ISRN Zoology

Volume 2012 (2012), Article ID 673050, 9 pages

http://dx.doi.org/10.5402/2012/673050

## Organ-Tissue Level Model of Resting Energy Expenditure Across Mammals: New Insights into Kleiber's Law

^{1}Obesity Research Center, St. Luke’s-Roosevelt Hospital, College of Physicians and Surgeons, Columbia University, New York City, NY 10025, USA^{2}Department of Statistics, Columbia University, New York City, NY 10027, USA^{3}Pennington Biomedical Research Center, Baton Rouge, LA 70808, USA

Received 25 April 2012; Accepted 5 August 2012

Academic Editors: A. Arslan, K. E. Ruckstuhl, and E. Tkadlec

Copyright © 2012 ZiMian Wang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

*Background.* Kleiber’s law describes the quantitative association between whole-body resting energy expenditure (REE, in kcal/d) and body mass (*M*, in kg) across mature mammals as REE . The basis of this empirical function is uncertain.
*Objectives.* The study objective was to establish an organ-tissue level REE model across mammals and to explore the body composition and physiologic basis of Kleiber’s law.
*Design.* We evaluated the hypothesis that REE in mature mammals can be predicted by a combination of two variables: the mass of individual organs/tissues and their corresponding specific resting metabolic rates. Data on the mass of organs with high metabolic rate (i.e., liver, brain, heart, and kidneys) for 111 species ranging in body mass from 0.0075 (shrew) to 6650 kg (elephant) were obtained from a literature review.
*Results. * predicted by the organ-tissue level model was correlated with body mass (correlation ) and resulted in the function , with a coefficient and scaling exponent, respectively, close to 70.0 and 0.75 () as observed by Kleiber. There were no differences between and calculated by Kleiber’s law; was correlated () with . The mass-specific , that is, , was correlated with body mass () with a scaling exponent −0.246, close to −0.25 as observed with Kleiber’s law. *Conclusion.* Our findings provide new insights into the organ/tissue energetic components of Kleiber’s law. The observed large rise in REE and lowering of REE/*M* from shrew to elephant can be explained by corresponding changes in organ/tissue mass and associated specific metabolic rate.

#### 1. Introduction

Resting energy expenditure (REE), defined as the whole-body energy expenditure under standard conditions, is the largest fraction of total energy expenditure. Body mass was applied early in exploring the quantitative association between REE and body composition. The best empirical fit between REE (in kcal/d) and body mass (*M*, in kg) from mouse to elephant with a ~330,000-fold difference in body size was derived by Kleiber [1, 2] and Brody [3],
Equation (1) is the well-known Kleiber’s law or power law, one of the most widely discussed rules in bioenergetics [2, 4]. Based on (1), Kleiber’s law can also be expressed in terms of mass-specific REE,
According to Kleiber’s law, small mammals (e.g., shrew) have lower REE but higher REE/*M* than do large mammals (e.g., elephant).

Although many investigators have attempted to clarify plausible mechanisms, a full understanding of Kleiber’s law is still uncertain and represents a knowledge gap in the studies of bioenergetics [12]. Primary questions remain what is the biological mechanism of the quantitative association between REE and body mass across mammals? and why is the REE/*M* ratio substantially smaller in animals with larger REE and body size?

The aim of the present study was to explore the potential physiologic and body composition basis of Kleiber’s law. Our group derived and validated an organ-tissue level REE model for humans in 1998 [13]. According to the model, the magnitude of human REE is determined by two variables, mass of all organs/tissues and their respective metabolic rates at rest [14]. Our hypothesis is that the established organ-tissue level REE model for adult humans is applicable in mature mammals. Specifically, we evaluate the REE-body mass associations at the organ-tissue level across mammalian species.

