A coupled system of PDEs and ODEs arising in electrocardiograms modelling

We study the well-posedness of a coupled system of PDEs and ODEs arising in the numerical simulation of electrocardiograms. It consists of a system of degenerate reaction-diffusion equations, the so-called bidomain equations, governing the electrical activity of the heart, and a diffusion equation governing the potential in the surrounding tissues. Global existence of weak solutions is proved for an abstract class of ionic models including Mitchell-Schaeffer, FitzHugh-Nagumo, Aliev-Panfilov and MacCulloch. Uniqueness is proved in the case of the FitzHugh-Nagumo ionic model. The proof is based on a regularisation argument with a Faedo-Galerkin/compactness procedure

Mathematical Modelling of Cardiac Rhythms in Health and Disease

Cardiac disease is the most common cause of death among the adult population worldwide and atrial fibrillation (AF) is the most common cardiac arrhythmia. The state of the art in AF treatment involves creating lesions of heart tissue through radiofrequency ablation. In this thesis, mathematical modelling techniques are developed to design decision support tools that could help a cardiologist determine the best location to ablate in clinic. Firstly, parameter optimisation methods are explored to adapt a model designed for the ventricles to the atria, and a novel technique is introduced to characterise pathways through parameter space from a healthy state to a diseased state using a multi-objective genetic algorithm. Next, I reproduce clinical signals recorded during AF ablation through the use of a phenomenological model of the cardiac action potential on a cylinder and show how this model can enable us to recover information lost in clinic to improve clinical decision. This is followed by introducing a more simplistic approach to the same problem, by characterising the electrical activity on the recording by a sine wave. Finally, the effectiveness of these two approaches is compared in the clinical setting by testing both as decision support tools. The emphasis of the approaches throughout the thesis is on developing techniques with clinical applicability. We demonstrate that lost information in clinic can affect the decision made by an experienced clinician, and that the mathematical modelling approaches developed in the thesis can significantly reduce the impact that this information loss can have on clinical decision making