a theoretical model of the cornea for use in studies of try

8/4/2019 A Theoretical Model of the Cornea for Use in Studies of try

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B U L L E T I N O FN IA TH EM A TIC A L B IO P H Y S IC SV O L U I~ IE 3 0 , 1 9 6 8

A T H E O R E T I C A L M O D E L O F T H E C O R N E A F O RU S E I N S T U D I E S O F T O N O M E T R u

9 C . C . M o w *T h e R A N D C o r p o r a t i o n ,Santa Monica , Ca l i fornia

A t h e o r e t i c a l m o d e l o f t h e c o r n e a b a s e d o n c o r n e a l d i m e n s i o n s a n d r e p o r t e d p r o p e r t i e s i sp r e s e n t e d i n th i s p a p e r . I t i s s h o w n t h a t b e c a u s e o f l a r g e d i f f e r en c e s i n t h e t h i c k n e s s e so f t h e B o w m a n ' s a n d D e s c e m e t ' s m e m b r a n e s a n d t h e s t r o m a , a n d b e c a u s e o f t h e r e p o r t e dl a r g e d i f f e r e n c e s i n t h e e l a s t i c p r o p e r t i e s o f t h e l a y e r s , a s a n d w i c h - s h e l l m o d e l i s a g o o da p p r o x i m a t i o n f o r t h e s t u d y o f c o r n e a l d e f o r m a t i o n . T h e t h e o r y i s a p p l i c a b l e f o r a p p l a n a -t i o n t o n o m e t r y . A s e t o f e q u i l i b r i u m e q u a t i o n s b a s e d o n R e i s s n e r ' s t h e o r y i s g i v e n .S h e ll p a r a m e t e r s w h i c h d e t e r m i n e t h e b e h a v i o r o f s h e l l s a re e x p r e s s e d i n t e r m s o f t h ec o r n e a l p r o p e r t i e s a n d d i m e n s i o n s . N u m e r i c a l e x a m p l e s w h i c h s h o w t h e e f fe c t s o f c o r n e a lp a r a m e t e r s o n t h e s t r e s s r e s u l t a n t s d u e t o i n t r a o c u l a r p r e s s u r e a r c a l so g iv e n .

I. Introduction. A m o n g t h e m a n y m e t h o d s f o r d i ag n o si n g g la u c o m a , t h e o n em o s t f r e q u e n t l y u s e d is t o n o m e t r y . T h e m e c h a n i c a l pr i n ci p l e o f t o n o m e t e r sh a s b e e n s t u d i e d i n d e t a i l ( G l o s te r , 1 9 6 6 a n d D r a e g e r , 1 9 66 ) a n d t h e r e f o r e w i l ln o t b e d i sc u s s e d h e re . R a t h e r , w e a r e c o n c e r n e d w i t h i d e n t i f y i n g s o m e o f t h ef e a t u r e s o f t h e c o r n e a t h a t m i g h t i n f l ue n c e t h e r e a d i n g s o f t h e t o n o m e t e r ( t h a ti s, c a u s e d i s p l a c e m e n t s i n t h e S e h i o t z t o n o m e t e r a n d a p p l i e d f o r c e i n t h ea p p l a n a t i o n t o n o m e t r y ) a n d h e n c e t h e i n t e r p r e t a t i o n o f t h e i n t r a o c u l a r p r e s s u re .

A b r i e f s u r v e y o f t h e o p h t h a l m i c l i t e r a t u r e s h o w e d t h a t n e a r l y a l l t h e e a r li e rt h e o r e ti c a l m o d e ls o f t h e e y e u s e d i n t h e s t u d y o f t o n o m e t r y h a v e a s s u m e dt h e e y e t o b e a s im p l e membrane s t r u c t u r e . G o l d m a n n ( 19 56 ) a d v a n c e d a

* A n y v i e w s e x p r e ss e d in th i s p a p e r a r e t h o s e o f t h e a u t h o r . T h e y s h o u l d n o t b e i n t e r p r e t e d a sr e f l e c t i n g t h e v i e w s o f t h e R A l q D C o r p o r a t i o n o r t h e o f fi ci al o p i n i o n o r p o l i c y o f a n y o f i ts g o v e r n -m e n t a l o r p r i v a t e r e s e a r c h s p o ns o rs . P a p e r s a r e r e p r o d u c e d b y T h e R A N D C o r p o r a t i o n a s ac o u r t e s y t o m e m b e r s o f i t s st af f .4 3 7


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4 3 8 C . C . M O W

mo d e l i n w h i ch t h e ey e w as a s s u m ed t o co n s i s t o f t w o co n cen t r i c mem b ran e ss ep a ra t ed b y i n co mp res si b l e f lu i ds , an d w i t h t h i s a s s u mp t i o n q u a l i t a t i v e l yd e t e r m i n e d t h e m e a n i n g o f a p p l a n a t i o n t o n o m e t r y . M o r e r e c e n tl y , S c h w a r tzet al. (1966) used a homogenous , i so t rop ie , sha l low-shel l theory to s tudy corneald e fo rma t i o n . H o w ev er , t h e re is a c lea r n eed fo r a b e t t e r mo d e l fo r u s e i n t h ean a l y s i s o f co rn ea l d e fo rma t i o n w h i ch i n t u rn w o u l d lead t o w ar d a b e t t e r u n d e r -s t a n d in g o f t o n o m e t r y .

T h i s p ap e r p re s en t s a mo re r ea l i s t i c mo d e l fo r t h e co rn ea w h i ch , h o p e fu l l y ,w i ll b e f lex ib le en o u g h t o co n t a i n m an y p a ram e t e r s o f t h e co rn ea an d s t il l b eamen ab l e t o an a l y s i s .

T h i s is t h e s o -ca ll ed " s an d w i ch s h e l l" mo d e l . I t s h o u l d b e n o t ed a t t h is p o i n tt h a t t h e " s an d w i ch s h e l l" t h e o ry w a s fi rs t d e r i v ed b y R e i s s n e r i n 1 94 9 an d h asb een i n ex i s t en ce fo r a l mo s t t w en t y y ea r s .

