The purpose of the presented investigations is to design, construct and establish the characteristic performance of the Jagiellonian Positron Emission Mammography (J-PEM), being designed for the detection and diagnosis of breast cancer. Its construction is based on a novel idea of PET tomography based on plastic scintillators and wavelength shifter (WLS), and a new concept of positronium imaging. We have prepared a simulation program based on Monte Carlo methods for optimizing the geometry and material of the J-PEM prototype. Here, we present the first results from the simulations and a brief review of the state of art of breast imaging modalities and their characteristics motivating our investigation.