Abstract

Fluorescence resonance energy transfer (FRET) is a nonradiative energy transfer process based on dipole-dipole interaction between donor and acceptor fluorophores that are spatially separated by a distance of a few nanometers. FRET has proved to be of immense value in the study of cellular function and the underlying cause of disease due to, for example, protein misfolding (of consequence in Alzheimer’s disease). The standard parameterization in intramolecular FRET is the lifetime and yield, which can be related to the donor-acceptor (DA) distance. FRET imaging has thus far been limited to in vitro or near-surface microscopy because of the deleterious effects of substantial scatter. We show that it is possible to extract the microscopic FRET parameters in a highly scattering environment by incorporating the FRET kinetics of an ensemble of DA molecules connected by a flexible or rigid linker into an optical diffusion tomography (ODT) framework. We demonstrate the efficacy of our approach for extracting the microscopic DA distance through simulations and an experiment using a phantom with scattering properties similar to tissue. Our method will allow the in vivo imaging of FRET parameters in deep tissue, and hence provide a new vehicle for the fundamental study of disease.

© 2009 Optical Society of America

1. INTRODUCTION

Fluorescence resonance energy transfer (FRET) results in a decrease in the lifetime and quantum yield of the donor in the presence of the acceptor [1, 2]. Both intramolecular and intermolecular FRET have been extensively used in the in vitro study of molecular activity related to cellular function and the cause of disease [3, 4, 5, 6, 7, 8, 9]. In intramolecular FRET, the donor and acceptor are connected by a linker and energy transfer is between a single donor and acceptor (fixed stoichiometry). The donor-acceptor (DA) distance is fixed for a rigid linker or is distributed over some (nm) range for a flexible linker [2]. Intramolecular FRET has been used to study protease activity [3], conformational changes in proteins [4], protein phosphorylation [5], biological phenomena like protein folding and denaturation [2, 8], and to measure intracellular calcium [6]. The study of protein folding and aggregation by FRET could be important in relation to diseases like Alzheimer’s and some cancers which have been linked to protein misfolding [10, 11]. In intermolecular FRET, the donor and acceptor reside on independent unlinked hosts and, therefore, there can be multiple acceptors for a single donor. Intermolecular FRET has been used to study protein-protein interactions [7].

The deleterious effects of substantial scatter precludes the direct imaging of molecular activity using FRET in deep tissue. Until now, the imaging of FRET in tissue has been limited to surgically removed thin tissue sections [12, 13] and near surface in vivo [14] multiphoton microscopy. While multiphoton or confocal microscopy allows the spatial mapping of FRET parameters [13], this is limited to weak scatter. Optical diffusion tomography (ODT), where the propagation of light in a scattering medium is modeled by a diffusion equation, facilitates deep tissue imaging [15, 16]. With the appropriate data set, ODT has been used to image fluorescence yield and lifetime [17]. Combining ODT with FRET would thus enable three-dimensional spatial mapping of FRET parameters and allow in-vivo deep-tissue imaging of molecular activity indicated by the microscopic DA distance or distance distribution.

Molecular beacons (MBs) [18] are similar in construction to FRET molecules, as they have a fluorophore linked to a quencher, and in-vivo images have been achieved for a mouse model [19] using techniques similar to fluorescence optical diffusion tomography (FODT) [17]. However, due to their particular construction, in which the fluorophore-quencher distance in MBs is much less than the DA distance in FRET probes, MBs behave fundamentally differently [20, 21] and provide no distance information.

We describe an approach to couple the kinetics of intramolecular FRET, with a rigid or flexible linker, with a model that describes multiply scattered light, thereby allowing the estimation of the microscopic FRET distance or distance distribution inside a scattering medium. To describe the light scatter, we use a diffusion model, which is valid at length scales large relative to the photon randomization length ls=(3D)1, where D is the diffusion coefficient, which in tissue is on the millimeter scale. Therefore, ODT length scales are not commensurate with FRET DA distances, making it unclear whether it would be possible to determine the DA distance from heavily scattered light. Combining FRET and ODT requires that we have an ensemble of DA pairs in order to ensure an adequate detector signal to noise ratio (SNR). Through simulations and a simple experiment, we show that it is possible to determine the microscopic FRET parameters. Specifically, the DA distance or distance distribution for rigid or flexible linkers, respectively, is reconstructed using the macroscopic ODT modality.

2. FRET-ODT PARAMETERIZATION

In the excited state and in the presence of an acceptor, the donor molecule can decay to the ground state radiatively by photon emission with rate kr and non-radiatively, by heat dissipation and energy transfer to an acceptor molecule, with rates knr and kF, respectively [1, 2]. If τD is the donor lifetime and ηD the donor quantum yield in the absence of an acceptor, then kF=τD1(R0rF)6, where rF is the distance between the donor and the acceptor and R0 is the DA distance where the energy transfer efficiency is 50%, τD1=kr+knr, and ηD=kr(kr+knr) [2]. In the presence of an acceptor, energy transfer results in a decreased donor lifetime, τDA, given by

1τDA=1τD[1+(R0rF)6]=kr+knr+kF
and a decreased donor quantum yield
ηDA=krkr+knr+kF.
For an ensemble of flexible linkers, the DA distance is commonly modeled by a probability distribution, which we assume to be Gaussian within the distance range [rmin,rmax] and given by p(rF)=(1Z)exp{a(rFb)2}, where Z=rminrmaxexp{a(rFb)2}drF is the partition function, b is the mean DA distance, and a is inversely proportional to the variance [2, 8].

For an ensemble of DA pairs connected by flexible linkers, in a highly scattering medium, we model the excitation and donor fluorescence photon propagation by the frequency domain coupled diffusion equations [17, 22]

