Abstract

Cerenkov luminescence tomography (CLT) was developed to reconstruct a three-dimensional (3D) distribution of radioactive probes inside a living animal. Reconstruction methods are generally performed within a unique framework by searching for the optimum solution. However, the ill-posed aspect of the inverse problem usually results in the reconstruction being non-robust. In addition, the reconstructed result may not match reality since the difference between the highest and lowest uptakes of the resulting radiotracers may be considerably large, therefore the biological significance is lost. In this paper, based on the minimization of a conformance error, a probability method is proposed that consists of qualitative and quantitative modules. The proposed method first pinpoints the organ that contains the light source. Next, we developed a 0-1 linear optimization subject to a space constraint to model the CLT inverse problem, which was transformed into a forward problem by employing a region growing method to solve the optimization. After running through all of the elements used to grow the sources, a source sequence was obtained. Finally, the probability of each discrete node being the light source inside the organ was reconstructed. One numerical study and two in vivo experiments were conducted to verify the performance of the proposed algorithm, and comparisons were carried out using the hp-finite element method (hp-FEM). The results suggested that our proposed probability method was more robust and reasonable than hp-FEM.

© 2014 Optical Society of America

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2013 (4)

R. Zhang, S. C. Davis, J. L. H. Demers, A. K. Glaser, D. J. Gladstone, T. V. Esipova, S. A. Vinogradov, and B. W. Pogue, “Oxygen tomography by Čerenkov-excited phosphorescence during external beam irradiation,” J. Biomed. Opt.18(5), 050503 (2013).
[CrossRef] [PubMed]

H. Yi, D. Chen, W. Li, S. Zhu, X. Wang, J. Liang, and J. Tian, “Reconstruction algorithms based on l1-norm and l2-norm for two imaging models of fluorescence molecular tomography: a comparative study,” J. Biomed. Opt.18(5), 056013 (2013).
[CrossRef] [PubMed]

Z. Hu, W. Yang, X. Ma, W. Ma, X. Qu, J. Liang, J. Wang, and J. Tian, “Cerenkov luminescence tomography of aminopeptidase N (APN/CD13) expression in mice bearing HT1080 tumors,” Mol. Imaging12(3), 173–181 (2013).
[PubMed]

J. L. Demers, S. C. Davis, R. Zhang, D. J. Gladstone, and B. W. Pogue, “Čerenkov excited fluorescence tomography using external beam radiation,” Opt. Lett.38(8), 1364–1366 (2013), http://www.opticsinfobase.org/vjbo/fulltext.cfm?uri=ol-38-8-1364&id=252787 .
[CrossRef] [PubMed]

2012 (6)

Z. Hu, X. Ma, X. Qu, W. Yang, J. Liang, J. Wang, and J. Tian, “Three-dimensional noninvasive monitoring iodine-131 uptake in the thyroid using a modified Cerenkov luminescence tomography approach,” PLoS ONE7(5), e37623 (2012).
[CrossRef] [PubMed]

Z. Hu, X. Chen, J. Liang, X. Qu, D. Chen, W. Yang, J. Wang, F. Cao, and J. Tian, “Single photon emission computed tomography-guided Cerenkov luminescence tomography,” J. Appl. Phys.112(2), 024703 (2012).
[CrossRef]

B. J. Beattie, D. L. J. Thorek, C. R. Schmidtlein, K. S. Pentlow, J. L. Humm, and A. H. Hielscher, “Quantitative modeling of Cerenkov light production efficiency from medical radionuclides,” PLoS ONE7(2), e31402 (2012).
[CrossRef] [PubMed]

Y. Xu, E. Chang, H. Liu, H. Jiang, S. S. Gambhir, and Z. Cheng, “Proof-of-concept study of monitoring cancer drug therapy with Cerenkov luminescence imaging,” J. Nucl. Med.53(2), 312–317 (2012).
[CrossRef] [PubMed]

D. Lj. Thorek, R. Robertson, W. A. Bacchus, J. Hahn, J. Rothberg, B. J. Beattie, and J. Grimm, “Cerenkov imaging - a new modality for molecular imaging,” Am J Nucl Med Mol Imaging2(2), 163–173 (2012).
[PubMed]

F. Kojima and J. S. Knopp, “Inverse problem for electromagnetic propagation in a dielectric medium using Markov chain Monte Carlo method,” Int. J. Innov. Comput., Inf. Control8(3), 2339–2346 (2012).

2011 (8)

J. C. Park, G. Il An, S. I. Park, J. Oh, H. J. Kim, Y. Su Ha, E. K. Wang, K. Min Kim, J. Y. Kim, J. Lee, M. J. Welch, and J. Yoo, “Luminescence imaging using radionuclides: a potential application in molecular imaging,” Nucl. Med. Biol.38(3), 321–329 (2011).
[CrossRef] [PubMed]

J. P. Holland, G. Normand, A. Ruggiero, J. S. Lewis, and J. Grimm, “Intraoperative imaging of positron emission tomographic radiotracers using Cerenkov luminescence emissions,” Mol. Imaging10(3), 177–186 (2011).

