## Abstract

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.

© Optical Society of America

## 1. Introduction

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 [2], 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 [3]. 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. [1], we generated synthetic data of Φ_{x,m} in a two dimensional, square domain of 4 × 4 cm^{2}. 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 Φ_{x,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}:

$$\phantom{\rule{10em}{0ex}}\frac{\mathrm{tan}\theta +\mathrm{tan}\left({0.1}^{*}G\right)}{1-\mathrm{tan}\theta \phantom{\rule{.2em}{0ex}}\mathrm{tan}\left({0.1}^{*}G\right)}=\frac{\mathit{img}\left({\Phi}_{x}^{1}\right)}{\mathrm{re}\left({\Phi}_{x}^{1}\right)}$$

$$\phantom{\rule{10em}{0ex}}\frac{\frac{\mathit{img}\left({\Phi}_{x}\right)}{\mathit{re}\left({\Phi}_{x}\right)}+\mathrm{tan}\left({0.1}^{*}G\right)}{1-\frac{\mathit{img}\left({\Phi}_{x}\right)}{\mathit{re}\left({\Phi}_{x}\right)}{}^{*}\mathrm{tan}\left({1.0}^{*}G\right)}=\frac{\mathit{img}\left({\Phi}_{x}^{1}\right)}{\mathrm{re}\left({\Phi}_{x}^{1}\right)}$$

$$\phantom{\rule{10em}{0ex}}\frac{\mathit{img}\left({\Phi}_{x}\right)+\mathit{re}{\left({\Phi}_{x}\right)}^{*}\mathrm{tan}\left({0.1}^{*}G\right)}{\mathit{re}\left({\Phi}_{x}\right)-\mathit{img}{\left({\Phi}_{x}\right)}^{*}\mathrm{tan}\left({0.1}^{*}G\right)}=\frac{\mathit{img}\left({\Phi}_{x}^{1}\right)}{\mathrm{re}\left({\Phi}_{x}^{1}\right)}$$

$$\phantom{\rule{10em}{0ex}}\mathit{img}\left({\Phi}_{x}^{1}\right)=\mathit{img}\left({\Phi}_{x}\right)-\mathit{re}{\left({\Phi}_{x}\right)}^{*}\mathrm{tan}\left({0.1}^{*}G\right)$$

$$\phantom{\rule{10em}{0ex}}\mathit{re}\left({\Phi}_{x}^{1}\right)=\mathit{re}\left({\Phi}_{x}\right)-\mathit{img}{\left({\Phi}_{x}\right)}^{*}\phantom{\rule{.2em}{0ex}}\mathrm{tan}\left({0.1}^{*}G\right)$$

$$\mathrm{Now}\phantom{\rule{.2em}{0ex}}\mathrm{our}\phantom{\rule{.2em}{0ex}}\mathrm{new}\phantom{\rule{.2em}{0ex}}\mathrm{fluence}\phantom{\rule{.2em}{0ex}}\mathrm{is}:\phantom{\rule{2em}{0ex}}{\Phi \prime}_{x}=\left(\mathit{re}\left({\Phi}_{x}^{1}\right),\phantom{\rule{.2em}{0ex}}\mathit{img}\left({\Phi}_{x}^{1}\right)\right)$$

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 [1]). 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
(*E _{x,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 μ_{axf} 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*.[3],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 [1]), 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 [4] 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
(τ_{b}=10 ns) within a background which possessed a
shorter lifetime (τ_{h}=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*. [5] and Jiang [8] 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
(τ_{h}=1 ns) within a background which possessed a
longer lifetime (τ_{h}=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 [11] using fluorescent contrast agents [12], 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.

## Acknowledgements

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.

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