differential diffuse optical tomography - physics & astronomy

Differential diffuse optical tomographyVasilis Ntziachristos12 Britton Chance2 and AG Yodh3

1Department of Bioengineering 2Department of BiochemistryBiophysics 3Department of Physics and Astronomy

University of Pennsylvania Philadelphia PA 19104-6089 USA

vasilismailmedupenneduhttp wwwmedupennedu~oisgoisghtml

Abstract We formulate a perturbative solution for the heterogeneousdiffusion equation which demonstrates how to use differential changes indiffuse light transmission to construct images of tissue absorption changesfollowing contrast agent administration The analysis exposesapproximations leading to an intuitive and simplified inverse algorithmshows explicitly why transmission geometries are less susceptible to errorthan the remission geometries and why differential measurements are lesssusceptible to surface artifacts These ideas about differential diffuseoptical tomography are not only applicable to tumor detection andcharacterization using contrast agents but also to functional activationstudies with or without contrast agents and multi-wavelengthmeasurements1999 Optical Society of AmericaOCIS codes (1106960) Tomography (1706960) Tomography (1703010) Imagereconstruction techniques (1705280) Photon migration (1703660) Light propagation intissues (1705270) Photon density waves (9999999) Contrast agents

References and links

1 BW Pogue ldquoFocus issue Biomedical diffuse optical tomography ndash Introductionrdquo Opt Express 4 230-230(1999) httpepubsosaorgopticsexpresstocv4n8htm

2 F Kelcz G Santyr ldquoGadolinium-Enhanced Breast MRIrdquo Critical Reviews in Diagnostic Imaging 36 287-338(1995)

3 F Barkhof J Valk ORHommes P Scheltens ldquoMeningeal Gd-DTPA enhancement in multiple-sclerosisrdquo Am JNeuroradiology 13 397-400 (1992)

4 C Tilcock ldquoDelivery of contrast agents for magnetic resonance imaging computed tomography nuclear medicineand ultrasoundrdquo Adv Drug Deliver Rev 37 33-51 (1999)

5 RP Kedar D Cosgrove VR McCready JC Bamber ER Carter ldquoMicrobubble contrast agent for colorDoppler US Effect on breast massesrdquo Radiology 198 679-686 (1996)

6 ML Melany EG Grant S Farooki et al ldquoEffect of US contrast agents on spectral velocities In vitro evaluationrdquoRadiology 211 427-431 (1999)

7 SE Thompson V Raptopoulos RL Sheiman MMJ McNicholas P Prassopoulos ldquoAbdominal helical CTMilk as a low-attenuation oral contrast agentrdquo Radiology 211 870-875 (1999)

8 R Jain S Sawhney P Sahni K Taneja M Berry ldquoCT portography by direct intrasplenic contrast injection anew techniquerdquo Abdominal Imaging 24 272-277 (1999)

9 LW Nunes MD Schnall SG Orel MG Hochman CP Langlotz CAReynolds MH Torosian ldquoCorrelationof lesion appearance and histologic findings for the nodes of a breast MR imaging interpretation modelrdquoRadiographics 19 79-92 (1999)

10 SG Orel MD Schnall VA Livolsi RH Troupin ldquoSuspicious Breast-Lesions - MR-Imaging With Radiologic-Pathological Correlationrdquo Radiology 190 485-493 (1994)

11 BW Pogue M Testorf T McBride U Osterberg K Paulsen ldquoInstrumentation and design of a frequency-domaindiffuse optical tomography imager for breast cancer detectionrdquo Opt Express 1 391-403 (1997)httpwwwopticsexpressorgoearchivesource2827htm

12 V Ntziachristos XH Ma B Chance ldquoTime-correlated single photon counting imager for simultaneous magneticresonance and near-infrared mammographyrdquo Rev Sci Instr 69 4221-4233 (1998)

13 V Ntziachristos AG Yodh MD Schnall B Chance ldquoComparison between intrinsic and extrinsic contrast formalignancy detection using NIR mammographyrdquo Proc SPIE 3597 565-570 (1999)

14 A Ishimaru Wave propagation and scattering in random media (New York Academic Press 1978)

(C) 1999 OSA 8 November 1999 Vol 5 No 10 OPTICS EXPRESS 23012606 - $1500 US Received September 07 1999 Revised November 06 1999

15 JB Fishkin E Gratton ldquoPropagation of photon-density waves in strongly scattering media containing an absorbingsemi-infinite plane bounded by a straight edgerdquo Opt Soc Am A 10 127-140 (1993)

16 AC Kak M Slaney Principles of Computerized Tomographic Imaging (New York IEEE Press 1988)17 SR Arridge P van der Zee M Cope DT Delpy ldquoReconstruction methods for infra-red absorption imagingrdquo

Proc SPIE 1431 204-215 (1991)18 MS Patterson B Chance BC Wilson ldquoTime Resolved Reflectance and Transmittance for the Noninvasive

Measurement of Tissue Optical Propertiesrdquo J Appl Opt 28 2331-2336 (1989)19 TJ Farrell MS Patterson B Wilson ldquoA diffusion-theory model of spatially resolved steady-state diffuse

reflectance for the non-invasive determination of tissue optical properties in-vivordquo Med Phys 19 879-888 (1992)20 RC Haskell LO Svaasand TT Tsay TC Feng MS McAdams BJ Tromberg ldquoBoundary conditions for the

diffusion equation in radiative transferrdquo J Opt Soc Am A 11 2727-2741 (1994)21 R Aronson ldquoBoundary conditions for diffusion of lightrdquo J Opt Soc Am A 12 2532-2539 (1995)22 MA OrsquoLeary DA Boas B Chance AG Yodh ldquoExperimental images of heterogeneous turbid media by

frequency-domain diffusing-photon tomographyrdquo Opt Lett 20 426-428 (1995)23 V Ntziachristos A Hielscher AG Yodh B Chance ldquoPerformance of perturbation tomography with highly

heterogeneous media under the P1 approximationrdquo in Biomedical Optics Advances in Optical Imaging PhotonMigration and Tissue Optics OSA Technical Digest CLEOEurope AMB3-1 211-213 (1999)

24 V Ntziachristos XH Ma AG Yodh B Chance ldquoMultichannel photon counting instrument for spatially resolvednear infrared spectroscopyrdquo Rev Sci Instr 70 193-201 (1999)

1 Introduction

The use of contrast agents in Diffuse Optical Tomography[1] (DOT) for disease diagnosticsand for probing tissue functionality follows established clinical imaging modalities such asMagnetic Resonance Imaging (MRI) [234] Ultrasound [456] and X-ray ComputedTomography[478] (CT) Contrast agent administration provides for accurate differenceimages of the same heterogenous tissue volume under nearly identical experimentalconditions This approach yields superior diagnostic information In MRI of breast cancerfor example it is the relative signal enhancement compared to the baseline image and thearchitectural features of this enhancement that aid in characterizing lesions as benign ormalignant [910] While architectural features are unlikely to improve breast cancerdiagnosis using DOT in current implementations due to the low resolution achieved [1112]the relative signal enhancement after contrast agent administration could significantlyincrease sensitivity and specificity