G. S. Mitchell, “In vivo Cerenkov luminescence imaging: a new tool for molecular imaging,” Phil. Trans. R. Soc. A369, 4605–4619 (2011).

F. Boschi, L. Calderan, D. D’Ambrosio, M. Marengo, A. Fenzi, R. Calandrino, A. Sbarbati, and A. E. Spinelli, “In vivo 18F-FDG tumour uptake measurements in small animals using Cerenkov radiation,” Eur. J. Nucl. Med. Mol. Imaging38(1), 120–127 (2011).
[CrossRef] [PubMed]

R. Robertson, M. S. Germanos, M. G. Manfredi, P. G. Smith, and M. D. Silva, “Multimodal imaging with (18)F-FDG PET and Cerenkov luminescence imaging after MLN4924 treatment in a human lymphoma xenograft model,” J. Nucl. Med.52(11), 1764–1769 (2011).
[CrossRef] [PubMed]

J. Zhong, J. Tian, X. Yang, and C. Qin, “Whole-body Cerenkov luminescence tomography with the finite element SP3 method,” Ann. Biomed. Eng.39(6), 1728–1735 (2011).
[CrossRef] [PubMed]

A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci.4(2), 618–650 (2011).
[CrossRef]

A. E. Spinelli, C. Kuo, B. W. Rice, R. Calandrino, P. Marzola, A. Sbarbati, and F. Boschi, “Multispectral Cerenkov luminescence tomography for small animal optical imaging,” Opt. Express19(13), 12605–12618 (2011), http://www.opticsinfobase.org/oe/fulltext.cfm?uri=oe-19-13-12605&id=218880 .
[CrossRef] [PubMed]

2010 (8)

H. Gao and H. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express18(3), 1854–1871 (2010).
[CrossRef] [PubMed]

C. Li, G. S. Mitchell, and S. R. Cherry, “Cerenkov luminescence tomography for small-animal imaging,” Opt. Lett.35(7), 1109–1111 (2010).
[CrossRef] [PubMed]

Z. Hu, J. Liang, W. Yang, W. Fan, C. Li, X. Ma, X. Chen, X. Ma, X. Li, X. Qu, J. Wang, F. Cao, and J. Tian, “Experimental Cerenkov luminescence tomography of the mouse model with SPECT imaging validation,” Opt. Express18(24), 24441–24450 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-24-24441 .
[CrossRef] [PubMed]

A. E. Spinelli, D. D’Ambrosio, L. Calderan, M. Marengo, A. Sbarbati, and F. Boschi, “Cerenkov radiation allows in vivo optical imaging of positron emitting radiotracers,” Phys. Med. Biol.55(2), 483–495 (2010).
[CrossRef] [PubMed]

J. A. Tropp and S. J. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE98(6), 948–958 (2010).
[CrossRef]

H. Liu, G. Ren, Z. Miao, X. Zhang, X. Tang, P. Han, S. S. Gambhir, and Z. Cheng, “Molecular optical imaging with radioactive probes,” PLoS ONE5(3), e9470 (2010).
[CrossRef] [PubMed]

A. Ruggiero, J. P. Holland, J. S. Lewis, and J. Grimm, “Cerenkov luminescence imaging of medical isotopes,” J. Nucl. Med.51(7), 1123–1130 (2010).
[CrossRef] [PubMed]

M. A. Pysz, S. S. Gambhir, and J. K. Willmann, “Molecular imaging: current status and emerging strategies,” Clin. Radiol.65(7), 500–516 (2010).
[CrossRef] [PubMed]

2009 (3)

J. S. Cho, R. Taschereau, S. Olma, K. Liu, Y. C. Chen, C. K. Shen, R. M. van Dam, and A. F. Chatziioannou, “Cerenkov radiation imaging as a method for quantitative measurements of beta particles in a microfluidic chip,” Phys. Med. Biol.54(22), 6757–6771 (2009).
[CrossRef] [PubMed]

R. Robertson, M. S. Germanos, C. Li, G. S. Mitchell, S. R. Cherry, and M. D. Silva, “Optical imaging of Cerenkov light generation from positron-emitting radiotracers,” Phys. Med. Biol.54(16), N355–N365 (2009).
[CrossRef] [PubMed]

R. Han, J. Liang, X. Qu, Y. Hou, N. Ren, J. Mao, and J. Tian, “A source reconstruction algorithm based on adaptive hp-FEM for bioluminescence tomography,” Opt. Express17(17), 14481–14494 (2009).
[CrossRef] [PubMed]

2008 (1)

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]

2007 (2)

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]

D. A. Mankoff, “A definition of molecular imaging,” J. Nucl. Med.48(6), 18N–21N (2007).
[PubMed]

2006 (1)

