[Paper Review] A mathematical model of atherosclerosis development in thin blood vessels and its asymptotic approximation

Taras Mel’nyk|arXiv (Cornell University)|Jan 1, 2017

Mathematical Biology Tumor Growth41 references2 citations

TL;DR

This paper proposes a refined mathematical model of atherosclerosis in thin blood vessels that incorporates recent experimental insights into macrophage heterogeneity. Using asymptotic analysis in thin domains, it justifies replacing the complex 3D reaction-diffusion system with a simplified 2D limit model, achieving high accuracy controlled by a small dimensionless parameter ε, thus enabling efficient simulation and analysis of disease progression.

ABSTRACT

Some existing models of the atherosclerosis development are discussed and a new improved mathematical model, which takes into account new experimental results about diverse roles of macrophages in atherosclerosis, is proposed. Using technic of upper and lower solutions, the existence and uniqueness of its positive solution are justified. After the nondimensionalisation, small parameters are found. Then asymptotic approximation for the solution is constructed and justified with the help of asymptotic methods for boundary-value problems in thin domains. The results argue for the possibility to replace the complex $3D$ (dimensional) mathematical model with the corresponding simpler $2D$ model with sufficient accuracy measured by these small parameters.

Motivation & Objective

• To develop a mathematically rigorous model of atherosclerosis that accounts for the dual roles of macrophages (pro- and anti-inflammatory) based on recent experimental findings.
• To address the limitations of existing models, which often oversimplify cellular dynamics or fail to incorporate key biological feedback mechanisms.
• To justify the use of a simplified 2D model in place of the original 3D problem by rigorously analyzing the asymptotic behavior as the intima thickness (ε) tends to zero.
• To establish existence and uniqueness of the positive solution for the proposed model using the upper-lower solution technique.
• To provide error estimates between the exact solution and the asymptotic approximation, ensuring the validity and accuracy of the simplified model.

Proposed method

• Formulates a 3D reaction-diffusion system with 11 coupled partial differential equations describing LDL transport, monocyte recruitment, macrophage differentiation, foam cell formation, and cytokine signaling.
• Applies nondimensionalization to identify small parameters (ε) related to the geometric thinness of the intima and the relative scales of diffusion and reaction rates.
• Employs the method of upper and lower solutions to prove existence and uniqueness of the positive solution for the initial-boundary value problem.
• Constructs a formal asymptotic approximation (Rε) for the solution in the thin domain as ε → 0, based on boundary layer and inner/outer expansion techniques.
• Justifies the asymptotic approximation by deriving error estimates between the exact solution and the approximating function, proving convergence in appropriate norms.
• Reduces the complex 3D problem in a thin tube domain to a simpler 2D limit problem on a rectangle, with the solution differing from the exact one by O(ε) in the L∞-norm.

Experimental results

Research questions

• RQ1 Can a 3D reaction-diffusion model of atherosclerosis be rigorously approximated by a simpler 2D model with controlled error as the intima thickness tends to zero?
• RQ2 How do the diverse functional roles of macrophages—particularly pro-inflammatory M1 and anti-inflammatory M2 subtypes—impact the stability and dynamics of the atherosclerotic process?
• RQ3 What is the long-term behavior of the solution as t → +∞, and does it converge to a steady state depending on the biochemical parameters?
• RQ4 How do the parameters of the limit problem influence the rate and potential blow-up of atherosclerotic plaque development?
• RQ5 Can free boundary problems be used to model the evolving intimal thickness and endothelial damage area, and how would this extend the current model?

Key findings

• The existence and uniqueness of a positive solution for the proposed 3D reaction-diffusion system are rigorously proven using the upper-lower solution method.
• After nondimensionalization, small parameters ε (representing the non-dimensional thickness of the intima) are identified, enabling asymptotic analysis.
• An asymptotic approximation Rε is constructed for the solution of the thin-domain problem, with the leading-order term derived from a limit problem on a 2D rectangle.
• The error between the exact solution and the asymptotic approximation is estimated as O(ε) in the L∞-norm, justifying the use of the 2D model with high accuracy.
• The 3D problem in the thin domain Cε is shown to be asymptotically equivalent to the 2D limit problem (4.13), allowing significant simplification of numerical and analytical studies.
• The results imply that the complex 3D model can be replaced by the simpler 2D model with an error bounded by ε, which is small in realistic biological scenarios.