Anisotropic mechanisms for multiphasic unipolar electrograms: simulation studies and experimental recordings.

The origin of the multiple, complex morphologies observed in unipolar epicardial electrograms, and their relationships with myocardial architecture, have not been fully elucidated. To clarify this problem we simulated electrograms (EGs) with a model representing the heart as an anisotropic bidomain with unequal anisotropy ratio, ellipsoidal ventricular geometry, transmural fiber rotation, epi-endocardial obliqueness of fiber direction and a simplified Purkinje network. The EGs were compared with those directly recorded from isolated dog hearts immersed in a conducting medium during ventricular excitation initiated by epicardial stimulation. The simulated EGs share the same multiphasic character of the recorded EGs. The origin of the multiple waves, especially those appearing in the EGs for sites reached by excitation wave fronts spreading across fibers, can be better understood after splitting the current sources, the potential distributions and the EGs into an axial and a conormal component and after taking also into account the effect of the reference or drift component. The split model provides an explanation of humps and spikes that appear in the QRS (the initial part of the ventricular EG) wave forms, in terms of the interaction between the geometry and direction of propagation of the wave front and the architecture of the fibers through which excitation is spreading.

Ozonation of blood during extracorporeal circulation. I. Rationale, methodology and preliminary studies.

We investigated whether exposure of blood ex-vivo to oxygen-ozone (O2-O3) through a gas exchanger is feasible and practical. We first evaluated the classical dialysis-type technique but we soon realized that semipermeable membranes are unsuitable because they are hydrophilic and vulnerable to O3. We therefore adopted a system with hydrophobic O3-resistant hollow fibers enclosed in a polycarbonate housing with a membrane area of about 0.5 m2. First we tested the system with normal saline, determining the production of hydrogen peroxide (H2O2) at O3 concentrations from 5 to 40 microg/ml. We then evaluated critical parameters by circulating swine blood in vitro; this revealed that heparin is not an ideal anticoagulant for this system. Finally, we performed several experiments in sheep and defined optimal anticoagulant dose (sodium citrate, ACD), priming solution, volume of blood flow per min, volume and concentration of O2-O3 mixture flowing countercurrent with respect to blood and the time necessary for perfusion in vivo. The biochemical parameters showed that an O3 concentration as low as 10 microg/ml is effective; this means that gas exchange and O3 reactivity are rapid and capable of inducing biological effects. The sheep showed no adverse effects even after 50 min of extracorporeal circulation at higher O3 concentrations (20 to 40 microg/ml) but the exchanger became less effective (low pO2 values) due to progressive clogging with cells.

Multiple components in the unipolar electrogram: a simulation study in a three-dimensional model of ventricular myocardium.

INTRODUCTION: For many decades, the interpretation of unipolar electrograms (EGs) and ECGs was based on simple models of the heart as a current generator, e.g., the uniform dipole layer, and, more recently, the "oblique dipole layer." However, a number of recent and old experimental data are inconsistent with the predictions of these models. To address this problem, we implemented a numerical model simulating the spread of excitation through a parallelepipedal myocardial slab, with a view to identifying the factors that affect the shape, amplitude, and polarity of unipolar EGs generated by the spreading wavefront. METHODS AND RESULTS: The numerical model represents a portion of the left ventricular wall as a parallelepipedal slab (6.5 x 6.5 x 1 cm); the myocardial tissue is represented as an anisotropic bidomain with epi-endocardial rotation of fiber direction and unequal anisotropy ratio. Following point stimulation, excitation times in the entire volume are computed by using an eikonal formulation. Potential distributions are computed by assigning a fixed shape to the action potential profile. EGs at multiple sites in the volume are computed from the time varying potential distributions. The simulations show that the unipolar QRS waveforms are the sum of a "field" component, representing the effect of an approaching or receding wavefront on the potential recorded by a unipolar electrode, and a previously unrecognized "reference" component, which reflects the drift, during the spread of excitation, of the reference potential, which moves from near the positive to near the negative extreme of the potential distribution during the spread of excitation. CONCLUSION: The drift of the reference potential explains the inconsistencies between the predictions of the models and the actual shapes of the EGs. The drift modifies the slopes of EG waveforms during excitation and recovery and can be expected to affect the assessment of excitation and recovery times and QRS and ST-T areas. Removing the drift reestablishes consistency between potential distributions and electrographic waveforms.

