The application of subspace preconditioned LSQR algorithm for solving the electrocardiography inverse problem

Regularization is an effective method for the solution of ill-posed ECG inverse problems, such as computing epicardial potentials from body surface potentials. The aim of this work was to explore more robust regularization-based solutions through the application of subspace preconditioned LSQR (SP-LSQR) to the study of model-based ECG inverse problems. Here, we presented three different subspace splitting methods, i.e., SVD, wavelet transform and cosine transform schemes, to the design of the preconditioners for ill-posed problems, and to evaluate the performance of algorithms using a realistic heart-torso model simulation protocol. The results demonstrated that when compared with the LSQR, LSQR-Tik and Tik-LSQR method, the SP-LSQR produced higher efficiency and reconstructed more accurate epcicardial potential distributions. Amongst the three applied subspace splitting schemes, the SVD-based preconditioner yielded the best convergence rate and outperformed the other two in seeking the inverse solutions. Moreover, when optimized by the genetic algorithms (GA), the performances of SP-LSQR method were enhanced. The results from this investigation suggested that the SP-LSQR was a useful regularization technique for cardiac inverse problems.     

Noninvasive Electrical Imaging of the Heart: Theory and Model Development

The aim of this work is to begin quantifying the performance of a recently developed activation imaging algorithm of Huiskamp and Greensite [IEEE Trans. Biomed. Eng. 44:433–446]. We present here the modeling and computational issues associated with this process. First, we present a practical construction of the appropriate transfer matrix relating an activation sequence to body surface potentials from a general boundary value problem point of view. This approach makes explicit the role of different Green's functions and elucidates features (such as the anisotropic versus isotropic distinction) not readily apparent from alternative formulations. A new analytic solution is then developed to test the numerical implementation associated with the transfer matrix formulation presented here and convergence results for both potentials and normal currents are given. Next, details of the construction of a generic porcine model using a nontraditional data-fitting procedure are presented. The computational performance of this model is carefully examined to obtain a mesh of an appropriate resolution to use in inverse calculations. Finally, as a test of the entire approach, we illustrate the activation inverse procedure by reconstructing a known activation sequence from simulated data. For the example presented, which involved two ectopic focii with large amounts of Gaussian noise (100 V rms) present in the torso signals, the reconstructed activation sequence had a similarity index of 0.880 when compared to the input source. © 2001 Biomedical Engineering Society. PAC01: 8719Nn, 8719Hh, 8710+e, 0210Yn, 0230Sa     

Accuracy of single-dipole inverse solution when localising ventricular pre-excitation sites: simulation study

Different factors are investigated that may affect the accuracy of an inverse solution that uses a single-dipole equivalent generator, in a standardised inhomogeneous torso model, when localising the pre-excitation sites. An anatomical model of the human ventricular myocardium is used to simulate body surface potential maps (BSPMs) and magnetic field maps (MFMs) for 35 pre-excitation sites positioned on the epicardial surface along the atrioventricular ring. The sites of pre-excitation activity are estimated by the single-dipole method, and the measure for the accuracy of the localisation is the localisation error, defined as the distance between the location of the best-fitting single dipole and the acutal site of pre-excitation in the ventricular model. The findings indicate that, when the electrical properties of the volume conductor and lead positions are precisely known and the ‘measurement’ noise is added to the simulated BSPMs and MFMs, the single-dipole method optimally localises the pre-excitation activity 20 ms after the onset of pre-excitation, with 0.71±0.28 cm and 0.65±0.30 cm using BSPMs and MFMs, respectively. When the standard torso model is used to localise the sites of onset of the pre-excitation sequence initiated in four individualised torso models, the maximum errors are as high as 2.6–3.0 cm (even though the average error, for both the BSPM and MFM localisations, remains within the 1.0–1.5 cm range). In spite of these shortcomings, it is thought that single-dipole localisations can be useful for non-invasive pre-interventional planning.     

Spatial resolution of body surface potential maps and magnetic field maps: A simulation study applied to the identification of ventricular pre-excitation sites

The spatial resolution of body surface potential maps (BSPMs) and magnetic field maps (MFMs) is investigated by means of an anatomically accurate computer model of the human ventricular myocardium. BSPMs and MFMs are calculated for the simulated activation sequences initiated at 35 pre-excitation sites located along the atrioventricular (AV) ring of the epicardium. Changes in the BSPMs and MFMs corresponding to different pre-excitation sites are quantified in terms of the correlation coefficient r. The spatial resolution (selectivity) for a given pre-excitation site is defined as the half-distance between those neighbouring locations at which morphological features of maps, in terms of r, become distinct (r<0.95). It is found that, at 28 ms after the onset of pre-excitation and with no noise added, this distance ±SD, for all sites along the AV ring for the 117-lead BSPMs, is 0.83±0.32 cm, and for the 64-lead and 128-lead MFMs it is 1.54±0.84 cm and 1.15±0.43 cm, respectively. The findings suggest that, when features of non-invasively recorded electrocardiographic and magnetocardiographic map patterns are used for identifying accessory pathways in patients suffering from WPW syndrome, BSPMs are likely to provide more detailed information for guiding the ablative treatment than MFMs. For some sites MFMs provide more information. Both modalities may provide additional assistance to the cardiologist in locating the site of the accessory pathway.     

