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Abstract

Allometric scaling relations, including the 3/4 power law for metabolic rates, are characteristic of all organisms and are here derived from a general model that describes how essential materials are transported through space-filling fractal networks of branching tubes. The model assumes that the energy dissipated is minimized and that the terminal tubes do not vary with body size. It provides a complete analysis of scaling relations for mammalian circulatory systems that are in agreement with data. More generally, the model predicts structural and functional properties of vertebrate cardiovascular and respiratory systems, plant vascular systems, insect tracheal tubes, and other distribution networks.
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REFERENCES AND NOTES

1
McMahon T. A., Bonner J. T., On Size and Life (Scientific American Library, New York, 1983);
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Schmidt-Nielsen K., Scaling: Why Is Animal Size so Important? (Cambridge Univ. Press, Cambridge, 1984);
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Patterson M. R., ibid.255, 1421 (1992).
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Hemmingsen A. M., Rep. Steno Mem. Hosp. (Copenhagen)4, 1 (1950);
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Mandelbrot B. B., The Fractal Geometry of Nature (Freeman, New York, 1977).
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Shinozaki K., et al., Jpn. J. Ecol.14, 97 (1964);
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12
The branching of a vessel at level k into nk smaller vessels (Fig. 1) is assumed to occur over some small, but finite, distance that is much smaller than either lk or lk+1. This relation is similar to that assumed in the Strahler method [A. N. Strahler, Trans. Am. Geophys. Union 34, 345 (1953); (11, 21)]. A generalization to nonuniform branching, where the radii and lengths at a given level may vary, is straightforward.
13
Normalization factors, such as M0, will generally be suppressed, as in Eq. 1. In general, all quantities should be expressed in dimensionless form; note, however, that this does not guarantee that they are size independent and scale as M0. For example, the Womersley number, ω of Eq. 8, although dimensionless, scales as M1/4.
14
This formula is not valid for plant vessel bundles because plants are composed of multiple parallel vessel elements. Their resistance is given by Z = 8µl/Ncπrc4, where l is the length of a single vessel element, rc is its radius, and Nc is their total number.
15
This relation holds for plant vessels from the roots to the leaves, but not within leaves [ M. J. Canney, Philos. Trans. R. Soc. London Ser. B 341, 87 (1993)].
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See, for example, A. S. Iberall, Math. Biosci.1, 375 (1967) and T. F. Sherman,J. Gen. Physiol. 78, 431 (1981), whichcontain summaries of earlier data; also M. Zamir, etal., J. Biomech. 25, 1303(1992) and J. K.-J. Li, Comparative Cardiovascular Dynamics ofMammals (CRC Press, Boca Raton, FL, 1996). Care must be takenin comparing measurements with prediction, particularly if averagesover many successive levels are used. For example, ifAk =+kπrk2is the total cross-sectional area at level k, thenfor the aorta and major arteries, where k < and the branching is area-preserving, wepredict A0 =Ak. Suppose, however,that the first K levels are grouped together. Then,if the resulting measurement givesK,area-preserving predictsK =KA0 (but notK =A0). It also predictsr03n1/2ΣNkrk3. Usingresults from M. LaBarbera [Science 249,992 (1990)], who used data averaged over the first 160 vessels(approximately the first 4 levels), gives, for human beings,A0 ≈ 4.90 cm2,K ≈ 19.98cm2, r03 ≈1.95 cm3, andΣNkrk3≈ 1.27 cm3, in agreement with areapreservation. LaBarbera, unfortunately, took the fact thatKA0 andr03 ≈ΣNkrk3as evidence for cubic rather than area-preserving branching. For smallvessels, where k > ,convincing evidence for the cubic law can be found in the analysis ofthe arteriolar system by M. L. Ellsworth, et al.,Microvasc. Res. 34, 168 (1987).
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Gehr P., et al., Respir. Physiol.44, 61 (1981).
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Bennett P. M., Harvey P. H., J. Zool.213, 327 (1987);
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24
This is reminiscent of the invariance of scaling exponents to details of the model that follow from renormalization group analyses, which can be viewed as a generalization of classical dimensional analysis.
25
J.H.B. is supported by NSF grant DEB-9318096, B.J.E. by NSF grant GER-9553623 and a Fulbright Fellowship, and G.B.W. by the Department of Energy.

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Science
Volume 276 | Issue 5309
4 April 1997

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Published in print: 4 April 1997

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Geoffrey B. West
Theoretical Division, T-8, Mail Stop B285, Los Alamos National Laboratory, Los Alamos, NM 87545, and The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA.
James H. Brown
Department of Biology, University of New Mexico, Albuquerque, NM 87131, and The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA.
Brian J. Enquist
Department of Biology, University of New Mexico, Albuquerque, NM 87131, and The Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501, USA.

Notes

* To whom correspondence should be addressed. E-mail: [email protected]

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