# The healthy and keratoconic human cornea: structure, imaging, mechanical characterization and modeling

Chloé Giraudet, Jean-Marc Allain

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Chloé Giraudet, Jean-Marc Allain. The healthy and keratoconic human cornea: structure, imaging, mechanical characterization and modeling. Revue ouverte d'ingénierie des systèmes d'information, 2023, 1, pp.1-55. hal-04259713

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# La cornée humaine saine et kératoconique : structure, imagerie, caractérisation mécanique et modélisation

## The healthy and keratoconic human cornea: structure, imaging, mechanical characterization and modeling.

Chloé Giraudet<fnref n="1" />,<fnref n="2" />, Jean-Marc Allain<fnref n="1" />,<fnref n="2" /> (jean-marc.allain@polytechnique.edu)

**RÉSUMÉ.** Cet article vise à faire un état des lieux de nos connaissances de la cornée humaine, sa structure et son comportement, en vue d'une modélisation mécanique. Il présente dans un premier temps l'organisation générale de la cornée, ainsi que ses macro et microstructures. Dans un second temps, nous détaillons les techniques de Tomographie par Cohérence Optique (OCT), qui sont utilisées en clinique et en recherche pour observer la cornée. Ensuite, nous présentons la caractérisation expérimentale des propriétés mécaniques de la cornée. Enfin, nous passons en revue les approches de modélisation qui ont été utilisées pour décrire la réponse élastique du tissu.

**ABSTRACT.** This paper aims to give a review of our knowledge on the human cornea, its structure, and its behavior, for mechanical modeling. It presents first the general organization of the cornea, as well as its macro and microstructures. In a second time, we detail the Optical Coherence Tomography (OCT) techniques, which are used in clinics and in research to observe the cornea. Then, we present the experimental characterization of the cornea mechanical properties. Finally, we review the modeling approaches that have been used to describe the elastic response of the tissue.

**MOTS-CLÉS.** Cornée, microstructure, imagerie, lamelles de collagène, OCT, DIC/DVC, caractérisation mécanique, modélisation mécanique, kératocone

**KEYWORDS.** Cornea, microstructure, imaging, collagen lamellae, OCT, DIC/DVC, mechanical characterization, mechanical modeling, keratoconus

## Introduction

Cornea is the most external tissue of the eye, responsible of our vision through its transparency and its shape. Pathological alteration of the cornea mechanical properties or structure may lead to a loss of vision, and thus have attracted biomechanical studies in the past, with a focus on the role of the collagen microstructure. We propose here a review of the literature on the biomechanics of the cornea. To do so, we will first describe the multiscale structure of cornea (Sec. 1.). Then, we present the main imaging technique, the Optical Coherent Tomography, which is used *in-vivo* or *ex-vivo* (Sec. 2.). Then, we present the mechanical characterizations, also *in-vivo* and *ex-vivo* (Sec. 3.). Finally, we will give some insights of the current models develops to describe the cornea at the scale of the tissue (Sec. 4.). In all section, we illustrate the effect of a pathological alteration through the case of the keratoconus: this disease, which couples mechanical weakening and structural alteration, has attracted the attention of the biomechanical community.

## 1. Structure of the cornea

### 1.1. Anatomy of the cornea

<footnote n="1">¹Laboratoire de Mécanique des Solides, CNRS, École Polytechnique, Institut Polytechnique de Paris,</footnote>

<footnote n="2">²Inria</footnote>
Macroscopic description

The cornea is the external part of the eye (Fig. 1.a-b). It is a transparent medium that transmits light inside the eye and protects it from external aggression. Accounting for 2/3 of the refractive power of the eye (around 40 diopters over 60), the cornea forms with the crystalline lens the ocular diopter, whose function is to focus the light on the retina. The cornea is attached to the eyeball through the sclera, the white tissue surrounding it (Fig. 1.a). Its posterior surface is in contact with the aqueous humor, a transparent liquid composed of 99% water (refractive index $n_{aqh} = 1.3375$ [Sardar et al., 2007]): see Fig. 1.c (anterior and posterior surfaces are the external and internal ones, respectively). Aqueous humor plays a key role in providing nutritional metabolic function to the lens and the cornea, and in regulating intraocular pressure (IOP), which is between 10 and 20 mmHg (1.3 and 2.7 kPa respectively) in healthy conditions [Hashemi, 2005].

<figure><img src="image_2.png" /><figcaption>**FIGURE 1.** a. Schematic view of the eye: the cornea is the external part of the eye, surrounded by the sclera and forming the optical diopter with the crystalline lens – [Created with BioRender.com]. b. Photo of an eye viewed from the front with the classical directions used in ophthalmology: Superior–Inferior in pink and Naso-Temporal in blue. c. Optical Coherence Tomography (OCT) image of a cornea defining the anterior and posterior surfaces (images b and c provided by Pr. Borderie from the 15-20 Hospital, Paris, France). The coordinate system is the same in all the scheme/images.</figcaption></figure>

Geometry

The corneal shape is close to a spherical cap. Typical dimensions of the cornea are around 1 cm in diameter, a thickness between approximately 500 µm in the central part and 600 µm in the periphery, and radii of curvature around 7.8 mm for the anterior surface and 6.5 mm for the posterior surface
[Mashige, 2013]. The geometry of the healthy cornea is regular and well described by a parametric quadratic equation [Gatinel et al., 2011]. Placing the apex of the cornea at the origin of a coordinate system with the z-axis oriented vertically and downwards (Fig. 1.c for the representation of the axes), the anterior and posterior surfaces of the cornea are described by the biconic function [Janunts et al., 2015]:

$$z(x, y, R_x, R_y, Q_x, Q_y) = z_0 + \frac{\frac{x^2}{R_x} + \frac{y^2}{R_y}}{1 + \sqrt{1 - (1+Q_x)\frac{x^2}{R_x^2} - (1+Q_y)\frac{y^2}{R_y^2}}}, \quad (1)$$

where $R_x$ and $R_y$ are the curvature radii of the flattest (x axis) and the steepest (y axis) meridians of the cornea, $Q_x$ and $Q_y$ are the associated asphericities. Note that the x and y directions can be rotated by an angle $\phi$ from the classical naso-temporal (N-T) and inferior-superior (I-S) axes (Fig. 1.b). Finally, $z_0$ is an arbitrary translation with respect to the z axis origin.

Considering this approximation of the geometry of the cornea, the local diopter of the cornea is computed using:

$$D = \frac{n_{aqh} - n_{air}}{R_{ant}} \quad (2)$$

with $n_{aqh} = 1.3375$ and $n_{air} = 1.0000$ the refractive index of the aqueous humor and the air respectively and $R_{ant}$ the local steepest curvature radius of the anterior surface.

## Mesoscopic organization

The cornea is composed of six layers (a histological section is presented Fig. 2) described hereafter from the most external layer to the most internal one.

<figure><img src="image_3.png" /><figcaption>**FIGURE 2.** Schematic view (left – [Schematic created with BioRender.com]) and histological section (right – from [Napoli et al., 2020]) of the cornea.</figcaption></figure>

The epithelium is the external part, composed of a few layers of epithelial cells (50-90 µm thick), and plays the role of a barrier against external aggression and a facilitator for the dispersion of the tears. Bowman's membrane is a non-cellular homogeneous layer (approximately 10 µm thick) composed of
collagen fibrils [Hayashi et al., 2002], which has the particularity to not regenerate after destruction. The stroma is the main part of the cornea and will be described in more detail in the following section (Sec. 1.2.). The fourth layer is Dua's layer, which was only discovered in 2013 [Dua et al., 2013]. It is around 15 µm thick and located between the stroma and Descemet's membrane. Its role is not yet clear. Descemet's membrane is a base membrane of the endothelium – and therefore plays the role of a joint between the stroma and the endothelium – and is mostly composed of a type VIII collagen network [Espana and Birk, 2020]. It grows over life from 3-4 µm thick at childhood to around 12 µm for adults [Bourne, 2003]. Finally, the endothelium is a single layer of endothelial cells approximately 5 µm thick. The endothelial density is between 3000 to 4000 cells/mm² for a new born and decreases down to approximately 2000 cells/mm² at old age [Bourne, 2003]. The endothelial cells do not regenerate so that when one of them dies, the surrounding cells enlarge to fill up the void. The endothelium acts like a pump between the aqueous humor and the cornea to maintain constant hydration – and with it transparency – of the tissue [Mergler and Pleyer, 2007]. It is commonly admitted that below a certain cell density (between 400 and 700 cells/mm² [Kaufman et al., 1998]), the endothelium does not longer fulfill its pumping role and then stromal edemas appear.

## 1.2. Stroma

The stroma represents about 90% of the cornea. It provides the cornea its mechanical strength and optical performance (transparency and optical power). This is due to the particular arrangement of collagen inside the tissue [Meek and Knupp, 2015, Ruberti et al., 2011]: the stroma is composed mainly of type I et V collagen fibrils arranged in lamellae anchored in a matrix of keratocytes (corneal fibroblasts) and proteoglycans [Espana and Birk, 2020]. Hereafter are described the multiple scales of collagen arrangement, from smallest (the triple helix) to largest (lamellae). The largest ones are detailed more precisely (see Fig. 9 for a summary of all scales), as they are consider as more important in the mechanical response.

### 1.2.1. Smallest scales of stromal collagen: the triple helix, the microfibril and the fibril of collagen

The triple helix of collagen (Fig. 3.a) is one of the smallest observable arrangement of collagen monomers, with a diameter of only 1.5 nm. X-ray diffraction is one of the techniques available to observe the triple helix of collagen. It has been first applied on kangaroo tails (Fig. 3.b) and led to a founding article on the organization of collagen at the smallest scale. It allows patterns of the arrangement of the collagen to be distinguished when looking at the different layers (e.g. alternation of black and white on the scattering gives a periodic pattern), but it is necessary to add amino acid composition and physicochemical data to obtain the classical 7/2-helical model (7 different amino acid per 2 turns of helix – Fig. 3.c) found in [Okuyama et al., 2006].

Multiple triple helices of collagen pack to form 4 nm diameter microfibrils of collagen which axial arrangement is represented in Fig. 4.a and c. Electron microscopy (Fig. 4.b from [Marchini et al., 1986]) and tomography (Fig. 4.c [Holmes et al., 2001]) allowed to observe the microfibrils of collagen of bovine cornea (orange boxes). They give access to the microfibril arrangement, which follows an alternation of overlap and gap zones with a period of 670 Å (Fig. 4.c).
<figure><img src="image_4.png" /></figure>

**FIGURE 3.** a. Schematic view of the triple helix of collagen [Created with BioRender.com]. b. X-ray diffraction scattering (from [Okuyama, 2008]) of the triple helix of collagen in a kangaroo tail. c. schematic representation of the monomer of the triple helix (from [Okuyama et al., 2006]).

<figure><img src="image_5.png" /></figure>

**FIGURE 4.** a. Schematic view of a microfibril of collagen [Created with BioRender.com]. b. Electron microscopy (from [Marchini et al., 1986] - 80000X). c. Electron tomography image (from [Holmes et al., 2001] – 20000X) of the microfibrils of collagen of a bovine cornea.
The microfibrils of collagen are themselves packed in fibrils of collagen (Fig. 5) of around 30 nm in diameter. Figure 5.b presents Transmission Electron Microscopy (TEM) images where the fibrils of collagen can be distinguished. TEM images also show that the directions of the fibrils of collagen alternate through the thickness of the cornea – from top to bottom three layers of fibrils of collagen. A keratocyte (corneal fibroblasts) can be distinguished on this image.

<figure><img src="image_6.png" /><figcaption>FIGURE 5. a. Schematic view [Created with BioRender.com] and b. TEM image (from [Komai and Ushikif, 1991]) of the fibrils of collagen of a human cornea. K = keratocyte. Scale bar: 1 µm.</figcaption></figure>

### 1.2.2. Collagen lamellae

Figure 5 shows not only the fibrils but their upper scale organization: they form lamellae, the largest scale of collagen arrangement in the cornea. Collagen lamellae are arranged like a plywood with an alternation of lamellae with different directions (Fig. 6.a). Scanning Electron Microscopy (SEM) and Second Harmonic Generation (SHG) microscopy images (Fig. 6.b and c) reveal the presence of the lamellae (red "L's" on the images). The SEM image also shows that the lamellae are intertwined rather than just superimposed on top of each other, whereas the SHG image points out that the angles between the different lamellae are not exactly 90°.

The scale of the lamellae is generally the one used in mechanical models of cornea, which are inspired from the extensively used models of materials reinforced by fibers (Sec. 4.). This leads to a terminology confusion, with the use of "fibers" and "lamellae" to describe the lamellae. We will try to stick to lamellae, but, from now, if "fiber" is used, it should be understood as collagen "lamellae". The lamellae are heterogeneously distributed in the cornea. To quantify this heterogeneity, X-ray (Fig. 7.a) and SHG (Fig. 7.b) images have been used.

X-ray imaging allows the scattering pattern of the collagen in the lamellae to be obtained, averaged over the thickness of the tissue [Meek and Quantock, 2001, Meek and Boote, 2009]. Fig. 7.a presents the results of X-ray analysis, with a polar plot (butterfly diagram) of the lamella direction at each point of measurement, which gives an in-plane information on the average orientation of the lamella through thickness. The radial extent of the polar plot shows the relative number of collagen lamellae aligned in a
<figure><img src="image_7.png" /><figcaption>FIGURE 6. a. Schematic view of the lamellae of collagen [Created with BioRender.com]. b. SEM image of the cornea (from [Komai and Ushikif, 1991]) – scale bar: 20 µm. c. SHG image of the cornea (from [Benoit et al., 2016]) – scale bar: 100 µm.</figcaption></figure>

particular direction, i.e. the longer the wing of the butterfly, the more lamellae along that direction. The width of the butterfly wings represents their scatter. The colors of the different butterflies indicate the factor with which the polar plots have been scaled down on the representation of Fig. 7.a. Two main observations are drawn from this analysis. First, the lamellae in the center are oriented in the naso-temporal (N-T) and superior-inferior (S-I) directions with an angle of approximately 90°, while in the peripheral zone, the lamellae are mostly oriented in the circumferential direction of polar coordinates. Second, fewer lamellae (in orange, i.e. with no scaling) are oriented in the central area than in the peripheral one (mostly in dark red, black and green so with a scale down factor of 2, 3 or 4 on the representation).

