Thermo-acoustic tomography is a hybrid method of biomedical imaging. It combines the good contrast of microwave imaging and high resolution of ultrasound imaging. A brief pulse of electromagnetic (EM) wave at the radio frequency is irradiated to slightly heat up the biological tissue of interest. The tissue expands due to the thermo-elastic effect; and it, hence, releases an ultrasound pressure propagating through the space. The pressure is then measured on an observation surface S. The initial pressure is roughly proportional to the EM energy absorption distribution, which contains helpful information for cancer detection. One, then, concentrates on reconstructing the initial pressure from the measured data. The mathematical model of thermo-acoustic tomography is the observability problem of wave equations.

In this talk, we will discuss several mathematical issues of thermo-acoustic tomography, such as inversion formulas, stability analysis, and speed determination. We will also briefly review some problems of quantitative thermo-acoustic tomography, whose goal is to extract physical parameters of the biological tissue from the EM energy absorption distribution.