The theory presented in this paper is motivated by clinical DOT measurements ofthe human breast [13] In these measurements optical data are obtained before and afterintravenous administration of the optical contrast agent Indocyanine Green (ICG) ICG is anabsorber and a fluorophore in the Near Infrared (NIR) here we focus on its absorptionfunction In principle DOT images of the breast before and after ICG administration may bereconstructed and subtracted However in practice a more robust approach derives images ofthe differential changes due to the extrinsic perturbation In the latter case experimentalmeasurements use the exact same geometry within a few minutes of one another therebyminimizing positional and movement errors and instrumental drift Furthermore the use ofdifferential measurements eliminates systematic errors associated with the different mediumrequired to calibrate operational parameters of the instrument or to provide a baselinemeasurement for independent reconstructions of the pre- and post ICG images In additionthe effect of surface absorbers such as hair or skin color variation is also minimized

The main analytical difficulties of the differential approach arise because the mediaare inhomogeneous Thus the total diffuse light field in the ICG perturbation problem doesnot separate into a homogeneous background field and a scattered field in a straightforwardway Furthermore the average background properties particularly the absorption change asa result of contrast agent administration

In this paper we derive a perturbation solution that can be used to yield differentialimages of induced absorption perturbations This is the primary effect of many molecularoptical contrast media that are administered in low concentration The solution is alsoapplicable for imaging the differential variations in tissue scattering In obtaining these


results we will expose the full consequences of inhomogeneous tissue volumes and identifythe conditions wherein these effects are small We demonstrate the superiority of the methodcompared to the typical perturbation approach using simulated data of contrast-enhancedbreast The method is also suitable for imaging differential changes as arise in measurementsof functional activity such as brain activation or muscle exercise (the only requirement is thatbaseline measurements of the same tissue volume are taken during the procedure) or inmeasurements at multiple wavelengths In the following analysis the term pre-ICG breast isused to indicate the ldquobaselinerdquo breast before the contrast agent injection and the term post-ICG breast marks the breast following contrast agent administration

2 Theory

Photon transport in highly scattering low absorbing media such as tissue can beapproximated as a diffusional process [14] For heterogeneous media the diffusion equationin the frequency domain is [15]

)()())( ()]()([ rcSrUracjrUrDrrrrr

minus=sdotminus+nablasdotnabla microω (1)

Here U( rr

) is the photon fluence [Wsdotcm-2] ω is the source modulation frequency c is thespeed of light in the medium [cmsdots-1] )(3)( rcrD s

rr

microprimeasymp is the medium diffusion coefficient

[cm2sdots-1] )(s rr

microprime the medium reduced scattering coefficient [cm-1] )(a rr

microprime the medium

absorption coefficient [cm-1] and )(rSr

the source term [Wsdotcm-3] In this work we consider

solutions of the heterogeneous diffusion equation in the frequency domain employing theperturbation approach [1617]

The first order perturbation expansion divides the absorption ()(a rr

microprime ) and diffusion

( )(rDr

prime ) coefficients of the pre-ICG breast into spatially varying ( )()(a rDrrr primeprime δmicroδ ) and background

components ( 00Da primeprimemicro ) ie )()( 0a rr aa

rr

microδmicromicro prime+prime=prime and )()( 0 rr DDDrr

primeprimeprime += δ Throughout this paper a

single prime denotes pre-ICG tissue volumes In the Rytov approximation [16] the total photondensity wave measured at position dr

r

due to a source at position srr

is written as the product

of two components ie

)()()(

0

ωφωω ds rrsc

dsds errUrrUrr

rrrrprime

prime=prime

(2)

where the scattered field )(sc ωφ ds rrrrprime is produced by heterogeneities ( )()(a rDr

rr

primeprime δmicroδ ) and

the incident field )(0 ωds rrUrr

prime is the field that would have been detected from the same

medium if these heterogeneities were not present These terms are incorporated into thediffusion equation whose formal solution can be expressed as an integral equation using theappropriate Greenrsquos function for the geometry implemented For simplicity we present thisanalysis for an infinite medium This theory however can be easily extended to other simplegeometries such as semi-infinite or slab using weights derived with the method of imagesources [18] and the appropriate extrapolated boundary condition [19 20 21]

The first order perturbative solution of the heterogeneous diffusion equation yields

rdrDDrrrWrDrrrWrr adssV

aadsadssc

rrrrrrrrrrr

)]()( )()([)( 0000 primeprimeprimeprime+primeprimeprimeprime=prime int δωmicromicroδωmicroωφ (3)

where )( sa WW primeprime represents the absorption (scattering) weight of the voxel at position rr

due

to a source at srr

and a detector at drr

In the infinite medium these weights are


ds

sd

rrki

rrkirrki

sd

dsadsa

e

ee

rrrr

rr

D

cDrrrW rr

rrrr

rrrr

rr

rrr

minusprime

minusprimeminusprime sdotsdotminussdotminus

minussdot

primeminus=primeprimeprime

20

00)(4

)(π

ωmicro (4)

ds

sd

rrki

s

rrki

d

rrki

dsadss

e

rr

e

rr

e

D

rrDrrrW

rr

rrrr

rrrrrr

rrr

minusprime

minusprimeminusprime

minusnablasdot

minusnabla

sdotprime

minus=primeprimeprime

20

00)(4

)(π

ωmicro (5)

0

0

D

ik a

prime+primeminus

=prime ωmicroυ

(6)

Following the administration of a contrast agent the background optical propertieschange The new post-ICG total field can be written in a similar form

)()()( 0

ωφωω ds rrscdsds errUrrU

rr

rrrr primeprimeprimeprime=primeprime (7)

Here )(sc ωφ ds rrrrprimeprime is the field component scattered from the post-ICG heterogeneities (ie

)()(a rDrrr

primeprimeprimeprime δmicroδ with respect to the new background optical properties 0a0D primeprimeprimeprimemicro ) and

)(0 ωds rrUrr

primeprime is the incident field obtained from the homogeneous background medium

with 0a0D primeprimeprimeprimemicro The first order perturbative solution of the heterogeneous diffusion equation

yields


aadsadssc

rrrrrrrrrrr

)]()( )()([)( 0000 primeprimeprimeprimeprimeprimeprimeprime+primeprimeprimeprimeprimeprimeprimeprime=primeprime int δωmicromicroδωmicroωφ (8)

where Wa (Ws) represents the absorption (scattering) weight of voxels at position rr

due to asource at sr

r

and for a detector at drr

ds

sd

rrki

rrkirrki

sd

dsadsa

e

ee

rrrr

rr

D

cDrrrW rr

rrrr

rrrr

rr

rrr

minusprimeprime

minusprimeprimeminusprimeprime sdotsdotminussdotminus

minussdot

primeprimeminus=primeprimeprimeprimeprimeprime

20

00)(4

)(π

ωmicro (9)

ds

sd

rrki

s

rrki

d

rrki

dsadss

e

rr

e

rr

e

D

rrDrrrW

rr

rrrr

rrrrrr

rrr

minusprimeprime

minusprimeprimeminusprimeprime

minusnablasdot

minusnabla

sdotprimeprime

minus=primeprimeprimeprimeprimeprime

20

00)(4

)(π

ωmicro (10)

