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Real-time method for fitting time-resolved reflectance and transmittancemeasurements with a Monte Carlo model

Pifferi, A; Taroni, P; Valentini, G; Andersson-Engels, Stefan

Published in:Applied Optics

DOI:10.1364/AO.37.002774

1998

Link to publication

Citation for published version (APA):Pifferi, A., Taroni, P., Valentini, G., & Andersson-Engels, S. (1998). Real-time method for fitting time-resolvedreflectance and transmittance measurements with a Monte Carlo model. Applied Optics, 37(13), 2774-2780.https://doi.org/10.1364/AO.37.002774

Total number of authors:4

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Real-time method for fitting time-resolvedreflectance and transmittance measurements witha Monte Carlo model

Antonio Pifferi, Paola Taroni, Gianluca Valentini, and Stefan Andersson-Engels

An efficient method is proposed for the evaluation of the absorption and the transport scattering coeffi-cients from a time-resolved reflectance or transmittance distribution. The procedure is based on alibrary of Monte Carlo simulations and is fast enough to be used in a nonlinear fitting algorithm. Testsperformed against both Monte Carlo simulations and experimental measurements on tissue phantomsshow that the results are significantly better than those obtained by fitting the data with the diffusionapproximation, especially for low values of the scattering coefficient. The method requires an a prioriassumption on the value of the anisotropy factor g. Nonetheless, the transport scattering coefficient israther independent of the exact knowledge of the g value within the range 0.7 , g , 0.9. © 1998Optical Society of America

OCIS codes: 170.6510, 300.6500, 290.7050.

1. Introduction

The knowledge of the optical properties of biologicaltissues is of great interest for clinical diagnostics aswell as for the optimization of therapeutic protocols.Among the techniques under study, time-resolved re-mittance spectroscopy ~either reflectance or transmit-tance! proved particularly appealing, since it allowsthe noninvasive evaluation of the absorption ~ma! andtransport scattering ~ms9! coefficients.1–3 The diffu-sion approximation to the radiative transport equa-tion has been successfully used to fit time-resolveddata obtained for long interfiber distances ~r . 1 cm!in highly scattering media ~ms9 . 20 cm21!,4,5 as itprovides a good description of the photon migrationafter a long path in the medium. However, the dif-fusion equation fails to predict the initial events of

When this research was performed, A. Pifferi and S. Andersson-Engels were with the Division of Atomic Physics, Lund Institute ofTechnology, P.O. Box 118, S-22100 Lund, Sweden. A. Pifferi iscurrently with the Department of Physics, Politecnico di Milano,Piazza Leonardo da Vinci 32, I-20133 Milano, Italy. P. Taroniand G. Valentini are with the Centro di Electtronica Quantistica eStrumentazione Electtronica-Consiglio Nazionale delle Ricerche,Politecnico di Milano, Piazza Leonardo da Vinci 32, I-20133, Mi-lano, Italy.

Received 9 January 1998.0003-6935y98y132774-07$15.00y0

© 1998 Optical Society of America

2774 APPLIED OPTICS y Vol. 37, No. 13 y 1 May 1998

photon migration,6–8 and other models should be con-sidered.

Very recently, interest has grown in the possibilityof using a Monte Carlo code to interpret photon-migration curves. In the case of a cw measurement,the value of ma, the scattering coefficient ms, and theanisotropy factor ~mean cosine of the scattering anglefor one scattering event! g can be inferred from alookup table of Monte Carlo simulations of the diffusereflectance and total and collimated transmittancethrough a turbid slab.9 Another approach has beenproposed for the analysis of coherent backscatteredphoton distributions. A single Monte Carlo simula-tion is performed, and the individual photon historiesrecorded. The photon angular distribution corre-sponding to any combination of the scattering and theabsorption coefficients is then reconstructed from thedatabase of photon histories, applying the Beer–Lambert factor to account for the absorption and aproper scaling factor to account for the scattering.10

In the case of a time-resolved measurement, theprevious solution is not effective since a high numberof individual photon histories must be processed toyield a statistically significant output curve. Kienleand Patterson11 and, independently, the authors12

have proposed a fast procedure for the construction ofa time-resolved reflectance distribution. The keyidea in Ref. 12 is to start from a library of simulatedphoton distributions obtained for a set of differentscattering coefficients and null absorption. The pho-

ton time distribution corresponding to any value of maand ms9 is then calculated when the two nearestcurves in the library are interpolated and the Beer–Lambert law is applied. This procedure is quite fastand can easily be implemented in a fitting algorithmfor the evaluation of the optical properties. The ma-jor differences between the methods proposed in Refs.11 and 12 are addressed in Section 6.

