In a companion paper1 we presented a model thatdescribes photon migration through a slab of diffusersand through a semi-infinite medium illuminated by apencil beam. In the scheme, a narrow collimatedpulsed light beam is normally incident upon the sur-face of a diffusing slab ~thickness s!; the pulse issupposed to be mathematically described by a Diracdelta function ~centered at the time t 5 0!, and it issupposed to be thin and collimated ~pencil beam!.The coordinate system was chosen with the origin inthe point in which light is entering the medium, withthe z axis along the direction of propagation of thecollimated pulse. The distance of the exit surfacefrom the pencil beam is indicated with r. The modelis used to predict the reflectance and the transmit-tance both in the time-dependent case and in the cwcase and is able to take into account reflections at theboundaries resulting from the refractive index mis-match. The model is obtained with the diffusion ap-
The authors are with the Dipartimento di Fisica dell’Universitadegli Studi di Firenze, Via Santa Marta 3, 50139 Firenze, Italy.
Received 15 April 1996; revised manuscript received 25 Novem-ber 1996.
4600 APPLIED OPTICS y Vol. 36, No. 19 y 1 July 1997
proximation ~DA!. The simplifying assumptionsnecessary to obtain the solutions presented in Ref. 1can be summarized: ~1! the diffuse specific inten-sity, due to many scattering events inside the diffus-ing medium, is assumed to be almost isotropic with asimple angular distribution; ~2! the variation of theflux vector over the time interval necessary to coveran effective mean free path z0 5 1yms9 is assumed tobe small with respect to the flux vector itself ~ms9 isthe transport or reduced scattering coefficient!; ~3!the pencil beam at the surface of the slab ~z 5 0! isreplaced with an isotropic source inside the slab atz 5 z0; and ~4! the exact boundary conditions cannotbe exactly satisfied with the simple angular distribu-tion assumed for the diffuse specific intensity.
The procedure followed to obtain the diffusionequation led us to define the diffusion coefficient asD 5 1y3ms9, i.e., independent of the absorption coef-ficient ma of the medium. The diffusion equation~DE! was solved with the extrapolated boundary con-dition, i.e., assuming the average diffuse intensityequal to zero at two extrapolated flat surfaces outsidethe turbid slab at a distance ze 5 2AD from thephysical boundaries of the slab. The coefficient Aaccounts for the effect of reflections resulting from therefractive index mismatch. With the method of im-age sources to account for the extrapolated boundarycondition, the solution of the DE was obtained for the
time-resolved reflectance and transmittance. Thetemporal point-spread function ~TPSF!, i.e., the prob-ability that a photon emitted by the source at the timet 5 0 crosses the surface element after a time t, at adistance r from the pencil beam, per unit of time andunit of area, was evaluated at z 5 0 and z 5 s. Byintegrating these solutions over the entire exit sur-face, we obtained the total time-resolved reflectanceand transmittance. By integrating the time-resolved responses, we obtained the relationships forthe cw source.
Owing to the approximations necessary to obtainthe DE, the formulas obtained are approximate solu-tions of the radiative transfer equation ~RTE! andtheir limit of validity should be checked. To validatethe predictions of the model, we developed a MonteCarlo ~MC! code to obtain solutions of the RTE inmany different physical and geometrical conditions,including the effect of reflections. In this researchthe results of the comparison between the numericalresults and the predictions of the analytical approx-imate model are presented. The comparison showedthat the analytical model, in spite of the approxima-tions necessary to obtain the solution, gives an excel-lent description of photon migration in a large rangeof conditions of interest for tissue optics. In partic-ular, for the cw source the formulas derived for thetransmittance and the reflectance give accurate re-sults ~within 10%! when the distance from the sourceis larger than approximately 10z0. For the time-dependent case the description of the received pulseis accurate ~within 10%! when the length of the tra-jectories followed by photons is longer than approxi-mately 15z0–30z0. Also the effect of reflections onthe boundary owing to the refractive index mismatch,causing both a spatial and temporal redistribution ofphotons, is well described for the whole range of val-ues of refractive index with physical interest.
A fitting procedure for obtaining the optical prop-erties of the medium from measurements of time-resolved reflectance or transmittance was developedwith the formulas obtained from the DE. The fittingprocedure was used on MC results to check the ap-plication of the model to the inverse problem.Within the applicability limits of the model previ-ously mentioned, the agreement between the resultsobtained by the inversion procedure and the expectedvalues is within 2% for ms9. For the absorption co-efficient the discrepancies with respect to the actualvalue are within 1024 mm21. These discrepanciesare not relevant for applications to tissue optics.Significantly different results for ms9 were obtainedwhen the model previously proposed by Patterson etal.,2 in which a more approximate boundary condition~the zero boundary condition! was used and reflec-tions were not taken into account, was used for theinversion procedure: The results obtained from thetime-resolved transmittance for the reduced scatter-ing coefficient may be overestimated by 10–20%, de-pending on the value of the relative refractive index.Significantly smaller differences, inverting the time-resolved reflectance, were observed. The discrepan-
cies decrease when the thickness of the slab isincreased.
The MC results were also used for obtaining infor-mation on the angular distribution of exiting photons.In particular, the effect of the limited angular field ofview of the receiver on the temporal distribution ofreceived photons was investigated together with theangular distribution of photons exiting from the dif-fusing medium. The numerical results showed thatthe shape of the temporal distribution of receivedphotons is not affected significantly by the receiverfield of view when the turbidity of the medium issufficiently high so that the DA can be applied. Con-cerning the angular distribution of exiting photons,significant differences were observed with respect tothe Lambertian surface. The differences also de-pend on the refractive index mismatch.
