*The paper is available here*: https://www.nature.com/articles/s41598-020-65417-y

Optical wavefront sensing using aberrometry is an **objective method** that allows mathematical reconstruction and analysis of lower and higher-order (degree) monochromatic aberrations of the eye. We believe that this objective method had the potential to be the new standard for **optimizing the correction of refractive errors** by converting aberrometry data to accurate sphero-cylindrical refractions.

The output of a typical wavefront analysis is **a list of coefficients**, each weighing a specific aberration mode. It is expected that the taxonomy of the higher-order aberrations is clinically relevant and equipped with interesting mathematical properties.

When a wavefront decomposition is performed, each mode is given a coefficient value. The higher the value of the coefficient, the higher the « weight » or effect of the considered mode on the total wavefront error. Wavefront reconstruction over a circular pupil can be achieved by adding some Zernike modes, each weighted by a coefficient whose value is specific to the wavefront of interest.

Various metrics can be derived from the wavefront coefficients analysis:

– The **low degree** component of the wavefront should correspond to or be highly correlated with the **spectacle correction** of the measured eye.

– The **higher degree** component should predict the **residual optical defects** that persist after correcting that eye with the best spectacles’ correction.

In general, there is a **difference** between the **subjective refraction** and the **objective refraction** predicted by the analysis of the wavefront from the value of the coefficients weighting the low-degree spherical and cylindrical defocus modes. Part of this discrepancy may be related to the influence of **high degree aberrations** on the subjective perception of the effect of a given low degree correction. The study of the interactions between the best subjective refraction and high degree aberrations is interesting for more than one reason: beyond the prediction of subjective refraction, it also aims to improve the **design of personalized refractive corrections** and adjust the component low degree refractive with respect to the possible presence of high degree aberrations. Current analyzes are hampered by the fact that the **Zernike classification** is not suitable for this situation, since certain **so-called high-degree aberration modes contain terms of low degree**.

**Machine learning** is already in use in Ophthalmology for image analysis in medical retina and glaucoma as well as regression tasks, notably in IOL calculations.

Our aim was to build and evaluate a set of **predictive machine learning models** to accurately and precisely objectively refract a patient using wavefront aberrometry with a new aberration decomposition and to evaluate the **relative importance of each aberration** in the prediction process for each vector.

**The accurate distinction** between lower and higher wavefront components is mandatory to accurately predict spectacle refraction (as well as accurate retinal image metrics, which was not investigated in this study).

However, Zernike polynomials have one major drawback for describing the ocular wavefront in an ophthalmic context: these polynomials do not accurately distinguish between Lower and Higher Order Aberrations. **Lower order terms are present within the expression of some Higher Order coefficients **so we can’t accurately predict sphero-cylindrical refraction or accurately define the effects Higher-Order Aberrations on the retinal image viewed.

Thus, the influence of the Zernike spherical aberration Z(4,0) on the spherical cylindrical refraction is essentially linked to the low degree component (defocus) contained in this mode. These characteristics do not allow us to isolate well the specific influence of the high degree component in r4.

The same goes for **high-degree astigmatism Z(4,+/-2)**, which **contains low-degree astigmatism**:

We have developed a solution to these problems based on new polynomial functions to better distinguish between low degree (LD) and high degree (HD) aberrations: the **LD/HD** modes expansion. **The core of this work has recently been published in the Journal of Optical Society of America (JOSA A), and some relevant clinical examples in the Journal of Refractive Surgery. **

We have created **new orthogonal and Normalized modes G(n,m) **to replace the high order Zernike « impure » modes. Their analytical expression no longer contains low order terms (2 or less), and their central portion is flat.

A wavefront error can be decomposed in the LD/HD classification as a **sum of weighted G(n,m) modes**. The g(n,m) coefficients can be **directly computed** from the Zernike coefficients z(n,m) without the need for a new wavefront acquisition.

The low order modes **G(2,0), G(2-2)**, and **G(2,2)** are orthogonal and their respective coefficients can be easily converted in the 3D dioptric vector space components of the refraction **M, J0**, and **J45**.

The remaining high degree component is devoid of low degree terms e.g. defocus and 2nd-degree astigmatism. This allows **unequivocal analysis of the influence of the high-order aberrations on the subjective spectacle low order correction**.

A total of 2890 electronic medical records of patients (6397 eyes) evaluated for refractive surgery at the Laser Vision Institute Noémie de Rothschild (Foundation Adolphe de Rothschild Hospital, Paris) were retrieved and analyzed. For each eye, the following data was available:

– **Non-cycloplegic subjective refractions**: Each refraction in** Sphere S**, **Cylinder C**, and **axis A** format was transformed into 3D dioptric vector space (**M, J0, J45**) where the three components are orthogonal.

