546 J. Opt. Soc. Am. A/Vol. 22, No. 3 /March 2005 Ripoll et al.

Experimental determination of photonpropagation in highly absorbing

and scattering media

Jorge Ripoll

Institute of Electronic Structure and Laser, Foundation for Research and Technology–Hellas,Heraklion, Crete, Greece

Doreen Yessayan, Giannis Zacharakis,* and Vasilis Ntziachristos

Laboratory for Bio-optics and Molecular Imaging, Center for Molecular Imaging Research, Massachusetts GeneralHospital, Harvard Medical School, Charlestown, Massachusetts 02129

Received August 2, 2004; accepted September 22, 2004

Optical imaging and tomography in tissues can facilitate the quantitative study of several important chro-mophores and fluorophores. Several theoretical models have been validated for diffuse photon propagation inhighly scattering and low-absorbing media that describe the optical appearance of tissues in the near-infrared(NIR) region. However, these models are not generally applicable to quantitative optical investigations in thevisible because of the significantly higher tissue absorption in this spectral region compared with that in theNIR. We performed photon measurements through highly scattering and absorbing media for ratios of theabsorption coefficient to the reduced scattering coefficient ranging approximately from zero to one. We exam-ined experimentally the performance of the absorption-dependent diffusion coefficient defined by Aronson andCorngold [J. Opt. Soc. Am. A 16, 1066 (1999)] for quantitative estimations of photon propagation in the low-and high-absorption regimes. Through steady-state measurements we verified that the transmitted intensityis well described by the diffusion equation by considering a modified diffusion coefficient with a nonlinear de-pendence on the absorption. This study confirms that simple analytical solutions based on the diffusion ap-proximation are suitable even for high-absorption regimes and shows that diffusion-approximation-based mod-els are valid for quantitative measurements and tomographic imaging of tissues in the visible. © 2005Optical Society of America

OCIS codes: 290.1990, 170.3660, 110.7050.

1. INTRODUCTIONThe diffusion approximation (DA)1 to the radiative trans-port equation1–3 has been widely used for modeling pho-ton propagation within biological media in the near-infrared (NIR) region where high photon penetration canbe achieved because of the low absorption of tissue in the650–1000-nm range.4,5 The DA is a robust model in thislow-absorption range and has been widely used since it of-fers practical solutions with good accuracy. There aremany works that deal with the limits of the DA (see forexample Refs. 6 and 7) or with the accurate expression forthe diffusion coefficient D.1,5,8–24 However, except forsome theoretical and general work such as that presentedin Refs. 23, 25, and 26, most results that deal either withexperiments or simulations are in any case studying rela-tively low absorption values, the absorption coefficient mabeing almost an order of magnitude smaller than the re-duced scattering coefficient ms8 . In this regime, the dif-ferences between the absorption-dependent andabsorption-independent diffusion coefficients are verysubtle, with values generally lower than the experimentalerror. Hence the most commonly used approximationsfor the diffusion coefficient, namely, D 5

13 ms8 and

D 513 (ms8 1 ma), have furnished very good experimental

results in the NIR, both in phantoms and in vivo.

1084-7529/2005/030546-06$15.00 ©

In this work we concentrate on studying the use of theDA in limiting cases where the absorption coefficient be-comes comparable with the scattering coefficient. Thisstudy becomes of practical interest for in vivo optical im-aging in the visible since it could allow quantitative inves-tigations of several important chromophores and fluoro-phores that absorb in the 400–600-nm range. Ofparticular importance has become the quantitative imag-ing of bioluminescent markers and fluorescent proteinsbecause of their the immense implications for the study ofgene expression, cell migration, and drug efficacy in deepin vivo animal tissue.27–37 While penetration depth issignificantly lower in the visible compared with that inthe NIR, recent results suggest that imaging in the visiblein small animals is feasible for surface and subsurfaceinvestigations.33–35

