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Opt Lett. Author manuscript; available in PMC 2010 August 2.

Published in final edited form as:

PMCID: PMC2913597

NIHMSID: NIHMS221206

We compare frequency-and time-domain formulations of deep-tissue fluorescence imaging of turbid media. Simulations are used to show that time-domain fluorescence tomography, implemented via the asymptotic lifetime-based approach, offers a significantly better separability of multiple lifetime targets than a frequency-domain approach. We also demonstrate experimentally, using complex-shaped phantoms, the advantages of the asymptotic time-domain approach over a Fourier-based approach for analyzing time-domain fluorescence data.

Optical technologies for noninvasive macroscopic fluorescence imaging in biological media utilize three main approaches: time domain (TD) using pulsed light sources [1–5], frequency domain (FD) using megahertz modulated sources [6–8], and continuous wave (CW) using steady state light sources [9–11]. Of these, the TD approach is the most comprehensive, since a short laser pulse (fs-ps) implicitly contains all the modulation frequencies, including the zero-frequency component. The tomographic analysis of TD data can, however, pose computational challenges. Several simplifications have been attempted, primarily using derived data types such as the fast Fourier transform (FFT) [5,7]. The FFT simplifies the TD forward problem but reduces it to a FD forward problem, identical to that for a genuine FD measurement. Alternatively, TD fluorescence data may also be analyzed by estimating lifetimes directly from the asymptotic region, followed by the separate inversion of the yield of each lifetime component [3,12]. The question arises as to how the FD forward problem compares with the asymptotic TD (ATD) approach, when lifetime sensitive targets are used. In this Letter, we address this question both with simulated and experimental TD data in the context of small animal imaging applications.

Consider a diffuse imaging medium of finite support Ω, embedded with fluorophores characterized by yield and lifetime distributions, *η* (**r**) and *τ*(**r**). In the FD approach, the forward problem takes the form [6]

$$\stackrel{~}{U}({\mathbf{r}}_{s},{\mathbf{r}}_{d},\omega )={\int}_{\Omega}{d}^{3}r{G}^{x}({\mathbf{r}}_{s},\mathbf{r},\omega ){G}^{m}({\mathbf{r}}_{d},\mathbf{r},\omega )F(\mathbf{r},\omega ).$$

(1)

Here, *G ^{x}* and

$${a}_{n}({\mathbf{r}}_{d},{\mathbf{r}}_{s})={\int}_{\Omega}{d}^{3}r{G}^{x}({\mathbf{r}}_{s},\mathbf{r},-i{\Gamma}_{n}){G}^{m}({\mathbf{r}}_{d},\mathbf{r},-i{\Gamma}_{n}){\eta}_{n}\left(\mathbf{r}\right).$$

(2)

Note that in Eq. (2), the Green’s functions are the same as that for the FD case in Eq. (1), but evaluated at an imaginary frequency of –*iΓ _{n}*. (see [12] for details). The difference between Eq. (1) and Eq. (2) is clear if we express

To compare the performance of the FD and ATD approaches, simulations were performed for a 2 cm thick infinite slab geometry, assuming 2 mm × 2mm × 2 mm size fluorescent targets (also the size of a single voxel) placed both laterally (Fig. 1) and axially (Fig. 2) with respect to the source-detector line of sight. The inclusions had distinct lifetimes of 0.5 and 1 ns. 29 sources and 21 detectors were arranged [Fig. 1(a)] along the planes at depths of *z*=0 cm and *z*=2 cm, respectively. The background medium was assumed to have values of ${\mu}_{s}^{\prime}=10\phantom{\rule{thinmathspace}{0ex}}{\mathrm{cm}}^{-1}$ for scattering and μ_{a}=0.1 cm^{−1} for absorption (at both excitation and emission wavelengths). The FD signal was simulated using Eq. (1) in the diffusion approximation, for *ω* =80 MHz, and reconstructed with Tikhonov regularization. The full TD signal was also simulated in the diffusion model with shot noise added, and the *a _{n}*(

Comparison of FD and ATD reconstructions with simulated data for laterally located fluorescent targets with lifetimes of 0.5 and 1 ns. (a) Cross-sectional (*x–y* plane) view of sources (asterisks) and detectors (circles) arranged in a transmission **...**

The simulations presented in Figs Figs11 and and22 clearly indicate the advantages of the ATD approach over the FD formalism for resolving lifetime contrast. We now experimentally demonstrate these results using a noncontact TD fluorescence tomography system, consisting of a Ti:Sapphire laser source for excitation and a time-gated intensified CCD camera for detection. Detailed aspects of the system are discussed in a separate publication [4]. A mouse-shaped phantom made of epoxy, ink, and TiO2 combination (${\mu}_{s}^{\prime}=10\phantom{\rule{thinmathspace}{0ex}}{\mathrm{cm}}^{-1}$, *μ _{a}*=0.1 cm

Experimental tomography results from a mouse phantom. (a) Photograph of mouse phantom used, with two inclusions. Inclusion A was filled with an aqueous solution of the fluorophore (0.5 ns lifetime), and inclusion B was filled with the fluorophore in 100% **...**

We have shown using simulations and experiments that an asymptotic multiexponential-analysis-based TD approach to fluorescence tomography has inherent advantages over the FD formalism of diffuse fluorescence when resolving discrete lifetimes present within a turbid medium. Fluorescence lifetime has been widely exploited in microscopy techniques with thin tissue sections (see, for example, [13]), and it is likely to serve as a key functional parameter for non-invasive diagnostic optical molecular imaging and drug discovery. The findings reported in this Letter have significant relevance for the design of lifetime-based optical molecular imaging systems. The results also strongly motivate further development of smart near-infrared molecular probes that specifically shift lifetime upon binding to disease targets of interest.

This work was supported by the National Institutes of Health (grants EB000768 and AG026240).

*OCIS codes:* 170.3880, 170.3650.

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