Engineering analysis of the J-loop posterior chamber lens

engineering analysis of the] -loop posterior chamber lens Alfred]. C hompff, Ph.D. Pasadena, California Optimum design of intraocular lenses is a major concern of both manufacturers and physicians. The objective of the present paper is to analyze the mechanical performance of an Intermedics Intraocular lens design and thereby obtain techniques to optimize design of presently manufactured lenses and new lens structures. The lens analyzed in this paper is the J-Ioopl posterior chamber lens (model 019). Preliminary results for the Simcoe-style posterior chamber lens (model 034) are also included. Both lenses have polymethylmethacrylate optics and loops made of polypropylene. These posterior chamber lenses may be implanted horizontally (Fig. 1) or vertically (Fig. 2). The undeformed lens has a length L; after implantation its

length is L - LlL. The compressive force needed for this change is called V; the gravitational pull on the optic is indicated by 2H. Of concern are the magnitude of loop pressure 2 on intraocular tissue and the amount of long-term sagging from gravity. To obtain analytical expressions for loop distortion the loop is divided into three structural members: an end-loaded cantilever BA, an end-loaded circular beam AC and an inactive circular beam CD (Fig. 1,2). Forces V and H act upon point C, resulting in deHections Xc and Yc (Fig. 3). These produce a moment M at point A, resulting in deHection yA (Fig. 4) and angle eA' Because the deformations involved are quite large, the effect of A on the system must also be considered (Fig. 5,6).

e

I


_________ I

1,~l.:-l'>L I I I

,

L---\ I

\

~I'

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,I

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C'

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' ' \

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I

I I I I \ I , \

,,

t.y_l-'

,I, ,I 'I I I 'I ' \ ,

I , I I I I I

,

,

'

\

----I

~'i:L C

-- 1~:.

c-

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L

L-l,>L

I I I

I

I

I

,,

~I

It.x_1

I

I

,

I I I

,

-

---

I

\-t-,

_---r c I I I I I

Lx Fig. l(Chompfl). Horizontal implantation of the J-loop lens, showing the length before (L) and after (L - AL) implantation.

i:;.,/+

W

,

I I

Fig. 2(Chompfl). Vertical implantation of the J-loop lens in the same coordinate system as shown in Fig. 1.

Supported by a grant from Intennedics Intraocular, Inc. Presented at the Third U.S. Intraocular Lens Symposium, Los Angeles, CA, March 26, 1980. Reprint requests to Dr. Chompff, Intennedics Intraocular, Inc., 3452 East Foothill Blt:d. #115, Pasadena, CA 91107. AM INTRA-OCULAR IMPLANT SOC J-VOL. 6, OCTOBER 1980

355

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-{Xc I I I

t--

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/ L _________ ----~~ I

H

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V "

-r-------- fi: de

MATERIALS AND METHODS

-;1':-==

C I : \ C ----

I

R.sin¢

where dL is the overall compression of the lens, dy is the perpendicular loop movement, I is the moment of inertia about the neutral axis of a cross-section through the fiber, and p is the radius of the fiber. With the use of the curves in Fig. 7 the amount of gravitational sag of the lens can be calculated using the equations of Appendix 1.

I

I

~\:

P \

¢

----

R

,,

"

~

Fig. 3(Chompff). End-loaded circular beam AC, for infinitesimal deformations.

~ ~AT-----

v

An Intermedics lens model 0l9C (#02535, 19 diopters) was selected for testing. This lens is intended for placement in the posterior chamber with loops placed in the ciliary body. Lens length L was 13.10 mm, loop fiber diameter was 0.14 mm, cantilever length was 2.45 mm and loop radius R was 1.50 mm. The weight of the lens in water was 2.6 mg, confirmed by a theoretical analysis for this diopter using a lens edge thickness of 0.28 mm and a specific gravity of 1.18 for polymethylmethacrylate. All dimensions were within the specifications for model 019C lenses. The lens was vertically immersed in water at 37.0°C and subjected to very small vertical loads (small weights on a loading pan) at 18-minute intervals. The corresponding deflections were measured with a sensitive linear variable displacement transducer in an instrument constructed in this laboratory.