Although X-ray analysis gives valuable information on the in-plane distribution of the lamellae, it does not give any insight into the out-of-plane distribution. To this end, SHG images (from [Winkler et al., 2011]) have been analyzed by Petsche and Pinsky [Petsche and Pinsky, 2013] to determine the out-of-plane dispersion of the collagen lamellae. They have shown that the maximum out-of-plane angle of the collagen lamellae varies with depth – decreasing from approximately 30° on the anterior part of the cornea to almost 0° on the posterior stroma – while no heterogeneity has been noted in the other two directions.

Those two analyses allow 3D maps of the orientation of the collagen lamellae in healthy corneas to be built. However, they remain limited to a very small number of cornea (around 1 or 2 per imaging approach method). In the case of pathological corneas such as keratoconus (see Sec. 1.3.), the arrangement of the collagen lamellae is altered, which affects the mechanical strength and the optical properties of the cornea, therefore causing a loss of vision.

### 1.2.3. Stromal striae

At tissue scale, "X-shape" linear structures appear both on Optical Coherence Tomography (OCT) images (Fig. 8) and on histological cuts [Napoli et al., 2020]. They are called Vogt's or stromal striae, and are detected in healthy and pathological corneas [Grieve et al., 2017]. They are associated with a wrinkle of the collagen lamellae, but their organisation remains not so well understood. The stromal striae become less visible as the intra-ocular pressure increases, meaning that these wrinkles fade with pressure (like a balloon that loses its ripples as it inflates). It suggests that the striae may have a protective
<figure><img src="image_8.png" /></figure>

**FIGURE 7.** a. In-plane and b. out-of-plane (right) distributions of the orientation of the collagen lamellae (width between 500-600 µm). The in-plane orientations have been determined by analyzing X-ray scattering observation (from [Aghamohammadzadeh et al., 2004, Boote et al., 2006]) (down scale factor: orange – 1, red – 1.5, dark red – 2, black – 3, green – 4, blue – 5), and the out-of-plane orientations use SHG images (from [Petsche and Pinsky, 2013, Winkler et al., 2011]).

role from damage induced by a sudden increase in pressure (when the eyes receive mechanical shocks for instance) [Grieve et al., 2017].

<figure><img src="image_9.png" /></figure>

**FIGURE 8.** Vogt striae in a human cornea on OCT imaging (top) and histology (bottom). The arrows indicate the most conspicuous 'X-shaped' structures. Scale bar: 100 µm (extracted from [Napoli et al., 2020]).

### 1.2.4. Summary of all scales

Figure 9 summarizes the different scales of the components of the cornea, especially the different scales of collagen found in the stroma.
<figure><img src="image_10.png" /><figcaption>**FIGURE 9.** Schematic view of the different scales of the components of the cornea. Top: eye scale. Red box: different layers composing the cornea – tissue scale. Blue box: collagen lamellae – micrometric scale. Violet box: fibrils of collagen – micro- to nanometric scale. Orange box: single microfibril of collagen – nanometric scale. Pink box: triple helix of collagen – nanometric scale. [Created with BioRender.com]</figcaption></figure>
1.3. The keratoconus

Keratoconus is an idiopathic pathology characterized by ectasia (corneal deformation) and progressive
non-inflammatory localized thinning of the cornea. In other words, the cornea changes its shape during
the progression of the pathology, from a quasi-spherical shape to an irregular conical one (Fig. 10). This
condition affects approximately 1 over 2000 to 500 new patients each year.

The keratoconus usually appears at the end of adolescence and affects both eyes, but not necessarily with the same severity nor in the same area of the eye. Although there is no clear origin of the disease, some risk factors have been identified such as family history (even if no specific gene has yet been identified), presence of allergy in the patient or regular eye rubbing [Najmi, 2019].

Four stages of the disease can usually be identified (Krumeich's classification [Naderan, 2017]): from
the least advanced stage (stage I or frustrated stage), with visual discomfort, decrease in visual acuity,
and astigmatism, to the most advanced stage (stage IV or severe stage), with thinning associated with a
loss of transparency, scars or opacities at the top of the cone, and risk of acute keratoconus. The evolution
from a stage to another is not systematic, not symmetric and not linear with time.

<figure><img src="image_11.png" /><figcaption>FIGURE 10. Schematic representation of a keratoconus: the shape of the cornea changes from a spherical to a conical one [Created with BioRender.com]</figcaption></figure>

Treating keratoconus involves two parallel tasks: improving the patient's vision and stopping the pro-
gression of the disease. To stop the progression, one can use a corneal cross-linking treatment. It is based
on the photo-polymerization of the collagen by riboflavin previously impregnated on the cornea. To im-
prove the patient's vision, the method is to use first glasses, then therapeutic contact lenses, and in case
of failure of those two approaches, corneal surgery. In the latter case, two options are considered: the
placement of intra-corneal rings or corneal transplant (only 1% of the patient with a known keratoconus
go through the latter). Intra-corneal rings are used to delay grafting. They are placed in the thickness of
the cornea to tighten the central cornea. This technology has become more common with the improve-
ment of femto-second lasers, which enable to dig tunnels with extreme precision in the cornea, through
which the surgeons introduce the intra-corneal rings. To treat the keratoconus, another approach can be
used, based on the "Excimer" laser treatment. It allows the cornea to be remodeled by photo-ablation
of the tissue and the visual accuracy of the patient to be increased, who will need a smaller visual cor-
rection. This technique is called LASIK surgery (Laser In Situ Keratomileusis) and is usually coupled
with cross-linking to stop the progression of the disease. Finally, when all those techniques have failed,
the only choice that remains is the graft. In the case of a keratoconus, surgeons perform a deep anterior
lamellar graft, which keeps the posterior part of the patient's cornea and only transplants the anterior part of the donor's cornea.

The change of shape of the cornea in case of a keratoconus comes with an alteration of the collagen lamellae. Figure 11 presents the orientation of the lamellae of collagen in one healthy and one stage III keratoconic cornea, taken at the central part of the cornea (Fig. 11.a and b). In the case of a healthy cornea, it confirms that the lamellae are mostly parallel to the surface of the cornea (greenish and blueish colors on Fig. 11.c) which is emphasized by the large peak near 0 on the histogram of out-of-plane orientations (Fig. 11.e). On the other hand, the keratoconic cornea presents more diverse colors than the control one (Fig. 11.d) – which is in agreement with the scatter in orientation angle in the middle stroma observed in the histograms on the right (Fig. 11.f). SHG images also show that the lamellae are curved in the case of the keratoconic cornea, while they are straight for healthy cornea. Those structural changes in the tissue lead to different mechanical responses of the corneal stroma, which will be studied in Sec. 4.2.

<figure><img src="image_12.png" /></figure>

**FIGURE 11.** Orientation distribution of collagen lamellae in control and stage III keratoconic corneas on samples located in the central part of the cornea (from Raoux et al., 2021). a, b. Location of the imaged samples. The anterior, middle and posterior stroma of the central part of the cornea are imaged. c, d: Typical out-of-plane orientation maps obtained from P-SHG images of transverse sections of the same control c. and keratoconic d. human corneas – scale bars: 100 µm – and e, f: histograms of the collagen out-of-plane orientation in the middle stroma of the e. control and f. keratoconic corneas displayed in c and d.

## 2. Optical Coherence Tomography (OCT) and Image analysis

### 2.1. Optical Coherence Tomography

A variety of imaging techniques have emerged in the literature regarding tomography, the imaging technique allowing to look inside a material, and especially for biological tissues imaging. X-ray computed
tomography, confocal microscopy, magnetic resonance imaging (MRI) or Optical Coherence Tomogra-
phy (OCT) are the most used in hospitals today. They each have their specificities (resolution, field of
application...) and are based on different theories. In the case of the eye, the reference technology is OCT,
which is widely used in clinics to image the retina and the cornea.

2.1.1. Principle of OCT

This section is largely based on [Drexler et al., 2014, Grieve, 2019, Davis, 2019]. It summarizes the
principle of the most widely used OCT systems.

General principle

Optical coherence tomography (OCT) systems are non-contact and non-invasive systems using light scattering (particularly the reflectivity) of the imaged sample. The incident light is partially reflected by the internal structures of the sample (Fig. 12). The characteristics of the reflected waves (amplitude and travel time) are then analyzed to compute a reflectivity map of the sample.

<figure><img src="image_13.png" /><figcaption>**FIGURE 12.** General principle of Optical Coherence Tomography (OCT) systems. The incident light is reflected by the internal structures of the samples and the reflected waves are analyzed to create the reflectivity map (adapted from [Davis, 2019]).</figcaption></figure>

Since it is not possible to capture the reflected waves directly with sensors (the speed of light being
too fast), another method – interferometry – has to be used. In the case of OCT systems, a Michelson
interferometer is used to image the sample, taking advantage of the low coherence property of the light
source to create localized interferences. Figure 13 presents the principle of the different OCT systems
commonly used in hospitals today.

First, the light is split using a beam splitter (BS in Fig. 13) between the reference and the sample arms. Then the signal is reflected both by the reference mirror and the reflective interfaces of the sample. The back-reflected signals create interferences when the optical path lengths are the same (within the coherence length) and then they are captured by a detector. With the optical path of the reference arm known, the interference signal is processed to generate images of the sample interfaces, and with that, reflectivity depth-profiles named A-scan. The axial (depth) resolution of the interferometer is determined
<figure><img src="image_14.png" /><figcaption>FIGURE 13. Optical Coherence Tomography (OCT) systems can be divided into Time Domain (TD) and Fourier Domain (FD) systems. These last ones can also be separated into two types, the spectrometer based and the swept source based systems (adapted from [Drexler et al., 2014]).</figcaption></figure>

by the so-called coherence length, which is classically defined by:

$$l_c = \frac{2 \ln 2}{\pi} \frac{\lambda_0^2}{\Delta\lambda} \qquad (1)$$

with $\lambda_0$ the central wavelength of the spectrum and $\Delta\lambda$ the width of the spectrum. Note that the wider the spectrum, the smaller the coherence length and therefore the better the resolution. A simplified version of the intensity at the output of the interferometer (as the sum of the reference and object arm intensities) is [Davis, 2019]:

$$I(z) = \frac{I_0}{4} [R_{inc}(z) + R_{coh}(z) + R_{ref} + 2\sqrt{R_{ref}R_{coh}(z)}V_{coh}(z) \cos(\Delta\phi(\lambda_0, z))] \qquad (2)$$

with $R_{coh}(z)$ the proportion of coherent back-scattered photons, $R_{inc}(z)$ the proportion of incoherent back-scattered photons, $R_{ref}$ the reflectivity of the reference surface, $\Delta\phi$ the phase shift between the two arms of the interferometer and $V_{coh}(z)$ the Gaussian envelope of the wave, centered on the plane of zero step difference.

The phase in Eq. (2) is given by:

$$\Delta\phi(\lambda_0, z) = \frac{2n\pi}{\lambda_0}z \qquad (3)$$

with $n$ the refractive index and $z$ the reference mirror position. This means that a change in intensity can be either obtained with a fixed wavelength $\lambda_0$ and a variable position $z$ (TD-OCT) or with a variable wavelength $\lambda_0$ and fixed position $z$ (FD-OCT) (Fig. 13).

## TD- and FD-OCT

TD-OCT systems use a broad-band light source (such as white light generated by optical fiber) with low coherence. This requires having very close optical paths between the two arms to obtain interferences. In this case, the position of the reference mirror changes ($z$ in Eq. (3)) to create the interference
with the different reflected interfaces of the sample and create an A-scan. Figure 14.a presents typical interferograms for two reflected interfaces of the sample. The envelope of the interferogram gives information on the material: the peak of intensity is related to the refractive power of the sample at that location. At the maximum intensity, constructive interferences appear: the image on the detector appears in white. Considering all the different intensity peaks, one can reconstruct the different gray levels of the A-scan. To distinguish two different interfaces of the sample, the two associated interferograms should be distinguishable. Thus, the coherence length of the source gives the limitation for the axial resolution of OCT. Once the A-scan is completed, a lateral sweeping (B-scan) is done to obtain a 2D image of the sample, and with a stack of 2D images, a 3D reconstruction of the sample can be obtained.

<figure><img src="image_15.png" /><figcaption>FIGURE 14. a. Typical interferograms of TD-OCT systems and b. typical spectrum of FD-OCT systems (adapted from [Davis, 2019]).</figcaption></figure>

In the case of FD-OCT, the reference arm does not move, but the wavelength changes. The spectrum of the back-scattered signal is analyzed in the frequency domain. Depending on the type of spectrum analysis, FD-OCT systems are split into two categories: spectrometer based (SD-OCT) and swept source based (SS-OCT) systems. In the first case, a broad-band light source is used and the reflected signal is diffracted with a diffraction grating or a prism, and detected by a 1D detector such as a CCD (charge coupled device), or the grating spectrometer is directly used as detector. In the case of the swept source based system, the source is a frequency scanning light source, in which case a simple photo detector (photo-diode) is sufficient to capture the interference at the end of the process. In both cases, the axial resolution is limited by the spectral bandwidth of the light source.
Figure 14.b presents the typical spectrum detected for a sample with two reflective structures, which presents grooves due to the interference between reference and reflective light. The modulation frequency is proportional to the optical path difference and therefore to the distance ($z_1$ or $z_2$ on Fig. 14.b) to the coherency plane. So different grooves mean different depths of the reflective structures. Fast Fourier Transform (FFT) of the spectrum allows three components (three peaks on the FFT of the spectrum Fig. 14.b) to be found, one of which is interesting: the positive peak (in $z_1$ and $z_2$ on Fig. 14.b) gives the corresponding depth of the refractive surface – its height gives the amplitude of the back-scattered signal and its width the signal range. For a real sample, the signal is way more complex as the system captures one A-scan (the whole depth profile) at a time, but extracting positive peaks of FFT allows all the positions of the reflective structures of the sample to be found.

