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

Part of: Mathematical biology in general Physiological, cellular and medical topics 30th Anniversary Special Issue

Published online by Cambridge University Press:  25 September 2020

M. BERTSCH Open the ORCID record for M. BERTSCH [Opens in a new window] ,
B. FRANCHI ,
L. MEACCI Open the ORCID record for L. MEACCI [Opens in a new window] ,
M. PRIMICERIO  and
M.C. TESI Open the ORCID record for M.C. TESI [Opens in a new window]
M. BERTSCH
Affiliation:
Dipartimento di Matematica, Università di Roma ‘Tor Vergata’, Via della Ricerca Scientifica 1, 00133 Roma, Italy, e-mail: bertsch@mat.uniroma2.it Istituto per le Applicazoni del Calcolo ‘M. Picone’, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma, Italy
B. FRANCHI
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy, e-mails: bruno.franchi@unibo.it; mariacarla.tesi@unibo.it
L. MEACCI
Affiliation:
Instituto de Ciências Matemáticas e de Computação, ICMC, Universidade de São Paulo, Avenida Trabalhador Sancarlense, 400, São Carlos (SP), CEP 13566-590, Brazil, e-mail: luca.meacci@usp.br
M. PRIMICERIO
Affiliation:
Istituto per le Applicazoni del Calcolo ‘M. Picone’, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma, Italy Dipartimento di Matematica ‘U. Dini’, Università degli Studi di Firenze, Viale Giovanni Battista Morgagni, 67/A, 50134 Firenze, Italy, e-mail: mario.primicerio@math.unifi.it
M.C. TESI
Affiliation:
Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy, e-mails: bruno.franchi@unibo.it; mariacarla.tesi@unibo.it
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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.

Keywords

Smoluchowski equationsmathematical models of Alzheimer’s diseaseamyloid cascade hypothesis

MSC classification

Primary: 92C50: Medical applications (general) 92B05: General biology and biomathematics
Secondary: 92C20: Neural biology

Information

Type
Papers
Information
European Journal of Applied Mathematics , Volume 32 , Special Issue 5: 30th Anniversary Special Issue , October 2021 , pp. 749 - 768
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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