## 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 ${l}_{s}={\left(3D\right)}^{-1}$, 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 ${k}_{F}=\tau _{D}{}^{-1}{({R}_{0}\u2215{r}_{F})}^{6}$, 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 ${\eta}_{D}={k}_{r}\u2215({k}_{r}+{k}_{nr})$ [2]. In the presence of an acceptor, energy transfer results in a decreased donor lifetime, ${\tau}_{\mathit{DA}}$, given by

*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]

*ω*$\left(\mathrm{rad}\phantom{\rule{0.2em}{0ex}}{\mathrm{s}}^{-1}\right)$, where subscripts

*x*and

*m*, respectively, denote the excitation and emission wavelengths ${\lambda}_{x}$ and ${\lambda}_{m}$, $\phi (r,\omega )$ $\left(\mathrm{W}\phantom{\rule{0.2em}{0ex}}{\mathrm{cm}}^{-2}\right)$ is the photon flux density at position

*r*inside the scattering medium, a point source ${S}_{x}(r;\omega )=\beta \delta \phantom{\rule{0.1em}{0ex}}(r-s)$ is assumed, with

*β*the modulation depth,

*c*is the speed of light in the medium, ${\mu}_{a}\left(r\right)$ $\left({\mathrm{cm}}^{-1}\right)$ is the absorption coefficient at position

*r*, $D\left(r\right)$ (cm) is the diffusion coefficient at position

*r*, and ${S}_{f}\phantom{\rule{0.1em}{0ex}}(r,\omega )$ is the source for the flexible (subscript

*f*) linker. Using the Fourier transform of the time-domain impulse response of the donor fluorophore, $[({\eta}_{\mathit{DA}}.{\mu}_{aD})\u2215{\tau}_{\mathit{DA}}].\mathrm{exp}(-t\u2215{\tau}_{\mathit{DA}})$, along with (1, 2),

*r*, ${\mu}_{aD}\left(r\right)$ is the absorption coefficient of the donor, and for notational simplicity, $\eta \left(r\right)={\eta}_{D}{\mu}_{aD}\left(r\right)$. Modulated light is necessary to determine all parameters in (5). For an ensemble of DA pairs connected by rigid linkers, $p\left({r}_{F}\right)$ reduces to a Dirac delta function, and the corresponding source term is ${S}_{g}(r;\omega )=\eta \left(r\right)\zeta \left({r}_{F}\left(r\right)\right)$, 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, ${\tau}_{\mathit{DA}}$, 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 ${\tau}_{\mathit{DA}}$, the distribution would be non-Gaussian. Having a Gaussian distribution makes it easier to calculate the integral in (5). Let ${g}_{x}({r}_{{s}_{k}},r;\omega )$ be the diffusion equation Green’s function for (3) at wavelength ${\lambda}_{x}$ and ${g}_{m}(r,{r}_{{d}_{m}};\omega )$ be the Green’s function for (4) at ${\lambda}_{m}$. Then, the donor fluorescence photon flux density (for flexible or rigid linker) at a detector positioned at ${r}_{{d}_{m}}$, due to a point source modulated at frequency

*ω*and positioned at ${r}_{{s}_{k}}$, is

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

*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

*M*detectors due to

*K*sources modulated at frequency ${\omega}_{q}$,

*Q*denotes the total number of modulation frequencies, and ${s}_{q}=[{s}_{{1}_{q}},\dots {s}_{{K}_{q}}]$ and ${d}_{q}=[{d}_{{1}_{q}},\dots {d}_{{M}_{q}}]$ are the source and detector coupling coefficient vectors respectively, at frequency ${\omega}_{q}$, for the

*K*sources and

*M*detectors, which are estimated jointly with the image ${x}_{F}$. For the simulations presented in the paper, ${s}_{k}=1+i0$ and ${d}_{m}=1+i0$ are used. Assuming a Gaussian shot-noise model, the data term is

