## Abstract

Two observations favor the presence of a lower mass-specific resting energy expenditure (REE/weight) in taller adult humans: an earlier report of height (H)-related differences in relative body composition; and a combined model based on Quetelet and Kleiber's classic equations suggesting that REE/weight∝H^{−0.5}. This study tested the hypothesis stating that mass-specific REE scales negatively to height with a secondary aim exploration of related associations between height, weight (W), surface area (SA), and REE. Two independent data sets (*n* = 344 and 884) were evaluated, both with REE measured by indirect calorimetry and the smaller of the two including fat estimates by dual-energy X-ray absorptiometry. Results support Quetelet's equation (W∝H^{2}), but Kleiber's equation approached the interspecific mammal form (REE∝W^{0.75}) only after adding adiposity measures to weight and age as REE predictors. REE/weight scaled as H^{∼(−0.5)} in support of the hypothesis with *P* values ranging from 0.17 to <0.001. REE and SA both scaled as H^{∼1.5}, and REE/SA was nonsignificantly correlated with height in all groups. These observations suggest that adiposity needs to be considered when evaluating the intraspecific scaling of REE to weight; that relative to their weight, taller subjects require a lower energy intake for replacing resting heat losses than shorter subjects; that fasting endurance, approximated as fat mass/REE, increases as H^{0.5}; and that thermal balance is maintained independent of stature by evident stable associations between resting heat production and capacity of external heat release. These observations have implications for the modeling of adult human energy requirements and associate with anthropological concepts founded on body size.

- body mass index
- stature
- body composition
- obesity

body size has a profound influence on mammalian metabolism (42), energetics (15, 37), and evolutionary biology (3). Height is one measure of body size in humans, and an array of life processes are linked with adult stature (21, 28, 31, 38).

Quetelet, in 1835, first made the empirical observation that adult weight is proportional to height (H) squared (33). Keys and his colleagues in 1972 (22) identified Quetelet's index, weight (W)/H^{2}, as the best measure of adiposity (i.e., %fat) among several evaluated weight-height indexes. Body mass index (BMI), as Keys et al. renamed Quetelet's index, is now used throughout most of the world as a measure of adiposity and health. Height as a phenotypic measure is thus quantitatively linked directly with variation in weight, body composition, and associated health-related outcomes in adults.

In a recent study (18), we extended the height-related observations of Quetelet to body compartments, including adipose tissue (AT), AT-free mass (ATFM), fat-free mass (FFM), skeletal muscle, bone, brain, liver, and residual mass as measured by whole body magnetic resonance imaging and dual-energy X-ray absorptiometry (DXA). Our findings indicate that AT, ATFM, FFM, skeletal muscle, and liver mass all scale to height with powers of ∼2, very similar to that of weight. This finding implies that as stature increases across subjects in the population, short and tall subjects have approximately the same percentage of weight as each of these five respective components. In contrast, we found that bone mass scaled to height significantly higher than weight with powers of ∼2.1, while brain mass either failed to correlate with height or scaled weakly to height with a power far less than 2. While our sample was relatively small and not randomly selected, the suggestion is that taller subjects have a larger percentage of their weight as structural bone and less as brain.

Since bone and brain represent extremes in tissue mass-specific metabolic rates (i.e., 2.3 vs. 240 kcal·kg^{−1}·day^{−1}) (10, 11), it would appear that short and tall subjects also differ in whole body mass-specific metabolic rate, defined as resting energy expenditure (REE)/W. Further support for this hypothesis is based on the combination of Quetelet's observation that W∝H^{2} (33) and Kleiber's law stating that REE in mammals is proportional to W^{0.75} (23): REE∝W^{0.75} → (H^{2})^{0.75} → H^{1.5}. This combination of two empirical biological models, one specific to humans and the other to mammals in general, predicts that mass-specific resting energy needs increase as H^{−0.5} and thus that taller human subjects have a lower REE for their weight than their shorter counterparts.