2. Description of the Monte Carlo Code
The MC method provides a physical simulation ofphoton migration and enables us to obtain the solu-tion of the RTE without any need for simplifyingassumptions. Therefore the solutions obtained canbe used as a standard reference. Details on the MCcode we used can be found in Refs. 3 and 4. Theeffect of reflections on the boundary, resulting fromthe refractive index mismatch, was taken into ac-count according to the Fresnel reflection coefficientfor unpolarized radiation. In both MC and DA, re-flections were evaluated only for the outgoing diffusedradiation ~reflection of the incident beam was notconsidered!, but they can be simply allowed for bymultiplying the reflectance and the transmittance by1 2 Rn, with Rn Fresnel reflection coefficient for nor-mal incidence. The useful photons ~photons whosetrajectories intersect the surface of the receiver withan incidence angle smaller than the receiver angularfield of view! were classified on the basis of the transittime. We obtained the estimation of the TPSF bydividing the number of useful photons receivedwithin any temporal window nui, by the number ofemitted photons, N, by the temporal window, by thearea of the receiver. The receiver was sufficientlysmall so that the irradiance could be considered con-stant on the receiver area. The accuracy of numer-ical solutions was evaluated with the followingrelationship for the standard deviation5:
s2Snui
N D 51
N2Snui 2nui
2
N D <nui
N2 . (1)
Photons were emitted at time t 5 0 on the surfaceof the slab ~z 5 0! along the z axis. Therefore theTPSF obtained by numerical simulations can be di-rectly compared with the analytical results withoutany normalization. Numerical simulations werecarried out for a nonabsorbing medium. The effectof absorption on the TPSF was introduced accordingto the RTE by multiplying the TPSF, referring to ma5 0 by exp~2mavt! with speed of light v in the diffus-ing medium.
The correctness of the routines used to determine
1 July 1997 y Vol. 36, No. 19 y APPLIED OPTICS 4601
the scattering points was checked, comparing somestatistical parameters with analytical formulas.6The scattering functions used for numerical simula-tions were evaluated by Mie theory, assuming amonodispersion of spheres. Since for a fixed value ofms9 the computation time is shorter when the asym-metry factor g is equal to zero, almost all simulationswere carried out with the scattering function thatreferred to very small spheres ~radius 5 0.001 mm!.
3. Comparison between Diffusion Approximation andMonte Carlo Results
In this section some comparisons between the resultsobtained from the formulas presented in Ref. 1 andresults obtained from Monte Carlo simulations arepresented.
A. Slab Geometry
In Fig. 1 a comparison of the transmittance of a slab40 mm thick with a reduced scattering coefficient ms95 0.5 mm21 is shown. The diffusing medium is non-absorbing and has a refractive index n2 5 1.4.Curves are presented for two values of the relativerefractive index: n 5 0.7143 and n 5 1.4 ~n 5 n2yn1,with n1 refractive index of the external medium!.The value n 5 1.4 is typical of a biological tissue–airinterface. Noiseless curves are those obtained fromEq. ~39! of Ref. 1. Data refer to r 5 0 ~surface ele-ment coaxial with the light beam!. In both cases theagreement between the simulations and the analyt-ical values is good. Equation ~39! of Ref. 1 is there-fore also able to describe correctly the effect of therefractive index mismatch. Note that the resultspresented in Fig. 1 and in subsequent figures wereobtained directly from numerical simulations or fromthe formulas, without any normalization.
In Fig. 2 a comparison for a nonabsorbing diffusingslab 40 mm thick with ms9 5 0.75 mm21, n2 5 1.4, andn 5 1.33 is shown. The two curves labelled R and T~both evaluated at r 5 30 mm! represent the reflec-tance and the transmittance obtained with the sim-
Fig. 1. Comparison of the results from the DE ~noiseless curves!and the MC simulations. The time-resolved transmittance at r 50 for a nonabsorbing slab 40 mm thick with ms9 5 0.5 mm21 and n2
5 1.4 is reported for two values of the relative refractive index, n 51.4 and n 5 0.7143.
4602 APPLIED OPTICS y Vol. 36, No. 19 y 1 July 1997
ulation and Eqs. ~36! and ~39! of Ref. 1. Once againthe noiseless curves are obtained from the DA. Theagreement is so good that curves that refer to the DAand to the numerical simulation are almost indistin-guishable.
In Fig. 3 a comparison is shown between numer-ical results and the predictions of Eq. ~41! of Ref. 1~noiseless curves! for the total time-resolved trans-mittance, i.e., the transmittance integrated on theentire exit surface. Results refer to a nonabsorb-ing slab 40 mm thick with ms9 5 0.5 mm21; n2 5 1.4;and two different values of the relative refractiveindex, n 5 1 ~no reflection! and n 5 1.4.
In Fig. 4 comparisons for the cw case are presentedfor the transmittance. A diffusing slab 40 mm thickwith ms9 5 0.5 mm21 was considered for two values ofthe relative refractive index, n 5 1 and n 5 1.4. Thefigure shows the cw transmittance versus the dis-tance from the light beam r for a nonabsorbing me-dium. The continuous curves represent the results
Fig. 2. Comparison of the results from the DE ~noiseless curves!and the MC simulations for the time-resolved reflectance R andtransmittance T for a nonabsorbing slab 40 mm thick with ms9 50.75 mm21, n2 5 1.4, and n 5 1.33. Both the transmittance andthe reflectance were evaluated at a distance r 5 30 mm from thelight beam.
