microcirculation; oxygen diffusion; endothelial cell; smooth muscle
cell; arterioles; nitric oxide
 |
INTRODUCTION |
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 (Mmt) 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, MNO, 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 Mmt. 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 Mmt by one or two orders of magnitude. In an attempt
to resolve this problem, we estimated the sensitivity of the
intravascular flux (Ji) to changes in M for
given intravascular (Pi) and perivascular (Po)
PO2 values and found that the
predited fluxes were nearly constant at
~10
6 ml O2 · cm2
· s1 for M spanning over four orders of magnitude. We
conjectured that precapillary Ji 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.
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RESULTS |
In this section, we first present the estimated Mmt
followed by MNO (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
Po. 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 Mmt. 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.
Mmt.
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 Mmt. In the mitochondria of
locomotory muscles of mammals running at their maximum aerobic capacity
(
O2max), 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 O2max 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 Mmt 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
MNO is at least an order of magnitude smaller than
Mmt.
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
MNO 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 MNO 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 MNO
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.
The maximum value of MNO (~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 MNO 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, MNO is at least an order of
magnitude lower than Mmt. 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 (Me)
and smooth muscle cell suspensions (Ms). 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.
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 MNO values presented in Table 1
are of the same order of magnitude or lower compared with
Me in suspension presented in Table 2. Bruttig and Joyner
(5) reported Me and Ms 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 Mmt. In Figure
1, we compiled all of the experimental values of M in the vascular wall listed in Tables 2 and 3.
Mmt is presented as a horizontal line.
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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).
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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
Mmt, 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 Mmt,
at least one order of magnitude higher than M in cell suspensions and
vascular segments and at least two orders of magnitude higher than
MNO.
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 Ji 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 Mmt. 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 Mmt or even Mmt + MNO.
For a cylindrical segment of blood vessel with a luminal diameter
d and length 
z, the diffusive loss of
O2 can be estimated as
![Q[Hb]C<SUB>b</SUB>&Dgr;<IT>S=&pgr;d&Dgr;zJ</IT><SUB>i</SUB>](/content/vol279/issue2/fulltext/H657/img001.gif)
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(1)
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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, Cb is the
O2 binding capacity of the hemoglobin, S is the saturation difference between the upstream and downstream points
along the vessel segment, and Ji is given per
unit area at the vessel wall (luminal surface) averaged over the vessel circumference (53). Ji 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
Ji. [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 Ji for the
data of Seiyama et al. (59, 60) since the authors had
already carried out these calculations. Pi values were
available for Tsai et al. (71) and Torres Filho et al.
(70) since these authors used the phosphorescence
quenching method to measure Pi; 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).
Physiological ranges for Po and M using a one-layer
diffusion model.
In a diffusion model of the vascular wall, the values of
Ji, Pi, Po, and M are
not independent; if Ji and Pi are
known from experiments, a range of possible values for Po
and M can be obtained subject to physical conditions
Po 
0, M 0; if Ji,
Pi, and Po are known, then M can be uniquely
determined. Therefore, to estimate the bounds of Po and M
for given values of other parameters, including Pi and
Ji, we solved the problem of O2
transport in the vascular wall. The vessel wall was modeled by an
axially symmetrical diffusion equation
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(2)
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where r is the radius.
The boundary conditions are applied at the inside and outside surfaces
of the vessel wall
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(3)
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where ri is the interior and
ro is the exterior radius of the vessel.
