Optical scattering properties ofsoft tissue: a discrete particle model

Joseph M. Schmitt and Gitesh Kumar

We introduce a micro-optical model of soft biological tissue that permits numerical computation of theabsolute magnitudes of its scattering coefficients. A key assumption of the model is that the refractive-index variations caused by microscopic tissue elements can be treated as particles with sizes distributedaccording to a skewed log-normal distribution function. In the limit of an infinitely large variance in theparticle size, this function has the same power-law dependence as the volume fractions of the subunitsof an ideal fractal object. To compute a complete set of optical coefficients of a prototypical soft tissue~single-scattering coefficient, transport scattering coefficient, backscattering coefficient, phase function,and asymmetry parameter!, we apply Mie theory to a volume of spheres with sizes distributed accordingto the theoretical distribution. A packing factor is included in the calculation of the optical cross sectionsto account for correlated scattering among tightly packed particles. The results suggest that the skewedlog-normal distribution function, with a shape specified by a limiting fractal dimension of 3.7, is a validapproximation of the size distribution of scatterers in tissue. In the wavelength range 600 # l # 1400nm, the diameters of the scatterers that contribute most to backscattering were found to be significantlysmaller ~ly4–ly2! than the diameters of the scatterers that cause the greatest extinction of forward-scattered light ~3–4l!. © 1998 Optical Society of America

OCIS codes: 170.6930, 350.4990, 290.0290.

1. Introduction

A growing number of applications of optical imagingand spectroscopy in medicine rely on measurement ofthe elastic scattering properties of tissue to detectunderlying pathology.1–4 Although it is recognizedthat the optical properties of tissue are related to itsmicrostructure and refractive index, the nature of therelationship is still poorly understood. Previous in-vestigations have focused on various aspects of thisrelationship, including the contribution of mitochon-dria to the scattering properties of the liver,5 thespatial variations in the refractive index of cells andtissue sections,6–8 and the diffraction properties ofsingle cells.9,10 Still lacking, however, is a quantita-tive model that relates the microscopic properties ofcells and other tissue elements to the scattering co-

Joseph M. Schmitt is with the Department of Electrical andElectronic Engineering, Hong Kong University of Science andTechnology, Clear Water Bay, Kowloon, Hong Kong. Gitesh Ku-mar is with the Perkin-Elmer Corporation, 50 Danbury Road,Wilton, Connecticut 06897-0208.

Received 18 August 1997; revised manuscript received 2 Decem-ber 1997.

0003-6935y98y132788-10$15.00y0© 1998 Optical Society of America

2788 APPLIED OPTICS y Vol. 37, No. 13 y 1 May 1998

efficients of bulk tissue. Ideally, such a modelshould be able to predict the absolute magnitudes ofthe optical scattering coefficients as well as theirwavelength and angle dependencies. Moreover, toassist in future efforts to invert measured data, themodel should give insight into how the scatteringproperties are influenced by the numbers, sizes, andarrangement of the tissue elements.

This paper presents a framework for a particulatemodel of soft tissue that satisfies at least a few ofthese requirements. We identify the major ele-ments in soft tissue responsible for the microscopicvariations in its refractive index and, to facilitatenumerical computations, treat the variations as dis-crete particles with statistically equivalent refractiveindices. A skewed form of the log-normal distribu-tion function is introduced to represent the size dis-tribution of the particles. In the limit of an infinitelylarge variance in the particle size, this distributionfunction describes the volume fractions of the sub-units of an ideal fractal object. Using Mie theoryapplied to a volume of spheres with diameters dis-tributed according to the theoretical distribution, wecalculate the wavelength dependence of the single-scattering coefficient, the transport-corrected scatter-ing coefficient, and the asymmetry parameter of atissue model. In addition to these more common op-

tical constants, the phase function and backscatter-ing coefficient are also calculated to study theinfluence of the microscopic composition of tissue onits angular scattering properties. Finally, we eval-uate a complete set of optical coefficients of soft tis-sues containing different proportions of connectivetissue fibers and make general observations aboutthe sizes of scatterers in tissue responsible for atten-uation and backscattering of a propagating lightbeam. The results suggest that the skewed log-normal distribution function, with a shape specifiedby the limiting fractal dimension, is a valid approxi-mation of the distribution of sizes of scatterers intissue.

2. Theory

Soft tissue is composed of tightly packed groups ofcells entrapped in a network of fibers through whichwater percolates. Viewed on a microscopic scale, theconstituents of the tissue have no clear boundaries.They appear to merge into a continuous structuredistinguished optically only by spatial variations inthe refractive index. To model such a complicatedstructure as a collection of particles, it is necessary toresort to a statistical approach.

