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.     

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     

Oxygen delivery to skeletal muscle fibers: effects of microvascular unit structure and control mechanisms

The number of perfused capillaries in skeletal muscle varies with muscle activation. With increasing activation, muscle fibers are recruited as motor units consisting of widely dispersed fibers, whereas capillaries are recruited as groups called microvascular units (MVUs) that supply several adjacent fibers. In this study, a theoretical model was used to examine the consequences of this spatial mismatch between the functional units of muscle activation and capillary perfusion. Diffusive oxygen transport was simulated in cross sections of skeletal muscle, including several MVUs and fibers from several motor units. Four alternative hypothetical mechanisms controlling capillary perfusion were considered. First, all capillaries adjacent to active fibers are perfused. Second, all MVUs containing capillaries adjacent to active fibers are perfused. Third, each MVU is perfused whenever oxygen levels at its feed arteriole fall below a threshold value. Fourth, each MVU is perfused whenever the average oxygen level at its capillaries falls below a threshold value. For each mechanism, the dependence of the fraction of perfused capillaries on the level of muscle activation was predicted. Comparison of the results led to the following conclusions. Control of perfusion by MVUs increases the fraction of perfused capillaries relative to control by individual capillaries. Control by arteriolar oxygen sensing leads to poor control of tissue oxygenation at high levels of muscle activation. Control of MVU perfusion by capillary oxygen sensing permits adequate tissue oxygenation over the full range of activation without resulting in perfusion of all MVUs containing capillaries adjacent to active fibers.     

Metabolic models of microcirculatory regulation.

The functions and integrity of body tissues are critically dependent on an adequate oxygen supply. Because the transport of oxygen to the cells is intimately linked to the microcirculation, the concept of microcirculation-metabolism coupling has received much attention. In essence, the metabolic theory of intrinsic control of the microcirculation states that microvascular tone is locally modulated to maintain adequate oxygen levels in the parenchymal cells. We propose a two-component control system for the regulation of tissue O2 delivery in accordance with metabolic needs. A precapillary sphincter control mechanism maintains tissue PO2 by governing the number of perfused capillaries. Functional capillary density in turn determines surface area available for diffusion and capillary-to-cell diffusion distance. On the other hand, the arteriolar control system modulates local blood flow in accordance with parenchymal O2 utilization and thereby minimizes changes in capillary PO2 when the O2 availability/demand ratio is decreased. We propose that the precapillary sphincters are more sensitive to changes in tissue PO2 than are the flow-regulating arterioles. Consequently, for mild stresses, adequate tissue oxygenation is maintained mainly by precapillary sphincter control of diffusion parameters without the need for changes in blood flow. However, as metabolic stresses become greater, blood flow regulation becomes the dominant factor in the control of tissue O2 delivery. Thus, by working in concert, the local mechanisms regulating microvascular resistance and effective capillary density provide a wide margin of safety against the development of cellular hypoxia.     

A compartmental model for oxygen transport in brain microcirculation in the presence of blood substitutes

A compartmental model is developed for oxygen (O(2)) transport in brain microcirculation in the presence of blood substitutes (hemoglobin-based oxygen carriers). The cerebrovascular bed is represented as a series of vascular compartments, on the basis of diameters, surrounded by a tissue compartment. A mixture of red blood cells (RBC) and plasma/extracellular hemoglobin solution flows through the vascular bed from the arterioles through the capillaries to the venules. Oxygen is transported by convection in the vascular compartments and by diffusion in the surrounding tissue where it is utilized. Intravascular resistance and the diffusive loss of oxygen from the arterioles to the tissue are incorporated in the model. The model predicts that most of the O(2) transport occurs at the level of capillaries. Results computed from the present model in the presence of hemoglobin-based oxygen carriers are consistent with those obtained from the earlier validated model (Sharan et al., 1989, 1998a) on oxygen transport in brain circulation in the absence of extracellular hemoglobin. We have found that: (a) precapillary PO(2) gradients increase as PO(2) in the arterial blood increases, P(50 p) (oxygen tension at 50% saturation of hemoglobin with O(2) in plasma) decreases, i.e. O(2) affinity of the extracellular hemoglobin is increased, the flow rate of the mixture decreases, hematocrit decreases at constant flow, metabolic rate increases, and intravascular transport resistance in the arterioles is neglected; (b) precapillary PO(2) gradients are not sensitive to (i) intracapillary transport resistance, (ii) cooperativity (n(p)) of hemoglobin with oxygen in plasma, (iii) hemoglobin concentration in the plasma and (iv) hematocrit when accounting for viscosity variation in the flow; (c) tissue PO(2) is not sensitive to the variation of intravascular transport resistance in the arterioles. We also found that tissue PO(2) is a non-monotonic function of the Hill coefficient n(p) for the extracellular hemoglobin with a maximum occurring when n(p) equals the blood Hill coefficient. The results of the computations give estimates of the magnitudes of the increases in tissue PO(2) as arterial PO(2) increases,P(50 p) increases, flow rate increases, hematocrit increases, hemoglobin concentration in the plasma increases, metabolic rate decreases, the capillary mass transfer coefficient increases or the intracapillary transport resistance decreases.     

