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     

Source-Field Relationships for Cardiac Generators on the Heart Surface Based on Their Transfer Coefficients

We consider three different types of equivalent sources over a closed surface enclosing all the electrical cardiac generators: the (in situ) potential, the (in situ) normal current density, and the (macroscopic) transmembrane potential on the heart surface. The last equivalent source, which behaves as a double layer, is derived from the bidomain (bisyncytia) model for anisotropic cardiac muscle. This model predicts that if ratios of intracellular to interstitial conductivity along all directions are equal, field potential can be calculated only using surface integrals. The volume integral arising from the tissue anisotropy of cardiac muscle vanishes in that case. For each type of source under study, we give the field potential in a bounded inhomogeneous volume conductor in the form of an integral equation. We also derive the conditions which the lead field (or the transfer coefficients) must satisfy. The in situ potential and normal current are related to the cardiac sources in a complex way, but their lead fields are independent of conductivity of heart muscle, whereas the transmembrane potential is directly involved as a source term, but the lead field depends on the anisotropy of the heart muscle.     

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.     

A simulation of cardiac action currents having curl

A digital simulation of a two-dimensional cardiac slice has been performed. It is stimulated at the center and an action potential propagates outward. An anisotropic bidomain model is used in which fast sodium physiology connects the intracellular and extracellular domains. For cases in which the inner asymmetry (expressed as longitudinal versus transverse electrical conductivity) is greater than the outer asymmetry, a current flow pattern is observed for which there is nonzero curl. Such a result explains recent observations of nonzero B_z magnetic field detected above a slab of tissue in the x-y plane. The current loop producing this field consists of outer domain current in the longitudinal direction flowing around in space and returning at the AP location in the transverse direction in the outer domain and then completing the loop in the longitudinal direction by passing distally through the AP in the inner domain where resistance is extremely low     

Effects of self-consistent electrostatic potential in quantum wells with several quantum confinement levels in high magnetic fields

The effect of self-consistent electrostatic potential on the spectrum of two-dimensional electron states in high magnetic fields is studied under the conditions where more than one quantum confinement subband is filled. The cases of magnetic field directed perpendicular to the quantum well plane and tilted magnetic field are considered. In the case of perpendicular magnetic field it is shown that two or more Landau levels that belong to different quantum confinement subbands can be degenerate in some ranges of concentrations (magnetic fields). The inclination of the magnetic field with respect to the growth direction produces opening of the energy gap between these levels; however, the gap as a function of concentration (magnetic field) remains almost constant in the same range of parameters.     

Electrical stimulation of cardiac tissue: a bidomain model withactive membrane properties

Numerical calculations simulated the response of cardiac muscle to stimulation by electrical current. The bidomain model with unequal anisotropy ratios represented the tissue, and parallel leak and active sodium channels represented the membrane conductance. The speed of the wavefront was faster in the direction parallel to the myocardial fibers than in the direction perpendicular to them. However, for cathodal stimulation well above threshold, the wavefront originated farther from the cathode in the direction perpendicular to the myocardial fibers than in the direction parallel to them, consistent with observations of a dog-bone-shaped virtual cathode made by Wikswo et al., (Circ. Res., vol.68, p.513-30, 1991). The model showed that the virtual cathode size and shape were dependent upon both membrane and tissue conductivities. Increasing the peak Na conductance or reducing the transverse intracellular conductivity accentuated the dog-bone shape, while the opposite change caused the virtual cathode to become more elliptical, with the major axis of the ellipse transverse to the fiber direction. A cathodal stimulus created regions of hyperpolarization that slowed conduction of the wavefront propagating parallel to the fibers. An anodal stimulus evoked a wavefront with a complex shape; activation originated from two depolarized regions 1 to 2 mm from the stimulus site along the fiber direction. The threshold current strength (0.5 ms duration pulse) for a cathodal stimulus was 0.048 mA, and for an anodal stimulus was 0.67 mA. When the model was modified to simulate the effect of electropermeabilization, which may be present, when the transmembrane potential reaches very large values near the stimulating electrode, the authors' qualitative conclusions remained unchanged     

Action Potential Collision in Heart Tissue-Computer Simulations and Tissue Expenrments

The electrical effects of action potential collision were studied using a computer simulation of one-dimensional action potential propagation and tissue experiments from isolated cardiac Purkinje strands and papillary muscles. The effects of collision, when compared to normal one-way propagation, included quantitative changes in all of the measured indexes of action potential upstroke and repolarization. These changes can be attributed to spatiotemporal changes in the net membrane current. Parameter sensitivity and analytic techniques identified five factors which determine the collision-induced decrease in action potential area: conduction velocity, action potential height, cable radius, specific intracellular resistivity, and the specific membrane resistance during action potential repolarization. The simulations demonstrated that collision effects were independent of inhomogeneity in action potential duration, the spatial extent of the collision effects was greater than the passive space constant, and certain simulated abnormal conditions (e. g., discontinuous propagation, ischemic tissue) increased the magnitude of the collision effects. The tissue experiments supported the simulations regarding the changes in action potential configuration directly at and on each side of the collision site. Elevated [K+]0 increased the changes in action potential duration in both tissue preparations. In papillary muscles, collision effects in the transverse direction were confined to a narrower region than collision effects in the longitudinal direction with no difference in the peak magnitude of the changes. Action potential collision is a common occurrence in the heart.     

