The role of A[Formula: see text] and Tau proteins in Alzheimer's disease: a mathematical model on graphs.

In this Note we study a mathematical model for the progression of Alzheimer's Disease in the human brain. The novelty of our approach consists in the representation of the brain as two superposed graphs where toxic proteins diffuse, the connectivity graph which represents the neural network, and the proximity graph which takes into account the extracellular space. Toxic proteins such as [Formula: see text] amyloid and Tau play in fact a crucial role in the development of Alzheimer's disease and, separately, have been targets of medical treatments. Recent biomedical literature stresses the potential impact of the synergetic action of these proteins. We numerically test various modelling hypotheses which confirm the relevance of this synergy.

Combined Model of Aggregation and Network Diffusion Recapitulates Alzheimer's Regional Tau-Positron Emission Tomography.

Background: Alzheimer's disease involves widespread and progressive deposition of misfolded protein tau (τ), first appearing in the entorhinal cortex, coagulating in longer polymers and insoluble fibrils. There is mounting evidence for "prion-like" trans-neuronal transmission, whereby misfolded proteins cascade along neuronal pathways, giving rise to networked spread. However, the cause-effect mechanisms by which various oligomeric τ species are produced, aggregate, and disseminate are unknown. The question of how protein aggregation and subsequent spread lead to stereotyped progression in the Alzheimer brain remains unresolved. Materials and Methods: We address these questions by using mathematically precise parsimonious modeling of these pathophysiological processes, extrapolated to the whole brain. We model three key processes: τ monomer production; aggregation into oligomers and then into tangles; and the spatiotemporal progression of misfolded τ as it ramifies into neural circuits via the brain connectome. We model monomer seeding and production at the entorhinal cortex, aggregation using Smoluchowski equations; and networked spread using our prior Network-Diffusion model. Results: This combined aggregation-network-diffusion model exhibits all hallmarks of τ progression seen in human patients. Unlike previous theoretical studies of protein aggregation, we present here an empirical validation on in vivo imaging and fluid τ measurements from large datasets. The model accurately captures not just the spatial distribution of empirical regional τ and atrophy but also patients' cerebrospinal fluid phosphorylated τ profiles as a function of disease progression. Conclusion: This unified quantitative and testable model has the potential to explain observed phenomena and serve as a test-bed for future hypothesis generation and testing in silico. Impact statement The presented aggregation-network-diffusion model exhibits all hallmarks of tau progression in human patients; it accurately captures not just the spatial distribution of empirical regional tau and atrophy but also patients' cerebrospinal fluid phosphorylated tau profiles. Thus, it serves to fill a theoretical gap between microscopic biophysical processes and empirical macroscopic measurements of pathological patterns in Alzheimer's disease. This unified quantitative and testable model has the potential to explain observed phenomena and serve as a test-bed for future hypothesis generation and testing in silico.