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Abstract
Fluorescence molecular tomography (FMT) is a promising molecular imaging technique for tumor detection in the early stage. High-precision multi-target reconstructions are necessary for quantitative analysis in practical FMT applications. The existing reconstruction methods perform well in retrieving a single fluorescent target but may fail in reconstructing a multi-target, which remains an obstacle to the wider application of FMT. In this paper, a novel multi-target reconstruction strategy based on blind source separation (BSS) of surface measurement signals was proposed, which transformed the multi-target reconstruction problem into multiple single-target reconstruction problems. Firstly, by multiple points excitation, multiple groups of superimposed measurement signals conforming to the conditions of BSS were constructed. Secondly, an efficient nonnegative least-correlated component analysis with iterative volume maximization (nLCA-IVM) algorithm was applied to construct the separation matrix, and the superimposed measurement signals were separated into the measurements of each target. Thirdly, the least squares fitting method was combined with BSS to determine the number of fluorophores indirectly. Lastly, each target was reconstructed based on the extracted surface measurement signals. Numerical simulations and in vivo experiments proved that it has the ability of multi-target resolution for FMT. The encouraging results demonstrate the significant effectiveness and potential of our method for practical FMT applications.
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1. Introduction
Fluorescence molecular tomography (FMT) is a promising non-invasive optical molecular imaging technology with strong specificity and sensitivity, which is based on the distribution of fluorescence in biological tissues [1–3]. It develops from two-dimensional (2D) qualitative imaging to three-dimensional (3D) quantitative research [4–6], and further expands the integration of stimulated fluorescence in the diagnosis and treatment of cancer [7,8], preclinical and clinical applications such as pharmacokinetics [9]. Compared to other Optical molecular imaging(OMI) technologies, FMT has the characteristics of low cost, safety, reliability, high signal strength, and flexible and reliable imaging [10]. It has developed rapidly in recent years and has become a research frontier and research hotspot for OMI technology [11].
With the application of FMT in preclinical and clinical studies of tumor imaging, FMT has the potential to be used for cancer staging studies. Generally, cancer staging takes into account the tumor size, the degree of invasion and the number of metastases for evaluating the clinical progression of tumor [12]. If using the optical molecular technique for cancer staging, tumors and lymph nodes can be regarded as multi-target. And metrics such as location (i.e. the spatial coordinates) and shape (i.e. the lesions area) can be used to assess the state of the cancer [13]. In addition, researchers found that background fluorescence is commonly observed in the process of FMT reconstruction [14]. From the perspective of optical molecular imaging, radiopharmaceuticals can be intaken by the tumor cells, so if there is a malignant lesion in certain parts of the organism, their images will be different from those of healthy tissue [15]. Since the background fluorescence will interfere with the quantitative analysis of tumor signal and brings certain challenges to tumor reconstruction, a more accurate diagnosis can be obtained by separating tumor regions from other organs with auto-fluorescence. Thus, multi-target (i.e. tumors, lymph nodes and Spontaneous light organs) imaging is very significant in pre-clinical and clinical studies of tumor region localization and cancer staging in FMT. However, accurately reconstructing multi-target is a highly challenging issue. On the one hand, the multi-target resolution in FMT is an ill-posed problem because of light scattering and absorption, and highly surface-weighted of FMT signal [16,17]. On the other hand, the interaction effect of multi-target, very complex surface-weighted FMT signal and the diversity of tumor biological characteristics make multi-target resolution more complicated. [18] Therefore, how to solve the multi-target resolution problem and enhance the reconstruction quality and stability is the main focus of this study.
In the previous study of FMT, many classical reconstruction algorithms have been proposed [19–21], including the Tikhonov regularization($l_{2}$-norm regularization) [22], the sparsity regularization with $l_{1}$-norm penalty function [23], the non-convex $l_{p}(0<p<1)$ regularization [24] and the total variation (TV) regularization [25]. Moreover, some prior information, like the optical properties of biological tissues and permissible regions, was incorporated into the reconstruction [26]. In the pre-clinical and clinical studies of FMT, multi-target reconstruction is more challenging than single-target reconstruction and has higher requirements on algorithm performance. Therefore, it is necessary to combine efficient reconstruction strategies to further improve algorithm performance. Pera proposed a strategy of combining spectral and temporal data, which enables high-throughput imaging of multiple fluorescent targets in bulk tissue [27]; Zhang used the data compression strategy in the multiple fluorophore reconstruction processes [28]; Wang introduced Longer near-infrared wavelengths within 1100–1400 nm ranges to improve the spatial resolution in multi-target reconstruction [2]; Wu applied the original synchronization-inspired clustering(OSC) algorithm to FMT for resolving multi-target from the reconstruction result [29]. These methods obtained good results when the edge-to-edge distance (EED) is relatively large and the intensity and densities of the fluorescent targets are the same. However, from the perspective of practical application, high-precision multi-target reconstruction results in complex situations are required, so it remains a challenge to resolve multiple targets reconstruction problem.
