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The validity of the Imbert-Fick law as applied to applanation tonometry
Exp. Eye Res. (1963) 2, 274-283
The
Validity
of the Imbert-Fick Law Applanation Tonometry
as applied
to
J. Gr.os'rlgn a>'J) E. S. I'].:I~KJ,','S
Department of lgx2)eri'mcnlal Ophthalmolqqy, l'nstitutc (f Ophlhahnology, U niversily of London, ],2.~gla~d (Received 21 May 1963) T h e p r i n c i p l e o f a p p l a n a t i o n t o n o m e b r y c o u l d be o x b ( : n d c d t o bhc ( l e t e r t n i n a b i o n o f o c u Inr rigidiLy a n d , p e r h a p s , faciliLy o f o u b f l o w , i?;obh p r o c e d u r e s w o u l d neeossit~at,o t h e f l a t t e n i n g o f r e l a t i v e l y l a r g o a r e a s o f c o r n e a , a n d t.he.ir vn.lidity w o u l d d c p o n d u p o n t h o a c c u r a c y o f c o m p u t a t , i o n s o f bho i n b r a o c u l a r p r e s s u r e a n d l,ho v o l u m e o f lluid d i s p l a c e d in t h e e y e . ] - t u m a n c o r n e a s w o r e m o u n t e d in a n al)pa.ralst,s no t h a b Lho r e h ~ b i o n s h i p b e t w e e n t h e i n b r a o c u l a . r p r e s s u r e , t h o a pplan~d,ing f o r c e a n d t h e a t ' c a o f c o r n e a f l a t t e n e d c o u l d be m e a s u r e d e x i ) e r i m e n t . a l l y . If, w a s f o u n d lamb t h i s r e l a t i o n s h i p d i d nob c o n f o r m e x a c t l y t o t h e I m bert,-.Fi c k b~w, t h e r e Imi ng so n m e vi(lcn ee that, s u r f a c e t e n s i o n a n d t h e f o r c e r e q u i r e d t o b o n d t h e c o r n e a l t i s s u e s wore r e s p o n s i b l e f o r t,his d e v i a t i o n from the expected behaviour. For the ma,jority of corno~s, at an applanation diameter o f td)oub 3 r a m , tam .[mbert,-li'iok law was ~MH)licablo, bul, at, l a r g e r ( l i a m e t o r s , t h e a r c ~ o f c o r n e a f l a t t e n e d w a s loss t h a n e x p e c t e d . I t is poinLed o u t t h a t a n a p l ) l a n a l A o n d i a m e t e r o f 3 m m is t ~ p p z ' o x i m a t o l y bht~l, u s e d in tlu; G o l d m a n n t o n o m o t e r . :l?ho o b s o r v e ( I v o l u m b s o f fluid d i s p l a c e d b y a p p l a n a t i o n w o r e in g e n e , ' M a g r e e m e n t w i t h t h e v a l u e s c a l c u l a t e d o n t h e a s s u m p t i o n t h a t t h e v o l u m e s c a n be re.garded as sl)llorical s e g m e n t s .
1. Introduction Applanation tonometry is based upon the principle illustrated in Fig. 1, which indicates lahat, when a flal5 surface is pressed wit, h a t'oree, W, against a spherical container having an internal pressure, P, equilibrium is attained w]mn PA = W, where A is
W
li'tO. I. Iq'inciplc ofapl>lnnal,ion I,onolnotry.
the area of flat~tening or applanal)ion. This simple rela, tionsl~ip, embodied in t,he Imberl>Fick law, holds only when the membrane bounding t)he sphe,'icM contaiJmr is dry, infinitely thin and perfectly flexible and elastic. ]3y contr,'tsl~, tt~e cornea has ~ wet; surface, a finite thickness and irrtperfecl) elasticity coupled with some viscous propert, ies. In the developrnenl) of t,he Ooldmann ,~pplanabion tonomet, er, special '27,1.
A PI~I,ANAq?ION
TONOMlg'PIKY
27,5
attent, ion was paid to two forces w h i c h could i n v a l i d a t e t h e simple r e l a t i o n s h i p given a.bove (C'old m a n n , 1 956; G o l d m a n n and Sehmid% 1.957, 1.961 ), t h e first, being t h e surface l~ellsion e x e r t e d by t h e film of tears a-nd t h e second t h e :force reqllired to bend t h e corneal tissue a.t th.e eiremnference of the a p p l a n a t e d area. G o l d m a n n c o n c l u d e d t h a t , for ~llO corneal ~rea a p p l a n a t e d b y his i n s t r u m e n t , t h e d i s t u r b i n g threes t e n d to nullify one a.notlmr and thai) the simple reh~.tionship between P, A a n d W is valid. lI is conelt,sion is s u p p o r t e d b y some experimenta.1 evidence. T h e r e are, however, two circumstanoes u n d e r wIfieh it is i m p o r t a n t to discover ttm reta;tionship 1)etween P , d a n d W wlten the d i e m e t e r of t h e a p p l a n a t e d area exceeds ;~-06 ram, which is t h a t produced by the G o l d m a , n t o n o m e t e r . The first of these coneerns the possibility of p e r f o r m i n g tonogra.phy nsing a p p l a n a t i o n .instead of i n d e n t a tion, whilst, the second relates to the me;~surement el" ocular r i g i d i t y b y aI)planation; b o t h procedures would involve fla.ttening r e l a t i v e l y large ureas of cornea i~ order to produce a d e q u a t e elevations of i n t r a o e u l a r p~:essure. No s a t i s f a c t o r y m e t h o d of a p p l a n a t i o n t o n o g r a p h y has as y e t been devised, b u t oeula.r r i g i d i t y has been measn r e d ~,sing the appla.na, t,ion principle (Moses a n d U?arkt~a.nen, 1959; Kruse, t.960). Both t o n o g c a p h y and t h e m e a s u r e m e n t of r i g i d i t y wo~ld i n v o l v e d e t e r m i n a t i o n s not, only of changes in pressure b~tt also o[' Mtera.tions in i n t r a o c u l a r v o l n m e produced b y corneal fla.ttening. [Phese v o l u m e t r i c changes t~ave usnMly been ea.leulated in a c c o r d a n c e with the el,coreticat relai,ionshil)s given in ]Yig. 2, b u t this implies t h a t ,
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= ~h~',ap-hi
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1fie. 2. Calenlat,lon o f v o l u m e o f l l u i d d i s p l a e e d b y al)plalmf~ion.
dltring applanat)ion, the original, s h a p e of t;he }~eriphera.1 region of t h e cornea is reI;a.ined so thaf5 I~tte v o l u m e of" fluid displaced can be cMculated as t h a t of a s])heric~l segment. ]~ven Lr this be brae, t h e r e is some d o u b t as to w h e t h e r r shoxlld be t a k e n as equ~l to t h e externM o r to the i n t e r n a l r a d i u s of corneal c u r v a t u r e . ~ x p e r i m . o n t a l eonfirma,tion of" t h e va.lidity 0[' the e;~leulated v o l u m e s of fluid displa.eed d u r i n g applaun.tion ,~t)pears to be laoking. '.l:hc t'olIowing investiga.tion was designed t h e r e f o r e to investiga'ce t h e r e l a t i o n s h i p bet)ween t h e m a g n i t u d e of cornea.l a.pplanat.ion, t h e iut,r~oeular press~tre and t h e v o l u m e of fluid dispirited b y a,pplana, tion. 2. M e t h o d s
The aplJ~ra.t.ua is show,1 di~grammat.icMly ila ~'ig. 8. The excised cornea or artificial cornea was clamped in a. perspex chamber connected to a, ma.nomet.er reservoir. A perspex disk suspended from a pivot could be a.pplied to the snrfa.ce of tl~e cornea. Side arms,
J. (.'I,()8TI~]I¢ A S I )
276
E. N, I ' i , ; I L K I N S
p i v o t e d ~tt' t,hc so,no pal,l,, h e m ~. s('~,.It~ ]mt~ a m l t~ c ( m n t e r w e i g h t ' t,o allow for ndjust'ing tim p e r s p c x pie.be so t'lmJ, ~)o l%rce was exerl.etl on the ¢'ornen }ml'o,'o bhc addibion o1" wt;igill.s to t,he so'Me p:tn. A gra.dua.te~[ t,ttbe coat,sin i]~g a mn;t I1 a itr-bu l,ble a.1lowell chq.ugos of vol urea t,o be m e a s u r e d . T h e alia.meter of t,he Ibtb~etmd a r e a wa.,~ men.su red by t n c a n s of v. slil;-l~mp m i c r o s c o p e wit.h a m c a s u r i ~ g aye-piece. E a c h scale ~iiviaiou of t,he eyt;-pieoe represent, ed 0-06 mtt~. l lttrnn.ri eornea.s~ were used a n d were o b b M n e d f r o m ent,oleaLe¢[ eyes t,hn.b ha.d bt:e~, r e j e c t e d a.s ttnsuit,~ble ~louor /nn.l;crial I'or kern.Loplasby. T h e cord,on., which w,.s excisetl t o g e t h e r w i t h a. s u r r o u n t l i n g rim of sclert~, w',.s c l a m p e d iu t,he perspu× c]m,,nl)~;r n,n~l, in s o m e experiment,s, w a s t.hen i m m c r m ; d i~, n. solubion of dext, ra.n unt.il it,s t h i c k n e s s lind been r e d , t e e d I.o tt v a l u e oorrespot~dit,g t:o that, of the living htttna.n eye. T h e p e r s p e x c h a m b e r w a s L h c ~ l oonneut, ed t.o l.he r e m a i n d e r of bhe aplm.ra.t.u.q qn,l l,he pres,m~rc w i t h i n it, was set, at, tl)o desired level wit,h the m a n o n m t , c r , n u d n. series o f weight,s :~thte.d t,o him acMe p a n , t,he al)plnnn.t, ion dit,.met, er beh~g measurc(I for en,:h weighU. Observat, ions were m a d e wit,h pressures in t,he clu~,u!:e:" of 20, 25, 30, 35, ,10 an,l 45 m m l l g nnd weight.s o1"2, 5, 7. 10, 12, 15, 20, 2-'.', anrl 3 0 g . • ,.