TD-OCT systems capture the A-scan pixel by pixel, whereas FD-OCT systems can capture one A-scan at a time. This gives the FD-OCT system a great advantage in speed and a better signal-to-noise ratio (SNR) [Drexler et al., 2014]. However, capturing the whole depth in one shot (A-scan) requires the lens to have a wide field of view to light up the whole measured depth of the sample. A compromise must be found between acquisition speed, SNR and field of view size. Today, most surgeons use FD-OCT, but this may change with the evolution of the different technologies.

### 2.1.2. Use of OCT in hospitals

*This section is mainly based on [Ramos et al., 2009, Venkateswaran et al., 2018]. It details the main uses of OCT in clinics, especially on the cornea. Most of the figures are clinical images where the scale bar is not provided.*

OCT in hospitals was first used to image the posterior chamber of the eye (especially retina) but it has proven to be a very effective tool to image the anterior chamber too, especially cornea. Ophthalmic OCT uses mainly infrared sources with wavelength typically between 800 and 900 nm for the retina and 1310 nm for the anterior chamber. With the emergence of FD-OCT systems, the scanning speed and axial resolution have increased due to the use of shorter wavelength. Commercial OCT systems achieve axial resolution of less than 5 µm for ultra-high resolution mode.

In hospitals, TD-OCT have been gradually replaced by FD-OCT systems to image anterior segments. For example it can help detect ocular surface lesion, diagnose pathologies (macular degeneration, keratoconus, tumor or retinal lesions), choose between different treatments (in particular different types of surgeries) or quantify some anterior segment parameters (such as the anterior chamber angle between the scleral spur and the iris). We will focus here on the imaging of the cornea and the detection of its pathologies, and particularly the keratoconus. Figure 15 presents a typical 2D OCT image of a normal cornea. Epithelium, stroma and endothelium are easily distinguished by the difference in contrast between the three regions of the cornea.

#### Detecting keratoconus

To detect Keratoconus, it is possible to use pachymetry, i.e. the map of corneal thicknesses over a 10 mm-diameter circle, obtained through OCT imaging. Figure 16.a presents a pachymetry of a kerato-
<figure><img src="image_16.png" /><figcaption>**FIGURE 15.** Optical Coherence Tomography (OCT) image of a normal cornea. Scale bar: 250 µm. (kindly provided by V. Borderie from 15-20 hospital).</figcaption></figure>

conic cornea. Central thinning is detected with a minimum thickness of 422 µm in the central zone (in red) of 2 mm in diameter, which may indicate a fruste keratoconus. Figure 16.b shows a 2D OCT scan of an other cornea presenting a scarring (the tissue appears whiter on the image in the affected area) and a thinning in the region pointed by the white arrow, also revealing a keratoconus. These two images show that pachymetry and 2D scans of corneas can be helpful to detect keratoconus.

<figure><img src="image_17.png" /><figcaption>**FIGURE 16.** a. Pachymetry (map of the corneal thicknesses) of a keratoconic cornea obtained using a Visante OCT (extracted from [Ramos et al., 2009]). b. OCT images of a keratoconic cornea showing a corneal scarring and thinning (arrow) (extracted from [Venkateswaran et al., 2018]).</figcaption></figure>

## Treatment adaptation

**Intrastromal rings** Figures 17.b-c present OCT cross-section images of keratoconic corneas after the implant of intrastromal corneal rings (Fig. 17.a for the picture of the eye implanted with the corneal rings). Ophthalmic surgeons use those images post-operative to verify the depth and position of the implantation (Fig. 17.c for a correct implantation). If the implants are too deep, they may perforate the anterior chamber (Fig. 17.b) and if they are not deep enough, it may lead to serious complications on both the stroma and/or the epithelium.
<figure><img src="image_18.png" /><figcaption>**FIGURE 17.** Intrastromal ring segments. a. Photograph of intrastromal ring segments (extracted from [Venkateswaran et al., 2018]). b-c. OCT images of intrastromal ring segments (arrow) (extracted from [Venkateswaran et al., 2018, Ramos et al., 2009]).</figcaption></figure>

**LASIK surgery** Figures 18.a shows an OCT scan after LASIK surgery. It shows a 198 µm flap – a cap created from the epithelium and a bit of stroma which is removed before LASIK surgery to access deeper stroma (where laser treatment is performed) and put back after the treatment – which was unexpectedly thick in this case and put the cornea at risk for ectasia. Here, OCT imaging helps the surgeon adapt the treatment rapidly. Figure 18.b presents an OCT scan of a cornea after cross-linking treatment. The arrow points toward the demarcation line between the non-penetrated and the penetrated tissue. In this case, OCT images help measure the penetration depth of the collagen cross-linking treatment, and to study the influence of this depth on the response of the patient to the treatment.

<figure><img src="image_19.png" /><figcaption>**FIGURE 18.** OCT images after laser surgery and cross-linking. a. LASIK flap of 198 µm and residual stromal bed at 215 µm, (extracted from [Ramos et al., 2009]). b. Corneal haze three days after corneal collagen cross linking. The arrows point at the demarcation line in the corneal haze (extracted from [Venkateswaran et al., 2018]).</figcaption></figure>

**Corneal transplant** Figure 19 presents OCT scans of a cornea after keratoplasty (corneal transplant). Subfigure 19.a shows a cornea and the graft after Descemet Stripping Automated Endothelial Keratoplasty (DSAEK), a type of lamellar transplant. The demarcation line between the graft and the original cornea is clearly visible on the OCT scan, allowing for a control of the interface and anticipating graft detachment. When the keratoplasty fails, OCT scan allows the graft edge detachment (red arrow on Fig. 19.b) to be rapidly localized and the necessary action for the proper recovery of the eye to be taken.
<figure><img src="image_20.png" /><figcaption>**FIGURE 19.** OCT images after keratoplasty. a. Post-operative OCT image of an attached DSAEK graft (arrow) (extracted from [Venkateswaran et al., 2018]). b. OCT scan of a cornea after transplant showing a detachment of the graft (extracted from [Ramos et al., 2009]).</figcaption></figure>

## Conclusion

In recent years, OCT has proven to be a very powerful tool for physicians. It allows both anterior and posterior chambers to be imaged (statically), pathologies such as keratoconus to be detected and the tissue pre- and post-operative to be imaged. The FD-OCT having a very high imaging speed, it opens a field of application on the characterization of the dynamics of the cornea in hospital. Finally, it can also be used to characterize the mechanics of the cornea using elastography coupled techniques (Sec. 3.2.2.). Although the resolution of OCT in hospitals may be lower than OCT used in the laboratory, clinical OCT images can be used to describe the geometry of a patient's cornea at physiological pressure, particularly via the pachymetry maps and especially in the case of keratoconus.

## 2.2. Image analysis and strain measurement techniques

Ophthalmologists use OCT techniques to measure and control the health of the eyes. Therefore, they produce a huge amount of images, which data are not fully exploited at the moment. New techniques, such as Optical Coherence Elastography (Sec. 3.2.2.), combining OCT with mechanical testing, have emerged to compute mechanical properties such as stiffness of the cornea. They rely mainly on two distinct types of image analysis: wavelength scanning interferometry and image correlation, introduced hereafter.

### 2.2.1. Phase contrast and Wavelength Scanning Interferometry (WSI)

*This part is mainly based on [Ruiz et al., 2004, Chakraborty and Ruiz, 2012].*

The phase contrast, or phase difference, technique is a method to compute the displacement in a material by comparing two interferogram-based images. Its suitability has been highlighted by Ruiz and colleagues [Ruiz et al., 2004, Chakraborty and Ruiz, 2012] using the technique of wavelength scanning interferometry (WSI). WSI is based on the same principles as the FD-OCT systems: a source with multiple wavelengths is used to image a sample at two different times. The source can be of different kinds: a low coherent source (like for OCT) associated with a prism, a swept source (considered the most expensive choice) or an optic splitter that divides power equally between the output channels. Interferences are
created using the different frequencies of the source. A Fourier Transform is then applied on the detected signal. Images could be then reconstructed, but it is the reconstruction of the phase between the two images $\Phi$ that is interesting to determine the displacement: it gives the phase difference field $\Phi(x)$ in the direction of the light propagation. The phase difference is due to the displacement of the material, and thus $\Phi$ can be related to the displacement in the direction of the light propagation. Combining three non co-planar sources, the phase shifts can be measured in the three different directions. The displacement vector $r$ is related to the phase difference vector through the relation $r: \Phi = S\Delta r$, which involves the sensitivity matrix $S$, which is a characteristic of the set-up (and thus can be known).

Therefore, it is possible to reconstruct displacement maps, as in Figure 20, which presents the three components $(u, v, w)$ of the displacement (rows) for different planes of the sample (columns) after an in-plane rotation and an out-of-plane tilt. Figures 20.a, c, e and f show a gradient in the $u$ and $v$ components, which is expected from the in-plane rotation. Figures 20.g and i present also an expected gradient from the out-of-plane tilt.

<figure><img src="image_21.png" /><figcaption>FIGURE 20. Cross sections of the measured 3D displacement field corresponding to an epoxy sample under in-plane rotation and out-of-plane tilt. The rows indicate the three displacement components $u$, $v$, and $w$. The columns show sections of the data volume on different planes. Displacements are in meters (Extracted from [Chakraborty and Ruiz, 2012]).</figcaption></figure>

The same principle of phase contrast will be applied with OCT systems to compute displacement fields in the case of Optical Coherence Elastography (Sec. 3.2.2.) [Wang and Larin, 2014, Ramier et al., 2020]. However, the main method in mechanics remains the Digital Image Correlation, which is presented in the next section.

### 2.2.2. Digital Image/Volume Correlation (DIC/DVC)

*This part is mainly based on [Pan et al., 2009, Buljac et al., 2018].*
Digital Image Correlation (DIC) is a commonly used technique to compute surface strain maps of a material under load. It became very popular with the evolution of digital camera technology with higher resolutions, and also with the evolution of computing power, allowing for faster computation on many images corresponding to the material loading stages. It is usually based on the tracking and matching of speckled samples between a reference and a deformed state. Figure 21 presents an example of the experimental methodology for a classical traction test on Achilles tendon (from [Luyckx et al., 2014]). First, the sample was dyed with methylene blue (Fig. 21.a). Then, it was speckled with white paint (Fig. 21.b) to give random contrast to the tissue. Finally, the longitudinal strain map was computed between two levels of load using DIC (Fig. 21.c).

<figure><img src="image_22.png" /><figcaption>FIGURE 21. Example of DIC on Achilles tendon. a. The Achilles tendon in the clamps dyed with methylene blue, b. speckled with white paint and c. longitudinal strain map computed by DIC (extracted from [Luyckx et al., 2014]).</figcaption></figure>

Principle

Figure 22 presents the principle of DIC. A correlation domain (framed in pink on Fig. 22) is composed of a certain number of pixels (here 9) associated with gray levels in the reference configuration. The goal is to track the evolution of this correlation domain in the deformed configuration, using the gray levels of the pixels in the so-called search domain of the deformed configuration. Denoting $f(\boldsymbol{x})$ and $g(\boldsymbol{x})$ the known gray levels of the pixels on the reference and deformed configurations respectively, image correlation relies on the existence of a displacement field $\boldsymbol{u}(\boldsymbol{x})$ such that:

$$g(\boldsymbol{x} + \boldsymbol{u}(\boldsymbol{x})) = f(\boldsymbol{x}) \qquad (4)$$

with $\boldsymbol{x}$ the location of the pixel. Figure 22 illustrates an idealized, simple, case of DIC (with a simple field $\boldsymbol{u}(\boldsymbol{x})$ of a vertical displacement leading to a vertical strain), which hides the complexity that entail actual experimental setups. First, the displacement $\boldsymbol{u}(\boldsymbol{x})$ can be of any values, resulting in the need to compute the displacement at the sub-pixel scale, to allow displacements of a non-integer pixels. Then,
to have a precise computation, a large range of image gray dynamic should be used. In particular in each correlation domain and between a correlation domain and its neighbors, the gray levels must be varied enough so that the pattern can be easily recognizable between the reference and the deformed configuration. In practice, it means that the quality of the image should be sufficient and the speckling of the sample should be randomly distributed, so that the contrast of the image can be sufficient to perform the correlation. Finally, even if sub-pixel gray level computation is done and contrast is good, images are naturally noisy. Even for two images taken at the same levels of load (i.e. where $u(x)$ should be 0), $f$ and $g$ are not perfectly equal and the difference between them is related to the noise coming from the experimental apparatus.

<figure><img src="image_23.png" /><figcaption>FIGURE 22. DIC Principle. The distinctive pattern of pixels of the reference correlation domain (pink square) will be compared using gray levels and correlation criterion with the deformed configuration pack of pixels. Then the correlation domain of the deformed configuration can be computed and with it the displacement between the two configurations.</figcaption></figure>

To compare $f$ and $g$ taking into account these issues, a correlation criterion is created, estimating the similarity of the correlation domain of the reference configuration with the corresponding area of pixels in the deformed configuration: the center of the correlation reference domain is searched in the deformed configuration within the search domain. A typical correlation criterion on a square correlation domain of $(2M + 1) \times (2M + 1)$ pixels is the zero-normalized cross-correlation criterion $C_{ZNCC}$ defined as:

$$C_{ZNCC} = \sum_{i=-M}^{M} \sum_{j=-M}^{M} \frac{[f(x_i, y_j) - \bar{f}][g(x'_i, y'_j) - \bar{g}]}{\Delta f \Delta g} \quad (5)$$

with $(x_i, y_j)$ and $(x'_i, y'_j)$ the positions of the same pixel in the reference and the deformed configurations, respectively,
$$ \bar{f} = \frac{1}{(2M+1)^2} \sum_{i=-M}^{M} \sum_{j=-M}^{M} f(x_i, y_j), \quad \bar{g} = \frac{1}{(2M+1)^2} \sum_{i=-M}^{M} \sum_{j=-M}^{M} g(x'_i, y'_j) $$

the mean values of the gray levels and

$$ \Delta f = \sqrt{\sum_{i=-M}^{M} \sum_{j=-M}^{M} [f(x_i, y_j) - \bar{f}]^2} \quad \text{and} \quad \Delta g = \sqrt{\sum_{i=-M}^{M} \sum_{j=-M}^{M} [g(x'_i, y'_j) - \bar{g}]^2} $$

A ZNCC criterion close to 1 corresponds to a very good correlation whereas a ZNCC criterion tending to -1 means a very poor correlation. Most DIC softwares allow a minimal acceptable value for the correlation criterion to be given, to consider that the correlation is good enough.

