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Abstract
Fluorescence molecular tomography can combine two-dimensional fluorescence imaging with anatomical information to reconstruct three-dimensional images of tumors. Reconstruction based on traditional regularization with tumor sparsity priors does not take into account that tumor cells form clusters, so it performs poorly when multiple light sources are used. Here we describe reconstruction based on an “adaptive group least angle regression elastic net” (AGLEN) method, in which local spatial structure correlation and group sparsity are integrated with elastic net regularization, followed by least angle regression. The AGLEN method works iteratively using the residual vector and a median smoothing strategy in order to adaptively obtain a robust local optimum. The method was verified using numerical simulations as well as imaging of mice bearing liver or melanoma tumors. AGLEN reconstruction performed better than state-of-the-art methods with different sizes of light sources at different distances from the sample and in the presence of Gaussian noise at 5–25%. In addition, AGLEN-based reconstruction accurately imaged tumor expression of cell death ligand-1, which can guide immunotherapy.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fluorescence molecular Imaging (FMI) infers the distribution of fluorescent probes inside biological tissues by detecting optical signals on the tissue surface. Its high spatial resolution and sensitivity make it useful for intraoperative imaging, such as during excision of tumors [1–3]. To increase the spatial information, two-dimensional structural imaging can be combined with anatomical information to infer the three-dimensional distribution of fluorescent probes [4,5]. This technique, called fluorescence molecular tomography (FMT), has been used to quantify expression of tumor proteins in three-dimensions, which creates new possibilities for personalized cancer staging and treatment [6].
The ill-posed mathematics underlying FMT [7,8] make traditional reconstruction inaccurate and sensitive to interference from noise and autofluorescence. Therefore FMT reconstruction is usually constrained by regularization based on prior information about tumor sparsity [9–13]. Most regularization approaches use Lp norm regularization, where p = (0, 2] [14], but this type of regularization is too sparse or too smooth for many in vivo situations [15], leading to artifacts at tumor boundaries or false optima when the sample is illuminated with multiple light sources.
More robust regularization can be achieved by combining the tumor sparsity-based Lp norm with additional prior information [16]. Examples include MAP estimation with structural priors for FMT [17], total variational norm regularization [18], Gaussian-weighted Laplace prior regularization [19], and group sparse norm regularization [20]. These approaches are unfortunately not perfect and can still lead to overly sparse or smooth FMT reconstructions.
An alternative regularization approach is the elastic net method, which balances the weights applied to L1 and L2 in order to mitigate excessive sparsity and smoothness. We have proposed an “adaptive parameter search elastic net” (APSEN) method to optimize the weights using the L0 norm and residual vector of the effective reconstruction [21]. However, these methods do not take into account that (1) the individual tumor cells containing fluorescent probe cluster into tumors, or (2) the fluorescence signals generated in response to stimulation by multiple light sources overlap and interfere with one another. These factors mean that elastic net regularization can lead to reconstructions with local discontinuities and anatomical inaccuracy.
To address these deficiencies, we propose here an “adaptive group least angle regression elastic net” (AGLEN) method for FMT reconstruction. The AGLEN method combines the elastic net approach with prior knowledge of tumor spatial structure, as well as interactive least angle regression. Different from the existing elastic net regularization strategy, we innovatively proposed a grouping strategy that utilizes tetrahedral mesh to describe the relationship between grouped subspaces without relying on hard prior of the tumor. By introducing the tetrahedrally meshed grouped subspace, we propose a novel group least angle regression method that replaces the conventional basis unit of previous works with a grouped subspace; by doing so, we obtain both spatially structured and sparse results in FMT reconstruction. Furthermore, considering that sparseness varies with tumor size and the number of tumors, we adaptively select the best result in parallel according to the residual vector with median smoothing to maximize robustness, and hence enhance the generality of the group least angle regression. We validated the AGLEN method using numerical simulations and in vivo images of mice bearing orthotopic liver or melanoma tumors expressing immune checkpoint programmed cell death ligand-1 (PD-L1) [1,2]. We showed that the AGLEN method led to significantly more accurate and robust reconstructions of fluorescence distribution in tumors than the fused least absolute shrinkage and selection operator method (FLM) [22], Nesterov's elastic net (N-EN) method [23], and our APSEN method. The results demonstrated that AGLEN significantly improved the reconstruction accuracy, robustness, and morphological recovery of the fluorescence distribution in the tumor area. The in vivo reconstruction results showed that AGLEN can provide more accurate results in the detection of tumor area and molecular information of the fluorescent probe and PD-L1 molecular expression.