Here we extend the method to include also time-resolved transmittance curves. Moreover, we checkthe accuracy of the procedure by evaluating experi-mental results, since up to now, to the best of ourknowledge, the method has been tested only on sim-ulations,11,12 and its validity for real experiments stillhas to be proved. Time-resolved transmittancemeasurements were performed on tissue phantomscovering a wide range of optical properties. In thesemeasurements a Ti:Sapphire laser was used as asource and a synchroscan streak camera was used forthe detection. The experimental curves were ana-lyzed both with the proposed Monte Carlo methodand with a standard solution of the diffusion equa-tion. The results were also compared with the testperformed on simulations for the same set of opticalproperties. Finally, tests on simulations were car-ried out also for the reflectance case.

2. Theory

In this section, we present a method to obtain thetime distribution R~ms, ma, r, t! of diffusely reflectedlight as a function of time t at the surface of a turbidmedium at a distance r from the injection point. Weused a cylindrical coordinate system with light inci-dent as a short pulse in the origin, and exploited somegeneral properties of time-resolved reflectance curvesthat, although well known, are recalled to appreciatethe limits of the method and some interesting impli-cations. The results are then extended to the eval-uation of the time-resolved transmittance T~ms, ma, r,d, t! through a slab of thickness d.

We assume that the medium is ~i! semi-infinite, ~ii!homogeneous, and ~iii! continuous. Moreover, weassume that we know the values of the refractionindex n and of the anisotropy factor g of the medium.

The first step consists of splitting the effects ofabsorption and scattering. Let us first suppose thatwe know the reflectance function R~ms0, ma 5 0, r, t!for a nonabsorbing white medium with a certain scat-tering coefficient ms0.

Furthermore, we use the fact that it is the scatter-ing properties that determine the photon path. Thescattering coefficient influences the length of the freepath between two subsequent interactions, whereasthe angular deviation of the photon is ruled by theanisotropy factor g. Therefore, if ms0 is multiplied bya factor k, the shape of the path will be unaffected,whereas the length of each step will be divided by thefactor k ~as shown in Fig. 1!. Thus one can derivethe reflectance function for other ms values by scalingR for ms0 by the factor k 5 msyms0:

R~ms, ma 5 0, r, t! 5 k3R~ms0, ma 5 0, kr, kt!, (1)

if both situations are characterized by the same gvalue.

Although the absorption simply selects the photonsthat ultimately survive, as illustrated in Fig. 2, itcannot alter the photon path. Therefore, if some ab-sorption is added to this medium, the correspondingreflectance function can be derived from Eq. ~1! withthe Beer–Lambert law. The diffuse reflectancefunction can thus be expressed

R~ms, ma, r, t! 5 k3R~ms0, ma 5 0, kr, kt!exp~2mavt!, (2)

where v 5 cyn is the speed of light in the medium.This approach is the same as used, for example, in therandom-walk model,13 and it has a general applica-bility, regardless of the theoretical model considered,as long as the hypotheses of the homogeneous and thecontinuous medium are satisfied.

By utilizing the relations in Eqs. ~1! and ~2!, onecan construct the distribution R~ms, ma, r, t! for everyset of optical coefficients starting from the singleMonte Carlo simulation of a nonabsorbing medium

Fig. 1. Scattering coefficient determines the distance betweentwo scattering sites, whereas the shape of the photon path is fixedsolely by the g value. The TPSF of a medium with a certain ms canbe derived when the TPSF of a medium with a different ms is scaledand if both are characterized by the same g value.

