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axis of a s t i g m a t i s m . We t r e a t e d a p a t i e n t w i t h g e n e r a l i z e d fibrosis s y n d r o m e w h o h a d a sig nificant a n d p e r s i s t e n t c h a n g e in r e f r a c t i v e er ror after s t r a b i s m u s s u r g e r y . A 2 - w e e k - o l d girl was first e x a m i n e d b y a pediatric ophthalmologist (M.G.) because she h a d m a r k e d b l e p h a r o p t o s i s . At that t i m e , the p a t i e n t h a d a +2.00 r e f r a c t i o n b i l a t e r a l l y . S h e was r e e x a m i n e d on n u m e r o u s o c c a s i o n s b e cause of her marked blepharoptosis and chinup p o s t u r e , a g e n e r a l i z e d d e c r e a s e in o c u l a r m o v e m e n t s w i t h i n f r a d u c t i o n o f the g l o b e s b i l a t e r a l l y , facial w e a k n e s s , and difficulty w i t h s w a l l o w i n g . At 2 y e a r s of a g e , c y c l o p l e g i c r e fraction h a d p r o g r e s s e d to R . E . : - 1 6 . 0 0 -1-10.00 X 1 0 5 a n d L . E . : - 1 6 . 0 0 -1-8.00 x 7 5 . R e p e a t r e f r a c t i o n m e a s u r e m e n t s e v e r a l w e e k s later w a s R . E . : - 1 6 . 0 0 -t-8.00 x 9 0 a n d L . E . : - 1 6 . 0 0 + 8.00 X 9 0 . The patient's refraction was inde p e n d e n t l y m e a s u r e d b y two a d d i t i o n a l p e d i a t ric o p h t h a l m o l o g i s t s , a n d t h e s e m e a s u r e m e n t s w e r e n e a r l y i d e n t i c a l to the i n i t i a l m e a s u r e m e n t . B e c a u s e o f the p a t i e n t ' s s i g n i f i c a n t c h i n up p o s t u r e a n d m a r k e d i n f r a d u c t i o n o f the globes, she underwent bilateral inferior rectus m u s c l e r e c e s s i o n o f 12 m m a n d total i n f e r i o r r e c t u s m u s c l e d i s i n s e r t i o n in e a c h e y e . F o r c e d d u c t i o n t e s t i n g at the t i m e o f the i n i t i a l o p e r a tion d i s c l o s e d significant r e s t r i c t i o n to e l e v a tion. Repeat refraction m e a s u r e m e n t s two m o n t h s after the o p e r a t i o n d i s c l o s e d a r e f r a c t i v e e r r o r of R . E . : - 1 2 . 0 0 +5.00 x 100 and L.E.: - 1 2 . 0 0 -1-5.00 X 9 0 . B e c a u s e o f the m a r k e d c h a n g e , r e f r a c t i o n w a s a g a i n m e a s u r e d at four m o n t h s a n d six m o n t h s after the final o p e r a t i o n . T h e m y o p i a a n d a s t i g m a t i c e r r o r h a d d e c r e a s e d to R . E . : - 6 . 5 0 - ^ 1 . 5 0 x 9 0 a n d L . E . : - 5 . 0 0 +2.00 X 1 0 5 . O n e y e a r later, v i s u a l a c u i t y was 2 0 / 5 0 in e a c h eye w i t h this r e f r a c t i o n . T h e findings in our p a t i e n t are u n u s u a l in t h e d e g r e e to w h i c h the r e f r a c t i v e e r r o r c h a n g e d after s t r a b i s m u s s u r g e r y . W h e r e a s o t h e r a u t h o r s have s p e c u l a t e d t h a t s m a l l e r , m o r e t r a n s i e n t c h a n g e s in p o s t o p e r a t i v e a s t i g m a t i s m a r e c a u s e d b y e y e l i d s w e l l i n g or by a c h a n g e in t h e p o s i t i o n of the eye r e l a t i v e to the e y e l i d s , the findings in o u r p a t i e n t s e e m to confirm t h a t larger c h a n g e s in a s t i g m a t i s m w i t h t h e i r l o n g e r d u r a t i o n a r e l i k e l y to r e s u l t from a v a r i a t i o n in the force t h a t the m u s c l e s e x e r t o n the globe.^ In this c a s e , the fibrotic a n d s h o r t e n e d i n f e r i o r recti m u s c l e s a p p e a r to h a v e b e e n the c a u s e o f the c o r n e a l s t e e p e n i n g at 9 0 d e g r e e s . T h e r e v e r s i b i l i t y o f the a s t i g m a t i s m after i n f e r i o r r e c tus m u s c l e d i s i n s e r t i o n l e n d s f u r t h e r s u p p o r t to
t h e s e m u s c l e s b e i n g r e s p o n s i b l e for the p r e o p e r a t i v e a s t i g m a t i s m in t h i s p a t i e n t .
References 1. Marshall, D.: Changes in refraction following operation for strabismus. Arch. Ophthalmol. 15:1020, 1 9 3 6 . 2. Thompson, W. E., and Reinecke, R. D.: The changes in refractive status following routine strabis mus surgery. J. Pediatr. Ophthalmol. Strabismus 17:372, 1980. 3. Fix, Α., and Baker, J. D.: Refractive changes following strabismus surgery. Am. Orthoptic J. 35:59, 1 9 8 5 . 4. Kushner, B. J.: The effect of oblique muscle surgery on the axis of astigmatism. J. Pediatr. Oph thalmol. Strabismus 23:277, 1986.
