Oxygen transport from
capillaries to exercising skeletal muscle is studied by use of a
Krogh-type cylinder model. The goal is to predict oxygen consumption
under conditions of high demand, on the basis of a consideration of
transport processes occurring at the microvascular level. Effects of
the decline in oxygen content of blood flowing along capillaries,
intravascular resistance to oxygen diffusion, and myoglobin-facilitated
diffusion are included. Parameter values are based on human skeletal
muscle. The dependence of oxygen consumption on oxygen demand,
perfusion, and capillary density are examined. When demand is moderate,
the tissue is well oxygenated and consumption is slightly less than
demand. When demand is high, capillary oxygen content declines rapidly
with axial distance and radial oxygen transport is limited by diffusion resistance within the capillary and the tissue. Under these conditions, much of the tissue is hypoxic, consumption is substantially less than
demand, and consumption is strongly dependent on capillary density.
Predicted consumption rates are comparable with experimentally observed
maximal rates of oxygen consumption.
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INTRODUCTION |
OXYGEN TRANSPORT FROM
BLOOD into tissue occurs by passive diffusion. The maximum
distance that oxygen can diffuse from a blood microvessel into
surrounding tissue decreases with increasing oxygen consumption rate
and is a few tens of micrometers in tissues with high oxygen demand,
such as heavily working skeletal muscle. For a tissue with a
given capillary density, the diffusion of oxygen to the mitochondria
where it is consumed is one of the factors limiting the maximal rate of
oxygen consumption (
O2 max). Observations of maximally working single muscles in humans have shown a
substantial oxygen saturation in the venous blood, in the range
15-30% (1, 28, 24, 25), showing that oxygen extraction is incomplete, reflecting diffusive limitation of oxygen transport (33). More complete extraction of the oxygen
available in the blood would result in very low values for the partial
pressure of oxygen (PO2) at the venous end of
capillaries, which would not provide a sufficient
PO2 gradient for oxygen diffusion. Clearly, both convective and diffusive limitations of oxygen transport are
important in determining O2 max.
A further factor that may limit the oxygen consumption of strongly
stimulated muscle is the maximal rate of turnover of mitochondrial enzymes in the Krebs cycle, which limits oxygen demand. Blomstrand et
al. (3) showed a correlation between the oxygen
consumption rate of a maximally working human leg muscle and the
maximal activity of the enzyme oxoglutarate dehydrogenase. Thus limited
oxygen demand, perfusion limitation, and diffusion limitation may all play a part in determining O2 max in
skeletal muscle.
At the microvascular level, the distribution of
PO2 in tissue surrounding microvessels in
heavily working muscle has been debated. In some analyses of oxygen
diffusion from blood to tissue (see below), it has been assumed that
the decline in PO2 occurs mainly in the tissue.
However, studies of dog gracilis muscle at
O2 max (11) showed low
values and small gradients of PO2 throughout
most of the tissue. The results were attributed to effects of the
particulate nature of blood on oxygen transport within capillaries and
to the facilitation of oxygen transport by myoglobin. Richardson et al.
(26) used magnetic resonance spectroscopy to measure
average myoglobin saturation in exercising human skeletal muscle and
concluded that average tissue PO2 is low during
maximal exercise. These studies suggested that
PO2 declines steeply in the radial direction
inside capillaries and within the first few micrometers outside
capillaries, with small gradients in the bulk of the tissue.
Since the classical work of Krogh (19), many
investigations of oxygen transport to skeletal muscle have used
theoretical models (22). Krogh's model is based on the
following assumptions: each capillary is the sole oxygen supply for a
surrounding cylindrical region of tissue, the
PO2 at the vessel wall is assumed to equal that
of the blood, the decline of PO2 along a
capillary is neglected, oxygen diffuses radially from the capillary,
and consumption is uniform in the tissue. In subsequent models
(21), several of these assumptions were relaxed. For
example, Blum (4) included axial decline in blood
PO2 and considered both constant and linear oxygen uptake kinetics. The importance of intravascular resistance to
oxygen transport from blood to tissue was shown by Hellums (14) and Hellums et al. (15). Theoretical
estimates of the minimum PO2 required to
achieve complete tissue oxygenation at maximal consumption in a variety
of muscles and species were obtained by Roy and Popel
(29).
Federspiel (9) developed a two-dimensional model of oxygen
delivery to a muscle fiber with a prescribed distribution of PO2 at its surface. Groebe and Thews
(12) considered a single cylindrical muscle fiber with
several adjacent capillaries in an effort to evaluate the effects of
model geometry on predicted PO2 profiles in
heavily working muscle. Both studies predicted relatively shallow
gradients of PO2 within muscle fibers, as
observed by Gayeski and Honig (11). Secomb and Hsu
(30) computed the PO2 at each
point in a finite tissue domain, taking into account the effects of all
microvessels within the domain. Their model showed that
PO2 levels in capillaries in resting skeletal
muscle can be strongly influenced by oxygen diffusion from arterioles at distances of 100 µm or more. As oxygen consumption rate increases, such relatively long-range effects become less important, and when
consumption is very high, the assumption of the Krogh model, that each
point in the tissue receives oxygen only from the nearest capillary,
becomes increasingly justified. Under such conditions, the use of a
model based on the Krogh cylinder concept is appropriate.
The goal of the present study is to use a theoretical model for oxygen
transport from capillaries to exercising skeletal muscle to predict the
dependence of oxygen consumption rate on oxygen demand, on the basis of
a consideration of transport processes occurring at the microvascular
level. A wide range of oxygen demand is considered, including
conditions of high oxygen demand in which a significant fraction of the
tissue becomes hypoxic (PO2 < 1 Torr).
None of the previous models cited above has analyzed the dependence of
oxygen consumption on demand under such conditions. Oxygen consumption
is assumed to depend on PO2 according to
Michaelis-Menten kinetics, and results are compared with those obtained
when zero-order consumption kinetics is assumed. Nonuniform oxygen
consumption due to mitochondrial clustering around the capillary is
considered. Effects of the decline of oxygen content in blood flowing
through capillaries, intravascular resistance to oxygen diffusion, and myoglobin-facilitated diffusion are included. As in the Krogh cylinder
model, the capillaries are assumed to be parallel and evenly spaced, so
that each capillary supplies oxygen to a surrounding uniform
cylindrical region of tissue. A range of capillary densities is considered.
