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An extension of an accommodation and convergence model of emmetropization to include the effects of illumination intensity
Ophthal. Physiol. Opt. Vol. 19, No. 2, pp. 112±125, 1999 # 1999 The College of Optometrists. Published by Elsevier Science Ltd All rights reserved. Printed in Great Britain 0275-5408/99 $19.00 + 0.00
PII: S0275-5408(98)00077-5
An extension of an accommodation and convergence model of emmetropization to include the effects of illumination intensity C. A. Blackie* and H. C. Howland Section of Neurobiology and Behavior, Cornell University, Ithaca, NY 14853-2702, USA Summary The Flitcroft (1998) emmetropization model incorporates a classical model of accommodation and convergence with blur-driven feedback control of eye growth. We have modified this model by incorporating the effects of illumination (with or without extensive near work) on accommodation, vergence, pupil diameter and emmetropization. In addition to replicating Flitcroft's results, we show that (1) decreased illumination and (2) a low convergence accommodation/ convergence (CAC) ratio exacerbate the progression of near-work-induced myopia. Our model further indicates that prescription of negative lenses, under these conditions, augments the advancement of myopia. # 1999 The College of Optometrists. Published by Elsevier Science Ltd. All rights reserved
Introduction
Tabarra, 1988; O'Leary and Millodot, 1979). Certainly, the prescription of positive and negative lenses is an example of environmental manipulation and the eect of these lenses has been investigated in chickens (Schaeel et al., 1990) and monkeys (Hung et al., 1995). Extensive near work, another environmental stimulus, is also believed to have an eect on refractive status (reviewed in Ong and Ciureda, 1997). The earliest documented association between near work and myopia is that of Kepler and dates back to the early 1600 s (Duke-Elder and Abrams, 1970). In concurrence, Ramazzini (1713) hypothesized that near work leads to `weakness of vision' and Donders (1864) ascribed the progression of myopia to accommodative stress. Some 80 years later, Duke-Elder (1949) stated that the onset and advancement of myopia could be explained by `extensive close work and general debility'. More recently, the association between near work and myopia has been con®rmed by numerous empirical studies (Goldschmidt, 1968; Young et al., 1969; Curtin, 1985; Framingham Ospring Eye Study Group, 1996). Near vision involves accommodation and, if binocular, convergence. These two processes, linked to each other by way of the ACA (accommodative convergence per unit of accommodation) and CAC (convergence accommodation per unit of convergence) ratios,
Emmetropization describes the disappearance of neonatal refractive errors during development (Wildsoet, 1997). The eye grows such that optical length matches physical length which results in formation of clear retinal images. In an evolutionary sense, correcting neonatal refractive errors to produce good vision is selectively advantageous. Thus, it is no surprise that humans are not the only animals to exhibit emmetropization. Studies on a variety of other animals provide evidence of this phenomenon (chicks; Wallman et al., 1978, guinea pigs; Lodge et al., 1994, cats; Gollender et al., 1979, tree shrews; Sherman et al., 1977 and monkeys; Wiesel and Raviola, 1977; Hung et al., 1995; Hung and Smith, 1996). As with any developmental process, emmetropization can be strongly in¯uenced by environmental stimuli, even to the extent that emmetropia is no longer reached (lens addition in chicks; Schaeel et al., 1990, lid suturing in monkeys; Wiesel and Raviola, 1977, lens addition in monkeys; Hung et al., 1995 and Hung and Smith, 1996, corneal opaci®cation and unilateral ptosis in humans; Gee and *Corresponding author. e-mail: [email protected]. Received: 5 August 1998 Revised form: 20 November 1998
112
An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland have been modeled by several authors (Morgan, 1968; Hung and Semmlow, 1980; Carroll, 1982; Schor, 1986). The Hung and Semmlow (1980) model of the static behavior of accommodation and vergence was employed by Flitcroft (1998) to include the eects of accommodation and vergence on emmetropization. Flitcroft's results provide quantitative veri®cation that myopic progression can be enhanced by excessive near
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work coupled with a high ACA ratio (Jiang et al., 1991), low tonic accommodation (Ebenholtz, 1983; McBrien and Millodot, 1987) and poor accommodative function (Gwiazda et al., 1993; Ong and Ciureda, 1997). The impact of the CAC ratio is not well understood and the evidence suggests that there is no signi®cant dierence in CAC ratios between dierent refractive groups (Rosen®eld and Gilmartin, 1987;
Fig. 1(a and b).
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Jones, 1990, Jiang, 1995). However, a more recent study proposes that late-onset myopia and a low CAC ratio may be correlated (Bobier, 1998). Flitcroft's model (1998) also demonstrates that, under the above conditions, the rate of myopic progression is accelerated by the prescription of negative lenses. Accurate accommodation and convergence for near viewing is a function of the optical quality of the stimulus and this depends, in part, on the illumination
of that stimulus (Johnson, 1976; Jiang et al., 1991). Under low photopic conditions the accommodation tends towards its tonic position (dark focus); however, vergence does not approach its tonic position (tonic vergence) until near scotopic conditions (Jiang et al., 1991). If the threshold for accurate accommodation (D) is indeed higher than that for vergence (MA), provided the illumination is above threshold for the vergence system, poor lighting will have a greater impact
Fig. 1(c and d). (continued overleaf).
An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland
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Figure 1. (a) The inclusive model: The simulink symbols and their descriptions are listed in Table 2. There are four individually labeled subsystems: the growth subsystem (Figure 1(b)), the accommodation-vergence subsystem (Figure 1(c)), the illumination subsystem (Figure1(d)) and the refraction subsystem (Figure1(e)). This diagram shows how each subsystem feeds into the inclusive model. Where possible, the parameters are written in full but for those that are abbreviated, the legend and the appendix provide the full name. (b) The growth subsystem receives several inputs from the inclusive model (Figure 1(a)). Angular blur ., Lens . and Ar . (accommodation response). This subsystem shows how angular blur, the accommodation response and the prescription of lenses affect growth of the eye over time. The outputs from this subsystem, Refraction Q, Vs Q (vergence response), Dioptric blur Q, feed back into the inclusive model. (c) The accommodation-vergence subsystem illustrates the combined effects of Angular blur ., Acg . (accommodative controller gain), Vcg . (vergence controller gain) and Vs . (vergence stimulus), in addition to other parameters, on the output Ar Q (accommodative response) which feeds into the inclusive model (Figure 1(a)). (d) The illumination subsystem showing the relationship between pupil diameter Q, Acg Q (accommodative controller gain), Vcg Q (vergence controller gain) and illumination. Each of the three outputs, pupil diameter, Acg and Vcg, feed into the inclusive model (Figure 1(a)). (e) The refraction subsystem showing the refraction regime. The input to this subsystem, the power of the eye, Refraction ., is an output of the growth subsystem (Figure 1(a) and (b)). The eye is refracted and the power of the lens prescribed. The output, Lens Q, feeds into the growth subsystem (Figure 1(b)) and contributes to the Angular blur driving accommodation and retinal growth. Prescription only occurs if the myopia exceeds 0.75 D.
on accommodative accuracy than on that of vergence. In addition, the pupil diameter is negatively correlated with the amount of light entering the pupil. As the illumination decreases, the pupil diameter increases to allow more light into the eye (Flamant, 1948). Our model suggests that people with larger pupils suer only slightly more from the eects of working in bad light (10ÿ1 cdmÿ2). This is most likely due to that fact that as pupil diameter increases, the depth of focus decreases, thus reducing the eect of a larger blur circle on the retina (Campbell, 1957). In expansion of previous models, speci®cally those of Hung and Semmlow (1980) and Flitcroft (1998), we (1) replicate Flitcroft's results, (2) include the CAC ratio in our analysis and (3) investigate the eect that illumination of the near environment has on the rate of myopic progression.
Methods The dierential equation model is comprised of three feedback loops, growth, accommodation and
convergence (Figure 1(a)±(c)). This creates a system with very fast (accommodation and vergence) and very slow dynamics (growth). To model this accurately, a variable-order multistep algorithm is used allowing for the rapid start-up followed by the steady-state response of the accommodation and vergence and for the slow gradual change in growth (Dabney and Harman, 1998). As the speed with which each loop reaches equilibrium is determined by its time constant, the growth loop has the longest of the three time constants. With the exception of the growth rate constant, the individual gain constants and their numerical values are those suggested by Hung and Semmlow (1980), Hung et al. (1996) and Flitcroft (1998) (see Table 1). The growth rate constant, Eg, was chosen to re¯ect a realistic value of emmetropization or progression of myopia. Approximating the change of refraction as a ®rst-order process, our value of Eg (0.025) resulted in a time constant of 1.6 years. This fell within the range of time constants of adjustments to spectacles found by Medina and Fariza (1993), and within the rates of myopic progression reviewed by
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Ophthal. Physiol. Opt. 1999 19: No 2 Table 1. Model parameters are derived by Hung et al. (1996). Abnormal values, suggested by Hung et al. (1996), Flitcroft (1998) and Hung (1998), are provided when used in simulations Constant Acg Vcg Abias Vbias ACA CAC
Description
Normal value
Accommodative controller gain Vergence controller gain Tonic accommodation Tonic vergence Accommodative convergence Convergence accommodation
Curtin (1985, Chap. 8). The dynamics of the accommodative and vergence loops are designed to create a relatively instantaneous response, thus mimicking a steady-state response on the growth rate time scale. Each loop is de®ned by a dierential equation. These are as follows: Growth :
d Pt Eg B dt
1
where Pt is the equivalent power of the eye in air in diopters (D) at time t, B is the blur signal in milliradians (see Eqs. (8), Figure 1(b) and Eg is the growth rate constant (see Appendix: table of parameters). The advantage of representing growth in terms of a change in the equivalent power of the eye in air as opposed to representing it as a change axial length is that this allows most of the units of the model to be diopters (D), which have the same dimensions as the other major unit of the model, meter-angles (MA). We are cognizant of the fact that these two forms of representation of growth (change in axial length and change in refractive power) are not equivalent; however, the dierence between them at this level of speci®city of the model was deemed insigni®cant. The growth loop is shown in the growth subsystem (see Figure 1(b)). Accommodation :
d Ai 1 ÿB Acg ÿ Ai 2 dt t
where Ai is an intermediate variable (see Figure 1(c)). t is a time constant; 1/t determines the speed at which the integrator reaches equilibrium, B is the blur signal (see Equation (8)) and Acg is the accommodative controller gain (see Appendix: table of parameters). At steady state as t 4 1: Ai ÿB Acg
3
Ar ÿB Acg Abias CAC Vi
4
and where Ar is the accommodative response (D), Vi is an intermediate variable de®ned in Equation (6) below,
10.00 150.00 0.61 0.29 0.80 0.37
D MA D/MA MA/D
Abnormal value 3.00 Ð 0.25 Ð 1.20 0.80 0.10
(late-onset myopes) (late-onset myopes) (late-onset myopes) (high) (young myopes)
CAC is the CAC ratio, and Abias is the tonic accommodation (D). Vergence :
d Vi 1 ÿ Vs VrVcg ÿ Vi 5 dt t
where Vi is an intermediate variable (see Figure 1(c)), t is a time constant; 1/t determines the speed at which the integrator reaches equilibrium, Vs is the vergence stimulus (MA), and is a negative number for any ®nite target, Vr is the vergence response (MA) and Vcg is the vergence controller gain (see Appendix: table of parameters). At steady state as t 4 1: Vi Vr VsVcg
6
Vr Vr VsVcg Vbias ACA Ai
7
and where Vr is the vergence response (MA), Vi is the intermediate variable de®ned above (D) (see Equation (3)), ACA is the ACA ratio, Vbias is the tonic vergence (MA), Vcg is the vergence controller gain and Vs is the vergence stimulus (MA) (see Appendix: table of parameters). The accommodativevergence loop is shown in the accommodative-vergence subsystem (see Figure 1(c)). Blur signal : B Vemm Lens Ar ÿ Pt Aspd 8 where Vemm (the emmetropic power of the eye) = 58D, Lens is the power of the lens prescribed where necessary (D), Ar is the accommodative response (D) (Equation (4)), As is the accommodative stimulus (D) (de®ned as a vergence which is negative for any ®nite target and generally equal to Vs) and pd is the pupil diameter (mm) (see Appendix: table of parameters). In other words, the initial conditions for the accommodative demand are equal to the sum of the refractive error (VemmÿPt), the amount of near work (As) and any additional spectacle lens (Lens). The
An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland
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Figure 2. The relationship between the accommodative controller gain, Acg, and the vergence controller gain, Vcg and luminance. Normal Acg 10, low Acg 3 in good light (102 cdmÿ2). The relationship between Acg and luminance is estimated using data from Johnson (1976). The relationship between Vcg and luminance estimated using data from Jiang et al. (1991).
