  Using a three stage model of hadron formation we calculate the change of the
transverse momentum distribution of hadrons produced in semi-inclusive deep
inelastic scattering (SIDIS) on nuclei. In the first stage after its
interaction with the virtual photon, the struck quark propagates quasi free in
the nuclear environment undergoing multiple collisions with nucleons. During
this stage it can acquire transverse momentum. In the second stage a prehadron
is formed which has a very small elastic cross section with the nucleons. In
the third stage the prehadron turns into a hadron. For HERMES energies,
prehadron elastic scatterings contribute little to $p_\perp$-broadening. The
acquired extra $\Delta p_\perp^2$ of hadrons can therefore be deduced entirely
from the first stage of quasi free quark propagation with quark-nucleon
collisions. We use this model to describe $\pi$-production on Ne, Kr, Xe and
compare with the most recent HERMES preliminary data.