*w*, ${\Vert w\Vert}_{\Lambda}^{2}={w}^{H}\Lambda w$ (where

*H*denotes Hermitian transpose), with ${\alpha}_{F}\u2215\left(2\Lambda _{{F}_{q}}{}^{-1}\right)$ the covariance matrix. We assume a shot noise model with $\Lambda _{{F}_{q}}{}^{-1}=\mathrm{diag}[\left|{y}_{{F}_{q}1}\right|,\dots \left|{y}_{{F}_{q}P}\right|]$ [23]. For the flexible linker, the prior density, $p\left({x}_{F}\right)$, assuming independence of the image vectors representing

*η*,

*a*or

*b*, is $p\left({x}_{F}\right)=p\left({x}_{\eta}\right).p\left({x}_{a}\right).p\left({x}_{b}\right)$. We use the generalized Gaussian Markov random field (GGMRF) prior [23]

*k*represents

*η*,

*a*or

*b*, ${x}_{{k}_{i}}$ denotes the $i\text{th}$ node of ${x}_{k}$, the set ${\mathcal{N}}_{k}$ consists of all pairs of neighboring nodes (in a 26 neighbor system), and ${b}_{i-j}$ is the weighting coefficient corresponding to the $i\text{th}$ and $j\text{th}$ nodes assigned to be inversely proportional to the node separation, under the condition that ${\sum}_{j}{b}_{i-j}=1$. Parameters

*ρ*(we use ${\rho}_{k}=2$) and

*σ*control the shape and scale of the distribution and $z\left(\rho \right)$ normalizes the density. The prior model for the rigid linker can be described in a similar way.

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

*V*is the (cubic) element volume associated with each node and the dependence of the FRET source term,

*S*, on the image ${x}_{F}$, is explicitly included by representing it as $S({r}_{j};\omega ,{x}_{F})$. Using (13), we rewrite (11) as

*M*and $k=[i\u2215M+1]$, where [.] is the integer part of $i\u2215M+1$, and ${S}_{{\omega}_{q}}\left({x}_{F}\right)={[S({r}_{1};{\omega}_{q},{x}_{F})\cdots S({r}_{N};{\omega}_{q},{x}_{F})]}^{T}$. Previously, we found that incorporating ${\alpha}_{F}$ 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, ${\left\{{\widehat{x}}_{F}\right\}}_{\mathit{MAP}}$, by performing a joint iterative minimization of $\mathrm{ln}\left[p\left({y}_{F}\right|{x}_{F})p\left({x}_{F}\right)\right]$ over ${\alpha}_{F}$, ${s}_{q}$, ${d}_{q}$ and ${x}_{F}$, leading to

*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 $i\text{th}$ node in the image of

*a*,

*b*or

*η*is

*k*represents either

*a*,

*b*or

*η*, ${\mathcal{N}}_{{k}_{i}}$ is the set of nodes neighboring node

*i*, and ${\left[{G}_{{\omega}_{q}}\right]}_{*\left(i\right)}$ denotes the $i\text{th}$ column of ${G}_{{\omega}_{q}}$.

