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Journal of Biomedical Optics 9(3), 632–647 (May/June 2004)Radiative transport in the delta-P1 approximation:accuracy of fluence rate and optical penetration depthpredictions in turbid semi-infinite mediaStefan A. CarpUniversity of California, IrvineDepartment of Chemical Engineeringand Materials ScienceandLaser Microbeam and Medical ProgramBeckman Laser InstituteIrvine, California 92697Scott A. PrahlOregon Medical Laser CenterProvidence St. Vincent Medical CenterPortland, Oregon 97225Vasan VenugopalanUniversity of California, IrvineDepartment of Chemical Engineeringand Materials ScienceandLaser Microbeam and Medical ProgramBeckman Laser InstituteIrvine, California 92697E-mail: vvenugop@uci.eduAbstract. Using the d-P1 approximation to the Boltzmann transportequation we develop analytic solutions for the fluence rate producedby planar (1-D) and Gaussian beam (2-D) irradiation of a homoge-neous, turbid, semi-infinite medium. To assess the performance ofthese solutions we compare the predictions for the fluence rate andtwo metrics of the optical penetration depth with Monte Carlo simu-lations. We provide results under both refractive-index matched andmismatched conditions for optical properties where the ratio of re-duced scattering to absorption lies in the range 0<(ms8/ma)<104. Forplanar irradiation, the d-P1 approximation provides fluence rate pro-files accurate to 616% for depths up to six transport mean free paths(l* ) over the full range of optical properties. Metrics for optical pen-etration depth are predicted with an accuracy of 64%. For Gaussianirradiation using beam radii r0>3l* , the accuracy of the fluence ratepredictions is no worse than in the planar irradiation case. For smallerbeam radii, the predictions degrade significantly. Specifically for me-dia with (ms8/ma)51 irradiated with a beam radius of r05l* , the errorin the fluence rate approaches 100%. Nevertheless, the accuracy ofthe optical penetration depth predictions remains excellent for Gauss-ian beam irradiation, and degrades to only 620% for r05l* . Theseresults show that for a given set of optical properties (ms8/ma), theoptical penetration depth decreases with a reduction in the beam di-ameter. Graphs are provided to indicate the optical and geometricalconditions under which one must replace the d-P1 results for planarirradiation with those for Gaussian beam irradiation to maintain ac-curate dosimetry predictions. © 2004 Society of Photo-Optical Instrumentation En-gineers. [DOI: 10.1117/1.1695412]Keywords: diffusion; photons; light; collimation.Paper 03047 received Apr. 17, 2003; revised manuscript received Jul. 28, 2003;accepted for publication Sep. 26, 2003.--t-le-.nsdis-cu-alima-htateddernim-s-helyteds in1 IntroductionMany biophotonics applications require knowledge of thelight distribution produced by illumination of a turbid tissuewith a collimated laser beam.1 Examples include photody-namic therapy, photon migration spectroscopy, and optoacoustic imaging. If one considers light propagating as a neutral particle, the Boltzmann transport equation provides anexact description of radiative transport.2 However, the Boltz-mann transport equation is an integrodifferential equation thaoften cannot be solved analytically. As an alternative, investigators have resorted to a variety of analytic and computationamethods, including Monte Carlo simulations, the adding-doubling method, and functional expansion methods.2–6 Eachof these methods possesses unique limitations. For exampwhile Monte Carlo simulations provide solutions to the Bolt-zmann transport equation that are exact within statistical unAddress all correspondence to Vasan Venugopalan, University of California—Irvine, Department of Chemical Engineering and Materials Science, 916 Engi-neering Tower, Irvine, California 92697. Tel: 949-824-5802; FAX: 949-824-2541; E-mail: vvenugop@uci.edu632 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3l,certainty, they require significant computational resources7–9While, numerical finite difference or finite element solutiofor the Boltzmann transport equation10 may involve less com-putational expenditure, they require spatial and angularcretizations of the computational domain that lead to inacracies that are often difficult to quantify. Finally functionexpansion methods, such as the standard diffusion approxtion ~SDA!, that express the angular distribution of the ligfield and the single-scattering-phase function as a truncseries of spherical harmonics are typically accurate only una limiting set of conditions.2,6,11–13Although the SDA provides only an approximate solutioto the Boltzmann transport equation, its computational splicity has proven valuable for applications in optical diagnotics and therapeutics. Unfortunately, the limitations of tSDA are significant and confine its applicability to highscattering media and to locations distal from both collimasources and interfaces possessing significant mismatche1083-3668/2004/$15.00 © 2004 SPIErleensan-a-ofcalen-thsnt ofp-lessno-theeir-ndsonely.,in-a-. Iney-on-altion,Radiative transport in the delta-P1 approximation . . .refractive index.11,14–16 Such conditions are not satisfied inmany biomedical laser applications and, over the past 15 yhybrid Monte Carlo–diffusion methods17,18 as well as thed-P1 , P3 , andd-P3 approximations have been proposed asimproved radiative transport models.1,6,19–25Our focus here isthe d-P1 ~or d -Eddington! model first introduced in 1976 byJoseph et al.26 and first applied to problems in the biomedicalarena independently by Prahl,23,27 by Star,6,24 and Star et al.25Many investigators in biomedical optics have studied theaccuracy of functional expansion methods. Groenhuis et aprovided one of the first comparative studies between MontCarlo and SDA predictions for the spatially resolved diffusereflectance produced by illumination of a turbid medium witha finite diameter laser beam.11 Later, Flock et al. providedanother comparison between Monte Carlo simulations and thSDA that focused primarily on optical dosimetry; specificallythe accuracy of fluence rate profiles and optical penetratiodepth predictions for planar irradiation of a turbid medium.28More recently, Venugopalan