Graves et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 231

Singular-value analysis and optimization ofexperimental parameters

in fluorescence molecular tomography

Edward E. Graves

Center for Molecular Imaging Research, Department of Radiology, Massachusetts General Hospital, HarvardMedical School, 149 13th Street, Room 5404, Charlestown, Massachusetts 02129

Joseph P. Culver

Photon Migration Laboratory, Department of Radiology, Massachusetts General Hospital, Harvard Medical School,149 13th Street, Charlestown, Massachusetts 02129

Jorge Ripoll

Institute for Electronic Structure and Laser, Foundation for Research and Technology–Hellas, P.O. Box 1527,71110 Heraklion, Greece

Ralph Weissleder and Vasilis Ntziachristos

Center for Molecular Imaging Research, Department of Radiology, Massachusetts General Hospital, HarvardMedical School, 149 13th Street, Room 5404, Charlestown, Massachusetts 02129

Received May 1, 2003; revised manuscript received August 15, 2003; accepted October 14, 2003

The advent of specific molecular markers and probes employing optical reporters has encouraged the applica-tion of in vivo diffuse tomographic imaging at greater spatial resolutions and hence data-set volumes. Thisstudy applied singular-value analysis (SVA) of the fluorescence tomographic problem to determine optimalsource and detector distributions that result in data sets that are balanced between information content andsize. Weight matrices describing the tomographic forward problem were constructed for a range of source anddetector distributions and fields of view and were decomposed into their associated singular values. Thesesingular-value spectra were then compared so that we could observe the effects of each parameter on imagingperformance. The findings of the SVA were then confirmed by examining reconstructions of simulated andexperimental data acquired with the same optode distributions as examined by SVA. It was seen that for a20-mm target width, which is relevant to the small-animal imaging situation, the source and detector fields ofview should be set at approximately 30 mm. Equal numbers of sources and detectors result in the best im-aging performance in the parallel-plate geometry and should be employed when logistically feasible. Thesedata provide guidelines for the design of small-animal diffuse optical tomographic imaging systems and dem-onstrate the utility of SVA as a simple and efficient means of optimizing experimental parameters in problemsfor which a forward model of the data collection process is available. © 2004 Optical Society of America

OCIS codes: 170.6960, 170.7050.

1. INTRODUCTIONTechnology for quantitative, three-dimensional in vivoimaging of fluorescence is of significant interest in light ofrecent advances in the development of novel fluorescentprobes and markers.1–5 Planar fluorescence reflectanceimaging is widely used for imaging fluorescence signa-tures in living animals6,7; however, the lack of depth reso-lution associated with this technique precludes quantita-tive assessment of all but the simplest subject geometries.Improvements in detection hardware, tomographictheory, and computing power have facilitated advances inapplications of diffuse optical tomography that use intrin-sic and extrinsic contrast in living subjects.8–10 Thistechnique has already been applied successfully toward invivo functional imaging, for example, in studies of cere-bral hemodynamics11,12 and of breast cancer.13,14 Dif-

1084-7529/2004/020231-11$15.00 ©

fuse optical tomography theory has recently been ex-panded to facilitate imaging of fluorescent agents.15–19

Recently the development of fluorescence molecular to-mography (FMT) has made possible the study of molecu-lar signatures in mouse disease models by resolving thebio-distribution and activation of fluorescent beacons.20

Reconstruction of tomographic data from diffusingsources involves the generation of a forward model thatpredicts the photon distribution striking the detectors fora given source location and medium. In most commonimplementations, the tomographic problem can be formu-lated in terms of a weight matrix W coupling the propertydistribution d to the measurements m as m 5 Wd. Thisexpression is then inverted so that the distribution d canbe recovered from the measurements m and the weightmatrix W. The number of elements in the weight matrix

2004 Optical Society of America

232 J. Opt. Soc. Am. A/Vol. 21, No. 2 /February 2004 Graves et al.

is given by the product of the number of detectors, thenumber of sources, and the number of voxels in the prop-erty distribution. Because of this, diffuse tomographicproblems may require large memory and computationalresources that can become unrealistic to generate and in-vert. This is especially true as higher spatial resolutionand better image fidelity are sought, since they necessi-tate the use of large measurement arrays and reconstruc-tion grids. A recently reported acquisition platform forFMT allows retrospective detector sampling at resolu-tions down to 0.15 mm, with the actual resolution of themeasurements sampled directly by a CCD camera.21

This acquisition scheme can generate as many as 1012

matrix elements assuming 32 sources and 50 3 503 25 voxels. Such systems greatly exceed current com-putational capacities for realistic applications. To maxi-mize the information content of the acquired measure-ments while minimizing the associated computationalexpense, optimization of not only the number of sources,detectors, and voxels but also their resolution and field ofview (FOV) must be performed.