Recording from a Single Motor Unit During Strong Effort

During strong voluntary effort it is rarely possible to identify the action potentials from single motor units. In large muscles the most selective recordings are obtained with bipolar wire electrodes. To elucidate this experimental finding we have calculated the extracellular field around a single muscle fiber from an intracellular muscle action potential. This model showed that the selectivity of a bipolar electrode is high provided: i) the diameter of the recording surfaces is less than half the diameter of the muscle fibers; ii) the center distance between the recording surfaces is of the same order or smaller than the diameter of the muscle fibers, and when iii) the center-line between the recording surfaces is oriented perpendicular to the direction of the muscle fibers.     

Periodic Conductivity as a Mechanism for Cardiac Stimulation and Defibrillation

This study examines the distribution of the transmembrane potential in the periodic strand of cardiac muscle established by configurations of sources similar to those arising during extracellular stimulation and defibrillation, during intracellular stimulation, and during propagation of action potential. The closed-form solution indicates that during extracellular stimulation with large current and during defibrillation, the periodic component of the transmembrane potential is very important. We postulate that this periodic component causes the depolarization or defibrillation in cardiac muscle, which is different from the depolarization mechanism for a continuous fiber. On the other hand, during propagation and intracellular stimulation, the periodic component only slightly modifies the monotonic decrease of the transmembrane potential, which suggests that the mechanism of propagation in discrete structures may be similar to that of the continuous fiber.     

A beam-splitting reflection grating

A metallic grating made of triangular grooves filled with penetrable material is considered. A plane wave normally incident and polarized perpendicular to the grooves produces three plane waves, one backreflected and two obliquely reflected, for certain ratios between groove dimensions and wavelength. This geometrical optics solution is exact. In particular, if the material filling the grooves is isorefractive to the surrounding medium, then the backreflected field is zero.     

FDTD Analysis of Electromagnetic Reflection by Conductive Plane Covered with Magnetized Inhomogeneous Plasmas

A novel finite-difference time-domain (FDTD) methodology which incorporates both anisotropy and frequency dispersion at the same time is developed for electromagnetic wave propagation in anisotropic magnetoactive plasmas in this paper. The numerical verification of the method are confirmed by computing the reflection and transmission of right-handed/left-handed circularly polarized (RCP/LCP) wave through a magnetized plasma layer, with the direction of propagation parallel to the direction of the biasing field. And, the right-handed / left-handed polarized wave reflection coefficients for electromagnetic signals normally incident upon a conductive plane covered with a layer of magnetized plasma are computed using the new FDTD method. The parabolic electron-number density profile varies only in the direction perpendicular to the plane. The function dependence of reflection coefficients on the number density, collision frequency and external magnetic field is studied.     

Characteristics of the electromagnetic radiation resulting from arbitrarily directed propagation of magnetostatic surface waves in an increasing transverse magnetic field

Radiation of the electromagnetic waves accompanying propagation of a magnetostatic surface wave with nonparallel directions of the phase and group velocities in a ferrite slab magnetized in its plane by a transverse linearly increasing magnetic field is analyzed using the model of accelerated motion of magnetic charges. It is shown that, at each point of the space surrounding the ferrite slab, the shape of the radiation pattern strongly depends on the angle between the direction perpendicular to the direction of propagation of the magnetic field and the direction of the wave vector of the magnetostatic surface wave.     

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.     

Construction and validation of a plunge electrode array for three-dimensional determination of conductivity in the heart.

The heart's response to electrical shock, electrical propagation in sinus rhythm, and the spatiotemporal dynamics of ventricular fibrillation all depend critically on the electrical anisotropy of cardiac tissue. Analysis of the microstructure of the heart predicts that three unique intracellular electrical conductances can be defined at any point in the ventricular wall; however, to date, there has been no experimental confirmation of this concept. We report the design, fabrication, and validation of a novel plunge electrode array capable of addressing this issue. A new technique involving nylon coating of 24G hypodermic needles is performed to achieve nonconductive electrodes that can be combined to give moderate-density multisite intramural measurement of extracellular potential in the heart. Each needle houses 13 silver wires within a total diameter of 0.7 mm, and the combined electrode array gives 137 sites of recording. The ability of the electrode array to accurately assess conductances is validated by mapping the potential field induced by a point current source within baths of saline of varying concentration. A bidomain model of current injection in the heart is then used to test an approximate relationship between the monodomain conductivities measured by the array, and the full set of bidomain conductivities that describe cardiac tissue.     

