The Electrocardiogram Restitution Portrait Quantifying Dynamical Electrical Instability in Young Myocardium

Novel methods need to be developed to detect electrical instability in children. The dynamical properties of action potential restitution play an important role in the development of instability leading to arrhythmias. A new method, the Restitution Portrait (RP), was developed to visualize and quantify these properties at the action potential level. Here, we apply the RP method using the activation-recovery interval (ARI) from the ECG to detect, in vitro, repolarization abnormalities in neonatal and preadolescent rabbit myocardium with drug-induced Long QT Syndrome (LQTS Type 1 or Type 2). The ECG was recorded during programmed endocardial pacing to record the RP. The ECG RP demonstrated significant changes in dynamical restitution components during drug-induced LQTS compared to baseline. This study shows that the ECG RP may be an important noninvasive diagnostic tool for detecting electrical instability in the young.

Condition for alternans and stability of the 1:1 response pattern in a "memory" model of paced cardiac dynamics

We analyze a mathematical model of paced cardiac muscle consisting of a map relating the duration of an action potential to the preceding diastolic interval as well as the preceding action potential duration, thereby containing some degree of "memory." The model displays rate-dependent restitution so that the dynamic and S1-S2 restitution curves are different, a manifestation of memory in the model. We derive a criterion for the stability of the 1:1 response pattern displayed by this model. It is found that the stability criterion depends on the slope of both the dynamic and S1-S2 restitution curves, and that the pattern can be stable even when the individual slopes are greater or less than one. We discuss the relation between the stability criterion and the slope of the constant-BCL restitution curve. The criterion can also be used to determine the bifurcation from the 1:1 response pattern to alternans. We demonstrate that the criterion can be evaluated readily in experiments using a simple pacing protocol, thus establishing a method for determining whether actual myocardium is accurately described by such a mapping model. We illustrate our results by considering a specific map recently derived from a three-current membrane model and find that the stability of the 1:1 pattern is accurately described by our criterion. In addition, a numerical experiment is performed using the three-current model to illustrate the application of the pacing protocol and the evaluation of the criterion.

Modelling induction of a rotor in cardiac muscle by perpendicular electric shocks

A strong, properly timed shock applied perpendicularly to a propagating wavefront causes a rotor in the canine myocardium. Experimental data indicate that the induction of this rotor relies on the shock exciting tissue away from the electrodes. The computational study reproduced such direct excitation in a two-dimensional model of a 2.7 x 3 cm sheet of cardiac muscle. The model used experimentally measured extracellular potentials to represent 100 and 150 V shocks delivered through extracellular electrodes. The shock-induced transmembrane potential was computed according to two mechanisms, the activating function and the unit-bundle sawtooth potential. The overall process leading to initiation of a rotor was the same in model and experiment. For the 100 V shock, the directly excited region extended 2.26 cm away from the electrode; the centre of the rotor ('critical point') was 1.28 cm away, where the electric field Ecr was 4.54 Vcm(-1). Increasing the shock strength to 150 V moved the critical point 1.02 cm further and decreased Ecr by 0.39 Vcm(-1). The results are comparable with experimental data. The model suggests that the unit-bundle sawtooth is responsible for the creation of the directly excited region, and the activating function is behind the dependence of Ecr on shock strength.

Existence of bistability and correlation with arrhythmogenesis in paced sheep atria

INTRODUCTION

Studies of the electrical dynamics of cardiac tissue are important for understanding the mechanisms of arrhythmias. This study uses high-frequency pacing to investigate the dynamics of sheep atria.

METHODS AND RESULTS

A 504-electrode mapping plaque was affixed to the right atrium in six sheep. Cathodal pacing stimuli were delivered to the center of the plaque. Pacing period (Tp) was decreased from 275 +/- 25 msec to 75 +/- 25 msec and then increased to 230 +/- 70 msec in steps of either 5 or 10 msec. In all 21 trials in six sheep, the atrium responded 1:1 at longer Tps and 2:1 at shorter Tps. As Tp was decreased, the response switched to 2:1 at a particular Tp. Conversely, as Tp was increased, the response switched back to 1:1 at a particular Tp. Over 21 trials, the 1:1-to-2:1 and 2:1-to-1:1 transitions occurred at 119.5 +/- 18.8 msec and 130.0 +/- 19.1 msec, respectively. This hysteretic behavior yielded bistability windows, 10.5 +/- 7.2 msec wide, wherein 1:1 and 2:1 responses existed at the same Tp. In 15 trials and in all animals, idiopathic wavefronts emanating from outside the mapped region passed through the mapped region. In 13 of those trials, the idiopathic wavefronts occurred at Tps within the bistability window or within 35 msec of its upper or lower limit.

