On a cold winter morning, twenty years back, as my friends and I sat for the class in the university, our professor shared an unique experience. He detailed us about his experience learning a new subject called biometry of eye, as he cared for his mother who developed cataract to undergo surgery. To the question, what is Biometry, our professor said in as many words, " Well from what I gathered, it seems that is an art of applying a combination of mathematics and statistics to our body (eye) ".
Little did I know, that I have to learn more, when destiny brought me closer to the field of applying science to biology. However, an understanding of the history of development of biometry formula, would help us to appreciate this science more. Lets take a walk down the history to understand more about modern biometry.
The first IOL, as we all know, was implanted by Sir Harold Ridley, post World War II. However the biometry machines found life no earlier than the late sixties or the early seventies. What this indicates is that the art of IOL implantation predates the science of biometry. The IOL power in the pre biometry days was a subject of individual judgement, though strategies like the Basic Refractive Error (BRE) was largely popular. In BRE, the surgeon would implant a 18.0 diopter lens to an emmetrope, but would deduct or add 1.25 D to the lens for every 1 D of myopia or hyperopia, respectively. This approach is therefore classified as Historical Approach.
As biometry machines appeared, a host of first generation theoretical formula got introduced. Notable of these are the Fyodorov, Collabander and Binkhorst. These formulas were theoretical because they were based on mathematical formula of Gaussian optics. They were two lens formula, which only took into consideration the axial length and corneal power of the eye. The first to apply theoretical vergence formula for IOL power calculation was Fyodorov (1) in 1967. Later, in early 1970's more and more formulas started appearing that would be based on theoretical vergence formula. The basic structure of these formulas were the same as the IOL formulas were all derived from theoretical vergence formula of Gaussian Optics.
To understand more on Gaussian Optics and theoretical vergence formula you can refer to my article Gaussian Optics, Gullstrand Eye & Theoretical & Ray Tracing formula on this website in the Biometry section (https://www.quickguide.org/post/all-about-gaussian-optics-gullstrand-eye-theoretical-and-ray-tracing-formula ).
Being based on Gaussian Optics, all these formulas were identical, but only differed in the theoretical value they used for retinal thickness and corneal refractive index. Ultrasound measures upto vitreoretinal interface or inner limiting membrane and then adds a correction factor to derive optical axial length (AL0), while corneal refractive index is used to determine effective corneal power (Ko),etc.
In Gaussian Optics, the effective power of cornea is referenced to the second principle plane of cornea (Ko) and the effective axial length signifies the distance from second principle plane of cornea (Ko ) to optical axial length (ALo) (see below image)
The second principle plane of cornea is approximately .05 mm in front of cornea (8) and it is important to derive the effective power of cornea in a theoretical formula. Thus, though ultrasound biometry measures from corneal epithelial vertex to vitreoretinal interface, the formulas convert this measurement to optical axial length(ALo), that is from second principle plane of cornea (Ko) to Retinal Pigment Epithelium (RPE). Binkhorst added a factor of .250 mm to account for retinal thickness, that is the distance from the vitreo retinal interface (Inner Limiting Membrane) to Retinal Pigment Epithelium. Holladay added .20 mm while Hoffer Q adds none(9).
The SRK T, a combination of regression and theoretical formula, uses a variable correction factor to account for retinal thickness to derive effective axial length for short and long axial length - with longer eyes having less correction factor.
Today, the recommended standardized conversion of measured ultrasonic axial length (ALu) to axial length optical (ALo) is, AL0 = ALu + .20 (2). This optical power of axial length is used in the theoretical vergence formula like Holladay I that is later explained. The ultrasonic axial length (ALu) is measured at a velocity of 1532 m/s for the aqueous and vitreous, and 1641 m/s for the lens. Thus in an average eye Olsen points(4) that the average velocity of ultrasonic sound for whole eye will be 1550 m/s.
Reflecting back to pre phacoemulsification time,of the many troubles associated back then with first generation theoretical formulas (Fyodorov, Binkhorst, etc), there was one bright point. The anterior chamber lenses were largely stable, unlike the later generation IOLs that were implanted in the capsular bag. As the axial stability of the lens was more stable (considering there was no bag contraction effect on anterior chamber lenses ), the need for more sophisticated formulas was not felt, till the advent of phacoemulsification. During this time with the first generation theoretical formulas like Fyodorov/ Binkhorst/ Collabander, ACD or Anterior Chamber Depth (Effective Lens Position is a term coined much later in the mid nineties) was a constant value added in the formulas. Also with these formulas, the value of the ACD represented true anatomic ACD, that is from the corneal endothelium to the iris plane.
