IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 2, FEBRUARY 1992
I35
Quantitative Analysis of Associated and Disassociated Phorias: Linear and Nonlinear Static Models George K. Hung, Senior Member, IEEE
Abstract-Ogle proposed two measures of oculomotor balance, called associated and disassociated phorias, which he assumed were equivalent. However, experimentally determined vaues of these phorias do not show a close correspondence. To analyze the rationale behind Ogle's assumption of equality, a linear static model was evaluated. It was found that indeed the linear model predicts an exact correspondence between associated and disassociated phorias. Thus, his assumption depended on the presence of a linear model. To account for the discrepancy between these two measures, a nonlinear static model, containing the dead space operators depth of field and Panum's fusional area, was evaluated. Four equations for fixation disparity were derived corresponding to the four combinations of deadspace operator outputs. It was found that only one of these four equally possible solutions for associated phoria correspondedto the disassociated phoria. This suggests that the variability in the four solutions may account for the scatter in the experimental data. The nonlinear model was analyzed further to determine its sensitivity to parameter changes and to show how such a model could generate the classical shape of the fixation disparity curve.
I. INTRODUCTION HE human near response consists of focusing of the lens in each eye, inward rotation of the two eyes, and pupillary constriction. ' The focusing, or accommodation system, senses blur of the retinal image and changes lens curvature to achieve retinal image clarity [3], [4]. The eye-turn, or vergence system, senses disparity between the retinal images in the two eyes and rotates the eyes to attain a single percept [5], [6]. Both systems are therefore feedback control systems [7], [8]. In addition, there are cross-link interactions between the two systems [7], [9][12]. That is, the accommodation system can drive vergence and the vergence system can drive accommodation. One important measure of the cross-link influence is the accommodative convergence to accommodation stimulus ratio (or AC/A)under monocular, or, vergence open-loop condition. A higher AC/A ratio, for example, indicates
T
Manuscript received February 13, 1991; revised July 25, 1991. This work was supported in part by National Institutes of Health Grant EY7519. The author is with the Department of Biomedical Engineering, Rutgers University, Piscataway, NJ 08855. IEEE Log Number 9104909. 'Pupillary constriction decreases the amount of light entering the eye and increases the depth of field [I]. However, under normal viewing conditions ( > 1.5 mm pupil diameter), the influence of pupil on accommodation response is small [2], and the pupillary feedback to accommodation can be considered to be open-looped.
a greater influence of accommodation on convergence. Values of AC /A outside of the normal range are taken as signs of oculomotor abnormality [ 131. Indeed, Schor and others have found that abnormal AC/A ratios may influence the adaptative properties of the near response system [91, [111. Ogle [ 141 proposed that a more informative measure of oculomotor insufficiency may be obtained by recording the vergence error, called fixation disparity, under the more natural binocular, or vergence closed-loop condition. Thus, in a group of subjects, the fixation disparity was measured for different vergence stimuli while the accommodative stimulus was held constant.* The resulting plots of fixation disparity versus vergence stimulus were examined in terms of the shapes of the curves. Ogle categorized these into four main types (Fig. 1). Type I was associated with lack of symptom, whereas Types 11, 111, and IV were associated with symptoms of discomfort [14]-[16]. He also defined two vergence measures, called phorias, under a constant accommodative stimulus condition. For monocular viewing, the vergence response was called the disassociated phoria. On the other hand, for binocular viewing, the vergence stimulus giving zero fixation disparity was called the associated phoria. The rationale for using the associated phoria was that this vergence stimulus value was assumed to indicate the point of oculomotor balance under binocular conditions. Ogle sought to determine whether the point of oculomotor balance is the same under binocular (associated phoria) and monocular (disassociated phoria) conditions. He therefore measured the associated and disassociated phorias in a number of individuals. The resulting plot of associated versus disassociated phoria showed data that were spread mainly parallel to a 45" line, although there was a relatively wide scatter and a bias in the associated esophoria direction (i.e., a relatively more eso3 associated than disassociated phoria) (Fig. 2) [ 141. Nevertheless, some of the data points did show an equality between associated and disassociated phoria. He attributed the extent of the scatter to such factors as insufficient effort of ac*For a clear description of the experimental setup and procedure, see [ 14, pp. 26-38]. 'Eso describes an angle of two eyes which is more convergent than absolute alignment with the target. On the other hand, exo describes a more divergent angle than absolute alignment.
0018-9294/92$03.00 0 1992 IEEE
136
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 2, FEBRUARY 1992 PIXATION DISPARITY
PIXATION DISPARITY
,
730
PRISM BASE-IN, 30
Y
I
1
-
4
2
_.................
VEI VE2
VE3 VE4
5
0
10
15 BO,
20
25
L j , ,
30
A
Fig. 5 . Nonlinear model simulation lines under closed-loop vergence. Upper pair of eso lines correspond to combinations 1 and 2 (VEl and VE2) in Table I. Lower pair of exo lines correspond to combinations 3 and 4 (VE3 and V U ) . Vertical dashed lines indicates where VE2 = - V U . The intersection with the horizontal line at zero fixation disparity gives the vergence stimulus value, or associated phoria. Simulation parameter values were based on experimental data (subj. JS) [7]. AS AD 2 .15
25 20
25
-"
k o
2
_____
1
20
-
25
-
30
-
[(l
,
,
,
10
15
20
25
30
A
Fig. 8. Increasing accommodative stimulus level raises the set of four model lines in the eso direction.
was expanded to include BI stimuli up to 30 A. The appropriate simulated fixation disparity points were determined as follows. For a large vergence stimulus value, the vergence error is positive (exo) (i.e., vergence stimulus is greater than the response) and the accommodative error is negative. Thus, the appropriate solution is the exo line VE3. On the other hand, for a large BI vergence stimulus value the opposite situation holds, where the vergence error is negative (eso) and the accommodative error is positive. In this case, the appropriate solution is the eso line VE2. The gradual transition in the intermediate range follows the diminution of PFA with decreased BO vergence stimulus amplitude, and the enlargement of PFA with increased BI stimulus amplitude. Therefore, the simulated fixation disparity curve in Fig. 9 starts from the large PFA (VD = 18 min of arc) exo line at a large BO value. It then progressed through the intermediate-PFA (VD = 6 min of arc) exo line at an intermediate BO value to the small-PFA (VD = 3 min of arc) exo line at a small BO value. The sequence is repeated but from small-to-large PFA eso lines as BI vergence stimulus is increased. The solid curve traces a fixation disparity plot (for subj. JS [7]) whose shape is similar to experimental Type I fixation disparity curves.
VD 18
B. Crossing at Zero Fixation Disparity It is reasonable to assume that the point where the exo line (VE3 or VE4) crosses the zero fixation disparity line to the eso line (VE1 or VE2) occurs when the errors are equal in magnitude and opposite in sign (see dashed vertical line in Fig. 5). The most likely candidates are VE3, applicable for large vergence and small accommodative stimuli, and VE2, applicable for small vergence and large accommodative stimuli. The equation for VE2 is given by
I
VE2 =
5
BO.