This model is used to investigate the roles played by oxygen demand,
perfusion limitation, and diffusion limitation in determining O2 max of skeletal muscle. Profiles of
PO2 in the tissue surrounding the capillary are
determined and are compared with measurements of tissue
PO2 in maximally stimulated skeletal muscle. The overall average oxygen consumption rate as a function of demand is
calculated and compared with experimental
O2 max values in human skeletal muscle.
Previous studies (29) have explored the relationship
between oxygen transport at the capillary level and
O2 max. However, no theoretical results
including all the factors considered here have previously been compared with experimentally determined oxygen consumption rates. The results are used to test the hypothesis that oxygen consumption rates in
maximally stimulated skeletal muscle can be predicted on the basis of
observed values of capillary density, blood perfusion, and oxygen
transport parameters in blood and in tissue.
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METHODS |
Governing equations.
The tissue is represented as an array of uniformly spaced cylinders
with capillaries along the axes. Each tissue cylinder is assumed to be
supplied with oxygen exclusively by the capillary within it (Fig.
1). Oxygen diffusion in the axial
direction is neglected, as evidenced by the fact that the
PO2 gradients are much steeper in the radial
direction than in the axial direction. Then Fick's law of diffusion
and conservation of mass lead to the following equation for oxygen
diffusion in muscle tissue
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(1)
|
where P is the partial pressure of oxygen at a radial distance
of r within the tissue cylinder, K (the Krogh
diffusion coefficient) is the product of the diffusivity and solubility
of oxygen in tissue, and M(P) is the oxygen consumption rate
per unit volume of the tissue cylinder. With Michaelis-Menten kinetics,
M(P) = M0 P/(P0 + P), where M0 is the oxygen demand, i.e., the
consumption when the oxygen supply is not limiting, and P0
is the PO2 at which consumption is half of the
demand. With zero-order kinetics, M =
M0 when P > 0 and M = 0 when
P = 0.
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Fig. 1.
Geometry of the Krogh cylinder-type model. Inner cylinder
represents the capillary. Outer cylinder corresponds to tissue
cylinder. Shaded area: example of hypoxic region under conditions of
high demand. Rt, tissue cylinder radius;
Rc, capillary radius; z, distance
along the capillary.
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At the blood-tissue interface, both PO2 and
oxygen flux must be continuous. Because of intravascular resistance to
oxygen diffusion, the PO2 at the blood-tissue
interface may be less than the average PO2
within the blood. This effect can be represented approximately
(15) by 
= Mt
[Pb 
P(Rc)] where
is the rate of oxygen diffusion from the capillary per unit length,
Mt is a mass transfer coefficient,
Pb is the average partial pressure of oxygen within the
blood, and P(Rc) is the partial pressure of
oxygen at the capillary wall, r = Rc. Equating this to the diffusive flux into the
tissue gives the following boundary condition at the capillary wall
![2&pgr;<IT>R</IT><SUB>c</SUB><IT>K </IT><FENCE><FR><NU>dP</NU><DE>d<IT>r</IT></DE></FR></FENCE><SUB>R<SUB>c</SUB></SUB><IT>=</IT>−<IT>M</IT><SUB>t</SUB>[P<SUB>b</SUB><IT>−</IT>P(<IT>R</IT><SUB>c</SUB>)]](/content/vol91/issue5/fulltext/2255/img002.gif)
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(2)
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The capillary walls are considered to be part of the tissue
region. It is assumed that no oxygen is exchanged across the outer
boundary of the tissue cylinder, so that
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(3)
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where Rt is the tissue cylinder radius.
The effects of myoglobin-facilitated diffusion of oxygen within the
tissue are included mathematically by defining the
myoglobin-facilitated PO2 (10) as
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(4)
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where DMb is the diffusion coefficient
for myoglobin, CMb is the total concentration of myoglobin,
SMb(P) is its oxygen saturation, and Vm is the
molar volume, and by replacing P with P* where it refers to tissue
PO2, i.e., in Eqs. 1 and 3 and the left-hand side of Eq. 2.
Clustering of mitochondria around capillaries has been observed
(16). Because oxygen consumption takes place almost
entirely within the mitochondria, such clustering would lead to
increased oxygen demand where the mitochondria are dense and very low
demand where they are sparse. Mitochondrial clustering is represented in the model as increased oxygen demand with Michaelis-Menten kinetics
in an inner cylindrical region around the capillary and no oxygen
consumption in the outer region. To show the effects of clustering
independent of changes in overall demand, the average demand in the
entire cylinder is held at a fixed level in these calculations.
As blood flows along the capillary, oxygen is extracted and the average
PO2 in the blood (Pb) declines. By
conservation of mass, the decline of convective oxygen flux must equal
, i.e.
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(5)
|
where 
is blood flow rate and z is distance
along the capillary. The oxygen content of blood is given by C = CB SHb(Pb), where CB is
the carrying capacity of blood at 100% saturation and the
oxyhemoglobin saturation is described using the Hill equation (22)
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(6)
|
where P50 is the PO2 at
which hemoglobin is 50% saturated and n is a constant. This
is a good approximation, except in cases of extremely rapid unloading
of oxygen by red blood cells. The red cell transit time assumed here is
much larger than the time constant for oxyhemoglobin dissociation, and
so the Hill equation is appropriate.
Parameter values.
Parameter values are chosen to represent blood flow in human skeletal
muscle under physiological conditions and are summarized in Table
1. At high flow rates, such as assumed
here, the decline in PO2 in arterioles is
small, and the PO2 entering the capillary is
assumed to equal that of the arterial blood, i.e., Pb = 100 Torr when z = 0. Experimentally determined values
for K in skeletal muscle range from 5 × 10
10 to 10 × 1010
(cm2/s)(cm3
O2 · cm3 · Torr1).
