One explanation for its very x-ray bright center is that the shock front expanded past small ISM clouds. They are now evaporating within the remnant's interior. Not only do they lower the temperature of the interior, but they supply much of the mass that is necessary for copious radiation. This model has been twice applied to W44. Another bit of physics to consider is thermal conduction within the remnant interior.
Thermal conduction removes heat from the center.
I will show some hydrodynamical simulation results from the application of the thermal conduction idea to W44.