Fast Posterior Estimation of Cardiac Electrophysiological Model Parameters via Bayesian Active Learning

Abstract

Probabilistic estimation of cardiac electrophysiological model parameters serves an important step toward model personalization and uncertain quantification. The expensive computation associated with these model simulations, however, makes direct Markov Chain Monte Carlo (MCMC) sampling of the posterior probability density function (pdf) of model parameters computationally intensive. Approximated posterior pdfs resulting from replacing the simulation model with a computationally efficient surrogate, on the other hand, have seen limited accuracy. In this study, we present a Bayesian active learning method to directly approximate the posterior pdf function of cardiac model parameters, in which we intelligently select training points to query the simulation model in order to learn the posterior pdf using a small number of samples. We integrate a generative model into Bayesian active learning to allow approximating posterior pdf of high-dimensional model parameters at the resolution of the cardiac mesh. We further introduce new acquisition functions to focus the selection of training points on better approximating the shape rather than the modes of the posterior pdf of interest. We evaluated the presented method in estimating tissue excitability in a 3D cardiac electrophysiological model in a range of synthetic and real-data experiments. We demonstrated its improved accuracy in approximating the posterior pdf compared to Bayesian active learning using regular acquisition functions, and substantially reduced computational cost in comparison to existing standard or accelerated MCMC sampling.

Keywords: Gaussian process; cardiac electrophysiological model; high-dimensional Bayesian optimization; probabilistic parameter estimation; variational autoencoder.

Figure 1

Workflow of the presented method. (A) A generative model of HD spatially varying tissue excitability of the 3D heart is trained offline. (B) The resulting generative model is embedded into Bayesian active learning to approximate the posterior pdf of model parameters using a small number of intelligently selected training points guided by the acquisition function.

Figure 2

Distribution of LD latent codes of the training data, color coded by (A) size of the abnormal tissue (the colors represent the percentage size of abnormal tissue). (B) Location of the abnormal tissue (the colors represent the 17 American Heart Association (AHA) segments of left ventricle).

Figure 3

(A) Comparison of estimated posterior pdf from different methods. (B) Comparison of computation cost from different methods.

Figure 4

Comparison of (A) DC, (B) RMSE, and (C) CC between estimated mean (blue) or mode (red) tissue excitability in comparison to the ground truth.

Figure 5

(A) The ground truth of tissue excitability. (B) Mean, mode, and standard deviation of tissue excitability estimated from presented method.

Figure 6

Illustrations of training points (blue dots) selected using variance based on the log-normal process (left), entropy based on the log-normal process (middle), and upper confidence bound (UCB) based on the Gaussian process (right).

Figure 7

Results of estimated tissue excitability from the presented method in 3D infarcts delineated from in vivo MRI images. Regions with low excitability (high θ values) correspond to infarct regions (0.5 = infarct core, 0.3–0.5 = gray zone). The red circles highlight non-transmural scars or gray zone.

Figure 8

Results of estimated tissue excitability from the presented method in real clinical data. (A) Voltage data from catheter map. (B) Mean, mode, and standard deviation estimated from multiple observations from different pacing sites. (C) Mean, mode, and standard deviation estimated from a single observation from one pacing site.