Mathematical modeling of laser linear thermal effects on the anterior layer of the human eye

Optics and Laser Technology 99 (2018) 72–80

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Mathematical modeling of laser linear thermal effects on the anterior layer of the human eye Sahar Rahbar, Mehrdad Shokooh-Saremi ⇑ Department of Electrical Engineering, Ferdowsi University of Mashhad, Mashhad 91779-48944, Iran

a r t i c l e

i n f o

Article history: Received 14 March 2017 Received in revised form 5 September 2017 Accepted 20 September 2017

Keywords: Human eye Excimer lasers Thermal distribution Cornea Numerical modeling

a b s t r a c t In this paper, mathematical analysis of thermal effects of excimer lasers on the anterior side of the human eye is presented, where linear effect of absorption by the human eye is considered. To this end, Argon Fluoride (ArF) and Holmium:Yttrium-Aluminum-Garent (Ho:YAG) lasers are utilized in this investigation. A three-dimensional model of the human eye with actual dimensions is employed and finite element method (FEM) is utilized to numerically solve the governing (Penne) heat transfer equation. The simulation results suggest the corneal temperature of 263 °C and 83.4 °C for ArF and Ho:YAG laser radiations, respectively, and show less heat penetration depth in comparison to the previous reports. Moreover, the heat transfer equation is solved semi-analytically in one-dimension. It is shown that the exploited simulation results are also consistent with those derived from the semi-analytical solution of the Penne heat transfer equation for both types of laser radiations. Ó 2017 Elsevier Ltd. All rights reserved.

1. Introduction Lasers with different wavelengths have been used greatly in various medical fields such as dentistry, gynecology, neurosurgery, dermatology, orthopedics, gastroenterology, angioplasty, cardiology, and particularly in ophthalmology and eye surgeries [1]. These lasers are mainly short pulsed type and able to provide higher intensity which is desirable for medical applications. The main reasoning behind the use of this type of lasers is that they can provide more control on the amount of energy irradiated by the laser, amount of energy dissipation and also more precision on the zones of the eye that are heated by the laser. Also, in LASIK (Laser in Situ Keratomileusis) and PRK (Photorefractive Keractomy) surgeries dealing with vision correction, short pulsed lasers are wellabsorbed by the corneal collagen of the eye in a more effective way and as a result the collagen is well-shrunk [2,3]. Absorption of laser light by different eye tissues leads to heat generation. However, due to the lack of blood circulation in most of the anterior part of the eye (especially cornea), the eye does not have the ability to reduce this temperature rise and hence damages to the tissues may occur. This issue should be taken seriously into account while employing laser in eye treatments. On the other hand, eye temperature measurement is not often possible by noninvasive methods;

⇑ Corresponding author. E-mail address: [email protected] (M. Shokooh-Saremi). https://doi.org/10.1016/j.optlastec.2017.09.033 0030-3992/Ó 2017 Elsevier Ltd. All rights reserved.

therefore, thermal analysis is required. This analysis should be accurate, reliable and repeatable. The human eye consists of three main parts. Outer layer includes sclera and cornea; the middle part consists of the iris, aqueous humor, and choroid; and the retina and vitreous humor are considered the inner section of the human eye [4]. Fig. 1 shows a normal eye with actual dimensions. The cornea, which has
0.5 mm thickness, plays the main role in refractive power of the eye [5]. The iris muscles by contraction help lens to change the focal distance which is known as eye’s accommodation [6]. Then, the beam of light hits the retina; chemical changes occur and consequently light energy converts into the nerve impulses. These nerve impulses are transmitted by the optic nerve to the brain where they are processed and the result is perceived as an image [7]. Measuring different parts of the eye’s temperature are common in the field of ophthalmology because any rise in temperature may indicate a kind of disease. Attempting to measure the temperature of the eye goes back to 1875. At that time, cornea’s surface temperature was measured 36.5° by Dohnberg [8]. Efforts continued until 1960; the temperature through infrared radiation was measured with an instrument called Bolometer. This measurement is based on electrically conductive media where infrared light is radiated into the environment by which changes in the peripheral resistance is measured. Then, the resistance changes are converted into the temperature which is monitored on the screen and represents the eye temperature [8]. One of the first researches on corneal

