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Dynamically consistent finite volume scheme for a bimonomeric simplified model with inflammation processes for Alzheimer's disease

Juan Barajas-Calonge, Mauricio A. Sepulveda Cortes, Nicolas Torres, Luis Miguel Villada

TL;DR

The paper develops a dynamically consistent finite volume scheme with a semi-implicit NSFD discretization for a four-PDE plus one-ODE model of Alzheimer's disease that couples Aβ monomers and oligomers with microglia and interleukins. It proves existence and convergence to admissible weak solutions while preserving the spatially homogeneous dynamics, positivity, boundedness, equilibria, and disease-free stability. The approach blends traditional FV spatial discretization with NSFD treatment of reaction terms to ensure structure-preserving properties and dynamic consistency with the SH model. Numerical experiments demonstrate chemotaxis-driven microglial responses and the formation of Turing patterns, validating robustness across geometries and time steps and highlighting potential for higher-order extensions.

Abstract

A model of progression of Alzheimer's disease (AD) incorporating the interactions of A$β$-monomers, oligomers, microglial cells and interleukins with neurons is considered. The resulting convection-diffusion-reaction system consists of four partial differential equations (PDEs) and one ordinary differential equation (ODE). We develop a finite volume (FV) scheme for this system, together with non-negativity and a priori bounds for the discrete solution, so that we establish the existence of a discrete solution to the FV scheme. It is shown that the scheme converges to an admissible weak solution of the model. The reaction terms of the system are discretized using a semi-implicit strategy that coincides with a nonstandard discretization of the spatially homogeneous (SH) model. This construction enables us to prove that the FV scheme is dynamically consistent with respect to the spatially homogeneous version of the model. Finally, numerical experiments are presented to illustrate the model and to assess the behavior of the FV scheme.

Dynamically consistent finite volume scheme for a bimonomeric simplified model with inflammation processes for Alzheimer's disease

TL;DR

The paper develops a dynamically consistent finite volume scheme with a semi-implicit NSFD discretization for a four-PDE plus one-ODE model of Alzheimer's disease that couples Aβ monomers and oligomers with microglia and interleukins. It proves existence and convergence to admissible weak solutions while preserving the spatially homogeneous dynamics, positivity, boundedness, equilibria, and disease-free stability. The approach blends traditional FV spatial discretization with NSFD treatment of reaction terms to ensure structure-preserving properties and dynamic consistency with the SH model. Numerical experiments demonstrate chemotaxis-driven microglial responses and the formation of Turing patterns, validating robustness across geometries and time steps and highlighting potential for higher-order extensions.

Abstract

A model of progression of Alzheimer's disease (AD) incorporating the interactions of A-monomers, oligomers, microglial cells and interleukins with neurons is considered. The resulting convection-diffusion-reaction system consists of four partial differential equations (PDEs) and one ordinary differential equation (ODE). We develop a finite volume (FV) scheme for this system, together with non-negativity and a priori bounds for the discrete solution, so that we establish the existence of a discrete solution to the FV scheme. It is shown that the scheme converges to an admissible weak solution of the model. The reaction terms of the system are discretized using a semi-implicit strategy that coincides with a nonstandard discretization of the spatially homogeneous (SH) model. This construction enables us to prove that the FV scheme is dynamically consistent with respect to the spatially homogeneous version of the model. Finally, numerical experiments are presented to illustrate the model and to assess the behavior of the FV scheme.

Paper Structure

This paper contains 25 sections, 15 theorems, 128 equations, 6 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Scope
  3. Outline of the paper
  4. Some properties of the continuous model
  5. Dynamics of the SH model
  6. Admissible weak solutions
  7. Finite Volume Discretization
  8. Space and time discretization meshes
  9. The finite volume scheme
  10. Existence of a solution for the finite volume scheme
  11. Discrete estimates
  12. Convergence
  13. Compactness argument
  14. Convergence analysis
  15. Convergence of the discrete plaque oligomers equation
  16. ...and 10 more sections

Key Result

Proposition 2.1

Let us assume that $\hat{m}\geq \frac{\lambda_M}{\sigma}$ and consider the rectangular region $\mathcal{R} = \prod_{j=1}^{5}[0,\beta_j]\subset \mathbb{R}^5$, where $\beta_1 = \frac{r_1}{\tau_0+\gamma_{\min}}(\frac{\tau_S \tau_1 \hat{m}}{\tau_2 \tau_3 d})^2$, $\beta_2 = \frac{\gamma_{\max} r_1}{\tau_

Figures (6)

  • Figure 1: Control volumes, centers and diamonds (in dashed lines).
  • Figure 2: (a) Brain-shaped domain and reference mesh with $2658$ triangles. (b) Circular domain and reference mesh with $2358$ triangles.
  • Figure 3: Numerical solutions for the five populations $u_i(t)$, $i=1,\dots,5$ for the SH model \ref{['eq:modelhomo']}, with $0\leq t\leq 200$ computed with NSFD (first column) and Euler (second column) numerical schemes with step sizes $\Delta t\in \{0.5,1.3,2\}$.
  • Figure 4: Numerical solutions of the concentrations of $A\beta$-oligomers $u_1(\boldsymbol{x},t)$ (first row) and microglial cells $u_4(\boldsymbol{x},t)$ (second row) for model \ref{['eq:model']} over the time interval $0 \leq t \leq 50$. Snapshots are shown at $t = 0$ (initial condition), $t = 1$, $t = 5$, $t = 10$, and $t = 50$.
  • Figure 5: Numerical solutions of the concentrations of $A\beta$-oligomers $u_1(\boldsymbol{x},t)$ (first row), microglial cells $u_4(\boldsymbol{x},t)$ (second row), and interleukins $u_5(\boldsymbol{x},t)$ (third row) for model \ref{['eq:model']} over the time interval $0 \leq t \leq 2000$. Snapshots are shown at $t = 0$ (initial condition), $t = 100$, $t = 500$, $t = 1000$, and $t = 2000$.
  • ...and 1 more figures

Theorems & Definitions (31)

  • Proposition 2.1
  • proof
  • Proposition 2.2
  • Proposition 2.3
  • Definition 2.4
  • Definition 3.1
  • Remark 1
  • Lemma 3.2: Maximum principle
  • proof
  • Lemma 3.3: Existence of discrete truncated problem
  • ...and 21 more