Since the early 19th century, the maximum rate at which an athlete can take up, transport, and use oxygen (V̇O_{2max} ) has been a focus of interest (^{3,7} ). In spite of reservations about the sensitivity of V̇O_{2max} to reflect training-induced improvements in endurance performance, it is still used to characterize cardiopulmonary function of athletes, and there are extensive guidelines on recommended protocols to produce this measure (^{4,26,29} ).

Attempts have been made to compare the V̇O_{2max} of elite athletes from a variety of sports to establish which mode of exercise elicits the greatest ability to take up, transport, and use oxygen (^{3,10} ). However, such comparisons have always been confounded by the known association between maximum oxygen uptake and body size.

It is well known that larger subjects have greater maximum oxygen uptake when V̇O_{2max} is expressed in the absolute units, liters (L·min^{−1} ). However, the most commonly reported measure of maximum oxygen uptake is a ratio standard in which mass-related V̇O_{2max} is expressed in the units (mL·kg^{−1} ·min^{−1} ). It is known that such ratio standards can mislead (^{16,24} ) and “over-scale” by converting the positive correlation between maximum oxygen uptake and body mass into a negative one (^{15} ). Thus, by comparing the maximum oxygen uptake of athletes expressed in either L·min^{−1} or mL·kg^{−1} ·min^{−1} , heavier athletes are being either advantaged or penalized, depending on units in which V̇O_{2max} is expressed, compared with their lighter counterparts. Clearly, a fairer comparison of maximum oxygen uptake needs to remove the effect of body mass before valid inference can be made as to which sport and/or mode of exercise requires its athletes to have the greatest maximum oxygen uptake.

There is a long history of demonstrations to show that physiological and performance variables are not linearly related to dimensions of the body (^{17,18,20,25} ); relationships tend to be nonlinear (^{13} ) and are said to be allometric (^{19} ). Allometric modeling expresses a physiological or performance variable y as a function of an anthropometric variable x by the equation y = a x ^{b} where a is the common multiplier and b the exponent. The constants a and b can be estimated by taking natural logarithms (ln) of x and y and then performing linear regression to produce an expression: ln(x ) = ln(a) + b · ln(x ) (^{13,19,28} ). It has been suggested that V̇O_{2} and V̇O_{2max} should be expressed as power function ratios where, from the surface law, body mass should be raised to the power 0.67 (^{15,19} ) or, with acknowledgment of elasticity in body structures, 0.75 (^{11} ). Moreover, a sample-specific exponent identified by appropriate log-log transformations can be used. Groups can be compared by means of these power function ratios or, alternatively, by log-linear ANCOVA (^{15} ). Such approaches assume that growth and development are accompanied by isometric changes in the size of the body and its segments. Alexander et al. (^{1} ) demonstrated clearly that in animals, such changes do not occur and that disproportionately large growth occurs for instance in the lower limb, especially the thigh.

Opportunities to explore allometric relationships in elite athletes are rare because of the difficulties in securing groups large enough to make such investigations meaningful. In this study, two large groups of elite sportsmen and women have been recruited as participants, and thus no such limitation was imposed. Hence, the purpose of the present study was to compare the maximum oxygen uptake (V̇O_{2max} ) of elite endurance athletes from a range of endurance sports and to identify and interpret the most appropriate methods to express V̇O_{2max} independent of body size.

METHODS
Participants (Study 1)
With local ethics committee and institutional procedures’ approval, 174 international-standard men and women athletes from seven sports in which endurance was an important component provided written informed consent and participated. The physical characteristics of participants (mean ± SD) are described in Table 1 .

TABLE 1: Physical characteristics (mean ± SD) of the men and women participants from study 1 by the various sports.

The athletes were part of either United Kingdom government funded sport science support program or the British Olympic Association’s similar scheme. Participants included world champions, world record holders, and Olympic gold medallists.

