Learning cardiac activation and repolarization times with operator learning

TL;DR

The study tackles the computational burden of cardiac electrophysiology by learning surrogate operators that map an applied stimulus to activation and repolarization time fields using Fourier Neural Operators (FNO) and Kernel Operator Learning (KOL). It evaluates these methods on 2D domains, 3D slabs, and a physiologic left-ventricle geometry, with data generated from Monodomain solves, comparing accuracy, training time, and inference speed. Results show that both approaches achieve sub-percent relative errors, with KOL generally outperforming FNO in accuracy and training time, especially on 3D unstructured meshes, while FNO offers faster per-query inference and can be extended to irregular geometries. These surrogates have strong potential to accelerate patient-specific simulations and facilitate clinical integration of rapid EP analyses, including cases where no PDE analog exists for the target map (repolarization).

Abstract

Solving partial or ordinary differential equation models in cardiac electrophysiology is a computationally demanding task, particularly when high-resolution meshes are required to capture the complex dynamics of the heart. Moreover, in clinical applications, it is essential to employ computational tools that provide only relevant information, ensuring clarity and ease of interpretation. In this work, we exploit two recently proposed operator learning approaches, namely Fourier Neural Operators (FNO) and Kernel Operator Learning (KOL), to learn the operator mapping the applied stimulus in the physical domain into the activation and repolarization time distributions. These data-driven methods are evaluated on synthetic 2D and 3D domains, as well as on a physiologically realistic left ventricle geometry. Notably, while the learned map between the applied current and activation time has its modelling counterpart in the Eikonal model, no equivalent partial differential equation (PDE) model is known for the map between the applied current and repolarization time. Our results demonstrate that both FNO and KOL approaches are robust to hyperparameter choices and computationally efficient compared to traditional PDE-based Monodomain models. These findings highlight the potential use of these surrogate operators to accelerate cardiac simulations and facilitate their clinical integration.

Figures (21)

• Figure 1: Schematic representation of the EP problem to address. In particular, we aim at reconstructing the activation/repolarization times of the cardiac tissue given the initial stimulus applied for 1 ms.
• Figure 2: Schematic architecture of Fourier Neural Operators (FNO).
• Figure 3: Kernel Operator Learning (KOL) diagram. Starting from the input function $a$, $A$ collects observations of the function at different collocation points through $\phi$. Then, the vector-valued $f^{\dagger}$ processes observations of the input into observations of the output $A = \phi(a)$. Finally, the reconstruction operator is applied to determine the output function $u = \chi(U)$.
• Figure 4: Grids adopted for the numerical simulations: (a) 2D grid, (b) 3D slab and (c) 3D unstructed ventricle.
• Figure 7: Example of FNO activation time prediction for the 3D case (acti with 2000 samples). The picture represents three slices of tissue: epicardium (top), middle (center) and endocardium (bottom).

Theorems & Definitions (1)

• Definition 1: Neural Operator