accommodative error is equal to the dierence between the accommodative stimulus and the accommodative response and this, multiplied by the pupil diameter, is the blur signal, B (angular blur, see Equation (15), Figure 1(a,b)). From Equations (3)±(8), one may write two simultaneous equations for Ar and Vr and solve them for the steady-state values of these variables in closed form. As the solution contains very many terms and does not shed any intuitive light on their behavior, we have omitted them here. The eect of illumination on accommodation was derived from the experimental data of Johnson (1976). One of the major ®ndings of this study was that as illumination decreases the accommodative accuracy decreases, slowly approaching the dark focus. We used the results of the measurements on Johnson's subjects (1976) to estimate the decline of the accommodative controller gain, Acg (see Figure 2. The value of this controller gain is calculated in the illumination subsystem (see Figure 1(d))Table 2. The illumination feeds into an interpolated look-up table and the estimated value for Acg is then multiplied by a constant to determine accommodative suciency. Thus Acg fa illKa
9
where Acg is the accommodative controller gain, fa(ill) is the function that de®nes the eect of illumination on
accommodation and Ka is the accommodative suciency constant (see Appendix: table of parameters). As myopes and, in particular, late-onset myopes are associated with a reduced accommodative function, this seems appropriate (Gwiazda et al., 1993; Ebenholtz, 1983; Ong and Ciureda, 1997, pp. 67, 69). Finally, Acg feeds into the inclusive model (see Figure 1(a)) and enters the accommodative loop (see Figure 1(c)). The relationship between illumination and vergence was derived from the experimental data of Jiang et al. (1991). Their results show that as illumination decreases the vergence accuracy decreases, slowly approaching the tonic vergence. We used the results of their measurements to estimate the decline of the vergence controller gain, Vcg (see Figure 2). Thus Vcg fv illKv
10
where Vcg is the vergence controller gain, fv(ill) is the function that de®nes the eect of illumination on vergence and Kv is the vergence suciency constant (see Appendix: table of parameters). The value of this controller gain is calculated in the illumination subsystem (see Figure 1(d)). The illumination feeds into an interpolated look-up table to obtain the estimated value for Vcg. This feeds into the inclusive model (see Figure 1(a)) and enters the vergence loop (see Figure 1(c)). The eect of illumination on pupil size was derived
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Ophthal. Physiol. Opt. 1999 19: No 2 Table 2. A list of the simulink symbols used in the model and their descriptions (Dabney and Harman, 1998)
from the experimental data of Flamant (1948). Fitting a polynomial to this data provided the relationship between pupil diameter and illumination (see Figure 3): y 1:945 kp ÿ Pg 1:218 x 0:07 x2
11
where y is the pupil diameter (mm); x is the log of the luminance (cdcmÿ2); kp is a constant used to increase the natural pupil size (for average pupil diameter kp=0; Pg is the pupil gain constant which could be used to simulate an abnormal response to a change in illumination (normal response Pg=1); R2=0.737; p < 0.0001. The illumination subsystem (see Figure 1(d)) shows how the pupil diameter is calculated. This feeds into the inclusive model (see Figure 1(a)). The eect of convergence on pupil size can be modeled by the addition of a near triad constant. The pupillary miosis that occurs due to convergence would be added to the miosis due to a decrease in illumination and the summed result would re¯ect the near
pupillary response. However, experiment with the model shows that the inclusion of this constant makes very little dierence to the overall outcome and it has been excluded. Due to the great disparity of time scales between accommodation, which occurs in a fraction of a second, and growth of the eye, which occurs over months, it is impractical to simulate the two processes in the same analog computer program. However, in a system such as ours it may be shown that the much slower growth system responds only to the average of the values of the rapidly changing near work driving function (``the method of averaging'', Guckenheimer and Holmes, 1983). Thus, a target of x diopters, viewed for a fraction of time, f, may be used to simulate the eects of a target of fx diopters, e.g. a 1.5 D accommodative stimulus observed 100% of the time is equivalent to 3 D accommodative stimulus observed 50% of the time. Refraction is shown in the refraction subsystem (see
An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland
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Figure 3. The relationship between pupil diameter and luminance. Data from Flamant (1948).
Figure 1(e)). The input to this section, Refraction, comes from the growth subsystem (see Figure 1(b)). The refraction is determined by subtracting the power of the eye (Pt, see Equation (1)) from the emmetropic 58 D, thus Refraction 58 ÿ Pt:
12
The amount of deviation from emmetropia is then compensated accordingly by the addition of a lens, provided that the myopia exceeds 0.75 D. Therefore Lens ÿRefraction Lens 0
if Refraction > 0:75D
13
if Refraction < 0:75D
14
The output from this section feeds back into the inclusive model and contributes to the blur signal (angular blur) driving the growth and accommodation loops (see Figure 1(a)±(c)). This model assumes that the eye growth mechanism is separate from the accommodative mechanism in the sense that it is only indirectly aected by the accuracy of the accommodative function, by way of the blur signal. This is in contrast to the assumption sometimes made by previous workers (e.g. Young (cited in Curtin, 1985, p. 121)) that the act of accommodation itself might in someway stimulate growth. As the growth signal is retinal (Wallman et al., 1987) and the accommodative signal is processed in the midbrain (Jaeger and Benevento, 1980), this assumption seems reasonable. Using the gain constants suggested by Hung and Semmlow (1980), Hung et al. (1996) and Flitcroft
(1998), the step application of spectacle lenses, at yearly intervals, is simulated for dierent lighting conditions and pupil diameters and for dierent percentages of near work. Due to the fact that late-onset myopes have a low accommodative gain (Ong and Ciureda, 1997, pp. 67, 69), they may experience more dramatic eects of performing extensive near work in poor. Using a reading card (Snellen chart), held at a distance of 30 cm, a light meter was used to measure the luminance (log luminance, cdmÿ2) of the near stimulus. Poor light (10ÿ1 cdmÿ2) was determined as the level of luminance in which the 20/20 line could just be read. This model was constructed in Simulink, a dynamic system simulation for MATLAB (Dabney and Harman, 1998).
Results Under conditions of reduced illumination and extended periods of near work, our model predicts an increase in the progression of myopia (Figures 4±6). The amount and rate of change in refraction over time depends on the parameters used in the simulation. For example: Normal subject (Figure 4) The parameters de®ning a normal subject are: Acg = 10, Vcg = 150, ACA ratio = 0.8, CAC ratio = 0.37, Abias=0.61, Vbias=0.29 (Hung et al., 1996; Flitcroft, 1998, Table 1). An accommodative
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Figure 4. Comparison between simulations for the following subject experiencing prolonged near lus ÿ3 D, observation time: 60% of a day): Acg = 10, Vcg = 150, ACA ratio = 0.8, CAC Abias=0.61, Vbias=0.29. Line 1 represents the effects of the near work in good light (102 cdmÿ2); ter = 4.1 mm (asymptote value = ÿ 0.32 D). Line 2 represents the effects of the near work (10ÿ1 cdmÿ2); pupil diameter = 6.3 mm (asymptote value = ÿ 0.60 D).
work (stimuratio = 0.37, pupil diamein bad light
Figure 5. Comparison between simulations for the following subject (late-onset myope) experiencing prolonged near work (stimulus ÿ3 D, observation time: 60% of a day): Acg = 3, Vcg = 150, ACA ratio = 1.2, Abias=0.25, Vbias=0.29. Line 1: late-onset myope reading in good light (102 cdmÿ2); pupil diameter = 4.1 mm, CAC ratio = 0.37 (asymptote value = ÿ 0.92 D). Line 2: late-onset myope reading in bad light (10ÿ1 cd mÿ2); pupil diameter = 6.3 mm, CAC ratio = 0.37 (asymptote value = ÿ 1.00 D). Line 3: late-onset myope reading in bad light (10ÿ1 cdmÿ2); pupil diameter = 6.3 mm, CAC ratio = 0.10 (asymptote value = ÿ 1.42 D). Line 4: late-onset myope reading in bad light (10ÿ1 cdmÿ2); pupil diameter = 6.3 mm, CAC ratio = 0.80 (asymptote value = ÿ 0.40 D). Note: the effects of an increased (0.8) or reduced (0.1) CAC ratio in good light are negligible and thus have not been included in the figure.
