Abstract. 1 Introduction

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A Powerful Finite Element for Analysis of Argon Laser Iridectomy: Influence of Natural Convection on the Human Eye G. Sbirlea, J.P. L'Huillier Laboratory ofadvanced Instrumentation and Robotics

1 A Powerful Finite Element for Analysis of Argon Laser Iridectomy: Influence of Natural Convection on the Human Eye G. Sbirlea, J.P. L'Huillier Laboratory ofadvanced Instrumentation and Robotics (L.I.R.A.), CE# - EA%4M, 2 7M c/w KoMCcray, Bf J J2 J, ^90 JJ, Angers Cedex, France Abstract A 3-D numerical model of the human eye is developed to study the conduction and convection heat transfer due to an argon laser iridectomy and to explain in which way the corneal burns and lens opacities depend on the laser parameters. The equations of mass, momentum and energy conservation, supplemented by boundary and initial conditions are discretized in three spatial dimensions and in time according to respectively the finite element method and to an implicite integration scheme. The contraction burn process was studied using shot of 0.5 s in duration, 0.5 mm in size and a power level of 0.4 W. The influence of the natural convection and the phenomena of cornea and lens overheating are presented and discussed. 1 Introduction The field of possible laser applications in intraocular microsurgery is considerably widened. During the past few years the argon laser has become an increasingly used modality for the treatement of angle-closure glaucoma through the production of peripheral iridotomy. This treatement may be necessary when the pupillary margin of the iris is adherent to the lens impairing the aqueous flow and causing a harmful rise of intraocular pressure (from 13 mm Hg to mm Hg). As a result, mechanical strains of tissue occur at sites of intrinsic weakness and contribute to damage the sensitive fibres in the optic nerve, e.g. Le Grand [1], Thompson [2]. To operate an iridectomy, the energy of the argon laser beam (488 nm nm wavelength) is focused on a small area of the iris. The heat source

68 Simulations in Biomedicine IV deposition is not only function of laser irradiation parameters but is also dependent upon the optical properties of the tissue.

2 68 Simulations in Biomedicine IV deposition is not only function of laser irradiation parameters but is also dependent upon the optical properties of the tissue. The process leads to vaporization and tissue removal and includes three steps : heating, boiling and ablation. Thus, a small opening in the iris is performed and the pressure between the anterior and posterior chambers of the eye is equalized. Previous studies have reported sophisticated theoretical models to explain and predict laser-induced thermal damage in the retina, e.g. Cain and Welch [3], Amara [4], and these ultimately rely on laboratory measurements for verification and refutation. To our knowledge, theoretical investigations devoted to the laser production of peripheral iridectomy have not yet been studied. However, complications such as corneal endothelial burns and lens opacities have been experienced and reported by Ritch [5] and Pollack [6] in case of CW argon laser long pulse settings (from 0.1 s to 0.5 s). The theoretical research described in this paper was undertaken in order to understand the primary effects of the laser energy immediately after absorption in the iris. The main absorption area for argon laser pulses is expected to be the melanin layer of the iris tissue (pigment epithelium P.E.). Computations are based upon a finite element method to predict the thermal effects induced on surrounding eye tissues, specifically on both cornea endothelium and lens by a CW argon laser iridectomy. To quantify these effects the intraocular temperature and the velocity field are calculated during the heating phase of the iris tissue. 2 Mathematical statement 2.1 Theoretical model To model mathematically the heat transport process during an iridectomy procedure the eye is divided in seven regions: cornea, aqueous humor, lens, iris, ciliary body, vitreous humor and retina. Each region is assumed to be homogeneous. The problem is formulated in three dimensional Cartesian coordinates (x,y,z), with the x axis constituting the pupillary axis. The unknown temperature T is a function of (x,y,z) in steady state case and of (x,y,z,t) in transient case. The aqueous and vitreous humors are assumed to be viscous, incompressible and Eulerian fluids The buoyancy force caused by density variation resulting from temperature rise between the ambient and the human body is modeled by Boussinesq approximation. It follows that the physical system governing the aqueous and vitreous humors flow is the representation of momentum, mass and energy conservations, e.g. FIDAP [7], Sacadura [8]:

3 Simulations in Biomedicine IV 69 +H (3) supplemented with temperature boundary conditions equations where the temperature is prescribed (4) and two initial condition equations which lie in the initial temperature and velocity distribution inside the eye (5): T =20 C The values of density p, specific heat at constant pressure c^, thermal conductivity k and dynamic viscosity ju are assumed constant within each region and independent against temperature variations. The thermal-physical constants values used in overall computations are given in table 1, e.g. Scott [10], Amara [4], Olza [9]. Equations (1) to (5) are expressed in terms of physical variables but a dimensionless formulation of the problem provides a measure of the relative importance of the terms in the equations and reduces the potentially large differences in orders of magnitude that may occur among these terms in field equations. 2.2 Numerical procedure The mathematical model presented in the previous section is treated with numerical procedures to obtain solutions to the temperature and velocity distributions of the human eye. The system of partial differential equations is discretized in three spatial dimensions according to the finite element method (FEM) and in time according to thefirstorder backward Euler algorithm with a variable increment option, maximum time increment and maximum increase in time increment per time step. The continuum region of the human eye is divided in seven main regions and every region in 8 nodes brick elements (see figure l(b)) as following : cornea in 1036 elements, aqueous humor in 6912 elements, lens in 3810 elements, iris in 2880 elements, ciliary body in 1312 elements, vitreous humor in elements and the retina in 2508 elements as shown in figure l(a). Within each element, the velocity, pressure and temperature fields are approximated by :

4 70 Simulations in Biomedicine IV where U,.(/), P and T are column vectors of element nodal point unknowns and #>, i// and $ are column vectors of the interpolation functions. Substitution of these approximations into the field equations yields a set of equations : (7) where R,, R^ and R^ are the residuals (errors) resulting from the approximations (6). The Galerkin form of the Method of Weighted Residuals is used to reduce these errors to zero by making the residuals orthogonal to the interpolation functions of each element. 2.3 Laser heat source In case of laser iridectomy procedure, the laser beam strikes on the anterior corneal surface, penetrates the aqueous humor and is absorbed in the melanin layer of the iris tissue which is assumed homogeneous and 0.2 mm thick. We considered a topflattype heat source H absorbed in the iris melanin layer according to Beer's law with a percent transmittance T, and an absorption coefficient a : H(x) - T, /o a exp(-a x) (8) where x is the distance from the irradiated surface of the sample and /<, is the irradiance at the iris surface and at the beam center. We assumed that the laser beam is entirely transmitted by the cornea and the aqueous humor: Tr - 1. This assumption overestimates the laser heat source. Typical values of the absorption coefficient available for the pigment epithelium such as cm"*, 832 cm and 900 cm"* have been reported by Cain [3], Polhamus [11] and Mainster [12] respectively. Among these an average value a = 800 cm"' was used. Because the absorption depth of the argon laser (I/a) on the iris is small compared to the laser beam radius, we assumed irradiance independent of the distance from the beam center. The irradiance is calculated as : /, =4f/,r(DL (9) The laser heat source is time dependent and can be described as: H(x,t) = 0 when the laser beam is "off' and H(x,t) = /o a exp(-a x) when the laser beam is "on".

