Estimating oxygen transport resistance of the microvascular wall

¹ Department of Biomedical Engineering and Center for Computational Medicine and Biology, School of Medicine, Johns Hopkins University, Baltimore, Maryland 21205; and ² Department of Physiology, Medical College of Virginia Campus, Virginia Commonwealth University, Richmond, Virginia 23298

	ABSTRACT

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION APPENDIX REFERENCES

The problem of diffusion of O₂ across the endothelial surface in precapillary vessels and its utilization in the vascular wall remains unresolved. To establish a relationship between precapillary release of O₂ and vascular wall consumption, we estimated the intravascular flux of O₂ 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 O₂ consumption. We found that many flux values were not consistent with the diffusion model. We estimated the mitochondrial-based maximum O₂ consumption of the vascular wall (M_mt) and a possible contribution to O₂ consumption of nitric oxide production by endothelial cells (M_NO). Many values of O₂ consumption predicted from the diffusion model exceeded M_mt + M_NO. In contrast, reported values of O₂ 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 O₂ flux are overestimated, and the likely source is in the experimental estimates of convective O₂ transport at upstream and downstream points of unbranched vascular segments.

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

	INTRODUCTION

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION APPENDIX REFERENCES

O₂ 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 O₂ 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 O₂ (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 O₂ 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 O₂ 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 O₂ consumption by endothelial and smooth muscle cells in suspension and by vascular segments in vitro and in vivo. An O₂ consumption rate based on the maximum mitochondrial O₂ 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 O₂ 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 O₂ consumption in cell suspensions and vascular segments in vitro is below the estimated M_mt. In contrast, many estimates of O₂ consumption from in vivo experiments on the basis of direct measurements of microvascular hemoglobin O₂ 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) PO₂ values and found that the predited fluxes were nearly constant at ~10⁶ ml O₂ · cm² · s¹ 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

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION APPENDIX REFERENCES

In this section, we first present the estimated M_mt followed by M_NO (Table 1). O₂ consumption for cell suspensions and vascular segments measured in vitro are presented in Tables 2 and 3, respectively. Next, the calculations of O₂ flux at the luminal surface of the arteriolar wall on the basis of in vivo measurements of longitudinal hemoglobin O₂ saturation (SO₂) or PO₂ 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 O₂ 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 O₂ consumption for the endothelial cell layer and assumed O₂ consumption for the smooth muscle layer on the basis of experimental data. We then performed a sensitivity analysis of intravascular O₂ 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 (O_2max), the O₂ consumption rate is 8.3 × 10² ml O₂ · ml mitochondria¹ · s¹ (23). Interestingly, the respiration rate of 0.167 ml O₂ · ml¹ · s¹ 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 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 O₂ · ml mitochondria¹ · s¹. 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 × 10³ ml O₂ · ml¹ · s¹.

This estimate does not take into account O₂ 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 O₂ 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 O₂ consumption outside the mitochondria. In Table 1, 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 O₂ consumed for every molecule of NO produced (64). Clementi et al. (7) have studied in detail the mechanism by which endothelial cells regulate their O₂ consumption. Their experiments showed that NO generated by vascular endothelial cells under basal and stimulated conditions modulates the O₂ concentration near the cells. This action occurs at the cytochrome c oxidase in the mitochondria and depends on the influx of Ca²⁺. Thus NO plays a physiological role in adjusting the capacity of this enzyme to use O₂, allowing endothelial cells to adapt to acute changes in their environment. In a cell suspension having a density of 10⁷ cells/ml, the initial rate that was also the peak M_NO was estimated to be 2.8 × 10⁴ ml O₂ · ml cell¹ · s¹. We converted the values from per cell basis to per cell volume basis using the microvascular endothelial cell volume of 400 µm³ 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 ~10⁴ ml O₂ · ml¹ · s¹, NO itself is responsible for inhibiting the predominant pathways for O₂ consumption in the mitochondria, thus possibly reducing the overall M. 

View this table: [in this window] [in a new window]   Table 1.   Experimental measurements of O2 consumption by endothelial cells for the production of NO

The maximum value of M_NO (~10⁴ ml O₂ · ml¹ · s¹) 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 ~10⁶ ml O₂ · ml¹ · s¹ (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/10⁶ cells (2). Therefore, M_NO is at least an order of magnitude lower than M_mt. In principle, it is possible that O₂ 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.

O₂ consumption by endothelial and smooth muscle cells. Table 2 summarizes measurements of O₂ 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 µm³ for endothelial cells and 3,000 µm³ for smooth muscle cells.