This paper is organized as follows. Section II of this paper introduces the forward model of FMT and AGLEN reconstruction algorithm. Section III presents the process and results of numerical simulations and in vivo experiments. Section IV discusses the proposed AGLEN method and draws conclusions for future research and application.
2. Methods
2.1 Forward problem in FMT
The propagation of photons at near-infrared wavelengths through biological tissue is most often described using the diffusion equation [24]:
The Robin-type boundary condition is applied to the boundary domain of Ω in order to solve the diffusion equation [24]:
2.2 Inverse problem in FMT
We propose AGLEN to solve the inverse problem in Eq. (3) by combining the elastic net method, least angle regression, and local spatial structure constraints (Algorithm 1). Elastic net regularization is defined as [21]:
Here we use least angle regression to solve Eq. (4) as described [27]. The system weight matrix A and N values of yi are taken from Eq. (3), where yi denotes the i-th term in the emitted light distribution Y. A and Y can be expanded into the form
2.2.1 Updating the path support set
AGLEN iteratively updates the path support set in order to improve use of local spatial structure constraints and increase the accuracy of reconstruction. During updating, the path support set from the previous iteration is combined with the appropriate node:
where Si and Si-1 are the path support sets in the i-th and (i - 1)-th iterations, and ji-1 is the new node selected in the (i - 1)-th iteration. The procedure merges the support set of the previous iteration with the selected node to calculate the optimal path of AGLEN in the new iteration as well as select the optimal node for the current state.2.2.2 Calculating the path solution
Least angle regression takes advantage of the piecewise linear continuity of the path to λ as defined by the “least absolute shrinkage and selection operator” (LASSO) [28]. Along this path, as λ decreases and more prediction nodes are added to the support set, correlation weakens between the support set and the residual. Based on the path support set from Eq. (7), the least-squares direction δ of the iterative AGLEN path is obtained as ${[\Delta ]_{{S^i}}} = \delta$, where Δ denotes the direction variable under the support set Si in the i-th iteration. Thus, the least-squares direction δ is computed as
2.2.3 Structuring groups
Given that tumors concentrate in certain areas of an in vivo image, one node is expanded into a set of four nodes comprising a tetrahedron, and tetrahedrons with adjacent nodes are assigned to the same group. AGLEN combines the set of nodes with the group support set to find the optimal solution. First, a new node set U is obtained from
where Ti-1 denotes the group support set based on the grouping strategy in the (i – 1)-th iteration. For each node u in new node set U, u is expanded to a group tetrahedron according to where G denotes the tetrahedral grid structure obtained based on the tetrahedral grouping strategy, Gl denotes the tetrahedral grid structure where node u is located, and T denotes the group support set. Equation (14) is used to assign a group support set, which in turn is used to calculate a solution and residual. The solution with the smallest residual is identified using the equations where (A*)T denotes a new matrix composed of column vectors corresponding to the group support set T in the system matrix A*. ${\tilde{X}_L}$ denotes the group solution, and rG denotes the group residual. To reduce the influence of noise and increase the robustness, the solution of each iteration is smoothed based on the median value of the intensity of the adjacent nodes, which is calculated as follows where Median(•) denotes each node xm in each group solution and ${\tilde{X}_L}$ takes the median value of the intensity of the adjacent nodes.2.2.4 Terminating the iterations
As a greedy optimization algorithm [19,20], AGLEN terminates when the number of iterations is Nmax or ${\|{r_G^i} \|_2}$ is less than tol = 1e-11.
3. Experiments and results
The reconstruction performance of AGLEN was assessed using numerical simulations and in vivo imaging of tumor-bearing mice. In addition, AGLEN was benchmarked against FLM, N-EN, and APSEN methods. All processing was carried out within MATLAB 2020a (MathWorks, USA) running on a desktop computer with an Intel Core i7-6700 CPU (3.40 GHz) and 16 GB RAM.
