Download Air Puff Induced Corneal Vibrations: Theoretical Simulations

BIOMECHANICS
Air Puff Induced Corneal Vibrations: Theoretical
Simulations and Clinical Observations
Zhaolong Han, PhD; Chen Tao, MD; Dai Zhou, PhD; Yong Sun, MD; Chuanqing Zhou, PhD;
Qiushi Ren, PhD; Cynthia J. Roberts, PhD
T
RESULTS: Numerical results showed that during the air
puff deformation, there would be vibrations along with
the corneal deformation, and the damping viscoelastic response of the cornea had the potential to reduce
the vibration amplitude. With consistent IOP, the overall vibration amplitude and inward motion depths were
smaller with a stiffer cornea.
he understanding of corneal biomechanical properties is important because they are associated with
corneal refractive surgeries and diagnosis and monitoring of some ocular diseases. In recent years, greater attention has been paid to the topic of corneal biomechanics.1-4
In vitro experimental studies such as stress–strain tensile
tests on corneal strips and inflation tests on the whole cornea
have shown the viscoelastic, hyperelastic, and anisotropic
features of both animal and human corneas.5-8 Electromagnetic radiation scanning approaches are able to demonstrate
how the collagen fibrils are spatially arranged within the
stroma tissue, which have a great impact on the corneal biomechanical performance.9 Meanwhile, the corneal biomechanical properties may vary between humans and animals,7
between people with different ages,7 between different ocular
diseases,10 and before and after clinical treatments.5 On the
other hand, the numerical simulation method is a powerful
methodology to study the mechanical behavior of the cornea.
For example, the finite element method has been successfully employed to carry out simulations such as keratoconic
corneas and response to ocular surgeries,11-20 showing great
potential in the prediction of corneal biomechanical and
refractive behaviors.
CONCLUSIONS: A kinematic viscoelastic model of the
cornea is presented to illustrate how the vibrations are
associated with factors such as corneal mass, viscoelasticity, and IOP. Also, the predicted corneal vibrations
during air puff deformation were confirmed by clinical
observation.
From the School of Biomedical Engineering (ZH, CT, YS, CZ) and School
of Naval Architecture, Ocean and Civil Engineering (ZH, DZ), Shanghai
Jiao Tong University, Shanghai, China; the Department of Ophthalmology,
The People’s Hospital of the Xinjiang Uygur Autonomous Region, Xinjiang,
China (YS); the Department of Biomedical Engineering, Peking University,
Beijing, China (QR); and the Department of Ophthalmology and Department
of Biomedical Engineering, The Ohio State University, Columbus, Ohio (CJR).
[J Refract Surg. 2014;30(3):208-213.]
Submitted: August 21, 2013; Accepted: December 3, 2013; Posted online:
February 28, 2014
ABSTRACT
PURPOSE: To investigate air puff induced corneal vibrations and their relationship to the intraocular pressure
(IOP), viscoelasticity, mass, and elasticity of the cornea
based on theoretical simulations and preliminary clinical
observations.
METHODS: To simulate the corneal movement during
air puff deformation, a kinematic viscoelastic corneal
model was developed involving the factors of corneal
mass, damping coefficient, elasticity, and IOP. Different
parameter values were taken to investigate how factors
would affect the corneal movements. Two clinical ocular instruments, CorVis ST (Oculus Optikgeräte GmbH,
Wetzlar, Germany) and the Ocular Response Analyzer
(ORA; Reichert, Inc., Buffalo, NY), were employed to
observe the corneal dynamical behaviors.
Supported by the National Basic Research Program of China (No. 2011CB707504,
and No. 2010CB933903), National Natural Science Foundation of China (No.
81171377, No. 11172174, No. 51078230, and No. 51278297), Program of
Shanghai Subject Chief Scientist (No. 13XD1402100), and SJTU Biomedical
Engineering Cross-Research Project (YG2011MS62).
Dr. Roberts is a consultant for Oculus Optikgeräte GmbH. The remaining
authors have no financial or proprietary interests in the materials presented
herein.
