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Calculation of retinal temperature distributions resulting from laser irradiation of the eye. I. Continuous lasers
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1976 Phys. Med. Biol. 21 616 (http://iopscience.iop.org/0031-9155/21/4/012) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 142.66.3.42 This content was downloaded on 02/10/2015 at 06:37
Please note that terms and conditions apply.
PHYS. MED.
BIOL., 1976, VOL. 21,
NO,
4, 616-630.
@
1976
Calculation of Retinal Temperature Distributions Resulting from Laser Irradiation of the Eye: I. Continuous Lasers C. B. WHEELER Plasma Physics Department, Imperial College, London SW7 2AZ, U.K.
Received 19 August 1975, infLnal form 10 M a r c h 1976 ABSTRACT.Thispaper develops a simple analytic expression for the maximum temperature rise attained in the retinaas a function of laser power, exposure duration, retinal absorption parameters andimage radius. It is assumed that thelaser beam has a Gaussian intensity distribution and is absorbed, following Beer's law, in both the pigment epithelium and the choroid layers of the eye. The h a 1 expression is mathematically accurate t o 1% and is valid for image radii ranging from 20 to 500 pm and for exposure times from 30 ms to infinity. Consequently, this analysis is ideally suited to the interpretation of laser threshold damage experiments using shuttered gas lasers operated in their uniphase mode. Finally, the problem of defining a safe exposure level to laser radiation is considered. However, the lack of available information on minimum retinal image diameter and maximum absorption coefficient permits only an estimate of this parameter to be made.
1. Introduction
Over the last decade the laser has found many applications in research and industry and the need for realistic safety criteria has become of paramount importanceas the number of personnel suffering possible exposure to the radiation increases. To this end, much experimental work on threshold damage to the eye has been carried out, embracing a wide range of laser parameters and exposure times. The most probable cause of threshold damage to biological material under laser irradiationis thermal enhancement of harmful irreversible biochemical reactionssuch asdenaturation of proteins or inactivation of enzymes. Integration of the equations of thermal-chemical reaction (Hu and Barnes 1970) shows that it is meaningful to define a critical temperature above which the damage process is complete. This critical temperature varies very slowly with time and is afunction of the biochemical reaction considered, generally lying between 10 and 30 'c above body temperature. Consequently, knowledge of the maximum temperature reached in the eye when exposed to laser radiation is of great importance in establishing whether or not threshold damage is thermal in nature. It is therefore very convenient, for the interpretation of thresholdexperiments, to have an analyticrelation expressing the maximum temperature attained in terms of t'he laser power, mode structure, beam diameter, exposure time and retinal parameters. It is the aim of this paper to establish theoretically such an analytic relation for threshold damage
RetinalTemperatureDistributions
from Laser Irradiation of Eye
617
to the retinausing as realistic a model of the eye as is mathematically feasible. Threshold damage is only zetinal in the wavelength range where the ocular media are sufficiently transparent to the incidentradiationandthislimits the analysis to lasers that function in the approximate wavelength interval 0.4-1.2 pm. A microscopic study of the macular region of thevertebrate fundus shows that there are two principal absorbing layers in the eye, the pigmentepitheliumadjacent tothe receptors andthe choroid behind it. I n each layer the agencies of absorption are melanin granules,particles of irregular shape with a characteristic dimension of about one micron. These granules are fairly closely packed in a thin band toward the anterior part of the pigment epithelium and more widely distributed throughout the thicker choroid layer.Thedensity of the granulesinthesetwolayersvaries considerably from eye to eye and any theoretical model must therefore contain theabsorptionparametersas working variables. Ifthe duration of the exposure to radiation is muchless than the timefor thermal relaxation between adjacent granules then thesegranules may be treated asindependent absorbers. For the granules in the pigment epithelium this time scale is of the order of a microsecond and an orderof magnitude greater for granules in theless densely populated choroid layer. On the other hand, for exposure times much greater than these relaxation times the respective layers may be treated as homogeneous absorbers. Consequently the theoretical approach is fairly straightforwardineach of these regimes andit is interesting to note thatthe Q-switched solid-state laser and the mechanically shuttered continuous laser the flashconveniently