Integrating Eq. 2 and applying boundary conditions Eq. 3, we get
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(4)
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With Eq. 4 we obtain an expression for
Ji
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(5)
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and the perivascular flux (Jo)
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(6)
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M can be expressed from mass balance using
Ji and Jo
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(7)
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Rearranging Eq. 7, we obtain Jo
in terms of Ji and M
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(8)
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If we require that the vessel wall be supplied with O2
from the lumen and not from the parenchymal side, then
Jo 0 in Eq. 8, and we obtain
the upper bound on the M, Mmax, when
Jo = 0, (i.e., all of the O2
diffusing from the lumen is consumed by the vascular wall)
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(9)
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Eliminating Jo from Eqs. 6 and 8 gives us Po in terms of Pi and
Ji
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(10)
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The term in square brackets can be shown to be >0. Since
Po 0, Eq. 10 yields
![<FENCE>r<SUB>i</SUB> ln(<IT>r</IT><SUB>o</SUB><IT>/r</IT><SUB>i</SUB>)<IT>J</IT><SUB>i</SUB><IT>−K</IT>P<SUB>i</SUB></FENCE> ≤ <FR><NU>M</NU><DE><IT>2</IT></DE></FR> <FENCE><FR><NU>(<IT>r</IT><SUP>2</SUP><SUB>o</SUB><IT>−r</IT><SUP>2</SUP><SUB>i</SUB>)</NU><DE><IT>2</IT></DE></FR></FENCE><IT>−r</IT><SUP>2</SUP><SUB>i</SUB> ln(<IT>r</IT><SUB>o</SUB><IT>/r</IT><SUB>i</SUB>)]](/content/vol279/issue2/fulltext/H657/img011.gif)
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(11a)
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This inequality can be satisfied in two ways. Case
1 is
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(11b)
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then
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(11c)
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where Mmax is given by Eq. 9,
Po min is the minimum possible Po, and
Po max is the maximum Po. Po min is obtained from Eq. 10 by setting M = 0 and is given
by
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(12)
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and the Po max is obtained by substituting Eq. 9 into Eq. 10
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(13)
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The schematic representing case 1 is shown in Fig.
2A. This case is
physiologically possible since both Po min and Po max are positive. It is important to note that the
Po increases as M increases. This appears to be
counterintuitive when compared with the conventional thinking in which
higher M results in a higher Ji and a
correspondingly lower PO2. Conventionally, e.g., under conditions of the Krogh tissue cylinder model,
Ji is proportional to tissue O2
consumption (Mt); thus an increase in Mt leads
to an increase in Ji (i.e., an increase in the
negative slope of the PO2 profile at the
luminal surface) and thus a decrease in Po. In contrast, in
the case under consideration, Ji 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
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(14a)
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case 2a then
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(14b)
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and, where Mmin is obtained from Eq. 10 for
Po = 0
![M<SUB>min</SUB><IT>=</IT><FR><NU><IT>2</IT>[<IT>r</IT><SUB>i</SUB> ln(<IT>r</IT><SUB>o</SUB><IT>/r</IT><SUB>i</SUB>)<IT>J</IT><SUB>i</SUB><IT>−K</IT>P<SUB>i</SUB>]</NU><DE><FENCE><FR><NU><IT>r</IT><SUP>2</SUP><SUB>o</SUB><IT>−r</IT><SUP>2</SUP><SUB>i</SUB></NU><DE><IT>2</IT></DE></FR><IT>−r</IT><SUP>2</SUP><SUB>i</SUB> ln(<IT>r</IT><SUB>o</SUB><IT>/r</IT><SUB>i</SUB>)</FENCE></DE></FR>](/content/vol279/issue2/fulltext/H657/img018.gif)
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(15)
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A schematic of case 2a is presented in Fig.
2B. The Po for M = 0 is negative, which is
not possible. Hence, a physiologically possible case would have to be
Po > 0, M > Mmin. The lower bound on the Po is a strict inequality, since it is not possible
to have Po = 0 for a finite flux leaving the wall.
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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.
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If the model parameters are such that Eqs. 9-10 yield
Po max < 0, then there is no solution to the
problem, i.e., there are no physical values of M and Po
that satisfy the above relationships; thus case 2b has no
solution if Po max < 0, which is represented in Fig.
2C. Note the increasing slope with respect to the horizontal axis of the PO2 profile at r = ri 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.
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 Po and M can be
found. In 9 out of 19 cases for arterioles shown in Table 5
(capillaries excluded), the minimum M (Mmin) exceeds
Mmt, and in 8 out of 17 cases Mmin also exceeds
Mmt + MNO (5.8 × 103
ml O2 · ml1 · s1).
Physiological ranges for Po 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 Ms on the basis
of published experimental values. The goal of the calculations was to
estimate the physiological limits of the Me and the
Po 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
Ms 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, Pi, and
Ji were taken from Table 4. The values of
Me are much higher than the corresponding computed values
of M in Table 5 and higher than Mmt. This is primarily
because we have fixed the Ms 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, Me 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 Ms 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 Po max < 0.
Sensitivity of Po 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 Pi, 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.
