Time resolved fluorescence imaging in diffuse media Anand T. N. Kumara , Jesse Skochd , Frank L. Hammond IIIb , Andrew K. Dunnc David A. Boasa and Brian J. Bacskaid a Athinoula A. Martinos Center for Biomedical Imaging, Massachusetts General
Hospital,Harvard Medical School, Charlestown MA 02129 b Univ. of Pennsylvania, Philadelphia PA. c Department of Biomedical Engineering, Univ. of Texas, Austin TX. d Alzheimer’s disease research unit, Dept. of Neurology, Massachusetts
General Hospital ABSTRACT We discuss the application of time domain fluorescence techniques to the recovery of targets embedded in several cm thick biological tissue. Considering the general time domain problem first, a singular value analysis is used to study the optimal use of multiple frequency components extracted from time domain data. Furthermore, a computationally efficient algorithm is presented to tomographically reconstruct fluorophore locations using their decay amplitudes and validated using phantom experiments. The reconstruction algorithm presented here has wide applicability for non-invasive, diagnostic fluorescence imaging in small animals and other biological systems, given that fluorescence lifetime is a sensitive indicator of local tissue environment and elementary interactions at the molecular level. Keywords: Fluorescence imaging, time resolved, lifetime sensing
1. INTRODUCTION Fluorescence imaging offers a potentially viable alternative to PET and MRI for diagnostic imaging of diseases and for drug development, given the increasing interest in the development of disease targeted fluorescent markers.1, 2 Currently, fluorescence optical tomographic technologies for 3-D imaging utilize either continuous wave or frequency modulated excitation light.3–5 Time domain (TD) techniques have also recently been employed,6–8 but the computational complexity involved in a full TD fluorescence tomographic reconstruction suggests an analysis in the frequency domain with multiple frequency components.9 Frequency domain (FD) approaches aim to recover the spatial distribution of fluorescence yield (product of the fluorophore concentration, extinction coefficient and quantum yield) and the lifetime within the biological medium of interest.3 Multiple lifetime components, if present, are invariably mixed with the diffuse component in a FD measurement, whereas a TD measurement allows their separation through multi-exponential fits. In this work, we analyze TD fluorescence signals using complex integration methods and present a rigorous algorithm for the separate 3−D yield reconstructions of multiple fluorophores with distinct lifetimes, using decay amplitudes extracted from TD data. The algorithm is shown to be applicable to biomedical imaging provided the fluorescence lifetimes are longer than the timescales for photon migration within the medium. Lifetime sensitive fluorescent dyes are widely employed in microscopy techniques such as fluorescence lifetime imaging(FLIM)2, 10 to provide high resolution 2−D spatial lifetime maps of thin tissue sections, thereby revealing the local micro-environment of the dye. The method presented here, on the other hand, offers a robust way to extend the application of lifetime sensitive fluorescent targets to 3−D in-vivo imaging in several cm thick turbid tissue. Address correspondence to A.T.N.Kumar:
[email protected] Optical Methods in Drug Discovery and Development, edited by Mostafa Analoui, David A. Dunn, Proc. of SPIE Vol. 6009, 60090Y, (2005) · 0277-786X/05/$15 · doi: 10.1117/12.630990
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2. TIME DOMAIN FLUORESCENCE THEORY 2.1. General expressions The expression for the time domain fluorescence signal is formulated in the Born approximation (which considers a single fluorophore absorption and emission event, ignoring re-emission of the absorbed fluorescence), for a source detector pair at rs , rd and a single fluorophore at medium point r as: UF (rs , rd , r, t) = η(r)
t 0
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dt2 Gm (rd − r, t − t1 ) exp(−(t1 − t2 )/τ (r))Φx (r − rs , t2 )
(1)