[Dx(r)φx(r,ω)][μax(r)iωc]φx(r,ω)=Sx(r;ω)
[Dm(r)φm(r,ω)][μam(r)iωc]φm(r,ω)=φx(r,ω)Sf(r;ω),
with exp(iωt) time dependence and circular modulation frequency ω (rads1), where subscripts x and m, respectively, denote the excitation and emission wavelengths λx and λm, φ(r,ω) (Wcm2) is the photon flux density at position r inside the scattering medium, a point source Sx(r;ω)=βδ(rs) is assumed, with β the modulation depth, c is the speed of light in the medium, μa(r) (cm1) is the absorption coefficient at position r, D(r) (cm) is the diffusion coefficient at position r, and Sf(r,ω) is the source for the flexible (subscript f) linker. Using the Fourier transform of the time-domain impulse response of the donor fluorophore, [(ηDA.μaD)τDA].exp(tτDA), along with (1, 2),
Sf(r;ω)=rminrmaxηDAμaD(r)1iωτDA(r)p(rF(r))drF(r)=rminrmaxη(r)ζ(rF(r))p(rF(r))drF(r),
upon setting ηDA=krτDA and ηD=krτD, where the distance-dependent τDA and ηDA are averaged over DA distance rF using the distribution p(rF), ζ(rF(r))=rF6(r).[(1iωτD(r))rF6(r)+R06]1, rF(r) is the DA distance at point r, μaD(r) is the absorption coefficient of the donor, and for notational simplicity, η(r)=ηDμaD(r). Modulated light is necessary to determine all parameters in (5). For an ensemble of DA pairs connected by rigid linkers, p(rF) reduces to a Dirac delta function, and the corresponding source term is Sg(r;ω)=η(r)ζ(rF(r)), where the subscript g indicates a rigid linker. We chose the DA distance and distance distribution to parameterize the FRET image, rather than donor lifetime in presence of the acceptor, τDA, because distance is a natural measure for FRET and, for the flexible linker, the probability distribution is Gaussian if the FRET image is parameterized in terms of distance, while for τDA, the distribution would be non-Gaussian. Having a Gaussian distribution makes it easier to calculate the integral in (5). Let gx(rsk,r;ω) be the diffusion equation Green’s function for (3) at wavelength λx and gm(r,rdm;ω) be the Green’s function for (4) at λm. Then, the donor fluorescence photon flux density (for flexible or rigid linker) at a detector positioned at rdm, due to a point source modulated at frequency ω and positioned at rsk, is
φ(rsk,rdm;ω)=gm(r,rdm;ω)S(r;ω)gx(rsk,r;ω)d3r,
where S() represents either Sf or Sg.

Assuming that μax, Dx, μam, Dm, and τD are known or have been reconstructed using a procedure we have previously described [17], then, for the flexible linker, the unknown parameters can be represented as xF=[η(r),a(r),b(r)]T, or for the rigid linker as xF=[η(r),rF(r)]T, where T is transpose. We use K optical sources modulated at Q frequencies for excitation and M detectors to measure the emitted donor fluorescence. To reconstruct xF and thus locate the ensemble inside the scattering medium, the measurements are incorporated in a Bayesian framework [17, 23], which facilitates a statistical approach for the reconstruction of xF.

3. RECONSTRUCTION

To reconstruct xF, the simulation domain is discretized into N nodes on a uniform 3-D grid, which transformed xF into the 3N×1 vector

xF=[η(r1)η(rN),a(r1)a(rN),b(r1)b(rN)]T
for the flexible linker, and the 2N×1 vector
xF=[η(r1)η(rN),rF(r1)rF(rN)]T
for the rigid linker, where the superscript T denotes the transpose operation.

Source and detector calibration coefficients [24] are explicitly introduced into the inversion. The inversion problem is couched in a Bayesian framework [17, 23], which allows us to incorporate a priori information as well as information about the physics of the FRET model into the maximum a posteriori (MAP) estimate

{x̂F}MAP=argmaxxF0,s,d{q=1Qp(yFq|xF,sq,dq)p(xF)},
where p(yFq|xF,sq,dq) is the data likelihood for the fluorescence signal measured from an ensemble of DA pairs connected by either flexible or rigid linker located inside a scattering medium, p(xF) is the prior model, yFq is a KM×1 vector representing the measurement at M detectors due to K sources modulated at frequency ωq, Q denotes the total number of modulation frequencies, and sq=[s1q,sKq] and dq=[d1q,dMq] are the source and detector coupling coefficient vectors respectively, at frequency ωq, for the K sources and M detectors, which are estimated jointly with the image xF. For the simulations presented in the paper, sk=1+i0 and dm=1+i0 are used. Assuming a Gaussian shot-noise model, the data term is
p(yFq|xF,sq,dq)=1(παF)P|ΛFq|1exp[yFqfFq(xF,sq,dq)ΛFq2αF],
where P=KM is the number of measurements, fFq(xF) is a KM×1 vector of the form
fFq(xF,sq,dq)=[s1qd1qφ(rs1,rd1;ω,xF)s1qdMqφ(rs1,rdM;ω,xF),s2qd1qφ(rs2,rd1;ω,xF)sKqdMqφ(rsK,rdM;ω,xF)]T,
αF is a scalar parameter that scales the noise variance, and, for an arbitrary vector w, wΛ2=wHΛw (where H denotes Hermitian transpose), with αF(2ΛFq1) the covariance matrix. We assume a shot noise model with ΛFq1=diag[|yFq1|,|yFqP|] [23]. For the flexible linker, the prior density, p(xF), assuming independence of the image vectors representing η, a or b, is p(xF)=p(xη).p(xa).p(xb). We use the generalized Gaussian Markov random field (GGMRF) prior [23]
p(xk)=1σkNz(ρk)exp(1ρkσρk{i,j}Nkbij|xkixkj|ρk),
where k represents η, a or b, xki denotes the ith node of xk, the set Nk consists of all pairs of neighboring nodes (in a 26 neighbor system), and bij is the weighting coefficient corresponding to the ith and jth nodes assigned to be inversely proportional to the node separation, under the condition that jbij=1. Parameters ρ (we use ρk=2) and σ control the shape and scale of the distribution and z(ρ) normalizes the density. The prior model for the rigid linker can be described in a similar way.

Numerically, φ(rsk,rdm;ω,xF) is calculated by discretizing (6), with S() being either Sf or Sg, as

φ(rsk,rdm;ω,xF)=j=1NV[gm(rj,rdm;ω)S(rj;ω,xF)gx(rsk,rj;ω)],
where V is the (cubic) element volume associated with each node and the dependence of the FRET source term, S, on the image xF, is explicitly included by representing it as S(rj;ω,xF). Using (13), we rewrite (11) as
fFq(xF,sq,dq)=GωqSωq(xF),
which is the product of a KM×N matrix and a N×1 vector, with Gωqij=skqdmqgx(rsk,rj;ωq) .gm(rj,rdm;ωq), m=i modulo M and k=[iM+1], where [.] is the integer part of iM+1, and Sωq(xF)=[S(r1;ωq,xF)S(rN;ωq,xF)]T. Previously, we found that incorporating αF into the inverse problem as an unknown for each image tends to improve the robustness and speed of convergence [25]. As a result, we obtain the MAP estimate, {x̂F}MAP, by performing a joint iterative minimization of ln[p(yF|xF)p(xF)] over αF, sq, dq and xF, leading to
({x̂F}MAP,α̂F,ŝ,d̂)=argminxF0,s,dminαF{1αFq=1QyFqGωqSωq(xF)ΛFq2+PlogαFlog[p(xF)]}.
As in our previous work, we use the iterative coordinate descent (ICD) algorithm, a Gauss-Seidel approach, along with a Golden Section Search, to estimate {x̂F}MAP from (15) [22]. In ICD, the cost function in (15) is optimized with respect to individual nodes in the image, where the image points are scanned in random order. For example, for the flexible linker, one update scan for xF consisted of updating all N nodes first with respect to a, then b and subsequently η. For the flexible linker, with all other image elements fixed, the ICD update for the ith node in the image of a, b or η is
x̂kiargminxki0{1α̂Fq=1QyFq[Gωq]*(i)S(ri;ωq,xF)ΛFq2+1ρkσkρkjNkibij|xkix̂kj|ρk},
where ← implies update, k represents either a, b or η, Nki is the set of nodes neighboring node i, and [Gωq]*(i) denotes the ith column of Gωq.