W. Han, W. Cong, and G. Wang, “Mathematical theory and numerical analysis of bioluminescence tomography,” Inverse Probl.22(5), 1659–1675 (2006).
[CrossRef]

2005 (3)

2004 (3)

G. Chen, Y. Wei, and Y. Xue, “The generalized condition numbers of bounded linear operators in Banach spaces,” J Aust. Math. Soc.76(2), 281–290 (2004).
[CrossRef]

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

M. Gurfinkel, T. S. Pan, and E. M. Sevick-Muraca, “Determination of optical properties in semi-infinite turbid media using imaging measurements of frequency-domain photon migration obtained with an intensified charge-coupled device,” J. Biomed. Opt.9(6), 1336–1346 (2004).
[CrossRef] [PubMed]

2003 (1)

T. F. Massoud and S. S. Gambhir, “Molecular imaging in living subjects: seeing fundamental biological processes in a new light,” Genes Dev.17(5), 545–580 (2003).
[CrossRef] [PubMed]

1998 (1)

A. El Badia and T. H. Duong, “Some remarks on the problem of source identification from boundary measurements,” Inverse Probl.14(4), 883–891 (1998).
[CrossRef]

1996 (1)

F. Santosa, “A level-set approach for inverse problems involving obstacles,” ESAIM Control Optim. Calc. Var.1, 17–33 (1996).
[CrossRef]

1995 (2)

K. Matsuura and Y. Okabe, “Selective minimum-norm solution of the biomagnetic inverse problem,” IEEE Trans. Biomed. Eng.42(6), 608–615 (1995).

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]

1992 (1)

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys.19(4), 879–888 (1992).
[CrossRef] [PubMed]

1986 (1)

S. Demko, “Condition numbers of rectangular systems and bounds for generalized inverses,” Linear Algebra Appl.78, 199–206 (1986).
[CrossRef]

Aghasi, A.

A. Aghasi, M. Kilmer, and E. L. Miller, “Parametric level set methods for inverse problems,” SIAM J. Imaging Sci.4(2), 618–650 (2011).
[CrossRef]

Ahn, S.

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]

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(11), 1779–1792 (1995).
[CrossRef] [PubMed]

Bacchus, W. A.

D. Lj. Thorek, R. Robertson, W. A. Bacchus, J. Hahn, J. Rothberg, B. J. Beattie, and J. Grimm, “Cerenkov imaging - a new modality for molecular imaging,” Am J Nucl Med Mol Imaging2(2), 163–173 (2012).
[PubMed]

Beattie, B. J.

D. Lj. Thorek, R. Robertson, W. A. Bacchus, J. Hahn, J. Rothberg, B. J. Beattie, and J. Grimm, “Cerenkov imaging - a new modality for molecular imaging,” Am J Nucl Med Mol Imaging2(2), 163–173 (2012).
[PubMed]

B. J. Beattie, D. L. J. Thorek, C. R. Schmidtlein, K. S. Pentlow, J. L. Humm, and A. H. Hielscher, “Quantitative modeling of Cerenkov light production efficiency from medical radionuclides,” PLoS ONE7(2), e31402 (2012).
[CrossRef] [PubMed]

Boschi, F.

A. E. Spinelli, C. Kuo, B. W. Rice, R. Calandrino, P. Marzola, A. Sbarbati, and F. Boschi, “Multispectral Cerenkov luminescence tomography for small animal optical imaging,” Opt. Express19(13), 12605–12618 (2011), http://www.opticsinfobase.org/oe/fulltext.cfm?uri=oe-19-13-12605&id=218880 .
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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).
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M. Gurfinkel, T. S. Pan, and E. M. Sevick-Muraca, “Determination of optical properties in semi-infinite turbid media using imaging measurements of frequency-domain photon migration obtained with an intensified charge-coupled device,” J. Biomed. Opt.9(6), 1336–1346 (2004).
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T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys.19(4), 879–888 (1992).
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J. L. Demers, S. C. Davis, R. Zhang, D. J. Gladstone, and B. W. Pogue, “Čerenkov excited fluorescence tomography using external beam radiation,” Opt. Lett.38(8), 1364–1366 (2013), http://www.opticsinfobase.org/vjbo/fulltext.cfm?uri=ol-38-8-1364&id=252787 .
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J. Zhong, J. Tian, X. Yang, and C. Qin, “Whole-body Cerenkov luminescence tomography with the finite element SP3 method,” Ann. Biomed. Eng.39(6), 1728–1735 (2011).
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Qu, X.

Z. Hu, W. Yang, X. Ma, W. Ma, X. Qu, J. Liang, J. Wang, and J. Tian, “Cerenkov luminescence tomography of aminopeptidase N (APN/CD13) expression in mice bearing HT1080 tumors,” Mol. Imaging12(3), 173–181 (2013).
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H. Liu, G. Ren, Z. Miao, X. Zhang, X. Tang, P. Han, S. S. Gambhir, and Z. Cheng, “Molecular optical imaging with radioactive probes,” PLoS ONE5(3), e9470 (2010).
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Rice, B. W.

