Computer localisation of detatched retinal tears for a scleral buckling procedure

N[ed. & Biol. Eng. & Comput., 1977, 15, 564--572

Computer Iocalisation of detatched retinal tears for a scleral buckling procedure M. S. Engleman S. E, Moskowitz* H .L . Zauberman Division of Applied Mathematics, Graduate School of Applied Science & Technology and Department of Opthalmology, Hebrew University--Hadassah Medical School, Jerusalem, Israel

Abstract--A mathematical model has been constructed for determining automatically the Iocalisation of a class of detached retinal breaks. I t includes a complete schematic eye in terms of the radius of curvature, centre of curvature and index of refraction. Computer control provides the surgeon with corrections from the apparent site for placement of an implant in a scleral buckling procedure. The arc length of settling back onto the choroid~ measured on the posterior surface of the sclera and starting from the extension of the line of sight to the retinal tear, is the sum of two displacements, one solely due to geometry, the other to retinal relaxation. The geometric contribution is paramount because the maximum distance of detachment is taken to be small compared with the radial distance from the centre of the globe. Vitreous traction affecting retinal relaxation is replaced by an estimated overall stretch, satisfying an observed chordal distance between eXtremal points and distance of the detachment. The retina is assumed elastically homogeneous and isotropic and subject to a uniform normal surface traction directed inward. Displacements are predicted by means of a nonlinear theory for large displacements of membranes, Representative computer results are given for the vertical meridian.

Keywords--Eye, Modelling

1 Introduction

THE scleral buckling procedure was developed by SCHEPENS et al., (1957, 1958) to seal retinal tears. An encircling band is used to obtain a permanent buckle produced by an implant, as shown in Fig. 1. This reduces vitreous traction on breaks located in the indented equatorial region, so that the detached portion of the retina is allowed to settle on the choroid effecting closure at the buckle site. Ophthalmoscopic examination determines whether the implant is correctly placed, a faulty placement requires adjustment. The purpose of this investigation is to provide the surgeon with computer control by which these positional errors are averted.

The most common cause for failure of a retinal detachment surgery is the failure to locate and seal the break. A chart is usually prepared by the surgeon of the fundus that maps the area of detachment, location of the break, and the configuration of the blood vessels. But false projection in cases of large detachment will lead the surgeon into error while attempting to seal the tear. Just before surgery, the sclera is exposed. Using the ophthalmoscope, the retinal break is located, and a mark is made on the sclera with diathermy at a

First received 13th September and in finalform l Oth November 1976

*Reprint requests: Prof. S. Moskowitz, Division of APP I/e~ /vi~the- matics, Graduate School of Applied Science & TechnoJOyY, ~e~rew University of Jerusalem, Israel

point overlying the break. To repair a retinal detachment it is essential first to create a watertight seal of the break, and second to drain the subretinal fluid sufficiently to allow the retina to settle on the choroid. The exudate from the choroid will create the seal and develop into a firm chorioretinal scar.

band

Fig. I Scleral buckling with encircling band and undermining for silicone implant polyester-fibre reinforcements of mattress sutures in scleral flaps are used

564 Med ica l & Biological Engineering & Computing September 1977

STINE (1934) recognised the inherenterror in the localisation of detached retinal tears, and pro- posed a graphical method for calculating the corresponding projected point on the sclera to which the tear will fall. It involved measurement of the three angles formed by the line of sight to the tear and anterior and posterior points on the boundary, and the optic axis. By using Snell's law and the schematic eye of COWAN (1928), a circular arc was struck between the intersection of the line of sight to the tear and reflected arc about the chord, connecting the endpoints of retinal detachment, and the predetached retinal location. This was taken to be the point of reattachment. A trigonometric formula continued the projection onto the scleral surface, and correction was reported in terms of arc length distance from the limbus.

There were serious deficiencies that limited the procedure's clinical use. First there was the in- accuracy of a graphical technique. Secondly, he reflected the minor circular arc connecting extremeties of detachment, in the meridional plane, about the included chord, the image representing an approximation to the actual detached retinal curve. Thirdly, he supposed retinal nondeformability for the required point projection onto the choroid. The present investigation corrects these short- comings.

2 Management of retinal detachment Retinal detachment is a separation of the retina from the pigment epithelium. Vitreous fibres adherent to the edge of a retinal break exert a pull that widens the tear. When this occurs fluid can enter the hole and strip the weakened retina off the pigment layer.

Photocoagulation is often used for detached breaks located near the posterior pole, for re- inforcing diathermy reaction and for giant tears. When photocoagulation is difficult, if not dangerous, to perform because of opacities or damaged sclera, and when breaks without detachment are located peripherially, cryoapplication is preferred (SCHEPENS, 1968). Diathermy, an older technique, is now con- fined to intermittently spaced breaks; whereas the more recent technique of laser is under clinical evaluation.