An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland
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Figure 6. Comparison between simulations for the following subjects experiencing prolonged near work (stimulus ÿ3 D, observation time: 60% of a day) in bad light (10ÿ1 cdmÿ2) and periodical refraction: Normal (Acg = 10, Vcg = 150, ACA ratio = 0.8, CAC ratio = 0.37, Abias=0.61, Vbias=0.29) and late-onset myope: Acg = 3, Vcg = 150, ACA ratio = 1.2, CAC ratio = 0.37, Abias=0.25, Vbias=0.29. R marks the time of refraction. Prescription of lenses will only occur once the myopia exceeds 0.75 D, hence no refraction was required for the simulation shown in line 1. Line 1: normal subject, the effects of the near work in bad light (10ÿ1 cdmÿ2; pupil diameter = 6.3 mm). Line 2: late-onset myope, the effects of the near work in bad light (10ÿ1 cdmÿ2; pupil diameter = 6.3 mm). Line 3: late-onset myope, the effects of the near work in bad light with naturally very large pupils (10ÿ1 cdmÿ2; pupil diameter = 9.3 mm). Note that natural pupil size has very little effect on the progression of myopia in this refraction regime.
stimulus of ÿ3 D, observation time: 60% of a day, is used in good light (102 cdmÿ2); line 1, and in bad light (10ÿ1 cdmÿ2); line 2, and the refraction is mapped over a period of one year. The result is that the refraction is more myopic (approximately 0.25 D more) when the near work is performed in bad light. Late-onset myope (Figure 5) The parameters de®ning a late-onset myope are: Acg = 3, Vcg = 150, ACA ratio = 0.8, CAC ratio = 0.37, Abias=0.61, Vbias=0.29 (Hung et al., 1996; Flitcroft, 1998). An accommodative stimulus of ÿ3 D, observation time: 60% of a day, is used in good light (102 cdmÿ2); line 1, and in bad light (10ÿ1 cdmÿ2); line 2, and the refraction is mapped over a period of one year. The result is that the refraction is more myopic (approximately 0.5 D more) when the near work is performed in bad light. Eects of the CAC ratio (Figure 5) Using the parameters for a late-onset myope: Acg = 3, Vcg = 150, ACA ratio = 0.8, Abias=0.61, Vbias=0.29, the CAC ratio was increased and decreased relative to the normal (0.37).
Accommodative stimulus is ÿ3 D, observation time: 60% of a day. Compared to a normal CAC ratio in good (102 cdmÿ2) and bad light (10ÿ1 cdmÿ2); lines 1 and 2, the model predicts that a reduced CAC ratio (0.1) has almost no impact on the rate of progression on myopia for a late-onset myope performing near work in good light (102 cdmÿ2) (not shown in Figure 5). However, the reduced CAC ratio has a de®nite eect on the late-onset myope performing the same task in bad light (10ÿ1 cdmÿ2). The amount of myopia is increased by approximately 0.25 D, line 3. The eect of an elevated CAC ratio (0.8 D/MA) is negligible in good light (102 cdmÿ2) but, compared to a normal CAC ratio in bad light (10ÿ1 cdmÿ2), the ®nal amount of myopia is reduced by approximately 0.5 D, line 4. Note that the initial rate of progression of myopia is the same as for a normal CAC ratio in bad light but this slows down signi®cantly over time. Furthermore, our model predicts that a high CAC ratio (0.8 D/MA) coupled with a high tonic vergence (0.6 MA), is not as eective in retarding the progression of myopia as a high CAC ratio with normal tonic vergence (0.29 MA). Reduced illumination exacerbates this relationship. The in¯uence of tonic vergence on myopic progression when coupled with a low CAC ratio (0.1 D/MA) is almost negligible. Thus, the
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Table 3. Normal, high and low values for the parameters are listed (Hung and Semmlow, 1980; Hung et al., 1996; Flitcroft, 1998; Hung, 1998). For each computation of angular blur only one of the parameters is altered at a time, thus the remaining parameters are held at their normal values. For those parameters having the greatest effect on angular blur the amount of angular blur is printed in bold type. Parameter
Normal value
Low value
Angular blur
High value
Angular blur
10.00 150.00 0.80 0.37 0.61 0.29 ÿ3.00 4.10 102
3.00 100 0.40 0.10 0.25 0.05 0.00 2.50 10ÿ1
ÿ0.5421 ÿ0.1912 ÿ0.1590 ÿ0.2247 ÿ0.2404 ÿ0.1790 0.0689 ÿ0.1871 ÿ0.8177
15.00 200.00 1.20 0.80 1.25 0.60 ÿ5.00 5.50 104
ÿ0.1288 ÿ0.1909 ÿ0.2392 ÿ0.0609 ÿ0.1033 ÿ0.2067 ÿ0.3644 ÿ0.1927 ÿ0.1842
Acg Vcg ACA CAC Abias (D) Vbias (D) Accomm. stimulus (D) Pupil diameter (mm) Log lum. (cdmÿ2)
predictive value of the CAC ratio on myopic progression, may depend on the tonic vergence when the CAC ratio is high.
Systematic analysis of angular blur as a function of the model parameters
Eect of periodic refraction (Figure 6) The increase in myopia is approximately 0.5 D for the normal subject; line 1, and ÿ3.75 D for the lateonset myope; line 2, both reading in bad light (10ÿ1 cdmÿ2), pupil diameters = 6.3 mm (accommodative stimulus is ÿ3 D, observation time: 60% of a day). For the late-onset myope with naturally large pupils (9.3 mm), the increase in myopia due to the increased pupil size is approximately 0.1 D; line 3. Clearly 9.3 mm is an extremely large pupil. This value was chosen to emphasize that natural pupil size has little eect on the rate of myopic progression. It may be shown that the dimensionless blur signal, B (See Equation (8)), is approximately equal to the angular blur, Dpd of the defocused object point on the retina. That is D pd1
1:337h b
running a simulation for a particular set of parameters.
15
where h is the diameter of the blur circle on the retina, b is the distance from the retina to the exit pupil, D is the dioptric defocus of the point, pd is the pupil diameter and 1.337 is the refractive index of the vitreous humor. The angular blur can be evaluated by setting the growth constant, Eg, to zero (see Equation (1)) and
There are eight model parameters that can be changed. Holding all other parameters stable except for one, their individual eects have been investigated (see Table 3). In addition, the angular blur is computed for a selection of parameter combinations (see Table 4). Table 3 shows the results of the individuals eects of each parameter on angular blur. The parameters with the greatest impact on the angular blur are the accommodative controller gain (cf. Hung, 1998), the CAC ratio, the tonic accommodation and the accommodative stimulus. Raising and lowering illumination aects the Acg, Vcg and pupil size so although it is included in the table it cannot be considered as an isolated parameter. The combined eects of these parameters is shown in Table 4 where we see the greatest amount of angular blur computed for the late-onset myope reading in bad light. These results show that performing near work in low light coupled with late-onset myopia (high ACA ratio, low tonic accommodation and poor accommodative function) increases the progression of myopia (Figure 5). In addition, under the same conditions, a high CAC ratio (depending on the tonic vergence) may retard, and low CAC ratio may exacerbate the progression of myopia (see Figure 5, lines 3 and 4).
Table 4. Angular blur is computed for four combinations of parameters Parameter (1) (2) (3) (4)
Acg Vcg ACA CAC Abias Vbias Acc. Stim Log lum. Pupil diam. Angular blur
Normal reading in good light 10.00 150 0.80 0.37 Normal reading in bad light 10.00 130 0.80 0.37 Late-onset myope reading in good light 3.00 150 1.20 0.37 Late-onset myope reading in bad light 3.00 130 1.20 0.37
0.61 0.61 0.25 0.25
0.29 0.29 0.29 0.29
ÿ3.00 ÿ3.00 ÿ3.00 ÿ3.00
2.00 ÿ1.00 2.00 ÿ1.00
4.10 6.30 4.10 6.30
ÿ0.1910 ÿ0.8177 ÿ0.8405 ÿ3.1150
An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland Prescribing negative lenses for those who experience extended periods of near-work-induced myopic blur results in a marked increase in the rate of progression of myopia (see Figure 6). People with larger pupils suffer only very slightly more from the eects of working in bad light (see Figure 6, line 3). This is most likely due to the fact that as pupil diameter increases the depth of focus decreases thus reducing the eect of a larger blur circle (Campbell, 1957).
Discussion In agreement with Flitcroft (1998), this model provides quantitative evidence of the eects of accommodative and oculomotor anomalies coupled with prolonged near work on the progression of myopia. There is extensive clinical support for these predictions. Poor accommodative performance is indicative of a low accommodative controller gain, Acg (Gwiazda et al., 1993; McBrien and Millodot, 1986). There is an association between accommodative hysteresis, low tonic accommodation, Abias, and lateonset myopes (McBrien and Millodot, 1987, 1988). An elevated ACA ratio is characteristic of myopic groups (Rosen®eld and Gilmartin, 1987; Jones, 1990) and, more recently, it appears that a low CAC ratio may also be associated with myopia (Bobier, 1998). Though our model predicts that the eect of the CAC ratio on myopic progression depends, to some extent, on the tonic vergence, Vbias, this eect is almost negligible for low CAC ratios. Indeed, if late-onset myopes do have low CAC ratios, tonic vergence may have little eect on the rate of myopic progression. However, in the case of a high CAC ratios the tonic vergence may in¯uence the rate of myopic progress. Flitcroft (1998) also reports the relationship between the CAC ratio and tonic vergence. The illumination level, aecting the near environment, is predicted to have an impact on the advancement of myopia, though empirical proof for this remains to be found. The prescription of negative lenses appears to enhance myopic progression, thus suggesting that more a conservative approach to refractive compensation be considered. We comment on possible prescriptive strategies below. Computation of the angular blur for the individual parameters and combination thereof reveals those parameters with the greatest in¯uence on angular blur (the accommodative controller gain, the CAC ratio, the tonic accommodation and the accommodative stimulus). As illumination has a strong eect on the accommodative function, it is not surprising it also has a large impact on the amount of angular blur. The eects of low illumination and near work on myopic progression predicted by the model are cause for concern. Of high risk are those with an already
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failing accommodative system (late-onset myopes), but even those with good accommodative system are prone to myopia when extended visual stress occurs in conjunction with large pupils and bad light (i.e. students taking notes in poorly lit lecture halls). A high CAC ratio may impede the progression of myopia, depending on the tonic vergence, hence a low CAC ratio may be indicative of a high-risk candidate for late-onset myopia. Importantly, the prescription of negative lenses for those suering from near-work-induced transient or permanent myopia simply increases the accommodative demand on a less ecient accommodative system compared to emmetropes and hyperopes. Our model predicts negative eects of post refractive compensation which appear to mimic the rapid progression of myopia observed in individuals who have been prescribed corrective lenses. If the fundamental postulates of the model are correct, namely that angular blur regulates the growth of the eye, and the accommodative gain is reduced in low light levels more than the gain of the growth feedback loop, then two conclusions immediately follow: (1) the prescriptive strategy should be one which minimizes hyperopic blur, i.e. situations where the image falls behind the retina and (2) near work should be performed in situations where the illumination is adjusted to minimize pupil diameter and maximize accommodative gain, i.e. ``in a good light''.
