  Supernova remnants (SNRs) are considered as being the sources of galactic
cosmic rays. In order to understand the origin, acceleration, and composition
of these cosmic rays, detailed knowledge of the physical conditions in the
local interstellar medium is needed. The shock interaction of SNRs with
molecular clouds that gives rise to strong molecular emission in the far-IR and
sub-mm wavelength regimes can be used as a highly valuable tracer of these
conditions. The application of MHD shock models in the interpretation of the
resulting line emission can yield information on the energetic and chemical
impact of supernova remnants. We have mapped two regions in the supernova
remnant W44 with the APEX telescope in ${}^{12}$CO (3-2), (4-3), (6-5), (7-6)
and ${}^{13}$CO (3-2). The extraction of integrated intensities on five
different positions, corresponding to local maxima of CO emission, allows to
compare these intensities to the outputs of a grid of models, which combine an
MHD shock code with a radiative transfer module based on the 'large velocity
gradient' approximation. We find that the observed CO line emission is
compatible with non-stationary shocks and a pre-shock density of $10^4$
cm${}^{-3}$. Our models furthermore allow to constrain shock ages, velocities,
the pre-shock magnetic field strength components perpendicular to the
line-of-sight, and the full ladder of CO transitions. Finally, our analysis can
be used to estimate the contribution of such SNRs to, e.g. the galactic energy
balance and the momentum-injection into the surrounding interstellar medium.
