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A detailed characterization of the trapped-proton-induced radiation environment on board Columbus and the International Space Station (ISS) has been carried out using the Geant4 Monte Carlo particle transport toolkit. Dose and dose equivalent rates, as well as penetrating particle spectra are presented. These results are based on detailed Geant4 geometry models of Columbus and ISS, comprising a total of about 1000 geometry volumes. Simulated trapped-proton dose rates are found to be strongly dependent on ISS altitude. Dose rates for different locations inside the Columbus cabin are presented, as well as for different models of the incident trapped-proton flux. Dose rates resulting from incident anisotropic trapped protons are found to be lower than, or equal to, those of omnidirectional models. The anisotropy induced by the asymmetric shielding distribution of Columbus/ISS is also studied. The simulated trapped-proton dose (equivalent) rates, averaged over different locations inside Columbus, are 120 muGy/d (154 muSv/d) and 79 muGy/d (102 muSv/d) for solar minimum and maximum conditions according to AP8 incident proton spectra and an ISS orbit of 380 km. The solar maximum dose rates are found to be of the same order as measurements in other modules in the present ISS.