Anisotropic Conductivity Distribution Research Articles

Low-frequency conductivity tensor imaging with a single current injection using DT-MREIT

Diffusion tensor-magnetic resonance electrical impedance tomography (DT-MREIT) is an imaging modality to obtain low-frequency anisotropic conductivity distribution employing diffusion tensor imaging and MREIT techniques. DT-MREIT is based on the linear relationship between the conductivity and water self-diffusion tensors in a porous medium, like the brain white matter. Several DT-MREIT studies in the literature provide cross-sectional anisotropic conductivity images of tissue phantoms, canine brain, and the human brain. In these studies, the conductivity tensor images are reconstructed using the diffusion tensor and current density data acquired by injecting two linearly independent current patterns. In this study, a novel reconstruction algorithm is devised for DT-MREIT to reconstruct the conductivity tensor images using a single current injection. Therefore, the clinical applicability of DT-MREIT can be improved by reducing the total acquisition time, the number of current injection cables, and contact electrodes to half by decreasing the number of current injection patterns to one. The proposed method is evaluated utilizing simulated measurements and physical experiments. The results obtained show the successful reconstruction of the anisotropic conductivity distribution using the proposed single current DT-MREIT.

Low-Frequency Conductivity Tensor Imaging of the Human Head In Vivo Using DT-MREIT: First Study.

We present the first in vivo images of anisotropic conductivity distribution in the human head, measured at a frequency of approximately 10 Hz. We used magnetic resonance electrical impedance tomography techniques to encode phase changes caused by current flow within the head via two independent electrode pairs. These results were then combined with diffusion tensor imaging data to reconstruct full anisotropic conductivity distributions in 5-mm-thick slices of the brains of two participants. Conductivity values recovered in this paper were broadly consistent with literature values. We anticipate that this technique will be of use in many areas of neuroscience, most importantly in functional imaging via inverse electroencephalogram. Future studies will involve pulse sequence acceleration to maximize brain coverage and resolution.

Evaluation of Numerical Techniques for Solving the Current Injection Problem in Biological Tissues.

Accurate computational modeling of electric fields in the human head has become important in clinical research to study or influence brain functionality. While existing numerical approaches have been evaluated against simple geometries with known closed form solutions, the relationship between these approaches in more complex geometries has not been studied. Here, we compare the three most commonly used approaches for bioelectric modeling: the finite element method (FEM), the finite difference method (FDM), and the boundary element method (BEM). Using both isotropic and anisotropic conductivity distributions, we construct and compare bioelectric models for a realistic head geometry. Our results suggest that both FEM and FDM are capable of accurately model voltages in the brain, while computations from BEM result in significantly larger errors, due to the increased simplicity and implicit model assumptions.

Practical Realization of Magnetic Resonance Conductivity Tensor Imaging (MRCTI)

Magnetic resonance conductivity tensor imaging (MRCTI) is an emerging modality which reconstructs images of anisotropic conductivity distribution within a volume conductor. Images are reconstructed based on magnetic flux density distribution induced by an externally applied probing current, together with a resultant surface potential value. The induced magnetic flux density distribution is measured using magnetic resonance current density imaging techniques. In this study, MRCTI data acquisition is experimentally implemented and anisotropic conductivity images of test phantoms are reconstructed using recently proposed MRCTI reconstruction algorithms.

Conductivity and scattering Q in GPR data: Example from the Ellenburger dolomite, central Texas

Total attenuation ([Formula: see text]) in ground-penetrating radar (GPR) data is a composite of intrinsic and scattering attenuations ([Formula: see text] and [Formula: see text]). For nonmagnetic materials, [Formula: see text] is a combination of the effects of real conductivity and dielectric relaxation. The attenuation for real conductivity [Formula: see text] in the GPR frequency band is a function of frequency while the dielectric relaxation is frequency-independent. These frequency behaviors allow separation of the attenuation types by attributing and fitting the [Formula: see text] decay shape with frequency to the conductivity, and by attributing the magnitude of [Formula: see text] to the sum of conductivity and dielectric relaxation attenuations at each frequency. Total attenuation is calculated from GPR data using spectral ratios, and [Formula: see text] is obtained by fitting a smooth lower bound to [Formula: see text]; the difference between [Formula: see text] and [Formula: see text] estimates the scattering contribution [Formula: see text]. Scatterer size spectra are evaluated using [Formula: see text] for 2D, and [Formula: see text] for 3D, propagation (where [Formula: see text] is wavenumber and [Formula: see text] is the scatterer size). We illustrate with 2D synthetic data and three field 2D crosshole profiles from an outcrop of an Ellenburger collapsed paleocave environment in central Texas. Between the three pairs of holes, we estimate the breccia sizes from the scattering spectra [Formula: see text]. To image the anisotropic electrical conductivity distributions, we use simultaneous iterative reconstruction tomography. There is a correlation between the low wavenumber features of the results of the current conductivity tomography and those in previous velocity tomography, and with surface data results that are predicted and calculated from GPR data attributes. Low- and high-conductivity zones tend to follow either the GPR facies distributions, lithological boundaries, or the larger of the fractures. Correlations are not visible where the breccias are finer because these tend to be more randomly oriented, and/or below the resolution of the GPR data.