In a common buckling procedure the sclera is dissected to form flaps leaving a thin scleral covering of the choroid. The flaps are closed after burying a silicone implant, which not only inhibits mi- gration but helps prevent infection. Silicone implants are used because they are nonabsorbable and soft, reducing the chance of pressure necrosis of under- lying tissues. A tough translucent layer of con- nective tissue eventually encases this type of implant which then compensates for the weakened sclera. The grooved implant 5-12ram wide, at times shaped to conform to the globe, is fitted

under the' flattened 2.5 mm silicone band. Pulling up on the band slackens vitreous traction, and brings the retina in apposition with the choroid at the site of the break.

The change in ocular pressure is controlled so that it does not exceed 5 mm Hg.

Usually the subretinal fluid is drained by per- forating posterior to the buckle, while fluid from the vitreous cavity is prevented from escaping.

- l imbus, L

/ /

Fig. 2 Localisation o f a detached retinal tear

Table 1. Values of parameters for schematic eye

Refract ive index air 1.000 cornea 1.336 aqueous humour 1.336 outer lens 1.386 core lens 1.430 vitreous humour 1 �9 336

Radius of curvature anterior surface of cornea posterior surface of cornea anterior surface of outer lens posterior surface of outer lens anterior surface of inner lens posterior surface of inner lens anterior surface of sclera posterior surface of sclera retina

mm 7.800 7.80O

10. 000 6.000 7.911 5.760

12.065 12.065 11.060

X co-ordinate of centre anterior surface of cornea posterior surface of cornea anterior surface of outer lens posterior surface of outer lens anterior surface of inner lens posterior surface of inner lens posterior surface of sclera anterior surface of sclera entrance pupil retina

mm - -3.650 --3" 650 - -1.650 10.750

--0-107 11.145

- -0.135 - -0 .875

8.900 0.000

Eccentr ic i ty corneal ellipse O. 500

Medical & Biological Engineering & Comput ing September 1977 565

Chorioretinal adhesion results from an in- flammatory exudative reaction produced in the choroid by diathermy transferred through the scleral cover left in undermining.

3 Mathematical model

A mathematical model was used for determining the localisation of detached breaks. The model and its assumptions will first be described and ex- plained. In addition to assumed values of three principal parameters of a schematic eye, the radius of curvature, centre of curvature and index of

refraction, there is a hypothesis regarding displacement during detachment.

The crystalline lens is a transparent, biconvex structure whose anterior and posterior surfaces meet at the equator. Although the nucleus contains minute opacities and concentric areas of different indices of refraction, two layers, each homogeneous and possessing a uniform index, were used. The meridional dependence of limbal diameter was taken as linear. Any adjustment of shape produced by a change in the intensity of light was neglected.

The centres of curvature of the various refracting

Table 2. Arc length P1 P2 from apparent location to recommended site for placement of implant, for vertical plane, given distance of detachment d in mm, and angle e in degrees formed by line of sight to tear and optic axis

Angle Distance of observed retinal tear from retina along line of sight

Degrees 0 mm 1 mm 2 mm 3 mm 4 mm 5 mm

1 0 ' 0 0 "0 0 . 0 0-1 0-1 0.1 3 0 . 0 0"1 0.1 0 . 2 0 "3 0 " 4 5 0 . 0 0"1 0 . 2 0 . 3 0 "4 0 - 6 7 0 - 0 0 "2 0 ' 3 0 . 4 0 ' 6 0 "8 9 0 . 0 0 "2 0 - 4 0 -5 0 ' 8 1 .1

