UNIVERSITY OF CINCINNATI

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1 UNIVERSITY OF CINCINNATI Date: I,, hereby submit this work as part of the requirements for the degree of: in: It is entitled: This work and its defense approved by: Chair:

2 Influence of Laser Parameters on Selective Retinal Photocoagulation for Macular Diseases A thesis submitted to the Division of Research and Advanced Studies of the University of Cincinnati in partial fulfillment of the requirements for the degree of MASTER OF SCIENCE in the Department of Mechanical, Industrial and Nuclear Engineering of the College of Engineering 005 by Pradeep Gopalakrishnan Bachelor of Engineering, PSG College of Technology, Coimbatore, India 001 Committee Chair: Dr. Rupak K. Banerjee

3 ABSTRACT Selective retinal photocoagulation uses short pulsed lasers to coagulate retinal pigment epithelium selectively, while sparing the photoreceptors. The problem associated with the usage of short pulsed laser is the difficulty in determining the correct dosimetry parameters. This study quantifies the influence of laser parameters and variation of thickness and melanin concentration of RPE over the therapeutic range. The laser-tissue interaction is numerically investigated by analyzing the transient temperature in ocular tissues during the treatment. The rate process analysis for thermal injury is employed to estimate the selective damage of retina. The contours of Arrhenius integral value (Ω/Ω max ) presented in this study show both the area and magnitude of damage caused by various laser parameters. Results reveal that the µs pulsed-laser with green wavelength and Gaussian profile is relatively more effective for selective retinal treatment. The repetition frequency of 100 Hz selectively damages RPE, while frequency of 1000 Hz produces damage to neural retina and choroid located within µm from RPE interface. However, the Ω max value for 1000 Hz is orders of magnitude (~ 000 times) greater than that for100 Hz. The amount of threshold energy required for 14 µm thick RPE is ~5% less than that for 10 µm thick RPE, while 6 µm thick RPE requires ~35% more energy.

4 TABLE OF CONTENTS ABSTRACT NOMENCLATURE Chapter 1: INTRODUCTION Influence of Wavelength 1. Influence of Pulse Width 1.3 Influence of Repetition Frequency Rate process analysis Chemical kinetics 5 Chapter : METHODOLOGY 9.1 Geometry 9. Governing Equation 10.3 Comparison of results for different flow rate 11.4 Heat Source Distribution 1.5 Boundary Conditions 15.6 Rate Process Analysis of Thermal Damage 16.7 Finite Volume Method 17.8 Numerical Using Alternating Direction Implicit method 18 Chapter 3: RESULTS 3.1 Validation 5 3. Effect of Wavelength Effect of Laser Beam Profile Effect of Pulse Width Effect of Repetition Frequency Effect of Thickness and Melanin Concentration of the RPE 43 Chapter 4: DISCUSSION 46 CONCLUSION 48 REFERENCES 49

5 LIST OF FIGURES Figure 1: Unimolecular process activation and denaturization 5 Figure : Two-dimensional meshing of the eye model used in the present laser-tissue interaction study. (A)Axi-symmetric eye model (B) Close up view of area of interest 9 Figure 3: Comparison of temperature for different vitreous outflow 1 Figure 4: Validation of boundary condition 15 Figure 5: Simple two-dimensional model of eye used for numerical simulation 19 Figure 6: Comparison of ADI scheme results with fluent results for CW laser 1 Figure 7: Definition of various laser terms used in terms of pulse width and pulse power Figure 8: Comparison of baseline temperature (lowest temperature at the end of each pulse) of present results with experimental results by Schule et al (004). The laser parameters are: wavelength - green, profile - top-hat, pulse width µs, energy/pulse µj, frequency Hz, number of pulses - 30, spot diameter µm. 5 Figure 9: Comparison of laser heat source profiles along the depth for all three wavelengths (57, 650, and 810 nm). The regions of neural retina, RPE and choroid are shown in the graph. The green wavelength has high absorption in all three layers followed by the red and infrared rays. 6 Figure 10: Predicted maximum temperature along the center line of the laser beam for all three wavelengths. Other laser parameters are: profile - top-hat, pulse width - µs, energy/pulse - 10 µj, frequency Hz, number of pulses Figure 11: Contours of Ω/Ω max for all three wavelengths at the end of 50 pulses. For the same laser power, the green wavelength produces maximum damage to RPE (high Ω max ), while infrared produces minimum damage. The contours of red and infrared rays show damage to neural retina and choroid while green produce selective damage to RPE. 30 Figure 1: Comparison of temperature along radial direction at the end of 50 pulses for Gaussian and top hat laser beam profiles. Gaussian profile results in high temperature at the center of the beam. 31 Figure 13: Contours of Ω/Ω max for Gaussian and top hat profile. Gaussian profile produces high value of Ω max, as its power at the center is twice that of average power. Other laser

