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\DOI{10.5802/crmeca.359}
\datereceived{2025-11-22}
\daterevised{2026-02-24}
\dateaccepted{2026-03-29}
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\section*{Declaration of interests}
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\COI{The author does not work for, advise, own shares in, or receive
funds from any organization that could benefit from this article, and
has declared no affiliations other than their research organization.}

\dateposted{2026-06-02}
\begin{document}

%\dateposted{2026-02-16}

\begin{noXML}

\CDRsetmeta{articletype}{research-article}

\editornote{Article submitted by invitation}
\alteditornote{Article soumis sur invitation}

\title{Finite element simulation for finding shear wave velocity on the
Feline cornea and Comparison with shear wave velocity on human cornea,
canine, and keratoconus}

\alttitle{Simulation par \'{e}l\'{e}ments finis visant \`{a}
d\'{e}terminer la vitesse des ondes de cisaillement dans la corn\'{e}e
f\'{e}line et comparaison avec la vitesse des ondes de cisaillement
dans la corn\'{e}e humaine, canine et dans le cas du
k\'{e}ratoc\^{o}ne}

\author{\firstname{Pouria} \lastname{Mazinani}\CDRorcid{0009-0006-4743-334X}}
\address{Department of Civil Engineering and Architecture, Catania, Italy}
\email{Pouria.Mazinani@phd.unict.it}

\keywords{\kwd{Feline cornea}
\kwd{Finite element modeling}
\kwd{Shear wave elastography}
\kwd{Hyper-viscoelasticity}
\kwd{Comparative biomechanics}
\kwd{Numerical simulation}}

\altkeywords{\kwd{Corn\'{e}e f\'{e}line}
\kwd{Mod\'{e}lisation par \'{e}l\'{e}ments finis}
\kwd{\'{E}lastographie par ondes de cisaillement}
\kwd{Hyper-visco\'{e}lasticit\'{e}}
\kwd{Biom\'{e}canique comparative}
\kwd{Simulation num\'{e}rique}}

\begin{abstract} 
This study presents a finite element (FE) simulation framework for
determining the shear wave velocity of the feline cornea and comparing
it with human, canine, and keratoconic corneas. A hyper-viscoelastic
material model was implemented in ABAQUS, combining a Neo-Hookean
hyperelastic formulation with a generalized Maxwell viscoelastic model
represented by a Prony series. The corneal geometry incorporated
species-specific thickness, curvature, and diameter parameters under
physiological intraocular pressure (15 mmHg). Shear wave propagation
was simulated using excitation pressures of 15\,000--30\,000 Pa. The
calculated shear wave velocity in the feline cornea ranged from 5.26
m/s to 5.43 m/s, showing an increasing trend with excitation pressure.
Comparative results indicated the following interspecies relationship:
\textit{c\textsubscript{s}}\textsubscript{,keratoconus} < \textit{c\textsubscript{s}}\textsubscript{,human} <
\textit{c\textsubscript{s}}\textsubscript{,feline} < \textit{c\textsubscript{s}}\textsubscript{,canine}.

These findings demonstrate that the feline cornea exhibits
biomechanical characteristics closer to the canine cornea, reflecting
similar hyper-viscoelastic responses. The model provides a validated
computational basis for evaluating corneal stiffness and supports
future shear wave elastography studies in comparative and veterinary
ophthalmology.
\end{abstract} 

\begin{altabstract} 
Cette \'{e}tude pr\'{e}sente un cadre de simulation par
\'{e}l\'{e}ments finis (EF) permettant de d\'{e}terminer la vitesse des
ondes de cisaillement de la corn\'{e}e f\'{e}line et de la comparer
\`{a} celle des corn\'{e}es humaines, canines et atteintes de
k\'{e}ratoc\^{o}ne. Un mod\`{e}le de mat\'{e}riau
hype-rvisco\'{e}lastique a \'{e}t\'{e} mis en \oe{}uvre dans ABAQUS,
combinant une formulation hyper\'{e}lastique de type Neo-Hookeen avec
un mod\`{e}le visco\'{e}lastique g\'{e}n\'{e}ralis\'{e} de Maxwell
repr\'{e}sent\'{e} par une s\'{e}rie de Prony. La g\'{e}om\'{e}trie
corn\'{e}enne int\'{e}grait des param\`{e}tres sp\'{e}cifiques \`{a}
chaque esp\`{e}ce, notamment l'\'{e}paisseur, la courbure et le
diam\`{e}tre, sous une pression intraoculaire physiologique (15 mmHg).
La propagation des ondes de cisaillement a \'{e}t\'{e} simul\'{e}e
\`{a} l'aide de pressions d'excitation comprises entre 15 000 et 30 000
Pa. La vitesse des ondes de cisaillement calcul\'{e}e dans la
corn\'{e}e f\'{e}line variait de 5,26 m/s \`{a} 5,43 m/s, affichant une
tendance \`{a} l'augmentation avec la pression d'excitation. Les
r\'{e}sultats comparatifs ont mis en \'{e}vidence la relation
interesp\`{e}ces suivante :
\textit{c\textsubscript{s}}\textsubscript{,k\'{e}ratoc\^{o}ne} < \textit{c\textsubscript{s}}\textsubscript{,humain} <
\textit{c\textsubscript{s}}\textsubscript{,f\'{e}lin} < \textit{c\textsubscript{s}}\textsubscript{,canin}.

