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\title{A numerical method for studying the oxygen supply to tissue in
hypertrophy}
\author{${\bf G.\; Simeonov^1\;\;}$ and $\;\;{\bf R.\;
Ivanova^2}$} \date{}
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\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}