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.     

Mathematical analysis of transport and exchange in skeletal muscle

A new approach to modelling microcirculatory transport and exchange is introduced and developed with specific reference to skeletal muscle. The objective is to describe the large-scale interaction of a great number of differently perfused capillary groups and to interpret phenomena observed in a whole organ in terms of processes occurring within individual capillaries. We consider fluid exchange and the associated macromolecular transport, the exchange of small metabolites carried in the plasma, and oxygen transport to tissue.     

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.     

The scar neovasculature after myocardial infarction in rats

A series of novel techniques, adapted from the field of tumor biology, were developed to quantify vascular structure and function and to explore the role of ANG II receptor AT1 in cardiac remodeling after myocardial infarction (MI). We examined the scar neovasculature at 1-4 wk post-MI in Sprague-Dawley rats with a view toward its ability to deliver and exchange oxygen. CD31 and DiOC7(3) staining was used to visualize anatomical vessels vs. those perfused. EF5/Cy3 immunohistochemical staining was used to quantify tissue hypoxia. We compared untreated controls with rats treated with losartan, an AT1 receptor antagonist. Our findings indicated that, at the infarct site, there was not only a 42-75% (1-4 wk post-MI) decrease in the number of anatomical vessels compared with controls but also a decrease in the fraction of perfused vessels from 70% in normal coronary vasculature to 48% at the infarct site. These changes were accompanied by progressive increases in diffusion distance and tissue hypoxia (100% increase in EF5/Cy3 staining at 4 wk post-MI). Losartan-treated rats exhibited a significantly less marked reduction in vascular perfusion and a significantly lesser extent of tissue hypoxia. Over the course of 4 wk post-MI, there is a reduction in coronary vasculature at the infarct site, the extent of which is attenuated by losartan. These findings implicate AT1 receptor upregulation, and perhaps angiotensin-related peptides, as being antiangiogenic.     

Oxygen diffusive shunts under conditions of heterogeneous oxygen delivery

Oxygen transport to a given site of tissue from capillaries closest to this site and from more distant capillaries is analyzed under conditions of heterogeneous flow. The analysis is based on a more general solution of the capillary-tissue oxygen transport problem obtained by the author (Popel, 1978). The results of calculations indicate that the oxygen tension in a region poorly supplied by the closest capillaries can be significantly increased due to oxygen diffusion from more distant capillaries.     

Cerebral regional capillary perfusion and blood flow after carbon monoxide exposure.

Alterations in regional cerebral blood flow (rCBF) and percent perfused capillaries (indicative of functional intercapillary distance) were determined in conscious male Long-Evans rats after reducing their blood O2-carrying capacity by exposing them to 1% CO for 12 min. rCBF was determined by the iodoantipyrine method. rCBF increased from a mean of 106 +/- 8 (SE) ml.min-1.100 g-1 before CO exposure to 173 +/- 14 ml.min-1.100 g-1 after CO exposure. There was a greater flow increase (126%) in the cerebral cortex than in the lower brain stem [pons (45%), medulla (39%)]. Presence of fluorescein isothiocyanate-labeled dextran identified the perfused capillaries before and after CO exposure. The volume fraction (Vv) and number/mm2 (Na) of all capillaries (perfused and nonperfused) in a given area of brain were determined after staining for alkaline phosphatase. The percent Vv and percent Na of perfused capillaries increased uniformly (from approximately 50% to approximately 80%) in all parts of the brain after CO exposure. In the presence of tissue hypoxia with undiminished plasma PO2, the brain vasculature allowed greater flow of blood while the microvasculature adjusted to reduce the diffusion distance for O2.     

Effect of hypoxia on percent of arteriolar and capillary beds perfused in the rat brain

The effects of moderate and severe hypoxia on quantitative regional morphometric indexes of the total and perfused arteriolar and capillary network were studied in the rat brain to determine whether diffusion distances were reduced in hypoxia. Fluorescein isothiocyanate (FITC)-labeled dextran was injected into the femoral vein of conscious control and hypoxic rats. After 20 s, the animal was decapitated and the head was frozen in liquid N2. Sections from eight brain regions were photographed to detect the perfused microvessels and then stained for alkaline phosphatase to visualize the total vascular network. There were significant increases in percent perfused arteriolar and capillary morphology between the two groups of hypoxic animals and control animals. In control rats, the percent of capillaries perfused averaged 45.6 +/- 0.6% (mean +/- SE). In moderate hypoxia 63.4 +/- 1.8% of the vessels were perfused and in severe hypoxia 89.4 +/- 0.1% were perfused. The percentage of arterioles perfused changed similarly. There were no significant differences in any index of total or percent perfused arteriolar or capillary morphometry among the regions within any group. During severe hypoxia, a greater percentage of the capillary reserves was utilized. These results demonstrate a uniform response to hypoxia in the capillary and arteriolar network of the conscious rat brain.     

