INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING Int. J. Numer. Meth. Biomed. Engng. 2012; 28:547–559 Published online 27 January 2012 in Wiley Online Library (wileyonlinelibrary.com). DOI: 10.1002/cnm.1489
Bio-heat transfer simulation of retinal laser irradiation Arunn Narasimhan* ,† and Kaushal Kumar Jha Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-600 036, India
SUMMARY Retinopathy is a surgical process in which maladies of the human eye are treated by laser irradiation. A two-dimensional numerical model of the human eye geometry has been developed to investigate transient thermal effects due to laser radiation. In particular, the influence of choroidal pigmentation and that of choroidal blood convection—parameterized as a function of choroidal blood perfusion—are investigated in detail. The Pennes bio-heat transfer equation is invoked as the governing equation, and finite volume formulation is employed in the numerical method. For a 500-μm diameter spot size, laser power of 0.2 W, and 100% absorption of laser radiation in the retinal pigmented epithelium (RPE) region, the peak RPE temperature is observed to be 103 ı C at 100 ms of the transient simulation of the laser surgical period. Because of the participation of pigmented layer of choroid in laser absorption, peak temperature is reduced to 94 ı C after 100 ms of the laser surgery period. The effect of choroidal blood perfusion on retinal cooling is found to be negligible during transient simulation of retinopathy. A truncated three-dimensional model incorporating multiple laser irradiation of spots is also developed to observe the spatial effect of choroidal blood perfusion and choroidal pigmentation. For a circular array of seven uniformly distributed spots of identical diameter and laser power of 0.2 W, transient temperature evolution using simultaneous and sequential mode of laser surgical process is presented with analysis. Copyright © 2012 John Wiley & Sons, Ltd. Received 13 May 2011; Revised 23 September 2011; Accepted 9 November 2011 KEY WORDS:
eye; bio-heat transfer; retinal laser surgery; choroidal blood perfusion; choroidal pigmentation
NOMENCLATURE English Symbols c specific heat, J kg1 K1 D distance from sclera/center-to-center distance between two consecutive spots, mm E evaporation rate of tear, W m2 h heat transfer coefficient, W m2 K1 Q laser power, W 000 Q heat generation rate, W m3 t time, ms T temperature, ı C Greek symbols " emissivity, non-dimensional unit outward normal, m ˛ absorptivity, % thermal conductivity, W m1 K1
*Correspondence to: Arunn Narasimhan, Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-600 036, India. † E-mail: [email protected] Copyright © 2012 John Wiley & Sons, Ltd.
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! perfusion rate, kg m3 s1 density, kg m3 Stefan–Boltzmann constant, W m2 K4 Subscripts a ambient b body bl blood c cornea s sclera t tissue 1. INTRODUCTION Laser surgery is routinely used in the treatment of a range of eye diseases such as retinal detachment, choroidal neovascularization, and macular degeneration. During laser irradiation, the pigmented epithelium and pigmented choroid absorb most of the laser energy that reaches the fundus. This causes local heating of the tissues at the fundus. Thus, the human eye has to physiologically cope with temperatures much higher than the body core temperature (37 ı C). For therapeutic purposes, laser irradiation should produce controlled thermal burns or coagulation within the eye. The photocoagulation temperature is around 60 ı C, and it is important to control irradiation such that a specific spot is heated to no more than this temperature. Overheating could cause serious damage to adjoining healthy regions by disrupting cellular mechanisms [1]. Even the choroidal blood flow that accounts for 85% of the blood flowing to the human eye [2] is insufficient to cool the retina during rapid laser-induced heating. Heating the spot to temperatures below 60 ı C, on the other hand, will render the treatment ineffective. Because of the sensitivity of the treatment to temperature evolution in tissues, it is important to understand the heating characteristics of laser surgery. A few direct in vivo experiments to measure temperature of eye tissues during laser irradiation have been reported in the past [3]. Such invasive techniques have now been rendered impractical because of high risk of eye damage and ethical constraints, and research is now confined to animal studies [4]. Computer simulations of such processes can help overcome this stalemate by predicting temperature evolution without need for invasive testing. Such simulations can assist in optimizing surgical conditions to reduce risk of eye damage. There have been a few reports of simulation studies on thermal effects of laser irradiation on the eye. A finite difference model of the eye was presented in Taflove and Brodwin [5] assuming thermal properties to be similar to water. Cain and Welch [6] presented a model incorporating bio-heat transfer equation and have also performed invasive experiments on the rabbit eye. Another similar numerical heat transport model for the rabbit eye was presented by Emery et al. [7] employing a finite element method of solution. Al-Badwaihy and Youssef [8] examined the thermal effects of microwave radiation on the steady-state temperature distribution of a rabbit eye geometry with an assumed combined (convection and radiation) heat transfer coefficient. Lagendijk [9] presented numerical models of rabbit and human eyes that approximated the shape of lens and other structures. An explicit forward difference scheme was used to simulate both transient and steady-state results. Temperature evolution was also studied experimentally in the rabbit eye, and the results were used to determine the heat transfer coefficient on corneal and scleral walls. These were extrapolated to human eye with thermal properties of the human eye. A two-dimensional finite element model of heat transport in the human eye was reported by Scott [10], using the bio-heat transfer equation. Steady-state temperature variation in the human eye exposed to infrared radiation was studied. Another two-dimensional finite element model by Scoot [11] calculated temperature change in the intra-ocular media of the human eye exposed to infrared radiation. This model considered both transient and steady-state solutions, showing that the temperature variation in the anterior segment of the eye can occur if an increase in evaporation from the anterior corneal surface and rapid blink factors are present simultaneously. Amara [12] presented a numerical thermal model of laser-ocular media interaction, which explains the effect of laser radiation on eye but does not attempt to relate to clinical applications Copyright © 2012 John Wiley & Sons, Ltd.