3.1 Performance indicators
The performance of AGLEN was assessed quantitatively using several performance indicators. One was position error (PE), which evaluates the accuracy of the average FMT reconstruction center [21]. PE measures the difference between the centers of the actual and reconstructed areas within a region of interest (ROI):
where Pa denotes the center coordinates of the actual area and Pr, the center coordinates of the reconstructed area. More accurate FMT reconstruction should have lower PE.Another indicator was relative intensity error (RIE), which evaluates the recovery of fluorescence yield [21]. RIE measures the difference in intensity between actual and reconstructed sources within an ROI:
where Ia denotes the actual fluorescence yield, and Ir the mean fluorescence yield of the reconstructed source. In the present study, we included in the reconstruction only nodes whose reconstructed intensity exceeded 30% of the maximum light intensity; other nodes were discarded as artifacts. FMT reconstruction that recovers more fluorescence yield should have RIE closer to 0.A third indicator was the Dice coefficient [22], which evaluates the morphological similarity between the reconstructed image and ground truth:
where XROI denotes the reconstructed intensity and S the actual fluorescence distribution in the ROI. More accurate FMT reconstruction should lead to Dice closer to 1.3.2 Performance assessment
3.2.1 Numerical simulations
A digital mouse was generated as described [30] with heart, lung, liver, kidney, muscle, and tumors whose radius and position were varied. The excitation light source and center of the fluorescence source were positioned on the same plane, and the fluorescence source was excited at four excitation angles (0, 90, 180 and 270°). Fluorescence yield was set to 0.5 mm-1, and the power of the excitation light source was set to 0.02 W. The fluorescence distribution on the phantom surface opposite to the excitation light source was captured over a field of view of 160°. To improve the computational efficiency, we used Amira 5.2 (Thermo Fisher Scientific, USA) to map the torso of the digital mouse to a uniform tetrahedral mesh. Digital mice were created with different numbers of tumors of different radii and position in order to verify the adaptability of AGLEN (Table 1). The optical parameters used in the simulations are shown in Table 2 [31].
Table 1. The Mesh Size and Tumor Locations in The Numerical Simulation
Table 2. Optical Absorption and Scattering Coefficients of Biological Tissues in in numerical simulations and in vivo experiments
In the first simulation, a single source with radius 1.0 mm was specified. This simulation aimed to assess whether AGLEN-based reconstruction could accurately recover tumor shape.
In the second simulation, three sources with radius 0.5 mm were specified. This simulation aimed to assess the positional accuracy of AGLEN-based reconstruction.
In the third simulation, two sources with radius 0.5 mm and edge-to-edge distances of 3 mm were specified, and Gaussian noise at 5, 15, or 25% was added. This simulation assessed whether AGLEN-based reconstruction was robust to noise.
In these simulations, AGLEN-based reconstruction was compared with reconstruction based on FLM, N-EN, or APSEN. Data were not normalized before reconstruction in order to calculate RIE more accurately.
3.2.2 In vivo experiments
In one tumor model, Huh7 hepatocellular cells (GeneChem, Shanghai, China) were implanted into the liver lobes of male BALB/c nude mice (Vital River Laboratory Animal Technology, Beijing, China) at 4 weeks of age. The animal handling procedures were mainly according to the guidelines of the Institutional Animal Care and Use Committee (Permit No: IA21-2203-24) at the Institute of Automation, Chinese Academy of Sciences. After 12 days, three animals with orthotopic liver tumors were imaged. The animals were injected with 2 nmol MMPSense 750 FAST (Perkin Elmer, MA, USA) via the tail vein. This fluorescent probe binds to matrix metalloproteinases, which are overexpressed in many types of cancers.
In a second tumor model, B16F10 melanoma tumor cells (American Tissue Culture Collections, Manassas, VA, USA) were implanted subcutaneously into the back of male C57BL/6 nude mice at 4 weeks of age (1 × 106 cells per mouse). One week after tumor cell implantation, 3 mice were injected with PD-L1-IRDy800CW imaging probe made in our lab, which has been proved to specifically bind to programmed cell death ligand-1 (PD-L1) expressed on the surface of tumor cells [1,12]. Another 3 mice were pre-injected with PD-L1 antibody (Bioxcell), and 1 h later, with the PD-L1-IRDy800CW imaging probe. At 6 h after injection with probes, mice were imaged as described [21] using the multi-modal tomography system developed by the Key Laboratory of Molecular Imaging of the Chinese Academy of Sciences [32]. After data acquisition, muscle, heart, lungs, liver, and bones were segmented and used to generate an in vivo mouse model. Multi-modal information was used to map the fluorescence image onto the anatomical surface of the mouse torso, after which data were discretized (Fig. 1). The optical parameters of the organs were the same as those used in the numerical simulations (Table 2).
