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A practical approach to lens implant power
a practical approach to lens implant power Robert C. Drews, M.D. St. Louis, Missouri
A number of articles have been written on the determination of lens implant power. Unfortunately their formulas are derived using different assumptions and yield a variety of answers from a given set of data. More importantly they require data that most of us cannot obtain, or cannot obtain with sufficient accuracy. Three basic parameters must be measured in order to predict lens implant power: (1) Corneal power (2) Aphakic chamber depth (distance from the cornea to the lens implant), and (3) Axial length of the eye. There are two other factors which are not usually mentioned in such discussions: (4) Distance between the principle planes of the lens implant, and (5) Astigmatic error. Corneal Power This is the one parameter which every ophthalmologist should be able to measure easily and with sufficient accuracy. But how long has it been since your keratometer or ophthalmometer was calibrated? When it reads 41.50 is that really 41.12 or 41.87? Satisfactory contact lens fitting is no criterion, especially when factors such as fitting "steeper than K" are used by some and not by others. But a keratometer does not measure "K": corneal power. It measures the radius of curvature and converts this to power by using a fictitious index of refraction for the cornea. This index mayor may not incorporate a correcting factor for the minus effect of the back of the cornea. Furthermore, the index arbitrarily used differs from one keratometer manufacturer to another. (See R. D. Binkhorst's discussion at the end of this paper.) Aphakic Chamber Depth Most of us cannot predict the postoperative depth of the anterior chamber. Galin makes a prediction by assuming the position of the chamber angle with respect to the observed limbus. He measures the diameter of the cornea at that level and calculates the aphakic chamber depth from this diameter and the corneal radius of curvature, but these relations are obviously subject to individual variation. Worst has worked on a gonio-photographic method of measurement but has not yet published this. He measures the From the Department of Ophthalmology, Washington University School of Medicine, Sf. Louis, Missouri. 170
distance from the apex of the cornea to a line joining the recess of the phakic chamber angle. Most others assume an average value in their tables. C. D. Binkhorst gives correction factors for cases in which the normal value may be assumed to be wrong, but you must make an educated guess about the correct value. R. D. Binkhorst assumes an average value for each type of lens: 3.2 3.5 3.7 2.8
mm mm mm mm
for for for for
iris clip iridocapsular or medallion Copeland Choyce
Axial Length of the Eye X-ray determination of the length of the eye is difficult, unavailable to most of us, and of barely adequate precision at best. Ultrasound is the best method today. Equipment must be calibrated, and ordinary ultrasound equipment yields an accuracy of only 1 mm; and 1 mm = 3 diopters! Only very high frequency ultrasound (10,-15,000 hertz) can yield measurements to the nearest 0.1 mm. Such equipment is prohibitively expensive and is therefore not available to most of us. Finally, no one has shown that consistently accurate results are possible with even the best of this equipment available today. Many ophthalmologists do not have access even to ordinary ultrasound equipment. When it is available, however, it can help the surgeon avoid drastic errors in unusual cases, giving him at least "ball park" limits. It is a tragedy to do an excellent implant and then find the patient is 10 or even 18 diopters myopic. (Such cases exist!) The Lens Implant: distance between principle planes The Choyce Mark VIII lens is convex-convex, and no current implant formula corrects specifically for this configuration. All other available lenses (including the Tennant) are convex-plano. The formulas of C.D. and R.D. Binkhorst, and Fyodorov (see Appendix) assume an optically thin lens. Colenbrander's formula and Hoffer's modification incorporate an arbitrary correction. Thijssen and van der Heijde incorporate factors for lens thickness, but they assume a convex-plano lens. None of these formulas are adequate to calculate the effect of the Troutman reversed-telephoto lens where a non-thin-Iens formula becomes truly significant. (Fortunately it has never been made.) Astigmatism The average astigmatic error present in several major implant series is 2.50 to 3.00 diopters. A highly accurate determination of lens implant power can be frustrated if the patient has a large astigmatic error
postoperatively. Control of corneal astigmatism presents a much more critical problem in patients with lens implants than those who are simply being fitted with cataract glasses. Theory vs. Practice Once the above data has somehow all been gathered with sufficient accuracy, the prediction of needed lens implant power is calculated from a formula; but which one? (See Appendix.) Each of these will yield a different prediction from a given set of data. Which is correct? Furthermore the choice of formula for anyone of these authors depends on goals: (1) Emmetropia (2) Aniseikonia (3) Matching the refractive error in the fellow eye (4) A power chosen to specifically benefit a given patient (-2.50 for an elderly bookworm) (5) Available stock lens implant power. A Practical Approach: Happiness If we choose as our goal patient happiness instead of an arbitrary numerical result, such as emmetropia or iseikonia, our choice of lens power can be broadened. Certain basic assumptions are made. We assume: (1) An 18 diopter anterior chamber lens implant will yield a patient's pre-cataractous refractive error. (This figure is for the iris clip lens. Iris plane lenses require more and the Choyce lens less.) (2) Postoperative refractive error in aphakia is predictable. With lens implants Jaffe is able to predict the postoperative refraction with an average error of only 0.68 diopters. Your own experience confirms this: the great majority of patients have a basic precataract refractive error of about 0 and after cataract surgery have a spherical equivalent error of about +10. The vast majority of your aphakic patients wear a spherical equivalent of + 10 to + 12, and you use temporary glasses in this power range almost exclusively. You can predict your own patient's postoperative aphakic correction, almost always within 2 diopters, by dividing his precataract refractive error by 2 and adding the result to 10. Thus a 4 diopter hyperope will be a + 12 aphake, and a 20 diopter myope will be approximately emmetropic after you remove his cataract. Errors in such predictions are seldom greater than 2 diopters except when the base refraction is a very large number. The word "seldom" means that once in a while there will be a larger error. These are patients who have an abnormal lens power of their own. There is no way to detect such patients without axial length measurement. Large refractive errors,
