Boundary Constraint-free Biomechanical Model-Based Surface Matching for Intraoperative Liver Deformation Correction

TL;DR

The paper tackles 3D-3D non-rigid registration in image-guided liver surgery by embedding a modified finite element model (FEM) directly into the surface-matching term, removing the need to specify zero boundary conditions. It stabilizes the stiffness matrix with diagonal loading (soft springs) so forces can be learned anywhere on the liver surface, and solves the optimization with an accelerated gradient method that uses an automatically determined step size. The method uses a soft correspondence matrix derived from a closest-point mapping and updates forces via a gradient-based scheme, ensuring the volumetric mesh preserves connectivity during deformation. Across in silico phantoms, in vitro phantoms, the Open-CAS dataset, and sparse data challenges, the approach achieves accurate registration with low target registration errors and robust performance, while remaining computationally practical (e.g., ~22 seconds for typical runs) and publicly releasing code and data for benchmarking.

Abstract

In image-guided liver surgery, 3D-3D non-rigid registration methods play a crucial role in estimating the mapping between the preoperative model and the intraoperative surface represented as point clouds, addressing the challenge of tissue deformation. Typically, these methods incorporate a biomechanical model, represented as a finite element model (FEM), into the strain energy term to regularize a surface matching term. We propose a 3D-3D non-rigid registration method that incorporates a modified FEM into the surface matching term. The modified FEM alleviates the need to specify boundary conditions, which is achieved by modifying the stiffness matrix of a FEM and using diagonal loading for stabilization. As a result, the modified surface matching term does not require the specification of boundary conditions or an additional strain energy term to regularize the surface matching term. Optimization is achieved through an accelerated gradient algorithm, further enhanced by our proposed method for determining the optimal step size. We evaluated our method and compared it to several state-of-the-art methods across various datasets. Our straightforward and effective approach consistently outperformed or achieved comparable performance to the state-of-the-art methods. Our code and datasets are available at https://github.com/zixinyang9109/BCF-FEM.

Figures (9)

• Figure 1: Overview of the proposed method. The process takes a biomechanical model generated from the volumetric liver model (typically segmented from a CT or MRI image) and an intraoperative point cloud. Force optimization leads to volumetric deformation, aligning the preoperative volume model with the intraoperative point cloud.
• Figure 2: The generation of an in silico phantom. The in silico phantom undergoes deformation with forces applied perpendicularly to the x-y plane, visualized by the deformed model in red. The cropped, deformed partial surface, represented as a red point cloud, constitutes the simulated intraoperative point cloud.
• Figure 3: The generation of a in vitro silicone phantom. The undeformed liver phantom is shown on the left, and the deformed one on the right, with wedges added under its posterior surface.
• Figure 4: Distributions of fiducial markers in in vitro silicone phantoms A (left) and B (right).
• Figure 5: Results of in silico phantom study. Force recovery with correct boundary conditions (ZBCs) and force locations (left). Examination of priors on registration results (middle). Optimization speed and accuracy through optimization step $\alpha$ update and Nesterov's acceleration (right).