We study the kinetics of prion fibril growth, described by the nucleated polymerization model analytically and by means of numerical experiments. The elementary processes of prion fibril formation lead us to a set of differential equations for the number of fibrils, their total mass, and the number of prion monomers. In difference to previous studies we analyze this set by explicitly taking into account the time-dependence of the prion monomer concentration. The theoretical results agree with experimental data, whereas the generally accepted hypothesis of constant monomer concentration leads to a fibril growth behavior which is not in agreement with experiments. The obtained size distribution of the prion fibril aggregates is shifted significantly toward shorter lengths as compared to earlier results, which leads to a enhanced infectivity of the prion material. Finally, we study the effect of filtering of the inoculated material on the incubation time of the disease.