**Purpose.**:
We generated a model of eye growth and tested it against an eye known to develop abnormally, one with a history of retinopathy of prematurity (ROP).

**Methods.**:
We reviewed extant magnetic resonance images (MRIs) from term and preterm-born patients for suitable images (*n* = 129). We binned subjects for analysis based upon postmenstrual age at birth (in weeks) and ROP history (“Term” ≥ 37, “Premature” ≤ 32 with no ROP, “ROP” ≤ 32 with ROP). We measured the axial positions and curvatures of the cornea, anterior and posterior lens, and inner retinal surface. We fit anterior chamber depth (ACD), posterior segment depth (PSD), axial length (AL), and corneal and lenticular curvatures with logistic growth curves that we then evaluated for significant differences. We also measured the length of rays from the centroid to the surface of the eye at 5° intervals, and described the length versus age relationship of each ray, *L*_{ray}(*x*), using the same logistic growth curve. We determined the rate of ray elongation, *E*_{ray}(*x*), from *L*_{ray} *dy*/*dx*. Then, we estimated the scleral growth that accounted for *E*_{ray}(*x*), *G*(*x*), at every age and position.

**Results.**:
Relative to Term, development of ACD, PSD, AL, and corneal and lenticular curvatures was delayed in ROP eyes, but not Premature eyes. In Term infants, *G*(*x*) was fast and predominantly equatorial; in age-matched ROP eyes, maximal *G*(*x*) was offset by approximately 90°.

**Conclusions.**:
We produced a model of normal eye growth in term-born subjects. Relative to normal, the ROP eye is characterized by delayed, abnormal growth.

^{1}a process influenced by intrinsic developmental signals and by visual experience.

^{2}However, a fundamental question in ophthalmology is specifically where and when the eye grows to transform from its neonatal to its adult shape. Having an accurate, mathematical model of the normal growth of the eye is essential to understanding the mechanisms that underpin typical and atypical ocular development. This is pertinent to the development of common conditions, such as myopia, and less common conditions, such as retinopathy of prematurity (ROP).

^{3,4}Paradoxically, myopia is typically associated with longer-than-average eyes, but, in ROP myopia, the eye frequently is short.

^{3–14}To be small and myopic, the anterior segment of the ROP eye must be of substantially higher-than-normal dioptric power. Relative to myopic adults born full-term, adults with the same degree of myopia and a history of ROP have eyes with increased corneal curvature (CC), increased lens thickness (LT), and shallower anterior chamber depth (ACD); among these features, the increased CC may be most responsible for the myopia.

^{12}

^{4,8,15}For instance, development of the cornea is arrested in the eye with a history of ROP, and so it may be that the eye simply does not lose optical power fast enough to emmetropize.

^{4,16}With respect to the posterior segment, the retina is one controller of eye growth (and, thus, refractive development).

^{17,18}Evidence from simian eyes

^{19,20}strongly indicates that it is the peripheral retina, in particular, which is most important to the process of emmetropization (although the evidence in humans is weaker

^{21}). Notably, the peripheral retinal vasculature is abnormal (or even absent) in eyes with ROP.

^{22}Severity of antecedent ROP is correlated positively with incidence and magnitude of refractive error

^{23–25}and with delays in eye growth, anterior and posterior.

^{16}Cycloplegic refractions indicate that the myopic shift in developing ROP eyes happens early and then persists.

^{23,24,26,27}Laser- (or cryo-) therapy, per se, does not seem to influence refractive outcome.

^{26–28}Other biometric measurements of the eye, such as axial length (AL), CC, LT, anterior segment length (ASL) to posterior segment depth (PSD) ratio, and so forth, also have been obtained from various ROP populations.

^{9,10,12,13,16,29–32}However, to our knowledge, no undertaking has yet been made to describe or model the development of these ocular components from infancy to adulthood in ROP eyes.