## 4. SIMULATION

The model problem we consider is the cubical tissue phantom of Fig. 1a with side $2.5\text{\hspace{0.17em}}\mathrm{cm}$, 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 $220\text{\hspace{0.17em}}\mathrm{MHz}$ ($20\text{\hspace{0.17em}}\mathrm{MHz}$ 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 ${\tau}_{D}=3.8\text{\hspace{0.17em}}\mathrm{ns}$ [26], and we use ${R}_{0}=4.7\text{\hspace{0.17em}}\mathrm{nm}$ [27]. We assume ${D}_{x}\approx {D}_{m}=0.027\text{\hspace{0.17em}}\mathrm{cm}$ and ${\mu}_{ax}\approx {\mu}_{am}=0.047\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$. For both rigid and flexible linker, we choose $\eta =0.025\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$, calculated from the quantum yield and absorption coefficient for EYFP, assuming a concentration of $250\text{\hspace{0.17em}}\mathrm{nM}$ for the FRET pair. The FRET pairs are contained within a centrally located sphere of radius $2.5\text{\hspace{0.17em}}\mathrm{mm}$. 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 $30\text{\hspace{0.17em}}\mathrm{dB}$. 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 ${r}_{\mathit{min}}=5\text{\hspace{0.17em}}\mathrm{\AA}$ and ${r}_{\mathit{max}}=3.5{R}_{0}$. 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 $2\text{\hspace{0.17em}}\mathrm{mm}$, 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 $488\text{\hspace{0.17em}}\mathrm{nm}$, emission $520\text{\hspace{0.17em}}\mathrm{nm}$), 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 $3\text{\hspace{0.17em}}\mathrm{mW}$
$488\text{\hspace{0.17em}}\mathrm{nm}$ argon-ion laser (Uniphase), an $x-y$ scanning mirror system, a Plexiglas box of size $8.8\text{\hspace{0.17em}}\mathrm{cm}\phantom{\rule{0.3em}{0ex}}\left(\mathrm{L}\right)\times 8.8\text{\hspace{0.17em}}\mathrm{cm}\phantom{\rule{0.3em}{0ex}}\left(\mathrm{H}\right)\times 3.4\text{\hspace{0.17em}}\mathrm{cm}\phantom{\rule{0.3em}{0ex}}\left(\mathrm{W}\right)$, a cylindrical plastic vial (length $3\text{\hspace{0.17em}}\mathrm{cm}$, inner-diameter $0.65\text{\hspace{0.17em}}\mathrm{cm}$, outer-diameter $0.75\text{\hspace{0.17em}}\mathrm{cm}$) to hold the FRET chemical, suspended from the lid of the Plexiglas box using an acrylic rod $1\text{\hspace{0.17em}}\mathrm{mm}$ in diameter and $1.5\text{\hspace{0.17em}}\mathrm{cm}$ in length, a $520\text{\hspace{0.17em}}\mathrm{nm}$ bandpass filter (Edmund Optics, $50.8\text{\hspace{0.17em}}\mathrm{mm}$ square) with FWHM $10\text{\hspace{0.17em}}\mathrm{nm}$ and peak transmission 45% [31], and a $105\text{\hspace{0.17em}}\mathrm{mm}$, f/2.8 lens (AF micro Nikkor, Nikon) to focus the $4.8\text{\hspace{0.17em}}\mathrm{cm}\times 4.8\text{\hspace{0.17em}}\mathrm{cm}$ 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 $10\text{\hspace{0.17em}}\mathrm{ml}$ of 20% Intralipid in $500\text{\hspace{0.17em}}\mathrm{ml}$ of deionized water. The plastic vial contained $1\text{\hspace{0.17em}}\mathrm{ml}$ of $3\text{\hspace{0.17em}}\mu \mathrm{M}$ solution of the FRET chemical in Intralipid, prepared by mixing $100\text{\hspace{0.17em}}\mu \mathrm{l}$ of a $0.18\text{\hspace{0.17em}}\mathrm{mM}$ stock solution of the FRET chemical with $5.9\text{\hspace{0.17em}}\mathrm{ml}$ 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 $150\text{\hspace{0.17em}}\mathrm{msec}$ 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 $\eta =15.6\times {10}^{-3}\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$. For the 0.4% Intralipid solution, we assume that all the absorption at ${\lambda}_{x}=488\text{\hspace{0.17em}}\mathrm{nm}$ is due to water, thereby giving ${\mu}_{{a}_{x}}\approx 2.5\times {10}^{-4}\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$ and ${\mu}_{{a}_{m}}\approx 3.2\times {10}^{-4}\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$ [32]. Prior experiments with Intralipid lead to ${D}_{x}\approx 0.051\text{\hspace{0.17em}}\mathrm{cm}$ and ${D}_{m}\approx 0.056\text{\hspace{0.17em}}\mathrm{cm}$ [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 $520\text{\hspace{0.17em}}\mathrm{nm}$, when excited at $488\text{\hspace{0.17em}}\mathrm{nm}$, as we found that compared to the emission of Bodipy-FL at $520\text{\hspace{0.17em}}\mathrm{nm}$, 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 $100\text{\hspace{0.17em}}\mu \mathrm{l}$ of a $10\text{\hspace{0.17em}}\mu \mathrm{M}$ stock solution of the donor BODIPY-FL with $3.9\text{\hspace{0.17em}}\mathrm{ml}$ of 0.4% (w/v) Intralipid, resulting in a concentration of ${C}_{D}=250\text{\hspace{0.17em}}\mathrm{nM}$. 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 $150\text{\hspace{0.17em}}\mathrm{msec}$ 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 ${\mu}_{{a}_{x}}\approx 2.5\times {10}^{-4}\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$, ${\mu}_{{a}_{m}}\approx 3.2\times {10}^{-4}\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$ [32], ${D}_{x}\approx 0.051\text{\hspace{0.17em}}\mathrm{cm}$ and ${D}_{m}\approx 0.056\text{\hspace{0.17em}}\mathrm{cm}$ [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 $250\text{\hspace{0.17em}}\mathrm{nM}$ solution was $13\times {10}^{-4}\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$, giving $\eta =15.6\times {10}^{-3}\text{\hspace{0.17em}}{\mathrm{cm}}^{-1}$ for the $3\text{\hspace{0.17em}}\mu \mathrm{M}$ 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]

