(Abridged) We have investigated two evolutionary scenarios advanced to explain the centrally-brightened X-ray morphology of the supernova remnant (SNR) W44: (1) a model involving the slow thermal evaporation of clouds engulfed by a supernova blast wave as it propagates though a clumpy interstellar medium (ISM), and (2) a hydrodynamical simulation of a blast wave propagating through a homogeneous ISM, including the effects of radiative cooling. Both models can have their respective parameters tuned to approximate the remnants morphology. The mean temperature of the hot plasma in W44 (~0.9 keV) as determined by our nonequilibrium ionization X-ray spectral analysis provides the essential key to discriminate between these scenarios. Based on the size (using the well established distance of 3 kpc) and temperature of W44, the dynamical evolution predicted by the cloud evaporation model gives an age for the SNR of merely 6500 yr. We argue that, because this age is inconsistent with the characteristic age (approx. 20000 yr) of the associated PSR 1853+01, this model cannot provide the explanation for the center-filled morphology. We favor the radiative-phase shock model since it can reproduce both the morphology and age of W44 assuming reasonable values for the initial explosion energy in the range 0.7E51 to 0.9E51 ergs and the ambient ISM density of between 3 and 4 cm**-3.
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