Facilitated diffusion: the elasticity of oxygen supply.

It is well established that the presence of oxygen-carrying proteins such as haemoglobin can facilitate the diffusion of oxygen through a solution. In this paper, it is shown that some properties of a facilitated flow are substantially different from those of unfacilitated flux, including especially the stability of the tension at which the oxygen arrives at the end of the diffusion path. The concept of the “output resistance” of the supply is introduced, and a facilitated pathway is shown to have a lowered resistance. A role for the storage capacity of the bound oxygen reservoir is also developed; it is shown that delivery oxygen tensions are stabilized against transient changes in oxygen demand. In treating the equations of facilitated diffusion, a simplified approach is used to take account of boundary layers in the solution where deviations from oxygen-protein equilibrium are significant. A measure of the thickness and importance of such boundary layers is calculated.     

Outer retinal anoxia during dark adaptation is not a general property of mammalian retinas

The purpose of this study was to measure the intraretinal oxygen distribution across the retina under conditions, which maximise outer retinal oxygen consumption. In particular, we looked for evidence of increased oxygen delivery from the choroid and the deep retinal capillary layer, and whether or not this was sufficient to avoid the development of intraretinal anoxia. Under dark-adapted conditions the photoreceptors need additional energy, at least part of which is derived from increased oxidative metabolism. In earlier studies in the cat retina it was revealed that dark adaptation could render some regions of the outer retina anoxic. The present study of the in vivo oxygen distribution across the rat retina in light and dark found no evidence of outer retinal anoxia in the dark. This was despite a mean increase of 52.6+/-11.4% (n=7) in outer retinal oxygen consumption in the dark. The mean value for the minimum outer retinal PO(2) in the dark was 5.2+1.2 mmHg. Oxygen delivery from both the choroid and the deep retinal capillary layer increased in the dark (P<0.01, and P<0.001, respectively). It is argued that the ability of the deep capillary layer to compensate for changes in oxygen demand in the outer retina is an important element in the maintenance of homeostasis in the retina. This is in addition to the role of the deep capillary layer in supplying oxygen to the highly consuming plexiform layers within the inner retina. These findings in the rat retina also demonstrate that intraretinal anoxia in the dark, is not, as implied by earlier work in the cat, a general feature of mammalian retinas.     

Mathematical modeling of oxygen transport in skeletal muscle

Inhomogeneous perfusion of capillary beds can result in large-scale diffusion of oxygen between distant portions of an organ. The conceptual model of a single capillary supplying oxygen to a surrounding concentric cylinder of tissue is not applicable to a consideration of such processes. An entirely different approach to the modeling of oxygen transport to tissue, with specific reference to the capillary beds of skeletal muscle, is presented here. This approach is intended to replace the theoretical Krogh cylinder model of capillary-tissue oxygen transport with a much more realistic model that takes into account inhomogeneities of capillary density, blood flow velocity, and oxygen concentration inherent in the micro-vasculature. The oxygen distribution in inhomogeneously perfused skeletal muscle is analyzed mathematically by defining an averaged concentration profile that neglects the fine-scale variation from capillary to capillary.     

Dosimetry in photodynamic therapy: oxygen and the critical importance of capillary density.

Recently published results of tumor response to various photoradiation protocols in photodynamic therapy appear to contradict accepted definitions of photodynamic dose. In this report, the failure of standard dosimetry models to predict therapeutic outcome is interpreted on the basis of PDT-induced oxygen consumption in tumors with relatively low capillary densities. Calculated estimates of oxygen consumption in photodynamic therapy are combined with the Krogh cylinder model of oxygen diffusion. It is shown that, for tissue volumes in which the intercapillary spacing is less than a specific critical distance, oxygen may be considered constant and unaffected by the therapy. Under these conditions, the 1O2 delivered to a given volume of tissue is spatially uniform and proportional to the number of photons absorbed by the sensitizer. When the intercapillary spacing exceeds the critical distance, the dose of 1O2 varies with radial distance from the capillary wall. In this situation, dose may no longer be considered simply in terms of the product of the photon fluence and the sensitizer absorption coefficient. Since fractionation will increase the 1O2 dose only to cells relatively remote from the capillary wall, the analysis further suggests that fractionating the radiation dose should result in an improved therapeutic ratio for photodynamic therapy.     

