Early-photon fluorescence tomography: spatial resolution improvements and noise stability considerations

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1444 J. Opt. Soc. Am. A/Vol. 26, No. 6 /June 2009 Leblond et al.

Early-photon fluorescence tomography:spatial resolution improvements and

noise stability considerations

Frederic Leblond,1,* Hamid Dehghani,2 Dax Kepshire,1 and Brian W. Pogue1

1Thayer School of Engineering, Dartmouth College, 8000 Cummings Hall, Hanover New Hampshire 03755, USA2School of Computer Science, University of Birmingham, Birmingham B15 2TT, UK

*Corresponding author: [email protected]

Received March 3, 2009; accepted April 27, 2009;posted April 30, 2009 (Doc. ID 108321); published May 27, 2009

In vivo tissue imaging using near-infrared light suffers from low spatial resolution and poor contrast recoverybecause of highly scattered photon transport. For diffuse optical tomography (DOT) and fluorescence moleculartomography (FMT), the resolution is limited to about 5–10% of the diameter of the tissue being imaged, whichputs it in the range of performance seen in nuclear medicine. This paper introduces the mathematical formal-ism explaining why the resolution of FMT can be significantly improved when using instruments acquiringfast time-domain optical signals. This is achieved through singular-value analysis of the time-gated inverseproblem based on weakly diffused photons. Simulations relevant to mouse imaging are presented showingthat, in stark contrast to steady-state imaging, early time-gated intensities (within 200 ps or 400 ps) can inprinciple be used to resolve small fluorescent targets (radii from 1.5 to 2.5 mm) separated by less than 1.5 mm.© 2009 Optical Society of America

OCIS codes: 260.2510, 170.6960, 170.6920, 170.3660, 170.3010.

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. INTRODUCTIONptical tomography refers to an ensemble of imagingethods designed to interrogate tissue using near-

nfrared light (NIR) transmission, where the transportrocess interacts with its microscopic components. Therere two main categories of tomographic imaging applica-ions, with each potentially probing different biologicalechanisms. Fluorescence molecular tomography (FMT)

s used to localize optical contrast associated with the ac-umulation or retention of fluorescent reporters, whichan be used to image specific cellular or organ-specificrocesses [1–16]. The second approach, based on trans-ission alone, is often referred to as diffuse optical to-ography (DOT), and is used to study how light interactsith tissue based on absorption contrast from molecules

uch as hemoglobin and water [17–19]. Both approachesse diffuse light measurement, which is known to lead to

ow resolution and blurry images. Typically, the reso-ution of optical tomography is limited to several millime-ers depending on the thickness of the tissue being im-ged, which is comparable to nuclear imaging resolution.ainly, there are three factors determining the reso-

ution: (1) the physical properties of light-matter trans-ort, (2) the instrument design, and (3) the image recon-truction method. The optimal combination of lightransport with time-gated data, optimal instrument de-ign, and an appropriate image reconstruction algorithman lead to dramatic improvements in image resolution.his subject is analyzed here through numerical analysistudies of the signal and inversion process.

Photons propagating through tissue are either ab-orbed or scattered, making it possible to model this with

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eneral bulk tissue interaction coefficients. The absorp-ion coefficient �a and scattering coefficient �s describeheir probability of interaction per unit length. The aver-ge distance covered by a photon between two scatteringites in tissue (mean free path) is typically near 100 �m,ut when isotropic scattering is approximated, then theransport (or reduced) scattering coefficient �s� is definedor approximate isotropic scattering. The transport scat-ering distance in tissue (or transport mean free path) ispproximately 0.5 mm to 1.0 mm. This sets a fundamen-al limit on the spatial resolution that can be attainedhen using diffuse optical imaging.The second consideration that can degrade this reso-

ution limit further is determined by the instrument de-ign. It has been shown that different combinations ofources and detectors will lead to different reconstructionmage quality [20]. In particular, singular-value analysisf the DOT and FMT forward model has been performedn order to design mathematical criteria that might besed to optimize the source–detector design [21–24]. How-ver, it is generally expected that an optimal configura-ion for optical tomography will be one which mimics-ray computed tomography (CT), where there is as muchircular symmetry as possible at the periphery of the tis-ue being imaged.

An inherent characteristic of optical tomography is theact that, even for excellent tissue sampling detection ge-metries, the inverse problem is ill-conditioned [25]. Inractice, this means that the problem is underdeter-ined, implying that a large number of solutions exist forgiven optical data set. By using an ideal tissue sampling

eometry, the symptomatic linear dependency of the mea-

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Leblond et al. Vol. 26, No. 6 /June 2009/J. Opt. Soc. Am. A 1445

urements is minimized, thereby improving the condition-ng of the problem. Then, a least-squares solver would bexpected to converge to a unique solution correspondingo a high-fidelity image for which the spatial resolution iset by the aforementioned fundamental limit imposed byiffuse imaging. However, this is not the case for diffuseptical tomography, as is well known, and will be ana-yzed further here. In reality, reconstructing optical im-ges is typically attained by finding a balance betweeninimization of the residual norm and the size of the so-

ution through regularization. Here, size of the solutionhould be understood broadly as representing a math-matical norm that can be used to include prior informa-ion of the problem. For example, the main trend in opti-al imaging consists of using structural anatomicalnformation from segmented CT or MR images as a wayo improve the conditioning of the optical imaging prob-em [26–30]. Another approach consists of using multi-pectral optical data to improve image quality [31,32].

An important consequence of the ill-posed nature of theroblem is that the inversion is hypersensitive to noise.herefore, the problem cannot be solved uniquely no mat-er how many diffuse measurements are added to theata vector, even if it becomes an overconditioned prob-em. Reducing the sensitivity to noise can be partially at-ained by using regularization methods or using differentata types. Here, the approach of choice consists of usingmodeling method that allows the user a choice to im-

rove the conditioning of the matrix inversion problemithout the use of spatial or spectral priors, hence reduc-

ng the sensitivity to noise. To do this, the optical tomog-aphy inversion is formulated with the ability to analyzehe effect of different optical data types that correspond toodified light transport paths. Niedre et al. [33] have

hown that image reconstructions based on so-calledarly photons lead to significant resolution improvementsn the FMT images when imaging lung tumors in mice inivo. The forward modeling approach they use is based onnalytic solutions to the radiative-transfer-equationRTE) [34,35]. In effect, this approach allows capture ofarly photons that are highly scattered in the forward di-ection and so deviate very little from the direction ofravel. For any given source–detector pair, the path ofravel of photons in the earliest time windows is mucharrower than the path that would have held for all timeates. Decreasing time gates directly translate into nar-ower more direct path photons, and this inherently im-roves the resolution of imaging with this data set. DOTpproaches have also been developed to reconstruct im-ges based on early photons associated with nonfluores-ent optical signals [36,37].