11 0 . 0 0 "3 0 "4 0 -7 0 ' 9 1 "3 13 0 ' 0 0 "3 0 - 5 0 "8 1 .1 1 "5 15 0 - 0 0 ' 4 0 - 6 0 . 9 1 '3 1 "7 17 0 ' 0 0 - 4 0 . 7 1 '0 1 .4 2 ' 0 19 0 "0 0 "5 0 "8 1 "1 1 "6 2 ' 2 21 0 "0 0 . 5 0 ' 8 1 .2 1 -7 2 "4 23 0 "0 0 , 6 0 ' 9 1 .3 1 .9 2 "6 25 0 ' 0 0 . 6 1 "0 1 .4 2 - 0 2 "8 27 0 - 0 0 . 7 1 "1 1 -5 2 . 2 3 "0 29 0 "0 0 . 7 1 "1 1 "6 2 . 3 3 '1 31 0 "0 0 .7 1 '2 1 .7 2 "4 3 "3 33 0 ' 0 0 . 8 1 '3 1 .8 2 . 6 3 -5 35 0 ' 0 0 . 8 1 "3 1 -9 2 -7 3 ' 7 37 0 "0 0 . 9 1 '4 2 - 0 2 - 8 3 "8 39 0 ' 0 0 . 9 1 "5 2.1 2 . 9 4 "0 41 0 "0 0 . 9 1 "5 2 "2 3.1 4.1 43 0 "0 1 "0 1 "6 2 ' 3 3 "2 4 " 2 45 0 - 0 1 ' 0 1 ' 6 2 . 4 3 . 3 4 "4 47 0 "0 1 .1 1 "7 2 . 5 3 . 4 4 "5 49 0 "0 1.1 1 "7 2 "5 3 . 5 4 "6 51 0 "0 1 .1 1 "8 2 "6 3 ' 6 4 . 7 53 0 "0 1 .2 1 "8 2 . 7 3 . 6 4 ' 8 55 0 . 0 1 '2 1 "9 2 .7 3 -7 4 " 9 57 0 ' 0 1 .2 1 '9 2 ' 8 3 "8 5 "0 59 0 . 0 1 "3 2 - 0 2 "8 3 . 9 5-1 61 0 "0 1 '3 2 "0 2 -9 3 . 9 5"1 63 0 ' 0 1 .3 2-1 2 ' 9 4 ' 0 5 "2 65 0 "0 1 .3 2.1 3 "0 4 . 0 5 ' 2 67 0 ' 0 1 "3 2"1 3 ' 0 4 '1 5 "3 69 0 "0 1 "4 2"1 3 '1 4 '1 5"3 71 0 "0 1 "4 2 ' 2 3"1 4.1 5 , 3 73 0 "0 1 "4 2"1 3"1 4.1 5 . 4 75 0 "0 1 "3 2"1 3"1 4"1 5 . 4 77 0 . 0 1 '3 2"1 3"1 4.1 5 . 4 79 0 ' 0 1 "3 2.1 3"1 4"1 5 . 3 81 0 "0 1 "3 2.1 3"0 4 '1 5 . 3 83 0 - 0 1 -3 2.1 3 - 0 4-1 5 .3 85 0 "0 1 "3 2.1 3 - 0 4 '1 5 . 3 87 0 ' 0 1 .2 2 . 0 3 "0 4 . 0 5 . 2 89 0 " 0 1 '2 2 .0 2 ' 9 4 ' 0 5 . 2

566 Medical & Biological Engineering & Computing September 1977

surfaces were taken to Iie on the geometric axis assumed coincident with the optic axis.

Each refracting surface is presumed spherical except for the anterior surface of the cornea which was assumed ellipsoidal.

Average values were ascertained from the pub- lished papers of POMERANTZEFF, et al. (1971) and of DRASDO and FOWLER (1974).

In the absence of folding, the displacement of the detached retina is assumed to take place in the meridional plane containing the line of sight to the break. There appears to be supportive

evidence of this contention in the case studies of BONIUK and BUTLER (1968) at autopsies of incidence and pathogenesis of predispostion to detachment.

The parameters, and their values, of the schematic eye used for the mathematical model are given in Table 1.

4 Computer localisation If the mechanical properties of vitreous traction, retina, and chorioretinal adhesion were known, an exact mathematical theory could possibly be

Table 3. Arc length LP3 from limbus to recommended site for placement of implant, for vertical p/ane, given distance of detachment d in mm, and angle 8 in degrees formed by line of sight to tear and optic axis


Degrees 0mm 1 mm 2 mm 3 mm 4 mm 5 mm

1 31 '1 31 "1 31 "1 31 "1 31 -0 31 '0 3 30 .5 30-5 30 .4 30"4 30 '3 30"2 5 30"0 29 ' 8 29"8 29 .7 29 .5 29 .4 7 29"4 29 .2 29.1 2 9 ' 0 28-8 28 ' 6 9 28 "8 28 ,6 28 .4 28-3 28 .0 27 "8

