Scalar Potential Finite Difference Method Research Articles

Electric field envelope focality in superficial brain areas with linear alignment montage in temporal interference stimulation

Temporal interference stimulation (TIS) uses two pairs of conventional transcranial alternating current stimulation (tACS) electrodes, each with a different frequency, to generate a time-varying electric field (EF) envelope (EFE). The EFE focality in primary somatosensory and motor cortex areas of a standard human brain was computed using newly defined linear alignment montages. Sixty head volume conductor models constructed from magnetic resonance images were considered to evaluate interindividual variability. Six TIS and two tACS electrode montages were considered, including linear and rectangular alignments. EFEs were computed using the scalar-potential finite-difference method. The computed EFE was projected onto the standard brain space for each montage. Computational results showed that TIS and tACS generated different EFE and EF distributions in postcentral and precentral gyri regions. For TIS, the EFE amplitude in the target areas had lower variability than the EF strength of tACS. However, bipolar tACS montages showed higher focality in the superficial postcentral and precentral gyri regions than in TIS. TIS generated greater EFE penetration than bipolar tACS at depths <5–10 mm below the brain surface. From group-level analysis, tACS with a bipolar montage was preferred for targets <5–10 mm in depth (gyral crowns) and TIS for deeper targets. TIS with a linear alignment montage could be an effective method for deep structures and sulcal walls. These findings provide valuable insights into the choice of TIS and tACS for stimulating specific brain regions.

Intercomparison of the Averaged Induced Electric Field in Learning-Based Human Head Models Exposed to Low-Frequency Magnetic Fields

Anatomical human models have been widely used in the assessment of induced field strength for low-frequency (LF) electromagnetic field exposure. One bottleneck is the assignment of a single electrical conductivity to all the voxels of the corresponding tissue. This simplification is known to cause computational artifact; therefore, a large reduction factor was considered in international guidelines and standards. Recently, head models with nonuniform conductivities generated using deep learning networks were proposed, and the effect on the reduction of staircasing artifacts was demonstrated. If the effectiveness of the models is confirmed for different models and codes, it would be useful to derive the relationship between the internal and external field strengths needed for setting the exposure limit. The Subcommittee 6 of the IEEE International Committee on Electromagnetic Safety Technical Committee 95 launched a working group to conduct the first intercomparison study of the induced electric field in learning-based head models exposed to LF magnetic fields. Seven international research groups have cooperated in this joint study. The highest relative difference (RD) in averaged electric fields was 23%, which is attributable to the difference caused the by scalar potential finite difference (SPFD) method and finite element method. Except for one group, the RDs in the 100 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> and 99 <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">th</sup> percentile values of the averaged electric field using the SPFD method with different solvers and codes were below 1%, indicating that the uncertainty due to different codes is sufficiently small under the same exposure scenarios. The findings would be informative for future revision of exposure limits and reduction factors in the exposure standard, which is closely related to computational uncertainty.

Analysis of Current Density Inside Human Body Using Geometric Multi-Grid Method

This paper describes the effectiveness of a geometric multi-grid (GMG) method in current density analysis using numerical human body models. The scalar potential finite difference (SPFD) method is used as a current analysis method inside a human body in the low-frequency domain, and studies have been conducted to solve faster large-scale linear equations made by the SPFD method. In the previous research, the block incomplete Cholesky conjugate gradients (ICCG) method is proposed as an effective method to solve linear equations faster. However, even though the block ICCG method is applied, many iterations are still needed. Therefore, in this research, the GMG method is considered as an effective solver for the problem. GMG method is developed and evaluated performances comparing with the block ICCG method with multi-color (MC) ordering in terms of computation time and the number of iterations. The results show that the number of iterations needed for GMG method is much smaller than that for the block ICCG and the ICCG with MC ordering. In addition, computation times are much shorter, depending on the number of threads and the number of coarse grids. Also, by using MC ordering, the scalability of the GMG method can be greatly improved.