An inverse association between mass-specific REE and height has important implications for the modeling of human energy metabolism and has relevance to fields such as physical anthropology and ecology, areas in which multiple “rules” and hypotheses are linked with body size, height, SA, and energy expenditure (5, 19, 24, 32–37). The aim of the present study was to test the hypothesis that mass-specific REE scales negatively to height in adult humans, implying that tall subjects have a lower metabolic rate at rest relative to their weight compared with their shorter counterparts. During the course of testing the study hypothesis, we evaluated classical models and associations that have received only modest recent attention, particularly in humans. A secondary study aim was to critically explore these models and associations with an emphasis on their relations to stature.

## METHODS

### Experimental Design and Rationale

#### Allometric models.

Two existing databases were used to sequentially examine three main questions related to the associations between REE, weight, and height in adults: How does weight scale to height? How does REE scale to weight and height? How does mass-specific REE (i.e., REE/W) scale to height? These three questions encompass the associations reported by Quetelet (33) and Kleiber (23) along with our conjecture that mass-specific REE scales negatively to height. Surface area, a function of both height and weight, is closely aligned with classical concepts in energy metabolism such as Rubner's law stating that REE∝W^{0.66} (25). We therefore evaluated a fourth related question: How does surface area scale to height?

The scaling of weight to height and REE to weight and height as summarized by the first two questions were evaluated using the classic allometric model (1) with *Y* as the dependent variable (e.g., weight), *X* as the predictor variable (e.g., height), β as the scaling exponent or power, α as the proportionality constant, and ε as a multiplicative error term. When expressed in logarithmic form, the allometric equation can be solved as (2) According to Quetelet, β is equal to ∼2 in *Eqs. 1* and *2* for weight (*Y*) scaled to height (*X*) (33). Similarly, Kleiber's law states that REE (*X*) scales to weight (*Y*) with a β value of ∼0.75 in mammals (23).

Mass-specific energy expenditure, as examined in the third question, is the heat produced per unit weight. The general allometric model as stated in *Eq. 1* can be written separately for energy expenditure and weight, both scaled to height: (3) (4) Therefore (5) When β1 and β2 are equal (i.e., when weight and energy expenditure scale the same to height), the value of β is 0, and a nonzero number raised to the power of zero equals 1. Thus there will be no association between mass-specific energy expenditure and height if the difference between β1 and β2 is at or near zero. If the difference between β1 and β2 in *Eq. 5* is not zero, mass-specific energy expenditure will scale positively or negatively to height, possibly significantly. If REE and weight scale differently to height, short and tall subjects will not have the same mass-specific energy expenditure.

The fourth question, the scaling of surface area to height, was examined in a manner similar to that of the first two questions by application of *Eq. 2*. We also examined the scaling of surface area/weight and REE/surface area to height by applying a model as outlined by *Eq. 5*.

### Subjects and Measurements

The hypothesis was examined using data from healthy adults of varying ethnicity. The first database was from subjects participating in multiple studies who were evaluated at the New York Obesity Research Center (NYORC) and who had indirect calorimetry studies for measurement of REE and DXA for evaluation of body fat (16). The second larger database included subjects evaluated at multiple centers and who also had REE measured by indirect calorimetry. The second database was reported as part of the National Academy of Sciences (NAS)-Institute of Medicine (IOM) Macronutrient Report (9), and a portion of this larger database was provided to us by Dr. Allison E. Black, formerly at Cambridge University, UK. Subjects evaluated in both databases were healthy adults without BMI restrictions who were at or over the age of 18 yr.

There are many equations available for estimating surface area (SA) from weight and height, and direct measurement is also now possible with three-dimensional photonic scanning (3DPS) (39). Surface areas of children and adults calculated from different prediction formulas are highly correlated with results from 3DPS with small mean differences (42, personal communication). Accordingly, we selected the simple Mosteller equation (27) for use in the present study, (6)

## STATISTICAL METHODS

Baseline subject demographic characteristics are reported as means ± SD in Table 1 and as mean powers ± SE in the text and Figs. 1–6. The statistical analyses were carried out using SPSS (SPSS for Windows, 11.5, SPSS, Chicago, IL).

Allometric model coefficients, α and β, were derived using least-squares multiple linear regression analysis and log-transformed data (20). In some cases dependent variables were also correlated with age, and we therefore included age as a predictor variable when appropriate. In addition, we developed a second set of allometric equations for young adults (≤35 yr) without age as a predictor variable using the larger NAS-IOM male and female groups. We empirically selected age 35 yr as the cutoff for this younger group as none of the developed models in this cohort included age as a significant predictor variable. Values for log α and β along with R and SEE for each of the developed regression models are presented in results.

