## Abstract

The problem of diffusion of O_{2} across the endothelial surface in precapillary vessels and its utilization in the vascular wall remains unresolved. To establish a relationship between precapillary release of O_{2} and vascular wall consumption, we estimated the intravascular flux of O_{2} on the basis of published in vivo measurements. To interpret the data, we utilized a diffusion model of the vascular wall and computed possible physiological ranges for O_{2} consumption. We found that many flux values were not consistent with the diffusion model. We estimated the mitochondrial-based maximum O_{2} consumption of the vascular wall (M_{mt}) and a possible contribution to O_{2} consumption of nitric oxide production by endothelial cells (M_{NO}). Many values of O_{2} consumption predicted from the diffusion model exceeded M_{mt} + M_{NO}. In contrast, reported values of O_{2}consumption for endothelial and smooth muscle cell suspensions and vascular strips in vitro do not exceed M_{mt}. We conjecture that most of the reported values of intravascular O_{2} flux are overestimated, and the likely source is in the experimental estimates of convective O_{2} transport at upstream and downstream points of unbranched vascular segments.

- microcirculation
- oxygen diffusion
- endothelial cell
- smooth muscle cell
- arterioles
- nitric oxide

o_{2}
diffusion from arterioles was demonstrated by Duling and Berne (12) and subsequently quantified by a number of researchers in different species and tissues, e.g., brain (13,27, 73), muscle (38, 39), liver and pancreas (59, 60), and mesentery (71). Analysis of the data by Kuo and Pittman (38, 39) showed that the experimentally observed rate of O_{2} transfer from the lumen into the wall was an order of magnitude higher than that predicted by a theoretical model (57). These results were corroborated by a more geometrically detailed model (58). To explain this discrepancy, Popel et al. (57) hypothesized a significantly higher (up to two orders of magnitude) in vivo tissue permeability to O_{2} (Krogh diffusion coefficient), but subsequent specially designed experiments did not confirm this hypothesis, although a significant correction of a factor of two was found (4, 45). Previous theoretical analyses of precapillary transport assumed an O_{2} consumption rate of the vascular wall (M) similar to that of resting muscle [on the basis of experiments with arterial smooth muscle (34)] or assumed it to be negligible (25, 56, 57, 76). Tsai et al. (71) proposed that the higher-than-expected O_{2}flux can be explained by high M, two orders of magnitude higher than that of the surrounding tissue in their experiments using the rat mesentery preparation.

In this study, we analyzed existing data from a variety of sources on O_{2} consumption by endothelial and smooth muscle cells in suspension and by vascular segments in vitro and in vivo. An O_{2} consumption rate based on the maximum mitochondrial O_{2} respiration rate (M_{mt}) was evaluated with the use of available data on mitochondrial content of endothelial and smooth muscle cells. We also examined one of the extramitochondrial pathways by analyzing experimental measurements of nitric oxide (NO) production in the cytosol of the endothelial cell and found that the O_{2} consumption rate for producing NO, M_{NO}, is at least an order of magnitude smaller than the maximum consumption associated with the mitochondrial pathway. We have shown that the O_{2} consumption in cell suspensions and vascular segments in vitro is below the estimated M_{mt}. In contrast, many estimates of O_{2} consumption from in vivo experiments on the basis of direct measurements of microvascular hemoglobin O_{2}saturation gradients and blood flow rate in single unbranched vessels exceed M_{mt} by one or two orders of magnitude. In an attempt to resolve this problem, we estimated the sensitivity of the intravascular flux (J_{i}) to changes in M for given intravascular (P_{i}) and perivascular (P_{o}) Po
_{2} values and found that the predited fluxes were nearly constant at ∼10^{−6} ml O_{2} · cm^{−2}· s^{−1} for M spanning over four orders of magnitude. We conjectured that precapillary J_{i} estimates reported by several laboratories are overestimated by as much as one or two orders of magnitude, and we discussed possible measurement artifacts that can account for the overestimates.

## RESULTS

In this section, we first present the estimated M_{mt}followed by M_{NO} (Table 1). O_{2} consumption for cell suspensions and vascular segments measured in vitro are presented in Tables 2 and 3, respectively. Next, the calculations of O_{2} flux at the luminal surface of the arteriolar wall on the basis of in vivo measurements of longitudinal hemoglobin O_{2} saturation (So
_{2}) or Po
_{2} gradients and blood flow rates in microvessels of different sizes in several tissues are presented in Table 4. One-layer and a two-layer diffusion models representing the vascular wall are used to evaluate the upper and lower bounds of M and P_{o}. We used these data to calculate physical bounds for O_{2} consumption in the vascular wall in Table 5 and showed that in most cases even lower bounds exceed the M_{mt}. We repeated the calculations for some of the cases for a two-layer model, in which we estimated the physical bounds of O_{2} consumption for the endothelial cell layer and assumed O_{2} consumption for the smooth muscle layer on the basis of experimental data. We then performed a sensitivity analysis of intravascular O_{2} flux over several orders of magnitude of M.

#### M_{mt}.

The mitochondrial volume content in capillary endothelial cells ranges between 2 and 5% for a variety of tissues (49). For smooth muscle cells, the mitochondrial volume content is ∼5% (61). If we know the maximum mitochondrial respiration rate, we can estimate the M_{mt}. In the mitochondria of locomotory muscles of mammals running at their maximum aerobic capacity (V˙o
_{2max}), the O_{2} consumption rate is 8.3 × 10^{−2} ml O_{2} · ml mitochondria^{−1} · s^{−1}(23). Interestingly, the respiration rate of 0.167 ml O_{2} · ml^{−1} · s^{−1} for mitochondria in hummingbird flight muscles is approximately twice that of mammals, and these mitochondria also contain twice the amount of oxidative enzymes (65). It is known that the respiration rate of mitochondria at V˙o
_{2max} is ∼80% of the maximum rate that can be achieved in suspensions with appropriate substrates (23). Thus the maximum respiration rate is estimated to be 0.1 ml O_{2} · ml mitochondria^{−1} · s^{−1}. Therefore, assuming a mitochondrial volume content of vascular wall of 5% as a maximum, the M_{mt} for vascular wall can be estimated to be 5 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1}.

