The effects of transmural transport in the microcirculation: a two gas species model

Retinal oxygen saturation: novel analysis method for the oxymap

PURPOSE

To use a novel image analysis approach to consider how oxygen saturation changes as a function of vessel width and distance from the nerve and between superior and inferior retinal hemifields.

METHODS

Ten images were acquired from one eye of 17 participants (mean [standard deviation] age, 28 [4] years; range, 22-38 years) using the Oxymap T1 retinal oximeter. Every pixel identified by the detection algorithm was extracted, and frequency histograms of retinal vessel oxygen saturation were plotted for each vessel diameter (70-170 μm). Histograms were fitted with two Gaussian models to identify peak arteriole and venule oxygen saturation. Mean (±standard error of the mean) arteriole and venule oxygen saturation at each vessel width were calculated. Data were also analyzed in (1) annuli of 100 μm centered on the optic nerve or (2) upper and lower hemifields demarcated by the center of the optic nerve.

RESULTS

Venous oxygen saturation was higher in smaller vessels than in larger vessels. Arterial oxygen saturation remained relatively constant with vessel width. Oxygen saturation was lower in veins nearer the optic nerve. The upper retinal hemisphere showed higher venous oxygen saturation compared with the lower hemifield.

CONCLUSIONS

The current objective analysis approach provides a more complete picture of retinal oxygen saturation at the posterior pole as a function of vessel width and retinal location.

A mathematical model of diffusional shunting of oxygen from arteries to veins in the kidney

To understand how arterial-to-venous (AV) oxygen shunting influences kidney oxygenation, a mathematical model of oxygen transport in the renal cortex was created. The model consists of a multiscale hierarchy of 11 countercurrent systems representing the various branch levels of the cortical vasculature. At each level, equations describing the reactive-advection-diffusion of oxygen are solved. Factors critical in renal oxygen transport incorporated into the model include the parallel geometry of arteries and veins and their respective sizes, variation in blood velocity in each vessel, oxygen transport (along the vessels, between the vessels and between vessel and parenchyma), nonlinear binding of oxygen to hemoglobin, and the consumption of oxygen by renal tissue. The model is calibrated using published measurements of cortical vascular geometry and microvascular Po2. The model predicts that AV oxygen shunting is quantitatively significant and estimates how much kidney V̇o2 must change, in the face of altered renal blood flow, to maintain cortical tissue Po2 at a stable level. It is demonstrated that oxygen shunting increases as renal V̇o2 or arterial Po2 increases. Oxygen shunting also increases as renal blood flow is reduced within the physiological range or during mild hemodilution. In severe ischemia or anemia, or when kidney V̇o2 increases, AV oxygen shunting in proximal vascular elements may reduce the oxygen content of blood destined for the medullary circulation, thereby exacerbating the development of tissue hypoxia. That is, cortical ischemia could cause medullary hypoxia even when medullary perfusion is maintained. Cortical AV oxygen shunting limits the change in oxygen delivery to cortical tissue and stabilizes tissue Po2 when arterial Po2 changes, but renders the cortex and perhaps also the medulla susceptible to hypoxia when oxygen delivery falls or consumption increases.

Microvascular oxygen tension in the rat mesentery

Longitudinal Po2profiles in the microvasculature of the rat mesentery were studied using a novel phosphorescence quenching microscopy technique that minimizes the accumulated photoconsumption of oxygen by the method. Intravascular oxygen tension (Po2, in mmHg) and vessel diameter ( d, in μm) were measured in mesenteric microvessels ( n = 204) of seven anesthetized rats (275 g). The excitation parameters were as follows: 7 × 7-μm spot size; 410 nm laser; and 100 curves at 11 pulses/s, with pulse parameters of 2-μs duration and 80-pJ/μm2energy density. The mean Po2(± SE) was 65.0 ± 1.4 mmHg ( n = 78) for arterioles ( d = 18.8 ± 0.7 μm), 62.1 ± 2.0 mmHg ( n = 38) at the arteriolar end of capillaries ( d = 7.8 ± 0.3 μm), and 52.0 ± 1.0 mmHg ( n = 88) for venules ( d = 22.5 ± 1.0 μm). There was no apparent dependence of Po2on d in arterioles and venules. There were also no significant deviations in Po2based on d (bin width, 5 μm) from the general mean for both of these types of vessels. Results indicate that the primary site of oxygen delivery to tissue is located between the smallest arterioles and venules (change of 16.3 mmHg, P = 0.001). In conclusion, oxygen losses from mesenteric arterioles and venules are negligible, indicating low metabolic rates for both the vascular wall and the mesenteric tissue. Capillaries appear to be the primary site of oxygen delivery to the tissue in the mesenteric microcirculation. In light of the present results, previously reported data concerning oxygen consumption in the mesenteric microcirculation can be explained as artifacts of accumulated oxygen consumption due to the application of instrumentation having a large excitation area for Po2measurements in slow moving and stationary media.