Procedures
Laboratory assessments were undertaken at the British Olympic Medical Center, Northwick Park Hospital, Harrow, UK, and De Montfort University Bedford, Bedford, UK. The athletes were fully accustomed to the procedures. Body mass was assessed to the nearest 0.05 kg by using beam balance scales (Avery Berkel, Dublin, Ireland; Herbert and Sons, Edmonton, UK) with the athletes wearing minimal clothing. By using British Association of Sport and Exercise Sciences criteria (^{4} ), V̇O_{2max} was determined during either continuous or discontinuous incremental tests to volitional exhaustion on motorized treadmills or rowing ergometers. Expired air was collected and analyzed by on-line breath-by-breath systems (CoVox Microlab, Exeter, UK; Jaeger EOS-Sprint, Market Harborough, UK; Mijnhardt Oxycon Champion, Bunnik, The Netherlands), which were calibrated immediately before and verified immediately after testing using gases of known concentration and syringes of known volume. Test-retest coefficients of variation (^{17} ) for V̇O_{2max} in the groups were less than 4%. The percentage of body fat was estimated using the techniques proposed by Durnin and Womersley (^{6} ).

Participants (Study 2)
With local ethics committee and institutional procedures’ approval, 106 men and 30 women from 11 sports or activities provided written informed consent and participated. Male subjects were part of a study designed to predict % fat in athletes (^{22} ). The physical characteristics of the participants (mean ± SD) are described in Table 2 . The men comprised runners, cyclists, swimmers, triathletes, racket players, upper-body athletes (rock climbers and kayakers), rugby players, rowers, strength athletes (power-lifters and bodybuilders), and noncompetitive “keep-fit” exercisers who undertook advanced circuit training. The women comprised cyclists, racket players, runners, rowers, noncompetitive exercisers, and dancers. All (except the keep-fit group and dancers) had competed at university or higher level and had trained for a minimum of 4 h·wk^{−1} , for a minimum of 3 yr. Of the entire group, 26 men and four women were international athletes, and of these, two men were Olympic athletes.

TABLE 2: Physical characteristics and corrected calf and thigh muscle girths (mean ± SD) of the men and women participants from study 2 by the various sports and activities.

Procedures
Anthropometric measures were made according to procedures recommended by the International Society for the Advancement of Kinanthropometry. Circumferences were measured using an anthropometric tape (Rockton, IL) to 0.1 cm at the thigh (midway between the inguinal crease and the proximal border of the patella) and the maximum calf on the right side of the body, or left if this was the preferred limb. Skinfolds were measured to 0.1 mm on the right side of the body using Harpenden calipers (British Indicators, Luton, UK) along these circumferences at anterior thigh and medial calf.

The calculation of corrected girths applies to upper and lower limbs, and used the principal assumptions outlined in Stewart et al. (^{23} ). The tissue boundaries were assumed to be circular and concentric. If the skin plus adipose tissue thickness is d, and the thigh girth is Gthigh, then the thigh muscle girth GTM is given by GTM = Gthigh − 2 pi·d (skinfold values are converted to cm for this calculation). If it is further assumed that the skinfold caliper reading S is twice the adipose tissue thickness, then GTM = Gthigh − pi·S. Physical characteristics and corrected calf and thigh muscle girths (mean ± SD) by the various sports and activities of the men and women participants from study 2 are described in Table 2 .

Statistical Methods
Study 1.
Differences in maximum oxygen uptake between “sport” and “sex” were compared using a two-way ANOVA. Maximum oxygen uptake was expressed in the absolute units, liters (L·min^{−1} ), as well as the ratio standard mass-related units (mL·kg^{−1} ·min^{−1} ). Comparisons of maximum oxygen uptake independent of body mass, age and body fat, were made using the ANCOVA , based on the proportional allometric model originally proposed by Nevill and Holder (^{13} ) and subsequently developed by Nevill and Holder (^{14} ). The adopted model for maximum oxygen uptake (V̇O_{2} ) is given by

The model can be linearized with a log-transformation, and ANCOVA can then be used to estimate the effects of sport and sex having controlled for differences in these confounding covariates of mass, age, age^{2} , and %fat. The transformed ANCOVA model becomes where the constant b_{0} is allowed to vary for each level of sport and sex. If either the ANOVA or ANCOVA main effect difference between sports was detected, pairwise comparisons were made using the Bonferroni adjustment for multiple comparisons.

To confirm the appropriateness of the allometric model (Equation 1 ), the maximum log-likelihood criterion, originally recommended by Cox (^{5} ) to compare models with different structural forms, was used to assess and compare the quality of fit of the log-transformed allometric ANCOVA model (Equation 2 ) with the equivalent additive ANCOVA model using untransformed data.

