## Abstract

The oxygen dependence of respiration in striated muscle in situ was studied by measuring the rate of decrease of interstitial Po_{2} [oxygen disappearance curve (ODC)] following rapid arrest of blood flow by pneumatic tissue compression, which ejected red blood cells from the muscle vessels and made the ODC independent from oxygen bound to hemoglobin. After the contribution of photo-consumption of oxygen by the method was evaluated and accounted for, the corrected ODCs were converted into the Po_{2} dependence of oxygen consumption, V̇o_{2}, proportional to the rate of Po_{2} decrease. Fitting equations obtained from a model of heterogeneous intracellular Po_{2} were applied to recover the parameters describing respiration in muscle fibers, with a predicted sigmoidal shape for the dependence of V̇o_{2} on Po_{2}. This curve consists of two regions connected by the point for critical Po_{2} of the cell (i.e., Po_{2} at the sarcolemma when the center of the cell becomes anoxic). The critical Po_{2} was below the Po_{2} for half-maximal respiratory rate (*P*_{50}) for the cells. In six muscles at rest, the rate of oxygen consumption was 139 ± 6 nl O_{2}/cm^{3}·s and mitochondrial *P*_{50} was *k* = 10.5 ± 0.8 mmHg. The range of Po_{2} values inside the muscle fibers was found to be 4–5 mmHg at the critical Po_{2}. The oxygen dependence of respiration can be studied in thin muscles under different experimental conditions. In resting muscle, the critical Po_{2} was substantially lower than the interstitial Po_{2} of 53 ± 2 mmHg, a finding that indicates that V̇o_{2} under this circumstance is independent of oxygen supply and is discordant with the conventional hypothesis of metabolic regulation of the oxygen supply to tissue.

- skeletal muscle
- respiratory rate
- interstitial Po
_{2} - oxygen disappearance curve
- phosphorescence quenching method
- cell Po
_{2}gradient

the coordination of oxygen demand and supply in skeletal muscle and in the heart is carried out by a mechanism not yet completely understood. Current cardiovascular texts propose the century-old hypothesis of metabolic control of capillary blood flow as the accepted theory of local autoregulation (review Ref. 38). According to this hypothesis, a decline of oxygen delivery leads to a decrease of intracellular Po_{2}, an evoked release of a metabolic vasodilator into the extracellular space, the dilation of arterioles, and, eventually, an increase in the flow velocity of blood and number of perfused capillaries. A key aspect of this model is the oxygen dependence of cellular metabolism, making the mitochondria or entire cell sensitive to an inadequate oxygen supply (11).

The oxygen dependence of respiration for isolated mitochondria and cells is represented by a hyperbolic curve empirically described by Hill's equation (*Eq. 10*) (29, 39, 46, 59, 60) with the parameters *V*_{m} (maximal respiratory rate), Hill coefficient = 1 to 1.4, and *P*_{50} (Po_{2} corresponding to a V̇o_{2} of one-half *V*_{m}). The curve shows the relative independence of the rate of respiration at high Po_{2}, and a strong dependence at low Po_{2}, while the point of transition of the two portions of the curve defines the critical Po_{2} *P*_{crit}.

It was shown in early studies (10, 22, 25, 29, 56) that the oxygen consumption of isolated mitochondria and cells remains relatively independent of the Po_{2} in their environment over a wide range of Po_{2}. A suspension of isolated mitochondria is insensitive to Po_{2} elevation >1 mmHg (10, 13, 14, 40, 60). Suspensions of isolated resting muscle cells show values of apparent *P*_{50} higher than those in mitochondria, yet much lower than the Po_{2} in venous blood, which is approximately equilibrated with the tissue around capillaries (2, 14, 24, 26, 35, 39). Thus, based on measurements made on isolated mitochondria and myocytes, mitochondria may not serve as oxygen sensors monitoring the physiological Po_{2} level in tissue because of their low critical Po_{2} (54).

Oxygen consumption by mitochondria depends on the rate of biochemical processes and the availability of substrates and oxygen (7, 10, 13, 14, 60). At the cellular level, the oxygen dependence of respiration is modulated by the cellular functional state and its capacity for oxygen transport. Diffusivity and solubility of oxygen, in concert with cell size and the spatial distribution of mitochondria, appear to be additional determinants of the oxygen dependence of respiration (2, 4, 21, 24, 37, 48, 53).

On the tissue/organ level, the external control of cell respiration (via contraction) and microcirculatory control of oxygen delivery appear in addition to the existing mechanisms of regulation at the levels of individual mitochondria and cells. It is also suggested that inhibiting cytochrome *c* oxidase with nitric oxide produces the contribution of intercellular regulation of muscle respiration by all tissue cells including the vascular endothelium (8, 12, 42). Thus the factors affecting the oxygen dependence of respiration in the tissue lead to a set of parameters *V*_{m}, *P*_{50}, and *P*_{crit} different from those obtained in isolated cells and mitochondria. The importance of the study of oxygen dependence of respiration in situ was well formulated by Wilson (54): “The oxygen dependence of cellular oxidative phosphorylation remains highly controversial. Quantitative knowledge of that dependence is critical for understanding of not only cellular biochemistry but also a wide range of physiological functions that help to regulate both metabolism and the oxygen delivery system. Is mitochondrial oxidative phosphorylation dependent on the oxygen pressures in normal tissues?” To answer this question, new approaches for the study of the oxygen dependence of respiration in living muscle in situ have been sought.

