Matrix Formula for Intraocular Lens Power Calculation

Transcription

1 Investigative Ophthalmology & Visual Science, Vol. 31, No. 2, February 1990 Copyright Association for Research in Vision and Ophthalmology Matrix Formula for Intraocular Lens Power Calculation Jean-Philippe Collide The matrix calculation was applied to Gaussian optics. Matrices were defined for a single diopter, a lens, an association of two centered systems having the same axis when the origins are taken at the vertices of the first and last refracting surfaces. The matrix formula then was used to calculate the power of the emmetropizing lens implant, and its results compared with the ones obtained by five other formulas. Invest Ophthalmol Vis Sci 31: ,1990 The objectives of the intraocular correction of the aphakic eye by means of an artificial lens implantation are the restoration of vision and the correction of any preexisting ametropia. The intraocular lens should not disturb the anatomic proportions of the operated eye in obtaining an optimal emmetropizing vision. Through a judicious method of intraocular lens power calculation, the surgeon can estimate, before the cataract extraction, what the refraction will be after the operation. Most formulas for intraocular lens power calculation use either an algebraic or a statistical method. I present here a new formula based on matrix calculations. This type of calculation has proven to be the most efficient method to study optical systems of two or more lenses. 1 Furthermore, the use of microcomputers has facilitated these matrix calculations. Material and Methods The eye is assimilated to a centered system having four surfaces separating areas of different indices of refraction. 2 " 6 These are the anterior and posterior corneal surfaces and the anterior and posterior lens surfaces. Applied to this centered system is the paraxial approximation, in which the rays of light stand at a small angle to the axis, and the angles of incidence on the surfaces are small. 2 ' 6 Biometry and Determination of the Ocular Parameters The measurement of the radius of curvature of the anterior surface of the cornea is made with a Java! From the Centre Hospitalier National d'ophtalmologie des Quinze-Vingts, Paris, France. Submitted for publication: March 20, 1989; accepted June 16, Reprint requests: Jean-Philippe Colliac, MD, Centre Hospitalier National d'ophtalmologie des Quinze-Vingts, 28 rue de Charenton, Paris, 75012, France. keratometer. 7 The mean of the two principal radii of curvature of the anterior surface of the cornea, measured preoperatively, is taken. The mean value of corneal thickness of the living eye is mm ± 0.039, measured with an optical pachymeter. 8 In clinical practice, it is.not possible to measure the radius of curvature of the posterior surface of the cornea. It is possible, however, to estimate its value from the measurement of the radius of curvature of the anterior surface of the cornea, as made with an ophthalmometer. The mean value of the radius of curvature of the anterior surface of the cornea is 7.65 mm ± The mean value of the radius of curvature of the posterior surface of the cornea is 6.46 mm ± The anterior and posterior radii of curvature of the cornea have the same standard deviation, ± The values of the anterior and posterior radii of curvature of the cocnea have the same dispersion around their mean, and Lowe and Clarck 9 ' found a highly significant correlation between them. Therefore, we can write: r 3 = (r, ) = r, H =- radius of curvature of the anterior surface of tke cornea (in millimeters) r 3 = nadius of curvature of the posterior surface of tfee cornea (in millimeters). Posterior corneal radii are approximately 1.19 mm lower than are anterior radii, in normal adult eyes. This approximation has been verified by Lowe and darck 9. For the matrix formula, the anterior chamber depth is measured from the posterior surface of the cornea to the anterior surface of the intraocular lens. Preoperative anterior chamber depth is measured with an A-scan ultrasound unit, 10 '" or with the optical method of Jaeger, with an attachment to a slit lamp. 12 Postoperative anterior chamber depth is an estimate. The preoperative anterior chamber depth, mea- 374

