simulation of fan-beam-type optical computed-tomography imaging of strongly scattering and weakly...

Simulation of fan-beam-type opticalcomputed-tomography imaging of stronglyscattering and weakly absorbing media

Yukio Yamada, Yasuo Hasegawa, and Yutaka Yamashita

Numerical simulations of the transmission of a light impulse through strongly scattering and weaklyabsorbing slab media and of fan-beam-type optical computed-tomography imaging for cylindrical mediaare presented. A hybrid calculation scheme of scattering by the Monte Carlo method is employed toobtain the temporal variation of transmittance of the light impulse through the media. A set ofprojection data is provided by temporally extrapolating the difference in the optical density between theabsorbing object and the nonabsorbing reference to the shortest time of flight. For the case of identicalscattering between the object and reference, the reconstructed image of the difference in the absorptioncoefficient has better accuracy and spatial resolution than those images by the time-gating method.

1. Introduction

Noninvasive measurement of the oxygenation statein living tissues by optical methods has been devel-oped and is becoming popular.1-3 The realization ofoptical computed-tomography (CT) imaging is ex-pected to be advanced technology for noninvasivemeasurement, but the difficulty resulting from thestrong scattering of light by living tissues has pre-vented its development. Fundamental investiga-tions have been conducted to realize optical-CT imag-ing by use of time-resolved spectroscopy. Chance etal.4 reported the usefulness of time-resolved spectros-copy for measuring the oxygenation state in thebrain, stating that the slopes of the decay curvesobtained by pulsed-light illumination can provide theabsorbance change. Delpy et al.,5 Wilson et al.,6 andNomura et al. 7 also used time-resolved spectroscopyto measure optical properties of tissues or to analyzethe fundamental phenomena of the light-tissue inter-action. Patterson et al. 8 analyzed the behavior oflight in tissues by use of the time-dependent diffusionapproximation, while Delpy et al. 5 and Hasegawa etal.9 applied the Monte Carlo method to simulate thetime-resolved transmittance. The results of No-

Y. Yamada and Y. Hasegawa are with the Mechanical Engineer-ing Laboratory, AIST-MITI 1-2 Namiki, Tsukuba, 305 Japan.Y. Yamashita is with the Hamamatsu University School of Medi-cine, 3600 Handa, Hamamatsu, 431-31 Japan.

Received 4 September 1992.

0003-6935/93/254808-07$06.00/0.© 1993 Optical Society of America.

mura et al.7 and Hasegawa et al.9 indicate that even instrongly scattering media, the Beer-Lambert lawholds when the path length of the scattered light ismicroscopically traced. This suggests that opti-cal-CT imaging might be possible if the path length ofthe scattered light is traced by time-resolved spectros-copy.

Recently some investigators have reported on thepossibility of optical-CT imaging experimentally andnumerically. Ito et al. 10 modified a conventionalX-ray CT scanner to an optical-CT scanner andsucceeded in obtaining an image of oxygenation changein a rat brain. Toida et al." tried to apply the opticalheterodyne method to detect the transmitted coher-ent component. Araki and Nashimoto12 developed adeconvolution technique using point spread functionsand succeeded in reconstruction of an optical-CTimage experimentally. Arridge et al.13 analyticallyand experimentally investigated a reconstructionmethod as an inverse problem using the time-dependent diffusion equation. Numerically Singeret al. 14 proposed a new algorithm for using the six-fluxmodel for radiation and an iterative method forreconstruction. Using Monte Carlo simulation, Ya-mada and Hasegawa15 proposed the use of temporalvariations of transmittance of a light impulse toobtain line integrals of the absorption coefficient.

This paper develops the method proposed by theprevious investigations and reports the results of thesimulation of light propagation and optical-CT imag-ing using the Monte Carlo method. The temporalprofile of transmitted light through a slab of scatter-

4808 APPLIED OPTICS / Vol. 32, No. 25 / 1 September 1993

ing and absorbing medium is analyzed first by ahybrid calculation scheme, the combination of realforward and approximated isotropic scattering. Forthe simulation of optical-CT imaging the previouslyproposed method 5 for generating projection data forthe conventional filtered backprojection is applied tofan-beam-type CT imaging. Finally the reconstruc-tured image of the difference in the absorption coeffi-cients between the object and the reference is shownto reflect the real profile with a satisfactory spatialresolution and accuracy. The result is also com-pared with CT images obtained by the time-gatingprocedure.