This estimate does not take into account O_{2} consumption outside the mitochondria. Jobsis (29) stated that extramitochondrial consumption (sometimes referred to as cyanide-insensitive consumption) could account for 10–15% of the total consumption. In the following section, we show that M_{NO} is at least an order of magnitude smaller than M_{mt}.

#### Consumption of O_{2} by EC for the production of NO.

The intense interest and rapid progress in the study of NO synthesis in biological tissues has developed primarily because *1*) NO synthesis has been found in a variety of cell types, *2*) NO regulates and affects physiological processes, and *3*) NO synthesis via the oxidation of l-arginine has been shown to involve unusual oxidative chemistry. The primary pathway for the production of NO is from l-arginine that is catalyzed by the enzyme NO synthase (NOS) (64). NOS is found in the cytosol of the endothelial cell and can hence represent a possible site for O_{2} consumption outside the mitochondria. In Table1, we have presented values of M_{NO} measured under different conditions. All values shown have been calculated from the amount of NO produced, with the use of the stoichiometric ratio of two molecules of O_{2} consumed for every molecule of NO produced (64). Clementi et al. (7) have studied in detail the mechanism by which endothelial cells regulate their O_{2} consumption. Their experiments showed that NO generated by vascular endothelial cells under basal and stimulated conditions modulates the O_{2}concentration near the cells. This action occurs at the cytochrome c oxidase in the mitochondria and depends on the influx of Ca^{2+}. Thus NO plays a physiological role in adjusting the capacity of this enzyme to use O_{2}, allowing endothelial cells to adapt to acute changes in their environment. In a cell suspension having a density of 10^{7} cells/ml, the initial rate that was also the peak M_{NO} was estimated to be 2.8 × 10^{−4} ml O_{2} · ml cell^{−1} · s^{−1}. We converted the values from per cell basis to per cell volume basis using the microvascular endothelial cell volume of 400 μm^{3} reported by Haas and Duling (20). Endothelial cell dimensions in large vessels presented by Levesque and Nerem (41) are consistent with those reported in Haas and Duling (20). For the remainder of the experiment, cell respiration was inhibited in parallel with the generation of NO. These results suggest that, whereas M_{NO}can be as high as ∼10^{−4} ml O_{2} · ml^{−1} · s^{−1}, NO itself is responsible for inhibiting the predominant pathways for O_{2} consumption in the mitochondria, thus possibly reducing the overall M.

The maximum value of M_{NO} (∼10^{−4} ml O_{2} · ml^{−1} · s^{−1}) is measured for bradykinin and shear stress-stimulated NO production as estimated from experiments (7, 19) and a mathematical model (72). The lowest values of M_{NO} are those in which NOS was not stimulated by an agonist and are ∼10^{−6} ml O_{2} · ml^{−1}· s^{−1} (2, 36, 40). NO production data reported as per milligram of protein (2, 19, 36) have been converted on a per milliliter basis by using the measured protein content in endothelial cells of 0.1 mg protein/10^{6} cells (2). Therefore, M_{NO} is at least an order of magnitude lower than M_{mt}. In principle, it is possible that O_{2} utilization in arteriolar endothelial cells in vivo, through some other pathway, is significantly higher than is presently believed, but to the best of our knowledge, no such pathway has been identified.

#### O_{2} consumption by endothelial and smooth muscle cells.

Table 2 summarizes measurements of O_{2} consumption rates for endothelial cells (M_{e}) and smooth muscle cell suspensions (M_{s}). The original sources present the consumption rates on a per cell basis. We converted these to a cell volume basis using the cell volume for microvascular endothelial cells and smooth muscle cells as estimated in Haas and Duling (20): 400 μm^{3} for endothelial cells and 3,000 μm^{3} for smooth muscle cells.

O_{2} consumption measurements made by Kjellstrom et al. (32) in cell cultures indicate a dependence on the source of the cell. Endothelial cell cultures from a bovine aortic cell line exhibit a respiration rate that is several times smaller than cultures derived from the rat pulmonary artery when grown in the same cell nutrient media. Data presented by Motterlini et al. (47) for O_{2} consumption on a cell count basis (per million cells) appear to indicate that smooth muscle cells consume more O_{2} than endothelial cells. However, this is not the case when the same data are presented in terms of cell volume, because the volume of a smooth muscle cell is ∼10 times larger than that of an endothelial cell. Note that M_{NO} values presented in Table 1are of the same order of magnitude or lower compared with M_{e} in suspension presented in Table 2. Bruttig and Joyner (5) reported M_{e} and M_{s} that were four to five orders of magnitude higher than the values presented in Table 2; values reported by Kuehl et al. (37) are two or three orders of magnitude higher. The source of this discrepancy is unknown.

Table 3 lists M for vascular segments presented on the basis of wet tissue volume. In most cases, data have been reported in terms of dry weight and we have converted them to wet tissue volume by first dividing the former by the reported percentage value of dry weight with respect to wet weight (20–27%) and then by multiplying it with the tissue density of 1.06 g/ml. M in vascular segments shown in Table 3 (9, 30, 33-35, 46, 48, 51,52) is generally lower than that in cell suspensions (17,32, 47, 63). In most cases, the vascular segments were devoid of the adventitial layer and consisted of only the intima-media region of the vessel wall, essentially containing endothelial and smooth muscle cells. Importantly, in both groups M is below the value of 5 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1}, corresponding to M_{mt}. In Figure1, we compiled all of the experimental values of M in the vascular wall listed in Tables 2 and 3. M_{mt} is presented as a horizontal line.

