Nonlinear reconstruction of absorption and fluorescence contrast from measured diffuse transmittance...

Nonlinear reconstruction of absorption andfluorescence contrast from measureddiffuse transmittance and reflectanceof a compressed-breast-simulating

phantom

Ronny Ziegler,1,2 Tim Nielsen,1,* Thomas Koehler,1 Dirk Grosenick,3

Oliver Steinkellner,3 Axel Hagen,3 Rainer Macdonald,3

and Herbert Rinneberg2,3

1Philips Research Europe–Hamburg, Röntgenstrasse 24, 22335 Hamburg, Germany2Department of Physics, Free University of Berlin, Arnimallee 14, 14195 Berlin, Germany

3Physikalisch-Technische Bundesanstalt, Abbestr. 2–12, 10587 Berlin, Germany

*Corresponding author: [email protected]

Received 21 October 2008; revised 19 July 2009; accepted 21 July 2009;posted 24 July 2009 (Doc. ID 102525); published 11 August 2009

We report on the nonlinear reconstruction of local absorption and fluorescence contrast in tissuelikescattering media from measured time-domain diffuse reflectance and transmittance of laser as wellas laser-excited fluorescence radiation. Measurements were taken at selected source–detector offsetsusing slablike diffusely scattering and fluorescent phantoms containing fluorescent heterogeneities.Such measurements simulate in vivo data that would be obtained employing a scanning, time-domainfluorescence mammograph, where the breast is gently compressed between two parallel glass plates, andsource and detector optical fibers scan synchronously at various source–detector offsets, allowing therecording of laser and fluorescence mammograms. The diffusion equations modeling the propagationof the laser and fluorescence radiation were solved in frequency domain by the finite element methodsimultaneously for several modulation frequencies using Fourier transformation and preprocessed ex-perimental data. To reconstruct the concentration of the fluorescent contrast agent, the Born approxi-mation including higher-order reconstructed photon densities at the excitation wavelength was used.Axial resolution was determined that can be achieved by various detection schemes. We show thatremission measurements increase the depth resolution significantly. © 2009 Optical Society of America

OCIS codes: 170.0110, 170.3830, 170.7050, 100.3010.

1. Introduction

Diffuse optical tomography (DOT) is a noninvasivemethod to image the optical properties of tissue.One of its potential applications is optical mammo-graphy, which might be employed clinically in thefuture [1]. However, imaging intrinsic optical proper-

ties of the female breast does not allow one to clearlydistinguish between malignant and benign tumors,i.e., intrinsic optical mammography suffers from in-sufficient specificity [2,3]. This situation might beimproved by using a near-infrared (NIR) contrastagent [4], the fluorescence of which can sensitivelybe detected above the autofluorescence backgroundbeing low in the NIR spectral range. Recently, thefeasibility of fluorescence DOT to detect breast can-cer was demonstrated using Indocyanine Green

0003-6935/09/244651-12$15.00/0© 2009 Optical Society of America

20 August 2009 / Vol. 48, No. 24 / APPLIED OPTICS 4651

(ICG) as contrast agent [5]. By employing the planargeometry with the breast being gently compressedbetween two parallel glass plates, optical mammo-grams were obtained by nonlinear reconstructionof the dye concentration from cw data recorded intransmission at various lateral source–detector off-sets. Another successful attempt to detect breast can-cer by fluorescence DOT was made employing atomographic fluorescence mammograph with cup-geometry and an ICG derivative (Omocyanine) asthe contrast agent [6]. Compared with the cup-geometry, the planar arrangement has the advan-tage of reduced and nearly constant breast thicknessand, thus, higher detection sensitivity for the weakfluorescence of the contrast agent. On the otherhand, it is generally difficult to achieve depth (axial)resolution from transmission data taken in parallelplane geometry [7]. Previously, the axial resolutionthat can be achieved for various source–detectorcombinations in parallel plane geometry was ana-lyzed theoretically by Markel and Schotland [8].As recently demonstrated by phantom measure-ments with a planar small animal imaging arrange-ment, the location of a fluorescent object close to thesource plane can be derived from an analysis of rawreflection data [9].Apart from fluorescence imaging based on diffusely

transmitted or reflected intensity, inhomogeneitiescontaining dyes of different fluorescence lifetimeswere imaged in diffusely scattering phantoms forsmall animal imaging. Furthermore, subcutaneoustumors in a mouse model were detected by fluores-cence lifetime imaging using a targeted near-infraredfluorescent probe [10].Likewise, aheterogeneity inanotherwise homogeneous breast phantom was recon-structed based on the different fluorescence life-times of the two different dyes that were containedin the target and in the background medium,respectively [11].In this paper, we investigate fluorescence imaging

of a breastlike phantom using the nontargeted dye(Omocyanine) that is known to enrich in tumors be-cause of differences in its pharmacokinetics betweennormal and diseased breast tissue [6]. Thus, dye ab-sorption and emitted fluorescence provides contrastbetween normal breast tissue and tumors, yetchanges in fluorescence lifetime due to changes inthe microenvironment are likely to be too small tobe detected, in particular, for the large breast thick-nesses encountered. We show that depth resolutionof the planar geometry can be increased by samplingdiffusely transmitted and, additionally, diffuselyreflected light at several lateral source–detectoroffsets. To this end, we present dye concentrationsobtained by three-dimensional nonlinear reconstruc-tion from experimental time-domain data of a breast-like phantom with a fluorescent lesion. Axialresolution obtained when using transmission dataonly is compared with resolution achieved from bothtransmission and remission data. The concentrationof the fluorescent contrast agent was obtained from

the Born approximation [12,13] of the fluorescencediffusion equation. To this end, the required densityof laser photons within the tissue was reconstructedin a nonlinear fashion using the Rytov approxima-tion [14]. Reconstruction of the dye concentrationwas performed by Fourier-transforming time-resolved data and by using the frequency data upto several hundred MHz, providing signal-to-noiseratios were sufficiently high. To cope with the largevariations in noise associated with different fre-quency components and source–detector combina-tions, we introduce an algebraic reconstructiontechnique (ART) based analysis [15] of DOT datausing a noise-weighted back projection method. Inthis way, noisy data can be incorporated into the re-construction scheme without the necessity to in-crease the regularization term. Hence, rather highcutoff frequencies can be chosen since significant dis-tortions of the reconstructed image due to noisy high-frequency data are avoided. A further improvementover existing reconstruction algorithms concerns theuse of a normalized image vector during the ARTiterations, allowing improvement of the separationof absorption and scattering by rescaling to a dimen-sionless image vector with comparable size of thenorm of both images.

In Section 2 we describe the laboratory setupthat was used to perform the phantom scans(Subsection 2.A). Subsequently, we explain the for-ward model that uses diffusion approximation to si-mulate the propagation of light in a turbid medium(Subsection 2.B), and we give a detailed descriptionof the reconstruction algorithm that has been usedfor our investigations (Subsections 2.C and 2.D).Preprocessing of experimental data is describedin Subsection 2.E. In Section 3 we illustrate recon-struction results and investigate improvements ofaxial resolution that can be achieved by includingmeasurements into the reconstruction that were ta-ken at additional source–detector offsets and datathat were collected in reflection geometry.

2. Method

A. Laboratory Setup

The measurements were accomplished with thelaboratory setup shown in Fig. 1. Its general featuresare very similar to those of the PTB clinical

Ti:SapphireLaser

Scanner

x

y z

Filter

GaAs−PMT

GaAs−PMT

Computer

TCSPC

Scan head withselected source−detector

offset

Fig. 1. (Color online) Laboratory setup for time-resolvedmeasurements in slab geometry (top viewwith coordinate system).TCSPC, time-correlated single photon counting.

