purpose. To describe a revised formula for the estimation of retinal trunk arteriole widths from their respective arteriolar branch widths that improves the summarizing of retinal arteriolar diameters.

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.

^{ 1 }Over the past decade, there has been great interest in the use of image-analysis techniques to describe vascular parameters of the retinal vascular branching network that can then be related to various systemic vascular conditions such as hypertension (Hubbard LD, et al.

*IOVS*1992;33:ARVO Abstract 804)

^{ 2 }

^{ 3 }

^{ 4 }Derived initially from the work of Parr and Spears

^{ 5 }

^{ 6 }and Hubbard et al.,

^{ 7 }summarizing measures of the retinal arteriolar diameters (central retinal arteriolar equivalent [CRAE]) and venular diameters (central retinal venular equivalent [CRVE]) and their quotient (arteriovenous ratio [AVR]) have been of use in large epidemiologic studies.

^{ 8 }

^{ 9 }

^{ 10 }

^{ 11 }

^{ 12 }The techniques used to combine individual retinal vessel widths into an equivalent central retinal vessel width are based on both theoretical and empiric formulas. Parr and Hubbard derived formulas for the CRAE and CRVE, respectively, by examining a prospective group of retinal images with branching points, calculating the relationship between individual trunk vessels and their respective branch vessels and using a root mean square deviation (RMSD) model that best fit the observed data. The resultant formula for the AVR was of great use for a large number of epidemiologic studies. However, Knudtson et al.

^{ 13 }pointed out the drawbacks of these formulas including the fact that they were not completely independent of the number of vessels measured and that, because they contained constants within the equations, they were dependent on the units with which the vessels were measured. Hence, Knudtson et al. developed new formulas for calculating the AVR, based on the concept of a BC, where:

^{ 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:

^{ 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 Zamir

^{ 16 }that incorporates the angle between the vessel branches.

^{ 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.

^{ 18 }

^{ 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 (ω) between retinal vessel branches to calculate the BC. Zamir calculates that the BC for a symmetrical dichotomous junction should be [2 · (cos ω + 1)]

^{1/2}.

*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 (ω), 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.

*R*= 0.275,

*P*= 0.002). A stepwise linear regression model was constructed with the BC as the dependent variable and the AI as the independent variable (Fig. 1 : BC = 0.78 + 0.63 · AI;

*P*= 0.002,

*N*= 125.

*R*= 0.022,

*P*= 0.81) nor the angle (ω) between the two arteriolar vessel branches (

*R*= 0.03,

*P*= 0.75) was associated with the arteriolar BC.

*R*= 0.03,

*P*= 0.76;

*R*= 0.05,

*P*= 0.63;

*R*= −0.59,

*P*= 0.58, respectively).

*R*= 0.22,

*P*= 0.041).

^{ 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.

^{ 13 }and the theoretical optimum value for a symmetrical dichotomous vessel junction, a better model of fit for the observed values of the BC can be achieved by relating it to the degree of asymmetry of the two vessel branches. This linear regression model equation was then tested on a different, independent dataset that was not used to derive the calculated formula, and we found the formula to be slightly superior, both to the formula using a fixed BC of 1.28 and to the theoretical formula devised by Zamir,

^{ 16 }which relates the BC to the angle between the two vessel branches.

^{ 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.

^{ 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 (α = 0.05, 1 − β = 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.

^{ 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.

^{ 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.

^{ 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.

^{ 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.

Training Group | Testing Group | t-Test (For Arteriolar Branches) | |||
---|---|---|---|---|---|

Arteriolar (n = 125) | Venular (n = 90) | Arteriolar Only (n = 86) | |||

Branching coefficient | 1.25 (1.18–1.32) | 1.22 (1.14–1.3) | 1.26 (1.17–1.34) | P = 0.86 | |

Asymmetry index | 0.74 (0.72–0.78) | 0.75 (0.71–0.78) | 0.78 (0.75–0.81) | P = 0.18 | |

Angle (deg) | 79 (76.2–82.5) | 75.8 (72.5–79.1) | 81.2 (77.9–84.5) | P = 0.46 | |

Degree of eccentricity (disc diameters) | 2.04 (1.9–2.19) | 2.4 (2.18–2.62) | 1.58 (1.42–1.75) | P = 0.0001^{*} |

*n*= Number of branching points.

**Figure 1.**

**Figure 1.**

Mean Difference between Calculated and Measured Widths | Range of Difference between Calculated and Measured Widths | Standard Deviation of Differences | |
---|---|---|---|

Linear regression model (BC = 0.78 + 0.63 × AI) | ± 2.16 | ± 0.01–10.58 | ± 1.83 |

Constant BC = 1 1.28 | ± 2.23 | ± 0.06–11.76 | ± 1.90 |

BC = (2×(cosω + 1)^{1/2} | 3.43 | ± 0.08–14.34 | ± 2.79 |