Objective To overcome the ill-posedness of the bioluminescence tomography (BLT) reconstruction problem and obtain stable reconstruction results, researchers combined different prior information and regularization techniques to design various reconstruction algorithms. Among them, biological tissue structure information, a permissible source region, multi-spectral measurement information, and light source distribution sparseness are priori information widely used in reconstruction. The reconstruction algorithm based on regularization is divided into convex and non-convex optimization methods according to whether the objective function is non-convex. Although the regularization models of these reconstruction algorithms are different, the regularization parameter play a significant role in the reconstruction process, which directly affects the reconstructed image quality. Thus, the selection of the optimal parameters has always been a challenging problem for research. In this study, we proposed a multi-spectral BLT reconstruction method based on primal dual active set with continuation (PDASC) algorithm. The proposed method combines the primal dual active set (PDAS) algorithm with continuity technology, which can automatically adjust the regularization parameter to obtain a globally optimal solution.
Methods In this study, the iterative algorithm, PDASC, contains inner and outer iterations. The inner iteration part is the PDAS algorithm, which determines the active set based on the primal and dual variables. It then updates the primal and dual variables by solving the least square problem of the active set. The outer iteration combines the continuity technology of the regularization parameter. In the PDASC algorithm, the stopping criterion in the continuity technology directly affects the determination of the regularization parameter. Thus, it is essential to select an appropriate stopping criterion. If the noise level is known, we can choose the deviation principle as the stopping criterion. However, it is not easy to accurately estimate the noise level in actual situations. Thus, we choose the Bayesian information criterion that can adjust the regularization parameter to control the size of the active set and obtain a globally optimal solution.
Results and Discussion To verify the performance of the proposed PDASC algorithm in BLT, we designed multiple sets of simulation experiments compared with PDAS and HTP algorithms on the digital mouse model. The proposed algorithm was further examined with a mouse in vivo experimental data. The simulation results of the non-homogeneous digital mouse model showed that the Dice coefficient based on the PDASC algorithm can reach or exceed 65%, the positioning error is within 0.9 mm, and the contrast noise ratio is greater than 16.52 in single and double light source experiments (Table 2 and Table 3). These quantitative indicators validate that PDASC algorithm has the smallest reconstruction error, the highest quality of the reconstructed image, and the best reconstruction results of the shape and volume of the real light source (Table 2 and Table 3). For the double-source case, the PDASC algorithm has the highest shape fit of the reconstructed image and the best source-resolving ability (Fig. 5). The performance of PDASC algorithm on the three indicators for the in vivo experiment is also consistent with the simulations (Table 4). Above results indicate that the proposed PDASC algorithm is promising in practical tumor detection applications.
Conclusions In this study, we proposed a multi-spectral BLT reconstruction algorithm based on primal dual active set with continuation. The proposed algorithm combines the PDAS with the continuation technology for the regularization parameter, which can automatically adjust the regularization parameter to obtain the global optimal solution. Besides, the use of multi-spectral information reduces the ill-posedness of reconstruction. Multiple sets of simulations on a digital mouse confirm the effectiveness and stability of the proposed algorithm. The in vivo experimental results show the potential of the algorithm in practical applications. Although the proposed algorithm is better than the compared algorithms, it cannot fit the shape or contour of the light source perfectly. With the continuous development of deep learning, deep imaging algorithms have also appeared in the field of optical molecular imaging. Most of these algorithms are currently based on end-to-end neural networks, such as K-nearest neighbor local connection network, 3D deep encoder-decoder network, and stacked auto encoders neural network. By establishing the nonlinear mapping relationships between surface fluorescence and light source distributions, the deviation caused by the simplified linear model is avoided. However, the size of the training dataset, which plays a significant role in the depth imaging algorithm will directly affect the reconstruction performance. Thus, the focus of our future study is to determine how to combine the model with the network to solve the ill-posedness of BLT reconstruction.
