The effects of high concentration of point defects in crystalline ZrSiO4 as originated by exposure to radiation, have been simulated using first principles density functional calculations. Structural relaxation and vibrational studies were performed for a catalogue of intrinsic point defects, with different charge states and concentrations. The experimental evidence of a large anisotropic volume swelling in natural and artificially irradiated samples is used to select the subset of defects that give similar lattice swelling for the concentrations studied, namely interstitials of O and Si, and the anti-site Zr(Si), Calculated vibrational spectra for the interstitials show additional evidence for the presence of high concentrations of some of these defects in irradiated zircon.