Ripoll, J.

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).
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D. Lj. Thorek, R. Robertson, W. A. Bacchus, J. Hahn, J. Rothberg, B. J. Beattie, and J. Grimm, “Cerenkov imaging - a new modality for molecular imaging,” Am J Nucl Med Mol Imaging2(2), 163–173 (2012).
[PubMed]

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J. P. Holland, G. Normand, A. Ruggiero, J. S. Lewis, and J. Grimm, “Intraoperative imaging of positron emission tomographic radiotracers using Cerenkov luminescence emissions,” Mol. Imaging10(3), 177–186 (2011).

A. Ruggiero, J. P. Holland, J. S. Lewis, and J. Grimm, “Cerenkov luminescence imaging of medical isotopes,” J. Nucl. Med.51(7), 1123–1130 (2010).
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B. J. Beattie, D. L. J. Thorek, C. R. Schmidtlein, K. S. Pentlow, J. L. Humm, and A. H. Hielscher, “Quantitative modeling of Cerenkov light production efficiency from medical radionuclides,” PLoS ONE7(2), e31402 (2012).
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J. S. Cho, R. Taschereau, S. Olma, K. Liu, Y. C. Chen, C. K. Shen, R. M. van Dam, and A. F. Chatziioannou, “Cerenkov radiation imaging as a method for quantitative measurements of beta particles in a microfluidic chip,” Phys. Med. Biol.54(22), 6757–6771 (2009).
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Silva, M. D.

R. Robertson, M. S. Germanos, M. G. Manfredi, P. G. Smith, and M. D. Silva, “Multimodal imaging with (18)F-FDG PET and Cerenkov luminescence imaging after MLN4924 treatment in a human lymphoma xenograft model,” J. Nucl. Med.52(11), 1764–1769 (2011).
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R. Robertson, M. S. Germanos, C. Li, G. S. Mitchell, S. R. Cherry, and M. D. Silva, “Optical imaging of Cerenkov light generation from positron-emitting radiotracers,” Phys. Med. Biol.54(16), N355–N365 (2009).
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Smith, P. G.

R. Robertson, M. S. Germanos, M. G. Manfredi, P. G. Smith, and M. D. Silva, “Multimodal imaging with (18)F-FDG PET and Cerenkov luminescence imaging after MLN4924 treatment in a human lymphoma xenograft model,” J. Nucl. Med.52(11), 1764–1769 (2011).
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Spinelli, A. E.

A. E. Spinelli, C. Kuo, B. W. Rice, R. Calandrino, P. Marzola, A. Sbarbati, and F. Boschi, “Multispectral Cerenkov luminescence tomography for small animal optical imaging,” Opt. Express19(13), 12605–12618 (2011), http://www.opticsinfobase.org/oe/fulltext.cfm?uri=oe-19-13-12605&id=218880 .
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F. Boschi, L. Calderan, D. D’Ambrosio, M. Marengo, A. Fenzi, R. Calandrino, A. Sbarbati, and A. E. Spinelli, “In vivo 18F-FDG tumour uptake measurements in small animals using Cerenkov radiation,” Eur. J. Nucl. Med. Mol. Imaging38(1), 120–127 (2011).
[CrossRef] [PubMed]

A. E. Spinelli, D. D’Ambrosio, L. Calderan, M. Marengo, A. Sbarbati, and F. Boschi, “Cerenkov radiation allows in vivo optical imaging of positron emitting radiotracers,” Phys. Med. Biol.55(2), 483–495 (2010).
[CrossRef] [PubMed]

Su Ha, Y.

J. C. Park, G. Il An, S. I. Park, J. Oh, H. J. Kim, Y. Su Ha, E. K. Wang, K. Min Kim, J. Y. Kim, J. Lee, M. J. Welch, and J. Yoo, “Luminescence imaging using radionuclides: a potential application in molecular imaging,” Nucl. Med. Biol.38(3), 321–329 (2011).
[CrossRef] [PubMed]

Tang, X.

H. Liu, G. Ren, Z. Miao, X. Zhang, X. Tang, P. Han, S. S. Gambhir, and Z. Cheng, “Molecular optical imaging with radioactive probes,” PLoS ONE5(3), e9470 (2010).
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J. S. Cho, R. Taschereau, S. Olma, K. Liu, Y. C. Chen, C. K. Shen, R. M. van Dam, and A. F. Chatziioannou, “Cerenkov radiation imaging as a method for quantitative measurements of beta particles in a microfluidic chip,” Phys. Med. Biol.54(22), 6757–6771 (2009).
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B. J. Beattie, D. L. J. Thorek, C. R. Schmidtlein, K. S. Pentlow, J. L. Humm, and A. H. Hielscher, “Quantitative modeling of Cerenkov light production efficiency from medical radionuclides,” PLoS ONE7(2), e31402 (2012).
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D. Lj. Thorek, R. Robertson, W. A. Bacchus, J. Hahn, J. Rothberg, B. J. Beattie, and J. Grimm, “Cerenkov imaging - a new modality for molecular imaging,” Am J Nucl Med Mol Imaging2(2), 163–173 (2012).
[PubMed]