A very attractive option for quantitative models of pho-ton propagation in the visible is to employ a modified dif-fusion coefficient that takes into account the photonpropagation differences between the visible and the NIR.This could allow the use of the same theoretical frame-work developed for the NIR to quantify and perform to-mographic imaging in the visible as well. We have foundthat a complete and manageable derivation for the diffu-sion coefficient is that presented in Ref. 25, where an

2005 Optical Society of America

Ripoll et al. Vol. 22, No. 3 /March 2005/J. Opt. Soc. Am. A 547

absorption-dependent expression is presented that isvalid for all values of ma . However, we realized that theexpression for D put forward in Ref. 25 may not be easilyapplied by those not familiar with the formal theoreticalformulation. Therefore we focused this work first on de-riving a more manageable expression for the diffusionequation that can be easily applied to the DA framework.Then we experimentally examined the limits of validity ofthis approach in regimes of increasing absorption. Webelieve that the work presented here will allow the appli-cation of DA models originally developed for the NIR tothe visible spectral window as well.

In Section 2 we present the absorption-dependent ex-pression for the diffusion coefficient for the isotropic andanisotropic scattering cases. In Section 3 we describe theexperiments and materials used. Results are presentedand discussed in Section 4. We conclude in Section 5.

2. ABSORPTION-DEPENDENT DIFFUSIONCOEFFICIENTSeveral expressions for the diffusion coefficient have beenderived in the past on the basis of different approximatesolutions of the transport equation.2,10,11,13–17,21–26,38–40

Making use of previous reported results, we intend topresent an expression for the diffusion coefficient that de-pends in a relatively simple manner on the three main op-tical properties used in in vivo experiments, namely, thereduced scattering coefficient ms8 , the absorption coeffi-cient ma , and the anisotropy factor g. We shall validatethis expression experimentally for regimes of both lowand very high absorption. For this derivation, we havefound that the expressions presented in Ref. 25 are easyto work with and yield accurate results when one uses theDA in the visible range.

Following Ref. 25, the correct derivation of D must beobtained from the value of the diffusion length, i.e., thedecay rate of the intensity in an infinite medium at latetimes compared with the transport mean free path.Within the DA the diffusion length Ld is defined as

Ld 5 AD/ma. (1)

A useful way to represent the absorption dependence is bywriting the diffusion coefficient as25

D 51

3~ms8 1 ama!, (2)

where a will generally depend on the absorption, scatter-ing, and anisotropy of the medium. In terms of the scat-tering coefficient ms , we recall that the reduced scatteringcoefficient is written as ms8 5 (1 2 g)ms . To derive thedifferent values for a we shall consider two main cases:(1) isotropic scattering ( g 5 0) and (2) anisotropic scat-tering ( g . 0).

A. Isotropic ScatteringFollowing Case et al.,39 the dispersion relation for Ld inthe case of isotropic scattering (i.e., g 5 0, ms 5 ms8) isgiven by

Ld

2lnS Ld 1 1/~ms 1 ma!

Ld 2 1/~ms 1 ma!D 5

1

ms. (3)

Values for Ld can easily be computed numerically fromEq. (3) and the corresponding value of D obtained fromEq. (1) as D 5 Ld

2ma . By analysis to first order the so-lution to Eq. (3) as in Ref. 25, the diffusion coefficient isfound to be

D 51

3~ms 1 0.2ma!, (4)

giving a value of a 5 0.2. We have checked numericallythe different values of a given by Eq. (3) for a large num-ber of values of absorption and scattering (for values ofma /ms from 0 to 1), and we have found that a 5 0.2 is agood approximation in general for the isotropic scatteringcase, showing deviations from this value of less than 5%.This expression, however, is not accurate when experi-ments in tissue are pursued because of the high anisot-ropy present [typical values of g in tissue range fromg 5 0.7 to 0.95 (Ref. 41)]. We shall therefore now con-sider the most general anisotropic scattering case.