To perform this optimization, a quantitative and simplemethod for assessing the efficacy of a given experimentalsetup is needed. Culver et al. forwarded the use ofsingular-value analysis (SVA) in the assessment of theutility of various experimental setups in diffuse opticaltomography.22 In their report, they applied singular-value decomposition (SVD) to weight matrices represent-ing various source and detector fields of view and demon-strated the effects of these parameters on imageresolution at experimental signal and noise levels. Thismethod of analyzing and optimizing an experimentalsetup is particularly attractive because it efficiently con-denses the information contained in the weight-matrixmodel of an experimental setup into a singular-valuespectrum, which can then be compared with spectra rep-resenting other experimental situations. This practiceprovides a single metric for the entire imaging domain.A similar study by Xu et al. has recently showcased howfiber placement can be optimized for a hybrid optical/magnetic resonance imaging system.23

In this paper we present a study of the effects of severalexperimental parameters on the imaging performance offluorescence tomography of small animals. The experi-mental parameters studied were the number and ar-rangement of detectors and sources, the resolution of thereconstruction, the FOV selected, and the medium’s opti-cal properties. The optimization problem is particularlyrelevant within this context, because improving imagingquality and resolution prompts the use of larger anddenser measurement and reconstruction arrays. Thepurpose of this study was to gain understanding and ob-tain rules for optimal acquisition and reconstruction pa-rameters in order to guide experimental work with smallanimals. The first portion of the study involved the gen-eration of weight matrices representing FMT setups withvariations in a single parameter or set of parameters.The singular-value spectra of these matrices were calcu-lated and compared. The conclusions of this analysiswere then tested by simulating fluorescence tomographydata at experimental signal and noise levels and recon-structing them by using various experimental param-

eters. The accuracy of the reconstructions was comparedwith that expected from the SVA. Finally, the results ofthe SVD study were applied in the reconstruction of aresolution phantom from experimental FMT data. Weconclude by discussing basic rules of thumb for optimalreconstruction performance, based on the quantitativeanalysis herein.

2. METHODS AND MATERIALSA. Weight Matrix FormulationGeneration of the forward problem was based on the nor-malized Born approximation,18 which relates averagephoton intensity measurements at the boundary of a dif-fuse body as a function of the fluorochrome distributionwithin the body, i.e.,

UnB~rs , rd! 5S0

U~rs , rd , kl1!• E S U~rs , r, kl1!

•

n~r!

Dl2 • G~rd 2 r, kl2!D d3r, (1)

where UnB(rs , rd) is the normalized Born average inten-sity, i.e., the ratio of the measured emission Ufl and exci-tation Uo average intensities measured at rd for a sourceat rs . U(rs , r, kl1) is the theoretically calculated aver-age photon intensity at the excitation wavelength l1 in-duced at position r by a source at position rs for a mediumwith a wave number kl1, and G(rd 2 r, kl2) is theGreen’s function that describes photon propagation at theemission wavelength l2 from a point r to the detector atlocation rd . Dl2 is the diffuse-medium diffusion coeffi-cient, and n(r) is the fluorochrome concentration at loca-tion r multiplied by the fluorescent yield. Finally, S0 is aunitless, experimentally determined calibration factorthat collectively accounts for the laser power and the un-known gain and attenuation factors of the system. Forweight-matrix generation, the volume of investigation isdiscretized into N voxels. Then Eq. (1) can be written asa summation of weights multiplied by the fluorochromedistribution at each of the discrete voxels i. For a singlesource–detector pair (s1 , d1), the discrete Eq. (1) can bewritten in vector form as

UnB~rs1 , rd1! 5 FS0U~rs1 , r1 , kl1!G~rd1 2 r1 , kl2!

U~rs1 , rd1 , kl1!Dl2

¯

S0U~rs1 , rN , kl1!G~rd1 2 rN , kl2!

U~rs1 , rd1 , kl1!Dl2 G• S n~r1!

]

n~rN!D . (2)

Additional source–detector pairs (measurements) areadded as further rows to the row vector on the right-handside of Eq. (2). For M source–detector pairs the system ofequations can be written in matrix form as

S UnB~rs1 , rd1!

]

UnB~rsM , rdM!D 5 F W11 ¯ W1N

] � ]

WM1 ¯ WMN

G • S n~r1!

