Analysis of countercurrent exchange between paired blood vessels.
AuthorFierro Murga, Leobardo.
Committee ChairSecomb, Timothy W.
MetadataShow full item record
PublisherThe University of Arizona.
RightsCopyright © is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author.
AbstractThroughout much of the blood circulatory system, supply vessels (arteries and arterioles) are situated adjacent to corresponding draining vessels (veins and venules), which flow in the opposite direction. In this dissertation, mathematical models are developed to describe diffusive exchange of heat, oxygen and inert gases between such paired countercurrent blood vessels and surrounding tissue. In preliminary analyses, exchange between a single vessel and surrounding tissue is considered. The concept of equilibration length is developed. Then a well-known solution for two-dimensional diffusion between two vessels situated in an infinite domain is presented. This provides a basis for developing semianalytic solutions for two vessels situated in a cylindrical tissue region with Dirichlet or zero-flux conditions at the outer boundary. A general approach is then developed for obtaining semianalytic solutions for domains with non-circular cross-sections, and applied to the case of a rectangular domain. The governing equations for paired blood vessels are then solved to obtain the axial variation of temperature or concentration for a variety of cases, including Dirichlet and zero-flux boundary conditions, with and without deposition or consumption of heat or gas. For the Dirichlet case, the equilibration length is compared to that for a single vessel, showing that equilibrium is achieved more rapidly when a single vessel is replaced by two vessels with the same diameter as the single vessel. For the zero-flux case, particular solutions to the full three-dimensional diffusion equation in the tissue are obtained from the two-dimensional solutions. The total transport (convective and diffusive) in the axial direction is evaluated, with and without consumption/deposition, and the results are interpreted in terms of an enhanced diffusivity. Finally, the complementary roles of convection and diffusion in mass and heat transport in the axial direction are considered. It is shown that as vessel diameter decreases, countercurrent exchange eventually results in a reduction of convective transport. Axial diffusion becomes significant at approximately the same range of diameters. This finding is interpreted in terms of the efficiency by which a branching network can transport heat and mass to its extremities.
Degree ProgramApplied Mathematics