– **Wavefront analysis**: obtained using the OPD-Scan III (Nidek, Gamagori, Japan) aberrometer specially configured to run using beta-software incorporating the **new series of LD/HD polynomials** up to the 6^{th} order.

A **machine learning approach using LD/HD polynomials** was developed for predicting the results of conventional, sphero-cylindrical refraction from wavefront aberrations.

**XGBoost** is an implementation of gradient boosted trees focused on performance and computer efficiency. It can perform both regression and classification tasks. It was chosen because of its recognized performances and its resistance to overfitting.

Feature importance analysis was achieved via the calculation of **SHAP** (SHapley Additive exPlanations) values for each model in order to determine the **most influential polynomials**.

Read this paper to learn how the best precision and accuracy were obtained, when all the novel polynomials coefficients were used as predictors, demonstrating the significant influence higher-order aberrations have on spectacle correction.

*I wish to thank the co-authors who have contributed to this work: Guillaume Debellemanière (MD), Radhika Rampat (MD), and my Ph.D. directors: Laurent Dumas (Ph.D.), and Jacques Malet (Ph.D.).*

*I also wish to thank the Nidek company and its R&D team for implementing this new method in a beta-software, now able to process wavefront data obtained with our OPD-Scan III wavefront sensor.*

**Novel wavefront decomposition method**

**Novel wavefront decomposition method: clinical analysis**

Vous y découvrirez comment les **frottements oculaires initient et entretiennent un processus physiopathologique** dont les conséquences physiques et biomoléculaires concourent à l’apparition puis l’évolution du tableau clinique et topographique du kératocône.

Ces données sont corroborées par la mise en évidence de nouveaux facteurs de risques comme la **position de sommeil**, et la remise en question du caractère ectasique de l’atteinte cornéenne! Cette méprise découle d’une interprétation erronée de la topographie cornéenne de courbure.

**Le kératocône n’est pas une ectasie d’origine mystérieuse et d’évolution capricieuse, mais une déformation d’origine primitivement mécanique à surface globalement constante et concomitante d’un amincissement focal de la région paracentrale du mur cornéen.**

L’arrêt de l’évolution du kératocône après simple éviction des facteurs mécaniques locaux valide **la théorie « No rub, no cone »** et ouvre de nouvelles perspectives pour une gestion éclairée et non invasive du kératocône: ces résultats font jaillir l’espoir de sa **prévention chez les sujets à risque**.

Lien vers le résumé de l’article: https://www.ncbi.nlm.nih.gov/pubmed/32050230

Version prépublication de l’article: Complications of cosmetic iris implants. El Chehab H, Gatinel D et coll. Journal of Cataract and Refractive Surgery 2019. pdf

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Objective assessments to quantify crystalline lens density are becoming indispensable for modern cataract surgeons, in particular when assessing early to moderate-stage cataracts, in which case clinical assessment might be less reliable, as well as when documenting cataract progression. Objective measurements are especially relevant given the recent focus on early lenticular changes, which can be visually significant for patients despite an unremarkable clinical examination and visual acuity test results.

We have tried to relate these properties to the **topographic study of diffractive gratings** measured directly on the surface of these IOLs. We hope these results can help clinicians better understand the optical reality of these lenses, which is sometimes blurred by commercial communication and marketing.

Skipping the mathematical details, one can apprehend the diffraction subtleties by learning some relatively simple rules. To illustrate some of the concepts discussed in this article, here are some explanatory diagrams that reflect some of the presented concepts and results. I hope that these illustrations will help the reader to become more familiar with the properties of diffractive IOLs, especially when it comes to chromatism.

Comparative analysis of the diffractive profile of two lenses can be very instructive as the diffractive profile has a strong impact on the optical properties of a given IOL. As an example, this figure shows two different bifocal IOL topologies: **Tecnis bifocal ZMB00** and the **Tecnis Symfony**. Topography measurements were acquired using an optical profilometer (Bruker Contour GTI).

Let’s leave aside the difference in height of the diffractive steps for the moment and concentrate on their **spacing**.

The pitch is inversely proportional to the addition power provided by the diffractive grating. From the depicted harmonic relation between the step widths of the two analyzed IOLs (2 steps on the Tecnis ZMB00 vs one step on the Technis Symfony) one can clearly conclude that the **diffractive structure of the Symfony resembles that of a bifocal IOL**, and that **the addition power of the Symfony is roughly half of that of the ZMB00**.

