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Ting Luo, Alberto De Castro, Lucie Sawides, Thomas Gast, Kaitlyn Sapoznik, Raymond Luval Warner, Stephen A Burns; Modeling retinal arterial diameters and velocities at bifurcations. Invest. Ophthalmol. Vis. Sci. 2017;58(8):1260. doi: https://doi.org/.
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© ARVO (1962-2015); The Authors (2016-present)
Murray’s law describes the vascular requirement of tissues based on minimizing energy requirements for blood distribution. We examine the predictions of a derivation of Murray’s law that includes the variation in viscosity of blood with vessel size (the Fåhræus-Lindqvist effect).
We derived the interaction of viscosity, vessel size, and blood velocity at vascular branches in four steps. 1) The energy cost of a vascular bifurcation includes the work to overcome friction between blood and vessel walls (Poiseuille’s law) and the energy requirement for maintaining blood volume. 2) The change in viscosity with blood vessel radius was incorporated into the energy cost equation (step 1) by fitting the viscosity function to vessel size. 3) The energy cost (step 2) was minimized to determine the relation between blood flow and radius. 4) Volume conservation (flow into a branching must equal the flow out, flow=velocity x cross sectional area) established the relation between velocity in parent and daughter branches.Size data were measured in 26 healthy subjects (203 arteriolar bifurcations) using confocal and multiply scattered light AOSLO images. Velocity data were from 3 normal subject (30 arteriolar bifurcations) using AOSLO temporal offset imaging. Size and velocity data were averaged and fit using the Matlab curve fitting toolbox.
This modeling predicts that the exponent in Murray’s law varies from 2 to 3 depending on vessel size. Vessels smaller than 40μm would have an average of 2.24, larger vessels would be close to the classical prediction of 3. The diameter measurements gave an exponent of 2.15 (1.89-2.41, 95% confidence interval) for vessels less than 40μm and an exponent of 2.74 (2.54-2.94) for vessels between 40 and 100μm. Velocity data for small vessels (<30μm) had an exponent of 2.56 (2.08-3.04). The large confidence interval for velocity precludes rejecting an exponent of 3, and arises from the variance of blood velocity measurements and the relatively small sample size. However the velocity data are consistent with the model in that the smaller size and slower velocity vessels had smaller fitting exponents.
This model may allow better understanding of the relation between the changes in vascular branching with diseases such as diabetes which effect the local viscosity of blood (which includes both hematological and other local factors, such as vascular walls) in the living human retina.
This is an abstract that was submitted for the 2017 ARVO Annual Meeting, held in Baltimore, MD, May 7-11, 2017.
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