11 28"2 28 ' 0 27 .8 27 .6 27-3 26 '9 13 27 ' 7 27-3 27"1 26"9 26 .6 26'1 15 27.1 26-7 26 .5 26 .2 25-8 25"3 17 26 ' 5 26"1 25 ' 8 25 .5 25-1 24-6 19 25"9 25-5 25-2 24-8 24-4 23-8 21 25 .4 24-8 24 .5 24.1 23 .6 23-0 23 24 .8 24.2 23 ' 9 23"5 22-9 22"2 25 24 .2 23 .6 23 .2 22 .8 22 .2 21 "5 27 23 '7 23-0 22 .6 22'1 21 .5 20 '7 29 23.1 22"4 22 .0 21 .5 20-8 20 .0 31 22 ' 5 21 "8 21 '4 2 0 ' 8 20.1 19"2 33 22 .0 21 -2 20.7 20.1 19 ' 4 18"5 35 21 .4 20 ' 6 20"1 19 .5 18 '7 17 '8 37 20"9 20"0 19-5 18"9 18.1 17"1 39 20 .4 19.5 18.9 18 '2 17 .4 16 '4 41 19 .8 18.9 18"3 17 .6 16.8 15 '7 43 19 '3 18"3 17"7 17 .0 16.1 15'1 45 18-8 17.7 17.1 16 .4 15 .5 14"4 47 18.2 17"2 16"5 15"8 14-9 13"8 49 17"7 16.6 16-0 15"2 14-3 13"1 51 17"2 16-1 15"4 14"6 13.7 12 '5 53 16.7 15 .6 14.9 14.1 13.1 11 '9 55 16"2 15 '0 14 '3 13~5 12.5 11 "3 57 15"7 14.5 13-8 12"9 11 .9 10"7 59 15.2 14"0 13.3 12 .4 11 .4 10"2 61 14"8 13"5 12"7 11 "9 10-8 9 ' 6 63 14-3 13"0 12 '2 11 "4 10.3 9.1 65 13.8 12-5 11 .7 10"9 9 .8 8 ' 6 67 13.4 12 ' 0 11 "3 10 .4 9 .3 8"1 69 12"9 11 '6 10 '8 9 ' 9 8 ,8 7"6 71 12.5 11.1 10.3 9 .4 8 .4 7"1 73 12.0 10.7 9"9 9 -0 7 .9 6"7 75 11 "6 10 '3 9 ' 5 8 ' 5 7 .5 6"2 77 11 "2 9"8 9"0 8"1 7 .0 5"8 79 10.8 9 .4 8 .6 7 .7 6 .6 5 ' 4 81 10.3 9"0 8"2 7 .3 6-2 5"0 83 9"9 8 .6 7"8 6"9 5 .8 4 ' 6 85 9 .5 8"3 7 .5 6 .5 5 .5 4"2 87 9.1 7"9 7.1 6"2 5.1 3"9 89 8 ' 7 7 -5 6"7 5"8 4 .7 3 ' 5

Medical & Biological Engineering & Computing September 1977 567

found by which the location of a detached retinal break can be predicted from the site of a lesion. Conversely, under traction relaxation produced by scleral buckling, and drainage of subretinal fluid, in which the retina is first brought into apposition with the choroid, then induced to fall back into place, a corresponding inverse mapping would predict settled locations from direct ob- servation of a detachment.

Our present state of knowledge of in vivo properties, however, is quite limited, and consequently this analysis is not yet within reach.. Instead, a

solution will be found by indirect means under additional assumptions.

Consider a point on the detached retina. Within its meridional plane, the arc-length_ displacement in settling back onto the choroid, measured on the posterior surface of the sclera and starting from the extension of the line of sight to the tear, can always be regarded as the sum of two displacements: one solely due to geometry, the other to retinal relaxation. The geometric contribution is paramount because the maximum distance of detachment is taken to be small compared with the radial

Table 4. Chordal length LP 3 from limbus to recommended site for placement of implant, for vertical plane, given distance of detachment d in ram, and angle 8 in degrees formed by line o f sight to tear and optic axis


Degrees 0 mm 1 mm 2 mm 3 mm 4 mm 5 mm

1 22"7 22 '7 22"6 22 '6 22"6 22"6 3 22"5 22"5 22 '5 22 '5 22.4 22"4 5 22"3 22"3 22 "3 22-2 22 "2 22 '2 7 22-2 22"1 22"1 22 '0 22"0 21 "9 9 22"0 21 -9 21 "8 21 "8 21 "7 21 "6

