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Methods for noninvasive imaging of electric function of the heart might become clinical standard procedure the next years. Thus, the overall procedure has to meet clinical requirements as an easy and fast application. In this paper, we propose a new electrode array which improves the resolution of methods for activation time imaging considering clinical constraints such as easy to apply and compatibility with routine leads. For identifying the body-surface regions where the body surface potential (BSP) is most sensitive to changes in transmembrane potential (TMP), a virtual array method was used to compute local linear dependency (LLD) maps. The virtual array method computes a measure for the LLD in every point on the body surface. The most suitable number and position of the electrodes within the sensitive body surface regions was selected by constructing effort gain (EG) plots. Such a plot depicts the relative attainable rank of the leadfield matrix in relation to the increase in number of electrodes required to build the electrode array. The attainable rank itself was computed by a detector criterion. Such a criterion estimates the maximum number of source space eigenvectors not covered by noise when being mapped to the electrode space by the leadfield matrix and recorded by a detector. From the sensitivity maps, we found that the BSP is most sensitive to changes in TMP on the upper left frontal and dorsal body surface. These sensitive regions are covered best by an electrode array consisting of two L-shaped parts of ~30 cm x 30 cm and ~20 cm x 20 cm. The EG analysis revealed that the array meeting clinical requirements best and improving the resolution of activation rime imaging consists of 125 electrodes with a regular horizontal and vertical spacing of 2-3 cm. Index Terms--Activation time (AT) imaging, effort gain analysis, electrocardiographam (ECG) mapping, electrode array, sensitivity analysis.

The aim of this study was to quantify the effect of cardiac anisotropy in the activation-based inverse problem of electrocardiography. Differences of the patterns of simulated body surface potential maps for isotropic and anisotropic conditions were investigated with regard to activation time (AT) imaging of ventricular depolarization. AT maps were estimated by solving the nonlinear inverse ill-posed problem employing spatio-temporal regularization. Four different reference AT maps (sinus rhythm, right-ventricular and septal pacing, accessory pathway) were calculated with a bidomain theory based anisotropic finite-element heart model in combination with a cellular automaton. In this heart model a realistic fiber architecture and conduction system was implemented. Although the anisotropy has some effects on forward solutions, effects on inverse solutions are small indicating that cardiac anisotropy might be negligible for some clinical applications (e.g., imaging of focal events) of our AT imaging approach. The main characteristic events of the AT maps were estimated despite neglected electrical anisotropy in the inverse formulation. The worst correlation coefficient of the estimated AT maps was 0.810 in case of sinus rhythm. However, all characteristic events of the activation pattern were found. The results of this study confirm our clinical validation studies of noninvasive AT imaging in which cardiac anisotropy was neglected. Index Terms--Activation time imaging, bidomain theory, electrical anisotropy, forward problem of electrocardiography, inverse problem of electrocardiography, spatiotemporal regularization.

We compare two source formulations for the electrocardiographic forward problem in consideration of their implications for regularizing the ill-posed inverse problem. The established epicardial potential source model is compared with a bidomain-theory-based transmembrane potential source formulation. The epicardial source approach is extended to the whole heart surface including the endocardial surfaces. We introduce the concept of the numerical null and signal space to draw attention to the problems associated with the nonuniqueness of the inverse solution and show that reconstruction of null-space components is an important issue for physiologically meaningful inverse solutions. Both formulations were tested with simulated data generated with an anisotropic heart model and with clinically measured data of two patients. A linear and a recently proposed quasi-linear inverse algorithm were applied for reconstructions of the epicardial and transmembrane potential, respectively. A direct comparison of both formulations was performed in terms of computed activation times. We found the transmembrane potential-based formulation is a more promising source formulation as stronger regularization by incorporation of biophysical a priori information is permitted. Index Terms--Bidomain theory, epicardial potential, forward problem of electrocardiography, inverse problem of electrocardiography, L-curve method, null space of electrocardiography, spatiotemporal regularization, Tikhonov regularization, transmembrane potential.

