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Abstract
Fluorescence molecular tomography (FMT) has been a promising imaging tool because it allows an accurate localizaton and quantitative analysis of the fluorophore distribution in animals. It, however, is still a challenge since its reconstruction suffers from severe ill-posedness. This paper introduces a reconstruction frame based on three-way decisions (TWD) for the inverse problem of FMT. On the first stage, a reconstruction result on the whole region is obtained by a certain reconstruction algorithm. With TWD, the recovered result has been divided into three regions: fluorescent target region, boundary region, and background region. On the second stage, the boundary region and fluorescent target region have been combined into the permissible region of the target. Then a new reconstruction on the permissible region has been carried out and a new recovered result is obtained. With TWD again, the new result has been classified into three pairwise disjoint regions. And the new fluorescent target region is the final reconstructed result. Both numerical simulation experiments and a real mouse experiment are carried out to validate the feasibility and potential of the presented reconstruction frame. The results indicate that the proposed reconstuction strategy based on TWD can provide a good performance in FMT reconstruction.
© 2018 Optical Society of America
1. INTRODUCTION
As a promising optical molecular imaging tool, fluorescence molecular tomography (FMT) has the advantages of high sensitivity, low cost, non-invasiveness, non-ionization, and so on [1,2]. It has gained a great deal of attention from researchers. Since it allows fast resolving of the three-dimensional visualization of fluorophore within small animals in vivo, FMT has been successfully applied in cancer diagnosis, drug development, and therapeutics assessment [1–4].
However, its reconstruction is a highly ill-posed inverse problem [5]. In order to obtain a stable and meaningful solution, researchers developed a variety of strategies for FMT. Regularization is typically used to tackle the inverse problem, such as well-known Tikhonov regularization [6,7] and sparisty regularization [8,9]. In [10], Jiang et al. have proposed a novel -norm optimization for FMT. The method not only achieves accurate and desirable fluorescent source reconstruction, but also demonstrates enhanced robustness to noise. In [11,12], the authors present a modified nonlinear anisotropic diffusion regularization method, which is called patch-based anisotropic diffusion with wavelet patch compression, for FMT reconstruction. This new method has the ability to preserve important features in the reconstructions. In [13], Mohajerani et al. have proposed a Fuzzy inference system for the minimization of FMT, which does not need to set the regularization parameters. In addition, some hybrid imaging systems, which combine other imaging modalities with FMT, have been designed. They provide priori information and further improve the reconstruction accuracy [6,7,14–18]. For example, anatomical information from the anatomical imaging modalities in the form of priors has also been incorporated into the inversion problem to improve the solutions. In particular, x-ray computed tomography (CT) has been combined with FMT, which is referred to as FMT-XCT [6,17,18]. The permissible region has also been employed to speed up the reconstruction and improve the quality of solutions [19–21]. However, the extraction of the permissible region is based on the two-decision theory; namely, the reconstruction region is divided into the permissible region and the non-permissible region. In this paper, we developed a reconstruction frame for FMT based on three-way decisions (TWD). This frame not only provides a permissible region but also allows a post-process of the recovered results. A numerical simulation experiment and an in vivo experiment on a small mouse are carried out to test the feasibility and potential of the reconstruction frame.
This paper is organized as follows. The inverse problem of FMT, TWD theory, and reconstruction frame based on TWD are presented in Section 2. In Section 3, a numerical simulation experiment and small animal experiment are conducted to test the frame. Finally, we discuss the results and present a conclusion in Section 4.