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3..Diagrammatic
rcpresont.abinn
~,F a
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scale
dispIoced Ioteral~y)
Areas of n.pph,m,~iun in squa.,:e ruiIIhnet'res were ca.lcula£ed from Lhe obs~,rved din.reeLers, ~ n d tJ~e v o l u m e d i s p l a c e d b y c;~oh 'q~plb;a.t'hm of ~ w e i g h t was me,qsured directly- by 6he m o v e m e n t ; of ~hc b u b b l e in i,he graduaLe(l bube. T h e e×~ernM r a d i u s of c o r u e a l curva.lmre w,~s m e a s u r e d wit, h Ifl~e Ja,vaI-Sch}o~z o])ht'halm o m e t e r nn,] corueM bbickness wa..~ dot.ermined by t h e met, hod of Mauvice 'lnti Giar,{iM (I95.]). Accordi~lg to the l n , b e r b - F i c k law, l~he ~re.q. "tppl,qu~ted. is p r o p o r L i o n ' d to the force a p p l i e d d i v i d e d by the p r e s s u r e , t~ud w h e n bhe area is m e a s u r e d iu s q u a r e millimet,re.% t h e force in g r a m s weighb a n d t'he p r e s s u r e in millimel~res o f m e r c u r y , t;he equ~tior, a s s u m e s bite
form:
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73-5
T h e o b s e r v e d v M u e s of A w e r e plott, ed a g a i n s t t.he known v a l u e s of tV/P a n d a regression liae c M c u l a t e d for t, he t)oinbs o b b ' d n c d . T h i s line could t h e n be c o m p a r e d wibh Lhe calcal a t e d line.
!
i
AI'I'I~ANATION
/rONOMI,]'I'I~,Y
277
.In order I~o tcst, l,hc w, lidit,y of t,hc met, hod, l,l,o expcvimenl~ was first; pot'_rorl-o.ed u.yitig a, I,hin r u l ) b o r iyiCll'ibl'It.iio iil ll]+l,oo of bho oorziott., Its it$ wlt.,'-t oxpooLod f , ] l a t milch ll, m e m b r a n e would nl)l)roxilnaLe inoro vloscly Lo bho l,hcorebica.l r e q u i r e m e n t s . Results
3.
lCubber m e m bra?~e T h e i'cmtli,s (Fig. ,1) s h o w t01mt, |,hurt is ~ l i n e a r rela, t;ionship b e t w e e n t h e o b s e r v e d +u'ca, o1' U.l)lJln.nabion, i1, ll,ntl HI/P. '.|'lie c.t~.leuh~l~c(I regression line intersect~s t:lle ordinat~ellb l.]ioro d i d crfotq,ly I)Mt~nccd tl, nd vorbioal, M'ter OOilt$1lOt; wil,ll t.ho IIlOilllJl'{iliO a. Shin.l[ fot'~'o was r e q u i r e d be (lot,aoh it~ fl'Olli ~Do litOlllbro, llO. \Ve.t.t.in~ t,he rllblJor illo1131)l'fl.llo I ) r o d u o o d a. sorlto\vhll, t wider soil, totel~ ill Lho rosulLs, wiii(:ll o l , ] i o r w i s o ,~iiowe(l tile SiilllO linoa.r i:olilLionsliij.l b o b w o o n i [ fl, lld W/P (Fig. 5~
I1~ i)rove(I il~q)os,llr\qil,ilro wti~;l'em¢, t;lio l'(']tl{.itJli,ullil.~ I)otwocil
]l~:ossuro it.lid ,'xi.'o{I, is
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[l.m(t~# (~fJrl#(>(I, (~) T h e r e h d i o , s h i p belween f o r c e , area c t , d ?Jressu're.. A l l n i n e c o r n e a s e x a m i n e d showed % l i i l c a r rel~l:.ionshi l) I)et.weoll I V I ] j 'til(l A . T l i i s (li(I iiol;, oorresl)On(1 Co ()lie l i n e e×i)ecl,cd t'rolll t h e ln)l)t;vt-lCic'k l a w , I)tll~ c o t l l d . Lle e x p r e s s e d [~s:
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I"lu. 5. I ( c l n t i o n ~ l l i l l I~ctwt,cn ,-I iiill~ I1"/1' I'l~l' wel. rulfitler i i l ( , n l b r n l l o . ' l ' h o chH.ted lino s h o w s I.ho rehll,ioll.~hill exist, t i e d I'l'~ml t i m I n l l l e r t . - I G e k Inw ( / 1 = 7 3 - 5 I1"/1~). T h o conl, i n u o u ~ l i n e is l,h o o l t l o u l a t e d a'egresaion tino fiw t h o t~b.~ea'vltl,iouls {.,I = 3" 19 q- 7 I .56 W / / ' ) .
(b) l',bl.me.~ o.f./I.,id displaced b!/,7?~btnation. I h e resulLs f o r s i x c o r n e a s a r e s h o w n a s "~. gr;i.pll ( F i g . 7). '.i:lle v o h u l t ( ; s c a . l c u h t t c d ill a c c o r d a n c c w i t h t h e e q u a t i o n r
1
•
17 = ~ h'-' ( 3 r - - h ) 3 ~,'c a.lso I~lol, t c d oil l,l,is g r a p h , l t s i n g v a l u e s f o r r o f 6, 7 a n d 8 m a n .
28,)
J.
(';I.()S'l'l,:l,'. ANI)
E. S.
Pt:~II.I';.IN,%'
I t c a n b e seen bha~ t~he v o l u m e s d i s p l a c e d b y a p p h m a t - i o n c o r r e s p o n d g e l : e r a l l y w i t h t h o s e d e r i v e d t h e o r e t i c a l l y . T h e rcsull;s f r o m s o m e cornea.s f i t t e d t h e c a l c u l a t e d c u r v e s bebl~er t h a n t h o s e !'rein o t h e r s , b u t t h e r e w a s n o ale~r i n d i c a t i o n t h a t t h i s w~Ls r e l a t e d to t h e c o r n e a ] t h i c k n e s s . Also, w h e n r e s u l t s f r o m i n d i v i d u a l c o r n e a s w e r e 60 ~
,
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.
