## A generalized hybrid algorithm for bioluminescence tomography |

Biomedical Optics Express, Vol. 4, Issue 5, pp. 709-724 (2013)

http://dx.doi.org/10.1364/BOE.4.000709

Acrobat PDF (2604 KB)

### Abstract

Bioluminescence tomography (BLT) is a promising optical molecular imaging technique on the frontier of biomedical optics. In this paper, a generalized hybrid algorithm has been proposed based on the graph cuts algorithm and gradient-based algorithms. The graph cuts algorithm is adopted to estimate a reliable source support without prior knowledge, and different gradient-based algorithms are sequentially used to acquire an accurate and fine source distribution according to the reconstruction status. Furthermore, multilevel meshes for the internal sources are used to speed up the computation and improve the accuracy of reconstruction. Numerical simulations have been performed to validate this proposed algorithm and demonstrate its high performance in the multi-source situation even if the detection noises, optical property errors and phantom structure errors are involved in the forward imaging.

© 2013 OSA

## 1. Introduction

*in vivo*biological process of small animals [1

1. C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. **4**, 235–260 (2002) [CrossRef] [PubMed] .

3. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**, 313–320 (2005) [CrossRef] [PubMed] .

1. C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. **4**, 235–260 (2002) [CrossRef] [PubMed] .

3. V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. **23**, 313–320 (2005) [CrossRef] [PubMed] .

1. C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. **4**, 235–260 (2002) [CrossRef] [PubMed] .

2. C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging **16**, 378–387 (2002) [CrossRef] [PubMed] .

5. A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia **2**, 491–495 (2000) [CrossRef] .

6. M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discovery **2**, 123–131 (2003) [CrossRef] .

*et al.*first introduced a bioluminescent tomography assembly into a multimodality tomography system to depict the internal bioluminescent source distribution upon the structures of object [4]. Hereafter, hundreds of research literatures have been reported focusing on the system development [4, 7

7. 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,” Phys. Med. Biol. **50**, 4225–4241 (2005) [CrossRef] [PubMed] .

9. G. Wang, H. Shen, Y. Liu, A. Cong, W. Cong, Y. Wang, and P. Dubey, “Digital spectral separation methods and systems for bioluminescence imaging,” Opt. Express **16**, 1719–1732 (2008) [CrossRef] [PubMed] .

10. G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express **14**, 7801–7809 (2006) [CrossRef] [PubMed] .

11. X. Ma, J. Tian, C. Qin, X. Yang, B. Zhang, Z. Xue, X. Zhang, D. Han, D. Dong, and X. Liu, “Early detection of liver cancer based on bioluminescence tomography,” Appl. Opt. **50**, 1389–1395 (2011) [CrossRef] [PubMed] .

12. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” Proc. SPIE **5535**, 335–351 (2004) [CrossRef] .

19. K. Liu, J. Tian, Y. Lu, C. Qin, X. Yang, S. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express **18**, 3732–3745 (2010) [CrossRef] [PubMed] .

20. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. **31**, 2289–2299 (2004) [CrossRef] [PubMed] .

16. S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol. **53**, 3921–3942 (2008) [CrossRef] [PubMed] .

17. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express **17**, 8062–8080 (2009) [CrossRef] [PubMed] .

19. K. Liu, J. Tian, Y. Lu, C. Qin, X. Yang, S. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express **18**, 3732–3745 (2010) [CrossRef] [PubMed] .

13. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express **13**, 6756–6771 (2005) [CrossRef] [PubMed] .

15. M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement.” Opt. Express **15**, 11095–11116 (2007) [CrossRef] [PubMed] .

21. Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol. **52**, 4497–4512 (2007) [CrossRef] [PubMed] .

24. M. A. Naser and M. S. Patterson, “Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region,” Biomed. Opt. Express **2**, 3179–3193 (2011) [CrossRef] [PubMed] .