In addition, we developed four models that examine the scaling of REE to weight as part of a more in-depth analysis of Kleiber's law in humans. First, we prepared the human counterpart of Kleiber's classic model (23) (*model 1*) as defined by *Eq. 3* including REE as dependent variable and weight as the main predictor variable (subjects age ≤35 yr in the NAS-IOM group) or with age as a potential additional covariate (i.e., in all subjects).

There is, however, a limitation to *model 1* when applied in adult human populations such as those in the present study. A larger adult weight (i.e., the model independent variable) may be secondary to greater stature and/or to a higher level of adiposity. With greater adult height, the proportion of weight as adipose tissue remains relatively stable. That is, both adipose tissue and weight scale similarly to H^{2} (18). In contrast, when greater weight is considered across subjects of similar height, adipose tissue comprises a large fraction (∼75%) of the additional body mass (13). Adipose tissue has a low mass-specific REE (i.e., ∼4.5 kcal·kg^{−1}·day^{−1}), lower than the components of adipose-tissue free mass (e.g., skeletal muscle, ∼13 kcal/kg; brain, ∼240 kcal·kg^{−1}·day^{−1}) (11), and adiposity thus may have a strong influence on the observed scaling of REE to weight. Kleiber's law with scaling of REE to W^{0.75} (23) reflects relations observed in nonobese mammals, or at least the “heavier” animals as observed across species are not necessarily the fatter members of their species. In contrast, adult humans, as might be observed in our study, are usually heavier because of greater adiposity in addition to potentially being taller. This phenomenon may thus “bias” the scaling of REE to weight as subjects with greater weight and adiposity have a lower mass-specific REE.

The scaling of REE to weight in the general adult population will thus reflect the combined relations between REE and stature-associated weight and adiposity-associated weight. We therefore developed three additional regression models (*models 2–4*) that examine the scaling of REE to weight. The first (*model 2*) includes BMI as an additional independent variable in an attempt to control for between-subject differences in weight accounted for by adiposity. In *models 2* and *3*, we replaced BMI with %fat and fat/H^{2} (41), respectively. Variance inflation factors were calculated for all models to check for the presence of multicollinearity among independent variables (1).

## RESULTS

### Baseline Group Characteristics

The baseline demographic information for the study groups is summarized in Table 1. The two groups, NYORC and NAS-IOM, had 344 subjects (147 men, 197 women) and 884 subjects (429 men, 455 women), subjects, respectively. The groups on average ranged in age from 39.5 to 51.6 yr and in BMI from 24.6 to 26.1 kg/m^{2}. The NAS-IOM groups were on average older (NAS-IOM vs. NYORC, men and women, both *P* < 0.001) and had a lower BMI (NAS-IOM vs. NYORC, men and women, both *P* < 0.001) than the corresponding NYORC groups. There were 164 men and 189 women age ≤35 yr in the NAS-IOM group.

### Allometric Analyses

The multivariate allometric regression models are summarized in Tables 2 and 3. The models formulated with height as a predictor variable are summarized in Table 2, and those with weight as a predictor variable are presented in Table 3. The univariate plots of allometric relations examining the three main study questions for subjects in the NAS-IOM group age ≤35 yr are presented in Fig. 1.

#### Weight vs. height.

Weight scaled to height in the NYORC and NAS-IOM men and women very close to a power of 2 (Table 2, Figs. 1 and 2), powers of height in the groups without age restriction ranging from 1.96 ± 0.32 in NYORC women to 2.24 ± 0.21 in NAS-IOM men. None of these observed powers differed significantly from a power of 2.0, and age was a significant covariate in all four models. There was no significant correlation between BMI (i.e., W/H^{2}) and height observed in any of the groups. Similar results were observed in men and women age ≤35 yr. These findings are consistent with Quetelet's classic observations that weight scales approximately as H^{2} (33) and that W/H^{2} is independent of height.

#### REE vs. weight.

REE scaled to weight (*model 1*) in the groups without age restriction with powers of 0.69 ± 0.06 and 0.53 ± 0.03 in the NYORC and NAS-IOM men with corresponding powers of 0.44 ± 0.04 and 0.46 ± 0.02 in the women (Table 3 and Fig. 3). Similar results were observed in subjects age ≤35 yr (Fig. 1).