The main aim of this paper is to explain why a combi-ation of appropriate light-transport modeling tech-iques, photon detection technology, and image recon-truction methods can lead to tomography withntrinsically higher spatial resolution as compared withhe majority of approaches that have been used in theast. The approach outlined is based on the FMT forwardodel constructed using a time-dependent finite-elementethod (FEM) solution to the diffusion equation. Theodel is used to show that the forward problem foreakly diffused photons produces a significantly im-

roved conditioning of the inversion matrix when com-ared with that associated with a steady-state diffuse sig-al. Evidence for this is provided through computation ofhe condition number of the forward problem matrix asell as through detailed singular-value analysis (SVA).he results of the SVA provide an intuitive explanation

or why the spatial resolution of optical tomography cane improved so dramatically. Those findings are sup-orted by actual image reconstructions performed forimulated fluorescence data of a multiple target phantomith different levels of contrast.Section 2 briefly presents the outline for time-domain

MT as well as a description of the preclinical instrumentotivating the detection geometry used in this work.hen, Section 3 presents the general formulation of the

ime-dependent optical tomography problem, with theerivation of the forward problem for time-gated diffuseduorescence signals and the inverse problem resolutionethods used for image reconstruction. Section 4 pre-

ents the main results of this paper in the form of a SVAf the problem with image recovery. The paper is con-luded in Section 5 with discussions pertaining to limita-ions and potential extensions of the results.

. BACKGROUND. Fan-Beam Detection of Time-Domain Signalshe simulation results presented in this paper are per-

ormed based on a detection geometry mimicking a newlyeveloped time-domain tomography system [38]. The sys-em utilizes a bed compatible for both FMT and x-ray CTnstruments for small animal studies. FMT imagesomplement the anatomical information from CT witholecular information pertaining to extracellular and in-

racellular processes highlighted by either endogenous orxogenous fluorophores. A schematic of the optical systems shown in Fig. 1(a). The system was designed to utilize aotating gantry, allowing use of a single source with a faneam configuration of photomultiplier tube (PMT) detec-ors that rotate around the surface of the specimen. Inhis configuration, fully noncontact excitation and detec-ion is achieved, and a flexible number of measurementsan be obtained. Five optical channels (labeled D1 to D5n the figure) use focused detection to collect the diffuseransmission of excitation and fluorescence signals fromhe surface of the specimen.

In this work, optical tomography resolution improve-ents are examined using data associated with time-

ated signals. The optical system design is based on time-orrelated single photon counting (TCSPC) instrumen-ation cards (Becker and Hickl, Berlin, Germany) and aaser diode driver module (PicoQuant, Berlin, Germany)ontrolling a 635 nm pulsed diode laser operating at0 MHz and delivered to the rotating gantry by fiber op-ics and focused through free space onto the animal tis-ue. Diffusely transmitted fluorescence and excitation sig-als are then collected from five lenses with an angulareparation of 22.5°. These couple through fibers to theransmission (Tr) and fluorescence (Fl) channels, respec-ively. The light is then collimated and spectrally sepa-ated using filters. Hamamatsu H7422P-50 PMTs detecthe incident photons and generate a single analog pulse

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or each detected photon. Fig. 1(b) shows a sampleemporal-point-spread function (TPSF) acquired with theptical system as well as the corresponding impulse-esponse-function (IRF). Time-referencing of the signalss accomplished by approximating the initial time whenhotons hit the specimen surface as the mean time of theRF. This approach allows sampling of the light into2.5 ps time bins, and while the temporal response func-ion of the laser through the system is close to 400 ps,alibration allows accurate sampling of data down to nearhe time resolution of the bins. Relative to photon detec-ion based on ICCD cameras, PMT-based detection has anncreased sensitivity due to internal gain factors that arene or two orders of magnitude larger.

As schematically illustrated in Fig. 1(a), the photonsaking up a time-domain signal can be divided into three

onceptual categories: (1) ballistic photons, (2) weaklycattered photons (WSP), and (3) diffused photons. Theallistic photons correspond to those particles that propa-ate from the source to the detector without goinghrough scattering events, while the WSPs are those pho-ons suffering only a limited number of scattering eventsrior to detection.As shown in the figure, the WSPs are associated with

ravel paths remaining relatively close to the direct line-

ig. 1. (Color online) Schematic representation of the optical in-trument. Shown is a single excitation source position and theve channels. Diffuse light signals at the surface of the specimenre collected using focalized detection. The detected signals arehen separated and directed to two sets of PMTs dedicated touorescence and excitation signals at each fiber channel. This de-ection geometry is that used for the simulations presented inhis paper. (b) Impulse-response-function (IRF) and sample fluo-escence time-resolved signal acquired for one of the channels.

f-sight between the source and the detector. They corre-pond to photons that were almost ballistic and ulti-ately propagated in a highly forward directed manner,

argely because each scattering event is anisotropic withhe highest probability of scatter being in the forward di-ection. In contrast to this, the diffuse photons are thosehat go through a large number of scattering events be-ore being detected. Light transport for the diffuse pho-ons can be modeled as a diffusion process. Indeed, it haseen shown that in the high scattering limit where �a�s and when light transport distances are large enough

ompared to the photon transport mean free path, theTE effectively reduces to the diffusion equation (see, forxample, [39]). Diffused photons are the main contribu-ors to a TPSF whenever those two physical conditionsre satisfied.Evidently, the smaller the number of scattering eventsphoton suffers, the earlier it can be picked up by one of

he detectors. For example, [34] shows, based on analyticolutions to the RTE, that the experimental signature ofallistic photons should be in the form of a prepulse in thePSF centered around tb=n d /c, where d is the distanceetween the source and the detector on the surface, n ishe index of refraction of the medium, and c is the speedf light. For all practical purposes, the contribution of bal-istic photons is negligible in situations relevant for tissuemaging [34,40]. However, there are situations where theontribution of WSPs can be relevant. For example, con-ider the case of small-animal imaging, which is the mainocus of study here. Typically, the absorption in small ani-als is rather large, and the distances between sources

nd detectors can be of the order of a few centimeters.oth of these facts make it unlikely that the diffusion ap-roximation to the RTE is valid for precisely modelingarly photons. Therefore, several of the time bins after tbould be dominated by highly forward directed WSPs.evertheless, in this paper light transport modeling isone by solving the diffusion equation, because it allowsn examination of the transition between diffuse andSP photons with continuous variation of the time bins

sed. The assumption made is that the conclusions thatre reached studying the forward problem for early dif-used photons time-gates are relevant for the WSPs, andt the very least, the trends observed are expected to bendicative of what would be seen if completed with a morexhaustive transport model. A discussion pertaining tohe generalization of the diffusion-based results and howhey relate to radiation transport modeling is presentedn section 5.

