Every sixth death in the world is due to cancer. This gives an indication of how important it is to detect cancer as early as possible in the progression of the disease. One way to address this problem is by use of acousto-electric tomography (AET). The method aims at reconstructing the electrical conductivity in the interior of a body without opening it. This comes in useful for cancer detection, as the conductivity of a tumor is different from organs and tissues. For AET the conductivity is reconstructed from information obtained by electrodes placed at the boundary of the body and information obtained by penetrating the body with acoustic waves.
The conductivity inside the object is linked to the model for the voltage potential inside some domain that is occupied by the body. The voltage potential in the domain can be modeled by a partial differential equation with boundary conditions. For the reconstruction problem it is important that we can guarantee that it is possible to reconstruct the conductivity (existence of a solution) and that the conductivity we find is the only solution (uniqueness of a solution). These criteria depend on the functions on the boundary of the domain. Also, the conductivity that solves the problem should continuously depend on these functions on the boundary (stability of the solution). We already investigated these three criterias for a two-dimensional domain in Euclidean space and now want to come up with some specific test problems, where we can reconstruct the conductivity. Also, out of pure mathematical interest we want to extend this analysis to curved spaces. We doubt that this analysis will help to cure cancer in the future, but it might help to get a better understanding of the reconstruction problem.