Sensitivity studies for imaging a spherical object embedded in a spherically symmetric, two-layer...

Sensitivity studies for imaging a spherical objectembedded in a spherically symmetric, two-layerturbid medium with photon-density waves

Yuqi Yao, Randall L. Barbour, Yao Wang, Harry L. Graber, and Jenghwa Chang

We present analytic expressions for the amplitude and phase of photon-density waves in stronglyscattering, spherically symmetric, two-layer media containing a spherical object. This layeredstructure is a crudemodel of multilayered tissues whose absorption and scattering coefficients lie withina range reported in the literature for most tissue types. The embedded object simulates a pathology,such as a tumor. The normal-mode-series method is employed to solve the inhomogeneous Helmholtzequation in spherical coordinates, with suitable boundary conditions. By comparing the total field atpoints in the outer layer at a fixed distance from the origin when the object is present and when it isabsent, we evaluate the potential sensitivity of an optical imaging system to inhomogeneities inabsorption and scattering. For four types of background media with different absorption andscattering properties, we determine the modulation frequency that achieves an optimal compromisebetween signal-detection reliability and sensitivity to the presence of an object, the minimum detectableobject radius, and the smallest detectable change in the absorption and scattering coefficients for a fixedobject size. Our results indicate that 112 enhanced sensitivity to the object is achieved when the outerlayer is more absorbing or scattering than the inner layer; 122 sensitivity to the object increases with themodulation frequency, except when the outer layer is the more absorbing; 132 amplitude measurementsare proportionally more sensitive to a change in absorption, phase measurements are proportionallymore sensitive to a change in scattering, and phase measurements exhibit a much greater capacity fordistinguishing an absorption perturbation from a scattering perturbation. r 1996 Optical Society ofAmerica

1. Introduction

It has been shown that the propagation of lightemitted from a sinusoidally varying intensity-modulated source in a strongly scattering medium isgoverned by the scalar Helmholtz wave equation.Such a wave is referred to as a photon-density wave1PDW21,2 or a diffuse photon-density wave 1DPDW2.3The mathematical analysis in the former instancestarts from the diffusion approximation to the Boltz-mann transport equation; the latter uses the diffu-

Y. Yao and Y. Wang are with the Department of Engineering,Polytechnic University, Brooklyn, New York 11201. R. L. Bar-bour is with the Department of Pathology and of Physiology andBiophysics, H. L. Graber is with the Department of Physiologyand Biophysics, and J. Chang is with the Department of Pathol-ogy, State University of New York Health Science Center, Brook-lyn, NewYork 11203.Received 23 May 1995; revised manuscript received 12 Septem-

ber 1995.0003-6935@96@040735-17$06.00@0r 1996 Optical Society of America

sion equation. Here, we use the name PDW. Animplication of these demonstrations is that lightpropagation can be analyzed in the frequency do-main with theories and methodologies developed forelectromagnetic 1EM2 and acoustic waves. The ma-jor difference between PDW and EM or acousticwaves is that wave number k of the PDW, whichdepends on the optical properties as well as on themodulation frequency, has a large imaginary part.Physically this implies an exponential attenuation ofthe wave’s amplitude as the wave propagates awayfrom the source, even for a nonabsorbing medium.Some important phenomena have been investigated,including the propagation of a PDW in an infinitemultiple-scattering medium4; diffraction and reflec-tion of a PDW from an absorbing or reflectingsemi-infinite plane bounded by a straight edge andimmersed in a uniform, infinite, strongly scatteringmedium2; refraction of a DPDW in piecewise homoge-neous turbid media3; scattering of a DPDW by aspherical object immersed in an infinite, highlyscattering medium5; and localization of a cylindrical

1 February 1996 @ Vol. 35, No. 4 @ APPLIED OPTICS 735

absorbing body by an interfering DPDW.6 If near-infrared 1NIR2 intensity-modulated illumination iscombined with frequency-resolved detection meth-ods, imaging of the optical properties of a humanhand7 and inhomogeneous spherical objects embed-ded in a scattering media have been attempted.8There are two general classes of problems regard-

ing the interaction of PDW’s with objects embeddedin a strongly scatteringmedium. In a direct scatter-ing problem 1DSP2, one computes the scattered fieldeverywhere, given the source location and opticalproperties and geometry of the test medium. In aninverse scattering problem 1ISP2, the goal is to derivethe properties of the test medium from the measuredscattered field, usually from measurements made atthe surface of the medium. It is the ISP that is ofinterest in clinical imaging. However, before onecan quantitatively characterize an object by analysisof the scattered PDW, one must first solve the DSPfor the given medium. This can be performed bycomputation of the numerical solutions to the trans-port or the diffusion equation. The solution of thediffusion equation is more tractable, especially whenMonte Carlo methods are used to compute solutionsto the transport equation. Analytic solutions to thediffusion equation are available for certain geom-etries, in particular the one considered here. Wehave adopted this approach to facilitate estimates ofvarious practical experimental parameters, such asthe required signal-to-noise ratio, the minimumdetectable object size, and the absorption- and scat-tering-coefficient changes for a specified detectorprecision. In addition, one can also investigate theeffects of varying the source and detector configura-tions to determine which are most sensitive to thepresence of an object.So far, most investigations of the DSP and ISP for

PDW’s have been limited to the case of an objectembedded within an otherwise homogeneous, infi-nite medium. However, biological tissues are inho-mogeneous media whose various components, suchas skin, fat, and muscle, are typically arranged inlayered structures.9 In the present work, we studyscattering from a spherical object embedded in thecenter of an infinite, two-layer spherically symmet-ric backgroundmedium. Figure 1 shows a sketch ofthe model medium that was examined. Dependingon the values chosen, the optical thickness of thesphere varied from approximately 75 to 105 trans-port mean-free path lengths 1tmfp2 31 tmpf 51@1µa 1 µ8s24. Assuming a value of 1 mm for the tmfpof NIR light in tissue, this corresponds to a tissuethickness of approximately 7.5–10.5 cm. The geom-etry and dimensions chosen are intended to repre-sent a crude model of multilayer tissue structuressuch as a breast or a head. For the latter, the outerlayer would represent the skin and the underlyingskull and the inner layer would correspond to thegray and the white matter, while the central objectwould simulate a pathology, such as a bleed in the

