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.

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. 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.

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.

A mathematical model of iron metabolism.

A mathematical model of iron metabolism is presented. It comprises the following iron pools within the body: transferrin-bound iron in the plasma, iron in circulating red cells and their bone marrow precursors, iron in mucosal, parenchymal and reticuloendothelial cells. The control exerted by a hormone, called erythropoietin, on bone marrow utilization of iron for hemoglobin synthesis is taken into account. The model so obtained consists of a system of functional differential equations of retarded type. Most model parameters can be estimated from radiotracer experiments, others can be measured and numerical values can be assigned to the remaining ones making few reasonable assumptions according to the available physiological knowledge. Iron metabolism behavior under different therapeutical treatments was stimulated. Model predictions were compared to experimental data collected in clinical routine.

An approach to inverse calculation of epicardial potentials from body surface maps.

The inverse problem of evaluating epicardial potentials from a knowledge of heart and torso geometry as well as body surface potentials is here formulated as a problem in control theory. As is well known, such an inverse problem is ill-posed and a regularization technique has been devised to overrun this difficulty. The resulting regularized problem is well-posed and requires the minimization of a cost function including, besides the square distance of any predicted surface potential distribution from the experimental one, a regularization term involving the second derivatives of the identified epicardial potentials. The results here presented were obtained on a model problem for a plane geometry. Surface potentials generated by multipoles and perturbated with a noise level reflecting both instrumentation and electrode placement uncertainties were fitted by the proposed method and 'epicardial potentials' were determined with a maximum sum square relative error of 15%. The results suggest that by introducing suited regularity constraints, the a priori difficulties inherent to the problem of computing epicardial potentials from torso potentials, can be overcome.