4652 APPLIED OPTICS / Vol. 48, No. 24 / 20 August 2009

mammograph [16]. However, data acquisition of thelaboratory setup is not fully automated. Further-more, since only two parallel detection channels wereavailable, measurements had to be carried outmostly sequentially. On the other hand, the labora-tory setup offers more flexibility in choosingsource–detector combinations.For the present phantom study, a Ti:sapphire laser

provided short (100 fs) pulses at λ ¼ 730nm incidenton the entrance face of the cuvette (z ¼ 0). Because ofthe open design of the laboratory setup, ambientlight contributed to the background of the detectedlaser and fluorescence signals. To minimize the influ-ence of ambient light, a rather high laser power(100mW) incident on the entrance window of thephantom was chosen and the transmitted lightwas subsequently attenuated correspondingly bysets of filters in front of the detectors to achievephoton count rates of about 1MHz, representing op-timum values with respect to scan time. Therefore,each set of raw data corresponding to a particularsource–detector offset had to be corrected for filtertransmittance before entering reconstruction.Data preprocessing is described in more detail inSubsection 2.E. Time-correlated single photon count-ing was employed to record temporal point spreadfunctions (TPSFs) of laser and fluorescence photons.To phantom simulate a compressed tumor-bearing

breast, we used a rectangular cuvette (25 × 25×6 cm3) filled with a scattering, absorbing, and fluor-escent liquid and containing a small scattering,absorbing, and fluorescent object, simulating a le-sion. Reflection measurements are prone to artifactsthat originate from reflections of the incident laserradiation at the front glass face of the cuvette andfrom light guiding effects within the glass pane. Suchlight has not or has only partially sampled the diffu-sely scattering medium inside the cuvette, and itsintensity may be several orders of magnitude largerthan that of the diffusely backscattered light [17]. Toavoid such problems, the entrance face of the cuvettewasmade from regularly perforated, blackened sheetmetal backed by a thin transparent plastic foil.The laser beam was scanned across the entrance

face of the cuvette from x ¼ −4 cm to x ¼ 4 cm, y ¼−4 cm to y ¼ 4 cm sampling 289 equidistant positionsat increments of 5mm. The 5mm pitch of the perfo-rated sheet metal is commensurate with the step sizeof the scan across the entrance face.Several offsets of the detector fiber with respect to

the source fiber were selected resulting in a total of17 source–detector combinations. At each source po-sition time-domain transmittance measurementswere carried out within 100ms using detectorsplaced at the opposite (exit) face (z ¼ 6 cm) of the cuv-ette with lateral (horizontal) source–detector offsetsof Δx ¼ �4, �3, �2, �1, and 0 cm. Offsets in y direc-tion were not included here, i.e., Δy ¼ 0 cm. Forremission measurements, detectors were placed onthe entrance face at offsets Δy ¼ 0 cm andΔx ¼ �4, �3, �2, and �1 cm. Only data collected

at detector positions inside the range −4 cm ≤ x ≤

4 cm were used for image reconstruction.Breast tissue was mimicked by a scattering and

absorbing liquid prepared by Philips ResearchEurope–Eindhoven. Optical properties of this scat-tering solution were deduced from time-resolved dif-fusive transmittance measurements. At 730nm, thediffusion coefficient D0 ¼ 1=ð3μ0s;0Þ amounted toabout D0 ¼ 0:032 cm and the absorption coefficientμchroma;0 to about 0:023 cm−1. A concentration of c0 ¼10nM of the Omocyanine [18] fluorescent dye (alsoknown as TSC), corresponding to an absorptioncoefficient of μdyea;0 ¼ 0:0016 cm−1, was added to thescattering fluid to simulate background fluorescenceas expected from measurements in tissue.

A lesion-simulating object was made from hollowthin-walled delrin twin cones (Fig. 2, left) filled withbackground scattering fluid and additional Omocya-nine fluorescent dye. The fluorescence decay timeand quantum efficiency of the Omocyanine dyeamounted to τ ≈ 500ps and η ≈ 0:1, respectively.

The equatorial diameters of the delrin twin conesamounted to 2 cm, corresponding to an outer lesionvolume of 2:1ml. The delrin twin cones were filledwith background scattering fluid and fluorescentdye was added to reach a concentration of 50nm,i.e., μspha ¼ μchroma;0 þ 5μdyea;0 ¼ 0:031 cm−1. This lesion-simulating heterogeneity was immersed in thescattering fluid and placed at three different posi-tions labeled A, B, and C at x ¼ y ¼ 0, z ¼ 1:5 cm,2:2 cm, and 3 cm, respectively (Fig. 2, right). Thedouble cone was suspended from a thin thread.The double cone was loaded by an additional weightsuspended from its lower tip by means of a thinthread to avoid buoyancy.

B. Forward Model

We use the frequency domain diffusion approxima-tion to model the propagation of excitation (laser)light through the turbid medium [19]:

∇ ·DðxÞ∇Φðx; xs;ωÞ − μaðxÞΦðx; xs;ωÞ −iωvΦðx; xs;ωÞ

¼ −q0ðx; xs;ωÞ; ð1Þ

Fig. 2. (Left) Photo of the lesion-simulating delrin twin cone filledwith background scattering fluid and additional Omocyaninefluorescent dye. (Right) Schematic top view of phantom indicatingthe three selected lesion positions A, B, and C inside the cuvette atx ¼ y ¼ 0 and z ¼ 1:5 cm, z ¼ 2:2 cm, and z ¼ 3 cm, respectively.The plane z ¼ 0 corresponds to the entrance face of the phantom.


whereΦðx; xs;ωÞ is the photon density per unit inter-val of angular frequency ω inside the slab, q0 is thesource term, v is the speed of light in the turbid med-ium, and DðxÞ and μaðxÞ ¼ μdyea ðxÞ þ μchroma ðxÞ repre-sent the diffusion and absorption coefficients,respectively. The absorption coefficient μaðxÞ consistsof the chromophore contribution μchroma ðxÞ and ofthe contribution of the fluorescent dye μdyea ðxÞ ¼ϵdyecðxÞ ln 10, where ϵdye is the molar extinction coef-ficient of the dye at the excitation (laser) wavelengthand cðxÞ is the dye concentration. The time-dependent photon density ~Φðx; xs; tÞ is expressed as

~Φðx; xs; tÞ ¼ ð1=2πÞZ þ∞

−∞

Φðx; xs;ωÞeþiωtdω: ð2Þ

When using a fluorescent dye with a fluorescencelifetime τ and quantum efficiency η, the propagationof the fluorescence light inside the tissue can be de-scribed by an analogous diffusion equation, wherethe source term is proportional to the laser photondensity and the dye concentration [20]:

∇ ·Df ðxÞ∇Φf ðx; xs;ωÞ − μfaðxÞΦf ðx; xs;ωÞ

−iωvΦf ðx; xs;ωÞ ¼ −

ημdyea ðxÞ1þ iωτ Φðx; xs;ωÞ: ð3Þ

The density of fluorescence photons per unit intervalof angular frequency is designated asΦf ðx; xs;ωÞ. Weconsider a single fluorescence wavelength ratherthan a fluorescence spectrum. Hence, the absorptionand diffusion coefficient at the fluorescence wave-length are represented by μfaðxÞ and Df ðxÞ, respec-tively. In the following, we set Df ðxÞ ¼ DðxÞ, whichis a good approximation because of the rather smallStokes shifts of the near-infrared dye used. Further-more, we neglect any re-emission of the fluorescenceby the dye itself and take the fluorescence quantumyield η and the lifetime τ as constant throughout thevolume.To simulate distributions of times of flight of laser

photons (i.e., TPSFs) and distributions of times of ar-rival of fluorescence photons, coupled-diffusionEqs. (1) and (3) were solved on a grid with an approx-imate cell size of ð2 × 2 × 2Þmm3 and additional localrefinement at the source and detector positions,using the deal.II finite-element library [21]. We useda modified Robin boundary condition [22] for thephoton density at the excitation wavelength:

�Φðx; xs;ωÞ þ 2

1þ K1 −K

n ·DðxÞ∇Φðx; xs;ωÞ�∂ΩðξÞ

¼ 0;

ð4Þ

where ξ is a point on the slab surface ∂Ω, n is the out-ward-pointing surface normal at ξ, and K is thereflectivity of the glass walls of the cuvette used inthe experiments.

