Thermal Modelling of the Human Eye Exposed to Laser Radiation

Transcription

1 Thermal Modelling of the Human Eye Exposed to Laser Radiation Mario Cvetković, Dragan Poljak, Andres Peratta Department of Power Engineering, University of Split, Croatia, Department of Electronics, University of Split, Croatia, Wessex Institute of Technology, UK, Abstract Nowadays, various lasers are widely used in medical surgery procedures involving the human eye. The most important issue with laser eye surgery is estimation of temperature rise in eye tissues due to absorption of high intensity laser radiation. Model of the human eye, using the finite elements method, is developed in order to study the temperature distribution in the human eye subjected to radiation by several lasers, covering the electromagnetic spectrum from ultraviolet to visible and infrared. The mathematical model is based on the space time dependent Pennes bioheat transfer equation supplemented with natural boundary condition equations for cornea and sclera. The results could prove to be useful in predicting the damage to intraocular tissues due to heating by laser radiation, but alongside the thermal effects, it could be used to evaluate other interaction mechanisms, such as photoablation, plasma induced ablation and photodisruption. I. INTRODUCTION In the last few decades lasers have become implemented in a number of medical surgery procedures. Laser systems, with wavelengths covering electromagnetic spectrum from ultraviolet to visible and infrared, are nowadays used on a daily basis. From increasing number of laser uses and spread of different types of eye surgeries, the need for quantitatively understanding of basic laser-tissue interaction has arisen. Due to this fact, a number of models were developed to predict the various processes inside the human eye, with the ultimate goal of helping the medical doctors in minimizing the possible damage to intracoular tissue while operating with laser. Light interacts with tissue in four different ways: transmission, reflection, scattering, and absorption. Among these, absorption is perhaps the most important process in modelling this interaction, since absorbed photon energy by tissue could be reemitted as radiant energy or transformed into heat. Blood flow is a key element in thermoregulation of the living organism. Human eye is a special case, since the blood flow cannot regulate the heating inside the ocular tissue, so, thermal model of human eye is required. In order to derive a model which describes laser thermal effects, heat transfer needs to be taken into account. In biological tissues, this is achieved by using the Pennes bioheat transfer equation, which takes into account different types of transfer processes. This model aims to provide the results for temperature distribution evolution inside the human eye, exposed to several laser wavelengths, 193 nm ArF excimer laser, nm Ruby laser, and 1064 nm Nd:YAG laser, widely used in ophthalmology operations, such as photorefractive keratectomy (PRK) and laser in situ keratomileusis (LASIK), posterior capsulotomy (after-cataract), peripheral iridotomy (treatment of glaucoma), and for retinal photocoagulation [1], [2], [3]. II. THE MATHEMATICAL STATEMENT A. Historical Background The first attempts to quantify the effects of electromagnetic radiation on temperature rise inside the eye were made by Taflove and Brodwin [4], and Emery et. al. [5], who computed the intraocular temperatures in the eye due to microwave radiation using the finite difference and finite elements method, respectively. Latter, Scott [6], [7] developed a 2D FEM model of heat transfer in human eye to study the cataract formation due to infrared radiation. Also, Lagendijk [8] used finite difference method to calculate the temperature distribution in human and animal eyes during hyperthermia treatment. Mainster made one of the earliest analysis of the thermal response of clear ocular media to infrared laser radiation [9], and Amara [10] developed a thermal model of the human eye exposed to laser radiation using the finite elements. Sbirlea and L Huillier [11] modeled the human eye exposed to argon laser using the 3D FEM, and Chua et. al. [12] studied the temperature distribution within the human eye subjected to a laser source. Recently, authors use even more powerful numerical methods to solve heat transfer problems in human eye. Dodig [13] used hybrid FEM/BEM to investigate the influence of high frequency time harmonic EM fields on the human eye and Peratta [14] modeled the human eye exposed to conductive keratoplasty procedure, using the 3D boundary element method. B. Human Eye Modelling Eye is a very complex optical system, although relatively small organ in the human body. This ovoid organ measures about 24 mm in length (along pupillary axis) and 23 mm in diameter. In modelling the eye, we have assumed it to be a solid structure of given dimensions, consisting of total of eight homogeneous tissues, namely, cornea, aqueous humour, cilliary body, lens, vitreous humour, retina, choroid and sclera.

