Discrete versus syncytial tissue behavior in a model of cardiacstimulation. II. Results of simulation

For pt. I see ibid., vol. 43, no. 12, p. 1129-40 (1996). The research presented here combines mathematical modeling and computer simulation in developing a new model of the membrane polarization induced in the myocardium by the applied electric field. Employing this new model termed the “periodic” bidomain model, the steady-state distribution of the transmembrane potential is calculated on a slice of cardiac tissue composed of abutting myocytes and subjected to two point-source extracellular current stimuli. The goal of this study is to examine the relative contribution of cellular discreteness and macroscopic syncytial tissue behavior in the mechanism by which the applied electric field alters the transmembrane potential in cardiac muscle. The results showed the existence of oscillatory changes in the transmembrane potential at cell ends owing to the local resistive inhomogeneities (gap-junctions). This low-magnitude sawtooth component in the transmembrane potential is superimposed over large-scale transmembrane potential excursions associated with the syncytial (collective) fiber behavior. The character of the cardiac response to stimulation is determined primarily by the large-scale syncytial tissue behavior. The sawtooth contributes to the overall tissue response only in regions where the large-scale transmembrane potential component is small     

A dynamic action potential model analysis of shock-inducedaftereffects in ventricular muscle by reversible breakdown of cellmembrane

To elucidate the subcellular mechanism underlying the aftereffects of high-intensity dc shocks, a small pore, which mimics reversible breakdown of the cell membrane (electroporation), was incorporated into the phase-2 Luo-Rudy (L-R) model of ventricular action potentials. The pore size was set to occupy 0.15%-4.25% of the total cell membrane during the 10-ms shock. The pore was assumed to decrease after the shock exponentially with a time constant of 100-1,400 ms to simulate resealing process. In normal myocytes, the pore formation results in a delay of repolarization of the shocked action potential, which is followed by prolonged depolarization and oscillation of membrane potential like early afterdepolarization (EAD). Time- and voltage-dependent changes in the delayed rectifier K+ currents (IKr, IKs) in combination with those of L-type Ca2+ current (ICa,(L)) and ion flux through the pore (I(pore)) are responsible for the potential changes. Spontaneous excitation from the oscillation depends on activation of ICa,(L). In myocytes overloaded with Na+ and Ca2+ secondary to 90% inhibition of Na+-K+ pump, the pore formation results in a delay of repolarization of the shocked action potential, which is followed by slower cyclic depolarization in response to spontaneous release of Ca2+ from the sarcoplasmic reticulum (SR). This delayed afterdepolarization-type oscillation is abolished by complete block of Ca2+ release from the SR. These findings suggest that high-intensity electric field application will cause arrhythmogenic responses through a transient rupture of sarcolemma with different subcellular events in ventricular cells under normal and pathological conditions.     

Propagation down a chain of excitable cells by electric field interactions in the junctional clefts: effect of variation in extracellular resistances, including a "sucrose gap" simulation

An electric field model for electrical transfer of excitation between contiguous excitable cells has been further developed by expanding the model to a chain of six cells, and examining the effect of changing the external resistances on propagation velocity. In this model, there is no requirement for low-resistance connections between the cells, and the major assumption is that the pre-and postjunctional membranes are ordinary excitable membranes. The electric field that develops in the narrow junctional cleft between contiguous cells during the rising phase of the action potential in the prejunctional membrane acts to depolarize the postjunctional membrane to threshold. Propagation occurred down the entire chain of cells at a constant velocity of about 17.1 cm/s. Raising the extracellular resistances (ROL and ROR) along the entire chain up to fourfold slowed propagation only slightly. However, when the radial cleft resistance (RJC) was varied concomitantly, then there was a marked slowing of propagation velocity, e.g., to 3.3 cm/s in 4.0 X resistance. There was an optimal RJC value for peak velocity. The lowering of ROL and ROR up to eightfold has almost no effect on velocity. Raising RJC, ROL, and ROR for the middle two cells, up to 3 times the normal value slowed propagation in the "sucrose-gap" region; raising the resistance to 4 times or higher blocked propagation. Hence, the electric field model allows successful transmission of excitation down a long chain of cells, not connected by low-resistance tunnels, at a constant velocity, and propagation velocity is dependent particularly on RJC.     

