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 Aβ monomers aggre-gating into oligomers and fibrils. Oligomers are amongst 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 oligomeric prions, PrP ol. In this work, we develop a model describing in vitro Aβ polymerization process. We include interactions between oligomers and PrP c , causing the misconformation of PrP c into PrP 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 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.