a structural model for the in vivo human cornea including collagen-swelling...

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ResearchCite this article: Cheng X, Petsche SJ, Pinsky

PM. 2015 A structural model for the in vivo

human cornea including collagen-swelling

interaction. J. R. Soc. Interface 12: 20150241.

http://dx.doi.org/10.1098/rsif.2015.0241

Received: 18 March 2015

Accepted: 17 June 2015

Subject Areas:biophysics, computational biology,

biomechanics

Keywords:cornea, hydration, collagen-swelling

interaction, proteoglycan, osmotic pressure,

pump-leak

Author for correspondence:Peter M. Pinsky

e-mail: [email protected]

Electronic supplementary material is available

at http://dx.doi.org/10.1098/rsif.2015.0241 or

via http://rsif.royalsocietypublishing.org.

& 2015 The Author(s) Published by the Royal Society. All rights reserved.

A structural model for the in vivo humancornea including collagen-swellinginteraction

Xi Cheng, Steven J. Petsche and Peter M. Pinsky

Department of Mechanical Engineering, Stanford University, Stanford, CA, USA

A structural model of the in vivo cornea, which accounts for tissue swelling

behaviour, for the three-dimensional organization of stromal fibres and for

collagen-swelling interaction, is proposed. Modelled as a binary electrolyte

gel in thermodynamic equilibrium, the stromal electrostatic free energy is

based on the mean-field approximation. To account for active endothelial

ionic transport in the in vivo cornea, which modulates osmotic pres-

sure and hydration, stromal mobile ions are shown to satisfy a modified

Boltzmann distribution. The elasticity of the stromal collagen network is

modelled based on three-dimensional collagen orientation probability distri-

butions for every point in the stroma obtained by synthesizing X-ray

diffraction data for azimuthal angle distributions and second harmonic-

generated image processing for inclination angle distributions. The model

is implemented in a finite-element framework and employed to predict

free and confined swelling of stroma in an ionic bath. For the in vivocornea, the model is used to predict corneal swelling due to increasing intra-

ocular pressure (IOP) and is adapted to model swelling in Fuchs’ corneal

dystrophy. The biomechanical response of the in vivo cornea to a typical

LASIK surgery for myopia is analysed, including tissue fluid pressure and

swelling responses. The model provides a new interpretation of the corneal

active hydration control (pump-leak) mechanism based on osmotic pressure

modulation. The results also illustrate the structural necessity of fibre incli-

nation in stabilizing the corneal refractive surface with respect to changes

in tissue hydration and IOP.

1. IntroductionIn the simplest terms, the cornea is a fibre-reinforced fluid membrane, which

resists the intraocular pressure (IOP) applied on its internal boundary. Its tensile

strength derives from the three-dimensional organization of collagen fibres and

its bulk properties derive from the interfibrillar fluid pressure. The fluid pressure

depends on both the IOP and the tissue osmotic pressure. The osmotic pres-

sure causes the cornea to have a tendency to swell by imbibing water from the

adjacent anterior chamber of the eye [1,2]. In the in vivo cornea, the osmotic

pressure is modulated by active ionic transport processes as a means to control

the level of tissue hydration, which is important for transparency and also affects

the mechanical behaviour of the tissue (cf. bending of a swollen and non-swollen

cornea). Ideally, the tissue fluid pressure, swelling effects (including active modu-

lation processes) and the local interaction of collagen fibres with the swelling

tissue, should be accounted for in any structural model based on first princi-

ples. Some important modelling concepts towards this end have already been

introduced, as described below. However, no fully three-dimensional and

comprehensive model has yet been presented. It is noteworthy that current

finite-element-based models for structural analysis of the cornea treat the inter-

fibrillar fluid as an incompressible or nearly incompressible elastic solid [3–6].

While this approach is convenient, it cannot describe swelling behaviour or

(steady-state) bulk compressibility. The goal of this work is to present a math-

ematical description of the corneal stroma as an electrolyte gel, characterizing

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12:20150241

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the fluid pressure, swelling behaviour and collagen-swelling

interaction. We show that such a model can extend predictive

accuracy in corneal biomechanics.

The cornea has five principal layers which, from anterior to

posterior, are the epithelium, Bowman’s, stroma, Descemet’s

and endothelium. Bowman’s and Descemet’s layers are

basement membranes for the cellular epithelium and endo-

thelium layers, respectively. The stroma occupies 90% of the

cornea’s thickness and like other soft, highly hydrated and

charged collagenous tissues such as cartilage and interverteb-

ral disc, is a polyelectrolyte gel consisting of a mixture of

interacting fluid, solid and ionic phases. Water is the principal

component of stroma which saturates the collagen solid phase,

solvates the ionic phase and accounts for about 78% of the

cornea by weight [7]. A small portion of the water is cellular

or bound to the stromal collagen [8], but much of the water

is free to flow within the stroma in response to gradients in

fluid pressure. The solid phase of corneal stroma comprises a

flexible collagen network, organized as fibrils within lamellae

and associated proteoglycans (PGs). Stromal PGs have sulfated

linear side chains of negatively charged disaccharide units

called glycosaminoglycans (GAGs) that are covalently bound

at one end to the PG core protein [9,10]. At normal pH, the

stromal fixed charge is almost entirely due to GAG ionization

[1]. The ionic phase includes dissolved salts, primarily Naþ

and Cl2, and metabolites such as C3H5O3� (lactate ion) and

HCO3� (bicarbonate ion).

The stromal mobile ions interact electrostatically with the

GAG fixed charges and form cloud-like distributions giving

rise to osmotic pressure. The aqueous humour filling the

anterior chamber produces the IOP, acts as an ionic bath for

the stroma and provides a reservoir of water that is available

for exchange with the stroma. Water transport across the per-

meable endothelial layer is driven by the osmotic pressure

difference between the stroma and the aqueous humour.

When the stromal ionic concentration exceeds the ionic con-

centration in the aqueous humour, a positive osmotic

pressure difference exists and the stroma will swell by

inflow of water from the anterior chamber until the ionic con-

centrations are equalized or the expansion is restrained by

some agent. By contrast, the anterior stroma is sealed by

the epithelial cellular layer, which is nearly impermeable to

water and ions (although permeable to O2 and CO2) [11].

In the in vivo cornea, the endothelial layer supports active

molecular mechanisms that produce an outward flux of

ions from the stroma into the aqueous humour. Our analysis

herein confirms that such active pumping reduces the stromal

ionic concentrations and hence lowers the osmotic pressure

[1]. The regulatory system and details of the molecular mech-

anisms responsible for active ion transport, which requires

Naþ, Kþ, ATPase and carbonic anhydrase activity to trans-

port HCO3�, Cl2 and possibly Naþ, are not yet fully

understood [12].

From a macroscopic perspective, the control of hydration

has been explained by a ‘pump-leak’ theory [12–14]. In the

pioneering work of [15], based on the phenomenological

membrane transport theory of Kedem–Katchalsky (KK)

[16], non-equilibrium fluid and ionic fluxes through the

corneal stroma and endothelium were modelled and time-

dependent swelling resulting from osmotic perturbations

were predicted. This work, and the related study by Ruberti &

Klyce [17], provides a theoretical confirmation that active

endothelial ion transport produces an osmotic gradient

across the endothelium which can modulate water flow and

swelling. Li & Tighe [18] later proposed an extended model

which considers multiple ionic species and Leung et al. [19]

have further extended the theory to include metabolites

(e.g. C3H5O3�) and metabolic reactions in a steady-state

model. These models employ a one-dimensional represen-

tation of the cornea, do not consider stromal fixed charge

or collagen interaction with swelling, and the swelling (i.e.

osmotic) pressure is not derived, but assumed as an empirical

function of tissue hydration based on measurements by

Hedbys & Dohlman [20]. In the important extension by

Bryant & McDonnell [21], a spherically symmetric (one-

dimensional) steady-state model based on the triphasic

theory of Lai et al. [22] was combined with KK theory for pas-

sive and active transport across the endothelium. This model

describes ionic interaction with fixed charges, fluid inter-

action with a solid elastic phase and ideal Donnan osmotic

pressure. While greatly simplifying the stromal elasticity,

this model does capture the full range of interactions in an

elegant triphasic framework. The above models are complex,

and it is not surprising that they do not agree for all predic-

tions, for example, on the sign of the stromal fluid pressure.

In this work, we build on these foundational efforts but

take an alternative modelling approach.

In the proposed model, conditions of thermodynamic

equilibrium are assumed to hold, rendering water and ionic

fluxes time-independent. Such an approach has utility when

the long-time, steady-state response of the cornea to disease

processes or surgical alteration is desired. While modelling

based on steady-state conditions does not allow the time

course of swelling to be described, it does avoid some of the

high complexity of non-equilibrium multiphasic theory and

the analytical simplicity of the resulting theory leads to a

practical and effective theory for finite-element-based general

structural analysis of the living cornea.

Modelled as an electrolyte gel, the free energy is taken to

be additively decomposed into various components which

characterize the behaviour of the tissue under general defor-

mations [23,24]. A key ingredient in this approach is the

statement of electrostatic free energy. For the ex vivo cornea

with no active ionic transport, we use the mean-field approxi-

mation of the Helmholtz free energy for a binary electrolyte

which measures the energy of the GAG-based fixed charge,

the osmotic energy of mobile ions in a Boltzmann distribution

and the dielectric free energy [25]. To account for active endo-

thelial ion transport in the in vivo cornea, we consider

thermodynamic equilibrium expressed in terms of ionic and

water electrochemical potentials and show that the stromal

mobile ions satisfy a modified Boltzmann distribution that

depends on the active ionic fluxes and endothelial ionic per-

meability. The theory easily takes into account the important

volume exclusion effects associated with collagen and kerato-

cyte populations [23] and the electrostatic free energy

corresponding to the modified Boltzmann ionic distribution

is readily found and used to predict the osmotic pressure in

the in vivo cornea. Under certain assumptions on the electro-

static potential, we are able to find analytical approximations

for the electrostatic free energy density and osmotic pressure

for the in vivo cornea. The electrostatic free energy functional

is convex in the volume dilation and provides the key vari-

ational ingredient for the finite-element formulation. The

approach can be easily implemented within a standard

finite-element framework using only the displacement field.



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Turning to collagen-swelling interaction, the influence of

collagen architecture on tissue swelling is readily demon-

strated in the ex vivo cornea. When a sample of human

corneal stroma is immersed in deionized water, the tissue

undergoes extreme swelling to many times its original

thickness by absorbing water from the surrounding bath.

However, the observed swelling is far from uniform, reflect-

ing the interaction of local swelling with the collagen

network. The swelling is observed to occur primarily in the

thickness direction and is most pronounced in the posterior

region of the stroma [26–29]. An analogous situation also

occurs in the in vivo swollen cornea [30]. This observed rigid-

ity of the anterior cornea with respect to swelling under

extreme hydration states has been ascribed to the specific

architecture of the collagen fibres [29,31].

