Propagation in cardiac tissue adjacent to connective tissue: two-dimensional modeling studies.

The conditions for activation transmission across a region of extracellular space was demonstrated in two-dimensional preparations with results consistent with those previously seen in the one-dimensional fiber studies. In addition, one sees changes in action potential morphology which occur in the tissue nearest the connective-tissue border as well as changes in conduction velocity along the border. These results hinge on an adequate representation of the connective-tissue region achieved by careful implementation of the boundary conditions in the intracellular and interstitial spaces and the expansion of the connective-tissue discretization to a "double-tier network" description. Through a series of simulations, a clear dependence on fiber orientation is illustrated in the efficacy to transmit activation. The collision of a front with an embedded connective-tissue region was also examined. The results revealed that fibers aligned normal to a planar stimulus would more greatly disrupt the advancement of a planar front. Such pronounced disruptions have been shown to be proarrhythmic in the literature. The increasing evidence of the ability of connective tissue to transmit activation has implications in understanding spread of activation through infarcted tissues and through the healthy ventricular wall in the presence of connective-tissue sheets.

Statistical analysis of signals from an intracavitary probe in a diseased heart.

A model study introduces the use of statistical signal processing to analyse the signals from an intracavitary probe. A complete derivation is given for the detection of one type of arrhythmogenic substrate, myocardial infarctions (MIs). Both the use of statistical signal processing and the detection of VT substrates, as opposed to activation maps, are unique. A quasi-stationary electromagnetic model with simplified geometry is presented. The model is used to simulate ventricular pacing in the presence of MI. The likelihood ratio is used for detection. A tabulation of the results from this model shows that an intracavitary probe can be used to detect MIs as small as 400 mm2 in 1 mV of noise with a detectability index of 0.495, where 0.5 indicates perfect detection. Sensitivity to noise can be reduced by analysing multiple heart beats. The results are only slightly affected by changing the probe from a cage frame design, which mechanically supports the electrodes on thin spokes, to a balloon design, which supports the electrodes on the surface of an insulating balloon.

Computer model analysis of the relationship of ST-segment and ST-segment/heart rate slope response to the constituents of the ischemic injury source.

The objective of the study was to investigate a proposed linear relationship between the extent of myocardial ischemic injury and the ST-segment/heart rate (ST/HR) slope by computer simulation of the injury sources arising in exercise electrocardiographic (ECG) tests. The extent and location of the ischemic injury were simulated for both single- and multivessel coronary artery disease by use of an accurate source-volume conductor model which assumes a linear relationship between heart rate and extent of ischemia. The results indicated that in some cases the ST/HR slope in leads II, aVF, and especially V5 may be related to the extent of ischemia. However, the simulations demonstrated that neither the ST-segment deviation nor the ST/HR slope was directly proportional to either the area of the ischemic boundary or the number of vessels occluded. Furthermore, in multivessel coronary artery disease, the temporal and spatial diversity of the generated multiple injury sources distorted the presumed linearity between ST-segment deviation and heart rate. It was concluded that the ST/HR slope and ST-segment deviation of the 12-lead ECG are not able to indicate extent of ischemic injury or number of vessels occluded.

Threshold variability in fibers with field stimulation of excitable membranes.

