# 17. Impact of Single Constant Optimization on the Precision of IOL Power Calculation

One of the primary factors influencing patient satisfaction after cataract surgery is the accurate determination of intraocular lens (IOL) power to achieve the desired postoperative refraction.

For optimal outcomes, these calculations need to be both **accurate** and **precise**. An accurate method will result in a **mean prediction error**—defined as the mean of the differences between the achieved and predicted refractions—being close to zero. This principle underpins the concept of « **zeroization**, » where adjustments are made to **minimize the mean prediction error** as much as possible.

Precision, in contrast, refers to the reliability or consistency of these calculations and is measured by the **standard deviation (SD)** of the prediction error. A lower SD signifies a more precise method, indicating that most errors are closely grouped around the mean.

These metrics are frequently utilized in clinical studies and research to evaluate the performance of various IOL power calculation formulas and to assess their accuracy and precision in predicting the correct lens power for cataract surgery.

This page is dedicated to the consequences of **formula optimization** performed in the traditional way (zeroing the mean error) on the precision (**SD**) of the prediction error. In real-life clinical practice, many variables contribute to the induction of refractive prediction error, which in turn can explain the variance (the square of the SD) of this error.

To give the takeaway message upfront: adjusting a positional constant (such as the A-constant, SF constant, a0 constant, etc.) improves **both** precision and accuracy when the prediction error is due to an incorrect estimation of the Effective Lens Position (**ELP**) or a mismeasurement of **axial length**—these are known as **positional errors**. In contrast, **optimizing a corneal power estimation error** using a positional constant leads to an **increase in the SD of the error**: even if the mean error is zeroed, the risk is an increase in the dispersion of the error.

## An instructive theoretical scenario

To illustrate this point, we can imagine a **theoretical scenario** where an IOL calculation formula makes** no error** for a series of implants with powers ranging from 0.5D to 35D. Now, let’s assume that the corneal power estimation has been **systematically overestimated by one diopter** in the spectacle plane due to a miscalibration of the biometer’s keratometer. What would be the resulting error?

The formula would predict the same implant position and calculate the IOL power required to achieve the target refraction. During postoperative refraction measurement, a hyperopic error of one diopter would be observed, corresponding to the effective corneal power deficit. The** mean prediction error** would be 1D, and the **standard deviation** (**SD**) of the prediction error would be **zero**.

Next, let’s imagine that **optimization** is performed by adjusting a** positional** constant. This constant would need to modify the predicted **position** of all implants in the** same** way (an increment is added to the position predicted by the formula, it was calculated at +0.86 mm in this example). This increment would then alter the predicted refraction for each eye so that the **recalculated mean error becomes zero**. However, this does not mean that the prediction error is minimized for each eye. Rather, it is the **final** mean that must be zero. Additionally, the axial displacement of an implant produces a refractive change proportional to the **square of its power**.

As a result, the constant adjustment would need to create a **negative** refractive prediction error (to compensate for the positive prediction error) in some eyes within the dataset (in the examples, eyes having received an IOL of 18.5 D or more). Some of the positive prediction errors would persist because it is not possible to adjust the refraction for eyes with low-power implants significantly. In the end, while the mean prediction error is zero and the formula appears accurate, it becomes **much less precise as the SD of the error increases from 0**.

This example is instructive but should not be overgeneralized regarding the mechanism of optimization. First, the dataset used in this example is not representative of the typical distribution of implant powers, which is generally centered around 21D. Second, the sources of prediction error in IOL calculation formulas are multifactorial.

This example highlights that** adjusting a positional constant cannot compensate for a portion of the prediction error (related to an incorrect estimation of corneal power) without degrading the precision of the power/post-operative refraction estimation, leading to an increase in the SD of the prediction error**.

The following figure shows the result of a similar experiment, conducted this time for a smaller error (0.50D), but with the addition of the prediction error representation for a series with a** typical distribution of implant power** (on the right).

These insights may help improve the optimization strategies for IOL power calculation formulas by highlighting the impact of the type of error being corrected and the appropriateness of the constant being adjusted.

The remainder of this page is dedicated to exploring these concepts in greater depth.

## Accuracy vs. Precision

In the context of intraocular lens (IOL) power calculation, **accuracy** refers to how close the predicted refractive outcome (the target refraction) is to the actual result after surgery. An accurate prediction means the average prediction error is near zero, indicating that, on average, the calculations are correct, but individual predictions may still vary.

