Methodologies based on optical radiation are widelyinvestigated both for monitoring and for imaging bi-ological tissue, since light at visible and near-infraredwavelengths is a noninvasive probe with a potentiallylarge number of applications. To develop methodol-ogies both for monitoring and for imaging, reliablemodels are necessary to describe photon migrationthrough highly scattering media. Light propagationthrough scattering media is described by the radia-tive transfer equation ~RTE!. The RTE is a complexintegrodifferential equation for which exact analyti-cal solutions are not known for problems of practicalinterest for tissue optics, and simplifying assump-tions are commonly assumed to obtain analytical for-mulas. With the diffusion approximation ~DA!,approximate solutions are obtained for homogeneousmedia with different boundary conditions ~infiniteand semi-infinite medium, slab, cylinder, sphere!.1–3
When this study was performed, all the authors were with theDipartimento di Fisica dell’Universita degli Studi di Firenze, ViaSanta Marta 3, 50139 Firenze, Italy. A Sassaroli and F. Martelliare now with the Mechanical Engineering Laboratory, AIST-MITI(Agency of Industrial Science and Technology—Ministry of Inter-national Trade and Industry), 1-2 Namiki, Tsukuba, Ibaraki 305,Japan. D. Contini is now with the Dipartimento di Energeticadell’Universita degli Studi di Firenze, Via Santa Marta 3, 50139Firenze, Italy.
Received 27 July 1998; revised manuscript received 27 July1998.
7392 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998
Quite similar solutions have been obtained with therandom-walk theory.4,5 For both the DA and therandom walk, the basic assumption is that photonsshould undergo a sufficiently large number of scat-tering events to produce an almost isotropic specificintensity. Solutions give an excellent description ofphoton migration apart from earlier received pho-tons, i.e., the ones received after a smaller number ofscattering events.
Approximate analytical solutions of the RTE forhomogeneous media are commonly used to deter-mine, by means of inversion procedures, the opticalproperties of tissue. Solutions for media containingsmall absorbing or scattering defects have also beenobtained with the DA6–10 or the random walk.11 Formore complex geometry or nonhomogeneous media,solutions become more and more complicated and theRTE is usually solved numerically. The two numer-ical methods commonly used are the finite-elementmethod12,13 ~FEM! and the Monte Carlo14–16 ~MC!method. The FEM can deal with complex geometryand provides solutions within a reasonably shortcomputation time. This method is commonly thebasis for complex methodologies of image reconstruc-tion for optical tomography.6,12 To limit computa-tion time and memory requirements, codes based onthe FEM often take the simplifying assumptions ofthe DA and are used to solve two-dimensional prob-lems,12 although codes to solve full three-dimensionalproblems have been developed.13 A finite-differencecode has also been used recently17 to calculate thefluence through a two-dimensional head model, solv-ing both the RTE and the diffusion equation.
tilpSohaoopc
dat
Alternatives to analytical or numerical investiga-tions are experiments on realistic phantoms. How-ever, it should be considered that these experimentsinvolve complex and expensive instrumentation, es-pecially when measurements in the time domain areused.18,19 Furthermore, as pointed out by Hall etal.,20 the experimental results may be significantlylimited by the inherent sources of systematic noise ofavailable systems. Therefore it may be desirable,even for a limited number of cases, to have solutionsof the RTE that are not affected by approximations toevaluate both the accuracy of approximate solutionsand the limits of methodologies based on optical ra-diation.
Numerical solutions of the RTE can be obtainedfrom MC simulations without the need of simplifyingassumptions, even when a complex geometry is con-sidered. The applicability of MC simulations of pho-ton migration through highly scattering media isessentially limited by the long computation time nec-essary to obtain statistically reliable results, evenwhen variance-reduction methods21 are used. Inhis paper a MC procedure that enables us to study,n a reasonably short time, some problems related toight propagation and imaging of biological tissue isresented. The proposed MC approach, described inection 2, is based on one full MC simulation only andn the use of two scaling relationships. This methodas been used for studying photon migration throughhighly diffusing slab. The time-resolved response
f a collinear source–receiver system, similar to thene usually investigated for optical mammogra-hy,18,19,22 has been evaluated for a large range ofonditions including ~1! the effect of one or more ab-
sorbing or scattering inhomogeneities, ~2! the effect ofabsorbing boundaries, and ~3! the effect of an absorb-ing surface inside the slab. The region of the me-dium through which photons migrating from thesource to the receiver travel has also been investi-gated in different situations. Examples of resultsare reported in Section 3 together with some compar-isons with experimental results.
