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In bioluminescence tomography (BLT), reconstruction of internal bioluminescent source distribution from the surface optical signals is an ill-posed inverse problem. In real BLT experiment, apart from the measurement noise, the system errors caused by geometry mismatch, numerical discretization, and optical modeling approximations are also inevitable, which may lead to large errors in the reconstruction results. Most regularization techniques such as Tikhonov method only consider measurement noise, whereas the influences of system errors have not been investigated. In this paper, the truncated total least squares method (TTLS) is introduced into BLT reconstruction, in which both system errors and measurement noise are taken into account. Based on the modified generalized cross validation (MGCV) criterion and residual error minimization, a practical parameter-choice scheme referred to as improved GCV (IGCV) is proposed for TTLS. Numerical simulations with different noise levels and physical experiments demonstrate the effectiveness and potential of TTLS combined with IGCV for solving the BLT inverse problem.

In recent years, molecular imaging has emerged as a promising tool in basic, preclinical and clinical research for monitoring a variety of molecular and cellular processes in living organisms [

The key problem of BLT is to reconstruct the bioluminescent source distribution inside a biological tissue from the optical signals detected on the body surface, which is a highly ill-posed inverse problem. By using numerical method such as finite element method (FEM), the inverse problem of BLT can be formulated into a nonsquare matrix equation, where the coefficient matrix is typically ill-conditioned [

The total least squares (TLS) method is a generalization of the least squares approximation method when the data in both sides of the matrix equation are perturbed [

In this paper, the aim of our study is to extend the BLT reconstruction to the case including both the measurement noise and the system errors. For this purpose, TTLS method combined with a practical scheme termed as improved GCV (IGCV) is proposed to solve the BLT inverse problem. In the next section, our methodology of solving the inverse problem in BLT is described. In Section

In general, light propagation in living subjects is mainly hindered by both tissue scattering and absorption [

Assuming that the BLT experiment is performed in a totally dark environment, the equation is subject to a Robin boundary condition [

Based on (

In order to obtain a stable solution, regularization methods are typically used for solving inverse problems [

As mentioned in the introduction, TLS method is designed for the case that both sides of the matrix equation are subject to errors. BLT inverse problem can be stated with TLS formulation as follows:

The TTLS algorithm used in this paper can be summarized as follows.

Compute the SVD of the augmented matrix

Select a truncation parameter

Partition the matrix

Then the TTLS solution is given by

In fact, the aim of TTLS regularization is to appropriately identify an optimal truncation level, and then to construct a truncated solution that can capture the essential features of the unknown true solution, without explicit knowledge about the true solution and even without a priori knowledge about the magnitude of the noise in the data. For this purpose, truncation level

MGCV criterion proposed by Sima in [

The property used for choosing the truncation parameter

As for the regularization problem in BLT, the choice of regularization parameter with classical GCV is by means of minimizing the GCV function:

According to the properties of filter factors mentioned above, for a

However, the regularization parameter directly identified by (

Use the MGCV criterion to get an initial truncation parameter

For

For all the local minimum points, compute the residual error

Thus, a proper truncation parameter

The experiments implemented in this section are to test the performance of TTLS combined with IGCV for BLT inverse problem. To demonstrate the effectiveness of the proposed scheme, we compare the following reconstruction algorithms: Tikhonov method with classical GCV (Tik-GCV), TTLS method with MGCV (TTLS-M), and TTLS method with the proposed IGCV (TTLS-I). The parameter-choice scheme of Tikhonov method is different from that of TTLS method because MGCV and IGCV are specially designed for TTLS. A similar scheme, namely, classical GCV, is adopted in Tikhonov method for comparison convenience. The qualities of the reconstruction are assessed by the following quantitative indices: relative residual error (RRE), reconstructed location error, and reconstructed source power. Here, RRE is used to depict the extent of the solution fitting the measured data and is defined as

In the numerical simulation, a 30 mm diameter and 30 mm high cylindrical mouse chest phantom is designed to evaluate the performance of the reconstruction method. The structure of the phantom is shown in Figure

Optical parameters of the heterogeneous phantom.

Material | ^{-1}) | ^{-1}) |
---|---|---|

Tissue | 0.007 | 1.031 |

Lung | 0.023 | 2.000 |

Heart | 0.011 | 1.096 |

Bone | 0.001 | 0.060 |

(a) A cross-section through two luminescent sources (S) in the left lung of a mouse phantom consisting of bone (B), heart (H), lungs (L), and tissue (T). (b) A 3D view of the permissible region.

In order to reduce the ill-posedness of the inverse problem, a priori information of the source permissible region (PR) is incorporated to our method, which is shown in Figure

Generally speaking, simulated data used in reconstruction algorithms for inverse problems often come from the numerical solution of the forward problem. To avoid the typical issue of

To comprehensively simulate the noise and system errors involved in real BLT experiment, the photon flux density

As discussed in Section

Quantitative results in single source case.

