Electric field induced in a spherical volume conductor from arbitrary coils: application to magnetic stimulation and MEG

A mathematical method is presented that allows fast and simple computation of the electric field and current density induced inside a homogeneous spherical volume conductor by current flowing in a coil. The total electric field inside the sphere is computed entirely from a set of line integrals performed along the coil current path. Coils of any closed shape are easily accommodated by the method. The technique can be applied to magnetic brain stimulation and to magnetoencephalography. For magnetic brain stimulation, the total electric field anywhere inside the head can be easily computed for any coil shape and placement. The reciprocity theorem may be applied so that the electric field represents the lead field of a magnetometer. The finite coil area and gradiometer loop spacing can be precisely accounted for without any surface integration by using this method. The theory shows that the steady-state, radially oriented induced electric field is zero everywhere inside the sphere for ramping coil current and highly attenuated for sinusoidal coil current. This allows the model to be extended to concentric spheres which have different electrical properties.     

Evaluation of multiple-sphere head models for MEG source localization

Magnetoencephalography (MEG) source analysis has largely relied on spherical conductor models of the head to simplify forward calculations of the brain's magnetic field. Multiple- (or overlapping, local) sphere models, where an optimal sphere is selected for each sensor, are considered an improvement over single-sphere models and are computationally simpler than realistic models. However, there is limited information available regarding the different methods used to generate these models and their relative accuracy. We describe a variety of single- and multiple-sphere fitting approaches, including a novel method that attempts to minimize the field error. An accurate boundary element method simulation was used to evaluate the relative field measurement error (12% on average) and dipole fit localization bias (3.5 mm) of each model over the entire brain. All spherical models can contribute in the order of 1 cm to the localization bias in regions of the head that depart significantly from a sphere (inferior frontal and temporal). These spherical approximation errors can give rise to larger localization differences when all modeling effects are taken into account and with more complex source configurations or other inverse techniques, as shown with a beamformer example. Results differed noticeably depending on the source location, making it difficult to recommend a fitting method that performs best in general. Given these limitations, it may be advisable to expand the use of realistic head models.     

Representation of bioelectric current sources using Whitney elements in the finite element method

Bioelectric current sources of magneto- and electroencephalograms (MEG, EEG) are usually modelled with discrete delta-function type current dipoles, despite the fact that the currents in the brain are naturally continuous throughout the neuronal tissue. In this study, we represent bioelectric current sources in terms of Whitney-type elements in the finite element method (FEM) using a tetrahedral mesh. The aim is to study how well the Whitney elements can reproduce the potential and magnetic field patterns generated by a point current dipole in a homogeneous conducting sphere. The electric potential is solved for a unit sphere model with isotropic conductivity and magnetic fields are calculated for points located on a cap outside the sphere. The computed potential and magnetic field are compared with analytical solutions for a current dipole. Relative difference measures between the FEM and analytical solutions are less than 1%, suggesting that Whitney elements as bioelectric current sources are able to produce the same potential and magnetic field patterns as the point dipole sources.     

On the influence of volume currents and extended sources on neuromagnetic fields: A simulation study

The influence of volume currents on the magnetic field is an important question in magnetoencephalography since the spherical volume conductor is still widely used for source localization. In theory, the magnetic field of a radial dipole in a homogeneous sphere is zero. In realistic models of the head, the field is suppressed when compared with a tangential dipole. To determine the influence of the volume currents, this suppression ratio (magnetic field of the radial dipole divided by the field of the tangential dipole) needs to be quantified. Large-scale finite element method models of the human head and the rabbit head were constructed and the suppression ratio was computed. The computed suppression value of 0.28 in the rabbit head was similar to the previously measured experimental value. In the human head, an average suppression ratio of 0.19±0.07 was found for different regions and depths in the gray matter. It was found that the computed magnetic field of radial sources varied significantly with the conductivities of the surrounding tissues where the dipole was located. We also modeled the magnetic field of an epileptic interictal spike in a finite element model of the rabbit head with a single dipole and with extended sources of varying length (1–8 mm). The extended source models developed were based on invasive measurements of an interictal spike within the rabbit brain. The field patterns of the small (1–2 mm) extended sources were similar to a single dipolar source and begin to deviate significantly from a dipolar field for the larger extended sources (6–8 mm).     