0

0

D

jk a

primeprime+primeprimeminus=primeprime ωmicroυ

(11)

Combining Eq2 with Eq7 we obtain the relative scattered field ie

primeprimeprime

sdotprimeprimeprime

=primeminusprimeprime=0

0lnU

U

U

Uscscsc φφφ

(12)

We will show that )(sc ωφ ds rrrr

can be attributed primarily to perturbations created

by the contrast agent injection )( ωds rrUrrprime and )( ωds rrU

rrprimeprime are the actual measurements on the


pre- and post- ICG breast respectively and )(0 ωds rrUrr

prime )(0 ωds rrUrr

primeprime can be determined from

the average optical properties of the pre-and post- ICG breast (see discussion section)During the administration of an absorption contrast agent the scattering properties

of tissue are not expected to change Therefore we assume 000 DDD =primeprime=prime and

)()( rDrDrr primeprime=prime δδ Substitution of Eq3 and Eq8 into Eq12 yields

rdrDrrrWrDrrrWrr aadsaV

aadsadssc

rrrrrrrrrrr

)]()( )()([)( 0000 microδωmicromicroδωmicroωφ primeprimeprimeprimeminusprimeprimeprimeprimeprimeprimeprimeprime= int (13)

Here we have also assumed )()( 0000 ωmicroωmicro DrrrWDrrrW adssadss primeprimeasympprimeprimeprimeprime rrrrrr This is a very good

approximation when the average absorption change due to the contrast agent is small or inthe transmission geometry (see Appendix and discussion for absorption variations below)

Let )(rICGa

r

δmicro be the total absorption perturbation due to the ICG injection that

includes both position-independent and position-dependent contributions Then)(a rrmicro primeprime can be

written

)()()()( 00a rrrr ICGaaaaa

rrrr

δmicromicroδmicromicroδmicromicro +prime+prime=primeprime+primeprime=primeprime (14)

so that)()()( 00 rrr a

ICGaaaa

rrr

microδδmicromicromicromicroδ prime++primeprimeprime=primeprime minus (15)

The quantity )()( 00 rr ICGaaa

rela

rr

δmicromicromicroδmicro +primeprimeprime= minus represents the position-dependent

absorption heterogeneities induced by the contrast agent The relative scattered field iscomputed by substitution of Eq 15 into Eq13 It depends on contrast agent inducedabsorption heterogeneities and on pre-ICG tissue absorption heterogeneities

intint primeprimeprimeprime+primeprime=V

aaaV

relaadssc rdrWWrdrWrr

rrrrrr

)() -()()( microδδmicroωφ (16)

The second integral in Eq16

int primeprimeprimeprime=V

aaa rdrWWSrr

)() -( microδ (17)

describes the influence of the pre-existing (intrinsic) absorption heterogeneity of the breaston the relative scattered field Since the intrinsic heterogeneity is weighted by the difference

aa WW primeprimeprime - the influence of this term can be quite small Using the analytical forms of the

weights (Eq4 and Eq 9) we can write out Eq17 explicitly ie

( )int primesdotminussdotminussdotminus

minussdotprimeprimesdot

primeprime= primeprimeminusprime

V

arRkki

sd

dsa rdre

rrrr

rrW

D

cS

rr

rrrr

rr

r

)(1)(4

)()(2

0

microδπ

(18)

where

dssd rrrrrrrRrrrrrrr

minusminusminus+minus=)( (19)

The term )()( rRkkierprimeprimeminusprime in Eq18 is approximately unity and Sasymp0 when the average absorption

increase due to the ICG injection is very small (ie kk primeprimeasympprime ) Usually however kk primeprimeneprime Forexample the recommended ICG dosage for humans (025mgkg) introduces an average microa

increase within the interval [0005-0015] cm-1 depending on breast vascularization [13] Fig


1a and b show the amplitude and phase of the term )()( rRkkierprimeprimeminusprime respectively for different

)(rRr

as a function of the post-ICG breast absorption coefficient for a source detector

separation sd rrrr

minus =6cm using the geometry of Figure 1c The background microa=005 cm-1 and

the background microsrsquo=10cm-1

Figure 1 (a) Amplitude of the term )()( rRkkierprimeprimeminusprime as a function of the average

absorption coefficient of the post-ICG breast (b) Phase of the term )()( rRkkierprimeprimeminusprime as a

function of the average absorption coefficient of the post-ICG breast (c) Testgeometry for calculations in (a) and (b)

The deviation of )()( rRkkierprimeprimeminusprime from unity increases for perturbations farther from the line

adjoining source and detector (ie as rrrr sd

rrrr

minus+minus grows larger than sd rrrr

minus when α

increases) However the probability for photons to pass through these ldquodistantrdquoperturbations decreases exponentially via the weight aW primeprime in the integrand of Eq18 Hence

accumulated contributions of the heterogeneities at large α are small Figure 2 plots thedeviations introduced into )( ωφ dssc rr

rr

by taking S=0 Figure 2a depicts the ratio of the

amplitude detected with no approximation to the amplitude detected assuming S=0Similarly Fig 2b depicts the phase shift between the phase detected with no approximationand the phase detected assuming S=0 The error is plotted for a single perturbation atdifferent positions α for the geometry depicted in Fig 1c The values assumed in Eq 16 were

)(ra

r

microδ prime =005 cm-1 )(rrel

a

r

δmicro =005 cm-1 and the background optical properties a0microprime =005cm-1

a0micro primeprime =005cm-1 and smicroprime =10cm-1

α

perturbation

source detector

rrdrr

minusrrsrr

minus(c)

ds rrrr

minus =6cm

005 0054 0058 006

Amplitude

00560052

α=15cm

α=1cm

α=05cm

(a)

005 0054 0058 0060

0004

0008

0012

0016

0020

0024

Phase Shift

00560052

absorption coefficient (cm-1)

α=15cm

α=1cm

α=05cm

(rad)(b)