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Fig. 1. The anatomy of the normal human eye with actual dimensions.

temperature change due to laser radiation was performed in 1979. During the initial investigation, the cornea has been modeled as one-dimensional slab and the temperature distribution has been determined for infrared radiation [9]. In 1999, the threshold value of argon fluoride laser’s energy to remove the corneal surface was investigated. In this study, the cornea was hit by different laser light energy densities, and the threshold value was set to 30 mJ/ cm2 for cornea [10]. Eye problems treatment related to refraction of light such as myopia, hyperopia and astigmatism with laser was introduced in 1997 [11]. Since then, a lot of studies have been done on the damaging effects of heat on the eye’s tissue. In [12], parts of the iris and sclera were considered as cellular and other parts of the cornea, aqueous humor, lens and vitreous were twodimensionally modeled and analyzed by FEM method. Four different models were considered to study inner and outer temperature of the eye. The conditions of the external ambient temperature, blood temperature and heat transfer coefficient of blood in different parts of the eye were taken into account. Finally, it was found that aqueous humor and blood flow in the choroid of the eye play an important role in moderating temperatures. In [13], steady state temperature was determined using a three-dimensional model. In [14], the temperature around the eye pupil in steady state was investigated with or without considering the metabolism and blood flow to the retina and in [15] the impact of warm environments such as sauna was studied on the eye health and vision. Sixty to seventy percent of light refraction in the eye is done by the cornea [16]. As Fig. 2 shows, in myopia or hyperopia, the cornea loses its normal shape and the beam of light is not properly focused on the retina to create a clear image. Therefore, ophthalmologists reshape the cornea surface by lasers to focus the beam of light again on the retina [11]. Two different kinds of lasers are used

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for corneal correction. The first one is the ArF laser with ultraviolet light radiation which is used for corneal reshape. The second one is Ho:YAG laser which is suitable for minimizing and shrinking the corneal surface. PRK, LASIK and LASEK (Laser Sub Epithelial Keratomileusis) are common ways to reshape the cornea. Transmission, reflection, scattering and absorption are the fundamental parameters affecting the light propagation. Among these, light absorption is the most important phenomena because it can raise the temperature of the media [17]. Hence, one of the most important issues before the eye surgery is estimating the temperature of the eye. The best temperature is the one that could cause the most desired effect and the least unwanted destruction. For example, in surgeries dealing with corneal curve reshape, the desired minimum temperature for warming up the surface of the cornea is 64 °C [18]. Unwanted destruction is not reversible in laser eye surgeries and that is why using software and mathematical models are very important to predict the temperature of the eye before any kind of laser surgery. In [19], thermal effects and heat transfer of blinking has been studied using FEM where it has been shown that blinking increases corneal and lens temperature. Also, it has been discussed that this model can be used in eye cancer diagnosis by searching for eye tissues with drastic rise in temperature. The same approach is applied in [20] to investigate thermal effect of physiological parameters such as blood perfusion, porosity, evaporation and environment temperature on sclera and cornea. Moreover, the parabolic and hyperbolic models of Penn’s general bio-heat equation are considered in [2] for analyzing heat temperature in cornea when exposed to high power ultra short pulsed laser. Then, these models have been solved by providing an analogy between the thermal and electrical systems using HSPICE program. Also, an experimental investigation (not mathematical) of thermal effects of laser irradiation on retina during photocoagulation by optical coherence tomography is studied [21]. In this paper we carry out a mathematical study, not in-vivo or ex-vivo study, on the linear thermal effects of human eye based on the Penn’s bio-heat transfer equation. To this end, first, the governing Penn’s bio-heat transfer equation is introduced. Next, the existing boundary conditions on the cornea and choroid representing the heat transfer from these layers to the adjacent layers are explained. Then, to study the thermal effects, eye is irradiated by a laser light (Gaussian beam) as an illuminating source. The absorption of this light by the eye tissues is considered to be linear. Using these boundary and illumination conditions, Penn’s equation is numerically solved using finite element method and the results of the simulations are discussed. In addition, the heat transfer equation is semi-analytical solved in one-dimension and the results are presented in order to verify the simulation results. Finally, the outcomes of the present research are compared with the previous studies.

