An efficient algorithm for the inverse problem in elasticity imaging by means of variational r-adaption

Area of Science:

• Computational mechanics
• Biomaterials science
• Medical imaging

Background:

• Accurate modeling of soft biological tissue mechanics is crucial for understanding disease progression and developing effective treatments.
• Finite element methods (FEM) are widely used but can be computationally expensive, especially for complex nonlinear problems like hyperelasticity.
• Detecting material interfaces, such as between healthy and diseased tissue, remains a challenge in current FEM approaches.

Purpose of the Study:

• To present a novel finite element formulation for efficient computation of stiffness distribution in soft biological tissues.
• To develop an adaptive mesh refinement strategy that effectively detects material interfaces and improves solution accuracy.
• To combine r-adaption and h-adaption into a variational hr-refinement algorithm for enhanced performance.

Main Methods:

• Consideration and iterative solution of the inverse problem of finite strain hyperelasticity.
• Implementation of a novel mesh adaption technique based on r-adaption (re-allocation of nodes).
• Integration of variational r-refinement with variational h-adaption (Arnold et al, 2010) to create a variational hr-refinement algorithm.

Main Results:

• The proposed r-adaption method effectively detects material interfaces between healthy and diseased tissue.
• The variational hr-refinement algorithm improves the accuracy of predicting deformation fields.
• The method achieves a good agreement between predicted and experimentally observed deformation fields.
• The computational efficiency is enhanced by using adaptive finite element methods.

Conclusions:

• The novel variational hr-refinement algorithm offers an efficient and accurate approach for analyzing soft biological tissues.
• This method successfully captures material interfaces and predicts deformation fields, aiding in the diagnosis and understanding of tissue pathologies.
• The variational consistency of the mesh adaption ensures improved numerical solution quality.