The effect of anisotropy on the potential distribution in biological tissue and its impact on nerve excitation simulations

We present a finite difference solution of the potential distribution associated with electrical current stimulation in an anisotropic in-homogeneous tissue environment and compare it to the isotropic case. The results demonstrate that there can be significant errors associated with the assumption of isotropic tissue properties in calculating the potential distribution along an axon in nerve excitation simulations. These errors can have a significant impact on predicted nerve fiber recruitment patterns when evaluating the efficacy of specific surface or intramuscular stimulus electrode configurations. The results of this study also suggest when a more comprehensive tissue model should be implemented in an electrode design study. Simulation results indicate that the isotropy assumption is worst under bipolar electrode stimulation as opposed to monopolar stimulation and that the bipolar error increases as the distance between electrodes decreases. In light of these results, it is concluded that in order to avoid large errors in the calculated potential distribution along an axon, the isotropy assumption should only be used when the transverse depth from the electrode to the nerve is relatively small.     

Modeling the field distribution in deep brain stimulation: the influence of anisotropy of brain tissue

The neurosurgical method of deep brain stimulation (DBS) is used to treat symptoms of movement disorders like Parkinson's disease by implanting stimulation electrodes in deep brain areas. The aim of this study was to examine the field distribution in DBS and the role of heterogeneous and anisotropic material properties in the brain areas where stimulation is applied. Finite element models of the human brain were developed comprising tissue heterogeneity and anisotropy. The tissue data were derived from averaged magnetic resonance imaging and diffusion tensor imaging datasets. Unilateral stimulation of the subthalamic nucleus (STN) was computed using an accurate model of an electrode used in clinical treatment of DBS extended with an encapsulation layer around the electrode body. Computations of anisotropic and isotropic brain models, which consider resistive tissue properties for unipolar and bipolar electrode configurations, were carried out. Electrode position was varied within an area around the stimulation center. Results have shown a deviation of 2% between anisotropic and isotropic field distributions in the vicinity of the STN. The sensitivity of this deviation referring to the electrode position remained small, but increased when the electrode position approached areas of high anisotropy.     

Magnetically induced currents in the canine heart: a finite elementstudy

A moderately detailed three-dimensional (3-D) finite element model of the conductive anatomy of a canine thorax was used to determine the fields and currents induced by a time-varying magnetic field that has been shown to cause irregular heart beats in canines. The 3-D finite element model of the canine thorax was constructed from CT scans and includes seven isotropic tissue conductivities and the anisotropic conductivity of skeletal muscle. The authors use this model to estimate the stimulation threshold associated with stimulation of the heart by the time-varying magnetic field of a figure-eight coil. Variants of the thoracic model were also constructed to examine the sensitivity of model results to variations in model size, shape, and conductive inhomogeneity and anisotropy. The authors' results show that myocardial fields were only mildly sensitive to thoracic size. However, model shape and conductive inhomogeneity and anisotropy substantially influenced the magnitude and distribution of myocardial fields and currents. The authors' results suggest that an induced peak field magnitude of ≈1 V/cm is required to stimulate the heart with the magnetic excitation simulated in this study     

Magnetic Measurements of Action Currents in a Single Nerve Axon: A Core-Conductor Model

Measurements have been performed on the medial giant axon of the crayfish in which both microelectrode recordings of the transmembrane action potential and magnetic recordings of the axial, intracellular action current were obtained at a single location along the nerve. This approach is unique because the combination of electric and magnetic techniques allows for independent measurements of transmembrane potential and intracellular current without assumptions regarding membrane capacitance or axonal resistances. The availability of both types of data recorded at a single location on the axon allows the core-conductor model to be solved explicitly for the internal axial resistance of the axon, the membrane capacitance, and the membrane conduction current density without the need to make a series of subthreshold measurements of passive cable properties. The extracellular magnetic measurements can be used both to determine the transmembrane action potential without the need to penetrate the nerve membrane with a microelectrode, and to obtain approximate values for the sodium and potassium peak conductances.     

