## Abstract

A previously developed Krogh-type theoretical model was used to estimate capillary density in human skeletal muscle based on published measurements of oxygen consumption, arterial partial pressure of oxygen, and blood flow during maximal exercise. The model assumes that oxygen consumption in maximal exercise is limited by the ability of capillaries to deliver oxygen to tissue and is therefore strongly dependent on capillary density, defined as the number of capillaries per unit cross-sectional area of muscle. Based on an analysis of oxygen transport processes occurring at the microvascular level, the model allows estimation of the minimum number of straight, evenly spaced capillaries required to achieve a given oxygen consumption rate. Estimated capillary density values were determined from measurements of maximal oxygen consumption during knee extensor exercise and during whole body cycling, and they range from 459 to 1,468 capillaries/mm^{2}. Measured capillary densities, obtained with either histochemical staining techniques or electron microscopy on quadriceps muscle biopsies from healthy subjects, are generally lower, ranging from 123 to 515 capillaries/mm^{2}. This discrepancy is partly accounted for by the fact that capillary density decreases with muscle contraction and muscle biopsy samples typically are strongly contracted. The results imply that estimates of maximal oxygen transport rates based on capillary density values obtained from biopsy samples do not fully reflect the oxygen transport capacity of the capillaries in skeletal muscle.

- oxygen transport
- Krogh cylinder model
- quadriceps muscle

maximal oxygen consumption (V̇o_{2 max}) in skeletal muscle is generally believed to be determined by the rate at which oxygen can be delivered to the mitochondria (40, 44). The supply of oxygen to mitochondria depends on convective transport in blood and on diffusive transport from blood to tissue. Both transport processes can restrict oxygen delivery to maximally exercising skeletal muscle (44). Under normoxic conditions, oxygen extraction from the blood is incomplete during maximal exercise (15, 16, 38, 43, 44), indicating that diffusive transport processes play a role in limiting the rate of oxygen consumption (44). The distance that oxygen must diffuse from capillaries to muscle tissue depends on muscle capillary density, defined as the number of capillaries per unit cross-sectional area of muscle. Previous theoretical studies (25) showed that for a given rate of convective oxygen supply, V̇o_{2 max} is strongly dependent on capillary density.

Measurements of human capillary density are difficult to obtain and, even when available, may not accurately represent oxygen transport capacity because of changes in muscle properties that occur during tissue preparation. The most usual method for measuring capillary density in humans is to visualize capillaries from a muscle biopsy sample with one of several different staining techniques. In a study comparing various staining methods, it was shown that capillary density measurements from serial sections of the same muscle can vary by as much as 19% depending on which staining technique is used (31). Another study showed that measurements may vary significantly depending on the depth from which the sample is taken and concluded that measurements from a single sample may be unreliable in the determination of capillary density (10). Swelling of muscle fibers as frozen tissue samples prepared for histochemical staining thaw on glass slides has also been shown to introduce errors in capillary density measurements (28). In six studies with biopsy samples from quadriceps muscle of healthy subjects with various levels of athletic training, capillary density values measured with histochemical staining techniques ranged from 123 to 515 capillaries/mm^{2} (8, 13, 20, 23, 26, 31). Both extreme values are from groups of healthy young (19–26 yr) male volunteers. The lowest value was found from hematoxylin and eosin-stained tissue (26). The highest value was from a group of sedentary subjects after 8 wk of training and was obtained with an amylase periodic acid Schiff (PAS) stain (20).

Electron microscopy has also been used to visualize capillaries in samples of muscle tissue. Its high resolution allows for clear identification of all capillaries. However, the hypertonic solution used during preparation of the tissue can lead to shrinkage of the muscle tissue (18). Measurements of capillary density found with electron microscopy on single biopsy samples from human quadriceps muscle range from 269 to 691 capillaries/mm^{2} with both extreme values obtained from healthy, untrained male subjects (47).

A Krogh-type model for predicting oxygen consumption in heavily working skeletal muscle based on a consideration of transport processes occurring at the microvascular level has been developed (25). Effects of the decline in oxygen content of the blood flowing along capillaries, intravascular resistance to oxygen diffusion, myoglobin-facilitated diffusion, and a right shift in the oxyhemoglobin saturation curve are included in the model. Parameter values are based on human skeletal muscle. Previously, the model was used to predict oxygen consumption rates in heavily working skeletal muscle with a range of capillary densities. Predicted consumption rates were compared with V̇o_{2 max} values measured during knee extensor exercise (1, 33). Andersen and Saltin (1) observed an average V̇o_{2 max} value of 35 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} in subjects who ranged from sedentary to endurance trained. With an assumed capillary density of 600 capillaries/mm^{2} and an oxygen demand of 80 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1}, the consumption predicted from parameters measured by Andersen and Saltin (1) was 36.1 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1}, close to the observed V̇o_{2 max} value. The subjects in the study by Richardson et al. (33) were all competitive endurance cyclists with an average V̇o_{2 max} of 60.2 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1}. Predicted consumption was ∼5% less than the measured value with a demand of 80 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1}, and capillary density of 1,000 capillaries/mm^{2}. This capillary density is higher than any measured value for capillary density in human skeletal muscle found in the literature.