**Precision**, on the other hand, relates to the **consistency** of the predictions. A precise IOL power calculation formula will have a** small spread** or variability in the prediction errors, meaning the results are clustered closely together, even if they are not centered on the target refraction (i.e., they may consistently miss the mark by a similar amount).

It is not clinically relevant for an implant power prediction formula to introduce a consistent positive or negative average error. A **systematic bias**, either positive or negative, can arise under various circumstances, such as a change in the implant model or alterations in the methods used to collect biometric data or estimate corneal power.

It is important to note that biometric calculation formulas have specific methods for predicting the effective lens position and estimating corneal power. For instance, the Holladay, Haigis, and Hoffer formulas use a **specific** keratometric index. These formulas partially compensate for discrepancies through their constants.

The **Mean Bias Error (MBE)** characterizes a formula’s accuracy, as it reflects how close the average prediction is to the actual refractive outcome. Zeroization improves the formula’s accuracy by reducing this bias. On the other hand, the **Standard Deviation (SD)** represents the formula’s precision, indicating the spread around the mean of refractive prediction errors. Precision, unlike accuracy, does not typically undergo any specific reduction procedure in clinical practice (learn more about these metrics).

However, we have proposed to change this paradigm with a new method intended to « optimize optimization« , and what follows justifies this approach.

This page is derived from a recently published article. We aimed to **evaluate the impact of zeroization on the standard deviation (SD) value**. To the best of our knowledge, no prior research had specifically examined the effects of reducing prediction errors resulting from systematic inaccuracies in various biometric parameters.

## Optimization of IOL Power Formulas: General Concepts

Let’s begin by reviewing some general concepts about the standard optimization procedure.

### Optimization: Historical Overview

The **SRK** formula introduced the concept of an adjustment constant. This formula is a regression-based, empirical formula:

IOL Power=**A constant**−2.5×AL−0.9×Km

Where **AL** represents the axial length (in mm) and **Km** is the average keratometry (in diopters). The mix of units highlights the empirical nature of this formula. The **A-constant** allows for a linear adjustment of the implant power to optimize the formula: an increase of one unit in the A constant corresponds to a 1 **diopter** (D) increase in the predicted **power**.

Thus, any variation in the A-constant has a** near-linear** effect on postoperative refraction, with a factor of 0.7, meaning that a 1D change in IOL power results in a 0.7D change in refractive outcome. For example, if an average prediction error of +0.50D is measured, it would be necessary to increase the A constant by approximately 0.50/0.7, or 0.71 units.

### Evolution with SRK/T Formula

During the development of their theoretical formula (**SRK/T**), the authors aimed to retain the concept of the A-constant. Still, they applied it in the context of an optically-based formula. In this new framework, adjustments were no longer made by varying the implant’s power but rather by its **predicted position (Effective Lens Position or ELP)**. The empirical A-constant values were typically around 118 for posterior chamber lenses, and a linear adjustment using a new constant (**ACD-constant**) was needed to convert variations in the empirical A constant into positional increments.

### Optimization of Theoretical Formulas

Most theoretical formulas developed alongside the SRK/T formula rely on an optical model in which an implant is placed at a predicted position from which the IOL power is calculated. These formulas possess a constant designed to optimize their performance by zeroing the mean prediction error, effectively eliminating the **Mean Bias Error (MBE)**. This optimization ensures that the formula provides an accurate refractive outcome by compensating for any systematic deviations in the prediction.

#### -Hoffer Q Formula: pACD (personalized Anterior Chamber Depth):

This is the primary constant used in the Hoffer Q formula. It represents an estimate of the effective lens position (ELP) and is typically personalized based on surgeon or lens manufacturer data. The pACD is specific to the formula and adjusts for the expected postoperative position of the intraocular lens (IOL).

#### -Holladay 1 Formula: Surgeon Factor (SF):

The primary constant for the Holladay 1 formula. The SF is used to adjust for the predicted effective lens position (ELP) based on the surgeon’s technique, IOL model, and other biometric factors.

**-Olsen formula: ** **C constant**:

This is the primary constant in the Olsen formula. It is used to estimate the Effective Lens Position (ELP), which is a critical factor in calculating the IOL power.