Frequency domain modeling of volume conduction of single musclefiber action potentials

An inhomogeneous frequency-dependent model of volume conduction in skeletal tissue is used to calculate a transfer function between injected membrane current and extracellular action potential in the frequency domain. This model accounts for tissue structure and microscopic electrical parameters (intra- and extracellular conductivities and membrane impedance) and represents the volume conductor by an extensive electrical network. Results obtained with the model are compared with those of a conventional homogeneous volume conductor model. The comparison shows a significant influence of tissue structure and microscopic electrical parameters close to the source. As a result, the transfer function close to the source is less sensitive to the frequency content and conduction velocity of the membrane current, compared with results of the homogeneous volume conductor approach     

The effect of morphological interdigitation on field couplingbetween smooth muscle cells

The electrical control activity (ECA) in the distal stomach, small intestine, and colon has been modeled by populations of coupled nonlinear oscillators. Coupling has traditionally been explained through gap junctions, but gap junctions alone are inadequate, as they are not always present or cannot account for the observed behavior. Coupling through extracellular electric fields has been proposed as another coupling path which may work instead of, or in conjunction with, gap junctions. A morphological structure, the interdigitation, is studied for its effect on fields produced by a spherical cell. Using boundary element methods, the potential produced by a cell and the transmembrane potential induced in an adjacent cell are considered. Computer simulation results indicate that the presence of an interdigitation between two neighboring cells produces a 60% increase in extracellular potential and a 50% increase in induced transmembrane voltage. The interdigitation length is the most important factor, with radius playing a very small part in determining peak values of potential and voltage. These interdigitation fields may be of appreciable magnitude with regard to coupling. Also, the upstroke phase of the ECA can play a major role in intercellular communication     

Reflection from a periodically perforated plane using a subsectional current approximation

The scattering from a zero thickness plane having finite sheet resistance and perforated periodically with apertures is calculated for arbitrary plane wave illumination. The surface current density within the unit cell is approximated by a finite number of current elements having rooftop spatial dependence. The transverse electric field is expressed in terms of the current, and the electric field boundary condition is satisfied in an integral sense over the conductor, generating a finite dimension matrix equation whose solution is the current density. Since the conductor shape is defined through the locations of subsectional current elements, arbitrary shaped apertures can be handled. The reflection coefficient and current distribution are calculated for square apertures in both perfectly conducting and resistive sheets, and for cross-shaped apertures. Finite resistivity is shown to cause the magnitude of the transverse magnetic (TM) reflection coefficient to decrease more rapidly and its phase to decrease less rapidly, as the angle of incidence approaches glancing. Through detailed plots of the current density, the current crowding around the apertures is made clearly evident.     

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.     

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     

不完全屏蔽腔体辐射耦合电场增强效应防护方法

针对武器装备机箱内部电磁辐射防护的技术需要,从不完全屏蔽腔体辐射耦合电场增强效应形成机理出发,对孔缝耦合及贯通导体耦合导致的屏蔽腔体内部局部电场增强效应防护方法进行了研究.仿真计算了屏蔽腔体内部加载吸波材料、腔体分区隔断以及贯通导体安装金属导管等防护方法对不完全屏蔽腔体电磁耦合的影响,研究结果表明:在屏蔽腔体内部加载吸波材料能够有效降低由于腔体谐振产生的电场增强效应,相同的吸波材料放置在强场位置防护效率会更高;采用分区隔断的屏蔽腔体能够提高腔体的谐振频率,大幅降低腔体内大部分位置的电磁耦合能力;贯通导体通过金属导管进入屏蔽腔体能够有效降低贯通导体的电磁耦合能力,削弱屏蔽腔体内部的电场增强效应,屏蔽腔体内部及外部的金属导管长度越长,其防护效果越明显.     