especially in myopes, are the most hazardous in this regard; and even the happiness zone in hyperopes looks wider than it may really be for this reason. Patients suspected of lenticular refractive error should be fitted with iris clip lenses close to 18 diopters. This is the way to avoid an unexpected high myopia in a (lenticular) hyperope, for example. This argument is another good reason for avoiding surgery on high myopes, and one can further generalize by urging caution in the use of intraocular lenses of unusual power. Our predictions are modified: (]) By patient needs: the sedentary elderly patient will have a primary need at near; in contrast the athletic patient needs accurate distance vision. (2) By the vision in the second eye (assuming fusional ability). Too much anisometropia must be avoided if the second eye has relatively good vision. Ideally, iseikonia should be the goal. The Following Data Must be Known: The basic refractive error (before cataract). Sometimes this can be assumed from the history (or lack of history) of glasses, especially before age 40. (Assumptions always carry a higher factor of risk however!) The Graph: We can now construct a graph which will aid in lens implant power selection. We plot four lines: (1) No change. (A lens that yields the patient's pre-cataract refractive error.) (2) Aniseikonia. (A lens that adds a minus two diopters to the patient's pre-cataractous refractive error will make him roughly aniseikonic at least in a lower range of powers.) (3) Emmetropia. Note that Shepard's line is given. A more accurate approximation is that of R. D. Binkhorst with a slope of 1: 1.25 (widening the "Happiness Zone"). These are now graphed in Table 1. Between these four lines lies a zone of choice of lens implant power which is most likely to leave the patient happy. Refractive Error
before cataract
+6
+5 +4
+3 +2
+1
-1
Power
-2 -3
-4 -5 -6
Table 1 (Drews, R.) 171
Modification Modification within the "happiness zone" is done as follows: (1) Move toward the "no change" line if the patient's other eye has good vision. (2) Move toward emmetropia if the patient is athletic, and move toward 2 diopters of myopia if the patient is sedentary. Note that the goal of emmetropia for the active patient will probably be frustrating for the surgeon. It is' hard to achieve sufficient accuracy especially in view of the problems with astigmatism. A golfer for example with one half diopter of error still needs glasses to track the golf ball accurately. Our goal should be to make the patient better but not too much better. Examples: Let's look at examples of using this graph. A generous probable range of error is the predicted refraction plus or minus two diopters.
Base Lens Implant Postop Refraction Power Refraction 0
-4
Postop Lens Implant Base Power Refraction Refraction +4
18
+4
Range
+2
+22
0
+24
172
-2
+2
18
Can't see without bifocals (and could be +4).
0
+2 to-2
All right, but may be +2 and can't see without bifocals .
20
-2
o to-4
Iseikonic : Error 0 to -4: this is the best range of error . Th is patient can read without glasses unless we are wrong and make him emmatropic , and that's all right too.
22
-4
+14
o
+2 to-2
Poor choice. If in error this could be a +2 now, and for the first time in the patient 's life he will be unable to see well at any distance without glasses.
+16
-2
o to-4
Good choice : this is an ideal amount of myopia, and the patient will be iseikonic.
+18
-4
-2 to-6 Poor choice : if in error could be -6. Patient may be happy but I won't .
+20
-6
-4 to-8 Use of the "standard" lens for this patient is a mistake .
-2 to-6 Too myopic without glasses and could be -6.
Comments
+4 to 0
Ibid , but better. May be the best choice for iseikonia if the other eye has close to 20/20 vision.
+2 to-2
Best choice, especially if the other eye has relatively poor vision. Note that the error range here is from +2 to -2.
o to-4
All right , but a big change for this patient, especially if we are wrong, and he turns out to be -4. He will be able to read without glasses, but if the other eye has fai r vision he may have problems because of his anisometropia .
Comments
+4 to 0
+6 to + 2 Patient will be happy (even if +6) espe-
cially if the visual acuity in the other eye is poor and anisei konia doesn't bother him; but I won't: this patient can't see well without glasses especially if the postoperative refraction turns out to be +6 instead of +4. +20
16
Range
Note that the range for physician happiness in patients who are high myopes becomes very narrow. On the other hand the patient is likely to be happy since he is used to being a high myope. The happiness zone is narrowest for emmetropes: be sure they understand they will need bifocals after surgery. Note also from the graph, the wide use of the standard 20 diopter lens for happiness: it can be used in all patients except those with more than 2 diopters of myopia. This does not of course guarantee that every patient with a lens implant power selected on the above basis will come out as predicted. Obviously there will be the occasional exception, but until better means are available to you, at least the above approach can give a reasonable way of deciding on lens implant power in your patients.
Summary The problem of choosing a lens implant of appropriate power can be solved happily by modifying the standard implant power according to the patient's basic refraction, with consideration for the patient's fellow eye and his refractive need. If ultrasound is available it certainly should be used (9), especially in cases where a patient's precataract refractive error is unknown. APPENDIX Formulas for Intraocular Lens Power 1. C. D. Binkhorst
Symbols Used in Appendix Corneal Thickness Axial Power of plus LO.L. AC Depth Length Radius Power Keratometer
CD. Binkhorst
RD. Binkhorst
Colen brander Hoffer F·G·L Thijssen van der Heijde
d Fe K Dc Pc Fe
v
CAr k d2 d, + d 2
a I A.L. a I b
1/fL D FL P Dp PLE F,
The following additional symbols appear m these formulae:
c.o. Binkhorst: 2. R. D. Binkhorst
= anterior focal length of the complete eye fe = anterior focal length of the cornea fE
D- 1336(4r-a) - (a - d)(4r - d)
fo d
3. Colenbrander FL
=
Q-
pseudophakos to the 2nd equivalent plane of the cornea.