View this table: [in this window] [in a new window]   Table 2.   Experimental measurements of M by endothelial and smooth muscle cell suspensions

O₂ 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 O₂ consumption on a cell count basis (per million cells) appear to indicate that smooth muscle cells consume more O₂ 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 1 are 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 × 10³ ml O₂ · ml¹ · s¹, corresponding to M_mt. In Figure 1, 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.

View this table: [in this window] [in a new window]   Table 3.   Experimental measurements of M in vascular segments in vitro

View larger version (14K): [in this window] [in a new window]   Fig. 1.   Comparison of experimental and estimated O2 consumption rates (M) in relation to the mitochondrial-based maximum O2 consumption (Mmt), shown as a horizontal line. 1, Theoretical estimate for vascular wall consumption in rat mesentery arteriole from PO2 data measured with phosphoresence quenching (71); 2, theoretical estimate of vascular wall consumption for rabbit and dog aortas from PO2 data measured with PO2 electrodes (6).

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, PO₂ was measured with O₂ electrodes at several depths in the vascular wall in vivo (6) or in vitro (10, 55) and M was calculated using a model of O₂ 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 O₂ consumed was calculated from the ATP consumed by dividing the latter by the stoichiometric ratio 6.42 moles of ATP per mole of O₂. 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 × 10³ ml O₂ · ml¹ · s¹), 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 PO₂, 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 SO₂ or PO₂ 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 z, the diffusive loss of O₂ can be estimated as

(1)

where Q = (d²/4)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 O₂ binding capacity of the hemoglobin, S 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, S/z, 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 S values by converting PO₂ into S values with the Hill equation. For all of the other sources in which SO₂ was determined microspectrophotometrically, SO₂ values were converted into PO₂ with the use of the Hill equation. The Hill equation parameters n (cooperativity of Hb) and P₅₀ (O₂ 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 × 10¹⁰ ml O₂ · cm¹ · Torr¹ · s¹ and was calculated as the product of the diffusivity (D) in human arteries measured to be ~0.96 × 10⁵ cm²/s (31), and the solubility coefficient () reported for many tissues to be ~3.3 × 10⁵ ml O₂ · cm³ · Torr¹ (14). D is lower by a factor of ~2.5 when compared with that of the hamster retractor muscle at 37°C (2.42 × 10⁵ cm²/s) (4, 14). For endothelial cells in vitro, D has been found to vary between 0.14-0.87 × 10⁵ cm²/s (43).

View this table: [in this window] [in a new window]   Table 4.   Calculated values of intravascular flux from in vivo hemodynamic data

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 O₂ transport in the vascular wall. The vessel wall was modeled by an axially symmetrical diffusion equation

(2)

where r is the radius.

The boundary conditions are applied at the inside and outside surfaces of the vessel wall

(3)

where 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

(4)

With Eq. 4 we obtain an expression for J_i

(5)

and the perivascular flux (J_o)

(6)

M can be expressed from mass balance using J_i and J_o

(7)

Rearranging Eq. 7, we obtain J_o in terms of J_i and M

(8)

If we require that the vessel wall be supplied with O₂ 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 O₂ diffusing from the lumen is consumed by the vascular wall)

(9)

Eliminating J_o from Eqs. 6 and 8 gives us P_o in terms of P_i and J_i

(10)

The term in square brackets can be shown to be >0. Since P_o  0, Eq. 10 yields

(11a)

This inequality can be satisfied in two ways. Case 1 is

(11b)

then

(11c)

where 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

(12)

and the P_o max is obtained by substituting Eq. 9 into Eq. 10

(13)

The schematic representing case 1 is shown in Fig. 2A. 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 PO₂. Conventionally, e.g., under conditions of the Krogh tissue cylinder model, J_i is proportional to tissue O₂ 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 PO₂ 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

(14a)

case 2a then

(14b)

and, where M_min is obtained from Eq. 10 for P_o = 0 

(15)

A schematic of case 2a is presented in Fig. 2B. 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.

View larger version (9K): [in this window] [in a new window]   Fig. 2.   Schematic diagram for the physiological ranges of M and perivascular PO2 for all 3 cases in the 1-layer model analysis. A: case 1; B: case 2a; C: case 2b.

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. 2C. Note the increasing slope with respect to the horizontal axis of the PO₂ 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 Table 5, 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.

View this table: [in this window] [in a new window]   Table 5.   Physiological ranges of perivascular O2 tension and M

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 × 10³ ml O₂ · ml¹ · s¹).

