Fernand A. Roberge

Reconstruction of propagated electrical activity with a two-dimensional model of anisotropic heart muscle.

The propagated electrical activity in normal anisotropic cardiac muscle is characterized by directionally dependent variations in the rising phase of the action potential. An important question concerns the relation between such variations and the propagation velocity and extracellular potentials. This problem was studied here in a sheet of cells, under conditions of uniform inrracellular anisotropic resistivity and constant electrical membrane properties, through a numerical solution of the two-dimensional propagation equation. The numerical solution implies a lumping of the cytoplasmic and intercellular resistances into an equivalent junctional resistance to form a distributed resistive network representing the intracellular domain. The interstitial space is assumed isotropic and unbounded, with a resistivity of 100 Ω ± cm. The electrical properties of the cell membrane are represented by a Beeler-Reuter model. The stimulus current is applied to a small area of the sheet, and attention is focussed on the stable propagated events occurring some 5 or 6 length constants away from the stimulation site. The numerical solution is a good approximation of a continuous uniform structure when the cell size is less than 10% of the length constant along both major axes. Conditions of non-uniform propagation, with directionally dependent variations in the maximum rate of rise and time constant of the foot of the action potential were simulated by increasing the cell size to 30% of the length constant in the transverse direction of the sheet. Our results indicate that the directional changes in the maximum rate of rise correspond to small modifications of the extracellular potentials, while the directional changes in time constant of the foot are associated with the propagation velocity. The maximum effects are observed along the transverse direction as follows: a 19% increase in maximum rate of rise corresponds to a decrease of about 6% in the peak-to-peak amplitude of the extracellular potential, and a 24% increase in time constant of the foot is associated with a decrease of about 7% in the propagation velocity. Under the conditions of the present study, however, the simulated directional changes in maximum rate of rise are smaller than those experimentally observed so the corresponding changes in the extracellular potentials are probably underestimated.

Simulation of two-dimensional anisotropic cardiac reentry: effects of the wavelength on the reentry characteristics.

A two-dimensional sheet model was used to study the dynamics of reentry around a zone of functional block. The sheet is a set of parallel, continuous, and uniform cables, transversely interconnected by a brick-wall arrangement of fixed resistors. In accord with experimental observations on cardiac tissue, longitudinal propagation is continuous, whereas transverse propagation exhibits discontinuous features. The width and length of the sheet are 1.5 and 5 cm, respectively, and the anisotropy ratio is fixed at approximately 4∶1. The membrane model is a mofified Beeler-Reuter formulation incorporating faster sodium current dynamics. We fixed the basic wavelength and action potential duration of the propagating impulse by dividing the time constants of the secondary inward current by an integerK. Reentry was initiated by a standard cross-shock protocol, and the rotating activity appeared as curling patterns around the point of junction (the q-point) of the activation (A) and recovery (R) fronts. The curling R front always precedes the A front and is separated from it by the excitable gap. In addition, the R front is occasionally shifted abruptly through a merging with a slow-moving triggered secondary recovery front that is dissociated from the A front and q-point. Sustained irregular reentry associated with substantial excitable gap variations was simulated with short wavelengths (K=8 andK=4). Unsustained reentry was obtained with a longer wavelength (K=2), leading to a breakup of the q-point locus and the triggering of new activation fronts.

Anisotropic conduction and functional dissociation of ischemic tissue during reentrant ventricular tachycardia in canine myocardial infarction.

We measured the conduction characteristics at the epicardial surface of the left anterior ventricular wall in the in situ canine heart before and 3 to 5 days (n = 9 dogs) after permanent occlusion of the left anterior descending coronary artery (LAD). During ventricular stimulation generating wavefronts conducted along the longitudinal or the transverse fiber direction, 61 unipolar electrograms were recorded with a fine-meshed plaque electrode. Before occlusion, the fastest conduction velocity was consistently found in a direction perpendicular to the nearby LAD segment (longitudinal direction), and the slowest velocity in a direction parallel to the LAD segment (transverse fiber direction). In 3- to 5-day-old infarct preparations, a layer of subepicardial muscle with 1 to 3 mm thickness survived over necrotic tissue. The velocities and directions of fast and of slow conduction measured in ischemic subepicardial muscle were not significantly different from preocclusion values during stimulation at a basic rate, but excitability was found to be depressed in response to premature stimuli. Premature impulses initiated in nonischemic myocardium and conducted into ischemic tissue in the longitudinal or in the transverse directions induced sustained (greater than 100 beats) monomorphic tachycardias during which figure-eight activation patterns were mapped with sock-array electrodes. During these tachycardias, the direction of the common reentrant wavefront of the figure-eight pattern was preferentially oriented along the longitudinal fiber direction, independently of the direction of the initiating impulse. When polymorphic beats were induced, tachycardia terminated spontaneously within 20 beats, or changed to a monomorphic pattern, as described above. In conclusion, the anisotropic organization of surviving subepicardial muscle overlying an infarct provides a spatial constraint that determines a preferential direction of reentrant propagation and may contribute to sustaining monomorphic tachycardia.