#### 2. Methods and Data Sources

##### 2.1. Organ-Tissue Level REE Model for Mature Mammals

The principle of the organ-tissue level REE model is that whole-body energy expenditure at rest reflects the total resting energy consumption of all organs and tissues. A mechanistic equation of the organ-tissue level REE model for mammals can be expressed as
Accordingly, a mass-specific REE (i.e., REE/*M*) model can be derived as
where *T* is the mass of individual organs/tissues of mature mammals; *T/M* is the fraction of body mass as individual organs/tissues; *K* is the specific metabolic rate of organs/tissues; is the organ/tissue number ( = 1, 2,…, ). Equations (3) and (4) demonstrate that the magnitude of REE (or REE/*M*) depends on both and (or /).

*The Values of Various Organs/Tissues across Mammalian Species*

Previous studies reported that the resting metabolic rates of homologous organs have smaller values in large animals compared to small animals [15]. The *in vitro * values for liver and kidney vary allometrically with body mass across mature mammals [16].

Four organs (i.e., liver, brain, heart, and kidneys) are particularly active in mammalian energy metabolism during resting conditions [7, 8]. The *in vivo * values of the four organs have been published for several mature mammals, including rat [5], rabbit, cat, dog [6], and human [7, 8]. For example, four human organs have high values (all in kcal/kg per day): 200 for liver, 240 for brain, and 440 for heart and kidneys. In contrast, the average values of the residuals are as low as 10.7 for humans [7].

Based on the information provided in Table 1, an exponential - function was derived for the four organs and residual mass [17],
where and are the organ/tissue-specific coefficient and scaling exponent, respectively. Although the exponent differs in various organs/tissues, all values (all *r* > 0.83) are negative, indicating that the values are smaller with greater body mass.

In the present study, the values of the four organs and residual mass were predicted by (5) for different mammals (Table 2). Small mammals have higher values than do large mammals. For example, liver’s value is 2533 kcal/kg per day for the shrew compared with 65 kcal/kg per day for the elephant. An important consideration is that the values are much higher in immature mammals than that in adult mammals, including humans [18].

*The T _{i} Values of Various Organs/Tissues across Mammalian Species.*

Because liver, brain, heart and kidneys are particularly active in resting mammalian energy metabolism, the following organ-tissue level model of body composition is applied, where the residual is the sum of body components with lower metabolic rate at rest, including skeletal muscle, adipose tissue, skeleton, blood, skin, lung, connective tissue, gastrointestinal tract, and spleen. Residual mass is calculated as body mass minus the sum of liver, brain, heart, and kidneys mass.

A literature search was performed to collect data on body mass and mass of the four organs (Table 2). The database contains 111 species distributed in 11 mammalian orders: artiodactyla, carnivora, didelphimorphia, diprotodontia, eulipotyphla, lagomorpha, perissodactyla, primates, proboscidea, rodentia, and scandentia. Most of the data () was obtained from a recent study of Navarrete et al. [11]. The mouse and dog (with body mass 20.42 kg) data were obtained from Martin and Fuhrman [9]; the rat, guinea pig, dog (with body mass 10 kg), sheep, hog, dairy cow, horse, steer, and elephant data were obtained from Elia [7]; the reference man and woman data were obtained from Snyder et al. [10]. All collected published data were used as is and no judgment on data quality was made.

*The Working REE Model across Mammalian Species*

Based on models (3) and (5), a working REE prediction model can be derived at the organ-tissue level for mature mammals,
Accordingly, a working mass-specific REE (i.e., REE/*M*) model can be derived as
where is body mass; is the mass of individual organs/tissues; is the fraction of body mass as individual organs/tissues; is the organ/tissue number ( = 1, 2,…, ); and are the organ/tissue-specific coefficient and scaling exponent, respectively.

##### 2.2. Statistical Analysis

Paired Student’s -tests were used to compare by (7) with calculated by Kleiber’s law (i.e., (1)) and to compare the by (8) with from (2). Simple linear regression analysis was used to find the association between and as well as between and . Scatter plots were applied to explore the association between (and ) and (and ). Data were analyzed by the Microsoft Excel version 5.0 (Microsoft Corporation, Redmond, WA). Statistical significance was set as 0.05.

