Abstract
In electrocardiography, the “classic” inverse problem is the reconstruction of electric potentials at a surface enclosing the heart from remote recordings at the body surface and an accurate description of the anatomy. The latter being affected by noise and obtained with limited resolution due to clinical constraints, a possibly large uncertainty may be perpetuated in the inverse reconstruction. The purpose of this work is to study the effect of shape uncertainty on the forward and the inverse problem of electrocardiography. To this aim, the problem is first recast into a boundary integral formulation and then discretised with a collocation method to achieve high convergence rates and a fast time to solution. The shape uncertainty of the domain is represented by a random deformation field defined on a reference configuration. We propose a periodic‐in‐time covariance kernel for the random field and approximate the Karhunen–Loève expansion using low‐rank techniques for fast sampling. The space–time uncertainty in the expected potential and its variance is evaluated with an anisotropic sparse quadrature approach and validated by a quasi‐Monte Carlo method. We present several numerical experiments on a simplified but physiologically grounded two‐dimensional geometry to illustrate the validity of the approach. The tested parametric dimension ranged from 100 up to 600. For the forward problem, the sparse quadrature is very effective. In the inverse problem, the sparse quadrature and the quasi‐Monte Carlo method perform as expected, except for the total variation regularisation, where convergence is limited by lack of regularity. We finally investigate an H1/2 regularisation, which naturally stems from the boundary integral formulation, and compare it to more classical approaches.
Highlights
Electrocardiographic recordings at the body surface, such as the standard 12-lead electrocardiogram (ECG), are a direct consequence of the electric activity of the heart
The most classical formulation of it is associated with a potential-based forward problem, which amounts to determining the body surface potential maps (BSPMs) on the chest from the electric potential at a surface enclosing the heart, e.g., the epicardium
As typical for inverse problems, the ECG imaging problem is severely ill-posed in the sense of Hadamard, since arbitrarily small perturbations of the BSPM, such as noise, may yield large variations in the epicardial potential
Summary
Electrocardiographic recordings at the body surface, such as the standard 12-lead electrocardiogram (ECG), are a direct consequence of the electric activity of the heart. Non-smooth regularisations such as Total Variation (TV) are a valid alternative to quadratic approaches in the presence of sharp gradients in the reconstructed data [8], but lead to a significantly more difficult solution of the inverse problem. In order to assess the effect of the shape uncertainty, we estimate the expectation and the variance of the chest potential (forward problem) and the pericardial potential (inverse problem) using the anisotropic sparse quadrature method from [28]. This approach is validated against the quasi-Monte Carlo method based on Halton points, see [32]. We validate the sparse quadrature and we quantify the effect of the shape uncertainty on the forward and inverse problems
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