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such as laser surgery of the eye. Thompson et al. [13] presented a numerical granule model of laser absorbance in retinal pigmented epithelium (RPE) considering spherical shape of the cone and receptors, which deviates from their realistic conical shapes. Furthermore, the study does not consider the bulk volume affected during laser surgery. Recently, Chua et al. [14] presented a numerical model to predict the temperature distribution within the human eye when subjected to laser source using only four ocular tissues along the central pupillary axis (see Figure 1). The initial temperature of the eye was assumed to be constant (37 ı C) throughout the eye. A complete geometry of the eye was also not considered. Flyckt et al. [15] reported the impact of choroidal convection heat transfer coefficient, the essential cooling effect due to the blood flow in the choroid. Using numerical simulation of heat transport along a simple three-dimensional geometry of the eye, they explored much higher values for the choroidal heat transfer coefficient from that of an earlier study by Lagendijk [9] involving one of the authors. A similar simple three-dimensional model was presented by Ng et al. [16], extending from their two-dimensional model [17]. These studies were restricted to steady-state conditions. The objective of the present study is to understand, through numerical simulations, the influence of choroidal blood perfusion and choroidal pigmentation on retinal temperature evolution during laser surgery in human eye. The Pennes bio-heat transfer equation is used to model heat transport. The simulations were carried out under transient conditions. A geometrically identical, full-scale axi-symmetric two-dimensional finite volume model of the human eye was developed. The twodimensional model involved irradiation of a single retinal spot. With the temperature results obtained using this model, a three-dimensional model was developed. Multi-spot irradiation is simulated using the three-dimensional model to resemble the actual laser surgical process. 2. MATHEMATICAL FORMULATION AND BOUNDARY CONDITIONS The Pennes bio-heat transfer equation was solved within the eye domain with necessary boundary conditions to understand the interaction between laser and tissue and the resulting temperature evolution. The Pennes bio-heat transfer equation [18] can be written as c
@T D r.rT / C S @t 000
P S D Q C mcT
(1)
(2)
Figure 1. Schematic of the physiology of the human eye. Modified from image at http://en.wikipedia.org/ wiki/Eye. Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Biomed. Engng. 2012; 28:547–559 DOI: 10.1002/cnm
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where ‘S ’ is the source term that includes effects of the volumetric heat generation due to laser irra000 000 diation .Ql /, metabolic heat generation .Qm /, and convection cooling due to choroidal blood perfusion .mcT P /. T is the lumped temperature difference between tissue and blood. The choroidal blood mass flow rate can be calculated as m P D! V
(3)
where ! is the blood perfusion rate and V is the volume of the choroid in the interior of eyeball. Suitable values for ! were taken from Flyckt et al. [15] and other earlier literature. In addition to the choroidal blood perfusion, a constant convection heat transfer coefficient of 65 W m2 K1 was assumed at sclera to include the influence of blood flow at the back of sclera. A constant heat transfer coefficient of 20 W m2 K1 at the cornea was also included to account for convection-based, evaporation-based, and radiation-based heat transfer at that region. As the frontal portion is absent in the three-dimensional model, an adiabatic boundary condition was evoked at the cut section (see Figures 1 and 2(c)). The initial temperature was assumed as 37 ı C, throughout the computational domain. 3. EYE GEOMETRY AND PROPERTIES Figure 1 shows a schematic cross section of the eye, divided into six regions for modeling purpose. The diameter of the eye, along the pupillary axis was taken as 24 mm. The vertical diameter was 23 mm. The posterior half of the human eyeball was taken to be almost spherical [19]. In the simulation, each region was assumed to be homogeneous, and the eye was assumed to be symmetrical about the pupillary axis. The retina has ten layers; the inner nine layers are neural retinas, and the outer layer comprises the RPE. RPE cells have multiple essential functions and serve as nurse cells for the retina. RPE absorbs and delivers nutrients to the neurosensory retina and transports the metabolic end products and waste to the choroid. The RPE cells have the pigment, melanin, which protects the photoreceptors from short-wavelength light damage and shields the sclera from scattered light. The thickness of RPE varies between 6 and 15 μm [1] and was assumed to be 10 μm in this study. The pigmented choroid was assumed to be 100 μm thick . The thickness of choroid including pigmented choroid was taken as 0.5 mm. The eye’s cooling mechanisms were assumed to be located at the surface of eyeball. Heat is lost from corneal surface to the surrounding air through radiation, convection, and evaporation. The choroid, present in between the sclera and the retina, facilitates blood flow at the back of the eye. Cooling or heating on the rest of the eye is achieved by the blood flow in the choroid. In the study, aqueous humor was assumed to be stagnant [7, 10]. Metabolic heat generation within the tissue was assumed to be negligible [20]. The values of the relevant thermo-physical material properties for all six regions mentioned in Figure 1 were collected from literature as tabulated in Table I. The values of , c, , and ˛ for the iris and ciliary body were assumed to be equal to that of aqueous humor [10]. An argon laser with power Q D 0.2 W irradiating a spot size of 500 μm was selected for the numerical simulation. The resultant heat generation percentages in different regions of the laser path are given in Table II. The spot size, laser power, and the reported heat generation percentage have been corroborated by several recent studies (spot size—[21, 22]; laser power—[15, 21, 22]). 4. NUMERICAL METHOD AND GRID INDEPENDENCE The two-dimensional and three-dimensional computational domains of human eye are shown in Figure 2. The models of human eye were created in the Cartesian coordinate system with the use of Gambitr 2.4.6. Quadrilateral finite volume (surface) elements were used to mesh the two-dimensional eye domain. With the temperature results of the two-dimensional model, a threedimensional model was constructed to mimic the realistic laser surgical process at the posterior of Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Biomed. Engng. 2012; 28:547–559 DOI: 10.1002/cnm
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Figure 2. (a) Two-dimensional (2D) computational domain with grid; (b) enlarged section of 2D computational domain with grid; (c) three-dimensional (3D) computational domain with grid; (d) enlarged section of 3D computational domain with grid and circular array of seven spots.
human eye (retinopathy) using a circular array of seven uniformly distributed spots (Figure 2(d)). The RPE region was meshed using hexahedral finite volume elements. The rest of the threedimensional domain was meshed using tetrahedral finite volume elements. Fluentr 6.3.26, which employs the finite volume method, was used to solve the discretized equations. The path of the laser beam was modeled with the user-defined function. Grid independence studies were performed for two-dimensional and three-dimensional models. With temperature results, grids with 51 447 and 2 996 189 cells were selected for the two-dimensional and three-dimensional models, respectively. Subsequently, with parameter values identical to those in the grid independence tests, a time independence test for the transient simulations was also performed. With the error in the maximum temperature in the eye domain, time steps of 0.01 and 0.5 ms were selected for two-dimensional and three-dimensional models, respectively. The convergence criteria was set as 109 for energy equation. The grid independence, time independence, and validation studies have been described in greater detail in Narasimhan et al. [23] and Narasimhan and Jha [24]. Copyright © 2012 John Wiley & Sons, Ltd.