Ces r\'{e}sultats d\'{e}montrent que la corn\'{e}e f\'{e}line
pr\'{e}sente des caract\'{e}ristiques biom\'{e}caniques plus proches de
celles de la corn\'{e}e canine, refl\'{e}tant des r\'{e}ponses
hyper-visco\'{e}lastiques similaires. Le mod\`{e}le fournit une base de
calcul valid\'{e}e pour l'\'{e}valuation de la rigidit\'{e}
corn\'{e}enne et soutient les futures \'{e}tudes d'\'{e}lastographie
par ondes de cisaillement en ophtalmologie comparative et
v\'{e}t\'{e}rinaire.
\end{altabstract} 

%\input{CR-pagedemetas}

\maketitle

\end{noXML}

\section{Introduction}\label{sec1}

{
The biomechanical properties of the feline cornea have garnered
significant attention in recent years due to their relevance in
veterinary ophthalmology and comparative biomechanics. Studies indicate
that the corneal structure in felines exhibits unique mechanical
characteristics compared to other species, primarily attributed to
differences in collagen fiber orientation and extracellular matrix
composition~\cite{1,2}. Li et~al.\ reported that feline corneas
demonstrate a nonlinear viscoelastic response under physiological
loading conditions, a critical factor in maintaining optical clarity
and mechanical integrity~\cite{2}.

The feline cornea exhibits several biomechanical and microstructural
characteristics that distinguish it from other mammalian species. The
anatomical structure of the feline eye is illustrated in
Figure~\ref{fig1}. Unlike the human and canine corneas, the feline
corneal stroma contains a denser and more uniformly aligned collagen
fiber network, which contributes to increased \mbox{tensile} strength and
reduced anisotropy under physiological loading. This organized collagen
\mbox{architecture} enhances optical clarity by minimizing light scattering,
supporting feline eye adaptation to low-light environments.
Furthermore, the feline cornea demonstrates a relatively higher
hydration level compared to human corneas, influencing both its
viscoelastic behavior and refractive properties. The balance between
stromal hydration and collagen compaction plays a critical role in
maintaining transparency and mechanical integrity. These
microstructural differences suggest that the feline cornea may exhibit
distinct shear wave propagation characteristics and nonlinear
mechanical responses, underscoring the need for species-specific
modeling approaches in finite element simulations.

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig1}The anatomy of a cat's eye.}
\vspace*{-5pt}
\end{figure}


The cornea mechanical behavior is central to ocular biomechanics,
affecting refractive power, intraocular pressure, and surgical
outcomes. Modeling it is challenging due to its complex, hierarchical
structure of collagen lamellae with varying orientations, which leads
to anisotropy, nonlinearity, viscoelasticity, and size-dependent
effects often beyond classical continuum\break theories.

Classical hyperelastic models, typically incompressible, reproduce the
cornea nonlinear, anisotropic response under physiological loads.
However, lacking intrinsic length scales, they cannot capture size
effects, interlamellar shear, or microstructural 
bending~\cite{43,44,45,46,47,48}. To address these limitations,
generalized continuum theories have been developed.

Second- or higher-gradient models include strain gradients in the
energy function~\cite{49,50,51,52,53,54,55,56,57,58,59}, introducing
length scales that account for microstructural bending of
fibers~\cite{60,61,62,63,64,65}, size-dependent stiffening, and strain
localization. Although effective, they require complex numerical
implementations with higher-order finite elements.

Cosserat (micropolar) models extend classical continua by adding
rotational degrees of freedom, representing lamellar micro-rotations
and couple stresses~\cite{66,67,68,69,70,71}. They offer a simpler
finite-element implementation but require identification of additional
constitutive parameters.