Sys. error | Meas. noise | Recon. method | Sys. error pattern | Regular. param. | RRE | Recons. power (nW) |
---|---|---|---|---|---|---|

Without errors | Without noises | Tik-GCV | N/A | 0.00946 | 0.0263 | 0.4284 |

TTLS-M | N/A | 75 | 0.0243 | 0.4661 | ||

TTLS-I | N/A | 78 | ||||

10% | Tik-GCV | N/A | 0.01437 | 0.0585 | 0.4096 | |

TTLS-M | N/A | 69 | 0.0517 | 0.4535 | ||

TTLS-I | N/A | 78 | ||||

20% | Tik-GCV | N/A | 0.02266 | 0.1083 | 0.3825 | |

TTLS-M | N/A | 65 | 0.0809 | 0.4641 | ||

TTLS-I | N/A | 69 | ||||

1% | 10% | Tik-GCV | Gaus. | 0.01847 | 0.0596 | 0.3502 |

Exp. | 0.02288 | 0.0705 | 0.3074 | |||

TTLS-M | Gaus. | 64 | 0.0590 | 0.3982 | ||

Exp. | 71 | 0.0562 | 0.3843 | |||

TTLS-I | Gaus. | 78 | ||||

Exp. | 82 | |||||

20% | Tik-GCV | Gaus. | 0.02844 | 0.1150 | 0.4295 | |

Exp. | 0.02281 | 0.0897 | 0.3432 | |||

TTLS-M | Gaus. | 52 | 0.1142 | 0.4410 | ||

Exp. | 73 | 0.0599 | 0.4755 | |||

TTLS-I | Gaus. | 64 | ||||

Exp. | 87 | |||||

5% | 10% | Tik-GCV | Gaus. | 0.03771 | 0.0850 | 0.3069 |

Exp. | 0.05016 | 0.1023 | 0.2760 | |||

TTLS-M | Gaus. | 65 | 0.0823 | 0.3498 | ||

Exp. | 53 | 0.1018 | 0.2960 | |||

TTLS-I | Gaus. | 94 | ||||

Exp. | 85 | |||||

20% | Tik-GCV | Gaus. | 0.05414 | 0.1626 | 0.2722 | |

Exp. | 0.06586 | 0.1871 | 0.2618 | |||

TTLS-M | Gaus. | 52 | 0.1536 | 0.3088 | ||

Exp. | 45 | 0.1758 | 0.2927 | |||

TTLS-I | Gaus. | 61 | ||||

Exp. | 109 |

Quantitative results in double source case.

Sys. error | Meas. noise | Recon. method | Sys. error pattern | Regular. param. | RRE | Recon. power (nW) | |

Without errors | Without noises | Tik-GCV | N/A | 0.00858 | 0.0252 | 0.4671 | 0.2677 |

TTLS-M | N/A | 76 | |||||

TTLS-I | N/A | 76 | |||||

10% | Tik-GCV | N/A | 0.01533 | 0.0586 | 0.4334 | 0.2148 | |

TTLS-M | N/A | 75 | |||||

TTLS-I | N/A | 75 | |||||

20% | Tik-GCV | N/A | 0.03001 | 0.1363 | 0.3633 | 0.1787 | |

TTLS-M | N/A | 59 | 0.1074 | 0.4425 | 0.2069 | ||

TTLS-I | N/A | 71 | |||||

1% | 10% | Tik-GCV | Gaus. | 0.01711 | 0.0560 | 0.4164 | 0.2448 |

Exp. | 0.02148 | 0.0618 | 0.4174 | 0.1974 | |||

TTLS-M | Gaus. | 74 | |||||

Exp. | 71 | ||||||

TTLS-I | Gaus. | 74 | |||||

Exp. | 71 | ||||||

20% | Tik-GCV | Gaus. | 0.03112 | 0.1248 | 0.3506 | 0.1891 | |

Exp. | 0.03034 | 0.1213 | 0.3688 | 0.1862 | |||

TTLS-M | Gaus. | 59 | 0.1023 | 0.4529 | 0.2042 | ||

Exp. | 50 | 0.1347 | 0.4085 | 0.2086 | |||

TTLS-I | Gaus. | 72 | |||||

Exp. | 58 | ||||||

5% | 10% | Tik-GCV | Gaus. | 0.04179 | 0.0969 | 0.2613 | 0.2347 |

Exp. | 0.05154 | 0.1151 | 0.3528 | 0.2427 | |||

TTLS-M | Gaus. | 63 | 0.0890 | 0.3194 | 0.2647 | ||

Exp. | 50 | 0.1192 | 0.3495 | 0.2576 | |||

TTLS-I | Gaus. | 81 | |||||

Exp. | 65 | ||||||

20% | Tik-GCV | Gaus. | 0.05077 | 0.1467 | 0.2859 | 0.1574 | |

Exp. | 0.06730 | 0.1858 | 0.3122 | 0.2480 | |||

TTLS-M | Gaus. | 51 | 0.1376 | 0.3713 | 0.1574 | ||

Exp. | 48 | 0.1741 | 0.3557 | 0.3304 | |||

TTLS-I | Gaus. | 87 | |||||

Exp. | 54 |

Regularization parameter determination in single-source case under measurement noise level of 10% and Gaussian system error level of 1%: (a) GCV function curve for Tikhonov, (b) MGCV function curve for TTLS, (c) illustration of the truncation parameter selection for TTLS with IGCV.