Fig. 1. The procedures of processing and reconstruction for in vivo experiments data. (1) The fluorescence collection system and the CT imaging system distribute orthogonally. (2) Denoise fluorescence data, segment CT data based on a threshold, and map processed data into the FEM tetrahedron based on coordinate. (3) Reconstruct 3D FMT and fuse with CT.
3.3 Results
3.3.1 Recovery of tumor shape in numerical simulation
The first simulation showed that AGLEN-based reconstruction faithfully recovered the tumor edge, with uniform overall signal intensity (Fig. 2). In fact, it was smoother than FLM-based reconstruction. Although the reconstructions based on N-EN or APSEN recovered the tumor edge more precise than AGLEN-based reconstruction, they showed spatial discontinuities near the edge.
Fig. 2. Reconstructions of a digital mouse bearing a tumor and illuminated with a single light source (first simulation). Reconstructions were carried out based on the indicated algorithms. The white circles in the plane view demarcate the actual fluorescence source.
Quantitative analysis showed that FLM led to an excessively convergent boundary and compression of the tumor shape, while N-EN and AGLEN accurately recovered the boundary (Table 3). AGLEN also recovered more fluorescence yield than FLM or N-EN.
Table 3. Quantitative comparison of reconstructions with a single source
3.3.2 Positional accuracy in numerical simulation
Next, we increased the light sources to three in order to test AGLEN performance when multiple sources give rise to mutually interfering signals. AGLEN accurately reconstructed the positions of all three sources (Fig. 3). FLM, in contrast, accurately reconstructed the positions of sources S1 and S2, but the position of source S3 was too close to S1. In addition, FLM led to low spatial resolution. N-EN and APSEN accurately reconstructed the positions of all three sources, but the source shapes were overly smooth, and the intensity of S3 was low. Quantitative analysis showed that AGLEN performed better than the previously published methods (Table 4).
Fig. 3. Reconstructions of a digital mouse bearing a tumor and illuminated with three light sources (second simulation). The white circles in the plane view demarcate the actual fluorescence source.
Table 4. Quantitative comparison of reconstructions with multi-sources
3.3.3 Robustness to noise in numerical simulation
We tested whether the ability of AGLEN to take full account of the correlation between local spatial structure and fluorescence yield during grouping as well as its median smoothing strategy rendered it robust to noise. Indeed, AGLEN performed better than the other methods as the level of Gaussian noise increased from 5 to 25% (Fig. 4, Table 5). We conducted repeated tests on multiple sets of simulation experiments, and found that, under the condition of fixed simulation experiment parameters, the qualitative and quantitative results obtained by the single light source, two light source and three light source reconstruction experiments obtained by each method are consistent.
Fig. 4. Reconstructions of a digital mouse bearing a tumor and illuminated with two light sources in the presence of Gaussian noise at 5, 15, or 25% (third simulation). The white circles in the plane view demarcate the actual fluorescence source.
Table 5. Quantitative Analysis of Dual-sources Anti-noise Ability Verification Results
3.3.4 Performance for in vivo FMT
In vivo experiments in which we mapped fluorescence images onto MRI images showed that all methods led to good three-dimensional reconstructions, but that AGLEN gave the most accurate results overall (Fig. 5). Quantitative analysis confirmed that AGLEN led to the most accurate reconstruction and appeared to be the most robust to noise (Table 6). FLM led to the shallowest reconstruction of all the methods.
Fig. 5. Reconstructions of living mice bearing single liver tumors, based on the indicated methods. In the plane view, red areas represent the reconstruction; blue areas, the ground truth; and purple areas, the intersection between the two.
Table 6. Quantitative Analysis of in vivo Reconstruction Results
3.3.5 Molecular specificity for in vivo FMT
To assess the ability of AGLEN to reconstruct fluorescent signals from specific molecules on the tumor surface, we imaged melanoma tumors in mice based on their abundant surface expression of PD-L1. To control for the specificity of imaging, we treated some animals with antibody against PD-L1 to block the binding of the probe. AGLEN-based reconstruction accurately recovered the distribution of PD-L1 on tumors when checked against the corresponding anatomical images and images obtained after pretreatment with antibody (Fig. 6(a)-(b)). We further verified the accuracy of the imaging signals by staining tissue sections with hematoxylin-eosin or immunofluoescence staining with antibody against PD-L1 (Fig. 6(c)-(d)).