The goal of the present study was to use this Krogh-type model to develop an alternative method for estimating capillary density in skeletal muscle. The model was used to estimate the minimum number of straight, evenly spaced capillaries per unit muscle cross-sectional area required to achieve a given observed oxygen consumption rate. Estimates are based on measurements of muscle oxygen consumption, arterial Po_{2}, and muscle blood flow made at V̇o_{2 max} during cycling or knee extensor exercise. The quadriceps and hamstring muscles are the primary muscles involved in cycling. Knee extensor exercise functionally isolates the quadriceps femoris muscle. Resulting estimates were compared with values obtained by counting capillaries in quadriceps muscle tissue with either histochemical staining techniques or electron microscopy. A simpler, spatially averaged model was also used to corroborate estimated capillary density values.

## METHODS

*Governing equations.* A Krogh-type cylinder model was used to represent oxygen diffusion in the tissue. The basic assumption of the Krogh model is that each capillary provides the sole oxygen supply to a cylindrical region of tissue centered on the capillary. As discussed below (see discussion), this approximation is justified under conditions of very high oxygen demand. The governing equation, based on Fick's law of diffusion and conservation of mass, is (1) The left-hand side of *Eq. 1* represents radial diffusion of oxygen, where P is Po_{2} at a radial distance of *r* within the tissue cylinder and *K* is the Krogh diffusion coefficient. Po_{2} gradients are much steeper in the radial than in the axial direction, and axial diffusion of oxygen is therefore neglected. The right-hand side of *Eq. 1* gives the oxygen consumption rate per unit volume of tissue. Oxygen consumption is assumed to obey Michaelis-Menten kinetics, where *M*_{0} is the oxygen demand, i.e., the consumption when the oxygen supply is not limiting, and P_{0} is the Po_{2} at which consumption is one-half of the demand. A modified form of *Eq. 1* is used to account for the effects of myoglobin-facilitated diffusion of oxygen within the tissue (25).

Intravascular resistance to oxygen diffusion is approximated as described by Hellums et al. (12). The boundary condition at the capillary wall is found by equating the transport of oxygen to the capillary wall to the diffusive flux into the tissue. The capillary walls are considered to be part of the tissue region. According to the assumptions of the Krogh cylinder model, no oxygen diffuses across the outer boundary of the tissue cylinder, and so dP/d*r* = 0 when *r* = *R*_{t}, the tissue cylinder radius.

The effects of axial decline in the average Po_{2} in the blood, P_{b}, are included. The oxygen content of blood is given by *C*_{B}S_{Hb}(P_{b}) + αP_{b}, where α is the solubility of oxygen in plasma and *C*_{B} is the carrying capacity of blood at 100% saturation. The oxyhemoglobin saturation, S_{Hb}(P_{b}), is described with the Hill equation (30). This is a good approximation for saturation levels from 20% to 80%, unless unloading of oxygen by red blood cells is extremely rapid. Red blood cell transit times calculated from given leg blood flow data and the geometry of the model range from 0.21 to 0.42 s. These values are ∼10 times larger than the time constant for oxyhemoglobin dissociation (46), and so use of the Hill equation is appropriate. In all cases considered here, hemoglobin saturation exceeds 80% in the upstream part of the capillary. In one case, it drops below 20% at the downstream end. To assess the possible errors resulting from use of the Hill equation over an extended range of saturations, an alternative representation for the oxyhemoglobin dissociation curve (the Easton fit; Ref. 4), valid for saturations ranging from 0% to 95%, was used. In the cases considered here, saturation exceeds 95% in no more than 4% of the capillary. Compared with corresponding results computed with the Hill equation, use of the Easton fit increased estimated capillary density values by no more than 3%. The Hill equation was therefore considered to provide a valid approximation to the oxyhemoglobin dissociation curve for all cases considered here.