NJ v - .00005
~ -v-
.00005
Fe
133600 82.05 L + 76.6 V - 5
1.336 1.336 - CAr - 5xlO-5 K 1000
5. Fyodorov - Galin - Linksz
Dp --
n - a Dc (a-k)
fL = focal length of the pseudophakos Colenbrander:
4. Hoffer (Colenbrander) p=
= posterior focal length of the cornea = distance from the 1st equivalent plane of the
(1- ~De)
= index of refraction of aqueous (vitreous) .00005 = assumed distance of 2nd principle point of NJ
the iris clip lens from its anterior surface.
Hoffer: L = thickness of the patient's lens (in microseconds)
V CAr
=
length of the vitreous cavity (in microseconds)
= corneal apex distance (corrected for radius of cornea)
6. Thijssen
ny PLE = 1 - d + d3 2
K • ny nL
7. van der Heijde
The confusion facing the practicing ophthalmologist concerning the choice of lens implant formulas is compounded by the lack of uniformity of the symbols used. The symbols appearing in the formulas above may be deciphered as follows:
=
spherical equivalent of the K reading in diopters.
Thijssen: nv
= index of refraction of vitreous
nA nL
= index of refraction of aqueous = index of refraction of lens
d3
,;
thickness of lens
van der Heijde: d3
= thickness of the lens implant
n2
= index of refraction of the lens implant
173
Approximations can be made by assuming average values for these parameters. In addition at least four different Nomograms have been published. ~1"""",~"""",~ """"~
""
"'~"""
..~.,
'~'"
"'~"""" ~"" ",,~,
'~"'"
~~
"~""'"
3000_'fA'
'"''
-
,.00 .101-00 -
.
.,., --
4&.00-"",.",.1."".".1 •• ", •••• 1""" 21 22 23 210
26
'11
•
29
Xl
31
''''''
«"
-
....
32
-
---------------
Fig. 4 (Drews, R.) The van der Heijde Nomogram 8.
"00
-
,.1 •• "" ••• 1"""",1 •• """,1""",.,1,.". ",I",.t.",I""" •.. I""", 25
"""
"HICl
-
-
l>OO
"00
-
-
1
;
"00 "00
-
.,.
"00
-
,,00 -
"""
-
:::~:..:~: :"-~~"'~"-" .
3000
'60>
~:~:!.'.·.····[F' ~' --1---" c . . ..._...:~ : ' .,
- noo
-
"00 "00 "00 "00 ,.00 -
"00
._d_ 01234S~7
<6(J()
10100
Fig. 1 (Drews, R.) The Binkhorst Nomogram 1 Radius corneae, mm
Finally the easiest instrument of all to use is the van der Heijde Biometer manufacturer by Medical Workshop. At a fraction of the cost of the cheapest programmable computer, this modified circular slide rule (Fig. 5) allows a very simple determination of the lens implant power once the keratometer reading and axial length of the eye are known. It makes no more assumptions than any of the other formulas which will allow determination of intraocular lens power from these parameters. It assumes, as is usually assumed in these formulas, that the pseudophakic depth of the anterior chamber will be 3.5mm (.0035m).
8.0
7.5
7.0
6. 5 LL4LfL'rl-'f4L+Lf--4L+44'-f'"'i.4LrL,L,<...,L,MLr4-<;""""'.,.--,-...,-,--r-' 25.0 35.0 20.0 30.0 Axial length, mm
Fig. 2 (Drews, R.) The Thijssen Nomogram 4. 27 26
.sE 25
~ 24 -
Q)
'0
23
.J::
g, 22
~
:'§ 21
,T
N::'~
~~ t--.~~~ 8:: r::: l'-r::: ~ t::: t:-- i"- f:::: ~ f::: t:, i'-.. Ii'f::: [:::: t:::t.:: ~1'-.'- , , ~~ r--- ~ 1'--- t, I I
19
i 11
I
i
I
I
. I
'. --. "
.,
"
"
..
.--,--
",
--
Optical
t-'::' ~~ :-:: 139 power
't--. of -- F:' . .::'. " ...~," . . .::. '43 cornea j-::t::: b--: --I- --1-1- I-+~,-..
....~.
15 17 19 21 23 FIG.3 vlo MEDEXPORT
25
Fig. 3 (Drews, R.) The Fyodorov Nomogram 174
Note the need in all of the formulas except Hoffer's to keep all lengths in meters.
I',
'-.-:--- ................ ...... "-~
o~tical power ot' intraocular lens (dioptrJa) 13
Fig. 5 (Drews, R.) The Medical Workshop Biometer by van der Heijde.
I
--. --
x
'" 20
The depth of the anterior chamber = 3mm
- -, --141
f 27 7
J~~ -1-' 1 29
In addition to the obvious differences in these formulas, there are hidden differences as well: Thijssen, for example, defines the corneal dioptic power to be used in his formula as 1 diopter less than the keratometric reading, to allow for the negative effect of the back surface of the cornea.