with η(r) and τ (r) referring to the (unknown) yield and lifetime distributions and the subscripts x, m referring to excitation and emission wavelengths, and where Φx (r, rs , ω) and Gm (rd , r, ω) are the Green’s function and fluence for the excitation light from a point source to the fluorophore and emission from the fluorophore to the detector respectively. While the above expression as it stands is a double convolution and might be computationally intensive, solving the problem in frequency domain offers several advantages. The standard frequency domain expression for the fluorescence signal is written as3, 5 : x (r, rs , ω) m (rd , r, ω) iΓ(r)η(r) Φ F (rs , rd , r, ω) = G (2) U ω + iΓ(r) m (rd , r, ω) are the Frequency domain Green’s function and fluence for the ex x (r, rs , ω) and G where Φ citation light from a point source to the fluorophore and emission from the fluorophore to the detector respectively. Note that the above expressions incorporates the fluorescence lifetime as a distribution within the medium. Typically, one can use a Fourier component of the time domain data to invert the fluorescence yield and lifetime. An advantage of doing the image reconstruction analysis in the frequency domain is the potential use of the reference measurements4, 11 for the frequency domain and CW DOT measurements. In addition to eliminating unknown source-detector coefficients (which otherwise need to be determined by fitting), the normalization can also eliminate the temporal instrument response functions like the finite gate width response of gated CCD cameras and fiber dispersion, which appear as detection-wavelength independent factors in the frequency domain. This can be often a huge advantage since the precise determination of the instrument response is difficult and can potentially affect the accuracy of the reconstructions. With the forward model written as the matrix equation m (rd , r, ω)Φ x (r, rs , ω), y = A.F , where y constitutes the measurements, the weight matrix is A = G and F = iΓ(r)η(r)/(ω + iΓ(r)). The inversion for the complex quantity F is carried out using Tichonov regularization: F = (AT A + λI)−1 AT y (3) It is easily verified that the phase of F directly yields the fluorescence lifetime distribution.3
2.2. Multiple frequency analysis The size of the experimental data sets in a typical time domain experiment with multiple time points (or equivalently multiple frequencies) and high source detector density can be huge, resulting in very large weight matrices. It is therefore essential to select the most relevant information to avoid redundant data sets and the resulting computational complexity. While it is obvious that source detector numbers and field of view will directly affect spatial resolution due to the spatial nature of their arrangement, the influence of modulation frequencies on the quality of the reconstruction is not immediately clear. It has been suggested that multiple modulation frequencies may help enhance the spatial resolution for localized objects.9 A singular value analysis of the sensitivity matrix A can be very helpful in
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selecting optimal modulation frequencies. SVD analysis has previously been used to study the optimal configuration of sources and detectors,12, 13 we are interested here on the choice of optimal modulation frequencies for a fixed optode configuration. If the weight matrix is first decomposed into its SVD components as A = U SV T , the unitary matrices U and V represent the modes in measurement and image spaces, while the diagonal components of the singular value matrix represent the extent to which these modes are coupled. Smaller singular values result in stronger coupling between the image space and detection space modes and large image noise for a given measurement noise.12 Thus, the larger the number of singular values above a predetermined noise threshold, the better the quality of the reconstruction. Substituting the SVD decomposition into the Tichonov inverse (Eq. 3), we have −1 T A y F = U (S 2 + λ)U T
(4)
Thus, the Tichonov regularization parameter λ can be viewed as the noise threshold for the singular values,13 the optimal choice of which will depend on the measurement noise, among other parameters. For the simulation, we consider a linear array of 11 sources and 13 detectors, spaced 1mm apart. The choice of a small source detector set is to keep the size of the weight matrix small, since we are mainly interested in multiple modulation frequencies. The weight matrix was generated for frequencies in 80MHz steps, (corresponding to the repetition rate of the Ti:Saph laser). In Fig. 1 we plot the the number of useful singular values for two choices of the noise threshold, as a function of the number of frequencies used. It is seen that the rate of change of the useful singular values dramatically slows down as the number of frequencies crosses about 5. While employing multiple frequency data sets, the extra computational burden must also be kept in mind (each frequency component essentially increases the data dimension of the weight matrix by the total number of source detector pairs). Also, as shown in Fig. 1 the number of singular values decreases when higher frequencies are employed. We show this for a choice of one and two modulation adjacent modulation frequencies. This analysis therefore suggests that the reconstruction may be enhanced using the first 1-5 modulation frequencies.
