1.

INTRODUCTION

Fluorescence molecular imaging (FMI) can label specific molecules or cells with targeted fluorescent molecular probes to detect physiological and pathological changes in vivo at the molecular or cellular levels in advance^1,2. However, FMI cannot quantitatively detect concentration changes of fluorescent molecular probes. Therefore, fluorescence molecular tomography (FMT) technology has been further presented. Owing to its high sensitivity and specificity, FMT has great application potential in tumor diagnosis, drug development, and treatment evaluation owing to its high sensitivity and specificity^3-5.

However, achieving accurate FMT reconstruction is a challenging task, which comes from the following reasons. Firstly, the simplified photon propagation model is difficult to accurately describe the propagation process of photons in biological tissues, which hinders the improvement of detection accuracy. Secondly, the inverse problem of FMT is severely ill-posed due to the strong scattering of photons in biological tissues. In addition, the inverse problem of FMT is also ill-conditioned because the measured values of the tissue surface are much smaller than the internal nodes, which makes the solution process vulnerable to system noise and model errors, resulting in poor noise immunity and low robustness.

In order to address the severe ill-posedness and ill-conditioned of the inverse problem to a large extent and enhance the reconstruction accuracy, various reconstruction algorithms have been presented in recent decades. On the one hand, feasible region methods^6-8 are proposed, i.e., the interior distribution of the fluorescent source is inferred from its distribution area on the surface of the object, thus reducing the number of unknowns in the equation to be solved. However, due to the subjective nature of the selection of feasible regions, this approach does not fully address the ill-posedness of the inverse problem. On the other hand, more constraints are added to introduce a priori knowledge to the reconstruction process. The constraints are usually implemented by introducing a regularization term into the objective function^9-13. The FMT reconstruction is converted into an objective function optimization problem with constraints, and the optimal solution of the reconstructed image is obtained through the optimization method. The commonly used penalty term includes the L2 norm⁹, L1 norm¹⁰, L0 norm¹¹, TV¹², group sparsity¹³, etc. These methods are usually effective only under certain assumptions; however, these methods typically involve complex iterative computations and time-consuming reconstruction processes. Therefore, it is crucial to explore more efficient methods or solutions to achieve accurate FMT reconstruction.

Recently, deep learning techniques with powerful generalization capabilities have shown great advantages in FMT reconstruction and have obtained more competitive reconstruction performance than traditional algorithms^14-16. Inspired by these works, we propose a novel multi-attention prior based residual encoder-decoder network for fast and accurate FMT reconstruction. First, the multi-attention mechanism can provide weighted a priori information to the fluorescence source, enabling MAP-REDN to effectively mitigate the ill-posedness to enhance the reconstruction accuracy. Second, since the direct reconstruction strategy is adopted, the complex iterative computation process in the traditional regularization-based algorithms can be avoided more effectively, thus greatly accelerating the reconstruction process. Experiments demonstrate that the proposed MAP-REDN method achieves faster and more accurate reconstructions than the cutting-edge methods.

2.

METHOD

2.1

Light propagation model

The light propagation process can be represented by the coupling of diffuse equation (DE) and Robin-type boundary conditions^17,18, which is defined by,

The above-detailed explanation can be found in References^13,15. Solving equation (1) with the finite element method (FEM) [19] in the discrete domain yields the following linear relationship,

where W is the system matrix, y is the boundary fluorescence measurement, and x denotes the distribution of fluorescent sources to be reconstructed in biological tissues.

Therefore, the FMT inverse problem is to reconstruct the fluorescence distribution x in equation (3). However, the inverse problem is ill-posed owing to the strong scattering effect and insufficient measurement data (i.e., y<<x). To alleviate the ill-posedness and obtain good reconstruction results in a robust manner, a regularization term is generally introduced as a constraint term to optimize the inverse problem, which is formulated as equation (3),

where λ denotes the regularization parameter used to balance the regularity term and data term.

2.2

MAP-REDN for FMT

In this section, we develop an deep learning method to establish a mapping relationship between surface photon measurements (y) and internal fluorescence source distribution (x), as shown in equation (4),

where f:R → R denotes the deep neural network, y denotes the surface fluorescence measurement signals.

The model structure of the MAP-REDN is shown in Figure 1. The designed multi-attention module is that the feature representations extracted from the network encoder are provided to the decoder module. We define the feature map extracted from the network encoder as , where D denotes the depth (i.e., the number of FMT projections), C represents the number of feature maps, and H, W represents the height and width of the FMT projections or feature maps, respectively. Feature compression and fusion are calculated as follows,

Figure 1.

