Universal Ventricular Coordinates (UVCs)

• Universal Ventricular Coordinates (UVCs) are a continuous, anatomically standardized system that assigns normalized apico–basal, transmural, and circumferential positions to the heart's biventricular mesh.
• The method solves Laplacian (harmonic) PDEs on finite-element meshes with Dirichlet boundary conditions to achieve bijective, gradient-consistent mappings grounded in anatomical landmarks.
• UVCs enable robust statistical shape analysis, precise digital twin registration, and the efficient implementation of neural implicit models for high-resolution 3D cardiac reconstruction.

Universal Ventricular Coordinates (UVCs) provide a continuous, anatomically standardized reference system for mapping, comparing, and modeling the human heart’s biventricular geometry. By solving Laplacian (harmonic) partial differential equations (PDEs) on detailed finite-element meshes of the left and right ventricles, UVCs offer bijective, normalized coordinates grounded directly in anatomical landmarks. This framework enables statistically robust shape analysis, precise registration for digital twin construction, and underpins neural implicit models for efficient, anatomically consistent mesh generation across individuals (Santvliet et al., 8 Jan 2025, Muffoletto et al., 22 Dec 2025).

1. Anatomical Parameterization and Key Definitions

The UVC system assigns normalized, continuous scalar fields to each point within a biventricular cardiac mesh, plus a discrete chamber tag. In the canonical formulation (Santvliet et al., 8 Jan 2025), the coordinate fields are:

• $z$ (apico–basal coordinate): Varies from 0 at the apex to 1 at the base (valve rings), parameterizing longitudinal position.
• $\rho$ (transmural coordinate): Runs from 0 (endocardium) to 1 (epicardium), encoding the radial (wall-thickness) direction.
• $\phi$ (circumferential coordinate): Encodes the angular position around the ventricles, running continuously from one septal insertion (e.g., $-\pi$) to the other ($+\pi$ in the LV, $[-\pi/2, +\pi/2]$ in the RV).
• $\nu$ (chamber tag): Discrete label +1 for RV, –1 for LV.

Variants such as the CobivecoX system (Muffoletto et al., 22 Dec 2025) generalize this to four principal coordinates: transventricular (LV/RV binary), transmural, circumferential/rotational, and apicobasal. All are constructed to satisfy bijectivity, normalization to $[0,1]$ (with small extensions around valves for continuity), and monotonic variation proportional to geodesic distances.

2. Mathematical and Numerical Formulation

The construction of UVCs is grounded in Laplace equations on the biventricular myocardium domain $\Omega$, with anatomy-driven Dirichlet boundary conditions:

• Apico–basal ($z(x)$):
  • $\rho$0 in $\rho$1 (typically $\rho$2)
  • $\rho$3 on apex patch, $\rho$4 on valve-ring base; zero-flux elsewhere
• Transmural ($\rho$5):
  • $\rho$6 in $\rho$7
  • $\rho$8 on endocardium, $\rho$9 on epicardium; zero-flux on base and apex
• Circumferential ($\phi$0):
  • $\phi$1 in $\phi$2
  • $\phi$3 on one septal boundary, $\phi$4 on the opposite; zero-flux elsewhere
  • $\phi$5 is obtained by rescaling $\phi$6 to the desired interval per ventricle
• Chamber tag ($\phi$7): Set from anatomical region labels; not a PDE.

Discretization employs linear tetrahedral finite elements—basis functions are “hats” on each tetrahedron, with assembly of a global sparse stiffness matrix $\phi$8. Dirichlet BCs are imposed via row/column elimination or strong enforcement, and the resulting linear systems $\phi$9 are solved using preconditioned conjugate-gradient or direct solvers. All fields being harmonic allows the reuse of solver routines with different boundary tags (Santvliet et al., 8 Jan 2025).

3. Anatomical Landmarking and Implementation Pipeline

Correct anatomical labeling is critical. The UVC pipeline requires:

• Mesh labeling of apex patches, valve-ring base planes (mitral and tricuspid), full endocardium and epicardium, and septal insertions.
• Watertight, topologically consistent meshes, preferably refined ($-\pi$0 mm resolution) to represent detailed structures.
• Identification of small or vanishing apex patches (using small caps where needed), and the imposition of zero-flux on valve ring rims.