For FMT multi-target reconstruction, if the aliased surface measurement data can be separated to obtain the photon distribution on the surface of each fluorophore signal, the individual reconstruction of each fluorophore can be achieved. Therefore, the problem of multi-target reconstruction can be transformed into a multiple single-target reconstruction problem by signal separation [30]. In mathematics, BSS is a technique to separate fluorophore signals from a set of mixed signals without knowing either the fluorophores or the mixing matrix and the only available information comes from the mixed observations [31,32], which has been widely used in biological or clinical research. For instance, in multiprobe biomedical imaging studies, BSS can effectively remove artifacts caused by simultaneous imaging of multiple biomarkers [33]; In the field of neurophysiology, BSS has become a powerful data analysis tool [34]; In the early detection of Alzheimer’s disease, BSS has been used to filter electroencephalogram(EEG) signals [35]; In real electrocardiogram (ECG) recordings, BSS addresses the problem of extracting atrial activity (AA) from atrial fibrillation (AF) [36]. Examples in other fields include spectral unmixing in remote sensing [31], analysis of multispectral astronomical images [37], and acoustic signal separation in telecommunications [38]. Therefore, BSS can provide a promising strategy to solve the multi-target resolution problem in FMT.
In this work, the BSS strategy was proposed to solve the multi-target resolving problem in FMT. Firstly, based on optical transmission theory and finite element method, the physical process of multi-target measurement signal generation was systematically analyzed. Secondly, multiple groups of superimposed measurement signals conforming to the conditions of BSS were constructed. Thirdly, an efficient nLCA-IVM algorithm [33] was used to construct the separation matrix, this algorithm finds the optimal demixing matrix by maximizing the volume of a solid region formed by the demixed source vectors regardless of whether the puresource sample assumption is valid or not. And the superimposed measurement signals were separated into the fluorophore measurement signals of each target. Where the least squares fitting method was combined with BSS to determine the number of fluorophores indirectly. Finally, the linear relationship between the measurement signal of each fluorophore and the distribution of the internal fluorophore were established, and the reconstruction algorithm was used to reconstruct each target separately. Then the multi-target reconstruction problem was transformed into multiple single-target reconstruction problems. Unlike previous studies [39,40], our work used BSS to separate the superimposed surface measurement signals, which is equivalent to preprocessing the collected data, and then reconstructing the single target. The main innovation of our work is to transform multi-target reconstruction into multiple single target reconstruction, focusing on reducing the difficulty of reconstruction. Our scheme is more suitable for multi-target reconstruction under complex conditions. In order to validate the performance of the BSS strategy, groups of comparison investigation were conducted on simulations and in vivo experiments. The results demonstrated the outstanding effectiveness of the BSS strategy in accurate location and morphology recovery for multi-target in FMT.
The rest of the paper is organized as follows: Section 2 introduces the method of multi-target reconstruction, focusing on the design of the BSS algorithm and its combination with FMT. Section 3 and Section 4 assess the proposed method with a series of numerical simulations experiments, and in vivo experiments. Finally, some discussions and conclusions have been made for this paper in Section 5.
