Fluorescence lifetime imaging by using time-gated dataacquisition

Vadim Y. Soloviev,1,* Khadija B. Tahir,2 James McGinty,2 Dan S. Elson,2 Mark A. A. Neil,2

Paul M. W. French,2 and Simon R. Arridge1

1Department of Computer Science, University College London, Gower Street, London WC1E 6BT, UK2Imperial College Photonics, Blackett Laboratory, Prince Consort Road, London SW7 2BW, UK

*Corresponding author: v.soloviev@cs.ucl.ac.uk

Received 14 February 2007; revised 4 July 2007; accepted 23 July 2007;posted 30 July 2007 (Doc. ID 80108); published 10 October 2007

The use of the time gating technique for lifetime reconstruction in the Fourier domain is a noveltechnique. Time gating provides sufficient data points in the time domain for reliable application of theFourier transform, which is essential for the time deconvolution of the system of the integral equationsemployed in the reconstruction. The Fourier domain telegraph equation is employed to model the lighttransport, which allows a sufficiently broad interval of frequencies to be covered. Reconstructed imagescontain enough information needed for recovering the lifetime distribution in a sample for any givenfrequency within the megahertz–gigahertz band. The use of this technique is essential for recoveringtime-dependent information in fluorescence imaging. This technique was applied in reconstruction of thelifetime distribution of four tubes filled with Rhodamine 6G embedded inside a highly scattering slab.Relatively accurate fluorescence lifetime reconstruction demonstrates the effectiveness and the potentialof the proposed technique. © 2007 Optical Society of America

OCIS codes: 290.0290, 290.7050, 170.0170, 170.3010.

1. Introduction

Lifetime imaging has many applications in biologyand medicine, for example, the lifetime of luminophorin tissue can provide functional information such aspH, enzyme, and oxygenation [1–3]. This paper ex-plores the potential of 3D fluorescence lifetime tomog-raphy utilizing wide-field time-gated image data.Fluorescence lifetime imaging and tomography inscattering media has previously been studied in thetime domain [4–6] but most of the reported workhas employed frequency domain lifetime imaging[7–12]. If the amplitude of the excitation light is sinu-soidally modulated, a phase shift between the ampli-tude of excitation light and the amplitude of re-emittedfluorescent light provides additional information,which allows reconstruction of lifetimes. Because typ-ical fluorescence lifetimes are short, the phase shiftbetween the excitation and fluorescent light is very

small for modulation frequencies below several mega-hertz. Therefore, lifetime reconstruction requiresusing higher frequencies. However, frequencies of am-plitude modulation of up to 1 GHz and higher are nottechnically achievable yet as well as not practical dueto exponential damping of the ac component with thefrequency. That is to say, the light transport at fre-quencies of the amplitude modulation in the gigahertzband and higher behaves almost identically to thesteady state case.

Time-domain data acquisition is an alternative tofrequency domain in fluorescence lifetime imaging. Afluorophor is excited from the boundary by an ultra-short pulse of light, and the fluorescent intensity isrecorded as a function of time on the surface of ascattering volume. Experimentally it is achievable byusing the time gating technique [13,14] when thelight is collected within a very short exposure time.Measured signals are then Fourier transformed togive complex functions, which form a data set forreconstruction. The very large interval of frequenciesmakes it possible to pick any frequency for recon-

0003-6935/07/307384-08$15.00/0© 2007 Optical Society of America

7384 APPLIED OPTICS � Vol. 46, No. 30 � 20 October 2007

struction. In addition, the experimental noise in thetime domain is mapped into higher frequencies,which results in much higher signal-to-noise ratio forfrequencies below a few gigahertz. In this study weapply this approach for experimental lifetime imagereconstruction. It should be mentioned that similarapproaches were suggested earlier but without exper-imental validations [15,16].

