A comparison of finite element and integral equation formulations for the calculation of electrocardiographic potentials--II

Electrocardiographic imaging in the atria

The inverse problem of electrocardiography or electrocardiographic imaging (ECGI) is a technique for reconstructing electrical information about cardiac surfaces from noninvasive or non-contact recordings. ECGI has been used to characterize atrial and ventricular arrhythmias. Although it is a technology with years of progress, its development to characterize atrial arrhythmias is challenging. Complications can arise when trying to describe the atrial mechanisms that lead to abnormal propagation patterns, premature or tachycardic beats, and reentrant arrhythmias. This review addresses the various ECGI methodologies, regularization methods, and post-processing techniques used in the atria, as well as the context in which they are used. The current advantages and limitations of ECGI in the fields of research and clinical diagnosis of atrial arrhythmias are outlined. In addition, areas where ECGI efforts should be concentrated to address the associated unsatisfied needs from the atrial perspective are discussed.

Scalable and Accurate ECG Simulation for Reaction-Diffusion Models of the Human Heart

Realistic electrocardiogram (ECG) simulation with numerical models is important for research linking cellular and molecular physiology to clinically observable signals, and crucial for patient tailoring of numerical heart models. However, ECG simulation with a realistic torso model is computationally much harder than simulation of cardiac activity itself, so that many studies with sophisticated heart models have resorted to crude approximations of the ECG. This paper shows how the classical concept of electrocardiographic lead fields can be used for an ECG simulation method that matches the realism of modern heart models. The accuracy and resource requirements were compared to those of a full-torso solution for the potential and scaling was tested up to 14,336 cores with a heart model consisting of 11 million nodes. Reference ECGs were computed on a 3.3 billion-node heart-torso mesh at 0.2 mm resolution. The results show that the lead-field method is more efficient than a full-torso solution when the number of simulated samples is larger than the number of computed ECG leads. While the initial computation of the lead fields remains a hard and poorly scalable problem, the ECG computation itself scales almost perfectly and, even for several hundreds of ECG leads, takes much less time than the underlying simulation of cardiac activity.

Cardiac anisotropy in boundary-element models for the electrocardiogram

The boundary-element method (BEM) is widely used for electrocardiogram (ECG) simulation. Its major disadvantage is its perceived inability to deal with the anisotropic electric conductivity of the myocardial interstitium, which led researchers to represent only intracellular anisotropy or neglect anisotropy altogether. We computed ECGs with a BEM model based on dipole sources that accounted for a “compound” anisotropy ratio. The ECGs were compared with those computed by a finite-difference model, in which intracellular and interstitial anisotropy could be represented without compromise. For a given set of conductivities, we always found a compound anisotropy value that led to acceptable differences between BEM and finite-difference results. In contrast, a fully isotropic model produced unacceptably large differences. A model that accounted only for intracellular anisotropy showed intermediate performance. We conclude that using a compound anisotropy ratio allows BEM-based ECG models to more accurately represent both anisotropies.

Elliptic Cylinder Geometry for Distinguishability Analysis in Impedance Tomography

Electrical impedance tomography (EIT) is a technique that computes the cross-sectional impedance distribution within the body by using current and voltage measurements made on the body surface. It has been reported that the image reconstruction is distorted considerably when the boundary shape is considered to be more elliptical than circular as a more realistic shape for the measurement boundary. This paper describes an alternative framework for determining the distinguishability region with a finite measurement precision for different conductivity distributions in a body modeled by elliptic cylinder geometry. The distinguishable regions are compared in terms of modeling error for predefined inhomogeneities with elliptical and circular approaches for a noncircular measurement boundary at the body surface. Since most objects investigated by EIT are noncircular in shape, the analytical solution for the forward problem for the elliptical cross section approach is shown to be useful in order to reach a better assessment of the distinguishability region defined in a noncircular boundary. This paper is concentrated on centered elliptic inhomogeneity in the elliptical boundary and an analytic solution for this type of forward problem. The distinguishability performance of elliptical cross section with cosine injected current patterns is examined for different parameters of elliptical geometry.

Electrical impedance tomography of complex conductivity distributions with noncircular boundary

Electrical impedance tomography (EIT) uses low-frequency current and voltage measurements made on the boundary of a body to compute the conductivity distribution within the body. Since the permittivity distribution inside the body also contributes significantly to the measured voltages, the present reconstruction algorithm images complex conductivity distributions. A finite element model (FEM) is used to solve the forward problem, using a 6017-node mesh for a piecewise-linear potential distribution. The finite element solution using this mesh is compared with the analytical solution for a homogeneous field and a maximum error of 0.05% is observed in the voltage distribution. The boundary element method (BEM) is also used to generate the voltage data for inhomogeneous conductivity distributions inside regions with noncircular boundaries. An iterative reconstruction algorithm is described for approximating both the conductivity and permittivity distributions from this data. The results for an off-centered inhomogeneity showed a 35% improvement in contrast from that seen with only one iteration, for both the conductivity and the permittivity values. It is also shown that a significant improvement in images results from accurately modeling a noncircular boundary. Both static and difference images are distorted by assuming a circular boundary and the amount of distortion increases significantly as the boundary shape becomes more elliptical. For a homogeneous field in an elliptical body with axis ratio of 0.73, an image reconstructed assuming the boundary to be circular has an artifact at the center of the image with an error of 20%. This error increased to 37% when the axis ratio was 0.64. A reconstruction algorithm which used a mesh with the same axis ratio as the elliptical boundary reduced the error in the conductivity values to within 0.5% of the actual values.

