In the framework of the planned upgrade of the IceCube Neutrino Observatory and a new generation neutrino telescope at the South Pole, new optical modules are being developed that are expected to significantly increase the sensitivity of the detector. One of these is the multi-PMT optical module (mDOM) with 24 3"-PMTs. As the deep ice at the South Pole has no intrinsic optical activity, the light produced by the modules themselves represents the dominant background. Two main sources of the dark rate are Cherenkov and scintillation photons produced by radioactive decays inside the module pressure vessel. This work presents detailed investigations of the underlying production mechanisms and their parameterisation based on measurements and simulations. With this parameterisation, the background response of the radioactive decays is estimated using a Geant4 simulation of the multi-PMT digital optical module.