## Abstract

The aim of this study was to critically examine the influence of body size on maximal oxygen uptake (V̇o_{2 max}) in boys and men using body mass (BM), estimated fat-free mass (FFM), and estimated lower leg muscle volume (Vol) as the separate scaling variables. V̇o_{2 max} and an in vivo measurement of Vol were assessed in 15 boys and 14 men. The FFM was estimated after percentage body fat had been predicted from population-specific skinfold measurements. By using nonlinear allometric modeling, common body size exponents for BM, FFM, and Vol were calculated. The point estimates for the size exponent (95% confidence interval) from the separate allometric models were: BM 0.79 (0.53–1.06), FFM 1.00 (0.78–1.22), and Vol 0.64 (0.40–0.88). For the boys, substantial residual size correlations were observed for V̇o_{2 max}/BM^{0.79} and V̇o_{2 max}/FFM^{1.00}, indicating that these variables did not correctly partition out the influence of body size. In contrast, scaling by Vol^{0.64} led to no residual size correlation in boys or men. Scaling by BM is confounded by heterogeneity of body composition and potentially substantial differences in the mass exponent between boys and men. The FFM is precluded as an index of involved musculature because Vol did not represent a constant proportion of FFM [Vol∝FFM^{1.45} (95% confidence interval, 1.13–1.77)] in the boys (unlike the men). We conclude that Vol, as an indicator of the involved muscle mass, is the most valid allometric denominator for the scaling of V̇o_{2 max} in a sample of boys and men heterogeneous for body size and composition.

- oxygen uptake
- allometry
- magnetic resonance imaging

in nature, maximal oxygen uptake (V̇o_{2 max}) is rarely attained, and most individuals function routinely at submaximal metabolic rates. However, V̇o_{2 max} is thought to be ecologically relevant and subject to selection (17). Furthermore, it sets the upper limit for relatively sustainable energy expenditure and, at least in adults, is inversely related to the risk of several chronic diseases (20).

A clear understanding of the development of V̇o_{2 max} between childhood and adulthood is confounded by variability in body size and composition. The question of the influence of body size on energy metabolism has occupied researchers for 100 years (8). Accounting properly for the influence of body size on V̇o_{2 max} requires detailed knowledge of size-function relationships. The most appropriate statistical model of size-structure and size-function relationships is provided by allometry (9). Huxley's simple allometric equation, *Y* = *a**X*^{b}, has been employed most frequently in scaling studies [where *Y* is the structural or functional variable of interest, *a* is the proportionality coefficient, *X* is the body size variable (usually mass), and *b* is the size exponent]. The resultant power function ratio *Y*/*X*^{b} is allegedly free from the confounding influence of body size.

To date, the most appropriate denominator for the power function ratio for scaling V̇o_{2 max} has not been established, thus limiting our understanding of the size-function relationship in the growing child (26, 27). Using body mass (BM) as a general index of body size, reported size exponents for children and adults are variable, ranging from 0.28 to 1.10 (5, 11, 15, 26, 38). However, as ∼90% of the oxygen passing through the lungs of a mammal exercising at V̇o_{2 max} is bound for a single sink in the skeletal muscle mitochondria (21), estimated fat-free mass (FFM) may be a more judicious choice as an indicator of body size (9, 33, 35). Indeed, it was suggested half a century ago that any scaling analysis in humans should be based ideally on FFM due to the often marked heterogeneity of body composition within human samples (13).

Although both BM and FFM have been used to scale V̇o_{2 max}, at best they only provide a surrogate measure of the metabolically active muscle mass during exercise that elicits V̇o_{2 max}. A direct quantification of the muscle, or at least a proportion of it, that is active during exercise should be used ideally to scale V̇o_{2 max}. Magnetic resonance imaging (MRI) is a noninvasive technique that can be used to provide an in vivo quantification of muscle volume (34, 36), which could further our understanding of the influence of body size on V̇o_{2 max} in the growing child. The number of studies that have attempted to quantify a proportion of the muscle volume that is active during dynamic exercise and then examined its relationship with V̇o_{2 max} in young people is rather sparse (e.g., Refs. 12, 15, 36). Moreover, the results from these studies vary because of differences in the methods adopted to estimate muscle volume and the statistical methods that have been used to scale the V̇o_{2 max} data by this body size variable. In light of the literature discussed above, the purpose of the present study was to investigate the most appropriate normalizing factor (i.e., denominator) for V̇o_{2 max} in children and adults of widely varying body size. To achieve this, nonlinear allometric modeling procedures were used to critically examine the interplay between BM, FFM, muscle volume, and V̇o_{2 max}.

