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Bioluminescence tomography (BLT) is a promising tool for studying physiological and pathological processes at cellular and molecular levels. In most clinical or preclinical practices, fine discretization is needed for recovering sources with acceptable resolution when solving BLT with finite element method (FEM). Nevertheless, uniformly fine meshes would cause large dataset and overfine meshes might aggravate the ill-posedness of BLT. Additionally, accurately quantitative information of density and power has not been simultaneously obtained so far. In this paper, we present a novel multilevel sparse reconstruction method based on adaptive FEM framework. In this method, permissible source region gradually reduces with adaptive local mesh refinement. By using sparse reconstruction with

In vivo bioluminescence imaging (BLI) is a low-cost, noninvasive, and valuable tool for studying physiological and pathological processes at cellular and molecular levels. This technology has been applied to various biological models to diagnose disease, monitor therapies, and facilitate drug development [

Mathematically, BLT is a severely underdetermined and ill-posed problem, which is mainly caused by insufficient measurement and the highly diffusive nature of the photon propagation in tissue [

From a computational perspective, the challenge in BLT, as in many other imaging modalities, is to reach the desirable resolution within acceptable computational cost. As an effective numerical method, finite element method (FEM) has been widely used in BLT reconstruction especially when the domain is arbitrary geometry [

Additionally, quantitative evaluation of the reconstructed density and power as well as accurate location is necessary in clinical or preclinical practice. For example, reconstructed total power can reflect the total tumor cell number, which is the basis for continuous monitoring but has gained less attention in most existing BLT studies so far. Especially, accurate quantitative information of density and power has not been simultaneously obtained so far. Note that the recovered density and power are associated with not only the mesh discretizations but also the regularization method used in the reconstruction. Because of the smooth characteristic of an

In the past few years, sparse regularization has been investigated in the area of compressed sensing (CS) for signal and image processing. According to the theory of CS, one can reconstruct a sparse or compressible signal from far fewer samples or measurements than what the Nyquist sampling theorem demands [

In this paper, inspired by CS, a novel sparse reconstruction method is proposed based on multilevel adaptive finite element framework for BLT. During the reconstruction process, the PSRs gradually shrink with adaptive local mesh refinement, which can effectively reduce the ill-posedness of BLT. In view of the characteristic of sparseness and undersampling in most BLT scenario, sparse regularization with

In Section

The radiative transfer equation (RTE) is regarded as the most accurate model for the light transport in tissue [

As a powerful tool, FEM has been widely used for solving diffusion equations, especially for solving domain with arbitrary geometries [

According to the surface photon distribution and anatomical information, the PSR can be identified as

To achieve the necessary resolution within acceptable computational cost, the domain

Let

The multilevel adaptive FEM based reconstruction algorithm includes the following three steps:

(

(

Ideally, we should conduct the mesh refinement based on rigorously derived error estimates [

When switching a coarser mesh

(

Note that, the reconstructed result at the previous mesh level not only guides mesh refinement and provides an initial value for the refined mesh, but also identifies the PSR for the subsequent reconstruction. Thus, the preliminary solution on the initial coarse mesh is very important.

For each mesh level, BLT reconstruction is carried out by solving problem (

However, due to the inherent characteristic of

The objective function in (

In this work, a truncated Newton interior-point method (TNIPM) is adopted at each mesh level to solve (

In TNIPM, the

And then, adding the constraints into the minimization problem (

Initialize the parameters: relative tolerance

1. Compute the search direction

(

2. Compute the step size

3. Update the iterate by

4. Construct a dual feasible point

5. Evaluate the duality gap

6.

7. Update

As suggested by [

At the end of this subsection, we summarize the multilevel

Initialize the parameter: set

Establish the linear system equation

Select those elements satisfy

mesh level;

Perform local mesh refinement and interpolate the new PSR to the next finer mesh;

We conducted a set of experiments with a numerical phantom model and a digital mouse model to validate the proposed multilevel

The qualities of the reconstruction are quantitatively assessed in terms of location error, relative error (RE) of source density and power. Here, the reconstructed power is estimated by computing the integral of the source density over its support domain, and the corresponding RE of density and power are calculated by

A cylindrical mouse chest phantom with 30 mm diameter and 30 mm height was employed to evaluate the performance of the ^{−1} and ^{−1 }for muscle, ^{−1} and ^{−1} for lung, ^{−1} and ^{−1} for heart, ^{−1} and ^{−1} for bone [

(a) Mouse chest phantom composed of muscle, lungs, heart, and bone, with one source in right lung. (b) The forward discretized mesh and the photon distribution on the surface. (c) The initial mesh used in the adaptive reconstruction, with average edge size 1.637 mm.