Correspondence: Chuanqing Zhou, PhD, School of Biomedical Engineering,
Shanghai Jiao Tong University, Shanghai 200240, China. E-mail: zhoucq@
sjtu.edu.cn
doi:10.3928/1081597X-20140212-02
208
Copyright © SLACK Incorporated
Corneal Vibrations/Han et al
In addition to studies that have focused on the static
state of the cornea as mentioned above, understanding
corneal biomechanics through a dynamic approach
is also important (eg, with vibration tonometers).21-25
In recent years, the Ocular Response Analyzer (ORA;
Reichert, Inc., Buffalo, NY), an ocular instrument that
can emit an air pulse directed to the corneal surface,
has been used to dynamically measure the intraocular
pressure (IOP) and investigate the corneal viscoelastic
response.26-30 Compared with the number of experimental studies, there is less theoretical work in these
aspects. Glass et al.31 developed a simplified model
to evaluate the corneal viscosity and elasticity in response to an air puff, but neither the nonlinearity of the
corneal elastic properties nor the effect of corneal mass
were taken into consideration. Elsheikh et al.32 used finite element analysis to simulate the corneal response
as a function of thickness, curvature, age, and true IOP.
Salimi et al.33 simulated the dynamic response of the
eye considering the fluid–structure interaction effect
by using the finite element method.
The current study aims at developing a numerical
model to investigate the dynamic responses of the cornea under air pulse pressure. This model includes the
factors of corneal mass, viscoelasticity, IOP, and the
external air pressure. Clinical observations are also
presented to support the numerical findings.
MATERIALS AND METHODS
Clinical Observation
Two ocular instruments, the ORA and the CorVis
ST (Oculus Optikgeräte GmbH, Wetzlar, Germany), a
new non-contact tonometer, were employed in the current study.
During measurement with the ORA, an air pulse is
delivered to the eye, which is in its initial state. Because
the air pressure on the cornea experiences an increasing
and then decreasing process, the cornea moves inward
and then backward to its initial curvature, passing
through the applanated state two times. The two pressures are recorded by an electro-optical collimation system, which can record the light signals reaching peak
values at the corneal applanation times when the cornea reaches the applanation position. The whole process lasts approximately 30 ms for each measurement.
The CorVis ST has the same working principle as
the ORA in that the corneal response to an air puff is assessed to determine IOP and corneal biomechanical parameters. However, the CorVis ST permits visualization
of the response of the cornea to an air impulse via a highspeed Scheimpflug camera capturing images at approximately 4,300 frames per second during the 30 ms air puff,
resulting in approximately 140 images per examination.
Journal of Refractive Surgery • Vol. 30, No. 3, 2014
Figure 1. (A) A illustration of the cornea under intraocular pressure (IOP)
and air pressure. (B) A kinematical model of the cornea under air puff
deformation. Here, m, k, and c represent the equivalent cornea mass, the
corneal elasticity coefficient, and the corneal damping coefficient, respectively. The sum of the external force on cornea F(t) equals the force from
IOP, FIOP minus the force from the Ocular Response Analyzer (Reichert,
Inc., Buffalo, NY) air pulse, Fair (t).
Biomechanical Model
A simplified kinematical differential equation could
be written as follows to describe the corneal movement
caused by the air pressure. The nonlinear dynamic
model is presented in Figure 1.
"
2
d y (t)
dy (t)
+c
+ ky (t) = F (t) = FIOP - Fair (t)
2
dt
dt
y (0) = FIOP /k
m
y (T) = FIOP /k
(1)
(2 )
(3)
In this model, m is the equivalent corneal mass involved in corneal movement. For a normal human eye,
the whole corneal mass is approximately 0.14 g with a
dimension of approximately 11.5 3 11.5 mm. According to the instrument instruction, the spatial radius
of the air pulse is approximately 1.5 mm, which may
imply that not all the corneal mass should be taken
into account. Here, we evaluate several equivalent corneal mass values in the computations as m = 0 g, m =
0.00433 g, m = 0.01 g, and m = 0.02 g, which accounts
for different ratios of the whole corneal mass.