Study 2
To establish whether calf and thigh muscle girth (MG) increased in proportion to body mass (kg), the following allometric model was fitted:

As with study 1, the model can be linearized with a log-transformation and ANCOVA used to compare leg muscle girths by sport and sex having controlled for differences in body mass, i.e., by allowing a_{i} to vary for each level of sport and sex using body mass as the covariate. Based on the principal of geometric similarity (^{19} ) (assuming participants’ bodies are the same shape but their body size dimensions vary/increase proportionally), both the calf and thigh muscle girths [linear dimensions (L) of body size] should increase in proportion to m ^{0.33} , where m denotes the participants’ body mass [a volumetric dimension (L^{3} ) of body size]. Hence, we anticipated that the mass exponents associated with the calf and thigh muscle girths, estimated using Equation 3 , should be approximately k_{2} = 0.33. Also based on the principle of geometric similarity, we could estimate that calf and thigh leg muscle volumes should increase in proportion to muscle girth (MG)^{3} , or from Equation 3 , (mass^{k2} )^{3} . MATH

RESULTS
Study 1.
Differences in maximum oxygen uptake, expressed in absolute units (L·min^{−1} ) and mass-related units (mL·kg^{−1} ·min^{−1} ) between the main effects sport and sex can be seen in Figures 1 and 2 , respectively.

FIGURE 1: Differences in maximum oxygen uptake (L·min−1) between sport and sex from study 1 (mean ± SD). * Heavyweight rowers’ mean V̇O2max (L·min−1) was greater than all the other sports (P < 0.001).

FIGURE 2: Differences in maximum oxygen uptake (mL·kg−1·min−1) between sport and sex from study 1 (mean ± SD). * Long-distance runners’ mean V̇O2max (mL·kg−1·min−1) was greater than all other sports except middle-distance runners (P < 0.01).

As anticipated by Nevill and Holder (^{13} ), the residuals from the ANOVA of maximum oxygen uptake (L·min^{−1} ) were not normally distributed, and a logarithmic transformation was required to correct this nonnormality. The two-way ANOVA of log_{e} (V̇O_{2} ) confirmed significant sport and sex main effects but no interaction. The Bonferroni pairwise comparisons between sports identified heavyweight rowers as having the greatest V̇O_{2max} [men = 6.5; (SD = 0.82) L·min^{−1} and women = 4.2 (SD = 0.40) L·min^{−1} ] that was greater than all the other sports (P < 0.001).

In contrast, the residuals from the two-way ANOVA of mass-related maximum oxygen uptake (mL·kg^{−1} ·min^{−1} ) were found to be acceptably normally distributed. The two-way ANOVA of V̇O_{2max} (mL·kg^{−1} ·min^{−1} ) also confirmed significant sport and sex main effects again with no interaction. However, on this occasion, the Bonferroni pairwise comparisons between sports identified the long-distance runners as having the greatest V̇O_{2max} [men = 77.4 (SD = 3.7) mL·kg^{−1} ·min^{−1} and women = 64.1 (SD = 4.6) mL·kg^{−1} ·min^{−1} ] that was significantly greater than all other sports except middle-distance runners whose mean V̇O_{2max} was only 1.44 (SEE = 1.39) mL·kg^{−1} ·min^{−1} below that of the long-distance runners.

The two-way ANCOVA of log_{e} (V̇O_{2} ) (Equation 2 ) confirmed significant sport and sex main effects but no interaction (Fig. 3 ).

FIGURE 3: Differences in adjusted maximum oxygen uptake (L·min−1) between sport and sex from study 1 (mean ± SD), adjustments made for the mean age (24.6 yr), body mass (67.8 kg), and body fat (16.7%). * Badminton and squash players’ mean adjusted V̇O2max were less that all other sports (P < 0.05).

The Bonferroni pairwise comparisons of the adjusted V̇O_{2max} between sports now identified the middle-distance runners as having the greatest adjusted V̇O_{2max} [men = 4.9 (SD = 0.58) L·min^{−1} and women = 4.3 (SD = 0.34) L·min^{−1} ] that was only significantly greater than badminton and squash players, i.e., there was no significant difference in adjusted V̇O_{2max} between the five groups middle-distance runners, long-distance runners, heavyweight rowers, lightweight rowers, and triathletes (P > 0.05). The ANCOVA of log-transformed V̇O_{2max} identified log_{e} (mass), age, and age^{2} as significant covariates. The body mass exponent was estimated as k = 0.94 (95% confidence interval from 0.79 to 1.08). The age and age^{2} slope parameters were b_{1} = 0.029 and b_{2} = −0.00048, respectively. Elementary differential calculus estimated V̇O_{2max} to peak at age = −b_{1} /(2·b_{2} ) = 0.029/0.00096 = 30.2 yr. The %body fat slope parameter was estimated as b_{3} = −0.0079.