2.3. Asymptotic analysis in the time domain The previous section employed a brute-force analysis of the time domain information using standard expressions in the frequency domain. However, experimental observations suggest alternative approaches to the problem. In fluorescence measurements with diffuse media, the fluorescence lifetimes are often revealed directly in the asymptotic tail of the decay signal. This suggests that fluorophore lifetimes within the biological medium are discrete rather than continuous, and can be directly inferred from the raw time domain decay using multi-exponential fits. Therefore, a model that incorporates prior knowledge of fluorophore lifetimes into the reconstruction is more appropriate. Considering a set of yield distributions ηn (r) with corresponding lifetimes τn = 1/Γn , the time domain fluorescence signal is written as (omitting scaling coefficients for simplicity): ∞ iΓn ηn (r) −iωt 3 Φx (r, rs , ω) UF (rd , rs , t) = dωe d r Gm (rd , r, ω) (5) ω + iΓn V −∞ n It is useful to examine the analytic nature (in the complex variable sense) of the integrand in Eq. 5 using the Greens functions for an infinite homogeneous medium. These are of the well known form 2 exp(ikx,m r)/4πDx,m with kx,m = (−vµax,m +iω)/Dx,m , where the subscripts x, m denote the excitation and emission wavelengths. The quantities µax,m , Dx,m (= v/3µsx,m ) and µsx,m denote, respectively, the absorption, diffusion and reduced scattering coefficients, and v is the velocity of light in the medium. It is evident from a direct inspection of these terms that the homogenous Green’s function and its spatial
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Figure 1. Singular value analysis of multiple frequency fluorescence weight matrix. The left panel shows the number of useful singular values as a function of the number of frequencies used, for two different choices of noise threshold (the higher threshold of 10−22 is more appropriate for typical experimental noise levels). The right panel shows the number of useful singular values for single and two-frequency weight functions, as a function of the modulation frequency.
derivatives are bi−valued owing to the square root in k. This implies branch points in the lower half plane at ω = −ivµax , −ivµam . In addition, the integrand in Eq. 5 possesses simple pole singularities distributed along the negative imaginary axis at ωn = −iΓn . On applying Cauchy’s integral theorem,14 UF separates into two parts, the first corresponding to fluorescence decay terms (arising from the residue at the simple poles) and the other corresponding to a diffuse photon density wave (arising from the integration on either side of the branch cut) : aF n (rd , rs ) exp [−Γn t] + aD (rd , rs , t) exp [−vµa t] . (6) UF (rd , rs , t) = n
where we have set µa = µax = µam , without any loss of generality of the results to follow. The amplitude aF n of the decay of the n − th fluorophore is readily obtained as the residue at the simple pole at −iΓn , and is in the form of a linear inverse problem for the yield distribution ηn (r) of the n − th fluorophore: aF n (rd , rs ) =
V
d3 rWn (rs , rd , r)ηn (r)
(7)
where the weight matrix for the inversion is given by x (r, rs , ω = −iΓn ). m (rd , r, ω = −iΓn )Φ W n = Γn G
(8)
The coefficient aD (second term in Eq. 6) is calculated as the contribution from the branch points and takes the following highly non-exponential form (for vµa > Γn , ∀n):
ηn (r) sin (ρsr + ρdr ) γ/D vΓn ∞ . (9) dγe−γt d3 r aD (rd , rs , t) = 2 D2 8π ρsr ρdr (γ + vµa − Γn ) V 0 n
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where ρsr = |rs − r| and ρdr = |rd − r|. The amplitudes for the case when Γn > vµa can also be similarly evaluated. We note that in the limit of instantaneous emission, i.e., Γn → ∞; ∀n, aD reduces to an analytical form15 for the temporal diffusion signal in the Born approximation from absorbing “perturbations” ηn (r). The above results were derived assuming a homogeneous infinite medium. But since a general solution for the inhomogeneous diffusion equation in a bounded volume may be written in terms of the homogeneous Green’s function and its normal derivatives at the boundary,16 it is plausible to assert that the complex plane structure of the integrand in Eq. 5 is reproduced for arbitrary inhomogeneous media. This implies that Eq. 7, which results from the contribution of the simple poles, can be generalized to arbitrary media by simply substituting the Greens function solutions of the heterogeneous diffusion equation with finite boundary models. Asymptotic behaviour of time resolved signals The inverse problem as expressed in Eqs. (7) and (8) and the subsequent generalization to arbitrary media enable the localization of multiple fluorophores from a lifetime analysis of asymptotic fluorescence decays. In order for these equations to be applicable to imaging in the presence of turbid tissue, it is first essential to determine the conditions when the first term of Eq. 6 is dominant, i.e., when the decay times of the measured signal are governed purely by the fluorescence lifetimes. It has been noted17 that the decay time of time resolved signals is affected by diffuse propagation effects for strongly scattering and weakly absorbing tissue, and for short fluorophore lifetimes. Direct inspection of Eq. 6 indeed suggests that when the absorption time scale τabs [= (vµa )−1 ] < Γ−1 n , the asymptotic behavior is primarily governed by the fluorescence decay. In order to quantify the influence of diffusive processes, we plot in Fig. 2, the decay time of simulated fluorescence