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Table I. Thermal properties of various tissues of the human eye. Medium
Property .W m1 K1 /
Cornea Aqueous humor Lens Vitreous humor RPE Sclera
0.580 [7] 0.580 [7] 0.400 [9] 0.603 [10] 0.603 [10] 0.603 [10]
c .J kg1 K1 /
.kg m3 /
˛ (%)
4178 [10] 3997 [10] 3000 [10] 4178 [10] 4178 [10] 4178 [10]
1050 [25] 1000 [10] 1050 [25] 1000 [10] 1000 [10] 1000 [10]
22 [21] 5 [26] 7 10 0
RPE, retinal pigmented epithelium.
Table II. Heat generation in all zones. Spot diameter D 500 μm
Laser power D 0.2 W
˛ (%)
Energy Absorbed (W)
22 5 7 10
0.0440 0.0078 0.0103 0.0138
Medium Cornea Aqueous humor Lens Vitreous humor
Volume .1012 / .m3 / 1.02 589.05 706.86 3118.03
000
Q .106 ) .W m3 / 430.94 13.24 14.68 4.42
Unpigmented Choroid RPE
100
0.1240
1.96
63174.78
1.96 19.60 194.39
31587.39 3158.74 0
Pigmented Choroid RPE Choroid Sclera
50 100 0
0.0620 0.0620 0
RPE, retinal pigmented epithelium.
5. TWO-DIMENSIONAL SIMULATIONS: RESULTS AND DISCUSSION Because the time duration of burning a single spot during laser surgery varies between 50 and 200 ms, the common being 100 ms followed by 100 ms cooling, transient simulations were performed for this time duration. For choroidal blood perfusion, two extreme conditions were considered—one with no blood perfusion in choroid and other with a maximum blood perfusion rate of 23.3 kg m3 s1 . Similarly, either a pigmented choroid of 100 μm or completely unpigmented choroid was considered. The total thickness of choroid including pigmented choroid was kept constant (500 μm). In case of pigmented choroid, the laser energy reaching to the retina is distributed uniformly between the RPE and the pigmented choroidal layer. The laser energy distribution among RPE and pigmented choroidal layer is presented in Table II. Figure 3 shows the transient temperature distribution along the pupillary axis of an eye under laser irradiation at 100 ms. Under all possible choroidal blood perfusion rates discussed earlier and unpigmented choroidal conditions, the peak temperature reached 103.1 ı C after 100 ms of laser irradiation. As expected, the location of peak temperature was at the center of the RPE. The corneal temperature reached a maximum of 43 ı C, low enough to not coagulate the corneal tissues. In spite of maximum blood perfusion rate of 23.3 kg m3 s1 in the choroid, no measurable change in peak Copyright © 2012 John Wiley & Sons, Ltd.
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Figure 3. Transient temperature results considering the effect of choroidal blood perfusion and pigmentation (a) along the laser path axis and (b) near the retinal region. RPE, retinal pigmented epithelium.
Figure 4. Effect of choroidal pigmentation on peak temperature during transient simulation.
temperature was seen during transient simulation of 100 ms. However, because of choroidal pigmentation, an almost 10 ı C decrease in peak temperature was observed after 100 ms of laser irradiation as shown in Figure 4. The location of peak temperature also shifted towards the choroid. Figure 5 shows the isotherms near the retinal and choroidal regions during 100 ms of laser irradiation for 10, 50, and 100 ms. Figure 5(a) represents unpigmented choroid, and Figure 5(b) shows the effect of choroidal pigmentation. During initial stage of laser irradiation, the temperature at the core region of RPE increased rapidly, after which it plateaued. The situation was similar when choroidal pigmentation was considered. However, the heat diffusion was more prominent in the later case. Copyright © 2012 John Wiley & Sons, Ltd.
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Figure 5. Isotherms near retinal region during transient simulations: (a) without choroidal pigmentation and (b) with choroidal pigmentation. RPE, retinal pigmented epithelium.
This helped in reducing the peak temperature as well as core thickness of the high-temperature zone to a moderate level. It can also be seen from Figure 5 that during 100 ms of laser irradiation, heat was not completely diffused into the entire choroidal region. Thus, transient simulations show no or negligible cooling effect on retina due to choroidal blood perfusion, opposed to that observed in steady-state simulations. It also obviates the independence of peak temperature from the boundary conditions at sclera during such laser surgical process. 6. SENSITIVITY ANALYSIS The constitutive relations used to determine the thermo-physical properties of biological material can involve uncertainties that are hard to locate. Considering this, a sensitivity analysis was performed by perturbing the thermal conductivity () and specific heat (c) in the range of ˙10% of the values reported in Table I, near the retinal region. Initially, steady-state simulations were performed. When specific heat of RPE was altered by ˙10%, no visible effect on peak temperature Copyright © 2012 John Wiley & Sons, Ltd.
