A qualitative model for aggregation and diffusion of $$\beta $$ -amyloid in Alzheimer’s disease

Achdou, Yves; Franchi, Bruno; Marcello, Norina; Tesi, Maria Carla

doi:10.1007/s00285-012-0591-0

Cited by 41 publications

(71 citation statements)

References 31 publications

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“…To the best of our knowledge, no model such as the one proposed here, has yet been advanced. There exist a variety of models specifically designed for Alzheimer's disease and their treatment, such as in [1,3,4,10,11,15,18,19,24,31,36]. Nevertheless, the prion protein has never been taken into account in the way we formulate here, and our model could helpful in designing new experiments and treatments..…”

Section: 2mentioning

confidence: 99%

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Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions

et al. 2013

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ABSTRACT. We introduce a mathematical model of the in vivo progression of Alzheimer's disease with focus on the role of prions in memory impairment. Our model consists of differential equations that describe the dynamic formation of β-amyloid plaques based on the concentrations of Aβ oligomers, PrP C proteins, and the Aβ-×-PrP C complex, which are hypothesized to be responsible for synaptic toxicity. We prove the well-posedness of the model and provided stability results for its unique equilibrium, when the polymerization rate of β-amyloid is constant and also when it is described by a power law.

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Section: 2mentioning

confidence: 99%

“…Hence, the solution f will be sought in the natural space L 1 (x 0 , +∞; xdx), since xdx measures the mass at any time. Our hypotheses for the system (1)(2)(3)(4) are…”

Section: A Power Law Polymerization Ratementioning

confidence: 99%

Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions

et al. 2013

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“…We propose a minimal model and show that the onset of AD is driven by a phase change from the soluble form of Aβ to its fibrillar form. The proposed model has the advantage of simplicity compared with the recent mathematical models [18,19,20,21]. A few other in-silico studies explained how the Aβ fibres form in the brain, using an agent-based model [22] or numerically solving the partial differential equations [21,23].…”

Section: Introductionmentioning

confidence: 99%

Mathematical model shows how sleep may affect amyloid β fibrillization

Hoore

Khailaie

Montaseri

et al. 2019

Preprint

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Deposition of amyloid β (Aβ) fibers in extra-cellular matrix of the brain is a ubiquitous feature associated with several neurodegenerative disorders, especially Alzheimer's disease (AD). While many of the biological aspects that contribute to the formation of Aβ plaques are well addressed at the intra-and inter-cellular level in short timescales, an understanding of how Aβ fibrillization usually starts to dominate at a longer timescale in spite of the presence of mechanisms dedicated to Aβ clearance, is still lacking. Furthermore, no existing mathematical model integrates the impact of diurnal neural activity as emanated from circadian regulation to predict disease progression due to a disruption in sleep-wake cycle. In this study, we develop a minimal model of Aβ fibrillization to investigate the onset of AD over a long time-scale. Our results suggest that the diseased state is a manifestation of a phase change of the system from soluble Aβ (sAβ) to fibrillar Aβ (fAβ) domination upon surpassing a threshold in the production rate of soluble Aβ. By incorporating the circadian rhythm into our model, we reveal that fAβ accumulation is crucially dependent on the regulation of sleep-wake cycle, thereby indicating the importance of a good sleep hygiene in averting AD onset. We also discuss potential intervention schemes to reduce fAβ accumulation in the brain by modification of the critical sAβ production rate.

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“…The progression of AD is mainly determined by the deterioration rate of the health state of the neurons, v = v x (a, t) ≥ 0. We use the notation v[f ] to stress its dependence on f : (1) ( 1] K(x, a, y, b) df y,t (b) dy + S(x, a, u 1 (x, t), . .…”

Section: Introductionmentioning

confidence: 99%

“…The quadratic terms (in u i ) model the aggregation of Aβ polymers, according to the Smoluchowski equations. We refer to [1,10] for an extensive discussion of the aggregation mechanism and the choice of the coagulation rates a i,j . The linear terms −σ m u m model the phagocytic activity of the microglia and other bulk clearance processes [14].…”

Section: Introductionmentioning

confidence: 99%

Well-Posedness of a Mathematical Model for Alzheimer's Disease

Bertsch¹,

Franchi²,

Tesi³

et al. 2018

SIAM J. Math. Anal.

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We consider the existence and uniqueness of solutions of an initial-boundary value problem for a coupled system of PDE's arising in a model for Alzheimer's disease. Apart from reaction diffusion equations, the system contains a transport equation in a bounded interval for a probability measure which is related to the malfunctioning of neurons. The main ingredients to prove existence are: the method of characteristics for the transport equation, a priori estimates for solutions of the reaction diffusion equations, a variant of the classical contraction theorem, and the Wasserstein metric for the part concerning the probability measure. We stress that all hypotheses on the data are not suggested by mathematical artefacts, but are naturally imposed by modelling considerations. In particular the use of a probability measure is natural from a modelling point of view. The nontrivial part of the analysis is the suitable combination of the various mathematical tools, which is not quite routine and requires various technical adjustments.

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A qualitative model for aggregation and diffusion of $$\beta $$ -amyloid in Alzheimer’s disease

Cited by 41 publications

References 31 publications

Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions

Alzheimer’s disease: analysis of a mathematical model incorporating the role of prions

Mathematical model shows how sleep may affect amyloid β fibrillization

Well-Posedness of a Mathematical Model for Alzheimer's Disease

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