Anisotropic Diffusion and Traveling Waves of Toxic Proteins in Neurodegenerative Diseases

TL;DR

This paper models the spread of toxic proteins in neurodegenerative diseases using anisotropic diffusion and coupled kinetics, revealing filamentary transmission and simplified FKPP models that capture key propagation features.

Contribution

It introduces coupled anisotropic diffusion models for toxic proteins and reduces them to FKPP-type systems, highlighting their propagation dynamics in neurodegenerative diseases.

Findings

Anisotropic diffusion leads to filamentary, one-dimensional transmission pathways.

Coupled models show interaction effects between healthy and toxic proteins.

Simplified FKPP models capture qualitative propagation features.

Abstract

Neurodegenerative diseases are closely associated with the amplification and invasion of toxic proteins. In particular Alzheimer's disease is characterized by the systematic progression of amyloid-$\beta$ and $\tau$-proteins in the brain. These two protein families are coupled and it is believed that their joint presence greatly enhances the resulting damage. Here, we examine a class of coupled chemical kinetics models of healthy and toxic proteins in two spatial dimensions. The anisotropic diffusion expected to take place within the brain along axonal pathways is factored in the models and produces a filamentary, predominantly one-dimensional transmission. Nevertheless, the potential of the anisotropic models towards generating interactions taking advantage of the two-dimensional landscape is showcased. Finally, a reduction of the models into a simpler family of generalized Fisher-Kolmogorov-Petrovskii-Piskunov (FKPP) type systems is examined. It is seen that the latter captures well the qualitative propagation features, although it may somewhat underestimate the concentrations of the toxic proteins.