*Oxygen consumption rate.* The radial Po_{2} profile within the tissue was calculated numerically at 100 discrete points along the length of the capillary. The consumption per unit length is found by numerically integrating the consumption per unit volume over the cross section of the tissue cylinder. The decline in blood oxygen content to the next nodal point is then computed according to conservation of mass. The corresponding P_{b} at that point is obtained by solving the equation giving the oxygen content of blood as a function of P_{b}. This procedure is repeated along the length of the capillary. The overall oxygen consumption rate is found by averaging the local consumption rate over the volume of the tissue cylinder. This value is generally less than the oxygen demand, because local consumption is less than demand in regions of low Po_{2}.

*Estimation of capillary density.* For each set of experimental data, the measured values of muscle blood flow, arterial Po_{2}, and hemoglobin concentration (C_{Hb}) were entered into the model. The capillary density is varied until computed maximal oxygen consumption matches the measured V̇o_{2 max} value, as shown schematically in Fig. 1. For example, the solid curve in Fig. 1 corresponds to data (39) in which arterial Po_{2} = 110.2 mmHg, hemoglobin concentration = 13.8 g/dl, and muscle blood flow = 7.0 l/min. Measured V̇o_{2 max} in this case is 43.2 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} (horizontal line). The intersection of this line with the curve gives the estimated capillary density, which for this example is 752 capillaries/mm^{2}.

*Parameter values.* Parameter values relating to oxygen transport in blood and tissue are chosen to represent human skeletal muscle under physiological conditions and are summarized in Tables 1 and 2. The oxygen content of blood depends on the oxyhemoglobin saturation and the oxygen-carrying capacity of the blood at full saturation. Under standard conditions, P_{50}, the Po_{2} at which hemoglobin is 50% saturated, is ∼26 mmHg for human blood (30). Numerous studies have shown that changes within the muscle during exercise, such as increases in temperature or decreases in pH, cause an increase in P_{50}, leading to a “right shift” of the oxyhemoglobin dissociation curve and thereby enhancing oxygen unloading. Endurance training has also been shown to cause an increase in P_{50} in trained muscle. Thomson et al. (42) observed a P_{50} of 38.8 mmHg in habitually active but not endurance-trained subjects during exercise. The effects of a right shift are included in the model by assuming that P_{50} increases linearly with distance traveled, from 26 mmHg at the capillary entrance to 39 mmHg at the venous end of the capillary (25). For comparison, calculations in which the dissociation curve is right-shifted along the entire length of the capillary, i.e., P_{50} = 39 mmHg throughout, were also considered. Other parameter values are *n*_{H} = 2.7, where *n*_{H} is the exponent in the Hill equation, and *C*_{B} = 1.39 × C_{Hb}. The hemoglobin concentration was measured in six of the nine data sets (Table 2). A value of C_{Hb} = 14.39 g/dl was assumed for the other three sets (30). The decline in Po_{2} in arterioles is assumed to be negligible when flow rates are high (41), as is the case here, and the Po_{2} entering the capillary is set equal to that of the arterial blood. Measured arterial Po_{2} values for each data set are given in Table 2.

Oxygen demand *M*_{0} can vary over a wide range in skeletal muscle. Experiments based on mitochondrial enzyme turnover imply that *M*_{0} ≤ 40 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} in maximally working muscle (3). However, measured oxygen consumption rates are higher than this in five of the nine studies considered. Because consumption cannot exceed demand, oxygen demand must exceed 40 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} in some cases. Measurements from Knight et al. (21) suggest that demand can exceed 69 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1}. A wide range of *M*_{0} values was therefore considered, with a reference value of 80 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} used to simulate maximal oxygen demand.

Capillary lengths *L* and average flow velocities *V̄*, have not been determined in exercising human skeletal muscle. However, the model results depend only on their ratio. The ratio *V̄*/*L* may be estimated from perfusion rate = , where *R*_{c} is capillary radius. Perfusion is determined from measured leg blood flow data, given in Table 2, by dividing by the mass of muscle active during knee extensor exercise. In *studies A–H*, active muscle mass is assumed to be 2.5 kg (32). In *study I*, Pedersen et al. (29) reported an estimated active muscle mass of 2.3 kg. A value *L* = 0.5 mm was chosen and, *V̄* was then calculated from the given leg perfusion rate for each study. The tissue cylinder radius *R*_{t} is determined from the capillary density *N*, where *N* = . The capillary radius *R*_{c} is assumed to be 2.5 μm, typical of mammalian skeletal muscle.

Intravascular resistance to oxygen diffusion depends on the mass transfer coefficient, which is given by *M _{t}* = π

*K*

_{pl}Sh, where

*K*

_{pl}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. For a diameter of 5 μm, Sh values ranging from 1.5 to 3 are consistent with available data. Here, Sh = 2.5 is assumed and

*K*

_{pl}= 8.3 × 10

^{–10}(cm

^{2}/s)(cm

^{3}O

_{2}·cm

^{–3}·mmHg

^{–1}) (12), giving

*M*

_{t}= 6.52 ×10

^{–9}(cm

^{2}/s)(cm

^{3}O

_{2}·cm

^{–3}·mmHg

^{–1}).