CONCLUSION

Bistability windows and idiopathic wavefronts were observed and found to be correlated with each other, suggesting a connection between bistability and arrhythmogenesis.

Increasing the computational efficiency of a bidomain model of defibrillation using a time-dependent activating function

Realistic simulations of the effects of strong shocks on cardiac muscle require solving the bidomain model, a continuum representation of cardiac tissue by a system of two reaction-diffusion equations. For two- and three-dimensional problems, the computations tend to take a prohibitively long time. This study develops a computationally efficient and accurate approximation of the bidomain model: a "reduced bidomain" model. The approximation is based on the fact that during a strong shock, the extracellular field in the muscle changes only slightly and, therefore, can be approximated by an activating function, following the concept introduced by Rattay (Rattay, F. Analysis of models for external stimulation of axons. IEEE Trans. Biomed. Eng. 33:974-977, 1986). The activating function used here is time-dependent and is computed using an iterative algorithm. The results show that in two spatial dimensions, the "reduced bidomain" model, as implemented in this study, cuts the computational cost by two orders of magnitude while preserving most properties of the "full bidomain" model. It faithfully represents the spatial pattern and the temporal development of the muscle polarization. Consequently, relative errors in the "defibrillation" threshold, the strength of the weakest shock that terminates all electrical activity within 100 ms, are below 10%.

Theoretical modeling of the effects of shock duration, frequency, and strength on the degree of electroporation

Electroporation is becoming an increasingly important tool for introducing biologically active compounds into living cells, yet the effectiveness of this technique can be low, particularly in vivo. One way to improve the success rate is to optimize the shock protocols, but experimental studies are costly, time consuming, and yield only an indirect measurement of pore creation. Alternatively, this study models electroporation in two geometries, a space-clamped membrane and a single cell, and investigates the effects of pulse duration, frequency, shape, and strength. The creation of pores is described by a first order differential equation derived from the Smoluchowski equation. Both the membrane and the cell are exposed to monophasic and biphasic shocks of varying duration (membrane, 10 micros-100 s; cell, 0.1 micros-200 ms) and to trains of monophasic and biphasic pulses of varying frequency (membrane, 50 Hz-4 kHz; cell, 200 kHz-6 MHz). The effectiveness of each shock is measured by the fractional pore area (FPA). The results indicate that FPA is sensitive to shock duration only in a very narrow range (membrane, approximately 1 ms; cell, approximately 0.25 micros). In contrast, FPA is sensitive to shock strength and frequency of the pulse train, increasing linearly with shock strength and decreasing slowly with frequency. In all cases, monophasic shocks were at least as effective as biphasic shocks, implying that varying the strength and frequency of a monophasic pulse train is the most effective way to control the creation of pores.

Modeling electroporation in a single cell. I. Effects Of field strength and rest potential

This study develops a model for a single cell electroporated by an external electric field and uses it to investigate the effects of shock strength and rest potential on the transmembrane potential V(m) and pore density N around the cell. As compared to the induced potential predicted by resistive-capacitive theory, the model of electroporation predicts a smaller magnitude of V(m) throughout the cell. Both V(m) and N are symmetric about the equator with the same value at both poles of the cell. Larger shocks do not increase the maximum magnitude of V(m) because more pores form to shunt the excess stimulus current across the membrane. In addition, the value of the rest potential does not affect V(m) around the cell because the electroporation current is several orders of magnitude larger than the ionic current that supports the rest potential. Once the field is removed, the shock-induced V(m) discharges within 2 micros, but the pores persist in the membrane for several seconds. Complete resealing to preshock conditions requires approximately 20 s. These results agree qualitatively and quantitatively with the experimental data reported by Kinosita and coworkers for unfertilized sea urchin eggs exposed to large electric fields.

