A discontinuous Galerkin method for the three-dimensional heterodimer model with application to prion-like proteins' dynamics
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866916332980666368 |
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| author | Antonietti, Paola F. Corti, Mattia Lorenzon, Giacomo |
| author_facet | Antonietti, Paola F. Corti, Mattia Lorenzon, Giacomo |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2407_16065 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A discontinuous Galerkin method for the three-dimensional heterodimer model with application to prion-like proteins' dynamics Antonietti, Paola F. Corti, Mattia Lorenzon, Giacomo Numerical Analysis Quantitative Methods 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. |
| title | A discontinuous Galerkin method for the three-dimensional heterodimer model with application to prion-like proteins' dynamics |
| topic | Numerical Analysis Quantitative Methods |
| url | https://arxiv.org/abs/2407.16065 |