Fig. 3. Comparison of the results from the DE ~noiseless curves!and the MC simulations for the total time-resolved transmittance.Data refer to a nonabsorbing slab 40 mm thick with ms9 5 0.5mm21, n2 5 1.4, and are reported for two values of the relativerefractive index, n 5 1.4 and n 5 1.
we obtained by using Eq. ~46! of Ref. 1, the marksrepresent the MC results. A good agreement be-tween analytical and numerical results is shown. Acomparison of numerical results for the mean pathlength with Eq. ~48! of Ref. 1 showed discrepancieswithin 1% for the whole range of r reported in Fig. 4.
Figure 5~a! reports the cw reflectance for the sameslab considered in Fig. 4 for n 5 1.4. In this case anabsorbing medium was considered with ma 5 0.01mm21 ~a value typical for biological tissue!. The dif-fusion results are obtained from Eq. ~45! of Ref. 1.Also this figure shows a good agreement betweenanalytical and numerical results, apart from the re-sults referring to the reflectance when r , 10 mm.When a surface element near the source is consid-ered, many of the received photons have not under-gone a sufficiently large number of scattering eventsto randomize the propagation as assumed by the DA.Also for small values of r the replacement of thepencil beam in z 5 0 with the isotropic source in z 5z0 5 1yms9, as was assumed to solve the DE, maycause discrepancies. To investigate this effect theMC simulation was repeated for an isotropic sourcein z 5 z0; these results are also reported in Fig. 5~a!.To better point out the discrepancies between theanalytical model and the numerical results, Fig. 5~b!reports the percentage of difference between the re-sults from the DE and the MC results. This figureshows that for r . 10 mm, i.e., at distances largerthan 5z0, the error is smaller than 10%. The agree-ment is better with results that refer to the pencilbeam. Also the results obtained in similar condi-tions for n 5 1 ~no reflection! were in a good agree-ment. In this case the agreement of analyticalresults was better with the isotropic source ~errorsmaller than 5% on the whole range investigated!;with respect to the pencil beam the error was smallerthan 10% only for r . 10z0. In Fig. 5~c! the resultsobtained from the DA for the mean path length @Eq.~47! of Ref. 1# have been compared with MC results.Also this quantity is well described by DE: for r .5z0 the differences are smaller than 5%.
Fig. 4. Comparison of the results from the DE and the MC sim-ulations ~marks! for a cw source. The transmittance is reportedfor a nonabsorbing slab 40 mm thick with ms9 5 0.5 mm21 for twovalues of the relative refractive index, n 5 1.4 and n 5 1.
The relationships for the cw transmittance and re-flectance obtained from the DE give accurate resultsfor sufficiently large values of r. Therefore they maybe used to invert cw measurements of reflectance or
Fig. 5. Comparison of the results from the DE and the MC sim-ulations for the reflectance of a slab 40 mm thick with ms9 5 0.5mm21, ma 5 0.01 mm21, and n 5 1.4: ~a! cw response versus thedistance from the light beam ~MC results are reported for both apencil beam at z 5 0 and an isotropic point source at z 5 z0!; ~b!percentage of difference between the DE and the MC results; ~c!comparison for the mean path length followed by received photonsfor the pencil beam.
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transmittance in order to determine the optical prop-erties of the diffusing medium when measurementsare carried out at a distance from the source largerthan approximately 10z0.
To investigate the limits of applicability of the DAto predict the time-resolved transmittance whensmall values of the reduced scattering coefficient areconsidered, Fig. 6 reports on a comparison with MCresults for a nonabsorbing slab 40 mm thick with ms95 0.2 mm21, n2 5 1.4, and n 5 1. The data refer tor 5 0. Together with the analytical results from Eq.~39! of Ref. 1, the figure depicts the numerical resultsfor the TPSF, referring to both the pencil beam andthe isotropic source. For the pencil beam the resultswere reported for two scattering functions with g 5 0and g 5 0.5. The MC results refer to only the scat-tered component of the received energy, i.e., the un-scattered component was not taken into account.
Fig. 6. Comparison of the results from the DE and the MC sim-ulations for a slab with moderate optical thickness: ~a! time-resolved transmittance at r 5 0 for a nonabsorbing slab 40 mmthick with ms9 5 0.2 mm21, n2 5 1.4, and n 5 1. To point out thedependence of earlier received photons on the single-scatteringproperties of the medium and on the characteristics of the source,the MC results are reported also for a scattering function with g 50.5 and for an isotropic source at z 5 z0; ~b! percentages of differ-ence between the DE and the MC results.
4604 APPLIED OPTICS y Vol. 36, No. 19 y 1 July 1997
Also in the figure the results that refer to the DE werereported only for times longer than the time neces-sary for unscattered photons to reach the exit surface.However, to point out the inadequacy of the DE todescribe earlier received photons, we note that theTPSF obtained by the DE predicts the receipt of pho-tons immediately after the emission of the pulse.The percentages of difference between analytical andnumerical results are shown in Fig. 6~b!. The agree-ment is good ~within 10%! with all the three numer-ical results for t . 500 ps, i.e., for photons that havefollowed trajectories longer than approximately 30z0.At short times there are significant differences. Thefirst-arrived photons underwent only a few scatteringevents and their histories depend strongly on thedetails of both the single scattering phase functionand the source. In particular, the contribution ofsingly scattered photons @proportional to p~q 50!exp~2mss!, with scattering function p~q 5 0! in theforward direction and scattering coefficient ms# is sig-nificantly larger for g 5 0 and accounts for the spikeat short times.