The stroma consists of 200–500 superposed sheet-like

lamellae. Meek et al. [32] quantified the orientation of lamellae

when viewed in the plane perpendicular to the optical axis

using X-ray diffraction. That study, and numerous others

since, described superior–inferior and nasal–temporal prefer-

ential collagen orientations in the central cornea and a

circumferential orientation near the limbus. These data have

been employed directly or indirectly in a number of conti-

nuum mechanics-based models to characterize the elastic

anisotropy induced by this ‘in-plane’ description of collagen

architecture, e.g. [3–6]. Abahussin et al. [33] have also per-

formed X-ray diffraction studies on isolated third-thickness

central stromal samples and showed that the percentage of

lamellae exhibiting preferential orientation is greatest in the

posterior stroma and reduces towards the anterior.

To investigate the inclination of lamellae as seen in

corneal cross-sections, Morishige et al. [34] used second

harmonic-generated (SHG) image stacks to reveal lamellae

exiting Bowman’s layer at steep angles (average 238 maxi-

mum 438) and extending up to 120 mm into the stromal

depth. It has long been thought that lamella inclination and

insertion into Bowman’s layer provides mechanical stabiliz-

ation of the refractive surface [26–29]. High-resolution

macroscopic imaging, which combines many SHG image

stacks to produce entire limbus-to-limbus stromal three-

dimensional cross-sections, has allowed some lamellae to be

traced over millimetres, even with branching and led to the

discovery of ‘bow string’ fibres that enter and exit Bowman’s

layer [35,36]. Recently, Winkler et al. [37] processed SHG

images to quantify inclination angle distributions in the

anterior stroma and found that lamellae inclination angles

satisfy a Gaussian distribution with inclination angles being

maximum at the anterior stroma and reducing linearly

towards the posterior.

In this work, we have synthesized X-ray scattering inten-

sity data and SHG image processing to describe the spatially

varying, three-dimensional orientation of lamellae at every

point in the human stroma, including through the depth.

The raw data are still incomplete in some regions and

extrapolation has been necessary, as described in §5. Never-

theless, the most significant features of collagen architecture

are now most probably accounted for, allowing a comprehen-

sive description of the elastic anisotropy of the stromal

collagen network and regional collagen-swelling interaction.

The proposed model is implemented in a finite-element

framework and employed to predict free and confined swel-

ling of stroma in an ionic bath. For the in vivo cornea, the

model is used to simulate corneal hydration changes which

occur as a result of increasing IOP, such as occurs in glau-

coma. Corneal swelling in Fuchs’ corneal dystrophy is

employed as an illustration of how the model can be adapted

to describe certain pathological conditions. Whenever the

cornea is altered by the surgical removal of tissue, the

tissue fluid pressure will change and a swelling response

can be anticipated. We analyse the biomechanical response of

the in vivo cornea to a typical LASIK surgery for myopia

and predict, for the first time, how the fluid pressure and

tissue swelling will respond. As a validation of the model,

the predicted biomechanical response of the cornea (based

on elastic deformation and swelling) is compared to extensive

clinical data.

2. Electrolyte gel in thermodynamic equilibrium2.1. Electrostatic free energyConsider a binary electrolyte gel, consisting of a mixture of

solid, fluid and ionic phases, immersed in an ionic bath at

constant electrostatic potential w0, hydrostatic pressure p0,

and ionic concentration C0þ ¼ C0� ¼ C0: The stroma can be

modelled as such a binary electrolyte of Naþ and Cl2. This

is justified by the fact that the molar concentration of these

ions accounts for about 85% of the total molarity of the aqu-

eous humour [18,19], which acts as an ionic bath. Noting that

all ionic species in the stroma have the same valence value of

21, we set the ionic concentration of Naþ and Cl2 in the aqu-

eous humour to C0þ ¼ C0� ¼ 150 mM and thereby account

for the total molarity of all solutes in the aqueous humour,

which is approximately 300 mM [18,19,23].

The reference state for the electrostatic free energy is taken

to be the normo-hydrated cornea (i.e. the physiological state)

which can be routinely measured. A unit cell reference con-

figuration is denoted V0, with volume V0, and with a fixed

charge concentration distribution modelled by the function

Cf0ðXÞ with X [ V 0: The motion maps V0 into the current

configuration V with deformation gradient F, considered

uniform over the unit cell by virtue of the small unit cell

size. The volume of V is V ¼ JV0 with J ¼ det F: The fixed

charge concentration distribution in the current configura-

tion is modelled by the function Cf(x, J ) with x [ V: The

fixed charge concentration functions Cf0(X) and Cf(x, J ) are

developed in §3.1.

The reference configuration V0 will generally not be a

stress-free configuration for the solid (collagen) phase1 and

it is necessary to introduce a stress-free reference configur-

ation for the solid phase, denoted S [3]. The deformation

gradient F0 from S to the current configuration V may be

multiplicatively decomposed such that F0 ¼ FF0, where F0

is the deformation gradient from S to V0. Algorithms for

finding S and F0 have been described, for example, by

Pinsky et al. [3] and Gee et al. [38].

The Gibbs free energy density, measured per unit refer-

ence volume, of the electrolyte gel in V may be additively

decomposed [24] into four components

Gðw, C, C0Þ ¼WelasticðC0Þ þWelectrolyteðw, CÞþWbathðCÞ þ Vexternal, ð2:1Þ

where w is the electrostatic potential and C ¼ FTF and

C0 ¼ F0TF0 are right Cauchy–Green deformation tensors.

The elastic strain energy density Welastic of the solid phase



12:20150241

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is associated with fibril stretching and matrix shearing and

must be based on the three-dimensional collagen organiz-

ation within the stroma. A detailed description of this term

is deferred to §5.

The mean-field approximation for the Helmholtz free

energy density Welectrolyte of a binary electrolyte in V is

given by [25]

Welectrolyteðw, CÞ ¼ 1

V0

ðV

"zfFCfðx, JÞwðxÞ

� 2RTC0

cosh

FRT

wðxÞ� �

� 1

!� ew

2jrwðxÞj2

#dV,

ð2:2Þ

where Cf(x, J ) is the fixed charge concentration in V and

measured in moles per litre, zf ¼ 21 is the fixed charge

valence value, and R, T, F and ew are the gas constant, temp-

erature, Faraday constant and dielectric permittivity of the

electrolyte solvent, respectively. The first term in the inte-

grand measures the free energy of the fixed charge. The

second term includes the excess mean concentration of

the mobile ions at any point in the electrolyte compared to

the bath and, after multiplication by RT, may be interpre-

ted as the osmotic work of introducing excess ions into

the neighbourhood of the fixed charges. The last term

ew=2jrwj2 ¼ E �D=2, where E ¼ rw is the electric field and

D ¼ ewE is the electric displacement, measures the dielectric

free energy. Observe that Welectrolyte is defined to be the

average electrostatic free energy density over the unit cell.

The average free energy density of the bath is

WbathðCÞ ¼ �1

V0

ðV(J)

V0

p0dV ¼ � p0ðJ � 1Þ: ð2:3Þ

Finally, Vexternal is the potential of the external loads, includ-

ing body forces and boundary tractions. The list of terms

appearing in the free energy expression (2.1) is not exhaus-

tive; for example, the free energies of mixing and

disassociation [39,40] can also be considered. However,

osmotic energy is dominant in highly hydrated gels [2,39]

and additional terms are neglected in the current theory.

2.2. Thermodynamic equilibriumThe condition dwWelectrolyte ¼ 0 corresponds to electrostatic

equilibrium and the Euler–Lagrange equation is the Poisson–

Boltzmann equation for the electrostatic potential on V,

�r2w ¼ Few

�Cf þX

i¼þ,�ziCi

!, ð2:4Þ

with ionic concentrations satisfying the Boltzmann distribution

Ci ¼ C0 exp �ziF

RT(w� w0)

� �, i ¼ þ,� : ð2:5Þ

The condition for mechanical equilibrium is found by set-

ting dC0Welastic þ dCWelectrolyte þ dCWbath ¼ 0: The variation of

the elastic free energy Welastic is taken with respect to C0 to

incorporate the solid phase prestress, as discussed above.

Noting that Welectrolyte and Wbath depend on C only through

J, it follows that:

@Welastic

@C0:dC0 þ

@Welectrolyte

@Jþ @Wbath

@J

� �@J@C

:dC ¼ 0: ð2:6Þ

The Euler–Lagrange equation of this variational form on V is

found to be

r � ðs� ðPos þ p0Þ1Þ ¼ 0, ð2:7Þ

where the effective Cauchy stress s is given in the standard

manner by

s ¼ f0� 2@Welastic

@C0

� �, ð2:8Þ

in which f0�ð�Þ ¼ ðJ0Þ�1F0ð�ÞF0T is the second-order push-

forward operator from S to V, with J0 ¼ det F0: The unit

cell osmotic pressure Pos in V is given by

Pos ¼ �@Welectrolyte

@J: ð2:9Þ

It is also seen that,

p0 ¼ �@Wbath

@J, ð2:10Þ

so that the electrolyte fluid pressure p is given by

p ¼ �@Welectrolyte

@J� @Wbath

@J¼ Pos þ p0: ð2:11Þ

2.3. Recovery of Donnan osmotic pressureTo illustrate the application of functional (2.2), consider the

special case in which the reference fixed charge density Cf0

is uniform over the unit cell. The fixed charge density in

the deformed configuration is then simply

CfðJÞ ¼ J�1Cf0: ð2:12Þ

For this problem, the (Donnan) electrostatic potential,

denoted ~w, is uniform over the unit cell and be may be

solved from (2.4) as

~w ¼ w0 þRTF

sinh�1 zfCf

2C0

� �: ð2:13Þ

Observing that jrewj ¼ 0 and introducing (2.13) into (2.2) and

with use of (2.12), the Helmholtz electrostatic free energy

density reduces to

Welectrolyte(J) ¼ Cf0RT sinh�1 Cf0

2 JC0

� �

� 2JRTC0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCf0

2 JC0

� �2

þ 1

s� 1

0@ 1A: ð2:14Þ

Employing the above equation in (2.9) yields

Pos ¼ 2RTC0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiCf

2C0

� �2

þ 1

s� 1

0@ 1A, ð2:15Þ

which recovers the classical Donnan osmotic pressure [41,42].


Table 1. Parameters of the electrolyte and collagen organization models.

parameter value

fcol, volume fraction of the collagen fibrils (%) 24.9a

fker, volume fraction of the keratocytes (%) 11.2a

fcoat, volume fraction of the PG-dense

coating (%)

13.3a

Cavg, average charge density of the corneal

stroma (mM)

38.6b

l, the charge partition parameter 0.65c

QþQ2, dimensionless parameter by the ionic

pumping effect

0.965d, 1.0e

m, matrix shear modulus (kPa) 5.0f

a, collagen fibril stiffness parameter (MPa) 0.03f

b, collagen fibril stiffness parameter 75.0f

p0, the hydrostatic pressure in bath solution

(mmHg)

0e, 15.0g

C0, bath concentration (mM) 150, 0.00015h

T, temperature (K) 298aValues derived from [8,23].bCalibrated value by Cheng & Pinsky [23].cValue estimated by Cheng & Pinsky [23].dValues calibrated for in vivo corneas (see §4).eValue for ex vivo corneas.fValues estimated by Petsche [44].gValues for in vivo corneas.hRepresentative of the ionic concentrations for aqueous humour anddeionized water.


