A Powerful Finite Element for Analysis of Argon · 2014. 5. 13. · discretized in three spatial...

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 of Advanced 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 convectionheat transfer due to an argon laser iridectomy and to explain in which way the corneal burnsand lens opacities depend on the laser parameters. The equations of mass, momentum andenergy conservation, supplemented by boundary and initial conditions are discretized in threespatial dimensions and in time according to respectively the finite element method and to animplicite integration scheme. The contraction burn process was studied using shot of 0.5 s induration, 0.5 mm in size and a power level of 0.4 W. The influence of the natural convectionand the phenomena of cornea and lens overheating are presented and discussed.

1 Introduction

The field of possible laser applications in intraocular microsurgery isconsiderably widened. During the past few years the argon laser has become anincreasingly used modality for the treatement of angle-closure glaucomathrough the production of peripheral iridotomy. This treatement may benecessary when the pupillary margin of the iris is adherent to the lens impairingthe aqueous flow and causing a harmful rise of intraocular pressure (from 13mm Hg to 20 - 22 mm Hg). As a result, mechanical strains of tissue occur atsites of intrinsic weakness and contribute to damage the sensitive fibres in theoptic nerve, e.g. Le Grand [1], Thompson [2].

To operate an iridectomy, the energy of the argon laser beam (488 nm -514.5 nm wavelength) is focused on a small area of the iris. The heat source

Transactions on Biomedicine and Health vol 4, © 1997 WIT Press, www.witpress.com, ISSN 1743-3525

68 Simulations in Biomedicine IV

deposition is not only function of laser irradiation parameters but is alsodependent upon the optical properties of the tissue. The process leads tovaporization and tissue removal and includes three steps : heating, boiling andablation. Thus, a small opening in the iris is performed and the pressure betweenthe anterior and posterior chambers of the eye is equalized.

Previous studies have reported sophisticated theoretical models to explainand predict laser-induced thermal damage in the retina, e.g. Cain and Welch [3],Amara [4], and these ultimately rely on laboratory measurements for verificationand refutation. To our knowledge, theoretical investigations devoted to the laserproduction of peripheral iridectomy have not yet been studied. However,complications such as corneal endothelial burns and lens opacities have beenexperienced and reported by Ritch [5] and Pollack [6] in case of CW argon laserlong pulse settings (from 0.1 s to 0.5 s).

The theoretical research described in this paper was undertaken in order tounderstand the primary effects of the laser energy immediately after absorptionin the iris. The main absorption area for argon laser pulses is expected to be themelanin layer of the iris tissue (pigment epithelium P.E.).

Computations are based upon a finite element method to predict the thermaleffects induced on surrounding eye tissues, specifically on both corneaendothelium and lens by a CW argon laser iridectomy. To quantify these effectsthe intraocular temperature and the velocity field are calculated during theheating phase of the iris tissue.

2 Mathematical statement

2.1 Theoretical model

To model mathematically the heat transport process during an iridectomyprocedure the eye is divided in seven regions: cornea, aqueous humor, lens, iris,ciliary body, vitreous humor and retina. Each region is assumed to behomogeneous. The problem is formulated in three dimensional Cartesiancoordinates (x,y,z), with the x axis constituting the pupillary axis. The unknowntemperature T is a function of (x,y,z) in steady state case and of (x,y,z,t) intransient case.

The aqueous and vitreous humors are assumed to be viscous, incompressibleand Eulerian fluids The buoyancy force caused by density variation resultingfrom temperature rise between the ambient and the human body is modeled byBoussinesq approximation.

It follows that the physical system governing the aqueous and vitreoushumors flow is the representation of momentum, mass and energyconservations, e.g. FIDAP [7], Sacadura [8]:


Simulations in Biomedicine IV 69

+H (3)

supplemented with temperature boundary conditions equations where thetemperature is prescribed (4) and two initial condition equations which lie in theinitial 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 regionand independent against temperature variations. The thermal-physical constantsvalues 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 adimensionless formulation of the problem provides a measure of the relativeimportance of the terms in the equations and reduces the potentially largedifferences in orders of magnitude that may occur among these terms in fieldequations.

2.2 Numerical procedure

The mathematical model presented in the previous section is treated withnumerical procedures to obtain solutions to the temperature and velocitydistributions of the human eye. The system of partial differential equations isdiscretized in three spatial dimensions according to the finite element method(FEM) and in time according to the first order backward Euler algorithm with avariable increment option, maximum time increment and maximum increase intime increment per time step.

The continuum region of the human eye is divided in seven main regions andevery region in 8 nodes brick elements (see figure l(b)) as following : cornea in1036 elements, aqueous humor in 6912 elements, lens in 3810 elements, iris in2880 elements, ciliary body in 1312 elements, vitreous humor in 24340 elementsand the retina in 2508 elements as shown in figure l(a).