C. Reconstruction Algorithm

Reconstruction of fluorescence is performed in twosteps. First, the absorption and scattering coeffi-cients are reconstructed simultaneously via a non-linear, iterative algorithm. After convergence of theiterative reconstruction of the absorption and scat-tering coefficients, the fluorescence reconstructionis carried out using the results of the former step.

The absorption and scattering reconstruction algo-rithmisbasedon theRytovapproximation [14]: anap-proximate solution of the diffusion equation based onfirst-order perturbation theory. The baseline for theperturbation isdefinedby the referencemeasurement~Φ0ðxdi

; xsi ; tÞ of the slab volumewith background opti-cal properties DðxÞ ¼ D0 ¼ const:, μaðxÞ ¼ μ0a ¼ μa;0chrom þ μa; 0dye ¼ const:, usinga temporalandspatialδ-like laser pulse, where μdyea;0 ¼ ϵdyec0 ln 10. Here, xsiand xdi

refer to the position of the source and detectorof combination i, respectively.

Using the Rytov approximation for each iterationstep σ of the nonlinear reconstruction algorithm, thechange of the absorption coefficient δμσaðxÞ and diffu-sion coefficient δDσðxÞ between the object measure-ment with the heterogeneity in place and thereference measurement is calculated by inversion of

ln�Φðxdi

; xsi ;ωÞΦ0ðxdi

; xsi ;ωÞΦsim

0 ðxdi; xsi ;ωÞ

Φsimσ ðxdi

; xsi ;ωÞ�

¼ −vZΩδμσaðxÞ

Gσðxdi; x;ωÞGσðx; xsi ;ωÞ

Gσðxdi; xsi ;ωÞ

dΩ

− vZΩδDσðxÞ∇Gσðxdi

; x;ωÞ∇Gσðx; xsi ;ωÞGσðxdi

; xsi ;ωÞ· dΩ; ð5Þ

where Gσðx; xsi ;ωÞ is the Green’s function of iterationσ ¼ 0; 1; 2; ::: at point x for a point source of laserphotons at xsi . The iteration σ ¼ 0 corresponds tothe homogeneous case. The Green’s functionGσðx; xsi ;ωÞ is obtained by solving Eq. (1) for thebackground medium with absorption and diffusioncoefficient:

μσaðxÞ ¼ μ0a þX

0≤m<σδμma ðxÞ;

DσðxÞ ¼ D0 þX

0≤m<σδDmðxÞ: ð6Þ

Here, μ0a and D0 are the optical properties of thehomogeneous medium used in the reference scan(see above), and the source is expressed as

qσ0ðx; xsi ;ωÞ ¼1vSσðx; xsiÞ; ð7Þ

for all ω with

Sσðx; xsiÞ ¼�

1

σsrcffiffiffiffiffiffi2π

p�

3expð−ðx − xσsiÞ2=2σ2srcÞ: ð8Þ


σsrc is the width of the Gaussian-blurred diffusesource, and xσsi is the modified source position [23],which is shifted into the slab volume in beam direc-tion by one reduced scattering length 1=μs;σ0ðxsiÞ. Theparameter σsrc was chosen to be 6mm in all recon-structions. The z position of the source changesslightly with each iteration step σ due to the updatedscattering properties of the medium. Calculating theGreen’s function after each image update takes non-linearity into account.The simulated data Φsim

σ ðxdi; xsi ;ωÞ is the solution

of Eq. (1) with the distribution μσaðxÞ and DσðxÞ ¼1=3μ0s;σðxÞ of the absorption and diffusion coefficient,and, in our case, is approximately given by theGreen’s function, i.e., Φsim

σ ðxdi; xsi ;ωÞ ≈ Gσðxdi

; xsi ;ωÞ.Sampling transmitted laser radiation and fluores-

cence radiation for a total of k source–detectorcombinations iði ¼ 1; :::; kÞ, integralEqs. (6)arediscre-tized for p frequency components ωqðq ¼ 1; :::;pÞ on afinite element (FE)gridofN vertices.This yieldsasys-tem of linear equations y ¼ Ab, with imageupdate vector b ¼ ðδμσaðx1Þ; :::; δμσaðxNÞ; δDσðx1Þ; :::;δDσðxNÞÞT and the 2kp component signal vector y,

y ¼

0BBBBBBBBBBBBBBBBBBBBBBBBBBB@

ℜ ln�

Φðxd1 ;xs1 ;ω1ÞΦ0ðxd1 ;xs1 ;ω1Þ

Φsim0 ðxd1 ;xs1 ;ω1Þ

Φsimσ ðxd1 ;xs1 ;ω1Þ

�

ℑ ln�

Φðxd1 ;xs1 ;ω1ÞΦ0ðxd1 ;xs1 ;ω1Þ

Φsim0 ðxd1 ;xs1 ;ω1Þ

Φsimσ ðxd1 ;xs1 ;ω1Þ

�

..

.

ℜ ln�

Φðxdk ;xsk ;ω1ÞΦ0ðxdk ;xsk ;ω1Þ

Φsim0 ðxdk ;xsk ;ω1Þ

Φsimσ ðxdk ;xsk ;ω1Þ

�

ℑ ln�

Φðxdk ;xsk ;ω1ÞΦ0ðxdk ;xsk ;ω1Þ

Φsim0 ðxdk ;xsk ;ω1Þ

Φsimσ ðxdk ;xsk ;ω1Þ

�

..

.

ℜ ln�

Φðxdk ;xsk ;ωpÞΦ0ðxdk ;xsk ;ωpÞ

Φsim0 ðxdk ;xsk ;ωpÞ

Φsimσ ðxdk ;xsk ;ωpÞ

�

ℑ ln�

Φðxdk ;xsk ;ωpÞΦ0ðxdk ;xsk ;ωpÞ

Φsim0 ðxdk ;xsk ;ωpÞ

Φsimσ ðxdk ;xsk ;ωpÞ

�

1CCCCCCCCCCCCCCCCCCCCCCCCCCCA

: ð9Þ

The ð2kpÞ × 2N system matrix A is given by

A ¼

0BBBBBBBBBBBBBB@

ℜaσ1ðx1;ω1Þ ::: ℜaσ

1ðxN ;ω1Þ ℜaσ1ðx1;ω1Þ ::: ℜaσ

1ðxN ;ω1Þℑaσ

1ðx1;ω1Þ ::: ℑaσ1ðxN ;ω1Þ ℑaσ

1ðx1;ω1Þ ::: ℑaσ1ðxN ;ω1Þ

..