2. RELATED WORK
A. Reconstruction Problem
Usually, the near-infrared photon propagation in biological tissues has the characteristics of high scattering and low absorption. In a continuous-wave form, two coupled diffusion equations, the lower-order approximation of the radiative transport equation, are used to describe the propagation of excitation and emission light in tissue [1,5]:
where is the domain of the position vector and is the excitation source position. Subscripts and denote excitation and emission light, respectively. is the diffusion coefficient, and is the absorption coefficient. denotes the photon density. is the fluorescent yield to be reconstructed, which is denoted as in the following part of this article. denotes the amplitude of the point excitation source. Here, the Robin-type boundary condition is utilized for the boundary conditions [22]. Equation (1) is solved using the finite element method and then the FMT problem can be linearized as the following weight matrix equation: where is the weight matrix, is the fluorescence intensity, and is the fluorescent target to be reconstructed. Detailed descriptions can be found in [13,19].In general, the regularized least-squares formation is utilized for the reconstruction as follows:
where is the regularization coefficient and . If , the Tikhonov regularization term is employed [6,7]. If , Eq. (3) is a sparsity regularization problem [8,9].B. Three-Way Decisions
TWD has attracted a growing interest in exploring various theories and models since it was proposed by Yao [23]. A TWD model is an extension of the traditional binary-decision model, which adds a third option. The idea of TWD is generated from decision-theoretic rough set (DTRS) [24]. Intuitively, the universe is divided into three pairwise disjoint regions: positive, negative, and boundary regions by two thresholds and of DTRS. The two-way decisions have only two kinds of actions: positive and negative. And the positive decision in the positive region makes the decision of acceptance. The negative decision in the negative region makes the decision of rejection. Compared to the two-way decision method including acceptance and rejection decisions only, TWD has an additional deferment decision, which is generated from the boundary region. As an effective way to deal with uncertain problems, TWD has been widely used in decision making. Jia and Shang have made a comparative study of TWD and two-way decisions on filtering spam email [25]. Li et al. proposed a TWD framework for cost-sensitive software defect prediction [26]. Savchenko has presented a fast multi-class recognition of piecewise regular objects based on sequential TWD and granular computing [27]. TWD has also been employed in multiple attribute group decision making [28]. For FMT reconstruction, there was no post-processing strategy for the recovered results. In general, the recovered images have spurious information which might mislead users and produce false information. The TWD idea implies that the reconstructed results can be further classified into three pairwise disjoint subparts: target (positive region), background (negative region), and boundary regions. The target region means it has a high probability to contribute to the fluorophore (fluorescent target). The background indicates that it makes very low probability contribution to the fluorescent target distribution. And the nodes in the boundary region either belong to the fluorescent target region or belong to the background region. More information needs to be added to make a clear decision. So a permissible region of a fluorescent target can be extracted by combining the target region with the background region. And an algorithm frame based on TWD for FMT is described in detail in Section 2.C.
C. Algorithm Frame Based on Three-Way Decisions for FMT
First, a whole region reconstruction is carried out and the recovered results are divided into three pairwise disjoint regions by two thresholds and based on three-way decisions. The nodes in the positive region are labeled as fluorescent target, the nodes in the negative are denoted as background, and the nodes in the boundary are deferred for further exam. The nodes in the boundary region either belong to the fluorescent target or belong to the background region. More information needs to be added to make a clear decision. Here, the information may be deduced from the known conditions or by other means. However, it can be sure that the measured data on the surface of the object are created by nodes in both the fluorescent target region and boundary region. So nodes in the two regions are considered as a permissible region of the fluorescent target. The background points, which make very low probability contribution in the fluorescent target distribution, are omitted. After this processing, the variable is limited to the permissible region. We define a vector , which is a columns vector composed of zero or unity elements, as follows:
Then, the and in Eq. (2) become and , and represents the operation that removes columns in matrix corresponding with the zero-element in column vector . Then the minimization problem of Eq. (3) is converted into the following form: where PR is the permissible region. On the second stage, a new reconstruction is carried out on the permissible region to obtain a new recovered result. With TWD again, new fluorescent target, background, and boundary regions are obtained by TWD. The fluorescent target region is the final reconstructed result. Figure 1 shows the flow chart of the proposed method for FMT.3. EXPERIMENTS
In this section, a numerical simulation experiment and real mouse experiment are implemented to test the performance of the reconstruction frame based on TWD.
A. Numerical Simulation Experiment
A non-uniform cylinder model is utilized to carry out the simulation experiments. The cylinder radius is 10 mm, with a high of 30 mm, which is composed of five organs: (1) muscle, (2) heart, (3) lungs, (4) liver, and (5) kidneys. The purpose of the small cylinder with a radius of 0.5 mm, high of 1.5 mm, is to imitate the fluorescent target. Its center coordinate is (0.0, 6.0, 15.0) mm, which is located in the lungs. The optical properties are the same as those presented in [19]. Figure 2 is the model of single fluorescent target reconstruction. The actual fluorescence yield is set to be . The fluorescent target was excited by 18 point sources at different positions in sequence, and the field of view of the detection with respect to each excitation source is 120°. For the inverse problem, a mesh with 5545 nodes and 30,636 tetrahedral elements is the reconstruction mesh. To evaluate the quality of the recovered images, localization error (LE) and normalized root mean square error (nRMSE) are adopted in this study [19]. In general, a high-quality reconstructed image possesses LE and nRMSE values close to 0.
