Patient-Specific Modeling of Corneal Refractive Surgery Outcomes and Inverse Estimation of Elastic Property Changes

Abstract

The purpose of this study is to develop a 3D patient-specific finite element model (FEM) of the cornea and sclera to compare predicted and in vivo refractive outcomes and to estimate the corneal elastic property changes associated with each procedure. Both eyes of a patient who underwent laser-assisted in situ keratomileusis (LASIK) for myopic astigmatism were modeled. Pre- and postoperative Scheimpflug anterior and posterior corneal elevation maps were imported into a 3D corneo-scleral FEM with an unrestrained limbus. Preoperative corneal hyperelastic properties were chosen to account for meridional anisotropy. Inverse FEM was used to determine the undeformed corneal state that produced <0.1% error in anterior elevation between simulated and in vivo preoperative geometries. Case-specific 3D aspheric ablation profiles were simulated, and corneal topography and spherical aberration were compared at clinical intraocular pressure. The magnitude of elastic weakening of the residual corneal bed required to maximize the agreement with clinical axial power was calculated and compared with the changes in ocular response analyzer (ORA) measurements. The models produced curvature maps and spherical aberrations equivalent to in vivo measurements. For the preoperative property values used in this study, predicted elastic weakening with LASIK was as high as 55% for a radially uniform model of residual corneal weakening and 65% at the point of maximum ablation in a spatially varying model of weakening. Reductions in ORA variables were also observed. A patient-specific FEM of corneal refractive surgery is presented, which allows the estimation of surgically induced changes in corneal elastic properties. Significant elastic weakening after LASIK was required to replicate clinical topographic outcomes in this two-eye pilot study.

Full text

Abhijit Sinha Roy
Cole Eye Institute,
Cleveland Clinic,
Cleveland, OH 44195
William J. Dupps, Jr.1
Cole Eye Institute,
Department of Biomedical Engineering,
and Transplant Center,
Surgery Institute,
Cleveland Clinic,
Cleveland, OH;
Department of Biomedical Engineering,
Case Western Reserve University,
Cleveland, OH 44195
e-mail: bjdupps@sbcglobal.net
Patient-Specific Modeling of
Corneal Refractive Surgery
Outcomes and Inverse Estimation
of Elastic Property Changes
The purpose of this study is to develop a 3D patient-specific finite element model (FEM)
of the cornea and sclera to compare predicted and in vivo refractive outcomes and to
estimate the corneal elastic property changes associated with each procedure. Both eyes
of a patient who underwent laser-assisted in situ keratomileusis (LASIK) for myopic
astigmatism were modeled. Pre- and postoperative Scheimpflug anterior and posterior
corneal elevation maps were imported into a 3D corneo-scleral FEM with an unre-
strained limbus. Preoperative corneal hyperelastic properties were chosen to account for
meridional anisotropy. Inverse FEM was used to determine the undeformed corneal state
that produced
⬍
0.1% error in anterior elevation between simulated and in vivo preop-
erative geometries. Case-specific 3D aspheric ablation profiles were simulated, and cor-
neal topography and spherical aberration were compared at clinical intraocular pres-
sure. The magnitude of elastic weakening of the residual corneal bed required to
maximize the agreement with clinical axial power was calculated and compared with the
changes in ocular response analyzer (ORA) measurements. The models produced curva-
ture maps and spherical aberrations equivalent to in vivo measurements. For the preop-
erative property values used in this study, predicted elastic weakening with LASIK was as
high as 55% for a radially uniform model of residual corneal weakening and 65% at the
point of maximum ablation in a spatially varying model of weakening. Reductions in ORA
variables were also observed. A patient-specific FEM of corneal refractive surgery is
presented, which allows the estimation of surgically induced changes in corneal elastic
properties. Significant elastic weakening after LASIK was required to replicate clinical
topographic outcomes in this two-eye pilot study. 关DOI: 10.1115/1.4002934兴
Keywords: cornea, computational model, laser-assisted in situ keratomileusis (LASIK),
hysteresis, biomechanics
1 Introduction
Laser-assisted in situ keratomileusis 共LASIK兲, the most com-
monly performed refractive surgery, involves the creation of a
lamellar flap followed by patterned photoablation of the underly-
ing stroma. In a recent large-scale review of LASIK outcomes
over a decade-long study period, most patients 共95%兲were highly
satisfied with their outcome. However, 5% expressed dissatisfac-
tion with the surgical outcome based on the low quality of life
scores that were attributed, in part to refractive regression, poor
night vision, and residual refractive error 关1兴. Modern excimer
laser systems provide options for wavefront-guided ablation pro-
files that, along with other technological advances such as eye
tracking, have reduced the tendency toward the induction of
higher order aberrations 共HOAs兲compared with previous conven-
tional LASIK technologies 关2–5兴. However, the predictability of
spherocylindrical refractive outcomes and the induction of HOAs
are persistent multifactorial problems that cannot be completely
explained by nonidealities in the laser-tissue interaction 关6–10兴,
nor can they be fully addressed with ablation profiles based solely
on preoperative wavefront data 关4,5,8兴.
Alterations in the biomechanical state of the cornea have been
proposed to affect the optical outcome of LASIK 关7–9兴. Concep-
tual biomechanical models presume a reduction in the material
strength of the cornea, owing to the combined effects of photoa-
blative lamellar disruption and flap creation 关7,8兴, but the magni-
tude of elastic weakening remains unknown. Quantification of this
weakening is important not only in relation to the unintended
biomechanically mediated spherocylindrical and higher order op-
tical effects 关3–5,11,12兴but also because of a presumed role in the
pathogenesis of postsurgical corneal ectasia.