Fig. 2. Absorption coefficient reduces the survival probability ofphotons for elapsing time. The TPSF of photons reemitted from aturbid medium can be derived when the TPSF of a medium withthe same scattering and no absorption is weighed.

1 May 1998 y Vol. 37, No. 13 y APPLIED OPTICS 2775

with a certain scattering ms0. Nevertheless, to beable to apply the scaling factor for any value of ms,without any other approximation, one must recordand process all the individual histories of photons.Since a high number of photons is needed to build atime-resolved curve with good statistics, the calcula-tion of a temporal point spread function ~TPSF! for agiven set of optical properties can still be too timeconsuming to be effectively used in an iterative fittingprocedure.

We followed an alternative approach, introducing afurther approximation. First, we built a library oftime-resolved reflectance curves for white media witha set of different scattering coefficients, processed allthe photon records of a unique Monte Carlo simula-tion, and applied Eq. ~1!. We followed this proce-dure only once to buildup the library for a certaingeometry. The curve produced by an arbitrary mscan be obtained from the linear interpolation of thetwo library curves corresponding to the scatteringcoefficients ms1 and ms2 nearest to ms ~with ms1 , ms ,ms2!. Then, we took into account the effect of theabsorption by applying the exponential factor, asshown in Eq. ~2!. Clearly, this procedure introducesan intrinsic approximation in the model related to thefinesse of the library.

This method can also be used to evaluate a time-resolved transmittance distribution. In this case,we consider a pulse of light normally incident on thesurface of a slab of thickness d, and we are interestedin the time distribution T~ms, ma, d, r, t! of the lightreemitted on the opposite side of the slab. Clearly,in this geometry, the hypothesis ~i! of the semi-infinite medium is no longer valid, and it is not pos-sible to scale the scattering coefficient as in Eq. ~1!.Therefore, compared with the reflectance case, thebuilding of the library takes longer, since curves fordifferent scattering coefficients must be simulatedindependently. Nevertheless, once the library isavailable, the reconstruction of the transmissioncurve requires the same computer time as for thereflectance case. Generally speaking, hypothesis ~i!allows an easier building of the library, but it is notstrictly necessary. The other tools, that is, the lin-ear interpolation and the Beer–Lambert factor to ac-count for the absorption, can be used in anygeometry.

In the beginning, we supposed that we knew thevalue of the anisotropy factor g and that it was thesame for the library curves as well as for the exper-imental data to be fitted. This requirement is hardto fulfill in a real situation, since the measurement ofg is generally difficult and g can vary among differentbiologic samples. Moreover, it is not practical tobuild a different library for every possible g. Nev-ertheless, in many instances, the optical parameter ofinterest is the transport scattering coefficient ms9 5ms~1 2 g!. Therefore a small indeterminacy in the gvalue could be acceptable as long as the correspond-ing variation on the fitted ms9 value can be neglected.This aspect is addressed in the Subsection 3.A.

It is useful to check the accuracy of the proposed

2776 APPLIED OPTICS y Vol. 37, No. 13 y 1 May 1998

Monte Carlo evaluation method in fitting a TPSF toextract the optical properties of turbid media againstthat provided by other models. This makes it pos-sible to compare the performances of the differentmethods. In particular, we consider the diffusionapproximation to the radiative transport theory,which is widely used to evaluate the optical proper-ties of turbid media.

In the case of a semi-infinite homogeneous contin-uous medium @i.e., hypotheses ~i! to ~iii!#, the solutionfor the reflected light, reemitted at a distance r, isgiven by1:

R~r, t! 5 ~4pDcyn!23y2zo t25y2

3 exp~2ma ctyn!expS2r2 1 zo

2

4DctynD , (3)

where z0 > ~ms9!21 and D 5 @3~ms9 1 ma!#21. This

expression was derived with the use of the zero-boundary condition, i.e., the assumption that the flu-ence rate is null on the surface.