102

104

106

108

1


Figure 2 (a) Amplitude deviation and (b) Phase shift introduced in the fieldmeasured in Eq 16 when S is assumed zero The calculation is done as a functionof the distance α for the geometry shown in Fig 1c assuming 200MHzbackground a0microprime =005cm-1 a0micro primeprime =005cm-1 smicroprime =10 cm-1 rel

aδmicro =005 cm-1 and

)(ra

r

microδ prime =001 cm-1

The simulation of Fig2 explicitly shows that the errors introduced because of theapproximation S=0 are very small for physiologically relevant optical properties (ierelatively small )(a r

rmicroδ prime ) provide the most probable photon paths The same behavior is

exhibited for the scattering weights as shown in the Appendix Eq16 thus becomes

rdrWU

U

U

Urr

V

relaadssc

rrrr

)(ln)(0

0 int primeprimecong

primeprimeprime

sdotprimeprimeprime

= δmicroωφ (20)

Our conclusions do not change when image sources are invoked to satisfy morecomplex boundary conditions such as semi-infinite or slab geometries In these cases

)()( rRkkierprimeprimeminusprime will appear in all the terms corresponding to image sources Note however that

the assumption that Sasymp0 is best suited for slab geometry where rrrr sd

rrrr

minus+minus asymp sd rrrr

minus for

the most probable photon paths This condition is not always true for reflectance geometry

Notice that the differential measurements are insensitive to surface artifacts such assmall skin absorbers and hair under a certain source or detector The term )(ra

r

microδ prime in Eq18could be used to approximate surface heterogeneities by taking r

r

to be close to mediumsurface near to the corresponding source or detector The influence of such terms is virtuallyzero since in such a geometry 0)( congrR

r

and subsequently S=0For image reconstruction Eq 20 is discretized into a sum of volume elements [22]

(voxels) and the scattered field is obtained at each modulation frequency ω for every source-detector pair ω For n voxels and m=otimesptimesq measurements where o is the number of sourcesp is the number of detectors and q is the number of frequencies employed the discretizationyields a set of coupled linear equations

sdot

primeprimeprimeprime

primeprimeprimeprime=

)(

)(

)(

)( 1

1

111111

1

nrela

rela

mna

ma

naa

qdpsomsc

dssc

r

r

WW

WW

rr

rr

r

M

r

L

MOM

L

rrM

rr

δmicro

δmicro

ωφ

ωφ

(21)

0 1 2 3 409985

0999

09995

1

10005 Amplitude change

distance α (cm)

(a)0 1 2 3 4

-12

-1

-08

-06

-04

-02

0x 10-3

distance α (cm)

Phase shift

(b)

(rad)


Inverting the weightsrsquo matrix determines the spatial map of absorption due to contrast agentinjection

3Results and Discussion

Although equations 20 and 21 resemble the result of typical perturbation analyses there arefundamental differences and constraints that must be considered when using them First theparameter imaged is the synthetic perturbation term )()( 00 rr ICG

aaa

rel

a

rr

δmicromicromicroδmicro +primeprimeprime= minus Secondly the

relative scattered field scφ depends both on the ratio UU primeprimeprime of the actual pre-ICG and post-

ICG measurements and the multiplicative term ))(exp(00 ds rrkkiUUrr

minussdotprimeprimeminusprime=primeprimeprime This term

expresses the change in the incident field due to the average absorption coefficient increaseof the post-ICG breast Its use in Eq12 leads to significant reconstruction improvements[23] Note that the term 00 UU primeprimeprime depends on | rrs

rr

minus | and not on )(rRr Therefore the arguments

that led on the elimination of S from Eq16 cannot be applied to this term since| rrs

rr

minus | )(rRr

gtgt The term 00 UU primeprimeprime can be analytically calculated for simple geometries such as

infinite semi-infinite or slab or calculated numerically for more complicated geometries ifwe know the average optical properties of the pre- and post- ICG breast

Using simulated data we have examined the performance of Eq20 to image thecontrast-enhanced breast as compared to

1) The typical Rytov approximation which assumes

primeprimeprime

=primeprimeU

Usc lnφ namely

rdrWU

Urr

V

ICG

aadssc

rrrr

)(ln)( int prime=

primeprimeprime

=primeprime δmicroωφ (22)

This comparison captures a critical feature of our work The classical perturbationapproach that does not consider the average absorption increase due to the extrinsiccontrast

2) Eq17 including the term S namely

rdrWSrrV

relaadssc

rrrr

)()( int primeprime=minus δmicroωφ (23)

This comparison investigates the effect of the approximation S=0 assumed inEq20

3) A difference image produced by subtracting the post-ICG image from the pre-ICGimage Both pre- and post- images were produced using Eq3 assuming ahomogeneous medium as baseline The optical properties of the homogeneousmedium were 0amicroprime =003cm-1 and 0smicroprime =8cm-1

In order to perform the comparisons we obtained two MRI coronal slices of a humanbreast one before and one after contrast enhancement Figure 3a depicts the T1-weighted MRimage This image depicts structure White regions correspond primarily to adipose (fatty)tissue while dark regions correspond to parenchymal (glandular) tissue Figure 3b depicts thesignal enhancement of the same T1-weighted image due to injection of the MRI contrastagent Gd-DTPA The Gd-DTPA enhancement is superimposed in color An infiltratingductal carcinoma (shown in yellow) demonstrated the highest signal enhancement Gd-DTPA and ICG have similar distribution patterns Here we assume that the Gd-DTPAdistribution reflects the ICG distribution


Figure 3 (a) T1-weighted MR coronal slice of a human breast (b) Gd-DTPAdistribution (in color) of the same coronal slice A ductal carcinoma appears inyellow

We converted the MR images to optical property maps separating four structuresbased on the image intensity information (by applying appropriate thresholds) The cancer isassumed having two states pre- and post- ICG contrast The structures selected and thecorresponding absolute optical properties are shown in TABLE I The optical properties arechosen to simulate breast optical properties as obtained from our breast clinicalmeasurements [13]

TABLE I

Absolute optical properties of the different structures used for the simulations

Structure microa (cm-1) microsrsquo (cm-1)

Adipose 003 8

Parenchymal 006 8

ICG-background 009 8

Pre-ICG Cancer 009 8

Post ICG Cancer 016 8

Scattering has been assumed constant for all structures for simplicity The resultingabsorption maps are shown in figure 4 The medium surrounding the breast was arbitrarilysimulated as a highly absorbing diffuse medium (microa =030cm-1 microsrsquo= 8cm-1) The averageabsorption of the pre- and post- ICG breast were found to be 0amicroprime =00473 cm-1

and 0amicro primeprime =00589 cm-1 so that average absorption increase due to the ICG is 00 aa micromicro primeprimeprime minus =00116

cm-1

Figure 4 Absorption maps used for the simulation (a) pre-ICG breast (b) post-ICG breast

The maps of figure 3 served as an input to a finite-differences implementation of thefrequency-domain diffusion equation The simulation assumed 7 sources and 21 detectors as

a b

0

255

000

015

030

16cm

6cm

a b


shown in Figure 5 The frequency employed was 200MHz No noise was added in theforward data

Figure 5 Sources and detector arrangement used for the simulation The regionreconstructed is outlined with a green double line