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was seen. When thermal conductivity of RPE alone was altered, a variation of ˙0.5 ı C was seen in peak temperature. Further altering the thermal conductivity of choroid by ˙10% led to a change of ˙2.5 ı C in peak temperature. The alteration in thermal conductivity of combined choroid and sclera by ˙10% led to a further change in ˙1.5 ı C in peak temperature. Thus, perturbation of ˙10% in thermo-physical properties led to a maximum variation of ˙2.5% in peak temperature. The effect of perturbation of thermo-physical properties ( and c) near the retinal regions during transient simulation are shown in Figure 6(a) and (b) for unpigmented and pigmented choroids, respectively. The spot was irradiated for 100 ms followed by a cooling period of 100 ms. When thermo-physical properties (, c) of only RPE region is perturbed within ˙10%, variation in peak temperature is observed to be less than ˙0.5 ı C. A variation in ˙10% in thermo-physical properties in and near the retinal region, which consist of vitreous humor, RPE, choroid and sclera, led to change in peak temperature within ˙7%. In a similar analysis, varying the choroidal blood perfusion rate in between 0% and 100%, did not yield any marked difference on the peak retinal temperature, as reported in [23]. 7. THREE-DIMENSIONAL SIMULATIONS: RESULTS AND DISCUSSION During laser surgery of human eye, a large number of spots are irradiated at the posterior region of the eye. The number of such spots ranges between 1500 and 1600. To simulate such surgical environment, a three-dimensional model was constructed on the basis of the temperature distribution obtained using the two-dimensional model of human eye irradiation. Initially, steady-state simulations using a circular array of seven uniformly distributed spots were performed. The same model
Figure 6. Effect of perturbation of thermo-physical properties W m1 K1 and c J kg1 K1 on transient peak temperature: (a) without choroidal pigmentation and (b) with choroidal pigmentation. Copyright © 2012 John Wiley & Sons, Ltd.
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was used to generate transient temperature evolution, which resembles more closely, the actual surgical process. As stated earlier, the typical duration of laser irradiation at a particular retinal spot during surgery is 100 ms followed by a 100-ms cooling period. Transient simulations were performed for 100 ms of heating and subsequent cooling of 100 ms using a circular array of seven uniformly distributed spots under conditions similar to steady-state simulations. All the seven spots were irradiated simultaneously. Figure 7 shows the isotherms resulting from the transient simulation of 10, 50, and 100 ms, considering the effect of choroidal pigmentation alone (Figure 7(b)), as choroidal blood perfusion does not affect the temperature evolution during transient heating of 100 ms. The peak temperature occurred at the center of the spots at the choroid–RPE interface and remained invariant for the all the seven spots of the circular array at a particular time. This is because, under transient conditions, when D D 0.75 mm, heat cannot diffuse to the neighboring spots well enough to cause premature damage to RPE cells during the 100-ms irradiation. Peak temperatures of the spots reached 100.5 ı C and 91 ı C for unpigmented and pigmented choroids, respectively, after 100 ms of laser irradiation. The temperature rise was steep during initial stage and then leveled off. The effect of choroidal pigmentation and blood flow during conventional mode of laser surgery in human eye was further studied by irradiating the spots of circular array in the sequence shown in Figure 2(d). Each spot in circular array was irradiated with laser for 100 ms. The laser was switched off for another 100 ms before irradiating the next spot in the array. The total time taken during such process for a circular array of seven uniformly distributed spots was thus 1400 ms. The resulting isotherms during sequential irradiation of spots are presented in Figure 8(a) and (b) for unpigmented
Figure 7. Isotherms at RPE–choroid interface during transient simulations and simultaneous heating: (a) without choroidal pigmentation and (b) with choroidal pigmentation. Copyright © 2012 John Wiley & Sons, Ltd.
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Figure 8. Isotherms at RPE–choroid interface during transient simulations and sequential heating: (a) without choroidal pigmentation and (b) with choroidal pigmentation.
and pigmented choroids, respectively. Isotherms to the level 43 ı C for odd number of spots (spots 1, 3, 5, and 7), at the end of 100, 500, 900, and 1300 ms are presented, respectively, for unpigmented choroid (Figure 8(a)) and pigmented choroid (Figure 8(b)). The peak temperature of spot 1 was the same under both simultaneous and sequential irradiation. There was a slight increase in peak temperatures ( 1 ı C) during the course of sequential irradiation, which can be attributed to preheating of spots due to diffusion during the sequential irradiation involving much larger time scales. Peak temperatures of spots 1, 3, 5, and 7 of circular array, during sequential heating, are presented in Figure 9. Both unpigmented and pigmented choroids are considered. From Figure 9, it can be observed that the peak temperatures drop down almost by 10 ı C in case of pigmented choroid for all the spots, even though the laser power remains constant. As observed earlier (Figure 8), a slight increase in peak temperature ( 1 ı C) can be seen between the first spot (spot 1) and the last spot (spot 7) during sequential heating for unpigmented as well as pigmented choroid. Copyright © 2012 John Wiley & Sons, Ltd.
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Figure 9. Transient peak temperature results of spots 1, 3, 5, and 7 during sequential heating.
8. CONCLUSIONS Finite volume formulation of the bio-heat transport equation was solved in a geometrically identical, axi-symmetric two-dimensional model of the human eye. With the use of the two-dimensional model, the effects of choroidal blood perfusion and choroidal pigmentation on temperature evolution inside the human eye during laser surgery in retina or retinopathy were investigated in detail for transient conditions. With the outcome of the two-dimensional model, a three-dimensional model of human eye was constructed. The transient effect of choroidal blood perfusion rate and choroidal pigmentation on temperature evolution inside the human eye, in particular in choroidal and retinal region of human eye, was studied by simulating the laser irradiation, simultaneously and sequentially, a circular array of uniformly distributed seven spot. From transient simulations that mimic the actual laser surgical process, the cooling effect of blood perfusion was found to be negligible. This is due to the larger time scale required for the heat to diffuse from retina to choroid compared with the duration of a single irradiation step (200–100 ms of heating and subsequent cooling of 100 ms). Choroidal pigmentation, however, had a significant cooling effect on the RPE region at the front because of participation of a larger volume of pigmented area near the retinal region in the absorption of laser energy. Irrespective of choroidal blood perfusion, reduction in peak temperature is observed because of choroidal pigmentation for the same laser power setting of 0.2 W used in the actual surgery. Transient simulations of simultaneous and sequential mode of laser surgical process confirm the same.
ACKNOWLEDGEMENTS
The authors thank the P. G. Senapathy Center for Computing Resources, IIT Madras, for providing computational time on their Vega Supercomputing cluster.