Use of Capillary Blood in Measurement of Arterial Po2

In this study we investigated the possibility of obtaining accurate values of arterial Po₂ from specimens of capillary blood stored in glass capillary tubes and measured in an oxygen microelectrode. It has been shown that Po₂ measurements made on the Radiometer oxygen microelectrode are as accurate as those made on the macroelectrode and that the storage of blood is as satisfactory in glass capillary tubes as in glass syringes. The important feature in obtaining accurate values for arterial Po₂ is the choice of the capillary bed and its method of preparation for sampling. If the ear lobe is massaged with thurfyl nicotinate (Trafuril) it is possible to obtain values of Po₂ from the capillary blood which are in close agreement with arterial Po₂ in normal, hyperoxic, and shocked vasoconstricted patients.     

Mathematical analysis of oxygen concentration in a two dimensional array of capillaries

 An approach is presented for modeling transport and exchange in skeletal muscle that can be used to analyze vascular beds consisting of a large number of interacting capillaries. First the oxygen concentration is determined in a functional unit consisting of a single capillary surrounded by a region of tissue in which a flux is prescribed on the outer boundary of the region. This flux, which is a result of the interaction among all of the capillaries comprising the vascular bed, is then found by matching the concentration along the borders between adjacent units. This leads to a system of ordinary differential equations for the oxygen concentration in the capillaries coupled with a system of algebraic equations for the fluxes. The method is illustrated by obtaining the oxygen concentration within an array of capillaries for the case when each capillary has a different initial concentration and for the case when each capillary has a different flow rate. Received: 12 June 2001 / Revised version: 18 April 2002 / Published online: 17 January 2003 Key words or phrases: Skeletal muscle – Transport – Microcirculation     

Integral mechanisms of regulation of oxygen delivery and consumption in human body

With new algorithms of pattern recognition three types of regulation of oxygen delivery and consumption were studied: at normal regulation, hypo- and hyperfunction of heart. It is shown that a surplus systemic blood flow results in increase of arterio-venous shunt flow. Insufficient blood flow results in increase of arterio-venous gradient of blood oxygen content due to the increase of gradient of oxygen content on both sides of capillary wall and to activating of vasomotor function of microcirculatory arterial bed. Quantitative estimations of shunt and capillary blood flow are obtained.     

Re-evaluating the Use of Voronoi Tessellations in the Assessment of Oxygen Supply from Capillaries in Muscle

The ability to characterise capillary supply plays a key role in developing effective therapeutic interventions for numerous pathological conditions, such as capillary loss in skeletal or cardiac muscle. However, quantifying capillary supply is fraught with difficulties. Averaged measures such as capillary density or mean inter-capillary distance cannot account for the local geometry of the underlying capillary distribution, and thus can only highlight a tissue wide, global hypoxia. Detailed tissue geometry, such as muscle fibre size, has been incorporated into indices of capillary supply by considering the distribution of Voronoi tessellations generated from capillary locations in a plane perpendicular to muscle fibre orientation, implicitly assuming that each Voronoi polygon represents the area of supply of its enclosed capillary. Using a modelling framework to assess the capillary supply capacity under maximal sustainable conditions in muscle, we theoretically demonstrate that Voronoi tessellations often provide an accurate representation of the regions supplied by each capillary. However, we highlight that this use of Voronoi tessellations is inappropriate and inaccurate in the presence of extensive capillary rarefaction and pathological variations in oxygen tension of different capillaries. In such cases, oxygen flux trapping regions are developed to provide a more general representation of the capillary supply regions, in particular incorporating the additional influences of heterogeneity that are absent in the consideration of Voronoi tessellations.     