_ _ _ _ _ 0.30 - 0.15
15-
20
0
6
-
,
30
+ ACG) . (VS - VD VCG + AD ACG . AC - VBIAS) - AD - ACG + VD VCG CA + ABIAS) - ACG AC] (1 + ACG) (1 + VCG) - ACG AC . VCG . CA
+ (-AS
*
*
*
*
*
(33)
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 2, FEBRUARY 1992
142 AS AD 2 .15
----- 18 -
I
401
...
"
30
!
~
1
I
30
For AD = 0.15 D, the difference is about 2 A baseout (i.e., in the overconverged, or eso direction). It can be shown that for the remaining two combinations, namely VE1 = -VE4 and VE2 = -VE3, the difference is about 1 A baseout. This suggests that a population sample should show a bias in the associated esophoria (base-out) direction. This indeed is seen in the data presented by Ogle et al. (Fig. 1) [14, pp. 109, 1111.
VD
20
10
0
30
20
10
(A)
El
BO
Fig. 9. Simulated fixation disparity curve starting from large-PFA exo line through a continuum of decreasing PFA exo lines as BO amplitude decreases. It then traces through a continuum of increasing PFA eso lines as BI amplitude increases. The resulting curve (solid) is similar in shape to the Ogle Type I fixation disparity curve.
The equation for VE3 was specified earlier in (31). Solving for VE2 = -VE3, or VE2 VE3 = 0 gives
+
vs(VE2 = -VE3)
=
1
+ ACG
1
+ ACG - ABIAS - AC].
(34)
Comparing (34) with (22) reveals that the term AD, which was summed with ABIAS in (22), is missing in (34). This is due to cancellation of the terms containing AD in (31) and (33). It turns out that the only combination which gives the same result as (22), or disassociated phoria, is VE2 and VE4. Thus, v s ( V E 2 = -VM) =
1
+ ACG
*
AC
- AS
ACG 1 ACG
+
1
(35)
+ ACG - AD - AC.
(36)
(ABIAS
+ AD)
*
AC
.
Therefore, in order to obtain a point where the associated phoria [VS(VE2= -vE4) in (35)] is equal to the disassociated phoria [VR in (22)], the system must move through VE4, where both accommodation and vergence are underdriven, to VE2, where accommodation is underdriven and vergence is overdriven. Clearly, this condition is not always attained since as other combinations are possible. The largest difference between associated and disassociated phorias occurs with the combination VE 1 and VE3. = -VE3) It can be shown that the difference between VS(VEI and the disassociated phoria [see (22)] is Differencevs,,,,
= -VE3)
- 2 -
1
*
ACG
IV. DISCUSSION A. Relationship Between Associated and Disassociated Phorias Although Ogle [14] did not state his ideas in control system terms, his notion of binocular oculomotor balance was deeply insightful. For if there were no deadspace operators, as in the linear model, the associated phoria would be equal to the disassociated phoria. This is because at zero-fixation disparity, the output of the vergence controller would be zero, which corresponds exactly to the condition of open-loop vergence. However, experimental measures of the phorias did not show such an exact correspondence. As mentioned previously, dynamic aspects of accommodation and vergence could not have accounted for the scatter in the plot of associated versus disassociated phoria. Instead, the scatter was attributed primarily to static parameters, such as the deadspace operators DOF and PFA, which gave four possible solutions. Thus, rather than giving totally unreasonable results, the experimental associated phoria technique gave values that were somewhat near the disassociated phoria, but was inherently prone to bias errors and variability. This was unfortunate because without knowing the results presented in this quantitative analysis, one would have been tempted to assume that through careful measurement and large number of trials, a distinct relationship between associated and disassociated phoria would reveal itself. This did not happen, as was documented by Ogle et al. [14] and supported by others [15], [17], [19], r201-
B. Determination of the A C / A Ratio The cross-link gain, AC, can be obtained by two different methods [ 141: first, in terms of the prism power and lens power required to give the same fixation disparity when both systems are close-looped, and second, in terms of accommodative convergence when vergence is openlooped. Judge [ 121 derived algebraic expressions for AC under these two conditions based on the linear model of Hung and Semmlow [7]. Since his derived equations were equivalent, he concluded that the available models were not able to account for the differences in the accommodative-convergence to accommodative stimulus (AC /A)8 ratios obtained experimentally under the two methods. However, in his analysis, he neglected to include two im'For the open-loop vergence case, a plot of disassociated phorias versus different accommodative stimuli gives the complete AC /A line.
143
HUNG: QUANTITATIVE ANALYSIS OF PHORIAS
portant parameters-tonic accommodation, ABIAS, and tonic vergence, VBIAS [7], [23]-[26]. It can be shown [27] that the accommodative convergence cross-link gain, AC, obtained using the fixation disparity method is
P AC=-. L
1 +ACG ACG
(37)
where P is the prism power (prism viewing only) which gives the same fixation disparity as the lens power L (lens viewing only). On the other hand, the AC obtained when vergence is open-looped can be shown to be AC =
VR - VBIAS L - ABIAS
1
+ ACG ACG
*
(38)
Equations (37) and (38) are the same (although the symbols were different) as (8) and (1 l), respectively, obtained by Judge [12], except for the absence of VBIAS and ABIAS in his equations. Indeed, if VBIAS and ABIAS are set to zero, the two equations can be shown to be equivalent, and Judge’s conclusions would be valid. However, experimentally determined tonic accommodation (ABIAS) and tonic vergence (VBIAS) are usually not zero, and (38) would hold in general. Therefore, contrary to Judge’s conclusion based on the simplified model, the more complete linear model predicts that the AC /A ratio determined by the two methods would not be the same. Indeed, this is supported by experimental data [14], [15], [17], [19], [20]. (This should not be confused with the analysis of the linear model in the present article, which shows that associated and disassociated phorias (not AC/A ratios) are expected to be the same.) There are similarities but important differences between the methods used to obtain AC /A and those for obtaining associated and disassociated phorias. For example, one similarity is that in both there is an overall comparison between a closed-loop measure involving fixation disparity and a measurement of vergence response when the vergence system is open-looped. An important difference is that the associated phoria is measured at zero fixation disparity, whereas the derived curve for AC/A is obtained by examining the two fixation disparity curves obtained under lens and prism viewing conditions, and matching the lenses and prisms that give the same (not necessarily zero) fixation disparity. Another important difference is that from the quantitative analysis above, the complete static linear model predicts equality between associated and disassociated phorias, but inequality between the AC /A determined by the derived curve and the open-loop vergence methods. Judge [12] pointed out that the addition of nonlinearities would not affect the result because the derived curve is obtained at the same fixation disparity under prism and lens power viewing conditions, and thus the vergence system would be performing at the same operating point, regardless of the nonlinearity. However, this is not neces-
sarily correct. Consider a dead-space operator, such as depth of field or Panum’s fusional area. Under dual-interactive feedback, the output of the operator may be on either side of deadspace, depending on the relative magnitudes of the accommodative and vergence drives. Thus, it is possible that certain points of the prism viewing fixation disparity curve result from operating at one side of the deadspace, whereas corresponding points of the lens viewing fixation disparity curve operate at the other side of the deadspace. This would lead to erroneous results. The only way to determine the effect of the nonlinearities is to formally introduce them into the model, as has been done in this article, and examining the resulting sets of solutions. Therefore, quantitative analysis of the static nonlinear model would predict inequality between associated and disassociated phorias, as well as between AC/A determined by the derived-curve and open-loop vergence methods. The latter holds because the linear model already predicts an inequality, and introducing nonlinearities will tend to increase rather than decrease the inequality.