Acknowledgements We thank Dr D. I. Flitcroft of Dublin University College, Professor G. K. Hung of Rutgers University and Professor R. Rand of Cornell University for thorough and critical review of the manuscript. For helpful comments and support we thank L. B. Peck, P. T. Starks and F. and A. Blackie. This project was funded by NIH Grant EY02994 to HCH.
References Bobier, W. R., Epp, K. L. and Stephen, C. M. (1998). Convergence accommodation is reduced in young myopes. In: ARVO Abstract Book. Campbell, F. (1957). The depth of ®eld of the human eye. Optica Acta 4, 157±164. Carroll, J. P. (1982). Control theory approach to accommodation and vergence. Am. J. Optom. Physiol. Opt. 59, 658±669. Curtin, B. J. (1985). The etiology of myopia. In: The Myopias: Basic Science and Clinical Management, Harper and Row, Philadelphia, pp. 61±151. Dabney, J. B. and Harman, T. L. (1998). The student edition of simulink; dynamic system stimulation for MATLAB, Version 2, Prentice Hall, Upper Saddle River, New Jersey. Duke-Elder, S. and Abrams, D. (1970). System of
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Ophthalmology. In: Ophthalmic Optics and Refraction, Henry Kimpton, London. Duke-Elder W.S., (1949). Textbook of Ophthalmology, Vol. 4, C.V. Mosby Co. London, pp. 4243±4413, Cited in Ong, E. and Ciureda K.J., 1997, Accommodation, Near Work and Myopia, Optometric Extension Program Foundation Inc., Santa Ana, CA. Donders FC, 1864, On the Anomalies of Accommodation and Refraction of the Eye, Translated by Moore WD, The New Symposium Society, London, Cited in Ong E, Ciureda KJ, 1997, Accommodation, Near Work and Myopia, Optometric Extension Program Foundation Inc., Santa Ana, CA. Ebenholtz, S. M. (1983). Accommodative hysteresis: a precursor for induced myopia. Invest. Ophthalmol. Vis. Sci. 24, 513±515. Flamant, F. (1948). Variation du diametre de la pupille de l'oeil en fonction de la brilliance. Rev. Opt. 27, 751±758. Flitcroft, D. I. (1998). A model of the contribution of oculomotor and optical factors to emmetropization and myopia. Vis. Res. 38, 2869±2879. Framingham Ospring Eye Study Group (1996). Familial aggregation and prevalence of myopia in the Framingham Ospring Eye Study. Arch. Ophthalmol. 114, 324±332. Gee, S. S. and Tabarra, K. F. (1988). Increase in ocular and axial length in patients of corneal opaci®cation. Ophthal mol. 95, 1276±8. Goldschmidt E., (1968). On the etiology of myopiaÐan epidemiology study, Copenhagen: Munksgaard, Acta. Ophthalmol. (Suppl)., 98, 1±172, Cited in Ong E, Ciureda KJ, 1997, Accommodation, Near Work and Myopia, Optometric Extension Program Foundation Inc., Santa Ana, CA. Gollender, M., Thorn, F. and Erickson, P. (1979). Development of axial ocular dimensions following eyelid suture in the cat. Vis. Res. 19, 221±223. Guckenheimer, J. and Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag, New York. Gwiazda, J., Thorn, F., Bauer, J. and Held, R. (1993). Myopic children show insucient accommodative response to blur. Invest. Ophthalmol. Vis. Sci. 34, 690±694. Hung, G. K. (1998). Sensitivity analysis of a stimulusresponse function of a static nonlinear accommodation model. IEEE Trans. Biomed. Eng. 45, 335±341. Hung, G. K. and Semmlow, J. L. (1980). Static behavior of accommodation and vergence: computer simulation of an interactive dual-feedback system. IEEE Trans. Biomed. Eng. 27, 439±447. Hung, G. K., Ciureda, K. J. and Rosen®eld, M. (1996). Proximal contribution to a linear static model of accumulation and vergence. Ophthal. Physiol. Opt. 16, 31± 41. Hung, L. F. and Smith, E. L. (1996). Extended-wear, soft, contact lenses produce hyperopia in young monkeys. Optom. Vis. Sci. 73 (9), 579±584. Hung, L. F., Crawford, M. L. J. and Smith, E. L. (1995). Spectacle lenses alter eye growth and refractive state in young monkeys. Nature Med. 1, 761±765. Jaeger, R. J. and Benevento, L. A. (1980). A horseradish peroxidase study of the innervation of structures of the eye. Invest. Ophthalmol. Vis. Sci. 19, 575±583. Jiang, B. C., Gish, K. W. and Leibowitz, H. W. (1991). Eect of luminance of the relation between accommodation and convergence. Optom. Vis. Sci. 68 (3), 220±225.
Jiang, B. C. (1995). Parameters of accommodative and vergence systems and the development of late-onset myopia. Invest. Ophthalmol. Vis. Sci. 36, 1737±1742. Johnson, C. (1976). Eects of luminance and stimulus distance on accommodation and visual resolution. J Opt. Soc. Am. 66, 138±142. Jones, R. (1990). Accommodative and convergence control system parameters are abnormal in myopia. Invest. Ophthalmol. Vis. Sci. (Suppl). 31, 81. Lodge, A., Peto, T. and McFadden, S. A. (1994). Form deprivation myopia and emmetropization in the guinea pig. Proc. Aust. Neurosci. Soc. 5, 123. McBrien, N. A. and Millodot, M. (1986). The eect of refractive error on the accommodative response gradient. Ophthalmol. Physiol. Opt. 6, 145±149. McBrien, N. A. and Millodot, M. (1987). The relationship between tonic accommodation and refractive error. Invest. Ophthalmol. Vis. Sci. 28, 997±1004. McBrien, N. A. and Millodot, M. (1988). Dierences in adaptation of tonic accommodation with refractive state. Invest. Ophthalmol. Vis. Sci. 29, 460±469. Medina, A. and Fariza, E. (1993). Emmetropization as a ®rst-order feedback system. Vis. Res. 33, 21±26. Morgan, M. W. (1968). Accommodation and vergence. Am. J. Optom. Arch. Am. Acad. Optom. 45, 417±454. O'Leary, D. J. and Millodot, M. (1979). Eyelid closure causes myopia in humans. Experimentia 35, 1478±9. Ong, E. and Ciureda, K. J. (1997). Accommodation, Near Work and Myopia, Optometric Extension Program Foundation Inc., Santa Ana, CA, pp. 67, 69, 76±93, 168± 174. Ramazzini B. (1713). Diseases of Workers, Translated by Wright WC, University of Chicago Press, Chicago, Cited in Owne DA, Wolfe Kelly K, Near work, visual fatigue and variations of oculomotor tonus, Invest. Ophthalmol. Vis. Sci., 28, 743±749. Rosen®eld, M. and Gilmartin, B. (1987). Synkinesis of accommodation and vergence in late-onset myopia. Am. J. Optom. Physiol. Opt. 64, 929±937. Schaeel, F., Troilo, D., Wallman, J. and Howland, H. C. (1990). Developing eyes that lack accommodation grow to compensate for imposed defocus. Vis. Neurosci. 4, 177±183. Schor, C. M. (1986). The Glenn A. Fry award lecture: adaptive regulation of accommodative vergence and vergence accommodation. Am. J. Optom. Physiol. Opt. 63, 587±609. Sherman, S. M., Norton, T. T. and Casagrande, V. A. (1977). Myopia in the lid-sutured tree shrew (Tupia glis). Brain Res. 124, 154±157. Wallman, J. and Adams, J. I. (1987). Developmental aspects of experimental myopia in chicks: susceptibility, recovery and relation to emmetropization. Vis. Res. 27, 1139±1163. Wallman, J., Turkel, J. and Trachtman, J. N. (1978). Extreme myopia produced by modest changes in early visual experience. Science 201, 1249±1251. Wiesel, T. N. and Raviola, E. (1977). Myopia and eye enlargement after neonatal lid fusion in monkeys. Nature (Lond.) 266, 66±68. Wildsoet, C. F. (1997). Active emmetropizationÐevidence for its existence and rami®cations for clinical practice. Ophthal. Physiol. Opt. 17 (4), 279±290. Young, F. A., Leary, G. A. and Baldwin, W. R. et al. (1969). The transmission of refractive errors within eskimo families. Am. J. Optom. Arch. Am. Acad. Optom. 46, 676± 685.
