ISRN Zoology

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

doi:10.5402/2012/673050

Research Article

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

ZiMian Wang,¹ Junyi Zhang,² Zhiliang Ying,² and Steven B. Heymsfield³

¹Obesity Research Center, St. Luke’s-Roosevelt Hospital, College of Physicians and Surgeons, Columbia University, New York City, NY 10025, USA

²Department of Statistics, Columbia University, New York City, NY 10027, USA

³Pennington 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.

Table 1: Estimated values of organ-tissue level components across mature mammals.

Species

Rat
0.48
870
470
968
685
33.7

Rabbit
2.5
590
—
—
—
—

Cat
3.0
420
—
—
—
—

Dog
10
380
370
—
679
—

Human
65
251
230
668
482
20.7

Human
70
200
240
440
440
10.7

Coefficient a

683.9
446.6
890.3
689.7
29.96

Scaling

exponent

−0.2677
−0.1423
−0.1181
−0.0833
−0.1667

value

0.975
0.961
0.855
0.848
0.827

Source of data. Rat [5], rabbit, cat, dog [6], and human [7, 8]. Specific resting metabolic rates for liver, brain, heart, and kidneys of various mammals are consistent with results given by the above references. Specific resting metabolic rates for the residuals are calculated from the above references.

Abbreviations: : body mass (in kg);   : specific resting metabolic rate of liver, brain, heart, kidney, and residuals, respectively (all in kcal/kg per day).

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].

Table 2: The organ-tissue level body composition, specific metabolic rates, and resting energy expenditure in 111 mammalian species.

Species

Sorex araneus
0.0075
2533
896
1586
1037
67.7
0.00038
0.00015
0.00011
0.00011
0.0068
1.8
1.8
245
238

Crocidura russula
0.010
2376
866
1542
1016
65.1
0.00055
0.00017
0.00008
0.00013
0.0086
2.3
2.1
237
224

Lasiurus borealis
0.014
2165
824
1480
987
61.4
0.00035
0.00017
0.00014
0.00011
0.013
2.0
2.8
148
205

Lasionycteris noctivagans
0.015
2115
814
1465
980
60.5
0.00033
0.00016
0.00016
0.00013
0.014
2.0
3.0
138
201

Mus musculus
0.015
2108
812
1463
979
60.4
0.00068
0.00036
0.00007
0.00028
0.014
2.9
3.0
196
200

Myodes glareolus
0.015
2087
808
1456
976
60.0
0.00067
0.00035
0.00010
0.00024
0.014
2.9
3.1
188
198

Microtus agrestis
0.015
2087
808
1456
976
60.0
0.00063
0.00039
0.00012
0.00017
0.014
2.8
3.1
183
198

Neomys fodiens
0.016
2080
807
1454
975
59.9
0.00055
0.00025
0.00014
0.00022
0.015
2.6
3.1
168
198

Blarina brevicauda
0.018
2011
792
1433
965
58.6
0.00093
0.00032
0.00018
0.00021
0.016
3.5
3.4
199
192

Apodemus sylvaticus
0.018
1997
789
1428
963
58.4
0.0011
0.00057
0.00014
0.00026
0.016
4.0
3.5
221
190

Microtus pinetorum
0.021
1925
774
1405
952
57.1
0.0011
0.00058
0.00015
0.00036
0.019
4.3
3.9
204
184

Peromyscus leucopus
0.022
1897
768
1396
947
56.6
0.0012
0.00074
0.00015
0.00030
0.020
4.5
4.0
204
182

Apodemus flavicollis
0.025
1835
755
1376
938
55.4
0.0010
0.00061
0.00018
0.00034
0.023
4.1
4.4
164
176

Nyctalus noctula
0.026
1824
752
1372
936
55.2
0.0005
0.00032
0.00037
0.00013
0.024
3.0
4.5
119
175

Microtus arvalis
0.027
1799
747
1364
932
54.7
0.0019
0.00039
0.00019
0.00055
0.024
5.7
4.7
213
173

Mouse
0.028
1781
743
1358
929
54.4
0.0018
0.00050
0.00016
0.00051
0.025
5.6
4.8
199
171

Gerbillus perpallidus
0.030
1748
735
1347
924
53.7
0.0010
0.00058
0.00013
0.00027
0.028
4.0
5.1
135
168

Mustela nivalis
0.032
1714
728
1335
918
53.1
0.0016
0.0018
0.00036
0.00043
0.028
6.3
5.3
196
165

Acomys minous
0.042
1593
700
1293
897
50.7
0.0009
0.0009
0.00018
0.00032
0.040
4.7
6.5
110
254

