Abstract

Bioluminescence tomography (BLT) is an effective molecular imaging (MI) modality. Because of the ill-posedness, the inverse problem of BLT is still open. We present a trust region method (TRM) for BLT source reconstruction. The TRM is applied in the source reconstruction procedure of BLT for the first time. The results of both numerical simulations and the experiments of cube phantom and nude mouse draw us to the conclusion that based on the adaptive finite element (AFE) framework, the TRM works in the source reconstruction procedure of BLT. To make our conclusion more reliable, we also compare the performance of the TRM and that of the famous Tikhonov regularization method after only one step of mesh refinement of the AFE framework. The conclusion is that the TRM can get faster and better results after only one mesh refinement step of AFE framework than the Tikhonov regularization method when handling large scale data. In the TRM, all the parameters are fixed, while in the Tikhonov method the regularization parameter needs to be well selected.

© 2010 Optical Society of America

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2010 (1)

B. Zhang, J. Tian, D. Liu, L. Sun, X. Yang, and D. Han, “A multithread based new sparse matrix method in bioluminescence tomography”, presented at Conference 7626 of SPIE on Medical Imaging, San Diego, USA, 13–18 February 2010.

2009 (6)

C. Qin, J. Tian, X. Yang, J. Feng, K. Liu, J. Liu, G. Yan, S. Zhu, and M. Xu, “Adaptive improved element free Galerkin method for quasi or multi spectral bioluminescence tomography,” Opt. Express 17, 21925–21934 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21925.
[CrossRef] [PubMed]

J. Feng, K. Jia, C. Qin, G. Yan, S. Zhu, X. Zhang, J. Liu, and J. Tian, “Three-dimensional Bioluminescence Tomography based on Bayesian Approach,” Opt. Express 17, 16834–16848 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16834.
[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. Express 17, 8062–8080 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062.
[CrossRef] [PubMed]

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally-resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation. Physics in Medicine and Biology,”  59, 6477–6493 (2009).

X. L. Cheng, R. F. Gong, and W. M. Han, “Numerical approximation of bioluminescence tomography based on a new formulation,” Journal of Engineering Mathematics 63, 121–133 (2009).
[CrossRef]

M. Chua and H. Dehghani, “Image reconstruction in diffuse optical tomography based on simplified spherical harmonics approximation,” Opt. Express 17, 24208–24223, (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-26-24208.
[CrossRef]

2008 (7)

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

W. Gong, R. Li, N. N. Yan, and W.B. Zhao, “An improved error analysis for finite element approximation of bioluminescence tomography,” Journal of Computational Mathematics 26, 297–309 (2008).

M. B. Unlu and G. Gulsen, “Effects of the time dependence of a bioluminescent source on the tomographic reconstruction,” Appl. Opt. 47, 799–806 (2008).
[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. Express 16, 15640–15654 (2008), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15640.
[CrossRef] [PubMed]

R. Weissleder and M. J. Pittet, “Imaging in the era of molecular oncology,” Nature 452, 580–589 (2008).
[CrossRef] [PubMed]

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 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. Bio. Mag. 27, 48–57 (2008).
[CrossRef]

2007 (3)

D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[CrossRef]

Y. Lv, J. Tian, H. Li, 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, 1–16 (2007).
[CrossRef]

V. Soloviev, “Tomographic bioluminescence imaging with varying boundary conditions,” Applied Optics 46, 2778–2784 (2007).
[CrossRef] [PubMed]

2006 (3)

M. K. So, C. J. Xu, A. M. Loening, S. S. Gambhir, and J. H. Rao, “Self-illuminating quantum dot conjugates for in vivo imaging,” Nature Biotechnol. 24, 339–343 (2006).
[CrossRef]

W. M. Han, W. X. Cong, and G. Wang, “Mathematical theory and numerical analysis of bioluminescence tomography,” Inverse Problems 22, 1659–1675 (2006).
[CrossRef]

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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211.
[CrossRef] [PubMed]

2005 (3)

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

Y. Wang and Y. Yuan, “Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems,” Inverse Problems 21, 821–838, (2005).
[CrossRef]

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]

2004 (3)