Modeling electroporation in a single cell. II. Effects Of ionic concentrations

This study expands a previously developed model of a single cell electroporated by an external electric field by explicitly accounting for the ionic composition of the electroporation current. The previous model with non-specific electroporation current predicts that both the transmembrane potential V(m) and the pore density N are symmetric about the equator, with the same values at either end of the cell. The new, ion-specific case predicts that V(m) is symmetric and almost identical to the profile from the non-specific case, but N has a profound asymmetry with the pore density at the hyperpolarized end of the cell twice the value at the depolarized end. These modeling results agree with the experimentally observed preferential uptake of marker molecules at the hyperpolarized end of the cell as reported in the literature. This study also investigates the changes in intracellular ionic concentrations induced around an electroporated single cell. For all ion species, the concentrations near the membrane vary significantly, which may explain the electrical disturbances observed experimentally after large electric shocks are delivered to excitable cells and tissues.

Effects of electroporation on the transmembrane potential distribution in a two-dimensional bidomain model of cardiac tissue

INTRODUCTION

Defibrillation shocks, when delivered through internal electrodes, establish transmembrane potentials (Vm) large enough to electroporate the membrane of cardiac cells. The effects of such shocks on the transmembrane potential distribution are investigated in a two-dimensional rectangular sheet of cardiac muscle modeled as a bidomain with unequal anisotropy ratios.

METHODS AND RESULTS

The membrane is represented by a capacitance Cm, a leakage conductance g(l) and a variable electroporation conductance G, whose rate of growth depends exponentially on the square of Vm. The stimulating current Io, 0.05-20 A/m, is delivered through a pair of electrodes placed 2 cm apart for stimulation along fibers and 1 cm apart for stimulation across fibers. Computer simulations reveal three categories of response to Io: (1) Weak Io, below 0.2 A/m, cause essentially no electroporation, and Vm increases proportionally to Io. (2) Strong Io, between 0.2 and 2.5 A/m, electroporate tissue under the physical electrode. Vm is no longer proportional to Io; in the electroporated region, the growth of Vm is halted and in the region of reversed polarity (virtual electrode), the growth of Vm is accelerated. (3) Very strong Io, above 2.5 A/m, electroporate tissue under the physical and the virtual electrodes. The growth of Vm in all electroporated regions is halted, and a further increase of Io increases both the extent of the electroporated regions and the electroporation conductance G.

CONCLUSION

These results indicate that electroporation of the cardiac membrane plays an important role in the distribution of Vm induced by defibrillation strength shocks.

Field stimulation of isolated chick heart cells: comparison of experimental and theoretical activation thresholds

INTRODUCTION

This study examines the accuracy of using membrane models to predict activation thresholds for chick heart cells during field stimulation.

METHODS AND RESULTS

Activation thresholds were measured experimentally in ten embryonic chick heart cells at 37 degrees C for stimulus durations 0.2 to 40 msec. Activation was assessed by observing the mechanical twitch of the cell. The heart cells ranged in diameter from 15.0 to 26.7 microm. Since the electric field required for activation depends on diameter, the thresholds were expressed as the maximum field-induced transmembrane potential, Vth = 1.5 a Eth, where a is the cell radius and Eth is the strength of the electric field at threshold. A cell model was created using a singular perturbation method and membrane models describing the ionic currents of a heart cell. The study used membrane models of Ebihara and Johnson (1980), Luo and Rudy (1991), Shrier and Clay (1994), and their combinations. The results show that for stimuli longer than 1 msec, theoretical activation thresholds were within one standard deviation of experimental thresholds. For shorter stimuli, the models failed to predict thresholds because of a premature deactivation of the sodium current. The modification of the m gates dynamics, so that they closed with a time constant of 1.4 msec, allowed to predict thresholds for all durations. The root mean square error between experimental and theoretical thresholds was 6.14%.

CONCLUSIONS

The existing membrane models can predict thresholds for field stimulation only for stimuli longer than 1 msec. For shorter stimuli, the models need a more accurate representation of the sodium tail current.

