Imaging of highly turbidmedia by the absorption method

Daniele Contini, Heather Liszka, Angelo Sassaroli, and Giovanni Zaccanti

The results of a study on imaging that is based on the absorptionmethod are presented. This method isbased on attenuation measurements carried out in the presence of a sufficiently high absorptioncoefficient by the use of a continuous-wave source. The benefit of absorption on image quality comesfrom the strong attenuation of photons traveling along long trajectories. When the absorptioncoefficient is increased, the received energy decreases, but the mean path length of received photonsdecreases. The effect of increasing the absorption coefficient is similar to that of decreasing the gatingtime when the time-gating technique is used. Experimental results showed that the spatial resolutionobtained with the absorption technique is similar to that obtained with the time-gating technique.Key words: Imaging, time gating, absorption method, spatial resolution, turbid media. r 1996

Optical Society of America

1. Introduction

Many research groups are studying the possibility ofperforming imaging of biological tissues with light inthe visible or the near-infrared region of the electro-magnetic spectrum. One of the most frequentlystudied problems is that of breast imaging. Breasttissue is a highly scattering media, and photonmigration occurs withmany scattering events; there-fore trajectories of photons may vary greatly fromstraight-line paths. Photons usually cover trajecto-ries many times longer than the geometric distancebetween source and receiver. The scattering phe-nomenon makes obtaining images of objects hiddeninside the turbid medium difficult, causing a de-crease in contrast and spatial resolution. In thepast, some methods to improve imaging have beendeveloped based on measurements in both the timedomain and in the frequency domain. For breastimaging, the most investigated technique, which isbased on measurements in the time domain, is thetime-gating technique. The time-gating techniqueinvolves discriminating between a few marginallydeviated photons and all other scattered light.When a laser pulse a few picoseconds long is relayedon a turbid medium, the scattering events cause a

The authors are with the Dipartimento di Fisica dell’Universitadi Firenze, Via Santa Marta 3, 50139 Firenze, Italy.Received 26 April 1995; revised manuscript received 7 August

1995.0003-6935@96@132315-10$10.00@0r 1996 Optical Society of America

temporal spread of the transmitted light. The pulseof received photons is described by the temporalpoint-spread function 1TPSF2, f 1t2, which representsthe probability density of receiving a photon at time tper unit time and per unit area of the receiver whena delta pulse is emitted at t 5 0. With the time-gating technique, the image reconstruction involvesonly earlier detected photons that are the leastdeviated from the laser beam axis. In this way, it ispossible to limit image blurring, recovering contrast,and spatial resolution. The spatial resolution ob-tainable with this method is ultimately limited bythe strong attenuation of energy received duringshort gating times. When ballistic photons can bedetected, the image quality is limited only by diffrac-tion limits. However, for strong scattering media,such as biological tissues, it becomes almost impos-sible tomeasure the received energy when the gatingtime becomes very short, if the thickness is largerthan ,5 mm.1–3 Recent studies have shown thatthe spatial resolution obtainable in typical physicaland geometric conditions for breast imaging is<0.2L1where L is the thickness of the slab2, i.e., 5–10mm.1–6 Recently Hebden and Delpy7 developed amethod to improve the spatial resolution further,which involves the use of all information carried bythe TPSF. They made a least-squares fit of theexperimental data to obtain an analytical formulafor the TPSF, and in this way the signal thatcorresponds to very short 120-ps2 gating times can beinferred.The possibility of using the natural effect of absorp-

1 May 1996 @ Vol. 35, No. 13 @ APPLIED OPTICS 2315

tion, which reduces preferentially multiply scatteredlight with respect to the ballistic component toobtain a better image quality has been recentlysuggested.8,9 In this paper we make a comparisonbetween the time-gating technique and the absorp-tion method, studying the advantages, drawbacks,and physical limits of these methods. In particularwe carried out experiments to evaluate the spatialresolution by using two methods, the first describedby Hebden4 and the second described by Mitic et al.5to make possible a direct comparison between datain the time domain and our results in the absorptiondomain. We also present the results of an imagingexperiment made with the absorption method to seethe effective improvement in the image quality.