I

~:

- - - - - - - - - --.!... - -

A

- - - - ---------'

.----i. - - - - - L .

1"""1

I

I

~Xc:­ I I

I

i~ __ t ___________ : I

I

HI

Fig. 4(Chompff). End-loaded cantilever AB, for infinitesimal deformations.

In the absence of gravitational pull, lens loop deflection may be represented by the two curves shown in Fig. 7 and by the following equations:

dy+ = YA + Yc I = 0.251Tp 4 356

(2) (3)

Fig. 5(Chompff). End-loaded curved beam AC, for finite deformations.

AM INTRA-OCULAR IMPLANT SOC J-VOL. 6, OCTOBER 1980

_________

=_:-:_~-.:::..J

B

RESULTS AND DISCUSSION Experimentally induced loop deformation is shown by the curve in Fig. 8. To compare this curve with the theoretical curve in Fig. 7, it was necessary to estimate the modulus E of the fiber from published data. However, the published values for fibers 3 are much too high for the present loop which has been subjected to a proprietary heat treatment during manufacture. The rigidity of the fiber after heat treatment should be about the same as that for injection-molded polypropylene, which is also a partially oriented polymer. Creep data for injection-molded polypropylene at 20°C and at various stress levels were plotted as "creep moduli" (E) (Fig. 9). These values for E were corrected for raising the temperature to 37°C and swelling the polymer in water (Appendix II). Substituting values of E corresponding to 18 minutes (log time = - 0.523) in Fig. 7 yields theoretical load-deflection curves at three different stress levels (Fig. 10). The agreement between theoretical prediction and experimental data is excellent. eOAD - ~EFLECT'ON

Fig. 6(Chompfl). End-loaded cantilever AB, for finite deformations.

(oXPEo.)

i 350!MG. FORCE

300 1

4,/ 2

24 X 10M

250

V k=E1

V

2

I 200t

r

o

1

,

.6 L

20

I

15

°1 I

100r

I

]L

16 2

k

0

12

8

0.4

0.8

1.2

'.6

2.0

2.4 M M

Fig. 8(Chompfl). Experimental load-deflection response of the lens in water at 37°C.

1

4 ---_~.6L

0·2

OA

0·6

OR

I1Y

0'8MM

Fig. 7(Chompfl). Theoretical load-deflection response of the lens in the absence offorce H, calculated for R = 1..50 mm and 1=2.45 mm. The loading parameter, k2 , represents the ratio VIEr, where E is the modulus of the fiber and I is the moment of inertia about the neutral axis of a cross-section through the fiber.

The time-dependent "sag" due to gravity in horizontal and vertical implants was predicted accurately using the equations of Appendix I and the experimental time-dependent moduli of Fig. 9. The results are shown in Fig. 11. It should be noted that the horizontal and vertical cases differ because the loops are open, yielding an asymmetric response. However, even the horizontal implant reaches a total sag of only 0.015 mm after 10,000 hours. The sensitivity of our instrument was tested by measuring the load-deflection characteristics of the much more compliant Simcoe-style posterior chamber lens (Fig. 12), intended for placement in the posterior chamber with its loops in the ciliary body. This

AM INTRA-OCULAR IMPLANT SOC J-VOL. 6, OCTOBER 1980

357

II

18 I"M .

w

~

11

OF f.'.ORi ZONTA L AND VER TI CA L

__

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if =

.,

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14

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0 9 ·0

SAG

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H RS.)

0~-------------------------------4~--~-a :0 15 20 10000 HRS e·2

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--~,. L OG

T ; ME

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Fig. 11 (Chompfl). Gravitational "sag" of horizontal and vertical implants over time periods up to 10,000 hours.

