A QUALITATIVE MODEL FOR AGGREGATION-FRAGMENTATION AND DIFFUSION OF β-AMYLOID IN ALZHEIMER’S DISEASE

In this paper we present a mathematical model for the aggregation, fragmentation and diffusion of Aβ amyloid in the brain affected by Alzheimer’s disease. The model is based on a classical discrete Smoluchowski aggregation-fragmentation equation modified to take diffusion into account. Ad Angelo con stima e affetto. Alzheimer’s disease (AD) is nowadays one of the most common late life dementia: current estimates of AD incidence are above 24 million of affected persons worldwide, a number that is expected to double every 20 years. Due to the consistent economic costs this will imply for the whole society, not to mention the disease caused to the affected patients and their families, it is clear that considerable efforts are made at all possible levels of research (medical, biological, pharmacological and even mathematical) to make any kind of feasible progress in the study of the disease. From the mathematical point of view, even if in recent years several models have been developed for the description and the study of pathologies such as tumors, the modeling for the study of AD is far less developed. Besides the classical approaches in vivo and in vitro, there has been an increasing interest toward the approach in silico, i.e. toward mathematical modeling and computer simulations. We refer for instance to [15], [3], and, first of all, to the remarkably exhaustive and deep paper [5]. It is important to stress that, despite the large number of experimental data that can be extracted from biomedical literature and incorporated in mathematical models as in [5], mathematical models do not currently have a “predictive” value; rather, they are what physicists call “toy models”, i.e. simplified formal models that can be used in order to test preliminary new theories, quickly identifying, for instance, the most relevant hypotheses or rejecting those less likely to lead to new insights. In this sense, qualitative models take a place beside more specific fully quantitative models, and can be used for reducing experimental costs or for overcoming structural difficulties. In this spirit, in our recent paper [1] we have provide an elementary mathematical model of the diffusion and agglomeration of the β-amyloid (Aβ hereafter) in the brain affected by Alzheimer’s disease (AD). For a detailed review of the current knowledge on the role of Aβ in AD (the so-called amyloid cascade hypothesis), we refer to [6]. Roughly speaking, Aβ is produced normally by the intramembranous proteolysis ∗The authors are supported by MURST, Italy, and by University of Bologna, Italy, funds for selected research topics.