A. Refraction-Index Variations: Particle Approximation

The results of earlier studies suggest that the tissueelements that contribute most to the local refractive-index variations are the connective tissue fibers ~bun-dles of elastin and collagen!, cytoplasmic organelles~e.g., mitochondria!, and cell nuclei.7 Figure 1 is ahypothetical index profile formed by measuring therefractive indices along a line in an arbitrary direc-tion through a volume of tissue composed of theseelements. The widths of the peaks in the index pro-file are proportional to the diameters of the elements,and their heights depend on the refractive index ofeach element relative to that of its immediate envi-ronment. Our goal here is to model the origin of theindex variations by a statistically equivalent volumeof discrete particles having the same index but dif-ferent sizes. The index profiles in Fig. 1 illustratethe nature of the approximation implied by thismodel.

We first define the average background index as

Fig. 1. Spatial variations of the refractive index of a soft biologicaltissue. A hypothetical index profile through several tissue ele-ments is shown along with the profile through a statistically equiv-alent volume of homogeneous particles. The indices of refractionlabeling the profiles are defined in Subsection 2.A.

the weighted average of refractive indices of the cy-toplasm and the interstitial fluid, nc and ns, as

n# bkg 5 fc nc 1 ~1 2 fc!ns, (1)

where fc is the fraction of the fluid in the tissue con-tained inside the cells. Estimated from the dis-solved fractions of proteins and carbohydrates in theintracellular and interstitial fluids, nc and ns are ap-proximately 1.36 and 1.34, respectively.9 Since ap-proximately 60% of the total fluid in soft tissue iscontained in the intracellular compartment, it followsfrom Eq. ~1! that n# bkg 5 0.4~1.34! 1 0.6~1.36! 5 1.352.Next we define the refractive index of a particle as thesum of the background index and the mean indexvariation,

n# p 5 n# bkg 1 ^Dn&, (2)

which can be approximated by another volume-weighted average,

^Dn& 5 ff~nf 2 ns! 1 fn~nn 2 nc! 1 f0~n0 2 nc!. (3)

Here subscripts f, n, and o refer to the fibers, nuclei,and organelles, which were identified above as themajor contributors to index variations. The terms inparentheses in this expression are the differencesbetween the refractive indices of the three types oftissue element and their respective backgrounds; themultiplying factors are the volume fractions of theelements in the solid portion of the tissue. The re-fractive index of the connective-tissue fibers is about1.47, which corresponds to 55% hydration of collagen,its main component.11 The nucleus and the cyto-plasmic organelles in mammalian cells that containsimilar concentrations of proteins and nucleic acids,such as mitochondria and the ribosomes, have refrac-tive indices that lie within a relatively narrow range~1.39–1.42!.12,13 However, other cytoplasmic inclu-sions, particularly pigment granules, can have muchhigher refractive indices.9 In this study we assumednn 5 n0 5 1.40 so that fn and f0 need not be distin-guished and Eq. ~3! can be written in a simpler formas

^Dn& 5 ff~nf 2 ns! 1 ~1 2 ff!~nn 2 nc!. (4)

This equation expresses the mean index variation interms of the fibrous-tissue fraction ff alone. Colla-gen and elastin fibers comprise approximately 70% ofthe fat-free dry weight of the dermis,14 45% of theheart,15 and 2–3% of the nonmuscular internal or-gans.16 Therefore, depending on tissue type, ff maybe as small as ;0.02 or as large as 0.7. For nf 2 ns5 1.47 2 1.34 5 0.13 and nn 2 nc 5 no 2 nc 5 1.40 21.36 5 0.04, the mean index variations that corre-spond to these two extremes are, according to Eq. ~4!,^Dn& 5 0.02~0.13! 1 ~1 2 0.02!~0.04! 5 0.042 and ^Dn&5 0.7~0.13! 1 ~1 2 0.7!~0.04! 5 0.10. Thereforethese calculations suggest that the mean amplitudeof the index variations in soft tissue lies between 0.04and 0.10.