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Z. Hu, W. Yang, X. Ma, W. Ma, X. Qu, J. Liang, J. Wang, and J. Tian, “Cerenkov luminescence tomography of aminopeptidase N (APN/CD13) expression in mice bearing HT1080 tumors,” Mol. Imaging12(3), 173–181 (2013).
[PubMed]

H. Yi, D. Chen, W. Li, S. Zhu, X. Wang, J. Liang, and J. Tian, “Reconstruction algorithms based on l1-norm and l2-norm for two imaging models of fluorescence molecular tomography: a comparative study,” J. Biomed. Opt.18(5), 056013 (2013).
[CrossRef] [PubMed]

Z. Hu, X. Chen, J. Liang, X. Qu, D. Chen, W. Yang, J. Wang, F. Cao, and J. Tian, “Single photon emission computed tomography-guided Cerenkov luminescence tomography,” J. Appl. Phys.112(2), 024703 (2012).
[CrossRef]

Z. Hu, X. Ma, X. Qu, W. Yang, J. Liang, J. Wang, and J. Tian, “Three-dimensional noninvasive monitoring iodine-131 uptake in the thyroid using a modified Cerenkov luminescence tomography approach,” PLoS ONE7(5), e37623 (2012).
[CrossRef] [PubMed]

J. Zhong, J. Tian, X. Yang, and C. Qin, “Whole-body Cerenkov luminescence tomography with the finite element SP3 method,” Ann. Biomed. Eng.39(6), 1728–1735 (2011).
[CrossRef] [PubMed]

Z. Hu, J. Liang, W. Yang, W. Fan, C. Li, X. Ma, X. Chen, X. Ma, X. Li, X. Qu, J. Wang, F. Cao, and J. Tian, “Experimental Cerenkov luminescence tomography of the mouse model with SPECT imaging validation,” Opt. Express18(24), 24441–24450 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-24-24441 .
[CrossRef] [PubMed]

R. Han, J. Liang, X. Qu, Y. Hou, N. Ren, J. Mao, and J. Tian, “A source reconstruction algorithm based on adaptive hp-FEM for bioluminescence tomography,” Opt. Express17(17), 14481–14494 (2009).
[CrossRef] [PubMed]

Tropp, J. A.

J. A. Tropp and S. J. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE98(6), 948–958 (2010).
[CrossRef]

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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).
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van Dam, R. M.

J. S. Cho, R. Taschereau, S. Olma, K. Liu, Y. C. Chen, C. K. Shen, R. M. van Dam, and A. F. Chatziioannou, “Cerenkov radiation imaging as a method for quantitative measurements of beta particles in a microfluidic chip,” Phys. Med. Biol.54(22), 6757–6771 (2009).
[CrossRef] [PubMed]

Vinogradov, S. A.

R. Zhang, S. C. Davis, J. L. H. Demers, A. K. Glaser, D. J. Gladstone, T. V. Esipova, S. A. Vinogradov, and B. W. Pogue, “Oxygen tomography by Čerenkov-excited phosphorescence during external beam irradiation,” J. Biomed. Opt.18(5), 050503 (2013).
[CrossRef] [PubMed]

Wang, E. K.

J. C. Park, G. Il An, S. I. Park, J. Oh, H. J. Kim, Y. Su Ha, E. K. Wang, K. Min Kim, J. Y. Kim, J. Lee, M. J. Welch, and J. Yoo, “Luminescence imaging using radionuclides: a potential application in molecular imaging,” Nucl. Med. Biol.38(3), 321–329 (2011).
[CrossRef] [PubMed]

Wang, G.

Wang, J.

Z. Hu, W. Yang, X. Ma, W. Ma, X. Qu, J. Liang, J. Wang, and J. Tian, “Cerenkov luminescence tomography of aminopeptidase N (APN/CD13) expression in mice bearing HT1080 tumors,” Mol. Imaging12(3), 173–181 (2013).
[PubMed]

Z. Hu, X. Chen, J. Liang, X. Qu, D. Chen, W. Yang, J. Wang, F. Cao, and J. Tian, “Single photon emission computed tomography-guided Cerenkov luminescence tomography,” J. Appl. Phys.112(2), 024703 (2012).
[CrossRef]

Z. Hu, X. Ma, X. Qu, W. Yang, J. Liang, J. Wang, and J. Tian, “Three-dimensional noninvasive monitoring iodine-131 uptake in the thyroid using a modified Cerenkov luminescence tomography approach,” PLoS ONE7(5), e37623 (2012).
[CrossRef] [PubMed]

Z. Hu, J. Liang, W. Yang, W. Fan, C. Li, X. Ma, X. Chen, X. Ma, X. Li, X. Qu, J. Wang, F. Cao, and J. Tian, “Experimental Cerenkov luminescence tomography of the mouse model with SPECT imaging validation,” Opt. Express18(24), 24441–24450 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-24-24441 .
[CrossRef] [PubMed]

Wang, L.

