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 $l_s={\left(3D\right)}^{-1}$, 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 $k_r$ and non-radiatively, by heat dissipation and energy transfer to an acceptor molecule, with rates $k_{nr}$ and $k_F$, respectively [1, 2]. If ${\tau }_D$ is the donor lifetime and ${\eta }_D$ the donor quantum yield in the absence of an acceptor, then $k_F=\tau _D{}^{-1}{(R_0∕r_F)}^6$, where $r_F$ is the distance between the donor and the acceptor and $R_0$ is the DA distance where the energy transfer efficiency is 50%, $\tau _D{}^{-1}=k_r+k_{nr}$, and ${\eta }_D=k_r∕(k_r+k_{nr})$ [2]. In the presence of an acceptor, energy transfer results in a decreased donor lifetime, ${\tau }_{\mathit{DA}}$, given by

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]

Assuming that ${\mu }_{a_x}$, $D_x$, ${\mu }_{a_m}$, $D_m$, and ${\tau }_D$ 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 $x_F={[\eta \left(r\right),a\left(r\right),b\left(r\right)]}^T$, or for the rigid linker as $x_F={[\eta \left(r\right),r_F\left(r\right)]}^T$, 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 $x_F$ 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 $x_F$.

3. RECONSTRUCTION

To reconstruct $x_F$, the simulation domain is discretized into N nodes on a uniform 3-D grid, which transformed $x_F$ into the $3N\times 1$ vector

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

Numerically, $\phi (r_{s_k},r_{d_m};\omega ,x_F)$ is calculated by discretizing (6), with $S(\cdot )$ being either $S_f$ or $S_g$, as

4. SIMULATION

The model problem we consider is the cubical tissue phantom of Fig. 1a with side $2.5\mathrm{cm}$, discretized on a $33\times 33\times 33$ uniform grid. A planar arrangement of 14 sources, with ten equally spaced modulation frequencies between 40 and $220\mathrm{MHz}$ ($20\mathrm{MHz}$ 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 ${\tau }_D=3.8\mathrm{ns}$ [26], and we use $R_0=4.7\mathrm{nm}$ [27]. We assume $D_x\approx D_m=0.027\mathrm{cm}$ and ${\mu }_{ax}\approx {\mu }_{am}=0.047{\mathrm{cm}}^{-1}$. For both rigid and flexible linker, we choose $\eta =0.025{\mathrm{cm}}^{-1}$, calculated from the quantum yield and absorption coefficient for EYFP, assuming a concentration of $250\mathrm{nM}$ for the FRET pair. The FRET pairs are contained within a centrally located sphere of radius $2.5\mathrm{mm}$. 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 $30\mathrm{dB}$. We used MUDPACK to generate the synthetic data [28, 29].

4A. Computation of Equation (5)

Determining ${\left\{{\widehat{x}}_F\right\}}_{\mathit{MAP}}$ involved multiple computations of the integral in (5). For given $R_0$, ω and ${\tau }_D$, piecewise fitting cubic polynomials and exponential functions to the real and imaginary parts of $\zeta \left(r_F\right)$ 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 $r_{\mathit{min}}=5\mathrm{\AA }$ and $r_{\mathit{max}}=3.5R_0$. 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 $r_F$ 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 $2\mathrm{mm}$, 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 $488\mathrm{nm}$, emission $520\mathrm{nm}$), 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 $3\mathrm{mW}$ $488\mathrm{nm}$ argon-ion laser (Uniphase), an $x-y$ scanning mirror system, a Plexiglas box of size $8.8\mathrm{cm}\left(\mathrm{L}\right)\times 8.8\mathrm{cm}\left(\mathrm{H}\right)\times 3.4\mathrm{cm}\left(\mathrm{W}\right)$, a cylindrical plastic vial (length $3\mathrm{cm}$, inner-diameter $0.65\mathrm{cm}$, outer-diameter $0.75\mathrm{cm}$) to hold the FRET chemical, suspended from the lid of the Plexiglas box using an acrylic rod $1\mathrm{mm}$ in diameter and $1.5\mathrm{cm}$ in length, a $520\mathrm{nm}$ bandpass filter (Edmund Optics, $50.8\mathrm{mm}$ square) with FWHM $10\mathrm{nm}$ and peak transmission 45% [31], and a $105\mathrm{mm}$, f/2.8 lens (AF micro Nikkor, Nikon) to focus the $4.8\mathrm{cm}\times 4.8\mathrm{cm}$ image of the scattering medium on a $512\times 512\text{\hspace{0.17em} pixel}$ 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 $10\mathrm{ml}$ of 20% Intralipid in $500\mathrm{ml}$ of deionized water. The plastic vial contained $1\mathrm{ml}$ of $3\mu \mathrm{M}$ solution of the FRET chemical in Intralipid, prepared by mixing $100\mu \mathrm{l}$ of a $0.18\mathrm{mM}$ stock solution of the FRET chemical with $5.9\mathrm{ml}$ of 0.4% (w/v) Intralipid. The chemical was dissolved in Intralipid to maintain tissue-like scattering properties.