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3. Application to the human cornea3.1. Fixed charge concentrationThe predominant corneal GAGs are keratan sulfate, dermatan

sulfate and chondroitin sulfate [9,10]. A detailed FACE2 [43]

analysis reveals that only a portion of the available GAG

monosaccharides is sulfated in the human cornea. It has

been estimated [23] that the average ionization fraction

(i.e. ratio of the charge per disaccharide unit to the num-

ber of groups that are ionizable) over all corneal GAGs is

favg ¼ 72.6%. If the unit cell within the normo-hydrated

stroma contains Ng GAG chains with average contour

length Lc and if b is the length of a monosaccharide unit,

the total GAG fixed charge in the unit cell is

Q ¼NgLce favg

b, ð3:1Þ

where e is the unit charge. The average fixed charge concen-

tration, measured in mM units, is then

Cavg ¼Q

FV0¼

ngLce favg

Fb, ð3:2Þ

where ng ¼ Ng=V0 is the average GAG number density and Fis the Faraday constant. Cavg has been measured indirectly

[1,2], and the value used in this study is taken from [23]

and given in table 1.

X-ray scattering studies under varying tissue hydration

[45] suggest that some portion of stromal PGs form a

charge-rich and water-binding PG-coating surrounding each

collagen fibril. The radius of the coating has been measured

to be rc ¼ 18.25 nm and this radius is insensitive to hydration

over a wide range. The existence of such a surface ultrastruc-

ture on the collagen fibrils is corroborated by image studies

from [46,47]. In addition, a theoretical study on transparency

by Twersky [48] proposed that collagen fibrils must be

centred in a transparent coating and the coated fibrils

occupy approximately 60% of the matrix volume, giving

rc ¼ 21.56 nm. Recent three-dimensional electron microscopy

reconstructions of corneal collagen and GAG chains [9] also

suggests that some GAG chains are in close association

with the fibrils while the remaining GAG chains have

random orientation in the interfibrillar fluid.

Given the regular arrangement of fibrils within a lamella,

the nanoscale unit cell is selected to be of the form shown in

figure 1c. It is a rectangular prism with cross-section as

shown in figure 1c and with axial axis aligned with the

collagen fibrils. Each fibril has a GAG-coating region. As

described below, fixed charge concentrations in the unit cell

will depend on various volume fractions, including the

collagen and coating regions.

Let the unit cell current configuration V be partitioned

into collagen fibril, coating and inter-coating domains

(figure 1c) such that V ¼ Vcoll < Vcoat < Vinter and where

vol(VÞ ¼ V ¼ JV0: We seek expressions for the fixed charge

density over the subregions of V as a function of dilation J.A parameter l is used to partition the total unit cell GAG

fixed charge Q into interfibrillar GAG fixed charge lQ and

coating GAG fixed charge (1 2 l)Q. The value of l ¼ 0.65

has been estimated from swelling pressure measurements

[23]. Note that because all GAG chains bind at one end to a

collagen fibril, the interfibrillar GAG chains must pass

through the GAG-coating region and their charges therefore

combine in that region. The coating and interfibrillar GAGs

are assumed to produce uniform charge concentration

distributions over their respective domains.

In determining the fixed charge concentrations as a func-

tion of macroscopic dilation J, account must be taken of

keratocyte volumes which are assumed to exclude GAG

fixed charge. Keratocytes are morphologically flattened fibro-

blasts that occur between lamellae [11]. They are considerably

larger than the unit cell and their effect is therefore intro-

duced in an average way (figure 1b). Stromal collagen and

keratocyte volume fractions are denoted fcol and fker,

respectively. Assuming that keratocyte cells dilate with the

tissue dilation [49] and assuming collagen fibrils are non-

swelling [1,50,51], the charge concentrations in the current

unit cell configuration V are then given by Ccoat and Cinter

in the coating and inter-coating regions, respectively, as

Ccoatðx, JÞ ¼ l

Jð1� fkerÞ � fcol

þ 1� l

fcoat

� �Cavg,

x [ Vcoat

ð3:3Þ

and

Cinterðx, JÞ ¼ l


� �Cavg, x [ Vinter, ð3:4Þ

where fcoat is the volume fraction of the coating region. Note

that the coating volume is independent of dilation J as

required by the experimental observations of Fratzl &



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Daxer [45]. In (3.3) and (3.4), Cavg ¼ Q/FV0 is the average

charge concentration in the reference configuration V0 given

by (3.2). It is easily verified that (3.3) and (3.4) conserve

unit cell total fixed charge under any volume dilation J.Volume fraction values used in this study are provided in

table 1.

Finally, assuming that keratocytes have the same mass

density as water, the unit cell volume dilation J can be related

to the stromal hydration Hw, defined as water weight per unit

dry (collagen) weight, as follows:

HwðJÞ ¼ðJ � frfcolÞrw

frfcolrcol

, ð3:5Þ

where rw ¼ 1 g cm23 and rcol ¼ 1.36 g cm23 are the mass

density of the water and dry collagen fibrils [51], respectively,

and fr is the relative volume fraction of collagen molecules

within the fibrils (which excludes the intrafibrillar bound

water) [1,50]. The value of fr is estimated to be 0.75 such

that the stromal hydration Hw is 3.2 at the normal condition

J ¼ 1 [1]. Therefore, equation (3.5) is rewritten as

HwðJÞ ¼ 3:94 J � 0:74: ð3:6Þ

The linearity is consistent with measurements [52,53].

It should also be noted that the estimated value fr ¼ 0.75 is

in very close agreement with the value 0.77 reported by

Goodfellow et al. [50].

3.2. Active endothelial ion transportThe corneal endothelium is a 5 mm thin cellular monolayer

located on the posterior surface of the cornea. It is permeable

to water, metabolic species, including glucose and lactate ion,

and other salt ions [11]. The endothelium regulates stromal

hydration by providing active ion transport across the endo-

thelium, from the stroma to the aqueous humour [1,12]. To

allow sufficient generality, we model steady and independent

anion and cation active transport and measure their effects

(singularly or combined) on osmotic pressure by deriving

a modified Boltzmann ion distribution for the stroma and

the corresponding electrostatic free energy density for the

in vivo cornea.

The endothelial layer is here modelled as an ideal mem-

brane in which the action of the ion pumps is represented

by independent active steady anion flux Ja2 and cation flux

Jaþ, transporting ions out of the stroma and into the aqueous

humour [21]. Because the endothelium is thin compared with

the stroma, we will assume transport across the endothelium

is one-dimensional. A coordinate z is introduced with z ¼ z0

at the endothelium–aqueous interface and z ¼ z* at the

endothelium–stroma interface. At equilibrium, the net ionic

fluxes J2 and Jþ, which result from both active and passive

transport, must vanish so that

Ji ¼ �Lidmi

dz� Jai ¼ 0, i ¼ þ,� , ð3:7Þ

where Li is the membrane permeability of ionic species i and

Jai . 0 is the steady active ionic flux across the endothelium,

from the stroma and into the aqueous humour. The ionic elec-

trochemical potential mi is given by

mi ¼ mrefi þ

RTMi

ln Ci þ ziFwMi

, i ¼ þ,� , ð3:8Þ

where Mi(i ¼ þ,2) are the cation and anion atomic weights,

respectively, mrefi are the ionic chemical potentials at the

reference state. Using (3.8) in (3.7) gives

1

Ci

dCi

dzþ zi

FRT

dw

dzþ 1

RTMiJai

Li¼ 0, i ¼ þ,� , ð3:9Þ

and adding both equations in (3.9) results in

1

Cþ

dCþdzþ 1

C�

dC�dz¼ � 1

RTMþJaþ

LþþM�Ja�

L�

� �: ð3:10Þ

Integrating (3.10) across the endothelium leads to

lnC�þC0þ ln

C��C0¼ � h

RTMþJaþ

LþþM�Ja�

L�

� �, ð3:11Þ

where h ¼ z� � z0 is the thickness of the endothelial layer, C�iare the ionic concentrations at z ¼ z� and C0 is the bath

(i.e. aqueous humour) ionic concentration. Observe that

the right-hand side of (3.11) is dimensionless as required.

Defining,

Qi ¼ exp � hRT

MiJai

Li

� �, i ¼ þ,� , ð3:12Þ

it follows that 0 � Qi � 1 and from (3.11),

C�þC�� ¼ QþQ�C20: ð3:13Þ

Finally, integrating (3.9) across the endothelium, and taking

the electrostatic potential in the aqueous w0 ¼ 0, gives

C�i ¼ QiC0 exp �ziFw�

RT

� �, ð3:14Þ

where w� is the electrostatic potential at z ¼ z*.

At thermodynamic equilibrium, the gradients of the ion

electrochemical potentials vanish and mi ¼ m�i , where mi is

the electrochemical potential at any point in the stroma, and

assuming that ionic concentrations are continuous across the

endothelial–stroma interface, this condition implies that

Ci ¼ C�i exp �ziF

RTðw� w�Þ

� �: ð3:15Þ

Introducing (3.14) into the above equation allows the stromal

ionic concentrations Ci to be expressed as

Ci ¼ QiC0 exp �ziFwRT

� �, ð3:16Þ

which generalizes the Boltzmann distribution (2.5). From this

result, it may be seen that at any point in the stroma, the ionic

concentrations Cþ and C2 must satisfy the condition

CþC� ¼ QþQ�C20: ð3:17Þ

3.3. Modified electrostatic free energyBased on (3.16), a modified Poisson–Boltzmann equation for

the electrostatic potential in the stroma is introduced as

�r2w ¼ Few

�Cf þX

i¼þ,�ziQiC0 exp �zi

FwRT

� � !, ð3:18Þ


stroma

epithelium

endothelium

IOP = 15 mmHg

K

(b)

(a)

(c)

Figure 1. (a) The cornea is a fibre-reinforced electrolyte gel that resists the IOP. The collagen lamellae mostly follow the corneal curvature except in the anteriorregion where the lamellae are interweaving and inclined relative to the corneal surface, and are seen to insert into Bowman’s layer. (b) An illustration of theorganization of the corneal stroma showing several lamellae and a keratocyte cell. Collagen fibrils within lamellae are found in parallel arrays following thefibril direction; keratocyte cells are found interspersed between adjacent lamellae. (c) Cross-section of an idealized unit cell representing the hexagonal collagenfibril lattice. The coating region around the central cylinder (representing the collagen fibril) and the irregular lines in between are illustrative of the coating GAGsand interfibrillar GAGs, respectively.


12:20150241

7


and the corresponding modified electrostatic free energy

density is

Wstroma ¼1

V0

ðV

"�FCfw� RTC0

Xi¼þ,�

Qi exp �ziFwRT

� �� 2

!