With the introduction of the polarographic method for measuring oxygen in tissues, it became possible to record the disappearance of oxygen caused by a momentary stoppage of blood flow. The interpretation of these curves was aimed at obtaining information on the rate of tissue respiration and its dependence on oxygen tension in the tissue (9, 34). The method was not widely used because of the complexity of accounting for the contribution of oxygenated blood and the limitations associated with the microelectrode technique of measuring oxygen.

The invention of the phosphorescence quenching method (PQM) paved the way for the measurement of Po_{2} in microscopic volumes of various organs (52, 61). Now one can record separate measurements of oxygen in the microvessels (3, 49, 62), in the interstitial fluid (44, 50, 58), and within individual muscle cells (45). Interstitial oxygen tension takes on an intermediate value between the intracapillary and intracellular Po_{2}, reflecting the current balance between rates of delivery and consumption of oxygen by muscle fibers. Furthermore, the interstitial oxygen tension is the Po_{2} on the surface of muscle cells, representing the boundary condition for the diffusion of oxygen into the cell. The critical Po_{2} of skeletal muscle in situ was determined for the first time in 1999 by recording the fall in interstitial Po_{2} caused by the rapid arrest of blood flow (36). As a criterion for the critical oxygen tension, workers used an increase of NADH fluorescence and a sharp change in the rate of decline in interstitial Po_{2}. In normally perfused resting muscle, the authors reported venular Po_{2} of 17.7 mmHg and a 3-mmHg Po_{2} decrease to the interstitial Po_{2} of 14.6 mmHg. Interstitial critical Po_{2} as defined by the two different criteria mentioned above was found to be in the range of 2.4–2.9 mmHg, which was slightly higher than that in isolated muscle fibers.

In our present work, we develop this approach by improving the quality of the Po_{2} measurements through reducing the artifact of oxygen consumption caused by the phosphorescence quenching method in a stationary fluid. Correction for the artifacts is done when calculating the oxygen disappearance curve (ODC) recorded in the interstitium. We also present a model for the interpretation of the dynamics of the interstitial Po_{2} decline due to oxygen consumption by muscle fibers to develop a new fitting model for the analysis of experimental data on the oxygen dependence of respiration in muscle.

## METHODS

In this study, we propose a method for the analysis of the ODCs in the interstitium of a thin skeletal muscle produced by the rapid pneumatic compression of the tissue. The measuring procedure for Po_{2} and V̇o_{2} in a muscle using the phosphorescent oxygen probe loaded into the interstitial space has been published before (18). However, previously we used only the initial part of an ODC to evaluate the respiration rate in the spinotrapezius muscle. In our present work, we have developed an approach for analysis of the entire ODC to determine the oxygen dependency of respiration of the muscle fibers in situ.

A thin planar muscle prepared for intravital microscopy (1) was placed between a thermo-stabilized sapphire plate and a gas barrier film. The interstitial space of the muscle was loaded with an albumin bound phosphorescent oxygen probe. Blood flow in the muscle was interrupted by rapidly inflating a bag of transparent film attached to the objective lens. Also, the removal of red blood cells (RBCs) from the muscle in the measuring volume was achieved and confirmed by microscopic observation. For the Po_{2} measurements, a brief light pulse (laser 532 nm, 15-ns duration, 1 pulse/s) was used to excite the probe inside a tissue disk of radius 300 μm.

In the following analysis, we use the flash number, *n*, as the independent variable instead of time. The index *n* = 0 denotes the variable before the onset of compression. The first flash after compression is denoted by *n* = 1. Under the conditions described above, the interstitial Po_{2} = *P*_{0} for normal blood flow in capillaries. Then, the rapid compression of the muscle removes RBCs from the vessels, leaving only physically dissolved oxygen in the tissue. From that moment, the interstitial Po_{2} inside the illuminated tissue disk is measured, thus forming the ODC data set (*P*_{n}; see Fig. 1).

The rate of Po_{2} change inside the sampled volume (*P*_{n}^{′}) depends on three components: first, the metabolic or cellular oxygen consumption component (*V*_{n}) which is the subject of interest; second, the photo-consumption by the method itself (*KP*_{n}); and third, the diffusion oxygen inflow from the surrounding tissue, proportional to the Po_{2} difference *Z*(*p*_{n} − *P*_{n}) at the boundary of the illuminated region. Here *p*_{n} is the Po_{2} outside the illuminated tissue disk at the moment of the *n*th flash, and the parameters *K* and *Z* are empirical coefficients of oxygen photo-consumption and inflow, respectively, which can be evaluated by fitting the experimental test data to the equations that follow. To account for all the factors influencing the measured rate of Po_{2} decrease, consider the equation:

The data set (*P*_{n}) is obtained from the experimental ODC, and the rate of Po_{2} drop (*P*_{n}^{′}) can be calculated by differentiating the ODC. The goal is to evaluate the rate of tissue respiration V̇o_{2} from the metabolic component *V*_{n}, which is calculated for a flash rate *F* = 1 Hz and the oxygen solubility in the muscle [α = 39 nl O_{2}/(cm^{3}·mmHg); Ref. 30] as:

We can simplify *Eq. 1* for the case when the metabolic component is absent (*V*_{n} = 0), for example, in a sample of dead tissue excised after the experiment (18). The ODC recorded in the sample under the same conditions of measurements as in vivo can be used for the evaluation of the coefficients *K* and *Z* and verification of the validity of assumptions underlying the model. In that case, the tissue outside the illuminated disk remains saturated with oxygen at an initial steady state Po_{2} of *p*_{n} = *P*_{0.}