*Estimation of capillary density by spatially averaged model.* An alternative, simpler method to estimate capillary density from measured V̇o_{2 max} was used to corroborate the results of the calculations described above and to aid in their interpretation. This method retains most of the assumptions of the original Krogh cylinder model. The Po_{2} at the blood-tissue interface, which in reality varies along the length of the capillary, is represented by a single effective average value, . The oxygen consumption rate *M* is assumed to be constant and equal to the measured V̇o_{2 max}. Furthermore, it is assumed that this is the maximum possible uniform consumption rate, i.e., the Po_{2} drops to zero at the outer boundary of the Krogh cylinder. The only other set parameters in the spatially averaged model are the Krogh diffusion coefficient *K* [= 9.4 × 10^{–10} (cm^{2}/s)(cm^{3}O_{2}·cm^{–3}·mmHg^{–1})] and *R*_{c} (= 2.5 μm). *R*_{t} is considered to be unknown.

*Equation 1*, with the right-hand side replaced by the constant *M*, and subject to the boundary conditions P = when *r* = *R*_{c} and dP/d*r* = 0 when *r* = *R*_{t}, can be solved explicitly (30). According to the assumption that the oxygen consumption rate is maximal, the additional boundary condition P = 0 when *r* = *R*_{t} is imposed. With this assumption, it can be shown that (2) This equation may be solved numerically to obtain *R*_{t}, and the resulting estimate of capillary density is *N* = .

## RESULTS

Figure 2 compares 26 measured values with 9 estimated capillary densities. No study could be found in the literature that includes measurements of both capillary density and the parameters needed to estimate capillary density by the present method. Therefore, no direct comparisons can be made between estimated and measured values. The measured values shown in Fig. 2 (open symbols) are from six experimental studies (Refs. 8, 13, 20, 23, 26, 31). In each of these studies, capillary density measurements were made from histochemically stained biopsies from the vastus lateralis muscle in the quadriceps femoris muscle group. The study by Hepple et al. (13) examined effects of training on the vastus lateralis muscle of healthy older subjects, 65–74 yr old. The other five studies involved healthy young subjects, 18–29 yr old, with various levels of training. Studies by Hepple et al. (Ref. 13; 6 values), Klausen et al. (Ref. 20; 4 values), and Melissa et al. (Ref. 26; 3 values), compared measurements from the same groups of subjects before and after training. Kuzon et al. (Ref. 23; 2 values) compared measurements from competitive athletes with those of sedentary subjects. Differences in skeletal muscle biopsy samples from black and white college-aged subjects were the focus of the study by Duey et al. (Ref. 8; 2 values). Qu et al. (Ref. 31; 5 values) examined the effects of five different staining techniques on measurements of capillary density. Measured values shown in Fig. 2 (closed symbols) were determined by Zumstein et al. (Ref. 47; 4 values) with electron microscopy of muscle biopsy samples of vastus lateralis muscle from 18 subjects ranging in age from 19 to 45 yr. Values from this study shown in Fig. 2 correspond to average values for healthy women and men and for highly trained women and men.

The estimated capillary densities shown in Fig. 2 and Table 2 are obtained from nine data sets found in seven previously published studies. All of these studies involve healthy college-aged subjects. Two of the data sets correspond to sedentary subjects (5, 37). One data set is from sedentary subjects after 9 wk of endurance training (37). Schaffartzik et al. (39) studied the effects of C_{Hb} on oxygen uptake in healthy individuals. Two data sets come from this study, one with normal hemoglobin content and one with elevated hemoglobin content. One data set is from Knight et al. (21). In these six data sets, measurements of leg blood flow, arterial Po_{2}, and leg V̇o_{2 max} were made during whole body cycling. The final three data sets come from measurements made during maximal knee extensor exercise (29, 32, 34). In the two studies by Richardson et al. (32, 34), subjects were elite bicyclists. Subjects in the Pedersen et al. study (29) were described as healthy.

Published values of capillary density shown in Fig. 2 are from quadriceps muscle of healthy subjects and range from 123 to 515 capillaries/mm^{2} (20, 26) with an average of 364 ± 110 capillaries/mm^{2} (mean ± SD) and a median of 416 capillaries/mm^{2}. The lowest value, 123 capillaries/mm^{2}, is from histochemically stained tissue of physically active subjects (26). The second and third lowest values, 129.6 and 135 capillaries/mm^{2}, are from this same group of subjects after 8 wk of aerobic training (26). The highest measured value, 514.8 capillaries/mm^{2}, is from tissue from a group of sedentary male students after 8 wk of aerobic training measured with an amylase PAS staining technique (20). The highest individual value reported, 691 capillaries/mm^{2}, was found by using electron microscopy on a biopsy from a healthy, untrained subject (47). The two measured values from groups of sedentary subjects (20, 23), 423 and 270.8 capillaries/mm^{2}, are not the lowest values, as one might expect. In fact, the higher of the sedentary measurements is greater than the average of all measured values.