7. 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,” Phys. Med. Biol. **50**, 4225–4241 (2005) [CrossRef] [PubMed] .

14. A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. **50**, 5421–5441 (2005) [CrossRef] [PubMed] .

16. S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol. **53**, 3921–3942 (2008) [CrossRef] [PubMed] .

17. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express **17**, 8062–8080 (2009) [CrossRef] [PubMed] .

25. H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. **35**, 4863–4871 (2008) [CrossRef] [PubMed] .

22. J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express **16**, 15640–15654 (2008) [CrossRef] [PubMed] .

26. Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express **14**, 8211–8223 (2006) [CrossRef] [PubMed] .

*et al.*successfully employed the expectation maximization (EM) algorithm and Landweber algorithm both in complete and partial measurement situations [12

12. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” Proc. SPIE **5535**, 335–351 (2004) [CrossRef] .

15. M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement.” Opt. Express **15**, 11095–11116 (2007) [CrossRef] [PubMed] .

*et al.*first established a linear relationship between the internal source distribution and the surface measurement, and adopted a modified Newton algorithm with the active set strategy to solve the problem [13

13. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express **13**, 6756–6771 (2005) [CrossRef] [PubMed] .

*et al.*introduced a multilevel adaptive finite element method into the reconstruction algorithm [26

26. Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express **14**, 8211–8223 (2006) [CrossRef] [PubMed] .

14. A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. **50**, 5421–5441 (2005) [CrossRef] [PubMed] .

16. S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol. **53**, 3921–3942 (2008) [CrossRef] [PubMed] .

**53**, 3921–3942 (2008) [CrossRef] [PubMed] .

17. Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express **17**, 8062–8080 (2009) [CrossRef] [PubMed] .

18. B. Zhang, X. Yang, C. Qin, D. Liu, S. Zhu, J. Feng, L. Sun, K. Liu, D. Han, X. Ma, X. Zhang, J. Zhong, X. Li, X. Yang, and J. Tian, “A trust region method in adaptive finite element framework for bioluminescence tomography,” Opt. Express **18**, 6477–6491 (2010) [CrossRef] [PubMed] .

*et al.*came up with different strategies for posterior estimation of the permissible source support, successively [21

21. Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol. **52**, 4497–4512 (2007) [CrossRef] [PubMed] .

23. M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express **1**, 512–526 (2010) [CrossRef] .

*et al.*developed a new BLT localization algorithm [19

19. K. Liu, J. Tian, Y. Lu, C. Qin, X. Yang, S. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express **18**, 3732–3745 (2010) [CrossRef] [PubMed] .

## 2. Formulation of BLT

27. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob. **15**, R41–R93 (1999) [CrossRef] .

28. F. Natterer and F. Wübbeling, *Mathematical Methods in Image Reconstruction* (SIAM, 2001) [CrossRef] .

27. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob. **15**, R41–R93 (1999) [CrossRef] .

28. F. Natterer and F. Wübbeling, *Mathematical Methods in Image Reconstruction* (SIAM, 2001) [CrossRef] .

**R**

^{3}be a bounded domain with its boundary denoted by Γ,

*u*

_{0}(

*x*) be the radiance at

*x*∈ Ω. BLT problem can be stated as follows:

*Given the measured outgoing radiance g*(

*x*)

*on*Γ

*, find a source distribution q*

_{0}(

*x*)

*with the corresponding diffusion approximation u*

_{0}(

*x*)

*such that*[15

15. M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement.” Opt. Express **15**, 11095–11116 (2007) [CrossRef] [PubMed] .

20. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. **31**, 2289–2299 (2004) [CrossRef] [PubMed] .

*μ*(

_{a}*x*) and

*μ*(

_{s}*x*) are absorption and scattering coefficients, respectively.