In the diffusion approximation, the expression ofthe transmittance through a homogeneous continu-ous slab of thickness d is given by1:

T~r, d, t! 5 ~4pDctyn!23y2t25y2

3 exp~2ma vt!expS2r2

4DctynD3 (

n51,k52n21

n5` H~kd 2 z0!expF2~kd 2 z0!2

4Dctyn G2 ~kd 1 z0!expF2~kd 1 z0!

2

4Dctyn GJ , (4)

where r is the distance between the collection pointand the direction of incidence. For practical appli-cations related to photon propagation in biologic me-dia, the summation in Eq. ~4! can be truncated at thefirst two dipoles ~n 5 1!, since higher-order termsyield negligible contributions.

In this study, we assumed a different expression forthe diffusion coefficient: D 5 ~3ms9!

21, independentof ma. In the derivation by Furutzu and Yamada,14

they assert that this expression is more appropriatethan the one normally used. The lack of dependenceon ma is also in agreement with the analysis above, asEq. ~2! holds for any theoretical model under thestated hypotheses. Since the information on ma iscarried exclusively by the factor exp~2mavt!, the dif-fusion coefficient must be exactly the same as for awhite medium with ma 5 0. In practice, since thediffusion approximation is valid only if ms9 .. ma, thenumerical difference between the two expressions isexpected to be small whenever the diffusion equationholds true. However, since the new expression for Dis not yet universally accepted, we also performedtests to analyze the simulated and the experimentaldata and used the conventional expression for D 5@3~ms9 1 ma!#21 in Eqs. ~3! and ~4!. The correspond-ing discrepancy in the fitted values of ma or ms9 in

Fig. 3. Experimental setup for the time-resolved transmittance measurements on tissue phantoms.

which the two expressions for D were used, was foundto be much smaller than the error due to the modelitself. Therefore only results obtained with D 5~3ms9!

21 are presented in the following sections.

3. Materials and Methods

A. Simulations

The library curves and the test simulation curveswere produced with a standard Monte Carlo code15

modified for temporal applications.16 For both re-flectance and transmittance, the corresponding li-braries were built with simulated data correspondingto 12 different values of ms9, starting from 1.0 cm21

and scaled in a geometric progression of ratio =2 ~i.e.,1.0, 1.4, 2.0, 2.8, 4.0, 5.7, 8.0, 11.3, 16.0, 22.6, 32.0,and 45.3 cm21!. When we followed the descriptionin Section 2, in the case of reflectance, all the curveswere obtained from a single simulation of 10 millionphotons. In the case of transmittance, all the librarycurves were simulated independently, with 10 mil-lion photons each, since the scaling factor was notapplicable.

To test the validity of the proposed method, wesimulated a total of 25 test reflectance curves and 25transmittance curves, combining 5 different trans-port scattering coefficients ~ms9 5 1.5, 3.0, 6.0, 12, and24 cm21! and 5 absorption coefficients ~ma 5 0.1, 0.2,0.4, 0.8, and 1.6 cm21!, always with an interfiberdistance r 5 1.0 cm ~reflectance! or a slab thicknessd 5 1.0 cm and r 5 0.0 cm ~transmittance!. Weobtained each of the test curves from simulations ofat least 1 million injected photons.

For both the library and the test curves, the valuesof g 5 0.80 and n 5 1.33 were assumed. Moreover,to evaluate the effect of an incorrect knowledge of theg value, we simulated two more sets of test reflec-tance curves, which corresponded to g 5 0.70 and g 50.90. The values of the scattering coefficient ms werechosen so as to yield the same values of ms9 as in theset of curves with g 5 0.80.

B. Experiments

The transmittance measurements were performed onphantoms composed of water solutions of Intralipid

~Pharmacia! and black ink ~Pelikan! by using a mode-locked Ti:Sapphire laser ~Coherent, Model Mira 900!and a streak camera ~Hamamatsu, Model C1587!, asshown in Fig. 3. The phantom solution was pouredinto a glass tank, 20 cm 3 20 cm wide, in properamount to form a 1.0-cm-thick slab. The laser wastuned at 790 nm with a pulse duration of '100 fs.The laser beam was focused on the upper surface ofthe phantom and the light emitted at the lower sur-face was collected by a 600-mm quartz fiber. Thelight was then focused on the entrance slit of thestreak camera, which was running in synchroscanmode. A small fraction of the laser beam was cou-pled to a 200-mm quartz fiber and directly fed into thestreak camera to account for eventual time drifts ofthe instrumentation. The instrumental transferfunction, measured with the same setup without anysolution in the tank, showed a FWHM , 30 ps.

