Ideal‐observer detectability in photon‐counting differential phase‐contrast imaging using a linear‐systems approach
Authors: Fredenberg Erik , Danielsson Mats , Stayman J. Webster , Siewerdsen Jeffrey H. , Åslund Magnus 
Published: August 2012

Purpose:To provide a cascaded‐systems framework based on the noise‐power spectrum (NPS), modulation transfer function (MTF), and noise‐equivalent number of quanta (NEQ) for quantitative evaluation of differential phase‐contrast imaging (Talbot interferometry) in relation to conventional absorption contrast under equal‐dose, equal‐geometry, and, to some extent, equal‐photon‐economy constraints. The focus is a geometry for photon‐counting mammography.Methods:Phase‐contrast imaging is a promising technology that may emerge as an alternative or adjunct to conventional absorption contrast. In particular, phase contrast may increase the signal‐difference‐to‐noise ratio compared to absorption contrast because the difference in phase shift between soft‐tissue structures is often substantially larger than the absorption difference. We have developed a comprehensive cascaded‐systems framework to investigate Talbot interferometry, which is a technique for differential phase‐contrast imaging. Analytical expressions for the MTF and NPS were derived to calculate the NEQ and a task‐specific ideal‐observer detectability index under assumptions of linearity and shift invariance. Talbot interferometry was compared to absorption contrast at equal dose, and using either a plane wave or a spherical wave in a conceivable mammography geometry. The impact of source size and spectrum bandwidth was included in the framework, and the trade‐off with photon economy was investigated in some detail. Wave‐propagation simulations were used to verify the analytical expressions and to generate example images.Results:Talbot interferometry inherently detects the differential of the phase, which led to a maximum in NEQ at high spatial frequencies, whereas the absorption‐contrast NEQ decreased monotonically with frequency. Further, phase contrast detects differences in density rather than atomic number, and the optimal imaging energy was found to be a factor of 1.7 higher than for absorption contrast. Talbot interferometry with a plane wave increased detectability for 0.1‐mm tumor and glandular structures by a factor of 3–4 at equal dose, whereas absorption contrast was the preferred method for structures larger than ∼0.5 mm. Microcalcifications are small, but differ from soft tissue in atomic number more than density, which is favored by absorption contrast, and Talbot interferometry was barely beneficial at all within the resolution limit of the system. Further, Talbot interferometry favored detection of “sharp” as opposed to “smooth” structures, and discrimination tasks by about 50% compared to detection tasks. The technique was relatively insensitive to spectrum bandwidth, whereas the projected source size was more important. If equal photon economy was added as a restriction, phase‐contrast efficiency was reduced so that the benefit for detection tasks almost vanished compared to absorption contrast, but discrimination tasks were still improved close to a factor of 2 at the resolution limit.Conclusions:Cascaded‐systems analysis enables comprehensive and intuitive evaluation of phase‐contrast efficiency in relation to absorption contrast under requirements of equal dose, equal geometry, and equal photon economy. The benefit of Talbot interferometry was highly dependent on task, in particular detection versus discrimination tasks, and target size, shape, and material. Requiring equal photon economy weakened the benefit of Talbot interferometry in mammography.