Numerical assessment of low-frequency dosimetry from sampled magnetic fields

Low-frequency dosimetry is commonly assessed by evaluating the electric field in the human body using the scalar potential finite difference method. This method is effective only when the sources of the magnetic field are completely known and the magnetic vector potential can be analytically computed. The aim of the paper is to present a rigorous method to characterize the source term when only the magnetic flux density is available at discrete points, e.g. in case of field measurements. The method is based on the solution of the discrete magnetic curl equation. The system is restricted to the independent set of magnetic fluxes and circulations of magnetic vector potential using the topological information of the computational mesh. The solenoidality of the magnetic flux density is preserved using a divergence-free interpolator based on vector radial basis functions. The analysis of a benchmark problem shows that the complexity of the proposed algorithm is linearly dependent on the number of elements with a controllable accuracy. The method proposed in this paper also proves to be useful and effective when applied to a real world scenario, where the magnetic flux density is measured in proximity of a power transformer. A 8 million voxel body model is then used for the numerical dosimetric analysis. The complete assessment is completed in less than 5 min, that is more than acceptable for these problems.

A Co-Simulation Scalar-Potential Finite Difference Method for the Numerical Analysis of Human Exposure to Magneto-Quasi-Static Fields

A two-step approach for the simulation of the exposure of a human body to magneto-quasi-static fields is presented. The co-simulation scalar-potential finite difference method is a calculation scheme, where a distribution of the magnetic vector potential inside the biological tissues is derived from a free space magnetic source field simulation using a tree-cotree gauging algorithm. With the obtained magnetic vector potential, the electric field strength inside the voxel representation of the biological tissues is calculated solving a high-dimensional discrete Poisson equation on a high-performance workstation with multiple graphics processing units (GPUs). An exposure scenario, including an inductive power transfer system, a car model, and an anatomical human body voxel model, is simulated using this approach, and the results are compared with a monolithic scaled-frequency finite difference time domain simulation. The maximum body-internal electric field strength (voxel average) is determined and compared with basic restrictions recommended by the International Commission on Non-Ionizing Radiation Protection.

Numerical Analyses of Transcranial Magnetic Stimulation Based on Individual Brain Models by Using a Scalar-Potential Finite-Difference Method

Recent studies show that transcranial magnetic stimulation (TMS) is effective for the treatment of neurological and psychiatric diseases and that the therapeutic effect varies significantly among patients. The distribution of the electric fields induced by TMS is affected by individual variability in the shape of the brain. Numerical analyses of the electric fields in each patient performed by using individual brain models are potentially valuable for maximizing the therapeutic effect if the calculated fields are correlated with the effect of TMS. However, most of the numerical simulations of TMS are carried out by using standard human models. In this paper, we develop a new TMS simulation system, which is able to calculate the electric field induced in the brain with consideration of a complex gyral folding pattern, based on a scalar-potential finite-difference (SPFD) method. An easy implementation and a high efficiency of calculation resulting in a reduced computation time are the advantages of the SPFD method. The simulation is first performed on a sphere conductor model and a cube conductor model with 0.75 mm voxels, and we evaluate its accuracy through comparison with a finite-element method simulation. Second, we perform the simulations using six individual brain models constructed by segmenting magnetic resonance images and calculate the intensity of the field induced in each subject when they are stimulated with the intensity of their resting motor threshold (RMT). As a result, we find that there is a diversity of not only the intensity of the RMT but also the induced electric field in their brain tissues; the difference between the maximum and minimum values of the induced field on the stimulation spot among the subjects is 97 V/m, while the average value of the induced field is 203 V/m. Finally, we discuss the potential of this software in practical use and in the research of TMS.

Evaluation of SAR in a human body model due to wireless power transmission in the 10 MHz band

This study discusses a computational method for calculating the specific absorption rate (SAR) due to a wireless power transmission system in the 10 MHz frequency band. A two-step quasi-static method comprised of the method of moments and the scalar potential finite-difference method are proposed. The applicability of the quasi-static approximation for localized exposure in this frequency band is discussed by comparing the SAR in a lossy dielectric cylinder computed with a full-wave electromagnetic analysis and the quasi-static approximation. From the computational results, the input impedance of the resonant coils was affected by the existence of the cylinder. On the other hand, the magnetic field distribution in free space and considering the cylinder and an impedance matching circuit were in good agreement; the maximum difference in the amplitude of the magnetic field was 4.8%. For a cylinder–coil distance of 10 mm, the difference between the peak 10 g averaged SAR in the cylinder computed with the full-wave electromagnetic method and our quasi-static method was 7.8%. These results suggest that the quasi-static approach is applicable for conducting the dosimetry of wireless power transmission in the 10 MHz band. With our two-step quasi-static method, the SAR in the anatomically based model was computed for different exposure scenarios. From those computations, the allowable input power satisfying the limit of a peak 10 g averaged SAR of 2.0 W kg−1 was 830 W in the worst case exposure scenario with a coil positioned at a distance of 30 mm from the chest.