A computational study of the effect of vasomotion on oxygen transport from capillary networks

The objective of this study was to investigate the effect of arteriolar vasomotion on oxygen transport from capillary networks. A computational model was used to calculate blood flow and oxygen transport from a simulated network of striated muscle capillaries. For varying tissue oxygen consumption rates, the importance of the frequency and amplitude of vasomotion-induced blood flow oscillations was studied. The effect of myoglobin on oxygen delivery during vasomotion was also examined. In the absence of myoglobin, it was found that when consumption is high enough to produce regions of hypoxia under steady flow conditions, vasomotion-induced flow oscillations can significantly increase tissue oxygenation and decrease oxygen transport heterogeneity. The largest effect was seen for low-frequency, high-amplitude oscillations (1.5-3 cycles min(-1), 90% of steady-state velocity). By contrast, at physiological tissue myoglobin concentrations, vasomotion did not improve tissue oxygenation. This unexpected finding is due to the buffering effect of myoglobin, suggesting that in highly aerobic muscles short-term storage of oxygen is more important than the possibility of increasing transport through vasomotion.     

A computational model of oxygen transport in skeletal muscle for sprouting and splitting modes of angiogenesis

Oxygen transport from capillary networks in muscle at a high oxygen consumption rate was simulated using a computational model to assess the relative efficacies of sprouting and splitting modes of angiogenesis. Efficacy was characterized by the volumetric fraction of hypoxic tissue and overall heterogeneity of oxygen distribution at steady state. Oxygen transport was simulated for a three-dimensional vascular network using parameters for rat extensor digitorum longus (EDL) muscle when oxygen consumption by tissue reached 6, 12, and 18 times basal consumption. First, a control network was generated by using straight non-anastomosed capillaries to establish baseline capillarity. Two networks were then constructed simulating either abluminal lateral sprouting or intraluminal splitting angiogenesis such that capillary surface area was equal in both networks. The sprouting network was constructed by placing anastomosed capillaries between straight capillaries of the control network with a higher probability of placement near hypoxic tissue. The splitting network was constructed by splitting capillaries from the control network into two branches at randomly chosen branching points. Under conditions of moderate oxygen consumption (6 times basal), only minor differences in oxygen delivery resulted between the sprouting and splitting networks. At higher consumption levels (12 and 18 times basal), the splitting network had the lowest volume of hypoxic tissue of the three networks. However, when total blood flow in all three networks was made equal, the sprouting network had the lowest volume of hypoxic tissue. This study also shows that under the steady-state conditions the effect of myoglobin (Mb) on oxygen transport was small.     

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 Po(2). The model predicts that AV oxygen shunting is quantitatively significant and estimates how much kidney Vo(2) must change, in the face of altered renal blood flow, to maintain cortical tissue Po(2) at a stable level. It is demonstrated that oxygen shunting increases as renal Vo(2) or arterial Po(2) 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 Vo(2) 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 Po(2) when arterial Po(2) changes, but renders the cortex and perhaps also the medulla susceptible to hypoxia when oxygen delivery falls or consumption increases.     

Cerebral blood flow regulation in REM sleep: a model for flow-metabolism coupling.

The pattern of metabolic and circulatory changes occurring during REM sleep in the whole brain is also observed at a regional level in different instances of functional activation. This pattern is characterized by an increase in metabolic rate, blood flow, glucose and oxygen uptake, the increase in glucose uptake generally exceeding oxygen uptake. A model of interpretation is presented, based on the assumption that substantial limitation to oxygen diffusion exists in the brain. According to the model, microregions lying at mid-distance between capillaries may become hypoxic, depending on metabolic rate and blood-cell PO2 difference. At increasing metabolic rates, O2 consumption in pericapillary microregions increases and the PO2 drop becomes steeper. As a consequence, in microregions far from capillaries a decrease in O2 availability occurs, in concomitance with the increase in metabolic rate, so that non-oxidative glucose metabolism develops locally. A similar spatial PO2 pattern forms in the case of arterial hypoxia, when capillary PO2, and then blood-cell PO2 difference, is reduced. The hypoxic microregions are the source of vasodilatatory messages, the consequent vasodilatation increasing average capillary PO2 and then favoring O2 diffusion to the tissue. Oxygen thus appears to be a better candidate than glucose as a mediator of blood flow-metabolism coupling. This is supported by its higher extraction fraction and by the fact that, in physiologic conditions, arterial hypoxia (and not hypoglycemia) acts on cerebral blood flow. Moreover, the diffusion capacity of glucose in the brain is higher than that of oxygen, so that diffusion limitation is more likely to occur for oxygen. The present model allows consistent organization of the stereotyped changes in cerebral blood flow and glucose and oxygen uptake occurring both in REM sleep and in other instances of brain activation.     