In single source test, we found that all the methods under consideration can detect the source with the same center location

Reconstructed results under measurement noise level of 10% and Gaussian system error level of 1% with TTLS + IGCV: (a)

In the double sources case, both of the two sphere sources located in the left lung are tested. The final reconstruction results are listed in Table

Reconstructed results in double source case under measurement noise level of 20% and system error level of 5%. (a), (b), and (c) separately show the

For BLT inverse problem, permission region is an effective way to regularize the solution by restricting the source distribution within a proper permissible region. In order to further test the proposed method, a ball shape permissible region of 10 mm in diameter is utilized, which is expressed as

Quantitative results for ball shape permission region

Sys. error | Meas. noise | Recon. method | Regular. param. | RRE | Recon. position (mm) and power (nW) | |||

1% | 10% | Tik-GCV | 0.02728 | 0.1720 | 0.2102 | 0.2509 | ||

TTLS-M | 19 | 0.1535 | 0.2308 | 0.2373 | ||||

TTLS-I | 30 | 0.1523 | ||||||

20% | Tik-GCV | 0.09397 | 0.4383 | 0.3633 | 0.3501 | |||

TTLS-M | 9 | 0.4288 | 0.3782 | 0.3651 | ||||

TTLS-I | 28 | 0.4226 | ||||||

5% | 10% | Tik-GCV | 0.09660 | 0.2350 | 0.3693 | 0.3784 | ||

TTLS-M | 9 | 0.2350 | 0.3752 | 0.3853 | ||||

TTLS-I | 18 | 0.2171 | ||||||

20% | Tik-GCV | 0.12804 | 0.4367 | 0.3549 | 0.3578 | |||

TTLS-M | 15 | 0.3865 | 0.3900 | 0.3653 | ||||

TTLS-I | 19 | 0.3532 |

It is shown in Table

A physical experiment was carried out to further investigate the performance of the proposed method. A cylindrical phantom of 45 mm height and 22.5 mm radius was designed to evaluate different methods. The phantom shown in Figure ^{-1} and 0.0138 mm^{-1}, respectively.

Physical phantom. (a) The homogeneous physical phantom; (b) The location of the single source in the phantom; (c) The cross-section of the phantom and the four directions of the CCD camera during data acquisition.

A scientific cooled back-illuminated CCD camera (PIXIS 2048B) is used to collect the outgoing photons from the phantom surface. The photon flux density from different angles can be acquired by rotating the stage under the phantom, as illustrated in Figure

The normalized surface measurement of the homogeneous phantom. (a), (b), (c), and (d) are left view, front view, right view, and back view of the cylindrical phantom on the CCD camera, respectively; (e) is the flux density on the surface of the cylindrical phantom after mapping from the CCD camera.

According to the photon flux density distribution on the phantom surface, the source permissible region is set as

Reconstruction results in physical phantom experiment.

Recon. method | Regular. param. | RRE | Recon. source position (mm) | AE (mm) |
---|---|---|---|---|

Tik-GCV | 0.00003 | 0.9513 | (7.64,4.42,27.18) | 3.71 |

TTLS-M | 40 | 0.9015 | 2.97 | |

TTLS-I | 45 |

Reconstructed results in physical experiment: (a)–(c) are the

BLT reconstruction is a highly ill-posed inverse problem where small measurement noise and system errors in the input data can produce large changes in the results. In addition, bioluminescence signals are generally very weak, thus the noise or errors will significantly affect the reconstruction quality. Regularization technique has played an important role in solving BLT inverse problem. And most of the previous works assume that there is only measurement noise, which affects the right-hand side of the system equations. However, the computed coefficient matrix

Simulations considering both system errors and measurement noise are conducted to investigate the performance of the proposed reconstruction method. Due to the lack of an accurate model to describe the system errors arising from multiple sources, commonly used Gaussian white noise and exponential noise are adopted to simulate the errors in matrix

Both the numerical simulations and physical experiments demonstrate the effectiveness of the proposed method. Tests with different noise levels show that TTLS with combined IGCV is able to produce much better reconstruction results than Tikhonov method, and TTLS combined with IGCV performs better than TTLS combined with MGCV, especially when both sides of the system equation are contaminated by measurement noise and system errors. Based on the experiments in this paper, we can draw a preliminary conclusion that TTLS combined with IGCV criterion is a potential reconstruction method for BLT inverse problem. Further investigation of the performance of the proposed method on animal experiments will be conducted in our future work.

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, 60532050, 30900334, the Beijing Municipal Natural Science Foundation of China under Grants no. 4071003, the CAS Scientific Research Equipment Develop Program (YZ0642, YZ200766), the Natural Science Basic Research Plan in Shaanxi Province of China under Grant no. 2009JQ8018, and the Science Foundation of Northwest University under Grant no. 09NW34.