## Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region |

Biomedical Optics Express, Vol. 2, Issue 11, pp. 3179-3193 (2011)

http://dx.doi.org/10.1364/BOE.2.003179

Acrobat PDF (4211 KB)

### Abstract

A reconstruction algorithm for bioluminescence tomography (BLT) has been developed. The algorithm numerically calculates the Green’s function at different wavelengths using the diffusion equation and finite element method. The optical properties used in calculating the Green’s function are reconstructed using diffuse optical tomography (DOT) and assuming anatomical information is provided by x-ray computed tomography or other methods. A symmetric system of equations is formed using the Green’s function and the measured light fluence rate and the resulting eigenvalue problem is solved to get the eigenvectors of this symmetric system of equations. A space can be formed from the eigenvectors obtained and the reconstructed source is written as an expansion of the eigenvectors corresponding to non-zero eigenvalues. The coefficients of the expansion are found to obtain the reconstructed BL source distribution. The problem is solved iteratively by using a permissible source region that is shrunk by removing nodes with low probability to contribute to the source. Throughout this process the permissible region shrinks from the entire object to just a few nodes. The best estimate of the reconstructed source is chosen that which minimizes the difference between the calculated and measured light fluence rates. 3D simulations presented here show that the reconstructed source is in good agreement with the actual source in terms of locations, magnitudes, sizes, and total powers for both localized multiple sources and large inhomogeneous source distributions.

© 2011 OSA

## 1. Introduction

1. 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**(3), 313–320 (2005). [CrossRef] [PubMed]

4. J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. **27**(5), 48–57 (2008). [CrossRef] [PubMed]

*in vivo*bioluminescence imaging does not give quantitative information about the distribution of the light source (corresponding to luciferase concentration) due to multiple light scattering and absorption in tissue. To correctly estimate the distribution of luciferase in tissue from the bioluminescence image, BLT algorithms need to be developed.

5. X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express **15**(26), 18300–18317 (2007). [CrossRef] [PubMed]

6. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. **31**(8), 2289–2299 (2004). [CrossRef] [PubMed]

6. G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys. **31**(8), 2289–2299 (2004). [CrossRef] [PubMed]

7. 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**(23), 5421–5441 (2005). [CrossRef] [PubMed]

9. H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett. **31**(3), 365–367 (2006). [CrossRef] [PubMed]

*a priori*information about permissible source regions [10

10. 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**(18), 6756–6771 (2005). [CrossRef] [PubMed]

12. H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys. **229**(13), 5246–5256 (2010). [CrossRef]

13. 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**(20), 15640–15654 (2008). [CrossRef] [PubMed]

*a priori*information about the source distribution can be beneficial for constraining the problem and improving the reconstruction. Different researchers have used sparsity regularization [14

14. 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**(10), 8062–8080 (2009). [CrossRef] [PubMed]

16. N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express **15**(21), 13695–13708 (2007). [CrossRef] [PubMed]

17. M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express **1**(2), 512–526 (2010). [CrossRef] [PubMed]

18. M. A. Naser and M. S. Patterson, “Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region,” Biomed. Opt. Express **2**(1), 169–184 (2011). [CrossRef] [PubMed]

*i.e.*power per unit volume) and sizes could not be distinguished.

19. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. **20**(2), 299–309 (1993). [CrossRef] [PubMed]

26. C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt. **12**(2), 024007 (2007). [CrossRef] [PubMed]

17. M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express **1**(2), 512–526 (2010). [CrossRef] [PubMed]

## 2. BLT reconstruction

27. S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. **15**(2), R41–R93 (1999). [CrossRef]

28. A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescence sources,” J. Opt. Soc. Am. A **24**(6), 1601–1608 (2007). [CrossRef]

*λ*; Ω and

*λ*is given by the absorption coefficient

*g*is the anisotropy factor.

*A*is derived from Fresnel’s law as [29

29. H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. **25**(6), 711–732 (2009). [CrossRef] [PubMed]

*η*is the tissue refractive index at the boundary while the air refractive index is assumed to be 1.