fit into theserespective time domains.However, lamp pumped dye laser and the passive solid-state lasers probably fall into the regime where neither the discrete or the continuum absorption process is applicable. This paper is concerned with exposure times of many microseconds, where both layers may be treated ashomogeneous absorbers for which the attenuation process is an exponentialdecay of intensitywithdistance,representedby Beer’s law. If the overall attenuation of the medium is small then the process approximates to uniform absorption in which all volume elements absorb an equal energy. However, both layers in the eye, by their very function, cause large attenuationand uniform absorptionisa poor approximation.The absorption coefficient of the layers is a function of the wavelength of the laser radiation and the total energy absorbed depends on the cross-section of the laser beam and the transverse energy distribution. A large number of laser applications, such as rangingand microwelding, utilize lasers operating in their uniphase(axial or longitudinal) mode for which the beam divergence is a minimum and thereforehas the capability of being focused t o a minimum spot size. Naturally it is this mode of operation that constitutes the greatest hazard to the relaxed eye, should the beam be viewed directly or by specular reflection. The incident beam intensity distribution in this case is circularly symmetricwitha Gaussian profile. After passage through the cornea and lens of the eye an image is formed on the retina, thesize of which is dependent on many factors. However, the radius of curvature of the fundus is always
618
C . B. Wheeler
severalorders of magnitudegreater thanboththe imageradiusandthe distances over which appreciable thermal conduction takes place. This enables the absorbing layers to be regarded as plane and immersed in a medium of infinite extent. Even with these valid simplifications the theoretical treatment is arduousandmanyauthorshave made furtherapproximations which necessarily maketheir models less acceptable. For example, Vos (1962) considers uniform absorptionwithina single layerforabeam of uniform intensity over a square cross-section. Felstead and Cobbold (1965) avoid the mathematical difficulties by resorting to an experimental technique involving an electrical analogue of the thermal properties of the retina in the form of a resistance-capacity network. Their technique enables a two-layer absorption process but the mesh is too coarse to represent the details of the absorption process or the beam profile. Hansen, Feigen andFine (1967) consider the steady-state case of uniform absorption within a single layer for a beam of uniform intensity over a circular cross-section. Peacock (1967) uses the true exponentialabsorptionina single layer of infinite depth. However, the incident laser beam is assumed to be of uniform intensity, infinite in area and the surface temperature only is evaluated. Hansen and Fine (1968) and Hayes and Wolbarsht (1968) treat the melanin granules as independent absorbers. The beam intensity profile is then of no consequence providing the maximum power density within the beam is known, and the greatest temperature rise takes place in the granules at the very frontof the pigment epithelium. However, the results of these analyses cannot be extrapolated to long time scales where granules, and indeed layers, cease to be independent. Clarke, Geeraets of and Ham (1969) consider exponential absorption within two layers for a beam uniform intensity over a circular cross-section. Their calculations are carried out by computer exclusively for the steady state. White, Mainster, Tips and Wilson (1970) give an exceedingly comprehensive treatment for exponential absorptionwithintwolayers that can include any beam profile andtime dependence. I n particular the technique permits differing thermal properties for the ocularmedia.However, the calculations are necessarily carried out numerically and considerable computer time is involved. Finally, Vassiliadis (1971) considers uniform absorption within two layers for a beam of uniform intensity over a circular cross-section. The model adopted for analysis here considers exponentialabsorption within two layers using the most hazardous beam profile, namely a Gaussian distribution. Many ophthalmologists investigating threshold damage use this beam profile and there is the added advantage that it is the only mode of laseroperationcapable of any degree of experimentalreproducibility.The ocular media are assumed identical in thermal properties and it is shown later, by considering the results of White et al. (1970)) that this is a valid approximation. It is then possible to obtain an analytic expression for the maximum temperatureattainedin a retina of prescribed absorption coefficients to a mathematicalaccuracy of betterthan 1% for exposuretimesin excess of about 30 ms. This time scale conveniently embraces the experimental investigations employing shuttered continuous gas lasers.