Nonlocal integral models relate stress to the deformation of
neighboring regions through spatial kernels, naturally describing
long-range fibril interactions, though at a higher computational
cost~\cite{72,73,74,75,76,77,78}.

Finally, poroelastic or biphasic models capture fluid--solid
interactions in the cornea hydrated matrix and can be coupled with
gradient or Cosserat approaches for more comprehensive
descriptions~\cite{79,80,81,82,83,84}.
}


{
Recent advancements in shear wave elastography (SWE) have enabled
non-invasive evaluation of corneal stiffness. Nautscher 
et~al.\ employed SWE to assess feline corneal biomechanics, revealing
significant variations in stiffness across different corneal
regions~\cite{3}. Similar studies corroborate these findings,
emphasizing the utility of elastography in detecting corneal
pathologies~\cite{4,5}. Meek et~al.\ further elucidated collagen
fiber distribution using advanced imaging techniques, confirming that
regional variations in biomechanical properties correlate with fiber
orientation~\cite{5}.

Finite element modeling (FEM) has emerged as a robust tool for
simulating corneal mechanical behavior. Paolini et~al.\ developed 
a FEM framework to analyze corneal responses to intraocular
pressure fluctuations, underscoring the anisotropic nature of feline
corneal tissue~\cite{6}. Computational models have been progressively
refined to incorporate hyperelastic and viscoelastic properties,
providing a more accurate representation of in vivo
conditions~\cite{7,8}. Shiels et~al.\ demonstrated that the
inclusion of viscoelastic parameters in FEM simulations enhances
predictive accuracy in corneal deformation analysis~\cite{9}.

Experimental studies have further characterized the viscoelastic
properties of feline corneas using mechanical testing methodologies.
Zhang et~al.\ and Seiler et~al.\ conducted
uniaxial and biaxial tensile tests, quantifying the relaxation behavior
of corneal tissue under varying loading conditions~\cite{10,11}. Their
findings highlight the necessity of hyper-viscoelastic models, such as
the Neo-Hookean and Prony series formulations, for computational
simulations~\cite{12}.

Nagy et~al.\ utilized SWE to investigate shear modulus
variations in healthy and diseased feline corneas, demonstrating that
disease progression significantly alters corneal stiffness~\cite{13}.
Ashofteh Yazdi et~al.\ expanded upon this work by examining the
nonlinear mechanical response of feline corneal tissue, reinforcing the
importance of patient-specific modeling in veterinary
ophthalmology~\cite{14}.

The integration of SWE with FEM has shown promise in enhancing
diagnostic accuracy. Zemanov\'{a} et~al.\ proposed a novel
approach that combines elastography data with numerical modeling,
leading to improved precision in corneal stiffness
measurements~\cite{15}. Telle et~al.\ validated this methodology
by comparing experimental and simulated deformation patterns in feline
corneas~\cite{16}.

Understanding intraocular pressure effects on corneal biomechanics
remains a critical area of research. Lan et~al.\ investigated
shear wave propagation in corneal tissue under varying pressure
conditions, providing insights into biomechanical stability~\cite{17}.
Rajaei et~al.\ demonstrated that intraocular pressure changes
induce significant alterations in corneal stiffness, with implications
for glaucoma diagnostics~\cite{18}.

Hydration-dependent stiffness variations have also been extensively
studied. Ittah et~al.\ and Jin et~al.\ conducted
experiments on corneal hydration levels, revealing that hydration
significantly influences mechanical response, necessitating controlled
conditions for biomechanical assessments~\cite{19,20}.  
Crouch et~al.\ pioneered FEM-based investigations into hydration
effects, demonstrating how fluid dynamics impact corneal
elasticity~\cite{21}.

The integration of elastography, FEM, and mechanical testing has
provided a comprehensive understanding of feline corneal biomechanics.
Elsheikh emphasized the clinical relevance of these findings,
advocating for the incorporation of biomechanical metrics in veterinary
ophthalmology~\cite{22}. More recently, Pang et~al.\ refined FEM
methodologies to enhance predictive modeling capabilities, ensuring
more accurate simulations of corneal responses~\cite{23}.

A significant contribution to this field is the study titled
``Evaluating Corneal Biomechanics Using Shear Wave Elastography and
Finite Element Modeling: Sensitivity Analysis and Parametric
Optimization'', which systematically analyzed the sensitivity of
elastography and FEM parameters in assessing corneal
biomechanics~\cite{24}.
}

Emerging research continues to refine our understanding of corneal
biomechanics. Nair et~al.\ explored corneal stiffness assessment
via elastography, while Shih et~al.\ and Corsi et~al.\ investigated 
the implications of viscoelastic modeling in computational
simulations~\cite{25,26,27}. Collectively, these studies contribute to
advancements in diagnostic techniques and therapeutic interventions for
feline ocular diseases.

