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.

Effects of rotational myocardial anisotropy in forward potential computations with equivalent heart dipoles.

Rotating fibers in the heart lead to a myocardium of inhomogeneous anisotropic conductivity. Besides affecting the activation isochrones, this anisotropy modifies the equivalent dipoles used in calculating extracardiac potentials, rendering them oblique rather than normal to the activation wavefront due to an added axial dipole component oriented along the fibers. Herein, however, consequences of the assumption usually made in forward potential calculations that the equivalent dipoles act in a myocardium that is homogeneous and isotropic are examined. A layered inner block representing the heart was placed inside an outer block representing an isotropic volume conductor. Fiber direction in the inner block rotated uniformly from layer to layer. Current dipoles of different orientations were placed in the inner block and the potentials calculated everywhere. Effects of the anisotropy of the inner block were gauged by computing an equivalent dipole that best fit the outer block surface potentials. For volume conductor conductivities close to that of the torso, the anisotropy diminished dipoles oriented along the fibers. Since the intraventricular blood masses in the heart also diminish such dipoles, these reductions of the axial component may explain the success of heart model simulations that ignore this component.

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

Computing the potentials on the heart's 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 stimulation 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 model's 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 model's 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.

A computer heart model incorporating anisotropic propagation. IV. Simulation of regional myocardial ischemia.

The main goal of this study was to simulate clinical body surface potential maps, recorded during percutaneous transluminal coronary angioplasty protocols, using a realistic geometry computer heart model. Other objectives were to address the question of reciprocal ST-segment changes observed in the 12-lead electrocardiogram during ischemia and to verify the hypothesis that the shortening of the QRS duration observed in left anterior descending (LAD) coronary artery occlusion may be explained by conduction delay in the septal His-Purkinje system. Simulation was achieved by first introducing into the heart model three transmural zones of mild, moderate, and severe ischemia for assumed occlusions in the LAD, left circumflex, and right coronary arteries. The heart model was then excited, in turn, with these three zones present for assumed occlusions in the LAD, left circumflex, and right coronary arteries. Myocardial conduction velocities in the regions of moderate and severe ischemia were assumed to be reduced to 75 and 50% of normal, respectively. Model action potentials in the mild, moderate, and severely ischemic zones were also altered to reflect known ischemic changes in these action potentials. Body surface potential maps and electrocardiograms were computed by placing the heart inside a numerical torso model. Simulated map patterns during both ST-segment and QRS were qualitatively similar to clinical maps. Reciprocal ST-segment depression was observed for all three occlusions in remote leads that did not overlie the ischemic zones. QRS shortening due to septal His-Purkinje conduction delay was verified. The simulation results attest to the model's ability to reproduce body surface potential distributions recorded following percutaneous transluminal coronary angioplasty protocols. The simulations also showed that reciprocal ST-segment changes occur as a natural consequence of the primary ischemic region and that there is no need to invoke a second region of ischemia. Finally, the model demonstrated that QRS shortening can occur in LAD occlusion despite a slowing of conduction down the septal His-Purkinje system.

A computer heart model incorporating anisotropic propagation. I. Model construction and simulation of normal activation.

Present-day computer models of the entire heart, capable of simulating the activation isochrones and subsequently the body surface potentials, focus on considerations of myocardial anisotropy. Myocardial anisotropy enters into play at two levels, first by affecting the spatial pattern of activation owing to faster propagation along cardiac fibers and second by altering the equivalent dipole sources used to calculate the surface potentials. The construction of a new and detailed model of the human heart is described, based on 132 transverse sections obtained following a computed tomography scan of a frozen human heart whose chambers were inflated with pressurized air. The entire heart anatomy was reconstructed as a three-dimensional array of approximately 250,000 points spaced 1 mm apart. Conduction in the thin-walled atria was assumed isotropic from the sinus node region to the atrioventricular node, where it was subject to a 50 ms delay. A two-tier representation of the specialized conduction system was used, with the initial segments of the left and right bundles represented by a system of cables that feeds to the second tier, which is a sheet of conduction tissue representing the distal Purkinje system. Approximately 1,120 "Purkinje-myocardium" junctions present at the terminations of the cables and sprinkled uniformly over the sheet, transmit the excitation to the ventricles. A stylized representation of myocardial fiber rotation was incorporated into the ventricles and the local fiber direction at each model point used to compute the velocity of propagation to its nearest neighbors. Accordingly, the activation times of the entire ventricular myocardium could be determined using the 1,120 or so Purkinje-myocardium junctions as start points. While myocardial anisotropy was considered in the ventricular propagation process, it was ignored in the computation of the equivalent dipole sources. Nevertheless, the computed electrocardiogram, vectorcardiogram, and body surface potential maps obtained with the new heart model properly positioned inside an inhomogeneous torso model were all within normal limits.

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 Dam's 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.

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.

Models of the electrical activity of the heart and computer simulation of the electrocardiogram.