2 No. 2 MATRIX FORMULA FOR IOL POWER CALCULATION / Collioc 375 sured from the cornea to the vertex of the cristalline lens, does not allow accurate estimation of the postoperative anterior chamber depth. Indeed, in the phakic eye, the anterior chamber depth depends above all on the thickness and the position of the cristalline lens. This method to estimate the pseudophakic anterior chamber depth is derived from the formula of Holladay et al. 13 Anatomic anterior chamber depth is equal to the height of the cupola (spherical segment), as given by the formula: 14 h = r - Vr 2 - d 2 /4 in which h is the height of the cupola in millimeters, or the distance from the posterior surface of the cornea to the anterior iris plane; r is the average radius of curvature of the cornea; and d is the length of the chord or diameter of the anterior chamber. To take into account the asphericity of the cornea and the shorter radius of curvature of the posterior surface of the cornea, compared to the anterior radius of curvature, we use the value of the anterior radius of curvature of the cornea to calculate the height of the cupola. The result is an approximation. The value of r is measured with a keratometer, but if r < 7 mm, then r = 7 mm. 13 The diameter (d, in millimeters) of the anterior chamber is calculated as: d = where 1 is the axial length of the eye in millimeters. If d > 13.5 mm, then d = 13.5 mm. 13 The pseudophakic anterior chamber depth is equal to the sum of the anatomic anterior chamber depth and the distance from the aphakic anterior plane of the iris to the vertex of the intraocular lens: ACD = r - Vr 2 - d 2 /4 + SF where ACD = anterior chamber depth, and SF is a constant, called the "surgeon factor," or the distance from the aphakic anterior iris plane to the vertex of the anterior surface of the intraocular lens. The surgeon factor depends on lens style, from a given manufacturer with a single surgeon. For a Shearing posterior chamber implant, the value of the SF is approximately mm. Axial length is measured with an A-scan ultrasound unit, and the results are read in millimeters The center thickness of the intraocular lens is determined by the manufacturer, and its value is estimated to be 0.7 mm The refractive indices are equal to (n 2 ) for the cornea and (n 4 ) for the aqueous humor. For the implant and the vitreous, they are equal to (n 6 ) and (n 8 ), 2 ' 6 respectively. Homographic Transformation Optically, there is a homographic relation between the object and the image: an object point and its image form a conjugate couple, and under the inverse return principle, the relationship remains the same if the object becomes image and the image object. This homographic relation is written in the general form: 6 ' 23 " 25 x' = ax + b ex + d (1-D The graph of this equation is represented by the two branches of an equilateral hyperbola. A proximity is the inverse of an abscissa. A reduced length is the quotient of a length by a refractive index, n, of the medium, where this length is located; x and x' are the reduced lengths of the projections of two conjugated points on the axis of a centered system. Those reduced lengths are counted respectively from any fixed origins O and O 2. 2 The coefficients a, b, c, and d are the Gaussian coefficients. 623 " 25 They depend on the system and on the choice of the Oi and O 2 origins. They are determined to within a constant factor because the relation between x and x' does not change when the numerator and denominator are multiplied by a same number. An arbitrary relation among a, b, c, and d can be made, in which: ad - be = 1 (1-2) We can prove that c is a power expressed in diopter units; that d and a are numbers; and that b is a length expressed in meters. Cardinal Elements The knowledge of the Gaussian coefficients allows the deduction of the cardinal elements of the system. Let x' be the reduced length of the image focus: and, as x -*- oo, X' = ax + b ex + d x' = - c Definition of a Matrix d c + - X (1-3) The matrix notation facilitates the calculations involved to determine the Gaussian coefficients. 6 ' 23 " 25 A matrix consists of mn numbers arranged in n col-

