Bayesian Calibration of Electrophysiology Models Using Restitution Curve Emulators

Abstract

Calibration of cardiac electrophysiology models is a fundamental aspect of model personalization for predicting the outcomes of cardiac therapies, simulation testing of device performance for a range of phenotypes, and for fundamental research into cardiac function. Restitution curves provide information on tissue function and can be measured using clinically feasible measurement protocols. We introduce novel "restitution curve emulators" as probabilistic models for performing model exploration, sensitivity analysis, and Bayesian calibration to noisy data. These emulators are built by decomposing restitution curves using principal component analysis and modeling the resulting coordinates with respect to model parameters using Gaussian processes. Restitution curve emulators can be used to study parameter identifiability via sensitivity analysis of restitution curve components and rapid inference of the posterior distribution of model parameters given noisy measurements. Posterior uncertainty about parameters is critical for making predictions from calibrated models, since many parameter settings can be consistent with measured data and yet produce very different model behaviors under conditions not effectively probed by the measurement protocols. Restitution curve emulators are therefore promising probabilistic tools for calibrating electrophysiology models.

Keywords: Bayesian; Gaussian processes; calibration; cardiology; electrophysiology; emulation; restitution; sensitivity analysis.

Figure 1

S2 restitution curves for S1: 600 ms for CV(S2) and APD(S2), colored by ERP(S1: 600), and plotted only for S2 > ERP(S1: 600) for clarity.

Figure 2

PCA components and means for CV(S2) curves and APD(S2) curves for S1: 600 ms.

Figure 3

R2 scores for S2 restitution curves from 5-fold cross-validation. Performance decreases with S2, although these validation scores include RCE prediction at S2 ≤ ERP(S1) corresponding to virtual regions of the curves where no measurements can be made.

Figure 4

Sensitivity indices of the coordinates f_c of the principal components Φ_c for the model parameters for S1: 600 ms. The total-effect indices are plotted semi-transparently, with the first-order indices (which contribute to the total-effect indices) overlaid with opaque shading.

Figure 5

Second-order interaction effects for S1: 600 ms, with text labels applied in cells where the effects are at least 0.01 (i.e., account for at least 1% of the overall variance).

Figure 6

ERP(S1) restitution curves, showing (top) principal components and mean, and (bottom) first-order and total-effect sensitivity indices for the coordinates of the principal components.

Figure 7

RCE predictions to explore the effects of τ_open on ERP(S1) across the parameter space.

Figure 8

The RCE prediction from maximum a posteriori (MAP) parameter estimates given noisy measurements for (left) CV(S2) and ERP(S1), (right) APD(S2) and ERP(S1), shown as light shaded regions representing RCE 95% confidence intervals. The orange dashed curves show these intervals including the observation error, also learned from MAP fitting. The noisy S2 restitution data are shown as crosses, while the red shaded bars represent observed intervals containing ERP: (top): bars horizontally span ERP(S1:600) interval; (bottom) bars vertically span ERP(S1) interval for several S1. The solid black lines in all plots represent the corresponding ground truth curves.

Figure 9

RCE predictions, shown as lightly shaded regions representing 95% confidence intervals, for 100 parameter samples from the posterior distribution given the same measurements shown in Figure 8 [black crosses are noisy S2 restitution data, red bars are observed ERP intervals, (left) MCMC with CV(S2) and ERP(S1) data, (right) MCMC with APD(S2) and ERP(S1) data].

Figure 10

The posterior parameter distribution for fits to CV(S2) and ERP(S1) measurements. The intersection of vertical and horizontal lines mark the true parameter value. The lower diagonal shows the density via hexbin plots, while the upper diagonal shows the log likelihood values for each sample plotted in order of increasing likelihood. The diagonals show the marginal histograms of each parameter.

Figure 11

The posterior parameter distribution for fits to APD(S2) and ERP(S1) measurements. The intersection of vertical and horizontal lines mark the true parameter value. The lower diagonal shows the density via hexbin plots, while the upper diagonal shows the log-likelihood values for each sample plotted in order of increasing likelihood. The diagonals show the marginal histograms of each parameter.

Figure 12

RCE predictions, shown as lightly shaded regions representing 95% confidence intervals, for 100 parameter samples from the posterior distribution given the same measurements shown in Figure 8 (black crosses are noisy S2 restitution data, red bars are observed ERP intervals). MCMC utilized CV(S2), APD(S2), and ERP(S1) data simultaneously, unlike in Figures 8, 9.

Figure 13

The posterior parameter distribution for calibration to CV(S2), APD(S2), and ERP(S1) measurements simultaneously. The intersection of vertical and horizontal lines mark the true parameter value. The lower diagonal shows the density via hexbin plots, while the upper diagonal shows the log likelihood values for each sample plotted in order of increasing likelihood. The diagonals show the marginal histograms of each parameter.

Figure 14

Principal components and means of CV(S1,S2) and APD(S1,S2) restitution surfaces, displayed as contour plots where dark/light colors represent low/high values, respectively. These surfaces are 2D analogues to the curves in Figure 2.