### Purpose

### Design

### Participants

### Methods

### Main Outcome Measures

### Results

### Conclusions

## Keywords

#### Abbreviations and Acronyms:

AREDS (Age-Related Eye Disease Study), GA (geographic atrophy), RPE (retinal pigment epithelium)^{1}

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- 1.Noncircular margins: Although sometimes starting as circular lesions, GA foci can grow into a variety of complex shapes.
- 2.Multifocality: GA lesions often comprise multiple foci.
- 3.Variations in global lesion growth rates: The global (area) rate at which GA lesions enlarge varies among eyes, with some lesions remaining relatively stable and others expanding rapidly.Although this variability is a topic of current investigation, lesion geometry
^{12}^{13}^{,}and choriocapillaris impairment^{14}^{15}^{, }^{16}^{, }^{17}^{, }have been implicated. - 4.Variations in local lesion growth rates: GA lesions do not grow uniformly along their margins.
^{10}^{,}^{12}^{,}Currently, it is not known what underlies anisotropic growth. - 5.Lesion merging: GA lesions often exhibit merging between different lesion foci (i.e., interfoci merging) and different segments of the same focus (i.e., intrafocus merging).

^{15}

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## Methods

Global and local growth rates | Global growth rates are single measurements describing how the entirety of a GA lesion expands. Local growth rates are collections of measurements, with each measurement describing how a segment (or point) on the baseline margin expands. In this article, we focus on global growth rates. |

Area-type and length-type growth rates | Area-type growth rates describe GA growth in units of area per time, whereas length-type growth rates describe GA growth in units of distance per time. In this article, we primarily focus on global length-type growth rates, which we denote by Λ. We use the notation $\stackrel{\u02c6}{\text{\Lambda}}$ to denote global length-type growth rate metrics, which we view as estimators of Λ. |

Atrophy-front growth model | The atrophy-front growth model describes GA growth as a margin-mediated expansion wherein lesion growth can be described by a growth field, v(x,t), where x is the fundus position and t is time. This model is stated mathematically in Appendix 1. |

Growth field | The growth field, v(x,t), is the local, geometry-independent rate of GA margin enlargement and has units of distance per time. Physiologically, the growth field captures aspects of the chorioretinal milieu that influence GA growth. Larger values of v correspond to faster lesion growths. |

Time-invariant vs. time-varying growth fields | Time-invariant growth fields do not change in time. In our atrophy-front growth model, this corresponds to v(x,t) = v(x); that is, the growth field v is independent of time. A growth field that is not time invariant is termed time varying. For simplicity, in this article, we primarily restrict our attention to time-invariant growth fields. |

Isotropic vs. anisotropic growth fields | Isotropic growth fields do not change with spatial position. In the atrophy-front growth model, this corresponds to v(x,t) = v(t); that is, v is independent of position. A growth field that is not isotropic is termed anisotropic. |

### Geographic Atrophy Growth Rate Measurement as a Problem of Estimating Growth Fields

*t*

_{b}, and a follow-up time,

*t*

_{f}=

*t*

_{b}+ Δ

*t*, where Δ

*t*is the intervisit time, the global area-type growth rate can be defined unambiguously as (

*A*(

*t*

_{f}) –

*A*(

*t*

_{b})) / Δ

*t*, where

*A*(

*t*) denotes the GA area at time

*t*. In contrast, no single notion of length-type growth rate exists. For example, depending on the application, it might be sensible to construct a length-type growth rate using closest extrinsic (e.g., Euclidean, that is, straight-line) distances,

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*v*(

*x*,

*t*) (Appendix 1). The growth field can be understood as a mathematical abstraction that encodes the state of the chorioretinal milieu, that is, the intracellular and extracellular environment of the choroid and retina, which includes RPE and photoreceptor integrity, as well as the choriocapillaris blood flow. Chorioretinal conditions leading to faster growth correspond to higher values of

*v*, and vice versa. In the wildfire analogy of GA growth, the growth field corresponds to the environmental conditions that, through interaction with the fire front, help to determine the fire’s spread.

