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1From the Princess Alexandra Eye Pavilion, Edinburgh, Scotland, United Kingdom; the 2Lions Eye Institute, Perth, Western Australia, Australia; the 3Manchester Royal Eye Hospital, Manchester, United Kingdom; and the 4Clinical Research Facility, Western General Hospital, Edinburgh, United Kingdom.
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Abstract |
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METHODS. A group of young, healthy individuals underwent retinal photography and arteriolar and venular branching points were identified. Vessel widths of the vessel trunks and their branches were determined. The relationship between the branching coefficient (BC; quotient of the area of the branch and trunk vessels) and the asymmetry index (AI) of the vessel branches was explored. The result was used to formulate a new BC. To test the new BC, a second group of young, healthy individuals also underwent retinal photography. Arteriolar branching points were identified, and the trunk and branch arteriolar widths were recorded. This "revised" BC was compared against the gold standard of the BC as a constant value (1.28), as well as a theoretical formula for the BC that includes the angle between the two vessel branches.
RESULTS. The BC of arterioles (but not venules) related to the AI (R = 0.275, P = 0.0001; BC arterioles = 0.78 ± 0.63 · AI). In the second group, the mean arteriolar trunk diameter was 15.56 pixels. The linear regression model for the arteriolar BC was superior to the BC constant of 1.28 (mean difference between estimated and calculated trunk vessel width was 2.16 vs. 2.23 pixels, respectively). The model based on the angle between the branch arterioles was the least accurate (3.43 pixels).
CONCLUSIONS. A revised formula has been devised for the arteriolar BC using a linear regression model that incorporates its relationship to the AI. Further studies using this refined formula to calculate the BC are needed to determine whether it improves the ability to detect smaller associations between the retinal vascular network and cardiovascular disease.
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In a young, normotensive population, they calculated a mean BC of 1.28 for arterioles and showed that the revised formulas was superior to the previously used Parr-Hubbard formulas.13 1.28 compares well with a theoretical BC of 1.26, derived from Murray’s law.14 The derivation of this theoretical value stems from the fact that across any vascular network, Murray calculated that:
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We performed a prospective study to ascertain the relationship between the degree of asymmetry at retinal vascular junctions (asymmetry index [AI] = branch 1/branch 2) and the BC.
Because calculations of the AVR, CRAE, and CRVE measure vessel branches 0.5 to 1.0 disc diameters from the optic disc margin, we further explored whether any relationship existed between how far from the optic disc the vessel branch occurred and the BC, as there may be a rationale for lower-order venules and arterioles being more predictive as risk factors for cardiovascular disease than larger vessels as they exhibit greater regulation via local factors, rather than autonomic control of the central retinal artery.15 In addition, we examined the influence of the angle between the branch vessels on the BC. Finally, we compared different formulas based on the BC in their predictive ability to calculate the trunk vessel diameters from the two branch vessel widths, including a technique described by Zamir16 that incorporates the angle between the vessel branches.
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Methods |
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All images were analyzed with a custom-written image-analysis program (written in the MatLab environment; The MathWorks, Natick, MA). Images were converted from color to grayscale and underwent contrast enhancement (contrast-limited, adaptive histogram equalization). Using the original color image, we identified arteriolar and venular branching points in each image. A point either side of the vessel was chosen to create a line perpendicular to the vessel. With microdensitometry, the profile of this created line was fitted to a single Gaussian curve. In addition to the selected line’s intensity profile, four other intensity profiles (2 and 4 pixels either side of the selected line) were automatically created by the computer image analysis program. Each intensity profile created was either rejected or accepted by the operator as being a good fit of the selected vessel’s profile. At least three of the five profiles had to be deemed a good fit before the vessel’s width could be calculated. In addition, the SD of the acceptable vessel intensity profiles had to be equal to or less than 10% of the mean width before the vessel width would be recorded. If it was greater than 10%, then the measurements would be rejected, and the vessel would be remeasured. The vessel width was calculated as the mean width of the intensity profiles at half the height of the intensity profile peak.17 Widths were calculated for both branches and for the trunk vessel of each evaluable branching point in each image. A portion of vessel segment that was fairly uniform in thickness and not obscured by other crossing vessels was chosen. In addition, the distance from the branching point of all vessels measured for each individual vascular junction was made as uniform as possible. Only vessel junctions that were of good photographic image quality were measured.
To determine the degree of eccentricity of each vessel junction from the optic disc, we measured the distance (in pixels) from the branching point to the nearest circumferential edge of the optic disc. In addition, we measured the vertical optic disc (also in pixels), and expressed the distance of the branching point from the edge of the optic disc in units of disc diameters.