Spread of excitation in 3-D models of the anisotropic cardiac tissue. III. Effects of ventricular geometry and fiber structure on the potential distribution.

In a previous paper we studied the spread of excitation in a simplified model of the left ventricle, affected by fiber structure and obliqueness, curvature of the wall and Purkinje network. In the present paper we investigate the extracellular potential distribution u in the same ventricular model. Given the transmembrane potential v, associated with the spreading excitation, the extracellular potential u is obtained as solution of a linear elliptic equation with the source term related to v. The potential distributions were computed for point stimulations at different intramural depths. The results of the simulations enabled us to identify a number of common features which appears in all the potential patterns irrespective of pacing site. In addition, by splitting the sources into an axial and conormal component, we were able to evaluate the contribution of the classical uniform dipole layer to the total potential field and the role of the superimposed axial component.

Spread of excitation in 3-D models of the anisotropic cardiac tissue. II. Effects of fiber architecture and ventricular geometry.

We investigate a three-dimensional macroscopic model of wave-front propagation related to the excitation process in the left ventricular wall represented by an anisotropic bidomain. The whole left ventricle is modeled, whereas, in a previous paper, only a flat slab of myocardial tissue was considered. The direction of cardiac fibers, which affects the anisotropic conductivity of the myocardium, rotates from the epi- to the endocardium. If the ventricular wall is conceived as a set of packed surfaces, the fibers may be tangent to them or more generally may cross them obliquely; the latter case is described by an "imbrication angle." The effect of a simplified Purkinje network also is investigated. The cardiac excitation process, more particularly the depolarization phase, is modeled by a nonlinear elliptic equation, called an eikonal equation, in the activation time. The numerical solution of this equation is obtained by means of the finite element method, which includes an upwind treatment of the Hamiltonian part of the equation. By means of numerical simulations in an idealized model of the left ventricle, we try to establish whether the eikonal approach contains the essential basic elements for predicting the features of the activation patterns experimentally observed. We discuss and compare these results with those obtained in our previous papers for a flat part of myocardium. The general rules governing the spread of excitation after local stimulations, previously delineated for the flat geometry, are extended to the present, more realistic monoventricular model.

Potential distributions generated by point stimulation in a myocardial volume: simulation studies in a model of anisotropic ventricular muscle.

INTRODUCTION: We present simulations of extracellular potential patterns elicited by delivering ectopic stimuli to a parallelepipedal slab of ventricular tissue represented as an anisotropic bidomain incorporating epi-endocardial fiber rotation. METHODS AND RESULTS: Simulations were based on an eikonal model that determines wavefront shapes throughout the slab at every time instant during the depolarization phase, coupled with an approximate model of the action potential profile. The endocardial face of the slab was in contact with blood and the composite volume was surrounded by an insulating medium. The effect of a simplified Purkinje network was also studied. RESULTS: (1) For all pacing depths, except endocardial pacing, a central negative area and two potential maxima were observed at QRS onset in all intramural planes parallel to the epicardium. In all planes, the axis joining the two maxima was approximately aligned with the direction of fibers in the plane of pacing. Endocardial pacing generated a different pattern, but only when blood was present; (2) During later stages of excitation, outflowing currents (from the wavefront toward the resting tissue) were always emitted, at all intramural depths, only from those portions of the wavefront that spread along fibers. At any given instant, the position of the two potential maxima in a series of planes parallel to the epicardium and intersecting the wavefront rotated as a function of depth, following the rotating direction of intramural fibers. Purkinje involvement modified the above patterns. CONCLUSION: Epicardial and endocardial potential maps provided information on pacing site and depth and on subsequent intramural propagation by reflecting the clockwise or counter-clockwise rotation of the deep positivity. Results may be applicable to epicardial and endocardial potential maps recorded at surgery or from endocavitary probes.