� ew

2jrwj2

#dV:

ð3:19Þ

It may be verified that this functional is stationary at the

solution of the modified Poisson–Boltzmann equation

(3.18), i.e. dwWstroma ¼ 0: It may also be observed that if the

active fluxes Ja2 and Jaþ increase, the osmotic free energy,

measured by the second term in the integrand, reduces.

3.4. Analytical approximation for osmotic pressureRecognizing that the fixed charge density Cf(x, J ) is piecewise

constant over the unit cell, and noting that the unit cell is large

compared to the Debye length [23], we employ a piecewise

constant Donnan potential ~w as an approximation for the

exact electrostatic potential w which is required to satisfy the

Poisson–Boltzmann equation (3.18). By assumption r~w ¼ 0,

and equation (3.18) may be used to solve for the Donnan poten-

tial. Recalling that the fixed charge density Cfðx,JÞ ¼ CcoatðJÞfor x [ Vcoat and Cfðx,JÞ ¼ CinterðJÞ for x [ Vinter, the

Donnan potentials over Vcoat and Vinter are given by

~wrðJÞ ¼RTF

ln �~CrðJÞQ�þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~C

2

r ðJÞQ2�þQþ

Q�

s0@ 1A,

r ¼ coat, inter,

ð3:20Þ

where the non-dimensional fixed charge concentrations ~Ccoat

and ~Cinter are defined by

~CrðJÞ ¼CrðJÞ2C0

, r ¼ coat, inter, ð3:21Þ

and where the fixed charge densities Ccoat(J ) and Cinter(J ) are

given by (3.3) and (3.4), respectively.

Employing potentials (3.20), the Helmholtz free energy

density (3.19) reduces to

WstromaðJÞ ¼X

r¼coat,inter

VrðJÞV0

"�FCrðJÞ~wrðJÞ � RTC0

Xi¼þ,�

Qi exp �ziF~wrðJÞ

RT

� �� 2

!#, ð3:22Þ

where Vcoat¼fcoatV0 and Vinter¼ðJð1� fkerÞ � fcol � fcoatÞV0:

It can be shown that Wstroma is convex in J. The

unit cell osmotic pressure then follows from (2.9) as

Pos ¼ �@Wstroma=@J, giving

PosðJÞ¼2ð1�fkerÞRTC0

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~C

2

interþQþQ�

q�1

� �þ2ð1�fkerÞfcoat

Jð1�fkerÞ�fcol

RTC0~Cinter(gsinh�1 ~Ccoat�gsinh�1 ~Cinter),

ð3:23Þ

in which a modified inverse hyperbolic sine function is

introduced to simplify the expression

gsinh�1ðxÞ ¼ ln xþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 þQþQ�

p� �: ð3:24Þ

The osmotic compressibility, defined as KosðJÞ ¼�@Pos=@J, is the core function in the stroma electrolyte

tensor Cstroma ¼ 4@2Wstroma=@C@C required for the finite-

element tangent operator (see the electronic supplementary


80

60

40

20

–20

–40

–60

–800.94 0.96 0.98 1.00

0

P os (

mm

Hg)

Q+Q–

ex v

ivo

in v

ivo

reduced osmotic pressure

increased ionicpumping

Figure 2. The in vivo osmotic pressure Pos at normal hydration (i.e. J ¼ 1) asa function of the ionic transport parameter QþQ2. The osmotic pressure Pos

reduces monotonically with increasing active ionic flux values (decreasingQþQ2). The calibrated value of QþQ2 is found to be 0.965 for thenormo-hydrated in vivo cornea. (Online version in colour.)


12:20150241

8


material for details) and is given by

KosðJÞ ¼ 2ð1� fkerÞ2


RTC0

~C2

interffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~C

2

inter þQþQ�q

þ 4ð1� fkerÞ2fcoat

½Jð1� fkerÞ � fcol�2RTC0

� (gsinh�1 ~Ccoat �gsinh�1 ~Cinter)~Cinter

þ 2ð1� fkerÞ2fcoat

½Jð1� fkerÞ � fcol�2RTC0

�~C

2

interffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~C

2

coat þQþQ�q �

~C2

interffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi~C

2

inter þQþQ�q

0B@1CA, ð3:25Þ

Equations (3.23)–(3.25) show that the osmotic pressure and

compressibility are dictated by the combined ionic transport

parameters QþQ2. These parameters must be estimated

indirectly since direct measurement is not currently feasible.

This is considered in the following section. In the ex vivocornea, no active transport occurs and Qþ ¼ Q2 ¼ 1. This

condition is employed in §6.1 to model in vitro free swelling

and swelling pressure experiments on isolated stroma

samples in ionic baths to provide evidence for the accuracy

of the proposed model.

4. Model calibration from imbibition pressureThe presented model requires the determination of QþQ2.

From (3.12), it is found that

QþQ� ¼ exp � hRT

MþJaþLþ

þM�Ja�L�

� �� , ð4:1Þ

the above equation implies that any increase in permeability,

described by Li, that is not balanced with a compensatory

increase in ionic pump function, described by Jai, will result

in stromal swelling and eventually clinical oedema [54].

The permeability and pumping rates have been estimated

via transient swelling experiments [55,56], but these are not

relevant for equilibrium conditions. Instead, we shall infer

the value of QþQ2 from measurement of the imbibition

pressure in rabbit.

Imbibition pressure is measured in the living cornea by

inserting a saline-filled cannula into the stroma [57]. At equi-

librium, a stable suction pressure is developed in the cannula

and is referred to as the imbibition pressure Pimb. As the

saline solution is in contact with the stroma in front of the

cannula tip, Donnan equilibrium is reestablished locally

[21] and equilibrium of water requires

mtipw ¼ mcan

w , ð4:2Þ

where mtipw ¼ mref

w þ ðp� RTP

iCiÞ=rw and mcanw ¼ mref

w þðPimb � 2RTC0Þ=rw are the water electrochemical potentials

of the stroma in front of the tip and of the saline solution

in the cannula, respectively. The stromal fluid pressure is pand Pimb is the hydrostatic (imbibition) pressure measured

in the cannula. Setting the local osmotic pressure difference

P ¼ RT(Cþ þ C� � 2C0), equation (4.2) implies

Pimb ¼ p�P: ð4:3Þ

As noted by Bryant & McDonnell [21], Q also corresponds to

the ex vivo osmotic pressure because the canula and ionic bath

have the same ionic concentration C0. In this case, equation

(4.3) states that imbibition pressure is the difference between

the stromal fluid pressure and the ex vivo osmotic pressure.

However, direct experimental in vivo measurement of imbibi-

tion pressure in rabbit [57] indicated that the following

relationship holds:

Pimb ¼ IOP� Ps, ð4:4Þ

where is the IOP and is the independently measured swelling

pressure in an ionic bath of concentration . As discussed in §1,

for the stroma it can be safely assumed that swelling and

osmotic pressure coincide and therefore equations (4.3) and

(4.4) allow the conclusion that in rabbit the stromal fluid

pressure is equal to the IOP, i.e. p ¼ IOP at normal hydration.

Furthermore, the measured Pimb was found to have insignif-

icant variation throughout the cornea [57]. It is therefore

reasonable to conclude that the fluid pressure is p ¼ IOP

everywhere in the normal in vivo rabbit cornea. Lacking

any alternative, we shall take this condition to also hold in

the human cornea.3

Recalling that equilibrium fluid pressure in the cornea is

given by (2.11), and setting p0 ¼ IOP, it follows that:

p ¼ IOPþ Pos: ð4:5Þ

However, by the above argument based on imbibition

pressure measurements, we must set p ¼ IOP. In this case,

equation (4.5) implies that in the normal in vivo cornea,

Pos � 0:

The condition Pos ¼ 0 is used to estimate QþQ2 as

follows. Figure 2 depicts the variation of Pos with QþQ2

at normal hydration, determined from (3.23). A nearly

linear relationship between Pos and QþQ2 is found. The

curve indicates that stromal osmotic pressure reduces

monotonically with increasing active ionic flux values. A

calibrated value of QþQ2 ¼ 0.965 is found for the normo-

hydrated in vivo cornea. As will be discussed in §6.2.1, the

positive slope of figure 2 is a necessary requirement for

hydration regulation.


offset D(X)

q q q

p p0

htotaltotal scatter area

FX-rayinterpolated total scatter

huniuniform scatter area

halign

aligned scatter area

2

q

p p02

q

p p02

p p02

p0 p2

rq (X, s, q)X-ray(b)

(a)

Figure 3. (a) The X-ray scattering intensity can be additively decomposed into uniform and aligned parts. (b) The distribution interpolated from surrounding scanpoints is offset based on the depth within the stroma to increase alignment in the posterior.


12:20150241

9


5. Collagen organization and stromal elasticity5.1. Angular averagingIn this section, we review the modelling of the stromal fibres

which contribute to the elastic strain energy density Welastic

appearing in (2.1). It is assumed that Welastic is additively

decomposed into an anisotropic part describing the strain

energy density of the three-dimensional collagen fibre network

and an isotropic part that gives a simple phenomenological

description of the small background shear stiffness of the

extracellular matrix such that

WelasticðC0Þ ¼WfibreðC0Þ þWshearðI01Þ: ð5:1Þ

Recall from §2.1, that C0 ¼ F0TF0 describes the deformation

measured from the stress-free configuration S, and F0 is

the deformation gradient from the stress-free configuration

S to the current configuration V. The invariant I01 ¼ tr C0

is used to describe an isotropic contribution based on the

unimodular right Cauchy–Green deformation tensor

C0 ¼ ðJ0Þ�2=3C0 and J0 ¼ det F0 [58].

The invariant I04 ¼ A � C0A ¼ ðl0AÞ2, which measures the

stretch along a fibre with direction A, is used to describe elas-

tic anisotropy. It is important to note that I04 depends on the

dilation through its dependence on C0, allowing the descrip-

tion of fibre stretching resulting from electrolyte swelling. The

form of WfibreðC0Þ is based on the use of angular integration

[3,59] such that

WfibreðC0Þ ¼ kWstretchðI04ÞlA ð5:2Þ

¼ 1

n

ðv

rðX, AÞWstretchðl0AÞdv, ð5:3Þ

where v is the unit sphere, r(X, A) is the distribution of fibre

directions, n ¼Ðv rðX, AÞdv is the normalization and the

fibre stretching energy Wstretchðl0AÞ is described below.