Method for guiding the ablation catheter to the ablation site: a simulation and experimental study

Radiofrequency catheter ablation (RCA) procedures for treating ventricular arrhythmias have evolved significantly over the past several years; however, the use of RCA has been limited due to the difficulty in identifying the appropriate site for ablation. In this study, we investigate the accuracy of a computer algorithm to guide the tip of an ablation catheter to the exit site of the scar tissue or the site of abnormal automaticity (the “target site”). This algorithm involves modeling the body surface potentials corresponding to the wavefront at the target site for ablation and current pulses generated from a pair of electrodes at the tip of the ablation catheter with a single equivalent moving dipole (SEMD) in an infinite homogeneous volume conductor. Despite the fact that the use of the homogeneous volume conductor introduces systematic error in the estimated compared to the true dipole location, we find that the systematic error had minor influence in the ability of the algorithm to accurately guide the tip of the ablation catheter to the ablation site and the overall error (1.9 ± 1.1 mm) in the left ventricle is adequate for RCA procedures. These results were verified, in saline tank studies in which the distance between the dipole due to the catheter tip and the dipole due to the target site was found to be 2.66 ± 0.52 mm. In conclusion, our algorithm to estimate the SEMD parameters from body surface potentials can potentially be a useful method to rapidly and accurately guide the catheter tip to the arrhythmic site during an RCA procedure.     

Comparison between electrocardiographic and magnetocardiographic inverse solutions using the boundary element method

The accuracy of imaging cardiac sources using electrocardiographic and magnetocardiographic signals is influenced by thoracic inhomogeneities, e.g. the lungs and cardiac blood masses. The effects is investigated of such inhomogeneities on the body-surface potential maps (BSPM) and magnetic-field maps (MFM) inverse solutions for a single moving dipole as the source model and a realistic torso model as the volume conductor, by employing a node-based boundary element method. Using the same number and placement of the body-surface potential and magnetic field leads, a comparison is obtained of the numerical accuracy of body-surface potential and magnetic field leads. The results show that, with no noise added, the body-surface potential solution is less sensitive to the exclusion of the inhomogeneities than the magnetic field solution. The influence of noise on the BSPM and MFM localization is comparable for x (left-right) and y (foot-head) oriented dipoles, and the BSPM localisation is more accurate than the MFM localisation for z (anterior-posterior) oriented dipoles.     

A Minimal Product Method and Its Application to Cortical Imaging

In order to reduce the spatial blurring effect due to the head volume conductor, cortical imaging technique (CIT) can be used to reconstruct the cortical potential distribution from the scalp potential measurement with enhanced spatial resolution. To overcome the ill-posed nature of the inverse problem, Tikhonov regularization (TIK) and truncated Singular Value Decomposition (TSVD) are commonly used by choosing the appropriate regularization parameter and truncation parameter, respectively. We have developed a minimal product method (MINP) to determine the regularization and truncation parameters. The present computer simulation and experimental results indicate that the MINP can be easily implemented in both TIK and TSVD with satisfactory performance, and suggest the potential applications of the MINP method in determining the corner of the L-curve.     

Non-invasive estimation of myocardial infarction by means of a heart-model-based imaging approach: A simulation study

In the study, a new myocardial infarction (MI) estimation method was developed for estimating Ml in the three-dimensional myocardium by means of a heart-model-based inverse approach. The site and size of Ml are estimated from body surface electrocardiograms by minimising multiple objective functions of the measured body surface potential maps (BSPMs) and the heart-model-generated BSPMs. Computer simulations were conducted to evaluate the performance of the developed method, using a single-site Ml and dual-site Ml protocols. The simulation results show that, for the single-site Ml, the averaged spatial distance (SD) between the weighting centres of the ‘true’ and estimated Mls, and the averaged relative error (RE) between the numbers of the ‘true’ and estimated infarcted units are 3.0±0.6/3.6±0.6 mm and 0.11±0.02/0.14±0.02, respectively, when 5μV/10μV Gaussian white noise was added to the body surface potentials. For the dual-site Ml, the averaged SD between the weighting centres of the ‘true’ and estimated Mls, and the averaged RE between the numbers of the ‘trus’ and estimated infarcted units are 3.8±0.7/3.9±0.7 mm and 0.12±0.02/0.14±0.03, respectively, when 5μV/10μV Gaussian white noise was added to the body surface potentials. The simulation results suggest the feasibility of applying the heart-model-based imaging approach to the estimation of myocardial infarction from body surface potentials.     