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[CrossRef]

R. Schultz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag. 23, 492–500 (2004).
[CrossRef]

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef] [PubMed]

2003 (1)

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nature Medicine 9, 123–128 (2003).
[CrossRef] [PubMed]

2002 (3)

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

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757–760 (2002).
[CrossRef] [PubMed]

D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–169 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=OPEX-10-3-159.
[PubMed]

1995 (2)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Meth. Prog. Biomed. 47, 131–146 (1995).
[CrossRef]

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, 1779–1792 (1995).
[CrossRef] [PubMed]

1993 (1)

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

1963 (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. 11, 431–441 (1963).
[CrossRef]

1944 (1)

K. Levenberg, “A method for the solution of certain nonlinear problems,” Quart. Appl. Math. 2, 164–168 (1944).

Alexandrakis, G.

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]

Arridge, S. R.

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, 1779–1792 (1995).
[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, 299–309 (1993).
[CrossRef] [PubMed]

Bachmann, M. H.

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

Bai, J.

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

Bao, S.

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

Boas, D.

Bremer, C.

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757–760 (2002).
[CrossRef] [PubMed]

Bruggen, N. van

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7, 591–607 (2008).
[CrossRef] [PubMed]

Chan, T. F.

Chatziioannou, A. F.

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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062.
[CrossRef] [PubMed]

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally-resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation. Physics in Medicine and Biology,”  59, 6477–6493 (2009).

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]

Cheng, X. L.

X. L. Cheng, R. F. Gong, and W. M. Han, “Numerical approximation of bioluminescence tomography based on a new formulation,” Journal of Engineering Mathematics 63, 121–133 (2009).
[CrossRef]

Chua, M.

Cong, A.

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[CrossRef]

Cong, W.

Y. Lv, J. Tian, H. Li, 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, 1–16 (2007).
[CrossRef]

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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211.
[CrossRef] [PubMed]

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[CrossRef]

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef] [PubMed]

Cong, W. X.

W. M. Han, W. X. Cong, and G. Wang, “Mathematical theory and numerical analysis of bioluminescence tomography,” Inverse Problems 22, 1659–1675 (2006).
[CrossRef]

Contag, C.

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

Culver, J.

Davis, S. C.

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

Dehghani, H.

Delpy, D. T.

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, 1779–1792 (1995).
[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, 299–309 (1993).
[CrossRef] [PubMed]

Dinkelborg, L. M.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7, 591–607 (2008).
[CrossRef] [PubMed]

Douraghy, A.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally-resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation. Physics in Medicine and Biology,”  59, 6477–6493 (2009).

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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062.
[CrossRef] [PubMed]

Duderstadt, J. J.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor analysis (Wiley, New York, 1976).

Dunn, A.

Feng, J.

Gambhir, S. S.

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7, 591–607 (2008).
[CrossRef] [PubMed]

M. K. So, C. J. Xu, A. M. Loening, S. S. Gambhir, and J. H. Rao, “Self-illuminating quantum dot conjugates for in vivo imaging,” Nature Biotechnol. 24, 339–343 (2006).
[CrossRef]

Gao, F.

D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[CrossRef]

Gong, R. F.

X. L. Cheng, R. F. Gong, and W. M. Han, “Numerical approximation of bioluminescence tomography based on a new formulation,” Journal of Engineering Mathematics 63, 121–133 (2009).
[CrossRef]

Gong, W.

W. Gong, R. Li, N. N. Yan, and W.B. Zhao, “An improved error analysis for finite element approximation of bioluminescence tomography,” Journal of Computational Mathematics 26, 297–309 (2008).

Gulsen, G.

Hamilton, L. J.

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor analysis (Wiley, New York, 1976).

Han, D.

B. Zhang, J. Tian, D. Liu, L. Sun, X. Yang, and D. Han, “A multithread based new sparse matrix method in bioluminescence tomography”, presented at Conference 7626 of SPIE on Medical Imaging, San Diego, USA, 13–18 February 2010.

Han, W. M.

