Theoretical Models of Microvascular Oxygen Transport to Tissue

Abstract

To improve understanding of microvascular O 2 transport, theoretical modeling has been pursued for many years. The large number of studies in this area attests to the complexities (biochemical, structural, hemodynamic) involved. This article focuses on theoretical studies from the last two decades and, in particular, on models of O 2 transport to tissue by discrete microvessels. A brief discussion of intravascular O 2 transport is first given, highlighting the physiological importance of intravascular resistance to blood-tissue O 2 transfer. This is followed by a description of the Krogh tissue cylinder model of O 2 transport by a single capillary, which is shown to remain relevant in modified forms that relax many of the original biophysical assumptions. However, there are many geometric and hemodynamic complexities that require the consideration of microvascular arrays and networks. Multi-vessel models are discussed which have shown the physiological importance of heterogeneities in vessel spacing, O 2 supply, red blood cell flow path, as well as interactions between capillaries and arterioles. These realistic models require sophisticated methods for solving the governing partial differential equations, and a range of solution techniques are described. Finally, the issue of experimental validation of microvascular O 2 delivery models is discussed, and new directions in O 2 transport modeling are outlined.

Modelling capillary oxygen supply capacity in mixed muscles: Capillary domains revisited

“…Although experimental studies of muscle tissue capillarity that are based on Voronoi polygons have revealed intricate details about the regulatory process of microvascular remodelling, there is yet to be a 3-dimensional theoretical exploration of capillary supply regions and their correlation with Voronoi polygons (Goldman, 2008). In contrast, such correlations have been explored in two dimensions only (e.g.…”

Image-based modelling of skeletal muscle oxygenation

“…The theory of oxygen transport to tissue was also presented and reviewed by Popel [10]. Jain [128,129] reviewed the transport of molecules (including oxygen) in the tumour interstitium and across the tumour vasculature, while [130] reviewed theoretical models of oxygen transport to tissue. In the review of [131] models of tumour oxygenation are discussed, which are similar to oxygenation of other soft tissues.…”

“…A number of mathematical models have been employed to investigate the influence of different factors on the permeability coefficient, but no uniform formula has been proposed to date. For more information see the review of Goldman [130].…”

“…P 50 is usually found to be 0.5-1.0 mmHg [130] and can be determined experimentally by methods described in [160]. For skeletal muscle, Al-Shammari and co-workers [141] took into account the fact that M 0 can differ between different muscle fibre types (I, IIa and IIb).…”

“…The permeability of the wall to oxygen molecules can be described by a mass transfer coefficient k, which depends on the blood haematocrit, the blood flow rate and the type of vessel, i.e. capillary, arteriole or venule, and their radius [130]. This boundary condition is given by À n v Á ðDrPÞ ¼ kðP 0 À PÞ, where n v is the unit normal vector to the vessel boundary, P and P 0 are the oxygen partial pressures at time t and at the onset (t ¼ 0), respectively, and D is the diffusion coefficient.…”