1 May 1998 y Vol. 37, No. 13 y APPLIED OPTICS 2789

B. Particle Size Distribution

Having defined the refractive indices of the scatteringparticles, we are left with the task of determiningtheir size distribution. The scattering centers in bi-ological tissue span a wide range of dimensions andtend to aggregate into complex forms suggestive offractal objects.17,18 Recently, the fractal propertiesof tissue have been modeled by a distribution of par-ticles whose number densities vary according to apower law.6,19 Although a power-law distribution isthe appropriate description of an ideal fractal object,the density functions of real objects rarely fit thesame power law over more than a couple of decades.20

In this study, we chose to employ a skewed logarith-mic function to represent the distribution of sizes ofparticles in tissue. This function is more plausibleon physical grounds, yet retains the essential char-acter of the fractal description.

Written in its generalized form, the skewed loga-rithmic distribution function is21

f ~x! 5 Cm xm expF2~ln x 2 ln xm!2

2sm2 G , (5)

where x is the distributed variable ~which can bechosen to be the diameter, area, or volume of parti-cles!, Cm is a normalizing factor, and the quantitiesxm and sm set the center and width of the distribu-tion, respectively. For m 5 21 and m 5 0, respec-tively, Eq. ~5! is called the logarithmic normaldistribution and zeroth-order logarithmic distribu-tion, respectively.21 Both distributions are used ex-tensively in particle-size analysis. Here we treatf ~x! as the volume fraction of particles of diameter dand rewrite Eq. ~5! in terms of a new set of variables:

h~d! 5Fv

Cmd32Df expF2

~ln d 2 ln dm!2

2sm2 G , (6)

with

Cm 5 smÎ2p dm42Df exp@~4 2 Df!

2sm2 y2#,

where h~d! is the volume fraction of particles with adiameter d and Fv is the total volume fraction of theparticles, given by

Fv 5 *0

`

h~d!dd. (7)

To establish a connection between the ideal fractaldistribution used in previous studies,6,7,19 we havewritten the exponent in Eq. ~6! in terms of the ~volu-metric! fractal dimension Df. Expressed in this way,the distribution takes on increasingly scale-invariant, or fractal, characteristics as its broadens.The parameter sm sets the fractal range over whichthe log–log slope of the distribution is approximatelyconstant. In the limit of an infinitely broad distri-bution of particle sizes,

limsm3`

h~d! < d32Df. (8)

2790 APPLIED OPTICS y Vol. 37, No. 13 y 1 May 1998

For 3 , Df , 4, this power-law relationship describesthe dependence of the volume fractions of the sub-units of an ideal mass fractal on their diameter d.22

For Df equal to 3 ~the Euclidean volume dimension!,Eq. ~6! reduces to the logarithmic normal distributiongiven by Eq. ~5!, with m 5 21.

C. Optical Coefficients.

If it is assumed that the waves scattered by the in-dividual particles in a thin slice of the modeled tissuevolume add randomly, then the scattering coefficientof the volume can be approximated as the sum of thescattering coefficients of the particles of a given di-ameter,

ms 5 (i51

M

ms~di!, (9)

where

ms~di! 5h~di!

vis~di!

and M is the number of particle diameters; ss~di! isthe optical cross section of an individual particle withdiameter di and volume vi. The volume-averagedphase function of the tissue slice is the sum of theangular-scattering functions of the individual parti-cles weighted by the product of their respective scat-tering coefficients,

P~u! 5(i51

M

ms~di!Pi~u!

(i51

M

ms~di!

. (10)

Similarly, the asymmetry parameter, a measure ofthe anisotropy of light scattering within the tissue, isgiven by

g 5(i51

M

ms~di!gi~di!

(i51

M

ms~di!

, (11)

where

gi~di! 5 *21

1

cos uPi~cos u!d~cos u!

is the mean cosine of the scattering angle of an indi-vidual particle of diameter di. The transport-corrected scattering coefficient, which is usedextensively to model optical diffusion in thick tissues,is defined in terms of the asymmetry parameter as mst5 ms~1 2 g!. The volume-averaged backscatteringcoefficient, a variable used in earlier studies to char-acterize the reflectivity of microscopic samples,23 isdefined here as the sum of the particle cross sections

weighted by their angular-scattering functions eval-uated at 180°,

mb 5 (i51

M h~di!

viss~di!Pi~180°!. (12)

Expressed in this manner, mb has units of mm21ysr,so that the product of mb and the thickness of thetissue slice yield the fraction of the incident irradi-ance backscattered per unit solid angle in the direc-tion opposite to the incident light.

