The Dose Estimation by Simulation of the ISS Radiation Environment (DESIRE) project aims to accurately calculate radiation fluxes and doses to astronauts inside the European Columbus module of the International Space Station using Geant4. Since Geant4 has not been previously used for this type of application it needs to be validated. This will be followed by a detailed evaluation of the incident radiation fields on ISS and culminate with the geometry modeling and full-scale flux and dose simulations for Columbus. Geant4 validation studies and comparisons to other tools are presented. These concern the angle and energy distributions of particles leaving irradiated targets and of energy depositions in the targets. Comparisons are made between simulations using different Geant4 physics models, experimental data, and other particle transport programs. Geant4 using the "Binary Cascade" model for inelastic nucleon reactions performs very well in these comparisons, but some issues with other models remain to be resolved if they are to be used for space radiation shielding applications. Results of Geant4 simulations of the transport of some relevant radiation field components through the hull of a simplified model of Columbus are also presented.