Electric fields induced in a spherical volume conductor by temporally varying magnetic field gradients

A homogeneous spherical volume conductor is used as a model system for the purpose of calculating electric fields induced in the human head by externally applied time-varying magnetic fields. We present results for the case where magnetic field gradient coils, used in magnetic resonance imaging (MRI), form the magnetic field, and we use these data to put limits on the rates of gradient change with time needed to produce nerve stimulation. The electric field is calculated analytically for the case of ideal longitudinal and transverse linear field gradients. We also show results from computer calculations yielding the electric field maps in a sphere when the field gradients are generated by a real MRI gradient coil set. In addition, the effect of shifting the sphere within each gradient coil volume is investigated. Numerical analysis shows similar results when applied to a model human head.     

磁感应方法检测脑组织电导率的FDTD仿真研究

为进一步优化和改进MIT（magnetic induction tomography，MIT）实验系统，采用FDTD（finite difference time domain）方法，对MIT单通道测罱系统进行仿真研究。根据天线理论，将激励线圈近似为螺线管，模拟为一系列电磁源。脑模型考虑颅骨、脑脊液、脑组织和水肿。在10MHz频率，测最与激励线圈同轴时，对自由空间、均匀脑模型、三层脑模型和含水肿的三层脑模型进行仿真计算，获得了上述四种情况下测量线圈上的电流值和激励电流与测量电流的相位差值。     

The role of model and computational experiments in the biomagnetic inverse problem

This is a review of the role of model and computational experiments in studies of the part of the biomagnetic inverse problem that deals with the determination of electrical sources in the body using magnetic measurements around the body. Results from modelling studies of the forward problem that are important for the inverse problem are also reviewed. An evaluation is made of the adequacy of various models of the body for use in the biomagnetic inverse problem. This evaluation indicates that simple torso models, e.g. a semi-infinite volume or sphere, are probably inadequate. The review of the modelling studies of the inverse problem includes the effects of noise, source modelling errors, body modelling errors and measurement errors on the accuracy of source localisation methods using magnetic measurements. Source modelling errors are caused by differences between an actual complex source in the body and the simple model of it used in most source localisation methods; body modelling errors are caused by differences between the actual body and a simple model of it. The review indicates that typical experimental noise will only cause significant source localisation errors for inverse solutions calculated using fewer than approximately ten measurement points; it also indicates that source modelling errors must be rather large to be detectable when typical experimental noise is present. In addition, the review indicates that many experimental measurement errors will not cause significant localisation errors. The effects of body modelling errors are largely unknown. Suggestions for further biomagnetic inverse problem research are given. These include the development of more realistic models of the body, the experimental verification of such models and source localisation methods, and the development of methods for detecting and localising distributed or multiple discrete sources.     

Sensitivity of EEG and MEG measurements to tissue conductivity

Monitoring the electrical activity inside the human brain using electrical and magnetic field measurements requires a mathematical head model. Using this model the potential distribution in the head and magnetic fields outside the head are computed for a given source distribution. This is called the forward problem of the electro-magnetic source imaging. Accurate representation of the source distribution requires a realistic geometry and an accurate conductivity model. Deviation from the actual head is one of the reasons for the localization errors. In this study, the mathematical basis for the sensitivity of voltage and magnetic field measurements to perturbations from the actual conductivity model is investigated. Two mathematical expressions are derived relating the changes in the potentials and magnetic fields to conductivity perturbations. These equations show that measurements change due to secondary sources at the perturbation points. A finite element method (FEM) based formulation is developed for computing the sensitivity of measurements to tissue conductivities efficiently. The sensitivity matrices are calculated for both a concentric spheres model of the head and a realistic head model. The rows of the sensitivity matrix show that the sensitivity of a voltage measurement is greater to conductivity perturbations on the brain tissue in the vicinity of the dipole, the skull and the scalp beneath the electrodes. The sensitivity values for perturbations in the skull and brain conductivity are comparable and they are, in general, greater than the sensitivity for the scalp conductivity. The effects of the perturbations on the skull are more pronounced for shallow dipoles, whereas, for deep dipoles, the measurements are more sensitive to the conductivity of the brain tissue near the dipole. The magnetic measurements are found to be more sensitive to perturbations near the dipole location. The sensitivity to perturbations in the brain tissue is much greater when the primary source is tangential and it decreases as the dipole depth increases. The resultant linear system of equations can be used to update the initially assumed conductivity distribution for the head. They may be further exploited to image the conductivity distribution of the head from EEG and/or MEG measurements. This may be a fast and promising new imaging modality.     