The coefficient of corneal elasticity, k, was varied between 38 and 50 to test how it affected corneal deformation and vibrations. The damping coefficient, c, reflects
the viscoelastic capacity of the corneal tissue. For comparison purpose, it was set at c = 0 and c = 0.0003 to investigate how the damping coefficient affects corneal motion.
F(t) is the sum of the external force on the cornea
model, and it can be divided into two parts: the force
from IOP, FIOP, and the force from the air pulse, Fair (t).
F (t) = FIOP - Fair (t)
(4)
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Corneal Vibrations/Han et al
Figure 2. (A) Ocular Response Analyzer
(ORA; Reichert, Inc., Buffalo, NY) measurement from a 27-year-old man’s eye. (B)
Comparison between the measured and
numerical fitted curves of air pressure versus time. IOPg = intraocular pressure with
Goldmann applanation tonometer; CH =
corneal hysteresis; IOPcc = corneal compensated IOP; CRF = corneal resistance
factor
where FIOP = IOP 3 A and Fair (t) = Pair(t) × Aair. During
calculations, the “A” represents the effective area for
the IOP. The IOP in this formula could be replaced by
the ORA reading, IOP measurement with Goldmann applanation tonometer (IOPg), and Aair = pr2, r = 1.5 mm is
the effective radius of the air pulse. Pair (t) is the air pressure, which is recorded in the ORA system. It changes
with time and can be fitted by the following function:
t - b1 2
t - b2 2
Pair (t) = a1 exp :-` c j D + a2 exp :-` c j D
1
2
( 5)
where a1, a2, b1, b2, c1, c2 are coefficients to be determined.
Although this formula was the expression chosen, any
other mathematical expression capable of having a good
fit for the air pulse pressure can also be used instead.
The function of time, y(t), represents the cornea position at time t. At t = 0, the initial state before each
measurement, the cornea is assumed to be in the position y (0) = FIOP/k. And at t = T, the end of the ORA
measuring process, the cornea returns to its initial position, so it is again set at y(T) = FIOP/k, where T = 30 ms.
All of the calculation codes were compiled based
on the Fortran 90 platform. The governing equation
with boundary conditions was solved by finite difference method by splitting the time interval [0, T] into
400 equal parts. The coefficients in (5) were computed
using the curve fitting tool box in Matlab R2007a
(Mathwork, Inc., Natick, MA).
RESULTS
Air Pulse Pressure
The air pressure profile used for simulations was
obtained from the ORA measurement of a 27-year-old
healthy man in China. The test is completed in a normal, experimental room temperature environment.
For the measurement of the ORA, IOPg = 13.8 mm
Hg, and corneal hysteresis = 11.5 mm Hg. The entire
time process lasted 30 ms, and the two applanation
210
points were found at 7.875 and 18.975 ms. The calculated maximum air pressure was approximately 43.5
mm Hg (Figure 2A).
Using the Curve Fitting Toolbox in the Matlab R2007a
software, the appropriate parameters were estimated
as a1 = 5149, b1 =0.01752, c1 = 0.00441, a2 = 3398, b2
= 0.01167, and c2 = 0.004311. The comparison between
the measured and calculated curves of air pulse pressure
versus time is shown in Figure 2B. Note that the pressure was described using the unit of Pa instead of mm Hg
used in the clinics, and time used the unit of “second.”
Corneal Vibrations
To investigate the corneal dynamic performances, we
obtained from the formulas the numerical movements
under a simulated IOP of 13.8 mm Hg using a finite difference numerical approach. The calculated results are
shown in Figure 3. In these graphs, several curves of
corneal movement under different considerations of m,
c, and k were plotted with the horizontal axis of time t.
According to the numerical simulations, the shapes
of the corneal motion curves, which changed with time
t, were similar overall to that of Fair (t) in Figure 2. However, there were some differences when corneal mass
m, elasticity k, and damping coefficient c were changed.
