A Mathematical Model for Amyloid-𝜷 Aggregation in the Presence of Metal Ions: A Timescale Analysis for the Progress of Alzheimer Disease

How Can We Use Mathematical Modeling of Amyloid-β in Alzheimer's Disease Research and Clinical Practices?

The accumulation of amyloid-β (Aβ) plaques in the brain is considered a hallmark of Alzheimer's disease (AD). Mathematical modeling, capable of predicting the motion and accumulation of Aβ, has obtained increasing interest as a potential alternative to aid the diagnosis of AD and predict disease prognosis. These mathematical models have provided insights into the pathogenesis and progression of AD that are difficult to obtain through experimental studies alone. Mathematical modeling can also simulate the effects of therapeutics on brain Aβ levels, thereby holding potential for drug efficacy simulation and the optimization of personalized treatment approaches. In this review, we provide an overview of the mathematical models that have been used to simulate brain levels of Aβ (oligomers, protofibrils, and/or plaques). We classify the models into five categories: the general ordinary differential equation models, the general partial differential equation models, the network models, the linear optimal ordinary differential equation models, and the modified partial differential equation models (i.e., Smoluchowski equation models). The assumptions, advantages and limitations of these models are discussed. Given the popularity of using the Smoluchowski equation models to simulate brain levels of Aβ, our review summarizes the history and major advancements in these models (e.g., their application to predict the onset of AD and their combined use with network models). This review is intended to bring mathematical modeling to the attention of more scientists and clinical researchers working on AD to promote cross-disciplinary research.

A mathematical model for the evaluation of iron transport across the blood-cerebrospinal fluid barrier in neurodegenerative diseases

Iron plays important roles in healthy brain but altered homeostasis and concentration have been correlated to aging and neurodegenerative diseases. Iron enters the central nervous system by crossing the brain barrier systems: the Blood- Brain Barrier separating blood and brain and the Blood-Cerebrospinal Fluid Barrier (BCSFB) between blood and CSF, which is in contact with the brain by far less selective barriers. Herein, we develop a two-compartmental model for the BCSFB, based on first-order ordinary differential equations, performing numerical simulations and sensitivity analysis. Furthermore, as input parameters of the model, experimental data from patients affected by Alzheimer's disease, frontotemporal dementia, mild cognitive impairment and matched neurological controls were used, with the aim of investigating the differences between physiological and pathological conditions in the regulation of iron passage between blood and CSF which can be possibly targeted by therapy.