The superiority of the allometric ANCOVA model was confirmed using the maximum log-likelihood quality-of-fit criterion. The maximum log-likelihood criterion for the log-transformed V̇O_{2max} model (Equation 2 ) described above was −28.34, much greater than the maximum log-likelihood criterion based on the equivalent additive linear model given by −59.89. Note that in these comparisons, the same number of predictor variables was used in both the log-transformed and nontransformed models.

Note that the fitted body mass exponent, k_{1} = 0.94, had a 95% confidence interval from 0.79 to 1.08. This interval precludes the two theoretical exponents of 0.67 and 0.75, but does included the ratio-standard mass exponent 1.0. To facilitate comparisons with other studies only, Table 3 describes mean V̇O_{2max} and SD, expressed as power function ratios in the units (mL·kg^{−0.67} ·min^{−1} ) and (mL·kg^{−0.75} ·min^{−1} ), by sport and sex.

TABLE 3: Mean and standard deviation (mean ± SD) V̇O2max (mL·kg−0.67·min−1) and (mL·kg−0.75·min−1) from study 1 by sport and sex.

Study 2.
To explain the unexpectedly high body mass exponent identified in study 1 (k_{1} = 0.94), we investigated whether the calf muscle girth and thigh muscle girth increased in proportion to body mass (kg) by using the allometric model (Equation 3 ). As described earlier in the Methods section, based on the assumption of geometric similarity, the body mass exponents should be approximately k_{2} = 0.33.

The ANCOVA failed to identify any significant differences between either sport or sex in calf or thigh muscle girths after having controlled for differences in body mass. The fitted body mass exponents (Equation 2 ) for the calf were k_{2} = 0.37 (95% confidence interval from 0.29 to 0.45) and for the thigh were k_{2} = 0.46 (95% confidence interval from 0.38 to 0.54). Note that the mass exponent for the calf muscle girth encompasses the theoretical value 0.33, but the mass exponent for the thigh muscle girth exceeds this theoretical value 0.33, predicted by geometric similarity. Based on these exponents and assuming geometric similarity, we can estimate that the calf and thigh muscle volume exponents (given by m ^{k2} ^{·3} ) will be proportional to m ^{1.11} and m ^{1.38} , respectively.

DISCUSSION
The men and women heavyweight rowers demonstrated the greatest V̇O_{2max} when expressed in the absolute units (L·min^{−1} ). Of course, this finding simply reflects the rowers’ greater body mass [men = 93.3 (SD = 6.0) women = 75.9 (SD = 7.2) kg] as seen in Table 1 . In contrast, when V̇O_{2max} was expressed as the traditional ratio standard (mL·kg^{−1} ·min^{−1} ), the lighter long-distance and, to a lesser extent, the middle-distance runners were found to have the greatest mass-related V̇O_{2max} . This result was anticipated by Nevill et al. (^{15} ), who demonstrated using log-linear multiple regression, that 5-k running speed was best predicted by speed (m·s^{−1} ) = 84.3·(V̇O_{2max} )^{1.01} ·mass^{−1.03} , i.e., the best predictor of 5-k run times recorded as a running speed (m·s^{−1} ) is almost exactly proportional to the ratio standard maximum oxygen uptake (L·min^{−1} ) divided by body mass (kg) or (mL·kg^{−1} ·min^{−1} ).

