Issue 
J. Eur. Opt. SocietyRapid Publ.
Volume 20, Number 1, 2024



Article Number  26  
Number of page(s)  11  
DOI  https://doi.org/10.1051/jeos/2024025  
Published online  26 June 2024 
Research Article
A method of fluorescence molecular tomographic reconstruction via the secondorder sensitivity matrix
School of Electronic and Information Engineering, Soochow University, Suzhou 215006, China
^{*} Corresponding author: zouwei@suda.edu.cn
Received:
6
January
2024
Accepted:
5
May
2024
Fluorescence molecular tomographic (FMT) reconstruction is commonly solved based on the Jacobian matrix, which is a firstorder sensitivity matrix. Basically, using the secondorder derivatives for iterative reconstruction can help improve the performance of convergence. In this paper, a reconstruction method of FMT based on the reduction of the secondorder sensitivity matrix is proposed. In addition, the strategy of detectors rotation is combined into the inverse reconstruction to further improve the reconstruction quality. The reconstructed results demonstrate that the proposed method accelerates the reconstruction with high precision of inverse solutions.
Key words: Optics / Fluorescence / Sensitivity matrix
© The Author(s), published by EDP Sciences, 2024
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1 Introduction
Biomedical imaging has gained a lot of attention due to the potential applications for clinical diagnosis and assessment of treatment. This technique can image the physical properties inside an object with measurements taken from around the object [1, 2]. More specifically, fluorescent molecular tomography (FMT) is an important optical imaging modality. The use of fluorescent agents in imaging diseased tissue has the potential for high specificity and contrast. Due to the advantages of safety, low cost, high sensitivity, and reliability, FMT has the important preclinical and clinical applications as pharmacokinetics. FMT can provide the functional information and therefore can be used to trace the pathological and physiological processes at the molecular level. Further, FMT can be used to quantify expression of tumor proteins for cancer assessment and treatment [3–11].
Since optical tomography remains a challenge, researchers have proposed many different methods [12]. Images of FMT are reconstructed based on the region methods [13, 14]. To tackle the illposedness of inverse problem, A priori sparsity is introduced [15]. L1norm regularization is used and the overrelaxation algorithm is improved [16]. To achieve higher image clarity, a shapebased method based on the cosinoidal level set is conceived [17]. In [18], an AIdriven tomography reconstruction approach is proposed to improve the efficiency for the optical tomography system. A computational segmentation approach with statistical inference is developed to identify the cell nucleus [19]. A deep neural network is designed for reconstruction of red blood cells [20]. A fast holographic measurement system with high resolution is presented [21]. A meshless approach is proposed based on the compactly supported basis functions [22]. A waveletbased method is proposed for FMT, where the principal component analysis is used for solving the inverse problem [23].
Generally, reconstruction for FMT involves the forward and the inverse problems [24]. The firstorder sensitivity matrix as Jacobian matrix is commonly used to solve the inverse problem. Actually, the secondorder derivatives as the Hessian matrix for iterative reconstruction can help improve the performance of convergence and increase the accuracy of image reconstruction [25]. Therefore, we incorporate the Hessian matrix into the process of reconstruction for FMT. Actually, solving largescale matrix equation is involved in the inverse problem of FMT. In this paper, the size of the Hessian matrix is reduced. Consequently, the Hessian of reduced size can help speed up solving the matrix equation involved in the inverse problem, and hence can accelerate the reconstruction process. In the section of Appendix, we prove that the columns and the rows of the Hessian can be removed in the iterative reconstruction. Hence, we can simplify the Hessian matrix by removing the columns and the rows. Additionally, the reconstructed results can be affected by the location or number of measurements. To further improve the reconstruction quality, the reconstruction for FMT is implemented based on a strategy of detectors rotation. The whole reconstruction is implemented using measurements taken from the detectors of different rotation angles. Measurements from the detectors with more angles are capable to offer more reconstruction information than those with the fix angles. Hence, the quality of reconstruction can be improved. Results demonstrate that the proposed method can significantly speed up the reconstruction process and improve the reconstruction quality.
2 Methods
2.1 Forward problem
For forward problem of FMT, the measurements are predicted according to the distributions of excitation light source and the optical parameters of the objects. The mathematical formulation of the forward problem is described primarily by the Maxwell’s equations. The radiative transfer equation (RTE) can be utilized to model the light propagation in tissues [26]. Further, the diffusion equation can be commonly employed to describe the light transportation due to the fact that the RTE involves a large amount of computation [27]. Two partial differential equations are usually employed for FMT to depict the light propagation. They are formulated by(1) (2)
