Abstract

A reconstruction method is presented for spectrally resolved three-dimensional bioluminescence tomography (BLT) in heterogeneous media using a level-set strategy. In order to reconstruct internal bioluminescent sources, a level-set strategy is utilized to quantitatively localize the distribution of bioluminescent sources. The results in numerical phantom experiments clearly show that the proposed method can tolerate different initial values and noise levels and, furthermore, can work credibly even when the number of phases (levels) is not known a priori. In addition, a mouse atlas reconstruction is employed to demonstrate the effectiveness of the proposed method in turbid mouse geometry. Finally, the physical experiment further evaluates the method’s potential in practical applications.

© 2010 Optical Society of America

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2010 (2)

2009 (3)

Y.-J. Lu, A. Douraghy, H. B. Machado, D. Stout, J. Tian, H. Herschman, and A. F. Chatziioannou, “Spectrally-resolved bioluminescence tomography with the third-order simplified spherical harmonics approximation,” Phys. Med. Biol. 59, 6477–6493 (2009).
[CrossRef]

J.-C. Feng, K.-B. Jia, C.-H. Qin, G.-R. Yan, S.-P. Zhu, X. Zhang, J.-T. Liu, and J. Tian, “Three-dimensional bioluminescence tomography based on Bayesian approach,” Opt. Express 17, 16834–16848 (2009).
[CrossRef] [PubMed]

S.-P. Zhu, J. Tian, G.-R. Yan, C.-H. Qin, and J.-C. Feng, “Cone beam micro-CT system for small animal imaging and performance evaluation,” Int. J. Biomed. Imaging 2009, Article ID 960573 (2009). http://www.hindawi.com/journals/ijbi/2009/960573/html.
[CrossRef] [PubMed]

2008 (6)

G.-R. Yan, J. Tian, S.-P. Zhu, Y.-K. Dai, and C.-H. Qin, “Fast cone-beam CT image reconstruction using GPU hardware,” J. X-Ray Sci. Technol. 16, 225–234 (2008).

H.-W. Li, X.-C. Tai, and S. I. Aanonsen, “Reservoir description by using a piecewise-constant level-set method,” J. Comput. Phys. 26, 365–377 (2008).

C.-H. Qin, J. Tian, X. Yang, K. Liu, G.-R. Yan, J.-C. Feng, Y.-J. Lv, and M. Xu, “Galerkin-based meshless methods for photon transport in the biological tissue,” Opt. Express 16, 20317–20333 (2008).
[CrossRef] [PubMed]

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

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

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

2007 (7)

Y.-J. Lv, J. Tian, W.-X. Cong, G. Wang, W. Yang, C.-H. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol. 52, 1–16 (2007).
[CrossRef]

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, 024007 (2007).
[CrossRef] [PubMed]

L. V. Wang and H. Wu, Biomedical Optics (Wiley-Interscience, (2007).
[PubMed]

A. Losnegård, O. Christiansen, and X.-C. Tai, “Piecewise-constant level-set method for 3D image segmentation,” in Scale Space and Variational Methods in Computer Vision, F.Sgallari, A.Murli, and N.Paragios, eds. (Springer-Verlag, 2007), pp. 687–696.
[CrossRef]

X.-C. Tai, and H.-W. Li, “A piecewise-constant level-set method for elliptic inverse problems,” Appl. Numer. Math. 57, 686–696 (2007).
[CrossRef]

H-J. Zhao, Fe. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. 12, 062107 (2007).
[CrossRef]

O. Christiansen and X.-C. Tai, “Fast implementation of piecewise-constant level-set methods,” in Proceedings of Image Processing Based on Partial Differential Equations, X.-C.Tai, K.-A.Lie, T.F.Chan, and S.Osher, eds. (Springer, 2007), pp. 289–308.
[CrossRef]

2006 (10)

G. Wang, H.-O. Shen, W.-X. Cong, S. Zhao, and G.-W. Wei, “Temperature-modulated bioluminescence tomography,” Opt. Express 14, 7852–7871 (2006).
[CrossRef] [PubMed]

W.-X. Cong and G. Wang, “Boundary integral method for bioluminescence tomography,” J. Biomed. Opt. 11, 020503 (2006).
[CrossRef] [PubMed]

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006).
[CrossRef] [PubMed]

M. Schweiger, S. R. Arridge, O. Dorn, A. Zacharopoulos, and V. Kolehmainen, “Reconstructing absorption and diffusion shape profiles in optical tomography by a level-set technique,” Opt. Lett. 31, 471–473 (2006).
[CrossRef] [PubMed]

J. Lie, M. Lysaker, and X.-C. Tai, “A binary level-set model and some applications to Mumford–Shah image segmentation,” IEEE Trans. Image Process. 15, 1171–1181 (2006).
[CrossRef] [PubMed]