C. Fixation Disparity Curve The fixation disparity curve provides a means for discriminating between asymptomatic and symptomatic individuals. It has been proposed that the most effective parameter is the curve shape 1141, 1161, 1181, 1191, 1281. The next most effective parameter is the slope near the zero fixation disparity crossover. A flatter slope (below - 1 .O min of arc/A) is more compatible with asymptomatic vision, whereas a steeper slope is associated with symptoms. The intercept at the ordinate (Y intercept) is also a useful parameter, especially for exophoric (or exhibiting exo disassicated phoria) individuals. The least useful diagnostic parameter has been shown to be the X intercept, or the associated phoria [ 161, [ 181, [ 191, 1281. The reasons that the shape of the curve is the most effective diagnostic parameter may be revealed upon closer examination of Fig. 9. As discussed above, the slope of the model eso and exo lines is dependent on the controller gains and crosslink gains of both accommodative and vergence subsystems. In addition, the separation between the eso and exo line is dependent on Panum’s fusional area. In other words, many model variables are involved in the generation of the fixation disparity curve. These parameters appear to be robust to variations in the stimulus. For example, the slope of the fixation disparity curves remain approximately the same even when they shift in the eso direction with increased accommodative stimulus [14, p. 721. Therefore, large deviations in one or more model variables would be required to alter significantly the shape of the fixation disparity curve. Conversely, a change in the shape of the fixation disparity curve indicates abnormality in binocular oculomotor control, leading to symptons of discomfort. The slope at zero fixation disparity is a useful diagnos-
~
,I,
144
,
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL.
tic parameter because it reflects the closeness of the eso and exo lines for small vergence stimulus angles (see Fig. 9). This in turn is related to the size of Panum’s fusional area. A large Panum’s fusional area indicates a “sloppiness’’ in bifixation ability. This will be manifested as symptoms of ocular discomfort. The Y intercept is also dependent on the eso and exo lines but is more variable than the slope as it is determined by both the absolute positions of the eso and exo lines and the slope of the curve in the transition region. The X intercept, or associated phoria, is the least effective diagnostic parameter because, as shown by the above quantitative analysis, there is an inherent variability in this measure. This is because there are a number of ways the crossover may take place from the exo to the eso lines. In addition, the separation between the eso and exo lines also influences the location where the crossover will occur. These factors all contribute to the variability in the associated phoria, which in turn results in a less effective diagnostic indicator of ocular discomfort [29], [30].
D. Relation of Model Lines to Ogle-Type Fixation Disparity Curves Qualitatively, the four types of Ogle fixation disparity curves (Fig. 1) can be generated from the different PFA eso and exo lines. The Type I curve is generated as shown in Fig. 9. It starts from the lower right or larger BO vergence stimulus portion of the curve, where the large PFA exo line is the appropriate solution (VD = 18, Fig. 9). As the BO vergence stimulus decreases, the c’urve moves to the intermediate-PFA exo line (VD = 6). As vergence stimulus approaches zero, the curve moves along the small-PFA exo line (VD = 3). Similarly, when the curve crosses the horizontal axis, it moves along the small PFA eso line, and continues through the medium to the large PFA eso lines as the BI stimulus is increased. Other types of curves can be generated by means of shift or rotation of the normal eso and exo lines (see Fig. 5 ) . For example, suppose the model lines are shifted by a large amount to the left. At large vergence stimulus Values, the exo (lower) line values would be too large and result in diplopia. Even at smaller BO stimulus values, the exo line magnitudes are still too large. However, at this point eso lines become viable solutions. Thus, the fixation disparity curve is formed by tracing points on the remaining eso lines. This results in a Type I1 shaped curve. If the leftward shift were somewhat smaller, then the flat portion of the curve may occur on the exo side. Since the Type I1 curve was formed by a shift in the BI direction, BO prisms should help to counterbalance the effect. Indeed, prism prescription has been quite successful in treating patients with Type I1 curves [ 161. On the other hand, suppose the model lines are shifted to the right. This is the opposite situation from the previous example. At large BI stimulus values, the eso (upper) line magnitudes are too large, resulting in diplopia. Even for intermediate BI values, the eso line magnitudes
39, NO. 2, FEBRUARY 1992
are still too large. However, at this point the exo lines become appropriate solutions. Therefore, the curve is formed by following the exo lines and tracing out the remainder of the curve in the BO direction. This results in a Type I11 shaped curve. Type IV curves can be simulated with shallow-sloped eso and exo lines (via counter-clockwise rotation of the lines). Unlike Types 1-111, in which the appropriate solutions are VE3 for BO and VE2 for BI prisms, respectively, the Type IV curve, with its shallower slope, contains four solutions that are more equally appropriate. Thus there is a greater fluctuation between the different solutions, which in turn results in poor oculomotor stability. Indeed, Ogle [14, p. 881 noted that “the curve represents a less stable oculomotor fusion relationship.. . It almost always is found in patients with symptoms associated with problems of fusion and accommodation. ”
E. Future Work While the analyses presented above quantified the relationship between associated and disassociated phoria, additional factors concerning the generation of the fixation disparity curve need to be investigated. Namely, by fine-tuning the model parameters, additional simulations could be performed to fit accurately the experimental curves for different individuals. Also, a probabilistic component could be added to the model to simulate the different paths generated by the various combinations of eso and exo lines near the cross-over point. Finally, clinical application of the different types of fixation disparity curves should be reexamined in light of this new model for the fixation disparity curve. V. SUMMARY A linear static model of the human near response system was used to quantify the relationship between two vergence measures, called phorias. At a constant accommodative stimulus level, the vergence response under open-loop vergence is called the disassociated phoria, whereas the vergence stimulus giving zero fixation disparity under close-loop vergence is called the associated phoria. Analysis of the linear model showed that there is an exact correspondence between these two measures. Yet experimentally determined associated and disassociated phorias do not show a close correspondence. To account for the discrepancy in the experimental data, a nonlinear static model containing the deadspace operators DOF and PFA was evaluated. The model output contained four solutions corresponding to the four combinations of deadspace operator outputs. The four model solutions can be considered as two pairs of fixation disparity lines, with one pair for exo and the other for eso disparity. Analysis of the zero-fixation disparity crossover indicated that only one of four possible solutions would give the correct diassociated phoria value, although the other solutions are at least as likely. In fact, the other combinations give bias errors in the associated
HUNG: QUANTITATIVE ANALYSIS OF PHORIAS
esophoria direction. It is proposed that the bias errors and the variability in the transition region resulted in the scatter seen in the population plot of associated versus disassociated phoria found in the literature. Sensitivity analysis of parameters indicated that increasing PFA increased the separation between the two pairs of model solution lines, whereas increasing DOF increased the gap between the lines within each pair. Also, increasing accommodative stimulus level shifted all four model lines upwards. The curved shape of the Type I experimental fixation disparity plot was found to be the result of the continual transition from the large PFA exo line, through medium and then to small-PFA exo lines as BO vergence stimulus decreased, which then crossed over to the small PFA eso line, continuing through the medium and finally reaching the large PFA eso line as BI vergence increased. In addition, it was proposed that the other three types of Ogle curves could be explained by shifts and rotations of the eso and exo lines. ACKNOWLEDGMENT The author wishes to thank Dr. J. L. Semmlow of the Department of Biomedical Engineering, Rutgers University, and Dr. K. J. Ciuffreda and Dr. M. Rosenfield of SUNY/State College of Optometry for helpful comments, and L. Sun of Rutgers University for graphical assistance.