The ultrasonic axial length (ALu) is measured at a velocity of 1532 m/s for the aqueous and vitreous, and 1641 m/s for the lens. Thus in an average eye Olsen points(4) that the average velocity of ultrasonic sound for whole eye will be 1550 m/s.
With advent of phacoemulsification, it was understood, that the capsular bag was the right place to implant an IOL. But challenge was the instability of the IOL implanted in the bag. Thus came a host of second generation formulas, notable among them SRK (regression based) formula, P=A-2.5L-.9K . The linear regression based formula like SRK made simple calculations for IOL power before powerful computers and calculators were available. Despite its many challenges, the SRK was revolutionary in the sense that for the first time it introduced the concept of A constant. The A constant helped to determine the lens position of the IOL after cataract surgery. The position of the IOL in the eye began to be termed as post operative ACD (pACD or ACDpost) and was determined by the distance from the corneal epithelium to the optical plane of IOL (for a thin lens formula).
To cut a long story short, the next ten years of Sanders, Retzalf, Kraff (SRK) were devoted to further improving the formula, that is determining more accurately the post operative lens position in eye largely by improving the constant based on individual eye features. Thus while the SRK II (regression based formula) modified the A constant with the individual Axial Length measurement, the third generation formula SRK T (combination of theoretical and regression formula) linked the same to the Keratometry reading and corneal height. Nevertheless, the original SRK formula, gives us an idea of how important and precise the axial length calculation needs to be, as any error on the axial length calculation is multiplied by almost three times, as reflected in the formula. If you want to access the SRK T formula, you can follow the link here : http://eyecalc.org/srk-t/
SRK T adapted Gaussian thin lens theoretical vergence formula on regression based data, thus making SRK T a combination of regression and theoretical formula. Expressing the motivation to further imporve the SRK II regression based formula, Sanders argued that regression-derived lens implant power formulas though have an excellent accuracy record, still, theoretical formulas, because they are based on physiologic optics, may be more accurate than regression formulas when extended past a given database as in unusually long or short eyes(9). Two aspects that SRK T introduced were - 1) accounted for retinal thickness (earlier explained) 2) predicted postoperative ACD through corneal height formula.
In predicting the postoperative ACD, SRK T adopts corneal height formula, which is a combination of corneal curvature, corneal width (White to White) and an off set factor to account for lens position from iris plane after implantation. Since the formula was developed in the early 1990s, there was very little means of knowing corneal width in an average practice, therefore, regression based formulas were applied to understand corneal width and then apply a theoretical formula to derive corneal height.
The height (H) of the corneal dome is computed from the corneal width (width) and the corneal curvature (r):
ACD (postoperative) = corneal thickness+ H + offset factor (refer diagram above)
where H is corneal height derived from corneal curvature and corneal width and the offset factor is to account for the distance from iris to IOL plane for an average eye ( offset = ACDconst - 3.336).
Holladay 1 IOL calculation formula was introduced in the late eighties as a third generation formula. Like SRK T, the improvement was about better prediction of the post operative ACD or lens position after surgery. To understand the lens position from corneal vertex, that is the location of optical plane of IOL with regard to corneal vertex, it relied on axial length and corneal curvature ( a steeper cornea will have a higher distance between the corneal vertex and the optical plane of IOL). To determine postoperative ACD, Holladay introduced his constant - Surgeon Factor (SF) which is the distance from the iris plane to the optical plane of IOL. Thus the post operative effective position of IOL is determined by not only the corneal curvautre and axial length, but also by the distance from iris plane to the optical plane of IOL.
The Surgeon Factor is determined by many factors - style/design of lens, capsulorhexis, wound closure, technician, etc. Thus it is best to optimize the Surgeon Factor based on individual surgeon's outcomes.