736 APPLIED OPTICS @ Vol. 35, No. 4 @ 1 February 1996

ventricles. For other tissues, the outer layer can beviewed as overlying fat or muscle. Each layer isassigned optical properties within the ranges of mosttissue types. The object is different from the sur-rounding layers in either its absorption or scatteringcoefficient. Here, we consider only the case of anobject located at the center of the inner layer so thatthe test medium is in fact a spherically symmetric,three-layer medium.An analytic solution for the total field in this

piecewise homogeneous structure, because of a sourcelocated on the surface of the outer layer, is derived bythe use of the normal mode-series 1NMS2 methodwith appropriate boundary conditions. For a givenset of background-medium properties, this solutionis then used to derive the total field on the surface ofthe outer layer for various modulation frequencies,object sizes, and object properties. The objectives ofthis study are to determine 112 the modulation fre-quency that achieves an optimal compromise be-tween signal detectability and sensitivity to thepresence of an object, 122 the minimum detectableobject radius for given properties of the object, 132 thesmallest detectable change in the absorption coeffi-cient for a given object radius, and 142 the smallestdetectable change in the scattering coefficient for agiven object radius. Four types of background me-dia are examined: type I, in which the outer layer ismore absorbing and both layers are equally scatter-ing; type II, in which the outer layer is less absorbingand both layers are equally scattering; type III inwhich the outer layer is more scattering and bothlayers are equally absorbing; and type IV in whichthe outer layer is less scattering and both layers areequally absorbing. In addition to identifying limitson detectability, we report several new findings thatmay prove useful in evaluating methods for imagingdense scattering media.In the following, we first derive the general solu-

tion for the total field for an arbitrary spherically

Fig. 1. Illustration of a spherical object embedded in a spheri-cally symmetric, two-layer, highly scattering infinite medium,along with the source and detector configuration.

symmetric, three-layer, infinite medium. We thenpresent results for the four types of backgroundmedia described above.

2. Problem Formulation and Solution

We consider a spherical object embedded in a spheri-cally symmetric, two-layer background medium.The geometry of this general three-layer structure isshown in Fig. 1. The radii of the object, the innerlayer, and outer layer are denoted by a, b, and c,respectively. We adopt a spherical coordinate sys-tem with its origin at the center of the inner sphere.The absorption and scattering coefficients of theouter layer, the inner layer, and the object aredenoted by µa1 and µ8s1, µa2 and µ8s2, and µa3 and µ8s3,respectively. The problem is to determine the totalfield at various points on the surface of the outerlayer, r 5 1r, u, f2, where r 5 c, that results from asinusoidally varying, intensity-modulated pointsource of light placed at specific point, r8 5 1r8, u8, f82,where r8 5 c on the same surface.For computational convenience, we neglect the

boundary effect attributable to light exiting themedium across the external boundary and assumeinstead that the outer layer extends infinitely be-yond the spherical surface where the source anddetectors are located. Previous studies5 that com-pared the solutions for finite and infinite mediashowed that the wave fronts in these two cases havesimilar shapes, except near the boundaries, and thatthe variations caused by an embedded object in theinner layer are unaffected by the boundaries of theouter layer. This is especially true when the detec-tor is opposite the source in a transmission geometry.Let G1r, r82 represent the photon density at r that

is due to a unit-strength point source at r8. ThenG1r, r82 is the Green’s function for the followingnonhomogeneous Helmholtz wave equation2,10,11:

=2Gj 1r, r82 1 kj2Gj 1r, r82 5 2d1r 2 r82,

for r [ Vj, j 5 1, 2, 3. 112

In Eq. 112, V1 refers to r$ b 1i.e., the outer layer2,V2 toa # r , b 1i.e., the inner layer2, and V3 to 0 # r , a1i.e., the object2. The squared complex wave numberis given by

kj2 5 2µajDj

1 iv

vjDj

, 122

with v 5 0 for dc and v fi 0 for ac and where vj is thespeed of a photon in Vj, Dj 5 1@31µaj 1 µ8sj2 is thediffusion coefficient, and µaj and µ8sj are the absorp-tion and scattering coefficients, respectively, in Vj.The wave number is, in general, complex and isdenoted by kj 5 bj 1 iaj, where the attenuation factor

aj and the phase factor bj are, respectively,

aj 5 Œµaj@2Dj 31 1 1v@nµaj22 1 141@2,

bj 5 Œµaj@2Dj 31 1 1v@nµaj22 2 141@2, 132

The Green’s function G is subject to the followingboundary conditions12,13:

112 The photon density G is continuous at r 5 band r 5 a:

G2 0 r5b 5 G1 0 r5b, G3 0 r5a 5 G2 0 r5a. 142

122 The photon current J is continuous at r 5 band r 5 a:

r · J2 0 r5b 5 r · J1 0 r5b, r · J3 0 r5a 5 r · J2 0 r5a. 152

132 G is bounded at r 5 0 and satisfies a radiationcondition at r= `.

The total field in the outer layer can, in general, beinterpreted as a superposition of the incident spheri-cal waveGi and a scattered spherical waveGs1, i.e.,

G11r, r82 5 Gi1r, r82 1 Gs11r, r82, r . b. 162

Similarly, the total field in the inner layer can berepresented as the sum of a transmitted sphericalwaveGt1 and a scattered fieldGs2:

G21r, r82 5 Gt21r, r82 1 Gs21r, r82, a , r , b. 172

The total field in the object consists of a transmittedspherical waveGt3:

G31r, r82 5 Gt31r, r82, r , a. 182

Each of the five waves in Eqs. 142–182 can beexpanded into a series of Legendre functions andspherical Bessel functions,14,15 as follows:

112 Outer layer:

Gi1r, r82 5 3ik14p on50

`

om50

n

12 2 dm212n 1 121n 2 m2!