signals for a range of optical properties (µa , µs ), assuming a slab diffusion model. It is clear from Fig. 2 that the influence of the absorption on the asymptotic decay time is intimately connected with the value of the scattering coefficient, with the absorption having a stronger effect for larger µs and Z. Also, an increase in medium thickness has a more pronounced effect on lifetime than a corresponding increase in scattering (maintaining the product µs × Z), as has also been noted recently.18 These results indicate that for medium thickness of several cm and physically reasonable reduced scattering (µs < 50cm−1 ), the fluorescence lifetime can be reliably extracted from the asymptotic tail of the TD signal provided the lifetimes satisfy τn > τabs (for heterogeneous media, τabs should be evaluated for the smallest absorption present in the medium). We note from the case of Z = 2, µs = 10/cm in Fig. 2 that the lifetime is recovered asymptotically even when τabs is much longer than the fluorophore lifetime of 0.5ns, indicating that the condition τn > τabs is relaxed for short propagation lengths. In what follows, it will be assumed that that we are in the regime of optical properties where the decay of the fluorescence signal is governed by the lifetime. Under this condition, it is clear that the first term of Eq. 6 is dominant, so that the asymptotic timedomain signal factorizes into a spatial and temporal part for each decay component. This implies that when only a single lifetime is present, the asymptotic decay rate is independent of source and detector x,m (rd , r, ω = −iΓ) are solutions to the CW diffusion equation (Helmholtz wave location. Also, since G equation) with a reduced absorption of µax,m (r) − Γn /v, the weight matrix Wn for the inverse problem in Eq. 8 is identical to that of a CW problem; a tomographic TD measurement with multiple lifetimes therefore separates asymptotically into equivalent CW “measurements”, each with only one lifetime component present. This fact vastly simplifies the computational complexity of the TD fluorescence problem for long times, which normally requires a double convolution.
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Figure 2. The asymptotic decay time of simulated TD fluorescence signals as a function of µa , for various combinations of µs and medium thickness in the transmission geometry. The fluorophore was located at the center of the slab, with a lifetime of 0.5ns. The circled line corresponds to the absorption time scale τabs = (vµa )−1 . The downward pointing arrows indicate the length scale associated with the fluorophore lifetime of 0.5ns.
3. PHANTOM EXPERIMENTS 3.1. Time domain fluorescence imaging system: sensitivity and linearity studies The experiments reported here are carried out using a time domain fluorescence imaging system consisting of a Spectra-Physics Ti:Sapph laser source (100fs pulse width, 80MHz rep. rate, 770nm excitation), and a gated intensified CCD camera from LaVision (500ps gate width, 560V gain, 100ms integration time). The light output from the laser was attenuated to a few mW and launched into a 200µm fiber using a fiber collimation package. The other end of the fiber was mounted on a computer controlled motorized translation stage placed below the dish containing the tubes. The light exiting the fiber was focused onto the surface of the object of interest using another collimation package, and served as the source. A key assumption in the fluorescence forward model as expressed in Eq. (1) is the linear dependence of the detected fluence on the fluorescence yield, which is a product of the fluorophore concentration, excitation coefficient (at the emission wavelength) and the quantum yield. Assuming that the dye excitation coefficient and quantum yield are independent of the dye concentration, the model assumption of linearity can be tested by measuring signals for various dye concentrations. Testing for linear behavior is essential for reliable quantitation of fluorophore concentrations in-vivo. In order to test for the linearity as well as to establish the threshold for the sensitivity, we performed phantom measurements with near-infrared dye IRdye800 from Li-Cor Biosciences with absorption and emission maxima at ≈ 770nm and 790nm. A petri dish was filled with intralipid (thickness 1.4cm) with optical properties µs =10
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cm−1 and µa = 0.2 cm−1 . A thin polypropylene tube of inner diameter 0.5mm was held immersed in the intra-lipid by two clips attached to an adjustable translation stage. Known concentrations IRdye were injected into the tube and the whole field CCD image was collected for all time gates. In Fig. 3 we show the peaks of the raw temporal data across a few pixel points on the camera image, for a set of 6 dye concentrations used. The raw camera images show that the imaging system is capable of detecting concentrations down to a about 100 nM (this will depend on the quantum yield). The dependence on the concentration is seen to be highly linear, suggesting that the system is capable of linear quantitation down to 100 nM. 1
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Figure 3. Demonstration of the linearity of the detected fluorescence signal with respect to fluorophore concentration. The peak of the time resolved curves are plotted for a set of points across the CCD image, as a function of dye concentration. The fluorescent dye was included in a thin tube immersed in a petridish, filled to 1.4cm with intra-lipid and ink solution (µs =10 cm−1 and µa = 0.2 cm−1 ).