X. L. Cheng, R. F. Gong, and W. M. Han, “Numerical approximation of bioluminescence tomography based on a new formulation,” Journal of Engineering Mathematics 63, 121–133 (2009).
[CrossRef]

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Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally-resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation. Physics in Medicine and Biology,”  59, 6477–6493 (2009).

Hiraoka, M.

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, 1779–1792 (1995).
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H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method,” Acad. Radiol. 11, 1029–1038 (2004).
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Jia, K.

Kumar, D.

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
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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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211.
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H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method,” Acad. Radiol. 11, 1029–1038 (2004).
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W. Gong, R. Li, N. N. Yan, and W.B. Zhao, “An improved error analysis for finite element approximation of bioluminescence tomography,” Journal of Computational Mathematics 26, 297–309 (2008).

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J. Tian, J. Bai, X.-P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Bio. Mag. 27, 48–57 (2008).
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J. Tian, J. Bai, X.-P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Bio. Mag. 27, 48–57 (2008).
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B. Zhang, J. Tian, D. Liu, L. Sun, X. Yang, and D. Han, “A multithread based new sparse matrix method in bioluminescence tomography”, presented at Conference 7626 of SPIE on Medical Imaging, San Diego, USA, 13–18 February 2010.

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Liu, K.

Liu, Y.

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Luo, J.

Lv, Y.

Machado, H. B.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally-resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation. Physics in Medicine and Biology,”  59, 6477–6493 (2009).

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R. Weissleder and M. J. Pittet, “Imaging in the era of molecular oncology,” Nature 452, 580–589 (2008).
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H. Dehghani, S. C. Davis, and B. W. Pogue, “Spectrally resolved bioluminescence tomography using the reciprocity approach,” Medical Physics 35, 4863–4871 (2008).
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M. K. So, C. J. Xu, A. M. Loening, S. S. Gambhir, and J. H. Rao, “Self-illuminating quantum dot conjugates for in vivo imaging,” Nature Biotechnol. 24, 339–343 (2006).
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R. Schultz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag. 23, 492–500 (2004).
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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, 1779–1792 (1995).
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M. K. So, C. J. Xu, A. M. Loening, S. S. Gambhir, and J. H. Rao, “Self-illuminating quantum dot conjugates for in vivo imaging,” Nature Biotechnol. 24, 339–343 (2006).
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Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally-resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation. Physics in Medicine and Biology,”  59, 6477–6493 (2009).

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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062.
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B. Zhang, J. Tian, D. Liu, L. Sun, X. Yang, and D. Han, “A multithread based new sparse matrix method in bioluminescence tomography”, presented at Conference 7626 of SPIE on Medical Imaging, San Diego, USA, 13–18 February 2010.

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B. Zhang, J. Tian, D. Liu, L. Sun, X. Yang, and D. Han, “A multithread based new sparse matrix method in bioluminescence tomography”, presented at Conference 7626 of SPIE on Medical Imaging, San Diego, USA, 13–18 February 2010.

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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062.
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J. Feng, K. Jia, C. Qin, G. Yan, S. Zhu, X. Zhang, J. Liu, and J. Tian, “Three-dimensional Bioluminescence Tomography based on Bayesian Approach,” Opt. Express 17, 16834–16848 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16834.
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Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally-resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation. Physics in Medicine and Biology,”  59, 6477–6493 (2009).

C. Qin, J. Tian, X. Yang, J. Feng, K. Liu, J. Liu, G. Yan, S. Zhu, and M. Xu, “Adaptive improved element free Galerkin method for quasi or multi spectral bioluminescence tomography,” Opt. Express 17, 21925–21934 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21925.
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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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15640.
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J. Tian, J. Bai, X.-P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Bio. Mag. 27, 48–57 (2008).
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Y. Lv, J. Tian, H. Li, 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, 1–16 (2007).
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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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211.
[CrossRef] [PubMed]

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method,” Acad. Radiol. 11, 1029–1038 (2004).
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Tung, C.

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757–760 (2002).
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Unlu, M. B.

Wang, G.