Model analyses of capillary growth and tissue oxygenation during hypoxia

Theoretical analyses were used to determine whether capillary growth is an adaptive response to hypoxia. Parameter values were obtained from models of transverse sections of muscles in which individual fibers were distributed in square-ordered arrays and capillaries were added to the perimeters of individual fibers in the arrays. Increasing the number of capillaries up to 2.0 per fiber increased hypoxic tolerance by 157% above that expected for a Krogh cylinder. However, increasing the number of capillaries from 2.0 to 4.0 per fiber increased hypoxic tolerance by only 18% and, assuming the entire perimeter of each fiber was perfused with blood, increased hypoxic tolerance by only 11% over the value obtained when capillary-to-fiber ratio was 4.0. Capillary growth during normal maturation may result in capillary-to-fiber ratios around 2.0, near the upper limit for producing marked changes in hypoxic tolerance. Therefore, capillary growth may not be an adaptive response to ambient hypoxia because there is little or no gas transport benefit derived from the additional capillaries.     

Capillary filtration coefficient is independent of number of perfused capillaries in cat skeletal muscle

The capillary filtration coefficient (CFC) is assumed to reflect both microvascular hydraulic conductivity and the number of perfused capillaries at a given moment (precapillary sphincter activity). Estimation of hydraulic conductivity in vivo with the CFC method has therefore been performed under conditions of unchanged vascular tone and metabolic influence. There are studies, however, that did not show any change in CFC after changes in vascular tone and metabolic influence, and these studies indicate that CFC may not be influenced by alteration in the number of perfused capillaries. The present study reexamined to what extent CFC in a pressure-controlled preparation depends on the vascular tone and number of perfused capillaries by analyzing how CFC is influenced by 1) vasoconstriction, 2) increase in metabolic influence by decrease in arterial blood pressure, and 3) occlusion of precapillary microvessels by arterial infusion of microspheres. CFC was calculated from the filtration rate induced by a fixed decrease in tissue pressure. Vascular tone was increased in two steps by norepinephrine (n = 7) or angiotensin II (n = 6), causing a blood flow reduction from 7.2 +/- 0.8 to at most 2.7 +/- 0.2 ml x min(-1) x 100 g(-1) (P < 0.05). The decrease in arterial pressure reduced blood flow from 4.8 +/- 0.4 to 1.40 +/- 0.1 ml x min(-1) x 100 g(-1) (n = 6). Vascular resistance increased to 990 +/- 260% of control after the infusion of microspheres (n = 6). CFC was not significantly altered from control after any of the experimental interventions. We conclude that CFC under these conditions is independent of the vascular tone and number of perfused capillaries and that variation in CFC reflects variation in microvascular hydraulic conductivity.     

Lignification of cell walls of infected cells in Casuarina glauca nodules that depend on symplastic sugar supply is accompanied by reduction of plasmodesmata number and narrowing of plasmodesmata

The oxygen protection system for the bacterial nitrogen‐fixing enzyme complex nitrogenase in actinorhizal nodules of Casuarina glauca resembles that of legume nodules: infected cells contain large amounts of the oxygen‐binding protein hemoglobin and are surrounded by an oxygen diffusion barrier. However, while in legume nodules infected cells are located in the central tissue, actinorhizal nodules are composed of modified lateral roots with infected cells in the expanded cortex. Since an oxygen diffusion barrier around the entire cortex would also block oxygen access to the central vascular system where it is required to provide energy for transport processes, here each individual infected cell is surrounded with an oxygen diffusion barrier. In order to assess the effect of these oxygen diffusion barriers on oxygen supply for energy production for transport processes, apoplastic and symplastic sugar transport pathways in C. glauca nodules were examined. The results support the idea that sugar transport to and within the nodule cortex relies to a large extent on the less energy‐demanding symplastic mechanism. This is in line with the assumption that oxygen access to the nodule vascular system is substantially restricted. In spite of this dependence on symplastic transport processes to supply sugars to infected cells, plasmodesmal connections between infected cells, and to a lesser degree with uninfected cells, were reduced during the differentiation of infected cells.