Alzheimer's disease and prion: analysis of an in vitro mathematical model

Abstract : Alzheimer's disease (AD) is a fatal incurable disease leading to progressive neuron destruction. AD is caused in part by the accumulation of Aβ monomers inside the brain, which have the faculty to aggregate into oligomers and fibrils. Oligomers are the most toxic structures as they can interact with neurons via membrane receptors, including PrP c proteins. This interaction leads to the misconformation of PrP c into pathogenic prions, PrP sc. In this work, we develop a model describing in vitro Aβ polymerization process. We include interactions between oligomers and PrP c , leading to the misconformation of PrP sc. The model consists of nine equations, including size structured transport equations , ordinary differential equations and delayed differential equations. We analyse the well-posedness of the model and prove the existence and uniqueness of solutions of our model using Schauder fixed point theorem and Cauchy-Lipschitz theorem. Numerical simulations are also provided to give an illustration of the profiles that can be obtained with this model.