Spatial and temporal frequency-dependent conductivities in volume-conduction calculations for skeletal muscle

Electrical impedance myography: A critical review and outlook

Electrical impedance myography (EIM) technology is finding application in neuromuscular disease research as a tool to assess muscle health. Correlations between EIM outcomes, functional, imaging and histological data have been established in a variety of neuromuscular disorders; however, an analytical discussion of EIM is lacking. This review presents an explanation for clinicians and others who are applying EIM and interpreting impedance outcomes. The background of EIM is presented, including the relation between EIM, volume conduction properties, tissue structure, electrode configuration and conductor volume. Also discussed are technical considerations to guide the reader to critically evaluate EIM and understand its limitations and strengths.

Sensor spacing affects the tissue impedance spectra of rabbit ventricular epicardium

This study was designed to test the hypothesis that a complex composite impedance spectra develops when stimulation and recording of cardiac muscle with sufficiently fine spatial resolution in a four-electrode configuration is used. With traditional (millimeter scale) separations, the ratio between the recorded interstitial central potential difference and total supplied interstitial current is constant at all frequencies. This occurs because the fraction of supplied current that redistributes to the intracellular compartment depends on effective membrane resistance between electrodes, which is low, to a much greater extent than effective membrane capacitance. The spectra should therefore change with finer separations at which effective membrane resistance increases, as supplied current will remain primarily interstitial at lower frequencies and redistribute between compartments at higher frequencies. To test this hypothesis, we built arrays with sensors separated (d) by 804 μm, 452 μm, and 252 μm; positioned those arrays across myocyte axes on rabbit ventricular epicardium; and resolved spectra in terms of resistivity (ρt) and reactivity (χt) over the 10 Hz to 4,000 Hz range. With all separations, we measured comparable spectra with predictions from passive membrane simulations that used a three-dimensional structural framework in which intracellular, interstitial, and membrane properties were prescribed based on the limited data available from the literature. At the finest separation, we found mean ρt at 100 Hz and 4,000 Hz that lowered from 395 Ω-cm to 236 Ω-cm, respectively, with maximal mean χt of 160 Ω-cm. This experimental confirmation of spectra development in whole heart experiments is important because such development is central to achieve measurements of intracellular and interstitial passive electrical properties in cardiac electrophysiological experiments using only interstitial access.

A structural framework for interpretation of four-electrode microimpedance spectra in cardiac tissue

Renewed interest in the four-electrode method for identification of passive electrical properties in cardiac tissue has been sparked by a recognition that measurements made with sensors in close proximity are frequency dependent. Therefore, resolution of four-electrode microimpedance spectra (4EMS) may provide an opportunity for routine identification of passive electrical properties for the interstitial and intracellular compartments using only interstitial access. The present study documents a structural framework in which the tissue resistivity (ρt) and reactivity (xt) that comprise spectra are computed using interstitial and intracellular microimpedance distributions that account for differences in compartment size, anisotropic electrical properties in each compartment and electrode separations. We used this framework to consider 4EMS development with relatively wide (d=1 mm) and fine (d=250 μm) electrode separations and sensors oriented along myocyte axes, across myocyte axes and intermediate between those axes.

Magnetic measurements of peripheral nerve function using a neuromagnetic current probe

The progress made during the last three decades in mathematical modeling and technology development for the recording of magnetic fields associated with cellular current flow in biological tissues has provided a means of examining action currents more accurately than that of using traditional electrical recordings. It is well known to the biomedical research community that the room-temperature miniature toroidal pickup coil called the neuromagnetic current probe can be employed to measure biologically generated magnetic fields in nerve and muscle fibers. In contrast to the magnetic resonance imaging technique, which relies on the interaction between an externally applied magnetic field and the magnetic properties of individual atomic nuclei, this device, along with its room-temperature, low-noise amplifier, can detect currents in the nano-Ampere range. The recorded magnetic signals using neuromagnetic current probes are relatively insensitive to muscle movement since these probes are not directly connected to the tissue, and distortions of the recorded data due to changes in the electrochemical interface between the probes and the tissue are minimal. Contrary to the methods used in electric recordings, these probes can be employed to measure action currents of tissues while they are lying in their own natural settings or in saline baths, thereby reducing the risk associated with elevating and drying the tissue in the air during experiments. This review primarily describes the investigations performed on peripheral nerves using the neuromagnetic current probe. Since there are relatively few publications on these topics, a comprehensive review of the field is given. First, magnetic field measurements of isolated nerve axons and muscle fibers are described. One of the important applications of the neuromagnetic current probe to the intraoperative assessment of damaged and reconstructed nerve bundles is summarized. The magnetic signals of crushed nerve axons and the determination of the conduction velocity distribution of nerve bundles are also reviewed. Finally, the capabilities and limitations of the probe and the magnetic recordings are discussed.