4. SIMULATION

The model problem we consider is the cubical tissue phantom of Fig. 1a with side 2.5cm, discretized on a 33×33×33 uniform grid. A planar arrangement of 14 sources, with ten equally spaced modulation frequencies between 40 and 220MHz (20MHz between measurements), is on the top face and 14 detectors are used on the bottom face. The molecules EYFP and HcRED1 are assumed as donor and acceptor, respectively, with τD=3.8ns [26], and we use R0=4.7nm [27]. We assume DxDm=0.027cm and μaxμam=0.047cm1. For both rigid and flexible linker, we choose η=0.025cm1, calculated from the quantum yield and absorption coefficient for EYFP, assuming a concentration of 250nM for the FRET pair. The FRET pairs are contained within a centrally located sphere of radius 2.5mm. We assume a FRET molecular concentration that is based on related work with a mouse tumor model [19], and tissue parameters for the light [16]. The detectors are assumed to be shot noise limited with an average SNR of 30dB. We used MUDPACK to generate the synthetic data [28, 29].

4A. Computation of Equation (5)

Determining {x̂F}MAP involved multiple computations of the integral in (5). For given R0, ω and τD, piecewise fitting cubic polynomials and exponential functions to the real and imaginary parts of ζ(rF) leads to analytical expressions for the integral in (5). This provided an efficient way to parameterize the FRET source. The limits of integration in (5) were chosen to be rmin=5Å and rmax=3.5R0. The accuracy of this approach is illustrated in Fig. 2.

4B. Simulation Results

Figure 1b shows the central slice of the discretized sphere (the isosurface plot in Fig. 1a gives a perspective view). Figures 1c, 1d show the reconstructed rF and η for the case of a rigid linker, indicating excellent accuracy. In Figs. 1e, 1f, we show the results for a and b with a flexible linker. Notice how well the true values are estimated. The key point is that we are able to determine the microscopic FRET parameters from the macroscopic ODT model with a realistic noise model, suggesting that the modality can be applied in vivo.

To emphasize the efficacy of our approach, consider the case in Fig. 3 of two spherical regions, each of radius 2mm, containing FRET DA pairs connected by flexible linkers having differing parameters (a and b), simulating different micro-environments. Figures 3a, 3c show the true images of a and b, respectively, while Figures 3b, 3d show the reconstructed results. The η image in Fig. 3e is determined quite well in the reconstructed image of Fig. 3f. We are thus able to accurately reconstruct the FRET distribution parameters, even if the FRET pairs are localized in two different regions with different distributions.

5. EXPERIMENT

To verify the model and simulations, a simple experiment was performed. The goal was to reconstruct the DA distance for an ensemble of FRET DA molecules located inside a scattering medium. The FRET chemical was comprised of the donor, 4,4-difluoro-5, 7-dimethyl-4-bora-3a,4a -diaza- s-indacene-3-propionic acid (BODIPY-FL: excitation 488nm, emission 520nm), the acceptor, tetraethyl rhodamine, and a structurally rigid peptide linker whose chemical structure is described elsewhere [30]. Figure 4 shows a schematic of the ODT measurement system, with a 3mW 488nm argon-ion laser (Uniphase), an xy scanning mirror system, a Plexiglas box of size 8.8cm(L)×8.8cm(H)×3.4cm(W), a cylindrical plastic vial (length 3cm, inner-diameter 0.65cm, outer-diameter 0.75cm) to hold the FRET chemical, suspended from the lid of the Plexiglas box using an acrylic rod 1mm in diameter and 1.5cm in length, a 520nm bandpass filter (Edmund Optics, 50.8mm square) with FWHM 10nm and peak transmission 45% [31], and a 105mm, f/2.8 lens (AF micro Nikkor, Nikon) to focus the 4.8cm×4.8cm image of the scattering medium on a 512×512  pixel Peltier cooled CCD Camera (PI-MAX, Roper Scientific). The Plexiglas box was filled with a suspension of 0.4% (w/v) Intralipid (Sigma-Aldrich). The 0.4% Intralipid solution was prepared by dissolving 10ml of 20% Intralipid in 500ml of deionized water. The plastic vial contained 1ml of 3μM solution of the FRET chemical in Intralipid, prepared by mixing 100μl of a 0.18mM stock solution of the FRET chemical with 5.9ml of 0.4% (w/v) Intralipid. The chemical was dissolved in Intralipid to maintain tissue-like scattering properties.

Using the xy scanning mirror system, the laser beam was scanned along the Plexiglas box surface to 19 pre-chosen source positions. For each source position, 100 frames with an exposure time of 150msec each were taken with the camera and added together. For reconstruction, the 512×512 image was down-sampled to 16×16, giving 256 detectors in total. Each detector is one pixel, and the detector pixels were uniformly spaced. The laser was not modulated. We determined η experimentally, as described below, and use η=15.6×103cm1. For the 0.4% Intralipid solution, we assume that all the absorption at λx=488nm is due to water, thereby giving μax2.5×104cm1 and μam3.2×104cm1 [32]. Prior experiments with Intralipid lead to Dx0.051cm and Dm0.056cm [33].

Using the procedure outlined in Sections 3, 5C, the acquired data was calibrated and the images were reconstructed. For reconstruction, the scattering medium was discretized into a 33×33×17 uniform grid. In the calculation for rF, we ignored the small contribution of rhodamine emission at 520nm, when excited at 488nm, as we found that compared to the emission of Bodipy-FL at 520nm, rhodamine’s signal was negligible. Also, due to the scattering medium, less than one μW was incident on the cuvette, which resulted in no observable photobleaching.

5A. Determination of η

A solution of the donor BODIPY-FL and Intralipid was prepared by mixing 100μl of a 10μM stock solution of the donor BODIPY-FL with 3.9ml of 0.4% (w/v) Intralipid, resulting in a concentration of CD=250nM. One ml of the donor-Intralipid solution was pipetted into a plastic vial (identical in size to the one described in the paper) and suspended inside the Plexiglas box containing 0.4% Intralipid. Using the same experimental setup as described above, the laser beam was scanned along the Plexiglas box surface to 26 pre-chosen source positions. For each source position, 100 frames with an exposure time of 150msec each were acquired with the camera and added together. For reconstruction, the 512×512 image was down-sampled to a 16×16, thus making 256 detectors in total. Each detector is one pixel, and the detector pixels were uniformly spaced. For reconstruction, the scattering medium was discretized into a 33×33×17 uniform grid. For the background optical properties, we assumed that for a 0.4% Intralipid solution, all the absorption was due to water, thereby giving μax2.5×104cm1, μam3.2×104cm1 [32], Dx0.051cm and Dm0.056cm [33]. Using fluorescence optical diffusion tomography (FODT) [17], η was reconstructed. We note that related fluorescence imaging work has been termed fluorescence molecular tomography (FMT) [19]. Figure 5 shows three slices along the xy plane for the reconstructed η. The estimated η in the reconstructed image for the 250nM solution was 13×104cm1, giving η=15.6×103cm1 for the 3μM solution used in the FRET imaging experiment.

