Computational methods for PDE constrained minimization problems in bio-mechanical imaging

Abstract

Elasticity imaging is a collection of techniques that are used to create images of mechanical properties of tissue. In elasticity imaging, the tissue is deformed, and the corresponding displacement field within the tissue is measured using standard imaging modalities such as ultrasound, optical coherence tomography, etc. Once the deformation field is obtained an inverse problem is solved in order to determine the spatial distribution of mechanical properties.

Present work focuses on developing new formulations to account for different types of data for quasi-static elasticity imaging of breast tissue. In addition to this, we have made an effort to make the elasticity imaging more robust.

This dissertation is focused on the solution of this inverse problem. In particular, we develop and implement techniques that can account for additional data (force and traction data, and three-dimensional data acquired on subjects) and techniques that are numerically more robust. The inclusion of force and traction data leads to quantitative elasticity imaging techniques that have been shown to improve the diagnostic capabilities of elasticity imaging. The application of elasticity imaging to three-dimensional clinical data leads to three-dimensional mechanical images of tumors that should be useful in surgical planning and training. Finally, the development of robust and stable finite element methods leads to numerical methods that can handle noise more effectively.

Disease changes the elastic properties of tissue. Further it is also affected by the elastic properties of tissue. Therefore measuring and quantifying the elastic properties of tissue can lead to techniques that are useful in the detection, diagnosis and treatment of different types of disease.