LDDMM stochastic interpolants: an application to domain uncertainty quantification in hemodynamics

1. Problem Statement

Numerical simulations of physical systems, particularly Computational Fluid Dynamics (CFD) for cardiovascular applications, rely heavily on the accuracy of the computational domain (the geometry). However, anatomical shapes derived from medical imaging (MRI/CT) are subject to domain uncertainties arising from:

• Measurement noise: Errors in image acquisition.
• Segmentation biases: Errors introduced during manual or automated processing to create surface and volume meshes.
• Inter-patient variability: Natural anatomical differences between individuals.

Traditional Uncertainty Quantification (UQ) methods often struggle with geometric uncertainties because anatomical shapes cannot be easily described by a low number of parameters. Existing approaches like Statistical Shape Models (SSMs) or Gaussian random fields often lack the ability to generate realistic, high-fidelity 3D shapes that preserve complex topological features and mesh quality required for high-order numerical solvers.

The Core Challenge: How to generate realistic, random perturbations of 3D anatomical shapes (specifically aortic geometries) that are mathematically consistent, preserve mesh quality for CFD simulations, and allow for the quantification of how shape variability affects hemodynamic biomarkers.

2. Methodology

The authors propose a novel framework combining Conditional Stochastic Interpolants with Large Deformation Diffeomorphic Metric Mapping (LDDMM).

A. Theoretical Framework: Conditional LDDMM Stochastic Interpolants

• Stochastic Interpolants: The method models the transformation between a source distribution (e.g., a template shape) and a target distribution (e.g., patient-specific shapes) as a continuous-time stochastic process. It learns a time-dependent drift (velocity field) $b_t$ that guides a stochastic differential equation (SDE) from the prior to the target.
• Handling Different Domains: Standard stochastic interpolants assume data exists on a fixed grid (like images). This paper extends the concept to random variables defined on distinct physical domains (different 3D meshes).
• The LDDMM Connection: To bridge different domains, the authors utilize ResNet-LDDMM registration.
  • They compute diffeomorphic (or bi-Lipschitz) registration maps $\phi_t$ between a template mesh and target meshes.
  • These maps are used to define a conditional drift $u^{cond}_t$. Instead of simply adding vectors (which is impossible on non-aligned meshes), the method uses the push-forward and pull-back operators induced by the registration map to transport fields between domains.
  • The drift is trained to minimize the difference between the learned velocity field and the LDDMM registration velocity field.

B. Generative Process

1. Latent Conditioning: Shapes are conditioned on a low-dimensional latent variable $c$ (e.g., centerline coordinates and inscribed sphere radii).
2. SDE Evolution: New shapes are generated by sampling a conditioning variable $c$ and evolving the SDE:
   $$dI_t = b_\theta(I_t, c)dt + \sigma_t dW_t$$
   starting from a template shape $I_0$.
3. Mesh Extension: A critical step involves extending the surface deformation (generated by the SDE) into the interior volume to create a valid computational mesh. The authors use an iterative non-linear elasticity solver combined with smoothing algorithms to ensure the resulting hexahedral mesh hierarchy is preserved (crucial for geometric multigrid solvers) and free of degenerate elements.

C. Application Setup

• Dataset: 1,261 synthetic aortic shapes generated via SSM from 228 patient MRI scans.
• Solver: High-order Discontinuous Galerkin (DG) solver (ExaDG) for incompressible Navier-Stokes equations with Windkessel boundary conditions.
• Task: Generate domain perturbations to perform forward UQ on hemodynamic biomarkers.

3. Key Contributions

1. Novel Generative Framework: Introduction of Conditional LDDMM Stochastic Interpolants, a method that unifies optimal transport-based registration (LDDMM) with modern generative modeling (Stochastic Interpolants) to handle data on variable 3D domains.
2. Domain-Aware Generation: The ability to generate random perturbations of 3D shapes defined on distinct meshes, overcoming the limitation of standard grid-based generative models.
3. Mesh Preservation Strategy: A robust algorithm for extending surface deformations to the interior volume while preserving the hierarchical structure of hexahedral meshes, enabling the use of efficient geometric multigrid preconditioners in CFD.
4. Comprehensive UQ Application: A systematic study quantifying the impact of geometric uncertainty on clinically relevant biomarkers in aortic blood flow, moving beyond simple parametric variations to full shape variability.

4. Numerical Results

The framework was tested on 30 base aortic geometries, generating perturbed batches with radius variations ($\alpha_r \in \{0.7, 1.0, 1.3\}$).

• Shape Generation: The model successfully generated realistic aortic shapes, including smooth interpolations between latent space points. It could generate perturbations of controlled magnitude, though it struggled with highly localized deformations (e.g., specific aneurysms) unless the training data explicitly contained such features.
• Biomarker Sensitivity:
  • Pressure and Velocity: Geometric uncertainty had a significant impact on pressure drops and velocity magnitudes. Narrower geometries (radius reduction) showed higher sensitivity and larger variations in pressure and velocity compared to wider geometries.
  • Wall Shear Stress (WSS): While absolute WSS values varied, the directional variability and sensitivity were comparable across different batches.
  • Oscillatory Shear Index (OSI): Sensitivity was comparable across batches and radius variations, suggesting OSI is a robust metric against geometric noise.
  • Secondary Flow: The Secondary Flow Degree (SFD) was less sensitive to geometric variations than pressure or velocity.
• Sample Size Study: Increasing the sample size from 10 to 100 perturbations per geometry showed that the smaller sample size (10) was sufficient to capture the statistical distributions of key quantities (mean pressure, velocity) with high fidelity, as measured by Wasserstein-1 distance.
• Mesh Quality: The proposed mesh extension algorithm successfully maintained valid volume meshes for the vast majority of generated shapes, with only ~5% of cases failing due to extreme radius reductions causing element degeneration.

5. Significance and Impact

• Clinical Relevance: This work provides a rigorous tool for Forward Uncertainty Quantification. It allows clinicians and researchers to assess how errors in medical image segmentation propagate to hemodynamic biomarkers (like pressure gradients or WSS), which are critical for diagnosing pathologies (e.g., aneurysms, stenosis) and planning interventions.
• Data Augmentation: The method enables the generation of synthetic 3D anatomical data, which is valuable for training deep learning models in medical imaging where data is scarce.
• Methodological Advancement: By bridging the gap between diffeomorphic registration (a staple in medical image analysis) and generative AI (a staple in modern machine learning), the paper opens new pathways for applying generative models to complex, non-Euclidean physical domains beyond simple image synthesis.
• Computational Efficiency: The preservation of mesh hierarchies ensures that these stochastic simulations can be run efficiently using high-order solvers, making large-scale UQ studies feasible.

In conclusion, the paper presents a mathematically sound and practically effective framework for quantifying the impact of anatomical shape uncertainty on blood flow simulations, offering a significant step forward in the reliability of patient-specific computational hemodynamics.