Asymptotic behavior and inverse problem in layered scattering media

24 J. Opt. Soc. Am. A/Vol. 21, No. 1 /January 2004 Tualle et al.

Asymptotic behavior and inverse problem inlayered scattering media

Jean-Michel Tualle, Ha Lien Nghiem, Dominique Ettori, Raphael Sablong, Eric Tinet, andSigrid Avrillier

Laboratoire de Physique des Lasers, Centre National de la Recherche Scientifique, Unite Mixte de Recherche (CNRSUMR 7538), Universite Paris 13, 99 avenue J.-B. Clement, 93430 Villetaneuse, France

Received March 24, 2003; revised manuscript received August 27, 2003; accepted September 8, 2003

The main challenge of noninvasive optical biopsy is to obtain an accurate value of the optical coefficients of anencapsulated organ (muscle, brain, etc.). The idea developed by us is that some interesting information couldbe deduced from the long-time behavior of the reflectance function. This asymptotic behavior is analyzed forlayered media in the framework of the diffusion approximation. A new method is derived to obtain accuratevalues for the optical parameters of the deepest layers. This method is designed to work in a specific long-time regime that is still within the scope of standard time-of-flight experiments but far from being included inthe mathematically defined asymptotic region. The limits of this method, linked to the cases where theasymptotic behavior is no longer governed by the deepest layer, are then discussed. © 2004 Optical Society ofAmerica

OCIS codes: 030.5620, 110.7050, 100.3190.

1. INTRODUCTIONThe real challenge of noninvasive optical biopsy is to ob-tain an accurate in vivo evaluation of the optical coeffi-cients of an embedded organ from reflectance measure-ments. There has recently been a lot of activity on lightpropagation and inverse problems in two-layered scatter-ing media.1–6 It appears from these studies that it is dif-ficult to obtain an accuracy better than 5% for the opticalcoefficients of the bottom layer.2,4 Moreover, the perfor-mance of such an inversion should considerably be de-creased when more than two layers are being considered.An interesting way to overcome these difficulties was re-cently proposed by Pifferi et al.,7 who used the knowledgeof the absorption spectra of the main absorbers present inthe medium to reconstruct the absorption coefficient.Their work concerns two-layered media, and very accu-rate results were obtained even with a very thick (1.2-cm)first layer. Their method, which gives encouraging re-sults, was, however, tested with a limited number of verycontrasted dies and cannot give any value for the reducedscattering coefficients.

It is a common assertion to say that another way to ob-tain the optical properties of an organ covered by layers oftissues is to select the light that stays a ‘‘long time’’ in themedium and therefore spends the most time in the deep-est layer: One would then obtain results that more spe-cifically depend on the deepest layer optical propertiesand not as much on the properties of the overlying layers.In fact, one main problem in that kind of approach is thatthe times that are experimentally considered long are stillfar from the mathematically defined asymptotic region.

This paper is devoted to the analysis of this problem inthe framework of the diffusion approximation. The long-time behavior of the reflectance function in scattering me-dia with N > 2 layers is explored, and a method is de-rived to obtain the optical coefficients of the deepest layer

1084-7529/2004/010024-11$15.00 ©

at realistic long times. This method is illustrated withdifferent two-layered media.

2. REFLECTANCE FUNCTION INSCATTERING MEDIA WITH N Ð 2 LAYERSLet us recall the well-known formulation of the diffusionapproximation8–13 in layered media: In the medium il-lustrated in Fig. 1, each layer i is characterized by an ab-sorption coefficient mai , a reduced scattering coefficientmsi8 , and a diffusion constant Di ; 1/(3msi8 ). The z axisis oriented toward the deepest layer, and the interface be-tween layer i and layer i 1 1 is located at z 5 li . Theincident laser pulse is introduced through a pointlikesource located at z 5 z0 .8 The diffuse photon fluencerate f satisfies the diffusion equation in each layer:

1

ci

]f

]t2 DiDf 1 maif 5 S, (1)

where ci ' c0 /ni is the light speed in layer i and wherethe source term can be written as S(r, z, t)5 d (r)d (t)d (z 2 z0) for one incident unit of energy; rdenotes the transverse plane coordinates.

Concerning the boundary conditions between two lay-ers, one can write the following at z 5 li (Refs. 1 and 10):

ni112 f~li

2! 5 ni2f~li

1!,

Di

]f

]z~li

2! 5 Di11

]f

]z~li

1!. (2)

For the free surface boundary condition, we can use theextrapolated boundary condition11–13:

f~2zb! 5 0, (3)

where zb is an extrapolated distance that takes into ac-count the index mismatch between air and medium 1.

