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A comparative study between the two-dimensional and three-dimensional human eye models

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Mathematical and Computer Modelling 48 (2008) 712–720www.elsevier.com/locate/mcm
A comparative study between the two-dimensional andthree-dimensional human eye models
Eddie-Yin-Kwee Ng
a,
∗
,1
, Ean-Hin Ooi
a
, U. Rajendra Archarya
b
a
School of Mechanical and Aerospace Engineering, College of Engineering, Nanyang Technological University, 50, Nanyang Avenue,Singapore 639798, Singapore
b
Department of Electronics and Computer Engineering, Ngee Ann Polytechnic, Singapore
Received 23 May 2007; received in revised form 29 October 2007; accepted 2 November 2007
Abstract
A comparative study between the ocular temperature predicted by both the two-dimensional and three-dimensional human eyemodels is carried out in this paper. The three-dimensional model is generated from the two-dimensional model reported in theauthors’ earlier work. The central corneal temperature for the three-dimensional model is found to be 34
.
48
◦
C compared to thetwo-dimensional model which has a value of 33
.
64
◦
C. Comparisons between the temperatures at various points inside the humaneye suggest that for some initial investigations, the use of a two-dimensional model may be sufﬁcient. In other cases such as the lossof thermal symmetry (asymmetrical boundary conditions, power deposition inside the eye), the use of a three-dimensional modelis preferable. Methods to improve the accuracy of the human eye model in terms of its anatomical and physiological features arealso suggested for future works to be considered.c
2007 Elsevier Ltd. All rights reserved.
Keywords:
Finite element method; Eye temperature; Three-dimensional; Isotherm; Bioheat transfer
1. Introduction
When developing mathematical models, we always aim to construct models that best resemble the actual system forthe purpose of minimizing inaccuracies. However, to perfectly simulate the actual system is always impossible, largelydue to limitations such as increased complexity of eye anatomy, where obtaining a unique solution is very difﬁcult,and the limitation of current computer technology to compute the highly complex modeling equations. Mathematicalmodels are thus accompanied by assumptions for the purpose of simplifying the problem. One such assumption is thatof working in a two-dimensional plane.
Abbreviations: k
: thermal conductivity;
T
: temperature;
T
amb
: ambient temperature;
T
bl
: blood temperature;
h
amb
: ambient convectioncoefﬁcient;
h
bl
: blood convection coefﬁcient;
E
: heat loss due to tear evaporation;
σ
: Stefan–Boltzmann constant;
ε
: emissivity;
n
: unit normalvector
∗
Corresponding author. Tel: +65 6790 4455; fax: +65 6791 1859.
E-mail address:
mykng@ntu.edu.sg (E.-Y.-K. Ng).
URL:
http://www.ntu.edu.sg/home/mykng (E.-Y.-K. Ng).1Adjunct National University Hospital (NUH) Scientist, Ofﬁce of Biomedical Research, Singapore.0895-7177/$ - see front matter c
2007 Elsevier Ltd. All rights reserved.doi:10.1016/j.mcm.2007.11.011
E.-Y.-K. Ng et al. / Mathematical and Computer Modelling 48 (2008) 712–720
713Fig. 1. The three-dimensional eye model in the present study.
