  A potential application for spectral computed tomography (CT) with
multi-energy-window photon-counting detectors is quantitative medical imaging
with K-edge contrast agents. Image reconstruction for spectral CT with such
contrast agents necessitates expression of the X-ray linear attenuation map in
at least three expansion functions, for example, bone/water/K-edge-material or
photo-electric- process/Compton-process/K-edge-material. The use of three
expansion functions can result in slow convergence for iterative image
reconstruction (IIR) algorithms applied to spectral CT. We propose a
block-diagonal step-preconditioner for use with a primal-dual iterative image
reconstruction framework that we have been developing for spectral CT. We
demonstrate the advantage of the new step-preconditioner on a sensitive
spectral CT simulation where the test object has low concentration of
Gadolinium (Gd) contrast agent and the X-ray attenuation map is represented by
three materials - PMMA, a soft-tissue equivalent, Aluminum, a bone equivalent,
and Gd.
