  This paper investigates both biomechanical-constrained non-rigid medical
image registrations and accurate identifications of material properties for
soft tissues, using physics-informed neural networks (PINNs). The complex
nonlinear elasticity theory is leveraged to formally establish the partial
differential equations (PDEs) representing physics laws of biomechanical
constraints that need to be satisfied, with which registration and
identification tasks are treated as forward (i.e., data-driven solutions of
PDEs) and inverse (i.e., parameter estimation) problems under PINNs
respectively. Two net configurations (i.e., Cfg1 and Cfg2) have also been
compared for both linear and nonlinear physics model. Two sets of experiments
have been conducted, using pairs of undeformed and deformed MR images from
clinical cases of prostate cancer biopsy.
  Our contributions are summarised as follows. 1) We developed a learning-based
biomechanical-constrained non-rigid registration algorithm using PINNs, where
linear elasticity is generalised to the nonlinear version. 2) We demonstrated
extensively that nonlinear elasticity shows no statistical significance against
linear models in computing point-wise displacement vectors but their respective
benefits may depend on specific patients, with finite-element (FE) computed
ground-truth. 3) We formulated and solved the inverse parameter estimation
problem, under the joint optimisation scheme of registration and parameter
identification using PINNs, whose solutions can be accurately found by locating
saddle points.