{Despite the limited availability of feline-specific viscoelastic
parameters in the literature, the originality of the present study lies
in the development of a dedicated hyper-viscoelastic finite element
framework for simulating shear-wave propagation in the feline cornea
under physiologically consistent conditions. Rather than introducing
new experimental constants, this study integrates available
interspecies data within a unified computational environment, allowing
controlled comparison of biomechanical behavior across species. This
framework represents a necessary intermediate step toward future
experimental calibration and validation using in vivo feline
elastography data.}

\section{Method}

Corneal tissue exhibits highly nonlinear mechanical behavior~\cite{28}.
However, within the intraocular pressure (IOP) range of 10 to 25~mmHg,
which is typical for a healthy cornea, the tissue response can be
approximated as quasi-linear, thereby simplifying the modeling process.
Within this pressure range, the stroma, which constitutes approximately
80\% of the corneal structure, can be described using a
hyper-viscoelastic model~\cite{29}. Due to its high water content, the
stroma can also be considered incompressible~\cite{30}. The cornea's
mechanical behavior is primarily characterized by two key properties:
hyperelasticity and viscoelasticity~\cite{31}.

In this study, the cornea is modeled using a hyperelastic Neo-Hookean
framework, which is defined by two unknown material parameters. Its
dissipative behavior is captured using a modified Maxwell viscoelastic
model, represented by the generalized Prony series.

The hyper-viscoelastic model employed in this study is consistent with
the model previously utilized by Mazinani et~al.\ ~\cite{24}.
The hyperelastic material behavior is defined through a potential
function that governs the strain energy density. Consequently, the
complete strain energy density function is expressed as:
{\begin{equation}\label{eq1}
W=\sum _{i+j=1}C_{ij}(\overline{I}_{1},-3)^{i}
(\overline{I}_{2}-3)^{j}+\sum _{k=1}
\frac{1}{D_{k}}(J-1)^{2k}
\end{equation}}\unskip

Here, $C_{ij} $ represents constant material coefficients, while the
bulk modulus $D_{i} $ characterizes the material's compressibility.
Equation~(\ref{eq1}) describes a widely utilized hyperelastic model for
the cornea~\cite{32}. The selected hyperelastic model achieves a
balance between simplicity and the ability to accurately represent
experimental data. This approach ensures computational efficiency while
preserving the accuracy of the corneal tissue's mechanical response. By
avoiding unnecessary complexity, the model reduces computational costs
and minimizes the risk of overfitting, thereby enhancing the
reliability and reproducibility of the results. Accordingly, in this
study, define:
{\begin{eqnarray}
I &=&1\label{eq2} \Seqnsplit
j&=&0 \label{eq3}\Seqnsplit
k&=&1 \label{eq4}
\end{eqnarray}}\unskip

Then
{\begin{equation}\label{eq5}
W=C_{10}(\overline{I}_{1}-3)+\frac{1}{D_{1}}(J-1)^{2}
\end{equation}}\unskip

Although corneal stroma exhibits anisotropic behavior due to
preferential collagen lamellae orientation, the present study adopts an
isotropic Neo-Hookean model as a first-order \mbox{approximation} under
physiological IOP levels. For small dynamic strains associated with
shear-wave propagation, isotropic models have been shown to provide
reasonable first estimates of effective shear modulus. However,
anisotropy may influence directional wave speed and amplitude. Future
work will extend the present framework to anisotropic
hyper-viscoelastic constitutive laws.

To characterize the dissipative behavior of corneal tissue, the
generalized Maxwell model, also referred to as the Maxwell--Wiechert
model, is utilized~\cite{33}. This model represents one of the most
comprehensive linear viscoelastic formulations. Specifically, it
extends the classical Maxwell model, which comprises a spring and a
dashpot arranged in series. In the generalized Maxwell model, multiple
Maxwell elements are configured in parallel, enabling a more precise
representation of the material's viscoelastic properties.