The solution of *Eq. 1* under these conditions predicts an exponential decline of Po_{2}:

In the presence of oxygen inflow across the boundary of the illuminated region of tissue, the ODC approaches an asymptotic Po_{2} (*P*_{a}) formed by equilibrium between the processes of oxygen photo-consumption and inflow:

When oxygen inflow is negligible, as in the case of an excitation area much larger than the area of detection, the Po_{2} asymptotically approaches zero and *Eq. 3* is transformed into:

In our previous work (18), we have shown that *Eq. 3* is a good fitting model of the ODC in nonrespiring tissue. In this special test, we have determined values for the coefficients *K* (= 4.1·10^{−3}) and *Z* (= 1.5·10^{−3}) for correction of measurements made in situ. These coefficients are dimensionless; however, since we have omitted the flash rate 1 Hz in the equations for simplicity, the dimension appears as [s^{−1}].

Oxygen dependence of respiratory rate (V̇o_{2} vs. Po_{2}) for muscle fibers in situ. In our present work, we employed the phosphorescent probe distributed in muscle interstitial (extracellular and extravascular) space. Rapid (∼0.1-s pressure elevation) application of external pressure to the tissue expels the RBCs from the vessels and makes the ODC independent of hemoglobin. That experimental situation opens the opportunity to recover the dependence of muscle V̇o_{2} on Po_{2} in the interstitial space, i.e., on the surfaces of the muscle fibers. In that case, the entire ODC from *P*_{0} to near zero Po_{2} level has to be analyzed. With rising flash number, *n*, the difference between external and internal Po_{2} (*p*_{n} − *P*_{n}) increases, so the contribution of oxygen inflow must be taken into account.

Following tissue compression Po_{2} in the tissue outside the illuminated spot decreases only due to tissue respiration (no photo-consumption), so that the rate of Po_{2} change is:

If coefficient Z is not small enough to ignore the oxygen inflow, then the *V*_{n} can be obtained through the iterative calculation:

Analysis of an ODC can be greatly simplified if the inflow contribution is negligible and the time course of the oxygen consumption rate after occlusion can then be expressed as:

This is possible when the illuminated spot is much larger than the region of detection but is not always acceptable because of the intention to avoid light exposure of adjacent sites in case of subsequent multiple measurements in the same muscle. Our data were collected in the presence of oxygen inflow, so that is why *Eq. 8* was used in the analysis.

The obtained *V*_{n} are separated from the artifacts and can be converted into V̇o_{2} according to *Eq. 2*. A plot of (V̇o_{2})_{n} vs. (*P*_{n}) values at sequential flashes represents the oxygen dependence of respiration for muscle fibers in situ (see Fig. 5), which can be fit with a sigmoid curve, described by Hill's equation, to evaluate the parameters *V*_{m}, maximal respiration rate for a collection of muscle fibers (hereafter the symbol *V* is used to designate the rate of oxygen consumption); *P*_{50}, oxygen tension for half-maximal respiration rate; and *a*, Hill coefficient:

However, this empirical approach gives only a limited understanding of the dependence of the rate of cell respiration on oxygen level. For that purpose, we developed a model to relate interstitial Po_{2} and the rate of mitochondrial respiration per unit volume of the cell.

Interpretation of the oxygen dependence curves. Since the oxygen probe is distributed in the interstitial space, it reports the Po_{2} on the surface of muscle fibers, at the sarcolemma, both during steady state and during the transient conditions of the ODC. Thus the curve relating respiration to the Po_{2} on the surface of muscle cells can be analyzed using an appropriate model. That model should take into account the respiratory dependence of microscopic intracellular volumes (related to the functional activity of mitochondria) and the Po_{2} gradient in cells produced by the transport resistance due to diffusion.

Our model is based on the assumption that all oxygen sinks (mitochondria) in the muscle fibers are identical to each other in their respiratory properties, which means they obey a hyperbolic equation (39, 40, 53, 60), written below in a normalized form:

where *v* is the local specific oxygen consumption (by an elementary volume); *V*_{M} is the maximal volume-specific O_{2} consumption, which is the same for the entire tissue (*V*_{m}) and for the elementary volumes inside the cells (*V*_{M}), so that we can set *V*_{m} = *V*_{M}; *p* is the local intracellular Po_{2}; and *k* is the local Po_{2} corresponding to the half-maximal respiration rate (i.e., *P*_{50} for mitochondria). We have attempted to explain the origin of the sigmoidal oxygen dependence of muscle cell respiration (*Eq. 10*) on the basis of the hyperbolic oxygen dependence of mitochondrial respiration (*Eq. 11*) and the intracellular gradient of Po_{2}. In a generalized muscle fiber (Fig. 2), the elementary volumes of the cell are depicted by concentric isobars. However, all these sinks in the muscle cells are localized in tissue volumes under different local oxygen tension *p* that creates heterogeneity in oxygen consumption rates inside the cell.

In our experiments, the parallel changes in interstitial Po_{2} and V̇o_{2} (*P* and *V*) were determined as values obtained at the sarcolemma (Fig. 2). There is no requirement of any special shape (circular, hexagonal, etc.) of the fiber cross-section; it can be quite natural. The only assumption is the existence of a Po_{2} gradient inside the muscle fibers, expressed as the difference, Δ = *P* − *P*_{c} between the surface (i.e., interstitial) Po_{2} = *P* and Po_{2} = *P*_{c} in the center of the fiber, i.e., the point in the fiber with the lowest Po_{2}.

