Comparison and testing of least-squares time domain inverse solutions in electrocardiography

The use of several mathematical methods for estimating epicardial ECG potentials from arrays of body surface potentials has been reported in the literature; most of these methods are based on least-squares reconstruction principles and operate in the time-space domain. In this paper we introduce a general Bayesian maximum a posteriori (MAP) framework for time domain inverse solutions in the presence of noise. The two most popular previously applied least-squares methods, constrained (regularized) least-squares and low-rank approximation through the singular value decomposition, are placed in this framework, each of them requiring the a priori knowledge of a ‘regularization parameter’, which defines the degree of smoothing to be applied to the inversion. Results of simulations using these two methods are presented; they compare the ability of each method to reconstruct epicardial potentials. We used the geometric configuration of the torso and internal organs of an individual subject as reconstructed from CT scans. The accuracy of each method at each epicardial location was tested as a function of measurement noise, the size and shape of the subarray of torso sensors, and the regularization parameter. We paid particular attention to an assessment of the potential of these methods for clinical use by testing the effect of using compact, small-size subarrays of torso potentials while maintaining a high degree of resolution on the epicardium.     

Computational issues of importance to the inverse recovery of epicardial potentials in a realistic heart-torso geometry

In vitro data from a realistic-geometry electrolytic tank were used to demonstrate the consequences of computational issues critical to the ill-posed inverse problem in electrocardiography. The boundary element method was used to discretize the relationship between the body surface potentials and epicardial cage potentials. Variants of Tikhonov regularization were used to stabilize the inversion of the body surface potentials in order to reconstruct the epicardial surface potentials. The computational issues investigated were (1) computation of the regularization parameter; (2) effects of inaccuracy in locating the position of the heart; and (3) incorporation of a priori information on the properties of epicardial potentials into the regularization methodology. Two methods were suggested by which a priori information could be incorporated into the regularization formulation: (1) use of an estimate of the epicardial potential distribution everywhere on the surface and (2) use of regional bounds on the excursion of the potential. Results indicate that the a posteriori technique called CRESO, developed by Colli Franzone and coworkers, most consistently derives the regularization parameter closest to the optimal parameter for this experimental situation. The sensitivity of the inverse computation in a realistic-geometry torso to inaccuracies in estimating heart position are consistent with results from the eccentric spheres model; errors of 1 cm are well tolerated, but errors of 2 cm or greater result in a loss of position and amplitude information. Finally, estimates and bounds based on accurate, known information successfully lower the relative error associated with the inverse and have the potential to significantly enhance the amplitude and feature position information obtainable from the inverse-reconstructed epicardial potential map.     

A new method for regularization of the inverse problem of electrocardiography.

The inverse problem of electrocardiography (specifically, that part concerned with the computation of the ventricular surface activation isochrones) is shown to be formally equivalent to the problem of identification and measurement of discontinuities in derivatives of body surface potentials. This is based on the demonstration that such measurements allow localization of the relative extrema of the ventricular surface activation map (given a forward problem solution), which in turn restricts the space of admissible solution maps to a compact set. Although the inverse problem and the problem of identifying derivative discontinuities are both ill-posed, it is possible that the latter may be more easily or justifiably resolved with available information, particularly as current methods for regularizing the inverse problem typically rely on a regularization parameter chosen in an a posteriori fashion. An example of the power of the approach is the demonstration that a recent Uniform Dipole Layer Hypothesis-based method for producing the ventricular surface activation map is largely independent on that hypothesis and capable in principle of generating maps that are very similar in a precise sense to those that would result from the usual epicardial potential formulation (assuming the latter were capable of producing intrinsic deflections in computed epicardial electrograms sufficiently steep to accurately compute the activation map). This is consistent with the preliminary success of the former method, despite the significant inaccuracy of its underlying assumption.     