In Figure 3A, IOP = 13.8 mm Hg and k = 43.724
(Unit: N/m), also showing no damping effect with c
= 0, the variation caused by different corneal masses
was depicted. When m = 0, there was no vibration; the
cornea only experienced the inward and outward process. At other corneal mass ratios, vibrations of similar
amplitude were found. Meanwhile, different vibrating
frequencies were observed in different groups. When
IOP and k are fixed, whereas c = 0, larger mass tends
to have a larger vibrating period and lower frequency.
In Figure 3B, the effect of corneal viscoelasticity was
tested when k was fixed as 43.724. At m = c = 0, there was
no induced vibration. At m = 0.00433, but c = 0, the vibrations occurred but the amplitude remained stable. When
m = 0.00433 and c = 0.0003, both the corneal mass and
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Corneal Vibrations/Han et al
Figure 3. The numerical results of corneal movements under different conditions: (A) k = 43.724, c = 0, and m varied, (B) k = 43.724 and m and
c varied, (C) m = 0.00433, c = 0, and k varied. Intraocular pressure measurement with Goldmann applanation tonometer = 13.8 mm Hg.
Figure 4. Human cornea deformation under an air pulse measured by CorVis ST (Oculus Optikgeräte GmbH, Wetzlar, Germany) at different time periods,
with a, b, and c in the inward direction and d, e, and f in the outward direction. Maximum deformation occurred between c and d.
damping effects were taken into account, and although
vibrations were observed, the amplitude became smaller
with the passage of time. This can imply that the damping ratio has the potential to reduce the vibrating amplitude by absorbing the energy of the air pulse pressure.
In Figure 3C, IOP = 13.8 mm Hg, m = 0.00433, c = 0,
and k was changed as 33.724, 38.724, and 43.724. It is
observed that lower k leads to larger inward depth and
is associated with larger vibrating amplitude.
Clinical Observation and Comparisons
The responses of corneal vibrations predicted from our
current model were validated from clinical observations
of a human cornea measured by the CorVis ST. Sample
images through the course of an air puff are shown in
Journal of Refractive Surgery • Vol. 30, No. 3, 2014
Figure 4. These images demonstrate how the cornea behaves under the air pulse stimulus. It is deformed inward and then outward to its initial position, which was
in agreement with the numerical investigations.
The time course of a single surface point through an air
puff is given in Figure 5A, which supports the existence
of corneal vibration behavior predicted by our simulated
results. In Figure 5B, the numerical results of IOPg = 13.8
mm Hg, m = 0.00433, c = 0, and k = 43.724 are presented
and compared. These parameters were adjusted based
on the clinical data. It was found that the two curves in
Figure 5 have many similar aspects: (1) the maximum
inward distance in Figure 5A was approximately 0.48
mm, the same order as in Figure 5B; (2) the corneal vibrations were observed, which could support the numerical
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Corneal Vibrations/Han et al
Figure 5. (A) Clinical measurement of the
position (Y axis) of one corneal surface point
tracked over time (X axis) during air puff
deformation using the CorVis ST (Oculus
Optikgeräte GmbH, Wetzlar, Germany). (B)
Numerical result at intraocular pressure
measurement with Goldmann applanation
tonometer = 13.8 mm Hg, m = 0.00433,
c = 0, and k = 43.724.
oscillation phenomenon in Figures 3 and 5B; and (3) between 15 and 20 ms, there are approximately 2.5 vibrating cycles in both curves. However, the difference lies in
that the clinical vibrations did not occur throughout the
entire course of deformation as predicted by the model,
but were isolated to the time period when the cornea was
in a concave state, between applanation points.
DISCUSSION
During the air pulse pressure from the ORA or CorVis
ST instruments, the cornea passes the applanation position twice, once in the inward direction during loading
and once in the outward direction during unloading. Our
numerical models additionally showed that there would
be vibrations along with the corneal movement. Based on
this analysis, it was suggested that the corneal vibrations
were mainly affected by corneal mass and elasticity k under a dynamic external force, and that the damping ability had the potential to reduce the vibrating amplitude.
