A Novel Objective Function for the Inverse Problem of Simultaneous Identification of Unknown Youngs Modulus and Boundary Conditions With Noisy and Partial Observation

Abstract

Abstract The biomechanical models of the surgical navigation systems need patient-specific elastic properties and boundary conditions to calculate the deformation of soft tissues. However, these information cannot be directly obtained in a real endoscopic surgical scenarios. Many studies have been carried out focusing on identification of unknown elastic parameters and boundary conditions. But they rarely estimate these unknown parameters together. For the convergence of unknown parameters with observation error, some methods add regularization terms to the objective functions which are complex and sometimes rely on experience. In this paper, a novel objective function based on the gradient of displacement field is proposed, which can ensure the convergence of unknown Young’s modulus and boundary conditions without additional regularization terms. A “two field” construction method, which can convert this ill-posed inverse problem to two well-posed problems, is introduced to construct this objective function. The sensitivity analysis is conducted based on approximate differential method and the Gauss-Newton (GN) method is applied to minimize the objective function and update unknown parameters. The proposed objective function is compared with a few classical objective functions in a series of numerical experiments based on the linear elastic finite element model. The results of numerical experiments show the advantages and feasibility of the proposed objective function.