D. Correlated Scattering

In the calculation of the total scattering concentra-tion of a mixture of particles, the usual assumption isthat the particles scatter independently. However,because correlated scattering is likely in fractal me-dia to be characterized by a high concentration ofsmall particles, this assumption may be violated intissues characterized by a high fractal dimension.Correlated scattering among particles with volumefractions higher than ;1% has been found to reducethe total scattering cross section when the diametersof the particles are much less than a wavelength.24

In contrast, correlated scattering has been found tobe insignificant among particles with diameterslarger than a wavelength for volume fractions below;10%.24 To account for interparticle correlation ef-fects, Twersky25 derived an expression for the pack-ing factor of a medium filled with a volume fraction hof subwavelength-diameter spheres:

Ws 5~1 2 h!4

~1 1 2h!2 . (13)

Ws can be regarded as the fraction by which the bulkoptical cross section diminishes as a result of corre-lated scattering among the spheres. Recently, Bas-com and Cobbold26 modified Eq. ~13! to account forpacking in a medium composed of scatterers withdifferent shapes. Their modified packing fraction is

Wp 5~1 2 h!p11

@1 1 h~p 2 1!#p21 , (14)

where p is a packing dimension that describes therate at which the empty space between scatterersdiminishes as the total number density increases.Although related, the packing and the fractal dimen-sions need not be the same. Packing of sphericalparticles is described well by a packing dimensionp 5 3, in which case Eq. ~14! reduces to Eq. ~13!. Onthe other hand, the packing of sheetlike and rod-shaped particles is characterized by packing dimen-sions that approach 1 and 2, respectively. Theelements of tissue have all of these different shapesand may exhibit cylindrical and spherical symmetrysimultaneously. For a medium of this type, thepacking dimension may lie anywhere between 1 and5.26 Adopting Eq. ~14! as a general description of thepacking fraction, we approximate the effective vol-

ume fraction of scatterers with the same diameter das

h9~d! 5 Wph~d! 5~1 2 h~d!!p11

@1 1 h~d!~p 2 1!#p21 h~d!.

(15)

When we calculate the optical coefficients using Eqs.~9!–~12!, the correlation-corrected distribution h9~d!replaces h~d!.

3. Numerical Evaluation of Optical Properties

The expressions developed in Subsection 2.C wereevaluated numerically to determine whether themodel yields credible estimates of the phase functionand the optical coefficients of tissue. Because of thewide range of particle diameters over which the sum-mations in Eqs. ~9!–~12! had to be evaluated, it wasnot practical to employ conventional discretized inte-gration techniques. Instead, we used ten meansphere diameters ranging from 50 nm to 25.6 mm inpowers of two to represent the particles in the tissuemodel and their distributions. A Gaussian distribu-tion ~FWHM 5 0.1di! of sphere sizes centered on themean diameters di was used to reduce the interfer-ence structure of the individual spheres. Figure 2illustrates the shape of the continuous distributionsfor two values of Df ~3.7 and 3.2! and the method bywhich the distributions were approximated by theten-sphere diameters. We chose to use spheres be-cause this shape is consistent with the statisticallyisotropic nature of the refractive-index model andpermits the application of Mie theory for calculationof the optical coefficients. However, it should be rec-ognized that tissues that contain aligned fiber layersare not represented well by an isotropic model be-cause their optical properties depend on the directionin which they are measured.

Fig. 2. Distribution of the volume fractions of spheres to whichMie theory was applied to calculate the scattering properties oftissue. Solid and dashed curves are the continuous correlation-corrected distributions given by Eq. ~15! for Df 5 3.7 and Df 5 3.2.Dotted curves are the distributions of spheres with diameters in-creasing in powers of two from 50 nm to 25.6 mm that represent theDf 5 3.7 distribution. The partial volume fractions of the sphereswere adjusted to conform to h9~d!, with their total volume fractionequal to Fv.

1 May 1998 y Vol. 37, No. 13 y APPLIED OPTICS 2791

For most of the simulations, the refractive indicesof the spheres and background medium were setequal to 1.42 and 1.352, respectively, to model a tis-sue with a refractive-index variation in the middle ofthe range calculated in Subsection 2.A. We adjustedthe volume fractions of the spheres to scale accordingto the skewed logarithmic distribution @Eq. ~6!#. Thetotal volume fraction occupied by the spheres wasestimated as Fv 5 1 2 Fw 2 Fpp, where Fw is theweight fraction of water in the tissue and Fpp is thefraction of organic solids in the combined interstitialfluid and cytoplasmic spaces. We chose Fw 5 0.75and Fpp 5 0.05 as representative of human soft tis-sue27 and set Fv 5 0.2 to obtain the results givenbelow in Section 4. The width parameter sm wasfixed at 2.0 to match the fractal scaling range ob-served in earlier experiments ~about one decade!,6with dm set equal to the geometric mean of the min-imum and maximum sphere diameters in microme-ters, dm 5 @~0.05!~25.5!#1y2 5 1.13. The correlation-corrected distribution @Eq. ~15!# with p 5 3, thepacking dimension for spheres, was applied in thenumerical calculations of the cross sections. Tocompute the bulk optical coefficients of the modeledtissue, we evaluated the optical cross sections in Eqs.~9!–~12! in Subsection 2.C numerically, using a Mie-scattering algorithm ~adapted from Ref. 28!.

