  Medical image registration is a critical task that estimates the spatial
correspondence between pairs of images. However, current traditional and
deep-learning-based methods rely on similarity measures to generate a deforming
field, which often results in disproportionate volume changes in dissimilar
regions, especially in tumor regions. These changes can significantly alter the
tumor size and underlying anatomy, which limits the practical use of image
registration in clinical diagnosis. To address this issue, we have formulated
image registration with tumors as a constraint problem that preserves tumor
volumes while maximizing image similarity in other normal regions. Our proposed
strategy involves a two-stage process. In the first stage, we use
similarity-based registration to identify potential tumor regions by their
volume change, generating a soft tumor mask accordingly. In the second stage,
we propose a volume-preserving registration with a novel adaptive
volume-preserving loss that penalizes the change in size adaptively based on
the masks calculated from the previous stage. Our approach balances image
similarity and volume preservation in different regions, i.e., normal and tumor
regions, by using soft tumor masks to adjust the imposition of
volume-preserving loss on each one. This ensures that the tumor volume is
preserved during the registration process. We have evaluated our strategy on
various datasets and network architectures, demonstrating that our method
successfully preserves the tumor volume while achieving comparable registration
results with state-of-the-art methods. Our codes is available at:
\url{https://dddraxxx.github.io/Volume-Preserving-Registration/}.