The central focus of this report is the evolution of transmembrane potentials following initiation of a point-source field stimulus, particularly when the stimulus is short and the stimulating electrode is close to the fiber. The transmembrane voltage threshold in response to a point-source field stimulus was determined in a numerical model of a single unmyelinated fiber. Both nerve (Hodgkin-Huxley) and cardiac (Ebihara-Johnson [1]) models of the fiber membrane were evaluated. A central question is whether it is possible to know in advance whether a stimulus of specific magnitude, duration, and location will result in a subsequent action potential. Such determination can be based on the membrane's "voltage threshold." In contrast to the commonly held view, the voltage threshold was found to vary markedly depending on the duration and location of the field stimulus. Voltage thresholds ranged from about 8 mV above baseline to more than 100 mV above baseline, the higher thresholds occurring with shorter stimuli and electrode locations closer to the membrane. A related question is whether the passive membrane response can be used as a tool in determining whether a subsequent action potential is elicited. If the answer is affirmative, this finding can be very useful, since passive properties are linear and thereby much simpler to evaluate than active ones. The results show that the passive response tracks active responses long enough to be a good estimator of subsequent action potential development. Examples show that the evaluation of Vm at 0.2-0.5 msec after stimulus initiation, times chosen on the basis of membrane characteristics, was a better predictor of subsequent excitation than was either initial transmembrane current or Vm at the time when the stimulus ends. Most of the circumstances analyzed here with electric field stimulation also appear likely to be valid with magnetic field stimulation.

Epicardial cardiac source-field behavior.

The accurate determination of the spatial distribution of cardiac electrophysiological state is essential for the mechanistic assessment of cardiac arrhythmias in both clinical and experimental cardiac electrophysiological laboratories. This paper describes three fundamental cardiac source-field relationships: 1) activation fields, 2) electrotonic fields, and 3) volume conductor fields. The three cases are described analytically and illustrated with experimentally obtained canine cardiac recordings that capitalize on a recently formulated technique for in vivo cardiac transmembrane current estimation.

Electric field stimulation of excitable tissue.

This paper examines the transmembrane voltage response of an unmyelinated fiber to a stimulating electric field from a point current source. For subthreshold conditions, analytic expressions for the transmembrane potential, vm, are developed that include the specific effects of fiber-source distance, h, and time from the onset of the stimulus, T. Suprathreshold effects are determined for two examples by extending the analytical results with a numerical model. The vm response is a complex evolution in time, especially for small h, that differs markedly from the "activating function." In general, the subthreshold response is a good predictor of the wave shape of the suprathreshold vm, but a poor predictor of its magnitude. The subthreshold response also is a good (but not a precise) predictor of the region where excitation begins.

Significance of inwardly directed transmembrane current in determination of local myocardial electrical activation during ventricular fibrillation.

Ventricular fibrillation (VF) is the principle cardiac rhythm disorder responsible for sudden cardiac death in humans. The accurate determination of local cardiac activation during VF is essential for its mechanistic elucidation. This has been hampered by the rapidly changing and markedly heterogeneous electrophysiological nature of VF. These difficulties are manifested when attempting to differentiate true propagating electrical activity from electrotonic signals and when identifying local activation from complex and possibly fractionated electrograms. The purpose of this investigation was to test the hypothesis that the presence of a balanced inwardly and outwardly directed transmembrane charge, obtained from the ratio of the inward to outward area under the cardiac transmembrane current curve (-/+ Im area), could reliably differentiate propagating from electrotonic deflections during VF. To test this hypothesis, we applied a recently described technique for the in vivo estimation of the transmembrane current (Im) during cardiac activation. A 17-element orthogonal epicardial electrode array was combined with an immediately adjacent optical fiber array to record electrical and optically coupled transmembrane potential signals during VF. Recordings were obtained during electrically induced VF in six dogs to determine the Im associated with activation and the time course of repolarization, as well as unipolar electrograms and bipolar electrograms recorded at multiple center-to-center interelectrode distances from 0.2 to 3 mm. Propagating local activations were associated with the presence of an easily identified inwardly directed Im, with a balanced inward and outward charge (-/+ Im area approximately 1.0). Electrotonic wave-forms lacked this inward Im (-/+ Im area approximately 0.0). Normal Na(+)-mediated inward currents were directly demonstrated to be responsible for some activations during VF.

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.

Boundary element analysis of the directional sensitivity of the concentric EMG electrode.