Methods for high-resolution anisotropic finite element modeling of the human head: Automatic MR white matter anisotropy-adaptive mesh generation

This study proposes an advanced finite element (FE) head modeling technique through which high-resolution FE meshes adaptive to the degree of tissue anisotropy can be generated. Our adaptive meshing scheme (called wMesh) uses MRI structural information and fractional anisotropy maps derived from diffusion tensors in the FE mesh generation process, optimally reflecting electrical properties of the human brain. We examined the characteristics of the wMeshes through various qualitative and quantitative comparisons to the conventional FE regular-sized meshes that are non-adaptive to the degree of white matter anisotropy. We investigated numerical differences in the FE forward solutions that include the electrical potential and current density generated by current sources in the brain. The quantitative difference was calculated by two statistical measures of relative difference measure (RDM) and magnification factor (MAG). The results show that the wMeshes are adaptive to the anisotropic density of the WM anisotropy, and they better reflect the density and directionality of tissue conductivity anisotropy. Our comparison results between various anisotropic regular mesh and wMesh models show that there are substantial differences in the EEG forward solutions in the brain (up to RDM = 0.48 and MAG = 0.63 in the electrical potential, and RDM = 0.65 and MAG = 0.52 in the current density). Our analysis results indicate that the wMeshes produce different forward solutions that are different from the conventional regular meshes. We present some results that the wMesh head modeling approach enhances the sensitivity and accuracy of the FE solutions at the interfaces or in the regions where the anisotropic conductivities change sharply or their directional changes are complex. The fully automatic wMesh generation technique should be useful for modeling an individual-specific and high-resolution anisotropic FE head model incorporating realistic anisotropic conductivity distributions towards more accurate analysis of bioelectromagnetic problems.

How the brain tissue shapes the electric field induced by transcranial magnetic stimulation

In transcranial magnetic stimulation (TMS), knowledge of the distribution of the induced electric field is fundamental for a better understanding of the position and extent of the stimulated brain region. However, the different tissue types and the varying fibre orientation in the brain tissue result in an inhomogeneous and anisotropic conductivity distribution and distort the electric field in a non-trivial way. Here, the field induced by a figure-8 coil is characterized in detail using finite element calculations and a geometrically accurate model of an individual head combined with high-resolution diffusion-weighted imaging for conductivity mapping. It is demonstrated that the field strength is significantly enhanced when the currents run approximately perpendicular to the local gyral orientation. Importantly, the spatial distribution of this effect differs distinctly between gray matter (GM) and white matter (WM): While the field in GM is selectively enhanced at the gyral crowns and lips, high field strengths can still occur rather deep in WM. Taking the anisotropy of brain tissue into account tends to further boost this effect in WM, but not in GM. Spatial variations in the WM anisotropy affect the local field strength in a systematic way and result in localized increases of up to 40% (on average ~7% for coil orientations perpendicular to the underlying gyri). We suggest that these effects might create hot spots in WM that might contribute to the excitation of WM structures by TMS. However, our results also demonstrate the necessity of using realistic nerve models in the future to allow for more definitive conclusions.

Influence of white matter anisotropic conductivity on EEG source localization: Comparison to fMRI in human primary visual cortex

The goal of this study was to experimentally investigate the influence of the anisotropy of white matter (WM) conductivity on EEG source localization. Visual evoked potentials (VEP) and fMRI data were recorded from three human subjects presented with identical visual stimuli. A finite element method was used to solve the EEG forward problems based on both anisotropic and isotropic head models, and single-dipole source localization was subsequently performed to localize the source underlying the N75 VEP component. The averaged distances of the localized N75 dipole locations in V1 between the isotropic and anisotropic head models ranged from 0 to 6.22+/-2.83mm. The distances between the localized dipole positions and the centers of the fMRI V1 activations were slightly smaller when using an anisotropic model (7.49+/-1.35-15.70+/-8.60mm) than when using an isotropic model (7.65+/-1.30-15.31+/-9.18mm). Anisotropic models incorporating realistic WM anisotropic conductivity distributions do not substantially improve the accuracy of EEG dipole localization in the primary visual cortex using experimental data obtained using visual stimulation. The present study represents the first attempt using a human experimental approach to assess the effects of WM anisotropy on EEG source analysis.