]

n~rN!D ,

(3)

Graves et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 233

where W ij is an element of the weight matrix W repre-senting the weight of voxel j to measurement i.

B. Singular-Value Decomposition and Noise ThresholdWeight matrices were decomposed according to W5 USVT, where U and V are orthonormal matrices

(U21 5 UT, V21 5 VT) and S is a diagonal matrix con-taining the singular values of W. Reformulating the for-ward problem of Eq. (3), UnB 5 Wn, as UTUnB 5 SVTn,we observe that the columns of U represent the detection-space modes of the weight matrix W, and the columns ofV represent the image-space modes. In this sense, thesingular values of W (the elements of S) denote the extentto which a given image-space mode is coupled to the cor-responding detection-space mode or, in other words, howeffectively a given image-space mode can be detected bythe experimental setup.

A noise threshold in the singular-value domain was de-termined from data acquired by using an experimentalparallel-plate tomographic system designed for the studyof small animals.21 Measurements acquired from asingle tube containing 1.0 mM Cy 5.5 fluorescent dye werereconstructed by computing the regularized inverse of theweight matrix, using the relation W̄21 5 WT(WWT

1 Il)21, where l is the regularization constant. The ef-fect of the regularization constant is to reduce the contri-bution of image-space modes with singular values lessthan l to the reconstructed distribution. Culver et al.24

observed that selection of a regularization constant basedon empirical assessment of reconstructed image qualityprovided results that were as good as or better than thoseobtained by selecting l with use of quantitativecriteria.25,26 Following this strategy, images were recon-structed with regularization constants of 1021, 1022,1023, 1024, 1025, and 1026, and image noise was quanti-fied for each by the norm of the reconstructed data. Itwas observed that for values of l less than 1024, imagenoise increased sharply and dominated the reconstructedimage. Therefore 1024 was chosen as a threshold foridentifying the portion of the singular-value spectra abovethe noise and therefore useful for imaging. This thresh-old is not a universal limit, as it depends on system noisespecifications and fluorescence strength, but neverthelessit allows for a generic specification metric of the experi-mental system employed and is useful for translating in-strumental performance to singular-value space. Thesensitivity of the SVAs to this cutoff was evaluated by re-peating the experiments below with a range of thresholdlevels.

C. Experimental SetsWeight matrices represented an experimental setup em-ploying a single row of sources, a single row of detectors,and a two-dimensional mesh as shown in Fig. 1. This ge-ometry was modeled after an existing three-dimensionalparallel-plate tomographic system,21 herein constrainedto two dimensions for computational simplicity. In thediscrete sense, this approximates the solution of Eq. (1)for a volume in which all contrast (i.e., fluorochrome) iscontained in a single plane. The dimensions used for thereconstructions’ mesh, 20 3 14 mm, are appropriate for

the imaging of mice. Three experiments were performedto assess imaging response as a function of source, detec-tor, and mesh distributions.

1. The number and FOV of the sources and detectorsemployed were varied while the mesh (41 3 29 elementsover a 20 3 14 mm area) was kept constant. The sourceand detector distributions in this experiment were ofidentical geometry for every setup examined.

2. The number and FOV of only the detectors werevaried while the source (6 sources over a 15-mm FOV)and mesh (41 3 29 elements over a 20 3 14 mm area)distributions were kept fixed. The symmetric experi-ment, where only the FOV and number of sources werevaried, was also executed.

3. The number of mesh points along the X directionwas varied while the source and detector distributions aswell as the mesh distribution in the Y direction were keptconstant. This step was performed once with source anddetector distributions of identical geometry (21 sources/detectors over a 30-mm FOV) and then again with adenser detector distribution (9 sources and 49 detectors,each over a 30-mm FOV).

Each of these three experiments was performed for threedifferent surrounding media, with diffusion path lengths@ld 5 (3mams8)

21/2# of 2.2, 4.3, and 8.6 mm, to assess theeffect of the surrounding medium on the information con-tent of the FMT setup. Detector readings were consid-ered to be point measurements, so that changes in detec-

Fig. 1. Diagram of the fluorescence molecular tomography ex-perimental setup used in the FMT analysis. A two-dimensionalmesh distribution was studied by use of linear source and detec-tor arrays in a parallel-plate configuration. The fields of viewand number of sources, detectors, and mesh points in X and Ywere varied to characterize system performance as a function ofthese parameters.

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tor number and/or FOV did not affect the sampling rangeof an individual detector element.

