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Haigis Formula
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Warren Hill, M.D.
IOL Intraocular Lens Power Calculations
The Haigis Formula
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Dr. Wolfgang Haigis
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Understanding
The Haigis Formula.
One of the final frontiers in ophthalmology is the consistently
accurate calculation of intraocular lens (IOL) power for all eyes.
When properly
"personalized," any of the modern IOL power calculation formulas will
do a good job for
normal axial lengths and normal central corneal powers. However, for very long or short eyes, or for eyes with very flat or very steep corneal powers, consistently accurate IOL power calculation has remained elusive.
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IOL constants and IOL power prediction
The present system of IOL constants works by simply moving the position of an IOL
power prediction curve for the utilized formula up or down.
For each formula, the shape of this power prediction curve is mostly fixed. The larger the IOL constant, the more IOL power each
formula will recommend for the same set of measurements. And the
smaller the IOL constant, the less IOL power the same formula will recommend for the same set of measurements.
It is essential to note that the shape of this curve
remains the same. Other than the lens constant, these formulas treat all IOLs as if they were the exactly same and make similar assumptions for all eyes regardless of individual differences.
In reality, two eyes with the exact same axial length and the same
keratometry may require completely different IOL powers. This is due to
two additional variables: the actual (not assumed) distance of the lens
from the cornea (known as the effective lens position) and the
individual geometry of each lens model. Commonly used
lens constants simply do not take this into account.
These include:
SRK/T formula — uses "A-constant"
Holladay 1 formula — uses "Surgeon Factor"
Holladay 2 formula —
uses "Anterior Chamber Depth" (ACD)
Hoffer Q formula —
uses "Anterior Chamber Depth" (ACD)
These standard IOL constants are mostly interchangeable. Knowing one,
it is possible to calculate another. In this way, surgeons can move
from one formula to another for the same intraocular lens implant. The shape of the power prediction curve generated by each formula remains the same no matter which IOL is being
used.
However, variations in keratometers, ultrasound machine settings and surgical
techniques (such as the creation of the capsulorrhexis) can all have an
impact on the refractive outcome as independent variables. "Personalizing" the lens constant for a given IOL and formula can be used to make global adjustments for a variety of practice-specific variables.
Also consider that 3rd generation 2-variable formulas (SRK/T, Hoffer Q and Holladay 1)
assume that the distance from the principal plane of the cornea to the
thin lens equivalent of the IOL is in part related to the axial
length. That is to say, short eyes will have more shallow anterior chambers and long eyes will always have deeper anterior chambers.
We
now know that this is not necessarily so. In reality, short eyes most
commonly have perfectly normal anterior chamber anatomy in the
pseudophakic state.
What these eyes do have is large lenses. Take out
the lens and the anterior chamber dimensions, 80% of the time, are not
all that different from an eye of normal axial length.
Think about when we do phaco for a patient with a short axial length and prior
angle closure — what does the resultant anatomy look like? It looks just like a normal eye; and that is why all 3rd generation 2-variable formulas have a limited axial length range of accuracy. The Holladay 1, for example, works
well for eyes of normal to moderately long axial lengths, while the Hoffer Q has been reported to work better for
shorter axial lengths.
A recent exception to all of this is the Haigis formula, which here in
North America comes as part of the IOL Master software package.
Rather than moving a fixed formula-specific IOL power prediction
curve up (more IOL power recommended) or down (less IOL power
recommended), the Haigis formula instead uses three constants (a0, a1
and a2) to set both the position and the shape of a power prediction
curve.
d = the effective lens position, where ...
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d = a0 + (a1 * ACD) + (a2 * AL) |
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ACD is the measured anterior chamber depth of the eye (corneal vertex
to the anterior lens capsule), and ... |
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AL is the axial length of the eye; the distance from the cornea vertex,
to the vitreoretinal interface. |
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*
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The a0 constant basically moves the power prediction curve up, or
down, in much the same way that the A-constant, Surgeon Factor, or
ACD does for the Holladay 1, Holladay 2, Hoffer Q and SRK/T formulas. |
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The a1 constant is tied to the measured anterior chamber depth. |
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The a2 constant is tied to the measured axial length. |
In this way, the value for d is determined by a function, rather than a
single number.