18. M. A. Naser and M. S. Patterson, “Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region,” Biomed. Opt. Express **2**(1), 169–184 (2011). [CrossRef] [PubMed]

*λ*obtained by discretizing the diffusion equation. Assuming that the source magnitude is not a function of wavelength or, equivalently, the source spectrum is known in the range of measurement, for multi-spectral measurements at the detectors on the object surface

*N*wavelengths in a normalized form can be given by

*d*is the detector index at the boundary

*R*is the source permissible region index which contains the index of nodes of the finite element mesh inside the domain Ω. This corresponds to the permissible source region which could be considered initially to be the whole domain Ω. The normalization of the Green’s function and light fluence rate by the maximum value of the light fluence rate at each wavelength improves the efficiency of the algorithm as shown before in [18

18. M. A. Naser and M. S. Patterson, “Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region,” Biomed. Opt. Express **2**(1), 169–184 (2011). [CrossRef] [PubMed]

9. H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett. **31**(3), 365–367 (2006). [CrossRef] [PubMed]

**2**(1), 169–184 (2011). [CrossRef] [PubMed]

*M*is the number of chosen eigenvectors. The coefficients

*R*corresponding to low source power as described before in [17

17. M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express **1**(2), 512–526 (2010). [CrossRef] [PubMed]

**2**(1), 169–184 (2011). [CrossRef] [PubMed]

*β*is the reduction factor. The reduction factor can be found from the initial and final number of nodes in the permissible region and the predefined number of iterations such that

*R*, and

## 4. Results and discussions

*z*-direction as shown in Fig. 1(b). The total number of detector readings at each wavelength is

*z*= 0) parallel to the upper and lower surfaces of the object. Figure 2(b) is a sagittal cross section that passes through the bowel and bone while Fig. 2(c) shows a coronal cross section passing through the two kidneys. Figure 2 shows 3 spherical bioluminescence sources of 5 mm and 3 mm diameter centered in the bone at (0, −7, 0) and two kidneys (−4.5, −2.5, 0) and (4.5, −2.5, 0), respectively.

**1**(2), 512–526 (2010). [CrossRef] [PubMed]

**1**(2), 512–526 (2010). [CrossRef] [PubMed]

*i.e*.

30. 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**(17), 4225–4241 (2005). [CrossRef] [PubMed]

31. S. A. Prahl, Oregon Medical Laser Clinic (2001), http://omlc.ogi.edu/spectra/index.html.

*R*was initially chosen to be the whole object except for nodes within one transport length

**2**(1), 169–184 (2011). [CrossRef] [PubMed]

^{3}, and the total powers of the three sources are 5.65, 5.65, and 40.71 nW, respectively. The transverse, sagittal, and coronal cross sections of the reconstructed sources in Figs. 3(d), (e), and (f) show good agreement with the actual sources in terms of source magnitude, total power and size. The total power of the reconstructed sources are 5.31, 5.49, and 41.4 nW. The source magnitude varies between 0.8 to 1.2 nW/mm

^{3}which is within 20% of the true value.

^{3}after the first iteration which includes a permissible region of 1050 nodes and at the optimized permissible region which includes 126 nodes and minimizes the objective function. After the first iteration, the reconstructed source powers in the two kidneys and bone are 3.37, 3.8, and 51.93 nW, respectively which are within 40% of their actual values. The source magnitudes and sizes cannot be reconstructed perfectly after the first iteration due to the degeneracy of the solution. When the size of the source permissible region becomes comparable to the size of the actual source, the objective function has its lowest value and good accuracy in reconstructing the source power, magnitude, and size can be obtained, as shown in Fig. 4(c). When the permissible region is further reduced, the size of the source permissible region becomes smaller than the size of the actual source. Hence, the objective function increases again and the reconstruction of the source power, magnitude and size becomes less accurate.

^{3}respectively, and the source in the bone is 1 nW/mm

^{3}. The three sources have total powers of 11.31, 16.96, and 40.71 nW. The reconstructed sources in Figs. 5(d), (e), and (f) have powers of 11.56, 18.81, and 38.92 nW respectively, which are within 11% of the true value. The source magnitudes are reconstructed within 20% of the true values. As a comparison with the result obtained using our previous algorithm in [18

**2**(1), 169–184 (2011). [CrossRef] [PubMed]

^{3}and 2 nW/mm

^{3}respectively and the source in the bone is 3 nW/mm

^{3}. The three sources have total powers of 5.65, 11.31, and 122.14 nW respectively. The reconstructed sources are shown in Figs. 6(d), (e), and (f). The reconstructed sources powers are 8.7, 10.94, and 120.14 nW respectively. The size and magnitude of the smallest source in the left kidney could not be reconstructed in this case. When there are large differences between source powers or there is a large inhomogeneous source distribution, the shrinking of the permissible region cannot be done correctly. The assumption that nodes with low source magnitude are least likely to be in the optimized source region is no longer true.