D. Analysis of Reconstructions of Simulated DataTo confirm that SVA trends predicted imaging perfor-mance, we performed reconstructions of data simulatedfor the different parameters tested. Simulations of theexcitation (Uo) and emission (Ufl) average intensity mea-surements for the test fluorochrome distribution shown inFig. 2 were generated according to

Uo~rs , rd! 5 U~rd 2 rs , kl1!, (4)

Ufl~rs1 , rd1! 5 @U~rs1 , r1 , kl1!G~rd1 2 r1 , kl2!

¯U~rs1 , rN , kl1!

3 G~rd1 2 rN , kl2!# • S n~r1!

]

n~rN!D . (5)

These simulations were performed with a source distribu-tion consisting of 121 sources equally spaced over a60-mm FOV and a detector distribution consisting of 401detectors with a spacing of 0.15 mm over a 60-mm FOV.The excitation measurements were then normalized to a

Fig. 2. Test fluorochrome distribution used to assess the experi-mental trends predicted by SVA. (a) For simulated reconstruc-tions, three-point fluorochromes equivalent to 1000 nM of Cy 5.5dye were aligned in a diagonal line from sources to detectors withcenter-to-center separations of 5.7 mm. (b) Experimental datawere acquired and reconstructed for a resolution phantom con-sisting of two parallel tubes, shown in relation to the source anddetector arrays.

typical experimental measurement count rate of 1.03 105 counts/s, and the emission (fluorescence) mea-

surements were normalized accordingly by the same fac-tor. A hybrid noise model, in which the standard devia-tion of the noise associated with a signal U is given by(0.012 3 U 1 3), was then added to the simulated high-resolution data. This noise model has been characteris-tic of the variability present in measurements and wasdetermined experimentally. The excitation and emissiondata were then subsampled with a variety of source anddetector numbers and fields of view, corresponding to ex-periment 1 described in the preceding subsection. Inver-sion was performed with 30,000 iterations of the methodof projections.27 Simulated fluorescence measurementswith count rates of less than 5 counts/s were excludedfrom the reconstruction process. An inversion techniqueother than SVD was used in order to demonstrate the factthat the effects observed are attributable to the informa-tion content of the experimental setup and are not appli-cable solely to SVD. As the model inverted in the recon-struction process is an ill-posed form of the radiativetransfer equation linearized through application of thediffusion approximation,28 local error minima may affectthe accuracy of the reconstruction, as discussed by Pierriand Tamburrino.29 These issues are not addressed inthis study, as they are beyond the scope of the SVA.

E. Analysis of Reconstructions of Experimental DataFinally, experimental FMT data collected from a resolu-tion phantom were reconstructed by using a variety of de-tector distributions, comparable to that performed in thestudy above. A resolution phantom was constructed, con-sisting of two parallel tubes 3 mm in diameter separatedby a gap of 1 mm as shown in Fig. 2(b). The tubes werefilled with a 1-mM solution of Cy 5.5 fluorescent dye andplaced in the center of a 15-mm wide parallel-plate imag-ing chamber21 filled with an intralipid and ink matchingfluid with a diffusion path length of 4.3 mm. Images ofthe photon distribution through the phantom were ac-quired at the fluorescence excitation and emission wave-length by use of six collinear sources each separated by 3mm. The measured excitation and emission images werethen corrected for background intensity and normalizedon the basis of the exposure time used for each acquisi-tion. These data were reconstructed repeatedly, and thenumber of measurements and the measurement FOV ex-tracted from the recorded images was varied in each in-stance, as was done in experiment 2 above. Each of thesedata sets was reconstructed by use of 10,000 iterations ofthe method of projections. As with the simulated data,fluorescence measurements with count rates of less than5 counts/s were not considered in the inversion process.

3. RESULTSA. Singular-Value Analysis of Weight MatricesAnalysis of singular values focused on determining thenumber of useful singular values above the noise thresh-old for each tomographic set examined herein. An ex-ample of singular-value spectra associated with weightmatrices representing 21 sources and 21 detectors spacedover a 10-, 20-, or 30-mm FOV are plotted on a logarith-

Graves et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 235

mic scale in Fig. 3. The experimentally determinedthreshold of 1024 is plotted as a horizontal dashed line.Subsequent figures 4–6 plot the number of singular val-ues found above the threshold as a function of the experi-mental parameters tested. This number represents ameasure of the useful information contained in the data.Repeated trials of the analyses with singular-value cutofflevels from 1021 to 1026 demonstrated that while the ab-solute number of useful singular values changed, thetrends in imaging performance as a function of experi-mental parameters were independent of the singular-value cutoff chosen within a range of experimentally rel-evant cutoff values from 1023 to 1026.

