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Vision Research, V. 36, No. 15, p. 2249-2251.
Nov. 24, 2015
Emmetropia approach dynamics with diurnal dual-phase cycling.
Greene PR, Brown OS, Medina AP, Graupner HB.
Vision Res. 1996 Aug;36(15):2249-51.
Emmetropia Approach Dynamics with Diurnal Dual-Phase Cycling
PETER R. GREENE,1 OTIS S. BROWN,2 ANTONIO P. MEDINA,3 HARRY B. GRAUPNER4 Received 28
February 1995; in revised form 23 August 1995; in final form 19 November 1995
Abstract
Numerical experiments are performed on a first order exponential response function
subjected to a diurnal square wave visual environment with variable duty cycle. The model is
directly applicable to exponential drift of focal status. A two-state square wave is employed
as the forcing function with high B for time H and low A for time L. Duty cycles of (1/3), (1/2)
and (2/3) are calculated in detail. Results show the following standard linear system
response: (1) Unless the system runs into the stops, the steady state equilibrium settling
level is always between A and B. The level is linearly proportional to a time-weighted average
of the high and low states. (2) The effective time constant t(eff) varies hyperbolically with
duty cycle. For DC = (1/3) and t\ = 100 days, the effective time constant is lengthened to 300
days. An asymptote is encountered under certain circumstances where t(eff) approaches
infinity. (3) Effective time constants and steady state equilibria are independent of square
wave frequency f, animal time constant t1, magnitude and sign of A & B, and diurnal
sequencing of the highs and lows. By presenting results on dimensionless coordinates, we
can predict the drift rates of some animal experiments. Agreement between theory and
experiment has a correlation coefficient r = 0.97 for 12 Macaca nemestrina eyes. Copyright ©
1996 Elsevier Science' Ltd.
Key Words. Animal experiments, Control theory, Dynamic response, Emmetropia, Focusing, Myopia
1To
whom all correspondence should be addressed at: B.G.K.T.Consulting, 153 Main Street, Huntington,New York, NY 11743,
U.S.A. Department of Biomedical Engineering, Homewood Campus, Johns Hopkins University, Baltimore, Maryland, U.S.A.
2McDonnell
3Jet
Douglas Co., Space Station Integration, Seabrook, Maryland, U.S.A.
Propulsion Laboratory, Optical Sciences, California Institute of Technology, Pasadena, California, U.S.A.
4B.G.K.T.
Consulting Stony Brook, New York, U.S.A. 11798
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INTRODUCTION
The visual phenomenon of emmetropization (i.e. approach to focused vision), myopia development,
and hyperopia development are three seemingly different processes which can be modeled with control
theory (Dorf, 1992; Brown & Berger, 1979; Greene & Guyton, 1986). Hyperopia may be a separate case
entirely (Guyton et al., 1989), perhaps because the system is near the control limits or "stops". Young (1961)
experimentally shows exponential approach to myopia with monkeys focusing at close range. Brown and
Young (1981) use a standard first order exponential response function with good results. Medina (1987) fits
ensemble data from 33,051 eyes with a second order response, i.e. exponential with cosine phase delay
term. Ku and Greene (1981) apply variable duty cycle square waves. Raviola and Wiesel (1985)
experimentally observe the exponential. Recent experimental work includes McBrien and Norton (1992),
Smith, Hung and Harworth (1994), and the special issue in Vision Research "Myopia".
All of the various theories assume 100% gazing at infinity, or 100% gazing at some intermediate distance.
However, the average human does not spend 100% of the time focused at the same distance. Instead, a
certain percentage of the day is spent, at T.V. distance, another percentage at book distance, another
percentage at blackboard distance, etc. Likewise, and approximately equivalent from the optics point of view,,
when reading or watching T.V., the individual may or may not use glasses, etc. Thus, given the distance, the
power of the lenses, and the time spent in the equivalent optical environment specifies an integral which is the
net optical impulse of the environmental effects.
There is uncertainty about the eye's behaviour during sleep, although several reports help with this
concern. Various aspects of control theory, accommodation, convergence, and dark focus are discussed by
Fincham (1962), Leibowitz and Owens (1975), Hung and Semmlow (1980), and Owens and Higgins (1983).
Results presented here are non-dimensional so that different time constants can be appreciated. For
instance, Schaeffel et al. (1988) and Wallman et at. (1978) show that chickens have a fast time constant, in
the order of 20-30 days, over which time 10 diopter changes in myopia or hyperopia occur. Goss and Criswell
(1981) summarize animal experiments as they apply to ametropia. Depending on environmental conditions,
changes can occur in either the myopic or hyperopic direction. Medina and Fariza (1993) show 1-2 year
effective time constants for some human subjects.
In this report, rather than dealing with the complications of several different environmental input levels, just
two are used, low level A and high level B, with variable duty cycle being the parameter of interest [see Fig.
