  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 [Y. Achdou et al., 2013] 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 [M. Bertsch at al., 2016] 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.
  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.