. Tissue Heterogeneities and Data Normalizationne of the key features associated with FMT and, in par-

icular, with the instrument described in Subsection 2.As the fact that it acquires light signals at two differentavelengths. Indeed, the consensus that seems to havemerged in FMT consists of using the so-called Born nor-alization approach [41]. This consists of reconstructing

mages based on raw fluorescence measurements normal-zed by a measurement acquired at the light source exci-ation wavelength. Formally, it can be shown that for ap-ropriate detection geometries this normalization processartially reduces the effect of high photon attenuation

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rom the fluorescence signal, thereby significantly reduc-ng the importance of precisely knowing the optical tissueroperties of the interrogated specimen [42–44]. This im-lies that when using Born-normalized data sets it iselatively safe to use a forward model assuming homoge-eous optical properties. An educated guess for the aver-ge values is then used based either on the literature orn average background values obtained by TPSF fittingrior to image reconstruction [45–47]. This approach con-iderably simplifies the procedure by making it easier toodel light propagation and by reducing the number of

nknowns to be reconstructed. In addition, these ratioata provide a normalization of the signal which reducesodeling errors when the diffusion model used does not

xactly mimic the tissue shape, or when the fibers usedor imaging do not have consistent contact with the tis-ue. This boundary error minimization provides an inher-nt stability to the signal used which is critical for routinepplication.The critical feature that is required for the Born ap-

roach to work is that the tissue region that is producinghe fluorescence signal should be similar to that sampledy the photons in the transmitted signal. In effect, thedeal geometry that satisfies this criterion is one wherehe signal acquisition is done in transmission across themaged specimen in a manner mimicking 360° x-ray ac-uisition as performed in CT devices [see Fig. 1(a)].In the remainder of this work, the assumption is made

hat the simulated input measurement vectors alwaysonsist of Born-normalized data. In the case of time-gatedignals, the fluorescence and transmitted signals prior toaking the ratio are assumed to be computed for the sameime-gates. As will be evidenced in Subsection 3.B, the tis-ue sampled by time-gates corresponding to early photonss much narrower than for conventional steady-state sig-als. This implies that the tissue volume sampled by fluo-escence and transmission early photons are then evenore similar, thereby further increasing the potential of

he Born ratio to reduce the impact of optical propertyeterogeneities and boundary errors when compared with

maging based on steady-state measurements.

. MODELING METHODS ANDLGORITHMS

n this section, the basic formalism to model time-ependent light transport in tissue is introduced withinhe scope of a tomographic approach. Then, this notations applied to develop the formalism for the forward prob-em associated with time-gated signals in FMT. The sec-ion ends with important features of the inverse problemesolution methods that emphasize those aspects relevanto improving the image spatial resolution.

. Formulation for Time-Dependent Opticalomographyn the diffusion approximation limit, the modeled opticalroperties consist of a family of local tissue parametersncluding the absorption coefficient �a and the reducedcattering coefficient �s� of the chromophores, the indexf refraction �n�, the absorption coefficient associated withuorophores �� F�, as well as the quantum yield Q and
a F
he lifetime of the fluorophores ��. It should be noted thatll these parameters will typically have a nontrivial spec-ral dependence in tissue. For example, in the case ofOT imaging it is usually assumed that the only vari-bles in the forward problem are �a�� and �s��. In thease of FMT, an approximation is usually made that thenly variable affecting the fluorescence signal is �a

F. Of-entimes, the lifetime is set to a constant close to that ofhe value of the main fluorophore under study, and theissue optical properties at the emission and excitationavelengths are assumed constant and sometimes equal

n value [48]. Significant work has been done assuminghat lifetime is the main variable of interest [49–51].owever, in the case examined here, the early time be-avior of the time-resolved pulsed laser signal is used ashe principal data type to set up the inverse problem.

Figure 2 shows a general tissue domain � where lightnjection is assumed to be in the form of a collimatedeam sourced either by a steady-state, a frequency-odulated, or a pulsed laser. For diffusion modeling pur-

oses, an isotropic time-dependent source term S�r , t� isnserted one reduced scattering distance �1/�s�� underhe tissue boundary �� at �r , t�. Light collection is per-ormed from a region including the point �r� , t��. Unlesstherwise noted, it is assumed that both illumination andight detection are performed locally. This is done for sim-licity in order not to obscure the main results with math-matical formalism. It should be noted that the numericalethod described can model arbitrary illumination and

etection geometries.The optical tomography forward problem is set up in

he form of an update equation predicting how much tis-ue property variation will affect a signal, namely,

�S� = �J=1

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here �J= ��a ,�s� ,�aF,n ,QF,�� collectively represents all

odeled tissue properties at voxels labeled J, and ��

ig. 2. (Color online) Schematic representation of a turbid me-ium � with boundary ��. The spatial distribution of opticalroperties is represented by � while a local perturbation overhis background is represented by ��. For modeling purposes,n isotropic source term S�r , t� is inserted under the tissue sur-ace at the place of entry of a collimated laser beam. The lightransport solutions between different space-time locations areabeled �.

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tands for the local variations responsible for predictedignal variations �S� over the signal S� associated withackground properties �J. AJ is the Jacobian for theransformation between �S� with respect to ��J. Effec-ively, an optical tomography method consists of using aarge number of measurements on �� to find those valuesf �J minimizing the mismatch between experimentaleasurements and signals that are predicted by theodel based on the update equation. For problems where

he matrix A depends nonlinearly on the properties tar-eted for reconstruction (as in DOT), nonlinear iterativeolvers are used, whereas if an approximation can beade to the effect that signal variations are linearly re-

ated to the reconstructed properties (as is the case forMT), simpler linear reconstruction techniques can besed [41].More specifically, the update equation associated with

ne optical measurement takes the form

�S��r��,t�� = N��

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�2��2�r�� − r��;t� − t��, �2�

here the ’s are mathematical operators, and �� is a so-ution to the diffusion equation at wavelength �. N is a

odel-dependent normalization constant, and the inte-ral runs over all space–time points potentially contribut-ng to the signal perturbation. For example, in the case ofOT the update equation takes the form

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here � is a solution to the diffusion equation, while G isGreen’s function. Here construction of the update equa-

ion is attained numerically evaluating the photon prob-bility distribution based on FEM [52,53]. Figure 3 illus-rates this in the case of the evaluation of the steady-stateacobian corresponding to perturbations in �a—the firsterm in �S� in Eq. (3). This also corresponds to theteady-state Jacobian used in FMT. In the case shown inig. 3, the local optical properties associated with eachode of the mesh were set based on a segmented CT im-ge of a mouse head shown in Fig. 3(d).Figure 3(a) showshe photon fluence ��r�� associated with the source term,hile Fig. 3(b) corresponds to the light sensitivity of theetector G�r� , r��, which effectively corresponds to a solu-ion to the diffusion equation with a delta-function sourceerm. Figure 3(c) illustrates the corresponding Jacobianhich, as shown in Eq. (3), is simply the product of theelds shown in Fig. 3(a) and Fig. 3(b).