^{10}

where

*A*(

*t*) is the lesion area at time

*t*. Note that the effective radius metric is simply a 1 / $\sqrt{\pi}$ scaled version of the square-root-of-area metric.

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^{,}

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*v*= Λ for isotropic, time-invariant growth fields; that is, under such conditions, ${\stackrel{\u02c6}{\text{\Lambda}}}_{ER}$ is a perfect estimator of length-type growth rate. For more complex lesion geometries, ${\stackrel{\u02c6}{\text{\Lambda}}}_{ER}$ becomes a worse estimator of Λ. For example, for lesion growths that are small relative to the baseline margin perimeter

*P*(

*t*

_{b}) and baseline area

*A*(

*t*

_{b}), it can be shown (Appendix 2) that Λ and ${\stackrel{\u02c6}{\text{\Lambda}}}_{ER}$ are related by:

where

*G*(

*t*) is the GA lesion geometry at time

*t*and circ(·) is the circularity operator, defined as: circ(

*G*(

*t*)) ≡ 4π

*A*(

*t*) /

*P*

^{2}(

*t*), which takes values between 0 and 1, inclusive. Note that this is the same definition of circularity as used by Domalpally et al

^{14}

*n*equal-radii circular foci undergoing isotropic growth:

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### Generating Datasets for Numerical Evaluation of Metric Accuracy

#### Simulating Lesion Growths and Computing Metric Accuracies

*t*. This propagation yields a sequence of GA margins, the last of which corresponds to the follow-up GA margin. Using the sequence of GA growth margins in conjunction with the simulated growth field, the ground truth global length-type growth rate Λ can be computed (via equation SI-3 of Appendix 1). Furthermore, mimicking the clinical situation, growth rate estimates $\stackrel{\u02c6}{\text{\Lambda}}$ can be computed using only the baseline and follow-up GA margins, via equation 1 (${\stackrel{\u02c6}{\text{\Lambda}}}_{ER}$) or equation 4 (${\stackrel{\u02c6}{\text{\Lambda}}}_{PA}$). Metric accuracy then is assessed by comparing the growth rate estimates $\stackrel{\u02c6}{\text{\Lambda}}$ with the ground truth growth rates Λ. Details of the numerical implementations of these steps are provided in Appendix 4. Note that, for simplicity, in all of our analyses we restrict our attention to time-invariant growth fields.

#### Simulations with Simplified Lesion Geometries and Growth Fields

*t*= 1 year follow-up intervals to have equal baseline areas of

*A*(

*t*

_{b}) = 6 mm

^{2}and to have equal length-type growth rates of Λ = 0.1 mm/year, the latter of which approximately corresponds to the reported mean of the 1-year perimeter adjusted growth rates of the AREDS dataset.

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#### Simulations with Observed Lesion Geometries and Random Growth Fields

^{19}

_{Λ}) and standard deviation (σ

_{Λ}) of the simulated global growth rates to those of the 1-year perimeter-adjusted growth rates of the AREDS dataset,

^{11}

_{Λ}≈ σ

_{Λ}≈ 0.1 mm/year. As described in Appendix 6, to augment our dataset, for each of the 38 baseline lesions, 100 random fields were generated, resulting in a total of 3800 baseline and follow-up margin pairs with which to evaluate metric accuracy for a given follow-up interval (Δ

*t*). Finally, we assessed metric accuracy at follow-up time intervals of Δ

*t*= 1, 3, and 5 years.

## Results

*t*= 1, 2, and 5 years). The statistics of these distributions match fairly well those of the perimeter-adjusted growth rates reported for the AREDS dataset, although the standard deviations are slightly smaller (approximately 0.07 mm/year for our simulations and approximately 0.10 for the AREDS dataset), presumably a consequence of the averaging involved in computing Λ (equation SI-3, Appendix 1). Figure 5 shows the accuracies of ${\stackrel{\u02c6}{\text{\Lambda}}}_{ER}$ and ${\stackrel{\u02c6}{\text{\Lambda}}}_{PA}$ evaluated on this semisimulated growth data. The perimeter-adjusted metric shows a marked improvement in accuracy compared with the effective radius metric (approximately 20 times improvement in average accuracy for the 1-year follow-up data). Notably, for nearly all simulations, the perimeter-adjusted metric has an absolute error of less than 0.05 mm/year. Nevertheless, for both metrics, errors tend to increase for increasing growth rates. Figure 6 shows representative growth simulation data, and Figure 7 shows plots of the mean per-eye absolute estimation errors as a function of several baseline lesion characteristics.

