In recent years great efforts have been made to de-termine the optical properties of biological tissue thatcan, in turn, be used to obtain knowledge of the phys-iological state of tissue. In almost all applications,models have been used that assume that the inves-tigated tissue is homogeneous, but this assumption isoften not valid. Instead, many parts of the bodysuch as skin, esophagus, stomach, intestine, bladder,and head have a layered tissue structure. Thus, it isincreasingly recognized that the results obtainedfrom homogeneous models must be interpreted care-fully1,2 and that the theoretical models must be im-proved.
Using the diffusion approximation to the transportequation,3 one can readily find solutions for a semi-
A. Kienle, N. Dognitz, R. Bays, G. Wagnieres and H. van denBergh are with the Institute of Environmental Engineering, SwissFederal Institute of Technology, CH-1015 Lausanne, Switzerland.M. S. Patterson is with the Department of Medical Physics, Ham-ilton Regional Cancer Centre and McMaster University, 699 Con-cession Street, Hamilton, Ontario, L8V 5C2 Canada.
Received 19 May 1997; revised manuscript received 15 Septem-ber 1997.
infinite and homogeneous turbid medium in thesteady-state, frequency, and time domains.4–6
These equations can be used to obtain the opticalproperties by applying nonlinear regression to exper-imental data. For the semi-infinite geometry it iseven possible to obtain the optical coefficients fromthe more exact transport equation using an approachthat is based on scaling data from a single MonteCarlo simulation.7 Monte Carlo simulations canalso be used to calculate the light propagation inlayered tissue,8 but determination of the optical co-efficients of two layers with this approach needs ahigh amount of computation time if used in an in-terative algorithm.9,10 Although the solution of thediffusion equation for two layers is more complexthan that for the semi-infinite medium, it offers muchmore rapid calculation than is possible with MonteCarlo simulations.
Several researchers have investigated the solutionof the diffusion equation for layered turbid media.Takatani and Graham11 and Schmitt et al.12 derivedanalytical formulas for the steady-state reflectanceby use of Green’s functions to solve the diffusionequation, while Dayan et al.13 applied Fourier andLaplace transforms to obtain expressions for thesteady-state and the time-resolved reflectance. Keij-zer et al.14 and Schweiger et al.15 employed a finite
1 February 1998 y Vol. 37, No. 4 y APPLIED OPTICS 779
element method and Cui and Ostrander16 used a fi-nite difference approach. A random walk model hasbeen developed by Nossal et al.17 However, theseresearchers did not compare their results to solutionsof the transport equation, and the possibility of de-riving the optical properties of the two layers fromtheir models has not been studied.
In this article we solve the diffusion equation usingthe Fourier transform approach for a two-layered tur-bid medium having a semi-infinite second layer.Unlike Dayan et al.,13 who introduced approxima-tions to obtain relatively simple expressions for thereflectance, we avoided any approximation by calcu-lating the reflectance using numerical integration.Moreover, the zero boundary condition was replacedby the more accurate extrapolated boundary condi-tion.
We compare these solutions to Monte Carlo simu-lations in the steady-state, frequency, and time do-mains. Furthermore, by fitting the solutions of thediffusion equation to reflectance data obtained fromthe same equations to which typical experimentalerrors have been added, we attempt to solve the in-verse problem and determine the optical properties ofthe two layers as well as the thickness of the firstlayer. To investigate whether the optical propertiescan also be derived if the solution of the diffusionequation is fitted to data from the more exact MonteCarlo method, nonlinear regressions were performedin the steady-state and frequency domains. For ex-perimental confirmation of these theoretical resultswe measured the absolute steady-state spatially re-solved reflectance on two-layered tissue phantoms.
We chose the values of the optical coefficients of thetwo layers and the thickness of the first layer of theinvestigated turbid media to resemble those that areespecially relevant for two potential applications ofthe layered model: Near-infrared spectroscopy formeasurements of cerebral oxygenation, where thethickness of the tissues above the brain is ;10 mm,18
and optical noninvasive glucose monitoring, whichhas been investigated by measuring the steady-statespatially resolved reflectance on the abdomen,19
where the thickness of the skin above the fat layer isapproximately 2 mm.
2. Theory
A. Diffusion Equation
We derive the solutions of the diffusion equation for atwo-layer medium for the steady-state reflectance~Subsection 2.A.1!, for the phase and modulation ofthe reflectance in the frequency domain ~Subsection2.A.2!, and for the time domain reflectance ~Subsec-tion 2.A.3!. The first layer of the two-layer mediumhas a thickness l and the second layer is semi-infinite.
1. Steady-State ReflectanceSimilar to Dayan et al.13 we assume that an infinitelythin beam is incident perpendicular onto the turbidtwo-layer medium and that the beam is scatteredisotropically in the upper layer at a depth of z 5 z0 5
780 APPLIED OPTICS y Vol. 37, No. 4 y 1 February 1998
1y~ms19 1 ma1!, where msi9 and mai are the reducedscattering and the absorption coefficients of layer i,respectively. The origin of the coordinate system isthe point where the beam enters the turbid mediumand the z coordinate has the same direction as theincident beam. The x and y coordinates lie on thesurface of the turbid sample and r 5 ~x2 1 y2!1y2.Thus, the steady-state diffusion equation becomes
D1DF1~r! 2 ma1F1~r! 5 2d~x, y, z 2 z0!, 0 # z , l,
(1)
D2DF2~r! 2 ma2F2~r! 5 0, l # z,(2)
where r 5 ~x, y, z!. Di 5 1y3~mai 1 msi9! and Fi arethe diffusion constant and the fluence rate of layer i,respectively.
We solve these equations by the following steps.First the equations are transformed to ordinary dif-ferential equations with the use of a two-dimensionalFourier transform
fi~z, s1, s2! 5 *2`
`
*2`
`
Fi~x, y, z!exp@i~s1x 1 s2y!#dxdy.
(3)
The derived equations are solved by use of the appro-priate boundary conditions, and finally the resultsare inverse Fourier transformed to obtain the solu-tion of Eqs. ~1! and ~2!. Using Eq. ~3!, we obtain fromEqs. ~1! and ~2!