#### 3. Results

##### 3.1. Organ-Tissue Level Body Composition of Mature Mammals

The data on body composition for 111 mammalian species are listed in Table 2. Body mass ranged from 0.0075 kg for the shrew (*Sorex araneus*) to 6650 kg for the elephant, with a ~887,000-fold difference in body size. The variability in the mass of high metabolic rate organs is very large across mammalian species. The sum of the four organs (liver, brain, heart, and kidneys) varied from 0.00075 kg for the shrew to 15.4 kg for the elephant, with a ~20,500-fold difference. However, the fraction of body mass as the four organs declined from 10.0% in the shrew to 0.23% in the elephant.

##### 3.2. Model-Predicted REE across Mammalian Species

Based on (7), the model-predicted energy expenditure (*K* × *T*) of individual organs/tissues was calculated as ranging from 0.0075 kg (shrew) to 6650 kg (elephant). The respective regression lines of (*K* × *T*) for liver, brain, heart, kidneys, and residual were derived and presented in Figure 1 and Table 3,

Based on (7), the model-predicted REE () was calculated (Table 2). The varies from 1.8 kcal/d in the shrew to 48000 kcal/d in the elephant. The is allometrically correlated with body mass across the 111 mammalian species (Figure 1), The scaling exponent between and body mass was 0.754 with a 95% CI (0.744, 0.764), close to 0.75 () for Kleiber’s law at the whole-body level.

Model-predicted was correlated with by Kleiber’s law (Figure 2), There was no significant difference between and across the 111 species (paired Student’s -test, ).

##### 3.3. Model-Predicted REE/M across Mammalian Species

The model-predicted was calculated by the working model (8), declining from 240 kcal/kg per day in shrew to 7.2 kcal/kg per day in elephant, that is, approximately 33 times greater in shrew than in elephant. The was allometrically correlated with body mass across the 111 species (Figure 3), The scaling exponent between and body mass was −0.246 with 95% CI (−0.256, −0.236), close to −0.25 as for Kleiber’s law at the whole-body level.

The model-predicted by (8) was correlatedwith calculated by (2) based on Kleiber’s law (Figure 4),
There were differences between and in the 111 mammals (paired Student’s -test, ). A Bland-Altman plot shows that there was no bias in REE/*M* differences, that is, [−], in relation to the average of the two REE/*M* values,

#### 4. Discussion

In 1932 Kleiber first described what has become known as the “mouse-to-elephant” curve by plotting REE versus body mass across mammals and reporting that REE scales as the three fourths (0.75) power of body mass. A number of mechanisms and models have been proposed over the ensuing eighty years to explain Kleiber’s law, including McMahon’s model [19], four-dimensional model [20], Economos model [21], mass transfer model [22], supply-side fractal model [23], resource-flow model [24], and Darveau’s model [25]. Up to now, however, there remains a lack of full agreement on the mechanistic basis of the 0.75 power of the REE-*M* relationship. The present study provides three new observations related to Kleiber’s law.

##### 4.1. New Insights into Kleiber’s Law

Our previous study indicated that ~40 body components are distributed between distinct but connected levels: whole body, organ tissue, cellular, molecular, and atomic [26]. In principle, REE can be expressed at whole body, organ tissue, cellular, and molecular levels, respectively [17, 27]. Kleiber’s law was established at the whole-body level to explore the quantitative association between REE and body mass.

The first contribution of the present study and our previous investigation [17, 27] was to develop an organ-tissue level REE model across mature mammals. This physiological model is based on the concept that whole-body REE is equal to the sum of energy consumption by all organs and tissues in the postabsorptive state. By using the reported data of 111 mammals, we derived the allometric equations of REE for the five organs and tissues (9). We found that the sum of the energy consumption of the five organs and tissues is equal to as defined by Kleiber’s law (Figure 1) with a scaling exponent of 0.754, similar to the value of 0.75 as observed by Kleiber.