High resolution EEG: a new model-dependent spatial deblurring method using a realistically-shaped MR-constructed subject's head model

This paper presents a new model-dependent method for the spatial deblurring of scalp-recorded EEG potentials based on boundary-element and cortical imaging techniques. This model-dependent spatial deblurring (MDSD) method used MR images for the reconstruction of the subject's head model, and a layer of 364 radially-oriented equivalent current dipoles as a source model. The validation of the MDSD method was performed on simulated potential distributions generated from equivalent dipoles oriented radially, obliquely, and tangentially to the head surface. Furthermore, this method was used to localize neocortical sources of human movement-related and somatosensory-evoked potentials. It was shown that the new MDSD method improved markedly the spatial resolution of the simulated surface potentials and scalp-recorded event-related potentials. The spatial information content of the scalp-recorded EEG potentials increased progressively by increasing the spatial sampling from 28 to 128 channels. These results indicate that the new method could be satisfactorily used for high resolution EEG studies.

Modeling current density distributions during transcutaneous cardiac pacing

We developed a two-dimensional finite element model of a cross-section of the human thorax to study the current density distribution during transcutaneous cardiac pacing. The model comprises 964 nodes and 1842 elements and accounted for the electrical properties of eight different tissues or organs and also simulated the anisotropies of the intercostal muscles. The finite element software employed was a version for electrokinetics problems of Finite Element for Heat Transfer (FEHT) and we assessed the effects upon the efficacy of transcutaneous cardiac pacing of several electrode placements and sizes. To minimize pain in the chest wall and still be able to capture the heart, we minimized the ratio, R, between the current density in the thoracic wall (which causes pain) and the current density in the heart wall (which captures the heart). The best placement of the negative electrode was over the cardiac apex. The best placement of the positive electrode was under the right scapula, although other placements were nearly as good. The efficiency of pacing increased as electrode size increased up to 70 cm2 and showed little improvement for larger areas. Between different configurations of the precordial electrodes V1, V2, ..., V6 the most efficient configuration to pace with was V1 and V2 positive and V5 and V6 negative. A more efficient configuration uses an auxiliary electrode located at the right subscapular region.

A generalized eigensystem approach to the inverse problem of electrocardiography

We develop a new approach to the ill-conditioned inverse problem of electrocardiography which employs finite element techniques to generate a truncated eigenvector expansion to stabilize the inversion. Ordinary three-dimensional isoparametric finite elements are used to generate the conductivity matrix for the body. We introduce a related eigenproblem, for which a special two-dimensional isoparametric area matrix is used, and solve for the lowest eigenvalues and eigenvectors. The body surface potentials are expanded in terms of the eigenvectors, and a least squares fit to the measured body surface potentials is used to determine the coefficients of the expansion. This expansion is then used directly to determine the potentials on the surface of the heart. The number of measurement points on the surface of the body can be less than the number of finite element nodes on the body surface, and the number of modes employed in the expansion can be adjusted to reduce errors due to noise.

Method to reduce blur distortion from EEG's using a realistic head model

A mathematical procedure, which we call "Deblurring," was developed to reduce spatial blur distortion of scalp-recorded brain potentials due to transmission through the skull and other tissues. Deblurring estimates potentials at the superficial cerebral cortical surface from EEG's recorded at the scalp using a finite element model of each subject's scalp, skull and cortical surface constructed from their magnetic resonance images (MRI's). Simulations indicate that Deblurring is numerically stable, while a comparison of deblurred data with a direct cortical recording from a neurosurgery patient suggests that the procedure is valid. Application of Deblurring to somatosensory evoked potential data recorded at 124 scalp sites suggests that the method produces a dramatic improvement in spatial detail, and merits further development.

Parallel computation of ECG fields

A parallel implementation of a finite difference model for computing the electric field of cardiac sources is presented. On a relatively inexpensive SIMD parallel computer, a full-forward solution is obtained in minutes, using accurate thoracic detail including anisotropy if required. Because the computation is based on a volume grid with constant size voxels, it readily accepts anatomical data from classified magnetic resonance imaging scans. By using a variation of the colored successive over-relaxation iteration, our finite difference model takes full advantage of the performance of massively parallel computers. Evaluations of the accuracy and performance of the model show the practicality of using specific anatomical models to recover the electrocardiographic field distributions for individual subjects. A relatively modest parallel machine is capable of assembling and computing a specific direct inverse solution from body surface potentials within an hour of measurement, assuming the magnetic resonance imaging classification has been previously completed.

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.