Here, K = 9.4 × 1010
(cm2/s)(cm3
O2 · cm3 · Torr1)
is assumed (2). To model the effects of myoglobin, the
following parameter values are assumed: DMb = 1.73 × 107 cm2/s (18);
CMb = 3.83 × 107
mol/cm3 (20); Vm = 2.24 × 104 cm3. The oxygen-myoglobin saturation
SMb(P) is represented by a Michaelis-Menten equation, with
half-saturation at a PO2 of
PMb50 = 3.2 Torr (26).
The oxygen content of blood depends on the oxyhemoglobin saturation and
the oxygen carrying capacity of the blood at full saturation. The
PO2 at which hemoglobin is 50% saturated,
P50, indicates hemoglobin's oxygen affinity. Under
standard conditions, P50 is ~26 Torr for human blood
(22). Numerous studies have shown that P50
increases during exercise, a "right shift" of the oxyhemoglobin
dissociation curve, enhancing oxygen unloading. Endurance training has
also been shown to cause an increase in P50 in trained
muscle. Thomson et al. (31) observed a P50 of 38.8 Torr in habitually active but not endurance-trained subjects during exercise. Here, a standard value of P50 = 26 Torr is assumed, and the effects of a right shift are examined by
assuming that P50 increases linearly with distance
traveled, from 26 Torr at the capillary entrance to 39 Torr at the
venous end of the capillary. Other parameter values are
n = 2.7 and CB = 0.2 cm3
O2/cm3 blood (22). If
P50 = 26 Torr and the solubility of oxygen in plasma
is 3 × 105 cm3
O2 · cm3 · Torr1,
the amount of free dissolved oxygen is <2% of the total amount of
oxygen in the blood when PO2 is between 15 and
100 Torr. Therefore, only hemoglobin-bound oxygen is considered.
M0 can vary over a wide range in skeletal
muscle. Experiments based on mitochondrial enzyme turnover imply that
M0 = 40 cm3
O2 · 100 cm3 · min1 in a maximally working
muscle (3), but observed consumption rates
(25) imply higher values. To show the effects of demand on
consumption, a range of M0 from 0 to 80 cm3 O2 · 100 cm3 · min1 is considered, and a
reference value of 40 cm3 O2 · 100 cm3 · min1 is used. The dependence
of oxygen consumption on PO2 is not precisely known. Here, Michaelis-Menten kinetics are assumed, with half-maximal consumption at a PO2 of P0 = 1 Torr. This value is in the range indicated by experimental studies
(6, 27). According to this assumption, oxygen consumption
is less than 50% of demand in hypoxic regions, where
PO2 <1 Torr. To assess the sensitivity of
the results to the Michaelis-Menten form of oxygen uptake kinetics and
the value chosen for P0, calculations were carried out
using zero-order kinetics, which corresponds to the limiting case of
Michaelis-Menten kinetics when P0 
0. To model possible
effects of mitochondrial clustering around capillaries, calculations
were carried with an oxygen demand of 80 cm3
O2 · 100 cm3 · min1 in a cylindrical region
around the capillary representing half of the total tissue volume and
zero outside this region, so that the average demand is equal to the
reference level, i.e., 40 cm3 O2 · 100 cm3 · min1.
In mammalian skeletal muscle, typical capillary diameters are in the
range of 4 to 8 µm. Here, a diameter of 5 µm is assumed, so
Rc = 2.5 µm. Capillary lengths,
L, and average flow velocities, 
, have not
been determined in exercising human skeletal muscle. However, the model
results depend only on their ratio, as shown by Eq. 5, which
is unchanged if z and are altered by the same factor. The ratio /L may be estimated from
the perfusion rate
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(7)
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Andersen and Saltin (1) determined an average
perfusion rate for the human quadriceps femoris muscle of 5.7 l/min
with an average muscle weight of 2.3 kg, i.e., a perfusion rate of 0.041 cm3
blood · cm3 · s1. For the
assumed capillary density and radius, this gives
/L = 4.5 s1. For purposes of
illustration, we choose L = 0.5 mm and = 2.25 mm/s, which are in the range of observed values in mammalian skeletal muscle. In the standard case, the tissue cylinder radius, Rt, is based on a capillary density in human
skeletal muscle tissue of 468 capillaries per square millimeter
(7). The Krogh tissue cylinder radius that gives the same
capillary density is Rt = 26 µm. However,
the training level of the subjects has been shown to have a significant
effect on capillary density. To compare model results with experimental
O2 max values, we consider a range of
capillary densities from 468 to 1,000 capillaries/mm2.
Intravascular resistance to oxygen diffusion depends on the mass
transfer coefficient, which is given by Mt = 
Kpl Sh, where Kpl is the Krogh diffusion coefficient in plasma
and Sh is the Sherwood number, a nondimensional constant
that depends on the oxygen transport process occurring within the
vessel and reflects the particulate nature of blood. Hellums et al.
(15) compiled theoretical and experimental data showing
the dependence of Sh on vessel diameter. For a diameter of 5 µm, Sh ranging from 1.5 to 3 are consistent with available
data. Here, Sh = 2.5 is assumed, except where noted,
and Kpl = 8.3 × 1010
(cm2/s) (cm3
O2 · cm3 · Torr1)
(15), giving Mt = 6.52 × 109 (cm2/s) (cm3
O2 · cm3 · Torr1).
Numerical procedures.