A fraction of tissue volume *f*, having a given Po_{2} = *p* (isobaric volumes), also has the same respiration rate *v* (Fig. 2). As a first approximation we can consider the distribution *f*(*p*) to be a Uniform (or Rectangular, Fig. 3) distribution having a width Δ = *P* − *P*_{c} and a density *f* = 1/Δ, meaning that the total volume is equal to unity and the probability density function can be applied to represent the tissue volume distribution as a function of Po_{2}. This approach also will allow us to define the first and second moments of this distribution, yielding its mean value and width. Our aim is the recovery of information on the properties of intracellular respiration by determining best fit parameters for experimental data points using the equations generated by the model.

The Uniform distribution of the intracellular volume based on Po_{2}, with density *f* = 1/Δ, is presented in the diagrams of Fig. 3. Interstitial Po_{2} = *P* is the right border of the cellular volume distribution on the oxygen tension *p* having width Δ. There are three possible physiological situations in the muscle fiber: *1*) normoxia, *P* > Δ; *2*) critical Po_{2}, *P* = Δ; and *3*) hypoxia, *P* < Δ. When *P* > Δ all isobaric volumes *f* in a cell have Po_{2} > 0 and participate in oxygen consumption. When *P* = Δ, *P*_{c} = 0 and *v* = 0 at the center of the fiber; this value of *P* is known as “critical.” For *P* < Δ, some deep volumes presented by the shaded region left of zero Po_{2} are excluded from respiration. Since negative Po_{2} values are impossible, that part of the cell volume also has Po_{2} = 0 and total *V* is the sum (or integral, see *Eq. 12*) of the oxygen consumption rates only in volumes having Po_{2} > 0.

The total oxygen consumption rate *V* is the sum of respiratory rates, *v*, of the isobaric volumes multiplied by their volume fractions (*f* = 1/Δ). Generally, using *Eq. 11*, the total oxygen consumption rate normalized to the maximal rate *V*_{M} can be written as:

This expression can be presented in a form convenient for integration:

The limits of integration of *Eq. 13* are different for each of the situations shown in Fig. 3, and the solutions for *V* are also different. The consumption curve (i.e., *V* as a function of *P*) for a generalized muscle fiber or tissue consists of two different regions that correspond to two different interstitial Po_{2} conditions:

The line formed by the points separating the two regions of *V*(*P*), that is *V* for *P = Δ* (middle plot), is described by the equation for the critical Po_{2}:

The equations obtained for the normoxic and hypoxic ranges (*Eqs. 14* and *15*) of interstitial Po_{2} can be used as fitting models for the analysis of experimental curves on the oxygen dependence of respiration, while *Eq. 16* may be applied for accurate evaluation of the critical Po_{2}.

*Equations 14–16* make it possible to predict the behavior of the oxygen dependence of cellular respiration for different ranges of the intracellular Po_{2} gradient and oxygen demand. The set of theoretical curves generated for different Δ are shown in Fig. 4. The curves are calculated for a set of parameters [*V*_{M} = 100 nl O_{2}/(cm^{3}·s), *k* = 10 mmHg, and Δ = 0, 5, 10, 20, 30, 40 mmHg] to demonstrate the effect of an intracellular oxygen gradient on the oxygen dependency of respiration. The first curve (Fig. 4, *curve 1*) is the oxygen dependence for mitochondria described by *Eq. 11*. This is the same relationship for a whole cell in the absence of an oxygen gradient due to intracellular diffusion resistance. When the different contributions of the diffusion resistance occur, the Po_{2} difference between the sarcolemma and core (Fig. 4, *curves 2–6*, Δ = 5–40 mmHg) leads to a sigmoidal appearance of the cellular respiration dependence on Po_{2}. This connection allows us to determine the parameters for the mitochondrial respiratory dependency on oxygen from the observed experimental oxygen dependency of oxygen consumption for whole cells. Each of the five solid curves (*2–6*) consists of two regions, a normoxic region described by *V*_{1} and a hypoxic region described by *V*_{2}, according to *Eqs. 14* and *15*, respectively. The dashed line (*curve 7*) corresponds to the situation (critical Po_{2}) described by *V*_{3} (*Eq. 16*), indicating the points separating the normoxic and hypoxic regions of the curves. The same curves plotted as a double-logarithmic plot (Fig. 4, *right*) demonstrate that the hypoxic regions are transformed into straight lines, which turn into hyperbolic lines above the *dashed line 7*, corresponding to the critical dependence, *V*_{3}. *Curve 1* represents the case when there is no Po_{2} difference between the cellular surface and the core, for example, in the case of zero diffusion resistance or a very thin cell. An increase in diffusion resistance or thickness of the cells leads to a proportional shift in *curve 7* to the right. The same effect is caused by an increase in *k*, which reflects a greater oxygen dependence of mitochondrial respiration.