Some of the formulas are unuseable as written: C. D. Binkhorst's for example. Hoffer's and Thijssen's formulas require the measurement of additional parameters such as the individual measurement of corneal thickness plus anterior chamber depth, lens thickness, and vitreous cavity length. Assumptions are then made about the index of refraction or speed of sound (Hoffer) of each of these media. Thijssen's formula can be used without these additional measurements only by assuming average values. The derivation of each of these formulas is dependent upon certain assumptions. The formulas differ, in most instances, because the assumptions were different. The result also differ for this reason. A sample calculation may suffice to illustrate this:
Hoffer used an HP6 5, programmed for Colenbrander's formula. He now prefers a second, technically much less difficult, formula, which has not yet been published. Thijssen correctly points out that formula's 5 and 6 transform into each other if the pseudophake thickness is taken to be zero. Once the differences in symbols are eliminated, formulas 2, 3, 5,6, and 7 can be transformed one into the other by dropping correction factors. In addition to a very careful point by point analysis of this paper with numerous suggest.ions (most of which have already been incorporated) R. D. Binkhorst added a discussion which I think is worthy of reproduction in toto in this Appendix:
Sample Calculation: Given: Corneal power = 42 diopters Axial length = 24 mm (.024 meters)
DISCUSSION Discussion (R. D. Binkhorst):
Assume: Pseudophakic AC depth (.0035 meters)
=
3.5 mm
Results: Intraocular Lens Power Formulas 1. 2. 3. 4. 5. 6.
C. D. Binkhorst R. D. Binkhorst Colenbrander Hoffer Fyodorov - Galin - Linksz Thijssen
can't use 18.53 18.05 can't use 17.98 17.40*
Nomograms 1. 2. 3. 4.
18.0 18.3 17.4 17.8
C. D. Binkhorst Thijssen Fyodorov van der Heijde
Biometer van der Heijde
17.8
Although most of the results cluster around 18 diopters, there is a half diopter variation to either side of this value. Each of the authors of the formulas quoted were contracted and asked to comment on this Appendix. C. D. Binkhorst replied that his published formula is for focal distance, and is very practical for calculations of iseikonia. For lens power he used the vergence formula of Colenbrander, but forbore in publishing this in order to afford Colen brander priority. He stresses that iseikonia is often a better goal than emmetropia.
Dr. Drews' practical approach has considerable merit. Happiness, however, is a state of mind and likely to change with circumstances. Specifically, the "happiness" with a standard power lens implant may yield to the true satisfaction which could be derived from the results of accurate lens power calculation. The long neglected pursuit of accuracy has just begun. Sophisticated equipment for ultrasonic biometry is becoming available for clinical use. The theory of intraocular lens power calculation has been developed. However, it remains for the results to prove the efficacy of the equipment as well as the validity of the theory. The formulas quoted by Dr. Drews are expressions of very basic optical principles. They appear confusing mainly because they h
*17.2 with 0.5 mm correction for retinal thickness. 175
keratometer is based on an assumption, a fictitious refractive index, used to convert this radius of curvature into diopters. The true refractive index of the cornea is 1.376, but to obtain an approximation of the true refracting power of the cornea from only the radius of the anterior surface a fictitious refractive index has to be used. The latter varies with the make of keratometer. Common values are 1. 33 7 5 (Haag-Streit, Bausch and Lomb), 1.336 (American Optical), and 1.332 (Gambs). A radius of 7.8 mm will thus read 43.27, 43.08, and 42.56 diopters respectively depending on which keratometer was used. These differences are hardly of importance in contact lens fitting and the determination of corneal astigmatism. They are not insignificant for intraocular lens power calculation. As an example Fyodorov's formula will be used for an average K-reading of 7.8 mm, an axial length of 23.65 mm, and a postoperative anterior chamber depth of 3.5 mm. Using the above refractive indices the implant power required to produce emmetropia would be 17.50, 17.75, or 18.40 diopters respectively. Such different answers from the same formula depending on the make of the keratometer used seems hardly desirable. This objection applies to all intraocular lens power formulas which use the dioptric power of the cornea. If the radius of curvature is used instead of the dioptric power, the confusion is eliminated, but the problem is not solved. The formula then has to contain the assumed refractive index of the cornea. The problem of which index to choose still remains, but at least the problem is now out in the open and no longer hidden in the instrument. The author's formula uses the average K-reading in mm. It assumes a refractive index of 1.333 (4/3). The correct value is unknown, simply because it has not been determined. With biometric A-scan ultrasonography, and from the postoperative refractions of implant patients, the correct value for clinical use can ultimately be established. Nevertheless, the choice of the value 4/3 has not been entirely arbitrary. It is within the range of values used for various keratometers. It is near the lower end of this range, and thus the formula yields higher implant powers than it would with the more commonly used value 1.3375. The following considerations have favored the choice: 1. It compensates for postoperative flattening of the cornea. 2. The margin of error is shifted toward the myopic side. 3. Previous reports have shown an average error on the hyperopic side. 176
In summary, Dr. Drews' analysis of the subject of intraocular lens power determination has its value for those who do not have access to sophisticated ultrasound equipment. In general, his views contribute much to the understanding of the problems at issue. After all, the pursuit of happiness is everyone's right. Richard D. Binkhorst References 1. Binkhorst, C. D. : Power of the pre-pupillary pseudo-
phakos. BJO, 56,332-337,1972. 2. Binkhorst, R. D.: The optical design of intraocular lens implants, 6, 17-31, 1975. 3. Colenbrander, M. D. : Calculation of the power of an iris clip lens for distance vision, BJO, 57,735- 1973 . 4. Thijssen, J. M. : In Shepard, D.: The Intraocular Lens M