2. Methods and eye model Since blood flow does not play a significant role in the thermal regulation process of the eye tissues, modeling and analyzing thermal effects of human eye is of great importance [22]. The heat distribution in the human eye’s tissues is modeled based on the Penne’s equation as follows [4]:

qt ct

Fig. 2. Sketch of the normal, myopia and hyperopia eye.

@Tt ¼ r  ðkt rTt Þ þ qb cb wb ðTb  Tt Þ þ Q m ; @t

ð1Þ

where qt and qb are the density of tissue and blood (kg=m3 ), respectively. ct and cb also show thermal coefficient of tissue and blood (J=kg  K). kt presents tissue heat transfer coefficient (W=mK) and Tt is the temperature of tissue in Kelvin. t is time in second. wb is

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blood flow rate (m3 =s), Tb is blood temperature in Kelvin and the last parameter is Q m representing external heat source (W=m3 ). The only vascular parts of the human eye are iris, choroid and sclera. However, the choroid and sclera do not play a role in human vision and as a result they are not engaged in the laser vision corrections study. Moreover, since iris is a small part of the eye we can ignore its blood flow term in the Penn’s equation for simplicity [23]. Thus, it is reasonable to ignore the blood flow related terms and as a result, Eq. (1) is reduced to:

@T qt ct t ¼ r  ðkt rTt Þ þ Q m : @t

ð2Þ

On the other hand, for steady state and without an external heat source the equation is:

r  ðkt rTt Þ ¼ 0:

ð3Þ

Two boundary conditions are required for solving the above equation. At the first boundary (the corneal surface), heat transfer from the cornea to the ambient medium is considered, taking into account tear and its surface evaporation:

k1

@T1 ¼ hamb ðT1  Tamb Þ þ reðT41  T4amb Þ þ E; @n

ð4Þ

where T1 is the corneal temperature in Kelvin, k1 is cornea heat transfer coefficient, hamb describes heat transfer coefficient between the eye and the environment (10W=m2 K), Tamb is the ambient tem
 perature (25 °C), r is the Boltzmann’s constant 5:67  108 m2wK4 , e shows the emission of heat by the eye (0.975) and E is ocular sur1 expresses the rate of change in face evaporation (40 W=m2 ). @T @n the cornea temperature (T1 Þ with respect to the normal vector of the surface. On the other hand, the effect of blood flow in the choroid and the associated heat transfer is defined as the second boundary condition:

k2

@T2 ¼ hbl ðT2  Tbl Þ; @n

ð5Þ

where T2 is the choroid temperature in Kelvin, k2 is choroid heat transfer coefficient, hbl is heat transfer coefficient between choroid and the adjacent tissues (65 W/m2K), Tbl is the blood temperature 2 in choroid (37 °C), and @T expresses the rate of changes in the @n choroid temperature (T2) with respect to the normal vector of the surface. Laser light is a Gaussian beam of low divergence with the intensity described in the cylindrical coordinates (r, z) and time (t) as 2r2

8t2

Iðr; z; tÞ ¼ I0 eax e w2 e s2 ;

where k is the wavelength of the light, f is the focal length of the lens which is set to 17 mm for a normal eye, and dp is the pupil diameter of the eye. Also, human eye pupil diameter varies between 1.5 and 8 mm depending on the ambient light [24]. Therefore, the laser light intensity on the retina is given by [4]: 2

Ir ¼ Ic

dp 2

dr

:

ð10Þ

In [23], accuracy of two and three-dimensional eye models has been studied. In this regard, 23 eye temperatures were examined and the calculated average was 34.36 °C. Then, the eye temperature in steady state without laser radiation were measured with acceptable accuracy; 33.64 °C and 34.48 °C were obtained for two and three-dimensional models, respectively. By comparing these temperatures with the average value, it was suggested in [23] that the three-dimensional model is more precise for future studies. Therefore, the three-dimensional eye model with real scales is utilized in our proposed investigation (see Fig. 3). Table 1 shows the values for heat transfer coefficient, heat capacity, density of each part of the eye and absorption coefficients in different wavelengths. 3. Simulation results In this section, simulation results using the three-dimensional eye model including steady state thermal distribution, cornea thermal distribution under the ArF and Ho:YAG laser radiations and penetration depth in the cornea are presented. To study the steady state condition, it is assumed that the patient is resting for about 15 min in a room with normal temperature to have a normal eye adaptation. Then, to commence the simulation, the three-dimensional model is divided into 331,548 pyramids, as shown in Fig. 3, for solving the heat transfer equation by the finite element method (FEM). Employing this method, Penne’s partial differential equation (PDE) is discretized and solved in smaller regions called finite elements. As it can be seen in Fig. 3, the cornea, choroid and sclera are placed at 0 mm, 23 mm and 24 mm on the x axis, respectively. Fig. 4 shows the steady state temperature distribution where heat external sources do not exist. It is clear from this figure that the cornea is the coldest and the choroid, because of its blood flow, is the warmest part of the eye. Now, this steady state result is used as the initial condition for further investigation of the thermal