#### Animal experiments.

The experimental protocol followed for these measurements was previously published in detail (18). All procedures were approved by the Institutional Animal Care and Use Committee of Virginia Commonwealth University. Six female Sprague-Dawley rats were initially anesthetized with a mixture of ketamine/acepromazine (72/3 mg/kg ip). Once femoral vein access was obtained, the animals received supplemental anesthesia as a continuous intravenous infusion of alfaxalone acetate (Alfaxan, Schering-Plough Animal Health, Welwyn Garden City, UK; ∼0.1 mg/kg/min). At the termination of an experiment, Euthasol (150 mg/kg iv, pentobarbital component; Delmarva, Midlothian, VA) was administered while the animal was under a surgical plane of anesthesia. The spinotrapezius muscle was used for measurement of interstitial Po_{2} and the surgical preparation was similar to the original description by Bailey et al. (1) and Gray (19). The muscle was placed on a thermo-stabilized (37°C) pedestal of the animal platform (17). The muscle was covered with gas barrier plastic film (Saran, Dow Corning, Midland, MI). An objective-mounted film airbag connected to a pressure controller allowed organ compression at 130 mmHg, which rapidly squeezed blood out of microvessels in the thin spinotrapezius muscle (15). Circular regions of muscle 600 μm in diameter and containing no large microvessels were selected for V̇o_{2} measurements. The Po_{2} was sampled once a second during 200 s of Po_{2} data collection in a reactive hyperemia-type protocol. Before rapid airbag inflation, the interstitial Po_{2} at normal tissue perfusion (i.e., baseline) was recorded for 30 s. This was followed by 90 s of muscle compression to arrest blood flow, after which the airbag was deflated for the remainder of the recording period (i.e., 80 s). This protocol was repeated at 3–11 different sites around the muscle, with 5- to 10-min intervals between measurements. Preparation quality and viability were confirmed by a return of interstitial Po_{2} to baseline between consecutive measurements. The measurement of Po_{2} with PQM has been described in detail previously (18). Respiration rates, *V*_{n}, were calculated according to *Eq. 8*. Each ODC was differentiated using a five-point differentiation smoothing function, after checking that this procedure had no effect on the fitting analysis. The Levenberg-Marquardt algorithm was used for Po_{2} calculations to fit the multiple phosphorescence decays (one Po_{2} value per second for 200 s) with a program put together using the LabView software platform (National Instruments, Austin, TX). Statistical calculations and parameter fitting were made with the Origin 7.0 software package. All data are presented as means ± SE (number of measurements).

## RESULTS

The oxygen disappearance curves were recorded at 34 sites in 6 spinotrapezius muscles with measurements at 3–11 sites per muscle. Curves obtained in the same muscle were aligned (time base “correction”) and averaged (see Fig. 1, as an example). Measures described previously were taken to reduce the artifact of oxygen photo-consumption, and its contribution at the normal interstitial Po_{2} was 0.6%. The effect of oxygen inflow into the detection area was noticeable at the lowest Po_{2} (accounting for 3.5% of the Po_{2} change). *Equation 8* was used, along with these measured values, to correct the oxygen disappearance curves. The resulting corrected curves were used to calculate the dependence of oxygen consumption on Po_{2}, which was then plotted and fit with Hill's equation (*Eq. 10*). The parameters recovered for the total data set were as follows: *V*_{m} = 120.9 ± 7.7 nl O_{2}/(cm^{3}·s); *P*_{50} = 11.1 ± 0.9 mmHg; and the exponent *a* = 2.0 ± 0.1.

For further analysis of the oxygen dependency of respiration, we used fitting *Eqs. 14* and *15* (see Fig. 5) to estimate the intracellular Po_{2} range Δ, *V*_{M}, and *k* . The parameters *V*_{M} , *k*, and Δ_{1} were determined first for the normoxic region of the curve (*Eq. 14*), which comprises most of its length; then the hypoxic region of the curve was fit (*Eq. 15*) at fixed *V*_{M} and *k* taken from the first procedure, to make a second estimation of the Po_{2} range, Δ_{2}. An example of such an analysis is shown in Fig. 5 (the same data set as in Fig. 1), where most of the points belong to the normoxic region of the curve described by *Eq. 14* and the low Po_{2} segment was fit with *Eq. 15*. A double-logarithmic plot facilitates finding the point of separation between the two regions of the overall curve; it could also be calculated using *Eq. 16*.

The set of curves averaged for each muscle was homogeneous, but the range in maximal and minimal *V*_{M} and *k* among the muscles was twofold (see Table 1). The average difference between the intracellular Po_{2} ranges calculated with *Eqs. 14* and *15* (i.e., Δ_{1} and Δ_{2}) was within 1 mmHg, and these data sets are well correlated (*R* = 0.87; *p* = 0.025). A high correlation was also found between *V*_{M} and *k* (*R* = 0.94; *p* = 0.0055), while the other parameter sets showed no significant correlation. It follows from the derivation of *Eq. 16* that the critical Po_{2} is equal to Δ, and, for the value obtained for Δ of 4–5 mmHg, the corresponding critical oxygen consumption is 21.2–25.2 nl O_{2}/(cm^{3}·s).

## DISCUSSION

Further improvement of the optical technique (PQM) to measure Po_{2} in living organs, including corrections for instrumental artifacts and incorporation of several significant technical innovations, made it possible to update the study of tissue respiration in situ previously made by Richmond et al. (36). To eliminate the intravascular phosphorescence signal, the oxygen probe was loaded directly by diffusion into the intercellular space of the thin muscle. To eliminate the influence of intravascular oxygen, the flow arrest was performed by pneumatic compression of the muscle, which squeezed RBCs out of the microvessels. The pressure in the air bag rapidly rose to a level above the systolic blood pressure and extrusion of blood from the compressed muscle was monitored with video microscopy. The diameter of the measuring area was increased to 600 μm (vs. 20 μm in Ref. 36) to include the interstitial space around 10 muscle fibers and make the diameter of the measuring volume similar to its depth. The larger sampling volume allowed us to reduce the excitation energy density and flash rate to 1.8 pJ/μm^{2} and *F* = 1 Hz (vs. 31 pJ/μm^{2} and 50 Hz in Ref. 36) and provided a phosphorescence decay signal with signal-to-noise ratio good enough for analysis of individual decays.