Figure 4 depicts the results of this analysis for experi-ment 1, in which the source and detector arrays were var-ied in conjunction while the mesh was kept fixed. It isapparent that increases in source and detector FOV ini-tially result in significant corresponding increases in thenumber of singular values above the noise threshold,which corresponds to a larger number of image-spacemodes available for reconstruction. This is true for arange of optode numbers marked with different symbols,as seen in Fig. 4(a). (Optode: an optical sampling ele-ment. In this context, either a photon source or detector.)However, the benefit is maximized at a FOV of approxi-mately 30 mm, and further increases lead to slight reduc-tions in available singular values. For very small optodenumbers, improvements in imaging performance due toincreases in source and detector FOV plateau quickly.More significant benefits are observed for larger optodenumbers, but the performance plateau at large fields ofview becomes more apparent. Figure 4(b) plots the cor-responding effects for varying diffusion length assuming21 sources and detectors. It is apparent that decreasesin the diffusion length of the medium increase the num-ber of useful singular values. The FOV for which imag-ing performance is maximized remains roughly constant,however.

Fig. 3. Singular-value analysis of the effects of detector andsource FOV and number for symmetric source and detector ar-rays. Singular-value spectra for weight matrices representingsetups with 21 sources and 21 detectors over a 10-mm (squares),20-mm (crosses), and 30-mm (triangles) FOV are shown. Theintersections of each spectrum with an empirically determinednoise threshold (1024) yielded the number of nonnoise, or useful,singular values for that experimental setup.

Figure 5(a) summarizes the results of experiment 2.The number of useful singular values as a function of de-tector FOV is plotted for detector arrays of 11, 21, 41, and81 elements, assuming a six-element source distribution.These curves suggest that for a fixed source array, imag-ing performance improves linearly with FOV at smallfields of view. For larger fields of view, the number ofuseful singular values breaks from linearity and plateaus.The FOV at which this deviation from linearity occurs ap-pears to be approximately 20 mm for each setup shown.The relation of this optimal detector FOV to the sourceFOV, in this case 15 mm, was then investigated by vary-ing the detector and source fields of view while the num-ber of sources and detectors was kept fixed at 9 and 49,respectively. These curves are shown in Fig. 5(b). Foreach source FOV selected, imaging performance peakedat a detector FOV of approximately 30 mm. Further-more, performance increases with increasing source FOVbut plateaus for source fields of view above 30 mm. Analmost identical set of curves was observed when sourcenumber and FOV were varied for a fixed detector arraywith a geometry identical to the fixed source array of ex-

Fig. 4. Singular-value analysis of the effects of detector andsource FOV and number for symmetric source and detector ar-rays. (a) Plots of the number of useful singular values, ex-tracted as shown in Fig. 3, versus source and detector FOV forsetups with 6 (squares), 11 (crosses), 21 (triangles), and 31(circles) sources and detectors. (b) Plots for setups involving 21sources and detectors interrogating media with diffusion lengthsof 2.2 mm (circles), 4.3 mm (triangles), and 8.6 mm (crosses).

236 J. Opt. Soc. Am. A/Vol. 21, No. 2 /February 2004 Graves et al.

periment 2. This is in agreement with the previously es-tablished symmetry of the tomographic problem with re-spect to sources and detectors.

Whereas the preceding experiments investigated imag-ing performance as a function of the experimental setupused, experiment 3 studied imaging performance in rela-tion to the reconstruction mesh FOV. The results ofvarying the reconstruction mesh in the X direction whilekeeping the mesh Y distribution fixed are shown in Fig. 6.Plots of the number of above-noise singular values as afunction of mesh X dimension are shown for situationswith equal (circles) and unequal (triangles) numbers ofsources and detectors. The numbers of measurementsfor each case are equal, however. In the symmetricsource and detector case, imaging performance reaches amaximum for approximately 30 mesh points in X and sub-sequently plateaus. Imaging performance is reduced inthe case where there are more detectors than sourcesrelative to the symmetric case. Also, the plateau in im-aging performance occurs at a larger number of meshpoints, approximately 35.

Fig. 5. Singular-value analysis of the effects of detector andsource FOV and number for asymmetric source and detector ar-rays. (a) Plots of the number of useful singular values versusdetector FOV for 11 (squares), 21 (crosses), 41 (triangles), and 81(circles) detectors for 6 sources over a 15-mm FOV. (b) Numberof useful singular values plotted against detector FOV, in thiscase for source fields of view of 10 (squares), 20 (crosses), 30 (tri-angles), and 40 (circles) mm. These plots were generated for 9sources and 49 detectors.