M e a n Visual Acuity Jack T. Holladay, M.D., and Thomas C. Prager, Ph.D. Department of Ophthalmology, Hermann Eye Cen ter, University of Texas Medical School at Houston. Inquiries to Jack T. Holladay, M.D., Hermann Center, 6411 Fannin, Houston, TX 77030.
Eye
Calculating mean visual acuity on a series of p a t i e n t s h a s b e e n d o n e i n c o r r e c t l y in m o s t s t u d i e s , w h i c h l e a d s to a s i g n i f i c a n t o v e r e s t i m a t e or u n d e r e s t i m a t e o f t h e t r u e m e a n v i s u a l a c u i t y . T h e b a s i c p r o b l e m r e l a t e s to t h e differ e n c e b e t w e e n the a r i t h m e t i c a n d g e o m e t r i c m e a n o f a set o f n u m b e r s . For the c o r r e c t m e a n visual acuity, calculating the geometric mean y i e l d s the p r o p e r v a l u e . M o d e r n v i s u a l a c u i t y c h a r t s a r e d e s i g n e d so t h a t the l e t t e r s i z e s on t h e c h a r t f o l l o w a g e o m e t r i c p r o g r e s s i o n (that is, a d v a n c e in u n i f o r m s t e p s on a l o g a r i t h m i c s c a l e ) . ' T h e I n t e r n a t i o n a l Council of O p h t h a l m o l o g y C o m m i t t e e on opto t y p e s a c c e p t e d the o r i g i n a l r e c o m m e n d a t i o n o f (¿reen^ to have t h e l e t t e r s i z e s c h a n g e b y 0 . 1 - l o g u n i t s t e p s , w h i c h is e q u i v a l e n t to l e t t e r s i z e s c h a n g i n g b y a factor o f 1 . 2 5 8 9 b e t w e e n l i n e s . ' T h i s s t a n d a r d led to t h e L o g M A R ( l o g a r i t h m o f the m i n i m u m a n g l e o f r e s o l u t i o n ) n o t a t i o n ' as p l o t t e d in the F i g u r e . The letter sizes between 2 0 / 1 0 and 2 0 / 2 0 0 p r o g r e s s in a l i n e a r f a s h i o n o n a l o g a r i t h m scale, and visual performance midway between
Vol. I l l , No. 3
Letters to the journal
Point A -- Patient 1 20/200 20/160 ^
20/125 -j-
Arittimetic Mean of Visual Angles or Snellen Acuity Denominators Underestimates True Mean Acuity Point Ε
H — 20/100
υ ^
20/80
-J 20/63 < «
20/50
^
20/40
+
Τωβ Geometric Mean
Point D
Ζ UJ 20^32
Aritfimetic Mean of (decimal Acuities or Snellen Fractions -Overestimates True Mean Acuity'
-I
¿
Point C
20/25
Ζ (O 20/20 20/16
Point Β - Patient 2 --
20/12.5 • • 20/10
I I I 1I I I I I I I I I 1 2 3 4 5 6 7 8 9 10 11 12 13 14
LINE
NUMBER
Figure (Holladay and Prager). The true geometric mean visual acuity betv^feen Patient 1 with visual acuity of 2 0 / 2 0 0 (Point A) and Patient 2 with visual acuity of 2 0 / 2 0 (Point B), is 2 0 / 6 3 (Point C). The incorrect arithmetic mean obtained by taking the average of the decimal visual acuities or Snellen fractions is 2 0 / 3 6 (Point D). The arithmetic mean obtained by taking the average of the Snellen visual acuity denominators or visual angles is 2 0 / 1 1 0 (Point E). Arithmetic means severely overestimate or underestimate the true geometric mean visual acuity.