_{2}-weighted head scans displaying images of the eyes. We reviewed only scans meeting certain acquisition requirements, including isometry in the transverse plane and maximum voxel size (2.5 mm in any dimension). From those, we selected only scans showing high-quality images of the eyes (e.g., motion artifacts, wrapping, or low contrast caused us to reject images).

*α*= 0.05) for statistical comparisons performed on this dataset.

*n*= 77) if they were born ≥37 weeks PMA, “Preterm” (

*n*= 23) if they were born ≤32 weeks PMA and had no history of ROP, and “ROP” (

*n*= 31) if they were born ≤32 weeks PMA and had a history of ROP noted in their medical charts. We excluded subjects born at intermediate ages (32–37 weeks PMA,

*n*= 31). Finally, we excluded from the analyses two ROP subjects who had ALs that were the longest observed in any group at any age because ROP eyes are typically shorter than normal and, thus, we felt that their eyes did not represent the typical developmental process in ROP (final

*n*= 129; ROP

*n*= 29). As summarized in the Table, our subjects were sent for MRI mainly because of injury, complications of preterm birth, seizures, or headache. Other reasons included hearing loss, developmental delay, and complications of cleft palate.

**Table**

^{33}The steps in the image analysis procedure are illustrated in Figure 1. In brief: (A) With input from the operator, the software rotated each image so that the plane of the ciliary body was parallel with the horizontal image axis. (B) The program automatically defined the positions of the cornea, lens, and retina by identifying troughs and peaks in the derivative of the intensity profile of a line drawn through the pupil–posterior pole axis of the eye. (C) From the relative positions of these features, we derived ACD (posterior cornea to anterior lens), ASL (anterior cornea to posterior lens), LT (anterior lens to posterior lens), PSD (posterior lens to retina), and AL (anterior cornea to retina). (D) We then identified and segmented edges representing the anterior and posterior cornea, anterior and posterior lens, retina, and sclera. We root mean square (RMS) error fit the corneal and lenticular surfaces with respective circles to generate measurements of curvature. (E) Then, we determined the arithmetic mean position of all pixels in the image of the eye (the centroid), defined by the vitreoretinal and aqueocorneal boundaries, and measured the length of rays, drawn every 5°, from the centroid to the surface. (F) Similarly, we measured the length of rays from the centroid to the margin of the lens.

**Figure 1**

**Figure 1**

*x*was always PMA,

*a*described the magnitude of the total developmental change in the parameter

*f*(

*x*),

*b*was related to the slope of the function,

*c*was the age at which the change in

*f*(

*x*) reached half of

*a*, and

*y*

_{0}represented the initial value of

*f*(

*x*). In the majority of cases, specified in the appropriate location in the Results, we fixed

*y*

_{0}at 0; otherwise, all parameters were free to vary.

*df*) afforded by more curves (i.e., “one curve for all data sets”). With individual eye data fit with each curve, this hypothesis can be tested using the formula where

*SS*

_{1}is the total sum of the squared deviations from individual points to the solitary fit,

*SS*

_{2}is the total sum of the squared deviations to the three respective fits,

*n*is the total number of subjects in the analysis (so 2

*n*would be the number of eyes),

*k*is the number of groups being tested, and

*p*is the number of parameters in Equation 1 that were allowed to freely vary (typically 3, as

*y*

_{0}was usually fixed at 0).

^{34}The main difference between this calculation and a typical ANOVA is that the error calculations are based upon the mean square (MS) difference from the fitted group curve (Equation 1) rather than from the group mean. However, since two eyes contributed by a subject are likely to be more similar than two eyes contributed by different subjects, the within-individual variability needed to be accounted for and, thus, we modified the

*F*ratio as follows: where

*SS*

_{R}is the sum of the squared intraocular (i.e., repeated-measures) differences from the group mean difference to the three respective fits. The inclusion of

*SS*

_{R}in the modified formula (Equation 3), means that the

*F*ratio now includes the ratio of the MS between treatments to the MS subject by treatments (Hays