Measurements made with electron microscopy, when averaged according to gender and training level, are not significantly different from average values obtained with histochemical staining techniques, as seen in Fig. 2. The average capillary density values found with electron microscopy are 418 and 442 capillaries/mm^{2} for healthy and trained subjects, respectively. These values are within the corresponding range of average values found with histochemical staining techniques, 123–492 capillaries/mm^{2} for healthy subjects and 130–515 capillaries/mm^{2} for trained subjects.

Our estimates of capillary density range from 458 to 1,468 capillaries/mm^{2}. Estimated capillary densities are, in general, much higher than measurements found in the literature, with an average of 761 ± 309 capillaries/mm^{2} (mean ± SD) and a median of 703 capillaries/mm^{2}. Parameter values from the study by Knight et al. (21) are significantly different from the other eight data sets. Excluding this study lowers the standard deviation in estimated capillary densities by nearly a factor of 2 and leads to an average of 673 ± 170 capillaries/mm^{2} (mean ± SD) and a median of 700 capillaries/mm^{2}. Only the two smallest estimated capillary densities, 459 and 463 capillaries/mm^{2}, fall within the range of average measured values shown in Fig. 2. These estimates correspond to measurements made on groups of sedentary subjects. The highest estimate, 1,468 capillaries/mm^{2}, corresponds to a group of trained male cyclists (21). The second-highest value, 937 capillaries/mm^{2}, is from a group of subjects with an elevated hemoglobin content caused by an 8-wk sojourn to an altitude of 3,801 m (39).

To assess the sensitivity of estimated capillary densities to the value chosen for oxygen demand, a wide range of *M*_{0} was considered, as shown in Fig. 3. Higher *M*_{0} values correspond to more complete extraction of oxygen. With a higher demand, a given oxygen consumption rate can be achieved with a lower capillary density. Because of resistance to oxygen transport both within the microvasculature and within the muscle tissue, demand must be significantly higher than consumption. In Fig. 3, results are therefore given for *M*_{0} values at least 10 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} greater than measured V̇o_{2 max}. Variation of *M*_{0} over a wide range gives relatively small changes in estimated capillary density, as seen in Fig. 3, so the uncertainty in *M*_{0} values does not lead to a large uncertainty in estimated capillary density. For all studies except Knight et al. (21), a 20% decrease in *M*_{0} from the reference value of 80 to 64 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} leads to at most a 14% increase in estimated capillary density and at least a 3.7% increase, with an average change of 6.4%. A 20% increase in *M*_{0} produces at most a 4.8% decrease in calculated capillary density for these eight studies. In the study by Knight et al. (21), measured consumption is 69.2 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1}, and a demand of 64 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} is not possible. In this case, a 20% increase in demand decreases estimated capillary density by 13%.

Estimates of capillary density obtained from the spatially averaged model are shown in Fig. 4. In this model, the effective Po_{2} at the blood-tissue interface, , is an unknown parameter. However, it must be much less than the arterial Po_{2} (of ∼100 mmHg) for two reasons. First, the sigmoidal shape of the oxyhemoglobin dissociation curve implies a rapid initial decline in blood Po_{2} as oxygen is extracted. Second, the intravascular resistance to oxygen diffusion implies that Po_{2} at the blood-tissue interface is less than the blood Po_{2}. Results for values from 20 to 40 mmHg are shown in Fig. 4, for consumption rates *M* corresponding to measured V̇o_{2 max} values ranging from 28.4 to 69.2 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} (Table 2). Figure 4 also includes corresponding estimates of capillary density obtained with the complete model. These estimates are generally within 12% of the results of the spatially averaged model, when a value = 28 mmHg is assumed. This level of is consistent with the above reasoning that must be considerably less than arterial Po_{2}. A larger deviation is evident for the result with the highest consumption (*study F* in Fig. 4). The extremely high capillary density estimated from the complete model in this case results from assuming an oxygen demand *M*_{0} that is close to the measured maximal consumption rate (80 vs. 69.2 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1}). The actual *M*_{0} may have been higher in this case, which would result in a lower estimated capillary density from the complete model, as indicated in Fig. 3. Results from the spatially averaged model are independent of *M*_{0} and depend only on measured V̇o_{2 max} values and assumed values of , *R*_{c}, and *K*.