2. Description of the Monte Carlo Procedure
The MC method provides a physical simulation ofphoton migration: A routine to generate randomnumbers uniformly distributed between 0 and 1 isused; the trajectories of emitted photons are chosenaccording to the statistical rules relevant for photonmigration through the diffusing medium. The dis-tance d traveled between two subsequent scatteringevents and the angles f and u identifying the orien-tation are obtained from the random numbers by useof the cumulative probability function. For a non-absorbing medium the distance d~w! correspondingto the random number w results: d~w! 5 2~1yms!ln~w!, where ms is the scattering coefficient.When a scalar scattering function is assumed, allazimuthal angles have the same probability, and
f~w! is simply given by f~w! 5 2pw, whereas thescattering angle u~w! is obtained from
*0
u~w!
2pp~u!sin udu 5 w,
where p~u! is the scattering function ~normalized to 1over the whole solid angle!. The trajectory of anyphoton is followed until it exits from the medium or itarrives at the detector. With the elementary MCmethod, the ratio between the number of useful pho-tons and the number of trajectories considered isused to evaluate the probability of receiving an emit-ted photon. From the length of the useful trajecto-ries it is also possible to evaluate the temporalresponse when the source emits a short pulse. Wecan evaluate trajectories by taking into account thescattering and the absorption properties in each partof the medium and the actual boundary conditions,including the effect of reflections. The main draw-back of the elementary MC method is the long com-putation time. To evaluate the attenuation with astandard deviation of 1%, 10,000 useful photons arenecessary. As an example, with reference to Fig. 1, inwhich the geometric scheme considered for transillu-mination imaging is described, ;5 h of computationtime are necessary to evaluate 10,000 useful trajecto-ries with a HP C160 workstation; a homogeneous non-absorbing slab with a thickness of s 5 40 mm, anasymmetry factor of g 5 0, and a reduced scatteringcoefficient of mso9 5 mso~1 2 g! 5 0.5 mm21 and aetector with a radius of 2 mm that can detect at allngles of incidence are assumed. The computationime rapidly increases when g or mso9 increases or
when the radius or the angular field of view of thereceiver decreases. The long computation time ismainly due to the small number of useful trajectories:only one trajectory in ;2700 in the above-mentionedexample. The largest part of the computation time isthus spent in evaluating nonuseful trajectories.
The use of the elementary MC method to studyproblems in which the response of the source–receiver system should be known for many differentpositions is thus strongly limited by the computationtime. As an example, to reconstruct the image ofinhomogeneities inside a slab of diffusers, a model
Fig. 1. Geometric scheme assumed for numerical simulations.
1 November 1998 y Vol. 37, No. 31 y APPLIED OPTICS 7393
lsttsm
sl
osp
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commonly used for a compressed breast, the responseshould be known for many hundreds of positions anddirect elementary MC simulations would take a pro-hibitively long time. To overcome, at least in part,this limit, a MC procedure based on only one full MCsimulation has been developed. The procedure canbe divided into three steps:
1. In the first step, an elementary MC code is usedfor the homogeneous nonabsorbing medium, and foreach received photon the coordinates of all the scat-tering points are stored. This step involves a longcomputation time ~typically 1 or more days! and aarge disk memory ~some hundreds of megabytes! totore the trajectories. We checked the correctness ofhe routines used to determine the trajectorieshrough the homogeneous medium by comparingome statistical parameters with exact analytical for-ulas.23
2. In the second step we evaluate the temporalresponse when one or more inhomogeneities are con-sidered, starting from the stored trajectories andmaking use of scaling relationships. The weight ofeach trajectory is assumed to be equal to the ratiobetween the probability of following the same trajec-tory after and before the introduction of the inhomo-geneity.