## Local and global DIC

Local DIC consists in correlating each correlation domain between its reference state and its deformed state. A global DIC consists in looking at the region of interest as a whole. One of the main advantage is that it ensures the continuity of the measured displacement. Also a regularization process can be added when looking at materials for which there is an a priori on the displacement. Local DIC/DVC has advantage of being determined at a controlled scale, without regularization or a priori on the material behavior. Although one can try to point strength/weakness of each method, both are used intensively in mechanics. We are going to present here the local approach, which is simpler to explain.

## From DIC to DVC

DIC is a surface method applied to 2D images. Historically, only such images provided enough contrast (due to the speckling for example). With the emergence of 3D imaging techniques in the 2000s however, such as X-ray tomography [Bay et al., 1999] or OCT [Fu et al., 2013], 3D images of materials with internal contrast are now available. The methodology of DIC can then be easily transposed to the third spatial dimension. Pixels become voxels, DIC becomes Digital Volume Correlation (DVC) and 2D cross-correlation algorithms become 3D. Similarly, WSI can be done in 3D. Thus, both offer the possibility to characterize the response of a material to loads. The use of 3D instead of 2D is interesting for complex materials, such as collagen-rich tissues (muscle, tendon, cornea...): the full deformation tensor can be measured. While WSI is more commonly used in optics laboratories, DVC is well implemented in the mechanics field.

Figure 23 presents the typical results for 3D displacements (A-C), 3D normal strain (D-F) and 3D shear strains (G-I) of chicken breast tissue under vertical compression obtained by DVC on 3D SS-OCT images [Meng et al., 2019]. During the test, muscle fibers were aligned along the x-direction. Results show an anisotropy in the mechanical deformation of the breast tissue: linear reduction along the z-direction, diagonal linear distribution along the x-direction and a near linear distribution along the y-direction.
<figure><img src="image_24.png" /><figcaption>FIGURE 23. Example of DVC on SS-OCT images of chicken breast tissue under vertical compression (6mm in both the width and length directions were scanned). (A-C) the 3D displacements in the x, y and z directions, respectively, (D-F) the 3D normal strains $ε_x$, $ε_y$ and $ε_z$ in the x, y and z directions respectively and (G-I) the 3D shear strains $ε_{yz}$, $ε_{zx}$ and $ε_{xy}$ (extracted from [Meng et al., 2019]).</figcaption></figure>

### 3. Mechanical characterization of the human cornea

In order to model the response of the cornea to pressure, one needs values for the mechanical parameters of the tissue, such as stiffness, viscous modulus... To do so, one of the newest experimental approach is called Elastography. It can be defined as any imaging process allowing to extract elasticity or stiffness of a material. It usually couples a mechanical exciter to create a perturbation in the tissue, with an imaging apparatus to measure the induced perturbation. Finally, to extract the mechanical properties of the material, one needs a model of the tissue (i.e. a constitutive law).

Even though it is usually easier to perform mechanical tests *ex-vivo* (repeatability of experimental conditions, sampling, availability of samples and the experimental apparatus, larger range of experimental conditions...), the cornea is a tissue that has the advantage of being accessible *in-vivo*. As a result, although the classical characterization tests of healthy tissues are usually performed *ex-vivo*, the characterization of pathological corneas can be done *in-vivo* thanks to new elastography techniques, which compare their properties with healthy ones.

#### 3.1. Ex-vivo mechanical characterization of cornea

This section presents the most common tests performed *ex-vivo* on corneas to measure and characterize the mechanical properties of the tissue. The inflation test is the experiment that mimics best the physiological action of pressure on the cornea. Strip extensometry has been used for decades to measure Young's modulus of all types of materials. Nanoindentation and acoustic radiation force elasticity microscopy
(ARFEM) are two more recent techniques for the evaluation of the Young's modulus, the latter having the advantage of being non destructive. Note that many of these tests apply only a small deformation of the tissue, leading to a linear relation between applied stress and strain. So, they can determine only a "tangent modulus", which is generally called Young's modulus.

### 3.1.1. Inflation test

#### Principle of the method

The inflation test is an experimental method to characterize the mechanical properties of cornea. It mimics the action of intraocular pressure on the tissue. Elsheikh and coauthors have studied the mechanical properties of porcine and human corneas using the experimental apparatus represented in Fig. 24 [Elsheikh and Anderson, 2005]. The cornea is attached to a pressure chamber by the sclera, and saline solution is injected under the cornea via the pressure chamber to put the tissue under pressure. The pressure in the chamber is monitored using a pressure sensor and the displacement of the apex of the cornea is measured via a laser displacement sensor. The biomechanics of the cornea is then analyzed using the pressure versus apex displacement curve.

<figure><img src="image_25.png" /><figcaption>**FIGURE 24.** Schematic view of the different components of the inflation test (extracted from [Elsheikh and Anderson, 2005]).</figcaption></figure>

#### Main results

Figure 25 presents the pressure with apical rise curves for six human corneas [Elsheikh et al., 2008]. At low pressure (under approximately 15 mmHg), the cornea exhibits a low stiffness. Then, with the increasing pressure, a sudden change of stiffness appears. Elsheikh and Anderson explain these two regimes by a domination of the mechanical response by the matrix (of low stiffness) at low pressure, and a domination by the collagen lamellae (high stiffness) at higher pressures [Elsheikh and Anderson, 2005].
Such behavior is similar phenomenologically to the mechanical response of other collagen-rich tissues
as tendon, skin or aorta [Fratzl et al., 2008].

<figure><img src="image_26.png" /><figcaption>**FIGURE 25.** Pressure with apical rise during inflation test for 6 human corneas (from [Elsheikh et al., 2008]).</figcaption></figure>

In the same article [Elsheikh et al., 2008], the authors also show that porcine and human corneas do respond the same way to pressure (low stiffness at low pressure and sudden change of stiffness at higher pressure) but that human corneas are stiffer than porcine ones, and also that human corneas stiffen with age.

Although the article of Elsheikh *et al.* proposes a first global mechanical behavior of the cornea using the displacement of the apex, it may seem reductive to consider that the whole biomechanics of the cornea is governed by what happens at the highest point, in particular in view of the complex microstructure. That's why other groups – particularly Boyce and coauthors [Boyce et al., 2008] – measured the displacement of the whole anterior surface of the cornea by stereo-correlation (a 3D displacement field measurement of a surface). Figure 26.a presents the results for the out-of-plane displacement (in the z-axis direction of Figure 26.a) at a particular pressure (60 mmHg). Figure 26.b presents the results for the out-of-plane displacement for various location over a loading cycle. Those maps show that the out-of-plane displacement is homogeneous for the central part of the bovine cornea at 60 mmHg and also relatively homogeneous over the loading cycle (the displacement at the different locations is similar).

It confirms that one could consider the apex displacement as a proxy for the anterior surface displacement of the cornea, at least for bovine eyes, when the surface geometry is known. But there is still a lack of information on the material behavior through thickness. Fu [Fu, 2014] tried to tackle this issue by measuring full-field displacement and strain maps using digital volume correlation (DVC) on OCT images of porcine corneas during an inflation test between 2 and 2.5 kPa (15–18.75 mmHg).

Figure 27.a shows displacement maps for a variation of pressure from 15 to 18.75 mmHg. The order of magnitude of the vertical displacement *u*<sub>y</sub> at the apex of the cornea seems to be in agreement with the apex displacement found by Elsheikh *et al.* [Elsheikh et al., 2008], i.e. approximately 0.01 mm. Figure 27.b shows the associated strain maps. A clear contraction (approximately -2% of ε<sub>yy</sub> strain) can be observed at the center of the cornea along the vertical direction (y-axis), while the cornea inflates in the peripheral zone (of approximately 2% also). ε<sub>xx</sub> strain maps show an extensional deformation along the x-axis and clear heterogeneity between the anterior and posterior parts (almost all of the deformation
<figure><img src="image_27.png" /></figure>

<figure><img src="image_28.png" /></figure>

FIGURE 26. a. Typical out-of-plane displacement at a pressure of 8 kPa (60 mmHg) for bovine cornea. b. Typical out-of-plane displacement at different locations during a loading cycle over a pressure range of 3.6–8.0 kPa (27–60 mmHg). (Both images are extracted from [Boyce et al., 2008]).

<figure><img src="image_29.png" /></figure>

FIGURE 27. a. Displacement and b. strain maps measured with DVC for the central z-slice of a porcine cornea under pressure from 2 to 2.5 kPa (15 to 18.75 mmHg) (extracted from [Fu, 2014] – Scale bar: 1 mm).
appears in the posterior stroma) while the cornea is submitted to shear strain in the periphery (and mostly
on the anterior part of the cornea), which may be due to the computation of the strain in Cartesian while
the cornea is curved (and thus would have a rather radial deformation).

Keratoconic corneas and treatment

To our knowledge, there is no mechanical characterization of a whole human cornea (healthy or patho-
logical) using inflation tests.

To test the effect of the cross-linking treatment, inflation tests have been performed on porcine corneas treated with a riboflavin solution (see Sec. 1.3.). Kling *et al.* [Kling et al., 2010] used a Scheimpflug corneal three-dimensional topographer (similar to an ultrafast OCT limited to 2D acquisition) to image the cornea. The Young’s modulus was determined using the classical Laplace’s law for the stress σ = *Rp*/(2*d* (with *R* the mean radii of curvature, *p* the applied pressure and *d* the corneal thickness)) and ε = Δ*R*/R for the strain. They show that the Young’s modulus in a cross-linked cornea is significantly higher (1.10 ± 0.30 kPa) than the one found in non-cross-linked eyes (0.69 ± 0.30 kPa). They also show that the cross-linking treatment influences the thickness of porcine corneas, especially 24 hours after the treatment. A strong dehydration was observed on porcine corneas with classical hyperosmolar riboflavin instillation, while swelling occurred for hyposmolar riboflavin treatment. In another study, Chang *et al.* [Chang et al., 2020] found that the shear modulus μ at a stress of 0.03 MPa shows an increase of 43% ± 24% in a cornea treated by cross-linking with respect to non-treated porcine corneas. One can conclude that the cross-linking process tends to stiffen the tissue and also to influence the thickness of the cornea.

3.1.2. Strip extensometry

Principle of the method

Strip extensometry is one of the most classical experiments designed to obtain mechanical proper-
ties of a material and especially the Young's modulus, which is obtained from the slope of the stress-
strain curve as E = σ/ε (with σ the axial stress and ε the axial strain). Elsheikh and Anderson
[Elsheikh and Anderson, 2005] have performed tensile tests until rupture on porcine cornea while Boyce
et al. have carried out creep tests on bovine cornea [Boyce et al., 2007]. Figure 28 presents the experi-
mental apparatus used by Boyce et al. (Fig. 28.a–d) and by Elsheikh and Anderson (Fig. 28.e–f). They
both cut a strip going through the apex diagonal out of a cornea (bovine and porcine respectively) and
applied uniaxial load on the coupon.

Main results

Figure 29 presents the stress-strain curve (a and c) and the load-extension curve (b) of the strip ex-
tensometry tests. In all the cases, the non-linear responses with two regimes is obtained, similar to the
inflation test response. Figure 29.c shows the comparison between the averaged stress-strain curve for
strip extensometry and inflation tests on porcine corneas. Strip test has a shorter first regime (of large
stretch at low stress) than the inflation test. Then the second part (large stress for large stretch) seems
<figure><img src="image_30.png" /><figcaption>FIGURE 28. a-d. Experimental apparatus used in (and extracted from) [Boyce et al., 2007]. a. Bovine cornea, b. and c. cut of the strip and d. tensile test. e. Schematic of the specimen and f. the strip extensometry test rig used in (and extracted from) [Elsheikh and Anderson, 2005].</figcaption></figure>

much more similar in term of rigidity.

Even though the stress-strain curves of the strip extensometry tests have a similar shape to those of the inflation test, three sources of errors due to the geometry of the cornea can be identified in the strip test. First, the cornea being almost spherical, the length of the anterior and posterior surfaces of the strip are not equal and so the usual area used has to be adapted to compute the Young's modulus [Elsheikh and Anderson, 2005] to take into account the non uniform stress distribution. Second, the corneal strip is flattened, so its initial curvature has to be taken into account in the computation of the Young's modulus, especially for pre-strain (extensional strain in the posterior surface and contractile strain in the anterior surface). Finally, the thickness is not homogeneous along the strip (the cornea is thinner in the central part than in the periphery) and this third geometrical effect has to be considered in the computation of the area. Anderson and Elsheikh [Elsheikh and Anderson, 2005] have built a model to correct the results of the strip extensometry test to match the inflation test results (the difference between the two sets is reduced to 5% in average) so that future work on the characterization of the mechanical properties of the cornea can be done using strip extensometry. Other sources of difference may be important, as the heterogeneity of stretch in the inflation test, contrary to the strip test, and so on.

## Keratoconic corneas and treatment

In the case of pathological corneas, there is only a few studies of strip extensometry comparing the Young's modulus of healthy and keratoconic human corneas. Andreassen and his colleagues [Andreassen et al., 1980] have shown that the Young's modulus of a keratoconic cornea is significantly lower than the one of a healthy cornea. However, Nash *et al.* [Nash et al., 1982] shows that this assertion is true for high level of stress but not at physiological pressure, where no significant difference is
<figure><img src="image_31.png" /><figcaption>**FIGURE 29.** a. Average tensile stress–strain response of bovine cornea tissue in the naso-temporal orientation at each of three different strain rates (extracted from [Boyce et al., 2007]). b. Load-elongation behavior of a selection of strip tests on porcine corneas and c. Material constitutive relationship obtained using both inflation and strip tests on porcine corneas (extracted from [Elsheikh and Anderson, 2005]).</figcaption></figure>

found between healthy and pathological corneas. Finally, Spoerl et al. [Spoerl et al., 1998] studied very early the effect of cross-linking by using strip extensometry on treated and non treated porcine specimens and showed that the cross-linking resulted in an increased stiffness. More recently, Wollensak et al. [Wollensak et al., 2003] performed extensometry tests on human and porcine corneas and showed that the Young's modulus was higher when the corneas have been treated with riboflavin (4.5 higher and 1.8 higher, respectively).