3 376 INVESTIGATIVE OPHTHALMOLOGY b VISUAL SCIENCE / February 1990 Vol. 31 umns and m rows. Using double-suffix notation, we denote a matrix A as: A = (ay) = an a^ aj n a 2t a 22 a 2n The individual term ajj is called an element of the matrix A, where i refers to the row and j to the column. A possesses m rows and n columns. If m = n, we have a square matrix of order n. A square matrix of order 2 is associated with a centered system. The four elements of this matrix are the Gaussian coefficients. Multiplication of Matrices Two matrices A and B may be multiplied in that order to form the product AB. The product C = AB is defined as possessing m rows and p columns. The product of two matrices of order 2 is a matrix of order 2 where the general element, c^ (in row i and column j) tallies with the multiplication of the ith row of the first matrix and the jth column of the second matrix. ["an ai 2 irbn b, 2 [a 2t a 22 J[b 21 b 22 AB = C ibn+ai 2 b 2 i anb 2 + ai 2 b 22 l = fen Ci 2 [a 2,bn + a 22 b 2i a 2) bi 2 + a 22 b 22 j [c 2, c 22 Multiplication of matrices is noncommutative, but it is associative. AB^BA (AB)C = A(BC) Matrix of Refraction for a Spherical Diopter Given a spherical diopter with D power, the origins are the principal points, which are merged with the diopter's vertex, r is the radius of curvature of a diopter that is separating two media of n and n' respective refractive indices. Thus, the diopter's power is: We have: D = n'-n (1-4) c = D (1-5) d = a=l (1-6) b = 0 (1-7) The matrix of refraction of this system is written: M = 1 01 fd b] D lj [c aj When we move the origin of the images of a reduced length b, the system which previously had a ls b l5 c h d, as coefficients, becomes a, b, c, d, such as: Hence,, a,x + b, x = x, - b = -- b CX + d We may write this as: - Ci6)x + (bi - di5) ax + b C X + di ex + d a = ai - c^ b = b, d,5 c = Ci d = d,, b.ir.i -5i_rd M = c, a, 0 l e a is a translatory matrix. 0 1 (1-8) (1-9) (1-10) (1-11) (1-12) Matrix of Transfer for a Lens Given a lens with thickness t and refractive index n: for the first refraction, origin Oi is merged with the vertex of the first diopter; for the second refraction, origin O 2 is merged with the vertex of the second diopter. M = i op -«p o D, 10 1 D 2 1 where b = t/n is the reduced thickness of the lens. The matrix of transfer of the lens is written: M = D, -bd 2 -b The formula to associate two diopters is: d b c a D, + D 2-5D,D 2 (1-13) Association of Two Centered Systems Having the Same Axis Given a first system ai, bi, Ci, di with Oi, O'i origins and a second system a 2, b 2, c 2, d 2 with O 2, O 2 origins, we can consider the resultant system a, b, c, d with O, O' origins. Through a suitable choice, we permit that O'i and O 2 are merged, the origin O of the object proximity is Oi, and the origin O' of the image proximity is O 2. We have: M = a.jl d 2 b 2 c 2 a 2 d b

4 No. 2 MATRIX FORMULA FOR IOL POWER CALCULATION / Collioc 377 and the Gaussian coefficients of the resulting system are: d = did 2 + b]c 2 c = C)d 2 + ajc 2 b = d)b 2 + bja 2 a = Cib 2 + aia 2 (1-14) (1-15) (1-16) (1-17) The matrix of the resulting system is equal to the product of the matrices of the constituent systems. In the matrix formula, matrix calculation is applied to Gaussian optics. Results The matrix formula allows one to determine the power of the emmetropizing implant. We have a planoconvex implant, which is the lens the more used; the anterior surface is convex, and the posterior surface is planar. 21 ' 2627 For this: (Fig. 1) ri = radius of curvature of the anterior surface of the cornea r 3 = radius of curvature of the posterior surface of the cornea r 5 = radius of curvature of the anterior surface of the planoconvex implant n 2 = refractive index of the cornea n 4 = refractive index of the aqueous humor n 6 = refractive index of the implant n 8 = refractive index of the vitreous humor Di = power of the anterior surface of the cornea D 3 = power of the posterior surface of the cornea D 5 = power of the anterior surface of the planoconvex implant D, = (n 2 - l)/r,, D 3 = (n 4 - n 2 )/r 3, D 5 = (n 6 - n 4 )/r 5 t 2 = thickness of the cornea t 4 = distance from the posterior surface of the cornea to the anterior surface of the implant, or pseudophakic anterior chamber depth A Fig. 