6 parameters are: wavelength - green, pulse width - µs, energy/pulse - 10 µj, frequency Hz, number of pulses Figure 14: Variation of pulse power and pulse-off duration required for different pulse widths in order to maintain constant average power of 0.01 W 33 Figure 15: Comparison of temperature for all three pulse widths µs, 0 µs, and 00 µs along the center line of the beam at the end of 1, 10, 5, and 50 pulses. The maximum temperature augmentation occurs within 10 pulses and further increase in number of pulses does not produce significant additional temperature rise. 34 Figure 16: Spreading of deposited energy after the pulse-on period for µs laser treatment. The o temperature of neural retina increases to 65 C at 50 µs after the end of 50 pulses. 35 Figure 17: Contours of Ω/Ω max for all pulse widths µs, 0 µs, and 00 µs. The µs pulse width has maximum Ω of.01e-1. The damage to neural retina and choroid is significant max in the cases of 0 µs, 00 µs, while the µs produce selective RPE damage. Other laser parameters are: wavelength - green, profile - top-hat, energy/pulse - 10 µj, frequency Hz, number of pulses Figure 18: Comparison of laser power and temperature for all for repetition frequencies of 100, 500 and 1000 Hz. (A) Variation of pulse-off duration and average power with repetition frequency. Same energy/ pulse of 10 µj is used for all repetition frequencies. (B) Transient temperature rise for all frequencies. The frequency of 1000 Hz shows strong cumulative effect in temperature with number of pulses. 39 Figure 19: Temperature along the center line of the laser beam for all repetition frequencies at the end of 50 pulses. The frequency of 1000 Hz produces highest temperature of ~ 110 C at RPE while frequencies of 500 and 100 HZ produces temperature of ~ 90 C and ~ 70 C, respectively. Figure 0: Contours of Ω/Ω o o for all repetition frequencies of 100, 500 and 1000 Hz. Frequency of 100 Hz produces selective damage to RPE, while 500 and 1000 Hz frequencies produces damage to neural retina located within µm from RPE interface. The value of Ω 1000 Hz. max max is very low (8.3e-3) for 100 Hz and high (14.08) for 37 o 40 4

7 Figure 1: Temperature along the center line of the laser beam for different RPE thicknesses (6, 10, 14 µm). The percentage absorptions of incident laser energy in RPE are 50, 47, and 35 for RPE thicknesses of 6, 10, and 14 µm, respectively. Other laser parameters are: wavelength - green, profile - top-hat, pulse width - µs, energy/pulse - 10 µj, frequency Hz, Number of pulses Figure : Contours of Ω/Ω max for all RPE thicknesses. The selectivity of thermal damage is unaffected by the change in RPE thickness. The 14 µm thick RPE has high Ω of.56e-1, while 6 µm thick RPE has a low value of 1.30e max value

8 LIST OF TABLES Table 1: The values of the thermo-physical properties of ocular tissues used in the numerical computations 11 Table : The values of optical properties of ocular tissues for green, red and infrared wavelengths. 14 Table 3: The value of pre-exponential factor and activation energy E for different temperatures used in the present study to compute the Arrhenius integral value Ω. 17 Table 4: Summary of runs. The case 1 is the validation of temperature with experimental results, respectively. Case serves as the reference baseline case. Laser parameters (in bold) are varied as shown in runs 3-11 to find their influence over the selectivity of the treatment. 4 a

9 NOMENCLATURE ρ Density of tissue (kg/m 3 ) C Specific heat capacity (J/Kg-K) T Temperature (K) S Laser heat source (W/m 3 ) K Thermal conductivity (W/m-K) µ a Absorption coefficient (1/m) µ s Scattering coefficient (1/m) φ Fluence rate (W/m ) R Radius of the laser beam (m) z Depth (m) r Radius (m) I 0 g E a A R u Ω Incident power of laser (W) Anisotropic factor Activation energy (J/mole) Frequency or Pre-exponential factor (1/s) Universal gas constant 8.3 (J/mole-K) Arrhenius integral value

10 Chapter 1: INTRODUCTION Laser retinal photocoagulation is used to treat a number of macular diseases such as agerelated macular degeneration (AMD), diabetic maculopathy, central serous retinopathy, choroidal neovascularization, etc. Most of the macular diseases are related to the decline in Retinal Pigment Epithelium (RPE) cell function. It has been suggested that cell proliferation and restoration of RPE barrier after the photocoagulation enhances the pumping mechanism, which produces the clinical effects of reducing subretinal fluid and improving macular pathology 1. During radiation treatment, the photons are absorbed in the tissues, which results in the excitation or increase in energy of the molecules. The increase in energy leads to a rise in temperature, which breaks the weak bonds in protein, enzymes and, thereby causing denaturation of proteins. Many proteins are affected in the temperature range of 50 o C to 80 o C. The degree of damage on a particular type of tissue depends on two factors, namely; magnitude and duration of temperature increase. Similar tissue damage can be produced either by long exposure to low temperature or by short exposure to high temperature. This reciprocity between exposure duration and temperature is the critical factor, which helped in the development of Selective Retinal Treatment (SRT). The SRT coagulates RPE selectively by short exposure to high temperatures. The radiation parameters such as wavelength, profile, pulse width, and repetition frequency govern the magnitude of temperature and its duration. Conventional laser photocoagulation treatment uses Continuous Wave (CW) laser with exposure time, in the range of ms and power, in the range of mw. The laser is primarily absorbed by the melanin granules in the RPE tissue. Since the exposure duration is 1

11 relatively long, the absorbed thermal energy spreads by diffusion into the neural retina and choroid destroying the photoreceptors. The laser scar expands postoperatively, which results in decrease in central visual sensitivity, epiretinal fibrosis, and reoccurrence of choroidal neovascularization. Alternatively, SRT uses short laser pulses, which minimizes the diffusion and hence spares neural retina. Many previous experimental studies substantiated the possibility of producing SRT with pulsed lasers and analyzed the effect of laser parameters to enhance the selectivity, which are briefly reviewed below. 1.1 Influence of Wavelength Mainster,3 studied the effect of wavelength based on absorption characteristics of melanin. They conferred the undesirable effects of blue wavelength and the advantage of using green or red lasers. Further, they recommended selection of wavelength based on type and severity of disease; near infrared rays are preferred over the visible lasers for severe subretinal hemorrhage, while visible rays are preferred for thin hemorrhage. However,many experimental studies 3 showed that the selection of wavelength for conventional CW photocoagulation is not a critical factor for better clinical outcome. 1. Influence of Pulse Width SRT using 5 µs laser pulses with pulse energy in the range of -10 µj at repetition rate of 500 Hz was first carried out in rabbits by Roider 4 with 110 mm retinal spot size. They analyzed both CW and pulsed lasers at various laser energies and exposure durations. They observed the proliferation of RPE cells after SRT, at the periphery of lesion towards the center of the irradiated site. Further, they showed that it is impossible to produce selective damage by CW laser, even with exposure duration in the order of few milliseconds.