2. Theory

The benefit of time gating on imaging is based on thefact that only photons traveling on short trajectoriesare used to build the image. Another possibility ofdecreasing the path length of received photons is tocarry out measurements in the presence of strongabsorption. Photons that undergo many scatteringevents, traveling on long trajectories in a purelyscattering medium, are greatly reduced when theabsorption effect is introduced. An improvement inimage quality, similar to the one obtained with shortgating times, is thus expected when attenuationmeasurements of a continuous-wave 1cw2 source arecarried out at high absorption coefficients. In thispaper the results of experiments conducted to studythe effect of absorption in improving contrast andspatial resolution of imaging of inhomogeneitieshidden inside a highly turbid medium are reported.The effect of absorption on photon migration can

be derived from the radiative transfer equation.10If f 1t, µa 5 02 is the TPSF for the nonabsorbingmedium, the TPSF when the absorption coefficient isµa is given by

f 1t, µa2 5 f 1t, µa 5 02exp12µavt2, 112

where v is the speed of light in the medium. Thetotal received power when a cw source is used isgiven by

P1µa2 5 P0 e0

`

f 1t, µa2dt

5 P0 e0

`

f 1t, µa 5 02exp12µavt2dt, 122

where P0 is the power emitted by the source.Equation 122 shows that the total received power isrelated to the TPSF for the nonabsorbing medium bythe Laplace transform. In principle, frommeasure-ments of total received power at different values ofµa, it is possible to obtain the same informationobtained from measurements in the time domain orin the frequency domain. The TPSF can be ob-tained by the inversion of the Laplace transform10–12;

2316 APPLIED OPTICS @ Vol. 35, No. 13 @ 1 May 1996

in other words, P1µa2 carries the same information asthe TPSF.From Eq. 122 it is possible to obtain a simple

formula for the mean path length, 7l 8, of photonsinside the turbid medium:

7l 8 5 v7t8 5 2d

dµa 5ln3P1µa2

P046

5P0v

P1µa2 e0`

tf 1t, µa2dt. 132

Equation 132 shows that it is possible to determine 7l 8from measurements with a cw source at two valuesof µa that approximate the derivative with the finitedifferentiation.A numerical analysis was carried out by a Monte

Carlo procedure. Details on the Monte Carlo codeused can be found in Refs. 13–15. Figure 1 reportsexamples of the TPSF evaluated with a Monte Carlosimulation for different values of µa: when µa in-creases, the received power decreases, but also 7l 8decreases. Figure 1 refers to the pulse transmittedthrough a slab of homogeneous diffusers and re-ceived by an open detector coaxial with the thin lightsource. The slab is 50 mm thick, the reducedscattering coefficient µs8 5 µs11 2 g2 1where µs is thescattering coefficient and g is the asymmetry factorof the scattering function2 is 0.4 mm21, g 5 0, and therefractive index n 5 1. The values of 7l 8 that corre-spond to the five curves in Fig. 1, referring to µa 50.0, 0.01, 0.03, 0.05, and 0.1 mm21, are 390.4, 202.9,134.3, 108.8, and 80.7 mm, respectively. Only thediffused radiationwas considered: the ballistic com-ponent is a negligible fraction with respect to thereceived diffuse radiation. Similar results are ex-pected for different values of g, provided that thevalue of µs8 is kept unchanged.To study the region of the diffusing medium

through which the received photons pass duringmigration from the source to the receiver, a Monte

Fig. 1. TPSF’s obtained by a Monte Carlo simulation for photonsmigrating through a slab with thickness L 5 50 mm, µs8 5 0.4mm21, and n 5 1 for different values of the absorption coefficient.The receiver was an open detector coaxial with the thin lightbeam. From the wider to the narrower curves, the values of µaare 0.0, 0.01, 0.03, 0.05, and 0.1 mm21.