2 0 / 2 0 0 ( l i n e 1) a n d 2 0 / 2 0 ( l i n e 1 1 ) is 2 0 / 6 3 ( l i n e 6 ) ( F i g u r e ) . T h i s v i s u a l a c u i t y o f 2 0 / 6 3 is the g e o m e t r i c m e a n o f t h e s e two v i s u a l a c u i t i e s . M a t h e m a t i c a l l y , t h e g e o m e t r i c m e a n is c a l c u l a t e d b y t a k i n g t h e l o g a r i t h m o f e a c h o f the s a m p l e v a l u e s , d e t e r m i n i n g t h e a v e r a g e o f the logarithm values, then taking the antilogarithm of this a v e r a g e . In this e x a m p l e , t h e l o g a r i t h m of 2 0 / 2 0 0 ( p o i n t A in t h e F i g u r e ) is - 1 . 0 a n d the log o f 2 0 / 2 0 ( p o i n t Β in t h e F i g u r e ) is 0. T h e
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a v e r a g e w o u l d b e - 0 . 5 0 , for w h i c h t h e a n t i l o g a r i t h m is d e c i m a l 0 . 3 2 or a S n e l l e n n o t a t i o n o f 2 0 / 6 3 ( p o i n t e in t h e F i g u r e ) . In our e x a m p l e , i f w e w e r e to c a l c u l a t e t h e inappropriate arithmetic m e a n using the deci m a l v a l u e s , as s u g g e s t e d m o s t r e c e n t l y b y V i l a C o r o a n d V i l a - C o r o , ^ w e w o u l d o b t a i n an aver a g e d e c i m a l v a l u e o f 0 . 5 5 ( [ 0 . 1 + 1 . 0 ] / 2 ) or 2 0 / 3 6 ( p o i n t D in t h e F i g u r e ) . T h i s m e t h o d overestimates the true g e o m e t r i c m e a n visual a c u i t y a n d m i n i m i z e s t h e c o n t r i b u t i o n o f the poor visual acuity samples. A s e c o n d i n c o r r e c t m e t h o d is to t a k e the arithmetic m e a n of the m i n i m u m angle of reso l u t i o n , w h i c h is e q u i v a l e n t to t a k i n g t h e aver age of the d e n o m i n a t o r s of the S n e l l e n nota t i o n . Pincus^ u s e d t h i s m e t h o d to d e t e r m i n e t h e a v e r a g e v i s u a l a c u i t y for a g i v e n r e f r a c t i v e er ror. U s i n g t h i s i n c o r r e c t m e t h o d in our p r e v i o u s example, the mean visual acuity would have been 2 0 / 1 1 0 ([200 + 2 0 ] / 2 ) . This arithmetic m e t h o d will s e v e r e l y u n d e r e s t i m a t e t h e a c t u a l g e o m e t r i c m e a n v i s u a l a c u i t y ( p o i n t Ε in the Figure). If one takes the g e o m e t r i c m e a n of the Snellen denominators, the m i n i m u m visual an g l e s of r e s o l u t i o n , or t h e S n e l l e n f r a c t i o n s or d e c i m a l s , t h e r e s u l t is t h e s a m e , 2 0 / 6 3 , w h i c h is t h e c o r r e c t r e s u l t . Fortunately, with the newer visual acuity c h a r t s (for e x a m p l e , B a j l e y - L o v i e , ETDRS, P E R K ) t h a t h a v e an e q u a l n u m b e r o f l e t t e r s o n a line and a constant geometric progression b e t w e e n l i n e s , t h e a c t u a l l i n e n u m b e r s are d i r e c t l y p r o p o r t i o n a l to t h e l o g a r i t h m o f t h e v i s u a l a c u i t y , as we h a v e s h o w n p r e v i o u s l y . ' W h e n t h e s e c h a r t s are u s e d , t h e g e o m e t r i c m e a n v i s u al a c u i t y is m o r e s i m p l y o b t a i n e d b y c a l c u l a t i n g the arithmetic m e a n of the n u m b e r of lines or l e t t e r s c o r r e c t , t h e n c o n v e r t i n g t h e r e s u l t to t h e corresponding Snellen visual acuity.' With m a n y s t a n d a r d p r o j e c t o r c h a r t s , h o w e v e r , in w h i c h the n u m b e r o f l e t t e r s o n e a c h l i n e are n o t e q u a l or t h e p r o g r e s s i o n is n o t c o n s i s t e n t (for e x a m p l e , t h e 2 0 / 1 2 5 a n d 2 0 / 1 6 0 l i n e s are m i s s ing a n d a 2 0 / 7 0 l i n e s h o u l d n o t b e p r e s e n t ) , t h i s s i m p l e m e t h o d m a y n o t b e u s e d a n d the l o g a r i t h m i c m e t h o d is n e c e s s a r y . Likewise, w h e n other statistical analyses are p e r f o r m e d on v i s u a l a c u i t y , s u c h as c o r r e l a t i o n c o e f f i c i e n t s or s t a n d a r d d e v i a t i o n s , t h e y must be calculated using the logarithm of the visual a c u i t y or l o g a r i t h m o f t h e v i s u a l a n g l e as S l o a n ' ' h a s s h o w n in h e r c o r r e l a t i o n s o f v i s u a l a c u i t y with refractive error. Care should be taken by a u t h o r s a n d r e v i e w e r s to a s s u r e t h a t t h e s e p r i n ciples are followed so that m e a n visual acuity
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p r e s e n t e d in a study is valid a n d c o m p a r a b l e to o t h e r studies. U n f o r t u n a t e l y , m a n y o f the m e a n visual a c u i t i e s in p u b l i s h e d r e p o r t s h a v e u s e d o n e o f t h e two i n c o r r e c t a r i t h m e t i c m e t h o d s , w h i c h m a k e m o s t c o m p a r i s o n s of m e a n v i s u a l acuity invalid.
References 1. Bailey, 1. L., and Lovie, J. E.: New design princi ples for visual acuity letter charts. Am. J. Optom. Physiol. Optics. 53:740, 1976. 2. Green, ] . : Notes on the clinical determination of the acuteness of vision including the construction and graduation of optotypes. Trans. Am. Ophthal mol. Soc. 10:644, 1905. 3. Vila-Coro, A. Α., and Vila-Coro, An. Α.: Mean visual acuity. Am. J . Ophthalmol. 107:564, 1989. 4. Pincus, M. H.: Unaided visual acuities correlat ed with refractive errors. Am. J. Ophthalmol. 29:853, 1946. 5. Holladay, J. T., and Prager, Τ. C.: Snellen equivalent for Bailey-Lovie acuity chart. Arch. Oph thalmol. 107:955, 1989. 6. Sloan, L. L.: Measurement of visual acuity. Arch. Ophthalmol. 45:704, 1 9 5 1 .