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Appendix Table of model variables, parameters and their descriptions Parameter Abias ACA Accommodative suciency, Ka Acg Ai Angular blur, B Ar As b B, angular blur CAC D Dioptric blur Eg fa(ill) fv(ill) Growth h kp Lens Log luminance Near work stimulus Pg Prism Pupil diameter, pd, y Pt R Refraction t Vbias Vcg Vemm Vergence suciency, Kv Vi Vr Vs x y
Description Tonic accommodation (D) Accommodative convergence per unit accommodation (MA/D) Constant used to model excessive or insucient accommodative response Accommodative controller gain* Intermediate variable in the accommodation loop (see equations (2) and (3)) The dioptric blur multiplied by the pupil diameter in mm (milliradians) Accommodative response (D) Accommodative stimulus (D) The distance from the retina to the exit pupil The dioptric blur multiplied by the pupil diameter in mm (milliradians) Convergence accommodation per unit convergence (D/MA) The dioptric defocus of a point target (D) Sum of the near work stimulus, Vemm, the lens and Ar minus the equivalent power of the eye (D) Growth rate constant* Eg=0.025 The function that de®nes the eect of illumination on accommodation The function that de®nes the eect of illumination on vergence The label for the growth integrator in Figure 1(a). Eye growth is integrated over time. This integrator is much slower than that of accommodation and vergence. Represented as equivalent power of the eye in air (D) The diameter of the blur circle on the retina Constant used to increase the natural pupil size (for average pupil diameter kp=0) Power of lens prescribed (D) The log of the luminance (cdmÿ2, cdcmÿ2) Accommodative demand of the target (D) Pupil gain constant* Amount of added prism (MA) Calculated for speci®c luminance (mm) The equivalent power of the eye in air at time t (D) Indicates time of refraction in Figure 6 Refraction. Appropriate lens is prescribed if myopia r0.75 D Time constant; 1/t determines the speed at which the integrator reaches equilibrium Tonic vergence (MA) Vergence controller gain* Emmetropic power of the eye (58 D) Constant used to model excessive or insucient vergence response Intermediate variable in the accommodation loop (see equations (5) and (6)) Vergence response (MA) Vergence stimulus (MA) The log of the luminance (cdcmÿ2) see Figure 1(d) The pupil diameter (mm) see Figure 1(d)
*All gain constants are dimensionless
PII: S0275-5408(98)00077-5
An extension of an accommodation and convergence model of emmetropization to include the effects of illumination intensity C. A. Blackie* and H. C. Howland Section of Neurobiology and Behavior, Cornell University, Ithaca, NY 14853-2702, USA Summary The Flitcroft (1998) emmetropization model incorporates a classical model of accommodation and convergence with blur-driven feedback control of eye growth. We have modified this model by incorporating the effects of illumination (with or without extensive near work) on accommodation, vergence, pupil diameter and emmetropization. In addition to replicating Flitcroft's results, we show that (1) decreased illumination and (2) a low convergence accommodation/ convergence (CAC) ratio exacerbate the progression of near-work-induced myopia. Our model further indicates that prescription of negative lenses, under these conditions, augments the advancement of myopia. # 1999 The College of Optometrists. Published by Elsevier Science Ltd. All rights reserved
Introduction
Tabarra, 1988; O'Leary and Millodot, 1979). Certainly, the prescription of positive and negative lenses is an example of environmental manipulation and the eect of these lenses has been investigated in chickens (Schaeel et al., 1990) and monkeys (Hung et al., 1995). Extensive near work, another environmental stimulus, is also believed to have an eect on refractive status (reviewed in Ong and Ciureda, 1997). The earliest documented association between near work and myopia is that of Kepler and dates back to the early 1600 s (Duke-Elder and Abrams, 1970). In concurrence, Ramazzini (1713) hypothesized that near work leads to `weakness of vision' and Donders (1864) ascribed the progression of myopia to accommodative stress. Some 80 years later, Duke-Elder (1949) stated that the onset and advancement of myopia could be explained by `extensive close work and general debility'. More recently, the association between near work and myopia has been con®rmed by numerous empirical studies (Goldschmidt, 1968; Young et al., 1969; Curtin, 1985; Framingham Ospring Eye Study Group, 1996). Near vision involves accommodation and, if binocular, convergence. These two processes, linked to each other by way of the ACA (accommodative convergence per unit of accommodation) and CAC (convergence accommodation per unit of convergence) ratios,
Emmetropization describes the disappearance of neonatal refractive errors during development (Wildsoet, 1997). The eye grows such that optical length matches physical length which results in formation of clear retinal images. In an evolutionary sense, correcting neonatal refractive errors to produce good vision is selectively advantageous. Thus, it is no surprise that humans are not the only animals to exhibit emmetropization. Studies on a variety of other animals provide evidence of this phenomenon (chicks; Wallman et al., 1978, guinea pigs; Lodge et al., 1994, cats; Gollender et al., 1979, tree shrews; Sherman et al., 1977 and monkeys; Wiesel and Raviola, 1977; Hung et al., 1995; Hung and Smith, 1996). As with any developmental process, emmetropization can be strongly in¯uenced by environmental stimuli, even to the extent that emmetropia is no longer reached (lens addition in chicks; Schaeel et al., 1990, lid suturing in monkeys; Wiesel and Raviola, 1977, lens addition in monkeys; Hung et al., 1995 and Hung and Smith, 1996, corneal opaci®cation and unilateral ptosis in humans; Gee and *Corresponding author. e-mail: [email protected]. Received: 5 August 1998 Revised form: 20 November 1998
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An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland have been modeled by several authors (Morgan, 1968; Hung and Semmlow, 1980; Carroll, 1982; Schor, 1986). The Hung and Semmlow (1980) model of the static behavior of accommodation and vergence was employed by Flitcroft (1998) to include the eects of accommodation and vergence on emmetropization. Flitcroft's results provide quantitative veri®cation that myopic progression can be enhanced by excessive near
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work coupled with a high ACA ratio (Jiang et al., 1991), low tonic accommodation (Ebenholtz, 1983; McBrien and Millodot, 1987) and poor accommodative function (Gwiazda et al., 1993; Ong and Ciureda, 1997). The impact of the CAC ratio is not well understood and the evidence suggests that there is no signi®cant dierence in CAC ratios between dierent refractive groups (Rosen®eld and Gilmartin, 1987;
Fig. 1(a and b).
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Jones, 1990, Jiang, 1995). However, a more recent study proposes that late-onset myopia and a low CAC ratio may be correlated (Bobier, 1998). Flitcroft's model (1998) also demonstrates that, under the above conditions, the rate of myopic progression is accelerated by the prescription of negative lenses. Accurate accommodation and convergence for near viewing is a function of the optical quality of the stimulus and this depends, in part, on the illumination
of that stimulus (Johnson, 1976; Jiang et al., 1991). Under low photopic conditions the accommodation tends towards its tonic position (dark focus); however, vergence does not approach its tonic position (tonic vergence) until near scotopic conditions (Jiang et al., 1991). If the threshold for accurate accommodation (D) is indeed higher than that for vergence (MA), provided the illumination is above threshold for the vergence system, poor lighting will have a greater impact
Fig. 1(c and d). (continued overleaf).
An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland
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Figure 1. (a) The inclusive model: The simulink symbols and their descriptions are listed in Table 2. There are four individually labeled subsystems: the growth subsystem (Figure 1(b)), the accommodation-vergence subsystem (Figure 1(c)), the illumination subsystem (Figure1(d)) and the refraction subsystem (Figure1(e)). This diagram shows how each subsystem feeds into the inclusive model. Where possible, the parameters are written in full but for those that are abbreviated, the legend and the appendix provide the full name. (b) The growth subsystem receives several inputs from the inclusive model (Figure 1(a)). Angular blur ., Lens . and Ar . (accommodation response). This subsystem shows how angular blur, the accommodation response and the prescription of lenses affect growth of the eye over time. The outputs from this subsystem, Refraction Q, Vs Q (vergence response), Dioptric blur Q, feed back into the inclusive model. (c) The accommodation-vergence subsystem illustrates the combined effects of Angular blur ., Acg . (accommodative controller gain), Vcg . (vergence controller gain) and Vs . (vergence stimulus), in addition to other parameters, on the output Ar Q (accommodative response) which feeds into the inclusive model (Figure 1(a)). (d) The illumination subsystem showing the relationship between pupil diameter Q, Acg Q (accommodative controller gain), Vcg Q (vergence controller gain) and illumination. Each of the three outputs, pupil diameter, Acg and Vcg, feed into the inclusive model (Figure 1(a)). (e) The refraction subsystem showing the refraction regime. The input to this subsystem, the power of the eye, Refraction ., is an output of the growth subsystem (Figure 1(a) and (b)). The eye is refracted and the power of the lens prescribed. The output, Lens Q, feeds into the growth subsystem (Figure 1(b)) and contributes to the Angular blur driving accommodation and retinal growth. Prescription only occurs if the myopia exceeds 0.75 D.
on accommodative accuracy than on that of vergence. In addition, the pupil diameter is negatively correlated with the amount of light entering the pupil. As the illumination decreases, the pupil diameter increases to allow more light into the eye (Flamant, 1948). Our model suggests that people with larger pupils suer only slightly more from the eects of working in bad light (10ÿ1 cdmÿ2). This is most likely due to that fact that as pupil diameter increases, the depth of focus decreases, thus reducing the eect of a larger blur circle on the retina (Campbell, 1957). In expansion of previous models, speci®cally those of Hung and Semmlow (1980) and Flitcroft (1998), we (1) replicate Flitcroft's results, (2) include the CAC ratio in our analysis and (3) investigate the eect that illumination of the near environment has on the rate of myopic progression.
Methods The dierential equation model is comprised of three feedback loops, growth, accommodation and
convergence (Figure 1(a)±(c)). This creates a system with very fast (accommodation and vergence) and very slow dynamics (growth). To model this accurately, a variable-order multistep algorithm is used allowing for the rapid start-up followed by the steady-state response of the accommodation and vergence and for the slow gradual change in growth (Dabney and Harman, 1998). As the speed with which each loop reaches equilibrium is determined by its time constant, the growth loop has the longest of the three time constants. With the exception of the growth rate constant, the individual gain constants and their numerical values are those suggested by Hung and Semmlow (1980), Hung et al. (1996) and Flitcroft (1998) (see Table 1). The growth rate constant, Eg, was chosen to re¯ect a realistic value of emmetropization or progression of myopia. Approximating the change of refraction as a ®rst-order process, our value of Eg (0.025) resulted in a time constant of 1.6 years. This fell within the range of time constants of adjustments to spectacles found by Medina and Fariza (1993), and within the rates of myopic progression reviewed by
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Ophthal. Physiol. Opt. 1999 19: No 2 Table 1. Model parameters are derived by Hung et al. (1996). Abnormal values, suggested by Hung et al. (1996), Flitcroft (1998) and Hung (1998), are provided when used in simulations Constant Acg Vcg Abias Vbias ACA CAC
Description
Normal value
Accommodative controller gain Vergence controller gain Tonic accommodation Tonic vergence Accommodative convergence Convergence accommodation
Curtin (1985, Chap. 8). The dynamics of the accommodative and vergence loops are designed to create a relatively instantaneous response, thus mimicking a steady-state response on the growth rate time scale. Each loop is de®ned by a dierential equation. These are as follows: Growth :
d Pt Eg B dt
1
where Pt is the equivalent power of the eye in air in diopters (D) at time t, B is the blur signal in milliradians (see Eqs. (8), Figure 1(b) and Eg is the growth rate constant (see Appendix: table of parameters). The advantage of representing growth in terms of a change in the equivalent power of the eye in air as opposed to representing it as a change axial length is that this allows most of the units of the model to be diopters (D), which have the same dimensions as the other major unit of the model, meter-angles (MA). We are cognizant of the fact that these two forms of representation of growth (change in axial length and change in refractive power) are not equivalent; however, the dierence between them at this level of speci®city of the model was deemed insigni®cant. The growth loop is shown in the growth subsystem (see Figure 1(b)). Accommodation :
d Ai 1 ÿB Acg ÿ Ai 2 dt t
where Ai is an intermediate variable (see Figure 1(c)). t is a time constant; 1/t determines the speed at which the integrator reaches equilibrium, B is the blur signal (see Equation (8)) and Acg is the accommodative controller gain (see Appendix: table of parameters). At steady state as t 4 1: Ai ÿB Acg
3
Ar ÿB Acg Abias CAC Vi
4
and where Ar is the accommodative response (D), Vi is an intermediate variable de®ned in Equation (6) below,
10.00 150.00 0.61 0.29 0.80 0.37
D MA D/MA MA/D
Abnormal value 3.00 Ð 0.25 Ð 1.20 0.80 0.10
(late-onset myopes) (late-onset myopes) (late-onset myopes) (high) (young myopes)
CAC is the CAC ratio, and Abias is the tonic accommodation (D). Vergence :
d Vi 1 ÿ Vs VrVcg ÿ Vi 5 dt t
where Vi is an intermediate variable (see Figure 1(c)), t is a time constant; 1/t determines the speed at which the integrator reaches equilibrium, Vs is the vergence stimulus (MA), and is a negative number for any ®nite target, Vr is the vergence response (MA) and Vcg is the vergence controller gain (see Appendix: table of parameters). At steady state as t 4 1: Vi Vr VsVcg
6
Vr Vr VsVcg Vbias ACA Ai
7
and where Vr is the vergence response (MA), Vi is the intermediate variable de®ned above (D) (see Equation (3)), ACA is the ACA ratio, Vbias is the tonic vergence (MA), Vcg is the vergence controller gain and Vs is the vergence stimulus (MA) (see Appendix: table of parameters). The accommodativevergence loop is shown in the accommodative-vergence subsystem (see Figure 1(c)). Blur signal : B Vemm Lens Ar ÿ Pt Aspd 8 where Vemm (the emmetropic power of the eye) = 58D, Lens is the power of the lens prescribed where necessary (D), Ar is the accommodative response (D) (Equation (4)), As is the accommodative stimulus (D) (de®ned as a vergence which is negative for any ®nite target and generally equal to Vs) and pd is the pupil diameter (mm) (see Appendix: table of parameters). In other words, the initial conditions for the accommodative demand are equal to the sum of the refractive error (VemmÿPt), the amount of near work (As) and any additional spectacle lens (Lens). The
An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland
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Figure 2. The relationship between the accommodative controller gain, Acg, and the vergence controller gain, Vcg and luminance. Normal Acg 10, low Acg 3 in good light (102 cdmÿ2). The relationship between Acg and luminance is estimated using data from Johnson (1976). The relationship between Vcg and luminance estimated using data from Jiang et al. (1991).
