Patient-Specific Geometry Parameterization

• Patient-specific geometry parameterization is the process of modeling anatomical structures from medical imaging, preserving critical morphology for analysis and simulation.
• Techniques such as deep learning segmentation, surface extraction, and atlas-based registration ensure robust mapping and mesh quality for finite element studies.
• Normalized coordinate systems and structured mesh generation enable direct quantitative comparisons and personalized diagnostics across patient populations.

Patient-specific geometry parameterization refers to the process of representing, modeling, and discretizing anatomical structures derived from medical imaging such that critical morphological information is preserved and ready for quantitative analysis, simulation, or statistical modeling—while capturing patient-specific variability. This parameterization fundamentally underpins the construction of computational models for personalized diagnostics, treatment planning, and risk prediction in fields such as cardiovascular biomechanics, musculoskeletal analysis, and organ-level finite element (FE) simulations. Across modalities, robust patient-specific parameterization ensures that geometric descriptors and physical quantities can be directly compared, aggregated, or analyzed across populations in a common reference or coordinate framework.

1. Image Acquisition and Anatomical Segmentation

The workflow begins with high-resolution 3D imaging—most commonly CT or MRI—followed by segmentation of the anatomical region of interest. Deep learning architectures such as nnU-Net (a U-shaped encoder–decoder with skip connections) have become standard for voxelwise segmentation of complex structures; for example, nnU-Net is trained on curated CT cohorts, with hybrid loss functions combining Dice loss and cross-entropy:

$$L = L_{\text{Dice}} + L_{\text{CE}}, \qquad L_{\text{Dice}} = 1 - \frac{2\sum_{i,j} y_{i,j} p_{i,j}}{\sum_{i,j} y_{i,j} + \sum_{i,j} p_{i,j}}$$

Segmentation outputs are typically reviewed and manually refined for anatomical @@@@1@@@@ (e.g., consensus editing in 3D Slicer (Chen et al., 16 Sep 2025)). These segmented masks encode the topology and gross shape, isolating input for further geometric processing.

2. Surface Extraction and Landmark-driven Representation

After segmentation, a geometric surface model is extracted. Techniques include marching cubes for explicit triangulated meshes or, alternatively, annotation of internal anatomical landmarks and centerlines that serve as a low-dimensional, topology-aware scaffold. For tubular structures (e.g., vessels), a dense centerline with per-point radii enables explicit parameterization:

$$P_{i,j} = r_i \cos \theta_j N_i + r_i \sin \theta_j B_i + C_i, \quad \theta_j = 2\pi j / n_{\theta}$$

with a Frenet frame $\{T_i, N_i, B_i\}$ computed at each centerline point $C_i$ (Chen et al., 16 Sep 2025). This approach bypasses noisy surface discretizations by encoding geometry through parametric skeletal descriptions, supporting robust downstream meshing.

Landmarks may include hinge points, end curves, or connectivity features for bifurcations; their identification is standard in vascular and musculoskeletal modeling for registering models across patients.

3. Mesh Generation: Structured and Unstructured Techniques

A paramount step is generating a computational mesh conforming to patient-specific anatomy. For FE and biomechanical simulations, structured hexahedral meshes are desirable for numerical accuracy and efficiency.

Template-driven Quad/Hex Mesh Synthesis

Parameterization converts segmentation masks and anatomical centerlines into structured surface meshes by deforming or fitting an initial template (often a tubular quad mesh):

1. Mesh nodes are laid out according to the parameterized $(s,\theta)$ space or along landmarks.
2. Mesh geometry is variationally optimized to minimize a composite meshing loss:

$$\text{meshing\_Loss} = \text{Chamfer}(S_{gt}, S_{temp}) + \text{mesh\_quality\_loss}(S_{temp})$$

Mesh quality loss includes penalties on element angles, surface normal consistency, and geometric fidelity, but avoids explicit smoothing (no Laplacian or NURBS regularization) (Chen et al., 16 Sep 2025).

1. For volumes, a uniform or anatomically informed wall thickness is swept from the luminal surface, ensuring all-hexahedral interior discretization.

