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Obesity Research Center, St. Luke’s-Roosevelt Hospital, Columbia University, College of Physicians and Surgeons, New York, NY 10025 and
Department of Biology, City College of New York, CUNY, New York, NY 10025
*
2To whom correspondence should be addressed. E-mail: zw28{at}columbia.edu
ABSTRACT
The relationship between resting energy expenditure (REE) (kJ/d) and body mass (M) (kg) is a cornerstone in the study of energy physiology. By expressing REE as a function of body mass observed across mammals, Kleiber formulated the now classic equation: REE = 293M0.75. The biological processes underlying Kleiber’s law have been a topic of long-standing interest and speculation. In the present report we develop a new perspective of Kleiber’s law by developing an organ-tissue level REE model consisting of five components: liver, brain, kidneys, heart and remaining tissues. The resting thermal output of each component is the product of the component’s specific resting metabolic rate (K) and mass (T). With increasing body size, the K values for all five components had negative exponents and were directly proportional to M-0.08--0.27, and all component T values were directly proportional to M0.76-1.01. The resulting exponents of the product (K x T) were M0.60-0.86 for the five components. Although the (K x T) values of individual components do not scale equally, their combined formula (286M0.76) is similar to that observed by Kleiber on the whole-body level. Modeling mammalian REE at the organ-tissue level provides new insights and pathways for future mechanistic explorations of REE–body composition relationships.
KEY WORDS: • Kleiber’s law • energy metabolism • body composition.
All living mammals expend energy for the maintenance of resting energy expenditure (REE)3,the thermic effect of feeding, and for physical activity. REE is usually the largest portion of total energy expenditure.
A major focus of REE research over the past 150 y is the
relationship between REE and body composition in mammals
(1
, 2
). As a fundamental physical characteristic, body mass
was applied early in the development of REE–body composition models.
Kleiber first surveyed REE estimates for mature mammals from rats to
steers with a 
2,800-fold difference in body size (3
).
By expressing heat production as a function of body mass, Kleiber found
an exponential relationship between REE (in kJ/d) and body mass (M) (in
kg). The best fit for the data were
 | (1) |
Several years later, Brody included additional mammalian species,
ranging from mice to elephants (4
). Although more species
were added to Kleiber’s database, extending the plot from rat-to-steer
to mouse-to-elephant, the exponent of Brody’s equation was nearly
identical with that of Eq. 1
,
 | (2) |
Based on his and other’s observations, Kleiber opted for the
so-called 0.75 rule for mature mammals (5
),
 | (3) |
Equations 1–3 are very similar, and Kleiber pointed out that the numerical differences in the scaling exponents and in the coefficients are not statistically significant.
Kleiber’ law is one of the most important and best-known laws in
bioenergetics (6
). The consistency of the REE-M
relation over so wide a range of body sizes and species suggests some
unique biological characters inherent within this relation. In the
following years, there was—and still is—considerable discussion
devoted to explaining Kleiber’s law. Although a number of hypotheses
have been proposed, there is not yet a fully agreed upon mechanistic
understanding of the 0.75 exponent observed across mature mammals
(1
, 2
, 7
).
One approach to examining Kleiber’s REE-M law is to construct REE
from individual components at the organ-tissue level. An
interesting question concerns whether the REE values for individual
organs and tissues scale to M0.75. Terroine and
Roche, Grafe, Kleiber, Krebs and others carried out early REE studies
of specific tissues in mammals (8
–11
), although questions
surrounded the interpretation of their observations because they were
usually carried out in vitro. A direct link between summated
organ-tissue REE and the whole-body
"M0.75" rule was therefore difficult to
establish (12
). Although in vivo information remains
limited, new studies over the past two decades provide the opportunity
to establish if Kleiber’s law can be constructed at the
organ-tissue body composition level. The aim of the present study
was to formulate Kleiber’s law on the organ-tissue body
composition level based on available in vivo metabolic data in mature
mammals.