accommodative error is equal to the dierence between the accommodative stimulus and the accommodative response and this, multiplied by the pupil diameter, is the blur signal, B (angular blur, see Equation (15), Figure 1(a,b)). From Equations (3)±(8), one may write two simultaneous equations for Ar and Vr and solve them for the steady-state values of these variables in closed form. As the solution contains very many terms and does not shed any intuitive light on their behavior, we have omitted them here. The eect of illumination on accommodation was derived from the experimental data of Johnson (1976). One of the major ®ndings of this study was that as illumination decreases the accommodative accuracy decreases, slowly approaching the dark focus. We used the results of the measurements on Johnson's subjects (1976) to estimate the decline of the accommodative controller gain, Acg (see Figure 2. The value of this controller gain is calculated in the illumination subsystem (see Figure 1(d))Table 2. The illumination feeds into an interpolated look-up table and the estimated value for Acg is then multiplied by a constant to determine accommodative suciency. Thus Acg fa illKa
9
where Acg is the accommodative controller gain, fa(ill) is the function that de®nes the eect of illumination on
accommodation and Ka is the accommodative suciency constant (see Appendix: table of parameters). As myopes and, in particular, late-onset myopes are associated with a reduced accommodative function, this seems appropriate (Gwiazda et al., 1993; Ebenholtz, 1983; Ong and Ciureda, 1997, pp. 67, 69). Finally, Acg feeds into the inclusive model (see Figure 1(a)) and enters the accommodative loop (see Figure 1(c)). The relationship between illumination and vergence was derived from the experimental data of Jiang et al. (1991). Their results show that as illumination decreases the vergence accuracy decreases, slowly approaching the tonic vergence. We used the results of their measurements to estimate the decline of the vergence controller gain, Vcg (see Figure 2). Thus Vcg fv illKv
10
where Vcg is the vergence controller gain, fv(ill) is the function that de®nes the eect of illumination on vergence and Kv is the vergence suciency constant (see Appendix: table of parameters). The value of this controller gain is calculated in the illumination subsystem (see Figure 1(d)). The illumination feeds into an interpolated look-up table to obtain the estimated value for Vcg. This feeds into the inclusive model (see Figure 1(a)) and enters the vergence loop (see Figure 1(c)). The eect of illumination on pupil size was derived
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Ophthal. Physiol. Opt. 1999 19: No 2 Table 2. A list of the simulink symbols used in the model and their descriptions (Dabney and Harman, 1998)
from the experimental data of Flamant (1948). Fitting a polynomial to this data provided the relationship between pupil diameter and illumination (see Figure 3): y 1:945 kp ÿ Pg 1:218 x 0:07 x2
11
where y is the pupil diameter (mm); x is the log of the luminance (cdcmÿ2); kp is a constant used to increase the natural pupil size (for average pupil diameter kp=0; Pg is the pupil gain constant which could be used to simulate an abnormal response to a change in illumination (normal response Pg=1); R2=0.737; p < 0.0001. The illumination subsystem (see Figure 1(d)) shows how the pupil diameter is calculated. This feeds into the inclusive model (see Figure 1(a)). The eect of convergence on pupil size can be modeled by the addition of a near triad constant. The pupillary miosis that occurs due to convergence would be added to the miosis due to a decrease in illumination and the summed result would re¯ect the near
pupillary response. However, experiment with the model shows that the inclusion of this constant makes very little dierence to the overall outcome and it has been excluded. Due to the great disparity of time scales between accommodation, which occurs in a fraction of a second, and growth of the eye, which occurs over months, it is impractical to simulate the two processes in the same analog computer program. However, in a system such as ours it may be shown that the much slower growth system responds only to the average of the values of the rapidly changing near work driving function (``the method of averaging'', Guckenheimer and Holmes, 1983). Thus, a target of x diopters, viewed for a fraction of time, f, may be used to simulate the eects of a target of fx diopters, e.g. a 1.5 D accommodative stimulus observed 100% of the time is equivalent to 3 D accommodative stimulus observed 50% of the time. Refraction is shown in the refraction subsystem (see
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Figure 3. The relationship between pupil diameter and luminance. Data from Flamant (1948).
Figure 1(e)). The input to this section, Refraction, comes from the growth subsystem (see Figure 1(b)). The refraction is determined by subtracting the power of the eye (Pt, see Equation (1)) from the emmetropic 58 D, thus Refraction 58 ÿ Pt:
12
The amount of deviation from emmetropia is then compensated accordingly by the addition of a lens, provided that the myopia exceeds 0.75 D. Therefore Lens ÿRefraction Lens 0
if Refraction > 0:75D
13
if Refraction < 0:75D
14
The output from this section feeds back into the inclusive model and contributes to the blur signal (angular blur) driving the growth and accommodation loops (see Figure 1(a)±(c)). This model assumes that the eye growth mechanism is separate from the accommodative mechanism in the sense that it is only indirectly aected by the accuracy of the accommodative function, by way of the blur signal. This is in contrast to the assumption sometimes made by previous workers (e.g. Young (cited in Curtin, 1985, p. 121)) that the act of accommodation itself might in someway stimulate growth. As the growth signal is retinal (Wallman et al., 1987) and the accommodative signal is processed in the midbrain (Jaeger and Benevento, 1980), this assumption seems reasonable. Using the gain constants suggested by Hung and Semmlow (1980), Hung et al. (1996) and Flitcroft
(1998), the step application of spectacle lenses, at yearly intervals, is simulated for dierent lighting conditions and pupil diameters and for dierent percentages of near work. Due to the fact that late-onset myopes have a low accommodative gain (Ong and Ciureda, 1997, pp. 67, 69), they may experience more dramatic eects of performing extensive near work in poor. Using a reading card (Snellen chart), held at a distance of 30 cm, a light meter was used to measure the luminance (log luminance, cdmÿ2) of the near stimulus. Poor light (10ÿ1 cdmÿ2) was determined as the level of luminance in which the 20/20 line could just be read. This model was constructed in Simulink, a dynamic system simulation for MATLAB (Dabney and Harman, 1998).
Results Under conditions of reduced illumination and extended periods of near work, our model predicts an increase in the progression of myopia (Figures 4±6). The amount and rate of change in refraction over time depends on the parameters used in the simulation. For example: Normal subject (Figure 4) The parameters de®ning a normal subject are: Acg = 10, Vcg = 150, ACA ratio = 0.8, CAC ratio = 0.37, Abias=0.61, Vbias=0.29 (Hung et al., 1996; Flitcroft, 1998, Table 1). An accommodative
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Figure 4. Comparison between simulations for the following subject experiencing prolonged near lus ÿ3 D, observation time: 60% of a day): Acg = 10, Vcg = 150, ACA ratio = 0.8, CAC Abias=0.61, Vbias=0.29. Line 1 represents the effects of the near work in good light (102 cdmÿ2); ter = 4.1 mm (asymptote value = ÿ 0.32 D). Line 2 represents the effects of the near work (10ÿ1 cdmÿ2); pupil diameter = 6.3 mm (asymptote value = ÿ 0.60 D).
work (stimuratio = 0.37, pupil diamein bad light
Figure 5. Comparison between simulations for the following subject (late-onset myope) experiencing prolonged near work (stimulus ÿ3 D, observation time: 60% of a day): Acg = 3, Vcg = 150, ACA ratio = 1.2, Abias=0.25, Vbias=0.29. Line 1: late-onset myope reading in good light (102 cdmÿ2); pupil diameter = 4.1 mm, CAC ratio = 0.37 (asymptote value = ÿ 0.92 D). Line 2: late-onset myope reading in bad light (10ÿ1 cd mÿ2); pupil diameter = 6.3 mm, CAC ratio = 0.37 (asymptote value = ÿ 1.00 D). Line 3: late-onset myope reading in bad light (10ÿ1 cdmÿ2); pupil diameter = 6.3 mm, CAC ratio = 0.10 (asymptote value = ÿ 1.42 D). Line 4: late-onset myope reading in bad light (10ÿ1 cdmÿ2); pupil diameter = 6.3 mm, CAC ratio = 0.80 (asymptote value = ÿ 0.40 D). Note: the effects of an increased (0.8) or reduced (0.1) CAC ratio in good light are negligible and thus have not been included in the figure.