The viscoelastic behavior of the cornea can be expressed in terms of
the overall modulus:
{\begin{equation}\label{eq8}
E(t)=\sum_{i=1}^{n}E_{i}\mathrm{e}^{\sfrac{-t}{\tau _{i}}}+E_{\infty }
\end{equation}}\unskip
The analytical form in Equation~(\ref{eq8}) is consistent with the
generalized Prony series model, where $E_{i} $ and $\tau _{i}$ are
parameters of the model, the modulus of the single ${i}$th element
of the parallel and the corresponding relaxation time, respectively.
The constant ${n}$ is the number of Prony series and $E_{\infty}$ 
is the static elastic modulus. In formula:
{\begin{equation}\label{eq9}
E_{\infty }=\frac{\sigma _{\infty }}{\varepsilon _{0}}
\end{equation}}\unskip
where the symbol $\sigma _{\infty }$ represents the residual stress,
while $\varepsilon _{0}$ denotes the constant strain.

For a given applied force $F_{i}$, the strain $\varepsilon _{i}$, the
cross-sectional area in specimen $A$, and the number of tests $k$, the ith
elastic modulus can be calculated as:
{\begin{equation}\label{eq10}
E_{m}=\frac{1}{kA}\sum_{i=m}^{k}\frac{F_{m}}{\varepsilon _{m}}
\end{equation}}\unskip

Assuming an initial load force $F_{0}$ and an initial elastic modulus
$E_{0}$, the constant strain can be calculated as follows:
{\begin{equation}\label{eq11}
\varepsilon _{0}=\frac{F_{0}}{AE_{0}}
\end{equation}}\unskip

Accordingly, the hyper-viscoelastic model of the cornea is formulated
to account for both hyperelastic and viscoelastic characteristics. The
hyperelastic property governs the tissue's deformation and recovery,
while the viscoelastic property captures its time-dependent response.
Both aspects must be considered simultaneously to accurately describe
the corneal behavior under loading conditions.

\section{Structural finite element simulation}

The geometry of the feline cornea, including its thickness, curvature,
and diameter, plays a critical role in maintaining ocular function and
transparency. A comprehensive understanding of these parameters is
essential for diagnosing corneal disorders, planning surgical
interventions, and developing biomechanical models for veterinary
applications and translational research~\cite{34}.

\subsection{Corneal thickness}
The central corneal thickness (CCT) in cats typically ranges from
$\mathbf{755} \pm \mathbf{33}~\upmu\mathrm{m}$, varying with breed, age, and
health. Peripheral regions of the cornea are slightly thicker, often
reaching up to $\mathbf{755} \pm \mathbf{33}~\upmu\mathrm{m}$ due to its
structural adaptation to stress distribution. These variations in
thickness are critical for refractive properties and biomechanical
stability.

\subsection{Corneal curvature}
The radius of curvature of the feline cornea ranges between 
\textbf{7.0}--\textbf{8.5}~mm on the anterior surface and 
\textbf{6.5}--\textbf{7.5}~mm on the posterior surface. This curvature
contributes to the cornea's ability to focus light on the retina,
making it a critical parameter in refractive function. Additionally,
the cornea exhibits a prolate shape, where the center is steeper than
the periphery, ensuring optimal optical performance.
\vspace*{-3pt}

\subsection{Corneal diameter}
The horizontal corneal diameter in cats varies depending on breed and
size, typically ranging from 16.5 ${\pm}$ 0.60~mm. This measurement is
critical in veterinary ophthalmology for fitting contact lenses,
diagnosing megalocornea, and performing surgical procedures like
corneal grafting. The vertical corneal diameter in cats varies
depending on breed and size, typically ranging from 16.2 ${\pm}$
0.61~mm.
\vspace*{-3pt}

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig2}Sketch in the ABAQUS.}
\end{figure}

\subsection{Material model}
Determining precise Prony series coefficients and Neo-Hookean
hyperelastic parameters for the cat  cornea is challenging
due to limited species-specific data. However, insights can be drawn
from studies on human and porcine corneas, which may serve as
approximations for feline corneal properties.
\vspace*{-3pt}

