Toroidal coil models for transcutaneous magnetic stimulation of nerves

A novel design of coils for transcutaneous magnetic stimulation of nerves is presented. These coils consist of a toroidal winding around a high-permeability material (Supermendur) core embedded in a conducting medium. Theoretical numerical calculations are used to analyze the effect of the design parameters of these coils, such as coil width, toroidal radius, conducting layer thickness and core transversal shape on the induced electric fields in terms of the electric field strength and distribution. Results indicate that stimulation of nerves with these coils has some of the advantages of both electrical and magnetic stimulation. These coils can produce localized and efficient stimulation of nerves with induced electric fields parallel and perpendicular to the skin similar to surface electrical stimulation. However, they retain some of the advantages of magnetic stimulation such as no risk of tissue damage due to electrochemical reactions at the electrode interface and less uncomfortable sensations or pain. The driving current is reduced by over three orders of magnitude compared to traditional magnetic stimulation, eliminating the problem of coil heating and allowing for long duration and high-frequency magnetic stimulation with inexpensive stimulators     

A volume-conduction analysis of magnetic stimulation of peripheralnerves

Magnetic stimulation is a method to study several nervous disorders as well as the intact nervous system in humans. Interest in magnetic stimulation of peripheral nerves has grown rapidly, but difficulties in locating the site of excitation have prevented it from becoming a routine clinical tool. It has been reasoned that the activating function of long and straight nerves is the first spatial derivative of the electric field component parallel to the nerves. Therefore, to predict the site of activation, one has to compute this field feature. We describe here an analytical mathematical model and investigate the influence of volume-conductor shape on the induced field, predictions of the site of activation are given for typical stimulation coil arrangements and these results are compared with experimental and literature data. Comparisons suggest that the activating function is not simply the spatial gradient of the induced electric field, but that other mechanisms are also involved. The model can be easily utilized in the search for more efficient coil constructions and improved placements with respect to the target nerves     

On the induced electric field gradients in the human body for magnetic stimulation by gradient coils in MRI

Prior theoretical studies indicate that the negative spatial derivative of the electric field induced by magnetic stimulation may be one of the main factors contributing to depolarization of the nerve fiber. This paper studies this parameter for peripheral nerve stimulation (PNS) induced by time-varying gradient fields during MRI scans. The numerical calculations are based on an efficient, quasi-static, finite-difference scheme and an anatomically realistic human, full-body model. Whole-body cylindrical and planar gradient sets in MRI systems and various input signals have been explored. The spatial distributions of the induced electric field and their gradients are calculated and attempts are made to correlate these areas with reported experimental stimulation data. The induced electrical field pattern is similar for both the planar coils and cylindrical coils. This study provides some insight into the spatial characteristics of the induced field gradients for PNS in MRI, which may be used to further evaluate the sites where magnetic stimulation is likely to occur and to optimize gradient coil design.     

A box coil for the stimulation of biological tissue and cells in vitro and in vivo by pulsed magnetic fields

An alternative coil system to the Helmoholtz coil-pair is described for the stimulation of biological tissue and cells: a relatively large box coil made of copper or aluminum sheet stock. The design is based on the principal determinant of the induced electric field, namely, the magnetic vector potential (A), in the equation, [formula: see text]. The second term in the equation is needed when boundaries of the conducting medium are in close proximity to the region of interest, such as in a culture dish. An electric surface charge builds up on the boundaries to generate an electric field which cancels [formula: see text] at the surface. The effectiveness of the new coil is demonstrated in a study of the outgrowth enhancement of axons from rat embryonic dorsal root ganglia.     

Focal magnetic stimulation of an axon

The induced electric field produced by a circular coil during magnetic stimulation of an axon is derived from Maxwell's equations. The foci and virtual cathodal and anodal regions are predicted as a function of coil radius and orientation. Two virtual anode and cathode pairs are predicted, one lying outside the coil's perimeter and predominant in the far field, and one lying within the perimeter of the coil which may stimulate the axon when the coil and nerve are in close proximity. When the coil is positioned tangent to the nerve, an orientation commonly used in clinical magnetic stimulation, the foci of the predominant cathode and anode pair are extremely sensitive to changes in coil placement. In addition, the radius of curvature of the activating function, a measure of the size of the virtual cathode at threshold, is predicted to decrease with decreasing coil diameter and distance to the nerve. These predictions may help explain observed variability in measurements of conduction velocity and latency during magnetic stimulation of peripheral axons     