When added to the REE prediction model, the BMI covariate (*model 2*) was nonsignificant in the NYORC men. BMI added significantly to the model in the three remaining groups with REE scaling to weight with powers of 0.73 ± 0.06 (NAS-IOM men), 0.66 ± 0.09 (NYORC women), and 0.64 ± 0.05 (NAS-IOM women) (Fig. 3).

When added to the REE prediction model in place of BMI, %fat (*model 3*) added significantly to the model in the two NYORC groups with REE scaling to weight with powers of 0.80 ± 0.07 (men) and 0.60 ± 0.06 (women) (Fig. 3). Last, when added to the REE prediction model in addition to weight and age, fat/H^{2} (*model 4*) added significantly to the model in the NYORC males and females with respective powers of 0.84 ± 0.08 and 0.69 ± 0.07 (Fig. 3). The variance inflation factors for *models 2–4* ranged from ∼1 to 7 (Table 4).

REE scaled to weight across all four models had lower powers in women than men as shown in Figs. 1 and 3.

#### REE vs. height.

The models describing REE scaled to height, after adjustment for age, are presented in Table 2, and the univariate models for subjects age ≤35 yr are presented in Fig. 1. REE scaled to height in the non-age-restricted groups with respective powers of 1.61 ± 0.33 and 1.58 ± 0.16 in NYORC and NAS-IOM men and 1.40 ± 0.21 and 1.32 ± 0.15 in NYORC and NAS-IOM women (Fig. 4). Results were similar in the subjects with age ≤35 yr (Fig. 1). None of these observed powers differed significantly from a power of 1.5.

Since REE scaled to height with powers approximating 1.5, we explored the potential of REE/H^{1.5} as a means of adjusting REE for between-individual differences in stature. There were no significant correlations between REE/H^{1.5} and height for any of the study groups.

#### Mass-specific energy expenditure vs. height.

Mass-specific REE scaled to height with powers in the non-age-restricted subjects of −0.38 ± 0.28 and −0.66 ± 0.17 in NYORC and NAS-IOM men, respectively (Table 2, Fig. 5); the *P* value for height in the NYORC model for men was nonsignificant at 0.17. The corresponding model for women had powers of −0.55 ± 0.27 and −0.67 ± 0.18, both of which were statistically significant (*P* = 0.04 and <0.01). Mass-specific REE in the subjects age ≤35 yr in the NAS-IOM groups scaled to height with powers of −0.89 ± 0.25 in men and −0.55 ± 0.25 in women, both of which were statistically significant (*P* < 0.01 and =0.03). None of the significant powers differed significantly from a power of −0.5.

#### Surface area evaluations.

Surface area scaled to height with powers of ∼1.5 in both men (NYORC, 1.50 ± 0.15; NAS-IOM, 1.62 ± 0.11) and women (1.48 ± 0.16; 1.49 ± 0.12) (Table 2). The ratio of surface area to weight scaled negatively to height with a power of ∼(−0.5) in men (−0.50 ± 0.15; −0.62 ± 0.11) and women (−0.48 ± 0.16; −0.49 ± 0.12). After controlling for age, REE/surface area was independent of height in all of the groups.

## DISCUSSION

### Hypothesis Validation

The present study findings support the hypothesis stating that mass-specific REE scales negatively to height in adult humans. Of four separate models used to test the hypothesis in non-age-restricted subjects, three were statistically significant, and one was nonsignificant (*P* = 0.17). The two corresponding models in subjects age ≤35 yr were also statistically significant.

Support for our hypothesis, in addition to the earlier body composition data that initiated the present investigation (17), was provided by combining the models reported by Quetelet (33) and Kleiber (23).

#### Quetelet's model.

Our findings strongly support Quetelet's model, W∝H^{2}, with the powers of height in adult subjects ranging from H^{1.96} to H^{2.24}. These observations are also consistent with earlier studies (6, 18), including Benn's analysis (2), showing that adult weight scales to height with powers of ∼2.