Adaptive local regularization methods for the inverse ECG problem

One of the fundamental problems in theoretical electrocardiography can be characterized by an inverse problem. We present new methods for achieving better estimates of heart surface potential distributions in terms of torso potentials through an inverse procedure. First, we outline an automatic adaptive refinement algorithm that minimizes the spatial discretization error in the transfer matrix, increasing the accuracy of the inverse solution. Second, we introduce a new local regularization procedure, which works by partitioning the global transfer matrix into sub-matrices, allowing for varying amounts of smoothing. Each submatrix represents a region within the underlying geometric model in which regularization can be specifically ‘tuned’ using an a priori scheme based on the L-curve method. This local regularization method can provide a substantial increase in accuracy compared to global regularization schemes. Within this context of local regularization, we show that a generalized version of the singular value decomposition (GSVD) can further improve the accuracy of ECG inverse solutions compared to standard SVD and Tikhonov approaches. We conclude with specific examples of these techniques using geometric models of the human thorax derived from MRI data.     

A model study of instability of the inverse problem in electrocardiography.

The inverse problem in electrocardiography is studied analytically using a concentric spheres model with no symmetry assumptions on the potential distribution. The mathematical formulation is presented, and existence and uniqueness of the solution are briefly discussed. Solution to the inverse problem is inherently very unstable. The magnitude of this instability is demonstrated using the derived analytical inverse solution for the spherical model. Regularization methods used to date are based on a regularization parameter that does not relate to any measurable physiological parameters. This paper presents a regularization method that is based on a parameter in the form of an a priori bound on the L2 norm of the inverse solution. Such a bound can be obtained from the theoretical estimates based on the measured values of the body surface potentials together with experimental knowledge about the magnitudes of the epicardial potentials. Based on the presented regularization, an exact form of the regularized solution and estimates of its accuracy are derived.     

改善心电图逆问题病态条件的两个有效方法

本文对应用数值方法求解三维心电图逆问题时出现的解的不稳定现象,提出了两种有效的解决办法,即提高电导率值和选用包含适当的正则因子的阻尼最小二乘法。文中应用有限元和边界元结合的方法,在一个包含各向异性导电性肌肉层的三维人体模型下进行了心外膜电位的重构计算。其结果证明这种方法对提高心电图逆问题的数值稳定性和解的精度非常有效。     

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.     

人体躯干模型中肺的存在对体表电位分布的影响

在所建三维人体躯干模型的基础上,给出了如何应用边界元方法对非均匀人体场域进行求解。在设定心外膜电位分别呈现为单偶极子和双偶极子时,求出相应非均匀场域中的体表电位分布,并将它们与相同情况下均匀场域的体表电位分布进行分析比较。结果表明：躯干模型中肺的存在虽然对体表电位中极值的大小和位置没有太大影响,但却较大程序地改变了整个体表电位的分布状况,具体地说,就是由于肺的存在使得体表电位值较均匀时的相对误差高     

Cardiac electric field at the period of depolarization and repolarization of the frog heart ventricle

Multichannel mapping of electrical field on heart ventricle epicardium and the body surface in frogs Rana esculenta and Rana temporaria was performed at periods of the ventricular myocardium depolarization and repolarization. The zone of the epicardium early depolarization is located on epicardium of the ventricle base posterior wall, while the late depolarization zone--on its apex and on the base anterior wall. The total vector of sequence of the ventricle epicardium depolarization is directed from the base to the apex. The zone of the early repolarization is located in the apical area, while that of the late one--in the area of the base. On the frog body surface the cardioelectric field with the cranial zone of negative and the caudal zone of positive potentials is formed before the appearance of the QRS complex on ECG. At the period of the heart ventricle repolarization the zone of the cardioelectric field negative potentials is located in the cranial, while that of the positive ones--in the body surface caudal parts. The cardioelectric field on the frog body surface at the periods of depolarization and repolarization of the ventricle myocardium reflects adequately the projection of sequence of involvement with excitation and of distribution of potentials on epicardium.     

Differences in regional capillary distribution and myocyte sizes in normal and hypertrophic rat hearts.

Data are reported which show significant regional capillary differences in left ventricular endocardium and epicardium of normal rats and of rats with hyperthyroid-induced cardiac hypertrophy. The epicardial region of control rats has 38% more capillaries than the endocardial region. Control endocardial myocytes are 62% larger in cross-sectional area than epicardial myocytes. Hypertrophic hearts exhibit regional differences in capillary density similar to those in the normal hearts, but there is an overall reduction of 12 and 17.5% in capillary density in both regions. The average cross-sectional area of myocytes increases 34.5% in the epicardium and 22.5% in the endocardium.     