ð6Þ

where I0 is the initial intensity of emitted light, w is the beam waist and s describes time duration. Part of the light, depending on the tissue absorption, is absorbed and heats up the tissue as described by

Q ðr; z; tÞ ¼ aIðr; z; tÞ;

ð7Þ

where Q(r, z, t) is the absorbed heat by the tissue. Laser light intensity on the cornea is:

Ic ¼

4P

pd2c

;

ð8Þ

where P is the laser power and dc is a beam diameter on the cornea. Moreover, beam diameter after passing through the lens of the eye is calculated by the following equation:

dr ¼ 2:44

kf ; dp

ð9Þ

Fig. 3. The meshing of the three-dimensional model of the human eye for FEM computations.

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S. Rahbar, M. Shokooh-Saremi / Optics and Laser Technology 99 (2018) 72–80 Table 1 Thermal properties of the human eye (see for example [4]). Eye components

Cornea Aqueous humor Iris Lens Vitreous humor Retina Choroid Sclera

Thermal specifications Thermal conductivity kðJ=m s KÞ

Specific heat capacity cðJ=kg KÞ

qðkg=m3 Þ

Density

að1=mÞ

Absorption coefficient at 193 nm

að1=mÞ

Absorption coefficient at 2090 nm

0.58 0.578 0.498 0.4 0.594 0.565 0.53 0.58

4178 3997 3340 3000 3997 3680 3840 4178

1076 1003 1040 1100 1009 1039 1060 1170

270,000 2228.3 2228.3 2558.4 542.7 6526.2 48475 28880

2923.8 2228.3 2228.3 2558.4 542.7 4370.3 6398.2 2923.8

etration depth for using this laser must be so small to prevent underlying layers damages.

3.2. Ho:YAG laser (2090 nm) radiation This type of laser is used in corneal transplant surgery. To warm up corneal surface, the minimum temperature should be 64 °C and in order to avoid the corneal weakness, it should be kept under 100 °C [18]. This laser emits 7 pulses on the corneal surface during 200 ls. For this laser, 150 W of peak power, 0.6 mm beam diameter on the cornea and 1.5 mm pupil diameter is taken into account [24]. The temperature change by this type of laser radiation is shown in Fig. 7. As seen, the temperature on the cornea is obtained to be 83.4 °C. Moreover, Fig. 8 shows the penetration depth of this laser on the cornea, which is about 0.167 mm.

4. Semi-analytical results Fig. 4. Steady state thermal distribution of eye where no laser radiation exists (steady-state condition).

distribution on the cornea under laser irradiation with two wavelengths. It should be noted that there is no explicit limits on the number and duration of irradiated pulses. In case of the limit on number of pulses, various parameters such as gender, age, dry eye, shape of cornea, eye anatomy and eye weakness are involved. Moreover, this limit is pertained to the type of surgery. Also, the limit on laser pulse width depends on the type of surgery. For example, in some surgeries such as Laser Thermo Keratoplasty (LTK) surgery, very short and high energetic pulsation (typical time periods are 200 ls and 50 ls) are used [2]. In this paper, the considered parameter values for the laser and eye parameters are typical values used in most of the studying the thermal effects during laser eye surgeries. 3.1. ArF laser (193 nm) radiation As already mentioned, ArF laser is widely used in reshaping the cornea. In this case, 15 ns laser radiation duration, 450 micro joules of energy, 0.28 mm beam diameter on the cornea and 1 mm pupil diameter are being taken into account (as proposed by [4]). To achieve a more accurate result, finer mesh on the cornea is considered. The temperature change by this laser radiation is shown in Fig. 5. As it can be seen, the highest temperature on the cornea is 263 °C which is high enough for melting intermolecular connections to reshape the corneal surface and meanwhile could be tolerated by the underlying tissues. Due to the low thickness of the cornea around 0.5464 mm, as Fig. 6 shows, it is clear that the pen-

