Comparing Accuracies of Length-Type Geographic Atrophy Growth Rate Metrics Using Atrophy-Front Growth Modeling

Abstract

Purpose: To compare the accuracies of the previously proposed square-root-transformed and perimeter-adjusted metrics for estimating length-type geographic atrophy (GA) growth rates.

Design: Cross-sectional and simulation-based study.

Participants: Thirty-eight eyes with GA from 27 patients.

Methods: We used a previously developed atrophy-front growth model to provide analytical and numerical evaluations of the square-root-transformed and perimeter-adjusted growth rate metrics on simulated and semisimulated GA growth data.

Main outcome measures: Comparison of the accuracies of the square-root-transformed and perimeter-adjusted metrics on simulated and semisimulated GA growth data.

Results: Analytical and numerical evaluations showed that the accuracy of the perimeter-adjusted metric is affected minimally by baseline lesion area, focality, and circularity over a wide range of GA growth rates. Average absolute errors of the perimeter-adjusted metric were approximately 20 times lower than those of the square-root-transformed metrics, per evaluation on a semisimulated dataset with growth rate characteristics matching clinically observed data.

Conclusions: Length-type growth rates have an intuitive, biophysical interpretation that is independent of lesion geometry, which supports their use in clinical trials of GA therapeutics. Taken in the context of prior studies, our analyses suggest that length-type GA growth rates should be measured using the perimeter-adjusted metric, rather than square-root-transformed metrics.

Keywords: AREDS, Age-Related Eye Disease Study; GA; GA, geographic atrophy; Geographic atrophy; Growth modelling; Growth rate; RPE, retinal pigment epithelium.

Figure 1

Illustration of the method for assessing growth rate metric accuracy. The geographic atrophy (GA) growth model (equation SI-2, Appendix 1) uses (A) baseline lesion data, either simulated or observed, and (B) a simulated growth field to produce (C) a sequence of GA margins. This sequence of GA margins, a portion of which is enlarged in (C.1), comprises simulated margins at different time points equally spaced between the baseline and follow-up visits. From this sequence of GA margins, (D) the baseline and follow-up margins can be extracted, mimicking clinically available GA growth data. Using the sequence of GA margins and the growth field data, the ground truth global length-type growth rate, Λ, can be computed (equation SI-3, Appendix 1). A growth metric $\widehat{\text{Λ}}$ uses the baseline and follow-up GA margins (D) to produce a growth rate measurement that can then be compared against the ground truth Λ.

Figure 2

Simple simulated lesion geometries and growth fields constructed to highlight key characteristics of global length-type growth rate metrics. Configurations 1 through 8 correspond to different lesion geometries and growth fields. All lesions were constructed to have baseline areas of 6 mm² and all growth fields were constructed to have a length-type growth rate of Λ = 0.1 mm/year, as measured over a 1-year follow-up time. Colors correspond to growth field values (see color bar). Note that all configurations except for configurations 4 and 7 have isotropic growth fields. Mathematical details of these configurations are provided in Appendix 5.

Figure 3

Accuracy of global length-type growth rate metrics as evaluated on the simple lesion geometries and growth fields of Figure 2. Note that although Figure 2 shows growth fields for a fixed growth rate of Λ = 0.1 mm/year, as described in the text, the growth fields were scaled to assess metric accuracy as a function of the growth rate Λ. The thick red line corresponds to a range of ± 6 μm, corresponding to the half-width of the 12-μm pixel size used for these simulations (Appendix 5). The solid and dashed lines correspond to the effective radius ${\widehat{\text{Λ}}}_{ER}$ and perimeter-adjusted ${\widehat{\text{Λ}}}_{PA}$ growth rates, respectively. For all plots, the intervisit time, Δt, was 1 year.

Figure 4

Lesion and growth field characteristics for the semisimulated metric analysis. A–E, Characteristics of the baseline lesion data for the 38 eyes with geographic atrophy. Gini-weighted focality, which adjusts for situations in which the foci have unequal perimeters, is described in Appendix 9. F, Probability distribution function (PDF)-weighted histograms of the ground truth global length-type growth rate Λ for the Δt = 1-, 3-, and 5-year simulations. Distribution means (μ_Λ) and standard deviations (σ_Λ) also are provided. As expected, the distributions, means, and standard deviations are very similar for the Δt = 1-, 3-, and 5-year simulations. For all boxplots, whiskers extend beyond the box edges to a maximum of ×1.5 the interquartile range; measurements beyond this range are indicated by crosses. Histogram bins were chosen using the Freedman-Diaconis rule.

Figure 5

Summary of effective radius and perimeter-adjusted metric accuracies as evaluated on the semisimulated geographic atrophy (GA) dataset. A, B, Density scatterplots of the estimated growth rates versus the ground truth growth rates, evaluated on 1-year intervisit times. For all panels, each marker represents a single growth simulation (i.e., 3800 markers per panel). Colors correspond to the relative density of measurement points, and marker sizes correspond to the Gini-weighted focality (see legend, far right; Gini-weighted focality is described in Appendix 9). C, D, Density scatterplots of the absolute estimation error, e, evaluated on 1-year intervisit times. The mean (μ_e$)$ and standard deviation (σ_e) of the absolute estimation errors are listed. E–H, I–L, Analogous panels, but corresponding to 2-year (E–H) and 5-year (I–L) intervisit times. The labels S1 through S12 correspond to representative cases that are explored in Figure 6. In particular, for each intervisit time, simulations generating absolute perimeter adjusted errors at the twenty-fifth (S1, S5, S9), fiftieth (S2, S6, S10), seventy-fifth (S3, S7, S11), and one-hundredth (S4, S8, S12) percentiles were selected.

Figure 6

Representative semisimulated geographic atrophy (GA) growth data, with panels S1 through S12 corresponding to the labels of Figure 5. Each column corresponds to simulations with absolute perimeter adjusted errors, e_PA, of the specified percentile, and each row corresponds to simulations of the specified intervisit time.

Figure 7

Scatterplots showing the mean per-eye absolute estimation errors as a function of baseline lesion descriptors for the semisimulated analyses. For all panels, each marker corresponds to the mean absolute error $\left|\widehat{\text{Λ}}-\text{ Λ}\right|$ for a single baseline lesion, where the average is computed over the 100 simulated random fields applied to that lesion (i.e., 38 green and 38 pink markers per panel, corresponding to the effective radius and perimeter-adjusted measurements, respectively). In columns 3 through 6, the dashed lines correspond to the errors predicted by equations 2 and 3. Gini-weighted focality is described in Appendix 9.