et al. presented analytic solutionfor radiative transport within thed-P1 approximation for in-finite media illuminated with a finite spherical source.19 Theaccuracy of these solutions was demonstrated by comparisowith experimental measurements made in phantoms overbroad range of optical properties. Spott and Svaasand reviewed a number of formulations of the diffusion approxima-tion (P1 , d-P1 , d-P3) for a semi-infinite medium illumi-nated with a collimated light source, and compared fluencerate and diffuse reflectance predictions with Monte Carlosimulations for optical properties representative ofin vivoconditions.16 Dickey et al.20,21 as well as Hull and Foster22have studied the improvements in accuracy offered by theP3approximation for predicting both fluence rate profiles andspatially resolved diffuse reflectance. These studies have cofirmed that thed-P1 approach can provide significant im-provements in radiative transport predictions relative to SDAwith minimal additional complexity.While these investigations have provided some indicationof the improved accuracy provided by thed-P1 approxima-tion relative to the SDA, none have offered a quantitativeassessment of its performance against a radiative transpobenchmark such as Monte Carlo simulations over a widerange of optical properties. Thus, it is difficult to establishapriori the loss of accuracy that one suffers when using thed-P1 approximation to determine fluence rate distributions oroptical penetration depths. Our objective is to provide a comprehensive quantitative assessment of theaccuracy of opticdosimetry predictions provided by thed-P1 approximationwhen a turbid semi-infinite medium is exposed to collimatedradiation. Here, we report on the variation of thed-P1 modelaccuracy with tissue optical properties and diameter of theincident laser beam.Specifically, we determined the fluence rate profiles predicted by thed-P1 approximation for semi-infinite mediawhen subjected to planar~1-D! or Gaussian beam~2-D! irra-diation. For comparison, we performed Monte Carlo simula-tions to provide ‘‘benchmark’’ solutions of the Boltzmanntransport equation for multiple sets of optical properties.While we include plots of diffuse reflectanceRd versus(ms8/ma) for planar irradiation, our focus is on the internallight distribution as represented by the spatial variation of the,.n--rtlfluence rate. Since it is cumbersome to display the variationfluence rate with depth for more than a few sets of optiproperties, we also examined predictions for the optical petration depth. Comparison of the optical penetration deppredicted by thed-P1 approximation with those derived fromMonte Carlo simulations enables a continuous assessmethe d-P1 model accuracy over a broad range of optical proerties. These results are presented within a dimensionframework to enable rapid estimation of the light distributioin a medium of known optical properties. Moreover, to prvide quantitative error assessment, we include plots ofdifference between thed-P1 and Monte Carlo estimates. Thvariation of these errors with tissue optical properties andradiation conditions provide much insight into the nature aorigin of the deficiencies inherent in thed-P1 approximationas well as other functional expansion methods.2 d -P1 Model Formulation and Monte CarloComputation2.1 d-P1 Approximation of the Single-ScatteringPhase FunctionThe basis of thed-P1 approximation to radiative transport ithe d-P1 phase function as formulated by Joseph et al.26pd2P1~v̂•v̂8!514p$2 f d @12~v̂•v̂8!#1~12 f !@113g* ~v̂•v̂8!#%, ~1!wherev̂ and v̂8 are unit vectors that represent the directiof light propagation before and after scattering, respectivIn Eq. ~1! f is the fraction of light scattered directly forwardwhich thed-P1 model treats as unscattered light. The remader of the light(12 f ) is diffusely scattered according tostandardP1 ~or Eddington! phase function with single scattering asymmetryg* . To determine appropriate values forfandg* , one must choose a phase function to approximatethis paper, we choose to provide results for the HenyGreenstein phase function, as it is known to provide a reasable approximation for the optical scattering in biologictissues29:pHG~v̂•v̂8!514p12g12@122g1~v̂•v̂8!1g12#3/2. ~2!Recalling that for a spatially isotropic medium, thenth mo-ment,gn , of the phase functionp(v̂•v̂8) is defined bygn52pE211Pn~v̂•v̂8!p~v̂•v̂8!d~v̂•v̂8!, ~3!wherePn is thenth Legendre polynomial, we determinef andg* by requiring the first two moments of thed-P1 phasefunction, g15 f 1(12 f )g* and g25 f , to match the corre-sponding moments of the Henyey-Greenstein phase funcwhich are given bygn5g1n . This yields the following expres-sions for f andg* :f 5g12 and g* 5g1 /~g111!. ~4!Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 633oima-mhlyt ben.tandt,n-er-tionitarearypo-i-aredCarp, Prahl, and VenugopalanFor simplicity, from this point forward we refer tog1 simplyas g and all d-P1 model results in this paper are shown forg50.9 unless noted otherwise.2.2 d-P1 Approximation of the RadianceIn a manner similar to the phase function, the radiance is alsseparated into collimated and diffuse components:L~r ,v̂ !5Lc~r ,v̂ !1Ld~r ,v̂ !, ~5!where r is the position vector andv̂ is a unit vector repre-senting the direction of light propagation.For irradiation with a collimated laser beam normally in-cident on the surface of a semi-infinite medium, the colli-mated radiance takes the formLc~r ,v̂ !512pE~r ,ẑ!d~12v̂• ẑ!, ~6!where ẑ is the direction of the collimated light within themedium, andE(r ,ẑ) is the complete spatial distribution ofcollimated light provided by the source. While the lateral spa-tial variation ofE(r ,ẑ) is given by the irradiance distributionof the incident laser beamE0(x,y), its decay with depth(z-dir! is governed by absorption and scattering within themedium. Specifically, loss of collimated light arises from bothabsorption and diffuse scattering. Noting that in thed-P1phase function only(12 f ) of the incident light is diffuselyscattered, the decay of the collimated light with depth willbehave as a modified Beer-Lambert law:E~r ,ẑ!5E0~x,y!~12Rs!exp$2@ma1ms~12 f !#z%5E0~x,y!