2. Monte Carlo Simulation of Light Transmission

A. Hybrid-Scattering Calculation by theMonte Carlo Method

The Monte Carlo method is frequently used in under-standing light propagation in strongly scatteringmedia such as living tissues.9" 6 When the scatter-ing pattern is highly forward directed, the isotropic-scattering approximation is often employed to sim-plify the computer coding and to shorten the CPUtime by use of the transport-scattering coefficientVu.' = (1 - g),uL, where pu andg are the real-scatteringcoefficient and the anisotropy parameter.' 7 Becausetypical values of g and p, are 0.9 and 10 mm-' forlivingtissues,' 8 pu' reduces to 1.0 mm-'. Thereforethe mean-scattering path length of approximatedisotropic scattering is - 1.0 mm, while that of realforward scattering is 0.1 mm. Because of thiselongation of the mean path length for the approxi-mated isotropic scattering, large numbers of unscat-tered transmitted photons are numerically observedfor homogeneously scattering slabs with an I =10-mm thickness as shown by curve To in Fig. 1.The incident beam is initially directed perpendicularto the slab surface. A 2-mm-diameter detector wasassumed to be located on the axis of the incident lightbeam. The time axis is converted from the total pathlength by dividing by the speed of light in water. Ifall the scattering is calculated by real forward scatter-

ing, the peak at the shortest time of flight woulddisappear, but the CPU time would be more than 10times that when the approximated isotropic scatter-ing is used. To avoid the unrealistic high peak at theshortest time of flight while keeping the CPU timeshort, we first treat several scattering events nf asforward scattering, and then the isotropic approxima-tion is adapted afterward. The results for nf = 0, 1,and 10 are shown by curves Tnf in Fig. 1. Even onefirst scattering event affects the behavior of the earlyarriving light. The high peak at the shortest time offlight disappears for nf > 5, and the profile of thetransmittance does not change for nf > 10. There-fore nf = 10 is the optimum number for the hybridcalculation in the conditions of Fig. 1. For a greaterp.u' and I than those in Fig. 1, the optimum numberfor nf might be smaller than 10 because the numberof scattering events increases before the photonstransmit through the slab, and thus it is easier for thephotons to lose the initial direction. For smaller pu,'and the optimum nf remains the same because nf =10 indicates that the forward scattering is physicallyapproximated by isotropic scattering after 10 forward-scattering events. In other words the initial direc-tion of the incident beam is almost lost after the first10 scattering events. Thus the hybrid calculationusing forward and isotropic scattering by the MonteCarlo method gives reasonable temporal profiles oftransmittance of the light impulse with a shorterCPU time.

B. Temporally Extrapolated Absorbance Method

Figure 2 shows the results of a hybrid calculation fortwo scattering slabs with and without absorption.Curve T10 is the same as that in Fig. 1, and curve Toais for the slab of added absorption with the absorptioncoefficient p.a = 0.10 mm-'. Because of the absorp-tion the transmittance of the absorbing slab is reduced.The difference in the optical density (or absorbance)between the two slabs is evaluated by taking thenatural logarithm of the ratio of T10 to T 0, i.e.,AOD = n(T1o/Toa). The increase in AOD with timeis due to the increase in the total path length, and thepolynomial fitting to AOD reduces to a regression line

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Fig. 1. Monte Carlo simulation results of relative transmittanceTn of light impulse through a scattering slab. Subscript n indi-cates the number of the forward-scattering events before theisotropic scattering.

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Fig. 2. Monte Carlo simulation results of relative transmittancesand the difference in the optical density between two homoge-neously scattering and absorbing slabs.