Theoretical modeling for radiofrequency ablation: state-of-the-art and challenges for the future

Radiofrequency ablation is an interventional technique that in recent years has come to be employed in very different medical fields, such as the elimination of cardiac arrhythmias or the destruction of tumors in different locations. In order to investigate and develop new techniques, and also to improve those currently employed, theoretical models and computer simulations are a powerful tool since they provide vital information on the electrical and thermal behavior of ablation rapidly and at low cost. In the future they could even help to plan individual treatment for each patient. This review analyzes the state-of-the-art in theoretical modeling as applied to the study of radiofrequency ablation techniques. Firstly, it describes the most important issues involved in this methodology, including the experimental validation. Secondly, it points out the present limitations, especially those related to the lack of an accurate characterization of the biological tissues. After analyzing the current and future benefits of this technique it finally suggests future lines and trends in the research of this area.

Cardiac microimpedance measurement in two-dimensional models using multisite interstitial stimulation

We analyzed central interstitial potential differences during multisite stimulation to assess the feasibility of using those recordings to measure cardiac microimpedances in multidimensional preparations. Because interstitial current injected and removed using electrodes with different proximities allows modulation of the portion of current crossing the membrane, we hypothesized that multisite interstitial stimulation would give rise to central interstitial potential differences that depend on intracellular and interstitial microimpedances, allowing measurement of those microimpedances. Simulations of multisite stimulation with fine and wide spacing in two-dimensional models that included dynamic membrane equations for guinea pig ventricular myocytes were performed to generate test data ( partial differentialphio). Isotropic interstitial and intracellular microimpedances were prescribed for one set of simulations, and anisotropic microimpedances with unequal ratios (intracellular to interstitial) along and across fibers were prescribed for another set of simulations. Microimpedance measurements were then obtained by making statistical comparisons between partial differentialphio values and interstitial potential differences from passive bidomain simulations (Deltaphio) in which a wide range of possible microimpedances were considered. Possible microimpedances were selected at 25% increments. After demonstrating the effectiveness of the overall method with microimpedance measurements using one-dimensional test data, we showed microimpedance measurements within 25% of prescribed values in isotropic and anisotropic models. Our findings suggest that development of microfabricated devices to implement the procedure would facilitate routine measurement as a component of cardiac electrophysiological study.

Measuring surface potential components necessary for transmembrane current computation using microfabricated arrays