2004 Optical Society of America

Tualle et al. Vol. 21, No. 1 /January 2004 /J. Opt. Soc. Am. A 25

For instance, we will consider in the following the casen1 5 1.4, which leads to zb 5 1.9/ms18 . The last condi-tion is, of course, f → 0 when z → `.

This problem can be solved by using the Fourier trans-form f in time and transverse plane coordinates, which isa function of the conjugate variables of t and r (v and k,respectively). Kienle et al.1 gave a solution in the case ofa two-layered medium. Let us extend the resolution tomultilayered media.

The general solution of the diffusion equation in me-dium i, without source term, is

f~z ! 5 Ai exp~2siz ! 1 Bi exp~siz !, (4)

where

si 5 S k2 1mai

Di1

jv

DiciD 1/2

.

The boundary conditions (2) become

ni112 @Ai exp~2sili! 1 Bi exp~sili!#

5 ni2@Ai11 exp~2si11li! 1 Bi11 exp~si11li!#,

Disi@2Ai exp~2sili! 1 Bi exp~sili!#

5 Di11si11@2Ai11 exp~2si11li! 1 Bi11 exp~si11li!#,

which can be written in the following form:

S Ai

BiD 5 PiS Ai11

Bi11D , (5)

where Pi is a 2 3 2 matrix depending on the optical coef-ficients. Consequently,

S A1

B1D 5 P1P2P3¯PN21S AN

BND , (6)

where BN 5 0 to satisfy the condition f → 0 when z→ `.

To include the source term, one only has1 to writef(z) 5 A0 exp(2s1z) 1 B0 exp(s1z) for z , z0 and f(z)5 A1 exp(2s1z) 1 B1 exp(s1z) for z0 , z , l1 with theboundary conditions

f~z01! 5 f~z0

2!,

Fig. 1. Layered medium.

]f

]z~z0

1! 2]f

]z~z0

2! 5 21/D1 , (7)

which gives a linear relationship between A0 , B0 and A1 ,B1 . The knowledge of AN therefore gives the solution ofthe problem in all layers (Appendix A).

The extrapolated boundary condition (3), which can berewritten as f(2zb) 5 A0 exp(s1zb) 1 B0 exp(2s1zb) 5 0,eventually allows us to obtain AN . The reflectance func-tion can then be deduced from8

R~v, k2! 5 2jz~0 ! 5 D1

]f

]z~0 ! (8)

and from an inverse Fourier transform by using the J0Bessel function:

R~r, t ! 51

4p2E

0

`

kdkE dvR~v, k2!J0~kr!exp~ jvt !.

(9)

We have here a quite general formula. The particularcase of a two-layered medium with n1 5 n2 can be solvedby using1

R 5 cosh~s1zb!

3D1s1 cosh@s1~l 2 z0!# 1 D2s2 sinh@s1~l 2 z0!#

D1s1 cosh@s1~l 1 zb!# 1 D2s2 sinh@s1~l 1 zb!#.

3. ASYMPTOTIC BEHAVIOR OF THEREFLECTANCE FUNCTION AT INFINITETIMESA numerical evaluation of Eq. (9) is not easy to perform,but its asymptotic behavior can be obtained from thesaddle-point method. Let us use in Eq. (9) the substitu-tion v ° s 5 sN . At fixed k, we have jv 5 DNcN(s2

2 k2) 2 maNcN and dv 5 22jDNcNsds. Using theseexpressions in Eq. (9) leads directly to

R~r, t ! 52 jDNcN

2p2exp~2maNcNt !

3 E0

`

kdkEG~k !

sdsR~s, k2!J0~kr!

3 exp@DNcN~s2 2 k2!t#, (10)

where G(k) is the hyperbolic contour s(v) in the complexplane at a given k for v varying from 2` to `. Thischoice for s leads to a privileged role for the optical coef-ficients of the deepest layer, as can be seen, for instance,from the exp(2maNcNt) term. The reasons for this choiceare developed in Appendix B. The only point that we willstress here is that with this choice, R(s, k2) is an analyticfunction of s. From this fact, it appears that the integra-tion over G(k) can be replaced by an integration over]01 2 j`, 01 1 j`@ , provided that R(s, k2) has no polesinside the contour defined in Fig. 2. On this integrationaxis, the integrand in Eq. (10) is proportional to the


Gaussian exp(DNcNs2t), which is all the more peaked ats 5 0 than t is high. It is therefore justified to use thefollowing development:

R~s, k2! 5 (p,q>0

Rp,qspk2q. (11)

Fig. 2. Integration contour. The right curve corresponds to hy-perbolic contour G(k), that is, to Re(s

2) 5 k2 1 ma /D; the leftpart is the imaginary axis ]01 2 j`, 01 1 j`@ . The integrationcontour is closed by arbitrary upper and lower curves that haveno contribution when extended to infinity.

