A 3D Deformation Mapping and Correction Framework for Expansion Microscopy

Abstract

Expansion microscopy (ExM) assumes biological specimens and the hydrogel expand isotropically, but distortions are often observed and are typically corrected in two dimensions, without addressing their three-dimensional structure or physical origin. Here, I treat the hydrogel as an independent material system and develop a quantitative framework to model its intrinsic deformation during expansion in 3D. Photoactivatable fluorescent fiducial beads were embedded throughout polyacrylamide gels and imaged pre- and post-expansion. I built a 3D registration pipeline based on polyharmonic spline modeling to reconstruct the full volumetric deformation field. Unlike existing slice-by-slice 2D correction methods, this framework captures out-of-plane distortions invisible to planar analysis and yields a continuous mathematical description of the deformation field. The results show that non-uniform expansion arises from intrinsic hydrogel properties rather than biological sample effects, allowing material-driven distortion to be quantitatively separated from specimen-specific contributions. Quantifying this baseline behavior enables separation of material-driven distortion from specimen-specific contributions and provides a framework for engineering ExM protocols with uniform expansion.