#### Kleiber's law.

Kleiber's law has been the subject of intense discussion and debate since first reported in 1932 (23). The exact power of weight in Kleiber's law was not the focus of our investigation but rather a component of hypothesis generation. Earlier investigators attribute variation around the “0.75” value to taxonomic affiliation, animal habitat, and foraging mode, duration of fasting, and aspects of thermal physiology (14, 43–47). White and Seymour (46), in their extensive review, suggest that REE after thermal adjustment scales to weight in mammals with a power (±95% confidence interval) of 0.686 ± 0.014. This debate centers on theories explaining Kleiber's law (1, 3, 14, 43–47), which diminished the importance of Rubner's earlier surface law implying that mammalian REE scales to weight with a power of 0.66 (25, 36, 44–47).

In the present study, when REE was scaled to weight, with adjustment for age, we observed powers in the range of 0.44–0.69, the lower values in the range observed in women. However, at least two weight effects are captured in this correlation, one related to stature and the other to adiposity. Accordingly, we adjusted for variation in adiposity across the weight range by adding BMI and body fat as covariates to our REE-weight model, and the observed powers then neared those for interspecific mammal comparisons (14), 0.60–0.84. Our analysis reveals yet another important factor to consider when studying the scaling of REE to weight, notably the complexities of interpretation arising when intraspecies assessments are carried out across individuals differing widely in adiposity.

REE still scaled to weight with consistently lower powers in females than males after adjusting within the sex groups for age and adiposity. This difference in REE scaling to weight is likely secondary to corresponding sex-related scaling differences for body composition and weight (4, 10, 16–18) along with hormonal effects (7, 8).

### Physiological Implications

#### REE∝H^{1.5}.

This model indicates that resting energy requirements increase with height, a process mediated by parallel increases in weight as H^{2} and REE as W^{<1}. The magnitude of these relations is evident from estimates provided by our models for adults who are at the extremes of height (±2 SD) in the US population (30): short and tall males (158 and 192 cm) differ in REE by 538 kcal/day and females (147 and 177 cm) by 330 kcal/day, short subjects at −2 SD requiring 27% and 21% less energy for replacing respective losses expended at rest than tall subjects at +2 SD. At the extreme end for group short stature, adult African pygmy men and women living in Zaire with mean respective heights of 147 and 139 cm (39) have resting energy requirements 706 and 391 kcal/day less than tall US subjects at +2 SD, differences of 35 and 27%.

Quetelet and Kleiber's equations were combined to generate the REE∝H^{1.5} model. We recognize that the respective powers of height and weight in these two equations are not exactly 2 and 0.75 as assumed, although our analyses (Table 2 and Fig. 4) provide strong support for the scaling of REE to height with an approximate power of 1.5.

Our observations indicated that REE/H^{1.5} was uncorrelated with height in all evaluated male and female groups. The REE-height index might thus serve as a useful measure when comparing energy expenditure at rest across subjects differing in height. The REE-height index is based on a similar concept to BMI (i.e., W/H^{2}) and height-normalized body composition indexes (e.g., FFM/ H^{2}) (41), which are the appropriate metrics used to compare weight and body composition in subjects differing in height.

Our study focused on REE, which is measurable in humans under highly controlled conditions and reflects the subject's endogenous resting metabolic processes. We avoided studying total energy expenditure (29) in the context of the present study hypothesis as our concepts were generated based on resting organ and tissue metabolic requirements (11, 17). Additionally, total energy expenditure is highly variable due to between-subject differences in physical activity levels (9), and we therefore focused on the well-defined resting component of maintenance energy requirements.

#### REE/W∝H^{−0.5}.

This model, based on the same assumptions used to generate the REE∝H^{1.5} model, affirms that mass-specific REE decreases as a function of height. Relative to their weight, the tall subjects at +2 SD in the previous example have a lower REE (males 24.4 and females 21.7 kcal·kg^{−1}·day^{−1}) than the short subjects at −2 SD (males 27.7 and females 24.2 kcal·kg^{−1}·day^{−1}) and the African pygmies (males 29.0 and females 25.5 kcal·kg^{−1}·day^{−1}). Tall subjects thus require a lower energy intake relative to their weight to replace fuels consumed at rest.

