Learning geometry-dependent lead-field operators for forward ECG modeling

TL;DR

A shape-informed surrogate model of the lead-field operator that serves as a drop-in replacement for the full-order model in forward ECG simulations, resulting in highly accurate ECG simulations and consistently outperforms the widely used pseudo lead-field approximation while preserving negligible inference cost.

Abstract

Modern forward electrocardiogram (ECG) computational models rely on an accurate representation of the torso domain. The lead-field method enables fast ECG simulations while preserving full geometric fidelity. Achieving high anatomical accuracy in torso representation is, however, challenging in clinical practice, as imaging protocols are typically focused on the heart and often do not include the entire torso. In addition, the computational cost of the lead-field method scales linearly with the number of electrodes, limiting its applicability in high-density recording settings. To date, no existing approach simultaneously achieves high anatomical fidelity, low data requirements and computational efficiency. In this work, we propose a shape-informed surrogate model of the lead-field operator that serves as a drop-in replacement for the full-order model in forward ECG simulations. The proposed framework consists of two components: a geometry-encoding module that maps anatomical shapes into a low-dimensional latent space, and a geometry-conditioned neural surrogate that predicts lead-field gradients from spatial coordinates, electrode positions and latent codes. The proposed method achieves high accuracy in approximating lead fields both within the torso (mean angular error 5°) and inside the heart, resulting in highly accurate ECG simulations (relative mean squared error <2.5%. The surrogate consistently outperforms the widely used pseudo lead-field approximation while preserving negligible inference cost. Owing to its compact latent representation, the method does not require a fully detailed torso segmentation and can therefore be deployed in data-limited settings while preserving high-fidelity ECG simulations.

Figures (9)

• Figure 1: Graphical representation of the developed pipeline. (1.) Mesh generation stage: we used torso and heart models from a statistical PCA atlas. To create joint models, we varied (A.) the first 10 principal components of the torso, (B.) the first 10 principal components of the heart, and (C.) heart rotation angles along anatomical axes: $\alpha_x$ along the LV--RV axis, $\alpha_y$ along the anterior-posterior axis, and $\alpha_z$ along the LV long axis. As a result, a set of geometries was obtained and used to generate training and test point clouds (2.), while a set of features (10 torso modes + 10 heart modes + 3 angles) was used as feature vectors describing each dataset (highlighted with an orange frame). (2.) Train/test point-cloud generation. (D.) Point cloud around and inside the torso used to train/test the DeepSDF model. (E.) Point cloud inside the torso used to train/test the $\nabla Z$ prediction model. (3.) Variation of unipolar electrode location on the anterior surface of the torso. (4.) Schematic representation of DeepSDF model used for generation of DeepSDF based shape codes (highlighted with a blue frame). (5.) Schematic representation of lead field gradient $\nabla Z$ prediction model.
• Figure 2: Computed lead field for a unipolar lead (left shoulder), visualized using streamlines representing the direction of $\nabla Z$. (A.) Streamlines shown throughout the entire torso. (B.) Close-up view of the heart region (the heart is shown in yellow); streamlines passing through the free wall of the left ventricle (LV) are highlighted in bold red. (C.) Enlarged view of the LV free wall (represented as a cylinder) with the corresponding gradient streamlines highlighted in bold. Note the change in the slope of the streamlines at the heart–torso interface, reflecting the change in direction of $\nabla Z$ across the boundary.
• Figure 3: Design of training samples for parametric models of the torso and heart. A Projection of a 10-dimensional parametric space onto a 3D cube, obtained by neglecting the last 7 dimensions. Black dots indicate the position of 100 sampling points selected using Latin hypercube sampling from a uniform distribution. B and C are cross-sections of the parametric space for the heart (B) and torso (C) showing the variations of the three principal components: PC1, PC2 in blue colours, PC1, PC3 in green colours.
• Figure 4: Chamfer distances (in mm) for the four SDF surfaces-torso, LV endocardium, RV endocardium, and epicardium, computed 80/20 training/validation geometries (left panel) and 10 test geometries (right panel). The color scale indicates the Chamfer distance value (in mm), with lighter colors corresponding to larger errors. For clarity, both the SDF surfaces and the geometry sets are sorted by increasing error.
• Figure 5: Streamline visualization of the lead-field gradient for a unipolar lead (left shoulder). (A) FEM-based lead-field gradient. (B) Predicted lead-field gradient. Smaller sub-panels show close-up views of the heart and the LV free wall. Streamlines intersecting the LV free wall are highlighted. The bend in the streamlines at the heart-torso interface (highlighted in yellow) reflects the change in conductivity across tissues.