Carlo code was developed in which, for any photonreceived by the detector, the coordinates of all thescattering points were stored. The stored trajecto-ries 1typically 15,000 trajectories2were used to obtainmaps that represent the density of scattering pointsinside the turbid medium. The maps were obtainedwhen the scattering points were projected on the x–zplane 1the z axis represents the direction of the thinlight beam2. The gray level was assumed to beproportional to the number of scattering points lyingin each square with a side of 1.25 mm. The accu-racy of the Monte Carlo code was checked withmeasurements carried out with the picosecond sys-tem of the European Laboratory for Non LinearSpectroscopy, in Florence. Examples of compari-sons between experimental and numerical resultscan be found in Ref. 16. In Fig. 2 some examples ofmaps that represent the density of scattering pointsare presented with density normalized to its maxi-mum value. These figures represent maps of scat-tering points for photons that arrive at the receiverwithin three different gating times 1Dtg2 and mapsthat refer to all the detected photons for threedifferent values of the absorption coefficient to makea comparison between the time-gating techniqueand the absorption methods. These maps showthat the region in which the photons travel becomesnarrower and narrower when µa is increased in amanner similar to the one that occurs when the

gating time is decreased in the time-gating technique.Similar effects are thus expected by an increase in µaor a decrease in Dtg. The narrowing of the spatialspread of photons inside the diffusing medium ex-plains the improvement in spatial resolution and incontrast.The performance of an imaging system can be

described by the spatial resolution. This quantitycan be obtained by the edge-response function 1ERF2,which is measured by the imaging of the abrupt edgeof an opaque surface embedded in a slab of turbidmedium. We investigated the resolution by usingtwo different methods: the first, the modulationtransfer function 1MTF2method, involves four steps4:

112 Obtain the line-spread function 1LSF2 as thederivative of the ERF.122 Obtain a least-squares fit of the LSF with a

Gaussian curve.132 Obtain the MTF as the modulus of the Fourier

transform of the LSF.142 Obtain the spatial resolution as the reciprocal

of the spatial frequency at which the MTF falls to1@10 of its maximum.

The procedure described at point 122 is necessary toreduce the effect of noise on the measured LSF.The second method, the ERF method, involves a

direct analysis of the ERF: the spatial resolution

Fig. 2. Examples of maps that represent the density of scattering points for photons reaching the receiver within different gating timesand for different values of the absorption coefficient. 1a2, 1b2, and 1c2 µa 5 0 and Dtg 5 3200, 625, and 250 ps, respectively; 1d2, 1e2, and 1f 2 totalreceived photons for µa 5 0.0, 0.016, and 0.048 mm21, respectively. L 5 50 mm, µs8 5 0.4 mm21, refractive index n 5 1. The receiverwas an open detector with a radius of 5 mm.

1 May 1996 @ Vol. 35, No. 13 @ APPLIED OPTICS 2317

1a2 1b2

Fig. 3. 1a2 Experimental setup, 1b2 scattering cell with the mask used for measurements of the ERF.

Dx is obtained when the line tangent to the ERF isconstructed at the edge position, and the distancebetween the intersections with the horizontal satura-tion lines of the ERF gives Dx.5

3. Experimental Results

The experiments were carried out with the experi-mental setup shown in Fig. 31a2. The light sourcewas a 35-mW He–Ne laser, and the received lightwas measured with a photomultiplier tube 1PMT2and a lock-in amplifier. The received light wasrelayed to the PMT by a 5-mm-wide optical-fiberbundle 1numerical aperture NA 5 0.42 coaxial withthe thin laser beam. An interference filter1FWHM 5 1 nm2 was placed in front of the PMT toreduce the ambient radiation and radiation that wasdue to fluorescence of the dye used. The scattering

1a2

1b2

2318 APPLIED OPTICS @ Vol. 35, No. 13 @ 1 May 1996

cell, shown in Fig. 31b2, was filled with a suspensionof Lipofundin S-20% in water to obtain the desiredreduced scattering coefficient µs8. Experimentswerecarried out with µs8 5 0.21, 0.474, and 1.08 mm21.The asymmetry factor was estimated to be g 5 0.42.We obtained this value by measuring the mean pathlength of the received photons by using Eq. 132 andcomparing it with the results obtained by using thediffusion approximation 1Eq. 15 of Ref. 172. A previ-ously calibrated dye 1brilliant Cresyl Blue Cer-tistain, supplied by Merck2 was added to reach thedesired absorption coefficient. The dye changes theabsorption of water, and the small quantities of dyeused in these experiments do not change appreciablythe scattering properties of the medium. This waschecked with both numerical analysis18 and experi-mental tests. The ERF was obtained by imaging