Electroporation and shock-induced transmembrane potential in a cardiac fiber during defibrillation strength shocks

Experimental studies have shown that the magnitude of the shock-induced transmembrane potential (Vm) saturates with increasing electric field strength. This study uses a mathematical model to investigate the effects of electroporation and membrane kinetics on Vm in a cardiac fiber. The model consists of the core conductor equation for a one-dimensional fiber, where excitability is represented by the Luo-Rudy dynamic model (1994-1995) and electroporation is described by a membrane conductance that increases exponentially with Vm squared. For shocks delivered during the plateau of an action potential, the model reproduces the experimentally observed saturation of Vm with a root mean square error of 4.27% and a correlation coefficient of 0.9992. For shocks delivered during diastole, the saturation of Vm is qualitatively reproduced even when the sodium and calcium channels are inactivated. Quantitative replication of the response to diastolic shocks is hindered by the choice of electroporation parameters (optimized for shocks delivered during the plateau) and differences in the membrane kinetics between model and experiment. The complex behavior of Vm during large shocks is due to a combination of electroporation, electrotonus, propagation, and active membrane kinetics. The modeling results imply that the experimentally observed saturation of Vm is due to electroporation of the lipid bilayer.

The role of spatial interactions in creating the dispersion of transmembrane potential by premature electric shocks

Strong electric shocks applied during the refractory period can initiate or terminate cardiac arrhythmias. To elucidate the underlying mechanism, Knisley et al. used rabbit papillary muscle in vitro to scan the refractory period of an action potential with shocks of different strengths. The resulting map of the shock-induced changes in the transmembrane potential (Vm) illustrates the substrate for the creation of rotors. Our study uses computer simulations to reproduce this experimental map. Three models (a space-clamped membrane, a single cell, and a one-dimensional fiber) were used to determine whether the observed map was caused by (i) the intrinsic dynamics of the membrane, (ii) the simultaneous depolarization and hyperpolarization of the opposite ends of each cell, or (iii) spatial interactions involving the whole muscle strand. The results show that the membrane and single cell models cannot reproduce the experimental map. The fiber model reproduces the shock-induced changes in Vm and demonstrates that they are caused by a propagating disturbance, which, depending on the coupling interval and the shock strength, can be a new action potential or an electrotonus and can arrive from the depolarized end or from both depolarized and hyperpolarized ends of the fiber. These results indicate that the induction of rotors in the heart may not be a direct effect of the electric field.

Effects of electroporation on transmembrane potential induced by defibrillation shocks

This study uses a one-dimensional model of cardiac strand to investigate the effects of electroporation on transmembrane potential (Vm) induced by defibrillation shocks. The strand is stimulated at the ends by extracellular electrodes. Its membrane, when exposed to large Vm, increases its conductance in a manner consistent with reversible electrical breakdown. Numerical simulations indicate that Vm increases proportionally to the shock strength only until the ends of the strand electroporate. Beyond this point, further increases in shock strength result in only a minor change in Vm. This arrest in the growth of Vm is caused by pores that develop in the cells immediately adjacent to the electrodes and that shunt part of the stimulating current directly into intracellular space. Consequently, only a fraction of the delivered current, Icr, gives rise to Vm; the current in excess of Icr divides itself proportionally between intra- and extracellular space and does not contribute to macroscopic Vm. Thus, electroporation has a beneficial effect: the formation of pores prevents the development of an excessively high Vm and limits the damage to the tissue. In contrast, electroporation does not affect the "sawtooth" component of Vm that reflects polarization of individual cells by electric field. These results indicate that electroporation does not impair the ability of the shock to reach the distant myocardium and may actually aid defibrillation by reducing nonuniformity of electrical conditions between regions close to the electrodes and in the bulk of tissue.

Choosing the optimal monophasic and biphasic waveforms for ventricular defibrillation

INTRODUCTION

The truncated exponential waveform from an implantable cardioverter defibrillator can be described by three quantities: the leading edge voltage, the waveform duration, and the waveform time constant (tau s). The goal of this work was to develop and test a mathematical model of defibrillation that predicts the optimal durations for monophasic and the first phase of biphasic waveforms for different tau s values. In 1932, Blair used a parallel resistor-capacitor network as a model of the cell membrane to develop an equation that describes stimulation using square waves. We extended Blair's model of stimulation, using a resistor-capacitor network time constant (tau m), equal to 2.8 msec, to explicitly account for the waveform shape of a truncated exponential waveform. This extended model predicted that for monophasic waveforms with tau s of 1.5 msec, leading edge voltage will be constant for waveforms 2 msec and longer; for tau s of 3 msec, leading edge voltage will be constant for waveforms 3 msec and longer; for tau s of 6 msec, leading edge voltage will be constant for waveforms 4 msec and longer. We hypothesized that the best phase 1 of a biphasic waveform is the best monophasic waveform. Therefore, the optimal first phase of a biphasic waveform for a given tau s is the same as the optimal monophasic waveform.

