Using two dimensional synthetic frequency-domain measurements, the inverse imaging problem is solved for absorption and fluorescence lifetime mapping with the truncated Newton’s optimization scheme developed in Part I of this contribution. Herein, we present reconstructed maps of absorption owing to a fluorophore from excitation and emission measurements which detail the presence of tissue heterogeneities characterized by tenfold increase in fluorescent contrast agent. Our results confirm that fluorescence provides superior mapping of heterogeneities over excitation measurements. Using emission measurements we then map fluorescent lifetime under conditions of tenfold uptake of contrast agent in tissue heterogeneities. The ability to map fluorescent quenching and lengthening of contrast agents facilitates the solution of the inverse problem and further improves the ability to reconstruct tissue heterogeneities.
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In Part I of this contribution, we have formulated the absorption and fluorescence inverse imaging problem as an optimization problem in which an interior map of tissue optical properties of absorption and fluorescence lifetime can be reconstructed from excitation and emission frequency-domain photon migration (FDPM) measurements. Since significant challenges for the non-linear and large scale reconstructions are further exaggerated by the physiological low contrast of absorption and scattering property differences between normal and diseased tissues , we have focused upon the development of absorption and fluorescence lifetime imaging following the administration of a fluorescent contrast agent. Previously, we have conducted FDPM experimental measurements to show that the FDPM contrast owing to fluorescence exceeds that possible by monitoring absorption with excitation FDPM measurements . One objective of this contribution is to compare the reconstruction of absorption cross section owing to a fluorescent agent, which has a tenfold preferential uptake in the tissue region of interest using excitation and emission FDPM measurements. Secondly, we focus upon the reconstruction of fluorescent lifetime for cases in which a tenfold preferential uptake exists within the tissue volume of interest is accompanied by a shortening and lengthening of a first order fluorescence decay time (or fluorescence lifetime). In the following, we describe the generation of the synthetic data set for input into the optimization scheme, and then the results for absorption and fluorescence lifetime imaging. Finally, we comment on the availability of fluorescent contrast agents for inducing optical contrast and the future work to adapt our inversion strategies to actual FDPM measurements.
2.0 Generation of synthetic data set using forward simulator
In order to test our inversion a truncated Newton’s optimization scheme developed in ref. , we generated synthetic data of Φx,m in a two dimensional, square domain of 4 × 4 cm2. For the forward simulation, the mesh contained 2209 internal and 192 boundary nodes with 4608 triangular elements. It should be noted that a uniform grid is used in this exercise whereas for future work we adapt a finer grid near the source points where the gradients are very high. Four sources of intensity modulated excitation light were simulated midpoint on each side of the domain with a total of 59 point detectors simulated equidistant along the domain periphery excluding positions occupied by one of the four sources and at each corner of the domain. Predictions of Φx,m were made at each detector for each of the four sources, each providing intensity modulation at 50, 100 and 150 MHz with unit depth of modulation and zero phase. For this simulated measurement configuration there were 4 × 59 = 236 simulated observations obtained for reconstruction of interior optical properties at the excitation and emission wavelengths. At the excitation wavelength, “heterogeneities” were to be detected based upon their absorption, and at the emission wavelength, based upon their fluorescence lifetime. Table I lists the background and “heterogeneity” optical properties, as well as the location and size of the simulated “heterogeneities” used for the generation of synthetic data for reconstruction. The simulator was coded in Fortran 77 and took 6 seconds on a SUN Ultrasparc 10 Workstation (200MHz).
To mimic measurement error, zero-mean, Gaussian noise corresponding to 1.0% standard error in fluence was added to the synthetic data sets by using the formula:
where G(0,1) is a Gaussian distribution with zero mean and unit variance; Z = 0.01 ; and Φ″x are the simulated data without noise.
To simulate 0.1 degree noise in phase, we compute the error in the final complex fluence, Φ′x:
The synthetic fluence for emission is similarly generated and employed as input to the iterative inversion algorithm.
A user specified double precision parameter ε is used as the convergence parameter within the inversion scheme (see section 3.1 of reference ). The iterative method is stopped if the length of the gradient vector is less than ε. The final images reported herein are the results of iterations until the length of the gradient ∥g∥ is less than ε = 10-9, chosen by trial and error.
3.0 Results and discussion
A coarse mesh with 225 internal nodes and 64 boundary nodes (total 289 nodes) is used for the inversion problems, whereas total 2401 nodes with uniform mesh is used for forward problem. Using the synthetic data sets described in Section 2, we demonstrate the truncated Newton method for reconstruction of absorption from excitation and fluorescence FDPM measurements as well as for reconstruction of lifetime from fluorescence FDPM measurements. For this mesh one function (Ex,m)evaluation took 9.1 seconds and one function and gradient calculation by reverse differentiation took 10.5 seconds.
3.1 Absorption imaging from synthetic excitation measurements
As described in Table I, Case I synthetic data set consisted of detecting three “heterogeneities” based upon a ten-fold increase in absorption as might occur upon the administration of a contrast agent. Figure 1(a) illustrates the actual distribution of absorption within the phantom with a background absorption μaxf value of 0.02 cm-1. Upon using the background absorption value as an initial starting guess, our attempt to reconstruct the spatial distribution of absorption with simulated measurement with noise was aborted after 25 iterations because the results remain unchanged. The result, illustrated in Figure 1(b), shows that the heterogeneities may be located and differentiated from the relatively uniform background. However, the differentiation of the heterogeneities from one another and the quantitation of their ten-fold increase in absorption are disappointing. The average values of the reconstructed absorption estimate of each of the three “heterogeneities” are plotted as a function of iteration in Figure 1(c). While oscillations in the parameter estimates occur in the initial iterations, the average values appear to smoothly approach an absorption value that underestimates their actual values. These numerical approximation errors may be due to inadequate discretization of the model geometry by the finite element method. Since the variations of the gradients near the source are very high, in future work we use a finer grid near the boundary. The choice of regularization schemes, finer mesh near the source point and optimal placement of sources and detectors should assist in the convergence.