I I . Bas i s for the Model . In the fo l lowing , we sha l l d i scuss the bas i s uponw h i ch t h e s an d w i ch - s h e l l mo d e l w as ch o s en fo r t h e co rn ea .

F i g u re 1 s h o w s a h o r i zo n t a l c ro s s s ec t io n o f an ey e , w i t h s o me o f t h e i m-p o r t a n t p a r t s l o ca t ed an d i den t if i ed . F i g u re 2 s h o w s t h e s t ru c t u re o f t h eco rn ea . I t i s ap p a ren t f ro m F i g u re 1 t h a t an ey e i s an ex t r em e l y co mp l ex


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MODEL OF THE CORNEA 439struc ture in the engineering sense. However, the scope of this st ud y is fairlylimited, and we will be concentr ating on the corneal portion of the eye sincethis is where tonometers are usually applied.

Description of the Cornea. The cornea is trans paren t. Seen from the front,it appears t o be elliptical (approximately 12 mm in the horizontal mer idi an and11 mm in the vertical), while fro m behind, it appears circular. This differenceis due to the fact that the sclera and conjunctiva overlap the cornea anteriorly,more above and below than laterally.

The normal radius of the cur vature of the anterior surface is 7.86 mm with ast and ard deviation of 0.26 ram. The posterior curva ture is ~ 7.00 ram.

Figure 2The cornea is slightly thicker along the periphery where it is ~ 1 mm thic kth an at the center where it is N0.58 ram. There are five microscopically

distinguishable layers which are described briefly (Wolff, 1948 and Maurice,1962) below:

1. The epi thel ium (outermost layer) is abou t 50 to 100 ~z (~z = 10 -3 mm)th ick an d consists of five or six layers of cells. No mechanical properties ofthe epithelium have been described in the ophthalmic literature.

2. Bowman's membrane is a fine layer of basement membrane which liesdirectl y under the basal cells of the epithelium. It is a thin, structureless sheetabou t 12 ~z thi ck and is described as an "elast ic" lamina.

6--B.M,B.


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4 4 0 C . C . M O W3. T h e s t r o m a ( s u b s t a n t i a l p r o p i a ) m a k e s u p a b o u t 9 0 p e r c e n t o f t h e t h i c k -

n e s s o f t h e c o r n e a . I t i s c o m p o s e d o f s h e e t s o f e o l la g e n e o us m a t e r i a l - - t h es t r o m a l l a m e l la e w h i c h li e p a r a l le l t o t h e s u rf a c e . T h e l a m e l la b a n d s a r ep a r a l le l t o e a c h o t h e r , b u t t h o s e o f a l t e r n a t i n g l a y e r s a r e o r t h o g o n a l t o e a c ho t h e r . T h e m e c h a n i c a l p r o p e r t ie s o f t h e s t ro m a , t h a t i s, s h e a r s t r e n g t h a n dc o h e s i v e s t r e n g th , h a v e b e e n d e s c r i b e d b y M a u r i c e ( 19 62 ) a n d a r e o f g r e a ti m p o r t a n c e . T h e s t r o m a h a s v i r t u a l l y n o r e s i s ta n c e t o s h e a r f o rc e s. I f a ni s o l a t e d c o rn e a is h e l d b e t w e e n f i ng e r a n d t h u m b , t h e t w o f a c e s m a y s li de u p o no n e a n o t h e r q u i t e fr e e ly . S t r o m a s e p a r a t e s r e a d i l y i n t o a c l e a v a g e p l a n ep a r a l le l t o i t s s u rf a c es . W h e n s t r e t c h e d i n a d i r e c t io n p e r p e n d i c u l a r t o t h ec o r n e a l s u rf a c e, s e c ti o n s o f s t r o m a d o n o t r u p t u r e i m m e d i a t e l y b u t r e t u r ne l a s t i c a l l y t o t h e i r o r i g in a l t h i c k n e s s . S c h w a r t z e t a l . ( 1 9 6 6 ) r e p o r t s t h e n o n -l i n ea r b e h a v i o r o f a n e x c i s e d c o r n e a u n d e r c o m p r e s s i o n a l t e s t ( t h a t is , w h e n t h ec o r n e a i s l o a d e d p e r p e n d i c u l a r t o i t s s u r fa c e , i t s d i s p l a c e m e n t i s n o t l i n e a r l yp r o p o r t i o n a l t o t h e l o a d a p p li e d) . F u r t h e r m o r e , i t w a s e s t i m a t e d b y S c h w a r t zt h a t t h e m o d u l u s o f e l a s t i c it y o f t h e s t r o m a i s a t l e a s t t w o o r th r e e o r d er s o fm a g n i t u d e l o w e r t h a n t h a t o f t h e o u t e r la y e r s ( B o w m a n ' s m e m b r a n e a n d t h ee p i t h e l i u m ) . ~

4 . D c s c e m e t ' s m e m b r a n e ( p o s te r i or e la s t ic m e m b r a n e ) d e s c r i b e d i n W o l f f a sa s t r o n g , s t r u c t u r e l e s s a n d v e r y r e s i s t a n t m e m b r a n e . I t is a b o u t 5 t o 1 0 ~ i nt h i c k n e s s .5 . T h e e n d o t h e l i u m i s t h e m o s t p o s t e r i o r l a y e r o f t h e c o r n e a a n d c o n si st s o fa s i n g l e l a y e r o f f l a t e p i t h e l iM - l i k e c e l l s a b o u t 5 f~ t h i c k .

A s c a n b e s e e n , t h e m e c h a n i c a l p r o p e r t i e s o f t h e f iv e c o rn e a l la y e r s d e s c r i b e da b o v e a r e q u i t e q u a l i t a t i v e r a t h e r t h a n q u a n t i t a t i v e , a n d t h e r e a p p e a r s t o b en o d e t a i le d i n f o r m a t i o n a v a i l a b l e o n th e s e p r o p e rt i e s . T h e r e is , h o w e v e r ,a m p l e i n f o r m a t i o n o n t h e o c u l a r r i g i d it y , K , a n e y e c o n s t a n t w h i c h is u s e d i nt o n o m e t r y f o r t h e d e t e r m i n a t i o n o f t h e r e l a t i o n s h ip o f p r e s s u re a n d v o lu m e,: ~b u t i t i s n o t u s e f u l f o r t h e d e t a i l e d m o d e l i n g o f a c o r n e a .

