determining the optical properties of turbid media by ...prahl/pubs/pdfx/prahl93a.pdfdetermining the...

Determining the optical properties of turbid mediaby using the adding-doubling method

Scott A. Prahl, Martin J. C. van Gemert, and Ashley J. Welch

A method is described for finding the optical properties (scattering, absorption, and scattering anisotropy)of a slab of turbid material by using total reflection, unscattered transmission, and total transmissionmeasurements. This method is applicable to homogeneous turbid slabs with any optical thickness,albedo, or phase function. The slab may have a different index of refraction from its surroundings andmay or may not be bounded by glass. The optical properties are obtained by iterating an adding-doubling solution of the radiative transport equation until the calculated values of the reflection andtransmission match the measured ones. Exhaustive numerical tests show that the intrinsic error in themethod is < 3% when four quadrature points are used.

1. Introduction

This paper introduces a practical way to determinethe optical properties of scattering and absorbingmaterials. These properties are obtained by repeat-edly solving the radiative transport equation until thesolution matches the measured reflection and trans-mission values. The advantages over existing meth-ods are increased accuracy and flexibility in modelingturbid samples with intermediate albedos, mis-matched boundary conditions, and anisotropic scatter-ing. The primary disadvantage is that this methodis entirely numerical. For brevity this method iscalled inverse adding-doubling (AD): inverse im-plies a reversal of the usual process of calculatingreflection and transmission from optical properties,and adding-doubling indicates the method used tosolve the radiative transport equation. The IADalgorithm and theory are described in this paper; anexperimental implementation is presented in thecompanion paper.'

The optical properties of biological tissue are impor-tant for photodynamic therapy and diagnostic tech-niques. 2 Typically optical properties are obtained by

S. A. Prahl is with the Department of Electrical Engineering andApplied Physics, Oregon Graduate Institute, Beaverton, Oregon97006. M. J. C. van Gemert is with the Laser Center, AcademicMedical Center, Amsterdam, The Netherlands. A. J. Welch iswith Biomedical Engineering, University of Texas at Austin,Austin, Texas 78712.

Received 8 January 19920003-6935/93/040559-10$05.00/0.© 1993 Optical Society of America.

using solutions of the radiative transport equationthat express the optical properties in terms of readilymeasurable quantities. 3 These solutions are eitherexact or approximate and correspond to the direct orindirect methods described by Wilson et al.4 Directmethods place stringent constraints on the sample tomatch the assumptions made for the exact solution.For example, the direct method used by Flock et al.5required thin samples in which multiple scatteringcould be ignored. Indirect methods relax the sampleconstraints but require approximations that are ofteninvalid for tissue samples (e.g., nearly isotropic scat-tering or no internal reflection at the boundaries).The theory used in indirect methods usually falls intoone of three categories: Beer's law, Kubelka-Munk,or the diffusion approximation.

Beer's law neglects scattering and is inappropriatefor thick scattering materials. The Kubelka-Munkmethod and variants6' 2 are still used 3" 4 but arelimited in their accuracy."" 15 Methods based on thediffusion approximation or a similar approximation(e.g., uniform radiances over the forward and back-ward hemispheres) tend to be more accurate than theKubelka-Munk method.16"17 Techniques using thediffusion approximation include pulsed photothermalradiometry,' 8 time-resolved spectroscopy,'9 radial re-flectance spectroscopy,20 weak localization,2' and aniterative technique that uses reflection and transmis-sion measurements.2 2 These methods remain popu-lar because they are easy to use, place relativelyminor constraints on the type of sample, and areamenable to analytic manipulation. However, thediffusion approximation assumes that the internal

1 February 1993 / Vol. 32, No. 4 / APPLIED OPTICS 559

radiance is nearly isotropic, and consequently it isinaccurate when scattering is comparable with absorp-tion.2 3

The adding-doubling method is a general, numeri-cal solution of the radiative transport equation.2 4

The adding-doubling method25 was chosen because itis sufficiently fast that iterated solutions are possibleon current microcomputers and sufficiently flexiblethat anisotropic scattering and internal reflection atthe boundaries may be included. Other accuratesolutions of the radiative transport equation such asChandrasekhar's X and Y functions,26 discrete ordi-nates,2 7 Monte Carlo models,2>30 invariant embed-ding,3 ' or successive orders24 are too slow or insuffi-ciently flexible to incorporate the necessary boundaryconditions needed for turbid materials with mis-matched boundaries.