In this paper the telegraph equation (TE) is em-ployed to model the light transport in turbid media inthe Fourier domain. The TE assumes finite propaga-tion speed and contains the well-known diffusion ap-proximation as a limiting case. The TE is moresuitable for such type of techniques due to its betteraccuracy for higher frequencies than the diffusionapproximation [17,18].

In terms of data acquisition, our approach employsnoncontact imaging, which may provide higher accu-racy than earlier tomographic schemes that use anumber of discrete light guides. For a noncontactimaging, a CCD camera is placed at some distancefrom the scattering volume. Each pixel of the CCDcamera absorbs photons escaping from the imagingsurface. We note that, although noncontact imagingprovides a large number of measurements acquiredfor every position of an excitation source (the typicalnumber of pixels is �105), each measurement dataset must be mapped onto the surface of a scatteringobject prior to image reconstruction by taking intoaccount the optical system and geometry [19,20].

This paper presents reconstruction of the quantumyield and the lifetime distribution from the experi-mental data set by use of the time gating techniquefor what we believe to be the first time. The timegating technique provides enough data points in thetime domain needed for the application of the Fouriertransform. The use of the Fourier transform is essen-tial for the time deconvolution of the system of theintegral equations employed for the lifetime recon-struction. The Fourier transform is more efficient andsuitable for the time gating technique than theLaplace transform suggested by Gao et al. [16]. Re-constructed complex valued functions depending onthe quantum yield and the lifetime contain all theinformation needed for recovering the lifetime distri-bution in a sample for any given frequency within themegahertz–gigahertz frequency band.

The paper is organized as follows. To make thepaper self-contained we briefly outline the basic the-ory of the light transport in highly scattering mediaand the associated inverse problem in the next sec-tion with an emphasis on experimentally observablequantities. A photon as a quantum object is not ob-servable unless it is absorbed. Quantities used as thedataset for the reconstruction are derived from theabsorbed energy of light by the CCD camera. A phan-tom and an experimental implementation of the timegating technique is described in Section 3. Section 4presents reconstruction results, discussions, and con-clusions.

2. Methodology

A. Experimentally Observable Quantities

A radiation field in space is described by the distri-bution of the intensity of radiation with respect tofrequency, to space, and to the direction of the ra-diant energy transfer. If we consider radiation as acollection of particles (photons), then the field canbe described by a photon distribution function f,which is, to a large extent, analogous to any otherparticle distribution function. Let f��, r, s, t�d�d3rdsbe the number of photons in the frequency interval��, � � d��, contained at the time t in the volumeelement d3r about the point r, and having a directionof motion within an element of solid angle ds �sin �d�d� about a unit vector s � �sin � cos �,sin � sin �, cos ��T, where � is the polar angle and � isazimuthal angle. Each photon possesses an energy h�and moves with the speed c, where h is the Planckconstant. Hence the quantity

I��r, s, t�d�ds � h�cf��, r, s, t�d�ds (1)

represents a radiant energy in the spectral intervald�, passing per unit time through a unit area, withdirections of energy propagation contained within theelement of solid angle ds, about the vector s. The areais allocated at the point r and oriented perpendicularto s. The quantity I� is called the spectral radiationintensity. The radiation field is essentially fully de-fined by specifying either the function I� or f. Theradiant energy of frequency � included in a unit spec-tral interval of frequency and contained in a unitvolume at the point r at the time t is termed thespectral radiant energy density (the energy densityfor short) and is given by

U��r, t� � h� ��4��

fds �1c ��4��

I�ds. (2)

The number of photons in the frequency interval d�exiting the unit area on the surface of a scatteringvolume per unit time within an element of solid angleds, i.e., the particle flux �mm�2 s�1�, is equal toc�n · s�fds, where n is the outward normal. Therefore,the number of photons dN escaping the scatteringvolume through the area dS within the time intervaldt and within an element of solid angle ds about s isgiven by dN � fd3rds, where d3r � �n · s�cdtdS. Next,let us assume that photons moving from the surfaceof the scattering volume within d3r are absorbed byan area dSp of the distant absorbing surface such asa CCD camera. Considering the reverse process wefind the number of photons absorbed by the area dSp

within dt and within dsp � �ds about the unit vectorsp � �s in the form

dNp � fd3rpdsp, d3rp � �np · sp�cdtdSp, (3)

20 October 2007 � Vol. 46, No. 30 � APPLIED OPTICS 7385

where np is outward normal to the absorbing surfaceand np · sp � 0.