## Discussion

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^{, }which often use smaller convenience samples. In any case, because the square-root and perimeter-adjusted metrics are comparable in terms of computational difficulty (equation 1 vs. equation 4), the barrier to using the perimeter-adjusted metric is somewhat lowered.

*t*).

^{10}

^{,}

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^{,}

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^{19}

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^{, }However, as mentioned previously, we do expect that modeling newly appearing foci will decrease the reported accuracies of the effective radius growth metric, particularly for the 5-year follow-up data.

## Conclusions

## Acknowledgments

## Supplementary Data

- Appendix 1

- Appendix 2

- Appendix 3

- Appendix 4

- Appendix 5

- Appendix 6

- Appendix 7

- Appendix 8

- Appendix 9

- Appendix 10

- Figure-S1

- Figure-S2

- Figure-S3

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## Article info

### Publication history

### Footnotes

*Supplemental material available at* www.ophthalmologyscience.org*.*

Disclosure(s):

All authors have completed and submitted the ICMJE disclosures form.

The author(s) have made the following disclosure(s): E.M.M.: Patent – VISTA-OCTA

N.K.W.: Consultant – Allegro, Regeneron, Apellis, Nidek, Stealth, Genentech, Astellas, Boehringer Ingelheim, Topcon; Lecturer – Nidek; Data Safety Monitoring Board or Advisory Board – Carl Zeiss Meditec; Equity owner – Gyroscope Therapeutics, Ocudyne; Officeholder – Gyroscope Therapeutics; Nonfinancial support – Optovue, Heidelberg Engineering, Nidek

G.G.: Financial support – Carl Zeiss Meditec

P.J.R.: Consultant – Annexon, Apellis, Bayer, Boehringer-Ingelheim, Carl Zeiss Meditec, Chengdu Kanghong Biotech, Ocunexus, Ocudyne, Regeneron, Unity Biotechnology; Financial support – Gyroscope Therapeutics, Stealth BioTherapeutics, Carl Zeiss Meditec, Iveric bio; Data Safety Monitoring Board or Advisory Board – Chengdu Kanghong Biotech; Equity owner – Apellis, Ocudyne, Valitor, Verana Health

J.G.F.: Financial support and Patent – Topcon; Royalties and Equity owner – Optovue; Lecturer – Bascom Palmer Eye Institute; Patent – VISTA-OCTA

Supported by the National Institutes of Health, Bethesda, Maryland (grant no.: R01-EY011289-35); Retina Research Foundation; Beckman-Argyros Award in Vision Research; Champalimaud Vision Award; Massachusetts Lions Eye Research Fund; Macula Vision Research Foundation; and Research to Prevent Blindness, Inc, New York, New York.

HUMAN SUBJECTS: Human subjects were included in this study. University of Miami Miller School of Medicine review board approved the study. All research complied with the Health Insurance Portability and Accountability Act (HIPAA) of 1996 and adhered to the tenets of the Declaration of Helsinki. All participants provided informed consent.

No animal subjects were included in this study.

Author Contributions:

Conception and design: Moult, Shi, Wang, Chen, Waheed, Gregori, Rosenfeld, Fujimoto

Analysis and interpretation: Moult, Shi, Wang, Chen, Waheed, Gregori, Rosenfeld, Fujimoto

Data collection: Moult, Shi, Wang, Chen, Waheed, Gregori, Rosenfeld, Fujimoto

Obtained funding: Fujimoto

Overall responsibility: Moult, Shi, Wang, Chen, Waheed, Gregori, Rosenfeld, Fujimoto

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