Neural Ordinary Differential Equation based Sequential Image Registration for Dynamic Characterization

TL;DR

We address dynamic characterization in medical image registration by extending Neural Ordinary Differential Equation-based registration (NODEO) to sequential data. The NODEO framework models the entire deformation as a non-autonomous ODE, with deformation evolving as $\frac{d \phi}{d t}(\mathbf{x}, t) = \mathcal{K} \mathbf{v}_{\theta}(\phi(\mathbf{x}, t), t)$ and $\phi_0 = \mathrm{Id}$, enabling continuous-time registration and trajectory extraction. It uses a velocity-field parameterization with flexible network representations (e.g., UNet) and loss components including $\mathcal{L}_{sim} = 1 - \text{NCC}(I,J)$, $\mathcal{L}_{Jdet}$, $\mathcal{L}_{mag}$, and $\mathcal{L}_{smt}$, ensuring diffeomorphism-like behavior and smooth flows. The method demonstrates competitive registration accuracy on 2D cardiac MRI (ACDC) and 3D brain MRI (ADNI), while providing a trajectory of deformations and enabling label propagation across sequences, thereby enhancing dynamic biological analysis and longitudinal biomarker tracking. Adaptive time stepping further improves robustness to irregular sampling and reduces measurement variability, making NODEO-DIR a practical tool for dynamic characterization in medical imaging.

Abstract

Deformable image registration (DIR) is crucial in medical image analysis, enabling the exploration of biological dynamics such as organ motions and longitudinal changes in imaging. Leveraging Neural Ordinary Differential Equations (ODE) for registration, this extension work discusses how this framework can aid in the characterization of sequential biological processes. Utilizing the Neural ODE's ability to model state derivatives with neural networks, our Neural Ordinary Differential Equation Optimization-based (NODEO) framework considers voxels as particles within a dynamic system, defining deformation fields through the integration of neural differential equations. This method learns dynamics directly from data, bypassing the need for physical priors, making it exceptionally suitable for medical scenarios where such priors are unavailable or inapplicable. Consequently, the framework can discern underlying dynamics and use sequence data to regularize the transformation trajectory. We evaluated our framework on two clinical datasets: one for cardiac motion tracking and another for longitudinal brain MRI analysis. Demonstrating its efficacy in both 2D and 3D imaging scenarios, our framework offers flexibility and model agnosticism, capable of managing image sequences and facilitating label propagation throughout these sequences. This study provides a comprehensive understanding of how the Neural ODE-based framework uniquely benefits the image registration challenge.

Figures (4)

• Figure 1: The qualitative results of label propagation via registration along sequence. The example shown here is patient 001 in ACDC dataset, which has 12 frames from ES to ED. The ground-truth segmentation labels are only available for frames ED and ES. In (a), we show original image sequence with ground truth segmentations of right ventricle (RV), left ventricle (LV) cavities, and the myocardium in the first row. The second row and the third row demonstrate qualitative comparison between pairwise registration, which conducts registration between $t$ and $t+1$ for $t=1, 2, ..., N-1$, where $N=12$, and sequential registration where the one-time registration processes the whole sequence. Here the frames are subsampled for visualization. In (b), we show the corresponding deformation field for both pairwise and sequantial registration.
• Figure 2: Comparison of hippocampal images between two disease stages: Amyloid-negative cognitively normal (A- CU) participants and Amyloid-positive Alzheimer's disease (A+ AD) participants. The A+ AD group typically exhibits smaller hippocampi and accelerated hippocampal volume loss compared to the A- CU group.
• Figure 3: Comparison of volumetric changes in the hippocampus between pairwise and sequential NODEO analyses in the Amyloid positive Alzheimer's disease (severe AD) group on ADNI. Volumes from each scan are normalized by the baseline volume, yielding proportions relative to the baseline volume expressed as percentages.
• Figure 4: Comparison of progression trajectories among subjects using fixed and adaptive time-step sizes NODEO, as well as between Amyloid-negative cognitively unimpaired and Amyloid-positive Alzheimer's disease groups. Spaghetti plots illustrating individual participant progressions are depicted with colorful lines for each hemisphere of the hippocampus.