Y. Lv, J. Tian, H. Li, 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, 1–16 (2007).
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W. M. Han, W. X. Cong, and G. Wang, “Mathematical theory and numerical analysis of bioluminescence tomography,” Inverse Problems 22, 1659–1675 (2006).
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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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211.
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W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
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H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method,” Acad. Radiol. 11, 1029–1038 (2004).
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Wang, L. H.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Meth. Prog. Biomed. 47, 131–146 (1995).
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Wang, L. H. V.

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

Wang, L. V.

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef] [PubMed]

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Y. Wang and Y. Yuan, “Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems,” Inverse Problems 21, 821–838, (2005).
[CrossRef]

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R. Weissleder and M. J. Pittet, “Imaging in the era of molecular oncology,” Nature 452, 580–589 (2008).
[CrossRef] [PubMed]

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

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nature Medicine 9, 123–128 (2003).
[CrossRef] [PubMed]

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757–760 (2002).
[CrossRef] [PubMed]

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J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7, 591–607 (2008).
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Xu, C. J.

M. K. So, C. J. Xu, A. M. Loening, S. S. Gambhir, and J. H. Rao, “Self-illuminating quantum dot conjugates for in vivo imaging,” Nature Biotechnol. 24, 339–343 (2006).
[CrossRef]

Xu, M.

C. Qin, J. Tian, X. Yang, J. Feng, K. Liu, J. Liu, G. Yan, S. Zhu, and M. Xu, “Adaptive improved element free Galerkin method for quasi or multi spectral bioluminescence tomography,” Opt. Express 17, 21925–21934 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21925.
[CrossRef] [PubMed]

Y. Lv, J. Tian, H. Li, 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, 1–16 (2007).
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Yan, G.

Yan, N. N.

W. Gong, R. Li, N. N. Yan, and W.B. Zhao, “An improved error analysis for finite element approximation of bioluminescence tomography,” Journal of Computational Mathematics 26, 297–309 (2008).

Yan, X.-P.

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

Yang, W.

Y. Lv, J. Tian, H. Li, 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, 1–16 (2007).
[CrossRef]

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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211.
[CrossRef] [PubMed]

Yang, X.

B. Zhang, J. Tian, D. Liu, L. Sun, X. Yang, and D. Han, “A multithread based new sparse matrix method in bioluminescence tomography”, presented at Conference 7626 of SPIE on Medical Imaging, San Diego, USA, 13–18 February 2010.

C. Qin, J. Tian, X. Yang, J. Feng, K. Liu, J. Liu, G. Yan, S. Zhu, and M. Xu, “Adaptive improved element free Galerkin method for quasi or multi spectral bioluminescence tomography,” Opt. Express 17, 21925–21934 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21925.
[CrossRef] [PubMed]

J. Tian, J. Bai, X.-P. Yan, S. Bao, Y. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Bio. Mag. 27, 48–57 (2008).
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Yuan, Y.

Y. Wang and Y. Yuan, “Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems,” Inverse Problems 21, 821–838, (2005).
[CrossRef]

Yuan, Y.x.

W. Sun and Y.x. Yuan, “Chapter 6 Trust-Region Methods and Conic Model Methods” in Optimization Theory and Methods: Nonlinear Programming (Springer US, 2006).

Zhang, B.

B. Zhang, J. Tian, D. Liu, L. Sun, X. Yang, and D. Han, “A multithread based new sparse matrix method in bioluminescence tomography”, presented at Conference 7626 of SPIE on Medical Imaging, San Diego, USA, 13–18 February 2010.

Zhang, X.

Zhao, H.

D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[CrossRef]

Zhao, W.B.

W. Gong, R. Li, N. N. Yan, and W.B. Zhao, “An improved error analysis for finite element approximation of bioluminescence tomography,” Journal of Computational Mathematics 26, 297–309 (2008).

Zheng, L. Q.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Meth. Prog. Biomed. 47, 131–146 (1995).
[CrossRef]

Zhu, F.

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef] [PubMed]

Zhu, S.