Atlas-based Elastic Registration (MMRep)

Elastic registration employs a $C^1$-diffeomorphic mapping $R$ that warps a generic “Atlas” mesh to the patient-specific surface:

$$R = r_N \circ \dots \circ r_1,\quad E[R_n] = \sum_{s\in S} d(R_n(s), D)$$

where each elementary deformation $r_n$ is localized to a grid node displacement and diffuses smoothly to neighbors via Hermite polynomials. Mechanical regularization is enforced using a proxy strain energy to prevent non-physical distortions. Local mesh repair (on element Jacobian and quality) ensures FE suitability and surface fidelity, enabling statistically meaningful population studies (Bucki et al., 2010).

4. Coordinate Systems and Parameter Domain Normalization

Critical for cross-patient comparison and statistical analysis is the mapping of patient-specific anatomy to a normalized parameter domain.

• For tubular or vessel-like structures, a two-parameter system $(s,\theta)\in[0,1]\times[0,2\pi]$ maps each mesh vertex to coordinates along the centerline and circumferentially.
• In cardiac applications, universal coordinates are extracted by solving Laplace equations on the surface mesh with Dirichlet constraints at anatomical boundaries, yielding intrinsic $(\alpha, \beta)\in[0,1]^2$ coordinates that normalize for pathlengths, aspect ratios, and boundary topology (Roney et al., 2018).
• For musculoskeletal tissue, region-based partitioning (e.g., via inflection points in distance-to-centroid distributions) allows per-region or per-vertex quantification (Paccini et al., 2023).

Standardized parameter domains enable homologous sampling of geometric and physical quantities—such as diameter, curvature, and wall thickness—with direct spatial registry across a cohort.

5. Extraction of Geometric Descriptors and Clinical Biomarkers

Once parameterization is established, geometric and functional descriptors are extracted:

• Diameter, curvature, torsion: Centerline-based computation with maximal inscribed sphere radii $r(s)$, derivatives for curvature $\kappa(s) = \|C''(s)\|/\|C'(s)\|^2$ (Chen et al., 16 Sep 2025).
• Wall thickness: Either assumed uniform (e.g., aorta) or locally enforced (e.g., cornea via per-node pachymetry) (Chen et al., 16 Sep 2025, Giraudet et al., 2021).
• Tissue and disease markers: For bone or cartilage, per-vertex mapping of greyscale (HU) values, local deformation metrics between follow-up and baseline, or partitioned region intensity statistics (Paccini et al., 2023).

Such descriptors are pivotal for predictive modeling (e.g., aneurysm rupture risk, degeneration quantification) and form the basis for subsequent finite element analyses or statistical shape modeling.

6. Integration with Simulation and Reduced-Order Modeling

Patient-specific parameterization is foundational for clinical simulation pipelines:

• Structured mesh topology enables physics-based FE simulation (e.g., via PyTorch-FEA for wall stress in aorta), with all nodes and elements registered in the common $(s,\theta)$ domain (Chen et al., 16 Sep 2025).
• The adoption of normalized parameter domains means that stress, strain, or flow metrics can be pooled across patients for statistical or machine learning analyses, ensuring comparability.
• In reduced-order and surrogate modeling, geometry parameterization is employed to compress and interpolate high-dimensional shape information, allowing efficient mapping from geometry to physical response fields with accelerations on the order of $10^5$-fold compared to full-order simulations (Ye et al., 2023, Du et al., 2022).

7. Challenges, Limitations, and Future Directions

Despite foundational advances, several challenges persist:

• Atlas dependence and anatomical variability: Atlas-based methods rely on the representativeness of the template mesh; extreme or pathological anatomies may require multi-atlas strategies or adaptive remeshing (Bucki et al., 2010).
• Landmarking and manual intervention: While deep learning automates much of segmentation, anatomical landmarking and mesh repair may involve expert input, limiting throughput.
• Parameter identifiability: Certain applications (e.g., multiscale corneal modeling) exhibit sensitivity to unobservable parameters; full-field imaging or tissue property measurement may be necessary for robust fitting (Giraudet et al., 2021).
• Generalizability: Universal coordinate systems and landmark-driven parameterizations are robust for many classes of geometries but may require adaptation for highly non-standard topologies or developmentally variant anatomies.

Continued standardization of pipeline steps, with emphasis on differentiable parameterizations, shared mesh topologies, and integration with statistical learning frameworks, is a key focus for scaling patient-specific modeling to large clinical populations. Advances in generative shape modeling (e.g., hierarchical NURBS diffusion, convolution surfaces) offer new avenues for unbiased synthesis and statistical representation of anatomical variability, further bridging clinical and computational domains (Du et al., 15 Jul 2025, Bošnjak et al., 2024).