. Time-Gated Fluorescence Molecular Tomographyubsection 3.A defined time-dependent update equations

or general optical tomography in cases where light trans-ort is modeled as a diffusive process, and this is now ex-ended within the case of a forward model for time-gatedMT signals.Fluorescence from tissue can be modeled as a two-step

rocess beginning with the propagation of light from a

ollimated source to the fluorescent molecular targets fol-owed by re-emission at a different wavelength. The firsttep is modeled by solving the differential equation

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here D=1/3��a+�s��. In Eq. (4), the photon fluence andhe optical properties are evaluated at the laser excitationavelength �x. Subsequent re-emission of light by theuorophores at a longer wavelength is modeled by a sec-nd coupled differential equation, where the source terms time-dependent corresponding to the exponential decayf the fluorescence excited by the photon field �x [54],

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here the photon fluence and the optical properties arevaluated at the re-emission wavelength �e. The summa-ion on the right-hand-side of Eq. (5) is included to ac-ount for the possibility of there being different fluores-ent species (l=1, . . . ,NF, where NF is the number ofpecies). The photophysical properties of these species arehe fluorescent quantum yield QF, the extinction coeffi-ient �F, the lifetime �, and the local concentration of fluo-ophores CF��a

F=QF�FCF�. The formal solution to the sys-em of Eqs. (4) and (5) is obtained using Greens’ theorem54],

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ig. 3. (Color online) Steady-state photon distribution associ-ted with a diffusive light source inserted one reduced scatteringistance under the tissue surface. (b) Steady-sate photon sensi-ivity distribution associated with a light collection point locatedn the surface of the mouse head. (c) Photon sensitivity distribu-ion associated with the source and detector location shown in (a)nd (b). (d) Segmented CT image (1, brain; 2, skull; 3, rest of tis-ues) used to project sources and detectors as well as to tag dif-erent anatomical regions with different optical properties.

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In order to bring the forward model Eq. (6) to a formhere it can be used as part of an inverse problem, an NVode FEM mesh of the tissue volume � is created fol-

owed by discretization of the spatial and time integrals.he spatial resolution is set by that of the created meshhile the temporal resolution is thenceforth set to �t10 ps in the numerical simulations. Assuming a largeumber of measurements Nm is collected, the forwardroblem associated with the signal in time-gate t� can beritten in the matrix form

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ponding to measurement k �k=1, . . . ,Nm� and voxel withpatial location ri �i=1, . . . ,NV�:

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Ge�r�i − r�k,t� − t��. �8�

imilarly, inspection of Eq. (8) shows that the Jacobianor one time bin, here t�, can be generalized to model theignal corresponding to any combination of time bins. Forxample, the forward model for a time-gate between t1nd t2 is obtained with the following substitution inq. (7):

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Riley et al. have shown in [55] that the use of local dataypes, such as the slope of the rising TPSF, lead to moretable FMT reconstructions with an approach where theorward model was based on an analytic expression for aimple geometry. The more general approach used herean be used to compute the Jacobian associated with theata type corresponding to the slope of the time-domainignal around t� located anywhere in the TPSF. This ismplemented by the following substitution into Eq. (7):

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The fluence distributions Ge and �x in Eq. (8) are com-uted by solving the time-dependent diffusion equationith the finite-element method. The solutions correspond

o NV probabilistic weights, one for each node in theesh. The steady-state limit of the Jacobian consists of

valuating Eq. (9) for t1=0 ns and t2=10 ns. The t210 ns time limit corresponds to the end point of theimulated time-window, where signals have decreased toero, as shown by inspection of the simulated TPSF inig. 6(a) below. Evaluating the Jacobian for one time bin

nvolves numerically performing two convolutions inime: one between the detector sensitivity function andhe exponential decay term of the fluorophore, and one be-

ween the resulting time-dependent expression and theime-dependent fluence produced from the light source.inally, time-gated forward models are evaluated by per-

orming the sum of the Jacobians for all time bins from t1o t2.

Figure 4 illustrates Jacobians numerically computedor different time gates with t1=0 ns and �=1 ns. Specifi-ally, the figure shows Jacobians associated with t250 ps, t2=150 ps, t2=300 ps and t2=10 ns. The first

hree time gates correspond to increasingly diffused pho-ons, while the last one effectively corresponds to theteady-state signal equivalent to that of a continuous-ave source. For a given source–detector pair, inspectionf Fig. 4 clearly shows that the tissue sampled by propa-ating photons is significantly reduced for early photonss compared with the highly scattered steady-state sig-al. Moreover, comparison of the Jacobians for homoge-eous and heterogeneous optical properties highlights theact that early photon signals appear less sensitive to theresence of optical absorption heterogeneities.

. Inverse Problem Resolution Algorithmsinear systems of equations can be solved based on itera-

ive regularization approaches such as those based ononjugate gradient methods. The end result then consistsf a sequence of iteration vectors xa, a=1,2,3. . ., that con-erge to the desired solution. Such methods are generallyreferable to direct methods when the coefficient matrixs so large that it is too time-consuming or too memory-emanding to work with an explicit decomposition of theacobian matrix A. In FMT, the linear inverse problem isften solved based on iterative regularization methodsnd, in this way, a regularized solution is computed. Here,e use the bound-constrained least-squares (BCLS) algo-

ithm [44,56] for solving problems corresponding to theeneric objective function problem:

argminCF s.t. l CF u

�ACF − ��2 + �2�LCF�2, �11�

here the NV-dimensional vectors l and u are lower andpper bounds on the optimization variables CF. The pa-ameter � is a regularization coefficient that is used toontrol the norm �LC �2 while balancing it against the re-

ig. 4. (Color online) Diffusion photon sensitivity distributionsor the imaging geometry represented in Fig. 2 for a givenource–detector pair. From left to right, sensitivity plots for in-reasingly large time gates are shown with the last image corre-ponding to a steady-state signal. The upper row shows distribu-ions computed assuming that the diffusive medium hasomogeneous optical properties, while the bottom row of imagesorresponds to a medium with heterogeneous optical properties.

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idual norm �ACF-��2. Typically, the matrix L can be en-ineered to play a wide variety of roles including the in-lusion of prior spatial information in the reconstructionrocess.Linear problems can also be solved using direct meth-

ds such as the singular-value decomposition (SVD) andts generalization to the matrix pair involved here (A, L).VD decompositions are useful because they allow one tonderstand the underlying problem in terms of vectors,llowing direct manipulation of the fundamental spectralnits from which tomography images are built. However,sing direct methods is usually not practical in tomogra-hy because most problems are associated with largeatasets. In fact, numerical algorithms performing de-ompositions of large matrices are memory-consumingnd usually cannot be run on individual processors. How-ver, here we are considering smaller datasets that can besed with SVD decomposition of the forward model ma-rices. This approach illustrates how the reconstructionsased on the SVD decomposition work and is used tohow generalizations of more conventional and practicalterative regularization methods. Hence, SVA is used onlyo gain an intuitive understanding of the underlying prob-em here.