A number of articles have been written on the determination of lens implant power. Unfortunately their formulas are derived using different assumptions and yield a variety of answers from a given set of data. More importantly they require data that most of us cannot obtain, or cannot obtain with sufficient accuracy. Three basic parameters must be measured in order to predict lens implant power: (1) Corneal power (2) Aphakic chamber depth (distance from the cornea to the lens implant), and (3) Axial length of the eye. There are two other factors which are not usually mentioned in such discussions: (4) Distance between the principle planes of the lens implant, and (5) Astigmatic error. Corneal Power This is the one parameter which every ophthalmologist should be able to measure easily and with sufficient accuracy. But how long has it been since your keratometer or ophthalmometer was calibrated? When it reads 41.50 is that really 41.12 or 41.87? Satisfactory contact lens fitting is no criterion, especially when factors such as fitting "steeper than K" are used by some and not by others. But a keratometer does not measure "K": corneal power. It measures the radius of curvature and converts this to power by using a fictitious index of refraction for the cornea. This index mayor may not incorporate a correcting factor for the minus effect of the back of the cornea. Furthermore, the index arbitrarily used differs from one keratometer manufacturer to another. (See R. D. Binkhorst's discussion at the end of this paper.) Aphakic Chamber Depth Most of us cannot predict the postoperative depth of the anterior chamber. Galin makes a prediction by assuming the position of the chamber angle with respect to the observed limbus. He measures the diameter of the cornea at that level and calculates the aphakic chamber depth from this diameter and the corneal radius of curvature, but these relations are obviously subject to individual variation. Worst has worked on a gonio-photographic method of measurement but has not yet published this. He measures the From the Department of Ophthalmology, Washington University School of Medicine, Sf. Louis, Missouri. 170
distance from the apex of the cornea to a line joining the recess of the phakic chamber angle. Most others assume an average value in their tables. C. D. Binkhorst gives correction factors for cases in which the normal value may be assumed to be wrong, but you must make an educated guess about the correct value. R. D. Binkhorst assumes an average value for each type of lens: 3.2 3.5 3.7 2.8
mm mm mm mm
for for for for
iris clip iridocapsular or medallion Copeland Choyce
Axial Length of the Eye X-ray determination of the length of the eye is difficult, unavailable to most of us, and of barely adequate precision at best. Ultrasound is the best method today. Equipment must be calibrated, and ordinary ultrasound equipment yields an accuracy of only 1 mm; and 1 mm = 3 diopters! Only very high frequency ultrasound (10,-15,000 hertz) can yield measurements to the nearest 0.1 mm. Such equipment is prohibitively expensive and is therefore not available to most of us. Finally, no one has shown that consistently accurate results are possible with even the best of this equipment available today. Many ophthalmologists do not have access even to ordinary ultrasound equipment. When it is available, however, it can help the surgeon avoid drastic errors in unusual cases, giving him at least "ball park" limits. It is a tragedy to do an excellent implant and then find the patient is 10 or even 18 diopters myopic. (Such cases exist!) The Lens Implant: distance between principle planes The Choyce Mark VIII lens is convex-convex, and no current implant formula corrects specifically for this configuration. All other available lenses (including the Tennant) are convex-plano. The formulas of C.D. and R.D. Binkhorst, and Fyodorov (see Appendix) assume an optically thin lens. Colenbrander's formula and Hoffer's modification incorporate an arbitrary correction. Thijssen and van der Heijde incorporate factors for lens thickness, but they assume a convex-plano lens. None of these formulas are adequate to calculate the effect of the Troutman reversed-telephoto lens where a non-thin-Iens formula becomes truly significant. (Fortunately it has never been made.) Astigmatism The average astigmatic error present in several major implant series is 2.50 to 3.00 diopters. A highly accurate determination of lens implant power can be frustrated if the patient has a large astigmatic error
postoperatively. Control of corneal astigmatism presents a much more critical problem in patients with lens implants than those who are simply being fitted with cataract glasses. Theory vs. Practice Once the above data has somehow all been gathered with sufficient accuracy, the prediction of needed lens implant power is calculated from a formula; but which one? (See Appendix.) Each of these will yield a different prediction from a given set of data. Which is correct? Furthermore the choice of formula for anyone of these authors depends on goals: (1) Emmetropia (2) Aniseikonia (3) Matching the refractive error in the fellow eye (4) A power chosen to specifically benefit a given patient (-2.50 for an elderly bookworm) (5) Available stock lens implant power. A Practical Approach: Happiness If we choose as our goal patient happiness instead of an arbitrary numerical result, such as emmetropia or iseikonia, our choice of lens power can be broadened. Certain basic assumptions are made. We assume: (1) An 18 diopter anterior chamber lens implant will yield a patient's pre-cataractous refractive error. (This figure is for the iris clip lens. Iris plane lenses require more and the Choyce lens less.) (2) Postoperative refractive error in aphakia is predictable. With lens implants Jaffe is able to predict the postoperative refraction with an average error of only 0.68 diopters. Your own experience confirms this: the great majority of patients have a basic precataract refractive error of about 0 and after cataract surgery have a spherical equivalent error of about +10. The vast majority of your aphakic patients wear a spherical equivalent of + 10 to + 12, and you use temporary glasses in this power range almost exclusively. You can predict your own patient's postoperative aphakic correction, almost always within 2 diopters, by dividing his precataract refractive error by 2 and adding the result to 10. Thus a 4 diopter hyperope will be a + 12 aphake, and a 20 diopter myope will be approximately emmetropic after you remove his cataract. Errors in such predictions are seldom greater than 2 diopters except when the base refraction is a very large number. The word "seldom" means that once in a while there will be a larger error. These are patients who have an abnormal lens power of their own. There is no way to detect such patients without axial length measurement. Large refractive errors,