1c2

Fig. 4. Examples of ERF’s measured for different values of theabsorption coefficient. µa increases from the wider to the nar-rower curve. 1a2 µs8 5 0.21 mm21, µa 5 0.0, 0.009, 0.045, 0.1, and0.193mm21; 1b2 µs8 5 0.474mm21, µa 5 0, 0.0036, 0.008, 0.013, and0.055 mm21; 1c2 µs8 5 1.08 mm21, µa 5 0, 0.00143, 0.0041, and 0.02mm21.

the edge 1x 5 0 6 1 mm2 of a black mask 1thickness, 1mm2 placed in the middle of the scattering cell1thickness L 5 50 mm2, which had black lateral wallsto avoid light propagation through the boundaries.The dimensions of the cell were 50 mm 3 140 mm 3140 mm, which were sufficient to avoid appreciableboundary effects. In our experiments the minimumdistance between the laser beam and the lateral wallof the cell was 45 mm. A Monte Carlo simulationcarried out with µs8 5 0.4 mm21 showed that, in thisgeometric condition, the effect of the totally absorb-ing walls is insignificant with respect to the case ofan infinitely extended slab: the reduction of thereceived power is smaller than 1%.In our experiments, the accuracy for distance

measurements was 1 mm and for µa it was between3% and 8%. The error in the ERF was between 2%and 4% in the unmasked zone. In the masked zone,for values of the ERF greater than 1@10 of themaximum signal, the error was between 4% and10%, and for values below, the error increases as thesignal-to-noise ratio becomes worse.Examples of ERF’s are shown for µs8 5 0.21 mm21

3Fig. 41a24, for µs8 5 0.474mm21 3Fig. 41b24, and for µs8 51.08 mm21 3Fig. 41c24 for different values of µa. TheGaussian curves obtained by the fitting of the deriva-tives of these experimental ERF curves are shown inFig. 5. The figures that refer to the ERF show thatthere is a shift of the normalized ERF at the maskposition 1x 5 0 6 1mm2 toward the value 0.5 when µaincreases. Correspondingly, figures that refer to

1a2

1b2

the LSF show that the apparent position of themask, the coordinate of the maximum of LSF, under-goes a shift toward the correct position 1x 5 0 6 1mm2. This behavior was also observed in measure-ments based on the time-gating technique and onfrequency-domain analysis.4,5,9 Figures 61a2, 61b2, and61c2 report the spatial resolution 1obtained with bothmethods2 versus the absorption coefficient for µs8 50.21, 0.474, and 1.08 mm21, respectively. Thesefigures show that the spatial resolution improveswhen the absorption coefficient increases. The spa-tial resolution obtained for the largest value of µaused with µs8 5 0.21 mm21 is quite similar to thatobtained by Hebden,4 who used the time-gatingtechnique under similar physical and geometric con-ditions. Measurements carried out for larger val-ues of the reduced scattering coefficient show thatwith the absorption method the spatial resolutioncan be improved by a factor of ,2 with respect to thecase of a nonabsorbing medium. This result issimilar to the one obtained by Mitic et al.5 who alsoused the time-gating technique.For a compressed breast with a typical thickness

between 30 and 60 mm, values for µs8 between 0.6and 1.2 mm21 and g . 0.9 are expected5,19,20 whichcorresponds to values for the reduced scatteringoptical thickness ts8 1ts8 5 µs8L2 of between 15 and 70.The experimental and numerical results reported inthis paper, which refer to ts8 between 10 and 54, canthus be considered as representative for a com-pressed breast. Our experiments were carried out

1c2

Fig. 5. Examples of LSF’s obtained by the fitting of the deriva-tives of the ERF’s shown in Fig. 4. µa increases from the wider tothe narrower curve. 1a2 µs8 5 0.21 mm21, µa 5 0.0, 0.009, 0.045,0.1, and 0.193 mm21; 1b2 µs8 5 0.474 mm21, µa 5 0, 0.0036, 0.008,0.013, and 0.055 mm21; 1c2 µs8 5 1.08 mm21, µa 5 0, 0.00143,0.0041, and 0.02 mm21.