Jaculus jaculus
0.048
1543
688
1275
888
49.7
0.0011
0.0012
0.00045
0.00029
0.045
5.6
7.2
117
150

Rhabdomys pumilio
0.050
1523
683
1267
885
49.3
0.0018
0.0006
0.00021
0.00041
0.047
6.2
7.4
123
148

Talpa europaea
0.051
1514
681
1264
883
49.2
0.0015
0.0010
0.00031
0.00036
0.048
6.1
7.6
118
147

Glaucomys volans
0.055
1486
675
1254
878
48.6
0.0029
0.0019
0.00056
0.00059
0.049
9.2
8.0
167
144

Arvicola terrestris
0.062
1441
664
1237
870
47.7
0.0026
0.0011
0.00028
0.00070
0.057
8.1
8.7
131
140

Glis glis
0.083
1330
636
1194
848
45.3
0.0032
0.0015
0.00048
0.00068
0.078
9.9
10.9
118
130

Tamias striatus
0.104
1254
616
1163
833
43.7
0.0029
0.0024
0.00066
0.00081
0.097
10.8
12.8
104
123

Octodon degus
0.129
1182
597
1134
818
42.1
0.0048
0.0019
0.00041
0.0011
0.121
13.3
15.1
103
117

Tupaia glis
0.141
1156
590
1122
812
41.5
0.0034
0.0034
0.00117
0.0011
0.132
13.6
16.1
96.9
114

Rat
0.150
1137
585
1114
808
41.1
0.0092
0.0023
0.00070
0.0014
0.136
19.3
16.9
129
112

Cebuella pygmaea
0.163
1112
578
1103
802
40.5
0.0135
0.0044
0.00086
0.0019
0.142
25.8
18.0
158
110

Rattus norvegicus
0.210
1039
558
1071
786
38.9
0.0092
0.0023
0.00087
0.0015
0.196
20.6
21.7
98.0
103

Cheirogaleus medius
0.231
1012
550
1059
779
38.3
0.0063
0.0028
0.00093
0.0010
0.220
18.1
23.3
78.4
101

Rat
0.250
991
544
1049
774
37.8
0.0120
0.0020
0.00094
0.0021
0.233
24.4
24.7
97.6
99.0

Mustela erminea
0.259
982
541
1044
772
37.5
0.0100
0.0057
0.0025
0.0023
0.238
26.2
25.4
101
98.1

Helogale parvula
0.260
981
541
1044
772
37.5
0.0111
0.0052
0.0015
0.0025
0.240
26.1
25.5
100
98.0

Sciurus vulgaris
0.275
967
537
1037
768
37.2
0.0055
0.0063
0.0017
0.0017
0.259
21.4
26.6
77.9
96.7

Callithrix jacchus
0.312
935
527
1022
760
36.4
0.0178
0.0073
0.0028
0.0029
0.281
35.8
29.2
115
93.7

Saguinus fuscicollis

lagonotus
0.330
920
523
1015
756
36.0
0.0144
0.0078
0.0033
0.0019
0.303
33.0
30.5
100
92.4

Rat
0.337
915
521
1012
755
35.9
0.0080
0.0019
0.0010
0.0023
0.324
22.7
31.0
67.3
91.9

Rat (Wistar)
0.390
880
511
995
746
35.1
0.0143
0.0019
0.0011
0.0028
0.370
29.8
34.6
76.3
88.6

Sciurus niger
0.412
867
507
989
743
34.7
0.0107
0.0075
0.0025
0.0030
0.389
31.3
36.0
75.8
87.4

Sciurus carolinensis
0.596
786
481
946
720
32.7
0.0164
0.0075
0.0028
0.0032
0.566
40.0
47.5
67.1
79.7

Saguinus oedipus
0.624
776
478
941
717
32.4
0.0209
0.0100
0.0037
0.0031
0.586
45.7
49.1
73.2
78.8

Mustela putorius
0.640
771
476
939
716
32.3
0.0288
0.0104
0.0048
0.0040
0.592
53.6
50.1
83.8
78.3

Leontopithecus chrysomelas
0.642
770
476
938
716
32.3
0.0189
0.0132
0.0038
0.0041
0.602
46.8
50.2
73.0
78.2

Guinea pig
0.800
726
461
914
703
31.1
0.0270
0.0047
0.0023
0.0056
0.760
51.5
59.2
64.3
74.0

Potorous tridactylus
0.809
724
460
913
702
31.0
0.0237
0.0114
0.0048
0.0062
0.763
54.8
59.7
67.8
73.8