A 3-D differential coil design for localized magnetic stimulation

A novel three-dimensional (3-D) differential coil has been designed for improving the localization of magnetic stimulation. This new coil design consists of a butterfly coil with two additional wing units and an extra bottom unit, both perpendicular to the plane of the butterfly coil. The wing units produce opposite fields to restrict the spread of induced fields while the bottom unit enhances the induced fields at the excitation site. The peak induced field generated by this new design is located at the center of the coil, providing an easy identification of the excitation site. The field localization of the new coil is comparable with that of much smaller coils but with an inductance compatible to current magnetic stimulators. Numerical computations based on the principles of electromagnetic induction and using a human nerve model were performed to analyze the induced fields and the stimulation thresholds of new coil designs. The localization of the coil design was assessed by a half power region (HPR), within which the magnitude of the normalized induced field is greater than 1/square root of 2. The HPR for a 3-D differential coil built is improved (decreased) by a factor of three compared with a standard butterfly coil. Induced fields by this new coil were measured and in agreement with theoretical calculations.     

Modeling of magnetic field stimulation of bent neurons

The authors consider a simple model of magnetic stimulation of a long bent neuron located in a semi-infinite volume conductor with a planar interface. It is shown that the stimulating coil characteristics (size, shape and location) and the neuron shape affect the location of the stimulation. The activating function, defined as the electric field derivative along the neuron, has two components. One component depends on the derivative of the electric field along the straight section of the neuron, and the other on the field magnitude. The maximal stimulation point is at the bent part of the nerve and its position depends on the nerve shape and coil parameters. The analysis also has shown a better performance (a stronger stimulus) for a double-circular (figure eight) coil than for a double-square coil     

Cylindrical tissue model for magnetic field stimulation of neurons:effects of coil geometry

An analytical solution for the electric field induced in a homogeneous cylindrical conductor under quasi-static conditions from current in a coil is used to model peripheral nerve stimulation with magnetic fields. A variety of coil geometries is analyzed in terms of the spatial derivative of the induced electric field along a long straight nerve, parallel to the axis, in a human-arm model. The results of these computations are found to be in general agreement with conclusions regarding the optimization of stimulating coils derived from analyses of the semi-infinite tissue model. The differences and their physical basis are pointed out     

An analytical model to predict the electric field and excitationzones due to magnetic stimulation of peripheral nerves

The main unknown factor in understanding magnetic stimulation of peripheral nerves is the distribution of the induced electric field. The authors have applied the so-called reciprocity theorem and developed an analytical model to compute the electric field and its spatial derivatives inside pseudocylindrical structures. The results can be used to predict the site of excitation in magnetic stimulation of peripheral nerves     

Magnetic stimulation of the motor cortex-theoretical considerations

The aim of this paper is to present a first approximation model for the computation of the electric fields produced in the brain tissues by magnetic stimulation. Results are given in terms of induced electric field and current density caused by coils of different radii and locations. Nontraditional coil locations and assemblies are also considered (multicoil arrangements). Model simulations show that a good control of the excitation spread can be achieved by proper positioning of the coil. It is also predicted that one of the major drawbacks of the technique, i.e., the poor ability to concentrate the current spread into a small brain area can be partially overcome by more effective coil positioning and/or assembly. Finally, some comparisons are made among the results obtained from electric and magnetic stimulation. This is thought to be of great help in the design of experiments aimed to understand the relative role of the different brain structures responsible for the motor response.     

Magnetic coil design considerations for functional magnetic stimulation

Our studies have demonstrated effective stimulation of the bladder, bowel, and expiratory muscles in patients with spinal cord injury using functional magnetic stimulation. However, one limitation of the magnetic coils (MC) is related to their inability to specifically stimulate the target tissue without activation of surrounding tissue. The primary goal of this study was to determine the governing parameters in the MC design, such as coil configuration, diameter, and number of turns in one loop of the coil. By varying these parameters, our approach was to design, construct, and evaluate the induced electric field distributions of two sets of novel MC's. Based on the slinky coil design, the first set of coils was constructed to compare their abilities in generating induced electric fields for focal nerve excitation. The second set of coils was built to determine the effect that changes in two parameters, coil diameter and number of turns in one loop, had on field penetration. The results showed that the slinky coil design produced more focalized stimulation when compared to the planar round coils. The primary-to-secondary peak ratios of the induced electric field from slinky 1 to 5 were 1.00, 2.20, 2.85, 2.62, and 3.54. We also determined that coils with larger diameters had better penetration than those with smaller diameters. Coils with less number of turns in one loop had higher initial field strengths; when compared to coils that had more turns per loop, initial field strengths remained higher as distance from the coil increased. In our attempt to customize MC design according to each functional magnetic stimulation application and patients of different sizes, the parameters of MC explored in this study may facilitate designing an optimal MC for a certain clinical application.     