##### 4.2. The Body Composition Basis of Kleiber’s Law

The second contribution of the present study was to provide the body composition basis of Kleiber’s law, revealing in the process the large difference in organ/tissue mass across mammals. The comprehensive body composition database across mammals presented in Table 2 includes body mass and the mass of four high-metabolic rate active organs (i.e., ) in 111 mammalian species. The absolute values of homologous organs increase with body mass. For instance, liver is only 0.00038 kg in the shrew compared with 6.30 kg in the elephant. The is thus the major contributor of the large differences in REE between species (e.g., shrew, 1.8 kcal/d and elephant 48,000 kcal/d; Table 2). In contrast, the values of homologous organs decrease with body mass. For example, liver’s value in the shrew of 0.0506 is much larger than that in the elephant 0.0009. The value is thus the major contributor to the large difference in REE/*M* between species, for example, shrew 245 kcal/kg per day versus elephant 7.2 kcal/kg per day (Table 2).

##### 4.3. Physiological Basis of Kleiber’s Law

The third contribution of the present study was to elucidate the physiological mechanism of Kleiber’s law: the values of homologous organs are not the same with larger values in smaller mammals.

There are several available methods to estimate values of individual organs and tissues. *In vitro* estimation of values from organ slices has been applied to determine values in mammals since the 1920s [16]. However, *in vitro* studies underestimate values [8]. More recently *in vivo* estimation approaches have been applied to estimate the values of organs [28, 29]. The oxygen consumption of an organ can be estimated by the arteriovenous differences in O_{2} concentration across the organ, combined with the assessment of blood flow perfusion of the organ. Oxygen consumption of the brain has been assessed since 1990s by positron emission tomography (PET) with 2-deoxy 2[^{18}F]fluoro-D-glucose [30]. Other isotopes have also been applied such as ^{11}C-acetate. Recently, oxygen consumption can be assessed by PET combined with computerized tomography (PET-CT) or magnetic resonance imaging (PET-MRI). These rapidly advancing methods provide new opportunities for estimating the values of organs and tissues in mammals.

As *in vivo* determination of values is still a technically demanding process, only a few studies have been reported for assessing the value of organs and tissues in mammals (Table 1). In this paper most values were predicted, rather than estimated. This is the major limitation of the present study. Further *in vivo* studies are needed to estimate the values of individual organs and tissues in >100 mammalian species. This area is a major challenge for completely validating the organ-tissue level REE model and Kleiber’s law.

Couture and Hulbert [16] estimated the values of liver and kidney in five mammals (mouse, rat, rabbit, sheep, and cattle). Although these authors used *in vitro* methods, the estimated values of homologous organs vary with body size, with allometric exponents −0.21 for liver and −0.12 for kidney, close to −0.2677 for liver and −0.0833 for kidney, observed in our *in vivo* study (Table 1).

Based on the observed *K*-*M* association [17, 27], we predicted the *K* values across mature mammals (Table 2). For example, liver’s value of 2533 kcal/kg per day in shrew is much larger than 65 kcal/kg per day in elephant. Therefore, the large difference in values across mammals is the physiological basis for Kleiber’s law.

In conclusion, although Kleiber’s law is a simple mathematical function, this classic empirical expression does not reflect the underlying body composition and physiologic mechanism. We derived and evaluated a novel organ-tissue level REE model that provides a fundamental linkage between REE (REE/*M*) and body size. The high REE/*M* in small mammals can be explained by two variables operating jointly: the high fraction of body mass as liver, brain, heart, and kidneys and their high specific resting metabolic rates.

#### Acknowledgments

Z. Wang was responsible for all aspects of the paper including study design, model development, data collection, and paper writing. Z. Ying and J. Zhang were responsible for statistical analysis and paper writing. S. B. Heymsfield provided consultation and assistance with paper writing. This project was supported by award DK081633 from the USA National Institute of Diabetes and Digestive and Kidney Diseases (NIDDK). The authors had no conflict of interests in any company or organization sponsoring this study. The content is the responsibility of the authors and does not necessarily represent the official views of the NIDDK or the National Institutes of Health (NIH).

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