. ... ..

. ...

ℜaσkðx1;ω1Þ ::: ℜaσ

kðxN ;ω1Þ ℜaσkðx1;ω1Þ ::: ℜaσ

kðxN ;ω1Þℑaσ

kðx1;ω1Þ ::: ℑaσkðxN ;ω1Þ ℑaσ

kðx1;ω1Þ ::: ℑaσkðxN ;ω1Þ

..

. ... ..

. ...

ℜaσkðx1;ωpÞ ::: ℜaσ

kðxN ;ωpÞ ℜaσkðx1;ωpÞ ::: ℜaσ

kðxN ;ωpÞℑaσ

kðx1;ωpÞ ::: ℑaσkðxN ;ωpÞ ℑaσ

kðx1;ωpÞ ::: ℑaσkðxN ;ωpÞ

1CCCCCCCCCCCCCCA

; ð10Þ

where the complex-valued sensitivity coefficients aregiven by

aσi ðx;ωÞ ¼ −v

Gσðxdi; x;ωÞGσðxdi

; xsi ;ωÞGσðx; xsi ;ωÞ

wðxÞ ð11Þ

for absorption and

aσi ðx;ωÞ ¼ −v

∇Gσðxdi; x;ωÞ ·∇Gσðx; xsi ;ωÞGσðxdi

; xsi ;ωÞwðxÞ ð12Þ

for scattering. Here, wðxÞ denotes the Voronoi cellvolume [24] associated with the vertex x.

When the convergence criterion of the nonlinearreconstruction is reached at iteration step σc, thefluorescence reconstruction is performed using thenormalized fluorescence signal [12,25]

Φf ðxdi; xsi ;ωÞ

Φðxdi; xsi ;ωÞ

¼ vηϵdye ln 101þ iωτ

ZΩcðxÞ

×Gσc

f ðxdi; x;ωÞGσcðx; xsi ;ωÞ

Gσcðxdi; xsi ;ωÞ

dΩ; ð13Þ

where the fluorescence photon density Φf ðxdi; xsi ;ωÞ

per unit interval of angular frequency ω is at the de-tector position xdi

and is a point source of laserphotons at xsi , and the Green’s function at the fluor-escence wavelengthGσc

f ðxdi; x;ωÞ is a point source at x

emitting at the fluorescence wavelength. In Eq. (15)we have assumed the quantum efficiency η and thefluorescence lifetime τ to be independent of locationx. Throughout our reconstructions we used theapproximation Gσc

f ðxdi; x;ωÞ ¼ Gσcðxdi

; x;ωÞ, henceignoring the change in optical properties between ex-citation and fluorescence wavelength. For the phan-tom experiments analyzed in this paper, thisassumption is well justified since optical propertieschange only slightly with wavelength in the spectralrange considered, and the Stokes shift of Omocya-nine is comparable to the width of its fluorescenceband. We note in passing that under the assumptionsmade [μfaðxÞ ¼ μaðxÞ, Df ðxÞ ¼ DðxÞ, and hence


Gσcf ðxdi

; x;ωÞ ¼ Gσcðxdi; x;ωÞ], Eq. (15) has a simple

physical meaning: setting μdyea ðxÞ ¼ ϵdyecðxÞ ln 10,the integral

vZΩμdyea ðxÞG

σcðxdix;ωÞGσcðx; xsi ;ωÞ

Gσcðxdi; xsi ;ωÞ

dΩ ð14Þ

corresponds to the relative increase ΔΦðxdi; xsi ;ωÞ=

Φðxdi; xsi ;ωÞ in laser transmission when absorption

of laser photons by the dye is neglected. It followsthat

Φf ðxdi; xsi ;ωÞ

Φðxdi; xsi ;ωÞ

¼ η1þ iωτ

ΔΦðxdi; xsi ;ωÞ

Φðxdi; xsi ;ωÞ

: ð15Þ

For fluorescence reconstruction, the systemmatrixsimplifies to

A ¼

0BBBBBBBBBBBBBBBB@

ℜaf1ðx1;ω1Þ ::: ℜaf

1ðxN ;ω1Þℑaf

1ðx1;ω1Þ ::: ℑaf1ðxN ;ω1Þ

..

. ...

ℜafkðx1;ω1Þ ::: ℜaf

kðxN ;ω1Þℑaf

kðx1;ω1Þ ::: ℑafkðxN ;ω1Þ

..

. ...

ℜafkðx1;ωpÞ ::: ℜaf

kðxN ;ωpÞℑaf

kðx1;ωpÞ ::: ℑafkðxN ;ωpÞ

1CCCCCCCCCCCCCCCCA

; ð16Þ

with the dye concentration sensitivity coefficientsgiven by the general expression

afi ðx;ωÞ ¼

vηϵdye ln 101þ iωτ

Gσcf ðxdi

; x;ωÞGσcðxdi; xsi ;ωÞ

Gσcðx; xsi ;ωÞwðxÞ;

ð17Þ

and the image vector b ¼ ðcðx1Þ; :::; cðxNÞÞT. UsingEq. (15), the signal vector of the fluorescencereconstruction is given by

y ¼

0BBBBBBBBBBBBBBBBBBB@

ℜΦf ðxd1 ;xs1 ;ω1ÞΦðxd1 ;xs1 ;ω1Þ

ℑΦf ðxd1 ;xs1 ;ω1ÞΦðxd1 ;xs1 ;ω1Þ

..

.

ℜΦf ðxdk ;xsk ;ω1ÞΦðxdk ;xsk ;ω1Þ

Φf ðxd1 ;xs1 ;ω1ÞΦðxd1 ;xs1 ;ω1Þ

..

.

ℜΦf ðxdk ;xsk ;ωpÞΦðxdk ;xsk ;ωpÞ

Φf ðxdk ;xsk ;ωpÞΦðxdk ;xsk ;ωpÞ

1CCCCCCCCCCCCCCCCCCCA

: ð18Þ

D. ART Reconstruction with Noise-Weighted Relaxationand Normalized Image Vector

We used the ART [15], which acts only on single rowsof

A ¼ ðAnjÞn∈I;j∈J ; ð19Þ

with I ¼ f1;…; 2kpg and J ¼ f1; :::; 2Ng for absorp-tion and diffusion reconstruction and J ¼ f1; :::;Ngfor fluorescence reconstruction, to solve for the image(update) vector [26]

b ¼ ðbjÞj∈J : ð20Þ

We reduce the difference between measurement dataand forward model data, Δyil ¼ yil −

Pj Ailjb

lj, at

every ART iteration step l. At each of these itera-tions, a random row il of the signal vector y is chosenwith associated source–detector combination i andangular frequency component ωq. Subsequently, allimage elements j ∈ J are updated by

blþ1j ¼ blj þ rði;ωqÞ

AiljΔyilPm Ailm

; ð21Þ

where m ¼ 1; :::; 2N for absorption and diffusion re-constructions, m ¼ 1; :::;N for fluorescence recon-structions, and r is a weighting factor. The ARTalgorithm uses each source–detector combination,each frequency component, and real and imaginarypart of the measurement data once, hence resultingin 2kp total iterations. To this end, the iterationprocess is initialized at l ¼ 0 with b ¼ 0.