Fig. 6. AGLEN-based reconstruction of living mice bearing a single melanoma tumor, based on fluorescence signal from a probe that binds to PD-L1 on the tumor surface. Animals were treated only with probe (“No block”) or first with antibody against PD-L1, followed by probe (“Block”). Fluorescence images were mapped onto anatomical images based on computed tomography (CT) and magnetic resonance imaging (MRI). (a) Representative FMT reconstructions of PD-L1 expression. (b-c) Average fluorescence intensity (AFI) and maximum fluorescence intensity (MFI) of the reconstruction. Results were normalized to the MFI of the “No block” reconstruction. (d) Tissue sections from two mice after staining with hematoxylin-eosin (left panel) or antibody against PD-L1 (right panel).
4. Discussion and conclusion
FMT is an important branch of optical molecular imaging. Compared with FMI, FMT provides 3-D molecular information and can achieve precise quantitative reconstruction. To further improve the quality of 3-D reconstruction, we proposed AGLEN method to achieve morphological FMT reconstruction with a higher accuracy of tumor regions. Recent studies showed that EN regularization which combined with conventional L1 and L2 norms balance the smoothness and sparsity of the FMT reconstruction results. In this study, we integrated the correlation of local spatial structure based on the LARS and utilized the greedy algorithm to optimize the reconstruction accuracy. Additionally, to reduce the influence of noise on the FMT reconstruction result, the median smoothing strategy was added to the intermediate iterative results, which helps obtain a robust optimal result.
To verify the performance of AGLEN, numerical simulations and in vivo experiments were performed, and the FLM, N-EN, and state-of-art APSEN were utilized for comparison. The experimental results showed that 1) AGLEN could restore the shape of the tumor more accurately and had a higher reconstruction accuracy. 2) In the anti-noise ability experiments, compared with other methods, AGLEN under Gaussian noise had more advantages in FMT reconstruction accuracy and robustness. 3) Particularly, the multi-source experiments and the anti-noise ability experiments proved that the AGLEN could have lower PE under the condition of mutual interference of multiple light sources, and its FMT reconstruction results were closest to the ground truth. 4) In vivo experiments have verified that the reconstruction results of AGLEN were closer to real tumors, which proved that AGLEN is practical in biomedical research. 5) Simulation experiments showed that AGLEN could restore more accurate in vivo fluorescence intensity and had been applied in in vivo information experiments, which showed that AGLEN could provide more accurate results in molecular information detection. All experimental results also verified that the EN regularization method based on group sparsity could significantly improve the performance of the morphological recovery and further expand the reconstruction strategy of FMT. Particular, the results of in vivo molecular information experiments can provide more non-invasive, dynamic, and comprehensive molecular expression information as the PD-L1, which may provide clinical guidance regarding immunotherapy.
In summary, we proposed an AGLEN method that can provide an accurate morphological FMT to reconstruct the tumor area. AGLEN integrated the local spatial structure correlation of group sparsity based on EN regularization. The LARS method was introduced to improve the accuracy of the FMT reconstruction results. The residual vector and median smoothing strategy were added to the iteration solution in AGLEN, which helped adaptively obtain a robust local optimal result. The reconstruction results showed that AGLEN had great advantages in reconstruction position accuracy and robustness. Additionally, AGLEN had a good improvement effect in fluorescence yield recovery and morphological reconstruction. In addition, it was proven that AGLEN has more advantages in practicality and detection of molecular information through the in vivo experiments. Future work will focus on optimizing the parameter selection method and calculation efficiency of AGLEN and exploring the feasibility of clinical application.
Funding
National Natural Science Foundation of China (62027901, 61901472, 81871514, 81227901, 81470083, 91859119, 61671449, 81527805, 81930053); Beijing Municipal Natural Science Foundation (7212207, 4232058); National Public Welfare Basic Scientific Research Program of Chinese Academy of Medical Sciences (2017PT32004, 2018PT32003); The Project of High-Level Talents Team Introduction in Zhuhai City (HLHPTP201703); Key Research and Development Program of Shandong (2022CXGC010501); .
Acknowledgments
The authors would like to thank the instrumental and technical support of Multimodal Biomedical Imaging Experimental Platform, Institute of Automation, Chinese Academy of Sciences for help identifying collaborators for this work.