~12Rs!exp@2~ma1ms* !z#, ~7!whereRs is the specular reflectance for unpolarized light,mais the absorption coefficient,ms is the scattering coefficient,andms* [ms(12 f ) is a reduced scattering coefficient. For acollimated beam traveling along thez axis that possesses ei-ther a uniform or Gaussian irradiance profile we can work incylindrical (r ,z) rather than Cartesian(x,y,z) coordinates. Inthis case, the collimated fluence rate is given bywc~r !5E4pLc~r ,v̂ !5E~r ,ẑ!5E0~r !~12Rs!exp~2m t* z!,~8!whereE0(r ) is the radial irradiance distribution of the inci-dent laser beam andm t* [ma1ms* .The diffuse radiance in Eq.~5! is approximated, as in theSDA, by the sum of the first two terms in a Legendre poly-nomial series expansion:Ld~r ,v̂ !514p E4pLd~r ,v̂ !dV134p E4pLd~r ,v̂8!~v̂8•v̂ !dV8514pwd~r !134pj ~r !•v̂ ~9!634 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3wherewd(r ) is the diffuse fluence rate andj ~r ! is the radiantflux.The improved accuracy offered by thed-P1 approximationstems from the addition of the Diracd function to both thesingle scattering phase function and the radiance approxtion. Thed function provides an additional degree of freedowell suited to accommodate collimated sources and higforward-scattering media. Thus the addition of thed functionrelieves substantially the degree of asymmetry that musprovided by the first-order term in the Legendre expansio62.3 Governing Equations and Boundary ConditionsSubstituting Eqs.~1!, ~6!, and~9! into the Boltzmann transporequation and performing balances in both the fluence ratethe radiant flux provides the governing equations in thed-P1approximation for a semi-infinite medium19:¹2wd~r !2meff2 wd~r !523ms* m trE~r ,ẑ!13g* ms* ¹E~r ,ẑ!• ẑ, ~10!j ~r !5213m tr@¹wd~r !23g* ms* E~r ,ẑ!ẑ#, ~11!where ms8[ms(12g) is the isotropic scattering coefficienm tr[(ma1ms8) is the transport coefficient, andmeff[(3mamtr)1/2 is the effective attenuation coefficient.Two boundary conditions are required to solve Eqs.~10!and ~11!. At the free surface of the medium, we require coservation of the diffuse flux component normal to the intface, which yields@wd~r !2Ah¹wd~r !• ẑ#uz50523Ahg* ms* E~r ,ẑ!uz50 ,~12!whereA5(11R2)/(12R1) andh52/3m tr . HereR1 andR2are the first and second moments of the Fresnel refleccoefficient for unpolarized light and are given byR152E01r F~n!ndn and R253E01r F~n!n2dn,~13!where n5v̂• ẑ, with ẑ defined as the inward pointing unvector normal to the surface. The details of this derivationprovided in Appendix A. Note that Eq.~12! represents anexact formulation for conservation of energy at the boundand avoids the approximations inherent in the use of extralated boundary conditions.30,31 The second boundary condtion requires the diffuse light field to vanish in regions faway from the source. Thus,wd~r !ur→`→0. ~14!2.4 Solutions for Planar and Gaussian BeamIrradiationThe total fluence rate is given by the sum of the collimatand diffuse fluence rates:w~r !5wc~r !1wd~r !. ~15!nceweeds.neral-isRadiative transport in the delta-P1 approximation . . .Fig. 1 Depiction of (a)planar and (b) Gaussian beam irradiation con-ditions.sd.or-by2.4.1 Collimated fluence rateFor either planar or Gaussian beam irradiation conditions, ashown in Fig. 1, the collimated fluence rate within the tissueis expressed in the formwc~r ,z!5E0~r !~12Rs!exp~2m t* z!. ~16!For planar irradiation,E0(r )5E0 while for Gaussian beamirradiation,E0(r )5E0 exp(22r 2/r 02), wherer 0 is the Gauss-ian beam radius, i.e., the radial location where the irradiafalls is 1/e2 of the maximum irradiance. Note thatE052P/pr 02, whereE0 denotes the peak irradiance andP is theincident power of the Gaussian laser beam. For generality,define a normalized collimated fluence ratew̄c asw̄c5wc~r ,z!E0~r !~12Rs!5exp~2m t* z!. ~17!2.4.2 Diffuse fluence rate for planar irradiationFor planar illumination the diffuse fluence rate is determinby solving Eq.~10! subject to the boundary conditions Eq~12! and ~14! and yieldswd~z!5E0~12Rs!@a exp~2m t* z!1b exp~2meffz!#,~18!wherea53ms* ~m t* 1g* ma!meff2 2m t*2 , ~19!andb52a~11Ahm t* !23Ahg* ms*~11Ahmeff!. ~20!The solution procedure is detailed in Appendix B. In a mananalogous to the collimated fluence rate, we define a normized diffuse fluence ratew̄d asw̄d~z!5wd~z!E0~12Rs!5a exp~2m t* z!1b exp~2meffz!.~21!2.4.3 Diffuse fluence rate for Gaussian beamirradiationFor Gaussian beam irradiation, the diffuse fluence rategiven bywd~r ,z!5E0~12Rs!E0`$g exp~2m t* z!1j exp@2~k21meff2 !1/2z#%J0~kr !kdk, ~22!whereg53ms* ~m t* 1g* ma!r 02 exp~2r 02k2/8!4~k21meff2 2m t*2!, ~23!j523g* ms* r 02 exp~2r 02k2/8!24g@~Ah!211m t* #4@~Ah!211~k21meff2 !1/2#,~24!and J0 is the zeroth-order Bessel function of the first kinThe solution procedure is detailed in Appendix C. The nmalized fluence rate for Gaussian beam irradiation is givenJournal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 635nce-tra-rst-en-ac-ion.o ara-g-Carp, Prahl, and Venugopalanw̄d~r ,z!5E0`$g exp~2m t* z!1j exp@2~k21meff2 !1/2z#%J0~kr !kdk. ~25!Numerical methods~MATLAB, MathWorks, Natick, Massa-chusetts! were employed to compute the definite integral inEqs.~22! and ~25!.2.5 Diffuse Reflectance for Planar IrradiationThe prediction of the diffuse reflectance provided by thed-P1approximation isRd52 j ~z!• ẑE0~12Rs!Uz50513m trE0~12Rs!F3g* ms* E0 exp~2mt* z!2dwd~z!dz GUz505w̄d~z!2A Uz50. ~26!2.6 Limiting CasesA unique feature of the solutions provided by thed-P1 ap-proximation is thatwd→0 in the limit of vanishing scattering,i.e., whenms8!ma . Thus in a medium where absorption isdominant m t* →ma and the total fluence rate is governedsolely by the collimated contribution, i.e.,lim(ms8 /ma)→0w~r ,z!5wc~r ,z!5E0~r !~12Rs!exp~2maz!.~27!Thus, unlike prevalent implementations of the SDA whereinthe collimated light source is replaced by a point sourceplaced at a depthz5(1/ms8) within the medium, thed-P1approximation correctly recovers Beer’s law in the limit of noscattering.For media in which scattering is dominant(ms8@ma orm t* @meff), the total fluence rate resulting from planar irradia-tion reduces tolim(ms8 /ma)→`w~z!5E0~12Rs!@~312A!exp~2meffz!22 exp~2m t* z!#. ~28!If we further consider this fluence rate in the far field~largez), Eq. ~28! reduces tolim(ms8 /ma)→`w~z!5E0~12Rs!