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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 . Unless the fluorescence lifetime is significantly longer than the intrinsic diffusion timescales (~10−9 s) , a robust recovery of the intrinsic lifetimes necessitates the inversion of coupled differential equations for fluorescence propagation in tissue , 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 at the excitation and at the emission wavelengths, with fluorophores described by yield distributions ηn(r) and lifetimes τn = 1/Γn. Using complex integration methods, it can be shown  that the TD fluorescence intensity at position rd and time t for point excitation at rs can be written as UF = ∑n UFn, where (for the case )
Here aFn = ∫Ω d3r(rs, rd, r, −iΓ)ηn(r) is the fluorescence decay amplitude of the nth lifetime species, valid for arbitrary heterogeneous transport media , where Ω is the medium volume and = x (rs, r, ω)m (rd, r, ω) is the FD weight function with x,m as the Green’s functions of the FD diffusion equation. For a homogeneous medium with ,
is the diffusive term arising from the branch points of the FD weight function , where Im refers to the imaginary part. Note that aDn involves at an imaginary frequency of −iγ, which is equivalent to a CW weight function with a negative absorption of μa(r) − γ/υ, while aFn involves at a reduced absorption of μa − Γn/υ.
The central point of this Letter is that aFn is independent of time, whereas aDn propagates similarly to intrinsic diffuse light through tissue . To see this, consider the spatial Fourier transform of UF w.r.t rd:
The distinct spatiotemporal behavior of aD and aF and the k-space transforms, ãD, and ãF, is illustrated in Fig. 1 using simulations with a diffusive slab model (μa = 0.2/cm, ) with a fluorescent inclusion (τ = 0.3 ns). The spatial profile of UF(rs, rd, t) expands with time, approaching aF asymptotically [Fig. 1(b)]. Figure 1(c) shows the separation of UF into a diffusive contribution, aD, with a spatial distribution that expands over time, and a time-independent contribution from aF. Correspondingly, Fig. 1(d) shows ŨF decomposed into a spatially narrowing ãD and a stationary ãF.
The rapid decrease of high spatial frequencies of ãD is similar to that of intrinsic diffuse signals , 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 (, μa ≈ 0.02/cm). The tube was filled with 100 µL of 1 µM IRdye800 (LI-COR Biosciences) in either water [τ = 0.4 ns, Fig. 2(b)] or glycerol solvents [τ = 0.72 ns, Fig. 2(c)], and excited in the transmission geometry with a Ti:sapphire laser at 790 nm. Detection was performed with a λ > 800 nm emission filter attached to an intensified CCD camera (PicostarHR, LAVision, Gmbh; 300 ps gatewidth, 560 V gain, 100 ps steps, 4 × 4 software binning). The excitation wavelength was chosen close to the filter passband to allow leakage of the diffuse excitation light into the filter, to demonstrate its subsequent elimination by spatial Fourier filtering. The full TD data was collected for two source positions 1 cm apart below the phantom, one directly below the fluorescent tube [S2, Figs. 2(b) and 2(c)], and the other 1 cm away from the tube [S1, Fig. 2(a)]. The CW (integrated TD) images [Figs. 2(d)–2(f)] do not distinguish either source positions or lifetimes in the tube within the phantom. Moreoever, the lifetime maps obtained from single-exponential fits to the asymptotic TD decay [Figs. 2(g)–2(i)] do not reflect the true lifetimes of the fluorophores due to the influence of tissue propagation on the TD decays and excitation. Note the increase in lifetime from the center of the image toward the edge, due to a larger propagation distance from the source. The signal for S1 was negligible for 770 nm excitation (not shown), while the lifetime map at 790 nm [Fig. 2(g)] was similar to that without any emission filter (not shown), confirming that the signal for the S1 case is diffuse excitation leakage, which, in the present case, is indistinguishable from the 0.4 ns dye [Fig. 2(h)].
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 S1 (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 ). Here we choose an annular ring [Fig. 3(d)] of the form f (kd) = exp[−(kx cos(θ) + ky sin(θ) − R)2/σ2], where and θ = tan−1[ky/kx] with R = 3 rad/cm and σ = 1 rad/cm. This eliminates the diffuse contribution from the low frequencies while also avoiding noisy data at high spatial frequencies. Subsequently, the spatial TD data is obtained as an inverse FFT:
Single-exponential fits to the asymptotic decays of [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., ∫ dtUfilt(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 S2 at (kx, ky) = (4, 3) rad/cm, whereas the corresponding diffuse excitation leakage for S1 decays at a much shorter rate (0.2 ns). Note the longer lifetime of ŨF (0, 0) for S2, reflecting the longer lifetimes near the edge of the spatial lifetime map in Fig. 2(h).
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 . 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  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.