The Role of the Sclera and Orbital Tissues in the Biomechanical Deformation Response of the Cornea and Whole Eye Under Loading B

Home , Cornea, Corneal epithelium, Orbit (anatomy)

The Role of the Sclera and Orbital Tissues in the Biomechanical Deformation Response

of the Cornea and Whole Eye Under Loading by Dynamic Scheimpflug Analyzer

Dissertation

Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy

in the Graduate School of The Ohio State University

Boihoan Audrey Nguyen, M.S.

Graduate Program in Biomedical Engineering

The Ohio State University

2019

Dissertation Committee

Cynthia J. Roberts, Ph.D, Advisor

Matthew A. Reilly, Ph.D., Co-advisor

Jun Liu, Ph.D., Committee Member

Copyrighted by

Boihoan Audrey Nguyen

2019

Abstract

The biomechanical behavior of the ocular and orbital tissues are important to

maintaining proper structure and function of the eye. The cornea and the sclera are the two

main components of the ocular shell, which are loaded by IOP and are responsible for maintaining the structure and therefore function of the eye. This work comprises of four

studies which explore the contribution of the sclera and the biomechanical response of the

eye.

The first study focused on the development of a finite-element model to explore the impact of varying scleral properties on the deformation response of the cornea to an air- puff. An axisymmetric model of the eye – consisting of a cornea, sclera, and vitreous humor

– loaded internally by intraocular pressure (IOP) and externally by an air-puff from a noncontact tonometer was generated in COMSOL 5.2a. Our results showed that increasing scleral stiffness (with constant corneal stiffness and IOP) resulted in decreasing displacement of the corneal apex, i.e. the cornea responded to the air-puff as if it had stiffer mechanical properties. The finite-element model also showed that apical displacement decreased nonlinearly with increasing IOP, which is consistent with literature reports. This study demonstrated that the sclera has an inseparable impact on the biomechanical deformation response of the cornea, and that the corneal response to an air-puff is the result of both corneal and scleral properties, in addition to IOP. ii

The second study further explores the impact of varying scleral properties on corneal deformation response in an ex vivo experiment. Our results showed that the corneal response to an air-puff is significantly impacted by scleral properties. Several dynamic corneal response (DCR) parameters exported by the CorVis ST noncontact tonometer were analyzed to compare the effect of stiffening the sclera while leaving the cornea untreated.

With stiffened sclera, changes in DCRs showed that the cornea had an apparently stiffer response to air-puff loading. Additionally, we showed that the Stiffness Parameter (SP) at

Highest Concavity parameter is sensitive to changes in scleral properties and changed significantly after scleral stiffening. The Stiffness Parameter at First Applanation (A1), which is known to be sensitive to corneal properties, showed no significant differences with scleral properties. The results of this study have important clinical implications, demonstrating corneal response is significantly affected by scleral properties and providing insights into potential response parameters for evaluating scleral biomechanics with a clinical device.

The third study focused on the development of an analytical model to determine the contribution of ocular and orbital tissues to the motion of the cornea and the whole-eye

under air-puff loading. During air-puff loading, the whole-globe experiences a rearward

displacement into the orbit while the cornea undergoes its deformation response. A one- dimensional model of centerline motion was developed to evaluate the relative contributions of ocular and orbital tissues to this motion. The resultant model was validated with data from ex vivo studies and showed that (1) orbital tissues were best represented by a Kelvin-Voigt viscoelastic solid, (2) the cornea must necessarily be viscoelastic, and was

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best represented as a 3-parameter standard linear solid, and (3) that the contribution of the

sclera must be considered to reproduce the timing of key events in in vivo motion, and that

the sclera is accurately modeled as a nonlinear stiffening spring.

The final study applied the analytical model to a clinical cohort of newly-diagnosed glaucoma subjects undergoing treatment with prostaglandin-analogs (PGA)s. While we originally hypothesized that measured whole-eye motion would increase as a result of potential atrophy of orbital fat, our results showed a significant decrease in measured whole-eye motion. The results of the analytical model showed that while there were no changes in orbital tissue properties, that the apparent paradox of decreasing whole-eye motion following PGA-treatment was the result of decreased corneal and scleral stiffness, in addition to a reduction in IOP. Calculated impulse of the cornea and globe showed that

PGA-treatment resulted in a significant increase in corneal impulse and decrease in globe impulse, demonstrating that a greater proportion of the kinetic energy from the air-puff is absorbed by the cornea and less kinetic energy is used in the rearward translation of the globe.

Overall, these studies have demonstrated that the biomechanical deformation response of the cornea to an air-puff is the result of the coupling of the corneal, scleral, and orbital tissue properties in addition to intraocular pressure. This work has provided insight into the relative contributions of ocular and orbital tissues to the deformation response of the cornea and globe, and may help to elucidate the changes in tissue properties in different disease states or treatment conditions.

Dedication

This document is dedicated to my parents, Tram Nguyen and Kieuhanh Ngo.

Acknowledgments

I would like to thank my advisor, Dr. Cynthia Roberts, for her guidance and support

of my graduate career. I will always be grateful that she took a chance on me and accepted

me into her lab, helping me to develop my skills and confidence as a researcher and making

all this possible. I would also like to thank my co-advisor, Dr. Matt Reilly, for being an

integral part of my research career, for his near-infinite patience, and for all of our conversations.

I would like to thank Dr. Jun Liu for serving on both my candidacy and dissertation committees, and for her helpful advice and insights on my research. I would also like to thank the members of the Liu lab for opening their lab space and our supportive collaborations. I would like to thank Dr. Alan Litsky, who has been an incredibly

supportive mentor to me since the beginning of my graduate studies. Thank you to all of

my colleagues in the Ophthalmic Engineering group for your friendship and camaraderie.

Finally, I would like to thank my friends (especially Monica Okon and Isabel

Fernandez), my family, my parents, and my love, Matthew Rudy. Words cannot express

how much your support has meant to me, and I am forever grateful to have you in my life.

Vita

The Ohio State University ...... Columbus, OH

Ph.D., Biomedical Engineering ...... 2013-Present

Master of Science, Biomedical Engineering ...... 2013-2017

Bachelor of Science, Biomedical Engineering ...... 2009-2013

Publications

Nguyen, B. A., Roberts, C. J., & Reilly, M. A. (2018). Biomechanical impact of the

sclera on corneal deformation response to an air-puff: a finite-element study. Frontiers in bioengineering and biotechnology, 6.

Fields of Study

Major Field: Biomedical Engineering

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Table of Contents

Abstract ...... ii

Dedication ...... v

Acknowledgments...... vi

Vita ...... vii

List of Tables ...... xii

List of Figures ...... xiv

Chapter 1. Introduction ...... 1

Structure and Function of the Eye ...... 1

CorVis ST ...... 4

Specific Aims ...... 9

Aim 1: Evaluate the biomechanical impact of the sclera on dynamic corneal

deformation response to an air-puff ...... 9

Aim 2: Describe the relative contribution of ocular and orbital tissues to corneal and

whole-eye motion...... 10

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Aim 3: Determine the impact of prostaglandin-analog (PGA) treatment on ocular

and orbital tissue properties and biomechanical response ...... 11

Chapter 2. Biomechanical impact of the sclera on corneal deformation response to an air- puff: a finite-element study ...... 12

Introduction ...... 12

Methods...... 13

Geometry and Mesh ...... 14

Estimation of Stress-Free Geometry ...... 16

Material Properties and Boundary Conditions ...... 17

Ex vivo Experiments ...... 19

Results ...... 21

Discussion ...... 27

Chapter 3: Biomechanical contribution of the sclera to dynamic corneal response in air- puff induced deformation in human donor eyes ...... 31

Introduction ...... 31

Materials and Methods ...... 32

Eye Preparation ...... 32

Experimental Setup ...... 33

Statistical Analyses ...... 36

Results ...... 38

Discussion ...... 41

Conclusions ...... 44

Chapter 4: Development and validation of a nonlinear viscoelastic model of corneal and

whole-eye motion under air-puff loading by a dynamic Scheimpflug analyzer ...... 46

Introduction ...... 46

Methods...... 48

Characteristic in vivo dataset ...... 48

Modeling whole-eye motion ...... 49

Modeling corneal and whole-eye motion ...... 51

Ex vivo validation ...... 53

Results ...... 55

Characteristic in vivo dataset ...... 55

Scleral and corneal validation ...... 59

Orbital fat validation ...... 62

Discussion ...... 63

Conclusion ...... 66

Chapter 5: A nonlinear viscoelastic model of corneal and whole-eye motion of prostaglandin-analog treated subjects under loading by dynamic Scheimpflug analyzer 67

Introduction ...... 67

Methods...... 68

Results ...... 70

Discussion ...... 75

Conclusion ...... 78

Chapter 6: Conclusions of the Study ...... 79

Bibliography ...... 82

List of Tables

Table 2.1 Summary of geometric parameters for finite element model...... 15

Table 2.2 Material properties of ocular tissues in the FE model...... 18

Table 2.3 Spatial measurements of air flow velocity at 4 mm from nozzle at peak

pressure...... 22

Table 2.4 Spatial measurements of air flow velocity along axis of symmetry at peak

pressure...... 23

Table 2.5 Sensitivity analysis of model dependence on vitreous body...... 23

Table 2.6 Summary of simulation results from FE model...... 25

Table 2.7 Summary of ex vivo results from human donor eyes (Nguyen et al., 2018). ... 25

Table 3.1 Select CorVis ST dynamic corneal response parameters and their description.

...... 37

Table 4.1 Results of ex vivo scleral stiffening validation studies. The parameter values

outputted by the model were tabulated pre- and post-scleral stiffening. Significant differences (p<0.05) were observed only in the non-linear stiffening spring term at all levels of IOP, validating that it represented the sclera...... 61

Table 4.2: Parameter values from ex vivo orbital tissues simulation study were tabulated.

There were large changes (>20%) in the magnitude of the model parameters ascribed to

orbital tissues. These large decreases were expected as the gelatin allowed for whole-eye xii

motion to occur while the rigid fixture restricted WEM. There were no substantial

changes (<10%) observed in the magnitudes of the model parameters describing cornea

or sclera...... 63

Table 5.1: CorVis ST DCRs describing first and second applanation (A1, A2) velocities

and whole-eye motion (WEM) following PGA-treatment. Statistical significance was determined if p<0.05 by paired t-test, and is denoted by ‡...... 72

Table 5.2: Parameter values for spring and dashpot elements in the model for PGA-

treated subjects at visit 1 (no treatment) and at visit 2 (1 month of PGA-treatment).

Statistical significance was determined if p<0.05 by paired t-test, and is denoted by ‡. . 72

Table 5.3: Select DCRs describing the deflection velocities of the cornea through

applanation, corneal shape parameters, and stiffness parameters (SP) at first applanation

(A1) and highest concavity (HC). Statistical significance was determined if p<0.05 by

paired t-test, and is denoted by ‡...... 73

Table 5.4: Calculated impulse for the cornea and globe ...... 74

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List of Figures

Figure 1.1: Ocular structure: cornea, sclera, iris, ciliary body, zonules, lens, optic nerve, retina, choroid, and vitreous and aqueous humors. (Liu et al., 2014) ...... 1

Figure 1.2: Scanning electron micrograph of stromal lamellae. Adjacent lamellae make large angles with each other. Lamellae sometimes split and interweave (arrowhead), particular toward the front of the stroma. Bar = 10 µm (Meek and Fullwood, 2001)...... 2

Figure 1.3: Orbit and eye appendages: orbit, orbital fat, and extraocular muscles (Liu et al. 2014)...... 3

Figure 1.4: CorVis ST dynamic Scheimpflug analyzer. (from Oculus Optikgerate GmbH)

...... 5

Figure 1.5: Sample images with markers illustrating first applanation (A1) length, radius of curvature at highest concavity, bending distance at highest concavity, maximum deformation amplitude at highest concavity, and A2 length. (Metzler et al., 2014)...... 6

Figure 1.6: Superimposed frames extracted from a single exam showing (A): Cornea in the Predeformation phase (pseudocolored blue), at maximal corneal deflection

(pseudocolored red), and at maximal whole eye movement (pseudoclored white); (B)

Cornea at maximum deflection (Highest Concavity) with illustration of displacement from predeformation anterior surface arc (blue lin); and (C) Correction for whole eye motion by aligning all corneal position to that at predeformation. (Roberts et al., 2017). . 7 xiv

Figure 1.7: The nine phases of corneal deformation response to an air-puff are superimposed over the plots of corneal deformation (blue) that includes whole eye motion (green), and pure corneal deflection (red), that is the difference between the other two. (Roberts et al., 2017)...... 8

Figure 2.1 (Left) 2D axisymmetric geometry of eye, consisting of cornea, sclera, and

vitreous, surrounded by an air region. The x-axis indicates radial distance from the line of rotational symmetry, and the y-axis indicates axial distance, where the undeformed corneal apex is at 0 mm. (Right) The mesh of the FE model, consisting of quadrilateral

and free triangular elements. The whole-eye geometry consisted of cornea, sclera, and

vitreous humor. The whole globe has a diameter of 24 mm, the cornea has a diameter of