~312A!exp~2meffz!for large z. ~29!Equation ~29! is equivalent to the fluence rate predictiongiven by the SDA.13 Thus, in the limit of high scattering, andaway from boundaries and collimated sources, the solutioprovided by thed-P1 approximation properly reduces to thatgiven by the SDA.636 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 32.7 Optical Penetration DepthApart from the fluence rate profiles and diffuse reflectanresults offered by thed-P1 approximation, we are also interested in its predictions for the characteristic optical penetion depth ~OPD! in the tissue. In Fig. 2, we display twovariations of the OPD that we consider in this study. The fipenetration depth metricD is simply the depth at which thefluence rate falls to1/e of the incident fluence rate after accounting for losses due to specular reflection. The second petration depth metricD int is the depth at which all but1/e ofthe power of the laser radiation has been absorbed aftercounting for losses due to both specular and diffuse reflectFor generality, we normalize both these metrics relative tcharacteristic length scale. We choose(1/meff) for this lengthscale as it is the traditional definition for the optical penettion depth32 and is the length scale over which the homoenous solution to Eq.~10! decays. Accordingly we defineFig. 2 Graphical depiction of optical penetration depths (a) D and (b)D int .---Aenen-luntsant ofd-on-Radiative transport in the delta-P1 approximation . . .D̄[meffD and D̄ int[meffD int . ~30!2.8 Monte Carlo SimulationsWe performed Monte Carlo simulations for planar and Gaussian beam irradiation of semi-infinite media under both refrac-tive index matched and mismatched conditions. For this purpose we employed code derived from the Monte Carlo Multi-Layer ~MCML ! package written by Wang et al.8,9 thatcomputes the 3-D fluence rate distribution and spatially resolved diffuse reflectance corresponding to irradiation with alaser beam possessing either uniform or Gaussian profiles.Henyey-Greenstein phase function was utilized with a singlescattering asymmetry coefficient ofg50.9 unless stated oth-erwise. This value ofg was chosen as it is representative ofmany biological tissues.29 To approximate planar irradiationconditions we used a beam with a uniform irradiance profilewith radius r 05200l * , where l * [(1/m tr) is the transportmean free path. For Gaussian beam illumination, we setr 0 tothe desired1/e2 radius of the laser beam. To provide sufficientspatial resolution a minimum of 100 grid points were con-tained within one beam radius. Between107 and23109 pho-tons were launched for each simulation and resulted in fluencrate estimates with relative standard deviation of less tha0.1%.3 Results and Discussion3.1 Planar IlluminationFigures 3~a! and 3~b! provide normalized fluence rate profilespredicted by thed-P1 approximation and Monte Carlo simu-lations under planar illumination conditions for0.3<(ms8/ma)<100 and relative refractive indicesn5(n2 /n1)51.0 and 1.4, respectively. Note that the profiles are plottedagainst a reduced depth that is normalized relative to thtransport mean free pathl * . These figures also provide theerror of thed-P1 predictions relative to the Monte Carlo es-timates.Overall, the performance of thed-P1 approximation is im-pressive. The fluence rate is predicted with an error of<12%over the full range of optical properties. In the far field, themodel performance is exceptional for large(ms8/ma), de-grades slightly when scattering is comparable to absorptio(ms8.ma), and improves again when absorption dominatesscattering(ms8/ma&0.3). This behavior is expected. For large(ms8/ma) the prevalence of multiple scattering enables the dif-fuse component of thed-P1 approximation to provide an ac-curate description of the light field. However, when scatteringis still significant but(ms8/ma) is reduced, the decay of thelight field occurs on a spatial scale intermediate to that predicted by diffusion, i.e.,exp(2meffz), and that predicted by thetotal interaction coefficient, i.e.,exp(2mt*z). This results in anerror between thed-P1 model and the Monte Carlo estimatesthat increases with increasing depth. This is seen most notabfor the case of(ms8/ma)51 for which the error is largest inthe far field. Finally, for highly absorbing media, the overallaccuracy of thed-P1 approximation improves again becausethe contribution of collimated irradiance to the total light fieldincreases markedly and is well described by the modifiedBeer-Lambertlaw of Eq.~7!.y In the near field, the accuracy of thed-P1 approximationdegrades with increasing(ms8/ma). The origin of this lies inthe fact that increases in scattering result in increased amoof light backscattered toward the surface. This leads toincrease in the angular asymmetry in the diffuse componenthe light field near the surface which is not accurately moeled by a radiance approximation that simply employs a cFig. 3 Normalized fluence rate w̄ versus reduced depth (z/l* ) as pre-dicted by the d-P1 approximation (solid curves) and Monte Carlosimulations (symbols) for planar illumination under refractive index (a)matched (n51.0) and (b) mismatched (n51.4) conditions. Profilesare shown for (ms8/ma)5100 (s), 10 (* ), 3 (L), 1 (3), and 0.3 (d) withg50.9. Lower plots show the percentage error of the d-P1 predictionsrelative to the Monte Carlo simulations using the same symbols as themain plot.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 637Carp, Prahl, and VenugopalanFig. 