The ill-posed nature of fluorescence tomographic image reconstruction means that it is highly susceptible to the effects of noise and numerical errors. To address this issue, a US research team has developed an image reconstruction scheme based on regularization, which involves adding a penalty function to encourage certain properties in the solution. The researchers, led by Richard Leahy from the University of Southern California (Los Angeles, CA) and Simon Cherry from the University of California, Davis, validated their procedure using simulated and experimental data from a mouse-shaped phantom (Phys. Med. Biol. 57 1459).

The most common regularization method for handling ill-posed inverse problems includes the L2 norm penalty. This penalty generates the minimum energy solution, which tends to be spread out in space. Fluorescent probes, on the other hand, typically localize within particular regions, such as tumours.

"The idea of regularization is to alleviate the ill-posedness by making use of additional a priori knowledge," explained Joyita Dutta, from the University of Southern California, currently a postdoctoral fellow at Massachusetts General Hospital (Boston, MA). "Fluorescent probes are often designed to accumulate within specific areas of interest, resulting in a spatial distribution that is sparse overall and smooth locally."

Based on this knowledge, Dutta and colleagues designed a regularization technique that includes a combination of L1 and total variation (TV) norm penalties. The L1 norm penalty suppresses spurious background signals and enforces sparsity (a small number of non-zero voxels and zero intensity elsewhere), while the TV penalty preserves local smoothness in the reconstructed images.

Solving the problem

Next, the researchers developed an optimization method to solve the resulting reconstruction problem. They first used the preconditioned conjugate gradient (PCG) algorithm to minimize the L1 and TV penalties. While this algorithm is straightforward to implement, the computational time per iteration is large. So they also examined a method based on the separable paraboloidal surrogates (SPS) algorithm, using ordered subsets to accelerate this approach.

Comparing the convergence speeds of the SPS algorithm with ordered subsets (OSSPS) and the PCG algorithm revealed that, for starting points far from the true solution, the OSSPS approach was much faster than PCG, while PCG was significantly faster near the solution. Thus the researchers employed a compound approach using 10 OSSPS iterations to initialize the optimization problem, followed by PCG to determine the final solution.

Dutta and co-workers validated their optimization method using simulated and experimental data from a mouse-shaped phantom containing two embedded fluorescent line sources. Surface fluorescence data were collected at an emission wavelength of 720 nm, using a 3D fluorescence molecular tomography set-up with a 650 nm excitation source.

Using the simulated data, the team reconstructed the embedded line sources using the L2, L1, TV and joint L1-TV penalties. They compared these four regularization schemes using a range of performance metrics: mean squared error (MSE); mean and standard deviation of the signal over the region-of-interest (ROI); mean and standard deviation of the background signal, and signal-to-background ratio (SBR).

Simulation results showed that the L1, TV and joint L1-TV penalties generated lower MSE values, stronger ROI mean signal levels, and higher SBR values than the L2 regularizer. The joint L1-TV approach generated a smooth solution with a sparse background. This scheme had the lowest MSE value, the least mean background signal level, and a lower background standard deviation than the L1 and TV penalties individually.

Next, the researchers reconstructed the experimental data using the four penalties. Overlaying the L1-TV reconstructed image against a CT image of the phantom revealed "a reasonable degree of overlap between the reconstructed fluorescence molecular tomography image and the ground truth revealed by the CT image".

For quantitative evaluation of the experimental results, the researchers computed Dice coefficients, which represent the similarity between the (blurred) CT image and the reconstructed fluorescence images. Results indicated that the L2 and joint L1-TV penalties performed consistently well, while the relative performance of the individual L1 and TV penalties varied.

The researchers concluded that L1 or TV regularization, used in combination or separately, leads to improvements in localizing fluorescent sources. Qualitatively, the joint L1-TV images had the most natural appearance in the simulation and phantom studies, but the quantitative studies did not identify a clear winner. In terms of computational demand, the joint L1-TV regularization penalty is more demanding than the L2 norm, but comparable to individual L1 and TV norm penalties.

The authors point out that even with the use of these regularizers, fluorescence molecular tomography struggles to produce 3D images with equivalent resolutions to small-animal PET or SPECT. Increasing the number of illumination/detection wavelengths, or using more optimal spatial patterns for illumination, may lead to further improvements.

Dutta also noted that the mathematical framework presented in this work is not restricted to fluorescence tomography. "It may be extended to iterative reconstruction for any modality where the images tend to be sparse and piecewise smooth simultaneously," she told medicalphysicsweb.

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