Previously, we introduced a whole-eye 2D finite element model
共FEM兲and evaluated the sensitivity of simulated LASIK out-
comes to preoperative corneal hyperelastic properties 关13兴. The
model, like others before it, made no allowance for a reduction in
intrinsic material properties after surgery. Most reported FE mod-
els of the preoperative and postoperative cornea have incorporated
this simplification and others that limit their ability to account for
clincially relevant astigmatic optical effects and higher order ab-
errations. These include analytical surfaces rather than clinically
derived topographies and 2D models that do not account for 3D
corneal asymmetry and asphericity 关13–17兴. Furthermore, ablation
profiles in previous studies 关15兴have been derived from the origi-
nal spherical Munnerlyn equation 关18兴without accounting for as-
tigmatic patterns or aspheric profiles in modern lasers 关19,20兴.
Another limitation of some models is the use of a fixed or re-
strained limbus boundary condition, which may lead to corneal
deformations that are inconsistent with in vivo behavior 关21–23兴.
Thus, an unrestrained limbus using a whole-eye model 关13兴or a
1Corresponding author.
Contributed by the Bioengineering Division of ASME for publication in the JOUR-
NAL OF BIOMECHANICAL ENGINEERING. Manuscript received February 11, 2010; final
manuscript received October 11, 2010; accepted manuscript posted November 2,
2010; published online December 22, 2010. Assoc. Editor: Victor H. Barocas.
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comparable corneoscleral model 关15兴is preferable in predictive
FEMs for approximating in vivo surgical outcomes such as
spherocylindrical refractive power and HOA.
The aims of this study were to 共1兲develop a computationally
efficient 3D in vivo 共patient-specific兲corneoscleral model using
clinical corneal geometry, 共2兲to compare computational corneal
power predictions to 1 week in vivo surgical outcomes of myopic
LASIK in a two-eye clinical pilot study, and 共3兲to estimate cor-
neal elastic property changes associated with each procedure. A
secondary aim was to assess the sensitivity of the model to two
different schemes for corneal elastic property change within the
residual bed of the treated zone: a uniform reduction and a non-
uniform reduction as a function of local ablation depth.
2 Methods
2.1 Geometry. The right and left eyes of a 35 year old female
patient who underwent LASIK for myopia with astigmatism at the
Cleveland Clinic Cole Eye Institute were investigated retrospec-
tively under an Institutional Review Board 共IRB兲approval for
chart review research 共Cleveland Clinic IRB No. 07-305兲. Preop-
erative and postoperative clinical characteristics are provided in
Table 1. The 3D model of the cornea was constructed using to-
mographic data from a commercial anterior segment imaging sys-
tem 共Pentacam, Oculus Optikgeräte GmbH, Germany兲. The x,y,
and zcoordinates from the elevation maps of the anterior and
posterior surfaces were interpolated using orthogonal Zernike
polynomials up to the sixth order, having 28 terms and with a 5
mm normalization radius. The root-mean-square errors, defined as
the square root of the mean of the sum of the difference between
the in vivo and Zernike predictions, for the elevation data were
1.65
␮
m and 1.32
␮
m for the right and left eyes, respectively.
The interpolated elevation data 共x,y, and z兲were used to obtain a
3D surface using a commercial computer aided drafting 共CAD兲
package 共PROENGINEER WILDFIRE, PTC, Needham, MD兲. Each
point along a radius was connected by a cubic spline to form a
curvilinear edge in 3D. Multiple edges that form the shape of a
corneal surface were then blended to obtain a 3D surface for the
anterior and posterior cornea. These surfaces 共anterior and poste-
rior兲were then joined at the limbus to form a 3D solid, represent-
ing the in vivo pre-LASIK cornea. To simulate an unrestrained
limbus, a scleral shell was extended from the cornea to an axial
length of 3.5 mm. The posterior borders of the sclera were re-
strained completely. The posterior surfaces of the cornea and
sclera were loaded with IOPcc 共Table 1兲obtained from the ocular
response analyzer 共ORA兲.IOP
cc is the cornea compensated in-
traocular pressure and is considered to be less sensitive to changes
in the corneal biomechanical properties and thickness after
LASIK compared with the Goldmann applanation 关24–28兴.
In the clinical setting, corneal topography is measured at a spe-
cific intraocular pressure 共IOP兲and is distinct from the unloaded
shape that would be obtained at an IOP of 0 mm Hg. To solve for
the undeformed state, a custom inverse model was developed us-
ing the commercial finite element 共FE兲analysis package ABAQUS
共Simulia Inc.兲and PYTHON scripting language. In the inverse
model, initial estimates of the unloaded shapes of the cornea and
sclera were loaded to clinical IOP and then compared with the in
vivo shape. The coordinates of the unloaded shape was then cor-
rected based on the difference between the coordinates of the in
vivo geometry and the FEM prediction. The resulting unloaded
shape was then loaded again to the same IOP. This procedure was
repeated until a user-specified tolerance of 0.1% 共the difference
between the coordinates of two FEM results from successive
simulations of loading the corneo-scleral model from zero to in
vivo IOP兲was achieved. The mesh for each model consisted of
linear, eight node 3D hexahedral elements.