–**Barrett Universal II formula: ****Lens Factor (LF)**:

This is the primary constant used in the Barrett formula, analogous to the A constant in other formulas. The **Lens Factor** takes into account the specific IOL design to estimate the Effective Lens Position (ELP) more accurately.

#### -Haigis Formula: a0, a1; a2

**• a0**: This constant adjusts the overall IOL power prediction, serving as a general offset for the formula.

**• a1**: This constant is related to the axial length (AL) of the eye and adjusts the prediction based on the eye’s length.

**• a2**: This constant is related to the anterior chamber depth (ACD) and adjusts the prediction based on the depth of the anterior chamber.

**Single-Optimized Haigis:**

In this approach,** only a0** is optimized, often using a lens constant provided by the manufacturer or based on the surgeon’s experience.

**Triple-Optimized Haigis:**

In this approach, all three constants—a0, a1, and a2—are optimized. This provides a more customized and potentially more accurate prediction, as it takes into account the specific biometric characteristics of the patient’s eye. In the triple-optimized version, a1 and a2 are also adjusted, which can be likened to retraining the formula. Other formulas likely use multiplicative constants for preoperative biometric parameters as well, but they do not provide the user with the ability to adjust these constants.

The** PEARL DGS formula** is a theoretical thick-lens formula that uses Artificial Intelligence to predict certain variables, such as the effective lens position (ELP). It utilizes the A constant to infer some data about the design of the considered implant but optimizes by incrementally adjusting the predicted position of the implant.

**Altering the value of a constant in a theoretical formula essentially means adjusting the predicted position of the implant rather than directly changing its power. This distinction is important because, in theoretical formulas, the constant is primarily used to refine the estimated Effective Lens Position (ELP). By modifying the constant, you are influencing where the formula predicts the IOL will sit within the eye, which in turn affects the final refractive outcome, but not by directly altering the IOL’s calculated power.**

## Consequences of Optimization on the SD

Holladay et al. provided compelling evidence that the standard deviation (SD) of the prediction error is the most accurate measure of surgical outcomes. This metric offers a reliable evaluation and can predict other indicators, such as the percentage of cases within a specific range (e.g., ±0.50), the mean absolute deviation, and the median. The SD is calculated **after** optimizing the constant to eliminate any systematic bias. The SD of the error is determined by the distance of each data point** from** the mean rather than the mean itself. As a result, the effect of « zeroing » the mean prediction error on the SD cannot be anticipated without further details since individual errors, not mean errors, influence the SD.

Our study will now focus on how specific factors can individually influence the changes in SD following the zeroization of predicted errors, providing new insights into the interplay between these biometric inaccuracies and overall formula precision.

In previous work, we introduced a method that facilitates the straightforward calculation of the optimal lens constant, which corrects for systematic biases in formula predictions.

Building on this work, we developed an analytical approach to identify the relationships and predict the effects of systematic errors in keratometry, axial length (AL) measurement, the estimation of the keratometric index, and the prediction of the effective lens position.

### Impact of Biometric Variable Alteration on Prediction Accuracy and Standard Deviation

In a recent study, we have conducted a series of theoretical simulations to evaluate the predictive accuracy of an IOL power calculation formula (Haigis single-optimized), which has previously shown a nearly zero prediction error (<0.05 diopters [D]) in a selected group of eyes. Before altering the value of one of the studied parameters, the formula used performs perfectly for all eyes in the dataset.

Our goal is to re-run these predictions after deliberately altering **one** of the input variables by a consistent positive or negative amount. This is meant to simulate a scenario where a new data acquisition method for this variable is introduced, such as using a different measurement instrument that might produce slightly different values from the same eyes.

For example, we might uniformly adjust the anterior corneal curvature radius by adding or subtracting a specific value from all radii.

By doing so, the formula will predict a **new postoperative refractive outcome**, which will deviate from the original prediction. This deviation represents the theoretical prediction error caused by inaccuracies in the modified biometric variable. Following this, we examine the effect of adjusting the optimization constant that compensates for the mean prediction error and** analyze its impact on the standard deviation (SD)** of this error.

### Analytical Method

The method used to study the impact of zeroization on the standard deviation is **relatively complex** for those who are not familiar with certain analytical and statistical concepts. However, beyond these theoretical developments, examples will be provided to illustrate and support the results of the formulas presented below (The method presented was published here).