The model structure of the proposed MAP-REDN.

where represents the fused features of Z₅ and Z₄, represents the fused features of Z₄ and Z₃, represent the corresponding compressed convolutions, represents the fused convolution of Z₅ and Z₄, and ⊕ represents the feature concatenation. The final output of the multi-branch attention prior module is calculated as follows:

where represents the fused convolution of Z₄ and Z₃, represents the fused output convolution.

2.3

Image quality evaluation

In this work, we use position error (PE), normalized mean square error (NMSE) and Dice to quantitatively assess the reconstruction performance of the MAP-REDN method,

here P_rec and P_true represent the reconstructed tumor centers and ground truth centers, respectively, X and represent the actual and reconstructed values, respectively, and |x| represents the number of voxels in the region X. The larger the Dice, the better the reconstruction performance, with 1 as the upper limit; while the smaller the PE and NMSE, the better the localization performance, with 0 as the lower limit.

3.

EXPERIMENTS AND RESULTS

To comprehensively assess the reconstruction capacity of MAP-REND in terms of accuracy, numerical simulation experiments were performed. The classical iteration-based L2-regularized, L1-regularized, and learning-based 3D-En-Decoder methods were utilized for comparison. The reconstruction results of the four different methods are shown in Figure 2.

Figure 2.

Geometry configuration of the numerical simulations. (a): The 3D display of the cylindrical phantom; (b): The 2D display of the cylindrical fluorfscent targets with the EED of 5mm.

Figure 2a displays the configurations of the numerical simulation experiments. A cylinder phantom model was put on a rotating stage, the axis of rotation is defined as the Z axis, and the bottom plane is set to Z=0 cm. A excitation light source is placed at a height of Z=0.75 cm to excite the fluorophore to be reconstructed. The fluorescence projection images were collected at 15° intervals for a full 360° view (i.e., 24 projection data were used for reconstruction). The cylindrical phantom had a height of 1.5 cm and a diameter of 3 cm. Two fluorescent targets (Figure 2a) with a diameter of 3 mm, a height of 5 mm and an edge-to-edge (EED) of 5 mm (Figure 2b) were put inside the cylinder phantom (Figure 2b). Furthermore, a level of 5% Gaussian noise was inserted into the measurement data to better mimic the real experimental situations.

Qualitative analysis is shown in Figures 3a-3d, including a 3D view of the entire imaged phantom and a sliced 2D view of the target localization plane. It can be seen that the L2-regularized method obtains a over-smoothed reconstruction result, which makes the reconstructed fluorescent target much larger than the real target with lower spatial resolution. Second, although the L1-regularized and 3D-En-Decoder methods can approximately reconstruct the tumor region, the two methods face the reconstruction problem of small discontinuous tumors located near the ground truth region. In contrast, the proposed MAP-REDN method achieves high morphological similarity and low position error. Furthermore, a series of evaluation metrics are used to quantitatively evaluate the reconstruction performance. As shown in Figure 3e, compared with the L2-regularized, L1-regularized and 3D-En-Decoder methods, the MAP-REDN has lower PE and NMSE and higher Dice. Therefore, quantitative analysis showed that MAP-REDN performed better in tumor localization and morphological similarity. Therefore, it can be concluded that the multi-attention mechanism is able to provide weighted prior information for the fluorescence sources, alleviating the severe ill-posed nature of the inverse problem and further improving the reconstruction performance.

Figure 3.

The qualitative and quantitative analysis of the four different reconstruction methods. (a)-(d): The white dotted circles represent the true fluorescent sources. The red polygons clustered around the red spheres in the 3D view are assumed to be reconstructed tumors; (e): The quantitative analysis results of the four different reconstruction methods.

4.

CONCLUSION AND DISCUSSION

In this work, we develop a novel MAP-REDN method to achieve fast and accurate FMT reconstruction. The motivation behind this method is that spatial feature mapping and reorganization can be enhanced by introducing a multi-attention mechanism to alleviate the severe ill-conditionedness and enhance the reconstruction accuracy. Experiments show that compared with the traditional L2-regularized, L1-regularized methods and the deep learning-based 3D-En-Decoder method, the MAP-REDN method has improved reconstruction accuracy, morphological similarity and spatial resolution. This method has great potential to provide a potential solution for FMT and alternative imaging models in pursuit of higher reconstruction accuracy.