4 Normalized fluence rate w̄ versus reduced depth (z/l* ) as pre-dicted by the d-P1 approximation (solid curves) and Monte Carlosimulations (symbols) for planar illumination under refractive index (a)matched (n51.0) and (b) mismatched (n51.4) conditions. Profilesare shown for g50 (s), 0.3 (* ), 0.7 (3), and 0.9 (d) with (ms8/ma)5100. Lower plots show the percentage error of the d-P1 predictionsrelative to the Monte Carlo simulations using the same symbols as themain plot.s.y ofthecat-stant and the first-order Legendre polynomial. Thed-P1model performs worse forn51.4because the refractive indexmismatch introduces internal reflection that further enhancethe angular asymmetry of the light field near the surface638 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3However, when scattering is less prominent, the accuracthe fluence rate profiles is not as strongly dependent onrefractive index mismatch because there is less light backstered toward the surface.Fig. 5 Normalized fluence rate w̄ versus reduced depth (z/l* ) as pre-dicted by the d-P1 approximation (solid curves) and Monte Carlosimulations (symbols) for planar illumination under refractive index (a)matched (n51.0) and (b) mismatched (n51.4) conditions. Profilesare shown for g50 (s), 0.3 (* ), 0.7 (3), and 0.9 (d) with (ms8/ma)51. Lower plots show the percentage error of the d-P1 predictionsrelative to the Monte Carlo simulations using the same symbols as themain plot.Radiative transport in the delta-P1 approximation . . .Fig. 6 Diffuse reflectance Rd versus (ms8/ma) as predicted by the d-P1approximation (solid curves) and MC simulations (d) for planar illu-mination under refractive index (a) matched (n51.0) and (b) mis-matched (n51.4) conditions. Lower plots show the percentage errorof the d-P1 predictions relative to the MC simulations.onrrorWe also examined the influence of the single scatteringasymmetry coefficientg on thed-P1 model predictions forfixed values of(ms8/ma). Figures 4~a! and 4~b! show thevariation of the normalized fluence rate profiles for0<g<0.9 and(ms8/ma)5100 for n51 and 1.4, respectively. Fig-ures 5~a! and 5~b! show these same results in media with(ms8/ma)51. In the highly scattering case, the effect ofg isseen most prominently in the near field due to its impactthe boundary condition used in thed-P1 approximation.However, the effect is small and results in changes of the eFig. 7 Normalized optical penetration depths D̄[meffD (s) and D̄ int[meffDint (d) versus (ms8/ma) as predicted by the d-P1 approximation(solid curves) and MC simulations (symbols) for planar illuminationunder refractive index (a) matched (n51.0) and (b) mismatched (n51.4) conditions. Lower plots show the percentage error of the d-P1predictions relative to the MC simulations.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 639le---eersur-sots.-forllerth-gustrateir-hefarldadythethehisenh-fed inre-theiornteans inis-Carp, Prahl, and Venugopalanbetweend-P1 and Monte Carlo~MC! estimates that do notexceed 4% relative to results found forg50.9. Note that for(ms8/ma)5100, the value ofg does not affect the predictionsin the far field as the SDA limit is applicable. As a result thedecay of the fluence rate profiles is governed byexp(2meffz)and is independent ofg for a fixed(ms8/ma). By contrast, for(ms8/ma)51, the variation ing affects the errors most promi-nently in the far field. This occurs because there is minimabackscattering due to the higher absorption in the mediumleading to a fluence rate profile whose decay is dependent og even for a fixed(ms8/ma). However, we again see that theeffect of g is minimal as the variations in the error are lessthan 7% even in the far field. Given that these error variationsare small and the fact that most soft biological tissues arstrongly forward scattering we show all remaining results fora value ofg50.9 ~Ref. 29!.Figures 6~a! and 6~b! present the variation of the diffusereflectanceRd with (ms8/ma) for n51.0 and 1.4, respectively.As in Fig. 3, there is good agreement for large(ms8/ma) in-dependent of the refractive index mismatch. Under indexmatched conditions, there is no internal reflection at the surface andRd is predicted with a relative error of68%. For arefractive index mismatch corresponding to a tissue-air interface, the model predictions degrade as(ms8/ma) is reduced.Specifically, relative errors exceed 15% for(ms8/ma),3.However, as(ms8/ma)→0 the model is bound to recover itsaccuracy since the diffuse component vanishes andRd→0 as(ms8/ma)→0. Moreover, for(ms8/ma),0.3 the amount of dif-fuse reflectance is negligible for all practical purposes. Thuswhile the relative error inRd may be large, the absolute erroris vanishingly small.To better characterize the variation in accuracy of thed-P1approximation with(ms8/ma) we examine the OPDs that char-acterize the fluence rate profiles. Figures 7~a! and 7~b! presentestimates for the normalized OPD metricsD̄[meffD andD̄ int[meffDint as predicted by thed-P1 approximation andMC predictions for1022<(ms8/ma)<104 under refractive in-dex matched(n51.0) and mismatched(n51.4) conditions,respectively.Note that under conditions of dominant absorption, i.e.,(ms8/ma)→0, meff→)ma . Thus both D̄ and D̄ int approach(1/ma)(meff)5) as(ms8/ma)→0. This result is confirmed inFigs. 7~a! and 7~b!. In the limit of high scattering, i.e.,(ms8/ma)→`, inspection of Eq.~29! reveals that the value ofD̄ is dependent on the refractive index mismatch through thboundary parameterA. Setting Eq.~29! equal toE0~12Rs!/eand solving we find thatD̄511ln~312A!. Thus, for(ms8/ma)→`, thed-P1 approximation predicts thatD̄→2.61 and 3.19for n51.0 and 1.4, respectively. By contrast, a similar analy-sis reveals thatD̄ int is not sensitive to the refractive indexmismatch andD̄ int→1 as (ms8/ma)→`. These asymptoticlimits predicted by thed-P1 model are confirmed by the re-sults shown in Figs. 7~a! and 7~b!. Overall thed-P1 predic-tions for the optical penetration depth are impressive andmatch the MC estimates to within64% over the entire rangeof (ms8/ma). The highest relative errors occur at(ms8/ma).1 as expected from the characteristics of the fluence rat640 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3nprofiles shown in Fig. 3. Better accuracy is observed forD̄(62%) than for D̄ int (64%). This is due to the strongeimpact that underestimation of the fluence rate near theface has on the determination ofD̄ int .3.2 Gaussian Beam IlluminationFigures 8~a! and 8~b! provide normalized fluence rate profilealong the beam centerline(r 50) as predicted by thed-P1approximation and MC simulations at(ms8/ma)5100 forbeam radii r 05100l * , 30l * , 10l * , 3l * , and 1l * with n51.0and 1.4, respectively. The errors of thed-P1 predictionsrelative to the MC estimates are shown below the mainplThe fluence rate along the beam centerline forr 05100l * dif-fers by less than60.5% from that produced by planar irradiation. For bothn51.0 and 1.4, thed-P1 approximationprovides good accuracy relative to the MC predictionsbeam radiir 0.3l * (617% in the near field,65% in the farfield!. However, the model accuracy degrades for smabeam radii and reaches625% for