4. Results and Discussion

A. Angular Scattering Function

Angular scattering functions of a typical soft tissuerepresented by the ten-sphere numerical model areplotted in Fig. 3. The magnitudes are plotted with-out normalization for a range of fractal dimensions ata fixed wavelength ~l 5 633 nm!. For comparison,the angular-scattering functions of brain and musclemeasured experimentally by Flock et al.29 and Van

Fig. 3. Angular-scattering functions calculated with the ten-sphere tissue model for Df 5 1, 3, 4, and 6. Measurements of theangular scattering functions of muscle and brain that were takenfrom published studies are shown for comparison. Model pa-rameters: n# bkg 5 1.352, n# p 5 1.42, Fv 5 0.2, dm 5 1.13, sm 5 2,p 5 3.

2792 APPLIED OPTICS y Vol. 37, No. 13 y 1 May 1998

der Zee et al.30 are shown in the same figure. Withthe magnitude of the ratio between the indices of theparticles and background fixed, the shape of P~u! at agiven wavelength is determined primarily by Df. AsDf increases, the contribution of smaller particles in-creases, leading to a shift in the scattered power fromthe forward to the backward hemisphere.

Our results substantiate the observation of Gele-bart et al.19 that the phase function of a volume ofspheres with a narrow distribution of diameters is apoor representation of the phase function of tissue.A wide range of sizes of scatterers must be used in themodel to fit the shape of experimentally measuredphase functions. For Df in the range 3.5–3.9, theshape of P~u! matches that of the phase functions ofthe brain and muscle well ~Fig. 3!. The only majordiscrepancy appears at angles close to the exact back-scattering direction ~u 5 180°!. At these angles, theamplitude of measured phase functions increasessharply, whereas the model curves remain almostflat. This sharp increase may be caused by the glo-ry,31 a phenomenon associated with very-large-sizescatterers ~.100 mm! not included in our numericalmodel, or it may be an experimental artifact causedby incompletely suppressed reflections at the surfaceof the tissue specimen. The similarity in the shapesof the experimental and the theoretical curvesthroughout most of the angular range supports thesupposition that particle sizes in tissue are distrib-uted like a fractal with a dimension in the range 3 ,Df , 4. Values of Df far outside of the fractal rangegive unrealistically flat or peaked phase functionscharacteristic of monodisperse distributions of Ray-leigh or Mie scatterers ~see curves Df 5 6 and Df 5 1in Fig. 3!.

B. Asymmetry Parameter

Figure 4 shows the computed magnitudes of the

Fig. 4. Magnitude and wavelength dependence of the asymmetryparameter of a model tissue for values of the limiting fractal di-mension Df between 3 and 4. The dependence is weakest for lowvalues of Df because large particles contribute more to the totalcross section. Model parameters: n# bkg 5 1.352, n# p 5 1.42, Fv 50.2, dm 5 1.13, sm 5 2, p 5 3.

asymmetry parameter g over the range of wave-lengths between 600 and 1300 nm. The magnitudeof g of the model tissue is nearly independent of wave-length for Df 5 3 and has a weak dependence onwavelength for the higher values of Df shown. Mostof the experimental measurements of g that havebeen made on biological tissues indicate that g ex-ceeds 0.9 in the visible and near-infrared spectralregions.32 Our model predicts g . 0.9 in these spec-tral regions for fractal dimensions below approxi-mately 3.7.

C. Scattering Coefficients

The scattering coefficients predicted by the tissuemodel are plotted in Fig. 5 as a function of wave-length for three different values of Df between 3 and4. As expected, the magnitude of ms increases as thefractal dimension decreases because the larger par-ticles, which have the largest optical cross sections,contribute relatively more to the total optical crosssection of the tissue. Our results show that fixingthe total volume fraction of particles and their refrac-tive indices places upper and lower bounds on themagnitude of the scattering coefficient. The modelpredicts that a tissue consisting of a narrow distribu-tion of large particles ~modeled by choosing smallvalues for Df and sm, and a large value for dm! wouldhave the highest value of ms in the visible spectralrange; a medium containing a narrow distribution ofparticles sizes with its mean skewed toward smalldiameters ~Df, sm large, and dm small! would havethe lowest value.