Basic SVD decomposition reformulates a forwardodel matrix A�Rmn into the form

A = U�VT = �i=1

n

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here U= �u1 , . . . ,un��Rmn and V= �v1 , . . . ,vn��Rnn

re orthonormal norm matrices, and where the diagonalatrix �=diag��1 , . . . ,�n� has non-negative diagonal ele-ents appearing in non-increasing order ��1��2� . . .�n�0�. The numbers �i are the singular values of A,hile the vectors ui and vi are the so-called image andata singular vectors of A, respectively. A critical aspect toolving a tomography inverse problem consists of findingethods allowing us to minimize as much as possible the

mpact of the intrinsic ill-conditioning of the forwardodel matrix. The degree of ill-conditioning can be used

s a measure of how much noise and intrinsic data-modelismatch propagate into the solutions [57]. To a large ex-

ent the degree of ill-posedness can be evaluated by study-ng the decay rate of the singular values, which in turnignificantly affects how noise propagates into the regu-arized solutions. Using the SVD decomposition, the solu-ion to the inverse problem can be rewritten as the sumver image singular vectors,

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here NSVD corresponds to the number of modes used touild the solution, and f�� is a regularization function.ssentially, this expression is a spectral decomposition of

he solution as a sum over image singular vectorseighted by a spectral coefficient. As a general rule, the

ingular values decrease with increasing values of the or-er i, and the spatial frequency of the modes ui increasesith i.

. RESULTSere the time-gated modeling approach presented in Sec-

ion 3 is used to show how reconstructing fluorescence im-ges based on early diffused photon signals can signifi-antly improve the intrinsically poor resolution of opticalomography.

. In silico Fluorescence Phantomsimulated optical data sets were generated for two cylin-rical phantoms, both containing three small fluorescentnclusions. As seen in Fig. 5 (left-most column), one nu-

erical phantom has infinite fluorescent contrast, andhe other one has a more realistic contrast-to-backgroundatio of 6 to 1. The phantoms were designed with featuresaking their use relevant in deriving conclusions appli-

able to mouse imaging. The radius of the phantom is R12.5 mm and the optical properties of the bulk diffusiveedium are �a=0.02 mm−1 and �s�=1 mm−1—both for

he excitation and the emission wavelengths. The fluores-ent inclusions have radii RI=2.5 mm, RII=2.0 mm, andIII=1.5 mm. The radii were chosen for consistency withnimal model tumors routinely imaged with FMT. Theize of the smallest inclusion was determined because itoughly corresponds to the fundamental spatial reso-ution limit set by diffusion theory. The centers-of-mass ofdjacent inclusions in the phantom are separated by aistance of �CM=2.5 mm, while the minimum distance be-ween the edges is �e=1 mm.

Forward model data were calculated with the 2D diffu-ion equation for a medium with homogeneous opticalroperties �a and �s� [53,58]. The choice of doing lightransport simulations in 2D was made for simplicity ando reduce simulation time. FMT matrices for three differ-nt time-gated signals were generated, namely, T200 ps0 ps–200 ps, T400 ps=0 ps–400 ps and the steady-statease TCW covering the full time window. As shown in Sub-ection 3.B, the FMT inverse problem for any type of dataan be cast in the matrix form

ig. 5. (Color online) Target images reconstructed with the it-rative regularization method BCLS. The first column corre-ponds to the target images used to generate synthetic data forifferent time-gates. In the upper images, the fluorescence con-rast is infinite, while in the lower ones the contrast is 6 to 1. Re-onstructed images shown are for weakly diffused photon time-ates 0–200 ps and 0–400 ps as well as for simulated steady-tate signal for comparison purposes. No noise was added in theimulated data for these reconstructions.

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�T = ATCF, �14�

here, in this case, T labels the time-gate under consid-ration. Simulated data are generated for each time-gatey multiplying the Jacobian, AT with the target fluores-ence images shown in Fig. 5. Then, statistical noise fol-owing a Gaussian distribution around the signal ampli-ude is added to the synthetic data vectors. Four levels ofoise are considered, namely, 0%, 1%, 5%, and 10%.

. Singular-Value Analysis and Spatial Resolutionmprovementxperimental data �T always contain noise, and the mostritical aspect of solving an optical tomography inverseroblem is controlling how much this noise is allowed toropagate into the reconstructed images. When using it-rative reconstruction methods, one or several regulariza-ion parameters are carefully chosen to allow convergenceoward those solutions that represent the better possibleompromise between minimization of the residual normnd minimization of noise propagation in the final solu-ion. An intuitive understanding of this is shown throughVA of the forward problem matrices for time-gated dataelative to steady-state.

As explained in subsection 3.C, reconstructed fluores-ence images can be built by summing image singularectors of increasingly high spatial frequencies. The spa-ial frequency of the modes increases with the order of theingular values i. Consequently, the number of modesSVD that are used to build the solution can effectively be

egarded as a parameter controlling the spatial resolutionf the resulting fluorescence images. For example, Fig.(a) shows the image singular vectors for i=1, 5 and 10 inhe case of FMT matrices for time-gates T200 ps and TCW.

hen forming a fluorescence image, those image singularectors ui are weighted against each other according tohe values taken by the spectral coefficient �vi

T�T� /�i. Asxplained further below, it is the behavior of those coeffi-
ients combined with the spatial frequency of the corre- t
ponding image singular modes that is setting an abso-ute spatial resolution limit to optical tomography.

Maximizing the spatial resolution of a fluorescence im-ge is attained by keeping as many high-spatial-requency image singular modes ui in the solution as pos-ible. This can be accomplished by truncating the sum inq. (13) for i=NSVD as large as possible. The problem is

hat noise in the data vector limits the number of modeshat can effectively be used when reconstructing an im-ge. Indeed, when noisy data are used the spectral coeffi-ients typically diverge for finite values of i. This implieshat if the corresponding modes are kept in the solution,hey will be the only ones contributing, potentially mak-ng it appear, incorrectly, that the solution has a very highesolution. However, all information pertaining to the un-erlying physical content of the image has been lost. Cut-ing off the divergent modes appropriately then amountso regularization of the solution.

Another possibility for obtaining a yet smoother regu-arized solution consists of introducing a smoothing func-ion f�� that is continuously decreasing from one to zeros the singular-value order increases. The inflexion pointf the smoothing function is then located around thealue where the spectral coefficients begin to misbehave.t can be shown analytically that this approach is equiva-ent to an iterative regularization Tikhonov method (forxample, see [57]). In fact, an important point here is thatny iterative method that is used to solve a linear inverseroblem can, in principle, be formulated in terms of aVD or a generalized SVD decomposition associated withspecific smoothing function. The reason this is so impor-

ant is that any conclusion that is derived based on SVAonsiderations can in principle be translated to otherore conventional approaches used to solve optical to-ography problems.To a large extent, the degree of ill-posedness of a prob-

em can be quantified by the decay rate of the singularalues [57]. The importance of the decay rate can be
raced back to the fact that the computation of the spec-
ig. 6. (Color online) Decay curve of singular values (log-scale) as a function of their order i for two FMT forward model matrices cor-esponding to weakly diffused photons (0–200 ps time-gate) as well as steady-state signal. The decay rate for the weakly diffuse photonss significantly less favorable to noise propagation in the images. (b) Illustration of some image-singular modes for the same two forward

odels. At the same order, the spatial frequency of the modes associated with weakly diffused photons is typically smaller than thosessociated with steady-state signal, again affording more leeway in reconstructing high-spatial-resolution images.