12 SRT in humans was performed by Roider 5 with 1.7 µs green laser at 500 Hz for retinal spot diameter of 160 µm. The SRT disrupts RPE without producing ophthalmoscopically visible white or grey lesions. The white or gray lesion, formed during CW photocoagulation, are due to the destruction of photoreceptors, which alters the scattering characteristics of the neural retina. They also showed effectiveness of SRT in treating the macular diseases like central serous retinopathy (CSR), drusen maculopathy, and diabetic maculopathy. Thus, it is evident that for these macular diseases, the therapeutic efficacy is only related with destruction of RPE and not with the destruction of photoreceptors. Autofluorescence imaging by Framme 6 after the SRT showed leakage from the irradiated site, which proved the damaging of RPE by short pulsed lasers. Influence of pulse width and pulse number was studied by Framme 7 with rabbits using 1.7 µs, 00 ns green laser at constant repetition rate of 100 Hz. They measured threshold energy, energy required to produce 50% of the damage, for various pulse widths and pulse numbers. Their analysis was based on the safety margin factor, which is defined as the ratio of threshold energies required to produce 84% of angiographically visible lesion and 16% of ophthalmoscopically visible lesion. They showed high safety margin factor for short pulse width and high number of pulses. Roider 8,9 and Brinkmann 10 studied the primary interaction mechanism involved in nanosecond laser treatment. They showed that the temperature of a single melanin granule increases above 100 o C, which results in the vaporization of cells. For pulse width in the range 3

13 of microseconds, the threshold energy is proportional to n -1/4, where n is the number of pulses. Nanosecond laser treatment by Brinkmann 10 showed the deviation of threshold energy requirement from n -1/4 relation. Thus, their results substantiated that other mechanisms such as evaporation and explosion are also involved during nanosecond laser treatment. 1.3 Influence of Repetition Frequency Effect of duty factor, defined as ratio of pulse-on time to the total pulse time, was studied by Berger 11 with a simplified two dimensional model of eye. They analyzed the thermal localization by comparing temperature of RPE (T RPE ) to that of the deep choroid (T ch ). Comparison of CW with pulsed laser showed better localization (higher T RPE /T ch ) for pulsed lasers with low duty factor (%). Mainster discussed the additive effect for different duty factors and explained the temperature increase based on the n -1/4 law. This study described lower irradiance requirement for higher frequencies, which leads to reduced thermomechanical damage and iatrogenic hemorrhage. Summarizing the past literature above shows that although there have been several experimental studies, the influences of laser parameters are not well quantified owing to the difficulty in measuring the actual temperature during the photocoagulation. Furthermore, variation in thickness and melanin concentration of RPE contributes to error in experimental studies. Similar laser power could cause suprathreshold damage for heavy pigmentation, which produces damage to photoreceptors or could lead to subthreshold damage for low pigmentation, which does not produce the required therapeutic damage. Another important factor to be 4

14 analyzed is the thermal diffusion after each laser pulse-on time, which can cause thermal damage to photoreceptors and choroid. The current study is aimed at finding the therapeutic range of laser parameters: wavelength, profile, pulse width, and repetition frequency by studying the temporal and spatial variations of temperature. Further, the impact of variation of melanin concentration over threshold energy requirement is analyzed. The selectivity of the treatment is analyzed using rate process model for thermal damage, based on unimolecular chemical kinetics, which is discussed in detail in the following section. Detailed derivation of the rate process model has been described in many of the earlier studies17, Rate process analysis Chemical kinetics K a C * Energy C K b Native K 3 H* H Denaturized State Figure 1: Unimolecular process activation and denaturization Chemical kinetic models can be used to describe the damage in tissue obtained with laser irradiation. Thermal damage in tissue is similar to unimolecular process (tissue constituents transition from the native state to the damaged state). In a typical reaction process, thermally 5

15 active reactants jump over an energy barrier to form products as illustrated in Fig. 1. There is a time lag exists between molecular activation and denaturization. During this time lag, the molecules may either denature or relax back to the native state, as illustrated in Fig. Here H is the internal energy difference between native and denaturized state. The relative barriers are such that in the thermal damage of tissue, H* is almost always smaller than H. Hence the probability of denaturized cell to get back to native format is almost zero. The rate of damage formation is then proportional to only those molecules which remain activated. For a unimolecular process in native state, C, having an activated state, C*, with velocity constants K a, K b, and K 3, C + C K a C + C* K b C* Damaged molecules (at rate K 3 ) For this process, the rate of disappearance of native state molecules, [C], is given by d[ C] = K 3 [ C*] (1.1) dt where the bracket is used to indicate molar concentration. Generally [C*] is neither known nor calculable; however, at sufficiently low concentration of C* the steady state principle asserts that for short lived activated states the rate of formation can be considered equal to rate of disappearance. So the activated state, [C*], forms at a rate Ka [C], relaxes back to the inactivated state at rate K b [C] [C*], and denatures at rate K 3 [C*]. Consequently and so K a [C] = K b [C] [C*] + K 3 [C*] (1.) [C*] = K a [C] / (K b [C] + K 3 ) 6

