Computational and numerical methods for bioelectric field problems

Quantifying and Visualizing Uncertainty for Source Localization in Electrocardiographic Imaging

Electrocardiographic imaging (ECGI) presents a clinical opportunity to noninvasively understand the sources of arrhythmias for individual patients. To help increase the effectiveness of ECGI, we provide new ways to visualize associated measurement and modeling errors. In this paper, we study source localization uncertainty in two steps: First, we perform Monte Carlo simulations of a simple inverse ECGI source localization model with error sampling to understand the variations in ECGI solutions. Second, we present multiple visualization techniques, including confidence maps, level-sets, and topology-based visualizations, to better understand uncertainty in source localization. Our approach offers a new way to study uncertainty in the ECGI pipeline.

Reducing Error in ECG Forward Simulations With Improved Source Sampling

A continuing challenge in validating electrocardiographic imaging (ECGI) is the persistent error in the associated forward problem observed in experimental studies. One possible cause of this error is insufficient representation of the cardiac sources; cardiac source measurements often sample only the ventricular epicardium, ignoring the endocardium and the atria. We hypothesize that measurements that completely cover the pericardial surface are required for accurate forward solutions. In this study, we used simulated and measured cardiac potentials to test the effect of different levels of spatial source sampling on the forward simulation. Not surprisingly, increasing the source sampling over the atria reduced the average error of the forward simulations, but some sampling strategies were more effective than others. Uniform and random distributions of samples across the atrial surface were the most efficient strategies in terms of lowest error with the fewest sampling locations, whereas “single direction” strategies, i.e., adding to the atrioventricular (AV) plane or atrial roof only, were the least efficient. Complete sampling of the atria is needed to eliminate errors from missing cardiac sources, but while high density sampling that covers the entire atria yields the best results, adding as few as 11 electrodes on the atria can significantly reduce these errors. Future validation studies of the ECG forward simulations should use a cardiac source sampling that takes these considerations into account, which will, in turn, improve validation and understanding of ECGI.

The effect of tissue anisotropy on the radial and tangential components of the electric field in transcranial direct current stimulation

Transcranial direct current stimulation (tDCS) is considered to be a promising technique for noninvasive brain stimulation and brain disease therapy. Recent studies have investigated the distribution of the electric field (EF) magnitude over gyri and sulci and the effect of tissue homogeneity with isotropic electrical conductivities. However, it is well known that the skull and white matter (WM) are highly anisotropic electrically, requiring investigations of their anisotropic effects on the magnitude and the directional components of the induced EF due to the high dependency between neuromodulation and the EF direction. In this study, we investigated the effects of the skull and WM anisotropy on the radial and tangential components of the EF via gyri-specific high-resolution finite element head models. For tDCS, three configurations were investigated: the conventional rectangular pad electrode, a 4(cathodes) +1(anode) ring configuration, and a bilateral configuration. The results showed that the skull anisotropy has a crucial influence on the distribution of the radial EF component. The affected cortical regions by the radial EF were reduced about 22 % when considering the skull anisotropy in comparison with the regions with the skull isotropy. On the other hand, the WM anisotropy strongly alters the EF directionality, especially within the sulci. The electric current tends to flow radially to the cortical surface with the WM anisotropy. This effect increases the affected cortical areas by the radial EF component within the sulcal regions. Our results suggest that one must examine the distribution of the EF components in tDCS, not just the magnitude of the EF alone.

Assessment of regularization techniques for electrocardiographic imaging

A widely used approach to solving the inverse problem in electrocardiography involves computing potentials on the epicardium from measured electrocardiograms (ECGs) on the torso surface. The main challenge of solving this electrocardiographic imaging (ECGI) problem lies in its intrinsic ill-posedness. While many regularization techniques have been developed to control wild oscillations of the solution, the choice of proper regularization methods for obtaining clinically acceptable solutions is still a subject of ongoing research. However there has been little rigorous comparison across methods proposed by different groups. This study systematically compared various regularization techniques for solving the ECGI problem under a unified simulation framework, consisting of both 1) progressively more complex idealized source models (from single dipole to triplet of dipoles), and 2) an electrolytic human torso tank containing a live canine heart, with the cardiac source being modeled by potentials measured on a cylindrical cage placed around the heart. We tested 13 different regularization techniques to solve the inverse problem of recovering epicardial potentials, and found that non-quadratic methods (total variation algorithms) and first-order and second-order Tikhonov regularizations outperformed other methodologies and resulted in similar average reconstruction errors.