1n 1 m2!4· 3hn1121k1r82 jn1k1r2Pn

m1cos u2Pnm1cos u82

3 cos m1f 2 f824, 192

Gs11r, r82 5 3ik14p on50

`

om50

n

Amn12 2 dm212n1 121n 2 m2!

1n 1 m2!4· 3hn1121k1r82hn1121k1r2Pn

m1cos u2Pnm1cos u82

3 cos m1f 2 f824. 1102


122 Inner layer:

Gt21r, r82 5 3ik14p on50

`

om50

n

Bmn12 2 dm212n 1 121n 2 m2!

1n 1 m2!4· 3hn1121k1r82 jn1k2r2Pn

m1cos u2Pnm1cos u82

3 cos m1f 2 f824, 1112

Table 1. Optical Properties of Background-Media Types I–IV

Medium

Outer Layer Inner Layer

Diameter1tmfp2

µa1cm212

µ8s

1cm212µa

1cm212µ8s

1cm212

Type I 0.08 10.0 0.04 10.0 80Type II 0.02 10.0 0.04 10.0 80Type III 0.04 20.0 0.04 10.0 105Type IV 0.04 5.0 0.04 10.0 75


Gs21r, r82 5 3ik14p on50

`

om50

n

Cmn12 2 dm212n 1 121n 2 m2!

1n 1 m2!4· 3hn1121k1r82 yn1k2r2Pn

m1cos u2Pnm1cos u82

3 cos m1f 2 f824. 1122

132 Object:

Gt31r, r82 5 3ik14p on50

`

om50

n

Dmn12 2 dm212n 1 121n 2 m2!

1n 1 m2!4· 3hn1121k1r82 jn1k3r2Pn

m1cos u2Pnm1cos u82

3 cos m1f 2 f824. 1132

In Eqs. 192–1132, Amn, Bmn, Cmn, and Dmn are theunknown expansion coefficients; jn, yn, and hn112 arethe spherical Bessel function, the spherical Neumanfunction, and the spherical Hankel function of the

Fig. 2. Plots of the amplitudes of the background field for types I–IV of the background media versus the detector angle for severalmodulation frequencies 1 f 5 0, for dc2. The optical properties of the background media are listed in Table 1.

first kind, respectively16;Pnm is the associated Legen-

dre function; dm 5 1 form 5 0; and dm 5 2 form $ 1.Substituting Eqs. 192–1132 into Eqs. 162–182 and using

appropriate boundary conditions, we find the coeffi-cients Amn in Eq. 1102 are the ratios of two 4 3 4determinants:

Amn 5

0A1* a12 a13 0

A2* a22 a23 0

0 a32 a33 a340 a42 a43 a44

00a11 a12 a13 0

a21 a22 a23 0

0 a32 a33 a340 a42 a43 a44

0, 1142

where the 14 nonvanishing elements ai j, A1*, andA2* are listed inAppendixA.In this paper, we use 0Gt 0 and Ft to represent the

magnitude and the phase, respectively, of the totalfield when the object is present, and 0Gb 0 and Fb,respectively, when the object is absent 1i.e., thebackground medium2. The total field for the back-ground medium is referred to as the backgroundfield. The difference between the total field in thepresence of an object and the background field is thescattered field from the object. The backgroundfield here plays the same role as the incident fielddoes when one studies the scattering that is due toan object embedded in a homogeneous background.In the following, we measure the sensitivity of adetector to a hidden object by the relative amplitudechange dG and the phase change dF. These aredefined as

dG 5 1 20Gt 0

0Gb 0, 1152

dF 5 Fb 2 Ft, 1162

respectively.

Fig. 3. Plots of the phase shift of the background field for types I–IV of the background media versus the detector angle for severalmodulation frequencies. The optical properties of the background media are listed in Table 1.


The absorption and scattering coefficients of thefour different test media are listed in Table 1. Thegeometries of these background media are identical,with the radii of the inner and outer layers being b 53 cm and c 5 4 cm, respectively. Further, theabsorption and scattering coefficients of the innerlayer are fixed at µa2 5 0.04 cm21 and µ8s2 5 10 cm21.Note that the background media differ only in theabsorption and scattering coefficients of the outerlayer. The diameters of these media, in units oftransport mean-free path lengths, are approxi-mately 80 for types I and II, and 100 and 70 for typesIII and IV, respectively. In some cases, variationsin object properties increase these values by up to anadditional 10 tmfp. As will be seen, these four typesof background media lead to somewhat differentconclusions with respect to the detector’s sensitivityto embedded objects.For each background medium, we try to answer

four questions: 112At which modulation frequency fis the amplitude 0Gb 0 of the background field largeenough to be measured reliably, while the changes dGand dF, which are due to the presence of an object,


are detectable as defined below? Based on thestudy of question 1, we select a modulation fre-quency for use in studying the other three problems.122 For a fixed modulation frequency and fixed objectproperties, what is the radius, amin, of the smallestdetectable object? 132 For a fixed modulation fre-quency and a fixed object size, what is the smallestdetectable relative change in absorption, da,min 51µa3 2 µa22@µa2? 142 Similarly, what is the smallestdetectable relative change in scattering, ds,min 51µ8s3 2 µ8s22@µ8s2? In answering these questions, wedefine a detectable signal as one for which theamplitude of the scattered field is no smaller than0Gmin 0 5 10210. In addition, the relative change inamplitude and the absolute change in phase must begreater than or equal to dG,min 5 0.1% and dF,min 50.1°, respectively. These two requirements lead to athird condition, which is that the magnitude of thebackground field must be no smaller than 0Gb,min 0 51027. The derivation of these values is based onstate-of-the-art optical instrumentation. Similar as-sumptions have been used by others working in thisfield.5,17

Fig. 4. Plots of the percent change in the amplitude versus the detector angle for different modulation frequencies, as attributed to an0.5-cm-radius object, with µa3 5 0.06 cm21 and µ8s3 5 10.0 cm21, embedded in types I and II background media and with µa3 5 0.04 cm21

and µ8s3 5 15.0 cm21 embedded in types III and IV background media.