The function r(X, A) may be based on fibre orientation

information obtained from X-ray diffraction and SHG

imaging as described in [6]. A fibre direction A can be

expressed in spherical coordinates with azimuthal angle

u [ ½0, 2p� and inclination angle f [ ½0, p�: As the anterior

and posterior surfaces have different curvatures, a local coor-

dinate system is introduced that varies pointwise through the

corneal thickness and matches both the anterior and posterior

curvatures at those limits; details can be found in [6]. Every

material point within the cornea has a unique angular distri-

bution with no perfect symmetry over quadrants. We set

rðX, AÞ ¼ rð�X, s, u, fÞ where �X is a point on the cornea

anterior surface, s [ ½0, 1� is the non-dimensional corneal

depth measured from the anterior surface, and assume,

rð�X, s, u, fÞ ¼ rX�rayu ð�X, s, uÞrSHG

f ð�X, s, fÞ, ð5:4Þ

where rX-rayu is based on X-ray diffraction data and rSHG

f is

based on SHG imaging. The current model generalizes

the approach in [6] in two respects: rX-rayu incorporates

depth-dependence of preferred fibre directions, and rSHGf

incorporates direct statistical measurement of inclination

data. These are briefly summarized below.

5.2. Azimuthal distribution rX-rayu

In [3], analytical distribution functions were introduced to

represent data from X-ray diffraction experiments by Meek

et al. [60]. Following [6], we have eliminated the need to

introduce analytical distribution functions and directly

employ the raw X-ray data. These data provide scattering

intensities versus orientation on a discrete grid of points

over the cornea’s anterior surface. At any point in the

cornea, the orientation distribution is obtained by interpolat-

ing the X-ray diffraction measurements taken at the nearest

four grid points using bilinear Lagrange functions. This

procedure ensures that any subtle variations in distribu-

tions found in the X-ray data are faithfully reproduced in

the model. At any scan point �X, the X-ray scattering data

FX�rayðuÞ, which corresponds to a measurement through

the entire thickness, can be analysed in the following way.


70

60

50

40

30

20

10

0

70

60

50

40

30

20

10

010 anterior central posterior

(0.3s + 0.16) × 100% (0.3s + 0.35) × 100%

per

cent

alig

ned

depth depth

13

23

10 anterior central posterior13

23

(b)(a)

Figure 4. Data from [33] of alignment percentage for each third in the central cornea is fit with a linear curve (a). The depth-dependence of alignment is approxi-mated from a full-thickness X-ray scan near the limbus by fitting with a line of the same slope (b). (Online version in colour.)


12:20150241

10


The total scatter htotal is the area under the curve,

htotal ¼ðp

0

FX-rayðuÞdu: ð5:5Þ

This can be additively decomposed into aligned and

uniform parts htotal ¼ halign þ huni where huni ¼ mp and

m ¼ min½FX-rayðuÞ� (figure 3a).

The percentage of all aligned fibres �nalign is then

�nalign ¼htotal �mp

htotal

: ð5:6Þ

The depth-dependence of aligned fibres may be intro-

duced based on the work of Abahussin et al. [33], who

performed X-ray diffraction studies on isolated third-

thickness stromal samples and characterized the variation

of scattering intensity with depth at the corneal centre.

They found the average percentage of total fibres exhibiting

preferred azimuthal directions valign was 22%, 31% and 42%

in the anterior, central and posterior thirds, respectively.

However, depth-dependent alignment was not measured at

the cornea’s periphery where there is significantly more

alignment. A linear best-fit of alignment as a function of

depth for the data is 0.3sþ1/6 with R2 ¼ 0.997. Based on

the slope of this function, a first approximation for the

depth-dependence of aligned fibres is nalignðsÞ ¼ 0:3sþ d:

For the central full-thickness X-ray scans in the current data-

base of implemented X-ray diffraction data (four donor

corneas, including a right and left cornea from the

same donor), we find �nalign � 31% for all subjects,

which corresponds well to the mid-thickness value measu-

red by Abahussin et al. [33]. Thus, to evaluate d we set

nalignð0:5Þ ¼ �nalign which gives d ¼ �nalign � 0:15: Then,

nalignð�X, sÞ ¼ �nalignð�XÞ þ 0:3s� 0:15 ð5:7Þ

defines the fraction of aligned fibres based on in-plane

position �X and depth s (figure 4).

In order to generate a depth-dependent distribution,

the full thickness distribution htotal is augmented by D(s)

such that nalignðsÞ ¼ halign=ðhtotal þ pDðsÞÞ (figure 3b). Solving

for D(s) gives

DðsÞ ¼ 1

p

halign

nalignðsÞ� htotal

!: ð5:8Þ

The final distribution will then be,

rX-rayu ð�X, s, uÞ ¼ FX-rayð�X, uÞ þ Dð�X, sÞ:

Incorporating the X-ray diffraction data in this manner

describes the depth-dependent anisotropy resulting from

the well-documented S–I and N–T preferred directions of

lamellae in the vicinity of the corneal vertex as well as the

circumferentially preferred orientations at the limbus [44].

5.3. Inclination distribution rSHGf

The presence of inclined fibres has been well-documented

with macroscale SHG imaging [34–36]. However, because

the angular distribution of inclined fibres was missing, an

approximation was proposed by Petsche & Pinsky [6] based

on inspection of the macroscale images and calibration

based on the depth-dependence of shear properties. Recently,

Winkler et al. [37] have processed SHG images to detect

and quantify the spatial distribution of inclined lamellae

and their distribution is employed in this study. Winkler

et al. [37] analysed half cross-sectional image stacks to

measure the inclination angle of every lamella in the anterior

half of the stroma. Lamellae were binned based on depth,

corneal quadrant and radial position. For each bin, a histo-

gram of fibre inclination was fit to a Gaussian distribution

for the full-width at half-maximum (FWHM), with the

mean fixed at 08 from the plane tangent to the anterior

surface (figure 5).

They found no significant differences between quadrants

and radial position but measured a significant variation with

depth. On the anterior half of the cornea, where analysis was

performed, a linear decrease of FWHM was found. In order

to complete the data through full thickness, it is assumed

that FWHM extrapolates linearly to zero at the posterior sur-

face. A best-fit line that forces no inclined lamellae at s ¼ 1

(R2 ¼ 0.87) was used to find FWHM as a function of depth

(figure 6).


150

100

50

0

s =

12

345678

910111213141516

(s, f

)SH

Gr f

f

0 p

112

inclination (°)

coun

ts

–20 –10 0 10 20

3s2p –

2p 3s

2p +

(b)

(a)

(c)

Figure 5. SHG images were processed by Winkler et al. [37] to obtain quan-titative depth-dependent inclination distributions. Images were scanned forlamella direction by first isolating a narrow section of the image and rotatingit so the anterior surface was horizontal. Then the image processing algor-ithm searched for fibres by scanning through the depth. Data wereprocessed to discard artefacts and then plotted in a histogram for eachdepth. A Gaussian distribution was fit to the data to define the FWHMvalue for each depth and radial position. The FWHM value was then usedto define the inclination distribution rSHG

f : (Online version in colour.)

16

14

12

10

8

6

4

2

00 0.25 0.50 0.75 1.00

14.677

full width half maximum

(1 – s)

depth

180p

Figure 6. FWHM values were estimated in the posterior half by fitting a lineartrend line that intersected 08 at the posterior surface. (Online version in colour.)


12:20150241

11


Using the one-to-one relationship between FWHM and

the standard deviation s this gives

sðsÞ ¼ 14:677

2ffiffiffiffiffiffiffiffiffiffiffi2 ln 2p p

180ð1� sÞ: ð5:9Þ

The inclination distribution therefore takes the form of

a Gaussian

rSHGf ðs, fÞ ¼ 1

sðsÞffiffiffiffiffiffi2pp exp �ðf� p=2Þ2

2s2ðsÞ

!, ð5:10Þ

where the mean is p/2 because f is the zenith angle [44].

5.4. Fibril and shear elasticityThe term Wstretchðl0AÞ in (5.3) represents the strain energy

density resulting from fibril stretching. In tension, collage-

nous tissues are characterized by a J-shaped stress–strain

curve resulting from unkinking of collagen molecules at

low strains followed by the stretching of stiffer covalent

bonds at high strains. However, they exhibit negligible com-

pressive stiffness due to buckling effects. To capture this

behaviour, Markert et al. [61] introduced a strain energy

function of the form

Wstretchðl0AÞ ¼a

bððl0AÞ

b � 1Þ � a ln l0A: ð5:11Þ

This model has been applied to the cornea by Studer et al. [5].

The model parameters a and b have been identified using

corneal inflation experimental data and optimal values

found to be a ¼ 30 kPa and b ¼ 75 [44]. These values satisfy

the constraints on a and b given by Markert et al. [61] and

render Wstretch convex in l0A:

The matrix shear term is modelled as simply as possible

and is assumed to be

WshearðI01Þ ¼m

2ðI01 � 3Þ, ð5:12Þ

where m ¼ 5 kPa is the matrix shear modulus which has

been measured experimentally by Petsche et al. [62] and

Sondergaard et al. [63].

Finally, the elasticity tensor Celastic ¼ 4@2WelasticðC0Þ=@C0@C0 and electrolyte tensor Celectrolyte ¼ 4@2ðWelectrolyteþWbathÞ=@C@C, needed for the finite-element tangent oper-

ator, are described in the electronic supplementary material.

5.5. Boundary conditionsThe in vivo cornea is loaded by the IOP at the endothelial–

aqueous interface (figure 1a). The anterior stroma is sealed

by the epithelium, which is a stratified squamous cellular

membrane that is composed of four to six cellular layers

and approximately 50 mm in thickness. In contrast to the

endothelium, epithelial cells exhibit tight junctions and the

epithelium is essentially impermeable to water and ions

under normal conditions [11]. As the mechanical stiffness of

both epithelium and endothelium is insignificant compared

to the stroma [11,64], the mechanical roles of these bounding

layers are reasonably ignored. Therefore, for modelling pur-

poses, the IOP is assumed to act directly on the stroma and

the posterior boundary condition is written as

ðs� p1Þ � n ¼�p0n, ð5:13Þ

where s is the effective Cauchy stress, p is the stromal fluid

pressure, p0 ¼ IOP is the hydrostatic pressure in the aqueous

humour and n is the normal unit vector. At the epithelium,

no external pressure is applied and then

ðs� p1Þ � n ¼ 0: ð5:14Þ

A full interpretation of the above boundary conditions is extre-

mely important since it provides insight into specific corneal

structural function, and in particular, the implied necessity of


porous piston

porous piston

2500

2000

1500

1000

500

0

confined swellingunconfined swellingOlsen & Sperling [52]Hedbys & Dohlman [20]

thickness, t (mm)

P s (

mm

Hg)

0.2 0.3 0.4 0.5 0.6 0.7

confined

unconfined

corneal button

corneal button

bath solution

bath solution

(b)(a) (i)

(ii)

Figure 7. (a) Illustrations of confined (i) and unconfined (ii) swelling pressure experiments for ex vivo cornea, respectively [20,52]. In both experiments, a mech-anical pressure needs to be applied in the transverse direction to maintain a specified thickness. (b) The predicted confined and unconfined swelling pressure Ps

versus corneal thickness t and their comparison with experimental measurements [20,52] over thickness range 0:2 mm � t � 0:75 mm: The swelling pressuredata have been extracted from the original papers by free graphics software PLOT DIGITIZER, and the hydration data H from [20] have been transformed to thickness tby the observed linear relation H ¼ 7.09t 2 0.44 [52,53]. (Online version in colour.)

