Hybrid inverse problems in molecular imaging
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Hybrid inverse problems refer to inverse problems where two partial differential equations of different types are coupled. Such problems appear in modern hybrid imaging modalities where we attempt to combine two different imaging modalities together to achieve imaging abilities that could not be achieved by either of the two modalities alone. This dissertation is devoted to the study of hybrid inverse problems in two molecular imaging modalities that are based on photoacoustics: the coupling of ultrasound imaging with optical tomography through photoacoustic effect to achieve high-resolution and high-contrast imaging of molecular functions of biological tissues. The first inverse problem we study here is related to quantitative two-photon photoacoustic tomography (TP-PAT). The mathematical problem here is to reconstruct coefficients in a semilinear diffusion equation from interior information on the solution of the PDE. We derive some uniqueness, non-uniqueness and stability results on the reconstruction problem under various circumstances. Moreover, we propose a few image reconstruction algorithms and perform numerical simulations using these algorithms to complement our theoretical analysis. The second inverse problem we study here arise in quantitative fluorescence photoacoustic tomography (fPAT). The objective is to reconstruct optical coefficients in a system of radiative transport equations from interior data on the solution to the system. We study the question of uniqueness and stability of reconstructions and develop some direct and iterative image reconstruction methods for the reconstruction of the quantum efficiency and the fluorescent absorption coefficient. We also perform numerical studies on the inverse problems for media with different absorption and scattering properties.