1. (A) Diopters of the pseudophakic eye. (B) Refractive indices, radii of curvature, thickness of the media. B Fig. 2. Schematic pseudophakic eye. t 6 = thickness of the implant 5 2 = t 2 /n 2, 5 4 = t 4 /n 4, b 6 = t 6 /n 6 1 = axial length of the eye 1' = distance from the posterior surface of the lens to the retina. The matrix of transfer of the lens implant between the vertex of the implant and the posterior surface of the implant is: 1 D 5 where oiri -i 10 1 di=l Ci = D 5 1 D 5 -d 6 D 5 Cj bi = -««a; = -6 6 D (2-1) (2-2) (2-3) (2-4) The matrix of transfer of the cornea between this anterior surface and the anterior surface of the implant is: 1 oin -fciri i oiri -5 4 i _ r^ ID, D 3 1N 0 1 c 2 In making the product of the 4 matrices: d 2 = D 3 c 2 = D, D,D 3 + D 3 a 2 = -5 4 Di D,D D D, b 2 a 2 (2-5) (2-6) (2-7) (2-8) If the Oi origin is the vertex V, of the anterior surface of the cornea, and the O 2 origin is the vertex V 7 of the posterior surface of the intraocular lens, the matrix of transfer of the pseudophakos and emmetropic eye (Fig. 2) between the corneal vertex and the

5 378 INVESTIGATIVE OPHTHALMOLOGY G VISUAL SCIENCE / Februory 1990 Vol. 31 posterior surface of the implant is obtained by making the following product: i oin -5 2 ][ i oiri -«4 ir 1 oin -5 D, 1JI0 I J p 3 IJIO I J[D 5 IJIO i d 2 dj c 2 dj d 2 c 2 a 2 j[cj d b c a anterior intraocular pseudochamber lens phakos d c b a i d 2 b; + b 2 a;l _ Id b j c 2 bj + a 2 a ; J [c a In accordance with Equation 1-3, the abscissa of the image focus is: c 2 bj + a 2 a; V 7 F' = n 8 - = n 8 (2-9) c c 2 d ; In place of a i5 b i5 c i5 dj we put their corresponding values given by Equations 2-4, 2-3, 2-2, 2-1. Hence we obtain: -c a 2 (-5 6 D 5 + 1) V 7 F' = n 8 c 2 + a 2 D 5 Let 1 be the axial length of the eye, and let 1' be the distance from the posterior surface of the lens to the retina. The condition for correction is: 1 = V,F\ or 1 = V,7 + V 7 F' (2-10) If 1' = V 7 F', we can write: -c a 2 (-5 6 D 5 (2-H) c 2 + a 2 D 5 The power of the emmetropizing implant is given by the formula: n 8 a 2 - n c 2 - c 2 r n a 2 + a 2 l' (2-12) In place of a 2, c 2, we must put their corresponding values given by Equations (2.8) and (2.6). 1' = 1 - t 2 - t 4 - t 6 All lengths are in meters. The accuracy of the matrix formula and five other formulas of intraocular lens power calculation was evaluated. Tested were three theoretic formulas: those of Binkhorst II, 26 ' 28 Colenbrander, 27 and Shammas, 29 ; The SRK II formula, 30 " 34 which is an empirical or regression formula; and the Holladay formula, 13 which is a combination of theoretic and empirical analysis. Data consisted of ten theoretic eyes with respective axial length ranges of mm, and with respective average keratometry ranges of D. The theoretic anterior chamber depth, from corneal vertex to implant vertex, was assumed to be known, and it ranged from 3.41 to 4.26 mm. All of the intraocular lenses were in the posterior chamber. The results of the five formulas and the matrix formula in obtaining emmetropia were compared; the results of the calculated powers are shown in Table 1. In Table 2 we compare the deviation of each formula from the matrix formula. Figure 3 is.a graphic representation of Table 2. Discussion The matrix formula is the most accurate formula, if the calculations are done in the limits of the geometric optics. As for the other formulas, the closeness of the matrix formula remains imperfect: there may be corneal and lens astigmatism; the lens axis does not pass through the center of corneal curvature; the fovea is not placed on the optic axis; and the real walk of the ray of light is not a paraxis walk, because the pupil is widely open. However, the paraxial approxi- Table 1. Clinical data often eyes and calculations of IOL power for emmetropia Eyes ) AL* Kf Rt ACD Formulas Binkhorst II Colenbrander Shammas Holladay (SF = +0.50) Holladay (SF = +0.57) SRK II (A = 116.8) SRKII >(A = 117.5) Matrix formula 22.61" * Axial length in millimeters. t Preoperative mean keratometry in diopters. $ Preoperative mean anterior radius of curvature of the cornea in millimeters. Anterior chamber depth from corneal vertex to implant vertex in millimeters. 11 IOL power in diopters.