The optical properties of Intralipid and black inkwere characterized performing a time-resolved reflec-tance measurement with an interfiber distance r 51.4 cm on a water solution containing 3 3 1022 ~solidfraction! Intralipid and 2 3 1025 ~vyv! ink. Underthese conditions, the measurement can be safely an-alyzed with the diffusion expression @Eq. ~4!#.5 Weobtained values corresponding to ms9 5 160 cm21 forthe batch solution of 20% Intralipid and ma 5 4.1 3104 cm21 for the pure ink. This preliminary char-acterization was exploited to produce a total of 25solutions with known optical properties, combining 5values of ms9 ~1.5, 3.0, 6.0, 12, and 24 cm21! with 5values of ma ~0.1, 0.2, 0.4, 0.8, and 1.6 cm21!.

C. Fitting Procedure and Data Analysis

The simulated curves were directly fitted to the an-alytical solution of the diffusion equation or evalu-ated with the suggested Monte Carlo method. Theexperimental curves were analyzed in the same way.In this case however, both the theoretical curves andthe simulated library curves were first convolutedwith the instrumental transfer function before theevaluation. In the case of transmittance measure-ments, the delay introduced by the glass wall of thetank was accounted for in the diffusion equation, andthe Monte Carlo library was built for a two-layered

1 May 1998 y Vol. 37, No. 13 y APPLIED OPTICS 2777

medium ~liquid phantom-glass!. For both simulatedand experimental data, the range of the fit includedall the points in the curve over a fixed threshold. Inparticular, the threshold was set to 2% of the peakvalue on the tail of the curve, whereas two differentthresholds of 2% and 80% were tested for the leadingedge. To achieve a faster and more robust fittingprocedure, we fixed the amplitude by normalizing thetheoretical curves to the area of the test curves and byvarying only two parameters, ma and ms9. The fittingprocedure was developed with the Microsoft ExcelSolver routine for nonlinear optimization.17 Thebest fit was obtained by minimizing the reduced x2.

The reconstruction of the theoretical expression~for both the diffusion and the Monte Carlo models!as well as the convolution procedure were compiled ina dynamic-link library, written in C, for computa-tional efficiency. The time needed to reconstruct acurve was almost the same with both models and wasnegligible compared with the convolution process. Afit with the diffusion model takes typically 4 s on astandard Intel 486DX2, 66 MHz personal computer.The Monte Carlo model requires 8 s of computer time,owing to the higher number of iterations needed forconvergence.

The accuracy of both models in fitting either thesimulated or the experimental curves was evaluatedin terms of the relative error ε, defined as

ε 5umf 2 meu

me, (5)

where me is the effective value ~of ma or ms9! used inthe simulations or calculated from the characteriza-tion of the samples, whereas mf is the correspondingfitted value.

4. Test against Simulations

In a first step, we evaluated the algorithm to extractthe optical properties of turbid media by fittingMonte Carlo-simulated diffuse reflectance or trans-mittance curves and by comparing the fitted opticalproperties to the ones used for the simulations. Fig-ure 4 shows the pattern of the relative error for thefitting of ma with reflectance and transmittancecurves @Figs. 4~a!–4~c! and Figs. 4~d!–4~f !, respective-ly# with either the diffusion expression or the MonteCarlo method. For the Monte Carlo, we used onlythe wider fitting range @Figs. 4~a! and 4~d!#, whereasfor the diffusion, which is not adequate to predictphoton propagation at early times,6 we tested bothranges, setting the first threshold on the remittancecurve to 2% @Figs. 4~b! and 4~e!# and 80% @Figs. 4~c!and 4~f !# of the peak value, respectively. The erroris expressed as a function of the optical propertiesconsidered in the simulations and represented in con-tour plots. Bilinear interpolation was performed onthe 25 values of ε, and areas of the plane ~ma, ms9! inwhich the error falls within different selected inter-vals were displayed with different shades: ε , 10%,white; 10% , ε , 20%, light gray; 20% , ε , 30%,dark gray; and ε . 30%, black, for both reflectance

