Most studies using diffusion-weighted MRI (DW-MRI) in Alzheimer's disease (AD) have focused their analyses on white matter (WM) microstructural changes using the diffusion (kurtosis) tensor model. Although recent works have addressed some limitations of the tensor model, such as the representation of crossing fibers and partial volume effects with cerebrospinal fluid (CSF), the focus remains in modeling and analyzing the WM. In this work, we present a brain analysis approach for DW-MRI that disentangles multiple tissue compartments as well as micro- and macroscopic effects to investigate differences between groups of subjects in the AD continuum and controls. By means of the multi-tissue constrained spherical deconvolution of multi-shell DW-MRI, underlying brain tissue is modeled with a WM fiber orientation distribution function along with the contributions of gray matter (GM) and CSF to the diffusion signal. From this multi-tissue model, a set of measures capturing tissue diffusivity properties and morphology are extracted. Group differences were interrogated following fixel-, voxel-, and tensor-based morphometry approaches while including strong FWE control across multiple comparisons. Abnormalities related to AD stages were detected in WM tracts including the splenium, cingulum, longitudinal fasciculi, and corticospinal tract. Changes in tissue composition were identified, particularly in the medial temporal lobe and superior longitudinal fasciculus. This analysis framework constitutes a comprehensive approach allowing simultaneous macro and microscopic assessment of WM, GM, and CSF, from a single DW-MRI dataset.