To investigate the limits of applicability of the DEto predict the time-resolved reflectance at short timesand at small distances from the source, we reportcomparisons in Fig. 7. Data refer to a nonabsorbingslab 40 mm thick with ms9 5 0.5 mm21, n2 5 1.4, andn 5 1.4. Figure 7~a! refers to r 5 4.2 and 10.2 mm;Fig. 7~b! to r 5 20.4 and 41.2 mm. Also in this casethe numerical results that refer to both the pencilbeam and the isotropic source have been reported.The comparison enables us to distinguish betweenthe discrepancies due to the assumption of an isotro-pic source and the other simplifying assumptions in-trinsic to the DA. Figures 7~c! and 7~d! report thepercentages of difference between the results fromthe DE and the numerical results for r 5 4.2 and 20.4mm. These figures show an excellent agreement forsufficiently long times. The mean error for longtimes remains within 63%. At very early times theDE overestimates the TPSF with respect to the nu-merical results, referring to both the pencil beam andthe isotropic source.
In Ref. 1 two relationships for the fraction of theincident energy reflected R and transmitted T werealso reported @Eqs. ~49! and ~50!#. Table 1 reportsexamples of comparisons with MC results for a slab40 mm thick with ms9 5 0.5 mm21. The results referto n 5 1 and n 5 1.4 for different values of theabsorption coefficient. A good agreement was ob-tained for both R and T. Although a good agreementwas expected for T since almost all photons transmit-ted through the slab are expected to have undergonemany scattering events, the good agreement for R isquite surprising since a significant contribution tothe total diffuse reflectance is given by photons thatexit near the source after a few scattering events @seeFigs. 5 and 7~a!# especially for large values of ma.The formulas for R and T can therefore also be usedto predict with good accuracy the fraction of energyabsorbed by the medium: 1 2 ~R 1 T!. For the
Fig. 7. Comparison between the results from the DE and the MC simulations for a nonabsorbing slab 40 mm thick with ms9 5 0.5 mm21,n2 5 1.4, and n 5 1.4: ~a!, ~b! examples of time-resolved reflectance for various values of r. The MC results are reported for both thepencil beam at z 5 0 and the isotropic point source at z 5 z0; ~c!, ~d! percentages of difference between the DE and the MC results for r5 4.2 and 20.4 mm, respectively.
semi-infinite medium the formula for the total reflec-tance reduces to
R 512
expF2S3ma
ms9D1y2GH1 1 expF2 4
3AS3ma
ms9D1y2GJ . (2)
The comparisons reported show clearly that theformulas presented in Ref. 1 are able to give correctresults in many different situations if we deal withthe radiation that has traveled inside the medium ona sufficiently long path: The discrepancy between
Table 1. Fraction Incident Energy Reflected and Transmitted: Comparison of DE and MC Simulationsa
aData refer to the slab 40 mm thick with ms9 5 0.5 mm21 for various values of the absorption coefficient for n 5 1 and n 5 1.4.bR is incident energy reflected.cT is incident energy transmitted.
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the analytical results and the true values is smallerthan 10% if the path followed is longer than 15–30effective mean free paths. Also the relationship forthe total reflectance, to which a significant contribu-tion is given by photons that exit near the source aftera few scattering events, gives accurate results.
The comparison of the DE with the MC results forthe isotropic point source and for the pencil beamshowed significant differences only at short times orat small distances from the source. When a signifi-cant refractive index mismatch is present, the agree-ment is better, with the results referring to the pencilbeam. Although we obtained the analytical resultsmodeling the pencil beam with an isotropic source,the comparison with MC results showed that in gen-eral there is no evidence of a better agreement withnumerical results that refer to the isotropic source.This indicates that the discrepancies observed aremainly due to the other simplifying assumptions nec-essary to obtain the DE. Farrel et al.7 solved the DEfor a semi-infinite medium with a cw source, alsoassuming a line distribution of isotropic sources withstrengths proportional to exp~2zms9! to better ap-proximate the pencil beam. However, they also ob-served that the simpler model with the single sourcegave a better agreement with the MC results.
In all MC simulations the standard deviation of theresults was calculated according to Eq. ~1!. In thefigures that represent the TPSF, error bars have notbeen reported for clarity, but the statistical noise ofthe curves gives an idea of the variance. In the sim-ulations for the cw response ~Figs. 4 and 5! the stan-dard deviation was always smaller than 2%.
B. Semi-Infinite Medium
In Fig. 8 a comparison between MC results and theDE that refers to the time-resolved reflectance is pre-sented for two different values of r. The analyticalresults were obtained with Eq. ~36! of Ref. 1, retain-ing only the term with m 5 0. The semi-infinitenonabsorbing medium has been chosen with ms9 5 0.5
Fig. 8. Comparison between the results from the DE ~noiselesscurves! and the MC simulations for the time-resolved reflectancefrom a semi-infinite nonabsorbing medium with ms9 5 0.5 mm21,n2 5 1.4, and n 5 1. Results are reported for two distances fromthe light beam.
4606 APPLIED OPTICS y Vol. 36, No. 19 y 1 July 1997
mm21, refractive index n2 5 1.4, and relative refrac-tive index n 5 1.4. In Fig. 9 the cw case is shown.The differences between analytical and numerical re-sults are smaller than 5% for r . 3z0. A good agree-ment between simulations and analytical results isthus shown both in the time-dependent case and inthe cw case.
The models derived from the DE for the reflectanceof a semi-infinite medium have been widely used toapproximate photon migration through an opticallythick slab because the formulas for the semi-infinitemedium are much more simple than for the slab ge-ometry. The higher-order terms in the series, rep-resenting the solutions of the DE for the slab, becomemore and more important, increasing the arrivaltime. For large values of t the TPSF’s are basicallydominated by the contribution of the sources placedfar outside the medium ~large values of the index min the formulas!; therefore it is not possible to indi-cate a value of the thickness of the slab s, startingfrom which this error is smaller than a prefixedthreshold for all values of t. However, in real casesthe TPSF’s are recorded in the experiments for a timeinterval on which the signal remains within 2–3 or-ders of magnitude with respect to the maximum.Equation ~45! of Ref. 1 shows that for a nonabsorbingmedium in the cw case, the difference between thereflectance for the semi-infinite medium and the fi-nite slab is smaller than 10% if the ratio between thedistance from the light beam r and the thickness ofthe slab s is smaller than 2y3, and the difference issmaller than 1% if rys , 1y4. The difference de-creases rapidly when an absorbing medium is consid-ered.