These technical improvements significantly reduced the photo-consumption of oxygen by this method (16, 18). That is why the interstitial Po_{2} in our experiments was significantly higher at rest: 53 mmHg vs. 15 mmHg in the study by Richmond et al. (36). Similar values of interstitial Po_{2} in skeletal muscles have been reported by other workers. Recent studies (57, 58) of interstitial oxygenation with the PQM using new oxygen probes found that the peak of the histogram of interstitial Po_{2} in mouse skeletal muscle corresponded to 41 mmHg. The interstitial Po_{2} measured near 1st, 2nd, and 3rd order arterioles in rat cremaster muscle varied between 51–29 mmHg (43). In the rat diaphragm muscle, average microvascular Po_{2} was normally ∼50 mmHg, which may also indicate similar Po_{2} in the interstitium (32). Peri-arteriolar Po_{2} for 2A arterioles in cat muscle, measured with a microelectrode, was found to be 52–40 mmHg, depending on the Po_{2} of the superfusate (6). In the rat spinotrapezius muscle, the Po_{2} values obtained with a microelectrode in the vicinity of venules were close to 50 mmHg (27). It should be noted that the reference volume of a polarographic electrode is not limited to the interstitial space but also includes the intracellular content having a lower Po_{2} than that in the interstitium. The PQM also opened the possibility to localize Po_{2} measurements in a selected compartment: intravascular, interstitial, or intracellular (23, 58).

Recording the ODCs in a stationary interstitial fluid requires a series of tens of light pulses, so the artifact of accumulated photo-consumption should be considered and corrected for. To accomplish this, a mathematical model of oxygen measurements in a microscopic volume of muscle was formulated and the contribution of photo-consumption and diffusional inflow of oxygen was determined and used to correct the data. In future experiments, the analysis can be simplified by increasing the size of the excitation area compared with the area of signal detection, which will make the contribution of oxygen inflow negligible.

Corrected data on the metabolic component of ODCs were converted to respiration rates and plotted against the corresponding values of Po_{2}, thus forming a scatter plot of oxygen dependency of muscle fiber respiration in situ. The data obtained were well approximated by Hill's equation (*Eq. 10*), which was used to determine the parameters *V*_{m} = 120.9 nl O_{2}/(cm^{3}·s), *P*_{50} = 11.1 mmHg, and the exponent *a* = 2.0. The sigmoidal curve describing the oxygen dependency of respiration does not contain a specific point indicating the critical Po_{2} associated with the appearance of an anoxic core in muscle fibers. This fact limits the usefulness of an empirical fitting model and points out the need for finding an analytical description of the oxygen dependence of cell respiration, based on knowledge of oxygen uptake by mitochondria and the intracellular oxygen gradient created by the diffusional influx of oxygen into a cell.

There are a number of studies on mathematical modeling of oxygen diffusion combined with its consumption within a tissue slice or a given cell geometry. These models are aimed at finding the shape of the Po_{2} profile in a flat sheet, sphere, or circular cylinder. The oxygen dependency of respiration is assumed to be constant (20) or possess a specific Michaelis-Menten (MM) kinetics (28, 33). The latter possibility (*Eq. 11*) is a good representation for the kinetics of mitochondrial respiration (53) sometimes being used with the caveat of “pseudo” MM kinetics. Many of the published models are presented in the form of numerical solutions and/or are applicable only to ideal geometric forms, which reduce their practical value for the analysis of experimental results. For this purpose, it is necessary to find a quantitative explanation, relating the properties of mitochondrial respiration (pseudo-MM kinetics) with a heterogeneous distribution of intracellular oxygen, which leads to a sigmoidal curve describing the collective oxygen dependency.

We have presented a curve of the collective oxygen dependency as a product of the oxygen consumption kinetics of individual oxygen sinks (pseudo-MM) and the cell volume distribution on the basis of Po_{2} isobars, given by a simple probability density function. This approach allowed us to describe the heterogeneity of the oxygen distribution inside a cell with two parameters: the Po_{2} on the cell surface *P* and the width of the intracellular Po_{2} distribution Δ, which arose from the combined diffusion and chemical reaction inside the cell. *P* and Δ have relatively straightforward physiological meanings and they can be converted into statistical moments of the intracellular Po_{2} distribution. We aimed to obtain fitting functions (*Eqs. 14*–*16*) that could be applied to experimental data to recover the parameters *P* and *Δ* and predict the shape of the oxygen dependency for oxygen consumption in a skeletal muscle. This approach has the potential to be extended to form a histogram-like model, in which several Uniform distributions with different weighting coefficients can be recovered by fitting the experimental points of the ODC. The first attempts at direct measurements of Po_{2} distributions within cardiomyocytes (31) showed that the distribution of mitochondrial Po_{2} may depend on the fraction of oxygen in the inspired gas mixture, and, therefore, knowledge of the characteristics of this distribution are necessary for understanding the functional state of cells.

