New mathematical models of inert gas transport through biological tissue in hyperbaric environments

Doctoral Thesis


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University of Cape Town

The thesis is concerned with a fundamental mathematical analysis of inert gas transport through biological tissue at a raised ambient partial pressure. Three basic time-scales of transport in tissue are defined and their relationship examined and compared with existing models, which shown to be usually inadequate in one or more ways. As a result three new mathematical models are proposed and solved both asymptotically and numerically. The first is applied to experimental data for non-perfused tissue which yields an improved value of the intracellular diffusion coefficient for nitrogen. An expression is also derived which should be useful in evaluating this constant and the volume fraction of extracellular fluid. The second embraces a number of current models and is applicable to perfused tissue. It should be useful in interpreting inert gas uptake curves. The model is applied to experimental data, and a source of possible error is discovered in using experimental non-asymptotic time constants. The third is a model which claims to resolve the controversy between the diffusion and perfusion theories of gas transport in tissue. The result is that in the large, diffusion is more important than perfusion, except in muscle tissue where they interact. Three different methods of numerical inversion of the Laplace Transform are compared and one is shown to be the most useful for solving gas uptake problems. The main result of the thesis is a contribution to the establishment of a mathematical basis for gas transport in various situations in the biological sphere.