The capillary is discretized into 100 points along its length. At each
point, Eqs. 1, 3, and 4 are solved
numerically to determine the radial profile of
PO2. The total consumption per unit length is
found by numerically integrating the consumption per unit volume over
the cross section of the tissue cylinder, i.e.
|
(8)
|
The decline in blood oxygen content to the next nodal point is
then computed using Eq. 5, and the corresponding
Pb at that point is obtained by solving Eq. 6. This procedure is repeated along the length of the
capillary. For each value of M0, the consumption rate averaged over the entire tissue cylinder, M(P), is
computed. In hypoxic regions, consumption falls below demand according
to Michaelis-Menten consumption kinetics, so average consumption may be
less than M0.
| |
RESULTS |
Figure 2 shows the predicted
variation of PO2 with position in the tissue
cylinder, including effects of intravascular resistance and
myoglobin-facilitated diffusion. The decline in the average blood
PO2 as it flows down the length of the
capillary is shown in Fig. 2A. Radial
PO2 profiles at three distances along the
capillary are shown in Fig. 2B. At each location along the
cylinder, PO2 declines with increasing distance
from the capillary. The PO2 in the tissue
adjacent to the capillary is substantially lower than the mean
PO2 within the capillary as a result of
intravascular resistance to radial oxygen transport. In reality,
intracapillary PO2 is continuous with the
tissue PO2 but varies with both position and
time as red blood cells traverse the capillary. For simplicity, only
the average intracapillary level is shown in Fig. 2B. At the
upstream end of the capillary (z = 0), the
PO2 remains relatively high throughout the
tissue, dropping from 78.9 Torr at the capillary wall to 35.5 Torr at
the outer boundary of the cylinder. However, PO2 declines rapidly with distance along the
capillary. At the midpoint (z = L/2), the
mean intravascular PO2 is 34.7 Torr, and the
tissue PO2 drops to 0.5 Torr at the outer edge
of the cylinder. At this point, much of the tissue is hypoxic, leading
to a reduced rate of oxygen consumption, according to Michaelis-Menten
kinetics. Beyond this point, intravascular PO2
declines more slowly, because less oxygen is being consumed per unit
capillary length.
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Fig. 2.
A: decrease in average blood
PO2 (Pb) with distance along
capillary. B: radial profiles of PO2
for M0 = 40 cm3
O2 · 100 cm3 · min1, at the upstream end
(z = 0), midpoint (z = L/2 = 0.25 mm) and downstream end (z = L = 0.5 mm) of the tissue cylinder, where
M0 is oxygen demand and L is
capillary length. Short horizontal lines represent
PO2 corresponding to mean oxygen content in the
vessel cross section. Vertical dotted lines show decline in
PO2 resulting from intravascular resistance to
oxygen diffusion.
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Figure 3 shows cumulative frequency
distributions of tissue PO2 and myoglobin
saturation, when demand is 40 cm3
O2 · 100 cm3 · min1. Figure 3A
indicates that a large fraction of tissue has low tissue
PO2. With 468 capillaries/mm2 and
P50 = 26 Torr, 37% of the tissue is hypoxic
(PO2 < 1 Torr) and more than 90% of the
tissue has a PO2 less than 12.5 Torr. Increasing capillary density to 600 capillaries/mm2 reduces
the fraction of tissue that is hypoxic to 20%. If a right shift in the
oxyhemoglobin dissociation curve is considered (by linearly increasing
P50 from 26 Torr at the arterial end to 39 Torr at the
venous end of the capillary) in addition to an increased capillary
density of 600 capillaries/mm2, 1.4% of the tissue is
hypoxic. The corresponding distributions of myoglobin
saturation are shown in Fig. 3B, indicating that saturation
is fairly evenly distributed over a wide range.
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Fig. 3.
Cumulative frequency distributions for
M0 = 40 cm3
O2 · 100 cm3 · min1. A: tissue
PO2. B: myoglobin saturation. Solid
lines represent the standard case with 468 capillaries/mm2.
Dotted lines correspond to a capillary density of 600 capillaries/mm2. Dashed lines correspond to a capillary
density of 600 per mm2 and a right shift of the
oxyhemoglobin dissociation curve [PO2 at which
hemoglobin is 50% saturated (P50) increasing linearly from
26 Torr at the capillary entrance to 39 Torr at the downstream end].
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The relationship between oxygen supply, demand, and consumption is
shown in Fig. 4. The rate of convective
oxygen supply to the tissue cylinder is 49.2 cm3
O2 · 100 cm3 · min1, and this represents an
upper limit to consumption. The demand is the rate at which oxygen
would be consumed if all the tissue was well oxygenated. At low levels
of demand (M0 < 10 cm3
O2 · 100 cm3 · min1), the consumption is
equal to the demand, but at higher levels, consumption falls short of
demand. The limited ability of oxygen to diffuse into the tissue
restricts the consumption that is achieved. Predicted consumption
decreases with increasing intravascular resistance, i.e., with
decreasing Sh. In the absence of intravascular resistance to
oxygen diffusion, a demand of 40 cm3
O2 · 100 cm3 · min1 results in a predicted
consumption rate of 30.3 cm3 O2 · 100 cm3 · min1. Inclusion of the
effects of intravascular resistance to oxygen transport further reduces
the predicted rate of consumption. For example, when the demand is 40 cm3 O2 · 100 cm3 · min1, the predicted
consumption is 24.7 cm3 O2 · 100 cm3 · min1 for the standard value
of intravascular resistance (Sh = 2.5). The effect of
varying the intravascular resistance is also shown in Fig. 4. The limit
Sh 
corresponds to the case of no intravascular resistance.
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Fig. 4.
Variation of oxygen consumption with demand. Straight
horizontal line: convective oxygen supply. Straight line through
origin: oxygen demand. Dashed curve: predicted consumption with no
intravascular resistance (Sh ). Solid curves:
predicted consumption including effects of intravascular resistance for
indicated values of Sherwood number (Sh). Standard case is
Sh = 2.5.
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To assess the sensitivity of the results to the assumed
Michaelis-Menten oxygen uptake kinetics, further calculations were carried out with the assumption that local consumption equals demand
until the PO2 falls to zero (zero-order
kinetics). The region of tissue that receives oxygen is smaller with
zero-order kinetics than with Michaelis-Menten, but the consumption
rate is higher. Overall, zero-order kinetics increase oxygen
consumption by 3% (from 24.7 to 25.5 cm3
O2 · 100 cm3 · min1) relative to
Michaelis-Menten kinetics when M0 = 40 cm3 O2 · 100 cm3 · min1.