The respiration rates for smooth muscle cell suspensions varied depending on the substrate in the nutrient media (17). The difference may be due to substrate participation at different points along the tricarboxylic acid cycle. This has also been observed for vascular segments in which the respiration rates of each layer exhibited a change when the substrate in the nutrient media changed (33).

In the above studies, M was measured directly by placing the vascular segments in a sealed chamber with oxygenated solution. In two studies, Po
_{2} was measured with O_{2} electrodes at several depths in the vascular wall in vivo (6) or in vitro (10, 55) and M was calculated using a model of O_{2} diffusion.

Experimental studies show an increase in M with vascular stimulation and contraction. These values may vary from approximately twice the resting muscle M (35, 51) to as much as 10 times (3). Values of M reported in Table 3 with the use of the data from Paul (50) have been calculated by estimating the isometric wall stress in a vessel having a blood pressure of 90 mmHg by multiplying the vessel radius by the arterial blood pressure and dividing the result with the corresponding vessel thickness (the Law of Laplace). Studies conducted by Paul (50) indicate a linear trend in ATP consumption with increasing isometric stress in the vessel wall. The corresponding ATP consumption was computed with the use of the relationship between ATP consumed and isometric force (50). On the basis of the respiration cycle, the O_{2} consumed was calculated from the ATP consumed by dividing the latter by the stoichiometric ratio 6.42 moles of ATP per mole of O_{2}. Note that the values of vascular tissue consumption are one to two orders of magnitude smaller than M_{mt}, except for the value for bovine cerebral microvessels in succinate (1.6 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1}), which is three times smaller. One of few available estimates of M presented by Tsai et al. (71) is an order of magnitude higher than M_{mt}, at least one order of magnitude higher than M in cell suspensions and vascular segments and at least two orders of magnitude higher than M_{NO}.

The experimental methods used to measure M in cell suspensions and vascular segments include microrespirometry, phosphorescence quenching, electron paramagnetic resonance oximetry, and polarography and generally compare well with each other. Differences in measured M may be due to different substrates, different temperature and Po
_{2}, vessel diameters, and sources of tissue samples.

#### Calculation of J_{i} from in vivo measurements in microvessels.

We showed that, with noted exceptions, in vitro measurements in cell suspensions and in vivo measurements in arteries and vascular segments yield values of M smaller than M_{mt}. We then analyzed in vivo data from different laboratories in different species and tissues from unbranched microvascular segments, on the basis of measurements of longitudinal hemoglobin SO_{2} or Po
_{2}gradients. We have shown that in most cases either the data cannot be interpreted in terms of a diffusion model or the predicted values of M exceed M_{mt} or even M_{mt} + M_{NO}.

For a cylindrical segment of blood vessel with a luminal diameter d and length Δz, the diffusive loss of O_{2} can be estimated as
Equation 1where Q = (πd^{2}/4)v is the volumetric blood flow rate, v is the mean velocity, [Hb] is the concentration of hemoglobin in the blood, C_{b} is the O_{2} binding capacity of the hemoglobin, ΔS is the saturation difference between the upstream and downstream points along the vessel segment, and J_{i} is given per unit area at the vessel wall (luminal surface) averaged over the vessel circumference (53). J_{i} can be calculated from *Eq. 1
* with all other parameters determined experimentally. In Table 4, we showed the microvascular parameters d, v, ΔS/Δz, and [Hb] for eight different sets of in vivo measurements necessary for calculating J_{i}. [Hb] for hamster and rat was calculated as the product of the discharge hematocrit and [Hb] in a single red blood cell (RBC) whose value is taken to be 19.58 mM (67). It was not necessary to calculate the J_{i} for the data of Seiyama et al. (59, 60) since the authors had already carried out these calculations. P_{i} values were available for Tsai et al. (71) and Torres Filho et al. (70) since these authors used the phosphorescence quenching method to measure P_{i}; in these studies, the authors determined ΔS values by converting ΔPo
_{2} into ΔS values with the Hill equation. For all of the other sources in which So
_{2} was determined microspectrophotometrically, So
_{2} values were converted into Po
_{2} with the use of the Hill equation. The Hill equation parameters *n* (cooperativity of Hb) and P_{50} (O_{2} tension corresponding to 50% saturation) for rat were taken from Altman and Dittmer (1) and those of the hamster retractor muscle from Ellsworth et al. (16). K for the vascular wall was estimated to be ∼3.17 × 10^{−10} ml O_{2} · cm^{−1} · Torr^{−1} · s^{−1} and was calculated as the product of the diffusivity (D) in human arteries measured to be ∼0.96 × 10^{−5} cm^{2}/s (31), and the solubility coefficient (α) reported for many tissues to be ∼3.3 × 10^{−5} ml O_{2} · cm^{3} · Torr^{−1}(14). D is lower by a factor of ∼2.5 when compared with that of the hamster retractor muscle at 37°C (2.42 × 10^{−5} cm^{2}/s) (4, 14). For endothelial cells in vitro, D has been found to vary between 0.14–0.87 × 10^{−5} cm^{2}/s (43).

#### Physiological ranges for P_{o} and M using a one-layer diffusion model.

In a diffusion model of the vascular wall, the values of J_{i}, P_{i}, P_{o}, and M are not independent; if J_{i} and P_{i} are known from experiments, a range of possible values for P_{o}and M can be obtained subject to physical conditions P_{o} ≥ 0, M ≥ 0; if J_{i}, P_{i}, and P_{o} are known, then M can be uniquely determined. Therefore, to estimate the bounds of P_{o} and M for given values of other parameters, including P_{i} and J_{i}, we solved the problem of O_{2}transport in the vascular wall. The vessel wall was modeled by an axially symmetrical diffusion equation
Equation 2where r is the radius.

The boundary conditions are applied at the inside and outside surfaces of the vessel wall
Equation 3where r_{i} is the interior and r_{o} is the exterior radius of the vessel.

