Collaborative Research: FRG: Inverse Problems in Transport Theory

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This award supports research by a group of applied and computational mathematicians and biomedical engineers who will analyze, implement, and test image reconstruction algorithms in the emerging fields of optical tomography and optical molecular imaging. These imaging techniques are based on using photons to probe tissues and deduce their optical properties and on using molecular markers, which seek out changes inside cells that are precursors for disease development, and emit radiation that is detectable outside the body. Mathematically, we will analyze linear and non-linear inverse problems in radiative transfer theory when only angularly averaged information is available at the domain boundary. We will then devise and implement robust and accurate algorithms that reconstruct optical properties of tissues and locations of source terms from practically available measurements. We will develop fast image reconstruction algorithms, based on analytic methods applicable in simple geometries, and will test them using large data sets of experimental data from a non-contact optical tomography system operated by the University of Pennsylvania group.brbrThis project brings together an interdisciplinary team of researchers at the interface of pure and applied mathematics, theoretical physics, and medicine in order to attack mathematical problems in optical imaging of biological systems. This research is at the forefront of efforts to achieve a molecular understanding of both basic biological processes and disease states. The investigators will combine their expertise in optical physics and the modeling of the propagation of light in highly-scattering media, such as biological tissue, with mathematical studies of inverse problems. The long term goal is to understand the mathematical structure of inverse problems for the so-called radiative transport equation, and to use this knowledge to develop computationally efficient methods for image reconstruction in optical tomography.