Deriving IOL effective power (IOLc) with vergence based formulas from Gaussian optics (Holladay I, SRK T, etc):
There are six variables in theoretical vergence based formulas
Determining optical corneal power (Ko)
Conversion of ultrasonic axial length to optical axial length described before (AL0)
IOL effective power determination (IOLc)
Effectie Lens Position or (previously ACD with second generation theoretical formulas)
Desired (targeted) post operative refraction (DPsot Rx)
Vertex distance of desired refraction
We want to find the appropriate IOL power (IOLc), therefore the equation as per Gaussian optics and standard vergence formula is
- (2)
That is, IOL power = vergence after the IOL - vergence in front of the IOL
Here 1336 is the refractive index of acqueous and vitreous (1.336 multiplied by 1000), 1000 is refractive index of air ( 1.0 multiplied by 1000 times), and the rest is explained above as per the six variables of the equation.
Determining the optical power of the Cornea from keratometry measurement:
All Keratometers measure only the front radius curvature of cornea (in mm) and then convert it to dioptic values with the Gaussian formula K=n-1/r (thin lens formula for determining Cornea as a single refracting surface) , where n is the Keratometry Index of cornea that the specific Keratometer use and 1 is the refractive index of air. This is explained in the article The Science Behind Keratometers For Biometry ( https://www.quickguide.org/post/the-science-behind-keratometers-for-biometry) To convert radius of curvature (r) into dioptric value usually most keratometers use 1.3375 as the Keratometry Index. The reason behind it is historical. The manufacturers of Keratometers were also the manufacturers of spectacle lenses. In spectacle lens industry, it was a common practice to label lens power as their back vertex power than paraxial power of secondary principle plane. This continued in the IOL industry too. In order to label the corneal power with keratometers they used the back vertex power and 1.3375 seemed to them more appropriate matching with Gullstrand model eye of front radius of curvature of cornea as 7.7.
The actual refractive index of cornea is however 1.376. Corneal power is based on the curvature of radius of anterior surface of the cornea since the curvature of posterior cornea is difficult to measure. Setting 7.8 mm as the value of the curvature of radius of anterior cornea and 1.3375 as the refractive index, the curvature radius of the back surface of the cornea will be 8.11 mm. It is well known that the radius of curvature of anterior cornea is higher than the posterior cornea. Therefore, different formulas adopt different refractive index of cornea further to rectify this problem (11).
The dioptric power of cornea (K) derived by Keratometer is converted by the IOL formulas into Koptical ( Ko ), that is power of cornea at the second principle plane. But in converting the same different formulas use different conversion factor or refractive index of cornea. Olsen & Haigis choose 1.3315(4), while Binkhorst chose 4/3 or 1.3333 which is also the value used in Holladay and SRK T (9) formulas. Olsern uses 1.3315 as he finds this more appropriate using Gullstrand ratio of model eyes as 7.7/6.8.
Holladay uses the value of 4/3 or 1.3333. If this is used then the formula for converting K to K0 is
- (2)
Next comes determining ELPo of effective lens position.
In the equation for determining effective IOLpower with vergence formula we have covered the components of determining optical axial length(ALo) and optical corneal power (Ko). Now we have to determine the ELPo. This can be done by reversing the formula for finding effective IOL power (IOLc). Finding ELP0 is possible when we have stabilized actual post operative refraction and the actual IOL power implanted. Holladay published his reverse solution for ELP0 with a detailed formula. In the interest of simplicity I am avoiding it in quickguideorg. But what is important to understand is that using the complex calculation in more than 2000 eyes he came up with a simple regression formula for determing ELP0.
Taking this equation forward Holladay found a relationship between Surgeon Factor (SF, the constant used in Holladay I) with ELPo.
Thus for a normal or average eye with average axial length and keratometry reading, the above relationship between the ELPo and Aconstant or surgeon factor (SF) is true.
A paper published by Holladay in 1996, however has slowly revolutionized the concept of lens power calculation. Holladay in a series of pioneering work, showed the world that some of the basic assumptions that were made earlier, were not scientific. Notable of this is that, the theoretical second and third generation formulas assume that a higher axial length would be associated with a higher anterior chamber depth, and hence the Effective Lens Position (ELP) further away from the cornea. This is not always true. To avoid this wrong assumption the Holladay II (Holladay IOL Consultant Software) incorporated seven variables - all in the interest of understanding the ELP more precisely. Holladay's famous nine types of eyes that relate to biometry and IOL power and lens position postoperatively is given below.
Improving accuracy of measurements
It is important that care is taken not to blindly follow measurement data from machines but to validate the biometry data. The validation guidelines are
Always do a two eye biometry measurement
Check both eye axial length difference less than .3 mm
If axial length is less than 22 mm or more than 25 mm, re measure
Rule out posterior staphyloma in case of long eyes. Do a B-scan
Re measure if corneal power is below 40 D or more than 47 D
Rule out dry eyes or irregularity of cornea if both eye astigmatism difference is more than 1 diopter.