especially in myopes, are the most hazardous in this regard; and even the happiness zone in hyperopes looks wider than it may really be for this reason. Patients suspected of lenticular refractive error should be fitted with iris clip lenses close to 18 diopters. This is the way to avoid an unexpected high myopia in a (lenticular) hyperope, for example. This argument is another good reason for avoiding surgery on high myopes, and one can further generalize by urging caution in the use of intraocular lenses of unusual power. Our predictions are modified: (]) By patient needs: the sedentary elderly patient will have a primary need at near; in contrast the athletic patient needs accurate distance vision. (2) By the vision in the second eye (assuming fusional ability). Too much anisometropia must be avoided if the second eye has relatively good vision. Ideally, iseikonia should be the goal. The Following Data Must be Known: The basic refractive error (before cataract). Sometimes this can be assumed from the history (or lack of history) of glasses, especially before age 40. (Assumptions always carry a higher factor of risk however!) The Graph: We can now construct a graph which will aid in lens implant power selection. We plot four lines: (1) No change. (A lens that yields the patient's pre-cataract refractive error.) (2) Aniseikonia. (A lens that adds a minus two diopters to the patient's pre-cataractous refractive error will make him roughly aniseikonic at least in a lower range of powers.) (3) Emmetropia. Note that Shepard's line is given. A more accurate approximation is that of R. D. Binkhorst with a slope of 1: 1.25 (widening the "Happiness Zone"). These are now graphed in Table 1. Between these four lines lies a zone of choice of lens implant power which is most likely to leave the patient happy. Refractive Error
before cataract
+6
+5 +4
+3 +2
+1
-1
Power
-2 -3
-4 -5 -6
Table 1 (Drews, R.) 171
Modification Modification within the "happiness zone" is done as follows: (1) Move toward the "no change" line if the patient's other eye has good vision. (2) Move toward emmetropia if the patient is athletic, and move toward 2 diopters of myopia if the patient is sedentary. Note that the goal of emmetropia for the active patient will probably be frustrating for the surgeon. It is' hard to achieve sufficient accuracy especially in view of the problems with astigmatism. A golfer for example with one half diopter of error still needs glasses to track the golf ball accurately. Our goal should be to make the patient better but not too much better. Examples: Let's look at examples of using this graph. A generous probable range of error is the predicted refraction plus or minus two diopters.
Base Lens Implant Postop Refraction Power Refraction 0
-4
Postop Lens Implant Base Power Refraction Refraction +4
18
+4
Range
+2
+22
0
+24
172
-2
+2
18
Can't see without bifocals (and could be +4).
0
+2 to-2
All right, but may be +2 and can't see without bifocals .
20
-2
o to-4
Iseikonic : Error 0 to -4: this is the best range of error . Th is patient can read without glasses unless we are wrong and make him emmatropic , and that's all right too.
22
-4
+14
o
+2 to-2
Poor choice. If in error this could be a +2 now, and for the first time in the patient 's life he will be unable to see well at any distance without glasses.
+16
-2
o to-4
Good choice : this is an ideal amount of myopia, and the patient will be iseikonic.
+18
-4
-2 to-6 Poor choice : if in error could be -6. Patient may be happy but I won't .
+20
-6
-4 to-8 Use of the "standard" lens for this patient is a mistake .
-2 to-6 Too myopic without glasses and could be -6.
Comments
+4 to 0
Ibid , but better. May be the best choice for iseikonia if the other eye has close to 20/20 vision.
+2 to-2
Best choice, especially if the other eye has relatively poor vision. Note that the error range here is from +2 to -2.
o to-4
All right , but a big change for this patient, especially if we are wrong, and he turns out to be -4. He will be able to read without glasses, but if the other eye has fai r vision he may have problems because of his anisometropia .
Comments
+4 to 0
+6 to + 2 Patient will be happy (even if +6) espe-
cially if the visual acuity in the other eye is poor and anisei konia doesn't bother him; but I won't: this patient can't see well without glasses especially if the postoperative refraction turns out to be +6 instead of +4. +20
16
Range
Note that the range for physician happiness in patients who are high myopes becomes very narrow. On the other hand the patient is likely to be happy since he is used to being a high myope. The happiness zone is narrowest for emmetropes: be sure they understand they will need bifocals after surgery. Note also from the graph, the wide use of the standard 20 diopter lens for happiness: it can be used in all patients except those with more than 2 diopters of myopia. This does not of course guarantee that every patient with a lens implant power selected on the above basis will come out as predicted. Obviously there will be the occasional exception, but until better means are available to you, at least the above approach can give a reasonable way of deciding on lens implant power in your patients.
Summary The problem of choosing a lens implant of appropriate power can be solved happily by modifying the standard implant power according to the patient's basic refraction, with consideration for the patient's fellow eye and his refractive need. If ultrasound is available it certainly should be used (9), especially in cases where a patient's precataract refractive error is unknown. APPENDIX Formulas for Intraocular Lens Power 1. C. D. Binkhorst
Symbols Used in Appendix Corneal Thickness Axial Power of plus LO.L. AC Depth Length Radius Power Keratometer
CD. Binkhorst
RD. Binkhorst
Colen brander Hoffer F·G·L Thijssen van der Heijde
d Fe K Dc Pc Fe
v
CAr k d2 d, + d 2
a I A.L. a I b
1/fL D FL P Dp PLE F,
The following additional symbols appear m these formulae:
c.o. Binkhorst: 2. R. D. Binkhorst
= anterior focal length of the complete eye fe = anterior focal length of the cornea fE
D- 1336(4r-a) - (a - d)(4r - d)
fo d
3. Colenbrander FL
=
Q-
pseudophakos to the 2nd equivalent plane of the cornea.