The total number of photons absorbed by dSp

within the time interval dt is obtained by summingall photons arriving at a pixel of the CCD camera atrp from the visible part of the emitting surface withinthe solid angle p [19]:

Np�t� � cdtdSp�p

w�sp, s0�f�r, sp, t��np · sp�dsp, (4)

where the weighting function w�sp, s0� describes theoptical system. An optical system consisting of a lensplaced at sufficient distance from the emitting sur-face can be approximated by a pinhole. A pinholeassumption results in a sharply peaked weightingfunction w�sp, s0� about a direction s0 along the lineconnecting each pixel of the CCD camera to the emit-ting surface through a pinhole. Therefore, one canbring out all slowly varying functions from the inte-gral in Eq. (4) at s0. Considering the direct relation-ship between the sample emission and the detectorsurfaces (which is parallel in our case), one can as-sume that w�sp, s0� depends only on cos 0 � sp · s0and, consequently, setting w � ��cos � cos 0� ob-tain

Np�t� � 2�cdtdSpf�r, s0, t��np · s0�. (5)

Because not all absorbed photons are counted, it isuseful to introduce a camera’s gating function g��t� inthe frequency interval ��, � � d��. The camera’s gat-ing function describes the camera’s response to the�-function input and has a dimension of inverse time.Thus the number of counted photons, denoted by Np�,is given by the time convolution of the number ofabsorbed photons, Np�t�, with the gating functionsg��t�. An assumption that g��t� has finite supportabout t allows the integration to be extended over theentire time domain:

Np��t� � ���

�

g��t � t��Np�t��dt�. (6)

By means of Eq. (1) this number is converted to theenergy Ep��t� � h�Np��t� recorded by a pixel of theCCD camera of area Sp within the finite but smallexposure time t and can be regarded as the observ-able quantity in the time domain,

Ep��t� � 2� t Sp�np · s0����

�

g��t � t��I��r, s0, t��dt�,

(7)

as well as in the Fourier domain (�),

Ep���� � 2� t Sp�np · s0�g����I��r, s0, ��. (8)

Technically, each pixel of the CCD camera recordsa signal proportional to Ep��t� at the finite number ofpoints t � tn. Recorded signals are then Fourier trans-formed to give complex functions Ep���n�, which formthe data set for the lifetime image reconstruction.

When the exposure time is very small the gatingfunction g� can be quite accurately approximated bythe step function defined as: g��t� � g0� t when td

� t � td � t and zero otherwise, where a dimension-less constant g0 describes how the absorbed energy istransformed to a recorded signal (assuming lineardependence), and td is the time delay with respect toexcitation time. The Fourier transform of the stepfunction is expressed in terms of a sinc function:

g���� � g0 exp��i��td � t�2��sinc�� t�2�. (9)

The exposure time in our experiments was set to t� 5 � 10�10 s. For the diffusion regime � �� 108 Hzand the delay time td � t the following holds:� t �� 1 and �td �� 1, and, therefore, g���� � g0. Forthe frequencies � � 108 Hz and higher the wholeexpression Eq. (9) must be used.