from Laser Irradiation of Eye
RetinalTemperatureDistributions
619
2. Mathematicalformulation
For the purposes of integration it is convenient to formulate the problem in Cartesiancoordinates. Consider amedium that is homogeneous inthermal properties, infinite in extent and initially all at the same temperature. If a t & ( X ’ , y‘, z’, t‘) applied, then, time t = 0 there is a heat generating function following Carslaw and Jaeger (1973), the temperature rise at any field point is given by the four-fold integration
T ( x ,y, Z , f )
=
Y’, z’, t’)
(8Kka n-*)-l.i’,”si’lQ(X’, x exp { - [ ( x- x’)2 X
(t - l‘)-+ dy’dx’
+ (y - y’)2+ ( z - z ’ ) 2 ] / 4 k ( t- t’))
dz’ dt‘.
(1)
The spatial integration here extends over the total source volume and K , E are respectively thethermalconductivityandthermal diffusivit’y of the medium. The present problem is basically concerned with heat generation by laser beam absorption within a thin layer that has the same thermal properties as the surrounding,perfectlytransparent,medium.Lett’he laserbeam be collimated and propagating in the positive Z direction with a Gaussian intensity distribut’ion in the transverse plane. If CL is the absorption coefficient for the radiation and the layer is of thickness 2p with the origin of cylindrical coordinates ( r ,z ) at’ the centre axial point, then the heat generating funct’ion is
where P is the total laser power incident on the layer and a is the radius a t of which the power hasfallen to exp ( - 1) of itsaxialvalue.Subst’itution eqn ( 2 ) into eqn (1) produces spatial integrations that are separable and which can be accomplishedanalytically.Thetemperature rise in response to a step function of radiation applied a t time t = 0 then reduces to the following single int’egral
+
x exp [u2- a2r2/(a2 a2 4u2)]
+
x {erf [u a(p - z)/2u]- erf [U - .(p
+ z)/2u]}du.
(3)
If theradiation is pulsed, of duration T , then t’he solution is obtainedby adding two opposing step function solutions of the above form with a relative delay T between them. The temperature rise in this case is therefore T ( r ,Z, t ) - T ( r ,Z , t - T )
where it is to be understood that the second term is zero for 0 t < T . 77
”
(4)
C. B. T17heeler
620 3.
Choice of retinal parameters
I n order to quantify various time scales in the theory and to illustrate the finalresult it is necessaq- to ascribenumericalvalues tothe para'meters introduced. The effective thicknesses and separation of the t'wo absorbing layers in the human eye are quoted by Vassiliadis (1971) as approximately four microns for the melanin band at the front of the pigment' epithelium a'nd thirt'y microns for the choroid, with an axial separation of thirty microns between the layer centres. White et al. (1970) give a table of the thermal properties of the ocular mediaextendingfrom the vitreous tothe sclera. All thestrata, withthe exception of the sclera can be ascribed the same conductivity and diffusivity to an accuracy of _+ 30q(,. For the sclera theseparametersaresome 60:; greater. White's computa,tions show that the effect of inhomogeneities on the temperatureinthe pigmentepitheliumabsorptionlayer is to raise the temperature at the front and to depress it at the rear. For t'he time scale of interest here the temperature error incurred by assuming homogeneity amounts to k 49,; overthisregion. In the following sections it is shown thatthe maximum retinal temperat'ure is located in the rear half of this layer and by making a suitable choice for the effective values of K and lc the assumption of homogeneity can lead to temperat'ure errors of less tha'n i 19:; in this region. The effective values assumed here are