Organ-tissue level modeling
Four metabolically active organs, brain, liver, kidneys and heart,
have high specific resting metabolic rates when compared with the
remaining less-active tissues, such as skeletal muscle, adipose
tissue, bone and skin (13
). Brain, liver, kidneys and
heart together account for 60% of REE in humans, even though the
four organs represent <6% of body mass. Our analysis concentrates on
the individual high metabolic rate organs, because the existing data
are now sufficient to develop an REE-M model based upon in vivo
experiments.
The fundamental REE-M relationship on the organ-tissue level
can be expressed as,
 |
 | (4) |
where i is the organ/tissue number; REEi is the REE of individual organs and tissues; Ki is the specific resting metabolic rate of individual organs and tissues; and Ti is the mass of individual organs and tissues. Equation 4 reveals that whole-body REE is determined by the K and T values of individual organs and tissues. In the following sections, the two REE determinants and their relationships with body mass will be discussed. We explore the relationships between K and body mass and between T and body mass across mature mammals and then apply the findings toward a formulation of Kleiber’s law at the organ-tissue level.
Mammalian organ and tissue-specific resting metabolic rate
The in vivo determination of organ and tissue K values
is a technically demanding process (14
, 15
). The most
reliable method involves in vivo measurement of arterio-venous
differences in oxygen concentration, together with simultaneous blood
flow measurements across the organ (12
–17
). Based on in
vivo measurements, there are several published reports that provide
empirical measurements of organ and tissue K values for
several mammals (Table 1
).
|
).
K values vary allometrically, not isometrically, with body
mass. An exponential function can be derived between specific resting
metabolic rate of individual organ/tissue and body mass across mature
mammals,
 | (5) |
where a is a constant and p is a scaling
exponent. Based on the information provided in Table 1
, exponential
equations were derived for the four organs (i.e., liver, brain, kidneys
and heart) and remaining mass (i.e., the difference between body mass
and the four organs) across mammals (all r > 0.83,
Table 2
). All P values for various organs and tissues are negative,
indicating that specific resting metabolic rates decrease as body mass
increases. For example, liver from a 70-g mouse produces energy at a
rate of 5,866 compared with 909 kJ/(kg · d) for liver from a 70-kg
human.
|
Mammalian organ and tissue mass
The proportion of body mass represented by individual organs and
tissues is not constant but varies with body mass across mammals. Organ
and tissue mass also varies allometrically, not isometrically, with
body mass (19
). Information taken from Calder
(19
) provides the exponential functions (all r
> 0.98) that relate organ and tissue mass to body mass among
mature mammals ranging in body size from mice to elephants (Table 2)
by
following the equation,
 | (6) |
where b is a constant and q is a scaling exponent.
Previous studies indicate that the q values differ for various organs and tissues. With increasing body mass, brain, liver and kidneys occupy a decreasing fraction of body mass (i.e., q < 1). In contrast, heart and remaining tissues are almost in direct proportion to body mass (i.e., q is 0.98 for heart and 1.01 for remaining tissues).
Organ-tissue level REE-M modeling
According to equations 5 and 6
, the fundamental REE-M model at
the organ-tissue level (Eq. 4)
can be re-written as,
 |
 | (7) |
The a, b, p and q
values of liver, brain, kidneys, heart and remaining tissues across
mature mammals are summarized in Table 2
. Based on all a,
b, p and q values, we calculated the
(a x b) and (p + q)
values and REEi values for the five components
across mature mammals. One can thus express whole-body REE as the
sum of REE-M functions of the five organs and tissues (Table 2)
,
 |
 |
 |
 | (8) |
To reduce this equation to the form of REE = c
x Md, we solved Eq. 8
for values of body
mass ranging from 0.01 to 1,000 kg in five steps and then regressed REE
on M using least-squares regression. The regression procedure was
simplified by taking the log of both sides of the equation to yield a
linear equation of the form
 | (9) |
The slope and intercept of this regression provide estimates for
parameters c and d of REE = c
x Md. The best resulting approximation of
Eq. 8
was,
 | (10) |
The correlation (i.e., r value) between REEs and body mass obtained from Eq. 10 was 0.999. Equation 10 is similar to Kleiber’s law observed at the whole-body level (equations 1–3) .