This study was designed to test the feasibility of using microfabricated electrodes to record surface potentials with sufficiently fine spatial resolution to measure the potential gradients necessary for improved computation of transmembrane current density. To assess that feasibility, we recorded unipolar electrograms from perfused rabbit right ventricular free wall epicardium ( n = 6) using electrode arrays that included 25-μm sensors fabricated onto a flexible substrate with 75-μm interelectrode spacing. Electrode spacing was therefore on the size scale of an individual myocyte. Signal conditioning adjacent to the sensors to control lead noise was achieved by routing traces from the electrodes to the back side of the substrate where buffer amplifiers were located. For comparison, recordings were also made using arrays built from chloridized silver wire electrodes of either 50-μm (fine wire) or 250-μm (coarse wire) diameters. Electrode separations were necessarily wider than with microfabricated arrays. Comparable signal-to-noise ratios (SNRs) of 21.2 ± 2.2, 32.5 ± 4.1, and 22.9 ± 0.7 for electrograms recorded using microfabricated sensors ( n = 78), fine wires ( n = 78), and coarse wires ( n = 78), respectively, were found. High SNRs were maintained in bipolar electrograms assembled using spatial combinations of the unipolar electrograms necessary for the potential gradient measurements and in second-difference electrograms assembled using spatial combinations of the bipolar electrograms necessary for surface Laplacian (SL) measurements. Simulations incorporating a bidomain representation of tissue structure and a two-dimensional network of guinea pig myocytes prescribed following the Luo and Rudy dynamic membrane equations were completed using 12.5-μm spatial resolution to assess contributions of electrode spacing to the potential gradient and SL measurements. In those simulations, increases in electrode separation from 12.5 to 75.0, 237.5, and 875.0 μm, which were separations comparable to the finest available with our microfabricated, fine wire, and coarse wire arrays, led to 10%, 42%, and 81% reductions in maximum potential gradients and 33%, 76%, and 96% reductions in peak-to-peak SLs. Maintenance of comparable SNRs for source electrograms was therefore important because microfabrication provides a highly attractive methods to achieve spatial resolutions necessary for improved computation of transmembrane current density.

Feasibility of cardiac microimpedance measurement using multisite interstitial stimulation

This study was designed to test the hypothesis that analyses of central interstitial potential differences recorded during multisite stimulation with a set of interstitial electrodes provide sufficient data for accurate measurement of cardiac microimpedances. On theoretical grounds, interstitial current injected and removed using electrodes in close proximity does not cross the membrane, whereas equilibration of intracellular and interstitial potentials occurs distant from electrodes widely separated. Multisite interstitial stimulation should therefore give rise to interstitial potential differences recorded centrally that depend on intracellular and interstitial microimpedances, allowing independent measurement. Simulations of multisite stimulation with fine (25 microm) and wide (400 microm) spacing in one-dimensional models that included Luo-Rudy dynamic membrane equations were performed. Constant interstitial and intracellular microimpedances were prescribed for initial analyses. Discrete myoplasmic and gap-junctional components were prescribed intracellularly in later simulations. With constant microimpedances, multisite stimulation using 29 total electrode combinations allowed interstitial and intracellular microimpedance measurements at errors of 0.30% and 0.34%, respectively, with errors of 0.05% and 0.40% achieved using 6 combinations and 10 total electrodes. With discrete myoplasmic and junctional components, comparable accuracy was maintained following adjustments to the junctions to reflect uncoupling. This allowed uncoupling to be quantified as relative increases in total junctional resistance. Our findings suggest development of microfabricated devices to implement the procedure would facilitate routine measurement as a component of cardiac electrophysiological study.

Surface electromyogram signal modelling

The paper reviews the fundamental components of stochastic and motor-unit-based models of the surface electromyogram (SEMG). Stochastic models used in ergonomics and kinesiology consider the SEMG to be a stochastic process whose amplitude is related to the level of muscle activation and whose power spectral density reflects muscle conduction velocity. Motor-unit-based models for describing the spatio-temporal distribution of individual motor-unit action potentials throughout the limb are quite robust, making it possible to extract precise information about motor-unit architecture from SEMG signals recorded by multi-electrode arrays. Motor-unit-based models have not yet been proven as successful, however, for extracting information about recruitment and firing rates throughout the full range of contraction. The relationship between SEMG and force during natural dynamic movements is much too complex to model in terms of single motor units.

Electrode systems for measuring cardiac impedances using optical transmembrane potential sensors and interstitial electrodes--theoretical design

The cardiac electrical substrate is a challenge to direct measurement of its properties. Optical technology together with the capability to fabricate small electrodes at close spacings opens new possibilities. Here, those possibilities are explored from a theoretical viewpoint. It appears that with careful measurements from a well-designed set of electrodes one can obtain structural conductivities, separating intracellular from interstitial values, and longitudinal from transverse. Resting membrane resistance also can be obtained.