Equation (1) models the excitation light propagation. Equation (2) depicts the propagation for the fluorescent light. In equations (1) and (2), ∇ denotes the gradient operator. S_{e} represents the source of excitation light. denotes the photon fluence. Subscripts e and m represent the parameters at the excitation and the fluorescent wavelength. The diffusion coefficients D_{e,m} can be defined as(3)
The decay coefficients k_{e,m} are defined by(4)
The emission source coefficient β is obtained by(5)
where µ_{aei} and µ_{ami} stand for the absorption coefficients of nonfluorescing chromophore. µ_{aef} and µ_{amf} denote the absorption coefficients of fluorophore. and represent reduced scattering coefficients. The fluorescence quantum efficiency is denoted by η. τ represents the fluorescent lifetime. c represents the speed of light i = (−1)^{1/2}.
In order to obtain the photon fluence, the Robin boundary conditions can be applied for describing the propagation of light on the tissue boundary(6) (7)where n represents the outer normal of the boundary. b_{e} and b_{m} denote the Robin boundary coefficients.
In the framework of the finite element method (FEM) [28–30], the domain Ω can be discretized consisting of P elements and N vertices. Thus, the forward problem can be rewritten as(8) (9)where A_{e,m} can be obtained by(10)
Here, Ω_{h} denotes the domain. Γ_{h} denotes the boundary. u_{i} (i = 1, ..., N) represent basis functions.
2.2 Inverse problem
Generally, the linear approximation is commonly used in conventional FMT reconstruction methods. Due to the high computational complexity of the higherorder derivatives involved in the iterative reconstruction, they are neglected in the largescale reconstruction [31, 32]. Actually, incorporating the Hessian matrix into the process of reconstruction for FMT can help improve the performance of convergence and increase the accuracy of image reconstruction.
In order to derive the framework for the inverse problem based on the Hessian matrix, the inverse reconstruction is formulated as follows(11)
Herein, M(x) denotes an objective function. y ∈ ℝ^{M} represents the measurement. F represents the forward operator and x ∈ ℝ^{N} stands for the optical parameters.
The above objective function can be expanded in a Taylor series with the firstorder term and secondorder term(12)
Herein, ∆x stands for the perturbation in optical parameters, ∇M(x) stands for the gradient of the objective function at x (13)
As the secondorder sensitivity matrix, the Hessian matrix H represents the secondorder partial derivative of M with respect to x. The N × N Hessian matrix can be achieved by:(14)
In order to tackle the illposedness of inverse problem for FMT, the regularization parameter λ is introduced [30]. Suppose M attains its extremum at x + Δx, and thus the inverse reconstruction based on the Hessian matrix can be expressed by:(15)
From equation (15), it can be seen that solving matrix equation is involved in the inverse problem. One of the major challenges in the reconstruction is its high computational complexity resulted from largerscale matrix manipulations. If the size of Hessian is reduced, computation requirements of solving the matrix equation will be reduced. Consequently, the reconstruction process can be accelerated.
2.3 Reduction of the Hessian matrix
During the process of iterative reconstruction of equation (15), the Hessian matrix H needs to be repeatedly calculated. Thus, the Hessian matrix is critically important for the reconstruction speed. Traditionally, the Hessian matrix can be achieved by direct derivation method, which is computationally expensive especially for largescale matrices. Hence, it is important to design a calculation method of Hessian matrix for fast reconstruction of FMT. Actually, if the Hessian matrix is simplified, the computing efficiency for inverse reconstruction can be enhanced.
In this paper, we can prove the column H_{i} of the Hessian is capable to be removed in the iterative reconstruction of solving equation (15) as H_{i} approaches 0 as in the section of Appendix A(1). Therefore, we can simplify the Hessian matrix by removing the columns. Consequently, the computation time for inverse reconstruction can be accelerated. The Hessian matrix is reduced through removing the jth column when the condition as below is satisfied(16)where H_{ij} represents the element of the Hessian matrix, and δ denotes a threshold.
Furthermore, we can also prove the row R_{i} in the Hessian is capable to be removed during the iterative reconstruction when R_{i} approaches 0 as in the section of Appendix A(2). Thus, the rows in the Hessian matrix are also be removed to further improve the computation efficiency of reconstruction when the following condition is satisfied,(17)
From equation (A14), it can be seen that the ith row in the Hessian is related to the ith component in ∇M(x). Therefore, removing the ith row in the Hessian means the ith element in ∇M(x) can be removed. Consequently, the product of H ^{T}∇M(x) is achieved. Thus, ∇M(x) is reduced with the reduction of rows in the Hessian. With the reduction of Hessian, solving the matrix equation involved in the inverse problem as in equation (15) can be accelerated, which is beneficial for speeding up the reconstruction process. Therefore, the computational complexity of inverse reconstruction is reduced.