B. Anisotropic ScatteringAn expression valid for the general anisotropic case wasderived in the late 1940s by Holte38 and presented in theform that we show here in Refs. 25 and 42. Since it is themost relevant form for in vivo experiments, we shall con-sider only the case of a Henyey–Greenstein phasefunction.43 In this case the hl variables presented in Ref.25 are expressed as hl 5 (2l 1 1)@1 2 glms /(ms 1 ma)#,where gl is the anisotropy factor to the power of l [notethat h0 5 ma /(ms 1 ma) and h1 5 3(ms8 1 ma)/(ms 1 ma)]. The expression for D with the expansiontruncated up to third order may be found by substitutingthis expression for hl in Eq. (14) of Ref. 25, bearing inmind that the diffusion length is expressed in this work inmean-free-path units [i.e., the variable n in Ref. 25 isn 5 Ld(ms 1 ma)]. This expression will not be presentedhere for the sake of clarity. However, a very simple ex-pression for D and a is reached if we truncate to firstorder the original expression of Holte,38,42 namely

D 51

3~ms8 1 ma!F1 2

4

5

ma

ms8~1 1 g ! 1 maG21

. (5)

Note that here we have chosen to express D in terms ofthe reduced scattering coefficient ms8 , which is the rel-evant quantity in scattering experiments in anisotropicmedia. One main difference between Eq. (5) and mostcommonly used expressions (namely, with a 5 0 ora 5 1) is that in this case the value of the anisotropy fac-tor g must be known a priori. From Eq. (5) we see thatthe diffusion coefficient depends nonlinearly on the scat-tering and the absorption properties of the medium. Theexpression for a is found from Eq. (5) and Eq. (2) as

a 5 1 24

5

ms8 1 ma

ms8~1 1 g ! 1 ma

. (6)

We have found that the deviation in the value of a whenconsidering higher-order contributions [i.e., Eq. (14) of

548 J. Opt. Soc. Am. A/Vol. 22, No. 3 /March 2005 Ripoll et al.

Ref. 25] in the derivation of D is less than 2%. Typicalvalues of a range from 0.2 to 0.6, being of the order of 0.5–0.55 in the case of tissue in the visible, assuming an an-isotropy factor of g ; 0.8. Note that for the isotropicscattering case g 5 0 we recover in Eq. (6) the a 5 0.2value obtained in Eq. (4). If the classical description ofthe diffusion coefficient is used (namely, with a 5 0 or a5 1), the diffusion approximation yields inaccurate re-sults in the cases of high absorption, as will be show be-low.

3. MATERIALS AND METHODSWe studied the performance of the absorption-dependentexpression of the diffusion coefficient, Eq. (5), with experi-ments performed with constant illumination. The ab-sorption coefficient values studied herein range approxi-mately from ma 5 0.005 cm21 to 7 cm21, simulating bloodabsorption in the NIR and in the visible,28 respectively.Two reduced scattering coefficient values were used,namely, ms8 5 6 cm21 and 12 cm21. To validate Eq. (5)experimentally we performed successive titrations ofknown amounts of ink in a medium of known scatteringproperties and spatially resolved the transmitted lightthrough a slab geometry. We then fitted the collected in-tensity data with the solution to the diffusion equation fora slab by using the method of images.5 Both measure-ments and numerical results are reported with referenceto the propagation of photons emitted by isotropic, point-like sources. In the following we detail the experimentalsetup and procedures used for these measurements.