Cardiac electric field at the period of depolarization and repolarization of the frog heart ventricle

Multichannel mapping of electrical field on heart ventricle epicardium and the body surface in frogs Rana esculenta and Rana temporaria was performed at periods of the ventricular myocardium depolarization and repolarization. The zone of the epicardium early depolarization is located on epicardium of the ventricle base posterior wall, while the late depolarization zone—on its apex and on the base anterior wall. The total vector of sequence of the ventricle epicardium depolarization is directed from the base to the apex. The zone of the early repolarization is located in the apical area, while that of the late one—in the area of the base. On the frog body surface the cardioelectric field with the cranial zone of negative and the caudal zone of positive potentials is formed before the appearance of the QRS complex on ECG. At the period of the heart ventricle repolarization the zone of the cardioelectric field negative potentials is located in the cranial, while that of the positive ones—in the body surface caudal parts. The cardioelectric field on the frog body surface at the periods of depolarization and repolarization of the ventricle myocardium reflects adequately the projection of sequence of involvement with excitation and of distribution of potentials on epicardium.     

Measuring surface potential components necessary for transmembrane current computation using microfabricated arrays

This study was designed to test the feasibility of using microfabricated electrodes to record surface potentials with sufficiently fine spatial resolution to measure the potential gradients necessary for improved computation of transmembrane current density. To assess that feasibility, we recorded unipolar electrograms from perfused rabbit right ventricular free wall epicardium (n = 6) using electrode arrays that included 25-microm sensors fabricated onto a flexible substrate with 75-microm interelectrode spacing. Electrode spacing was therefore on the size scale of an individual myocyte. Signal conditioning adjacent to the sensors to control lead noise was achieved by routing traces from the electrodes to the back side of the substrate where buffer amplifiers were located. For comparison, recordings were also made using arrays built from chloridized silver wire electrodes of either 50-microm (fine wire) or 250-microm (coarse wire) diameters. Electrode separations were necessarily wider than with microfabricated arrays. Comparable signal-to-noise ratios (SNRs) of 21.2 +/- 2.2, 32.5 +/- 4.1, and 22.9 +/- 0.7 for electrograms recorded using microfabricated sensors (n = 78), fine wires (n = 78), and coarse wires (n = 78), respectively, were found. High SNRs were maintained in bipolar electrograms assembled using spatial combinations of the unipolar electrograms necessary for the potential gradient measurements and in second-difference electrograms assembled using spatial combinations of the bipolar electrograms necessary for surface Laplacian (SL) measurements. Simulations incorporating a bidomain representation of tissue structure and a two-dimensional network of guinea pig myocytes prescribed following the Luo and Rudy dynamic membrane equations were completed using 12.5-microm spatial resolution to assess contributions of electrode spacing to the potential gradient and SL measurements. In those simulations, increases in electrode separation from 12.5 to 75.0, 237.5, and 875.0 microm, which were separations comparable to the finest available with our microfabricated, fine wire, and coarse wire arrays, led to 10%, 42%, and 81% reductions in maximum potential gradients and 33%, 76%, and 96% reductions in peak-to-peak SLs. Maintenance of comparable SNRs for source electrograms was therefore important because microfabrication provides a highly attractive methods to achieve spatial resolutions necessary for improved computation of transmembrane current density.     

Electrotonic load triggers remodeling of repolarizing current Ito in ventricle

A change in activation sequence electrically remodels ventricular myocardium, causing persistent changes in repolarizing currents (T-wave memory). However, the underlying mechanism for triggering activation sequence-dependent remodeling is unknown. Optical action potentials were mapped with high resolution from the epicardial surface of the arterially perfused canine wedge preparation (n = 23) during 30 min of baseline endocardial stimulation, followed by 40 min of epicardial stimulation, and, finally, restoration of endocardial stimulation. Immediately after the change from endocardial to epicardial stimulation, phase 1 notch amplitude of epicardial cells was attenuated by 74 +/- 8% (P < 0.001) compared with baseline and continued to diminish during the period of epicardial pacing, suggesting progressive remodeling of the transient outward current (Ito). When endocardial pacing was restored, notch amplitude did not immediately recover but remained attenuated by 23 +/- 10% (P < 0.001), also consistent with a remodeling effect. Peak Ito current measured from isolated epicardial myocytes changed by 12 +/- 4% (P < 0.025), providing direct evidence for Ito remodeling occurring on a surprisingly short time scale. The mechanism for triggering remodeling of Ito was a significant reduction (by 14 +/- 4%, P < 0.001) of upstroke amplitude in epicardial cells during epicardial stimulation. Reduction in upstroke amplitude during epicardial pacing was explained by electrotonic load on epicardial cells by fully repolarized downstream endocardial cells. These data suggest a novel mechanism for triggering electrical remodeling in the ventricle. Electrotonic load imposed by a change in activation sequence reduces upstroke amplitude, which, in turn, attenuates Ito according to its known voltage-dependent properties, triggering downregulation of current.     

