Abstract

Parameterizing the bioluminescent source globally in Gaussians provides several advantages over voxel representation in bioluminescence tomography. It is mathematically unique to recover Gaussians [Med. Phys. 31(8), 2289 (2004)] and practically sufficient to approximate various shapes by Gaussians in diffusive medium. The computational burden is significantly reduced since much fewer unknowns are required. Besides, there are physiological evidences that the source can be modeled by Gaussians. The simulations show that the proposed model and algorithm significantly improves accuracy and stability in the presence of Gaussian or non- Gaussian sources, noisy data or the optical background mismatch. It is also validated through in vivo experimental data.

© 2010 OSA

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2010 (4)

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55(4), 947–962 (2010).
[PubMed]

J. Liu, A. Li, A. E. Cerussi, and B. J. Tromberg, “Parametric diffuse optical imaging in reflectance geometry,” IEEE Sel. Top. Quantum Electron. 16, 555–564 (2010).

H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation Part 1: l1 regularization,” Opt. Express 18(3), 1854–1871 (2010).
[PubMed]

H. Gao and H. K. Zhao, “Multilevel bioluminescence tomography based on radiative transfer equation part 2: total variation and l1 data fidelity,” Opt. Express 18(3), 2894–2912 (2010).
[PubMed]

2009 (1)

H. Gao and H. K. Zhao, “A fast forward solver of radiative transfer equation,” Transp. Theory Stat. Phys. 38, 149–192 (2009).

2007 (2)

W. Cong, H. Shen, A. Cong, Y. Wang, and G. Wang, “Modeling photon propagation in biological tissues using a generalized Delta-Eddington phase function,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5 Pt 1), 051913 (2007).
[PubMed]

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 86741 (2007).
[PubMed]

2006 (5)

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–8 (2006).

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220, 441–470 (2006).

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

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express 14(17), 7801–7809 (2006).
[PubMed]

2005 (4)

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[PubMed]

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[PubMed]

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005).
[PubMed]

2004 (2)

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

X. Gu, Q. Zhang, L. Larcom, and H. Jiang, “Three-dimensional bioluminescence tomography with model-based reconstruction,” Opt. Express 12(17), 3996–4000 (2004).
[PubMed]

2003 (1)

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology 299, 566 (2003).

2002 (1)

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging 16(4), 378–387 (2002).
[PubMed]

1999 (1)

A. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).

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(17), 4225–4241 (2005).
[PubMed]

Arridge, A. R.

A. R. Arridge, “Optical tomography in medical imaging,” Inverse Probl. 15, R41–R93 (1999).

Bading, J. R.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[PubMed]

Cerussi, A. E.

J. Liu, A. Li, A. E. Cerussi, and B. J. Tromberg, “Parametric diffuse optical imaging in reflectance geometry,” IEEE Sel. Top. Quantum Electron. 16, 555–564 (2010).

Chatziioannou, A. F.

G. Alexandrakis, F. R. Rannou, and A. F. Chatziioannou, “Tomographic bioluminescence imaging by use of a combined optical-PET (OPET) system: a computer simulation feasibility study,” Phys. Med. Biol. 50(17), 4225–4241 (2005).
[PubMed]

Chaudhari, A. J.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[PubMed]

Cheng, J.

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

Cherry, S. R.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[PubMed]

Cong, A.

W. Cong, H. Shen, A. Cong, Y. Wang, and G. Wang, “Modeling photon propagation in biological tissues using a generalized Delta-Eddington phase function,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5 Pt 1), 051913 (2007).
[PubMed]

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[PubMed]

Cong, W.

W. Cong, H. Shen, A. Cong, Y. Wang, and G. Wang, “Modeling photon propagation in biological tissues using a generalized Delta-Eddington phase function,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5 Pt 1), 051913 (2007).
[PubMed]

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 86741 (2007).
[PubMed]

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express 14(17), 7801–7809 (2006).
[PubMed]

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–8 (2006).

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[PubMed]

Contag, C. H.

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging 16(4), 378–387 (2002).
[PubMed]

Conti, P. S.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[PubMed]

Coquoz, O.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

Darvas, F.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[PubMed]

Davis, S. C.

Dehghani, H.

Durairaj, K.

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–8 (2006).

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express 14(17), 7801–7809 (2006).
[PubMed]

Gao, H.

Gu, X.

Han, W.

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

Henry, M.

Hoffman, E.

Hoffman, E. A.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology 299, 566 (2003).

Jiang, H.

Jiang, M.

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[PubMed]

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

Jiang, S.

Klose, A. D.

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220, 441–470 (2006).

Kumar, D.

Kuo, C.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

Larcom, L.

Larsen, E. W.

A. D. Klose and E. W. Larsen, “Light transport in biological tissue based on the simplified spherical harmonics equations,” J. Comput. Phys. 220, 441–470 (2006).