### 3.1.3. Nanoindentation

#### Principle of the method

The test of indentation consists in penetrating a tip (the indenter) in a flat surface of the tested material, and monitoring the height of penetration (h in Fig. 30) as a function of the applied load (P on Fig. 30). The mechanical properties – such as hardness – can then be determined.

#### Main results

Dias and Ziebarth [Dias and Ziebarth, 2013], and Last et al. [Last et al., 2012] measured the Young's modulus of corneal stroma using atomic force microscopy (AFM) as nanoindentor. Dias and Ziebarth found that the posterior stromal elasticity is 39.3% of the anterior stromal elasticity, with an anterior Young's modulus of 281 ± 214 kPa. Last et al., on their side, measured an anterior stromal Young's
<figure><img src="image_32.png" /><figcaption>**FIGURE 30.** *The principle of nanoindentation (extracted from [den Toonder et al., 2005]).*</figcaption></figure>

modulus of 33.1 ± 6.1 kPa. The difference in the order of magnitude may come from experimental protocols, especially the preparation of the exposition of the anterior stroma. Eberwein and coauthors [Eberwein et al., 2014] used an indenter (called Bioindenter) to measure the properties of cornea, limbus and sclera. Figure 31 presents the typical force with displacement curve obtained on the three tissues. Young's modulus measured was approximately 19 kPa for the central cornea, 10k Pa in the limbal region and 17 kPa in the scleral region.

<figure><img src="image_33.png" /><figcaption>**FIGURE 31.** *Typical force with displacement curves obtained using nanoindentation for the cornea (blue), the limbus (red) and the sclera (green) (extracted from [Eberwein et al., 2014]).*</figcaption></figure>

## Keratoconic corneas and treatment

The same groups used nanoindentation to measure the elasticity of healthy corneas after cross-linking. Dias *et al.* [Dias et al., 2015] showed that porcine corneas treated with riboflavin were stiffer and less viscous than the control ones. Nohava *et al.* [Nohava et al., 2018] used the Bioindenter on human corneas and measured that the stiffness of the central cornea is two times higher after riboflavin treatment. In both articles, the authors show that the region of higher stiffness is centered around the riboflavin application and does not extend significantly towards the periphery.
### 3.1.4. Acoustic Radiation Force Elasticity Microscopy (ARFEM)

One of the most recent techniques used to measure the mechanical properties of the cornea (and other soft tissues) is the acoustic radiation force elasticity microscopy (ARFEM). Figure 32 presents the setup used in Mikula et al. [Mikula et al., 2014, Mikula et al., 2018]. Details of the experimental apparatus can be found in [Mikula et al., 2014]. Only the main components are described below.

<figure><img src="image_34.png" /><figcaption>FIGURE 32. ARFEM setup (extracted from [Mikula et al., 2014]).</figcaption></figure>

A femtosecond laser generates a micro-bubble inside the sample with one pulse. Then an acoustic force (low-frequency, high-intensity) displaces the bubble while an ultrasound probe (high-frequency, low-intensity) monitors the displacement of the bubble. The Young's modulus is related to the displacement of the bubble using the following relationship [Mikula et al., 2014]:

$$E = Ia/2cx_{max}, \qquad (1)$$

where *I* is the acoustic intensity, *a* the bubble radius, *c* the speed of sound in the medium, and *x*<sub>max</sub> the maximum displacement of the bubble.

With this technique Mikula et al. managed to measure the Young's modulus in healthy [Mikula et al., 2016] and keratoconic [Mikula et al., 2018] corneas. They found that the anterior central part is stiffer ($E = 4.2 \pm 1.2$ kPa and $E = 1.67 \pm 0.44$ kPa for healthy and keratoconic corneas, respectively) than the posterior central part of the cornea ($E = 2.3 \pm 0.7$ kPa and $E = 0.97 \pm 0.3$ kPa for healthy and keratoconic corneas, respectively). They also show that keratoconic corneas are significantly less stiff than healthy tissues.

### 3.1.5. Summary of ex-vivo mechanical characterization without microscopic imaging

To conclude, although all these techniques give quite different results in terms of numerical values, significant trends emerge with respect to ex-vivo measurements.
Figure 33 presents a summary of the previous results on characteristic stiffness measured using the different techniques presented, which definition varies across experiments (see Table 3.1 at the end of this section for a summary of all mechanical measures). First, in all cases, porcine corneas (in pink on Fig. 33) are globally less stiff than human corneas (in blue on Fig. 33). Then, keratoconic corneas (△ on Fig. 33) are significantly softer than healthy tissue (□ on Fig. 33), but cross-linked corneas (○ on Fig. 33) are significantly stiffer than healthy tissue. The lack of rigidity of pathological corneas can thus be counterbalanced by the increase of stiffness brought by cross-linking treatment. Finally, anterior stroma (middle blue and pink) is stiffer than posterior stroma (light blue and pink) either for humans and pigs. However, no study compared the measure on the whole cornea to those on anterior and/or posterior parts only.

Based on Figure 33, we conclude that it is difficult to choose a single value for a given mechanical property of the cornea. The scatter across the results available in the literature is very large: the scale of Figure 33 being logarithmic, the range of possible values comprises almost four orders of magnitude for human corneas. Furthermore, it shows that there are discrepancies between the anterior and posterior parts and that local information on the whole organ is therefore missing.

<figure><img src="image_35.png" /><figcaption>FIGURE 33. Ex-vivo measured corneal stiffness coefficient using the different methods previously presented: inflation test (Sec. 3.1.1.), strip extensometry (Sec. 3.1.2.), nano-indentation (Sec. 3.1.3.) and ARFEM technique (Sec. 3.1.4.). Notation: W = whole cornea, A = anterior stroma, P = posterior stroma.</figcaption></figure>

The inflation test is the most physiological one. It has enable to show that (i) the cornea presents two very distinct regimes of mechanical responses depending on the applied pressure, (ii) cross-linking can be a suitable treatment approach for keratoconus because it stiffens the weakened pathological corneas.
However, the characterization remains limited at the moment, with partial measures of the stretch: at best on the external surface but generally limited to the apex displacement, despite the complexity of the tissue and of the imposed loading.

Strip extensometry is easier to perform. However, the naturally-bend shape of the human cornea make its analysis quite complex: the transition from plane properties (of the strip) to 3D properties (of the half-sphere) is not obvious. On top of that, these assays have been limited to strip going through the cornea apex. So, they may miss the effect of the heterogeneity in the lamellae distribution. That is why this test has become less popular recently.

Nano-indentation and ARFEM are recent techniques that allow direct measurements of the Young's modulus of the anterior and posterior cornea. They have the advantage of being in principle quite simple to implement, and not very invasive than. They also allows to test the cornea at different depths, working on cutted tissues. However, they are limited to small deformations, which means that only Young's modulus can be determined and not the full non-linear stress-strain curve. Also, they are very sensitive to the preparation of the surface (flatness, humidity, etc.); This explains partly the high variability in their results.

### 3.1.6. Mechanical testing coupled with imaging of the microstructure

Two founding papers have studied the response of the cornea to mechanical load whilst imaging the microstructure: one using Second Harmonic Generation (SHG) microscopy coupled with inflation test [Benoit et al., 2016] and one using X-ray diffraction coupled with strip extensometry [Bell et al., 2018]. In these papers, macroscopic strains are compared to the distribution of the lamellae during the mechanical tests.

#### Macroscopic strain response

Figure 34.a.1 presents the scan positions used for the X-ray analysis and Figure 34.b.1 the different volumes of interest for the SHG study, in particular Stack#2 (2PEF) that was used to perform the strain analysis.

Figures 34.a.2 and b.3 present the local strain results measured by DIC. Figure 34.a.2 shows the computed parallel strain on the strip cut along the vertical meridian at 5% of strain during tensile test [Bell et al., 2018]. Figure 34.b.3 illustrates the strain on the corneal surface between the loading steps of the inflation test along the two preferential directions of the collagen lamellae $\phi_1$ and $\phi_2$ presented in Figure 34.b.2 [Benoit et al., 2016]. Figure 34.a.2 shows a relatively symmetric distribution of strain with respect to the center of the sample with a bigger parallel strain in the center (around 7%) than in its closest periphery (around 3-4% at positions of 6 and 10 mm). The end of the strip is subject to higher strain, up to 10%. Normal strain maps ($\epsilon_{11}$ and $\epsilon_{22}$ in Fig. 34.b.3) and shear strain maps ($\epsilon_{12}$) show, respectively, the progressive stretching and shearing of the cornea along dominant orientations with increasing pressure. The strain appears rather uniform, leading to homogeneous deformations at the sub-millimeter scale.
<figure><img src="image_36.png" /><figcaption>FIGURE 34. a.1. Scan positions used for the X-ray analysis during tensile test. a.2. Parallel strain computed by DIC at 5% of strain – scale bar 11% (red) to 2% (blue) (extracted from [Bell et al., 2018]). b.1. Volumes of interest considered in the SHG study. b.2. Main direction of the lamellae used to compute b.3. the local normal ($ε_{11}$ and $ε_{22}$) and shear ($ε_{12}$) strains (extracted from [Benoit et al., 2016]).</figcaption></figure>
Figure 35 presents the results on lamellae orientation during the two loading tests. Figure 35.a shows the analysis for the tensile test and Figure 35.b for the inflation test. WAXS analysis (Fig 35.a) shows the polar plot of the aligned collagen lamellae. It shows a clear alignment of the lamellae along the stretching direction with increasing strain. The anisotropy value, representing the ratio of maximum parallel collagen to maximum orthogonal collagen peaks, is clearly increasing in the middle of the sample, which leads to a rearrangement of the collagen lamellae along the stretch direction. However, the pattern of the macroscopic strain at 5% is not completely recovered in these WAXS results. In particular, there is no clear difference between the center of the sample and its nearest periphery (around 6 and 10 mm). Figure 35.b.1 presents the results of SHG image analysis determining the distribution of the orientation probability of the lamellae on the whole thickness of the cornea and on the anterior, middle and posterior stroma with increasing pressure. It shows first that the main orientation remains the same during the whole experiment and, second, that the probability distribution is changing with pressure, especially in the middle and posterior stroma. Figures 35.b.2 and b.3 detail the percentage of lamellae oriented along the two main directions (with a margin of ± 15°) in the same zones of the cornea with increasing pressure. It is interesting to note that the proportion of lamellae oriented along the main direction $\varphi_1$ decreases with pressure, whereas the proportion of lamellae oriented along the second main direction $\varphi_2$ slightly increases, as if the lamellae reoriented themselves to be less anisotropically distributed. More particularly, this reorientation happens in the middle and posterior stroma, contrary to the collagen lamellae of the anterior stroma which tend to keep their quasi-isotropic orientation. Again, no direct link can be established between the observed macroscopic strain and the response of the collagen lamellae to increasing pressure.

## Summary

These experiments have shown that collagen lamellae reorganize themselves when the cornea is submitted to mechanical load, in the stretch direction for the tensile test, and with a flip in orientation during inflation test, but in all cases, although the fraction of lamellae changes between the two main orientations, these main directions do not change. However, there is no clear and simple relationship between the reorientation of the lamellae and the macroscopic strain measured by DIC. For this reason, microscopic observations are interesting and suitable when it comes to understanding a part of the response of the cornea, but they do not fully explain the behavior of the tissue.

## 3.2. *In-vivo* mechanical characterization of cornea

Even though *ex-vivo* experiments can give insight into the mechanical properties, *in-vivo* properties are the one ophthalmologists look at to detect pathology in hospitals. In the past few years, two main methods have emerged to measure the response of the cornea to intraocular pressure: Brillouin optical microscopy and optical coherence elastography (OCE). Although it has been shown that these methods can be used *in-vivo*, their clinical development is not yet complete and the authors who attempted to use it so far only have results on a very limited number of patients, if any.
<figure><img src="image_37.png" /><figcaption>FIGURE 35. a. Polar plot of the lamellae orientation drawn from the X-ray analysis (extracted from [Bell et al., 2018]). b.1. Distribution of the orientation probability of the collagen lamellae on the whole thickness of the cornea and on the anterior, middle and posterior stroma. $\varphi_1$ and $\varphi_2$ represent the two main orientations. b.2. and b.3. Proportions of collagen lamellae oriented along $\varphi_1$ and $\varphi_2$ (± 15°) (extracted from [Benoit et al., 2016]).</figcaption></figure>
### 3.2.1. Brillouin optical microscopy

#### Principle of the method

Brillouin optical microscopy is a non-invasive technique used to characterize the tissue. It has been used for the first time on human by Scarcelli et al. [Scarcelli et al., 2012]. The principle is presented in Figure 36 and briefly explained hereafter. A near-infrared laser light is focused on the sample and creates a local thermodynamic acoustic wave. The Doppler Brillouin frequency shift $\Omega$ is then analyzed by a confocal spectrometer and can be directly linked to the real part of longitudinal viscoelastic modulus $M'$ [Scarcelli et al., 2014] using the relationship $M' = \rho\lambda^2\Omega^2/4n^2$ with $\rho$ the density of the material, $\lambda$ the optical wavelength and $n$ the refractive index. At low frequencies (around 1 Hz), the variation of the longitudinal modulus $M'$ can be easily linked to the variation of the classical Young's modulus $E'$ using this relationship: $\delta E'/E' = (1/a)\delta M'/M'$, with $(1/a)$ a conversion factor dependent on the material.