*D*(

*x*) = 1/[3(

*μ*(

_{a}*x*) +

*μ*′

*(*

_{s}*x*))] is diffusion coefficient with

*μ*′

*= (1 −*

_{s}*η̄*)

*μ*being the reduce scattering coefficients and

_{s}*η*̄(0 ≤

*η*̄ ≤ 1)) the anisotropy parameter.

*N*source notes

*M*detectors

13. W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express **13**, 6756–6771 (2005) [CrossRef] [PubMed] .

**m**

_{1},

**m**

_{2}, ···,

**m**

*] ∈*

_{N}**R**

^{M}^{×}

*denotes the system matrix with*

^{N}**m**

*being the*

_{i}*i*th column of it, which can be calculated based on the nodal basis functions

*ϕ*(

_{i}*x*) for the source

*q*

_{0}(

*x*) [25

25. H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. **35**, 4863–4871 (2008) [CrossRef] [PubMed] .

29. S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. **34**, 8026–8037 (1995) [CrossRef] [PubMed] .

**q**= (

*q*

_{1},

*q*

_{2}, ···,

*q*)

_{N}*∈*

^{T}*R*is the discrete distribution of the source with

^{N}*q*=

_{i}*q*

_{0}(

*x*) being the nodal value of

_{i}*q*

_{0}(

*x*), and

**b**∈

**R**

*being the measurement on the boundary.*

^{M}*λ*is the regularization parameter,

*G*(

**q**) is the regularization term. In some of BLT algorithms, the source support constraint has been well used as a substitute for the regularization term.

## 3. Reconstruction of BLT

### 3.1. EM, Landweber and modified Newton algorithm

**EM algorithm**[12

12. M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” Proc. SPIE **5535**, 335–351 (2004) [CrossRef] .

**15**, 11095–11116 (2007) [CrossRef] [PubMed] .

28. F. Natterer and F. Wübbeling, *Mathematical Methods in Image Reconstruction* (SIAM, 2001) [CrossRef] .

**1**∈

**R**

*is a vector with only 1’s components.*

^{M}**Landweber algorithm**[12

**5535**, 335–351 (2004) [CrossRef] .

**15**, 11095–11116 (2007) [CrossRef] [PubMed] .

30. M. Piana and M. Bertero, “Projected landweber method and preconditioning,” Inverse Prob. **13**, 441–463 (1997) [CrossRef] .

32. C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Prob. **18**, 441–453 (2002) [CrossRef] .

**Modified Newton algorithm**[13

**13**, 6756–6771 (2005) [CrossRef] [PubMed] .

26. Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express **14**, 8211–8223 (2006) [CrossRef] [PubMed] .

### 3.2. Graph cuts algorithm

**18**, 3732–3745 (2010) [CrossRef] [PubMed] .

34. Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Mach. Intell. **26**, 1124–1137 (2004) [CrossRef] .

35. V. Kolmogorov and R. Zabin, “What energy functions can be minimized via graph cuts?” IEEE Trans. Pattern Anal. Mach. Intell. **26**, 147–159 (2004) [CrossRef] [PubMed] .

*θ*is the constant term of the energy,

_{const}*θ*(·) are the unary terms which denote the weight of the edges between the source nodes and the terminals, and

_{i}*θ*(·, ·) are the pairwise terms which denote the weight of the edges between the internal source nodes. The concrete forms of

_{ij}*θ*,

_{const}*θ*and

_{i}*θ*depend on the regularization term

_{ij}*G*(

**q**). When

*L*

^{2}regularization is chosen, the forms of

*θ*,

_{const}*θ*and

_{i}*θ*are given in [19

_{ij}**18**, 3732–3745 (2010) [CrossRef] [PubMed] .

*G*(

**q**) = ||

**q**

*||*

_{L1}, i.e. the

*L*

^{1}regularization is chosen,

*θ*,

_{const}*θ*and

_{i}*θ*are as follows Other kinds of regularizations, such as total variation (TV) regularization, can also be chosen for BLT.