For an inhomogeneity that has the same scatteringproperties of the slab, but with a different absorptioncoefficient, the weight ~w! of the photon is changedaccording to the Lambert–Beer law:
w 5 exp~2maili 2 maolo!, (1)
where mai and mao are the absorption coefficients in-ide and outside the inhomogeneity, respectively, andi and lo are the lengths of the path followed inside
and outside the inhomogeneity, respectively.When a scattering inhomogeneity is introduced, the
weight is obtained from the probability that a photonhas to undergo a scattering event within a volumeelement subtended by dl at distance l from the previ-us one and to be scattered with an angle u within theolid angle dV. For a nonabsorbing medium thisrobability is given by exp~2msl !msdlp~u!dV. There-
fore, when a scattering inhomogeneity is introducedthat has the same absorption coefficient and the samescattering function as the surrounding medium, theweight of the photon is given by
w 5 Smsi
msoDKi
exp@2~msi 2 mso!li#, (2)
where li is the length of the path, Ki is the number ofscattering events undergone inside the inhomogene-ity, and msi and mso are the scattering coefficients ofthe inhomogeneity and the surrounding medium, re-spectively. A similar scaling relationship has beenpreviously used to study light propagation throughthe atmosphere.24,25
Equations ~1! and ~2! are exact scaling relation-
394 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998
ships and can be combined to vary both the scatteringand the absorption coefficients of one or more inho-mogeneities. The weight and the length of each tra-jectory have been used to evaluate the temporalresponse and the corresponding standard deviation.This step takes a short time ~typically 1 min! withrespect to the one taken for the full MC simulation instep 1. The key point is that the computation time isnow spent to work on only useful trajectories.
In this step, it is quite simple and quick to accountalso for the effect of absorbing boundaries or of anabsorbing surface embedded in the slab, like the oneused to measure the edge-response function ~ERF!:Each trajectory that intersects the surface or theboundary should be deleted from useful trajectories.Furthermore, the knowledge of the scattering pointsenables us to evaluate also the probability that re-ceived photons have to pass through different parts ofthe medium. This probability, proportional to thedensity of scattering points, can be used to visualizephoton migration by means of maps.
3. The temporal responses obtained in step 2 can beused as the starting point to evaluate the effect ofinhomogeneities, absorbing boundaries, or surfaceson photon migration. On the temporal responses weusually perform a three-parameter least-squares fitby using the solution of the diffusion equation for thehomogeneous medium.26,27 An error equal to thestandard deviation evaluated in step 2 is assumed onMC results for the fitting procedure. The fitting pro-cess essentially involves deriving an amplitude fac-tor, the absorption coefficient maf, and the reducedscattering coefficient msf9 for an homogeneous slabthat would produce a temporal response as close aspossible to that obtained for the nonhomogeneousslab. This procedure should be considered as amethod to summarize, with three parameters only,the information contained by the temporal responseand also to filter the statistical fluctuations on MCdata. For this purpose, even values of maf and msf9that, if considered as optical properties of a medium,are not meaningful ~for example, maf , 0! are accept-able. Although the solution used for the fitting re-fers to the homogeneous medium, the x2 test showedthat a set of parameters that fit well the numericalresults is usually obtained even when large inhomo-geneities are considered.
The parameters obtained from the fit can be di-rectly used to obtain an image of the inhomogeneities,as suggested by Cubeddu et al.18 and by Hebden.19
Furthermore, the noiseless analytical form of thetemporal response can be easily used to evaluate theattenuation for a cw source, the mean time of flight,or the energy received even during a short gatingtime. The response in the frequency domain can beobtained from the Fourier transform.
Since photon migration in highly scattering mediais influenced mainly by ms9 and very little by thescattering function, for reducing the computationtime and the disk memory required for storing thetrajectories, all MC simulations whose results are
tmdtbK
np
a4D
reported in Section 3 were carried out with the scat-tering function for Rayleigh scatterers for whichg 5 0. With respect to the results that refer to largervalues of g, discrepancies are expected only near thesource or for photons arriving with a short time offlight.
Procedures based on a single MC simulation havebeen previously used to study photon migration inhighly scattering media.28–30 Graaff et al.28 haveused a scaling relationship to obtain the transmit-tance and the reflectance for a homogeneous slab:The relationship enables us to change only the valueof the single-scattering albedo, a 5 msy~ms 1 ma!, buthe extinction coefficient, me 5 ms 1 ma, should re-ain the same for which the MC simulation has been
one. For evaluating the new weight for each pho-on leaving the slab, it is sufficient to store the num-er of interactions within the turbid medium.ienle and Patterson29 and Pifferi et al.30 have in-
stead used a similarity relationship. This relation-ship enables us to change the values of both ma andms, but can be applied to only the homogeneous infi-nitely extended medium and to the semi-infinite me-dium.