For reconstruction purposes the region of interest (indicated in Fig 5 as a greendouble line) was segmented into 35x25 voxels The inversion was performed using theAlgebraic Reconstruction Technique (ART) [16] with relaxation parameter λ=01Convergence was assumed when an additional 100 iterations did not change the result morethan 01 The simulated image and the reconstructed )(rrel

a

r

δmicro for the three cases examined

are shown in Figure 6

Figure 6 Reconstruction results imaging the region indicated with the greendouble line on Fig5 assuming an infinite slab geometry a)The reconstructed imageusing Eq20 b)The reconstructed image using Eq22 c)The reconstructed imageusing Eq23 d) The result of subtracting an image of the post-ICG breast(reconstructed relative to a homogeneous baseline medium) from an image of thepre-ICG breast (reconstructed relatively to the same baseline medium)The opticalproperties of the homogeneous medium were 0amicroprime =003cm-1 and smicroprime = 8cm-1

1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro


The superior performance of Eq20 compared to the typical perturbative formulation(Eq 22) can be evaluated by examining Fig6a and b Although both methods resolve thecancerous lesion with comparable positional and size accuracy the typical formulation(Fig6b) yields several strong artifacts close to the boundaries These artifacts illustrate thatin the presence of distributed absorption the perturbative method converges preferentially tolocalized regions of high absorption This is often true when inverting underdeterminedsystems Eq20 on the other hand removes the ldquoaverage absorption increaserdquo from themeasurement vector Therefore Fig 6a images weaker perturbations introduced by the ICGinjection relative to the average absorption increase Since by construction the perturbationmethod works especially well for weak perturbations [16] it is expected that the use of Eq20 will more accurately image the heterogeneous medium The same behavior is expected fora Born-type [16] perturbative formulation We note that Fig6b images the )(rICG

a

r

δmicro and notthe )(rrel

a

r

δmicro as in Fig 6a and 6c Therefore it is reasonable that the reconstructed value forcancer in Fig 6b is higher than the value reconstructed in Fig6a and 6c The difference inreconstructed values equals approximately the average absorption increase in the post-ICGbreast ( 00 aa micromicro primeprimeprime minus =00116 cm-1)

Figure 6c has been produced after correcting the measurement vector with S insteadof setting it to zero as in Fig 6a Only minor differences exist between the two images as hadbeen predicted in Figure 2 In this simulation the pre-ICG cancer had a contrast of 21 to theaverage pre-ICG background value This contribution has most likely resulted in the minordifferences observed between the two images especially in the structures close to theboundaries

Figure 6d is the result of subtracting an image of the post-ICG breast from an imageof the pre-ICG breast Here the )(rICG

a

r

δmicro is imaged The magnitude of the cancer is slightlyoverestimated and its size is significantly overestimated Similarly to Fig 6b strong artifactsappear close to the boundary A distributed absorption is also reconstructed which does notcorrespond to the ICG distribution and is also an artifact Compared to the other approachesthe subtraction yields the most artifacts

In these simulations the average optical properties were known by simpleintegration over the optical property map In our clinical implementation [13] the averageoptical properties of the pre-ICG breast are calculated by fitting the experimental time-domain data obtained to the appropriate solution of the diffusion equation for the geometryused Furthermore an algorithm has been developed [24] that calculates thedifference 00 aa micromicro primeprimeprime minus (necessary to calculate both 00 UU primeprimeprime and )(rICG

a

r

δmicro ) with an accuracy of the

order of 10-3 cm-1To conclude we have described a formulation of perturbation theory which is

particularly well suited for image reconstructions of differences in the absorption propertiesof tissues as a result of optical contrast agent administration Importantly these resultsenable the extraction of differential contrast agent absorption even within media that areheterogeneous in the absence of the contrast agent Our primary result is an intuitiveequation which is valid over a large range of conditions We have shown explicitly whatthese corrections are and how these corrections can be included in more careful analysesThe results should be applicable for a broad range of other DOT applications whereinbaseline and ldquostimulatedrdquo measurements are available particularly functional imaging inbrain and muscle


AcknowledgementsThe authors acknowledge useful discussions with Bruce Tromberg Monica Holboke JoeCulver and Deva Pattanayak This work was supported in part by the National Institutes ofHealth grant no RO1-CA75124 and in part by the National Institutes of Health grant noRO1- CA60182


APPENDIX

In this section we show that the ratio of the scattering weight )( 00 ωmicro DrrrW adss primeprimeprimeprime rrr

to

)( 00 ωmicro DrrrW adss primeprime rrr

is sufficiently small so that when subtracting Eq3 from Eq8 to produce

Eq13 the scattering terms cancel out Instead of expressing all the terms analytically weplot the ratio

)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss

primeprimeprimeprimeprimeprimeprime

=rrr

rrr

(A1)

for the geometry of Fig1a and for the same absorption variation of the post ICG-breastFigure A1 shows that the amplitude and phase of R remains very small for the most probablephoton paths (ie when the distance α is small) For higher α R increases but similarly tothe arguments that reduced Eq16 to Eq20 the contribution of terms for higher α isweighted less Therefore for the small absorption changes considered here and for smallvariations of the diffusion coefficient )()( rDrD

rr primeprime=prime δδ the scattering terms can be ignored

when reconstructing the absorption perturbation only due to contrast agent injection Whenintroducing boundaries the assumption of small α compared to the source-detector distancefor the most probable photon paths works better for transmittance geometry

Figure A1 (a) Amplitude and (b) Phase of R as a function of the average absorption coefficient of thepost-ICG breast plotted for the geometry of Fig 1c

005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)

absorption coefficient (cm-1)absorption coefficient (cm-1)


15 JB Fishkin E Gratton ldquoPropagation of photon-density waves in strongly scattering media containing an absorbingsemi-infinite plane bounded by a straight edgerdquo Opt Soc Am A 10 127-140 (1993)

16 AC Kak M Slaney Principles of Computerized Tomographic Imaging (New York IEEE Press 1988)17 SR Arridge P van der Zee M Cope DT Delpy ldquoReconstruction methods for infra-red absorption imagingrdquo

Proc SPIE 1431 204-215 (1991)18 MS Patterson B Chance BC Wilson ldquoTime Resolved Reflectance and Transmittance for the Noninvasive