Although the solution presented above can be usedto derive the total field at any point for an arbitrarysource position, we present only the distribution ofthe total field on the surface of the outer layer at r5 cas a result of a source located on the same surface atr8 5 c, then f 5 u 5 0°. Further, because ofspherical symmetry, only detector responses on ahalf-circle defined by r 5 c, f 5 0°, and u 5 0° 2 180°are calculated. The detector at u 5 180° is oppositethe source on the other side of the sphere. Usingthe solution above, we have computed the back-ground fields for different background media andmodulation frequencies, and, for a fixed backgroundand a fixed modulation frequency, the total fields fordifferent object sizes and object properties. In thenext section, for each background medium, we firstplot the amplitude 0Gb 0 and the phase shift Fb of thebackground field 1Fig. 22. We then show the rela-tive changes in amplitude dG and phase shift dF thatare due to an embedded object 1Fig. 32. Both sets offigures are drawn for different modulation frequen-cies and are used to derive the answer to the firstquestion. Next, we show the dG and df plots for

different object sizes and object properties 1Figs. 4and 52. These figures are meant to answer the otherthree questions. All plots are drawn as functions ofthe detector angle u to evaluate the spatially depen-dent detector sensitivity to a hidden object. Wehave also calculated the minimum detectable objectsize and the relative changes in absorption andscattering for the detector angle that has the maxi-mum sensitivity 1i.e., the maximum change in ampli-tude or phase shift2. The reader can derive theanswers from the presented plots for other detectorangles.In our calculations, we assumed that the indices of

refraction ni of the different layers are n1 5 n2 5 n3 51.4, following Bolin et al.18 The speeds of light inthese layers are, correspondingly, v1 5 v2 5 v3 52.14 3 1010 cm@s.

3. Results

A. Influence of the Modulation Frequency

To determine the optimal modulation frequency fordifferent background media, we have evaluated the

Fig. 5. Plots of the phase change in degrees versus the detector angle for different modulation frequencies, as attributed to a0.5-cm-radius object, with µa3 5 0.06 cm21 and µ8s3 5 10.0 cm21, embedded in types I and II background media, and with µa3 5 0.04 cm21

and µ8s3 5 15.0 cm21, embedded in types III and IV background media.


amplitude and phase of the total field, with andwithout the object present, for several frequencies:f 5 0, 100, 200, 500MHz. The value f 5 0 applied todc. In Figs. 2 and 3 we show the amplitude andphase distributions of the background field as afunction of the detector angle at these modulationfrequencies for each of the four background mediatested. As expected, the amplitude of the back-ground field falls as the detector angle increases at afixed frequency or as the frequency increases at afixed detector angle. For example, for the type-Imedium at detector angle u 5 180°, the amplitude is3.5 3 1027 when f 5 0. This value falls to 1027 <0Gb,min 0 when f 5 200 MHz and to 8 3 1029 , 0Gb,min 0when f 5 500 MHz. The amplitude is greatest for adc incident field. The background-field amplitudesfor type-I and type-III media are approximately10–100-fold smaller than those for types II and IVbecause of the stronger absorption or scattering inthe outer layer. Figure 3 shows an increased phaseshift with an increasing detector angle or a highermodulation frequency.Figures 4 and 5 show the relative amplitude and

phase change caused by a 0.5-cm-radius object with


µa3 5 0.06 cm21 and µ8s3 5 10.0 cm21 that is embeddedin types-I and -II background media, and the sameobject but with µa3 5 0.04 cm21 and µ8s3 5 15.0 cm21,embedded in types-III and -IV background media.It can be seen that the relative amplitude and phasechanges are small when the detector angle is ,60°.This is because the background field is much stron-ger than the scattered field at these angles. Atlarger detector angles, the changes are more visibleand vary in a nonlinear manner with the detectorangle. In most cases the change in amplitude in-creases approximately monotonically with the detec-tor angle. For the type-I background medium, weobserve that the amplitude sensitivity decreaseswith an increasing modulation frequency and the dccomponent has the greatest amplitude sensitivity.Interestingly, nearly the opposite trend is seen in allother background-media types. An unusual obser-vation is the occurrence of a peak in the relativeamplitude change for type-IV media at a 500-MHzmodulation frequency and large detector angles.Overall qualitatively, we observed that, whereasthere is some dependence of the amplitude responseson the type of background medium as a function of

Fig. 6. Percent change in amplitude plotted versus the detector angle for different absorption coefficients µa3 of an object with a radius of0.5 cm and µ8s3 5 10.0 cm21, embedded in the types I–IV background media. The modulation frequency is 200 MHz.