Disclosures
The authors declare no conflicts of interest.
Data availability
The data sets and the raw code are available from the corresponding author upon request.
References
1. Y. Du, Y. Jin, W. Sun, J. Fang, J. Zheng, and J. Tian, “Advances in molecular imaging of immune checkpoint targets in malignancies: current and future prospect,” Eur. Radiol. 29(8), 4294–4302 (2019). [CrossRef]
2. M. T. Berninger, P. Mohajerani, M. Kimm, S. Masius, X. Ma, M. Wildgruber, B. Haller, M. Anton, A. B. Imhoff, V. Ntziachristos, T. D. Henning, and R. Meier, “Fluorescence molecular tomography of DiR-labeled mesenchymal stem cell implants for osteochondral defect repair in rabbit knees,” Eur. Radiol. 27(3), 1105–1113 (2017). [CrossRef]
3. Y. Du, Y. Qi, Z. Jin, and J. Tian, “Noninvasive imaging in cancer immunotherapy: The way to precision medicine,” Cancer Lett. 466, 13–22 (2019). [CrossRef]
4. Y. An, J. Liu, G. Zhang, S. Jiang, J. Ye, C. Chi, and J. Tian, “Compactly Supported Radial Basis Function-Based Meshless Method for Photon Propagation Model of Fluorescence Molecular Tomography,” IEEE Trans. Med. Imaging 36(2), 366–373 (2017). [CrossRef]
5. P. Zhang, G. Fan, T. Xing, F. Song, and G. Zhang, “UHR-DeepFMT: Ultra-High Spatial Resolution Reconstruction of Fluorescence Molecular Tomography Based on 3-D Fusion Dual-Sampling Deep Neural Network,” IEEE Trans. Med. Imaging 40(11), 3217–3228 (2021). [CrossRef]
6. A. Ale, V. Ermolayev, E. Herzog, C. Cohrs, M. H. de Angelis, and V. Ntziachristos, “FMT-XCT: in vivo animal studies with hybrid fluorescence molecular tomography-X-ray computed tomography,” Nat. Methods 9(6), 615–620 (2012). [CrossRef]
7. J. C. Baritaux, K. Hassler, and M. Unser, “An efficient numerical method for general L(p) regularization in fluorescence molecular tomography,” IEEE Trans. Med. Imaging 29(4), 1075–1087 (2010). [CrossRef]
8. Y. Liu, S. Jiang, J. Liu, Y. An, G. Zhang, Y. Gao, K. Wang, and J. Tian, “Reconstruction method for fluorescence molecular tomography based on L1-norm primal accelerated proximal gradient,” J. Biomed. Opt. 23(8), 1–11 (2018). [CrossRef]
9. W. Zou, E. Fang, J. Wang, and X. Pan, “Fluorescent molecular tomographic reconstruction via compensating for modelling error,” J. Mod. Opt. 66(19), 1904–1912 (2019). [CrossRef]
10. G. Hongbo, H. Xiaowei, L. Muhan, Z. Zeyu, H. Zhenhua, and T. Jie, “Weight Multispectral Reconstruction Strategy for Enhanced Reconstruction Accuracy and Stability With Cerenkov Luminescence Tomography,” IEEE Trans. Med. Imaging 36(6), 1337–1346 (2017). [CrossRef]
11. Y. An, H. Meng, Y. Gao, T. Tong, C. Zhang, K. Wang, and J. Tian, “Application of machine learning method in optical molecular imaging: a review,” Sci. China Inf. Sci. 63(1), 111101 (2020). [CrossRef]
12. Y. An, C. Bian, D. Yan, H. Wang, Y. Wang, Y. Du, and J. Tian, “A Fast and Automated FMT/XCT Reconstruction Strategy Based on Standardized Imaging Space,” IEEE Trans. Med. Imaging 41(3), 657–666 (2022). [CrossRef]
13. P. Mohajerani and V. Ntziachristos, “An Inversion Scheme for Hybrid Fluorescence Molecular Tomography Using a Fuzzy Inference System,” IEEE Trans. Med. Imaging 35(2), 381–390 (2016). [CrossRef]
14. E. Edjlali and Y. Bérubé-Lauzière, “Lq−Lp optimization for multigrid fluorescence tomography of small animals using simplified spherical harmonics,” J. Quant. Spectrosc. Radiat. Transfer 205, 163–173 (2018). [CrossRef]
