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**Computational Methods**

A 14 x 14 mm2 domain was discretized into a regular mesh of 0.05 x 0.05 mm2 finite elements with discontinuities of scale and arrangement similar to those observed in patchy fibrosis (Kawara, Derksen et al. 2001; de Bakker, Stein et al. 2005) in a 10 x 4 mm2 region at the centre (Figure 2.1) The bidomain equations were solved (Austin, Trew et al. 2006) on this domain with a modified Luo-Rudy action potential model (Luo and Rudy 1994).

Figure 2.1 Cardiac activation and propagation through an electrically uniform and isotropic domain (14 x 14 mm2) with a 10 x 4 mm2 region containing patchy inexcitable voids. Transmembrane potentials are shown 1 and 30 ms after the first bipolar stimulus: anode (A), cathode (C). Action potentials from a typical pacing protocol are shown for points above (1), within (2) and below (3) the heterogeneous region.

**Computer Modeling of Electrical Activation and Diffusion**

This paradigm of electrical flow, outlined in the sections below, was implemented by a computer program developed at the Department of Engineering Science at the University of Auckland and Auckland Bioengineering Institute known as CMISS (Continuum Mechanics, Image analysis, Signal processing and System Identification). CMISS is a mathematical modeling environment that allows the application of finite element analysis, boundary element and collocation techniques to a variety of complex bioengineering problems. The ODE integrators that were used to solve the cellular activation model (Section 2.2.1.1) and a formulation of the Bidomain equations (Section 2.2.1.2) have been previously implemented in this framework (Austin, Trew et al. 2006).

**Myocardial Activation Model**

The formulations and foundations of modeling the electrical activity of cardiac cells were established in a series of experiments by Hodgkin and Huxley involving the nature of electrical flux through the cell membrane of a squid giant nerve fibre (Hodgkin and Huxley 1952; Hodgkin and Huxley 1952; Hodgkin and Huxley 1952; Hodgkin and Huxley 1952; Hodgkin and Huxley 1952; Hodgkin and Huxley 1952; Hodgkin, Huxley et al. 1952). These works presented the idea that electrical behaviour of physiologic tissue may be represented by an equivalent electrical circuit (Figure 2.2). The membrane voltage (Vm) can then be found by solving a series of ordinary differential equations with parameters that match voltageclamp data.

Figure 2.2 Equivalent circuit of a small patch of membrane of a squid giant axon derived by Hodgkin and Huxley. Im, represent current through ion channels (Na, K, Cl, Leak), which is associated with the particular ion‟s conductance (g‟s). The Nernst equilibrium voltage is modelled for each ion as the E, which restricts current flow, where appropriate, until the transmembrane voltage (Vm) becomes larger. Membrane capacitance is assumed to be constant and is represented by Cm. The electrical behaviour of a single cell is described by the following equation: Structural Heterogeneity and Electrical Instability where is the voltage change over time, Iion is the sum of all transmembrane ionic currents, Istim is the sum of externally applied currents, and Cm is the cell capacitance per unit area. Ionic currents for each ion species are defined as the product of the permeability coefficient (gion), the difference between the transmembrane voltage (Vm) and the ion‟s equilibrium potential (Eion), and the probability that the ion gate is open, defined by the gating variable (n)

CHAPTER ONE. INTRODUCTION

1.1 CARDIAC ANATOMY

1.2 ELECTRICAL ACTIVATION IN THE HEART

1.3 VENTRICULAR ARRHYTHMIAS 9

1.4 DETERMINANTS OF ELECTRICAL CONDUCTION

1.5 MYOCARDIAL INFARCTION

1.6 DYNAMIC INSTABILITY

1.7 MEASURING ELECTRICAL EVENTS IN THE HEART

1.8 AIMS OF THE THESIS

CHAPTER TWO. STRUCTURAL HETEROGENEITY ALONE IS A SUFFICIENT SUBSTRATE FOR

DYNAMIC INSTABILITY AND ALTERED RESTITUTION

2.1 INTRODUCTION

2.2 COMPUTATIONAL METHODS

2.3 IMPACT OF FIXED SUBSTRATE ON CV AND APD DURING FAST PACING

CHAPTER THREE. OVINE MODEL OF MYOCARDIAL INFARCTION

3.1 EXPERIMENTAL MODEL OF MYOCARDIAL INFARCTION

3.2 IN VIVO HEART PREPARATION

3.3 EXVIVO IMAGING AND IMAGE SEGMENTATION

3.4 MYOCARDIAL AND INFARCT TOPOLOGY

3.5 INTRAMURAL ELECTROGRAMS

CHAPTER FOUR. INTRAMURAL PACING AND HIGH DENSITY MAPPING REVEALS THE

ELECTRICAL CHARACTERISTICS OF THE INFARCT BORDER ZONE

4.1 INTRODUCTION

4.2 METHODS

4.3 ELECTRICAL CHARACTERISTICS OF THE INFARCT BORDER ZONE

4.4 THREE DIMENSIONAL SPREAD OF ELECTRICAL ACTIVATION IN INFARCT BORDER ZONE

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Arrhythmic Substrates in Structural Heart Disease