We observed that the scaling law for particle sizesused in the model results in a remarkably simpledependence of the total scattering coefficient on thewavelength: ms ; l22Df. The exponent of thispower law is one less than the limiting fractal dimen-sion of the volume-fraction distribution @Eq. ~6!#.

Fig. 5. Magnitude and wavelength dependence of the scatteringcoefficient ms of a model tissue for values of the limiting fractaldimension Df between 3 and 4. Dashed curves are plots of thepower-law function ms ' l22Df versus wavelength. Notice thegood fit of the curves for Df 5 3.5 and Df 5 4.0. Model parameters:n# bkg 5 1.352, n# p 5 1.42, Fv 5 0.2, dm 5 1.13, sm 5 2, p 5 3.

The fitted curves in Fig. 5 illustrate the accuracy ofthis approximation for different values of Df. Thedeviation evident for Df 5 3 results from the correc-tion for correlated scattering @Eq. ~15!#, which re-duces the effective cross sections of the smallestdiameter particles.

In contrast to ms, the transport-corrected scatteringcoefficient, mst, has a relatively weak dependence onDf. Figure 6 shows that the magnitude of mst pre-dicted by the model is between 1 and 2 mm21

throughout most of the visible and near-infraredrange for 3 # Df # 4. The implication of this resultis that mst is relatively insensitive to the shape of thedistribution of particle sizes; its magnitude is deter-mined, for the most part, by the total volume fractionand refractive indices of the particles. Although themst-versus-l curves are fitted well by power-lawcurves, the log–log slope of the curves does not appearto have a simple relationship with Df.

The magnitude and the wavelength dependence ofmb are shown in Fig. 7. Like mst, mb has a weakdependence on Df. However, in contrast to thesmooth decline in the magnitude of mst wavelength,mb has a nonmonotonic dependence, first decliningsteeply in the visible region and then leveling off to anearly constant value in the near infrared. This be-havior is a consequence of the strong wavelength de-pendence of the scattering cross section of thesubwavelength particles that contribute most to thetotal backscattering coefficient ~see discussion inSubsection 4.E!.

D. Optical Properties of a Soft Tissue Model:Comparison with Measured Values

Table 1 summarizes the optical properties predictedby the model at three wavelengths ~l 5 633, 800, and1300 nm! for a soft tissue containing different dry-

Fig. 6. Magnitude and wavelength dependence of the scatteringcoefficient mst of a model tissue for values of the limiting fractaldimension Df between 3 and 4. The dependence of mst on wave-length is weaker compared with that of ms ~see Fig. 5!, and itslog–log slope does not have a simple dependence on Df. Modelparameters: n# bkg 5 1.352, n# p 5 1.42, Fv 5 0.2, dm 5 1.13, sm 52, p 5 3.

1 May 1998 y Vol. 37, No. 13 y APPLIED OPTICS 2793

weight fractions of connective tissue fibers ~ ff 5 0.03,0.3, and 0.7!. The coefficients ms, mst, mb, and g in thetable were computed for Df 5 3.7, the fractal dimen-sion that yielded values that correspond best overallto available measurements. This value of Df is alsoclose to the fractal dimensions estimated by Fourieranalysis of phase-contrast micrographs of fresh tis-

Fig. 7. Magnitude and wavelength dependence of the backscat-tering coefficient mb of the model tissue for values of the limitingfractal dimension Df between 3 and 4. Model parameters: n# bkg

5 1.352, n# p 5 1.42, Fv 5 0.2, dm 5 1.13, sm 5 2, p 5 3.

2794 APPLIED OPTICS y Vol. 37, No. 13 y 1 May 1998

sue sections in Ref. 6. ~Note that the fractal dimen-sions given in this reference are given as areadimensions and therefore are one less than the volu-metric dimensions used in this study.! Unfortu-nately, the accuracy of the predicted coefficientscannot be established at present because measure-ments of all four coefficients on living tissue are notavailable. However, measurements of the opticalproperties of excised tissue made by a number ofresearchers over the past decade have been compiledby Cheong et al.32 Table 2 lists a set of coefficientsselected from this compilation. The correspondenceof these coefficients with the model results in Table 1is discussed below.

