The amyloid cascade hypothesis and Alzheimer’s disease: A mathematical model

Abstract

The paper presents a conceptual mathematical model for Alzheimer’s disease (AD). According to the so-called amyloid cascade hypothesis, we assume that the progression of AD is associated with the presence of soluble toxic oligomers of beta-amyloid. Monomers of this protein are produced normally throughout life, but a change in the metabolism may increase their total production and, through aggregation, ultimately results in a large quantity of highly toxic polymers. The evolution from monomeric amyloid produced by the neurons to senile plaques (long and insoluble polymeric amyloid chains) is modelled by a system of ordinary differential equations (ODEs), in the spirit of the Smoluchowski equation. The basic assumptions of the model are that, at the scale of suitably small representative elementary volumes (REVs) of the brain, the production of monomers depends on the average degradation of the neurons and in turn, at a much slower timescale, the degradation is caused by the number of toxic oligomers. To mimic prion-like diffusion of the disease in the brain, we introduce an interaction among adjacent REVs that can be assumed to be isotropic or to follow given preferential patterns. We display the results of numerical simulations which are obtained under some simplifying assumptions. For instance, the amyloid cascade is modelled by just three ordinary differential equations (ODEs), and the simulations refer to abstract 2D domains, simplifications which can be easily avoided at the price of some additional computational costs. Since the model is suitably flexible to incorporate other mechanisms and geometries, we believe that it can be generalised to describe more realistic situations.

A Neuron-Glial Model of Exosomal Release in the Onset and Progression of Alzheimer's Disease

“…The diffusion of IP 3 concentration in the IP c cell can stimulate Ca 2+ release from the endoplasmic reticulum in response to the activation of the firing of action potentials by external stimuli. Increased concentrations of P c pathways may then encourage the transition to a low-threshold activated L-type Ca 2+ current, causing additional disturbance of intracellular Ca 2+ homeostasis which is a key phenomenon for AD (Bertsch et al, 2020 ).…”

A mathematical model of the Alzheimer’s Disease biomarker cascade demonstrates statistical pitfalls in identifying neurobiological surrogates of cognitive reserve

“…differential equation models, to model AD. However, some of these studies focused on specific aspects of the disease process whereas most arguably used conceptually as well as mathematically more elaborate models [4, 12, 19–21]. In contrast, for the present study, we aimed to demonstrate a statistical artifact through a mathematical model that was as simple as possible in order to be able to argue that it may be present in more complex conceptions of AD as well.…”

A sensitivity analysis of a mathematical model for the synergistic interplay of Amyloid beta and tau on the dynamics of Alzheimer's disease

“…Although this may seem an over-simplification, in fact it is not because, as far as the results of the model are concerned, considering a more detailed description of the soluble oligomers with their precise lengths, such as e.g. in [2], has proved not to add significant information [3,6]. The coefficient ε in front of the time derivatives is due to the relationship (1), in particular to the fact that, on the longer time scale t, the rate at which agglomeration and diffusion of Aβ take place is as high as 1…”