The theory of TWD proposed by Yao is constructed based on the notions of the acceptance, rejection, and deferment, which can be directly generated by the three regions of probabilistic rough sets. For the fluorescent target recognition of FMT, the detail of representing TWD is summarized in Table 1. The cost function matrix of TWD is shown in Table 2.
Table 1. TWD for Fluorescent Target Recognition of FMT
Table 2. Cost Function Matrix for Fluorescent Target Prediction Based on TWD
And the threshold parameters and () are defined by the following formula: , . This means that these two parameters are determined by the costs function. In this case, we obtained the following rules:
In the FMT problem, we only obtained the fluorescent yield of each node. The normalized value of the fluorescent yield is considered as conditional probability. However, the cost function of FMT reconstruction is unknown, and there are no other methods to the pair of thresholds in current three-way decision theory. Thus, we cannot use the above formula to compute the two thresholds. For simplicity, we define and in this work as 20% and 80%, respectively, which are two values often used for FMT reconstruction.In this work, we considered and in Eq. (3). When , gradient projection for sparse reconstruction (GPSR) is usually employed to solve the sparsity regularization problem [29]. If , conjugate gradient least squares (CGLS) is utilized for the Tikhonov regularization problem [30]. Incomplete variables truncated conjugate gradient (IVTCG), which was developed in our previous work, has also been successfully applied in FMT [30]. To make a comparative study on the reconstruction, this work showed the recovered results by these three methods combined with TWD and without TWD.
Figure 3 and Table 3 show results of single target reconstruction. Figures 3(a), 3(d), and 3(g) show the 2D views of the recovered results by IVTCG, CGLS, and GPSR with no post-processing, respectively. Figures 3(c), 3(f), and 3(i) present the results by , , and . Figures 3(b), 3(e), and 3(h) are the results by IVTCG, CGLS, and GPSR with artificial threshold. In general, when the original recovered results of FMT, containing spurious information, are directly provided to users without post-processing, it is likely to mislead users. In [29], we presented the recovered results by selecting nodes with a threshold of 70% of the maximum value by manual experience. And in [31], we developed an adaptive threshold method for recovered images of FMT. Both of these two strategies can only remove the artifacts with smaller values, but they cannot modify the recovered results; please see Table 3. However, TWD not only provides a permissible region of target, which further improves the recovered results (see Table 3), but also removes the spurious information. According to Fig. 3 and Table 3, the location errors of the fluorescent target are smaller by using a permissible region extracted by TWD. And nRMSEs are smaller than those of methods with no TWD.
Fig. 3. First row shows the 2D views () of the reconstructed results by IVTCG with no post-processing, threshold and . Second row shows the 2D views () by CGLS with no post-processing, threshold and . Third row shows the 2D views () of results by GPSR with no post-processing, threshold and .
Table 3. Quantitative Analysis of Single Target Reconstruction Results
A relationship between the recovered results with TWD and the number of excitation points (3, 6, and 9) has been presented. Table 4 shows that the accuracy is improved with the increase of the number of excitation points. This is mainly because the scale of fluorescence measurement increases when the number of excitation points raises, which reduces the ill-posedness of the inverse problem to a certain extent. And nine excitation points can obtain satisfactory reconstruction results.
Table 4. Quantitative Analysis of the Reconstruction Results of under Different Excitation Points
Because of the severe ill-posedness of the inverse problem, it is difficult to get a stable solution. In order to study the effect of noise on the reconstruction results, a comparison test of 5%, 10%, 15%, 20%, and 25% noise levels is designed to verify the stability of the method. The different levels of the Gaussian noise are added on the surface measurements by the formula , where is the surface measure data and is the noise level parameter (5%, 10%, 15%, 20%, and 25%). And is a random error generated by a MATLAB function randn; its standard deviation is , where is the norm of and Num is the number of surface measured nodes. This noise is similar to the noise caused by the dark current of the CCD camera in the physical experiment. The quantitative analysis is shown in Fig. 4. After adding different proportions of noise, the location errors and nRMSEs have a slight fluctuation, which indicates that the has good stability and noise immunity. In addition, it is obvious that the proposed algorithm provides better recovered results with smaller LE and nRMSE according to Fig. 4.
