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 ~100 K in the post-shock environment by collisions with energetic electrons and ions. The ratio of the 70 to 24 mum fluxes is a diagnostic of the dust temperature, which in turn is a sensitive function of the plasma density. We perform two-dimensional hydrodynamic simulations of a Type Ia supernova expanding into a density gradient in the ISM, and find that the overall round shape of the remnant is still easily achievable, even for explosions into significant gradients. The best match with hydrodynamical simulations is achieved if Tycho is located at a large distance in a medium with a mean preshock density of ~0.2 cmsup -3. Such preshock densities are obtained for highly (50%) porous ISM grains.