If two eye IOL power difference is more than i diopter for emmetropia, do check measurements
The last five years has been very exciting in the ophthalmic world. We have seen the introduction of a host of new age laser biometry, topographers, tomography. The world of IOL power calculation has not failed us to live up to the expectation of the patient, and the many scientific discoveries. The Olsen formula, Barrett Universal II, the papers published by Wang & Koch for high myopia patients, the Hill RBF, centroid SIA, are all revolutionary concepts. The efficacy of the Barrett formula lies in this, that it takes into account the changes in the principles plane of refraction of an IOL, as you move from the higher to the lower diopter of the manufacturer.
ELP/ post surgery lens position prediction by modern formulas
A study by Sverker Norrby(5), points that estimation of post operative lens position remains the most important causative factor for refractive surprises. While two variable second and third generation formulas depended on axial length and corneal height to predict post operative lens position, newer generation formulas additionally take into account pre operative ACD and lens thickness like Olsen or Barrett Universal II formulas. In as study by Olsen in 2006, pre operative ACD remains second in importance after axial length to predict the ELP or post operative lens position. If you have a refractive surprise in the patient, it is important you do a psuedophakic measurement to determine the post operative lens position. You can use my own regression based calculator in the CALCULATOR section the uses pre ACD and optional lens thickness to help you match predicted ACD with actual pseudophakic ACD - Predicted post operative ACD calculator to match ELP (https://www.quickguide.org/post/acd-calculator). Remeber the thumb rule - in an average eye, a .1 mm shift in lens position can lead to .2 dioptre shift in refractive outcome.
Classification of Formulas:
Let us now classify these formulas into order so that we can be clear on our approach for IOL power calculation.
Despite the many improvements that we have seen in Biometry in the last few years, achieving the target refraction is still eluding in many a practice. This may be because, achieving the target refraction and a smiling face is not just a factor of science, but an art - a combination of surgical skill and biological factors of the patient post operatively. Notable among these variables are the rhexis and the capsular contraction of the individual patient. While we perfect the science, the art remains an art. Let us hope for the best art.
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Update: The ESCRS has lauched a free biometry online tool wherein they have integrated all the major new age calculators in one webpage. In a single webpage you can now enter patient data once and compare IOL power for emmetropia across seven calculators. I would encourage you to visit this page https://iolcalculator.escrs.org/ and make best use of the services. Here is a screenshot of the webpage to give you an idea
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Reference:
1. Fedorov SN, Kolinko AI, Kolinko AI: Estimation of optical power of the intraocular lens. Vestnk Oftalmol80(4):27-31, 1967
2. Standardizing constants for ultrasonic biometry....Jack Holladay, JCRS Vol 23, Nov 1997
3. A three-part system for refining intraocular lens power calculations. Jack Holladay, et al, J Cataract Refract Surg 1988 Jan;14(1):17-24. doi: 10.1016/s0886-3350(88)80059-2.
4. Calculation of Intra Ocular Lens Power: A review ; Acta Ophthalmol. Scand. 2007: 85: 472–485
5.Sources of error in intraocular lens power calculation Sverker Norrby,J Cataract Refract Surg
. 2008 Mar;34(3):368-76
6. Pursuing perfection in intraocular lens calculations: I. Logical approach for classifying IOL calculation formulas ; JCRS Volume 43 Issue 6 June 2017
7. Total Corneal Power Estimation, Li Wang et al, Investigative Ophthalmology & Visual Science March 2011, Vol.52, 1716-1722.
8. Unfortunate Discrepencies: Sverker Norrby Journal of Cataract & Refractive Surgery 24(4):p 433-434, April 1998. | DOI: 10.1016/S0886-3350(98)80276-9
9. Development of the SRK/T intraocular lens implant power calculation formula; Journal of Cataract & Refractive Surgery 16(3):p 333-340, May 1990
11. Chapter 24 Intraocular Lens Power Calculations for High Myopia, Renyuan Chu, Jinhui Dai, IOL POWER CALCULATIONS, Second Edition, JAYPEE, Editors Ashok Garg, JT LIN, Robert Latkany, Jerome Bovet, Wolffang Haigis