Technical Briefs

Murray's Law in Elastin Haploinsufficient (Eln+/−) and Wild-Type (WT) Mice

[+] Author and Article Information
Bradley A. Sather

Department of Biomedical Engineering,
Saint Louis University,
3507 Lindell Blvd.,
Saint Louis, MO 63103

Daniel Hageman

Department of Biomedical Engineering,
Case Western Reserve University,
Cleveland, OH 44106

Jessica E. Wagenseil

Department of Biomedical Engineering,
Saint Louis University,
3507 Lindell Blvd.,
Saint Louis, MO 63103
e-mail: jwagense@slu.edu

1Corresponding author.

Contributed by the Bioengineering Division of ASME for publication in the JOURNAL OF BIOMECHANICAL ENGINEERING Manuscript received June 25, 2012; final manuscript received November 15, 2012; accepted manuscript posted November 28, 2012; published online December 5, 2012. Assoc. Editor: Kevin D. Costa.

J Biomech Eng 134(12), 124504 (Dec 05, 2012) (3 pages) doi:10.1115/1.4023093 History: Received June 25, 2012; Revised November 06, 2012; Accepted November 28, 2012

Using either the principle of minimum energy or constant shear stress, a relation can be derived that predicts the diameters of branching vessels at a bifurcation. This relation, known as Murray's Law, has been shown to predict vessel diameters in a variety of cardiovascular systems from adult humans to developing chicks. The goal of this study is to investigate Murray's Law in vessels from mice that are haploinsufficient for the elastin protein (Eln+/−). Elastin is one of the major proteins in the blood vessel wall and is organized in concentric rings, known as lamellae, with smooth muscle cells (SMCs) around the vessel lumen. Eln+/− mice have an increased number of lamellae, as well as smaller, thinner vessels. It is possible that due to decreased amounts of elastin available for vessel wall remodeling during development and in adulthood, Eln+/− vessels would not follow Murray's Law. We examined vessel bifurcations in six different physiologic regions, including the brain, heart, epidermis, ceocum (or cecum), testes, and intestines, in Eln+/− mice and wild-type (WT) littermates. All vessels were between 40 and 300 μm in diameter. We found that the diameters of both Eln+/− and WT vessels have an average of 13% error from the diameters predicted by Murray's Law, with no significant differences between genotypes or physiologic regions. The data suggest that vessels are optimized to follow Murray's Law, despite limitations on the proteins available for growth and remodeling of the vessel wall.

Copyright © 2012 by ASME
Your Session has timed out. Please sign back in to continue.