10 10

9

8

7

6

5

4

3

2

1

8

6

4

2

0

t m/t 0

(m

m)

J

depthC0 (mM)0 0.2 0.4 0.6 0.8 1.010–3 10–1 10 103 105

(b)(a)with lamella inclinationwithout lamella inclination

C0 = 150 mM

C0 = 0.00015 mM

Figure 8. (a) The effect of lamella inclination on stromal swelling ratio during free swelling of the sample in a bath ionic concentration range of 1023 – 105 mM. At eachvalue of C0, the swelling ratio is calculated as the ratio of the swollen thickness tm to the original thickness t0 at which the swelling pressure is zero. (b) Predicteddepth-dependent swelling at physiological bath concentration (C0 ¼ 150 mM) and deionized water (C0 ¼ 150 � 1026 mM). (Online version in colour.)


12:20150241

12


fibre inclination in the anterior (such that n � sn = 0Þ needed to

balance the stromal fluid pressure at the anterior corneal

surface. A detailed discussion is presented in §7.

6. Results6.1. Swelling in the ex vivo stroma6.1.1. Confined and unconfined swelling pressureFinite-element predictions for ex vivo swelling pressure were

compared to experimental measurements on a sample of iso-

lated corneal stroma immersed in a bath solution (figure 7a).

Stromal swelling pressure is the mechanical pressure required

to maintain a specified thickness of the tissue at a specified

bath ionic concentration. Ex vivo conditions (no active ion

transport) are modelled by setting the ionic transport func-

tion QþQ2 ¼ 1 in equation (3.23). The stroma sample

was modelled as a 7 mm diameter cylinder with an initial

thickness of 0.5 mm and immersed in an ionic bath of con-

centration C0 ¼ 150 mM [20,52]. The finite-element mesh

employed 2500, 27-node (tri-quadratic) hexahedral elements.

Fixed displacement boundary conditions were applied on the

bottom surface and prescribed transverse displacements were

applied to the top surface to model changes in sample

thickness. The equilibrium swelling pressure at any sample

thickness t is computed as the total reaction force on the

top surface divided by the surface area. In order to model

confined [20] and unconfined [52] experimental measure-

ments, edge boundary conditions were taken as zero radial

displacement or free, respectively (figure 7a). Predictions

are compared with experimental measurements [20,52] in

figure 7b. The results show good agreement for the full

range of experimental measurements with t varying from

0.75 mm (swollen) to 0.2 mm (highly compressed), with the

corresponding swelling pressure values varying over two

orders of magnitude. Swelling pressure predictions for the

confined and unconfined cases are nearly identical; this is

expected because lateral expansion of the sample is restricted

by the collagen fibrils and, as a consequence, stromal volume

change in these tests is predicted to be associated with

changes in sample thickness [1].

6.1.2. Free swelling of a stromal sampleThe free swelling of the above described cylindrical sample of

ex vivo stroma is modelled to investigate the role of the col-

lagen organization in mediating the swelling response.

Results for edge-confined free swelling of the stroma

sample are shown in figure 8a (solid curve), which depicts


600 1000

800

600

400

200

500

400

300

200

100

–100

0

Ksexp = 39.12J–4.48 kPa

Kos

(kP

a)

P os (

mm

Hg)

dilation J0.5 1.0 1.5

dilation J0.5 1.0 1.5

(b)(a)predictionOlsen & Sperling [52]

Q+Q– = 1.0

Q+Q– = 0.930

Q+Q– = 0.965

Olsen & Sperling [52]Hedbys & Dohlman [20]

Figure 9. (a) The computed osmotic pressure Pos versus volume dilation J for three representative values of QþQ2 and comparison with swelling pressure measure-ments [20,52]. (b) The predicted osmotic compressibility Kos versus volume dilation J and its comparison with experimental measurements [52]. The measuredmodulus is given by K exp ¼ 39:12J�4:48 kPa, which is computed by the power law fit function for swelling pressure Pexp

s ¼ 7:56t�3:48 mmHg from [52].(Online version in colour.)


12:20150241

13


the ratio of swollen thickness to original thickness (swelling

ratio) versus bath ionic concentration C0. For dilute bath sol-

utions, the model predicts that the stroma sample will swell

to approximately four times its original thickness, which

may be compared to experimental measurements in de-

ionized water [28,29] in which human corneas were observed

to swell to approximately three times their original thickness.

For concentrated bath solutions, the abundance of ions results

in ionic shielding of the fixed charges, reduction in osmotic

pressure and minimal swelling. Both limiting states are

captured by the swelling predictions.

It has been observed in free swelling experiments that the

anterior third of the stroma remains virtually unswollen with

most swelling taking place in the deeper stroma [28,29].

We used the model to investigate the depth-dependence

of stromal swelling at two bath ionic concentrations C0 ¼

150 mM and 150 � 1026 mM, corresponding to the physio-

logical state and de-ionized water, respectively. The results are

presented in figure 8b which depicts the profile of local volume

dilation J across the depth of the stroma (J ¼ 1 corresponding

to the normo-hydrated state). At C0 ¼ 150 mM, the local

volume dilation varies from 1.5 in the anterior to 1.8 in the pos-

terior. Massive swelling occurs when C0 ¼ 150 � 1026 mM and

volume dilation varies from 1.8 in the anterior to 10.0 in the pos-

terior. The highly non-uniform swelling across the corneal

thickness agrees qualitatively with experimental observations

[28,29]. Comparison of the two results in figure 8b indicates

that the stromal anterior third thickness is predicted to maintain

its thickness for any bath concentration, indicating the rigidity of

this region with respect to extreme hydration changes.

A theoretical study was also undertaken to confirm

the importance of lamellae inclination with respect to

anterior stromal rigidity. The above study was repeated

while constraining lamellae to have zero inclination. This is

accomplished by replacing the Gaussian inclination distri-

bution given by (5.10) with rSHGf ðs, fÞ ¼ dðf� p=2Þ, where

d is the Dirac delta. In this case, all lamellae throughout the cor-

neal depth will have zero inclination. As shown in figure 8a(dashed curve), for dilute bath solutions the stroma sample

now swells to almost 10 times its original thickness. In fact,

the only mechanical constraint preventing infinite swelling is

the work done by the matrix shear stiffness which bounds

the swelling. As expected, the model still predicts little

swelling when the bath ionic concentration is high.

6.2. Swelling in the in vivo cornea6.2.1. Effect of active endothelial ion transportRecall that the active endothelial ionic transport term

(equation (4.1)) satisfies 0 � QþQ� � 1, with QþQ2 ¼ 1 cor-

responding to no active transport and QþQ2 ¼ 0.965

corresponding to normal active transport in the in vivocornea. Predicted values of osmotic pressure Pos (equation

(3.23)) are plotted in figure 9a for these two values of the

ionic transport function and also for QþQ2 ¼ 0.930 which

corresponds to hyper-active ionic transport. Osmotic pressure

is reduced with reducing values of QþQ2. For QþQ2 ¼ 1, the

osmotic pressure is positive for all dilation J and the tissue

will tend to swell. For the normal in vivo cornea with

QþQ2 ¼ 0.965, it is seen that a reduction in dilation from

the physiological condition (J ¼ 1) will result in positive stro-

mal osmotic pressure which will produce a tendency to swell.

On the other hand, an increase in dilation will result in negative

stromal osmotic pressure and produce a tendency to de-swell.

Consider the curve for the hyper-active QþQ2 ¼ 0.930. In this

case, positive or negative deviations in dilation from J ¼ 1 will

always result in negative osmotic pressure and produce a ten-

dency to de-swell. These conditions illustrate the hydration

regulation mechanism of active endothelial ion transport.

Increasing active transport shifts the osmotic pressure–dilation

curve downwards in figure 9a. Predicted values of osmotic

compressibility Kos (equation (3.25)) versus dilation J are

shown in figure 9b for no active transport. Values closely

match measurements reported in [52].

6.2.2. Swelling of a cornea with Fuchs’ dystrophyFuchs’ dystrophy is usually characterized by morphological

changes in endothelial cells or by an accelerated loss of endo-

thelial cells [11,54]. In this situation, the cornea will swell due

to increasing endothelial permeability, decreasing active ion

flux, or both mechanisms simultaneously. The effect of

these pathological changes on stromal ionic concentrations

is described by the ionic transport function QþQ2 (equation

(4.1)). Because the absolute values of endothelial ionic

permeability Lþ and L2 and active ionic fluxes Jaþ and Ja2

have limited clinical significance, swelling effects due to

relative changes in these parameters are considered. For sim-

plicity and without loss of generality, we consider active

anion transport only and set Qþ ¼ 1. Let L�n and Ja�n be


700

670

640

610

580

550

5201.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 0.25 0.40 0.55 0.70 0.85 1.00

CC

T (

mm)

L–p/L–n Ja–p/Ja–n

(b)(a)

Figure 10. The predicted swollen CCT in Fuchs’ dystrophy when (a) the endothelial ionic permeability is increased up to a factor of 4.5 (1 � L�p=L�n � 4:5) withnormal active ionic flux (Ja�p=Ja�n ¼ 1) and (b) the active ionic flux rate is reduced up to a factor of 0.2 (0:2 � Ja�p=Ja�n � 1) with normal anionicpermeability (L�p=L�n ¼ 1). (Online version in colour.)

53.0

–58.5

–170.0

uz (mm)

Figure 11. Fringe plot of the vertical displacement field uz at Ja�p=Ja�n ¼ 0:22 for a cornea with Fuchs’ dystrophy. The anterior surface of the cornea deformsmuch less than that of the posterior surface. It is noted that the sclera elements are distorted at the limbus. This artefact is due to the abrupt transition fromswelling to non-swelling tissue, and may be removed by defining a transition zone between the cornea and sclera.


12:20150241

14


the anion membrane permeability and active anion flux rate

of the normal cornea, respectively, and consider pathological

increases in ionic permeability such that 1 � L�p=L�n � 1

and reductions active flux such that 0 � Ja�p=Ja�n � 1: For

any value of L�p=L�n and Ja�p=Ja�n, the value of Q�p can

be found through equation (4.1)

ln Q�p ¼Ja�p

Ja�n� L�n

L�p

� �ln Q�n, ð6:1Þ

where Q�n ¼ 0:965 is the calibrated value for normal cornea

as determined in §4.

The cornea was modelled geometrically with a central

thickness of 520 mm and the anterior and posterior surfaces

defined as spherical surfaces with radii of 7.87 and 6.7 mm,

respectively. Part of the sclera was also modelled with

anterior and posterior surfaces defined with radii of 12.00

and 11.38 mm, respectively. Because the sclera swells much

less than the cornea [8], its bulk behaviour was modelled as

a (non-swelling) compressible neo-Hookean material, as

described by Petsche & Pinsky [6], and the limbal corneal

collagen fibre elasticity was extended to the scleral tissue.

