(Left panel) Schematic of the generic absorption-contrast system that was used as gold standard in the study and with geometry loosely based on the Philips MicroDose Mammography system. It is equipped with photon-counting silicon strip detectors that are scanned across the object to acquire an image. (Right panel) Schematic of the Talbot interferometer with equal geometry as the generic absorption-contrast system. A beam splitter (phase grating) illuminated by an x-ray source induces interference fringes in the x-direction. The fringes are displaced by phase gradients in an object, which can be located either before or after the beam splitter. A fine-pitch analyzer grating can be used to demodulate the high-frequency fringes into lower frequencies so that the fringe displacement and hence the object phase gradient can be measured by the coarser detector elements (phase stepping). To cover the full field-of-view, the strip detectors are scanned in the y-direction. The figure is not to scale.
Thickness-normalized phase- and absorption-contrast signals (k × δ and μ) as a function of energy and for average breast tissue. The absorption-contrast signal is additionally divided into photoelectric and Compton-scattering components, i.e., μ = μ PE + μ C .
Target shapes: (Top) A sphere. (Bottom) The target that was used for wave-propagation simulations [Eq. (46)] with a constant derivative in the phase-contrast direction.
(Left) 2D plot of the noise-power spectrum (NPS) in the Talbot interferometer and in the generic absorption-contrast system. The phase-contrast NPS has a dependence, but is flat in the y direction. The absorption-contrast NPS is flat in both systems and both directions, but slightly higher in Talbot interferometry because of grating absorption. (Center) Axial plot of the NPS in the x direction. Wave-propagation simulations (markers) are compared to analytical results (lines). (Right) The axial modulation-transfer function (MTF), which is expected to be equal in all cases.
Detectability index as a function of photon energy for four different 200-μm-diameter spherical targets: (1) a tumor structure, (2) a glandular structure, (3) a microcalcification (MC), and (4) an air cavity. The optimal energies for phase and absorption contrast are indicated with circles and squares, respectively.
The impact of target size and material on phase and absorption contrast for (1) a tumor structure, (2) a glandular structure, (3) a microcalcification (MC), and (4) an air cavity. The detectability benefit ratio at optimal energy and equal dose for phase over absorption contrast () is plotted as a function of target diameter.
The squared signal template (F 2) for a 0.4-mm spherical tumor structure, and the squared signal-difference-to-noise ratio (SDNR2) for phase and absorption contrast. The plot is in radial coordinates and the signal template was multiplied with 2πf ϱ so that an integral over the product of these plots illustrates calculation of the detectability index in Eq. (21).
The detectability benefit ratio at optimal energy and equal dose for phase over absorption contrast () as a function of target diameter. The impact of “sharpness” on detection tasks is illustrated by two kinds of target shapes: (1) A sphere represents smooth targets and (2) a pillbox represents sharp or spiculated targets. The difference between detection and discrimination is illustrated by two different tasks: (1) Discrimination between two spheres with a factor of 2 difference in size, and (2) discrimination between a pillbox and a sphere.
The squared signal templates (F 2) for the detection and discrimination tasks in Fig. 8. As in Fig. 7, the signal templates were multiplied with 2πf ϱ to convert from Cartesian to radial coordinates. The plots are normalized for convenient comparison.
Simulated images that illustrate the impact of target size on phase and absorption contrast. (Left panel) Tumor structures/spicula with diameters 100–500 μm. (Right panel) Solid tumors with diameters 1–5 mm. Images with Talbot phase contrast and generic absorption contrast are in the top and bottom rows of the respective panels. The targets are zoomed to equal display size within each panel.
Simulated images that illustrate the impact of target material on phase and absorption contrast. (Left column) A 300-μm-diameter air cavity; (Right column) A 300-μm-diameter microcalcification. Phase and absorption-contrast images are in the top and bottom rows of the respective columns.
(Left) Simulated phase-step modulation () as a function of the spectrum energy resolution () and normalized projected source size (equivalent to source-grating fill factor, Γ0). (Center) Modulation along the axes of the surface plot (markers) compared to analytical results (lines) for perfect energy resolution, a point source, and a 4-μm source. (Right) Relative photon economy in phase and absorption contrast () as a function of modulation with dotted and dashed lines for two different spectrum widths (“wide” and “narrow”).
The detectability benefit ratio at equal dose and equal photon economy () as a function of target size. The targets and tasks were picked from Fig. 8.
Glossary of variables and symbols.
Parameters used for the plane- and spherical-wave geometries.
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