which are very close to the values for pure water. The amount of energy absorbed by t'he two layers depends on many factors, particularly on the wavelength of the incident radiation and the pigmentation densitywithinthelayers.Forexample,themeasurements of Geeraets, Williams,Chan, Ham, Guerry and Schmidt (1962) on 24 human eyes, both white and negroid, show that' at thehelium-neon laser wavelength of 0.63 unl the pigmentepitheliummayabsorbbetween 13 and 467; and the choroid between 8 and 55% of the incident'cornealenergy. At longerwavelengths the absorptions presented by Geeraets are questionable since t'hey are inferred fromtransmissionmeasurementsanddonot' consider ba'ckscatterfrom the fundus. Campbell and Alpern (1962) and VOS, Munnik a'nd Boogaa'rd (1965) havemeasured the spectralreflectivity of the human fundus in vivo. They observe a value of about 5?/o over t'he range 0.40 to 0.55 pm, followed by a rapid rise with increase in wavelength, attaining about 40% at the ruby laser wavelengt'h of 0.69 pm. I n principle it is possible to correctforreflection losses by using thetheory of Kubelkaand Munk (1931) that considers simultaneousbackscatterandabsorptionwithinapigment'edlayer.At the other end of the visible spectrum there is considerable absorption in the human (1972) and macularpigment that is not allowed for by Geeraets.Ruddock Reading and Weale (1974) havemeasured the spectraltransmission of this pigmentbytotallydifferenttechniquesandfound a significant departure from unity in the wavelength ra'nge 0.40-0.55 pm. For example,only 200;
RetinalTemperatureDistributionsfromLaserIrradiation
of Eye
621
of the incident corneal energy is transmitted to the retina at the argon laser atthe wavelength of the helium-neon wavelength of 0.49 pm.However, laser the fundal reflection is small, the macular absorption negligible and for the purpose of illustrating these calculations it is assumed that each retinal layer absorbs 3576 of the incident corneal energy with 15% of the incident energy absorbed in the media in front of the retina and the remaining 150;6 absorbed in the sclera behind the choroid. Applying the exponential absorption law then gives the requirement that 2 4 is 0.52 for the pigment with the above epithelium and 1.2 for the choroid. I n conjunction estimates for the absorbinglayerthicknesses, 215, the inferredabsorption coefficients, CY,of the pigmentepithelium and the choroid arerespectively 1300 and 400 cm-l. Providing the calculationshereareappliedonly to K , k and CY as thresholddamage, it is valid to regardtheparameters independent of timeduringthe exposure since thetemperature rise is probably less than 30 'c. The remaining parameter to be considered is the size of the image on the retina. I n particular the minimum size is of great importance from the hazard point of view since the energydensityon the retina is thenamaximum. However, experimental results indicate that t'he minimum diameter depends on the technique of measurement. If the eyebehaved as aperfectoptical system then the size of the retinal image produced by an accurately collimated laser beam would be diffraction limited to the order of 8 pm in diameter (the full diameter of the Airydisc).TheexperimentsreportedbyWestheimer (1963), based on an examination of the human fundal reflection, show that the minimum retinal image diameter is about three times the theoretical value and he concludes thatthe asphericity of the cornea is responsible. More recently Gubisch (1967) evaluated retinal point-image profiles using measured modulationt'ransferfunctions for thehuman eye.Hefound that apupil diameter of 3.8 mm produced the smallest retinal image which was very close to the diffract'ion limit. Direct ophthalmic measurement using low divergence laser beams again gives retinal diameters in excess of the theoretical value. For example, Bresnick, Frisch, Powell, Landers, Holst and Dallas (1970) and Frisch,Beatrice and Holsen (1971) measuredimagediametersforrhesus monkeys irradiated by uniphaseargon lasers using ophthalmoscopes calibrated directly in the image plane by the intraocular wire technique. They obtained minimum values between 40 and 50 pm which were a factor of three greater than the theoretical diameters based on beam divergence and focal length. It is possible that pupildilation,generalanaesthetic,etc., influence these animal measurements ; furthermore, Jones, Fairchild and Spyropoulos (1968) question the resolving power of conventionalophthalmoscopes.The calculat'ions presented here are intended for the interpretation of threshold damage measurements and the range of diameters considered will be restricted to that covered by ophthalmologists, namely from 40 to 1000 pm. Sincediffraction effects arenotdominantit is assumed thatthe image intensitydistributionalwayshas the sameGaussian form asthe incident laser beam.
C. B. Wheeler
622 4.