5B. Computation of R0

The Förster distance R0 for the donor BODIPY-FL and acceptor rhodamine was calculated from [2]

R06=9000ln(10)κ2ηD128π5Nn4λminλmaxfD(λ)εA(λ)λ4dλ,
where κ2 is the relative orientation of the donor and acceptor transition dipole moments, assumed to be κ2=23, N is Avogadro’s number, n is the solvent refractive index, which was assumed to be n=1.33 (the refractive index of water, because it is the main solvent), ηD is the donor quantum yield in the absence of the acceptor, which for BODIPY-FL is ηD=0.9 [34], fD(λ) is the normalized fluorescence spectrum of the donor, and εA(λ) is the molar absorption coefficient of the acceptor at wavelength λ. The integral in (17) was calculated between λmin=475nm and λmax=649nm, giving R0=5.8nm.

5C. Calibration for Experimental Data

Measurement calibration is necessary to account for source and detector coupling to the scattering medium. For accurate image reconstruction, a two step calibration procedure was followed. In the first step, a baseline measurement, ykmbase, for the kth source and mth detector, was made at the excitation wavelength (488nm) with only Intralipid in the plastic vial and no filter in front of the CCD camera lens. Using the optical properties, i.e., D and μa for 0.4% Intralipid at 488nm, described above, synthetic data ykmsyn was computed using a numerical solution to (3). Uncalibrated fluorescence data, ykmuncal, obtained from the experiment described above, was calibrated using

ykmcal=ykmuncal.ykmsynykmbase,
where ykmcal is the calibrated data for the kth source and mth detector. Thus, for our experiment, Q=1, yF1=ycal and ω1=0, because unmodulated light is used. As Eq. (18) does not fully characterize the source and wavelength-dependent detector arrangement, we found it necessary to explicitly introduce source and detector coupling coefficients into the inversion procedure [24].

5D. Experiment Results

Figure 6 shows three slices along the xy plane of the true discretized location and shape of the plastic vial inside the scattering medium. Also shown is an isosurface image of the plastic vial plotted at a DA distance of rF=3.6nm, which was obtained from an another experiment (see Subsection 5E, performed to verify that we accurately estimated the DA distance in the presence of scatter), in which we imaged the plastic vials in the absence of the scattering medium. Figure 7 shows the reconstructed image for the DA distance. As can be seen from the images, the reconstructed object is very close to Fig. 6 in terms of both object shape, object location and the reconstructed DA distance. Also shown in Fig. 7 is an isosurface plot for the reconstructed DA distance at rF=3.4nm.

5E. Experiment to Verify Reconstruction of rF

To verify whether the DA distance had been estimated accurately, we performed another experiment in which the plastic vials containing the 250nM solution of donor and Intralipid and the 3μM solution of the FRET chemical and Intralipid were imaged without the scattering medium with the same experimental setup shown in Fig. 4. Figure 8 shows the images for each vial. The detected power is proportional to the intensity in counts and is

PI.dS,
where I is the intensity in counts and dS represents an area element. Using (19), the DA distance can be estimated from
PDPDA=ηDCD(ηDCDA)[1+(R0rF)6],
where PD and PDA are the detected powers for the donor-only and FRET samples, respectively, CD is the concentration of the donor, and CDA is the concentration of the FRET chemical. This gives
rF=R0(CDAPDPDACD1)16,
which was calculated using the area highlighted in Fig. 8a, 8b. For the images, PDγ7.3×108  Watt and PDAγ4.2×108W, where γ is a proportionality constant (Watts/counts) which cancels out when PD and PDA are substituted in (20), and R0=5.8nm, yielding rF=3.6nm. This is very close to the reconstructed DA distance in the presence of scatter. In the calculation for rF, we ignored the small contribution of rhodamine emission at 520nm, when excited at 488nm.

6. CONCLUSION

We have shown that it is possible to image the microscopic DA distance or distance distribution of a FRET chemical inside a scattering medium by incorporating FRET kinetics in an ODT framework. Through simulations, we showed the efficacy of our approach to distinguish macroscopic regions having differing ensemble parameters. This emulates a situation where multiple tumors or organs are imaged.

The macroscopic FRET regions used in the simulations and experiment are also representative of Alzheimer’s, where the onset of senility occurs with a total senile plaque density of approximately 80mm2 [35] (716mm3). Assuming a plaque dimension of 50μm, this gives a minimum plaque volume of 3.5×103cm3 in a 1cm3 tissue volume, commensurate with the FRET volume length scale we considered. The simple tissue model experiment performed demonstrates that the microscopic DA distance can be determined in a heavily scattering environment, leading to the conclusion that it should be possible to apply our approach in vivo. The resulting three-dimensional spatial maps of the DA distance could be used for quantitative deep tissue in vivo FRET imaging, thus expanding the study of disease through FRET. For example, using fluorescence lifetime imaging microscopy (FLIM) together with FRET, Bacskai et al. [36] showed a distribution of lifetimes within individual senile plaques (characteristic features of Alzheimer’s formed by large deposits of amyloid—β peptides), which implied that amyloid—β has different conformations within the plaque. Although the size of an individual plaque (order of 10μm) is smaller than the resolution of ODT, because senile plaques exist as an aggregate, with appropriate DA labeling, an in vivo image of a distance distribution representative of amyloid—β could be extracted, as we have described. Deep tissue FRET imaging should also facilitate the study of cancers related to protein misfolding [8, 10, 11, 37]. Our results suggest that, with similar SNRs (integration times), we should be able to detect smaller FRET regions or lower concentrations. A more sophisticated experiment involving small animals like mice would further establish the efficacy of our approach.

ACKNOWLEDGMENTS

We acknowledge funding from the National Science Foundation under award CCR-0431024.

 

Fig. 1 (a) The rF=4.2nm isosurface plot of the true image for a fixed linker. Also shown are the locations of the sources (top circles) and detectors (bottom circles) used to produce the simulated data. (b) Discretized spherical geometry with radius 2.5mm (with 33×33×33 image resolution). For a rigid linker with rF=4.2nm and η=0.025: (c) reconstructed rF, (d) reconstructed η. For a flexible linker with a=5.42×1017m2 and b=5.45nm, with the geometry in (b): (e) reconstructed a, (f) reconstructed b. The average detector SNR was 30dB.

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Fig. 2 Piecewise polynomial and exponential fitting of the real and imaginary parts of ζ(rF) when R0=4.7nm and τD=3.8ns: (a) real part, and (b) imaginary part, for a modulation frequency of 80MHz.