Finite element modeling of electromagnetic signal propagation in a phantom arm

Improving the control of artificial arms remains a considerable challenge. It may be possible to graft remaining peripheral nerves in an amputated limb to spare muscles in or near the residual limb and use these nerve-muscle grafts as additional myoelectric control signals. This would allow simultaneous control of multiple, degrees of freedom (DOF) and could greatly improve the control of artificial limbs. For this technique to be successful, the electromyography (EMG) signals from the nerve-muscle grafts would need to be independent of each other with minimal crosstalk. To study EMG signal propagation and quantify crosstalk, finite element (FE) models were developed in a phantom-arm model. The models were validated with experimental data collected by applying sinusoidal excitations to a phantom-arm model and recording the surface electric potential distribution. There was a very high correlation (r > 0.99) between the FEM data and the experimental data, with the error in signal magnitude generally less than 5%. Simulations were then performed using muscle dielectric properties with static, complex, and full electromagnetic solvers. The results indicate that significant displacement currents can develop (> 50% of total current) and that the fall-off of surface signal power varies with how the signal source is modeled. Index Terms-Control, electromyography (EMG), finite element (FE), modeling, prosthesis.

A novel approach for precise simulation of the EMG signal detected by surface electrodes

We propose a new electromyogram generation and detection model. The volume conductor is described as a nonhomogeneous (layered) and anisotropic medium constituted by muscle, fat and skin tissues. The surface potential detected in space domain is obtained from the application of a two-dimensional spatial filter to the input current density source. The effects of electrode configuration, electrode size and inclination of the fibers with respect to the detection system are included in the transfer function of the filter. Computation of the signal in space domain is performed by applying the Radon transform; this permits to draw considerations about spectral dips and clear misunderstandings in previous theoretical derivations. The effects of generation and extinction of the action potentials at the fiber end plate and at the tendons are included by modeling the source current, without any approximation of its shape, as a function of space and time and by using again the Radon transform. The approach, based on the separation of the temporal and spatial properties of the muscle fiber action potential and of the volume conductor, includes the capacitive tissue properties.

Surface EMG models: properties and applications

After a general introduction on the kind of models and the use of models in the natural sciences, the main body of this paper reviews potential properties of structure based surface EMG (sEMG) models. The specific peculiarities of the categories (i) source description, (ii) motor unit structure, (iii) volume conduction, (iv) recording configurations and (v) recruitment and firing behaviour are discussed. For a specific goal, not all aspects conceivable have to be part of a model description. Therefore, finally an attempt is made to integrate the 'question level' and the 'model property level' in a matrix providing direction to the development and application of sEMG models with different characteristics and varying complexity. From this overview it appears that the least complex are models describing how the morphological muscle features are reflected in multi-channel EMG measurements. The most challenging questions in terms of model complexity are related to supporting the diagnosis of neuromuscular disorders.

Temperature-controlled and constant-power radio-frequency ablation: what affects lesion growth?

Radio-frequency (RF) catheter ablation is the primary interventional therapy for the treatment of many cardiac tachyarrhythmias. Three-dimensional finite element analysis of constant-power (CPRFA) and temperature-controlled RF ablation (TCRFA) of the endocardium is performed. The objectives are to study: 1) the lesion growth with time and 2) the effect of ground electrode location on lesion dimensions and ablation efficiency. The results indicate that: a) for TCRFA: i) lesion growth was fastest during the first 20 s, subsequently the lesion growth slowed reaching a steady state after 100 s, ii) positioning the ground electrode directly opposite the catheter tip (optimal) produced a larger lesion, and iii) a constant tip temperature maintained a constant maximum tissue temperature; b) for CPRFA: i) the lesion growth was fastest during the first 20 s and then the lesion growth slowed; however, the lesion size did not reach steady state even after 600 s suggesting that longer durations of energy delivery may result in wider and deeper lesions, ii) the temperature-dependent electrical conductivity of the tissue is responsible for this continuous lesion growth, and iii) an optimal ground electrode location resulted in a slightly larger lesion and higher ablation efficiency.