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ral coefficients involves evaluating the ratio 1/�i, whichends to become very large for small singular values. Ifhe singular values are decreasing rapidly, then the spec-ral coefficients weighting the image singular vectors willave a tendency to blow up for small values of i, therebyompromising the stability of the solution. Consequently,he sensitivity to noise is going to be more important forases where the decay rate of the singular values isarger. Figure 6(b) shows the decay curves of the singularalues for two different time-gates, namely T200 ps andCW. Inspection of this log-scale figure clearly reveals that

he decay rate associated with the steady-state forwardodel is significantly larger than that associated with

arly photons. This implies that the solutions for earlyhoton data are more stable. This also means that the di-ergence of the spectral coefficients will typically occur forarger values of i in the case of early photon forward mod-ls when compared to steady-state. Therefore, more high-requency image singular modes can be used in a givenarly-photon solution, potentially leading to improvedesolution when compared with steady-state-based recon-tructions.

As mentioned earlier, for a given singular order i, thepatial frequency of a mode appears higher for early pho-ons than it does for steady-state [Fig. 6(a)]. This furtherllustrates there is an increase in spatial resolution formages reconstructed based on early-photon signals. Fi-ally, it can also be shown based on perturbation boundheorems [57] that the error committed when solving annverse problem using a truncated SVD (at i=NSVD)—dueo noise and bad modeling—is proportional to the ratio1/�Nsvd

. Again, this emphasizes the importance of insur-ng that the singular-value decay rate is as slow as pos-ible when choosing a forward model.

A measure that is often used to quantify the degree ofll-conditioning of a system of linear equations is the con-ition number of the Jacobian, which consists of the ratioetween the largest and the smallest singular values.inear systems associated with large condition numberatrices typically have a large number of linear depen-

encies, which makes them more susceptible to noise.igure 7(a) shows a transmission TPSF that was gener-

ig. 7. (Color online) Illustration of the FMT forward model man the signal: (a) simulated time-domain signal, (b) graph showinhere t is the x-axis value on the graph.

ted numerically for the in silico phantom described inubsection 4.A for the coaxial detection channel in Fig.(a). Then, Fig. 7(b) shows the condition number of theorresponding forward model matrix AT for differentime-gate sizes. The time-gates considered here start at1=0 ps. The x-axis on the graph corresponds to increas-ng values of t2, that is, increasingly large time-gates withhe largest one �t2=2 ns� effectively corresponding to theteady-state signal. Inspection of Fig. 7(b) shows that theegree of ill-posedness as measured by the condition num-er increases exponentially as a function of the time-gateize. This is certainly consistent with the results derivedn Subsection 4.B based on a singular-value decomposi-ion approach. However, analyzing the ill-conditioning ofhe optical tomography problem solely based on the con-ition number is not sufficient to gain a clear intuitive un-erstanding of the basic mechanisms at play.

. Fluorescence Reconstructions with Weaklyiffused Photonseconstruction results are now presented based on simu-

ations performed with the in silico phantoms describedn Subsection 4.A. As mentioned earlier, different levels ofoise were added to the time-gated data prior to inversion

ncluding 0%, 1%, 5%, and 10%. A discussion relating tooise propagation in early-photon time-gates is presentedt the end of this section, placing these values in a moreppropriate experimental context.Throughout this section, inverse problem resolution re-

ults are presented only for tomography images obtainedith the BCLS solver described in Subsection 3.C. It was

ound that the reconstruction results obtained with theCLS solver are consistent with expectations gained byerforming the SVA. Indeed, every reconstruction weave performed with BCLS was also performed using theruncated SVD approach as well as the SVD method withikhonov smoothing function. The trends, in terms of sta-ility to noise and spatial resolution improvements, wereaintained for all approaches. We are limiting the pre-

entation to BCLS results partly for conciseness andartly because of the improved quality of the fluorescencemages. In this case, our main criterion for judging the

ndition number as a function of the time-gates that are includedcondition number of the matrix for time-gates from 0 ns to t ns,

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uality of an image is based on a qualitative assessmentf the number of artifacts present in the solutions.

Part of the reason BCLS images are of better qualityight be the non-negativity constraint that is imposed

hrough the merit function in Eq. (11)—l=0 and v=�. In-eed, it has been observed repeatedly in the past thatMT images reconstructed using non-bound-constrained

terative regularization methods have a tendency to im-ue images with nonphysical negative intensity values.ll-conditioned systems of linear differential equations aressociated with a high degree of linear dependencies,hich implies that for a given data vector there exist a lotf candidate solutions with comparable residual norms.mong that large number of solutions composing the de-enerate landscape, non-bound-constrained solvers seemo favor those solutions containing a subset of negative in-ensity values. Imposing convergence toward positive-efinite solutions is therefore also found to indirectly im-rove the conditioning of the inverse problem.In what follows the regularization parameter � is set to

ero in the objective function Eq. (11). The only regular-zation parameter used in BCLS is the tolerance factor it-elf, which corresponds to the targeted convergence valuef the residual norm of the solution. This regularizationarameter is henceforth referred to as �. Comparisonith direct SVD-based inversion methods provides evi-ence that � plays a role similar to that of the truncationrder �NSVD� in the singular-value analysis presented inubsection 4.B. In this case though, smaller values of �

ead to solutions with higher intrinsic spatial resolution.lso, the more noise there is in the data vector, the largerneeds to be to prevent noise from propagating into and

ominating the solution. This is in accord with the intui-ive SVA arguments presented in Subsection 4.B: solu-ions with a larger residual norm are built with a smallerumber of high-spatial-frequency modes than solutionsaving a smaller residual norm.Figure 5 shows reconstructions performed for different

ime-gates when no noise (0%) was added to the in silicoata vectors. In a way, since these reconstructions wereithout any noise, the results can be interpreted as theest possible result based on the detection geometryhown in Fig. 1(a): 32 sources, 5 detectors per source. Inhe infinite fluorescent contrast case, clear spatial reso-ution improvements can be seen to have been achievedy using a time-gate consisting of photons as weakly dif-used as possible. The improvement is quite dramaticince steady-state imaging cannot be used to resolve theargets, while individual fluorescent sources can be dis-inguished on the image reconstructed based on the200 ps time-gate. Clear image improvements are alsooted for the T400 ps time-gate when compared with theteady-state case. Indeed, the fluorescence intensity forhe smallest inclusion �RIII=1.5 mm� is correctly recov-red for T400 ps yet it is not when using TCW.

