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1 Biomedical Engineering, University of California, Irvine, Irvine, CA, USA
* To whom correspondence should be addressed. E-mail: gkassab{at}uci.edu.
The branching pattern and vascular geometry of biological tree structure is complex. Here we show that the design of all vascular trees for which there exists morphometric data in the literature (e.g., coronary, pulmonary; vessels of various skeletal muscles, mesentery, omentum and conjunctiva) obeys a set of scaling laws which are based on the hypothesis that the cost of construction of the tree structure and operation of fluid conduction are minimized. The laws consist of scaling relationships between 1) length and vascular volume of tree, 2) lumen diameter and blood flow rate in each branch and 3) diameter and length of vessel branches. The exponent of the diameter-flow rate relation is not necessarily equal to 3.0 as required by Murray's law but depends on the ratio of metabolic-to-viscous power dissipation of the tree of interest. The major significance of the present analysis is to show that the design of various vascular trees of different organs and species can be deduced on the basis of minimum energy hypothesis and conservation of energy under steady state conditions. The present study reveals the similarity of natures scaling laws that dictate the design of various vascular trees and the underlying physical and physiological principles.
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