Oxygen gradients in the microcirculation

As arterialized blood transits from the central circulation to the periphery, oxygen exits through the vessel walls driven by radial oxygen gradients that extend from the red blood cell column, through the plasma, the vessel wall, and the parenchymal tissue. This exit determines a longitudinal gradient of blood oxygen saturation whose extent is inversely related to the level of metabolic activity of the tissue, being small for the brain and considerable for skeletal muscle at rest where hemoglobin is only half-saturated with oxygen when blood arrives to the capillaries. Data obtained by a variety of methods show that the oxygen loss is too great to be explained by diffusion alone, and oxygen gradients measured in the arteriolar wall provide evidence that this structure in vivo is a very large oxygen sink, and suggests a rate of oxygen consumption two orders of magnitude greater than seen in in vitro studies. Longitudinal gradients in the capillary network and radial gradients in surrounding tissue also show a dependence on the metabolic rate of the tissue, being more pronounced in brain than in resting skeletal muscle and mesentery. Mean PO2 values increase from the postcapillary venules to the distal vessels of this network while radial gradients indicate additional oxygen loss. This circumstance may be due to pathways with higher flow having higher oxygen content than low flow pathways as well as possible oxygen uptake from adjacent arterioles. Taken together, these newer findings on oxygen gradients in the microcirculation require a reexamination of existing concepts of oxygen delivery to tissue and the role of the capillaries in this process.

Venular-arteriolar communication in the regulation of blood flow

Muscle blood flow is regulated to meet the metabolic needs of the tissue. With the vasculature arranged as a successive branching of arterioles and the larger, >50 microm, arterioles providing the major site of resistance, an increasing metabolic demand requires the vasodilation of the small arterioles first then the vasodilation of the more proximal, larger arterioles. The mechanism(s) for the coordination of this ascending vasodilation are not clear and may involve a conducted vasodilation and/or a flow-dependent response. The close arteriolar-venular pairing provides an additional mechanism by which the arteriolar diameter can be increased due to the diffusion of vasoactive substances from the venous blood. Evidence is presented that the venular endothelium releases a relaxing factor, a metabolite of arachidonic acid, that will vasodilate the adjacent arteriole. The stimulus for this release is not known, but it is hypothesized that hypoxia-induced ATP release from red blood cells may be responsible for the stimulation of arachidonic release from the venular endothelial cells. Thus the venous circulation is in an optimal position to monitor the overall metabolic state of the tissue and thus provide a feedback regulation of arteriolar diameter.

Incorporating O2-Hb reaction kinetics and the Fåhraeus effect into a microcirculatory O2-CO2 transport model

The influence of O2-Hb reaction kinetics and the Fåhraeus effect on steady state O2 and CO2 transport in cat brain microcirculation was investigated using our refined multicompartmental model. The most important model predictions include: 1) capillaries are the sites in the microcirculation where the effect of O2-Hb kinetics is most pronounced; 2) while there is only a small difference between equilibrium and actual oxygen saturation, this effect is not negligible; 3) O2-Hb kinetics tends to make the PO2 level at the venous entrance higher than in venules; 4) the influence of the Fåhraeus effect leads to a lower tissue PO2 level than in venules and the outlet vein. The resultant decline in tissue PO2 may lead to a decrease in O2 consumption rate and extraction ratio; 5) although the Fåhraeus effect changes the ratio between arteriolar flux and capillary flux, incorporating the Fåhraeus effect and O2-Hb kinetics into the simulation does not change our previous conclusion, that most of the O2 and CO2 exchange takes place at the capillary level; 6) in general, influences of O2-Hb kinetics and Fåhraeus effect are synergistic; 7) a model that excludes these two mechanisms might overestimate the tissue oxygenation level especially during severe hypoxia.