15. D. Zhu and C. Li, “Nonconvex regularizations in fluorescence molecular tomography for sparsity enhancement,” Phys. Med. Biol. 59(12), 2901–2912 (2014). [CrossRef]
16. P. Zhang, C. Ma, F. Song, G. Fan, Y. Sun, Y. Feng, X. Ma, F. Liu, and G. Zhang, “A review of advances in imaging methodology in fluorescence molecular tomography,” Phys. Med. Biol. 67(10), 10TR01 (2022). [CrossRef]
17. G. Zhang, X. Cao, B. Zhang, F. Liu, J. Luo, and J. Bai, “MAP estimation with structural priors for fluorescence molecular tomography,” Phys. Med. Biol. 58(2), 351–372 (2013). [CrossRef]
18. J. Dutta, S. Ahn, C. Li, S. R. Cherry, and R. M. Leahy, “Joint L1 and total variation regularization for fluorescence molecular tomography,” Phys. Med. Biol. 57(6), 1459–1476 (2012). [CrossRef]
19. H. Meng, K. Wang, Y. Gao, Y. Jin, X. Ma, and J. Tian, “Adaptive Gaussian Weighted Laplace Prior Regularization Enables Accurate Morphological Reconstruction in Fluorescence Molecular Tomography,” IEEE Trans. Med. Imaging 38(12), 2726–2734 (2019). [CrossRef]
20. L. Kong, Y. An, Q. Liang, L. Yin, Y. Du, and J. Tian, “Reconstruction for Fluorescence Molecular Tomography via Adaptive Group Orthogonal Matching Pursuit,” IEEE Trans Biomed Eng 67(9), 2518–2529 (2020). [CrossRef]
21. H. Wang, C. Bian, L. Kong, Y. An, Y. Du, and J. Tian, “A Novel Adaptive Parameter Search Elastic Net Method for Fluorescent Molecular Tomography,” IEEE Trans. Med. Imaging 40(5), 1484–1498 (2021). [CrossRef]
22. S. Jiang, J. Liu, G. Zhang, Y. An, H. Meng, Y. Gao, K. Wang, and J. Tian, “Reconstruction of Fluorescence Molecular Tomography via a Fused LASSO Method Based on Group Sparsity Prior,” IEEE Trans Biomed Eng 66(5), 1361–1371 (2019). [CrossRef]
23. Y. Liu, J. Liu, Y. An, and S. X. Jiang, “Novel Regularized Sparse Model for Fluorescence Molecular Tomography Reconstruction,” International Conference on Innovative Optical Health Science 0245 (2017). [CrossRef]
24. K. M. Tichauer, R. W. Holt, F. El-Ghussein, S. C. Davis, K. S. Samkoe, J. R. Gunn, F. Leblond, and B. W. Pogue, “Dual-tracer background subtraction approach for fluorescent molecular tomography,” J. Biomed. Opt. 18(1), 016003 (2013). [CrossRef]
25. W. Bangerth and A. J. I. P. Joshi, “Adaptive finite element methods for the solution of inverse problems in optical tomography,” Inverse Probl. 24(3), 034011 (2008). [CrossRef]
26. Y. Liu, J. Liu, Y. An, and S. Jiang, Novel Regularized Sparse Model For Fluorescence Molecular Tomography Reconstruction (2017).
27. B. Efron, T. J. Hastie, I. M. Johnstone, and R. J. Tibshirani, “Least angle regression,” Euclid 32, 407–499 (2004). [CrossRef]
28. S. Rosset and J. Zhu, “Piecewise Linear Regularized Solution Paths,” Ann. Statist. 35(3), 1012–1030 (2007). [CrossRef]
29. M. N. Tabassum and E. Ollila, “Sequential adaptive elastic net approach for single-snapshot source localization,” J. Acoust. Soc. Am. 143(6), 3873–3882 (2018). [CrossRef]
30. B. Dogdas, D. Stout, A. F. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Physics in Medicine and Biology 52(3), 577–587 (2007). [CrossRef]
31. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Physics in Medicine and Biology 50(17), 4225–4241 (2005). [CrossRef]
32. M. Liu, H. Guo, H. Liu, Z. Zhang, C. Chi, H. Hui, D. Dong, Z. Hu, and J. Tian, “In vivo pentamodal tomographic imaging for small animals,” Biomed. Opt. Express 8(3), 1356–1371 (2017). [CrossRef]