The single-beat reconstruction of electrical cardiac sources from body-surface electrocardiogram data might become an important issue for clinical application. The feasibility and field of application of noninvasive imaging methods strongly depend on development of stable algorithms for solving the underlying ill-posed inverse problems. We propose a novel spatiotemporal regularization approach for the reconstruction of surface transmembrane potential (TMP) patterns. Regularization is achieved by imposing linearly formulated constraints on the solution in the spatial as well as in the temporal domain. In the spatial domain an operator similar to the surface Laplacian, weighted by a regularization parameter, is used. In the temporal domain monotonic nondecreasing behavior of the potential is presumed. This is formulated as side condition without the need of any regularization parameter. Compared to presuming template functions, the weaker temporal constraint widens the field of application because it enables the reconstruction of TMP patterns with ischemic and infarcted regions. Following the line of Tikhonov regularization, but considering all time points simultaneously, we obtain a linearly constrained sparse large-scale convex optimization problem solved by a fast interior point optimizer. We demonstrate the performance with simulations by comparing reconstructed TMP patterns with the underlying reference patterns. Index Terms--Convex quadratic optimization, inverse problem of electrocardiography, L-curve method, spatiotemporal regularization, Tikhonov regularization, transmembrane potential.

Linear approaches like the minimum-norm least-square algorithm show insufficient performance when it comes to estimating the activation time map on the surface of the heart from electrocardiographic (ECG) mapping data. Additional regularization has to be considered leading to a nonlinear problem formulation. The Gauss-Newton approach is one of the standard mathematical tools capable of solving this kind of problem. To our experience, this algorithm has specific drawbacks which are caused by the applied regularization procedure. In particular, under clinical conditions the amount of regularization cannot be determined clearly. For this reason, we have developed an iterative algorithm solving this nonlinear problem by a sequence of regularized linear problems. At each step of iteration, an individual L-curve is computed. Subsequent iteration steps are performed with the individual optimal regularization parameter. This novel approach is compared with the standard Gauss-Newton approach. Both methods are applied to simulated ECG mapping data as well as to single beat sinus rhythm data from two patients recorded in the catheter laboratory. The proposed approach shows excellent numerical and computational performance, even under clinical conditions at which the Gauss-Newton approach begins to break down. Index Terms--Activation time imaging, inverse problem, L-curve method, noninvasive electrocardiography, tikhonov regularization.

Activation time (AT) imaging from electrocardiographic (ECG) mapping data has been developing for several years. By coupling ECG mapping and three-dimensional (3-D) + time anatomical data, the electrical excitation sequence can be imaged completely noninvasively in the human heart. In this paper, a bidomain theory-based surface heart model AT imaging approach was applied to single-beat data of atrial and ventricular depolarization in two patients with structurally normal hearts. In both patients, the AT map was reconstructed from sinus and paced rhythm data. Pacing sites were the apex of the right ventricle and the coronary sinus (CS) ostium. For CS pacing, the reconstructed AT pattern on the endocardium of the right atrium was compared with the CARTO map in both patients. The localization errors of the origins of the initial endocardial breakthroughs were determined to be 6 and 12 mm. The sites of early activation and the areas with late activation were estimated with sufficient accuracy. The reconstructed sinus rhythm sequence was in good qualitative agreement with the pattern previously published for the isolated Langendorff-perfused human heart. Index Terms--Activation time map, atrium, electrocardio-graphic (ECG) mapping, human heart, inverse problem, magnetic resonance imaging, modeling, ventricle.

