Cardiac memory phenomenon, time-fractional order nonlinear system and bidomain-torso type model in electrocardiology

Aziz Belmiloudi; Aziz Belmiloudi

doi:10.3934/math.2021050

AIMS Mathematics

2021, Volume 6, Issue 1: 821-867. doi: 10.3934/math.2021050

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Cardiac memory phenomenon, time-fractional order nonlinear system and bidomain-torso type model in electrocardiology

Aziz Belmiloudi ^,

Institut de Recherche Mathématique de Rennes (IRMAR), Université Européenne de Bretagne, 20 avenue Buttes de Coësmes, CS 70839, 35708 Rennes Cédex 7, France

Received: 11 June 2020 Accepted: 11 October 2020 Published: 02 November 2020
MSC : 35R11, 34A08, 35K57, 35B30, 92C50, 35Q92

Cardiac memory, also known as the Chatterjee phenomenon, refers to the persistent but reversible T-wave changes on ECG caused by an abnormal electrical activation pattern. After a period of abnormal ventricular activation in which the myocardial repolarization is altered and delayed (such as with artificial pacemakers, tachyarrhythmias with wide QRS complexes or ventricular pre-excitation), the heart remembers and mirrors its repolarization in the direction of the vector of "abnormally" activated QRS complexes. This phenomenon alters patterns of gap junction distribution and generates changes in repolarization seen at the level of ionic-channel, ionic concentrations, ionic-current gating and action potential. In this work, we propose a mathematical model of cardiac electrophysiology which takes into account cardiac memory phenomena. The electrical activity in heart through torso, which is dependent on the prior history of accrued heartbeats, is mathematically modeled by a modified bidomain system with time fractional-order dynamics (which are used to describe processes that exhibit memory). This new bidomain system, that I name "it memory bidomain system", is a degenerate nonlinear coupled system of reaction-diffusion equations in shape of a fractional-order differential equation coupled with a set of time fractional-order partial differential equations. Cardiac memory is represented via fractional-order capacitor (associate to capacitive current) and fractional-order cellular membrane dynamics. First, mathematical model is introduced. Afterward, results on generalized Gronwall inequality within the framework of coupled fractional differential equations are developed. Next, the existence and uniqueness of solution of state system are proved as well as stability result. Further, some preliminary numerical applications are performed to show that memory reproduced by fractional-order derivatives can play a significant role on key dependent electrical properties including APD, action potential morphology and spontaneous activity.
- fractional-order dynamics,
- heart-torso coupling,
- cellular heterogeneity,
- integral inequality,
- weak solution,
- cardiac memory,
- memory induced T-wave,
- ionic models
Citation: Aziz Belmiloudi. Cardiac memory phenomenon, time-fractional order nonlinear system and bidomain-torso type model in electrocardiology[J]. AIMS Mathematics, 2021, 6(1): 821-867. doi: 10.3934/math.2021050

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Abstract

Cardiac memory, also known as the Chatterjee phenomenon, refers to the persistent but reversible T-wave changes on ECG caused by an abnormal electrical activation pattern. After a period of abnormal ventricular activation in which the myocardial repolarization is altered and delayed (such as with artificial pacemakers, tachyarrhythmias with wide QRS complexes or ventricular pre-excitation), the heart remembers and mirrors its repolarization in the direction of the vector of "abnormally" activated QRS complexes. This phenomenon alters patterns of gap junction distribution and generates changes in repolarization seen at the level of ionic-channel, ionic concentrations, ionic-current gating and action potential. In this work, we propose a mathematical model of cardiac electrophysiology which takes into account cardiac memory phenomena. The electrical activity in heart through torso, which is dependent on the prior history of accrued heartbeats, is mathematically modeled by a modified bidomain system with time fractional-order dynamics (which are used to describe processes that exhibit memory). This new bidomain system, that I name "it memory bidomain system", is a degenerate nonlinear coupled system of reaction-diffusion equations in shape of a fractional-order differential equation coupled with a set of time fractional-order partial differential equations. Cardiac memory is represented via fractional-order capacitor (associate to capacitive current) and fractional-order cellular membrane dynamics. First, mathematical model is introduced. Afterward, results on generalized Gronwall inequality within the framework of coupled fractional differential equations are developed. Next, the existence and uniqueness of solution of state system are proved as well as stability result. Further, some preliminary numerical applications are performed to show that memory reproduced by fractional-order derivatives can play a significant role on key dependent electrical properties including APD, action potential morphology and spontaneous activity.

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