We present a method for phase retrieval in propagation-based x-ray imaging, based on the contrast transfer and transport of intensity equation approaches. We show that the contrast transfer model does not coincide with the transport of intensity in the limit of small propagation distances, and we derive a new model that alleviates this problem. Using this model, we devise an algorithm to retrieve the phase from slowly varying samples that is valid beyond the limit of small distances. We show its utility by imaging in three dimensions a biological sample that causes both strong absorption and phase shift.

Bone strength and failure are increasingly thought to be due to ultrastructural properties, such as the morphology of the lacuno-canalicular network, the collagen fiber orientation and the mineralization on the nanoscale. However, these properties have not been studied in 3D so far. Here we report the investigation of the human bone ultrastructure with X-ray phase nanotomography, which now provides the required sensitivity, spatial resolution and field of view. The 3D organization of the lacuno-canalicular network is studied in detail over several cells in osteonal and interstitial tissue. Nanoscale density variations are revealed and show that the cement line separating these tissues is hypermineralized. Finally, we show that the collagen fibers are organized as a twisted plywood structure in 3D.