To establish the validity of the Uniform distribution to describe the heterogeneity of intracellular Po_{2}, let us compare the radial profiles of Po_{2} in the case of a muscle fiber in the form of a circular cylinder of radius *R*. As a simple example, we consider the conventional case of constant, uniform oxygen consumption V̇o_{2} and *p*(*r*) > *0* throughout the fiber. For this situation, the radial dependence of Po_{2} is:

where Do_{2} is the diffusion coefficient and α is the solubility of oxygen. From this equation, the volume fraction of the fiber contained within radius r is related to Po_{2} at this radius by

where the Po_{2} at the center of the fiber (*r = 0*) is *P*_{c} *= P* − V̇o_{2}*R*^{2}/*4*Do_{2}*α*. The Po_{2} volume density function, *f*(*p*), for this situation is given by its definition, *f*(*p*) = *4α* Do_{2}/V̇o_{2}*R*^{2}. Note that the right hand side of this equation is 1/(*P* − *P*_{c}) or *1*/Δ. This is exactly the value of *f*(*p*) used for the Uniform distribution in *Eq. 12*. For the MM kinetics used to describe the Po_{2} dependence of mitochondrial oxygen consumption in our model (*Eq. 11*), the Po_{2} profile will still be parabolic to a good approximation and thus the Uniform distribution given by *f*(*p*) = *1*/Δ will be appropriate.

A parabolic profile is the typical result for oxygen diffusion/consumption in a cylindrical fiber (20) and has been repeatedly confirmed in experiments on isolated muscle cells (47, 48). However, the observation of a parabolic Po_{2} profile does not necessarily require correspondence with Hill's model (20) in which the oxygen consumption by elementary cell volumes is independent of the Po_{2}. The diffusion coefficient Do_{2} can be calculated according to Hill's model as (4, 20, 45):

Calculations based on the values of parameters at the critical Po_{2} (Table 1) gives Do_{2} = 0.25·10^{−6} cm^{2}/s, which is much smaller than literature values (2, 5, 30). An explanation of this discrepancy lies in the inapplicability of Hill's model to the situation in real cells in which respiration is dependent on oxygen tension over wide limits (53, 56). This wider Po_{2} dependency range is described by Wilson et al. (53, 56) such that changes in the concentrations of various intracellular metabolic factors work together to maintain a relatively constant oxygen consumption in the face of decreasing Po_{2}. However, below a critical Po_{2}, changes in the concentrations of these substances are not able to work together to maintain oxygen consumption and it begins to fall. An additional factor to consider is the significant difference between the shape of muscle cells and a circular cylinder. Replacing the square of the radius by the cross-sectional area (45, 51) in the calculation of the diffusion coefficient using Hill's model is incorrect.

It should be noted that the proposed model is shape-independent and based on the assumption of intracellular heterogeneity in Po_{2}, which can be described by a Uniform distribution defined by the two parameters *P* and Δ. Mathematical solutions of the model formulated by *Eq. 12* for the three situations of cellular oxygenation–normoxic, hypoxic, and critical–are represented by the sigmoidal composite curve consisting of two regions connected at the point of critical Po_{2}. On a double-logarithmic plot, the low Po_{2} region (*Eq. 15*) appears as a straight line in contrast to the hyperbolic region (*Eq. 14*; Fig. 4, *right*). Remarkably, this property of the oxygen dependence curves was discovered earlier and used to determine the critical Po_{2} in experiments with isolated muscle cells (4). The resulting *Eqs. 14*–*16* do not have a formal resemblance to Hill's equation, although the resulting sigmoidal curves are obviously similar to it but depend only on the difference of Po_{2} between the surface and center of the muscle fibers (Fig. 4). By accounting for the oxygen dependence of mitochondrial respiration, we obtained a description of their collective effect at the cellular level, which somewhat changes the understanding of critical Po_{2} and the oxygen dependency of cellular respiration. The actual critical Po_{2}, corresponding to zero Po_{2} at the cell core, can be even lower than *P*_{50} for small Δ, although the oxygen dependence of respiration extends to much higher Po_{2} (see Fig. 5).

The parameters recovered by fitting the experimental oxygen dependency curves (i.e., V̇o_{2} vs. Po_{2}) with *Eqs. 14*–*16* are presented in Table 1. Relatively small differences were observed in the asymptotic values of *V*_{M} from the two models we considered (121 nl O_{2}/cm^{3}·s from *Eq. 10* and 139 nl O_{2}/cm^{3}·s from *Eqs. 14*–*16*). Practically no differences were found between the *P*_{50} = 11.1 mmHg obtained for muscle fibers using Hill's equation (*Eq. 10*) and *k* = 10.5 mmHg for mitochondrial respiration. It is well known that the *P*_{50} for coupled isolated mitochondria under a sufficient concentration of ATP is ∼0.5–1 mmHg, while in the presence of an uncoupler, *P*_{50} is <0.03 mmHg (14, 60). It has also been shown that diffusion limitations approximately double the value of *P*_{50} in isolated cells (26, 39, 53). The oxygen dependence of respiration in isolated mitochondria and cells is usually studied with vigorous stirring to reduce the contribution of diffusion resistance (14, 60). For cells in organs and tissues, convective effects are limited to blood flow through nearby microvessels, while both interstitial fluid and sarcoplasm are essentially stationary in a resting striated muscle. There is a possibility that the *P*_{50} value is dependent on the diffusional resistance to oxygen transport between the capillary to mitochondria, and this may be part of the explanation as to why the oxygen dependence of respiration extends to >30 mmHg (53, 55). The question of the extent to which diffusion of oxygen determines the oxygen dependence of cellular respiration in situ is extremely important, but poorly understood.