]2
]z2 f1~z, s! 2 a12f1~z, s! 5 2
1D1
d~z 2 z0!, 0 # z , l,
(4)
]2
]z2 f2~z, s! 2 a22f2~z, s! 5 0, l # z,
(5)
where ai2 5 ~Dis
2 1 mai!yDi and s2 5 s12 1 s2
2.The following boundary conditions were employed
to solve Eqs. ~4! and ~5!:
f1~2zb, s! 5 0, (6)
f2~`, s! 5 0, (7)
f1~l, s!
f2~l, s!5
n12
n22 5 1, (8)
D1
]f1~z, s!
]z Uz5l 5 D2
]f2~z, s!
]z Uz5l. (9)
Equation ~6! states the extrapolated boundary condi-tion for the tissue–air boundary.5 Previously wecompared different possible boundary conditions for asemi-infinite medium and found that Eq. ~6! providesthe best agreement between the diffusion equation
solution and Monte Carlo simulations.6 The quan-tity zb equals
zb 51 1 Reff
1 2 Reff2D1. (10)
Reff represents the fraction of photons that is inter-nally diffusely reflected at the boundary. Reff wascalculated according to Haskell et al.,5 who found thatReff equals 0.493 for a refractive index n of 1.4, whichis representative of measured tissue data. In Eq. ~8!we assumed that the refractive index ni is the samefor the first and the second layer. We solved Eqs. ~4!and ~5! by using Eqs. ~6!–~9! and by considering theappropriate treatment of the Dirac function in Eq.~4!. The results for f1~z, s! are
f1~z, s! 5sinh@a1~zb 1 z0!#
D1a1
3D1a1 cosh@a1~l 2 z!# 1 D2a2 sinh@a1~l 2 z!#
D1a1 cosh@a1~l 1 zb!# 1 D2a2 sinh@a1~l 1 zb!#
2sinh@a1~z0 2 z!#
D1a1, 0 # z , z0, (11)
f1~z, s! 5sinh@a1~zb 1 z0!#
D1a1
3D1a1 cosh@a1~l 2 z!# 1 D2a2 sinh@a1~l 2 z!#
D1a1 cosh@a1~l 1 zb!# 1 D2a2 sinh@a1~l 1 zb!#,
z0 , z , l,(12)
where we assumed that l . z0, and for f2~z, s! we get
f2~z, s! 5sinh@a1~zb 1 z0!#exp@a2~l 2 z!#
D1a1 cosh@a1~l 1 zb!# 1 D2a2 sinh@a1~l 1 zb!#.
(13)
The two-dimensional Fourier inversion of Eqs. ~11!–~13! is given by
Fi~r, z! 51
~2p!2 *`
*`
fi~z, s!exp@2i~s1x 1 s2y!#ds1ds2
51
2p *0
`
fi~z, s!sJ0~sr!ds, (14)
where J0 is the Bessel function of zeroth order. Weperformed this inverse transform numerically by ap-plying Gauss’s formula.20 To check the obtained re-sults, the Simpson formula for numerical integrationwas also programmed. The spatially resolved reflec-
tance R~r! is calculated as the integral of the radianceover the backward hemisphere5
R~r! 5 *2p
dV@1 2 Rfres~u!#
31
4p FF1~r, z 5 0! 1 3D1
]F1~r, z 5 0!
]zcos uG
3 cos u, (15)
where Rfres~u! is the Fresnel reflection coefficient for aphoton with an incident angle u relative to the normalto the boundary. For a refractive index n 5 1.4, Eq.~15! gives6
R~r! 5 0.118F1~r, z 5 0! 1 0.306D1
]
]zF1~r, z!uz50. (16)
In Subsection 4.C a semi-infinite model is also used tofit the steady-state spatially resolved reflectancefrom two-layered media. Equation ~8! from Ref. 6 isapplied, which is identical to Eq. ~16! from this articlefor ma1 5 ma2 and ms19 5 ms29.
2. Frequency Domain ReflectanceIn the frequency domain method the source is sinu-soidally modulated at frequency f. Thus the mea-sured signal at the detector is also sinusoidal but theoscillation is delayed and the modulation is reduced.The interesting quantities in the frequency domainare the phase angle u between the source and thedetected signal and the modulation M:
u 5 tan21 Im@R~r, v!#
Re@R~r, v!#, (17)
M 5 HIm@R~r, v!#2 1 Re@R~r, v!#2
Re@R~r, v 5 0!#2 J1y2
, (18)
where v 5 2pf.The real and imaginary parts of the reflectance
R~r, v! can be obtained by using the formula for R~r!in Subsection 2.A.1, but ai has now to be computedwith ai
2 5 ~Dis2 1 mai 1 jvyc!yDi. The velocity of
light in the medium is c and j 5 ~21!1y2.In this article the investigated quantities for mea-
surements in the frequency domain are the phaseand steady-state reflectance because the use of thesequantities is often superior to the use of the phaseand modulation for the determination of the opticalcoefficients.21,22 In Subsection 4.C we also use asemi-infinite model to fit the phase and the steady-state reflectance from two-layered media. Equa-tions ~1! and ~3! from Ref. 22 are applied, which areidentical to Eqs. ~15! and ~16! of this article for ma1 5ma2 and ms19 5 ms29.
3. Time Domain ReflectanceThe time domain reflectance R~r, t! can be derived bythe Fourier and Laplace transforming Eq. ~11!.13 Adifferent approach has been used here. The real and
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imaginary parts of the reflectance in the frequencydomain R~r, v!, ~see Subsection 2.A.2! were calcu-lated at many frequencies. To obtain the time do-main reflectance these data were fast Fouriertransformed. We checked these results by compar-ing R~r, t! for a semi-infinite turbid medium usingms19 5 ms29 and ma1 5 ma2 in Eqs. ~11! and ~12! withthe time domain reflectance obtained from the diffu-sion equation for a semi-infinite medium @Eq. ~8! inRef. 6#.