Current induced by TE excitation on a conducting cylinder located near the planar interface between two semi-infinite half-spaces

An analysis is presented for determining the current induced by a known transverse electric excitation on a perfectly conducting cylinder located near the planar interface separating two semi-infinite, homogeneous half-spaces of different electromagnetic properties. The conducting cylinder of general cross section is of infinite extent and the excitation is transverse electric to the cylinder axis. Two types of integral equations, the magnetic field integral equation and the electric field integral equation, are formulated, and the Green's functions for the integral equations are derived in an appendix. Numerical solution methods for solving the integral and integrodifferential equations are presented. For a strip parallel or perpendicular to the interface, a circular cylinder, and a rectangular cylinder, data are presented and discussed for selected parameters, including the case of a cylinder resting on the interface.     

The transient electromagnetic field of an electric line sourceabove a plane conducting earth

The pulsed electromagnetic radiation from an electric line source above a conducting earth is investigated theoretically. The modified Cagniard method is used to derive closed-form expressions for the electric and magnetic field anywhere above the conducting earth. Numerical results are presented for the electric field for different points of excitation and observation above the earth, as well as for different values of the earth's material parameters, i.e. permittivity and electrical conductivity. It is shown that the effect of the conductivity is of significant importance to the calculation of interfering transient electromagnetic fields in the presence of a conducting earth     

Effects of induced electric fields on finite neuronal structures: asimulation study

An analysis is presented of magnetic stimulation of finite length neuronal structures using computer simulations. Models of finite neuronal structures in the presence of extrinsically applied electric fields indicate that excitation can be characterized by two driving functions: one due to field gradients and the other due to fields at the boundaries of neuronal structures. It is found that boundary field driving functions play an important role in governing excitation characteristics during magnetic stimulation. Simulations indicate that axons whose lengths are short compared to the spatial extent of the induced field are easier to excite than longer axons of the same diameter. Simulations also indicate that independent cellular dendritic processes are probably not excited during magnetic stimulation. Analysis of the temporal distribution of induced fields indicates that the temporal shape of the stimulus waveform modulates excitation thresholds and propagation of action potentials     

Computation of electric and magnetic stimulation in human head using the 3-D impedance method

A comparative, computational study of the modeling of transcranial magnetic stimulation (TMS) and electroconvulsive therapy (ECT) is presented using a human head model. The magnetic fields from a typical TMS coil of figure-eight type is modeled using the Biot-Savart law. The TMS coil is placed in a position used clinically for treatment of depression. Induced current densities and electric field distributions are calculated in the model using the impedance method. The calculations are made using driving currents and wave forms typical in the clinical setting. The obtained results are compared and contrasted with the corresponding ECT results. In the ECT case, a uniform current density is injected on one side of the head and extracted from the equal area on the opposite side of the head. The area of the injected currents corresponds to the electrode placement used in the clinic. The currents and electric fields, thus, produced within the model are computed using the same three-dimensional impedance method as used for the TMS case. The ECT calculations are made using currents and wave forms typical in the clinic. The electrical tissue properties are obtained from a 4-Cole-Cole model. The numerical results obtained are shown on a two-dimenaional cross section of the model. In this study, we find that the current densities and electric fields in the ECT case are stronger and deeper penetrating than the corresponding TMS quantities but both methods show biologically interesting current levels deep inside the brain.     

