  {\it Spitzer} images of Tycho's supernova remnant in the mid-infrared reveal
limb-brightened emission from the entire periphery of the shell and faint
filamentary structures in the interior. As with other young remnants, this
emission is produced by dust grains, warmed to $\sim 100$ K in the post-shock
environment by collisions with energetic electrons and ions. The ratio of the
70 to 24 $\mu$m fluxes is a diagnostic of the dust temperature, which in turn
is a sensitive function of the plasma density. We find significant variations
in the 70/24 flux ratio around the periphery of Tycho's forward shock, implying
order-of-magnitude variations in density. While some of these are likely
localized interactions with dense clumps of the interstellar medium, we find an
overall gradient in the ambient density surrounding Tycho, with densities 3-10
times higher in the NE than in the SW. This large density gradient is
qualitatively consistent with the variations in the proper motion of the shock
observed in radio and X-ray studies. Overall, the mean ISM density around Tycho
is quite low ($\sim 0.1-0.2$ cm$^{-3}$), consistent with the lack of thermal
X-ray emission observed at the forward shock. We perform two-dimensional
hydrodynamic simulations of a Type Ia SN expanding into a density gradient in
the ISM, and find that the overall round shape of the remnant is still easily
acheivable, even for explosions into significant gradients. However, this leads
to an offset of the center of the explosion from the geometric center of the
remnant of up to 20%, although lower values of 10% are preferred. The best
match with hydrodynamical simulations is achieved if Tycho is located at a
large (3-4 kpc) distance in a medium with a mean preshock density of $\sim 0.2$
cm$^{-3}$. Such preshock densities are obtained for highly ($\ga 50$%) porous
ISM grains.