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.

Anisotropic conductivity imaging with MREIT using equipotential projection algorithm

Magnetic resonance electrical impedance tomography (MREIT) combines magnetic flux or current density measurements obtained by magnetic resonance imaging (MRI) and surface potential measurements to reconstruct images of true conductivity with high spatial resolution. Most of the biological tissues have anisotropic conductivity; therefore, anisotropy should be taken into account in conductivity image reconstruction. Almost all of the MREIT reconstruction algorithms proposed to date assume isotropic conductivity distribution. In this study, a novel MREIT image reconstruction algorithm is proposed to image anisotropic conductivity. Relative anisotropic conductivity values are reconstructed iteratively, using only current density measurements without any potential measurement. In order to obtain true conductivity values, only either one potential or conductivity measurement is sufficient to determine a scaling factor. The proposed technique is evaluated on simulated data for isotropic and anisotropic conductivity distributions, with and without measurement noise. Simulation results show that the images of both anisotropic and isotropic conductivity distributions can be reconstructed successfully.

Reconstruction of the anisotropic complex conductivity distribution of a human thigh in three dimensions

In electric impedance tomography (EIT), a set of electrodes measures surface potentials resulting from known currents injected into the target region. Using a correct model of the target region, these potentials can be simulated very accurately, too. Since injected currents spread out in three dimensions, simulations in three dimensions are inevitable. In addition, the anisotropic nature of electric material properties of biological tissue should not be neglected to obtain appropriate results. The aim of this work is to reconstruct the anisotropic complex conductivity distribution of a human thigh. Pseudomeasurements making up for EIT data were available. A finite element model of a human thigh based on magnetic resonance images was generated. A classical Gauss-Newton algorithm with regularization was used to match the real material values and the ones fed into the simulation. Results for a simple, but unrealistic isotropic case as well as for a complicated anisotropic case are presented and discussed

Identification of current density distribution in electrically conducting subject with anisotropic conductivity distribution

Current density imaging (CDI) is able to visualize a three-dimensional current density distribution J inside an electrically conducting subject caused by an externally applied current. CDI may use a magnetic resonance imaging (MRI) scanner to measure the induced magnetic flux density B and compute J via the Ampere law . However, measuring all three components of B = (Bx, By, Bz) has a technical difficulty due to the requirement of orthogonal rotations of the subject inside the MRI scanner. In this work, we propose a new method of reconstructing a current density image using only Bz data so that we can avoid the subject rotation procedure. The method utilizes an auxiliary injection current to compensate the missing information of Bx and By. The major advantage of the method is its applicability to a subject with an anisotropic conductivity distribution. Numerical experiments show the feasibility of the new technique.

Reconstruction of the anisotropic complex conductivity distribution in 3D

PurposeThe aim of the work is to reconstruct the anisotropic complex conductivity distribution with the common Gauss‐Newton algorithm.Design/methodology/approachA cubic region with anisotropic material properties is enclosed by a larger cube with isotropic material properties. Numerical simulations are done with tetrahedral nodal finite elements of second‐order.FindingsIt can be shown that it is possible to reconstruct anisotropic complex conductivity distribution if the starting values are chosen sufficiently close to the true values of the complex conductivity.Originality/valueIn this paper, the anisotropic electric conductivity and the anisotropic permittivity are reconstructed in 3D.