Acad. Radiol. (1)

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, “A mouse optical simulation enviroment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method,” Acad. Radiol. 11, 1029–1038 (2004).
[CrossRef] [PubMed]

Annu. Rev. Biomed. Eng. (1)

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

Appl. Opt. (1)

Applied Optics (1)

V. Soloviev, “Tomographic bioluminescence imaging with varying boundary conditions,” Applied Optics 46, 2778–2784 (2007).
[CrossRef] [PubMed]

Comput. Meth. Prog. Biomed. (1)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Meth. Prog. Biomed. 47, 131–146 (1995).
[CrossRef]

IEEE Eng. Med. Bio. Mag. (1)

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

IEEE Trans. Med. Imag. (1)

R. Schultz, J. Ripoll, and V. Ntziachristos, “Experimental fluorescence tomography of tissues with noncontact measurements,” IEEE Trans. Med. Imag. 23, 492–500 (2004).
[CrossRef]

Inverse Problems (2)

Y. Wang and Y. Yuan, “Convergence and regularity of trust region methods for nonlinear ill-posed inverse problems,” Inverse Problems 21, 821–838, (2005).
[CrossRef]

W. M. Han, W. X. Cong, and G. Wang, “Mathematical theory and numerical analysis of bioluminescence tomography,” Inverse Problems 22, 1659–1675 (2006).
[CrossRef]

Journal of Computational Mathematics (1)

W. Gong, R. Li, N. N. Yan, and W.B. Zhao, “An improved error analysis for finite element approximation of bioluminescence tomography,” Journal of Computational Mathematics 26, 297–309 (2008).

Journal of Engineering Mathematics (1)

X. L. Cheng, R. F. Gong, and W. M. Han, “Numerical approximation of bioluminescence tomography based on a new formulation,” Journal of Engineering Mathematics 63, 121–133 (2009).
[CrossRef]

Med. Phys. (2)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 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, 1779–1792 (1995).
[CrossRef] [PubMed]

Medical Physics (1)

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

Nat. Med. (1)

V. Ntziachristos, C. Tung, C. Bremer, and R. Weissleder, “Fluorescence molecular tomography resolves protease activity in vivo,” Nat. Med. 8, 757–760 (2002).
[CrossRef] [PubMed]

Nat. Rev. Drug Discov. (1)

J. K. Willmann, N. van Bruggen, L. M. Dinkelborg, and S. S. Gambhir, “Molecular imaging in drug development,” Nat. Rev. Drug Discov. 7, 591–607 (2008).
[CrossRef] [PubMed]

Nature (1)

R. Weissleder and M. J. Pittet, “Imaging in the era of molecular oncology,” Nature 452, 580–589 (2008).
[CrossRef] [PubMed]

Nature Biotechnol. (2)

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

M. K. So, C. J. Xu, A. M. Loening, S. S. Gambhir, and J. H. Rao, “Self-illuminating quantum dot conjugates for in vivo imaging,” Nature Biotechnol. 24, 339–343 (2006).
[CrossRef]

Nature Medicine (1)

R. Weissleder and V. Ntziachristos, “Shedding light onto live molecular targets,” Nature Medicine 9, 123–128 (2003).
[CrossRef] [PubMed]

Opt. Express (7)

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), http://www.opticsinfobase.org/abstract.cfm?URI=oe-14-18-8211.
[CrossRef] [PubMed]

C. Qin, J. Tian, X. Yang, J. Feng, K. Liu, J. Liu, G. Yan, S. Zhu, and M. Xu, “Adaptive improved element free Galerkin method for quasi or multi spectral bioluminescence tomography,” Opt. Express 17, 21925–21934 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-24-21925.
[CrossRef] [PubMed]

J. Feng, K. Jia, C. Qin, G. Yan, S. Zhu, X. Zhang, J. Liu, and J. Tian, “Three-dimensional Bioluminescence Tomography based on Bayesian Approach,” Opt. Express 17, 16834–16848 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-19-16834.
[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. Express 17, 8062–8080 (2009), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-17-10-8062.
[CrossRef] [PubMed]

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), http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-16-20-15640.
[CrossRef] [PubMed]

D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–169 (2002), http://www.opticsinfobase.org/abstract.cfm?URI=OPEX-10-3-159.
[PubMed]

