Oxygen Diffusion from Capillary Layers with Concurrent Flow

August Krogh's theory of muscle microvascular control and oxygen delivery: a paradigm shift based on new data

August Krogh twice won the prestigious international Steegen Prize, for nitrogen metabolism (1906) and overturning the concept of active transport of gases across the pulmonary epithelium (1910). Despite this, at the beginning of 1920, the consummate experimentalist was relatively unknown worldwide and even among his own University of Copenhagen faculty. But, in early 1919, he had submitted three papers to Dr Langley, then editor of The Journal of Physiology in England. These papers coalesced anatomical observations of skeletal muscle capillary numbers with O₂ diffusion theory to propose a novel active role for capillaries that explained the prodigious increase in blood-muscle O₂ flux from rest to exercise. Despite his own appraisal of the first two papers as "rather dull" to his friend, the eminent Cambridge respiratory physiologist, Joseph Barcroft, Krogh believed that the third one, dealing with O₂ supply and capillary regulation, was"interesting". These papers, which won Krogh an unopposed Nobel Prize for Physiology or Medicine in 1920, form the foundation for this review. They single-handedly transformed the role of capillaries from passive conduit and exchange vessels, functioning at the mercy of their upstream arterioles, into independent contractile units that were predominantly closed at rest and opened actively during muscle contractions in a process he termed 'capillary recruitment'. Herein we examine Krogh's findings and some of the experimental difficulties he faced. In particular, the boundary conditions selected for his model (e.g. heavily anaesthetized animals, negligible intramyocyte O₂ partial pressure, binary open-closed capillary function) have not withstood the test of time. Subsequently, we update the reader with intervening discoveries that underpin our current understanding of muscle microcirculatory control and place a retrospectroscope on Krogh's discoveries. The perspective is presented that the imprimatur of the Nobel Prize, in this instance, may have led scientists to discount compelling evidence. Much as he and Marie Krogh demonstrated that active transport of gases across the blood-gas barrier was unnecessary in the lung, capillaries in skeletal muscle do not open and close spontaneously or actively, nor is this necessary to account for the increase in blood-muscle O₂ flux during exercise. Thus, a contemporary model of capillary function features most muscle capillaries supporting blood flow at rest, and, rather than capillaries actively vasodilating from rest to exercise, increased blood-myocyte O₂ flux occurs predominantly via elevating red blood cell and plasma flux in already flowing capillaries. Krogh is lauded for his brilliance as an experimentalist and for raising scientific questions that led to fertile avenues of investigation, including the study of microvascular function.

Calculation of oxygen pressures in tissue with anisotropic capillary orientation. II. Coupling of two-dimensional planes

The mathematical descriptions of oxygen transport in terms of two-dimensional diffusion in planes perpendicular to the capillaries can be coupled for an ensemble of planes to yield a three-dimensional description. In tissue with a distinct orientation of the oxygen supplying structures, the capillaries, this leads to a description of oxygen pressure in a whole tissue volume. Muscle tissue is an example of such tissue. The method allows calculation of oxygen pressure (pO2) at any location in the volume. Tissue volume oxygen status can be assessed by constructing a pO2 histogram, gathering pO2's at a large number of locations in the tissue.

A Green's function method for analysis of oxygen delivery to tissue by microvascular networks

A theoretical model is formulated for analyzing oxygen delivery from an arbitrary network configuration of cylindrical microvessels to a finite region of tissue. In contrast to models based on the classical Krogh cylinder approach, this model requires no a priori assumptions concerning the extent of the tissue region supplied with oxygen by each vessel segment. Steady-state conditions are assumed, and oxygen consumption in the tissue is assumed to be uniform. The nonlinear dissociation characteristics of oxyhemoglobin are taken into account. A computationally efficient Green's function approach is used, in which the tissue oxygen field is expressed in terms of the distribution of source strengths along each segment. The utility of the model is illustrated by analyses of oxygen delivery to a cuboidal tissue region by a single segment and by a six-segment network. It is found that the fractional contribution of the proximal segments to total oxygen delivery increases with decreasing flow rate and metabolic rate.

Buffering of muscle tissue PO2 levels by the superposition of the oxygen field from many capillaries

High resolution measurement in both exercised skeletal and cardiac tissue made radially outward from capillaries and longitudinally parallel to capillaries by Gayeski and Honig (1986, 1986a,b) and Honig and Gayeski (1987) indicate shallow variation of tissue PO2 and the absence of strong causal relation between the PO2 at a point and the proximity of that point to the nearest active capillary. Proposed as a model for the analysis of this tissue PO2 distribution, so contrary to the expectations of Krogh type models, are a class of multicellular tissue cylinder models. Each cylinder is penetrated by many parallel capillaries. In order to better represent the natural irregularities of the skeletal and cardiac tissue both with regard to radial placement and the stagger of the capillary inlets, the following types of models both with and without yoglobin are examined: regular square arrays where the capillary PO2 levels are random, uniform capillary PO2 levels but random capillary positions, and those with both the capillary PO2 levels and the positions are random. The results of the model calculations show that the superposition of the oxygen diffusion fields of all the capillaries produce a tissue PO2 distribution with the properties: (1) lower tissue PO2 levels than those predicted by Krogh theory, (2) significant non-local contributions to the PO2 at a point in the tissue which greatly reduces this correlation between PO2 at a point and its proximity to an active capillary, (3) shallow transcellular PO2 variation.