^{3}in the center of the organ and a uniform “background” of 1 nW/mm

^{3}elsewhere. The total powers of the actual and reconstructed sources integrated over the kidney volume are 53.67 and 53.71 nW. The reconstructed source in Figs. 7(c) and 7(d) is in good agreement with the actual source in terms of size and total power. The inhomogeneity of the source distribution could be reconstructed and the high spot in the center of the kidney could be distinguished from the background. The magnitude of the reconstructed source at the center of the hot spot is within 20% of its actual value. As a comparison with the result obtained using our previous algorithm in [18

**2**(1), 169–184 (2011). [CrossRef] [PubMed]

^{3}which is 5 times that of the surrounding background. The total powers of the actual and reconstructed sources are 70.63 and 70.8 nW. However, the inhomogeneity of the source could not be reconstructed correctly in this case. The uniform background shows a smaller size and a higher magnitude, which varies from 1.5 to 2.5 nW/mm

^{3}, than the actual source. The objective function curve with respect to the number of source nodes for the reconstructed source shown in Fig. 8 does not show a deep valley with a clear global minimum as shown in Fig. 4(a). Instead, the curve is slowly increasing after the global minimum indicating that different source distributions can generate the same error and hence are indistinguishable. Therefore, shrinking the permissible region reduces the possible solutions, and if the permissible region equals the actual source size, a unique solution can be obtained for uniform and low inhomogeneity source distributions as shown in Figs. 3, 5, and 7. However, for high inhomogeneity source distributions as in Figs. 6 and 8, the solution is still not unique and different size and magnitude sources that have the same total power can generate the same light fluence rate at the object boundary. One way to improve the uniqueness of the solution is to increase the available data points by doing measurements at more wavelengths.

## 5. Conclusion

## Acknowledgments

## References and links

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

2. | R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology |

3. | J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. |

4. | J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. |

5. | X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express |

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

7. | 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. |

8. | 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. |

9. | H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett. |

10. | 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 |

11. | X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging |

12. | H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys. |

13. | 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 |

14. | 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 |

15. | P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt. |

16. | N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express |

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

18. | M. A. Naser and M. S. Patterson, “Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region,” Biomed. Opt. Express |

19. | S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. |

20. | M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys. |

21. | H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt. |

22. | A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging |

23. | X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express |

24. | N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. |

25. | D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol. |

26. | C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt. |

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

28. | A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescence sources,” J. Opt. Soc. Am. A |

29. | H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. |

30. | 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. |

31. | S. A. Prahl, Oregon Medical Laser Clinic (2001), http://omlc.ogi.edu/spectra/index.html. |

**OCIS Codes**

(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: September 27, 2011

Revised Manuscript: October 27, 2011

Manuscript Accepted: October 25, 2011

Published: October 26, 2011

**Citation**

Mohamed A. Naser and Michael S. Patterson, "Bioluminescence tomography using eigenvectors expansion and iterative solution for the optimized permissible source region," Biomed. Opt. Express **2**, 3179-3193 (2011)