B. Radiative Transfer Equation and Its Approximations

We employ the TE approximation to the radiativetransfer equation for modeling the light transport inthe turbid medium. The TE is the result of conse-quent simplifications made to the radiative transferequation according to the conditions assumed to bevalid for the studied physical system [21,22]. In theFourier domain the radiative transfer equation de-scribing the evolution of the intensity I� due to anultrashort pulse of light at t0 reads as

�s, �I�� � ��0I� ��s

4� ��4��

p�s� · s�I��s��ds�

� p�ss · s�S0, (10)

S0 � �0N0h���r � r0�exp��i��t0 � �r � r0��c��,(11)

��r � r0� ��tr�r�

4��r � r0�2 exp� �0

�r�r0�

�tr�r��dl�, (12)

where N0 is the number of injected photons. Thescattering coefficient is denoted by �s and the ab-sorption by �a; �tr � �s � �a is the transport coeffi-cient (1��tr � l, where l is the free path length); �0� �s��tr is the albedo of a single scattering event; �0is given by

�0 � �tr�1 � i��c�tr�, (13)

and p�s� · s� is the phase function. The unit vector ss

originates at the point source and directs toward theobservation point r � �x, y, z�T, while the coordinatesof a point source are denoted by r0 � �x0, y0, z0�T. The

7386 APPLIED OPTICS � Vol. 46, No. 30 � 20 October 2007

integration in Eq. (12) is performed along the straightline connecting points r0 and r.

The source term in Eq. (11) describes only the scat-tered (diffusive) photons incoming from the point-likesource, while ballistic photons are excluded [23].Thus, for the perfectly absorbing medium �0 � 0 aswell as for the nonscattering medium l � �, thesource term S0 disappears. The amplitude modulatedlight transport model is obtained by convolving thesource term, Eq. (11), with a function governing theamplitude modulation, say ��t0�, with respect to t0[i.e., exp��i�t0� ) ����]. The physical meaning of thesource term in Eq. (11) becomes clear in the timedomain:

S0�t� � �0N0h��0�r � r0���t � t0 � �r � r0��c�,(14)

�0�r � r0� � �tr exp���tr�r � r0���4��r � r0�2, (15)

where a constant �tr is assumed in the neighborhoodof r0. The function �0�r � r0� in Eq. (14) can be re-garded as a distribution function of ballistic photonsscattered for the first time. Its first moment in infinitespace gives unity. The second moment gives 1��tr,i.e., the mean free path l. The � function in Eq. (14)describes the time delay of ballistic photons arrivingat r. In a highly scattering medium with low absorp-tion the source term S0 can be appreciably simplified.First, at the distances where �tr|r � r0| �� 1 thedistribution function �0�r � r0� is replaced with ��r� r0�. Then, assuming that all injected photons arescattered immediately at t0 in the neighborhood of r0the time delayed � function is replaced by ��t � t0�.Therefore, the source function in the Fourier domaindescribing the pulse of light of duration 0 � t0 � �tcan be approximated by

S0 � �0N0h���r � r0�exp��i��t�2�sinc���t�2�.

Taking �t � 10 ps and considering frequencies belowseveral gigahertz we can simplify the source functioneven further by replacing exp��i��t�2�sinc���t�2�by 1.

The telegraph equation approximation to the radi-ative transfer results by retaining the first two termsin the Fourier power series of the phase function andis obtained by taking the first two moments of theradiative transfer equation. If the energy flux H isdefined as

H��r, �� � ��4��

I��r, s, ��sds, (16)

then the system of two coupled differential equationsin the Fourier domain (�) reads as

� · H� � cqU� � S0, H� � ��c�3�1��U�, (17)

q � �a�1 � i��c�a�, �1 � �0 � �sg, (18)

and g is the average cosine. By eliminating the en-ergy flux from Eq. (17), the Helmholtz equation, theFourier image of the telegraph equation, is obtained.The diffusion approximation (DA) results by neglect-ing i��c�tr in �1.

