The standard bidomain model is based on the fact that electrically active cardiomyocytes are present uniformly everywhere in the heart. While this is a reasonable assumption for healthy hearts, it fails in some pathological cases where significant alterations in the tissue structure occur, for example fibrosis in infarct scars, or due to ageing. The effects of the non-excitable regions, that we call the inclusions, are often taken into account through ad-hoc rough tuning of the model. In this work, we introduce a mathematically rigorous way to derive the modified model from a microscopic description of the inclusions. More precisely we show that both shape and volume fraction of the inclusions affect the conductivity tensors in the bidomain model. We develop a numerical toolkit for determining the modified conductivities, based on shape and size of the diffusive inclusions. We perform 2D and 3D numerical studies of the effects of the inclusions on the conductivities by testing several basic shapes for the inclusions and varying their volume fraction. We observe the change in the anisotropy ratios for both intracellular and extracellular conductivity tensors. Additionally, in some cases we observe the change in the principal direction of the propagation. Finally, we apply our model to a rat heart model desgined from HiResMRI. Using image analysis softwares we asses the structural properties of the tissue, that we use then as parameters in the modified model. Simulating both the standard and modified model in the whole heart, we observe the differences in the the signal propagation.