Ramesh M. Gulrajani

A Comparison of Monodomain and Bidomain Reaction-Diffusion Models for Action Potential Propagation in the Human Heart

A bidomain reaction-diffusion model of the human heart was developed, and potentials resulting from normal depolarization and repolarization were compared with results from a compatible monodomain model. Comparisons were made for an empty isolated heart and for a heart with fluid-filled ventricles. Both sinus rhythm and ectopic activation were simulated. The bidomain model took 2 days on 32 processors to simulate a complete cardiac cycle. Differences between monodomain and bidomain results were extremely small, even for the extracellular potentials, which in case of the monodomain model were computed with a high-resolution forward model. Propagation of activation was 2% faster in the bidomain model than in the monodomain model. Electrograms computed with monodomain and bidomain models were visually indistinguishable. We conclude that, in the absence of applied currents, propagating action potentials on the scale of a human heart can be studied with a monodomain model

Selecting the corner in the L-curve approach to Tikhonov regularization

The performance of two methods for selecting the corner in the L-curve approach to Tikhonov regularization is evaluated via computer simulation. These methods are selecting the corner as the point of maximum curvature in the L-curve, and selecting it as the point where the product of abcissa and ordinate is a minimum. It is shown that both these methods resulted in significantly better regularization parameters than that obtained with an often-used empirical Composite REsidual and Smoothing Operator approach, particularly in conditions where correlated geometry noise exceeds Gaussian measurement noise. It is also shown that the regularization parameter that results with the minimum-product method is identical to that selected with another empirical zero-crossing approach proposed earlier.

A new method for regularization parameter determination in the inverse problem of electrocardiography

Computing the potentials on the hearts epicardial surface from the body surface potentials constitutes one form of the inverse problem of electrocardiography. An often-used approach to overcoming the ill-posed nature of the inverse problem and stabilizing the solution is via zero-order Tikhonov regularization, where the squared norms of both the surface potential residual and the solution are minimized, with a relative weight determined by a so-called regularization parameter. This paper looks at the composite residual and smoothing operator (CRESO) and L-curve methods currently used to determine a suitable value for this regularization parameter, t, and proposes a third method that works just as well and is much simpler to compute. This new zero-crossing method selects t such that the squared norm of the surface potential residual is equal to t times the squared norm of the solution. Its performance was compared with those of the other two methods, using three simulation protocols of increasing complexity. The first of these protocols involved a concentric spheres model for the heart and torso and three current dipoles placed inside the inner sphere as the source distribution. The second replaced the spheres with realistic epicardial and torso geometries, while keeping the three-dipole source configuration. The final simulation kept the realistic epicardial and torso geometries, but used epicardial potential distributions corresponding to both normal and ectopic activation of the heart as the source model. Inverse solutions were computed in the presence of both geometry noise, involving assumed erroneous shifts in the heart position, and of Gaussian measurement noise added to the torso surface potentials. It was verified that in an idealistic situation, in which correlated geometry noise dominated the uncorrelated Gaussian measurement noise, only the CRESO approach arrived at a value for t. Both L-curve and zero-crossing approaches did not work. Once measurement noise dominated geometry noise, all three approaches resulted in comparable t values. It was also shown, however, that often under low measurement noise conditions none of the three resulted in an optimum solution.

A computer heart model incorporating anisotropic propagation. III. Simulation of ectopic beats.