11 21 '8 21 "7 21 "6 21 '5 21 '4 21 '3 13 21 '5 21 "4 21 "3 21 "2 21 "1 20"9 15 21 "3 21 '2 21 -1 20"9 20"8 20"6 17 21 '1 20"9 20 '8 20"6 20"4 20"2 19 20"8 20"6 20 '5 20 '3 20"1 19 '8 21 20 '6 20.3 20 '2 20"0 19"7 19"3 23 20 '3 20.0 19"8 19"6 19"3 18"9 25 20.0 19.7 19'5 19"2 18"9 18"5 27 19-7 19 '4 19'1 18-9 18"5 18"0 29 19"4 19 '0 18"8 18 '5 18"1 17"5 31 19'1 18-7 18 '4 18'1 17 '6 17"1 33 18"8 18"3 18"0 17'7 17"2 16"6 35 18"5 18"0 17.6 17 "2 16"7 16"1 37 18-1 17.6 17.2 16 '8 16 '3 15 '6 39 17"8 17-2 16"8 16"4 15"8 15'1 41 17.4 16-8 16.4 15"9 15"3 14"5 43 17.1 16.4 16.0 15"5 14"8 14 '0 45 16"7 16 '0 15 '6 15"0 14 '4 13 '5 47 16-4 15 '6 15 '2 14.6 13"9 13 '0 49 16"0 15 '2 14.7 14"1 13.4 12"5 51 15.7 14.8 14.3 13'7 12"9 12"0 53 15 '3 14"4 13.9 13.2 12.4 11 '4 55 14"9 14.0 13.4 12 '8 11 "9 10 '9 57 14.5 13"6 13"0 12 '3 11 "5 10.4 59 14.2 13"2 12.6 11 '9 11.0 9 '9 61 13"8 12"8 12"2 11 '4 10"5 9"4 63 13.4 12"4 11 .7 10"9 10 '0 8.9 65 13-1 12"0 11 "3 10"5 9"5 8.4 67 12.7 11 .5 10"9 10 '0 9'1 7"9 69 12"3 11 .1 10.4 9"6 8"6 7.5 71 11 -9 10'7 10"0 9"2 8"2 7 '0 73 11 '5 10.3 9"6 8"8 7 '8 6 '6 75 11 '2 9.9 9 '2 8"4 7 '4 6 '2 77 10"8 9"6 8 '8 8-0 6.9 5-8 79 10"4 9-2 8 ' 4 7.6 6 '5 5 '4 81 10"0 8 '8 8'1 7.2 6"2 5"0 83 9"7 8"5 7.7 6-8 5 '8 4"6 85 9"3 8"1 7"3 6"5 5 '4 4"2 87 8"9 7"7 7"0 6.1 5.1 3"9 89 8"6 7"4 6"6 5"8 4"7 3"5


distance from the centre of the globe. This implies that the unknown surface traction affecting relaxation can be replaced by an estimated overall retinal stretch, not exceeding 10% of the observed chordal distance between the extremal points of detachment, and fulfilling the observed distance of detachment. The reattached retinal state is presumed identical to that before detachment, except for the local buckled area produced by an implant. In preference to the problem of reattachment, we may equivalently analyse the question of original detachment. In arriving at the detached state, the retina, a membrane assumed elastically homogeneous and isotropic, is subject to a uniform normal surface traction directed inward. Dis- placements are predicted by means of the GREEN and ADKINS (1970) nonlinear theory for large displacement of plane membranes, modified for a spherical choroidal surface.

The input subroutine of the computer program calls for information on the radius of curvature, centre of curvature and index of refraction. If such data are not available, quantities from Table 1 are substituted. To complete the schematic eye, the program then asks for the meridional plane in which the retinal break lies. The vertical meridian is used, corrected by interpolating between extreme values, if the actual meridian is unknown. Finally two parameters are introduced, 0 < 0 < n/2 and 0 < d < 5mm, the angle formed by the line of sight to the tear and the optic axis, and the maximum distance of detachment, respectively.

An incoming light ray is traced by the computer through the various faces of refraction, and continued past the retinal break to the posterior scleral surface, point P1 of Fig. 2. Next, a ray from the centre of the globe, passing through the tear, is extended to the posterior surface to form point P2 of Fig. 2. Arc length Pt P2 then represents the correction in 1ocalisation, from an apparent location, based on considerations of optics and geometry of the schematic eye. The remaining correction, arc length P2 P3, is attributed to retinal relaxation. Point P3 becomes the recommended site for placement of the implant. The Appendix contains a mathematical derivation of the system of differential equations that is integrated simul- taneously through the observed distance, at small intervals, to reveal the detached retinal configuration.

5 Results

Representative computer results are given in the accompanying tables for the vertical meridan, Relative to the recommended site for placement of the implant, arc length from the apparent location, arc length from the limbus and chordal length from the limbus are contained in Tables 2, 3 and 4, respectively.

6 Errors

Numerical experimentation indicated that for detachments less than 4 mm and offsets not exceeding 25% of the chordal distance between extremal points of detachment, the error in localisation was less than 10% of the arc length from the apparent site of an implant.

Sensitivity to changes in the parametric data for the schematic eye was not clinically significant.

The truncation error introduced at each stage of the numerical integration arises from the omission of terms of order three and higher in the Taylor- series expansion of dependent variables and their first two derivatives. This error is of course cumu- lative and was tested by reducing the interval of integration from the undeformed state to ensure numerical precision stability. Single precision was adequate to control the round-off error in the CDC Cyber 70 computer.

Acknowledgment--A portion of this work was prepared in partial fulfilment of the requirements for a Master of Science Degree in Applied Mathematics.

References BONIUK, M. and BUTLER, F. C. (1968) An autopsy study

of lattice degeneration, retinal breaks, and retinal pits in MCPHERSON, A. (Ed.): New and controversial aspects o f retinal detachment. Harper & Row, 59-75.

COWAN, A. (1928) Ophthalmic optics. Davies. DRASDO, N. and FOWLER, C. W. (1974) Nonlinear

projection of the retinal image in a wide-angle schematic eye. Brit. J. Ophthal. 58, 709.

