Forward and inverse potential field solutions for cardiac strands of cylindrical geometry

A dual coordinate system finite difference method for forward and inverse solutions of the volume conductor problem in neurology

Finite difference methods for the volume conductor problem have used a single coordinate system for the mesh and made approximations of Laplace's equation. This method is simple but has two major problems. Firstly, to deal with boundary conditions properly, the normal potential gradient at the boundary must be known. However it is complicated to compute at a curved surface point. Secondly, for an inverse solution the equation on a curved boundary is difficult to reverse since more than one inner mesh node appears in the approximation equation for each surface point. The new method developed in this paper is a dual coordinate system. One system serves as a frame mesh, the other is a sub-coordinate system in which surface points become mesh points (regular nodes). The equation at each surface point is then directly reversible since only one inner point appears in the equation. The forward solution is applied to both centric and eccentric bone models and uses the conventional successive over-relaxation (SOR) method. Noise is added to this solution for input to the inverse procedure which is a direct step-in non-iterative method. Low pass filtering was effective in reducing the effects of noise. In the examples given, only one coordinate subsystem is used but, for complex shape boundaries, multiple subsystems would be necessary.

Spatial- and frequency-domain ring source models for the single-muscle fibre action potential

In the paper, single-fibre models for the extracellular action potential are developed that will allow the potential to be evaluated at an arbitrary field point in the extracellular space. Fourier-domain models are restricted in that they evaluate potentials at equidistant points along a line parallel to the fibre axis. Consequently, they cannot easily evaluate the potential at the boundary nodes of a boundary-element electrode model. The Fourier-domain models employ axial-symmetric ring source models, and thereby provide higher accuracy than the line source model, where the source is lumped into a line source at the centre of the fibre. In the paper, new spatial models are developed based on elliptic integrals. These models employ axial-symmetric ring source models, and therefore are more accurate than the line source model. In the analysis, dual transform pairs are identified. Numerical examples including anisotropy show that the spatial models require extreme care in the integration procedure owing to the singularity in the weighting functions. With adequate sampling, the spatial models can evaluate extracellular potentials with high accuracy.

Limitations of approximate solutions for computing the extracellular potential of single fibers and bundle equivalents

The mathematical description of the extracellular field generated by activity in an excitable fiber in an unbounded volume conductor will depend on assumptions made about the sources and the source-field relationship. This paper examines and compares the rigorous and conventional approximate solutions of Laplace's equation used to evaluate the extracellular potential of a single, cylindrical fiber. The single fiber is considered as both a prototypical element (such as a nerve or muscle fiber) and an elementary model of an entire multicellular preparation (e.g., nerve bundle or Purkinje strand). The effects of the fiber radius, the intracellular and extracellular conductivities, and the shape and extent of the source function (either the transmembrane potential or the intracellular potential) on the solutions are discussed. The results show that, in general, the approximate solutions are unsatisfactory for computing the surface extracellular potential when the single fiber is used to represent a large bundle (greater than 300 microns).

Estimating cardiac transmembrane activation and recovery times from unipolar and bipolar extracellular electrograms: a simulation study

A model of one-dimensional action potential propagation was used to compare activation times and recovery times measured from simulated unipolar and bipolar electrograms with the activation and recovery times measured from simulated transmembrane action potentials. Theory predicts that the intrinsic deflection--the time of the maximum negative slope of the unipolar electrogram QRS complex--corresponds to the time of maximum positive slope of action potential depolarization. Similarly, the time of the maximum positive slope of the unipolar electrogram T wave corresponds to the time of maximum negative slope of action potential repolarization. This study showed that the difference between the unipolar electrogram activation time and the action potential activation time and the difference between the unipolar electrogram recovery time and the action potential recovery time were small during ideal conditions of uniform propagation in a long cable. Nonideal conditions, however, were associated with activation time differences in excess of 1.8 msec and recovery time differences in excess of 30 msec (243 msec in certain conditions). Nonideal conditions that had a major influence were changes in activation sequence, propagation in a short cable, and propagation through regions of nonuniform coupling resistance and/or nonuniform membrane properties. Nonideal conditions that had a smaller influence were variations in distance from the measurement site to the simulated tissue surface, nonzero reference potentials, and the addition of distant events. Recovery time differences were more sensitive to the nonideal conditions than were activation time differences, and both depended on the action potential shape. When distant events significantly contributed to the unipolar electrogram waveform, the time differences when bipolar electrograms were used were less than those when unipolar electrograms were used; however, under other conditions, the time differences were comparable. Results showed that activation times and especially recovery times measured from electrograms can be greatly affected by conditions independent of changes in the underlying action potential waveforms.