## DISCUSSION

In the present method, a Krogh-type model is used to estimate capillary density based on measured V̇o_{2 max} values. The Krogh cylinder model (22) is a highly simplified representation of oxygen transport processes occurring in the microcirculation (30), and it is necessary to consider whether these simplifications could lead to significant errors in the estimation of capillary density. By including intravascular resistance to oxygen transport and taking account of the axial decline in Po_{2}, the present model avoids two important simplifications made in Krogh's original analysis. The original Krogh model assumed a constant uniform rate of oxygen consumption in tissue, whereas Michaelis-Menten dependence of oxygen consumption on Po_{2} is assumed here (*Eq. 1*). The present model neglects effects of oxygen diffusion in the axial direction. To test whether this introduces significant errors, further calculations were carried out using the same governing equations but including axial diffusion, using a finite-element method (FlexPDE; PDE Solutions, Antioch, CA). For parameter values in the range considered here, inclusion of axial diffusion increased predicted consumption by no more than 0.5%, resulting in at most a 1.6% decrease in estimated capillary densities. In the present model, oxygen demand is assumed to be uniform throughout all muscle fibers. In reality, maximal oxygen consumption may be reached at submaximal levels of muscle activation, with some fibers having relatively low demand. Such nonuniform demand would be expected to increase the capillary density needed to support a given overall consumption rate, because the consumption in the active fibers must then be higher than the overall rate, and correspondingly steeper gradients in Po_{2} are required for diffusive oxygen delivery.

A fundamental assumption of the Krogh model is that capillaries are equally spaced and each capillary represents the sole oxygen supply to a surrounding cylindrical tissue region. In reality, capillaries are not evenly spaced and the regions that they supply are not necessarily cylindrical. Moreover, significant oxygen losses may occur in arterioles (9), and capillaries may in some places take up rather than deliver oxygen (11). Theoretical simulations of three-dimensional oxygen diffusion (41) confirmed the importance of these phenomena in resting muscle but showed that they become less significant as the rates of consumption and blood flow increase. At the very high consumption rates considered here, available oxygen is consumed within a short radial distance of each vessel, on the order of tens of micrometers (25). Therefore, the fraction of the tissue that can be supplied by arterioles, which are spaced hundreds of micrometers apart, is limited and each capillary can supply only a relatively slender region of adjacent tissue, as assumed in the Krogh model.

To further explore the effects of arteriolar oxygen supply on estimated capillary density values, we performed calculations assuming that 10% of the total oxygen consumed is supplied by diffusion from arterioles. In this case, the amount of oxygen that capillaries must deliver for a given overall consumption rate is decreased, implying a decrease in estimated capillary density. However, the oxygen content of the blood entering the capillaries is also reduced, because of loss of oxygen from the arterioles, implying an increase in estimated capillary density. The combination of the effects of oxygen loss from arterioles corresponding to 10% of the total oxygen consumption reduces estimated capillary density an amount ranging from 7.4% (*study C*) to 16% (*study F*).

In reality, capillaries do not have identical flow rates, as assumed in the model. To estimate the effect of heterogeneous perfusion on estimated capillary densities, we considered a configuration consisting of two identical cylinders, one with flow decreased by one-third and the other with flow increased by one-third. For given capillary densities, predicted consumption rates, averaged over the two cylinders, were reduced by ∼3% in most cases (Table 2, *studies C–E* and *G–I*) and estimated capillary densities for given consumption rates increased by ∼5%. In *study F*, larger changes were found, whereas in *studies A* and *B* the numerical method failed in the capillary with reduced flow. In general, including heterogeneous perfusion would result in small increases in estimated capillary density values.

Estimated capillary densities depend on assumed values of several oxygen transport parameters. The effects of varying these parameters are shown in Table 3. Because estimated capillary densities are much higher than measured values, parameter values have been modified only in the direction that reduces estimated capillary density. Early measurements of the diffusivity of myoglobin (*D*_{Mb}) made in protein solutions gave *D*_{Mb} = 8.0 × 10^{–7} cm^{2}/s, a much higher value than that used here. If calculations are repeated with this higher diffusivity, estimated capillary densities are reduced by 5.4–8.2%, depending on which data set is used for the remaining parameters. Increasing *D*_{Mb} reduces the average capillary density from 761 to 708 capillaries/mm^{2}, but estimated values are still considerably higher than measured values shown in Fig. 2. An increase in the value used for the concentration of myoglobin (C_{Mb}) will also lower estimated values of capillary density. However, even if C_{Mb} is increased by a factor of 10, estimated capillary densities are still much higher than measured values and range from 381 to 1,299 capillaries/mm^{2}.