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Figure 6. Comparison between simulations for the following subjects experiencing prolonged near work (stimulus ÿ3 D, observation time: 60% of a day) in bad light (10ÿ1 cdmÿ2) and periodical refraction: Normal (Acg = 10, Vcg = 150, ACA ratio = 0.8, CAC ratio = 0.37, Abias=0.61, Vbias=0.29) and late-onset myope: Acg = 3, Vcg = 150, ACA ratio = 1.2, CAC ratio = 0.37, Abias=0.25, Vbias=0.29. R marks the time of refraction. Prescription of lenses will only occur once the myopia exceeds 0.75 D, hence no refraction was required for the simulation shown in line 1. Line 1: normal subject, the effects of the near work in bad light (10ÿ1 cdmÿ2; pupil diameter = 6.3 mm). Line 2: late-onset myope, the effects of the near work in bad light (10ÿ1 cdmÿ2; pupil diameter = 6.3 mm). Line 3: late-onset myope, the effects of the near work in bad light with naturally very large pupils (10ÿ1 cdmÿ2; pupil diameter = 9.3 mm). Note that natural pupil size has very little effect on the progression of myopia in this refraction regime.
stimulus of ÿ3 D, observation time: 60% of a day, is used in good light (102 cdmÿ2); line 1, and in bad light (10ÿ1 cdmÿ2); line 2, and the refraction is mapped over a period of one year. The result is that the refraction is more myopic (approximately 0.25 D more) when the near work is performed in bad light. Late-onset myope (Figure 5) The parameters de®ning a late-onset myope are: Acg = 3, Vcg = 150, ACA ratio = 0.8, CAC ratio = 0.37, Abias=0.61, Vbias=0.29 (Hung et al., 1996; Flitcroft, 1998). An accommodative stimulus of ÿ3 D, observation time: 60% of a day, is used in good light (102 cdmÿ2); line 1, and in bad light (10ÿ1 cdmÿ2); line 2, and the refraction is mapped over a period of one year. The result is that the refraction is more myopic (approximately 0.5 D more) when the near work is performed in bad light. Eects of the CAC ratio (Figure 5) Using the parameters for a late-onset myope: Acg = 3, Vcg = 150, ACA ratio = 0.8, Abias=0.61, Vbias=0.29, the CAC ratio was increased and decreased relative to the normal (0.37).
Accommodative stimulus is ÿ3 D, observation time: 60% of a day. Compared to a normal CAC ratio in good (102 cdmÿ2) and bad light (10ÿ1 cdmÿ2); lines 1 and 2, the model predicts that a reduced CAC ratio (0.1) has almost no impact on the rate of progression on myopia for a late-onset myope performing near work in good light (102 cdmÿ2) (not shown in Figure 5). However, the reduced CAC ratio has a de®nite eect on the late-onset myope performing the same task in bad light (10ÿ1 cdmÿ2). The amount of myopia is increased by approximately 0.25 D, line 3. The eect of an elevated CAC ratio (0.8 D/MA) is negligible in good light (102 cdmÿ2) but, compared to a normal CAC ratio in bad light (10ÿ1 cdmÿ2), the ®nal amount of myopia is reduced by approximately 0.5 D, line 4. Note that the initial rate of progression of myopia is the same as for a normal CAC ratio in bad light but this slows down signi®cantly over time. Furthermore, our model predicts that a high CAC ratio (0.8 D/MA) coupled with a high tonic vergence (0.6 MA), is not as eective in retarding the progression of myopia as a high CAC ratio with normal tonic vergence (0.29 MA). Reduced illumination exacerbates this relationship. The in¯uence of tonic vergence on myopic progression when coupled with a low CAC ratio (0.1 D/MA) is almost negligible. Thus, the
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Table 3. Normal, high and low values for the parameters are listed (Hung and Semmlow, 1980; Hung et al., 1996; Flitcroft, 1998; Hung, 1998). For each computation of angular blur only one of the parameters is altered at a time, thus the remaining parameters are held at their normal values. For those parameters having the greatest effect on angular blur the amount of angular blur is printed in bold type. Parameter
Normal value
Low value
Angular blur
High value
Angular blur
10.00 150.00 0.80 0.37 0.61 0.29 ÿ3.00 4.10 102
3.00 100 0.40 0.10 0.25 0.05 0.00 2.50 10ÿ1
ÿ0.5421 ÿ0.1912 ÿ0.1590 ÿ0.2247 ÿ0.2404 ÿ0.1790 0.0689 ÿ0.1871 ÿ0.8177
15.00 200.00 1.20 0.80 1.25 0.60 ÿ5.00 5.50 104
ÿ0.1288 ÿ0.1909 ÿ0.2392 ÿ0.0609 ÿ0.1033 ÿ0.2067 ÿ0.3644 ÿ0.1927 ÿ0.1842
Acg Vcg ACA CAC Abias (D) Vbias (D) Accomm. stimulus (D) Pupil diameter (mm) Log lum. (cdmÿ2)
predictive value of the CAC ratio on myopic progression, may depend on the tonic vergence when the CAC ratio is high.
Systematic analysis of angular blur as a function of the model parameters
Eect of periodic refraction (Figure 6) The increase in myopia is approximately 0.5 D for the normal subject; line 1, and ÿ3.75 D for the lateonset myope; line 2, both reading in bad light (10ÿ1 cdmÿ2), pupil diameters = 6.3 mm (accommodative stimulus is ÿ3 D, observation time: 60% of a day). For the late-onset myope with naturally large pupils (9.3 mm), the increase in myopia due to the increased pupil size is approximately 0.1 D; line 3. Clearly 9.3 mm is an extremely large pupil. This value was chosen to emphasize that natural pupil size has little eect on the rate of myopic progression. It may be shown that the dimensionless blur signal, B (See Equation (8)), is approximately equal to the angular blur, Dpd of the defocused object point on the retina. That is D pd1
1:337h b
running a simulation for a particular set of parameters.
15
where h is the diameter of the blur circle on the retina, b is the distance from the retina to the exit pupil, D is the dioptric defocus of the point, pd is the pupil diameter and 1.337 is the refractive index of the vitreous humor. The angular blur can be evaluated by setting the growth constant, Eg, to zero (see Equation (1)) and
There are eight model parameters that can be changed. Holding all other parameters stable except for one, their individual eects have been investigated (see Table 3). In addition, the angular blur is computed for a selection of parameter combinations (see Table 4). Table 3 shows the results of the individuals eects of each parameter on angular blur. The parameters with the greatest impact on the angular blur are the accommodative controller gain (cf. Hung, 1998), the CAC ratio, the tonic accommodation and the accommodative stimulus. Raising and lowering illumination aects the Acg, Vcg and pupil size so although it is included in the table it cannot be considered as an isolated parameter. The combined eects of these parameters is shown in Table 4 where we see the greatest amount of angular blur computed for the late-onset myope reading in bad light. These results show that performing near work in low light coupled with late-onset myopia (high ACA ratio, low tonic accommodation and poor accommodative function) increases the progression of myopia (Figure 5). In addition, under the same conditions, a high CAC ratio (depending on the tonic vergence) may retard, and low CAC ratio may exacerbate the progression of myopia (see Figure 5, lines 3 and 4).
Table 4. Angular blur is computed for four combinations of parameters Parameter (1) (2) (3) (4)
Acg Vcg ACA CAC Abias Vbias Acc. Stim Log lum. Pupil diam. Angular blur
Normal reading in good light 10.00 150 0.80 0.37 Normal reading in bad light 10.00 130 0.80 0.37 Late-onset myope reading in good light 3.00 150 1.20 0.37 Late-onset myope reading in bad light 3.00 130 1.20 0.37
0.61 0.61 0.25 0.25
0.29 0.29 0.29 0.29
ÿ3.00 ÿ3.00 ÿ3.00 ÿ3.00
2.00 ÿ1.00 2.00 ÿ1.00
4.10 6.30 4.10 6.30
ÿ0.1910 ÿ0.8177 ÿ0.8405 ÿ3.1150
An extension of an accommodation and convergence model: C. A. Blackie and H. C. Howland Prescribing negative lenses for those who experience extended periods of near-work-induced myopic blur results in a marked increase in the rate of progression of myopia (see Figure 6). People with larger pupils suffer only very slightly more from the eects of working in bad light (see Figure 6, line 3). This is most likely due to the fact that as pupil diameter increases the depth of focus decreases thus reducing the eect of a larger blur circle (Campbell, 1957).
Discussion In agreement with Flitcroft (1998), this model provides quantitative evidence of the eects of accommodative and oculomotor anomalies coupled with prolonged near work on the progression of myopia. There is extensive clinical support for these predictions. Poor accommodative performance is indicative of a low accommodative controller gain, Acg (Gwiazda et al., 1993; McBrien and Millodot, 1986). There is an association between accommodative hysteresis, low tonic accommodation, Abias, and lateonset myopes (McBrien and Millodot, 1987, 1988). An elevated ACA ratio is characteristic of myopic groups (Rosen®eld and Gilmartin, 1987; Jones, 1990) and, more recently, it appears that a low CAC ratio may also be associated with myopia (Bobier, 1998). Though our model predicts that the eect of the CAC ratio on myopic progression depends, to some extent, on the tonic vergence, Vbias, this eect is almost negligible for low CAC ratios. Indeed, if late-onset myopes do have low CAC ratios, tonic vergence may have little eect on the rate of myopic progression. However, in the case of a high CAC ratios the tonic vergence may in¯uence the rate of myopic progress. Flitcroft (1998) also reports the relationship between the CAC ratio and tonic vergence. The illumination level, aecting the near environment, is predicted to have an impact on the advancement of myopia, though empirical proof for this remains to be found. The prescription of negative lenses appears to enhance myopic progression, thus suggesting that more a conservative approach to refractive compensation be considered. We comment on possible prescriptive strategies below. Computation of the angular blur for the individual parameters and combination thereof reveals those parameters with the greatest in¯uence on angular blur (the accommodative controller gain, the CAC ratio, the tonic accommodation and the accommodative stimulus). As illumination has a strong eect on the accommodative function, it is not surprising it also has a large impact on the amount of angular blur. The eects of low illumination and near work on myopic progression predicted by the model are cause for concern. Of high risk are those with an already
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failing accommodative system (late-onset myopes), but even those with good accommodative system are prone to myopia when extended visual stress occurs in conjunction with large pupils and bad light (i.e. students taking notes in poorly lit lecture halls). A high CAC ratio may impede the progression of myopia, depending on the tonic vergence, hence a low CAC ratio may be indicative of a high-risk candidate for late-onset myopia. Importantly, the prescription of negative lenses for those suering from near-work-induced transient or permanent myopia simply increases the accommodative demand on a less ecient accommodative system compared to emmetropes and hyperopes. Our model predicts negative eects of post refractive compensation which appear to mimic the rapid progression of myopia observed in individuals who have been prescribed corrective lenses. If the fundamental postulates of the model are correct, namely that angular blur regulates the growth of the eye, and the accommodative gain is reduced in low light levels more than the gain of the growth feedback loop, then two conclusions immediately follow: (1) the prescriptive strategy should be one which minimizes hyperopic blur, i.e. situations where the image falls behind the retina and (2) near work should be performed in situations where the illumination is adjusted to minimize pupil diameter and maximize accommodative gain, i.e. ``in a good light''.