Spread of excitation in a myocardial volume: simulation studies in a model of anisotropic ventricular muscle activated by point stimulation.

INTRODUCTION: The purpose of this study was to present simulations of excitation wavefronts spreading through a parallelepipedal slab of ventricular tissue measuring 6.5 x 6.5 x 1.0 cm. METHODS AND RESULTS: The slab incorporates the anisotropic properties of the myocardium including the transmural counterclockwise fiber rotation from epicardium to endocardium. Simulations were based on an eikonal model that determines excitation times throughout the ventricular wall, which is represented as an anisotropic bidomain. Excitation was initiated by delivering ectopic stimuli at various intramural depths. We also investigated the effect of a simplified Purkinje network on excitation patterns. Excitation wavefronts in the plane of pacing, parallel to epicardial-endocardial surfaces, were oblong with the major axis approximately oriented along the local fiber direction, with bulges and deformations due to attraction from rotating fibers in adjacent planes. The oblong intersections of the wavefront with planes at increasing distance from pacing plane rotated clockwise or counterclockwise, depending on pacing depth, but wavefront rotation was always less than fiber rotation in the same plane. For all pacing depths, excitation returned toward the plane of pacing. Return occurred in multiple, varying sectors of the slab depending on pacing depth, and was observed as close as 6 mm to the pacing site. CONCLUSION: Curvature of wavefronts and collision with boundaries of slab markedly affected local velocities. Shape and separation of epicardial isochrones and spatial distribution of epicardial velocities varied as a function of site and depth of pacing. When the Purkinje network was added to the model, epicardial velocities revealed the subendocardial location of the Purkinje-myocardial junctions. Considerable insight into intramural events could be obtained from epicardial isochrones. If validated experimentally, results may be applicable to epicardial isochrones recorded at surgery.

Spreading of excitation in 3-D models of the anisotropic cardiac tissue. I. Validation of the eikonal model.

In this work we investigate, by means of numerical simulations, the performance of two mathematical models describing the spread of excitation in a three dimensional block representing anisotropic cardiac tissue. The first model is characterized by a reaction-diffusion system in the transmembrane and extracellular potentials v and u. The second model is derived from the first by means of a perturbation technique. It is characterized by an eikonal equation, nonlinear and elliptic in the activation time psi(x). The level surfaces psi(x) = t represent the wave-front positions. The numerical procedures based on the two models were applied to test functions and to excitation processes elicited by local stimulations in a relatively small block. The results are in excellent agreement, and for the same problem the computation time required by the eikonal equation is a small fraction of that needed for the reaction-diffusion system. Thus we have strong evidence that the eikonal equation provides a reliable and numerically efficient model of the excitation process. Moreover, numerical simulations have been performed to validate an approximate model for the extracellular potential based on knowledge of the excitation sequence. The features of the extracellular potential distribution affected by the anisotropic conductivity of the medium were investigated.

Models of the spreading of excitation in myocardial tissue.

We consider a macroscopic model of the excitation process in the anisotropic myocardium involving the transmembrane, extracellular, and extracardiac potentials v, ue, and u0. The model is described by a reaction-diffusion (R-D) system, and the component v exhibits a front-like behavior reflecting the features of the excitation process. In numerical simulations, the presence of a moving excitation layer imposes severe constraints on the time and space steps to achieve stability and accuracy; consequently, application of the model is very costly in terms of computer time. An approximate model has been derived from the R-D system by means of a singular perturbation technique, and it is described by an eikonal equation, nonlinear and elliptic, in the activation time psi (x). Larger space steps are possible with this equation. From psi (x), we can derive, for a given instant t, the transmembrane potential v and subsequently, by solving an elliptic problem, we can compute the corresponding extracellular and extracardiac potentials ue and u0. The results of the R-D and the eikonal models applied to a portion of the ventricular wall are in excellent agreement; moreover, the eikonal model requires only a small fraction of the computer time needed by the R-D system. Therefore, for large-scale simulations of the excitation process, only the eikonal model has been used, and we investigate its ability to cope with complex situations such as front-front collisions and related potential patterns.