Epicardial epithelial-to-mesenchymal transition in injured heart

Cre-LoxP-mediated genetic lineage trace has been used to illuminate the cell fate of progenitor cells in vivo. Application of this strategy to the epicardium, a sheet of cells covering the surface of heart, revealed that it dynamically participates in both heart development and postnatal heart repair and regeneration. After myocardial infarction, epicardial cells undergo epithelial-to-mesenchymal transition (EMT) and mainly adopt myofibroblast, fibroblast and smooth muscle cell fates. Here we present the wholemount images that map epicardial EMT following myocardial infarction, taking advantage of an inducible epicardial Cre line and a double fluorescence reporter. While remote epicardium retained its epithelial cell shape, reactivated epicardium in the infarcted region showed significant EMT. This image supports active involvement of the epicardium in repair and regeneration of infarcted myocardium.     

T wave alternans in an in vitro canine tissue model of Brugada syndrome

Macroscopic T wave alternans (TWA) associated with increased occurrence of ventricular arrhythmias has been reported in patients with Brugada syndrome. However, the mechanisms in this syndrome are still unclear. We evaluated the hypothesis that TWA in Brugada syndrome was caused by the dynamic instability and heterogeneity of action potentials (APs) in the right ventricle. Using an optical mapping system, we mapped APs on the epicardium or transmural surfaces of 28 isolated and arterially perfused canine right ventricular preparations having drug-induced Brugada syndrome (in micromol/l: 2.5-15 pinacidil, 5.0 terfenadine, and 5.0-13 pilsicainide). Bradycardia at cycle length (CL) of 2,632 +/- 496 ms (n = 19) induced alternating deep and shallow T waves in the transmural electrocardiogram. Compared with the shallow T waves, deep T waves were associated with epicardial APs having longer durations and larger domes. Adjacent regions having APs with alternating domes, with constant domes, and without domes coexisted simultaneously in the epicardium and caused TWA. In contrast to the alternating epicardial APs, midmyocardial and endocardial APs did not change during TWA. Alternans could be terminated by rapid (CL: 529 +/- 168 ms, n = 7) or very slow (CL: 3,000 ms, n = 7) pacing. The heterogeneic APs during TWA augmented the dispersion of repolarization both within the epicardium and from the epicardium to the endocardium and caused phase 2 reentry. In this drug-induced model of Brugada syndrome, heterogeneic AP contours and dynamic alternans in the dome of right ventricular epicardial, but not midmyocardial or endocardial, APs caused TWA and heightened arrhythmogenicity in part by increasing the dispersion of repolarization.     

Cardiac electric field on the epicardial and body surfaces during the period of depolarization and repolarization of ventricular myocardium in pikes

Body surface and ventricular epicardial potential distributions during the electrocardiographic QRST interval were studied in pikes with the aid of potential mapping. The earliest epicardial activation was observed at the posterior base near the atrioventricular orifice. The areas of the earliest repolarization were found at the apex and the posterior base, whereas the area of the latest repolarization was detected at the anterior base. In the initial period of the QRS, the minimum was developed in the middle third of the right lateral body surface, and the maximum in the middle third of the ventral body surface. The body surface potential distribution during the ST-Twas characterized by the clear-cut negative potential zone in the cranial ventral area with the rest of the body surface having positive potentials, a pattern being largely unchanged throughout the period of the T-wave. The ventricular epicardial repolarization sequence differed from the activation sequence. The ventricular epicardial depolarization and repolarization sequences as well as epicardial potential distributions are expressed in the cardiac electric field on the body surface during the QRS and ST-T complexes.     

