Anisotropic deformation analysis and registration field equilibrium smoothing method in expansion microscopy

Background and Objective Expansion microscopy is an innovative technique that employs swellable polymers to physically enlarge biological specimens, enabling high-resolution imaging beyond traditional optical diffraction limits. However, specimen expansion may result in non-uniform, anisotropic deformation across the sample dilation process. A method to quantify and correct this anisotropic expansion is required. Herein, we used cultured cells, HeLa (a cancer cell line) with stained microtubules, followed by 4X expansion. Lattice lightsheet microscopy captured 3D pre- and post-expansion images on the identical cell. The present imaging analysis method is based on these two sets of 3D images. Methods To address anisotropic deformation, we implemented both rigid and B-spline-based non-rigid image registration algorithms for effective 3D image warping. Furthermore, we developed a numerical technique, the registration field equilibrium smoothing method, which modifies the deformation field obtained from image registration to ensure compliance with physical laws. Results Our technique integrates the finite element method with the Lagrange multiplier method, guaranteeing that the deformation field conforms to linear elastic material equilibrium equations. Conclusions This research not only offers a method for examining anisotropic deformation in specimens processed via expansion microscopy but also enables direct analysis of mechanical strain distribution in biological samples. Furthermore, this numerical methodology could be applicable to larger samples.