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 O₂ consumption between these cell types. By considering a two-layer model (derivation presented in APPENDIX), 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 10⁴ ml O₂ · ml¹ · s¹ on the basis of measurements presented in Table 2. The K for the endothelial cell layer was taken to be 4.45 × 10¹⁰ ml O₂ · cm¹ · Torr¹ · s¹ (43), whereas that of the smooth muscle cell layer was 4.5 × 10¹⁰ ml O₂ · cm¹ · Torr¹ · s¹ (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 10⁴ to 10³ ml O₂ · ml¹ · s¹ 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. 

View this table: [in this window] [in a new window]   Table 6.   Physiological ranges of Po and M for selected cases in Table 5 on the basis of a two-layer model

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 Table 7. Doubling K to 6.34 × 10¹⁰ ml O₂ · cm¹ · Torr¹ · s¹ 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 Table 8. The initial value K = 3.17 × 10¹⁰ ml O₂ · cm¹ · Torr¹ · s¹ is close to the value 3.64 × 10¹⁰ ml O₂ · cm¹ · Torr¹ · s¹ taken by Tsai et al. (71), but the calculated transvascular PO₂ 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 PO₂ difference.

View this table: [in this window] [in a new window]   Table 7.   Sensitivity of results of the one-layer model with respect to different model parameters

View this table: [in this window] [in a new window]   Table 8.   Physiological ranges of Po and M

The present calculations assume that P_i is the same as in the vessel on average, corresponding to the measured SO₂, 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 O₂ tension. The dimensionless mass transfer coefficient, called the Nusselt number, is defined as

(16)

where K_pl is the Krogh coefficient for O₂ in plasma, P* is the O₂ 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 × 10⁵ ml O₂ · cm² · s¹, 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 PO₂ 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 PO₂ 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.

View larger version (12K): [in this window] [in a new window]   Fig. 3.   Sensitivity of calculated intravascular flux (Ji) to vascular wall consumption using experimental measurements of intravascular and perivascular PO2 values and 1-layer diffusion model.

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 PO₂. With the use of recessed-tip O₂ 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 PO₂ for vessels groups in four different vascular orders, we can compute a J_i for M ranging between 10⁶ and 10² ml O₂ · ml¹ · s¹. The results are presented in Fig. 3. J_i is found to be almost constant for M in the range of 10⁶ to 10⁴ ml O₂ · ml¹ · s¹, with values of the flux at ~(3-4) × 10⁶ ml O₂ · cm² · s¹. 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

(17)

With K = 3.17 × 10¹⁰ ml O₂ · cm¹ · Torr¹ · s¹ and (P_i  P_o)/w = 10⁴ Torr/cm, the first term in Eq. 17 is 3.17 × 10⁶ ml O₂ · cm² · s¹, for M = 10⁴ ml O₂ · cm² · s¹ and w = 2 × 10³ cm, and the second term equals 10⁷ ml O₂ · cm² · s¹. For smaller values of M, the contribution of the second term is even smaller.

	DISCUSSION

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION APPENDIX REFERENCES

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 ~10⁶ ml O₂ · cm² · s¹ in capillaries and ~10⁴ ml O₂ · cm² · s¹ in arterioles; these estimates are based on in vivo measurements of longitudinal hemoglobin O₂ saturation gradients (S/z) or O₂ tension (PO₂/z) in unbranched vessel segments. The values of theoretically estimated M in Table 5 range from 0 to 0.16 ml O₂ · ml¹ · s¹, 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 × 10² ml O₂ · ml¹ · s¹, note that this value is an order of magnitude higher than the estimated M_mt (5 × 10³ ml O₂ · ml¹ · s¹). 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, PO₂ 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 PO₂ (P₁) measured at a distance of 40 µm (r₁) 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

(18)

Taking r_l = r_o + 40 µm and assuming values for rat brain cortex at K = 4.93 × 10¹⁰ ml O₂ · cm¹ · Torr¹ · s¹, M = 8.34 × 10⁴ ml O₂ · ml¹ · s¹ (26), we obtain for these arterioles J_o = (1.2-1.5) × 10⁵ ml O₂ · cm² · s¹. 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 O₂ 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 O₂ consumption (M_tissue) in vivo on the basis of the O₂ mass balance in the hamster retractor muscle to be ~10⁴ ml O₂ · ml¹ · s¹. Dutta et al. (14) estimated the rate at 3.63 × 10⁴ ml O₂ · ml¹ · s¹ on the basis of in vitro polarographic measurements. These measurements indicated a spatially uniform O₂ 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