^{35}formula 13.21.4). We evaluated

*F*with the difference between the number of parameters in the multiple and single curve scenarios as

*df*

_{numerator}(e.g., multiple curve scenario = 3 curves × 3 free parameters = 9; single curve scenario = 3 free parameters;

*df*

_{numerator}= 9 − 3 = 6), and the number of subjects less the number of groups (e.g.,

*df*

_{denominator}=

*n*−

*k*= 129 − 3 = 126). The combined use of Equation 3 in the calculation of

*F*and of subjects instead of eyes in the calculation

*df*

_{denominator}results in an appropriately increased threshold for statistical significance that offsets the putatively decreased variability in the sample inherent in our repeated-measures design.

^{35}Where statistical significance was attained, we concluded that a “different curve for each data set” was appropriate. In those cases, we performed post hoc pairwise comparisons, following the same procedure, to detect which of the three respective curves differed from which others. We made the threshold for statistical significance more stringent (

*α*= 0.01) for these pairwise post hoc tests.

*Δ*

_{normal}) by subtracting the Term group's growth curve from every respective measured value. We tested for significant group differences in the abnormality data by two-factor (Group × Eye), repeated-measures ANOVA. Where a significant Group effect was detected, we evaluated intergroup differences, pairwise, using Tukey's Honestly Significant Difference (HSD) post hoc statistical test (SigmaPlot 11.2; Systat Software, Inc., San Jose, CA, USA).

*L*

_{ray}, as a function of age by fit of Equation 1 (

*L*

_{ray}[

*x*] =

*f*[

*x*]); we fixed

*y*

_{0}at 0 when fitting the eye data, but allowed it to vary freely when fitting the lens data (as noted in the Results, below, LT did not increase during the ages observed in our study). So that the position of the lens could be tracked relative to the eye, we also fit the

*X*and

*Y*offsets of the lens centroid, relative to the eye centroid, using Equation 1 with all parameters free to vary. We next calculated the rate of elongation,

*E*

_{ray}, from the derivative of

*L*

_{ray}:

*L*

_{ray}at 5° intervals, we produced 72 elongation curves, which we denoted by their angle from the axis of the eye. Below, we use

*θ*and

*φ*to reference these 72 spots on the eye or lens. Since

*L*

_{ray}approximates the radius of a circle, the scleral growth that would increase the circumference of the eye at a rate of

*E*

_{ray},

*E*

_{circ}, is:

*θ*,

*G*(

_{θ}*x*), and at point

*θ*+ 180°, both contribute nothing to

*E*

_{ray}at

*θ*; conversely, half of the scleral growth at

*θ*± 90° contributes to

*E*

_{ray}at

*θ*(the other half contributes to

*E*

_{ray}at

*θ*+ 180°), and growth at intermediate angles contributes intermediate amounts. A practical consequence of this is that growth at the equator would serve only to make eyes longer along the axis, whereas growth at the axis would only make eyes wider at the equator; growth at intermediate eccentricities would make eyes longer and wider. Therefore, to calculate the growth occurring at any age (

*x*) and any position (

*θ*) around the globe,

*G*, we referenced the values of

^{′}_{θ}*E*

_{circ}at every point around the eye (

*φ*) and scaled them based upon their relationship to

*θ*using the formula

*θ*and

*φ*indicate points (in our case, every 5°) around the globe. The point at which growth is being measured is

*θ*. It is measured by evaluating the contributions of

*E*

_{circ}at every other point,

*φ*.