2.4 Reconstruction with the strategy of detectors rotation
Basically, the detectors are placed at the fixed positions in inverse reconstruction. The location or number of measurements can also affect the reconstructed results. In order to enhance the reconstruction efficiency, an approach of detectors rotation is combined into the reconstruction. In this approach, the measurements taken from the original detectors are used for reconstruction in the first iteration. Then the measurements taken from the rotated detectors with a certain angle are used for next iteration. The whole reconstruction is implemented using measurements taken from the detectors of different rotation angles. During the process of reconstruction, measurements taken from the detectors with more angles are capable to offer more reconstruction information than those with the fix angles. Hence, the quality of reconstruction can be improved. In this paper, we set the rotation angle as a half of the angle between the adjacent detectors. Figure 1 depicts the strategy of detectors rotation.
Figure 1 Illustration of the approach of detectors rotation. (a) Detectors before the rotation. (b) Detectors after the rotation. 
3 Results and discussion
To validate the proposed method, we first use the simulated phantom with one inclusion as shown in Figure 2 to test the performance. The simulated input data is used to assess the performance of the algorithm. Four excitation light sources and fifteen detectors are evenly placed around of the phantom, where the triangle denotes the excitation light source and the circle denotes the detector. The inverse reconstruction is performed with the grid containing 212 triangular elements as shown in Figure 3 [23].
Figure 2 Simulated phantom of one inclusion. 
Figure 3 Reconstruction grid for one inclusion. 
Table 1 lists the details for optical parameters of the reconstruction phantom. To quantitatively evaluate the reconstruction results, the mean square error (MSE) is introduced(18)
Optical parameters for oneinclusion phantom.
where the superscript rea stands for the real values of the optical parameters and rec stands for the reconstructed optical parameters. To choose the threshold δ, another parameter is introduced as(19)
where ε is set as 0.05 in this work.
The reconstructed results of μ_{aef} of oneinclusion phantom using the proposed approach and that using the Jacobian matrix are depicted in Figure 4(a) and 4(b), respectively. We see the proposed method yields the reconstruction result with improved quality comparatively to the method based on the Jacobian matrix. Table 2 lists the performance of reconstructions in terms of the computational time and MSE for quantitative validation. The reconstructions are implemented based on the computer with CPU at 2.6 GHz and 1 GB RAM. We remark the computational time of the proposed approach is faster than that of the Jacobian matrix method. Additionally, the MSE of our approach is smaller than that of the Jacobian matrix method. Therefore, the proposed method accelerates the inverse reconstruction process and achieves high precision.
Figure 4 Reconstructed results of absorption coefficient μ_{aef} of oneinclusion phantom. (a) Reconstructed result using the proposed approach. (b) Reconstructed result using the Jacobian matrix. 
Method performance of phantom of one inclusion.
To validate the performance of the strategy of detectors rotation, Figure 5(a) and 5(b) provide the results without the strategy of detectors rotation and that with the proposed method, respectively. Table 3 provides the quantitative comparisons of different reconstructions. It is observed that the approach of detectors rotation improves the performance for reconstruction.
Figure 5 Reconstructed results of absorption coefficient of oneinclusion phantom. (a) Reconstructed result without the strategy of detectors rotation. (b) Reconstructed result using the proposed approach. 
Method performance of phantom of one inclusion.
Figure 6 shows the simulated phantom, which contains two inclusions with different shapes. Four excitation light sources are uniformly located on the boundary of this phantom. The measurements are achieved from fifteen detectors around the phantom. The triangle denotes the excitation light source and the circle denotes the detector. The reconstruction is implemented based on the grid consisting of 264 triangular elements, which is described in Figure 7 [23]. Table 4 lists the values of optical parameters of the phantom.
Figure 6 Simulated phantom with two inclusions. 
Figure 7 Reconstruction grid for two inclusions. 
Optical parameters for twoinclusion phantom.
The reconstructed image using our approach is provided in Figure 8(a) and the result from the method based on the Jacobian matrix is given in Figure 8(b). We see the quality of image can be enhanced by our method. Table 5 presents the quantitative comparisons of different reconstruction methods. It can be seen that the improvement in reconstruction accuracy is achieved using the proposed approach. In addition, we can see that the computational time of the proposed approach is less than that using the Jacobian matrix method. Thus, our approach improves the computation efficiency with high quality.
