Atrial Rotor Dynamics Under Complex Fractional Order Diffusion

Abstract

The mechanisms of atrial fibrillation (AF) are a challenging research topic. The rotor hypothesis states that the AF is sustained by a reentrant wave that propagates around an unexcited core. Cardiac tissue heterogeneities, both structural and cellular, play an important role during fibrillatory dynamics, so that the ionic characteristics of the currents, their spatial distribution and their structural heterogeneity determine the meandering of the rotor. Several studies about rotor dynamics implement the standard diffusion equation. However, this mathematical scheme carries some limitations. It assumes the myocardium as a continuous medium, ignoring, therefore, its discrete and heterogeneous aspects. A computational model integrating both, electrical and structural properties could complement experimental and clinical results. A new mathematical model of the action potential propagation, based on complex fractional order derivatives is presented. The complex derivative order appears of considering the myocardium as discrete-scale invariant fractal. The main aim is to study the role of a myocardial, with fractal characteristics, on atrial fibrillatory dynamics. For this purpose, the degree of structural heterogeneity is described through derivatives of complex order γ = α + jβ. A set of variations for γ is tested. The real part α takes values ranging from 1.1 to 2 and the imaginary part β from 0 to 0.28. Under this scheme, the standard diffusion is recovered when α = 2 and β = 0. The effect of γ on the action potential propagation over an atrial strand is investigated. Rotors are generated in a 2D model of atrial tissue under electrical remodeling due to chronic AF. The results show that the degree of structural heterogeneity, given by γ, modulates the electrophysiological properties and the dynamics of rotor-type reentrant mechanisms. The spatial stability of the rotor and the area of its unexcited core are modulated. As the real part decreases and the imaginary part increases, simulating a higher structural heterogeneity, the vulnerable window to reentrant is increased, as the total meandering of the rotor tip. This in silico study suggests that structural heterogeneity, described by means of complex order derivatives, modulates the stability of rotors and that a wide range of rotor dynamics can be generated.

Commentary: Atrial Rotor Dynamics Under Complex Fractional Order Diffusion

“…Whereas the fractional Laplacian (Figure 1A ) correctly replicates for all α the circular propagation patterns observed on isotropic cardiac monolayers as the simplest yet inhomogeneous in-vitro model of cardiac tissue (Badie and Bursac, 2009 ; Bian et al, 2014 ; Molitoris et al, 2016 ), the fractional Riesz operator (Figure 1B ) induces increasingly larger curvature artifacts on wavefront conduction for decreasing α. Such curvature artifacts indeed translate into the results of Ugarte et al ( 2018 ), as evidenced by their square-like spiral wavefronts and rotor trajectories. Given the well-known curvature-related modulation of conduction velocity and therefore wavefront-waveback interactions (Fast and Kléber, 1997 ; Comtois and Vinet, 1999 ; Comtois et al, 2005 ; Kadota et al, 2012 ), their results on vulnerability to re-entry and associated rotor biomarkers thus must be cautiously interpreted.…”

“…It is nevertheless relevant to note that more squared propagation patterns have been reported in both optical mapping (Koura et al, 2002 ; de Diego et al, 2011 ) and computational (He and Liu, 2010 ) studies. This was however under marked anisotropic conduction, not accounted in the isotropic model by Ugarte et al ( 2018 ). In addition, fractional Riesz operators have been also used in modeling electrical propagation (Liu et al, 2013 , 2015 ; Zeng et al, 2014 ).…”

“…However, an important limitation of Ugarte et al ( 2018 ) is that their proposed model is not consistent with the fractional Laplacian in which the authors base their analysis. Taking β = 0 for simplicity, Equations (2), (3) then reduce to…”

“…(A) Fractional Laplacian (Bueno-Orovio et al, 2014b ). (B) Fractional Riesz operator (Ugarte et al, 2018 ). Diffusion coefficients were optimized to match standard diffusion conduction velocity at the center of 5 cm fiber strands.…”

Proliferation of Fibroblast Modulates the Action Potential Duration Dispersion: An Atrial Fibrosis Model Using Fractional Diffusion

“…Fractional calculus generalizes the classical integer order derivatives for real or complex orders [6]. Recent works in cardiac electrophysiology modeling account the fractional derivative order as the degree of structural heterogeneity in healthy ventricular [7] tissue and in atrial remodeled tissue due to chronic AF [8]. Thus, the aim of this work is to assess properties of a fractional order diffusion operator under the presence of cardiomyocytes and fibroblasts, as a model of fibrosis.…”