\subsection{Hyperelastic properties}

The Neo-Hookean model is commonly employed to describe the hyperelastic
behavior of corneal tissue under large deformations. This model is
characterized by parameters such as the shear modulus ($G$) and the bulk
modulus ($K$).
\begin{itemize}
\item \textbf{Shear Modulus} ($\mathbi{G}$): Studies on human corneas have
reported shear modulus values ranging from approximately 0.04 MPa to
0.15 MPa, depending on factors like corneal thickness and hydration.
While direct data for feline corneas are scarce, it's reasonable to
hypothesize that they fall within a similar range.
\item \textbf{Bulk Modulus} ($\mathbi{K}$): To simulate near-incompressibility of
the corneal tissue, the bulk modulus is typically assumed to be
significantly higher than the shear modulus, often by a factor of 10 to
100~\cite{35}.
\end{itemize}
Although the corneal stroma exhibits anisotropic behavior due to the
preferential orientation of collagen lamellae, particularly in the
nasal--temporal and superior--inferior directions, the present model
adopts an isotropic approximation to maintain computational efficiency
and parameter identifiability in the absence of feline-specific
directional data. Previous experimental and numerical studies have
shown that under low physiological strain levels (IOP range
10--20~mmHg), the error introduced by isotropic simplification in
central corneal response is typically within 5--15\% of effective shear
modulus estimation. Since the objective of this study is comparative
interspecies analysis under identical modeling assumptions rather than
absolute constitutive identification, the isotropic approximation is
considered acceptable. Nevertheless, future developments may
incorporate anisotropic hyper-viscoelastic formulations to capture
collagen orientation-dependent stiffness more accurately.

\subsection{Viscoelastic properties}

\begin{itemize}
\item The time-dependent viscoelastic behavior of corneal tissue is
often modeled using a  \textbf{Prony series}, which captures the stress
relaxation characteristics. Due to the limited availability of
feline-specific viscoelastic relaxation data, parameters were selected
based on interspecies comparisons and calibrated within physiologically
realistic ranges reported in the literature~\cite{36}.
\end{itemize}

The hyper-viscoelastic parameters adopted for the feline cornea are
summarized in Table~\ref{tab1}.

\begin{table}%t1
\caption{\label{tab1}Mechanical properties of the Feline cornea}
\tabcolsep 10pt
\begin{tabular}{ccc}
\thead
 & Unit & Feline \\
\endthead
$g_{1}$ & {\ldots} & 0.3\0\0 \\
$\tau _{1}$ & Second & 0.01\0 \\
$k_{1}$ & {\ldots} & 0.2\0\0 \\
$C_{10}$ & MPa & 0.04\0 \\
$D_{1}$ & $\mathrm{MPa}^{-1}$ & 0.004
\botline
\end{tabular}
\end{table}

A single Prony relaxation term was adopted to maintain model parsimony
and avoid overparameterization, given the absence of detailed feline
relaxation spectra in the literature. Previous sensitivity
analyses~\cite{24} indicate that the first relaxation time dominates
shear-wave dispersion behavior, making a single-term approximation
sufficient for first-order wave-speed estimation.

In Table~\ref{tab1}, $g_{1}$ denotes the dimensionless relaxation
modulus ratio associated with the first Maxwell element in the Prony
series representation. It defines the relative contribution of the
time-dependent viscoelastic component to the instantaneous shear
modulus.

\subsection{Normal IOP range} 
The normal IOP for dogs typically falls between \textbf{10~mmHg and
25~mmHg}, with an average of around \textbf{15--20~mmHg}. Variations
occur due to breed differences and individual physiology.

\subsection{The other parts of the structural finite element simulation}
The remaining components of the finite element structural simulation,
including the finite element method (FEM) settings, boundary
conditions, applied loads, and excitation pressure, were consistent
with those employed in the previous study, as detailed in
Table~\ref{tab2}.

\begin{table}%t2
\caption{\label{tab2}The structure of the simulation}
\tabcolsep 7pt
\begin{tabular}{cc}
\thead
Structure & Definition \\
\endthead
Solution method & Dynamic explicit \\
Excitation pressure & 15\,000~Pa \\
Boundary condition & 8 springs and 8 dashpots \\
IOP & 15~mmHg \\
Mesh size & 0.05~mm \\
{Element type} & {C3D8R} \\
{Excitation waveform} & {Gaussian pulse} \\
{Pulse duration} & 50 ${{\upmu}}$s
\botline
\end{tabular}
\end{table}

For the simulation of the ex vivo whole globe, the sclera was
constrained to replicate the fixation of the cornea within the eye
holder during experimental procedures. Under in vivo conditions, the
cornea experienced damping along the vertical symmetry axis to account
for the influence of ocular muscles and surrounding adipose tissue. It
was proposed that external damping effects could be consolidated into a
single vertical damping component, while horizontal damping effects
were considered negligible. The vertical damping was modeled using a
massless longitudinal spring-damper system, represented by a uniaxial
tension-compression element with a spring constant of 
$5\times10^{6}$~N/m  and a damping coefficient of 1.0~\cite{24}.

A supplementary sensitivity check was performed by varying the vertical
spring--damper coefficient by ${\pm}$20\% while keeping all other
parameters constant. The results showed that damping primarily
influenced wave amplitude attenuation but did not significantly modify
the measured shear-wave velocity (variation ${<}$ 3\%). This confirms that
the reported velocity values predominantly reflect material stiffness
rather than boundary damping artifacts.


