Reconstruction of Excitation Waves from Mechanical Deformation using Physics-Informed Neural Networks

TL;DR

Synthetic data sets were created, consisting of 2D excitation waves coupled to an isotropic and linearly deforming elastic medium and it is shown that for both focal and spiral patterns, the underlying excitation waves can be reconstructed accurately.

Abstract

Non-invasive assessment of the electrical activation pattern can significantly contribute to the diagnosis and treatment of cardiac arrhythmias, due to faster and safer diagnosis, improved surgical planning and easier follow-up. One promising path is to measure the mechanical contraction via echocardiography and utilize this as an indirect way of measuring the original activation pattern. To solve this demanding inversion task, we make use of physics-informed neural networks, an upcoming methodology to solve forward and inverse physical problems governed by partial differential equations. In this study, synthetic data sets were created, consisting of 2D excitation waves coupled to an isotropic and linearly deforming elastic medium. We show that for both focal and spiral patterns, the underlying excitation waves can be reconstructed accurately. We test the robustness of the method against Gaussian noise, reduced spatial resolution and projected tri-planar data. In situations where the data quality is heavily reduced, we show how to improve the reconstruction by additional regularization on the wave speed. Results on the optimization of hyperparameters are also discussed. Our findings suggest that physics-informed neural networks hold the potential to solve sparse and noisy bio-mechanical inversion problems and may offer a pathway to non-invasive assessment of certain cardiac arrhythmias.

Figures (11)

• Figure 1: Time series of synthetic data generated as outlined in the Methods section. Two focal sources were initiated by inserting external voltage inside a circular region. The resulting transmembrane voltage wave produces a similar active tension wave, slightly delayed and broadened. The active tension then instantaneously induces deformation in the elastic medium.
• Figure 2: Schematic overview of one iteration of the optimization process in the NC-PINN. The neural network consist of a densely connected architecture of $N_l$ hidden layers with $N_n$ nodes each. Inputs are spatiotemporal coordinates $(x,y,t)$, outputs are deformation components $U_x$, $U_y$ and active tension $T_a$. Colours indicate the points (coordinates) used for each loss term. The network is updated after the total loss is minimized with respect to the network parameters $\theta$ (weights and biases of the network).
• Figure 3: (a) Overview of the EIK-PINN to enhance the reconstruction of the local activation time (LAT) $\tau$. Colours indicate the points (coordinates) used for each loss term. (b) Schematic overview of the two-step optimization (NC+EIK) in case both elastic and wave propagation constraints are used. The standard NC-PINN is used to convert deformation $\vec{U}$ into a first estimate of active tension $\hat{T}_a$. From this field, the LAT is calculated, and optimized subject to the eikonal relation (Eq. \ref{['eik-eq']}) via a second PINN, depicted in panel (a). The emerging $\tau$ can either inter- or extrapolate the solution or overwrite initially difficult regions, after which it is converted back to $T_a$ via Eq. \ref{['gaussian-eq']}.
• Figure 4: NC-PINN results for the reconstruction of an active tension wave from mechanical displacement data using the optimization hyperparameters in Table \ref{['tab: hyperparameters']}. Panels (a) and (b) show the true and predicted spatial fields $U_x$, $U_y$ and $T_a$ as well as their differences, at time $t=30$. Panels (c) and (d) visualise true and predicted snapshots of $T_a$ over time and their difference.
• Figure 5: Effects of varying hyperparameters in optimizing the NC-PINN on all available data. Reported fsim values are averaged over $5$ independent runs, while the shaded area represents the standard deviation. Panels (a) and (b) show the influence on the fsim score for the number of layers $N_l$ and number of neurons $N_n$. All curves seem to reach a plateau phase, while their converging point depends on the pattern type. Panel (c) shows the influence of the weighting parameter $\alpha_{nc}$, while $\alpha_{data} = 1$. Both curves indicate an optimal value around $10^2$.