## MATERIALS AND METHODS

#### Participants.

Fifteen boys [age 12.3 yr (SD 0.3)] and 14 men [age 25.4 yr (SD 4.4)] volunteered to participate in the study that was approved by the University Research Ethics Committee. The boys were recruited from a local secondary school, whereas the men were university students and staff personnel. All of the participants indicated that they were involved in a variety of recreational and competitive sporting activities. The boys' parents and the men gave their written, informed consent, whereas the boys assented to the procedures following an extensive familiarization period. Completion of a pretest health questionnaire indicated that all participants were asymptomatic and apparently healthy, with no stated contraindications to exercise.

Before any data were collected, the boys completed a structured familiarization session to become accustomed with treadmill running, reduce apprehension, and minimize metabolic measurement variability. Each boy completed between 20 and 30 min of walking and running on the treadmill at various speeds (4–10 km/h) and gradients (0–8%). All participants (boys and men) were familiar with the procedures before data collection commenced. A self-assessment of secondary sexual characteristics by the boys was used to estimate physical maturity. The boys used drawings of the five stages of genitalia and pubic hair development to provide this information (23). The parents were asked to assist the boys with this assessment by *1*) discussing the schematic illustrations with them and *2*) comparing their son's genital and pubic hair development with the schematics and accompanying written descriptions. Five and 10 boys were at *stages 2* and *3* for genital development, respectively (median = 3), whereas 6 and 9 boys were at *stages 2* and *3* for pubic hair development, respectively (median = 3) (31). This would suggest that the boys were in the early to mid stages of puberty.

#### Anthropometric measurements.

Stature was measured to the nearest 0.01 m (Seca stadiometer 208) and BM to the 0.1 kg (Seca balance beam 710), with participants wearing only their running shorts and t-shirt. Skinfold thickness was measured to the nearest 0.2 mm using Harpenden callipers (John Bull, St. Albans, UK). All measurements were taken from the right hand side of the body, with the median of three measurements calculated as the fold thickness. Triceps and subscapular skinfold thickness were used to calculate the boys' percent body fat (%BF) using maturation-, race-, and sex-specific equations (29). The men's %BF was calculated using the sum of four sites (triceps, biceps, subscapular, and suprailiac) (14). The same experienced investigator took all of the skinfold measurements. The intra- and interinvestigator technical error of measurement in our laboratory using these sites with these populations are 3.8% and 5.7%, respectively. Therefore, our measures meet the criteria for acceptability for skilled anthropometrists of <5% and <7.5%, respectively (25). By using BM and %BF, it was possible to estimate FFM using the formula:

#### Treadmill protocol.

In a temperature-controlled laboratory (19–22°C), each participant completed an incremental exercise protocol on a motorized treadmill (Woodway, Ergo ELG2) designed to elicit V̇o_{2 max}. Following an initial warm-up, participants ran at a fixed speed with the treadmill gradient being raised 1% each minute until volitional physical exhaustion (2). No holding of the handrail was permitted during the test and strong verbal encouragement was given in the latter stages of the test. The starting speed was chosen to ensure exhaustion occurred between 8 and 12 min [10.0 min (SD 1.5)].