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Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Biomed. Engng. 2012; 28:547–559 DOI: 10.1002/cnm
Bio-heat transfer simulation of retinal laser irradiation Arunn Narasimhan* ,† and Kaushal Kumar Jha Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-600 036, India
SUMMARY Retinopathy is a surgical process in which maladies of the human eye are treated by laser irradiation. A two-dimensional numerical model of the human eye geometry has been developed to investigate transient thermal effects due to laser radiation. In particular, the influence of choroidal pigmentation and that of choroidal blood convection—parameterized as a function of choroidal blood perfusion—are investigated in detail. The Pennes bio-heat transfer equation is invoked as the governing equation, and finite volume formulation is employed in the numerical method. For a 500-μm diameter spot size, laser power of 0.2 W, and 100% absorption of laser radiation in the retinal pigmented epithelium (RPE) region, the peak RPE temperature is observed to be 103 ı C at 100 ms of the transient simulation of the laser surgical period. Because of the participation of pigmented layer of choroid in laser absorption, peak temperature is reduced to 94 ı C after 100 ms of the laser surgery period. The effect of choroidal blood perfusion on retinal cooling is found to be negligible during transient simulation of retinopathy. A truncated three-dimensional model incorporating multiple laser irradiation of spots is also developed to observe the spatial effect of choroidal blood perfusion and choroidal pigmentation. For a circular array of seven uniformly distributed spots of identical diameter and laser power of 0.2 W, transient temperature evolution using simultaneous and sequential mode of laser surgical process is presented with analysis. Copyright © 2012 John Wiley & Sons, Ltd. Received 13 May 2011; Revised 23 September 2011; Accepted 9 November 2011 KEY WORDS:
eye; bio-heat transfer; retinal laser surgery; choroidal blood perfusion; choroidal pigmentation
NOMENCLATURE English Symbols c specific heat, J kg1 K1 D distance from sclera/center-to-center distance between two consecutive spots, mm E evaporation rate of tear, W m2 h heat transfer coefficient, W m2 K1 Q laser power, W 000 Q heat generation rate, W m3 t time, ms T temperature, ı C Greek symbols " emissivity, non-dimensional unit outward normal, m ˛ absorptivity, % thermal conductivity, W m1 K1
*Correspondence to: Arunn Narasimhan, Heat Transfer and Thermal Power Laboratory, Department of Mechanical Engineering, Indian Institute of Technology Madras, Chennai-600 036, India. † E-mail: [email protected] Copyright © 2012 John Wiley & Sons, Ltd.
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! perfusion rate, kg m3 s1 density, kg m3 Stefan–Boltzmann constant, W m2 K4 Subscripts a ambient b body bl blood c cornea s sclera t tissue 1. INTRODUCTION Laser surgery is routinely used in the treatment of a range of eye diseases such as retinal detachment, choroidal neovascularization, and macular degeneration. During laser irradiation, the pigmented epithelium and pigmented choroid absorb most of the laser energy that reaches the fundus. This causes local heating of the tissues at the fundus. Thus, the human eye has to physiologically cope with temperatures much higher than the body core temperature (37 ı C). For therapeutic purposes, laser irradiation should produce controlled thermal burns or coagulation within the eye. The photocoagulation temperature is around 60 ı C, and it is important to control irradiation such that a specific spot is heated to no more than this temperature. Overheating could cause serious damage to adjoining healthy regions by disrupting cellular mechanisms [1]. Even the choroidal blood flow that accounts for 85% of the blood flowing to the human eye [2] is insufficient to cool the retina during rapid laser-induced heating. Heating the spot to temperatures below 60 ı C, on the other hand, will render the treatment ineffective. Because of the sensitivity of the treatment to temperature evolution in tissues, it is important to understand the heating characteristics of laser surgery. A few direct in vivo experiments to measure temperature of eye tissues during laser irradiation have been reported in the past [3]. Such invasive techniques have now been rendered impractical because of high risk of eye damage and ethical constraints, and research is now confined to animal studies [4]. Computer simulations of such processes can help overcome this stalemate by predicting temperature evolution without need for invasive testing. Such simulations can assist in optimizing surgical conditions to reduce risk of eye damage. There have been a few reports of simulation studies on thermal effects of laser irradiation on the eye. A finite difference model of the eye was presented in Taflove and Brodwin [5] assuming thermal properties to be similar to water. Cain and Welch [6] presented a model incorporating bio-heat transfer equation and have also performed invasive experiments on the rabbit eye. Another similar numerical heat transport model for the rabbit eye was presented by Emery et al. [7] employing a finite element method of solution. Al-Badwaihy and Youssef [8] examined the thermal effects of microwave radiation on the steady-state temperature distribution of a rabbit eye geometry with an assumed combined (convection and radiation) heat transfer coefficient. Lagendijk [9] presented numerical models of rabbit and human eyes that approximated the shape of lens and other structures. An explicit forward difference scheme was used to simulate both transient and steady-state results. Temperature evolution was also studied experimentally in the rabbit eye, and the results were used to determine the heat transfer coefficient on corneal and scleral walls. These were extrapolated to human eye with thermal properties of the human eye. A two-dimensional finite element model of heat transport in the human eye was reported by Scott [10], using the bio-heat transfer equation. Steady-state temperature variation in the human eye exposed to infrared radiation was studied. Another two-dimensional finite element model by Scoot [11] calculated temperature change in the intra-ocular media of the human eye exposed to infrared radiation. This model considered both transient and steady-state solutions, showing that the temperature variation in the anterior segment of the eye can occur if an increase in evaporation from the anterior corneal surface and rapid blink factors are present simultaneously. Amara [12] presented a numerical thermal model of laser-ocular media interaction, which explains the effect of laser radiation on eye but does not attempt to relate to clinical applications Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Biomed. Engng. 2012; 28:547–559 DOI: 10.1002/cnm
