This work presents a novel mathematical framework to model biological tissue atrophy driven by the diffusion of a biological agent, with a focus on neurodegenerative diseases. We introduce and validate a mathematical and computational model consisting of a Fisher–Kolmogorov equation, which describes species diffusion and is commonly used to represent prion-like processes, coupled with an elasticity equation that accounts for tissue mass loss. The key innovation of the model lies in the coupling mechanism: a logistic law that governs the progressive reduction of the medium’s mass. The proposed framework is applied to the study of Alzheimer’s disease, where it captures both the spatial spread of misfolded tau-proteins and the associated brain atrophy that characterizes the disease. To address the numerical challenges posed by the coupled system, we employ a Discontinuous Galerkin method for spatial discretization and a Crank–Nicolson scheme for time integration. We detail the mathematical formulation of the model, analyze its main properties, and describe the numerical approach. Convergence studies are presented to validate the implementation, followed by numerical simulations that demonstrate the model’s ability to reproduce key features of Alzheimer’s disease onset and progression.