In this section, the previously proposed simulation results are going to be verified using a semi-analytical solution of the Penne’s heat transfer equation. For this purpose, two boundary conditions on the cornea and choroid are applied and then Penne’s equation is solved to derive the steady state distribution so that the initial condition of Penne’s PDE is established. Then, a linear approximation of the nonlinear terms in boundary condition is also used as a part of the further efforts towards finding a closed-form expression for the initial condition. This approximation is next shown to provide results that are very close to the non-approximated numerical results and hence suggest a good precision while giving a closedform expression. The Penne’s equation can be simplified in order to verify the simulation results. Since the changes of temperature in the direction of eye pupil are important, which is placed on the x direction, this dimension is kept and y and z dimensions are removed. Accordingly, Eq. (1) can be written as

qt ct

@Tðx; tÞ @ 2 Tðx; tÞ ¼ kt þ qb cb wb ðTb  Tðx; tÞÞ þ Q m : @t @x2

ð11Þ

By ignoring the blood flow related terms, we have:

qt ct

@Tðx; tÞ @ 2 Tðx; tÞ ¼ kt þ Q m; @t @x2

ð12Þ

and f or the steady state case, the equation reduces to

@ 2 Tðx; tÞ ¼ 0: @x2

ð13Þ

To solve the above equation, two boundary conditions are required as stated before:

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Fig. 5. Thermal distribution of the eye under the ArF laser radiation. (a) Meshing model. (b) Simple thermal distribution.

Fig. 6. Penetration depth under ArF laser radiation.

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(2) On the cornea (x ¼ 0 mm):


 4 k1 a ¼ E þ hamb ðb  Tamb Þ þ er b  T4amb :

ð18Þ

Taylor linear approximation of T4 ðx; tÞ  T4amb around Tamb can be written as:

T4 ðx; tÞ  T4amb ¼ 4T3amb ðTðx; tÞ  Tamb Þ:

ð19Þ

By applying this linear approximation to Eq. (18), the following equation is obtained:

k1 a ¼ E þ hamb ðb  Tamb Þ þ 4erT3amb ðb  Tamb Þ
 ¼ E  hamb Tamb  4erT4amb þ hamb þ 4erT3amb b;

ð20Þ

where a and b are calculated from the linear system of the following equations:

ðk2 þ 24hbl Þa þ hbl b ¼ hbl Tbl ;
  k1 a þ hamb þ 4erT3amb b ¼ hamb Tamb þ 4erT4amb  E;

Fig. 7. Thermal distribution under the Ho:YAG laser radiation.

(1) For the first boundary condition, surface of the cornea, the eye’s heat, ambient temperature, tears and vector perpendicular in x direction are considered:


 @Tðx; tÞ k1 ¼ hamb ðTðx; tÞ  Tamb Þ þ er T4 ðx; tÞ  T4amb þ E; @x x ¼ 0 mm: ð14Þ (2) For the second boundary condition, blood flow in the choroid is defined, where the vector perpendicular is at the end of the eye and in þx direction:

k2

@Tðx; tÞ ¼ hbl ðTðx; tÞ  Tbl Þ; @x

x ¼ 24 mm:

ð15Þ

To achieve the primary heat distribution, Eq. (13) is to be solved subject to boundary equations (14) and (15). The general solution of the Eq. (13) is given by

TðxÞ ¼ ax þ b

ð16Þ

Now, considering the boundary conditions on pupil axis, choroid and cornea, coefficients a and b are extracted as:

which results in:

h
i  hamb þ 4erT3amb ðTbl  Tamb Þ þ E
 a¼ k1 hbl þ ðk2 þ 24hbl Þ hamb þ 4erT3amb
 k1 hbl Tbl þ Tamb ðk2 þ 24hbl Þ hamb þ 4erT3amb  Eðk2 þ 24hbl Þ
 b¼ : k1 hbl þ ðk2 þ 24hbl Þ hamb þ 4erT3amb hbl