Computed basis functions for finite element analysis based on tomographic data

In bioelectromagnetics, the structures in which the electromagnetic field is to be computed are sometimes defined by a fine grid of voxels (3-D cells) whose tissue types are obtained by tomography. A novel finite element method is proposed for such cases. A simple, regular mesh of cube elements is constructed, each containing the same, integer number of voxels. There may be several different tissues present within an element, but this is accommodated by computing element basis functions that approximately respect the interface conditions between different tissues. Results are presented for a test model of 128 (3) voxels, consisting of nested dielectric cubes, driven by specified charges. The electrostatic potential computed with the new method agrees well with that of a conventional finite element code: the rms difference along the sample line is 1.5% of the highest voltage. Results are also presented for the potential due to a current dipole placed in a brain model of 181 × 217 × 181 voxels, derived from MRI data. The new method gives potentials that are different to those obtained by treating each voxel as an element by 1% of the peak voltage, yet the global finite element matrix has a dimension which is more than 50 times smaller.

Selective neural activation in a histologically derived model of peripheral nerve

Functional electrical stimulation (FES) is a general term for therapeutic methods that use electrical stimulation to aid or replace lost ability. For FES systems that communicate with the nervous system, one critical component is the electrode interface through which the machine-body information transfer must occur. In this paper, we examine the influence of inhomogeneous tissue conductivities and positions of nodes of Ranvier on activation of myelinated axons for neuromuscular control as a function of electrode configuration. To evaluate these effects, we developed a high-resolution bioelectric model of a fascicle from a stained cross-section of cat sciatic nerve. The model was constructed by digitizing a fixed specimen of peripheral nerve, extruding the image along the axis of the nerve, and assigning each anatomical component to one of several different tissue types. Electrodes were represented by current sources in monopolar, transverse bipolar, and longitudinal bipolar configurations; neural activation was determined using coupled field-neuron simulations with myelinated axon cable models. We found that the use of an isotropic tissue medium overestimated neural activation thresholds compared with the use of physiologically based, inhomogeneous tissue medium, even after controlling for mean impedance levels. Additionally, the positions of the cathodic sources relative to the nodes of Ranvier had substantial effects on activation, and these effects were modulated by the electrode configuration. Our results indicate that physiologically based tissue properties cause considerable variability in the neural response, and the inclusion of these properties is an important component in accurately predicting activation. The results are used to suggest new electrode designs to enable selective stimulation of small diameter fibers.

Finite-element-based discretization and regularization strategies for 3-D inverse electrocardiography

We consider the inverse electrocardiographic problem of computing epicardial potentials from a body-surface potential map. We study how to improve numerical approximation of the inverse problem when the finite-element method is used. Being ill-posed, the inverse problem requires different discretization strategies from its corresponding forward problem. We propose refinement guidelines that specifically address the ill-posedness of the problem. The resulting guidelines necessitate the use of hybrid finite elements composed of tetrahedra and prism elements. Also, in order to maintain consistent numerical quality when the inverse problem is discretized into different scales, we propose a new family of regularizers using the variational principle underlying finite-element methods. These variational-formed regularizers serve as an alternative to the traditional Tikhonov regularizers, but preserves the L(2) norm and thereby achieves consistent regularization in multiscale simulations. The variational formulation also enables a simple construction of the discrete gradient operator over irregular meshes, which is difficult to define in traditional discretization schemes. We validated our hybrid element technique and the variational regularizers by simulations on a realistic 3-D torso/heart model with empirical heart data. Results show that discretization based on our proposed strategies mitigates the ill-conditioning and improves the inverse solution, and that the variational formulation may benefit a broader range of potential-based bioelectric problems.

The influence of age and skull conductivity on surface and subdermal bipolar EEG leads

Bioelectric source measurements are influenced by the measurement location as well as the conductive properties of the tissues. Volume conductor effects such as the poorly conducting bones or the moderately conducting skin are known to affect the measurement precision and accuracy of the surface electroencephalography (EEG) measurements. This paper investigates the influence of age via skull conductivity upon surface and subdermal bipolar EEG measurement sensitivity conducted on two realistic head models from the Visible Human Project. Subdermal electrodes (a.k.a. subcutaneous electrodes) are implanted on the skull beneath the skin, fat, and muscles. We studied the effect of age upon these two electrode types according to the scalp-to-skull conductivity ratios of 5, 8, 15, and 30 : 1. The effects on the measurement sensitivity were studied by means of the half-sensitivity volume (HSV) and the region of interest sensitivity ratio (ROISR). The results indicate that the subdermal implantation notably enhances the precision and accuracy of EEG measurements by a factor of eight compared to the scalp surface measurements. In summary, the evidence indicates that both surface and subdermal EEG measurements benefit better recordings in terms of precision and accuracy on younger patients.