Temperature distribution for a single layer
For sufficiently small time scales the integrand of eqn (3) can be greatly simplified. Theradialfactorcan be takenoutside the integralprovided 4u2
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My IOPscience
Calculation of retinal temperature distributions resulting from laser irradiation of the eye. I. Continuous lasers
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1976 Phys. Med. Biol. 21 616 (http://iopscience.iop.org/0031-9155/21/4/012) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 142.66.3.42 This content was downloaded on 02/10/2015 at 06:37
Please note that terms and conditions apply.
PHYS. MED.
BIOL., 1976, VOL. 21,
NO,
4, 616-630.
@
1976
Calculation of Retinal Temperature Distributions Resulting from Laser Irradiation of the Eye: I. Continuous Lasers C. B. WHEELER Plasma Physics Department, Imperial College, London SW7 2AZ, U.K.
Received 19 August 1975, infLnal form 10 M a r c h 1976 ABSTRACT.Thispaper develops a simple analytic expression for the maximum temperature rise attained in the retinaas a function of laser power, exposure duration, retinal absorption parameters andimage radius. It is assumed that thelaser beam has a Gaussian intensity distribution and is absorbed, following Beer's law, in both the pigment epithelium and the choroid layers of the eye. The h a 1 expression is mathematically accurate t o 1% and is valid for image radii ranging from 20 to 500 pm and for exposure times from 30 ms to infinity. Consequently, this analysis is ideally suited to the interpretation of laser threshold damage experiments using shuttered gas lasers operated in their uniphase mode. Finally, the problem of defining a safe exposure level to laser radiation is considered. However, the lack of available information on minimum retinal image diameter and maximum absorption coefficient permits only an estimate of this parameter to be made.
1. Introduction
Over the last decade the laser has found many applications in research and industry and the need for realistic safety criteria has become of paramount importanceas the number of personnel suffering possible exposure to the radiation increases. To this end, much experimental work on threshold damage to the eye has been carried out, embracing a wide range of laser parameters and exposure times. The most probable cause of threshold damage to biological material under laser irradiationis thermal enhancement of harmful irreversible biochemical reactionssuch asdenaturation of proteins or inactivation of enzymes. Integration of the equations of thermal-chemical reaction (Hu and Barnes 1970) shows that it is meaningful to define a critical temperature above which the damage process is complete. This critical temperature varies very slowly with time and is afunction of the biochemical reaction considered, generally lying between 10 and 30 'c above body temperature. Consequently, knowledge of the maximum temperature reached in the eye when exposed to laser radiation is of great importance in establishing whether or not threshold damage is thermal in nature. It is therefore very convenient, for the interpretation of thresholdexperiments, to have an analyticrelation expressing the maximum temperature attained in terms of t'he laser power, mode structure, beam diameter, exposure time and retinal parameters. It is the aim of this paper to establish theoretically such an analytic relation for threshold damage
RetinalTemperatureDistributions
from Laser Irradiation of Eye
617
to the retinausing as realistic a model of the eye as is mathematically feasible. Threshold damage is only zetinal in the wavelength range where the ocular media are sufficiently transparent to the incidentradiationandthislimits the analysis to lasers that function in the approximate wavelength interval 0.4-1.2 pm. A microscopic study of the macular region of thevertebrate fundus shows that there are two principal absorbing layers in the eye, the pigmentepitheliumadjacent tothe receptors andthe choroid behind it. I n each layer the agencies of absorption are melanin granules,particles of irregular shape with a characteristic dimension of about one micron. These granules are fairly closely packed in a thin band toward the anterior part of the pigment epithelium and more widely distributed throughout the thicker choroid layer.Thedensity of the granulesinthesetwolayersvaries considerably from eye to eye and any theoretical model must therefore contain theabsorptionparametersas working variables. Ifthe duration of the exposure to radiation is muchless than the timefor thermal relaxation between adjacent granules then thesegranules may be treated asindependent absorbers. For the granules in the pigment epithelium this time scale is of the order of a microsecond and an orderof magnitude greater for granules in theless densely populated choroid layer. On the other hand, for exposure times much greater than these relaxation times the respective layers may be treated as homogeneous absorbers. Consequently the theoretical approach is fairly straightforwardineach of these regimes andit is interesting to