Assessment of the motor unit architecture based on concentric electrode motor unit potentials requires a thorough understanding of the recording characteristics of the concentric EMG electrode. Previous simulation studies have attempted to include the effect of EMG electrodes on the recorded waveforms by uniformly averaging the tissue potential at the coordinates of one- or two-dimensional electrode models. By employing the boundary element method, this paper improves earlier models of the concentric EMG electrode by including an accurate geometric representation of the electrode, as well as the mutual electrical influence between the electrode surfaces. A three-dimensional sensitivity function is defined from which information about the preferential direction of sensitivity, blind spots, phase changes, rate of attenuation, and range of pick-up radius can be derived. The study focuses on the intrinsic features linked to the geometry of the electrode. The results show that the cannula perturbs the potential distribution significantly. The core and the cannula electrodes measure potentials of the same order of magnitude in all of the pick-up range, except adjacent to the central wire, where the latter dominates the sensitivity function. The preferential directions of sensitivity are determined by the amount of geometric offset between the individual sensitivity functions of the core and the cannula. The sensitivity function also reveals a complicated pattern of phase changes in the pick-up range. Potentials from fibers located behind the tip or along the cannula are recorded with reversed polarity compared to those located in front of the tip. Rotation of the electrode about its axis was found to alter the duration, the peak-to-peak amplitude, and the rise time of waveforms recorded from a moving dipole.

In vivo estimation of cardiac transmembrane current.

The ionic currents that cross the myocardial membrane during cardiac activation have a corresponding return path in the extracellular space. The transmembrane current (Im) during activation of cardiac cells in situ has previously been envisioned only in mathematical models. We have developed a remarkably simple in vivo technique that incorporates an electrode array with cellular dimensions to continuously estimate the extracellular counterparts of cardiac Ims. Mathematical modeling was performed for uniform plane wave propagation to clarify the biophysical basis and underlying assumptions inherent in this approach. Five-element electrode arrays incorporating 75-microns-diameter silver electrodes with center-to-center distances of 210 microns were experimentally verified to provide spatially sufficient samples for voltage gradient determinations of myocardial activation. Similar results were obtained with 25-microns-diameter electrodes at a center-to-center spacing of 65 microns. An estimate of Im was obtained from the derivative of the magnitude of the voltage gradient of the measured interstitial potentials. The inward component of Im generated by normal Na+ channel activation at 37 degrees C was measured in vivo to be less than 1 msec in duration, consistent with previously known voltage-clamp and simulation results. Intravenous KCl bolus injection was used to demonstrate the voltage-dependent depression of Na(+)-mediated Im in vivo, culminating in either severely depressed Na(+)-mediated or Ca(2+)-mediated activations. Normal Na(+)-, depressed Na(+)-, and possibly Ca(2+)-mediated currents can be recorded in vivo using this technique.

Active response of a one-dimensional cardiac model with gap junctions to extracellular stimulation.

To study the response of cardiac tissue to electrical stimulation, a one-dimensional model of cardiac tissue has been developed using linear core-conductor theory and the DiFrancesco-Noble model of Purkinje tissue. The cable lies in a restricted extracellular medium and includes a representation of the junctional resistances known to interconnect cardiac cells. Two electrode geometries are considered: (a) a semi-infinite cable with a monopolar electrode at the end of the cable and (b) a terminated cable with one electrode at each end of the cable. In a series of simulations, stimuli of varying magnitude and polarity are applied at three different times during the plateau of the action potential. The results at the stimulus site show that the action potential duration may either decrease or increase in response to the stimulus, depending on the polarity and application time of the stimulus. The spatial behaviour of the cable in response to the stimulus indicates that sites greater than 200 cells from the stimulating electrode are not affected by the stimulus.

Electrophysiological interaction through the interstitial space between adjacent unmyelinated parallel fibers.

