Smoluchowski Equation with Variable Coefficients in Perforated Domains: Homogenization and Applications to Mathematical Models in Medicine

Franchi, Bruno; Lorenzani, Silvia

doi:10.1007/978-3-319-52742-0_4

Cited by 11 publications

(22 citation statements)

References 15 publications

(21 reference statements)

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“…x ∈ Ω the function y → A x (y, t) is injective for all t ∈ [0, T ]. Below we shall reformulate problem (6)- (8) in terms of the characteristics, but before doing so we prove the following result.…”

Section: The Characteristicsmentioning

confidence: 91%

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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Section: The Characteristicsmentioning

confidence: 91%

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

Bertsch¹,

Franchi²,

Tesi³

et al. 2018

SIAM J. Math. Anal.

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“…Most of these models have focused on macroscale phenomena, with only few models accounting for their counterparts at levels of single cells and ion exchangers on the cellular membrane (Mauri et al, 2016 ; Sala et al, 2019a ; Sacco et al, 2020 ). As a further note, to date, only few mechanism-driven models have included the dynamics of Aβ (Craft et al, 2002 ; Puri and Li, 2010 ; Das et al, 2011 ; Kyrtsos and Baras, 2015 ; Franchi and Lorenzani, 2016 , 2017 ). However, these models have been developed in the context of the brain and their applicability to the eye remains to be assessed.…”

Section: Overview Of Mechanism-driven Models Of the Eye And Their mentioning

confidence: 99%

Neurodegenerative Disorders of the Eye and of the Brain: A Perspective on Their Fluid-Dynamical Connections and the Potential of Mechanism-Driven Modeling

et al. 2020

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Neurodegenerative disorders (NDD) such as Alzheimer's and Parkinson's diseases are significant causes of morbidity and mortality worldwide. The pathophysiology of NDD is still debated, and there is an urgent need to understand the mechanisms behind the onset and progression of these heterogenous diseases. The eye represents a unique window to the brain that can be easily assessed via non-invasive ocular imaging. As such, ocular measurements have been recently considered as potential sources of biomarkers for the early detection and management of NDD. However, the current use of ocular biomarkers in the clinical management of NDD patients is particularly challenging. Specifically, many ocular biomarkers are influenced by local and systemic factors that exhibit significant variation among individuals. In addition, there is a lack of methodology available for interpreting the outcomes of ocular examinations in NDD. Recently, mathematical modeling has emerged as an important tool capable of shedding light on the pathophysiology of multifactorial diseases and enhancing analysis and interpretation of clinical results. In this article, we review and discuss the clinical evidence of the relationship between NDD in the brain and in the eye and explore the potential use of mathematical modeling to facilitate NDD diagnosis and management based upon ocular biomarkers.

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“…Finally, the papers [7,23,24,22] are dedicated to the transition from the microscopic scale to the macroscopic scale, through different mathematical procedures: [7] adapts arguments from modern Boltzmann-type kinetic theory for multi-agent systems [50], while [23,24,22] rely on homogenisation procedures (in [23,24], neurons are assumed to be periodically distributed, whereas [22] introduces randomness of the distributions of neurons and the onset of the disease).…”

Section: Mathematical Modellingmentioning

confidence: 99%

“…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.…”

Section: Mathematical Modellingmentioning

confidence: 99%

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

Bertsch

Franchi²,

Meacci³

et al. 2020

Eur. J. Appl. Math

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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.

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Smoluchowski Equation with Variable Coefficients in Perforated Domains: Homogenization and Applications to Mathematical Models in Medicine

Cited by 11 publications

References 15 publications

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

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

Neurodegenerative Disorders of the Eye and of the Brain: A Perspective on Their Fluid-Dynamical Connections and the Potential of Mechanism-Driven Modeling

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

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