Resistivity and phase in localized BIA

We describe a system for highly reproducible non-invasive rf impedance measurements as a function of position along body segments such as the thigh. Results are reported for mainly healthy male and female subjects ranging in age from 19 to 65 and in body-mass index from 15 to 40. A principal conclusion is that the phase of the impedance falls monotonically with increasing distance from the knee, with average values substantially above what is found using standard, whole-body bioelectrical impedance analysis (BIA). To compensate for thigh shape, the data are further analysed using an anatomical model based on reasonable approximations for the distributions of muscle, fat and bone, yielding values of the effective resistivity for current flow parallel to the muscle fibres. The phase and resistivity results are discussed with reference to the whole-body BIA study of maintenance haemodialysis patients by Chertow et al, and in regard to possible physiological correlations observed in this work.

Modeling the effects of electric fields on nerve fibers: influence of tissue electrical properties

The effects of anisotropy and inhomogeneity of the electrical conductivity of extracellular tissue on excitation of nerve fibers by an extracellular point source electrode were determined by computer simulation. Analytical solutions to Poison's equation were used to calculate potentials in anisotropic infinite homogeneous media and isotropic semi-infinite inhomogeneous media, and the net driving function was used to calculate excitation thresholds for nerve fibers. The slope and intercept of the current-distance curve in anisotropic media were power functions of the ratio and product of the orthogonal conductivities, respectively. Excitation thresholds in anisotropic media were also dependent on the orientation of the fibers, and in strongly anisotropic media (sigma z/sigma xy > 4) there were reversals in the recruitment order between different diameter fibers and between fibers at different distances from the electrode. In source-free regions of inhomogeneous media (two regions of differing conductivity separated by a plane boundary), the current-distance relationship of fibers parallel to the interface was dependent only on the average conductivity, whereas in regions containing the source the current-distance relationship was dependent on the individual values of conductivity. Reversals in recruitment order between fibers at different distances from the electrode and between fibers of differing diameter were found in inhomogeneous media. The results of this simulation study demonstrate that the electrical properties of the extracellular medium can have a strong influence on the pattern of neuronal excitation generated by extracellular electric fields, and indicate the importance of tissue electrical properties in interpreting results of studies employing electrical stimulation applied in complex biological volume conductors.

Modeling of surface myoelectric signals--Part I: Model implementation

The relationships between the parameters of active motor units (MU's) and the features of surface electromyography (EMG) signals have been investigated using a mathematical model that represents the surface EMG as a summation of contributions from the single muscle fibers. Each MU has parallel fibers uniformly scattered within a cylindrical volume of specified radius embedded in an anisotropic medium. Two action potentials, each modeled as a current tripole, are generated at the neuromuscular junction, propagate in opposite directions and extinguish at the fiber-tendon endings. The neuromuscular junctions and fiber-tendon endings are uniformly scattered within regions of specified width. Muscle fiber conduction velocity and average fiber length to the right and left of the center of the innervation zone are also specified. The signal produced by MU's with different geometries and conduction velocities are superimposed. Monopolar, single differential and double differential signals are computed from electrodes placed in equally spaced locations on the surface of the muscle and are displayed as functions of any of the model's parameters. Spectral and amplitude variables and conduction velocity are estimated from the surface signals and displayed as functions of any of the model's parameters. The influence of fiber-end effects, electrode misalignment, tissue anisotropy, MU's location and geometry are discussed. Part II of this paper will focus on the simulation and interpretation of experimental signals.

Effect of skin electrode location on radiofrequency ablation lesions: an in vivo and a three-dimensional finite element study

INTRODUCTION

OBJECTIVES

To assess the effect of skin electrode location on radiofrequency (RF) ablation lesion dimensions and energy requirements.

BACKGROUND

Little is known about the effects of skin electrode location on RF ablation lesion dimensions and efficiency.

