Shannon Entropy of Gray Matter Eigenmodes: A Novel Biomarker for Alzheimer's Disease and Heterogeneous MCI Trajectories

Background: Current Alzheimer's disease (AD) diagnostics rely on late–stage cognitive assessments or invasive biomarkers. Neuroimaging offers non–invasive alternatives, but single–modality approaches (structural atrophy or functional connectivity) face limitations in sensitivity and specificity for early detection. We introduce entropy and temperature, novel structure–function coupling (SFC) biomarkers based on gray matter eigenmodes, to quantify cortical disorganization in early AD. Methods: We analyzed multimodal MRI (structural + resting–state fMRI) and amyloid-PET data from two independent cohorts (BABRI: N=135; ADNI: N=275), including cognitively normal (CN), mild cognitive impairment (MCI), and AD participants. We computed Shannon entropy by projecting fMRI time series onto individual structural eigenmodes, and temperature based on eigenmode–based reconstruction of functional connectivity. These indexes were evaluated for diagnostic classification (SVM), prediction of amyloid-β (Aβ) burden (regression), and stratification of MCI subtypes (reversed/stable/progressed). Results: Entropy was significantly elevated in AD compared to CN and MCI participants across both cohorts (Δ=8-21%, p<0.001), with left–hemisphere entropy showing superior diagnostic accuracy (AUC=0.901 for CN vs. MCI; AUC=0.873 for CN vs. AD). Right–hemisphere and global entropy robustly predicted Aβ deposition (error reduction: 38.7–42.1% vs. baseline, p<0.01). Entropy stratified MCI into distinct subtypes: progressors (MCI→AD) showed higher entropy and Aβ than stable/reverted MCI (p<0.001) and exhibited a biphasic entropy trajectory during conversion. Temperature indices did not show significant differences across diagnostic groups. Discussion: Entropy derived from gray matter eigenmodes emerges as a sensitive and dual-purpose biomarker for AD diagnosis and pathological prediction. Its hemispheric asymmetry (left: optimal classification; right: Aβ prediction) and ability to detect nonlinear MCI progression offer mechanistic insights for early intervention.