3. Examples of Numerical Results and Comparisonswith Experiments
A. Spatial Spread due to Multiple Scattering
For visualizing the spatial spread due to many scat-tering events undergone by received photons, thedensity of scattering points is used. Examples aregiven in Fig. 2 for ~1! an infinitely extended homoge-
eous slab, ~2! a totally absorbing plane boundarylaced 8 mm from the light beam, and ~3! a totally
Fig. 2. Probability that received photons have to pass through~bottom row! results. Left column, infinitely extended homogeneothe light beam; right column, a totally absorbing spherical inhomois 40 mm thick with mao 5 0.0003 mm21 and mso9 5 0.5 mm21.
bsorbing spherical inhomogeneity with a radius ofmm and the center at x 5 210 and z 5 20 mm.ata refer to a slab 40 mm thick with mao 5 0.0003
mm21 and mso9 5 0.5 mm21. These maps have beenobtained by projection of the scattering points on thex–z plane ~the z axis corresponds to the light beam!for 30,000 useful trajectories. Each map is scaledseparately to its maximum value. For better visu-alization the contour levels are given. The figureshows that in the center of the homogeneous slab thebroadening of the light beam is approximately equalto the thickness of the slab. Therefore an inhomo-geneity may affect the received signal also when it isplaced at a significant distance from the light beam,and this causes a blurred image.
Figure 2 also gives examples of experimental resultsobtained under conditions similar to the ones assumedfor numerical simulations ~mso9 5 0.5 6 0.05 mm21
and mao 5 0.0003 6 0.0001 mm21!. To measure theprobability that received photons have to pass throughdifferent regions of the slab, we evaluated the relativevariation of transmittance when a totally absorbingcylinder ~diameter 1.5 mm!, parallel to the y axis, wasintroduced inside a scattering cell containing a suspen-sion of Liposyn in water. Measurements were carriedout with the experimental setup described in Ref. 31.The agreement between numerical and experimentalresults is good.
B. Effect of a Totally Absorbing Boundary
Examples of results referring to the effect of a totallyabsorbing boundary on photon migration are given inFig. 3. The figure gives the values of the amplitudefactor, the absorption coefficient maf, and the reduced
ent regions of the slab: numerical ~top row! and experimentalab; middle column, a totally absorbing plane boundary 8 mm fromity with radius r 5 4 mm at x 5 210 and z 5 20 mm. The slab
differus slgene
1 November 1998 y Vol. 37, No. 31 y APPLIED OPTICS 7395
m
1
w
7
scattering coefficient msf9 of the homogeneous mediumthat best fits the temporal responses obtained from theMC procedure for different distances of the light beamfrom a plane boundary. A slab 40 mm thick was as-sumed, and 30,000 useful trajectories were used.Each figure shows the results for two values of mso9:0.5 and 1 mm21. The error bars are shown only for
so9 5 0.5 mm21. The results show that the boundaryhas a significant effect when the distance from thebeam becomes smaller than the thickness of the slab.The comparison between the results for mso9 5 0.5 and
mm21 shows almost the same values for the ampli-tude factor and almost the same relative variations formsf9. The variation of maf for mso9 5 0.5 mm21 is ap-proximately twice the variation for mso9 5 1 mm21.The results shown in Fig. 3 refer to a nonabsorbingslab. The corresponding results for an absorbing slabare exactly the same for the amplitude factor and formsf9, whereas the values of maf are shifted by mao.
C. Edge-Response Function
Measurements of the ERF are often used to evaluatethe spatial resolution of an imaging system based ontransillumination measurements.31–33 Measure-ments of the ERF have been simulated by evaluation
Fig. 3. Effect of a plane absorbing boundary: The values of thethree parameters that best fit the temporal response are shownversus the distance of the light beam from the boundary. Theresults are given for mso9 5 0.5 and 1 mm21 for a slab 40 mm thick
ith mao 5 0.
396 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998
of the temporal response for different distances of thecollinear source–receiver system with respect to theedge of a totally absorbing surface inside the slab.Examples of results are given in Fig. 4, in whichsimulations of the ERF corresponding to photons re-ceived during different gating times are shown;30,000 useful trajectories were used. For each gat-ing time the attenuation was evaluated by integra-tion of the temporal response obtained at step 2, i.e.,before the least-squares fit. The spatial resolutioncan be evaluated from the modulation transfer func-
Fig. 5. Images of two totally absorbing spheres: comparisonamong experimental ~first row! and numerical ~second row! re-sults. The images have been generated by the cw attenuation~left panel! and the mean path length ~right panel!. The sphereon the right is in the center of the slab; the other is at z 5 10 mm.s 5 40 mm, mso9 5 1 mm21, and mao 5 0.0025 mm21.