In modeling the heat transfer inside the human eye, most of the models available in the literature were constructedin two dimensions (see References [1–4] for examples). Considering the
x
–
y
plane (see Fig. 1) to be a typicaltwo-dimensional plane, a two-dimensional model assumes the heat transfer in the
z
-direction to be negligible. Thisassumption is valid if the eye structure is relatively thin or inﬁnitely long in the
z
-direction such that temperaturechanges along this axis are negligible. The human eye with the inclusion of the optic nerve however, is farfrom having the mentioned anatomy and the variation in the
z
-axis is not to be assumed negligible in reality.Although predicted ocular surface temperature (OST) from these models (see Scott [3], Amara [4] and Ng and
Ooi [5] for examples) had shown reasonable agreement with measured values from other literatures, it is unwise
to assume that the models are accurate. This is largely due to the limitation of infrared thermography which onlyprovide temperature on the corneal surface for comparison. If a three-dimensional model of the human eye canbe developed and the temperature variations in the
z
-axis can be shown to be small compared to the
x
- and
y
-axes, this may then provide a better justiﬁcation for the use of a two-dimensional model with a single set of parameters and symmetric boundary conditions for lesser CPU and RAM resources (though this may not be so criticalnowadays).This paper presents the development of a three-dimensional heat transfer eye model in Cartesian coordinates. Thecurrent study is a follow-up to a recently published two-dimensional eye model [5] which would hereafter be referredto as the two-dimensional model as opposed to the three-dimensional model in this study. A section in this paper isdedicated to the investigation of the temperature change in the
z
-axis at different locations for the purpose of sheddingmore information regarding the use of two-dimensional approximations to simulate the human eye. A complete three-dimensional simulation is necessary when dealing with asymmetric boundary conditions.
2. Model development
The graphical image and the procedures for constructing the two-dimensional model can be found in [5]. Thethree-dimensional model is generated by revolving the two-dimensional model without the optic nerve, 180
◦
aroundits horizontal pupiliary axis. The cylindrical optic nerve which is assumed to have similar properties as the sclera isthen merged with the sclera. In the actual eye, two additional layers can be found beneath the sclera
i
.
e
. the choroidand the retina. These two layers are thin compared to the sclera and are thus, combined with the sclera and optic nerveand treated as one homogeneous region. Fig. 1 illustrates the three-dimensional model of the human eye where thegeometrical center of the cornea (GCC) is depicted by the thick line across the corneal surface. The GCC line shownin Fig. 1 is for post-processing purposes.
714
E.-Y.-K. Ng et al. / Mathematical and Computer Modelling 48 (2008) 712–720
Table 1Thermal properties of the human eyeDomains Sub-domain index,
i
Thermal conductivity
(
W m
−
1
K
−
1
)
ReferencesCornea 1 0.58 Emery et al. [1]Aqueous humor 2 0.58 Emery et al. [1]Iris 3 1.0042 Cicekli [8]Lens 4 0.40 Lagendijk [9]
Vitreous 5 0.603 Scott [3]
Sclera 6 1.0042 Cicekli [8]
2.1. Governing equation
One of the more widely used equation for modeling transfer of heat inside biological systems is the Pennes bioheatequation[6]whichcontainsthemetabolicheatgenerationandbloodperfusionterm.Inthecurrentstudy,themetabolicheat generation is neglected based on the fact that the human eye comprises mainly water. Consequently, tissuesresponsible for metabolic heat generation are minimal.Since the human eye is modeled as an organ isolated from the human head, we can afford to drop the bloodperfusion term and account for the thermal effects of blood in the boundary condition. More information is given inthe next section.For a steady state solution, the bioheat equation reduces to the classical heat conduction equation which is writtenas
∇
(
k
i
∇
T
i
)
=
0 (1)with
i
: sub-domains 1, 2, 3, 4, 5 and 6. The variable
k
is the thermal conductivity and
T
is the temperature. Thesubscript
i
denotes the index for each sub-domain as shown in Table 1. The thermal conductivity for each sub-domain
may be found in the literature and they are tabulated in Table 1.
2.2. The boundary conditions
The boundary conditions used in this study are similar to the ones used for the two-dimensional model. They arereproduced here in Eqs. (2) and (3). The ﬁrst boundary condition is written on the scleroid surface. We assume the
human eye to be embedded in a homogeneous surrounding anatomy which is at body core temperature. Consequently,the transfer of heat from the surrounding to the eye may be described by a single heat transfer coefﬁcient such that [9]
−
k
∂
T
∂
n
=
h
bl
(
T
−
T
bl
)
;
at
Γ
1
(2)where
h
bl
is the blood convection coefﬁcient,
T
bl
is blood temperature which we have assumed as body coretemperature and
n
is the unit normal vector.