The IAD method consists of the following steps:(1) Guess a set of optical properties. (2) Calculatethe reflection and transmission by using the adding-doubling method. (3) Compare the calculated valueswith the measured reflection and transmissions.(4) Repeat until a match is made. The set of opticalproperties that generates reflection and transmissionvalues matching the measured values is taken as theoptical properties of the sample. The results ob-tained with the IAD method are accurate for alloptical properties and can be made arbitrarily preciseat the cost of increased computation time. Further-more, by avoiding an analytical solution, it is possibleto incorporate the necessary corrections for measure-ments made directly with integrating spheres32 (seeSection 3.F). Such corrections are usually quiteawkward to implement because the magnitude of thecorrection depends on the optical properties of thesample measured.

2. Adding-Doubling Method

A. Definitions

The optical properties of a turbid medium are charac-terized by the absorption coefficient pha, the scatteringcoefficient pu,, and the single-scattering phase func-tion p(O). The reciprocal of the absorption (scatter-ing) coefficient is the average distance a photon willtravel before being absorbed (scattered) by the me-dium. Two dimensionless quantities that character-ize light propagation in a turbid medium are thealbedo a and the optical thickness T. These aredefined as

asI 1¾P. + I.'

T = d([L8 + Ia ),

where d is the physical thickness of the slab.The single-scattering phase function p(O) describes

the amount of light scattered at an angle 0 from theincoming direction. The phase function is oftenexpressed in terms of the cosine of this scatteringangle v = cos 0. The phase function is normalized so

that its integral over all directions is unity:

JI1p(v)dw = 2w J p(v)dv = 1,4Tr -~~~~~1

where do is a differential solid angle. The functionalform of the phase function in tissue is usually notknown. However, research by Jacques et al. 33 andYoon et al.34 have shown that a Henyey-Greensteinfunction approximates single-particle light scatteringin human dermis and aorta at 633 nm. Consequent-ly this phase function is used in all calculations in thispaper:

1 1-g 2

P( = 4r (1 + g2 - 2gv)&2

The Henyey-Greenstein phase function depends ononly the anisotropy coefficient g, defined as

~~~~~~~~1g = f p(v)vdwo = 2 f p(v)vdv.

Consequently g is the average cosine of the scatteringangle. When g = 0, scattering is equally probable inall directions. Typical values for tissues in the redregion of the spectrum are g 0 0.8,2 which is moder-ately forwardly directed.

Reflection and transmission are relative to thenormal irradiance on the sample surface and varybetween zero and one. The total reflection RT is alllight specularly reflected and backscattered by thesample. The total transmission TT is all the lightthat passes through the sample and includes any lighttraveling through the sample without being scattered.This unscattered light is denoted T, because forcollimated irradiance on a sample the light passingdirectly through the sample is often called the colli-mated transmission. For a nonabsorbing sampleRT+ TT= 1.

B. Theory

The doubling method was introduced by van de Hulstfor solving the radiative transport equation in a slabgeometry.35 The advantages of the adding-doublingmethod are that only integrations over angle arerequired, physical interpretation of results can bemade at each step, the method is equivalent forisotropic and anisotropic scattering, and results areobtained for all angles of incidence used in theintegration.3 6 The disadvantages are that (a) it isslow and awkward for calculating internal fluences,(b) it is suited to a layered geometry with uniformirradiation, and (c) it is necessary that each layer havehomogeneous optical properties. For determiningoptical properties using only reflection and transmis-sion, internal fluences are not needed, so disadvan-tage (a) is not a problem. Disadvantages (b) and (c)place restrictions on the sample geometry-the sam-ples must be uniformly illuminated, homogeneousslabs. The adding-doubling method is well suited to

560 APPLIED OPTICS / Vol. 32, No. 4 / 1 February 1993

iterative problems because it provides accurate totalreflection and transmission calculations with rela-tively few quadrature points. The method is fast forsmall numbers of quadrature points, and conse-quently iteration is practical.

In all calculations that follow the following assump-tions are made: the distribution of light is indepen-dent of time, samples have homogeneous opticalproperties, the sample geometry is an infinite plane-parallel slab of finite thickness, the tissue has auniform index of refraction, internal reflection atboundaries is governed by Fresnel's law, and the lightis unpolarized. A nonabsorbing layer with a differ-ent index of refraction may be present at the bound-aries (glass slide).

The doubling method assumes that the reflectionR(v, v') and transmission T(v, v') for light incident atan angle v and exiting at an angle v' is known for onelayer. The reflection and transmission of a slab thatis twice as thick is found by juxtaposing two identicalslabs and adding the reflection and transmissioncontributions from each slab.25 The reflection andtransmission for an arbitrary slab are calculated firstby finding the reflection and transmission for a thinstarting slab with the same optical properties (e.g., byusing single scattering) and then by doubling untilthe desired thickness is reached. The adding methodextends the doubling method to dissimilar slabs.Thus slabs with different optical properties can beplaced adjacent to one another to simulate layeredmedia or internal reflection caused by index-of-refraction differences.

C. Internal Reflection and Boundary Conditions

Internal reflection. at the boundaries (caused by mis-matched indices of refraction) was included in thecalculation by adding an additional layer for eachmismatched boundary. The reflection and transmis-sion of this layer equaled the Fresnel reflection andtransmission for unpolarized light incident on a planeboundary between two transparent media with thesame indices of refraction. If r(vi) is the unpolarizedFresnel reflection for light incident from a mediumwith the index of refraction n1 on a medium with anindex of refraction n2 at an angle from the normalwith the cosine equal to vi, the reflection and transmis-sion operators for the boundary slab are

Rbndry(vi, v>) = r(v) ij-2; i'

1 - r(vi) n, 2Tbndry(Vi, 1V) 2v, n~fo) 8i

where bij is the Kronecker delta-function. The squareof the ratio of the indices of refraction is due to the n2

law of radiance,37 which accounts for the difference inradiances across an index-of-refraction mismatch.The factor of 2vi is included for uniformity with vande Hulst's definition of the reflection function.2 4

Note finally that the transmission and reflection ofthe boundary layer were zero and one, respectively,for light that is incident at angles exceeding the

critical angle for total internal reflection. Both oper-ators are diagonal because light is specularly reflectedand the angle of incidence equals the angle of reflec-tion.

By sandwiching a tissue sample between glass orquartz plates, we can minimize the usual irregulari-ties in the tissue surface, and the Fresnel reflection isa good approximation. To account for a nonabsorb-ing boundary with a different index of refraction, wemust include all the multiple internal reflections.2

For example, if vi is the cosine of the angle ofincidence from the turbid slab (index ns) onto a glassslide (index ng), the cosine of the angle inside the glassslide (vg) is determined by using Snell's law:

ng(1 - vg2 )"2 = nJ(1 -vil) ' Vi < v,