Leahy, R. M.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[PubMed]

Li, A.

J. Liu, A. Li, A. E. Cerussi, and B. J. Tromberg, “Parametric diffuse optical imaging in reflectance geometry,” IEEE Sel. Top. Quantum Electron. 16, 555–564 (2010).

Li, Y.

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

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

Liu, J.

J. Liu, A. Li, A. E. Cerussi, and B. J. Tromberg, “Parametric diffuse optical imaging in reflectance geometry,” IEEE Sel. Top. Quantum Electron. 16, 555–564 (2010).

Liu, Y.

Lv, Y.

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 86741 (2007).
[PubMed]

McCray, P.

McLennan, G.

Meinel, J.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology 299, 566 (2003).

Moats, R. A.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[PubMed]

Patterson, M. S.

Paulsen, K. D.

Pogue, B. W.

Qian, X.

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express 14(17), 7801–7809 (2006).
[PubMed]

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–8 (2006).

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(17), 4225–4241 (2005).
[PubMed]

Rice, B.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

Ross, B. D.

C. H. Contag and B. D. Ross, “It’s not just about anatomy: in vivo bioluminescence imaging as an eyepiece into biology,” J. Magn. Reson. Imaging 16(4), 378–387 (2002).
[PubMed]

Shen, H.

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55(4), 947–962 (2010).
[PubMed]

W. Cong, H. Shen, A. Cong, Y. Wang, and G. Wang, “Modeling photon propagation in biological tissues using a generalized Delta-Eddington phase function,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5 Pt 1), 051913 (2007).
[PubMed]

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express 14(17), 7801–7809 (2006).
[PubMed]

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–8 (2006).

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

Sinn, P.

Smith, D. J.

A. J. Chaudhari, F. Darvas, J. R. Bading, R. A. Moats, P. S. Conti, D. J. Smith, S. R. Cherry, and R. M. Leahy, “Hyperspectral and multispectral bioluminescence optical tomography for small animal imaging,” Phys. Med. Biol. 50(23), 5421–5441 (2005).
[PubMed]

Suter, M.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology 299, 566 (2003).

Tian, J.

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 86741 (2007).
[PubMed]

Tromberg, B. J.

J. Liu, A. Li, A. E. Cerussi, and B. J. Tromberg, “Parametric diffuse optical imaging in reflectance geometry,” IEEE Sel. Top. Quantum Electron. 16, 555–564 (2010).

Troy, T.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

Wang, G.

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55(4), 947–962 (2010).
[PubMed]

W. Cong, H. Shen, A. Cong, Y. Wang, and G. Wang, “Modeling photon propagation in biological tissues using a generalized Delta-Eddington phase function,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5 Pt 1), 051913 (2007).
[PubMed]

Y. Lv, J. Tian, W. Cong, and G. Wang, “Experimental study on bioluminescence tomography with multimodality fusion,” Int. J. Biomed. Imaging 2007, 86741 (2007).
[PubMed]

G. Wang, W. Cong, K. Durairaj, X. Qian, H. Shen, P. Sinn, E. Hoffman, G. McLennan, and M. Henry, “In vivo mouse studies with bioluminescence tomography,” Opt. Express 14(17), 7801–7809 (2006).
[PubMed]

G. Wang, H. Shen, K. Durairaj, X. Qian, and W. Cong, “The first bioluminescence tomography system for simultaneous acquisition of multi-view and multi-spectral data,” Int. J. Biomed. Imaging 2006, 1–8 (2006).

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

W. Cong, G. Wang, D. Kumar, Y. Liu, M. Jiang, L. Wang, E. Hoffman, G. McLennan, P. McCray, J. Zabner, and A. Cong, “Practical reconstruction method for bioluminescence tomography,” Opt. Express 13(18), 6756–6771 (2005).
[PubMed]

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

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology 299, 566 (2003).

Wang, L.

Wang, L. V.

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. Meinel, “Development of the first bioluminescent CT scanner,” Radiology 299, 566 (2003).

Wang, Y.

W. Cong, H. Shen, A. Cong, Y. Wang, and G. Wang, “Modeling photon propagation in biological tissues using a generalized Delta-Eddington phase function,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 76(5 Pt 1), 051913 (2007).
[PubMed]

Zabner, J.

Zhang, Q.

Zhao, H. K.

Zhou, T.

G. Wang, X. Qian, W. Cong, H. Shen, Y. Li, W. Han, K. Durairaj, M. Jiang, T. Zhou, and J. Cheng, “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).

Zwarg, D.

C. Kuo, O. Coquoz, T. Troy, D. Zwarg, and B. Rice, “Bioluminescent tomography for in vivo localization and quantification of luminescent sources from a multiple-view imaging system,” Mol. Imaging 4, 370 (2005).