Integrating *Eq. 2
* and applying boundary conditions *Eq.3,* we get
Equation 4With *Eq. 4
* we obtain an expression for J_{i}
Equation 5and the perivascular flux (J_{o})
Equation 6M can be expressed from mass balance using J_{i} and J_{o}
Equation 7Rearranging *Eq. 7,* we obtain J_{o}in terms of J_{i} and M
Equation 8If we require that the vessel wall be supplied with O_{2}from the lumen and not from the parenchymal side, then J_{o} ≥ 0 in *Eq. 8,* and we obtain the upper bound on the M, M_{max}, when J_{o} = 0, (i.e., all of the O_{2}diffusing from the lumen is consumed by the vascular wall)
Equation 9Eliminating J_{o} from *Eqs. 6
* and *
8
* gives us P_{o} in terms of P_{i} and J_{i}
Equation 10The term in square brackets can be shown to be >0. Since P_{o} ≥ 0, *Eq. 10
* yields
Equation 11aThis inequality can be satisfied in two ways. *Case 1* is
Equation 11bthen
Equation 11cwhere M_{max} is given by *Eq. 9,*P_{o min} is the minimum possible P_{o}, and P_{o max} is the maximum P_{o}. P_{o min}is obtained from *Eq. 10
* by setting M = 0 and is given by
Equation 12and the P_{o max} is obtained by substituting *Eq.9
* into *Eq. 10
*
Equation 13The schematic representing *case 1* is shown in Fig.2
*A.* This case is physiologically possible since both P_{o min} and P_{o max} are positive. It is important to note that the P_{o} increases as M increases. This appears to be counterintuitive when compared with the conventional thinking in which higher M results in a higher J_{i} and a correspondingly lower Po
_{2}. Conventionally, e.g., under conditions of the Krogh tissue cylinder model, J_{i} is proportional to tissue O_{2}consumption (M_{t}); thus an increase in M_{t} leads to an increase in J_{i} (i.e., an increase in the negative slope of the Po
_{2} profile at the luminal surface) and thus a decrease in P_{o}. In contrast, in the case under consideration, J_{i} is fixed when M varies and the above argument is not applicable. Similar behavior will also be seen in the next two cases. *case 2* is
Equation 14a
*case 2a* then
Equation 14band, where M_{min} is obtained from *Eq. 10
* for P_{o} = 0
Equation 15A schematic of *case 2a* is presented in Fig.2
*B.* The P_{o} for M = 0 is negative, which is not possible. Hence, a physiologically possible case would have to be P_{o} > 0, M > M_{min}. The lower bound on the P_{o} is a strict inequality, since it is not possible to have P_{o} = 0 for a finite flux leaving the wall.

If the model parameters are such that *Eqs. 9-10
* yield P_{o max} < 0, then there is no solution to the problem, i.e., there are no physical values of M and P_{o}that satisfy the above relationships; thus *case 2b* has no solution if P_{o max} < 0, which is represented in Fig.2
*C.* Note the increasing slope with respect to the horizontal axis of the Po
_{2} profile at r = r_{i} as one moves from *case 1* to*2a* to *2b* in Fig. 2. This suggests that an in vivo measurement with a very high flux is most likely to fall under*case 2*.

All three cases are represented in the data analysis in Table5, in which the entries represented by blanks are those of *case 2b*. The values for the wall thickness, w, necessary for these calculations were chosen as follows. Swain and Pittman (67) optically measured w and d for different order arterioles in the hamster retractor muscle and found the following correlation: w = 0.24d + 0.39 μm. We use this relationship to estimate w for all blood vessels with diameters > 25 μm in Table 4. On the basis of measurements from Haas and Duling (20), a w of 6.5 μm (6 μm for a single layer of smooth muscle cells and 0.5 μm for a layer of endothelial cells) was used for our calculations in Table 5 for all arterioles with d < 25 μm. The capillary wall was assumed to consist of a layer of endothelial cells with a thickness of 0.5 μm.

Clearly, there is a problem in interpreting many of the experiments listed in Table 5 (13 out of 32) since they fall under *case 2b* in which no physical values of P_{o} and M can be found. In 9 out of 19 cases for arterioles shown in Table 5(capillaries excluded), the minimum M (M_{min}) exceeds M_{mt}, and in 8 out of 17 cases M_{min} also exceeds M_{mt} + M_{NO} (5.8 × 10^{−3}ml O_{2} · ml^{−1} · s^{−1}).

#### Physiological ranges for P_{o} and M with the use of a two-layer diffusion model.

In the one-layer model calculations, we lumped the endothelial and smooth muscle cells together, thus neglecting a possible difference in O_{2} consumption between these cell types. By considering a two-layer model (derivation presented in
), we show that taking these differences into account does not resolve the problem of interpreting the in vivo data. We chose M_{s} on the basis of published experimental values. The goal of the calculations was to estimate the physiological limits of the M_{e} and the P_{o} on the basis of the in vivo measurements presented in Table 4. In Table 6, we present calculations for three arteriolar cases from Table 4. The M_{s} is taken to be 10^{−4} ml O_{2}· ml^{−1} · s^{−1} on the basis of measurements presented in Table 2. The K for the endothelial cell layer was taken to be 4.45 × 10^{−10} ml O_{2} · cm^{−1} · Torr^{−1} · s^{−1} (43), whereas that of the smooth muscle cell layer was 4.5 × 10^{−10} ml O_{2} · cm^{−1}· Torr^{−1} · s^{−1} (1). Values of inner diameter, thickness, P_{i}, and J_{i} were taken from Table 4. The values of M_{e} are much higher than the corresponding computed values of M in Table 5 and higher than M_{mt}. This is primarily because we have fixed the M_{s} that occupy most of the vascular wall to a value that is two orders of magnitude lower than the values of M in Table 5. As a result, M_{e} that occupies a smaller volume fraction of the vascular wall is in general two orders of magnitude higher than the values of M in Table 5. Increasing the value of M_{s} from 10^{−4} to 10^{−3} ml O_{2} · ml^{−1} · s^{−1} has little effect on the results in Table 6. Because of the thin endothelial cell layer, we also do not encounter *case 2b* in Table 5 where P_{o max} < 0.