4. Inversion Procedures for the TemporalPoint-Spread Function to Determine the OpticalProperties
Fitting procedures of the TPSF have been usedwidely to extract information on the optical proper-ties of a turbid medium. Usually an analytical for-mula derived from the DA is fitted to a set of
Fig. 9. Comparison between the results from the DE and the MCsimulations for the cw reflectance from a semi-infinite nonabsorb-ing medium with ms9 5 0.5 mm21 and n 5 1.4. The results referto a pencil beam.
measured data. The relevant fitting parameters arethe reduced scattering coefficient ms9 and the absorp-tion coefficient ma. In this section the results of afitting procedure based on the formulas reported inRef. 1 are presented. A set of TPSF’s in differentphysical and geometrical conditions has been gener-ated with MC simulations in order to produce data tobe fitted with the formulas derived from the DA.The fitting procedure was implemented with aLevenberg–Marquardt algorithm8 in which the fit-ting parameters were ms9, ma, and, in some cases, anamplitude factor a. For example, for fitting thetransmittance from a diffusing slab, the fitting func-tion was Eq. ~39! of Ref. 1:
Tfit~r, t! 5 a
expS2mavt 2r2
4DvtD2~4pDv!3y2t5y2
3 (m52`
1` 3z1,m expS2z2
1,m
4DvtD2z2,m expS2
z22,m
4DvtD4, (3)
with
z1,m 5 s~1 2 2m! 2 4mze 2 z0(4)
z2,m 5 s~1 2 2m! 2 ~4m 2 2!ze 1 z0.
The error assumed for the numerical results was 1standard deviation, evaluated according to Eq. ~1!.When MC data are used the amplitude factor is notstrictly necessary as a fitting parameter, but it be-comes necessary when the procedure is applied toexperimental data, usually given in arbitrary units,or to TPSF obtained for a receiver with a limited fieldof view. We therefore assumed a 5 1 in the fittingprocedures, apart from the case of data that refer toa limited field of view, presented in Section 5. Fitshave also been carried out with the amplitude factoras a fitting parameter, but the final results are veryclose to those obtained through assuming a 5 1 al-though in the case of three-parameters fitting a sig-nificant correlation was observed especially betweenthe choices of a and ma. Typical values for the cor-relation coefficients were 0.93 6 0.06 between a andma, 0.88 6 0.08 between a and ms9, and 0.80 6 0.14between ma and ms9. Significantly smaller values~,0.1! were obtained for the correlation coefficientbetween ma and ms9 when a 5 1 was assumed.
In Fig. 10 an example of fit is presented for thetime-resolved transmittance through a nonabsorbingslab 40 mm thick with ms9 5 0.5 mm21 and refractiveindex n2 5 1.4. Data refer to r 5 0 ~coaxial receiver!.The results for two values of the relative refractiveindex n 5 1.4 and n 5 0.7143 are reported. Thevalues of the reduced scattering coefficient obtainedfrom the fit are 0.497 mm21 for n 5 1.4 and 0.501mm21 for n 5 0.7143. The values obtained for the
absorption coefficient are 22.9 3 1025 mm21 for n 51.4 and 24 3 1025 mm21 for n 5 0.7143.
The values of ms9 and ma, obtained by the fit of theTPSF with Eq. ~3! for light transmitted at r 5 0through a nonabsorbing slab 40 mm thick, are shown~as diamond shapes! in Fig. 11 for various values ofthe relative refractive index. The horizontal dashedlines represent the values assumed for simulations:ms9 5 0.5 mm21 and ma 5 0. In Figs. 12~a! and 12~b!the results of a similar analysis are presented for thesame slab regarding the reflectance. Equation ~36!of Ref. 1 was used with r 5 40 mm: the exit surfaceelement was at approximately 20 effective mean freepaths from the beam axis. The results of ms9 ob-tained from both transmittance and reflectance arewithin 2% of the actual value used in MC simula-tions. For ma the differences are smaller than 0.0001mm21.
During recent years some models for solving theDE for the slab and for the semi-infinite mediumhave been presented. In general, these models differin the boundary conditions used. The simplestmodel is the one presented by Patterson et al.2 inwhich the average diffused intensity is set equal tozero on the real boundary of the medium ~zero bound-ary condition!. With this boundary condition thereis no way of taking into account the effect of a refrac-tive index mismatch. Comparisons among the solu-tions obtained with the zero boundary condition andmore accurate boundary conditions ~partial currentboundary condition, extrapolated boundary condi-tion! are reported for the reflectance for a semi-infinite medium9,10: small differences in the shapeof the time-resolved reflectance were observed whendifferent boundary conditions were used. Solutionsof the DE based on the zero boundary condition arecommonly used also for the fitting procedures neces-sary for obtaining the optical properties from mea-
Fig. 10. Examples of fits performed on the time-resolved trans-mittance obtained with MC simulations. Data are reported fortwo values of the relative refractive index and refer to r 5 0 for anonabsorbing slab 40 mm thick with ms9 5 0.5 mm21 and n2 5 1.4.The noiseless curves are obtained from the DE with the opticalparameters ms9 5 0.497 mm21 and ma 5 24 3 1025 mm21 for n 50.7143, ms9 5 0.501 mm21 and ma 5 22.9 3 1025 mm21 for n 5 1.4,coming from the fitting procedure.