Curr. Med. Imaging Rev. (1)

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Figures (9)

Fig. 1
Fig. 1

Reconstructions with single Gaussian inclusion. (a)-(d) are the phantoms with 0°, 45°, 90° and 135° rotations respectively; (e)-(h) are from the voxel-based BLT; (i)-(l) are from Gaussian-based BLT. Please see Table 1 for the true and recovered parameters. Please note that the maximum intensity recovered via the voxel-based BLT is only up to 20% of the true value. The results show that Gaussian-based BLT not only localizes the inclusion better than, but also quantitatively more accurate than the voxel-based BLT.

Fig. 2
Fig. 2

Single-inclusion reconstructions with various shapes. (a)-(d) are the phantoms with circular, square, triangular and rectangular inclusion respectively; (e)-(h) are from the voxel-based BLT; (i)-(l) are from Gaussian-based BLT. Please see Table 2 and 3 for the true and recovered parameters. Please note that the maximum intensity recovered via the voxel-based BLT is only up to 20% of the true value. The results show that Gaussian-based BLT not only localizes the inclusion better than, but also quantitatively more accurate than the voxel-based BLT.

Fig. 3
Fig. 3

Comparison of boundary measurements from Gaussian sources and non-Gaussian sources. (a)-(c) are from non-Gaussian sources with side length ratio of 1:1, 1:2 and 2:1. (d)-(f) are from Gaussian sources with the radius ratio of 1:1, 1:2 and 2:1. (g)-(i) plot the boundary measurements originating counterclockwise from x-axis, in which the red curve are from Gaussian sources and the blue curves are from non-Gaussian sources. Please note that the data are normalized so that the absolute sum is 1.

Fig. 4
Fig. 4

Single-inclusion reconstructions with different Gaussian noise levels. The phantom is as the same as (a) in Fig. 2 with single circular inclusion; (a)-(d) are from the voxel-based BLT with 5%, 10%, 20% and 30% noise level respectively; (e)-(h) are correspondingly from Gaussian-based BLT. Please see Table 2 and 4 for the true and recovered parameters. The results show that Gaussian-based BLT is more robust to the noise than the voxel-based BLT.

Fig. 5
Fig. 5

Single-inclusion reconstructions with different mismatch in optical background. The phantom is as the same as (a) in Fig. 2 with single circular inclusion; (a)-(d) are from Gaussian-based BLT with −10%, −30%, −50%, −70% error in optical background; (e)-(h) are from Gaussian-based BLT with + 10%, + 30%, + 50%, + 70% error in optical background. Please see Table 2 and 5 for the true and recovered parameters. Please also note that the reconstruction results with the voxel-based BLT are not presented since it does not localize the inclusion satisfactorily even in the case without optical background error. The results show that Gaussian-based BLT is robust to the mismatch error of optical background in the sense that the inclusion can still be localized well despite of the shifting of centers due to the systematic discrepancy between the true values and the values used in the reconstruction.

Fig. 6
Fig. 6

Multiple-inclusion reconstructions with 5% Gaussian noise level. (a) is the phantom with two elliptical inclusions.; (b) is from the voxel-based BLT; (c) is from Gaussian-based BLT. Please see Table 6 for the true and recovered parameters. The results show again that Gaussian-based BLT not only localizes the inclusion better than, but also quantitatively more accurate than the voxel-based BLT.

Fig. 7
Fig. 7

Multiple-inclusion reconstructions with 5% Gaussian noise level. (a) is the phantom with three circular inclusions.; (b) is from the voxel-based BLT; (c) is from Gaussian-based BLT without minimal-separation constraint; (d) is from Gaussian-based BLT with the minimal-separation constraint. Please see Table 7 for the true and recovered parameters. The results show that Gaussian-based BLT with proper geometric constraints is in general able to separate multiple inclusions better than the voxel-based BLT.

Fig. 8
Fig. 8

Combinatorial optimization with the varying n for G-BLT. (a) is the phantom with three Gaussian inclusions.; (b)-(f) are the reconstructed images from n = 1, 2, 3, 4, 5 respectively.

Fig. 9
Fig. 9

In vivo mouse studies on Gaussian-based BLT. (a) shows two reconstructed bioluminescent sources with dominant power on two kidneys respectively; (b) is the reconstructed bioluminescent isosurface of the intensity value 7; (c) is the histological verification with two tumors at the same locations on the dissected kidneys. Please see Table 8 for the recovered parameters.