T h e c o r n e a h a s t h e f o l lo w i n g t y p i c a l d i m e n s i o n s ( se e F i g . 3 ):1 . R a d i u s o f c u r v a t u r e o f p o s t e r i o r s u r f a c e , R ~ ~ 7 .0 m m2 . l ~ a d iu s o f c u r v a t u r e o f a n t e r i o r s u r f a c e , R~ ~ 7 .0 m m

t T h i s is a n i m p o r t a n t f a c t o r in t h e c h o i ce o f t h e s a n d w i c h m o d e l f o r t h e c o rn e a . A n y e v i d e n ce ,e i t h e r s u p p o r t i n g o r c o n t r a d i c t i n g t h e s t a t e m e n t , w o u l d b e w e l c o m e d b y t h e a u t h o r .$ T h e c o e f f i c ie n t o f o c u l a r r i g i d i t y , K , i s i m p o r t a n t f o r t h e d e t e r m i n a t i o n o f t h e p r e s s u r e -v o l u m er e l a t i o n s h i p o f a n e y e o n l y i n a m a c r o s c o p i c s e n s e . I t i s d e f i n e d a s :K = I n Pt/PoV t

w h e r e p t i s t h e i n t r a o c u l a r p r e s s u r e w h e n t h e t o n o m e t e r i s a p p l ie d ; P o i s t h e u n d i s t u r b e d i n rp r e s s u r e ; a n d V t i s t h e v o l u m e o f f lu i d d i s p l a c e d b y t h e t o n o m e t o r .


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MODE L OF TH E COR17EA 441

3. Horizont al radius, r o H 6.0 mm4. Combined thickness of epit helium and Bowm an's

t * H 5 0 ~ l O O5. Thickness of Bowman 's membrane, t o ~ 106. Combined thickness of I)escemet's memb rane and

t~ ~ 1 0 - 1 57. Thickness o f th e s troma, h c H 700 N 8008. T he mean radius of cu rv atu re ,/ ~ H 7.45 mm

membrane,

endothelium,

r a ~ ra

Figure 3C h a r a c t e r i s t i c S h e l l P a r a m e t e r s . Using the above information about the

cornea, we shall now examine some of the characteristic paramete rs t ha t arenorma lly used to characterize a shell structur e. The parameter s used toclassify the t ype o f shell are (1) the ratio of shell thic knes s to the mea n radiusof curvature, i//~, and (2) the ratio of the shell height to the base diameter,H / 2 r o ( a param eter used to determine the shallowness of the shell). Using thedimensions given above, we find th at a typi cal cornea has the following valuesof t//~ and H / 2 r o :

i 0.86- - -- 0.115R 7.45H 12r--~ ~


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442 C. C. MOW

w h e r e t i s t h e t o t a l t h i c k n e s s o f t h e c o r n e a , ~ = R o - R ~. F r o m t h e s e v a l u e s ,i t c a n b e s e e n t h a t t h e c o r n e a i s n o t a m e m b r a n e - t y p e s he ll , n o r i s i t a v e r y t h i nshe l l ( u sua l ly des cr i be d a s a she l l w i th ~ //~ < -2!6), no r i s i t a sha l low she l l( u s u a l l y d e s c r i b e d a s a s h e l l w i t h H / 2 R o < } ) (Re is sner , 1946 ) .

I n a d d i t i o n t o t h e g e o m e t r i c p a r a m e t e r s o f t h e c o r ne a , w e m u s t c o n s id e r a l sot h e d i f fe r e n t p ro p e r t i e s o f t h e v a r i o u s l a y e r s, t h e n o n l i n e a ri t y , a n d t h e v i s c o -e l a s ti c b e h a v i o r o f t h e c o r n ea .

I t b e c o m e s r e a d i ly a p p a r e n t t h a t n o o ne m o d e l t h a t w e c a n c o n s t ru c t w i lli n c l u d e a ll o f t h e s e g e o m e t r i c a n d p h y s i c a l p r o p e r t i e s a n d s ti ll b e a m e n a b l e t oa n a l y si s . H o w e v e r , w e b e l i e v e t h a t R e i s s n e r ' s s an d w i c h - sh e l l t h e o r y w il la d e q u a t e l y t a k e i n to a c c o u n t m o s t o f t h e p r o p e r t ie s o f t h e c o r ne a a n d c a n t h u sb e u s e d to d e v e l o p a s a t i s f a c t o r y m o d e l . T o a p p l y th i s t h e o r y , w e h a v e m a d et h e f o ll o w i ng a s s u m p t i o n s :

1. T h e c o r n e a is c o m p o s e d o f t h r e e l a y e r s ( t w o f a c e la y e r s a n d o n e c o r e l a y e r )2 . T h e p r o p e r t i e s o f t h e t w o f a c e l a y e r s d if fe r fr o m t h o s e o f t h e c o re la y e r3 . T h e v a l u e o f t // ~ i s sm a l l4 . T h e t w o f a c e l a y e r s h a v e t h e s a m e t h i c k n e s s5 . T h e m a t e r i a l o f t h e c o r n e a i s e l a s t ic a n d l i n e a rW e s h a l l M s o u s e t h e f o l lo w i n g p r o p e r t i e s o f t h e c o r n e a , a s d i s c u s s e d e a r l ie r :1. T h e r a t i o o f t h i c k n e s s o f o u t e r o r in n e r l a y e r t o s t r o m a , to. J h c ~,, 0( 1 0 - t ~ 10 -2 )2 . T h e r a t i o o f u m o d u l u s o f t h e s t r o m a t o t h e f a c e l a y e r s ( S c h w a rt z ,

1966) E c / E f , , , 0(10 -2 - 10 -3 )3 . T h e c o r e l a y e r ( s t r o m a ) a p p e a r s t o h a v e n o i n - p la n e s h e a r r e s i s t a n c e