REFERENCES [I] F. W. Campbell, “The depth of field of the human eye,” Opr. Acta., vol. 4, pp. 157-164, Dec. 1957. [2] H. Ripps, N. B. Chin, J. M. Siegel, and G. M. Breinin, “Effect of pupil size on accommodation, vergence, and AC/A ratio,” Invest. Ophthal. Vis. Sci., vol. 1, pp. 127-135, Feb. 1962. [3] E. F. Fincham, “The accommodative reflex and its stimulus,” Brit. J. Ophrhal., vol. 35, pp. 381-393, July 1951. [4] G. A. Fry, “Blur of the retinal image,” Brit. J. Physiol. Opt., vol. 12, pp. 130-152, June 1955. [SI J. Westheimer and A. M. Mitchell, “Eye movement responses to convergence stimuli,” Arch. Ophthal., vol. 5 5 , pp. 848-856, June 1956. [6] M. W. Morgan, “Accommodation and vergence,” Amer. J. Optom. Arch. Amer. Acad. Oprom., vol. 45, pp. 415-454, July 1968. [7] G. K. Hung and J. L. Semmlow, “Static behavior of accommodation and vergence: computer simulation of an interactive dual-feedback system,” IEEE Trans. Biomed. Eng., vol. BME-27, pp. 439-447, Aug. 1980. [8] G. K. Hung and J . L. Semmlow, “A quantitative theory of control sharing between accommodative and vergence controllers, ” IEEE Trans. Biomed. Eng., vol. BME-29, pp. 364-370, May, 1982. [9] C. M. Schor, “Fixation disparity and vergence adaptation,” in Vergence Eye Movemenrs: Basis and Clinical Aspecrs, C. M. Schor and K. J . Ciuffruda, Eds. Butterworths, Boston, MA, 1983, pp. 465516. [IO] C. M. Schor and J. Kotulak, “Mutual interaction between accommodation and convergence are reduced by tonic adaptation,” in Adaprative Processes in Visual and Oculomotor Systems, E. L. Keller and D.S. Zee, Eds. New York: Pergamon, 1986, pp. 135-143. [I I] C. M. Schor and D. Homer, “Adaptative disorders of accommodation and vergence in binocular dysfunction,” Ophthal. Physiol. Optics, vol. 9, pp. 264-268, 1989. [12] S . J . Judge, “Can current models of accommodation and vergence control account for the discrepancies between AC /A measurements made by the fixation disparity and phoria methods?,” Vis. Res., vol. 25, pp. 1999-2001, 1985.
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I. M. Borish, Clinical Refraction, 3rd ed. Chicago, IL: Professional Press, 1970. K. N. Ogle, T. G. Martens, and J. A. Dyer, Oculomoror Imbalance in Binocular Vision and Firadon Disparity. Philadelphia, PA: Lea and Febiger, 1967, pp. 9-119. D. B. Carter, “Parameters of fixation disparity,” Amer. J . Physiol. Opt., vol. 57, pp 610-617, Sept. 1980. J . E. Sheedy, “Fixation disparity analysis of oculomotor imbalance,” Amer. J. Opromer. Physiol. Opr., vol. 57, pp. 632-639, Sept. 1980. J. J. Saladin and J. E. Sheedy, “Population study of fixation disparity, heterophoria, and vergence,” Amer. J. Optomer. Physiol. Opr., vol. 55, pp. 744-750, NOV.1978. J. E. Sheedy and J. J . Saladin, “Phoria, vergence, and fixation disparity in oculomotor problems,” Amer. J. Opromer. Physiol. Opt., vol. 54, pp. 474-478, July 1977. -, “Association of symptoms with measures of oculomotor deficiencies,” Amer. J. Opromer. Physiol. Opt., vol. 55, pp. 670-676, Oct. 1978. B. Wicks and R. London, “Analysis of binocular visual functions using tests made under binocular conditions,” Amer. J. Opromer. Physiol. Opt., vol. 64, pp. 227-240, Apr. 1987. M. Schapero, D. Cline, and H. W. Hofstetter. Dictionary of Visual Science. Philadelphia, PA: Chilton Book, 1968. K. N. Ogle, Researches in Binocular Vision. New York: Hafner, 1972, pp. 64-93. H. W. Leibowitz and D. A. Owens, “Anomalous myopias and the intermediate dark focus of accommodation,” Science, vol. 189, pp. 646-648, 1975. T. L. Amerson and D.H. Mershon, “Time-of-day variations in oculomotor function: I. Tonic accommodation and tonic vergence, ” Ophrhal. Physiol. Opt., vol. 8, pp. 415-422, Oct., 1988. B. Gilmartin, R. E. Hogan, and S. M. Thompson, “The effect of timolol maleate on tonic accommodation, tonic vergence and pupil diameter,” Invest. Ophrhal. Vis. Sci., vol. 25, pp. 763-770, 1984. M. Rosenfield and K. J. Ciuffreda, “Distance heterophoria and tonic vergence,” Oprom. Vis. Sci., vol. 67, pp. 667-669, 1990. G. K. Hung, “Linear model of accommodation and vergence can account for discrepancies between AC /A measures using the fixation disparity and phoria methods,” Ophrhal. Physiol. Opr., vol. 11, pp. 275-278, July 1991. C. R. Payne, J. D. Grisham, and K. L. Thomas, “A clinical evaluation of fixation disparity,” Amer. J. Opromer. Physiol. Opt., vol. 51, pp. 88-90, Feb. 1974. L. D. Pickwell, J . M. Gilchrist, and J . Hesler, “Comparison of associated heterophoria measurements using the Mallett test for near vision and the Sheedy disparometer,” Ophthal. Physiol. Opr., vol. 8, pp. 19-25, Jan. 1988. J. Cooper, J. Feldman, D. Horn, and C. Dibble, “Reliability of fixation disparity curves,” Amer. J. Optomer. Physiol. Opr., vol. 5 8 , pp. 960-964, NOV.1981.
George K. Hung (M’82-SM’90) was born in Shanghai, China in 1947. He received the B.S. degree in mechanical engineering, the M.S. degree in bioengineering, and the Ph.D. degree in physiological optics from the University of California, Berkeley, in 1970, 1971, and 1977, respectively. He joined the faculty at Rutgers University, Piscataway, NJ, in 1978, where he is currently an Associate Professor of Biomedical Engineering. He has published extensively in the areas of experimentation and modeling of ;he human accommddation and vergence systems. In 1986, he was a Visiting Scholar at the Shanghai Institute of Physiology, where he investigated the pupillary control mechanism. His current research interests include voluntary and ptoximal accommodation and vergence, brain activity during eye movements using single photon emission camputed tomography (SPECT), measuement of speed of noninertial shifts in focal attention, application of human eye movement control to machine vision, suppression in the vergence system, and modeling of eye movement control systems. Dr. Hung is a member of Sigma Xi and Association for Research in Vision and Ophthalmology.