A canonical implementation (NumeriCor’s CardioTwin/meshtool) follows the outlined PDE solves, with global node interpolation for mapping Cartesian coordinates to and from UVCs. Explicit sampling is performed at regular $-\pi$1 grids to enable anatomical correspondence—e.g., $-\pi$2 points sampled at 80 $-\pi$3-levels, 160 $-\pi$4’s on LV, and 80 on RV for surface correspondence (Santvliet et al., 8 Jan 2025).

4. Pointwise Correspondence and Shape Model Construction

Once each mesh is endowed with UVC coordinates, pointwise anatomical correspondence is established by mapping uniform $-\pi$5 tuples to mesh elements and barycentric interpolation. Resulting point clouds on different hearts are Procrustes-aligned for statistical analysis.

Principal component analysis (PCA) is then performed on these vectors for statistical shape modeling (SSM), enabling:

• Dimensionality reduction and quantification of anatomical variation
• Bias/error assessment (mean Euclidean reconstruction error, explained variance by principal components, sensitivity to cohort size, and sex bias)
• Generation of synthetic, anatomically accurate geometries for digital twin or simulation studies

Mean leave-one-out cross-validation (LOOCV) errors, pointwise error mapping, and explained variance curves serve as standard metrics for SSM quality (Santvliet et al., 8 Jan 2025).

5. Geometric Interpretation of Coordinate Surfaces

UVC scalar fields define anatomically meaningful submanifolds:

• $-\pi$6=const: Cross-sections parallel to the mitral–tricuspid plane, spanning apex to base.
• $-\pi$7=const: Nested shell layers between endocardial and epicardial surfaces.
• $-\pi$8=const: Helical sheets initiated from septal insertions, wrapping each ventricle.
• $-\pi$9: Binary partitioning delineating LV and RV.

This parameterization grants monotonic, bijective, and continuous mapping throughout the myocardium, supporting tasks such as fiber assignment, region annotation, and mesh registration (Santvliet et al., 8 Jan 2025, Muffoletto et al., 22 Dec 2025).

6. Neural Implicit Representations and UVCs

In recent frameworks such as Neural Implicit Heart Coordinates (NIHCs) (Muffoletto et al., 22 Dec 2025), UVCs serve as input coordinates for multi-layer perceptrons (MLPs) jointly trained to decode 3D geometry from sparse clinical segmentations. In NIHC:

• UVCs from a template mesh (UK Biobank mean shape) are transferred to every patient’s fitted mesh.
• The neural decoder learns a shared function, mapping UVCs to Cartesian points with a latent code $+\pi$0 encoding patient-specific shape.
• This architecture enables efficient, resolution-independent 3D reconstruction:
  • Achieves mean Euclidean surface errors of $+\pi$1 mm (diseased) and $+\pi$2 mm (healthy), with inference times $+\pi$3–$+\pi$4 s (vs $+\pi$5 s for diffeomorphic pipelines).
  • Ensures anatomical consistency across sparse, misaligned, or noisy input contours.
  • Supports mesh decoding at arbitrary output resolution directly from the continuous UVC space.

The UVC-based system thus underpins both explicit mesh correspondence and implicit anatomical regularization in data-driven shape inference.

7. Quality Assurance, Limitations, and Recommended Practices

Robustness and consistency of UVC parameterizations are evaluated using:

• LOOCV mesh reconstruction errors, pointwise error mapping, PCA explained variance curves, and sex-bias analysis.
• Validation checks for harmonic field monotonicity and gradient regularity.
• Practical recommendations include localized Laplacian smoothing of sampled surfaces, coarsening $+\pi$6-samples near the apex to avoid oversampling, and exclusive preparameterization of meshes before shape modeling or fiber assignment (Santvliet et al., 8 Jan 2025).

A plausible implication is that the generalization and reproducibility of digital twin models in both classical statistical and neural settings critically depend on high-quality UVC-based correspondence.

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References:

• Integrating anatomy and electrophysiology in the healthy human heart: Insights from biventricular statistical shape analysis using universal coordinates
• Neural Implicit Heart Coordinates: 3D cardiac shape reconstruction from sparse segmentations