B. Analysis of Reconstructions of Simulated DataThe effects observed through SVA above were then com-pared with results of reconstructions of data simulatedfor the fluorochrome distribution shown in Fig. 2. Figure7 shows reconstructions of simulated data with use ofsource and detector numbers and fields-of-view identicalto those used for the SVA presented in Fig. 4(a). Becauseof the threshold used to exclude fluorescence measure-ments of low intensity, analogous to that performed withexperimental data, the percentage of measurements usedfor reconstruction varied as a function of FOV, rangingfrom 100% at a 10-mm source/detector FOV to 52% at a40-mm source/detector FOV for a medium diffusionlength of 4.3 mm. The accuracy of the reconstructionswas analyzed quantitatively by computing the root-mean-square deviation from the actual (simulated) fluoro-chrome distribution for each of the reconstructed datasets. These data are presented in Fig. 8. A decrease inthe error of the reconstruction is observed with an in-crease in the source and detector fields of view, corre-sponding to an improvement in imaging performance.The error is minimized at a FOV of 30 mm for the 31-optode case and then increases slightly when the FOV isincreased to 40 mm. These trends in reconstruction er-ror mirror the imaging performance of each experimentalsetup predicted by SVA, shown in Fig. 4(a). Simulationswere also performed for medium diffusion lengths of 2.2and 8.6 mm. Imaging performance, in terms of resolu-tion of the three objects and object shape, was improvedfor the same experimental setup by using media withshorter diffusion lengths, and the trends in the error ofthe reconstruction paralleled those observed in Fig. 8.

C. Analysis of Reconstructions of Experimental DataFigure 9 shows the results of reconstructions of experi-mental FMT data in which varying detector distributionswere used. From each row of reconstructions, it is appar-ent that changing the detector FOV influences the shapeof the reconstructed objects. Reconstructions for fields ofview of 10–20 mm show focii that are elongated and tilted

Fig. 6. Singular-value analysis of the effects of mesh FOV andnumber of elements for symmetric and asymmetric source anddetector arrays. Circles show the number of useful singular val-ues versus mesh X FOV for 21 sources and 21 detectors over a20-mm FOV. Triangles are plotted for the corresponding analy-sis of mesh X FOV and number of points for 9 sources and 49 de-tectors over a 20-mm FOV.

Graves et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 237

Fig. 7. Analysis of the effects of detector and source FOV and number for symmetric source and detector arrays, using reconstructionsof simulated data. Reconstructions of the fluorochrome distribution shown in Fig. 2 over a 41 3 29 element mesh with FOV 203 14 mm are given for varying source and detector number and FOV. These reconstructions were performed for an optical medium

with diffusion length 4.3 mm.

outward, whereas reconstructions at a detector FOV of 40mm show focii that are tilted inward. The reconstruc-tions for a detector FOV of 30 mm exhibit approximatelycircular focii. Altering the number of detectors used inthe reconstruction and correspondingly the detector reso-lution appears to affect the resolving power of the setup.The individual tubes are not clearly resolved in recon-

structions employing 11 detectors but begin to becomedistinct focii as the number of detectors is increased to 21and 41. As with the simulated reconstructions, thetrends in the experimental FMT reconstructions also re-flect the SVAs, in this case the plots shown in Fig. 5. Aswas seen with the reconstructions of simulated data, thepercentage of data points above the chosen threshold de-

238 J. Opt. Soc. Am. A/Vol. 21, No. 2 /February 2004 Graves et al.

creased with increasing FOV, from 100% at a detectorFOV of 10 mm to 69% at a FOV of 40 mm.

4. DISCUSSIONWith advances in detection hardware and reconstructionmethodology as well as the development of sensing opticalprobes, FMT is poised to assume a prominent role in themedical imaging arsenal. In order to most effectively ap-ply this imaging modality, this study investigated the ex-perimental parameters, focusing on the source and detec-tor distributions, that maximize the quality andinformation content of diffuse tomographic data obtainedfrom imaging geometries relevant to small-animal sub-jects. This was achieved by constructing weight matricesdescribing photon propagation through a target medium,as is done in the solution of the experimental tomographicproblem, for a range of experimental setups. These ma-trices were then decomposed into their corresponding sin-gular values, which were compared in terms of the extentto which they lie above an experimentally determinednoise threshold. The validity of this measure of experi-mental utility was demonstrated by comparing the imag-ing performance expected from SVA for a given experi-mental setup to that observed in reconstructions ofsimulated and real experimental data with the same setof parameters.