16 Further based on overall reaction velocity K, the change of [C] is given as d[ C] = K [ C] (1.3) dt There are two limiting cases; first the concentration of remaining undamaged material [C] may be large enough that the deactivation at K b dominates the K 3 pathway, this results in first order process with overall rate constant (from Eq. 1. and Eq. 1.3) K = K 3 K a /K b (1.4) Second, if the remaining undamaged material concentration [C] is small, K 3 >> K b [C] K = Ka/ K 3 [C] (1.5) which results in second order process, d[ C] dt = (1.6) K [ C] where for simplicity, the [C] dependence has been removed from K = Ka/ K 3 [C]. Since at low concentration of [C] the damage process is saturated the first case is considered for thermal damage. Eq. 1.3 is a Bernoulli differential equation with the solution [ ] C(τ ) = C(0) exp Kdt (1.7) The value of reaction velocity K(s -1 ) is given by K = exp (- G*/RT) * RT/Nh (1.8) where R universal gas constant N Avogadro s number h Planck s constant G* Gibbs free energy = H* - T S* 7

17 and H* = E a - RT The value of entropy change is not calculable and hence determined by experiments. From Eq. 1.8 the value of K is K = A exp (-E a /RT) (1.9) where A = (RT/Nh) exp( S*/R) and E a H* (Since H* >> RT) From Eq. 1.8 and 1.9, the following parameter known as Arrhenius integral value is obtained t C(0) Ω( t) = ln = A [ Ea RuT t dτ C t exp / ( )] (1.10) ( ) 0 where frequency factor, A, and the activation energy are related to activation enthalpy H* and activation entropy S*. 8

18 Chapter : METHODOLOGY.1 Geometry Figure : Two-dimensional meshing of the eye model used in the present laser-tissue interaction study. (A)Axi-symmetric eye model (B) Close up view of area of interest 9

19 The geometrical description of the eye, given by Charles 1 is used to create an axisymmetric model of a complete eye as shown in Fig. A. This model consists of nine different eye compartments: cornea, lens, iris, aqueous humour, vitreous humour, neural retina, RPE, choroid, and sclera. The thickness of neural retina, RPE, choroid and the sclera are key parameters for the determination of correct laser heat source. The thickness of retina ranges from 500 µm near the optic nerve region to 50 µm near ora serrata 13. Similarly, the thicknesses of choroid and sclera have maximum values near optic nerve and minimum values near the equator 13,14. Hence, the average thicknesses of 10 µm, 190 µm, 50 µm, and 700 µm are used for RPE, neural retina, choroid, and sclera, respectively 15. The close up view of RPE, neural retina and choroid regions is shown in Fig. B.. Governing Equation The calculation of laser-tissue interaction is carried out by solving the following transient heat conduction equation, with temporal and spatial dependent laser heat source, throughout the various regions of ocular tissues using finite volume discretization technique: ( ρct ) t = ( k T ) + S( z, r, t) (.1) where k is the thermal conductivity (W/m-K), ρ is the density (kg/m 3 ), C is the specific heat capacity (J/kg-K), and S is the volumetric laser source term (W/m 3 ), which varies spatially and temporally (discussed in detail in Section.4). The thermo-physical properties of each ocular tissue are critical for the evaluation of actual temperature for laser photocoagulation. The properties of ocular tissues based on previous study 13 are given in Table 1. The properties are similar to that of water as all tissues are 10

20 composed mainly of water. It is very difficult to measure the thermal conductivity of choroid owing to the blood perfusion through it. Thus, the properties of choroid are chosen to be similar to that of retina. Properties Vitreous Humor 13 Retina 13 Choroid Sclera 13 Thermal Conductivity (W/m-K) Density (kg/m 3 ) Specific heat capacity (J/kg-K) Table 1: The values of the thermo-physical properties of ocular tissues used in the numerical computations.3 Comparison of results for different flow rate In normal eye, vitreous flows through retina at a flow rate of 0.14 µl/min 16. In order to delineate possible cooling effect by the vitreous, an initial steady state computation is conducted by including convection terms in the governing equation. Fig. 3 shows the comparison of results with no convection and with ocular flow rate and times the normal flow. The temperature is reduced by o C for times the normal flow rate. Hence, the convection due to ocular hydrodynamics is neglected for all the results presented in this study. 11

21 Figure 3: Comparison of temperature for different vitreous outflow.4 Heat Source Distribution The heat generation due to laser-tissue interaction depends on two parameters namely, fluence rate and the absorption coefficient. It is given by, S( r, z) = µ a ( z) φ( r, z) (.) where µ is the absorption coefficient, which depends on type of tissue and incident laser a wavelength, and φ ( r, z) is the fluence rate (W/m ). The fluence rate depends on many parameters, including the laser intensity, diameter, and the optical properties of tissues such as scattering and absorption. If absorption dominates scattering ( µ a >> µ s ), the fluence rate is governed by Beer-Lambert law as: I 0 φ = exp πr ( z) µ a for µ a >> µ s (.3) 1

