  Deformable Image Registration (DIR) plays a significant role in quantifying
deformation in medical data. Recent Deep Learning methods have shown promising
accuracy and speedup for registering a pair of medical images. However, in 4D
(3D + time) medical data, organ motion, such as respiratory motion and heart
beating, can not be effectively modeled by pair-wise methods as they were
optimized for image pairs but did not consider the organ motion patterns
necessary when considering 4D data. This paper presents ORRN, an Ordinary
Differential Equations (ODE)-based recursive image registration network. Our
network learns to estimate time-varying voxel velocities for an ODE that models
deformation in 4D image data. It adopts a recursive registration strategy to
progressively estimate a deformation field through ODE integration of voxel
velocities. We evaluate the proposed method on two publicly available lung 4DCT
datasets, DIRLab and CREATIS, for two tasks: 1) registering all images to the
extreme inhale image for 3D+t deformation tracking and 2) registering extreme
exhale to inhale phase images. Our method outperforms other learning-based
methods in both tasks, producing the smallest Target Registration Error of
1.24mm and 1.26mm, respectively. Additionally, it produces less than 0.001\%
unrealistic image folding, and the computation speed is less than 1 second for
each CT volume. ORRN demonstrates promising registration accuracy, deformation
plausibility, and computation efficiency on group-wise and pair-wise
registration tasks. It has significant implications in enabling fast and
accurate respiratory motion estimation for treatment planning in radiation
therapy or robot motion planning in thoracic needle insertion.