note thatthe Q-switched solid-state laser and the mechanically shuttered continuous laser the flashconveniently fit into theserespective time domains.However, lamp pumped dye laser and the passive solid-state lasers probably fall into the regime where neither the discrete or the continuum absorption process is applicable. This paper is concerned with exposure times of many microseconds, where both layers may be treated ashomogeneous absorbers for which the attenuation process is an exponentialdecay of intensitywithdistance,representedby Beer’s law. If the overall attenuation of the medium is small then the process approximates to uniform absorption in which all volume elements absorb an equal energy. However, both layers in the eye, by their very function, cause large attenuationand uniform absorptionisa poor approximation.The absorption coefficient of the layers is a function of the wavelength of the laser radiation and the total energy absorbed depends on the cross-section of the laser beam and the transverse energy distribution. A large number of laser applications, such as rangingand microwelding, utilize lasers operating in their uniphase(axial or longitudinal) mode for which the beam divergence is a minimum and thereforehas the capability of being focused t o a minimum spot size. Naturally it is this mode of operation that constitutes the greatest hazard to the relaxed eye, should the beam be viewed directly or by specular reflection. The incident beam intensity distribution in this case is circularly symmetricwitha Gaussian profile. After passage through the cornea and lens of the eye an image is formed on the retina, thesize of which is dependent on many factors. However, the radius of curvature of the fundus is always
618
C . B. Wheeler
severalorders of magnitudegreater thanboththe imageradiusandthe distances over which appreciable thermal conduction takes place. This enables the absorbing layers to be regarded as plane and immersed in a medium of infinite extent. Even with these valid simplifications the theoretical treatment is arduousandmanyauthorshave made furtherapproximations which necessarily maketheir models less acceptable. For example, Vos (1962) considers uniform absorptionwithina single layerforabeam of uniform intensity over a square cross-section. Felstead and Cobbold (1965) avoid the mathematical difficulties by resorting to an experimental technique involving an electrical analogue of the thermal properties of the retina in the form of a resistance-capacity network. Their technique enables a two-layer absorption process but the mesh is too coarse to represent the details of the absorption process or the beam profile. Hansen, Feigen andFine (1967) consider the steady-state case of uniform absorption within a single layer for a beam of uniform intensity over a circular cross-section. Peacock (1967) uses the true exponentialabsorptionina single layer of infinite depth. However, the incident laser beam is assumed to be of uniform intensity, infinite in area and the surface temperature only is evaluated. Hansen and Fine (1968) and Hayes and Wolbarsht (1968) treat the melanin granules as independent absorbers. The beam intensity profile is then of no consequence providing the maximum power density within the beam is known, and the greatest temperature rise takes place in the granules at the very frontof the pigment epithelium. However, the results of these analyses cannot be extrapolated to long time scales where granules, and indeed layers, cease to be independent. Clarke, Geeraets of and Ham (1969) consider exponential absorption within two layers for a beam uniform intensity over a circular cross-section. Their calculations are carried out by computer exclusively for the steady state. White, Mainster, Tips and Wilson (1970) give an exceedingly comprehensive treatment for exponential absorptionwithintwolayers that can include any beam profile andtime dependence. I n particular the technique permits differing thermal properties for the ocularmedia.However, the calculations are necessarily carried out numerically and considerable computer time is involved. Finally, Vassiliadis (1971) considers uniform absorption within two layers for a beam of uniform intensity over a circular cross-section. The model adopted for analysis here considers exponentialabsorption within two layers using the most hazardous beam profile, namely a Gaussian distribution. Many ophthalmologists investigating threshold damage use this beam profile and there is the added advantage that it is the only mode of laseroperationcapable of any degree of experimentalreproducibility.The ocular media are assumed identical in thermal properties and it is shown later, by considering the results of White et al. (1970)) that this is a valid approximation. It is then possible to obtain an analytic expression for the maximum temperatureattainedin a retina of prescribed absorption coefficients to a mathematicalaccuracy of betterthan 1% for exposuretimesin excess of about 30 ms. This time scale conveniently embraces the experimental investigations employing shuttered continuous gas lasers.