A. Experimental SetupWe employed a previously described tomographic scannerdeveloped for small animal imaging.32 The setup con-sists of a slablike container, which we filled with an ab-sorbing and scattering liquid. A set of 32 100-mm multi-mode fibers (Thorlabs Inc., Newton, N.J.) are mounted ona sliding plate so that the slab thickness can be variable.The plate was coated with a highly absorbing black paintto reduce unwanted reflections. Light propagating in thechamber is collected in transmittance through a glasswindow, through which we focused a CCD camera (RoperScientific, Princeton Instruments, Trenton, N.J.) using aNikkor, 60-mm f/2.8 lens (Nikon, Melville, N.Y.). Thelight source employed was a 672-nm cw diode laser (B&WTek, Newark, Del.) with adjustable light power reaching amaximum of 100 mW. The incidence signal is distributedto the illuminating fibers via a programmable opticalswitch (Dicon FiberOptics, Berkeley, Calif.). For most re-sults presented, a single fiber position located at the cen-ter of the imaging area was used.

For all the experiments performed, laser power waskept constant. To maximize the signal-to-noise ratio overthe large dynamic range of photon intensities considered,appropriate neutral-density filters were placed in front ofthe CCD camera lens and fine-tuned by adjusting the ex-posure time at each measurement.

B. Scattering and Absorbing MediaTo control the scattering properties of our medium, weemployed a diluted Intralipid–water solution (FreseniusKabi Clayton, L.P., Clayton, N.C.) using different concen-

trations, 1% and 0.5%. The absorbing properties of thesolution were changed by adding appropriate amounts ofIndia ink. Two different sets of measurements were con-sidered. The first set consisted of a total of 12 ink titra-tions ranging from 0 to 100 ppm that were added to a so-lution with constant Intralipid concentration of 0.5%,with expected reduced scattering coefficient ofms8 ; 6 cm21. The second set consisted of repeating theprevious experiment for Intralipid concentrations of 1%,which yields an expected reduced scattering coefficient ofms8 ; 12 cm21. The optical properties of the dye-free, In-tralipid solution were determined from time-resolved andsteady-state measurements, calibrating the relationshipbetween the concentration of Intralipid and the reducedscattering coefficient. The relationship between the con-centration of India ink and the absorption coefficient wascontrasted with a calibrated U3000 Hitachi Instrumentsspectrophotometer. In all cases presented, the anisot-ropy factor used was g 5 0.8.44

C. Data FittingFor the values of ms8 considered, the diffusing mediumwas sufficiently wide to act as an infinitely extended me-dium in the x and y directions, even for the smallest val-ues of ma . The thickness of the plate was fixed atL 5 1 cm. The theory used was the solution to the DA ina slab geometry by the method of images5 with 101 imagedipoles to ensure convergence. The boundary conditionswere applied through an extrapolated distance lext de-fined as45

lext 5 CndD, (7)

where the coefficient Cnd takes into account the differencein index of refraction45 between the Intralipid solutionwith index of refraction n 5 1.33 and air, yielding valuesof Cnd 5 5 (note that in the index-matched caseCnd 5 2). Note further that in the high-absorption casethe extrapolated distance will also be absorption-dependent. The expression for D used in Eq. (7) and inthe slab solution to the DA was that given in Eq. (5).Since we are dealing with cw measurements, only onequantity may be extracted from the fitting procedure. Inall cases we assumed that the scattering properties of themedium were known, and thus fitted the best value of mawith a multidimensional, unconstrained, nonlinear mini-mization algorithm (Nelder–Mead simplex method46). Inall cases, excellent agreement between the fit and the ex-periment was reached.

4. RESULTS AND DISCUSSIONFigure 1 depicts results from the fitting process, plottingthe retrieved ma versus the ink concentration in parts permillion by use of three different expressions for the diffu-sion coefficient presented, namely, the absorption-dependent Eq. (5), the expression D 5

13 (ms8 1 ma)

(a 5 1), and the absorption-independent expressionD 5

13 ms8 (a 5 0). A linear fit to the results obtained

with Eq. (5) is also plotted. All measurements shown areretrieved at constant scattering background(ms8 5 6 cm21).