Anisotropic resistivity tomography

Geophysical tomographic techniques have the potential to remotely detect and characterize geological features, such as fractures and spatially varying lithologies, by their response to signals passed through these features. Anisotropic behaviour in many geological materials necessitates the generalization of tomographic methods to include anisotropic material properties in order to attain high-quality images of the subsurface. In this paper, we present a finite element (FE) based direct-current electrical inversion method to reconstruct the conductivity tensor at each node point of a FE mesh from electrical resistance measurements. The inverse problem is formulated as a functional optimization and the non-uniqueness of the electrical inverse problem is overcome by adding penalty terms for structure and anisotropy. We use a modified Levenberg–Marquardt method for the functional optimization and the resulting set of linear equation is solved using pre-conditioned conjugate gradients. The method is tested using both synthetic and field experiments in cross-well geometry. The acquisition geometry for both experiments uses a cross-well experiment at a hard-rock test site in Cornwall, southwest England. Two wells, spaced at 25.7 m, were equipped with electrodes at a 1 m spacing at depths from 21–108 m and data were gathered in pole–pole geometry. The test synthetic model consists of a strongly anisotropic and conductive body underlain by an isotropic resistive formation. Beneath the resistive formation, the model comprises a moderately anisotropic and moderately conductive half-space, intersected by an isotropic conductive layer. This model geometry was derived from the interpretation of a seismic tomogram and available geological logs and the conductivity values are based on observed conductivities. We use the test model to confirm the ability of the inversion scheme to recover the (known) true model. We find that all key features of the model are recovered. However, the inversion model is smoother than the true model and the difference in absolute value of anisotropy and conductivity between features is slightly underestimated. Using an anisotropic conductivity distribution aggravates the problem of non-uniqueness of the solution of the inverse electrical problem. This problem can be overcome by applying appropriate structural and anisotropy constraints. We find that running a suite of inversions with varying constraint levels and subsequent examination of the results (including the inspection of residual maps) offers a viable method for choosing appropriate numerical values for the imposed constraints. Inversion of field data reveals a strongly anisotropic subsurface with marked spatial variations of both magnitude of anisotropy and conductivity. Average conductivities range from 0.001 S m−1 (= 1000 Ω m) to 0.003 S m−1 (= 333 Ω m) and anisotropy values range from 0 per cent to more than 300 per cent. As an independent test of the reliability of the structures revealed by anisotropic electric tomography, anisotropic seismic traveltime tomograms were calculated. We find a convincing structural agreement between the two independently derived images. Areas of high electric anisotropy coincide with seismically anisotropic areas and we observe an anticorrelation between electric conductivity and seismic velocity. Both observations are consistent with anisotropy anomalies caused by fracturing or layering.

A second-order finite element algorithm for solving the three-dimensional EEG forward problem

A finite element algorithm has been developed to solve the electroencephalogram (EEG) forward problem. A new computationally efficient approach to calculate the stiffness matrix of second-order tetrahedral elements has been developed for second-order tetrahedral finite element models. The present algorithm has been evaluated by means of computer simulations, by comparing with analytic solutions in a multi-spheres concentric head model. The developed finite element method (FEM) algorithm has also been applied to address questions of interest in the EEG forward problem. The present simulation study indicates that the second-order FEM provides substantially enhanced numerical accuracy and computational efficiency, as compared with the first-order FEM for comparable numbers of tetrahedral elements. The anisotropic conductivity distribution of the head tissue can be taken into account in the present FEM algorithm. The effects of dipole eccentricity, size of finite elements and local mesh refinement on solution accuracy are also addressed in the present simulation study.

Anisotropic resistivity inversion

We present an inversion method for 3D electrical imaging in media with aninhomogeneous and anisotropic conductivity distribution. The conductivitydistribution is discretized via finite elements and is described by a second-ordertensor at each finite element node. The inversion method is formulated as afunctional optimization with an error functional containing terms measuringdata misfit and model covariance by means of smoothness, anisotropyand deviation from a starting model. Including the model covarianceinformation overcomes the problem of ill-posedness at the expense oflimiting the allowed models to the class of models which are compatible withthe provided model covariance information. The discretized form of theerror functional is minimized by a Levenberg–Marquardt type methodusing an iterative preconditioned conjugate gradient solver. The use ofan iterative solver allows one to bypass the actual computation of theJacobian or an inverse system matrix. The use of a memory efficient iterativesolver together with the implementation on parallel computers allowslarge-scale inverse problems, comprising several hundred thousand nodes withhundreds of sources and receivers, to be solved. The new method is testedusing computer-generated data from two- and three-dimensional syntheticmodels. For each inversion a choice of penalty parameters, gauging thelevel of model covariance information imposed, has to be made and thelevel of regularization required is hard to estimate. We find that runninga suite of inversions with varying penalty parameters and subsequentexamination of the results (including inspection of residual maps) offers a viablemethod for choosing appropriate numerical values for the penalty levels. Inthe applications we found the inversion process to be highly non-linear.Inversion models from intermediate steps of the iterative inversion showstructure in places that do not exhibit structure in the true model andonly at later iterations do anomalies move to the correct location in themodelling domain. This result indicates that linearized inversions that fail tore-linearize during the inversion process will fail to find meaningful inversionimages. The inversion images achieved using the new method recoverthe important features of the true models, including the approximatemagnitudes of the conductivity anomalies and the magnitudes and directions ofanisotropy anomalies. The inversion images are generally ‘blurred’, that issharp edges are smoothed, and the recovered magnitudes of conductivity,anisotropy and anisotropy direction are generally under-estimated.

