X-ray tomography is an invaluable tool in biomedical imaging. It can deliver the three-dimensional internal structure of entire organisms as well as that of single cells, and even gives access to quantitative information, crucially important both for medical applications and for basic research1, 2, 3, 4. Most frequently such information is based on X-ray attenuation. Phase contrast is sometimes used for improved visibility but remains significantly harder to quantify5, 6. Here we describe an X-ray computed tomography technique that generates quantitative high-contrast three-dimensional electron density maps from phase contrast information without reverting to assumptions of a weak phase object or negligible absorption. This method uses a ptychographic coherent imaging approach to record tomographic data sets, exploiting both the high penetration power of hard X-rays and the high sensitivity of lensless imaging7, 8, 9. As an example, we present images of a bone sample in which structures on the 100 nm length scale such as the osteocyte lacunae and the interconnective canalicular network are clearly resolved. The recovered electron density map provides a contrast high enough to estimate nanoscale bone density variations of less than one per cent. We expect this high-resolution tomography technique to provide invaluable information for both the life and materials sciences.
Figures at a glance
Figure 1: Experimental set-up and sample. 
a, Schematic of experimental set-up. The X-ray beam (X) impinges on the pinhole (P), which creates a localized illumination on the sample (S). Diffraction patterns are recorded with a two-dimensional pixellated detector D. One out of the 704 diffraction patterns per projection is shown. b, Projection image of the bone specimen as seen on a scintillator screen imaged with a video microscope used for alignment purposes. The scan points indicated by the black dots cover a rectangular area of 40 µm × 32 µm and are placed on concentric circles starting from the centre. c, Scanning electron micrograph of the specimen. Scale bars, 10 µm.
Figure 2: Projection images reconstructed from ptychographic data. 
a, Reconstructed amplitude of the complex object transmission function normalized with respect to air. Artefacts are visible that are attributed to fluctuations in the X-ray intensity. b, Phase part of the complex transmission function. c, Phase after linear-ramp correction and unwrapping. d, Profile along red line in panel a, revealing a low signal-to-noise ratio. e, Profile along red line in panel b, illustrating the wrapping of the phase into the range (−π, π]. f, Profile along red line in panel c. Scale bars, 5 µm.
Figure 3: 3D rendering of the tomographic reconstruction. 
a,Volume rendering with the bone matrix in translucent colours to show osteocyte lacunae (L) and the connecting canaliculi (C). b, Isosurface rendering of the lacuno-canalicular network obtained by segmenting the corresponding peak in the density on histogram shown in Fig. 4c. Morphological analysis of tomographic reconstructions is most often based on this type of segmentation, which is independent of the absolute scale of the density. Long edges of 3D scale bars, 5 µm.
Figure 4: Result of tomographic phase reconstruction. 
a, Cut parallel to the rotation axis through the reconstructed volume. The phase values have been converted to quantitative electron density ne (linear greyscale). The labelled structures are air (A), bone matrix (B), canaliculi (C), gallium coating (G), which is a result of focused ion beam preparation, and osteocyte lacuna (L). b, Cut perpendicular to the rotation axis. The two large dark areas are osteocyte lacunae, and small dark dots are sections through individual canaliculi. The slight variations in the shades of grey in a and b indicate inhomogeneity in the bone density at the submicrometre scale. c, Histogram of electron density values in the reconstructed volume (500 equally sized bins for ne values between −0.2 and 1.3 Å−3). The labels correspond to the ones in panel a and indicate the grey values that can be associated with the aforementioned features. d, Comparison of the bone peak (label B) of the histogram for two cubic sub-volumes of 1 µm3, indicated by the red and blue boxes in a and b. At the micrometre scale, the detection threshold of density fluctuations is slightly less than 0.001 Å−3 or about 0.2% of the mean bone density. Scale bars in a and b, 5 µm.