M. Chua and H. Dehghani, “Image reconstruction in diffuse optical tomography based on simplified spherical harmonics approximation,” Opt. Express 17, 24208–24223, (2009), http://www.opticsinfobase.org/abstract.cfm?URI=oe-17-26-24208.
[CrossRef]

Phys. Med. Biol. (2)

Y. Lv, J. Tian, H. Li, 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, 1–16 (2007).
[CrossRef]

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]

presented at Conference 7626 of SPIE on Medical Imaging, San Diego, USA (1)

B. Zhang, J. Tian, D. Liu, L. Sun, X. Yang, and D. Han, “A multithread based new sparse matrix method in bioluminescence tomography”, presented at Conference 7626 of SPIE on Medical Imaging, San Diego, USA, 13–18 February 2010.

Proc. SPIE (2)

W. Cong, D. Kumar, Y. Liu, A. Cong, and G. Wang, “A practical method to determine the light source distribution in bioluminescent imaging,” Proc. SPIE 5535, 679–686 (2004).
[CrossRef]

D. Qin, H. Zhao, Y. Tanikawa, and F. Gao, “Experimental determination of optical properties in turbid medium by TCSPC technique,” Proc. SPIE 6434, 64342E (2007).
[CrossRef]

Quart. Appl. Math. (1)

K. Levenberg, “A method for the solution of certain nonlinear problems,” Quart. Appl. Math. 2, 164–168 (1944).

SIAM J. Appl. Math. (1)

D. W. Marquardt, “An algorithm for least-squares estimation of nonlinear parameters,” SIAM J. Appl. Math. 11, 431–441 (1963).
[CrossRef]

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Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally-resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation. Physics in Medicine and Biology,”  59, 6477–6493 (2009).

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Figures (7)

Fig. 1.
Fig. 1.

Heterogeneous cylindrical numerical phantom with single source (a), consisted of muscle (white), bone (black), heart (pink), lungs (green), liver (yellow) and a ball source (blue) in the right lung. Homogeneous cube phantom with single source (b) and double sources (c). Those blue cylinders in sub figures (b) and (c) denote the light source.

Fig. 2.
Fig. 2.

Reconstruction results comparison between Tikhonov method (sub figures (a) to (d)) and TRM (sub figures (e) to (l)) in single source heterogeneous cylindrical numerical phantom case. Sub figures (a), (e) and (i) are 3D views; (b), (f) and (j) are front views; (c), (g) and (k) are side views; (d), (h) and (l) are top views. Sub figures (i) to (l) are zoom in images of sub figures (e) to (h) around the real source, respectively. The blue ball in each sub figure denotes the real source and the red tetrahedron denotes the reconstructed source with the maximum density. For concision, only the real source and the reconstructed source are displayed.

Fig. 3.
Fig. 3.

Overview of our imaging system that consists of a CCD camera, a camera holder, a translation stage and a rotation stage [8].

Fig. 4.
Fig. 4.

Reconstruction results comparison between Tikhonov method (sub figures (a) to (d)) and TRM (sub figures (e) to (l)) in single source homogeneous cube phantom case. Sub figures (a), (e) and (i) are 3D views; (b), (f) and (j) are front views; (c), (g) and (k) are side views; (d), (h) and (l) are top views. Sub figures (i) to (l) are zoom in images of sub figures (e) to (h) around the real source, respectively. The blue cylinder in each sub figure denotes the real source and the red tetrahedron denotes the reconstructed source with the maximum density.

Fig. 5.
Fig. 5.

Reconstruction results comparison between Tikhonov method (sub figures (a) to (d)) and TRM (sub figures (e) to (l)) in double sources homogeneous cube phantom case. Sub figures (a), (e) and (i) are 3D views; (b), (f) and (j) are front views; (c), (g) and (k) are side views; (d), (h) and (l) are top views. Sub figures (i) to (l) are zoom in images of sub figures (e) to (h) around the real source, respectively. The blue cylinder in each sub figure denotes the real source and the red tetrahedron denotes the reconstructed source with the maximum density.

Fig. 6.
Fig. 6.

Sub figure (a) is the mesh used in the reconstruction procedure. The mesh consists 5 tissues, including the heart in blue, the bone in red the lung in yellow, the liver in green and the muscle in gray. Sub figure (b) is the 3D bioluminescence mapping result from 2D bioluminescence photos.

