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

Helal, Mohamed; Hingant, Erwan; Praly, Laurent; Webb, Glenn

doi:10.1007/s00285-013-0732-0

Cited by 55 publications

(61 citation statements)

References 40 publications

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“…We conclude the Introduction by mentioning earlier mathematical models which deal with some aspects of AD: A β polymerization [40], A β plaque formation and the role of prions interacting with A β [41, 42], linear cross-talk among brain cells and A β [43], and the influence of SORLA on AD progression [44, 45]. …”

Section: Introductionmentioning

confidence: 99%

Mathematical model on Alzheimer’s disease

2016

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BackgroundAlzheimer disease (AD) is a progressive neurodegenerative disease that destroys memory and cognitive skills. AD is characterized by the presence of two types of neuropathological hallmarks: extracellular plaques consisting of amyloid β-peptides and intracellular neurofibrillary tangles of hyperphosphorylated tau proteins. The disease affects 5 million people in the United States and 44 million world-wide. Currently there is no drug that can cure, stop or even slow the progression of the disease. If no cure is found, by 2050 the number of alzheimer’s patients in the U.S. will reach 15 million and the cost of caring for them will exceed $ 1 trillion annually.ResultsThe present paper develops a mathematical model of AD that includes neurons, astrocytes, microglias and peripheral macrophages, as well as amyloid β aggregation and hyperphosphorylated tau proteins. The model is represented by a system of partial differential equations. The model is used to simulate the effect of drugs that either failed in clinical trials, or are currently in clinical trials.ConclusionsBased on these simulations it is suggested that combined therapy with TNF- α inhibitor and anti amyloid β could yield significant efficacy in slowing the progression of AD.

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Section: Introductionmentioning

confidence: 99%

Mathematical model on Alzheimer’s disease

2016

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“…Nowadays, Alzheimer's disease is the most common form of senile dementia with enormous socio-economic implications. In recent years, besides in vivo and in vitro experimental models, there has been an increasing interest in mathematical modeling and computer simulations (the so-called in silico approach) [1,6,16,20,24,41], in order to better understand the mechanisms for the onset and the evolution of AD. It is largely accepted that Aβ peptide (especially in soluble form) has a substantial role in the process of synaptic degeneration leading to neuronal death and eventually to dementia (the so-called amyloid cascade hypothesis [28]).…”

Section: Introductionmentioning

confidence: 99%

From a Microscopic to a Macroscopic Model for Alzheimer Disease: Two-Scale Homogenization of the Smoluchowski Equation in Perforated Domains

Franchi

Lorenzani

2016

J Nonlinear Sci

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In this paper, we study the homogenization of a set of Smoluchowski's discrete diffusion-coagulation equations modeling the aggregation and diffusion of β-amyloid peptide (Aβ), a process associated with the development of Alzheimer's disease. In particular, we define a periodically perforated domain Ω ǫ , obtained by removing from the fixed domain Ω (the cerebral tissue) infinitely many small holes of size ǫ (the neurons), which support a non-homogeneous Neumann boundary condition describing the production of Aβ by the neuron membranes. Then, we prove that, when ǫ → 0, the solution of this micro-model two-scale converges to the solution of a macro-model asymptotically consistent with the original one. Indeed, the information given on the microscale by the non-homogeneous Neumann boundary condition is transferred into a source term appearing in the limiting (homogenized) equations. Furthermore, on the macroscale, the geometric structure of the perforated domain induces a correction in that the scalar diffusion coefficients defined at the microscale are replaced by tensorial quantities.

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“…The technique used, implying hypotheses on the regularity of the coefficients, is based on the works on Lifshitz-Slyozov (LS) equation in [6] and LS with encounters (coagulation) in [5]. Techniques also adapted in [22] for biological polymers. The LS equation is a size structured model for clusters (polymers or more general) formation by addition-depletion of monomers [30], while LS with encounters also take into account merging clusters .…”

Section: 3mentioning

confidence: 99%

Derivation and mathematical study of a sorption-coagulation equation

Hingant

Sepúlveda

2015

Nonlinearity

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This work is devoted to the derivation and the matematical study of a new model for water-soluble polymers and metal ions interactions, which are used in chemistry for their wide range of applications. First, we motivate and derive a model that describes the evolution of the configurational distribution of polymers. One of the novelty resides in the configuration variables which consider both, the size of the polymers and the quantity of metal ions they captured through sorption. The model consists in a non-linear transport equation with a quadratic source term, the coagulation. Then, we prove the existence of solutions for all time to the problem thanks to classical fixed point theory. Next, we reformulate the coagulation operator under a conservative form which allows to write a finite volume scheme. The sequence of approximated solutions is proved to be convergent (toward a solution to the problem) thanks to a L 1 − weak stability principle. Finally, we illustrate the behaviour of the solutions using this numerical scheme and we intend to discuss on the long-time behaviour. 9 3. Numerical approximation 13 3.1. A conservative truncated formulation 13 3.2. The numerical scheme and convergence statement 15

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

Cited by 55 publications

References 40 publications

Mathematical model on Alzheimer’s disease

Mathematical model on Alzheimer’s disease

From a Microscopic to a Macroscopic Model for Alzheimer Disease: Two-Scale Homogenization of the Smoluchowski Equation in Perforated Domains

Derivation and mathematical study of a sorption-coagulation equation

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