Erinaceus europaeus
0.950
693
450
896
693
30.2
0.0496
0.0043
0.0055
0.0089
0.881
74.1
67.3
78.0
70.9

Sylvilagus floridanus
0.972
689
448
893
691
30.1
0.0320
0.0079
0.0048
0.0063
0.921
61.9
68.5
63.7
70.5

Ondatra zibethicus
0.991
686
447
891
690
30.0
0.0260
0.0047
0.0030
0.0058
0.952
55.1
69.5
55.6
70.2

Saimiri boliviensis
1.00
683
446
890
690
29.9
0.0194
0.0290
0.0065
0.0067
0.941
64.8
70.2
64.6
69.9

Martes foina
1.41
624
425
855
670
28.3
0.0349
0.0190
0.0098
0.0073
1.335
81.0
90.4
57.6
64.3

Mephitis mephitis
1.45
619
424
852
669
28.2
0.0174
0.0098
0.0060
0.0066
1.409
64.2
92.5
44.3
63.8

Trichosurus vulpecula
1.55
608
420
845
665
27.9
0.0332
0.0127
0.0090
0.0135
1.482
83.3
97.2
53.7
62.7

Martes martes
1.60
603
418
842
663
27.7
0.0379
0.0205
0.0108
0.0088
1.525
88.6
99.7
55.3
62.2

Cebus apella
1.75
589
412
833
658
27.3
0.0493
0.0508
0.0134
0.0104
1.626
112
107
64.2
60.9

Eulemur macaco macaco
1.88
578
408
827
655
27.0
0.0778
0.0242
0.0091
0.0142
1.750
119
112
63.4
59.8

Chrotagale owstoni
1.96
571
406
822
652
26.8
0.0441
0.0233
0.0116
0.0128
1.868
103
116
52.3
59.2

Vulpes corsac
2.08
563
403
817
649
26.5
0.0356
0.0341
0.0217
0.0088
1.975
110
121
52.8
58.3

Lemur catta
2.08
563
403
817
649
26.5
0.0729
0.0228
0.0117
0.0112
1.956
119
121
57.3
58.3

Eulemur fulvus fulvus
2.50
535
392
799
639
25.7
0.0434
0.0225
0.0118
0.0095
2.413
110
139
43.8
55.7

Felis silvestris
2.57
531
390
796
638
25.6
0.0502
0.0381
0.0103
0.0154
2.459
122
142
47.6
55.3

Didelphis virginiana
2.63
528
389
794
636
25.5
0.1573
0.0083
0.0121
0.0229
2.433
172
145
65.5
54.9

Aonyx cinerea
2.68
526
388
793
635
25.4
0.1064
0.0359
0.0151
0.0306
2.487
165
146
61.5
54.7

Leopardus geoffroyi
3.10
505
380
779
628
24.8
0.0584
0.0321
0.0160
0.0307
2.963
147
164
47.4
52.8

Lepus europaeus
3.34
495
376
772
624
24.5
0.0904
0.0148
0.0289
0.0185
3.186
162
173
48.6
51.8

Dasyprocta punctata
3.40
493
375
771
623
24.4
0.1088
0.0228
0.0363
0.0213
3.211
182
175
53.5
51.5

Potos flavus
3.92
474
368
758
616
23.9
0.1657
0.0311
0.0211
0.0144
3.688
203
195
51.8
49.7

Dasyprocta azarae
4.10
469
365
754
613
23.7
0.0935
0.0238
0.0304
0.0227
3.930
182
202
44.5
49.2

Varecia rubra
4.20
466
364
752
612
23.6
0.0722
0.0357
0.0181
0.0224
4.052
170
205
40.4
48.9

Alouatta sara
4.40
460
362
747
610
23.4
0.0812
0.0565
0.0240
0.0099
4.228
181
213
41.1
48.3

Monkey
4.50
457
361
745
609
23.3
0.110
0.0420
0.0230
0.0210
4.304
196
216
43.5
48.1

Martes pennanti
4.79
450
357
740
605
23.1
0.113
0.0412
0.0274
0.0211
4.588
204
227
42.7
47.3

Trachypithecus vetulus
5.00
445
355
736
603
22.9
0.090
0.0720
0.0192
0.0154
4.803
199
234
39.8
46.8

Lutrogale perspicillata
5.10
442
354
734
602
22.8
0.152
0.0622
0.0485
0.0485
4.789
263
238
51.6
46.6

Chlorocebus pygerythrus
5.30
438
352
731
600
22.7
0.089
0.0808
0.0426
0.0121
5.076
221
245
41.7
46.1

Lutra lutra
5.33
437
352
731
600
22.7
0.255
0.0478
0.0514
0.0611
4.910
314
245
58.9
46.1