Measurement of Tissue Optical Propertiesrdquo J Appl Opt 28 2331-2336 (1989)19 TJ Farrell MS Patterson B Wilson ldquoA diffusion-theory model of spatially resolved steady-state diffuse

reflectance for the non-invasive determination of tissue optical properties in-vivordquo Med Phys 19 879-888 (1992)20 RC Haskell LO Svaasand TT Tsay TC Feng MS McAdams BJ Tromberg ldquoBoundary conditions for the

diffusion equation in radiative transferrdquo J Opt Soc Am A 11 2727-2741 (1994)21 R Aronson ldquoBoundary conditions for diffusion of lightrdquo J Opt Soc Am A 12 2532-2539 (1995)22 MA OrsquoLeary DA Boas B Chance AG Yodh ldquoExperimental images of heterogeneous turbid media by

frequency-domain diffusing-photon tomographyrdquo Opt Lett 20 426-428 (1995)23 V Ntziachristos A Hielscher AG Yodh B Chance ldquoPerformance of perturbation tomography with highly

heterogeneous media under the P1 approximationrdquo in Biomedical Optics Advances in Optical Imaging PhotonMigration and Tissue Optics OSA Technical Digest CLEOEurope AMB3-1 211-213 (1999)

24 V Ntziachristos XH Ma AG Yodh B Chance ldquoMultichannel photon counting instrument for spatially resolvednear infrared spectroscopyrdquo Rev Sci Instr 70 193-201 (1999)

1 Introduction

The use of contrast agents in Diffuse Optical Tomography[1] (DOT) for disease diagnosticsand for probing tissue functionality follows established clinical imaging modalities such asMagnetic Resonance Imaging (MRI) [234] Ultrasound [456] and X-ray ComputedTomography[478] (CT) Contrast agent administration provides for accurate differenceimages of the same heterogenous tissue volume under nearly identical experimentalconditions This approach yields superior diagnostic information In MRI of breast cancerfor example it is the relative signal enhancement compared to the baseline image and thearchitectural features of this enhancement that aid in characterizing lesions as benign ormalignant [910] While architectural features are unlikely to improve breast cancerdiagnosis using DOT in current implementations due to the low resolution achieved [1112]the relative signal enhancement after contrast agent administration could significantlyincrease sensitivity and specificity

The theory presented in this paper is motivated by clinical DOT measurements ofthe human breast [13] In these measurements optical data are obtained before and afterintravenous administration of the optical contrast agent Indocyanine Green (ICG) ICG is anabsorber and a fluorophore in the Near Infrared (NIR) here we focus on its absorptionfunction In principle DOT images of the breast before and after ICG administration may bereconstructed and subtracted However in practice a more robust approach derives images ofthe differential changes due to the extrinsic perturbation In the latter case experimentalmeasurements use the exact same geometry within a few minutes of one another therebyminimizing positional and movement errors and instrumental drift Furthermore the use ofdifferential measurements eliminates systematic errors associated with the different mediumrequired to calibrate operational parameters of the instrument or to provide a baselinemeasurement for independent reconstructions of the pre- and post ICG images In additionthe effect of surface absorbers such as hair or skin color variation is also minimized

The main analytical difficulties of the differential approach arise because the mediaare inhomogeneous Thus the total diffuse light field in the ICG perturbation problem doesnot separate into a homogeneous background field and a scattered field in a straightforwardway Furthermore the average background properties particularly the absorption change asa result of contrast agent administration

In this paper we derive a perturbation solution that can be used to yield differentialimages of induced absorption perturbations This is the primary effect of many molecularoptical contrast media that are administered in low concentration The solution is alsoapplicable for imaging the differential variations in tissue scattering In obtaining these



2 Theory




Here U( rr


rr


[cm2sdots-1] )(s rr








( )(rDr



rr




r




)()()(

0

ωφωω ds rrsc

dsds errUrrUrr

rrrrprime

prime=prime

(2)


rr







aadsadssc

rrrrrrrrrrr



due

to a source at srr




ds

sd

rrki

rrkirrki

sd

dsadsa

e

ee

rrrr

rr

D

cDrrrW rr

rrrr

rrrr

rr

rrr

minusprime


minussdot


20

00)(4

)(π

ωmicro (4)

ds

sd

rrki

s

rrki

d

rrki

dsadss

e

rr

e

rr

e

D

rrDrrrW

rr

rrrr

rrrrrr

rrr

minusprime


minusnablasdot

minusnabla

sdotprime


20

00)(4

)(π

ωmicro (5)

0

0

D

ik a

prime+primeminus

=prime ωmicroυ

(6)


)()()( 0


rr



)()(a rDrrr


)(0 ωds rrUrr



yields


aadsadssc

rrrrrrrrrrr




r


ds

sd

rrki

rrkirrki

sd

dsadsa

e

ee

rrrr

rr

D

cDrrrW rr

rrrr

rrrr

rr

rrr

minusprimeprime


minussdot


20

00)(4

)(π

ωmicro (9)

ds

sd

rrki

s

rrki

d

rrki

dsadss

e

rr

e

rr

e

D

rrDrrrW

rr

rrrr

rrrrrr

rrr

minusprimeprime


minusnablasdot

minusnabla

sdotprimeprime


20

00)(4

)(π

ωmicro (10)

0

0

D

jk a


(11)


primeprimeprime

sdotprimeprimeprime


0lnU

U

U

Uscscsc φφφ

(12)













aadsadssc

rrrrrrrrrrr




Let )(rICGa

r



written


rrrr



ICGaaaa

rrr



rela

rr




aaaV


rrrrrr




aaa rdrWWSrr

)() -( microδ (17)







V

arRkki

sd

dsa rdre

rrrr

rrW

D

cS

rr

rrrr

rr

r

)(1)(4

)()(2

0

microδπ

(18)

where








)(rRr


separation sd rrrr








rrrr


minus when α



rr



)(ra

r


a

r



α

perturbation

source detector

rrdrr

minusrrsrr

minus(c)

ds rrrr

minus =6cm

005 0054 0058 006

Amplitude

00560052

α=15cm

α=1cm

α=05cm

(a)

005 0054 0058 0060

0004

0008

0012

0016

0020

0024

Phase Shift

00560052


α=15cm

α=1cm

α=05cm

(rad)(b)


102

104

106

108

1




)(ra

r





rdrWU

U

U

Urr

V

relaadssc

rrrr

)(ln)(0


primeprimeprime

sdotprimeprimeprime

= δmicroωφ (20)




rrrr


minus for



r


r


r



sdot



)(

)(

)(

)( 1

1

111111

1

nrela

rela

mna

ma

naa

qdpsomsc

dssc

r

r

WW

WW

rr

rr

r

M

r

L

MOM

L

rrM

rr

δmicro

δmicro

ωφ

ωφ

(21)

0 1 2 3 409985

0999

09995

1


distance α (cm)

(a)0 1 2 3 4

-12

-1

-08

-06

-04

-02

0x 10-3

distance α (cm)

Phase shift

(b)

(rad)





aaa

rel

a

rr






rr



rr

minus | )(rRr





primeprimeprime

=primeprimeU

Usc lnφ namely

rdrWU

Urr

V

ICG

aadssc

rrrr

)(ln)( int prime=

primeprimeprime




rdrWSrrV

relaadssc

rrrr








TABLE I



Adipose 003 8

Parenchymal 006 8






cm-1



a b

0

255

000

015

030

16cm

6cm

a b





a

r




1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro



a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)