Symmetric multilayer megampere X-pinch

Raising the power of X-ray emission from an X-pinch by increasing the pinch current to the megampere level requires the corresponding increase in the initial linear mass of the load. This can be achieved by increasing either the number of wires or their diameter. In both cases, special measures should be undertaken to prevent the formation of a complicated configuration with an uncontrolled spatial structure in the region of wire crossing, because such a structure breaks the symmetry of the neck formed in the crossing region, destabilizes plasma formation, and degrades X-ray generation. To improve the symmetry of the wire crossing region, X-pinch configurations with a regular multilayer arrangement of wires in this region were proposed and implemented. The results of experiments with various symmetric X-pinch configurations on the COBRA facility at currents of ∼1MA are presented. It is shown that an X-pinch with a symmetric crossing region consisting of several layers of wires made of different materials can be successfully used in megampere facilities. The most efficient combinations of wires in symmetric multilayer X-pinches are found in which only one hot spot forms and that are characterized by a high and stable soft X-ray yield.     

Analysis of oxygen exchange between arterioles and surrounding capillary-perfused tissue.

A theoretical model is used to analyze oxygen transport in a three-dimensional tissue region containing an arteriole surrounded by an array of capillaries in planes perpendicular to the arteriole. Convective removal of oxygen from the vicinity of the arteriole by nearby capillaries is shown to increase diffusive oxygen loss from the arteriole. This effect depends on the locations of the capillaries, particularly those nearest to the arteriole. The arteriolar oxygen efflux is comparable to that predicted by a previous model which used a continuum approach, but the efflux does not increase with increasing perfusion as rapidly as predicted by the continuum model. Even a small capillary flow rate strongly influences the oxygen field surrounding the arteriole.     

A numerical model for the oxygenation of blood in lung capillaries--effect of nth order one-step kinetics of oxygen uptake by haemoglobin

A numerical model is described for the oxygenation of blood in lung capillaries by considering the transport mechanisms of molecular diffusion, convection and the facilitated diffusion due to the presence of haemoglobin. In order to represent the oxygen dissociation curve accurately in the model, the nth order one-step kinetics of oxygen uptake by haemoglobin has been used. The resulting system of coupled, non-linear partial differential equations is solved numerically. It is shown that the blood is required to traverse a larger distance in the capillary before becoming fully oxygenated with nth order one-step kinetics in comparison to first-order one-step kinetics.     

Estimation of capillary density in human skeletal muscle based on maximal oxygen consumption rates

A previously developed Krogh-type theoretical model was used to estimate capillary density in human skeletal muscle based on published measurements of oxygen consumption, arterial partial pressure of oxygen, and blood flow during maximal exercise. The model assumes that oxygen consumption in maximal exercise is limited by the ability of capillaries to deliver oxygen to tissue and is therefore strongly dependent on capillary density, defined as the number of capillaries per unit cross-sectional area of muscle. Based on an analysis of oxygen transport processes occurring at the microvascular level, the model allows estimation of the minimum number of straight, evenly spaced capillaries required to achieve a given oxygen consumption rate. Estimated capillary density values were determined from measurements of maximal oxygen consumption during knee extensor exercise and during whole body cycling, and they range from 459 to 1,468 capillaries/mm2. Measured capillary densities, obtained with either histochemical staining techniques or electron microscopy on quadriceps muscle biopsies from healthy subjects, are generally lower, ranging from 123 to 515 capillaries/mm2. This discrepancy is partly accounted for by the fact that capillary density decreases with muscle contraction and muscle biopsy samples typically are strongly contracted. The results imply that estimates of maximal oxygen transport rates based on capillary density values obtained from biopsy samples do not fully reflect the oxygen transport capacity of the capillaries in skeletal muscle.     

