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The morphology and the distribution of material observed in SNRs reflect the interaction of the SN blast wave with the ambient environment, the physical processes associated with the SN explosion and the internal structure of the progenitor star. IC 443 is a MM SNR located in a quite complex environment: it interacts with a molecular cloud in the NW and SE areas and with an atomic cloud in the NE. In this work we aim at investigating the origin of the complex morphology and multi-thermal X-ray emission observed in SNR IC 443, through the study of the effect of the inhomogeneous ambient medium in shaping its observed structure, and the exploration of the main parameters characterizing the remnant. We developed a 3D HD model for IC 443, which describes the interaction of the SNR with the environment, parametrized in agreement with the results of the multi-wavelength data analysis. We performed an ample exploration of the parameter space describing the initial blast wave and the environment, and the surrounding clouds. From the simulations, we synthesized the X-ray emission maps and spectra and compared them with actual X-ray data collected by XMM-Newton. Our model explains the origin of the complex X-ray morphology of SNR IC 443 in a natural way, being able to reproduce, for the first time, most of the observed features, including the centrally-peaked X-ray morphology (characteristic of MM SNRs) when considering the origin of the explosion at the position where the PWN CXOU J061705.3+222127 was at the time of the explosion. In the model which best reproduces the observations, the mass of the ejecta and the energy of the explosion are $\sim 7 M_\odot$ and $\sim10^{51}$ erg, respectively. From the exploration of the parameter space, we found that the density of the clouds is $n>300$ cm$^{-3}$ and that the age of SNR IC 443 is $\sim8000$ yr.