B. Monte Carlo Simulations
The solutions of the diffusion equation are compared~Subsection 4.A! and fitted ~Subsection 4.C! to MonteCarlo simulations. The principles of Monte Carlosimulation of photon transport in turbid media havebeen described elsewhere,23,24 so that we mentiononly the salient features of our Monte Carlo program.A pencil photon beam was normally incident onto thetwo-layered turbid medium. For calculation of thescattering angle we used the Henyey-Greenstein25
phase function. The refractive index was equal to1.4 for both layers. The Monte Carlo simulationswere performed in the time domain. The steady-state reflectance and the phase were calculated fromthe time domain reflectance using the fast Fouriertransform. For the Monte Carlo simulations in thetime domain a spatial resolution of 0.5 or 1.0 mm anda temporal resolution of 2.5 ps for t , 100 ps and of10 ps for 100 ps , t , 4000 ps were chosen for scoringthe reflectance. This ensures that all essential in-formation about R~r, t! is obtained from the simula-tions, and errors are avoided when the steady-stateand frequency domain reflectance are calculated.The anisotropy factor g was chosen to be 0.8, becausethe variation in g between 0.8 and 1 does not changethe reflectance significantly as long as ms9 is con-stant.7 For investigations of the turbid media withthicknesses of the first layer of 6 and 10 mm, R~r, t!was calculated for different absorption coefficients inthe second layer from only one simulation by scoringthe lengths of the photon paths in the second layerand applying Beer’s law.26,7
For the nonlinear regression a combination of thegradient search method and the method of lineariz-ing the fitting function was used.27 The logarithm ofthe absolute reflectance was fitted in the nonlinearregressions for both the investigations in the timedomain and in the steady-state domain. For the in-vestigations in the frequency domain, relative phasevalues ~the phase difference determined at adjacentdistances! and relative steady-state reflectance ~theratio determined at adjacent distances! were used.Equal weights for all data points were applied in thefitting procedure.
3. Materials and Methods
Absolute steady-state spatially resolved reflectancemeasurements were performed in a similar way tothose described in detail previously.28 Briefly, lightfrom a He–Ne laser at l 5 543 nm or at l 5 612 nmwas incident approximately perpendicular onto the
782 APPLIED OPTICS y Vol. 37, No. 4 y 1 February 1998
tissue phantoms. The diffusely reflected photonswere imaged onto a CCD camera and the intensityvalues were recorded. From these, the spatially re-solved reflectance was calculated. To make absolutemeasurements it is necessary to calibrate the detec-tor response. This was done by measuring the re-flectance profile for a homogeneous Liposyn phantomwith known optical properties. By measuring thesource power and calculating the theoretical reflec-tance, we determined the detector response.
The measurements of the spatially resolved reflec-tance were made on two-layered tissue phantoms.The principles of the manufacturing process are de-scribed in Ref. 29. The basic medium is a two-component transparent silicone that cures at roomtemperature. The refractive index of the silicone isclose to 1.4.29 Before the base and the catalyst weremixed, the scattering and absorbing particles wereadded. We used polystyrene spheres with a 2.5-mmdiameter dispersed in ethanol as scattering materialand graphite powder ~also dispersed in ethanol! asabsorbing material. These components were mixedin the silicone and heated at ;100 °C for severalhours to remove the ethanol. If the ethanol is in-completely removed, bubbles can be formed duringthe curing process.
Two two-layered phantoms were manufactured,one consisting of a 2-mm-thick and the other of a6-mm-thick first layer. Two silicon mixtures weremade, each with different optical properties. One ofthese was used for the first layer of phantom 1 and forthe second layer of phantom 2. The other mixturewas used for the first layer of phantom 2 and thesecond layer of phantom 1 ~see Fig. 1!. Therefore, itwas possible to determine the optical properties of thelayers separately by measuring the absolute spatiallyresolved reflectance at the sides of the second layersof the phantoms. In these measurements the firstlayer does not influence the light propagation, andthus the solution of the diffusion equation for a semi-infinite medium can be applied. The lateral dimen-sions of the smaller of the two phantoms were 9 cm 39 cm and the height was 6 cm. This is large enoughto eliminate any influence of the lateral boundaries.
4. Results
In this section we first present results comparing thesolutions of the diffusion equation derived in Subsec-tion 2.A to Monte Carlo simulations in the steady-state, frequency, and time domains ~Subsection 4.A!.This is followed by two subsections in which the pos-sibility of extracting information about the two-layer
Fig. 1. Scheme of the manufactured phantoms.
medium predominantly by measurements in thesteady-state and frequency domains is studied. Theoptical coefficients of the two-layered medium weredetermined by fitting the diffusion solutions to reflec-tance data calculated with the diffusion equation~Subsection 4.B! or with the Monte Carlo method~Subsection 4.C!. Finally, we present the results ob-tained from the measurements of the absolutesteady-state spatially resolved reflectance from two-layered phantoms ~Subsection 4.D!.
A. Comparison with Monte Carlo Simulations
In our comparison of the solutions of the diffusionequation to Monte Carlo simulations, the reflectanceis compared for media having optical coefficients thatare used in Subsection 4.C. Figure 2 shows thesteady-state reflectance R~r! for ma1 5 0.02 mm21, ms195 1.3 mm21, ma2 5 0.01 mm21, and ms29 5 1.2 mm21
or ms29 5 0.7 mm21. The thickness of the first layeris l 5 2 mm. In general, R~r! calculated with thediffusion equation is close to the Monte Carlo data~the differences are smaller than 7% for distancesgreater than 1.25 mm!. This is not the case for dis-tances close to the source, where it is known that thediffusion approximation is not valid. The differentbehavior of the reflectance for the media with differ-ent ms29 can be explained as follows. For small dis-tances ~r , 2 mm! the influence of the second layer isminimal and therefore the reflectance is similar.For long distances ~r . 8 mm! the reflectance fromthe turbid medium with ms29 5 0.7 mm21 is greaterthan from that with ms29 5 1.2 mm21, because thesmaller reduced scattering coefficient of the secondlayer allows the light to travel further away from theincident beam. For intermediate distances thegreater reflectance for the medium with ms29 5 1.2mm21 is caused by the enhanced remitted photonsfrom the second layer due to the higher reduced scat-tering coefficient of the second layer.
Fig. 2. Comparison of the steady-state spatially resolved reflec-tance calculated with Eq. ~16! ~lines! to Monte Carlo simulations~symbols!. The optical parameters of the two-layered turbid me-dia are ma1 5 0.02 mm21, ms19 5 1.3 mm21, ma2 5 0.01 mm21, andms29 5 1.2 mm21 ~solid curve, crosses! or ms29 5 0.7 mm21 ~dashedcurve, circles!. The thickness of the first layer is l 5 2 mm andn 5 1.4.