Renal oxidative stress, oxygenation, and hypertension

Hypertension is closely associated with progressive kidney dysfunction, manifested as glomerulosclerosis, interstitial fibrosis, proteinuria, and eventually declining glomerular filtration. The postulated mechanism for development of glomerulosclerosis is barotrauma caused by increased capillary pressure, but the reason for development of interstitial fibrosis and the subsequently reduced kidney function is less clear. However, it has been hypothesized that tissue hypoxia induces fibrogenesis and progressive renal failure. This is very interesting, since recent reports highlight several different mechanisms resulting in altered oxygen handling and availability in the hypertensive kidney. Such mechanisms include decreased renal blood flow due to increased vascular tone induced by ANG II that limits oxygen delivery and increases oxidative stress, resulting in increased mitochondrial oxygen usage, increased oxygen usage for tubular electrolyte transport, and shunting of oxygen from arterial to venous blood in preglomerular vessels. It has been shown in several studies that interventions to prevent oxidative stress and to restore kidney tissue oxygenation prevent progression of kidney dysfunction. Furthermore, inhibition of ANG II activity, by either blocking ANG II type 1 receptors or angiotensin-converting enzyme, or by preventing oxidative stress by administration of antioxidants also results in improved blood pressure control. Therefore, it seems likely that tissue hypoxia in the hypertensive kidney contributes to progression of kidney damage, and perhaps also persistence the high blood pressure.     

Numerical simulation of oxygen delivery to muscle tissue in the presence of hemoglobin-based oxygen carriers

This work represents a culmination of research on oxygen transport to muscle tissue, which takes into account oxygen transport due to convection, diffusion, and the kinetics of simultaneous reactions between oxygen and hemoglobin and myoglobin. The effect of adding hemoglobin-based oxygen carriers (HBOCs) to the plasma layer of blood in a single capillary surrounded by muscle tissue based on the geometry of the Krogh tissue cylinder is examined for a range of HBOC oxygen affinity, HBOC concentration, capillary inlet oxygen tension (pO(2)), and hematocrit. The full capillary length of the hamster retractor muscle was modeled under resting (V(max) = 1.57 x 10(-4) mLO(2) mL(-1) s(-1), cell velocity (v(c)) = 0.015 cm/s) and working (V(max) = 1.57 x 10(-3) mLO(2) mL(-1) s(-1), v(c) = 0.075 cm/s) conditions. Two spacings between the red blood cell (RBC) and the capillary wall were examined, corresponding to a capillary with and without an endothelial surface layer. Simulations led to the following conclusions, which lend physiological insight into oxygen transport to muscle tissue in the presence of HBOCs: (1) The reaction kinetics between oxygen and myoglobin in the tissue region, oxygen and HBOCs in the plasma, and oxygen and RBCs in the capillary lumen should not be neglected. (2) Simulation results yielded new insight into possible mechanisms of oxygen transport in the presence of HBOCs. (3) HBOCs may act as a source or sink for oxygen in the capillary and may compete with RBCs for oxygen. (4) HBOCs return oxygen delivery to muscle tissue to normal for varying degrees of hypoxia (inlet capillary pO(2) < 30 mmHg) and anemia (hematocrit < 46%) for the hamster model.     

Unloading oxygen in a capillary vessel under a pathological condition

In this work, we study theoretically the unloading of oxygen from a hemoglobin molecule to the wall of a typical capillary vessel, considering that the hemoglobin under pathological conditions, obeys the rheological Maxwell model. Based on recent experimental evidences in hypertension, we consider that the red blood cells (RBCs) are composed by a single continuous medium in contrast with the classical particulate or discrete RBC models, which are only valid under normal physiological conditions. The analysis considers the hemodynamic interactions between the plasma and the hemoglobin, both circulating in a long horizontal capillary. We apply numerical and analytical methods to obtain the main fluid-dynamic characteristics for both fluids in the limit of low Reynolds and Womersley numbers. A diffusion boundary layer formulation for the oxygen transport in the combined plasma-hemoglobin core region is presented. The main aspects derived are the time and spatial evolution of the membrane. The hemoglobin and plasma velocities and the pressure distributions are shown. For the oxygen unloading the results are the oxy-hemoglobin saturation, the oxygen flux and the oxygen concentration in the cell-free plasma layer. The volume fraction of red blood cells and the Strouhal number have a great influence on the hemodynamic interactions.     

Adaptations in skeletal muscle capillarity following changes in oxygen supply and changes in oxygen demands.

