Neural Implicit Heart Coordinates

• Neural Implicit Heart Coordinates are continuous neural fields that represent cardiac anatomy, motion, and function using MLP-based mappings.
• They enable resolution-agnostic reconstruction from sparse data and facilitate accurate cardiac mesh extraction and functional analysis.
• NIHCs integrate physics-informed training and positional encodings to achieve robust spatio-temporal modeling for clinical and computational applications.

Neural Implicit Heart Coordinates (NIHCs) constitute a unifying paradigm for representing heart anatomy, motion, and function as continuous, differentiable fields using neural implicit functions. These coordinate systems, realized through multilayer perceptrons (MLPs), model complex cardiac structures and dynamics by mapping spatial and temporal coordinates—optionally conditioned on additional variables—directly to physiologically meaningful quantities. NIHCs enable resolution-agnostic synthesis and analysis of cardiac anatomy, facilitate robust reconstruction from sparse data, and allow spatiotemporally continuous queries for clinical or computational modeling.

1. Mathematical Foundations and Coordinate Systems

NIHCs generalize implicit neural functions to cardiac domains, providing continuous mappings from spatial (and often temporal) coordinates to physiologically relevant fields. Several formulations have been established:

• Spatio-temporal Neural Distance Field: $$f_\theta : \mathbb{R}^3 \times \mathbb{R} \times \mathcal{C} \rightarrow \mathbb{R}$$, where, for a 3D point $p = (x, y, z)$, time $t \in [0,1]$, and conditioning $c$ (demographic/clinical variables), the network predicts the signed distance $\hat{d} = f_\theta(p, t, c)$ to the cardiac surface (Sørensen et al., 2024). The cardiac surface at phase $t$ is the zero level set $\{ p : f_\theta(p, t, c) = 0 \}$.
• Physiological Cardiac Coordinates: In shape reconstruction tasks, NIHCs coincide with Universal Ventricular Coordinates (UVCs), assigning each 3D anatomical point a standardized quadruple $$\Phi(p) = (\varphi_1, \varphi_2, \varphi_3, \varphi_4) \in [0, 1]^4$$, encoding transventricular, transmural, rotational, and apicobasal axes (Muffoletto et al., 22 Dec 2025).
• Motion and Displacement Fields: For capturing 3D myocardial motion, one models a continuous mapping $f_\theta(x, y, z, t)$ returning both local occupancy (intensity) $O(X, t)$ and instantaneous displacements $m_{t \rightarrow t\pm 1}(X, t)$, enabling full 6D (space-time + motion) field reconstruction (Shen et al., 2023).
• Functional or Modality-Specific Mappings: Other instantiations include intensity fields for MRI (complex-valued), or displacement fields parameterized by material coordinates for strain analysis, $f_\theta(X, Z)$ (Bell et al., 10 Sep 2025).

Coordinate normalization and pre-alignment to anatomical frames (e.g., cardiac axes) ensure bijective and physiologically consistent representation across subjects and time.

2. Neural Implicit Architectures and Encodings

NIHC models rely on deep neural networks—typically MLPs—to approximate the requisite mappings. Common architectural features include:

Task Domain	Input Encoding	MLP Structure	Output Quantity
Shape SDF/occupancy (Sørensen et al., 2024)	Positional encoding (sin/cos), latent codes	8 layers × 512 (ReLU), skip conn.	Signed distance $\hat{d}$
Universal Coordinates (Muffoletto et al., 22 Dec 2025)	UVCs $\Phi(p)$, per-case latent code	8 resblocks × 128 (ReLU)	Segmentation/logits, geometry
Myocardial Motion (Shen et al., 2023)	Raw (x, y, z, t) (normalized)	SIREN MLP streams × 4 layers	Occupancy, 3D disp. vectors
MRI Intensity (Kunz et al., 2023)	Fourier features $\gamma(x, y, t)$	7 layers × 512 (ReLU), norm.	Real/Imag. image intensity
Displacement/Strain (Bell et al., 10 Sep 2025)	(x, y, \text{normalized}), latent Z	4 layers × 256 (sine, FiLM)	2D displacements

Position, time, and (optionally) conditioning variables are embedded using sin/cos positional encodings, raw normalized coordinates, or SIREN-style activations as appropriate for the task. Per-case latent vectors or demographic encoders disentangle variation due to anatomy, sequence, or external variables.

3. Training Paradigms and Loss Functions

Day-to-day training protocols for NIHCs depend on application but share commonalities:

• Point-wise Supervision: SDF, occupancy, motion, or intensity targets at random (x, y, z, t) samples guide function approximation (Shen et al., 2023, Muffoletto et al., 22 Dec 2025, Sørensen et al., 2024).
• Auto-decoder Latent Structure: For generative models, two concatenated latent vectors represent demographic (z_c) and residual (z_r) factors, with dropout on z_r to enforce conditioning (Sørensen et al., 2024).
• Loss Terms:
  • Clamped point-wise losses (e.g., L1 on SDF within surface band) (Sørensen et al., 2024).
  • Physics-informed or cycle consistency losses for motion fields—enforcing conservation, bidirectional path consistency, and cardiac cycle periodicity (Shen et al., 2023).
  • Regularization/priors on latent vectors, e.g., Mahalanobis or L2 penalties for auto-decoder codes (Muffoletto et al., 22 Dec 2025, Sørensen et al., 2024).
  • Direct Data Consistency: For MRI, losses ensure the reconstructed signal (after forward model) matches measured k-space data (Kunz et al., 2023).
  • Auxiliary losses: segmentation and mesh regression (Dice, BCE, Chamfer, RMSE) for explicit anatomical labeling (Muffoletto et al., 22 Dec 2025).