O. Dorn and D. Lesselier, “Level-set methods for inverse scattering,” Inverse Probl. 22, R67–R131 (2006).
[CrossRef]

J. Lie, M. Lysaker, and X.-C. Tai, “A variant of the level-set method and applications to image segmentation,” Math. Comput. 75, 1155–1174 (2006).
[CrossRef]

W.-X. Cong, D. Kumar, L. V. Wang, and G. Wang, “A Born-type approximation method for bioluminescence tomography,” Med. Phys. 33, 679–686 (2006).
[CrossRef] [PubMed]

Y.-J. Lv, J. Tian, G. Wang, W.-X. Cong, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express 14, 8211–8223 (2006).
[CrossRef] [PubMed]

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

2005 (7)

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

A. H. Hielscher, “Optical tomographic imaging of small animals,” Curr. Opin. Biotechnol. 16, 79–88 (2005).
[CrossRef] [PubMed]

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

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

E. T. Chung, T. F. Chan, and X.-C. Tai, “Electrical impedance tomography using level-set representation and total variational regularization,” J. Comput. Phys. 205, 357–372 (2005).
[CrossRef]

H. Zhao, T. C. Doyle, O. Coquoz, F. Kalish, B. W. Rice, and C. H. Contag, “Emission spectra of bioluminescent reporters and interaction with mammalian tissue determine the sensitivity of detection in vivo,” J. Biomed. Opt. 10, 041210 (2005).
[CrossRef]

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

2004 (3)

R. L. Burden and J. D. Faires, Numerical Analysis (Wadsworth, 2004).

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

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

2003 (4)

H. Herschman, “Molecular imaging: looking at problems, seeing solutions,” Science 302, 605–608 (2003).
[CrossRef] [PubMed]

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

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence CT scanner,” Radiology 229, 566 (2003).

S. Osher and R. P. Fedkiw, Level Sets Methods and Dynamic Implicit Surfaces (Springer, 2003).

2002 (1)

L. A. Vese and T. F. Chan, “A multiphase level-set framevork for image segmentation using the Mumford and Shah model,” Int. J. Comput. Vis. 50, 271–293 (2002).
[CrossRef]

2001 (2)

T. F. Chan and L. Vese, “An active contour model without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[CrossRef]

L. Zöllei, A. Yezzi, and T. Kapur, “A variational framework for joint segmentation and registration,” in IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (IEEE Computer Society, 2001), pp. 44–51.

2000 (1)

O. Dorn, E. Miller, and C. Rappaport, “A shape reconstruction method for electromagnetic tomography using adjoint fields and level sets,” Inverse Probl. 16, 1119–1156 (2000).
[CrossRef]

1997 (1)

T. F. Chan and X.-C. Tai, “Augmented Lagrangian and total variation methods for recovering discontinuous coefficients from elliptic equations,” in Proceedings of Computational Science for the 21st Century, M.Bristeau, G.Etgen, W.Fitzgibbon, J.L.Lions, J.Periaux, and M.F.Wheeler, eds. (Wiley, 1997), pp. 597–607.

1995 (1)

D. Adalsteinsson and J. A. Sethian, “A fast level-set method for propagating interfaces,” J. Comput. Phys. 118, 269–277 (1995).
[CrossRef]

1994 (1)

X-D. Liu, S. Osher, and T. Chan, “Weighted essentially non-oscillatory schemes,” J. Comput. Phys. 115, 200–212 (1994).
[CrossRef]

1991 (1)

T. Lu, P. Neittaanmäki, and X.-C. Tai, “A parallel splitting up method and its application to Navier–Stokes equations,” Appl. Math. Lett. 4, 25–29 (1991).
[CrossRef]

1988 (2)

C-W. Shu and S. Osher, “Efficient implementation of essentially non-oscillatory shock capturing schemes,” J. Comput. Phys. 83, 439–471 (1988).
[CrossRef]

S. Osher and J. A. Sethian, “Fronts propagating with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations,” J. Comput. Phys. 79, 12–49 (1988).
[CrossRef]

Aanonsen, S. I.

H.-W. Li, X.-C. Tai, and S. I. Aanonsen, “Reservoir description by using a piecewise-constant level-set method,” J. Comput. Phys. 26, 365–377 (2008).

Adalsteinsson, D.

D. Adalsteinsson and J. A. Sethian, “A fast level-set method for propagating interfaces,” J. Comput. Phys. 118, 269–277 (1995).
[CrossRef]

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]

Antich, P. P.

N. V. Slavine, M. A. Lewis, E. Richer, and P. P. Antich, “Iterative reconstruction method for light emitting sources based on the diffusion equation,” Med. Phys. 33, 61–68 (2006).
[CrossRef] [PubMed]

Arridge, S. R.