Robin type boundary conditions for the TE or DAfollow from the approximate expression of the inten-sity,

I� �c

4�U� �

34�

s · H�, (19)

by taking into account Eq. (17) together with theconditions I�n · s � 0�|�V � 0 on the surface of thescattering volume, where n is the surface’s outwardnormal. Making use of Eq. (19) at the boundary of thescattering volume �V one can rewrite Eq. (8) in theform:

Ep���� � c t Sp�np · s0�g����U��r, ��. (20)

C. Inverse Problem: the Time Domain Versus the FourierDomain

An assumption that all nonlinear effects in the emis-sion light transport are negligible makes the mathe-matical description of the emission photon transportanalogous to the excitation one except for differentvalues of optical parameters and a source function.The number of created fluorescent photons N�* [in-stead of N0 in Eq. (11)] is proportional to the timeconvolution of the absorbed excitation energy density��a��s� U� and the � response of the fluorophor:

q�*�r�, t � t�� � q0H�t � t���0

�

F�r�, ���exp��t � t�

�� �d��

��,

(21)

where H�t � t�� is the Heaviside step function, q0 isthe quantum yield, � is the lifetime, and F�r�, �� is thelifetime distribution function. Thus, the number offluorescence photons created by a volume d3r�, eachpossessing an energy h�*, is given in the form of aconvolution:

N�*�r�; t� ��a

�sh�d3r��

t0

t

q�*�r�, t � t��

� U��r�, r0; t� � t0�dt�. (22)

The lower limit of the integral in Eq. (22) can beextended over the entire history of the system, i.e., to��, because U��t� � t0� � 0 when t� � t0, and theupper limit can be extended to due to properties ofthe Heaviside function.

For the further analysis it is convenient to intro-duce the Green function. The fluorescent Green func-tion in the time domain G�*�r, r�; t � t�� is found byapplying the inverse Fourier transform to the solu-

20 October 2007 � Vol. 46, No. 30 � APPLIED OPTICS 7387

tion of the boundary value problem,

��1�3�1*��G�* � q*G�* � ��s*��r � r��exp��i�t��,(23)

subject to Robin boundary conditions. The parame-ters �1*, �s*, and q* are the fluorescent analogs of �1,�s, and q. The scattering coefficient �s* appearing inthe source function expresses the fact that only scat-tered photons are considered. Note that the Greenfunction defined by Eq. (23) has a dimension mm�3

s�1 in the time domain and mm�3 in the Fourierdomain. The number of scattered fluorescence pho-tons N�*��r, t� is given by the time convolution of theGreen function with the number of created fluores-cent photons:

N�*��r; t� � d3r�t0

t

N�*�r�, t��G�*�r, r�; t � t��dt�, (24)

where the integration can be extended over the entiretime domain. Insertion of Eq. (22) into Eq. (24) to-gether with Eqs. (5)–(7) results in a system of Fred-holm integral equations of the first kind:

Ep��r; t� � c t Sp�np · s0���*����V

d3r���a��s�

������

�

dt�dt�dt�g�*�t � t��q�*�r�; t� � t��

� G�*�r, r�; t� � t��U��r�, r0; t� � t0�, (25)

where p runs through all pixels of images taken byCCD camera; r0 runs through all source positions;and g�*�t� is the gating function in the frequency in-terval ��*, �* � d�*�. An additional time convolutionwill be present in Eq. (25) in the case of a sufficientlylong excitation pulse. The inverse problem formu-lated as the reconstruction of the quantum yield andthe lifetime in the time domain on the basis of asystem of equations of Eq. (25) type is almost intrac-table due to the triple time convolution. That is tosay, the inverse problem is three-dimensional intime. However, the system becomes appreciably sim-pler in the Fourier domain where it reads as

Ep��r; �� � c t Sp�np · s0���*����V

d3r���a��s�

� G�*�r, r�; ��U��r�, r0; ��Q0�r�; ��,(26)

Q0�r; �� � g�*���q0�r��0

�

F�r, ���d��

1 � i���. (27)

An assumption that the lifetime distribution functionis sharply peaked at �(r) results in

Q0�r; �� �g�*���q0�r�1 � i���r�

, (28)

where �F����d�� � 1 has been used.The system of Eq. (26) is further discretized and

solved numerically in the Fourier domain on theadaptive Cartesian grid [15,24] by use of the conju-gate gradient method with early termination of theiterative process, which has a regularization effect.The number of iterations is found empirically fromthe discrepancy principle [25].