(Maur ice , 1962 )U s i n g t h e a b o v e a s s u m p t i o n s a n d t h e p h y s i c a l p r o p e r t i e s o f t h e c o rn e a , w e

f in d t h a t t h e c o r n e a c a n b e a p p r o x i m a t e d b y R e i s s n e r' s sa n d w i c h s h e ll t h e o r yw h i c h i s b a s e d o n t h e f o ll o w i ng a s s u m p t i o n s :

1 . t / R < < 12 . ( E ~ t ) / ( E c h ) >> 13 . t /h


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M O D E L O F T H E C O R N E A 443

m e r r y , t h e n , b e c a u s e o f t h e s m a l l d e f o r m a t i o n i n v o l v e d i n a p p l a n a t io n , t h ec o r n e a m a y v e r y w e l l b e d e f o r m e d w i t h i n t h e l i n e a r p o r t i o n o f i t s f o r c e -v e r s u s -d i s p l a c e m e n t c u r v e . w A n d f i n al ly , t h e v i s c o e l a s ti c b e h a v i o r o f t h e c o r n e a m i g h tn o t b e i m p o r t a n t i n t o n o m e t r y b e c a u s e o f t h e s h o r t l o ad i n g d u r a t i o n t h a tt o n o m e t r y r e q u ir e s . I]

H a v i n g d e t e r m i n e d t h a t t h e u s e o f R e i s s n e r' s s a n d w i ch - s h el l m o d e l f o r t h ec o r n e a a p p e a r s t o b e ju s t if ie d , w e s h a ll n e x t p r e s e n t t h e b a s i c e q u a t i o n s t h a tc a n b e u s e d t o d e t e r m i n e t h e e ff ec ts o f t h e c o rn e a l p a r a m e t e r o n t h e d e f o r m a t i o no f t h e c o r n e a. A l s o, w e s ha l l s h o w t h e e f f e c t o f t h e c o r n e a l p a r a m e t e r o n t h es t re s s e s d u e t o i n t r a o e u l a r p r e s s u r e in t h e v a r i o u s l a y e r s o f t h e c o r n e a .

I I I . ;Basic Equat ions . I n t h i s s e c t io n w e s h a l l p r e s e n t t h e e q u a t i o n s o fe q u i l ib r i u m f o r t h e s a n d w i c h - t y p e s h el l a n d t h e n r e la t e t h e p a r a m e t e r s i n t h es h e ll t h e o r y t o t h o s e o f t h e c o r n e a.

U s i n g t h e t h e o r y f o r a s a n d w i c h s h e ll o f r e v o l u t i o n , t h e d e r i v a t i o n o f w h i c hi s g i v e n b y R e i s s n e r ( 19 4 9) , i f w e a s s u m e t h a t t h e c o r n e a i s a s e g m e n t e r a s p h e r ea n d t h e l o a d s ar e a x ia l l y s y m m e t r i c , t h e n t h e e q u a t i o n o f e q u i l ib r i u m a n d t h er e l a t i o n s h i p b e t w e e n d i s p l a c e m e n t a n d s t r e s s f o r a s a n d w i c h s h e l l i n s p h e r i c a lp o l a r c o o r d i n a t e s a p p l y .

T h e e q u a t i o n s o f e q u i l i b r i u m f o r a s a n d w i c h s h e l l i n ( r, 0 , ~ ) c o o r d i n a t e s( s e e F i g . 4 ) a r e

d(s in q~N ~ ) N o d ( si n ~ ) ( ~ )R d~o R d-----~ + si n ~ + p ~ = 0 (1)d( sin ~0Qo) sin ~ [ No + HO] + s in ~ / = 0 (2 )R d~0 \ R !

d ( s in ~ M r ) M s d ( si n ~ )R d~0 R d- - -~ s in ~( m o - Qo) = 0 (3)T h e y a r e r e l a te dh e r e P o , q a n d m o a r e t h e s t re s s r e s u l t a n t s o f e x t e r n a l l o a d s .

t o t h e e x t e r n a l l y a p p l i e d l o a d s a s f o l l o w s :g + + poz (4)

w h e r e Po ~ a n d p ~ a r e e x t e r n a l l o a d s a c t i n g i n t h e ~o d i r e c t i o n o n t h e u p p e r a n dl o w e r m e m b r a n e s , a n d (= 1 + - - 7 ) 1 q ,

w These points can be clarified by e xperiment s.II Viscoelastici ty can be incorpora ted into th e layers w ithou t undue difficulties; it is moreimpor tant to know just what viscoelastic model is appropriate for the cornea before incorporatingit into the shell theory.


8/17

4 4 4 C . C . M O W

de~.O

( a ) S e o m e t ry

/ , ( b ) C o o r d i n a t e s a n d d im e n s i o n s

e

/A ~ ? . / . y , c , S,re,,re,u,tan,,,n,oceN ~ / q ~ ~ _ _ ~ _ _ _ _ . ~ X / - , l a y e r s a n d c o r e la y e r

N r N r e#.,

F i g u r e 4


9/17

MODEL OF THE CORNE A 445

w h e r e qu a n d qz a re n o r m a l l o a d s a c t in g o n u p p e r a n d l o w e r m e m b r a n e s a n d

T h e r a d ia l n o r m a l s tr e ss i n th e m i d d l e p l a n e i s r e l a te d t o t h e m o m e n t s a n d s b ye q u a t i o n (7 ), w h i c h is p e c u l i a r t o s a n d w i c h - s h e l l t h e o r y :

1arrm q- (h q - t ) t l ( M e + M ~) - s = 0 ( 7 )w h e r e

s = - ~ 1 + ~ / q u - 1 2 R qz " (8 )T h e r e l a t i o n s h i p b e t w e e n s tr e s s a n d d i s p l a c e m e n t r e l a t i o n s h i p f o r t h e s h e l l i s

g i v e n b y t h e f o l l o w i n g e q u a t i o n s :Q ~ d w u

(h + t )Gc = f i + R dcp R (9 )( - F 'd ~ w (h + t___~) )(1 + 89 N r - (v - 89 JVe = C * d u + -R + 1 2 R E c q ( 10 )