22 where R is the radius of laser spot (m), I 0 is the incident laser power (W) and z is the depth of the irradiated material (m). For materials whose scattering is of the order of absorption, the fluence rate predicted by Beer-Lambert law is inaccurate 17,18. Welch 17 showed significance of alternate fluence rate based on combined scattering and absorption characteristics. They suggested the fluence rate for considerable scattering as: φ I πr [ ( µ + (1 g) z] for µ ~ = 0 exp µ a s ) a µ s (.4) where g anisotropy factor, which is the cosine of average scattering angle. The fluence rate, given by Eq..4, assumes that the light scattered in the direction, other than the direction of the collimated beam, is negligible. Since ocular tissues possess highly forward scattering characteristics (anisotropy factor, g, ranges from 0.85 to 0.97), the assumption holds good for ocular tissues. In order to predict the accurate heat source values, the scattering and absorption characteristics of all ocular tissues namely vitreous humour, neural retina, RPE, choroid and sclera are incorporated. The values of µ a, µ s and anisotropy factor g (Table ) are taken from the in-vitro results from Hammer 19. They employed double-integrating sphere technique and inverse Monte Carlo simulation to calculate the optical properties of ocular tissue. The variation of laser intensity along radial direction depends on the profile of the beam. For top-hat profile, i.e., uniform intensity along radial direction, the heat source for a particular ocular tissue is: S I 0 µ a ( z) = q ( z, r, t) = δ on exp[ ( µ a + (1 g ) µ s ) z] (.5) πr 13

23 where δ on is a multiplication factor used to simulate pulsed laser, which has a value of one or zero corresponding to on and off duration, respectively. For Gaussian profile, variation of incident intensity along the radial direction is also included as follows: S I µ ( ) = r (,, ) 0 a z q z r t δ on exp µ (.6) πr [ ( µ + (1 ) ) ] exp a g s z R = where r is the actual radius (m). Ocular tissue Wavelength λ (nm) Absorption µ (mm -1 ) a Scattering µ (mm -1 ) s Anisotropy factor (g) Neural retina RPE Choroid Sclera 57 (Green) (red) (infrared) Table : The values of optical properties of ocular tissues for green, red and infrared wavelengths. 14

24 .5 Boundary Conditions Thermal boundary conditions are required on all sides of the domain to solve the diffusion equation. At the axis zero heat flux condition (symmetry) is imposed: T n = 0 (along the axis) (.7) The actual boundary conditions at the ends of sclera and cornea are not known. In previous numerical analysis by Amera 13, convective heat transfer or mixed type boundary conditions were used at the ends of sclera and cornea. Further, they have analyzed the effect of boundary conditions by applying wide range of convection heat transfer coefficients. Their results showed the insignificant effect of boundary conditions over the solution. This is because of the significant difference in length scale associated with the eye geometry (in mm) and the size of the laser spot (in µm). Moreover, since total time exposure is in the range of a few hundred milliseconds, the domain behaves as infinite length in the z-direction. Figure 4: Validation of boundary condition 15

25 In order to validate the applicability of the boundary condition, as an initial step, two limiting boundary conditions viz., constant temperature (Dirichlet B.C.) and zero heat flux (Newmann B.C) are analyzed. The maximum temperature obtained by the two cases differs only by 0.0%. Thus, the zero heat flux boundary condition is applied for all calculations presented in this study: T n = 0 (along the periphery of sclera and cornea) (.8).6 Rate Process Analysis of Thermal Damage The rate process analysis for thermal damage was first reported by Mortiz and Henriques 0. In their work, the thermal damage to tissue was quantified using a single parameter, Ω, which is calculated from an Arrhenius integral as follows: t Ω( t) = A exp[ E / R T ( t) ] dτ (.9) 0 a u where A is the frequency or pre-exponential factor (1/s), and t is the total heating time (s), E a is the activation energy barrier (J/mole), R u is the universal gas constant (8.3 J/mole-K) and T is the absolute temperature (K). The effect of both magnitude and duration of temperature increase is included in the Arrhenius integral value (Ω), as it involves integration of temperature over time. Thus, the value of Ω is a measure of the total thermal damage caused by the laser treatment. 16

26 Reference Temperature range (K) Pre-exponential factor A (s -1 ) Activation energy E a ( J/mole) Takata et al < T Bringruber et al 1 33 < T Table 3: The value of pre-exponential factor and activation energy E a for different temperatures used in the present study to compute the Arrhenius integral value Ω. The rate process model of thermal damage is based on first order unimolecular chemical kinetics. The values of E a and A photocoagulation treatment were determined by many previous experimental studies 1-. Welch 18 showed that the activation energy remains constant for a wide range of temperatures, while the pre exponential factor increases with temperature. The activation energy and the frequency factor of retina for different temperature ranges used in this study are shown in Table 3. Experimental results by Takata 3 showed that the temperature rise of up to 6 o C produces negligible damage and hence zero value is used for pre-exponential factor A for temperatures less than 316 K. Experimental investigation by Bringruber 1 was conducted at temperature of 350 K. However, in the present numerical study the values of E a and A from Bringruber 1 are used for temperatures above 33K. Trapezoidal rule is employed to compute the value of Ω (Eq..9) and the mean temperature is used to select the values of E a and A..7 Finite Volume Method The finite volume mesh is generated in Gambit.1.0 using Cooper scheme with ~60,000 cells of quadrilateral elements. Heat conduction equation is solved using finite volume method. 17

27 Double precision coupled solver with second-order time implicit scheme is employed to discretize the governing equations. A second order upwind scheme is adapted for the energy equation. Convergence criterion for the energy equation is kept at 10-6, which is an order of magnitude lower than recommended values. Mesh independency is checked by increasing the number of elements in the RPE region by 0% over the current mesh and both the results are compared. The finer mesh showed less than 1% difference in temperature. For detailed numerical technique on Finite Volume Method our earlier publications 4,5 on laser and x-ray heating may be referred to..8 Numerical Using Alternating Direction Implicit method In addition to modeling using Fluent, the heat conduction equation is solved by developing numerical solution using Alternating Direction Implicit (ADI) scheme. The results obtained by ADI scheme is then compared with the Fluent results in order to validate the heat source applied using User Defined Function of Fluent. Due to axi-symmetry of the problem, the governing equation for laser photocoagulation is Two-dimensional unsteady heat equation in cylindrical coordinates T 1 1 T T q = r + + α t r r r z k (.10) where, α - Thermal diffusivity of the material (m /sec) and q 3 Volumetric Heat Source in W/m = I 0 exp(-z/l L att att ) 18