2 transfer [22]. In Pennes model, the rate of tissue temperature increase is given by the sum of the net heat conduction into the tissue, metabolic heat generation, and the heating (cooling) effects due to arterial blood flow: Fig. 1. Parameters required for modelling thermal interaction. From [15] The spatial extent and degree of tissue damage depends primarily on laser parameters such as wavelength, power density, exposure time, spot size, and repetition rate, but also on optical tissue properties like absorption and scattering coefficients, and thermal tissue properties such as heat capacity and thermal conductivity needs to be taken into account. Fig. 1 depicts the use of important parameters required for modelling thermal interaction such is the laser-eye case. Heat generation inside given tissue is determined by laser parameters and optical tissue properties, but even when no external sources are present, parameters such as volumetric perfusion rate of blood W b and the internal volumetric heat generation Q m, i.e. metabolic rate, needs to be taken into account. These are always present in the organism, and are responsible for the constant temperature of body. On the other hand, the transport of heat to the surrounding tissues is solely characterized by the tissue thermal conductivity k and specific heat capacity C. Data on these tissue parameters are given in Table I, taken primarily from [16]. Coefficient of absorption is the most important among optical tissue properties. In biological tissues, absorption is mainly due to the presence of water molecules, proteins, pigments, and other macromolecules, and is governed by law frequently known as Beer s or Lambert Beer s law. The absorption coefficient strongly depends on the wavelength of the incident laser radiation as seen from Fig. 2. In the infrared region of the spectrum, it can be primarily atributed to water molecules, while proteins and pigments are main absorbers in the UV and visible range of the spectrum [17], [18], [19]. Values for absorption coefficent of various eye tissue are given in Table I, taken from [20]. C. Heat Transfer in Human Eye The mathematical model of the human eye is based on the Pennes bioheat transfer equation [21], since it still is the foundation for all mathematical analysis in the field of bioheat ρc T t = (k T ) + W bc pb (T a T ) + Q m + H (1) Bioheat equation is extended with the new term H representing heat generated inside the tissue due to some external source. In our case, this source is laser radiation, but it could be any other source of electromagnetic radiation. This equation is supplemented with natural boundary condition equations for cornea, sclera and domain inside the eye, respectively: n = h c (T T amb ) + σɛ ( T 4 Tamb 4 ) Γ1 (2) n = h s (T T a ) Γ 2 (3) n = 0 Γ 3 (4) where k is specific tissue thermal conductivity given in Table I, h c heat transfer coefficient of cornea, h s heat transfer coefficient of sclera, σ Stefan Boltzmann constant, ɛ emissivity of the corneal surface, T amb temperature of the ambient air anterior to the cornea, and T a arterial blood temperature taken to be 36.7 C. The value for h c is taken to be 14 W/m 2 C, h s is 65 W/m 2 C, emissivity of the cornea is 0.975, and the value of ambient temperature is 30 C. Eqs. (2) and (3) describe the thermal exchange between cornea and surrounding air due to convection and radiation, and thermal exchange between sclera and ocular globe due to convection only, respectively. Second term on the right handside of Eq. (2) is approximated by σɛt 3 F (T T amb), where T F is the melting temperature of the substance [10]. The value for T F is taken to be 100 C. Representation of the whole eye domain with associated boundaries are depicted on Fig. 3. Fig. 2. Absorption coefficient of cornea and retina Fig. 3. Illustration of the domain meshed with triangular elements. Notice the two boundaries: Γ 1 on cornea and Γ 2 on sclera

