represents the time for both ventricular depolarization and repolarization to occur, and therefore roughly estimates the duration of an average ventricular action potential.
This interval can range from 0. At high heart rates, ventricular action potentials shorten in duration, which decreases the QT interval. Because prolonged QT intervals can be diagnostic for susceptibility to certain types of tachyarrhythmias, it is important to determine if a given QT interval is excessively long. In practice, the QT interval is expressed as a "corrected QT QTc " by taking the QT interval and dividing it by the square root of the R-R interval interval between ventricular depolarizations.
This allows an assessment of the QT interval that is independent of heart rate. Normal corrected Q-c intervals are 0. There is no distinctly visible wave representing atrial repolarization in the ECG because it occurs during ventricular depolarization.
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