Resolution strategies for the finite-element-based solution of the ECG inverse problem

Successful employment of numerical techniques for the solution of forward and inverse ECG problems requires the ability to both quantify and minimize approximation errors introduced as part of the discretization process. Our objective is to develop discretization and refinement strategies involving hybrid-shaped finite elements so as to minimize approximation errors for the ECG inverse problem. We examine both the ill-posedness of the mathematical inverse problem and the ill-conditioning of the discretized system in order to propose strategies specifically designed for the ECG inverse problem. We demonstrate that previous discretization and approximation strategies may worsen the properties of the inverse problem approximation. We then demonstrate the efficacy of our strategies on both a simplified and a realistic 2-D torso model.

Two hybrid regularization frameworks for solving the electrocardiography inverse problem

In this paper, two hybrid regularization frameworks, LSQR-Tik and Tik-LSQR, which integrate the properties of the direct regularization method (Tikhonov) and the iterative regularization method (LSQR), have been proposed and investigated for solving ECG inverse problems. The LSQR-Tik method is based on the Lanczos process, which yields a sequence of small bidiagonal systems to approximate the original ill-posed problem and then the Tikhonov regularization method is applied to stabilize the projected problem. The Tik-LSQR method is formulated as an iterative LSQR inverse, augmented with a Tikhonov-like prior information term. The performances of these two hybrid methods are evaluated using a realistic heart-torso model simulation protocol, in which the heart surface source method is employed to calculate the simulated epicardial potentials (EPs) from the action potentials (APs), and then the acquired EPs are used to calculate simulated body surface potentials (BSPs). The results show that the regularized solutions obtained by the LSQR-Tik method are approximate to those of the Tikhonov method, the computational cost of the LSQR-Tik method, however, is much less than that of the Tikhonov method. Moreover, the Tik-LSQR scheme can reconstruct the epcicardial potential distribution more accurately, specifically for the BSPs with large noisy cases. This investigation suggests that hybrid regularization methods may be more effective than separate regularization approaches for ECG inverse problems.

Truncated total least squares: a new regularization method for the solution of ECG inverse problems

The reconstruction of epicardial potentials (EPs) from body surface potentials (BSPs) can be characterized as an ill-posed inverse problem which generally requires a regularized numerical solution. Two kinds of errors/noise: geometric errors and measurement errors exist in the ECG inverse problem and make the solution of such problem more difficulty. In particular, geometric errors will directly affect the calculation of transfer matrix A in the linear system equation AX = B. In this paper, we have applied the truncated total least squares (TTLS) method to reconstruct EPs from BSPs. This method accounts for the noise/errors on both sides of the system equation and treats geometric errors in a new fashion. The algorithm is tested using a realistically shaped heart-lung-torso model with inhomogeneous conductivities. The h-adaptive boundary element method [h-BEM, a BEM mesh adaptation scheme which starts from preset meshes and then refines (adds/removes) grid with fixed order of interpolation function and prescribed numerical accuracy] is used for the forward modeling and the TTLS is applied for inverse solutions and its performance is also compared with conventional regularization approaches such as Tikhonov and truncated single value decomposition (TSVD) with zeroth-, first-, and second-order. The simulation results demonstrate that TTLS can obtain similar results in the situation of measurement noise only but performs better than Tikhonov and TSVD methods where geometric errors are involved, and that the zeroth-order regularization is the optimal choice for the ECG inverse problem. This investigation suggests that TTLS is able to robustly reconstruct EPs from BSPs and is a promising alternative method for the solution of ECG inverse problems.

Patient-specific volume conductor modeling for non-invasive imaging of cardiac electrophysiology

We propose a general workflow to numerically estimate the spread of electrical excitation in the patients' hearts. To this end, a semi-automatic segmentation pipeline for extracting the volume conductor model of structurally normal hearts is presented. The cardiac electrical source imaging technique aims to provide information about the spread of electrical excitation in order to assist the cardiologist in developing strategies for the treatment of cardiac arrhythmias. The volume conductor models of eight patients were extracted from cine-gated short-axis magnetic resonance imaging (MRI) data. The non-invasive estimation of electrical excitation was compared with the CARTO maps. The development of a volume conductor modeling pipeline for constructing a patient-specific volume conductor model in a fast and accurate way is one essential step to make the technique clinically applicable.

