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Patient-specific Modeling of the Heart: Estimation of Ventricular Fiber Orientations
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Published on: January 8, 2013

Understanding cardiac alternans: a piecewise linear modeling framework.

R Thul1, S Coombes

  • 1School of Mathematical Sciences, University of Nottingham, Nottingham, Nottinghamshire NG7 2RD, United Kingdom. ruediger.thul@nottingham.ac.uk

Chaos (Woodbury, N.Y.)
|January 5, 2011
PubMed
Summary
This summary is machine-generated.

Cardiac alternans, a precursor to fibrillation, involves beat-to-beat alternations in action potential duration and calcium cycling. This study simplifies complex cardiac models to mathematically analyze alternans, revealing their emergence via period-doubling instabilities.

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Area of Science:

  • Computational Biology
  • Cardiac Electrophysiology
  • Nonlinear Dynamics

Background:

  • Cardiac alternans, characterized by alternations in action potential duration and intracellular calcium cycling, are linked to cardiac fibrillation.
  • Physiologically detailed models have been developed to study myocyte response to pacing, including alternans.
  • Mathematical analysis of these models often involves approximation techniques to derive low-dimensional maps.

Purpose of the Study:

  • To analyze the dynamical behaviors of the Shiferaw-Karma cardiac model by identifying key organizing elements.
  • To construct a simplified, piecewise linear caricature of the Shiferaw-Karma model for systematic mathematical analysis.
  • To investigate the emergence and properties of cardiac alternans using a novel mathematical framework.

Main Methods:

  • Identification of 'switches' as key organizing elements in the Shiferaw-Karma model's nonlinear dynamics.
  • Construction of a piecewise linear caricature of the model with switching manifolds.
  • Formulation of calcium cycling dynamics and computation of periodic orbits using a stroboscopic map without approximation.
  • Analysis of alternans emergence via period-doubling instability and spatial pattern analysis using a generalized master stability approach.

Main Results:

  • The Shiferaw-Karma model's dynamics are organized by a set of switches, simplifying its nonlinear behavior.
  • A piecewise linear caricature preserves physiological interpretation while enabling systematic mathematical analysis.
  • Cardiac alternans emerge via a period-doubling instability, which can be tracked using physiologically relevant parameters.
  • Coupling with a spatially extended model reveals spatially varying patterns of alternans, analyzed using a generalized master stability approach.

Conclusions:

  • A simplified, piecewise linear model effectively captures the essential dynamics of cardiac alternans and their underlying mechanisms.
  • The study provides a framework for systematic mathematical analysis of cardiac alternans, linking cellular dynamics to macroscopic phenomena.
  • This approach facilitates a deeper understanding of the instabilities leading to cardiac arrhythmias and informs potential therapeutic strategies.