28 The governing equation is scaled using following non-dimensional parameter t * αt =, L T k( T Ti =, '' I L * ) 0 * r = r L, * z = z L, q * = Lq I 0 By substituting above variable, the equation (.10) becomes '' I 0L α T kl α t In non dimensional form '' * I 0L * T T = r + * * L k r * r * r z * * * * + q T t * * 1 = * r r * r * T r * * T + z * * + q * (.11) Where * q = * L exp(-z L/L L att att ) The Fig. 5 shows the simple two-dimensional model used for the present study. The figure also shows various ocular regions. R = 1000 µm Vitreous Humour Neural Retina R P E Choroid Sclera 1000 µm r = 0 µm z = 0 µm 00 µm 195 µm 50 µm 150 µm 10 µm z= 800 µm Figure 5: Simple two-dimensional model of eye used for numerical simulation 19

29 The ADI Method involves splitting in two direction and hence only tridiagonal systems of linear algebraic equations must be solved. During the first step the tridiagonal matrix is solved for each i row of the grid point (r-direction) and during the second step it is solved for each j row of grid points (z-direction). The ADI Method is second-order accurate with a T.E. of O ( ) ( ) ( ) [ ],, y x t. First Step: Sweeping along r-direction j i n j i n j i n j i n j i n j i n j i n j i n j i n j i n j i Q z T T T r T T T r T T r t T T, 1, 1 1, 1 1, 1, 1 1, 1 1, 1 1, 1 1, 1, 1, ) ( ) ( = Second Step: Sweeping along z-direction j i n j i n j i n j i n j i n j i n j i n j i n j i n j i n j i Q z T T T r T T T r T T r t T T, 1, 1 1, 1 1,, 1, 1, 1, 1,, 1, ) ( ) ( = As discussed in section.4, due to semi-infinite nature of the problem insulated boundary condition is applied on all boundaries. 0 0 = z= z T Left boundary = z= z T Right boundary 0 0 = r= r T Symmetry axis = r= z T Top boundary 0

30 Figure 6: Comparison of numerical solution with fluent results for CW laser Fig 6 shows the comparison of temperature obtained by fluent and numerical code developed using ADI schemes. The maximum temperatures obtained by two methods are identically same. This validates the user defined function used in the fluent. 1

31 Chapter 3: RESULTS The success of selective retinal photocoagulation depends primarily on the laser parameters. The results from this study quantitatively show the influence of laser parameters such as wavelength, laser profile, pulse width, repetition frequency on the selective damage of RPE. In addition, the variation of threshold energy for different RPE thickness is also analyzed. The summary of different computations performed is listed in Table 4. Figure 7: Definition of various laser terms used in terms of pulse width and pulse power

32 A base line computation (case ) is carried out with the following laser parameters: wavelength - 57 nm (green), profile - top-hat, pulse width - µs, energy/pulse - 0 µj, frequency Hz, and number of pulses - 50, exposure duration -100 ms and average power - 10 mw. The spot diameter of 100 µm is held fixed in all computations. The definition of laser parameters used in the study such as energy/pulse, repetition frequency and average power, in terms of pulse width and pulse power, are given in Fig. 6. The effect of laser parameters is studied by varying the corresponding values (Table 4) from the base line case. The transient temperature and the Arrhenius integral value, Ω are obtained from the cases. The temperature profile along the center of the laser is analyzed to study the deposition and diffusion of thermal energy. The ratio of maximum temperature at RPE (T RPE ) to the temperature at the neural retina (T NR ), 5 µm away from RPE-neural retina interface, is used to help quantify the selectivity. Since Ω max differs with variation of laser parameters, the value Ω/ Ω max is used to study the selective damage of RPE. A higher value Ω/ Ω max at neural retina signifies more damage to photoreceptors, while a lower value represents selective damage of RPE. 3

33 Case Runs Wavelength (nm) Profile Pulse width (µs) Energy /pulse (µj) Repetition frequency (Hz) No of pulses Exposure duration (ms) Average power (mw) RPE Thickness (µm) 1 Validation 57 Top hat Base case 57 Top hat Top hat Wavelength Top hat Profile 810 Gaussian Pulse 57 7 duration 57 Top hat Top hat Repetition 57 Top hat Frequency 57 Top hat RPE 57 Top hat thickness 57 Top hat Table 4: Summary of runs. The case 1 is the validation of temperature with experimental results, respectively. Case serves as the reference baseline case. Laser parameters (in bold) are varied as shown in runs 3-11 to find their influence over the selectivity of the treatment. 4

34 3.1 Validation The results of this present calculation are validated with the in-vivo temperature measurements taken by Schule 6 using optoacoustic methods. They conducted an experiment by applying 30 pulses of 1.7 µs green laser at 100 Hz with 100 µj pulse energy. The base line temperature, defined as temperature at the end of each pulse-off cycle, was measured during the treatment. The calculation for the present study is conducted with the same laser parameters (Table 4: case 1) with initial temperature of 34 o C. Fig. 7 shows the variation of the base line temperature with number of pulses for experimental 6 and present study. A log-fit curve to the experimental result is also shown in Fig. 7. Figure 8: Comparison of baseline temperature (lowest temperature at the end of each pulse) of present results with experimental results by Schule et al (004). The laser parameters are: wavelength - green, profile - top-hat, pulse width µs, energy/pulse µj, frequency Hz, number of pulses - 30, spot diameter µm. 5