#### Measurement of V̇o_{2 max}.

Heart rate was monitored continuously via radio telemetry (Polar Accurex Plus, Kempele, Finland) as the participants ran on the treadmill. Expired air samples were collected into 150-liter Douglas bags in each successive minute until the termination of the test. Oxygen and carbon dioxide concentrations in each Douglas bag were analyzed using a paramagnetic oxygen analyzer and an infrared carbon dioxide analyzer (Servomex 1400, Sussex, UK) calibrated against gases of known concentration before each test. Volume of expired air was determined using a dry gas meter (Harvard, Kent, UK). For each sample, expired oxygen (V̇o_{2}), expired carbon dioxide, minute ventilation, and respiratory exchange ratio were calculated.

To verify an exhaustive effort, each participant had to satisfy at least two of the following criteria on termination of the treadmill test due to volitional exhaustion: *1*) a plateau in V̇o_{2} (≤2.1 ml·kg^{−1}·min^{−1}) with an increase in treadmill gradient, *2*) a maximum heart rate ≥95% age-predicted maximum (220 − age), and *3*) respiratory exchange ratio ≥ 1.00 for the boys and ≥ 1.10 for the adult men. In addition, all of the participants demonstrated overt signs of extreme physical exertion, such as hyperpnea, sweating, facial flushing and grimacing, and unsteady gait, at the end of the test. According to these criteria, all participants demonstrated a valid V̇o_{2 max} (Table 1).

#### Measurement of leg muscle volume.

Sequential axial plane MRI scans of the lower right leg using a fixed 0.2-T magnetic imaging scanner (E-scan, ESAOTE Biomedica, Genova, Italy) were used to measure leg muscle volume (Vol). Participants lay supine for the duration of the scans. Starting from the proximal head of the tibia (determined via ultrasonography), three to four contiguous 7-cm scans were performed distally toward the soleus-Achilles tendonous junction. A 7-cm scan above the proximal head of the tibia allowed the identification of the lateral and medial gastrocnemius (GM) insertions at the posterior of the knee joint. A T1 weighted three-dimensional isotropic profile with parameters set at time to echo = 16 ms, repetition time = 38 ms, field of view = 180 × 180 mm, matrix = 256 × 192 pixels was used for each scan. The borders of each scan were marked using cod liver oil tablets interpolated every 7 cm to prevent duplicate slice measurements. Each axial scan was divided into 7 × 1 cm slices. For every alternate 1-cm slice, the mean of three muscle anatomical cross-sectional areas measurements was calculated by tracing around the muscle using imaging software (NIH version 1.61/ppc, National Institute of Health, Bethesda, MD). By multiplying each of the anatomical cross-sectional areas by the slice thickness (1 cm), with a further duplication to account for the missing slices during the analysis, Vol (ml) was estimated. Due to limitations of positioning the distal calf under the magnet, a second-order polynomial regression of the available data was used for each participant to extrapolate the remaining muscle volume to the soleus-Achilles junction (24). To calculate the error implicit to estimating the distal portion of the soleus in a subsample of the groups (*n* = 10), direct measurement of the entire GM volume was compared with the GM volume obtained when the distal third of the muscle was estimated from the regression procedure described above. By using this method, the typical error of the predicted results compared with the direct measurement was 2.8% of the measured Vol. All scans were analyzed by the same experienced investigator with an intraobserver variability of <3%.

#### Allometric modeling procedures.

All statistical analyses were conducted using SPSS version 11.5 (Chicago, IL). Standard descriptives [means (SD)] were used to characterize the groups of boys and men. Working in the arithmetic space defined by the original raw *X* and *Y* variables (1), we solved all model parameters by an iterative nonlinear protocol using the Levenberg-Marquardt algorithm, according to the following model: where X is the body size variable of interest, and “group” is a dummy variable coded 0 for boys and 1 for men. The above model derives a size exponent common to both groups. This process relies on the assumption that the slope of the relationship between the body size variable and the dependent variable does not differ substantially between groups. This assumption was tested by the addition of a group × *X* interaction term to the above model. A nonsignificant interaction was revealed for all three body size variables, which implies homogeneity of regression slopes, that is, that a common size exponent can be applied appropriately to both boys and men. Age and physical maturation were entered initially into each allometric model as potential covariates, but neither made a substantial contribution to the regression model. All regression model assumptions (7) were satisfied. Power function ratios (*Y*/*X*^{b}) were constructed for each of the three scaling denominators from the equation derived from the above model by taking antilogs of the coefficient a (for boys) and the constant plus the group coefficient (*a* + *c*) for men. The association between these power function ratios and the scaling denominators were then calculated using Pearson's product-moment correlations. If the allometric model has been successful in partitioning out the influence of body size, then the correlation between the derived power function ratio and the size variable should approach zero, that is, there should be little or no residual size correlation (32). Correlation coefficients that do not approach zero, regardless of whether they are statistically significant, would suggest that the power function ratio has not been completely successful in rendering V̇o_{2 max} independent of body size.