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surements of reflectance11 or transmittance12–14
carried out on diffusing slabs. However, for the dif-fusing slab the solutions based on the zero boundarycondition show significant differences with respect tothe ones based on the extrapolated boundary condi-tion, in both the total received energy and the shapeof the TPSF. The differences in the shape are par-ticularly significant for the transmittance, as shownin Fig. 3 of Ref. 1: significant differences were ob-served in both the position of the maximum and theslope at long times.
For evaluation of the accuracy of the formulasbased on the zero boundary condition, a three-parameters fit of the MC results was also carried outwith Eqs. ~12! and ~13! of Ref. 2, describing the re-flectance and the transmittance, respectively. Theresults obtained for ms9 and ma are also reported inFigs. 11 and 12 ~as square shapes!. For the ampli-tude factor, values that range from 1.5 to 2 for thetransmittance and from 1.7 to 2.4 for the reflectance
Fig. 11. Results of the two-parameters fitting procedure carriedout on the time-resolved transmittance at r 5 0 to determine the~a! reduced scattering coefficients, ~b! absorption. The TPSF’swere obtained with MC simulations for a nonabsorbing slab 40 mmthick with ms9 5 0.5 mm21 and n2 5 1.4 for different values of therelative refractive index. Diamond shapes and square shapesrepresent the results obtained by means of fitting with Eq. ~39!from Ref. 1 and Eq. ~13! from Ref. 2, respectively. The error bars~reported only when they are larger than the mark! indicate 1standard deviation.
4608 APPLIED OPTICS y Vol. 36, No. 19 y 1 July 1997
were obtained. The values of ms9 obtained with thePatterson et al.2 formula for the time-resolved trans-mittance are overestimated by about 10% formatched refractive index conditions; the differenceincreases when the refractive index mismatch in-creases: for n 5 1.4—a value typical of the tissue–air interface—ms9 is overestimated by approximately20%. The discrepancy decreases when the reducedoptical thickness of the slab ms9s increases. The er-ror is approximately 10% under typical conditions formeasurements on breast, in which ms9 ' 1 mm21 ands is between 40 and 60 mm. The discrepancy alsodecreases when the distance of the receiver from thebeam axis increases. The results for ma showed dif-ferences within 0.001 mm21. This discrepancy maybe significant when measurements are carried out ontissues with small absorption, as is the case of breasttissue at near-infrared wavelengths.12
Smaller differences are observed on ms9 when thePatterson et al.2 model is used on time-resolved re-
Fig. 12. Results of the two-parameters fitting procedure carriedout on the time-resolved reflectance at r 5 40 mm for determiningthe ~a! reduced scattering coefficients, ~b! absorption. The TPSF’swere obtained by MC simulations for a nonabsorbing slab 40 mmthick with ms9 5 0.5 mm21 and n2 5 1.4 for various values of therelative refractive index. Diamond shapes and square shapesrepresent the results obtained by means of fitting with Eq. ~36!from Ref. 1 and Eq. ~12! from Ref. 2, respectively. The error bars~reported only when they are larger than the mark! indicate 1standard deviation.
flectance. In this case the differences in the TPSF’srefer especially to the intensity and to the slope atlong times. These differences are expected to causediscrepancies, especially in the amplitude factor andin ma. Data reported in Fig. 12, referring to r 5 s 540 mm and ms9 5 0.5 mm21, show discrepancies in ms9within 5% almost independent of the refractive indexmismatch. The absorption coefficient was underes-timated by ;0.001 mm21 when n 5 1.4. The dis-crepancies become smaller when the ratio rysdecreases.
In Figs. 11 and 12 the error bars, which are calcu-lated by adding and subtracting 1 standard deviationto the value obtained from the fit, are reported onlywhen larger than the marks. The standard devia-tion was evaluated from the diagonal elements of thecovariance matrix and it was obtained by assumingfor the MC data an error equal to 1 standard devia-tion of the simulation.
To investigate the accuracy of the inversion proce-dure when it is used to invert measurements carriedout near the source, Table 2 reports the results of fitson the time-resolved reflectance at various distancesfrom the light beam for a nonabsorbing slab 40 mmthick with ms9 5 0.5 mm21. The results obtainedwith the models described in Refs. 1 and 2 are pre-sented. Also these results confirm that the moreaccurate diffusive model gives better results for alarge range of distances. The error in ma is within0.0002 mm21 even at r 5 10.2 mm > 5z0, i.e., atapproximately 5 transport mean free paths from thelight beam. At this distance the value of ms9 is over-estimated by about 20%. However, this error be-comes significantly smaller if the fit is done whenonly photons that have followed trajectories longerthan 30z0 are considered. These results are similarto results reported by Hielscher et al.10 for the semi-infinite medium. Comparisons between DE and MCresults for the time-resolved reflectance were alsoreported in Refs. 9, 15–17.
The agreement between the analytical curves ob-tained from fitting procedures and the MC curveswas in general very good: The values of the reducedchi square ~xred
2! were smaller than 1.3 for the TPSF
Table 2. Optical Properties Obtained from Fitting Proceduresa
r ~mm!b
Fit with Eq. ~36!of Ref. 1
Fit with Eq. ~12!of Ref. 2
ms9 ~mm21! ma ~mm21! ms9 ~mm21! ma ~mm21!
65.7 0.496 4 3 1025 0.48 21.3 3 1023
41.2 0.497 2 3 1025 0.49 21.1 3 1023
20.4 0.531 24 3 1025 0.63 4 3 1024
10.2 0.70 2 3 1024 0.93 1.8 3 1023
8.1 0.852 8.5 3 1024 1.129 3.2 3 1023
6.2 0.99 2.3 3 1023 1.34 3.2 3 1023
4.2 1.50 8.3 3 1023 1.91 5 3 1023
2.1 3.60 4.6 3 1022 3.92 1.4 3 1022
aFor the fits of time-resolved reflectance for a nonabsorbing slab40 mm thick with ms9 5 0.5 mm21 and n 5 1.4.
bSource–receiver distance.
that referred to distances from the source larger than15z0. The value of xred
2 increases rapidly for smallerdistances if earlier received photons also are consid-ered in the fitting procedure.