I35
Quantitative Analysis of Associated and Disassociated Phorias: Linear and Nonlinear Static Models George K. Hung, Senior Member, IEEE
Abstract-Ogle proposed two measures of oculomotor balance, called associated and disassociated phorias, which he assumed were equivalent. However, experimentally determined vaues of these phorias do not show a close correspondence. To analyze the rationale behind Ogle's assumption of equality, a linear static model was evaluated. It was found that indeed the linear model predicts an exact correspondence between associated and disassociated phorias. Thus, his assumption depended on the presence of a linear model. To account for the discrepancy between these two measures, a nonlinear static model, containing the dead space operators depth of field and Panum's fusional area, was evaluated. Four equations for fixation disparity were derived corresponding to the four combinations of deadspace operator outputs. It was found that only one of these four equally possible solutions for associated phoria correspondedto the disassociated phoria. This suggests that the variability in the four solutions may account for the scatter in the experimental data. The nonlinear model was analyzed further to determine its sensitivity to parameter changes and to show how such a model could generate the classical shape of the fixation disparity curve.
I. INTRODUCTION HE human near response consists of focusing of the lens in each eye, inward rotation of the two eyes, and pupillary constriction. ' The focusing, or accommodation system, senses blur of the retinal image and changes lens curvature to achieve retinal image clarity [3], [4]. The eye-turn, or vergence system, senses disparity between the retinal images in the two eyes and rotates the eyes to attain a single percept [5], [6]. Both systems are therefore feedback control systems [7], [8]. In addition, there are cross-link interactions between the two systems [7], [9][12]. That is, the accommodation system can drive vergence and the vergence system can drive accommodation. One important measure of the cross-link influence is the accommodative convergence to accommodation stimulus ratio (or AC/A)under monocular, or, vergence open-loop condition. A higher AC/A ratio, for example, indicates
T
Manuscript received February 13, 1991; revised July 25, 1991. This work was supported in part by National Institutes of Health Grant EY7519. The author is with the Department of Biomedical Engineering, Rutgers University, Piscataway, NJ 08855. IEEE Log Number 9104909. 'Pupillary constriction decreases the amount of light entering the eye and increases the depth of field [I]. However, under normal viewing conditions ( > 1.5 mm pupil diameter), the influence of pupil on accommodation response is small [2], and the pupillary feedback to accommodation can be considered to be open-looped.
a greater influence of accommodation on convergence. Values of AC /A outside of the normal range are taken as signs of oculomotor abnormality [ 131. Indeed, Schor and others have found that abnormal AC/A ratios may influence the adaptative properties of the near response system [91, [111. Ogle [ 141 proposed that a more informative measure of oculomotor insufficiency may be obtained by recording the vergence error, called fixation disparity, under the more natural binocular, or vergence closed-loop condition. Thus, in a group of subjects, the fixation disparity was measured for different vergence stimuli while the accommodative stimulus was held constant.* The resulting plots of fixation disparity versus vergence stimulus were examined in terms of the shapes of the curves. Ogle categorized these into four main types (Fig. 1). Type I was associated with lack of symptom, whereas Types 11, 111, and IV were associated with symptoms of discomfort [14]-[16]. He also defined two vergence measures, called phorias, under a constant accommodative stimulus condition. For monocular viewing, the vergence response was called the disassociated phoria. On the other hand, for binocular viewing, the vergence stimulus giving zero fixation disparity was called the associated phoria. The rationale for using the associated phoria was that this vergence stimulus value was assumed to indicate the point of oculomotor balance under binocular conditions. Ogle sought to determine whether the point of oculomotor balance is the same under binocular (associated phoria) and monocular (disassociated phoria) conditions. He therefore measured the associated and disassociated phorias in a number of individuals. The resulting plot of associated versus disassociated phoria showed data that were spread mainly parallel to a 45" line, although there was a relatively wide scatter and a bias in the associated esophoria direction (i.e., a relatively more eso3 associated than disassociated phoria) (Fig. 2) [ 141. Nevertheless, some of the data points did show an equality between associated and disassociated phoria. He attributed the extent of the scatter to such factors as insufficient effort of ac*For a clear description of the experimental setup and procedure, see [ 14, pp. 26-38]. 'Eso describes an angle of two eyes which is more convergent than absolute alignment with the target. On the other hand, exo describes a more divergent angle than absolute alignment.
0018-9294/92$03.00 0 1992 IEEE
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IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 2, FEBRUARY 1992 PIXATION DISPARITY
PIXATION DISPARITY
,
730
PRISM BASE-IN, 30
Y
I
1
-
4
2
_.................
VEI VE2
VE3 VE4
5
0
10
15 BO,
20
25
L j , ,
30
A
Fig. 5 . Nonlinear model simulation lines under closed-loop vergence. Upper pair of eso lines correspond to combinations 1 and 2 (VEl and VE2) in Table I. Lower pair of exo lines correspond to combinations 3 and 4 (VE3 and V U ) . Vertical dashed lines indicates where VE2 = - V U . The intersection with the horizontal line at zero fixation disparity gives the vergence stimulus value, or associated phoria. Simulation parameter values were based on experimental data (subj. JS) [7]. AS AD 2 .15
25 20
25
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2
_____
1
20
-
25
-
30
-
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,
,
,
10
15
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25
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Fig. 8. Increasing accommodative stimulus level raises the set of four model lines in the eso direction.
was expanded to include BI stimuli up to 30 A. The appropriate simulated fixation disparity points were determined as follows. For a large vergence stimulus value, the vergence error is positive (exo) (i.e., vergence stimulus is greater than the response) and the accommodative error is negative. Thus, the appropriate solution is the exo line VE3. On the other hand, for a large BI vergence stimulus value the opposite situation holds, where the vergence error is negative (eso) and the accommodative error is positive. In this case, the appropriate solution is the eso line VE2. The gradual transition in the intermediate range follows the diminution of PFA with decreased BO vergence stimulus amplitude, and the enlargement of PFA with increased BI stimulus amplitude. Therefore, the simulated fixation disparity curve in Fig. 9 starts from the large PFA (VD = 18 min of arc) exo line at a large BO value. It then progressed through the intermediate-PFA (VD = 6 min of arc) exo line at an intermediate BO value to the small-PFA (VD = 3 min of arc) exo line at a small BO value. The sequence is repeated but from small-to-large PFA eso lines as BI vergence stimulus is increased. The solid curve traces a fixation disparity plot (for subj. JS [7]) whose shape is similar to experimental Type I fixation disparity curves.
VD 18
B. Crossing at Zero Fixation Disparity It is reasonable to assume that the point where the exo line (VE3 or VE4) crosses the zero fixation disparity line to the eso line (VE1 or VE2) occurs when the errors are equal in magnitude and opposite in sign (see dashed vertical line in Fig. 5). The most likely candidates are VE3, applicable for large vergence and small accommodative stimuli, and VE2, applicable for small vergence and large accommodative stimuli. The equation for VE2 is given by
I
VE2 =
5
BO.
