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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2009

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, "Spectrallyresolved bioluminescence tomography with the third-order simplified spherical harmonics approximation," Phys. Medicine Bio. 59, 6477−6493 (2009).

X. L. Cheng, R. F. Gong, and W. M. Han, "Numerical approximation of bioluminescence tomography based on a new formulation," J. Engin. Math. 63, 121−133 (2009).
[CrossRef]

2008

H. Dehghani, S. C. Davis, and B. W. Pogue, "Spectrally resolved bioluminescence tomography using the reciprocity approach," Medical Phys. 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," J. Comp. Math. 26, 297−309 (2008).

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]

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]

2007

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

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]

2005

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

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

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

2003

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

2002

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]

1995

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

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

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

1944

K. Levenberg, "A method for the solution of certain nonlinear problems," Quart. Appl. Math. 2, 164−1681944).

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]

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]

Chatziioannou, A. F.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, "Spectrallyresolved bioluminescence tomography with the third-order simplified spherical harmonics approximation," Phys. Medicine Bio. 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," J. Engin. Math. 63, 121−133 (2009).
[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]

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, "A mouse optical simulation environment (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]

Davis, S. C.

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

Dehghani, H.

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

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, "Spectrallyresolved bioluminescence tomography with the third-order simplified spherical harmonics approximation," Phys. Medicine Bio. 59, 6477−6493 (2009).

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," J. Engin. Math. 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," J. Comp. Math. 26, 297−309 (2008).

Gulsen, G.

Han, W. M.

X. L. Cheng, R. F. Gong, and W. M. Han, "Numerical approximation of bioluminescence tomography based on a new formulation," J. Engin. Math. 63, 121−133 (2009).
[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]

Herschman, H.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, "Spectrallyresolved bioluminescence tomography with the third-order simplified spherical harmonics approximation," Phys. Medicine Bio. 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).
[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]

Hoffman, E. A.

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

Jacques, S. L.

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]

Levenberg, K.

K. Levenberg, "A method for the solution of certain nonlinear problems," Quart. Appl. Math. 2, 164−1681944).

Li, H.

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]

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

Li, R.

W. Gong, R. Li, N. N. Yan, and W. B. Zhao, "An improved error analysis for finite element approximation of bioluminescence tomography," J. Comp. Math. 26, 297−309 (2008).

Li, Y.

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]

Liang, W.

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]

Loening, A. M.

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]

Lu, Y.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, "Spectrallyresolved bioluminescence tomography with the third-order simplified spherical harmonics approximation," Phys. Medicine Bio. 59, 6477−6493 (2009).

Lv, Y.

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]

Machado, H. B.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, "Spectrallyresolved bioluminescence tomography with the third-order simplified spherical harmonics approximation," Phys. Medicine Bio. 59, 6477−6493 (2009).

Marquardt, D. W.

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

Ntziachristos, 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]

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

Pittet, M. J.

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

Pogue, B. W.

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

Qin, C.

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]

Qin, D.

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]

Rannou, F. R.

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]

Rao, J. H.

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]

Ripoll, J.

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

Schultz, R.

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

Schweiger, 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).
[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]

So, M. K.

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]

Soloviev, V.

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

Stout, D.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, "Spectrallyresolved bioluminescence tomography with the third-order simplified spherical harmonics approximation," Phys. Medicine Bio. 59, 6477−6493 (2009).

Tanikawa, Y.

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]

Tian, J.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, "Spectrallyresolved bioluminescence tomography with the third-order simplified spherical harmonics approximation," Phys. Medicine Bio. 59, 6477−6493 (2009).

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]

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]

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

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).
[CrossRef] [PubMed]

Unlu, M. B.

van Bruggen, N.

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]

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).
[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]

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

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).
[CrossRef]

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 environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method," Acad. Radiol. 11, 1029−1038 (2004).
[CrossRef] [PubMed]

Wang, 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]

Weissleder, R.

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]

Willmann, J. K.

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]

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.

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]

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," J. Comp. Math. 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]

Yang, X.

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]

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]

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," J. Comp. Math. 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 environment (MOSE) to investigate bioluminescent phenomena in the living mouse with the Monte Carlo Method," Acad. Radiol. 11, 1029−1038 (2004).
[CrossRef] [PubMed]

Acad. Radiol.

H. Li, J. Tian, F. Zhu, W. Cong, L. V. Wang, E. A. Hoffman, and G. Wang, "A mouse optical simulation environment (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.

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.

Applied Optics

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

Comput. Meth. Prog. Biomed.

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.

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.

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

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]

J. Comp. Math.

W. Gong, R. Li, N. N. Yan, and W. B. Zhao, "An improved error analysis for finite element approximation of bioluminescence tomography," J. Comp. Math. 26, 297−309 (2008).

J. Engin. Math.

X. L. Cheng, R. F. Gong, and W. M. Han, "Numerical approximation of bioluminescence tomography based on a new formulation," J. Engin. Math. 63, 121−133 (2009).
[CrossRef]

Med. Phys.

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 Phys.

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

Nat. Med.

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.

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

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

Nature Biotechnol.

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

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

Phys. Med. Biol.

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]

Phys. Medicine Bio.

Y. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, "Spectrallyresolved bioluminescence tomography with the third-order simplified spherical harmonics approximation," Phys. Medicine Bio. 59, 6477−6493 (2009).

Proc. SPIE

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.

K. Levenberg, "A method for the solution of certain nonlinear problems," Quart. Appl. Math. 2, 164−1681944).

SIAM J. Appl. Math.

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

Other

M. J. D. Powell, "A new algorithm for unconstrained optimization," in Nonlinear Programming, J. B. Rosen, O. L. Mangasarian, and K. Ritter, eds. (Academic Press, New York, 1970), 31−65.

M. J. D. Powell, "Convergence properties of a class of minimization algorithms," in Nonlinear Programming, O. L. Mangasarian, R. R. Meyer, and S. M. Robinson, eds. (Academic Press, New York, 1975), 1−27.

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

S. S. Rao, The finite element method in engineering (Butterworth-Heinemann, Boston, 1999).

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

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.

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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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.
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V. Isakov, Inverse Problems for Partial Differential Equations (Springer-Verlag, New York, 1998).

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.
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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, 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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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]

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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)

Tables Icon

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

Tables Icon

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

Tables Icon

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

Tables Icon

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

Tables Icon

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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