Increasing the value of *K* would decrease estimated capillary density. For instance, doubling *K* would lead to estimated capillary densities of 339–1,125 capillaries/mm^{2}, closer to, but still higher than, the range of measured values. However, the assumed value *K* = 9.4 ×10^{–10} (cm^{2}/s)(cm^{3}O_{2}·cm^{–3}·mmHg^{–1}) is already among the highest found in the literature, so the assumption of a much higher value would contradict most existing data on this point. For a vessel diameter of 5 μm, Sh values ranging from 1.5 to 3.0 are consistent with available data (12). An increase in Sh, which corresponds to a decrease in intravascular resistance, will lower estimated values. Increasing Sh from 2.5 to 3.0 decreases the average estimated capillary density from 761 to 720 capillaries/mm^{2}, still nearly twice the average of measured values.

Hemoglobin P_{50} values as high as 38.8 mmHg have been measured in active subjects during exercise. Increasing P_{50} to 39 mmHg throughout the vessel also lowers estimated capillary densities, but again not enough to account for the differences between measured and estimated values. Experimentally determined values for P_{0}, the Po_{2} at which consumption is half of demand, range from 0.05 to 1.25 mmHg (6, 36). If calculations are repeated with constant consumption, i.e., P_{0} = 0 mmHg, estimates decrease on average by 6% and range from 437 to 1,253 capillaries/mm^{2}.

Exact capillary geometries are not known for the muscles used here. Human capillary diameters typically range from 4 to 8 μm. If the value of *R*_{c} is increased from 2.5 to 4 μm, estimated capillary densities range from 385 to 1,169 capillaries/mm^{2}. Results from the model depend only on the ratio *V̄*/*L*. For any given *L, V̄* is calculated with the measured perfusion values, and the model is thus insensitive to changes in *L*.

The model estimates are based on four experimentally determined parameters: arterial Po_{2}, leg blood flow, leg V̇o_{2 max}, and, when given, C_{Hb}. All measurements have some degree of error associated with them. It is therefore important to determine the sensitivity of the model to possible errors in the input parameters. For given measurements of C_{Hb} and arterial Po_{2} in the studies used here, the standard error is on average ∼5% of the measured values. The standard error for measurements of V̇o_{2 max} and blood flow are closer to 10% of the measured values. For comparison, each of these input parameters for each of the studies was varied independently in the model by ±10%. The model is least sensitive to values of arterial Po_{2}. A 10% change in either direction causes at most a 2% change in estimated capillary densities. A 10% increase in the values of either blood flow or C_{Hb} decreases estimated capillary density by at most 8.2%. A 10% decrease in either of these measurements changes estimated capillary density by as little as 5.8% and as much as 14%. Because errors in the measurements of C_{Hb} are generally smaller than 10%, results of the model are fairly insensitive to such errors. Errors in blood flow measurements are slightly larger than errors in C_{Hb} measurements, and the model is therefore slightly more sensitive to the flow measurement errors. The model is most sensitive to errors in the measurement of V̇o_{2 max}. In the case of Knight et al. (21), a 10% increase in V̇o_{2 max} increases estimated capillary density by 75%. However, the increased value of V̇o_{2 max}, 76.12 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1}, suggests that a demand higher than 80 cm^{3}O_{2}·(100 cm^{3})^{–1}·min^{–1} would be needed to achieve this consumption. In the other studies, a 10% increase in V̇o_{2 max} increases estimated capillary density by 16–21%. The model is slightly less sensitive to reductions in V̇o_{2 max}. A 10% decrease leads to an average decrease in estimated capillary density of 16%.

Capillary density measurements found in the literature are significantly lower than estimated capillary densities found by using the model. The arguments in the preceding paragraphs suggest that this discrepancy is unlikely to result entirely from inherent limitations of the model or inaccuracies in the estimation of oxygen transport parameters. Therefore, it is relevant to examine the possibility that current experimental techniques underestimate actual capillary densities in skeletal muscle.

Studies have shown that results from current histochemical staining methods are sensitive to the type of stain used to visualize the capillaries (31) as well as to the location from which the sample is taken (10). Electron microscopy has been used as an alternative method for visualizing capillaries (47). Several studies have shown large differences in fiber cross-sectional area measurements when comparing tissue fixed for electron microscopy with tissue frozen for histochemical staining (7, 14, 18, 28). These differences are most likely due to a combination of fiber shrinkage during fixation for electron microscopy and swelling when frozen sections prepared for histochemical staining thaw on glass slides. This suggests that electron microscopy may overestimate true capillary density values whereas measurements made with histochemical staining techniques may be underestimates. However, average capillary density values obtained with electron microscopy are not significantly higher than values obtained with histochemical staining techniques, as shown in Fig. 2, suggesting that both experimental techniques may underestimate true capillary density values.

