Three-dimensional Neuroretinal Rim Thickness and Visual Fields in Glaucoma: A Broken-stick Model

Abstract

I. Analysis of the structure-function relationship in the linear scale with the broken-stick model We also applied the broken-stick model to the VF data and the MDB structural parameters both in the linear scale. When the VF TD data was transformed to the linear scale, the observation that "the structurefunction relationship displayed a plateau, with neuroretinal rim thickness values being unrelated to VF values" was not readily visualized in some of quadrants/ sectors (Figure S1). Further statistical evaluation shows that the slopes above the tipping point are not significantly different from 0, as shown in Table S1. We added in a model-agnostic lowess smoothing curve to compare with the fitted broken-stick model on the data of unlogged VF TD against the MDB. The broken stick model fit didn't match with the lowess fit for the Temporal-Inferior region and the Inferior quadrant. Furthermore, the breakpoint didn't exist for the nasal and temporal quadrants and the TS and NI octants, as shown in the table below where the pvalues are greater than 0.05 (Table S1). For the breakpoints which did exist (i.e. the global value, superior and inferior quadrants, TI and NS sectors), the fitted breakpoints in the linear scale were different for the breakpoints obtained in the logarithmic fits and not encompassed by their respective confidence intervals from the logarithmic scale. As a final remark, indeed the question of whether the structure-function relationship between the VF data and the neuroretinal rim data should be established as a logarithmic-linear scale or as a linear-linear scale has been actively studied in the past two decades. 1-4 A variety of models have been employed to describe the structure-function relationship in both the linear and curvilinear regimes. With this rich history of research on this topic behind us, we primarily presented the data by looking at the VF in the logarithmic scale (or in its native "untransformed" form) since it is more clinically utilized. Also, the logarithmic scale helps to distinguish among the moderate and severe glaucomatous patients.

LIMBARE: an Advanced Linear Mixed-effects Breakpoint Analysis with Robust Estimation Method with Applications to Longitudinal Ophthalmic Studies

“…The segmented mixed effects regression model with P breakpoints φ 1 ,…, φ P can be written as where β = ( β 0 , β 1 , β 2 ) t corresponds to the intercept and random effects of x ijk , z ijk respectively, Ψ – ( β 1 , β 1 + ξ 1 ,…, β 1 + ξ 1 + ξ 2 +… + ξ P ) t are the P + 1 slopes (association effects) for each of the segments between P breakpoints. γ i ~ N (0,Σ γ ) and γ ′ ij ~ N (0,Σ γ ′ ) are the random effects for covariate S ij and S′ ijk .ε ijk ~ N (0, σ 2 ) are the independent individual errors. The model will shrink to the standard linear mixed-effects model when no breakpoint is detected (i.e., P = 0).…”

“…An example for its use in ophthalmology is the association between structural and functional measurements in glaucoma. Numerous studies demonstrated varying level of association between structural measurements obtained by OCT, such as average retinal nerve fiber layer (RNFL) thickness, and functional measurements obtained with visual field tests, at different stages of disease severity [1][2][3][4][5][6] .…”

Detecting disease progression in mild, moderate and severe glaucoma

“…Increasingly rapid progression of VF loss can subsequently be encountered as RNFL thinning plateaus. This asymmetric structure-then-function pattern of progression has been termed the ‘broken stick model’; Liu et al [38] found that detectible visual field loss began after the neuroretinal rim reached one-third normal thickness. Thus, a finding of OCT RNFL and/or macular progression should be noted and treated even in the absence of VF change, as such changes often predict future VF progression [39 ▪ ].…”