3. Materials and Experimental Methods

A. Phantom

For this experiment a solid phantom of slab geom-etry 120 � 75 � 40 mm shown in Fig. 1 was builtusing a solution of epoxy (MY753) and hardener(XD716) from Robnor Resins Ltd. mixed with thescattering particles �TiO2� [26,27]. The resin has arefractive index of 1.547, the TiO2 particles (TirantiLtd., London), has an average diameter of 0.25 �mand its refractive index is 2.58. The following valuesof optical parameters were measured [28]: �s �2.0016 mm�1; g � 0.5580; �a � 0.0008 mm�1. Weassume that the optical parameters of the medium

Fig. 1. Phantom and the positions of tubes filled with Rhoda-mine 6G.

7388 APPLIED OPTICS � Vol. 46, No. 30 � 20 October 2007

with respect to excitation and fluorescence are thesame. This assumption is reasonable considering thecorresponding wavelength, which results in the dif-ference between the corresponding optical parame-ters being indistinguishable from the experimentalerror. The slab was drilled with four wells each of8 mm diameter. These wells had the same opticalproperties but mixed with Rhodamine 6G dye as thefluorophor.

B. Experimental Setup and Method

The illumination was provided by a commercial su-percontinuum source (SC450-2, Fianium Ltd.), pro-ducing a continuum from 450 to 2000 nm with aspectral density up to 5 mW�nm at a repetition rateof 50 MHz. An electronically tunable spectral selec-tion arm [29] was incorporated in the setup to selectan appropriate excitation wavelength. The diagramof the experimental setup is shown in Fig. 2. Briefly,following angular dispersion at the prism, P, thecontinuum was collimated by an achromatic lens(f � 180 mm, 01LAO139, Melles Griot) positioned onefocal length from the prism. A motorized triangular-shaped aperture located directly in front of a mirror,M3, provided spectral selection and bandwidth con-trol (horizontal and vertical adjustment, respec-tively). After reflection back through the lens-prismarrangement (removing the angular dispersion), theselected spectral range was picked off by a knife-edgemirror M4. To efficiently excite the fluorescent cylin-ders, 25 mW at a wavelength of 525 � 15 nm wasdirect toward the sample using M5, resulting in a“source” diameter of �5 mm with a pulse duration of�10 ps.

Time-gated diffuse transmitted fluorescence im-ages were acquired using a gated optical intensifier

(GOI, Kentech Instruments Ltd.) optically coupled toa CCD (OrcaER, Hamamatsu Photonics). The GOIcan produce time gates ranging in duration from�100 ps up to 1 ns at repetition rates kilohertz to1 GHz. For the acquisitions described here the gatewidth was set to 600 ps with the GOI triggered at therepetition rate of the laser. The transmitted fluores-cence signal was imaged onto the photocathode of theGOI through a 550 nm long pass filter (550GY25,Comar) using a camera lens (CC21211, Cosmicar�Pentax), imaging 3.0 � 2.3 cm of the back surface ofthe sample.

During an acquisition, 70 time-gated images at 200ps intervals, controlled by delay electronics (TriggerDelay Unit, Kentech Instruments Ltd.), were re-corded, covering a sampling extent of 13.8 ns as in-dicated in Fig. 3. The gain voltage and integrationtime were fixed at 950 v and 0.1 s on the GOI andCCD, respectively. The sample was then translatedin 4 mm intervals to generate data sets for a numberof source positions, ranging from 8 mm either side ofthe first and last fluorescent cylinders.

