Discretization Error Analysis and Adaptive Meshing Algorithms for Fluorescence Diffuse Optical Tomography in the Presence of Measurement Noise

Quantitatively accurate fluorescence diffuse optical tomographic (FDOT) image reconstruction is a computationally demanding problem that requires repeated numerical solutions of two coupled partial differential equations and an associated inverse problem. Recently, adaptive finite element methods have been explored to reduce the computation requirements of the FDOT image reconstruction. However, existing approaches ignore the ubiquitous presence of noise in boundary measurements. In this paper, we analyze the effect of finite element discretization on the FDOT forward and inverse problems in the presence of measurement noise and develop novel adaptive meshing algorithms for FDOT that take into account noise statistics. We formulate the FDOT inverse problem as an optimization problem in the maximum a posteriori framework to estimate the fluorophore concentration in a bounded domain. We use the mean-square-error (MSE) between the exact solution and the discretized solution as a figure of merit to evaluate the image reconstruction accuracy, and derive an upper bound on the MSE which depends upon the forward and inverse problem discretization parameters, noise statistics, a priori information of fluorophore concentration, source and detector geometry, as well as background optical properties. Next, we use this error bound to develop adaptive meshing algorithms for the FDOT forward and inverse problems to reduce the MSE due to discretization in the reconstructed images. Finally, we present a set of numerical simulations to illustrate the practical advantages of our adaptive meshing algorithms for FDOT image reconstruction.