Fig. 3. Convergence of the development in Eq. (12), using themedium described in Table 1 with l 5 1 cm, at r 5 0. ln R`

2 ln Rn is plotted versus time for different values of the order nof the development and with a logarithmic scale.

Table 1. Optical Coefficients of a Two-LayeredMedium for the Simulation of a Neonate Heada

Layer

Coefficient

ms8 (cm21) ma (cm21) n

1 15 0.15 1.42 10 0.10 1.4

a Ref. 16.

In the following, we will use ma , D, and c instead of maN ,DN , and cN . Inserting Eq. (11) into Eq. (10) and inte-grating over ]01 2 j`, 01 1 j`@ leads to

R~r, t ! 5 2DcA2p

2p2 H (n>2p11,q>0

R2p11,q~2p 1 1 !!!

3 ~2Dct !2p~21 !p1qDTq

1 O~t2n21!J ~2Dct !25/2 expS 2r2

4Dct2 mact D ,

where DT is the transverse Laplacian. From this expres-sion, we deduce that

R~r, t ! 5 F (n>p>2q>0

Rp,qr2qt2p

1 O~t2n21!G t25/2 expS 2r2

4Dct2 mact D

(12)

or

ln R~r, t ! 5 25

2ln t 2

r2

4Dct2 mact

1 (n>p>2q>0

Cp,qr2qt2p 1 O~t2n21!.

(13)

At long times, the reflectance function therefore con-verges toward the solution of a semi-infinite geometry,with the optical coefficients of the deepest layer. This de-velopment is similar to the result obtained by Mochiet al.14 in two dimensions with r 5 0.

An example is presented in Fig. 3 to illustrate the con-vergence of Eq. (12), using the medium described in Table1, with l 5 1 cm and r 5 0. ln R` 2 ln Rn ' DR/R isplotted versus time for different values of the order n ofthe development with a logarithmic scale. As can beseen in this figure, the order n 5 0 leads to a more than10% error for t , 9 ns; the order n 5 1 is over this valuefor t , 3 ns. As the physical accessible values lie ratherbetween 0 and 3 ns, some refinements are needed in orderto exploit Eq. (13).

4. BEHAVIOR OF THE REFLECTANCEFUNCTION AT PRACTICAL LONG TIMESIn practical situations, the asymptotic behavior is ana-lyzed in some interval around a relatively high value t0 .In that case, we need access to a development around t0 ,which can slightly differ from Eq. (13). To analyze this,we set s 5 ju in Eq. (10). Taking into account thatR(0, k2) gives no contribution in Eq. (10), so that R(s, k2)can be replaced by R(s, k2) 2 R(0, k2), we arrive at (Ap-pendix C)


R~r, t ! 5 2a5/2p25/2 exp~2ar2 2 mact !

3 E0

`

RdRE2`

`

dZR~Z, R!~1 2 2aZ2!

3 J0~2jarR!exp@2a~R2 1 Z2!#, (14)

where a 5 1/(4Dct) and where

R~Z, R! 5 E dud2kDc

jpu@R~s, k2! 2 R~0, k2!#

3 exp~ jk – R 1 jZu!.

Setting a 5 a0 1 da in the integrand of Eq. (14) leadsto a development in rqdap that is very similar to Eq. (12).Let us state explicitly this development to first order:

R~r, t !

5 2a5/2p25/2 exp~2ar2 2 mact !

3 E RdRdZ R~Z, R!exp@2a0~R2 1 Z2!#

3 $~1 2 2a0Z2! 2 @2Z2 1 ~1 2 2a0Z2!

3 ~R2 1 Z2!#da 1 R2~1 2 2a0Z2!a2r2

2 R2@2Z2 1 ~1 2 2a0Z2!~R2 1 Z2!#a02r2da%. (15)

Fig. 4. Parameters (a) E and (b) F calculated from Eq. (15) ver-sus t0 5 1/(4Dca0) in the case of the medium of Table 1 with l5 1 cm. The values C1,0 and C2,1 of the development (13) areshown as dashed lines.

