Enabling forward uncertainty quantification and sensitivity analysis in cardiac electrophysiology by reduced order modeling and machine learning

Abstract

We present a new, computationally efficient framework to perform forward uncertainty quantification (UQ) in cardiac electrophysiology. We consider the monodomain model to describe the electrical activity in the cardiac tissue, coupled with the Aliev-Panfilov model to characterize the ionic activity through the cell membrane. We address a complete forward UQ pipeline, including both: (i) a variance-based global sensitivity analysis for the selection of the most relevant input parameters, and (ii) a way to perform uncertainty propagation to investigate the impact of intra-subject variability on outputs of interest depending on the cardiac potential. Both tasks exploit stochastic sampling techniques, thus implying overwhelming computational costs because of the huge amount of queries to the high-fidelity, full-order computational model obtained by approximating the coupled monodomain/Aliev-Panfilov system through the finite element method. To mitigate this computational burden, we replace the full-order model with computationally inexpensive projection-based reduced-order models (ROMs) aimed at reducing the state-space dimensionality. Resulting approximation errors on the outputs of interest are finally taken into account through artificial neural network (ANN)-based models, enhancing the accuracy of the whole UQ pipeline. Numerical results show that the proposed physics-based ROMs outperform regression-based emulators relying on ANNs built with the same amount of training data, in terms of both numerical accuracy and overall computational efficiency.

Keywords: artificial neural network regression; cardiac electrophysiology; reduced basis method; reduced order modeling; sensitivity analysis; uncertainty quantification.

FIGURE 1

FOM solutions of the monodomain system coupled with the Aliev‐Panfilov ionic model for different values of the parameters, at t = 520 ms and t = 800 ms. The great variability of the solution with respect to the parameters μ ₁, μ ₂, μ ₃ is clearly visible: from left to right, tissue refractoriness, sustained reentry and non‐sustained reentry cases are reported

FIGURE 2

Graphical sketch of the input–output map. Input parameters (top) are related with the time interval between S1 and S2 stimuli (μ ₁), the radius of the circular region in which S2 is applied (), and the tissue recovery properties, modeled by the coefficient ε ₀ affecting the ionic model (μ ₃). The output quantity of interest y(μ) represents the deviation of the activation map from a reference value, evaluated as a scalar index and acts as a classifier: small values of the output are related with tissue refractoriness, intermediate values with non‐sustained reentry, large values with sustained reentry. Being dependent of the transmembrane potential u(μ), the output y(μ) can be evaluated exploiting one of the proposed models: a reduced order model (ROM), an ANN‐based emulator (MLP), a ROM corrected with an ANN‐based emulator for the ROM error (ROM+MLP)

FIGURE 3

Activation maps computed from the FOM solution for different choices of (h, Δt) = (1.0 mm,0.5 ms), (0.5 mm,0.25 ms), and (0.25 mm,0.125 ms) (from left to right) and different parameter values describing sustained reentry, tissue refractoriness, or non‐sustained reentry (from top to bottom)

FIGURE 4

ROM solutions of the monodomain system coupled with the Aliev‐Panfilov ionic model for different values of the parameters, at t = 520 ms and t = 800 ms

FIGURE 5

Singular values decay and cumulative expressed variance for the solution (top) and the nonlinear term (bottom) snapshot sets, after their clustering obtained through the k‐means algorithm. The query to extremely accurate ROM might be computationally demanding due to the high number of retained POD modes

FIGURE 6

Relative errors on solution obtained with ROMs of increasing accuracy. For decreasing POD tolerances, ROMs of increasing dimensions allow us to obtain decreasing error norms on the solution. The (H ¹(Ω)) computed norm for the solution error sums errors on both the solution and its spatial derivatives. Note that the ROMs with POD tolerances 10⁻⁵ (resp., 10⁻⁴, 10⁻³) are 5 (resp., 4, 1.7) times slower than the ROM with POD tolerance 10⁻²

FIGURE 7

MLP construction of the correction of the ROM output QoI. Left: convergence of the training in terms of MSE over the epochs on the training, validation, and test sets. Right: testing performance of the MLP correction of the ROM output

FIGURE 8

MLP construction of the ANN‐based input–output surrogate. Left: convergence of the training procedure in terms of MSE over the epochs on the training, validation, and test sets. Right: testing performance of the MLP output

FIGURE 9

Accuracy of the MLP and ROM+MLP models in terms of mean squared error (MSE) over the test set, plotted as a function of the size N _offline of training+validation sample. The ROM+MLP is more accurate than the MLP (in terms of MSE error), and definitely outperforms the MLP for N _offline > 1000. The MSE error provided by the ROM is also reported

FIGURE 10

Main effect plots obtained with the six considered models, showing the output mean (with respect to μ _∼i) as a function of each input μ _i, i = 1,2,3. From top to bottom, from left to right: FOM, ROM; MLP (100); MLP (1700); ROM+MLP (100), ROM+MLP (1700). Main effect plots are useful to visualize if different levels of each μ _i affect the output differently

FIGURE 11

Output QoI distributions resulting from forward propagation obtained with the ROM, the ROM+MLP (100) and the MLP (100) models (hence, in presence of a small dataset size), compared with the FOM output QoI distribution

FIGURE 12

Output QoI distributions resulting from forward propagation obtained with the ROM+MLP (1700) and the MLP (1700) models (hence, in presence of a large dataset size), compared with the FOM output QoI distribution

FIGURE 13

Comparison between online and offline costs entailed by the proposed models. Performing forward UQ requires N _mc evaluations of the output QoI, while GSA requires (p + 2)N _s evaluations of the output QoI. For the case at hand, N _mc = 1000, p = 3 and N _s = 10⁴

FIGURE 14

Electric potential computed on a 3D template left atrium geometry for different parameter values. The input–output setting described for the two‐dimensional case can represent a first step towards the classification of different conditions also on more complex configurations