where v, is the cosine of the critical angle for totalinternal reflection. When ri = r(vi) is defined as theunpolarized Fresnel reflection coefficient for lightpassing from the slab into glass and rg = r(vg) is forlight passing from glass to air, the net reflectioncoefficient that should be used in Eqs. (1) is

Irl(vi) + rg(vg) - 2r1(vi)rg(vg)

r(v) = 1 - r1(vi)rg(vg)

1

if vi < v,

if vi > v,

This value for r(v) accounts for the extra reflectionswithin the glass slide. Finally, since light is re-fracted at the boundary, we must be sure that theincident and reflected fluxes are identified with theproper angles.

The reflection and transmission functions for thethin starting layers were obtained by the diamondinitialization method.38 The optical thickness forthe starting layers was based on the smallest quadrat-ure angle as suggested by Wiscombe.39 The Henyey-Greenstein phase function was used for all calcula-tions. We avoided phase-function renormalizationby always using the 8 - M method, which facilitatesaccurate calculations with highly anisotropic phasefunctions. 4 0

The adding-doubling method is based on the nu-merical integration of functions with quadrature:

fl

N

f (v, v')dv' = Y Hkf(xk)-k=1

The quadrature points Xk and weights Hk are chosenso that the integral is approximated exactly for apolynomial of order 2N - 1 (or possibly 2N - 2,depending on the quadrature method). Using Nquadrature points (Gaussian quadrature) is equiva-lent to the spherical harmonic method of orderPN-1, 4 1 i.e., four quadrature points correspond to theP3 method. The choice of quadrature methods isdescribed in Section 3.

The total internal reflection caused problems bychanging the effective range of integration. Usuallyadding-doubling integrals range from 0 to 1, since


the angle varies from ir/2 to 0 and therefore the cosinevaries from 0 to 1. We used numerical quadrature incalculating the integrations, and the quadrature an-gles were optimized for this range. If the cosine ofthe critical angle is denoted by v,, then for a boundarylayer with total internal reflection the effective rangeof integration is reduced down to v, to 1 (because therest of the integration range is now 0). To maintainintegration accuracy, we separated the integral intotwo parts, and each is evaluated by quadrature overthe specified subrange:

A(v, v')B(v', v")dv'

= J A(v, v')B(v', v")dv' + A(v, v')B(v', v")dv'.

Here A(v, v') and B(v, v') represent reflection ortransmission functions, and, if either is identicallyzero for v < v, then the integration range is reduced.The calculations in this paper used Gaussian quadrat-ure42 for the range from 0 to v, and thereby calcula-tions at both end points were avoided. (In particularthe angle v = 0 is avoided, which may cause divisionby zero.) Radau quadrature was used for the rangefrom v to 1, because one quadrature angle may bespecified.43 Normal irradiance corresponds to v = 1,and if this angle is specified, interpolation betweenquadrature points is not needed to obtain reflectionand transmission for normal unscattered irradiance.Interpolation can be a significant source of error.The number of quadrature points are divided evenlybetween 0 to v and v to 1. Since the quadraturemethods work well with even numbers of quadraturepoints, this dictates that the quadrature points shouldbe chosen in multiples of four. Radau quadraturewas used when there was no critical angle.

3. Iteration Process

The iteration process consists of finding optical prop-erties that generate the measured reflection andtransmission values. This section begins by show-ing that a unique solution to the inverse problemexists, and then the simplifications that are necessarywhen one or more of the experimental measurementsis missing are described. Finally three componentsof the iteration process are given: (1) the functionthat defines the distance the calculated values arefrom the measure values, (2) the initial set of opticalproperties guessed, and (3) the algorithm used tominimize this function.

A. Uniqueness

The iteration method implicitly assumes that a uniquecombination of the albedo, the optical depth, and theanisotropy is determined by a set of reflection andtransmission measurements. Clearly, if a sample isso thick that an accurate unscattered transmissionmeasurement cannot be made, there are more un-knowns than observations. Even when all the mea-

surements are available, it is not obvious that aunique set of optical properties (a, r, g) exists for anyset of measurements (RT, TT, TC). For example, in-creasing the albedo of a sample will increase its totalreflection and decrease its total transmission-but sowill increasing the optical thickness of the sample.Uniqueness is demonstrated for two cases: fixedunscattered transmission and fixed scattering anisot-ropy. The former is representative when (RT, TT,Tc) are all known; the latter applies when the unscat-tered transmission measurement is unavailable and afixed value for the scattering anisotropy must beassumed.

The dependence of the total transmission and totalreflection on the anisotropy and albedo is shown inFig. 1. The bold (a, g) grid was computed with theunscattered transmission fixed at 10% (Tc = 0.1),and the boundaries of the sample were matched withits environment. The intersection of the measuredtotal reflection and total transmission grid linesdetermines a unique albedo and anisotropy.