2 Theory




Here U( rr


rr


[cm2sdots-1] )(s rr








( )(rDr



rr




r




)()()(

0

ωφωω ds rrsc

dsds errUrrUrr

rrrrprime

prime=prime

(2)


rr







aadsadssc

rrrrrrrrrrr



due

to a source at srr




ds

sd

rrki

rrkirrki

sd

dsadsa

e

ee

rrrr

rr

D

cDrrrW rr

rrrr

rrrr

rr

rrr

minusprime


minussdot


20

00)(4

)(π

ωmicro (4)

ds

sd

rrki

s

rrki

d

rrki

dsadss

e

rr

e

rr

e

D

rrDrrrW

rr

rrrr

rrrrrr

rrr

minusprime


minusnablasdot

minusnabla

sdotprime


20

00)(4

)(π

ωmicro (5)

0

0

D

ik a

prime+primeminus

=prime ωmicroυ

(6)


)()()( 0


rr



)()(a rDrrr


)(0 ωds rrUrr



yields


aadsadssc

rrrrrrrrrrr




r


ds

sd

rrki

rrkirrki

sd

dsadsa

e

ee

rrrr

rr

D

cDrrrW rr

rrrr

rrrr

rr

rrr

minusprimeprime


minussdot


20

00)(4

)(π

ωmicro (9)

ds

sd

rrki

s

rrki

d

rrki

dsadss

e

rr

e

rr

e

D

rrDrrrW

rr

rrrr

rrrrrr

rrr

minusprimeprime


minusnablasdot

minusnabla

sdotprimeprime


20

00)(4

)(π

ωmicro (10)

0

0

D

jk a


(11)


primeprimeprime

sdotprimeprimeprime


0lnU

U

U

Uscscsc φφφ

(12)













aadsadssc

rrrrrrrrrrr




Let )(rICGa

r



written


rrrr



ICGaaaa

rrr



rela

rr




aaaV


rrrrrr




aaa rdrWWSrr

)() -( microδ (17)







V

arRkki

sd

dsa rdre

rrrr

rrW

D

cS

rr

rrrr

rr

r

)(1)(4

)()(2

0

microδπ

(18)

where








)(rRr


separation sd rrrr








rrrr


minus when α



rr



)(ra

r


a

r



α

perturbation

source detector

rrdrr

minusrrsrr

minus(c)

ds rrrr

minus =6cm

005 0054 0058 006

Amplitude

00560052

α=15cm

α=1cm

α=05cm

(a)

005 0054 0058 0060

0004

0008

0012

0016

0020

0024

Phase Shift

00560052


α=15cm

α=1cm

α=05cm

(rad)(b)


102

104

106

108

1




)(ra

r





rdrWU

U

U

Urr

V

relaadssc

rrrr

)(ln)(0


primeprimeprime

sdotprimeprimeprime

= δmicroωφ (20)




rrrr


minus for



r


r


r



sdot



)(

)(

)(

)( 1

1

111111

1

nrela

rela

mna

ma

naa

qdpsomsc

dssc

r

r

WW

WW

rr

rr

r

M

r

L

MOM

L

rrM

rr

δmicro

δmicro

ωφ

ωφ

(21)

0 1 2 3 409985

0999

09995

1


distance α (cm)

(a)0 1 2 3 4

-12

-1

-08

-06

-04

-02

0x 10-3

distance α (cm)

Phase shift

(b)

(rad)





aaa

rel

a

rr






rr



rr

minus | )(rRr





primeprimeprime

=primeprimeU

Usc lnφ namely

rdrWU

Urr

V

ICG

aadssc

rrrr

)(ln)( int prime=

primeprimeprime




rdrWSrrV

relaadssc

rrrr








TABLE I



Adipose 003 8

Parenchymal 006 8






cm-1



a b

0

255

000

015

030

16cm

6cm

a b





a

r




1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro



a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)



ds

sd

rrki

rrkirrki

sd

dsadsa

e

ee

rrrr

rr

D

cDrrrW rr

rrrr

rrrr

rr

rrr

minusprime


minussdot


20

00)(4

)(π

ωmicro (4)

ds

sd

rrki

s

rrki

d

rrki

dsadss

e

rr

e

rr

e

D

rrDrrrW

rr

rrrr

rrrrrr

rrr

minusprime


minusnablasdot

minusnabla

sdotprime


20

00)(4

)(π

ωmicro (5)

0

0

D

ik a

prime+primeminus

=prime ωmicroυ

(6)


)()()( 0


rr



)()(a rDrrr


)(0 ωds rrUrr



yields


aadsadssc

rrrrrrrrrrr




r


ds

sd

rrki

rrkirrki

sd

dsadsa

e

ee

rrrr

rr

D

cDrrrW rr

rrrr

rrrr

rr

rrr

minusprimeprime


minussdot


20

00)(4

)(π

ωmicro (9)

ds

sd

rrki

s

rrki

d

rrki

dsadss

e

rr

e

rr

e

D

rrDrrrW

rr

rrrr

rrrrrr

rrr

minusprimeprime


minusnablasdot

minusnabla

sdotprimeprime


20

00)(4

)(π

ωmicro (10)

0

0

D

jk a


(11)


primeprimeprime

sdotprimeprimeprime


0lnU

U

U

Uscscsc φφφ

(12)













aadsadssc

rrrrrrrrrrr




Let )(rICGa

r



written


rrrr



ICGaaaa

rrr



rela

rr




aaaV


rrrrrr




aaa rdrWWSrr

)() -( microδ (17)







V

arRkki

sd

dsa rdre

rrrr

rrW

D

cS

rr

rrrr

rr

r

)(1)(4

)()(2

0

microδπ

(18)

where








)(rRr


separation sd rrrr








rrrr


minus when α



rr



)(ra

r


a

r



α

perturbation

source detector

rrdrr

minusrrsrr

minus(c)

ds rrrr

minus =6cm

005 0054 0058 006

Amplitude

00560052

α=15cm

α=1cm

α=05cm

(a)

005 0054 0058 0060

0004

0008

0012

0016

0020

0024

Phase Shift

00560052


α=15cm

α=1cm

α=05cm

(rad)(b)


102

104

106

108

1




)(ra

r





rdrWU

U

U

Urr

V

relaadssc

rrrr

)(ln)(0


primeprimeprime

sdotprimeprimeprime

= δmicroωφ (20)




rrrr


minus for



r


r


r



sdot



)(

)(

)(

)( 1

1

111111

1

nrela

rela

mna

ma

naa

qdpsomsc

dssc

r

r

WW

WW

rr

rr

r

M

r

L

MOM

L

rrM

rr

δmicro

δmicro

ωφ

ωφ

(21)

0 1 2 3 409985

0999

09995

1


distance α (cm)

(a)0 1 2 3 4

-12

-1

-08

-06

-04

-02

0x 10-3

distance α (cm)

Phase shift

(b)