6 No. 2 MATRIX FORMULA FOR IOL POWER CALCULATION / Collioc 079 Table 2. Comparison of deviations from matrix formula in prediction of implant power AL* Eyes Formulas Binkhorst II Colenbrander Shammas Holladay (SF = +0.50) Hol'.aday (SF = = +0.57) SRK II (A = 116.8) SRK II (A = 117.5) / 20^ f JO * Axial length in millimeters. f IOL power calculated by each formula minus the IOL power calculated by the matrix formula, in diopters. mation and the assimilation of the eye to a dioptric centered system is accurate enough for studying the optics of the motionless eye and his correction. In the first part of the following discussion, the data or preoperative measurements is analyzed, and in the second part, the theoretical accuracy of the matrix formula and five other formulas is evaluated. Data Analysis In the Binkhorst II, Colenbrander, Shammas, Holladay, and SRK II 5 formulas, the cornea is assumed to be represented as a thin lens with a single refractive surface. The keratometer measures the anterior radius of curvature of the cornea, expressed in millimeters, which is then translated into diopters by considering the entire power of the cornea to be at its anterior surface; an arbitrary index of refraction of the interface is adopted. For the usual keratometers, this corneal index of refraction is , and the relationship between the keratometry reading (K, in diopters) and the value of the corneal radius (R, in millimeters) is: K = 337.5/R With the matrix formula, we take into account the thickness of the cornea, with a refractive index of The real value of the power of the cornea is about 1 diopter less than the reading values. 6 A small amount of postoperative astigmatism does not alter the mean of the keratometric readings. 35 ' 36 There is no statistical correlation between the central thickness of the cornea and the refraction. 8 Therefore, we can take a fixed value of 0.55 mm for the corneal thickness. For the radius of curvature of the cornea, an error of 0.02 mm affects the postoperative refraction by 0.10 diopter. In a phakic eye, the average speed of sound along the optical axis is 1550 m/sec, and in an aphakic eye the sound velocity is 1532 m/sec. 37 In a phakic cataractous eye the speed is 1556 m/sec. 38 It is more accurate to use an immersion technique than to put the transducer directly on the cornea. 39 I do not add a correction factor for retinal thickness, like some authors do, and I advise against it. With our Biophysic Ophthascan A-scan unit (Biophysic Medical, Pleasant Hill, CA), the mean axial length of 117 eyes was found to be mm (±1.35 SD), the same as that measured by Hoffer with a Kretz 7500 MA immersion A-scan unit (KT Medical, Houston, TX)." 40 However, with the applanation Sonometric DBR unit (Sonometrics Systems, New York, NY), Binkhorst found a mean axial length of mm. 41 This is the reason why some authors add to the measured axial length a retinal thickness factor of 0.25 mm or 1 - Fig. 3. Relationship between matrix formula and five other formulas. a c3 I Shammas Binkhorst II Holladay SRK Colenbrander Axial length (mm) 28 30

7 380 INVESTIGATIVE OPHTHALMOLOGY b VISUAL SCIENCE / February 1990 Vol mm. 13 Ophthalmic ultrasound units use frequencies ranging from 6 to 20 MHz. With a frequency of 15 MHz and a velocity of 1550 m/s, the wavelength is approximately 0.1 mm (1550/ ) and the theoretic resolution is 0.1 mm. For the axial length, an error of 0.1 mm affects the postoperative refraction by 0.25 diopter." 1819 ' For an anterior chamber lens, the values of the postoperative anterior chamber depths recommended by Shammas 10 and Hoffer" are 2.57 mm and 2.8 mm, respectively. For a posterior chamber lens, the value is 3.5 mm." These pseudophakic anterior chamber depths are measured from the posterior surface of the cornea to the anterior surface of the intraocular lens. However, the mean values of Shammas and Hoffer do not apply to an individual patient. The true value of pseudophakic anterior chamber depth can be determined only postoperatively, when the calculations have already been made. This is the reason why we estimate the pseudophakic anterior chamber depth with a method derived from the formula of Holladay et al. 13 In all of the theoretic formulas described previously, the anterior chamber depth is an estimated measure from the anterior vertex of the cornea to the anterior vertex of the artificial lens; in the matrix formula, however, the anterior chamber depth is an estimated measure from the posterior surface of the cornea. Ideally, it should be necessary to locate accurately, with a B-scan ultrasound unit, the position of the frontal plane, which goes through the iridocorneal angle for an anterior chamber implant, and through the iridociliary angle for a posterior