Infectious Crystalline K e r a t o p a t h y After Relaxing Incisions Marilyn C. Kincaid, M.D., Bradley D. Fouraker, M.D., and David J. Schanzlin, M . D .
MarcJi, 1991
A 75-year-old woman underwent penetrating k e r a t o p l a s t y in the left e y e for a p h a k i c b u l l o u s k e r a t o p a t h y in July 1 9 8 9 . P o s t o p e r a t i v e v i s u a l a c u i t y was p o o r b e c a u s e o f r e s i d u a l a s t i g m a tism, r e d u c e d s o m e w h a t b y s e l e c t i v e s u t u r e r e m o v a l b y p h o t o k e r a t o s c o p y . H o w e v e r , 11 di o p t e r s o f a s t i g m a t i s m r e m a i n e d , with the s t e e p m e r i d i a n at 1 5 0 d e g r e e s . In M a y 1 9 9 0 , two r e l a x i n g i n c i s i o n s w e r e p l a c e d b y the m e t h o d d e s c r i b e d by Lindquist.^ T h e p a t i e n t w a s given 1% p r e d n i s o n e e y e d r o p s four t i m e s per day, a n d v i s u a l a c u i t y i m p r o v e d to 20/70. Three m o n t h s after this p r o c e d u r e , r o u n d e d c r y s t a l l i n e - a p p e a r i n g infiltrates w e r f § e e n on b o t h s i d e s o f o n e o f t h e i n c i s i o n s (Fig. 1 ) . V i s u a l acuity h a d d e c r e a s e d to 2 0 / 3 0 0 . S h e w a s treat ed w i t h t o b r a m y c i n / d e x a m e t h a s o n e e y e d r o p s e v e r y two h o u r s for 72 h o u r s , w i t h s l i g h t i m p r o v e m e n t in v i s i o n . T h e a n t i b i o t i c was t h e n c h a n g e d to c h l o r a m p h e n i c o l . The patient underwent repeat penetrating k e r a t o p l a s t y . T h e c o r n e a l b u t t o n w a s p l a c e d in r o u t i n e 1 0 % buffered f o r m a l i n for l i g h t m i c r o s c o p y . B o t h of the r e l a x i n g i n c i s i o n s w e r e i d e n t i fied, a n d the b u t t o n w a s b i s e c t e d p e r p e n d i c u l a r to t h e s e i n c i s i o n s . M i c r o s c o p i c a l l y , c o r n e a l e p i t h e l i u m l i n e d the e n t i r e d e p t h of o n e i n c i s i o n . T h e o p p o s i t e i n c i sion was p a r t i a l l y l i n e d b y e p i t h e l i u m . G r a m positive cocci were present within stroma adja c e n t to the e p i t h e l i a l l i n i n g for a l m o s t the e n t i r e d e p t h of the o p p o s i t e i n c i s i o n , c o r r e s p o n d i n g to the infiltrates s e e n c l i n i c a l l y . T h e b a c t e r i a a l s o e x t e n d e d away from the i n c i s i o n e d g e s w i t h i n p r e - D e s c e m e t ' s s t r o m a (Fig. 2 ) . M o r e t h a n 17 c a s e s o f i n f e c t i o u s c r y s t a l l i n e
Bethesda Eye Institute, Departments of Ophthalmol ogy (M.C.K., B.D.F., D.J.S.) and Pathology (M.C.K.), St. Louis University School of Medicine. Inquiries Institute,
to Marilyn C. Kincaid, M.D., Bethesda 3655 Vista Ave., St. Louis, MO 63ΊΊ0.
Eye
I n f e c t i o u s c r y s t a l l i n e k e r a t o p a t h y is c h a r a c t e r i z e d c l i n i c a l l y by a b r a n c h i n g , c r y s t a l l i n e infiltrate within c o r n e a l s t r o m a , with m i n i m a l or n o i n f l a m m a t i o n , a n d n o n e c r o s i s or e d e m a . ' In a p a t i e n t d e s c r i b e d b y G o r o v o y a n d a s s o c i a t e s , ' t h e d i a g n o s i s o f a b a c t e r i a l infiltrate w a s n o t m a d e u n t i l the c o r n e a w a s e x a m i n e d h i s t o logically. Many of these cases have occurred after p e n e t r a t i n g k e r a t o p l a s t y . It w o u l d s e e m r e a s o n a b l e that the d e e p , b u t n o t p e r f o r a t i n g , i n c i s i o n s m a d e in t h e c o r n e a for k e r a t o r e f r a c tive p u r p o s e s m i g h t a l s o i n a d v e r t e n t l y b e c o m e c o l o n i z e d in a s i m i l a r m a n n e r .
F i g . 1 (Kincaid, Fouraker, and Schanzlin). Clinical appearance of the infiltrate (arrows) along one of the relaxing incisions. The other incision is free of infil trates.
