A hierarchy of blood vessel models, Part II: 3D-3D to 3D-1D and 1D
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912489448407040 |
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| author | Ohm, Laurel Strikwerda, Sarah |
| author_facet | Ohm, Laurel Strikwerda, Sarah |
| contents | 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 $ε^{1/6}|\logε|$, where $ε$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_13330 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A hierarchy of blood vessel models, Part II: 3D-3D to 3D-1D and 1D Ohm, Laurel Strikwerda, Sarah Analysis of PDEs Biological Physics Fluid Dynamics Tissues and Organs 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 $ε^{1/6}|\logε|$, where $ε$ 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. |
| title | A hierarchy of blood vessel models, Part II: 3D-3D to 3D-1D and 1D |
| topic | Analysis of PDEs Biological Physics Fluid Dynamics Tissues and Organs |
| url | https://arxiv.org/abs/2507.13330 |