11 mm, and the central corneal thickness (CCT) is 500 μm. The thickness of the sclera

ranged from 400 to 1,000 μm...... 14

Figure 2.2 A convergence study showed that a mesh containing >20,000 elements was

sufficient to achieve discretization errors which were significantly less than normal

experimental variations. This mesh density, shown in Figure 2.1, was therefore adopted

for all subsequent simulations...... 16

Figure 2.3 (Left) The undeformed geometry of the eye, where the interior boundary of

the ocular tunic is highlighted in blue to indicate where the negative IOP will be applied

to estimate the stress-free geometry of the eye. (Center) The resultant stress-free state of

the eye (in blue) compared to the original undeformed geometry prior to loading by the

negative IOP (in black). (Right) Von Mises stresses in the residually-stressed state of the

eye when loaded by IOP, where the black lines represent the geometry of the stress-free

state of the eye prior to loading by IOP...... 17

Figure 2.4: (Left) Experimental setup for ex vivo studies on human donor eyes, showing a

whole globe in the purpose-designed mount in front of the CorVis ST. (Right) A view of

the human donor eye showing the 22-gauage needle inserted into the anterior chamber,

used to set and maintain intraocular pressure...... 20

Figure 2.5 Simulated CorVis ST air-puff velocity profile (left) and pressure profile

(right), which serves as the load on the anterior boundary of the globe. The simulation

profiles agree within 2% of experimental measurements using hot-wire anemometry from

Roberts et al. (2017)...... 22

Figure 2.6 Resultant von Mises stresses and deformation for a globe with scleral-to-

corneal ratio of Young’s moduli of 1.5 and IOP of 10 mmHg (left), scleral-to-corneal

ratio of 1.5 and IOP of 40 mmHg (center), and a globe with scleral-to-corneal ratio of 4.0

and IOP of 40 mmHg (right)...... 24

Figure 2.7 Contour plot of maximum apical displacement predicted by the finite-element model as a function of IOP and scleral-to-corneal ratio of Young’s moduli. The biomechanical response is non-linear with both increasing IOP and increasing scleral-to- corneal ratio...... 26

Figure 2.8 Comparison of simulation results from FE model for several levels of scleral-

to-corneal ratio of Young’s Modulus (SCRs) against ex vivo data. Error bars represent

95% confidence intervals...... 27

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Figure 3.1 (Left) Experimental setup for ex vivo studies on human donor eyes, showing

a whole globe in the purpose-designed mount in front of the CorVis ST. (Right) A view

of the human donor eye showing the 22-gauge needle inserted into the anterior chamber,

used to set and maintain IOP (Nguyen et al., 2018) ...... 34

Figure 3.2 (Left) Excised strip of sclera mounted in a (Right) commercial testing fixture.

A humidified chamber, not pictured, surrounded the test fixture to maintain tissue

hydration...... 35

Figure 3.3 Average tensile stress-strain results for glutaraldehyde-treated and control sclera with error bars representing standard deviation. Statistically significant (p<0.05) differences were observed at 4, 5, and 6% strain using paired t test, denoted by * ...... 38

Figure 4.1 Characteristic in vivo dataset of corneal (red) and whole-eye motion (green)

under air-puff loading (dashed gray). While the cornea is deforming from a convex to a

concave orientation, the whole eye undergoes posterior translation at constant velocity.

Once the cornea has reached maximum concavity and can no longer deform despite

increasing load from the air-puff, the WEM rapidly increases. Maximum corneal motion

occurs prior to maximum air-puff pressure, and the cornea undergoes an oscillatory

motion before recovery. Maximum whole-eye motion occurs near full recovery of the

cornea. (Roberts et al., 2017) ...... 49

Figure 4.2 Representations of the peri-ocular tissues as (a) purely elastic or (b) Kelvin-

Voight (KV)-viscoelastic ...... 50

Figure 4.3 Schematic representation of the sequence of viscoelastic models investigated.

In all cases, the peri-ocular tissues were considered as KV-viscoelastic. The cornea was

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modeled as (a) linearly elastic, (b) KV-viscoelastic, (c) Maxwell-viscoelastic, (d) 3- element linear viscoelastic solid, and (e) 3-element linear viscoelastic solid in parallel with a non-linear spring representing lateral confinement by the sclera...... 52

Figure 4.4: (Left) Experimental setup for ex vivo studies on human donor eyes, showing

a whole globe in the purpose-designed mount in front of the CorVis ST. (Right) A view

of the human donor eye showing the 22-gauge needle inserted into the anterior chamber,

used to set and maintain IOP (Nguyen et al., 2018)...... 54

Figure 4.5: Human donor eye mounted in an acrylic holder filled with 2.4% gelatin to

simulate the bony orbit geometry and orbital fat mechanical properties, respectively. ... 55

Figure 4.6 Results of WEM only simulation where the peri-ocular tissues are represented

as (a) purely elastic or (b) KV-viscoelastic...... 56

Figure 4.7 Comparison of in vivo experimental measurements and model predictoins of

WEM (above) and corneal displacement (below) where (a, b) cornea is represented as purely elastic and where (c, d) cornea is represented as a 3-parameter linear viscoelastic

solid...... 57

Figure 4.8 Comparison of simulation results where (a, b) the non-linear stiffening spring

(representing the sclera) is absent and (c, d) where the sclera is present...... 58

Figure 4.9: Results of ex vivo validation studies, in which the sclera of one eye was

stiffened with 4% glutaraldehyde. Significant (p<0.05) changes were observed only in the

non-linear stiffening spring term (validating that it represents the sclera) before and after

scleral stiffening on donor eyes, denoted by *. There were no changes observed in the

three parameters ascribed to representing corneal properties...... 60

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Figure 4.10: Results of ex vivo orbital tissues study. The WEM and cornea motion

profiles of a human donor eye mounted in the rigid fixture (blue), in 2.4% gelatin in a

simulated orbit (pink), and the characteristic published in vivo subject (black) are shown

here for comparison...... 62

Figure 5.1: Representation of the eye as a 2 degree-of-freedom mass-spring-damper system (Nguyen et al. 2018B) ...... 69

Figure 5.2: Simulation results for one eye of one in vivo PGA-treated subject, showing

the fit of the model to corneal and whole-eye motion before and after PGA-treatment. . 71

Figure 5.3: A comparison of the relationship between the corneal damping parameter (b2)

and IOP for the analytical model and each of four tonometric modalities: (a) CorVis ST

(bIOP), (b) ORA (IOPcc), (c) DCT, and (d) GAT...... 75

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Chapter 1. Introduction

Structure and Function of the Eye

The human eye is a complex organ responsible for vision. Light passes through the transparent cornea, through the pupil where it is focused by the lens onto the retina, where photoreceptors interpret the stimulus into a signal that can be transmitted to the visual cortex in the brain via the optic nerve (Lens et al., 2008; Ethier et al., 2004)

Figure 1.1: Ocular structure: cornea, sclera, iris, ciliary body, zonules, lens, optic nerve, retina, choroid, and vitreous and aqueous humors. (Liu et al., 2014)

The main structural components of the eye are the clear, transparent cornea and the white, opaque sclera which make up the ocular tunic (Figure 1.1). These collagenous tissues are loaded by IOP and are responsible for maintaining proper structure (and therefore proper function) of the eye (Meek, 2008; Watson and Young, 2004, Komai and Ushiki, 1991).

The transparent cornea is a collagenous tissue responsible for approximately two- thirds of the refractive power of the eye (Lens et al., 2008; Elsheikh et al., 2008; Nishida and Saika, 2010). Its mechanical strength comes from the organization of collagen fibrils into lamellae (Figure 1.2), which run parallel to the corneal surface in combination with cross-links between lamella (Meek 2008; Komai and Ushiki, 1991),

and its optical transparency is due to the uniformity and highly ordered configuration of

collagen fibrils throughout its structure (Meek, 2008; Nishida and Saika, 2010; Knupp et

al., 2009).

The sclera comprises the vast majority of the ocular shell, is contiguous with the

cornea, serves as the principal load-bearing tissue of the eye (Coudrillier et al., 2015), and is important to maintaining the relatively rigid shape of the globe (Watson and Young,

2004; Nishida and Saika, 2010). The sclera also serves as the insertions for the six extraocular muscles (Figure 1.3). It is stiffer than the cornea and its mechanical strength also stems from the lamellar structure of collagen (Coudrillier et al., 2015; Watson and

Young, 2004; Meek, 2008). The collagen organization of the sclera is more interwoven

and heterogeneous compared to the cornea, and the size and arrangement of collagen fibrils

is less uniform (Meek, 2008; Watson and Young, 2004, Komai and Ushiki, 1991).

Figure 1.3: Orbit and eye appendages: orbit, orbital fat, and extraocular muscles (Liu et

al. 2014).

Intraocular pressure is maintained by balance of aqueous humor production and

outflow in the anterior chamber of the eye (Acott et al., 2014; Johnson, 2006, Weinreb and

Khaw, 2004). It is an important biomarker in several ocular pathologies, including glaucoma (Weinreb and Khaw, 2004) and ocular hypertension (Copt et al., 1999). Multiple tonometric devices have been developed to measure IOP in a clinical setting and include

contact – such as Goldmann applanation tonometry, the “gold standard” for IOP measurement (Wessels and Oh, 1990; Kotech et al., 2010) and noncontact modalities – such as air-puff tonometry (Luce, 2005; Hong et al., 2013). However, IOP measurement has been shown to be influenced by the structure (Doughty and Zaman, 2000; Bhan et al.,

2002) and properties of the cornea (Liu and Roberts, 2005; Pepose et al., 2007), and

procedures which alter the sclera may introduce error and measurement artifact (Fukasaku

and Marron, 2001; Hipsley et al., 2017). IOP is therefore a confounding variable in the

assessment of ocular biomechanics.

CorVis ST

Often biomechanical studies of ocular tissues are conducted using destructive ex

vivo methods, such as strip testing (Hatami-Marbini, 2014; Buzard and Hoeltzel, 1991;

Elsheikh et al., 2010; Geraghty et al., 2012; Palko et al., 2016 ). However, these methods

may alter fibril orientation and hydration of tissues, and may not be able to fully reproduce

physiological conditions (Hatami-Marbini, 2014; Ruberti et al., 2011, Elsheikh and

Anderson, 2005). Finite-element models have been explored as a way to simulate

physiological conditions, loading, and the effect of refractive surgeries (Roy et al., 2014; 4

Seven and Dupps, 2013; Elsheikh, 2010), or to employ inverse-modeling to determine mechanical properties of tissues (Nguyen and Boyce, 2011;, Kling et al., 2014). However, there is a demonstrated need for clinical in vivo assessment of biomechanics of ocular and orbital tissues. One such device that can provide clinical assessment of the biomechanical response of ocular tissues is the CorVis ST, a dynamic Scheimpflug analyzer and noncontact tonometer.

The CorVis ST is a noncontact tonometer with high speed Scheimpflug camera, and captures 140 images at a frame rate of 4430 Hz of the corneal cross-section as it undergoes loading by the air-puff (Figure 1.4). Image analysis is performed and dynamic corneal response (DCR) parameters are exported for further analysis (Figure 1.5).

Figure 1.4: CorVis ST dynamic Scheimpflug analyzer. (from Oculus Optikgerate

GmbH)

The CorVis ST not only evaluates the corneal deformation response to an air-puff, but also measures the rearward displacement of the globe, or the whole-eye motion (WEM) by tracking the displacement of the corneal periphery (Figure 1.6).

Figure 1.6: Superimposed frames extracted from a single exam showing (A): Cornea in the Predeformation phase (pseudocolored blue), at maximal corneal deflection

(pseudocolored red), and at maximal whole eye movement (pseudoclored white); (B)

The motion of the cornea and the whole-eye over the duration of the CorVis ST air-puff is dynamic and complex (Figure 1.7) Briefly, as the air-puff loads the anterior part of the eye, the cornea and whole eye initiate their backward motion (Ingoing Convex Phase), move through first applanation (A1), and into the Ingoing Concave Phase. During these phases, the whole-eye motion is slow and linearly increasing rearward displacement. The cornea experiences a continuous deflection until it is no longer able to deform and reaches the Oscillation Phase.

Because the air-puff pressure continues to increase after the cornea has reached is

maximum deflection, the whole-eye motion transitions into a rapid nonlinearly increasing motion. As the air-puff pressure begins to decrease, the cornea begins to recover, however the whole-eye still displaces backward until second applantion (A2). After this point, the whole-eye motion begins to recover. The interaction between and respective contributions

of ocular and orbital tissues to this dynamic deformation response may provide new

insights into clinical assessment of ocular biomechanics.

Specific Aims

The goal of this work is to determine the relative contribution of ocular and orbital

tissues to the biomechanical deformation response of the cornea under air-puff loading.

This goal will be accomplished by the following specific aims:

Aim 1: Evaluate the biomechanical impact of the sclera on dynamic corneal deformation

response to an air-puff

The biomechanical impact of varying scleral properties on the dynamic corneal deformation response to an air-puff will be evaluated with a finite-element model and accompanying ex vivo study. Paired human donor eyes will be secured in a custom whole- globe mount, and the CorVis ST will be used to load the eyes and evaluate the corneal response at multiple levels of IOP. One eye within each pair will be randomly selected to have its sclera stiffened with a fixative, and the corneal biomechanical response will be

evaluated to evaluate the effect of scleral stiffening. A finite-element model of the ex vivo

donor eye setup will be created in order to explore the impact of varying degrees of scleral

stiffness and IOP on the deformation response of the cornea to air-puff loading.