_{ij}*E*(

**q**) can be coverted to a graph-representable function and a directed graph can be constructed for it [19

**18**, 3732–3745 (2010) [CrossRef] [PubMed] .

35. V. Kolmogorov and R. Zabin, “What energy functions can be minimized via graph cuts?” IEEE Trans. Pattern Anal. Mach. Intell. **26**, 147–159 (2004) [CrossRef] [PubMed] .

36. P. L. Hammer, P. Hansen, and B. Simeone, “Roof duality, complementation and persistency in quadratic 0–1 optimization,” Math. Program. **28**, 121–155 (1984) [CrossRef] .

34. Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Mach. Intell. **26**, 1124–1137 (2004) [CrossRef] .

### 3.3. Performance of above algorithms

## 4. Generalized hybrid algorithm

*first-order*algorithm, and the other group is the

*second-order*algorithm.

### 4.1. Algorithm description

*first-order*algorithms with

*second-order*algorithms to accelerate the calculation.

*first-order*algorithm with low computational cost needs to be pre-executed to provide the source number and total power for it. And being well adjusting the iteration factors, the iteration scheme involving these two algorithms can be built up in source support computation step (indicated as the right dashed box in Fig. 1) to make the support shrink to some stable regions. Then to the step of source distribution computation, all the gradient-based algorithms are appropriate as the source support is given. Considering the speed of algorithm convergence, one

*second-order*algorithm is first executed to obtain the primary source distribution, and one

*first-order*algorithm is executed to update the source distribution successively. Furthermore, based on the multilevel meshes concept, we also build up an iteration scheme here (indicated as the left dashed box in Fig. 1) to descend the scale of source discretization step by step to improve the performance of BLT reconstruction.

*L*

_{1},

*L*

_{2},... ,

*L*

_{K0}for the internal sources, with

*K*0 the maximum number of mesh refinement.

### 4.2. Relevant issues

#### 4.2.1. Estimation of source number and source power

*first-order*algorithm to preliminarily obtain a source distribution, the source total power could be easily estimated by integrating the source distribution. Then using the region growing algorithm, the source number could also be estimated [37]. At first the local extreme points of the source distribution were taken as the seed points, and after region growing we took the number of disjoint regions as the source number.

#### 4.2.2. Source deblurring

#### 4.2.3. Related criteria in iterations

## 5. Numerical simulations

### 5.1. Simulation settings

#### 5.1.1. Phantom and optical properties

**13**, 6756–6771 (2005) [CrossRef] [PubMed] .

**15**, 11095–11116 (2007) [CrossRef] [PubMed] .

#### 5.1.2. Source setting

#### 5.1.3. Algorithm setting

*L*

^{1}regularization is employed in the graph cuts algorithm. In fact, numerical simulations have demonstrated that the differences among the reconstructed results respectively based on

*L*

^{1},

*L*

^{2}and TV regularization are in a negligible level. That because the graph cuts algorithm is only adopted in the support computation step, and the differences among these regularizations are dramatically suppressed by the coarse mesh used in that step.

*first-order*algorithm to compute source distribution and provide source number and total power for graph cuts algorithm in the support computation step of the hybrid algorithm. In the source distribution computation step, the modified Newton algorithm is first applied to obtain the primary source distribution, and then the Landweber algorithm is applied to update the result.

*K*

_{0}is set to 2 manually in our simulations.

#### 5.1.4. Source discretization and forward process

*x*, the nodal basis function is Where Δ

_{i}*x*is the spatial resolution. There are two levels of meshes for the internal sources in our hybrid algorithm, for the coarse mesh Δ

_{i}*x*= 3mm, and for the fine mesh Δ

_{i}*x*= 1mm. The system matrices corresponding to the meshes are computed in advance and stored before source reconstruction procedure.