The average value of ms measured at 633 nm on thedifferent soft tissue types shown in Table 2 is 22mm21, the same as the value that we found for themodel tissue with ff 5 0.3, a proportion of connectiontissue fibers about midway between the physiologicalextremes estimated in Subsection 2.A ~ ff 5 0.02 andff 5 0.7!. The strong dependence of the magnitudeof the scattering coefficients on ff underscores thesensitivity of scattering in tissue to the mean varia-tion in the refractive index. In accordance with themodel results, the measurements indicate that themagnitude of ms measured on the same tissue de-creases substantially with wavelength. For exam-ple, Parsa et al.33 measured a decrease in ms of ratliver from 14 mm21 at 633 nm to 4.4 mm21 at 1300

Table 1. Optical Coefficients of Model Tissues with Three Different Dry-Weight Fiber Fractions ~ff !, for Df 5 3.7

Coefficient

Wavelength ~nm!

633 800 1300

ff 5 0.03 ff 5 0.3 ff 5 0.7 ff 5 0.03 ff 5 0.3 ff 5 0.7 ff 5 0.03 ff 5 0.3 ff 5 0.7

ms ~mm21! 10.5 22.4 40.2 6.9 14.6 27.4 2.9 6.3 11.9mst ~mm21! 0.80 2.0 4.5 0.57 1.4 3.2 0.30 0.75 1.65mb ~mm21ysr! 0.08 0.22 0.50 0.05 0.13 0.31 0.03 0.09 0.20g 0.92 0.91 0.89 0.92 0.90 0.88 0.90 0.88 0.86

Table 2. Published Optical Properties of Tissues at Selected Wavelengths

Tissue Type ms ~mm21! mst ~mm21! mb ~mm21ysr! g Wavelength ~nm! Reference

Aorta ~human! 31.6 4.1 — 0.87 633 Yoon et al.35

17.1–31.0 2.6–3.7 — 0.81–0.91 633 Keijzer et al.34

23.3 2.3 — 0.9 1320 Essenpreis40

Aorta ~rat! 14.0–22.0 — 0.16 6 0.1 — 800 Schmitt38

11.0—20.0 — 0.05 6 0.05 — 1300 Schmit38

Cartilage ~human! 24.6 — — 0.95 633 Qu et al.36

Liver ~rabbit! 19.0 0.89 — 0.934 633 Beek et al.39

Liver ~rat! 14.4 0.72 — 0.95 633 Parsa et al.33

9.7 0.58 — 0.94 800 Parsa et al.33

4.4 0.40 — 0.91 1300 Parsa et al.33

Myocardium ~rabbit! 16.0–19.1 1.1 — 0.93–0.94 633 Beek et al.39

Skin, dermis ~pig! 28.9 2.1 — 0.93 633 Beek et al.39

25.4 1.4 — 0.95 790 Beek et al.39

nm. Their measured values of ms are close to thosecalculated for the model tissue with ff between 0.03and 0.3, and the calculated and measured scatteringcoefficients have nearly the same power-law depen-dencies on wavelength: ms ' l21.7 ~predicted! ver-sus ms ' l21.8 ~measured!.

The calculated value of g for the model tissue at 633nm is approximately the same as the value measuredfor the aorta by Keijzer et al.34 and Yoon et al.,35 butthe measured values of g in Table 2 are somewhatsmaller for the rest of the tissues shown. Theangular-scattering properties of tissues may reflectthe relative contributions of the nuclei and thesmaller organelles, such as the mitochondria.9 Theslow decline with wavelength in the magnitude of gpredicted by the model is substantiated by the mea-surements of Parsa et al.33 on rat liver ~Table 2!, butother investigators have found that g may also in-crease with wavelength.36,37 The wavelength de-pendence of g is difficult to measure experimentallybecause the narrow forward-scattering peak of thephase function of tissue ~see Fig. 3! makes averagingover angles prone to error.

The two measured values of mb listed in Table 2were derived from the average magnitude of interfer-ence signals recorded in an earlier study by anoptical-coherence microscope viewing an excisedspecimen of the aorta in the reflection mode.38 Theaverage backscattered powers, measured at l 5 800nm and l 5 1300 nm relative to the incident powers,were found to equal 242 and 247 dB. Taking intoaccount the thickness of the sample volume ~20 mm!and the numerical aperture of the microscope ~0.04!,we calculated mean values of mb equal to 0.16 and0.05 mm21ysr, respectively, at l 5 800 nm and l 51300 nm. These values agree fairly well with thebackscattering coefficients of the model tissue calcu-lated at the same wavelengths for ff 5 0.3 ~mb 5 0.13mm21ysr at l 5 800 nm and mb 5 0.088 mm21ysr atl 5 1300 nm!. However, because of the interferencenoise in the measured signals, the precision of themeasured values is poor. Additional studies need tobe carried out to test the validity of the calculatedbackscattering coefficients.

