Biomechanics-Informed Non-Rigid Medical Image Registration With Elasticity Theories

Biomechanical modelling of soft tissue provides a method for constraining medical image registration, such that the estimated spatial transformation is considered biophysically plausible. Existing methods either directly optimize the loss function containing the biomechanical-constrained regularization term over deformations, which takes much computational time, or are trained using biomechanically plausible data generated via finite element simulation, which is cumbersome. This work first instantiates the recently-proposed physics-informed neural networks (PINNs) to 3D elastic models that are used to establish the partial differential equations (PDEs) representing physics laws of biomechanical constraints to be satisfied. The registration algorithm that aligns point sets considering PINN-imposed biomechanics (i.e., the forward problem) is then formulated. In addition, the inverse problem and its algorithm of physical parameter (i.e., material property) estimation along with the registration are also formulated and developed. We carefully compare linear and nonlinear elasticity theories' capabilities in solving both tasks of forward registration and inverse physical parameter identification under PINNs respectively. Furthermore, two specific network configurations that leverage one common branch or two individual branches to predict deformation vectors and biomechanical states are also constructed and compared. The proposed PINNs-based registration approaches have been extensively evaluated with three experiments, that is single and multiple patient MRI-US registration using clinical MRI-US pairs, and registration using pairs of undeformed MR images from clinical cases of prostate cancer biopsy and deformed counterparts with finite-element-computed ground-truth deformation. Results demonstrate that the proposed methods achieve state-of-the-art performances compared to biomechanical-model-based and learning-based registration approaches, and the biomechanical constraints of soft tissues have been successfully warranted after registration. The codes are available at https://github.com/ZheMin-1992/Registration_PINNs.