Electric Fields in Bone Marrow Substructures at Power-Line Frequencies

Bone marrow is known to be responsible for leukemia. In order to study the hypothesis relating power-line frequencies electromagnetic fields and childhood leukemia from a subcellular perspective, two models of bone marrow substructures exposed to electric field are computed numerically. A set of cancellous bone data obtained from computed tomography scan is computed using both the finite element method (FEM) and scalar potential finite difference method. A maximum electric field enhancement of 50% is observed. Another model of bone marrow stroma cells is implemented only in FEM using thin film approximation. The transmembrane potential (TMP) change across the gap junctions is found to range from several to over 200 microV. The two results suggest that imperceptible contact currents can produce biologically significant TMP change at least in a limited number of bone marrow stroma cells.

Peripheral Nerve Stimulation by Gradient Switching Fields in Magnetic Resonance Imaging

A heterogeneous model of the human body and the scalar potential finite difference method are used to compute electric fields induced in tissue by magnetic field exposures. Two types of coils are considered that simulate exposure to gradient switching fields during magnetic resonance imaging (MRI). These coils producing coronal (y axis) and axial (z axis) magnetic fields have previously been used in experiments with humans. The computed fields can, therefore, be directly compared to human response data. The computed electric fields in subcutaneous fat and skin corresponding to peripheral nerve stimulation (PNS) thresholds in humans in simulated MRI experiments range from 3.8 to 5.8 V/m for the fields exceeded in 0.5% of tissue volume (skin and fat of the torso). The threshold depends on coil type and position along the body, and on the anatomy and resolution of the human body model. The computed values are in agreement with previously established thresholds for neural stimulation.

Evaluation of Nonuniform 60-Hertz Magnetic-Field Exposures for Compliance with Guidelines

Magnetic-field exposures are considered in compliance with guidelines if they do not cause the induced electric field or current density to exceed basic restrictions that are based on possible adverse biological responses. Magnetic-field guidelines provide induction models for extrapolating from external field exposures to basic restrictions and vice versa. However, the uniform-field exposures used in these models do not reflect the nonuniform fields often encountered in actual high-field exposures. The purposes of this study were to investigate the relationships between external magnetic-field exposures and induced electric fields in nonuniform 60-hertz fields and to present a method for evaluating the compliance of such exposures with guidelines. Induction factors provide the induced electric field per unit of incident magnetic field. They represent a means of extrapolating from external field exposure to a peak induced electric field. Uniform and nonuniform field induction factors were computed for homogeneous ellipses and ellipsoids, and for an anatomically correct heterogeneous human model. Computations were carried out for three orthogonal uniform fields and for related nonuniform fields at varying distances from three line sources. Analytic expressions were used to compute induced peak electric fields for homogeneous models in uniform fields. A scalar-potential finite-difference method computed induced quantities for all models at 3.6-mm resolution in uniform and nonuniform fields. Equivalent uniform magnetic fields that produce the same peak electric field as a nonuniform field with a known maximum field were derived from the induction factors. To evaluate a nonuniform field exposure for compliance, the equivalent uniform field for the exposure is estimated based on the magnitude of the maximum surface field and the distance from the line source. Compliance is achieved if the equivalent uniform magnetic field is below the magnetic-field limit. Equivalent uniform magnetic-field exposures are computed for two actual electric utility tasks, as examples. *Deceased

Electric fields in the human body due to electrostatic discharges.

Electrostatic discharges (ESDs) occur when two objects at different electric potentials come close enough to arc (spark) across the gap between them. Such discharges may be either single-event or repetitive (e.g., 60 Hz). Some studies have indicated that ESDs may be a causative factor for health effects in electric utility workers. Moreover, a hypothesis has recently been forwarded imperceptible contact currents in the human body may be responsible for health effects, most notably childhood leukemia. Numerical modeling indicates that the electric fields in human tissue resulting from typical contact currents are much greater than those induced from typical exposures to electric and magnetic fields at power line frequencies. Numerical modeling is used here to compute representative spark-discharge dosimetry in a realistic human adult model. The frequency-domain scalar potential finite difference method is applied in conjunction with the Fourier transform to assess electric fields in selected regions and tissues of interest in the body. Electric fields in such tissues as subcutaneous fat (where peripheral nerves may be excited), muscle and bone marrow are of the order of kilovolts per meter in the lower arm. The pulses, however, are of short duration (approximately 100 ns).