A simulation study of the combined thermoelectric extracellular stimulation of the sciatic nerve of the Xenopus laevis: the localized transient heat block

The electrical behavior of the Xenopus laevis nerve fibers was studied when combined electrical (cuff electrodes) and optical (infrared laser, low power sub-5?mW) stimulations are applied. Assuming that the main effect of the laser irradiation on the nerve tissue is the localized temperature increase, this paper analyzes and gives new insights into the function of the combined thermoelectric stimulation on both excitation and blocking of the nerve action potentials (AP). The calculations involve a finite-element model (COMSOL) to represent the electrical properties of the nerve and cuff. Electric-field distribution along the nerve was computed for the given stimulation current profile and imported into a NEURON model, which was built to simulate the electrical behavior of myelinated nerve fiber under extracellular stimulation. The main result of this study of combined thermoelectric stimulation showed that local temperature increase, for the given electric field, can create a transient block of both the generation and propagation of the APs. Some preliminary experimental data in support of this conclusion are also shown.     

Prediction of magnetically induced electric fields in biologicaltissue

There are many potential medical applications in which it is desirable to noninvasively induce electric fields. One such application that serves as the backdrop of this work is that of stimulating neurons in the brain. The magnetic fields necessary must be quite high in magnitude, and fluctuate rapidly in time to induce the internal electric fields necessary for stimulation. Attention is focused on the calculation of the induced electric fields commensurate with rapidly changing magnetic fields in biological tissue. The problem is not a true eddy current problem in that the magnetic fields induced do not influence the source fields. Two techniques are introduced for numerically predicting the fields, each employing a different gauge for the potentials used to represent the electric field. The first method employs a current vector potential (analogous to A in classical magnetic field theory where DEL x A = B) and is best suited to two-dimensional (2-D) models. The second represents the electric field as the sum of a vector plus the gradient of a scalar field; because the vector can be determined quickly using Biot Savart (which for circular coils degenerates to an efficient evaluation employing elliptic integrals), the numerical model is a scalar problem even in the most complicated three dimensional geometry. These two models are solved for the case of a circular current carrying coil near a conducting body with sharp corners.     

Analysis of magnetic stimulation of a concentric axon in a nervebundle

The authors present an analysis of magnetic stimulation of an axon located at the center of a nerve bundle. A three-dimensional axisymmetric volume conductor model is used to determine the transmembrane potential response along an axon due to induced electric fields produced by a toroidal coil. The authors evaluate four such models of an axon located in: (1) an isotropic nerve bundle with no perineurium, (2) an anisotropic nerve bundle without a perineurium, (3) an isotropic nerve bundle surrounded by a perineurium, and (4) an anisotropic nerve bundle surrounded by a perineurium. The transmembrane polarization computed along an axon for the above four models is compared to that for an axon located in an infinite homogenous medium. These calculations indicate that a nerve bundle with no sheath has little effect on the transmembrane potential. However, the presence of a perineurium around the nerve bundle and anisotropy in the bundle significantly affects the shape of the transmembrane response. Therefore, during magnetic stimulation, nerve bundle anisotropy and the presence of perineurium must be taken into account for calculation of stimulus intensities for threshold excitation     

The response of a spherical heart to a uniform electric field: abidomain analysis of cardiac stimulation

A mathematical model describing electrical stimulation of the heart is developed, in which a uniform electric field is applied to a spherical shell of cardiac tissue. The electrical properties of the tissue are characterized using the bidomain model. Analytical expressions for the induced transmembrane potential are derived for the cases of equal anisotropy ratios in the intracellular and interstitial (extracellular) spaces, and no transverse coupling between fibers. Numerical calculations of the transmembrane potential are also performed using realistic electrical conductivities. The model illustrates several mechanisms for polarization of the cell membrane, which can be divided into two categories, depending on if they polarize fibers at the heart surface only or if they polarize fibers both at the surface and within the bulk of the tissue. The latter mechanisms can be classified further according to whether they originate from continuous or discrete properties of cardiac tissue     

Excitation of piezoelectric device through resonant helical coil antenna-like structure

A resonant helical coil antenna-like structure has been explored for wireless excitation of piezoelectric devices. The basic idea behind this wireless excitation of piezoelectric device is electromagnetic resonance along with piezoelectric resonance. The analytical studies reveal that the maximum excitation occurs in the piezoelectric device when the operating frequency of the system coincides with its mechanical resonant frequency. It has been seen that the intensity of piezoelectric stimulation depends on the frequency of operation, air gap, relative position, electric load, and the generated electric field strength by the helical coil antenna-like structure. The analytical results are verified with the measured experimental results, and are found to be in well agreement. By adopting this proposed wireless excitation system the free actuation of the piezoelectric devices can be enabled as opposed to the confined motion for various potential applications.