## RESULTS AND DISCUSSION

#### Participant characteristics.

The physical and physiological characteristics of the boys and men are shown in Table 1. The men were significantly taller and heavier and had greater FFM and Vol, but there was no difference in %BF compared with the boys. When Vol was expressed per kilogram BM, the values were similar (24.1 boys vs. 26.0 men ml/kg), but relative to FFM, the boys' Vol was slightly lower (28.3 vs. 30.9 ml/kg). Collectively, the participants were relatively heterogeneous for BM (31.4–97.9 kg), FFM (27.9–80.1 kg), and Vol (648–2,416 ml).

#### Allometric modeling.

The regression output for the allometric models is summarized in Table 2. The common group exponents for each scaling denominator are shown, together with the exponents for boys and men separately. The separate group size exponents do not differ substantially between boys and men for the FFM and Vol variables, underlining the appropriateness of applying a common group exponent. For BM, the point estimate of the exponent for boys (0.96) appears larger than that for men (0.73). Although the group × BM interaction term in the combined allometric model indicated that these exponents were not significantly different, this test is obviously influenced by sample size, and the difference may be real and important. Hence, the common group exponent for BM appears less robust than for the other two body size variables. In addition to the above issue, for reasons clearly articulated below, we believe that BM is not an appropriate scaling denominator for normalizing V̇o_{2 max} in boys and men.

The correlation coefficients between the power function ratios and the scaling denominators are shown in Table 3. For the men, the scaled variables were all clearly size independent, with the point estimates for the correlation coefficient all negligible and approaching zero. However, for the boys, the correlations for BM and FFM represent potentially substantial residual size correlation, suggesting that these scaled variables may not be independent of body size. The smallest practically significant effect size for the correlation coefficient in this context is 0.1 (10). In the boys, the point estimate for the correlation between the power function ratio and the body size variable exceeded this threshold for both BM and FFM, suggesting meaningful residual size correlations. From the observed correlation coefficients in this sample of boys, the probability that the population correlation coefficient exceeds the smallest practically significant effect of *r* = 0.1 may be calculated, given the sampling variability (6). For FFM, the probability that the population correlation between the power function ratio and BM exceeds 0.1 is 0.80. Hence, there is an 80% chance that the obtained residual size correlation of 0.33 is practically meaningful. For BM, there is a 54% chance that the observed residual size correlation of 0.13 is practically important. Based on this analysis, both BM and FFM, therefore, appear to be invalid allometric scaling denominators in the boys. In contrast, the residual size correlation for Vol in the boys was negligible and approaching zero, suggesting that the influence of body size had been partitioned out successfully by the V̇o_{2 max}/Vol^{b} power function ratio. Collectively, these results suggest that only volume correctly partitions out the influence of body size for both boys and men.