_{i}^{8}rays were traced for each source. The results show that the measurements obtained by FEM are in good agreement with that obtained by MC method, and compared with MC, the FEM had an excellent computation efficiency. The results also demonstrate the diffusion approximation equation is a proper approximation to RTE in weakly absorbing and highly scattering media, and FEM is capable of solving the forward problem of BLT efficiently with a high accuracy in this case.

### 5.2. Algorithm comparison

22. J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express **16**, 15640–15654 (2008) [CrossRef] [PubMed] .

*x*,

*y*and

*z*directions respectively; RMS (Root Mean Square) denotes the root mean square of LE, i.e. the distance between the centers of the actual and reconstructed sources. PE (Power Error) denotes the error of the reconstructed source power; RPE (Relative Power Error) denotes the relative error of the reconstructed source power; Source Size denotes the radius of reconstructed source; Computation Time denotes the time cost of the reconstruction except the time of computing the system matrix.

*s*.

22. J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express **16**, 15640–15654 (2008) [CrossRef] [PubMed] .

**16**, 15640–15654 (2008) [CrossRef] [PubMed] .

**16**, 15640–15654 (2008) [CrossRef] [PubMed] .

**16**, 15640–15654 (2008) [CrossRef] [PubMed] .

### 5.3. Consideration of influence factors involved

#### 5.3.1. Detection noise

^{4}, and add a zero-mean Gaussian noise with standard deviation

*σ*, Gaussian(0,

*σ*), to the image.

*σ*= 10, 20, 50, 100), respectively. The numerical results demonstrate that the reconstructed results are slightly influenced by Gaussian noise even if the standard deviation is up to 100. The location and power errors of the reconstructed source slightly increase with the increasing of the Gaussian noise. Table 5 shows the quantitative results obtained by the hybrid algorithm with the measurement being corrupted by Gaussian(0,100). In this case the location error is within 0.07mm, and the power error is within 3% which is only slightly different from the quantitative results in the ideal case. The reconstructed result is not given here as it is very similar to the result illustrated in Figs. 3(a) and 3(b).

#### 5.3.2. Optical property error

8. G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Effect of optical property estimation accuracy on tomographic bioluminescence imaging: simulation of a combined optical-PET (OPET) system,” Phys. Med. Biol. **51**, 2045–2053 (2006) [CrossRef] [PubMed] .

#### 5.3.3. Phantom structure error

#### 5.3.4. Mixed influence

## 6. Conclusions and discussions

24. M. A. Naser and M. S. Patterson, “Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region,” Biomed. Opt. Express **2**, 3179–3193 (2011) [CrossRef] [PubMed] .

## Acknowledgments

## References and links

1. | C. H. Contag and M. H. Bachmann, “Advances in in vivo bioluminescence imaging of gene expression,” Annu. Rev. Biomed. Eng. |

2. | C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging |

3. | V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole-body photonic imaging,” Nat. Biotechnol. |

4. | G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology |

5. | A. Rehemtulla, L. D. Stegman, S. J. Cardozo, S. Gupta, D. E. Hall, C. H. Contag, and B. D. Ross, “Rapid and quantitative assessment of cancer treatment response using in vivo bioluminescence imaging,” Neoplasia |

6. | M. Rudin and R. Weissleder, “Molecular imaging in drug discovery and development,” Nat. Rev. Drug Discovery |

7. | 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,” Phys. Med. Biol. |

8. | G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Effect of optical property estimation accuracy on tomographic bioluminescence imaging: simulation of a combined optical-PET (OPET) system,” Phys. Med. Biol. |

9. | G. Wang, H. Shen, Y. Liu, A. Cong, W. Cong, Y. Wang, and P. Dubey, “Digital spectral separation methods and systems for bioluminescence imaging,” Opt. Express |

10. | G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express |

11. | X. Ma, J. Tian, C. Qin, X. Yang, B. Zhang, Z. Xue, X. Zhang, D. Han, D. Dong, and X. Liu, “Early detection of liver cancer based on bioluminescence tomography,” Appl. Opt. |

