A mathematical model demonstrating the role of interstitial fluid flow on the clearance and accumulation of amyloid β in the brain

Abstract

A system of partial differential equations is developed to describe the formation and clearance of amyloid β (Aβ) and the subsequent buildup of Aβ plaques in the brain, which are associated with Alzheimer's disease. The Aβ related proteins are divided into five distinct categories depending on their size. In addition to enzymatic degradation, the clearance via diffusion and the outflow of interstitial fluid (ISF) into the surrounding cerebral spinal fluid (CSF) are considered. Treating the brain tissue as a porous medium, a simplified two-dimensional circular geometry is assumed for the transverse section of the brain leading to a nonlinear, coupled system of PDEs. Asymptotic analysis is carried out for the steady states of the spatially homogeneous system in the vanishingly small limit of Aβ clearance rate. The PDE model is studied numerically for two cases, a spherically symmetric case and a more realistic 2D asymmetric case, allowing for nonuniform boundary conditions. Our investigations demonstrate that ISF advection is a key component in reproducing the clinically observed accumulation of plaques on the outer boundaries. Furthermore, ISF circulation serves to enhance Aβ clearance over diffusion alone and that non-uniformities in ISF drainage into the CSF can lead to local clustering of plaques. Analysis of the model also demonstrates that plaque formation does not directly correspond to the high presence of toxic oligomers.

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

“…They use suitable Smoluchowski-type equations to describe the diffusion and agglomeration of soluble Aβ-oligomers of different lengths in small portion of the cerebral parenchyma, of the size of the soma of a single neuron (from 4 to 100 µm) and the formation of plaques, identified with insoluble assemblies of very long polymers. Some other phenomena were also included in the model, such as fragmentation of long polymers [25] and clearance of Aβ from the CSF [11].…”

“…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.…”

“…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. They use suitable Smoluchowski-type equations to describe the diffusion and agglomeration of soluble Aβ-oligomers of different lengths in small portion of the cerebral parenchyma, of the size of the soma of a single neuron (from 4 to 100 µm) and the formation of plaques, identified with insoluble assemblies of very long polymers.…”

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.…”

“…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. They use suitable Smoluchowski type equations to describe the diffusion and agglomeration of soluble Aβ -oligomers of different lengths in small portion of the cerebral parenchyma, of the size of the soma of a single neuron (from 4 to 100 µm), and the formation of plaques, identified with insoluble assemblies of very long polymers.…”