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Many pathologies are related to hemodynamic changes occurring at the microvascular level, where small vessels pierce the tissue, perfusing it with blood. Since there is a large number of vessels of small caliber, it is impractical to model the fluid flow through each one of them separately, as it is done in the case of large arteries using, for example, the Navier–Stokes equations. As an alternative, tissue perfusion is modeled here via three‐dimensional (3D) partial differential equations (PDEs) for fluid flow through deformable porous media, where blood vessels are modeled as pores within a deformable solid representing the tissue. Since it is known that the local perfusion is related to the systemic features of surrounding blood circulation, we couple the PDE system with a zero‐dimensional (0D) lumped circuit model, obtained by the analogy between fluid flows in hydraulic networks and current flowing in electrical circuits. An important feature in this multiscale 3D–0D coupling is the specification of interface conditions between the 3D and the 0D parts of the system. In this article, we focus on two types of interface conditions driven by physical considerations, and compare the behavior of the solutions for the two different scenarios.
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This content will become publicly available on March 31, 2026
Analysis of a Multiscale Interface Problem Based on the Coupling of Partial and Ordinary Differential Equations to Model Tissue Perfusion
In biomechanics, local phenomena, such as tissue perfusion, are strictly related to the global features of the surrounding blood circulation. In this paper, we propose a heterogeneous model where a local, accurate, 3D description of tissue perfusion by means of fluid flows through deformable porous media equations is coupled with a systemic, 0D, lumped model of the remainder of the circulation, where the fluid flow through a vascular network is described via its analog with a current flowing through an electric circuit. This represents a multiscale strategy, which couples an initial boundary value problem to be used in a specific tissue region with an initial value problem in the surrounding circulatory system. This PDE/ODE coupling leads to interface conditions enforcing the continuity of mass and the balance of stresses across models at different scales, and careful consideration is taken to address this interface mismatch. The resulting system involves PDEs of mixed type with interface conditions depending on nonlinear ODEs. A new result on local existence of solutions for this multiscale interface coupling is provided in this article.
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- PAR ID:
- 10639180
- Publisher / Repository:
- SIAM
- Date Published:
- Journal Name:
- Multiscale Modeling & Simulation
- Volume:
- 23
- Issue:
- 1
- ISSN:
- 1540-3459
- Page Range / eLocation ID:
- 1 to 24
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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