This dissertation presents a mathematical analysis of oxygen convection, diffusion, and consumption in a uniform and multi-capillary muscle tissue region. Capillaries are tiny vessels connecting arterioles with venules and forming networks throughout the body. The diffusion of the substrate, such as oxygen, from the microcirculation through capillaries and its consumption by tissue cells are basic in human physiology. Blockage or shortage of blood in one or more capillaries due to thrombosis or systemic hypoperfusion could lead to pathological conditions such as stroke. However, the transient process of capillary-tissue oxygen transport is poorly understood. The object of this thesis is to develop a mathematical and computationally efficient model that estimates transient oxygen concentration in three-dimensional striated muscles (e.g. cardiac muscle), which is a four-dimensional unsteady convection-diffusion-consumption moving boundary problem. In particular, we aim to simulate the consequential stages of pathological conditions such as hyperoxia or hypoxia, by determining oxygen concentration levels and its transport in order to better understand and predict adverse health effects. Our computational method is based on a compartmental modeling concept. It is simple, efficient and can be applied to boundaries of arbitrary shape. In Chapter 1 the problem is introduced. In Chapter 2 the discrete compartmental modeling is described. In Chapter 3 the discrete compartmental method is developed for 1D and 2D problems. In Chapter 4 the method is extended to 3D problems and is used to simulate the development and the recovery process of anoxia. In Chapter 5 theoretical analysis of the method is given. Chapter 6 contains discussions, conclusions, and possible future work.
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