METHODS AND RESULTS

We tested these hypotheses in two animal experiments. Part I: Defibrillation thresholds were determined for monophasic waveforms in eight dogs. For tau s of 1.5 msec, waveforms were truncated at 1, 1.5, 2, 2.5, 3, 4, 5, and 6 msec. For tau s of 3 msec, waveforms were truncated at 1,2,3,4,5,6, and 8 msec. For tau s of 6 msec, waveforms were truncated at 2,3,4,5,6,8, and 10 msec. For waveforms with tau s of 1.5, leading edge voltage was not significantly different for the waveform durations of 1.5 msec and longer. For waveforms with tau s of 3 msec, leading edge voltage was not significantly different for waveform durations of 2 msec and longer. For waveforms with tau s of 6 msec, there was no significant difference in leading edge voltage for the waveforms tested. Part II: Defibrillation thresholds were determined in another eight dogs for the same three tau s values. For each value of tau s, six biphasic waveforms were tested: 1/1, 2/2, 3/3, 4/4, 5/5, and 6/6 msec. For waveforms with tau s of 1.5 msec, leading edge voltage was a minimum for the 2/2 msec waveform. For waveforms with tau s of 3 msec, leading edge voltage was a minimum for the 3/3 msec waveform. For waveforms with tau s of 6 msec, leading edge voltage was a minimum and not significantly different for the 3/3, 4/4, 5/5, and 6/6 msec waveforms.

CONCLUSIONS

The model predicts the optimal monophasic duration and the first phase of a biphasic waveform to within 1 msec as tau s varies from 1.5 to 6 msec: for tau s equal to 1.5 msec, the optimal monophasic waveform duration and the optimal first phase of a biphasic waveform is 2 msec, for tau s equal to 3.0 msec, the optimal duration is 3 msec, and for tau s equal to 6 msec, the optimal duration is 4 msec. For both monophasic and biphasic waveforms, optimal waveform duration shortens as the waveform time constant shortens.

Response of a single cell to an external electric field

The response of a cell to an external electric field is investigated using dimensional analysis and singular perturbation. The results demonstrate that the response of a cell is a two-stage process consisting of the initial polarization that proceeds with the cellular time constant (< 1 microseconds), and of the actual change of physiological state that proceeds with the membrane time constant (several milliseconds). The second stage is governed by an ordinary differential equation similar to that of a space-clamped membrane patch but formulated in terms of intracellular rather than transmembrane potential. Therefore, it is meaningful to analyze the physiological state and the dynamics of a cell as a whole instead of the physiological states and the dynamics of the underlying membrane patches. This theoretical result is illustrated with an example of an excitation of a cylindrical cell by a transverse electric field.

Strength-interval curves in canine myocardium at very short cycle lengths

While ventricular electrophysiological properties have been intensively studied at normal heart rates, little is known about these properties at the very short cycle lengths (approximately 100 msec), which are present in ventricular fibrillation. We examined refractoriness in the right ventricles of six dogs at stimulation intervals of 80 to 300 msec. Starting at 300 msec, the basic (S1) cycle length was decremented by 10 msec each beat to 200, 150, or 125 msec. A 1-msec premature (S2) stimulus of 1, 5, 10, or 20 mA was then introduced. The S1-S2 interval was decremented until capture was lost. The refractory period was considered to be the shortest interval that captured the heart for each S2 strength. Only pacing episodes that did not induce fibrillation were included. Strength-interval curves maintained the same hyperbolic shape but shifted to very short refractory periods as the S1-S1 interval was decreased. At the shortest S1-S1 intervals, premature stimuli were capable of capturing the heart without inducing ventricular fibrillation for S1-S2 intervals as short as 83 +/- 3 msec. Thus, decremental rapid pacing can produce refractory periods shorter than the cycle length during ventricular fibrillation. This finding suggests that there is no need to postulate a discontinuous jump to new electrophysiological properties or relationships at the onset of fibrillation, but that the capability for fibrillation is an integral part of normal electrophysiological parameters when they are pushed to values that do not occur normally. The results of this study should be useful in the further development of active membrane models and cellular automata models of cellular electrical behavior.