Fig. 4. Examples of the ERF: numerical and experimental re-sults. The different curves refer to MC results and show theenergy received within different gating times: 70, 100, 200, 500,1000 ps and cw attenuation ~upper to lower curve!. The symbolswith the error marks refer to cw experimental results. Data referto s 5 40 mm, mso9 5 0.5 mm, and mao 5 0.0003 mm21. Theopaque surface was in the center of the slab.
ar
w
a
tion obtained from the ERF or directly from the slopeof the ERF.32–34 The figure shows that the slope in-creases when the gating time decreases, indicating animproved spatial resolution. In Fig. 4 the results ofcw measurements carried out in conditions similar tothe ones assumed for the numerical simulation ~mso9 50.55 6 0.1 mm and mao 5 0.0003 6 0.0001 mm21! arelso shown. There is a good agreement with the cor-esponding numerical results.
D. Imaging of Absorbing and Scattering Inhomogeneities
Examples of results are shown in Figs. 5 and 6. Fig-ure 5 shows a comparison among numerical and ex-perimental results for two totally absorbing spheres ofradius r 5 4 mm with the centers at C1 [ ~212, 0, and20 mm! and C2 [ ~12, 0, and 20 mm! in a 40-mm-thickslab with mso9 5 1 mm21 and mao 5 0.0025 mm21.The sphere on the right was in the center of the slaband the other at z 5 10 mm. We generated the im-ages by displaying with a gray scale the variation of
Fig. 6. Images of two spheres ~radius, r 5 5 mm! in the center ofsphere with its center at C1 [ ~216, 0, and 20 mm! has msi9 5 0.4nd mai 5 0.01 mm21. From left to right, the images have been
phase delay for a modulation frequency of 100 MHz ~top row!; msf9 aand attenuation for three different gating times, 20, 100, and 400
the cw attenuation and the mean path length evalu-ated for 33 3 33 different positions of the collinearsource–receiver system. 30,000 useful trajectorieswere used. Numerical and experimental results arein good agreement: for both, the variation of cw at-tenuation was of '50%, whereas the mean path lengthvaried by '20 mm over 370 mm. To measure themean path length by use of a cw experimental setup,we used the method of adding absorption.31
Another example of numerical results is given inFig. 6 to compare the effect of a scattering and anabsorbing defect. The figure shows the images of twospheres of radius r 5 5 mm in the center of a slab withs 5 40 mm, mso9 5 0.5 mm21, and mao 5 0.005 mm21
obtained from 200,000 useful trajectories. The firstsphere has the center at C1 [ ~216, 0, and 20 mm!
ith msi9 5 0.4 mm21, and mai 5 mao; the second one isat C2 [ ~0, 0, and 20 mm! with msi9 5 mso9 and mai 50.01 mm21. We generated the images by displayingthe variation of the chosen parameter with a linear
with s 5 40 mm, mso9 5 0.5 mm21, and mao 5 0.005 mm21. The1 and mai 5 mao; the one in C2 [ ~0, 0, and 20 mm! has msi9 5 mso9ined with the following: cw attenuation, mean path length, andplitude factor maf obtained from the least-squares fit ~middle row!;ottom row!.
a slabmm2
obtand amps ~b
1 November 1998 y Vol. 37, No. 31 y APPLIED OPTICS 7397
tt
2
ususEefts
Hatrttawtwiwbhi
ao
7
gray scale. The brightest and the darkest pixels cor-respond to the maximum and the minimum values,respectively. From left to the right, the images wereobtained with cw attenuation, mean path length, andphase delay for a modulation frequency of 100 MHz~top row!; msf9 and amplitude factor maf obtained fromhe least-squares fit ~middle row!; and attenuation forhree different gating times: 20, 100, and 400 ps ~bot-
tom row!. We evaluated the energy received duringthe short gating times by using the procedure sug-gested by Hebden and Delpy,35 i.e., from the analyticalexpression of the temporal response obtained from theleast-squares fit. Figure 6 shows that the scatteringsphere is more easily detectable than the absorbingsphere: It is clearly visible in each image with a sizequite similar to the actual one. The absorbing sphereis visible with reasonable clarity only in images gen-erated with the cw attenuation, the amplitude factor,and the longer gating time. As expected, the imagesobtained from the mean path length and from thephase delay are almost identical. With respect to thesurrounding medium, the scattering sphere appearsbrighter in images obtained from cw and gating time,darker in other images. The absorbing sphere ap-pears darker when visible. From a comparisonamong images obtained with different parameters it isthus possible to distinguish a scattering inhomogene-ity from an absorbing inhomogeneity.