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0 40
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0.50
060
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0.70
0"60
~I _
Appl~noth*gforce (g wt ) P - Intr¢oculer pressure (rnmHg) F I e . 6. :Relnt, i o n s h i p [)obw(2ol:l ~'t a n d I I ' / P f o r e x c i s e d h u m a n c o r n e a , h a v i n g a. t h i c k n e s s o f 0-55 ,urn. T h e dotted line shows the r01atioL~ship expected f r o m the Imlut~t-t-Fid¢ law (.4----~73.5 IV~P). '.l'h6con. tinuous ]i2*c is the cah;ulatcd t'cgrcssion line for the observations ( A = 0 - 6 7 + 53.1 IV/P).
i
i
i
/" = 6 r a m
60-
/'--7ram
50--~mm
r
_~ 4 0 -
--
¢2
3O-.
o
20-
q)
E %
0
..a~ 3
4
S
I
1
I
I
6
7
8
9
Observed diameter o f applanation (ram) FIG, 7. V o l u m e o f t l u i d di.uplaced b y o~pplanation. R e s u l t s f o r s i x h u m a n c o r n e a s . T i l e t h r e e c o n t i n u o u s l i n es s h o w t i l e v o l u m e s c a l c u l a t e d a.q s p h e r i c a l s e g m e n t s f o r c o r n e a l r a d i i o f 0, 7 a n d 8 r a m .
AI'PbANA'.i?ION
q'ON03I,
le,'.I'tt, Y
281
c o n s i d e r e d in re,lation to t h e m e ~ s u r e d ra.dixm of e x t e r n a l ¢mrvatxtre mad t h i c k n e s s , it w~s i m p o s s i b i e to d e t e r m i n e w h e t h e r t h e o b s e r v e d v o l u n m s a g r e e d b e t t e r w i t h those calettla.l:ed usi~g t~l~e e x t e r n a l r a d i u s as ~, or w i t h t h o s e u s i n g tire interns.1 r a d i u s as r. 4.
Discussion
T h e r e s u l t s o b t ~ i n e d ill t h e e x p e r i m e n t s on r u b b e r m e m b r a n e s w e r e of v a h l e in a s s e s s i n g t h e a c c u r a c y of m e a s u r e n t e n t s of the a.l~planated ~rea.s. ~\rit]l t h e d r y m e m b r a n e , i t was found t h a t , a.pa.rt f r o m a snmll e r r o r a t t r i b u t a b l e t o e l e c t r o s t a t i c a t t r a c t,ion betweeJl t h e pla.t.e a n d tile m e m b r a n e , t h e e x p e c t e d r e l a t i o n s h i p b e t w e e n pressure, a r e a a n d force was o b e y e d . (Ja.refill a d j u s t m m ~ t of t]xe s l i t - l a m p was n e e d e d to ensu re gha.t t h e d i a m e t e r , r a t h e r t h a n a. c h o r d , of t h e a p p l a n a t e d circle w a s mea.sured, b u t t h e restllts strongly s a g g e s t t h a t t h e p o i n t s a t w h i c h t h e c u r v e d m e m b r a n e m e t t h e fla.t pla.te were defined a c c u r a t e l y . I t was ex]?eoted t h a t t h e m e a s u r e m e n t s m i g h t be iess aeealrate w i t h t h e c o r n e a since t h e a n g l e b e t w e e n tlle p l a t e a n d tile c o r n e a w o u l d be o c c u p i e d b y Huid. Neve-ctheless, t h e resltlts obta.ine& w i t h t h e w e t r u b b e r m e m b r a n e , a l t h o u g h s h o w i n g m o r e s c a t t e r , did nob d i k e r signifiea.ntly f r o m t h o s e for t h e d r y m e m b r a n e , s u g g e s t i n g ttaat t h e p r e s e n c e of fluid b e t w e e n t h e p l a t e a n d tlle al?plana.ted s u r f a c e h a d l i t t l e et~eet, on a c c u r a c y o f m e a s u r e m e n t s of t h e f l a t t e n e d area.. ]~'or r e a s o n s g i v e n in t h e p r e v i o u s section, it was n o t possible to o b t a i l l reiia.blc m e a s u r e m e n t s o[7 the arrtmlnt of t2uid d i s p l a c e d b y a p p l a n a t i o n of the r u b b e r m e m brane. T h e r e s u l t s for all. nine eornca.s e x a m i n e d i n d i c a t e d tha.t t.h.ere was a l~near r e l a t i o n ship b e t w e e n A a n d I V / P , h a v i n g t h e [7orm A = (~ + bW/P. l~or s e v e n cornea.s (.~, was l?ositive, indieatit~g t h a t a. sin,M1 ~trea of f l a t t e n i n g w a s 1)resen~ w i t h no w e i g h t in t h e scale p a n . ~l'his reseval~led t h e r e s u l t for t h e d r y r u b b e r m e m b r a n e w h e r e e l e c t r o s t a t i c attra.etion b e t w e e n ~he m e m b r a n e a n d elm p l a t e p r o v i d e d ~m e x p l a n a t i o n o[7 t h i s finding. I n t h e case of t h e cornea., a n a n a l o g o u s e x p l a n a t i o n w o u l d be t h a t stlr[itce t e n s i o n ga.ve rise to a. force a d d i t i o n a l to t h a t a p p l i e d b y t h e w e i g h t s in t h e scale l?a]x of t h e al)para.t, us. T h e r e s u l t s for all n i n e cornea.s g a v e v a l u e s of /) lower t k a n e x p e c t e d tk'om t k e p. [robert-l} ~ek. law so tlm.t.-, for t h e cornea.s h a v i n g p o s i t i v e v a l u e s of c, (i.e. t)he m a j o r i t y of cornea.s exa.mi~ed), t h e r e w a s a n i n t e r s e c t i o n b e t w e e n t h e line c a l c u l a t e d from t h e o b s e r v e d v a l u e s of A a n d W / P a n d t.be line drawI~ in accords.nee w i t h t h e [ m h e r t .IPick ].aw. F o r v a l u e s of tV/] ) g r e a t e r thatx a t t h e p o i n t of interseetio2~, t[~e areas of corne,% fla.ttened were t h u s s m a l l e r tlxa.n exj.~eeted, f r e t s t h e t h e o r e t i c a l r e l a t i o n s h i p , ,% f i n d i n g for w h i c h t w o e x p l a n a t i o n s can be ~d r a n t e d , q?he first p o s s i b i l i t y is theft; A s h o u l d c o r r e s p o n d , n o t to t h e m e a s u r e d a r e a of api?lana.tion on t h e e x t e r n a l s u r N c e of t h e cornea, b~,tt, to t h e a r e a of t h e i n t e r n a l surface t h a t b e c o m e s f l a t t e n e d . ~.fhis e x p l a n a t i o n appea,rs untena.ble, h o w e v e r , since it is difficult to believe t t t a t tt~e f l a t t e n i n g of t h e i n t e r n a l s u r f a c e co~fld be g r e a t e r Is]tlLIt t h a t of t h e e x t e r n a l snr.face. 'Phe s e c o n d p o s s i b i l i t y is t h a t p a r t of t h e aj?.p l a n a t i n g force was e x p e n d e d in b e n d i n g t h e c o r n e a l t i s s a e a t t h e c~rcumferenee of t h e ~lattened area. ~.l?his m o r e l i k e l y e x p l a n a t i o n w o u I d ha.re been s t r e n g t h e n e d if t h e r e h a d b e e n a consistez~t d e p e n d e n c e of t h e v a l u e of b u p o n t h e e o r n e M t h i c k n e s s ; this, h o w e v e r , w a s n o t t h e ease in t h e small m n ~ b e r of t h i c k c o r n e a s s t u d i e d . 2kt t h i s po~r~t w h e r e t h e line c a l c u l a t e d .from t h e e x p e r i m e n t a l r e s u l t s i n t e r s e c t e d g}~e line d e r i v e d t h e o r e t i c a l l y , t h e o b e e r v e d r e l a t i o n s h i p b e t w e e n pressure, area. a n d force w a s in a g r e e m e n t w i t h t h e I m b e r t ~ F i e k l~w, a n d t h i s p o i n t c o r r e s p o n d e d to a
282
J . GLOS'I?]~.I~ A N D
1~. 8, P . E I I K [ N S