To handle the variations in measurement noisethat are associated with transmission and reflectionmeasurements, with different source–detector com-binations i (source–detector offsets), and frequencycomponents ωq, we introduce a noise-weighted backprojection [27]. Hence, the update for each image ele-ment is proportional to the difference between mea-surement data and forward model data, but is scaledby the relaxation factor r:

rði;ωqÞ ¼σerr0

σerrði;ωqÞ; ð22Þ

where σerrði;ωqÞ is the noise of the angular frequencycomponent ωq at the source–detector combination i.In contrast to analogous approaches that have beentested in computed tomography, where onlymarginalenhancements in reconstructed images are shown[27], noise-weighted back projection can improveand stabilize reconstructions in diffuse opticaltomography significantly, as noise can vary by ordersof magnitude between frequency components ωq orsource–detector combinations i used duringreconstruction.

Since the reconstruction inputs are the real andimaginary parts of


ln�Φðxdi

; xsi ;ωqÞΦsim0 ðxdi

; xsi ;ωqÞΦ0ðxdi

; xsi ;ωqÞΦsimσ ðxdi

; xsi ;ωqÞ�

ð23Þ

and Φf ðxdi; xsi ;ωqÞ=Φðxdi

; xsi ;ωqÞ, we estimate thecorresponding noise σerrði;ωqÞ by analyzing datafrom a homogeneous region of the object scan andput it in relation to the lowest signal noise σerr0at ω ¼ 0.Rather than solving y ¼ Ab, we searched for the

minimum of f‖A0b0 − y‖2 þ λreg‖b0‖2g in the absorp-tion and scattering reconstruction to achieve regular-ization by the parameter λreg [28]. The new matrixA0 ¼ ðA0

ijÞi∈I;j∈J and the new vector b0 ¼ ðb0jÞj∈J arecalculated by

A0ij ¼ Aijγj; ð24Þ

b0j ¼ bj=γj; ð25Þ

with γj ¼ Δμ⋆a þ μ0a for j ≤ N and γj ¼ ΔD⋆ þD0 forj > N for absorption and diffusion reconstructionand γj ¼ 1 for fluorescence reconstruction. Insteadof scaling the rows of the sensitivity matrix to im-prove its conditioning as was presented in [29], thenormalization of vector b makes the image vectorb0 dimensionless and effectuates that updates ofthe image during each ART back projection are dis-tributed equally for the absorption and the scatteringimage. Since the norm of b0 for j ≤ N and j > N are ofcomparable size, the regularization term λreg hascomparable effects on the absorption and scatteringimages.The values Δμ⋆a and ΔD⋆ are the estimated (or a

priori known) absorption and scattering coefficientdeviations of the homogeneous medium comparedwith the heterogeneity and accelerate the conver-gence of the nonlinear reconstruction algorithm.Since we pretended to have no prior knowledge ofthe lesion contrast, we assume Δμ⋆a ¼ ΔD⋆ ¼ 0 forall of the following reconstructions, while λreg hasbeen chosen by hand for each reconstructionseparately.A schematic view of the nonlinear reconstruction

algorithm is given in Fig. 3. The complete algorithmcan be summarized in the following pseudocode:Algorithm: nonlinear reconstruction

Step 1: Collection of measurement data for theobject scan, ~Φðxdi

; xsi ; tÞ, i.e., with the heterogeneityin place, and for the reference scan (homogeneousmedium), ~Φ0ðxdi

; xsi ; tÞ, of all source–detector combi-nations. Fourier transformation and additionalpreprocessing of measurement data if needed (seeSubsection 2.E).Step 2: Simulation of the reference scan data,

Φsim0 ðxdi

; xsi ;ωqÞ, for all source–detector combinationsi and all angular frequency components ωq used inthe reconstruction, by solving Eq. (1) for a homoge-

neous medium with the absorption coefficient μ0aand the diffusion coefficient D0 of the reference scan.

Step 3: Calculation of the Green’s functionsGσðxdi

; x;ωqÞ, respectively Gσðx; xsi ;ωqÞ for all detec-tor positions xdi

, all source positions xsi, and all mod-ulation frequency components ωq. Simulated dataΦsim

σ ðxdi; xsi ;ωqÞ are calculated by the forward model

with the reconstructed spatial distributions μσaðxÞand DσðxÞ for each source–detector combination iand angular frequency ωq. This calculation can beomitted for σ ¼ 0 because, in this case, the resultis equal to Φsim

0 ðxdi; xsi ;ωqÞ, corresponding to the

reference scan.Step 4: The results obtained in the previous steps

[Gσðx; xsi ;ωqÞ, Gσðxdi; x;ωqÞ, Φsim

0 ðxdi; xsi ;ωqÞ, and

Φsimσ ðxdi

; xsi ;ωqÞ] and the experimental data[Φ0ðxdi

; xsi ;ωqÞ and Φðxdi; xsi ;ωqÞ] are used to solve

Eq. (6) via ART to reconstruct the difference of opti-cal properties δμσaðxÞ and δDσðxÞ (see Eqs. (11)–(14)).

Step 5:When a given stopping criterion is reached(e.g., the norm of the reconstructed image updatevector b is lower than a given limit), fluorescencereconstruction is started in Step 6, and σc ¼ σ þ 1.If the stopping criterion has not been reached, thespatial distribution of optical properties inside thereconstructed volume is updated according toEq. (8). The iteration number is increased and calcu-lations are continued at Step 3.

Step 6: Using the reconstructed absorption μσca ðxÞand diffusion coefficient DσcðxÞ, the Green’s functionGσcðx; xsi ;ωqÞ is calculated for each source and

Fig. 3. Scheme of the nonlinear reconstruction of the spatial dis-tribution of optical properties and fluorescent dye concentration,initialized with a homogeneous distribution at iteration σ ¼ 0.Numbers in circles correspond to the detailed description of thesteps of the reconstruction algorithm given in the text.


detector position. Setting Gσcf ðx; xsi ;ωqÞ ¼ Gσc

ðx; xsi ;ωqÞ, the fluorescent dye concentration isreconstructed according to Eq. (13) via ART (seeEqs. (16)–(18)).

When using experimental phantom data, fiveiterations (σc ¼ 5) were needed to reconstruct ab-sorption coefficients. Convergence of the nonlinearabsorption and scattering reconstruction is beyondthe scope of the present paper and will be discussedin a forthcoming publication.

E. Preprocessing and Image Reconstructionfrom Time-Domain Data

Before the acquired measurement data can be usedin the image reconstruction, several steps of datapreprocessing are needed. When measuring in timedomain, a large amount of data are collected. Usingthe complete data set of all collected TPSFs is notpractical during the reconstruction process becausethe resulting linear system of equations is too largeand cannot be solved on today’s hardware. To reducethe amount of data, one can either use temporal fil-ters [30] or carry out the reconstruction in the fre-quency domain, as is performed in the following.For this purpose, time-domain data have to beFourier transformed and frequency components withsufficiently high signal-to-noise ratios have to beselected.After discrete Fourier transformation of the

measured time-domain data at the laser andfluorescence wavelength to frequency domain, threefrequencies (ω1 ¼ 0MHz, ω2 ¼ 2π · 117MHz, andω3 ¼ 2π · 234MHz) were selected for image recon-structions. As a result of the noise-weighted ARTalgorithm used, higher frequencies could be incorpo-rated into the reconstruction without distorting theimage. Nonetheless, frequency components ω4 ¼ 2π ·351MHz and above were ignored because of their lowsignal-to-noise ratios that would prevent significantinfluence of the corresponding measurement data onthe reconstructed image, yet causing additional cal-culation time.As was mentioned in Subsection 2.A, measure-

ments were carried out sequentially using differentcombinations of optical filters for the reference andobject scans as well as for different source–detectoroffsets. Therefore, for each source–detector offsetΔxdi;si ¼ xdi

− xsi , we select data of the object scan,i.e., an area in the transmission or reflection imagethat is not affected by the presence of the heteroge-neity. The scaling factor Rdi;si that takes experimen-tal conditions into account, e.g., filter transmittanceor detector efficiency, is defined as

Rdi;si ¼ðΦmeasðxdi

; xsi ;ωqÞÞave;NA

ðΦmeas0 ðxdi

; xsi ;ωqÞÞave; ð26Þ

where the average over the object scan data in thenominator is taken over a subset (area) of data notaffected (NA) by the presence of the heterogeneity.