In Fig. 2 the dependence of the total transmissionand the total reflection on the reduced albedo andreduced optical thickness is shown. Scattering inthe medium is assumed to be isotropic (g = 0), andthe index of refraction is 1.4.44 Again, any nonzeroreflection and transmission measurement yields aunique value for the reduced scattering and absorp-tion coefficient. Figure 2 is quite useful for obtain-ing quick estimates of the optical properties of asample. For example, if RT = 0.4 and TT = 0.2, thena 0.96 and = 7. If the thickness of the sample is0.4 mm and the scattering anisotropy is assumed tobe 0.8, using the similarity relations (see below) willfixtheopticalpropertiesat(Pap g8,g) (0.7mm-', 17mm-', 0.8).

C. Simplification

The three measurements usually available are thetotal reflection, the total transmission, and the unscat-

a0

.E7o

H

0 0.2 0.4 0.6

Total Reflection

0.8

Fig. 1. Total reflection and total transmission of an index-matched slab (n = 1) as a function of the albedo a and anisotropygfor a fixed unscattered transmission value of 10%. Each point onthe bold (a, g) grid corresponds to a unique (RT, TT) pair.


needed. Several metrics were tried, but one basedon a sum of relative errors worked best:

..0

H

0EH

0.6

0.4

0.2

0.0

I R - Rme IM(a T g) - Rme + 10-6

T'= 12

0 0.2 0.4 0.6 0.8

Total Reflection

Fig. 2. Total reflection and total transmission of a slab as afunction of the reduced albedo a' and reduced optical thicknessT'. Isotropic scattering is assumed as well as an index of refrac-tion mismatch (n = 1.4). Each point on the (a', T') grid corre-sponds to a unique (RT, TT) pair.

tered transmission. It is assumed that the unscat-tered reflection from each surface is equal to theFresnel reflection for the unpolarized light that isnormally incident on the sample. From the unscat-tered transmission Tc, we may immediately calculatethe optical thickness T by solving

§1_ (1 - r8l)(1 - r 2)exp(-'r)

U 1 - rlr 82 exp(-2r)

where r81 and ri 2 are the primary reflections for lightthat is normally incident on the front and backsurfaces of the slab. If glass slides are present, thespecular reflection coefficients rj1 and r 2 should in-clude multiple internal reflections in the slide (seeSubsection 2.C). Once the optical thickness is known,only the albedo and anisotropy remain to be deter-mined. They are varied until the calculated reflec-tion and transmission match the measured values.

When the only measurements available are thetotal reflection and transmission, as might happen fora thick sample in which an accurate unscatteredtransmission measurement cannot be made, only twooptical parameters can be determined. Typically avalue for the scattering anisotropy is assumed, andthe albedo and optical thickness are calculated basedon this assumed value. The accuracy of say fluencecalculations when these values are used (based onsimilarity) is unknown.

If only one measurement is available, the sample isusually too thick for a transmission measurement tobe made. In this case the optical thickness of thesample is assumed to be infinite, and again a fixedvalue for the scattering anisotropy is chosen. Thereduced albedo can now be calculated. This is asimple one-parameter minimization problem, andconvergence is robust and rapid.

C. Metric

When the reflection and transmission for a particularset of optical properties are calculated, a definition ofhow far these values are from those measured is

I Tc - Tme I

Tmeas + 10-6

Here Rmeas and Tme, are the measured reflection andtransmission for scattered light. The factor of 10-6is included to prevent division by zero when thereflection or transmission is zero. Note that themeasurement errors for reflection and transmissionare assumed to be equal for this metric. If this is nottrue, suitable modifications are necessary.

D. Inverse Method

The iteration method uses an N-dimensional minimi-zation algorithm based on the downhill simplexmethod of Nelder and Mead.45 One implementationof this method (AMOEBA) varies the parameters from-mc to o.42 Since the anisotropy, albedo, and opticaldepth have fixed ranges, they are transformed into acomputation space. For example, the transforma-tion function for the albedo is

2a - 1acomp = a(1 - a)

Thus, as acomp varies from -cc to cc, the albedo a variesfrom 0 to 1. The transformation for the anisotropyg(which varies from -1 to 1) is

ggcomp =1 +

The transformation for the optical thickness v (whichvaries from 0 to o) is

bcomp = ln(T).

We can easily invert these relations to obtain rela-tions for the optical properties in terms of the compu-tation values. We made all calculations by using thereal values: the transformed values were used onlyby the simplex method for choosing the next iterationpoint. Typical convergence was in 20-30 iterations.

E. Initial Values

The starting set of optical properties affects both therapidity of convergence and the convergence to thecorrect values. Clearly, with a better first guess,fewer subsequent iterations are needed. Poorguesses have the added disadvantage that the minimi-zation algorithm may converge to a relative ratherthan the global minimum. Since the global mini-mum corresponds to the unique solution describedabove, local minima must be guarded against. Fortu-nately they are easily detected by examining themagnitude of the minimum. If the magnitude ex-ceeds a small tolerance (typically 10-3), the minimumis a local one and the iteration process must berestarted with a better initial guess. If necessary,


0.8n = 1.4

an

r _Yi[R,(1 - Y2Rd) + Y3Rcd](l - y4Rd) + YlY34Td(Tcd + PY5Tc)(1 - Y4 Rd)(l

one or two restarts suffice to reach the global mini-mum.

Generating an initial set of approximately correctoptical properties for any reflection and transmissioncombination is difficult. We can use the similarityrelations46 to facilitate picking starting values byrelating (a, T, g) to the reduced optical properties(a', T', g' = 0):