METHODS AND RESULTS

Temperature-controlled ablation at 60 degrees C for 60 seconds was performed in six sheep. Paired lesions were created in the lateral, anterior, posterior, and septal walls of both the ventricles. For group 1 lesions, the skin electrode was positioned directly opposite the catheter tip (optimal). For group 2 lesions, we used either the standard posterior location or an anterior location if the posterior skin electrode location was used for group 1. Group 1 lesions were 5.8+/-0.8 mm deep and 9.3+/-1.9 mm wide, compared with 4.6+/-1.0 mm deep and 7.7+/-1.9 mm wide group 2 lesions (P < or = 0.001). Group 1 lesion dimensions also had less variability. A finite element model was used to simulate temperature-controlled ablation and to study the effect of skin electrode locations on lesion dimensions, ablation efficiency, and blood heating. The optimal location was 1.6 times more efficient, and the volume of blood heated to > or = 90 degrees C was 0.005 mm3 for optimal versus 2.2 mm3 for the nonoptimal location.

CONCLUSION

Optimal skin electrode placement: (1) creates deeper and larger lesions; (2) reduces lesion size variability; and (3) decreases blood heating.

Anisotropy of human muscle via non-invasive impedance measurements

Combining non-invasive 50 kHz impedance measurements with a mathematical model for the underlying structure, we obtain in vivo values for the transverse and longitudinal conductivities of the muscles of the human thigh and for the (isotropic) conductivity of the covering skin-fat layer. Results for a healthy male subject are in acceptably good agreement with those obtained elsewhere on surgically exposed or freshly excised animal tissue and with 'global' measurements on humans. Also, measurements using rotatable probes reveal orientations of underlying muscle fibres via minima in resistance versus angle curves. The results suggest potentially useful methods for studying muscle properties in clinical and physiological research.

Analysis of magnetic stimulation of a concentric axon in a nerve bundle

In this paper, we 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. We 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 homogeneous medium. These calculations indicate that a nerve bundle with no sheath has little effect on the transmembrane potential. However, the presence of a perinerium 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.

Magnetic stimulation of axons in a nerve bundle: effects of current redistribution in the bundle

Recently, we developed a model of magnetic stimulation of a concentric axon in an anisotropic nerve bundle. In that earlier paper, we considered a single axon surrounded by a nerve bundle represented as a homogeneous anisotropic monodomain medium. In this paper we extend our previous calculations to examine excitation of axons within a nerve bundle without neglecting the presence of other axons in the nerve bundle. A three-dimensional axial symmetry volume conductor model is used to determine the transmembrane potential response along an axon due to induced electric fields produced by a toroidal coil. Our principal objective is to examine the effect of current redistribution to other axons in the bundle on excitation characteristics. We derive the transmembrane potential along an axon for two currently available models of current redistribution: the biodomain model and the spatial--frequency monodomain model. Results indicate that a reduction in the transmembrane potential along an axon due to the presence of other nerve fibers in the bundle is observed. Axons located at the periphery of a nerve bundle have lower thresholds and different excitation sites compared with axons located near the center of a nerve bundle.

Muscle electric activity. I: A model study on the effect of needle electrodes on single fiber action potentials

Needle recorded electromyographic signals can be expected to be influenced by the presence of the needle, the electrical double layer at the metal-electrolyte interface, and by an edematous layer around the needle electrode. The magnitude of each of these effects is derived from a cylinder symmetrical volume conductor model. Analytical solutions of Laplace's equation have been derived. These are used for simulating single muscle fiber action potentials (SFAPs) recorded by a typical single fiber electrode. The results indicate that there is no short-circuiting effect, in spite of the presence of a highly conducting needle shaft, which is due to the high impedance of the electrical double layer. The insulating properties of the double layer cause the SFAP amplitudes to increase, when the muscle fiber passes the electrode at the side of the leading-off point. The edematous layer counteracts this increase depending on the thickness and the conductivity of this layer. Only slight SFAP wave-form changes are found.

Muscle electric activity. II: On the feasibility of model-based estimation of experimental conditions in electromyography

From regular one-channel registrations of single muscle fiber action potential no measures on the recording configuration can be derived. When multichannel recordings are made, experimental parameters such as the distance between muscle fiber and needle electrode can be estimated. With the help of a volume conductor model, the single fiber activity at each of the electrodes can be predicted as a function of the recording conditions. Within known physical and physiological constraints such a model approach can be inverted (the inverse model) and used to estimate basic experimental conditions. From eight simultaneous single fiber action potential recordings we estimated (a) the distance between fiber and needle, (b) the axial position of the needle with respect to the muscle fiber, (c) a factor related to the muscle tissue anisotropy, and (d) a factor combining the muscle fiber diameter and the effective muscle tissue conductivity. With the help of a model describing the influence of the needle shaft it is made plausible that the needle inhomogeneity influences the results of the proposed procedure.