35 The temperature predicted by present study is higher than the experimental study and the average deviation is 3.5% and maximum deviation is within 8%. The deviation may be due to local variation of RPE thickness and melanin concentration. Further, the experimental results 6 have a standard deviation of 17% due to the usage of material constants of porcine RPE. 3. Effect of Wavelength Figure 9: Comparison of laser heat source profiles along the depth for all three wavelengths (57, 650, and 810 nm). The regions of neural retina, RPE and choroid are shown in the graph. The green wavelength has high absorption in all three layers followed by the red and infrared rays. The melanin granules in RPE have strong absorption characteristics for wavelengths in the visible range, and its absorption decreases with increase in wavelength. The influence of 6

36 wavelength is studied by analyzing three frequently used wavelengths (Table 4: cases, 3, and 4) namely green (57 nm), red (650 nm), and infrared (810 nm), with similar laser parameters as that of the base line case. The thermal energy deposited as depicted by the heat source profiles given by Eq..5 are shown in Fig. 8. For all the three wavelengths, the heat generation is high in RPE and few orders of magnitude lower in neural retina and choroid. The main advantage of green laser is the high absorption ~47% of incident energy in RPE, while it is ~ 4% and ~ 5% for red and infrared rays respectively. The advantages of red and infrared rays over green rays are that the value of heat source at interface of RPE and neural retina is low and absorption in the neural retina is about 5 times less than that of green rays. Figure 10: Predicted maximum temperature along the center line of the laser beam for all three wavelengths. Other laser parameters are: profile - top-hat, pulse width - µs, energy/pulse - 10 µj, frequency Hz, number of pulses

37 Fig. 9 shows comparison of the resulting temperatures at the end of the 50 th pulse along the centre of the laser beam. The green laser produces highest temperature of around 90 o C at RPE, while infrared produces lowest temperature of around 60 o C, for the same laser energy. The temperature at the neural retina and choroid is also higher for the green laser. However, the value of T RPE /T NR is 1.71 for green, 1.63 for red, and 1.36 for infrared lasers. The higher temperature ratio means relatively better selective damage of RPE. Contours of Ω/Ω max for three wavelengths are shown in Fig. 10. The actual value of Ω can be obtained by multiplying the contour value with Ω max, which is also shown in the figure. The location of RPE is from -5 to +5 µm. The contours for green wavelength show that the thermal damage is limited to RPE, as the value of Ω/Ω max is 0.01 at neural retina and choroid. The contours of red and infrared show the significant damage beyond RPE to neural retina and choroid. The value of Ω/Ω max at neural retina is 0.05 and 0.4 for red and infrared, respectively. Hence, despite the advantage of a low heat source at interface of RPE-neural retina and lower absorption in neural retina, the red and infrared produces thermal damage to neural retina. Thus, the fraction of energy absorbed in the RPE is the main criteria for selectivity of the treatment. Further, the value of Ω max for green laser is.01e-1, while it is 4.5e- and 3.16e-4 for red and infrared, respectively. Thus, the threshold energy required for green laser is lower than that of red and infrared rays. Another important factor to be considered is the amount of transmitted energy (unabsorbed laser energy leaving the sclera), which can produce damage to the optic nerves. The percentage transmission is very high for infrared rays. For green laser 8

38 almost 70% of incident energy is absorbed whereas it is 65% and 50% for red and infrared rays, respectively, which can damage the optic nerves. Thus, based on selective effect, lower threshold energy requirement, and transmission criteria, the green laser is preferred over red infrared lasers. 9

39 Ω/Ω max Ω max =.01E-1 Ω max = 4.5E- Ω max = 3.16E-4 Figure 11: Contours of Ω/Ω max for all three wavelengths at the end of 50 pulses. For the same laser power, the green wavelength produces maximum damage to RPE (high Ω max ), while infrared produces minimum damage. The contours of red and infrared rays show damage to neural retina and choroid while green produce selective damage to RPE. 30

40 3.3 Effect of Laser Beam Profile The intensity pattern of laser along radial direction depends on the Transverse Electro- Magnetic (TEM) operating mode of the laser. The fundamental mode (TEM00) produces a Gaussian profile, while the multimode produces a near uniform intensity profile (top-hat), over the spot diameter. In this study, the effect of the laser profile on the therapeutic effect is analyzed by comparing computations with two different laser profiles: Gaussian and top-hat (Table 4: cases and 5). Figure 1: Comparison of temperature along radial direction at the end of 50 pulses for Gaussian and top hat laser beam profiles. Gaussian profile results in high temperature at the center of the beam. The Gaussian profile produces a higher temperature of ~ 100 o C at the center of the beam, while the top-hat profile produces a temperature of ~ 80 o C as shown in Fig. 11. This is due to the fact that the intensity at the center of the Gaussian profile is twice than that of the top-hat profile for the same laser power. The contours of Ω/Ω max for the two profiles are shown in Fig. 31

41 1. The contour for the beam with the Gaussian profile show selective damage in the radial direction. The Ω max values are.01e-1 and 1744 for the top-hat and the Gaussian profiles, respectively. Since Ω max is much higher for the Gaussian distribution, it requires lower threshold energy than that for the top-hat. Thus, the Gaussian profile is preferable to achieve selective damage along radial direction with lower threshold energy. Ω/Ω max Ω max =.01E-1 Ω max = Figure 13: Contours of Ω/Ω max for Gaussian and top hat profile. Gaussian profile produces high value of Ω max, as its power at the center is twice that of average power. Other laser parameters are: wavelength - green, pulse width - µs, energy/pulse - 10 µj, frequency Hz, number of pulses

