Influence of head tissue conductivity in forward and inverse magnetoencephalographic simulations using realistic head models

The influence of head tissue conductivity on magnetoencephalography (MEG) was investigated by comparing the normal component of the magnetic field calculated at 61 detectors and the localization accuracy of realistic head finite element method (FEM) models using dipolar sources and containing altered scalp, skull, cerebrospinal fluid, gray, and white matter conductivities to the results obtained using a FEM realistic head model with the same dipolar sources but containing published baseline conductivity values. In the models containing altered conductivity values, the tissue conductivity values were varied, one at a time, between 10% and 200% of their baseline values, and then varied simultaneously. Although changes in conductivity values for a single tissue layer often altered the calculated magnetic field and source localization accuracy only slightly, varying multiple conductivity layers simultaneously caused significant discrepancies in calculated results. The conductivity of scalp, and to a lesser extent that of white and gray matter, appears especially influential in determining the magnetic field. Comparing the results obtained from models containing the baseline conductivity values to the results obtained using other published conductivity values suggests that inaccuracies can occur depending upon which tissue conductivity values are employed. We show the importance of accurate head tissue conductivities for MEG source localization in human brain, especially for deep dipole sources or when an accuracy greater than 1.4 cm is needed.     

Effect of conductivity uncertainties and modeling errors on EEGsource localization using a 2-D model

This paper presents a sensitivity study electroencephalography-based source localization due errors in the head-tissue conductivities and to errors in modeling the conductivity variation inside the brain and scalp. The study is conducted using a two-dimensional (2-D) finite element model obtained from a magnetic resonance imaging (MRI) scan of a head cross section. The effect of uncertainty in the following tissues is studied: white matter, gray matter, cerebrospinal fluid (CSF), skull, and fat. The distribution of source location errors, assuming a single-dipole source model, is examined in detail for different dipole locations over the entire brain region. We also present a detailed analysis of the effect of conductivity on source localization for a four-layer cylinder model and a four-layer sphere model. These two simple models provide insight into how the effect of conductivity on boundary potential translates into source location errors; and also how errors in a 2-D model compare to errors in a three-dimensional model. Results presented in this paper clearly point to the following conclusion: unless the conductivities of the head tissues and the distribution of these tissues throughout the head are modeled accurately, the goal of achieving localization accuracy to within a few millimeters is unattainable     

Conventional and reciprocal approaches to the inverse dipole localization problem of electroencephalography

Forward transfer matrices relating dipole source to surface potentials can be determined via conventional or reciprocal approaches. In numerical simulations with a triangulated boundary-element three-concentric-spheres head model, we compare four inverse electroencephalogram (EEG) solutions: those obtained utilizing conventional or reciprocal forward transfer matrices, relating in each case source dipole components to potentials at either triangle centroids or triangle vertices. Single-dipole inverse solutions were obtained using simplex optimization with an additional position constraint limiting solution dipoles to within the brain region. Dipole localization errors are presented in all four cases, for varying dipole eccentricity and two different values of skull conductivity. Both conventional and reciprocal forward transfer matrices yielded inverse dipole solutions of comparable accuracy. Localization errors were low even for highly eccentric source dipoles on account of the nonlinear nature of the single-dipole solution and the position constraint. In the presence of Gaussian noise, both conventional and reciprocal approaches were also found to be equally robust to skull conductivity errors.     

Approximating dipoles from human EEG activity: the effect of dipolesource configuration on dipolarity using single dipole models

Dipolarity is the goodness-of-fit of the observed potential distribution with one calculated using specific assumptions about the source of the electrical potential distribution. The authors used computer simulations to examine the effect of different distributions of sources on their resulting dipolarity values. Electric dipoles were placed in a head-shaped model with uniform conductivity using four different dipole configurations (randomly oriented dipoles, a uniform dipole disk layer, a dipole disk layer with uniformly distributed holes, or one with randomly oriented dipoles). The best-fitting single dipole for each configuration was calculated and the dipolarity was computed as the mean squared error of the electrical potential distributions generated by the actual dipole configuration and by the best-fitting single dipole. The simulations show that: 1) a smooth dipole layer with or without holes gives dipolarities above 99.5% even when extended over areas as large as 1256 mm²; 2) randomly oriented dipoles under a smooth dipole layer also give dipolarities above 99.5%; and 3) randomly oriented and distributed dipoles, even if contained in a small portion of the total area, give dipolarities below 93.0%. These simulations show that inhomogeneity (holes) within a dipole disk layer per se do not lower dipolarity; rather, it is the random orientation and distribution of these dipoles which lowers dipolarity. Furthermore, dipolarity is not lowered by such randomly oriented and distributed dipoles when they are beneath a dipole disk layer     