BM may not adequately reflect the metabolically active skeletal muscle mass, due to heterogeneity of body composition across the sample. Moreover, the use of BM as a gross surrogate of active muscle mass may be especially problematic in humans (compared with scaling studies in quadrupeds) because, in bipeds, running at V̇o_{2 max} does not involve whole-body musculature (19). Theoretically, the confounding factor of variability in body composition within the sample may potentially be addressed, in part, by scaling by estimated FFM. However, scaling by FFM relies on the assumption that the involved, metabolically active muscle mass at V̇o_{2 max} represents a constant proportion of FFM across the sample. We tested this assumption for the present study. In men, Vol is proportional to FFM^{0.98} (95% confidence interval: 0.57–1.40), indicating that leg volume comprises a relatively constant fraction of FFM across the size range. In boys, however, Vol is proportional to FFM^{1.45} (95% confidence interval: 1.13–1.77), revealing that the proportion of FFM comprised of Vol is not constant and increases as a function of increasing body size (Fig. 1). This disproportionate relationship between Vol and estimated whole body FFM reflects the change in the proportion of lower extremity muscle mass relative to total muscle mass. At birth, it represents ∼40% increasing to ∼55% at full biological maturity (22). Furthermore, as boys enter puberty, there is a dramatic increase in muscle mass relative to other lean tissues. The self-assessment of secondary sexual characteristics suggests that the boys in our study were in the early to mid stages of puberty, which would account for the relationship between Vol and FFM (Fig. 1). This reinforces the fact that FFM cannot be used legitimately as the scaling denominator to normalize the V̇o_{2 max} of boys and men in this study.

The BM relative and absolute V̇o_{2 max} scores for the boys and adults are similar to those reported previously using treadmill ergometry (4, 28). We used MRI to measure Vol in vivo; unlike anthropometric techniques, these data are not confounded by subcutaneous fat or bone tissue. It was not possible to measure the volume of a larger proportion of the active musculature in our study due to the size of the scanner. However, other studies using MRI proton relaxation times, muscle glycogen depletion, and EMG recordings provide convincing evidence that the lower leg musculature is involved substantially during walking and running exercise, with a muscle activation higher (particularly for the plantar flexors and dorsiflexors; Ref. 18) or comparable to the thigh muscles including quadriceps, hamstrings, adductors, and gluteals (e.g., Ref. 30). Although we acknowledge that the lower leg volume represents a relatively small proportion of the total muscle volume involved during running exercise that elicits V̇o_{2 max}, the plantar flexors and dorsiflexors play a key role in locomotion, in propelling the body forward during the push phase, and for clearing the foot off the surface during the swing phase. In fact, during the step cycle, the EMG activities of the tibialis anterior and the gastrocnemii and soleus are two- to threefold longer than those of the vastus lateralis and biceps femoris (18). Besides, in the present study, we are concerned primarily with the nature of the allometric relationship between muscle volume and V̇o_{2 max}. Making the reasonable assumption of a proportional relationship between leg volume and the total volume of involved musculature in boys and men indicates that the value of the scaling exponent *b* in the allometric relationship would be unaffected by the absolute muscle volume included in the model. Hence, for the purpose of deriving a valid allometric relationship to correctly partition out the influence of body size on V̇o_{2 max}, we believe that the Vol is adequately representative of the involved musculature during running in bipeds.

Armstrong and colleagues (3, 5) used multilevel regression modelling in a large group of boys and girls to determine what factors might explain longitudinal changes in absolute peak V̇o_{2} during growth and maturation. A sample-specific BM exponent of 0.88 (SE 0.04), which is similar to the common exponent in our study (Table 2), was given when an estimate of body composition was included in the model along with biological maturity, sex, and chronological age. The authors concluded that the change in FFM was the most important factor when considering the progressive divergence in peak V̇o_{2} between the boys and girls as they aged from 11 to 17 yr. More specifically, they speculated that greater relative changes in muscle mass would facilitate the use of oxygen and enhance venous return via the peripheral muscle pump, thus augmenting stroke volume in the boys (3). When a measurement of active muscle mass is available, our data do not support the notion that FFM per se is the most important factor to consider when scaling V̇o_{2 max}. However, in the absence of a muscle size measurement, this supposition is understandable.

