A Multivariable Mathematical Model of Conductivity, β-Amyloid and T-Protein Dynamics in Alzheimer’s Disease Progression

Alzheimer’s disease (AD) affects over 55 million individuals worldwide, yet no transformative disease-modifying therapies exist. Mathematical modelling provides a powerful framework to elucidate complex disease mechanisms, predict therapeutic outcomes, and enable precision medicine—capabilities urgently needed where multiscale spatiotemporal processes defy experimental analysis alone. We developed a mechanistic spatiotemporal model coupling four AD hallmarks: β-amyloid (Aβ) accumulation, T-protein (T-p) aggregation, neuroinflammation and electrical conductivity decline. Formulated as non-linear partial differential equations (p.d.es) on a 3-dimensional biological interpretation of non-linear terms (the ellipsoidal brain domain with biologically grounded parameters), the model was solved using eigenfunction expansion, Fourier analysis and numerical methods. Therapeutic interventions were simulated through mechanistically motivated parameter modifications and validated against longitudinal biomarker data from major cohort studies. Simulations reveal Aβ-initiated spatiotemporal cascades originating in the hippocampus and spreading radially at 0.15–0.20 cm/year, with T-pathology emerging after 2–3 years. Conductivity decline accelerates upon T-onset (year 5–7), reflecting the transition to symptomatic disease. Multimodal intervention at early symptomatic stages reduces peak Aβ by 36% and inflammation by 52% and preserves 41% more conductivity than untreated controls. Sensitivity analysis identifies Aβ production and inflammatory regulation as critical therapeutic targets, with dose–response curves demonstrating linear efficacy relationships. This biologically grounded framework explicitly links molecular pathology to functional decline, enabling patient-specific trajectory prediction through parameter calibration. The model establishes a foundation for precision medicine applications including individualized prognosis, optimal treatment timing and virtual clinical trial design, advancing quantitative systems biology of neurodegeneration.