电镀法制备CoNiMnP永磁薄膜研究

采用电子能谱仪(EDS)与振动样品磁强计(VSM),研究了电流密度及镀液pH值对电镀CoNiMnP永磁薄膜矫顽力的影响。结果表明:CoNiMnP永磁薄膜具有明显的垂直各向异性,且随电流密度和镀液pH值的增大,CoNiMnP薄膜的矫顽力呈先增大后降低的变化;当电流密度为5×10–3A/cm2、镀液pH值为5时,CoNiMnP薄膜垂直方向的矫顽力Hcj达到最大值80kA·m–1。     

Magnetostatic wave propagation in YIG double layers

Calculations are presented for the magnetostatic surface wave propagation characteristics in single-crystal yttrium-iron-garnet (YIG) double layers with arbitrary direction of magnetization. The induced uniaxial magnetic anisotropy field is assumed to be different in the two layers; hence, the magnetization in one layer is aligned at an angle with respect to the magnetization direction in the other layer. The magnetostatic field interactions between layers depend on the angle between the two magnetization directions and on the separation between the two YIG layers. The wave propagation directions and time delays in each layer can be strongly affected by the use of an applied magnetic field and the magnetostatic coupling between the two layers, as well as by the uniaxial anisotropy energy in each layer     

Finite-difference time-domain analysis of gyrotropic media. I.Magnetized plasma

When subjected to a constant magnetic field, both plasmas and ferrites exhibit anisotropic constitutive parameters. For electronic plasmas this anisotropy must be described by using a permittivity tensor in place of the usual scalar permittivity. Each member of this tensor is also very frequency dependent. A finite-difference time-domain formulation which incorporates both anisotropy and frequency dispersion, enabling the wideband transient analysis of magnetoactive plasma, is described. Results are shown for the reflection and transmission through a magnetized plasma layer, with the direction of propagation parallel to the direction of the biasing field. A comparison to frequency-domain analytic results is included     

Deep entry of defibrillating effects into homogeneous cardiac tissue

A simple, idealized mathematical model of cardiac tissue is used to show that electrical fields applied with the intent of defibrillating the heart can be effective deep within (that is, many space constants into) cardiac tissue, even in cases when the tissue is assumed to have completely homogeneous electrical properties. This conclusion is drawn from the analysis of the two eigenmodes present in the model, which have fundamentally different characteristics. One mode decays very rapidly with space, implying that the associated membrane potential is only present with appreciable amplitude within a few space constants of the tissue surface. The other mode, however, is not directly dependent on the value of the space constant, and allows deep penetration of the membrane potential and, by implication, its associated defibrillating effects. For deep membrane potentials to be generated by this mechanism, the intracellular and extracellular resistivity anisotropy ratios must be unequal, as is typically the case in cardiac tissue. The model also predicts that this mechanism is most effective for a given applied field strength when the electrode size and separation, or spatial features of the externally applied field at the heart surface, are characterized by scalelengths that are commensurate with approximately two times the heart wall thickness.     

Magnetostatic Wave Propagation in a Finite YIG-Loaded Rectangular Waveguide

The propagation of magnetostatic waves (MSW) in a waveguide partially loaded with a low-loss ferrite slab is investigated theoretically. The most common low-loss ferrite material used for MSW propagation is epitaxial yttrium iron garnet (YIG). A YIG slab is placed inside and along the guide and not in contact with the sidewalls of the wavegnide. The dc magnetic field is assumed to be parallel to the YIG slab and perpendicular to the direction of propagation. Using the integral equation method, the dispersion relation is found to be an infinitely large determinant equal to zero. Proper truncation of this determinant and numerical analysis to find its roots are carried out in this work. It is seen that in order to obtain high values of group time delay, the YIG slab must be narrow and placed at the bottom of the guide. On the other hand, to maximize the device bandwidth, a narrow YIG slab positioned at the top inside surface of the waveguide is preferred. It is also noticed that there exists a tradeoff between the time delay and the device bandwidth and that maximization of one property leads to a poor value in the other. Thus, some design compromises should be made. It is also observed that the frequency range of operation of the device can be adjusted by an external magnetic bias field. This property of tuning the device to operate in any frequency range adds an extra dimension of flexibility to the operation and also to the design of these devices.