Aim 2: Describe the relative contribution of ocular and orbital tissues to corneal and whole-eye motion

The respective contributions of ocular and orbital tissues to corneal and whole-eye motion during air-puff loading will be determined through the development and validation of an analytical nonlinear viscoelastic model of centerline motion. The cornea and globe under air-puff loading can be represented as a 2 degree-of-freedom mass-spring-damper system. The system of equations of motion for this model were derived and solved numerically. The values of constants representing the stiffness or cornea and viscosity of the spring and dashpot elements were optimized to reduce the cumulative error between the simulated motion and in vivo motion of the cornea and whole-eye exported by the

CorVis ST. The model representation of ocular and orbital tissues will be validated with data from ex vivo studies. This model may provide insight into biomechanical response of the cornea and whole-eye under air-puff loading and may elucidate the contributions and interactions of ocular and orbital tissues to this complex dynamic motion.

Aim 3: Determine the impact of prostaglandin-analog (PGA) treatment on ocular and orbital tissue properties and biomechanical response

The nonlinear viscoelastic model of centerline motion will be applied to an in vivo cohort of newly-diagnosed glaucoma subjects undergoing treatment with prostaglandin- analogs (PGAs) to elucidate impact of PGA treatment on ocular and orbital tissue properties and response. The model results may provide new insights into the effect of prostaglandins on corneal and scleral properties, as well as properties of orbital fat. The change in the motion of the cornea and whole-globe due to the coupling of altered corneal and scleral properties with reduction in intraocular pressure may be interpreted more clearly using this analytical model.

Chapter 2. Biomechanical impact of the sclera on corneal deformation response to an air-

puff: a finite-element study

Introduction

Biomechanical markers are being explored to improve screening, diagnosis, and

management of diseases such as keratoconus and glaucoma (Liu and Roberts, 2005;

Elsheikh et al., 2009; Ruberti et al., 2011; Coudrillier et al., 2012; Tang and Liu, 2012;

Hon and Lam, 2013; Metzler et al., 2014; Girard et al., 2015; Sinha Roy et al., 2015; Ariza-

Gracia et al., 2016; Roberts, 2016). Biomechanical properties of the sclera are often determined via ex vivo mechanical strip testing or inflation tests (Coudrillier et al., 2012;

Geraghty et al., 2012; Girard et al., 2015; Pandolfi and Boschetti, 2015; Nguyen, 2016).

However, this destructive method depends on a multitude of factors including tissue hydration and strain rate, and there is a wide range of reported values in the literature on the value of Young's modulus of the sclera (Coudrillier et al., 2012). There is a demonstrated need for a clinical tool to determine the in vivo biomechanical behavior of ocular tissues, including how the cornea and the sclera interact.

It has been shown that the biomechanical response of the cornea under air-puff

deformation is significantly affected by its boundary properties (Elsheikh, 2010; Metzler

et al., 2014). Metzler et al. showed that the human cornea behaves more stiffly in the case

of a corneoscleral button mounted on a rigid artificial anterior chamber (simulating stiffer

scleral material properties) as opposed to a cornea that remains connected to an intact whole

globe at physiologically normal values of IOP. Their results suggest that the scleral

properties have an important impact on the deformation response of the cornea under air-

puff loading (Metzler et al., 2014).

Two commercial devices have been developed to assess the biomechanical

deformation response of the cornea using an air-puff as the non-destructive load, and they

have been used to provide clinical insight into the in vivo biomechanical behavior in

pathological conditions (Luce,2005; Ambrósio et al.,2013). The data produced by these

devices are often interpreted as purely corneal response without considering the

contribution of the sclera. Therefore, our purpose is to develop a whole-eye finite-element model that utilizes a well-characterized air-puff as the load to investigate the impact of varying scleral properties on corneal deformation response. This model will be validated using ex vivo experimental data obtained using human donor eyes.

Methods

The CorVis ST (Oculus Optikgeräte, GmbH, Wetzlar, Germany) utilizes a high-

speed camera with Scheimpflug geometry at a frame rate of 4,300 Hz to acquire 140 images

of the cornea along a single horizontal meridian, providing dynamic corneal deformation

response parameters, and visualization of the cornea as it deforms (Roberts, 2014).

Additionally, the air-puff of the CorVis ST has been shown to be consistent and repeatable

(Roberts et al., 2017), and for these reasons was chosen as the load for the FE model. The 13

ability to apply an accurate and precise load while monitoring deformation is the hallmark

of mechanical testing.

Geometry and Mesh

A simplified axisymmetric FE model of the human eye was developed in COMSOL

Multiphysics 5.2a (COMSOL Inc.; Burlington, MA) based on average dimensions (Figure

2.1, Table 2.1).

Figure 2.1 (Left) 2D axisymmetric geometry of eye, consisting of cornea, sclera, and

and free triangular elements. The whole-eye geometry consisted of cornea, sclera, and

vitreous humor. The whole globe has a diameter of 24 mm, the cornea has a diameter of

11 mm, and the central corneal thickness (CCT) is 500 μm. The thickness of the sclera ranged from 400 to 1,000 μm.

Table 2.1 Summary of geometric parameters for finite element model.

Tissue Geometric Parameter Parameter Value Central corneal thickness 500 µm (Elsheikh and Wang, 2007) Cornea Anterior radius of curvature 8.0 mm (Vojniković et al., 2013) Posterior radius of curvature 6.8 mm (Vojniković et al., 2013) Thickness at equator 400 µm (Norman et al., 2010) Sclera Thickness at posterior pole 1,000 µm (Norman et al., 2010) Radius of curvature 12 mm (Norman et al., 2010)

The geometry was meshed using the built-in meshing capabilities of COMSOL.

The central air region and ocular tunic were meshed with mapped 12,837 quadrilateral elements, with 3 elements through the thickness of the cornea and sclera. The remaining air region domains were meshed with 7,453 free triangular elements. The globe interior was meshed using free triangular elements and boundary layers to create a highly refined mesh at the interface of the ocular tunic and the vitreous body and more accurately model corneal deformation response. A convergence study was undertaken to identify the mesh density required to ensure that the solution was nearly independent of meshing parameters

(Figure 2.2).

Figure 2.2 A convergence study showed that a mesh containing >20,000 elements was sufficient to achieve discretization errors which were significantly less than normal experimental variations. This mesh density, shown in Figure 2.1, was therefore adopted for all subsequent simulations.

Estimation of Stress-Free Geometry

The eye is naturally under tension due to loading by intraocular pressure (IOP); therefore the unloaded or stress-free state of the eye was first estimated (Elsheikh et al.,2013). A negative pressure equal to the IOP was applied to the interior boundaries of the ocular tunic to generate the unloaded state. The resultant stress-free geometry was then loaded by IOP to determine the residually-stressed state of the eye (Figure 2.3).

Figure 2.3 (Left) The undeformed geometry of the eye, where the interior boundary of

the ocular tunic is highlighted in blue to indicate where the negative IOP will be applied

to estimate the stress-free geometry of the eye. (Center) The resultant stress-free state of

the eye (in blue) compared to the original undeformed geometry prior to loading by the

negative IOP (in black). (Right) Von Mises stresses in the residually-stressed state of the

eye when loaded by IOP, where the black lines represent the geometry of the stress-free state of the eye prior to loading by IOP.

This residually-stressed geometry was used as the starting point for the multiphysics FE simulation.

Material Properties and Boundary Conditions

The multiphysics model included the air-puff from the CorVis ST, which acted as

the anterior load on the eye. The air-puff velocity and pressure profiles acting on the

anterior cornea were simulated using turbulent k-ω shear stress transport fluid flow physics. 17

The maximum velocity at the nozzle outlet was set to 140.2 m/s based on data from the characterization performed by Roberts et al. (2017). The remaining edges of the air region were set to be open boundaries, such that the air flow is not affected by the arbitrary region boundaries. The resultant pressure profile from the air-puff simulation was applied as the load on the anterior cornea to simulate corneal deformation. The central 1.5 mm radius of the posterior sclera was held fixed.

The material properties of the ocular tunic were described by an isotropic, nearly incompressible, neo-Hookean constitutive model with values estimated from the literature

(Hamilton and Pye, 2008; Heys et al., 2001; Ng and Ooi, 2006; McKee et al., 2011). The vitreous was modeled as a linear elastic solid. Table 2.2 summarizes the material properties utilized in the FE model. The air-puff region surrounded the eye, which was fixed at the back to replicate the mounted whole-globe setup used in the ex vivo validation studies.

Table 2.2 Material properties of ocular tissues in the FE model.

Cornea Sclera Vitreous Air Young’s modulus 1.5 (Hamilton and 2.25, 3.0, 4.5, – – [MPa] Pye, 2008; McKee 6.0 (McKee et et al., 2011) al., 2011) Poisson ratio 0.49 0.49 0.49 – Bulk modulus 50.68 76.01 0.375 (Heys – [MPa] et al., 2001) Shear modulus – – 0.75 (Heys et [Pa] al., 2001) Density [kg/m3] 1,050 (Ng and Ooi, 1,100 (Ng and 1,000 (Heys 1.1855 2006) Ooi, 2006) et al., 2001) Viscosity [Pa*s] – – 18.6E-6

The Young's modulus of the cornea was fixed at a representative value from the literature while the Young's modulus of the sclera was varied to examine its relative effect on corneal deformation. Intraocular pressure (IOP) was varied from 10 to 40 mmHg in 10 mmHg increments. The properties of air, cornea, and vitreous were kept constant between simulations.

In the ex vivo setup of human donor studies, which was replicated by the finite element model, the peak of corneal motion/deformation is in phase with the peak of the air-puff loading curve. As the loading rate (dP/dt) approaches 0, this aligns with the maximum apical displacement as measured by the CorVis ST. We therefore assumed, as a first approximation, the cornea is in a quasi-static equilibrium such that a static model will capture this particular aspect of the air-puff-induced deformation.

Ex vivo Experiments

The CorVis ST was used to acquire data on the biomechanical deformation response of the corneas of 12 pairs of human donor eyes (65 ± 11.4 years, 8 male/4 female).

The donor eye study is considered exempt from review by the university. In each pair, one eye was randomly selected to have its sclera stiffened by crosslinking treatment with glutaraldehyde, while the fellow eye served as a control. The globe was immersed for 30 min in 4% glutaraldehyde in Dulbecco's phosphate-buffered saline (DPBS) (Sigma-

Aldrich, St. Louis, Missouri, USA) just below the level of the limbus, leaving a visible gap such that the cornea remained untreated. Glutaraldehyde visibly stains ocular tissues

yellow after a short period of time. The ocular tissues were visually inspected to ensure

that the cornea remained unaffected by the glutaraldehyde treatment.

Figure 2.4: (Left) Experimental setup for ex vivo studies on human donor eyes, showing

a whole globe in the purpose-designed mount in front of the CorVis ST. (Right) A view

of the human donor eye showing the 22-gauage needle inserted into the anterior chamber,

used to set and maintain intraocular pressure.

The CorVis ST was used to load the eye and quantify the resulting corneal

deformation response (Figure 2.4). Each eye was secured in a custom whole-globe mount using shallow sutures in the sclera. A 22-gauge needle attached to a saline column was inserted into the anterior chamber of the eye to set and maintain IOP. The column was set to specified heights to generate IOPs corresponding to 10, 20, 30, and 40 mmHg. At least

3 examinations were performed at each pressure step, with DPBS dripped onto the cornea

between examinations to maintain hydration. All data were acquired within 48 h post- mortem.

Results

The CorVis ST air-puff velocity profile and corresponding pressure profile were

simulated in COMSOL 5.2a (Figure 2.5). The spatial velocity profile agreed within 2% of

experimental measurements by hot wire anemometry (Roberts et al., 2017) along the

centerline/axis of symmetry as well as the central corneal region (Tables 2.3, 2.4). The

coordinate system for the model has the undeformed corneal apex at the origin, where r

indicates the radial distance from the axis of symmetry, and z indicates the axial distance

from the undeformed corneal apex. The air-puff pressure is concentrated at the central

corneal region.

Figure 2.5 Simulated CorVis ST air-puff velocity profile (left) and pressure profile

(right), which serves as the load on the anterior boundary of the globe. The simulation profiles agree within 2% of experimental measurements using hot-wire anemometry from

Roberts et al. (2017).

Table 2.3 Spatial measurements of air flow velocity at 4 mm from nozzle at peak pressure.

Radial distance, Simulation Hot wire (Roberts Error r [mm] [m/s] et al. 2017) [m/s] [%] 0.75 120.496 120.734 −0.196 1.50 50.601 50.594 0.015

Table 2.4 Spatial measurements of air flow velocity along axis of symmetry at peak pressure.

Axial distance, Simulation Hot wire (Roberts Error z [mm] [m/s] et al. 2017) [m/s] [%] 11 140.524 140.218 0.218 9 138.394 137.725 0.486 7 136.440 137.609 −0.850 5 135.186 136.697 −1.105 3 134.616 135.337 −0.533 1 134.178 133.238 0.706

The simulated pressure profile was applied as the boundary load on the anterior boundary of the eye. The resulting stress magnitudes were highest at the posterior surface of the corneal apex, which experiences the greatest total deformation and strain (Figure

2.6). The stresses are also higher at the posterior sclera where the eye was fixed in this simulation. There is an increase in the magnitude of stress near the area of the limbus

(interface of cornea and sclera) due to the dissimilar stiffnesses between cornea and sclera in the model. A sensitivity analysis was performed to show model independence from the vitreous body properties (Table 2.5).

Table 2.5 Sensitivity analysis of model dependence on vitreous body.