The a0, a1 and a2 constants are derived by multi-variable regression
analysis from a large sample of surgeon and IOL-specific outcomes for a
wide range of axial lengths and anterior chamber depths. The resulting a0, a1 and a2 constants are such that they
closely match actual observed results for a specific surgeon and the
individual geometry of an intraocular lens implant. This means that a
portion of the mathematics of the Haigis formula is individually
adjusted for each surgeon/IOL combination. Dr. Wolfgang Haigis gets
high marks for this innovative approach.
The Haigis formula IOL constants will appear different than what we are
normally used to seeing, as they interact with the ACD and the AL.
Recall that 3rd generation 2-variable formula lens constants all
basically represent the same thing, which is an attempt to predict the
distance from the principal plane of the cornea to the thin lens
equivalent of the IOL. In the parley of IOL mathematics, this is known
as "d." The Haigis constants, when viewed all together, also determine
this distance, but calculate it in a new and more flexible manner.
"d" for the five formulas commonly in use are:
SRK/T d = A-constant
Hoffer Q d = pACD
Holladay 1 d = Surgeon Factor
Holladay 2 d = ACD
Haigis d = a0 + (a1 * ACD) + (a2 * AL)
The key to highly accurate IOL power calculations is being able to
correctly predict "d" for any given patient and IOL.
One way is to measure the ACD, lens thickness and axial length, and
then force the formula to make adjustments based on previous
observations from some large research data set. This is probably what
the Holladay 2 formula does, adding or subtracting power from a
Holladay 1-type IOL power prediction based on prior observations of
ACD, AL, LT, Rx, corneal diameter, etc.
The calculation data base for
the Holladay 2 formula is obviously substantial, as the Holladay 2 formula
works exceptionally well. We've used it for eyes as short as 16 mm and
as long as 38 mm. Dr. Holladay deserves high marks for what must have been painstaking research and excellent science.
Another way is to look at actual observed outcomes and adjust "d" for
measured axial lengths and anterior chamber depths. This can be done
by multi-variable regression analysis.
Now we're back to:
d = a0 + (a1 * ACD) + (a2 * AL)
The following example uses two different sets of actual regression analysis derived
Haigis constants for two intraocular lenses with the same SRK/T
A-constant of 118.40.
Lens #1 is a single piece acrylic IOL with a
positive shape factor and lens #2 is a biconvex 3-piece PMMA IOL with
10° per mm of posterior haptic angulation. At first glance (as we're
used to looking at an A-constant, SF, or ACD) these two sets of Haigis
constants look completely different. However, they simply represent a
similar power prediction curves with a slightly different shape that
takes into account the differences in lens geometry between these two
IOLs.
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Len #1
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Len #2
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a0 = -1.441
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a0 = 1.274
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a1 = 0.064
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a1 = 0.189
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a2 = 0.261
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a2 = 0.128
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Let's look at three patients:
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Patient 1
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Patient 2
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Patient 3
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AL = 28.25 mm
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AL = 23.45 mm
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AL = 21.25 mm
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ACD = 3.45 mm
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ACD = 3.25 mm
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ACD = 2.75 mm
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Plugging into our little formula, we get for "d":
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Patient 1
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Patient 2
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Patient 3
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Lens #1
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6.15
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4.89
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4.28
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Lens #2
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5.54
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4.89
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4.51
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What this shows is that in the setting of axial myopia, the Haigis
formula will call for a little more power for Lens #1 than for Lens #2.
For axial emmetropes, both constants will give the same IOL power. And
for axial hyperopes, the Haigis formula will call for a little less
power for Lens #1 than for Lens #2. This illustrates is the fact that by
regression analysis it is possible to embed information regarding
differences in geometry of the two IOLs within the three Haigis formula
lens constants.
All of this gives the Haigis formula a new level of mathematical
flexibility not yet before seen in ophthalmology. As the a0, a1 and a2
Haigis constants for the more commonly used IOLs become established,
and the Haigis formula begins to be included with ultrasound machines,
this formula will understandably gain in popularity.
