Physics-Informed Prostate MR–US Registration by Biomechanical Fields Prediction Network

Embedding biomechanical priors into medical image registration is essential to ensure physiologically credible soft-tissue motion. However, the optimal strategy for enforcing these priors within learning-based models re mains under-explored. In this work, we propose the Biome chanical Fields Prediction Physics-Informed Neural Net work (BFP-PINN) as a unified framework for prostate MR TRUS point-set registration. To systematically investigate the impact of different physics-embedding mechanisms, we instantiate and compare three distinct strategies within this backbone: (i) Deformation, which predicts displacement constrained by strong-form Navier-Cauchy residuals; (ii) Strain, which predicts both displacement and strain; and (iii) StressStrain, which jointly predicts displacement, stress, and strain. This controlled study isolates the effects of predicting intermediate physical fields versus direct reg ularization. Extensive experiments on simulated and clinical datasets show that the Deformation strategy provides the best overall balance between geometric accuracy and biomechanical plausibility in the evaluated setting. These findings suggest that while coupled-field predictions are theoretically rigorous, the direct residual constraint offers superior optimization stability for clinical registration tasks. Source codes are available at https://github.com/ Msx00/PINNs-for-Point-Set-Registration.git.