J

2

(H RS)

Fig. 9(Chompfl). Nonlinear tensile creep modulus of polypropylene saturated in water at 37°C, at four different stress levels. Creep data for injection-molded polypropylene at 20°C and various stress levels were obtained from the Modern Plastics Encyclopedia , vol. 55, 1978, p. 574. 1 350

LOAD- DEFLEC TION

( T HEO R.

AND

EXPER.l

CD

M G. FO RC E

300

250

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200

150

100

SO

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O ~

o

________________________ 0.4

0.8

1. 2

~

1.6

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2.4 MM

Fig. lO(Chompfl). Comparison be tween the experimental loaddeflection curve and the theore tical curve, calculated at three dil~ ferent stress levels .

lens had a total length of 12.7 mm and a loop fib er diameter of 0.19 mm. The load-deflection curve of the Simcoe-style lens, measured under the same conditions as described above, is shown in Fig. 13 together with that of the J-Ioop lens. The instrument responds to load changes of less than 2-mg force. The slopes of the curves at the origin show that even with a fiber diameter of 0.14 mm the J-Ioop lens is eight times more rigid than the Simcoe-style lens with a fiber diameter of 0.19 mm. Applying Equation (3), a J-Ioop lens with a fiber diameter of 0.19 mm can be shown to be 27 times more rigid than the Simcoe lens. 358

z Fig. 12(Chompfl). The Simcoe-style posterior chambe r l ens.

For a vertically implanted Simcoe-style lens with a fiber diameter of 0.19 mm, the gravitational sag after 10,000 hours can be estimated as only 0.069 mm using an equation analogous to Equation (17) of Appendix 1. Similar calculations for a fiber diameter of 0.14 mm yield a sag due to gravity of about 0.24 mm afte r 10,000 hours. Thus, from the designer's point of view it is clear that a J-Ioop lens with a fiber diameter of 0.19 mm is much too rigid, whereas a Simcoe-style lens with a fib er diameter of 0.14 mm is much too flexible, since a 0.24-mm sag is considered too much .

AM INTRA-OCULAR IMPLANT SOC J-VOL. 6, OCTOBER 1980

LOA)- DEFLECTION iEXPER.)

350

MG. FORCE /

J
APPENDIX I Using Castigliano's first theorem the deflections yp and xp and the change in slope 8 p at any point p on a beam can be calculated by the following equations:

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250

o

200

yp =

V

i

150

100

I

50

0

2.0

M EI

~ .(

SIMCOE- OOPS

I

0

P

~

2.4 MM

Fig. 13(Chompfl). Comparison between the experimental loaddeflection curves of the ]-loop lens and the Simcoe-style lens.

i

M EI

P

M EI

x dx

(4)

y dy

(5)

dx

(6)

where M is the moment at point p. For example, the moment at point A' in Fig. 5 is Therefore, it is concluded that the optimal loop fiber diameter is 0.14 mm for a J-loop lens and 0.19 mm for a Simcoe-style lens. The theoretical and experimental techniques developed here are valuable tools in the design and optimization of new intraocular lenses.

SUMMARY A detailed mathematical analysis of the stresses and strains occurring during and after horizontal or vertical implantation of a J-Ioop posterior chamber lens showed that vertically implanted lenses are almost twice as resistant to gravitational forces than are horizontal implants, and that gravitational "sag" for either lens position is less than 0.02 mm. Similar analyses using the Simcoe-style posterior chamber lens showed that this lens requires a greater loop fiber diameter than does the more rigid J-loop lens.

MA = VR sin
+ HR(l-cos
which must be equal to the moment at point A' in Fig. 6. Use of the stability conditions for an eccentrically compression-loaded cantilever yields the following equations: MA ( YA=y-

MAk 8A=-V

1 cos kl

-~

1.

2.

3. 4.