A compartmental model for oxygen-carbon dioxide coupled transport in the microcirculation

We present a multicompartmental model for an oxygen-carbon dioxide transport system. The compartmental equations and their lumped parameters are derived through space averaging of the corresponding distributed model. The model can predict compartmental distributions of oxygen and carbon dioxide partial pressures, oxygen-hemoglobin saturation, and pH. Other unique features include the effects of the radial distribution of partial pressures and the difference in metabolic rates between vessel wall and tissue. A model for the cat brain, based on this formulation, is compared with results of experiments and with two types of earlier models: one without space averaging and one without carbon dioxide transport. The results suggest that space averaging the convective terms significantly affects the behavior of the model. This is consistent with conclusions from our earlier oxygen-only model. Our observations also demonstrate, however, significant differences between the results from the oxygen-carbon dioxide model and the oxygen-only model. For instance, at low blood flow rates or at low level of oxygen input, predicted oxygen partial pressures can differ by as much as 30% between the two models. Results obtained from the present model are supported by available experimental findings.

A model of brain arteriolar oxygen and carbon dioxide transport during anemia

Existing experimental and theoretical evidence suggests that precapillary diffusion of O2 and CO2 occurs between arterioles and tissue under normal physiologic conditions. However, limited information is available on arteriolar gas transport during anemia. With use of a mathematical model of an arteriolar network in brain tissue, anemic hematocrits of 35, 25, and 15% were modeled to determine the effect of anemia on the exchange, the change in the equilibrium tissue O2 and CO2 tensions, and the increase in blood flow needed to restore tissue oxygenation. We found that the blood PO2 exiting the network fell from 66 mm Hg normally to 48 mm Hg during the severest anemia. Concurrently, the equilibrium tissue O2 tensions dropped from 44 to 23 mm Hg. For CO2 the exit blood PCO2 was 58 mm Hg for a 15% hematocrit, an increase of 4 mm Hg from the normal value, and equilibrium tissue PCO2 increased from 56 to 61 mm Hg. Blood flow increases from normal values necessary to offset the effects of the decreased O2 delivery to the tissue were 26, 86, and 222%, respectively, for hematocrits of 35, 25, and 15%. We compared our model results with recent experimental studies that have suggested that the amount of O2 diffusion is much higher than predicted values. We found that these experimental O2 gradients are three to four times larger than theoretical.

A theoretical model of gas transport between arterioles and tissue

A theoretical model of CO2 and O2 diffusion between arterioles and tissue was developed to determine if significant transport could occur in precapillary vessels. There is increasing evidence, both theoretical and experimental, that such exchange does occur. Using a model in which CO2 and O2 were coupled through the Bohr and Haldane effects, we quantified the radial and axial transport. We also examined the roles of axial diffusion in the arteriole wall and tissue and capillary structure on the transport. Capillary arrangements investigated included capillaries independent of the arteriole with the entering capillary PCO2 or PO2 equal to a constant, and capillaries branching off along the length of the arteriole with the entering capillary partial pressure equal to the arteriole partial pressure at the given axial location. We found that for CO2 in arterioles with an inner diameter ranging from 200 to 22 microns, the exiting blood was 6 to 45% of the way to complete equilibrium with the surrounding tissue, respectively. For O2, the range was 8 to 25%, respectively. We also determined that axial diffusion in the arteriole wall and tissue has little effect on the transport and that capillary structure can alter tissue PCO2 by as much as 12 mm Hg in the smallest arteriole, but has little effect on O2 transport.

Effect of convection in capillaries on oxygen removal from arterioles in striated muscle

Based on experimental data that show the presence of significant oxygen saturation gradients in precapillary arterioles, it has been suggested that the in vivo permeability to oxygen of resting striated muscle may be significantly higher than the corresponding in vitro value obtained in unperfused tissue samples (Popel et al., 1989b, Adv. expl. Med. Biol. 247, 215). The present study performs two analyses to further compare theoretical predictions with experimental data obtained under control conditions and during hemodilution and hemoconcentration. First, it is shown that, in principle, a capillary-perfused tissue layer with a thickness of a few hundred microns is necessary to convectively carry the experimentally determined amount of oxygen released by precapillary arterioles under control and hemodiluted conditions. This capacity to convect oxygen depends strongly on the resting tissue oxygen tension. Second, a more general version of a previous model (Weerappuli & Popel, 1989, J. Biomech. Eng. 111, 24) is used to examine whether changes made in the model parameters within the physiological range of values can explain the experimentally measured flux. The results show that the theoretical predictions can be made compatible with experimental observations if the in vivo permeability of perfused tissue to oxygen is assumed to be one to two orders of magnitude higher than the in vitro value. Furthermore, the predicted in vivo permeability for perfused tissue surrounding an arteriole varies with the arteriolar luminal oxygen tension and flow. This may be due to simplifying approximations made in the model or possible experimental artifacts. Alternatively, it could also be speculated that this variability indicates the flow dependency of the permeability of perfused tissue to oxygen.

