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# Novel Polynomial Decomposition: Clinical Examples

## CLINICAL EXAMPLES: TAKING OUR THEORY TO YOUR CLINICAL PRACTICE!

### To aid understanding of the application of a novel wavefront decomposition method and polynomials(+read the JOSA A paper), we have created further clinical examples including clinical scenarios you will come across in your practice.

Conversion Zernike to LD HD (Low Degre / High Degree): proper analytical methods can be implemented to convert a Zernike coefficient expansion into a new LD-HD coefficient expansion.

### Current wavefront sensors (read more about wavefront sensing) express the wavefront as a sum of Zernike weighted coefficients (read more about Zernike polynomials). From these, one can calculate the values of the GM coefficients in the new basis called « LD-HD » for Low Degree / High Degree. Such calculation requires the rearrangement or reorganization within and between the low vs high degree terms which are parceled or combined in the high order modes.

fl (for f low) is obtained after collecting all the low order terms (those present in the low order modes, and those which contaminate the HO modes. Subtracting fl to the total wavefront f enables to obtain the « pure » HO WF component (fh). This pure HO component, called fh, is decomposed using the new series of orthogonal modes.

## EXAMPLE 1

### The eye is emmetropic in photopic conditions (20/15 uncorrected visual acuity) but perceives large halos in mesopic conditions.

The vergence map displays the local dioptric power variations throughout the pupil area (for mesopic conditions).

### The following figure displays the Zernike vs New LD/HD wavefront coefficients:

Comparison between the Zernike vs LD/HD (GM for Gatinel-Malet) coefficients

### A vectorial depiction of the wavefront decomposition between the Zernike vs LD-HD methods is shown:

Vectorial depiction of total Wavefront decomposition : Zld : sum of Zernike low order, Zhd: sum of Zernike high order, Gld: sum of Gatinel-Malet low order, Ghd: sum of Gatinel-Malet high order. The Zernike reconstruction tends to underestimate the role of the high order within the total wavefront error.

### The comparison between the refraction and retinal image predicted by the two wavefront methods for the best spectacle correction is shown here:

The comparison between the refraction and retinal image predicted by the two wavefront methods (Z: Zernike, GM: Gatinel Malet)

## EXAMPLE 2:

### The Zernike predicted refraction is biased by the defocus coefficients which correspond to artefactual compensation for low order terms embedded in the Z40 mode:

Refraction prediction discrepancy between wavefront decomposition methods: Zernike vs LD/HD with Gatinel Malet (GM) coefficients.

### The comparison between the wavefront coefficients and their predicted metrics is instructive:

Wavefront coefficients comparison and derived metrics between Zernike and LD HD methods.

### The subjective refraction is +0.50 (-5 x 75°) and the best corrected visual acuity is 20/20.

Keratoconus: discrepancy between the Zernike vs LD/HD refraction and derived metrics.

### Here is the comparison between the wavefront coefficients charts:

Coefficients comparison between Zernike and LDHD coefficients wavefront decomposition methods.

## RELEVANCE TO Q-VALUE AND ASPHERICAL/CUSTOMIZED PHOTOABLATIONS:

### The first graph allows to compare the variations of the defocus coefficients (Δz20 vs Δg20):

Comparison of the theoretical variations in the defocus coefficients (Δz20 vs Δg20) after the deliverance of a custom-Q hyperopic correction (+3 D) for various target asphericities (toward increased prolateness). Note the stability of the g20 defocus coefficient which is, as opposed to the z20 defocus coefficient, robust to the variations of the corneal asphericity.

### The second graph allows to compare the variations of the spherical aberration coefficients:

Variations of the spherical aberration coefficient after customization of the asphericity in hyperopic correction : Δz40 vs Δg40.

## CONSEQUENCES:

### In this example, the refractive correction is +2.50D, the initial asphericity is Q=0, the target asphericity is also 0 (no intended change in corneal asphericity). This correction incurs an expected variation in the defocus (noted c4 in a single index scheme) and spherical aberration (noted c12 in a single index scheme) coefficients.

Effect on the Zernike coefficients of a customized aspheric correction of +2.50 D and no intended change in the corneal asphericity.

### Now, let us examine the effect of the change in target asphericity (target Q is now -0.6). The refractive correction is unchanged. It results in an increase in the expected change in Zernike spherical aberration (as expected) but also a large change in the Zernike defocus (which is NOT clinically expected as no change in the planned refraction correction was made).

The modification of the corneal asphericity (toward increased prolateness) results in a large variation of the Zernike defocus coefficient, although the intended defocus correction is unchanged (+2.50 D).

### – Secondly, because of the presence of a defocus term in the Zernike spherical aberration mode, an adjustment of magnitude -0.89 D (added to the sphere obtained from subjective refraction) must be made to the corrected refractive sphere (see the « Measured SPHERE » in the REFRACTIVE PARAMETERS box).

The list of single scheme index Zernike polynomials corresponds to the predicted change induced by a customized topographic ablation.

### This figure displays a 3-D plot of the ablation profile which aims at correcting higher order aberrations of corneal origin (mainly positive spherical aberration Z40 and trefoil Z3+/-3 modes). Note the central dip, caused by the parasitic defocus term embedded in the Zernike spherical aberration mode, which explains the unwanted induction of some myopic defocus (the overall profile grossly looks like a hyperopic ablation profile, not like a « pure higher order » correction profile which should be paraxially null or constant).

3-D view of the custom higher-order profile of ablation.

## CONCLUSION:

### This may promote a better understanding and adoption of these cutting edge or customized techniques by the surgeons, and avoid unnecessary steps in treatment programming.

#### See other relevant examples and material :

Pre and Post LASIK aberrations

PresbyLASIK

Keratoconus

Wavefront sensing

Zernike polynomials

### 2 réponses à “Novel Polynomial Decomposition: Clinical Examples”

1. Len Zheleznyak dit :

Hi Damien,

This is super interesting work. I look forward to learning more about it. There are some topics in my life that took several introductions before I really understood it. For example, Fourier transforms and physical optics was like that for me as a student. This will also probably take several redundant explanations. But just because something is difficult, does not mean it is unimportant. Quite the contrary. Do you plan to present this at Wavefront Congress or ARVO this year?

Take care,
Len

2. Dr Damien Gatinel dit :

Hi Len, thank you for your encouraging comments. I have presented an introduction to this work last year in Denver at the Wavefront Congress and ARVO (as a paper communication). I have submitted the continuation of this (clinical applications related to refraction prediction) to be presented at 2019 ARVO… I hope to see you there!