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

TL;DR

This paper develops and analyzes a hierarchy of blood vessel models, establishing convergence from detailed 3D models to simplified 1D models, with rigorous proofs and explicit convergence rates, to better understand blood perfusion modeling.

Contribution

It introduces a novel 3D-1D Darcy--Poiseuille model with boundary conditions, proves its well-posedness, and establishes convergence to a 1D Green's function model as vessel radius shrinks.

Findings

The 3D-1D model is well-posed.

The 1D model converges to the 3D-1D model with rate proportional to .5||.5|log|.

Explicit Green's function expression for exterior blood pressure.

Abstract

We propose and analyze a family of models describing blood perfusion through a tissue surrounding a thin blood vessel. Our goal is to rigorously establish convergence results among 3D-3D Darcy--Stokes, 3D-1D Darcy--Poiseuille, and 1D Green's function methods commonly used to model this process. In Part I, we propose a 3D-1D Darcy--Poiseuille system where the coupling across the permeable vessel surface involves an angle-averaged Neumann boundary condition coupled with a geometrically constrained Robin boundary condition. We show that this model is well-posed and moreover limits to a 1D Green's function model as the maximum vessel radius $\epsilon\to 0$. In the 1D model, the exterior blood pressure is given by an explicit Green's function expression involving the interior blood pressure. The interior pressure satisfies a novel 1D integrodifferential equation in which the integral term incorporates the effects of the exterior pressure and the vessel geometry. Much of this paper is devoted to analyzing this integrodifferential equation. Using the \emph{a priori} bounds obtained here, we show that the solution to the 1D model converges to the 3D-1D solution with a rate proportional to $\epsilon^{1/2}|\log\epsilon|$. In Part II [Ohm \& Strikwerda, arXiv preprint July 2025], we rely on the 1D estimates to show that both the 1D and 3D-1D models converge to a coupled 3D-3D Darcy-Stokes system as $\epsilon\to 0$, thereby establishing a convergence chain among all hierarchy levels.