2.2 Material Properties. The cornea is an anisotropic tissue
that shows stress stiffening at higher strains due to a complex
collagen fibrillar distribution. Fibril-oriented material properties
have been used in some recent FEM studies 关14,16,17兴, while
orthotropic linear elastic material properties have been used in
others 关15兴. In this study, a spatially dependent, hyperelastic ma-
terial property formulation has been used to simulate the elastic
properties of the cornea. Elsheikh et al. 关29兴measured corneal
elastic properties along the horizontal, oblique, and vertical me-
ridians of ex vivo human corneas and observed that the vertical
and oblique meridian were most and least stiff, respectively 共Fig.
1共a兲兲. While Elsheikh et al. 关29兴used uniaxial strip testing to
quantify properties along meridians, similar meridional differ-
ences were demonstrated experimentally by Dupps et al. 关30兴,
using a validated nondestructive ultrasound based method of wave
speed measurement 关31兴in intact corneas with physiologic whole-
globe boundary conditions. For the purposes of the current study,
each of the experimentally derived stress versus strain curves 共Fig.
1共a兲兲 were fit to a reduced polynomial material model, W
=C10共I1−3兲+C20共I1−3兲2, where W is the strain energy potential
and I1is the strain invariant. C10 and C20 are hyperelastic con-
stants obtained from the fitting of the experimental stress versus
strain data and were determined for each of the three meridia. The
magnitude of C10 and C20 along the other meridia were then in-
terpolated using sinusoidal functions 共one each for C10 and C20兲,
C=a⫻sin2共theta兲+b⫻sin共theta兲+c, where theta is the meridian
and a, b, and c are the constants obtained from the regression. At
theta=0 共the horizontal meridian兲, C obtained from the above
function will yield the same hyperelastic constant values C10 and
C20 obtained for the horizontal meridian ex vivo data. A sinusoidal
function was chosen to ensure a smooth gradient in C10 and C20
across all meridians of the cornea. The reduced polynomial form
and the sinusoidal function were implemented using user-defined
subroutines in ABAQUS. The sclera was also modeled using the
reduced polynomial form as an isotropic and hyperelastic mate-
rial. Scleral elastic properties were assumed to be three times the
stiffness of the vertical meridian of the cornea 关13兴.
2.3 Ablation Profile. The patient underwent wavefront-
optimized 共Allegretto Wave®Eye-Q Excimer Laser System, Al-
con Laboratories, Fort Worth, TX兲ablation for myopia with astig-
matism in both eyes. Details of the ablation parameters and flap
thickness are provided in Table 2. The mathematical formulation
Table 1 Preoperative and post-LASIK characteristics of the
right and left eyes
Right eye Left eye
Preoperative Post-LASIK Preoperative Post-LASIK
Manifest
refraction 共D兲
−5.00+0.75
⫻120 deg
−0.25+0.50
⫻20 deg
−5.50+0.50
⫻87 deg
−0.25+0.50
⫻165 deg
Best corrected
visual acuity 20/20 20/25 20/20 20/25
CCT
共
␮
m兲
603 - 604 -
SimK 共D兲41.25/42.87
at 112 deg
38.50/39.00
at 160 deg
42.37/43.50
at 78 deg
38.00/38.50
at 90 deg
CH
共mm Hg兲
10.7 7.5 10.5 8.3
CRF
共mm Hg兲
10.5 7.8 10.3 8.0
IOPg
共mm Hg兲
15.3 15.1 15.3 14.5
IOPcc
共mm Hg兲
15 18.7 14.8 16.5
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for calculating the ablation depth as a function of the radius and
meridian has been described for an aspheric profile 关19兴and was
used to generate the initial ablation profile for the programmed
treatments. The ablation depth function was then modified to ac-
count for loss of ablation efficiency from a non-normal incidence
of the laser beam away from the center of the cornea 关9兴. The
ablation depth in millimeters was subtracted from the correspond-
ing zcoordinate of the anterior surface of the unloaded corneal
shape obtained from inverse modeling, as described previously.
The resulting unloaded shape was then loaded to the clinically
measured IOPcc obtained 1 week after surgery, and the post-
LASIK geometry was calculated in ABAQUS and compared with 1
week clinical geometry outcomes.
2.4 Estimation of Magnitude of Elastic Weakening After
LASIK. To model the effect of tissue removal on corneal material
properties in post-LASIK cornea, the elastic properties 共C10 and
C20兲were reduced through the full thickness encompassed by the
ablation zone diameter 共including the region of the flap and the
residual stromal bed兲to a magnitude that produced the best three-
dimensional match to the in vivo postoperative anterior surface
axial power. It was assumed that the elastic properties of the flap
region were the same as the elastic properties of the postoperative
residual corneal bed. The parameter used for comparison was the
ratio of axial power calculated by FEM to the axial power in vivo.
Thus, the ratio would be 1 if axial power calculated by FEM
equals the axial power in vivo at a given point on the anterior
surface. It is most likely that the elastic weakening following
LASIK is nonhomogenous 共spatially varying兲, though the exact
distribution of variation is not known. Therefore, two methods
were adopted to model the weakened cornea post-LASIK:
共i兲Uniform reduction. In the central 6.5 mm optical zone, the
hyperelastic constants 共C10 and C20兲were reduced through
the postoperative corneal thickness by a constant factor F
in all meridians. Therefore, the post-LASIK elastic prop-
erties of the cornea were modeled as C10-post =C10-pre⫻F.
This approach assumed negligible elastic property changes
in the transition zone peripheral to the optical zone.