5 3 Results 3.1 Steady state case Simulations in Biomedicine IV 71 To simulate the heat transfer in the human eye during an argon laser iridectomy, temperature distribution and velocity field in steady case and without heat source are necessary. The eye is submitted to natural convection in vitreous and aqueous humors due to temperature difference setted as boundary conditions between the human body Tb= 37 C and the ambient 7^=20 C. The Reynolds number value is equal to 722 and the flow is then assumed to be laminar. The figure 2 gives the temperature distribution along the pupillary axis in cases of conduction and both conduction and convection heat transfer. We can see that the temperature decreases continually from retina to cornea according to a conduction heat transfer only. When both conduction and natural convection phenomena are taken into account, a great part of the intra-ocular temperature distribution becomes flat. 3.2 Transient case The solution of the transient state problem is obtained by including the laser energy volume density as heat source term in the energy equation. For both conduction and convection heat transfer the equations of energy, mass and momentum conservation associated with the boundary conditions represented by the ambient and human body temperatures and the initial conditions given by the steady state temperature distribution and velocity field are solved. We used the irradiation parameters encountered for a contraction burn in case of glaucoma treatement, e.g. Ritch [5]: 0.5 mm spot size, 0.5 s duration and 0.4 W laser beam power. This type of burn is designed to contract the iris tissue and not to penetrate it. Thus, the vaporization and the tissue removal are not treated. The iridectomy site is placed superiorly at 4.33 mm of the pupillary center and at 1:00 o'clock position (see figure 3). The temperature shape given by a 0.5 s argon laser exposition and 1.5 s of relaxation period is depicted in figure 4. For a laser beam setting of 0.4 W the temperature on different nodes of the iridectomy site increases rapidly and reaches a maximum peak of 370 C at the surface of the iris and at the beam center. After the laser source has been turned-off, the temperature decreases. The same shape of the temperature curve can be observed on the aqueous humor with a maximum peak of 170 C (seefigure4(b)). As depicted in figure 4(c), on the cornea-iris angle, precisely on the cornea endothelium nodes by the side of the iridectomy site, the temperature increases during the relaxation phase. The same phenomena is observed on the lens, behind the iris by the side of the iridectomy site (see figure4(d)). The velocity field in the aqueous and vitreous humors by the side of the iridectomy site after s of argon laser exposition and 0.3 s of relaxation

72 Simulations in Biomedicine IV time are depicted respectively in figures 5(a) and 5(b). The velocities vary from 0.254 mm/s to 2.03 mm/s.

6 72 Simulations in Biomedicine IV time are depicted respectively in figures 5(a) and 5(b). The velocities vary from mm/s to 2.03 mm/s. 4 Discussion and Conclusion As we can see in figure 2, the convection currents induce an important modification of the temperature distribution. Along the pupillary axis the temperatures are higher on vitreous humor, lens and aqueous humor in case of both conduction and convection heat transfer that in case of single conduction. The 3-D computation results of the steady conduction heat transfer agree well with those obtained by Scott [10] and Amara [4] and based on a 2-D simulation. Thus, we used the temperature distribution and velocityfieldissued for the steady state simulation as initial conditions for the transient state case. In transient state case we noted 4-5 C of temperature increase on the cornea endothelium by the side of the iridectomy site during the relaxation phase. After 1.5 s of relaxation time the temperature has not yet decreased (see figure 4(c)) The same phenomena may be observed on the lens, behind the iris and by the side of the iridectomy site with a temperature increase of 3-4 C (see figure During the relaxation phase, an important increase of the velocities on the cornea-iris angle and between lens and iris is observed and a vortex appears (see figures 5(a) and 5(b)). In this work we presented the preliminary results of a 3-D numerical model using thefiniteelement method to compute the temperature distribution and the velocity field in the ocular media in case of argon laser iridectomy. The contraction burn in glaucoma treatement is studied with the following settings: 0.5 mm laser spot size, 0.5 s duration and 0.4 W laser beam power. The results show that the natural convection in the ocular media has a great influence on temperature distribution. An overheating of lens and cornea endothelium by the side of the iridectomy site is observed. That may be an explanation of corneal burns and lens opacities reported by ophtalmologists as postoperative complications of argon laser iridectomy. At present we are using the same theoretical model and numerical procedure to study the pathological human eye in case of angle closure glaucoma. The geometry of the diseased eye will allow us to estimate the real importance of the phenomena already noted. In addition, further investigations should be performed to determine the optimum time lapses between successive shots, using a combination of much shorter exposures and moderate energy levels. Acknowledgements The authors are greatful to Fluent France, specially to Mr. Gerard de Neuville, managing director, for equipement support and to Mr. Olivier SANTAL, fluid dynamics engineer for scientific support.