Once this equation is generated, we can expand the term on the right to obtain a more interpretable expression:

The generated equations **split** the expected SD value after zeroing the mean error into three terms (A, B, and C). Terms A and B always have a positive value, with B proportional to the mean prediction error. Only term C (which can take negative values) has the potential to **reduce** the SD value.

It is not easy to predict in advance the impact of an error source on the sign of C. Practical investigations are preferable to study the impact of zeroing on compensating for measurement or estimation errors in biometric variables.

### Example 1: Prediction Error caused by Effective Lens Position (ELP)

Let’s begin by examining the impact of zeroing an error caused solely by an** incorrect** prediction of the Effective Lens Position (ELP). In this theoretical experiment, we start with a formula (Haigis single optimized) that performs perfectly for all the eyes in a dataset, with a prediction error of less than 0.05 D for each eye. As a result, the mean error is also below 0.05 D.

In this context, we know the power of the IOLs that were placed and the postoperative prediction error, which was measured to be less than 0.05 D in all cases. To calculate the error induced by an incorrect estimation of the ELP, we increment the a0 constant that helps predict the ELP value (which had previously ensured the desired refraction in these eyes without prediction errors) and **calculate the error induced for each eye by the displacement of the IOL**. This is not exactly equivalent to the real-life situation where the formula predicts a power intended to achieve the desired target refraction. However, the induced error is transferable since the equations are reversible.

So, we can suppose that the formula is applied to calculate the power of an implant whose geometry is slightly different (the optic plane is shifted by a certain amount, the same for all implants, regardless of their power). This introduces an error that will likely be proportional to the implant’s power. We can explore the impact of zeroing this error on the evolution of the variance (the square of the standard deviation, SD) in two ways: by using the classic formula that gives the value of variance (the mean of the sum of the squares of deviations from the mean), and by using our formula, which is based on the relationship between the implant’s displacement and the induced refractive error. This formula breaks the postoperative variance into three distinct terms.

#### Visual Representation of ELP Prediction Error

We begin by graphically representing the consequences of a prediction error in the ELP (the same error applied to all eyes in the dataset).

When the formula underestimates the ELP (i.e., the predicted position is smaller than the actual observed position), the predicted power will be weaker, leading to a positive mean refractive error. Conversely, when the formula overestimates the ELP (i.e., the predicted position is larger than the actual observed position), the predicted power will be stronger, resulting in a negative prediction error (myopic surprise).

#### Consequences of Zeroing on the SD

What is the impact of zeroing the mean error on the SD? For each value of the ELP error, an adjustment of the constant a0 is made to zero out the formula, and the value of the SD of the prediction error is calculated. This SD is then estimated using the formula based on the variance calculation through the sum of the terms (A + B + C).

We can calculate this in two ways (predicted vs. measured) and explain it through the analysis of the variations of the A, B, and C terms in our formula:

When the error in the ELP value increases, the values of terms A and B increase exponentially, as expected, while term C **decreases** exponentially. As a result, the SD is minimized (very close to zero) when the mean-induced prediction error is zeroed.

Is this surprising? Not really, because a positional adjustment** perfectly and appropriately** corrects the initial error (the ELP prediction error). Thus, we return to the ideal situation, where no residual error remains.

#### Covariance Analysis: Ei and (P² + 2KP)

This conclusion is corroborated by studying the** covariance** between Ei (the individual refractive error) and (P² + 2KP), where P represents the IOL power and K is the keratometry. If we plot a graph of the prediction errors induced by an underestimated ELP (which leads to a negative mean prediction error), we notice that the larger the value of (P² + 2KP), the more negative the induced error (indicating a** strong, negative** covariance).

The product of the negative mean error and the negative covariance is positive, and thus, the sign of C is negative.

Had we plotted the graph for an overestimated ELP, the mean error would be positive, and the correlation (covariance) between Ei and (P² + 2KP) would also be positive, again ensuring a negative value for C.

In this example, we demonstrate that **zeroing the mean error caused by an incorrect ELP prediction minimizes the standard deviation**. This is expected because the zeroing is achieved by adjusting the effective lens position, which essentially corrects the initial positioning error.

We can address another potential source of error: the estimation of corneal power. Again, we will start with a « perfect » Haigis formula and then simulate the impact of **modifying the keratometric index value** used for corneal power measurement. For each variation, we will evaluate the induced mean prediction error (MBE), the SD of this error, and then analyze its **evolution during optimization** by adjusting the positional constant a0.

