In early stages of various pulmonary diseases, such as emphysema and fibrosis, the change in X-ray attenuation is not detectable with absorption-based radiography. To monitor the morphological changes that the alveoli network undergoes in the progression of these diseases, we propose using the dark-field signal, which is related to small-angle scattering in the sample. Combined with the absorption-based image, the dark-field signal enables better discrimination between healthy and emphysematous lung tissue in a mouse model. All measurements have been performed at 36 keV using a monochromatic laser-driven miniature synchrotron X-ray source (Compact Light Source). In this paper we present grating-based dark-field images of emphysematous vs. healthy lung tissue, where the strong dependence of the dark-field signal on mean alveolar size leads to improved diagnosis of emphysema in lung radiographs.