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Fig. 3 A phantom having two spheres, each of radius 2mm, containing DA molecules connected by a flexible linker. (a) True image of a. For the top right sphere, a=3.5×1017m2, and for bottom left sphere, a=7.45×1017m2. (b) Reconstructed a. (c) True image of b. For top right sphere, b=5.95nm, and for bottom left sphere, b=3.65nm. (d) Reconstructed b. (e) True image of η. For top right and bottom left spheres, η=0.025. (f) Reconstructed η. The average detector SNR=30dB.

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Fig. 4 (a) and (b) Experimental setup: a 3mW 488nm argon-ion laser (Uniphase), an xy scanning mirror system, a Plexiglas box of size 8.8cm (L)×8.8cm (H)×3.4cm (W), a cylindrical plastic vial (length 3cm, inner-diameter 0.65cm, outer-diameter 0.75cm), to hold the FRET chemical, suspended from the lid of the Plexiglas box using an acrylic rod 1mm in diameter and 1.5cm in length, a 520nm narrow bandpass filter (Edmund Optics) with FWHM 10nm [31], and a 105mm, f/2.8 lens (AF micro Nikkor, Nikon) to focus the 4.8cm×4.8cm image of the scattering medium on a 512×512  pixel Peltier cooled CCD Camera (PI-MAX, Roper Scientific).

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Fig. 5 Slices along the xy plane of the reconstructed image of η for the plastic vial containing the donor mixed with Intralipid. (a) Slice at z=0.2125cm. (b) Slice at z=0.0cm. (c) Slice at z=0.2125cm. Reconstructed η0.0013cm1.

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Fig. 6 Slices along the xy plane of the true rF, true location and shape of the plastic vial suspended inside the Intralipid scattering medium (see Fig. 3 for xyz axes orientation). (a) Slice at z=0.2125cm. (b) Slice at z=0.0cm. (c) Slice at z=0.2125cm. (d) Isosurface plot of the DA distance at rF=3.6nm, which is the distance estimated without the scattering medium (see Section 5E).

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Fig. 7 Slices along the xy plane of the reconstructed image of the DA distance. Expected rF=3.6nm. (a) Slice at z=0.2125cm. (b) Slice at z=0.0cm. (c) Slice at z=0.2125cm. (d) Isosurface plot of the reconstructed DA distance at rF=3.4nm.

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Fig. 8 Acquired image of plastic vials without the scattering medium. (a) Image of vial containing donor Bodipy-FL mixed with Intralipid at concentration CD=250nM. The total detected power is assumed to be the integral of the intensity in the dotted rectangle, which gives PD=γ7.3×108W. (b) Image of the vial containing the FRET chemical mixed with Intralipid at concentration CDA=3μM. The detected power, using the intensity in the dotted rectangle, is PDA=γ4.2×108W. Using (20) and R0=5.8nm gives rF3.6nm

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1. T. Förster, “Zwischenmolekulare energiewanderung und fluoreszenze,” Ann. Phys. 2, 55 (1948). [CrossRef]  

2. J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 2nd ed. (Kluwer Academic, 1999).

3. R. D. Mitra, C. M. Silva, and D. C. Youvan, “Fluorescence resonance energy transfer between blue-emitting and red-shifted excitation derivatives of green fluorescent protein,” Gene 173, 13–17 (1996). [CrossRef]   [PubMed]  

4. Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380–383 (1998). [CrossRef]   [PubMed]  

5. M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287–294 (2002). [CrossRef]   [PubMed]  

6. A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881–887 (1997).

7. N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547–552 (1998). [CrossRef]   [PubMed]  

8. E. Haas, “The study of protein folding and dynamics by determination of intramolecular distance distributions and their fluctuations using ensemble and single-molecule FRET measurement,” Chem. Phys. 6, 858–870 (2005). [CrossRef]  

9. K. Truong and M. Ikura, “The use of FRET imaging technology to detect protein-protein interactions and protein conformational changes in vivo,” Curr. Op. Struct. Biol. 11, 573–578 (2001). [CrossRef]  

10. C. Dobson, “The structural basis of protein folding and its links with human disease,” Phil. Trans. R. Soc. Lond. B 356, 133–145 (2001). [CrossRef]  

11. A. Bullock and A. Fersht, “Rescuing the function of mutant p53,” Nat. Rev. Cancer 1, 1 (2001). [CrossRef]  

12. J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347–356 (2003). [CrossRef]   [PubMed]  

13. R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283–291 (2006). [CrossRef]   [PubMed]  

14. D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215–222 (2005). [CrossRef]   [PubMed]  

15. A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, and R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95–97 (2002). [CrossRef]  

16. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1–R43 (2005). [CrossRef]   [PubMed]  

17. A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081–3094 (2003). [CrossRef]   [PubMed]  

18. S. Tyagi and F. Kramer, “Molecular beacons: Probes that fluoresce upon hybridization,” Nat. Biotechnol. 14, 303–308 (1996). [CrossRef]   [PubMed]  

19. V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757–760 (2002). [CrossRef]   [PubMed]  

20. S. Bernacchi and Y. Mely, “Exciton interaction in molecular beacons: a sensitive sensor for short range modifications of the nucleic acid structure,” Nucleic Acids Res. 29, e62 (2001). [CrossRef]   [PubMed]  

21. S. Marras, F. Kramer, and S. Tyagi, “Efficiencies of fluorescence resonance energy transfer and contact-mediated quenching in oligonucleotide probes,” Nucleic Acids Res. 30, e122 (2002). [CrossRef]   [PubMed]  

22. A. Milstein, J. Stott, S. Oh, D. Boas, R. Millane, C. Bouman, and K. Webb, “Fluorescence optical diffusion tomography using multiple-frequency data,” J. Opt. Soc. Am. A 21, 1035–1049 (2004). [CrossRef]  

23. J. C. Ye, K. J. Webb, C. A. Bouman, and R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400–2412 (1999). [CrossRef]  

24. S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, and K. J. Webb, “Source-detector calibration in three-dimensional Bayesian optical diffusion tomography,” J. Opt. Soc. Am. A 19, 1983–1993 (2002). [CrossRef]  

25. J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909–922 (2001). [CrossRef]   [CrossRef]  

26. P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151–156 (2000). [CrossRef]   [PubMed]  

27. www.microscopyu.com/tutorials/java/fluorescence/fpfret/index.html.

28. J. C. Adams, “Mudpack: Multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113–146 (1989). [CrossRef]  

29. J. C. Adams, Multigrid Software for Elliptic Partial Differential Equations (National Center for Atmospheric Research, Boulder, Colorado, 1991).

30. J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872–13877 (2006). [CrossRef]   [PubMed]  

31. http://www.edmundoptics.com/onlinecatalog/displayproduct.cfm?productid=1903.

32. G. Hale and M. Querry, “Optical constants of water in the 200nmto200μm wavelength region,” Appl. Opt. 12, 555–563 (1973). [CrossRef]   [PubMed]  

33. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907–5925 (2008). [CrossRef]   [PubMed]  

34. http://probes.invitrogen.com/handbook/sections/0104.html.

