Proton computed tomography (pCT) is an evolving tomographic imaging modality with applications in proton and ion therapy. The method allows direct reconstruction of relative stopping power of patient tissues in a 3D-fashion. The pCT collaboration has built first experimental prototypes of pCT scanning systems  and has developed approaches to reconstruct proton CT images based on registering the coordinates and water equivalent path length (WEPL) of individual protons traversing the scanned volume. From these data one reconstructs the object boundary (hull) and initial image based on filtered back projection (FBP), calculates a most likely path (MLP) for each proton, and improves the initial image iteratively by solving a large linear system of equations of the form Ax = b using an iterative projection algorithm .
Schultze, B., Karbasi, P., Giacometti, V., Plautz, T., Schubert, K. E. & Schulte, R. W. (2015). Reconstructing highly accurate relative stopping powers in proton computed tomography. 2015 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS/MIC) (pp. 1-3). United States: IEEE.