Parameter Estimation and Reduced-order Modeling in Electrocardiology Open Access

Yang, Huanhuan (2015)

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Computational modeling of electrocardiology (EC) has a great potential for use in improved diagnosis and prognosis of cardiac arrhythmia. However, recent computational methods usually suffer from three major limitations that hinder their clinical use: lack of efficient model personalization strategies; high computational demand from the EC solver; lack of good trade-off between the simplification of cellular ionic models and the demand on keeping sufficient biophysical details. This thesis aims at solving above challenges. The principal part is on the estimation of cardiac conductivities that parameterize the bidomain/monodomain model--the current standard model for simulating cardiac potential propagation. We consider a variational approach by regarding the parameters as control variables to minimize the mismatch between computed and measured potentials. The existence of a minimizer of this misfit function is proved. We significantly improve the numerical approaches in the literature by resorting to a derivative-based optimization method with the settlement of some challenges due to discontinuity. Our numerical results is mainly in 3D, on both idealized and real geometries. We demonstrate the reliability and stability in presence of noise and with an imperfect knowledge of other model parameters. We then focus on the computational cost reduction for the inverse conductivity problem. The Proper Orthogonal Decomposition (POD) approach is taken for forward model reduction, along with the Discrete Empirical Interpolation Method (DEIM) for tackling nonlinearity. In the application of this POD-DEIM combination, we obtain a rather small set of samples by sampling the parameter space based on polar coordinates and densifying the "boundary layer" of the sample space utilizing Gauss-Lobatto nodes. The computational effort is finally reduced by at least 90% in conductivity estimation. The last part is developing a data-driven approach to the reduction of state-of-the-art cellular models in atrial electrophysiology. The reduced model predicts cellular action potentials (AP) in a simple form but is effective in capturing the physiological complexity of the original model. We start from an AP manifold learning, and continue with a regression model construction. The reduced cellular model drastically improves the performance of tissue-level atrial electrophysiological modeling and enables almost real-time computations.

Table of Contents

1 Introduction. 1

1.1 Clinical significance of electrocardiological modeling. 1

1.2 Some challenges in computational electrocardiology. 3

1.3 Thesis outline. 4

2 Mathematical models in electrocardiology. 7

2.1 Heart function and electrical activity. 7

2.1.1 Anatomy and function of the heart. 7

2.1.2 Electrical activity of cardiac myocytes. 10

2.1.3 Electrical activity of the heart. 14

2.2 Electrocardiological models in the cellular level. 16

2.2.1 Biophysics-based ionic models. 18

2.2.2 Reduced ionic models. 21

2.2.3 Phenomenological ionic models. 26

2.3 Electrocardiological models in the tissue level. 27

3 Cardiac conductivity estimation by a variational approach. 33

3.1 Introduction. 33

3.2 Variational formulation of the inverse problem. 36

3.3 Numerical solver. 38

3.3.1 Time and space discretization. 38

3.3.2 Computation of derivatives of discontinuous terms. 42

3.3.3 Interplay between optimization and time discretization. 48

3.4 Numerical results. 51

3.4.1 Two-dimensional test. 51

3.4.2 Three-dimensional tests. 54

3.5 Chapter conclusions and developments. 61

4 Existence analysis for the inverse conductivity problem. 65

4.1 Preliminary. 65

4.2 The existence proof. 67

5 Reduced-order modeling for cardiac conductivity estimation. 73

5.1 Introduction. 73

5.2 The full-order monodomain inverse conductivity problem. 77

5.3 Model order reduction for nonlinear system. 82

5.3.1 Proper Orthogonal Decomposition (POD). 84

5.3.2 Discrete Empirical Interpolation Method (DEIM). 88

5.4 POD-DEIM for the inverse conductivity problem. 91

5.4.1 POD-DEIM on the monodomain model. 91

5.4.2 The reduced monodomain inverse conductivity problem. 95

5.5 Numerical results. 98

5.5.1 POD-DEIM on the forward problem. 98

5.5.2 Domain of effectiveness (DOE) of the reduced basis. 101

5.5.3 POD-DEIM on the inverse problem . 104

5.6 Chapter conclusions. 112

6 Reduced-order modeling for atrial electrophysiology by a data-driven approach. 115

6.1 Introduction. 115

6.2 Methods. 117

6.2.1 AP manifold learning for dimensionality reduction. 118

6.2.2 Regression model. 120

6.2.3 Application to tissue-level atrial EP modeling. 123

6.3 Experiments and results. 125

6.3.1 Model parameter selection and sampling. 125

6.3.2 PCA versus LLE for manifold learning. 127

6.3.3 Regression model construction. 128

6.3.4 Application to tissue-level EP modeling. 133

6.3.5 Restitution study by diastolic interval change. 136

6.4 Discussion and conclusion. 138

7 Conclusions. 139

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