This paper reviews models of the electrical activity of the heart and their use for simulations of the surface electrocardiogram. As such, it focuses on the forward problem of electrocardiography. The paper starts with an overview of the biophysical background that forms the underpinning of most heart models and surface potential computations. The basic volume-conductor equations and the different techniques used to solve them, the bidomain characterization of the myocardium, propagation in cardiac tissue, and equivalent source formulations for the electrical activity of the heart are all reviewed. Following this, a review of heart models conceived for cardiac rhythm simulations is presented. These models do not represent the heart geometry accurately, and hence ECG simulations with such models are, at best, first approximations. Next, realistic-geometry heart models, but without propagation, are described. Since the activation isochrones in such models are fixed, they are essentially suited only for the study of normal activation or for ischemia and infarction simulations. Finally, realistic-geometry models which use element-to-element propagation are described. Because of their ability to alter activation patterns, such models may also be used to study conduction disturbances and arrhythmias. Surface potential simulations realized with both types of realistic-geometry models are also reviewed. The paper concludes with a section on heart models of the future.

The inverse problem in electrocardiography: solutions in terms of equivalent sources.

This paper reviews those inverse electrocardiographic solutions that compute the electrical activity of the heart in terms of equivalent sources such as multipoles or multiple dipoles, as opposed to more realistic source formulations such as epicardial potentials. It treats, in succession, inverse solutions in terms of a single fixed-location dipole, a multipole series, moving dipoles, and, finally, multiple fixed-location dipoles. For each category of solution, simulation studies, animal experiments, and work involving human subjects are reviewed. Finally, more recent work that seeks to compute the cardiac activation isochrones, from the time integrals of the torso potentials during the QRS complex of the electrocardiogram, is described. The paper concludes with a discussion on the future of inverse electrocardiographic solutions in terms of equivalent sources.

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.

Surface integration and least-squares procedures for the inverse recovery of cardiac multipole components.

A simulation study was performed to evaluate different recovery procedures for computing the multipole components of the cardiac electrical activity. A series of dipolar potential distributions was first generated on a realistic numerical model of the human torso. Then, different procedures based on surface integration (SI) and least-squares (LS) minimization were used to compute the multipole components. The parameters of a single moving dipole (SMD) computed from the estimated multipoles were compared with those of the original dipole source. For a finite and homogeneous simulation as well as recovery medium, the results showed that SI employing the potentials over all 1216 surface elements of the torso model was not affected by the various numerical approximations used to perform the integration (e.g., rms error for the SMD position, p = 0.7 mm). By integrating the potentials with truncated capping surfaces at the neck and the waist, the recovery errors increased (p = 2.1 mm). Sampling the potentials at 63 sites, followed by interpolation over the rest of the torso surface, severely affected the SI results for the SMD (p = 6.4 mm), as compared with LS minimization using also 63 values (p = 0.9 mm). With lungs and intraventricular blood masses in the simulation medium but a finite and homogeneous recovery medium, SI was less effective (p = 10.8 mm) than LS (p = 8.6 mm). Adequate compensation for the effects of lungs was obtained by including regions of lower electrical conductivity in the recovery medium for LS, and by a correction matrix for SI. In general, LS gave better results than SI, but with a higher initial computation time.

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 patient's 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 patient's 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 simulation study of the effects of torso inhomogeneities on electrocardiographic potentials, using realistic heart and torso models.

The effects of torso inhomogeneities on electrocardiographic potentials were investigated via computer stimulation, using a 23-dipole heart model placed within a realistically shaped human torso model. The transfer coefficients relating the individual dipoles to the torso surface potentials, as well as the body surface potential maps, the vectorcardiogram, and the 12-lead electrocardiogram resulting due to normal activation of the heart model, were calculated for each of the following torso conditions: homogeneous, homogeneous + skeletal muscle layer, homogeneous + muscle layer + lungs, and homogeneous + muscle layer + lungs + intraventricular blood masses. The effects of each inhomogeneity were deduced by comparing results before and after its inclusion. For individual dipole transfer coefficients we confirm the validity of the "Brody effect," whereby the high conductivity blood masses augment radially oriented dipoles and diminish tangentially oriented ones. With regard to the vectorcardiogram , the electrocardiogram, and the body surface potential maps, the major qualitative effects were an augmentation of the head-to-foot component of the vectorcardiogram due to the lungs, and a smoothening of notches in the electrocardiogram (temporal filtering) and of isopotential contours in the body surface potential maps (spatial filtering) with a consequent loss of information, due to the blood masses, muscle layer, and, to a lesser extent, the lungs. Besides the above qualitative effects of the inhomogeneities, there were also large quantitative effects on the surface potentials, namely, magnitude increases due to the blood masses and magnitude decreases due to the muscle layer, that--if unaccounted for--could compromise the inverse solution of these potentials for the cardiac dipole sources.