In Figs. 3 and 4 the steady-state spatially resolvedreflectance for two-layered media with thicknesses ofthe first layer of 6 and 10 mm, respectively, areshown. The optical parameters are ms19 5 1.3mm21, ma1 5 0.005 mm21, ms29 5 1.0 mm21, and ma25 0.01 mm21 or ma2 5 0.022 mm21. For these pa-rameters the solutions of the diffusion equation arealso close to the Monte Carlo simulations. For smalldistances the curves for different ma2 are similar,because the reflectance is not influenced by the opti-cal properties of the second layer. Because of thegreater absorption coefficient of the second layer ofthe medium with ma2 5 0.022 mm21 the reflectance issmaller at large distances values compared with theturbid medium with ma2 5 0.01 mm21. The distancefrom the source where the reflectance of the mediumwith the greater ma2 has a noticeablely smaller re-
Fig. 3. Comparison of the steady-state spatially resolved reflec-tance calculated with Eq. ~16! ~lines! to Monte Carlo simulations~symbols!. The optical parameters of the two-layered turbid me-dia are ms19 5 1.3 mm21, ma1 5 0.005 mm21, ms29 5 1.0 mm21, andma2 5 0.01 mm21 ~solid curve, circles! or ma2 5 0.022 mm21
~dashed curve, crosses!. The thickness of the first layer is l 5 6mm and n 5 1.4.
Fig. 4. Comparison of the steady-state spatially resolved reflec-tance calculated with Eq. ~16! ~curves! to Monte Carlo simulations~symbols!. The optical parameters of the two-layered turbid me-dia are ms19 5 1.3 mm21, ma1 5 0.005 mm21, ms29 5 1.0 mm21, andma2 5 0.01 mm21 ~solid curve, circles! or ma2 5 0.022 mm21
~dashed curve, crosses!. The thickness of the first layer is l 5 10mm and n 5 1.4.
1 February 1998 y Vol. 37, No. 4 y APPLIED OPTICS 783
flectance than the medium with the smaller ma2 isgreater when the top layer is thicker ~compare Fig. 3with Fig. 4!. These figures show also that the dif-ference between the reflectance for media of differentma2 is greater for l 5 6 mm than for l 5 10 mm.Consequently, one has to measure at greater dis-tances to obtain the optical coefficients from mediahaving a thicker first layer.
For the determination of the optical coefficients inthe frequency domain ~Subsections 4.B and 4.C! themeasurement of the steady-state reflectance and thephase is assumed. Figure 5 compares the phase cal-culated with Eq. ~17! with that obtained from MonteCarlo simulations. The optical coefficients and thethickness of the first layer are those that have beenused in Fig. 2. The modulation frequency is f 5 195MHz. The phase values from the turbid mediumwith the greater reduced scattering coefficient in thesecond layer are greater than those from the othermedium. The phase obtained from the diffusionequation has a systematically lower value than theMonte Carlo data. Similar differences can also beobserved with the solutions of the diffusion equationfor the semi-infinite geometry. In Subsections 4.Band 4.C we use the phase difference determined atdifferent distances to determine the optical coeffi-cients. In this way a considerable part of these dif-ferences is canceled out.
To compare the time-resolved reflectance fromthese two-layered media the solution of the diffusionequation was obtained by calculating the real partand the imaginary part of R~r, v! at 512 differentfrequencies ~97.7 MHz, 195 MHz, . . . , 50 GHz! andby fast Fourier transforming these data. The reflec-tance at r 5 9.75 mm can be seen in Fig. 6. Theresults calculated with the diffusion theory arewithin the statistical errors of the Monte Carlo datafor times longer than t 5 0.2 ns. The differences atshorter times are caused by the failure of the diffu-
Fig. 5. Comparison of the phase versus distance calculated withEq. ~17! ~curves! to Monte Carlo simulations ~symbols!. The op-tical parameters of the two-layered turbid media are n 5 1.4, ma1
5 0.02 mm21, ms19 5 1.3 mm21, ma2 5 0.01 mm21, and ms29 5 1.2mm21 ~solid curve, crosses! or ms29 5 0.7 mm21 ~dashed curve,circles!. The thickness of the first layer is l 5 2 mm and themodulation frequency is f 5 195 MHz.
784 APPLIED OPTICS y Vol. 37, No. 4 y 1 February 1998
sion approximation for photons that have been min-imally scattered. The time-resolved reflectance forthe medium with ms29 5 0.7 mm21 calculated withthe solution proposed by Dayan et al. is also shown inthe figure @long dashed curve, Eqs. ~12! and ~25! inRef. 13#. It can be seen that this solution does notmatch the Monte Carlo data owing to the approxima-tions made in its derivation.
B. Determination of the Optical Coefficients fromNonlinear Regressions to Solutions of the DiffusionEquation
To investigate what information can be obtained frommeasurements of the reflectance from a two-layeredmedium in the steady-state, frequency, and time do-mains if no approximations in the theoretical descrip-tion are made, we used the solutions presented inSubsection 2.A for nonlinear regressions to data thatwere calculated with the same equations and towhich typical experimental errors were added. Er-rors of 1% and 0.1° were assumed for the measure-ment of the steady-state reflectance and the phase.We found that for measurements of the absolutesteady-state spatially resolved reflectance it is possi-ble to determine ms9 and ma for both layers if thethickness of the first layer is known and the distancerange of the measurements of R~r! is suitable. Thisconfirms the results that we obtained in a previousstudy10 where Monte Carlo simulations were used orthe solution of the diffusion equation described bySchmitt et al.12 For a first layer thickness of 2 mm,measurement errors in R~r! in the range of 1% resultin errors of typically 10–20% in ma1, ms29, ma2,whereas ms19 can be determined with an accuracy ofbetter than 5%. Uncertainties in l of the order of10% give similar errors in the determination of the
Fig. 6. Comparison of the time-resolved reflectance calculatedwith the diffusion theory ~curves! to Monte Carlo simulations~symbols!. The optical parameters of the two-layered turbid me-dia are n 5 1.4, ma1 5 0.02 mm21, ms19 5 1.3 mm21, ma2 5 0.01mm21, and ms29 5 1.2 mm21 ~solid curve, crosses! or ms29 5 0.7mm21 ~dashed curve, circles!. The thickness of the first layer isl 5 2 mm, and the distance is r 5 9.75 mm. The time-resolvedreflectance calculated with the solution proposed by Dayan et al. isalso shown for the medium with ms29 5 0.7 mm21 ~long dashedcurve!.
optical coefficients. If the thickness of the first layeris not known, i.e., if five parameters are fitted, it ispossible to get reasonable values of the optical coef-ficients and l when the starting parameters of thenonlinear regression are not too far from the realones. We note that these quantitive data depend onthe number of points at which R~r! is measured.~Here, typically, reflectance data at approximately 20or 40 distances located between r 5 1 mm and r 5 20mm were used.!