2. Materials and methods
2.1 Separation of multi-target measurement signals
The diffusion equation (DE), a simplified first-order diffusion approximation model of the radiative transfer equation(RTE), has been widely used in FMT [41]. In a continuous wave (CW) FMT imaging system, the photon propagation process can be described as coupled diffusion equations
2.2 Construction of the extracted measurement signals based on BSS
BSS deals with a real-world situation where neither the fluorophores nor the mixing matrix is known and the only available information comes from the mixed observations. A statistical latent variables model is often used to rigorously define BSS:
- • $\mathbf{\Phi } =(\Phi _{1} ,\ldots,\Phi _{p} )^{T}$ is an aliasing observation matrix of $p\times M$, the superposition of light emitted by multiple fluorophores in FMT can be regarded as the mixed observation value $\Phi$ of BSS,
- • $\boldsymbol{W} =[\omega _{\mathrm {i}^{(\mathrm {j})}} ]_{p\times k}\in \mathbb {R}^{p\times k}$ is an unknown mixing matrix,
- • $\boldsymbol{s} =(s_{1} ,\ldots,s_{k})^{T}$ is the fluorophore matrix of $k\times M$, representing the surface measurement signal of each original signal, which is equivalent to Eq. (7) in FMT condition.
It is often further assumed that the dimensions of $\Phi$ and s are equal, that is, p=k, and we made this assumption in the rest of the paper, which means the number of separated fluorophores should be consistent with the number of excitation points; Each row in $\Phi =\left (\Phi _1, \ldots, \Phi _p\right )^T$ contains p fluorophore information ${\,s}_1,{\,s}_2,\ldots,{\,s}_p$, which is consistent with the FMT multi-target signal aliasing situation. Therefore, the physical process of multi-target measurement signal generation satisfies the conditions for BSS.
For $\Phi$, our goal is to obtain the extracted surface light distribution information $z$ of each fluorophore through the separation algorithm, and then reconstruct each fluorophore separately.
- • $\boldsymbol{z}=(z_{1} ,\ldots,z_{p})^{T}$ is the extracted (demixed) fluorophore point,
- • $\boldsymbol{B}=[B_{\mathrm {i}^{(\mathrm {j})}}]_{p\times p}\in \mathbb {R}^{p\times p}$ is a real demixing matrix,
- • $\boldsymbol{P}$ is a permutation matrix, meaning that the extracted fluorophore point $z$ is equivalent to the true fluorophore point s up to a permutation.
During the specific separation process, our goal herein is to design the demixing matrix B, and based on that, we can solve for the unknowns W and s.
2.3 Blind source separation algorithm: nLCA-IVM
In this paper, the nLCA-IVM algorithm was used to design the demixing matrix (Algorithm 1). This algorithm finds the optimal demixing matrix by maximizing the volume of a solid region formed by the demixed source vectors regardless of whether the puresource sample assumption is valid or not. In other words, nLCA-IVM requires no assumption on source independence or zero correlations and obtains positive or zero results. In addition, the algorithm does not need parameter adjustment, is insensitive to initial conditions, and has high computational efficiency. Therefore, by introducing this algorithm into multi-target separation, the results obtained will be relatively better.
It can be seen from the relevant literature that the optimal separation matrix can be obtained by maximizing the absolute value of its determinant [33]. Therefore, the BSS problem can be transformed into the following non-convex optimization problem:
Algorithm 1. The nLCA-IVM Algorithm
Carry out algebraic cofactor expansion for any row of B, take the $i^{th}$ row as an example, denote $B_{i}^{T}=\left [B_{\mathrm {i}^{(\mathrm {1})}}, B_{\mathrm {i}^{(\mathrm {2})}}, \ldots, B_{\mathrm {i}^{(\mathrm {N})}}\right ]$, Eq. (10) is transformed into the following non-convex maximization problem:
During the implementation of the algorithm, we select the optimal solution by updating the matrix each iteration until convergence, as shown in Algorithm 1.
2.4 Determination of fluorophore number based on least square fitting of the BSS model
Since the number of source signals is necessary for the BSS algorithm, we need to determine the number of fluorophores in advance. It can be seen from Eq. (9) that the smaller the difference between the values of z and $B\Phi$, the closer the extracted (demixed) fluorophore signal is to the real source signal. In this section, the least squares fitting method was used to judge the difference between the values of z and $B\Phi$. From the perspective of likelihood function, the occurrence probability of each sample is as follows:
2.5 Reconstruction based on surface measurement signals after BSS
After separation, since the surface measurement signal of each fluorophore is still the result of the combined action of multiple excitation lights after BSS, for the $j^{th}$ fluorophore, the FMT model can be expressed as follows:
So far, we have successfully transformed the multi-target reconstruction problem into multiple single-target reconstruction problems. After the number of light sources is determined by the least square method, we can reconstruct each target based on the extracted surface measurement signals, and we get $x^{(\mathrm {j})}=(x^{(\mathrm {j})}[1] ,\ldots,x^{(\mathrm {j})}[N] )^{T}$, where j = 1, 2,…, p. Then we combine and display the reconstruction results of each target in a coordinate system.