A lower relative metabolic rate and longer “fasting endurance” appears to favor animal survival, a topic of study in mammalian evolutionary biology (26). One theory explaining selection for greater animal body size in environments that are seasonal is that larger animals within the same taxonomic group have a greater fasting endurance than their smaller counterparts (5, 24, 26). Fasting endurance, defined as energy stores/REE (5), is greater in larger animals that have a similar proportion of weight as energy stores but a lower mass-specific REE since REE/W in mammals scales as W^{−0.25}. Greater fasting endurance confers a survival advantage during periods of severe food shortage when activity levels are near basal (24, 26). The survivors of severe seasonal food shortages would then be able to maximally restore body energy content and reproduce with less competition when conditions improve.

Fat is the primary human energy store, and both total body adipose tissue and fat scale in proportion to H^{2} (18). Since REE scales as H^{1.5}, fasting endurance in humans is approximately proportional to H^{0.5}. These estimates suggest that taller humans can survive a fast longer than their shorter counterparts and that a low mass-specific REE may have contributed to selection for greater stature in seasonal environments during periods of human evolution.

A related hypothesis has been advanced for African pygmies who subsist in hot and humid tropical rainforest environments (39) and who, by our estimates, have a high mass-specific REE and thus substantially shorter fasting endurance time than tall subjects. According to this hypothesis (39), the low energy requirement and short fasting endurance time of pygmies are adaptations to a chronic low-energy yielding environment in which the limited food resources are steadily available and not subject to large seasonal variation.

### Stature and Thermal Physiology

Humans and other mammals vary in body size and morphology according to climate, and these within-species “ecogeographical” effects are captured in Bergmann and Allen's rules (35). Colder climates are associated with larger size, notably a wider body, shorter extremities, and lower SA/W with reduced heat loss. Tall subjects have a lower SA/W than short subjects, so the question arises if there are height-related differences in the relations between SA with capacity for external heat release and endogenous heat production through REE.

We can examine this question by combining Quetelet and Mosteller's equations [W∝H^{2} and SA∝ (W × H)^{0.5}] to estimate how surface area scales to height in adults: SA∝H^{1.5}. Our results for surface area models presented in Table 2 suggest this conjecture has reasonable validity with powers of height ranging from 1.48 to 1.62. We also did not find a significant correlation between REE/SA and height in a relatively large subject sample consisting of two independent cohorts. Thus, with greater height, both REE and surface area are smaller relative to weight, and a stable relationship is maintained independent of stature between resting heat production and the external surface area available for heat release. This evident balance of heat-producing and heat-releasing capabilities would appear to be an important feature allowing adult humans living within a geographic region (35) to vary in height with prevailing nutritional conditions without the requirement for other major thermoregulatory adaptations.

These observations remain to be integrated in future studies with variation in SA/W influencing factors such as body “width” and extremity morphology as components of Bergmann and Allen's rules (21). Several thermoregulatory and climatic theories attempt to explain the adaptive value of the small body size observed in African pygmies (34), and it would be interesting to test the applicability of our models at the lower end of adult stature and in children.

In conclusion, body size has pronounced effects on human physiology and metabolism. As permitted by the experimental data available to us and by exploiting classical available models for use in the present context, we have linked height with weight, surface area, and REE (Fig. 6). Our central focus was on the relationship between REE and height, and our results strongly support the hypothesis that REE/W scales negatively to height. This finding is consistent with the observation that taller subjects may have less metabolically active tissue relative to weight compared with shorter subjects (18). The stature-related effects on REE are of the same magnitude as those associated with variation in race (12), aging (10), and the development of obesity (44). On the path to examining the main study hypothesis, we explored a controversial area involving the scaling of REE to weight (14), and we demonstrated the sensitivity of this analysis to adiposity in the population under study. Finally, we have established that variation in human stature is accompanied by stable associations between resting heat production and capacity for external heat release. These findings have implications for wide-ranging areas in the study of human biology.

## GRANTS

Funding for the present study was provided in part by National Institute of Diabetes and Digestive and Kidney Diseases Grant PO1-DK-42618.

## Acknowledgments

S. B. Heymsfield, Principal Investigator, participated in design, data collection, analysis, and article preparation. D. Childers, J. Beetsch, D. B. Allison, and A. Pietrobelli participated in design, analysis, and article preparation.

## Footnotes

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- Copyright © 2007 the American Physiological Society