GREEN, A. E. and ADKINS, J. E. (1970) Theory of elastic membranes in Large elastic deformations. Clarendon Press, 161-169.

POMERANTZEFF, O., GOVIGNON, J. and SCHEVENS, C. L. (1971) Wide angle optical model of the human eye. Am. J. Ophthal. 3, 815.

SCHEPENS, C. L., OKAMURA, I. D. and BROCKHURST, R. J. (1957) The scleral buckling procedures. Pt. 1 - Surgical techniques and management. Arch. Ophthal. 58, 797-811.

SCHEPENS, C. L., OKAMURA, I. D. and BROCKHURST, R. J. (1958) The scleral buckling procedures. Pt. 2- Technical difficulties of primary operations. Arch. Ophthal. 60, 84-92.

SCHEPENS, C. L., OKAMVRA, I. D. and BROCICIJURSX, R. J. (1958) The scleral buckling procedures. Pt. 3-- Technical difficulties of reoperations. Arch. Ophthal. 60, 1003-1012.

SCHEPENS, C. L. (1968) Indications for photocoagulation, laser, and cryoapplications: a summary in McPHERSON, A. : New and controversial aspects o f retinal detachment. Harper & Row, 275-279.

SX~NE, G. H. (1934) Tables for accurate retinal localiza- tion. Am. J. Ophthal. 17, 314.


8. Appendix

Membrane theory for retinal relaxation

The retina is represented as a thin membrane of uniform thickness ho. It forms a spherical cap of radius R with axis of symmetry x3, before detachment or prior to deformation. In terms of average material properties, the membrane can be regarded as incompressible, isotropic and homogeneous. Its middle surface after deformation under a uniform external pressure p is generated by revolving a plane curve about axis Y3 in its plane, which remains coincident with xa. The curve becomes a meridian. It does not intersect itself or the axis of symmetry except at the apex, so there is only one edge which is clamped in chorioretinal adhesion. Further, singularities are neglected, that is the effect of the retinal break on the deformed shape of the membrane is not considered significant.

A point whose cylindrical polar co-ordinates are (p, 0, xa), located in the middle surface before deformation, is carried over to a point (r, 0, y~) in the deformed middle surface. Arc length measured along a meridian before deformation is denoted by % while the arc length along the corresponding meridian after deformation "s given by E.

On account of axial symmetry, the principal directions of strain at a point coincide with the meridian, latitude and unit normal. The principal extension ratios in these directions are

d# a, = ~ . . . . . . . . . . . (1)

r az = - - �9 . . . . . . . . . . (2)

P

aa = ( t l 2~2)-* . . . . . . . . . (3)

Since AI, t2, An, r, E and , /are independent of 0, 0 can be chosen the independent variable. From geometry of a spherical cap

drl _ R ( R 2 _ p 2 ) - ~ (4) d o . . . . . . . .

ds e _ R ( R 2 _ p2)- , ;~ . . . . . . . (5) dp

Let (7"1, T2) be the physical stress resultants in the directions of the meridian and latitude at a point in the deformed surface. Then

T~ = 4ho ),a (2,~ 2 - As 2) (~/~, a W \

+a2"2-~2) (6)

/'2 = 4ho ,~a(a2z--~a2, ( ~ +&'ac~W'~) (7)

where

I1 ~ A12-~-A22~-A32

/2 : A 1 - 2 - ~ A 2 - 2 - ~ - A 3 - 2

are the invariants. For the strain energy function W, the Mooney form for incompressible rubber-like materials is convenient:

W = { I ' ( I ~ - - 3 ) + ( I 2 - - 3 ) } C 2 . . . . . (8)

r ~ c ~ / c 2 is the ratio of elastic constants. The value F = 0.1 is assumed.

Denote the principal curvatures at a point in the deformed middle surface by ( ~ , xz). The equations of equilibrium reduce to

_ _ = T dr d ( r l r) (9) d~: 2~-~ . . . . . . . .

'q TI+K2 Tz = --p . . . . . . . . (10)

The Codazzi equations yield only one nontrivial relation

d dr d ( (K2 r) =- , q ~ - . . . . . . . . (11)

If the curvature ~ of a plane curve in the deformed middle surface, given by

d 2 r/dE 2 Iq (12)

t \ d E ] J

is introduced into the preceding relation and the resulting equation integrated, we obtain

~ r = {1- (d~VI ~ \ ~ ] j . . . . . . . (13)

which fulfils the condition dr/dE = 1 at r = 0 for l~z[ < ~ . Alternatively,

d 2 r K1 K2 r - - d~ z . . . . . . . . (14)

By combining eqns. 9 and 10 with 11 and then integrat- ing, the product of curvature ~2 and stress resultant 7"1 can be found:

k ~ i q = a ' ~ . . . . . . . . - ~ p T (15)