35. L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326–355 (1998). [CrossRef]   [PubMed]  

36. B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368–375 (2003). [CrossRef]   [PubMed]  

37. C. Dobson, “Protein folding and misfolding,” Nature 426, 884–890 (2003). [CrossRef]   [PubMed]  

References

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  • |

  1. T. Förster, “Zwischenmolekulare energiewanderung und fluoreszenze,” Ann. Phys. 2, 55 (1948).
    [CrossRef]
  2. J. R. Lakowicz, Principles of Fluorescence Spectroscopy, 2nd ed. (Kluwer Academic, 1999).
  3. R. D. Mitra, C. M. Silva, and D. C. Youvan, “Fluorescence resonance energy transfer between blue-emitting and red-shifted excitation derivatives of green fluorescent protein,” Gene 173, 13-17 (1996).
    [CrossRef] [PubMed]
  4. Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380-383 (1998).
    [CrossRef] [PubMed]
  5. M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
    [CrossRef] [PubMed]
  6. A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).
  7. N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
    [CrossRef] [PubMed]
  8. E. Haas, “The study of protein folding and dynamics by determination of intramolecular distance distributions and their fluctuations using ensemble and single-molecule FRET measurement,” Chem. Phys. 6, 858-870 (2005).
    [CrossRef]
  9. K. Truong and M. Ikura, “The use of FRET imaging technology to detect protein-protein interactions and protein conformational changes in vivo,” Curr. Op. Struct. Biol. 11, 573-578 (2001).
    [CrossRef]
  10. C. Dobson, “The structural basis of protein folding and its links with human disease,” Phil. Trans. R. Soc. Lond. B 356, 133-145 (2001).
    [CrossRef]
  11. A. Bullock and A. Fersht, “Rescuing the function of mutant p53,” Nat. Rev. Cancer 1, 1 (2001).
    [CrossRef]
  12. J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
    [CrossRef] [PubMed]
  13. R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
    [CrossRef] [PubMed]
  14. D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
    [CrossRef] [PubMed]
  15. A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, and R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95-97 (2002).
    [CrossRef]
  16. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
    [CrossRef] [PubMed]
  17. A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081-3094 (2003).
    [CrossRef] [PubMed]
  18. S. Tyagi and F. Kramer, “Molecular beacons: Probes that fluoresce upon hybridization,” Nat. Biotechnol. 14, 303-308 (1996).
    [CrossRef] [PubMed]
  19. V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757-760 (2002).
    [CrossRef] [PubMed]
  20. S. Bernacchi and Y. Mely, “Exciton interaction in molecular beacons: a sensitive sensor for short range modifications of the nucleic acid structure,” Nucleic Acids Res. 29, e62 (2001).
    [CrossRef] [PubMed]
  21. S. Marras, F. Kramer, and S. Tyagi, “Efficiencies of fluorescence resonance energy transfer and contact-mediated quenching in oligonucleotide probes,” Nucleic Acids Res. 30, e122 (2002).
    [CrossRef] [PubMed]
  22. A. Milstein, J. Stott, S. Oh, D. Boas, R. Millane, C. Bouman, and K. Webb, “Fluorescence optical diffusion tomography using multiple-frequency data,” J. Opt. Soc. Am. A 21, 1035-1049 (2004).
    [CrossRef]
  23. J. C. Ye, K. J. Webb, C. A. Bouman, and R. P. Millane, “Optical diffusion tomography using iterative coordinate descent optimization in a Bayesian framework,” J. Opt. Soc. Am. A 16, 2400-2412 (1999).
    [CrossRef]
  24. S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, and K. J. Webb, “Source-detector calibration in three-dimensional Bayesian optical diffusion tomography,” J. Opt. Soc. Am. A 19, 1983-1993 (2002).
    [CrossRef]
  25. J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909-922 (2001).
    [CrossRef]
  26. P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
    [CrossRef] [PubMed]
  27. www.microscopyu.com/tutorials/java/fluorescence/fpfret/index.html.
  28. J. C. Adams, “Mudpack: Multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113-146 (1989).
    [CrossRef]
  29. J. C. Adams, Multigrid Software for Elliptic Partial Differential Equations (National Center for Atmospheric Research, Boulder, Colorado, 1991).
  30. J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
    [CrossRef] [PubMed]
  31. http://www.edmundoptics.com/onlinecatalog/displayproduct.cfm?productid=1903.
  32. G. Hale and M. Querry, “Optical constants of water in the 200 nmto200 μm wavelength region,” Appl. Opt. 12, 555-563 (1973).
    [CrossRef] [PubMed]
  33. R. Michels, F. Foschum, and A. Kienle, “Optical properties of fat emulsions,” Opt. Express 16, 5907-5925 (2008).
    [CrossRef] [PubMed]
  34. http://probes.invitrogen.com/handbook/sections/0104.html.
  35. L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
    [CrossRef] [PubMed]
  36. B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368-375 (2003).
    [CrossRef] [PubMed]
  37. C. Dobson, “Protein folding and misfolding,” Nature 426, 884-890 (2003).
    [CrossRef] [PubMed]

2008

2006

J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
[CrossRef] [PubMed]

R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
[CrossRef] [PubMed]

2005

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

E. Haas, “The study of protein folding and dynamics by determination of intramolecular distance distributions and their fluctuations using ensemble and single-molecule FRET measurement,” Chem. Phys. 6, 858-870 (2005).
[CrossRef]

2004

2003

J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
[CrossRef] [PubMed]

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

C. Dobson, “Protein folding and misfolding,” Nature 426, 884-890 (2003).
[CrossRef] [PubMed]

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, “Fluorescence optical diffusion tomography,” Appl. Opt. 42, 3081-3094 (2003).
[CrossRef] [PubMed]

2002

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757-760 (2002).
[CrossRef] [PubMed]

A. B. Milstein, S. Oh, J. S. Reynolds, K. J. Webb, C. A. Bouman, and R. P. Millane, “Three-dimensional Bayesian optical diffusion tomography with experimental data,” Opt. Lett. 27, 95-97 (2002).
[CrossRef]

M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
[CrossRef] [PubMed]

S. Marras, F. Kramer, and S. Tyagi, “Efficiencies of fluorescence resonance energy transfer and contact-mediated quenching in oligonucleotide probes,” Nucleic Acids Res. 30, e122 (2002).
[CrossRef] [PubMed]

S. Oh, A. B. Milstein, R. P. Millane, C. A. Bouman, and K. J. Webb, “Source-detector calibration in three-dimensional Bayesian optical diffusion tomography,” J. Opt. Soc. Am. A 19, 1983-1993 (2002).
[CrossRef]

2001

J. C. Ye, C. A. Bouman, K. J. Webb, and R. P. Millane, “Nonlinear multigrid algorithms for Bayesian optical diffusion tomography,” IEEE Trans. Image Process. 10, 909-922 (2001).
[CrossRef]