Algorithm 2. Multi-target reconstruction strategy based on BSS
Since BSS strategy establishes a general multi-target reconstruction model, the reconstruction algorithm is not the focus of this paper. In the following experiments, except for the algorithm feasibility experiment, we used the classic fast iterative shrinkage threshold algorithm (FISTA) as the specific reconstruction algorithm.
The main steps of multi-target reconstruction strategy based on BSS of surface measurement signals are given in Algorithm 2.
3. Experiments settings
3.1 Numerical simulations
A series of numerical simulations based on the trunk of the digital mouse [42] model was conducted to evaluate the performance of BSS. The digital mouse model was discretized into a uniform tetrahedral mesh consisting of 13064 nodes and 67649 tetrahedral elements for reconstruction, as shown in Fig. S2 of the Supplement 1. The mouse was segmented into six tissues (heart, lung, liver, stomach, kidney and muscle). Their relevant optical properties are in Table S1 of the Supplement 1, where the excitation wavelength is $650nm$ and the emission wavelength is $670nm$.
3.1.1 Feasibility experiments
In order to verify the algorithm feasibility of the BSS strategy, FISTA and OMP algorithm were tested in the dual-fluorophore feasible experiment. Two spherical fluorescent fluorophores with $1.2mm$ radius with a distance of 5$mm$ were placed in the liver, the centers were $(16.1, 8.0, 16.5)$ and $(23.5, 8.0, 16.5)$, respectively.
3.1.2 Robustness experiments
In order to simulate more complex multi-target reconstruction scenarios, four groups of numerical simulations were conducted to verify the robustness of the BSS strategy against the variations in spatial distance, intensity, size, and fluorophore numbers. Specifically, Group 1 is to test the minimum fluorophore separation which can be successfully distinguished by the BSS strategy. Two spherical fluorophores with EED of 3mm and 2mm were placed in the mouse abdomen, respectively. In this case, the targets were with the same volume and equal intensity. Group 2 and group 3 are to reflect the ability of BSS to overcome the interaction effect for accurately reconstructing fluorophores with different volumes and intensities, respectively. In group 2, the intensity ratios of the two fluorophores were set to 1:5 and 1:10, respectively. The rest of the source settings were as same as the feasibility experiment. In group 3, the radius of the No.1 fluorophore was fixed to 1 mm, while the radius of the No.2 fluorophore was set to 1.5mm and 2mm, respectively. Group 4 emphasizes the ability of the BSS strategy to separate more than two fluorophores. The number of targets was set to three and four with equal EED separation, intensity and, volume. The detailed source parameters in robustness experiments are shown in Table S3 of the Supplement 1.
3.2 In-vivo imaging experiment
Two groups of abdominal fluorophore implantation experiment was conducted on two 6-week-old female BALB/C nude mice. An integrated FMT/micro-CT system was used to acquire optical and anatomical structure data [17]. The in vivo experiments met the ethical requirements of animals, and all experimental procedures were under the approval of the Animal Ethics Committee of the Northwest University of China.
In both sets of experiments, the glass tubes with 0.9-mm inner radius and 7.5-mm height were implanted into the abdomen of the anesthetized mice. The tubes were injected with Cy5.5 solution(with an extinction coefficient of about 0.019$mm^{-1}\mu M^{-1}$ and quantum efficiency of 0.23 at the peak excitation wavelength of 671 nm)to serve as the fluorescent targets. According to the published paper, the fluorescent yield of Cy5.5 is 0.0402 $mm^{-1}$ [43]. In the dual fluorophores experiment, from the structural data scanned by micro-CT, the actual centers of the targets were determined at $(49.4, 57.6, 16.2mm)$ and $(58.2, 54.6, 12.6mm)$, while two excitation sources located at (60.0, 60.0, 14.0mm) and (45.0, 60.0, 14.0mm), respectively. In the three fluorophores experiments, the centers of the targets were placed at $(58.5, 62.2, 16.7mm)$, $(59.2, 48.4, 17.7mm)$ and $(62.1, 52.1, 18.3mm)$, while three excitation sources located at (63.0, 64.0, 17.5mm), (63.0, 55.0, 17.5mm) and (63.0, 46.0, 17.5mm), respectively.