Proteles cristata
5.40
435
351
730
599
22.6
0.182
0.0399
0.0906
0.0243
5.063
288
248
53.4
45.9

Agouti paca
5.46
434
351
729
599
22.6
0.140
0.0321
0.0176
0.0222
5.248
217
250
39.7
45.8

Macaca nigra
5.60
431
350
726
598
22.5
0.095
0.1052
0.0239
0.0186
5.357
227
255
40.5
45.5

Puma yagouaroundi
5.90
425
347
722
595
22.3
0.116
0.0430
0.0296
0.0391
5.673
235
265
39.9
44.9

Hylobates concolor
6.55
414
342
713
590
21.9
0.293
0.1378
0.0582
0.0352
6.026
363
287
55.3
43.8

Prionailurus viverrinus
7.30
402
337
704
584
21.5
0.160
0.0529
0.0335
0.0559
6.998
289
311
39.6
42.6

Macropus agilis
7.70
396
334
700
582
21.3
0.203
0.0308
0.0602
0.0463
7.360
317
324
41.1
42.0

Lontra canadensis
7.90
393
333
697
581
21.2
0.255
0.0425
0.0541
0.0747
7.474
354
330
44.8
41.8

Dolichotis patagonum
8.43
387
330
692
578
21.0
0.158
0.0365
0.0651
0.0360
8.134
310
346
36.8
41.1

Symphalangus syndactylus
8.50
386
329
691
577
21.0
0.294
0.1430
0.0515
0.0437
7.968
388
348
45.7
41.0

Colobus guereza
9.75
372
323
680
571
20.5
0.171
0.0865
0.0370
0.0233
9.432
323
386
33.2
39.6

Felis chaus
9.80
371
323
680
570
20.5
0.153
0.0497
0.0483
0.0819
9.467
346
388
35.3
39.6

Lynx canadensis
10.0
369
322
678
569
20.4
0.158
0.0826
0.0388
0.0549
9.666
340
394
34.0
39.4

Dog
10.0
369
322
678
569
20.4
0.420
0.0750
0.0850
0.0700
9.350
468
394
46.8
39.4

Arctictis binturong
10.3
366
320
676
568
20.3
0.915
0.0605
0.0752
0.0513
9.198
621
402
60.3
39.1

Hystrix indica
11.3
358
316
669
564
20.0
0.255
0.0407
0.0562
0.0524
10.85
388
430
34.5
38.2

Theropithecus gelada
11.4
357
316
668
563
20.0
0.236
0.1409
0.0772
0.0380
10.91
419
434
36.8
38.1

Pudu puda
13
345
310
658
557
19.6
0.206
0.0616
0.0505
0.0199
12.56
380
476
29.5
36.9

Gazella gazella
15
331
304
647
550
19.1
0.327
0.0793
0.120
0.0406
14.43
508
534
33.8
35.6

Castor fiber
15.6
328
302
644
549
19.0
0.345
0.0489
0.044
0.0783
15.05
485
548
31.1
35.2

Macaca arctoides
15.9
326
301
642
548
18.9
0.241
0.1180
0.061
0.0500
15.40
472
556
29.7
35.1

Lynx lynx
17.5
318
297
635
543
18.6
0.264
0.0943
0.093
0.0795
16.97
529
599
30.3
34.2

Capreolus capreolus
20
307
292
625
537
18.2
0.480
0.100
0.160
0.0800
19.18
668
662
33.4
33.1

Cuon alpinus
20.0
307
292
625
537
18.2
0.346
0.116
0.158
0.0764
19.30
631
662
31.5
33.1

Dog
20.4
305
291
624
537
18.1
0.447
0.096
0.153
0.0920
19.6
665
672
32.6
32.9

Mandrillus sphinx
23.0
295
286
615
531
17.8
0.331
0.168
0.076
0.0499
22.4
616
735
26.8
32.0

Papio hamadryas
23.3
295
285
614
531
17.7
0.392
0.174
0.103
0.0803
22.5
670
741
28.8
31.9

Zalophus californianus
34.0
266
270
587
514
16.6
1.274
0.310
0.168
0.2059
32.0
1161
986
34.1
29.0

Hydrochaeris hydrochaeris
34.0
266
270
587
514
16.6
0.696
0.084
0.104
0.1035
33.0
872
986
25.6
29.0

Canis lupus chanco
38.0
258
266
579
509
16.3
0.971
0.140
0.303
0.2069
36.4
1163
1071
30.6
28.2