Dr. Haigis is a PhD, rather than an MD, and the Head of the Biometry
Department at the University of Wurzburg Eye Hospital and the Users Group for Laser Interference Biometry (ULIB). As such, he brings to this exercise the formal training of a mathematician and
physicist, to facilitate our
understanding of the essentially non-linear relationship between IOL power,
ACD, Ks and axial length.
Click here to go to our download page for a free Excel spreadsheet you
can use to derive your own set of a0, a1 and a2 Haigis constants and a
set of instructions for submitting this data to Dr. Hill in North
America or Dr. Haigis in Europe.
As an original innovation, the Haigis formula holds out the promise of a new level of mathematical flexibility for increasing the accuracy
of all IOL power calculations.
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2008-09-13This statement would be wrong, because the formula actually is good but the input data was definitely bad. To prevent this from happening, it is reasonable to go back to the actually measured parameters - namely radii of curvature - because they should be identical irrespective of the instrument used. Once radii are obtained, they can be re-converted into Ks again, this time using 1.3375. Thus, the formulas get what they want (namely Ks from 1.3375 sources) and can process these Ks in whichever way they want. So, to prevent IOL formulas from being miscredited by wrong input data in a world where there is more than one keratometer index, a 2-stage procedure seems reasonable: 1. start out from radii of curvature or convert back from Ks to radii making allowance for the calibration of the source instrument, 2. use the keratometer index (1.3375) which the IOL formula expects to have been used during keratometry. How the IOLMaster handles the K problem The IOLMaster makes use of the above approach. The index of the keratometry source has to be input under 'options-setup-program-keratometer-refractive index' (user manual, page 20; cf Fig.2). In IOLMaster instruments for the US market, the default setting is 1.3375. For other countries, the factory-set value is 1.332. Fig.2: If, in the IOLMaster, Ks are manually entered in diopters, the refractive index set here (under 'options-setup-program-keratometer-refractive index') is used to convert Ks into radii. Each IOL formula in the IOLMaster by itself makes sure internally that the correct conversion is subsequently applied for power calculations. This setting, however, is only relevant if K readings are manually entered in diopters. In case the IOLMaster keratometry is used for IOL calculation, no problems occur. Problems can only show up if 1. Ks are manually entered in diopters, and, at the same time, 2. the index under 'options-setup-program-keratometer-refractive index' has not been choosen properly i.e. according to the index the external keratometer actually uses. To further illustrate the situation: problems will e.g. occur in the following cases: - if a German surgeon has not changed the default index setting (1.332) and enters Ks e.g. from a Javal type keratometer (index=1.3375) - if an American surgeon has not changed the default index setting (1.3375) and enters Ks e.g. from a Gambs keratometer (index=1.332) - if someone has fiddled around with the index setting and Ks are entered in diopters - if Ks are entered from different K sources when no allowance is made for the individual K source indices. No problems will arise - if the German has not changed the default index setting (1.332) and enters Ks from a keratometer with an index of 1.332, or - if the American has not changed the default index setting (1.3375) and enters Ks from a keratometer with an index of 1.3375, or - the IOLMaster keratometry is used, i.e. no Ks are entered manually. How do ultrasound systems handle the K problem ? Ultrasound devices mostly do not distinguish between different 'K modes' but usually assume an index of 1.3375 to hold. To check out the effect of different K sources you may deliberately produce an approximate 0.8 D difference relative to an ultrasound device by proceeding as follows: 1. set the index to 1.332 under 'options-setup-program-keratometer-refractive index' in your IOLMaster, 2. enter Ks manually in diopters, 3. compare results on the IOLMaster and the A-scan. Therefore, for comparison purposes with most A-scan equipment in the United States, in the IOLMaster's setup menu. correct index in your IOLMaster Which formula, which equipment is affected by the K problem ? The described problem will affect all biometry devices, all IOL formulas, and, likewise, all computer programs for IOL calculations if Ks are entered in diopters. If whoever enters Ks originating e.g. from a Haag-Streit keratometerinto any K-accepting IOL program - running in the IOLMaster, in any A-scan, on any computer - he will stand a good chance to receive different IOL power than when he had measured the patient with a Javal type instrument. In the IOLMaster, however, this problem can be overcome as has been discussed in the foregoing. The difficulties described reflect a basic problem between primary and secondary measurement parameters and certainly contribute to reasons confusing comparisons of IOL formula performance.
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