In the frequency domain it is also, in principle,possible to determine ms9 and ma of the two layers andthe thickness of the first layer. However this is trueonly if the start parameters of the nonlinear regres-sions are not too far from the real ones. In addition,it was found that the convergence of the nonlinearregression was often slow. We also investigated thedetermination of ms9 and ma when the thickness of thefirst layer was known. We found that it is sufficientto measure the phase and steady-state reflectance atthree distances to obtain the correct optical coeffi-cients. Here it was assumed that the phase differ-ence and the steady-state reflectance ratio aredetermined between the first and the second and be-tween the second and the third distances. We addedtypical errors for the measurements of the phase andthe intensity and found that the obtained optical co-efficients are quite stable. This was also the casewhen the thickness of the first layer was known onlywithin an error of 10%. For example, for l 5 6 mmand measurement positions at 7.5, 13.5, and 19.5 mmand with optical coefficients similar to those in Sub-section 4.A, the errors in the derived absorption and
Fig. 7. Estimated absorption coefficients of the first ~ma1*, cross-es! and second layer ~ma2*, open circles! determined by nonlinearregressions of Eq. ~16! to Monte Carlo data are shown versus thetrue absorption coefficient of the second layer used in the MonteCarlo simulations ~ma2!. The optical parameters of the MonteCarlo simulations are ms19 5 1.3 mm21, ma1 5 0.02 mm21, ms29 51.0 mm21, and ma2 is varied between ma2 5 0.0025 mm21 and ma2
5 0.02 mm21. The thickness of the first layer is l 5 2 mm. Thelines indicate the correct values. Also shown are the absorptioncoefficients ~ma*! obtained from nonlinear regressions to the two-layer Monte Carlo data using a diffusion solution that assumes asemi-infinite medium ~solid circles!. Reflectance data at dis-tances r 5 1.25, 1.75, . . . , 17.75 mm were used in the nonlinearregression.
reduced scattering coefficients were ;10% or less as-suming measurement errors of 0.1° in the phase and1% in the steady-state reflectance. Similar errorswere obtained when assumptions of l 5 5.4 mm or l 56.6 mm were used in the nonlinear regression to re-flectance data generated for a medium with l 5 6 mm.
In the time domain it was found that it is possibleto determine ms9 and ma for both layers from absolutetime-resolved reflectance measured at one distance ifthe thickness of the first layer is known.
C. Determination of the Optical Coefficients fromNonlinear Regressions to Monte Carlo Simulations
To determine the optical properties with the use ofnonlinear regressions to Monte Carlo data, we con-centrate here on the steady-state domain and fre-quency domain reflectance, because the time domaincalculations are relatively time-consuming. We de-scribed above the possibility of obtaining not only theoptical parameters of the two layers but also thethickness of the first layer if the theoretical modelcontains no approximations ~a solution of the diffu-sion equation was fitted to data from this solutionplus noise!. However, if the solution of the diffusionequation is used to fit Monte Carlo data or experi-ments, a systematic error is introduced. In this caseit is difficult to obtain good estimates of all five pa-rameters. Thus, in the following it is assumed thatthe thickness of the first layer is known.
1. Steady-State DomainWe first present the optical coefficients obtained byfitting the solution of the diffusion equation to Monte
Fig. 8. Estimated reduced scattering coefficients of the first~ms19*, crosses! and second layer ~ms29*, open circles! determined bynonlinear regressions of Eq. ~16! to Monte Carlo data are shownversus the absorption coefficient of the second layer used in theMonte Carlo simulations ~ma2!. The optical parameters of theMonte Carlo simulations are ms19 5 1.3 mm21, ma1 5 0.02 mm21,ms29 5 1.0 mm21, and ma2 is varied between ma2 5 0.0025 mm21
and ma2 5 0.02 mm21. The thickness of the first layer is l 5 2 mm.The lines indicate the correct values. Also shown are the reducedscattering coefficients ~ms9*! obtained from nonlinear regressions tothe two-layer Monte Carlo data using a diffusion solution thatassumes a semi-infinite medium ~solid circles!. Reflectance dataat distances r 5 1.25, 1.75, . . . , 17.75 mm were used in the non-linear regression.
1 February 1998 y Vol. 37, No. 4 y APPLIED OPTICS 785
Carlo data for a two-layered medium consisting of a2-mm-thick first layer. The optical coefficients usedin the Monte Carlo simulations are ms19 5 1.3 mm21,ma1 5 0.02 mm21, ms29 5 1.0 mm21, and ma2 is variedbetween ma2 5 0.0025 mm21 and ma2 5 0.02 mm21.Figures 7 and 8 show the absorption and reducedscattering coefficients, respectively, which were ob-tained from the nonlinear regressions. Thesegraphs show that it is possible to obtain the fouroptical coefficients from absolute spatially resolvedreflectance measurements by using the solution ofthe diffusion equation in the nonlinear regression.The reduced scattering coefficient of the first layercan be determined quite accurately, while ma1, ms29,and ma2 show differences of typically 10–30% com-pared with the correct values. These deviations arecaused by both the statistical uncertainty in theMonte Carlo results and the systematic errors intro-duced by the diffusion approximation. To demon-strate the former, ma2* and ms29* obtained fromnonlinear regression to three independent MonteCarlo simulations calculated with ma2 5 0.01 mm21
are shown in Figs. 7 and 8. About five million pho-tons were used in the simulations resulting in anerror in R~r! of ;2% at distances greater than r 5 15mm. This results in an uncertainty in the determi-nation of ma2 and ms29 of ;10–20%. In contrast, thereduced scattering and absorption coefficients deter-mined from the nonlinear regression using a solutionof the diffusion equation for a semi-infinite mediumshow a change smaller than 1% when different MonteCarlo simulations are fitted. However, the derivedoptical coefficients give little information about the
Fig. 9. Estimated reduced scattering coefficients of the first~ms19*, crosses! and second layer ~ms29*, open circles! determined bynonlinear regressions of Eq. ~16! to Monte Carlo data are shownversus the reduced scattering coefficient of the second layer used inthe Monte Carlo simulations ~ms29!. The optical parameters of theMonte Carlo simulations are ma1 5 0.02 mm21, ms19 5 1.3 mm21,ma2 5 0.01 mm21, and ms29 is varied between ms29 5 0.7 mm21 andms29 5 1.2 mm21. The thickness of the first layer is l 5 2 mm.The lines indicate the correct values. Also shown are the reducedscattering coefficients ~ms9*! obtained from nonlinear regressions tothe two-layer Monte Carlo data using a diffusion solution thatassumes a semi-infinite medium ~solid circles!. Reflectance dataat distances r 5 1.25, 1.75, . . . , 19.75 mm were used in the non-linear regression.