Using the $x-y$ 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 $150\mathrm{msec}$ each were taken with the camera and added together. For reconstruction, the $512\times 512$ image was down-sampled to $16\times 16$, 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 $\eta =15.6\times {10}^{-3}{\mathrm{cm}}^{-1}$. For the 0.4% Intralipid solution, we assume that all the absorption at ${\lambda }_x=488\mathrm{nm}$ is due to water, thereby giving ${\mu }_{a_x}\approx 2.5\times {10}^{-4}{\mathrm{cm}}^{-1}$ and ${\mu }_{a_m}\approx 3.2\times {10}^{-4}{\mathrm{cm}}^{-1}$ [32]. Prior experiments with Intralipid lead to $D_x\approx 0.051\mathrm{cm}$ and $D_m\approx 0.056\mathrm{cm}$ [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 $33\times 33\times 17$ uniform grid. In the calculation for $r_F$, we ignored the small contribution of rhodamine emission at $520\mathrm{nm}$, when excited at $488\mathrm{nm}$, as we found that compared to the emission of Bodipy-FL at $520\mathrm{nm}$, rhodamine’s signal was negligible. Also, due to the scattering medium, less than one $\mu \mathrm{W}$ 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 $100\mu \mathrm{l}$ of a $10\mu \mathrm{M}$ stock solution of the donor BODIPY-FL with $3.9\mathrm{ml}$ of 0.4% (w/v) Intralipid, resulting in a concentration of $C_D=250\mathrm{nM}$. 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 $150\mathrm{msec}$ each were acquired with the camera and added together. For reconstruction, the $512\times 512$ image was down-sampled to a $16\times 16$, 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 $33\times 33\times 17$ 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 ${\mu }_{a_x}\approx 2.5\times {10}^{-4}{\mathrm{cm}}^{-1}$, ${\mu }_{a_m}\approx 3.2\times {10}^{-4}{\mathrm{cm}}^{-1}$ [32], $D_x\approx 0.051\mathrm{cm}$ and $D_m\approx 0.056\mathrm{cm}$ [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 $x-y$ plane for the reconstructed η. The estimated η in the reconstructed image for the $250\mathrm{nM}$ solution was $13\times {10}^{-4}{\mathrm{cm}}^{-1}$, giving $\eta =15.6\times {10}^{-3}{\mathrm{cm}}^{-1}$ for the $3\mu \mathrm{M}$ solution used in the FRET imaging experiment.

5B. Computation of $R_0$

The Förster distance $R_0$ for the donor BODIPY-FL and acceptor rhodamine was calculated from [2]

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, $y_{km}^{\mathit{base}}$, for the $k\text{th}$ source and $m\text{th}$ detector, was made at the excitation wavelength $\left(488\mathrm{nm}\right)$ 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 ${\mu }_a$ for 0.4% Intralipid at $488\mathrm{nm}$, described above, synthetic data $y_{km}^{\mathit{syn}}$ was computed using a numerical solution to (3). Uncalibrated fluorescence data, $y_{km}^{\mathit{uncal}}$, obtained from the experiment described above, was calibrated using

5D. Experiment Results

Figure 6 shows three slices along the $x-y$ 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 $r_F=3.6\mathrm{nm}$, 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 $r_F=3.4\mathrm{nm}$.

5E. Experiment to Verify Reconstruction of $r_F$

To verify whether the DA distance had been estimated accurately, we performed another experiment in which the plastic vials containing the $250\mathrm{nM}$ solution of donor and Intralipid and the $3\mu \mathrm{M}$ 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

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 $80{\mathrm{mm}}^{-2}$ [35] $\left(716{\mathrm{mm}}^{-3}\right)$. Assuming a plaque dimension of $50\mu \mathrm{m}$, this gives a minimum plaque volume of $3.5\times {10}^{-3}{\mathrm{cm}}^3$ in a $1{\mathrm{cm}}^3$ 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 $10\mu \mathrm{m}$) 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 $r_F=4.2\mathrm{nm}$ 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 $2.5\mathrm{mm}$ (with $33\times 33\times 33$ image resolution). For a rigid linker with $r_F=4.2\mathrm{nm}$ and $\eta =0.025$: (c) reconstructed $r_F$, (d) reconstructed η. For a flexible linker with $a=5.42\times {10}^{17}{\mathrm{m}}^{-2}$ and $b=5.45\mathrm{nm}$, with the geometry in (b): (e) reconstructed a, (f) reconstructed b. The average detector SNR was $30\mathrm{dB}$.