3 TABLE I THERMAL AND OPTICAL PROPERTIES OF VARIOUS HUMAN EYE TISSUES Internal volumetric Thermal conductivity Tissue type Volumetric perfusion Specific heat Abs.coef. Abs.coef. Abs.coef. rate W b heat generation Q m k capacity C (1064nm) (694.3nm) (193nm) Vitreous humour Lens Aqueous humour Cornea Sclera Cilliary body Choroid Retina D. Modelling the Laser Source Energy density H(r, z, t), absorbed by the eye tissue at the n th node with cylindrical coordinates (r,z), is given by a product H(r, z, t) = αi(r, z, t) (5) where α is the wavelength dependent absorption coefficient of the specific tissue, given in Table I, and I is the irradiance of the n th node, given by ) ) I(r, z, t) = I 0 exp ( 2r2 w 2 αz exp ( 8t2 τ 2 (6) where I 0 is the incident value of intensity, w is the beam waist, and τ is the pulse duration. For a given laser parameters, the irradiance at the cornea is calculated from I c = 4P d 2 (7) cπ where P is laser power and d c is a beam diameter on the cornea. Taking into account the focusing action of the lens, diameter of the image and irradiance on the retina is calculated from and d r = 2.44 λf d p (8) I r = I c d 2 p where λ is laser wavelength, f focal distance of the lens, and d p diameter of pupil opening. From these values, we now interpolate the intermediate values for the beam width, irradiance and energy density along the beam path. The parameters for Ruby laser, used in our calculations were; pulse duration 1 ms, laser power 0.15 W, beam diameter on the cornea 8 mm, and pupil diameter 7.3 mm. The parameters for Nd:YAG laser were; pulse duration 1 ms, laser power 0.16 W, beam diameter on the cornea 8 mm, and pupil diameter 8.9 mm. Laser pulse was sampled in ten 0.1 ms steps, and in each of these steps, calculation of temperature distribution has been done. Following the laser cut off, for the next five steps (2 s each), tissue cooling was simulated. ArF excimer laser was simulated by 10 successive, 14 ns long pulses, of 2 J/cm 2 energy density each, beam diameter on the cornea 1 mm, and laser off time between pulses of 10 µs. d 2 r (9) E. Numerical Method Analytical solution of the Pennes bioheat transfer equation is limited to a few simple geometries with high degree of symmetry, but using the Finite Elements Method we are able to solve problems on complex geometries such as human eye. The equation (1) is discretized in two spatial dimensions and solved using the weak formulation and the Galerkin Bubnov procedure. A total number of 21,595 triangular elements and 11,094 nodes were generated using the GID 7.2 mesh generator. Algorithm written in MATLAB was used for processing part. First, the equation is solved for the steady-state case, i.e. when no external sources are present, and latter, these results will be used as initial conditions in the time domain analysis with included external source, i.e. laser radiation. A. Steady-State Case III. RESULTS The results for the steady-state case temperature distribution in the human eye along the pupillary axis are shown on Fig. 4. Steady state case results, used latter as initial condition for time domain analysis, are in a good agreement with number of papers [10], [6], [12], [23], [24]. One can note on Fig. 4 the steady rise of temperature from anterior to posterior parts of the eye, with steapest slope being within lens tissue. Reason for this is lowest value for thermal conductivity of lens, compared to other eye tissues. Fig. 4. Steady-state case temperature distribution along the eye pupillary axis. Length from anterior of cornea to posterior of sclera equals 24 mm.

