Summary: Alzheimer’s disease (AD) is a fatal incurable disease leading to progressive neuron destruction. AD is caused in part by the accumulation in the brain of \(A\beta\) monomers aggregating into oligomers and fibrils. Oligomers are amongst the most toxic structures as they can interact with neurons via membrane receptors, including \(\mathrm{PrP}^{\mathrm c}\) proteins. This interaction leads to the misconformation of \(\mathrm{PrP}^{\mathrm c}\) into pathogenic oligomeric prions, \(\mathrm{PrP}^{\mathrm{ol}}\).
We develop here a model describing in vitro \(A\beta\) polymerization process. We include interactions between oligomers and \(\mathrm{PrP}^{\mathrm c}\), causing the misconformation of \(\mathrm{PrP}^{\mathrm c}\) into \(\mathrm{PrP}^{\mathrm{ol}}\). 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 the solution of our model using Schauder fixed point and Cauchy-Lipschitz theorems. Numerical simulations are also provided to some specific profiles.