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such as laser surgery of the eye. Thompson et al. [13] presented a numerical granule model of laser absorbance in retinal pigmented epithelium (RPE) considering spherical shape of the cone and receptors, which deviates from their realistic conical shapes. Furthermore, the study does not consider the bulk volume affected during laser surgery. Recently, Chua et al. [14] presented a numerical model to predict the temperature distribution within the human eye when subjected to laser source using only four ocular tissues along the central pupillary axis (see Figure 1). The initial temperature of the eye was assumed to be constant (37 ı C) throughout the eye. A complete geometry of the eye was also not considered. Flyckt et al. [15] reported the impact of choroidal convection heat transfer coefficient, the essential cooling effect due to the blood flow in the choroid. Using numerical simulation of heat transport along a simple three-dimensional geometry of the eye, they explored much higher values for the choroidal heat transfer coefficient from that of an earlier study by Lagendijk [9] involving one of the authors. A similar simple three-dimensional model was presented by Ng et al. [16], extending from their two-dimensional model [17]. These studies were restricted to steady-state conditions. The objective of the present study is to understand, through numerical simulations, the influence of choroidal blood perfusion and choroidal pigmentation on retinal temperature evolution during laser surgery in human eye. The Pennes bio-heat transfer equation is used to model heat transport. The simulations were carried out under transient conditions. A geometrically identical, full-scale axi-symmetric two-dimensional finite volume model of the human eye was developed. The twodimensional model involved irradiation of a single retinal spot. With the temperature results obtained using this model, a three-dimensional model was developed. Multi-spot irradiation is simulated using the three-dimensional model to resemble the actual laser surgical process. 2. MATHEMATICAL FORMULATION AND BOUNDARY CONDITIONS The Pennes bio-heat transfer equation was solved within the eye domain with necessary boundary conditions to understand the interaction between laser and tissue and the resulting temperature evolution. The Pennes bio-heat transfer equation [18] can be written as c
@T D r.rT / C S @t 000
P S D Q C mcT
(1)
(2)
Figure 1. Schematic of the physiology of the human eye. Modified from image at http://en.wikipedia.org/ wiki/Eye. Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Biomed. Engng. 2012; 28:547–559 DOI: 10.1002/cnm
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where ‘S ’ is the source term that includes effects of the volumetric heat generation due to laser irra000 000 diation .Ql /, metabolic heat generation .Qm /, and convection cooling due to choroidal blood perfusion .mcT P /. T is the lumped temperature difference between tissue and blood. The choroidal blood mass flow rate can be calculated as m P D! V
(3)
where ! is the blood perfusion rate and V is the volume of the choroid in the interior of eyeball. Suitable values for ! were taken from Flyckt et al. [15] and other earlier literature. In addition to the choroidal blood perfusion, a constant convection heat transfer coefficient of 65 W m2 K1 was assumed at sclera to include the influence of blood flow at the back of sclera. A constant heat transfer coefficient of 20 W m2 K1 at the cornea was also included to account for convection-based, evaporation-based, and radiation-based heat transfer at that region. As the frontal portion is absent in the three-dimensional model, an adiabatic boundary condition was evoked at the cut section (see Figures 1 and 2(c)). The initial temperature was assumed as 37 ı C, throughout the computational domain. 3. EYE GEOMETRY AND PROPERTIES Figure 1 shows a schematic cross section of the eye, divided into six regions for modeling purpose. The diameter of the eye, along the pupillary axis was taken as 24 mm. The vertical diameter was 23 mm. The posterior half of the human eyeball was taken to be almost spherical [19]. In the simulation, each region was assumed to be homogeneous, and the eye was assumed to be symmetrical about the pupillary axis. The retina has ten layers; the inner nine layers are neural retinas, and the outer layer comprises the RPE. RPE cells have multiple essential functions and serve as nurse cells for the retina. RPE absorbs and delivers nutrients to the neurosensory retina and transports the metabolic end products and waste to the choroid. The RPE cells have the pigment, melanin, which protects the photoreceptors from short-wavelength light damage and shields the sclera from scattered light. The thickness of RPE varies between 6 and 15 μm [1] and was assumed to be 10 μm in this study. The pigmented choroid was assumed to be 100 μm thick . The thickness of choroid including pigmented choroid was taken as 0.5 mm. The eye’s cooling mechanisms were assumed to be located at the surface of eyeball. Heat is lost from corneal surface to the surrounding air through radiation, convection, and evaporation. The choroid, present in between the sclera and the retina, facilitates blood flow at the back of the eye. Cooling or heating on the rest of the eye is achieved by the blood flow in the choroid. In the study, aqueous humor was assumed to be stagnant [7, 10]. Metabolic heat generation within the tissue was assumed to be negligible [20]. The values of the relevant thermo-physical material properties for all six regions mentioned in Figure 1 were collected from literature as tabulated in Table I. The values of , c, , and ˛ for the iris and ciliary body were assumed to be equal to that of aqueous humor [10]. An argon laser with power Q D 0.2 W irradiating a spot size of 500 μm was selected for the numerical simulation. The resultant heat generation percentages in different regions of the laser path are given in Table II. The spot size, laser power, and the reported heat generation percentage have been corroborated by several recent studies (spot size—[21, 22]; laser power—[15, 21, 22]). 4. NUMERICAL METHOD AND GRID INDEPENDENCE The two-dimensional and three-dimensional computational domains of human eye are shown in Figure 2. The models of human eye were created in the Cartesian coordinate system with the use of Gambitr 2.4.6. Quadrilateral finite volume (surface) elements were used to mesh the two-dimensional eye domain. With the temperature results of the two-dimensional model, a threedimensional model was constructed to mimic the realistic laser surgical process at the posterior of Copyright © 2012 John Wiley & Sons, Ltd.