4 B. Time Domain Analysis 1) nm Ruby Laser: Maximum temperature of C, for Ruby laser, is obtained on a node on sclera, as shown on Fig. 5, detailing the temperature distribution around posterior part of the eye. Same figure on the right shows also the temperature evolution on selected nodes. Using similar setup parameters, Amara [10] calculated maximum temperature increase of 97 C, independent of laser exposure time, while we believe that exposure time is very important parameter, and significantly influence the temperature rise in human eye. Also, we noticed that author switched values for tissue density and specific heat capacity. 2) 1064 nm Nd:YAG Laser: For Nd:YAG laser, maximum temperature of C, is obtained on the same node on sclera, as shown on Fig. 6, detailing the temperature distribution around posterior part of the eye. Same figure on the right shows also the temperature evolution on selected nodes. Same paper [10], reported maximum temperature increase of 74 C. One should expect the higher temperature values for the Ruby laser, due to higher absorption coefficient values of tissues on that wavelength, compared to those of Nd:YAG, but Nd:YAG calculation used larger value for diameter of pupillary opening, which significantly affects the amount of light entering the eye. Table II depicts the effect of pupillary beam diameter on temperature rise on nodes number 1985, 10421, 10537, and 10913, situated on the lens, vitreous humour, vitreo-retinal boundary, chorio-retinal boundary and sclera, respectively. Values are calculated using the setup for Ruby laser. TABLE II THE EFFECT OF THE PUPIL BEAM DIAMETER ON THE TEMPERATURE DISTRIBUTION ON THE SELECTED NODES IN THE EYE d p (mm) No.1985 No No No No Animated results of evolution for the whole eye temperature distribution could be found at [25]. 3) 193 nm ArF Excimer Laser: Results for ArF excimer laser setup show high temperature rise in the anterior part of the eye, especially cornea. This is important since excimer lasers are used in surgical operations that reshape the surface of cornea by process of photoablation. We obtained from our calculation a steady temperature rise to approximately 200 C, due to 10 succesive laser pulses, as shown on Fig. 7, on the right. On the same figure, on the left, temperature distribution in the whole eye, can be seen. It has been known that lens functions as an optical filter for UV radiation, and our results support this statement. While temperature of the anterior parts of the eye rise steaply, crystalline lens seams to prevent the significant rise of temperature in posterior parts of the eye, thus acting as a heat shield for this particular wavelength. Another reason for such behavior is due to high absorption coefficient of cornea, as seen from Fig. 2, i.e. due to it s high value, almost all UV radiation is absorbed in the corneal layer. IV. CONCLUSION Thermal model of the human eye exposed to laser radiation, based on the space time dependent Pennes bioheat transfer equation, has been developed to calculate the temperature distribution. Calculation has been done for wavelengths of 193 nm ArF excimer, nm Ruby, and 1064 nm Nd:YAG lasers, covering the electromagnetic spectrum from ultraviolet, visible to infrared. Results show maximum temperatures of C and C in posterior part of eye, for Ruby and Nd:YAG wavelength, respectively, while temperature around 200 C is obtained for the ArF laser. Analysis have shown that pupillary opening significantly affects the maximum temperature achieved in the eye. Also, results for UV wavelength confirm that lens functions as an optical filter, thus acting as a heat shield for posterior eye tissues. REFERENCES [1] K. Thompson, Q. Ren, and J. Parel, Therapeutic and diagnostic application of lasers in ophthalmology, Proceedings of the IEEE, vol. 80, no. 6, [2] J. Parrish and T. Deutsch, Laser photomedicine, IEEE Journal of Quantum Electronics, vol. 20, no. 12, pp , [3] M. W. Berns, L. Chao, A. W. Giebel, L. H. Liaw, J. Andrews, and B. VerSteeg, Human corneal ablation threshold using the 193-nm ArF excimer laser, Investigative Ophthalmology & Visual Science, vol. 40, no. 5, pp , Apr [4] A. Taflove and M. Brodwin, Computation of the electromagnetic fields and induced temperatures within a model of the microwave-irradiated human eye, Microwave Theory and Techniques, vol. 23, pp , [5] A. Emery, P. Kramar, A. Guy, and J. Lin, Microwave induced temperature rises in rabbit eyes in cataract research, Journal of Heat Transfer, vol. 97, pp , [6] J. Scott, A finite element model of heat transport in the human eye, Physics in Medicine and Biology, vol. 33, no. 2, pp , [7], The computation of temperature rises in the human eye induced by infrared radiation, Physics in Medicine and Biology, vol. 33, no. 2, pp , [8] J. Lagendijk, A mathematical model to calculate temperature distributions in human and rabbit eyes during hyperthermic treatment, Physics in Medicine and Biology, vol. 27, no. 11, pp , [9] M. A. Mainster, Ophthalmic applications of infrared lasers thermal considerations, Investigative Ophthalmology & Visual Science, vol. 18, no. 4, pp , Apr [10] E. Amara, Numerical investigations on thermal effects of laser-ocular media interaction, International Journal of Heat and Mass Transfer, vol. 38, no. 13, pp , [11] G. Sbirlea and J. L Huillier, A powerful finite element for analysis of argon laser iridectomy - influence of natural convection on the human eye, Transactions on BIomedicine and Health, vol. 4, pp , Apr [12] K. J. Chua, J. C. Ho, S. K. Chou, and M. R. Islam, On the study of the temperature distribution within a human eye subjected to a laser source, International Communications in Heat and Mass Transfer, vol. 32, pp , [13] H. Dodig, EM and thermal modelling of the human eye, Master s thesis, Wessex Institute of Technology, September [14] A. Peratta, Electromagnetic modelling of human eye exposed to Conductive Keratoplasty, WIT Transactions in Biomedicine and Health, vol. 12, no. VII, pp , Jul [15] M. H. Niemz, Laser-Tissue Interactions: Fundamentals and Applications, Third, Enlarged Edition. Springer-Verlag, Berlin, [16] S. C. DeMarco, G. Lazzi, W. Liu, J. D. Weiland, and M. S. Humayun, Computed SAR and thermal elevation in a 0.25-mm 2-D model of the human eye and head in response to an implanted retinal stimulator Part I: Models and methods, IEEE Transactions on Antennas and Propagation, vol. 51, no. 9, pp , [17] W. Makous and J. Gould, Effects of lasers on the human eye, IBM Journal of Research and Development, vol. 12, no. 3, pp , 1968.

5 Fig. 5. Detail of temperature distribution around posterior part of the eye, and temperature evolution of nodes in vitreous, vitreo-retinal boundary, chorio-scleral boundary and sclera for 1 ms exposure and relaxation after source cancelling, achieved with nm Ruby laser Fig. 6. Detail of temperature distribution around posterior part of the eye, and temperature evolution of nodes in vitreous, vitreo-retinal boundary, chorio-scleral boundary and sclera for 1 ms exposure and relaxation after source cancelling, achieved with 1064 nm Nd:YAG laser Fig. 7. Detail of temperature distribution in the whole eye and maximum temperature evolution of a point in anterior part of the eye, due to 10 pulses of 193 nm ArF excimer laser [18] J. Krauss, C. Puliafito, and R. Steinert, Laser interactions with the cornea, Survey of Ophthalmology, vol. 31, no. 1, pp , [19] L. Carroll and T. R. Humphreys, Laser-tissue interactions, Clinics in Dermatology, vol. 24, pp. 2 7, [20] M. Cvetkovic, Analysis of the Temperature Distribution to the Human Eye Exposed to Laser Radiation, 2008, Wessex Institute of Technology, Master Thesis, to be published. [21] H. H. Pennes, Analysis of tissue and arterial blood temperatures in the resting human forearm. 1948, Journal of Applied Physiology, vol. 85, no. 1, pp. 5 34, Jul [22] W. J. Minkowycz, E. M. Sparrow, and J. Y. Murthy, Handbook of Numerical Heat Transfer, Second Edition. John Willey & Sons, Inc., New York, [23] E. Ng and E. Ooi, FEM simulation of the eye structure with bioheat analysis, Computer Methods and Programs in Biomedicine, vol. 82, pp , [24] E. Ooi, W. Ang, and E. Ng, Bioheat transfer in the human eye: A boundary element approach, Engineering Analysis with Boundary Elements, vol. 31, no. 6, pp , [25] mcvetkov/mfield/softcom2008/.