The diffusive resistance to oxygen transport depends on the capillary
density, N. For a given level of oxygen demand, increasing N decreases the amount of oxygen that each capillary must
deliver, and so a smaller PO2 gradient is
needed to produce the necessary flux. Furthermore, higher N
corresponds to a smaller Rt and therefore a
shorter distance over which oxygen must diffuse. Consequently, predicted oxygen consumption increases substantially with increasing N, as shown in Fig. 5.
Increasing N from 468 to 800 capillaries/mm2
increases predicted consumption by 33%, to 32.8 cm3
O2 · 100 cm3 · min1 when
M0 = 40 cm3
O2 · 100 cm3 · min1.
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Fig. 5.
Variation of oxygen consumption with demand, showing
effect of varying capillary density (N). Standard case is
N = 468.
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A conceptual model has previously been proposed (33) in
which O2 max is determined by the
condition that the PO2 at the venous end of
each capillary equals the minimum PO2 required for complete oxygenation of the adjacent tissue. This concept is
explored in Fig. 6, which shows the
variation of predicted mean intravascular PO2
at the venous end of capillaries with oxygen consumption rate (solid
line). Because the perfusion rate is assumed to be fixed, venous
PO2 drops with increasing consumption rate. The
shape of this curve is dictated by conservation of mass and does not
depend on the distribution of tissue PO2. The
predicted consumption that can be achieved when demand is 40 cm3 O2 · 100 cm3 · min1 corresponds to a venous
PO2 of 25.5 Torr (or 24.2 Torr if mitochondrial clustering is assumed). The dashed curve in Fig. 6 shows the minimum blood PO2 that is required to supply oxygen to
the outer boundary of the tissue cylinder (r = Rt) if a uniform consumption rate is assumed.
The intersection of the two curves represents the point at which the
blood PO2 at the downstream end of the
capillary is equal to the minimum PO2 required
to meet MO. This implies that the maximum consumption rate
that could be achieved assuming a uniform consumption rate with no
hypoxia throughout the tissue cylinder is 19.0 cm3
O2 · 100 cm3 · min1. According to the
present model, significantly higher overall consumption rates can be
achieved if demand is increased and a portion of the tissue is
permitted to become hypoxic.
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Fig. 6.
Variation of venous PO2 with
oxygen consumption rate. Solid curve and symbols: model predictions;
 , M0 = 40 cm3
O2 · 100 cm3 · min1;  , with
clustering;  , M0 = 80 cm3 O2 · 100 cm3 · min1. Dashed curve:
intravascular PO2 required to ensure
PO2 > 0 at outer boundary of tissue
cylinder, assuming uniform consumption throughout the cylinder.
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Predicted rates of oxygen consumption, under several different sets of
assumptions, are compared in Fig. 7 with
experimentally measured O2 max values.
Experimental results are indicated by solid bars, and the other bars
represent theoretical predictions. For each, the shaded region shows
consumption when M0 = 40 cm3
O2 · 100 cm3 · min1, the hatched region
shows the additional consumption when mitochondrial clustering near the
capillary is assumed, and the white region shows the additional
consumption when M0 is increased to 80 cm3 O2 · 100 cm3 · min1 throughout the tissue.
The leftmost bars correspond to the case in which effects of
intravascular resistance and myoglobin-facilitated diffusion are
neglected. Subsequent bars show the cumulative results of successively
including intravascular resistance, myoglobin-facilitated diffusion, a
right shift in the oxyhemoglobin dissociation curve (P50
increasing linearly from 26 Torr at the entrance to 39 Torr at the
venous end of the capillary), and increased capillary density (as
discussed below).
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Fig. 7.
Comparison of experimentally determined maximal rate of
oxygen consumption (O2 max) with
corresponding consumption rates predicted by the model. A:
Anderson and Saltin (1). B: Richardson et al.
(25). Solid bars (Exp.) show experimental results. Other
bars show theoretical predictions, as follows. Shaded regions:
consumption with M0 = 40 cm3
O2 · 100 cm3 · min1. Hatched regions:
additional consumption with mitochondrial clustering. Open regions:
additional consumption when M0 = 80 cm3 O2 · 100 cm3 · min1. Leftmost bars show
predictions of basic model. Subsequent gray bars show predictions with
successive, cumulative inclusion of other factors: +IVR, with
intravascular resistance; +Myo, with myoglobin-facilitated diffusion;
+RS, with right-shifted oxyhemoglobin dissociation curve; +HD, with
increased capillary density (A: N = 600 capillaries/mm2, B: N = 1,000 capillaries/mm2). See text for further explanation.
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The predictions shown in Fig. 7A correspond to the results
of Andersen and Saltin (1) for the human quadriceps
muscle, with a perfusion rate of 0.041 cm3
blood · cm3 · s1,
corresponding to an oxygen supply of 49.2 cm3
O2 · 100 cm3 · min1. The third bar (+Myo) in
Fig. 7A, including effects of intravascular resistance and
myoglobin-facilitated diffusion, corresponds to the standard case shown
in Figs. 2, 4, and 5. Inclusion of intravascular resistance reduces
predicted consumption significantly, from 29.9 to 24.3 cm3
O2 · 100 cm3 · min1 when
M0 = 40 cm3
O2 · 100 cm3 · min1 (model and +IVR bars).
Myoglobin-facilitated diffusion leads to only a small (<2%) increase
in consumption to 24.7 cm3 O2 · 100 cm3 · min1 for
M0 = 40 cm3
O2 · 100 cm3 · min1 (+Myo bar). This level
of consumption is substantially lower than the reported
O2 max value of 35.0 cm3
O2 · 100 cm3 · min1 (1). A
right shift in the oxyhemoglobin dissociation curve (+RS bar) increases
consumption by almost 14%. Predicted consumption with
M0 = 80 cm3
O2 · 100 cm3 · min1 is then 13% less than
the observed O2 max. To obtain consumption levels close to the observed
O2 max, a capillary density higher than
468 capillaries/mm2 must be assumed. For example,
increasing capillary density to 600 capillaries/mm2 (+HD
bar) leads to predicted consumption rates of 32.1 cm3
O2 · 100 cm3 · min1 for
M0 = 40 cm3
O2 · 100 cm3 · min1, 33.8 cm3
O2 · 100 cm3 · min1 with mitochondrial
clustering, and 36.1 cm3 O2 · 100 cm3 · min1 when
M0 = 80 cm3
O2 · 100 cm3 · min1, which are close to the
observed level.