This expression differs from Eq. (12) only through thelast term, which has the form r2/t and can influence theresult obtained for the diffusion coefficient D. In our ex-perience, however, this term can be neglected if, first, R issufficiently peaked to 0 and, second, if a0 is small enough.The first condition requires that there be no pole ofR(s, k2) in the vicinity of the pure imaginary axis, in or-der for R to be relatively smooth on this axis, and for R tobe a peaked function. To fulfill the second condition, onehas to choose t0 as high as possible with a reasonablesignal-to-noise ratio. The validity domain of this ap-proximation will be discussed in Section 8. Starting fromthis hypothesis and taking the logarithm of Eq. (15) up tofirst order leads to

ln R~r, t ! ' R0 25

2ln t 2

r2

4Dct2 mact 1

E

t1 F

r2

t2.

(16)

The parameters E and F calculated from Eq. (15) arerepresented in Fig. 4 versus t0 5 1/(4Dca0) in the case ofthe medium of Table 1 with l 5 1 cm. The correspondingvalues C1,0 and C2,1 of the development (13) are shown asdashed lines. One can see in Fig. 4(a) that in the 1–3-nsrange, the value of E is significantly different from itsasymptotic value C1,0 . The coefficient F in Fig. 4(b) doesnot differ as much from C2,1 .

One way to obtain the optical coefficients of the deepestlayer with experimental results obtained for a space do-main near r 5 0 and a time domain near a chosen time t0is to fit the logarithm of the reflectance function by usingEq. (16) with five adjustable parameters: R0 , mac, Dc,E, and F. From the result of this fit, one can extract maand ms8 .

5. IMPLEMENTATION OF THE INVERSEPROBLEMThe logarithm of the reflectance function in Eq. (16) de-pends linearly on the parameters, and the inverse prob-lem reduces to a simple linear regression: If y is a vectorof size M collecting all the experimental data for differentpositions in time and space and p a vector that representsthe five parameters, we can write y ; Jp (where theJacobian J is defined by Jij 5 ]yi /]pj) and the variance

s 2 5 M21iy 2 Jpi2

is minimized by

p 5 @ tJJ#21 tJy.

The inverse problem can therefore be solved through asimple matrix inversion, leading to a unique result, with-out the problem of the choice of initial values. If we as-sume that the variance is almost constant on the small re-gion considered for the linear regression (i.e., ^dytdy&5 s 2I, with y 5 y0 1 dy, p 5 p0 1 dp, and y0 5 Jp0),the covariance matrix reads as

^dptdp& 5 s 2@ tJJ#21

and the square root of the diagonal elements of the cova-riance matrix corresponds to the average statistical error


Fig. 5. Logarithm of the time-resolved reflectance function ob-tained from a Monte Carlo simulation with the medium of Table1 and l 5 5 mm for six detectors chosen with an equal spacing in[1.14 cm, 2.49 cm]. The fit, performed for 1500 ps , t, 2000 ps, is presented together with a zoom of this time inter-val.

Fig. 6. Results obtained for (a) ma and (b) ms8 for different valuesof l using the medium described in Table 1. The squares corre-spond to the results obtained with a simple semi-infinite fit (E5 F 5 0), and the circles correspond to the results obtainedwith relation (16).

on the different parameters. One can also define the cor-relation between two parameters as Corr( pi , pj)5 ^dpidpj&/(^dpi

2&^dpj2&)1/2.

To test our inversion procedure, we numerically evalu-ate some reflectance functions with a Monte Carlosimulator,15 using a Henyey–Greenstein phase functionwith an anisotropy factor g 5 0.8. The random walkersare collected at a distance r and at time t in a ring of sizeDr 5 137 mm and Dt 5 20 ps. The result is then nor-malized to one incident photon and to a pixel size of Dr3 Dr 3 Dt. The fits were performed for 1.14 cm , r, 2.49 cm, which corresponds to an array of 100 detec-tors of size Dr 3 Dr, and for 1500 ps , t , 2000 ps (i.e.,t0 5 1750 ps), with a time resolution of 20 ps. Such asetup can correspond to the use of a streak camera.

Let us begin with the medium described in Table 1,which can be used to describe the neonate head.16 Wepresent in Fig. 5 a part of the result obtained from theMonte Carlo simulation for a thickness l 5 5 mm for thefirst layer. We plot on this figure the logarithm of thetime-resolved reflectance function corresponding to sixdetectors chosen with an equal spacing in [1.14 cm, 2.49cm] (detector numbers 0, 20, 40, 60, 80, and 100). Thehighest curve obviously corresponds to the detector placedat r 5 1.14 cm. We represent too the fit performed for

Fig. 7. Results obtained (circles) for (a) E and (b) F, assumingcorrect values for ma and ms8 , for different values of l. Thecurves with open circles correspond to the results obtained withrelation (16).