a _ a(1-g)1 - ag

The inverse relations are

a'a -g+a'g

'= (1 - ag)T.

a'-r'gT = ' + g- gNow only two parameters need to be found (a', T') fora combination (RT, TT). For example, originally theKubelka-Munk relations were used to obtain thesevalues for initial guesses.47 Unfortunately these val-ues were often worse than using just a fixed guess(a', T', g) = (0.5, 1, 0.2) to begin all iterations. Agood starting set of starting values is based on a crudefit of the reflection and transmission values of Fig. 2.The formula for the reduced albedo is

1

1

(1 - 4Rd-TT 2

1 - TTJ4 (1 -Rd- TT29 - TT

if Rd < 0.1f TT

i > 0. 1.

The formula for the reduced optical thickness is

-In TT ln(0.05)

Tl = In RT21+5(Rd+TT)

ifRd < 0.1

if Rd > 0.1.

Once we obtain r by using Eq. (3), and a' and r' havebeen calculated as above, Eqs. (1) and (2) can be usedto generate a single set of starting values (a, , g).This is the only time the similarity relations are usedin the LAD method.

F. Integrating-Sphere Correction

Total transmission and reflection are usually mea-sured with integrating spheres. Interaction be-tween the sample and spheres makes the detectedsignal no longer proportional to the sample reflectionor transmission.3 2 For example, when a double-integrating-sphere arrangement is used, the poweron the wall of the reflectance sphere Pr normalized tothe incident power P is32

- 6Rd) - Y4Y6Td

where the yi values depend on only the geometry andreflectivity of the integrating spheres, and various Rand T values correspond to different types of reflec-tion and transmission by the sample. A similarformula holds for the normalized power on the trans-mission sphere wall. Once the integrating sphereshave been characterized (i.e., the yi values are known),we can calculate the normalized powers on the wallsof the integrating sphere by using reflection andtransmission values obtained with the adding-doubling method. The IAD algorithm calculates thenormalized powers in the spheres and matches themto the detected powers (rather than matching reflec-tion and transmission directly).

4. Accuracy of the MethodThis section addresses the question: If RT, TT, andTC are known exactly, what is the maximum possibleerror in the derived optical properties caused by theinverse adding-doubling method? The error analy-sis must be made numerically, because analyticalexpressions for light propagation in anisotropic mediawith mismatched boundaries are not available. Thenumerical tests are designed to find the accuracy ofthe inverse algorithm as functions of both the opticalproperties and the reflection and transmission of thesample. Implicit are checks on (a) the convergenceof the inverse algorithm to the global minimum, (b)the termination criterion for stopping the iterationprocedure, (c) the choice of starting parameters, and(d) the effect of using small numbers of quadraturepoints. This last test is particularly important, sincethe method would be useless (too slow) if small errorscould be achieved only by using many quadraturepoints.

Generating a set of accurate testing values pre-sented a problem, since accurate tabulated values formismatched boundaries with anisotropic scatteringcould not be found in the literature. Consequently,after demonstrating that our adding-doubling imple-mentation reproduced the published values formatched boundaries24 46 exactly, we used Monte Carloto verify a few mismatched cases. This confirmedthe boundary condition algorithm. Finally, we com-pared 36- and 48-quadrature-point adding-doublingcalculations. The 36-point calculation was alwayscorrect to at least 0.1%. The 48-point calculationwas used for the test data and was constrained toinclude only samples with reduced optical thicknessesof > 0.25 and optical thicknesses of < 32. The ratio-nale was that when T' < 0.25 either the sample doesnot multiply scatter and therefore a simpler algo-rithm should be used, or the sample does multiplyscatter, but because of the highly anisotropic natureof each scattering event, the separation of Tc from


Pr

scattered light would be extremely difficult. Thesecond criterion ( < 32) ensured that the sufficientcollimated transmitted light passed through the sam-ple so that it could be measured Tc > exp(-32) 10-14]. The IAD method works for any optical thick-ness, but it becomes progressively less accurate out-side this range.

All samples assume slabs with an index of refrac-tion of 1.4 relative to the environment. Results forthe index-matched samples indicate that the error inthe LAD method is approximately half of those pre-sented here for the mismatched case. This is causedby total internal reflection of light at quadratureangles that are greater than the critical angle (one-half of the total number, see Subsection 2.C). Theslabs are not sandwiched by glass: calculations witha sample between glass slides do not alter the mis-matched (n = 1.4) results significantly and are omit-ted for brevity.

In all calculations the scattering phase functionused was the Henyey-Greenstein phase function.Despite the limited experimental evidence that thisphase function is appropriate for tissues, errors aris-ing from an inappropriate phase function should besmall, since different phase functions with equivalentg values and different higher-order moments gener-ate nearly equal reflection and transmission values.46

The absolute error is defined as the differencebetween the true or accurate values and those calcu-lated with the inverse adding-doubling method:

Aa' = latrue' - acalc I (5)

The relative percentage error is the absolute errordivided by the true value and multiplied by 100. Theabsolute errors in the scattering and absorptioncoefficients are proportional to the physical thicknessof the sample and consequently are not particularlyuseful. The relative errors for [La and pu can also beobtained despite the fact that only dimensionlessoptical parameters are used by noting that the physi-cal thickness cancels:

AN.tu

11at= 100[1 -

= 100(1 -

(1 - acalc)Tcalc1

(1 -atrue)Ttrue

aceTca \

atrueTtrue

B. Variation with Optical Properties

The sensitivity of the inverse adding-doubling methodto the optical properties of a sample was evaluated byusing a data set consisting of reflection and transmis-sion values for 11 reduced albedos spaced so that(1 - atrueT)/2 = (0, 0.1, . . ., 1), 10 anisotropies spacedsimilarly (0, 0.19, 0.36, 0.51, 0.64, 0.75, 0.84, 0.91,0.96, 0.99), and 14 reduced optical depths (listed inTable 1). Thus for each pair (a', T') there were 10different values of g that all corresponded to more orless equal values of RT, TT. There was a total of 1540

Table 1. Maximum Errors in the Calculated Reduced Optical Thickness' for any Albedo or Anisotropy as a Function of the True Reduced

Optical Thicknessa

Maximum MaximumAbsolute Errors Relative Errors

T' Diffusion 4 8 Diffusion 4 8

/4 0.04 0.02 0.01 16 9 43/8 0.03 0.02 0.01 9 6 31/2 0.03 0.02 0.01 5 5 23/4 0.03 0.03 0.01 3 3 1

1 0.04 0.03 0.01 4 3 13'2 0.07 0.03 0.01 5 2 1

2 0.2 0.04 0.02 11 2 .83 0.2 0.04 0.02 7 1 .64 0.3 0.05 0.02 8 1 .66 0.6 0.1 0.04 10 1 .68 0.9 0.2 0.05 11 2 .6

12 1.5 0.3 0.09 12 2 .716 2 0.4 0.1 13 2 .7

aThe calculated vales were obtained by using a 8-Eddingtonapproximation (diffusion) or the LAD method with four or eightquadrature points.

different combinations of atrue, Ttrue, gtrue. For eachcombination accurate reflection and transmission(RT, TT, TC) values were calculated. The optical prop-erties were found by using the LAD method with fourand eight quadrature points. Diffusion equationresults were obtained by using the same iterationalgorithm but by replacing the adding-doubling calcu-lation with a 8-Eddington diffusion approxima-tion. 22,48

The spacing in the reduced albedo was based on theobservation that reflection and transmission are quitesensitive to changes in the reduced albedo when it isnear unity. The anisotropy was chosen to vary in asimilar manner for the same reason. Completelyforward scattering (g = 1) was omitted because itcorresponds to the nonscattering case. The nonscat-tering (a = 0) and nonabsorbing (a = 1) cases wereboth included. The range of the reduced opticalthickness was chosen because of the natural advan-tage of making calculations for optical thicknessesthat vary by a factor of 2 when the adding-doublingtechnique is used.

The variation in IAD accuracy with reduced opticalthickness is tabulated in Table 1 and displayed in Fig.3. The maximum relative error for a fixed T' is thegreatest error in a calculated value of i' for any albedoand scattering anisotropy. The relative error de-creases as the number of quadrature points increases.When 1 < r' < 16 the maximum relative error whenfour quadrature points are used is < 2%. Surprising-ly the diffusion approximation works best for T' 1but when only the relative error is considered.Table 1 shows that the maximum absolute error AT'monotonically increases with T' for the diffusionapproximation as well as the four- and eight-quadrat-ure-point IAD calculations.

The variation in the relative error in the scattering


Diffusiona 10

0.1 1 10

Reduced Optical Depth '

Fig. 3. Maximum error in the calculated reduced optical thicknessT' for any anisotropy and any albedo by using the diffusionapproximation and the IAD method with four and eight quadraturepoints.

coefficient was evaluated for changes in the reducedoptical thickness, the reduced albedo, and the scatter-ing anisotropy when only four quadrature pointswere used. In Fig. 4 the maximum relative error inthe calculated scattering coefficient is plotted as afunction of the three dimensionless quantities a', ',and g. As expected the accuracy in determining lincreases as the scattering increases (i.e., as a' -> 1).The increased scattering implies more uniform inter-nal radiance distributions, which in turn are moreaccurately approximated than highly anisotropic inter-nal radiance by four-quadrature-point distributions.Errors in the scattering coefficient do not dependstrongly on either the reduced optical thickness or thescattering anisotropy. The maximum relative errornever exceeds 6% for positive albedos.

10*

.50

t)

04

ea3E

10 "'

10:

10:

0.1

a'_-. 0,2 0.4 0.6 0.8 1t

g 0,2 0,4 0,6 0,8

0

B. Variation with Reflection and Transmission

The accuracy of the inverse adding-doubling methodvaries with the reflection and transmission of thesample. To quantify the maximum error for specificreflection and transmission values another paired setof atrue, Ttrue, 9true and RT, TT, TC were generated. Inthis set the total reflection and total transmissionwere spaced evenly (in 0.05 increments). By permit-ting the unscattered transmission to take on differentvalues, 10 different sets of atrue, 'true, true wereobtained for each pair (RT, TT). The maximum rela-tive error in the absorption coefficient, the scatteringcoefficient, and the scattering anisotropy was calcu-lated by using the IAD method with four quadraturepoints.

