Markov chain Monte Carlo for Bayesian inference of the non-conducting region in intra-atrial reentrant tachycardia

Abstract

We present a Bayesian approach to estimate the parameters of mathematical models of cardiac electrophysiology with quantified uncertainty. Such models capture the dynamics of the electrical signal that coordinates the muscle cell contraction in the heart wall and can support cardiac arrhythmia treatment. We consider an illustrative case motivated by a cardiac arrhythmia, namely, by intra-atrial reentrant tachycardia. We estimate a low-dimensional geometrical parameter that describes the boundary of an electrically non-conducting region in the heart tissue from synthetic electrical measurements outside of the tissue. Instead of relying on a deterministic fit for this region, we estimate a posterior distribution on the geometrical parameter using Bayesian inference that captures the uncertainty due to measurement errors. We propose a likelihood based on a set of quantities that characterize the data for improved accuracy. To efficiently approximate the posterior distribution, we propose a compressed likelihood function and an adapted Metropolis-Hastings (MH) algorithm. We obtain an algorithm that strongly decreases the number of samples by using an adaptive proposal strategy. Our algorithm also gives attention to the impact of discretization errors on inference outcomes, as these introduce artificial discontinuities in the posterior if not properly addressed. We account for discretization errors in the likelihood and in the accept-reject step of our adapted MH algorithm to improve the robustness of our estimates and to further increase the sampling efficiency. All of these elements combined give us a method that efficiently estimates the non-conducting parameters with uncertainty. We perform several experiments with different amounts of measurement noise and illustrate how this translates into the posterior distributions.

Figures (12)

• Figure 1: The colors annotate the arrival time of the electrical wave from early (red) to late (dark blue) under IART. The grey area is a surgical scar and does not yield electrical activity. We observe reentry around this scar. Adapted from kahle2021. IVC = inferior vena cava.
• Figure 2: Left: the IART behavior occurs inside a 2D tissue slab with a non-conducting elliptical region, with $D=0$ in equation \ref{['eq:model']} in this region, and a spiral wave as the initial condition. Right: the geometrical parameter that describes the non-conducing region. The plot on the left takes $\vartheta=[9.0~\text{mm},9.0~\text{mm},0.0~\text{rad}]$
• Figure 3: Left: the $\text{PENTARAY}^\text{®}$ NAV ECO High Density Mapping Catheter (Biosense Webster, Diegem, Belgium) consists of 5 arms with 4 electrodes (grey) on each. Right: the location of the electrodes is projected on the 2D plane of the tissue. Endocardial measurements are made at the blue locations for all experiments, with additional measurements at the red locations for the experiment with two catheter locations.
• Figure 4: Plots of the compressed likelihood as a function of the long radius $a$ for an inference experiment that only estimates $a$, with ground truth $a_{true}=10.0$ mm. We set the conduction slowing to $\gamma=0.1$ and the likelihood variance to $\Sigma_{\varepsilon'}=\mathop{\mathrm{diag}}\nolimits(0.1,1.0,1.0,\dots,1.0)$. Left: the compressed likelihood using different resolutions $\Delta x$. Right: the discretized compressed likelihood for different $\gamma$ with $\Delta x=0.5$ mm fixed.
• Figure 5: In shape estimation problems, the numerical solution of the forward model for the shape $\vartheta$ consists of a discontinuous meshing step to generate a mesh $\mu$, that models the shape $\vartheta$, and the FEM step to solve the discretized model over this mesh.