Murray, C. D., 1926, “The Physiological Principle of Minimum Work: I. The Vascular System and the Cost of Blood Volume,” Proc. Natl. Acad. Sci. U.S.A., 12(3), pp. 207–214. [CrossRef] [PubMed]
Zamir, M., 1977, “Shear Forces and Blood Vessel Radii in the Cardiovascular System,” J. General Physiol., 69(4), pp. 449–461. [CrossRef]
Sherman, T. F., 1981, “On Connecting Large Vessels to Small. The Meaning of Murray's Law,” J. General Physiol., 78(4), pp. 431–453. [CrossRef]
van der Waal, E. C., Mintz, G. S., Garcia-Garcia, H. M., Bui, A. B., Pehlivanova, M., Girasis, C., Serruys, P. W., van der Giessen, W. J., and Weissman, N. J., 2009, “Intravascular Ultrasound and 3D Angle Measurements of Coronary Bifurcations,” Catheter. Cardiovasc. Intervent., 73(7), pp. 910–916. [CrossRef]
Hutchins, G. M., Miner, M. M., and Boitnott, J. K., 1976, “Vessel Caliber and Branch-Angle of Human Coronary Artery Branch-Points,” Circ. Res., 38(6), pp. 572–576. [CrossRef] [PubMed]
Taber, L. A., Ng, S., Quesnel, A. M., Whatman, J., and Carmen, C. J., 2001, “Investigating Murray's Law in the Chick Embryo,” J. Biomech., 34(1), pp. 121–124. [CrossRef] [PubMed]
Wilson, T. A., 1967, “Design of the Bronchial Tree,” Nature, 213(5077), pp. 668–669. [CrossRef] [PubMed]
LaBarbera, M., 1990, “Principles of Design of Fluid Transport Systems in Zoology,” Science, 249(4972), pp. 992–1000. [CrossRef] [PubMed]
McCulloh, K. A., Sperry, J. S., and Adler, F. R., 2003, “Water Transport in Plants Obeys Murray's Law,” Nature, 421(6926), pp. 939–942. [CrossRef] [PubMed]
Wolinsky, H., and Glagov, S., 1967, “A Lamellar Unit of Aortic Medial Structure and Function in Mammals,” Circ. Res., 20(1), pp. 99–111. [CrossRef] [PubMed]
Li, D. Y., Faury, G., Taylor, D. G., Davis, E. C., Boyle, W. A., Mecham, R. P., Stenzel, P., Boak, B., and Keating, M. T., 1998, “Novel Arterial Pathology in Mice and Humans Hemizygous for Elastin,” J. Clin. Invest., 102(10), pp. 1783–1787. [CrossRef] [PubMed]
Faury, G., Pezet, M., Knutsen, R. H., Boyle, W. A., Heximer, S. P., McLean, S. E., Minkes, R. K., Blumer, K. J., Kovacs, A.Kelly, D. P., Li, D. Y., Starcher, B., and Mecham, R. P., 2003, “Developmental Adaptation of the Mouse Cardiovascular System to Elastin Haploinsufficiency,” J. Clin. Invest., 112(9), pp. 1419–1428. [CrossRef] [PubMed]
Wagenseil, J. E., Nerurkar, N. L., Knutsen, R. H., Okamoto, R. J., Li, D. Y., and Mecham, R. P., 2005, “Effects of Elastin Haploinsufficiency on the Mechanical Behavior of Mouse Arteries,” Am. J. Physiol. Heart Circ. Physiol., 289(3), pp. H1209–H1217. [CrossRef] [PubMed]
Treuting, P. M., and Dintzis, S. M., 2012, Comparative Anatomy and Histology, A Mouse and Human Atlas, Academic Press, Waltham, MA.
Greenwald, S. E., 2007, “Ageing of the Conduit Arteries,” J. Pathology, 211(2), pp. 157–172. [CrossRef]
Kamiya, A., Bukhari, R., and Togawa, T., 1984, “Adaptive Regulation of Wall Shear Stress Optimizing Vascular Tree Function,” Bull. Math. Biol., 46(1), pp. 127–137. [CrossRef] [PubMed]
Painter, P. R., Eden, P., and Bengtsson, H. U., 2006, “Pulsatile Blood Flow, Shear Force, Energy Dissipation and Murray's Law,” Theor. Biol. Med. Model., 3, p. 31. [CrossRef] [PubMed]
Kassab, G. S., and Fung, Y. C., 1995, “The Pattern of Coronary Arteriolar Bifurcations and the Uniform Shear Hypothesis,” Ann. Biomed. Eng., 23(1), pp. 13–20. [CrossRef] [PubMed]
Langille, B. L., 1993, “Remodeling of Developing and Mature Arteries: Endothelium, Smooth Muscle, and Matrix,” J. Cardiovasc. Pharmacol., 21(Suppl. 1), pp. S11–S17. [CrossRef] [PubMed]


Grahic Jump Location
Fig. 1

Representative image of the vessels in the testes of an Eln+/− mouse showing the lines drawn for the average diameter measurements

Grahic Jump Location
Fig. 2

Average ratio α for WT and Eln+/− vessels in different physiologic regions. The ratio is near the predicted value of 1 from Murray's Law, with no significant differences between genotypes or regions. The number of vessel images used for the ratio calculation for each region and genotype is listed on the columns.

Grahic Jump Location
Fig. 3

Composite graph of all ratio α measurements and the average for all of the data combined. The average for all the data is α = 1.13 ± 0.30.

Grahic Jump Location
Fig. 4

Relative vessel diameters and the fit to (a) Eq. (3) and (b) Eq. (4) for all of the data combined. Fitted values are b = 2.72 (R2 = 0.99) and 2.76 (R2 = 0.99), respectively, which is near the predicted value of 3 from Murray's Law.



Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging and repositioning the boxes below.

Related Journal Articles
Related eBook Content
Topic Collections

Sorry! You do not have access to this content. For assistance or to subscribe, please contact us:

  • TELEPHONE: 1-800-843-2763 (Toll-free in the USA)
  • EMAIL: asmedigitalcollection@asme.org
Sign In