Wang, L. V.

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]

Wang, X.

H. Yi, D. Chen, W. Li, S. Zhu, X. Wang, J. Liang, and J. Tian, “Reconstruction algorithms based on l1-norm and l2-norm for two imaging models of fluorescence molecular tomography: a comparative study,” J. Biomed. Opt.18(5), 056013 (2013).
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G. Chen, Y. Wei, and Y. Xue, “The generalized condition numbers of bounded linear operators in Banach spaces,” J Aust. Math. Soc.76(2), 281–290 (2004).
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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]

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J. C. Park, G. Il An, S. I. Park, J. Oh, H. J. Kim, Y. Su Ha, E. K. Wang, K. Min Kim, J. Y. Kim, J. Lee, M. J. Welch, and J. Yoo, “Luminescence imaging using radionuclides: a potential application in molecular imaging,” Nucl. Med. Biol.38(3), 321–329 (2011).
[CrossRef] [PubMed]

Willmann, J. K.

M. A. Pysz, S. S. Gambhir, and J. K. Willmann, “Molecular imaging: current status and emerging strategies,” Clin. Radiol.65(7), 500–516 (2010).
[CrossRef] [PubMed]

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys.19(4), 879–888 (1992).
[CrossRef] [PubMed]

Wright, S. J.

J. A. Tropp and S. J. Wright, “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE98(6), 948–958 (2010).
[CrossRef]

Xu, H.

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]

Xu, Y.

Y. Xu, E. Chang, H. Liu, H. Jiang, S. S. Gambhir, and Z. Cheng, “Proof-of-concept study of monitoring cancer drug therapy with Cerenkov luminescence imaging,” J. Nucl. Med.53(2), 312–317 (2012).
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G. Chen, Y. Wei, and Y. Xue, “The generalized condition numbers of bounded linear operators in Banach spaces,” J Aust. Math. Soc.76(2), 281–290 (2004).
[CrossRef]

Yang, W.

Z. Hu, W. Yang, X. Ma, W. Ma, X. Qu, J. Liang, J. Wang, and J. Tian, “Cerenkov luminescence tomography of aminopeptidase N (APN/CD13) expression in mice bearing HT1080 tumors,” Mol. Imaging12(3), 173–181 (2013).
[PubMed]

Z. Hu, X. Chen, J. Liang, X. Qu, D. Chen, W. Yang, J. Wang, F. Cao, and J. Tian, “Single photon emission computed tomography-guided Cerenkov luminescence tomography,” J. Appl. Phys.112(2), 024703 (2012).
[CrossRef]

Z. Hu, X. Ma, X. Qu, W. Yang, J. Liang, J. Wang, and J. Tian, “Three-dimensional noninvasive monitoring iodine-131 uptake in the thyroid using a modified Cerenkov luminescence tomography approach,” PLoS ONE7(5), e37623 (2012).
[CrossRef] [PubMed]

Z. Hu, J. Liang, W. Yang, W. Fan, C. Li, X. Ma, X. Chen, X. Ma, X. Li, X. Qu, J. Wang, F. Cao, and J. Tian, “Experimental Cerenkov luminescence tomography of the mouse model with SPECT imaging validation,” Opt. Express18(24), 24441–24450 (2010), http://www.opticsinfobase.org/oe/abstract.cfm?uri=oe-18-24-24441 .
[CrossRef] [PubMed]

Yang, X.

J. Zhong, J. Tian, X. Yang, and C. Qin, “Whole-body Cerenkov luminescence tomography with the finite element SP3 method,” Ann. Biomed. Eng.39(6), 1728–1735 (2011).
[CrossRef] [PubMed]

Yi, H.

H. Yi, D. Chen, W. Li, S. Zhu, X. Wang, J. Liang, and J. Tian, “Reconstruction algorithms based on l1-norm and l2-norm for two imaging models of fluorescence molecular tomography: a comparative study,” J. Biomed. Opt.18(5), 056013 (2013).
[CrossRef] [PubMed]

Yoo, J.

J. C. Park, G. Il An, S. I. Park, J. Oh, H. J. Kim, Y. Su Ha, E. K. Wang, K. Min Kim, J. Y. Kim, J. Lee, M. J. Welch, and J. Yoo, “Luminescence imaging using radionuclides: a potential application in molecular imaging,” Nucl. Med. Biol.38(3), 321–329 (2011).
[CrossRef] [PubMed]

Zabner, J.

Zhang, R.