#### Sensitivity of P_{o} and M to input parameters with the use of the one-layer model.

To see the effect of input parameters on the overall results, we varied K, the P_{i}, and the thickness of the vascular wall and presented the results in Table7. Doubling K to 6.34 × 10^{−10} ml O_{2} · cm^{−1}· Torr^{−1} · s^{−1} results in a larger number of in vivo measurements falling under *case 1* (twice as many). Since doubling the value of K has the greatest effect on the in vivo results, we presented the physiological bounds for some of the in vivo measurements in Table8. The initial value K = 3.17 × 10^{−10} ml O_{2} · cm^{−1}· Torr^{−1} · s^{−1} is close to the value 3.64 × 10^{−10} ml O_{2} · cm^{−1} · Torr^{−1} · s^{−1} taken by Tsai et al. (71), but the calculated transvascular Po
_{2} difference of 11.4 Torr in Table 5 is less than the experimentally observed value of 18.1 Torr. Increasing K only further decreased the transvascular Po
_{2} difference.

The present calculations assume that P_{i} is the same as in the vessel on average, corresponding to the measured So
_{2}, whereas in reality there exists a gradient in P_{i}. At present, radial P_{i} gradients can be determined theoretically using mass transfer coefficients estimated for arteriolar-sized vessels by Hellums et al. (21). The mass transfer coefficient is used to represent the ratio of the J_{i} and the corresponding driving force in O_{2} tension. The dimensionless mass transfer coefficient, called the Nusselt number, is defined as
Equation 16where K_{pl} is the Krogh coefficient for O_{2} in plasma, P* is the O_{2} tension in equilibrium with the mixed mean hemoglobin saturation, P_{i}and J_{i} are measured at the vessel wall, and r_{i} is the internal radius of the vessel. On the basis of the calculations of Hellums et al. (21), for an arteriole with a diameter of 23 μm and flux of 6 × 10^{−5} ml O_{2} · cm^{−2} · s^{−1}, the radial P_{i} drop, P* − P_{i}, would be 66 Torr. Performing similar calculations with flux values from Table 4 for corresponding diameters yielded similar values. Obviously, these Po
_{2} drops are unrealistically high since they are greater than the measured P_{i} values in Table 5. On the other hand, if we use an order of magnitude lower flux value, consistent with those calculated from transvascular Po
_{2} measurements presented in Fig.3 and described in *Sensitivity of*J_{i}
* to wall consumption rates,* we obtain P* − P_{i} of ∼5–6 Torr. Hence, we chose a drop of 5 Torr in the P_{i} for our sensitivity analysis calculations. When the P_{i} is decreased by 5 Torr, the number of measurements falling under *case 1* and *case 2a* reduced by one and two, respectively, whereas those of *case 2b* increased correspondingly by three.

Changes in wall thickness only slightly affected the number of in vivo measurements falling under *case 1*, whereas the distribution of cases falling under *cases 2a* and *2b* changed to a larger extent. Hence, the results are most sensitive to a 100% increase in K, although no experimental measurements are available in support of such a high vascular wall K.

#### Sensitivity of J_{i} to wall consumption rates.

The sensitivity of J_{i} with respect to M can also be analyzed with the use of *Eq. 10
* and available experimental data on transvascular Po
_{2}. With the use of recessed-tip O_{2} microelectrodes, Duling et al. (13) measured the P_{i} and P_{o} for cat pial microvessels with internal diameters ranging from 22 μm to 230 μm and found the gradient (P_{i} − P_{o})/ w ≈ 1 Torr/μm for vessels of all sizes. Knowing the internal and external vessel diameter and the corresponding Po
_{2} for vessels groups in four different vascular orders, we can compute a J_{i}for M ranging between 10^{−6} and 10^{−2} ml O_{2} · ml^{−1} · s^{−1}. The results are presented in Fig. 3. J_{i} is found to be almost constant for M in the range of 10^{−6} to 10^{−4} ml O_{2} · ml^{−1} · s^{−1}, with values of the flux at ∼(3–4) × 10^{−6} ml O_{2} · cm^{−2} · s^{−1}. Thereafter, J_{i} increases by a factor of two when M increases to its respective M_{max} value. The relative sensitivity of J_{i} to M can be explained using *Eq. 5.*For small ratios of w/R_{i}, *Eq. 5
* can be expressed as
Equation 17With K = 3.17 × 10^{−10} ml O_{2} · cm^{−1} · Torr^{−1} · s^{−1} and (P_{i}− P_{o})/ w = 10^{4} Torr/cm, the first term in *Eq. 17
* is 3.17 × 10^{−6} ml O_{2} · cm^{−2} · s^{−1}, for M = 10^{−4} ml O_{2} · cm^{−2} · s^{−1} and w = 2 × 10^{−3} cm, and the second term equals 10^{−7} ml O_{2} · cm^{−2} · s^{−1}. For smaller values of M, the contribution of the second term is even smaller.

## DISCUSSION

We have shown in Table 5 that for 30 reported sets of measurements of arteriolar J_{i}, 13 cases cannot be explained using physically realistic parameters; for the remaining ones, in nine cases M_{min} > M_{mt}. We discuss possible reasons for the overestimation of J_{i}.