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AMERICAN JOURNAL OF OPHTHALMOLOGY
axis of a s t i g m a t i s m . We t r e a t e d a p a t i e n t w i t h g e n e r a l i z e d fibrosis s y n d r o m e w h o h a d a sig nificant a n d p e r s i s t e n t c h a n g e in r e f r a c t i v e er ror after s t r a b i s m u s s u r g e r y . A 2 - w e e k - o l d girl was first e x a m i n e d b y a pediatric ophthalmologist (M.G.) because she h a d m a r k e d b l e p h a r o p t o s i s . At that t i m e , the p a t i e n t h a d a +2.00 r e f r a c t i o n b i l a t e r a l l y . S h e was r e e x a m i n e d on n u m e r o u s o c c a s i o n s b e cause of her marked blepharoptosis and chinup p o s t u r e , a g e n e r a l i z e d d e c r e a s e in o c u l a r m o v e m e n t s w i t h i n f r a d u c t i o n o f the g l o b e s b i l a t e r a l l y , facial w e a k n e s s , and difficulty w i t h s w a l l o w i n g . At 2 y e a r s of a g e , c y c l o p l e g i c r e fraction h a d p r o g r e s s e d to R . E . : - 1 6 . 0 0 -1-10.00 X 1 0 5 a n d L . E . : - 1 6 . 0 0 -1-8.00 x 7 5 . R e p e a t r e f r a c t i o n m e a s u r e m e n t s e v e r a l w e e k s later w a s R . E . : - 1 6 . 0 0 -t-8.00 x 9 0 a n d L . E . : - 1 6 . 0 0 + 8.00 X 9 0 . The patient's refraction was inde p e n d e n t l y m e a s u r e d b y two a d d i t i o n a l p e d i a t ric o p h t h a l m o l o g i s t s , a n d t h e s e m e a s u r e m e n t s w e r e n e a r l y i d e n t i c a l to the i n i t i a l m e a s u r e m e n t . B e c a u s e o f the p a t i e n t ' s s i g n i f i c a n t c h i n up p o s t u r e a n d m a r k e d i n f r a d u c t i o n o f the globes, she underwent bilateral inferior rectus m u s c l e r e c e s s i o n o f 12 m m a n d total i n f e r i o r r e c t u s m u s c l e d i s i n s e r t i o n in e a c h e y e . F o r c e d d u c t i o n t e s t i n g at the t i m e o f the i n i t i a l o p e r a tion d i s c l o s e d significant r e s t r i c t i o n to e l e v a tion. Repeat refraction m e a s u r e m e n t s two m o n t h s after the o p e r a t i o n d i s c l o s e d a r e f r a c t i v e e r r o r of R . E . : - 1 2 . 0 0 +5.00 x 100 and L.E.: - 1 2 . 0 0 -1-5.00 X 9 0 . B e c a u s e o f the m a r k e d c h a n g e , r e f r a c t i o n w a s a g a i n m e a s u r e d at four m o n t h s a n d six m o n t h s after the final o p e r a t i o n . T h e m y o p i a a n d a s t i g m a t i c e r r o r h a d d e c r e a s e d to R . E . : - 6 . 5 0 - ^ 1 . 5 0 x 9 0 a n d L . E . : - 5 . 0 0 +2.00 X 1 0 5 . O n e y e a r later, v i s u a l a c u i t y was 2 0 / 5 0 in e a c h eye w i t h this r e f r a c t i o n . T h e findings in our p a t i e n t are u n u s u a l in t h e d e g r e e to w h i c h the r e f r a c t i v e e r r o r c h a n g e d after s t r a b i s m u s s u r g e r y . W h e r e a s o t h e r a u t h o r s have s p e c u l a t e d t h a t s m a l l e r , m o r e t r a n s i e n t c h a n g e s in p o s t o p e r a t i v e a s t i g m a t i s m a r e c a u s e d b y e y e l i d s w e l l i n g or by a c h a n g e in t h e p o s i t i o n of the eye r e l a t i v e to the e y e l i d s , the findings in o u r p a t i e n t s e e m to confirm t h a t larger c h a n g e s in a s t i g m a t i s m w i t h t h e i r l o n g e r d u r a t i o n a r e l i k e l y to r e s u l t from a v a r i a t i o n in the force t h a t the m u s c l e s e x e r t o n the globe.^ In this c a s e , the fibrotic a n d s h o r t e n e d i n f e r i o r recti m u s c l e s a p p e a r to h a v e b e e n the c a u s e o f the c o r n e a l s t e e p e n i n g at 9 0 d e g r e e s . T h e r e v e r s i b i l i t y o f the a s t i g m a t i s m after i n f e r i o r r e c tus m u s c l e d i s i n s e r t i o n l e n d s f u r t h e r s u p p o r t to
t h e s e m u s c l e s b e i n g r e s p o n s i b l e for the p r e o p e r a t i v e a s t i g m a t i s m in t h i s p a t i e n t .
References 1. Marshall, D.: Changes in refraction following operation for strabismus. Arch. Ophthalmol. 15:1020, 1 9 3 6 . 2. Thompson, W. E., and Reinecke, R. D.: The changes in refractive status following routine strabis mus surgery. J. Pediatr. Ophthalmol. Strabismus 17:372, 1980. 3. Fix, Α., and Baker, J. D.: Refractive changes following strabismus surgery. Am. Orthoptic J. 35:59, 1 9 8 5 . 4. Kushner, B. J.: The effect of oblique muscle surgery on the axis of astigmatism. J. Pediatr. Oph thalmol. Strabismus 23:277, 1986.