http://www.opticsinfobase.org/boe/abstract.cfm?URI=boe-2-11-3179

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### References

- 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(3), 313–320 (2005). [CrossRef] [PubMed]
- R. Weissleder and U. Mahmood, “Molecular imaging,” Radiology219(2), 316–333 (2001). [PubMed]
- J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov.7(7), 591–607 (2008). [CrossRef] [PubMed]
- J. Tian, J. Bai, X. P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag.27(5), 48–57 (2008). [CrossRef] [PubMed]
- X. Song, D. Wang, N. Chen, J. Bai, and H. Wang, “Reconstruction for free-space fluorescence tomography using a novel hybrid adaptive finite element algorithm,” Opt. Express15(26), 18300–18317 (2007). [CrossRef] [PubMed]
- G. Wang, Y. Li, and M. Jiang, “Uniqueness theorems in bioluminescence tomography,” Med. Phys.31(8), 2289–2299 (2004). [CrossRef] [PubMed]
- 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(23), 5421–5441 (2005). [CrossRef] [PubMed]
- 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(14), 3921–3942 (2008). [CrossRef] [PubMed]
- H. Dehghani, S. C. Davis, S. Jiang, B. W. Pogue, K. D. Paulsen, and M. S. Patterson, “Spectrally resolved bioluminescence optical tomography,” Opt. Lett.31(3), 365–367 (2006). [CrossRef] [PubMed]
- 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. Express13(18), 6756–6771 (2005). [CrossRef] [PubMed]
- X. He, J. Liang, X. Qu, H. Huang, Y. Hou, and J. Tian, “Truncated total least squares method with a practical truncation parameter choice scheme for bioluminescence tomography inverse problem,” Int. J. Biomed. Imaging2010, 291874 (2010). [CrossRef] [PubMed]
- H. Huang, X. Qu, J. Liang, X. He, X. Chen, D. Yang, and J. Tian, “A multi-phase level set framework for source reconstruction in bioluminescence tomography,” J. Comput. Phys.229(13), 5246–5256 (2010). [CrossRef]
- 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. Express16(20), 15640–15654 (2008). [CrossRef] [PubMed]
- 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. Express17(10), 8062–8080 (2009). [CrossRef] [PubMed]
- P. Mohajerani, A. A. Eftekhar, J. Huang, and A. Adibi, “Optimal sparse solution for fluorescent diffuse optical tomography: theory and phantom experimental results,” Appl. Opt.46(10), 1679–1685 (2007). [CrossRef] [PubMed]
- N. Cao, A. Nehorai, and M. Jacobs, “Image reconstruction for diffuse optical tomography using sparsity regularization and expectation-maximization algorithm,” Opt. Express15(21), 13695–13708 (2007). [CrossRef] [PubMed]
- M. A. Naser and M. S. Patterson, “Algorithms for bioluminescence tomography incorporating anatomical information and reconstruction of tissue optical properties,” Biomed. Opt. Express1(2), 512–526 (2010). [CrossRef] [PubMed]
- M. A. Naser and M. S. Patterson, “Improved bioluminescence and fluorescence reconstruction algorithms using diffuse optical tomography, normalized data, and optimized selection of the permissible source region,” Biomed. Opt. Express2(1), 169–184 (2011). [CrossRef] [PubMed]
- S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys.20(2), 299–309 (1993). [CrossRef] [PubMed]
- M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: boundary and source conditions,” Med. Phys.22(11), 1779–1792 (1995). [CrossRef] [PubMed]
- H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element-based algorithm and simulations,” Appl. Opt.37(22), 5337–5343 (1998). [CrossRef] [PubMed]
- A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction scheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging18(3), 262–271 (1999). [CrossRef] [PubMed]
- X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express12(17), 3996–4000 (2004). [CrossRef] [PubMed]
- N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys.33(1), 61–68 (2006). [CrossRef] [PubMed]
- D. C. Comsa, T. J. Farrell, and M. S. Patterson, “Quantification of bioluminescence images of point source objects using diffusion theory models,” Phys. Med. Biol.51(15), 3733–3746 (2006). [CrossRef] [PubMed]
- C. Kuo, O. Coquoz, T. L. Troy, H. Xu, and B. W. Rice, “Three-dimensional reconstruction of in vivo bioluminescent sources based on multispectral imaging,” J. Biomed. Opt.12(2), 024007 (2007). [CrossRef] [PubMed]
- S. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl.15(2), R41–R93 (1999). [CrossRef]
- A. D. Klose, “Transport-theory-based stochastic image reconstruction of bioluminescence sources,” J. Opt. Soc. Am. A24(6), 1601–1608 (2007). [CrossRef]
- H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng.25(6), 711–732 (2009). [CrossRef] [PubMed]
- 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(17), 4225–4241 (2005). [CrossRef] [PubMed]
- S. A. Prahl, Oregon Medical Laser Clinic (2001), http://omlc.ogi.edu/spectra/index.html .

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