Tables (9)

Tables Icon

Table 1 Tests with single Gaussian source. The Gaussian phantom has ρ = 1, xc = 5, yc = 0, rx = 2, ry = 1. The rotation angles are 0°, 45°, 90°, 135°respectively for Test 1-4. In all tests, the total energy of the phantom E is approximately 6.285. Please notice that the result from Case 2 can be understood according to the non-unique representation of the Gaussian

Tables Icon

Table 2 The parameters for single-object simulations with inclusions of various shapes. E is the total source energy; ρ is the maximal source intensity; xc and yc are x- and y-coordinate of the center of the inclusion; rx and ry are maximum distances from the center to the boundary along x and y direction respectively

Tables Icon

Table 3 The reconstructed parameters for single-object simulations with inclusions of various shapes. Please see Table 2 for the corresponding true values

Tables Icon

Table 4 The reconstructed parameters for single-circular-object simulations with different Gaussian noise levels. Please see Table 2 for the corresponding true values

Tables Icon

Table 5 Gaussian-based BLT for single-circular-object simulations with different mismatch in optical background. Please see Table 2 for the corresponding true values

Tables Icon

Table 6 Multiple-Gaussian-inclusion simulation with Inclusion 1 [the left ellipse in Fig. 6(a)] and Inclusion 2 [the right ellipse in Fig. 6(a)]. The total energy E of the phantom is 12.570; the recovered total energy from G-BLT is 12.596 while that from V-BLT is 11.582

Tables Icon

Table 7 Multiple-inclusion simulations. Three circular inclusions of 1mm diameter are located at (0, 0), (4, 0) and (8, 0) respectively with the unit source intensity. The total energy E is 9.364; the recovered total energy from G-BLT is 9.290 while that from V-BLT is 8.282. Please note that the presented values are from G-BLT with minimal-separation constraints as in Eq. (7.3). Please also notice that three inclusions are assumed in the initial guess for the reconstruction with G-BLT although there are actually three inclusions. After the reconstruction, 3 out of 4 inclusions have distinguished peak values

Tables Icon

Table 8 Combinatorial optimization with the varying n for G-BLT. Three Gaussian inclusions are located at (0, 5), (4.33, −2.5) and (−4.33, −2.5) respectively with ρ = 1, rx = 1 and ry = 2. The number of Gaussians n is treated as a variable in this combinatorial problem. The value of n in the reconstruction is set to 1 as the initial guess and iteratively increased until the solution converges. The following table records the reconstructed values for n = 1, 2, 3, 4 and 5. Please notice that the reconstruction should have terminated at n = 3 according to the given criterion if it were not for illustration purpose. The total energy E is 18.843; the recovered total energy is 21.164, 20.861, 18.879, 18.908 and 18.912 respectively. The measurement difference d is 0.167, 0.132, 0.00315, 0.00322 and 0.00327 respectively

Tables Icon

Table 9 In vivo study using Gaussian-based BLT. Please also notice that four inclusions are assumed in the reconstruction although there are actually two tumors from the histological verification. The reconstructed result also shows exactly two inclusions with the distinguished values, i.e. Inclusion 1 and 2 in the following table

Equations (21)

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q = i = 1 N q i 1 τ i ,
( D φ ) + μ a φ = q φ + 2 A D φ n ^ = 0
F Φ = Q
f = D φ n ^ = φ / 2 A .
q ( x , y , z ) = i = 1 n ρ i E x p ( X i T R i T Σ i R i X i ) .
X G = arg min X G 1 2 i = 1 M [ P i T φ ( X G ) f i ] 2 + R ( X G ) .
X min G < X < X max G .
r x < c x y r y , r y < c y z r z , r z < c z x r x .
c x ( r x , i + r x , j ) 2 + c y ( r y , i + r y , j ) 2 + c z ( r z , i + r z , j ) 2 < ( x i x j ) 2 + ( y i y j ) 2 + ( z i z j ) 2 .
g k ( X G ) < 0.
R ( X G ) = k ln [ g k ( X G ) ] .
X n + 1 = X n + arg min δ X 1 2 | | J δ X ( f f n ) | | 2 2 + R ( X n + δ X ) .
x = arg min x 1 2 | | J x b | | 2 2 + R ( X n + x ) .
min x L ( x , t ) = min x 1 2 | | J x b | | 2 2 1 t k ln [ g k ( X n + x ) ] .
x 2 L ( x n , t n ) δ x = x L ( x n , t n ) x n + 1 = x n + s δ x t n + 1 = μ t n .
J i j = P i T φ x j = k = 1 N q k x j P i T φ q k = k = 1 N q k x j J 0 , i k ,
J i j = P i T ( k = 1 N q k x j φ q k ) = P i T k = 1 N q k x j [ F 1 ( Q q k ) ] = P i T [ F 1 ( k = 1 N q k x j Q q k ) ] .
J i j = [ ( F T ) 1 P i ] T ( k = 1 N q k x j Q q k ) .
2 R k = ( R k ) T ( R k ) .
x L ( x , t ) = J T b + 1 t k R k x 2 L ( x , t ) = J T J + 1 t k ( R k ) T ( R k ) .
X = arg min X 1 2 | | J 0 X f | | 2 2 + λ | | X | | 2 2 .

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