2778 APPLIED OPTICS y Vol. 37, No. 13 y 1 May 1998

and transmittance measurements. The MonteCarlo method can be successfully used to fit thecurves with an error generally ,10% for all the op-tical parameters tested. As clearly illustrated inFig. 4, fits with the wider range with the diffusionapproximation solution results in more inaccurateestimations of the optical properties, yielding ε values.30% for ms9 , 10–15 cm21. The behavior is dras-tically improved, but only in reflectance, if a largepart of the leading edge of the curves is disregarded.In fact, in this case, the error is ,10% for most of theoptical coefficients considered. On the contrary, fortransmittance measurements, the estimated valuesbecome even more inaccurate when the reduced fit-ting range is used.

Figure 5 shows the corresponding relative errorobtained for ms9. Again, the Monte Carlo methodprovides accurate results with an error mostly ,10%for all combinations of optical properties studied bothin reflectance and in transmittance @Figs. 5~a! and5~d!#. The diffusion model is clearly less adequate toevaluate the scattering properties with high accuracy

Fig. 4. Relative error on the fitted ma for ~a!–~c! simulated reflec-tance or ~d!–~f ! transmittance curves. The method used for thefitting is ~a! and ~d! the Monte Carlo, ~b! and ~e! the diffusion withthe full fitting range, or ~c! and ~f ! the diffusion with the reducedfitting range.

Fig. 5. Relative error on the fitted ms9 for ~a!–~c! simulated reflec-tance or ~d!–~f ! transmittance curves. The method used for thefitting is ~a! and ~d! the Monte Carlo, ~b! and ~e! the diffusion withthe full fitting range, or ~c! and ~f ! the diffusion with the reducedfitting range.

for both fitting ranges considered. It may be usefulto note that the error is always an overestimation ofms9.

In the case of reflectance measurements, additionalvalues of the transport scattering coefficient and theinterfiber distance had already been tested, and sim-ilar results were obtained.12

To test the dependence of the fitted values of ma andms9 on the exact knowledge of g in the Monte Carlomodel, we used the same library ~obtained with g 50.80! to fit a set of test simulations that we producedstarting from a different g value. Two values for gwere tested, a lower one ~g 5 0.70! and a higher one~g 5 0.90!. In both cases, the corresponding relativeerror on the fitted ma or ms9 was rather small, being,10% for most of the points. Therefore a limited~but for practical applications realistic! inaccuracy onthe g value results in only a small error of the esti-mated optical parameters.

5. Test against Experiments

The ability of the Monte Carlo method to estimate theoptical parameters from experimental data wastested on 25 time-resolved transmittance curves mea-sured on phantoms with known optical properties.

Using for the fitting procedure the Monte Carlomethod or the diffusion expression with the two dif-ferent fitting ranges, we show in Fig. 6 the contourplot of the relative error on ma. The Monte Carlomethod @Fig. 6~a!# provides a good estimation of mawith an error generally ,10% for ms9 . 3–5 cm21.For lower values of ms9, the experimental measure-ment is probably so close to the instrumental transferfunction that we cannot extract reliable informationby fitting the curve. The error in the estimation ofma with the diffusion equation is much higher for bothfitting ranges, allowing reliable measures only for thehighest scattering values ~ms9 . 10–15 cm21!.

The error in the estimation of ms9 is shown in Fig.7. The pattern is quite similar to the one observedfor ma, and, again, use of the Monte Carlo methodgreatly reduces the error in the estimations of theoptical parameters compared with fitting with thediffusion solution. In particular, for the MonteCarlo ε , 10% for ms9 . 5 cm21 for the diffusion withthe wider fitting range ε . 30% for ms9 , 15 cm21, andwith the limited range, ε is mostly .30% for ms9 , 5

Fig. 6. Relative error on the fitted ma for experimental transmit-tance curves for which ~a! the Monte Carlo, ~b! the diffusion withthe full fitting range, or ~c! the diffusion with the reduced fittingrange were used.

cm21, but in only a few points it reduces to valueslower than 20%.

