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**|**HHS Author Manuscripts**|**PMC3732578

Opt Lett. Author manuscript; available in PMC 2013 August 4.

Published in final edited form as:

Opt Lett. 2013 May 1; 38(9): 1440–1442.

PMCID: PMC3732578

NIHMSID: NIHMS482022

Anand T. N. Kumar, Athinoula A. Martinos Center for Biomedical Imaging, Department of Radiology, Massachusetts General Hospital, Harvard Medical School, Charlestown, Massachusetts 02129, USA (Email: ude.dravrah.hgm.rmn@ramukna);

It is demonstrated that high spatial frequency filtering of time domain fluorescence signals can allow efficient detection of intrinsic fluorescence lifetimes from turbid media and the rejection of diffuse excitation leakage. The basis of this approach is the separation of diffuse fluorescence signals into diffuse and fluorescent components with distinct spatiotemporal behavior.

The temporal decay time constant (lifetime) of fluorescence from a turbid medium, such as tissue can, in general, depend on the intrinsic absorption and scattering coefficients [1]. Unless the fluorescence lifetime is significantly longer than the intrinsic diffusion timescales (~10^{−9} s) [2], a robust recovery of the intrinsic lifetimes necessitates the inversion of coupled differential equations for fluorescence propagation in tissue [3], which can be ill-posed. In this Letter, it is shown that direct recovery of shorter *in vivo* lifetimes is possible with time domain (TD) measurements, by exploiting a unique phenomenon, namely the separation of diffuse fluorescence into diffuse and pure fluorescent decay terms that exhibits distinct spatiotemporal responses. In the spatial frequency domain (FD), the diffuse term decays at a rate proportional to the spatial frequency, tissue absorption, and scattering (analogous to the temporal propagation of intrinsic diffuse light [4,5]). However, the decay of the fluorescence term is independent of spatial frequency or optical properties but reflects the characteristic lifetime of the fluorophore in the tissue environment. The fluorescence therefore remains significant at high spatial frequencies, where the diffuse term is rapidly eliminated. This observation has important implications for macroscopic lifetime imaging in turbid media. In particular, we show experimentally that it allows direct detection of lifetimes shorter than the intrinsic diffuse timescales, and discrimination of fluorescence from diffuse excitation leakage through emission filters, a common problem encountered in fluorescence imaging.

Consider a diffuse medium with optical properties $({\mu}_{a}^{x}(\mathbf{r}),{\mu}_{s}^{x}(\mathbf{r}))$ at the excitation and $({\mu}_{a}^{m}(\mathbf{r}),{\mu}_{s}^{m}(\mathbf{r}))$ at the emission wavelengths, with fluorophores described by yield distributions η_{n}(**r**) and lifetimes τ_{n} = 1/Γ_{n}. Using complex integration methods, it can be shown [1] that the TD fluorescence intensity at position **r**_{d} and time *t* for point excitation at **r**_{s} can be written as *U _{F}* = ∑

$${U}_{Fn}({\mathbf{r}}_{s},{\mathbf{r}}_{d},t)=-{a}_{Dn}({\mathbf{r}}_{s},{\mathbf{r}}_{d},t)+{a}_{Fn}({\mathbf{r}}_{s},{\mathbf{r}}_{d}){e}^{-{\mathrm{\Gamma}}_{n}t}.$$

(1)

Here *a _{Fn}* = ∫

$${a}_{Dn}={\int}_{\mathrm{\Omega}}{\mathrm{d}}^{3}r[\frac{1}{\pi}{\int}_{\upsilon {\mu}_{a}}^{\infty}\mathrm{d}\gamma \frac{\mathrm{I}\mathrm{m}[\mathrm{({\mathbf{r}}_{s},{\mathbf{r}}_{d},\mathbf{r},-i\gamma )]}\gamma -{\mathrm{\Gamma}}_{n}}{{e}^{-\gamma t}}$$

(2)

is the diffusive term arising from the branch points of the FD weight function [1], where Im refers to the imaginary part. Note that *a _{Dn}* involves at an imaginary frequency of −

The central point of this Letter is that *a _{Fn}* is independent of time, whereas

$${\u0168}_{Fn}({\mathbf{r}}_{s},{\mathbf{k}}_{d},t)=-{\xe3}_{Dn}({\mathbf{r}}_{s},{\mathbf{k}}_{d},t)+{\xe3}_{Fn}({\mathbf{r}}_{s},{\mathbf{k}}_{d}){e}^{-{\mathrm{\Gamma}}_{n}t}.$$