The arbitrary constant of integration k is evaluated by means of conditions of symmetry at p = 0

A1 = ?'2 = a

T I = T 2 = T

Eqns. 9 and 10 yield k = 0, hence

2~cT = --p

There are nine unknown functions of #: ,X~, )t2, A3; 7'1, T2; ~q, •:; E and r. They can be determined from eqns. 1, 2, 3, 6, 7, 9, 10, 11, 13 or 14. The algorithm for integration, however, follows ADKINS and RIVLIN 0952). Instead of treating this nonlinear system of equations, 1st- and 2nd-order derivatives of the dependent variables are found for insertion into a Taylor-series expansion. If Ap is made sufficiently small, numerical convergence can be realised by neglecting 3rd-order terms. For example, the stress resultant T I (p+Ap) is determined from T I ( p ) by using

d T l ( p ) A + a d : T I ( p ) TI (p+Ap) = TI (p )+ ~ p ~ - ~ f AP 2

0 ~ p ~< c. The numerical integration is continued until


A2 -- 1. Given the chordal length of detachment e, a scaling process is ultimately applied which multiplies all linear dimensions by the factor c/b, where b is the value of p when Z2 reaches unity.

To determine the shape of the detached retina Y3 -- Y3(p), we proceed to adjoin

(dy3~ 2 = ( d ~ 2 {dr~ 2 dp ] \ d o / -- \ dp] . . . . . (16)

with

dr dp -- A,(1 -- ~22 r2) ~ R ( R 2 - p2)-~r

and eqn. 5. Numerical decoupling of derivatives is achieved by the

following order of equations:

A3 = (A1 A2) -~ . . . . . . . . . . . (17)

TI/=A3(A12--A32)(l~-A22 r ) . . . . . . (18)

T2'=Aa(A22--Aa2)(l+A~ 2 r ) . . . . . . (19)

r = A2 p . . . . . . . . . . . . (20)

dr dp -- A1(1- -m22r2)§ . . . . . (21)

dA2- - 1 ( d r - - A 2 ) p # O (22) dp p dpp . . . . . . .

d~2 1 (xs-- dr d~ = T K2) d p ' r # O . . . . . . (23)

d T l ' 1 (7"1'-- , dr dp r T2)d~p , r # O . . . . (24)

dA: _ dT( __{3A32 A12@(A12_t_A32)A22 r} A~3 dA2

do A2 do

dp (A12-1-3A32)(1 @A2 2 r )

(25) ~ d A 2 dTt" _ {(A22+3Aa2)( 1 +Az2 1~)} _ _

do " d o

@{3A32--A22-]-(A22@A32)a, 2 r} A3 dA2 al do

(26)

d ~ 1 t d r / d r ; d~: I dp T ( t/41 dp-~ Af g2 dp-~ +T2t dpJ (27)

d 2 r ~1 ~c2 rpRA12 p dr 1 dr dAt dp2 -- R 2 p2 -~ R 2 _ p2 dp -~- A 1 dp dp

(28)

d2A2 1 (d2r 2dA2] do 2 P \ dp 2 -- ~pp ] , p # O . . . . t29)

d 2 ~c2 1 . d~r [dK1 2d~c2~ dr I d~-p 2 = T {(K~ + r # O

�9 ( 3 0 )

d 2 T / 1 { d 2 r dp 2 r (7",' -- Tz') ~p2-

( d T / d T 2 ' ) d r } + 2 dp dp dTp ' r # O . (31)

d 2 TI ' a9

d 2 A1 A1 dp2 dp 2 -- Aa (A12+3Aa2)(l+A22 I') . . . . . (32)

(I) : (3A32--AI2-[-(At2+A32)A22 I 1} A3 d2 A2 A2 dp z

- - {12(1+A22 r ) A1 ~ A33 \~p//dAl~21]

+2{-- (h~ 2 +9),a2) + A22(),~ 2 --3Aa2)r}Aa a ~ dA2do

+2A~{A~2(a, 2-6A,2)- r} [ dA~ ~dT/

d 2 T ' 2 _ {(A22+3Aa2)(l+at2 r)} A3 d2 A2 dp 2 A 2 dp 2

+{3A32 A22@(A22@Aa2)),I2 i ~} A3 d 2 At A1 dp 2

--12~32(1+A, 2 I') ~23 ~ {dh2~ 2

+2{-- (ha2 +9ha2)+ ax2(h22--3Aa2)r}

�9 2 dA, dA2 x aa -d7 dp

(dA,V +2Aa3{A22(a22-6Aa2) - r ) \ ~ p ]