#### Estimates of J_{i} and M on the basis of in vivo measurements.

The average arteriolar J_{i} in Table 4 varies between ∼10^{−6} ml O_{2} · cm^{−2} · s^{−1} in capillaries and ∼10^{−4} ml O_{2} · cm^{−2}· s^{−1} in arterioles; these estimates are based on in vivo measurements of longitudinal hemoglobin O_{2} saturation gradients (ΔS/Δz) or O_{2} tension (ΔPo
_{2}/ Δz) in unbranched vessel segments. The values of theoretically estimated M in Table 5 range from 0 to 0.16 ml O_{2} · ml^{−1} · s^{−1}, with a tendency to increase with increasing vascular diameter; this range is shown in Fig. 1 together with in vitro and in vivo data.

In all of the studies, except Ref. 71, the P_{o}was not measured, and therefore we were only able to estimate possible ranges of P_{o} and M. However, since Tsai et al. (71) measured both P_{i} and P_{o} using the phosphoresence decay method, they were able to estimate M at 6.5 × 10^{−2} ml O_{2} · ml^{−1} · s^{−1}, note that this value is an order of magnitude higher than the estimated M_{mt} (5 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1}). Looking at the overall picture shown in Fig. 1 and Table 5, we hypothesize that there is a problem with most microvascular measurements listed in Table 4. We discuss possible reasons in*Possible reasons for the overestimation of intravascular flux*.

In another study, Po
_{2} on the surface of rat brain cortex microvessels was measured (73). At normoxia, P_{o} values for arterioles of diameters 7–70 μm fall with decreasing diameter, (P_{o} = 55.7 + 0.45d Torr), and the tissue Po
_{2} (P_{1}) measured at a distance of 40 μm (*r*
_{1}) from the wall of the arterioles averaged 35.5 Torr (74). Using*Eq. 2
* with boundary conditions P = P_{o} at r = r_{o} and P = P_{l} at r = r_{l}, we obtain the J_{o} at r = r_{o}
Equation 18Taking r_{l} = r_{o}+ 40 μm and assuming values for rat brain cortex at K = 4.93 × 10^{−10} ml O_{2} · cm^{−1} · Torr^{−1} · s^{−1}, M = 8.34 × 10^{−4} ml O_{2} · ml^{−1} · s^{−1}(26), we obtain for these arterioles J_{o} = (1.2–1.5) × 10^{−5} ml O_{2} · cm^{−2} · s^{−1}. Note that the P_{o} values were measured only 3–4 μm from the lumen (the microelectrode was pressed against the wall). Thus values estimated above should lie between true J_{i} and J_{o}.

#### Distribution of O_{2} consumption between vascular and parenchymal cells.

It is instructive to delineate, on the basis of the results presented in Table 5, possible M vs. those in parenchyma. Swain and Pittman (67) estimated the tissue O_{2}consumption (M_{tissue}) in vivo on the basis of the O_{2} mass balance in the hamster retractor muscle to be ∼10^{−4} ml O_{2} · ml^{−1}· s^{−1}. Dutta et al. (14) estimated the rate at 3.63 × 10^{−4} ml O_{2} · ml^{−1} · s^{−1} on the basis of in vitro polarographic measurements. These measurements indicated a spatially uniform O_{2} consumption as the electrode penetrated into the muscle; thus it is unlikely that vascular consumption was significantly different from the parenchymal cell consumption.

As a theoretical possibility we can consider a more general case. M_{tissue} can be estimated from the relationship
Equation 19where φ is the vascular volume fraction (strictly speaking, in estimates of φ the volume of blood has to be subtracted from the total tissue volume), M_{vasc} is the vascular O_{2}consumption, and M_{parench} is the parenchymal O_{2}consumption. If the in vivo M_{vasc} is similar to that of the parenchyma, then the estimates for the M_{parench} would not change from the value of ∼10^{−4} ml O_{2}· ml^{−1} · s^{−1} (14, 67), since the vascular tissue content of most tissues is ∼5% (77). However, if the M_{vasc} is high, say 50% of M_{mt}, i.e., M_{vasc} = 2.5 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1}, then applying *Eq. 19
* for φ = 0.05, we obtain M_{tissue} = 1.25 × 10^{−4} ml O_{2} · ml^{−1} · s^{−1}, with M_{parench} = 0. Thus in this example the vascular wall consumes all of the O_{2}. If we assume M_{tissue} = 10^{−4} ml O_{2} · ml^{−1} · s^{−1}, then M_{parench} = 0.53 × 10^{−4} ml O_{2} · ml^{−1} · s^{−1}, i.e., 50% of the O_{2} is consumed by the vasculature and 50% by the parenchymal cells. This possibility, that a substantial fraction of O_{2} released from the arterioles is consumed within the vascular wall, was raised in Tsai et al. (71). However, polarographic measurements in several tissues (14) do not point to significant spatial nonuniformities of O_{2} consumption.

We can now consider existing experimental evidence of M_{vasc}in vitro and in vivo and relate it to the consumption of M_{parench}. The data on vascular segments in vitro in Table 3show that the values of O_{2} consumption in microvessels are generally higher than in large vessels, with the maximum value of 1.6 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1} obtained for cerebral microvessels with succinate as substrate (66). Note that this value is below the M_{mt}.

In a series of papers, Clark and co-workers (77) reported a substantial increase of rat hindlimb M following vasoconstriction, and they initially attributed it to M_{vasc}, thus introducing the “hot pipes” concept. However, in subsequent publications the authors put forward alternative explanations of the results, including functional blood flow shunts and changes of muscle fiber metabolism (68, 69).

Marshall and Davies (44) found a 2.5-fold increase in M in hindlimbs of chronically hypoxic rats compared with normoxic control rats, from 0.7 × 10^{−4} to 1.8 × 10^{−4} ml O_{2} · ml^{−1} · s^{−1}. On the basis of their results, using *Eq.19
* and assuming that the entire increase in the organ consumption with stimulation is attributed to the vasculature and that a vascular volume fraction of 3.4% remains unchanged during the period of chronic hypoxia, one would estimate the difference between the M_{vasc} for rats exposed to chronic hypoxia and normoxic controls of 3.2 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1}. An upper bound for the M_{vasc} in the control case could be obtained by attributing the entire M to the vasculature (2.1 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1}). Then a lower and an upper bound for the M_{vasc} in chronic hypoxia would be 3.2 × 10^{−3} and 5.3 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1}, respectively.