Theory of oxygen transport to tissue

This review focuses on the theory of oxygen transport to tissue and presents the state of the art in mathematical modeling of transport phenomena. Results obtained with the classic Krogh tissue-cylinder model and recent advances in mathematical modeling of hemoglobin-oxygen kinetics, the role of hemoglobin and myoglobin in facilitating oxygen diffusion, and the role of morphologic and hemodynamic heterogeneities in oxygen transport in the microcirculation are critically discussed. Mathematical models simulate different parts of the pathway of oxygen molecules from the red blood cell, through the plasma, the endothelial cell, other elements of the vascular wall, and the extra- and intracellular space. Special attention in the review is devoted to intracapillary transport, which has been the subject of intensive theoretical research in the last decade. Models of pre- and postcapillary oxygen transport are also discussed. Applications to specific organs and tissues are reviewed, including skeletal muscle, myocardium, brain, lungs, arterial wall, and skin. Unresolved problems and major gaps in our knowledge of the mechanisms of oxygen transport are identified.

Effect of Heterogeneous Oxygen Delivery on the Oxygen Distribution in Skeletal Muscle

Calculations of the oxygen distribution in resting and contracting skeletal muscle are presented, based on a mathematical model and experimental data obtained on the hamster cremaster muscle [Klitzman et al., Microvasc. Res. 25:108-131 (1983)]. The model considers a slab of tissue penetrated by a regular square array of capillaries with concurrent flow. The intracapillary resistance to oxygen transport is neglected. The capillary red blood cell flux and capillary inlet oxygen tension are assumed random variables following certain probability distributions. The sensitivity of the tissue and intracapillary PO2 to variations of these probability distributions are investigated. The mean tissue PO2 decreases as the dispersion of the random variables increases, provided that their mean values remain constant. Hypoxic regions appear gradually, especially in the case of contracting muscle, as the dispersion increases. The effect of the number of capillaries in the sample on the resultant distribution of oxygen is studied systematically. The calculated tissue PO2 histograms are compared with previously reported PO2 distributions obtained experimentally for resting and contracting skeletal muscle.

Oxygen transport in resting and contracting hamster cremaster muscles: experimental and theoretical microvascular studies

Intravital microscopy of the superfused cremaster muscle was used to measure the density, diameter, length, hematocrit, red cell velocity, and red cell flux in capillaries of the pentobarbital-anesthetized hamster. Oxygen microelectrodes were used to measure oxygen tension (Po2) at a position 75-100 micrometers deep in the muscle between the venous ends of capillaries and, very importantly, at the superfusate-muscle interface. These parameters were measured in resting and contracting muscles and under three values of superfusate Po2: low (8mm Hg), medium (40 mm Hg), and high (75 mm Hg). These data were complete enough to be useful input parameters in a recently developed mathematical model of oxygen transport in exposed tissue (A. S. Popel, 1981, Math. Biosci. 55, 231-246). The model indicated that with high superfusate Po2, oxygen was supplied to the resting muscle almost exclusively from the superfusate because of the vasoconstriction and reduced blood flow. Oxygen consumption of the resting muscle was estimated to be 0.4 ml O2/100 ml tissue X min, assuming muscle oxygen consumption was uniform and independent of Po2 above 1 mm Hg. The estimated rise in oxygen consumption with exercise was four to eight times resting muscle values, which agrees with previously published data. Also, the model predicted an inlet capillary Po2 of 27 mm Hg with a low superfusate Po2, which is consistent with the few available direct measurements. The model emphasized that with measurement of the Po2 at the superfusate-tissue interface, the complex O2 transport effects of the superfusate can be accurately characterized. Measurement of this and other parameters of the model leads to a potentially useful prediction of the Po2 distribution within tissues under a variety of conditions.

Mathematical Modeling of Oxygen Transport Near a Tissue Surface: Effect of the Surface PO2

The effect of the surface oxygen tension on the oxygen-tension distribution in the tissue is considered. A mathematical formulation of the problem is presented, and the solutions are obtained numerically using a finite-element method. It is shown that the oxygen tension at the surface of the tissue may significantly affect the oxygen-tension distribution in layers of tissue situated within several intercapillary distances below the surface.