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.

Potential and current distributions in a cylindrical bundle of cardiac tissue

The intracellular and interstitial potentials associated with each cell or fiber in multicellular preparations carrying a uniformly propagating wave are important for characterizing the electrophysiological behavior of the preparation and in particular, for evaluating the source contributed by each fiber. The aforementioned potentials depend on a number of factors including the conductivities characterizing the intracellular, interstitial, and extracellular domains, the thickness of the tissue, and the distance (depth) of the field point from the surface of the tissue. A model study is presented describing the extracellular and interstitial potential distribution and current flow in a cylindrical bundle of cardiac muscle arising from a planar wavefront. For simplicity, the bundle is considered as a bidomain. Using typical values of conductivity, the results show that the intracellular and interstitial potential of fibers near the center of a very large bundle (greater than 10 mm) may be approximated by the potentials of a single fiber surrounded by a limited extracellular space (a fiber in oil), hence justifying a core-conductor model. For smaller bundles, the peak interstitial potential is less than that predicted by the core-conductor model but still large enough to affect the overall source strength. The magnitude of the source strength is greatest for fibers lying near the center of the bundle and diminishes sharply for fibers within 50 microns of the surface.

Extracellular currents and potentials of the active myelinated nerve fiber

This paper is concerned with the accurate and rapid calculation of extracellular potentials and currents from an active myelinated nerve fiber in a volume conductor, under conditions of normal and abnormal conduction. The neuroelectric source for the problem is characterized mathematically by using a modified version of the distributed parameter model of L. Goldman and J. S. Albus (1968, Biophys. J., 8:596-607) for the myelinated nerve fiber. Solution of the partial differential equation associated with the model provides a waveform for the spatial distribution of the transmembrane potential V(z). This model-generated waveform is then used as input to a second model that is based on the principles of electromagnetic field theory, and allows one to calculate easily the spatial distribution for the potential everywhere in the surrounding volume conductor for the nerve fiber. In addition, the field theoretic model may be used to calculate the total longitudinal current in the extracellular medium (I0L(z)) and the transmembrane current per unit length (im(z)); both of these quantities are defined in connection with the well-known core conductor model and associated cable equations in electrophysiology. These potential and current quantities may also be calculated as functions of time and as such, are useful in interpreting measured I0L(t) and im(t) data waveforms. An analysis of the accuracy of conventionally used measurement techniques to determine I0L(t) and im(t) is performed, particularly with regard to the effect of electrode separation distance and size of the volume conductor on these measurements. Also, a simulation of paranodal demyelination at a single node of Ranvier is made and its effects on potential and current waveforms as well as on the conduction process are determined. In particular, our field theoretic model is used to predict the temporal waveshape of the field potentials from the active, non-uniformly conducting nerve fiber in a finite volume conductor.

Electrically silent magnetic fields

There has been a significant controversy over the past decade regarding the relative information content of bioelectric and biomagnetic signals. In this paper we present a new, theoretical example of an electrically-silent magnetic field, based on a bidomain model of a cylindrical strand of tissue generalized to include off-diagonal components in the conductivity tensors. The physical interpretation of the off-diagonal components is explained, and analytic expressions for the electrical potential and the magnetic field are found. These expressions show that information not obtainable from electrical potential measurements can be obtained from measurements of the magnetic field in systems with conductivity tensors more complicated than those previously examined.