R. Zhang, S. C. Davis, J. L. H. Demers, A. K. Glaser, D. J. Gladstone, T. V. Esipova, S. A. Vinogradov, and B. W. Pogue, “Oxygen tomography by Čerenkov-excited phosphorescence during external beam irradiation,” J. Biomed. Opt.18(5), 050503 (2013).
[CrossRef] [PubMed]

J. L. Demers, S. C. Davis, R. Zhang, D. J. Gladstone, and B. W. Pogue, “Čerenkov excited fluorescence tomography using external beam radiation,” Opt. Lett.38(8), 1364–1366 (2013), http://www.opticsinfobase.org/vjbo/fulltext.cfm?uri=ol-38-8-1364&id=252787 .
[CrossRef] [PubMed]

Zhang, X.

H. Liu, G. Ren, Z. Miao, X. Zhang, X. Tang, P. Han, S. S. Gambhir, and Z. Cheng, “Molecular optical imaging with radioactive probes,” PLoS ONE5(3), e9470 (2010).
[CrossRef] [PubMed]

Zhao, H.

Zhong, J.

J. Zhong, J. Tian, X. Yang, and C. Qin, “Whole-body Cerenkov luminescence tomography with the finite element SP3 method,” Ann. Biomed. Eng.39(6), 1728–1735 (2011).
[CrossRef] [PubMed]

Zhu, S.

H. Yi, D. Chen, W. Li, S. Zhu, X. Wang, J. Liang, and J. Tian, “Reconstruction algorithms based on l1-norm and l2-norm for two imaging models of fluorescence molecular tomography: a comparative study,” J. Biomed. Opt.18(5), 056013 (2013).
[CrossRef] [PubMed]

Am J Nucl Med Mol Imaging (1)

D. Lj. Thorek, R. Robertson, W. A. Bacchus, J. Hahn, J. Rothberg, B. J. Beattie, and J. Grimm, “Cerenkov imaging - a new modality for molecular imaging,” Am J Nucl Med Mol Imaging2(2), 163–173 (2012).
[PubMed]

Ann. Biomed. Eng. (1)

J. Zhong, J. Tian, X. Yang, and C. Qin, “Whole-body Cerenkov luminescence tomography with the finite element SP3 method,” Ann. Biomed. Eng.39(6), 1728–1735 (2011).
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Figures (19)

Fig. 1
Fig. 1

Outline of the probability method. (a) a rough positioning procedure. (a1) a mouse region bounded in [0,40]×[0,30]×[0,40] . (a2) the region of interest (ROI) cropped from the mouse region and bounded in [0,40]×[0,30]×[0,7] . (a3) the procedure for searching for the permissible region (PR). (a4) the resulting PR located in one of the organs inside the ROI. (b) a transformation of the inverse problem into the forward problem using the region growing method. Images (b1)-(b4) show examples of four grown sources after region growth. Images (b5)-(b8) show the normalized calculated surface photon density (CSPD) calculated by each corresponding grown source. (b9) shows the original measured surface photon density (MSPD) after normalization. (c) probability reconstruction of each element node being the light source inside the organ. The probability is dyed with a pseudo-color after interpolation.

Fig. 2
Fig. 2

Flowchart of the region growing algorithm.

Fig. 3
Fig. 3

Heterogeneous phantom. (a) with a single-source in the right lung. (b) measured surface photon density (MSPD) distribution after normalization.

Fig. 4
Fig. 4

The robustness with regards to β . (a) conformance error ε c of the probability method as a function of β , where the cropped interval of the ROI was [3,3] . (b) distance error ε d of hp-FEM as a function of β , where the PR was bounded in a box region I x × I y × I z =[10,10]×[10,10]×[3,3] .

Fig. 5
Fig. 5

Four results of region growth. The sub-images (a)-(d) on the first line are the grown sources in the right lung. (e) is the original MSPD after normalization. Correspondingly, the sub-images (f)-(i) are the normalized CSPD resulting from (a)-(d).

Fig. 6
Fig. 6

Reconstructed optimum sources when a noise is not or is added to the MSPD. (a) reconstructed optimum source composed of fifteen tetrahedrons without noise. (b) reconstructed optimum source, which is composed of 32 tetrahedrons when 5% random noise was added.

Fig. 7
Fig. 7

Extraction of the conformance errors during probability reconstruction. (a) visualization of the conformance error sequence { ε j } , where the horizontal axis represents the subscript of { ε j } , the vertical axis shows the values of { ε j } , and the data between l 1 =7.40× 10 3 and l 2 =1.52× 10 2 constitute ε . (b) error histogram of data ε and ε N(0.0128, 0.0024 2 ) .

Fig. 8
Fig. 8

Reconstructed results of the probability method, where β=0.9 , and the ROI is cropped in [3,3] along the z-axis. (a), (b) are the reconstructed PS. (c) normalized surface photon density calculated using APDD. (d)-(f) are the slices, which are perpendicular to the x, y, and z-axes at the probability core (3.2,6.7,0.6) mm respectively.

Fig. 9
Fig. 9

Reconstructed results of hp-FEM, where β=0.95 , and the PR is the region bounded in [10,10]×[10,10]×[3,3] . (a) normalized surface photon density calculated using the reconstructed sources. (b) reconstructed source with a distance error of 1.96 mm. (c) enlarged source, where the normalized photon density is dyed with a pseudo-color after interpolation.