(3)

The distinct spatiotemporal behavior of *a _{D}* and

(a) Simulation geometry indicating the source (x) and detectors (o) with a single fluorophore (τ = 0.3 ns) at the center. (b) Normalized TD fluorescence signal *U*_{F} as a function of **r**_{d} at various times. Inset shows *U*_{F} (blue) at the central detector **...**

The rapid decrease of high spatial frequencies of *ã _{D}* is similar to that of intrinsic diffuse signals [5], and suggests that both of these contributions can be minimized in fluorescence signals by spatial filtering, as we demonstrate using a simple phantom experiment. A small tube was placed near the bottom of a 1.75 cm thick intralipid phantom (${\mu}_{s}^{\prime}\approx 22/\text{cm}$, μ

Measurement setup up with a 1.75 cm thick intralipid phantom excited at 790 nm at (a) *S*_{1} and (b),(c) *S*_{2}, and detected with a λ > 800 nm filter. The tube with IRdye800 is shown schematically (color depicts true lifetime) in (b) with water **...**

A spatial 2D fast Fourier transform (FFT) in MATLAB (The Mathworks Inc.,) was applied to the 2D spatial TD data thresholded at 3% of the maximum intensity, resulting in the full detector spatial frequency (*k*-space) TD data for all delays. The *k*-space lifetime maps (Fig. 3) obtained from single-exponential fits to the decay portion of the *k*-space TD data show distinct behavior of the diffuse excitation [Fig. 3(a)] and fluorescence signals [Figs. 3(b) and 3(c)]; while the *k*-space lifetime for source *S*_{1} (diffuse excitation/leakage) continuously decreases toward higher *k*’s, the lifetime for fluorescence approaches the true fluorophore lifetimes of 0.4 ± 0.01 ns and 0.72 ± 0.02 ns. It is plausible to apply a high-frequency filter in *k* space to extract the intrinsic fluorescence. The choice of the appropriate filter will depend on rate of decay of the intrinsic lifetime maps (which depends on the tissue optical properties [7]). Here we choose an annular ring [Fig. 3(d)] of the form *f* (**k**_{d}) = exp[−(*k _{x}* cos(θ) +

$${U}_{F}^{\text{filt}}({\mathbf{r}}_{s},{\mathbf{r}}_{d},t)=\int \frac{{\mathrm{d}}^{2}{k}_{d}}{{(2\pi )}^{2}}f({\mathbf{k}}_{d}){\u0168}_{F}({\mathbf{r}}_{s},{\mathbf{k}}_{d},t){e}^{i{\mathbf{k}}_{d}\xb7{\mathbf{r}}_{d}}.$$

(4)

Single-exponential fits to the asymptotic decays of ${U}_{F}^{\text{filt}}$ [Figs. 2(m)–2(o)] recover the true lifetimes [Figs. 2(b) and 2(c)] to within 5% and clearly distinguish the true fluorescence from diffuse excitation leakage [Fig. 2(m)], while the CW components, viz., ∫ d*tU*^{filt}(*t*) [Figs. 2(j)–2(l)] do not distinguish the three cases. Figure 4 shows a sample of raw TD data for a detector above the source and the spatial Fourier components for the 0.4 ns case. While the raw TD signals are similar for excitation leakage (black) and fluorescence (blue), the true lifetime of 0.4 ± 0.01 ns is recovered for source *S*_{2} at (*k _{x}, k_{y}*) = (4, 3) rad/cm, whereas the corresponding diffuse excitation leakage for

In summary, an approach to extract fluorescence lifetimes from diffuse media was presented, based on spatial Fourier filtering of time-resolved data. The key principle is that the diffuse component of fluorescence signals decays at a rate that increases with spatial frequency, while the pure fluorescence component always decays at the intrinsic lifetime. This observation offers a powerful way to detect the presence of scattering or excitation leakage in biological lifetime measurements [8]. While only a single lifetime was considered in the experiments, the formalism presented here, is readily applicable to multiple lifetimes, and can also be extended to tomographic lifetime multiplexing [2] in the spatial Fourier domain. A novel aspect of this work is that the entire spectrum of detector side *k*-space amplitudes is obtained from the raw TD data with a simple FFT, without the need for complicated modulation techniques [5,9]. However, the extension of this approach to modulated excitation [5,7,9] can offer a powerful new approach for high-throughput tomographic lifetime imaging and will be considered in future work.

The author thanks Simon Arridge for useful discussions. This work was supported by the National Institutes of Health, grant R01 EB015325.

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