Effective boundary conditions for syncytial tissues

This study derives effective boundary conditions for potentials and currents on the interface between syncytial tissue and a surrounding volume conductor. The derivation is based on an idealized representation of the syncytium as a network of interconnected cells arranged periodically in space. The microscopic model of an interface assumes that the extracellular fluid is in direct contact with the outside volume conductor and that the inside of the cells is separated from the outside by the membrane. From this microscopic model, a homogenization process and boundary layer analysis derive effective boundary conditions applicable to macroscopic volume-averaged potentials. These effective boundary conditions call for the extracellular potential and current density to be continuous with the potential and current density in the volume conductor, and for the intracellular current to vanish. Hence, the long-debated appropriate boundary conditions for the bidomain model are established     

Effects of metallization lay-out on turn-off failure of modern power bipolar transistors

A new simulation tool is presented which is able to describe the behaviour of the modern cellular power BJTs. It is based on SPICE circuit simulation of a rather complex circuit where each cell is described by a single transistor and all of the cells are interconnected by a resistive network. The tool is able to predict current and electric field distribution over the chip during device turn-off. The effects of metallization lay-out on failure of cellular BJTs are studied. It is demonstrated both experimentally and numerically, for the first time, that current crowding over the chip of these devices is related to different storage times of each cell and not to the emitter depolarization as it is usually assumed for devices of traditional design.     

Effects of the propagation velocity of a surface depolarizationwave on the extracellular potential of an excitable cell

Extracellular electric fields have been proposed as a mechanism for electrical coupling between excitable cells. This study deals with the extracellular potential produced by an isolated excitable spherical cell due to a traveling depolarization wave on the cell's surface. Both uniform and nonuniform propagation velocity profiles are considered. Using boundary element methods, the extracellular potential was computed. The polarity of the extracellular potential was found to be space-dependent. The peak extracellular potential increased when a) the propagation velocity decreased, b) the rise time of the depolarization decreased, and c) the extracellular resistivity increased     

Moving dipole inverse solutions using realistic torso models

A noniterative numerical solution for the potentials on the surfaces of a piecewise homogeneous volume conductor due to a current dipole is described. This forward solution has been used in electric and magnetic single moving dipole (SMD) inverse solutions that employ a torso volume conductor model whose boundaries are specified numerically. Thus, the volume conductor model used by the inverse solutions need not be limited to simple geometric shapes; torso models of realistic shape can be used.     

Construction of the Dirichlet to Neumann Boundary Operator for Triangles and Applications in the Analysis of Polygonal Conductors

This paper introduces a fast and accurate method to investigate the broadband inductive and resistive behavior of conductors with a nonrectangular cross section. The presented iterative combined waveguide mode (ICWM) algorithm leads to an expansion of the longitudinal electric field inside a triangle using a combination of parallel-plate waveguide modes in three directions, each perpendicular to one of the triangle sides. This expansion is used to calculate the triangle's Dirichlet to Neumann boundary operator. Subsequently, any polygonal conductor can be modeled as a combination of triangles. The method is especially useful to investigate current crowding effects near sharp conductor corners. In a number of numerical examples, the accuracy of the ICWM algorithm is investigated, and the method is applied to some polygonal conductor configurations.      

The activating function for magnetic stimulation derived from athree-dimensional volume conductor model

A three-dimensional volume conductor model of magnetic stimulation is proposed that relates transmembrane potential of an axon to the induced electric field in a uniform volume conductor. This model validates assumptions used to derive a one-dimensional cable model of magnetic stimulation (Roth & Basser, IEEE Trans. Biomed. Eng., vol. 37, pp. 588-597, 1990) of unmyelinated axons. The three-dimensional volume conductor model reduces to this one-dimensional cable equation forced by the activating function, -delta EzA/delta z.     

