Inverse problems from medical imaging

Oregon State University, Corvallis OR

Investigators

Abstract

Thermoacoustic and photoacoustic tomography are among the emerging techniques in medical imaging which seek to combine the high sensitivity of tissue interaction with electromagnetic radiation and the high spatial resolution of ultrasound. They give rise to a variety of inverse problems for wave type equations where the unknowns may be the initial conditions or the coefficients. Some of the goals of this project are to develop analytic tools to understand the dependence of the boundary wave field on the coefficients, and algorithms to recover tissue parameters. To this end, the investigator and colleagues study inversion with variable sound speed, develop more refined models for the boundary measurement process incorporating possible reflections, and study the structure of the range of the forward operator to enhance reconstruction from limited data. A second class of problems addressed in this project relate to an older tool from tomography, namely single photon emission computed tomography. A decade ago, there were significant advances in the theory of the attenuated x-ray transform in two dimensions. In this project, the investigator and colleagues study some limited data problems for SPECT (single-photon emission computerized tomography) in three dimensional space. The problems addressed in this proposal arise in medical imaging where the overarching problem is to obtain diagnostically significant information about what is happening inside the body without having to open it. However, problems which are mathematically similar arise in other areas such as geological exploration, non-destructive testing in an industrial setting, and finding the location and orientation of buried objects (e.g. nuclear waste at Hanford). For each such area of application and new measurement technology, there are usually new mathematical questions to be answered to find effective and reliable methods to reconstruct the unknown properties of medium from external measurements. This project addresses problems arising in two areas: the first is to adapt theoretical advances in two dimensional reconstruction theory for the transform of single photon emission tomography to the three dimensional setting of clinical scanners; the second is to develop theory and algorithms for reconstruction in thermoacoustic and photoacoustic tomography. These are among a new class of hybrid imaging methods which hold the possibility of providing information on biological activity which is complementary to the structural information obtained from traditional scanning techniques.