With the advent of catheter ablation procedures, it has become an important goal to predict noninvasively the site of origin of ventricular tachycardia. Site classifications based on the observed body surface potential maps (BSPMs) during ventricular endocardial pacing, as well as on the patterns of the QRS integrals of these maps, have been suggested. The goals of this study were to verify these maps and their QRS integral patterns via simulation using a computer heart model with realistic geometry and to determine whether the model could improve clinical understanding of these ectopic patterns. Simulation was achieved by initiating excitation of the heart model at different endocardial sites and their overlying epicardial counterparts. This excitation propagated in anisotropic fashion in the myocardium. Retrograde excitation of the models His-Purkinje conduction system was necessary to obtain realistic activation durations. Simulated BSPMs, computed by placing the heart model inside a numerical torso model, and their QRS integrals were close to those observed clinically. Small differences in QRS integral map patterns and in the positions of the QRS integral map extrema were noted for endocardial sites in the left septal and anteroseptal regions. The simulated BSPMs during early QRS for an endocardial site and its epicardial counterpart tended to be mirror images about the zero isopotential contour, exchanging positive and negative map regions. The simulation results attest to the models ability to reproduce accurately clinically recorded body surface potential distributions obtained following endocardial stimulation. The QRS integral maps from endocardial sites in the left septal and anteroseptal regions were the most labile, owing to considerable cancellation effects. Conventional BSPMs can be useful to help distinguish between endocardial and epicardial ectopic sites.

Application of the single moving dipole inverse solution to the study of the wolff-parkinson-white syndrome in man

The single moving dipole (SMD) inverse solution was performed in 28 patients with the Wolff-Parkinson-White preexcitation syndrome to see if the calculated position of the SMD during the initial delta wave could indicate the site of the underlying accessory pathway. This site was first estimated to be at one of eight locations around the atrioventricular ring, from the patients QRS and ST segment body surface potential maps, as has been described by others. Next, SMD parameters were calculated during the delta wave so as to approximate, on a numerical torso model, the patients body surface potential map. Visualization of the calculated position of the SMD around the atrioventricular ring was done by projecting it on a plane parallel to this ring. This plane corresponded to the most basal transverse section of a heart model present in the torso model. One limitation was the use of non-varying heart and torso models for all patients. As a result, the SMD technique lacked the precision to separate accessory pathway sites into eight atrioventricular locations. However it was capable of distinguishing between patients belonging to the larger classes of right-sided, posterior, and left-sided preexcitation, formed by combining adjacent atrioventricular accessory pathway locations. With more accurate heart and torso models, it may be possible to increase SMD resolution so as to locate accessory pathway sites deep within the heart. This would represent an advantage over the surface potential map approach which only identifies the site of earliest epicardial breakthrough associated with the accessory pathway.

A computer heart model incorporating anisotropic propagation. II. Simulations of conduction block.

This study describes the simulation of the more common types of conduction blocks with a computer model of the heart incorporating anisotropic propagation. The rationale was to test the model as to its ability to simulate these blocks by physiologically justifiable adjustments of the conduction system alone. The complete blocks were generated by simply blocking conduction totally at selected sites in the proximal conduction system, and the incomplete blocks by slowing down the conduction velocity in the proximal system. Also simulated were the left fascicular blocks and the bilateral blocks. All simulated electrocardiograms, vectorcardiograms, body surface potential maps, and epicardial isochrones for these blocks were similar to clinically observed data, with the exception of the left posterior hemiblock, which was slightly atypical. This could be because such blocks are usually accompanied by other cardiac pathologies not included in our simulations. The model also supports van Dams observation that during left bundle branch block the passage of activation from right to left occurs via slow myocardial activation with no evidence of a local delay due to a septal barrier. Finally, the model suggests that a left bundle branch block with a normal frontal plane QRS axis may simply represent a case of an incomplete left bundle block, whereas one that exhibits a left axis QRS deviation in the frontal plane represents a more severe complete left bundle branch block.

Simulation of QRST integral maps with a membrane-based computer heart model employing parallel processing

The simulation of the propagation of electrical activity in a membrane-based realistic-geometry computer model of the ventricles of the human heart, using the governing monodomain reaction-diffusion equation, is described. Each model point is represented by the phase 1 Luo-Rudy membrane model, modified to represent human action potentials. A separate longer duration action potential was used for the M cells found in the ventricular midwall. Cardiac fiber rotation across the ventricular wall was implemented via an analytic equation, resulting in a spatially varying anisotropic conductivity tensor and, consequently, anisotropic propagation. Since the model comprises approximately 12.5 million points, parallel processing on a multiprocessor computer was used to cut down on simulation time. The simulation of normal activation as well as that of ectopic beats is described. The hypothesis that in situ electrotonic coupling in the myocardium can diminish the gradients of action-potential duration across the ventricular wall was also verified in the model simulations. Finally, the sensitivity of QRST integral maps to local alterations in action-potential duration was investigated.