Γ
1
is given by the surface of the sclera which is illustrated in Fig. 2.The second boundary condition is written on the corneal surface where three heat loss mechanisms take place.These losses are the heat transfer due to convection and radiation and the heat loss as a result of tear evaporation fromthe corneal surface. Mathematically, this may be written as
−
k
∂
T
∂
n
=
E
+
h
amb
(
T
−
T
amb
)
+
σε
T
4
−
T
4amb
;
at
Γ
2
(3)where the ﬁrst, second and third terms on the right-hand side are the heat loss due to tear evaporation, convection andradiation respectively,
E
is heat loss due to evaporation of tears,
h
amb
is the ambient convection coefﬁcient and
T
amb
isambient temperature. Variables
σ
and
ε
are the Stefan–Boltzmann constant and corneal emissivity respectively.
Γ
2
isdepicted by the corneal surface which is shown in Fig. 3. More information regarding the boundary conditions can be
obtained from [5]. The values for the control parameters used in this study are the same as that for the two-dimensionalmodel (homogeneous and isotropic thermal properties within each layer) and they are summarized in Table 2.
E.-Y.-K. Ng et al. / Mathematical and Computer Modelling 48 (2008) 712–720
715Fig. 2. Boundary condition
Γ
1
on the sclera surface.Fig. 3. Boundary condition
Γ
2
on the corneal surface.Table 2Control parameters used in the present studyControl parameters Value ReferencesBlood temperature,
T
bl
(
◦
C
)
37 Ng and Ooi [5]
Ambient temperature,
T
amb
(
◦
C
)
25 Ng and Ooi [5]
Emissivity of cornea,
ε
0.975 Mapstone [11]Blood convection coefﬁcient,
h
bl
(W m
−
2
K
−
1
) 65 Lagendijk [9]
Ambient convection coefﬁcient,
h
amb
(W m
−
2
K
−
1
) 10 Emery et al. [1]Evaporation rate,
E
(W m
−
2
) 40 Scott [3]
Stefan–Boltzmann constant,
σ (
W m
−
2
K
−
4
)
5
.
67
×
10
−
8
Incroprera and Dewitt [18]
3. Numerical methodology
The governing equation (1) along with its respective boundary conditions (2 & 3) are solved numerically using theﬁnite element method. Computations are carried out with the help of the commercialized software package, COMSOLMultiphysics 3.2 [7] which is a partial differential calculator utilizing the ﬁnite element method. Computationsare conducted on a personal computer with a processor speed of 2.41 GHz and a RAM of 512 MB. The three-dimensional model is meshed using tetrahedral elements and the temperature along each surface is approximatedusing the Lagrange quadratic polynomial.
716
E.-Y.-K. Ng et al. / Mathematical and Computer Modelling 48 (2008) 712–720
Fig. 4. The meshed three-dimensional model with 91,666 tetrahedral elements.Fig. 5. A typical temperature distribution of the three-dimensional normal eye model.
In investigating the dependency of predicted results on the number of elements, it is found that beyond 55,000elements, the predicted results showed very minor change (mesh invariant test). In order to obtain an accurate result,sub-domains such as the cornea which are relatively thin compared to other sub-domains are ﬁnely meshed. To avoidthe increase of the total number of elements used from this procedure, element sizes for larger sub-domains such asthe vitreous, lens and aqueous humor are increased. These procedures are done by changing the maximum elementsize and the element growth parameter in the COMSOL Multiphysics 3.2 workspace. The total number of elementsgenerated is 91,666. The meshed structure of the three-dimensional model is shown in Fig. 4.
4. Numerical results
Fig. 5 shows the steady state temperature distribution inside the three-dimensional eye model. The warmer regionis located at the posterior of the eye where heat from the surrounding anatomy srcinates while the cooler region ison the cornea surface where heat losses occur. The lowest temperature is 34
.
48
◦
C and is located at the center of thecornea, which hereafter, is referred to as the central corneal temperature (CCT).The temperature distribution along the horizontal pupiliary axis for the three-dimensional model is plotted inFig. 6 along with the results from the two-dimensional model and those found in Scott [3]. In Fig. 7, the temperature

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