Figure 5 shows a contour plot of the maximumrelative error in the calculated absorption coefficientas a function of total reflection and total transmission.The maximum relative error in the absorption coeffi-cient (assuming a > 0 or RT + TT < 1) was 10%.The RT + TT = 0 case is omitted because I.a = 0 forconservative scattering and the relative error is infi-nite in this case. For the majority ofRT, TT combina-tions the maximum error is 2-3%. The error isgreatest when the total transmission is highest, whichcorresponds to samples that absorb only a smallfraction of the light.

Figure 6 shows a contour plot of the maximumrelative error in the scattering coefficient as a func-tion of the total reflection and total transmission.In contrast to the error in the absorption coefficientthe error for the scattering coefficient is greatestwhen little light is reflected by the sample. Theerror drops sharply with increasing reflection values.The maximum error is always < 2% for any nonzerovalue of [L,. The p, = 0 case is omitted for the samereason that conservative scattering is avoided in Fig.5. The larger relative errors seen in Fig. 4 allcorrespond to the very small reflection values indi-cated by the black area in Fig. 6.

Figure 7 shows a contour plot of the maximum

100.

r.0

coF-r_"I

1,0

0.1 1 10

Fig. 4. Maximum relative error in the calculated scatteringcoefficient as a function of the reduced albedo a', the reducedoptical thickness T', and the scattering anisotropy g by using theIAD method with four quadrature points and mismatched bound-ariesn = 1.4.

0 20 40 60 80 100

Reflection

Fig. 5. Relative error in the calculated absorption coefficient as afunction of reflection and transmission. The AD method wasused with four quadrature points, and the slab had mismatchedboundaries.


100

1-2%

80 Maximum Relative

Error in jt

*° 60

9 40

0.1120

<0.1%

0 20 40 60 80 100

Reflection

Fig. 6. Relative error in the calculated scattering coefficient as afunction of reflection and transmission. The IAD method wasused with four quadrature points, and the slab had mismatchedboundaries.

absolute error in the scattering anisotropy as afunction of total reflection and total transmission.The error is always < 0.03 for all the cases in whichthe scattering coefficient is nonzero. When the scat-tering coefficient is zero, no scattering takes place andany measurement will not give information on theshape of the scattering event. These cases corre-spond to total reflection values that are equal to theunscattered reflected light.

Typically the errors resulting from the LAD methodwill be much smaller than those resulting fromexperimental uncertainties. For example, assumethat the measurements RT = 0.264, TT = 0.261, andTc = 5.91 x 10-5 are known with a 1% uncertainty.The relative errors resulting from the IAD methodare 0.04% for ps, 1% for p.a, and 0.1% for g.However, perturbing the reflection and transmissionvalues by 1% and re-solving for the optical propertiesresult in errors of 0.4% for p.,, 17% for [1a, and 0.4%for g or approximately an order of magnitude largerthan the errors inherent in the IAD method itself.Such error estimates are necessary because of the

1000.01-0.02

80 _ Maximum Relative80 r \ Error in g

*° 60<0.01

40

.01-0.02_20

00 20 40 60 80 100

Reflection

Fig. 7. Relative error in the calculated scattering anisotropy as afunction of reflection and transmission. The IAD method wasused with four quadrature points, and the slab had mismatchedboundaries.

nonlinear relationship between reflection and trans-mission and the intrinsic optical properties. A pri-mary goal of this study was to develop an accuratemethod of computing optical properties, so that suchestimates can be made without introducing errorscaused by the calculation itself.

In practice the two most common sources of experi-mental errors are (1) loss of light from the edges of thesample (which thereby invalidates the one-dimen-sional assumptions and underestimating both RT andTT) and (2) collecting scattered light in the unscat-tered transmission measurement (which thereby over-estimates Tc). The first problem manifests itself inspuriously high absorption values, and the latter caseleads to optical properties that depend on the thick-ness of the sample. These problems are discussed atlength in the experimental implementation of thistechnique.'

5. Conclusions

The inverse adding-doubling method translates totalreflection, total transmission, and unscattered trans-mission measurements into scattering, absorption,and scattering anisotropy values. The method isapplicable when the one-dimensional radiative trans-port equation adequately describes the propagation oflight through the sample. Using only four quadrat-ure points, the IAD method generates optical proper-ties (a, pLs, g) that are accurate to 2-3% for mostreflection and transmission values. Higher accuracyis achieved by using more quadrature points, but itrequires increased computation time. The validityof the IAD method for samples in which (a, = ) isespecially important since other methods (e.g., thosebased on the diffusion approximation) are known tofail in this regime. Furthermore both anisotropicphase functions and Fresnel reflection at boundariesare accurately approximated, and therefore the IADmethod is well suited to measurements involvingbiological tissues sandwiched between glass slides.A computer implementation of the inverse adding-doubling method is available from the authors.

This work was supported by the Office of NavalResearch Contract N00014-86-K-0117 and the Na-tional Institutes of Health grant 5ROlAR25395-11.

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