In the study by Richardson et al. (25), the
perfusion in the quadriceps muscle was 0.0642 cm3
blood · cm3 · s1,
corresponding to an oxygen supply of 75.7 cm3
O2 · 100 cm3 · min1. Predictions for this
higher perfusion level are shown in Fig. 7B. Relative to
Fig. 7A, oxygen supply is increased by ~54%, but the
increase in consumption is less, ~12% for the case including effects
of intravascular resistance and myoglobin (gray +Myo bar). Inclusion of
intravascular resistance, myoglobin-facilitated diffusion, and right
shifting the oxyhemoglobin dissociation curve lead to similar effects
to those seen in Fig. 7A. In this case, even when a right
shift is included, the predicted consumption rates (31.4 cm3 O2 · 100 cm3 · min1 when
M0 = 40 cm3
O2 · 100 cm3 · min1, 33.4 cm3
O2 · 100 cm3 · min1 with clustering, and
34.9 cm3 O2 · 100 cm3 · min1 when
M0 = 80 cm3
O2 · 100 cm3 · min1) fall far short of the
reported O2 max, which is 60.2 cm3 O2 · 100 cm3 · min1 (25).
According to the model, such high oxygen consumption can only be
achieved by a combination of high demand and high N.
Clearly, this reported consumption rate cannot be achieved with a
demand of 40 cm3 O2 · 100 cm3 · min1. If N is
increased from 468 to 1,000 capillaries/mm2 and a demand of
40 cm3 O2 · 100 cm3 · min1 is considered, nearly
all of the tissue is well oxygenated, and the predicted consumption,
38.8 cm3 O2 · 100 cm3 · min1, nearly equals demand.
As seen in the +HD bar, under these conditions, mitochondrial
clustering has practically no effect on oxygen consumption, and
predicted consumption still falls well short of the reported O2 max value. However, if an increased
N is coupled with a demand of 80 cm3
O2 · 100 cm3 · min1, the resulting
consumption is 57.4 cm3 O2 · 100 cm3 · min1, close to the observed
O2 max.
| |
DISCUSSION |
Factors determining oxygen consumption rate.
When M0 is high in skeletal muscle, the rate of
oxygen consumption depends not only on the demand, but also on the rate
of convective delivery and on the diffusive processes occurring within capillaries and in the surrounding tissue. The present analysis shows
that all these factors have significant effects on the consumption rate
that is achieved. Clearly, consumption cannot exceed either the demand
of the muscle or the convective supply (Fig. 4). When demand is low,
all tissue has sufficient oxygen, and predicted consumption equals
demand. As demand increases, the limited rate of oxygen diffusion leads
to low oxygen levels in the outer regions of the tissue cylinder, as
shown schematically in Fig. 1. This oxygen deficiency causes the
average consumption within the cylinder to fall below the demand.
Further increases in demand, beyond ~40 cm3
O2 · 100 cm3 · min1, lead to only slight
increases in consumption (Fig. 4).
As shown previously (14), inclusion of intravascular
resistance to oxygen diffusion significantly lowers the predicted
consumption rate (Figs. 4 and 7, +IVR). Thus both intravascular
resistance and resistance to diffusion within the tissue are important
factors in determining consumption. With increased capillary density, the size of each tissue cylinder is reduced, reducing the diffusion resistance. Furthermore, for a given consumption rate, each capillary has to deliver less oxygen, reducing the effect of diffusion resistance on oxygen delivery. This leads to marked increases in predicted consumption (Figs. 5 and 7, +HD). At higher Ns, consumption
increases with demand beyond 40 cm3
O2 · 100 cm3 · min1 before leveling off
(Fig. 5).
The role of myoglobin in oxygen delivery has been controversial.
Gayeski and Honig (11) considered that myoglobin was an important factor in oxygen transport within muscle fibers. Conley et
al. (5) used measurements of myoglobin saturation in vivo made with magnetic resonance spectroscopy to support the opinion that
myoglobin may play a role in facilitating oxygen transport in skeletal
muscle but did not quantify the effects of myoglobin on oxygen
transport. In a recent review, Jürgens et al. (17) concluded that myoglobin plays at most a minor role in oxygen transport
under physiological conditions. A similar conclusion was reached by Roy
and Popel (29). According to the present analysis,
inclusion of myoglobin-facilitated diffusion has relatively small
effects on overall oxygen transport. It leads to shallower PO2 gradients, lower intravascular
PO2 values, higher PO2
levels in the outer part of the cylinder where tissue is hypoxic, and a
slight (<2%) net increase in consumption (Fig. 7, +Myo). To understand this result, it is helpful to consider the facilitation pressure (Pf) = DmbCmbVm/K,
which represents the contribution of myoglobin to oxygen transport in
terms of an effective additional partial pressure of oxygen available
to drive diffusive transport (10). For the parameter
values used here (DMb = 1.73 × 107 cm2/s), Pf = 1.6 Torr,
which is much less than the capillary PO2 even
at the venous end. This result reflects the fact that the diffusivity
of myoglobin in muscle is two orders of magnitude less than the
diffusivity of oxygen. Therefore, myoglobin-facilitated oxygen
transport would not be expected to have much effect. Some earlier
models were based on the diffusivity of myoglobin measured in protein
solutions (18), DMb = 8 × 107 cm2/s. Assuming this value gives
Pf = 7.3 Torr and leads to a 5% increase in the oxygen
consumption rate when M0 = 40 cm3 O2 · 100 cm3 · min1.