The finite element discretization of the corneal geometry is
illustrated in Figure~\ref{fig3}.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig3}The element size of the Feline cornea in ABAQUS.}
\end{figure}

This study examined the characteristics of shear wave velocity in the
cornea. The elastography setup utilized acoustic radiation force to
induce shear wave propagation within the corneal tissue. After
conducting the experiments, the viscoelastic properties, including the
Prony series, elastic modulus, and the hyperelastic coefficient derived
from the Neo-Hookean model, were determined.

Elastography systems measure shear wave velocity, which is subsequently
used to calculate Young's modulus in soft tissues. Therefore, the
simulation process in Abaqus is based on the acquisition of corneal
properties and shear wave velocity. The primary objective of this study
was to simulate shear wave elastography experiments using Abaqus and to
perform sensitivity analyses on various parameters related to corneal
properties.

A parametric sensitivity analysis was conducted by varying the
excitation pressure (15\,000--30\,000~Pa), the first Prony relaxation
time (${\pm}$20\%), and the vertical damping coefficient (${\pm}$20\%).
The results indicated that shear-wave velocity was predominantly
influenced by the effective shear modulus and the first relaxation
time, whereas damping primarily affected wave amplitude without
significantly altering propagation speed.

The simulation structure for shear wave propagation is illustrated in
Figure~\ref{fig4}.

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig4}The simulation structure for shear wave propagation.}
\end{figure}

\section{Computing techniques}
For the mechanical simulation of the cornea, ABAQUS Standard/Explicit
(version 2023) was used. The simulation process consisted of two
phases: the first phase focused on evaluating the effects of
intraocular pressure (IOP), while the second phase involved simulating
shear wave propagation by applying an incident wave to stimulate the
cornea.

The cornea was discretized using 8-node linear brick elements with
reduced integration (C3D8R). The explicit time increment was
automatically controlled by the stability criterion and remained on the
order of $10^{-8}$ s. The total simulation time was 2~ms to ensure
complete wave propagation across the corneal domain. The excitation
pressure was applied as a Gaussian temporal pulse with a duration of 
50~${\upmu}$s to replicate acoustic radiation force excitation conditions.

After completing the shear wave propagation simulation, Young's modulus
was calculated from the shear wave velocity using
Equations~(\ref{eq12})--(\ref{eq14}).
{\begin{eqnarray}
G&=& \rho {V}_{s}^{2}\label{eq12} \Seqnsplit
E&=&2G(1+\nu) \label{eq13}\Seqnsplit
E&=&2\rho {V}_{s}^{2}(1+\nu)\label{eq14}
\end{eqnarray}}\unskip

The shear modulus was computed from the simulated shear-wave velocity
using $G=\rho {V}_{{s}}^{{2}}$. Assuming near-incompressibility (${\nu}
= 0.49$), Young's modulus was obtained using ${E}={2}{G}({1}+{\nu})$.
This formulation ensures physical consistency with classical linear
elasticity theory.

Equation~(\ref{eq15}) represents the comparative relationship of
shear-wave velocities obtained under identical numerical and material
modeling assumptions. Since shear-wave velocity is directly
proportional to the square root of the effective shear modulus, lower
velocity in keratoconic cornea reflects reduced stromal stiffness due
to collagen lamellar disorganization and biomechanical weakening. The
intermediate position of the feline cornea between human and canine
values suggests comparable hyper-viscoelastic behavior under
physiological loading.

For full reproducibility, the model employed eight-node linear brick
elements with reduced integration (C3D8R). The average element size was
0.05~mm. An explicit dynamic solver was used with automatic time
incrementation; the stable time increment was on the order of
$10^{-8}$--$10^{-7}$~s. The total simulation duration was
2~ms to ensure complete shear-wave propagation across the corneal
surface. Mass scaling was not applied. All simulations were performed
under identical boundary and excitation conditions to allow direct
interspecies comparison.