To calculate the bifurcation angle (
), a region of interest within the grayscale enhanced digital image was magnified in a separate window, and the angle between the two daughter arterioles was calculated according to the cosine rule.18
This dataset was then used to formulate a new calculation of the BC, based on a step-wise, least-squares linear regression model. Variables that were considered for inclusion in this model were the AI of the retinal vessel widths (AI), the eccentricity (in disc diameters) of the branching point, and the angle () between the branch vessels.
Finally, another group of normal, nonhypertensive individuals with no prior history of cardiovascular disease and aged between 18 and 40 years were recruited. This "testing" group also had mydriatic retinal photography performed. A 50° color photograph of the fundus centered on the optic disc was taken, using a fundus camera (Topcon, Tokyo, Japan) with a digital camera (Nikon, Tokyo, Japan), set at a different resolution (3008 x 1960 pixels). Again, all images were stored in TIFF format. This "testing" group underwent the same image analysis measurements as the first group. The ability of the BC calculated using the linear regression model from the first group to predict trunk vessel arteriolar measurements from the two branch arteriolar vessels in this "testing" group was compared with a constant BC of 1.28 calculated by Knudtson et al.13 In addition, we compared the two predictive formulas given earlier in the article for summarizing retinal vessel widths with a theoretical formula based on work by Zamir,16 which incorporates the angle () between retinal vessel branches to calculate the BC. Zamir calculates that the BC for a symmetrical dichotomous junction should be [2 · (cos + 1)]1/2.
Statistical Analysis
Inspection of the four variables (BC, AI, angle and degree of eccentricity) revealed normal distributions and hence were subject to parametric analysis. To justify using multiple branching point measurements from individual fundi, we compared the within-group (each fundus) variation of the BC with the between-group variation, by using the intraclass correlation coefficient (ICC). Student’s t-test was used to compare means of the variables between the "training" and "testing" groups. Pearson’s correlation coefficient was used for all bivariate correlations. Step-wise linear regression was used to model a formula for the BC as the dependent variable, with the AI, the angle between the branch vessels (), and the degree of eccentricity as the independent variables. The plot of the residuals of the linear regression model were inspected to confirm random scattering of the residual Y values about X = 0 and reasonable homoscedasticity. Nonlinear regression models were evaluated but were an inferior fit to a linear model. Statistical significance was set at P < 0.05.
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Results |
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For the second "testing" group, 22 fundus images of 11 individuals (7 women; mean age, 26 years; range, 21–32) were analyzed. These 22 fundal images provided 86 evaluable arteriolar branching points.
Table 1 shows the mean and 95% CI for both the arterioles and venules in the "training" group of the BC, AI, angle at vessel bifurcation, and degree of eccentricity. In addition, it shows the mean and 95% CI for the arterioles of the "testing" group.
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) between the two arteriolar vessel branches (R = 0.03, P = 0.75) was associated with the arteriolar BC.
Venular Branching Points.
Neither the AI, the degree of eccentricity, nor the angle () was associated with the BC for the venular junctions (R = 0.03, P = 0.76; R = 0.05, P = 0.63; R = –0.59, P = 0.58, respectively).
Testing Group
Because there was no association between the venular BC and any of the variables from the "training" group, only arteriolar vascular junctions were analyzed in the "testing" group. As in the "training" group, the only parameter associated with the BC was the AI (R = 0.22, P = 0.041).
The predictive ability of the regression model for the BC formed from the "training" group to determine the trunk diameter from the two branch vessel diameters was compared with the constant BC of Knudtson et al.13 (1.28), as well as the theoretical formula of the BC by Zamir.16 This predictive ability of the three models for the BC was compared by using a least-squares technique.
The mean difference between the calculated and measured arteriolar widths for the three different formulas to predict the trunk diameter from the two branch diameters is summarized in Table 2 below (mean arteriolar trunk width = 15.56 pixels).
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Discussion |
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This relationship of the BC to the AI was also observed by Parr and Spears,5 when they developed their empiric formula to calculate the trunk arteriole from retinal branch arterioles. However, rather than try to alter the BC with this observation, they developed a new formula that required a conversion of measured units into micrometers (necessary because of the presence of a constant within their formulated equation). Knudtson et al.13 pointed out the limitations of this approach, and the distinct advantages of using a dimensionless parameter such as the BC. However, we believe that rather than describe the BC as a constant value, relating it in terms of the AI may improve its accuracy in predicting arteriolar trunk vessel widths from the respective branches. This approach retains the advantages of having a dimensionless BC, as the AI is a ratio and therefore remains dimensionless. In addition we deliberately used a different camera set-up, degree of field of view, and resolution of retinal photographs between both the "training" group that was used to derive the linear model and the "testing" group used to test the formulated model. The calculation of the BC should be independent of all these factors. A relationship between the AI and the BC should also be expected from theoretical formulas, as the optimum BC of 1.26 derives from the assumption of a symmetrical, dichotomous branching.