4. Results and Discussions

Quantum yield was reconstructed from the steadystate case � � 0. Slice at y � 25 mm is shown in Fig.4. The overall spatial accuracy of the reconstructionis �2–3 mm. The spatial resolution of the tube closestto the camera is the best, while the farthest tube

Fig. 2. (Color online) Experimental setup. M1–M5, mirrors; P,prism; F, long pass filter. Mirror M3 has a triangular-shaped slitpositioned directly in front of it. Sample is mounted on a motorizedtranslation stage to produce different source positions.

Fig. 3. (Color online) Graph showing the average image intensityas a function of time-gated image delay, including gated images atdelays of (a) 0, (b) 5, and (c) 8.6 ns.

Fig. 4. Slice y � 25 mm showing the reconstructed quantum yielddistribution at � � 0.

20 October 2007 � Vol. 46, No. 30 � APPLIED OPTICS 7389

appears to be smaller and slightly closer to the cam-era than it should be. This implies that spatial reso-lution significantly improves if other sides of thephantom were imaged. The true value of the quan-tum yield in the tubes was 0.95. Therefore, the quan-tum yield was found with approximately 10%–20%error depending on the target’s location.

Lifetime can be reconstructed independently fromthe quantum yield. Thus, having found the complexfunction Q for any given frequency we exclude thequantum yield from equations

Re Q � q0

Re g�*��� � �� Im g�*���1 � ����2 , (29)

Im Q � q0

Im g�*��� � �� Re g�*���1 � ����2 , (30)

and obtain

� �AB�

B2�2 � �, (31)

where

A � Im g�*���Re Q � Re g�*���Im Q, (32)

B � Re g�*���Re Q � Im g�*���Im Q, (33)

and � is a regularization parameter. Note that, hav-ing found �, this procedure allows one to recover thequantum yield by use of Eq. (29) or Eq. (30).

Reconstructed images of the function Q at � �1 GHz are shown in Fig. 5. Note that both Re Q andIm Q are negative, which was caused by the timedelay td � 1 ns in the gating function. The lifetimeshown in Fig. 6 was recovered from Eq. (31). The truevalue of the lifetime in the tubes was 4.05 � 10�9 s.As a result, the reconstruction of the lifetime distri-

bution was achieved with relatively good accuracywith approximately 10%–20% error.

In summary we have demonstrated the feasibilityof fluorescence lifetime imaging experimentally onthe basis of a time-gating technique. The time-gatingtechnique provides information about fluorophordistributed inside the studied object in the nanosec-ond time scale, which facilitates the reconstructionalgorithm and allows the fluorescence lifetime distri-bution to be recovered with relative ease and accept-able accuracy. It provides enough data points in thetime domain needed for application of the Fouriertransform. The use of the Fourier transform is es-sential for the time deconvolution of the system ofthe integral equations employed in the reconstruc-tion. The application of the telegraph equationallows a sufficiently broad interval of frequencies tobe covered. Reconstructed image, which is a com-plex function, contains enough information neededfor recovering the lifetime distribution in a samplefor any given frequency within the megahertz–gigahertz band. This technique was applied in recon-structing the quantum yield and lifetimes of fourtubes filled with Rhodamine 6G embedded inside ahighly scattering slab. Relatively accurate recon-struction demonstrates the effectiveness and the po-tential of the proposed technique.

This work was supported in part by a DTI Beaconproject and an EU Framework VI project: IntegratedTechnologies for in vivo Molecular Imaging LSHG-CT-2003-503259.

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Fig. 5. Reconstructed real and imaginary parts of the functionQ �r; �� at � � 1 GHz.

Fig. 6. Reconstructed lifetime distribution from real and imagi-nary parts of the function Q �r; �� at � � 1 GHz.

7390 APPLIED OPTICS � Vol. 46, No. 30 � 20 October 2007

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