from Laser Irradiation of Eye
RetinalTemperatureDistributions
619
2. Mathematicalformulation
For the purposes of integration it is convenient to formulate the problem in Cartesiancoordinates. Consider amedium that is homogeneous inthermal properties, infinite in extent and initially all at the same temperature. If a t & ( X ’ , y‘, z’, t‘) applied, then, time t = 0 there is a heat generating function following Carslaw and Jaeger (1973), the temperature rise at any field point is given by the four-fold integration
T ( x ,y, Z , f )
=
Y’, z’, t’)
(8Kka n-*)-l.i’,”si’lQ(X’, x exp { - [ ( x- x’)2 X
(t - l‘)-+ dy’dx’
+ (y - y’)2+ ( z - z ’ ) 2 ] / 4 k ( t- t’))
dz’ dt‘.
(1)
The spatial integration here extends over the total source volume and K , E are respectively thethermalconductivityandthermal diffusivit’y of the medium. The present problem is basically concerned with heat generation by laser beam absorption within a thin layer that has the same thermal properties as the surrounding,perfectlytransparent,medium.Lett’he laserbeam be collimated and propagating in the positive Z direction with a Gaussian intensity distribut’ion in the transverse plane. If CL is the absorption coefficient for the radiation and the layer is of thickness 2p with the origin of cylindrical coordinates ( r ,z ) at’ the centre axial point, then the heat generating funct’ion is
where P is the total laser power incident on the layer and a is the radius a t of which the power hasfallen to exp ( - 1) of itsaxialvalue.Subst’itution eqn ( 2 ) into eqn (1) produces spatial integrations that are separable and which can be accomplishedanalytically.Thetemperature rise in response to a step function of radiation applied a t time t = 0 then reduces to the following single int’egral
+
x exp [u2- a2r2/(a2 a2 4u2)]
+
x {erf [u a(p - z)/2u]- erf [U - .(p
+ z)/2u]}du.
(3)
If theradiation is pulsed, of duration T , then t’he solution is obtainedby adding two opposing step function solutions of the above form with a relative delay T between them. The temperature rise in this case is therefore T ( r ,Z, t ) - T ( r ,Z , t - T )
where it is to be understood that the second term is zero for 0 t < T . 77
”
(4)
C. B. T17heeler
620 3.
Choice of retinal parameters
I n order to quantify various time scales in the theory and to illustrate the finalresult it is necessaq- to ascribenumericalvalues tothe para'meters introduced. The effective thicknesses and separation of the t'wo absorbing layers in the human eye are quoted by Vassiliadis (1971) as approximately four microns for the melanin band at the front of the pigment' epithelium a'nd thirt'y microns for the choroid, with an axial separation of thirty microns between the layer centres. White et al. (1970) give a table of the thermal properties of the ocular mediaextendingfrom the vitreous tothe sclera. All thestrata, withthe exception of the sclera can be ascribed the same conductivity and diffusivity to an accuracy of _+ 30q(,. For the sclera theseparametersaresome 60:; greater. White's computa,tions show that the effect of inhomogeneities on the temperatureinthe pigmentepitheliumabsorptionlayer is to raise the temperature at the front and to depress it at the rear. For t'he time scale of interest here the temperature error incurred by assuming homogeneity amounts to k 49,; overthisregion. In the following sections it is shown thatthe maximum retinal temperat'ure is located in the rear half of this layer and by making a suitable choice for the effective values of K and lc the assumption of homogeneity can lead to temperat'ure errors of less tha'n i 19:; in this region. The effective values assumed here are
which are very close to the values for pure water. The amount of energy absorbed by t'he two layers depends on many factors, particularly on the wavelength of the incident radiation and the pigmentation densitywithinthelayers.Forexample,themeasurements of Geeraets, Williams,Chan, Ham, Guerry and Schmidt (1962) on 24 human eyes, both white and negroid, show that' at thehelium-neon laser wavelength of 0.63 unl the pigmentepitheliummayabsorbbetween 13 and 467; and the choroid between 8 and 55% of the incident'cornealenergy. At longerwavelengths the absorptions presented by Geeraets are questionable since t'hey are inferred fromtransmissionmeasurementsanddonot' consider ba'ckscatterfrom the fundus. Campbell and Alpern (1962) and VOS, Munnik a'nd Boogaa'rd (1965) havemeasured the spectralreflectivity of the human fundus in vivo. They observe a value of about 5?