The asymptotic lifetime-based algorithm was validated on the TD system using the mouse phantom described in Section III-C. The two inclusions in the phantom were filled with a NIR dye (3,3′-Diethylthiatricarbocyanine) with absorption and emission maxima near 755 and 790 nm. To achieve lifetime contrast, two solutions were prepared with 0.1 mg of the dye dissolved in 50-mL distilled water and glycerol solutions. The lifetimes of the dye in aqueous and glycerol solvents were measured to be near 0.5 and 1 ns, respectively. The increase in fluorescence lifetime with viscosity reflects the reduced ability of the dye molecules in their excited state to relax through rotational modes, which in turn reduces the nonradiative decay rate [74
The longer lived dye solution in glycerol was injected in inclusion B and the shorter lived aqueous dye solution of the dye in inclusion A (see ). The mouse phantom was placed in the imaging setup shown in and the data were collected in the transillumination geometry for 21 source positions and 93 assigned detectors on the camera image (see and ). Before dye injection into the two inclusions, the incident fields Ux,x and Um,m were measured for excitation at 755 and 790 nm, respectively. The fluorescence data was collected after dye injection using a 790–10 nm width band pass filter (CVI – laser), for excitation at 755 nm. Typical values for the other experimental parameters were intensifier gain: 500 V, gatewidth: 300 ps, and CCD integration time: 100 ms. In , a representative sample of the DFTR for a single S–D pair is plotted, along with the IRF and the temporal response of the excitation light. The initial time t0 determined from the IRF is also indicated (see Section III-B).
Fig. 5 (a) Representative TD data of the impulse response (dotted line), diffuse excitation signal, Ux,x at 755 nm (dashed line) and diffuse fluorescence signal Ux,m at 790 nm (dots) for a single S–D pair. The vertical arrow indicates t0 determined from (more ...)
The global DFTR (see Section III-E) for a single S–D measurement with the mouse phantom is shown in (inset). The dramatic improvement in the noise statistics of the global DFTR over that for a single SD pair is evident from . Although the global DFTR can recover two independent lifetimes, further improvement in the robustness of the fit can be obtained if one of the lifetimes is known in advance, as in a controlled phantom experiment as reported here. This is also possible in practical in vivo applications, if the lifetime of the dye within a control animal can be estimated separately. Considering this scenario, the global fit was performed with the shorter lifetime fixed at 0.5 ns, and the second lifetime was recovered as τ2 = 0.95 ns, as expected. also shows the raw data and the linear fit for a representative DFTR for a single S–D pair, along with the individual contributions from the shorter and longer lived components.
The S–D coupling coefficients and the background absorption were fit using (10
) and (11
), as described in Section III-F. A homogenous background absorption was assumed as a first-pass approach, although the formalism presented here can readily incorporate the effect of background absorption due to fluorophores themselves. For the mouse phantom used here, where the top inclusion (A) is located further along the source to detector axis than the bottom inclusion (B) (), the homogenous (or linear) approximation primarily amounts to a neglect of preabsorption of the excitation light to A by any fluorophores present in inclusion B, and the reabsorption of the fluorescence emitted from B by the fluorophores present in A. Since the inclusions are small and the fluorophores are effectively nonscattering, these two effects are expected to only alter the relative intensities of the fluorophore strengths in the inclusions and not their localization (see below), which is the main concern of this paper. Moreover, the actual volume of dye present within the inclusion was not known precisely due to imperfections in the phantom.
The decay amplitudes for the two lifetime components were inverted to recover the fluorescence yield localizations as prescribed in (5
). The set of amplitude coefficients of each life-time component for all S–D pairs formed the data vector for the inversion of the yield distributions η1
) and η2
) using (14
). Although the Monte-Carlo weight functions were calculated for the full set of 413 000 medium voxels (0.125 mm3
volume) of the original mouse segmentation (see Section III-F), the inversion was performed with a reduced weight matrix of 50 000 voxels (1 mm3
) for computational ease. The reconstruction results are shown in as 3-D images overlayed with the surface boundary of the phantom. The images are iso-surface maps at 95% peak intensity. The regularization parameter λ was near unity for the reconstructions shown and was chosen based on an empirical assessment of image quality. The yield reconstruction for the shorter lifetime of 0.5 ns is seen to have a shallower depth than that for the longer lifetime of 0.95 ns, as expected. shows the reconstructions performed using the CW component of the entire TD data [(8
)]. It is clear that the CW reconstruction fails to resolve the two targets, and is biased towards the inclusion containing the longer lifetime. This bias could be attributed to the fact that the longer lived fluorescence of the dye in the glycerol solution leads to a higher CW intensity than the shorter lived one, since it loses less energy through non-radiative channels [57
], assuming identical concentration and excitation conditions. This fact is also evident from the expression for the CW forward model in (8
), where the contribution of yield for each lifetime component is scaled by the lifetime. Also, note from the raw time resolved signal in that the shorter lived component has a much weaker contribution to the net fluorescence decay than the longer lived component. We note here that this bias could not possibly arise from any sensitivity variations between the regions occupied by the two inclusions, since they were located approximately symmetrically with respect to the sources and detectors.
