Adaptive finite element methods for increased resolution in fluorescence optical tomography

Fluorescence optical tomography is an emerging tool for molecularly based medical imaging. In order to provide the required accuracy and resolution for imaging interior fluorescent yield and/or lifetime within the tissue, accurate experimental measurements as well as efficient and accurate numerical algorithms are needed. Herein, we present a new adaptive finite element approach to the inverse imaging problem that is able to significantly increase the resulting image resolution and accuracy, by (i) using finer meshes for the parameter estimation where the dye concentration varies significantly, (ii) using finer meshes for the fluence prediction where gradients are significant, while (iii) choosing coarse meshes in other locations. The nonlinear iterative optimization scheme is formulated in function spaces, rather than on a fixed grid. Each step is discretized separately, thus allowing for meshes that vary from one nonlinear step to the next. Furthermore, by employing adaptive schemes in the optimization, only the discretization level of the final mesh defines the achievable resolution, while the initial steps can be performed on coarse, cheap meshes. Using this technique, we can significantly reduce the total number of unknowns, which not only stabilizes the ill-posedness of the inverse problem, but also adapts the location and density of unknown parameters to achieve higher image resolution where it is needed. Specifically, we use an a posteriori error criterion to iteratively and adaptively refine meshes for both the forward and inverse problems based on derivatives of excitation and emission fluences as well as the sought parameter. We demonstrate this scheme on synthetically generated data similar to available experimental measurements.