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.

Effective defibrillation in pigs using interleaved and common phase sequential biphasic shocks

Previous studies have shown that low internal defibrillation thresholds (DFTs) can be attained by using two pairs of electrodes and combining biphasic shocks with sequential timing. The purpose of this two-part study was to test the defibrillation efficacy of two new shock sequences, an interleaved biphasic, and a common phase sequential biphasic, that utilized two pairs of electrodes and were developed from the concept of sequential biphasic shocks. In the first part, defibrillation catheters were placed in the right ventricle and the superior vena cava of six anesthetized pigs. A small patch electrode was placed on the LV apex through a subxiphoid incision and a cutaneous patch was placed on the left thorax. The mean DFT energies for the interleaved biphasic (5.2 +/- 0.4 J) and the common phase sequential biphasic waveforms (5.4 +/- 0.4 J) were substantially less (P < 0.0001) than those for either the sequential monophasic (10.6 +/- 1.0 J) or single biphasic waveforms (9.0 +/- 1.0 J). In the second study, which used nine anesthetized pigs, the importance of phase reversal was demonstrated by the finding that the DFT energy of a common phase sequential biphasic shock (6.2 +/- 0.4 J) was much less than a common phase sequential monophasic shock (17.9 +/- 1.3 J, P < 0.0001); furthermore, the average DFT for four common phase sequential biphasic configurations (5.7 +/- 0.2 J) was much less than for a configuration that was similar except that current flow was not reversed in one phase so that no biphasic effect was present (19.7 +/- 1.2 J). The efficacy of common phase sequential biphasics was comparable to that of sequential biphasics. The effectiveness of sequential biphasics, interleaved biphasics, and common phase sequential biphasics is possibly due to two mechanisms: (A) an increase in the potential gradient during a later phase in regions that were low during the first phase, and (B) the exposure of most of the myocardium to a biphasic shock that reduces the minimum extracellular potential gradient needed to defibrillate.

Propagation Versus Delayed Activation During Field Stimulation of Cardiac Muscle

This modeling study seeks to explain the experimentally detected delay between the application of an electric field and the recorded response of the transmembrane potential. In this experiment, conditions were deliberately set so that the field should excite all cells at once and so that no delay should be caused by a propagating wave front. The explanation of the observed delay may lie in the intrinsic properties of the membrane. To test this hypothesis, the strength latency curves were determined for three cases: (1) for a membrane patch model, in which the membrane is uniformly polarized and its intrinsic properties can be studied; (2) for the cardiac strand directly excited by the electric field; and (3) for the cardiac strand excited by a propagating wave front. The models of the membrane patch and the directly excited strand yield excitation delays that are comparable to those observed experimentally in magnitude and in the overall shape of the strength latency curves. The delays resulting from propagation are, in general, dependent on the position along the strand, although for some positions the strength latency curves for propagation are similar to those obtained from the directly activated strand and from the patch model. Therefore, the delay in excitation does not necessarily imply the presence of propagating wave fronts and can be attributed to intrinsic membrane kinetics.

Potential distribution in three-dimensional periodic myocardium--Part II: Application to extracellular stimulation

Modeling potential distribution in the myocardium treated as a periodic structure implies that activation from high-current stimulation with extracellular electrodes is caused by the spatially oscillating components of the transmembrane potential. This hypothesis is tested by comparing the results of the model with experimental data. The conductivity, fiber orientation, the extent of the region, the location of the pacing site, and the stimulus strength determined from experiments are components of the model used to predict the distributions of potential, potential gradient, and the transmembrane potential throughout the region. Next, assuming that a specific value of the transmembrane potential is necessary and sufficient to activate fully repolarized myocardium, the model provides an analytical relation between large-scale field parameters, such as gradient and current density, and small-scale parameters, such as transmembrane potential. This relation is used to express the stimulation threshold in terms of gradient or current density components and to explain its dependence upon fiber orientation. The concept of stimulation threshold is generalized to three dimensions, and an excitability surface is constructed, which for cardiac muscle is approximately conical in shape. The numerical values of transmembrane potential and stimulation thresholds calculated using asymptotic analysis are in agreement with the results of animal experiments, confirming the validity of this approach to study the electrophysiology of periodic cardiac muscle.