Int. J. Numer. Meth. Biomed. Engng. 2012; 28:547–559 DOI: 10.1002/cnm
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Figure 2. (a) Two-dimensional (2D) computational domain with grid; (b) enlarged section of 2D computational domain with grid; (c) three-dimensional (3D) computational domain with grid; (d) enlarged section of 3D computational domain with grid and circular array of seven spots.
human eye (retinopathy) using a circular array of seven uniformly distributed spots (Figure 2(d)). The RPE region was meshed using hexahedral finite volume elements. The rest of the threedimensional domain was meshed using tetrahedral finite volume elements. Fluentr 6.3.26, which employs the finite volume method, was used to solve the discretized equations. The path of the laser beam was modeled with the user-defined function. Grid independence studies were performed for two-dimensional and three-dimensional models. With temperature results, grids with 51 447 and 2 996 189 cells were selected for the two-dimensional and three-dimensional models, respectively. Subsequently, with parameter values identical to those in the grid independence tests, a time independence test for the transient simulations was also performed. With the error in the maximum temperature in the eye domain, time steps of 0.01 and 0.5 ms were selected for two-dimensional and three-dimensional models, respectively. The convergence criteria was set as 109 for energy equation. The grid independence, time independence, and validation studies have been described in greater detail in Narasimhan et al. [23] and Narasimhan and Jha [24]. Copyright © 2012 John Wiley & Sons, Ltd.
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Table I. Thermal properties of various tissues of the human eye. Medium
Property .W m1 K1 /
Cornea Aqueous humor Lens Vitreous humor RPE Sclera
0.580 [7] 0.580 [7] 0.400 [9] 0.603 [10] 0.603 [10] 0.603 [10]
c .J kg1 K1 /
.kg m3 /
˛ (%)
4178 [10] 3997 [10] 3000 [10] 4178 [10] 4178 [10] 4178 [10]
1050 [25] 1000 [10] 1050 [25] 1000 [10] 1000 [10] 1000 [10]
22 [21] 5 [26] 7 10 0
RPE, retinal pigmented epithelium.
Table II. Heat generation in all zones. Spot diameter D 500 μm
Laser power D 0.2 W
˛ (%)
Energy Absorbed (W)
22 5 7 10
0.0440 0.0078 0.0103 0.0138
Medium Cornea Aqueous humor Lens Vitreous humor
Volume .1012 / .m3 / 1.02 589.05 706.86 3118.03
000
Q .106 ) .W m3 / 430.94 13.24 14.68 4.42
Unpigmented Choroid RPE
100
0.1240
1.96
63174.78
1.96 19.60 194.39
31587.39 3158.74 0
Pigmented Choroid RPE Choroid Sclera
50 100 0
0.0620 0.0620 0
RPE, retinal pigmented epithelium.
5. TWO-DIMENSIONAL SIMULATIONS: RESULTS AND DISCUSSION Because the time duration of burning a single spot during laser surgery varies between 50 and 200 ms, the common being 100 ms followed by 100 ms cooling, transient simulations were performed for this time duration. For choroidal blood perfusion, two extreme conditions were considered—one with no blood perfusion in choroid and other with a maximum blood perfusion rate of 23.3 kg m3 s1 . Similarly, either a pigmented choroid of 100 μm or completely unpigmented choroid was considered. The total thickness of choroid including pigmented choroid was kept constant (500 μm). In case of pigmented choroid, the laser energy reaching to the retina is distributed uniformly between the RPE and the pigmented choroidal layer. The laser energy distribution among RPE and pigmented choroidal layer is presented in Table II. Figure 3 shows the transient temperature distribution along the pupillary axis of an eye under laser irradiation at 100 ms. Under all possible choroidal blood perfusion rates discussed earlier and unpigmented choroidal conditions, the peak temperature reached 103.1 ı C after 100 ms of laser irradiation. As expected, the location of peak temperature was at the center of the RPE. The corneal temperature reached a maximum of 43 ı C, low enough to not coagulate the corneal tissues. In spite of maximum blood perfusion rate of 23.3 kg m3 s1 in the choroid, no measurable change in peak Copyright © 2012 John Wiley & Sons, Ltd.
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Figure 3. Transient temperature results considering the effect of choroidal blood perfusion and pigmentation (a) along the laser path axis and (b) near the retinal region. RPE, retinal pigmented epithelium.
Figure 4. Effect of choroidal pigmentation on peak temperature during transient simulation.
temperature was seen during transient simulation of 100 ms. However, because of choroidal pigmentation, an almost 10 ı C decrease in peak temperature was observed after 100 ms of laser irradiation as shown in Figure 4. The location of peak temperature also shifted towards the choroid. Figure 5 shows the isotherms near the retinal and choroidal regions during 100 ms of laser irradiation for 10, 50, and 100 ms. Figure 5(a) represents unpigmented choroid, and Figure 5(b) shows the effect of choroidal pigmentation. During initial stage of laser irradiation, the temperature at the core region of RPE increased rapidly, after which it plateaued. The situation was similar when choroidal pigmentation was considered. However, the heat diffusion was more prominent in the later case. Copyright © 2012 John Wiley & Sons, Ltd.
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Figure 5. Isotherms near retinal region during transient simulations: (a) without choroidal pigmentation and (b) with choroidal pigmentation. RPE, retinal pigmented epithelium.
This helped in reducing the peak temperature as well as core thickness of the high-temperature zone to a moderate level. It can also be seen from Figure 5 that during 100 ms of laser irradiation, heat was not completely diffused into the entire choroidal region. Thus, transient simulations show no or negligible cooling effect on retina due to choroidal blood perfusion, opposed to that observed in steady-state simulations. It also obviates the independence of peak temperature from the boundary conditions at sclera during such laser surgical process. 6. SENSITIVITY ANALYSIS The constitutive relations used to determine the thermo-physical properties of biological material can involve uncertainties that are hard to locate. Considering this, a sensitivity analysis was performed by perturbing the thermal conductivity () and specific heat (c) in the range of ˙10% of the values reported in Table I, near the retinal region. Initially, steady-state simulations were performed. When specific heat of RPE was altered by ˙10%, no visible effect on peak temperature Copyright © 2012 John Wiley & Sons, Ltd.
