Geometrical measurement of cardiac wavelength in reaction-diffusion models

Abstract

The dynamics of reentrant arrhythmias often consists in multiple wavelets propagating throughout an excitable medium. An arrhythmia can be sustained only if these reentrant waves have a sufficiently short wavelength defined as the distance traveled by the excitation wave during its refractory period. In a uniform medium, wavelength may be estimated as the product of propagation velocity and refractory period (electrophysiological wavelength). In order to accurately measure wavelength in more general substrates relevant to atrial arrhythmias (heterogeneous and anisotropic), we developed a mathematical framework to define geometrical wavelength at each time instant based on the length of streamlines following the propagation velocity field within refractory regions. Two computational methods were implemented: a Lagrangian approach in which a set of streamlines were integrated, and an Eulerian approach in which wavelength was the solution of a partial differential equation. These methods were compared in 1D/2D tissues and in a model of the left atrium. An advantage of geometrical definition of wavelength is that the wavelength of a wavelet can be tracked over time with high temporal resolution and smaller temporal variability in an anisotropic and heterogeneous medium. The results showed that the average electrophysiological wavelength was consistent with geometrical measurements of wavelength. Wavelets were however often shorter than the electrophysiological wavelength due to interactions with boundaries and other wavelets. These tools may help to assess more accurately the relation between substrate properties and wavelet dynamics in computer models.

Geometric Wavelength for Monitoring Dynamic Changes in Repolarization Caused by Time-Varying Parameters

“…For each action potential, conduction velocity and action potential duration were measured; electrophysiological wavelength (EWL) was defined as their product. EWL values were assigned to the times of repolarization of the corresponding action potentials to create a time series of wavelengths [5].…”

“…Mathematical approaches have been proposed to calcu-late a geometrical wavelength (GWL) that measures the actual spatial extent of depolarization wavelets, thus offering a more complete description of the dynamics [5]. In a uniform isotropic medium, GWL is essentially equivalent to EWL, but has the advantage of providing an instantaneous measure.…”

“…The challenge is then to monitor the time evolution of wavelength. Dupraz et al [5] investigated the monitoring of wavelength in tissues with functional heterogeneities. In this paper, we explore the effect of time-dependent repolarization-related parameter variations on wavelength using reentries in a simple ring model of acetylcholine (ACh) release and degradation.…”

Nonlinear diffusion and thermo-electric coupling in a two-variable model of cardiac action potential

“…The resulting set of equations constitute a nonlinear reaction-diffusion (RD) system in which spatial propagation of electrical potentials are coupled with evolution equations describing the ion-channel, exchanger and pump gate dynamics 39 . On these bases, spatio-temporal irregularities in excitable media have been studied from several points of view 4,55 and multiple approaches have been proposed to understand and control spiral waves dynamics in heterogeneous active media 5,6, 19,46,47,49,54 .…”

“…PACS numbers: 87. 19 Systemic temperature is kept approximately constant through many delicate physiological feedbacks. In the heart, temperature changes greatly affect the features of the excitation wave and occur because of pathological conditions, during surgery or because of various unfortunate events.…”