7 References Simulations in Biomedicine IV Le Grand Y, El Hage SG. Physiological Optics, Springer Series in Optical Sciences, New York, Thompson KP, Ren QS, Pare! JM. Therapeutic and Diagnostic Application of lasers in ophtalmology. Proceedings of the IEEE 1992; Vol. 80, 6: Cain C.P., Welch A J Measured and predicted laser-induced temperature rises in the rabbit fundus, Investigative Ophtalmology, 1974, 13, Amara EH. Numerical investigations on thermal effects of laser-ocular media interaction, IntJ Heat Mass Transfer, 1995; 38, 13: Ritch R, Solomon IS. Laser treatement of glaucoma, J Ophtalmic Lasers, , MO:CV MOSBY, 3d Ed, St Louis, Pollack IP. Use of argon laser to produce iridotomies, Trans Am Ophtalmol Soc, 1979,77, Fluid Dynamics Analysis Package FIDAP 7.6, Fluid Dynamics International, Inc. Evanston, Sacadura JF. Initiation aitx transfer ts thermiques, Technique et Documentation, Paris, Olza A, Taillard F, Vautravers E, Diethelm JC Tables numeriques et formulaires, Spes S.A., Vevey, Scott JA. A finite element model of heat transport in the human eye, Phys MedBiol, 1988, 33, 2: Polhamus G.D., Welch A J Threshold lesion temperatures in argon laser. Irradiated Rabbit Eyes, J of Heat Transfer, 1975, 8, Mainster M.A., White T.J. Allen R.G. Spectral dependance of retinal damage produced by intense light sources, J Optical Soc Am, , 6:

8 74 Simulations in Biomedicine IV Table 1. Thermal-physical constants values in the ocular media Eye region Cornea Aqueous humor Lens Ciliary body Iris Vitreous humor Retina * [W/m K] P [kg/m'] *P [J/kg K] H [kg/ms] Pr ["Kl Nomenclature Cp specific heat at constant pressure gi gravitational force vector H heat generation k thermal conductivity p fluid pressure Phs laser power / time T temperature l'a ambient temperature % human body temperature iti Eulerian fluid velocity components *i Cartesian coordinates a absorption coefficient of the tissue PT thermal volume expansion coefficient Si,- Kronecker delta S^ = 1 if i = j and 8^= 0 O^ laser spot diameter T, percent transmittance of tissue // dynamic viscosity P density

9 Simulations in Biomedicine IV 75 (a) (b) Figure 1: (a) Three dimensional meshing of the human eye (b) Example of 8 nodes brick element.

10 76 Simulations in Biomedicine IV o-o-o o o-o-o-o o< Temperature, T [ C] A Conduction ^*^*S^ -o Conduction and Convection 1 i Pupillary axis, x [mm] Figure 2: Steady temperature distribution along the pupillary axis laser beam Figure 3: Laser spot position and representative nodes of temperature computation (see figure 4)

11 370 T Simulations in Biomedicine IV 77 Node 133 Node 135 Node 118 Node 141 (a) 1-0, ,0 Time, t [s] i 1,5 2,0 (b) 0,0 Figure 4: Transient temperature distribution (a) on iris and (b) aqueous humor for 0.4 W laser beam power, 0.5 mm spot size and 0.5 s time exposure

12 Simulations in Biomedicine IV 0,0 29 0,0 Figure 4: Transient temperature distribution (c) on cornea and (d) lens for 0.4 W laser beam power, 0.5 mm spot size and 0.5 s time exposure

13 Simulations in Biomedicine IV 79 \ \ (a) (b) \ \ \ \, \ \, AAn ' - N \ V- O- \\\\^' X.', \ \ >, I / //, Mil/, \ \ \ \ i i i / / u/// i.'.\v.,' ->m Figure 5: Velocity field (a) after s of laser irradiation and (b) 0.3 s of relaxation; 0.5 mm spot size, 0.4 W laser beam power and 0.5 s time exposure.

Thermal modelling of the human eye exposed to infrared radiation of 1064 nm Nd:YAG and 2090 nm Ho:YAG lasers

Environmental Health Risk V 221 Thermal modelling of the human eye exposed to infrared radiation of 1064 nm Nd:YAG and 2090 nm Ho:YAG lasers M. Cvetković 1, A. Peratta 2 & D. Poljak 1 1University of Split,