/o over t'he range 0.40 to 0.55 pm, followed by a rapid rise with increase in wavelength, attaining about 40% at the ruby laser wavelengt'h of 0.69 pm. I n principle it is possible to correctforreflection losses by using thetheory of Kubelkaand Munk (1931) that considers simultaneousbackscatterandabsorptionwithinapigment'edlayer.At the other end of the visible spectrum there is considerable absorption in the human (1972) and macularpigment that is not allowed for by Geeraets.Ruddock Reading and Weale (1974) havemeasured the spectraltransmission of this pigmentbytotallydifferenttechniquesandfound a significant departure from unity in the wavelength ra'nge 0.40-0.55 pm. For example,only 200;
RetinalTemperatureDistributionsfromLaserIrradiation
of Eye
621
of the incident corneal energy is transmitted to the retina at the argon laser atthe wavelength of the helium-neon wavelength of 0.49 pm.However, laser the fundal reflection is small, the macular absorption negligible and for the purpose of illustrating these calculations it is assumed that each retinal layer absorbs 3576 of the incident corneal energy with 15% of the incident energy absorbed in the media in front of the retina and the remaining 150;6 absorbed in the sclera behind the choroid. Applying the exponential absorption law then gives the requirement that 2 4 is 0.52 for the pigment with the above epithelium and 1.2 for the choroid. I n conjunction estimates for the absorbinglayerthicknesses, 215, the inferredabsorption coefficients, CY,of the pigmentepithelium and the choroid arerespectively 1300 and 400 cm-l. Providing the calculationshereareappliedonly to K , k and CY as thresholddamage, it is valid to regardtheparameters independent of timeduringthe exposure since thetemperature rise is probably less than 30 'c. The remaining parameter to be considered is the size of the image on the retina. I n particular the minimum size is of great importance from the hazard point of view since the energydensityon the retina is thenamaximum. However, experimental results indicate that t'he minimum diameter depends on the technique of measurement. If the eyebehaved as aperfectoptical system then the size of the retinal image produced by an accurately collimated laser beam would be diffraction limited to the order of 8 pm in diameter (the full diameter of the Airydisc).TheexperimentsreportedbyWestheimer (1963), based on an examination of the human fundal reflection, show that the minimum retinal image diameter is about three times the theoretical value and he concludes thatthe asphericity of the cornea is responsible. More recently Gubisch (1967) evaluated retinal point-image profiles using measured modulationt'ransferfunctions for thehuman eye.Hefound that apupil diameter of 3.8 mm produced the smallest retinal image which was very close to the diffract'ion limit. Direct ophthalmic measurement using low divergence laser beams again gives retinal diameters in excess of the theoretical value. For example, Bresnick, Frisch, Powell, Landers, Holst and Dallas (1970) and Frisch,Beatrice and Holsen (1971) measuredimagediametersforrhesus monkeys irradiated by uniphaseargon lasers using ophthalmoscopes calibrated directly in the image plane by the intraocular wire technique. They obtained minimum values between 40 and 50 pm which were a factor of three greater than the theoretical diameters based on beam divergence and focal length. It is possible that pupildilation,generalanaesthetic,etc., influence these animal measurements ; furthermore, Jones, Fairchild and Spyropoulos (1968) question the resolving power of conventionalophthalmoscopes.The calculat'ions presented here are intended for the interpretation of threshold damage measurements and the range of diameters considered will be restricted to that covered by ophthalmologists, namely from 40 to 1000 pm. Sincediffraction effects arenotdominantit is assumed thatthe image intensitydistributionalwayshas the sameGaussian form asthe incident laser beam.
C. B. Wheeler
622 4.
Temperature distribution for a single layer
For sufficiently small time scales the integrand of eqn (3) can be greatly simplified. Theradialfactorcan be takenoutside the integralprovided 4u2