the modulation frequency, no consistent pattern isevident.The change in phase due to the presence of the

object is shown in Figs. 5. These data show thatwhen an absorption-coefficient gradient exists 1types-Iand -II background media2, a peak in the phase plotis seen at intermediate detector angles 1120°–140°2.The amplitude and position of this peak vary withthe frequency of modulation, with higher positivevalues observed for higher frequencies. For type-IImedia we also observed that the algebraic sign of thephase change becomes negative at a certain detectorangle, which is a function of the modulation fre-quency. A qualitatively different trend is observedfor background media having a scattering-coefficientgradient 1types III and IV2. Little to no intermedi-ate peak is seen and, at larger detector angles andhigher modulation frequencies, the phase changesare increasingly negative.A quantitative comparison of the responses ob-

served for different background media reveal anunexpected result: The amplitude and phasechanges seen in type-I and type-IIImedia are greater,

respectively, than those seen for type-II and type-IVbackground media. The former two media havelarger absorption- and scattering-coefficient values,respectively, in the outer layer, than do Types-II and-IV media. Note that the difference in amplitudesensitivity is also seen for a dc source. This findingfor the type-III media is particularly noteworthy if itis considered that the diameter of this medium isalmost 50% greater than that of the type-IVmedium.Thus, we are observing greater sensitivity to theobject even though the absolute amplitude of emerg-ing light is significantly less than for the type-IVmedium, e.g., approximately 14 times less at u 5

180° 3see Figs. 21c2 and 21d24. Possible explanations,described in more detail in Section 4, include aselective suppression of detected photons that re-main in the outer layer, the trapping of photons inthe inner layer, or a combination of these.Based on the criterion that the amplitude of the

background field should be greater than 0Gb,min 0 5

1027, the modulation frequencies that achieve theoptimal compromise between signal strength and

Fig. 7. Phase change in degrees plotted versus the detector angle for different absorption coefficients µa3 of an object with a radius of 0.5cm and µ8s3 5 10.0 cm21, embedded in types I–IV background media. The modulation frequency is 200 MHz.


Table 2. Optimal Modulation Frequency, Minimum Detectable Object Radius, and Absorption and Scattering Variations for BackgroundMedia Types I–IV

MediumType

fopt1MHz2

Detectable Signal Change Object Radiusa Absorption and Scattering Variationb

SignalType

Amountof Change u

amin1cm2 u

da,min1%2 u

ds,min1%2

I 200 dG $0.1% 180° 0.22 180° 3.75 180° 2.0200 dF $0.1° 140° 0.40 140° 25.0 180° 5.0

II 300 dG $0.1% 180° 0.25 180° 5.0 180° 4.0300 dF $0.1° 180° 0.50 180° 50.0 180° 4.0

III 200 dG $0.1% 180° 0.20 130° 12.5 180° 2.5200 dF $0.1° 180° 0.23 130° 37.5 180° 4.0

IV 500 dG $0.1% 180° 0.25 140° 20.0 180° 4.5500 dF $0.1° 180° 0.30 140° 50.0 180° 8.0

ada 5 50%; f 5 200 MHz.ba 5 0.5 cm; f 5 200 MHz.

relative response are approximately 200, 300, 200,and 500 MHz for types-I–IV background media,respectively. For comparison purposes, however,we chose a frequency of 200 MHz for all the back-ground media in the following studies.


B. Effect of Variations in the Absorption Coefficient of theObjectThe results shown in Figs. 6 and 7 show the com-puted sensitivities to a change in the object’s absorp-tion coefficient as a function of the detector angle for

Fig. 8. Percent change in amplitude plotted versus the detector angle for different scattering coefficients µ8s3 of an object with a radius of0.5 cm and µa3 5 0.04 cm21, embedded in types I–IV background media. The modulation frequency is 200 MHz.

each of the background media tested. For thesestudies the size of the object and its scatteringcoefficient are fixed at a 5 0.5 cm and µ8s 5 10 cm21,respectively. In each case, the absorption coeffi-cient of the object was varied over a range ofapproximately 0.04–0.1 cm21. At increasing detec-tor angles and µa values, the change in amplitudeincreases monotonically in a nonlinear fashion.Figures 7 show that an absorption perturbationproduces a peak in the phase plot at intermediatedetector angles 1120°–140°2. The amplitude and po-sition of this peak vary with the object’s µa, with themaximum absolute phase shift increasing as µaincreases. For type-II media we also observe thatat detector angles .140°, the algebraic sign of thephase change becomes negative and is a function ofµa. Further inspection shows, interestingly, the ex-istence of an isosbestic point at u < 145°, indicatingthat at this angle the detector is insensitive to achange in the absorption coefficient of the object.1In spectroscopy the term isosbestic point refers to awavelength at which variations in the ratio of twocompounds in equilibrium do not produce any changein the net absorbance. We borrowed the term here

because we observed that at a specific angle themeasured phase is independent of the value of theobject’s absorption coefficient in one instance or of itsradius in another.2 A quantitative comparison ofthe phase shift and the relative changes in ampli-tude for the different media reveals a greater sensi-tivity to absorption perturbations in the object fortypes-I and -III media. By interpolating the data inthese plots, we determine that the smallest detect-able absorption perturbation varies from approxi-mately 0.0015 to 0.005 cm21 13.75–12.5%2 for ampli-tude measurements and from 0.01 to 0.02 cm21

125–50%2 for phase measurements, depending on theproperties of the background medium. See Table 2.

C. Effect of Variations in the Scattering Coefficient of theObject

The results in Figs. 8 and 9 show the computedsensitivities to a change in the object’s scatteringcoefficient as a function of the detector angle for eachbackground medium tested. Note that for thesestudies the size of the object and its absorptioncoefficient are fixed at a 5 0.5 cm and 0.04 cm21,respectively. In each case the scattering coefficient

Fig. 9. Phase change in degrees plotted versus the detector angle for different scattering coefficients µ8s3 of an object with a radius of 0.5cm and µa3 5 0.04 cm21, embedded in types I–IV background media. The modulation frequency is 200 MHz.