The effects of a right shift of the oxyhemoglobin dissociation curve
and mitochondrial clustering can also increase consumption, as
indicated in Fig. 7. A right shift of the oxyhemoglobin dissociation curve increases blood PO2 at a given saturation
and therefore increases the PO2 gradient for
oxygen diffusion into tissue. Simulating a right shift by linearly
increasing P50 from 26 to 39 Torr along the length of the
capillary leads to a significantly increased consumption (Fig. 7, +RS).
Additional simulations in which P50 varied smoothly from 26 to 39 Torr, but with nonlinear dependence on distance traveled, showed
very similar results. Effects of mitochondrial clustering are
represented here by assuming that consumption occurs only in a region
surrounding each capillary representing half of the tissue volume. For
an average consumption rate 40 cm3
O2 · 100 cm3 · min1, clustering leads to a
significant increase in consumption (Fig. 7, hatched areas) whenever
hypoxic regions are present. When N is increased to 1,000 capillaries/mm2 (Fig. 7B, +HD), virtually all
the tissue is well oxygenated at a uniform demand of 40 cm3
O2 · 100 cm3 · min1 and clustering has
almost no effect.
Model predictions are sensitive to N, which is equal to the
capillary-to-fiber ratio divided by the average fiber cross-sectional area. Changes in N can be caused by changes in the number of
capillaries per fiber or in fiber size. Because the model assumes that
each capillary supplies oxygen exclusively to the surrounding cylinder of tissue, regardless of the number of fibers intersected by that cylinder, the results are not sensitive to the capillary-to-fiber ratio
or the fiber size per se, but only to N (i.e.,
capillaries/mm2).
Some effects that are not considered in this model may also influence
consumption rates. Only oxygen diffusion from capillaries is
considered. In resting muscle, a significant fraction of the M0 can be met by oxygen diffusion from
arterioles. However, this fraction decreases with increasing
M0 (30). In the model, the contribution of oxygen dissolved in the plasma to convective delivery is neglected. Additional calculations including this component were
carried out, showing that predicted consumption increased by <1.1%.
Capillaries are lined with a glycocalyx or endothelial surface layer
0.5-1 µm thick that can exclude flowing red blood cells
(23). This layer may increase intravascular
resistance to oxygen diffusion, but its effects are unknown. The
assumed Sherwood number, Sh = 2.5, was based on
theoretical estimates that neglect such effects (15).
Effects of nonuniform capillary spacing and flow.
The Krogh-type model assumes that all capillaries in the tissue are
identical, with the same flow in each and uniform spacing between them.
In reality, capillaries are not evenly spaced, and this heterogeneity
may lead to reduced oxygen delivery, because capillaries in more
densely perfused regions may flow through regions of higher
PO2 where less oxygen is extracted. Under
conditions of very high demand, however, even relatively densely
perfused areas can be assumed to contain a significant amount of
hypoxic tissue, particularly near the downstream end of each capillary, so that extraction from each capillary is not greatly affected by the
presence of near neighbors. According to this argument, the effect of
nonuniform spacing on average consumption is expected to be small when
demand is very high. Heterogeneity in capillary length or flow rate
could also affect the results. To estimate the effects of variations in
capillary lengths, two cylinders were considered, one 
longer
than the standard case and the other shorter, and both with the same flow as in the standard case. The consumption rate was averaged over the two cylinders and compared with the standard single-cylinder case. Similarly, to investigate the effects of uneven
flow distribution, two cylinders of the same length were considered,
one with higher flow than the standard case and the other
with lower flow. In both cases the result was a slight
decrease in overall average consumption, <2.1%.
Relationship to venous PO2.
According to the conceptual model of Wagner (33), for a
given perfusion rate, the PO2 at the venous end
of capillaries declines with increasing consumption, whereas the
PO2 required to achieve tissue oxygenation
increases. Wagner suggested that O2 max is reached when venous PO2 falls to the minimum
needed for complete tissue oxygenation. This condition is indicated
graphically in Fig. 6 by the intersection of the two curves. Wagner
defines O2 max as the point at which
venous blood PO2 is equal to the
PO2 required to fully meet the
M0 of the tissue, i.e., no hypoxic regions are present. However, as shown in Fig. 6, significantly higher overall oxygen consumption rates can be achieved if oxygen demand increases beyond the level at which hypoxia first appears. With further increases
in demand, consumption in the well-oxygenated regions increases,
whereas the hypoxic regions spread and consumption decreases there
according to Michaelis-Menten kinetics. The decrease in consumption in
hypoxic regions is more than compensated for by the increased
consumption in well-oxygenated regions, and the net effect is an
increase in consumption. For example, the curves in Fig. 6 intersect
when demand is 19 cm3 O2 · 100 cm3 · min1 and consumption is 17.9 cm3 O2 · 100 cm3 · min1. Increasing demand to 40 cm3 O2 · 100 cm3 · min1 results in a
significantly higher consumption rate, 24.7 cm3
O2 · 100 cm3 · min1. Therefore, Wagner's
conceptual model, if applied quantitatively, may underestimate the
maximal oxygen consumption rate. Similarly, the predictions of
end-capillary PO2 at
O2 max by Roy and Popel
(29), which were mainly in the range 20-30 Torr, were based on the assumption that no hypoxia is present and correspond to
the intersection of the two curves in Fig. 6. According to the present
model, their approach may overestimate end-capillary PO2 at O2 max
by several millimeters of mercury.
Comparison with observations of tissue oxygen levels.
Gayeski and Honig (11) used cryospectrophotometry to
measure myoglobin saturation of dog gracilis muscle rapidly frozen at
O2 max. Measured myoglobin saturation
levels were between 8% and 40%, corresponding to
PO2 values of 0.5 Torr and 3.5 Torr, based on
PMb50 = 5.3 Torr for dog muscle. They reported shallow
PO2 gradients of <0.1 Torr/µm and low,
nonzero PO2 values throughout the tissue at
distances greater than 3 µm from the nearest capillary. They
concluded that most of the drop in PO2 between
red blood cells in a capillary and an adjacent muscle fiber occurs
outside the fiber or within a few micrometers inside the fiber. These
results were stated to contradict classical theories of oxygen
transport to tissue such as the Krogh model and were attributed to the
effects of intravascular resistance and myoglobin facilitation. In
subsequent studies, Voter and Gayeski (32) showed that the
sampling region of the earlier studies was larger than assumed by
Gayeski and Honig (11), implying that the oxygen gradients
may have been larger than originally estimated. However, these studies
supported the previous conclusion that oxygen gradients are shallow
with low PO2 throughout the tissue at maximal consumption.