In the simulations, the phantom was discretized into a fine tetrahedral-element mesh to generate the synthetic measurements on the surface using FEM. To simulate the noise involved in real BLT experiment, 10% random Gaussian noise was added to synthetic measurements.

Firstly, reconstruction for a single source target was attempted. A solid spherical source with 0.5 mm radius was centered at (9.5 mm, 1 mm, 15 mm) inside the right lung. The initial power source was 0.5236 nano-Watts, and the power density was 1 nano-Watts/mm^{3}. The forward mesh of the phantom consisted of 11288 nodes and 62069 tetrahedral elements with 10832 boundary elements. Figure

PSR strategy was incorporated to the reconstruction algorithm to decrease the ill-posedness of BLT. As

The reconstruction was carried out using the proposed algorithm. The maximum mesh level was set to 4. The reconstructed results with regularization on multilevel adaptive meshes are shown in Figures

Reconstruction results in single source case on different mesh levels.

Mesh | Regular. | Number | Location center | Error | Density^{3}) | RE of | Recon. | RE of |
---|---|---|---|---|---|---|---|---|

1 | 3623 | 9.42,1.24,15.02 | 0.25 | 0.0434 | 95.66% | 0.3315 | 36.69% | |

3623 | 9.43,−0.16,14.55 | 1.24 | 0.0418 | 95.82% | 0.2896 | 44.69% | ||

2 | 3924 | 9.42,1.24,15.02 | 0.25 | 0.0942 | 90.58% | 0.4515 | 13.77% | |

3960 | 9.18,0.43,15.26 | 0.70 | 0.0522 | 94.78% | 0.3396 | 35.14% | ||

3 | 4242 | 9.42,1.24,15.02 | 0.25 | 0.3105 | 68.95% | 0.4537 | 13.35% | |

4435 | 9.83,0.96,15.42 | 0.54 | 0.1227 | 87.73% | 0.4185 | 20.07% | ||

4 | 4910 | 9.42,1.24,15.02 | 0.25 | 1.0056 | 5.6% | 0.4663 | 10.94% | |

5232 | 9.40,0.54,15.26 | 0.54 | 0.3739 | 62.61% | 0.3700 | 29.34% |

Reconstruction results in single source case on different mesh level, where the actual source is drawn as a red sphere. (a)–(d) are the isosurface of the reconstructed density by regularization from initial level to the final level, respectively. (e)–(h) are the corresponding results by regularization method, with a threshold of 50% of the maximum value.

The reconstructed source positions by

Mesh evolution in the single source case and the regularized solutions on different mesh levels. The green mesh denotes the local region around the regularized solution in PSR; the black sphere is the actual source. (a), (b), and (c) are the reconstruction by

It is noted that the quantitative information of source density and power is remarkably enhanced as the mesh became finer due to the multilevel meshes strategy. The final REs of density and power in

As aforementioned, compared with

Comparison of the regularized solutions on the initial coarse mesh.

In order to investigate the spatial resolution capability of the proposed multilevel reconstruction method, we performed a multisource simulation experiment. Beside the spherical source located in right lung, two spatially close sources were added to the previous phantom with their centers at (−9 mm, −1.5 mm, 15 mm) and (−9 mm, 1.5 mm, 15 mm), respectively. The two sources located in left lung were 2 mm apart. The size, density, and power of each source were the same as in the single source case. The initial PSR was-same those that of single source case in this experiment. The final quantitative reconstruction results and the comparison with the actual sources are summarized in Table

Quantitative results and the comparison with the actual sources in multisource case.