6. Discussion

With both the simulations and the measurements weaimed to explore the region where the diffusion equa-tion is more likely to fail, that is, for a low value of theinterfiber distance ~r # 1 cm or d # 1 cm! or of thetransport scattering coefficient ~ms9 # 10 cm21!. Un-der these conditions, the Monte Carlo method pro-vides better results than the diffusion approximationand is adequate to evaluate the optical properties aslong as the experimental curve is broad enough withrespect to the instrumental transfer function. Thismakes the limits of good applicability somehow de-pendent on the performance of the particular instru-mental setup used to perform the measurements.Far from these limits, that is, whenever the earlyevents of photon migration have no crucial effect,even the diffusion equation holds safely and can beprofitably used to interpret the experimental data.

The fitting method is quite insensitive to the choiceof the initial values of ma and ms9 for both simulatedand experimental data. An initial guess of ma 5 1cm21 and ms9 5 10 cm21 is appropriate for reachingthe best fit in most cases. Only for very low scatter-ing values ~ms9 , 3 cm21! the lowest x2 was obtained,thus lowering the initial ms9 to 1 cm21. As a generalrecipe, we start the fit using ms9 5 10 cm21. If thefitted ms9 is , 4 cm21, we repeat the fit starting fromms9 5 1 cm21.

The method proposed here for generating a photonmigration time distribution differs from the one pre-sented by Kienle and Patterson.11 In that study, theMonte Carlo code was used to simulate a uniquetime-resolved reflectance distribution correspondingto a fixed ms09 and a time scale t, disregarding theinformation on the individual photon histories.Thus the scaling relation of Eq. ~1! could be applied toproduce a set of curves corresponding to discrete val-ues of ms9, r, and t. Therefore the TPSF’s corre-sponding to any value of ms9, r, and t were obtainedwhen the set of reflectance curves with respect toboth the radial position and the time were interpo-lated. We followed a different approach, building alibrary of Monte Carlo simulations corresponding tothe desired values of r and t and different values ofms9. Then, the TPSF was generated when the li-

Fig. 7. Relative error on the fitted ms9 for experimental transmit-tance curves for which ~a! the Monte Carlo, ~b! the diffusion withthe full fitting range, or ~c! the diffusion with the reduced fittingrange were used.

1 May 1998 y Vol. 37, No. 13 y APPLIED OPTICS 2779

brary curves were interpolated only over ms9. Thelibrary curves must be tailored to the specific exper-imental setup in use; nevertheless, because we in-tended to apply the fitting method also to time-resolved transmittance curves for which Eq. ~1! doesnot hold, we were forced to choose this approach. Inthe case of reflectance measurements, we can greatlyease the library buildup since the library curves canbe generated for the desired r and t by applying Eq.~1! if all the individual photon histories have beenrecorded ~as in Ref. 10!. Compared with the methodpresented by Kienle and Patterson,11 our approach isless accurate ~they showed an error of ,3%, althoughonly on simulations!. On the other hand, it has awider applicability and does not require a rather crit-ical interpolation of the library curves in the timedomain.

7. Conclusions

We have tested a fast method to analyze time-resolved reflectance or transmittance distributionswith a library of Monte Carlo simulations. Com-pared with the solution of the diffusion equation, themethod improves the accuracy of the fitting of bothms9 and ma for a small distance between the injectionand the collection fibers and for low scattering values.The method requires an a priori assumption on the gvalue, although a limited discrepancy between thereal g and the assumed one causes little changes onthe fitted ms9.

The Monte Carlo code used in this study was mod-ified for temporal applications by Roger Berg. Fi-nancial support by the European Commission underthe grant ERBCHBGCT940657 as well as by theSwedish Natural Research Council and the SwedishResearch Council for Engineering Sciences is grate-fully acknowledged.

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