Numerical integration in the reconstruction of cardiac action potentials using Hodgkin-Huxley-type models

A comparison between traditional numerical integration methods and a new hybrid integration method for the reconstruction of action potential activity is presented, using a mathematical model of the cardiac Purkinje fiber (MNT model). It is shown that the hybrid integration method reduces importantly the overall computation time required for solving the Hodgkin-Huxley differential equations describing membrane electrical events. To accomplish this, the particular form of the gating variable equations is exploited to reformulate the step-by-step computation. In this way, the time increment can be made much larger compared with traditional methods when the membrane potential changes slowly. A mathematical analysis of the hybrid integration method is presented also, together with a numerical verification of its performance both for the propagated and nonpropagated membrane action potential. It is shown that the local error, that is the error arising at each integration step, and the cumulative integration error are strictly controlled by the membrane potential offset. Using the MNT model, the nonpropagated cardiac Purkinje action potential can be reconstructed in real time with an accuracy of 1% for the potential and 5% for the time of occurrence of its main features. In reconstructing propagated events, the hybrid integration method allows computation time savings by a factor of 10 or more compared to accurate Runge-Kutta schemes.

Directional characteristics of action potential propagation in cardiac muscle. A model study.

Propagation of an elliptic excitation wave front was studied in a two-dimensional model of a thin sheet of cardiac muscle. The sheet model of 2.5 x 10 mm consisted of a set of 100 parallel cables coupled through a regular array of identical transverse resistors. The membrane dynamics was represented by a modified Beeler-Reuter model. We defined the charging factor (CF) to represent by a single number the proportion of input current used to charge the membrane locally below threshold and showed that CF is inversely correlated with the time constant of the foot of the action potential (tau foot) during propagation on a cable. A safety factor of propagation (SF) was also defined for the upstroke of the action potential, with SF directly correlated with the maximum rate of depolarization (Vmax) and, for cablelike propagation, with propagation velocity. Propagation along the principal longitudinal axis of the elliptic wave front is cablelike but, in comparison with a flat wave front, transverse current flow provides a drag effect that somewhat reduces the propagation velocity, Vmax, SF, and CF. With a longitudinal-to-transverse velocity ratio of 3:1 or more, the wave front propagating along the principal transverse axis is essentially flat and is characterized by multiple collisions between successive pairs of input junctions on a given cable; Vmax, SF, and CF are larger than for longitudinal propagation, but CF is no longer correlated with tau foot. There are transient increases in propagation velocity and Vmax with distance from the stimulation site along both principal axes until stablized values are achieved, and a similar transient decrease in tau foot. Away from the principal axes, the action potential characteristics change progressively along the elliptic wave front.(ABSTRACT TRUNCATED AT 250 WORDS)

The dynamics of sustained reentry in a ring model of cardiac tissue

This paper describes the dynamics of circus movement around a fixed obstacle, using a one-dimensional continuous and uniform ring model of cardiac tissue to simulate sustained reentry. The membrane ionic current is simulated by a modified Beeler-Reuter formulation in which the kinetics of the fast sodium current were updated using more recent voltageclamp data. Changes in the ring length are used to modify the dynamics of reentry. Reentry is stable if the ring length (X) exceeds a critical value (Xcrit) and complete block occurs ifX is below a minimum (Xmin). Irregular sustained reentry is observed at intermediate ring lengths, as a narrow range of aperiodic reentry nearXcrit, and a larger range of quasi-periodic reentry at shorter ring lengths. The basic pattern of irregular reentry is an alternation between long and short cycle length, action potential duration (APD), diastolic interval (DIA), wavelength, and excitable gap. In aperiodic reentry cycle length variations are small,APD andDIA fluctuations are of medium amplitude, and conduction velocity over the whole pathway is essentially constant during successive turns. Much larger fluctuations in these various quantities occur during quasi-periodic reentry, and they increase in size asX approachesXmin. The complexity of quasiperiodic reentry patterns is related to three factors: the slope of theAPD versus DIA relation, which is greater than 1, the existence of a zone of slow conduction on the ring when the excitable gap becomes quite short, and the occurrence of triggered waves of secondary repolarization and excitability recovery. In the present model, quasi-periodic reentry with triggered secondary recovery covers most of the range of ring lengths, giving rise to sustained irregular reentry. There is very close agreement between our simulation results and experimental data obtained on rings of cardiac tissue.