Ripoll et al. Vol. 22, No. 3 /March 2005/J. Opt. Soc. Am. A 549

As may be seen from Fig. 1, all expressions give a lin-ear dependence for low values of absorption up toma , 0.3ms8 . As mentioned in Section 1 most of the pre-viously reported experimental work on the subject ex-plored only this domain and compared the experimentalresults with the solutions of the diffusion equation ob-tained with D for values of a 5 0 and a 5 1 [see Eq. (2)].However, for high absorption the values obtained with theabsorption-independent expression clearly deviate non-linearly from the expected ones, yielding errors as high as70%. As absorption increases it is expected that this er-ror will also increase nonlinearly. The values obtainedwhen using a 5 1 show a much smaller deviation thanwith a 5 0, but still clearly deviate from a linear depen-dence. On the other hand the absorption-dependent ex-pression proposed here and in Ref. 25 retains accuracyeven for absorption values that are higher than the re-duced scattering coefficient. This result supports the va-lidity of the expression for D shown in Eq. (5) for allranges of absorption studied (ma 5 0.02 cm21 to 7 cm21)in the presence of anisotropic scattering ( g 5 0.8 in thiscase). Values of a obtained in these measurements [seeEqs. (2) and (6)] ranged from a 5 0.555 to 0.52. Resultssimilar to those presented in Fig. 1 were obtained for thems8 5 12-cm21 case, and are summarized in Table 1.There the deviations from the linear dependence are

Fig. 1. Retrieved absorption coefficient versus ink concentrationfor a slab of width 1 cm with ms8 5 6 cm21 for the casesa(ma , ms8) [see Eq. (6)] (solid circles), a 5 0 [see Eq. (2)](squares), and a 5 1 [see Eq. (2)] (triangles). The solid curverepresents a linear fit of the a(ma , ms8) data.

smaller as a result of the smaller ratio of absorption to re-duced scattering coefficient achieved.

The deviation from the expected linear dependence ofthe retrieved absorption coefficient versus the ink titra-tions is presented in Table 1. The findings tabulated arethe result of fitting the data to a linear expression of theform ma 5 B1@ink#, where @ink# is the ink concentration,and to a second-order polynomial of the form ma 5 A1 B1@ink# 1 B2@ink#2. The second-order coefficient B2in this case reveals the curvature of the data and is there-fore indicative of the nonlinearity present in the recov-ered absorption values. For the linear-fit case wepresent the standard deviation (STD) from the fit and thecorrelation coefficient r. In the second-order-fit case wepresent the factor B2 . These results are presented forthe same cases exposed in Fig. 1, namely, the absorption-and scattering-dependent a shown in Eq. (6) for a 5 0and for a 5 1 in the ms8 5 6-cm21 and ms8 5 12-cm21

cases. These results are a clear indication of the validityof Eq. (6), retrieving very accurate linear fits for the de-pendence between ink titration and absorption coefficient.

As mentioned in Section 2 one drawback of using Eq.(6) is that the anisotropy factor g must be known a priori.This factor is generally not known, especially when per-forming optical tomography in vivo. However, what isknown is that the values of g expected in tissue rangefrom g 5 0.7 to 0.95.41 To study the effect that an over-estimation or underestimation of g will have in the re-trieved absorption coefficient, we plot in Fig. 2 the devia-tion from the fitted values for g 5 0.8 in thems8 ; 12-cm21 case with g 5 0.7, g 5 0.9, or g 5 0.95.As a reference we also include the STD of the data whenusing Eq. (6) for seven different source positions [the cen-ter source and six sources equidistant (0.35 cm) from thecenter] as a measure of the error in retrieving the absorp-tion coefficient. This figure clearly shows that the errorintroduced when using a different value of g that is stillwithin the values found in tissue is much smaller thanthe experimental error obtained when fitting for the ab-sorption coefficient.