E. Dominant Particle Sizes for Attenuation andBackscattering

An attractive feature of the tissue model is that itpermits identification of the sizes of the particles intissue that contribute most to attenuation and back-scattering of light at a given wavelength. Figure 8shows the contributions of each sphere size to thetotal scattering coefficient and backscattering coeffi-cient in the ten-sphere model with Df 5 3.7 and ff 50.3. The widths of the distributions mb~di! andms~di! indicate the wide range of particle sizes thatare responsible for scattering in tissue. A conve-nient measure of the size of the particles that con-tribute most to extinction at a given wavelength isthe centroid, ^d&ext 5 ¥i dims~di!y¥i ms~di!; defined ina similar way, the centroid ^d&bk 5 ¥i dimb~di!y¥imb~di! is a convenient measure of the size of the par-

ticles that contribute most to backscattering. Thecalculated centroids of the distributions in Fig. 8 forl 5 633 nm are ^d&ext 5 2.4 mm and ^d&bk 5 0.29 mm.At the longer wavelength, l 5 1300 nm, the centroidsshift toward larger values, ^d&ext 5 4.0 mm and ^d&bk5 0.38 mm. From these results we conclude that theparticles in tissues with diameters between ly4 andly2 are the dominant backscatterers, whereas theparticles that cause the greatest extinction offorward-scattered light have diameters between 3and 4l. An implication of this finding that pertainsto optical-coherence microscopy and other thick-tissue imaging modalities is that large particles thatattenuate a focused probe beam strongly may limitthe penetration of the beam, yet backscatter tooweakly to be seen. Conversely, small particles thatbackscatter strongly may cause little attenuation of afocused beam and therefore have a negligible effecton its penetration.

5. Summary and Conclusions

In this study we have developed a micro-opticalmodel that explains most of the observed scatteringproperties of soft biological tissue. The model treatsthe tissue as a collection of isotropic scattering par-ticles whose volume fractions are distributed accord-ing to a skewed log-normal distribution @Eq. ~6!#modified by a packing factor @Eq. ~14!# to account forcorrelated scattering among densely packed parti-cles. Ratios between the refractive indices of the

Fig. 8. Contributions of the different sizes of spheres in the ten-sphere model to the total scattering coefficient ms and backscatter-ing coefficient mb of the model tissue for ~a! l 5 633 nm and ~b! l5 1300 nm. The calculated values of the centroids of the distri-butions ms~d! and mb~d! are labeled on the x axes as ^d&ext and ^d&bk,respectively. Model parameters: Df 5 3.7, n# bkg 5 1.352, n# p 51.42, Fv 5 0.2, dm 5 1.13, sm 5 2, p 5 3.

1 May 1998 y Vol. 37, No. 13 y APPLIED OPTICS 2795

particles and background were derived from statisti-cal arguments about the relative proportions of themicroscopic elements of tissue and were found torange from 1.39y1.352 to 1.452y1.352, which corre-spond to dry-weight fractions of fibers between 0.03and 0.3. After evaluating the model by applying Mietheory to a collection of spheres with a wide range ofsizes, we found a set of parameters for the distribu-tion and packing of the particles ~Fv 5 0.2, Df 5 3.7,sm 5 2, dm 5 1.13, p 5 3! that yields credible esti-mates of the scattering coefficients and asymmetryparameters of representative soft tissues. Applyingthis model, we observed the following: ~1! as an op-tical medium, tissue is represented best by a volumeof scatterers with a wide distribution of sizes, ~2!fixing the total volume fraction of particles and theirrefractive indices places upper and lower bounds onthe magnitude of the scattering coefficient, ~3! thescattering coefficient decreases with wavelength ap-proximately as ms ; l22Df for 600 # l # 1400 nm,where Df is the limiting fractal dimension, and ~4!scatterers in tissue with diameters between ly4 andly2 are the dominant backscatterers; the scatterersthat cause the greatest extinction of forward-scattered light have diameters between 3l and 4l.

We hope that the model developed in this study willprovide a starting point for further exploration of themicro-optics of tissue. Remaining problems includethe influence of the arrangement of tissue elementson coherent optical scattering and the origins of thevariability in the scattering coefficients of normal andpathological tissues.

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