A physics-informed neural network for improving surface reconstruction of intracranial saccular aneurysms via variational membrane equilibrium

Technical Summary: Physics-Informed Neural Network for Intracranial Saccular Aneurysm Reconstruction

Problem Statement
Intracranial saccular aneurysms (ISAs) pose a significant health risk, yet current patient-specific rupture risk assessment is limited by the fidelity of surface reconstruction from clinical imaging (CTA/MRA). Conventional reconstruction pipelines rely on mathematical smoothing techniques, such as L-curve criteria and discrete Laplacian filtering, to suppress imaging noise. However, these methods indiscriminately treat high-curvature regions as noise, thereby erasing genuine pathological features like rupture-prone blebs while failing to distinguish them from non-physical concave artifacts induced by hardware limitations (e.g., partial volume effects) and post-processing algorithms. These geometric artifacts introduce spurious stress concentrations in subsequent biomechanical simulations, corrupting rupture risk diagnostics. Furthermore, existing approaches lack a physical mechanism to differentiate between mathematically smooth surfaces and those that satisfy the biomechanical equilibrium required for a pressurized vascular wall.

Methodology
The authors propose a Physics-Informed Neural Network (PINN) framework that integrates B-spline surface representations with variational mechanics to reconstruct ISA geometries. The workflow proceeds as follows:

1. Data Acquisition and Parameterization: Clinical MRA/CTA data is segmented to extract a discrete point cloud. This unstructured data is mapped into a structured parametric domain $(u, v)$ using a normal-driven alignment and grid sampling algorithm, resulting in a 3-channel spatial tensor.
2. Baseline Generation: A low-resolution NURBS surface is fitted to the data to generate macro control points. These are upsampled via bicubic interpolation to create a high-resolution baseline ($P_{base}$) for fine-scale refinement.
3. Manifold-Consistent CNN: A Convolutional Neural Network (CNN) acts as a nonlinear ansatz to predict a control point displacement field ($d$). Unlike standard CNNs, this architecture employs a "Manifold-Consistent Padding" scheme. This custom padding respects the topological constraints of the closed vascular surface: it enforces periodicity at the circumferential seam ($u=0, 1$), handles singular pole convergence at the dome apex ($v=0$), and applies cubic extrapolation at the open neck boundary ($v=1$).
4. Physics-Informed Loss Function: The network is optimized via backpropagation using a composite loss function ($L_{total}$) that balances three components:
   • Data Fidelity ($L_{data}$): Minimizes the mean squared error between the reconstructed NURBS surface and the measured point cloud.
   • Variational Equilibrium ($L_{equil}$): Embeds the stationarity condition of total potential energy ($\delta\Pi = 0$) as a physics-informed constraint. This term enforces Laplace membrane equilibrium ($\kappa_1 T_1 + \kappa_2 T_2 = P$), ensuring the reconstructed surface can sustain transmural pressure. This effectively filters out non-physical concave artifacts (where mean curvature $H < 0$) while preserving genuine high-curvature features that satisfy mechanical equilibrium.
   • Tangential Suppression ($L_{tang}$): Penalizes control point displacement along the tangent plane, ensuring updates occur strictly in the normal direction to effect physical shape changes rather than mere reparameterization.

Key Contributions
The paper identifies three primary contributions:

1. Biomechanically Grounded Artifact Discrimination: The framework replaces the mathematical L-curve criterion with a variational equilibrium loss ($\delta\Pi = 0$). This allows the system to distinguish between imaging artifacts (which cannot sustain transmural pressure) and genuine pathological features (blebs) based on physical admissibility rather than geometric smoothness.
2. Manifold-Consistent CNN Ansatz: The development of a specialized CNN architecture that preserves the closed-surface topology of vascular geometries through custom padding operations, addressing the discontinuities inherent in flattening 3D manifolds into 2D parametric grids.
3. Laplace Equilibrium-Driven Reconstruction: The establishment of a reconstruction pipeline that filters noise while preserving diagnostically critical high-curvature features, resulting in a surface consistent with the mechanical homeostasis of the aneurysm wall.

Results
The framework was validated on a patient-specific clinical dataset of an unruptured ISA.

• Artifact Removal: Iterative optimization successfully eliminated non-physical concave artifacts (negative mean curvature) present in the initial constrained fitting baseline. By iteration 3000, the curvature field was strictly positive ($H > 0$), consistent with a pressurized membrane.
• Risk Map Optimization: The resulting rupture risk map, derived from the stress ratio ($\sigma_R$), showed a spatially coherent pattern. High-risk zones were concentrated at the dome apex, aligning with established clinical observations and intraoperative findings.
• Comparison with Conventional Methods: In contrast, the conventional L-curve smoothing method produced scattered, incoherent risk hotspots across the vessel wall. Quantitative analysis revealed that any application of L-curve smoothing ($\beta > 0$) drastically altered the curvature at high-risk locations, with no viable parameter window to recover original high-curvature features. The PINN approach maintained the integrity of these features while removing noise.

Significance and Claims
The authors claim that this work establishes a computational biomarker foundation for patient-specific rupture risk assessment. By integrating deep learning with the fundamental governing equations of aneurysm wall mechanics, the framework provides a method to reconstruct artifact-free geometries directly from routine CTA/MRA data without invasive measurements. The resulting risk maps are consistent with neurosurgical observations (rupture at the dome apex), offering a more reliable tool for preoperative planning and surveillance than current mathematical smoothing techniques. The study bridges geometric modeling and mechanical analysis, treating the reconstructed shape as the equilibrium configuration of the aneurysm wall under deformation.

Limitations and Future Directions
The authors acknowledge that the current model assumes an isotropic elastostatic membrane, neglecting wall thickness variation, anisotropic collagen architecture, and pulsatile hemodynamic loading. Validation was qualitative, relying on neurosurgical experience rather than direct comparison with confirmed rupture sites, partly due to the rarity of side-wall aneurysm ruptures compared to bifurcation sites. Future work aims to extend the framework to multi-patch NURBS for complex multilobed morphologies, incorporate fluid-structure interaction (FSI), and enable longitudinal tracking of weak point evolution.