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Understanding The Haigis Formula.

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.

The  Haigis Formula for IOL Calculations
Dr. Wolfgang Haigis

One of the final frontiers in ophthalmology is the consistently accurate calculation of intraocular lens 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.

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"

Hoffer Q formula — uses "Anterior Chamber Depth"

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 ...

    d = a0 + (a1 * ACD) + (a2 * AL)
    ACD is the measured anterior chamber depth of the eye (corneal vertex to the anterior lens capsule), and ...
    AL is the axial length of the eye; the distance from the cornea vertex, to the vitreoretinal interface.

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.
  * The a1 constant is tied to the measured anterior chamber depth.
  * 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.

Cataract Surgery Arizona
Len #1 Len #2
a0 = -1.441 a0 = 1.274
a1 = 0.064 a1 = 0.189
a2 = 0.261 a2 = 0.128

Let's look at three patients:
Cataract Surgery Arizona
Patient 1 Patient 2 Patient 3
AL = 28.25 mm AL = 23.45 mm AL = 21.25 mm
ACD = 3.45 mm ACD = 3.25 mm ACD = 2.75 mm

Plugging into our little formula, we get for "d":
Cataract Surgery Arizona
  Patient 1 Patient 2 Patient 3
Lens #1 6.15 4.89 4.28
Lens #2 5.54 4.89 4.51
Cataract Surgery Arizona

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.

For a free Excel spreadsheet you can use to derive your own set of a0, a1 and a2, follow this link: Haigis Formula IOL Constants. Be sure to download the instructions for submitting this data to Dr. Hill in North America or Dr. Haigis in Europe.


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