1500 ps , t , 2000 ps, together with a zoom of this timeinterval. The six curves look like lines in this small re-gion; as a line is described by two parameters, and asthese parameters depend on the position r, one can expectat least four parameters from this set of curves. The fifthparameter comes from the small curvature of thesecurves. Of course, the statistical error on parameters Eand F can be relatively important, but as these param-eters have only a corrective effect on the values of the op-tical coefficients, this procedure can lead to a reasonablestatistical error for the absorption and reduced scatteringcoefficients. The result presented in Fig. 5 was obtainedby using 109 random walkers. The link between thisnumber and the signal-to-noise ratio is, however, not

Fig. 8. Results obtained for (a) ma and (b) ms8 for different valuesof l using the medium described in Table 2. The squares corre-spond to the results obtained with a simple semi-infinite fit (E5 F 5 0), and the circles correspond to the results obtainedwith relation (16).

Table 2. Optical Coefficients of a Two-LayeredMedium for the Simulation of the Fat/Muscle

Succession

Layer

Coefficient

ms8 (cm21) ma (cm21) n

1 15 0.026 1.42 10 0.15 1.4

trivial, as many technical procedures are implied in orderto improve this ratio.15 We measure a standard devia-tion s 5 1.51% for the logarithm of the reflectance func-tion when using our model (we have s 5 1.54% with thesemi-infinite model). This corresponds to a signal-to-noise ratio of 65 for the reflectance function @R/dR; 1/(d ln R)#. As one needs to detect 652 ; 4000photons in order to have such a signal-to-noise ratio, andas the average attenuation factor is approximately 10210,we deduce that 4 3 1013 photons are needed for thesource. If, furthermore, the numerical aperture of thedetection is limited, for instance with a ratio aperture of4, this number has to be increased to 53 1015 photons, which corresponds to an irradiation ofapproximately 1 mW with a 1-s on-chip integration time.The numerical simulations presented here are thereforecompatible with realistic experimental conditions.

The results obtained for ma and ms8 for different valuesof l are shown in Figs. 6(a) and 6(b), respectively. Thesquares correspond to the results obtained with a simplesemi-infinite fit (E 5 F 5 0), and the circles correspondto the results obtained with relation (16). As the semi-infinite fitting procedure requires only three parameters,the statistical error is very low and the error bars will notbe represented in that case. We obtained a clear im-provement of the accuracy by using our method. The pa-rameters ma and ms8 are weakly correlated when ma is

Fig. 9. Results obtained (circles) for (a) E and (b) F, assumingcorrect values for ma and ms8 , for different values of l. Thecurves with open circles correspond to the results obtained withrelation (16).


highly correlated with E and ms8 with F [we have the fol-lowing, for instance, for l 5 5 mm: Corr(ma , ms8)5 0.16, Corr(ma , E) 5 20.98, Corr(ms8 , F) 5 0.99,Corr(ma , F) 5 0.16, and Corr(ms8 , E) 5 20.3]. Thishigh correlation between ma , E and ms8 , F is of course notin contradiction with the fact that E and F have only acorrective effect on the values of the optical coefficients.The values for E and F, obtained from a fit that assumesthe correct values for ma and ms8 , are presented in Figs.7(a) and 7(b), respectively (circles), and compared withthe values calculated from Eq. (15). We observe a com-plete agreement for E and a small systematic error for F.The deviation from theory comes from the large time win-dow used for the fit. Reducing this window, however, in-troduces fatal undesirable noise.

We reproduce this numerical experiment by using an-other two-layered medium, described in Table 2. Theonly change concerns the absorption coefficient, which isnow lower in the upper layer than in the deepest one.This case approaches the situation of the fat/muscle suc-cession (the scattering coefficients of fat and muscle are infact estimated17 to be lower than the values in Table 2,but the case considered here is less advantageous for ourmethod). The results obtained for ma and ms8 are shownin Figs. 8(a) and 8(b), respectively, and the values for Eand F in Figs. 9(a) and 9(b), respectively. Here again weobserve an improvement when using our method, but theerror increases very quickly when l . 4 mm. A jumpappears on the theoretical curves in Fig. 9 after l. 5 mm, so that for higher thickness the theory com-pletely disagrees with fitted values. This behavior istypically linked to the presence of a pole in the contour ofFig. 2.

6. INFLUENCE OF POLESPoles account for diffusion waves guided in the layeredstructure (Appendix A). They induce crucial modifica-tions in the change of the integration contour in Eq. (10).If sPi denotes the poles lying inside the contour of Fig. 2,one can write

EG~k !

sds R~s, k2!exp~Dcs2!