*Δ*

_{minor}(

*φ*,

*θ*) is the “minor angle” between a

*φ*and

*θ*pair, and we calculated it as shown in Equation 6b. Figure 2 illustrates the relationships between

*θ*,

*φ*, and

*Δ*

_{minor}for the same 10 positions of

*φ*and two different positions of

*θ*. For an example from the figure, the contribution that

*E*

_{circ}at position 220 makes when

*G*′ is evaluated at position 315 is

*E*

_{circ,220}(

*x*) × 85/360 because

*Δ*

_{minor}(220,315) = 85. By applying Equations 6a and 6b at every

*x*(birth to 10 years corrected age) to every

*θ*(0°–355° in 5° steps) and referencing

*E*

_{circ}at every

*φ*(0°–355° in 5° steps), we derived a model of where and when the eye grows. Notably, this measure of scleral growth,

*G*, is proportional at each

^{′}_{θ}*θ*, but increases in absolute value with the number of contributing points,

*φ*. However, it can be normalized to the actual growth, in millimeters, around the sclera,

*G*(

_{θ}*x*), using the following correction factor:

**Figure 2**

**Figure 2**

*G*÷ 9, in our analysis, because

^{′}_{θ}*φ*was in 5° increments.

^{36}(for ages up to 4 years) and the Collaborative Longitudinal Evaluation of Ethnicity and Refractive Error (CLEERE) study (for ages 6–14 years).

^{37}Encouragingly, our Term-born sample appeared to be following the normal course of emmetropization with respect to SE.

**Figure 3**

**Figure 3**

**Figure 4**

**Figure 4**

**Figure 5**

**Figure 5**

*y*

_{0}fixed at 0 in all cases) revealed that a different curve for each group provided a significantly better model than one curve for all groups for ACD (

*F*= 9.87;

*df*= 6,126;

*P*= 6.53 · 10

^{−9}), PSD (

*F*= 4.03;

*df*= 6,126;

*P*= 0.000997), and AL (

*F*= 3.10;

*df*= 6,126;

*P*= 0.00722); note, however, that the fitting process did not converge on a solution for LT for any group. For the three successful fits (ACD, PSD, AL), the pairwise post hoc evaluations found that, on the one hand, the Term and Preterm curves did not significantly differ, in any case. On the other hand, the Term and ROP curves did significantly differ, in every case. The Preterm curve also differed from the ROP curve for ACD only (not PSD or AL). The most notable shift was of the

*c*parameter to the right, indicating that the development of these features of the ROP eye is delayed.

*F*= 13.5;

*df*= 2,126;

*P*= 4.86 · 10

^{−6}), LT (

*F*= 9.79;

*df*= 2,126;

*P*= 0.000111), PSD (

*F*= 16.6;

*df*= 2,126;

*P*= 4.14 · 10

^{−7}), and AL (

*F*= 13.1;

*df*= 2,126;

*P*= 6.65 · 10

^{−6}). The HSD detected differences only between the ROP group and the Term and Preterm subjects; there were two homogenous subsets: “ROP” and “Term and Preterm.” In other words, Term and Preterm were indistinguishable. The direction of the effect in the ROP group was shorter for ACD, PCD, and AL. and longer for LT.

*y*

_{0}fixed at 0; for the fit to the ratio,

*y*

_{0}was free to vary. In all cases, the curve comparison indicated that a different curve for each group was appropriate: CC (

*F*= 9.46;

*df*= 6,126;

*P*= 1.44 · 10

^{−8}), ALC (

*F*= 12.1;

*df*= 6,126;

*P*= 1.04 · 10

^{−10}), PLC (

*F*= 8.78;

*df*= 6,126;

*P*= 5.42 · 10

^{−8}), and ASL/PSD (

*F*= 4.221;

*df*= 6,126;

*P*= 0.000663). Pairwise post hoc evaluation detected that the Term curves differed from the ROP curves, but not from the Preterm curves, for all four parameters. Furthermore, with the exception of CC, the Preterm curve also was distinct from the ROP curve (i.e., for CC only, the Preterm and ROP data could be satisfactorily fit by a single curve).

**Figure 6**

**Figure 6**

*F*= 11.1;

*df*= 2,126;

*P*= 3.63 · 10

^{−5}), ALC (

*F*= 9.99;

*df*= 2,126;

*P*= 9.42 · 10

^{−5}), PLC (

*F*= 8.83;

*df*= 2,126;

*P*= 0.000257), and ASL/PSD (

*F*= 8.03;

*df*= 2,126;

*P*= 0.000524). In each case, HSD testing once again detected two homogenous subsets: ROP, and Term and Preterm.