Vitreous shear modulus Apical displacement Percent change from 7.5 Pa [Pa] [mm] [%] 0.75000 −0.81888 0.147 7.5000 −0.81768 – 75.000 −0.81002 0.937

Figure 2.6 Resultant von Mises stresses and deformation for a globe with scleral-to-

corneal ratio of Young’s moduli of 1.5 and IOP of 10 mmHg (left), scleral-to-corneal ratio of 1.5 and IOP of 40 mmHg (center), and a globe with scleral-to-corneal ratio of 4.0 and IOP of 40 mmHg (right).

The corneal deformation response produced by the model matched the expected

trend (Table 2.6). For each value of IOP tested, increasing the ratio of scleral to corneal

Young's moduli resulted in decreasing maximum apical displacement (Figure 2.7). Further,

increasing the value of IOP while keeping other material properties constant also resulted

in decreasing maximum apical displacement, which is consistent with with the results of

our ex vivo experiments (Table 2.7). Figure 2.8 shows that the effective stiffness of the

cornea increases with higher IOP, resulting in decreased maximum apical displacement in

both the FE model and ex vivo results. The simplified Neo-Hookean approximation may

contribute to the why the model does not completely bound ex vivo observations.

Table 2.6 Summary of simulation results from FE model.

IOP Scleral-to-corneal ratio Max apical displacement [mmHg] [MPa/MPa] [mm] 1.5 −1.079 2 −0.954 10 3 −0.818 4 −0.743 1.5 −0.940 2 −0.818 20 3 −0.689 4 −0.569 1.5 −0.848 2 −0.733 30 3 −0.620 4 −0.549 1.5 −0.786 2 −0.682 40 3 −0.598 4 −0.492

Table 2.7 Summary of ex vivo results from human donor eyes (Nguyen et al., 2018).

IOP [mmHg] Group Max apical displacement [mm] Mean ± Std. Dev. 10 Control −1.494 ± 0.191 Treated −1.216 ± 0.148 20 Control −0.992 ± 0.114 Treated −0.891 ± 0.088 30 Control −0.706 ± 0.068 Treated −0.663 ± 0.071 40 Control −0.520 ± 0.051 Treated −0.487 ± 0.051

Figure 2.7 Contour plot of maximum apical displacement predicted by the finite- element model as a function of IOP and scleral-to-corneal ratio of Young’s moduli. The biomechanical response is non-linear with both increasing IOP and increasing scleral-to- corneal ratio.

Figure 2.8 Comparison of simulation results from FE model for several levels of scleral-

to-corneal ratio of Young’s Modulus (SCRs) against ex vivo data. Error bars represent

95% confidence intervals.

Discussion

Scleral stiffness significantly influenced corneal apical displacement in both

simulations and ex vivo experiments. Thus, with increasingly stiff sclera, the greater the

limitation on corneal deformation to an air-puff at physiologic levels of IOP. The ex vivo

results are consistent with the trends observed by Metzler et al., where the untreated globe

undergoes stress stiffening at higher IOP and behaves similarly to the treated globe. This

has important clinical implications, where the corneal biomechanical response is often

attributed solely to the corneal properties. The maximum apical displacement of the cornea

decreased non-linearly with increasing IOP, which is consistent with the principles of 27 tonometry and literature reports (Metzler et al., 2014). Because traditional methods for biomechanical evaluation are limited to ex vivo studies, finite-element (FE) models have been explored as a way to evaluate in vivo properties and response (Elsheikh, 2010; Girard et al., 2015; Pandolfi and Boschetti, 2015; Ariza-Gracia et al., 2016). In published models that simulate corneal deformation by an air-puff, some do not include the sclera (Elsheikh et al., 2009, 2013; Kling et al., 2014; Lago et al., 2014; Pandolfi and Boschetti, 2015; Sinha

Roy et al., 2015; Bekesi et al., 2016; Simonini and Pandolfi, 2016). The inseparable impact of the sclera to limiting corneal deformation response requires a whole-eye model. Further, the simulated load must be accurate because the biomechanical response is load-dependent.

This FE model, while simple, achieved qualitative agreement with ex vivo experiments. Several aspects of this result give insights which can be used to inform other models. Several models of air puff-induced deformation are comprised of only the cornea, with a variety of scleral boundary conditions applied at the limbus (Elsheikh et al., 2009, 2013; Kling et al., 2014; Lago et al., 2014; Pandolfi and Boschetti, 2015; Sinha

Roy et al., 2015; Bekesi et al., 2016; Simonini and Pandolfi, 2016). Our findings indicate that this simplifying representation of the scleral influence on the limbus is inaccurate and introduces systematic errors in these models. Non-contact tonometry devices estimate IOP by observing the response of the cornea; our findings demonstrate that clinical interpretations which neglect scleral contributions to the corneal response to an air-puff may lead to incorrect conclusions. This is particularly important in conditions where the sclera is altered, such as prostaglandin treatment in glaucoma (Toris et al., 2008; Alm and

Nilsson, 2009)—which increases scleral permeability and uveoscleral outflow—and

28 progressive myopia (McBrien and Gentle, 2003; Harper and Summers, 2015)—in which the sclera experiences significant collagen remodeling. Reductions in IOP have been reported in some procedures that alter the sclera to treat presbyopia; this is likely artifact from the altered sclera, and there is little or no actual change in IOP (Fukasaku and

Marron, 2001; Hipsley et al., 2017).

The present model accurately replicated the velocity and pressure profiles of the

CorVis ST air-puff. Similar to the work done by Elsheikh et al. the model also estimated the stress-free state of the eye and generated the residually-stressed state before deformation to more accurately describe the stresses in the cornea and sclera and the resultant biomechanical response (Elsheikh et al., 2013). Finally, a significant advantage of this model is that the simulation results are qualitatively validated with data from ex vivo human donor eyes, in an experimental setup that was replicated closely in the FE model.

One limitation of this model is that it does not account for the dynamic time- dependent response of the cornea to the air-puff. The cornea is a soft biological tissue and therefore exhibits viscoelasticity in its biomechanical response. To account for the non- linear behavior of the tissue, we chose to consider the ocular tissues as hyperelastic, resulting in a non-linear response to increasing IOP. The use of an incompressible neo-

Hookean constitutive model is a simplification: the cornea and sclera are complex structures with significant anisotropy (Nguyen, 2016). In this case, the model effectively assumed that the secant modulus is constant regardless of the magnitude of deformation. It is therefore not surprising that the experimental response shows a much larger dependence

29 on SCR and behaves in a strain-stiffening manner. Inclusion of more accurate material models, such as Mooney-Rivlin, will significantly improve the model's ability to make quantitative predictions. Another limitation of this model it does not capture the dynamic nature of the air-puff test. Despite this simplification, the steady-state approximation produced results which qualitatively matched those from ex vivo experiments on human donor tissues. Future iterations of the model will include the dynamic response of the cornea to an air-puff.

The finite-element model presented in this paper demonstrates that scleral material properties have an important impact on the biomechanical deformation response of the cornea in air-puff induced deformation. Namely, the stiffer the sclera, the greater the limitation on corneal deformation. This may have important clinical implications. Often in the clinic, the observed biomechanical deformation response of the cornea is attributed solely to the material properties of the cornea, as well as IOP. However, it is clear from these simulations and experiments that the deformation response of the same cornea varies significantly with varying scleral properties. This suggests that when looking at air-puff induced deformation, that the observed biomechanical response is a result of the combination of both corneal and scleral material properties, in addition to IOP.

Chapter 3: Biomechanical contribution of the sclera to dynamic corneal response in air-

puff induced deformation in human donor eyes

Introduction

Elevated intraocular pressure (IOP) is an important diagnostic marker in ocular diseases such as glaucoma (Weinreb and Khaw, 2004) and ocular hypertension (Copt et al., 1999). Understanding how the biomechanical behavior of ocular structures influence

IOP measurement may assist in the diagnosis, management, and treatment of ocular diseases (Liu and Roberts, 2005; Elsheikh et al., 2009; Metzler et al., 2014; Girard et al.,

2015; Tejwani et al., 2015; Ariza-Gracia et al., 2016; Roberts, 2016). IOP is estimated by either contact tonometry (such as Goldmann Applanation Tonometry) or noncontact tonometry (such as air-puff tonometers) which apply a load to the cornea and estimate the

IOP based on assumptions about the structure and mechanical properties of the cornea (Liu and Roberts, 2005; Luce, 2005, Ruberti et al., 2011). It has also been shown that the thickness and mechanical properties of the cornea can influence tonometric readings (Liu and Roberts, 2005; Teiwani et al., 2015; Broman et al., 2007), and neglecting patient-to- patient variability in the mechanical response of the cornea would lead to measurement error. Several devices utilize various algorithms to account for the contributions of corneal

stiffness, thickness, and age (Joda et al., 2016, Ruberti et al., 2011). The limitation in these

algorithms is the underlying assumption that the corneal response to a mechanical load is

solely attributable to the cornea’s structural and material properties, without taking into

account the contribution of the sclera.

It has been shown that affecting the boundary conditions of the cornea (Metzler et

al. 2014; Elsheikh, 2010; Nguyen et al., 2018 will impact the corneal biomechanical

response to an applied load. Substantially fewer studies have explored the impact of the

sclera on clinical tonometry measurements as compared to the cornea, and to our

knowledge, a method for clinical evaluation of scleral properties has yet to be developed.

The purpose of this study was to explore the impact of varying scleral properties on the

corneal biomechanical response under air-puff induced deformation in human donor

eyes.

Materials and Methods

Eye Preparation

Human donor eyes (40 paired eyes from 20 donors) from the Central Ohio Lions

Eye Bank were obtained within 24 hours post-mortem, so experiments could be completed within 48 hours. Exclusion criteria included gross defects on the corneal epithelium and corneal surgeries other than bilateral cataracts. On enucleation, the anterior chamber of each eye was injected via 22-guage needle with 0.5-1.0 mL of 12.5-15% dextran (Sigma-

Aldrich, St. Louis, Missouri, USA) dissolved in Dulbecco’s phosphate-buffered saline

(DPBS; Sigma-Aldrich) prior to submersion in 35 mL of the same solution. Eyes were then

refrigerated until the central corneal thickness was measured to be 550-600 microns by ultrasound pachymeter (generally 12-14 hours).

Experimental Setup

One eye from each matched pair was randomly selected to have its sclera stiffened by crosslinking treatment with glutaraldehyde, while the fellow eye served as a control for uniaxial testing. The globe was immersed for 30 minutes in 4% glutaraldehyde in

Dulbecco’s phosphate-buffered saline (DPBS) solution just below the level of the limbus, such that the cornea remained untreated. Corneal hydration was maintained using a humidifier.

A dynamic Scheimpflug analyzer (CorVis ST with research software 1.3b1716;

Oculus, Wetzlar, Germany) was used to load the eye and quantify the resulting corneal

deformation response (Figure 3.1). Each eye was secured in a custom whole-globe mount

using shallow sutures in the sclera. A 22-gauge needle attached to a saline column was inserted into the anterior chamber of the eye to set and maintain the IOP. The saline column was set to specified heights to generate IOPs corresponding to 10, 20, 30, and 40 mmHg.

At least 3 examinations were performed at each pressure step, and DPBS was dripped onto

33 the cornea between examinations to maintain hydration and reflectivity. The order of eyes during testing was randomized, and measurements were taken pre- and post-treatment.

Figure 3.1 (Left) Experimental setup for ex vivo studies on human donor eyes, showing a whole globe in the purpose-designed mount in front of the CorVis ST. (Right) A view of the human donor eye showing the 22-gauge needle inserted into the anterior chamber, used to set and maintain IOP (Nguyen et al., 2018)

Simple uniaxial testing was performed to quantify the effect of the stiffening protocol on the mechanical properties of the sclera. Nasotemporal strips from the sclera were excised along the central meridian of the eye. Strip dimensions were measured using a digital caliper before the strips were mounted in the testing apparatus (Figure 3.2;

Rheometrics Systems Analyzer, RSA III; TA Instruments, New Castle, Delaware, USA).

A humidified chamber was constructed around the testing apparatus to maintain tissue hydration.

Figure 3.2 (Left) Excised strip of sclera mounted in a (Right) commercial testing fixture.

A humidified chamber, not pictured, surrounded the test fixture to maintain tissue hydration.

A strip testing protocol was adapted from a study by Metzler et al. (2016) and is described briefly herein. Tissues were preconditioned by manually cycling the load 5 times from 0.5 to 5 gram-force (gf). The strip was then loaded to 2.0 gf for 1 minute prior to sinusoidal tension with strain amplitude of 0.15% and frequency of 1.0 Hz. The tissue was unloaded to 0.5 gf for 5 minutes before applying a tensile ramp up to 6% strain at a strain rate of 0.1% per second.

The tensile ramp data were fit with an incompressible, isotropic, neo-Hookean model to estimate the Young’s modulus E of the sclera. This formulation predicts a force- displacement relationship 35

( ) = + (1) 𝑬𝑬𝑬𝑬 𝒖𝒖 +𝟏𝟏 𝑭𝑭 𝒖𝒖 �𝟏𝟏 − 𝟐𝟐� 𝟑𝟑 𝑳𝑳 𝒖𝒖 �𝟏𝟏 � where A is the unloaded cross-sectional area, u is the change𝑳𝑳 in sample length, and L is the

original length of sample between clamps. E was estimated by fitting this model equation

to the experimentally-measured data using a least-squares approach.