1 September 1993 / Vol. 32, No. 25 / APPLIED OPTICS 4809

as shown in Fig. 2. The gradient of the line isproportional to the difference in the absorption coeffi-cient. Now the extrapolated value of AOD to theshortest time of flight tmin (expressed as AODmin inFig. 2) becomes the line integral of the difference inthe absorption coefficient along the incident beamaxis.' 5 The actual value of AODmin in Fig. 2 is 1.015,which is 1.5% greater than the ideal value of lAg0 =10.0 * 0.100 = 1.000. If the scattering coefficients ofthe two slabs are different from each other, the curveis no longer linear but has a curvature, particularly inthe early arriving period.9 For the purpose of theconventional CT reconstruction procedure, line inte-grals of physical properties must be obtained as theprojection data. Strong scattering of light by tissuesmakes it difficult for optical-CT imaging to obtain theline integrals. If the transmittances of a light pulsecan be measured at the shortest time of flight by adetector with picosecond resolution, the line integralsmay be obtained and this may be possible for thintissues.'9 However, the transmittances at the short-est time of flight are too small to be measured asshown in Figs. 1 and 2 because of the strong scatter-ing in tissues. Nevertheless the temporally extrapo-lated difference in the absorbance to the shortest timeof flight AODmin can be obtained and can be used asthe projection data in the CT reconstruction algo-rithm. At this moment the strong scattering infor-mation has been eliminated, and the absorptioninformation has been extracted by taking the tempo-ral extrapolation of the difference in the absorbance(optical density). This method is called the tempo-rally extrapolated absorbance method (TEAM).

3. Simulation of Fan-Beam-Type Optical-CT Imaging

A. Model of Fan-Beam-Type Optical-CT Imaging

The object of optical-CT imaging is modeled as shownin Fig. 3 It is a 10-mm-diameter cylinder containinga scattering medium. The object has a coaxial cylin-drical portion containing a scattering and weakly

absorbing medium. The reference is the same as theobject except that it has no absorbing inner cylinder.The object and reference are not necessarily twodifferent cylinders but a single cylinder irradiated bylight beams of two different wavelengths; the mediuminside the coaxial cylinder of the object is scatteringbut not absorbing at one wavelength, and it is scatter-ing and weakly absorbing at the other wavelength.One constraint is that the scattering coefficients atthe two wavelengths must be the same as in Section2. This model can be considered to simulate livingtissues irradiated by near-IR light beams. The trans-port-scattering coefficient of the medium is given as1.0 mm-' for the whole region with the anisotropicparameter g = 0.9, and the absorption coefficient ofthe inner cylinder of the object is given as 0.1 mm-'.

Around the cylinder 35 detectors are located withthe increment of A = 5, and each detector has arectangular detecting area with 0.4-mm-circumferen-tial and 0.8-mm-axial lengths.

A light impulse is incident on the object andreference. The goal of the simulation is to recon-struct a CT image of the difference in the absorptioncoefficient between the object and reference. Likethe first-generation x-ray CT scanner of the translate-rotate type, detectors can be located on the light-beam axis and traversed laterally together with theincident light beam as in the previous paper.'5 Onecan remember that the medium is highly scatteringand the initial direction of the incident beam is almostlost after several scattering events as described inSection 2. This means that the incident beam prop-agates into all directions isotropically afterward.Therefore only a small portion of transmitted lightcan be detected by the translate-rotate-type detectingsystem. Also this disadvantage makes the CPU timeof simulation significantly long because only thephotons hitting the detector on the line of incidenceare counted and most of the transmitted photons aresimply discarded. To utilize this strong scattering,

I OBJECT IREFERENCE

Detectors Detectors

Ught Impulse Ught Impulse

Fig. 3. Simulation model of fan-beam-type, optical-CT imaging. Tle object has a coaxial scattering and absorbing inner cylinder, whilethe reference contains only a scattering medium.


one can measure the transmitted light at any locationsimultaneously as with modern fan-beam-type x-rayCT scanners. Many high-speed optical detectors canbe located around the cylinder to detect the transmit-ted light, and in the simulation all the photons hittingthe detectors are counted to save CPU time.

B. Generation of Projection Data by TEAM

For both the object and reference, temporal varia-tions of the transmitted light intensity at each detec-tor are obtained by the Monte Carlo simulation.Then the differences in the optical density betweenthe object and reference AOD(t; 4)) are calculated, andTEAM is employed to obtain AOD(tmin; 4)). Thesevalues of AOD(tmin; 4)) are the line integral of thedifference in the absorption coefficient between theobject and reference and provide the projection datato be used in the reconstruction procedure of theconventional CT algorithm.

In the Monte Carlo simulation the hybrid calcula-tion method was employed to avoid the unrealisticsharp peak at the shortest time of flight. The num-ber of the totally launched photons was - 7 x 108.The time step of counting transmitted photons was0.89 ps, corresponding to the path-length incrementof 0.20 mm.