of the object was varied over a range of approxi-mately 10–20 cm21. A qualitative examination ofthe amplitude and phase plots reveals little differ-ence in the responses seen for different media. Inall cases, an increase in the scattering coefficient ofthe object produces a nonlinear, monotonic increasein the relative amplitude change, with larger differ-ences seen at larger detector angles. A qualita-tively similar response is seen in the phase data.In this case, the phase change becomes increasinglymore negative at larger detector angles and forlarger values of the object’s scattering coefficient.By linearly interpolating the data for other interme-diate values of µ8s3, we have found that the smallestdetectable change, ds,min, at u 5 180°, varies from 0.2to 0.45 cm21 12–4.5%2 for amplitude measurementsand from 0.4 to 0.8 cm21 14–8%2 for phase measure-ments. See Table 2.A comparison of the responses in the phase and

amplitude data produced by a perturbation in absorp-tion 1Figs. 6 and 72 to a perturbation in scattering1Figs. 8 and 92 shows important qualitative andquantitative differences. Qualitatively, we observe


that, unlike the trends seen in the amplitude data,the phase data show distinctly different responses toabsorption- and scattering-coefficient perturbations.A quantitative comparison reveals that amplitudemeasurements are proportionally more sensitive to achange in absorption, whereas phase measurementsare proportionally more sensitive to a change inscattering. In absolute terms, however, we ob-served that in all cases amplitudemeasurements aremore sensitive than phase measurements. Furtherinspection reveals that the trends seen in the phasedata can be further influenced by differences in theoptical properties of the outer layer. A comparisonof the phase data obtained from type-I and type-IIImedia shows that an absorption perturbation causesa greater phase change for a type-I medium, whereasa scattering perturbation causes a larger phase shiftfor a type-III medium.

D. Effect of Variations in the Object Diameter

The results shown in Figs. 10 and 11 show thecomputed sensitivities to a change in the object’s sizeas a function of the detector angle for each of the

Fig. 10. Percent change in amplitude plotted versus the detector angle for different radii a of an object with µa3 5 0.06 cm21 and µ8s3 5

10.0 cm21, embedded in types I and II background media and with µa3 5 0.04 cm21 and µ8s3 5 15.0 cm21, embedded in types III and IVbackground media. The modulation frequency is 200 MHz.

Fig. 11. Phase change in degrees plotted versus the detector angle for different radii a of an object with µa3 5 0.06 cm21 and µ8s3 5 10.0cm21, embedded in types I and II background media, and µa3 5 0.04 cm21 and µ8s3 5 15.0 cm21, embedded in types III and IV backgroundmedia. The modulation frequency is 200 MHz.

background media tested. For these studies, theabsorption and scattering coefficients of the objectare fixed: for types-I and -II media, µa3 5 0.06 cm21

and µ8s3 5 10.0 cm21, respectively; for types-III and-IV media, µa3 5 0.04 cm21 and µ8s3 5 15.0 cm21.Thus, for a specified object size, only µa3 is perturbedin Types-I and -II media, and only µ8s3 is perturbed intypes-III and -IV media. The radius of the objectwas varied from approximately 0.2 to 1.0 cm. Aswas the case for the other perturbations, inspectionof the amplitude data reveals few or no qualitativedifferences in the responses seen for the differentmedium types. At increasing detector angles andobject sizes, the change in amplitude increasesmono-tonically in a nonlinear fashion. A similar compari-son for the phase data, however, shows a qualita-tively different response. For types-I and -II media,a perturbation in object size produces a peak in thephase plot at intermediate detector angles, 120°–140°. The amplitude and position of this peak varywith the object size, with higher positive valuesobserved for larger radii. In fact, the responses

seen for both the amplitude and the phase data fortypes-I and -II media to a perturbation in object sizeappear qualitatively almost indistinguishable fromthose produced by a perturbation in the absorptioncoefficient 1see Figs. 5 and 62. A qualitatively simi-lar response is observed for types-III and -IV mediawith respect to the changes seen in the amplitudeand phase plots for a perturbation in scattering 1seeFigs. 7 and 82. Interestingly, we also observe anisosbestic point in the phase data at u < 145° for thetype-II medium. These findings demonstrate that,when there is a gradient in only one parameter, thistype of measurement does not readily distinguishbetween a change in the object’s intrinsic properties1i.e., the coefficients2 and a change in its extrinsicproperties 1i.e., the volume2. Interpolation of theplotted data shows that the smallest detectableperturbation in the object radius varies from approxi-mately 0.2 to 0.25 cm for amplitude data and from0.23 to 0.5 cm for phase data, depending on theproperties of the background medium. See Table 2.


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4. Discussion and Conclusion

In this paper we have presented a detailed sensitiv-ity study for the PDW for imaging a sphericallysymmetric, two-layer, highly scattering, infinite me-dium containing a spherical object. The changes inthe amplitude and the phase of the total field thatare caused by the object have been calculated asfunctions of the modulation frequency, the objectradius, and the differences between the absorptionand scattering coefficients of the object and thebackground. For each of the four different types ofbackground media, we have derived the optimalmodulation frequency, the smallest detectable objectsize, and the minimal detectable differences in ab-sorption and scattering coefficients. The results aresummarized in Table 2. The quantitative resultsare based on certain assumptions concerning theprecision of commercially available optical detectors.For a different set of detector precisions, one caneasily derive corresponding results from the plotspresented in Figs. 4–11.