Self-supervision is prominent: motion and cycle constraints, or explicit physics priors, replace the need for dense annotation or paired deformation fields (Shen et al., 2023, Bell et al., 10 Sep 2025).

4. Inference, Reconstruction, and Clinical Utility

At inference, NIHCs offer fully continuous field evaluation—enabling structure extraction, motion analysis, and functional quantification at arbitrary spatiotemporal sampling:

• Mesh and Surface Generation: The zero level-set of SDF fields is efficiently extracted using Marching Cubes, with vertex coordinates reconstructed from NIHCs or decoded from high-dimensional representations (Sørensen et al., 2024, Muffoletto et al., 22 Dec 2025).
• Latent Code Fitting: For new patients, per-case latent codes are optimized (or inferred via encoder networks) to fit sparse input data (e.g., contours, image pairs) under the trained NIHC model, eliminating the need for explicit mesh registration or dense labeling (Muffoletto et al., 22 Dec 2025, Bell et al., 10 Sep 2025).
• Motion and Strain Quantification: Continuous displacement fields allow direct computation of dense trajectories, strain tensors, and functionally relevant clinical metrics (e.g., global circumferential/radial strain, volume change across cardiac cycle) (Bell et al., 10 Sep 2025, Shen et al., 2023, Sørensen et al., 2024).
• Sparse-to-Dense Reconstruction: High-fidelity 3D cardiac meshes are reconstructed from minimal input (2D segmentations), with accuracy on real clinical data reaching mean surface errors of 2.51±0.33 mm (diseased) and 2.3±0.36 mm (healthy), robust to slice sparsity and moderate segmentation noise (Muffoletto et al., 22 Dec 2025).

Evaluation against existing pipelines shows substantial improvements in speed and resolution-agnostic reconstruction, with inference times reduced by ~5× (5–15 s vs. 60 s for mesh fitting) and the ability to recover fine structures such as valve planes.

5. Applications Across Modalities and Physiological Domains

NIHCs have enabled advancements across multiple cardiac imaging and modeling domains:

• Patient-Specific 3D Anatomy: Anatomically coherent, high-resolution mesh generation for biophysical modeling (e.g., mechanics, electrophysiology, CFD) (Muffoletto et al., 22 Dec 2025).
• Spatio-temporal Generative Modeling: Conditional synthesis of dynamic heart anatomies for demographic-controlled cohort simulation; investigation of how clinical variables affect cardiac structure and motion (Sørensen et al., 2024).
• Motion and Strain Analysis: Fast, accurate quantification of myocardial deformation and functional metrics from tagging MRI or echocardiography (Bell et al., 10 Sep 2025, Shen et al., 2023).
• Image Reconstruction: Unsupervised, resolution-adaptive MRI video reconstruction—enabling recovery of fine temporal and spatial cardiac dynamics without ECG or large databases (Kunz et al., 2023).
• Surgical and Clinical Planning: Mapping of fiber orientation, scar, and surgical landmarks into a common coordinate system for interventions and therapy planning (Muffoletto et al., 22 Dec 2025).

The standardized, continuous representation fosters interoperability across datasets and modalities, and enables mapping between anatomical, functional, and clinical reference frames.

6. Limitations, Robustness, and Future Research

NIHC-based approaches demonstrate robustness to input sparsity, moderate segmentation noise, and varying acquisition parameters, but several limitations and areas for refinement remain:

• Runtime and Computational Cost: In domains such as MRI reconstruction, evaluation of dense coordinate fields remains costly (e.g., 8.5 h for 4 s/225-frame series on RTX 6000 GPUs), due to the need to sample the MLP at every spatial location and frame (Kunz et al., 2023).
• Sensitivity to Segmentation Quality: While robust to moderate noise, domain gap persists when input segmentations are highly misaligned; valve-plane regions and RV epicardium present localized reconstruction errors (exceeding 5–10 mm in extremes) (Muffoletto et al., 22 Dec 2025).
• Anatomical Extremes: Generalization may degrade for cases far outside the training distribution or with severe morphological alterations.
• Scalability and Multi-structure Extensions: While UVC-based NIHCs generalize anatomically within ventricles, full extension to atria, valves, or whole-cycle time-dependent fields remains an area of ongoing research (Muffoletto et al., 22 Dec 2025, Sørensen et al., 2024).

Future directions include integration of automated pre-alignment, explicit noise modeling, region-aware inference, and multi-domain fusion (CT/echo) for universal cardiac field representations.

7. Significance and Impact in Cardiac Modeling

NIHCs unify and standardize the representation of complex cardiac structure and function as continuous, queryable neural fields. By enabling rapid, anatomically consistent reconstruction from sparse data, NIHCs facilitate next-generation patient digital twins, cohort simulators, and functional modelers. Their modularity and adaptability—in supporting demographic conditioning, motion tracking, or multimodal fusion—position NIHCs as a foundational tool for computational cardiology, scalable population analytics, and emerging clinical decision-support systems (Muffoletto et al., 22 Dec 2025, Sørensen et al., 2024, Shen et al., 2023, Bell et al., 10 Sep 2025, Kunz et al., 2023).