<figure><img src="image_38.png" /><figcaption>FIGURE 36. a. Brillouin microscopy setup (extracted from [Scarcelli et al., 2012]) and b. Brillouin microscopy principle (extracted from [Yun and Chernyak, 2018]).</figcaption></figure>

#### Main results on healthy and pathological cases

So far, Brillouin microscopy has not been extensively used to measure the mechanical properties of corneas. A few examples exist however, first *ex-vivo* [Scarcelli et al., 2013] on healthy and keratoco-nic corneas and then *in-vivo* [Scarcelli et al., 2014, Shao et al., 2019, Seiler et al., 2019]. Figure 37 presents the typical sagittal curvature (in Diopter), thickness and Brillouin shift maps for a 53-year-old person with normal corneas (Fig. 37.a) and a 40-year-old patient with advanced keratoconus (Fig. 37.b) [Scarcelli et al., 2015]. *In-vivo* data reveal homogeneous Brillouin shift in healthy corneas while kera-toconic corneas present an heterogeneous dispersion of the shift. Moreover a statistically significant decrease of Brillouin shift (and with it a decrease of stiffness) was observed in keratoconic corneas compared to healthy corneas [Scarcelli et al., 2015].
<figure><img src="image_39.png" /><figcaption>**FIGURE 37.** Typical sagittal curvature (in Diopter), thickness and Brillouin shift $\Omega$ maps for a. a 53-year-old patient with normal corneas and b. a 40-year-old patient with advanced keratoconus (extracted from [Scarcelli et al., 2015]).</figcaption></figure>

## Summary

Brillouin microscopy being a non-invasive method showing significant differences in mechanical elasticity of normal and pathological corneas is a promising tool to detect early stage keratoconus. However, it is a relatively new technique that only a few people have expertise on, and is therefore not used in hospitals yet. An interesting point will be to determine precisely the scale which is probed with this method, which may be very small: is the lamellae, the fibrils or even smaller structure?

### 3.2.2. Optical coherence elastography (OCE)

#### Principle of the method

Optical coherence elastography is a class of imaging techniques based on an elastography system, whose imaging device is an Optical Coherence Tomography (OCT) microscope (Sec. 2.1.). Kennedy *et al.* and Larin and Sampson reviewed the different techniques of OCE [Kennedy et al., 2014, Larin and Sampson, 2017].

Here the focus is on the methods actually applied to the cornea. The mechanical exciter can be of two main kinds: a compression load (Fig. 38.a) or an acoustic wave perturbation (Fig. 38.b). The two mechanical perturbations are associated with different constitutive laws, which relate the Young's modulus to a characteristics of the response to the perturbation (the observed strain $\epsilon_l$ – Fig. 38.a or the phase velocity $c_p$ – Fig. 38.b). In the first case, tissue displacement can be obtained using one of two methods: the OCT speckle-tracking method or the phase-resolved OCT approach [Wang and Larin, 2015]. Similarly, phase velocity can be derived from the phase of the complex OCT signals in the case of acoustic wave
excitation (details on the computation can be found in [Wang and Larin, 2015]).

<figure><img src="image_40.png" /><figcaption>**FIGURE 38.** Loading schemes and elasticity estimation for the two OCE techniques presented: (a) Compression: a mechanical load is applied and the corresponding strain is measured $\epsilon_l$. (b) Surface Acoustic Wave: an acoustic wave is generated in the tissue and its phase velocity $c_p$ is measured (extracted from [Kennedy et al., 2014]).</figcaption></figure>

## Main results on healthy and pathological cases

Ford *et al.* demonstrated [Ford et al., 2011] that OCE based on speckle tracking can be used on corneas to produce 2D maps of heterogeneous displacement, using IOP as a mechanical load (Fig. 39.a). They used a 2D cross-correlation algorithm to compute the displacement. The observed vertical displacement is consistent with the axial displacement they imposed (20 µm), and the horizontal displacement shows heterogeneity among the central region and the periphery. In a following study, they used the same speckle tracking procedure to compute the displacement resulting from the application of a flat lens on the corneas, in a clinical study with healthy (21 eyes of 12 patients) and keratoconic (15 eyes of 12 patients) corneas [De Stefano et al., 2020]. A biomechanical indicator was created to compare anterior and posterior mechanical responses. They showed that the ratio of anterior to posterior surface stiffness was smaller in keratoconic corneas, creating a novel method, clinically applicable, for the detection of the pathology.

Nahas *et al.* [Nahas et al., 2013] also used speckle-tracking to reveal the presence of two zones in the
response of porcine corneas *ex-vivo* at 1% mean strain of deformation (induced by a piston system). Figure 39.b shows clearly a difference in the 2D vertical strain maps (computed by digital volume correlation) between the epithelium (1.6% of strain – red color) and the other regions of the cornea (less than 1% of strain). However, this technique has not yet been used *in-vivo*.

<figure><img src="image_41.png" /><figcaption>FIGURE 39. Computation of OCE results. a. Horizontal and vertical displacement maps (middle and bottom) computed by a 2D cross-correlation algorithm corresponding to the cornea imaged by OCT (top) and submitted to axial displacement generated by IOP (extracted from [Ford et al., 2011]). b. Lamb phase velocity computed during air-puff experiment on rabbit eye (extracted from [Wang and Larin, 2014]). c. FF-OCT cross-sectional image of porcine cornea superimposed with the corresponding computing strain maps (in %) under a 1% strain application (extracted from [Nahas et al., 2013]). d. Young's modulus for untreated (d.1) and cross-linking-treated (d.2) porcine corneas at 15 (d.1.1 and d.2.1), 20 (d.1.2 and d.2.2) and 25 mmHg (d.1.3 and d.2.3) (extracted from [Singh et al., 2017]).</figcaption></figure>

Wang and Larin [Wang and Larin, 2014] used the clinical air-puff test to create dynamical loading on the cornea, and used spectral analysis of Lamb wave propagation combined with phase signal to compute the displacement of the rabbit eye, first *ex-vivo*. They have highlighted the presence of four regions of different elasticity (refining Nahas et al. study [Nahas et al., 2013]): the epithelium, the anterior stroma, the posterior stroma and the innermost region. Figure 39.c shows the Lamb velocity computed during the experiment. The four regions appear (in accordance with the phase velocity profile
[Wang and Larin, 2014]): the epithelium corresponds to the dark red, the anterior stroma goes from red to green, the posterior stroma from green to blue, and the innermost region is in purple. Lan *et al.* [Lan et al., 2021] then conducted a clinical trial on 18 healthy eyes of 9 patients to assess the mechanical properties of human cornea *in-vivo*. A microliter air-pulse stimulator was used to generate corneal displacements, during which the wave velocity was recorded. Corneal elastic speed was found to increase with IOP and to be positively correlated with central corneal thickness. Ramier *et al.* [Ramier et al., 2020] used a similar technique to measure the shear modulus on 12 human corneas *in-vivo* using Rayleigh wave theory instead of Lamb wave theory. The shear modulus was found to be of 72 ± 14 kPa and decreasing with age.

Singh *et al.* [Singh et al., 2017] managed to obtain Young's modulus maps using an air-pulse delivery system while monitoring the intraocular pressure and imaging porcine corneas *ex-vivo* with an SS-OCT. Figure 39.d presents Young's modulus maps at 15, 20 and 25 mmHg for healthy and CXL-treated corneas. They showed that the technique they adopted was able to quantify changes in Young's modulus between healthy and treated corneas and that cross-linked corneas were stiffer than untreated ones. This technique remains to be applied *in-vivo*.

OCE can also be coupled to acoustic/ultrasound methods to determine corneal elasticity. For example Qu *et al.* [Qu et al., 2016] have coupled Acoustic Radiation Force (ARF – see Sec. 3.1.4.) to an OCE system to compare healthy and cross-linked rabbit corneas *ex-vivo*. They showed that the elastic wave shift dropped for cross-linked corneas, corresponding to a change of stiffness in the tissue. A few years later, they used Lamb wave model to compute the wave velocity resulting from the ultrasound excitation on rabbit eyes *in-vivo* [Jin et al., 2019]. They demonstrated in those two papers the interest of the methodology to assess corneal elasticity *in-vivo*.

## Summary

Similarly to Brillouin microscopy, OCE can be used to detect a pathology associated to a mechanical defect. The main limitation to determine actual values of mechanical properties in the case of cornea is that the mechanical excitation may lead to large deformations. Thus, they cannot be captured by simple linear models that are usually used in OCE. Also, most times the tissue is assumed to be homogeneous, which is not the case for cornea, especially for pathological ones. As a result, the models used to describe the mechanical response of the cornea may have to be adapted to tackle these issues.

### 3.2.3. Conclusion of *in-vivo* mechanical characterization of the cornea

To conclude, even if all the authors have not yet applied their methods *in-vivo* on human, it is only a question on *when* and not *how*. Brillouin microscopy and OCE are very promising techniques to determine mechanical properties of the cornea *in-vivo*, if not quantitatively, at least qualitatively. To this end, they can be used to detect pathology by comparing the measured data with healthy corneas. But as promising as they are, they cannot be used to measure the parameters that are used in mechanical modeling. The table 3.1 summarizes the measures obtained with these *in-vivo* approaches, as well as with the *ex-vivo* ones.
<table><caption>TABLEAU 3.1. Summary of the different techniques and data measured for the mechanical characterization of the cornea.</caption><thead><tr><th>Scale</th><th>Imaging Technique</th><th>Data measured</th><th>Typical measure</th><th>Nature of the cornea</th><th>Local / Global measure</th><th>Stress applied</th><th>Stiffness coefficient</th><th>In-vivo / Ex-vivo</th></tr></thead><tbody><tr><td rowspan="4">Tissue scale</td><td>Laser</td><td>Apex displacement (VS pressure)</td><td>~1/10 mm</td><td>Human</td><td>Local – on surface</td><td>Global</td><td>σ = A(e<sup>Bε</sup> - 1), E = Cp + D</td><td>Ex-vivo</td></tr><tr><td>Stereo-correlation</td><td>Surface displacement (VS pressure)</td><td>~1/10 mm</td><td>Bovine</td><td>Local – on surface</td><td>Global</td><td>NA</td><td>Ex-vivo</td></tr><tr><td>Strain / stress gauge + camera</td><td>Strip displacement / strain (VS force applied)</td><td>~1/10 mm / ~%</td><td>Porcine / bovine</td><td>Global &amp; Local – on surface</td><td>Global</td><td>E = σ/ε</td><td>Ex-vivo</td></tr><tr><td>Brillouin microscopy</td><td>Brillouin shift</td><td>~GHz</td><td>Human</td><td>Local – through thickness</td><td>Global</td><td>M = ρλ<sup>2</sup> Ω<sup>2</sup> / 4n<sup>2</sup>, δE / E = (1/a) δM / M</td><td>In-vivo</td></tr><tr><td rowspan="4">Meso-scale</td><td>OCT / OCE</td><td>3D displacement / strain</td><td>~10 microns / ~%</td><td>Porcine / rabbit</td><td>Local – through thickness</td><td>Global</td><td>Based on wave theory or elasticity</td><td>Ex-vivo / In-vivo</td></tr><tr><td>Scheimpflung 3D topographer</td><td>Corneal deformation – curvature radius</td><td>~1/10 mm</td><td>Porcine</td><td>Local – on surface</td><td>Global</td><td>σ = Rp / 2d, ε = ΔR / R</td><td>Ex-vivo</td></tr><tr><td>AFM as nanoindenter</td><td>Anterior / posterior elasticity</td><td>10's kPa</td><td>Human</td><td>Local – on multiple surfaces</td><td>Local</td><td>F = 4E√R / (3(1 - v<sup>2</sup>)) D<sup>3/2</sup></td><td>Ex-vivo</td></tr><tr><td>Bio-indenter</td><td>Young's modulus</td><td>10's kPa</td><td>Human</td><td>Local – on surface</td><td>Local</td><td>F = 4E√R / (3(1 - v<sup>2</sup>)) D<sup>3/2</sup></td><td>Ex-vivo</td></tr><tr><td rowspan="3">Micro-scale</td><td>SHG</td><td>Distribution of the lamellae orientation</td><td></td><td>Human</td><td>Local – through thickness</td><td>Global</td><td>NA</td><td>Ex-vivo</td></tr><tr><td>X-rays (WAXS)</td><td>Mean orientation of the lamellae</td><td></td><td>Human</td><td>Local – averaged on thickness</td><td>Global</td><td>NA</td><td>Ex-vivo</td></tr></tbody></table>
# 4. Mechanical modeling of the human cornea

We have shown in the previous sections that we have access to the geometry and/or to the micro-structure of the cornea. Similarly, it is possible to determine, at least to some extend, the strain in the cornea. However, data are sparse due to limited numbers of cornea tested. This is particularly true for the keratoconic corneas. As it is interesting to be able to predict the consequences of a change in the microstructure, in the geometry or in the mechanical properties (either pathological or due a treatment), models of cornea have been developed in the past decades.

## 4.1. Mechanical response of cornea

The stroma is considered to be the main contributor to the mechanical response of cornea. In parallel, there is also a lack of data on the other layers of cornea. As a result, "stroma" and "cornea" will be used indifferently in the framework of mechanical modeling. Similarly, fibrils or fibers are sometimes used instead of lamellae, as the models are inspired from ones for fiber-enrich composites.

### 4.1.1. Constitutive behavior of cornea

Cornea is usually considered as a quasi-incompressible material reinforced by fibers – the two families of lamellae can be considered each as a family of fibers – which may sustain very large deformations. As such, it is generally modeled as a hyperelastic material [Alastrué et al., 2006, Pandolfi and Holzapfel, 2008, Gefen et al., 2009, Studer et al., 2010, Simonini and Pandolfi, 2015, Giraudet et al., 2022] – with the possible addition of viscous terms to introduce history dependence and dissipation [Nguyen et al., 2008, Kling et al., 2014, Whitford et al., 2018] when considering the static or dynamic cases, respectively. Dynamic cases are usually used to model air-puff clinical tests. Here the emphasis will be placed on the static response as it is simpler but already quite complex. In this case, the associated strain energy density $\psi$ is then split into three contributions:

$$ \psi = \psi^{iso} + \psi^{vol} + \psi^{lam}, \qquad (1) $$

with an isotropic part $\psi^{iso}$ corresponding to the matrix, the keratocytes and the randomly distributed lamellae, a volumetric part $\psi^{vol}$ penalizing any change of volume and an anisotropic part $\psi^{lam}$, taking into account the mechanical role of oriented lamellae.