At the junction of the cornea and sclera, there is an abrupt

interface of swelling and non-swelling tissue, which should

be replaced by use of a gradual transition (although we

have not done so). Homogeneous Dirichlet boundary con-

ditions are applied to fix the periphery of the scleral

section. The solution proceeds by first solving for the

normal cornea4 with QþQ� ¼ 0:965, and then solving a

sequence of problems in which QþQ� ¼ Q�p is systematically

varied to simulate changes in endothelial permeability and

active anionic flux.

Figure 10a shows the predicted swollen central corneal

thickness (CCT) for endothelial ionic permeability increased

with 1 � L�p=L�n � 4:5 and with normal active ionic flux

Ja�p=Ja�n ¼ 1: Figure 10b shows the predicted swollen CCT

for active ionic flux rate reduced up to a factor of 0.2, so

that 0:2 � Ja�p=Ja�n � 1, with normal anionic permeability

L�p=L�n ¼ 1: In both cases, the cornea swells. The predicted

maximum swollen CCT of approximately 690 mm lies in the

range of clinical observation (500–1000 m) measured for

patients with Fuchs’ dystrophy [30].

The model predicts that at any value of the swollen CCT,

the anterior surface deforms much less than the posterior sur-

face, with swelling concentrated in the posterior region. This

prediction agrees with the clinical observation of Brunette

et al. [30] that the anterior surface of the cornea is nearly

normal among patients with Fuchs’ dystrophy, whereas the

posterior surface shows significant change. Figure 11 pro-

vides a fringe plot of axial (vertical) displacements due to

swelling resulting from a 78% reduction in active ion trans-

port, Ja�p=Ja�n ¼ 0:22: The results again suggest the stability

of the anterior surface with respect to swelling resulting

from the presence of inclined lamellae.

6.2.3. Swelling due to changes in intraocular pressureAn example of corneal oedema with an intact and functional

endothelium occurs in acute glaucoma. In this case, elevated

IOP combined with normal stromal osmotic pressure can

create an increase in corneal thickness [65]. Likewise, corneal

thickening is reported in patients with ocular hypertension

[66,67]. We emphasize that we are considering the equili-

brium solution that is achieved after long-time exposure to


p = IOP

p = IOP sel

sel

Epi.

Endo.

Figure 12. Illustration of the stress state in the normal living cornea. A nearlyuniform fluid pressure exists in the stroma at the magnitude of the IOP. The in-plane lamellae are responsible for resisting such pressure in the lateral direction.In the transverse direction, the applied IOP at the posterior surface balanceswith the fluid pressure. At the anterior surface, the inclined lamellae insertinto Bowman’s layer and act as anchors resisting the stromal fluid pressureapplied to the epithelium. (Online version in colour.)

680

640

600

560

520

IOP (mmHg)

CC

T (

mm)

15 30 45 60 75 90 105 120

with lamella inclinationwithout lamella inclination

Figure 13. The predicted CCT response to increased IOP from 15 to120 mmHg. The absence of inclined lamellae results in a much more swollencornea. (Online version in colour.)


12:20150241

15


increased IOP. It is very likely that the short-time non-

equilibrium response of the cornea is that of elastic thinning.

In fact, if we model an increase in IOP and do not allow the

fluid pressure to equilibrate, the model does predict small

corneal thinning. However, persistent elevated IOP produces

an increased pressure gradient which drives more fluid across

the endothelium, creating oedema of the stroma (and epi-

thelium), as seen in acute angle-closure glaucoma [65,68].

This is undoubtedly a simplified view of a complex problem

but it is supported by predictions from the current model.

There are two competing mechanisms with regard to cor-

neal swelling. The increased (mechanical) IOP loading at the

endothelial–aqueous humour interface acts to thin the

cornea, but an increase in stromal fluid pressure produces a

countering ‘inflation’ effect. At Bowman’s layer, the stromal

fluid pressure must be balanced by forces in the inclined

lamellae, because there is no external pressure loading at

that boundary which can balance the stromal fluid pressure

(figure 12). As shown in figure 13 (solid curve), at

44.5 mmHg, the CCT is predicted to swell by approximately

15 mm, which is in agreement with clinical data [65] in which

the CCT was measured to be 575 and 593 mm for a group of

patients who had normal IOP (average 15.8 mmHg) in one

eye and high IOP (average 44.5 mmHg) in the other eye,

giving a thickness difference of 18 mm. In the simulation of

increasing IOP, the active ionic pumping was maintained at

a normal level throughout. It is important to note that other

measurements, performed on eyes with retinal detachment

or non-human species, or performed within a relatively

short timescale, have concluded an opposite trend [69–71].

To illustrate the importance of lamella inclination in

resisting the IOP, the cornea was reanalysed with lamella

inclination constrained to be zero, as described in §6.1.2. In

this case, the response shown in figure 13 (dashed curve)

includes a significant further increase in swelling.

6.2.4. Biomechanical behaviour after LASIK surgeryLASIK (laser in-situ keratomileusis) is commonly used to cor-

rect refractive errors including myopia and hyperopia. An

anterior circular hinged flap of about 150 mm thickness is

first created using a femtosecond laser. The flap is then

folded back to expose the stromal bed which is reshaped by

excimer laser ablation. The procedure is completed by

returning the flap to conform with the ablated profile.

Although laser ablation creates a new central corneal curvature

with the goal of achieving emmetropia, the biomechanical

response of the residual (significantly thinned) stromal tissue

is an important but challenging factor that should be con-

sidered [72]. In this study, we assume that active

endothelium ionic pumping is functioning normally and use

the proposed model to investigate the biomechanical response

of the cornea, including stromal fluid pressure and local swel-

ling, resulting from a typical and representative myopic

correction.

The geometry shown in figure 14 is for the pre-operative

cornea and defines the shape of the normo-hydrated cornea

and surgical profile while the cornea is supporting an IOP

of 15 mmHg. The ablation depth profile was determined

using theory presented by Munnerlyn [73] for an ablation

radius of curvature of 8.847 mm (corresponding to an intended

correction of approx. 25 dioptres). The CCT is 520 mm, and the

maximum ablation depth of 87 mm combines with the 150 mm

flap to produce a residual central stromal thickness of 283 mm.

With this ablation profile, the residual stromal thickness is

54.4% at the cornea centre and gradually increases to 71.2%

at the periphery of the flap. The flap is returned after

ablation (where negative osmotic pressure holds it in place by

suction) but it is no longer mechanically integrated with the

stroma and can be easily separated at any time after surgery

[72]. LASIK therefore effectively removes that part of the

stroma in which lamella inclination is most pronounced (the

anterior third).

Because the collagen fibril distribution is not symmetric

about any axis, a full 3608 model is required. The eye is sup-

ported in the eye socket by fatty tissue and stabilized by six

muscles responsible for eye movement. However, it is not

necessary to extend the sclera to the posterior pole; numerical

experiments were conducted and it was determined that

extending the scleral tissue for an arc length of 2 mm was suf-

ficient, at which point the tissue can be ‘clamped’. The finite-

element model consists of 7008 27-node hexahedral elements.

The solution was obtained in two phases. In the presurgical

phase, the normal cornea was loaded by an IOP of 15 mmHg

and a global fixed-point algorithm was employed to obtain

the collagen prestress. With this prestress, the normal cornea

is in equilibrium under the IOP and is in the pre-operative geo-

metry of figure 14. In the postsurgical phase, the elements

corresponding to the flap and ablation region are removed


Cposterior Cablation

Canteriord1

d2

F

T

A

O

t

Figure 14. Illustration of the flap geometry and ablation profile in the LASIK surgery designed for an intended correction of approximately 25 dioptres. Theanterior flap diameter F ¼ 9 mm, transition zone diameter T ¼ 6.16 mm, sculpted surface diameter A ¼ 6 mm and optic zone diameter (for least-squares cur-vature extraction) O ¼ 5 mm. The CCT is 520 mm, and the maximum ablation depth of 87 mm combines with the 150 mm flap thickness to produce a residualcentral stromal thickness of d2 ¼ 283 mm.

Table 2. Comparisons of the curvature change coefficient C, postsurgicalcorneal radius Rpost and spherical equivalent SE calculated by FEM andmeasured by Ortiz et al. [74].

C Rpost (mm) SE (1/m)

Ortiz et al.—mech 23.2%+ 1.9% 8.516 23.638

Ortiz et al.—laser 21.6%+ 1.3% 8.685 24.411

FEM 23.12% 8.571 23.892


12:20150241

16


from the mesh while maintaining the IOP and normal ionic

pumping. The model is no longer in equilibrium and the

new deformed state is obtained by Newton’s method.

We first compare the predicted biomechanical response to

clinical measurements. Ortiz et al. [74] made measurements of

changes in corneal anterior curvature and correlated these to

the surgical radius of curvature in the ablation profile. The

case study analysed 85 eyes with myopia or myopic astigma-

tism and subdivided them into two groups: 44 with flaps

created by microkeratome (‘mechanical’) and 41 with

flaps created by femtosecond laser (‘laser’). The biomechani-

cal response to LASIK was evaluated by means of the

coefficient of curvature change defined by

C ¼Rpost � RS

RS� 100,

where Rpost is the postsurgical corneal radius at one month

and RS is the surgical radius imposed on the stromal bed

by ablation. The postsurgical corneal radius was measured

by topography over a central 5 mm diameter central zone.

Likewise, the postsurgical corneal radius of the finite-element

model used least-squares fitting over a 5 mm diameter zone.5

The mean value of C found by Ortiz et al. [74] is compared to

the predicted value found by the present model in table 2.

Ortiz et al. [74] also performed a statistical analysis for a

linear fit function between Rpost and RS and found

Rpost ¼ 0:68� RS þ 2:5 ðr2 ¼ 0:85Þ for the ‘mechanical’ group

and Rpost ¼ 0:88� RS þ 0:9 ðr2 ¼ 0:85Þ for the ‘laser’ group.

Using the model value RS ¼ 8:847, the fit functions for Rpost

are compared to the predicted value in table 2. Finally, the

spherical equivalent SE ¼ 337:5=Rpost � 337:5=Rpre measures

the power change in dioptres achieved by changing the presur-

gical radius Rpre to the postsurgical radius Rpost. Clinical values

for SE with Rs ¼ 8.847 are compared to the predicted value6 in

table 2. It may be seen that the finite-element model predictions

agree closely with the clinical data.

We analysed the fluid pressure and swelling response of

the postsurgical cornea. The fluid pressure is given by

p ¼ Pos þ p0 (equation (2.11)), where Pos is the osmotic

pressure and p0 is the bath pressure (IOP) of 15 mmHg.

Because most inclined lamellae have been removed by

surgery, it may be anticipated that the fluid pressure must

approach zero in the anterior region of the residual stroma.