Scattering cross section of composite conducting and lossydielectric bodies

An E-field integral equation for the computation of the radar cross section of finite composite conducting and lossy inhomogeneous dielectric bodies is presented. The equivalence principle is used to replace all conducting bodies by an equivalent surface electric current, and the dielectric is replaced by an equivalent volume polarization current. The respective boundary conditions on the dielectric and the conductor are utilized to solve for the electric current on the entire structure. Also the augmented conjugate gradient method is presented for the solution of extremely large systems of equations that arise in the present problem. Finally, typical results are presented to illustrate the potential of this method     

Full-Periphery Surface Impedance for Skin-Effect Approximation in Electric Field Integral Equation

A new surface impedance model for RL-extraction in lossy two-dimensional (2-D) interconnects of rectangular cross section is presented. The model is derived directly from the volumetric electric field integral equation under the approximation of the unknown volumetric current density as a product of the exponential factor describing the skin-effect and the unknown surface current density on the conductor's periphery. By proper accounting for the coupling between the boundary elements situated on the top and bottom surfaces of conductor with the elements located on the side-walls, the model maintains accuracy from dc to multi-GHz frequencies as well as for conductors with both large and small thickness/width ratios.     

Distribution of electrical stimulation current in a planar multilayer anisotropic tissue.

This study analytically addresses the problem of neuromuscular electrical stimulation for a planar, multilayer, anisotropic model of a physiological tissue (referred to as volume conductor). Both conductivity and permittivity of the volume conductor are considered, including dispersive properties. The analytical solution is obtained in the 2-D Fourier transform domain, transforming in the planes parallel to the volume conductor surface. The model is efficient in terms of computational cost, as the solution is analytical (only numerical Fourier inversion is needed). It provides the current distribution in a physiological tissue induced by an electrical current delivered at the skin surface. Three representative examples of application of the model are considered. 1) The simulation of stimulation artefact during transcutaneous electrical stimulation and EMG detection. Only the effect of the volume conductor is considered, neglecting the other sources of artefact (such as the capacitive coupling between the stimulating and recording electrodes). 2) The simulation of the electrical current distribution within the muscle and the low-pass filter effect of the volume conductor on sinusoidal stimulation currents with different stimulation frequencies. 3) The estimation of the amplitude modulated current distribution within the muscle for interferential stimulation. The model is devoted to the simulation of neuromuscular stimulation, but the same method could be applied in other fields in which the estimation of the electrical current distribution in a medium induced by the injection of a current from the boundary of the medium is of interest.     

A model analysis of aftereffects of high-intensity DC stimulationon action potential of ventricular muscle

The mechanism for aftereffects of high-intensity dc stimulation on ventricular muscle was studied by using Beeler-Reuter's action potential model. A leak conductance (G_pore maximal value from 40 to 80 μS for 1 cm² of membrane), which mimics reversible dielectric breakdown of the cell membrane by the shock, was incorporated into the model. To simulate resealing process, G_pore was assumed to decrease after the shock exponentially at a time constant (τ_pore) of 5-50 s. The simulation results are qualitatively consistent with the authors' experimental observations in guinea pig papillary muscle (Amer. J. Physiol., vol. 267, p. H248-58, 1994); they include prolonged depolarization, diastolic depolarization or oscillation of membrane potential leading to a single or multiple spontaneous excitation. The phase-independence and shock intensity-dependence can also be reproduced. Analysis of current components has revealed that: (1) a large inward leak current (l_leak) is responsible for the prolonged depolarization (2) time-dependent decay of outward current (I_X1) in combination with I_leak and slow inward current (I_s) results in diastolic depolarization or oscillation of membrane potential; (3) spontaneous excitation depends on an activation of I_s. These findings support the authors' hypothesis that strong shocks (>15 V/cm) will produce abnormal arrhythmogenic responses in ventricular muscle through a transient rupture of sarcolemmal membrane