Fitting procedures were carried out also on TPSFsthat refer to ma . 0. Because the diffusion coefficientis independent of the absorption coefficient in ourformulas, the results we obtained for ms9 were iden-tical to the ones we obtained when ma 5 0, and for mathe difference with respect to the actual value wasidentical to the difference obtained for ma 5 0.
5. Effect of a Limited Receiver Field of View andAngular Distribution of Outcoming Photons
All the formulas presented in Ref. 1 refer to all thephotons that exit with any exit angle. A receiverwith an angular field of view h 5 90° ~open detector!is thus necessary for measuring the quantities rep-resented by the formulas. However, when measure-ments are carried out, the value of h is usuallysignificantly smaller. To investigate the effect of alimited field of view, we carried out MC simulationsfor receivers with different values of h.
In Fig. 13 the time-resolved transmittance for acoaxial receiver ~r 5 0! for three different values of his shown together with the fitting results. Thecurves refer to h 5 90°, 60°, and 30°. Noiselesscurves are the ones from the three-parameters fitwith Eq. ~3!; the others represent MC simulations.The chosen slab was 40 mm thick with ms9 5 0.5mm21, ma 5 0, refractive index n2 5 1, and relativerefractive index n 5 1. To compare the shape of theTPSF’s that refer to different values of h, we dividedthe TPSF’s by the TPSF that refers to h 5 90°. Theratios that resulted were almost independent of timeapart from small values of t ~t , 500 ps! for which, onthe other hand, the DA does not work. Although thetotal received power changes with h, the shape of theTPSF is not changed significantly. Therefore the
Fig. 13. Effect of the receiver angular field of view h on the TPSF.The three curves refer to h 5 90°, 60°, and 30° ~from upper to lowercurve, respectively!. The transmittance for a coaxial receiver wasobtained by a MC simulation for a nonabsorbing slab 40 mm thickwith ms9 5 0.5 mm21 and n2 5 n 5 1. The curves obtained froma three-parameters fitting procedure ~noiseless curves! are alsoreported.
1 July 1997 y Vol. 36, No. 19 y APPLIED OPTICS 4609
curves can be fitted with Eq. ~3!: The fitting algo-rithm can recover the effect of the limited field of viewof the receiver, decreasing the amplitude factor a.In Fig. 14~a! the values of the reduced scatteringcoefficient obtained from the fitting procedure arepresented for various values of h. Results that referto the absorption coefficient are shown in Fig. 14~b!.Data refer to the same slab described for Fig. 13.
In Fig. 15 results for the total time-resolved trans-mittance are shown. The MC curves are presentedtogether with the fitted curves obtained from Eq. ~41!of Ref. 1 for h 5 90° and 4°. Data refer to the sameslab as described for Fig. 13 with n2 5 1.4 and n 5 1.A complete set of fitted results for the total time-resolved transmittance for this slab was generatedfor h between 90° and 0.6°. The agreement betweenthe true values of the coefficients ~ms9 5 0.5 mm21 andma 5 0! and the ones obtained from the fit was verygood ~within 4% for ms9 and smaller than 0.0004mm21 for ma for the whole range of h considered!.The physical explanation for this result is that theshape of the pulse remains almost the same even
Fig. 14. Results obtained for the ~a! reduced scattering coefficient,~b! absorption coefficient from the three-parameters fitting proce-dure carried out on the time-resolved transmittance. The resultsare reported for various values of h and refer to the same slab thatwas described for Fig. 13. The horizontal dashed lines representthe actual values of the optical parameters used for MC simula-tions. The error bars indicate 1 standard deviation.
4610 APPLIED OPTICS y Vol. 36, No. 19 y 1 July 1997
when h 5 0.6°, so that the fitting algorithm can fit thedata, changing the amplitude factor to match the realintensity, while the values of the optical coefficientsthat determine the shape of the pulse remain almostunchanged.
Monte Carlo simulations were also used for study-ing the effect of h on the cw transmittance at r 5 0.Figure 16 shows the relative transmittance f ~h! 5T~h!yT~h 5 90! for a nonabsorbing slab 40 mm thickwith ms9 5 0.5 mm21 for two values of the relativerefractive index @T~h! represents the cw transmit-tance at r 5 0 for a receiver with an angular field ofview equal to h#. The behavior expected for a Lam-bertian surface is also reported: fL~h! 5 sin2~h!.The figure shows that the angular distribution ofoutcoming photons is affected significantly by the re-fractive index mismatch and differs significantly from
Fig. 15. Effect of the receiver angular field of view on the totaltime-resolved transmittance. The curves refer to h 5 90° and 4°~upper and lower curve, respectively! and were obtained by a MCsimulation for a nonabsorbing slab 40 mm thick with ms9 5 0.5mm21, n2 5 1.4, and n 5 1. The curves obtained from a three-parameters fitting procedure ~continuous noiseless curves! are alsoreported.
Fig. 16. Effect of the receiver angular field of view h on the re-ceived energy: The relative transmittance f ~h!, evaluated by MCsimulations ~marks!, is reported for a nonabsorbing slab 40 mmthick with ms9 5 0.5 mm21. The results are reported for twovalues of the relative refractive index. The dashed curve repre-sents the behavior expected for a Lambertian surface. The solidcurve was obtained with the model described in Ref. 18.
that of a Lambertian surface. Numerical simula-tions were repeated for the same slab and geometrybut for ms9 5 1 mm21. The results for f ~h! werealmost indistinguishable from the results obtainedfor ms9 5 0.5 mm21, showing that f ~h!, at least forsufficiently large values of the slab thickness, is al-most independent of the reduced scattering coeffi-cient.
