\makeatletter \@ifundefined{HCode} {\documentclass[screen,CRMECA,Unicode,biblatex,published]{cedram} \addbibresource{crmeca20250942.bib} \newenvironment{noXML}{}{}% \def\tsup#1{$^{{#1}}$} \def\tsub#1{$_{{#1}}$} \def\stbfrac#1#2{{(#1)}/{(#2)}} \def\sbfrac#1#2{{#1}/{(#2)}} \def\sfrac#1#2{{#1}/{#2}} \def\thead{\noalign{\relax}\hline} \def\endthead{\noalign{\relax}\hline} \def\tbody{\noalign{\relax}\hline} \def\xsection{} \def\mathbi#1{\text{\textbf{\textit{#1}}}} \def\xtbody{\noalign{\relax}} \def\tabnote#1{\vskip4pt\parbox{.95\linewidth}{#1}} \def\xxtabnote#1{\vskip4pt\parbox{.96\linewidth}{#1}} \usepackage{upgreek} \usepackage[T1]{fontenc} \usepackage{multirow} \newcounter{runlevel} \let\MakeYrStrItalic\relax \def\jobid{crmeca20250942} \def\nrow#1{\@tempcnta #1\relax% \advance\@tempcnta by 1\relax% \xdef\lenrow{\the\@tempcnta}} \def\morerows#1#2{\nrow{#1}\multirow{\lenrow}{*}{#2}} \def\textemail#1{\href{mailto:#1}{#1}} \csdef{Seqnsplit}{\\} \def\botline{\\\hline} %\graphicspath{{/tmp/\jobid_figs/web/}} \graphicspath{{./figures/}} \def\0{\phantom{0}} \def\mn{\phantom{$-$}} \def\refinput#1{} \def\back#1{} \let\Lbreak\break \allowdisplaybreaks \def\xmorerows#1#2{{#2}} \DOI{10.5802/crmeca.359} \datereceived{2025-11-22} \daterevised{2026-02-24} \dateaccepted{2026-03-29} \ItHasTeXPublished } {\documentclass[crmeca]{article} \usepackage{upgreek} \usepackage[T1]{fontenc} \def\CDRdoi{10.5802/crmeca.359} \def\centering{} \def\newline{\unskip\break} \def\hyperlink#1#2{#2} \def\hypertarget#1#2{#2} \def\sfrac#1#2{{#1}/{#2}} \def\sbfrac#1#2{{#1}/{(#2)}} \def\stbfrac#1#2{{(#1)}/{(#2)}} \def\xxtabnote#1{\tabnote{#1}} \def\href#1#2{\url[#1]{#2}} \newcommand\@coi{} \newcommand\COI[1]{\gdef\@coi{#1}} \newcommand\printCOI{\ifx\@coi\@empty\else% \section*{Declaration of interests} \@coi\fi } } \makeatother \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}}