12. | M. Jiang and G. Wang, “Image reconstruction for bioluminescence tomography,” Proc. SPIE |

13. | W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. V. Wang, E. A. Hoffman, G. McLennan, P. B. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express |

14. | A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. |

15. | M. Jiang, T. Zhou, J. Cheng, W. Cong, and G. Wang, “Image reconstruction for bioluminescence tomography from partial measurement.” Opt. Express |

16. | S. Ahn, A. J. Chaudhari, F. Darvas, C. A. Bouman, and R. M. Leahy, “Fast iterative image reconstruction methods for fully 3D multispectral bioluminescence tomography,” Phys. Med. Biol. |

17. | Y. Lu, X. Zhang, A. Douraghy, D. Stout, J. Tian, T. F. Chan, and A. F. Chatziioannou, “Source reconstruction for spectrally-resolved bioluminescence tomography with sparse a priori information,” Opt. Express |

18. | B. Zhang, X. Yang, C. Qin, D. Liu, S. Zhu, J. Feng, L. Sun, K. Liu, D. Han, X. Ma, X. Zhang, J. Zhong, X. Li, X. Yang, and J. Tian, “A trust region method in adaptive finite element framework for bioluminescence tomography,” Opt. Express |

19. | K. Liu, J. Tian, Y. Lu, C. Qin, X. Yang, S. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express |

20. | G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. |

21. | Y. Lv, J. Tian, W. Cong, G. Wang, W. Yang, C. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol. |

22. | J. Feng, K. Jia, G. Yan, S. Zhu, C. Qin, Y. Lv, and J. Tian, “An optimal permissible source region strategy for multispectral bioluminescence tomography,” Opt. Express |

23. | M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express |

24. | M. A. Naser and M. S. Patterson, “Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region,” Biomed. Opt. Express |

25. | H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Med. Phys. |

26. | Y. Lv, J. Tian, W. Cong, G. Wang, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express |

27. | S. R. Arridge, “Optical tomography in medical imaging,” Inverse Prob. |

28. | F. Natterer and F. Wübbeling, |

29. | S. R. Arridge and M. Schweiger, “Photon-measurement density functions. Part 2: Finite-element-method calculations,” Appl. Opt. |

30. | M. Piana and M. Bertero, “Projected landweber method and preconditioning,” Inverse Prob. |

31. | M. Jiang and G. Wang, “Convergence studies on iterative algorithms for image reconstruction,” IEEE Trans. Med. Imaging |

32. | C. Byrne, “Iterative oblique projection onto convex sets and the split feasibility problem,” Inverse Prob. |

33. | P. E. Gill, W. Murray, and M. H. Wright, |

34. | Y. Boykov and V. Kolmogorov, “An experimental comparison of min-cut/max-flow algorithms for energy minimization in vision,” IEEE Trans. Pattern Anal. Mach. Intell. |

35. | V. Kolmogorov and R. Zabin, “What energy functions can be minimized via graph cuts?” IEEE Trans. Pattern Anal. Mach. Intell. |

36. | P. L. Hammer, P. Hansen, and B. Simeone, “Roof duality, complementation and persistency in quadratic 0–1 optimization,” Math. Program. |

37. | M. Sonka, V. Hlavac, and R. Boyle, |

**OCIS Codes**

(100.3190) Image processing : Inverse problems

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.3880) Medical optics and biotechnology : Medical and biological imaging

(170.6960) Medical optics and biotechnology : Tomography

**ToC Category:**

Image Reconstruction and Inverse Problems

**History**

Original Manuscript: February 21, 2013

Revised Manuscript: March 29, 2013

Manuscript Accepted: March 29, 2013

Published: April 10, 2013

**Citation**

Shengkun Shi and Heng Mao, "A generalized hybrid algorithm for bioluminescence tomography," Biomed. Opt. Express **4**, 709-724 (2013)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-4-5-709

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