Curtis et al. (11) showed that removing the endothelium in the dog hindlimb decreases hindlimb M by 35%. Assuming that the decrease is attributed entirely to the M_{e} and assuming the endothelium to be 1% of tissue volume, one would estimate the M_{e} to be 3.8 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1} on the basis of the O_{2} uptake of the whole hindlimb of 1.1 × 10^{−4} ml O_{2} · ml^{−1} · s^{−1}. Note that the above values (11, 44) are based on rather crude assumptions, the validity of which have not been demonstrated; thus the estimates have to be considered with extreme caution.

Tsai et al. (71) estimated M_{vasc} of rat mesentery to be 6.5 × 10^{−2} ml O_{2}· ml^{−1} · s^{−1}, with M_{parench} estimated directly from Po
_{2} gradients at 2.4 × 10^{−4}ml O_{2} · ml^{−1} · s^{−1}, i.e., the ratio of the consumption rates of ∼280. Note that the estimated value of M_{vasc} is 13 times higher than M_{mt}. If one assumes a vascular volume fraction of 5%, using the above numbers, one would conclude that the vasculature consumes 94% of the O_{2}.

To summarize the evidence, the question of vascular O_{2}consumption in vivo remains controversial. Additional experimental measurements are necessary to resolve this important question.

#### Possible reasons for the overestimation of intravascular flux.

A common feature of all of the measurements listed in Table 4 is that the J_{i} was estimated as the difference between convective inflow and outflow of O_{2} in unbranched segments of microvessels, mostly arterioles. It is clear from Tables 5 and 6that results from different laboratories, employing slightly different implementations of similar techniques, are quite consistent in that, apparently, they all, or at least most of them, yield overestimates of J_{i}. What are the possible reasons why such overestimates could be made?

The calculation of convective oxygen flow, Q_{O2}
^{C}, at an intravascular site uses the following equation
Equation 20where d is the luminal diameter, v is the RBC velocity averaged over the lumen, and So
_{2} is determined spectrophotometrically. Several assumptions are implicit in this calculation. They are axisymmetric flow in a vessel of circular cross-section and the appropriateness and accuracy of using values of [Hb] and So
_{2} determined spectrophotometrically over the centerline of the vessel. A number of these assumptions have been evaluated previously (8, 15,53), and most authors agree that each one falls short in one respect or another. For instance, the shape of the vessel lumen may not be circular; flow may not be axisymmetric and the calculation for the average velocity might not properly take into account the shape of the velocity profile; RBCs are unlikely to be uniformly distributed across the lumen, especially near a bifurcation; and So
_{2} is most likely nonuniform due to continuous diffusion of O_{2} across the vessel wall. The microspectrophotometric techniques used to measure [Hb] (42) and So
_{2} (54) provide average values along a narrow vertical path through the center of the vessel; the larger the nonuniformity in the luminal values, the less accurately these space-averaged numbers estimate the true luminal average. The Po
_{2} values measured in Ref.71 may also be subject to this limitation. Since there should be steep Po
_{2} gradients within the lumen if there is a large J_{i}, the phosphorescence signal used to determine Po
_{2} must be analyzed using a method that takes such gradients into account (18). In addition, using Po
_{2}values to estimate So
_{2} in the steep part of the O_{2} dissociation curve can give large uncertainties in predicted So
_{2}. The RBC velocity in each case has been measured using an on-line photosensor and the cross-correlation technique of Wayland and Johnson (75).

The diffusive J_{i} is calculated from the difference in convective fluxes at upstream and downstream sites, so that it is important to have accurate estimates of the latter fluxes. How could the J_{i} be overestimated? If we suppose that the observed convective flux differs from the true flux by a systematic error, δ^{C}, which could have a different magnitude at the two measurement sites, then the observed diffusive flux will overestimate the true diffusive flux if the systematic error at the upstream site is consistently larger than that at the downstream site. In this case we can express the observed diffusive flux (J_{obs}
^{D}) relative to the true flux (J_{true}
^{D}) as
Equation 21where (δ_{in}
^{C} − δ_{out}
^{C}) is the positive systematic error in Q_{O2obs}
^{C} and L is the length of the vascular segment. Typically, the diffusive loss in O_{2} amounts to only a 1–2% decrease in the convective O_{2} flux, so that the term Q_{O2obs}
^{C}/(*J*
_{true}
^{D}πdL) in *Eq. 21
* should lie between 50 and 100. If the difference in systematic error of convective flux (δ_{in}
^{C} − δ_{out}
^{C}) in *Eq. 21
* is 5–10% of the true value, then the J_{obs}
^{D}, could be an order of magnitude higher than the true value.

Why might a systematic error in convective O_{2} flux be higher at the upstream end of an unbranched segment? Typically, the upstream site is close to the previous bifurcation, hence the luminal distribution of RBCs and RBC velocity profile might not have reached a stable configuration and the cross-sectional shape might not be circular. This could lead to a nonaxisymmetric distribution of velocity, [Hb], and So
_{2} at the upstream site, so that the product of the factors derived from center-line measurements might not yield an accurate estimate of the true convective flux in *Eq. 20.*