5 E2j`101

j`101

sds R~s, k2!exp~Dcs2!

1 2pj(sPi

Re s@R~sPi , k2!#exp~DcsPi2 t !, (17)

where Re s@R(sPi , k2)# stands for the residue of functionR(s, k2) at sPi .

In the example of the medium described by Table 2, apole appears on the real axis inside the contour of Fig. 2when the thickness l of the first slab passes a criticalvalue. In fact, the pole crosses the contour for (AppendixD)

lc 1 zb 5p

2 S D1

ma2 2 ma1D 1/2

. (18)

Note that this implies that ma1 , ma2 , which is theparticularity of Table 2. This numerically gives lc5 5.4 mm, which corresponds to the transition length ob-served in Section 5. In fact, we observe in Figs. 8 and 9 adecrease of the accuracy even for l , lc , which is due tothe fact that the pole approaches the imaginary axis, sothat approximations in Eq. (15) and relation (16) are nolonger valid. This pole can be written as

sP 5 s0 1 bk2 1 o~k2!

or sP2 5 s0

2 1 2s0bk2 1 o8~k2!. (19)

A rough approximation of s0 and b is (Appendix D)

s0 'p2D1

4~lc 1 zb!2D2

~l 2 lc!, b 'D2 2 D1

2D2~lc 1 zb!.

The point is that s0 . 0, so that the exponential termon the right-hand side of Eq. (17) is diverging with in-creasing time, and the residue is in fact the dominantterm:

Fig. 10. Results obtained (circles) for (a) meff and (b) ms,eff8 usingEq. (21) on the medium of Table 2 for different values of l, to-gether with their values corrected by using Eqs. (22) (squares).


Etk

sds R~s, k2!exp~Dcs2t !

' 2pj Re s@R~sP , k2!#exp~DcsP2 t !. (20)

Reporting this approximation in Eq. (10) leads to

R~r, t ! 5Dc

pexp~2mact !E

0

`

kdk Re s@R~k2!#

3 exp@Dco8~k2!t#J0~kr!

3 exp$Dc@s02 1 ~2bs0 2 1 !k2#t%.

Here again the Gaussian term becomes very thin atlong times, so that we can make the approximation:Re s@R(k2)#exp@o8(k2)t# ' Re s@R(0)#, leading to

R~r, t ! }1

texpF2~ma 2 Ds0

2!ct 2r2

4~1 2 2bs0!DctG .

We therefore have

ln R~r, t ! 5 R0 2 ln t 2 meff ct 2r2

4Deff ct, (21)

with

meff 5 ma 1 Ds02, Deff 5 D~1 2 2bs0!. (22)

We see in Eq. (21) that the 5/2 factor before ln t disap-pears and that the values of the absorption coefficient andthe diffusion constant are somewhat modified. Wepresent in Figs. 10(a) and 10(b) the results obtained onthe medium of Table 2 for the effective coefficients meffand ms,eff8 5 1/(3Deff), respectively, using Eq. (21), to-gether with the values for ma and ms8 deduced from Eqs.(22). We find that Eqs. (22) allows a relatively goodagreement, using, however, an a priori knowledge of themedium through the values of s0 and b.

Table 3. Optical Coefficients of a Four-LayeredMedium for the Simulation of a Neonate Head

Medium

Coefficient

ms8 (cm21) ma (cm21) Thickness (mm) n

Skin 20 0.13 2 1.45Skull 15 0.15 5 1.45CSF 0.1 0.01 0.5 1.33Brain 10 0.05–0.1–0.15 — 1.4

7. EXAMPLE WITH A MULTILAYEREDMEDIUMThe considerations on the reflectance function developedin this paper are not limited to two-layered media and canbe applied to a multilayered structure with the same limi-tations concerning the position of the poles of the reflec-tance function. We present here a test of our procedureperformed on the four-layered media described in Table 3.We use in this table the data of Table 1 corresponding tothe neonate head, and we add one layer for the skin18 andone layer for the cerebrospinal fluid (CSF). The resultsobtained with our procedure are presented in Table 4 forthree different values of the absorption coefficient of thedeepest layer and are compared with the results obtainedwith a simple semi-infinite fit. Our method leads to asignificant improvement of the accuracy for both the ab-sorption and reduced scattering coefficients. These per-formances will be of course deteriorated when the thick-ness of the different layers is increased, depending on thecomplex behavior of the poles of the reflectance function.

