Green's function methods for analysis of oxygen delivery to tissue by microvascular networks

Delivery of oxygen to tissue is an essential function of the circulatory system. The distance that oxygen can diffuse into oxygen-consuming tissue is small, and so tissue oxygenation is critically dependent on the spatial arrangement of microvessels in tissue. Theoretical methods have been developed to simulate the spatial distribution of oxygen levels in tissue surrounding a network of microvessels. Here, numerical methods based on a Green's function approach are presented, for realistic three-dimensional network geometries derived from observations of skeletal muscle, brain, and tumor tissues. Relative to finite-difference methods, the Green's function approach reduces the number of unknowns in the numerical formulation and allows rapid computations even for complex vascular geometries. Generally, the boundary conditions on the exterior of the computational domain are not known. Imposition of a no-flux boundary condition can lead to exaggerated estimates of the extent of hypoxia in the tissue region. A new version of the method is described that avoids this problem and can be applied to arbitrarily shaped tissue domains.