r 05 l * . This is expectedgiven that the diffusion model breaks down when lengscales comparable tol * are considered.Figures 9~a! and 9~b! provide results for the more challenging case of(ms8/ma)51. Due to the reduced scatterindispersion that occurs in media of higher absorption, one mconsider much smaller beam diameters before the fluenceprofiles along the center differ noticeably from the planarradiation case. Specifically, for(ms8/ma)51, the fluence ratealong the beam centerline forr 0530l * differs by less than60.5% from that produced by planar irradiation. Forr 0.3l * , errors in the fluence rate predictions provided by td-P1 model relative to the MC estimates are63% in thenear field and622% in the far field. However, forr 05 l * ,the fluence rate is overestimated by nearly 100% in thefield. While a 100% error may appear striking, one shounotice that this occurs once the fluence rate has alredropped by more than two orders of magnitude relative tosurface value. Thus, while the percentage error is large,error with respect to the overall energy balance is small. Tlarge relative error for small beam radii is not surprising givthe great difficulty that low-order functional expansion metods have in modeling the light field whenms8.ma . In the farfield, the accuracy of thed-P1 model is nearly independent othe refractive index for the same reasons as those discussSec. 3.1.Figures 10~a! and 10~b! provide the normalized OPDD̄along the beam centerline for Gaussian irradiation as pdicted by thed-P1 model and MC simulations for1022<(ms8/ma)<104 and beam radiir 051 – 100l * with n51.0and 1.4, respectively. Corresponding results forD̄ int are pre-sented similarly in Figs. 11~a! and 11~b!. The OPDs deter-mined in the 1-D case are included for comparison as arecorresponding relative errors. The expected limiting behavfor (ms8/ma)→0 is identical to that in the planar irradiatiocase and thus bothD̄ and D̄ int converge to). For large(ms8/ma) the decay of the fluence rate with depth for finibeam illumination occurs on a spatial scale smaller thexp(2meffz) because as the incident laser beam propagatethe medium, optical scattering results in significant lateral dRadiative transport in the delta-P1 approximation . . .Fig. 8 Normalized fluence rate along the beam centerline w̄(r50)versus reduced depth (z/l* ) as predicted by the d-P1 approximation(solid curves) and MC simulations (symbols) for Gaussian beam illu-mination under refractive index (a) matched (n51) and (b) mis-matched (n51.4) conditions. Profiles are shown for (ms8/ma)5100with r05100l* (s), 30l* (* ), 10l* (L), 3l* (3), 1l* (d), and g50.9. Lower plots show the percentage error of the d-P1 predictionsrelative to the MC simulations.-heestpersion from the high fluence region along the beam centerline to the periphery. ThusD̄, D̄ int→0 as(ms8/ma)→`. Thed-P1 predictions forD̄ andD̄ int track the MC estimates well,with errors of less than64% in D̄ and620% in D̄ int for thesmallest beam radius studied(r 05 l * ). Once again, the larg-est errors occur forms8.ma and D̄ is predicted more accu-rately thanD̄ int . Both of these features are consistent with tfluence rate profiles shown in Figs. 8 and 9 where the largerrors are observed close to the surface(z,2l * ) and forms8.ma .Fig. 9 Normalized fluence rate along the beam centerline w̄(r50)versus reduced depth (z/l* ) as predicted by the d-P1 approximation(solid curves) and MC simulations (symbols) for Gaussian beam illu-mination under refractive index (a) matched (n51.0) and (b) mis-matched (n51.4) conditions. Profiles are shown for (ms8/ma)51 withr0530l* (s), 10l* (L), 3l* (3), 1l* (d), and g50.9. Lower plotsshow the percentage error of the d-P1 predictions relative to the MCsimulations.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 641Carp, Prahl, and VenugopalanFig. 12 (a) Color contour plot of the normalized fluence rate w̄(r,z) as predicted by both the d-P1 approximation (solid contours and color) and MCsimulations (dashed contours) for Gaussian beam irradiation with r053l* in media with (ms8/ma)5100 for g50.9 under refractive index mis-matched conditions (n51.4); and (b) relative error between d-P1 approximation and MC simulations.Fig. 13 (a) Color contour plot of the normalized fluence rate w̄(r,z) as predicted by both the d-P1 approximation (solid contours and color) and MCsimulations (dashed contours) for Gaussian beam irradiation with r053l* in media with (ms8/ma)53 for g50.9 under refractive index mismatchedconditions (n51.4); and (b) relative error between d-P1 approximation and MC simulations.642 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3Radiative transport in the delta-P1 approximation . . .Fig. 10 Normalized optical penetration depth D̄ versus (ms8/ma) aspredicted by the d-P1 approximation (solid curves) and MC simula-tions (symbols) along the beam centerline for Gaussian beam illumi-nation for g50.9 with r05100l* (s), 30l* (* ), 10l* (L), 3l* (3), and1l* (d) under refractive index (a) matched (n51.0) and (b) mis-matched (n51.4) conditions. The optical penetration depth for planarillumination predicted by the d-P1 approximation is plotted as adashed curve. Lower plots shows the percentage error of the d-P1predictions relative to the MC simulations.n-MC.Figure 12~a! provides a color contour plot representing the2-D fluence rate distribution for a Gaussian beam of radiusr 053l * with (ms8/ma)5100andn51.4. The solid isofluencerate contours and the color map correspond to the predictioprovided by thed-P1 approximation while the dashed isofluence rate contours represent predictions given by thesimulations. Figure 12~b! provides the 2-D distribution of therelative errors between thed-P1 predictions and the MCsimulations. Thus, thed-P1 and MC contours shown in FigFig. 11 Normalized optical penetration depth D̄ int versus (ms8/ma) aspredicted by the d-P1 approximation (solid curves) and MC simula-tions (symbols) along the beam centerline for Gaussian illuminationfor g50.9 with r05100l* (s), 30l* (* ), 10l* (L), 3l* (3), and 1l*(d) under refractive index (a) matched (n51.0) and (b) mismatched(n51.4) conditions. The optical penetration depth for planar illumi-nation predicted by the d-P1 approximation is plotted as a dashedline. Lower plots show the percentage error of the d-P1 predictionsrelative to the MC simulations.