In the CT volume acquisition process, the tube voltage and tube current were set as 40kVp and 300mA, respectively. The rotating stage rotated 360$^{\circ }$ with 1$^{\circ }$ intervals to capture the X-ray projection images. The fluorescent images and white light data were collected by a highly sensitive EMCCD camera. Before optical acquisition, the EMCCD camera was cooled to -80$^{\circ }$C to reduce the effects of thermal noise. In the fluorescent acquisition process, the exposure time was set to 1.5 seconds, while for white light, the exposure time was set to 0.25 seconds
Prior to the FMT reconstruction, some essential preprocessing operations were carried out. Firstly, the raw CT data was converted into 3D volume data via the Feldkamp-Davis-Kress (FDK) algorithm for obtaining anatomical structural information. Secondly, the CT data were discretized and segmented into six components, including heart, lungs, liver, kidneys, stomach and muscle by using Amria 5.2 (Amria, Visage Imaging, Australia). Lastly, two groups of mouse models were discretized into 17995 nodes and 92390 tetrahedral units and 18237 nodes and 93527 tetrahedral units, respectively, for 2D-3D surface energy mapping and FMT reconstruction.
3.3 Algorithm comparison and evaluation index
To investigate the performance of the BSS strategy, we compared it with the method of direct reconstruction without separation. To be specific, OMP and FISTA were chosen in the feasible experiment to demonstrate the applicability of BSS to different algorithms. Since the FISTA outperforms OMP, we choose FISTA as the representative algorithm in the other experiments. And the regularization parameters for the FISTA algorithm were determined by $L-Curve$ method. location error(LE) and Dice coefficient were used to quantitatively evaluate the accuracy of the reconstruction in both fluorophore location and shape recovery, which are introduced in the Supplement 1.
4. Experiment results
4.1 Numerical simulation results
4.1.1 Feasible experiments
Figure 1 and Table 1 show the results of the feasible experiment. It can be seen from Fig. 1 (c) that when the number of fluorophores is set equal to the number of real fluorophores, $\min f(x)$ reaches the minimum value. Therefore, the least square fitting method can be used to determined the number of sources indirectly. Figure 1 (f) and (g) show the poor results obtained by FISTA and OMP directly. From the transverse view, OMP produces only one fluorophore with a big location deviation in the X-axis. Hence, OMP has the largest location errors and the smallest Dice with poor morphological recovery performance from the quantitative results. Although FISTA produces a better result than OMP, the two fluorophores are connected, and the location accuracy and morphological recovery ability are still poor from the quantitative results. In comparison, combined with the proposed BSS strategy, both FISTA+BSS and OMP+BSS yield a significant improvement in results as shown in Fig. 1 (d3) and (e3). FISTA+BSS achieves the relatively small LE and largest Dice, and OMP+BSS obtain considerable results as well. It indicates that the BSS strategy is feasible and does not depend on the reconstruction algorithm.
Fig. 1. Results of the feasible experiments. (a1)-(a2) The surface light distribution of dual-fluorophore before BSS under two different excitations; (b1)-(b2) The surface light distribution of dual-fluorophore after BSS; (c) The least squares fitting estimation of source numbers. (d1)-(d2) Reconstruction results of (b1)-(b2) using FISTA algorithm; (d3) 3D view and transverse view of the reconstruction results by FISTA+BSS; (e1)-(e2) Reconstruction results of (b1)-(b2) using OMP algorithm; (e3) 3D view and transverse view of the reconstruction results by OMP+BSS; (f)-(g) 3D view and transverse view of the reconstructed results by FISTA and OMP, respectively. In the 3D view, the actual and reconstructed fluorescent fluorophore are delineated with red meshes and green areas, respectively. Meanwhile, in transverse view, the black circles indicate the actual positions of the fluorescent fluorophore in the slice of $z=16.5mm$.
