Shape of my heart: Cardiac models through learned signed distance functions

TL;DR

The paper addresses the challenge of patient-specific cardiac modeling by proposing a Lipschitz-regularized DeepSDF that encodes four cardiac surfaces (LV/RV endocardium and epicardium) in a single network $S_ heta: \mathbb{R}^{3+d} \rightarrow \mathbb{R}^4$, with latent codes $\boldsymbol{z} \in \mathbb{R}^d$ (where $d=64$) for shape identity. It trains on a library of 44 bi-ventricular shapes and uses a loss $\mathcal{L}(\theta,Z)$ with a Gaussian prior on the latent codes, augmented by a Lipschitz regularization term to produce a stable, smooth latent space, while encoding the surfaces via explicit signed distances. Shape completion from partial data is achieved by optimizing the latent code in a MAP-like objective, accommodating noisy measurements through an iterative noise estimator and a noise-dependent weighting $\beta$, enabling reconstruction from sparse endocardial data and electroanatomical mapping. The method demonstrates competitive or superior reconstruction quality compared with state-of-the-art surface reconstruction methods, shows robustness to sparsity and noise, and enables cross-modality inference, with potential impact on digital twins and time-resolved cardiac modeling.

Abstract

The efficient construction of anatomical models is one of the major challenges of patient-specific in-silico models of the human heart. Current methods frequently rely on linear statistical models, allowing no advanced topological changes, or requiring medical image segmentation followed by a meshing pipeline, which strongly depends on image resolution, quality, and modality. These approaches are therefore limited in their transferability to other imaging domains. In this work, the cardiac shape is reconstructed by means of three-dimensional deep signed distance functions with Lipschitz regularity. For this purpose, the shapes of cardiac MRI reconstructions are learned to model the spatial relation of multiple chambers. We demonstrate that this approach is also capable of reconstructing anatomical models from partial data, such as point clouds from a single ventricle, or modalities different from the trained MRI, such as the electroanatomical mapping (EAM).

Figures (9)

• Figure 1: Left: a schematic representation of the employed DeepSDF. Right: comparison of ground truth meshes (LV in red, RV in blue) with reconstruction on the training dataset. The reconstructed meshes are color-coded with the signed distance to the ground truth mesh in mm.
• Figure 2: Left: mean and standard deviation of the Chamfer distance (CD) for different numbers of input points $n$ and different levels of noise $\xi$ (lowest value in bold magenta, second lowest value in teal and italic; short version of \ref{['tab:results']} in the appendix). The asterisk denotes the non Lipschitz regularized version of our network. Right: reconstruction of both endocardia from points on the LV endocardium (gt mesh in grey, reconstruction color-coded with implicit distance).
• Figure 3: Reconstruction of the LV endocardium from the EAM data. We compare the SSM reconstruction (left panel) to our approach for an optimal noise level estimation of $\bar{\sigma}=3.368$ mm (right panel). We also report the absolute distance from the geometry provided by the EAM system.
• Figure 4: The figure shows the generation of heart models via interpolation and extrapolation of latent code vectors. In detail, we take two latent code representations $\mathbf{z}_1,\mathbf{z}_2$ from the training dataset and decode the linear combinations $t \mathbf{z}_1 +(1-t)\mathbf{z}_2$. We present the reconstructed endocardia.
• Figure 5: We reconstruct points in the latent space that are sampled from a zero-mean Gaussian distribution with covariance $\sigma^2I$.