3.2 Absorption imaging from synthetic fluorescence measurements
As described by Sevick et al.,reconstruction from excitation measurements is inherently disadvantaged by the small influence of light-absorbing heterogeneities on the excitation signal measured at the interface. Figure 2(a) represents the reconstruction of the ten-fold absorption owing to fluorophore in three heterogeneities from the synthetic fluorescence FDPM measurements as described by case II in Table I. Figure 2(b) shows the corresponding average values within each of the heterogeneities as a function of iteration. As expected from the superior contrast offered by fluorescence over absorption and the linearity of emission equation (see Eqn (2) of ref ), the convergence upon an absorption map from fluorescence measurements occurs faster and more accurately than from excitation measurements. Reconstruction of absorption from excitation requires recalculation of the global matrix at each iteration while the global matrix in absorption or lifetime reconstructions from emission remains unchanged throughout the calculation.
3.3 Lifetime imaging from synthetic fluorescence FDPM measurements
While the contrast owing to absorption may provide localization of tissue disease, the discrimination of diseased tissues based upon changes in lifetime of exogenous fluorescent contrast agents has been demonstrated in endoscopic applications  and proposed for optical tomography [5–7]. In the following two studies, we demonstrate the truncated Newton method for demonstrative cases of fluorophore quenching (shortening of fluorophore lifetime) and enhanced activated fluorophore stability (i.e., lengthening of fluorophore lifetime) within tissue “heterogeneities.”
3.3.1 Imaging based upon lengthening of fluorophore lifetime
Figure 3(a) depicts the actual lifetime map of the Case III study listed in Table I in which three heterogeneities with ten-fold uptake of fluorescent dye exhibit a lengthening of lifetime (τb=10 ns) within a background which possessed a shorter lifetime (τh=1 ns). With an initial starting guess equal to the background lifetime, the reconstructed maps of lifetime are shown for 50MHz (Figure 3(b)); 100 MHz (Figure 3(c)); and 150 MHz (Figure 3(d)). The quality of the reconstruction map seems to improve with increasing modulation frequency and Figure 4(a) through (c) suggests more rapid convergence at increased modulation frequencies. It is noteworthy that using the Levenberg-Marquardt approaches, Paithankar, et al.  and Jiang  report only reconstruction involving fluorophore quenching and were unable to successfully reconstruct heterogeneities with longer lifetimes within a background of short lifetime. Two additional observations of the results can be made:(i) while the recovered values of the absorption from excitation measurements underpredict their local absorption coefficient (0.05 as opposed to 0.20 cm-1), the recovered values of lifetime appear to be closer to the actual values (7 instead of 10 ns); (ii) while the absorption imaging from excitation FDPM measurements (Figure 1) did not achieve convergence after 25 iterations, convergence of fluorescence lifetime imaging was achieved in 20–30 iterations. As described below similar results were obtained for fluorescence lifetime imaging involving fluorophore quenching.
3.3.2 Imaging based upon fluorophore quenching
Figure 5(a) depicts the actual lifetime map of the Case IV study listed in Table I in which three heterogeneities with ten-fold uptake of fluorescent dye exhibit a shortening of lifetime (τh=1 ns) within a background which possessed a longer lifetime (τh=10 ns). We assume all other parameters are known. With an initial starting guess equal to the background lifetime, the reconstructed maps of lifetime are shown for 50MHz (Figure 5(b)); 100 MHz (Figure 5(c)); and 150 MHz (Figure 5(d)). The quality of the reconstruction map again seems to improve with increasing modulation frequency and Figures 6(a) through (c) again suggests rapid convergence at increased modulation frequencies and greater efficiency than can be shown with absorption imaging.
4.0 Conclusions and Future Work
In this contribution, we demonstrate the truncated Newton’s optimization scheme for absorption and fluorescence optical tomography. We have presented a method to calculate the gradient of error function based on reverse differentiation method in a finite element method based scheme. This method is programmed by hand so that the overhead problems usually associated with automatic differentiation do not occur. It is shown that a function and the gradient calculations are cheaper than three function evaluations using reverse differentiation. Our work confirms that reconstruction of absorption owing to a contrast agent is enhanced if fluorescence measurements are made over excitation measurements. In addition, the ability to detect heterogeneities with a tenfold uptake of dye that experiences lengthening and shortening of fluorescence lifetime is demonstrated. In practice, FDPM optical tomography must be accomplished with a substantive number of source-detector measurements and for three-dimensional geometries [9,10]. While work continues to develop three-dimensional multi-pixel FDPM measurements  using fluorescent contrast agents , the development of effective inversion strategies to handle large data sets at minimal computational cost and storage burden is paramount. We believe the continued development of gradient-based optimization schemes, such as presented herein, are necessary for stable recovery of interior optical property maps for medical imaging.
This work is supported in part by the National Institutes of Health Awards (R01CA67176 and K04CA68374) and Department of Defense Army Medical Command (RP951661). We would like to thank D. Hawrysz and J. Lee for their careful reading of this paper.
References and links
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