POD-Enhanced Deep Learning-Based Reduced Order Models for the Real-Time Simulation of Cardiac Electrophysiology in the Left Atrium

Abstract

The numerical simulation of multiple scenarios easily becomes computationally prohibitive for cardiac electrophysiology (EP) problems if relying on usual high-fidelity, full order models (FOMs). Likewise, the use of traditional reduced order models (ROMs) for parametrized PDEs to speed up the solution of the aforementioned problems can be problematic. This is primarily due to the strong variability characterizing the solution set and to the nonlinear nature of the input-output maps that we intend to reconstruct numerically. To enhance ROM efficiency, we proposed a new generation of non-intrusive, nonlinear ROMs, based on deep learning (DL) algorithms, such as convolutional, feedforward, and autoencoder neural networks. In the proposed DL-ROM, both the nonlinear solution manifold and the nonlinear reduced dynamics used to model the system evolution on that manifold can be learnt in a non-intrusive way thanks to DL algorithms trained on a set of FOM snapshots. DL-ROMs were shown to be able to accurately capture complex front propagation processes, both in physiological and pathological cardiac EP, very rapidly once neural networks were trained, however, at the expense of huge training costs. In this study, we show that performing a prior dimensionality reduction on FOM snapshots through randomized proper orthogonal decomposition (POD) enables to speed up training times and to decrease networks complexity. Accuracy and efficiency of this strategy, which we refer to as POD-DL-ROM, are assessed in the context of cardiac EP on an idealized left atrium (LA) geometry and considering snapshots arising from a NURBS (non-uniform rational B-splines)-based isogeometric analysis (IGA) discretization. Once the ROMs have been trained, POD-DL-ROMs can efficiently solve both physiological and pathological cardiac EP problems, for any new scenario, in real-time, even in extremely challenging contexts such as those featuring circuit re-entries, that are among the factors triggering cardiac arrhythmias.

Keywords: bidomain equations; cardiac electrophysiology; deep learning; isogeometric analysis; left atrium; proper orthogonal decomposition; reduced order modeling.

Figure 1

Starting from the FOM solution u_h(t; μ), the intrinsic coordinates $V^T{u}_h(t;\mu )$ are computed through rSVD; their approximation ${\tilde{u}}_N(t;\mu )$ is provided by the neural network as output, so that the reconstructed solution ${\tilde{u}}_h(t;\mu )$ is recovered through the rPOD basis matrix. In particular, the intrinsic coordinates $V^T{u}_h(t;\mu )$ are provided as input to block (A), which returns as output ${\tilde{u}}_n(t;\mu )$. The same parameter instance (t; μ) enters block (B), which provides as output u_n(t; μ), and the error between the low-dimensional vectors is accumulated. The minimal coordinates u_n(t; μ) are given as input to block (C), which returns the approximated intrinsic coordinates ${\tilde{u}}_N(t;\mu )$. Then, the reconstruction error is computed.

Figure 2

Test 1.1: FOM solution (top) and POD-DL-ROM one (center), with n = 2 and N = 256, along with the relative error ϵ_k for u_h (bottom-left) and u_e,h (bottom-right), for the testing-parameter instance ${\mu }_{test}=0.0143{\Omega }^{-1}{\text{cm}}^{-1}$ at t = 150 ms.

Figure 3

Test 1.1: Relative error ϵ_k, averaged with respect to the spatial coordinates, for the transmembrane (A) and the extracellular (B) potentials, for the testing-parameter instance ${\mu }_{test}=0.0143{\Omega }^{-1}{\text{cm}}^{-1}$, over time.

Figure 4

Test 1.2: FOM solution (left) and POD-DL-ROM one (right), with n = 2 and N = 256, for the testing-parameter instance ${\mu }_{test}=0.0295{\Omega }^{-1}{\text{cm}}^{-1}$ at t = 52.8 ms (A) and t = 112 ms (B).

Figure 5

Test 1.2: FOM solution (left) and POD-DL-ROM one (right), with n = 2 and N = 256, for the testing-parameter instance ${\mu }_{test}=0.0295{\Omega }^{-1}{\text{cm}}^{-1}$ at t = 52.8 ms (A) and t = 112 ms (B).

Figure 6

Test 1.2: FOM and POD-DL-ROM, with n = 2 and N = 256, APs for the testing-parameter instance μ_test = 0.0295 Ω⁻¹cm⁻¹.

Figure 7

Test 2: Parameter space (dark magenta region) and portion of domain affected by the stimulus (light magenta region).

Figure 8

Test 3: FOM (left) and POD-DL-ROM (center), with n = 4 and N = 256, ACs and relative error ϵ_k (right), for the testing-parameter instances μ_test = (1.7168, −0.353198, −1.70097) cm (A) and μ_test = (1.43862, −0.803806, −1.43678) cm (B).

Figure 9

Test 3: FOM (left) and POD-DL-ROM (center) solutions, the latter obtained with n = 4 and N = 256 (A), and n = 4 and N = 1, 024 (B), together with ϵ_k (right), for the testing-parameter instance μ_test = (0.2508, 0.7932, 1.66) cm at t = 316.4 ms.

Figure 10

Test 3: Possible sites of S2 stimulus applications in the case of re-entry dynamics (magenta region) (A) and including both re-entry and non-re-entry dynamics (magenta region) (B). The coordinates of the points belonging to the highlighted region are the input parameters.

Figure 11

Test 3: FOM (left) and POD-DL-ROM (center) solutions, the latter obtained with n = 4 and N = 1, 024, together with ϵ_k (right), for the testing-parameter instances μ_test = (0.3162, 0.8638, 0.6864) cm (A) and μ_test = (0.2508, 0.7932, 0.8895) cm (B) at t = 300.8 ms.