v a l u e o f / / e q u a l to 6-1 r a m ' , w h e n c a l c u l a t e d as t h e m e a n from t h e r e s u l t s of five e o r n e a e h a v i n g t h i c k n e s s e s w i t h i n t h e r a n g e f o u n d in h e a l t h y l i v i n g l m m a n corneas. T h e d i a m e t e r c o r r e s p o n d i n g to t h i s a r e a is 2'8 m m , w h i c h is r e m a r k a b l y close to t h e d i a m e t e r (3.06 ram) of t h e area a p p l a n a t e d b y t h e G o l d m a n n t o n o m e t e r . T h e s e r e s u l t s t h u s SUlyply e x c e l l e n t eonfirn~ation of t h e a p p l i c a b i l i t y of t h e Imbert-Ir'ick l a w to d e t e r m i n a t i o n s of i n t r a o e u l a r p r e s s u r e w i t h t:he Goldmam~ i n s t r u m e n t . F o r ap.planated a r e a s h a v i n g d i a m e t e r s s i g n i f i c a n t l y g r e a t e r or less t b a n 3 tmn, t.he I m b e r t - F i c k la,w c a n n o t be used for t h e a c c u r a t e d e r i v a t i o n of t h e illtra.omdar pressure. Moreover, it would seem f r o m t h e r e s u l t s of gifts s t u d y t h a t t h e m o d e of d e p a r t n r e from t h e h n b e r t - F i e k l a w v a r i e s f r o m cornea, to cornea, a l t h o u g h f u r t h e r w o r k is r e q u i r e d to e l u c i d a t e t h e f a c t o r s w h i c h d e t e r m f i m t h e d e v i a t i o n f r o m t h e "dmoretical r e l a t i o n s h i p . I t seems, t h e r e f o r e , t h a t s o m e c a u t i o n is n e e d e d in t h e i n t e r p r o r a t i o n of t h e r e s u l t s of a p p l a n a t i o n t o n o m e t r y w h e n r e l a t i v e l y large a r e a s of c o r n e a are f l a t t e n e d ; a n d t h i s w o u l d a p p l y , as p o i n t e d o u t earlier, to m e t h o d s of detern~ining octdar r i g i d i t y or f a c i l i t y of mttflow using t h e a p p l a n a t i o n principle. S o m e i d e a o.f t h e m a g n i t u d e of t h e error t h a t w o u l d be i n v o l v e d in measurirlg o c u l a r r i g i d i t y b y a p p l a n a t i a l g r e l a t i v e l y large a r e a s of c o r n e a m a y be o b t a i n e d in t.he followi n g waW. L e t us s u p p o s e t h a t the p r e s s u r e is 15 lllnl l-Ig in a n e y e h a v i n g a. coelt~cient of o c u l a r r i g i d i t y of 0.0215 a n d t h a t ~he c o r n e a is f l a t t e n e d so t h a t t h e d i a m e t e r of t h e a t ) p l a n a t e d a r e a is 6 ram. I f we t a k e r e q u a l to 7.5 ram, t h e v o h t m e of flnid disp l a c e d b y a p p l a n a t i o n an.xomlts to 9"2 m m a. B y a p p l y i n g F r i e d e n w a l d ' s e q u a t i o n for o c u l a r r i g i d i t y , t h e n e w level of intra.oeular p r e s s u r e (Pt) p r o d u c e d b y t h i s apl?lanation c a n be c a l c u l a t e d thus. log Pt -- log 15 = 0,0215 9.2 ~,Vtsenee
Pt
= 23.7 m m H g
T h e force n e c e s s a t W to p r o d u c e t h i s d e g r e e of" a.pl?la.nati.on will be g i v e n b y t h e relationship: =
t-04
+
62-5
W
if we t a k e t h e m e a n v a l u e s of a a n d b f o u n d for t i m " t h i n " c o r n e a (Table JI). Therefore ~:Vhence
rr = W
=
1 - 0 4 - ~ - 6 2 " 5 23-7 19,32 g wg.
I f i t were a s s u m e d t h a t t h e I m b e r t - F i e k l a w w a s va.Iid, ~hen t h e i a g r a o c u l a r p r e s s u r e d u r i n g a p p l a n a t i o n w o u l d h a v e b e e n d e r i v e d in a c c o r d a n c e w i t h t h e e q l m t i o n A
= 73-5
i.e. 971-
W
P, 73.5
x 10-32
Pt
T h u s P t w o u l d h a v e b e e n e v a l u a t e d as 26.8 marl H g i n s t e a d of t h e t r u e v a l u e of
A.P I ' I ~ A N A q : I 0 N
2.. O N,O M
Jg ']'
11"5:
283
23'7 m m ~[-Ig. i f t h i s e r r o n e o u s vMue /'or Pr. w e r e u s e d in dor~vJng t h e coet~icient of ocuLar r i g i d i t y , we woxtld e a l c u l a t e t h e l a t t e r t h u s : K
~-
log 2.6.8 9;2
-
log 15 =
0.0274,
w h e r e a s t h e t r u e v M u e oP 1i7 was 0"021 5. T h e errors Jn t~ and/( h~we b e e n ealcuh~ted .[br v,~rious intraocuh~r p r e s s u r e s trod d i a m e t e r s of a p p h m a t i o n a n d are g i v e n in T a b l e IY. This shows t.hat t h e errors in Pt increase as t h e d i a m e t e r s of a p p l a n a t i o n increase, w h e r e a s t h e c o n v e r s e is t r u e f o r the errors .in t h e e o e i t i e i e n t of ocuhtr r i g i d i t y . TA £~L~; I V .l)i a I i1 c l,et' el + a plda ila Ied are~t
(ran,)
]~t (ram H g ) calculated froth I m b e r t . l~'iok law lllhlklS t r u e 1)+ tq,'= I0 m m H~ P,*= 15 m m 14g J~,*,=20 m m
4
0"9
5
1.'4
6 7 8
2"1 3"3 6"0
l .3 2-1 3.1
1 -7 2.8 4-2
4.9
6-7
9.0
13. I
1( c a l c u l a t e d u.sing b n l ) e r t Fick law (~l'Lie N = 0 " 0 2 LS)
Hg
1~o~'= 15 m m H g 0-040 0-033 0-027 0 -025 0-024
* J~o is Lho intratlcular t_~reasuro before a p p l ~ n a t i o n . I" T h e s e values are given for .P,-- I5 lllln H g o n l y since t h e y differed o n l y slightly f r o m t h o s e eMculated for Po= 10 m m f i g a.nd for P o = 2 0 m m Hg.
O b s E r v a t i o n s o f t h e v o l u m e s of fluid displaced, b y a p p l a n a . t i o n s h o w e d t h a t t h e s e a p p r o x i m a t e d g e n e r a l l y t o t h e v o l u m e s calcttla.ted as s p h e r i c M s e g m e n t s , u s i n g t h e e x t e r n a l r a d i u s of c o r n e a l o u r v ~ u r e in tlxe c a l c u l a t i o n . I f w e b e a r in m i n d t h e p h y s i c a l p r o p e r t i e s of ~he c o r n e a l s t r o m a , i t s e e m s i m p r o b a b l e t h a t t h e a . p p l a n a t e d eornee~ does, m fact, 1Tare t h e s h a p e of a sphericM s e g m e n t f r o m w h i c h a i \ t r t h e r s e g m e n t h a s b e e n sliced o{t. I P m ' t h e r m o r e , t h e i n t e r n a l r a d i u s of e o r n e M c u r v a t u r e m i g h t b e c o n s i d e r e d m o r e a p p l i c a b l e t o t h e cale u l a t i o n of t h e v o h t m e t h a n t h e ex'cernal radius. This s e c o n d p o i n t was e x a . m i n e d for t h o s e c o r n e a s o n w h i c h m e a s u r e m e n t s of e x t e r n a l ellrvatvre a n d thickness h a v e been made, b~t the e x p e r i m e n t a l results did n o t allow a n y d e f i n i t e c o n c l u s i o n to b e r e a c h e d , T h e i m p r e s s i o n gah~ed w a s t h a t t h e u s u a I m e t h o d of c a l c u l ' t t i n g vohm~es of a p p l a n a t i o n d i d n o t l e a d t o gross h m c c u r a e i e s . A Ct,2 N O~.VL ]g D G M E I N T
We should like to a c k n o w l e d g e the a.ssistarme of Mr. P. I'ta.rdy S m i t h with s o m e of the experi meJ ~ta.1 work. J { E ]?F, R E N C E , S
C l o l d m a n n , I-:]2. ( 1 9 5 6 ) . I n Gla~tcoma, T r a n s . 2nd Conf. 1957, ed. b y F. W . N a w e l l , p. I 6 7 . J o s i M t M a c s : Jr. l r o u n d a t i o n , N e w Y o r k . G o l d m a n n , ]-L a n d S c h m i d t , T . (1957). Ophthalm.oloffica, 2?asel 134, 221. G o l d n m n n , H . a n d S o h m i d t , T. (1961). Ophthal~nologica., Basel 1 4 1 , 4 4 1 . I ( r u s e , ~.V. (1 960). v. C,rae.fcs ,'Itch. Ophlhal, 162, 78. M a u r i c c , J). M. a n d G i a r d i n i , A. A. (1951). Brit. J . Ophthal. 35. 169. M o s e s , [~.. A. a n d T a r k k a n e n , A. (1959). Amer. J. Opht/+al. 47, ,557.