In optical tomography, it is notoriously more difficult toeconstruct finite contrast targets [59]. The bottom row ofmages in Fig. 5 shows the 0% noise reconstructions forhe multiple targets phantom where the contrast associ-ted with the inclusions is 6 to 1. Comparison with the in-nite contrast results (upper row of images in Fig. 5)hows that the reconstructed images are in general much

ore diffuse for finite contrast values even when early-hoton time-gates are considered. Nonetheless, we findhat although they cannot be distinguished, all three tar-ets can be seen when using a weakly diffused signal cor-esponding to the 0 ps–200 ps time-gate. The TCW imagehows a single diffuse blob with its highest intensity lo-ated around the center of mass of the largest inclusionRI=2.5 mm�. Then, comparison with the T200 ps and400 ps images shows that decreasing the size of the time-ate allows the other inclusions to start taking form. Ino way should these results be interpreted as a funda-ental limit of early photon optical tomography. In fact,

mproving tissue sampling by increasing the number ofeasurements is likely to further improve the quality of

he fidelity of the tomography images.

. Noise Propagation in Time-Domain Signalsn the scope of early diffused photon simulations, it is im-ortant to understand the limitations of early times win-ows in terms of inherent noise limits on imaging throughhick tissues. The stochastic noise associated withCSPC-based PMT detection can be modeled by adding toach time bin a random value according to a Poisson dis-ribution with the mean corresponding to the total num-er of photon counts in each time bin. Then, the stochasticoise in each bin is associated with the SNR as

SNR = �N, �15�

here N is the number of photons collected for a givenime bin of the TPSF. For example, Fig. 8(b) shows twoifferent simulated TPSFs for which stochastic noise wasdded. Therefore, increasing the total number of photonsollected to build a TPSF increases the overall SNR of theignal. Often, the quality of a TPSF in terms of stochasticoise is measured by the number of counts at the curveeak corresponding to where the signal is maximal.

ig. 8. (Color online) Stochastic noise propagation in time-gatedPSF signals. The lower graph shows two curves (peak count of00) where noise following a Poisson distribution with mean Nnumber of counts in individual time bins) was added. Time-ates T200 ps, T400 ps, and TCW are highlighted. The upper imagehows how noise propagates into the time-gates for differentPSF peak counts. The x-axis for both pictures corresponds tohe end-point of the time-gate �t2�. The initial point of the time-ates was always t =0 ns.
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enceforth, this quantity is referred to as the peak countf a TPSF. The experimental parameters that can be var-ed in order to improve the SNR for a given data set arehe laser power and the illumination/collection integra-ion time when acquiring each measurement. Imaginghrough thick and highly absorbing tissue requires in-reasing either or both of these instrument parameters.

Figure 8(a) shows how stochastic noise propagates intoifferent time-gate sizes for different levels of photoneak count, namely, 102, 103, and 104. The variable on the-axis of the figure corresponds to t2. The initial time inhe gate always corresponds to t1=0 ns. Inspection of theraph shows that the exponential signal increase that isypical of early time bins of a TPSF leads to an exponen-ial increase of the noise propagating in the early time-ates. From a practical point of view, it is therefore ex-ected that the product of laser integration time andower required to obtain good SNR datasets will also in-rease exponentially as the size of the time-gates de-reases. However, the modeled signal here is purely diffu-ive and it is likely that in situations involving mousemaging, for example, there will be a non-negligible WSPomponent in the signal improving the count rate. This isentioned here to clarify that the results shown in Fig.

(a) should be used only as an example illustrating therends associated with considering different count levels.

Based on the noise analysis presented above, thehorter the early-photon time-gate is, the smaller the cor-esponding SNR will be. This can be problematic for to-ography since the relevant inverse problem methods

end to be very sensitive to noise. The optimization gamehat needs to be played when considering early photon to-ography consists of acquiring signals that have a good

nough SNR so the benefits discussed earlier—stability tooise and improved spatial resolution—can be preservedt a reasonable cost in terms of laser power and integra-ion time.

Figures 9 and 10 show more realistic reconstruction re-

ig. 9. (Color online) Infinite contrast target image recon-tructed with the iterative regularization method BCLS for dif-erent levels of noise. The first column corresponds to the targetmage used to generate synthetic data for different time-gates.econstructed images shown are for weakly diffused photon

ime-gates 0–200 ps and 0–400 ps as well as for simulatedteady-state signal for comparison purposes. Three different lev-ls of noise are shown: 1%, 5%, 10%.

ults where different levels of noise are added to theimulated data vectors. In agreement with the 0% noiseesults (Fig. 5), improvements in imaging fidelity areound for reconstruction based on weakly diffused signals.erhaps the most salient feature of both the infinite and 6

o 1 contrast images is that the steady-state images arearticularly sensitive to noise. Indeed, all TCW imageshat were reconstructed with more than 1% noise haveost all information relating to the original image, both inerms of target localization and recovered fluorescence in-ensity. Only in the case of 1% added noise can the ap-roximate center of mass of the larger inclusion be recov-red. In this case, it would be very difficult to predict thathere are other smaller inclusions.

There is another interesting observation that can beade by inspecting the 1% and 5% noise rows in Fig. 9. In

his case, the quality of the reconstructions for both200 ps and T400 ps appear to be able to localize the centerf mass of all inclusions as well as to recover the correctuorescence intensity. This is a sharp difference with theesults obtained with the steady-state signal. On thether hand, the information content in the 10% noise im-ges for the early-photon time-gates in Fig. 9 is essen-ially equivalent to what is contained in the 1% TCW time-ate. In other words, it does not appear that doing early-hoton imaging with 10% noise would significantlymprove the results obtained with steady-state imaging at% noise level. Those findings for the infinite contrastases can essentially be transposed to the 6:1 contrast re-ults shown in Fig. 10. The only exception is that in theresence of a fluorescence background the fidelity of themages obtained with T200 ps is significantly improvedhen compared with those corresponding to T400 ps.

. DISCUSSION. Spatial Resolution and Sensitivity to Noisesing arguments derived from inverse problem consider-tions, the analysis here illustrates why an instrument

ig. 10. (Color online) 6 to 1 contrast target image recon-tructed with the iterative regularization method BCLS for dif-erent levels of noise. The first column corresponds to the targetmage used to generate synthetic data for different time-gates.econstructed images shown are for weakly diffused photon

ime-gates 0–200 ps and 0–400 ps as well as for simulatedteady-state signal for comparison purposes. Three different lev-ls of noise are shown: 1%, 5%, 10%.

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esigned with tomographic fan-beam detection geometrynd single-photon-counting technology can significantlyncrease the spatial resolution of FMT if the data type ishosen appropriately. This is an important prospect be-ause optical tomography is a modality that has tradition-lly been plagued with low resolution, low contrast recov-ry, and ultimately low predictive power, when used as atand-alone imaging modality.