Likewise, in the denominator, data are averaged ofthe reference scan. It follows that the corrected data,

ðΦðxdi; xsi ;ωqÞÞ

ðΦ0ðxdi; xsi ;ωqÞÞ

¼ 1Rdi;si

Φmeasðxdi; xsi ;ωqÞ

Φmeas0 ðxdi

; xsi ;ωqÞ; ð27Þ

enter Eqs. (5) and (9).For the fluorescence contrast reconstruction, in-

strumental factors have to be taken into accountby a similar approach. Before reconstruction of themeasured fluorescence data was started, four prepro-cessing steps were applied to the raw fluorescencedata:

1. At each angular frequency ωq and source–detector combination iðxdi

; xsiÞ selected, the datameasured at the fluorescence wavelength was nor-malized by the corresponding measured data atthe laser wavelength, yielding Φmeas

f ðxdi; xsi ;ωqÞ=

Φmeasðxdi; xsi ;ωqÞ, eliminating the corresponding

Fourier components of the instrumental responsefunction;

2. Fortheangularfrequenciesandsource–detectorcombinationsselected,simulateddataweregeneratedin frequencydomainat the laser [Φsim

0 ðxdi; xsi ;ωqÞ] and

fluorescence [Φf0simðxdi; xsi ;ωqÞ] wavelengths by

solving Eqs. (1) and (3) in 3D for a homogeneous slab;3. To correct fluorescence scan data for experi-

mental factors, we set

�Φmeasf ðxdi

; xsi ;ωqÞΦmeasðxdi

; xsi ;ωqÞ�

ave;NA¼ Rf

di;si

Φsimf ðxdi

; xsi ;ωqÞΦsim

0 ðxdi; xsi ;ωqÞ

;

ð28Þ

and calculate the scaling factor Rfdi;si

, which takes ex-perimental conditions into account. The average onthe left-hand side is taken over a subset of measureddata (area) that is unaffected by the presence of theheterogeneity, Φsim

f0 ðxdi; xsi ;ωqÞ, and Φsim

0 ðxdi; xsi ;ωqÞ

refers to simulated data of the homogeneous mediumwithout the heterogeneity being present. The cor-rected data,

Φf ðxdi; xsi ;ωqÞ

Φðxdi; xsi ;ωqÞ

¼ 1

Rfdi;si

Φmeasf ðxdi


; xsi ;ωqÞ; ð29Þ

enter into Eqs. (13) and (18).4. To improve image contrast [31], the mean

homogeneous background value

Φsimf0 ðxdi

; xsi ;ωqÞ=Φsim0 ðxdi

; xsi ;ωqÞ ð30Þ

is subtracted form the corrected data, yielding


Φf ðxdi; xsi ;ωqÞ

Φðxdi; xsi ;ωqÞ

−Φsim

f0 ðxdi; xsi ;ωqÞ

Φsim0 ðxdi

; xsi ;ωqÞ

¼ 1

Rfdi;si

�Φmeasf ðxdi


; xsi ;ωqÞ

−

�Φmeasf ðxdi


; xsi ;ωqÞ�

avg;NA

�

¼ vηϵdye ln 101þ iωqτ

ZΔcðxÞ

×Gσc

f ðxdi; xsi ;ωqÞGσcðx; xsi ;ωqÞGσcðxdi

; xsi ;ωqÞdΩ; ð31Þ

where ΔcðxÞ ¼ cðxÞ − c0 is the difference between thedye concentration and the background concentrationvalue. To present absolute values cðxÞ rather thanonly differences ΔcðxÞ between the object and thereference fluid, the optical properties of the back-ground scattering liquid, especially the homo-geneous background dye concentration c0, weretaken as a priori knowledge. These optical propertiescan be deduced by an analysis of the reference scandata. Reconstruction of ΔcðxÞ using Eq. (31) is car-ried out analogously as described for the reconstruc-tion of cðxÞ using Eq. (13).

3. Results

To investigate to what extent the inclusion of offsetmeasurements taken in transmission and reflectiongeometry will improve results, we compare recon-structions using different restricted sets of source–detector combinations. In each case, the pulsedsource scanned across the lesion at 5mm increments.From these measurements, we performed recon-structions using data sets representing three differ-ent setups [32], each sketched in Fig. 4: (left)projection-shadow geometry corresponding to zero-offset data taken in transmission; (middle) slabfan-beam geometry, where detectors measure intransmission (y ¼ 0, z ¼ 6 cm) at offsets Δx ¼ �4,�3,…, 0 cm; and (right) slab reflection and transmis-sion geometry, where additionally the remission is

measured at the entrance face (z ¼ 0) within the y ¼0 plane at offsets Δx ¼ �4, �3, …, �1 cm. Nonlinearreconstructions up to iteration number σ ¼ 4 werecarried out on the volume of interest Ω ¼ 18×12 × 6 cm3. The FE grid consisted of approximately51; 500 vertices. The Voronoi cell volume amountedto approximately wðxÞ ¼ 0:125 cm3 throughout thevolume of interest but was chosen smaller at sourceand detector positions, i.e., wðxsiÞ ¼ wðxdi

Þ ¼0:005 cm3. The regularization parameter was chosenas λreg ¼ 0:02 for reconstructions of the absorptionand reduced scattering coefficients and as λreg ¼ 0for reconstructions of fluorescent dye concentrations,while in all cases ΔD⋆ ¼ Δμ⋆a ¼ 0.

The resulting dye concentrationsΔcðxÞ of the threedifferent reconstructions are illustrated in Fig. 5using the corresponding source–detector combina-tions shown in Fig. 4. In all three cases, the slicesfor y ¼ 0 (upper row) and x ¼ 0 (lower row) are given.The images are presented in min =max scaling. Theouter hull of the twin-cone that was positioned atx ¼ y ¼ 0, z ¼ 2:2cm is outlined as a circle in theseimages. As had been predicted previously [33], usingonly zero-offset transmittance data does not provideany axial definition of the heterogeneity. Hence, inthe resulting images (Fig. 5, left) the axial lesion po-sition cannot be reconstructed. Using transmittancedata with offsets along the x direction improves axialdefinition (Fig. 5, middle), although the lesionposition is not reconstructed correctly but shifted to-wards the entrance face. Including additional remis-sion measurements into the reconstruction enhancesaxial definition of the lesion, and its center is recon-structed at the correct position (see Fig. 5, right).