The
Validity
of the Imbert-Fick Law Applanation Tonometry
as applied
to
J. Gr.os'rlgn a>'J) E. S. I'].:I~KJ,','S
Department of lgx2)eri'mcnlal Ophthalmolqqy, l'nstitutc (f Ophlhahnology, U niversily of London, ],2.~gla~d (Received 21 May 1963) T h e p r i n c i p l e o f a p p l a n a t i o n t o n o m e b r y c o u l d be o x b ( : n d c d t o bhc ( l e t e r t n i n a b i o n o f o c u Inr rigidiLy a n d , p e r h a p s , faciliLy o f o u b f l o w , i?;obh p r o c e d u r e s w o u l d neeossit~at,o t h e f l a t t e n i n g o f r e l a t i v e l y l a r g o a r e a s o f c o r n e a , a n d t.he.ir vn.lidity w o u l d d c p o n d u p o n t h o a c c u r a c y o f c o m p u t a t , i o n s o f bho i n b r a o c u l a r p r e s s u r e a n d l,ho v o l u m e o f lluid d i s p l a c e d in t h e e y e . ] - t u m a n c o r n e a s w o r e m o u n t e d in a n al)pa.ralst,s no t h a b Lho r e h ~ b i o n s h i p b e t w e e n t h e i n b r a o c u l a . r p r e s s u r e , t h o a pplan~d,ing f o r c e a n d t h e a t ' c a o f c o r n e a f l a t t e n e d c o u l d be m e a s u r e d e x i ) e r i m e n t . a l l y . If, w a s f o u n d lamb t h i s r e l a t i o n s h i p d i d nob c o n f o r m e x a c t l y t o t h e I m bert,-.Fi c k b~w, t h e r e Imi ng so n m e vi(lcn ee that, s u r f a c e t e n s i o n a n d t h e f o r c e r e q u i r e d t o b o n d t h e c o r n e a l t i s s u e s wore r e s p o n s i b l e f o r t,his d e v i a t i o n from the expected behaviour. For the ma,jority of corno~s, at an applanation diameter o f td)oub 3 r a m , tam .[mbert,-li'iok law was ~MH)licablo, bul, at, l a r g e r ( l i a m e t o r s , t h e a r c ~ o f c o r n e a f l a t t e n e d w a s loss t h a n e x p e c t e d . I t is poinLed o u t t h a t a n a p l ) l a n a l A o n d i a m e t e r o f 3 m m is t ~ p p z ' o x i m a t o l y bht~l, u s e d in tlu; G o l d m a n n t o n o m o t e r . :l?ho o b s o r v e ( I v o l u m b s o f fluid d i s p l a c e d b y a p p l a n a t i o n w o r e in g e n e , ' M a g r e e m e n t w i t h t h e v a l u e s c a l c u l a t e d o n t h e a s s u m p t i o n t h a t t h e v o l u m e s c a n be re.garded as sl)llorical s e g m e n t s .
1. Introduction Applanation tonometry is based upon the principle illustrated in Fig. 1, which indicates lahat, when a flal5 surface is pressed wit, h a t'oree, W, against a spherical container having an internal pressure, P, equilibrium is attained w]mn PA = W, where A is
W
li'tO. I. Iq'inciplc ofapl>lnnal,ion I,onolnotry.
the area of flat~tening or applanal)ion. This simple rela, tionsl~ip, embodied in t,he Imberl>Fick law, holds only when the membrane bounding t)he sphe,'icM contaiJmr is dry, infinitely thin and perfectly flexible and elastic. ]3y contr,'tsl~, tt~e cornea has ~ wet; surface, a finite thickness and irrtperfecl) elasticity coupled with some viscous propert, ies. In the developrnenl) of t,he Ooldmann ,~pplanabion tonomet, er, special '27,1.
A PI~I,ANAq?ION
TONOMlg'PIKY
27,5
attent, ion was paid to two forces w h i c h could i n v a l i d a t e t h e simple r e l a t i o n s h i p given a.bove (C'old m a n n , 1 956; G o l d m a n n and Sehmid% 1.957, 1.961 ), t h e first, being t h e surface l~ellsion e x e r t e d by t h e film of tears a-nd t h e second t h e :force reqllired to bend t h e corneal tissue a.t th.e eiremnference of the a p p l a n a t e d area. G o l d m a n n c o n c l u d e d t h a t , for ~llO corneal ~rea a p p l a n a t e d b y his i n s t r u m e n t , t h e d i s t u r b i n g threes t e n d to nullify one a.notlmr and thai) the simple reh~.tionship between P, A a n d W is valid. lI is conelt,sion is s u p p o r t e d b y some experimenta.1 evidence. T h e r e are, however, two circumstanoes u n d e r wIfieh it is i m p o r t a n t to discover ttm reta;tionship 1)etween P , d a n d W wlten the d i e m e t e r of t h e a p p l a n a t e d area exceeds ;~-06 ram, which is t h a t produced by the G o l d m a , n t o n o m e t e r . The first of these coneerns the possibility of p e r f o r m i n g tonogra.phy nsing a p p l a n a t i o n .instead of i n d e n t a tion, whilst, the second relates to the me;~surement el" ocular r i g i d i t y b y aI)planation; b o t h procedures would involve fla.ttening r e l a t i v e l y large ureas of cornea i~ order to produce a d e q u a t e elevations of i n t r a o e u l a r p~:essure. No s a t i s f a c t o r y m e t h o d of a p p l a n a t i o n t o n o g r a p h y has as y e t been devised, b u t oeula.r r i g i d i t y has been measn r e d ~,sing the appla.na, t,ion principle (Moses a n d U?arkt~a.nen, 1959; Kruse, t.960). Both t o n o g c a p h y and t h e m e a s u r e m e n t of r i g i d i t y wo~ld i n v o l v e d e t e r m i n a t i o n s not, only of changes in pressure b~tt also o[' Mtera.tions in i n t r a o c u l a r v o l n m e produced b y corneal fla.ttening. [Phese v o l u m e t r i c changes t~ave usnMly been ea.leulated in a c c o r d a n c e with the el,coreticat relai,ionshil)s given in ]Yig. 2, b u t this implies t h a t ,
J
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= ~h~',ap-hi
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t
1fie. 2. Calenlat,lon o f v o l u m e o f l l u i d d i s p l a e e d b y al)plalmf~ion.