Mathematical model for the computation of alveolar partial pressure of carbon monoxide

A mathematical model is formulated for computing alveolar partial pressure of carbon monoxide (PACO) from that in the atmospheric air. The model takes into account parameters like inspired/expired air flow rates, diffusion capacity of the lung, concentration of CO in the atmospheric air, blood flow rate and the non-linear CO dissociation curve. The effect due to the presence of O2 in the blood on CO dissociation curve is also incorporated. It is shown that for a given atmospheric CO concentration, PACO increases exponentialy with time and attains asymptotic value. Alveolar PCO increases further with the increase in the atmospheric CO concentration. The model can also be used to compute carboxyhaemoglobin levels in the blood as a function of exposure time and the results are comparable with the CFK equation and the values measured experimentally.

A simple model for prediction of oxygen transport rates by flowing blood in large capillaries

A simple model has been developed for simulation of oxygen transport to and from blood flowing in conduits of the diameter of arterioles and larger (greater than or equal to 20 microns). The basis is the large capillary model (LCAP) of P.K. Nair, et al. 1989 which has been validated experimentally. Detailed calculations of the oxygen concentration distribution reveal that the dominant resistance to oxygen transport is distributed in the plasma. Relatively little resistance is present within or in the immediate vicinity of the red cells. On the basis of these findings, LCAP was simplified from four simultaneous nonlinear partial differential equations (PDEs) to one PDE by (1) assuming chemical equilibrium within the red blood cells, (2) neglecting intracellular and extracellular boundary layer resistances, and (3) incorporating transport in the cell-free region adjacent to the capillary wall into the boundary conditions. The simplified model is much easier to apply mathematically to new situations. A comparison between LCAP and the simpler model shows that they give virtually the same predictions, and the predictions agree well with experimental measurements. The model is predictive in that all the parameters are determined from the literature or from independent measurements. Thus it should be useful in studies of physiological significance, as well as in design and analysis of extracorporeal blood oxygenators.

Counter-current blood flow in tissues: protection against adverse effects

In hypoxia, the tissue counter-current can thus, by virtue of the Bohr effect, increase tissue Po2 and thus tissue oxygenation. In hyperoxia, on the other hand, the counter-current system, acting as a diffusion shunt, can protect the tissue against adverse O2-toxic effects. It thus appears, that the counter-current system is advantageous for O2 supply to tissues.

Prediction of oxygen transport rates in blood flowing in large capillaries

A mathematical model has been developed to predict oxygen transport to and from blood flowing in tubes of the diameter of arterioles and larger (approximately 20 microns and larger). The resistance to oxygen transport in red cell suspensions is much higher than that of a comparable homogeneous hemoglobin solution. The increased resistance is associated with encapsulation of the hemoglobin in the red cells. Yet, somewhat paradoxically, for large capillaries relatively little resistance is within or in the immediate vicinity of the red cells. The great majority of the resistance is shown to be distributed in the plasma. Predictions of oxygen uptake and release are shown to be in excellent agreement with results of measurements taken on red cell suspensions flowing in capillaries of 27- and 100-microns diameter. The model seems to be the first for oxygen transport in flowing blood that is validated by detailed comparison with experimental results. It is a predictive model in that all parameters in the model are determined from independent measurements or from the literature.

Oxygen supply of the blood-free perfused guinea-pig brain in normo- and hypothermia measured by the local distribution of oxygen pressure

The O2 supply of the blood-free perfused brain cortex of the guinea pig was investigated by measuring polarographically the local distribution of tissue PO2 at 18 degrees C, 24 degrees C, and 37 degrees C. The perfusion was performed in situ, using a medium equilibrated by a gas mixture of 95% O2 and 5% CO2. Papaverine was added to prevent vasoconstriction during hypothermia. To avoid measuring artefacts thin micro electrodes with a small sharpened tip of ca. 4 microns in diameter were used and a special puncturing technique was applied. The experimental results indicate the presence of a large variation of local tissue PO2. Local mean PO2 increased up to a depth of 1000 microns, reached a plateau, and then decreased towards 3000 microns. This demonstrates that the O2 supply changes in dependence of the distance of the brain surface. This may partly be caused by the special vascularization pattern of the brain cortex. As it follows from the PO2 histograms, at 24 degrees C the tissue layer between 0-2000 microns (layer I) was well supplied with oxygen, whereas at the same time the layer between 2001-3000 microns (layer II) was hypoxic. At 37 degrees C, both layers were hypoxic, but layer III showed the more pronounced tissue hypoxia. To obtain a sufficient oxygen supply the temperature had to be reduced below 24 degrees C to sufficiently decrease tissue O2 consumption: at 18 degrees C, there was no sign of hypoxia any more. In comparison with the PO2 histogram of the tissue the PO2 histogram of the pial surface was shifted to higher PO2 values.(ABSTRACT TRUNCATED AT 250 WORDS)