The present model shares some significant features with these
observations. It predicts that a substantial fraction of the tissue is
hypoxic, with low nonzero PO2 and shallow
gradients. With the standard assumptions, 68% of the tissue has a
PO2 of <3.5 Torr and 37% has a
PO2 of <1 Torr, as shown in Fig. 3A.
PO2 gradients are <1 Torr/µm in 84% of the
tissue and <0.1 Torr/µm in 32% of the tissue. Although a wide range
of myoglobin saturation is predicted, as shown in Fig. 3B,
much of the tissue is in the approximate range that was found by Honig
and Gayeski (11). The main factors contributing to these
behaviors are evident in Fig. 2. Intravascular resistance to oxygen
diffusion leads to a significant radial decline in
PO2 within capillaries,
PO2 declines rapidly with distance traveled
along the capillary, and PO2 declines with
radial distance within 5-10 µm of the capillaries. Low gradients occur in hypoxic regions, because consumption is low in those regions.
As already pointed out, myoglobin-facilitated diffusion is not an
important factor.
Comparison with measured values of
O2 max.
Oxygen consumption depends on M0. On the basis
of the experimental work of Blomstrand et al. (3) and
Richardson et al. (25), a range of
M0 from 0-80 cm3
O2 · 100 cm3 · min1 was considered. In the
absence of a measured value for M0, no unique
prediction of O2 max can be made.
However, predicted consumption increases slowly with
M0 in the range from 40 to 80 cm3
O2 · 100 cm3 · min1 (Fig. 4). Therefore,
consumption rates for M0 values in this range
can be considered as approximate predictions of
O2 max.
In experiments using knee extensor exercise, oxygen consumption can be
measured in a single quadriceps muscle to determine O2 max, independent of systemic
limitations on oxygen delivery. Results using this approach (1,
25) are compared in Fig. 7 with model predictions. With standard
parameter values (N = 468 capillaries/mm2,
P50 = 26 Torr, including intravascular resistance and
myoglobin), the model underestimates the measured
O2 max values. However, the values of
some key parameters were not measured in the experimental studies, and
it is of interest to examine what values of these parameters would lead
to consumption rates close to measured
O2 max values.
Andersen and Saltin (1) observed an average
O2 max of 35 cm3
O2 · 100 cm3 · min1 in subjects who ranged
in training from sedentary to endurance trained. In the corresponding
model results (Fig. 7A), when a right shift in the
oxyhemoglobin dissociation curve is included by linearly increasing
P50 from 26 Torr at the arterial end to 39 Torr at the
venous end of the capillary, consumption with
M0 = 40 cm3
O2 · 100 cm3 · min1 increases from 24.7 to
28.0 cm3 O2 · 100 cm3 · min1 but is still less than
the measured value. With mitochondrial clustering and a right shift,
the consumption is 29.9 cm3 O2 · 100 cm3 · min1, and when
M0 = 80 cm3
O2 · 100 cm3 · min1 the consumption with the
right shift is 30.6 cm3 O2 · 100 cm3 · min1. Assuming a higher
capillary density, N = 600 capillaries/mm2,
leads to a predicted consumption of 32.1 cm3
O2 · 100 cm3 · min1 when
M0 = 40 cm3
O2 · 100 cm3 · min1, close to the measured
value. Therefore, the observed O2 max can be accounted for by modifying the standard model to include a right
shift of the oxyhemoglobin dissociation curve and a slightly higher
capillary density.
The subjects in the experiments of Richardson et al. (25)
were competitive endurance bicyclists, and
O2 max values with a mean of 60.2 cm3 O2 · 100 cm3 · min1 were obtained, much
higher than any model predictions with a capillary density of 468 capillaries/mm2. However, if capillary density is increased
to 1,000 capillaries/mm2 and a right shift is included,
with M0 = 80 cm3
O2 · 100 cm3 · min1, a consumption rate of
57.4 cm3 O2 · 100 cm3 · min1 is predicted, close to
the measured value. Endurance training has been shown to have many
effects on skeletal muscle, including increasing N and
mitochondrial number and causing an even more pronounced right shift in
the oxyhemoglobin dissociation curve during exercise when compared with
untrained subjects. The highest N in human skeletal muscle
found in the literature was 810 capillaries/mm2
(8). N estimated by staining methods may
substantially underestimate the true N (13).
According to the model results, a combination of these effects,
including a marked increase in N, must be present in the
muscles of the subjects studied by Richardson et al. (25) to account for the very high muscle-specific
O2 max levels observed.
In conclusion, a theoretical model for oxygen transport to tissue has
been developed on the basis of the tissue cylinder concept of Krogh
(19) but incorporating recent information on oxygen transport mechanisms and parameters. According to the model, oxygen consumption rates under conditions of high demand depend on both convective and diffusive limitations on oxygen delivery. Intravascular resistance to oxygen diffusion significantly restricts delivery, whereas myoglobin-facilitated diffusion in tissue plays a negligible role. A right shift of the oxyhemoglobin dissociation curve can significantly enhance delivery, and consumption rates are sensitively dependent on N. When demand is high, much of the tissue is
predicted to be hypoxic, with low PO2
gradients, as observed experimentally.
Predicted O2 max are comparable in
magnitude to experimentally observed
O2 max values. Andersen and Saltin (1) considered subjects whose level of physical activity
ranged from sedentary to endurance trained. The average value
determined for these subjects can be predicted with this model using
capi
TOP
ABSTRACT
INTRODUCTION