Statistical Analyses

Dynamic corneal response (DCR) parameters were exported by the CorVis ST

analysis software for each exam. Paired t-tests compared the change in DCRs following

either stiffening or control treatment, and multiple analysis of co-variance (MANCOVA)

was performed for selected delta DCRs as dependent variables with log(IOP) and scleral

Young’s modulus as independent co-variates. The log(IOP) was used in order to normalize

the distribution of residuals. Because our study focuses on a small subset of planned

comparisons, a correction for multiple t-tests was not applied in order to avoid type II error

(Armstrong, 2014). Statistical significance was determined if p < 0.05. The Cohen effect

size were calculated for each DCR parameter to evaluate the size of the effect of IOP,

Scleral Modulus, or the interaction effect.

A subset of DCRs were selected for further statistical analysis: peak distance (PD),

highest concavity (HC) deformation amplitude (DA), DA Ratio 2 mm, integrated inverse

radius, and stiffness parameter-applanation 1 (SP-A1) and stiffness parameter-highest concavity (SP-HC). Peak distance refers to the distance between the two highest points of

the cornea at highest concavity. Deformation amplitude ratio is the ratio of deformed amplitude of the corneal apex and the average deformation either 2 mm or 1 mm on either side of the apex, respectively. The integrated inverse radius, denoted as Integrated Radius in the export, is the integrated sum of inverse radius of curvature between first and second applanation. Finally, the stiffness parameters are calculated as a ratio of load to displacement. Displacement for SP-A1 is between the undeformed apex and the position of first applanation, and displacement for SP-HC is between the position of first applanation and highest concavity. SP-A1 and SP-HC were calculated using the equations developed and described by Roberts et al., 2017. These parameters are summarized in

Table 3.1.

Table 3.1 Select CorVis ST dynamic corneal response parameters and their description.

Parameter Name [unit] Description Peak Distance [mm] Distance between corneal peaks at highest concavity HC Deformation Amplitude [mm] Deformation amplitude at highest concavity Radius [mm] Radius of curvature at highest concavity DA Ratio Max (2 mm) Deformation amplitude ratio at 2 mm from corneal apex Integrated Inverse Radius [mm-1] Integrated inverse radius of curvature SP-A1 Stiffness parameter at first applanation SP-HC Stiffness parameter at highest concavity

Results

Tensile strip testing ramp data confirmed that the glutaraldehyde-treated sclera had a stiffer biomechanical behavior than the matched controls. The control and treated scleral

Young’s modulus were found to be significantly different, and were determined to be

2.5±2.4 MPa and 7.6±8.4 MPa, respectively. Paired t-tests showed a statistically significant

(p<0.05) increase in scleral stiffness at 4, 5, and 6% strain (Figure 3.3).

Paired t-tests showed statistically significant (p<0.05) decreases after scleral stiffening in

peak distance and HC deformation amplitude for all tested levels of IOP (Table 3.2). There 38

were also significant decreases in Integrated Inverse Radius only at IOP of 10 mmHg. Of

the two stiffness parameters, only SP-HC was affected. SP-HC was higher post-scleral-

stiffening at all levels of IOP while there was no significant difference observed for SP-A1

at any level of IOP.

Several important relationships were elucidated via MANCOVA analyses (Table

3.3). It was shown that the HC deformation amplitude and SP-HC had a statically significant relationship with both IOP and scleral Young’s modulus. While SP-A1 did vary significantly with IOP, there was no significant relationship with scleral Young’s modulus.

Peak distance and radius were significantly correlated with scleral Young’s modulus.

While integrated inverse radius was found to significantly correlate with scleral modulus, this was driven by the response at 10 mmHg and there were no differences at all other levels of IOP. Additionally, while DA Ratio max (2mm) was found to vary significantly with IOP, paired t-test analysis showed no differences at each tested level of IOP.

Table 3.2: Paired t-test results for DCRs at each tested level of IOP. Statistical significance was determined if p < 0.05, denoted by †.

IOP 10 IOP 20 CorVis ST DCR Control Glut 4 Sclera Mean Δ Control Glut 4 Sclera Mean Δ Peak Dist. [mm] 5.838 ± 0.283 5.398 ± 0.333 -0.527 ± 0.411† 4.965 ± 0.194 4.668 ± 0.295 -0.313 ± 0.394† HC Def. Amp. 1.441 ± 0.184 1.182 ± 0.159 -0.317 ± 0.243† 0.982 ± 0.094 0.879 ± 0.091 -0.112 ± 0.134† [mm] Radius [mm] 5.974 ± 0.572 6.123 ± 0.735 0.380 ± 0.755† 6.872 ± 0.759 6.588 ± 0.792 -0.076 ± 0.841 DA Ratio Max 4.491 ± 0.603 4.236 ± 0.672 -0.174 ± 0.702 3.853 ± 0.497 3.640 ± 0.439 -0.178 ± 0.513 (2mm) Integrated 9.961 ± 1.196 9.700 ± 1.236 -0.786 ± 1.438† 7.752 ± 1.081 7.809 ± 1.191 -0.087 ± 0.838 Radius [mm^-1] SP-A1 99.448 ± 17.385 102.039 ± 16.153 2.591 ± 15.993 127.609 ± 15.925 127.480 ± 19.689 -0.128 ± 16.558 SP-HC 9.709 ± 2.847 13.016 ± 3.982 3.306 ± 2.939† 19.847 ± 4.929 23.840 ± 4.568 3.993 ± 2.055† IOP 30 IOP 40 CorVis ST DCR Control Glut 4 Sclera Mean Δ Control Glut 4 Sclera Mean Δ Peak Dist. [mm] 4.126 ± 0.206 3.889 ± 0.276 -0.217 ± 0.366† 3.388 ± 0.196 3.175 ± 0.242 -0.185 ± 0.335† HC Def. Amp. 0.712 ± 0.061 0.662 ± 0.065 -0.041 ± 0.084† 0.530 ± 0.050 0.492 ± 0.047 -0.030 ± 0.062† [mm] Radius [mm] 6.509 ± 0.758 6.297 ± 0.880 -0.176 ± 0.935 6.908 ± 1.141 6.985 ± 1.292 0.222 ± 1.523 DA Ratio Max 3.434 ± 0.330 3.342 ± 0.371 -0.054 ± 0.439 3.033 ± 0.367 2.840 ± 0.396 -0.121 ± 0.487 (2mm) Integrated 6.492 ± 1.041 6.374 ± 1.280 -0.086 ± 0.986 4.799 ± 1.062 4.458 ± 1.218 -0.360 ± 1.225 Radius [mm^-1] SP-A1 142.966 ± 15.484 145.485 ± 25.129 2.519 ± 19.770 141.683 ± 18.567 144.668 ± 25.093 2.985 ± 15.272 SP-HC 34.504 ± 7.488 38.985 ± 6.529 4.481 ± 7.182† 54.655 ± 14.970 62.889 ± 25.093 8.234 ± 16.024†

Table 3.3 MANCOVA analysis for DCR parameters, with delta responses as the dependent variable, and with log(IOP) and Scleral Young’s modulus as independent covariates. Cohen effect sizes are shown, and statistical significance from MANCOVA was determined if p<0.05 denoted by †.

CorVis ST DCR log(IOP) Scl Mod (E) log(IOP)*Scl Mod(E) Peak Dist. [mm] 0.6019 0.0386† 0.0949 HC Deformation Amp. [mm] 0.2542† 0.0680† 0.0335 Radius [mm] 2.7522 0.1690† 0.2479† DA Ratio Max (2mm) 0.9032† 0.0204 0.0755 Integrated Radius [mm^-1] 0.1979 0.0307† 0.0308 SP-A1 2.909† 0.0019 0.112 SP-HC 3.7287† 0.0149† 0.0368

Discussion

The response of the cornea to a dynamic load such as the CorVis ST air-puff is the result of both corneal and scleral material properties, in addition to loading by IOP. The deformation response of the cornea is greatly impacted by the loading due to IOP, which generates tension in the collagen microstructure of both cornea and sclera. During air-puff induced deformation, the cornea deforms through applanation, then continues to deform and becomes concave. This concave state will result in displacement of fluid, i.e. aqueous humor, in the anterior chamber. The volume of fluid displacement will be limited by the scleral biomechanical properties, such that the greatest influence of the sclera on corneal deformation response is at highest concavity. With stiffer sclera, elastic stresses that are transmitted from the cornea to the limbus will experience additional stresses, and stress wave propagation through the sclera will dissipate less energy, potentially returning more 41

stress in the sclera and other ocular tissues and resulting in increased resistance to corneal

deformation (Esposito et al., 2015).

Decreases in HC DCRs at low IOP (peak distance, radius of curvature, and HC

deformation amplitude) showed that after stiffening only the sclera, the cornea will behave

as if it has a higher apparent stiffness under the same load. This is a result of the altered

boundary condition at the limbus, where the stiffened sclera is less compliant and will have

greater resistance to the displaced fluid in the anterior chamber (Nguyen et al., 2018). This

may lead to misinterpretation of corneal response in ocular disease states or treatments that

alter the sclera, such as progressive myopia or topical prostaglandin-analogs for glaucoma

management.

At all levels of IOP, the highest concavity parameters of HC Deformation

Amplitude, Peak Distance and SP-HC were significantly different with treatment, indicating a stiffer response despite only stiffening the sclera and leaving the cornea unaltered. In contrast, at all levels of IOP, there were no significant differences in DA

Ratio Max (2mm) or SP-A1 with scleral stiffening, indicating these parameters are primarily corneal parameters and are not influenced by scleral properties. We expected larger differences between eyes having stiffened and untreated sclerae at HC over those at

A1, because at HC the cornea has reached its maximum deformation and therefore has induced aqueous displacement in the anterior chamber. At the same time, scleral resistance to corneal deformation is maximized. A1 occurs very early in the CorVis ST loading curve, where the corneal deformation is very small compared to HC, and the contribution of the sclera to limiting corneal motion is minimal due to the miniscule amount of displaced

aqueous humor. Therefore parameters derived at A1 would be less sensitive to scleral

properties and primarily indicative of corneal properties.

The two stiffness parameters SP-HC and SP-A1 are indicative of the stiffness of the cornea. SP-A1 is determined as the load on the cornea over the displacement to A1, where SP-HC is the same load over the deflection displacement beyond applanation to HC

(difference of HC deformation amplitude and A1 deformation amplitude). SP-A1 only varied significantly with IOP and was independent of scleral stiffening in paired eyes.

Since first applanation occurs very early and there is minimal contribution of the sclera to limiting corneal deformation, the SP-A1 results suggest that there were no differences in corneal properties. SP-A1 has been shown to be sensitive to changes in corneal properties

(Roberts et al., 2017), and the findings of our study are consistent. SP-HC was shown to vary significantly with both IOP and estimated scleral stiffness, increasing at all levels of

IOP. These results show that differences in scleral stiffness are detectable in observing the change in SP-HC parameter, which may be clinically useful as a diagnostic tool.

Only at 10mmHg, the two parameters based on radius of curvature at highest concavity – radius and integrated inverse radius – were significantly different indicating a stiffer response. At all other levels of IOP, these two parameters were not different, indicating they are also primarily corneal parameters and not influenced by scleral properties unless the IOP is at the lower limit of what is considered physiologic. A limitation of this study was that other physiologic levels of IOP between 10mmHg and

20mmHg were not included. However, studies have shown that integrated inverse radius and radius are sensitive corneal parameters that can detect subtle differences in corneal

43 response, such as the difference between standard surface ablation and an “extra” procedure where accelerated crosslinking was performed immediately after surface ablation (Lee et al, 2017).

The results of this study may have important clinical implications. The corneal biomechanical response evaluated by the Corvis ST is often attributed solely to the properties of the cornea. However, this study shows that the corneal deformation response under air-puff induced deformation is significantly impacted by varying the properties of the sclera and leaving the cornea untreated. Specifically, the stiffer the sclera, the greater will be the limitation on corneal deformation. It is important to consider, especially in vivo, that the biomechanical deformation response of the cornea is due to contributions from both the cornea and sclera, in addition to the effects of IOP. Some parameters have been shown to be more sensitive to scleral properties while others have been shown to be unaffected by scleral properties, which may be useful in clinical interpretation.

Conclusions

It has been shown that scleral properties influence corneal biomechanical deformation response to an air puff. Specifically, a stiffer sclera will limit corneal motion by resisting displacement of fluid when the cornea becomes concave, and thus the resulting response might be misinterpreted as a stiffer cornea. The study results suggest that there may be a need to account for scleral properties in the assessment of corneal biomechanical response and the estimation of IOP under air-puff loading. Through careful study, we may be able to evaluate the respective contributions of cornea, sclera, and IOP to the dynamic 44 corneal deformation response to give a better understanding of the eye’s overall susceptibility to glaucoma and other ocular diseases.

Chapter 4: Development and validation of a nonlinear viscoelastic model of corneal and

whole-eye motion under air-puff loading by a dynamic Scheimpflug analyzer

Introduction

The biomechanical properties of ocular tissues drive their response to a load (Glass et al., 2008; Lee et al., 2017), such as the loading from intraocular pressure (IOP) or from an air-puff of a non-contact tonometer. Ocular disease states, treatments, or surgeries which impact the biomechanical properties of the tissues will necessarily also impact their biomechanical response (Roberts, 2002; Dupps and Wilson, 2006; Ruberti et al., 2011).