( u d ( s i n ~ ) w ( h - r - t ) )(1 + 8 9 0 - ( v - 8 9 o = C * s i n v ~ + ~ + ~ R - E ~ q (1 1 )( I + 2 ) M _ ( v _ 2 t) M e = D , ( d f l 1 )+ E - - ~ s ( 1 2 )

( n i l_ d ( s i n ~ ) 1 )(1 + 2 ,) M o - ( v - 2 ) M ~ = D * S i ~o R d ~ + E - ~ s ( 13 )w h e r e

C * = 2 t E sD * = 8 9 ( h + t ) 2 E f

; ~ = l ( h + t ) t E~2 R 2 E c

a n d R i s t h e r a d i u s o f t h e c u r v a t u r e ; t i s t h e t h i c k n e s s o f t h e m e m b r a n e ; h ist h e t h i c k n e s s o f t h e c o r e ; u a n d w a r e t h e d i s p l a c e m e n t s i n t h e ~0 a n d r d i r e c t io n s ,r e s p e c t iv e l y ; v i s t h e P o i s s o n r a t i o ; E I a n d E c a r e t h e Y o u n g ' s m o d u l u s o f t h ef a c e m e m b r a n e a n d c o r e , r e s p e c t i v e l y ; G c is t h e s h e a r m o d u l u s o f t h e c o re ; a n df l i s t h e s l o p e o f t h e d e f l e c t i o n .


10/17

446 C. C. MOW

E q u a t i o n s ( 1 ) t h r o u g h ( 3 ) a n d e q u a t i o n s ( 9 ) t h r o u g h ( 1 3 ) p r o v i d e e i g h te q u a t i o n s t o s o l v e t h e e i g h t u n k n o w n s g i v e n b el o w .

1 . T h e d i r e c t - st r e s s r e s u l t a n t i n t h e 0 d i r e c t i o n :N e = 1 + 2 R ] -h70u 1 2 R ] Ne~

(N e u a n d N oz d e n o t e t h e d i r e c t- s tr e s s r e s u l t a n t s i n t h e u p p e r a n d l o w e r m e m -b r a n e s , r e s p e c t i v e l y )

2 . T h e d i r e c t s t re s s r e s u l t a n t i n t h e ~o d i r e c t i o n :h + t ( 1 h _ + t ]

N o = (1 + 2R ] -N ~ u + 2R ] N ~(1V ou a n d N ~z a r e de f i n e d s i m i l a r l y t o N e u a n d N 0z a bo ve )

3 . T h e t r a n s v e r s e - s h e a r - s t r e s s r e s u l t a n t :

( a r ~ a n d a ~ l a r e t h e s h e a r in g s tr es se s a c ti n g a t t h e u p p e r a n d l o w e r m e m b r a n e ,r e s p e c t i v e l y )

4 . T h e s tr e s s c o u p l e f o r t h e c o m p o s i t e s he l l:M 8 h + t h t ~ - ( 1 h + t~ Noz= 2 [ ( 1 + - - 2 R - ] N ~ " 2 - R ]

5 . T h e s l o p e o f t h e d e f l e c ti o n , f l6 . T h e d i s p l a c e m e n t i n t h e ~ a n d r d i r e ct io n s , u a n d w .I V . Stresses in the Cornea Due to Intraocular Pressure. I t i s a p p a r e n t t h a t

i n o r d e r t o d e t e r m i n e t h e v a lu e s o f t h e e i g h t u n k n o w n s l is te d i n S e c t io n I I I , w em u s t s o lv e th e e i g h t si m u l t a n e o u s e q u a t i o n s - - a t a s k w h i c h e n ta il s m u c h e f fo r t.H o w e v e r , it is a s im p l e m a t t e r t o u s e t h e s e e q u a t i o n s t o d e t e r m i n e t h e s t r es se sd u e t o t h e i n t r a o c u l a r p r e s s u r e in t h e v a r i o u s la y e r s o f t h e c o r n e a .

F o r t h i s p u r p o s e , w e s h a ll a s su m e t h a t t h e e y e is a c o m p l e t e sp h e r e s u b j e c t e dt o a l o a d o f o n l y t h e i n t r a o c u l a r p r e s s u r e , p c . T h e c o m p l e t e - sp h e r e a s s u m p t i o nis s a ti sf ie d if t h e s tr e ss e s o b t a i n e d a r e u s e d o n l y t o e s t i m a t e t h o s e n e a r t h e c e n t r a lp o r t i o n o f t h e c o r n e a . T h e r e s u l ts w i l l n o t b e v a l i d , h o w e v e r , f o r s t re s s e s n e a rt h e s u lc u s s cl er a , b e c a u s e o f th e d i f fe r e n c es i n t h e p r o p e r t i e s o f t h e c o r n e a a n dt h e s c l e r a .

A s a r e s u l t o f t h e a b o v e a s s u m p t i o n , a l l q u a n t i ti e s a r e i n d e p e n d e n t o f ~ ;


11/17

MODEL OF TH E CORNEA 447t h e r e f o r e , w e c a n s o lv e f o r t h e u n k n o w n b y a s e t o f a l g e b r a ic e q u a t io n s . S i n c et h e o n l y l o a d i s qz, t h e n

2o~, = O, qu = O, m ~ = O.B y r e a s o n s o f s y m m e t r y , fl, u a n d / 9 + a r e z e r o a l s o ( fl i s t h e s lo p e o f t h e d e f l e c t i o n ;u , t h e d i s p l a c e m e n t in t h e W d i r e c ti o n ; a n d p ~ , t h e t r a n s v e r s e s h e a r i n t h ed i r e c t i o n ) .

F r o m e q u a t i o n (1),N ~ , = N o = N o . ( 1 4 )

S u b s t i t u t i o n o f e q u a t i o n (1 4) in t o e q u a t io n S ( 2 ) g i v e sN O - - P ~ (15)2

w h i c h is t h e f a m i l ia r e x p r e ss i o n fo r t h e t e n s i o n i n a s p h e r e s u b j e c t e d t o i n t e r n a lp r e s s u r e P o . W e s h a l l d i s c u s s t h i s r e s u l t l a t e r i n t h i s p a p e r .