M e a n Visual Acuity Jack T. Holladay, M.D., and Thomas C. Prager, Ph.D. Department of Ophthalmology, Hermann Eye Cen ter, University of Texas Medical School at Houston. Inquiries to Jack T. Holladay, M.D., Hermann Center, 6411 Fannin, Houston, TX 77030.
Eye
Calculating mean visual acuity on a series of p a t i e n t s h a s b e e n d o n e i n c o r r e c t l y in m o s t s t u d i e s , w h i c h l e a d s to a s i g n i f i c a n t o v e r e s t i m a t e or u n d e r e s t i m a t e o f t h e t r u e m e a n v i s u a l a c u i t y . T h e b a s i c p r o b l e m r e l a t e s to t h e differ e n c e b e t w e e n the a r i t h m e t i c a n d g e o m e t r i c m e a n o f a set o f n u m b e r s . For the c o r r e c t m e a n visual acuity, calculating the geometric mean y i e l d s the p r o p e r v a l u e . M o d e r n v i s u a l a c u i t y c h a r t s a r e d e s i g n e d so t h a t the l e t t e r s i z e s on t h e c h a r t f o l l o w a g e o m e t r i c p r o g r e s s i o n (that is, a d v a n c e in u n i f o r m s t e p s on a l o g a r i t h m i c s c a l e ) . ' T h e I n t e r n a t i o n a l Council of O p h t h a l m o l o g y C o m m i t t e e on opto t y p e s a c c e p t e d the o r i g i n a l r e c o m m e n d a t i o n o f (¿reen^ to have t h e l e t t e r s i z e s c h a n g e b y 0 . 1 - l o g u n i t s t e p s , w h i c h is e q u i v a l e n t to l e t t e r s i z e s c h a n g i n g b y a factor o f 1 . 2 5 8 9 b e t w e e n l i n e s . ' T h i s s t a n d a r d led to t h e L o g M A R ( l o g a r i t h m o f the m i n i m u m a n g l e o f r e s o l u t i o n ) n o t a t i o n ' as p l o t t e d in the F i g u r e . The letter sizes between 2 0 / 1 0 and 2 0 / 2 0 0 p r o g r e s s in a l i n e a r f a s h i o n o n a l o g a r i t h m scale, and visual performance midway between
Vol. I l l , No. 3
Letters to the journal
Point A -- Patient 1 20/200 20/160 ^
20/125 -j-
Arittimetic Mean of Visual Angles or Snellen Acuity Denominators Underestimates True Mean Acuity Point Ε
H — 20/100
υ ^
20/80
-J 20/63 < «
20/50
^
20/40
+
Τωβ Geometric Mean
Point D
Ζ UJ 20^32
Aritfimetic Mean of (decimal Acuities or Snellen Fractions -Overestimates True Mean Acuity'
-I
¿
Point C
20/25
Ζ (O 20/20 20/16
Point Β - Patient 2 --
20/12.5 • • 20/10
I I I 1I I I I I I I I I 1 2 3 4 5 6 7 8 9 10 11 12 13 14
LINE
NUMBER
Figure (Holladay and Prager). The true geometric mean visual acuity betv^feen Patient 1 with visual acuity of 2 0 / 2 0 0 (Point A) and Patient 2 with visual acuity of 2 0 / 2 0 (Point B), is 2 0 / 6 3 (Point C). The incorrect arithmetic mean obtained by taking the average of the decimal visual acuities or Snellen fractions is 2 0 / 3 6 (Point D). The arithmetic mean obtained by taking the average of the Snellen visual acuity denominators or visual angles is 2 0 / 1 1 0 (Point E). Arithmetic means severely overestimate or underestimate the true geometric mean visual acuity.