Examples of this are (1) the altered or increased corneal deformation observed in keratoconus eyes as a result of localized thinning and weakening in the cornea (Touboul et al., 2007; Vinciguerra et al., 2016), (2) changes in corneal deformation following LASIK, where the structure and therefore biomechanical properties of cornea are altered (Frings et al., 2015; Yang et al., 2015), and (3) an apparently stiffer corneal deformation response with stiffer sclerae (Nguyen et al., 2018), where stiffer properties of sclera have been associated with glaucoma (Coudrillier et al., 2012; Girard et al., 2015). There is a demonstrated need for clinical evaluation of in vivo properties of ocular tissues, in order to understand how they contribute to and interact in various disease states, and how they change as a result of a particular treatment.

One clinical device which can evaluate the biomechanical response of the eye to an

air-puff is the CorVis ST (Oculus Optikgerate GmbH, Wetzlar, Germany). The device utilizes an air-puff in combination with high-speed Scheimpflug camera to capture images

of the cornea as it undergoes its deformation, then performs image analysis to derive

parameters that describe this deformation. In addition to measuring corneal deformation,

the CorVis ST also measures the displacement of the whole globe, referred to as whole-

eye motion (WEM). This whole-eye motion (WEM) is measured by tracking the displacement of the peripheral parts of the cornea on the edges of the imaging window. In vivo, while the cornea is deforming from a convex to a concave orientation, the whole eye undergoes posterior translation at constant velocity (Roberts et al., 2017). Once the cornea has reached maximum concavity and can no longer deform despite increasing load from the air-puff, the WEM rapidly increases. Recently, changes in WEM have been associated with changes in the properties of orbital tissues due to thyroid orbitopathy or Graves ophthalmopathy (Leszczynska et al., 2017, Hwang et al., 2019).

In this study, we developed an analytical model of corneal and whole-eye motion during CorVis ST loading which provides an objective method to evaluate the respective contributions of ocular and orbital tissues to this dynamic in vivo response. This model may also help to elucidate how different disease states or treatments might impact particular tissues and their biomechanical properties. We present in this study the development and ex vivo validation of a nonlinear viscoelastic model of corneal and whole- eye motion during loading by the CorVis ST.

Methods

Characteristic in vivo dataset

The dynamic corneal response parameters of interest that were outputted by the

research software were: (1) the air-puff pressure which loads the cornea, (2) the corneal

deformation (measured as the total displacement of the corneal apex and includes WEM),

(3) the whole-eye motion (measured as the posterior displacement of the whole globe), and

(4) the corneal deflection (calculated as the difference between corneal deformation and

WEM). The corneal deformation and whole-eye motion profiles from a published characteristic in vivo CorVis ST dataset were selected for the development of the analytical model (Roberts et al., 2017), and are described below (Figure 4.1). A key feature of in the timing of events is that the maximum corneal deflection (corneal motion without whole- eye motion) occurs prior to the maximum air-puff pressure. Other models of CorVis ST motion do not capture this plateauing of corneal motion, instead having the peak of corneal deflection in phase with the peak of air pressure (Sinha Roy et al., 2016).

Figure 4.1 Characteristic in vivo dataset of corneal (red) and whole-eye motion (green) under air-puff loading (dashed gray). While the cornea is deforming from a convex to a concave orientation, the whole eye undergoes posterior translation at constant velocity.

Once the cornea has reached maximum concavity and can no longer deform despite increasing load from the air-puff, the WEM rapidly increases. Maximum corneal motion occurs prior to maximum air-puff pressure, and the cornea undergoes an oscillatory motion before recovery. Maximum whole-eye motion occurs near full recovery of the cornea. (Roberts et al., 2017)

Modeling whole-eye motion

To generate an analytical model of centerline motion of the cornea and globe, the eye was considered as a system of inertial masses, linear elastic springs, and viscous dampers loaded by the CorVis ST air-puff. The first iteration of the model only considered

WEM and aimed to characterize the orbital tissues, such as the orbital fat and ocular

muscles. The mass of the globe was set as 7.5 grams (Bron 1997) and the orbital tissues

were represented by either a purely elastic spring component (Figure 4.2a) or a Kelvin-

Voigt (KV) viscoelastic element (Figure 4.2b). The system of equations of motion were

derived and solved numerically using MATLAB (The MathWorks, Inc.; Natick, MA). The

values of constants representing the stiffness and viscosity of spring and dashpot elements

were optimized to reduce the cumulative error between the measured in vivo WEM and

the simulated WEM. The linear KV-viscoelastic representation of orbital tissues, which is consistent with numerical and finite-element models in the literature (Cirovic et al., 2006;

Pfann et al., 1995), sufficiently reproduced the in vivo WEM and was selected for all future iterations of the analytical model.

(a)

(b)

Figure 4.2 Representations of the peri-ocular tissues as (a) purely elastic or (b) Kelvin-

Voight (KV)-viscoelastic

Modeling corneal and whole-eye motion

A second mass, representing the cornea, was added to the model to allow

simultaneous simulation of both corneal and whole-eye motion. The masses of the cornea and whole globe were estimated to be 0.05 grams (Clayson, 2017) and 7.45 grams, respectively, maintaining the total eye mass of 7.5 grams. The cornea was modeled in several ways, ranging from a linear elastic solid to a 3-parameter linear viscoelastic solid

(Figure 4.3a-d). This model predicted most features of the in vivo corneal motion and

WEM accurately, but failed to reproduce the plateau in corneal deformation at maximum concavity. The motion of the cornea reaches its limit before the peak of the air-puff in vivo.

Therefore, the final model included a non-linear stiffening spring with stiffness ( ), �𝟒𝟒 given by 𝒌𝒌 𝒖𝒖

(1) ( ) = 𝒌𝒌𝟒𝟒 �𝟒𝟒 𝒌𝒌 𝒖𝒖 𝒄𝒄 𝒈𝒈 �𝒖𝒖 − 𝒖𝒖 in parallel with the 3-element model representing the cornea to represent the constraining

effect of the sclera (Figure 4.3e). The non-linear stiffening spring is described by Equation

1, where uc and ug represent the displacement of the cornea and globe, respectively.

(a)

(b)

(c)

(d)

(e)

Figure 4.3 Schematic representation of the sequence of viscoelastic models investigated.

In all cases, the peri-ocular tissues were considered as KV-viscoelastic. The cornea was

The system of equations of motion (Equations 2 – 6) for this model were derived and solved numerically using MATLAB. The values of constants representing the stiffness and viscosity of the spring and dashpot elements were optimized to reduce the cumulative error between the simulated motion and in vivo motion of both the cornea and the whole eye. The forces through the KV-viscoelastic element, the 3-element linear viscoelastic 52 solid, and the non-linear stiffening spring element were represented as FKV, F3LV, and

FNL, respectively.

= + ( ) (2)

𝑲𝑲𝑲𝑲 𝟏𝟏 𝒈𝒈 𝟏𝟏 𝒈𝒈 𝑭𝑭 𝒌𝒌 (�𝒖𝒖 +� 𝒃𝒃) 𝒗𝒗 (3) = + 𝟐𝟐 𝟐𝟐 𝟑𝟑 𝟐𝟐 𝒂𝒂𝒂𝒂𝒂𝒂 𝟑𝟑𝟑𝟑𝟑𝟑 𝟑𝟑 𝒄𝒄 𝒈𝒈 𝒃𝒃 𝒌𝒌 𝒌𝒌 𝒄𝒄 𝒈𝒈 𝒃𝒃 𝒅𝒅𝑭𝑭 𝑭𝑭 𝒌𝒌 �𝒖𝒖 − 𝒖𝒖 � 𝟐𝟐 �𝒗𝒗 − 𝒗𝒗 � − 𝟐𝟐 = 𝒌𝒌 𝒌𝒌 𝒅𝒅𝒕𝒕 (4)

𝑭𝑭𝑵𝑵𝑵𝑵 𝒌𝒌𝟒𝟒�𝒖𝒖𝒄𝒄 − 𝒖𝒖𝒈𝒈 = = + (5)

� 𝑭𝑭𝒈𝒈𝒙𝒙 𝒎𝒎𝒈𝒈 ∗ 𝒂𝒂𝒈𝒈 𝑭𝑭𝟑𝟑𝟑𝟑𝟑𝟑 𝑭𝑭𝑵𝑵𝑵𝑵 − 𝑭𝑭𝑲𝑲𝑲𝑲 = = ( + ) (6)

� 𝑭𝑭𝒄𝒄𝒙𝒙 𝒎𝒎𝒄𝒄 ∗ 𝒂𝒂𝒄𝒄 𝑭𝑭𝒂𝒂𝒂𝒂𝒂𝒂 − 𝑭𝑭𝟑𝟑𝟑𝟑𝟑𝟑 𝑭𝑭𝑵𝑵𝑵𝑵

These constitutive equations were consistent for all investigated models, where the values of certain parameters would be reduced to zero in simpler models.

Ex vivo validation

A subset of data (n = 16 pairs) from previous human donor eye experiments

(Nguyen et al., 2019) were used in the validation of the model and its characterization of ocular tissues. Briefly, paired human donor globes were used for ex vivo experiments, where the sclera of one eye was stiffened with 4% glutaraldehyde and the sclera of the fellow eye was left untreated. The corneas for both eyes were also left untreated. The globes were mounted in a rigid fixture to restrict whole-eye motion, and the IOP set by a saline column (Figure 4.4). The corneal biomechanical response and motion were

53 evaluated with the CorVis ST at pressures of 10, 20, 30 and 40 mmHg, and the corneal motion profiles were exported for use in the simulation.

Figure 4.4: (Left) Experimental setup for ex vivo studies on human donor eyes, showing a whole globe in the purpose-designed mount in front of the CorVis ST. (Right) A view of the human donor eye showing the 22-gauge needle inserted into the anterior chamber, used to set and maintain IOP (Nguyen et al., 2018).

In a separate proof-of-concept experiment, a single human donor eye was first mounted in the rigid holder – representing extremely stiff orbital tissues – then potted in

2.4% gelatin (Jangjai, 2015) in a purpose-designed holder (Sponsel et al., 2011) to simulate normal human orbital tissues in a bony orbit (Figure 4.5). The IOP was set to 20 mmHg for each setup. The CorVis ST evaluated the corneal biomechanical response and exported the corneal and whole-eye motion profiles to use in the simulation.

Figure 4.5: Human donor eye mounted in an acrylic holder filled with 2.4% gelatin to simulate the bony orbit geometry and orbital fat mechanical properties, respectively.

Results

Characteristic in vivo dataset

It was shown that the orbital tissues were well represented by a Kelvin-Voigt viscoelastic element, and the simulated WEM had good agreement with the in vivo data

(Figure 4.6). The optimized parameter values describing the orbital tissues were utilized as the initial guess for future iterations of the model.

Figure 4.6 Results of WEM only simulation where the peri-ocular tissues are represented as (a) purely elastic or (b) KV-viscoelastic

When the cornea was represented as linearly elastic, the simulated corneal motion

demonstrated poor fit with the in vivo data and showed oscillations due to underdamping

(Figure 4.7a, 4.7b). These oscillations show that viscous damping must be considered to

accurately reproduce the corneal response to dynamic loading, and that the viscoelasticity

of the cornea cannot be neglected under this short time scale. The simulated corneal motion

improved dramatically once a viscous element was added, and the 3-parameter viscoelastic

solid representation was ultimately selected to represent the cornea (Figure 4.7c, 4.7d).

Figure 4.7 Comparison of in vivo experimental measurements and model predictoins of

WEM (above) and corneal displacement (below) where (a, b) cornea is represented as purely elastic and where (c, d) cornea is represented as a 3-parameter linear viscoelastic solid.

Figure 4.8 Comparison of simulation results where (a, b) the non-linear stiffening spring

(representing the sclera) is absent and (c, d) where the sclera is present.

Unlike the motion observed in vivo, the peak of corneal motion in the model

simulations was in phase with the peak of the air-puff pressure load. There was also a curvature mismatch near the peak of corneal motion, where in vivo there is an observed oscillatory motion after plateauing. The addition of the non-linear stiffening spring element resulted in shifting the peak of corneal motion to occur before the peak of the load, which is observed in vivo (Figure 4.8).

Scleral and corneal validation

Paired t-test analysis was used to compare model parameter values before and after scleral-stiffening, and significant differences were found if p < 0.05. Following scleral stiffening with glutaraldehyde, the model showed that only the non-linear stiffening spring parameter value significantly changed, suggesting that the non-linear stiffening spring represented the contribution of the sclera to limiting corneal motion, and may also include any non-linear response within the cornea (Figure 4.9). Meanwhile, the three parameters that make up the standard linear solid element showed no significant changes at all tested levels of IOP, suggesting that the biomechanical properties of cornea may be ascribed to these three parameters.

For corneal parameters in the model, the mean difference between treated and untreated sclera groups was < 2% (Table 4.1). At each level of IOP, the scleral parameter in the model increased after glutaraldehyde treatment, suggesting that the model can detect the change in scleral stiffness. The difference between treated and untreated scleral stiffness parameter values decreased as a function of IOP, which is consistent with literature reports (Metzler et al., 2014). The only parameter that was shown to be strongly dependent on IOP was the corneal damping parameter, b2, which may be useful in interpretation of the measurement of IOP.

Figure 4.9: Results of ex vivo validation studies, in which the sclera of one eye was stiffened with 4% glutaraldehyde. Significant (p<0.05) changes were observed only in the non-linear stiffening spring term (validating that it represents the sclera) before and after scleral stiffening on donor eyes, denoted by *. There were no changes observed in the three parameters ascribed to representing corneal properties.

