The inverse source problem arising in Photoacoustic Tomography

The project addresses fundamental analytical questions on the acoustic wave
equation. The questions arise in, but are not limited to the context of
Photoacoustic Tomography (PAT).

PAT is a medical imaging technique that combines the high contrast in
electromagnetic absorption, say between healthy and cancerous tissue, with the
high resolution of ultrasound. The existing PAT image reconstruction methods
based on time-reversal arguments do not work when photoacoustic measurements
are obtained by using an array of detectors that reflects acoustic waves.
However, the use of detector arrays significantly speeds up the data
acquisition in practice. One of the main objectives of the project is to
develop a comprehensive analytical theory for PAT in a reflecting cavity
formed by detector arrays.

The mathematical results to be obtained have applications beyond PAT in the
fields of Control Theory for Partial Differential Equations and Inverse
Problems. These applications will be explored. In particular, the results
allow us to understand better the stability properties of hyperbolic inverse
boundary value problems. These problems occur in a large number of
applications including medical ultrasound tomography, seismology, oceanology,
process monitoring and non-destructive testing. In terms of these
applications, our focus is on seismic imaging.

The analytical theory for Photoacoustic Tomography (PAT) in a reflecting
cavity to be developed will immediately benefit the practitioners of this
imaging technique. Collaboration with the experimentalists in the
Photoacoustic Imaging Group in the Department of Medical Physics and
Bioengineering at University College London (UCL) guarantees that the results
of the project will directly improve the use of PAT in clinical and
preclinical imaging. Via this pathway, the results will translate, for example,
to the UCL Centre for Advanced Biomedical Imaging, which has a small animal
PAT scanner that is used in a variety of in vivo preclinical imaging studies.

We will prove new stability results on the hyperbolic inverse boundary value
problems. The mathematical theory of these problems has not been applied to
real-life measurements. However, motivated by the increased computational
power, there is an emerging interest in the theory in the field of seismic
imaging. Collaboration with the Geo- Mathematical Imaging Group at Purdue
University offers a pathway for the results to disseminate to the
practitioners of seismic imaging.