1 May 1996 @ Vol. 35, No. 13 @ APPLIED OPTICS 2319

with a medium with g 5 0.42, which is below thevalue of the asymmetry factor expected for biologicaltissue 1g . 0.92. In the experiments carried outwith large values of the reduced scattering coefficient1µs8 5 0.474 and µs8 5 1.08 mm212, light propagationcan be described with a diffusion regime in which thepropagation does not depend on µs and g separatelybut only on µs8 5 µs11 2 g2. Maybe the reducedoptical thickness of 10.5 1which corresponds toµs8 5 0.21 mm212 is not large enough to describe fullylight propagation with the diffusion approximation,

1a2

1b2

2320 APPLIED OPTICS @ Vol. 35, No. 13 @ 1 May 1996

as a detectable component of ballistic photons istransmitted. In this case photon migration maydepend on both µs and g. However, a numericalsimulation carried out for this value of ts8 in thegeometric conditions of our experiment showed thatthe received multiply scattered light overcomes theballistic component by more than 2 orders of magni-tude, even for the largest value of µa considered,which indicates that the contribution of ballisticphotons to the ERF is not significant. Because theenergy received during very short gating times is

1c2

Fig. 6. Spatial resolution versus µa. The spatial resolutionsevaluated with both the MTF method and the ERF method arereported. Data refer to L 5 50 mm and 1a2 µs8 5 0.21 mm21, 1b2µs8 5 0.474 mm21, 1c2 µs8 5 1.08 mm21.

1a2 1b2

Fig. 7. 1a2 Scheme of the object used for the imaging experiment, 1b2 object placed into the scattering cell.

Fig. 8. Examples of imaging obtained with the absorption method. L 5 50 mm, µs8 5 0.6 mm21. 1a2Nonabsorbing medium 1µa 5 02, 1b2µa 5 0.028 mm21.

expected to depend on the single-scattering proper-ties of the medium,1 attenuation that corresponds tohigh values of µa is expected to depend also on theasymmetry factor.

4. Imaging Experiment

The experimental results presented in Section 3have shown how the spatial resolution improveswhen the absorption method is used. To demon-strate the improvement on image quality, an experi-ment was carried out in which the image of aphantom was obtained by a confocal scanning. Theexperimental setup used was the same as thatdescribed in Fig. 31a2. The object that was imaged issimilar to that described in Ref. 7 and is shown inFig. 71a2: it consists of three opaque and threetransparent spheres, 7.2 mm in diameter, suspendedin the cell shown in Fig. 31b2. The scattering me-dium was an aqueous suspension of 0.305-µm-diameter latex microspheres with g 5 0.65 and µs8 50.6 mm21 at the He–Ne wavelength. This value ofµs8 is not far below the mean value expected for thehuman breast. The object was placed into the scat-tering cell at an angle of 45° to the input and theoutput faces, as shown in Fig. 71b2. Measurementsinvolved translation of the phantom horizontallyand vertically across the laser beam in 2.5-mm stepsand recording the total received power at eachposition. The total horizontal displacement was 40mm and the total vertical displacement was 37.5mm, which resulted in 272 separate measurementsfor each value of µa. Imaging was repeated fordifferent values of the absorption coefficient of themedium. In Fig. 81a2 the image that refers to thenonabsorbing medium 1µa 5 02 is presented, whereasFig. 81b2 shows the image obtained with µa 5 0.028mm21. We obtained the images by subtracting from

each value the minimum received power and bynormalizing the values so that the values occupiedthe full range of available gray levels; the imageswere interpolated to remove the edges betweenadjacent pixels.In the image for the nonabsorbing medium it is

possible to see only a white cloud that corresponds tothe transparent spheres and a black cloud thatcorresponds to the black spheres. The image thatrefers to µa 5 0.028 mm21 reveals all three transpar-ent spheres and two of the black spheres with abetter contrast with respect to the image obtainedfor the nonabsorbing medium. There is a big im-provement in the image quality when the absorptioncoefficient is increased.