_ _ _ _ _ 0.30 - 0.15
15-
20
0
6
-
,
30
+ ACG) . (VS - VD VCG + AD ACG . AC - VBIAS) - AD - ACG + VD VCG CA + ABIAS) - ACG AC] (1 + ACG) (1 + VCG) - ACG AC . VCG . CA
+ (-AS
*
*
*
*
*
(33)
IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, VOL. 39, NO. 2, FEBRUARY 1992
142 AS AD 2 .15
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401
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1
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For AD = 0.15 D, the difference is about 2 A baseout (i.e., in the overconverged, or eso direction). It can be shown that for the remaining two combinations, namely VE1 = -VE4 and VE2 = -VE3, the difference is about 1 A baseout. This suggests that a population sample should show a bias in the associated esophoria (base-out) direction. This indeed is seen in the data presented by Ogle et al. (Fig. 1) [14, pp. 109, 1111.
VD
20
10
0
30
20
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(A)
El
BO
Fig. 9. Simulated fixation disparity curve starting from large-PFA exo line through a continuum of decreasing PFA exo lines as BO amplitude decreases. It then traces through a continuum of increasing PFA eso lines as BI amplitude increases. The resulting curve (solid) is similar in shape to the Ogle Type I fixation disparity curve.
The equation for VE3 was specified earlier in (31). Solving for VE2 = -VE3, or VE2 VE3 = 0 gives
+
vs(VE2 = -VE3)
=
1
+ ACG
1
+ ACG - ABIAS - AC].
(34)
Comparing (34) with (22) reveals that the term AD, which was summed with ABIAS in (22), is missing in (34). This is due to cancellation of the terms containing AD in (31) and (33). It turns out that the only combination which gives the same result as (22), or disassociated phoria, is VE2 and VE4. Thus, v s ( V E 2 = -VM) =
1
+ ACG
*
AC
- AS
ACG 1 ACG
+
1
(35)
+ ACG - AD - AC.
(36)
(ABIAS
+ AD)
*
AC
.
Therefore, in order to obtain a point where the associated phoria [VS(VE2= -vE4) in (35)] is equal to the disassociated phoria [VR in (22)], the system must move through VE4, where both accommodation and vergence are underdriven, to VE2, where accommodation is underdriven and vergence is overdriven. Clearly, this condition is not always attained since as other combinations are possible. The largest difference between associated and disassociated phorias occurs with the combination VE 1 and VE3. = -VE3) It can be shown that the difference between VS(VEI and the disassociated phoria [see (22)] is Differencevs,,,,
= -VE3)
- 2 -
1
*
ACG
IV. DISCUSSION A. Relationship Between Associated and Disassociated Phorias Although Ogle [14] did not state his ideas in control system terms, his notion of binocular oculomotor balance was deeply insightful. For if there were no deadspace operators, as in the linear model, the associated phoria would be equal to the disassociated phoria. This is because at zero-fixation disparity, the output of the vergence controller would be zero, which corresponds exactly to the condition of open-loop vergence. However, experimental measures of the phorias did not show such an exact correspondence. As mentioned previously, dynamic aspects of accommodation and vergence could not have accounted for the scatter in the plot of associated versus disassociated phoria. Instead, the scatter was attributed primarily to static parameters, such as the deadspace operators DOF and PFA, which gave four possible solutions. Thus, rather than giving totally unreasonable results, the experimental associated phoria technique gave values that were somewhat near the disassociated phoria, but was inherently prone to bias errors and variability. This was unfortunate because without knowing the results presented in this quantitative analysis, one would have been tempted to assume that through careful measurement and large number of trials, a distinct relationship between associated and disassociated phoria would reveal itself. This did not happen, as was documented by Ogle et al. [14] and supported by others [15], [17], [19], r201-
B. Determination of the A C / A Ratio The cross-link gain, AC, can be obtained by two different methods [ 141: first, in terms of the prism power and lens power required to give the same fixation disparity when both systems are close-looped, and second, in terms of accommodative convergence when vergence is openlooped. Judge [ 121 derived algebraic expressions for AC under these two conditions based on the linear model of Hung and Semmlow [7]. Since his derived equations were equivalent, he concluded that the available models were not able to account for the differences in the accommodative-convergence to accommodative stimulus (AC /A)8 ratios obtained experimentally under the two methods. However, in his analysis, he neglected to include two im'For the open-loop vergence case, a plot of disassociated phorias versus different accommodative stimuli gives the complete AC /A line.
143
HUNG: QUANTITATIVE ANALYSIS OF PHORIAS
portant parameters-tonic accommodation, ABIAS, and tonic vergence, VBIAS [7], [23]-[26]. It can be shown [27] that the accommodative convergence cross-link gain, AC, obtained using the fixation disparity method is
P AC=-. L
1 +ACG ACG
(37)
where P is the prism power (prism viewing only) which gives the same fixation disparity as the lens power L (lens viewing only). On the other hand, the AC obtained when vergence is open-looped can be shown to be AC =
VR - VBIAS L - ABIAS
1
+ ACG ACG
*
(38)
Equations (37) and (38) are the same (although the symbols were different) as (8) and (1 l), respectively, obtained by Judge [12], except for the absence of VBIAS and ABIAS in his equations. Indeed, if VBIAS and ABIAS are set to zero, the two equations can be shown to be equivalent, and Judge’s conclusions would be valid. However, experimentally determined tonic accommodation (ABIAS) and tonic vergence (VBIAS) are usually not zero, and (38) would hold in general. Therefore, contrary to Judge’s conclusion based on the simplified model, the more complete linear model predicts that the AC /A ratio determined by the two methods would not be the same. Indeed, this is supported by experimental data [14], [15], [17], [19], [20]. (This should not be confused with the analysis of the linear model in the present article, which shows that associated and disassociated phorias (not AC/A ratios) are expected to be the same.) There are similarities but important differences between the methods used to obtain AC /A and those for obtaining associated and disassociated phorias. For example, one similarity is that in both there is an overall comparison between a closed-loop measure involving fixation disparity and a measurement of vergence response when the vergence system is open-looped. An important difference is that the associated phoria is measured at zero fixation disparity, whereas the derived curve for AC/A is obtained by examining the two fixation disparity curves obtained under lens and prism viewing conditions, and matching the lenses and prisms that give the same (not necessarily zero) fixation disparity. Another important difference is that from the quantitative analysis above, the complete static linear model predicts equality between associated and disassociated phorias, but inequality between the AC /A determined by the derived curve and the open-loop vergence methods. Judge [12] pointed out that the addition of nonlinearities would not affect the result because the derived curve is obtained at the same fixation disparity under prism and lens power viewing conditions, and thus the vergence system would be performing at the same operating point, regardless of the nonlinearity. However, this is not neces-
sarily correct. Consider a dead-space operator, such as depth of field or Panum’s fusional area. Under dual-interactive feedback, the output of the operator may be on either side of deadspace, depending on the relative magnitudes of the accommodative and vergence drives. Thus, it is possible that certain points of the prism viewing fixation disparity curve result from operating at one side of the deadspace, whereas corresponding points of the lens viewing fixation disparity curve operate at the other side of the deadspace. This would lead to erroneous results. The only way to determine the effect of the nonlinearities is to formally introduce them into the model, as has been done in this article, and examining the resulting sets of solutions. Therefore, quantitative analysis of the static nonlinear model would predict inequality between associated and disassociated phorias, as well as between AC/A determined by the derived-curve and open-loop vergence methods. The latter holds because the linear model already predicts an inequality, and introducing nonlinearities will tend to increase rather than decrease the inequality.