dltring applanat)ion, the original, s h a p e of t;he }~eriphera.1 region of t h e cornea is reI;a.ined so thaf5 I~tte v o l u m e of" fluid displaced can be cMculated as t h a t of a s])heric~l segment. ]~ven Lr this be brae, t h e r e is some d o u b t as to w h e t h e r r shoxlld be t a k e n as equ~l to t h e externM o r to the i n t e r n a l r a d i u s of corneal c u r v a t u r e . ~ x p e r i m . o n t a l eonfirma,tion of" t h e va.lidity 0[' the e;~leulated v o l u m e s of fluid displa.eed d u r i n g applaun.tion ,~t)pears to be laoking. '.l:hc t'olIowing investiga.tion was designed t h e r e f o r e to investiga'ce t h e r e l a t i o n s h i p bet)ween t h e m a g n i t u d e of cornea.l a.pplanat.ion, t h e iut,r~oeular press~tre and t h e v o l u m e of fluid dispirited b y a,pplana, tion. 2. M e t h o d s
The aplJ~ra.t.ua is show,1 di~grammat.icMly ila ~'ig. 8. The excised cornea or artificial cornea was clamped in a. perspex chamber connected to a, ma.nomet.er reservoir. A perspex disk suspended from a pivot could be a.pplied to the snrfa.ce of tl~e cornea. Side arms,
J. (.'I,()8TI~]I¢ A S I )
276
E. N, I ' i , ; I L K I N S
p i v o t e d ~tt' t,hc so,no pal,l,, h e m ~. s('~,.It~ ]mt~ a m l t~ c ( m n t e r w e i g h t ' t,o allow for ndjust'ing tim p e r s p c x pie.be so t'lmJ, ~)o l%rce was exerl.etl on the ¢'ornen }ml'o,'o bhc addibion o1" wt;igill.s to t,he so'Me p:tn. A gra.dua.te~[ t,ttbe coat,sin i]~g a mn;t I1 a itr-bu l,ble a.1lowell chq.ugos of vol urea t,o be m e a s u r e d . T h e alia.meter of t,he Ibtb~etmd a r e a wa.,~ men.su red by t n c a n s of v. slil;-l~mp m i c r o s c o p e wit.h a m c a s u r i ~ g aye-piece. E a c h scale ~iiviaiou of t,he eyt;-pieoe represent, ed 0-06 mtt~. l lttrnn.ri eornea.s~ were used a n d were o b b M n e d f r o m ent,oleaLe¢[ eyes t,hn.b ha.d bt:e~, r e j e c t e d a.s ttnsuit,~ble ~louor /nn.l;crial I'or kern.Loplasby. T h e cord,on., which w,.s excisetl t o g e t h e r w i t h a. s u r r o u n t l i n g rim of sclert~, w',.s c l a m p e d iu t,he perspu× c]m,,nl)~;r n,n~l, in s o m e experiment,s, w a s t.hen i m m c r m ; d i~, n. solubion of dext, ra.n unt.il it,s t h i c k n e s s lind been r e d , t e e d I.o tt v a l u e oorrespot~dit,g t:o that, of the living htttna.n eye. T h e p e r s p e x c h a m b e r w a s L h c ~ l oonneut, ed t.o l.he r e m a i n d e r of bhe aplm.ra.t.u.q qn,l l,he pres,m~rc w i t h i n it, was set, at, tl)o desired level wit,h the m a n o n m t , c r , n u d n. series o f weight,s :~thte.d t,o him acMe p a n , t,he al)plnnn.t, ion dit,.met, er beh~g measurc(I for en,:h weighU. Observat, ions were m a d e wit,h pressures in t,he clu~,u!:e:" of 20, 25, 30, 35, ,10 an,l 45 m m l l g nnd weight.s o1"2, 5, 7. 10, 12, 15, 20, 2-'.', anrl 3 0 g . • ,.
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View wilh slil.-iamp of area of cornea fl,a~.te,~ed by perspex disk (measurinq Fro.
3..Diagrammatic
rcpresont.abinn
~,F a
l~l~ar'atus.
scale
dispIoced Ioteral~y)
Areas of n.pph,m,~iun in squa.,:e ruiIIhnet'res were ca.lcula£ed from Lhe obs~,rved din.reeLers, ~ n d tJ~e v o l u m e d i s p l a c e d b y c;~oh 'q~plb;a.t'hm of ~ w e i g h t was me,qsured directly- by 6he m o v e m e n t ; of ~hc b u b b l e in i,he graduaLe(l bube. T h e e×~ernM r a d i u s of c o r u e a l curva.lmre w,~s m e a s u r e d wit, h Ifl~e Ja,vaI-Sch}o~z o])ht'halm o m e t e r nn,] corueM bbickness wa..~ dot.ermined by t h e met, hod of Mauvice 'lnti Giar,{iM (I95.]). Accordi~lg to the l n , b e r b - F i c k law, l~he ~re.q. "tppl,qu~ted. is p r o p o r L i o n ' d to the force a p p l i e d d i v i d e d by the p r e s s u r e , t~ud w h e n bhe area is m e a s u r e d iu s q u a r e millimet,re.% t h e force in g r a m s weighb a n d t'he p r e s s u r e in millimel~res o f m e r c u r y , t;he equ~tior, a s s u m e s bite
form:
IV A .-= i)
x
73-5
T h e o b s e r v e d v M u e s of A w e r e plott, ed a g a i n s t t.he known v a l u e s of tV/P a n d a regression liae c M c u l a t e d for t, he t)oinbs o b b ' d n c d . T h i s line could t h e n be c o m p a r e d wibh Lhe calcal a t e d line.
!
i
AI'I'I~ANATION
/rONOMI,]'I'I~,Y
277
.In order I~o tcst, l,hc w, lidit,y of t,hc met, hod, l,l,o expcvimenl~ was first; pot'_rorl-o.ed u.yitig a, I,hin r u l ) b o r iyiCll'ibl'It.iio iil ll]+l,oo of bho oorziott., Its it$ wlt.,'-t oxpooLod f , ] l a t milch ll, m e m b r a n e would nl)l)roxilnaLe inoro vloscly Lo bho l,hcorebica.l r e q u i r e m e n t s . Results
3.
lCubber m e m bra?~e T h e i'cmtli,s (Fig. ,1) s h o w t01mt, |,hurt is ~ l i n e a r rela, t;ionship b e t w e e n t h e o b s e r v e d +u'ca, o1' U.l)lJln.nabion, i1, ll,ntl HI/P. '.|'lie c.t~.leuh~l~c(I regression line intersect~s t:lle ordinat~e
I1~ i)rove(I il~q)os,
]l~:ossuro it.lid ,'xi.'o{I, is
11ol6,
[l.m(t~# (~fJrl#(>(I, (~) T h e r e h d i o , s h i p belween f o r c e , area c t , d ?Jressu're.. A l l n i n e c o r n e a s e x a m i n e d showed % l i i l c a r rel~l:.ionshi l) I)et.weoll I V I ] j 'til(l A . T l i i s (li(I iiol;, oorresl)On(1 Co ()lie l i n e e×i)ecl,cd t'rolll t h e ln)l)t;vt-lCic'k l a w , I)tll~ c o t l l d . Lle e x p r e s s e d [~s:
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\:'~lucs o f a a.l~(I b f , r i n d i v i ( l u q l eornca+s wore d c t c r m i n e d b y c a l c u l a b i n g t h e l i n e a r re,g r u s s i t , i o1" A on IV/IL a n d are givel~ ill TM)le I., a. t y p i c a l result~ b e i n g i l l u s b r a t e d in Fig. O. TA lii,E l .
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I"lu. 5. I ( c l n t i o n ~ l l i l l I~ctwt,cn ,-I iiill~ I1"/1' I'l~l' wel. rulfitler i i l ( , n l b r n l l o . ' l ' h o chH.ted lino s h o w s I.ho rehll,ioll.~hill exist, t i e d I'l'~ml t i m I n l l l e r t . - I G e k Inw ( / 1 = 7 3 - 5 I1"/1~). T h o conl, i n u o u ~ l i n e is l,h o o l t l o u l a t e d a'egresaion tino fiw t h o t~b.~ea'vltl,iouls {.,I = 3" 19 q- 7 I .56 W / / ' ) .
(b) l',bl.me.~ o.f./I.,id displaced b!/,7?~btnation. I h e resulLs f o r s i x c o r n e a s a r e s h o w n a s "~. gr;i.pll ( F i g . 7). '.i:lle v o h u l t ( ; s c a . l c u h t t c d ill a c c o r d a n c c w i t h t h e e q u a t i o n r
1
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28,)
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(';I.()S'l'l,:l,'. ANI)
E. S.
Pt:~II.I';.IN,%'
I t c a n b e seen bha~ t~he v o l u m e s d i s p l a c e d b y a p p h m a t - i o n c o r r e s p o n d g e l : e r a l l y w i t h t h o s e d e r i v e d t h e o r e t i c a l l y . T h e rcsull;s f r o m s o m e cornea.s f i t t e d t h e c a l c u l a t e d c u r v e s bebl~er t h a n t h o s e !'rein o t h e r s , b u t t h e r e w a s n o ale~r i n d i c a t i o n t h a t t h i s w~Ls r e l a t e d to t h e c o r n e a ] t h i c k n e s s . Also, w h e n r e s u l t s f r o m i n d i v i d u a l c o r n e a s w e r e 60 ~
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Appl~noth*gforce (g wt ) P - Intr¢oculer pressure (rnmHg) F I e . 6. :Relnt, i o n s h i p [)obw(2ol:l ~'t a n d I I ' / P f o r e x c i s e d h u m a n c o r n e a , h a v i n g a. t h i c k n e s s o f 0-55 ,urn. T h e dotted line shows the r01atioL~ship expected f r o m the Imlut~t-t-Fid¢ law (.4----~73.5 IV~P). '.l'h6con. tinuous ]i2*c is the cah;ulatcd t'cgrcssion line for the observations ( A = 0 - 6 7 + 53.1 IV/P).
i
i
i
/" = 6 r a m
60-
/'--7ram
50--~mm
r
_~ 4 0 -
--
¢2
3O-.
o
20-
q)
E %
0
..a~ 3
4
S
I
1
I
I
6
7
8
9
Observed diameter o f applanation (ram) FIG, 7. V o l u m e o f t l u i d di.uplaced b y o~pplanation. R e s u l t s f o r s i x h u m a n c o r n e a s . T i l e t h r e e c o n t i n u o u s l i n es s h o w t i l e v o l u m e s c a l c u l a t e d a.q s p h e r i c a l s e g m e n t s f o r c o r n e a l r a d i i o f 0, 7 a n d 8 r a m .