The Po_{2} difference, Δ, and critical Po_{2} estimated with *Eqs. 14* and *15* yielded close results and all six pairs of values are well correlated. In this regard, one may consider the possible distortion of the curve of oxygen dependency through interference caused by the presence of myoglobin. Due to the very low *P*_{50} for myoglobin (2.39 mmHg at 37°C and pH = 7.0; Ref. 41), it is highly saturated at normal Po_{2}, so that the effect on oxygen dependency should occur only at low Po_{2}. If the effect of myoglobin is not negligible, then the difference in the observed values of Δ_{1} and Δ_{2} would be expected to be significant, but they are not (Table 1). The final resolution of this issue will require experiments in which the influence of muscle myoglobin has been eliminated; however, close agreement between Δ_{1} and Δ_{2} indicates the marginal impact of myoglobin in the spinotrapezius muscle.

The definitions of critical Po_{2} are different for mitochondria and cells. The contribution of diffusion resistance to *P*_{crit} in isolated mitochondria is negligible due to their small size, while for the whole cell it can be the determining factor at a high level of metabolism. In a muscle, a sharp increase in NADH fluorescence reports mitochondrial anoxia, while an abrupt change in the rate of decline of extracellular Po_{2} corresponds to *P*_{crit} for the myocytes (35, 36). Critical oxygen tension in the cells of the spinotrapezius muscle was measured in isolated cells and in situ, and *P*_{crit} in isolated cells was 1.25 mmHg, somewhat lower than the in situ value of 2.9 mmHg (35, 36). According to our data, a *P*_{crit} of 4 to 5 mmHg is close to these values but too low for involvement of the critical Po_{2} in oxygen sensing by resting myocytes. However, due to diffusion limitations, the oxygen dependency of respiration extends to the range of physiological oxygen pressure in the interstitium (53, 56) or 53 mmHg in the present study. It should be noted that the sensitivity of the respiratory rate to oxygen is small at this Po_{2}, but it may increase with increasing intracellular differences of Po_{2} (right shift in Fig. 4) caused by an augmentation in metabolic activity or cell diameter.

Taking advantage of the range of variability of the parameters obtained in six muscles, we assessed the connections among them and found that *V*_{M} and *k* are strongly correlated. This correlation indicates a self-similarity of the oxygen dependence curves for various rates of metabolism. In that case, curves with different *V*_{M} are located to the right of the line passing through the origin with а slope equal to the diffusion resistance of the cell (V̇o_{2}/Po_{2}). Given the small number of muscles studied, this relationship can be considered only hypothetically possible. Later, this phenomenon can be studied with greater precision, considering the ability of muscles to increase their maximum oxygen consumption many fold. In the proposed model, *V*_{M} is assumed to be the same for different values of Δ, which simplifies the analysis but limits its applicability. Clearly, a significant increase in Δ is the result of increased respiration rate, and, in the analysis of future experiments with stimulated oxygen consumption, the physical relationship between *V*_{M} and Δ will be taken into account.

In conclusion, we have developed an approach to study the oxygen dependence of respiration in a skeletal muscle in situ, using Po_{2} measurements in interstitial fluid made with phosphorescence quenching microscopy and rapid pneumatic compression of the tissue. The metabolic component of the oxygen disappearance curve was used to construct a plot of oxygen dependency of cell respiration, which was analyzed using a model for oxygen consumption developed for the situation of heterogeneous Po_{2}. The model predicted a number of properties for the oxygen dependence of cellular respiration associated with the existence of a respiratory-induced Po_{2} gradient in cells: *1*) the dependence has a sigmoidal shape with an increasing rightward *P*_{50} shift with increasing intracellular Po_{2} gradient; *2*) the dependence is described by two different functions, which represent normoxic and hypoxic regions of the model, the graphs of which are connected at the point for the critical Po_{2} of the cell; and *3*) at physiological values of the intracellular Po_{2} gradient, the critical Po_{2} for the cells is below their *P*_{50}.

Above the critical Po_{2} or critical oxygen delivery, as usually understood, most published studies demonstrate that oxygen consumption is independent of oxygen delivery. Our analysis showed that, although the critical cellular Po_{2} is much lower than the physiological oxygen tension in the interstitium for resting muscle, the oxygen dependency of cellular respiration may reach high Po_{2} values. To what extent the respiratory oxygen dependency of muscle fibers determines their ability to serve as oxygen sensors in the regulation of oxygen delivery can be established in future experiments applying this novel method to the situation of enhanced oxygen consumption caused by muscle stimulation and uncoupling of oxidative phosphorylation.

## GRANTS

This research is supported by National Heart, Lung, and Blood Institute Grants HL-18292 and HL-79087.

## DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the author(s).

## AUTHOR CONTRIBUTIONS

Author contributions: A.S.G. conception and design of research; A.S.G. performed experiments; A.S.G. analyzed data; A.S.G. and R.N.P. interpreted results of experiments; A.S.G. prepared figures; A.S.G. drafted manuscript; A.S.G. and R.N.P. edited and revised manuscript; A.S.G. and R.N.P. approved final version of manuscript.

- Copyright © 2012 the American Physiological Society

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