Due to corneal viscoelasticity (damping coefficient c), the
amplitude of the vibrations became smaller with time t.
In addition, a stiffer cornea results in reduced inward
depth and lower vibrating amplitude, as shown in Figure
3C. The numerical results to some extent agree with the
observation by CorVis ST that there are vibrations along
the corneal inward and outward movements. However,
the time course is different, in that the CorVis ST shows
the vibrations occurring while the cornea is in a concave
state, and not throughout the entire course of the air puff.
As reflected by both the clinical observation in Figure
5A and the numerical results in Figures 3C and 5A, the
corneal performance is associated with two aspects: the
overall inward-outward movement and the vibrations.
The overall movement is mainly caused by the air pressure applied to the cornea. The vibrations may reflect
the characteristics of the natural frequency of the cornea. As shown in Figure 5, there are approximately 2.5
vibrating cycles between 15 and 20 ms (5 ms). So, it is
estimated that the corneal natural frequency is in the
order of approximately  = 2p / (0.005 / 2.5) = 1,000 p.
212
However, it should be recognized that the numerical model developed in this study is only a simplified
one and, therefore, is not able to distinguish the convex and concave states of corneal shape. Therefore, the
model has some limitations that should be discussed.
First, the corneal dynamic model is constructed in one
dimension. For the real cornea, three dimensional models
are required to simulate the convex and concave states,
with their effect on vibrations. However, describing the
three-dimensional air pulse, the corneal dynamic responses, and the IOP and aqueous humor changes is a
complicated process. The parameters are difficult to determine and have to be estimated; the fluid–structure
interaction algorithm is needed and the pre-stress effect
within the corneal tissue caused by the IOP should be
considered. Meanwhile, a large number of computations
is required. Despite these difficulties, this issue is still regarded as valuable and worthy of further investigations.
Second, the corneal mass, m, and the viscoelasticity
coefficient, c, were adjusted and changed to test how
they would affect the results. They can only show some
overall characteristics of the cornea. Third, it is assumed
in the expressions of FIOP = IOP 3 A and Fair (t) = Pair
(t) 3 Aair that the IOP, A, and Aair are unchanged during the whole measuring process. Nevertheless, due to
the complicated interaction effects between the air pulse
and the cornea tissue, and between the cornea tissue and
the aqueous humour within as short as 30 ms, it is still
unclear whether these parameters remain unchanged.
Fourth, comparing the vibrating cycles in Figure 5, it is
found that the corneal vibrating frequencies between the
numerical and the measured data have a good match.
But the parameters are adjusted through trial and error
processes. As we understand, this vibrating frequency is
also the interacting results of the air, aqueous humour,
IOP, and corneal tissue. It should be stated that the current model is unable to give a precise prediction of such
a complicated coupled system. A more detailed mathematical analysis is required in the further investigation.
A nonlinear, viscoleastic dynamic corneal model was
presented that predicts corneal vibrations during air puff
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Corneal Vibrations/Han et al
deformation, which is a novel prediction with supporting
clinical observations. The corneal mass m, the damping
coefficient c, the coefficient of elasticity k, intraocular
force FIOP and the air force Fair (t) were involved in our
multi-parameter model, where k and FIOP were related
to IOP. According to our results, the corneal vibrations
in the deformation were mainly caused by the corneal
mass m and elasticity k under a dynamic external force.
The damping viscoelastic coefficient c had the potential
to reduce the amplitude of the vibrations. Clinical observations by the CorVis ST support the cornea vibrating
performance found in our numerical models. Our results
would be helpful to understand the viscoelastic and dynamical characteristics of the corneal mechanics during
an ORA measurement, as well as with the CorVis ST.
AUTHOR CONTRIBUTIONS
Study concept and design (CT, QR, ZH); data collection (CT, YS,
ZH); analysis and interpretation of data (CZ, CJR, DZ, ZH); drafting the manuscript (CT, ZH); critical revision of the manuscript (CZ,
CJR, DZ, QR, YS); obtaining funding (CZ, DZ, QR); administrative,
technical, or material support (QR)
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