Luciano Guerri

A mathematical procedure for solving the inverse potential problem of electrocardiography. analysis of the time-space accuracy from in vitro experimental data☆

Abstract The inverse potential problem of electrocardiography leads to a Cauchy problem for an elliptic operator and is strongly ill posed. Its solution must be determined by some regularization technique in which a parameter controls the amount of regularization of the solution. Therefore the choice of this smoothing parameter is important for achieving the best accuracy attainable given the discrete approximation errors and the noise level on the data. A regularized inverse procedure is applied to data from an in vitro experiment, and a new criterion for the choice of a quasi-optimal value of the smoothing parameter is described. The performance of this criterion is investigated, and a detailed analysis of the accuracy of the results is carried out. This analysis concerns both the recovered epicardial maps (space analysis) and the ECGs (time analysis).

Finite element approximation of regularized solutions of the inverse potential problem of electrocardiography and applications to experimental data

SummaryThis paper investigates the problem of estimating the electric potential distribution in proximity of the heart from potential data given on the body and is here reformulated as a control problem in terms of a «transfer» operator and stabilized by means of a suitable regularization operator. The numerical approximation by means of the finite element method of the regularized problem is investigated; convergence results and error estimates are established.The numerical procedures for the computation of the approximate «transfer» operator and for the solution of the least square problem approximating the regularized control problem are described. The performance of several criteria for the best choice of the regularization parameter is analyzed. Finally the numerical inverse procedure, extensively tested using test functions and data fromin vitro experiments, is applied to human data.

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.

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.

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.

Spread of Excitation in a Myocardial Volume:.: Simulation Studies in a Model of Anisotropic Ventricular Muscle Activated by Point Stimulation

Spread of Excitation in a Myocardial Volume. Introduction: The purpose of this study was to present simulations of excitation wavefronts spreading through a parallelepipedal slab of ventricular tissue measuring 6.5 × 6.5 × 1.0cm.

Spread of excitation in 3-D models of the anisotropic cardiac tissue. 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.

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

Potential Patterns in a 3‐D Cardiac Depolarization Model, 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 epiendocardial fiber rotation.

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

AbstractThe 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.

ACCURATE COMPUTATION OF ELECTROGRAMS IN THE LEFT VENTRICULAR WALL

An integral representation of electrograms is used for large scale simulations in an anisotropic model of the whole left ventricle. Numerical artifacts, like spurious oscillations or peaks, may appear if the computational grid is not fine enough. To avoid an excessive increase in the number of elements and nodes, we present a numerical procedure based on a combination of: a special refinement of the grid, a sub-element technique, a nonlinear interpolation and a split form of the integral. The numerical simulations show that in this way it is possible to suppress the numerical artifacts thus allowing an accurate computation of electrograms in any point inside or outside the myocardial wall.