*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), ${\eta}_{D}$ is the donor quantum yield in the absence of the acceptor, which for BODIPY-FL is ${\eta}_{D}=0.9$ [34], ${f}_{D}\left(\lambda \right)$ is the normalized fluorescence spectrum of the donor, and ${\epsilon}_{A}\left(\lambda \right)$ is the molar absorption coefficient of the acceptor at wavelength

*λ*. The integral in (17) was calculated between ${\lambda}_{\mathit{min}}=475\text{\hspace{0.17em}}\mathrm{nm}$ and ${\lambda}_{\mathit{max}}=649\text{\hspace{0.17em}}\mathrm{nm}$, giving ${R}_{0}=5.8\text{\hspace{0.17em}}\mathrm{nm}$.

#### 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 $\left(488\text{\hspace{0.17em}}\mathrm{nm}\right)$ 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 $488\text{\hspace{0.17em}}\mathrm{nm}$, 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 ${r}_{F}=3.6\text{\hspace{0.17em}}\mathrm{nm}$, 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 ${r}_{F}=3.4\text{\hspace{0.17em}}\mathrm{nm}$.

#### 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 $250\text{\hspace{0.17em}}\mathrm{nM}$ solution of donor and Intralipid and the $3\text{\hspace{0.17em}}\mu \mathrm{M}$ 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

where*I*is the intensity in counts and $dS$ represents an area element. Using (19), the DA distance can be estimated from

*γ*is a proportionality constant (Watts/counts) which cancels out when ${P}_{D}$ and ${P}_{\mathit{DA}}$ are substituted in (20), and ${R}_{0}=5.8\text{\hspace{0.17em}}\mathrm{nm}$, yielding ${r}_{F}=3.6\text{\hspace{0.17em}}\mathrm{nm}$. This is very close to the reconstructed DA distance in the presence of scatter. In the calculation for ${r}_{F}$, we ignored the small contribution of rhodamine emission at $520\text{\hspace{0.17em}}\mathrm{nm}$, when excited at $488\text{\hspace{0.17em}}\mathrm{nm}$.

## 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 $80\text{\hspace{0.17em}}{\mathrm{mm}}^{-2}$ [35] $\left(716\text{\hspace{0.17em}}{\mathrm{mm}}^{-3}\right)$. Assuming a plaque dimension of $50\text{\hspace{0.17em}}\mu \mathrm{m}$, this gives a minimum plaque volume of $3.5\times {10}^{-3}\text{\hspace{0.17em}}{\mathrm{cm}}^{3}$ in a $1\text{\hspace{0.17em}}{\mathrm{cm}}^{3}$ 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 $10\text{\hspace{0.17em}}\mu \mathrm{m}$) 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.

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