2 0 / 2 0 0 ( l i n e 1) a n d 2 0 / 2 0 ( l i n e 1 1 ) is 2 0 / 6 3 ( l i n e 6 ) ( F i g u r e ) . T h i s v i s u a l a c u i t y o f 2 0 / 6 3 is the g e o m e t r i c m e a n o f t h e s e two v i s u a l a c u i t i e s . M a t h e m a t i c a l l y , t h e g e o m e t r i c m e a n is c a l c u l a t e d b y t a k i n g t h e l o g a r i t h m o f e a c h o f the s a m p l e v a l u e s , d e t e r m i n i n g t h e a v e r a g e o f the logarithm values, then taking the antilogarithm of this a v e r a g e . In this e x a m p l e , t h e l o g a r i t h m of 2 0 / 2 0 0 ( p o i n t A in t h e F i g u r e ) is - 1 . 0 a n d the log o f 2 0 / 2 0 ( p o i n t Β in t h e F i g u r e ) is 0. T h e
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a v e r a g e w o u l d b e - 0 . 5 0 , for w h i c h t h e a n t i l o g a r i t h m is d e c i m a l 0 . 3 2 or a S n e l l e n n o t a t i o n o f 2 0 / 6 3 ( p o i n t e in t h e F i g u r e ) . In our e x a m p l e , i f w e w e r e to c a l c u l a t e t h e inappropriate arithmetic m e a n using the deci m a l v a l u e s , as s u g g e s t e d m o s t r e c e n t l y b y V i l a C o r o a n d V i l a - C o r o , ^ w e w o u l d o b t a i n an aver a g e d e c i m a l v a l u e o f 0 . 5 5 ( [ 0 . 1 + 1 . 0 ] / 2 ) or 2 0 / 3 6 ( p o i n t D in t h e F i g u r e ) . T h i s m e t h o d overestimates the true g e o m e t r i c m e a n visual a c u i t y a n d m i n i m i z e s t h e c o n t r i b u t i o n o f the poor visual acuity samples. A s e c o n d i n c o r r e c t m e t h o d is to t a k e the arithmetic m e a n of the m i n i m u m angle of reso l u t i o n , w h i c h is e q u i v a l e n t to t a k i n g t h e aver age of the d e n o m i n a t o r s of the S n e l l e n nota t i o n . Pincus^ u s e d t h i s m e t h o d to d e t e r m i n e t h e a v e r a g e v i s u a l a c u i t y for a g i v e n r e f r a c t i v e er ror. U s i n g t h i s i n c o r r e c t m e t h o d in our p r e v i o u s example, the mean visual acuity would have been 2 0 / 1 1 0 ([200 + 2 0 ] / 2 ) . This arithmetic m e t h o d will s e v e r e l y u n d e r e s t i m a t e t h e a c t u a l g e o m e t r i c m e a n v i s u a l a c u i t y ( p o i n t Ε in the Figure). If one takes the g e o m e t r i c m e a n of the Snellen denominators, the m i n i m u m visual an g l e s of r e s o l u t i o n , or t h e S n e l l e n f r a c t i o n s or d e c i m a l s , t h e r e s u l t is t h e s a m e , 2 0 / 6 3 , w h i c h is t h e c o r r e c t r e s u l t . Fortunately, with the newer visual acuity c h a r t s (for e x a m p l e , B a j l e y - L o v i e , ETDRS, P E R K ) t h a t h a v e an e q u a l n u m b e r o f l e t t e r s o n a line and a constant geometric progression b e t w e e n l i n e s , t h e a c t u a l l i n e n u m b e r s are d i r e c t l y p r o p o r t i o n a l to t h e l o g a r i t h m o f t h e v i s u a l a c u i t y , as we h a v e s h o w n p r e v i o u s l y . ' W h e n t h e s e c h a r t s are u s e d , t h e g e o m e t r i c m e a n v i s u al a c u i t y is m o r e s i m p l y o b t a i n e d b y c a l c u l a t i n g the arithmetic m e a n of the n u m b e r of lines or l e t t e r s c o r r e c t , t h e n c o n v e r t i n g t h e r e s u l t to t h e corresponding Snellen visual acuity.' With m a n y s t a n d a r d p r o j e c t o r c h a r t s , h o w e v e r , in w h i c h the n u m b e r o f l e t t e r s o n e a c h l i n e are n o t e q u a l or t h e p r o g r e s s i o n is n o t c o n s i s t e n t (for e x a m p l e , t h e 2 0 / 1 2 5 a n d 2 0 / 1 6 0 l i n e s are m i s s ing a n d a 2 0 / 7 0 l i n e s h o u l d n o t b e p r e s e n t ) , t h i s s i m p l e m e t h o d m a y n o t b e u s e d a n d the l o g a r i t h m i c m e t h o d is n e c e s s a r y . Likewise, w h e n other statistical analyses are p e r f o r m e d on v i s u a l a c u i t y , s u c h as c o r r e l a t i o n c o e f f i c i e n t s or s t a n d a r d d e v i a t i o n s , t h e y must be calculated using the logarithm of the visual a c u i t y or l o g a r i t h m o f t h e v i s u a l a n g l e as S l o a n ' ' h a s s h o w n in h e r c o r r e l a t i o n s o f v i s u a l a c u i t y with refractive error. Care should be taken by a u t h o r s a n d r e v i e w e r s to a s s u r e t h a t t h e s e p r i n ciples are followed so that m e a n visual acuity
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p r e s e n t e d in a study is valid a n d c o m p a r a b l e to o t h e r studies. U n f o r t u n a t e l y , m a n y o f the m e a n visual a c u i t i e s in p u b l i s h e d r e p o r t s h a v e u s e d o n e o f t h e two i n c o r r e c t a r i t h m e t i c m e t h o d s , w h i c h m a k e m o s t c o m p a r i s o n s of m e a n v i s u a l acuity invalid.
References 1. Bailey, 1. L., and Lovie, J. E.: New design princi ples for visual acuity letter charts. Am. J. Optom. Physiol. Optics. 53:740, 1976. 2. Green, ] . : Notes on the clinical determination of the acuteness of vision including the construction and graduation of optotypes. Trans. Am. Ophthal mol. Soc. 10:644, 1905. 3. Vila-Coro, A. Α., and Vila-Coro, An. Α.: Mean visual acuity. Am. J . Ophthalmol. 107:564, 1989. 4. Pincus, M. H.: Unaided visual acuities correlat ed with refractive errors. Am. J. Ophthalmol. 29:853, 1946. 5. Holladay, J. T., and Prager, Τ. C.: Snellen equivalent for Bailey-Lovie acuity chart. Arch. Oph thalmol. 107:955, 1989. 6. Sloan, L. L.: Measurement of visual acuity. Arch. Ophthalmol. 45:704, 1 9 5 1 .