(33)

d ~ ~ 1 [2 d ~ d T / d ~ T / d ~ ,:~ dp z TI" ~ dp dp t~ :<1 ~ -~- T2" dp 2

+ 2 ~ p dT2" _ d2 T2" l de ~-~2 d,o2 ] . . . . (34)

dy3 __ A1 KzrR(R2- -p2) -~ . . . . . . . (35) dp

d2ya (rRAa d1r dr d A ~ dp 2 -- -- ~ +K2 RAt dpp + r R K 2 ~ ]

X (R 2 - p2)-t-+2K2 rRA1 p(R 2 - p2)-{- (36)

At p = r = 0

dr -dp = a . . . . . . . . . . . . (37)


d 2 r _ dA~ dA 2 d T ( _ dT2' d,,q d.'r = 0 d 2 TI ' dp 2 do dp dp dp do dp dp 2

(38) (39a)-(39f)

d 2 ,~1 dp 2

K2k3{(2--3o -) -22F}

. . . . (40)

4 ( 1 + ~3--6-) ( I+A2F) d 2 T a " - - 3 dp 2

d2 ~2 dp 2 T"

d2A2 1 [ d 2 A t - - K 2 2 ] do 2 = T \ ~ ~3 ~_ ~ / . . . . . (41) d~,q _ d z K 2

(42)

d 2 T~' dp 2 . . . . . . . . . (43)

,< d2 T / dp 2 (44)

. . . . . . . . . �9 (45)

Localisation par ordinateur des morceaux de r6tine ddtach6s en vue d'une op6ration d'agrafage sur la schl6rotique

Sommaire--Un module math6matique a 6t6 r~alis6 pour d6term/ner automatiquement l'emplacement d'un type de ruptures r6tiniennes d6tach6es. I1 s'agit d'un oeil sch6matique complet tenant compte du rayon et du centre de courbure ainsi que de l'indice de r6fraction. Le eontr61e par ordinateur permet au chirurgien de corriger le site apparent lors du placement d'un implant dans une op6ration d' agrafage sur la schl6rotique. La longueur de l'arc de retomb6e sur la choroide, mesur6e sur la surface post6rieure de la schl6rotique en commenqant ~t partir du prolongement de l'axe de vision jusqu'~t la rupture r6tinienne, et la somme de deux d6placements: un uniquement dO/t la g6om6trie, l'autre b. la relaxation de la r6tine. Le d6placement g6om6trique est l'616ment essentiel, car la distance maximale de la partie d6tach6e est consid6r6e comme faible compar6e 5. la distance radiale depuis le centre du globe oculaire. La traction de l 'humeur vitr6e affectant la relaxation r6tinienne est remplac6e par une traction globale correspondant ~t une distance chordale observ6e entre des points extremes et la distance du d6tachement. La r6tine est suppos6e 61astiquement homog6ne et isotrope et est soumise ~t une traction normale uniforme de sa surface vers l'int6rieur. Les d6placements soot pr6dits ~ l'aide d'une th6orie non lin6aire pout les forts d6placements de membrane, Les r6sultats d'ordinateur repr6sentatif soot donn6s pour le m6ridien vertical.

Computer-Lokalisierung abgel6ster Netzhautrisse ffir das sklerale Knickverfahren

Zusammenfassung--Es wurde ein mathematisches Modell zur automatischen Lokalisierung einer Klasse abgel6ster Netzhautrisse konstruiert. Es schliegt ein komplettes schematisches Auge mit Krtimmungs- halbmesser, Krtimmungsmittelpunkt und Brechungsindex ein. Die Computer-Steuerung liefert dem Chirurgen Korrekturen von dem augenscheinlichen Sitz ftir die Anbringung eines Transplantats in einem skleralen Knickverfahren. Die Bogenlange der Wiederabsetzung auf die Gefiil3haut, gemessen auf der hinteren Fliiche der Sklera und beginning vonder Verl~.ngerung der Sichtlinie zum Netzhau- tril3, ist die Summe yon zwei Verlagerungen: eine ausschlieBlich infolge der Geometrie, die andere infolge der Netzhauterholung. Der geometrische Beitrag ist am wichtigsten, weil der Maximalabstand der Abl6sung in Vergleich mit dem Radialabstand yon der Mitre des Augapfels als gering angenommen wird. Ein hyalines Ziehen, das die Netzhauterholung beeintrS, chtigt, wird durch eine geschS.tzte Gesamtstreckung ersetzt, wodurch ein beobachteter chordaler Abstand zwischen Extremalpunkten und Abstand der Abl6sung befriedigt wird. Die Netzhaut wird als elastisch homogen und isotrop und einer gleichf6rmigen, normalen Oberfl~ichenziehung nach innen ausgesetzt angenommen. Verlagerungen werden mittels einer nichtlinearen Theorie ftir groge Verlagerungen yon Membranen vorhergesagt. FOr den vertikalen Meridian werden reprhsentative Computer-Ergebnisse vermittelt.


Documents

Computer localisation of detatched retinal tears for a scleral buckling procedure