(rad)





aaa

rel

a

rr






rr



rr

minus | )(rRr





primeprimeprime

=primeprimeU

Usc lnφ namely

rdrWU

Urr

V

ICG

aadssc

rrrr

)(ln)( int prime=

primeprimeprime




rdrWSrrV

relaadssc

rrrr








TABLE I



Adipose 003 8

Parenchymal 006 8






cm-1



a b

0

255

000

015

030

16cm

6cm

a b





a

r




1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro



a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)










aadsadssc

rrrrrrrrrrr




Let )(rICGa

r



written


rrrr



ICGaaaa

rrr



rela

rr




aaaV


rrrrrr




aaa rdrWWSrr

)() -( microδ (17)







V

arRkki

sd

dsa rdre

rrrr

rrW

D

cS

rr

rrrr

rr

r

)(1)(4

)()(2

0

microδπ

(18)

where








)(rRr


separation sd rrrr








rrrr


minus when α



rr



)(ra

r


a

r



α

perturbation

source detector

rrdrr

minusrrsrr

minus(c)

ds rrrr

minus =6cm

005 0054 0058 006

Amplitude

00560052

α=15cm

α=1cm

α=05cm

(a)

005 0054 0058 0060

0004

0008

0012

0016

0020

0024

Phase Shift

00560052


α=15cm

α=1cm

α=05cm

(rad)(b)


102

104

106

108

1




)(ra

r





rdrWU

U

U

Urr

V

relaadssc

rrrr

)(ln)(0


primeprimeprime

sdotprimeprimeprime

= δmicroωφ (20)




rrrr


minus for



r


r


r



sdot



)(

)(

)(

)( 1

1

111111

1

nrela

rela

mna

ma

naa

qdpsomsc

dssc

r

r

WW

WW

rr

rr

r

M

r

L

MOM

L

rrM

rr

δmicro

δmicro

ωφ

ωφ

(21)

0 1 2 3 409985

0999

09995

1


distance α (cm)

(a)0 1 2 3 4

-12

-1

-08

-06

-04

-02

0x 10-3

distance α (cm)

Phase shift

(b)

(rad)





aaa

rel

a

rr






rr



rr

minus | )(rRr





primeprimeprime

=primeprimeU

Usc lnφ namely

rdrWU

Urr

V

ICG

aadssc

rrrr

)(ln)( int prime=

primeprimeprime




rdrWSrrV

relaadssc

rrrr








TABLE I



Adipose 003 8

Parenchymal 006 8






cm-1



a b

0

255

000

015

030

16cm

6cm

a b





a

r




1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro



a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)




)(rRr


separation sd rrrr








rrrr


minus when α



rr



)(ra

r


a

r



α

perturbation

source detector

rrdrr

minusrrsrr

minus(c)

ds rrrr

minus =6cm

005 0054 0058 006

Amplitude

00560052

α=15cm

α=1cm

α=05cm

(a)

005 0054 0058 0060

0004

0008

0012

0016

0020

0024

Phase Shift

00560052


α=15cm

α=1cm

α=05cm

(rad)(b)


102

104

106

108

1




)(ra

r





rdrWU

U

U

Urr

V

relaadssc

rrrr

)(ln)(0


primeprimeprime

sdotprimeprimeprime

= δmicroωφ (20)




rrrr


minus for



r


r


r



sdot



)(

)(

)(

)( 1

1

111111

1

nrela

rela

mna

ma

naa

qdpsomsc

dssc

r

r

WW

WW

rr

rr

r

M

r

L

MOM

L

rrM

rr

δmicro

δmicro

ωφ

ωφ

(21)

0 1 2 3 409985

0999

09995

1


distance α (cm)

(a)0 1 2 3 4

-12

-1

-08

-06

-04

-02

0x 10-3

distance α (cm)

Phase shift

(b)

(rad)





aaa

rel

a

rr






rr



rr

minus | )(rRr





primeprimeprime

=primeprimeU

Usc lnφ namely

rdrWU

Urr

V

ICG

aadssc

rrrr

)(ln)( int prime=

primeprimeprime




rdrWSrrV

relaadssc

rrrr








TABLE I



Adipose 003 8

Parenchymal 006 8






cm-1



a b

0

255

000

015

030

16cm

6cm

a b





a

r




1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro



a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)





)(ra

r





rdrWU

U

U

Urr

V

relaadssc

rrrr

)(ln)(0


primeprimeprime

sdotprimeprimeprime

= δmicroωφ (20)




rrrr


minus for



r


r


r



sdot



)(

)(

)(

)( 1

1

111111

1

nrela

rela

mna

ma

naa

qdpsomsc

dssc

r

r

WW

WW

rr

rr

r

M

r

L

MOM

L

rrM

rr

δmicro

δmicro

ωφ

ωφ

(21)

0 1 2 3 409985

0999

09995

1


distance α (cm)

(a)0 1 2 3 4

-12

-1

-08

-06

-04

-02

0x 10-3

distance α (cm)

Phase shift

(b)

(rad)





aaa

rel

a

rr






rr



rr

minus | )(rRr





primeprimeprime

=primeprimeU

Usc lnφ namely

rdrWU

Urr

V

ICG

aadssc

rrrr

)(ln)( int prime=

primeprimeprime




rdrWSrrV

relaadssc

rrrr








TABLE I



Adipose 003 8

Parenchymal 006 8






cm-1



a b

0

255

000

015

030

16cm

6cm

a b





a

r




1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro



a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)






aaa

rel

a

rr






rr



rr

minus | )(rRr





primeprimeprime

=primeprimeU

Usc lnφ namely

rdrWU

Urr

V

ICG

aadssc

rrrr

)(ln)( int prime=

primeprimeprime




rdrWSrrV

relaadssc

rrrr








TABLE I



Adipose 003 8

Parenchymal 006 8






cm-1



a b

0

255

000

015

030

16cm

6cm

a b





a

r




1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro



a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)





TABLE I



Adipose 003 8

Parenchymal 006 8






cm-1



a b

0

255

000

015

030

16cm

6cm

a b





a

r




1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro



a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)






a

r




1cm

035cm9cm

ba

c d0001002003004005006

(cm-1)rel

aδmicro c

0

004

006

008

010

(cm-1)ICG

aδmicro

(cm-1)aδmicro ICG

0

002

004

006

008

-0020001002003004005006

(cm-1)rel

aδmicro



a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)




a

r


a

r




a

r



a

r







APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)





APPENDIX


to




)(

)(

00

00

ωmicroωmicro

DrrrW

DrrrWR

adss

adss


=rrr

rrr

(A1)





005 0052 0054 0056 0058 006092

094

096

098

1

005 0052 0054 0056 0058 006-008

-006

-004

-002

0Amplitude

α=05cm

α=1cm

α=15cm(a)

Phase

α=05cm

α=1cm

α=15cm

(rad)

(b)



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