Source | Actual | Recon. | Location | Recon. density^{3}) | RE of density | Recon. power | RE of power |
---|---|---|---|---|---|---|---|

Source-1 | (9.5,1,15) | (9.42,1.24,15.02) | 0.49 | 1.06 | 6% | 0.4916 | 6.12% |

Source-2 | (−9, 1.5,15) | (−9.29,1.5,15.06) | 0.30 | 0.5713 | 42.87% | 0.4416 | 15.66% |

Source-3 | (−9, −1.5,15) | (−9.30, −1.46,15.08) | 0.31 | 0.6205 | 37.95% | 0.4584 | 12.45% |

Incorporating PSR into the reconstruction algorithm, the proposed method can always accurately distinguish these sources at different mesh levels. The reconstruction results in Figure

Reconstruction results in multiple sources case. (a)–(c) The isosurface of the reconstruction by the proposed method on the first and the last level, respectively. (b)–(d), The corresponding transverse view of the reconstruction at

The numerical experiment with a 3D digital mouse atlas was also performed to further demonstrate the performance of the proposed reconstruction method on a real animal-shaped model. A mouse atlas of CT and cryoSection data was employed to provide anatomical information [^{3}, respectively.

Optical properties for the atlas organs region.

Material | Muscle | Lung | Heart | Liver | Kidney | Stomach |
---|---|---|---|---|---|---|

^{−1}] | 0.23 | 0.35 | 0.11 | 0.45 | 0.12 | 0.21 |

^{−1}] | 1 | 2.3 | 1.1 | 2 | 1.2 | 1.7 |

A 3D digital mouse model. (a) The torso of the mouse model with a cylindrical source in the liver. (b) Forward mesh and photon distribution on surface.

This torso model was discretized into tetrahedral-element mesh to generate the synthetic measurements on the boundary. The forward mesh consisted of 112795 elements and 21277 nodes, as shown in Figures

It took about 120 seconds to complete the multilevel

Reconstruction results for 3D atlas model on different mesh level.

Mesh level | Mesh size | Number | Location center | Error (mm) | Density^{3}) | Power |
---|---|---|---|---|---|---|

1 | 1.6833 | 2382 | (17.85,6.22,14.85) | 0.61 | 0.0026 | 0.0636 |

2 | 1.3087 | 2641 | (17.85,6.22,14.85) | 0.61 | 0.0309 | 0.1951 |

3 | 0.8277 | 3033 | (17.99,6.31,15.88) | 0.49 | 0.3080 | 0.3677 |

4 | 0.6181 | 3908 | (17.99,6.31,15.88) | 0.49 | 1.0231 | 0.6518 |

Reconstruction results of 3D digital mouse model on different mesh level. (a)–(d) are the 3D view of the results by the proposed method from the first level to the fourth level, respectively. (e)–(h) are the corresponding XY view of these results, where the small black circles indicate the real sources.

In this paper, we present a sparse reconstruction method based on multilevel adaptive FEM and evaluated its performance in numerical simulation. Numerical simulation results suggest that the

It is well known that the density as well as position and shape of reconstructed source are significantly affected by the degree of discretization [

In the existing adaptive FEM based reconstruction methods, although the source density can be remarkably improved as the mesh became finer, the reconstructed power tends to decline. The reconstruction results by using

We observed that relatively accurate power and density can be simultaneously recovered when the mesh dimension is commensurate to the source size by the proposed method. There are two key points contributing to the superior performance of the proposed reconstruction method: (

The experiment on a mouse-shaped model with heterogeneous optical properties demonstrates the potentiality for animal experiments. Physical phantom and

This work is supported by the Program of the National Basic Research and Development Program of China (973) under Grant no. 2006CB705700, the Cheung Kong Scholars and Innovative Research Team in University (PCSIRT) under Grant no.IRT0645, the Chair Professors of Cheung Kong Scholars Program of Ministry of Education of China, CAS Hundred Talents Program, the National Natural Science Foundation of China under Grant no. 30873462, 30900334, the Shaanxi Provincial Natural Science Foundation Research Project under Grant no. 2009JQ8018, the Fundamental Research Funds for the Central Universities, and the Science Foundation of Northwest University under Grant no. 09NW34.