AI'PbANA'.i?ION
q'ON03I,
le,'.I'tt, Y
281
c o n s i d e r e d in re,lation to t h e m e ~ s u r e d ra.dixm of e x t e r n a l ¢mrvatxtre mad t h i c k n e s s , it w~s i m p o s s i b i e to d e t e r m i n e w h e t h e r t h e o b s e r v e d v o l u n m s a g r e e d b e t t e r w i t h those calettla.l:ed usi~g t~l~e e x t e r n a l r a d i u s as ~, or w i t h t h o s e u s i n g tire interns.1 r a d i u s as r. 4.
Discussion
T h e r e s u l t s o b t ~ i n e d ill t h e e x p e r i m e n t s on r u b b e r m e m b r a n e s w e r e of v a h l e in a s s e s s i n g t h e a c c u r a c y of m e a s u r e n t e n t s of the a.l~planated ~rea.s. ~\rit]l t h e d r y m e m b r a n e , i t was found t h a t , a.pa.rt f r o m a snmll e r r o r a t t r i b u t a b l e t o e l e c t r o s t a t i c a t t r a c t,ion betweeJl t h e pla.t.e a n d tile m e m b r a n e , t h e e x p e c t e d r e l a t i o n s h i p b e t w e e n pressure, a r e a a n d force was o b e y e d . (Ja.refill a d j u s t m m ~ t of t]xe s l i t - l a m p was n e e d e d to ensu re gha.t t h e d i a m e t e r , r a t h e r t h a n a. c h o r d , of t h e a p p l a n a t e d circle w a s mea.sured, b u t t h e restllts strongly s a g g e s t t h a t t h e p o i n t s a t w h i c h t h e c u r v e d m e m b r a n e m e t t h e fla.t pla.te were defined a c c u r a t e l y . I t was ex]?eoted t h a t t h e m e a s u r e m e n t s m i g h t be iess aeealrate w i t h t h e c o r n e a since t h e a n g l e b e t w e e n tlle p l a t e a n d tile c o r n e a w o u l d be o c c u p i e d b y Huid. Neve-ctheless, t h e resltlts obta.ine& w i t h t h e w e t r u b b e r m e m b r a n e , a l t h o u g h s h o w i n g m o r e s c a t t e r , did nob d i k e r signifiea.ntly f r o m t h o s e for t h e d r y m e m b r a n e , s u g g e s t i n g ttaat t h e p r e s e n c e of fluid b e t w e e n t h e p l a t e a n d tlle al?plana.ted s u r f a c e h a d l i t t l e et~eet, on a c c u r a c y o f m e a s u r e m e n t s of t h e f l a t t e n e d area.. ]~'or r e a s o n s g i v e n in t h e p r e v i o u s section, it was n o t possible to o b t a i l l reiia.blc m e a s u r e m e n t s o[7 the arrtmlnt of t2uid d i s p l a c e d b y a p p l a n a t i o n of the r u b b e r m e m brane. T h e r e s u l t s for all. nine eornca.s e x a m i n e d i n d i c a t e d tha.t t.h.ere was a l~near r e l a t i o n ship b e t w e e n A a n d I V / P , h a v i n g t h e [7orm A = (~ + bW/P. l~or s e v e n cornea.s (.~, was l?ositive, indieatit~g t h a t a. sin,M1 ~trea of f l a t t e n i n g w a s 1)resen~ w i t h no w e i g h t in t h e scale p a n . ~l'his reseval~led t h e r e s u l t for t h e d r y r u b b e r m e m b r a n e w h e r e e l e c t r o s t a t i c attra.etion b e t w e e n ~he m e m b r a n e a n d elm p l a t e p r o v i d e d ~m e x p l a n a t i o n o[7 t h i s finding. I n t h e case of t h e cornea., a n a n a l o g o u s e x p l a n a t i o n w o u l d be t h a t stlr[itce t e n s i o n ga.ve rise to a. force a d d i t i o n a l to t h a t a p p l i e d b y t h e w e i g h t s in t h e scale l?a]x of t h e al)para.t, us. T h e r e s u l t s for all n i n e cornea.s g a v e v a l u e s of /) lower t k a n e x p e c t e d tk'om t k e p. [robert-l} ~ek. law so tlm.t.-, for t h e cornea.s h a v i n g p o s i t i v e v a l u e s of c, (i.e. t)he m a j o r i t y of cornea.s exa.mi~ed), t h e r e w a s a n i n t e r s e c t i o n b e t w e e n t h e line c a l c u l a t e d from t h e o b s e r v e d v a l u e s of A a n d W / P a n d t.be line drawI~ in accords.nee w i t h t h e [ m h e r t .IPick ].aw. F o r v a l u e s of tV/] ) g r e a t e r thatx a t t h e p o i n t of interseetio2~, t[~e areas of corne,% fla.ttened were t h u s s m a l l e r tlxa.n exj.~eeted, f r e t s t h e t h e o r e t i c a l r e l a t i o n s h i p , ,% f i n d i n g for w h i c h t w o e x p l a n a t i o n s can be ~d r a n t e d , q?he first p o s s i b i l i t y is theft; A s h o u l d c o r r e s p o n d , n o t to t h e m e a s u r e d a r e a of api?lana.tion on t h e e x t e r n a l s u r N c e of t h e cornea, b~,tt, to t h e a r e a of t h e i n t e r n a l surface t h a t b e c o m e s f l a t t e n e d . ~.fhis e x p l a n a t i o n appea,rs untena.ble, h o w e v e r , since it is difficult to believe t t t a t tt~e f l a t t e n i n g of t h e i n t e r n a l s u r f a c e co~fld be g r e a t e r Is]tlLIt t h a t of t h e e x t e r n a l snr.face. 'Phe s e c o n d p o s s i b i l i t y is t h a t p a r t of t h e aj?.p l a n a t i n g force was e x p e n d e d in b e n d i n g t h e c o r n e a l t i s s a e a t t h e c~rcumferenee of t h e ~lattened area. ~.l?his m o r e l i k e l y e x p l a n a t i o n w o u I d ha.re been s t r e n g t h e n e d if t h e r e h a d b e e n a consistez~t d e p e n d e n c e of t h e v a l u e of b u p o n t h e e o r n e M t h i c k n e s s ; this, h o w e v e r , w a s n o t t h e ease in t h e small m n ~ b e r of t h i c k c o r n e a s s t u d i e d . 2kt t h i s po~r~t w h e r e t h e line c a l c u l a t e d .from t h e e x p e r i m e n t a l r e s u l t s i n t e r s e c t e d g}~e line d e r i v e d t h e o r e t i c a l l y , t h e o b e e r v e d r e l a t i o n s h i p b e t w e e n pressure, area. a n d force w a s in a g r e e m e n t w i t h t h e I m b e r t ~ F i e k l~w, a n d t h i s p o i n t c o r r e s p o n d e d to a
282
J . GLOS'I?]~.I~ A N D
1~. 8, P . E I I K [ N S