Table 1. Quantitative results of feasible experiments
4.1.2 Robustness experiments
As can be seen from the first column of Fig. 2, $\min f(x)$ in the four groups of experiments all reached the minimum value when the number of fluorophores was set equal to the actual number of fluorophores. Figure 2 (a1)-(a6) show the results of group 1. As is known, the reconstruction challenges increase with decreasing EED. It can be seen that FISTA+BSS can distinguish two fluorophores with EED of 3mm and 2mm clearly, as shown in Fig. 2 (a2) and (a4). Meanwhile, the LE and Dice of FISTA+BSS are both satisfactory, as shown in Table 2. However, the reconstructed image of FISTA are blurred with some artifacts, as shown in Fig. 2 (a1) and(a3), which resulted in worse Dice and LE.
Figure 2 (b1)-(b6) show the results of group 2. Direct using of FISTA cannot distinguish the two targets with high intensity difference (intensity ratio of 1:5 and 1:10), which can only reconstruct one fluorophore with higher intensity, as shown in Fig. 2 (b1) and (b3). However, FISTA+BSS can clearly reconstruct two targets even for the intensity ratio is 1:10, as shown in Fig. 2 (b4). More quantitative information of the results are presented in Table 3.
Fig. 2. Results of the robustness experiments. The first column is the least squares fitting estimation of source numbers; The second column is the 3D view and transverse view of the reconstructed results by FISTA; The third column is the 3D view and transverse view of the reconstructed results by FISTA+BSS.
Table 2. Quantitative results of robustness experiments with the variations of spatial distance.
Table 3. Quantitative results of robustness experiments with the variations of intensity.
Figure 2 (c1)-(c6) show the results of group 3. It can be observed from Fig. 2 (c2) and (c4) that the results by FISTA + BSS have less spreading and are better localized around the corresponding true center, while FISTA cannot distinguish the two targets, especially when the ratio of the radius reaches 1:2, as shown in Fig. 2 (c3). The quantitative indexes in Table 4 further indicate that FISTA+BSS has stronger resolution and better positioning accuracy for fluorophores with different sizes compared with FISTA. The DICE of FISTA+BSS are as high as 60% and 71%, and the LE values are also relatively reasonable.
Table 4. Quantitative results of robustness experiments with the variations of size.
The results of group 4 are presented in Fig. 2(d1)-(d6) and Table 5. It can be seen that direct reconstruction using FISTA yields poor results. In transverse views of the results for both three-targets and four-targets cases, only two areas are recovered with big location deviation in x-axis, as shown in Fig. 2(d1) and (d3). Even worse, the Dice of FISTA is reduced to 27% with a significant shape information losing in four-targets case. However, the offset of the reconstructed area by FISTA+BSS is much smaller than FISTA. Moreover, in the extreme case where the fluorophores number is 4 and EED is 2mm, each target can still be clearly reconstructed by FISTA+BSS, as shown in Fig. 2(d4).
Table 5. Quantitative results of robustness experiments with the variations of fluorophores number
All these results demonstrate the good robustness of the least square fitting method and the outstanding spatial resolving capability of FISTA+BSS under various multi-target settings.
4.2 In vivo experiments results
Figure 3 shows the in vivo results for the two groups of experiments(dual and triple targets). The reconstructed images in the CT coordinate system were merged with the CT slice. The comparison results show that: (1) Compared with FISTA, algorithms based on BSS strategy are able to restrict reconstructed artifact to some degree which are consistent with the actual target area in CT, that is, better localized around the glass tube; (2) For some fluorophores with close distance, traditional algorithms could not distinguish them. Fortunately, the BSS strategy had a good performance in spatial resolution. The detailed evaluation results are listed in Table 6. Similar to the numerical studies, FISTA+BSS performed better than FISTA in terms of location accuracy and spatial aggregation. The LE of FISTA+BSS was as low as 0.19 mm in the dual-fluorophore experiment, and the DICE was as high as 68% in the three-fluorophore experiment. However, for FISTA, the reconstructed target areas tended to be connected together and the location deviation mainly manifested in the x-axis, leading to the large LE and smaller Dice. These results further revealed the significance and effectiveness of the BSS strategy in obtaining the morphology and location tracking of the in vivo fluorescence probe distribution.
Fig. 3. The results of in vivo imaging experiments. (a)-(b) 3D-view and images fused by the transverse view and CT image of the dual-fluorophore reconstructed results before and after BSS.(c)-(d) 3D-view and images fused by the transverse view and CT image of the three fluorophores reconstructed results before and after BSS. In the 3D view, the actual and reconstructed fluorescent fluorophore were delineated with red meshes and green areas, respectively. In fused images which include coronal view (C) and sagittal view (S), the green curve is the contour of cross-sectional images, the flare area is the reconstruction result, and the red dotted line outlines the actual location of the fluorophores.
