Microscopic and macroscopic models for the onset and progression of Alzheimer's disease

Abstract

Abstract. In the first part of this paper we review a mathematical model for the onset and progression of Alzheimer's disease (AD) that was developed in subsequent steps over several years. The model is meant to describe the evolution of AD in vivo. In [1] we treated the problem at a microscopic scale, where the typical length scale is a multiple of the size of the soma of a single neuron. Subsequently, in [2] we concentrated on the macroscopic scale, where brain neurons are regarded as a continuous medium, structured by their degree of malfunctioning.In the second part of the paper we consider the relation between the microscopic and the macroscopic models. In particular we show under which assumptions the kinetic transport equation, which in the macroscopic model governs the evolution of the probability measure for the degree of malfunctioning of neurons, can be derived from a particle-based setting.The models are based on aggregation and diffusion equations for β Amyloid, a protein fragment that healthy brains regularly produce and eliminate. In case of dementia Aβ monomers are no longer properly washed out and begin to coalesce forming eventually plaques. 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; ii) neuron-to-neuron prion-like transmission.In the microscopic model we consider basically mechanism i), modelling it by a system of Smoluchowski equations for the amyloid concentration (describing the agglomeration phenomenon), with the addition of a diffusion term as well as of a source term on the neuronal membrane. At the macroscopic level instead we model processes i) and ii) 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 the 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.Even though we deliberately neglected many aspects of the complexity of the brain and the disease, numerical simulations are in both cases (microscopic and macroscopic) in good qualitative agreement with clinical data.

Mathematical Modeling of Protein Misfolding Mechanisms in Neurological Diseases: A Historical Overview

“…Note that Bertsch’s modeling approach is the first study that attempts to build a bridge between the microscopic and macroscopic processes that characterize the impact of Aβ aggregation on the onset and clinical progression of AD. Although still insufficient, some effort has been already put on checking the mathematical correctness and internal consistency of that model ( 86 , 87 ).…”

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

“…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.…”

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

“…In [3,4] a macroscopic mathematical model was proposed which describes the onset and evolution of Alzheimer's disease (AD). This model is meant to mirror the so-called Amyloid Cascade Hypothesis [11,15,21], coupled with the spreading of the disease through neuron-to-neuron transmission (prionoid hypothesis [5,23]).…”

“…In order to clarify the structure of our equations and the choice of our assumptions, let us sketch a gist of their biological background. We refer to [3,4] for a complete description of the model and an account of the most recent biomedical literature. The model focusses on the role of the polymer beta-amyloid, in particular its toxic soluble isoform Aβ 42 .…”

“…For more details on the model we refer to [3], which also contains some two-dimensional simulations and qualitative comparison with experimental data. A derivation of the transport equation for f from a Boltzmann-type kinetic approach is contained in [4].…”