共ii兲Radially nonuniform reduction. This method assumes that
a greater ablation depth produces a greater local reduction
in elastic property coefficients. In the central 9 mm zone
encompassing both the optical and transition zones, elastic
properties were reduced in proportion to the ablation depth
at each point. The factor that was multiplied with the pre-
operative elastic property coefficients to determine the
postoperative local coefficients was given by F=共⌬E−1兲
⫻共t/CAD兲+1, where CAD is the central ablation depth, t
is the ablation depth at a particular point, and ⌬E is the
fraction of the original elastic coefficients to which the
postoperative elastic coefficients were reduced at the cen-
ter of ablation. With this construct, the maximum elastic
weakening was located at the center of the ablation zone
and decreased progressively toward the peripheral cornea.
In the above methods, the change in elastic properties was cal-
culated based on pre-LASIK properties obtained by Elsheikh et al.
关29兴who used specimens from older patients. Since the true elas-
tic properties of the eyes used in the present study were unknown
and the patients in this study were younger than those tested by
Elsheikh et al. 关29兴, a simple sensitivity analysis was performed
by decreasing the elastic properties of the cornea and sclera by
25% from the baseline values and maintaining a scleral-corneal
elasticity ratio of 3:1.
Four replicate ORA measurements for each eye and at each
time point were obtained before and after LASIK. Estimated re-
ductions in elastic properties using the uniform and nonuniform
reduction methods were compared with changes in corneal hys-
teresis 共CH兲and corneal resistance factor 共CRF兲.
2.5 Computation of Spherical Aberration of the Anterior
Surface of Cornea. The spherical aberration of the in vivo and
FEM anterior corneal surfaces was compared across a central 10
mm diameter zone to include central, paracentral, and peripheral
corneal effects. A diameter of 10 mm was chosen to include the
entire ablation zone of a 9 mm diameter and to avoid numerical
noise associated with Zernike polynomials at the edges of the
analysis zone. The mathematical formulation to calculate the ab-
Fig. 1 „a…Stress-strain relationships along three meridia ob-
tained from the experimental data of Elsheikh et al. †29‡and „b…
a 3D corneoscleral model with finite element mesh and a super-
imposed map of displacements resulting from loading the un-
deformed pre-LASIK model „determined from inverse FEM…to
clinically measured preoperative IOP. The paracentral and pe-
ripheral cornea exhibits greater displacements than the central
zone „in mm….
Table 2 LASIK treatment parameters for the left and right eyes
OD OS
Programed correction 共D兲
−4.75+0.75
⫻120 deg
−5.25+0.50
⫻87 deg
Optical zone/total ablation
zone diameter 共mm兲
6.5/9 6.5/9
Programed flap thickness 共
␮
m兲100 100
Flap thickness by ultrasound
subtraction pachymetry 共
␮
m兲
129 115
Programed central ablation
depth 共
␮
m兲
71 78
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erration at any point on the anterior surface of the cornea was
previously described 关32兴. The aberrations of the entire 10 mm
zone were analyzed using Zernike polynomials. The fourth order
term 共C40 using the double index scheme兲in the Zernike analysis
was designated as the spherical aberration 关32兴.
3 Results
Comparisons of in vivo and FEM optical predictions are pre-
sented as ratios of the axial power predicted by FEM to the axial
power in vivo at the same location on the anterior cornea surface.
In Fig. 1共b兲, a displacement contour is shown along with the finite
element mesh for the right eye when stressed with IOPcc from the
unloaded to the in vivo preoperative state. The central corneal
displacement 共⬃33
␮
m兲was much lower than the displacement
of the paracentral 共⬃55
␮
m兲and peripheral cornea. This dis-
placement distribution is consistent with results from a previous
whole-eye analysis 关13兴. Fig. 2 shows the maximum principal
strain when the model is loaded with physiologic IOP. The maxi-
mum and minimum strains were localized along the oblique and
vertical meridia, respectively. At the limbus and surrounding
sclera, the strain distribution was more circumferentially oriented
due to the assumed isotropy in these regions. This strain pattern
was consistent with past studies that have used preferred collagen
fibril orientation material models 关14,16,17兴. The means of repli-
cate CH, CRF, IOPg共Goldmann-equivalent IOP兲, and IOPcc mea-
surements are reported in Table 1 for the pre- and post-LASIK
states.
3.1 Right Eye Geometric Results. The anterior surface axial
power measured clinically prior to LASIK is shown in Fig. 3共a兲.
Figure 3共b兲depicts the ratio of the preoperative axial power de-
rived from inverse FEM to the preoperative in vivo axial power.
In Fig. 3共c兲, the ratio map describes the agreement of the FEM-
derived axial power to the in vivo axial power 1 week after
LASIK, with the assumption that no change in elastic properties
occurs. Figure 3共d兲shows a similar plot of the ratio, with the
assumption that LASIK causes a uniform reduction in elastic
properties to 0.45 times their preoperative value in the central 6.5
mm optical zone. Table 3 summarizes each model’s fit to the in
vivo data as the mean ratio⫾sd for the central 3 mm zone, the
3–6 mm diameter zone, and the 6–10 mm diameter zone. The
mean agreement of the preoperative inverse model prediction to
the clinical axial power map is within 0.5%, with a standard de-
viation of less than 0.15%. This implies that the no-load 共zero
IOP兲configuration of the corneo-scleral model predicted by the
inverse model, when loaded to physiological IOP, produces a cor-
neal curvature equivalent to patient-specific in vivo
measurements.