Forward problem solution of electromagnetic source imaging using a new BEM formulation with high-order elements.

Representations of the active cell populations on the cortical surface via electric and magnetic measurements are known as electromagnetic source images (EMSIs) of the human brain. Numerical solution of the potential and magnetic fields for a given electrical source distribution in the human brain is an essential part of electromagnetic source imaging. In this study, the performance of the boundary element method (BEM) is explored with different surface element types. A new BEM formulation is derived that makes use of isoparametric linear, quadratic or cubic elements. The surface integration is performed with Gauss quadrature. The potential fields are solved assuming a concentric three-shell model of the human head for a tangential dipole at different locations. In order to achieve 2% accuracy in potential solutions, the number of quadratic elements is of the order of hundreds. However, with linear elements, this number is of the order of ten thousand. The relative difference measures (RDMs) are obtained for the numerical models that use different element types. The numerical models that employ quadratic and cubic element types provide superior performance over linear elements in terms of accuracy in solutions. Assuming a homogeneous sphere model of the head, the RDMs are also obtained for the three components (radial and tangential) of the magnetic fields. The RDMs obtained for the tangential fields are, in general, much higher than those obtained for the radial fields. Both quadratic and cubic elements provide superior performance compared with linear elements for a wide range of dipole locations.     

A new magnetic resonance electrical impedance tomography (MREIT) algorithm: the RSM-MREIT algorithm with applications to estimation of human head conductivity

We have developed a new magnetic resonance electrical impedance tomography (MREIT) algorithm, the RSM-MREIT algorithm, for noninvasive imaging of the electrical conductivity distribution using only one component of magnetic flux density. The proposed RSM-MREIT algorithm uses the response surface methodology (RSM) algorithm for optimizing the conductivity distribution through minimizing the errors between the measured and calculated magnetic flux densities. A series of computer simulations has been conducted to assess the performance of the proposed RSM-MREIT algorithm to estimate electrical conductivity values of the scalp, the skull and the brain tissue, in a three-shell piecewise homogeneous head model. Computer simulation studies were conducted in both a spherical and realistic-geometry head model with a single variable (the brain-to-skull conductivity ratio) and three variables (the conductivity of the brain, the skull, and the scalp). The relative error between the target and estimated head conductivity values was less than 12% for both the single-variable and three-variable simulations. These promising simulation results demonstrate the feasibility of the proposed RSM-MREIT algorithm in estimating electrical conductivity values in a piecewise homogeneous head model of the human head, and suggest that the RSM-MREIT algorithm merits further investigation.     

Biomechanics of Traumatic Brain Injury: Influences of the Morphologic Heterogeneities of the Cerebral Cortex

Traumatic brain injury (TBI) can be caused by accidents and often leads to permanent health issues or even death. Brain injury criteria are used for assessing the probability of TBI, if a certain mechanical load is applied. The currently used injury criteria in the automotive industry are based on global head kinematics. New methods, based on finite element modeling, use brain injury criteria at lower scale levels, e.g., tissue-based injury criteria. However, most current computational head models lack the anatomical details of the cerebrum. To investigate the influence of the morphologic heterogeneities of the cerebral cortex, a numerical model of a representative part of the cerebral cortex with a detailed geometry has been developed. Several different geometries containing gyri and sulci have been developed for this model. Also, a homogeneous geometry has been made to analyze the relative importance of the heterogeneities. The loading conditions are based on a computational head model simulation. The results of this model indicate that the heterogeneities have an influence on the equivalent stress. The maximum equivalent stress in the heterogeneous models is increased by a factor of about 1.3–1.9 with respect to the homogeneous model, whereas the mean equivalent stress is increased by at most 10%. This implies that tissue-based injury criteria may not be accurately applied to most computational head models used nowadays, which do not account for sulci and gyri.     

