This work investigates the attenuation of a proton beam to reconstruct the map of the linear attenuation coefficient of a material which is mainly caused by the inelastic interactions of protons with matter. Attenuation proton computed tomography (pCT) suffers from a poor spatial resolution due to multiple Coulomb scattering (MCS) of protons in matter, similarly to the conventional energy-loss pCT. We therefore adapted a recent filtered back-projection algorithm along the most likely path (MLP) of protons for energy-loss pCT (Rit et al 2013) to attenuation pCT assuming a pCT scanner that can track the position and the direction of protons before and after the scanned object. Monte Carlo simulations of pCT acquisitions of density and spatial resolution phantoms were performed to characterize the new algorithm using Geant4 (via Gate). Attenuation pCT assumes an energy-independent inelastic cross-section, and the impact of the energy dependence of the inelastic cross-section below 100 MeV showed a capping artifact when the residual energy was below 100 MeV behind the object. The statistical limitation has been determined analytically and it was found that the noise in attenuation pCT images is 411 times and 278 times higher than the noise in energy-loss pCT images for the same imaging dose at 200 MeV and 300 MeV, respectively. Comparison of the spatial resolution of attenuation pCT images with a conventional straight-line path binning showed that incorporating the MLP estimates during reconstruction improves the spatial resolution of attenuation pCT. Moreover, regardless of the significant noise in attenuation pCT images, the spatial resolution of attenuation pCT was better than that of conventional energy-loss pCT in some studied situations thanks to the interplay of MCS and attenuation known as the West-Sherwood effect.