In the post-LASIK case, with the assumption of unchanged
corneal elastic properties, the mean value of the ratio was
0.9773⫾0.0145 共sd兲in the central 3 mm diameter zone. Recall
that the post-LASIK FEM geometry is derived by the application
of the simulated case-specific ablation to the undeformed model
and loading of the resulting geometry to the clinical IOP measured
after LASIK. The FEM predicted a slightly flatter cornea 共ratio
⬍1兲than the in vivo post-LASIK outcome. When the elastic
properties in the optical zone were uniformly reduced to a level
that minimized model error 共0.45 times the preoperative value兲,
the average ratio in the central 3 mm diameter zone increased to
1.0012⫾0.015. In the paracentral zone 共3–6 mm兲, there was an
improvement in the mean value of the ratio from 0.9888 to
1.0092, with the reduction in elastic properties. The results indi-
cate that a 55% reduction 共共1−0.45兲⫻100兲in corneal elastic
properties within the diameter of the optical zone is required to
reproduce the in vivo topographic outcome.
In the nonuniform reduction method, it was assumed that the
magnitude of elastic property change is a function of ablation
depth, with the maximum reduction at the center of a myopic
ablation. The optimization routine described in Sec. 2 resulted in a
nonuniform elastic property reduction model with ⌬E=0.35 共rep-
resenting a 65% central reduction in elastic properties relative to
preoperative properties兲for the right eye 共Fig. 5共a兲兲. The distribu-
tion of elastic property reduction was defined by the ablation pro-
file. For ⌬E=0.35, the mean value of the ratio in the central 3 mm
zone was 1.0006⫾0.014 共mapped in Fig. 5共c兲and summarized by
zone in Table 3兲. Mean values of the ratio in the other zones were
similar to those obtained with a uniform 55% reduction in elastic
properties. Table 3 also lists the values of the ratios⫾sd for the
conditions where the cornea and sclera elastic properties were
decreased by 25%. Optimization yielded a weakening by 50% and
⌬E=0.40, with the uniform and nonuniform model, respectively,
having only a 5% difference from the original material property
assumption.
3.2 Left Eye Geometric Results. Figure 4共a兲shows the in
vivo axial power before LASIK. Figure 4共b兲depicts the ratio of
the preoperative axial power derived from the inverse FEM to the
preoperative in vivo axial power. In Fig. 4共c兲, the ratio map de-
scribes the agreement of the FEM-derived axial power to the in
vivo axial power 1 week after LASIK, with the assumption that
there are no changes in elastic properties. Figure 4共d兲shows a
similar plot of the ratio, with the assumption that LASIK reduces
the elastic properties of the central 6.5 mm optical zone to 0.6
times the preoperative values. Table 4 summarizes each model’s
fit to the in vivo data as the mean ratio⫾sd for the central 3 mm
zone, the 3–6 mm diameter zone, and the 6–10 mm diameter
zone. The mean agreement of the preoperative inverse model pre-
diction to the clinical axial power map is within 0.5%, with a
standard deviation of less than 0.2%. As with the right eye, the
zero-load configuration of the corneo-scleral model predicted by
the inverse FEM routine, when loaded to physiological IOP, pro-
duced a corneal curvature equivalent to the in vivo measurements.
In the post-LASIK case, with the assumption of unchanged
elastic properties, the mean value of the ratio was 0.9887⫾0.021
in the central 3 mm diameter zone. As in the right eye, the FEM
overestimated the flattening effect of LASIK when corneal elastic
properties were assumed to be unaffected by the surgery. When
the elastic properties in the optical zone were uniformly reduced
to a level that minimized model error 共0.6 times the preoperative
values兲, the average ratio in the central 3 mm diameter zone in-
creased to 1.0002⫾0.02. In the paracentral zone 共3–6 mm兲, there
was an improvement in the mean value of the ratio from 0.9888 to
1.0092, with this reduction in stiffness, and differences of less
Fig. 2 Maximum principal strain under applied IOP. The black
dashed circle demarcates the corneal border. Differences in the
strain are due to encoding experimentally derived meridional
variations in hyperelastic properties of the cornea. Peak strains
are predicted by the FEM along the oblique meridia.
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Table 3 Average ratios of FEM-derived/in vivo axial power in different zones of the right cornea before and after LASIK, with
assumptions of no surgically induced reduction, uniform reduction throughout the optical zone, and ablation-depth dependent
reduction throughout the optical and transition zones. The table also lists the values of the ratios when the elastic properties of
the cornea and sclera were reduced by 25%.