Sheep
52
237
255
558
496
15.5
0.960
0.106
0.280
0.160
50.5
1274
1356
24.5
26.1

Reference women
58
200
240
440
440
10.4
1.40
1.20
0.240
0.275
54.9
1365
1471
23.5
25.4

Human
60
200
240
440
440
10.4
1.70
1.30
0.320
0.250
56.4
1490
1509
24.8
25.2

Reference man
70
200
240
440
440
10.4
1.80
1.40
0.330
0.310
66.2
1666
1694
23.8
24.2

Panthera tigris altaica
75
215
242
535
481
14.6
1.10
0.342
0.305
0.4246
72.8
1750
1784
23.3
23.8

Hog
125
188
225
503
461
13.4
1.60
0.12
0.350
0.260
123
2267
2617
18.1
20.9

Dairy cow
488
130
185
429
412
10.7
6.46
0.40
1.88
1.16
478
7304
7268
15.0
14.9

Horse
600
123
180
418
405
10.3
6.70
0.67
4.25
1.66
587
9449
8486
15.7
14.1

Steer
700
118
176
411
400
10.1
5.00
0.50
2.30
1.00
691
8973
9526
12.8
13.6

Elephant
6650
65
128
315
331
6.9
6.30
5.70
2.20
1.20
6635
48054
51548
7.2
7.8

Source of and data: mouse and dog ( = 20.42 kg) from Marting and Fuhrman, 1955 [9]; rat, guinea pig, dog ( = 10 kg), sheep, hog, dairy cow, horse, steers, and elephant from Elia, 1992 [7]; reference man and woman from Snyder et al., [10]; rest species from Navarrete et al., 2011 [11].

Abbreviations: : body mass (in kg); , and : Specific resting metabolic rate of liver, brain, heart, kidney, and residuals, respectively (all in kcal/kg per day); , and : mass of liver, brain, heart, kidney, and residuals, respectively (all in kg); : resting energy expenditure predicted by the organ-tissue level REE model (in kcal/day); : resting energy expenditure calculated by Kleiber’s Law (in kcal/day); : the ratio of resting energy expenditure by the organ-tissue level REE model to body mass (in kcal/kg per day); : the ratio of resting energy expenditure calculated by Kleiber’s Law to body mass (in kcal/kg per day).

The Ti 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,

Table 3: The -M and (K × )-M functions for five organs/tissues across 111 mammals.

(K-) function
(K × )- function

Liver
= 683.9 × −0.2677  

× = 19.56 × 0.6046 

Brain
= 446.6 × −0.1423  

× = 4.82 × 0.6446  

Heart
= 890.3 × −0.1181  

× = 5.16 × 0.8137  

Kidneys
= 689.7 × −0.0833  

× = 4.35 × 0.7441  

Residual
= 29.96 × −0.1667  

× = 28.16 × 0.8402  

Abbreviations: : body mass (in kg); and : specific resting metabolic rate of liver, brain, heart, kidney and residual, respectively (all in kcal/kg per day).

Figure 1: The (K × T) values (in kcal/d) of individual organs/tissues predicted by (9) on the ordinate versus body mass (M, in kg) on the abscissa across 111 mammalian species. The (K × T) values of liver, brain, heart, kidneys, and residuals are shown as the five thin lines. The resting energy expenditure () predicted by (7) is represented as the thick line (upper). All (K × T) and M values are on a logarithmic scale. The and M are significantly correlated: = 66.33 × ; .

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, ).

Figure 2: The resting energy expenditure (, in kcal/d) calculated by Kleiber’s law (1) on the ordinate versus the resting energy expenditure (, in kcal/d) predicted by the organ-tissue level model (7) on the abscissa across 111 mammalian species. Both and are shown on a logarithmic scale. The and are significantly correlated: = 1.0644 × ; . The line of identity is shown.

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.

Figure 3: The ratio of resting energy expenditure to body mass (i.e., in kcal/kg per day) predicted by the organ-tissue level model (8) on the ordinate versus body mass (M, in kg) on the abscissa across 111 mammalian species. Both and M are shown on a logarithmic scale. The and M are significantly correlated: = 66.33 × ; .

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,

Figure 4: The ratio of resting energy expenditure to body mass (i.e., , in kcal/kg per day) calculated by Kleiber’s law (2) on the ordinates versus the ratio of resting energy expenditure to body mass (i.e., , in kcal/kg per day) predicted by (8) on the abscissa across 111 mammalian species. Both and are shown on a logarithmic scale. The and are significantly correlated: = 1.0644 × ; . The line of identity is shown.

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 O₂ 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[¹⁸F]fluoro-D-glucose [30]. Other isotopes have also been applied such as ¹¹C-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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