A theoretical framework of x-ray dark-field tomography

X-ray mammography is currently the most prevalent imaging modality for screening and diagnosis of breast cancers. However, its success is limited by the poor differentiation between healthy and diseased tissues in the mammogram. A potentially prominent imaging modality is based on the significant difference of x-ray scattering behaviors between tumor and normal tissues. Driven by major practical needs for better x-ray imaging, exploration into contrast mechanisms other than attenuation has been active for decades, e. g., in terms of scattering, which is also known as dark-field tomography. This paper provides a preliminary theoretical study of x-ray dark-field tomography (XDT) assuming the spectral x-ray detection technology. For XDT, the modified Leakeas-Larsen equation (MLLE) is an appropriate approximation of the radiative transfer equation (RTE) for a highly forward-peaked medium with small but sufficient amounts of large-angle scattering. Properties of the MLLE are studied, such as existence of a unique solution and positivity of the solution. MLLE and its discrete analogues can be solved naturally with an iteration procedure, and convergence of the iteration procedure is shown. XDT, as an inverse parameter problem with MLLE as the forward model, is then studied. Numerical discretization schemes of MLLE and the associated XDT are introduced. Simulation results are reported on several numerical examples for MLLE and for XDT. The paper concludes with some remarks on research topics for further study of XDT.