Although the improvement in predictability of the linear regression model in determining trunk arteriolar widths compared with the technique by Knudtson et al.13 was modest, even a small improvement may have significant advantages in the ability to detect cardiovascular associations with retinal vascular geometrical measurements such as the AVR in larger epidemiologic studies. Although the difference between the linear regression model and the technique by Knudtson et al. was not statistically significant (P = 0.287), power calculations (
= 0.05, 1 – ß = 0.8) estimate a sample size of approximately 11,000 to be able to detect this difference as significant. The linear regression model developed in this study improves on the predictability of trunk arteriolar widths by approximately 3% when compared with the method described by Knudtson et al., who showed that their revised calculations for the AVR led to tighter CIs for cardiovascular associations, when compared with the original Parr-Hubbard formula. Further large epidemiologic studies are needed to determine whether this revised BC model is able to reveal smaller associations between retinal vascular changes and cardiovascular disease.
We chose a much larger field of view than either Knudtson et al.13 or Parr and Spears,5 6 because we also wished to explore whether the degree of eccentricity of the retinal junctions influenced the calculated BC. Although there was a significant difference between the "training" group and the "testing" group in degree of eccentricity of the arteriolar junctions, no relationship between the degree of eccentricity and BC was found. Thus, we can conclude that the calculated BC from peripheral arteriolar junctions is no different from more central arteriolar junctions. In addition, we performed a subanalysis on only the larger arterioles (>15 pixels) to confirm that the appropriateness of the linear regression model over a constant BC of 1.28 persists in this group. Thus, this model appears valid in calculating the AVR using only the six largest arterioles and venules within 0.5 to 1.0 disc diameters of the edge of the optic disc, as recommended by Knudtson et al.
Of the three BC-based formulas tested in this study, the principle based on the theoretical formula by Zamir,16 using the angle between the two branch vessels, was the least accurate. We found no association between the angle between the two branches and the BC for either arterioles or venules. Zamir’s formula has been proposed as an optimal system of vessel branching designed to minimize drag in a vascular network. Our findings suggest that the optimal principle of minimum work (minimum energy requirements) across the retinal vascular network as devised by Murray plays a more prominent role in a healthy, young, normotensive population.
It is interesting that there was no relationship between the BC of venules and the AI. This finding suggests that venules may exhibit different optimal principles than arterioles, possibly due to their different physiological function, although this is not entirely clear. Our mean venular BC (1.22) was also different from the mean BC calculated by Knudtson et al. (1.11).13 The lack of association between the BC and AI for venules also emphasizes that there is no intrinsic mathematical relationship between these entities that makes association inevitable, thus providing further evidence of the significance of the arteriolar association.
Based on our findings, we would recommend calculating the CRAE using the same methodology as Knudtson et al.,13 except incorporating our linear regression model for the BC, rather than a constant. For the CRVE, as no relationship exists between the AI and the BC, we recommend using a constant BC as before.
In determining the degree of eccentricity of the retinal vascular junctions, this study makes the assumption that the average optic disc size between individuals approximates to the same to measure in terms of optic disc diameters. It also assumes that in peripheral vascular junctions, the greater degree of ocular curvature does not unduly influence the relative widths between the vessel trunks and branches. In addition, a limitation of the study is the assumption that all the individuals photographed were normotensive, as no blood pressure measurements were taken. It is possible that some of the subjects had undiagnosed hypertension or other cardiovascular disease.
In conclusion, we describe a new equation to calculate the BC of retinal arterioles in terms of the asymmetry of the two vessel branches, based on an empirically derived linear regression model. In a dataset of retinal photographs from a young, healthy, normotensive population that was not used to derive the new formula, the model appeared superior to using a constant value for the BC. This further revision to summarizing retinal arteriolar widths may improve our ability to detect smaller associations between cardiovascular disease and retinal vascular network geometry.
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Footnotes |
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Disclosure: N. Patton, None; T. Aslam, None; T. MacGillivray, None; B. Dhillon, None; I. Constable, None
The publication costs of this article were defrayed in part by page charge payment. This article must therefore be marked "advertisement" in accordance with 18 U.S.C. 
1734 solely to indicate this fact.
Corresponding author: Niall Patton, Princess Alexandra Eye Pavilion, Chalmers Street, Edinburgh EH3 9HA, UK; niallpatton{at}hotmail.com.
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