Modeling And Algorithm Of Fluorescence Imaging Based On Inverse Problem For Partial Differential Equations

Fluorescence imaging is a target-specific molecular imaging technology using absorption coefficient of fluorophore,and fluorescence imaging in biological sciences is useful and powerful for examining fixed and living specimen,which is of great significance to the non-destructive detection of physiological tissues.The fluorescence diffuse tomography involves two physical processes,namely,the excitation photons and the emission ones.And it aims to reconstruct the absorption coefficient of fluorophore in the biological tissue,which is mathematically governed by a coupled diffusion system with respect to the excitation field and emission one.Due to the nonlinear dependence of the excitation and emission field on the absorption coefficient and the coupling of the excitation and emission field,such an inverse problem is completely nonlinear.The main work consists of the following three parts.Firstly,we reformulate the nonlinear inverse problem of fluorescence imaging as an optimization problem,with a nonconvex objective functional.It is based on the general transport diffusion equation model satisfied by the excitation field and scattering field.Then the existence of minimizers is proved with a rigorous analysis.Also,the objective functional is Fr?0)chet derivable with respect to the unknown absorption coefficient.The second part proposes an iterative algorithm for solving the minimizer of the functional.And a rigorous analysis on the convergence property of the iterative scheme is provided.In this paper,we consider an approximate optimization problem of quadratic functions in each iteration.This algorithm proposed avoids the error caused by neglecting the dependence of the excitation field on the unknown absorption coefficient directly,and it takes advantage of the quadratic functional,so that the reconstruction accuracy is improved with less computational cost.Numerical implementations are presented to show the validity of our proposed scheme in the third part.The forward problem is solved by finite element method.In the iterative algorithm of inverse problem,the formula for the gradient of the cost functional is derived via an adjoint problem,and the conjugate gradient method is applied to construct the numerical solutions.Finally,we present some numerical examples to show the efficiency of the proposed algorithms for both accurate input data and input data with noise.