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was seen. When thermal conductivity of RPE alone was altered, a variation of ˙0.5 ı C was seen in peak temperature. Further altering the thermal conductivity of choroid by ˙10% led to a change of ˙2.5 ı C in peak temperature. The alteration in thermal conductivity of combined choroid and sclera by ˙10% led to a further change in ˙1.5 ı C in peak temperature. Thus, perturbation of ˙10% in thermo-physical properties led to a maximum variation of ˙2.5% in peak temperature. The effect of perturbation of thermo-physical properties ( and c) near the retinal regions during transient simulation are shown in Figure 6(a) and (b) for unpigmented and pigmented choroids, respectively. The spot was irradiated for 100 ms followed by a cooling period of 100 ms. When thermo-physical properties (, c) of only RPE region is perturbed within ˙10%, variation in peak temperature is observed to be less than ˙0.5 ı C. A variation in ˙10% in thermo-physical properties in and near the retinal region, which consist of vitreous humor, RPE, choroid and sclera, led to change in peak temperature within ˙7%. In a similar analysis, varying the choroidal blood perfusion rate in between 0% and 100%, did not yield any marked difference on the peak retinal temperature, as reported in [23]. 7. THREE-DIMENSIONAL SIMULATIONS: RESULTS AND DISCUSSION During laser surgery of human eye, a large number of spots are irradiated at the posterior region of the eye. The number of such spots ranges between 1500 and 1600. To simulate such surgical environment, a three-dimensional model was constructed on the basis of the temperature distribution obtained using the two-dimensional model of human eye irradiation. Initially, steady-state simulations using a circular array of seven uniformly distributed spots were performed. The same model
Figure 6. Effect of perturbation of thermo-physical properties W m1 K1 and c J kg1 K1 on transient peak temperature: (a) without choroidal pigmentation and (b) with choroidal pigmentation. Copyright © 2012 John Wiley & Sons, Ltd.
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was used to generate transient temperature evolution, which resembles more closely, the actual surgical process. As stated earlier, the typical duration of laser irradiation at a particular retinal spot during surgery is 100 ms followed by a 100-ms cooling period. Transient simulations were performed for 100 ms of heating and subsequent cooling of 100 ms using a circular array of seven uniformly distributed spots under conditions similar to steady-state simulations. All the seven spots were irradiated simultaneously. Figure 7 shows the isotherms resulting from the transient simulation of 10, 50, and 100 ms, considering the effect of choroidal pigmentation alone (Figure 7(b)), as choroidal blood perfusion does not affect the temperature evolution during transient heating of 100 ms. The peak temperature occurred at the center of the spots at the choroid–RPE interface and remained invariant for the all the seven spots of the circular array at a particular time. This is because, under transient conditions, when D D 0.75 mm, heat cannot diffuse to the neighboring spots well enough to cause premature damage to RPE cells during the 100-ms irradiation. Peak temperatures of the spots reached 100.5 ı C and 91 ı C for unpigmented and pigmented choroids, respectively, after 100 ms of laser irradiation. The temperature rise was steep during initial stage and then leveled off. The effect of choroidal pigmentation and blood flow during conventional mode of laser surgery in human eye was further studied by irradiating the spots of circular array in the sequence shown in Figure 2(d). Each spot in circular array was irradiated with laser for 100 ms. The laser was switched off for another 100 ms before irradiating the next spot in the array. The total time taken during such process for a circular array of seven uniformly distributed spots was thus 1400 ms. The resulting isotherms during sequential irradiation of spots are presented in Figure 8(a) and (b) for unpigmented
Figure 7. Isotherms at RPE–choroid interface during transient simulations and simultaneous heating: (a) without choroidal pigmentation and (b) with choroidal pigmentation. Copyright © 2012 John Wiley & Sons, Ltd.
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Figure 8. Isotherms at RPE–choroid interface during transient simulations and sequential heating: (a) without choroidal pigmentation and (b) with choroidal pigmentation.
and pigmented choroids, respectively. Isotherms to the level 43 ı C for odd number of spots (spots 1, 3, 5, and 7), at the end of 100, 500, 900, and 1300 ms are presented, respectively, for unpigmented choroid (Figure 8(a)) and pigmented choroid (Figure 8(b)). The peak temperature of spot 1 was the same under both simultaneous and sequential irradiation. There was a slight increase in peak temperatures ( 1 ı C) during the course of sequential irradiation, which can be attributed to preheating of spots due to diffusion during the sequential irradiation involving much larger time scales. Peak temperatures of spots 1, 3, 5, and 7 of circular array, during sequential heating, are presented in Figure 9. Both unpigmented and pigmented choroids are considered. From Figure 9, it can be observed that the peak temperatures drop down almost by 10 ı C in case of pigmented choroid for all the spots, even though the laser power remains constant. As observed earlier (Figure 8), a slight increase in peak temperature ( 1 ı C) can be seen between the first spot (spot 1) and the last spot (spot 7) during sequential heating for unpigmented as well as pigmented choroid. Copyright © 2012 John Wiley & Sons, Ltd.
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Figure 9. Transient peak temperature results of spots 1, 3, 5, and 7 during sequential heating.
8. CONCLUSIONS Finite volume formulation of the bio-heat transport equation was solved in a geometrically identical, axi-symmetric two-dimensional model of the human eye. With the use of the two-dimensional model, the effects of choroidal blood perfusion and choroidal pigmentation on temperature evolution inside the human eye during laser surgery in retina or retinopathy were investigated in detail for transient conditions. With the outcome of the two-dimensional model, a three-dimensional model of human eye was constructed. The transient effect of choroidal blood perfusion rate and choroidal pigmentation on temperature evolution inside the human eye, in particular in choroidal and retinal region of human eye, was studied by simulating the laser irradiation, simultaneously and sequentially, a circular array of uniformly distributed seven spot. From transient simulations that mimic the actual laser surgical process, the cooling effect of blood perfusion was found to be negligible. This is due to the larger time scale required for the heat to diffuse from retina to choroid compared with the duration of a single irradiation step (200–100 ms of heating and subsequent cooling of 100 ms). Choroidal pigmentation, however, had a significant cooling effect on the RPE region at the front because of participation of a larger volume of pigmented area near the retinal region in the absorption of laser energy. Irrespective of choroidal blood perfusion, reduction in peak temperature is observed because of choroidal pigmentation for the same laser power setting of 0.2 W used in the actual surgery. Transient simulations of simultaneous and sequential mode of laser surgical process confirm the same.
ACKNOWLEDGEMENTS
The authors thank the P. G. Senapathy Center for Computing Resources, IIT Madras, for providing computational time on their Vega Supercomputing cluster.
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