Figure 8 Reconstructed results of absorption coefficient μ_{aef} of twoinclusion phantom. (a) Reconstructed result using the proposed approach. (b) Reconstructed result using the Jacobian matrix. 
Method performance of phantom of two inclusions.
The reconstruction result without the approach of detectors rotation and that with the proposed approach are described in Figure 9(a) and 9(b). The performance of reconstruction from different methods is described in Table 6. We can observe that improvements in reconstruction quality are gained with the strategy of detectors rotation.
Figure 9 Reconstructed results of absorption coefficient of twoinclusion phantom. (a) Reconstructed result without the strategy of detectors rotation. (b) Reconstructed image using the proposed approach. 
Method performance of phantom of two inclusions.
In order to investigate the effect of reducing the size of Hessian matrix, reconstructions based on the proposed method and without reduction of the Hessian matrix are performed. The comparison of performance is listed in Table 7. It can be seen that the method of reduction of Hessian matrix can significantly speed up the reconstruction process at the expense of a small reduction in reconstruction accuracy.
Comparison of performance for phantom of two inclusions.
To further evaluate the robustness of the algorithms, Gaussian noise with a signaltonoise ratio of 10 dB is added to the simulated data. Table 8 shows the reconstruction performance of different reconstruction methods. It can be seen that our approach is able to achieve the reconstruction results with high quality.
Comparison of performance for phantom of two inclusions.
4 Conclusions
In this paper, an efficient method for image reconstruction of FMT based on the reduction of Hessian matrix is developed. Further, the strategy of detectors rotation is combined into the inverse reconstruction to enhance the quality of reconstruction. We see from the reconstruction results that the proposed method can considerably accelerate the reconstruction of FMT. In addition, the proposed method improves the accuracy for the inverse solutions.
Funding
This work was supported by Suzhou Science and Technology Planning Project (Grant No. SKJY2021044), Natural Science Foundation of Jiangsu Province, China (Grant No. BK20130324, BK20171249), Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) (Grant No. 20123201120009), and Natural Science Foundation of the Jiangsu Higher Education Institutions of China (Grant No. 12KJB510029).
Conflicts of interest
The authors declare that they have no conflict of interest.
Data availability statement
The datasets generated during the current study are available from the corresponding author on reasonable request.
Author contribution statement
Conceptualization, W.Z.; methodology, W.Z and J.W.; software, W.Z.; validation, W.Z. and J.W.; formal analysis, W.Z. and J.W.; writing—original draft preparation, W.Z.; writing—review and editing, W.Z; supervision, J.W.
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Appendix A
1 Proof
stand for the perturbation of optical parameters and the Hessian, which are substituted into equation (15). Then, we can obtain(A1)
Equation (A1) can be written as follows(A2)
From equation (A2), the ith equation can be formulated by(A3)
The limit of ∆x_{i} as H_{i} tends towards 0 is obtained by(A5)
Equation (A1) can be rewritten as follows(A6)
When H_{i} approaches 0, we have(A7)
Equation (A9) can be rewritten by(A10)
From equations (A1) and (A12), we see the ith column H_{i} in the Hessian is capable to be removed as H_{i} tends towards 0 for solving equation (15).
2 Proof
Substituting them into equation (15), we can obtain(A13)
From equation (A13), we have(A14)
When R_{i} approaches 0, we have(A15)
Equation (A16) can be rewritten by(A17)
From equations (A13) and (A17), we see that the ith row R_{i} in the Hessian is capable to be removed as R_{i} tends towards 0 for solving equation (15).
All Tables
All Figures
Figure 1 Illustration of the approach of detectors rotation. (a) Detectors before the rotation. (b) Detectors after the rotation. 

In the text 
Figure 2 Simulated phantom of one inclusion. 

In the text 
Figure 3 Reconstruction grid for one inclusion. 

In the text 
Figure 4 Reconstructed results of absorption coefficient μ_{aef} of oneinclusion phantom. (a) Reconstructed result using the proposed approach. (b) Reconstructed result using the Jacobian matrix. 

In the text 
Figure 5 Reconstructed results of absorption coefficient of oneinclusion phantom. (a) Reconstructed result without the strategy of detectors rotation. (b) Reconstructed result using the proposed approach. 

In the text 
Figure 6 Simulated phantom with two inclusions. 

In the text 
Figure 7 Reconstruction grid for two inclusions. 

In the text 
Figure 8 Reconstructed results of absorption coefficient μ_{aef} of twoinclusion phantom. (a) Reconstructed result using the proposed approach. (b) Reconstructed result using the Jacobian matrix. 

In the text 
Figure 9 Reconstructed results of absorption coefficient of twoinclusion phantom. (a) Reconstructed result without the strategy of detectors rotation. (b) Reconstructed image using the proposed approach. 

In the text 
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