The effects of changes in oxygen supply and oxygen demands on fiber cross-sectional areas, capillary densities and capillary to fiber ratios were determined in three skeletal muscles of rat. The muscles examined were the vastus lateralis, soleus, and diaphragm. Reduced oxygen supply was produced by subjecting rats to ambient hypoxia, and increased oxygen demands were produced by subjecting rats to low ambient temperatures or treatment with thyroxin. Capillaries were visualized by injecting fluorescent dyes into the circulation. Muscles were quick frozen at resting lengths to preserve normal fiber geometry and were subsequently sectioned on a cryostat. All of the muscles sampled from animals in the experimental groups had elevated capillary densities. However, capillary to fiber ratios were not increased significantly in any muscle, for any experimental condition. Thus, all of the observed differences in capillarity were due to changes in the intrinsic rate of muscle fiber growth. Further, the relations of capillary density and capillary to fiber ratio to fiber area were the same as those obtained during normal maturation, suggesting that capillary growth is closely linked to the intrinsic rate of fiber growth.     

The effect of different blood sampling sites and analyses on the relationship~ between exercise intensity and 4.0 mmol ·1? blood lactate concentration

The aim of the study was to examine whether the difference in lactate concentration in different blood fractions is of practical importance when using blood lactate as a test variable of aerobic endurance capacity. Ten male firefighters performed submaximally graded exercise on a cycle ergometer for 20-25 min. Venous and capillary blood samples were taken every 5 min for determination of haematocrit and lactate concentrations in plasma, venous and capillary blood. At the same time, expired air was collected in Douglas bags for determination of the oxygen consumption. A lactate concentration of 4.0 mmol.l-1 was used as the reference value to compare the oxygen consumption and exercise intensity when different types of blood specimen and sampling sites were used for lactate analysis. At this concentration the exercise intensity was 17% lower (P less than 0.01) when plasma lactate was compared to venous blood lactate, and 12% lower (P less than 0.05) when capillary blood lactate was used. Similar discrepancies were seen in oxygen consumption. The results illustrated the importance of standardizing sampling and handling of blood specimens for lactate determination to enable direct comparisons to be made among results obtained in different studies.     

Pulmonary surfactant layers accelerate O2 diffusion through the air-water interface

During respiration, it is accepted that oxygen diffuses passively from the lung alveolar spaces through the respiratory epithelium until reaching the pulmonary capillaries, where blood is oxygenated. It is also widely assumed that pulmonary surfactant, a lipid-protein complex secreted into alveolar spaces, has a main surface active function, essential to stabilize the air-liquid interface, reducing in this way the work of breathing. The results of the present work show that capillary water layers containing enough density of pulmonary surfactant membranes transport oxygen much faster than a pure water phase or a water layer saturated with purely lipidic membranes. Membranes reconstituted from whole pulmonary surfactant organic extract, containing all the lipids plus the hydrophobic surfactant proteins, permit also very rapid oxygen diffusion, substantially faster than achieved by membranes prepared from the surfactant lipid fraction depleted of proteins. A model is proposed suggesting that protein-promoted membrane networks formed by pulmonary surfactant might have important properties to facilitate oxygenation through the thin water layer covering the respiratory surface.     

A theoretical model for oxygen transport in skeletal muscle under conditions of high oxygen demand.

Oxygen transport from capillaries to exercising skeletal muscle is studied by use of a Krogh-type cylinder model. The goal is to predict oxygen consumption under conditions of high demand, on the basis of a consideration of transport processes occurring at the microvascular level. Effects of the decline in oxygen content of blood flowing along capillaries, intravascular resistance to oxygen diffusion, and myoglobin-facilitated diffusion are included. Parameter values are based on human skeletal muscle. The dependence of oxygen consumption on oxygen demand, perfusion, and capillary density are examined. When demand is moderate, the tissue is well oxygenated and consumption is slightly less than demand. When demand is high, capillary oxygen content declines rapidly with axial distance and radial oxygen transport is limited by diffusion resistance within the capillary and the tissue. Under these conditions, much of the tissue is hypoxic, consumption is substantially less than demand, and consumption is strongly dependent on capillary density. Predicted consumption rates are comparable with experimentally observed maximal rates of oxygen consumption.