Abstract:
In this thesis, the forward problem of electrocardiography is investigated from a cellular level
through to potentials on the surface of the torso. This integrated modelling framework is based
on three spatial scales. At the smallest spatial resolution, several cardiac cellular models are
implemented that are used to represent the underlying cellular electrophysiology. A bidomain
framework is used to couple multiple individual cells together and this provides a mathematical
model of the myocardial tissue. The cardiac geometry is described using finite elements with
high order cubic Hermite basis function interpolation. An anatomically based description of the
fibrous laminar cardiac microstructure is then defined relative to the geometric mesh. Within the
local element space of the cardiac finite elements, a fine collocation mesh is created on which the
bidomain equations are solved. Each collocation point represents a continuum cell and contains
a cellular model to describe the local active processes. This bidomain implementation works in
multiple coordinate systems and over deforming domains, in addition to having the ability to
spatially vary any parameter throughout the myocardium. On the largest spatial scale the passive
torso regions surrounding the myocardium are modelled using a generalised Laplace equation
to describe the potential field and current flows. The torso regions are discretised using either
finite elements or boundary elements depending on the electrical properties of each region.
The cardiac region is coupled to the surrounding torso through several methods. A traditional
dipole source approach is implemented that creates equivalent cardiac sources through the summation
of cellular dipoles. These dipoles are then placed within a homogeneous cardiac region
and the resulting potential field is calculated throughout the torso.
Two new coupling techniques are developed that provide a more direct path from cellular activation
to body surface potentials. One approach assembles all of the equations from the passive
torso regions and the equations from the extracellular bidomain region into a single matrix
system. Coupling conditions based on the continuity of potential and current flow across the
myocardial surfaces are used to couple the regions and therefore solving the matrix system
yields a solution that is continuous across all of the solution points within the torso. The second
approach breaks the large system into smaller subproblems and the continuity conditions are
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imposed through an iterative approach. Across each of the myocardial surfaces, a fixed point
iteration is set up with the goal of converging towards zero potential and current flow differences
between adjacent regions.
All of the numerical methods used within the integrated modelling framework are rigorously
tested individually before extensive tests are performed on the coupling techniques. Large scale
simulations are run to test the dipole source approach against the new coupling techniques. Several
sets of simulations are run to investigate the effects of using different ionic current models,
using different bidomain model simplifications, and the role that the torso inhomogeneities play
in generating body surface potentials.
The main question to be answered by this study is whether or not the traditional approach of
combining a monodomain heart with an equivalent cardiac source in a two step approach is
adequate when generating body surface potentials. Comparisons between the fully coupled
framework developed here and several dipole based approaches demonstrate that the resulting
sets of signals have different magnitudes and different waveform shapes on both the torso and
epicardial surface, clearly illustrating the inadequacy of the equivalent cardiac source models. It
has been found that altering the modelling assumptions on each spatial scale produces noticeable
effects. At the smallest scale, the use of different cell models leads to significantly different body
surface potential traces. At the next scale the monodomain approach is unable to accurately
reproduce the results from a full bidomain framework, and at the largest level the inclusion of
different torso inhomogeneities has a large effect on the magnitude of the torso and epicardial
potentials. Adding a pair of lungs to the torso model changes the epicardial potentials by an
average of 16% which is consistent with the experimental range of between 8 and 20%. This
provides evidence that only a complex, coupled, biophysically based model will be able to
properly reproduce clinical ECGs.