Quantitative susceptibility mapping exploits magnetic susceptibility to probe biological tissue. The obtained quantitative information carries additional clinical value, e.g. for the diagnosis of neurodegenerative diseases. However, standard protocols have long acquisition times which impedes their use in clinical routine. Here, we propose a compressed sensing acquisition based on a Cartesian spiral-phyllotaxis readout scheme to drastically reduce the total acquisition times from 20 minutes to 2 minutes for a 1mm isotropic susceptibility map of the whole brain. A preliminary qualitative and quantitative validation is performed on healthy subjects.