Within each element, the velocity, pressure and temperature fields areapproximated by :



where U ,.(/), P and T are column vectors of element nodal point unknowns

and #>, i// and $ are column vectors of the interpolation functions. Substitutionof 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 interpolationfunctions of each element.

2.3 Laser heat source

In case of laser iridectomy procedure, the laser beam strikes on the anteriorcorneal surface, penetrates the aqueous humor and is absorbed in the melaninlayer of the iris tissue which is assumed homogeneous and 0.2 mm thick.

We considered a top flat type heat source H absorbed in the iris melanin layeraccording to Beer's law with a percent transmittance T, and an absorptioncoefficient 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 laserbeam is entirely transmitted by the cornea and the aqueous humor: Tr - 1 . Thisassumption overestimates the laser heat source. Typical values of the absorptioncoefficient available for the pigment epithelium such as 637.6 cm"*, 832 cm and900 cm"* have been reported by Cain [3], Polhamus [11] and Mainster [12]respectively. Among these an average value a = 800 cm"' was used. Becausethe absorption depth of the argon laser (I/a) on the iris is small compared to thelaser beam radius, we assumed irradiance independent of the distance from thebeam 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) = 0when the laser beam is "off' and H(x,t) = /o a exp(-a x) when the laser beam is"on".



3 Results

3.1 Steady state case

To simulate the heat transfer in the human eye during an argon laser iridectomy,temperature distribution and velocity field in steady case and without heatsource are necessary.

The eye is submitted to natural convection in vitreous and aqueous humorsdue to temperature difference setted as boundary conditions between the humanbody Tb= 37°C and the ambient 7 =20°C. The Reynolds number value is equalto 722 and the flow is then assumed to be laminar.

The figure 2 gives the temperature distribution along the pupillary axis incases of conduction and both conduction and convection heat transfer. We cansee that the temperature decreases continually from retina to cornea accordingto a conduction heat transfer only. When both conduction and naturalconvection phenomena are taken into account , a great part of the intra-oculartemperature distribution becomes flat.

3.2 Transient case

The solution of the transient state problem is obtained by including the laserenergy volume density as heat source term in the energy equation. For bothconduction and convection heat transfer the equations of energy, mass andmomentum conservation associated with the boundary conditions representedby the ambient and human body temperatures and the initial conditions given bythe steady state temperature distribution and velocity field are solved.

We used the irradiation parameters encountered for a contraction burn incase of glaucoma treatement, e.g. Ritch [5]: 0.5 mm spot size, 0.5 s durationand 0.4 W laser beam power. This type of burn is designed to contract the iristissue and not to penetrate it. Thus, the vaporization and the tissue removal arenot treated.

The iridectomy site is placed superiorly at 4.33 mm of the pupillary centerand 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 ofrelaxation period is depicted in figure 4. For a laser beam setting of 0.4 W thetemperature on different nodes of the iridectomy site increases rapidly andreaches a maximum peak of 370° C at the surface of the iris and at the beamcenter. After the laser source has been turned-off, the temperature decreases.The same shape of the temperature curve can be observed on the aqueoushumor with a maximum peak of 170° C (see figure 4(b)).

As depicted in figure 4(c), on the cornea-iris angle, precisely on the corneaendothelium nodes by the side of the iridectomy site, the temperature increasesduring the relaxation phase. The same phenomena is observed on the lens,behind the iris by the side of the iridectomy site (see figure 4(d)).

The velocity field in the aqueous and vitreous humors by the side of theiridectomy site after 0.015 s of argon laser exposition and 0.3 s of relaxation



time are depicted respectively in figures 5(a) and 5(b). The velocities vary from0.254 mm/s to 2.03 mm/s.

4 Discussion and Conclusion

As we can see in figure 2, the convection currents induce an importantmodification of the temperature distribution. Along the pupillary axis thetemperatures are higher on vitreous humor, lens and aqueous humor in case ofboth conduction and convection heat transfer that in case of single conduction.

The 3-D computation results of the steady conduction heat transfer agreewell with those obtained by Scott [10] and Amara [4] and based on a 2-Dsimulation. Thus, we used the temperature distribution and velocity field issuedfor 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 corneaendothelium by the side of the iridectomy site during the relaxation phase. After1.5 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 theside 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 thecornea-iris angle and between lens and iris is observed and a vortex appears (seefigures 5(a) and 5(b)).

In this work we presented the preliminary results of a 3-D numerical modelusing the finite element method to compute the temperature distribution and thevelocity field in the ocular media in case of argon laser iridectomy. Thecontraction 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 greatinfluence on temperature distribution. An overheating of lens and corneaendothelium by the side of the iridectomy site is observed. That may be anexplanation of corneal burns and lens opacities reported by ophtalmologists aspostoperative complications of argon laser iridectomy.