A. Comparison of Background-Media Types I–IV

A comparison of the corresponding figures for thedifferent background media permits the followinginteresting points to be observed:

1. Increasing µa or µ8s in the outer layer results inan enhanced sensitivity to the embedded object.The suggestion that enhanced sensitivity to an em-bedded object can result from the presence of a morestrongly absorbing or scattering outer layer haspotentially important implications for experimentalstudies. Should this effect prove to be significant inpractice, one could easily consider placing a balloonfilled with a strongly absorbing or scattering me-dium around the test medium being investigated.The balloon could be constructed to permit opticalfibers to be in direct contact with a target tissue and,in doing so, act to trap photons that would ordinarilyhave exited the medium at the air–tissue interfaceby reflecting them back into the medium.2. More information is gained about the inhomo-

geneity if frequency-domain light-detection tech-niques are used to acquire data for image reconstruc-tion. To obtain a good spatial resolution, one mustincrease the modulation frequency. The optimalmodulation frequency depends on the background-medium properties. For all media except type I, thesensitivity of the detected light to the embeddedobject increases with an increasing modulation fre-quency.3. Whereas the absolute sensitivity of the ampli-

tude measurements is greater than that of the phasemeasurements to changes in the absorption andscattering properties of an embedded object, theformer measurements are relatively insensitive tothe type of perturbation. Phase measurements, onthe other hand, exhibit a much greater ability todistinguish absorption perturbations from scatter-ing perturbations.


4. Amplitude measurements are proportionallymore sensitive to absorption changes, whereas phasemeasurements are proportionally more sensitive toscattering changes.5. Evidence of degenerate detector responses was

observed under several of the conditions tested.This phenomenon may contribute to the difficulty ofquantification of the absolute optical coefficients oftissue by the use of simplified measurement andanalysis schemes.

A possible physical interpretation of some thesefindings is depicted in Fig. 12. Illustrated are twoclasses of photon paths: those that interact withthe object at some point along their path 1class I2, andthose that do not interact but can still enter thedetector 1class II2. Point 1 above is explained if werecognize that increasing µa in the outer layer prefer-entially removes class-II photons because this groupincludes those having the longest paths in the outerlayer. This removal serves to increase the fractionof detected photons that interact with the object,thus enhancing the sensitivity to the object. Asomewhat similar explanation holds for the observedeffect of increased scattering in the outer layer on theobject’s detectability. Here, preferential removal ofclass-II photons occurs because, as scattering in-creases, the probability that scattered photons willultimately enter the detector is reduced. In es-sence, once photons are removed, it becomes moredifficult for the scattered photons to come back. Analternative explanation is that the presence of a

Fig. 12. Illustration of two classes of photon paths. Class-Iphotons 1dashed photon paths2 interact with the object 1darkestcircle, at the center2 at some point along their path. Class-IIphotons 1solid photon paths2 do not interact with the object but canstill enter the detector.

more-scattering outer layer could serve to trap thosephotons that enter the inner layer, permitting greaterinteraction with the object.The findings listed under point 2 above regarding

the frequency dependence of the object’s detectabil-ity on the background-medium type follow from thevariation that can be reasonably expected in detectorresponse with the changing overall 1i.e., averagedover the medium2 attenuation. Thus, becausetypes-I and -III media attenuate the propagatingsignal more in the outer layer than do types II andIV, it can be expected that, for a specified sensitivitylimit, higher modulation frequencies will remainmeasurable for the latter media types. This findingis independent of the photon class and is instead afunction of detector sensitivity. We note, however,that this explanation cannot completely describe theobservations. In particular, we point out that un-like types-II–IV media, the type I medium exhibitsthe greatest amplitude sensitivity to the object atzero frequency 1dc2.In point 3 above, two observations are made: For

a specified perturbation, amplitude measurementsexhibit a greater absolute sensitivity, whereas phasemeasurements can distinguish between absorptionand scattering perturbations. Phenomenologically,this means that a specified perturbation 1absorption,scatter, or object size2 influences the number ofphotons reaching a detector more than it does theirtime of arrival. The observation regarding the dif-ferential response of phase data is not unexpected.It occurs because the changes in scattering influencethe mean time of photon arrival at a detector quitedifferently than do the changes in absorption.Point 4 is explained as follows: Changes in the

scattering coefficient influence the paths of essen-tially all photons migrating through a specifiedregion. Corresponding changes in absorption, onthe other hand, preferentially influence those pho-tons having the longest paths. Therefore, it can beexpected that phase measurements, which are pro-portional to the mean path length, will be moresensitive to variations in the scattering coefficient.The observation that amplitude measurements arepreferentially sensitive to variations in the absorp-tion coefficient can be explained if we consider that,whereas absorption removes photons, scatteringcauses their redistribution, allowing some to ulti-mately enter the detector.In Figs. 71b2 and 111b2, an isosbestic point in the

phase data is clearly evident. In light of the geom-etry of the media studied, we believe that thisobservation may be a special case. Basically, be-cause the object is centrally located, increases inobject size, for example, will decrease the time ofarrival of the PDW at the object–inner-layer bound-ary in the hemisphere closest to the source andincrease the arrival time in the hemisphere oppositethe source. For a centrally located object, thesedifferences might be expected to cancel at a particu-lar point. If this argument is correct, an isosbestic

point would be less likely to occur if the object was offcenter. For media containing multiple embeddedobjects such as tissue, however, isosbestic pointsmight well occur as a result of the expected multipleinteractions.It is worthwhile to examine the results summa-

rized in Table 2. Contrary to our initial expecta-tions, the optimal sensitivity to the studied perturba-tions is not always associated with detectors locatedopposite the source 1u 5 180°2. Instead, in manyinstances we observed the optimal sensitivity atintermediate angles in the forward hemisphere190° , u # 180°2. We interpret this finding to indi-cate that, because of multiple scattering that resultsfrom the presence of two internal boundaries, partialcancellation of the scattered PDW’s can occur in aposition-dependent manner. The magnitude of thiseffect varies with the composition of the backgroundmedium, as the characteristics of scattered wavesproduced at the outer-layer–inner-layer boundaryvary.As noted above, we observed different sensitivities

for the amplitude and phase data. In essentially allcases studied, amplitude measurements were moresensitive than phase measurements to perturba-tions in object size or absorption or scattering coeffi-cients. As mentioned above, this difference indi-cates that such changes influence the number ofarriving photons more than it does their meanarrival time. Interestingly, this difference in sensi-tivity seemed more pronounced for perturbations inabsorption than in scattering or object size. For thecases studied, we find that, with a 50% contrast inabsorption between the object and the inner layer,the smallest detectable object size is approximately 4mm in diameter. As all media studied have opticalcoefficients in the range expected for most tissues,we believe that centrally located objects having thiscontrast and size should be detectable for tissuehaving a thickness of approximately 7.5–10.5 cm.Detection of even smaller off-center objects should bepossible. Also shown in Table 2 are the contrastlimits for a 1-cm-diameter, centrally located object.Generally, we observe a greater sensitivity to frac-tional changes in the scattering coefficient. Over-all, the minimum perturbations required to producea measureable change in a detector response arequite small. We also list the optimal frequency forthe greatest sensitivity to the introduced perturba-tions, which varies between 200 and 500 MHz.