E q u a t i o n s (3 ) a n d (7 ) a r e s i m p l i fi e d t o t h e f o l l o w i n g r e l a t io n s h i p s :M o ( 1 6 )M ~ = M e = M o , arrm = S 2 R ( h + t )

F r o m e q u a t i o n ( 8 ) ,s = - - P-2~ (1 7)2

U s i n g e q u a t i o n s ( 1 2 ) a n d ( 1 3 ) w e o b t a i n1 ( h + t ) R A

M ~ = - 2 1 + 2 A - v po (18)1 -- v ) (19)P o 1 2 2 - - V "Yrrm ~ 2 -{-

E q u a t i o n s (15 ), ( 18) a n d (19) g i ve t h e ov e ra l l - s t r e s s r e s u l t a n t s i n t h e c o m po s i t es he l l. Th e d i r e c t - s t r e s s r e s u l t a n t i s Ms o g i ve n , t h a t i s, N o = ( p o R / 2 ) w h i c hs h o u l d n o t b e s u r p r is i n g s in c e t h e c o m p o s i t e s he ll m u s t a ls o s a t is f y t h e o v e r a l le q u i li b r iu m c o n d it io n . H a v i n g d e t e r m i n e d t h e a b o v e q u a n ti t i e s, w e m a y u s et h e d e f i n i ti o n f o r N , M t o d e t e r m i n e t h e s t re s se s i n t h e m e m b r a n e s . I n ad i m e n s i o n le s s f o r m t h e y a r e a s f o ll ow s :

N o ~ ( l h + t ) 22 R ( 1 - 1 ,)= - - R p o = + ( 2 0 )

4 ( 1 + 2 R !+ t] ( 1 + 2 2 - v )


12/17

448 C .C. MOWh + t ~

- R p o - 4 + 2 a -, = ~rrm 1 1 9 (22)r r m p o = " 2 -- k ! 1 +

I n a d d i t i o n t o t h e s e s t r e s s r e s u l t a n t s a n d t h e s t r e ss a t m i d p l a n e , w e c a n al sod e t e r m i n e t h e r a d i a l d i s p l a c e m e n t w , a n d t h e s t r e ss d i s t r i b u t io n a c ro s s t h es t r o m a : w (1 - v ) R pow * . . . . (23)R 4 t E I

_~O-'ff ~rrm - - - - - ~ h 1 - 1 - - 9 (24)E q u a t i o n s (2 0) t h r o u g h ( 24 ) c a n n o w b e u s e d to c o m p u t e t h e s t r e s s e s i n t h em e m b r a n e s , t h e r a d ia l d i s p l a c e m e n t s a n d t h e v a r i a t i o n o f t h e r a d i a l n o r m a ls t r e s s a c ro s s th e t h i c k n e s s o f t h e c o r n e a .

I t m i g h t b e o f i n t e r e s t h e r e t o r e c a p i t u l a t e t h e m e a n i n g s o f t h e p a r a m e t e r st h a t i n f lu e n c e t h e s t r e s s e s i n t h e p r e s e n t c o n t e x t . I n e q u a t i o n s ( 2 0) t h r o u g h(2 4) t h e i m p o r t a n t p a r a m e t e r s a r e

1. T h e r a t i o o f o n e - h a l f t h e t o t a l t h i c k n e s s o f t h e c o r n e a t o t h e m e a n r a d i u so f c u r v a t u r e o f t h e c o r ne a , h * = ( h t ) / 2 R2 . T h e P o i s s o n r a t i o o f t h e c o r n e a , v , w h i c h l i es i n t h e r a n g e 0 < v _< 0 .5

3. A d i m e n s i o n l e s s p a r a m e t e r , 2 , d e t e r m i n e d b y t h e d i m e n s i o n s o f t h em e m b r a n e s a n d s t r o m a a s w e l l a s th e r a t i o o f t h e Y o u n g ' s m o d u l u s o f t h em e m b r a n e a n d s t r o m a , ~ = 8 9 + t ) ( E I / R 2 E c )

4 . T h e t h i c k n es s o f t h e m e m b r a n e , t5 . T h e t h i c k n e s s o f t h e s t r o m a , h .T h e o r d e r - o f - m a g n i t u d e e s t i m a t e s f o r t h e s e p a r a m e t e r s a c c o r d i n g t o S e c t i o n

I I , a r e a s f o ll o w s : F o r a t y p i c a l c o r n e a/ ~ ~ - 7 . 4 3 _+ 0 . 2 6 r a m , o r 7 . 1 7 < R


13/17

MODEL OF THE CORNEA 449

Using the above quantities, we find th at ~ has the following rat her large rangeof values:

7.2 10 -S < ~ < 1.18 x 10 -1 .It appears th at the dimensions of the cornea are well known, bu t infor matio nconcerning the elastic properties of the cornea is grossly lacking.

V . N u m e r i c a l C o m p u t a t i o n . The preceding section has presented a set ofequations th at can be used to compute stresses in the vari ous layers of thecornea as well as the range of values for the par ameters th at we have esti matedusing the informa tion available. Wit h these values we obtai ned the resultsgiven below for No* iV*u, (~* and w*.

Parts (a) and (b) of Figure 5 show the stress resultant s in the up per and lowermembrane , respectively. The values of h* used are 0.036 and 0.056, represent-ing the upper and lower bounds o f the ratios of thickness to radius of curvature.Poisson ratios, v, of 0.1 and 0.5 are used, with 0.1 representing a highly com-pressible medi um and 0.5 an incompressible medium. Values of A are0.0076 _< ~i < 0.118, the en tire range es timated in the preceding section .