Fig. 10
Fig. 10

The robustness with regards to β . (a) conformance error ε c of the probability method as a function of β , where the cropped interval of the ROI is [0,7] . (b) distance error ε d of hp-FEM as a function of β , where the PR is a box region I x × I y × I z = [0,40]×[0,30]×[0,7] .

Fig. 11
Fig. 11

Reconstructed optimum sources when noise was or was not added to the MSPD. (a) two reconstructed sources. (b) reconstructed optimum source composed of five tetrahedrons, when 5% random noise was added. (c) reconstructed optimum source composed of two tetrahedrons without noise.

Fig. 12
Fig. 12

Extraction of conformance errors during probability reconstruction. (a) visualization of the conformance error sequence { ε j } , where the horizontal axis represents the subscript of { ε j } , and the vertical axis shows the values of { ε j } . The data between l 1 =4.86× 10 2 and l 2 =7.56× 10 2 constitute ε . (b) error histogram of data ε , where ε N(0.0607, 0.0048 2 ) .

Fig. 13
Fig. 13

Reconstructed results of the probability method, where β=0.9 , and the ROI is cropped in [0,7] along the z-axis. (a) and (b) are the reconstructed PS. (c) normalized surface photon density calculated by APDD. (d)-(f) are the slices perpendicular to the x, y, z-axes at the probability core of (19.06,25.79,4.38) mm respectively.

Fig. 14
Fig. 14

Reconstructed results of hp-FEM, where β=0.9 , and the PR is the region bounded in the region [0,40]×[0,30]×[3,8] . (a) normalized surface photon density calculated from the reconstructed sources. (b) reconstructed source. (c) enlarged source, where the photon density is dyed with a pseudo-color after interpolation.

Fig. 15
Fig. 15

Conformance errors for ε c vs. β values showing the robustness of β , where the ROI is confined in the cropping interval [22,28] .

Fig. 16
Fig. 16

Reconstructed optimum sources when noise was or was not added to MSPD. (a) reconstructed optimum source composed of 49 tetrahedrons without noise. (b) and (c) are the two optimum sources composed of 60 and 76 tetrahedrons respectively, when two different 5% random noise were added.

Fig. 17
Fig. 17

Extraction of conformance errors during probability reconstruction. (a) visualization of the conformance error sequence { ε j } , where the horizontal axis represents the subscript of { ε j } , and the vertical axis shows the values of { ε j } . The data between l 1 =4.76× 10 2 and l 2 =9.80× 10 2 constitute ε . (b) error histogram of data ε , where ε N(0.0728, 0.0102 2 ) .

Fig. 18
Fig. 18

Reconstructed results of the probability method, where β=0.7 , and the ROI is cropped in [22,28] along the z-axis. (a) and (b) are the front and side perspective views of the source, which is dyed in blue. (c) original MSPD after normalization. (d) normalized surface photon density calculated by APDD. (e) is the reconstructed PS. (f)-(h) are the slices perpendicular to the x, y, z-axes at the probability core of (20.0,17.6,24.4) respectively.

Fig. 19
Fig. 19

Reconstructed results of hp-FEM, where β=0.7 , and the PR is the region bounded in [14,25]×[15,21]×[22,28] . (a) and (b) are the front and side perspective views of the source, which is dyed in blue. (c) normalized surface photon density calculated from the reconstructed sources. (d) is the reconstructed source.

Tables (7)

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Table 1 Optical parameters of the heterogeneous phantom.

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Table 2 Conformance errors of the probability method to the cropped interval of the ROI when β = 0.9. CI stands for the cropped interval.

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Table 3 Distance errors of hp-FEM to the PR when β = 0.9. The PR of hp-FEM is a cylindrical region confined in a box region I x × I y × I z , where I x and I y are the interval [-10,10].

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Table 4 Optical parameters of the biological tissues, where μ s =(1g) μ s (x) is the reducing scattering coefficient.

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Table 5 Conformance errors of the probability method for the cropped interval of the ROI when β = 0.9. CI stands for a cropped interval.

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Table 6 Distance error of hp-FEM for the PR when β = 0.9. The PR of hp-FEM is bounded in a box region I x × I y × I z , where I x and I y are the interval [0,40] and [0,30] respectively.

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Table 7 Conformance errors of the probability method for the cropped interval of the ROI when β = 0.7. CI stands for a cropped interval.

Equations (6)

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A S p = Φ m ,
A i S p i = Φ m ,
ε i 1cos< A i S 1 p i , Φ m >=min (for  i = 1,2, K 1 ),
1cos<A S p , Φ m >=min,
p i = N( N i ) / maxN( N i ) ,
p(ξ=c)= (p( ξ 1 =c),p( ξ 2 =c),,p( ξ n =c)) T = ( p 1 , p 2 ,, p n ) T .

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