8. DISCUSSION AND CONCLUSIONWe have shown in this paper that it is possible to deducewith a high accuracy the optical coefficients of an organcovered by scattering layers from time- and space-resolved reflectance measurements, provided that thethickness of the upper layers is not too high. The case ofthicker covering layers was also analyzed; in that case, weproved that the asymptotic behavior of the reflectancefunction is poorly correlated to the properties of the deep-est layer, since the former layers behave as wave guides.Therefore additional information about the encapsulatingmedium is needed in that regime to understand theasymptotic behavior.

A criterion giving a limit between these two regimeswas given in the case of a two-layered medium, but ofcourse it will be hard to generalize such a criterion tomultilayered media. The simplest method to determinethe regime corresponding to a given medium is to test theinversion method proposed in this paper on simulateddata. Such tests will allow the determination of regionsin which our method is applicable. One then has to knowonly whether the biological medium lies in this region.The optical properties of the deepest layer of such a me-dium can then be deduced without any additional infor-mation.

Much more work is therefore needed now to exploit thismethod in biomedical applications. One can process asexplained in the paragraph above, or one can use anotherinvestigation method, such as the systematic use of this

Table 4. Results Obtained from Simulations Performed with the Data of Table 3

ma,Brain (cm21)

Five-Parameter Fit Semi-Infinite Fit

ms8 (cm21) ma (cm21) ms8 (cm21) ma (cm21)

0.05 9.85 6 0.5 0.052 6 0.005 11.39 6 4 3 1022 0.0585 6 4 3 1024

0.1 9.8 6 0.6 0.102 6 0.007 11.7 6 5 3 1022 0.102 6 5 3 1024

0.15 10.35 6 0.3 0.150 6 0.0035 12.08 6 2 3 1022 0.142 6 2 3 1024


method with several patients, at several locations foreach patient, and study the reproducibility of the results,depending on the patient and on the location of the sen-sor. An interesting approach would be to have a dis-crimination test, relying for instance on the use of differ-ent values for t0 , but this is beyond the scope of thispaper.

APPENDIX ALet us set

P1P2P3¯PN 5 Fa b

c dG .We can then rewrite Eq. (6), taking into account that BN5 0:

A1 5 aAN , B1 5 cAN .

Equation (7) directly leads to

A0 5 A1 2exp~s1z0!

2s1D15 aAN 2

exp~s1z0!

2s1D1,

B0 5 B1 1exp~2s1z0!

2s1D15 cAN 1

exp~2s1z0!

2s1D1.

The extrapolated boundary condition (3) then gives

@a exp~s1zb! 1 c exp~2s1zb!#AN

[ Den~s, k2!AN 5sinh@s1~z0 1 zb!#

s1D1, (A1)

and we finally have

R~s, k2! 5 2s1D1~A0 2 B0!

5a exp~2s1z0! 1 c exp~s1z0!

Den~s, k2!cosh~s1zb!.

(A2)

If we cancel the source term by setting Eq. (A1) equal to0, we can have nontrivial solutions if Den(s, k2) 5 0;from Eq. (A2), such guided-wave solutions correspond tothe poles of R.

APPENDIX BTo begin, we note that the resolution procedure for R (Sec-tion 2) involves only exponential functions and resolu-tions of linear systems, so that R is an analytic function ofs1 , s2 ,...,sN . The point now is that for any layer i, N, the substitution si ° 2si corresponds to the substi-tution Ai ↔ Bi . This substitution has no other influenceon the equations and leaves R invariant. So R is an evenfunction of s1 , s2 ,...,sN21 and is therefore an analyticfunction of s1

2,...,sN212 . However, R is not an even func-

tion of sN .Let us consider the more general substitution v ° s

5 (h 1 jgv)1/2 at fixed k, where h and g are positive realnumbers. We have, instead of Eq. (10),

R~r, t ! 52 j

2gp2E

0

`

kdkEG~h!

sds R~s, k2!J0~kr!exp@~s2

2 h!t/g#. (B1)

As si2 5 k2 1 (mai /Di) 1 (s2 2 h)/(gDici) are obviously

analytic functions of s, the only thing to do to ensure theanalyticity of R(s, k2) is to check the analyticity of sN5 @(s2 1 s)/(gDNcN)#1/2, where s 5 gcN(DNk2 1 maN)2 h is real.

If the square-root function is defined for complex num-bers with an argument in ]2p, p[, that is, with a cut inthe complex plane located on the real axis at ]2`, 0[, sN isanalytic in the half-plane Res . 0 for s > 0, and the in-tegration on G(k) can be replaced by an integration on]01 2 j`, 01 1 j`@ . But if s . 0, we have sN(s)5 sN(2s) for any s on @01 2 jAs, 01 1 jAs#; R is thenan even function of s and Eq. (B1) is 0 on that segment,leading for any n . 0 to

R~r, t ! 5 Or~t2n!exp@2~ma 2 dm!t# ~dm . 0 !.

