Quantitative evaluation of the superposition principle at specific locations of the combined samples.

<p>The positions from which the curves were taken correspond to locations on the wooden skewers, where either one of them were penetrated by the x-ray beam or both after each other. They are marked in <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0061268#pone-0061268-g003" target="_blank">fig. 3</a> by the corresponding symbols. For the parallel oriented samples the single layer data were taken from separate scans of the single samples. To test the validity of the model derived from the simulation results fit curves were calculated for the single samples shown in gray. The predicted superposition signals resulting from the fit parameters and the model are shown in orange color. The direction-dependent curves for the superposition of both samples coincide with the predictions drawn from the model and both single curves for every relative orientation between the two wooden skewers.</p

Experimental realization of the simulated scenario of differently oriented layers on top of each other.

<p>Two wooden skewers were put on top of each other and examined with respect to different relative orientations. Because the fibers of the wood form highly oriented substructures parallel to one skewer's central axis each skewer strongly scatters in a well-defined direction perpendicular to its central axis. The color scale has been adapted to increase the contrast for the transmission images. Compared to the rather weak absorption of the material the dark-field signal clearly predominates for this kind of sample. Shown in the middle row, the mean value of the dark-field signal with respect to the rotation simply is the product of the mean values of each skewer, as it is for the transmission depicted above. In the bottom row the anisotropy images encode the direction of the structures in the color and the degree of anisotropy in the brightness. For completely parallel orientation of the two skewers () the superposition shows the same orientation. For an angle of between them, the resulting signal depends on the amount of material penetrated by the beam. Wherever both thicknesses are equal the measured orientation is exactly the circular mean of both individual orientations. Otherwise intermediate orientations appear wherever the penetrated thickness of one of both skewers is larger than for the other. For an angle of and equal thicknesses the anisotropy signal completely cancels out. Length of the scalebar: .</p

Directional dark-field simulation results for three different relative orientations of two strongly oriented sample layers.

<p>The upper row and left column both show the dark-field signal with respect to the orientation of two sample layers containing 250 cylinders each. The cylinders were randomly distributed over a plane perpendicular to the x-ray beam. We examined different relative orientations between the layers . For each relative orientation the plots show the dark-field signals of the separate layers and both of them together. Clearly visible is the dependence for each single layer and the superposition of both of them. For the single layer results we calculated the corresponding fit curves shown as solid lines. The product of both fit functions is shown as well as a solid line. It perfectly agrees with the simulation results for both layers together, although in the upper left plot the variance of the sum signal with respect to the actual microstate of the cylinder ensemble is quite large. In the lower right plot the first sample layer contained only 125 cylinders to simulate a layer of half the thickness. The shape of the signals does not change. From these results we derived that the superposition signal is simply the product (logarithmic scale) of the two single layers. As a simple consequence for a relative angle of the oscillation almost completely vanishes and the whole sample appears nearly perfectly isotropic.</p

A clinically relevant example for the application of the derived model.

<p>Here the two sample layers consist of human femoral bone cut in cubes. Both cubes were each measured separately with the beam penetrating cube 1 in the anterior-posterior and cube 2 in the medial-lateral direction, and then once again when both cubes were placed on top of each other but oriented in the same way. The upper image row shows the resulting transmission images and the lower row the measured orientation and anisotropy. As before, the anisotropy images encode the direction of the structures in the color and the degree of anisotropy in the brightness. Depending on the local amount of anisotropy in both separate samples the local direction of the combined sample is either dominated by one of the components (regions marked with circle and rectangle) or the anisotropy is strongly reduced because the average structure orientations are perpendicular to each other between both cubes.</p