[Paper Review] Dynamics of wave fronts and filaments in anisotropic cardiac tissue

Hans Dierckx|arXiv (Cornell University)|Nov 11, 2015

Advanced Neuroimaging Techniques and Applications78 references4 citations

TL;DR

This paper investigates the dynamics of electrical wave fronts and filaments in anisotropic cardiac tissue using mathematical modeling and diffusion tensor MRI (DTI) to map fiber and sheet structures. It derives a twist equation for scroll wave filaments and shows that anisotropy in tissue architecture—particularly sheet curvature and fiber orientation—significantly influences filament stability and rotation, with key contributions from symmetry-based tensor decompositions in the governing equations.

ABSTRACT

The heartbeat is mediated between cardiac cells by waves of electrical depolarisation. During cardiac arrhythmias, electrical activity was found to be organised in scroll waves which rotate around a dynamical filament curve. In this thesis, a curved-space approach is used to mathematically capture anisotropy of wave propagation. We derive for the first time the covariant laws of motion for traveling wave fronts and scroll wave filaments in anisotropic excitable media such as cardiac tissue. We show that locally varying anisotropy yields non-zero Riemann tensor components, which may alter the stability of scroll wave filaments. The instability of scroll wave filaments has been linked to transition from ventricular tachycardia to fibrillation.

Motivation & Objective

• To understand how anisotropic tissue architecture—specifically fiber and sheet organization—affects the stability and dynamics of electrical wave fronts and filaments in cardiac tissue.
• To develop a mathematical framework for modeling scroll wave filaments in anisotropic media, incorporating structural constraints from diffusion tensor MRI (DTI).
• To investigate the role of geometric and structural symmetries in the twist equation governing filament motion, particularly in relation to Pauli matrix tensor decompositions.
• To quantify how local tissue architecture, including fiber angle and sheet curvature, influences the rotation and drift of scroll wave filaments.
• To validate the model using numerical simulations and analytical solutions derived from symmetry-restricted tensor decompositions in higher-order corrections to the wave dynamics.

Proposed method

• Employed diffusion tensor MRI (DTI) to map the three-dimensional orientation of myofibers and myocardial sheets in cardiac tissue, providing structural input for the model.
• Formulated a twist equation for scroll wave filaments in anisotropic media, derived from the reaction-diffusion system with structural constraints.
• Applied group-theoretic methods to decompose rank-4 and rank-6 tensors in the twist equation using Pauli matrices, enforcing index exchange symmetries (e.g., (23), (12)(3456)).
• Used irreducible tensor representations to identify independent components of the twist tensor, reducing the complexity of the dynamics under rotational symmetry.
• Incorporated higher-order corrections to the angular frequency of scroll waves via symmetry-restricted tensor decompositions, including terms like $ T_{000}{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{0} $ and $ T_{s0s}({oldsymbol{ ho}}^{1}igotimes{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{1} + {oldsymbol{ ho}}^{3}igotimes{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{3}) $.
• Validated the model by comparing analytical solutions with numerical simulations of filament motion under various anisotropy conditions.

Experimental results

Research questions

• RQ1 How does the anisotropic organization of cardiac tissue—specifically fiber and sheet structure—affect the stability and motion of scroll wave filaments?
• RQ2 What is the role of geometric symmetries in the twist equation governing filament dynamics in anisotropic media?
• RQ3 How do higher-order corrections to the angular frequency of scroll waves depend on the tensor structure of the medium’s anisotropy?
• RQ4 Which tensor components in the twist equation are physically relevant under rotational and index-exchange symmetries?
• RQ5 To what extent can diffusion tensor MRI (DTI) data be used to predict filament drift and rotation in realistic cardiac tissue models?

Key findings

• The twist equation for scroll wave filaments in anisotropic tissue is significantly influenced by the local orientation of myocardial sheets and fibers, with curvature and anisotropy inducing drift and rotation.
• Under (23) index exchange symmetry, the tensor $ 2{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{0} $ coincides with $ {oldsymbol{ ho}}^{1}igotimes{oldsymbol{ ho}}^{1} + {oldsymbol{ ho}}^{3}igotimes{oldsymbol{ ho}}^{3} $, reducing the number of independent components in the twist tensor.
• For rank-6 tensors with (12)(3456) symmetry, only three independent isotropic components remain after enforcing commutation with the rotation generator $ {oldsymbol{ ho}} $, including $ T_{000}{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{0} $ and two mixed-symmetry terms.
• The decomposition reveals that terms like $ rac{T_{s0}}{4}({oldsymbol{ ho}}^{1}igotimes({oldsymbol{ ho}}^{1}igotimes{oldsymbol{ ho}}^{0} + {oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{1}) + ext{cyclic} ) $ are essential for capturing symmetric contributions to filament dynamics.
• The presence of $ {oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{0} $ and $ {oldsymbol{ ho}}^{1}igotimes{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{1} + {oldsymbol{ ho}}^{3}igotimes{oldsymbol{ ho}}^{0}igotimes{oldsymbol{ ho}}^{3} $ in the decomposition indicates that isotropic and symmetric anisotropic contributions dominate the filament dynamics.
• The model predicts that structural anisotropy, particularly sheet curvature, can stabilize or destabilize filaments depending on the local orientation of the fiber and sheet planes, with implications for reentrant arrhythmias.