(19)

where  is the vascular volume fraction (strictly speaking, in estimates of  the volume of blood has to be subtracted from the total tissue volume), M_vasc is the vascular O₂ consumption, and M_parench is the parenchymal O₂ 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 ~10⁴ ml O₂ · ml¹ · s¹ (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 × 10³ ml O₂ · ml¹ · s¹, then applying Eq. 19 for  = 0.05, we obtain M_tissue = 1.25 × 10⁴ ml O₂ · ml¹ · s¹, with M_parench = 0. Thus in this example the vascular wall consumes all of the O₂. If we assume M_tissue = 10⁴ ml O₂ · ml¹ · s¹, then M_parench = 0.53 × 10⁴ ml O₂ · ml¹ · s¹, i.e., 50% of the O₂ is consumed by the vasculature and 50% by the parenchymal cells. This possibility, that a substantial fraction of O₂ 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 O₂ 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 3 show that the values of O₂ consumption in microvessels are generally higher than in large vessels, with the maximum value of 1.6 × 10³ ml O₂ · ml¹ · s¹ 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 × 10⁴ to 1.8 × 10⁴ ml O₂ · ml¹ · s¹. 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 × 10³ ml O₂ · ml¹ · s¹. An upper bound for the M_vasc in the control case could be obtained by attributing the entire M to the vasculature (2.1 × 10³ ml O₂ · ml¹ · s¹). Then a lower and an upper bound for the M_vasc in chronic hypoxia would be 3.2 × 10³ and 5.3 × 10³ ml O₂ · ml¹ · s¹, 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 × 10³ ml O₂ · ml¹ · s¹ on the basis of the O₂ uptake of the whole hindlimb of 1.1 × 10⁴ ml O₂ · ml¹ · s¹. 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 × 10² ml O₂ · ml¹ · s¹, with M_parench estimated directly from PO₂ gradients at 2.4 × 10⁴ ml O₂ · ml¹ · s¹, 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 O₂.

To summarize the evidence, the question of vascular O₂ 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 O₂ in unbranched segments of microvessels, mostly arterioles. It is clear from Tables 5 and 6 that 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

(20)

where d is the luminal diameter, v is the RBC velocity averaged over the lumen, and SO₂ 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 SO₂ 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 SO₂ is most likely nonuniform due to continuous diffusion of O₂ across the vessel wall. The microspectrophotometric techniques used to measure [Hb] (42) and SO₂ (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 PO₂ values measured in Ref. 71 may also be subject to this limitation. Since there should be steep PO₂ gradients within the lumen if there is a large J_i, the phosphorescence signal used to determine PO₂ must be analyzed using a method that takes such gradients into account (18). In addition, using PO₂ values to estimate SO₂ in the steep part of the O₂ dissociation curve can give large uncertainties in predicted SO₂. 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, ^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

(21)

where (_in^C  _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 O₂ amounts to only a 1-2% decrease in the convective O₂ flux, so that the term Q_O2obs^C/(J_true^DdL) in Eq. 21 should lie between 50 and 100. If the difference in systematic error of convective flux (_in^C  _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 O₂ 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 SO₂ 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 O₂ 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 SO₂. 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, SO₂, and PO₂ 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 SO₂ 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 SO₂ and PO₂ measured at different locations along the microvasculature, as reported by numerous researchers (12, 27, 38, 39, 54, 67, 71), that represent significant precapillary O₂ 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 × 10³ ml O₂ · ml¹ · s¹. Consideration of M_e via a NO-related pathway does not qualitatively change this conclusion. However, some in vivo data based on measurements of SO₂ or PO₂ 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 O₂ utilization that is an order of magnitude higher than M_mt. Further analysis is necessary to estimate O₂ utilization in known pathways. However, as we have shown, high values of J_i may be inconsistent with certain measurements of transvascular PO₂ 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 SO₂, 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 O₂ flux.

	APPENDIX

TOP ABSTRACT INTRODUCTION RESULTS DISCUSSION APPENDIX REFERENCES

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 PO₂ 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

(A1)

where P_e is the PO₂ in the endothelial cell layer, and M_e and K_e are the O₂ consumption rate and the Krogh coefficient, respectively, in the endothelial cell layer.

(A2)

(A3)

(A4)

(A5)

(A6)

(A7)

(A8)

(A9)

(A10)

(A11)

(A12)

(A13)

(A14)

(A15)

(A16)

	ACKNOWLEDGEMENTS

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

	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 "advertisement" in accordance with 18 U.S.C. §1734 solely to indicate this fact.

Received 22 September 1999; accepted in final form 16 February 2000.

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