Analysis
diameter
共mm兲
Average values of ratio⫾sd Average values of ratio⫾sd
Pre-LASIK Post-LASIK
Post-LASIK
cornea and sclera: 25% weaker
Inverse
model
No elastic
property
reduction
Uniform
elastic
property
reduction
共55%兲
Nonuniform
elastic
property
reduction
共⌬E=0.35兲
No elastic
property
reduction
Uniform
elastic
property
reduction
共50%兲
Nonuniform
elastic
property
reduction
⌬E=0.40
0–3 1.0041⫾0.0006 0.9773 ⫾0.0145 1.0012 ⫾0.0150 1.0006⫾0.0141 0.9831⫾0.0143 1.0074 ⫾0.0147 1.004 ⫾0.0138
3–6 1.0028 ⫾0.0006 0.9888⫾0.0154 1.0092⫾0.0153 1.0055 ⫾0.0145 0.9941 ⫾0.015 1.0151⫾0.015 1.0119 ⫾0.0143
6–10 0.9997 ⫾0.0013 1.0258⫾0.0256 1.0348⫾0.0219 1.0318 ⫾0.0200 1.0306 ⫾0.0256 1.0396⫾0.0220 1.0368 ⫾0.0224
Fig. 3 „a…Clinical preoperative anterior axial power map of the right eye in diopters, „b…map of the ratio of the preoperative
axial power predicted from the inverse FEM to the in vivo axial power for the right eye, „c…maps of the ratio „post-LASIK
FEM/post-LASIK in vivo…of the postoperative anterior axial power assuming there was no change in elastic properties after
LASIK, and „d…uniformly reduced elastic properties throughout the optical zone „diameter of 6.5 mm…after LASIK in the
right eye
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Table 4 Average ratios of FEM-derived/in vivo axial power in different zones of the left cornea before and after LASIK, with
assumptions of no surgically induced reduction, uniform reduction throughout the optical zone, and ablation-depth dependent
reduction throughout the optical and transition zones. The table also lists the values of the ratios when the elastic properties of
the cornea and sclera were reduced by 25%.
Analysis
diameter
共mm兲
Average values of ratio⫾sd Average values of ratio⫾sd
Pre-LASIK Post-LASIK
Post-LASIK
cornea and sclera: 25% weaker
Inverse
model
No elastic
property
reduction
Uniform
elastic
property
reduction
共40%兲
Nonuniform
elastic
property
reduction
共⌬E=0.50兲
No elastic
property
reduction
Uniform
elastic
property
reduction
共40%兲
Nonuniform
elastic
property
reduction
⌬E=0.50
0–3 1.0044⫾0.0011 0.9887 ⫾0.0205 1.0002 ⫾0.0210 1.0005⫾0.0205 0.9891⫾0.0201 1.0047 ⫾0.0208 1.0013 ⫾0.0202
3–6 1.003 ⫾0.0008 0.9983⫾0.0176 1.0083⫾0.0177 1.0083 ⫾0.0173 0.9992 ⫾0.0173 1.0129⫾0.0175 1.0102 ⫾0.0169
6–10 0.9995 ⫾0.0017 1.0326⫾0.0210 1.0375⫾0.0213 1.0363 ⫾0.0211 1.0362 ⫾0.0244 1.0400⫾0.0220 1.036 ⫾0.0218
Fig. 4 „a…Clinical preoperative anterior axial power map of the left eye in diopters, „b…map of the ratio of the preoperative
axial power predicted from the inverse FEM model to the in vivo axial power for the left eye, „c…maps of the ratio „post-
LASIK FEM/post-LASIK in vivo…of the postoperative anterior axial power assuming there was no change in elastic proper-
ties after LASIK, and „d…uniformly reduced elastic properties throughout the optical zone „diameter of 6.5 mm…after LASIK
in the left eye
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than 1% were seen in the peripheral zones between the no-change
and uniform-change stiffness models. The results indicate that in
the left eye, a 40% reduction 共共1−0.60兲⫻100兲in corneal elastic
properties within the diameter of the optical zone was required to
reproduce the clinical topographic changes.
With a nonuniform post-LASIK stiffness reduction, the optimi-
zation produced ⌬E=0.5, representing a 50% maximum central
reduction in elastic properties relative to preoperative properties
for the right eye 共Fig. 5共b兲兲. The mean value of the axial power
ratio in the central 3 mm zone was 1.0005⫾0.0205 共mapped in
Fig. 5共d兲and summarized by zone in Table 4兲. Mean values of the
ratio in the other zones were similar to those obtained with a
uniform 40% elastic property reduction. Table 4 also lists the val-
ues of the ratios⫾sd for the conditions where the cornea and
sclera elastic properties were decreased by 25%. Optimization of
the FEM prediction to the post-LASIK outcome yielded a weak-
ening by 40% and ⌬E=0.50 with the uniform and nonuniform
models, respectively. Unlike the right eye, altering the cornea and
sclera properties did not alter the estimate of corneal weakening
post-LASIK.
3.3 Spherical Aberration: Pre- and Post-LASIK. Spherical
aberration of the cornea for both pre- and post-LASIK was evalu-
ated over a diameter of 10 mm. Figure 6共a兲shows the spherical
aberration in right eye before and after LASIK. LASIK caused an
increase in spherical aberration from 0.011
␮
m to 0.017
␮
min
vivo. The FEM predicted an increase in spherical aberration from
0.012
␮
m to 0.018
␮
m when there was no change in elastic
properties of the cornea after LASIK. With the nonuniform reduc-
tion model, the post-LASIK aberration predicted by the FEM in-
creased slightly to 0.019
␮
m. Figure 6共b兲shows the spherical
aberration in the left eye before and after LASIK. Similar results
for spherical aberration were obtained for the left eye. The change
in spherical aberration predicted by the FEM with ablation-depth
dependent reductions in elastic properties was similar to that pre-
dicted by a uniform reduction in elastic properties. It should be
noted that in all simulations, the depth of the tissue is removed
and the flap thickness itself undergoes a negligible change as the
cornea is loaded from zero to physiological IOP in the FEM and
therefore does not contribute significantly to curvature change.
