Stability and Discretization Techniques for Elliptic Inverse Parameter Problems in Elastography - Application to Breast Tumor Detection

Explore stability and discretization techniques for elliptic inverse parameter problems in elastography, focusing on breast tumor detection. Delve into the challenges of recovering the shear modulus of biological tissues from internal displacement data, and learn about a novel Galerkin approach that constructs finite dimensional operators invertible with stability in the L^2 norm. Examine the Reverse Weak Formulation of linear elasticity equations and discover how well-chosen finite element spaces can satisfy generalized discrete inf-sup conditions. Gain insights into quantitative error estimates for the inverse problem and understand the efficiency of a method that doesn't require iterative resolution of the forward problem or smoothness hypotheses. Witness the application of these techniques through numerical examples, experimental data, and in vivo experiments from elasto-static stimulations of breast tumors in this comprehensive seminar presented by Laurent Seppecher from École Centrale de Lyon.

Intro
The reduced elastography problem
Elastography from internal data
Quasi-static deformation of a phantom
Quasi-static elastography
Reconstruct the corresponding shear modulus
An hybrid (multi-physics) imaging method
Inversion step 1: recover the displacement
Inversion step 2: recover the shear modulus
Available inversion algorithms (1)
Available inversion algorithms (2)
Current chalenges for medical elastography
A general equation
The Reverse Weak Formulation
Reverse Weak Formulation discretization
Approximation of the operator
Questions
A model problem: discretization
Inf-sup constant for the operator T
Generalized inf-sup constant
Discrete inf-sup constant
Upper semi-continuity of the inf-sup constant
honeycomb finite element
Inverse gradient problem: behavior of 8(T)
Reconstruction for the honeycomb
In vivo quasistatic elastography