\documentstyle[12pt]{article} \textwidth 16cm \textheight 24.7cm \topmargin -1.45cm \oddsidemargin -0.2cm \evensidemargin -0.2cm \title{A numerical method for studying the oxygen supply to tissue in hypertrophy} \author{${\bf G.\; Simeonov^1\;\;}$ and $\;\;{\bf R.\; Ivanova^2}$} \date{} \pagestyle{empty} \begin{document} \maketitle $$^1{\it Department\; of\; Fluid\; Mechanics}\; and\;\> ^2{\it Department\; of\; Biomechanics}$$ $${\it of\; Systems\; and\; Processes,\; Institute\; of\; Mechanics,\; Acad.\; G.\; Bontchev\; St.,\; Bl.\; 4}$$ $${\it Bulgarian\; Academy\; of\; Sciences,\; 1113\; Sofia,\; Bulgaria}$$ \par \par The oxygen tension field in the muscle $\>$ tissue in hypertrophy is characterized by presence of low level zones. The modelling of the oxygen transport-consumption processes under these conditions asks special attention to be paid to the adequate presentation of the sink term in the diffusion equation as well as to its linearization when respective numerical method is designed. An essential requirement to both is negative oxygen tension to be avoided by any means. \par A numerical method for studying the oxygen supply to tissue which satisfies this requirement is developed. The used model is based on Krogh tissue cylinder as a microcirculatory unit. It consists of two differential equations. The first one is two-dimensional diffusion equation for oxygen tension in the tissue, $P$, with a sink term which corresponds to Michaelis-Menten kinetics of oxygen consumption and non-linearly depends on $P$. The second is one-dimensional equation of convective oxygen transport through the capillary described in terms of averaged over its cross section blood oxygen concentration. The last is a nonlinear function of the blood oxygen tension, $\overline{P}$. Both equations conjugate at the capillary wall. \par The proposed numerical procedure is based on the "embeding" the steady-state problem under consideration into an unsteady one and on a modification of the alternating direction implicit method due to Samarskii. Because of the nonlinearity of the consumption rate term and necessity to conjugate both solutions, $P$ and $\overline{P}$, at each "time" step an iterative procedure is carried out. The computational process is followed up until a steady-state solution is reached. \par Several linearizations of the consumption rate term were tested under hard physiological conditions - hypoxia in midwall region of hypertrophied rat heart with reduced blood flow. The test computational conditions were rather severe, too. Three linearizations were examined more systematically with respect to the positivity of the solution. The linearization substituting the curve of consumption rate dependence on $P$ by the secant which connects the considered point on it with the zero oxygen tension point is recognized to be the most appropriate. \par Samples of the obtained oxygen tension distribution in the blood as well as along the outer edge of the Krogh tissue cylinder are presented. \end{document} 