Parameter-robust preconditioners for a cell-by-cell poroelasticity model with interface coupling

Abstract

This paper presents a scalable and robust solver for a cell-by-cell poroelasticity model, describing the mechanical interactions between brain cells embedded in extracellular space. Explicitly representing the complex cellular shapes, the proposed approach models both intracellular and extracellular spaces as distinct poroelastic media, separated by a permeable cell membrane which allows hydrostatic and osmotic pressure-driven fluid exchange. Based on a three-field (displacement, total pressure, and fluid pressure) formulation, the solver leverages the framework of norm-equivalent preconditioning and appropriately fitted norms to ensure robustness across all material parameters of the model. Scalability for large and complex geometries is achieved through efficient Algebraic Multigrid (AMG) approximations of the preconditioners' individual blocks. Furthermore, we accommodate diverse boundary conditions, including full Dirichlet boundary conditions for displacement, which we handle efficiently using the Sherman-Morrison-Woodbury formula. Our theoretical analysis is complemented by numerical experiments demonstrating the preconditioners' robustness and performance across various parameters relevant to realistic scenarios. A large scale example of cellular swelling on a dense reconstruction of the mouse visual cortex highlights the method's potential for investigating complex physiological processes such as cellular volume regulation in detailed biological structures.

Figures (8)

• Figure 1: Left) illustration of the geometrical setting: $\Omega_i$ describes the intracellular spaces, separated from the extracellular space $\Omega_e$ by the membrane $\Gamma$. The interface normal $\mathbf{n}$ points from the intra- into the extracellular domain; right) rendering of a realistic tissue reconstruction from the mouse brain, containing 200 cells in a tissue cube with $20\,\mu$m sidelength.
• Figure 1: Number of MinRes iterations of system \ref{['eq:total_pressure']} with preconditioner \ref{['eq:prec_naiv']} on a unit square geometry. The number of iterations is capped at 250.
• Figure 1: Number of MinRes iterations for test cases on a unit square with exact inverses of the preconditioner blocks computed via Cholesky factorization: Mixed displacement boundary conditions ($|\Gamma_d| >0$ and $|\Gamma_t| >0$) with preconditioner \ref{['eq:TP_preconditioner']} (left); and full displacement Dirichlet boundary conditions ($\partial \Omega = \Gamma_d$) with preconditioner \ref{['eq:TP_preconditioner_L20proj']} (right).
• Figure 2: Number of MinRes iterations for test cases on a unit cube with mixed displacement boundary conditions ($|\Gamma_d| >0$ and $|\Gamma_t| >0$) and preconditioner \ref{['eq:TP_preconditioner']}, with exact inverses of the individual blocks (left), and algebraic multigrid approximations (right).
• Figure 3: Number of MinRes iterations for test cases on a unit cube with full Dirichlet displacement boundary conditions ($\Gamma_d = \partial \Omega$) and preconditioner \ref{['eq:TP_preconditioner_L20proj']}, with exact inverses of the individual blocks (left), and algebraic multigrid approximations (right).

Theorems & Definitions (14)

• Example 2.1: Naïve preconditioner
• Definition 3.1: Fitted norms hong2023new
• Theorem 3.2: hong2023new, Th. 2.14
• Theorem 3.3: Parameter robust stability of the continuous total pressure formulation
• Proof 1
• Remark 3.4
• Theorem 3.5: Parameter robust stability - homogeneous Dirichlet displacement boundary conditions
• Proof 2
• Remark 3.6
• Remark 3.7: Stability in the limit of incompressible constituents