ð22Þ After applying the previously given values of eye and blood parameters in Section 2, the numerical values of a and b are obtained as

a ¼ 0:664777256;

ð17Þ

b ¼ 21:039992535:

ð23Þ

On the other hand, the exact non-approximated values of the coefficients a and b can be calculated through direct solving of (17) and (18) without linearization of Eq. (18), which releases the following results:

a ¼ 0:66476164;

(1) On the choroid (x ¼ 24 mm):

k2 a ¼ hbl ð24a þ b  Tbl Þ:

ð21Þ

b ¼ 21:0399421:

ð24Þ

As a result, the linear approximation could be used unconditionally to find the steady state thermal distribution.

Fig. 8. Penetration depth under Ho:YAG laser radiation.

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Fig. 9. The thermal distribution of the eye under ArF laser radiation, where x axis is drawn from 0 to 24 mm (eye diameter), time axis (t) indicates the duration of laser radiation and temperature axis (T) shows the eye temperature. (a) Three-dimensional view. (b) Front view.

Fig. 10. The thermal distribution of the eye under Ho:YAG laser radiation, where x axis is drawn from 0 to 24 mm (eye diameter), time axis (t) indicates the duration of laser radiation and temperature axis (T) shows the eye temperature. (a) Three-dimensional view. (b) Front view.

S. Rahbar, M. Shokooh-Saremi / Optics and Laser Technology 99 (2018) 72–80

The external heat source distribution is also expressed in x and t dimensions, which can be described as 8t2

Q ðx; tÞ ¼ aI0 eax e s2 :

ð25Þ

Light intensity on the cornea and retina are exactly as given before. Now the characteristics of each laser are applied individually as described in Section 3. For solving Penne’s equation for a given type of laser radiation, Eq. (25) is inserted into (12). This results the Penne’s PDE to be simplified as a differential equation in terms of the temperature distribution which depends on time and the position on the corneal axis. This PDE can be obtained first by numerically solving through FEM and then interpolating the numerical solutions using the Hermite method to find the final solution. In our eye model, corneal axis (x axis) is drawn from 0 to 24 mm (due to the eye diameter) in all figures. Also, time axis (t) indicates the duration of laser radiation and temperature axis (T) shows the eye temperature. 4.1. Verification of the simulation results for ArF laser radiation Fig. 9 shows that the maximum temperature occurs just on the cornea at x ¼ 0 mm without penetrating to the inner layers of the eye. As it can be seen from the figure, the temperature has an exponential behavior with a maximum of 280 °C occurring at x ¼ 0 mm after 15 ns. The achieved temperature for the cornea is very close to the simulation result presented in the previous section. There exists a difference of around 17 °C in the simulation and numerical achieved temperatures. This may be mainly due to not considering y and z coordinates in the computation, higher accuracy of meshes in our simulations, and the use of different methods. Accordingly, Fig. 9 confirms the accuracy of the simulation. 4.2. Verification of the simulation results for Ho:YAG laser radiation Fig. 10 shows the thermal distribution of the eye under Ho:YAG laser radiation. It signifies that the maximum temperature occurs at x ¼ 0 mm as confirmed by the simulation result. Additionally, the thermal distribution is quite exponential and the temperature increases to the maximum of 78 °C during 0.2 ms of laser radiation. This figure also indicates a low penetration depth. The small difference in the ultimate value of temperature (which is less than 5°) is due to various solving approaches. Based on the explanations mentioned above, the results of our study are fairly verified. 5. Discussion The thermal distribution and penetration depth for two practical lasers were provided in the previous section. For further discussion of the results achieved by the ArF laser radiation, the above presented results are compared with those obtained in [25,22] and section 14 of [4]. Referring to [25], the approximate proposed temperature of 175 °C and high penetration depth is not counted as a favorable result. Also, [22] provides an acceptable temperature of 207 °C, however, it does not present an acceptable penetration depth. The maximum temperature occurs in the aqueous humor and seems to be disputable and considered as unwanted destruction. As it can be seen in [4], a 548 °C temperature is also demonstrated along with an acceptable and fine penetration depth; however, this temperature is too high to be tolerated by cornea’s tissue. By comparing the results of the present study with the results from [25,22,4], one may conclude that the accuracy of the implemented model, its temperature precision, the low penetration depth and accuracy of the simulations are much more desirable.