The isotropic part depends generally on the first three invariants $I_1$, $I_2$ and $I_3$ or reduced invariants ($\bar{I}_1 = I_1\bar{I}_3^{-1/3}$, $\bar{I}_2 = I_2\bar{I}_3^{-2/3}$) and is usually chosen as a Mooney-Rivlin or neo-Hookean like energy density [Alastrué et al., 2006, Pandolfi and Holzapfel, 2008, Studer et al., 2010, Simonini and Pandolfi, 2015, Giraudet et al., 2022]:

$$ \psi^{iso}(\xi, \mathbf{C}) = \frac{1}{2}C_1(\bar{I}_1 - 3) + \frac{1}{2}C_2(\bar{I}_2 - 3) \qquad (2) $$

with $C_1$ and $C_2$ two parameters of the model characterizing the isotropic part of the tissue.

The volumic part penalizes any change of volume and can have several expression, function only of $J$:
typically, it can be based on the work of P. Ciarlet and G. Geymonat [Ciarlet and Geymonat, 1982].

$$ \psi^{vol}(\xi, \mathbf{C}) = \begin{cases} K \log^2(J) & [\text{Alastrué et al., 2006, Pandolfi and Holzapfel, 2008}] \\ \frac{1}{D}(J-1)^2 & [\text{Studer et al., 2010}] \\ \frac{1}{4}K(J^2 - 1 - 2\log(J)) & [\text{Simonini and Pandolfi, 2015}] \end{cases} \quad (3) $$

with K the bulk modulus (or the corresponding $1/D$). Minimizing the energy $\psi^{vol}$ leads to $J = 1$, so that the volume of the material does not change. So, to obtain a quasi-incompressible material, most of the models consider a high bulk modulus compared to the other moduli of the energy density. A notable exception is the model of Studer et al. [Studer et al., 2010], which has a bulk modulus of the same order of magnitude as the isotropic material constant.

For the anisotropic part, there are two classical ways of taking into account the contribution of fibers in biological tissues [Holzapfel et al., 2019]:

- the 'generalized structure tensor' (GST) approach, which is widely used in Holzapfel-like models for many different tissues reinforced by fibers such as arteries or other blood vessels [Gasser et al., 2006, Alastrué et al., 2008, Bols et al., 2013, Holzapfel et al., 2015], myocardium [Holzapfel and Ogden, 2009] or skin [Jor et al., 2011] and cornea [Pandolfi and Holzapfel, 2008, Montanino et al., 2018, Whitford et al., 2015];

- the 'angular integration' (AI) approach, used for arteries [Sacks, 2003, Billiar and Sacks, 2000], cartilage [Ateshian et al., 2009, Pierce et al., 2016] or cornea [Studer et al., 2010, Nguyen and Boyce, 2011], with 2D data and with 3D data [Petsche and Pinsky, 2013, Giraudet et al., 2022].

The anisotropic energy density used in GST models is usually based on the fourth and sixth invariants ($I_4$ and $I_6$ respectively, considering each family of fiber). Calling $M_{01}$ and $M_{02}$ the mean fiber directions on the reference configuration, the reduced invariants are defined as follow: $\bar{I}_4 = M_{01} \cdot \bar{C} \cdot M_{01}$ and $\bar{I}_6 = M_{02} \cdot \bar{C} \cdot M_{02}$, with $\bar{C} = J^{2/3} \bar{C}$. The associated energy density takes the following form (or a variation there of) [Alastrué et al., 2006, Pandolfi and Holzapfel, 2008]:

$$ \psi^{lam}(\xi, \mathbf{C}) = \frac{k_{11}}{2k_{21}}\{\exp[k_{21}(\bar{I}_4 - 1)^2] - 1\} + \frac{k_{12}}{2k_{22}}\{\exp[k_{22}(\bar{I}_6 - 1)^2] - 1\}. \quad (4) $$

An exponential function is generally used to mimic the mean response of the collagen lamellae, as they are considered to behave like collagen fibrils, with a low stiffness response at low strain and a rapid change of response leading to a high stiffness at high strain (Fig. 40).

In particular, it allows to have a quasi null response of the collagen lamellae when they are crimped (toe region in Fig. 40). It is also consistent with the pressure with apex displacement results from inflation tests [Elsheikh et al., 2008]. Considering the whole cornea, a parameter ($\kappa$ in [Pandolfi and Holzapfel, 2008]) can be added to take into account the variation of the scatter of the fibers in the tissue. In this approach, at each point of computation, the mean and the scatter of the fiber direction are assessed, and a global mechanical contribution is computed based on these two parameters, for each family of fibers.

In the AI case, the approach is reversed. An elementary response for collagen lamellae $\delta\psi^{lam}$ is consi-
<figure><img src="image_42.png" /></figure>

FIGURE 40. Collagen fibril response to tension (from [Fratzl et al., 2008]). At low strain, the fibrils are crimped and thus oppose practically no resistance to tension (toe region). In the transitional regime (heel region), fibrils tend to align one next to the other and to straighten. They begin to oppose resistance to tension. In the linear elastic range region, fibrils are not crimped anymore – their stiffness can be measured using a tension test along the direction of the fibrils.

dered, and the total mechanical response on a particular location $\psi^{lam}(\xi, C)$ is the integral over the unit sphere $\omega$ (for the most complete 3D model [Petsche and Pinsky, 2013]) of the elementary contribution weighted by the density distribution $\rho(\xi, \theta, \phi)$ of the fiber at this location,

$$ \psi^{lam}(\xi, C) = \int_{\omega} \rho(\xi, \theta, \phi) \delta\psi^{lam}(\xi, \theta, \phi) d\omega. \qquad (5) $$

In that case also, all the models found in the literature on cornea for the elementary mechanical response of lamellae $\delta\psi^{lam}$ are based on exponential (or logarithmic) or bi-slope functions:

$$ \delta\psi^{lam}(\xi, \theta, \phi) = \begin{cases} \frac{\alpha}{\beta}[\exp(\beta\{\lambda_f^2 - 1\}) - \beta\lambda_f^2] & [\text{Nguyen and Boyce, 2011}] \\ \frac{\alpha_1}{2\alpha_2}[\exp(\alpha_2\{I_4 - 1\}^2) - 1] & [\text{Petsche and Pinsky, 2013}] \\ \frac{\mu}{\gamma}(\bar{I}_4^{\gamma/2} - 1) - \mu \ln(\bar{I}_4^{1/2}) & [\text{Studer et al., 2010}] \\ \frac{1}{2}kl_u(\frac{\lambda}{\lambda_u} - 1)_+^2 & [\text{Giraudet et al., 2022}] \end{cases} \qquad (6) $$

with

$$ \left\{ \begin{array}{l} \lambda_f^2 = I_4 = \lambda^2 = a \cdot C \cdot a \\ \bar{I}_4 = a \cdot \bar{C} \cdot a \\ a(\theta, \phi) \quad \text{the local integrating direction} \end{array} \right. \qquad (7) $$

In the case of Studer *et al.* [Studer et al., 2010], they use a modified version of the polynomial Odgen law [Markert et al., 2005] to include one direction of anisotropy, giving a logarithmic formula which cannot be linked directly to the collagen fibril response of Figure 40.
Whether using GST or AI approaches, these models do not take into account the energy due to water exchange, except for [Cheng et al., 2015], so that the action of the endothelium is not considered. Also, validations of these models are based on very few experiments – as the quantity of data on cornea is limited – measuring apex displacement ([Elsheikh et al., 2008, Lombardo et al., 2014] for the human cornea) or 3D displacement of the anterior surface ([Boyce et al., 2008] for bovine cornea), and performed exclusively on healthy subjects.

### 4.1.2. Density distribution of fibers

The 3D density distribution of fibers is not available for the whole cornea. However, X-ray scattering (Fig. 7 – [Aghamohammadzadeh et al., 2004, Boote et al., 2006]) allows a 2D mean representation of the oriented lamellae to be obtained. Two ways of incorporating these data are found in the literature. The first option is to look at the data at each measurement point – an example is given Figure 41. It is well known that the intensity at each point of measurement $I_{exp}$ of X-ray scattering can be represented by a function $I_m$,

$$I_m(\phi|\kappa_{ip,1}, \kappa_{ip,2}, \mu_1, \mu_2) = I_{iso} + C_1VM_{ip}(\phi|\kappa_{ip,1}, \mu_1) + C_2VM_{ip}(\phi|\kappa_{ip,2}, \mu_2), \quad (8)$$

depending on $I_{iso}$, a constant component representing the isotropic part of the measure and $C_1$ and $C_2$, the measures of the number of oriented lamellae in each direction at the point of measure. The distributions of lamellae is described by Von Mises distributions:

$$VM_{ip}(\phi|\kappa_{ip}, \mu) = \frac{e^{\kappa_{ip} \cos(2(\phi-\mu))}}{C(\kappa_{ip})}, \quad (9)$$

where $\mu$ is the main direction of the fiber family, $\kappa_{ip}$ represents a measure of the scatter of the lamellae, and $C(\kappa_{ip})$ is a normalization factor.

<figure><img src="image_43.png" /><figcaption>FIGURE 41. Determination of the 2D density distribution. a. Polar plot of the direction of the lamella [Aghamohammadzadeh et al., 2004], b. zoom on a particular measurement point, and c. experimental and corresponding optimized intensity (extracted from [Giraudet et al., 2022]).</figcaption></figure>

Minimizing the error $|I_m - I_{exp}|$ at each point of measurement allows the parameters linked to the lamellae to be found: $C_i$ the relative number of oriented lamellae, $\mu_i$ their main direction and $\sigma_{ip,i}$ their
scatter. Afterwards, the data are interpolated on the whole cornea (i.e. at the nodes, for finite element modeling).

The second technique is based on a model by Pinsky et al. [Pinsky et al., 2005] (inspired by X-ray images), which gives the probability density function $\Phi(R, \phi|\theta)$, with $(R, \phi)$ being the polar coordinate of the location (the center of the cornea being the center of the coordinate system and the N-T and S-I directions being the x- and y-axes):

$$ \Phi(R, \phi|\theta) = \begin{cases} \Phi_1(R, \phi|\theta) = \cos^{2n}(\theta) + \sin^{2n}(\theta) + c_1 & \text{for } R \in [0, 4] \\ \Phi_2(R, \phi|\theta) = \sin^{2n}(\theta - \phi) + c_2 & \text{for } R \in [5.5, 6.5] \end{cases} \quad (10) $$

with a linear combination of the 2 functions for $R \in [4, 5.5]$ to smoothly transition from one to the other. Figure 42 presents the results for the case used by Studer et al. [Studer et al., 2010] with $c_1 = 0.45$, $c_2 = 0.72$ and $n = 4$ and the linear combination $\Phi(R, \phi|\theta) = w_1(R)\Phi_1(R, \phi|\theta) + (1-w_1(R))\Phi_2(R, \phi|\theta)$ in between with $w_1 = \frac{1}{2}\{\cos(\frac{\pi}{1.5}[5.5-R])+1\}$.

<figure><img src="image_44.png" /><figcaption>FIGURE 42. Probability distribution map, computed using Eq. (10), with $c_1 = 0.45$ and $c_2 = 0.72$. Transitional zone corresponds to radius between 4 and 5.5mm (dashed circles) [Studer et al., 2010].</figcaption></figure>

Both approaches give information on the so called in-plane orientation of the lamellae, but it gives no information on the out-of-plane distribution. The development of SHG microscopy imaging of cornea made it possible to have a model for the third dimension. Winkler et al. [Winkler et al., 2013] imaged the cornea and Petsche and Pinsky [Petsche and Pinsky, 2013] built a model of the distribution on the third dimension. SHG images (Fig. 43 – middle) are studied to model the out-of-plane scatter of the lamellae.
The authors show that there is an exponential dependency of the cut-off-angle $\psi_{coa}$ with normalized depth $s$:

$$ \psi_{coa}(s) = 28.6^{\circ} \frac{e^{3.19(1-s)} - 1}{e^{3.19} - 1}, \qquad (11) $$

with a maximum cut-off-angle of 28.6°. No lateral heterogeneity has been measured. Then the out-of-plane distribution of the fibers can be modeled either as a Von Mises distribution

$$ \mathrm{VM}_{t}(\theta|\kappa_{t}, \nu) = \frac{e^{\kappa_{t} \cos(2\theta)}}{C(\kappa_{t})} (C(\kappa_{t}) \text{ being a normalization factor}), $$

taking into account the depth variation of the maximum out-of-plane angle $\psi(s)$ to characterize the change of the out-of-plane scatter $\kappa_t$ in thickness, or as a coupled Heaviside function (as in [Petsche and Pinsky, 2013])

$$ \rho^{SHG}(s, \phi) = H[\phi - \{\frac{\pi}{2} - \psi_{coa}(s)\}] - H[\phi - \{\frac{\pi}{2} + \psi_{coa}(s)\}], $$

with $\phi$ the local inclination angle.

