A hierarchy of blood vessel models, Part II: 3D-3D to 3D-1D and 1D

TL;DR

This paper develops a rigorous mathematical framework linking detailed 3D blood flow models with simplified 1D models, providing convergence rates and analysis for blood perfusion modeling around small vessels.

Contribution

It establishes a convergence analysis between complex 3D-3D models and simplified 1D models for blood flow, including explicit rates and handling of vanishing vessel radii.

Findings

Derived convergence rate proportional to ε^{1/6} |log ε|

Proved estimates for 1D integrodifferential models

Analyzed the limit as vessel radius approaches zero

Abstract

We propose and analyze a hierarchy of three models of blood perfusion through a tissue surrounding a thin arteriole or venule. Our goal is to rigorously link 3D-3D Darcy--Stokes, 3D-1D Darcy--Poiseuille, and 1D Green's function methods commonly used to model this process. Here in Part II, we consider the most detailed level, a 3D-3D Darcy-Stokes system coupled across the permeable vessel surface by mass conservation and pressure/stress balance conditions. We derive a convergence result between the 3D-3D model and both the 3D-1D Darcy--Poiseuille model and 1D Green's function model proposed in Part I [Ohm \& Strikwerda, arXiv preprint July 2025] at a rate proportional to $\epsilon^{1/6}|\log\epsilon|$, where $\epsilon$ is the maximum vessel radius. The rate is limited by the inclusion of a degenerate endpoint where the vessel radius vanishes, i.e. becomes indistinguishable from a capillary. Key to our proof are \emph{a priori} estimates for the 1D integrodifferential model obtained in Part I.