Since p0 is constant, this implies that the tissue will swell

and the osmotic pressure will become negative (see also

figure 9a). Fluid pressure fringes are shown in the cutaway

half-mesh of figure 15a and local volume dilation fringes

are shown in 15b. Figure 15a shows that the fluid pressure

reduces to approximately half of its normal value in the abla-

tion region.7 The local swelling shown in figure 15b is

necessary to produce a negative osmotic pressure. It is con-

centrated in the upper region of the residual stroma,

particularly where the ablation is deepest and lamella incli-

nation is lost. It is observed that the swelling is modest,

around 3.5% maximum and decreasing with depth. This is

so because only small swelling is needed to produce negative

osmotic pressure of the order of 15 mmHg (figure 9a). This

will add some contribution to the biomechanical response

in LASIK, but it is probably secondary to the elastic response.

However, if IOP increases, then the swelling increases signifi-

cantly, as shown in table 3.

7. DiscussionThe proposed model follows the general theory for hydrophi-

lic and highly swollen polyelectrolyte gels introduced by

Katchalsky & Michaeli [24] combined with the membrane

transport theory of Kedem & Katchalsky [16]. The main

work was the characterization of the stromal electrolyte and


18.6

15.7

12.8

9.9

7.0

1.050

1.038

1.026

1.014

1.002

0.990

4.1

p (mmHg)

dilation J(b)

(a)

Figure 15. Fringe plots of (a) the fluid pressure and (b) volume dilation of a postsurgical cornea after LASIK designed for an intended correction of 25 dioptres.

Table 3. Predicted volume dilation of the anterior residual stroma afterLASIK for three values of IOP.

IOP (mmHg) dilation J

15 1.031

30 1.086

45 1.147


12:20150241

17


elastic free energies, including the modelling of modifications

due to active endothelial ionic transport. The resulting model

predicts the ex vivo (no active endothelial ionic transport)

osmotic pressure and osmotic compressibility with good

accuracy compared to experimental measurements (as

described in §6.1.1) and provides some confidence in the gen-

eral framework. Previous equilibrium modelling of corneal

swelling by Bryant & McDonnell [21] was based on numeri-

cal solution of the spherically symmetric (one-dimensional)

fluid and ionic continuity equations in a multiphasic theory

[22,75]. The current energy-based approach is equivalent

(see the electronic supplementary material) in that it implies

satisfaction of the same governing equations but it offers

some distinct advantages.

The energy approach provides an average (or homogen-

ized) energy-equivalent value of osmotic pressure over a

unit cell (see equation (2.9)). This is useful for consistently

embedding nanoscale effects, for example, the distribution

of GAG charge into collagen coating and non-coating

regions, in a macroscopic theory. By contrast, the triphasic

theory produces a pointwise varying expression for the

osmotic pressure. The energy approach is also convenient

for describing the deviation of the stromal electrolyte from

ideal conditions, including electrolyte volume exclusion

effects associated with the collagen fibrils and stromal cell

(keratocyte) population. Also noteworthy is the fact that the

energy formulation is a natural approach for finite-element

approximation which can be based directly on the variatio-

nal forms provided by the free energy terms. The proposed

formulation only requires solving for the displacement

field, which gives practical convenience in finite-element

implementation. The model contains numerous physical

constants and parameters (see reference values in table 1),

but most are well characterized and all simulations pres-

ented used these reference values—no problem-dependent

parameters appear in the theory.

Modelling the influence of active endothelial ionic fluxes

on the biophysics of corneal tissue hydration in a fully three--

dimensional context presents a challenge. In this approach,

we have analytically integrated the phenomenological KK

[16] membrane transport equations to describe the membrane

osmotic gradient as a function of the active ionic fluxes and

endothelial ionic permeability.8 This leads to a modified

Boltzmann ionic distribution in the stroma and correspond-

ing electrostatic free energy. The effects are contained in the

introduced ionic transport parameters Qþ and Q2 which cor-

respond to cation and anion transport, respectively.

Interestingly, for a binary electrolyte the influence of active

ionic transport appears as a function of the combined



12:20150241

18


parameter QþQ2. The model suggests that active endothelial

ionic transport reduces osmotic pressure and hence swelling

pressure (these are essentially equivalent in the case of the

highly hydrated cornea) and the pump-leak mechanism can

be explained as active pumping producing a downward

shift of the swelling pressure–hydration curve as discussed

in §6.2.1 and illustrated in figure 9a.

Precise quantification of the fluid and osmotic pressures

remains an open question for the human in vivo cornea, as

direct measurements are not currently feasible. Conventional

analysis [15,18,19] predicts a negative fluid pressure throughout

the cornea. In this study, osmotic calibration based on imbibition

pressure measurements for rabbit indicate positive fluid pressure

at normal hydration, which is in agreement with [21]. The cali-

bration suggests that stromal fluid pressure must be close to

IOP and osmotic pressure must be close to zero. It is interesting

to analyse how the forces acting at the corneal bounding surfaces

are balanced under these conditions. The posterior stroma

boundary condition, equation (5.13), may be rewritten in the

direction normal to the corneal posterior surface as

sel� p0�Pos ¼� p0, where sel is the collagen effective stress

component and p0 denotes the IOP. As the posterior collagen

fibres are parallel to the corneal surface, they can provide negli-

gible force in the transverse direction and so sel � 0: Thus, it

may be concluded that the in vivo osmotic pressure Pos must be

zero at the posterior stroma—in agreement with the calibration.

The situation is quite different at the epithelial layer. Here

there is no external pressure loading to balance the stromal

fluid pressure (figure 12). Inclined lamellae insert into

Bowman’s layer [35] where they must act to resist the stromal

fluid pressure applied to the epithelium. The mechanical bal-

ance equation across the epithelium, equation (5.14), written

in the normal direction is sel � p0 � Pos ¼ 0: Thus, the cor-

neal anterior boundary could not be stably maintained

without the contribution from inclined fibres and which

leads to the following hypothetical question—if lamella incli-

nation was absent, how would the boundary condition then

be satisfied? This in essence is what occurs in LASIK where

the inclined lamellae are largely removed by flap creation

and ablation. In this case, the effective stress will become

zero in the normal direction sel � 0 and the in vivo cornea

must swell until a negative osmotic pressure is generated so

that Pos ¼ � p0: This is precisely what the model predicts in

LASIK as described in §6.2.4 (although only approx. 3%

swelling is needed to achieve Pos¼� p0).

We have calibrated this model for the normal living

cornea by assuming that GAG fixed charge is not depth-

dependent; this leads to a hydration state that is also uniform

with depth. It has been observed by Castoro et al. [76] that the

anterior stroma may be slightly less hydrated than the pos-

terior. The water content of these two stromal layers is

measured to be 75% and 79% for rabbit, respectively [77].

Furthermore, biochemical studies on bovine corneas have

shown that posterior stroma has a higher ratio of keratan

sulfate to dermatan sulfate [76]. For human corneas, the

depth-dependence of GAG types is unknown. Given that

the hydration gradient is small through the corneal depth

(less than 5%), it seems justifiable to employ a uniform

fixed charge distribution as a first approximation. However,

this can be easily modified as new information becomes

available. Likewise, flexible GAG chains, the linear macro-

molecular structures formed from disaccharide units, may

contribute an entropic elastic energy [23]. Each GAG chain

is covalently bound at one end to the PG core protein and

so is anchored to a collagen fibril (and thereby providing

fixed charges). It has been proposed [9,10] that two or

more GAG chains can link to form antiparallel duplex struc-

tures bridging between neighbouring collagen fibrils and

can transmit entropic-based forces deriving from thermal

motions. Such contributions can be modelled through appro-

priate free energy expressions but are considered to be a

second-order effect and neglected in this study.

The stromal electrolyte free energy (equation (3.22)) is

convex in the dilation J and represents a rational basis for

modelling the osmotic behaviour of the tissue. Indeed we

have shown that the model is capable of simulating massive

swelling and the associated hydration change in the case of

stroma immersed in an ionic bath. The predictions also

show that swelling interaction with the collagen network

leads to the observed anisotropy of swelling (primarily

through the thickness direction) and inhomogeneity (most

swelling occurs in the posterior cornea) which strongly sup-

ports the idea that the anterior cornea exhibits rigidity with

respect to changes in hydration [29,31]. Measurements for

in vivo swelling are not precise, but model predictions

appear to agree qualitatively in all presented cases, and quan-

titatively for the rather subtle problem of swelling under

increasing IOP. The model has provided predictions for swel-

ling under Fuchs’ dystrophy and should also be adaptable to

modelling other diseases of the cornea such as macular cor-

neal dystrophy, where the GAG concentration is found to

be abnormally low.

The model supports the necessity of lamellae inclination

for stability of the corneal anterior (refractive) surface

[29,31]. When lamella inclination is removed synthetically

by adjusting the fibre distributions, the model predicts an

enormous increase in swelling in a stroma sample immersed

in de-ionized water (§6.1.2). Theoretically, such swelling

would become infinite, but in this model is limited by the

small matrix shear stiffness. The structural role of lamella

inclination has been examined here for the first time and

the results suggest that this aspect of the collagen architecture

is extremely important for understanding swelling behaviour

and how the stromal fluid pressure is equilibrated.

Finally, it is remarked that clinical swelling (oedema),

which results from hydration changes in the pathological

cornea, can occur under a wide variety of circumstances.

For example, during hypoxia due to contact lens wear, lactate

ion production increases and results in oedema [19,78]. The

underlying mechanism has been identified as the accumu-

lation of lactate acid due to increasing anaerobic respiration

[78]. Extension of the model to include these multiple species

and their interactions through metabolic reactions for aerobic

and anaerobic respiration is currently under development.

Authors’ contributions. X.C. and P.M.P. designed the research and devel-oped the mathematical modelling of the electrolyte; S.J.P. and P.M.P.developed the mathematical modelling of the stromal elasticity; X.C.and P.M.P. performed the numerical analysis and interpretation; allauthors wrote the paper jointly.

Competing interests. We declare we have no competing interests.

Funding. This research was supported by the Stanford University Bio-XInterdisciplinary Initiatives Program (IIP). X.C. received a StanfordGraduate Fellowship (SGF) from Stanford University. S.J.P. receiveda Bio-X Graduate Student Fellowship from Stanford University.


rsif.royalsocietypublishing.org

19


Endnotes1For example, in application to the in vivo cornea, collagen fibres inV0 are prestressed by the action of the IOP.2Fluorophore-assisted carbohydrate electrophoresis.3If future studies produce a different conclusion regarding human stro-mal fluid pressure, including variation through depth, the value ofQþQ2 can be re-calibrated without any change to the general theory.4This requires iterative solution for the configuration S and collagenprestress that equilibrates the IOP—see §2.5The model analysed the biomechanical (elastic and swelling)response of the residual stroma. The flap was not analysed and

the postsurgical anterior radius of the cornea was takento be that of the postsurgical ablation profile radius, foundby least-squares fitting, extended by a constant flap thickness of150 mm.6The mean clinical and finite-element model values of Rpre were 7.8and 7.87 mm, respectively—the value Rpre ¼ 7.8 mm was used in cal-culating SE for consistency.7Because p0 is constant at 15 mmHg (0.002 MPa), we have Pos ¼ p20.002,and the fringe plot can be interpreted in terms of osmotic pressure.8Water permeability does not appear because at equilibrium waterflux will be zero.

J.R.Soc.

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