C. Fixation Disparity Curve The fixation disparity curve provides a means for discriminating between asymptomatic and symptomatic individuals. It has been proposed that the most effective parameter is the curve shape 1141, 1161, 1181, 1191, 1281. The next most effective parameter is the slope near the zero fixation disparity crossover. A flatter slope (below - 1 .O min of arc/A) is more compatible with asymptomatic vision, whereas a steeper slope is associated with symptoms. The intercept at the ordinate (Y intercept) is also a useful parameter, especially for exophoric (or exhibiting exo disassicated phoria) individuals. The least useful diagnostic parameter has been shown to be the X intercept, or the associated phoria [ 161, [ 181, [ 191, 1281. The reasons that the shape of the curve is the most effective diagnostic parameter may be revealed upon closer examination of Fig. 9. As discussed above, the slope of the model eso and exo lines is dependent on the controller gains and crosslink gains of both accommodative and vergence subsystems. In addition, the separation between the eso and exo line is dependent on Panum’s fusional area. In other words, many model variables are involved in the generation of the fixation disparity curve. These parameters appear to be robust to variations in the stimulus. For example, the slope of the fixation disparity curves remain approximately the same even when they shift in the eso direction with increased accommodative stimulus [14, p. 721. Therefore, large deviations in one or more model variables would be required to alter significantly the shape of the fixation disparity curve. Conversely, a change in the shape of the fixation disparity curve indicates abnormality in binocular oculomotor control, leading to symptons of discomfort. The slope at zero fixation disparity is a useful diagnos-
~
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tic parameter because it reflects the closeness of the eso and exo lines for small vergence stimulus angles (see Fig. 9). This in turn is related to the size of Panum’s fusional area. A large Panum’s fusional area indicates a “sloppiness’’ in bifixation ability. This will be manifested as symptoms of ocular discomfort. The Y intercept is also dependent on the eso and exo lines but is more variable than the slope as it is determined by both the absolute positions of the eso and exo lines and the slope of the curve in the transition region. The X intercept, or associated phoria, is the least effective diagnostic parameter because, as shown by the above quantitative analysis, there is an inherent variability in this measure. This is because there are a number of ways the crossover may take place from the exo to the eso lines. In addition, the separation between the eso and exo lines also influences the location where the crossover will occur. These factors all contribute to the variability in the associated phoria, which in turn results in a less effective diagnostic indicator of ocular discomfort [29], [30].
D. Relation of Model Lines to Ogle-Type Fixation Disparity Curves Qualitatively, the four types of Ogle fixation disparity curves (Fig. 1) can be generated from the different PFA eso and exo lines. The Type I curve is generated as shown in Fig. 9. It starts from the lower right or larger BO vergence stimulus portion of the curve, where the large PFA exo line is the appropriate solution (VD = 18, Fig. 9). As the BO vergence stimulus decreases, the c’urve moves to the intermediate-PFA exo line (VD = 6). As vergence stimulus approaches zero, the curve moves along the small-PFA exo line (VD = 3). Similarly, when the curve crosses the horizontal axis, it moves along the small PFA eso line, and continues through the medium to the large PFA eso lines as the BI stimulus is increased. Other types of curves can be generated by means of shift or rotation of the normal eso and exo lines (see Fig. 5 ) . For example, suppose the model lines are shifted by a large amount to the left. At large vergence stimulus Values, the exo (lower) line values would be too large and result in diplopia. Even at smaller BO stimulus values, the exo line magnitudes are still too large. However, at this point eso lines become viable solutions. Thus, the fixation disparity curve is formed by tracing points on the remaining eso lines. This results in a Type I1 shaped curve. If the leftward shift were somewhat smaller, then the flat portion of the curve may occur on the exo side. Since the Type I1 curve was formed by a shift in the BI direction, BO prisms should help to counterbalance the effect. Indeed, prism prescription has been quite successful in treating patients with Type I1 curves [ 161. On the other hand, suppose the model lines are shifted to the right. This is the opposite situation from the previous example. At large BI stimulus values, the eso (upper) line magnitudes are too large, resulting in diplopia. Even for intermediate BI values, the eso line magnitudes
39, NO. 2, FEBRUARY 1992
are still too large. However, at this point the exo lines become appropriate solutions. Therefore, the curve is formed by following the exo lines and tracing out the remainder of the curve in the BO direction. This results in a Type I11 shaped curve. Type IV curves can be simulated with shallow-sloped eso and exo lines (via counter-clockwise rotation of the lines). Unlike Types 1-111, in which the appropriate solutions are VE3 for BO and VE2 for BI prisms, respectively, the Type IV curve, with its shallower slope, contains four solutions that are more equally appropriate. Thus there is a greater fluctuation between the different solutions, which in turn results in poor oculomotor stability. Indeed, Ogle [14, p. 881 noted that “the curve represents a less stable oculomotor fusion relationship.. . It almost always is found in patients with symptoms associated with problems of fusion and accommodation. ”
E. Future Work While the analyses presented above quantified the relationship between associated and disassociated phoria, additional factors concerning the generation of the fixation disparity curve need to be investigated. Namely, by fine-tuning the model parameters, additional simulations could be performed to fit accurately the experimental curves for different individuals. Also, a probabilistic component could be added to the model to simulate the different paths generated by the various combinations of eso and exo lines near the cross-over point. Finally, clinical application of the different types of fixation disparity curves should be reexamined in light of this new model for the fixation disparity curve. V. SUMMARY A linear static model of the human near response system was used to quantify the relationship between two vergence measures, called phorias. At a constant accommodative stimulus level, the vergence response under open-loop vergence is called the disassociated phoria, whereas the vergence stimulus giving zero fixation disparity under close-loop vergence is called the associated phoria. Analysis of the linear model showed that there is an exact correspondence between these two measures. Yet experimentally determined associated and disassociated phorias do not show a close correspondence. To account for the discrepancy in the experimental data, a nonlinear static model containing the deadspace operators DOF and PFA was evaluated. The model output contained four solutions corresponding to the four combinations of deadspace operator outputs. The four model solutions can be considered as two pairs of fixation disparity lines, with one pair for exo and the other for eso disparity. Analysis of the zero-fixation disparity crossover indicated that only one of four possible solutions would give the correct diassociated phoria value, although the other solutions are at least as likely. In fact, the other combinations give bias errors in the associated
HUNG: QUANTITATIVE ANALYSIS OF PHORIAS
esophoria direction. It is proposed that the bias errors and the variability in the transition region resulted in the scatter seen in the population plot of associated versus disassociated phoria found in the literature. Sensitivity analysis of parameters indicated that increasing PFA increased the separation between the two pairs of model solution lines, whereas increasing DOF increased the gap between the lines within each pair. Also, increasing accommodative stimulus level shifted all four model lines upwards. The curved shape of the Type I experimental fixation disparity plot was found to be the result of the continual transition from the large PFA exo line, through medium and then to small-PFA exo lines as BO vergence stimulus decreased, which then crossed over to the small PFA eso line, continuing through the medium and finally reaching the large PFA eso line as BI vergence increased. In addition, it was proposed that the other three types of Ogle curves could be explained by shifts and rotations of the eso and exo lines. ACKNOWLEDGMENT The author wishes to thank Dr. J. L. Semmlow of the Department of Biomedical Engineering, Rutgers University, and Dr. K. J. Ciuffreda and Dr. M. Rosenfield of SUNY/State College of Optometry for helpful comments, and L. Sun of Rutgers University for graphical assistance.
