This thesis work originated from a collaborative project with J. Greenleaf and M. Fatemi at the Ultrasound Research Laboratory at the Mayo Clinic. The main objective of the project was to develop a full simulation infrastructure for the assessment and design of Ultrasound Vibro-Acoustography (UVA) systems. While significant results had already been obtained for simplified (purely acoustic) models of UVA (where the human tissue is modeled as fluid-like material), there were certainly possibilities for further improvements.
In this connection, the work in this thesis focuses on the derivation of more realistic viscoelastic UVA models that extend the acoustic models, and on the development of efficient numerical algorithms to resolve the resulting mathematical problems.
The first part of the thesis reviews the fundamental background of acoustic and viscoelastic wave propagation, followed by a detailed description of UVA models. A variety of numerical schemes for resolving the mathematical models are briefly discussed and their advantages and limitations are reviewed. This review suggests that accelerated integral equation methods might constitute the most suitable approach to the problem at hand.
The second part of the thesis presents the derivation of the relevant boundary integral equations (BIEs) formulation and introduces an efficient numerical scheme with high order accuracy for solving the problem. Numerical experiments are provided to demonstrate the validity of the proposed scheme. The thesis ends with some conclusions and suggestions for future work.
University of Minnesota Ph.D. dissertation. September 2011. Major: Mathematics. Advisor: Dr. Fernando Reitich. 1 computer file (PDF); vii, 60 pages, appendices A-B.
Integral equation methods for the simulation of viscoelastic ultrasound vibro-acoustography..
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