Infectious Crystalline K e r a t o p a t h y After Relaxing Incisions Marilyn C. Kincaid, M.D., Bradley D. Fouraker, M.D., and David J. Schanzlin, M . D .
MarcJi, 1991
A 75-year-old woman underwent penetrating k e r a t o p l a s t y in the left e y e for a p h a k i c b u l l o u s k e r a t o p a t h y in July 1 9 8 9 . P o s t o p e r a t i v e v i s u a l a c u i t y was p o o r b e c a u s e o f r e s i d u a l a s t i g m a tism, r e d u c e d s o m e w h a t b y s e l e c t i v e s u t u r e r e m o v a l b y p h o t o k e r a t o s c o p y . H o w e v e r , 11 di o p t e r s o f a s t i g m a t i s m r e m a i n e d , with the s t e e p m e r i d i a n at 1 5 0 d e g r e e s . In M a y 1 9 9 0 , two r e l a x i n g i n c i s i o n s w e r e p l a c e d b y the m e t h o d d e s c r i b e d by Lindquist.^ T h e p a t i e n t w a s given 1% p r e d n i s o n e e y e d r o p s four t i m e s per day, a n d v i s u a l a c u i t y i m p r o v e d to 20/70. Three m o n t h s after this p r o c e d u r e , r o u n d e d c r y s t a l l i n e - a p p e a r i n g infiltrates w e r f § e e n on b o t h s i d e s o f o n e o f t h e i n c i s i o n s (Fig. 1 ) . V i s u a l acuity h a d d e c r e a s e d to 2 0 / 3 0 0 . S h e w a s treat ed w i t h t o b r a m y c i n / d e x a m e t h a s o n e e y e d r o p s e v e r y two h o u r s for 72 h o u r s , w i t h s l i g h t i m p r o v e m e n t in v i s i o n . T h e a n t i b i o t i c was t h e n c h a n g e d to c h l o r a m p h e n i c o l . The patient underwent repeat penetrating k e r a t o p l a s t y . T h e c o r n e a l b u t t o n w a s p l a c e d in r o u t i n e 1 0 % buffered f o r m a l i n for l i g h t m i c r o s c o p y . B o t h of the r e l a x i n g i n c i s i o n s w e r e i d e n t i fied, a n d the b u t t o n w a s b i s e c t e d p e r p e n d i c u l a r to t h e s e i n c i s i o n s . M i c r o s c o p i c a l l y , c o r n e a l e p i t h e l i u m l i n e d the e n t i r e d e p t h of o n e i n c i s i o n . T h e o p p o s i t e i n c i sion was p a r t i a l l y l i n e d b y e p i t h e l i u m . G r a m positive cocci were present within stroma adja c e n t to the e p i t h e l i a l l i n i n g for a l m o s t the e n t i r e d e p t h of the o p p o s i t e i n c i s i o n , c o r r e s p o n d i n g to the infiltrates s e e n c l i n i c a l l y . T h e b a c t e r i a a l s o e x t e n d e d away from the i n c i s i o n e d g e s w i t h i n p r e - D e s c e m e t ' s s t r o m a (Fig. 2 ) . M o r e t h a n 17 c a s e s o f i n f e c t i o u s c r y s t a l l i n e
Bethesda Eye Institute, Departments of Ophthalmol ogy (M.C.K., B.D.F., D.J.S.) and Pathology (M.C.K.), St. Louis University School of Medicine. Inquiries Institute,
to Marilyn C. Kincaid, M.D., Bethesda 3655 Vista Ave., St. Louis, MO 63ΊΊ0.
Eye
I n f e c t i o u s c r y s t a l l i n e k e r a t o p a t h y is c h a r a c t e r i z e d c l i n i c a l l y by a b r a n c h i n g , c r y s t a l l i n e infiltrate within c o r n e a l s t r o m a , with m i n i m a l or n o i n f l a m m a t i o n , a n d n o n e c r o s i s or e d e m a . ' In a p a t i e n t d e s c r i b e d b y G o r o v o y a n d a s s o c i a t e s , ' t h e d i a g n o s i s o f a b a c t e r i a l infiltrate w a s n o t m a d e u n t i l the c o r n e a w a s e x a m i n e d h i s t o logically. Many of these cases have occurred after p e n e t r a t i n g k e r a t o p l a s t y . It w o u l d s e e m r e a s o n a b l e that the d e e p , b u t n o t p e r f o r a t i n g , i n c i s i o n s m a d e in t h e c o r n e a for k e r a t o r e f r a c tive p u r p o s e s m i g h t a l s o i n a d v e r t e n t l y b e c o m e c o l o n i z e d in a s i m i l a r m a n n e r .
F i g . 1 (Kincaid, Fouraker, and Schanzlin). Clinical appearance of the infiltrate (arrows) along one of the relaxing incisions. The other incision is free of infil trates.