42 3.4 Effect of Pulse Width The effect of pulse width in selective damage was successfully demonstrated by a number of experiments 4-7. The main supporting argument for the selection of short-pulsed laser is that it reduces the thermal diffusion to the surrounding tissue, and thus, causes selective damage of RPE. However, the energy diffusing after the pulse-on can increase the temperature of neural retina and choroid. The current study evaluates the effectiveness of short-pulsed laser in terms of temperature and the Arrhenius integral value. Figure 14: Variation of pulse power and pulse-off duration required for different pulse widths in order to maintain constant average power of 0.01 W Three pulse widths, 0, and 00 µs (Table 4: cases, 6, and 7) with the same pulse energy of 0 µj at 500 Hz are analyzed. The variation of laser power and the pulse-off duration for all pulse widths are shown in Fig. 13. The temperature profiles along the central axis at the end of different number of pulses are shown in Fig. 14. For all pulse widths, the maximum temperature is nearly achieved within the first 10 pulses and only moderate temperature rise occurs between the 10 th and 50 th pulses. The maximum temperature of 90 o C at the RPE is obtained for the µs pulse width, while the 00 µs pulse width causes low temperature of 70 o C. 33

43 Figure 15: Comparison of temperature for all three pulse widths µs, 0 µs, and 00 µs along the center line of the beam at the end of 1, 10, 5, and 50 pulses. The maximum temperature augmentation occurs within 10 pulses and further increase in number of pulses does not produce significant additional temperature rise. 34

44 For µs pulse, there is no significant conduction occurring during the pulse-on duration and thus, the thermal energy is confined to the RPE. This, in turn, results in minimum temperature of ~ 50 o C at the neural retina for µs pulse. In case of 00 µs pulse width, higher thermal diffusion of energy into neural retina is observed. This results in higher temperature of ~ 70 o C in significant portion (till ±5 µm) of the neural retina. Figure 16: Spreading of deposited energy after the pulse-on period for µs laser treatment. The temperature of neural retina increases to 65 o C at 50 µs after the end of 50 pulses. The diffusion after the end of the µs pulse-on duration is studied by analyzing the transient temperature profiles along the center axis at different times after the 50 th pulse. Fig. 15 shows increase in neural retina and choroid temperature from 50 o C to 65 o C at around 50 µs after the end of the 50 th pulse. Thus, even for µs pulse width, diffusion after the pulse increases the temperature of neural retina. However, the temperature drops from 65 o C to 60 o C at around 00 35

45 µs after the end of the 50 th pulse. As mentioned earlier, the damage depends on both magnitude and duration of temperature increase. Thus, the value of Ω needs to be analyzed to find selectivity of the treatment. Fig. 16 shows contours of Ω/Ω max for all pulse widths. The values of Ω max are.01e-1, 5.06e-, and 8.47e-3 for, 0, and 00 µs, respectively. The Ω/Ω max contours for 00 µs shows the significant damage to neural retina and choroid and the value of Ω/Ω max is 0.4 at neural retina. For 0 µs pulse width, the damage occurs till ±5 µm from the interface of RPE-neural retina. For µs, the damage occurs only to the RPE and the value of Ω/Ω max is less than 0.01 in the neural retina and choroid regions. Thus, despite the temperature rise in the neural retina after the pulse-on, the µs pulse produces selective damage. This is due to high temperature at RPE (Fig. 14) and short duration of temperature increase in neural retina (Fig 15). The short-pulsed laser effectively produces selective damage to the RPE, sparing the neural retina and choroid. 36

46 Ω/Ω max Ω max =.01E-1 Ω max = 5.067E- Ω max = 8.47E-3 Figure 17: Contours of Ω/Ω max for all pulse widths µs, 0 µs, and 00 µs. The µs pulse width has maximum Ω max of.01e-1. The damage to neural retina and choroid is significant in the cases of 0 µs, 00 µs, while the µs produce selective RPE damage. Other laser parameters are: wavelength - green, profile - top-hat, energy/pulse - 10 µj, frequency Hz, number of pulses

47 3.5 Effect of Repetition Frequency The repetition frequency or duty factor is another important parameter in selective retinal photocoagulation treatment. Previous studies by Mainster used repetition frequency in the range of 100 to 1000 Hz. In this study, the influence of repetition frequency is analyzed by simulating three cases (Table 4: cases, 8, and 9) with repetition frequencies of 100, 500, and 1000 Hz. In all the cases, 50 pulses of µs pulse width are used with constant energy per pulse of 0 µj. Fig 17A shows the variation of pulse-off duration and average power for all repetition frequencies. The transient temperature at the center of RPE for all frequencies is shown in Fig. 17B. The maximum temperature at the end of each pulse remains same for 100 Hz and does not show any cumulative effect, while the high frequency 1000 Hz show a strong cumulative effect in the temperature increase. In case of 100 Hz, the pulse-off cycle duration is sufficiently long, so that the temperature of RPE drops to the initial temperature before the next pulse cycle. For 1000 Hz, the pulse-off cycle duration is much shorter, and does not allow enough time for dissipation of thermal energy in the interim period. This results in augmentation of thermal energy and cumulative increase in temperature to a value of ~ 110 o C in RPE at the end of the 50 pulses. 38

48 Figure 18: Comparison of laser power and temperature for all for repetition frequencies of 100, 500 and 1000 Hz. (A) Variation of pulse-off duration and average power with repetition frequency. Same energy/ pulse of 10 µj is used for all repetition frequencies. (B) Transient temperature rise for all frequencies. The frequency of 1000 Hz shows strong cumulative effect in temperature with number of pulses. 39