Use of surface potential spectral characteristics for solving the inverse problem in electroneurography

The changes in the power spectra of single-fibre extracellular action potentials (SFEAPs) generated in an infinite anisotropic frequency-dependent volume conductor, which occurred as a result of alterations in the propagation velocity v and duration T(in) of the intracellular action potential (IAP) were analytically determined. Effects of the temporal and spatial dispersions of almost synchronously activated fibres on the power spectrum of compound extracellular potentials (CEPs) were analysed for different shapes and sizes of the activated fibres' territory. It was found that, as a result of desynchronisation in the fibres' activation, dips existed in the CEP power spectra and that the frequencies of the dips depended on the degree of desynchronisation but did not depend on the velocity. It was shown that the hypothetical power spectrum of compound IAP was sensitive to the variations in the desynchronisation in the fibres' activation and in the risetime and duration of IAP even at a great fibre electrode distance typical for surface recordings.

Potential distribution and single-fibre action potentials in a radially bounded muscle model

In modelling the electrical behaviour of muscle tissue, we used to employ a frequency-dependent volume conductor network model, which was infinitely extended in all directions. Equations in this model could be solved using a finite-difference approach. The most important restriction of this model was the fact that no boundary effects could be incorporated. Analytical models of muscle tissue normally do not have this disadvantage, but in those models the microscopic structure of muscle tissue cannot be taken into account. In the paper, we present a combined numerical/analytical approach, which enables the study of potential distributions and SFAPs in simulated microscopic muscle tissue in which the influence of the muscle boundary has been considered. We considered muscle models with radii of 1.5 mm and 10 mm. Both models were compared with an unbounded network model. In the model with a radius of 1.5 mm we varied the position of the active fibre relative to the muscle surface. It appeared that in most cases the presence of a boundary had a considerable effect on the potential distribution. An increase in the peak-to-peak value of the SFAP amplitude up to 300 per cent was noticed when the active fibre was positioned 500 microns beneath the muscle surface in a model with a radius of 1.5 mm.

A comparison of two boundary conditions used with the bidomain model of cardiac tissue

In the bidomain model, two alternative sets of boundary conditions at the interface between cardiac tissue and a saline bath have been used. It is shown that these boundary conditions are equivalent if the length constant of the tissue in the direction transverse to the fibers is much larger than the radius of the individual cardiac cells. If this is not the case, the relative merits of the two boundary conditions are closely related to the question of the applicability of a continuum model, such as the bidomain model, to describe a discrete multicellular tissue.

A mathematical model for calculating the vector magnetic field of a single muscle fiber

A mathematical model is described for calculating the volume-conducted magnetic field from active muscle fibers in an anisotropic bundle. With earlier models, the azimuthal magnetic field of a nerve bundle was calculated and the results were compared with the fields measured by toroidal pickup coils. The present model is capable of evaluating all three of the magnetic field components and is thus applicable for analyzing SQUID magnetometer recordings of fields from a muscle bundle. The component of the magnetic field parallel to the fiber axis is more than an order of magnitude smaller than either of the other two components. The amplitude of the magnetic signal is strongly dependent upon the anisotropy of the muscle bundle, the intracellular conductivity, the radius of the muscle fiber, the radius of the muscle bundle, and the location of the fiber in the muscle bundle. The peak-to-peak amplitude of the single-muscle-fiber action field increases linearly with increasing intracellular conductivity, as the square of the radius of the muscle fiber, and exponentially with the distance between the location of the fiber and the center of the bundle.

