Intraspecific scaling laws of vascular trees

A fundamental physics-based derivation of intraspecific scaling laws of vascular trees has not been previously realized. Here, we provide such a theoretical derivation for the volume–diameter and flow–length scaling laws of intraspecific vascular trees. In conjunction with the minimum energy hypothesis, this formulation also results in diameter–length, flow–diameter and flow–volume scaling laws. The intraspecific scaling predicts the volume–diameter power relation with a theoretical exponent of 3, which is validated by the experimental measurements for the three major coronary arterial trees in swine (where a least-squares fit of these measurements has exponents of 2.96, 3 and 2.98 for the left anterior descending artery, left circumflex artery and right coronary artery trees, respectively). This scaling law as well as others agrees very well with the measured morphometric data of vascular trees in various other organs and species. This study is fundamental to the understanding of morphological and haemodynamic features in a biological vascular tree and has implications for vascular disease.

[1]  B. Masters,et al.  Fractal pattern formation in human retinal vessels , 1989 .

[2]  S. Heymsfield,et al.  The reconstruction of Kleiber's law at the organ-tissue level. , 2001, The Journal of nutrition.

[3]  J. Bassingthwaighte,et al.  Fractal Branchings: The Basis of Myocardial Flow Heterogeneities? , 1990, Annals of the New York Academy of Sciences.

[4]  James H. Brown,et al.  A General Model for the Origin of Allometric Scaling Laws in Biology , 1997, Science.

[5]  G S Kassab,et al.  Topology and dimensions of pig coronary capillary network. , 1994, The American journal of physiology.

[6]  Jie Zheng,et al.  Roles of myocardial blood volume and flow in coronary artery disease: an experimental MRI study at rest and during hyperemia , 2010, European Radiology.

[7]  B. Mandelbrot,et al.  Fractals: Form, Chance and Dimension , 1978 .

[8]  B L Langille,et al.  Arterial bifurcations in the cardiovascular system of a rat , 1983, The Journal of general physiology.

[9]  Tatsuhisa Takahashi,et al.  Quantitative assessments of morphological and functional properties of biological trees based on their fractal nature. , 2007, Journal of applied physiology.

[10]  D. Stoyan,et al.  Mandelbrot, B. B., Fractals: Form, Chance, and Dimension. San Francisco. W. H. Freeman and Company. 1977. 352 S., 68 Abb., $14.95 , 1979 .

[11]  B. Fenton,et al.  Microcirculatory model relating geometrical variation to changes in pressure and flow rate , 1981, Annals of Biomedical Engineering.

[12]  G Cumming,et al.  Morphometry of the Human Pulmonary Arterial Tree , 1973, Circulation research.

[13]  J. Kozłowski,et al.  West, Brown and Enquist's model of allometric scaling again: the same questions remain , 2005 .

[14]  Y. Huo,et al.  The scaling of blood flow resistance: from a single vessel to the entire distal tree. , 2009, Biophysical journal.

[15]  J. Wübbeke,et al.  Three-dimensional image analysis and fractal characterization of kidney arterial vessels , 1992 .

[16]  K. Horsfield,et al.  Morphometry of pulmonary veins in man , 2007, Lung.

[17]  Yunlong Huo,et al.  A scaling law of vascular volume. , 2009, Biophysical journal.

[18]  G S Kassab,et al.  Morphometry of pig coronary arterial trees. , 1993, The American journal of physiology.

[19]  James H. Brown,et al.  The fourth dimension of life: fractal geometry and allometric scaling of organisms. , 1999, Science.

[20]  G S Kassab,et al.  Morphometry of the dog pulmonary venous tree. , 1993, Journal of applied physiology.

[21]  Jan Kozłowski,et al.  Is West, Brown and Enquist's model of allometric scaling mathematically correct and biologically relevant? , 2004 .

[22]  B Dawant,et al.  Analysis of vascular pattern and dimensions in arteriolar networks of the retractor muscle in young hamsters. , 1987, Microvascular research.

[23]  T R Nelson,et al.  Modeling of lung morphogenesis using fractal geometries. , 1988, IEEE transactions on medical imaging.

[24]  G. Kassab.  Scaling laws of vascular trees: of form and function. , 2006, American journal of physiology. Heart and circulatory physiology.

[25]  Y C Fung,et al.  Morphometry of cat pulmonary venous tree. , 1983, Journal of applied physiology: respiratory, environmental and exercise physiology.

[26]  Sabee Molloi,et al.  In vivo validation of the design rules of the coronary arteries and their application in the assessment of diffuse disease. , 2002, Physics in medicine and biology.

[27]  G. Coppini,et al.  Hypoxia- or hyperoxia-induced changes in arteriolar vasomotion in skeletal muscle microcirculation. , 1991, The American journal of physiology.

[28]  R. Yen,et al.  Morphometry of the human pulmonary vasculature. , 1996, Journal of applied physiology.

[29]  Sakae Shibusawa,et al.  Modelling the branching growth fractal pattern of the maize root system , 1994, Plant and Soil.

[30]  M. Labarbera.  Principles of design of fluid transport systems in zoology. , 1990, Science.

[31]  G S Kassab,et al.  Diameter-defined Strahler system and connectivity matrix of the pulmonary arterial tree. , 1994, Journal of applied physiology.

[32]  Ghassan S Kassab,et al.  Scaling of myocardial mass to flow and morphometry of coronary arteries. , 2008, Journal of applied physiology.

[33]  J. Lawton,et al.  Fractal dimension of vegetation and the distribution of arthropod body lengths , 1985, Nature.

[34]  Yunlong Huo,et al.  Diameter asymmetry of porcine coronary arterial trees: structural and functional implications. , 2008, American journal of physiology. Heart and circulatory physiology.

[35]  B Dawant,et al.  Quantitative analysis of arteriolar network architecture in cat sartorius muscle. , 1987, The American journal of physiology.

[36]  Y C Fung,et al.  Morphometry of cat's pulmonary arterial tree. , 1984, Journal of biomechanical engineering.

[37]  E. vanBavel,et al.  Branching patterns in the porcine coronary arterial tree. Estimation of flow heterogeneity. , 1992, Circulation research.

[38]  Y. Huo,et al.  Capillary Perfusion and Wall Shear Stress Are Restored in the Coronary Circulation of Hypertrophic Right Ventricle , 2007, Circulation research.

[39]  A. Pries,et al.  Topological structure of rat mesenteric microvessel networks. , 1986, Microvascular research.