K. Truong and M. Ikura, “The use of FRET imaging technology to detect protein-protein interactions and protein conformational changes in vivo,” Curr. Op. Struct. Biol. 11, 573-578 (2001).
[CrossRef]

C. Dobson, “The structural basis of protein folding and its links with human disease,” Phil. Trans. R. Soc. Lond. B 356, 133-145 (2001).
[CrossRef]

A. Bullock and A. Fersht, “Rescuing the function of mutant p53,” Nat. Rev. Cancer 1, 1 (2001).
[CrossRef]

S. Bernacchi and Y. Mely, “Exciton interaction in molecular beacons: a sensitive sensor for short range modifications of the nucleic acid structure,” Nucleic Acids Res. 29, e62 (2001).
[CrossRef] [PubMed]

2000

P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
[CrossRef] [PubMed]

1999

1998

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Y. Suzuki, T. Yasunaga, R. Ohkura, T. Wakabayashi, and K. Sutoh, “Swing of the lever arm of a myosin motor at the isomerization and phosphate-release steps,” Nature 396, 380-383 (1998).
[CrossRef] [PubMed]

N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
[CrossRef] [PubMed]

1997

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

1996

R. D. Mitra, C. M. Silva, and D. C. Youvan, “Fluorescence resonance energy transfer between blue-emitting and red-shifted excitation derivatives of green fluorescent protein,” Gene 173, 13-17 (1996).
[CrossRef] [PubMed]

S. Tyagi and F. Kramer, “Molecular beacons: Probes that fluoresce upon hybridization,” Nat. Biotechnol. 14, 303-308 (1996).
[CrossRef] [PubMed]

1989

J. C. Adams, “Mudpack: Multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113-146 (1989).
[CrossRef]

1973

1948

T. Förster, “Zwischenmolekulare energiewanderung und fluoreszenze,” Ann. Phys. 2, 55 (1948).
[CrossRef]

Adams, J.

A. Miyawaki, J. Llopis, R. Heim, J. McCaffery, J. Adams, M. Ikura, and R. Tsien, “Fluorescent indicators for, Ca2+ based on green fluorescent proteins and calmodulin,” Nature 388, 881-887 (1997).

Adams, J. C.

J. C. Adams, “Mudpack: Multigrid portable fortran software for the efficient solution of linear elliptic partial differential equations,” Appl. Math. Comput. 34, 113-146 (1989).
[CrossRef]

J. C. Adams, Multigrid Software for Elliptic Partial Differential Equations (National Center for Atmospheric Research, Boulder, Colorado, 1991).

Aelst, L.

R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
[CrossRef] [PubMed]

Allen, R.

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Asano, T.

M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
[CrossRef] [PubMed]

Bacskai, B.

B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368-375 (2003).
[CrossRef] [PubMed]

Bartoli, M.

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

Baty, J.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Berg, L.

L. Berg, D. W. McKeel, J. P. Miller, M. Storandt, E. H. Rubin, J. C. Morris, J. Baty, M. Coats, J. Norton, A. M. Goate, J. L. Price, M. Gearing, S. S. Mirra, and A. M. Saunders, “Clinicopatholigic studies in cognitively healthy aging and Alzheimer disease,” Arch. Neurol. 55, 326-355 (1998).
[CrossRef] [PubMed]

Bernacchi, S.

S. Bernacchi and Y. Mely, “Exciton interaction in molecular beacons: a sensitive sensor for short range modifications of the nucleic acid structure,” Nucleic Acids Res. 29, e62 (2001).
[CrossRef] [PubMed]

Berry, G.

N. Mahajan, K. Linder, G. Berry, G. Gordon, R. Heim, and B. Herman, “Bcl-2 and bax interactions in mitochondria probed with green fluorescence protein and fluorescence resonance energy transfer,” Nat. Biotechnol. 16, 547-552 (1998).
[CrossRef] [PubMed]

Boas, D.

Boas, D. A.

Bouman, C.

Bouman, C. A.

Bourg, N.

D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
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P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
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D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
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P. Schwille, S. Kummer, A. Heikal, W. Moerner, and W. Webb, “Fluorescence correlation spectroscopy reveals fast optical excitation-driven intramolecular dynamics of yellow fluorescent proteins,” Proc. Natl. Acad. Sci. U.S.A. 97, 151-156 (2000).
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D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
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R. D. Mitra, C. M. Silva, and D. C. Youvan, “Fluorescence resonance energy transfer between blue-emitting and red-shifted excitation derivatives of green fluorescent protein,” Gene 173, 13-17 (1996).
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B. Bacskai, J. Skoch, G. Hickey, R. Allen, and B. Hyman, “Fluorescence resonance energy transfer determinations using multiphoton fluorescence lifetime imaging microscopy to characterize amyloid-beta plaques,” J. Biomed. Opt. 8, 368-375 (2003).
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R. Yasuda, C. Harvey, H. Zhong, A. Sobczyk, L. Aelst, and K. Svoboda, “Supersenstive ras activation in dendrites and spines revealed by two-photon fluorescence lifetime imaging,” Nature Neurosci. 9, 283-291 (2006).
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D. Stockholm, M. Bartoli, G. Sillon, N. Bourg, J. Davoust, and I. Richard, “Imaging calpain protease activity by multiphoton FRET in living mice,” J. Mol. Biol. 346, 215-222 (2005).
[CrossRef] [PubMed]

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J. Mills, J. Stone, D. Rubin, D. Melon, D. Okonkwo, and A. P. G. Helm, “Illuminating protein interactions in tissue using confocal and two-photon excitation fluorescent resonance energy transfer microscopy,” J. Biomed. Opt. 8, 347-356 (2003).
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V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757-760 (2002).
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S. Marras, F. Kramer, and S. Tyagi, “Efficiencies of fluorescence resonance energy transfer and contact-mediated quenching in oligonucleotide probes,” Nucleic Acids Res. 30, e122 (2002).
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M. Sato, T. Ozawa, K. Inukai, T. Asano, and Y. Umezawa, “Fluorescent indicators for imaging protein phosphorylation in single living cells,” Nat. Biotechnol. 20, 287-294 (2002).
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J. Yang, H. Chen, I. Vlahov, J. Cheng, and P. Low, “Evaluation of disulfide reduction during receptor-mediated endocytosis by using FRET imaging,” Proc. Natl. Acad. Sci. U.S.A. 103, 13872-13877 (2006).
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Chem. Phys.

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[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

(a) The r F = 4.2 nm isosurface plot of the true image for a fixed linker. Also shown are the locations of the sources (top circles) and detectors (bottom circles) used to produce the simulated data. (b) Discretized spherical geometry with radius 2.5 mm (with 33 × 33 × 33 image resolution). For a rigid linker with r F = 4.2 nm and η = 0.025 : (c) reconstructed r F , (d) reconstructed η. For a flexible linker with a = 5.42 × 10 17 m 2 and b = 5.45 nm , with the geometry in (b): (e) reconstructed a, (f) reconstructed b. The average detector SNR was 30 dB .