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 643--iaellri.rtrertheofthetheres,taintoia-ura-aysres-ec-sionnst isdic-e-en-ncetheeCarp, Prahl, and Venugopalan12~a! provides some indication of the errors in penetrationdepth that one makes when using thed-P1 approximation,while Fig. 12~b! provides the errors in the actual optical do-simetry.The quality of thed-P1 predictions are excellent; the errorin the fluence rate relative to the MC estimates never exceed20% and is less than 10% over the vast majority of the domain. In the axial direction, the maximum errors occur in thenear field close to the boundary, while in the radial direction,they occur along the beam centerline. This is expected because it is at these locations where the spatial gradients anangular asymmetry of the light field are greatest. Figures13~a! and 13~b! provide plots under identical irradiation con-ditions for a turbid medium with(ms8/ma)53. In Fig. 13~a!we see similar errors in the location of the isofluence ratecontours when comparing thed-P1 approximation relative tothe MC predictions. However, in Fig. 13~b!,we observe adifferent spatial pattern and magnitude of the fluence rate errors incurred when using thed-P1 approximation rather thana MC estimate. As in Fig. 12~b!, the maximum errors in theradial direction occur along the beam centerline. However, inthe axial direction, the maximum errors reside in the far fieldand appear to be increasing with depth. This is similar tothe planar irradiation case and occurs because the spatscale for the decay of the fluence rate with depth liesbetweenexp(2meffz) andexp(2mt*z); thereby leading to poorpredictions by thed-P1 approximation in the far field underthese conditions. It is important to note that examination ofd-P1 predictions at radial locations away from the centerlinereveals equivalent, if not better, accuracy in both fluence ratprofiles and OPD metrics. For example, for Gaussian beamradii r 0.3l * , the errors in bothD̄ and D̄ int at the radiallocation r 5r 0 are<5 and<8%, respectively, over the fullrange of(ms8/ma). This result is consistent with the errors ofthe full fluence rate distributions shown in Figs. 12 and 13.3.3 Gaussian Beam versus Planar IrradiationTreatmentAs is evident from the results, the use of laser beams of smadiameter significantly alters the fluence rate profile and opticapenetration depth. For example, Gaussian irradiation of a medium with (ms8/ma)5100 using a beam radius ofr 053l *results in a fluence rate that is only;50% of that achievedusing planar illumination. Moreover, the reduction in bothfluence rate and OPD for decreasing beam diameters is moprominent in media with large(ms8/ma) because the scatteringenhances lateral dispersion of the collimated radiation~Figs.8–13!. However, the Gaussian beam expressions are a bmore formidable than those for the case of planar irradiationAs a result, for simplicity and convenience, it may be usefulto determine the conditions under which the results of a planairradiation analysis provides sufficiently accurate predictionsalong the centerline of a Gaussian beam. This may obviate thneed to use the more complex expressions correspondingGaussian beam irradiation in some cases.Figure 14 provides these results in the form of a contouplot showing the percentage difference between the fluencrate predictions given by thed-P1 approximation for Gauss-ian beam irradiation along the centerline compared to planairradiation as a function of both normalized beam radius644 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3s-dll-eteo(r 0 / l * ) and optical properties(ms8/ma). Contours are pro-vided for differences of 1, 3, 10, and 30% forn51.0 ~solidcontours! and 1.4~dashed contours!, respectively. These re-sults indicate that as absorption becomes more dominant,centerline fluence rate profiles produced by laser beamssmaller diameter can be adequately approximated usingplanar irradiation predictions. This can also be seen inOPD results shown earlier in Figs. 10 and 11. In these figuwe observed that for a given beam radius, there is a cervalue of (ms8/ma) above which the OPDs correspondingGaussian irradiation drop below the OPDs for planar irradtion. We note that this value of(ms8/ma) becomes lower assmaller beam diameters are used. Note also that the inacccies incurred in using the planar irradiation results are alwlower for the index-matched case. This is because the pence of a refractive index mismatch results in internal refltion at the tissue-air interface that enhances lateral disperof the light field. This additional source of dispersion hastethe need for the use of a radiative transport model thageometrically faithful to the irradiation conditions.4 ConclusionWe have shown that thed-P1 approximation to the Boltz-mann transport equation provides remarkably accurate pretions of light distribution and energy deposition in homogneous turbid semi-infinite media. Examination of thfunctional expressions involved in thed-P1 approximationreveals proper asymptotic behavior in the limits of absorptioand scattering-dominant media. Comparison of the fluerate and optical penetration depth predictions given byd-P1 approximation with MC simulations demonstrate thgreater fidelity and accuracy of thed-P1 model relative to thestandard diffusion approximation.Fig. 14 Contours for the error incurred in predicting fluence rate pro-files along the centerline of a Gaussian laser beam of normalizedradius r0 /l* as a function of (ms8/ma) when using d-P1 predictions forthe planar irradiation case for n51.0 (dashed) and n51.4 (solid).lne-rlsientofingashtalof-Radiative transport in the delta-P1 approximation . . .The availability of an analytic light transport model pro-viding accurate optical dosimetry predictions is an invaluabletool for the biomedical optics community. By providing ourresults in terms of dimensionless quantities, they can be useto rapidly estimate the fluence rate distributions and opticapenetration depths generated by a wide range of irradiatioconditions and tissue optical properties. Thus beyond a greattheoretical understanding of the significant gains to be realized through the use of thed-P1 approximation over the stan-dard diffusion approximation, these figures provide the bio-medical optics community with charts that can be used forapid lookup and estimation of light-transport related quanti-ties.5 Appendix A Derivation of Surface BoundaryConditions in the d -P1 ApproximationThe governing equations of thed-P1 approximation are~seeSec. 