<figure><img src="image_45.png" /><figcaption>FIGURE 43. Extraction of the 3D density distribution (from [Petsche and Pinsky, 2013]). The SHG images allow a model to be built for the out-of-plane angle $\psi$ of the lamellae with depth.</figcaption></figure>

The table 4.2 gives a summary of the different models in the literature cited here.
<table><caption>TABLEAU 4.2. Summary of the energy functions and parameters values used in the last paper on modeling of the cornea as a hyperelastic material only.</caption><thead><tr><th></th><th>Notation</th><th>Reference</th><th>Expression</th><th>Value of the coefficients</th><th>Comments</th></tr></thead><tbody><tr><td rowspan="7">Isotropic matrix</td><td rowspan="7">ψ<sup>iso</sup></td><td>[Alastrué et al., 2006]</td><td>$\frac{C_1}{2}(\bar{I}_1 - 3) + \frac{C_2}{2}(\bar{I}_2 - 3)$</td><td>$C_1 = 5 \text{ kPa}$<br>$C_2 = 0 \text{ kPa}$</td><td rowspan="7">The neo-Hookean is mostly used to represent the isotropic matrix, but the Mooney-Rivlin energy allows to take into account second order response, that may exists in the cornea (no experimental data available)</td></tr><tr><td>[Pandolfi et al., 2008]</td><td>$\frac{c}{2}(\bar{I}_1 - 3)$</td><td>$c = 60 \text{ kPa}$</td></tr><tr><td>[Simonini et al., 2008]</td><td>$\frac{\mu_1}{2}(\bar{I}_1 - 3) + \frac{\mu_2}{2}(\bar{I}_2 - 3)$</td><td>$\mu_1 = 60 \text{ kPa}$<br>$\mu_2 = -10 \text{ kPa}$</td></tr><tr><td>[Studer et al., 2010]</td><td>$\frac{C_{10}}{2}(\bar{I}_1 - 3)$</td><td>$C_{10} = 60 \text{ kPa}$</td></tr><tr><td>[Nguyen et al., 2011]</td><td>$\frac{\mu}{2}(\bar{I}_1 - 3)$</td><td>$\mu \sim 26.5 \text{ kPa}$</td></tr><tr><td>[Petsche et al., 2013]</td><td>$\frac{\mu}{2}(\bar{I}_1 - 3)$</td><td>$\mu = 5.59 \text{ kPa}$</td></tr><tr><td>[Giraudet et al., 2022]</td><td>$\kappa_1(\bar{I}_1 - 3) + \kappa_2(\bar{I}_2 - 3)$</td><td>$\kappa_1$ and $\kappa_2$ distributed over the cornea ~MPa</td></tr><tr><td rowspan="10">Quasi-incompressibility</td><td rowspan="10">ψ<sup>vol</sup></td><td>[Alastrué et al., 2006]</td><td>$\frac{1}{D}(\ln(J))^2$</td><td>$D = 13.33 \text{ MPa}^{-1}$</td><td rowspan="10">As cornea is considered as a quasi-incompressible tissue, the volumetric part of the energy function is here to ensure that the volume is preserved (J~1), by penalizing using a relatively huge bulk modulus (at least one order of magnitude higher than the coefficient of the isotropic part)</td></tr><tr><td>[Pandolfi et al., 2008]</td><td>$K(\log(J))^2$</td><td>$K = 5.5 \text{ MPa}$</td></tr><tr><td>[Simonini et al., 2008]</td><td>$\frac{1}{4}K(J^2 - 1 - 2 \log(J))$</td><td>$K = 5.5 \text{ MPa}$</td></tr><tr><td>[Studer et al., 2010]</td><td>$\frac{1}{D}(J - 1)^2$</td><td>NK</td></tr><tr><td>[Nguyen et al., 2011]</td><td>$\frac{1}{4}K(I_3 - \ln(I_3) - 1)$</td><td>NK</td></tr><tr><td>[Petsche et al., 2013]</td><td>$\frac{\lambda}{4}(J^2 - 1) - (\frac{\lambda}{2} + \mu)\ln(J)$</td><td>$\lambda = 500 \text{ kPa}$</td></tr><tr><td>[Giraudet et al., 2022]</td><td>$K(J^2 - 1 - 2 \log(J))$</td><td>$K = 80 \text{ kPa}$</td></tr><tr><td>[Alastrué et al., 2006]</td><td>$\frac{k_1}{2k_2}\{\exp[k_2(\bar{I}_4 - 1)^2] - 1\} + \frac{k_3}{2k_4}\{\exp[k_4(\bar{I}_6 - 1)^2] - 1\}$</td><td>$k_1 = 102.643 \text{ MPa}$<br>$k_2 = 102.643$<br>$k_3 = 0.004852 \text{ MPa}$<br>$k_4 = 102.643$</td></tr><tr><td>[Pandolfi et al., 2008]</td><td>$\frac{k_{11}}{2k_{21}}\{\exp[k_{21}(\bar{I}_4^* - 1)^2] - 1\} + \frac{k_{12}}{2k_{22}}\{\exp[k_{22}(\bar{I}_6^* - 1)^2] - 1\}$<br>$\bar{I}_i^* = \kappa_i \bar{I}_i + (1 - 3\kappa_i)\bar{I}_i$</td><td>$k_{11} = 20 \text{ kPa}$<br>$k_{21} = 400$<br>$k_{12} = 20 \text{ kPa}$<br>$k_{22} = 400$</td></tr><tr><td>[Simonini et al., 2008]</td><td>$\frac{k_{11}}{2k_{21}}\{\exp[k_{21}(\bar{I}_{4,1}^* - 1)^2]\}\{1 + K_1^* \sigma_{\bar{I}_{4,1}}^2\} + \frac{k_{12}}{2k_{22}}\{\exp[k_{22}(\bar{I}_{4,2}^* - 1)^2]\}\{1 + K_2^* \sigma_{\bar{I}_{4,2}}^2\}$</td><td>$k_{11} = 40 \text{ kPa}$<br>$k_{21} = 200$<br>$k_{12} = 40 \text{ kPa}$<br>$k_{22} = 200$</td></tr><tr><td>ψ<sup>lam</sup> (GST)<br>or<br>δψ<sup>lam</sup> (AI)</td><td>$\frac{\mu_m}{\gamma_m}\{\bar{I}_4^* - 1\} - \mu_m \ln(\bar{I}_4)^{\frac{1}{2}} + \frac{\mu_k}{\gamma_k}\{\bar{I}_6^* - 1\} - \mu_k \ln(\bar{I}_6)^{\frac{1}{2}}$</td><td>$\mu_m = 24 \text{ MPa}$<br>$\gamma_m = 0.13$<br>$\mu_k = 95 \text{ MPa}$<br>$\gamma_k = 0.08$</td></tr><tr><td>[Studer et al., 2010]</td><td>$\frac{\alpha}{\beta}\{\exp[\beta(\lambda_f^2 - 1)] - \beta\lambda_f^2\}$<br>$\lambda_f^2 = \frac{\alpha}{\beta}\frac{C}{a}\frac{a}{\alpha} = (\cos(\theta), \sin(\theta), 0)$</td><td>$\alpha \sim 0.12 \text{ kPa}$<br>$\beta \sim 91$</td></tr><tr><td>[Nguyen et al., 2011]</td><td>$\frac{\alpha_1}{2\alpha_2}\{\exp[\alpha_2(\bar{I}_4 - 1)^2] - 1\}$</td><td>$\alpha_1 = 638 \text{ kPa}$<br>$\alpha_2 = 312$</td></tr><tr><td>[Petsche et al., 2013]</td><td>$\frac{\alpha_1}{2\alpha_2}\{\exp[\alpha_2(\bar{I}_4 - 1)^2] - 1\}$</td><td>$\alpha_1 = 638 \text{ kPa}$<br>$\alpha_2 = 312$</td></tr><tr><td>[Giraudet et al., 2022]</td><td>$\frac{1}{2}k\lambda_u l_0 (\frac{\lambda}{\lambda_u} - 1)^2 + t_u l_0 \lambda$<br>$\lambda^2 = \frac{r_0}{\lambda_0} \frac{C}{r_0}$</td><td>$k\lambda_u l_0 \sim 100-1000 \text{ kPa}$<br>$t_u = 0$</td></tr></tbody></table>
### 4.1.3. Validation of models on inflation test

Most models are validated on inflation tests, against the data of Elsheikh et al. [Elsheikh et al., 2008] (see Sec. 3.1.1.), since they are the only available data on human corneas. Finite element (FE) methods are used to compute the mechanical response of cornea to intraocular pressure. Even though the mechanical behavior of cornea and the boundary conditions can change from one author to another, the principle remains the same. The tissue is fixed to the sclera and the intraocular pressure is applied to the posterior surface of the cornea. No other forces are applied. A constitutive model has to be provided. The geometry of cornea is described by the biconic function and usually adapted to real patient ones. A spatial discretization is performed to create the mesh used in the FE code. Figure 44 presents a typical scheme (Fig. 44.a) and mesh (Fig. 44.b) used to implement the corneal model in a FE code [Gefen et al., 2009].

<figure><img src="image_46.png" /><figcaption>**FIGURE 44.** FE modeling of an inflation test. a. Model description of the forces applied to the cornea (IOP) and of the boundary conditions. b. Typical mesh used to describe the geometry of the tissue (extracted from [Gefen et al., 2009]).</figcaption></figure>

Figure 45 presents the results for four different cases of modeling: one model with AI approach and only in-plane distribution of the lamellae (a.1 and a.2 – [Studer et al., 2010]), one model with GST approach and only in-plane distribution of the lamellae (b – [Pandolfi and Holzapfel, 2008]) and two models with AI approach and 3D distribution of the lamellae (c – [Petsche and Pinsky, 2013] and d – [Giraudet et al., 2022]). Figures 45.a.1, a.2, c and d are comparing simultations to data from Elsheik et al. study [Elsheikh et al., 2008] on human cornea. Fig. 45.b is comparing simulation to data from [Elsheikh and Anderson, 2005] on porcine corneas. It shows that all the models can be calibrated to fit the data of pressure with apex displacement and for both age groups in the first case. The parameters found in the validation process provide insight into the mechanical behavior of healthy cornea ex-vivo but they may not be unique and should be adapted for pathological cases.

## 4.2. The case of the keratoconus

The case of keratoconus has been studied numerically through inflation tests by several authors [Gefen et al., 2009, Carvalho et al., 2009, Norouzpour and Mehdizadeh, 2012, Pandolfi et al., 2019, Giraudet et al., 2022] considering different hypotheses based on the same assumptions: in keratoconic corneas a change of geometry is observed and the stiffness is decreased in the cone region with respect to the rest of the cornea (Sec. 3.2.). Gefen et al. [Gefen et al., 2009], Carvalho et al. [Carvalho et al., 2009] and Giraudet et al. [Giraudet et al., 2022] studied both the thinning of the region of the cone and the reduction of some mechanical properties of cornea (meridian elastic modulus and shear modulus perpendicular to corneal surface). While Gefen et al. [Gefen et al., 2009] shows that the main ingredient to
<figure><img src="image_47.png" /><figcaption>FIGURE 45. Results of inflation test simulations compared with Elsheikh et al. [Elsheikh et al., 2008, Elsheikh and Anderson, 2005] data. a.1 and a.2. Results extracted from [Studer et al., 2010] for two age groups (a.1. 65–79 years old corneas and a.2. 80–95 years old corneas) – model with AI approach and only in-plane distribution of the lamellae compared to [Elsheikh et al., 2008]. b. Results extracted from [Pandolfi and Holzapfel, 2008] – model with GST approach and only in-plane distribution of the lamellae compared to data from [Elsheikh and Anderson, 2005]. c. Results extracted from [Petsche and Pinsky, 2013] – model with AI approach and 3D distribution of the lamellae (with different cases for the out-of-plane distribution) compared to data from [Elsheikh et al., 2008]. d. Results extracted from [Giraudet et al., 2022] – model with 3D AI approach, with different cases for the pre-elongation of the lamellae.</figcaption></figure>
keratoconus is the change in shape of cornea [Gefen et al., 2009], Carvalho *et al.* [Carvalho et al., 2009] and Giraudet *et al.* [Giraudet et al., 2022] state that the change of local mechanical properties is needed to generate a keratoconus. Figure **46** shows the maximum tissue displacement (Fig. **46.a**) and maximal diopter (Fig. **46.b**) with increasing IOP for normal and keratoconic cases from Gefen *et al.* study [Gefen et al., 2009]. The maximum displacement is more than three times greater in keratoconic corneas than in healthy tissues. For the diopter, while there is no change with pressure for healthy cornea, an increase of 8 diopters is found for keratoconic cases between 10 and 25 mmHg. In Giraudet *et al.* study, the change in diopter is also seen in the case of the keratoconus, while smaller (around 2 diopters as in [McMonnies and Boneham, 2010]). Figure **46.c** shows the results in variation of the SimK (representative diopter of the cornea) during an inflation test between 15 and 30 mmHg. It clearly showed that a change in geometry is not enough to have the expected change in diopter, but there is a need of mechanical weakness. Pandolfi *et al.* [Pandolfi et al., 2019] adopted another approach. Starting from a healthy

<figure><img src="image_48.png" /><figcaption>**FIGURE 46.** a. Maximum corneal tissue displacement with IOP for healthy and keratoconic cases. b. Maximum dioptric power with IOP for normal and keratoconic corneas (extracted from [Gefen et al., 2009]). c. Variation of the SimK (a diopter indicator) with IOP for healthy (top) and keratoconic (bottom) geometries, for cornea with or without mechanical weakness (extracted from [Giraudet et al., 2022]).</figcaption></figure>

geometry, the authors created a network of fibrils and cross links (Fig. 47.a.2) to represent the lamellae of collagen and their links to each other (glycosaminoglycans or other proteoglycans), and finally decreased the mechanical strength of each part of the fibrils unit cell (Fig. 47.a.1). Looking at the displacement results of Fig. 47.b.1–4, one must conclude that the weakening of the intra-cross link (in orange
<figure><img src="image_49.png" /></figure>

**FIGURE 47.** a.1. Unit cell of the fibrils and cross links. Fibrils are divided into two sets: fiber set 1 (F1) in blue and fiber set 2 (F2) in red. Two types of cross-links are defined: cross-links between the fibrils of the same set (inter-link: C1 and C2) are in green and between the fibrils of two different sets (intra-link: C12) in orange. a.2. Fibrils and cross-link networks. b.1-4. Comparison of the deformed shape and displacement field between b.1. a healthy cornea, b.2. a cornea weakened only in the F1 and F2 fibril stiffness, b.3. a cornea weakened only in the C1 and C2 cross-link stiffness, and b.4. a cornea weakened only in the C12 cross-link stiffness.

on Fig. 47.a.1–2) gives a high deformability of the cornea and may lead to thinning and bulging of the tissue in the zone of the cone, as found in keratoconus.

Keratoconus has been studied numerically by various authors with different results regarding the explanation for the inception of the disease: the thinning of the zone (geometrical factor) or the decreasing of the fibrils stiffness (mechanical weakening) were considered and found to be relevant. While the geometry found for keratoconic corneas are consistent with actual cases of keratoconus, there is however no comparison of quantitative parameters with real data.

**5. Conclusion**

This paper presented an overview of the knowledge on cornea for mechanical modeling. It first presented it macro and microstructure (Sec. 1.), then detailed the OCT techniques used to image the cornea (Sec. 2.). Then, it showed the results of the different experiments performed on the cornea to measure its mechanical parameters (Sec. 3.) and finally how they can be used to model human cornea as an hyperelastic material reinforced by fibers (Sec. 4.). This paper did not present the viscous behavior of the cornea as it only focused on static experiments, but it has to be taken into account when modeling dynamic experiments such as air-puff.

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