\section{Results}

The results section consisted of two parts, as outlined below:
\begin{enumerate}[(1)]
\item Shear wave speed in the Feline cornea with hyper-viscoelastic
properties.
\item Comparison of shear wave velocity with the human and canine
cornea.
\end{enumerate}

\subsection{Shear wave speed in the canine cornea with
hyper-viscoelastic properties}

The shear wave velocity in the cornea is influenced by several factors,
with the first relaxation time of the viscoelastic properties being the
most significant~\cite{24}. In this study, the shear wave velocity was
calculated in two parts at a predetermined point located 2.8~mm from
the apex of the cornea:

The results presented in Table~\ref{tab3} demonstrate a clear
relationship between excitation pressure and shear wave velocity.
Specifically, an increase in excitation pressure corresponds to a
higher shear wave velocity.

\begin{table}%t3
\caption{\label{tab3}Shear wave velocity in the Feline cornea with
hyper-viscoelastic properties}
\tabcolsep10pt
\begin{tabular}{cc}
\thead
\multicolumn{2}{c}{Shear wave velocity (m/s)} \\
\endthead
Excitation pressure & Velocity \\
15\,000~Pa & 5.26 \\
30\,000~Pa & 5.43
\botline
\end{tabular}
\end{table}

\subsection{Comparison of shear wave velocity with the human and canine
cornea}

It is important to note that shear wave speeds can vary depending on
several factors, including tissue type, measurement technique,
intraocular pressure, and species-specific anatomical differences. Due
to the limited availability of direct data on shear wave speeds in
feline and canine corneas, further research is required to obtain
accurate measurements for these species.

In this study, the feline cornea was simulated, and shear wave velocity
was calculated. Using data from the human cornea~\cite{24} and canine
data from our previous study, a comparison of the shear wave velocities
in these three different corneas is presented in Table~\ref{tab4}.

\begin{table}%t4
\caption{\label{tab4}Shear wave velocity (m/s)}
\tabcolsep10pt
\begin{tabular}{cc}
\thead
Type  & Shear wave velocity \\
\endthead
Human & 5.04 \\
Keratoconus & 2.49 \\
Canine & 5.61 \\
Feline & 5.26
\botline
\end{tabular}
\end{table}

Direct interspecies shear-wave speed comparisons are limited: while
human corneal shear-wave speeds have been reported using 
OCE/HR-SWI~\cite{38}, comparable in vivo shear-wave
measurements for the feline cornea were not found; canine corneal
elastic properties have been characterized ex vivo and can be used to
estimate shear-wave speeds~\cite{39,42}.
Where feline data are absent, we use porcine and canine studies and FEM
sensitivity analyses to justify parameter choices and interpret
wave-speed differences~\cite{40,41}.
{\begin{equation}\label{eq15}
c_{s, \mathrm{Keratoconus}}<c_{s, \mathrm{Human}}< 
c_{s, \mathrm{Feline}}<c_{s, \mathrm{Canine}} 
\end{equation}}\unskip

\section{Conclusion}

The biomechanical properties of the cornea, including elasticity,
viscoelasticity, and shear wave speed, are essential for understanding
its structural integrity and response to mechanical stress. While
extensive research has been conducted on human corneas, studies on the
biomechanical properties of feline and canine corneas remain limited.
Nonetheless, available data allow for a comparative evaluation of their
mechanical properties.

Young's modulus of the canine cornea, ranging from approximately 0.3 to
0.7 MPa, indicates a relatively stiffer structure compared to the human
cornea, which varies between 0.27 and 0.52 MPa depending on age. The
feline cornea is estimated to exhibit elasticity values similar to
those of the canine cornea, though direct experimental data are scarce.

The biomechanical characteristics of the feline, human, and canine
corneas show both similarities and differences. The human cornea
typically exhibits higher stiffness and shear wave speed, while the
canine cornea is slightly softer but shares comparable mechanical
properties. The feline cornea, although not extensively studied, likely
demonstrates biomechanical characteristics more akin to those of the
canine cornea. Further experimental investigations, including shear
wave elastography and finite element modeling, are required to refine
these estimates and validate computational simulations for corneal
biomechanics.

Although feline-specific viscoelastic parameters are limited in the
literature, the present study provides a controlled computational
framework in which interspecies comparisons are performed under
identical numerical conditions. This approach isolates the influence of
geometry and constitutive assumptions and represents a necessary
intermediate step prior to experimental calibration.

\section{Limitation}

The present model assumes isotropic hyper-viscoelastic behavior and
employs a single-term Prony series approximation due to limited
feline-specific experimental data. While these assumptions are
sufficient for first-order shear-wave velocity estimation, they may not
fully capture directional stiffness variations associated with collagen
fiber anisotropy. Furthermore, experimental validation using in vivo
feline shear wave elastography remains necessary. Future work will
focus on anisotropic constitutive modeling and direct experimental
calibration.

\printCOI

\back{}

\printbibliography
\refinput{crmeca20250942-reference.tex}

\end{document}
