A discontinuous Galerkin method for the three-dimensional heterodimer model with application to prion-like proteins' dynamics
<|reference_start|>A discontinuous Galerkin method for the three-dimensional heterodimer model with application to prion-like proteins' dynamics: Neurocognitive disorders, such as Alzheimer's and Parkinson's, have a wide social impact. These proteinopathies involve misfolded proteins accumulating into neurotoxic aggregates. Mathematical and computational models describing the prion-like dynamics offer an analytical basis to study the diseases' evolution and a computational framework for exploring potential therapies. This work focuses on the heterodimer model in a three-dimensional setting, a reactive-diffusive system of nonlinear partial differential equations describing the evolution of both healthy and misfolded proteins. We investigate traveling wave solutions and diffusion-driven instabilities as a mechanism of neurotoxic pattern formation. For the considered mathematical model, we propose a space discretization, relying on the Discontinuous Galerkin method on polytopal/polyhedral grids, allowing high-order accuracy and flexible handling of the complicated brain's geometry. Further, we present a priori error estimates for the semi-discrete formulation and we perform convergence tests to verify the theoretical results. Finally, we conduct simulations using realistic data on a three-dimensional brain mesh reconstructed from medical images.<|reference_end|>
@article{antonietti2024a,
    title={A discontinuous Galerkin method for the three-dimensional heterodimer
  model with application to prion-like proteins' dynamics},
    author={Paola F. Antonietti, Mattia Corti and Giacomo Lorenzon},
    journal={arXiv preprint arXiv:2407.16065},
    year={2024},
    archivePrefix={arXiv},
    eprint={2407.16065},
    primaryClass={math.NA cs.NA q-bio.QM}
}