(8)

tan kl

(9)

Successive application of Equations (4) and (5) to Fig. 5 gives Yc and xc, leading to the following results:

Horizontal implants. Substituting Equation (7) in Equation (9) yields e A - R k tan (kl) [cose A ± (1- sine A)HIV ]

REFERENCES Barraquer J: Anterior chamber plastic lenses: results of and conclusions from five years experience. Trans Ophthalmol Soc UK 79:393, 1959 Olson RJ, Morgan KS, Kolodner H: The Shearing-style intraocular lens and the posterior chamber. Am Intra-Ocular Implant Soc] 5:338, 1979 Listner GJ: Polypropylene monofilament sutures. U.S. Patent #3,630,205; 1971 Williams ML, Landel RF, Ferry JD: The temperature dependence of relaxation mechanisms in amorphous polymers and other glass-forming liquids. ] Am Chem Soc 77:3701, 19,55

(7)

=

0

(10)

where R, E, I and H are all known. Thus, for each value of V the angle 8 A can be obtained by solving Equation (10) graphically. Then for each set of V and 8 A values the loop deflections Llx and Lly can be calculated from Equations (ll) and (12):

Llx+,_

=

±

1(1 -

Si~:A

)

± 0.5 k2R3 (0.51T-8A -sin8Acos 8A)

+ p2R3 (1- sin 8 A - 0.5 cos 28 A) AM INTRA-OCULAR IMPLANT SOC J-VOL. 6, OCTOBER 1980

(ll) 359

+R (

-

1

[ cos 8 A ± (1- sin8 A) HiV ]

cos kl

± k2R3 (1- sin 8 A - 0.5 cos 2 8 A)

+ p2R3 (0.7571' -1.58 A - 2 cos 8 A + 0.5 sin 8 A cos 8 A) (12)

where the ± sign has been used to combine the results for the left loop (Ax+, Ay+ ) and the right loop (Ax_, Ay _ ). The "sag" (oy) is then obtained from the following equation: H EI

oy where p2

=

HIE!.

Vertical implants . V and 8 A values are obtained by graphing the relationship shown in Equation (14). 8A - Rk tan (kl) cos 8A = 0

(14)

These values are then substituted into the following equations: dx +, _

~

± I (I -

Ay+ _

=

+R (

,

s~:A ) 1

cos kl

± 0.5 k2 R' (0.5.. -

e

A -

sine cose A

A)

+ Sx (15)

2 - 1 ) cos 8 A ± k R3 (1- sin 8 A - 0.5 cos2 8 A)

+ oy,

(16)

where the plus sign applies to the lower loop, which is more compressed than the upper loop (minus sign). Since Ox and oy constitute only minor perturbations , the "sag" (ox) may be obtained as follows: H EI

Ox =

where

~

(17)

= VlEI. Stress-strain curves can then be calculated according to Equations (1) and (2) of the text.

APPENDIX II A change in temperature causes polymers to "shift" their time-dependent moduli to longer or shorter times or frequencies. This is conveniently calculated from the semiempirical equation 4 :

(18) where log aT is the horizontal shift along the logarithmic time axis, Tg is the glass transition temperature of the polymer and C 1 and C 2 are constants given by: C1 =

1 2.303 fg

-~--

For insufficiently known systems fg may assume the value of 0.025 and (Xf may assume the value of 4.8 x lO- 4 /°C. The glass transition temperature for polypropylene is -lOoC . Substitution of these values in Equation (18) yields -1.892 for the horizontal shift from 20°C to 37°C. If the polymer absorbs water it will become plasticized. This effect on the moduli can be accounted for by the vertical shift in Fig. 9. Experimental data from this laboratory show that the modulus of polypropylene (Prolene) suture is reduced due to swelling in water by a factor of about two-thirds. The corresponding logarithmic value is -0.l76 . 360

AM INTRA-OCULAR IMPLANT SOC J-VOL. 6, OCTOBER 1980