Hydrogeological aspects of fault zones on various scales in theRoer Valley Rift System

The impact of faults on the groundwater flow system in the Roer Valley Rift System (RVRS) is demonstrated with examples from outcrop scale to regional scale. Faults in the RVRS can form strong barriers to horizontal groundwater flow as well as enhanced vertical groundwater flow paths at the same location. The strongly anisotropic hydraulic conductivity distribution within fault zones has important implications for the modeling of groundwater flow in sedimentary aquifer systems that are cut by faults. In this study, the hydraulic behavior of fault zones is studied at different scales. An outcrop study over the Geleen Fault zone shows deformation mechanisms as particulate flow and clay smearing in great detail. Qualitative and quantitative image analysis allows for an estimate of the micro-scale variation of the hydraulic properties within a fault zone. Additional core-plug measurements indicate that the damage zone around fault zones may form preferential flow paths. On a larger scale, observations over the Peel Boundary fault near the village of Uden also indicate that vertical groundwater flow close to the fault is enhanced, which results in a discharge of the underlying aquifers at the location of the fault zone. Finally, on a regional scale, hydraulic head patterns around the lignite mining areas in Germany show the importance of faults and the variation of their hydraulic properties to regional groundwater flow patterns.

A validation of MRI cortical surface rendering of the human post-mortem brain

In transcranial magnetic stimulation (TMS), knowledge of the distribution of the induced electric field is fundamental for a better understanding of the position and extent of the stimulated brain region. However, the different tissue types and the varying fibre orientation in the brain tissue result in an inhomogeneous and anisotropic conductivity distribution and distort the electric field in a non-trivial way. Here, the field induced by a figure-8 coil is characterized in detail using finite element calculations and a geometrically accurate model of an individual head combined with high-resolution diffusion-weighted imaging for conductivity mapping. It is demonstrated that the field strength is significantly enhanced when the currents run approximately perpendicular to the local gyral orientation. Importantly, the spatial distribution of this effect differs distinctly between gray matter (GM) and white matter (WM): While the field in GM is selectively enhanced at the gyral crowns and lips, high field strengths can still occur rather deep in WM. Taking the anisotropy of brain tissue into account tends to further boost this effect in WM, but not in GM. Spatial variations in the WM anisotropy affect the local field strength in a systematic way and result in localized increases of up to 40% (on average ~ 7% for coil orientations perpendicular to the underlying gyri). We suggest that these effects might create hot spots in WM that might contribute to the excitation of WM structures by TMS. However, our results also demonstrate the necessity of using realistic nerve models in the future to allow for more definitive conclusions.

Disturbed gyrification and reduced volume of the prefrontal lobe in male schizophrenics—A morphometric postmortem study

In transcranial magnetic stimulation (TMS), knowledge of the distribution of the induced electric field is fundamental for a better understanding of the position and extent of the stimulated brain region. However, the different tissue types and the varying fibre orientation in the brain tissue result in an inhomogeneous and anisotropic conductivity distribution and distort the electric field in a non-trivial way. Here, the field induced by a figure-8 coil is characterized in detail using finite element calculations and a geometrically accurate model of an individual head combined with high-resolution diffusion-weighted imaging for conductivity mapping. It is demonstrated that the field strength is significantly enhanced when the currents run approximately perpendicular to the local gyral orientation. Importantly, the spatial distribution of this effect differs distinctly between gray matter (GM) and white matter (WM): While the field in GM is selectively enhanced at the gyral crowns and lips, high field strengths can still occur rather deep in WM. Taking the anisotropy of brain tissue into account tends to further boost this effect in WM, but not in GM. Spatial variations in the WM anisotropy affect the local field strength in a systematic way and result in localized increases of up to 40% (on average ~ 7% for coil orientations perpendicular to the underlying gyri). We suggest that these effects might create hot spots in WM that might contribute to the excitation of WM structures by TMS. However, our results also demonstrate the necessity of using realistic nerve models in the future to allow for more definitive conclusions.

Anatomically constrained electrical impedance tomography for three-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 will 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.