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

Abstract

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.

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

“…A macroscopic model was proposed in [5,8]. The authors couple the set of truncated Smoluchowski equations already used in [1] to a kinetic-type transport equation that models the spreading of neuronal damage, including the possibility of spreading through neuron-to-neuron prion-like transmission.…”

“…As far as we know, Murphy and Pallitto [45,49] were the first ones who used Smoluchowski equations to describe Aβ-agglomeration, starting from an in vitro approach. More recently, a systematic approach to the modelling of Aβ-agglomeration and the formation of senile plaques was carried on in a series of papers [1,25,5,8,7,23,24,22,11,14]. In [1,25,11], the authors consider a model at microscopic scale.…”

Stationary distribution of a stochastic Alzheimer's disease model

“…Helal et al 17 introduced a mathematical model of the in vivo progression of AD and established the well‐posedness of the model and the stability for its unique equilibrium. Bertsch et al 18 investigated the well‐posedness of another mathematical model for AD. Bertsch et al 19 studied a mathematical model for the onset and progression of AD.…”

A two-component model of the red blood cell membrane and other mathematical models in medicine

“…As far as we know, Murphy and Pallitto (MURPHY;PALLITTO, 2000;MURPHY, 2001) were the first ones who used Smoluchowski equations to describe Aβagglomeration, starting from an in vitro approach. More recently, a systematic approach to the modelling of Aβ -agglomeration and the formation of senile plaques was carried on in a series of papers (ACHDOU et al, 2013;TESI, 2012;BERTSCH et al, 2018;LORENZANI, 2016;LOREN-ZANI, 2017;HEIDA;LORENZANI, 2019;TSENG;WARD, 2019;CRAFT;WEIN;SELKOE, 2002). In (ACHDOU et al, 2013;TESI, 2012;TSENG;WARD, 2019) the authors consider a model at microscopic scale.…”

“…A macroscopic model was proposed in BERTSCH et al, 2018). The authors couple the set of truncated Smoluchowski equations already used in (ACHDOU et al, 2013) to a kinetic-type transport equation that models the spreading of neuronal damage, including the possibility of spreading through neuron-to-neuron prion-like transmission.…”