A code for hadrontherapy treatment planning with the voxelscan method

A code for the implementation of treatment plannings in hadrontherapy with an active scan beam is presented. The package can determine the fluence and energy of the beams for several thousand voxels in a few minutes. The performances of the program have been tested with a full simulation.     

Mapping EEG-potentials on the surface of the brain: A strategy for uncovering cortical sources

Summary This paper describes a uniform method for calculating the interpolation of scalp EEG potential distribution, the current source density (CSD), the cortical potential distribution (cortical mapping) and the CSD of the cortical potential distribution. It will be shown that interpolation and deblurring methods such as CSD or cortical mapping are not independent of the inverse problem in potential theory. Not only the resolution but also the accuracy of these techniques, especially those of deblurring, depend greatly on the spatial sampling rate (i.e., the number of electrodes). Using examples from simulated and real (64 channels) data it can be shown that the application of more than 100 EEG channels is not only favourable but necessary to guarantee a reasonable accuracy in the calculations of CSD or cortical mapping. Likewise, it can be shown that using more than 250 electrodes does not improve the resolution.This study was supported by grants from the Deutsche Forschungsgemeinschaft to Thomas Elbert and Brigitte Rockstroh. The authors would like to thank Dr. Dorothy Charbonnier.     

A Novel Interpolation Method for Electric Potential Fields in the Heart during Excitation

In mapping the electrical activity of the heart, interpolation of electric potentials plays two important roles. First, it permits the estimation of potentials in regions that could not be sampled or where signal quality was poor, and second, it supports the construction of isopotential lines and surfaces for visualization. The difficulty in developing robust interpolation techniques for cardiac applications lies in the abrupt change in potential in the vicinity of the activation wave front. Despite the resulting nonlinearities in spatial potential distributions, simple linear interpolation methods are the current standard and the resulting errors due to aliasing can be large if electrode spacing does not lie on the order of 0.5–2 mm—the thickness of the activation wave front. We have developed a novel interpolation method that is based on two observations specific to the spread of excitation in the heart: (1) that propagation velocity changes smoothly within a region large enough to contain several measurement electrodes and (2) that electrogram morphology varies very little in the neighborhood of each sample point except for a time shift in the potential wave forms. The resulting interpolation scheme breaks the interpolation of one highly nonlinear variable—extracellular potential—into two separate interpolations of variables with much less drastic spatial variation—activation time and electrogram morphology. We have applied this method to potentials originally recorded at 1.5 mm spacing and then subsampled at a range of densities for testing of the interpolation. The results based both on reconstruction of isopotential contour maps and statistical comparison showed significant improvement of this novel approach over standard linear techniques. The applications of the new method include improved determination of electrophysiological parameters such as spatial gradients of potential and the path of cardiac activation and recovery, estimation of electrograms at desired locations, and visualization of electric potential distributions. © 1998 Biomedical Engineering Society. PAC98: 8790+y, 0260Ed, 8710+e     

Orthogonal expansions: their applicability to signal extraction in electrophysiological mapping data

The applicability of orthogonal expansions (singular-value decomposition, Karhunen-Loève transform and principal-component analysis) for the purpose of identifying source distributions associated with definite electrophysiological events in the heart and brain is explored with a current dipole source model. By definition, the expansion eigenvectors are orthogonal, and as such will extract the features of one specific source only if all other secondary signals are orthogonal to that first source. The number of significant eigenvectors can be related to the number of original components forming a signal, but there is not a one-to-one correspondence between these eigenvectors and the individual components. Furthermore, many eigenvectors may be needed to faithfully represent even a single source, if that source is nonstationary. We conclude that generally it would be inappropriate to ascribe any physiological significance to the data resulting from such expansions.     

Characterization of the Highly Nonlinear and Anisotropic Vascular Tissues from Experimental Inflation Data: A Validation Study Toward the Use of Clinical Data for <Emphasis Type="Italic">In-Vivo</Emphasis> Modeling and Analysis

We study whether an inverse modeling approach is applicable for characterizing vascular tissue subjected to various levels of internal pressure and axial stretch that approximate in-vivo conditions. To compensate for the limitation of axial-displacement/pressure/diameter data typical of clinical data, which does not provide information about axial force, we propose to constrain the ratio of axial to circumferential elastic moduli to a typical range. Vessel wall constitutive behavior is modeled with a transversely isotropic hyperelastic equation that accounts for dispersed collagen fibers. A single-layer and a bi-layer approximation to vessel ultrastructure are examined, as is the possibility of obtaining the fiber orientation as part of the optimization. Characterization is validated against independent pipette-aspiration biaxial data on the same samples. It was found that the single-layer model based on homogeneous wall assumption could not reproduce the validation data. In contrast, the constrained bi-layer model was in excellent agreement with both types of experimental data. Due to covariance, estimations of fiber angle were slightly outside of the normal range, which can be resolved by predefining the angles to normal values. Our approach is relatively invariant to a constant or a variable axial response. We believe that it is suitable for in-vivo characterization.