The electric field induced in the brain by magnetic stimulation: a 3-D finite-element analysis of the effect of tissue heterogeneity and anisotropy

We investigate the effect of tissue heterogeneity and anisotropy on the electric field and current density distribution induced in the brain during magnetic stimulation. Validation of the finite-element (FE) calculations in a homogeneous isotropic sphere showed that the magnitude of the total electric field can be calculated to within an error of approximately 5% in the region of interest, even in the presence of a significant surface charge contribution. We used a high conductivity inclusion within a sphere of lower conductivity to simulate a lesion due to an infarct. Its effect is to increase the electric field induced in the surrounding low conductivity region. This boost is greatest in the vicinity of interfaces that lie perpendicular to the current flow. For physiological values of the conductivity distribution, it can reach a factor of 1.6 and extend many millimeters from the interface. We also show that anisotropy can significantly alter the electric field and current density distributions. Either heterogeneity or anisotropy can introduce a radial electric field component, not present in a homogeneous isotropic conductor. Heterogeneity and anisotropy are predicted to significantly affect the distribution of the electric field induced in the brain. It is, therefore, expected that anatomically faithful FE models of individual brains which incorporate conductivity tensor data derived from diffusion tensor measurements, will provide a better understanding of the location of possible stimulation sites in the brain.     

Influence of anisotropic conductivity on EEG source reconstruction: investigations in a rabbit model

The aim of our work was to quantify the influence of white matter anisotropic conductivity information on electroencephalography (EEG) source reconstruction. We performed this quantification in a rabbit head using both simulations and source localization based on invasive measurements. In vivo anisotropic (tensorial) conductivity information was obtained from magnetic resonance diffusion tensor imaging and included into a high-resolution finite-element model. When neglecting anisotropy in the simulations, we found a shift in source location of up to 1.3 mm with a mean value of 0.3 mm. The averaged orientational deviation was 10 degree and the mean magnitude error of the dipole was 29%. Source localization of the first cortical components after median and tibial nerve stimulation resulted in anatomically verified dipole positions with no significant anisotropy effect. Our results indicate that the expected average source localization error due to anisotropic white matter conductivity is within the principal accuracy limits of current inverse procedures. However, larger localization errors might occur in certain cases. In contrast, dipole orientation and dipole strength are influenced significantly by the anisotropy. We conclude that the inclusion of tissue anisotropy information improves source estimation procedures.     

Calculation of electric fields in a multiple cylindrical volume conductor induced by magnetic coils

A method is presented for calculating the electric field, that is induced in a cylindrical volume conductor by an alternating electrical current through a magnetic coil of arbitrary shape and position. The volume conductor is modeled as a set of concentric, infinitely long, homogeneous cylinders embedded in an outer space that extends to infinity. An analytic expression of the primary electric field induced by the magnetic coil, assuming quasi-static conditions, is combined with the analytic solution of the induced electric scalar potential due to the inhomogeneities of the volume conductor at the cylindrical interfaces. The latter is obtained by the method of separation of variables based on expansion with modified Bessel functions. Numerical results are presented for the case of two cylinders representing a nerve bundle with perineurium. An active cable model of a myelinated nerve fiber is included, and the effect of the nerve fiber's undulation is shown.     

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     

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

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

Electrical stimulation of cardiac tissue by a bipolar electrode ina conductive bath