Figures 4(a), 4(b), 4(c) show typical temporal varia-tions of the transmittances and differences in theoptical density at the locations of 4) = 0, 400, and 600.The abscissa of the figures is the excess time of flightte, which is the total time of flight t subtracted by theshortest time of flight tin. T and To indicate therelative transmittances for the reference and object,respectively, and the attenuation observed at To isdue to the inner absorbing cylinder of the object.The differences in the optical density AOD in Fig. 4are slightly curved. At 4 = 600 the curvature isclearly seen at early time because the early arrivingphotons do not interact with the inner absorbingmedium to show a negligible value of AOD, but laterarriving photons interact with it to give increasingAOD.

From these curves, AOD(tmin; ) is obtained byTEAM described above. Figure 5 shows the profileof AOD(tnin; 4)) by the solid curve. The dashed curvefor the nonscattering case is the difference in theabsorption coefficient between the object and refer-ence multiplied by the segment in the inner cylinderof the line connecting the incident point and eachdetector. The curve obtained by TEAM is very closeto the nonscattering true curve, although some differ-ence is observed at - 4 = 300 where the straight beampasses the boundary of the sharp change of theabsorption coefficient in the transverse direction.This curve of OD(tmin; ) is used for the conven-tional algorithm of the CT reconstruction procedure.

C. Reconstructed Profile of the Difference in theAbsorption Coefficient

The reconstruction procedure used in this paper isthe filtered backprojection with the interpolationconversion technique from the fan beam to the

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Fig. 4. Relative transmittances and differences in the opticaldensity at (a) = 0, (b) = 40°, and (c) = 60°. The abscissa isthe excess time of flight te, which is time subtracted by the shortesttime of flight.

parallel beam. The reconstructed image of the differ-ence in the absorption coefficient between the objectand reference is shown in Fig. 6 with the true imagefor comparison. Because of the uncertainty induced

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Fig. 6. Reconstructed image of the difference in the absorptioncoefficient between the object and reference (lower image). Theupper image is the true one.

by the statistical nature of the Monte Carlo method,the reconstructed image has some noise in the hill ofthe absorption coefficient. Also the gradient of theperiphery of the hill, which illustrates the spatialresolution, is less sharp than the true image. Butboth the accuracy of the height of the hill and thespatial resolution of the obtained image are satisfac-tory. Thus we have succeeded in reconstructing theimage of the difference in the absorption coefficientbetween the object and reference, which scatter lightstrongly and absorb light weakly.

3. Comparison with Another Method and Discussion

A. Comparison with Time-Gating Method

The time-gating method has been referred to' 9 toobtain the unscattered component approximately.It is possible to evaluate TEAM images by comparingthem with time-gating images. The AOD profile,which is obtained by time gating and used in thereconstruction procedure, is calculated by the follow-ing equation:

AOD(4)) = lnj| T/(t; t To(t; 4)dt],tn tmn

where At = tg - tin is the gating period. Figure 7shows the profiles of AOD for At = 1.8, 4.4, 44, and177 ps together with the true profile. AOD for shortgating periods is close to the true value but morenoisy than that obtained by TEAM in Fig. 5. AODterms for long gating periods are much larger thanthe true profile and have long tails in the large )

True \4,4ps0.00I I J 1' 4 _"_ L ~

0 15 30 45 60 75 90

0 (deg)

Fig. 7. Profiles of the difference in the optical density obtained bythe time-gating method for the gating periods of 1.8, 4.4 (bold), 44,and 177 ps (solid curves). The broken curve is the true profile forthe nonscattering case.

region. The reconstructed time-gating images areshown in Fig. 8. Because of large relative errors oftransmittances in the very early arriving period, theimage for At = 1.8 ps is extremely noisy. The imagefor At = 4.4 ps is closest to the true image among thetime-gating images, but its spatial resolution is lowerthan that by TEAM in Fig. 6. The images for At =44 and 177 ps are shown to have a very low spatialresolution. The image for At = 177 ps might almostcorrespond to temporally unresolved measurement.

B. Discussion

The TEAM image of Fig. 6 has a less sharp edge in theprofile of the absorption coefficient than the trueimage, which leads to a lower spatial resolution.This difference comes from the fact that the value ofAOD(tnin; 4)) at 4) = 30° is 0.143, while it is zero forthe true profile in Fig. 5. At 4 = 30° the lineconnecting the incident point and the detector centeris just tangent to the inner absorbing cylinder. SoAOD(tmin; 4)) should be zero ideally. But because thedetector has a width of 0.4 mm circumferentially, thedetector makes a pencil of solid angle viewed from theincident point as shown in Fig. 3, and half of thepencil goes through the absorbing cylinder. Themaximum length of the segment in the absorbingcylinder cut by the pencil is 2.62 mm, which gives0.262 for the difference in the optical thickness.This degrades the spatial resolution of the image, andthis type of degradation will always be associated withsharp changes in absorption coefficient profiles.

