Radioembolization with 90Y-microspheres is used as a treatment for non-resectable liver cancer. 90Y is mainly a β- emitter but a few β+ particles are also emitted. It enables to quantify the amount of radioactivity in the body using PET imaging, which could be especially useful for dosimetry purpose. Yet, the reconstructed images are very noisy due to the limited amount of collected data with 90Y, and the usual reconstruction algorithms have positive bias in regions with low activity. In this context, we propose to combine two complementary approaches recently published, both using the Alternating Direction Method of Multipliers (ADMM) algorithm, within a nested ADMM. The first one allows for negative values in the image by enforcing the non-negativity in the projection space only, hence reducing the bias. The second one intends to lower the noise in the image by adding the constraint that the reconstructed image is the output of a Deep Image Prior (DIP) network.