In a previous publication, we have demonstrated that the Symfony, claimed to be an Extended Depth of Focus (EDoF) lens has a **diffractive structure providing a power addition of +1.75 D in green light**. Again, knowing that the power addition of a diffractive lens is theoretically fixed by the spacing between two consecutive diffractive steps, the number of rings observed on the Symfony is in accordance with the theory, as **9 rings** were been utilized in the case of the EDoF lens, versus **more than 20** for the bifocal Tecnis.

Another striking difference is the **height of the diffractive steps** between these two implants. The Symfony lens has **higher steps**, which are labeled with the term « **echelette** » by the manufacturer a semantic choice which could be motivated by the need to differentiate this diffractive grating from competing multifocal diffractive IOLs.

The height of the steps governs the **amount of light energy distributed** on each focus * for a given wavelength*, and the variation of the refraction index between the IOL and its surrounding medium. The energy associated with a particular focus is linked to the

Diffractive optical elements have **opposite color dispersion characteristics (i.e different longitudinal chromatic aberration – LCA)** than refractive lenses: for a considered focus, the short-wavelength (blue) come into focus **after** the long-wavelength (red).

The IOL material has a higher refractive index than the aqueous humor; hence, any incremental thickness of the IOL causes some **optical path delay **or equivalently**, a shift in the phase of the considered light wave**. For a given wavelength of interest, a phase difference of 2 Pi corresponds to a full-wave difference. If the **height** of a diffractive step is adjusted so that the phase shift is equal to **2 Pi** (one exact wavelength of interest of optical path delay), then all the light energy corresponding to this wavelength will be directed to the **1****st**** order foci**.

However, one should note at this stage that a 2 Pi phase shift scenario **only applies to the wavelength of interest**, which would be presumably of green color (peak of retinal sensitivity) in the context of IOL design. Longer wavelength (between green and the red extremity of the visible spectrum) undergo a phase shift slightly less than 2 Pi and thus have some energy diffracted in the 0th order, in addition to the 1st order. Conversely, shorter wavelengths (between green and the violet extremity of the visible spectrum) have some energy diffracted in the 2nd order of diffraction as they undergo a phase shift of more than 2 Pi.

Diffracting light into one single diffraction order is not very interesting in the case of a multifocality, as the purported goal of the diffractive design is to split incoming light into several foci. As one can guess, when the height of the step does **not** match a 2 Pi phase shift, incoming light will be diffracted in **multiple consecutive diffraction orders**.

When the phase delay introduced by the height of the diffractive step is longer than 2Pi for the wavelength of interest, the main split occurs between the 1st and 2nd orders of diffraction. Such design incurs higher steps than that of a diffractive IOL diffracting light of the design wavelength in the 0th and 1st orders of diffraction.

The **extinction of the 0****th**** order** of diffraction provides the opportunity to design diffractive multifocal lenses for which all the incoming light is diffracted into the 1st and higher orders of diffraction. In these orders, the focal distance of diffracted colors is dictated by diffractive chromatism (nth order diffraction red foci has more power than nth order blue foci), which is opposite to refractive chromatism. This “diffractive chromatism” can be used to compensate for the “refractive chromatism” of the monofocal IOL carrier alone or even the combined refractive chromatism of the cornea and the lens, a claim of the Symfony lens.

The **higher steps of the Symfony bifocal EDoF** lens are indeed conceived to diffract green light in the** 1****st** and **2****nd** order of diffraction (phase shift > 2 Pi for green light). In comparison, the **lower height** of the **Tecnis ZMB00 bifocal diffractive steps** is designed to split green light evenly between the **0****th** and **1****st** orders of diffraction.

These striking difference in the diffractive design explain the differences in the optical behavior and chromatism of the two lenses:

The design of the Symfony implies **a reduction of the base power of the monofocal carrier**, by an amount equal to the addition power of the 1st order of diffraction! The sum of these two powers provides the nominal power of the IOL, for distance vision correction (ex: in a labeled « 22D » power IOL, the monofocal carrier contributes to 20.25D, and the 1st order to 1.75D). The chromatic aberration at this far focus will be dominated by the diffractive nature of the light focusing mechanism, and could compensate for the refractive chromatism brought the cornea.

The improvement of vision quality by LCA reduction has not yet been demonstrated. Although the results were not extrapolated to clinical relevance, we hope that this study still offers the reader a new performance metric to characterize multifocal IOLs and their different foci.

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