Acknowledgements We thank Dr D. I. Flitcroft of Dublin University College, Professor G. K. Hung of Rutgers University and Professor R. Rand of Cornell University for thorough and critical review of the manuscript. For helpful comments and support we thank L. B. Peck, P. T. Starks and F. and A. Blackie. This project was funded by NIH Grant EY02994 to HCH.
References Bobier, W. R., Epp, K. L. and Stephen, C. M. (1998). Convergence accommodation is reduced in young myopes. In: ARVO Abstract Book. Campbell, F. (1957). The depth of ®eld of the human eye. Optica Acta 4, 157±164. Carroll, J. P. (1982). Control theory approach to accommodation and vergence. Am. J. Optom. Physiol. Opt. 59, 658±669. Curtin, B. J. (1985). The etiology of myopia. In: The Myopias: Basic Science and Clinical Management, Harper and Row, Philadelphia, pp. 61±151. Dabney, J. B. and Harman, T. L. (1998). The student edition of simulink; dynamic system stimulation for MATLAB, Version 2, Prentice Hall, Upper Saddle River, New Jersey. Duke-Elder, S. and Abrams, D. (1970). System of
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Ophthalmology. In: Ophthalmic Optics and Refraction, Henry Kimpton, London. Duke-Elder W.S., (1949). Textbook of Ophthalmology, Vol. 4, C.V. Mosby Co. London, pp. 4243±4413, Cited in Ong, E. and Ciureda K.J., 1997, Accommodation, Near Work and Myopia, Optometric Extension Program Foundation Inc., Santa Ana, CA. Donders FC, 1864, On the Anomalies of Accommodation and Refraction of the Eye, Translated by Moore WD, The New Symposium Society, London, Cited in Ong E, Ciureda KJ, 1997, Accommodation, Near Work and Myopia, Optometric Extension Program Foundation Inc., Santa Ana, CA. Ebenholtz, S. M. (1983). Accommodative hysteresis: a precursor for induced myopia. Invest. Ophthalmol. Vis. Sci. 24, 513±515. Flamant, F. (1948). Variation du diametre de la pupille de l'oeil en fonction de la brilliance. Rev. Opt. 27, 751±758. Flitcroft, D. I. (1998). A model of the contribution of oculomotor and optical factors to emmetropization and myopia. Vis. Res. 38, 2869±2879. Framingham Ospring Eye Study Group (1996). Familial aggregation and prevalence of myopia in the Framingham Ospring Eye Study. Arch. Ophthalmol. 114, 324±332. Gee, S. S. and Tabarra, K. F. (1988). Increase in ocular and axial length in patients of corneal opaci®cation. Ophthal mol. 95, 1276±8. Goldschmidt E., (1968). On the etiology of myopiaÐan epidemiology study, Copenhagen: Munksgaard, Acta. Ophthalmol. (Suppl)., 98, 1±172, Cited in Ong E, Ciureda KJ, 1997, Accommodation, Near Work and Myopia, Optometric Extension Program Foundation Inc., Santa Ana, CA. Gollender, M., Thorn, F. and Erickson, P. (1979). Development of axial ocular dimensions following eyelid suture in the cat. Vis. Res. 19, 221±223. Guckenheimer, J. and Holmes, P. (1983). Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields, Springer Verlag, New York. Gwiazda, J., Thorn, F., Bauer, J. and Held, R. (1993). Myopic children show insucient accommodative response to blur. Invest. Ophthalmol. Vis. Sci. 34, 690±694. Hung, G. K. (1998). Sensitivity analysis of a stimulusresponse function of a static nonlinear accommodation model. IEEE Trans. Biomed. Eng. 45, 335±341. Hung, G. K. and Semmlow, J. L. (1980). Static behavior of accommodation and vergence: computer simulation of an interactive dual-feedback system. IEEE Trans. Biomed. Eng. 27, 439±447. Hung, G. K., Ciureda, K. J. and Rosen®eld, M. (1996). Proximal contribution to a linear static model of accumulation and vergence. Ophthal. Physiol. Opt. 16, 31± 41. Hung, L. F. and Smith, E. L. (1996). Extended-wear, soft, contact lenses produce hyperopia in young monkeys. Optom. Vis. Sci. 73 (9), 579±584. Hung, L. F., Crawford, M. L. J. and Smith, E. L. (1995). Spectacle lenses alter eye growth and refractive state in young monkeys. Nature Med. 1, 761±765. Jaeger, R. J. and Benevento, L. A. (1980). A horseradish peroxidase study of the innervation of structures of the eye. Invest. Ophthalmol. Vis. Sci. 19, 575±583. Jiang, B. C., Gish, K. W. and Leibowitz, H. W. (1991). Eect of luminance of the relation between accommodation and convergence. Optom. Vis. Sci. 68 (3), 220±225.
Jiang, B. C. (1995). Parameters of accommodative and vergence systems and the development of late-onset myopia. Invest. Ophthalmol. Vis. Sci. 36, 1737±1742. Johnson, C. (1976). Eects of luminance and stimulus distance on accommodation and visual resolution. J Opt. Soc. Am. 66, 138±142. Jones, R. (1990). Accommodative and convergence control system parameters are abnormal in myopia. Invest. Ophthalmol. Vis. Sci. (Suppl). 31, 81. Lodge, A., Peto, T. and McFadden, S. A. (1994). Form deprivation myopia and emmetropization in the guinea pig. Proc. Aust. Neurosci. Soc. 5, 123. McBrien, N. A. and Millodot, M. (1986). The eect of refractive error on the accommodative response gradient. Ophthalmol. Physiol. Opt. 6, 145±149. McBrien, N. A. and Millodot, M. (1987). The relationship between tonic accommodation and refractive error. Invest. Ophthalmol. Vis. Sci. 28, 997±1004. McBrien, N. A. and Millodot, M. (1988). Dierences in adaptation of tonic accommodation with refractive state. Invest. Ophthalmol. Vis. Sci. 29, 460±469. Medina, A. and Fariza, E. (1993). Emmetropization as a ®rst-order feedback system. Vis. Res. 33, 21±26. Morgan, M. W. (1968). Accommodation and vergence. Am. J. Optom. Arch. Am. Acad. Optom. 45, 417±454. O'Leary, D. J. and Millodot, M. (1979). Eyelid closure causes myopia in humans. Experimentia 35, 1478±9. Ong, E. and Ciureda, K. J. (1997). Accommodation, Near Work and Myopia, Optometric Extension Program Foundation Inc., Santa Ana, CA, pp. 67, 69, 76±93, 168± 174. Ramazzini B. (1713). Diseases of Workers, Translated by Wright WC, University of Chicago Press, Chicago, Cited in Owne DA, Wolfe Kelly K, Near work, visual fatigue and variations of oculomotor tonus, Invest. Ophthalmol. Vis. Sci., 28, 743±749. Rosen®eld, M. and Gilmartin, B. (1987). Synkinesis of accommodation and vergence in late-onset myopia. Am. J. Optom. Physiol. Opt. 64, 929±937. Schaeel, F., Troilo, D., Wallman, J. and Howland, H. C. (1990). Developing eyes that lack accommodation grow to compensate for imposed defocus. Vis. Neurosci. 4, 177±183. Schor, C. M. (1986). The Glenn A. Fry award lecture: adaptive regulation of accommodative vergence and vergence accommodation. Am. J. Optom. Physiol. Opt. 63, 587±609. Sherman, S. M., Norton, T. T. and Casagrande, V. A. (1977). Myopia in the lid-sutured tree shrew (Tupia glis). Brain Res. 124, 154±157. Wallman, J. and Adams, J. I. (1987). Developmental aspects of experimental myopia in chicks: susceptibility, recovery and relation to emmetropization. Vis. Res. 27, 1139±1163. Wallman, J., Turkel, J. and Trachtman, J. N. (1978). Extreme myopia produced by modest changes in early visual experience. Science 201, 1249±1251. Wiesel, T. N. and Raviola, E. (1977). Myopia and eye enlargement after neonatal lid fusion in monkeys. Nature (Lond.) 266, 66±68. Wildsoet, C. F. (1997). Active emmetropizationÐevidence for its existence and rami®cations for clinical practice. Ophthal. Physiol. Opt. 17 (4), 279±290. Young, F. A., Leary, G. A. and Baldwin, W. R. et al. (1969). The transmission of refractive errors within eskimo families. Am. J. Optom. Arch. Am. Acad. Optom. 46, 676± 685.
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Appendix Table of model variables, parameters and their descriptions Parameter Abias ACA Accommodative suciency, Ka Acg Ai Angular blur, B Ar As b B, angular blur CAC D Dioptric blur Eg fa(ill) fv(ill) Growth h kp Lens Log luminance Near work stimulus Pg Prism Pupil diameter, pd, y Pt R Refraction t Vbias Vcg Vemm Vergence suciency, Kv Vi Vr Vs x y
Description Tonic accommodation (D) Accommodative convergence per unit accommodation (MA/D) Constant used to model excessive or insucient accommodative response Accommodative controller gain* Intermediate variable in the accommodation loop (see equations (2) and (3)) The dioptric blur multiplied by the pupil diameter in mm (milliradians) Accommodative response (D) Accommodative stimulus (D) The distance from the retina to the exit pupil The dioptric blur multiplied by the pupil diameter in mm (milliradians) Convergence accommodation per unit convergence (D/MA) The dioptric defocus of a point target (D) Sum of the near work stimulus, Vemm, the lens and Ar minus the equivalent power of the eye (D) Growth rate constant* Eg=0.025 The function that de®nes the eect of illumination on accommodation The function that de®nes the eect of illumination on vergence The label for the growth integrator in Figure 1(a). Eye growth is integrated over time. This integrator is much slower than that of accommodation and vergence. Represented as equivalent power of the eye in air (D) The diameter of the blur circle on the retina Constant used to increase the natural pupil size (for average pupil diameter kp=0) Power of lens prescribed (D) The log of the luminance (cdmÿ2, cdcmÿ2) Accommodative demand of the target (D) Pupil gain constant* Amount of added prism (MA) Calculated for speci®c luminance (mm) The equivalent power of the eye in air at time t (D) Indicates time of refraction in Figure 6 Refraction. Appropriate lens is prescribed if myopia r0.75 D Time constant; 1/t determines the speed at which the integrator reaches equilibrium Tonic vergence (MA) Vergence controller gain* Emmetropic power of the eye (58 D) Constant used to model excessive or insucient vergence response Intermediate variable in the accommodation loop (see equations (5) and (6)) Vergence response (MA) Vergence stimulus (MA) The log of the luminance (cdcmÿ2) see Figure 1(d) The pupil diameter (mm) see Figure 1(d)
*All gain constants are dimensionless