In Fig. 16 the curve that corresponds to the angulardistribution obtained with the formula presented byFirbank et al. @Eq. ~13! of Ref. 18# for the radiance atthe surface has also been reported. A good agree-ment was observed with MC results that refer to n 51. The formula was obtained through evaluatingthe radiation coming from each volume element of thediffusing medium that reaches the exit surface with-out being scattered or absorbed. The radiance in-side the diffusing medium was assumed to beisotropic, and the photon density was assumed tovary linearly with depth into the medium.
6. Discussion and Conclusions
In Ref. 1 a complete set of formulas derived from DAfor dealing with photon migration through a slab andthe semi-infinite medium were presented. In thispaper the correctness of these solutions is analyzedby means of comparisons with MC simulations.This analysis has shown that the formulas give anexcellent description of photon migration and can cor-rectly take into account the effect of a refractive indexmismatch between the diffusing medium and the ex-ternal medium in a large range of conditions. Theformulas for the time-resolved reflectance and trans-mittance give results in excellent agreement with theMC results for sufficiently long times: for photonsthat have followed trajectories longer than 15–30transport mean free paths, the discrepancy betweenthe DE and the numerical results for the TPSF issmaller than 10%. Below this threshold the approx-imation starts to break down for two different rea-sons: ~1! the medium interposed between sourceand exit surface is not turbid enough to assure acomplete diffusive regime, i.e., the contribution ofphotons scattered only a few times is no longer neg-ligible; ~2! the distance from the source is not largeenough to replace the pencil beam with the isotropicpoint source. Figures 6 and 7 show this clearly.The comparison with MC simulation for an isotropicpoint source, such as the one assumed to solve theDE, has shown that the agreement is generally notbetter; this indicates that the discrepancies observedat short times are mainly due to the other simplifyingassumptions.
Also the formulas derived for the cw response andfor the mean path length are in excellent agreementwith MC results: When the distance from the sourceis larger than 5z0–10z0, the differences are smallerthan 10% for the total energy and smaller than 5% forthe mean path length. Even the relationships forthe fraction of the incident energy reflected andtransmitted give accurate results. These relation-
ships can therefore be used to predict the fraction ofenergy absorbed by the medium.
Inversion procedures based on the formulas for theTPSF’s were also developed for determining the op-tical properties of the medium from time-resolvedmeasurements. These procedures were used to in-vert the MC results: The values obtained for ms9 andma were in excellent agreement with the actual val-ues ~values within 2% for ms9 and within 0.0001mm21 for ma! when the distance from the source islarger than 20z0. Good results were obtained alsofor smaller distances if only the part of the TPSF thatcorresponds to photons that have traveled more than30 reduced mean free paths is considered.
The model derived by Patterson et al.2 for solvingthe DE with the zero boundary condition has alsobeen used to fit the MC results. This model is usedwidely to invert time-resolved measurements. Theresults obtained from the time-resolved reflectance atr 5 40 mm for a slab with s 5 40 mm, ms9 5 0.5 mm21
and various values of n showed discrepancies in ms9within 5% and in ma within 0.001 mm21. The dif-ferences in ma may be significant when measure-ments are carried out on turbid media with a smallabsorption, as is the case for breast tissue at 800 nm,for which values of ma between 0.0017 and 0.0045mm21 are reported.12 The discrepancies becomesmaller when the ratio rys decreases.
The results from the time-resolved transmittanceshow larger differences: For ms9s 5 20 the values ofms9 are overestimated by '10% for n 5 1 and increasewhen n Þ 1. For n 5 1.4 the error is '20%. There-fore the results obtained from time-resolved mea-surements of transmittance may be overestimatedsignificantly if the effect of the refractive index mis-match is not taken into account properly. The dis-crepancies decrease when ms9s increases.
Monte Carlo simulations were used to investigatethe effect of a limited angular field of view of thereceiver. The numerical results showed that whenthe DA is valid the shape of the TPSF remains almostunchanged when the angular field of view h is de-creased. The results obtained for ms9 from a three-parameters fit were within 4% of the actual value for0.6 , h , 90°. The values of ma were within 0.0004mm21. Therefore the values of ms9 and ma are notaffected significantly by the limited field of view of thereceiver.
The angular distribution of photons that exit fromthe diffusing slab was also studied. The numericalresults showed that the angular distribution is af-fected significantly by the value of the relative refrac-tive index and differs significantly from thedistribution of a Lambertian surface.
The formulas obtained from the DE refer to a slabinfinitely extended. Monte Carlo simulations werealso used to investigate the effect of the boundarywhen a finite slab is considered. The results showedthat the effect of the boundary is negligible if mea-surements of reflectance or transmittance are carriedout at a distance from the boundary that is largerthan the thickness of the slab.
1 July 1997 y Vol. 36, No. 19 y APPLIED OPTICS 4611
Part of the results presented here were obtained atthe Blackett Laboratory, Imperial College of Science,Technology and Medicine in London, where D. Con-tini spent a period of seven months funded by theEuropean Community through the Human Capitaland Mobility Project ~contract ERB-CHRX-CT93-0335!. The authors thank J. C. Dainty ~ImperialCollege! for support and useful discussions. Wealso thank P. Bruscaglioni for useful suggestionsand discussions. Part of the research was sup-ported by Consiglio Nazionale delle Ricerche grant95.01135.02.
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