Electric fields in the human body resulting from 60-Hz contact currents.

Contact currents occur when a person touches conductive surfaces at different potentials and completes a path for current flow through the body. Such currents provide an additional coupling mechanism to that, due to the direct field effect between the human body and low-frequency external fields. The scalar potential finite difference method, with minor modifications, is applied to assess current density and electric field within excitable tissue and bone marrow due to contact current. An anatomically correct adult model is used, as well as a proportionally downsized child model. Three pathways of contact current are modeled: hand to opposite hand and both feet, hand to hand only, and hand to both feet. Because of its larger size relative to the child, the adult model has lower electric field and current-density values in tissues/unit of contact current. For a contact current of 1 mA [the occupational reference level set by the International Commission on Non-ionizing Protection (ICNIRP)], the current density in brain does not exceed the basic restriction of 10 mA/m2. The restriction is exceeded slightly in the spine, and by a factor of more than 2 in the heart. For a contact current of 0.5 mA (ICNIRP general public reference level), the basic restriction of 2 mA/m2 is exceeded several-fold in the spine and heart. Several microamperes of contact current produces tens of mV/m within the child's lower arm bone marrow.

High-resolution magnetic field numerical dosimetry for live-line workers

Numerical modeling provides one means for the realistic assessment of the electric field and current density levels induced in the bodies of live-line utility workers during occupational exposure to 60 Hz nonuniform magnetic fields. An efficient scalar potential finite difference method is applied to anatomically-derived high-resolution heterogeneous human body conductivity models, to provide such estimates for four representative posture and exposure scenarios. Dosimetric data are presented in terms of volumetric maximum and root mean square fields induced in various tissue groups by unit current. The results scale linearly with total source current, which allows the data to be applied to arbitrary current levels.

Induced current densities from low-frequency magnetic fields in a 2 mm resolution, anatomically realistic model of the body

This paper presents calculations of current density in a fine-resolution (2 mm) anatomically realistic voxel model of the human body for uniform magnetic fields incident from the front, side and top of the body for frequencies from 50 Hz to 10 MHz. The voxel phantom, NORMAN, has a height of 1.76 m and a mass of 73 kg. There are 8.3 million voxels in the body differentiated into 37 tissue types. Both the impedance method and the scalar potential finite difference method were used to provide mutual corroboration. Results are presented for the current density averaged over in muscle, heart, brain and retina.

A comparison of 60 Hz uniform magnetic and electric induction in the human body

High-resolution computations of induced fields are used to assess equivalent source levels for human exposure to uniform low-frequency electric and magnetic fields. These results pertain to 60 Hz foot-to-head electric excitation of the body in three positions with respect to a ground plane, and to magnetic excitation by three orthogonal source orientations. All computations are based on an anatomically derived human body model composed of 1736 873 cubic voxels with 3.6 mm edges. The data for magnetic excitation are computed using a scalar potential finite difference (SPFD) method, while those for electric excitation are computed using a hybrid method based on the SPFD method coupled with a quasistatic finite difference time domain code. The data are analysed in two ways, using an induced current density threshold of . Firstly, the various field strengths required to produce a whole-body average current density magnitude equal to the threshold are derived for each configuration, and the associated current density levels in various organs and tissues are presented. It is found that the average current density magnitude values in at least one tissue group can be up to 3 (5) times greater than the whole-body average under electric (magnetic) excitation, and that the associated maximum values can be up to 46 (28) times greater than the whole-body average under electric (magnetic) excitation, for at least one source/body configuration. Secondly, the data are analysed from the opposite point of view, in which the source levels required to induce average or maximum induced current density magnitudes at the threshold level in specific tissue groups are determined. Evaluations such as the present one should prove useful in the development of protection standards, and are also expected to aid in the understanding of results from various animal and tissue culture studies.