Electromagnetic fields inside a lossy, multilayered spherical head phantom excited by MRI coils: models and methods

The precise evaluation of electromagnetic field (EMF) distributions inside biological samples is becoming an increasingly important design requirement for high field MRI systems. In evaluating the induced fields caused by magnetic field gradients and RF transmitter coils, a multilayered dielectric spherical head model is proposed to provide a better understanding of electromagnetic interactions when compared to a traditional homogeneous head phantom. This paper presents Debye potential (DP) and Dyadic Green's function (DGF)-based solutions of the EMFs inside a head-sized, stratified sphere with similar radial conductivity and permittivity profiles as a human head. The DP approach is formulated for the symmetric case in which the source is a circular loop carrying a harmonic-formed current over a wide frequency range. The DGF method is developed for generic cases in which the source may be any kind of RF coil whose current distribution can be evaluated using the method of moments. The calculated EMFs can then be used to deduce MRI imaging parameters. The proposed methods, while not representing the full complexity of a head model, offer advantages in rapid prototyping as the computation times are much lower than a full finite difference time domain calculation using a complex head model. Test examples demonstrate the capability of the proposed models/methods. It is anticipated that this model will be of particular value for high field MRI applications, especially the rapid evaluation of RF resonator (surface and volume coils) and high performance gradient set designs.     

A sensor-weighted overlapping-sphere head model and exhaustive head model comparison for MEG

The spherical head model has been used in magnetoencephalography (MEG) as a simple forward model for calculating the external magnetic fields resulting from neural activity. For more realistic head shapes, the boundary element method (BEM) or similar numerical methods are used, but at greatly increased computational cost. We introduce a sensor-weighted overlapping-sphere (OS) head model for rapid calculation of more realistic head shapes. The volume currents associated with primary neural activity are used to fit spherical head models for each individual MEG sensor such that the head is more realistically modelled as a set of overlapping spheres, rather than a single sphere. To assist in the evaluation of this OS model with BEM and other head models, we also introduce a novel comparison technique that is based on a generalized eigenvalue decomposition and accounts for the presence of noise in the MEG data. With this technique we can examine the worst possible errors for thousands of dipole locations in a realistic brain volume. We test the traditional single-sphere model, three-shell and single-shell BEM, and the new OS model. The results show that the OS model has accuracy similar to the BEM but is orders of magnitude faster to compute.     

Improved procedure for neuromagnetic localisation

We describe an improved algorithm for localising equivalent sources of biomagnetic fields in the human brain. The algorithm is an improvement over the sphere model in that it considers two distinct surfaces: an ellipsoid, to model the region of the skull on which the sensors are placed, and a sphere as the medium in which the current dipole model is considered. This allows us to easily correct the formula of the magnetic field in order to take better account of the true position of the sensor with respect to the subject's head.     

Suppression of Interference and Artifacts by the Signal Space Separation Method

Multichannel measurement with hundreds of channels oversamples a curl-free vector field, like the magnetic field in a volume free of sources. This is based on the constraint caused by the Laplace's equation for the magnetic scalar potential; outside of the source volume the signals are spatially band limited. A functional solution of Laplace's equation enables one to separate the signals arising from the sphere enclosing the interesting sources, e.g. the currents in the brain, from the magnetic interference. Signal space separation (SSS) is accomplished by calculating individual basis vectors for each term of the functional expansion to create a signal basis covering all measurable signal vectors. Because the SSS basis is linearly independent for all practical sensor arrangements, any signal vector has a unique SSS decomposition with separate coefficients for the interesting signals and signals coming from outside the interesting volume. Thus, SSS basis provides an elegant method to remove external disturbances. The device-independent SSS coefficients can be used in transforming the interesting signals to virtual sensor configurations. This can also be used in compensating for distortions caused by movement of the object by modeling it as movement of the sensor array around a static object. The device-independence of the decomposition also enables physiological DC phenomena to be recorded using voluntary head movements. When used with properly designed sensor array, SSS does not affect the morphology or the signal-to-noise ratio of the interesting signals.     

Brain evoked potential topographic mapping based on the diffuse approximation

Evoked potential mapping is a convenient technique to describe brain electrical activity using pictorial representation. A new interpolation method based on the diffuse approximation is applied to represent evoked potential distribution over the skull. The method retains most of the attractive features of the finite-element method but does not require explicit elements. In the simulation examples, the human head is assumed to be a single-layer sphere with homogeneous conductivity, and Ary eccentricity transformation is considered to approximate the more realistic three-shell model. The patterns shown in the computed maps suggest the ability of the proposed method to extract coherent information from the data from different electrodes. In the application protocol, visual evoked potentials are used to test the method with a realistic head shape.     