786 APPLIED OPTICS y Vol. 37, No. 4 y 1 February 1998
true optical coefficients. As we reported in an ear-lier study, in some cases the estimate is not evenbetween the true values for the top and bottom lay-ers.10
Figure 9 shows the reduced scattering coefficientsderived from nonlinear regressions to Monte Carloreflectance data from a two-layered medium havingthe same optical parameters in the first layer as thecase discussed above. However, ma2 5 0.01 mm21
was now held constant and ms29 was varied betweenms29 5 0.7 mm21 and ms29 5 1.2 mm21. The thick-ness of the first layer was 2 mm. The reduced scat-tering coefficient of the first layer can be determinedwith errors less than 5%, whereas the estimates ofms29 are ;0.15 mm21 too low. The figure indicatesthat the errors arising from the diffusion approxima-tion are greater than those produced from the statis-tical uncertainty of the Monte Carlo results. Theerrors in determining ma1 and ma2 are smaller than10% and 30%, respectively ~data not shown!.
Investigations of the turbid media with thicknessesof the first layer of 6 and 10 mm showed that theerrors in deriving the optical coefficients by use ofabsolute steady-state spatially resolved reflectancewere greater than for l 5 2 mm. We found for mediawith l 5 6 mm, ms19 5 1.3 mm21, ma1 5 0.005 mm21,ms29 5 1.0 mm21, and ma2 between ma2 5 0.01 mm21
and ma2 5 0.026 mm21, that ms19 can be derivedwithin an error of 3% and that the errors in the othercoefficients are approximately 50% when reflectancevalues for distances up to r 5 30 mm are used. Formedia with the same optical coefficients and l 5 10
Fig. 10. Estimated reduced scattering coefficients of the first~ms19*, crosses! and second layer ~ms29*, open circles! determined bynonlinear regressions of Eqs. ~16! and ~17! to Monte Carlo data areshown versus the reduced scattering coefficient of the second layerused in the Monte Carlo simulations ~ms29!. The optical parame-ters of the Monte Carlo simulations are ma1 5 0.02 mm21, ms19 51.3 mm21, ma2 5 0.01 mm21, and ms29 is varied between ms29 5 0.7mm21 and ms29 5 1.2 mm21. The thickness of the first layer is l 52 mm. The lines indicate the correct values. The frequency do-main reflectance at distances r 5 3.75, 6.75, 9.75 mm were used inthe nonlinear regression and the modulation frequency is f 5 195MHz. Also shown are the reduced scattering coefficients ~ms*!obtained from nonlinear regressions to the two-layer Monte Carlodata using a diffusion solution that assumes a semi-infinite me-dium ~solid circles!.
mm the errors in the optical coefficients of the secondlayer are even greater.
2. Frequency DomainFigure 10 shows the reduced scattering coefficientestimated from frequency domain data for the sameMonte Carlo simulations that have been used in Fig.9. The phase difference and the steady-state reflec-tance ratio between r 5 3.75 mm and r 5 6.75 mmand between r 5 6.75 mm and r 5 9.75 mm are usedin the nonlinear regression. The modulation fre-quency is f 5 195 MHz, as in all results for thissubsection. Also shown is the reduced scattering co-efficient derived by employing a solution for the semi-infinite medium. In this case the phase differenceand the steady-state reflectance ratio between r 53.75 mm and r 5 9.75 mm are used in the nonlinearregression.
Whereas ms19 can be determined less accuratelythan in the steady-state domain, better estimates ofms29 are obtained. The errors in determining ma2 aretypically smaller than 10% and ma1 can be deter-mined with an error of typically 30–40% ~data notshown!. The reduced scattering coefficient deter-mined from the semi-infinite model is usually be-tween ms19 and ms29.
Figures 11 and 12 show ma2 derived from nonlinearregressions to Monte Carlo reflectance data from two-layered turbid media with first layer thicknesses of 6and 10 mm, respectively. The optical parameters ofthe Monte Carlo simulations are ms19 5 1.3 mm21,
Fig. 11. Estimated absorption coefficients of the second layer~ma2*, open circles, pluses! determined by nonlinear regressions ofEqs. ~16! and ~17! to Monte Carlo data are shown versus theabsorption coefficient of the second layer used in the Monte Carlosimulations ~ma2!. The optical parameters of the Monte Carlosimulations are ms19 5 1.3 mm21, ma1 5 0.005 mm21, ms29 5 1.0mm21, and ma2 is varied between ma2 5 0.01 mm21 and ma2 50.025 mm21. The thickness of the first layer is l 5 6 mm. Theline indicates the correct values. Also shown is the absorptioncoefficient ~ma*, crosses, boxes! obtained from nonlinear regres-sions to the two-layer Monte Carlo data when a diffusion solutionthat assumes a semi-infinite medium is used. The frequency do-main reflectance at distances r 5 7.5, 13.5, 19.5 mm were used inthe nonlinear regression and the modulation frequency is f 5 195MHz. Two independent Monte Carlo simulations were per-formed.
ma1 5 0.005 mm21, ms29 5 1.0 mm21, and ma2 isvaried between ma2 5 0.01 mm21 and ma2 5 0.026mm21 ~ma2 5 0.022 mm21 in Fig. 12!. The data fromtwo independent Monte Carlo simulations are shownin Fig. 11. The reflectance for the media with dif-ferent ma2 values were calculated from one MonteCarlo simulation as explained in Subsection 2.B.Frequency domain reflectance data at r 5 7.5, 13.5,and 19.5 mm were used in the analysis. The absorp-tion coefficients derived from a semi-infinite modelare also shown. In this case the phase differenceand the reflectance ratio between r 5 7.5 mm and r 519.5 mm ~r 5 29.5 mm in Fig. 12! are used in thenonlinear regression.