Wireless and inductively powered implant for measuring electrocardiogram

The development of an active implantable device for measuring electrocardiogram (ECG) is presented. The study is a part of a project which aims at developing implantable ECG instrumentation with wireless data and power transfer ( http://www.ele.tut.fi/tule ). The developed implant presented here has all the measurement electronics as well as power and data communication instrumentation included. The implant itself contains no battery, while power for the implant is transferred electromagnetically from an external reader device. The results of testing the implant attached on the body surface and in vitro in a water container are also presented. The developed system was also successfully tested in in vivo measurements, which were conducted on four cows with an implantation time of 24 h. The in vivo testing of implant in cows was conducted by a veterinarian in supervised conditions under approved animal experiment licence.

Contribution of the left anterior myocardium to the body surface potentials in case of apical ectopic beat

The present paper describes a study where effects of anterior myocardium on body surface potentials were investigated. The study combines numerical lead field analysis combined with cardiac automata model. Electric fields are calculated with finite difference method in a 3-D model of male thorax. The cardiac activation applied in the study is an ectopic beat originating in the apex. The correlations and mean differences between signal generated by anterior segments of left ventricle and signal generated by both ventricles were analysed for 117 leads. The results show that there are leads which have high correlation (>0.9) with low the relative mean difference (<0.2) between the signals generated by anterior segments and signals generated by whole ventricles. These electrode locations would be optimal to monitor the activation of anterior segments when ectopic beats originate in apex.

Finite difference and lead field methods in designing implantable ECG monitor

To minimize time-consuming and expensive in vitro and in vivo testing, information regarding the effects of implantation and the implants on measurements should be available during the designing of active implantable devices measuring bioelectric signals such as electrocardiograms (ECG). Modeling offers a fairly inexpensive and effective means of studying and demonstrating the effects of implantation on ECG measurements prior to any in vivo tests, and can thus provide the designer with valuable information. Finite difference model (FDM) and lead field approaches offer straightforward and effective modeling methods supporting the designing of active implantable ECG devices. The present study demonstrates such methods in developing and studying ECG implants. They were applied in demonstrating the effects of implant dimensions and of electrode implantation on the measurement sensitivity of the ECG device. The results of the simulations indicated that the interelectrode distance is the factor of the implant design determining the lead sensitivity. Other parameters related implant dimensions and shape have minor effect on the morphology of the ECG or on the average sensitivity of the measurement. This is shown for example when the interelectrode distance was reduced to 1/3 of original the average lead sensitivity decreased by 69.1% while larger relative changes in other dimensions produced clearly smaller changes. It was also observed here that implanting the electrodes deeper under the skin has major effects on the local sensitivities in heart muscle and thus affect to the morphology of the ECG. The study indicated also that non-conducting medium (i.e. implant insulated body) between the electrodes increases the sensitivity on heart muscle compared to cases where only electrodes are implanted.

Modeling the current distribution during transcranial direct current stimulation

OBJECTIVE

To investigate the spatial distribution of the magnitude and direction of the current density in the human head during transcranial direct current stimulation (tDCS).

METHODS

The current density distribution was calculated using a numerical method to implement a standard spherical head model into which current was injected by means of large electrodes. The model was positioned in 'MNI space' to facilitate the interpretation of spatial coordinates.

RESULTS

The magnitude and direction of the current density vector are illustrated in selected brain slices for four different electrode montages. Approximately half of the current injected during tDCS is shunted through the scalp, depending on electrode dimension and position. Using stimulating currents of 2.0 mA, the magnitude of the current density in relevant regions of the brain is of the order of 0.1 A/m2, corresponding to an electric field of 0.22 V/m.

CONCLUSIONS

Calculations based on a spherical model of the head can provide useful information about the magnitude and direction of the current density vector in the brain during tDCS, taking into account the geometry and position of the electrodes. Despite the inherent limitations of the spherical head model, the calculated values are comparable to those used in the most recent in vitro studies on modulation of neuronal activity.

SIGNIFICANCE

The methodology presented in this paper may be used to assess the current distribution during tDCS using new electrode montages, to help optimize montages that target a specific region of the brain or to preliminarily investigate compliance with safety guidelines.

Semiautomatic volume conductor modeling pipeline for imaging the cardiac electrophysiology noninvasively

In this paper we present an approach for extracting patient individual volume conductor models (VCM) using volume data acquired from Magnetic Resonance Imaging (MRI) for computational biology of electrical excitation in the patient's heart. The VCM consists of the compartments chest surface, lung surfaces, the atrial and ventricular myocardium, and the blood masses. For each compartment a segmentation approach with no or little necessity of user interaction was implemented and integrated into a VCM segmentation pipeline to enable the inverse problem of electrocardiography to become clinical applicable. The segmentation pipeline was tested using volume data from ten patients with structurally normal hearts.