In clinical electrocardiography, the zero-potential is commonly defined by the Wilson central terminal. In the electrocardiographic forward and inverse problem, the zero- potential is often defined in a different way, e.g., by the sum of all node potentials yielding zero. This study presents relatively simple to implement techniques, which enable the incorporation of the Wilson Terminal in the boundary element method (BEM) and finite element method (FEM). For the BEM, good results are obtained when properly adopting matrix deflation for modeling the Wilson terminal. Applying other zero-potential-definitions, the obtained solutions contained a remarkable offset with respect to the reference defined by the Wilson terminal. In the inverse problem (nonlinear dipole fit), errors introduced by an erroneous zero-potential-definition can lead to displacements of more than 5 mm in the computed dipole location. For the FEM, a method similar to matrix deflation is proposed in order to properly consider the Wilson central terminal. The matrix obtained from this manipulation is symmetric, sparse and positive definite enabling the application of standard FEM-solvers. Index Terms--Bioelectric phenomena, electrocardiography, numerical analysis, source localization.

We compare two source formulations for the electrocardiographic forward problem in consideration of their implications for regularizing the ill-posed inverse problem. The established epicardial potential source model is compared with a bidomain-theory-based transmembrane potential source formulation. The epicardial source approach is extended to the whole heart surface including the endocardial surfaces. We introduce the concept of the numerical null and signal space to draw attention to the problems associated with the nonuniqueness of the inverse solution and show that reconstruction of null-space components is an important issue for physiologically meaningful inverse solutions. Both formulations were tested with simulated data generated with an anisotropic heart model and with clinically measured data of two patients. A linear and a recently proposed quasi-linear inverse algorithm were applied for reconstructions of the epicardial and transmembrane potential, respectively. A direct comparison of both formulations was performed in terms of computed activation times. We found the transmembrane potential-based formulation is a more promising source formulation as stronger regularization by incorporation of biophysical a priori information is permitted. [ABSTRACT FROM AUTHOR]

Methods for noninvasive imaging of electrical function of the heart seem to become a clinical standard procedure the next years. Thus, the overall procedure has to meet clinical requirements as easy and fast application. In this study we propose a new electrode array meeting clinical requirements such as easy to apply and compatibility with routine leads. Within body surface regions of high sensitivity, identified in a prior, information content based study, the number of required electrodes was optimized using effort-gain plots. These plots were generated by applying a so called type one detector criterion. The optimal array was selected from a set of 12 electrode arrays. Each of them consists of two L-shaped regular spaced parts. The optimal array was found by comparing several layouts and electrode densities to the electrode array we use for clinical studies. It consists of 125 electrodes with a regular spacing between 2cm and 3cm.

Methods for noninvasive imaging of electric function of the heart might become clinical standard procedure the next years. Thus, the overall procedure has to meet clinical requirements as easy and fast application. In this study we propose a new electrode array which improves the information content in the ECG map, considering clinical constraints such as easy to apply and compatibility with routine leads. A major challenge is the development of an electrode array which yields a high information content even for a large interindividual variation in torso shape. For identifying regions of high information content we introduce the concept of a locally applied virtual electrode array. As a result of our analysis we constructed a new electrode array consisting of two L-shaped regular spaced parts and compared it to the electrode array we use for clinical studies upon activation time imaging. We assume that one side effect caused by the regular shape and spacing of the new array be that the reconstruction of electrodes placed on the patients back is simplified. It may be sufficient to record a few characteristic electrode positions and merge them with a model of the posterior array.

The effect of different atrial electrical activation sequences (sinus rhythm and atrial flutter circling in the right atrium) on the body surface potentials is investigated in this study. A realistic volume conductor model consisting of atria, lungs, chest and blood masses is generated from image stacks recorded by magnetic resonance imaging. The electrical sources-the transmembrane potentials-within the atrial volumetric model are simulated for different atrial rhythms employing a cellular automaton capable of considering different parameters depending on the specific properties of the tissues. The potentials on the torso surface are computed applying the finite element method for solving the differential equations derived from the bidomain theory. Both the simulated atrial activation patterns and the computed torso potentials for atrial sinus rhythm and atrial flutter are in qualitatively and quantitatively good agreement with data measured in humans. The simulation of body surface potentials generated by different electrical activation sequences in the atria or ventricles allows testing and assessing noninvasive imaging of cardiac electrophysiology, as both the potentials on the body surface and the reference activation in the heart are available.