Fig. 4. Reconstruction results of IVTCG and methods under different proportions of noise levels. (a) The location errors with different noise levels. (b) The nRMSEs with different noise levels.
B. Real Implanted Experiments
We further assess the performance of the TWD with in vivo small animal experimental data. The experiment data was collected from the FMT/micro-CT dual-mode system on adult nude mice; the mice were scanned with micro-CT [30]. We show the torso section of the mice with a height of 41 mm in Fig. 5(a). The main organs of the experimental mice included muscle, lung, heart, stomach, liver, and kidney organs. The optical parameters of the organs are consistent with the literature [30]. The purpose of the cylinder, implanted at (20.2, 27.8, and 7.4 mm), is to mimic the fluorescence target. Figure 5(b) shows the vertical view of the fluorescent target at . It is 6.4 mm away from the surface along the -axis and 5.0 mm away from the surface along the -axis. The depth of the fluorescent target in tissue is about 4.7 mm, as shown in Fig. 5(b). The cylinder contains Cy5.5 solution at a concentration of 4000 nM. Figure 6 and Table 5 display the results. The reconstruction frame based on TWD made an improvement on results from the third column images of Fig. 6. The LE and nRMSE were smaller with the proposed reconstruction frame in Table 5.
Fig. 6. First row shows the 2D views () of the reconstructed results by IVTCG with no post-processing, threshold and . Second row shows the 2D views () by CGLS with no post-processing, threshold and . Third row shows the 2D views () of results by GPSR with no post-processing, GPSR+artificial threshold and .
Table 5. Quantitative Analysis of Experimental Reconstruction Results of Real Mice
4. DISCUSSION AND CONCLUSIONS
This work presents a reconstruction frame for FMT based on TWD. Different from the traditional two decisions, the two threshold values of TWD divide the recovered results into three pairwise disjoint regions: the target region, the boundary region, and the background region. Intuitively, the target region is the fluorescent target region which we are interested in. The nodes in the boundary region either belong to the fluorescent target or do not belong to it. If one wants to know the exact classification, more information needs to be added. However, it is natural to combine the boundary region with the target region to form a permissible region of fluorescent target. Then, a new reconstruction can be carried out on the permissible region. Here, the TWD provides a permissible region for reconstruction. This is the first time that we used TWD. In theory, the permissible region can reduce the number of variables, which reduces the ill-posedness of the problem and improves the image quality. For the recovered results in the permissible region, three new pairwise disjoint regions would be obtained by TWD, namely, the target region, the boundary region, and the background region. This is the second time that we used TWD. From the reconstruction frame, it is known that the TWD idea has been used twice. The first time is to extract the permissible region. The second time is to post-process the recovered results, which provides the target clearly.
To test the performance of the reconstruction frame based on the TWD, simulaiton experiments and in vivo experiments are carried out in the work. The IVTCG, GPSR, and CGLS are employed as reconstruction methods. The experiments demonstrated that the recovered images with the proposed frame are better than those of ones without the proposed strategy according to LE and nRMSE.
In TWD theory, the threshold values and are very important because they determine the division of the three regions. The threshold parameters are related to cost functions, which are unknown in FMT. Then, we cannot give the calculation formula of the threshold parameters in the current TWD theory. With the development of the theory, multiple ways will be developed for them. In this work, they are defined as 20% and 80% by experience, respectively. Our future work will be focused on investigating how to determine the pair of thresholds, which is important to both three-way decision theory and the FMT reconstruction problem.
It is challenging work to get a shape of the target in FMT. But, according to the TWD, the boundary region should contain the boundary of the target, which provides a possible way to delineate the three-dimensional boundary of the target. The shape of the target can be used to guide the radiation therapy further, which is a meaningful issue. This is also our future work.
Funding
Natural Science Foundation of Shaanxi Province (2017JM1036, 2017JQ6006, 2017JQ6017, 2018JQ6099); China Postdoctoral Science Foundation (2016M602851); National Natural Science Foundation of China (NSFC) (11571012, 61701403, 61772019).
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