Effect of hypothermia on sequence of the ventricle epicardium repolarization in rabbits]

In anaesthetised rabbits at normal body temperature, the earliest ventricles' epicardial recovery occurs at the heart apex and adjacent left ventricle's surface whereas the latest one occurs at the epicardium of the right ventricle's base. A decrease in the mediastinum temperature to 32 degrees C reversed the recovery sequence. Following the cooling of the heart, the longest prolongation of the activation-recovery interval occurred at the heart apex area and the lowest one--at the right ventricle base.     

MicroRNA-processing enzyme Dicer is required in epicardium for coronary vasculature development

The epicardium is a sheet of epithelial cells covering the heart during early cardiac development. In recent years, the epicardium has been identified as an important contributor to cardiovascular development, and epicardium-derived cells have the potential to differentiate into multiple cardiac cell lineages. Some epicardium-derived cells that undergo epithelial-to-mesenchymal transition and delaminate from the surface of the developing heart subsequently invade the myocardium and differentiate into vascular smooth muscle of the developing coronary vasculature. MicroRNAs (miRNAs) have been implicated broadly in tissue patterning and development, including in the heart, but a role in epicardium is unknown. To examine the role of miRNAs during epicardial development, we conditionally deleted the miRNA-processing enzyme Dicer in the proepicardium using Gata5-Cre mice. Epicardial Dicer mutant mice are born in expected Mendelian ratios but die immediately after birth with profound cardiac defects, including impaired coronary vessel development. We found that loss of Dicer leads to impaired epicardial epithelial-to-mesenchymal transition and a reduction in epicardial cell proliferation and differentiation into coronary smooth muscle cells. These results demonstrate a critical role for Dicer, and by implication miRNAs, in murine epicardial development.     

Cardioelectric field on the atrial surface in pigs

The form and distribution of extracellular cardioelectric potentials and the sequence of the excitation wave propagation on epicardium of the pig atria were studied by the method of multichannel synchronous cardioelectrotopography. The studies have shown that in pig the excitation wave breaks on epicardium of the right atrium at the base of the upper vena cava. Negative initial atrial complexes are registered in this area. Two fronts of excitation wave spread from the zone of initial epicardial activation: one--to upper segments of dorsal and ventral sides of the right atrium, the second--to inter-atrial septum. The excitation wave comes to the left atrium with a delay relative to the beginning of depolarization of the right atrium. On account of the successive movement of the front of the excitation wave from pacemaker the two-phase potentials are formed on greater part of the epicardium of the pig atria. The lower part of the auricle of the left atrium is depolarized in atrial epicardium in the last turn. The sequence of excitation wave propagation in atrial epicardium close to ravenous (dog) and ungulate (sheep) animals and man is typical for the pig, but herewith the differences in time of covering the atria with excitation do exist.     

Repolarization gradients in the intact heart: transmural or apico-basal?

Controversies regarding the genesis of the T wave in the electrocardiogram and the role of midmural M cells in the intact heart include: In normal, intact canine and human hearts there is no significant transmural gradient in repolarization times. The T wave results primarily from apico-basal differences in repolarization times. Also, in the intact heart there is no midmural region of prolonged action potential duration. This contrasts with isolated preparations, such as the wedge preparation or myocardial slices or disaggregated myocytes in which M cells, with action potentials longer than those of endocardial and epicardial myocardium, can be found. This disparity in action potential duration probably results from partial uncoupling of myocardial cells in the regions where measurements are made, e.g., the cut surface of a wedge preparation. In regions of a wedge where cellular coupling is normal, or in isolated myocardial bundles or sheets, no evidence for M cells is detected. In some wedge preparations, a drug-induced large transmural repolarization gradient, involving M cells, can lead to Torsade de Pointes, possibly caused by so-called phase two reentry. In contrast, when a gradient of repolarization times of more than 100?ms was created in intact hearts, no evidence for reentry was found and no spontaneous arrhythmias occurred. In conclusion, in the intact heart, M cells appear not to contribute to repolarization gradients and arrhythmias. Furthermore, no significant repolarization gradients between endocardium and epicardium exist. The T wave in the body surface electrocardiogram is caused by apico-basal and anterior-posterior differences in repolarization times.