Identification of Young’s Modulus and Equivalent Spring Constraint Boundary Conditions of the Object With Incomplete Displacement Boundary Conditions

Abstract

In endoscopic surgery, the surgical navigation needs to calculate the internal deformation of the soft tissue by biomechanical model which needs to determine the elastic properties and boundary conditions. However, these information cannot be obtained accurately in a real operation scenario. For example, only a limited portion of a liver surface can be observed in a hepatic surgery under endoscope while its elastic properties remain unknown. In addition, simple boundary conditions such as fixed constraints and free-force constraints are not physically adequate to simulate the elastic effect of ligaments attached to the liver. Biomechanical models of the soft tissue have been thoroughly studied in recent years. In these studies, boundary conditions play an important role in identification of elastic properties for mechanical model based methods. But they rarely combine these unknown conditions together to construct the model, and instead set boundary conditions or elastic properties as known for simplification. In this paper, we present a novel method to identify the Young’s modulus and equivalent spring constraint boundary conditions of a partially observed soft object with incomplete boundary conditions. The spring constraint boundary condition is applied to alternate the conventional displacement boundary conditions (e.g. free constraint and fixed constraint) and an inverse algorithm based on the standard finite element method (FEM) and Gauss-Newton (GN) method is developed, which takes external forces and displacements of observable nodes as inputs. A series of numerical simulation experiments are implemented and the analysis of simulation results show the feasibility of the proposed method.

Identification of Young’s modulus and equivalent spring constraint boundary conditions of the soft tissue with locally observed displacements for endoscopic liver surgery

“…The spring constraint boundary condition is a generalization of fixed and free constraints, which can be regarded as attaching a spring to the constrained node on the surface of a FEM model. The fixed constraint and free constraints can be captured with very large and infinitesimal spring stiffness respectively (Xu and Lei 2020). Considering the elatisity of ligaments attached to an organ, the spring constraint boundary condition is more suitable than a fixed constraint boundary condition in a soft tissue model.…”

“…Barrett et al discovered a procedure of determing parameters of a simplified ligament model (Barrett and Callaghan 2018). In our previous work (Xu and Lei 2020), a novel method was proposed to identify unknown Young's modulus and equivalent spring constraint boundary conditions. However, the input noise will seriously affect the robustness of the algorithm, and the diversity of stiffness values of equivalent spring constraint boundary conditions is not considered.…”