Fig. 2
Fig. 2

Piecewise polynomial and exponential fitting of the real and imaginary parts of ζ ( r F ) when R 0 = 4.7 nm and τ D = 3.8 ns : (a) real part, and (b) imaginary part, for a modulation frequency of 80 MHz .

Fig. 3
Fig. 3

A phantom having two spheres, each of radius 2 mm , containing DA molecules connected by a flexible linker. (a) True image of a. For the top right sphere, a = 3.5 × 10 17 m 2 , and for bottom left sphere, a = 7.45 × 10 17 m 2 . (b) Reconstructed a. (c) True image of b. For top right sphere, b = 5.95 nm , and for bottom left sphere, b = 3.65 nm . (d) Reconstructed b. (e) True image of η. For top right and bottom left spheres, η = 0.025 . (f) Reconstructed η. The average detector SNR = 30 dB .

Fig. 4
Fig. 4

(a) and (b) Experimental setup: a 3 mW 488 nm argon-ion laser (Uniphase), an x y scanning mirror system, a Plexiglas box of size 8.8 cm ( L ) × 8.8 cm ( H ) × 3.4 cm (W), a cylindrical plastic vial (length 3 cm , inner-diameter 0.65 cm , outer-diameter 0.75 cm ), to hold the FRET chemical, suspended from the lid of the Plexiglas box using an acrylic rod 1 mm in diameter and 1.5 cm in length, a 520 nm narrow bandpass filter (Edmund Optics) with FWHM 10 nm [31], and a 105 mm , f/2.8 lens (AF micro Nikkor, Nikon) to focus the 4.8 cm × 4.8 cm image of the scattering medium on a 512 × 512   pixel Peltier cooled CCD Camera (PI-MAX, Roper Scientific).

Fig. 5
Fig. 5

Slices along the x y plane of the reconstructed image of η for the plastic vial containing the donor mixed with Intralipid. (a) Slice at z = 0.2125 cm . (b) Slice at z = 0.0 cm . (c) Slice at z = 0.2125 cm . Reconstructed η 0.0013 cm 1 .

Fig. 6
Fig. 6

Slices along the x y plane of the true r F , true location and shape of the plastic vial suspended inside the Intralipid scattering medium (see Fig. 3 for xyz axes orientation). (a) Slice at z = 0.2125 cm . (b) Slice at z = 0.0 cm . (c) Slice at z = 0.2125 cm . (d) Isosurface plot of the DA distance at r F = 3.6 nm , which is the distance estimated without the scattering medium (see Section 5E).

Fig. 7
Fig. 7

Slices along the x y plane of the reconstructed image of the DA distance. Expected r F = 3.6 nm . (a) Slice at z = 0.2125 cm . (b) Slice at z = 0.0 cm . (c) Slice at z = 0.2125 cm . (d) Isosurface plot of the reconstructed DA distance at r F = 3.4 nm .

Fig. 8
Fig. 8

Acquired image of plastic vials without the scattering medium. (a) Image of vial containing donor Bodipy-FL mixed with Intralipid at concentration C D = 250 nM . The total detected power is assumed to be the integral of the intensity in the dotted rectangle, which gives P D = γ 7.3 × 10 8 W . (b) Image of the vial containing the FRET chemical mixed with Intralipid at concentration C DA = 3 μ M . The detected power, using the intensity in the dotted rectangle, is P DA = γ 4.2 × 10 8 W . Using (20) and R 0 = 5.8 nm gives r F 3.6 nm

Equations (21)

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1 τ DA = 1 τ D [ 1 + ( R 0 r F ) 6 ] = k r + k n r + k F
η DA = k r k r + k n r + k F .
[ D x ( r ) φ x ( r , ω ) ] [ μ a x ( r ) i ω c ] φ x ( r , ω ) = S x ( r ; ω )
[ D m ( r ) φ m ( r , ω ) ] [ μ a m ( r ) i ω c ] φ m ( r , ω ) = φ x ( r , ω ) S f ( r ; ω ) ,
S f ( r ; ω ) = r min r max η DA μ a D ( r ) 1 i ω τ DA ( r ) p ( r F ( r ) ) d r F ( r ) = r min r max η ( r ) ζ ( r F ( r ) ) p ( r F ( r ) ) d r F ( r ) ,
φ ( r s k , r d m ; ω ) = g m ( r , r d m ; ω ) S ( r ; ω ) g x ( r s k , r ; ω ) d 3 r ,
x F = [ η ( r 1 ) η ( r N ) , a ( r 1 ) a ( r N ) , b ( r 1 ) b ( r N ) ] T
x F = [ η ( r 1 ) η ( r N ) , r F ( r 1 ) r F ( r N ) ] T
{ x ̂ F } MAP = arg max x F 0 , s , d { q = 1 Q p ( y F q | x F , s q , d q ) p ( x F ) } ,
p ( y F q | x F , s q , d q ) = 1 ( π α F ) P | Λ F q | 1 exp [ y F q f F q ( x F , s q , d q ) Λ F q 2 α F ] ,
f F q ( x F , s q , d q ) = [ s 1 q d 1 q φ ( r s 1 , r d 1 ; ω , x F ) s 1 q d M q φ ( r s 1 , r d M ; ω , x F ) , s 2 q d 1 q φ ( r s 2 , r d 1 ; ω , x F ) s K q d M q φ ( r s K , r d M ; ω , x F ) ] T ,
p ( x k ) = 1 σ k N z ( ρ k ) exp ( 1 ρ k σ ρ k { i , j } N k b i j | x k i x k j | ρ k ) ,
φ ( r s k , r d m ; ω , x F ) = j = 1 N V [ g m ( r j , r d m ; ω ) S ( r j ; ω , x F ) g x ( r s k , r j ; ω ) ] ,
f F q ( x F , s q , d q ) = G ω q S ω q ( x F ) ,
( { x ̂ F } MAP , α ̂ F , s ̂ , d ̂ ) = arg min x F 0 , s , d min α F { 1 α F q = 1 Q y F q G ω q S ω q ( x F ) Λ F q 2 + P log α F log [ p ( x F ) ] } .
x ̂ k i arg min x k i 0 { 1 α ̂ F q = 1 Q y F q [ G ω q ] * ( i ) S ( r i ; ω q , x F ) Λ F q 2 + 1 ρ k σ k ρ k j N k i b i j | x k i x ̂ k j | ρ k } ,
R 0 6 = 9000 ln ( 10 ) κ 2 η D 128 π 5 N n 4 λ min λ max f D ( λ ) ε A ( λ ) λ 4 d λ ,
y km cal = y km uncal . y km syn y km base ,
P I . d S ,
P D P DA = η D C D ( η D C DA ) [ 1 + ( R 0 r F ) 6 ] ,
r F = R 0 ( C DA P D P DA C D 1 ) 1 6 ,

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