NJ v - .00005
~ -v-
.00005
Fe
133600 82.05 L + 76.6 V - 5
1.336 1.336 - CAr - 5xlO-5 K 1000
5. Fyodorov - Galin - Linksz
Dp --
n - a Dc (a-k)
fL = focal length of the pseudophakos Colenbrander:
4. Hoffer (Colenbrander) p=
= posterior focal length of the cornea = distance from the 1st equivalent plane of the
(1- ~De)
= index of refraction of aqueous (vitreous) .00005 = assumed distance of 2nd principle point of NJ
the iris clip lens from its anterior surface.
Hoffer: L = thickness of the patient's lens (in microseconds)
V CAr
=
length of the vitreous cavity (in microseconds)
= corneal apex distance (corrected for radius of cornea)
6. Thijssen
ny PLE = 1 - d + d3 2
K • ny nL
7. van der Heijde
The confusion facing the practicing ophthalmologist concerning the choice of lens implant formulas is compounded by the lack of uniformity of the symbols used. The symbols appearing in the formulas above may be deciphered as follows:
=
spherical equivalent of the K reading in diopters.
Thijssen: nv
= index of refraction of vitreous
nA nL
= index of refraction of aqueous = index of refraction of lens
d3
,;
thickness of lens
van der Heijde: d3
= thickness of the lens implant
n2
= index of refraction of the lens implant
173
Approximations can be made by assuming average values for these parameters. In addition at least four different Nomograms have been published. ~1"""",~"""",~ """"~
""
"'~"""
..~.,
'~'"
"'~"""" ~"" ",,~,
'~"'"
~~
"~""'"
3000_'fA'
'"''
-
,.00 .101-00 -
.
.,., --
4&.00-"",.",.1."".".1 •• ", •••• 1""" 21 22 23 210
26
'11
•
29
Xl
31
''''''
«"
-
....
32
-
---------------
Fig. 4 (Drews, R.) The van der Heijde Nomogram 8.
"00
-
,.1 •• "" ••• 1"""",1 •• """,1""",.,1,.". ",I",.t.",I""" •.. I""", 25
"""
"HICl
-
-
l>OO
"00
-
-
1
;
"00 "00
-
.,.
"00
-
,,00 -
"""
-
:::~:..:~: :"-~~"'~"-" .
3000
'60>
~:~:!.'.·.····[F' ~' --1---" c . . ..._...:~ : ' .,
- noo
-
"00 "00 "00 "00 ,.00 -
"00
._d_ 01234S~7
<6(J()
10100
Fig. 1 (Drews, R.) The Binkhorst Nomogram 1 Radius corneae, mm
Finally the easiest instrument of all to use is the van der Heijde Biometer manufacturer by Medical Workshop. At a fraction of the cost of the cheapest programmable computer, this modified circular slide rule (Fig. 5) allows a very simple determination of the lens implant power once the keratometer reading and axial length of the eye are known. It makes no more assumptions than any of the other formulas which will allow determination of intraocular lens power from these parameters. It assumes, as is usually assumed in these formulas, that the pseudophakic depth of the anterior chamber will be 3.5mm (.0035m).
8.0
7.5
7.0
6. 5 LL4LfL'rl-'f4L+Lf--4L+44'-f'"'i.4LrL,L,<...,L,MLr4-<;""""'.,.--,-...,-,--r-' 25.0 35.0 20.0 30.0 Axial length, mm
Fig. 2 (Drews, R.) The Thijssen Nomogram 4. 27 26
.sE 25
~ 24 -
Q)
'0
23
.J::
g, 22
~
:'§ 21
,T
N::'~
~~ t--.~~~ 8:: r::: l'-r::: ~ t::: t:-- i"- f:::: ~ f::: t:, i'-.. Ii'f::: [:::: t:::t.:: ~1'-.'- , , ~~ r--- ~ 1'--- t, I I
19
i 11
I
i
I
I
. I
'. --. "
.,
"
"
..
.--,--
",
--
Optical
t-'::' ~~ :-:: 139 power
't--. of -- F:' . .::'. " ...~," . . .::. '43 cornea j-::t::: b--: --I- --1-1- I-+~,-..
....~.
15 17 19 21 23 FIG.3 vlo MEDEXPORT
25
Fig. 3 (Drews, R.) The Fyodorov Nomogram 174
Note the need in all of the formulas except Hoffer's to keep all lengths in meters.
I',
'-.-:--- ................ ...... "-~
o~tical power ot' intraocular lens (dioptrJa) 13
Fig. 5 (Drews, R.) The Medical Workshop Biometer by van der Heijde.
I
--. --
x
'" 20
The depth of the anterior chamber = 3mm
- -, --141
f 27 7
J~~ -1-' 1 29
In addition to the obvious differences in these formulas, there are hidden differences as well: Thijssen, for example, defines the corneal dioptic power to be used in his formula as 1 diopter less than the keratometric reading, to allow for the negative effect of the back surface of the cornea.