2(a) and (b)]. The results probably can be generalized to three or more environments, but we have not tried
that yet. The highs and lows of the diurnal cycle can be thought of as the nears and fars of the environment.
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MATERIALS AND METHODS
As per Brown and Berger (1979) and Greene and Guyton (1986), we employ here a simple first order
exponential to model the refractive drift of the optical system over time when the effective optical environment
changes:
R(t) = B - (B - A) * [1 - exp( -t/t1 ) ]
(1)
Here, R(t) is the effective refractive state of the visual system, B is the refractive state at time t < 0, and A is
the new equivalent visual environment to which the system is exposed, step-wise, at t = 0. Animal time
constant fl is taken to be a nominal 100 days, as supported by the experiments of Young (1961), Brown and
Young (1981), Raviola and Wiesel (1985), and Greene and Guyton (1986).
As shown in Fig. 1(b), this control system will respond with a decaying sawtooth when the environment is
applied as a variable percentage square wave. The "high" phase of the square wave is denoted Phase I, and
the "low" phase is denoted Phase II. We use the following basic iterative equations during these phases:
Phase I: R(t) = R1(t1) [ Rl (ti) - A] * [ l – exp (- t/t1 ) ]
(2)
Phase II: R(t) = R2 ( tj ) [B - R2 ( tj ) ] * [ l - exp( -t / tl ) ]
(3)
where [ tj = ( ti + 1 ) ].
Then, to complete the iterative recursion procedure, we advance to the next time increment with boundary
and initial conditions:
(4)
B.C.
I.C.
Rl =
R2
t=0.
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RESULTS AND DISCUSSION
Figure 2(a) shows the strong effect duty cycle has on the effective time constant of the system. Figure 2(b)
shows that the equilibrium settling level is simply a time-weighted average of the highs and lows of the applied
square wave environment. Figure 3, with animal data from Young (1961) arid a time constant t\ of 100 days,
correlates at the level r = 0.97 for 12 age matched Macaca nemestrina eyes from six adolescent monkeys.
Theory agrees with experiment plus or minus 0.06 diopters, i.e. with greater accuracy than one can currently
measure experimentally. Thus, we say with confidence the drift dynamics are exponential.
Applications
As a practical example of these equations, say you are observing a 4 diopter refractive state R(t) subjected
to a 3D and 4D square wave visual environment with duty cycle of (1/3). With a normal time constant of 100
days, this system will take approx. 300 days to reach the (1/e) point, hence the origin of the effective time
constant. With a re-scaling of the system boundary conditions, based on the time-weighted average of the
applied environment, the time constant re-scales to 100 days. Independent of duty cycle, an exponential
response always results. Time constant is always preserved. But, as a practical matter, the system takes
longer to influence with low impulse cycles.
In terms of practical real-time applications, notice that 4-6 h per day one way or the other, i.e. a fraction of
one duty cycle, can make the difference between 1 year and 3 years in terms of achieving an objective level.
Ku and Greene (1981) present similar time-weighted average results appropriate during the initial dynamic
stage (i.e. t < to) for variable duty cycle loading of rabbit eyes; in this case, theory and experiment agree with
correlation r = 0.91.
Focusing during sleep is a matter of experimental interest. Events during the night are not known. We can
assume, based on preliminary experiments (see Introduction), that the eye resorts to a compensatory offset,
or "dark focus" value of 0.5 D of hyperopic reserve. The net optical impulse of the sleep cycle should
approximately balance the environmental impulse accumulated during the waking hours to maintain steady
state level. If this is not the case, theory predicts that R(t) will drift to a new level. In the absence of
experimental data while the subject is unconscious, we can invent a Conservation of Impulse law to indirectly
subsume the night time activities. Results presented here are sufficiently general to accept any level .
Note added after proof. Another practical application of this mathematical model is predicting the effect of
bifocal or progressive addition lenses ( PALs ) on the progression rate of myopia. In this case, the difference
between the near field ( A diopt.) and far field ( B diopt. ) values of the environment are reduced in magnitude
by approximately 2 diopters, the value of the ( + ) reading Add in the lower segment of the frame. Theory
predicts this should slow the rate of myopia progression [12] [19].
General results
Effective time constants and steady state equilibrium levels are independent of square wave frequency /,
animal time constant fl, magnitude and sign of the high and low states A and B, and even random diurnal
sequencing of the highs and lows. Presenting results on dimensionless coordinates allows prediction of
different animals with different time constants in variable environments. In general, the observed and
calculated diopter drift dynamics are consistent with standard control theory and first order linear system
response.
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REFERENCES
[ n.b. – References are in Pub Med format. Click on Title to view Abstract (and Article) ]
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21. Acknowledgement—This work was supported in part by NIH/NEI Research Grant EY-05013.
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