Biophysical models are a promising means for interpreting diffusion weighted magnetic resonance imaging (DW-MRI) data, as they can provide estimates of physiologically relevant parameters of microstructure including cell size, volume fraction, or dispersion. However, their application in cardiac microstructure mapping (CMM) has been limited. This study proposes seven new two-compartment models with combination of restricted cylinder models and a diffusion tensor to represent intra- and extracellular spaces, respectively. Three extended versions of the cylinder model are studied here: cylinder with elliptical cross section (ECS), cylinder with Gamma distributed radii (GDR), and cylinder with Bingham distributed axes (BDA). The proposed models were applied to data in two fixed mouse hearts, acquired with multiple diffusion times, q-shells and diffusion encoding directions. The cylinderGDR-pancake model provided the best performance in terms of root mean squared error (RMSE) reducing it by 25% compared to diffusion tensor imaging (DTI). The cylinderBDA-pancake model represented anatomical findings closest as it also allows for modelling dispersion. High-resolution 3D synchrotron X-ray imaging (SRI) data from the same specimen was utilized to evaluate the biophysical models. A novel tensor-based registration method is proposed to align SRI structure tensors to the MR diffusion tensors. The consistency between SRI and DW-MRI parameters demonstrates the potential of compartment models in assessing physiologically relevant parameters.