It is to be noted that as ~ increases, that is, either E I / E c or ( t / R ) ( h + t / R )getting larger, the stress resultant in the lower membrane tends to have ahigher value, while the converse is true for the upper membrane . An increasingE ~ / E c implies tha t the m embranes are more difficult to stretch; therefore, moreforce is needed to stretch the lower membr ane, and, consequently, less force istra nsm itt ed to the upper membrane. The same reasoning can be applied to thecase when [ ( t /R ) ( h + t ) / 2 R ] is large. Most of the increase in ~ however, is duet o E I / E c. For all values of 2, we note that N*~ is always greater than No~.The effect of 2 on the transverse norm al stress in the mid-plane and the varia tionof the normal stress across the stro ma are shown in parts (a) and (b) of Figure 6,respectively. It is to be observed th at as A increases, o~m decreases; the reasonfor this is the same as above, th at is, the inner me mbrane is carryin g more load,thus less force is being transmitted across the stroma to the outer membrane.The transverse normal stress is shown to decrease linearly across the strom a wi ththe highest value occurring at t he inn er-membrane side of the stroma.

The displacement is shown to be linearly proportional to the intraocularpressure. Bu t if equation (23) is expressed in terms of volume changes instea dof displacements, it becomes

A p 4 E r ( . ~ ) A V (25)3 ( 1 - v ) Vo

where l ip is the incremental pressure change, and A V is the change of volume.If we denote K* as a cons tant t ha t relates change of pressure to change of


14/17

4 50 C . C . M O W0 . 5 0

@0 . 2 50 . 2 00 . 1 50 . 1 00 . 0 5

0 0

h * = 0 . 0 3 6N o l

v = 0 . 5

- 0 . 1

( a )

0 . 1N o u 0 . 5

I I I0 . 0 5 O . 10

X

0 . 2 50 . 2 00 . 1 50 . l00 . 0 5

0

I . = 0 . 0 5 6N o ~ . ~' = 0 . 5

" " - - 0 . 1

- ' ~ ' ' " - - ~ 0 . 1N o u ~ " " - - - " - - - - " " - - '- - - - - 0 . 5

(b )I I I0 . 0 5

F i g u r e 50 . I 0

v o l u m e , i t i s s e en t h a t K * is a c t u a l l y d e p e n d e n t u p o n t h e o r ig i na l v o l u m e o f t h ee y e . 8 2 T h i s p h e n o m e n o n h a s b e e n o b s e r v e d a n d r e p o r t e d b y M a m m a r e l l a( 1 9 6 5 ) .

82Here , K* is not th e u sual ocular r igidi ty as defined by Freidenwald; ra ther , K * = E f t /( 1 - - ~ ) R V o .


15/17

~ I O D E L O F T H E C O R N E A 451

0.50

/ Crrrm~- -~ -o

0.25

-h* = 0 . 0 3 6- - - - - ~ ~ ~_o,- - ~ - " - ~ .o ~ . ~ - ~ o ~

( a )M idplane norma l s tressversus X

f I0 . 0 5 O . lO lm . . .y

h c ~

h C2

~

(b )

Var ia t ion o f n o r m a l stress ac ross c o r n e a( fo r h : 0 . 0 5 6 , v : O . 5 , X : 0 .0 0 7 6 )

~ ; ~ / - ~ r r~-~Gr r m

I0.30 I I 1 I0.40 O. 50 0.60 O. 70(~ )F i g u r e 6


16/17

452 C. C. MOW

E q u a t i o n (2 3) c a n a l s o b e r e a r r a n g e d i n t o t h e f o ll o w i n g f o r m :

F o r t h e p u r p o s e o f c o m p a r i n g t h e p r e s e n t r e s u l ts w i t h t h o s e o f S c h w a r t z ( 19 66 ),w e u s e t h e e x p e r i m e n t a l v a l u e o f Ap/Aw = 1 / 0 . 0 0 4 f r o m t h a t r e f e r e n c e ,a s s u m i n g v = 0 .1 , 0. 3 a n d 0 .5 , a n d t h e v a l u e s o f R a n d t f r o m t h e p r e v i o u ss e o ti o n. W e f i nd t h a t

( 2 1 . 0 x 104 m m H gE r T ~ 1 6 . 4 1 04 m m H g

( 1 1 . 7 x 104 m m H gO' = 0.1 )( ~ = 0 . 3 )( v = 0 . 5 ) .

T h e v a l u e s o f E ~ o b t a i n e d a r e v e r y c l o s e t o t h e u p p e r - b o u n d v a l u e o f t h eY o u n g ' s m o d u l u s e s t i m a t e d b y S c h w a r tz .

VI. Concluding Remarks. T h i s p a p e r h a s p r e s e n t e d a p a r~ o f a s t u d y o n t h em e c h a n i c s o f d e f o r m a t i o n o f t h e c o r n e a u s i n g a s a n d w i c h - s h e ll t h e o r y . T h i st h e o r y h a s s e v e ra l a d v a n t a g e s s u c h a s i ts a b i l i ty t o t a k e i n to a c c o u n t t h ev a r i o u s m e c h a n i c a l p r o p e r t i e s a n d d i m e n s i o n s o f t h e l a y e r s o f t h e c o r n ea . I na d d i t i o n , i t p r o v i d e s r e s u l t s t h a t w i ll y i e l d i n f o rm a t i o n o n h o w t h e s t re s s e s a r ed i s t r ib u t e d b e t w e e n t h e m e m b r a n e s a n d t r a n s m i t t e d a c ro s s t h e s t ro m a . T h el i m i t a t i o n s o n t h e t h e o r y a r e t h a t i t is r e s t r i c t e d t o s m a l l d e f o r m a t i o n a n d t h em a t e r i a l s c o n s i d e r e d m u s t b e li n ea r .