REFERENCES [I] F. W. Campbell, “The depth of field of the human eye,” Opr. Acta., vol. 4, pp. 157-164, Dec. 1957. [2] H. Ripps, N. B. Chin, J. M. Siegel, and G. M. Breinin, “Effect of pupil size on accommodation, vergence, and AC/A ratio,” Invest. Ophthal. Vis. Sci., vol. 1, pp. 127-135, Feb. 1962. [3] E. F. Fincham, “The accommodative reflex and its stimulus,” Brit. J. Ophrhal., vol. 35, pp. 381-393, July 1951. [4] G. A. Fry, “Blur of the retinal image,” Brit. J. Physiol. Opt., vol. 12, pp. 130-152, June 1955. [SI J. Westheimer and A. M. Mitchell, “Eye movement responses to convergence stimuli,” Arch. Ophthal., vol. 5 5 , pp. 848-856, June 1956. [6] M. W. Morgan, “Accommodation and vergence,” Amer. J. Optom. Arch. Amer. Acad. Oprom., vol. 45, pp. 415-454, July 1968. [7] G. K. Hung and J. L. Semmlow, “Static behavior of accommodation and vergence: computer simulation of an interactive dual-feedback system,” IEEE Trans. Biomed. Eng., vol. BME-27, pp. 439-447, Aug. 1980. [8] G. K. Hung and J . L. Semmlow, “A quantitative theory of control sharing between accommodative and vergence controllers, ” IEEE Trans. Biomed. Eng., vol. BME-29, pp. 364-370, May, 1982. [9] C. M. Schor, “Fixation disparity and vergence adaptation,” in Vergence Eye Movemenrs: Basis and Clinical Aspecrs, C. M. Schor and K. J . Ciuffruda, Eds. Butterworths, Boston, MA, 1983, pp. 465516. [IO] C. M. Schor and J. Kotulak, “Mutual interaction between accommodation and convergence are reduced by tonic adaptation,” in Adaprative Processes in Visual and Oculomotor Systems, E. L. Keller and D.S. Zee, Eds. New York: Pergamon, 1986, pp. 135-143. [I I] C. M. Schor and D. Homer, “Adaptative disorders of accommodation and vergence in binocular dysfunction,” Ophthal. Physiol. Optics, vol. 9, pp. 264-268, 1989. [12] S . J . Judge, “Can current models of accommodation and vergence control account for the discrepancies between AC /A measurements made by the fixation disparity and phoria methods?,” Vis. Res., vol. 25, pp. 1999-2001, 1985.
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I. M. Borish, Clinical Refraction, 3rd ed. Chicago, IL: Professional Press, 1970. K. N. Ogle, T. G. Martens, and J. A. Dyer, Oculomoror Imbalance in Binocular Vision and Firadon Disparity. Philadelphia, PA: Lea and Febiger, 1967, pp. 9-119. D. B. Carter, “Parameters of fixation disparity,” Amer. J . Physiol. Opt., vol. 57, pp 610-617, Sept. 1980. J . E. Sheedy, “Fixation disparity analysis of oculomotor imbalance,” Amer. J. Opromer. Physiol. Opr., vol. 57, pp. 632-639, Sept. 1980. J. J. Saladin and J. E. Sheedy, “Population study of fixation disparity, heterophoria, and vergence,” Amer. J. Optomer. Physiol. Opr., vol. 55, pp. 744-750, NOV.1978. J. E. Sheedy and J. J . Saladin, “Phoria, vergence, and fixation disparity in oculomotor problems,” Amer. J. Opromer. Physiol. Opt., vol. 54, pp. 474-478, July 1977. -, “Association of symptoms with measures of oculomotor deficiencies,” Amer. J. Opromer. Physiol. Opt., vol. 55, pp. 670-676, Oct. 1978. B. Wicks and R. London, “Analysis of binocular visual functions using tests made under binocular conditions,” Amer. J. Opromer. Physiol. Opt., vol. 64, pp. 227-240, Apr. 1987. M. Schapero, D. Cline, and H. W. Hofstetter. Dictionary of Visual Science. Philadelphia, PA: Chilton Book, 1968. K. N. Ogle, Researches in Binocular Vision. New York: Hafner, 1972, pp. 64-93. H. W. Leibowitz and D. A. Owens, “Anomalous myopias and the intermediate dark focus of accommodation,” Science, vol. 189, pp. 646-648, 1975. T. L. Amerson and D.H. Mershon, “Time-of-day variations in oculomotor function: I. Tonic accommodation and tonic vergence, ” Ophrhal. Physiol. Opt., vol. 8, pp. 415-422, Oct., 1988. B. Gilmartin, R. E. Hogan, and S. M. Thompson, “The effect of timolol maleate on tonic accommodation, tonic vergence and pupil diameter,” Invest. Ophrhal. Vis. Sci., vol. 25, pp. 763-770, 1984. M. Rosenfield and K. J. Ciuffreda, “Distance heterophoria and tonic vergence,” Oprom. Vis. Sci., vol. 67, pp. 667-669, 1990. G. K. Hung, “Linear model of accommodation and vergence can account for discrepancies between AC /A measures using the fixation disparity and phoria methods,” Ophrhal. Physiol. Opr., vol. 11, pp. 275-278, July 1991. C. R. Payne, J. D. Grisham, and K. L. Thomas, “A clinical evaluation of fixation disparity,” Amer. J. Opromer. Physiol. Opt., vol. 51, pp. 88-90, Feb. 1974. L. D. Pickwell, J . M. Gilchrist, and J . Hesler, “Comparison of associated heterophoria measurements using the Mallett test for near vision and the Sheedy disparometer,” Ophthal. Physiol. Opr., vol. 8, pp. 19-25, Jan. 1988. J. Cooper, J. Feldman, D. Horn, and C. Dibble, “Reliability of fixation disparity curves,” Amer. J. Optomer. Physiol. Opr., vol. 5 8 , pp. 960-964, NOV.1981.
George K. Hung (M’82-SM’90) was born in Shanghai, China in 1947. He received the B.S. degree in mechanical engineering, the M.S. degree in bioengineering, and the Ph.D. degree in physiological optics from the University of California, Berkeley, in 1970, 1971, and 1977, respectively. He joined the faculty at Rutgers University, Piscataway, NJ, in 1978, where he is currently an Associate Professor of Biomedical Engineering. He has published extensively in the areas of experimentation and modeling of ;he human accommddation and vergence systems. In 1986, he was a Visiting Scholar at the Shanghai Institute of Physiology, where he investigated the pupillary control mechanism. His current research interests include voluntary and ptoximal accommodation and vergence, brain activity during eye movements using single photon emission camputed tomography (SPECT), measuement of speed of noninertial shifts in focal attention, application of human eye movement control to machine vision, suppression in the vergence system, and modeling of eye movement control systems. Dr. Hung is a member of Sigma Xi and Association for Research in Vision and Ophthalmology.