However, in an attempt to identify the form of exercise or sporting activity that elicits the greatest V̇O_{2max} independent of body mass (as well as other confounding variables), the ANCOVA of log transformed V̇O_{2max} could not distinguish between the five endurance sports of middle-distance runners, long-distance runners, heavyweight rowers, lightweight rowers, and triathletes. Observed differences were not significant (P > 0.05). Only the mass-adjusted V̇O_{2max} values of the two racket sports, badminton and squash, were less than the other five pure endurance sports (P < 0.001 and P < 0.05, respectively). The allometric ANCOVA model estimated the body mass exponent as 0.94 (95% confidence intervals from 0.79 to 1.08), which is significantly higher than either of the anticipated body mass exponents 0.67 or 0.75 previously reported (^{3,9,12,13,15,19} ), although not dissimilar to that (1.01) reported by Sjödin and Svedenhag (^{21} ) using young trained male runners. We acknowledge that the 95% confidence interval for mass in this study does indeed encompass the ratio-standard parameter 1.0, suggesting a possible linear association between V̇O_{2max} (L·min^{−1} ) and body mass. However, a closer inspection of Figure 2 clearly demonstrates that the ratio standard favors the lighter long- and middle-distance runners and thus fails to render V̇O_{2max} (mL·kg^{−1} ·min^{−1} ) completely independent of body mass. The results from study 2 provide a plausible explanation for these inflated exponents.

Alexander et al. (^{1} ) observed that the proximal leg muscle mass of a wide range of mammals had a greater proportion of muscle mass to body mass in larger mammals, m ^{1.1} . The results of study 2 suggest that the volume of calf muscle mass increases at approximately the same rate, m ^{1.11} , but the volume of thigh muscle mass of the sportsmen and women increases at an even greater rate, in proportion to m ^{1.38} . If we assume that V̇O_{2max} of nonathletes increases in proportion to body mass (m ^{2/3} ) and that the thigh leg muscles are making a major contribution to the participants’ V̇O_{2max} performance, then the larger athletes will consume a similar disproportionate increase in maximum oxygen uptake given by not dissimilar to the mass exponent 0.94, associated with V̇O_{2max} of elite athletes identified by the ANCOVA in study 1. We recognize that drawing the above conclusions, based on data from two independent studies, is a limitation of the present work. Clearly, future research should attempt to collect measurements of maximum oxygen uptake and estimates of leg muscle mass on the same individuals to help confirm the above assumptions and associations between V̇O_{2max} and leg muscle mass.

It is important to recognize the value of adopting the allometric ANCOVA model (Equation 1 ) compared with alternative additive/linear ANCOVA models (^{8,24,27} ). The superiority was confirmed when the maximum log-likelihood criteria for the log-transformed ANCOVA analyses were greater than the analyses based on the equivalent additive linear models (not log-transformed). Because the maximum likelihood criteria were greater, the error variance will automatically be less. Consequently, any inferences drawn from the results based on the log-transformed model (Equation 2 ) are more powerful with reduced probability of type II errors and hence more likely to lead to correct conclusions when comparing differences in V̇O_{2max} between groups of sportsmen and women.

Although postural muscles of the upper and lower leg may be common to all activities, strength training triggers adaptations including hypertrophy and force-velocity characteristics according to movement and activity type according to the principle of specificity (^{2} ). Despite this, no significant differences were found in calf and thigh corrected girths between sporting groups after adjustment for body size, reinforcing the validity of this methodology. In contrast, comparison of anthropometric relationships of upper-body morphology between sporting groups is more problematic due to differential contribution of specific muscle groups to performance. Thus, upper body modeling may be inappropriate using such a variety of sporting groups and has been omitted from the present analysis.

The ANCOVA also identified significant age and age^{2} parameters as b_{1} = 0.029 and b_{2} = −0.00048, respectively, from which we can estimate that the V̇O_{2max} of elite athletes will peak at approximately 30 yr. This is consistent with the training that is required to develop endurance capacity, whereas sprinters in general perhaps owe more to genetic endowment than subsequent training practices. Also noteworthy is the observation that V̇O_{2max} declines at the rate of 0.79% for each additional percentage of body fat of elite endurance athletes.

In conclusion, the results of the present studies have identified a disproportionate increase in both calf and thigh leg muscle mass (m ^{1.11} and m ^{1.38} , respectively) and a similar disproportionate increase in V̇O_{2max} found to be in proportion to m ^{0.94} . Having controlled for these differences in body mass, the ANCOVA of log transformed V̇O_{2max} could not distinguish between the five sports, middle-distance runners, long-distance runners, heavyweight rowers, lightweight rowers, and triathletes. Only the V̇O_{2max} of two racket sports, badminton and squash, were significantly below the other five pure endurance sports.

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