It can be seen from Fig. 11 that the absorptioncoefficient of the second layer can be derived quiteaccurately. The errors increase with an increasingabsorption coefficient, probably because fewer de-tected photons have passed through the second layer.The results for the two different Monte Carlo simu-lations show that the errors at large ma2 values aredue to the statistical uncertainty in the Monte Carloresults. For these simulations approximately 10million photons were used, which resulted in uncer-tainties in the phase of ;0.1° and in the steady-statereflectance of ;1% at r 5 19.75 mm. The errors inthe derived ms19, ms29, and ma1 are smaller than 1%,10%, and 15%, respectively, ~figure not shown!.
In Fig. 12 the results of one independent MonteCarlo simulation are shown. The absorption coeffi-cients of the second layer can be derived within anerror of 20%. The errors in ms19, ms29, and ma1 aresmaller than 1%, 20%, and 20%, respectively, ~figure
Fig. 12. Estimated absorption coefficients of the second layer~ma2*, solid circles! determined by nonlinear regressions of Eqs.~16! and ~17! to Monte Carlo data are shown versus the absorptioncoefficient of the second layer used in the Monte Carlo simulations~ma2!. The optical parameters of the Monte Carlo simulations arems19 5 1.3 mm21, ma1 5 0.005 mm21, ms29 5 1.0 mm21, and ma2 isvaried between ma2 5 0.01 mm21 and ma2 5 0.022 mm21. Thethickness of the first layer is l 5 10 mm. The line indicates thecorrect values. Also shown is the absorption coefficient ~ma*, opencircles! obtained from nonlinear regressions to the two-layer MonteCarlo data when a diffusion solution that assumes a semi-infinitemedium is used. The frequency domain reflectance at distances r5 7.5, 18.5, 29.5 mm were used in the nonlinear regression and themodulation frequency is f 5 195 MHz.
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Table 1. Optical Coefficients of the Two-Layered Medium ~Phantom 1! Having a First Layer Thickness of l 5 2 mm Derived from Measurements onthe Side of Phantom 1 and 2 ~Semi-Infinite Media! and on the Top ~Two-Layered Media! at 543 and 612 nm
not shown!. Although 50 million photons were usedfor these Monte Carlo simulations, the uncertaintiesare quite large at distances of approximately 25–30mm. Therefore, the errors in the derived opticalproperties are probably mainly due to the uncertain-ties in the Monte Carlo results. Figures 11 and 12also show that the relative increase in the absorptioncoefficient estimated with the semi-infinite model ismuch smaller than the true increase in ma2.
D. Determination of the Optical Coefficients fromMeasurements on Phantoms
As described in Section 3, the optical properties of thetwo-layer phantoms were determined from measure-ments of the absolute spatially resolved reflectanceon the side of the second layer of these phantoms faraway from the first layer so that a semi-infinite me-dium could be assumed. For the second layer ofphantom 2 ~same material as the first layer of phan-tom 1! we obtained at l 5 543 nm ~l 5 612 nm! ms9 50.52 mm21, ma 5 0.024 mm21 ~ms9 5 0.50 mm21, ma5 0.025 mm21!; and for the second layer of phantom1 ~same material as the first layer of phantom 2! ms95 1.05 mm21 and ma 5 0.009 mm21 ~ms9 5 1.05mm21, ma 5 0.008 mm21! ~see Table 1!. The mea-surement of phantom 2 at l 5 543 nm is shown inFig. 13 ~lower dashed curve! and that for phantom 1in Fig. 14 ~upper dashed curve!. Also shown are thetheoretical curves obtained from the nonlinear re-
Fig. 13. Measurements of the spatially resolved absolute reflec-tance at the top of phantom 1 ~two-layered, upper dashed curve!and at the side of phantom 2 ~semi-infinite, lower dashed curve!.Also shown are the nonlinear regression to the semi-infinite mea-surement ~lower solid curve! and the theoretical curve of the two-layer measurement calculated from the known optical coefficients~upper solid curve!. A He–Ne laser at l 5 543 nm was used as alight source.
788 APPLIED OPTICS y Vol. 37, No. 4 y 1 February 1998
gression ~corresponding solid curves!. Figures 13and 14 show also the measurements of the spatiallyresolved reflectance on the top surface of the two-layer medium for phantom 1 ~upper dashed curve ofFig. 13! and phantom 2 ~lower dashed curve of Fig.14! and the corresponding theoretical curves ~solidcurves!. The theoretical curves were calculated us-ing the solutions of the two-layer diffusion equation ofSection 2 and the optical coefficients derived from thesemi-infinite measurements. This means no param-eter has been fitted in this case. It can be seen in thefigures that the measurements agree qualitativelywith the theoretical predictions. However, the mea-sured reflectance of the two-layered phantoms is inboth cases lower than the theoretical prediction.Similar results have been obtained for the measure-ments at l 5 612 nm ~data not shown!. We alsoperformed Monte Carlo simulations for the two-layered phantoms by using as input optical proper-ties determined from the measurements in semi-infinite geometry. It was found that the results ofthe simulations were a bit closer to the experimentaldata, especially at distances of ;2 mm, but the mea-sured reflectance was still smaller than the theoret-ical reflectance.
The effect of the second layer on the spatially re-solved reflectance is evident in Figs. 13 and 14. Inthe case of phantom 1 the second layer causes anincrease of the reflectance for all distances compared
Fig. 14. Measurements of the spatially resolved absolute reflec-tance at the top of phantom 2 ~two-layered, lower dashed curve!and at the side of phantom 1 ~semi-infinite, upper dashed curve!.Also shown are the nonlinear regression to the semi-infinite mea-surement ~lower solid curve! and the theoretical curve of the two-layer measurement calculated from the known optical coefficients~upper solid curve!. A He–Ne laser at l 5 543 nm was used as alight source.
with the semi-infinite case; this is because the secondlayer has a greater reduced scattering coefficient anda smaller absorption coefficient so that more photonsare remitted back to the first layer. In contrast, thespatially resolved reflectance of the second phantomdecreases compared with the semi-infinite case, be-cause of the greater absorption of the second layer.The decrease is greater at long distances, whereasalmost no difference can be seen at small distances.Because the first layer of phantom 2 is relativelythick ~l 5 6 mm!, only a small number of the photonsremitted at small distances have propagated throughthe second layer.