Table 4.1 Results of ex vivo scleral stiffening validation studies. The parameter values

Cornea Maxwell Spring (k2) [N/m] Pre Post % Change IOP 10 38.728 ± 9.182 39.055 ± 9.483 0.714 ± 2.009 IOP 20 44.911 ± 14.875 44.888 ± 15.325 -0.332 ± 4.544 IOP 30 45.533 ± 16.672 45.315 ± 16.192 -0.268 ± 2.999 IOP 40 42.718 ± 14.181 43.091 ± 14.442 0.844 ± 4.370

Cornea Maxwell Damper (b2) [N*s] Pre Post % Change IOP 10 4.614 ± 2.490 4.139 ± 2.857 1.952 ± 3.452 IOP 20 0.947 ± 0.410 0.952 ± 0.407 0.865 ± 4.108 IOP 30 0.134 ± 0.128 0.135 ± 0.127 1.191 ± 2.869 IOP 40 0.022 ± 0.016 0.021 ± 0.015 -0.793 ± 5.009

Cornea Parallel Spring (k3) [N/m] Pre Post % Change IOP 10 42.824 ± 8.808 42.969 ± 9.007 0.273 ± 2.126 IOP 20 63.068 ± 20.366 61.889 ± 18.554 -1.087 ± 4.644 IOP 30 56.439 ± 20.303 56.629 ± 20.540 0.163 ± 2.830 IOP 40 44.875 ± 22.216 45.123 ± 22.209 -0.991 ± 9.009

Sclera Non-linear Stiffening Spring (k4) [N/m] Pre Post % Change IOP 10 3.150 ± 1.782 9.222 ± 3.189 277.298 ± 266.295† IOP 20 3.567 ± 0.863 4.972 ± 0.731 45.566 ± 35.741† IOP 30 3.730 ± 0.401 4.095 ± 0.672 9.732 ± 13.466† IOP 40 5.105 ± 0.536 5.463 ± 0.712 7.386 ± 12.609†

Orbital fat validation

The corneal and whole-eye motion profiles from a human donor globe mounted in a rigid fixture and then in 2.4% gelatin were exported for analysis with the mathematical model (Figure 4.10). We hypothesized that the softer gelatin would reduce the value of the orbital stiffness parameter and that the parameters describing the sclera and cornea would remain unchanged.

Figure 4.10: Results of ex vivo orbital tissues study. The WEM and cornea motion

profiles of a human donor eye mounted in the rigid fixture (blue), in 2.4% gelatin in a

simulated orbit (pink), and the characteristic published in vivo subject (black) are shown

here for comparison.

The model fitting showed large reductions in the orbital spring and dashpot parameter

values, 24.8% and 71.8% respectively, between the rigid fixture and the gelatin (Table

4.2). The change in parameter values for the springs and dashpot representing corneal and

scleral properties showed <10% change, and were not substantially different between the

two ocular supports.

Table 4.2: Parameter values from ex vivo orbital tissues simulation study were tabulated.

There were large changes (>20%) in the magnitude of the model parameters ascribed to

orbital tissues. These large decreases were expected as the gelatin allowed for whole-eye

motion to occur while the rigid fixture restricted WEM. There were no substantial

changes (<10%) observed in the magnitudes of the model parameters describing cornea

or sclera.

Rigid Fixture 2.4% Gelatin % Change Orbital Spring (k1) [N/m] 434.910 326.957 -24.822 Orbital Damper (b1) [N*s] 7.881 2.223 -71.793 Cornea Maxwell Spring (k2) [N/m] 76.732 82.323 7.286 Cornea Maxwell Damper (b2) [N*s] 4.411 4.054 -8.093 Cornea Parallel Spring (k3) [N/m] 14.824 14.001 -5.552 Sclera Non-Linear Spring (k4) [N/m] 3.970 3.685 -7.179

Discussion

Several numerical and finite element studies have suggested that the viscoelasticity

of the cornea may be neglected under air-puff loading (Sinha Roy et al., 2015, Simonini and Pandolfi, 2016). This approach is predicated on the assumption that dynamic loading may be accurately represented by purely elastic deformations. However, the present analysis demonstrates that viscous contributions are essential during dynamic loading, that 63

the viscoelasticity is an important component of corneal biomechanical response and must

be considered under these loading conditions. The poor fit and significant oscillations in

the simulated corneal motion indicated that a purely elastic cornea is significantly

underdamped on the timescales involved in air-puff loading. This may be seen most easily

by examining the equation of motion for a linear viscoelastic Kelvin-Voigt element

(Equation 7),

= + (7)

𝑭𝑭𝑲𝑲𝑽𝑽 𝒌𝒌�𝒌𝒌 𝝁𝝁�𝝁𝝁 𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆𝒆 𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗𝒗

where the elastic contribution depends on the displacement u but is independent of the

velocity v, whereas the viscous force is proportional to the viscosity (the viscosity µ is the

proportionality constant). Thus, increasing the velocity will always increase the viscous

contribution to the total force F. In quasi-static loading, v ~ 0 such that the elastic forces

are much larger than the viscous forces. However, at high loading rates, the viscous

contribution becomes increasingly important rather than less important.

There are several important aspects of corneal and whole-eye motion that are well

described by this simple model. The model closely matched the displacement behaviors of

the corneal apex and the whole eye; specifically, even when the cornea begins to recover

and move anteriorly, the whole eye continues to move posteriorly due to inertial forces,

with maximum magnitude near second (outgoing) applanation (A2).

One of the most important events that was successfully recreated by the model was that the peak of corneal motion in vivo precedes the peak of the air-puff load profile; 64

reproducing this feature in the model required inclusion of a non-linear stiffening spring

element representing the constraint on corneal lateral expansion by the sclera. It has been

shown experimentally that the sclera has a significant contribution to limiting corneal

deformation under air-puff loading by a dynamic Scheimpflug analyzer (Metzler et al.,

2014; Nguyen et al., 2018, Nguyen et al., 2019). The addition of the non-linear stiffening

spring element resulted in shifting the peak of corneal motion to occur before the peak of

the load. In vivo, the cornea reaches a point where it can no longer deform (i.e. where the

sclera limits further corneal deformation). It is at this point that the whole-eye motion

transitions from slow, linearly increasing to rapid, non-linearly increasing posterior

motion. The model was able to successfully recreate these important events and transitions

that are observed in vivo.

The representation of ocular tissue properties as mechanical springs and dashpots

was validated via several ex vivo experiments. In an experiment where only the sclera was

chemically stiffened, the model parameters showed significant changes only in the values

of non-linear stiffening spring representing the sclera. The three model parameters describing corneal properties did not significantly change following scleral stiffening.

However, the non-linear stiffening spring may also include the non-linear response of the

cornea, and further study is required to clarify whether the non-linear stiffening spring

represents purely scleral response. And in a separate experiment where the simulated

orbital tissues were varied, the model resulted in large changes only in the parameters that

were ascribed to orbital fat. These ex vivo experimental results strongly support the model

representation of ocular and orbital tissues.

Conclusion

This simple but powerful mathematical model, which appropriately considers the

biomechanical behavior of cornea, sclera, and orbital tissues, may be useful in

interpretation of in vivo changes in corneal and whole-eye motion in different disease states or following treatments which alter properties of these tissues.

Chapter 5: A nonlinear viscoelastic model of corneal and whole-eye motion of

prostaglandin-analog treated subjects under loading by dynamic Scheimpflug analyzer

Introduction

Glaucoma is a blinding diseased that is associated with increased intraocular pressure (IOP) (Weinreb and Khaw, 2004). IOP is the only modifiable risk factor in glaucoma, and a common first line of treatment is topical prostaglandin-analogs (PGAs) due to their effectiveness of reducing measured IOP to preserve vision (Toris et al., 2008).

The primary mechanism of action of PGAs is to increase uveoscleral outflow by increasing scleral permeability (Toris et al., 2008; Alm and Nilsson, 2009), which may result in a lower measured IOP and a less stiff sclera (Sagara et al., 1999).

The measured reduction in IOP following PGA-treatment may mask changes in ocular and orbital tissue properties (i.e. changes to stiffness of the cornea, sclera, and orbital fat). It has been shown that corneal properties significantly influence IOP as measured by

Goldmann tonometry (Liu and Roberts, 2005). It has also been shown that the biomechanical response of the cornea is impacted by the properties of the sclera (Nguyen et al., 2018A).

The CorVis ST (Oculus Optikgerate GmbH, Wetzlar, Germany) is a noncontact tonometer and high-speed Scheimpflug camera that allows for clinical assessment of

corneal biomechanical response in vivo. The device outputs dynamic corneal response

(DCR) parameters to describe the motion of the cornea during air-puff loading, as well as the motion of the whole-eye. Recently, it has been shown that changes to orbital tissue properties such as swelling in Graves orbitopathy (Leszcynska et al., 2018; Hwang et al.,

2019) resulted in changes in measured whole-eye motion.

PGA-treatment is known to cause atrophy of peri-orbital fat (Jayaprakasam and

Ghazi-Houri, 2010; Park et al, 2011; Tan and Berke, 2013), therefore we hypothesize that there will be orbital fat atrophy following PGA treatment and resulting increase in measured whole-eye motion under loading by CorVis ST.

Methods

An analytical nonlinear viscoelastic model of centerline corneal and whole-eye motion had previously been developed and validated, and is briefly described herein

(Nguyen et al., 2018B). The model considers the eye as a 2 degree-of-freedom mass- spring-damper system loaded by the air-puff from the CorVis ST (Figure 1). The cornea and globe are represented as inertial mass elements. Orbital tissues are represented as a

Kelvin-Voigt (KV) viscoelastic solid, the cornea was represented as a 3-parameter standard linear solid, and the sclera was represented as nonlinear stiffening spring element.

Figure 5.1: Representation of the eye as a 2 degree-of-freedom mass-spring-damper system (Nguyen et al. 2018B)

The system of equations of motion for this model were derived and solved numerically using MATLAB (The MathWorks, Inc.; Natick, Ma). The forces through the

KV-viscoelastic element, the 3-element standard linear solid, and the non-linear stiffening spring are represented as FKV, FSLS, and FNL, respectively (Equations 1 – 5). Displacement of the cornea and the globe are denoted as uc and ug, respectively, with velocities denoted

as vc and vg, respectively.

= + ( ) (1)

𝑲𝑲𝑲𝑲 𝟏𝟏 𝒈𝒈 𝟏𝟏 𝒈𝒈 𝑭𝑭 𝒌𝒌 (�𝒖𝒖 +� 𝒃𝒃) 𝒗𝒗 (2) = + 𝒃𝒃𝟐𝟐 𝒌𝒌𝟐𝟐 𝒌𝒌𝟑𝟑 𝒃𝒃𝟐𝟐 𝑑𝑑𝐹𝐹𝑎𝑎𝑎𝑎𝑎𝑎 𝑭𝑭𝑺𝑺𝑺𝑺𝑺𝑺 𝒌𝒌𝟑𝟑�𝒖𝒖𝒄𝒄 − 𝒖𝒖𝒈𝒈� �𝒗𝒗𝒄𝒄 − 𝒗𝒗𝒈𝒈� − 𝒌𝒌𝟐𝟐 𝒌𝒌2 𝑑𝑑𝑑𝑑 (3) = 𝑘𝑘4 𝐹𝐹𝑁𝑁𝑁𝑁 �𝑢𝑢𝑐𝑐 − 𝑢𝑢𝑔𝑔� �𝑢𝑢𝑐𝑐 − 𝑢𝑢𝑔𝑔 = = + (4)

� 𝐹𝐹𝑔𝑔𝑥𝑥 𝑚𝑚𝑔𝑔 ∗ 𝑎𝑎𝑔𝑔 𝐹𝐹𝑆𝑆𝑆𝑆𝑆𝑆 𝐹𝐹𝑁𝑁𝑁𝑁 − 𝐹𝐹𝐾𝐾𝐾𝐾 = = ( + ) (5)

� 𝐹𝐹𝑐𝑐𝑥𝑥 𝑚𝑚𝑐𝑐 ∗ 𝑎𝑎𝑐𝑐 𝐹𝐹𝑎𝑎𝑎𝑎𝑎𝑎 − 𝐹𝐹𝑆𝑆𝑆𝑆𝑆𝑆 𝐹𝐹𝑁𝑁𝑁𝑁

From a separate study, 10 eyes from 5 human subject (1 female, 4 male, age 55.8 ±

4.6 years) were selected to have their measured corneal and whole-eye motion profiles analyzed by the model (Roberts et al., 2018). The subjects were selected by having at least a 10 mmHg drop in IOP as measured by Goldmann applanation tonometry in order to ensure compliance with PGA treatment protocol. IOP was also measured for each subject using the CorVis ST (bIOP), Ocular Response Analyzer (ORA, Reichert Technologies,

Depew, NY, USA) (IOPcc), and PASCAL Dynamic Contour Tonometer (DCT, Ziemer,

Port, Switzerland).

The values of constants representing the stiffness or cornea and viscosity of the spring and dashpot elements were optimized to reduce the cumulative error between the simulated motion and in vivo motion of the cornea and whole-eye exported by the CorVis

ST. To evaluate the proportion of air-puff force acting on the cornea as compared to the whole-eye, the impulse for each mass element was calculated. This was done by integrating the net force acting on either the cornea or globe over time, as described by the system of equations of motion.

Parameter values outputted by the model and select dynamic corneal response

(DCR) parameters were compared before and after PGA-treatment using paired t-tests.

Statistical significance was determined if p<0.05.