chamber implant, measured from the posterior surface of the cornea. There is a mean flattening of the cornea after cataract surgery, estimated at about 0.16 dt. 45 ~ 46 Hoffer disregards this few significant flattening of the cornea for his lens power calculation." For the anterior chamber depth, an error of 0.1 mm affects the postoperative refraction by 0.15 diopter. The center thickness of the intraocular lens is determined by the manufacturer and depends on optic diameter and configuration. Its value is mm. 22 It is desirable to have a fixed value for the center thickness of the intraocular lens. The estimate of the posterior corneal radius and the postoperative anterior chamber depth are still the limiting factors for the formulas, even in the matrix method. However, the most significant factor leading to great postoperative refractive error is mismeasurement, and not any inaccuracy of the formula. 13 ' 47 ' 48 Formulas The matrix formula is assumed to be an exact mathematic model in the study of Gaussian optics. According to Figure 3, we can say that the graphs of the Holladay, 13 SRK II, 30 " 34 and Colenbrander 27 formulas draw parallels to the graph of the matrix formula, which means that these formulas perform well to within a constant factor. For the given example often eyes, the constant was optimized. With the Colenbrander formula, this constant is about diopters added to the calculated power. For the SRK II formula, constant A is determined by an analysis of the postoperative refraction; its value is For the Holladay formula the constant, or SF, 13 is about mm. The calculation of Holladay's surgical factor is based on a reverse solution of the formula to calculate the resultant refraction of an eye with a lens of a known power. The input variables are the stabilized postoperative refraction, the preoperative corneal and axial measurements, and the intraocular lens power. This calculation is done with a number of more than ten patients, with a given surgeon, and with the same lens. For Holladay, the SF is the sum of the distance between the anterior iris plane and the effective optical plane of the intraocular lens added to a constant. With the matrix formula, the SF is a measurement: it is the distance from the anterior iris plane to the anterior surface of the intraocular lens. With about ten eyes, it can be determined in measuring the postoperative anterior chamber depth with an ultrasound unit. To a good approximation, however, its value is about 0.25 mm for a posterior chamber lens. The knowledge of the constants for the Colenbrander, Holladay, and SRK II formulas allows an accuracy comparable to that of the matrix formula. The Binkhorst II 26>28 and Shammas 29 formulas undervalue the implant power for eyes shorter than 25.5 mm and 23.5 mm, respectively. The Shammas formula is accurate for eyes of axial length between 23 and 26.3 mm. The Binkhorst II formula works well in the case of long eyes, with axial length greater than 25 mm. The Holladay formula, the SRK II formula, and the matrix formula also are accurate only if the SF and A constants are known. Theoretically, with the Holladay, SRK II, and matrix formulas, the errors in prediction of implant power are insignificant, even for the short and long eyes. Clinically, the inaccuracy of the techniques of measurement affects the postoperative refraction by approximately 0.50 diopter. The postoperative refractive errors are mismeasurements. A BASIC language computer program can be used to calculate the matrix formula. With matrix calculations, we have established theoretic formulas that allow us to determine the power of the intraocular lens for emmetropia or ametropia, or to correct a preexisting ametropia. We can predict preoperatively

8 No. 2 MATRIX FORMULA FOR IOL POWER CALCULATION / Collioc 381 what the refraction will be in an eye implanted with a lens of known power. Key words: intraocular lens, power calculation, matrix calculation, matrix formula, theoretic formula References 1. Brouwer W: Matrix methods in optical instrument design. New York, W.A. Benjamin, 1964, pp Gullstrand A: Die Dioptrik des Auges. In Handbuch der Physiologischen Optik, 3rd ed, Von Helmholtz H, editor. Hamburg/Leipzig, L. Vors, 1909, pp Von Helmoltz H: Handbuch der Physiologischen Optik. Leipzig, L. Vors, 1867, pp Tron EJ: The optical elements of the refractive power of the eye. In Modern Trends in Ophthalmology, Ridley F and Sorsby A, editors. 1940, pp Hecht E: Optics. Reading, Massachusetts, Addison-Wesley, 1987, pp LeGrand Y: Optique physiologique: Tome 1. La dioptrique de Pceil et sa correction, 2nd ed. Paris, Editions de la Revue d'optique, 1965, pp , , Javal E: Memoires d'ophtalmometrie. 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