The present calculations assume that Pi is the same as in
the vessel on average, corresponding to the measured
SO2, whereas in reality there exists a gradient
in Pi. At present, radial Pi 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
Ji and the corresponding driving force in
O2 tension. The dimensionless mass transfer coefficient,
called the Nusselt number, is defined as
|
(16)
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where Kpl is the Krogh coefficient for
O2 in plasma, P* is the O2 tension in
equilibrium with the mixed mean hemoglobin saturation, Pi
and Ji are measured at the vessel wall, and
ri 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 Pi drop, P* 
Pi,
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 Pi 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
Ji to wall consumption rates, we obtain P* Pi of ~5-6 Torr. Hence, we chose a drop of 5 Torr
in the Pi for our sensitivity analysis calculations. When
the Pi 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.
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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.
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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 Ji to wall consumption
rates.
The sensitivity of Ji 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 Pi and Po for
cat pial microvessels with internal diameters ranging from 22 µm to
230 µm and found the gradient (Pi Po)/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 Ji
for M ranging between 106 and 102 ml
O2 · ml1 · s1.
The results are presented in Fig. 3. Ji 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,
Ji increases by a factor of two when M increases to its respective Mmax value. The relative sensitivity of
Ji to M can be explained using Eq. 5.
For small ratios of w/Ri, Eq. 5 can be expressed
as
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(17)
|
With K = 3.17 × 1010 ml
O2 · cm1 · Torr1 · s1 and (Pi Po)/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 |
We have shown in Table 5 that for 30 reported sets of measurements
of arteriolar Ji, 13 cases cannot be explained
using physically realistic parameters; for the remaining ones, in nine
cases Mmin > Mmt. We discuss possible
reasons for the overestimation of Ji.
Estimates of Ji and M on the basis of in
vivo measurements.
The average arteriolar Ji 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 Po
was not measured, and therefore we were only able to estimate possible
ranges of Po and M. However, since Tsai et al.
(71) measured both Pi and Po 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 Mmt (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,
Po values for arterioles of diameters 7-70 µm fall
with decreasing diameter, (Po = 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 = Po at
r = ro and P = Pl at
r = rl, we obtain the
Jo at r = ro
|
(18)
|
Taking rl = ro + 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
Jo = (1.2-1.5) × 105 ml O2 · cm2 · s1. Note that the Po 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
Ji and Jo.
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 (Mtissue) 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.
Mtissue 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), Mvasc is the vascular O2
consumption, and Mparench is the parenchymal O2
consumption. If the in vivo Mvasc is similar to that of the
parenchyma, then the estimates for the Mparench 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 Mvasc is high, say 50%
of Mmt, i.e., Mvasc = 2.5 × 103 ml O2 · ml1 · s1, then applying Eq. 19 for = 0.05, we obtain Mtissue = 1.25 × 104 ml
O2 · ml1 · s1,
with Mparench = 0. Thus in this example the vascular
wall consumes all of the O2. If we assume
Mtissue = 104 ml O2 · ml1 · s1, then
Mparench = 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 Mvasc
in vitro and in vivo and relate it to the consumption of
Mparench. 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 Mmt.
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 Mvasc, 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
Mvasc for rats exposed to chronic hypoxia and normoxic
controls of 3.2 × 103 ml O2 · ml1 · s1. An upper bound for the
Mvasc 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 Mvasc 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 Me and assuming the
endothelium to be 1% of tissue volume, one would estimate the
Me 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 Mvasc of rat
mesentery to be 6.5 × 102 ml O2
· ml1 · s1, with
Mparench 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 Mvasc is 13 times higher than
Mmt. 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 Ji 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
Ji. What are the possible reasons why such
overestimates could be made?
The calculation of convective oxygen flow,
QO2C, at an intravascular site uses the
following equation
![Q<SUP>C</SUP><SUB>O2</SUB><IT>=</IT>(<IT>&pgr;d<SUP>2</SUP>/4</IT>)<IT>v</IT>[Hb]<IT>C</IT><SUB>b</SUB>S<SC>o<SUB>2</SUB></SC>](/content/vol279/issue2/fulltext/H657/img023.gif)
|
(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 Ji, 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 uncertaintie
TOP
ABSTRACT
INTRODUCTION