Application of the single moving dipole inverse solution to the study of the wolff-parkinson-white syndrome in man

The single moving dipole (SMD) inverse solution was performed in 28 patients with the Wolff-Parkinson-White preexcitation syndrome to see if the calculated position of the SMD during the initial delta wave could indicate the site of the underlying accessory pathway. This site was first estimated to be at one of eight locations around the atrioventricular ring, from the patients QRS and ST segment body surface potential maps, as has been described by others. Next, SMD parameters were calculated during the delta wave so as to approximate, on a numerical torso model, the patients body surface potential map. Visualization of the calculated position of the SMD around the atrioventricular ring was done by projecting it on a plane parallel to this ring. This plane corresponded to the most basal transverse section of a heart model present in the torso model. One limitation was the use of non-varying heart and torso models for all patients. As a result, the SMD technique lacked the precision to separate accessory pathway sites into eight atrioventricular locations. However it was capable of distinguishing between patients belonging to the larger classes of right-sided, posterior, and left-sided preexcitation, formed by combining adjacent atrioventricular accessory pathway locations. With more accurate heart and torso models, it may be possible to increase SMD resolution so as to locate accessory pathway sites deep within the heart. This would represent an advantage over the surface potential map approach which only identifies the site of earliest epicardial breakthrough associated with the accessory pathway.

Representation of cardiac electrical activity by a moving dipole for normal and ectopic beats in the intact dog.

We evaluated the ability of a computer procedure to locate cardiac electrical activity from body surface potentials during normal and ectopic beats in six intact dogs. The location of the dipole that best reconstructed the signals recorded from 26 thoracic electrodes was computed by using a torso model. This model was designed from geometrical measurements made on the first dog. Chronically implanted subepicardial electrodes produced ectopic foci at three known locations: apex, and right and left ventricles. For the normal QRS complex, the dipole started in the middle of the septum, moved upward, then to the left, downward, and back to the right; it remained stationary at the level of the base during repolarization. After ectopic stimulation the dipole started its course at a distance of 1.9 ± 0.8 (SD) cm from the stimulus site and then traversed the heart, moving away from the ectopic site; during the T wave, it roughly followed the QRS path, but at a slower speed. The speed of the dipole ranged between 0 and 6 m/sec, and its orientation often did not coincide with the direction of its path. The location of the dipole appeared to be close to the corresponding wavefronts when these were unique and dipolar, especially during the early and terminal portions of the QRS. The results show that the initial location of the dipole can give the approximate position of an ectopic focus in vivo, and that the trajectory of the dipole during QRS can portray the passage of an ectopic beat across the heart. Circ Res 46: 415-425, 1980

Structural complexity effects on transverse propagation in a two-dimensional model of myocardium

A thin sheet of cardiac tissue was modeled as a set of resistively coupled excitable cables with membrane dynamics described by the modified Beeler Reuter model. Transverse connections have a resistance R/sub n/ and are regularly distributed with a spacing Delta on any given cable, to provide alternating input and output junctions. Flat wave longitudinal propagation corresponds to propagation along a single continuous cable since all units of the network are functionally isolated due to the absence of transverse current flow. Overall, the behavior of the network model is in good agreement with available structural and electrophysiological data on myocardium. The network topology allows parameters governing propagation to be addressed more easily and avoids very large and computationally costly matrices.<<ETX>>

Localization of cardiac ectopic activity in man by a single moving dipole. Comparison of different computation techniques

The accuracy of different computation techniques for the non-invasive localization of cardiac ectopic activity was evaluated. Body surface potentials were recorded from 63 leads in 14 patients with implanted pacemakers. The location, orientation and magnitude of a single moving dipole (SMD) were computed from the first eight terms of a truncated multipole expansion estimated from the body surface potentials. The SMD trajectories obtained during the QRS complex were plotted along with the heart outlines and pacing leads obtained independently from chest x-rays. The origin of the SMD trajectories was compared to the position of the pacing lead to evaluate the accuracy of the SMD. The optimum computation technique used a least-squares (LS) estimation of the multipole expansion truncated at 15 multipoles, in conjunction with a torso model that included regions of lower conductivity representing the lungs. With this method, the SMD trajectories originated near the pacing lead (25 +/- 12 mm) and adequately represented the progression of the ectopic wavefront across the entire heart silhouette. With the LS techniques using 8 or 24 multipoles, the spans of the trajectories were respectively too short, or too long to cover the heart, and the average distance between the SMD at QRS onset and the pacing lead was larger. With a surface integration technique, the SMD-pacing lead distances were similar, both for a finite homogeneous torso model with a fixed geometry, as well as for torso models adapted to the torso geometry of each patient. The SMD was found adequate to represent the progression of an ectopic wavefront, and to localize its origin in man.