At present we are using the same theoretical model and numerical procedureto study the pathological human eye in case of angle closure glaucoma. Thegeometry of the diseased eye will allow us to estimate the real importance of thephenomena already noted. In addition, further investigations should beperformed 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, fluiddynamics engineer for scientific support.



References

1. Le Grand Y, El Hage SG. Physiological Optics, Springer Series in OpticalSciences, New York, 1980.

2. Thompson KP, Ren QS, Pare! JM. Therapeutic and Diagnostic Application oflasers in ophtalmology. Proceedings of the IEEE 1992; Vol. 80, 6: 838-860.

3. Cain C.P., Welch A J Measured and predicted laser-induced temperaturerises in the rabbit fundus, Investigative Ophtalmology, 1974, 13, 60-70.

4. Amara EH. Numerical investigations on thermal effects of laser-ocular mediainteraction, IntJ Heat Mass Transfer, 1995; 38, 13: 2479-2488.

5. Ritch R, Solomon IS. Laser treatement of glaucoma, J Ophtalmic Lasers,650-689, MO:CV MOSBY, 3d Ed, St Louis, 1989.

6. Pollack IP. Use of argon laser to produce iridotomies, Trans Am OphtalmolSoc, 1979,77,674-706.

7. Fluid Dynamics Analysis Package FIDAP 7.6, Fluid Dynamics International,Inc. Evanston, 1996.

8. Sacadura JF. Initiation aitx transfer ts thermiques, Technique etDocumentation, Paris, 1978.

9. Olza A, Taillard F, Vautravers E, Diethelm JC Tables numeriques etformulaires, Spes S.A., Vevey, 1978.

10. Scott JA. A finite element model of heat transport in the human eye, PhysMedBiol, 1988, 33, 2: 227-241.

11. Polhamus G.D., Welch A J Threshold lesion temperatures in argon laser.Irradiated Rabbit Eyes, J of Heat Transfer, 1975, 8, 457-462.

12. Mainster M.A., White T.J. Allen R.G. Spectral dependance of retinaldamage produced by intense light sources, J Optical Soc Am, 1970. 60, 6: 848-855.



Table 1. Thermal-physical constants values in the ocular media

Eye region

CorneaAqueous humorLensCiliary bodyIrisVitreous humorRetina

*[W/m°K]

0.580.580.400.580.580.6030.628

P[kg/m']

1050100010501000100010001000

*P[J/kg°K]4178399730003997399741784190

H[kg/ms]

0.001

0.001

Pr["Kl

0.0002

0.0002

Nomenclature

Cp specific heat at constant pressuregi gravitational force vectorH heat generationk thermal conductivityp fluid pressurePhs laser power/ timeT temperaturel'a ambient temperature% human body temperatureiti Eulerian fluid velocity components*i Cartesian coordinatesa absorption coefficient of the tissuePT thermal volume expansion coefficientSi,- Kronecker delta S^ = 1 if i = j and 8^

O^ laser spot diameter

T, percent transmittance of tissue// dynamic viscosityP density

= 0



(a)

(b)

Figure 1: (a) Three dimensional meshing of the human eye (b) Example of 8nodes brick element.



38

36 o-o-o—o—o-o-o-o o<

Temperature, T [°C] 34 -

32

30 -

28

26 -

24 -

22 -

90

* *S

-A— Conduction

-o— Conduction and Convection

— 1 i

10 15

Pupillary axis, x [mm]

20 25

Figure 2: Steady temperature distribution along the pupillary axis

laser

beam

Figure 3: Laser spot position and representative nodes of temperaturecomputation (see figure 4)



370 T

(a)

Node 133

Node 135

Node 118

Node 141

0,0 0

1 -

5 1,0

Time, t

i

1,5

[s]

2,0

0,0

(b)

Figure 4: Transient temperature distribution (a) on iris and (b) aqueous humorfor 0.4 W laser beam power, 0.5 mm spot size and 0.5 s time exposure


Simulations in Biomedicine IV

0,0

29

0,0

Figure 4: Transient temperature distribution (c) on cornea and (d) lens for0.4 W laser beam power, 0.5 mm spot size and 0.5 s time exposure



\ \(a)

(b)

\ \\ \,\

\, AAn' - N \ V-

O- • \\\\ 'X

• • • . ' , i .'.\v.

,' ->m

\ \ >,

• I / //,M i l / ,

\ \ i i i\ \ i / /

u///

Figure 5: Velocity field (a) after 0.015 s of laser irradiation and (b) 0.3 s ofrelaxation; 0.5 mm spot size, 0.4 W laser beam power and 0.5 s time exposure.


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