Lead field computation for the electrocardiographic inverse problem--finite elements versus boundary elements

In order to be able to solve the inverse problem of electrocardiography, the lead field matrix (transfer matrix) has to be calculated. The two methods applied for computing this matrix, which are compared in this study, are the boundary element method (BEM) and the finite element method (FEM). The performance of both methods using a spherical model was investigated. For a comparable discretization level, the BEM yields smaller relative errors compared to analytical solutions. The BEM needs less computation time, but a larger amount of memory. Inversely calculated myocardial activation times using either the FEM or BEM computed lead field matrices give similar activation time patterns. The FEM, however, is also capable of considering anisotropic conductivities. This property might have an impact for future development, when also individual myocardial fiber architecture can be considered in the inverse formulation.

Mapping human brain function with MEG and EEG: methods and validation

We survey the field of magnetoencephalography (MEG) and electroencephalography (EEG) source estimation. These modalities offer the potential for functional brain mapping with temporal resolution in the millisecond range. However, the limited number of spatial measurements and the ill-posedness of the inverse problem present significant limits to our ability to produce accurate spatial maps from these data without imposing major restrictions on the form of the inverse solution. Here we describe approaches to solving the forward problem of computing the mapping from putative inverse solutions into the data space. We then describe the inverse problem in terms of low dimensional solutions, based on the equivalent current dipole (ECD), and high dimensional solutions, in which images of neural activation are constrained to the cerebral cortex. We also address the issue of objective assessment of the relative performance of inverse procedures by the free-response receiver operating characteristic (FROC) curve. We conclude with a discussion of methods for assessing statistical significance of experimental results through use of the bootstrap for determining confidence regions in dipole-fitting methods, and random field (RF) and permutation methods for detecting significant activation in cortically constrained imaging studies.

The electric field induced in the brain by magnetic stimulation: a 3-D finite-element analysis of the effect of tissue heterogeneity and anisotropy

We investigate the effect of tissue heterogeneity and anisotropy on the electric field and current density distribution induced in the brain during magnetic stimulation. Validation of the finite-element (FE) calculations in a homogeneous isotropic sphere showed that the magnitude of the total electric field can be calculated to within an error of approximately 5% in the region of interest, even in the presence of a significant surface charge contribution. We used a high conductivity inclusion within a sphere of lower conductivity to simulate a lesion due to an infarct. Its effect is to increase the electric field induced in the surrounding low conductivity region. This boost is greatest in the vicinity of interfaces that lie perpendicular to the current flow. For physiological values of the conductivity distribution, it can reach a factor of 1.6 and extend many millimeters from the interface. We also show that anisotropy can significantly alter the electric field and current density distributions. Either heterogeneity or anisotropy can introduce a radial electric field component, not present in a homogeneous isotropic conductor. Heterogeneity and anisotropy are predicted to significantly affect the distribution of the electric field induced in the brain. It is, therefore, expected that anatomically faithful FE models of individual brains which incorporate conductivity tensor data derived from diffusion tensor measurements, will provide a better understanding of the location of possible stimulation sites in the brain.

On modeling the Wilson terminal in the boundary and finite element method

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.

Computing and visualizing electric potentials and current pathways in the thorax

The long-term goal of electrocardiography is to relate electric potentials on the body surface with activities in the heart. Many previously reported studies have focused on direct links between heart and body surface potentials. The goals of this study were first to validate computational methods of determining volume potentials and currents with high-resolution experimental measurements and then to use interactive visualization of thoracic currents to understand features of the electrocardiographic fields from measured cardiac sources. We developed both simulation and experimental studies based on a realistic shaped torso phantom containing an isolated, perfused dog heart. Interventions included atrial pacing, single pacing and simultaneously pacing at multiple locations on the ventricles. Simulated torso volume potentials closely matched measured potentials in the torso-tank preparation (mean correlation coefficients of 0.95). Simulation further provided a means of estimating the current field in the torso from the computed torso volume potentials and the local geometric and conductive properties of the medium. Applying these techniques to the torso electric fields under a variety of pacing conditions, we have further demonstrated that thoracic current can provide many insights into the relationship between heart surface potential and body surface potentials. Specifically, we have shown that geometric factors including cardiac source configuration and location play an important role in determining to what extent electric activity in the heart is directly visible on the body surface electrocardiogram. The computation and visualization toolkit we developed in this study to explore current fields associated with cardiac events may provide new insights into electrocardiology.