Theoretical evaluation of accuracy in position and size of brain activity obtained by near-infrared topography

Near-infrared (NIR) topography can obtain a topographical distribution of the activated region in the brain cortex. Near-infrared light is strongly scattered in the head, and the volume of tissue sampled by a source-detector pair on the head surface is broadly distributed in the brain. This scattering effect results in poor resolution and contrast in the topographic image of the brain activity. In this study, a one-dimensional distribution of absorption change in a head model is calculated by mapping and reconstruction methods to evaluate the effect of the image reconstruction algorithm and the interval of measurement points for topographic imaging on the accuracy of the topographic image. The light propagation in the head model is predicted by Monte Carlo simulation to obtain the spatial sensitivity profile for a source-detector pair. The measurement points are one-dimensionally arranged on the surface of the model, and the distance between adjacent measurement points is varied from 4 mm to 28 mm. Small intervals of the measurement points improve the topographic image calculated by both the mapping and reconstruction methods. In the conventional mapping method, the limit of the spatial resolution depends upon the interval of the measurement points and spatial sensitivity profile for source-detector pairs. The reconstruction method has advantages over the mapping method which improve the results of one-dimensional analysis when the interval of measurement points is less than 12 mm. The effect of overlapping of spatial sensitivity profiles indicates that the reconstruction method may be effective to improve the spatial resolution of a two-dimensional reconstruction of topographic image obtained with larger interval of measurement points. Near-infrared topography with the reconstruction method potentially obtains an accurate distribution of absorption change in the brain even if the size of absorption change is less than 10 mm.     

Thermodynamics of movable inductively heated seeds for the treatment of brain tumors

A thermodynamic study is presented of temperature distributions created by an inductively heated 6-mm-diam Ni sphere imbedded in vivo and in vitro into porcine brain tissue. This study was performed in support of the development of a system that creates localized heat-induced lesions in deep-seated brain tumors. In this system, a magnetic "seed" will be remotely repositioned within the brain by an externally produced magnetic field. Convective effects of a hot moving seed will produce a different thermodynamic situation than that arising from an array of static implants. In this work, a study is presented of part of the expected change, in which a static sphere is heated to high temperature. Measurements were made of the temporal and spatial dependence of the temperature rise in the vicinity of the heated sphere, in vivo in four animals and in one that was euthanized immediately prior to experimentation. These results are used for parameter estimation with a theoretical model based on a point source solution to a form of the thermal diffusion equation, i.e., the "bioheat transfer equation." With this model thermal distributions from a power source of arbitrary geometry can be found using appropriate integration methods, and the method has widespread applicability. Estimates of blood flow rates, tissue thermal conductivity, and seed power absorption were found using the parameter estimation algorithm. The estimated blood perfusion exhibits a step increase following the first heating in multiple heating experiments. Thermal conductivity estimated using data from the nonperfused (in vitro) animal is 0.6 W/m degrees C. Seed power absorption is estimated correspondingly to be 0.9 W, a result confirmed independently with calorimetry. Statistical uncertainty is established for the radial decrease of the tissue temperature rise created by this method. This result allows estimation of a cell death boundary uncertainty of 0.6 mm, caused by fluctuations in power delivered to the seed, uncertainty in the temperature probe placements, and thermal properties such as blood perfusion and tissue thermal conductivity.     

基于头部组织不同电导率特性的阻抗成像正问题仿真研究

目的 通过建立5层有限元真实头模型,研究了各层组织非均质和颅骨、脑白质各向异性电特性对电阻抗成像问题中电磁场分布的影响.方法 对头部各组织建立4种电导率分布模型:均质分布、非均质分布以及颅骨和脑白质各向异性电导率模型;通过正问题数值求解得到不同模型下的磁场分布和电场分布,并通过定量的统计分析研究非均质和各向异性电导率特...     

A theoretical comparison of electric and magnetic stimulation of the brain

We present a theoretical comparison of the electric field produced in the brain by three modalities of transcranial stimulation of the cortex: magnetic stimulation, bifocal electric stimulation, and unifocal electric stimulation. The primary focus of this comparison is the focality and direction of the electric fields produced. A three-sphere model is used to represent the scalp, skull, and brain. All electric fields are calculated numerically. For magnetic stimulation we consider only a figure-of-eight coil. We find that magnetic stimulation produces the most focal field, while unifocal electric produces the least. Fields produced during magnetic stimulation are parallel to the head surface, while fields produced during electric stimulation have components both parallel and perpendicular to the head surface. The electric field produced by magnetic stimulation is shown to be insensitive to the skull conductivity, while that produced by electric stimulation is very sensitive to it.