4 Discussion
In this study, we presented a novel set of FEM methods for a
patient-specific computational simulation of corneal refractive
surgery and performed a pilot comparison of model predictions to
in vivo LASIK outcomes. The model allows an analysis of the
biomechanical changes associated with a specific LASIK treat-
Fig. 5 Patterns of radially nonuniform reduction in elastic properties in „a…the right eye „⌬E=0.35…and „b…the left eye
„⌬E=0.5…, where ⌬E represents the factor multiplied with the preoperative property values to give the postoperative values
and was determined for each case through an optimization process designed to maximize agreement between clinical and
simulated postoperative anterior surface axial powers. Ratio of the predicted to actual anterior axial power „post-LASIK
FEM/post-LASIK in vivo…, assuming an ablation-depth dependent reduction in elastic properties within the ablation zone for
„c…the right eye and „d…the left eye.
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ment plan and their impact on the accuracy of the postoperative
optical result. In addition, the model was used to estimate the
magnitude of corneal elastic property reductions based on clinical
topographic changes between the preoperative and 1 week post-
operative examinations. In both eyes, a reduction in corneal elas-
tic properties or weakening of the cornea from the preoperative
state resulted in more accurate axial power estimates by the FEM
compared with in vivo. These results suggest that the anterior
corneal flattening associated with a myopic correction is overesti-
mated if elastic properties are assumed to be as high after surgery
as before surgery; conversely, weakening of the cornea, which
favors less biomechanical flattening by allowing more forward-
directed central corneal displacement after LASIK, is required to
produce the best match to the clinical response. The tendency
toward overcorrection due to excessive biomechanical flattening
with stiff corneas and undercorrection in corneas with lower elas-
tic properties is consistent with the results from our previous
whole-eye model of LASIK 关13兴and supports the hypothesis that
a myopic undercorrection may be a clinical marker of weaker
corneal elastic properties.
The current approach improves upon previous FEM-based stud-
ies that have not incorporated patient-specific clinical measure-
ments of IOP and corneal geometry 共including pan-corneal thick-
ness data兲, have not included the corneo-scleral limbus, have not
modeled the effects of corneal meridional elastic property varia-
tion, have not accounted for nonidealities in photoablative effi-
ciency during ablation, or have not allowed for a change in cor-
neal elastic properties after LASIK. This study combines the
relevant patient-specific clinical information into a model in
which a specific surgical algorithm can be simulated to predict the
postoperative outcome and to determine the magnitude of elastic
property change within the treatment zone, which is required to
achieve the best fit to actual clinical outcome. The ablation depth
dependent reduction yielded similar optical outcomes after
LASIK compared with a uniform reduction method 共Tables 3 and
4兲. It is likely that the changes in elastic properties after LASIK
are heterogeneous since ablation depth varies spatially according
to the ablation profile. The ability of the nonuniform elastic prop-
erty reduction method to replicate in vivo outcomes in this pilot
study strongly supports this mathematical approach to expressing
spatial elastic property change as a function of LASIK ablation
parameters in future studies.
Previous FEM studies have not attempted to quantify patient-
specific changes in corneal elastic properties after refractive sur-
gery with ORA. Recent clinical studies with the ORA have shown
that both CH and CRF decrease after LASIK 关33–37兴. While CH
and CRF capture aspects of ocular biomechanical behavior that
are not simple equivalents of elastic modulus 关38,39兴, these stud-
ies support the notion of decreased corneal elastic properties after
LASIK. In this study, despite similar attempted refractive correc-
tions and similar preoperative CH and CRF values in both eyes,
CH decreased by 30% in the right eye and 21% in the left, and
CRF decreased by 26% in the right eye and 22% in the left. The
FEM results also suggested an asymmetric material property re-
sponse with a greater weakening of the right eye 共55%兲than the
left 共40%兲, with the baseline pre-LASIK properties. Both findings
suggest that the right eye underwent a greater amount of weaken-
ing. Moderate to weak correlations have been found between CH
and CRF and the change in central corneal thickness 共CCT兲
关26,34,35兴or ablation volume 关40兴in myopic surgery. Deeper
ablations and thicker flaps presumably result in greater reductions
in corneal elastic properties due to the severance of a proportion-
ate number of collagen lamellae 关8,41兴. Though symmetric
100
␮
m thick flaps were attempted with the femtosecond laser,
an unintended 14
␮
m difference in ultrasonically measured cen-
tral flap thickness 共Table 2兲between the eyes could be a factor
influencing the differential response to similar LASIK procedures.
The role of CH and CRF as surrogates for classical elastic prop-
erties in computational modeling is unclear and requires further
investigation.
This study incorporates spatially varying elastic structures of
the cornea using data from ex vivo uniaxial tests at different me-
ridians. Dupps et al. used a nondestructive method to measure the
surface ultrasound wave speed along the horizontal and vertical
meridians of the cornea in whole-eye human donor globes 关30兴.