Localization of a dipolar source in a skull phantom: realisticversus spherical model

The dependence of the neuromagnetic source localization accuracy on the volume conductor model was studied by the analysis of measured magnetic fields generated by tangentially oriented dipoles in a realistically shaped skull phantom. When using a homogeneous sphere model in the localization procedure, the errors were found to increase from about 3 mm to about 9 mm when the distance between the dipoles and the inner surface of the skull increased from 1 cm to 3 cm, whereas when using a true, realistic model in the inverse procedure the localization errors were only about 2-3 mm, independent of dipole depth. To account for the realistic geometry of the inner surface of the skull, the Boundary Element Method, based on a surface discretization in terms of about 300 triangles, proved to be sufficient. In addition to these analyses of experimental data, simulations were carried out to study the localization errors in the case of the spherical approximation for a dipole orientation changing from tangential to radial. For the latter orientation, errors of up to a few centimeters were found     

Anatomically constrained electrical impedance tomography forthree-dimensional anisotropic bodies

As shown previously for two-dimensional geometries, anisotropy effects should not be ignored in electrical impedance tomography (EIT) and structural information is important for the reconstruction of anisotropic conductivities. Here, we describe the static reconstruction of an anisotropic conductivity distribution for the more realistic three-dimensional (3-D) case. Boundaries between different conductivity regions are anatomically constrained using magnetic resonance imaging (MRI) data. The values of the conductivities are then determined using gradient-type-algorithms in a nonlinear-indirect approach. At each iteration, the forward problem is solved by the finite element method. The approach is used to reconstruct the 3-D conductivity profile of a canine torso. Both computational performance and simulated reconstruction results are presented together with a detailed study on the sensitivity of the prediction error with respect to different parameters. In particular, the use of an intracavity catheter to better extract interior conductivities is demonstrated     

Comparative simulation of excitation and body surfaceelectrocardiogram with isotropic and anisotropic computer heart models

Comparative simulations between isotropic and anisotropic computer heart models were conducted to study the effects of myocardial anisotropy on the excitation process of the heart and on body surface electrocardiogram. The isotropic heart model includes atria, ventricles, and a special conduction system, and is electrophysiologically specified by parameters relative to action potential, conduction velocity, automaticity, and pacing. The anisotropic heart model was created by incorporating rotating fiber directions into the ventricles of the isotropic heart model. The orientation of the myocardial fibers in the ventricles of the model was gradually rotated counterclockwise from the epicardial layer to the endocardial layer for a total rotation of 90°. The anisotropy of conduction velocity and intracellular electric conductivity was included in the simulation. Comparative simulations of the normal heart, LBBB, and RBBB showed no significant differences between the two models in the excitation processes of the whole heart or in the body surface electrocardiograms. However, it was easier to induce ventricular fibrillation in the anisotropic model than in the isotropic model. The comparative simulation is useful for investigating the effects of myocardial anisotropy at the whole heart level and for evaluating limitations of the isotropic heart model     

General approach to first-order error prediction in rigid point registration

A general approach to the first-order analysis of error in rigid point registration is presented that accommodates fiducial localization error (FLE) that may be inhomogeneous (varying from point to point) and anisotropic (varying with direction) and also accommodates arbitrary weighting that may also be inhomogeneous and anisotropic. Covariances are derived for target registration error (TRE) and for weighted fiducial registration error (FRE) in terms of covariances of FLE, culminating in a simple implementation that encompasses all combinations of weightings and anisotropy. Furthermore, it is shown that for ideal weighting, in which the weighting matrix for each fiducial equals the inverse of the square root of the cross covariance of its two-space FLE, fluctuations of FRE and TRE are mutually independent. These results are validated by comparison with previously published expressions and by simulation. Furthermore, simulations for randomly generated fiducial positions and FLEs are presented that show that correlation is negligible (correlation coefficient < 0.1) in the exact case for both ideal and uniform weighting (i.e., no weighting), the latter of which is employed in commercial surgical guidance systems. From these results we conclude that for these weighting schemes, while valid expressions exist relating the covariance of FRE to the covariance of TRE, there are no measures of the goodness of fit of the fiducials for a given registration that give to first order any information about the fluctuation of TRE from its expected value and none that give useful information in the exact case. Therefore, as estimators of registration accuracy, such measures should be approached with extreme caution both by the purveyors of guidance systems and by the practitioners who use them.     