Figures 4 and 5(a) demonstrate the effectiveness of SVAfor optimizing experimental setups in optical tomographyof turbid media. The data on the number of useful sin-gular values are given as a function of FOV, with multipleplots shown for different numbers of sources and/or detec-tors. The reason for this mode of presentation is practi-cal. Image-reconstruction time is largely a function ofthe number of measurements and the number of points inthe reconstruction mesh (and therefore by extension, thesize of the weight matrix to be inverted). Specific diffusetomographic applications will select a data-set volume, in

Fig. 8. Image error for reconstructions of simulated data withsymmetric source and detector arrays. The reconstruction er-ror, in terms of deviation from the test fluorochrome distributionshown in Fig. 2, for the reconstructions in Fig. 7 (diffusion length4.3 mm) is plotted as a function of source/detector FOV, withseparate traces for reconstructions with different numbers ofsources and detectors.

terms of number of measurements and mesh size, thatwill reconstruct in a reasonable amount of time given theavailable computing resources. This corresponds to theselection of an imaging performance curve representingthe relevant number of optodes and mesh points in Figs.4–6, which can then be used to select an optimal FOV.Careful comparison of Figs. 4(a) and 5(a) reveal that forthis parallel-plate configuration, symmetric source anddetector configurations generate measurements with thehighest information content. This is explicitly demon-strated in Fig. 6, in which imaging performance for a sym-metric source and detector arrangement is seen to be su-perior to an arrangement with more detectors thansources, although with the same total number of mea-surements. These optimizations are subject, however, tologistical constraints such as the specific availability ofthe number of sources or detectors. For example, fiber-based systems result in a fixed number of measurementsthat need to be optimized given the volume considered.In many cases it may be more practical to have more de-tectors than sources since parallel operation can be moreeasily achieved on the photon detection side. Such issuescan result in situations similar to that of experiment 2,where either the source or the detector distribution isasymmetrically dense.

An important distinction to make is whether the ben-efits in imaging performance obtained from increasing thenumber of measurements, shown in Figs. 4 and 5, are at-tributable to increases in signal-to-noise ratio because ofsignal-averaging effects or whether these results demon-strate a fundamental increase in the information contentof the recorded measurements. This question was ad-dressed by recalculating the data from experiments 1 and2, with a singular-value threshold normalized by thesquare root of the number of measurements for eachweight matrix. At large (.20 mm) fields of view thesecurves demonstrated the same trends as the correspond-ing plots in Figs. 4 and 5, whereas for small fields of viewthe curves for setups with different numbers of measure-ments all converged to approximately the same trace.This suggests that at a small FOV, increases in the num-ber of measurements acquired improve data quality sim-ply through increased overall signal-to-noise ratio. Asthe FOV is increased, the improvement in data qualitycan no longer be explained by signal averaging, suggest-ing that the actual information content of the data set isimproved. With this validation in hand, we can statethat the superiority of parallel-plate symmetric sourceand detector distributions relative to source- or detector-heavy setups is due to their ability to most effectively andcompletely obtain angular projections from a target vol-ume. Similarly, the results of experiment 2 (Fig. 5) dem-onstrate that increasing the detector number and FOV fora fixed source distribution improves the information con-tent of the measurements associated with asymmetric ge-ometries. These findings support shifts from small-data-set FMT implementations using fiber optic sources anddetectors30 to large-data-set FMT implementations usingsetups with scalable numbers of detectors and/orsources.21

For practical reasons our analysis is restricted to theconsideration of parallel-plate configurations with all

Graves et al. Vol. 21, No. 2 /February 2004 /J. Opt. Soc. Am. A 239

sources located in one plane and all detectors in another.This facilitates the acquisition of large data sets throughhigh-resolution detector and/or source sampling. How-ever, more sophisticated arrangements of sources and de-tectors may result in a different optimization. Singular-value analysis of detector-heavy parallel-plateimplementations demonstrated that such setups may besufficient when detectors are placed on both sides of thetarget (data not shown). Xu et al. recently presented asimilar SVA of source and detector optimization for imag-ing of rat brains.23 Singular-value spectra were calcu-lated for Jacobian matrices representing eight source andeight detector fibers arranged in two configurationsaround the head of an adult rat. In one case, the fibers

were equally spaced around the circumference of thehead; in the other case, the fibers were placed predomi-nantly near the brain. The findings of the SVA suggestedthat comparable information content is obtained from thetwo setups.