The wave speed was significantly higher along the vertical merid-
ian than the horizontal meridian. This finding is similar to the
uniaxial test observations reported by Elsheikh et al. 关29兴but with
an in situ corneal configuration with physiologic boundary condi-
tions, stressed with physiologic IOP, and subject to biaxial stretch-
ing. The same surface wave technique was subsequently validated
to be nearly linearly proportional to the elastic modulus measured
by extensiometry 关31兴. While these observations do not eliminate
the possibility of error in the translation of uniaxial test to in situ
behavior, they do suggest that the spatial variation of corneal elas-
tic properties is preserved to some degree under both experimental
conditions and that the modeling error associated with the ex-
trapolation is likely to be lower as a result. Further, cohesive ten-
sile strength measurements in human corneas after flap creation
have suggested a mean reduction in one analog of corneal elastic
strength of 72%, though variability was noted 关42兴. Our estimates
of material property reduction in this model are similar, though
perhaps lower due to the fact that flap thicknesses in the previous
study 关42兴were greater than the femtosecond laser flaps modeled
here. This provides an experimental validation of the magnitude
of elastic property change estimated by the present FEMs. The
sensitivity analysis where the elastic properties of the cornea and
Fig. 6 Comparison of the corneal first-surface spherical aber-
ration calculated from in vivo measurement and FEM prediction
for „a…the right eye and „b…the left eye before and after LASIK:
FEM preoperative, FEM1postoperative „unchanged elastic
properties…, FEM2postoperative „uniform reduction in elastic
properties…, and FEM3postoperative „nonuniform reduction in
elastic properties…
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sclera were reduced keeping the cornea-sclera elasticity ratio con-
stant suggests that at least in these two cases, the model is rela-
tively insensitive to the pre-LASIK hyperelastic property esti-
mates. The model still predicted a greater degree of weakening in
the right eye than the left when a lower range of elastic property
coefficients 共corneal and sclera properties reduced by 25%兲were
tested.
This study utilizes ex vivo measurements of spatial corneal
elastic properties to estimate the changes in corneal elastic prop-
erty in both eyes. This study also treats the effects of flap creation,
photoablative tissue removal, and wound healing as aggregate
biomechanical insults to the full thickness of the postoperative
cornea within the ablation zone diameter. In modeling this elastic
property change as a generalized zone of injury and repair without
a separate modeling of the flap, the changes in properties within
the flap, within the residual stromal bed, and at the flap interface
have been combined into a single parameter. Modeling the dis-
crete contributions of the flap and tissue to the overall elastic
property change is possible with the current model but would
require spatially resolved knowledge of the differential mechani-
cal property effects of LASIK within and deep to the flap. Con-
tinued development of in vivo techniques for characterizing cor-
neal material properties in three dimensions 关43兴will be important
for patient-specific surgical optimization and avoidance of ex vivo
estimates since relatively fine control of flap dimensions and ex-
cimer laser photoablation algorithms is possible. However, until
such data is available, a zonal injury approach to the problem
allows for optimization of a single material property parameter
and is adequate for retrospectively assessing the biomechanical
impact of LASIK and for producing accurate predictions of post-
operative optical outcomes with a model informed only by preop-
erative clinical measurements and the planned surgical algorithm.
Another limitation of this study is that the assumed corneal-scleral
relationship can affect the no-load geometry estimated by the in-
verse FEM procedure. Since in this study, the same no-load con-
figuration was used for pre- and post-LASIK 共with tissue removal
simulated in the post-LASIK case兲, it was possible to express the
change in elastic properties as a percentage value of the preopera-
tive elastic properties. While the results of cohesive tensile
strength testing 关42兴and the sensitivity analysis performed in this
study provide some support for the elastic property change esti-
mates in these simulations, future studies will need to assess the
sensitivity of these models to the range of variations in corneal-
scleral elastic properties likely to be encountered in patients
across a range of ages. Large scale modeling will also allow for
the study of the variation in elastic property changes across a wide
range of ablation parameters to better assess the impact of such
variables on the risk of corneal ectasia.
In addition to material properties and ablation profile, the IOP
may influence the shape of the cornea before and especially after
corneal surgery and is a potential limitation in patient-specific
modeling. The accuracy of the transcorneal IOP measurement is
uncertain due to reduced thickness, weakening of the cornea
关24–26兴, and differences in device measurement methods 关26–28兴.
Reduced thickness and weakening lead to underestimated IOP
关24兴. Studies have compared devices commonly used to measure
changes in IOP from pre- to postsurgery 关25–28兴. Goldmann IOP
tends to decrease after ablation due to reduced thickness and
weakening, whereas IOP measured by dynamic contour tonometry
共Ziemer Ophthalmic Systems AG, Switzerland兲is relatively unaf-
fected by thickness 关25–28兴. Because the IOPcc measured by the
ORA is only weakly correlated with CCT 关26,33兴and is relatively
unaffected by LASIK 关26,34兴, we used it as a practical proxy for
actual intraocular loading pressure before and after LASIK.
In summary, we present a method for patient-specific 3D com-
putational modeling of corneal refractive surgery that provides
representations of clinical corneal shape changes in LASIK and
allows inverse estimation of the surgical impact on corneal elastic
properties. The FEMs generated in this study take less than 10
min to solve and do not require significant computational re-
sources. Further improvements in meshing size and automation of
the 3D model generation can further reduce the solution time. The
accuracy of the models will depend on the quality of the clinical
geometry data supplied, and the results could therefore be device-
specific. While the sensitivity of model accuracy to this potential
error is unknown, the use of data obtained from the same instru-
ment before and after surgery minimizes potential confounding
effects. Ultimately, an accurate FE model of the corneoscleral
complex that can be populated with patient-specific clinical data
affords the opportunity to simulate and perhaps refine surgical
results in any corneal surgery that affects the cornea’s biome-
chanical state.
Acknowledgment
This study was supported in part by NIH Grant Nos.
K12RR023264 and 1KL2RR024990, Challenge and Unrestricted
Grants from Research to Prevent Blindness to the Department of
Ophthalmology of the Cleveland Clinic Lerner College of Medi-
cine of Case Western Reserve University, and the National Kera-
toconus Foundation/Discovery Eye Foundation. W.D. is a recipi-
ent of a Research to Prevent Blindness Career Development
Award.
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