v a l u e o f / / e q u a l to 6-1 r a m ' , w h e n c a l c u l a t e d as t h e m e a n from t h e r e s u l t s of five e o r n e a e h a v i n g t h i c k n e s s e s w i t h i n t h e r a n g e f o u n d in h e a l t h y l i v i n g l m m a n corneas. T h e d i a m e t e r c o r r e s p o n d i n g to t h i s a r e a is 2'8 m m , w h i c h is r e m a r k a b l y close to t h e d i a m e t e r (3.06 ram) of t h e area a p p l a n a t e d b y t h e G o l d m a n n t o n o m e t e r . T h e s e r e s u l t s t h u s SUlyply e x c e l l e n t eonfirn~ation of t h e a p p l i c a b i l i t y of t h e Imbert-Ir'ick l a w to d e t e r m i n a t i o n s of i n t r a o e u l a r p r e s s u r e w i t h t:he Goldmam~ i n s t r u m e n t . F o r ap.planated a r e a s h a v i n g d i a m e t e r s s i g n i f i c a n t l y g r e a t e r or less t b a n 3 tmn, t.he I m b e r t - F i c k la,w c a n n o t be used for t h e a c c u r a t e d e r i v a t i o n of t h e illtra.omdar pressure. Moreover, it would seem f r o m t h e r e s u l t s of gifts s t u d y t h a t t h e m o d e of d e p a r t n r e from t h e h n b e r t - F i e k l a w v a r i e s f r o m cornea, to cornea, a l t h o u g h f u r t h e r w o r k is r e q u i r e d to e l u c i d a t e t h e f a c t o r s w h i c h d e t e r m f i m t h e d e v i a t i o n f r o m t h e "dmoretical r e l a t i o n s h i p . I t seems, t h e r e f o r e , t h a t s o m e c a u t i o n is n e e d e d in t h e i n t e r p r o r a t i o n of t h e r e s u l t s of a p p l a n a t i o n t o n o m e t r y w h e n r e l a t i v e l y large a r e a s of c o r n e a are f l a t t e n e d ; a n d t h i s w o u l d a p p l y , as p o i n t e d o u t earlier, to m e t h o d s of detern~ining octdar r i g i d i t y or f a c i l i t y of mttflow using t h e a p p l a n a t i o n principle. S o m e i d e a o.f t h e m a g n i t u d e of t h e error t h a t w o u l d be i n v o l v e d in measurirlg o c u l a r r i g i d i t y b y a p p l a n a t i a l g r e l a t i v e l y large a r e a s of c o r n e a m a y be o b t a i n e d in t.he followi n g waW. L e t us s u p p o s e t h a t the p r e s s u r e is 15 lllnl l-Ig in a n e y e h a v i n g a. coelt~cient of o c u l a r r i g i d i t y of 0.0215 a n d t h a t ~he c o r n e a is f l a t t e n e d so t h a t t h e d i a m e t e r of t h e a t ) p l a n a t e d a r e a is 6 ram. I f we t a k e r e q u a l to 7.5 ram, t h e v o h t m e of flnid disp l a c e d b y a p p l a n a t i o n an.xomlts to 9"2 m m a. B y a p p l y i n g F r i e d e n w a l d ' s e q u a t i o n for o c u l a r r i g i d i t y , t h e n e w level of intra.oeular p r e s s u r e (Pt) p r o d u c e d b y t h i s apl?lanation c a n be c a l c u l a t e d thus. log Pt -- log 15 = 0,0215 9.2 ~,Vtsenee
Pt
= 23.7 m m H g
T h e force n e c e s s a t W to p r o d u c e t h i s d e g r e e of" a.pl?la.nati.on will be g i v e n b y t h e relationship: =
t-04
+
62-5
W
if we t a k e t h e m e a n v a l u e s of a a n d b f o u n d for t i m " t h i n " c o r n e a (Table JI). Therefore ~:Vhence
rr = W
=
1 - 0 4 - ~ - 6 2 " 5 23-7 19,32 g wg.
I f i t were a s s u m e d t h a t t h e I m b e r t - F i e k l a w w a s va.Iid, ~hen t h e i a g r a o c u l a r p r e s s u r e d u r i n g a p p l a n a t i o n w o u l d h a v e b e e n d e r i v e d in a c c o r d a n c e w i t h t h e e q l m t i o n A
= 73-5
i.e. 971-
W
P, 73.5
x 10-32
Pt
T h u s P t w o u l d h a v e b e e n e v a l u a t e d as 26.8 marl H g i n s t e a d of t h e t r u e v a l u e of
A.P I ' I ~ A N A q : I 0 N
2.. O N,O M
Jg ']'
11"5:
283
23'7 m m ~[-Ig. i f t h i s e r r o n e o u s vMue /'or Pr. w e r e u s e d in dor~vJng t h e coet~icient of ocuLar r i g i d i t y , we woxtld e a l c u l a t e t h e l a t t e r t h u s : K
~-
log 2.6.8 9;2
-
log 15 =
0.0274,
w h e r e a s t h e t r u e v M u e oP 1i7 was 0"021 5. T h e errors Jn t~ and/( h~we b e e n ealcuh~ted .[br v,~rious intraocuh~r p r e s s u r e s trod d i a m e t e r s of a p p h m a t i o n a n d are g i v e n in T a b l e IY. This shows t.hat t h e errors in Pt increase as t h e d i a m e t e r s of a p p l a n a t i o n increase, w h e r e a s t h e c o n v e r s e is t r u e f o r the errors .in t h e e o e i t i e i e n t of ocuhtr r i g i d i t y . TA £~L~; I V .l)i a I i1 c l,et' el + a plda ila Ied are~t
(ran,)
]~t (ram H g ) calculated froth I m b e r t . l~'iok law lllhlklS t r u e 1)+ tq,'= I0 m m H~ P,*= 15 m m 14g J~,*,=20 m m
4
0"9
5
1.'4
6 7 8
2"1 3"3 6"0
l .3 2-1 3.1
1 -7 2.8 4-2
4.9
6-7
9.0
13. I
1( c a l c u l a t e d u.sing b n l ) e r t Fick law (~l'Lie N = 0 " 0 2 LS)
Hg
1~o~'= 15 m m H g 0-040 0-033 0-027 0 -025 0-024
* J~o is Lho intratlcular t_~reasuro before a p p l ~ n a t i o n . I" T h e s e values are given for .P,-- I5 lllln H g o n l y since t h e y differed o n l y slightly f r o m t h o s e eMculated for Po= 10 m m f i g a.nd for P o = 2 0 m m Hg.
O b s E r v a t i o n s o f t h e v o l u m e s of fluid displaced, b y a p p l a n a . t i o n s h o w e d t h a t t h e s e a p p r o x i m a t e d g e n e r a l l y t o t h e v o l u m e s calcttla.ted as s p h e r i c M s e g m e n t s , u s i n g t h e e x t e r n a l r a d i u s of c o r n e a l o u r v ~ u r e in tlxe c a l c u l a t i o n . I f w e b e a r in m i n d t h e p h y s i c a l p r o p e r t i e s of ~he c o r n e a l s t r o m a , i t s e e m s i m p r o b a b l e t h a t t h e a . p p l a n a t e d eornee~ does, m fact, 1Tare t h e s h a p e of a sphericM s e g m e n t f r o m w h i c h a i \ t r t h e r s e g m e n t h a s b e e n sliced o{t. I P m ' t h e r m o r e , t h e i n t e r n a l r a d i u s of e o r n e M c u r v a t u r e m i g h t b e c o n s i d e r e d m o r e a p p l i c a b l e t o t h e cale u l a t i o n of t h e v o h t m e t h a n t h e ex'cernal radius. This s e c o n d p o i n t was e x a . m i n e d for t h o s e c o r n e a s o n w h i c h m e a s u r e m e n t s of e x t e r n a l ellrvatvre a n d thickness h a v e been made, b~t the e x p e r i m e n t a l results did n o t allow a n y d e f i n i t e c o n c l u s i o n to b e r e a c h e d , T h e i m p r e s s i o n gah~ed w a s t h a t t h e u s u a I m e t h o d of c a l c u l ' t t i n g vohm~es of a p p l a n a t i o n d i d n o t l e a d t o gross h m c c u r a e i e s . A Ct,2 N O~.VL ]g D G M E I N T
We should like to a c k n o w l e d g e the a.ssistarme of Mr. P. I'ta.rdy S m i t h with s o m e of the experi meJ ~ta.1 work. J { E ]?F, R E N C E , S
C l o l d m a n n , I-:]2. ( 1 9 5 6 ) . I n Gla~tcoma, T r a n s . 2nd Conf. 1957, ed. b y F. W . N a w e l l , p. I 6 7 . J o s i M t M a c s : Jr. l r o u n d a t i o n , N e w Y o r k . G o l d m a n n , ]-L a n d S c h m i d t , T . (1957). Ophthalm.oloffica, 2?asel 134, 221. G o l d n m n n , H . a n d S o h m i d t , T. (1961). Ophthal~nologica., Basel 1 4 1 , 4 4 1 . I ( r u s e , ~.V. (1 960). v. C,rae.fcs ,'Itch. Ophlhal, 162, 78. M a u r i c c , J). M. a n d G i a r d i n i , A. A. (1951). Brit. J . Ophthal. 35. 169. M o s e s , [~.. A. a n d T a r k k a n e n , A. (1959). Amer. J. Opht/+al. 47, ,557.