The variations of different parameters used to ob-tain the images shown in Fig. 6 are quite small withrespect to the values for the homogeneous medium.Values were within 23.9% and 12.9% for cw atten-
Fig. 7. For the same conditions of Fig. 6: the parameters msf9nd maf for y 5 0 are shown together with the standard deviationbtained from the fitting procedure.
398 APPLIED OPTICS y Vol. 37, No. 31 y 1 November 1998
uation, 21.5% and 10.0% for mean path length,0.38° and 0.0° for phase delay, 23.7% and 11.8%
for amplitude factor, 21.6% and 11.6% for maf,22.7% and 10.2% for msf9, 23.5% and 134% for 20-psgating time, 23.5% and 120% for 100-ps gating time,and 23.% and 17% for 400-ps gating time. Varia-tions are in some cases of the same order of the stan-dard deviation on the parameters obtained from thenumerical procedure, as shown by the results in Fig.7. For the same conditions of Fig. 6, the figureshows the values of msf9 and maf versus x for y 5 0,together with the standard deviation obtained fromthe fitting procedure. Figures 6 and 7 were obtainedwith 200,000 useful trajectories. The same exercisewas repeated with 30,000 trajectories: The stan-dard deviation on the results was ;2.5 times largerthan for 200,000 trajectories, but the images werequite similar.
The images obtained are quite regular even whenvariations are comparable with the standard devia-tion: This can be due to the fact that, when thescaling relationships are used, the effect of inhomo-geneities is evaluated on a fixed set of trajectories andthus the errors are not completely uncorrelated.Therefore small variations can be statistically signif-icant even when smaller than the error evaluated onthe unscaled results. The small variations on theparameters used to obtain images, apart from imagesreferring to short gating times, involve low values forthe contrast. Accurate measurements are thus nec-essary to obtain similar images from experiments onphantoms.
4. Conclusions
A MC procedure developed to study photon migrationthrough nonhomogeneous media has been presented.When two scaling relationships, Eqs. ~1! and ~2!, are
sed, the temporal response when scattering or ab-orbing inhomogeneities are introduced can be eval-ated in a short time from the results of theimulation carried out for the homogeneous medium.xamples of applications to the imaging of defectsmbedded into a diffusing slab, a model usually usedor optical mammography, have been reported in Sec-ion 3. Comparisons with experimental resultshowed the correctness of the results obtained.Equations ~1! and ~2! are exact scaling relations.owever, their range of applicability is limited by theccuracy of the results obtained with a finite set ofrajectories. The standard deviation on the tempo-al response increases when the number of usefulrajectories decreases or when the range of valueshat the weighting factor can assume increases. Forbsorbing perturbations, according to Eq. ~1!, theeight of the photon depends on only the length of the
rajectory inside the inhomogeneity. In any case# 1, and in the extreme case of totally absorbing
nhomogeneities, for trajectories impinging on them,5 0. For absorbing inhomogeneities, absorbing
oundaries, or absorbing surfaces, reliable resultsave been obtained for a large range of conditions,
ncluding small defects that have small contrast, also
bacarhpaotswt
10. P. N. den Outer, Th. M. Nieuwenhuizen, and Ad. Lagendijk,
1
2
2
2
2
2
starting from a set of 30,000 useful trajectories. Ex-amples of results have been reported in Ref. 27.
For scattering inhomogeneities, Eq. ~2! shows thatw strongly depends on not only the length of thetrajectory, but also on the number of scatteringevents undergone within the inhomogeneity. Thisinvolves larger variations of the weighting factor, andthe noise on the scaled results can rapidly increasewith respect to the noise on the unscaled results.The range of applicability of the scaling relationshipdepends on many factors, such as the volume, thescattering coefficient of the inhomogeneity ~the num-er of scattering events increases almost proportion-lly both with the volume and with the scatteringoefficient of the inhomogeneity!, and the positionnd the number of trajectory simulated. A generalule to determine a priori the accuracy of the resultsas not been identified; however, the errors on thearameters from the fitting procedure ~Fig. 7! give uscriterion to evaluate the reliability of the results
btained. Examples reported in Ref. 36 showedhat, starting from 30,000 trajectories, reliable re-ults can be obtained for a spherical inhomogeneityith r 5 5 mm within a slab 40 mm thick for varia-
ions of ms within 20%. These variations, althoughsmall, are of practical interest for optical mammog-raphy.
This research was partially supported by EuropeanCommunity contract BMH4-CT96-1658 and by theIstituto Nazionale di Fisica della Materia.
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