Some of the formulas are unuseable as written: C. D. Binkhorst's for example. Hoffer's and Thijssen's formulas require the measurement of additional parameters such as the individual measurement of corneal thickness plus anterior chamber depth, lens thickness, and vitreous cavity length. Assumptions are then made about the index of refraction or speed of sound (Hoffer) of each of these media. Thijssen's formula can be used without these additional measurements only by assuming average values. The derivation of each of these formulas is dependent upon certain assumptions. The formulas differ, in most instances, because the assumptions were different. The result also differ for this reason. A sample calculation may suffice to illustrate this:
Hoffer used an HP6 5, programmed for Colenbrander's formula. He now prefers a second, technically much less difficult, formula, which has not yet been published. Thijssen correctly points out that formula's 5 and 6 transform into each other if the pseudophake thickness is taken to be zero. Once the differences in symbols are eliminated, formulas 2, 3, 5,6, and 7 can be transformed one into the other by dropping correction factors. In addition to a very careful point by point analysis of this paper with numerous suggest.ions (most of which have already been incorporated) R. D. Binkhorst added a discussion which I think is worthy of reproduction in toto in this Appendix:
Sample Calculation: Given: Corneal power = 42 diopters Axial length = 24 mm (.024 meters)
DISCUSSION Discussion (R. D. Binkhorst):
Assume: Pseudophakic AC depth (.0035 meters)
=
3.5 mm
Results: Intraocular Lens Power Formulas 1. 2. 3. 4. 5. 6.
C. D. Binkhorst R. D. Binkhorst Colenbrander Hoffer Fyodorov - Galin - Linksz Thijssen
can't use 18.53 18.05 can't use 17.98 17.40*
Nomograms 1. 2. 3. 4.
18.0 18.3 17.4 17.8
C. D. Binkhorst Thijssen Fyodorov van der Heijde
Biometer van der Heijde
17.8
Although most of the results cluster around 18 diopters, there is a half diopter variation to either side of this value. Each of the authors of the formulas quoted were contracted and asked to comment on this Appendix. C. D. Binkhorst replied that his published formula is for focal distance, and is very practical for calculations of iseikonia. For lens power he used the vergence formula of Colenbrander, but forbore in publishing this in order to afford Colen brander priority. He stresses that iseikonia is often a better goal than emmetropia.
Dr. Drews' practical approach has considerable merit. Happiness, however, is a state of mind and likely to change with circumstances. Specifically, the "happiness" with a standard power lens implant may yield to the true satisfaction which could be derived from the results of accurate lens power calculation. The long neglected pursuit of accuracy has just begun. Sophisticated equipment for ultrasonic biometry is becoming available for clinical use. The theory of intraocular lens power calculation has been developed. However, it remains for the results to prove the efficacy of the equipment as well as the validity of the theory. The formulas quoted by Dr. Drews are expressions of very basic optical principles. They appear confusing mainly because they h
*17.2 with 0.5 mm correction for retinal thickness. 175
keratometer is based on an assumption, a fictitious refractive index, used to convert this radius of curvature into diopters. The true refractive index of the cornea is 1.376, but to obtain an approximation of the true refracting power of the cornea from only the radius of the anterior surface a fictitious refractive index has to be used. The latter varies with the make of keratometer. Common values are 1. 33 7 5 (Haag-Streit, Bausch and Lomb), 1.336 (American Optical), and 1.332 (Gambs). A radius of 7.8 mm will thus read 43.27, 43.08, and 42.56 diopters respectively depending on which keratometer was used. These differences are hardly of importance in contact lens fitting and the determination of corneal astigmatism. They are not insignificant for intraocular lens power calculation. As an example Fyodorov's formula will be used for an average K-reading of 7.8 mm, an axial length of 23.65 mm, and a postoperative anterior chamber depth of 3.5 mm. Using the above refractive indices the implant power required to produce emmetropia would be 17.50, 17.75, or 18.40 diopters respectively. Such different answers from the same formula depending on the make of the keratometer used seems hardly desirable. This objection applies to all intraocular lens power formulas which use the dioptric power of the cornea. If the radius of curvature is used instead of the dioptric power, the confusion is eliminated, but the problem is not solved. The formula then has to contain the assumed refractive index of the cornea. The problem of which index to choose still remains, but at least the problem is now out in the open and no longer hidden in the instrument. The author's formula uses the average K-reading in mm. It assumes a refractive index of 1.333 (4/3). The correct value is unknown, simply because it has not been determined. With biometric A-scan ultrasonography, and from the postoperative refractions of implant patients, the correct value for clinical use can ultimately be established. Nevertheless, the choice of the value 4/3 has not been entirely arbitrary. It is within the range of values used for various keratometers. It is near the lower end of this range, and thus the formula yields higher implant powers than it would with the more commonly used value 1.3375. The following considerations have favored the choice: 1. It compensates for postoperative flattening of the cornea. 2. The margin of error is shifted toward the myopic side. 3. Previous reports have shown an average error on the hyperopic side. 176
In summary, Dr. Drews' analysis of the subject of intraocular lens power determination has its value for those who do not have access to sophisticated ultrasound equipment. In general, his views contribute much to the understanding of the problems at issue. After all, the pursuit of happiness is everyone's right. Richard D. Binkhorst References 1. Binkhorst, C. D. : Power of the pre-pupillary pseudo-
phakos. BJO, 56,332-337,1972. 2. Binkhorst, R. D.: The optical design of intraocular lens implants, 6, 17-31, 1975. 3. Colenbrander, M. D. : Calculation of the power of an iris clip lens for distance vision, BJO, 57,735- 1973 . 4. Thijssen, J. M. : In Shepard, D.: The Intraocular Lens M