3.2. Demonstration of lifetime based tomography In order to demonstrate the lifetime based tomography algorithm, discussed above, we used IRdye800, with lifetime altered by the use of different viscous environments. Two polypropylene tubes were placed with a 4.5mm vertical separation in a petri dish filled with intralipid+ink solution, with the dye mixed in aqueous and glycerol solvents. The data were collected in a transmission geometry, with The full temporal fluorescence signal collected at 830nm using a bandpass filter. A line of 41 sources and detectors across the tube, and placed 1 mm apart were employed (detectors were assigned on the full field camera image). The lifetime components were determined directly from the decay portion of a subset of the data with high SNR to be 0.5ns and 0.8ns, and the amplitude of each component was then estimated for all the source detector pairs using a linear fit with known fixed lifetimes. Figure 4 shows the x-z slice of 3−D reconstructions using the amplitudes in Eq. 7, compared with the standard frequency domain reconstruction (both in the diffusion model). The separation of the tubes is seen to be accurately obtained using the lifetime based approach. The reason for larger point spread function
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in the reconstruction of the deeper tube is not immediately clear, but could possibly be attributed to the Born approximation which ignores fluorescence re-emission. It is seen that the frequency domain reconstruction is once again less successful in resolving the axially located inclusions, with the peak of the reconstruction located in-between the true depths of the tubes. (b)
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Figure 4. Demonstration of lifetime based tomography with experimental data. (a) shows the measurement geometry, with two tubes (0.5mm inner diameter) immersed at depths 1.2cm and 0.7cm in an intralipid + ink solution(µa = 0.1/cm, µs = 10/cm) in a phantom, and filled with IRdye800 (Li-Cor Biosciences) mixed in distilled water (giving a lifetime of 0.8ns) and glycerol (lifetime 0.5ns) solutions (b) Reconstructions of fluorescence yield using the frequency domain diffusion model. (c) and (d) are reconstructions using the amplitudes of the 0.5ns and the 0.8ns decay components in Eq. 6. The ’+’ signs are the true fluorophore locations for reference.
4. SUMMARY We have explored the application of time resolved fluorescence tomography for 3-D localization of targets within a diffuse medium. Viewing the problem in frequency domain, we used a singular value analysis to estimate the the optimal number of frequency components, and showed that the first few (∼ 5-6) Fourier components provide most of the useful information in a time domain fluorescence signal. Further simplification of the forward problem is achieved when we employ an asymptotic time domain analysis based on extracting the lifetimes directly from the raw data. The information content in the asymptotic decay portion of the time domain signal was shown, using an analysis of the complex spectral behavior of diffuse fluorescence signal, to be no more than that of a CW (zero frequency) weight matrix. An outcome of this analysis is a novel algorithm to localize multiple fluorescent targets based on their lifetimes. This method achieves superior localization by first obtaining the lifetimes directly from time domain measurements using multi-exponential fits, followed by a separate reconstruction of the fluorophore localization for each lifetime component. In frequency domain experiments, the fluorescence lifetime can only be estimated after performing tomographic reconstructions, which are generally ill-posed. In addition, when multiple lifetimes are present, the reconstructed phase is the sum of contributions from all the lifetimes present, thereby necessitating procedures that fit to dispersion relations to extract the individual lifetimes.19 Time resolved measurements, on the other hand, allow a clean and direct separation of the fluorescent and diffuse components based on their distinct temporal responses. The method reported here can be considered as a natural extension of FLIM
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techniques to imaging fluorescent targets within turbid biological tissue. The increasing interest in the development of lifetime sensing fluorescent molecular probes makes the approach presented here a potentially highly relevant tool for optical imaging in non-invasive disease diagnostics and for drug discovery and development.
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