title: Alzheimer's disease: a mathematical model for onset and progression.
abstract: In this article we propose a mathematical model for the onset and progression of Alzheimer's disease based on transport and diffusion equations. We regard brain neurons as a continuous medium and structure them by their degree of malfunctioning. Two different mechanisms are assumed to be relevant for the temporal evolution of the disease: i) diffusion and agglomeration of soluble polymers of amyloid, produced by damaged neurons and ii) neuron-to-neuron prion-like transmission. We model these two processes by a system of Smoluchowski equations for the amyloid concentration, coupled to a kinetic-type transport equation for the distribution function of the degree of malfunctioning of neurons. The second equation contains an integral term describing the random onset of the disease as a jump process localized in particularly sensitive areas of the brain. Our numerical simulations are in good qualitative agreement with clinical images of the disease distribution in the brain which vary from early to advanced stages.
authors.affiliation: Dipartimento di Matematica, Università di Roma 'Tor Vergata', Via della Ricerca Scientifica 1, 00133 Roma, Italy and Istituto per le Applicazoni del Calcolo 'M. Picone', Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma, Italy.
authors.forename: Michiel
authors.initials: M
authors.lastname: Bertsch
authors.affiliation: Department of Mathematics, University of Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy.
authors.forename: Bruno
authors.initials: B
authors.lastname: Franchi
authors.affiliation: S.C. Neurologia, Azienda Ospedaliera Santa Maria Nuova-IRCCS, Viale Risorgimento 80, 42123 Reggio Emilia, Italy.
authors.forename: Norina
authors.initials: N
authors.lastname: Marcello
authors.affiliation: Department of Mathematics, University of Bologna, Piazza di Porta S. Donato 5, 40126 Bologna, Italy.
authors.forename: Maria Carla
authors.initials: MC
authors.lastname: Tesi
authors.affiliation: Istituto per le Applicazoni del Calcolo "M. Picone", Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185 Roma, Italy.
authors.forename: Andrea
authors.initials: A
authors.lastname: Tosin
mesh_terms: D000544:Alzheimer Disease / Q000209:etiology* / Q000378:metabolism / Q000473:pathology; D016229:Amyloid beta-Peptides / Q000737:chemistry / Q000378:metabolism; D001921:Brain / Q000378:metabolism / Q000473:pathology; D018450:Disease Progression; D006801:Humans; D055641:Mathematical Concepts; D008954:Models, Biological; D009474:Neurons / Q000378:metabolism; D066263:Protein Aggregation, Pathological
chemical_list: D016229:Amyloid beta-Peptides
keywords: Alzheimer's disease; Smoluchowski equations; numerical simulations; transport and diffusion equations