A three-dimensional (3-D) computer simulation of the electrical stimulation of passive cardiac tissue from a bipolar electrode placed within a conductive bath is presented. Through the bidomain model, the syncytial and anisotropic properties of cardiac tissue are taken into account; tissues with equal anisotropy and no transverse coupling are also considered. The membrane is represented by a capacitor and passive resistor in parallel. Located within an isotropic bath, the bipolar electrode is oriented either perpendicular or parallel to the tissue surface. For anisotropic tissue with a small cathode-tissue separation, the tissue surface is highly depolarized under the cathode with the depolarization persisting a considerable distance from the electrode in the transverse fiber direction. Adjacent to this region in the longitudinal direction, areas of hyperpolarization exist. At large distances from the cathode, the tissue surface is hyperpolarized in all directions when the electrode axis is perpendicular to the tissue. In the parallel case, surface depolarization creates buried regions of hyperpolarization. For the perpendicular configuration, the ratio of the steady-state maximum depolarization to steady-state maximum hyperpolarization, an estimate of the ratio of anodal to cathodal threshold, decreases rapidly with increasing cathode-tissue separation. In the parallel case, the depth of the conductive bath significantly affected the transmembrane potential distribution in the tissue. The use of a 3-D model more realistically simulates real-life electrical stimulation (such as stimulation with an implantable pacemaker) and provides insight into the effect of the volume conductor adjacent to the tissue     

Distortion of an electromagnetic pulse carrier propagating in an anisotropic homogeneous plasma

Transient signal propagation in anisotropic, homogeneous cold plasmas is considered. The anisotropy is caused by an external longitudinal magnetic field. Using the saddlepoint method of integration, an approximate expression for the electric field is obtained. The envelope of a distorted rectangular pulse carrier is plotted for various magnetic fields, including the isotropic case.     

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

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

Three-dimensional segmentation and interpolation of magnetic resonance brain images

The authors propose a method for the 3-D reconstruction of the brain from anisotropic magnetic resonance imaging (MRI) brain data. The method essentially consists in two original algorithms both for segmentation and for interpolation of the MRI data. The segmentation process is performed in three steps. A gray level thresholding of the white and gray matter tissue is performed on the brain MR raw data. A global white matter segmentation is automatically performed with a global 3-D connectivity algorithm which takes into account the anisotropy of the MRI voxel. The gray matter is segmented with a local 3-D connectivity algorithm. Mathematical morphology tools are used to interpolate slices. The whole process gives an isotropic binary representation of both gray and white matter which are available for 3-D surface rendering. The power and practicality of this method have been tested on four brain datasets. The segmentation algorithm favorably compares to a manual one. The interpolation algorithm was compared to the shaped-based method both quantitatively and qualitatively.     

Prediction of neural excitation during magnetic stimulation using passive cable models

A method for predicting neural excitation during magnetic stimulation using passive cable models has been developed. This method uses the information of the threshold capacitor voltage for magnetic stimulation coils to determine the equivalent excitation thresholds for the passive transient (PT) and passive steady-state (PSS) cable models as well as for the activating function. The threshold values for the PT, PSS models, and the activating function vary only with the pulsewidth of the stimulus for a variety of coils at different locations and orientations. Furthermore, the excitation threshold for the PSS model is also independent of axon diameter and best fitted to a simple mathematical function. By comparing the transmembrane potential of the PSS model with the corresponding threshold, the prediction of excitation during magnetic stimulation can be made. Similarly, it is also possible to predict excitation using the PT model and the activating function with the corresponding thresholds provided. By taking advantage of the weighted pulsewidth, this method can even predict the excitation for stimuli with various waveforms, greatly simplifying the determination of neural excitation for magnetic stimulation.     

Influence of a Constant Magnetic Field on Nervous Tissues: I. Nerve Conduction Velocity Studies

A study of the biomagnetic response of the circumesophageal connective of the lobster has been undertaken in order to investigate the influence of a constant magnetic field upon (i) the nerve impulse conduction velocity of the entire trunk, and (ii) the membrane potentials and the transmembrane currents of the giant axon of this nerve, under voltage-clamp conditions using the double sucrose gap technique. In this first paper, the results of the conduction velocity experiments are reported. They show that there is no significant effect of a 1.2 T magnetic field upon conduction velocity of the isolated nerve of the lobster, in either parallel or perpendicular configurations with respect to the field. The results of the biomagnetic experiments on the giant axon of the lobster under current-clamp and voltage-clamp conditions will be reported later.     

