Multiscale mathematical modelling of structural heterogeneities in cardiac electrophysiology

Abstract : In this thesis we addressed two problems in mathematical modelling of propagation of electrical signals in the heart: tissue scale propagation with presence of tissue heterogeneities and cell scale propagation with non-linear gap junctions. The standard model used in cardiac electrophysiology is the bidomain model. It is an averaged model derived from the microscopic properties of the tissue. The bidomain model assumes that the electrically active myocytes are present uniformly everywhere in the heart. While this is a reasonable assumption for healthy hearts, it fails in some pathological cases where significant changes in the tissue structure occur, for example in ischaemic and rheumatic heart disease, inflammation, hypertrophy, or infarction. These tissue heterogeneities are often taken into account through an ad-hoc tuning of model parameters. The first aim of this thesis consisted in generalizing the bidomain equations to the case of structural heart diseases. We assumed a periodic alternation of healthy (bidomain model) and altered (diffusive inclusion) tissue patches. Such a model may be simulated directly, at the high computational cost of a very fine discretisation. Instead we derived a homogenized model at the macroscopic scale, using a rigorous two-scale analysis. We recovered a bidomain-type model with modified conductivity coefficients, and performed a 2D numerical verification of the convergence of the microscopic model towards the homogenized one. In the second part we quantified the effects of different shapes and sizes of diffusive inclusions on the effective conductivity coefficients and their anisotropy ratios in 2D and 3D. Additionally, we ran simulations on 2D patches of tissue with modified conductivity coefficients. We observed changes in the propagation velocity as well as in the shape of the depolarization wave-front. In the third part, based on high-resolution MR images of a rat heart we simulated 3D propagations with the homogenized model. Using image analysis software tools we assessed the structural properties of the tissue, that we used afterwards as parameters inthe homogenized model. In the last part of this thesis, we studied the effects of non-linear gap junction channels on the signal propagation at the cell scale. In existing models, the gap junction channels, if modelled, are assumed to have a linear behaviour, while from experimental data we know that they have a time- and voltage-dependent non-linear behaviour. Firstly, we stated a non-linear 0D model for the gap junctional current, and secondly fitted the model to available experimental data. Finally, we proposed a 2D mathematical model that describes the electrical interaction of cardiac myocytes on the cell scale. It accounts for the gap junctional current as "the direct link" between the adjacent cells.
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Submitted on : Monday, February 27, 2017 - 10:33:58 PM
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Anđela Davidović. Multiscale mathematical modelling of structural heterogeneities in cardiac electrophysiology . Mathematics [math]. Université de Bordeaux, 2016. English. ⟨tel-01478145v1⟩

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