A model for compound action potentials and currents in a nerve bundle. I: The forward calculation

We describe a model for the Compound Action Currents (CACs) and Compound Action Potentials (CAPs) produced by a peripheral nerve bundle in vitro. The Single Fiber Action Currents (SFACs) and the extracellular Single Fiber Action Potentials (SFAPs) are calculated using a generalized volume conduction model. Frequency-dependent conductivities, variations in the intracellular action potentials with recording temperature and axon conduction velocity, and the effects of axonal myelination are incorporated into the volume conduction calculation. We demonstrate how the propagation distance and the recording radius affect the simulated Compound Action Signals (CASs) of various nerve bundles. We also demonstrate how the frequency-dependent and -independent conductivities affect the CASs simulated by our model. For this simulation, some of the parameters for the nerve bundles and Conduction Velocity Distributions (CVDs) were obtained from the literature. In accompanying papers, we use the simulated CASs to investigate the effects of variations in the model parameters on the CVDs predicted by our inverse model.

Potential distribution in three-dimensional periodic myocardium--Part I: Solution with two-scale asymptotic analysis

The use of two-scale asymptotic analysis allows development of a model of the steady-state potential distribution in three-dimensional cardiac muscle preserving the underlying cellular network. The myocardium is modeled as a periodic structure consisting of cylindrical cells embedded in extracellular fluid and connected by longitudinal and side junctions. The method is applicable to cardiac muscle of arbitrary extent since the periodicity of the tissue is dealt with analytically, and thus numerical computations require no more resources than a continuous volume conductor problem. The asymptotic analysis approach reveals that the potential in a periodic myocardium consists of two components. The large-scale component provides the baseline for the total solution and can be determined from the anisotropic monodomain model associated with the original periodic problem. The method provides the formula for calculating the conductivity of the equivalent monodomain model on the basis of cell geometry and conductivity distribution in the cardiac tissue. The small-scale component reflects the periodicity of the underlying structure and oscillates with periods determined by the dimensions of cardiac cells. The magnitude of these oscillations depends upon the gradient of the large-scale component. During stimulation with extracellular electrodes, the small-scale component determines both the shape and the magnitude of the transmembrane potential, while the influence of the large-scale component is negligible. Hence, the small-scale component merits closer attention in pacing and defibrillation studies, especially since the model based on two-scale asymptotic analysis provides an effective means of its computation.

Modelling the periodicity of cardiac muscle

The intractable problem of modelling cardiac muscle of arbitrary extent while preserving cellular structure has been solved using an analytical rather than numerical approach with a method called two-scale asymptotic analysis. In this method, the myocardium was modelled as a collection of bundles arranged periodically in space and connected by junctions, and the distribution of the steady-state potential and current density was determined. The potential both along and across fibers was found to contain a distinct periodic component that determines the transmembrane potential. The magnitude of the transmembrane potential depends on the gradient of applied potential, the dimensions of the bundles, and their internal conductivity. Current flows primarily in the extracellular space, and the extracellular pathway also determines the apparent conductivity of cardiac muscle.

Interpretation of skeletal muscle four-electrode impedance measurements using spatial and temporal frequency-dependent conductivities

Spatial and temporal frequency-dependent conductivities are used to interpret four-electrode conductivity measurements on skeletal muscle. The model qualitatively explains the observed dependence of the experimental data on the temporal frequency of the injected current, the angle between the electrode array and the fibre direction, and the distance between the electrodes.

The electrical potential produced by a strand of cardiac muscle: a bidomain analysis

Analytic expressions are derived relating the transmembrane potential to the intracellular, interstitial and external potentials in a cylindrical strand of cardiac muscle lying in a saline bath. The bidomain model is used to account for the anisotropy and interstitial space in the tissue. The implications of this model for interpreting potential data from strands of cardiac muscle are discussed.

A comparison of two models for calculating the electrical potential in skeletal muscle

We compare two models for calculating the extracellular electrical potential in skeletal muscle bundles: one a bidomain model, and the other a model using spatial and temporal frequency-dependent conductivities. Under some conditions the two models are nearly identical. However, under other conditions the model using frequency-dependent conductivities provides a more accurate description of the tissue. The bidomain model, having been developed to describe syncytial tissues like cardiac muscle, fails to provide a general description of skeletal muscle bundles due to the non-syncytial nature of skeletal muscle.