Precapillary O2 loss and arteriovenous O2 diffusion shunt are below limit of detection in myocardium

1. Mean intracellular PO2 is much lower than mean venous PO2 in subepicardium. 2. The drop in Hb saturation between aorta and terminal arterioles is within the 5% error of our method. 3. Arteriolar O2 has no effect on saturation in paired countercurrent venules in myocardium. 4. Saturation in coronary venules is independent of venule diameter and indistinguishable from saturation in macroscopic epicardial veins. 5. Since diffusive O2 shunting is negligible and PO2 is approximately linearly related to saturations over the observed range, mean coronary venous PO2 should closely approximate mean-end capillary PO2. 6. O2 mass transport from blood to tissue requires a steep PO2 gradient between the capillary and the surface of a tissue cell.

A compartmental model for oxygen transport in brain microcirculation

A compartmental model is formulated for oxygen transport in the cerebrovascular bed of the brain. The model considers the arteriolar, capillary and venular vessels. The vascular bed is represented as a series of compartments on the basis of blood vessel diameter. The formulation takes into account such parameters as hematocrit, vascular diameter, blood viscosity, blood flow, metabolic rate, the nonlinear oxygen dissociation curve, arterial PO2, P50 (oxygen tension at 50% hemoglobin saturation with O2) and carbon monoxide concentration. The countercurrent diffusional exchange between paired arterioles and venules is incorporated into the model. The model predicts significant longitudinal PO2 gradients in the precapillary vessels. However, gradients of hemoglobin saturation with oxygen remain fairly small. The longitudinal PO2 gradients in the postcapillary vessels are found to be very small. The effect of the following variables on tissue PO2 is studied: blood flow, PO2 in the arterial blood, hematocrit, P50, concentration of carbon monoxide, metabolic rate, arterial diameter, and the number of perfused capillaries. The qualitative features of PO2 distribution in the vascular network are not altered with moderate variation of these parameters. Finally, the various types of hypoxia, namely hypoxic, anemic and carbon monoxide hypoxia, are discussed in light of the above sensitivity analysis.

A Mathematical Model of Countercurrent Exchange of Oxygen Between Paired Arterioles and Venules

A mathematical model is formulated for diffusive countercurrent exchange of oxygen between paired arterioles and venules. A closed form solution of the problem is obtained by linearizing the nonlinear oxyhemoglobin dissociation curve at the inlet P_O2 in the vessel. The closed form solution is compared with the corresponding numerical solution of the nonlinear problem. Under normal conditions, longitudinal gradients of venular P_O2 are found to be small. Examples are presented where the model predicts significant gradients of venular P_O2 when the blood flow rate in the venule is several times smaller than that in the arteriole.

Carbon dioxide exchange across the walls of arterioles: implication for the location of the medullary chemoreceptors

The location of the medullary chemoreceptors is not conclusively established. The original experiments, which were believed to suggest a shallow surface location in the ventrolateral medulla, have been questioned because substances, particularly CO2, applied on the surface of the medulla could diffuse into small arterioles. Because the whole tissue blood flow is supplied by surface arterioles, they could transport substances from the surface into the tissue to the respiratory centers. We studied simple transport equations describing movement of CO2 in arterioles bathed by rapidly flowing cerebrospinal fluid (CSF) and arterioles in tissue perfused by capillaries. Substantial exchange of CO2 could occur across the arteriole wall for all expected sizes of vessels when the partial pressure of CO2 at the outside wall was determined by CSF. When an arteriole is surrounded by tissue, only vessels with inside diameters (ID) less than or equal to 50 micron will exchange substantial amounts of CO2 but the smallest arterioles may be nearly in equilibrium with the tissue. The CO2 gradient in tissue around the arteriole will extend approximately 1 mm. Our simple theoretical description of CO2 transport in arterioles predicts substantial exchange in precapillary vessels. CO2 picked up by the smallest surface arterioles when the medulla is perfused at a high rate with CSF will not stay in the blood past the putative depth of the chemoreceptors. In arterioles greater than 30 micron, however, the CO2 could be carried to the respiratory centers.