If all the scattering is calculated by the isotropicscattering, many photons transmit straight to thedetectors to produce high peaks of the transmittancesat the shortest time of flight as shown by curve To inFig. 1. This unscattered component does not gothrough the inner absorbing cylinder even for theangle 4) = 30°, and the value of AOD(tmin; ) = 30')becomes closer to the true value zero than thatobtained by the hybrid-scattering calculation as inFig. 5. The resultant image (not shown here) has ahigher spatial resolution than that in Fig. 6, but thisis unrealistic as explained above, and one should be


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careful about this unrealistic higher spatial resolu-tion.

The peaks at the shortest time of flight also im-prove the images by the time-gating method. In theprevious paper 5 all the scattering events were treatedas isotropic, and high peaks of the transmittances atthe shortest time of flight were observed. Actuallyin that paper the time-gating image with 4.4-psgating gave a reasonable image equivalent to theTEAM image. But the time-gating image presentedin this paper is more degraded because of the greatererror associated with the smaller transmittance inthe early arriving time.

The constraint for TEAM is that the scatteringcoefficient of the object must be identical with that ofthe reference as described above. Practically thismust hold for one living body irradiated by twodifferent wavelengths in the near-IR range for measur-ingthe oxygenation state in tissue. Generally speak-ing the scattering coefficient varies with wavelength,but its dependence is much weaker than that of theabsorption coefficient. From an approximate analy-sis of slab transmittance, 9 the error of AODmin causedby the change in the scattering coefficient betweenthe object and reference can be expressed approxi-mately by 3(iu', obj - its', ref)/4(-L.2 - Sal)- Twotypical wavelengths are 780 and 805 nm, which areused in the instrument monitoring hemoglobin oxy-genation.20 Within this narrow wavelength rangethe variation in the scattering coefficient is believed tobe small compared with the variation in the absorp-tion coefficient. Thus, as far as these two wave-lengths are used, the error will be small enough for

TEAM to be valid, although it cannot be estimatedaccurately because of the deficiency of reliable data ofthe scattering coefficient.

The size of the model is still small for practicalclinical use, and the model is simple in geometry.This is due to the long CPU time needed to generatetransmittances with satisfactory accuracy for obtain-ing AOD(tmin; 4)) for greater size and more complexmodels. For greater size models the accuracy ofAOD(tmin; 4)) will reduce because the number of earlyarriving photons will be smaller. For more complexmodels the interaction by neighboring absorbing vol-umes may reduce the accuracy of AOD(tmin; () unlessthe time resolution is small enough to reveal a narrowspace between them. But CPU time for more com-plex models will be impractically long because theincident point must be rotated around asymmetriccylinders. Other calculation methods may be devel-oped to show TEAM to be valid for greater size andcomplex models. Also TEAM should be experimen-tally verified by use of the combination of a picosec-ond light pulse and a very fast light detector.

4. Conclusions

By use of the Monte Carlo method the behavior oflight pulses incident on strongly scattering and weaklyabsorbing tissues has been simulated, and optical-CTimaging of simple geometry has been obtained withthe use of the proposed method. The following isconcluded:

(1) Without the long CPU time associated withhighly forward-directed scattering, the combination


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of forward and isotropic scattering in the Monte Carlomethod produces reasonable temporal variation oftransmitted light through strongly scattering andweakly absorbing media.

(2) The difference in the optical density (or absor-bance) extrapolated to the shortest time of flight isnumerically shown to provide the line integral of thedifference in the absorption coefficient along the lineconnecting the incident and detecting points forfan-beam-type, optical-CT imaging.

(3) The projection data generated by TEAM pro-vide a set of data for the reconstruction procedure ofthe conventional fan-beam-type CT algorithm.

(4) The reconstructed image of the difference inthe absorption coefficient between the object andreference has satisfactory spatial resolution and accu-racy.

(5) The time-gating method with a very shortgating period produces images with quality less thanor comparable with the TEAM image, but for thelonger gating period the reconstructed images arebadly degraded.

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simulation of fan-beam-type optical computed-tomography imaging of strongly scattering and weakly...

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