Regularized Kelvinlet Functions to Model Linear Elasticity for Image-to-Physical Registration of the Breast

Image-guided surgery requires fast and accurate registration to align preoperative imaging and surgical spaces. The breast undergoes large nonrigid deformations during surgery, compromising the use of imaging data for intra-operative tumor localization. Rigid registration fails to account for nonrigid soft tissue deformations, and biomechanical modeling approaches like finite element simulations can be cumbersome in implementation and computation. We introduce regularized Kelvinlet functions, which are closed-form smoothed solutions to the partial differential equations for linear elasticity, to model breast deformations. We derive and present analytical equations to represent nonrigid point-based translation ("grab") and rotation ("twist") deformations embedded within an infinite elastic domain. Computing a displacement field using this method does not require mesh discretization or large matrix assembly and inversion conventionally associated with finite element or mesh-free methods. We solve for the optimal superposition of regularized Kelvinlet functions that achieves registration of the medical image to simulated intraoperative geometric point data of the breast. We present registration performance results using a dataset of supineMRbreast imaging from healthy volunteers mimicking surgical deformations with 237 individual targets from 11 breasts. We include analysis on themethod's sensitivity to regularized Kelvinlet function hyperparameters. To demonstrate application, we perform registration on a breast cancer patient case with a segmented tumor and compare performance to other image-to-physical and image-to-image registrationmethods. We show comparable accuracy to a previously proposed image-to-physical registration method with improved computation time, making regularized Kelvinlet functions an attractive approach for image-to-physical registration problems.