Article

# Oxygen diffusion in large single-celled organisms

Bulletin of Mathematical Biology (Impact Factor: 1.39). 01/1969; 31(2):327-340. DOI: 10.1007/BF02477010

**ABSTRACT**

The functional relationship between the oxygen uptake rate of a spherical, single cell organism and the external oxygen tension

is shown to be related to the dependence of the specific oxygen consumption rate, that is, the consumption rate of an infinitesimal

volume element of cellular material, on the external oxygen tension. Analytical solutions of the governing steady state diffusion

equation are obtained by dividing the system into three regions, an inner region of the sphere in which oxygen consumption

rate depends upon oxygen tension, an outer region of the sphere in which oxygen consumption rate is constant (independent

of oxygen tension), a nonconsuming membrane over the sphere that offers only resistance to oxygen diffusion, and an infinite

region outside the sphere and membrane supplying oxygen to the system. The solutions show the oxygen tension as a function

of position inside the spherical cell for a variety of system parameters.

is shown to be related to the dependence of the specific oxygen consumption rate, that is, the consumption rate of an infinitesimal

volume element of cellular material, on the external oxygen tension. Analytical solutions of the governing steady state diffusion

equation are obtained by dividing the system into three regions, an inner region of the sphere in which oxygen consumption

rate depends upon oxygen tension, an outer region of the sphere in which oxygen consumption rate is constant (independent

of oxygen tension), a nonconsuming membrane over the sphere that offers only resistance to oxygen diffusion, and an infinite

region outside the sphere and membrane supplying oxygen to the system. The solutions show the oxygen tension as a function

of position inside the spherical cell for a variety of system parameters.

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**ABSTRACT:**Oxygen diffusion in a spherical cell with Michaelis-Menten oxygen uptake kinetics is re-examined and the results of a recent paper by Lin corrected. An extension of this model to include external diffusion is made and its effect is shown to be significant. A model which attempts to model the nucleus as a central sphere which does not consume any oxygen is also investigated. Finally, a perturbation solution for a small Michaelis constant is developed. - [Show abstract] [Hide abstract]

**ABSTRACT:**Diffusion problem with variabale diffusion coefficient in a spherical biological system is investigated. Also included in this study is the biological reaction of the Michaelis-Menten type. The problem formulated consists of a highly nonlinear differential equation which, however, can be efficiently solved by the orthogonal collocation method on a digital computer. The effects of the dimensionless governing parameters on the transient and steady state concentration responses are parametrically examined for the diffusion system with and without biological reaction. - [Show abstract] [Hide abstract]

**ABSTRACT:**A combined oxygen consumption kinetics, used previously by other investigators, is utilized to calculate oxygen transport into heterogeneous tissue. When the oxygen partial pressure in the tissue decreases below a critical value, the oxygen respiration rate inside the cell decreases linearly with the partial pressure. Theoretical calculations of oxygen penetration depths into tissue show that the combined oxygen consumption kinetics gives considerably deeper penetration than zero-order consumption. The increase of the penetration depth is of the order of half the intercapillary distance in tissue. The results show that the form of the assumed oxygen consumption model is important in determining the depth of penetration into tissue, particularly in hypoxia.

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