Features of organ-tissue level REE-M modeling
The present study shows that the classic whole body–level Kleiber equation can be reformulated as the integrated REE-M function for five organ-tissue level components, liver, brain, kidneys, heart and remaining tissues. In this section, we describe some of the features of the organ-tissue level REE-M function.
Scaling of individual organ and tissue.
Although whole body-level analysis shows that REE across mammals scales
to M0.75, it was not clear
whether the REE values for individual organs and tissues also scale to
M0.75. Our organ-tissue level model revealed
that the REE of the evaluated organs and tissues individually do not
scale to M0.75. Of the five organs and tissues,
only kidneys have a (p + q) value close to 0.75.
The (p + q) values were <0.75 for liver and
brain and >0.75 for heart and remaining tissues (Table 2)
. However,
although the scaling exponents were between 0.60 and 0.86 for the five
components, their combination yielded a similar REE-M function as
observed on the whole-body level.
Scaling of four organs as a whole.
Using the methods described in solving for eq. 8
, we derived an
exponential REE-M function for the sum of four high metabolic rate
organs (i.e., liver, brain, kidneys, and heart). The best approximation
for the REE sum of the four organs is,
 | (11) |
The exponent of eq. 11 appears to be much lower than that of the remaining tissues (i.e., REErem = 117.4 x M0.84).
It is often assumed that metabolically active components (i.e., the
four evaluated organs) consume more energy than the remaining
less-active tissues at rest. However, this may not be true for very
large mammals. According to equations 10 and 11
, the ratio of REE for
the four organs to whole-body REE can be calculated as
 |
 | (12) |
Equation 12
indicates that the four metabolically active organs
account for 68% of REE for a mammal with 0.1 kg body mass, and this
ratio decreases to 34% for a 1,000 kg body mass mammal. In other
words, the four organs consume two-thirds of REE in mammals
weighing 0.1 kg, but only consume one-third of REE in mammals
weighing 1,000 kg.
Two determinants of Kleiber’s law.
Organ-tissue level REE-M modeling demonstrates that Kleiber’s law is determined by two functions: the K value-body mass function and the T value-body mass function across mammals. According to Kleiber’s law, the mass-specific REE of a mammal with a 0.1 kg body mass is 10 times that of a mammal weighing 1,000 kg. Our organ-tissue modeling further indicates that this is attributable to two factors: organs and tissues have lower K values as body size increases, and the metabolically active organs constitute a smaller percentage of body mass in larger size mammals.
Future studies of Kleiber’s law.
In the present study we were able to reconstruct Kleiber’s law on the organ-tissue level. However, there are still many unanswered questions, and future research related to Kleiber’s law on the organ-tissue level needs to explore additional issues. Specifically, when body mass increases across mammals, why are all p values negative? Why do various organs and tissues have different p and q values? Further studies are also needed to measure in vivo K values of individual organs and tissues in species that vary widely in body mass. Finally, cells within organs and tissues are the unique source of body heat production. Unfortunately, presently there exists very little information upon which cellular level REE model can be based. Acquiring this information by use of newly developed in vivo technologies remains an important future research goal.
ACKNOWLEDGMENTS
We thank James Greenberg of Brooklyn College for carefully reading the manuscript and contributing many useful comments and suggestions.
FOOTNOTES
1 Supported by National Institute of Health Grant
PO1 DK 42618. 
3 Abbreviations used: REE, resting energy
expenditure.
Manuscript received 17 May 2001. Revision accepted 9 August 2001.
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3. Kleiber, M. (1932) Body size and metabolism. Hilgardia 6:315-353.
4. Brody, S. (1945) Bioenergetics and Growth, with Special Reference to the Efficiency Complex in Domestic Animals 1945 Reinhold New York. .
5. Kleiber, M. (1961) The Fire of Life: An Introduction to Animal Energetics 1961 Wiley New York. .
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