5. CONCLUSIONSBy using a highly sensitive experimental setup, we couldaccurately measure intensities in low-absorbing andhighly absorbing media. We explored variations of up toma ' ms8 by use of ink titrations in a medium with knownscattering properties. Fitting results that we obtainedby use of the modified diffusion coefficient of Eq. (5) vali-

Table 1. Fitting Resultsa

a

ms8 . 6 cm21 ms8 . 12 cm21

r STD B2 r STD B2

a(ma , ms8) 0.99985 0.04304 20.0033 0.99976 0.06822 20.0008a 5 0 0.99913 0.47791 0.036 0.99892 0.16832 0.008a 5 1 0.99628 0.09322 20.0085 0.99925 0.06924 20.005

a Results for fitting the retrieved values versus the ink concentration to a linear fit ma 5 B1@ink# or to a second-order polynomial ma 5 A 1 B1@ink#1 B2@ink#2. STD represents the standard deviation of the data from the linear fit and r is the correlation coefficient. B2 is an indication of the curvatureof the data, i.e., the nonlinearity present in the retrieved absorption values.

550 J. Opt. Soc. Am. A/Vol. 22, No. 3 /March 2005 Ripoll et al.

dated the use of the diffusion approximation in situationsof high absorption. The main conclusion of this experi-ment is that the diffusion equation is satisfactorily accu-rate for modeling light propagation through homoge-neous, highly absorbing, highly anisotropic mediaprovided that an absorption-dependent expression for thediffusion coefficient of the type D 5

13 @ms8 1 ama# is used.

We have shown that the expression for D presented hereis an excellent approximation even when high absorption(ma ' ms8) and high anisotropy ( g 5 0.8) are considered.Even though this expression for the diffusion coefficientneeds a priori knowledge of the anisotropy factor, we haveshown that the variation introduced for the range of val-ues found in tissue is much smaller than the experimen-tal error introduced when retrieving the absorption coef-ficient. On the basis of the results presented, we believethat use of a general value g 5 0.8 will not impair the ac-curacy of the absorption-dependent expression for D,while retaining its simplicity.

It is interesting to note that the experiments confirmthat any value of a can be interchangeably used for ab-sorption coefficients of up to 1 cm21 and 1.5 cm21 corre-sponding to ma /ms8 ratios of 1:7 and 1:5, respectively.However, the exact selection of a will play a role in quan-tification for higher absorption values (i.e., greater ma /ms8ratios). Our findings indicate that quantification errorsas high as 70% or even more are to be expected in thepresence of high absorption if not using the absorption de-pendent coefficient of Eq. (5) is not used but previouslydefined solutions for D are used instead. From prelimi-nary time-resolved studies (results not shown) we haveseen that these results are general and will also hold fortime-resolved experiments. We intend now to translatediffuse optical tomography principles47 for performing to-mography in the visible wavelength range by using the

Fig. 2. Variation of the retrieved absorption coefficient for dif-ferent anisotropy values ( g 5 0.7, 0.8, 0.9, and 0.95) versus inkconcentration for a slab of width 1 cm with ms8 5 12 cm21. Re-sults presented correspond to the difference between g 5 0.7 andg 5 0.8 (squares), g 5 0.9 and g 5 0.8 (solid circles), andg 5 0.95 and g 5 0.8 (triangles). For comparison we show thestandard deviation when averaging the retrieved absorption co-efficient for seven different source positions (solid squares) in theg 5 0.8 case.

absorption-dependent diffusion coefficient presented here.We believe that the implications of this work will be of im-portance, specifically in the area of tissue imaging andcharacterization in the visible.

ACKNOWLEDGMENTSThis research was supported in part by the National In-stitutes of Health grants RO1 EB 000750-1 and R33 CA91807. J. Ripoll acknowledges support from EuropeanUnion Integrated Project Molecular Imaging LSHG-CT-2003-503259.

J. Ripoll and V. Ntziachristos may be reached bye-mail at jripoll@iesl.forth.gr and vasilis@helix.mgh.harvard.edu, respectively.

*Present address, Institute of Electronic Structure andLaser, Foundation for Research and Technology–Hellas,Heraklion, Crete, Greece.

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