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In this paper, we analyse the process of misfolding and spreading of both<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-synuclein and Amyloid-<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b2<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>, related to Parkinson\u2019s and Alzheimer\u2019s diseases, respectively. We introduce and analyze a positivity-preserving numerical method for the discretization of the Fisher-Kolmogorov equation, modelling accumulation and spreading of prionic proteins. The proposed approximation method is based on the discontinuous Galerkin method on polygonal and polyhedral grids for space discretization and on<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\vartheta -$$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mrow><mml:mi>\u03d1<\/mml:mi><mml:mo>-<\/mml:mo><\/mml:mrow><\/mml:math><\/jats:alternatives><\/jats:inline-formula>method time integration scheme. We prove the existence of the discrete solution and a convergence result where the Implicit Euler scheme is employed for time integration. We show that the proposed approach is structure-preserving, in the sense that it guarantees that the discrete solution is non-negative, a feature that is of paramount importance in practical application. The numerical verification of our numerical model is performed both using a manufactured solution and considering wavefront propagation in two-dimensional polygonal grids. Next, we present a simulation of<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\alpha $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b1<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>-synuclein spreading in a two-dimensional brain slice in the sagittal plane. The polygonal mesh for this simulation is agglomerated maintaining the distinction of white and grey matter, taking advantage of the flexibility of PolyDG methods in the mesh construction. Finally, we simulate the spreading of Amyloid-<jats:inline-formula><jats:alternatives><jats:tex-math>$$\\beta $$<\/jats:tex-math><mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\"><mml:mi>\u03b2<\/mml:mi><\/mml:math><\/jats:alternatives><\/jats:inline-formula>in a patient-specific setting by using a three-dimensional geometry reconstructed from magnetic resonance images and an initial condition reconstructed from positron emission tomography. Our numerical simulations confirm that the proposed method is able to capture the evolution of Parkinson\u2019s and Alzheimer\u2019s diseases.<\/jats:p>","DOI":"10.1007\/s10915-024-02581-7","type":"journal-article","created":{"date-parts":[[2024,6,20]],"date-time":"2024-06-20T19:01:40Z","timestamp":1718910100000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Structure Preserving Polytopal Discontinuous Galerkin Methods for the Numerical Modeling of Neurodegenerative Diseases"],"prefix":"10.1007","volume":"100","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-7014-972X","authenticated-orcid":false,"given":"Mattia","family":"Corti","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Francesca","family":"Bonizzoni","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Paola F.","family":"Antonietti","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,6,20]]},"reference":[{"key":"2581_CR1","unstructured":"Brett, M., Markiewicz, C., Hanke, M., C\u00f4t\u00e9, M., Cipollini, B., McCarthy, P., Cheng, C.: NiBabel 4.0.0: access a cacophony of neuro-imaging file formats. 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