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.

Transmural activations and stimulus potentials in three-dimensional anisotropic canine myocardium

Epicardial and endocardial pacing are widely used, yet little is known about the three-dimensional distribution of potentials generated by the pacing stimulus or the spread of activation from these pacing sites. In six open-chest dogs, simultaneous recordings were made from 120 transmural electrodes in 40 plunge electrodes within a 35 X 20 X 5-mm portion of the right ventricular outflow tract during epicardial and endocardial pacing at a strength of twice diastolic threshold and at 1 mA. The magnitude of extracellular potentials generated by the stimulus and the activation times were compared in regions proximal (less than 10-12 mm) and distal to the pacing site. Local fiber orientation was histologically determined at each recording electrode. For endocardial pacing, endocardial potentials were larger than epicardial potentials only in the proximal region (p less than 0.001); while in the distal region, epicardial potentials were larger (p less than 0.001), and endocardial activation occurred earlier than epicardial activation for both regions (p less than 0.001). For epicardial pacing, epicardial potentials were larger than endocardial potentials in both regions (p less than 0.001), and epicardial activation occurred earlier only in the proximal region (p less than 0.02), while endocardial activation occurred before epicardial activation in the distal region (p less than 0.01). In planes of recording electrodes parallel to the epicardium and endocardium, the initial isochrones were elliptical with the major axes of the ellipses along the mean fiber orientation between the pacing site and recording plane rather than along the local fiber orientation in the recording plane. Thus, the ellipses in each plane rotated with respect to each other so that in three dimensions the activation front was helicoid, yet the twist of the helix was less than that of the corresponding transmural rotation of fibers. For pacing from the right ventricular outflow tract, we conclude that beyond 10-12 mm from endocardial and epicardial pacing sites epicardial stimulus potentials in both cases are larger than endocardial potentials because of resistivity differences inside and outside the heart wall and activation in both cases is primarily endocardial to epicardial because of rapid endocardial conduction, and we conclude that the initial spread of activation is helicoid and determined by transmural fiber direction.

Extracellular field required for excitation in three-dimensional anisotropic canine myocardium

It is not known how well potential gradient, current density, and energy correlate with excitation by extracellular stimulation in the in situ heart. Additionally, the influence of fiber orientation and stimulus polarity on the extracellular thresholds for stimulation expressed in terms of these factors has not been assessed. To answer these questions for myocardium in electrical diastole, extracellular excitation thresholds were determined from measurements of stimulus potentials and activation patterns recorded from 120 transmural electrodes in a 35 X 20 X 5-mm region of the right ventricular outflow tract in six open-chest dogs. Extracellular potential gradients, current densities, energies, and their components longitudinal and transverse to the local fiber orientation at each recording site were calculated from the stimulus potentials produced by 3-msec constant-current stimuli. The resulting values in regions directly excited by the stimulus field were compared with the values in regions not directly excited but activated by the spread of wavefronts conducting away from the directly excited region. Magnitudes of 3.66 mA/cm2 for current density, 9.7 microJ/cm3 for energy, and 804 mV/cm for potential gradient yielded minimum misclassifications of 8%, 13%, and 17%, respectively, of sites directly and not directly excited. A linear bivariate combination of the longitudinal (l) and transverse (t) components of the potential gradient yielded 7% misclassification (threshold ratio t/l of 2.88), and linear combination of corresponding current density components yielded 8% misclassification (threshold ratio t/l of 1.04). Anodal and cathodal thresholds were not significantly different (p = 0.39). Potential gradient, current density, and energy strength-duration curves were constructed for pulse durations (D) of 0.2-20 msec. The best fit hyperbolic curve for current density magnitude (Jm) was Jm = 3.97/D + 3.15, where Jm is in mA/cm2, and D is in msec. Thus, for stimulation during electrical diastole 1) both current density magnitude and longitudinal and transverse components of the potential gradient are closely correlated with excitation, 2) the extracellular potential gradient along cardiac cells has a lower threshold than across cells, while current density thresholds along and across cells are similar, 3) anodal and cathodal thresholds are approximately equal for stimuli greater than or equal to 5 mA, and 4) the extracellular potential gradient, current density, and energy excitation thresholds can be expressed by strength-duration equations.