Measured values of capillary density shown in Fig. 2 were all obtained from muscle biopsy samples. Muscle fibers in biopsy samples are highly contracted, leading to kinking of capillaries and to an overestimate of fiber cross-sectional area (47). Both of these factors result in an underestimate of actual capillary density values. Fiber contraction can be accounted for by normalizing fiber cross-sectional area to the sarcomere length, but this was not done in any of the studies shown in Fig. 2. Capillary length density *J*_{v}, i.e., the capillary length per unit volume of tissue, is an alternative measure of tissue capillarity that is insensitive to changes in tissue samples due to muscle fiber contraction. The relationship between capillary density *N* and *J*_{v} is given by *J*_{v} = c(k)*N*, where c(k) is a dimensionless parameter related to the degree of capillary tortuosity in the tissue sample, which reflects the degree of contraction within the fibers. When all capillaries are perfectly straight and parallel to the preferred orientation of the tissue, as in Krogh-type cylinder models, c(k) = 1 and *J*_{v} = *N*. When the orientation of the capillaries is completely random, c(k) = 2. Values of c(k) found in the literature for mammalian skeletal muscle range from 1.05 to 1.73 (19, 24, 45). If a value of c(k) = 1.73 (24), representing the greatest observed effects of tortuosity, is used to account for muscle contraction in the measured capillary density values shown in Fig. 2, corresponding values of *J*_{v} range from 213 to 891 mm^{–2}. Estimated values of *J*_{v} found by using the model are equal to estimated capillary density values and therefore range from 459 to 1,468 mm^{–2}. Comparing values of *J*_{v} estimated from measured capillary density values to model results shows that the discrepancy between measured and predicted capillary density values can be reconciled in large part by accounting for muscle fiber contraction present in biopsy samples.

Studies (20) have shown that capillary density increases significantly with endurance training and remains high in trained individuals. Trained subjects, in general, also have higher measured V̇o_{2 max} values. As expected, estimated capillary densities for trained subjects tend to be higher. Figure 2 shows that the lowest values of estimated capillary density clearly correspond to groups of sedentary subjects. Furthermore, the study by Roca et al. (37) gives data for a group of subjects before and after training. With these values, the estimated capillary density was much higher after training than before, 697 and 459 capillaries/mm^{2}, respectively, corroborating the experimental findings that capillary density is increased with training. However, an increase in capillary density with training is much less evident in the measured values. The two measurements of capillary density from groups of sedentary subjects are not the lowest values, as one would expect. In fact, one of these measurements, 423 capillaries/mm^{2}, is actually higher than the mean measured value of 364 capillaries/mm^{2}. One of the lowest values, 129.6 capillaries/mm^{2}, corresponds to a group of healthy subjects after 8 wk of training. In both measured and estimated values there is considerable overlap between groups of healthy subjects and groups of trained subjects. This can be partially accounted for by the fact that groups of healthy subjects contain individuals of various training levels and can include endurance athletes.

The present method for estimation of capillary density is not simple to implement and depends on the estimation of a substantial number of parameters. The alternative spatially averaged method described above provides a useful corroboration of the results of the complete model and confirms its general conclusion, i.e., that relatively high capillary densities are required to explain observed maximal oxygen consumption rates. This conclusion depends only on the assumed values of the capillary radius and the Krogh diffusion coefficient and on the fact that capillary Po_{2} declines rapidly along capillaries. The spatially averaged model provides an easy method to estimate capillary density from observed V̇o_{2 max} values, if a value of is assumed. According to the data considered here, estimates obtained from *Eq. 2* with a value of 28 mmHg are close to those obtained with the complete model, except at the highest consumption level.

In summary, estimates of capillary density in human skeletal muscle have been derived from measured values of maximal oxygen consumption with a theoretical, Krogh-type model. The model provides a method for determining the minimum number of straight, evenly spaced capillaries required to achieve a given oxygen consumption rate in maximally exercising skeletal muscle. Estimated capillary densities are generally much higher than measured values found in the literature. This discrepancy can be explained partly by accounting for muscle contraction in biopsy samples. These results suggest that estimates of maximal oxygen transport rates based on capillary density measurements obtained from muscle biopsy samples underestimate the oxygen transport capacity of capillaries in skeletal muscle.

## DISCLOSURES

This work was supported by National Heart, Lung, and Blood Institute Grants HL-07249 and HL-70657 and National Science Foundation Grant 9870659.

## Footnotes

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- Copyright © 2003 by the American Physiological Society