Estimation of number of independent brain electric sources from the scalp EEGs

In electromagnetic source analysis, many source localization strategies require the number of sources as an input parameter (e.g., spatio-temporal dipole fitting and the multiple signal classification). In the present study, an information criterion method, in which the penalty functions are selected based on the spatio-temporal source model, has been developed to estimate the number of independent dipole sources from electromagnetic measurements such as the electroencephalogram (EEG). Computer simulations were conducted to evaluate the effects of various parameters on the estimation of the source number. A three-concentric-spheres head model was used to approximate the head volume conductor. Three kinds of typical signal sources, i.e., the damped sinusoid sources, sinusoid sources with one frequency band and sinusoid sources with two separated frequency bands, were used to simulate the oscillation characteristics of brain electric sources. The simulation results suggest that the present method can provide a good estimate of the number of independent dipole sources from the EEG measurements. In addition, the present simulation results suggest that choosing the optimal penalty function can successfully reduce the effect of noise on the estimation of number of independent sources. The present study suggests that the information criterion method may provide a useful means in estimating the number of independent brain electrical sources from EEG/MEG measurements.     

An investigation of the importance of myocardial anisotropy infinite-element modeling of the heart: methodology and application to theestimation of defibrillation efficacy

Finite-element (FE) modeling has been widely used in studies of bioelectric phenomena of tissues, including ventricular defibrillation. Most FE models, whether built from anatomical atlases or subject-specific tomographic images, treat the myocardium as an isotropic tissue. However, myocardium has been experimentally shown to have significant anisotropy in its resistivities, although myocardial fiber directions are difficult to measure on a subject-specific basis. In this paper, we: 1). propose a method to incorporate a widely known myocardial fiber direction model to a specific individual and 2). assess the effects of myocardial anisotropy on myocardial voltage gradients computed for a study of implantable defibrillators. The thoracic FE model was built from CT images of a young pig, and the myocardial fiber structures were incorporated via elastic mapping. Our results demonstrate a good mapping of geometry between the source and target hearts with an average root-mean-square error of less than 2.3 mm and a mapped fiber pattern similar to those known to exist in vivo. With the mapped fiber information, we showed that the estimated minimal myocardial voltage gradient over 80% of the myocardium differs by less than 10% between using an isotropic and anisotropic myocardial models. Thus, myocardial anisotropy is expected to have only a small effect on estimates of defibrillation threshold obtained from computed voltage gradients. On the other hand, anisotropy may be essential if defibrillation efficacy is analyzed by transmembrane voltage of the myocardial cells.     

Modeling the effects of electric fields on nerve fibers: influenceof tissue electrical properties

The effects of anisotropy and inhomogeneity of the electrical conductivity of extracellular tissue on excitation of nerve fibers by an extracellular point source electrode were determined by computer simulation. Analytical solutions to Poison's equation were used to calculate potentials in anisotropic infinite homogeneous media and isotropic semi-infinite inhomogeneous media, and the net driving function was used to calculate excitation thresholds for nerve fibers. The slope and intercept of the current-distance curve in anisotropic media were power functions of the ratio and product of the orthogonal conductivities, respectively. Excitation thresholds in anisotropic media were also dependent on the orientation of the fibers, and in strongly anisotropic media (sigma z/sigma xy > 4) there were reversals in the recruitment order between different diameter fibers and between fibers at different distances from the electrode. In source-free regions of inhomogeneous media (two regions of differing conductivity separated by a plane boundary), the current-distance relationship of fibers parallel to the interface was dependent only on the average conductivity, whereas in regions containing the source the current-distance relationship was dependent on the individual values of conductivity. Reversals in recruitment order between fibers at different distances from the electrode and between fibers of differing diameter were found in inhomogeneous media. The results of this simulation study demonstrate that the electrical properties of the extracellular medium can have a strong influence on the pattern of neuronal excitation generated by extracellular electric fields, and indicate the importance of tissue electrical properties in interpreting results of studies employing electrical stimulation applied in complex biological volume conductors.     

