Parameter-robust formulation and preconditioning of poroelasticity equations for brain modelling
Abstract
Mathematical modelling of brain parenchyma mechanics and fluid dynamics is a powerful tool to better understand clearance mechanisms, and investigate mechanistic hypotheses that cannot be verified with in vivo experiments. Single and multiple network poroelasticity theory (MPET) can be used to model the behaviour of different types of porous media. In addition, MPET has been used in the past decade to understand better how the different fluid compartments exchange mass in the brain and, more generally, the brain’s clearance process. Nonetheless, the MPET equations applied to brain modelling present several numerical and modelling challenges. Therefore, in the articles collected in this thesis, an analysis of the system of equations from a numerical and computational viewpoint using both theoretical proofs and practical numerical experiments is presented. In particular, we present parameter-robust formulations and preconditioners for the MPET equations in order to solve the system in an efficient and accurate maner. In addition, brain parenchyma pulsatility is modelled via linear elasticity and single network poroelasticity equations in a realistic human brain domain.List of papers
Paper I. J. J. Lee, E. Piersanti, K.-A. Mardal, M. E. Rognes. “A mixed finite element method for nearly incompressible multiple-network poroelasticity”. In: SIAM Journal on Scientific Computing. Vol. 41, no. 2 (2019), DOI: 10.1137/18M1182395. The article is included in the thesis. Also available at: https://doi.org/10.1137/18M1182395 |
Paper II. E. Piersanti, M. E. Rognes, K.-A. Mardal. “Parameter robust preconditioning for multi-compartmental Darcy equations”. In: Numerical Mathematics and Advanced Applications ENUMATH 2019 Vol. 139, (August 2020), DOI: 10.1007/978-3-030-55874-1_69. The article is not available in DUO due to publisher restrictions. The published version is available at: https://doi.org/10.1007/978-3-030-55874-1_69 |
Paper III. E. Piersanti, J. J. Lee, T. Thompson, K.-A. Mardal. “Parameter robust preconditioning by congruence for multiple-network poroelasticity”. In: SIAM Journal on Scientific Computing. Vol. 43, no. 4 (2021), DOI: 10.1137/20M1326751. The article is included in the thesis. Also available at: https://doi.org/10.1137/20M1326751 |
Paper IV. E. Piersanti, M. E. Rognes, V. Vinje. “Are brain displacements and pressures within the parenchyma induced by surface pressure differences? A computational modelling study”. (bioRxiv DOI: 10.1101/2022.09.07.506967). Published in: PLoS ONE 18(12): e0288668. DOI: 10.1371/journal.pone.0288668.The paper is included in the thesis. Also available at: https://doi.org/10.1371/journal.pone.0288668 |