**Figure 7**

**Figure 7**

**Figure 8**

**Figure 8**

^{38}and our model estimated an almost identical amount of growth, 0.69 mm (22.26–22.94 mm), over the same age range; indeed, our ALs are in good agreement with extant data from preterm to adult ages.

^{39}For another example, it is recognized that the cornea flattens early, relative to the growth of the eye

^{39}; this same pattern is clearly evidenced in our data (Fig. 6) wherein CC is shown to saturate early, relative to AL. For additional examples: From age 6 to 12 years, mean ACD measured in CLEERE was approximately 3.11 mm (after subtracting the thickness of the cornea, 0.5 mm

^{40}), and our value is approximately 2.93 mm (Fig. 4); at the same ages, CLEERE measures LT at approximately 3.45 mm, and our value is approximately 3.29 mm (Fig. 4)

^{38}; lens power is thought to fall precipitously during the first few years of life, largely due to a change in the lens refractive gradient, but also due to a monotonic (and nearly constant) change in curvature,

^{41}and our fits to Term (and Preterm) ALC and PLC followed this same course.

^{41}so we would not capture saturation in our sample from pediatric radiology records. With respect to eye growth and emmetropization, our finding that early, rapid growth is mainly at the equator is most consistent with the data that suggest that it is mainly the peripheral retina that mediates these processes; later, smaller-scale refinement of refractive state may be mediated by the fovea.

^{37,38,42–44}

^{45}Thus, the magnitude of the ROP-induced changes to the structure of the lens could be as large as or larger than the changes to the cornea while having a lesser impact on refractive state. Our data suggest that this is the case.

^{24,26,27}Those subjects show a shift to myopia at early ages. Studies of other cohorts have found that the ROP eye is characterized by a peculiar delay in growth at early ages.

^{16}Our model confirmed both of these observations. We do, however, recognize that a major limitation of our mathematical modeling procedure is that it assumes that the growth of the eye follows a sigmoidal function. This probably is a relatively safe assumption over the period of time modeled within our study, because it is not until approximately 10 years of age that significant incidence of school age myopia occurs; school age myopia is something quite distinct from myopia of prematurity. We imagine that a second developmental curve might be needed to describe certain parts of the development of the posterior of the globe as it develops school age myopia. Staphylomata are an additional important cause of myopic refractive state.

^{46–49}Thus, while the asymptotes of our sigmoidal curve (the

*a*parameter) predict an “ultimate” eye, it is unlikely that this prediction will be an accurate estimate of the average shape of the eye at the end of life.

^{50–52}Moreover, such directionally-appropriate changes in eye growth are observed even if the optic nerve or accommodation system is ablated, indicating that the retina is an important controller of ocular development.

^{17,18}Apparently, retinal control of eye growth is an (almost) entirely local phenomenon (although there is evidence from chickens of a subtle role for feedback from the brain

^{53}).

^{54–56}The abnormally avascular peripheral ROP retina may, therefore, underpin the abnormalities that our model detected in the ROP eye's growth.

^{33}In many respects, the human data described herein closely parallel, at least as reasonably as is possible between two such strikingly anatomically different eyes (the rat eye consists mostly of lens), the data obtained during the course of that rat work. Thus, the “ROP rat” eye may represent a convenient model in which to study the peculiarities of ROP ocular development. More broadly, modeling the growth of normally and abnormally developing animal eyes can provide a basis for determining the molecular mechanisms that mediate eye growth and refractive development.

**R.J. Munro**, None;

**A.B. Fulton**, None;

**T.Y.P. Chui**, None;

**A. Moskowitz**, None;

**R. Ramamirtham**, None;

**R.M. Hansen**, None;

**S.P. Prabhu**, None;

**J.D. Akula**, None

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