These results are supported by profiles of the ab-sorption coefficient and the fluorescent dye concen-tration along the z axis through the center of theheterogeneity (x ¼ y ¼ 0) located at three positionsA, B, and C (see Subsection 2.A). The profiles wereobtained by three-dimensional reconstruction fromtime-domain transmittance and remittance data asdescribed above, selecting three equidistantangular frequencies ω1 ¼ 0, ω2 ¼ 2π · 117MHz, andω3 ¼ 2π · 234MHz.

x

0 cm

3 cm

6 cm

z

O 2 cm

+4 cm−4 cm

headscan

+4 cm−4 cm

(a) (b) (c)Fig. 4. Schematic view of different source–detector geometries with a lesion near the entrance face of the slab, i.e., at position x ¼ y ¼ 0,z ¼ 2:2 cm (position B, Fig. 2). The source and the detector fibers are scanned in tandem keeping source–detector offsets fixed. The step sizeamounted to 5mm, sampling a total of 17 source positions across the front face. (a) Projection-shadow geometry. (b) Slab fan-beam geo-metry. (c) Slab reflection and transmission geometry.


Reconstructed absorption coefficients and fluores-cent dye concentrations are shown in Fig. 6, left andright, respectively. Line profiles along z are given forall three positions: A (no symbol), B (stars), and C(triangles). Full lines correspond to reconstructedtransmittance and reflectance data, dashed lineswere obtained from transmittance data only. Ascan be seen from Fig. 6, inclusion of remittance databesides transmittance data generally narrows lineprofiles (solid lines) of the absorption coefficient,and fluorescent dye concentration compared to pro-files (dashed lines) obtained from transmittance dataalone thus improving axial resolution. Furthermore,all line profiles based on transmittance data aloneexhibit maxima that are further shifted from the truelesion position toward the entrance face compared tothe maxima of the line profiles based on both remit-

tance and transmittance data. In particular, themaxima of the fluorescent dye concentrations recon-structed from remittance and transmittance data(solid line, solid line with stars) coincide with thetrue lesion positions A and B, while for position Cthe corresponding maximum is shifted only slightlyfrom the true position toward the entrance face. Theline profiles (solid lines in Fig. 6, right) allow the se-paration of all three lesion positions. In contrast, cor-responding line profiles (solid lines) of the absorptioncoefficient (Fig. 6, left) do not allow the separation ofpositions A and B. The normalized Born approxima-tion used to reconstruct fluorescent dye concen-trations involves transmittance and remittancemeasurements at the excitation and fluorescencewavelength and, hence, contains more informationthan measurements at the laser wavelength only.

Fig. 5. Reconstruction of three-dimensional dye concentration ΔcðxÞ ¼ cðxÞ − c0 of fluorescent dye of the delrin twin cone indicated bycircle and located at position B ðx ¼ y ¼ 0; z ¼ 2:2 cmÞ. Dye concentration ΔcðxÞ is given in min =max scaling in the x–z plane (y ¼ 0, toprow) and in the y–z plane (x ¼ 0, bottom row) through the center of the lesion. The three images of each row correspond to the source–detector combinations shown in Fig. 4: (left) projection-shadow geometry, (middle) slab fan-beam geometry, and (right) reflection andtransmission geometry. Line profiles along x ¼ y ¼ 0 and absolute values for the middle and right image are shown in Fig. 6.

0 1 2 3 4 5 6

2.4

2.5

2.6

2.7

2.8

2.9

3

A B C

µ a / 10

−3 m

m−

1

z / cm

Pos APos BPos C

0 1 2 3 4 5 60

20

40

60

80

100

A B C

c / n

M

z / cm

Pos APos BPos C

(a) (b)

Fig. 6. Line profiles of three-dimensional reconstruction of (left) μaðx ¼ 0; y ¼ 0; zÞ and of fluorescent dye concentration cðx ¼ 0; y ¼ 0; zÞfor the three different twin-cone positions A (no symbol), B (star), and C (triangle). Full lines correspond to reconstructions using bothremittance (z ¼ 0) and transmittance (z ¼ 6 cm) data; dashed lines are obtained from transmittance data only. The horizontal dash-dot lineshows the background value μ0a of the absorption coefficient (left) and the background dye concentration c0 (right), respectively. The verticalarrows indicate the three positions (A, B, C) of the heterogeneity.


Generally, line profiles are skewed, exhibiting a ra-pid drop towards the entrance face (source plane)and a gradual decline towards the exit face (detectorplane), since diffuse reflectance was recorded on theentrance face only. Furthermore, some of the line pro-files illustrated in Fig. 6 exhibit minima toward theexit face even falling below the reference values μ0aand c0, respectively, as can be seen most clearly forposition A in Fig. 6, left. Besides the ill-posednessof the inverse problem, such artifacts probably de-monstrate limitations in the data sets used for recon-struction, in particular, the limited angular samplingand deficiencies in the forward model used, neglect-ing for example the spectral dependance of the ab-sorption and diffusion coefficients.The reconstructed maximal absorption coefficient

and the maximal dye concentration are dependenton lesion position, as is the reconstructed axial exten-sion of the object. In each case, the maximum de-creases towards the center of the cuvette, whilethe reconstructed volume increases due to the lowerspatial resolution at larger depths of the object. Atpositions A and B, the absolute value of the recon-structed dye concentration is too large. Because ofuncertainties in the quantum yield η and lifetimeof the Omocyanine dye, such errors are to be ex-pected. Furthermore, while improving the overallimage quality, the chosen values of rði;ωqÞ have someeffect on the reconstructed values. It is certainlychallenging to achieve the correct absolute dye con-centration from phantom experiments, let alone fromin vivo data.

4. Summary and Conclusions

We reported on several phantom measurements re-levant for fluorescence optical mammography andnonlinearly reconstructed absorption and fluores-cence contrast of lesion-simulating heterogeneitiesusing the Rytov approximation (absorption contrast)and the normalized Born approximation (fluores-cence contrast). A rectangular cuvette filled with afluorescent scattering solution having tissuelike op-tical properties simulated an average-sized femalebreast gently compressed between two plates. Asmall object (2:1ml) was placed at selected positionsinside the phantom. The lesion-simulating objectcontained the same background scattering solution,yet the concentration of the fluorescent dye was fivetimes higher compared to the background medium.Time-domain transmission and remission measure-ments were carried out at the excitation and fluores-cence wavelength for a large number of sourcepositions sampled across the cuvette’s entrance faceand at each source position for a small number of lat-eral source–detector offsets. Using the algebraic re-construction technique, reconstructions of theabsorption and fluorescence contrast were performedin the Fourier domain for a small number of angularfrequencies using data normalized to the homoge-neous (background) part of the phantom. We ex-tended the ART algorithm by a noise-weighted

relaxation term to incorporate noisy data into the re-construction. This way, high-frequency componentsof the measured TPSFs can be used during the recon-struction without distorting the image. Additionally,the image vector was normalized to achieve a com-parable regularization effect for the scattering andabsorption image. Transmission measurements ta-ken at selected source–detector offsets improve axialresolution of the reconstructed absorption and fluor-escence contrast to a certain degree. By includingreflectance data at the excitation and fluorescencewavelength into the reconstruction besides trans-mittance data, depth resolution is significantlyimproved. Results of scanning time-domain fluores-cence mammography will be enhanced when trans-mittance and reflectance measurements are carriedout on either side of the compressed breast.

This work was supported in part by the GermanFederal Ministry of Education and Research, grantNo. 13N8774 (“Fluoromamm”). The authors thankBayer Schering Pharma AG for providing the fluor-escent Omocyanine dye and Philips Research Europe—Eindhoven for supplying the scattering fluid andthe delrin twin cone.

References1. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent

advances in diffuse optical imaging,” Phys. Med. Biol. 50,R1–R43 (2005).

