On fluorescence imaging: The diffusion equation model and recovery of the absorption coefficient of fluorophores

To quantify fluorescence imaging of biological tissues, we need to solve an inverse problem for the coupled radiative transfer equations which describe the excitation and emission fields in biological tissues. We begin by giving a concise mathematical argument to derive coupled diffusion equations with the Robin boundary condition as an approximation of the radiative transfer system. Then by using this coupled system of equations as a model for the fluorescence imaging, we have a nonlinear inverse problem to identify the absorption coefficient in this system. The associated linearized inverse problem is to ignore the absorbing effect on the excitation field. We firstly establish the estimates of errors on the excitation field and the solution to the inverse problem, which ensures the reasonability of the model approximation quantitatively. Some numerical verification is presented to show the validity of such a linearizing process quantitatively. Then, based on the analytic expressions of excitation and emission fields, the identifiability of the absorption coefficient from the linearized inverse problem is rigorously analyzed for the absorption coefficient in the special form, revealing the physical difficulty of the 3-dimensional imaging model by the back scattering diffusive system.