Mathematical modeling of the excitation process in myocardial tissue: influence of fiber rotation on wavefront propagation and potential field.

In our macroscopic model the heart tissue is represented as a bidomain coupling the intra- and extracellular media. Owing to the fiber structure of the myocardium, these media are anisotropic, and their conductivity tensors have a principal axis parallel to the local fiber direction. A reaction-diffusion system is derived that governs the distribution and evolution of the extracellular and transmembrane potentials during the depolarization phase of the heart beat. To investigate frontlike solutions, the system is rescaled and transformed into a system dependent on a small parameter. Subsequently a perturbation analysis is carried out that yields zero- and first-order approximations called eikonal equations. The effects of the transmural fiber rotation on wavefront propagation and the corresponding potential field, elicited by point stimulations, are investigated by means of numerical simulations.

Wavefront propagation in an activation model of the anisotropic cardiac tissue: asymptotic analysis and numerical simulations.

In this paper we present a macroscopic model of the excitation process in the myocardium. The composite and anisotropic structure of the cardiac tissue is represented by a bidomain, i.e. a set of two coupled anisotropic media. The model is characterized by a non linear system of two partial differential equations of parabolic and elliptic type. A singular perturbation analysis is carried out to investigate the cardiac potential field and the structure of the moving excitation wavefront. As a consequence the cardiac current sources are approximated by an oblique dipole layer structure and the motion of the wavefront is described by eikonal equations. Finally numerical simulations are carried out in order to analyze some complex phenomena related to the spreading of the wavefront, like the front-front or front-wall collision. The results yielded by the excitation model and the eikonal equations are compared.

Oblique dipole layer potentials applied to electrocardiology.

We study the properties of the potential field generated by an oblique dipole layer. This field arises, for instance, in describing the potential elicited by a depolarization wavefront spreading in the myocardium when a dependence of the potential on the cardiac fiber orientation is introduced. The representation of cardiac bioelectric sources by means of an oblique dipole layer leads to a mathematical structure which generalizes the classical solid angle theory used in electrocardiology, which has been challenged by recent experimental evidence, and links models previously proposed with a view to adequately reproduce the potential observed in experiments. We investigate also the relationship between our model and an intracellular current model and we derive potential jump formulae for some models which account for the anisotropic structure of the myocardium. The potential generated by an oblique dipole layer is considered both for unbounded and bounded domains. In the latter case an integral boundary equation is derived and we study its solvability. A numerical procedure for solving this integral equation by means of the finite element method with collocation is outlined.

Potential fields generated by oblique dipole layers modeling excitation wavefronts in the anisotropic myocardium. Comparison with potential fields elicited by paced dog hearts in a volume conductor.

The potential distribution in a homogeneous, cylindrical volume conductor surrounding an isolated paced dog heart was first measured and then calculated by using a mathematical model that stimulates an anisotropic excitation wavefront spreading through the heart muscle. The study was performed with a view to establish to what extent the anisotropy of cardiac generators affects the potential field in the extra-cardiac conducting media at a great distance from the heart. The model considers an oblique dipole layer on the wavefront which, assuming axial symmetry of the electrical properties of the fibers, can be viewed as the superposition of an axial and transverse dipole layer. These layers are, respectively, parallel and perpendicular to the local fiber due to such an oblique distribution is also equivalent to the sum of the potentials generated, respectively, by a normal and an axial dipole layer. In this form, the model generalizes the classical, uniform double layer model, upon which the solid angle theory is based, by adding to it an axial component. The features of the measured potential fields, which could not be interpreted on the basis of the solid angle theory, were satisfactorily reproduced by the model, at least on a qualitative basis. The results clearly showed the dominant role played by the axial component of the potential field even at a considerable distance from the heart.