Simulation of propagation along a cylindrical bundle of cardiactissue. II. Results of simulation

Previous evaluations of the cylindrical bidomain model of a bundle of cardiac tissue, have been obtained by using an analytic function for the transmembrane potential and assuming the activating wavefront through the bundle cross section is planar. In this paper, nonlinear membrane kinetics are introduced into the bidomain membrane and equal anisotropy ratios are assumed, permitting the transmembrane potential to be computed and its behavior examined at different depths in the bundle and for different values of conductivity and bundle diameters. In contrast with single fiber models, the bundle model reveals that the shape of the action potential is influenced by tissue resistivities. In addition, the steady-state activation wavefront through the cross-section perpendicular to the long axis of the bundle is not planar and propagates with a velocity that lies between that of a single fiber in an unbounded volume and a single fiber in a restricted extracellular space. In general, the bundle model is shown to be significantly better than the classical single fiber model in describing the behavior of real cardiac tissue.     

Formulation générale de la méthode spectrale pour l’étude de structures planaires multicouches à tenseurs de permittivité et de perméabilité diagonaux

The materials usually used in microwave integrated circuits are often assumed isotropic. However, in certain cases anisotropy is introduced unintentionally during the manufacturing process, or deliberately in order to obtain non reciprocal devices, radar absorbers, and so on., or serves to improve circuit performances. In several cases, neglecting the anisotropy of certain substrates induces errors in integrated-circuit design. Hence the characteristics of planar structures containing anisotropic layers must be accurately described in order to secure the circuit design and improve the CAD models. On the other hand, the measurement of dielectric or magnetic anisotropy of materials at microwaves frequencies is of great interest for several applications and as such the planar structures on anisotropic layers can be used in this domain. Several methods enable the propagation characteristics to be calculated for a large number of structures such as microstrip, coplanar waveguides, and slotlines. The spectral domain technique (SDT) is one of the fastest. For anisotropic substrates, the formulation of the spectral domain method can be very difficult and it depends on the form of the relative tensors z and t. The main difficulty in considering anisotropic layers is to obtain the Green matrix of the spectral domain technique. The aim of this paper is to extent the SDT for 'planar lines on anisotropic (electric and/or magnetic) substrates. A generalized formulation for all diagonal and t is presented and used to calculate the propagation parameters for several planar lines.     

Correspondence between the location of evoked potential generators and sites of maximal sensitivity to stimulation

The potential recorded by a set of electrodes as an action potential traverses a small axonal segment is proportional to the transmembrane potential produced during stimulation of that axon segment by the same set of recording electrodes, under certain circumstances. First, the membrane must have a constant thickness which is so small that the difference between the surface area of the inner and outer surfaces is minimal. Second, all media must be linear. Third, there must be a monotonically increasing relation between the mean transmembrane potential induced by a stimulus and the maximum transmembrane potential. Fourth, as each axon segment depolarizes, the transmembrane current and change in membrane potential during this time are same. This principle remains true for magnetic stimulation and recording as long as currents generated at the boundaries between regions of differing conductivity outside the axon contribute minimally to the field at the axon. This allows the identification of the point at which an action potential generates a maximal extracellular potential as the point that is stimulated with the lowest threshold.     

On Radial Transmission Line Representations of Electromagnetic Fields in an Anisotropic Moving Medium

The problems of electromagnetic waves in moving isotropic or uniaxial mediums have been dealt with by numerous authors. Chawla and Unz considered the fields in a moving anisotropic plasma, and Chen and Cheng analyzed waves in an isotropic plasma in a moving dielectric medium. In this note we consider electromagnetic fields in a moving anisotropic medium and propose the network formulation of electromagnetic fields in the moving medium in the radial cylindrical coordinate. The method is an extension of the transmission line representation of electron beams on infinite magnetic fields. We can apply these results to the cases of any magnetic field intensity and, further, solve the complex problems for a stationary anisotropic plasma by a similar method.