Results

The simulation sufficiently fit the measured corneal and whole-eye motion curves from the CorVis ST for each eye before and after PGA-treatment (Figure 2). It was 70 observed for all eyes that there was a significant (p<0.05) decrease in measured whole-eye motion (Table 1).

WEM only - Visit 1 WEM Only - Visit 2 250 250 Ex vivo WEM Ex vivo WEM 1.6 Optimized Sol't; 1.6 Optimized Sol't; Peak Air Pressure Peak Air Pressure 200 200 1.4 1.4 Air-Puff Pressure Air-Puff Pressure

1.2 1.2 150 150 1 1

0.8 0.8 100 100

0.6 0.6

0.4 50 0.4 50 Pressure (mmHg) Pressure (mmHg) Displacement (mm) Displacement (mm) 0.2 0.2

0 0 0 0

-0.2 -0.2 -50 -50 -0.4 -0.4 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Time (ms) Time (ms)

Corneal Deflection only - Visit 2 Corneal Deflection only - Visit 1 250 250 Ex vivo Corneal Deflection Ex vivo Corneal Deflection 1.6 Optimized Sol't; 1.6 Optimized Sol't; Peak Air Pressure Peak Air Pressure 200 200 1.4 Air-Puff Pressure 1.4 Air-Puff Pressure

1.2 1.2 150 150 1 1

0.8 0.8 100 100 0.6 0.6

0.4 50 0.4 50 Pressure (mmHg) Pressure (mmHg) Displacement (mm) Displacement (mm) 0.2 0.2 0 0 0 0

-0.2 -0.2 -50 -50 -0.4 -0.4 0 5 10 15 20 25 30 35 0 5 10 15 20 25 30 35 Time (ms) Time (ms) Figure 5.2: Simulation results for one eye of one in vivo PGA-treated subject, showing the fit of the model to corneal and whole-eye motion before and after PGA-treatment.

Table 5.1: CorVis ST DCRs describing first and second applanation (A1, A2) velocities

and whole-eye motion (WEM) following PGA-treatment. Statistical significance was determined if p<0.05 by paired t-test, and is denoted by ‡.

CorVis ST Parameter Visit 1 Visit 2 % Change A1 Velocity [m/s] 0.10 ± 0.02 0.13 ± 0.02 38.20 ± 25.59 ‡ A2 Velocity [m/s] -0.19 ± 0.04 -0.25 ± 0.03 38.20 ± 31.29 ‡ WEM Max [mm] 0.30 ± 0.04 0.26 ± 0.05 -11.08 ± 10.18 ‡ WEM Max [ms] 22.42 ± 0.92 22.69 ± 0.50 1.32 ± 3.07

The model simulation however, showed no statistical difference in the parameter

values describing the orbital tissues (Table 2). The simulation results showed significant

decreases in scleral and corneal stiffness, and a significant increase in corneal damping

parameter (strongly associated with the decrease in measured IOP).

Table 5.2: Parameter values for spring and dashpot elements in the model for PGA-

treated subjects at visit 1 (no treatment) and at visit 2 (1 month of PGA-treatment).

Statistical significance was determined if p<0.05 by paired t-test, and is denoted by ‡.

Model Elements Visit 1 Visit 2 % Change k1 342.23 ± 55.54 375.81 ± 57.24 11.01 ± 14.54 Orbital Tissue Parameters b1 2.18 ± 0.55 2.62 ± 0.64 24.93 ± 36.96 k2 (Series Spring) 118.19 ± 29.91 86.01 ± 11.00 -23.52 ± 18.69 ‡ Corneal Parameters b2 0.16 ± 0.12 3.53 ± 0.50 4.8e3 ± 5.6e3 ‡ k3 (Parallel Spring) 41.79 ± 21.61 54.05 ± 11.62 115.74 ± 236.44 ‡ Scleral Parameter k4 3.80 ± 0.77 2.00 ± 0.23 -45.49 ± 11.19

When comparing select CorVis ST DCRs, DA Ratio 2mm and Integrated Inverse

Radius both significantly increase, while SP-A1 significantly decreases suggesting a less stiff cornea after PGA-treatment (Table 3). SP-HC, which may be sensitive to scleral properties, showed significant decrease following PGA-treatment. Deflection velocities

(corneal velocity where whole-eye motion is removed) at A1 and A2 were both significantly increased.

Table 5.3: Select DCRs describing the deflection velocities of the cornea through applanation, corneal shape parameters, and stiffness parameters (SP) at first applanation

(A1) and highest concavity (HC). Statistical significance was determined if p<0.05 by paired t-test, and is denoted by ‡.

CorVis ST Parameter Visit 1 Visit 2 Mean % Change A1 Deflection Velocity 0.095 ± 0.028 0.132 ± 0.032 38.28‡ A2 Deflection Velocity -0.240 ± 0.060 -0.339 ± 0.047 41.18‡ DA Ratio 2 mm 3.922 ± 0.346 4.712 ± 0.498 20.16‡ Integrated Inv. Radius 6.497 ± 0.715 8.178 ± 1.063 25.88‡ SP A1 140.036 ± 16.379 113.508 ± 21.832 -18.94‡ SP HC 17.286 ± 5.167 8.530 ± 3.602 -50.65‡ SP A1/SP HC 8.767 ± 2.35 14.227 ± 2.539 62.27‡

In order to compare the amount of force or kinetic energy acting on each mass over

the duration of the air-puff loading cycle, the impulse was calculated for both the cornea

and the globe mass elements (Equation 6, 7). When comparing the calculated impulse for

cornea and globe, it was shown that the globe impulse significantly decreased while the

corneal impulse significantly increased (Table 4). The change in impulse suggests that a 73 greater proportion of the air-puff force is redistributed to corneal motion and deformation as compared to the globe.

(6) = ( ) = ( ) = = 𝜕𝜕𝜕𝜕 𝐼𝐼 � 𝐹𝐹 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑚𝑚 � 𝑎𝑎 𝑡𝑡 𝑑𝑑𝑑𝑑 𝑚𝑚 � 𝑑𝑑𝑑𝑑 𝑚𝑚𝑚𝑚 1𝜕𝜕 𝜕𝜕 (7) = = = 2 2 𝐾𝐾𝐾𝐾 � 𝐼𝐼 𝐼𝐼𝐼𝐼 𝑚𝑚 � 𝑣𝑣𝑣𝑣𝑣𝑣 𝑚𝑚 𝑣𝑣

Table 5.4 : Calculated impulse for the cornea and globe

Calculated Impulse Visit 1 Visit 2 % Change Globe [kg*m/s] -1.65e-4 ± 3.93e-5 -1.33e-4 ± 2.72e-5 -18.34 ± 15.20 ‡ Cornea kg*m/s] -4.73e-7 ± 1.47e-7 -5.72e-7 ± 1.54e-7 25.71 ± 30.08 ‡

Figure 5.3: A comparison of the relationship between the corneal damping parameter

(b2) and IOP for the analytical model and each of four tonometric modalities: (a) CorVis

ST (bIOP), (b) ORA (IOPcc), (c) DCT, and (d) GAT.

The corneal viscous parameter (b2) has been previously shown to be strongly

correlated with IOP in the ex vivo validation studies. The simulation values of b2 for the

PGA-treated subjects and their relationship with measured IOP is presented in Figure 3.

Discussion

The measured whole-eye motion of PGA-treated subjects decreased significantly following treatment, which may be misinterpreted as a stiffening or stiffer response of the 75 orbital tissues. Literature reports suggest that there is peri-orbital fat atrophy following

PGA use and we thus expected an increase in WEM due to potential atrophy of orbital fat.

However, the opposite was found in that WEM significantly decreased. This seeming paradox can be explained using the analytical model which showed that there were no significant changes in the parameter values describing orbital tissue properties (k1, b1) following PGA-treatment. The source of the decreased WEM was found to be due to coupling of the reduced corneal stiffness, scleral, stiffness and IOP. When comparing calculated impulse (change in momentum) the cornea and globe, it was shown that the corneal impulse increased and globe impulse decreased following PGA-treatment. This change in impulse shows that a greater proportion of the kinetic energy from the CorVis

ST air-puff is absorbed by the cornea following PGA-treatment, and that less of the air- puff force is used in the rearward translation of the globe. This complex and dynamic interaction explains why the measured whole-eye motion decreases despite no changes to the properties of the orbital tissues from fitting with the analytical model.

Significant changes were found in corneal and scleral model parameters. The nonlinear stiffening spring stiffness parameter significantly decreased, suggesting a less stiff sclera. This is consistent with increased scleral permeability and uveoscleral outflow, which results in a mechanically softer sclera. Additionally, the stiffness parameter at highest concavity (SP-HC), which has been shown to be sensitive to scleral properties, showed a significant decrease following PGA use and also suggests a less stiff sclera

(Nguyen et al., 2019).

The increased corneal viscous parameter (b2) has been shown to be strongly

associated with changes in IOP in ex vivo validation studies (Nguyen et al, 2018) and the

significant increase observed for the PGA-treated subjects is similarly associated with the reduction in IOP. When comparing the relationship between this parameter and IOP for the model and several tonometric modalities, we observe that the IOPcc measured by the ORA is most closely aligned with the model. However, we note the limitation in comparing the b2 relationship from the model, which was determined with ex vivo donor eyes, and the b2 relationship with a small sample of human subjects.

Only one of the spring terms in the cornea saw a significant reduction (k2), while the other did not significantly change following PGA-treatment. To evaluate whether this reduction in model stiffness parameters suggested a less stiff cornea after PGA use, we looked at the change in several corneal shape parameters outputted by the CorVis ST. It was observed that both the Integrated Inverse Radius and Deformation Amplitude (DA)

Ratio at 2 mm significantly increased, indicating a less stiff cornea. The stiffness parameter at first applanation (SP-A1), which has been shown to be sensitive to corneal properties, showed a significant decrease with PGA use, also indicating a less stiff cornea.

The simulation results strongly suggest that the observed reduction in whole-eye motion in PGA-treated subjects is due to the coupling of reduced corneal stiffness, scleral stiffness and IOP. The reduction in whole-eye motion may be misinterpreted as a stiffening or fibrosis of the orbital fat.

Conclusion

An analytical nonlinear viscoelastic model of corneal and whole-eye motion was applied to the motion profiles of 5 prostaglandin-naïve subjects undergoing first time treatment with prostaglandin-analogs. The parameter values output by the model were compared before and 1 month after treatment. An unexpected significant decrease in measured WEM, despite no changes in orbital tissue model parameters, was determined to be the result of decreased corneal stiffness, scleral stiffness, and reduction of IOP. The calculated impulse of cornea and globe showed a greater amount of kinetic energy from the CorVis ST air-puff being redirected to corneal deformation and motion instead of rearward translation of the globe. The model showed decreased corneal and scleral stiffness, and analysis of DCRs supported that there were changes to ocular tissue properties following PGA-treatment. This analytical model provides new insights into the impact of prostaglandin-analog treatment on ocular tissues, and elucidates the complex interaction of ocular and orbital tissues and their contributions to the motion of the cornea and globe under air-puff loading with clinical use of prostaglandin-analog treatment in glaucoma.

Chapter 6: Conclusions of the Study

The results of these studies have provided significant insights and contributions to

the understanding of the biomechanical behavior of ocular and orbital tissues. The overall

conclusions are presented below:

Aim 1: Evaluate the biomechanical impact of the sclera on dynamic corneal deformation

response to an air-puff

The results of the finite element model showed that increasing scleral stiffness (with constant corneal stiffness and IOP) resulted in decreasing displacement of the corneal apex, i.e. the cornea responded to the air-puff as if it had stiffer mechanical properties. The finite-

element model also showed that apical displacement decreased nonlinearly with increasing

IOP, which is consistent with literature reports. This study demonstrated that the sclera has

an inseparable impact on the biomechanical deformation response of the cornea, and that

the corneal response to an air-puff is the result of both corneal and scleral properties, in

addition to IOP.

The results of the ex vivo studies of scleral stiffening showed that the corneal

response to an air-puff is significantly impacted by scleral properties. Several dynamic

corneal response (DCR) parameters exported by the CorVis ST noncontact tonometer were

analyzed to compare the effect of stiffening the sclera while leaving the cornea untreated.

With stiffened sclera, changes in DCRs showed that the cornea had an apparently stiffer 79 response to air-puff loading. Additionally, we showed that the Stiffness Parameter (SP) at

Aim 2: Describe the relative contribution of ocular and orbital tissues to corneal and whole-eye motion

The resultant analytical model was validated with data from ex vivo studies and showed that (1) orbital tissues were best represented by a Kelvin-Voigt viscoelastic solid,

(2) the cornea must necessarily be viscoelastic, and was best represented as a 3-parameter standard linear solid, and (3) that the contribution of the sclera must be considered to reproduce the timing of key events in in vivo motion, and that the sclera is accurately modeled as a nonlinear stiffening spring.

Aim 3: Determine the impact of prostaglandin-analog (PGA) treatment on ocular and orbital tissue properties and biomechanical response

The results of applying the analytical model showed that while there were no changes in orbital tissue properties, that the apparent contradiction of decreasing whole- eye motion following PGA-treatment was the result of decreased corneal and scleral stiffness, in addition to a reduction in IOP. Calculated impulse of the cornea and globe

80 showed that PGA-treatment resulted in a significant increase in corneal impulse and decrease in globe impulse, demonstrating that a greater proportion of the kinetic energy from the air-puff is absorbed the cornea and less kinetic energy is used in the rearward translation of the globe.

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