Fig. 7.
Fig. 7.

Reconstruction results comparison between Tikhonov method (sub figures (a) to (d)) and TRM (sub figures (e) to (l)) in single source heterogeneous nude mouse case. Sub figures (a), (e) and (i) are 3D views; (b), (f) and (j) are front views; (c), (g) and (k) are side views; (d), (h) and (l) are top views. Sub figures (i) to (l) are zoom in images of sub figures (e) to (h) around the real source, respectively. The blue cylinder in each sub figure denotes the real source and the red tetrahedron denotes the reconstructed source with the maximum density. For concision, only the real source and the reconstructed source are displayed.

Tables (6)

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Table 1. Optical parameters of different tissues of the heterogeneous cylindrical phantom

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Table 2. Reconstruction results comparison between Tikhonov method and TRM in single source heterogeneous cylindrical numerical phantom case.

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Table 3. Reconstruction results comparison between Tikhonov method and TRM in single source homogeneous cube phantom case

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Table 4. Reconstruction results comparison between Tikhonov method and TRM in double sources homogeneous cube phantom case

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Table 5. Optical parameters of the nude mouse

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Table 6. Reconstruction results comparison between Tikhonov method and TRM in single source heterogeneous nude mouse case

Equations (24)

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· ( D ( x , λ ) ) Φ ( x , λ ) + μ a ( x , λ ) Φ ( x , λ ) = S ( x , λ ) ( x Ω )
Φ ( x , λ ) + 2 A ( x ; n , n ) D ( x , λ ) ( v ( x ) · Φ ( x , λ ) ) = 0 ( x Ω )
Ω ( D ( x , λ ) ( Φ ( x , λ ) ) · ( Ψ ( x , λ ) ) + μ a ( x , λ ) Φ ( x , λ ) Ψ ( x , λ ) ) d x +
Ω 1 2 A ( x ; n , n ) Φ ( x , λ ) Ψ ( x , λ ) d x = Ω S ( x , λ ) Ψ ( x , λ ) d x ( Ψ ( x , λ ) H 1 ( Ω ) )
{ k ij ( l ) = Ω D ( x ) ( φ i ( l ) ( x ) ) · ( φ j ( l ) ( x ) ) d x c ij ( l ) = Ω μ a ( x ) φ i ( l ) ( x ) φ j ( l ) ( x ) d x b ij ( l ) = Ω φ i ( l ) ( x ) φ j ( l ) ( x ) / ( 2 A ( x ; n , n ) ) d x s ij ( l ) = Ω S i ( l ) φ i ( l ) ( x ) φ j ( l ) ( x ) d x
Φ l meas = A l S l P
f l ( S l P ) = ∣∣ A l S l P Φ l meas ∣∣ 2 2
min x f ( x ) = { ∣∣ Ax b ∣∣ 2 2 }
q k ( s ) = f ( x k ) + g k T s + 1 2 s T G k s ,
x k + 1 = x k + s k .
Ω k = { x : ∣∣ x x k ∣∣ Δ k }
{ x k + s ∣∣ s Δ k }
min q k ( s ) = f ( x k ) + g k T s + 1 2 s T B k s
s . t . s Δ k
Ared k = f ( x k ) f ( x k + s k )
Pred k = q k ( 0 ) q k ( s k )
r k = Ared k Pred k ,
θ = g k s k 2 [ f ( x k + s k ) f ( x k ) g k s k ]
Dis tan ceError = ( x x 0 ) 2 + ( y y 0 ) 2 + ( z z 0 ) 2 ,
RelativeError = S reconstruct S real S real
PS = { ( x , y , z ) 13 < z < 17 , ( x , y , z ) Right Lung }
PS = { ( x , y , z ) 6.5 < x < 14.5 , 6.5 < y < 14.5,6.5 < z < 14.5 }
PS = { ( x , y , z ) 5 < x < 15 , 5 < y < 15,8 < z < 13 }
PS = { ( x , y , z ) 18 < x < 27 , 13 < y < 19,3 < z < 11 }

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