In recent years, interstellar dust has been detected by spacecraft within a few AU from the Sun. The first analysis of observed data revealed that certain properties such as size distribution and average mass of interstellar dust grains found close to the Sun differ from those of classical interstellar grains. In particular, detected grains seem to be heavier at small distances from the Sun than in the interstellar space. It was concluded that the characteristics of interstellar dust grains are modified by some process as they approach the vicinity of the Sun. In this paper for the first time we simulate the dynamics of dust grains in the interstellar medium and the heliosphere and demonstrate that they experience a filtration process in the region of the heliopause. The main result of this process is the exclusion of small submicron-sized grains from the incoming stream. We estimate the cutoff range for the process to lie between 0.1 and 0.2 &mgr;m, which corresponds to 10-16-10-17 kg dust grains. Although this result is consistent with satellite observations, we show that it alone does not explain interstellar grain data that were collected in situ in the interplanetary space. ¿ 2000 American Geophysical Union