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For the Ho:YAG laser, the current results are compared with those presented in [27,26]. The reported results which propose temperature values about 70 °C and 63 °C, respectively, are very close to the ones obtained in our simulation, but clearly, the heat penetration depth is greater than the thickness of the cornea and the maximum temperature is achieved on aqueous humor instead of the cornea. Thus, the results seem not to be very desirable due to possible damages to adjacent eye tissues. Comparing with the present study, the demonstrated penetration depth of Fig. 9 shows a fine depth which is limited to the surface of the cornea. Accordingly, our simulation results show perfect penetration depth and acceptable maximum temperature in a predetermined position. In addition, the presented simulation was verified by solving the Penne’s equation semi-analytically.

6. Conclusion In this paper, mathematical modeling and analysis of the linear thermal effects of laser radiation on the anterior part of the human eye were studied. To this end, first, a three-dimensional model of the human eye structure was introduced for which the output results provided much more precision compared to the ones presented by two-dimensional models. Second, the Penne’s heat transfer equation was introduced and the human eye parameters in this equation were presented. This equation was applied to the three-dimensional eye model and numerically solved using finite element method in order to extract the thermal distribution and the penetration depth of the corneal tissue under ArF and Ho: YAG laser radiations. For numerical analysis of the Penne’s equation, the steady state temperature distribution of the normal eye was derived as the initial condition. Moreover, to obtain a closed form expression of the steady state distribution, a linear approximation was used, which proposed an accurate result compared to the non-approximated one. Finally, it was shown that the numerical results using the Hermite interpolation technique, confirm the accuracy of the obtained simulation results to an acceptable extent under both types of laser radiations and the penetration depth of the presented results are less than the one suggested by previous reports. References [1] M.H. Niemz, Laser-Tissue Interactions, Fundamentals and Applications, 3rd ed., Springer, 2007. [2] A.M. Gheitaghy, B. Takabi, M. Alizadeh, Modeling of ultrashort pulsed laser irradiation in the cornea based on paraboloic and hyperboloic heat equations using electrical analogy, Int. J. Modern Phys. C 25 (9) (2014) 1450039-1– 1450039-17. [3] E.M. Ahmed, F.J. Barrera, E.A. Early, M.L. Denton, C.D. Clark, Maxwell’s equations-based dynamic laser-tissue interaction model, Comput. Biol. Med. 43 (12) (2013) 2278–2286. [4] E.Y.K. Ng, J.H. Tan, U.R. Acharya, J.S. Suri, Human Eye Imaging and Modeling, CRC Press, 2012. [5] H. Gross, in: Handbook of Optical Systems (Vol. 4 Survey of Optical Instruments), WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim, Germany, 2008, pp. 1–88. [6] K. Rogers, The Eye: The Physiology of Human Perception, first ed., Britannica Educational Publishing, NY, 2011. [7] D. Atchison, G. Smith, Optics of the Human Eye, second ed., Elsevier Science Limited, Edinburgh, Germany, 2002. [8] R. Mapstone, Measurement of corneal temperature, Exptl. Eye Res. 7 (2) (1968) 237–242. [9] M.A. Mainster, Ophthalmic applications of infrared lasers-thermal considerations, Investig. Ophthalmol. Visual Sci. (IOVS) 18 (4) (Apr. 1979) 414–420. [10] M.W. Berns, L. Chao, A.W. Giebel, L.H. Liaw, J. Andrews, B. Versteeg, Human corneal ablation thresholdusing the 193-nm ArF excimer laser, Investig. Ophthalmol. Visual Sci. (IOVS) 40 (5) (Apr. 1999) 826–830. [11] E.E. Manche, J.D. Carr, W.W. Haw, P.S. Hersh, Excimer laser refractive surgery, West. J. Med. (WJM) 169 (1) (Jul. 1998) 30–38. [12] M. Shafahi, K. Vafai, Human eye response to thermal disturbances, J. Heat Transf. 133 (011009) (Jan. 2011) 1–6.

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