To our knowledge, this is the first study to appropriately examine the scaling of V̇o_{2 max} in a heterogeneous sample of boys and men using measurements of muscle volume in vivo. However, some other studies have examined the relationship between V̇o_{2 max} and body size in young people, albeit using different body size measures or when comparing males and females (5, 11, 12, 15, 36, 37). Using MRI to measure thigh muscle volume in 16 prepubertal boys, Welsman et al. (36) reported that this body size parameter was related to absolute peak V̇o_{2} (*r* = 0.80, *P* < 0.01). Furthermore, an analysis of covariance revealed an exponent of *b* = 0.55 (SE 0.08) when using thigh muscle volume as the scaling variable. Although this point estimate is within the 95% confidence interval that we found when scaling V̇o_{2 max} using Vol (Table 2, combined model), interstudy differences in the samples and muscle volume measurements preclude a direct comparison. In two separate studies involving late adolescent girls (15–17 yr old) and pre- to early pubertal girls (8–10 yr old), respectively, Eliakim et al. (15, 16) explored the relationships between exercise training, thigh Vol, maturation, and peak V̇o_{2}. Results from both longitudinal and cross-sectional analyses were reported (15, 16). When the data for all 84 girls were pooled together, the BM exponent for peak V̇o_{2} was only 0.28 (SE 0.07). This value is extremely low compared with the 0.79 (SE 0.30) in the present study but also relative to all other studies with this population. An exponent of only 0.28 suggests that, as BM increases within the sample of girls, the concomitant change in peak V̇o_{2} is proportionally much smaller. Although the difference between changes in BM and peak V̇o_{2} fits with most other allometric analyses of this relationship (i.e., b < 1), the rate of change is far lower than expected. It would appear that the difference in thigh Vol from 9 to 16 yr of age was smaller than expected (∼8%), and it was suggested that this is the most likely cause for the unusual finding (15). Group differences between the two maturity groups were not found when aerobic capacity was expressed per thigh Vol using the ratio standard (ml·ml^{−1}·min^{−1}; Ref. 15). However, our results suggest that this comparison may not control appropriately for differences in Vol. The scaling exponent for thigh muscle volume in Eliakim et al.'s study (15) was very similar to the Vol value for the boys in our study (0.69 vs. 0.64, respectively), albeit from different muscles. This finding lends additional support to our contention that the leg volume is adequately representative of the involved musculature in exercise at V̇o_{2 max}. Unfortunately, a comparison of the power function ratios for peak V̇o_{2} using the exponent was not made between the 9- and 16-yr-old girls, thus precluding an indirect comparison with our results.

We do not purport that our data are adequate for determining a precise or optimum value of the size exponent that may be generalized to the population, an endeavour that has been termed the “search for the Holy Grail” (27). The observed point estimates for the exponents represent the “best guess” of what the population exponent may be. The 95% confidence intervals surrounding these estimates are interpreted commonly as the likely range within which the population exponent will lie. Clearly, with a larger sample, the precision of estimation of the population exponent would improve. In the present study, the precision (confidence interval half-width) for the common muscle volume exponent was 0.24. If a much greater precision were desired (e.g., 0.1), the sample size requirements would be inflated approximately fivefold. However, given that our primary aim was to shed light on the most appropriate body size denominator for the scaling of V̇o_{2 max}, rather than to define the precise population value of the exponent, the cost-to-benefit ratio of a much larger study would seem high. We believe, therefore, that our sample size is adequate for addressing the specific research question posed.

In conclusion, in a heterogeneous sample of boys and men, we have demonstrated clearly that an estimate of a proportion of the involved musculature is a more valid allometric scaling denominator than either BM or FFM for properly partitioning out the influence of body size on V̇o_{2 max}. Further research is required, quantifying a larger proportion of active muscle that is utilized during exercise, to confirm our findings.

## Acknowledgments

Current address of A. Barker: Children's Health and Exercise Research Centre, School of Health and Sport Sciences, University of Exeter, Heavitree Road, Exeter, EX1 2LU, UK.

Current address of J. Thom: School of Sport, Health and Exercise Sciences, University of Wales Bangor, George Building, Holyhead Road, Bangor, North Wales, LL57 2PX.

## Footnotes

The costs of publication of this article were defrayed in part by the payment of page charges. The article must therefore be hereby marked “

*advertisement*” in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

- Copyright © 2006 the American Physiological Society