The influence of interstitial or extracellular potentials on propagation usually has been ignored, often through assuming these potentials to be insignificantly different from zero, presumably because both measurements and calculations become much more complex when interstitial interactions are included. This study arose primarily from an interest in cardiac muscle, where it has been well established that substantial interstitial potentials occur in tightly packed structures, e.g., tens of millivolts within the ventricular wall. We analyzed the electrophysiological interaction between two adjacent unmyelinated fibers within a restricted extracellular space. Numerical evaluations made use of two linked core-conductor models and Hodgkin-Huxley membrane properties. Changes in transmembrane potentials induced in the second fiber ranged from nonexistent with large intervening volumes to large enough to initiate excitation when fibers were coupled by interstitial currents through a small interstitial space. With equal interstitial and intracellular longitudinal conductivities and close coupling, the interaction was large enough (induced Vm approximately 20 mV peak-to-peak) that action potentials from one fiber initiated excitation in the other, for the 40-microns radius evaluated. With close coupling but no change in structure, propagation velocity in the first fiber varied from 1.66 mm/ms (when both fibers were simultaneously stimulated) to 2.84 mm/ms (when the second fiber remained passive). Although normal propagation through interstitial interaction is unlikely, the magnitudes of the electrotonic interactions were large and may have a substantial modulating effect on function.

The effect of cellular discontinuities on the transient subthreshold response of a one-dimensional cardiac model.

Previous studies have examined the influence of the gap-junction discontinuity on the steady-state response of a cardiac cable to electrical defibrillation. It is important to understand when steady-state conditions may be assumed. For this reason, the transient, subthreshold behavior of a discontinuous cardiac cable is examined in this study. The behavior of the cable reflects two characteristics: 1) the continuous nature of the entire cable and 2) the isolated behavior of individual cells imposed by the junction discontinuity. The results show two effective time constants of activation: a large time constant corresponding to the time constant of a continuous cable of equivalent length, and a small time constant reflecting the rapid activation of an isolated cell. The rapid activation establishes a voltage gradient across each cell of the cable with one end of the cell hyperpolarized and the opposite end depolarized. This pattern of hyperpolarization and depolarization reaches a maximum value in approximately 3 microseconds and may play an important role in the mechanism of defibrillation.

The transient subthreshold response of spherical and cylindrical cell models to extracellular stimulation.

The effect of extracellular stimulation on excitable tissue is evaluated using analytical models. Primary emphasis is placed on the determination of the rate of rise of the membrane potential in response to subthreshold stimulation. Three models are studied: 1) a spherical cell in a uniform electric field, 2) an infinite cylindrical fiber with a point source stimulus, and 3) a finite length cable with sealed ends and a stimulus electrode at each end. Results show that the rate of rise of the transmembrane potential was more rapid than the step response of a space-clamped membrane for all geometries considered. The response of the cylindrical fiber to extracellular stimulation is compared to previously reported studies of the cylindrical fiber response to intracellular stimulation. It is found that the location of the stimulus has little effect on the infinite fiber response. For terminated cables, however, an accurate model of stimulus response must discriminate between intracellular and extracellular stimulation.

The Brody effect revisited.

This paper reexamines the Brody effect, both in the far-field and in the near-field approximation. It stresses the fact that near an inhomogeneity the Brody factor is not a constant but a function of space. A full documentation of this function for realistic values of the inhomogeneity as relevant to electrocardiography is included. The existence of a zone having "anomalous" Brody factors is demonstrated. Moreover, the importance for this problem of the zero reference point is stressed.

One-dimensional model of cardiac defibrillation.

The response of a single strand of cardiac cells to a uniform defibrillatory shock assuming steady-state linear conditions is examined. It is argued that the effect of this current is quantitatively described by the induced transmembrane potential even under passive conditions. The characteristics of the single strand are those that would exist if the heart was a system of equivalent parallel pathways from apex to base. It is shown that essentially every cell is both hyperpolarized and depolarised from the shock by an amount proportional to the stimulus intensity and the intercellular junctional resistance. For physiological values of model parameters the evaluated depolarisations are consistent with levels necessary to affect electrophysiological behaviour.