Fig. 6 3-D reconstructions of TD fluorescence data using the asymptotic and CW components. (a) Asymptotic lifetime based reconstructions of the DFTR from the mouse phantom injected with the 0.5 ns dye solution in the top inclusion (A in ) and the 1 ns (more ...)
The asymptotic TD reconstructions were compared with CT images of the mouse phantom (Section III-C), to verify accuracy. Coregistration was performed between the high-resolution (0.125 mm3) image of the 3-D camera and the CT image, using an affine-transformation with fiducial points chosen from identifiable features on the outline of the images. Once the transformation matrix was determined, the optical images were rotated to the CT-image plane after being first interpolated to revert to the original resolution of the 3-D camera image. The overlayed images in show that the peak of the asymptotic yield reconstructions match the true locations of the inclusions as recovered by the CT image reasonably well. In future work with phantoms, we will use more well defined fiducials common to the optical and CT images to improve the accuracy of the coregistration and to better quantify the accuracy of the asymptotic reconstructions.
Fig. 7 Co-registration of the optical reconstructions with the CT reconstructed images of the mouse phantom. Two sagittal planes containing the centroid of each inclusion are shown. The CT image is shown in gray scale, with the mouse background visible as dark (more ...)
For comparison, separate measurements were performed with fluorophores occupying only one inclusion at a time, viz., with the glycerol solution occupying B with A empty, and with the aqueous solution occupying A with B empty. shows a vertical slice of the depth dependent yield for all cases, at the X, Y location of the maxima of the corresponding reconstructed yields. Also shown in the figure are the CT recovered centroids of the inclusions, along the Z-direction. The optical reconstructions are seen to match the locations recovered by CT. It is also seen that the reconstructions with either inclusion A or B filled with the short or the long lifetime fluorophores nearly match the corresponding yield localizations recovered from the combined data set. This indicates that the neglect of background fluorophore absorption has not affected the yield localization in a significant way. However, it is also noted from (legend) that the strength of the reconstructed yield for inclusion A was approximately a factor of 3 smaller than that for inclusion B, although equal concentrations of dye were used in both. This could be attributed to several factors. The homogeneous assumption could underestimate the yield distribution of inclusion A more than that of B. This is due to the fact that preabsorption of the excitation light by fluorophores in B occurs near the peak of the fluorophore absorption, whereas the reabsorption of the fluorescence from B by the fluorophores in A occurs much further from the absorption maximum. In addition, the presence of glycerol increases the lifetime (as discussed earlier in this section), and could also alter the extinction coefficient, which was not taken into account in this work. Finally, there was also an experimental uncertainty in that the volume of fluorophore solution in each inclusion was not known accurately, due to a slow leakage of the dye from the inclusions back to the injection tubes. A more controlled experiment is needed to verify the quantitation accuracy of the reconstructions and will be carried out in future work.
also shows that the widths of the asymptotic yield reconstructions (> 1 cm at 80% of the peak) are much larger than the center-to-center separation between inclusions (≈ 6 mm). This illustrates the key advantage of the asymptotic approach: it can resolve lifetime-sensitive targets that are well within the resolution limits imposed by diffuse optics (which determines the widths of the individual distributions). This is made possible since the yield distribution in the asymptotic approach is a distinct
image for each lifetime, unlike the CW method which recovers the sum of the two yield distributions. It is also noteworthy that the lateral (i.e., along the length of the mouse, or Y-axis in and ) center-to-center separation of the inclusions was near 1.6 mm, which is more than a factor of 2 smaller than what was previously achieved (4.3 mm) using free space CW measurements [8
The asymptotic TD yield reconstructions will be affected by errors in lifetime estimation. An error in estimating lifetimes affects the yield reconstructions indirectly through the corresponding error in the decay amplitudes. The problem of fitting noisy decay data with multiple exponentials is complex and has been dealt with extensively in the literature related to microscopic FLIM (see, for example, [77
]). The global analysis employed in this work offers a robust approach to this problem, since the determination of lifetime is not subject to the highly noisy data sets from the individual measurement pairs. The stability of the global analysis was verified as follows. The biexponential fit for the global analysis was performed with one lifetime having a range of 0.4–0.6 ns, which is ±20% with respect to the true value of the short lifetime component of 0.5 ns. This mimics the scenario where the a priori
lifetime is not known accurately. For each case, the second lifetime component was determined from the global fit, and found to be within the range, 0.9–1 ns, which is a ≈ ±6% error with respect to the true value of 0.95 ns. Subsequently, the amplitude coefficients for both lifetimes were determined for each S–D pair using a linear fit as described in Section III-E. Tomography was then performed for each case, and the maximum error in depth localization of the short lived component (inclusion A) was estimated to be 1 mm. The error in the localization of the long lived component (inclusion B) was well within the 1-mm voxel resolution used, and was estimated from the relative variation in the lifetime of this component (±6%) compared to that of the shorter component (±20%) to be near 0.3 mm. These results indicate the robustness of lifetime based tomography using a global lifetime estimation approach, given that a ±20% error bar in lifetime estimation is well above what we can typically achieve experimentally. In future work, we will present a more detailed study of the influence on the lifetime based reconstructions, of various other factors such as the noise statistics of the data collected, the values of the lifetimes involved, their separation, the point at which the asymptotic data is truncated, and the fluorophore distribution.