The Inverse Problem of Thermoacoustic Tomography in Attenuating Media

Abstract

Thermoacoustic tomography is a developing medical imaging technique that combines the propagation of electromagnetic and ultrasound waves with the purpose of producing a high contrast and high resolution internal image of a specific part of the body. The underlying physics involved in this technique naturally divides its theoretical analysis into two different problems corresponding to each type of waves. One of them is the inverse thermoacoustic tomography problem which studies the propagation of ultrasound across biological tissues and aims to obtain internal information related with the initial source of the acoustic waves, from data acquired on transducers placed outside the body of interest. A natural difficulty in the analysis of ultrasound propagation is the attenuation effect caused by different tissues, in particular in the case of biological bodies. Its study is of course of great interest for real applications of this technology and it has received considerable attention in the last few years. In this thesis, I review some previous results for the thermoacoustic problem in the absence of acoustic dissipation, and in the second part, I analyze the inverse problem of thermoacoustic tomography in the presence of two types of attenuation models, local and nonlocal in time, and provide reconstruction formulas for both cases under some assumptions on the geometry of the problem. New results on uniqueness and stability of the inverse problem in the latter case are also established.