2!:¹2wd~r !23mam trwd~r !523ms* m trE~r ,ẑ!13g* ms* ¹E~r ,ẑ!• ẑ ~31!j ~r !5213m tr@¹wd~r !23g* ms* E~r ,ẑ!ẑ#, ~32!where r is the position in the medium,ẑ is the unit vectorcolinear with the direction of the collimated source,E(r ,ẑ) isthe irradiance distribution of the collimated source,ma is theabsorption coefficient,m tr[ma1ms8 is the transport coeffi-cient with ms8 being the isotropic scattering coefficient,g* isthe single scattering asymmetry coefficient of theP1 portionof the d-P1 phase function, andms* [ms(12 f ) is a reducedscattering coefficient. Selection off and g* depends on theselection of the phase function as described in Sec. 2.1.Two boundary conditions are required to solve Eq.~31!.Requiring conservation of the diffuse flux component normato the interface, we obtain6,23Ev̂• ẑ>0Ld~r ,v̂ !~v̂• ẑ!dv̂5Ev̂• ẑ,0Ld~r ,v̂ !r F~2v̂• ẑ!~2v̂• ẑ!dv̂,~33!where ẑ is the inward-pointing surface normal, andr F(2v̂• ẑ) is the Fresnel reflection coefficient for unpolarizedlight. The preceding condition can be described in words aequating the amount of diffuse light that travels upward(v̂• ẑ,0) and gets internally reflected at the interface withthe amount of diffuse light traveling downward(v̂• ẑ>0)from the interface.Substituting the approximation for the diffuse fluence rategiven by Eq. ~9! and using Eq.~32! to eliminate j ~r !, weobtain the following form for the surface boundary conditionin the d-P1 approximation:@wd~r !2Ah¹wd~r !• ẑ#uz50523Ahg* ms* E~r ,ẑ!uz50 ,~34!drwhere A5(11R2)/(12R1) and h52/3m tr . This result isidentical to that provided by Eq.~12!. HereR1 andR2 are thefirst and second moments of the Fresnel reflection coefficfor unpolarized light, as given by Eq.~13!.Note that in many implementations of the SDA,A is ap-proximated instead byA'(11R1)/(12R1). While this isstrictly incorrect, it results in slightly better approximationsthe fluence rate in the near field at the expense of providworse fluence rate approximations in the far field as wellviolating conservation of energy when integrating the ligfield over the entire volume. The following cubic polynomiprovides an estimate forA5(11R2)/(12R1) that typicallydiffers from the exact value by less than 1%:23A~n!520.13755n314.3390n224.90366n11.6896.~35!6 Appendix B Solution of the d -P1Approximation for Planar Illuminationof a Semi-Infinite MediumFor planar illuminationthe source term is given byE~z,v̂ !5E0~12Rs!exp~2m t* z!d~12v̂• ẑ!, ~36!whereE0 is the irradiance,v̂ is the unit direction vector, andẑ is the inward pointing unit vector normal to the surfacethe medium and is colinear with thez coordinate axis. Sub-stituting Eq.~36! into Eq.~10!, we obtain the governing equation for a planar geometry:d2wd~z!dz2 23mam trwd~z!523ms* ~m t* 1g* ma!E0~12Rs!exp~2m t* z!.~37!The boundary conditions for the 1-D case reduce toS wd2Ahdwd~z!dz D Uz50523Ahg* ms* E0~12Rs!,~38!wd~z!uz→`→0. ~39!The solution to Eq.~37! satisfying the Eqs.~38! and ~39! iswd~z!5E0~12Rs!@a exp~2m t* z!1b exp~2meffz!#,~40!wherea53ms* ~m t* 1g* ma!meff2 2m t*2 ~41!andb52a~11Ahm t* !23Ahg* ms*~11Ahmeff!. ~42!These results are identical to that provided by Eqs.~18! to~20!.Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 3 645.hCarp, Prahl, and Venugopalan7 Appendix C Solution of the d -P1Approximation for Gaussian Beam Illuminationof a Semi-Infinite MediumThe source term for a Gaussian beam profile is given byE~r ,z!5E0~12Rs!exp~2m t* z!expS 22r 2r 02 D , ~43!wherer 0 is the1/e2 beam radius, andE052P/(pr 02), whereP is the power of the laser beam. The governing equation incylindrical coordinates has the form1r]]r S r]wd~r ,z!]r D1]2wd~r ,z!]z2 2meff2 wd~r ,z!523ms* ~m t* 1g* ms* !E~r ,z!, ~44!subject to the boundary conditions:S wd2Ah]wd]z D Uz50523Ahg* ms* E~r ,z!uz50 , ~45!]wd~r ,z!]r Ur 5050, ~46!wd~r ,z!uz→`→0, ~47!wd~r ,z!ur→`→0. ~48!The solution procedure begins by assuming that bothwd(r ,z) and the right-hand side of Eq.~44! can be written asHankel transforms of two functionsf (k,z) and u(k,z), re-spectively, i.e.,E0`f ~k,z!J0~kr !kdk5wd~r ,z! ~49!andE0`u~k,z!J0~kr !kdk523ms* ~m t* 1g* ma!E0~12Rs!3exp~2m t* z!expS 22r 2r 02 D , ~50!whereJ0 is the zeroth-order Bessel function of the first kind.Substituting Eqs.~49! and ~50! into Eq. ~44! we obtain1r]]r F r]]r E0`f ~k,z!J0~kr !kdkG1]2]z2 E0`f ~k,z!J0~kr !kdk2meff2 E0`f ~k,z!J0~kr !kdk5E0`u~k,z!J0~kr !kdk.~51!We note that the first term of Eq.~51! appears in the Bessel’sequation:646 Journal of Biomedical Optics d May/June 2004 d Vol. 9 No. 31r]]rr]]rJ0~kr !1k2J0~kr !50, ~52!for which J0 is a solution. Thus Eq.~51! can be rewritten byadding and subtractingk2J0(kr) on the left-hand side of Eq~52!, which yieldsE0`~2k22meff2 ! f ~k,z!J0~kr !kdk1E0` ]2]z2 J0~kr ! f ~k,z!kdk5E0`u~k,z!J0~kr !kdk. ~53!Using a table of Hankel transforms,33 u(k,z) can be chosensuch that Eq.~50! is satisfied, namely,]2]z2 f ~k,z!2~k21meff2 ! f ~k,z!523ms* ~m t* 1g* ma!E0~12Rs!r 0243expS 2r 02k28 Dexp~2m t* z!. ~54!The boundary conditions in(k,z) space are obtained througHankel transformation of Eqs.~45! to ~48!:F ]]zf ~k,z!21Ahf ~k,z!GUz50534g* ms* E0~12Rs!r 023expS 2r 02k28 D , ~55!andf ~k,z!uz→`→0. ~56!Solving the Eq.~54! for f (k,z) and substitution of theresults into Eq.~49! gives the following form forwd(r ,z):wd~r ,z!5E0~12Rs!E0`$g exp~2m t* z!1j exp@2~k21meff2 !1/2z#%J0~kr !kdk, ~57!whereg53ms* ~m t* 1g* ma!r 02 exp~2r 02k2/8!4~k21meff2 2m t*2!~58!andj523g* ms* r 02 exp~2r 02k2/8!24g@~Ah!211m t* #4@~Ah!211~k21meff2 !1/2#.~59!These results are identical to that provided by Eqs.~22! to~24!.f,hfu-ind es-u-ortngi--y infabl-IEhtndton,’’ewn,-y,’’ofd-neorys,’’sueRadiative transport in the delta-P1 approximation . . .AcknowledgmentsWe thank Fre´déric Bevilacqua, Carole Hayakawa, ArnoldKim, and Jerry Spanier for helpful and stimulating discus-sions. 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## Perguntas dessa disciplina

UNIEURO

UNOPAR

UNINASSAU PARANAÍBA

ESTÁCIO

ESTÁCIO