### Example 2: Systematic Keratometric Index Value Increment

The value of the keratometric index used for estimating corneal power varies depending on the formula used. A biometer equipped with keratometer measures not the corneal power itself but the average apical radius of curvature (in mm). This value is then converted into keratometric power (in diopters) using a keratometric index (typically nk=1.3375). When an IOL calculation formula uses a different value (e.g., nk=1.3315 for the Haigis formula), the corneal power is estimated based on this index and the average curvature radius.

In all cases, these keratometric index values provide an approximation based on an **assumed constant** ratio between the anterior and posterior corneal surfaces (we have published the formula linking these parameters). Simulating the error caused by a deviation in the keratometric index from an ideal situation allows us to assess the impact of a systematic average deviation on the accuracy and precision of a biometric calculation formula.

The value of nk was adjusted in increments that ranged from −0.005 to +0.005, with each step involving a change of ±0.001:

We observe that an overestimation of the keratometric index value results in a positive prediction error (hyperopia), which is expected, as the formula underestimates the IOL power needed to achieve the target refraction. The standard deviation (SD) of the error appears to increase exponentially; however, when examining the actual SD values, they **remain low and relatively constant** (unlike what was observed in the previous example).

This is explained by the induction of a relatively consistent refractive variation across all eyes in the dataset, as the impact of a change in the keratometric index produces a refractive shift that is relatively insensitive to the value of the average apical curvature radius.

#### Analysis of the Consequences of Zeroing the Mean Error

Let’s now examine the consequences of zeroing the mean error (this adjustment is made for each increment of the keratometric index, and the SD of the error distribution, now centered on zero, is calculated):

We observe that zeroing, achieved through positional adjustment, leads to an** exponentially increasing** SD. This result is striking and warrants further analysis.

Even though the value of term C is negative, the magnitude of C is too small to offset the increase in terms A and B observed during the zeroing process.

The study of the covariance between the induced prediction error and (P² + 2KP) explains these small values of C:

Looking at the graph, we observe that there is no clear relationship or pattern between Ei and P² + 2KP, indicating that the variables are not correlated. This lack of visible correlation suggests that the covariance between them is likely low, as covariance measures the degree to which two variables move together. In this case, the scattered nature of the data points implies that changes in P² + 2KP have little to no consistent impact on Ei, supporting the presumption of a low covariance. This low correlation is expected because a corneal power error related to an inappropriate keratometric index value induces a **relatively constant prediction error**, regardless of the power of the implant used. This can be easily understood if we imagine the refractive consequences of placing a +1 D contact lens (which modifies the corneal diopter power): it induces a constant change in refraction, regardless of the eye concerned. In other words, the correlation between the error in estimating corneal power and the power of the implanted IOL is very low, resulting in a reduced value for term C. To be precise, the effect of the refractive index change is not entirely constant, as its impact varies slightly with the value of the average corneal curvature radius used for calculating corneal vergence. However, this variation is too small to create a strong enough correlation between the induced prediction error and the value of (P² + 2KP).

### Impact of other variables

Equivalent results would be observed with an error in estimating the corneal curvature radius (see the referenced article for more detailed results). This outcome stems from the low correlation between the refractive error induced by the discrepancy in estimated corneal power and the value of (P² + 2KP) in the eyes studied.

In the experiment involving **axial length** (a systematic error leading to a non-zero mean bias error, which is then zeroed by adjusting a positional constant), we would observe that zeroing **neither significantly reduces nor increases the SD** of the prediction error. Using a** positional** constant to adjust the mean error addresses an initial **positional error** (the estimated position of the retinal plane), which has a certain degree of correlation with the implant power (the induced refractive error increases with the power of the implant).

This demonstrates that while zeroing is effective—meaning it improves both accuracy (by eliminating the mean prediction error) and precision (by controlling the standard deviation, SD)—for positional errors where some correlation exists between the prediction error and implant power, it **may not be as effective in cases like corneal power estimation errors**. In such cases, the induced error remains largely independent of implant power, making it difficult to simultaneously achieve both improved accuracy and control over SD through zeroing alone.

These results have led to considering an alternative to the traditional optimization technique, which is presented here.