B. Implications of Results for Imaging Thick Tissues

Anumber of laboratories, including ours, are investi-gating the possibility of computing tomographic im-ages of thick tissues on the basis of the detection ofhighly scattered optical photons.8,19–22 Whereas werecognize that the conditions examined here arehighly idealized, we believe our results should pro-vide an estimate of the image quality that might beachieved. Real media such as breast tissue arehighly inhomogeneous and have nonuniform bound-


aries. Air–tissue interfaces can be treated as per-fect absorbers. Variations in tissue geometry, e.g., acompressed versus an uncompressed state, can leadto additional light losses. Taken together, it can beexpected that, for tissues having an optical thicknessequivalent to the media examined here, greaterlosses will be experienced. As a result, the accuratedetection of these signals will require correspond-ingly higher source intensities. The sensitivity val-ues reported in Table 2 correspond to an input-source intensity of 1014 photons@s, or 26.5 µW at 750nm. Source intensities of 26.5 mW 31017 photons@sfor a beam cross-sectional area of, say, 2 mm 1<1W@cm224 are easily tolerated at NIR frequencies bymost nonocular tissues. Thus, we believe there isconsiderable room to accommodate the expectedlosses. This possible intensity range suggests thatthe sensitivity values listed in Table 2 may underes-timate the achievable values to the extent thatsensitivity is photon limited. Given that these val-ues represent only a fraction of the absorption andscattering coefficients of most tissues, it is temptingto suggest that the results obtained here support theprospect of detecting relatively small changes in theoptical coefficients for deeply buried anomalies bythe use of frequency–domain methods. We recog-nize, however, that other sources of noise exist, inparticular those arising frommotion artifacts.Recently Moon and Reintjes23 have reported that,

for media thicker than <35 tmfp, image resolutionwith diffuse light degrades approximately linearly,with a scale dependence of R 5 0.2d, where R is theresolution and d is the distance between the objectand detector. Note that this relation is a function ofonly distance and thus holds even for a perfectabsorber. Comparison of this value with those ofthe media examined here 1<70–100 tmfp2 indicatesthat the smallest object located in the center thatcould be resolved is 0.7–1.0 cm in size. Coinciden-tally this value is on the higher end of the detectionlimits computed here. Taken together, and with thepossible differences arising from the different sourceconditions used here neglected, these findings indi-cate that, for a single source and single projection,objects of this size and the contrast values listed inTable 2, which are quite small, might be resolved.We wish to emphasize, however, that the conclusionof Moon et al.,23 while valid, does not pertain toresolution limits achievable by the use of tomo-graphic-imaging schemes. Indeed, Graber et al.24have recently reported experimental results demon-strating edge-detection limits of ,1 mm for recon-structed images of a single inclusion 1a black rod2located in the center of a cylindrical vessel having anoptical thickness of approximately 150–200 tmfp.This result was obtained from tomographic measure-ments with a time-independent laser source operat-ing at 720 nm. Assuming a transport mean freepath length value of 1 mm, these data yield aresolution 15–20 times greater than the theoreticallimit achievable from a single projection.


The reason why tomographic methods are capableof yielding improved resolution is that the multipro-jection algorithms of the type used by Graber et al.24basically represent a spatial-filtering technique.The best one can learn from a single projection isthat an object lies somewhere along a particularpath. Essentially, one knows where the object is1i.e., along the path2 but not where it is not 1i.e., whichportions along the path it is not located2. In amultiple-scattering medium, this path becomesblurred, causing a loss of resolution. With multipleprojections, however, one is essentially able to deter-mine where the object is not located. More for-mally, this filtering effect proceeds directly from thestructure of the Jacobian matrix.19We view the results presented here as being

moderately encouraging regarding the prospect fordeveloping practical imaging schemes for studyingthick tissue structures. We wish to emphasize thatmany important issues remain to be addressed.Of particular concern is how best to deal with thedifficult issue of ill posedness in the inverse problem.One approach we have recently begun to explore isthe use of priors derived from magnetic-resonancedata.25 We choose the magnetic-resonance tech-nique because of the excellent contrast it affords forsoft tissues. By assigning optical coefficients toappropriately segmented images, one can derivepotentially accurate reference states.

Appendix A.

The 14 nonvanishing elements appearing in Eq. 1142are as follows:

x A1* 5 jn1k1b2,x A2*5 k1D1 j8n 1k1b2,x a11 5 2hn1121k1 b2,x a12 5 jn1k2b2,x a13 5 yn1k2b2,x a21 5 2k1D1hn11281k1b2,x a22 5 k2D2 j8n1k2b2,x a23 5 k2D2y8n 1k2b2,x a32 5 jn1k2a2,x a33 5 yn1k2a2,x a34 5 2 jn1k3a2,x a42 5 k2D2y8n1k2a2,x a43 5 k2D2y8n1k2a2,x a44 5 2k3D3 j8n1k3a2.

This work was supported in part by the NationalInstitutes of Health under grant RO1-CA59955, byOffice of Naval Research grant N000149510063, andby the New York State Science and TechnologyFoundation.

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