Modelling the influence of ion channel distributions in cardiac arrhythmias

Abstract

Understanding cardiac electrophysiology in general and the role of ion channels in particular, is a problem of practical significance. Limitations on experimental investigations have necessitated the development of theoretical and computer models. In this thesis, a theoretical perspective informed by published experimental observations and computer modelling tools is adopted. Myocyte organisation and distribution of ion channels in the cardiac ventricular tissue greatly influence its mechanical and electrical behaviour. Computer simulations of these systems are computationally-expensive, nontrivial and limit model based investigations. A novel algorithm that substantially reduces the computational expenses under specific conditions is developed. The integration of multiple models to reproduce the cardiac behaviour across spatial and temporal scales, introduces non-trivial computing and the mathematical issues. Using an analytic approach that partially resolves these issues, novel models of the passive cardiac-muscle-tissue, extracellular excitation-propagation and stability of the tissue are developed. A rational approach to model deformation of cellular aggregations is presented together with relevant proofs. A remarkable relationship, that the minimisation of local mean angular distortion and Dirichlet energy are identical (up to a measure) is presented. It is anticipated that the results presented in this thesis will initiate further investigation to expand the proposed models and increase understanding the dynamics and control of the heart.