Table 6. Quantitative results of in vivo experiments
5. Discussion and conclusion
As a promising preclinical imaging technique, FMT has been paid much attention in imaging theories, acquisition equipment and biomedical applications. At present, resolving multiple tumors and determining their areas are significant problems for preclinical and clinical research. However, because of the severe ill-posedness of inverse problems, the accuracy of multi-target reconstruction remains limited in many biomedical applications. In this study, we proposed a multi-target reconstruction strategy based on BSS of surface measurement signals to convert multi-target reconstruction problem into multiple single-target reconstruction problem, which greatly improves the reconstruction quality. In the implementation, we chose the efficient non-negative BSS algorithm nLCA-IVM to construct the separation matrix. Although, the specific BSS algorithm is not the focus of research, the performance of source separation algorithm will seriously affect the reconstruction results. Therefore, more efficient BSS algorithms will be further investigated in future.
To validate the performance of the proposed multi-target reconstruction strategy, we performed numerical simulations and in vivo experiments. It should be emphasized that since the energy values of the reconstruction results of the two fluorophores after BSS differed greatly, we normalized them respectively so that they could be displayed under the same threshold. Firstly, feasibility experiments demonstrate the applicability and generality of BSS to different reconstruction algorithms. The BSS strategy does not depend on a specific reconstruction algorithm. Combined with BSS, the performance of the traditional algorithm can be improved significantly; Robustness experiments results illustrate that: (1) Dual-fluorophore with EED of 2mm can be distinguished by BSS clearly; (2) BSS can distinguish dual-fluorophore with an intensity ratio of 1:10, which is difficult for general algorithms and multi-target reconstruction strategy; (3) Compared with traditional reconstruction methods, the ability of BSS to overcome the interaction effect for reconstructing fluorophores with different volumes has a great improvement; (4) Even in the extreme case where four fluorophores were reconstructed simultaneously at such a small EED(2mm), the proposed strategy still performed well. Secondly, two groups of abdominal fluorophore implantation experiments were conducted to assess the in vivo resolving power of the BSS strategy. All results further verified the improvement of the BSS strategy in obtaining the morphology and location tracking of the in vivo fluorophores.
It should be noted that this work mainly focuses on multi-target separation and reconstruction. For single target reconstruction, BSS strategy is not required, and efficient reconstruction algorithm can be used directly. Although BSS has achieved much outstanding results in multi-target reconstruction, it still has many deficiencies that need to be addressed. First of all, the first-order approximation of the radiative transfer equation presents some important limitations, such as a decline in the accuracy of light propagation modeling near the fluorophores or close to the boundaries. In addition, the BSS algorithm needs to be executed with a priori information about the known number of fluorophores, which has been a difficult problem in the field of BSS. The least squares fitting method is used to determine the number of fluorophores indirectly in this work. Although the estimation value of this method is accurate, the calculation cost is high due to the need to traverse and test traverse and test within a range of the number of light sources. Therefore, we still need to explore more efficient methods to get the number of sources automatically. For example, we can adaptively cluster the fuzzy information of surface measurement data and search for the best cluster number, that is, the number of fluorophores. However, for more complex conditions, it may still be a challenging problem to rely only on "methods" to determine the number of sources. More clinical information and assistance from experts may be helpful in such cases.
In conclusion, a BSS strategy was introduced into the reconstruction of multiple fluorescent targets for FMT, transforming the multi-target reconstruction problem into multiple single-target reconstruction problem. The results of numerical simulations and in vivo experiments demonstrated that the BSS strategy could significantly enhance the multi-target resolution capability. The performance of the BSS on multi-target reconstruction will be further investigated with tumor-bearing mice in our future work. We believed that this strategy will facilitate the development of multi-target resolving in theoretical studies and the preclinical applications in FMT.
Funding
National Natural Science Foundation of China (12271434, 61901374, 61906154, 61971350, 62271394).
Disclosures
The authors declare no potential conflict of interests.
Data availability
Data underlying of the in-vivo results presented in this paper are not publicly available at this time, but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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