This is not astonishing, as exp(2dmt) 5 O(t2n), but it isuseless. So we need to choose s 5 0, that is, s5 (gDNcN)1/2sN 5 sN (since g is a simple scaling factorthat can be set as g 5 1/DNcN) and the deepest layerthen plays a central role, as expected.

APPENDIX CWe have already seen that R(s, k2) can be replaced byR(s, k2) 2 R(0, k2) in Eq. (10). Let us rewrite Eq. (10)by using an integration over the whole k plane (with-out using the cylindrical symmetry) and insert*d (u 2 u8)d (k 2 k8)du8d2k8; we obtain

R~r, t ! 52Dc

4p3exp~2mact !E d2kd2k8dudu8 s21

3 @R~s, k2! 2 R~0, k2!#

3 d ~u 2 u8!d ~k 2 k8!u82

3 exp@2Dc~u82 1 k82!t 2 jk8–r#;

using

d ~u 2 u8!d ~k 2 k8! 51

8p3EE exp@ jZ~u 2 u8!

1 jR • ~k 2 k8!#dZd2R,

we get two different contributions:

E d2kdu1

ju@R~s, k2! 2 R~0, k2!#exp@ jZu 1 jR–k#

[p

DcR~Z, R!,

E d2k8du8 u82

3 exp@2Dc~u82 1 k82!t 2 jk • ~r 1 R! 2 ju8Z#.


We recognize in this second contribution the Fouriertransform of a Gaussian with

E du exp~2Dcu82t 2 ju8Z!

5 S p

Dct D1/2

expS 2Z2

4Dct D ,

E du8 u82 exp~2Dcu82t 2 ju8Z!

5 2S p

Dct D1/2 d2

dZ2expS 2

Z2

4Dct D5 2S p

Dct D1/2S Z2

~Dct !22

1

2Dct D expS 2Z2

4Dct D ,

E dk82 exp@2Dck82t 2 jk8 • ~R 1 r!#

5p

DctexpS 2

R2 1 r2 1 2R – r

4Dct D .

Note that the term R – r is the only one that breaks thecylindrical symmetry, so that writing d2R 5 RdRdf andR – r 5 Rr cos f, we can directly integrate over f:

E df expS j2jRr

4Dctcos f D 5 2pJ0S 2jRr

4Dct D .

Setting a 5 1/(4Dct) and putting all things togetherleads directly to Eq. (14).

APPENDIX DLet us consider the poles of

R 5 cosh~s1zb!

3D1s1 cosh@s1~l 2 z0!# 1 D2s2 sinh@s1~l 2 z0!#

D1s1 cosh@s1~l 1 zb!# 1 D2s2 sinh@s1~l 1 zb!#.

If we write l* 5 l 1 zb , these poles satisfy

coth@s1~sP!l* # 5 2D2sP

D1s1~sP!, (D1)

where we recall that

s1~sP! 5 FD2

D1sp

2 1 k2S 1 2D2

D1D 1

ma1 2 ma2

D1G1/2

.

It appears numerically that a first pole on the real axiscrosses the imaginary axis for a critical thickness of thefirst layer. Let us indeed set sP 5 0: Equation (D1) canhave a solution (for k 5 0) only when ma1 , ma2 . Inthat case, we have

cothF jlc* S ma2 2 ma1

D1D 1/2G 5 2j cotF lc* S ma2 2 ma1

D1D 1/2G

5 0,

leading to Eq. (18). Now if l* 5 lc* 1 dl, we expect thatsP ' 0 and s1(sP) ' s1(0), so that Eq. (D1) becomes

2sinh22@s1~0 !lc* #s1~0 !dl 5 2D2sP

D1s1~0 !

or

sP 5D1

D2s1

2~0 !dl 5p2D1

4lc*2D2

~l 2 lc!.

In the same manner, we can follow the effect of k by set-ting

s1~sp! ' s1~0 ! 1k2

2s1~0 !S 1 2

D2

D1D ,

2sinh22@s1~0 !lc* #k2

2s1~0 !S 1 2

D2

D1D lc*

5 2D2sP

D1s1~0 !⇒ sP 5

k2

2 S 1 2D1

D2D lc* .

Corresponding author Jean-Michel Tualle may bereached by e-mail, [email protected]; phone,33-1-49404092; or fax, 33-1-49403200.

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Asymptotic behavior and inverse problem in layered scattering media