E v e n w i t h t h e s e r e s tr i c ti o n s , w e f e el t h a t t h e t h e o r y c a n b e a p p l i e d t oa p p l a n a t i o n t o n o m e t r y b e c a u s e t h e d i s p la c e m e n t i n a p p l a n a t i o n t o n o m e t r y iss m a l l a n d c o n s e q u e n t l y t h e l i n e a r it y a s s u m p t i o n f o r t h e c o r n e a i s p e r h a p s n o t ab a d o n e. A t t e m p t s a r e be i n g m a d e t o r e d u c e t h e g i v e n se t o f e q u a t io n s( eq s . (1 ) t h r o u g h ( 3) a n d (9 ) t h r o u g h ( 13 )) t o s o l v e t h e a p p l a n a t i o n -t o n o m e t r y p ro b l e m . I t is t o b e n o t e d , h o w e v e r , t h a t i n a p p l a n a t i o n t o n o m e t r yt h e p r o b l e m i s t h e s o - c a ll e d " i n v e r s e p r o b l e m . " T h a t i s , w e a r e g i v e n a d i s-p l a c e m e n t w h i le t r y i n g t o d e t e r m i n e t h e l o a d t h a t c a u s e d i t. T h e u s u a l s he llp r o b l e m i s t o d e t e r m i n e t h e d i s p l a e e m e n t f o r a g i v e n l o a d . F o r e x a m p l e , t h ec r i te r i on f o r t h e G o l d m a n n a p p l a n a t i o n t o n o m e t e r i s t h a t i t m u s t f l a t te n a na r e a 3 . 06 m m i n d i a m e t e r . T h u s , i n e f f e ct , w e a r e p r e s c r i b i n g d i s p l a c e m e n t so v e r a p o r t i o n o f t h e c o r n e a . T h e q u a n t i t y t h a t w e s e e k i n t h is e a s e i s t h e l o a dr e q u i r e d t o f l a t t e n t h is a r e a ; w e m u s t t h e n d e t e r m i n e t h e r e l a t io n s h i p o f t h isl o a d t o t h e i n t r a o c u l a r p r e s s u r e . I t is a p p a r e n t f r o m e q u a t i o n s ( 1) t h r o u g h ( 1 3)t h a t t h e l o a d r e q u i r e d t o f l a t te n a g i v e n a r e a v eil] d e p e n d o n t h e i n t r a o e u l a rp r e s s u re , a s r e l a t e d t h r o u g h e q u a t i o n (5 ), a n d t h e c o r n e a l p a r a m e t e r s C * , D *a n d h. T h e c o n t r i b u t i o n m a d e b y C * , D * a n d ~ t o th e a p p l a n a t i o n l o a d h a s


17/17

M O D E L O F T H E C O R N EA 4 53t o b e s e p a r a t e d f r o m t h e a p p l i e d l o a d b ef o r e w e c a n r e l a t e t h e a p p l i e d l o a d t ot h e i n t r a o c u l a r p r e s s u r e .

I n c o n cl u si o n , w e w i sh t o e m p h a s i z e t h a t t h e m o d e l p r e s e n t e d h e r e c a n o n l yb e c o n s i d e r e d a s a n a t t e m p t t o p r o v i d e a b e t t e r m o d e l f o r t h e c o rn e a . T h ec h o ic e o f t h i s m o d e l w a s b a s e d p r i m a r i l y o n s o m e q u a l i t a t i v e f a c t o r s a n di s o la t e d q u a n t i t a t i v e d a t a w h i c h a r e g i v e n i n S e c t i o n 2 . T h e v a l i d i t y o f t h em o d e l w i ll b e p r o v e n o n l y w h e n w e c a n a c t u a l l y d e t e r m i n e q u a n t i t a t i v e l y t h em e c h a n i c a l p r o p e r t i e s o f t h e B o w m a n ' s a n d D e s c e m e n t ' s m e m b r a n e s a n dt h e p r o p e r t ie s o f t h e s t r o m a . I t is h o p e d t h a t t h i s p a p e r w il l s t i m u l a t e s o m ei n t e r e s t a n d w i l l p e r h a p s r e s u l t i n so m e c o m m e n t s o n t h e p l a u s i b i l i t y o f t h em o d e l .

L I T E R A T U R EDra eger , JSrg . 1966 . To n o me t r y : P h y s i c a l Fu n d a m e n t a l s , De v e l o pme n t o f Me t h o d s a n dCl in ica l App l ica t ion . N e w Y o r k : H a f n e r P u b l i sh i n g C o m p a n y , I n c .Glos te r , J . 1966 . To n o me t r y a n d To n o g r ap h y . L o n d o n : J . & A . C h u r c h il l, L t d .G o l d m a n n , H . 1 95 6. " A p p l a n a t i o n T o n o m e t r y , G l a u c o m a , " Trans . Second ConJerenceF . W . I q e w e l l ( e d . ) , s p o n s o r e d b y t h e J o s i a h M a e y J r . F o u n d a t i o n , N e w Y o r k .M a m m a r e l l a , E . , a n d M . M a l on e . 1 96 5. " T h e P r e s s u r e -V o l u m e R e l a t i o n i n T o n o m e t r y . "Ophthalmologica, 149, 81-89.M a u ri ce , D . M . 1 9 6 2 . Th e Cornea and Schlera, Th e Ey e, V o l . I , H u g h D a v s o n ( ed .) .

N e w Y o r k a n d L o n d o n : A c a d e m i c P r e s s .R e i s s n e r , E . 1 94 6. "S t r e s s a n d S m a l l D i s p l a c e m e n t o f S h a l l o w S p h e r i c a l S h e l l s ."J . M a t h . P h y s . , 25, 80-85.1949 (March) . S m a l l Be n d i n g a n d S tr e tc h in g o f S a n d w i c h - Ty p e S h e ll s, IqACAT N 1 8 3 2 .S c h w a r tz , N a t h a n J . , S . M a c K a y a n d J . L . S a c k m a n . 1 96 6. " A T h e o r e t ic a l a n dE x p e r i m e n t a l S t u d y o f t h e M e c h a n i ca l B e h a v i o u r o f t h e C o r n e a W i t h A p p l i c a ti o n t ot h e M e a s u r e m e n t o f I n t r a o c u l a r P r e s s u r e . " Bu l l . Ma t h . B i o p h y s i c s , 28, 585-643.W olf f , Eug ene . 1948 . Th e A n a t o m y o f th e Ey e a n d Or bit. P h i l a d e l p h i a a n d T o r o n t o :T h e B l a k i s to n C o m p a n y . RECEIVED 12-18-67

a theoretical model of the cornea for use in studies of try

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