The approach used to estimate J_{i} relies on accurate values to calculate convective O_{2} flux. High accuracy is especially critical in this approach, since the differences are typically small (∼1%) relative to the values that are subtracted. Under the ideal controlled conditions of calibration, the accuracy of the methods used to determine the parameters of *Eq.20
* is quite good, as shown previously by various investigators. However, when these methods are applied in vivo under conditions that can deviate substantially from those used in calibration studies, their accuracy is unknown and difficult to determine. It thus becomes important to search for the source(s) of potential systematic errors that could arise during in vivo applications of the methods used to measure velocity, [Hb], and So
_{2}. It appears likely then that a crucial aspect to consider is the impact of intravascular nonuniformities in luminal shape and velocity and in the distributions of RBCs, So
_{2}, and Po
_{2} on the accuracy of estimates of J_{i}. If the assumptions inherent in the use of*Eq. 20
* are violated to a large enough degree, then it might be necessary to replace this simple expression with a more complicated, but more accurate, form in which the product of luminal distributions in velocity, [Hb], and So
_{2} is integrated over the luminal cross-section. Full-diameter profiles of these variables have been reported (15), but the analysis of these profiles has not progressed to the point where luminal variations (i.e., radial dependencies in an axisymmetric situation) can be determined with confidence. It appears that future progress in this area will require that the existing methods be further developed to include information that can be interpreted in terms of intravascular profiles or that methods requiring less restrictive assumptions than those listed above must be developed and/or applied (e.g., use of fluorescently-labeled RBCs for RBC flow determination).

It is important to emphasize that the above arguments apply only to determinations of J_{i} in unbranched segments that require subtraction of nearly equal quantities. The values of So
_{2} and Po
_{2} measured at different locations along the microvasculature, as reported by numerous researchers (12, 27, 38, 39, 54, 67, 71), that represent significant precapillary O_{2} transport are not in question.

#### Conclusions.

In vitro experiments on endothelial and smooth muscle cell suspensions and vascular segments show that M of the vessel wall does not exceed the estimated M_{mt} of 5 × 10^{−3} ml O_{2} · ml^{−1} · s^{−1}. Consideration of M_{e} via a NO-related pathway does not qualitatively change this conclusion. However, some in vivo data based on measurements of So
_{2} or Po
_{2} gradients in unbranched segments yield values that are significantly higher than M_{mt} by an order of magnitude or more. A possibility remains that some other extramitochondrial pathways are associated with O_{2}utilization that is an order of magnitude higher than M_{mt}. Further analysis is necessary to estimate O_{2} utilization in known pathways. However, as we have shown, high values of J_{i} may be inconsistent with certain measurements of transvascular Po
_{2} difference. The likely place to look for the source of discrepancy is in the microcirculatory measurements in arterioles. All measurements in the arterioles use similar techniques; the interpretation of these measurements, such as the neglect of nonuniform velocity, [Hb], and So
_{2}, and possibly the vascular wall K, may need to be reconsidered. It is important to note that capillary measurements do not suffer from the same potential artifacts due to the single-file nature of flow and give reasonable values of O_{2} flux.

## Acknowledgments

We thank Drs. Christopher G. Ellis and Mary L. Ellsworth for critical comments and discussion of the manuscript.

## Appendix

#### Derivation of the two-layer model.

We formulated the two-layer model in which the vascular wall is represented as two separate layers: an endothelial cell layer and a smooth muscle cell layer. For a given vascular wall, the endothelial cell layer is assumed to be ∼0.5-μm thick, with the rest modeled as the smooth muscle layer. K and M are different in each layer. The Po
_{2} profile and flux are assumed to be continuous across the endothelial cell and smooth muscle cell interface. Similar to the one-layer model, the governing equation of diffusion in the endothelial cell layer, can be represented as
Equation A1where P_{e} is the Po
_{2} in the endothelial cell layer, and M_{e} and K_{e} are the O_{2} consumption rate and the Krogh coefficient, respectively, in the endothelial cell layer.

The corresponding diffusion equation for the smooth muscle cell layer is
Equation A2where P_{s} is the Po
_{2} in the smooth muscle cell layer and M_{s} and K_{s} are the O_{2} consumption rate and the Krogh coefficient, respectively, in the smooth muscle cell layer.

The boundary conditions are
Equation A3where r_{i} is the radius and P_{i}is the Po
_{2} at the luminal side of the endothelial cell layer
Equation A4where r_{o} is the radius and P_{o}the Po
_{2} at the outside of the smooth muscle cell layer, and
Equation A5where r_{m} is the radius of the interface between the endothelial and smooth muscle cell layers.

Applying these boundary conditions, we get
Equation A6where
Equation A7
Equation A8
Equation A9
Equation A10
J_{i} is expressed as
Equation A11Similarly J_{o} is obtained as
Equation A12Substituting the value of A_{1} in *Eq. EA7
*into *Eq. EA11,* we obtain
Equation A13This gives the P_{o} for a given J_{i} and P_{i}. The P_{o min}can be obtained by setting M_{e} to 0 in *Eq. EA13.*If the value of P_{o} turns out to be negative, a value that is not physiologically possible, we have M_{e min} > 0. We obtain M_{e min} by setting P_{o} to 0 and solving for M_{e} in *Eq. EA13.*

To get the M_{e max} and P_{o max}, we impose a no-flux condition (J_{o} = 0) on the outer vessel wall in *Eq. EA12.* Substituting the value of A_{3} into *Eq. EA12
* for j_{o} = 0, we get
Equation A14
*Eq. EA14
* corresponds to the case of maximum wall consumption rate, and solving it with *Eq. EA13
* gives us P_{o max} and M_{e max}.

Hence, the different cases for M_{e} and P_{o} can be summarized as Case 1
Equation A15and Case 2
Equation A16

## Footnotes

This project was supported by National Institutes of Health Grant HL-18292.

Address for reprint requests: A. S. Popel, Dept. of Biomedical Engineering and Center for Computational Medicine and Biology, School of Medicine, Johns Hopkins Univ., Baltimore, MD 21205 (E-mail: apopel{at}jhu.edu). Address for other correspondence to A. Vadapalli at the same address (E-mail: arjun{at}bme.jhu.edu).

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

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- Copyright © 2000 the American Physiological Society