5. Summary and Conclusions

The advantage of the absorption method with re-spect to time gating is that the experimental setupnecessary for cw measurements is simple and muchless expensive than the ultrafast laser and detectornecessary for time-gating measurements. In prin-ciple, the spatial resolution obtainable with thistechnique can be arbitrarily reduced by an increasein the absorption coefficient, but in practice it islimited by the minimum detectable signal level thatdetermines the maximum usable value of µa. If amore sensitive detection system is used, such as aphoton counter, measurements can also be carriedout at larger values of µa and the spatial resolutionmay be further improved. As opposed to the use oftime gating, the use of absorption in order to sepa-rate earlier received photons also involves an attenu-ation of useful photons. This attenuation may be atleast partially compensated for by the use of adetector with a large area. In our experiments areceiver with a diameter equal to 5 mm and a NA of

1 May 1996 @ Vol. 35, No. 13 @ APPLIED OPTICS 2321

0.4 were used. A small receiver area 1typical slitarea 0.1 mm22 should be used to obtain a hightemporal resolution whenmeasurements are carriedout with a streak camera.The spatial resolution versus the attenuation of

the received power that is due to the absorptioneffect is shown in Fig. 9 for measurements carriedout at µs8 5 0.21 mm21. The figure shows that, toobtain an improvement in the spatial resolution of afactor of 4, it is necessary to increase the attenuationby more than 5 orders of magnitude. Similar re-sults were obtained with the time-gating tech-nique1,2,4,5: only a small portion of the transmittedlight carries the information for a good-quality image.

Fig. 9. Spatial resolution versus the attenuation that is due toabsorption. Data refer to measurements carried out at µs8 5 0.21mm21.

1a2

1b2

2322 APPLIED OPTICS @ Vol. 35, No. 13 @ 1 May 1996

Time-gating and absorption techniques are able toseparate this small portion of photons with similareffectiveness, but the spatial resolution is ultimatelylimited by physical limits regarding the detection ofsmall power levels. With our experimental setup,measurements were carried out up to an attenuationfactor of 109 with respect to the emitted power.However, when a more efficient detection system isused it is reasonable to extend measurements up toan attenuation factor of 1015. This attenuationfactor corresponds to a detection threshold of ,100photons per second when a 35-mW source at theHe–Ne wavelength is used. To evaluate the spatialresolution that can be expected under these condi-tions, a Monte Carlo simulation was carried out toevaluate the mean path length and the attenuationthat is due to a slab with thickness L 5 50 mm, µs8 5

0.474 mm21, and g 5 0.42 when a receiver with anangular field of view of 25° was considered. Theresults are shown in Fig. 10. Figure 101a2 shows theattenuation versus the absorption coefficient for areceiver with an area of 10 mm2. In the figure thehorizontal dotted line represents the attenuationthat corresponds to the threshold of detection de-fined above, whereas the vertical dotted line corre-sponds to the maximum value of µa used in ourexperiment carried out at µs8 5 0.474 mm21. Thevalues expected for the mean lengthening Dl of thetrajectories 1Dl is the mean path length minus thegeometric thickness of the slab2 are presented in Fig.101b2 versus µa. The vertical dotted line at µa 5

1c2

Fig. 10. Results of a Monte Carlo simulation for L 5 50 mm,µs8 5 0.474 mm21, g 5 0.42, n 5 1.33, and a receiver with anangular field of view of 25°. 1a2 Attenuation versus µa for areceiver of 10 mm2; the vertical dotted line corresponds to themaximum value of µa used in our measurements, and the horizon-tal dotted line corresponds to the threshold of detection thatcorresponds to ,100 photons per second. 1b2 Mean lengthening1Dl2 versus µa; the vertical dotted line corresponds to the maxi-mum value of µa used in our experiment, and the circle corre-sponds to the value of µa that corresponds to the threshold ofdetection. 1c2Attenuation versus the gating time for a receiver of0.1 mm2.