Equation 16 was used to fit the spatially resolvedabsolute reflectance measurements from phantom 1,and the results are summarized in Table 1. Theerrors in the reduced scattering coefficients rangefrom 6% to 24% whereas those of the absorption co-efficients are between 0% and 88%. Because of theexperimental errors the results are worse than thoseobtained in Subsection 4.C. The nonlinear regres-sion to the measurements of phantom 2 did not resultin reasonable optical coefficients, because, as ex-plained in Subsection 4.C.1, the greater thickness ofthe first layer results in greater errors in the derivedoptical coefficients. This is especially true for theseexperiments because the reflectance could be accu-rately measured only at distances to 14 mm.
Both measurements on the side ~semi-infinite case!of the phantoms show values of R~r! at small dis-tances ~r # 1 mm! that are much greater than thetheoretical ones. This is caused by specular reflec-tion, because the surfaces are not very flat at thesides of the phantoms. This effect is removed whenmeasuring at the top of the phantoms ~see Figs. 13and 14!, which are much smoother.
5. Conclusions
The solutions of the diffusion equation for a two-layered turbid medium presented in this article havebeen derived with the Fourier transform approachproposed by Dayan et al.13 However, we did not in-troduce any approximation, and, furthermore, wesolved the equations employing the extrapolatedboundary condition. It has been shown for a semi-infinite medium in the steady-state, frequency, andtime domains that this boundary condition results inexpressions of the reflectance that are closer to MonteCarlo simulations than those based on the zeroboundary condition used by Dayan et al.6 The com-parisons of the reflectance in the steady-state, fre-quency, and time domains showed that the derivedsolutions are close to the results obtained from two-layer Monte Carlo simulations and much better thanthose derived from Dayan et al. Thus, for many ap-plications the reflectance ~and probably the fluencerate in the tissue! calculated with these equations areexact enough to replace the time-consuming MonteCarlo simulations. Nonlinear regressions of thesesolutions of the diffusion equation to reflectance dataobtained from the same solutions to which typicalexperimental noise were added showed in the steady-
state and frequency domains that the absorption andthe reduced scattering coefficients of the two layersand the thickness of the first layer can, in principle,be obtained. However, additional systematic errors,caused by the diffusion approximation ~when com-pared with experiments or Monte Carlo simulations!or by the experimental apparatus, will deterioratethe results.
Although the solutions of the diffusion equation forthe reflectance from a two-layered medium are quiteclose to the results of the transport theory, the errorsin determining the optical properties caused by thisapproximation are greater than in the semi-infinitecase. Therefore, it would be advantageous to have asolution of the transport equation for a two-layer me-dium fast enough to be used for determination of theoptical properties. We have concentrated in this ar-ticle on the determination of the absorption and re-duced scattering coefficients when the thickness ofthe first layer is known. This information can oftenbe obtained from general anatomy or can be deter-mined from other methods in specific cases, e.g., fromultrasound. In the frequency domain it was foundthat the four optical coefficients can be determinedfrom relative measurements of the phase and steady-state reflectance at three distances. Fantini et al.30
developed an apparatus with which it is possible toperform accurate and fast measurements of thephase and steady-state reflectance at four distances.Their aim was to determine the oxygen saturation inthe muscle, for example, on the forearm by use of asemi-infinite model. However, these measurementscan be influenced by the overlying subcutaneous fatlayer and the skin. If the thickness of the tissuesabove the muscle can be determined, the two-layermodel presented here could be applied and shouldlead to improved results.
It was shown by fitting the solutions of the diffu-sion equation to results obtained from Monte Carlodata that the optical parameters are obtained withdifferent accuracy. For example, for a layer thick-ness of l 5 2 mm, ms19 could be derived within 5%from measurements in the steady-state domain,whereas ms29 could be more accurately determined inthe frequency domain compared with the steady-state domain. Thus one can choose the method ac-cording to the problem that has to be investigated.For example, if the optical coefficients of the skin arerequired, frequency domain measurements ~at leastif performed at relatively small frequencies! do notseem to add useful informations. Investigations forgreater thicknesses of the first layers showed that inthese cases measurements in the frequency domainare superior to those performed in the steady-statedomain. It was observed that the optical coefficientsobtained from the two-layered model are more influ-enced by statistical errors than those derived withthe semi-infinite model. This might be a serious dis-advantage of the two-layer approach when used formonitoring physiological variables.
The results obtained in this article can a priori be
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used only for two-layered turbid media. Biologicaltissue often has a multilayered structure. It mustbe carefully investigated to determine if it is possibleto condense several tissue layers to one in order to usethe two-layered model. This point is crucial for theapplication of the two-layered model for in vivo mea-surements. Alternatively, it is possible to extendthe solution of the diffusion equation presented hereto three or more layers in a straightforward manner,but the inverse problem becomes more difficult.
A major question in near-infrared spectroscopy ofthe adult head is whether the probing volume of themeasurements is limited to the skull and skin layersor if it is possible to obtain information from thebrain. Measurements on the adult head are limitedby signal-to-noise to distances of approximately r 5 5cm, and the layers above the brain are ;10 mmthick.31 Thus, this situation is comparable withthose for which results are presented in Figs. 11 or12. These figures show that the use of a semi-infinite model results in apparent changes in the ab-sorption coefficient that are much smaller than thereal changes in ma2. For the medium with l 5 10mm ~Fig. 12!, the derived absorption coefficient in-creases by 0.0022 mm21 when ma2 increases from ma25 0.01 mm21 to ma2 5 0.022 mm21. Thus the realchange in the absorption coefficient is more than fivetimes greater than the derived change. Therefore, amultilayer model could potentially improve the mea-surement of hemodynamics in the adult brain.
The investigations in the time domain showed thatit is possible to obtain the optical coefficients of thetwo layers from a measurement of the absolute re-flectance at a single distance if the thickness of thefirst layer is known. Measurements in the time do-main contain more information than those in thesteady-state or frequency domains ~when performedat a single frequency! and may, therefore, offer thegreatest potential for the determination of the opticalproperties of multilayered media.
This research was supported by the National Insti-tutes of Health grant PO1-CA43892 and the SwissPriority Program Optics 2 ~Project 423!. AlwinKienle is grateful for a postdoctoral scholarship fromthe German Research Society ~Deutsche Forschungs-gemeinschaft!.
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