Analysis of Arterial Branching in the Retina

Abstract

Abstract: : Purpose. To determine if arterial branch angles and relative branch blood vessel (bv) diameters in the retina fit mathematical relations derived from Poiseulle’s law (Murray’s law). Methods: Digital images of single arteries in the retina were obtained by lasering bv at entrance onto the retina, releasing fluorescein from liposomes. Data were 25 well–resolved images of complete artery fills. Segment length between branches of retinal arteries; trunk and daughter branch diameters; and branch angles of pairs at branch points were measured on the images using image analysis software. Each measurement of the bv in the images was repeated at three separate sessions for analysis. Results: Murray’s law is r₀^x = r₁^x + r₂^x, where r₀ = the radius of the mother bv, r₁ = larger branch radius and r₂ = smaller branch radius, with the value 3 being the theoretical estimate and values close to x = 3 reported for different tissues (range 2.7–3.0). Murray’s law shows that the cube of the mother artery radius equals the sum of the cubes of the radii of the daughter arteries. The Murray’s law exponent we found was 2.8335 with 95% CI of 2.75–2.95. We analyzed lengths of successive branches from the main arterial trunk (1^st order branches) and the lengths of branches that arose from 1^st order branches (2^nd order branches). The relationship of the 1^st order branch lengths to the branch number along the main trunk was linear with negative slope. When 2^nd order branches were added, a quadratic regression was found. Vascular lumen width was measured in pixels. We used a method to compare branch diameters with mother arteries with diameters expressed in non–dimensional form. The diameters of larger (d₁) and smaller (d₂) branches from a mother artery (d₀) are expressed in terms of ratios: d₁/d₀ = 1/[1 + (d₂/d₁)³]^1/3 d₂/d₀ = (d₂/d₁)/[1 + (d₂/d₁)³]^1/3 . Goodness of fit of these calculated values to our observed data was statistically significant. Conclusions: The value around 3 we found for Murray’s law exponent indicates that bv grow such that a minimum value is maintained for work done in moving blood and against local shear forces from blood flow on the walls of arterial bifurcations. The relationship of large and small branches relative to the mother branch in a retinal artery bifurcation obeys the functions predicted if branch sizes and angles give minimum resistance to flow. The order of branch lengths of first order branches from primary retinal artery trunks is a linear function of distance along the main trunk. Second order branches depart from this relationship and represent a transition to a distributive vascular network from a conveying network.

Keywords: retina • imaging/image analysis: non-clinical • blood supply