2. H. Rinneberg, D. Grosenick, K. T. Moesta, H. Wabnitz, J.Mucke, G. Wübbeler, R. Macdonald, and P. Schlag, “Detectionand characterization of breast tumors by time-domainscanning optical mammography,” Opto-electron. Rev. 16,147–162 (2008).

3. S. P. Poplack, T. D. Tosteson, W. A. Wells, B. W. Pogue, P. M.Meaney, A. Hartov, C. A. Kogel, S. K. Soho, J. J. Gibson, andK. D. Paulsen, “Electromagnetic breast imaging: results of apilot study in women with abnormal mammograms,” Radiol-ogy (Oak Brook, Ill.) 243, 350–359 (2007).

4. D. J. Hawrysz and E. M. Sevick-Muraca, “Developmentstoward diagnostic breast cancer imaging using near-infraredoptical measurements and fluorescent contrast agents,”Neoplasia 2, 388–417 (2000).

5. A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S.R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensionalin vivo fluorescence diffuse optical tomography of breast can-cer in humans,” Opt. Express 15, 6696–6716 (2007).

6. R. Ziegler, “Modeling photon transport and reconstruction ofoptical properties for perfromance assessment of laser andfluorescence mammographs and analysis of clinical data,”Ph.D. dissertation (Free University of Berlin, 2008), http://www.diss.fu‑berlin.de/diss/receive/FUDISS_thesis_000000005928.

7. T. Dierkes, D. Grosenick, K. T. Moesta, M. Möller, P. M. Schlag,H. Rinneberg, and S. Arridge, “Reconstruction of opticalproperties of phantom and breast lesion in vivo from paraxialscanning data,” Phys. Med. Biol. 50, 2519–2542 (2005).

8. V. A. Markel and J. C. Schotland, “Symmetries, inversionformulas, and image reconstruction for optical tomography,”Phys. Rev. E 70, 056616 (2004).

9. M. Brambilla, L. Spinelli, A. Pifferi, A. Torricelli, andR. Cubbedu, “Time-resolved scanning system for doublereflectance and transmittance fluorescence imaging ofdiffusive media,” Rev. Sci. Instrum. 79, 013103 (2008).


10. R. E. Nothdurft, S. V. Patwardhan,W. Akers, Y. Ye, S. Achilefu,and J. P. Culver, “In vivo fluorescence lifetime tomography,”J. Biomed. Opt. 14, 024004 (2009).

11. R. Roy, A. Godavarty, and E.M. Sevick-Muraca, “Fluorescence-enhanced three-dimensional lifetime imaging: a phantomstudy,” Phys. Med. Biol. 52, 4155–4170 (2007).

12. V. Ntziachristos and R. Weissleder, “Experimental three-dimensional fluorescence reconstruction of diffuse media byuse of a normalized Born approximation,” Opt. Lett. 26, 893–895 (2001).

13. E. Scherleitner and B. G. Zagar, “Optical tomography imagingbasedonhigherorderBornapproximationofdiffusephotonden-sitiywaves,”IEEETrans.Instrum.Meas.54,1607–1611(2005).

14. A. Ishimaru, Wave Propagation and Scattering in RandomMedia (Academic Press, 1978).

15. Y. Censor, “Row-action methods for huge and sparse systemsand their applications,” SIAM Rev. 23, 444–466 (1981).

16. D. Grosenick, K. T. Moesta, M. Möller, J. Mucke, H. Wabnitz,B. Gebauer, C. Stroszczynski, B. Wassermann, P. M. Schlag,and H. Rinneberg, “Time-domain scanning optical mam-mography: I. Recording and assessment of mammograms of154 patients,” Phys. Med. Biol. 50, 2429–2449 (2005).

17. O. Steinkellner, A. Hagen, C. Stadelhoff, D. Grosenick,R. Macdonald, H. Rinneberg, R. Ziegler, and T. Nielsen,“Recording of artifact-free reflection data with a laser andfluorescence scanning mammograph for improved axialresolution,” in Biomedical Optics/Digital Holography andThree-Dimensional Imaging/Laser Applications to Chemical,Security and Environmental Analysis on CD-ROM (OpticalSociety of America, 2008), BMD 45.

18. C. Perlitz, K. Licha, F.-D. Scholle, B. Ebert, M. Bahner, P.Hauff, K. T. Moesta, and M. Schirner, “Comparison of twotricarbocyanine-based dyes for fluorescence optical imaging,”J. Fluoresc. 15, 443–454 (2005).

19. S. R. Arridge, “Optical tomography in medical imaging,”Inverse Probl. 15, R41–R93 (1999).

20. D. Y. Paithankar, A. U. Chen, B. W. Pogue, M. S. Patterson, andE. M. Sevick-Muraca, “Imaging of fluorescent yield and life-time from multiply scattered light reemitted from randommedia,” Appl. Opt. 36, 2260–2272 (1997).

21. W. Bangerth, R. Hartmann, and G. Kanschat, “deal.II—ageneral-purpose object-oriented finite element library,”ACM Trans. Math. Softw. 33, 24 (2007).

22. M. Schweiger, S. R. Arridge, M. Hiraoka, and D. Delpy, “Thefinite element method for the propagation of light in scatter-ing media: boundary and source conditions,” Med. Phys. 22,1779–1792 (1995).

23. S. R. Arridge, Schweiger, M. Hiraoka, and D. T. Delpy, “A finiteelement approach for modeling photon transport in tissue,”Med. Phys. 20, 299–309 (1993).

24. G. Voronoi, “Nouvelles applications des paramètres continus àla théorie des formes quadratiques,” J. Reine Angew. Math.133, 97–178 (1907).

25. A. Soubret, J. Ripoll, and V. Ntziachristos, “Accuracy of fluor-escent tomography in the presence of heterogeneities: study ofthe normalized Born ratio,” IEEE Trans. Med. Imaging 24,1377–1386 (2005).

26. T.Nielsen,B.Brendel, R. Ziegler,M. vanBeek, F.Uhlemann,C.Bontus, and T. Köhler, “Linear image reconstruction for a dif-fuse optical mammography system in a non-compressed geo-metry using scattering fluid,” Appl. Opt. 48, D1–D13 (2009).

27. T. Köhler, R. Proksa, and T. Nielsen, “SNR-weighted ARTapplied to transmission tomography,” in Nuclear ScienceSymposium Conference Record (IEEE, 2003), pp. 2739–2742.

28. A. Dax, “On row relaxationmethods for large constrained leastsquares problems,” SIAM J. Sci. Comput. 14, 570–584 (1993).

29. Y. Pei, H. L. Graber, and R. L. Barbour, “Normalized-constraint algorithm for minimizing inter-parameter cross-talk in DC optical tomography”Opt. Express 9, 97–109 (2001).

30. M. Schweiger and S. R. Arridge, “Application of temporalfilters to time resolved data in optical tomography,” Phys.Med. Biol. 44, 1699–1717 (1999).

31. A. Soubret and V. Ntziachristos, “Fluorescence moleculartomography in the presence of background fluorescence,”Phys. Med. Biol. 51, 3983–4001 (2006).

32. B. W. Pogue, T. O. McBride, U. L. Osterberg, and K. D. Paulsen,“Comparison of imaging geometries for diffuse optical tomog-raphy of tissue,” Opt. Express 4, 270–286 (1999).

33. V. A. Markel and J. C. Schotland, “Scanning paraxial opticaltomography,” Opt. Lett. 27, 1123–1125 (2002).


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Nonlinear reconstruction of absorption and fluorescence contrast from measured diffuse transmittance...

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