Mathematics of thermoacoustic tomography

Computerized tomography has had a huge impact on medical diagnostics. Numerous methods of tomographic medical imaging have been developed and are being developed (e.g., the ‘standard’ X-ray, single-photon emission, positron emission, ultrasound, magnetic resonance, electrical impedance, optical) [62, 67, 84–86]. The designers of these modalities strive to increase the image resolution and contrast, and at the same time to reduce the costs and negative health effects of these techniques. However, these goals are usually rather contradictory. For instance, some cheap and safe methods with good contrast (like optical or electrical impedance tomography) suffer from low resolution, while some high-resolution methods (such as ultrasound imaging) often do not provide good contrast. Recently researchers have been developing novel hybrid methods that combine different physical types of signals, in the hope of alleviating the deficiencies of each of the types, while taking advantage of their strengths. The most successful example of such a combination is the thermoacoustic tomography (TAT) [69, 70, 95]. Albeit not yet a common feature in clinics, TAT scanners are actively researched, developed and already manufactured, for instance by OptoSonics, Inc. (http://www.optosonics.com/), founded by the pioneer of TAT, R. Kruger. After a substantial effort, major breakthroughs have been achieved in the last few years in the mathematical modeling of TAT. The aim of this article is to survey this recent progress and to describe the relevant models, mathematical problems and reconstruction procedures arising in TAT and to provide references to numerous research publications on this topic.