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Fig. 2 Piecewise polynomial and exponential fitting of the real and imaginary parts of $\zeta \left(r_F\right)$ when $R_0=4.7\mathrm{nm}$ and ${\tau }_D=3.8\mathrm{ns}$: (a) real part, and (b) imaginary part, for a modulation frequency of $80\mathrm{MHz}$.

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Fig. 3 A phantom having two spheres, each of radius $2\mathrm{mm}$, containing DA molecules connected by a flexible linker. (a) True image of a. For the top right sphere, $a=3.5\times {10}^{17}{\mathrm{m}}^{-2}$, and for bottom left sphere, $a=7.45\times {10}^{17}{\mathrm{m}}^{-2}$. (b) Reconstructed a. (c) True image of b. For top right sphere, $b=5.95\mathrm{nm}$, and for bottom left sphere, $b=3.65\mathrm{nm}$. (d) Reconstructed b. (e) True image of η. For top right and bottom left spheres, $\eta =0.025$. (f) Reconstructed η. The average detector $\mathrm{SNR}=30\mathrm{dB}$.

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Fig. 4 (a) and (b) Experimental setup: a $3\mathrm{mW}$ $488\mathrm{nm}$ argon-ion laser (Uniphase), an $x-y$ scanning mirror system, a Plexiglas box of size $8.8\mathrm{cm}$ $\left(\mathrm{L}\right)\times 8.8\mathrm{cm}$ $\left(\mathrm{H}\right)\times 3.4\mathrm{cm}$ (W), a cylindrical plastic vial (length $3\mathrm{cm}$, inner-diameter $0.65\mathrm{cm}$, outer-diameter $0.75\mathrm{cm}$), to hold the FRET chemical, suspended from the lid of the Plexiglas box using an acrylic rod $1\mathrm{mm}$ in diameter and $1.5\mathrm{cm}$ in length, a $520\mathrm{nm}$ narrow bandpass filter (Edmund Optics) with FWHM $10\mathrm{nm}$ [31], and a $105\mathrm{mm}$, f/2.8 lens (AF micro Nikkor, Nikon) to focus the $4.8\mathrm{cm}\times 4.8\mathrm{cm}$ image of the scattering medium on a $512\times 512\text{\hspace{0.17em} pixel}$ Peltier cooled CCD Camera (PI-MAX, Roper Scientific).

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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 $z=-0.2125\mathrm{cm}$. (b) Slice at $z=0.0\mathrm{cm}$. (c) Slice at $z=0.2125\mathrm{cm}$. Reconstructed $\eta \approx 0.0013{\mathrm{cm}}^{-1}$.

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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 $z=-0.2125\mathrm{cm}$. (b) Slice at $z=0.0\mathrm{cm}$. (c) Slice at $z=0.2125\mathrm{cm}$. (d) Isosurface plot of the DA distance at $r_F=3.6\mathrm{nm}$, which is the distance estimated without the scattering medium (see Section 5E).

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Fig. 7 Slices along the $x-y$ plane of the reconstructed image of the DA distance. Expected $r_F=3.6\mathrm{nm}$. (a) Slice at $z=-0.2125\mathrm{cm}$. (b) Slice at $z=0.0\mathrm{cm}$. (c) Slice at $z=0.2125\mathrm{cm}$. (d) Isosurface plot of the reconstructed DA distance at $r_F=3.4\mathrm{nm}$.

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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 $C_D=250\mathrm{nM}$. The total detected power is assumed to be the integral of the intensity in the dotted rectangle, which gives $P_D=\gamma 7.3\times {10}^8\mathrm{W}$. (b) Image of the vial containing the FRET chemical mixed with Intralipid at concentration $C_{\mathit{DA}}=3\mu \mathrm{M}$. The detected power, using the intensity in the dotted rectangle, is $P_{\mathit{DA}}=\gamma 4.2\times {10}^8\mathrm{W}$. Using (20) and $R_0=5.8\mathrm{nm}$ gives $r_F\approx 3.6\mathrm{nm}$

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