Farm Science Reporter Vol. 6 No. 1

<p>(a) <i>Ev</i> as function of diameter ratio for different length ratios;(b) <i>Ev</i> as function of length ratio for different diameter ratios.</p

Data_Sheet_1_Scaling Laws of Flow Rate, Vessel Blood Volume, Lengths, and Transit Times With Number of Capillaries.pdf

<p>The structure-function relation is one of the oldest hypotheses in biology and medicine; i.e., form serves function and function influences form. Here, we derive and validate form-function relations for volume, length, flow, and mean transit time in vascular trees and capillary numbers of various organs and species. We define a vessel segment as a “stem” and the vascular tree supplied by the stem as a “crown.” We demonstrate form-function relations between the number of capillaries in a vascular network and the crown volume, crown length, and blood flow that perfuses the network. The scaling laws predict an exponential relationship between crown volume and the number of capillaries with the power, λ, of 4/3 < λ < 3/2. It is also shown that blood flow rate and vessel lengths are proportional to the number of capillaries in the entire stem-crown systems. The integration of the scaling laws then results in a relation between transit time and crown length and volume. The scaling laws are both intra-specific (i.e., within vasculatures of various organs, including heart, lung, mesentery, skeletal muscle and eye) and inter-specific (i.e., across various species, including rats, cats, rabbits, pigs, hamsters, and humans). This study is fundamental to understanding the physiological structure and function of vascular trees to transport blood, with significant implications for organ health and disease.</p

Evolution parameter for a turbulent model.

<p>(a) <i>Ev</i> as function of diameter ratio for different length ratios; (b) <i>Ev</i> as function of length ratio for different diameter ratios.</p

Box and whisker plot of the evolution parameter for various species and organs.

<p>The median, mean and SD values are shown in the figure, the box edges represent the 25th and 75th percentiles, the whiskers extend to the most extreme data points not considered outliers. Outliers (star symbols) are larger than P₇₅+1.5(P₇₅-P₂₅) or smaller than P₂₅−1.5(P₇₅-P₂₅), where P₇₅ and P₂₅ are the 75th and 25th percentiles, respectively.</p

Evolution parameter for non-Newtonian pseudo-plastic fluids with n = 0.5.

<p>(a) <i>Ev</i> as function of diameter ratio for different length ratios; (b) <i>Ev</i> as function of length ratio for different diameter ratios.</p

The least-squares of exponents in shape factors and , and corresponding <i>Ev</i> for different species and organs.

<p>RCA, right coronary artery; LAD, left anterior descending artery; LCx, left circumflex artery; PA, pulmonary artery; PV, pulmonary vein; SKMA, skin muscle arteries; SMA, sartorius muscle arteries; MA, mesentery arteries; OV, omentum veins; BCA, bulbular conjunctiva arteries; RMA, retractor muscle artery; BCV, bulbular conjunctiva vein; , number of total generation in the respective vascular trees <a href="http://www.plosone.org/article/info:doi/10.1371/journal.pone.0116260#pone.0116260-Gan1" target="_blank">[32]</a>.</p><p>The least-squares of exponents in shape factors and , and corresponding <i>Ev</i> for different species and organs.</p

The evolution parameter as function of shape factors.

<p>The range of shape factors is obtained from experimental measurement of vascular trees.</p