Bayesian estimates of error bounds for EEG source imaging

Given a set of electrical potential measurements at the surface of the head, localizing the sources of the electrical activity is an inherently ill-posed problem. Bayesian methods can be used to specify prior information to constrain the possible source solutions. We show that Bayesian analysis can also provide a means for characterizing system noise levels, estimating the "error bars" surrounding source localization results, and estimating the information about brain processes conveyed by dense sensor array electroencephalographic (EEG) recordings. This method is, in principal, applicable to any linear model of EEG or magnetoencephalographic (MEG) processes. A series of simulations demonstrated the internal consistency of our method, the robustness to noise levels, and the limitations of accurate source localization with large numbers of sources.     

The effect of artifact rejection by signal-space projection on source localization accuracy in MEG measurements

The consequences of artifact suppression by means of signal-space projection on dipole localization accuracy for magnetoencephalography measurements are studied. Approximate analytical formulas, equivalent to the Cramer-Rao bound, are presented and verified by Monte Carlo simulations which relate the increase of localization error for individual coordinates to the similarity of the artifact field and respective (contravariant) quadrupole fields obtained by differentiating the dipole field with respect to its origin. The expressions simplify significantly for dipoles placed below the center of the measuring system giving rise to highly symmetric field patterns. Formulas are presented both for single- and for multiple-artifact rejection. As illustrative examples artifact fields are constructed which a) lead to highly decreasing signal-to-noise ratio and goodness-of-fit (GOF), while the localization error is unaffected for all coordinates and b) lead to an increase of localization error while the SNR and the GOF stays constant. Finally, the rich structure of localization error increase is demonstrated for a class of artifact fields originating from artifact current dipoles.     

Radiation from a dipole near a general anisotropic layer

The radiation properties of a dipole source located near a gyrotropic layer are investigated analytically. Both electric and magnetic anisotropy of the most general form are assumed. Fourier-transform domain field representations in conjunction with matrix analysis techniques are used to facilitate the analysis. Transmission phenomena through the general anisotropic layer are investigated by examining the radiation patterns at the far-field region. The analysis is also used to derive the response of the anisotropic layer to an incident plane wave. In this case, the transmission and reflection coefficient matrices are obtained     

A new method to derive white matter conductivity from diffusion tensor MRI

We propose a new algorithm to derive the anisotropic conductivity of the cerebral white matter (WM) from the diffusion tensor MRI (DT-MRI) data. The transportation processes for both water molecules and electrical charges are described through a common multicompartment model that consists of axons, glia, or the cerebrospinal fluid (CSF). The volume fraction (VF) of each compartment varies from voxel to voxel and is estimated from the measured diffusion tensor. The conductivity tensor at each voxel is then computed from the estimated VF values and the decomposed eigenvectors of the diffusion tensor. The proposed VF algorithm was applied to the DT-MRI data acquired from two healthy human subjects. The extracted anisotropic conductivity distribution was compared with those obtained by using two existing algorithms, which were based upon a linear conductivity-to-diffusivity relationship and a volume constraint, respectively. The present results suggest that the VF algorithm is capable of incorporating the partial volume effects of the CSF and the intravoxel fiber crossing structure, both of which are not addressed altogether by existing algorithms. Therefore, it holds potential to provide a more accurate estimate of the WM anisotropic conductivity, and may have important applications to neuroscience research or clinical applications in neurology and neurophysiology.      

