The Puri-Li kinetic model is modified to include neuronal calcium ion homeostasis to study the effect of calcium ions on the production of amyloid-β peptides (Aβ), microglia, and astroglia during the pathogenesis of Alzheimer's disease (AD). This is carried out by solving the modified Puri-Li model under steady-state conditions. The derived expressions show that the inclusion of calcium ions has altered the steady-state populations of Aβ, microglia, and astroglia. The calcium ions activate the synthesis of Aβ which in turn increases the calcium ions entering the cytoplasm of the neuronal cells, thus creating a positive loop. The study also shows that as AD progresses, the inclusion of calcium ions enhances the production of microglia and astroglia. Examination of the steady-state solutions of microglia and astroglia shows that equilibrium conditions are achieved by microglia and astroglia destroying neurons. These model results are in agreement with experimental findings, which show a feed back loop between calcium ion levels and Aβ; population increase in microglia, astroglia during AD; and microglia, astroglia acting as inflammatory cells producing toxins to destroy neurons during AD. Increased production of Aβ, microglia, and astroglia resulting from increased levels of calcium ions suggests that controlling the calcium ion levels could present a therapeutic strategy to combat AD.
Keywords: Alzheimers disease; Astroglia; Calcium ions; Mathematical model; Microglia.