Estimated brain age from magnetic resonance image (MRI) and its deviation from chronological age can provide early insights into potential neurodegenerative diseases, supporting early detection and implementation of prevention strategies. Diffusion MRI (dMRI), a widely used modality for brain age estimation, presents an opportunity to build an earlier biomarker for neurodegenerative disease prediction because it captures subtle microstructural changes that precede more perceptible macrostructural changes. However, the coexistence of macro- and micro-structural information in dMRI raises the question of whether current dMRI-based brain age estimation models are leveraging the intended microstructural information or if they inadvertently rely on the macrostructural information. To develop a microstructure-specific brain age, we propose a method for brain age identification from dMRI that minimizes the model's use of macrostructural information by non-rigidly registering all images to a standard template. Imaging data from 13,398 participants across 12 datasets were used for the training and evaluation. We compare our brain age models, trained with and without macrostructural information minimized, with an architecturally similar T1-weighted (T1w) MRI-based brain age model and two state-of-the-art T1w MRI-based brain age models that primarily use macrostructural information. We observe difference between our dMRI-based brain age and T1w MRI-based brain age across stages of neurodegeneration, with dMRI-based brain age being older than T1w MRI-based brain age in participants transitioning from cognitively normal (CN) to mild cognitive impairment (MCI), but younger in participants already diagnosed with Alzheimer's disease (AD). Approximately 4 years before MCI diagnosis, dMRI-based brain age yields better performance than T1w MRI-based brain ages in predicting transition from CN to MCI.