孔径接收下各向异性海洋湍流UWOC系统误码分析

为了研究孔径接收对各向异性海洋湍流条件下水下无线光通信(UWOC)系统误比特率的影响, 系统采用高斯光束传输, 接收端通过孔径接收, 在脉冲位置调制方式下通过各向异性海洋湍流信道。引入各向异性海洋湍流结构常数, 通过对闪烁的形成原理和各向异性海洋湍流条件下闪烁系数的分析, 数值模拟得到了在不同接收孔径和各向异性因子下, 海洋湍流参量、传输距离、雪崩光电二极管(APD)平均增益和调制阶数对系统误比特率的影响。结果表明, 相同各向异性因子和海洋湍流参量下, 大孔径接收能有效提升系统误比特率性能; 相同孔径直径和海洋湍流参量下, 各向异性因子越大, 系统通信性能越好; 均方温度耗散率、温度和盐度对海洋功率谱变化贡献的比值较小, 湍流动能耗散率、动力粘度较大以及传输距离越短, 系统误码性能越好; APD增益为100或150时, 系统通信性能最佳; 调制阶数M=8时, 系统通信性能最佳, M>64时, 系统误比特率变化程度几乎饱和。该研究为UWOC系统平台搭建和性能估计提供了参考。     

Review of Anisotropic Terahertz Material Response

Anisotropy is ubiquitous in solids and enhanced in low-dimensional materials. In response to an electromagnetic wave, anisotropic absorptive and refractive properties result in dichroic and birefringent optical phenomena both in the linear and nonlinear optics regimes. Such material properties have led to a diverse array of useful polarization components in the visible and near-infrared, but mature technology is non-existent in the terahertz (THz). Here, we review several novel types of anisotropic material responses observed in the THz frequency range, including both linear and circular anisotropy, which have long-term implications for the development of THz polarization optics. We start with the extreme linear anisotropy of macroscopically aligned carbon nanotubes, arising from their intrinsically anisotropic dynamic conductivity. Magnetically induced anisotropy will then be reviewed, including the giant Faraday effects observed in semiconductors, semimetals, and two-dimensional electron systems.     

Far field radiation from an arbitrarily oriented Hertzian dipole in the presence of a layered anisotropic medium

Far field radiation from an arbitrarily oriented Hertzian dipole for two-layered uniaxially anisotropic medium with a tilted optic axis is treated analytically by using the dyadic Green's function of the problem when the dipole is placed over or embedded in a two-layered uniaxially anisotropic medium. The radiation fields are evaluated using the steepest descent method. Parameter studies including anisotropy, layer thickness and dipole location are performed to investigate the effects of changing different variables on the radiation fields. Results of this work can be applied in microstrip circuits and antennas.     

White matter fiber tractography via anisotropic diffusion simulation in the human brain

A novel approach to noninvasively tracing brain white matter fiber tracts is presented using diffusion tensor magnetic resonance imaging (DT-MRI). This technique is based on successive anisotropic diffusion simulations over the human brain, which are utilized to construct three dimensional diffusion fronts. The fiber pathways are determined by evaluating the distance and orientation from the fronts to their corresponding diffusion seeds. Synthetic and real DT-MRI data are employed to demonstrate the tracking scheme. It is shown that the synthetic tracts are accurately replicated, and several major white matter fiber pathways can be reproduced noninvasively, with the tract branching being allowed. Since simulating the diffusion process, which is truly a physical phenomenon reflecting the underlying architecture of cerebral tissues, makes full use of the diffusion tensor data, including both the magnitude and orientation information, the proposed approach is expected to enhance robustness and reliability in white matter fiber reconstruction.     

Current dipole localization with an ideal magnetometer system

The goal of the study was to explore the most fundamental aspects of a magnetoencephalography (MEG)-based dipole source analysis. For that purpose, a MEG measurement with an ideal magnetometer system (providing the radial component of the magnetic field as a continuous function) is considered. The analytical formulas derived for the variances and covariances of the parameter estimation errors, validated by means of Monte Carlo simulations, allow quantitative predictions in terms of dipole depth, radius and span of the magnetometer system, signal-to-noise (SNR) ratio, and other parameters. A negative correlation exists between radial coordinate and longitudinal component of the moment (perpendicular to radial direction, same plane as actual dipole moment and center of sphere), whereas the other parameters are independent. The standard deviations of the 5 dipole parameters show fundamental differences with respect to their asymptotic behavior for deep dipoles: If the root mean square (rms) value of the magnetic field is kept constant (moment with depth-dependent amplitude), the error for the transverse coordinate (perpendicular to radial and longitudinal coordinate) is proportional to the distance R between dipole and center of sphere, the errors for the other dipole coordinates, and the relative error for the transverse component of the dipole moment are constant, and the relative error for the longitudinal component of the dipole moment follows a 1/R law