It is useful to note that by formulating the inverseproblem as W̄21 5 WT(WWT 1 Il)21, the computationalexpense of adding either mesh points or measurementscan be alleviated.24 This is accomplished by reducing thedimensions of the matrix to be inverted from Nvox3 NsNd to either NsNd 3 NsNd or Nvox 3 Nvox , whereNs , Nd , and Nvox are the number of sources, detectors,and mesh points, respectively. Therefore the time tosolve the inverse problem can be made independent of ei-

Fig. 9. Analysis of the effects of detector FOV and number, using reconstructions of experimental data. FMT data were acquired fortwo tubes filled with 1000 nM Cy 5.5 fluorescent dye, separated by a gap of 1 mm, immersed in a medium with diffusion length 4.3 mm.Reconstructions are shown for a range of detector fields of view and numbers, which were sampled from the acquired high-resolutionimages. The mesh used contained 41 3 29 elements with a FOV of 20 3 14 mm.

240 J. Opt. Soc. Am. A/Vol. 21, No. 2 /February 2004 Graves et al.

ther the number of measurements or the number of ele-ments of the reconstruction mesh. This obviates thequestion of which component, measurements or meshpoints, has the greater effect on the quality of the recon-structed data. From the simulations presented in Figs.4–6, it appears that increasing the measurement densityhas a more pronounced effect on the singular-value spec-tra of the weight matrix. That imaging performance pla-teaus with increasing mesh density can be seen in Fig. 6,whereas Figs. 4 and 5 demonstrate improvements in thenumber of useful singular values for increases in detectornumber and FOV up to the largest distributions studied.

In the simulations we assumed that all the measure-ments had equivalent signal-to-noise levels (e.g.,constant-percent noise). This assumption affects the op-timization results. As an example, consider increasingthe detector FOV while keeping the number of detectorsconstant. With the constant-percent-noise model, a lossof imaging performance (in singular-value terms) withlarge fields of view is most likely due to decreases in sam-pling density. For the experimental data, the benefits ofincreasing the detector FOV are also limited by the loss ofsignal-to-noise ratio for measurements beyond the edgesof the target volume. This limitation is typically ad-dressed during reconstruction by removing measurementvalues with low signal-to-noise ratio from the inversionprocess. As described above, this can result in reducednumbers of available measurements for the same numberof detector points at larger detector fields of view.

These data offer guidelines for the design of small-animal, parallel-plate diffuse tomographic images. First,the optimal source and detector fields of view should beslightly larger than the width of the target volume to pro-vide the most effective angular sampling. For the 20-mmtarget studied here, the optimal source and detector fieldsof view were approximately 30 mm [Fig. 4(a)]. Further-more, the optimal source and detector fields of view ap-pear to be independent of the number of sources and de-tectors employed. Second, it is generally more useful toincrease the number of measurements acquired than toincrease the number of elements of the reconstructionmesh in order to improve image quality. Third, for afixed number of measurements, symmetric source and de-tector arrays provide the most effective imaging setup, afactor that should considered appropriately when design-ing an imaging chamber. With these rules of thumb inmind, we conclude that an optimal FMT imaging setupfor small-animal imaging involves a source and detectorFOV of 30 mm along the width of the animal, an optodesampling density of 1 optode/mm, and a 20-mm voxelmesh FOV with resolution 0.5 mm/voxel. It is useful tonote that similar analyses performed for larger targets,up to 5.0 3 4.7 cm, exhibited the same trends and opti-mal configurations as for the small-animal imaging di-mensions (data not shown). These recommendations,based on SVA, were corroborated by evaluation of recon-structions of both simulated and experimental data gen-erated with similar geometries. These results demon-strate that SVA of the FMT problem is an effective andefficient means of assessing the effects of individual ex-perimental parameters and identifying optimal values forthese parameters.

ACKNOWLEDGMENTSThe authors gratefully acknowledge the contributions ofRalf Schulz, Anabela da Silva, Doreen Yessayan, and An-dreas Yulliano to this work. This work was supported inpart by National Institute of Health grants P50 CA86355, R24 CA 92782, RO1 EB 000750-1, R21 CA 91807,and T32 CA 79443.

Address correspondence to Vasilis Ntziachristos,Center for Molecular Imaging Research, MassachusettsGeneral Hospital, 149 13th Street, Room 5404,Charlestown, Massachusetts 02129; e-mail, vasilis@helix.mgh.harvard.edu; phone, 617-726-5788; fax, 617-726-5708.

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