An assessment of variable thickness and fiber orientation of the skeletal muscle layer on electrocardiographic calculations

This paper assesses the effectiveness of including variable thickness and fiber orientation characteristics of the skeletal muscle layer in calculations relating epicardial and torso potentials. A realistic model of a canine torso which includes extensive detail about skeletal muscle layer thickness and fiber orientation is compared with two other uniformly anisotropic models: one of constant thickness and the other of variable thickness. First, transfer coefficients are calculated from the model data. Then torso potentials for each model are calculated from the transfer coefficients and measured epicardial potentials. The comparison of calculated and observed torso potentials indicates that a simple model consisting of a uniformly anisotropic skeletal muscle layer of 1.0-1.5 cm constant thickness significantly improves the model. However, if photographic slices of the canine torso are used to introduce more detailed data about the variation in skeletal muscle thickness and fiber orientation into the model, the agreement and between calculated and measured torso potentials decreased, although a finite element mesh of over 5000 nodes was used to describe the skeletal muscle in the more detailed model. One source of error increase was considered to be due to numerical discretization and could be reduced with a much finer mesh or by utilizing higher order polynomials to represent the potential distribution within each finite element. However, the results presented in this paper show that high precision computation (64-bit word length) on the mainframe IBM 3081 with an attached FPS-164 gives a slow rate of improvement with reduced discretization intervals and that utilizing higher order polynomials within each finite element gives an even slower rate of improvement.(ABSTRACT TRUNCATED AT 250 WORDS)

Epidural spinal cord stimulation: calculation of field potentials with special reference to dorsal column nerve fibers

The effect of electrical stimulation with several electrode combinations on nerve fibers with different orientations in the spinal cord was investigated by computing the steady-state field potentials and activating functions. At first an infinite homogeneous model was used while secondly the spinal cord and its surrounding tissues were modeled as an inhomogeneous anisotropic volume conductor. The effect of mediodorsal epidural stimulation was calculated. It was concluded that with cathodal stimulation, mediodorsally in the epidural space, longitudinal fibers are depolarized, but dorsoventral ones are hyperpolarized. With anodal stimulation the opposite will occur. It was found that parameters substantially affecting the potential distribution in the dorsal columns are the conductivity of the white matter and the width and the conductivity of the csf layer.

The finite element method for a realistic head model of electrical brain activities: preliminary results

In order to model the brain's electrical activity realistically, the finite element method has been used to compute the potential distribution due to a current dipole. This approach has the advantage over the boundary element method of being able to consider anisotropies of the different conducting sub-volumes. The forward solution has been evaluated in the particular case of a three-layer concentric sphere isotropic head model of the head where an analytical formula is known. The errors on the dipole position and orientation have been estimated in the inverse problem procedure.

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 modeling study of nerve fascicle stimulation

A nerve stimulation model has been developed, incorporating realistic cross-sectional nerve geometries and conductivities. The potential field in the volume conductor was calculated numerically using the variational method. Nerve fiber excitation was described by the model of McNeal. Cross-sectional geometries of small monofascicular rat common peroneal nerve and multifascicular human deep peroneal nerve were taken as sample geometries. Selective stimulation of a fascicle was theoretically analyzed for several electrode positions: outside the nerve, in the connective tissue of the nerve, and inside a fascicle. The model results predict that the use of intraneural or even intrafascicular electrodes is necessary for selective stimulation of fascicles not lying at the surface of the nerve. Model predictions corresponded with experimental results of Veltink et al. on intrafascicular and extraneural stimulation of rat common peroneal nerve and to results of McNeal and Bowman on muscle selective stimulation in multifascicular dog sciatic nerve using an extraneural multielectrode configuration.

The potential gradient field created by epicardial defibrillation electrodes in dogs

Knowledge of the potential gradient field created by defibrillation electrodes is important for the understanding and improvement of defibrillation. To obtain this knowledge by direct measurements, potentials were recorded from 60 epicardial, eight septal, and 36 right ventricular transmural electrodes in six open-chest dogs while 1 to 2 V shocks were given through defibrillation electrodes on the right atrium and left ventricular apex (RA. V) and on the right and left ventricles (RV .LV). The potential gradient field across the ventricles was calculated for these low voltages. Ventricular fibrillation was electrically induced, and ventricular activation patterns were recorded after delivering high-voltage shocks just below the defibrillation threshold. With the low-voltage shocks, the potential gradient field was very uneven, with the highest gradient near the epicardial defibrillation electrodes and the weakest gradient distant from the defibrillation electrodes for both RA. V and RV .LV combinations. The mean ratio of the highest to the lowest measured gradient over the entire ventricular epicardium was 19.4 +/- 8.1 SD for the RA. V combination and 14.4 +/- 3.4 for the RV .LV combination. For both defibrillation electrode combinations, the earliest sites of activation after unsuccessful shocks just below the defibrillation threshold were located in areas where the potential gradient was weak for the low-voltage shocks. We conclude that there is a markedly uneven distribution of potential gradients for epicardial defibrillation electrodes with most of the voltage drop occurring near the electrodes, the potential gradient field is significant because it determines where shocks fail to halt fibrillation, and determination of the potential gradient field should lead to the development of improved electrode locations for defibrillation.