Localization of cardiac ectopic activity in man by a single moving dipole. Comparison of different computation techniques

The accuracy of different computation techniques for the non-invasive localization of cardiac ectopic activity was evaluated. Body surface potentials were recorded from 63 leads in 14 patients with implanted pacemakers. The location, orientation and magnitude of a single moving dipole (SMD) were computed from the first eight terms of a truncated multipole expansion estimated from the body surface potentials. The SMD trajectories obtained during the QRS complex were plotted along with the heart outlines and pacing leads obtained independently from chest x-rays. The origin of the SMD trajectories was compared to the position of the pacing lead to evaluate the accuracy of the SMD. The optimum computation technique used a least-squares (LS) estimation of the multipole expansion truncated at 15 multipoles, in conjunction with a torso model that included regions of lower conductivity representing the lungs. With this method, the SMD trajectories originated near the pacing lead (25 +/- 12 mm) and adequately represented the progression of the ectopic wavefront across the entire heart silhouette. With the LS techniques using 8 or 24 multipoles, the spans of the trajectories were respectively too short, or too long to cover the heart, and the average distance between the SMD at QRS onset and the pacing lead was larger. With a surface integration technique, the SMD-pacing lead distances were similar, both for a finite homogeneous torso model with a fixed geometry, as well as for torso models adapted to the torso geometry of each patient. The SMD was found adequate to represent the progression of an ectopic wavefront, and to localize its origin in man.

A comparative evaluation of three different approaches for detecting body surface isopotential map abnormalities in patients with myocardial infarction

Three approaches for detecting abnormalities in body surface potential maps recorded from patients with myocardial infarction were evaluated. The maps are generated from 26 simultaneously recorded unipolar electrocardiograms. All three approaches detect the deviations in certain parameters from control values determined from 50 normal subjects. The first approach emphasizes qualitative deviations in the trajectories of the surface potential map extrema during QRS and correctly classified all but one infarct in a test group comprising 30 normals and 30 cases of myocardial infarction. The second approach classifies a test subject as abnormal if any one of his 26 lead waveforms deviates appreciably at any instant during QRS from the mean waveform for the particular lead plus or minus two standard deviations, these being determined from the control group. This method, while correctly identifying all infarcts, resulted in a large number of false positives, misclassifying 22 of 30 normals. A final method was to obtain an instant by instant plot of the correlation coefficient between the mean surface potential map during QRS for the 50 normals and that of the subject being tested. Test cases were classified as abnormal if any correlation coefficient value fell below an envelope determined from the correlation coefficient plots obtained by correlating the maps of all 50 normals with their own mean. Twenty-nine normals and 26 infarcts were correctly classified. On the basis of these results, the first approach is superior to the other two for detecting surface potential map abnormalities in patients with myocardial infarction.

Improving Tikhonov regularization with linearly constrained optimization: Application to the inverse epicardial potential solution

Two methods to improve on the accuracy of the Tikhonov regularization technique commonly used for the stable recovery of solutions to ill-posed problems are presented. These methods do not require a priori knowledge of the properties of the solution or of the error. Rather they exploit the observed properties of overregularized and underregularized Tikhonov solutions so as to impose linear constraints on the sought-after solution. The two methods were applied to the inverse problem of electrocardiography using a spherical heart-torso model and simulated inner-sphere (epicardial) and outer-sphere (body) potential distributions. It is shown that if the overregularized and underregularized Tikhonov solutions are chosen properly, the two methods yield epicardial solutions that are not only more accurate than the optimal Tikhonov solution but also provide other qualitative information, such as correct position of the extrema, not obtainable using ordinary Tikhonov regularization. A heuristic method to select the overregularized and underregularized solutions is discussed.