0.055 mm21 corresponds to the maximum value of µaused in our experiment at µs8 5 0.474 mm21, and thecircle refers to the maximum usable value of µa1which corresponds to the threshold of detection2.The value of Dl that corresponds to µa 5 0.055 mm21

is 68 mm, whereas the value that corresponds to themaximum usable value of µa is 22 mm. This valueof Dl corresponds to a mean time of ,100 ps. Thevalue of Dl can thus be decreased by a factor of 3 withrespect to the value reached in our measurements.A simple model2 shows that the spatial resolution isexpected to be proportional to the square root of Dl,and thus an improvement in the spatial resolution ofa factor of ,1.7 with respect to that obtained in ourexperiment can be expected when themeasurementsare carried out at the threshold of detection.For a comparison with the time-gating technique,

Fig. 101c2 shows the attenuation versus the gatingtime for a receiver with an area of 0.1 mm2. Thisfigure shows that an attenuation of 1015 is obtainedwhen Dtg is ,20 ps. These considerations concern-ing the time-gating technique refer to an idealexperimental setup that is able to perform a perfecttemporal gate, but in practice measurements carriedout at very short gating times can be stronglyinfluenced by the impulsive response of the experi-mental apparatus.Considerations concerning the minimum spatial

resolution obtainable involve a small fraction ofphotons following trajectories with a small lengthen-ing with respect to the straight line and are expectedto depend also on the asymmetry factor g of thediffusers.1Some of the results 1results in Figs. 1 and 92 refer to

small values of µs, values that are sufficiently smallso that the ballistic component can be easily detectedby the use of simple spatial and angular filtering.However, both numerical and experimental resultsrefer to a receiver 1diameter 5 mm, NA $ 0.42 forwhich the diffuse component of the received radia-tion overcomes the ballistic component of more than2 orders of magnitude, even in the most unfavorablecase considered. In the conditions considered inour experiments, the diffuse component does notdepend significantly on g, apart from very shortvalues of t 1t , 300 ps2; the results presented can alsobe applied to diffusing media that have larger valuesof g, as expected for biological tissues.The results have been presented for a slab of

diffusers as a simple model for a breast that issqueezed during a mammography. The idea of us-ing the absorption technique for breast imaging doesnot involve the adding of a dye, but rather the use ofa tunable laser source to choose the optimal value ofµa in each measure to obtain the best spatial resolu-tion; the optimal value is themaximumvalue compat-ible with the sensitivity of receiving system. Thepossibility of using this technique for breast imagingis thus conditioned by having a range of wavelengthsfor which both the absorption coefficient and thecontrast between healthy and nonhealthy tissues

are sufficiently large. It is thus ultimately relatedto the actual optical properties of the breast. In thevisible and the near-infrared regions of the electro-magnetic spectrum, biological tissues are highlydiffusing and weakly absorbing media. For thehealthy human breast, the value of the reducedscattering coefficient µs8 ranges from 0.6 to 1.2mm21.5,19,20 To our knowledge, there are only a fewresults on the dependence of the absorption coeffi-cient on wavelength. Results recently published byHeusmann et al.21 on the attenuation in transillumi-nation of the female breast for wavelengths between550 and 1050 nm seem to show values of µa that aretoo small to use the absorption method for breastimaging. Our results showed that an absorptioncoefficient that is equal to ,0.04 mm21 is necessaryto obtain the optimum spatial resolution when a slab50 mm thick, which has a reduced scattering coeffi-cient of µs8 5 1mm21 1typical values for a compressedbreast2, is considered. However, a better knowledgeof the optical properties of the breast on a largerrange of wavelengths is necessary to understandultimately if the proposed technique may be usefulfor breast imaging.

The authors thank Costantino Blumetti andMichele Fedi for their collaboration in collectingexperimental data. This research was funded byConsiglio Nazionale delle Ricerche, Progetto Finaliz-zato Tecnologie Elettroottiche, grant 93.02061.PF65.

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