The main conclusions of this work are with respect tohe use of early-time photons in the FMT inverse prob-em. Better conditioned inversion can be gained withime-gated data as compared with steady-state data. Aseful numerical measure of this behavior is found to behe condition number of the Jacobian. For cases relevanto small-animal imaging, this number was found to growxponentially with increasing size of the time-gates. Forxample, in Fig. 7 the condition number going from T200 pso the steady-state forward model matrix increases by aactor of 450. Reconstructions using synthetic data from aealistic in silico phantom were performed and used tohow that, in accord with condition number analysis, thearly-photon fluorescence tomography problem is signifi-antly less sensitive to noise than its steady-state coun-erpart. It is observed that this increased stability tooise is accompanied by an improved intrinsic spatial res-lution. In fact, early-photon tomography was shown toave the potential to resolve small fluorescence inclusionsaving their center of mass separated by a distance of.5 mm. More precisely, as shown in Fig. 5, inclusionsaving a distance between their edges �e=1 mm could beesolved in ideal conditions when using early-photonime-gates with no stochastic noise added. Even in suchdeal situations, steady-state imaging is incapable of re-olving the targets to a level that is comparable witharly-photon imaging.

The prognostic analysis based on the condition numberf the Jacobian matrix is insufficient to gain a full under-tanding of the underlying principles behind the en-anced spatial resolution and limited noise sensitivity as-ociated with early-photon fluorescence tomography. Inarticular, the analysis does not allow one to disentanglehe cause of two main aspects affecting the quality of to-ography images, namely, sensitivity to noise and intrin-

ic spatial resolution. Therefore, as a complement toondition-number considerations, a singular-value analy-is of the time-gated FMT inverse problem was per-ormed.

As a general rule, the spatial resolution of a recon-tructed optical image is related to the number and thepatial frequency of the image-singular mode vectors thatan be used to build the solution. An analysis of themage-singular vectors associated with inverse problemsor different time-gates was performed. In general, it wasound that for the same singular order (i) the spatial fre-uency of the image vectors was usually larger for smallerime-gate matrices. This implies that if one were to use anite number of singular modes to reconstruct an imageNSVD�, then the resolution associated with early time-ate images would be improved compared with largerime-gates and, in particular, with steady-state imaging.

However, one might argue that if the number of modeshat are kept in the solution is not limited but kept as

arge as is possible—for the cases of NmNV matrices,his number is min�Nm,NV�—then the resolution of early-hoton images and steady-state images could be equiva-ent. However, as illustrated in Fig. 5, this is found not toe the case. Part of the reason for this might have beenxplained by the fact that the decay rate of the singularalues associated with the steady-state problem is mucharger when compared with the early time-gated case [seeig. 6(b)]. This implies that, even in the absence of noise,he high-spatial-frequency components of the SVD solu-ions will have an increased tendency to diverge becausef divisions by small numbers 1/�i, where i is large. Di-erging high-frequency modes should not, of course, beart of the solution because they would dominate over theower-frequency modes in which much of the physical in-ormation is contained. Inspection of Fig. 6(b) shows thathe smallest singular value for the 0 ps–200 ps time-gates about two orders of magnitude larger than that corre-ponding to the steady-state signal. This makes the high-patial-frequency components potentially more sensitiveo both noise in the data vector as well as to the machinerecision value. Indeed, were the singular values associ-ted with higher-frequency modes smaller than the ma-hine precision, then those modes would diverge and beendered nonaccessible for image reconstruction. How-ver, for the particular inverse problems considered inhis work, the smallest singular values that were consid-red are always safely above the machine precision value.ssues relating to machine precision divergence will bencountered when considering denser data sets leading toank-deficient inverse problems that are associated withigh content of redundant information.Based on this argument, the increased intrinsic spatial

esolution of early-photon tomography can be thought ofathematically as being due to there being more high-

requency modes available for image reconstruction. Simi-arly, a combination of slower singular-value decay ratend higher spatial frequency of the corresponding image-ingular vectors can be used to explain the improved sta-ility to noise associated with early-photon tomography.

. Generalization to Weakly Scattered Photonshe main challenge in early-photon imaging consists ofsing time-domain detection technology to collect signalsssociated with those photons that have traveled alongaths that are as straight as possible between the sourcend the detector [33,36,37]. Ideally, the ballistic compo-ent of the signal should be used in order to maximally

mprove on the poor spatial resolution of diffuse opticalmaging. This is expected, of course, based on our intu-tion developed working with x-ray CT systems. However,or small-animal optical imaging, sampled tissue volumesre usually such that the ballistic component of the signals negligible.

Typically then, as described in Subsection 2.A, early-hoton signals are composed of a mixture of WSPs and ofeakly diffused photons. The main difference between

hese two types of particles is that the WSPs are highlyorward directed while the weakly diffused photons cane modeled as a purely diffusive process. The key ques-ion one needs to answer when devising a model for earlyhotons consists in determining how much of the signal is

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omposed of weakly scattered photons compared witheakly diffused photons. Their relative proportion is go-

ng to be affected by the thickness of tissue being imageds well as by the magnitude of the absorption. Also, in-reasing the size of the early-photon time-gate will reducehe proportion of WSP up to a point where diffused pho-ons dominate the signal. Evidence is provided in [33]hat the T200 ps time-gate contains a large fraction of

SPs in the case of an application that images lung tu-ors with FMT.In [33], light transport for WSPs is modeled with ana-

ytic solutions obtained from the cumulant approximationo the RTE [35]. This allows the authors to build forwardodels with photon sensitivity functions that are thinner,ith more of the photon weights localized along the direct

ine of sight between sources and detectors. This is to beontrasted with the diffusion model used in the work herehere the photon sensitivity distributions remained com-aratively broad even in the case of the 0 ps–50 ps time-ate, as shown in Fig. 4. This is simply a manifestation ofhe fact that WSPs and ballistic photons are not modeledy solutions to the diffusion equation. Of course, the fron-ier between WSPs and weakly diffused photons is not aiscrete one, and there is a smooth continuation betweeniffusion-based solutions and RTE-based solutions. Theesults found here in terms of improved spatial resolutionnd increased stability to noise apply as well to inverseroblems associated with RTE-based light transportodeling.In conclusion, there are two types of information a fluo-

escence tomography approach is expected to provide. Ide-lly, the method should afford localization of small targetsnd it should quantify their relative fluorescence intensi-ies. Often, appropriately calibrated FMT instrumentsre able to measure only the total fluorescence emittedrom a tissue without providing clear delineation of theuorescence sources such as tumors. Here, we havehown how and why the spatial resolution and quantita-ive power of FMT can be significantly improved by opti-izing tissue sampling and using time-domain technol-

gy to acquire early-photon signals.

CKNOWLEDGMENTShe authors thank Guobin Ma, Niculae Mincu, and Nico-

as Robitaille from ART Inc., Montreal, Canada, for usefuliscussions. This work was supported by the National In-titutes of Health (NIH) grant R01CA120368.

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Early-photon fluorescence tomography: spatial resolution improvements and noise stability considerations