Synchrotron-based tomographic imaging often suffers from artifacts that are caused by faulty assumptions and irregularities in the imaging system. For example, most reconstruction models assume that the source intensity and the detector response are known and static, but in practice, both are unknown, may vary over time, and must be estimated from measurements. To mitigate this uncertainty, we propose an extended reconstruction model in which the combination of the source intensity and detector response is modeled as a low-rank matrix that must be estimated along with the image. We demonstrate the advantages of the proposed model in challenging experimental setups through some numerical examples.