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

Bai, J.

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

Bao, S.-L.

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

Burden, R. L.

R. L. Burden and J. D. Faires, Numerical Analysis (Wadsworth, 2004).

Chan, T.

X-D. Liu, S. Osher, and T. Chan, “Weighted essentially non-oscillatory schemes,” J. Comput. Phys. 115, 200–212 (1994).
[CrossRef]

Chan, T. F.

E. T. Chung, T. F. Chan, and X.-C. Tai, “Electrical impedance tomography using level-set representation and total variational regularization,” J. Comput. Phys. 205, 357–372 (2005).
[CrossRef]

L. A. Vese and T. F. Chan, “A multiphase level-set framevork for image segmentation using the Mumford and Shah model,” Int. J. Comput. Vis. 50, 271–293 (2002).
[CrossRef]

T. F. Chan and L. Vese, “An active contour model without edges,” IEEE Trans. Image Process. 10, 266–277 (2001).
[CrossRef]

T. F. Chan and X.-C. Tai, “Augmented Lagrangian and total variation methods for recovering discontinuous coefficients from elliptic equations,” in Proceedings of Computational Science for the 21st Century, M.Bristeau, G.Etgen, W.Fitzgibbon, J.L.Lions, J.Periaux, and M.F.Wheeler, eds. (Wiley, 1997), pp. 597–607.

Chatziioannou, A. F.

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

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, 5421–5441 (2005).
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Y.-J. Lv, J. Tian, G. Wang, W.-X. Cong, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express 14, 8211–8223 (2006).
[CrossRef] [PubMed]

W.-X. Cong, D. Kumar, L. V. Wang, and G. Wang, “A Born-type approximation method for bioluminescence tomography,” Med. Phys. 33, 679–686 (2006).
[CrossRef] [PubMed]

W.-X. Cong and G. Wang, “Boundary integral method for bioluminescence tomography,” J. Biomed. Opt. 11, 020503 (2006).
[CrossRef] [PubMed]

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

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

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence CT scanner,” Radiology 229, 566 (2003).

Wang, L. V.

L. V. Wang and H. Wu, Biomedical Optics (Wiley-Interscience, (2007).
[PubMed]

W.-X. Cong, D. Kumar, L. V. Wang, and G. Wang, “A Born-type approximation method for bioluminescence tomography,” Med. Phys. 33, 679–686 (2006).
[CrossRef] [PubMed]

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

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

G. Wang, E. A. Hoffman, G. McLennan, L. V. Wang, M. Suter, and J. F. Meinel, “Development of the first bioluminescence CT scanner,” Radiology 229, 566 (2003).

Wei, G.-W.

Weisslder, R.

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

Weissleder, R.

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

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

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[CrossRef]

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C.-H. Qin, J. Tian, X. Yang, K. Liu, G.-R. Yan, J.-C. Feng, Y.-J. Lv, and M. Xu, “Galerkin-based meshless methods for photon transport in the biological tissue,” Opt. Express 16, 20317–20333 (2008).
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J. Tian, J. Bai, X.-P. Yan, S.-L. Bao, Y.-H. Li, W. Liang, and X. Yang, “Multimodality molecular imaging,” IEEE Eng. Med. Biol. Mag. 27, 48–57 (2008).
[CrossRef] [PubMed]

Yang, W.

Y.-J. Lv, J. Tian, W.-X. Cong, G. Wang, W. Yang, C.-H. Qin, and M. Xu, “Spectrally resolved bioluminescence tomography with adaptive finite element: methodology and simulation,” Phys. Med. Biol. 52, 1–16 (2007).
[CrossRef]

Y.-J. Lv, J. Tian, G. Wang, W.-X. Cong, J. Luo, W. Yang, and H. Li, “A multilevel adaptive finite element algorithm for bioluminescence tomography,” Opt. Express 14, 8211–8223 (2006).
[CrossRef] [PubMed]

Yang, X.

Yezzi, A.

L. Zöllei, A. Yezzi, and T. Kapur, “A variational framework for joint segmentation and registration,” in IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (IEEE Computer Society, 2001), pp. 44–51.

Zabner, J.

Zacharopoulos, A.

Zhang, Q.

Zhang, X.

Zhao, H.

H. Zhao, T. C. Doyle, O. Coquoz, F. Kalish, B. W. Rice, and C. H. Contag, “Emission spectra of bioluminescent reporters and interaction with mammalian tissue determine the sensitivity of detection in vivo,” J. Biomed. Opt. 10, 041210 (2005).
[CrossRef]

Zhao, H.-K.

Zhao, H-J.

H-J. Zhao, Fe. Gao, Y. Tanikawa, and Y. Yamada, “Time-resolved diffuse optical tomography and its application to in vitro and in vivo imaging,” J. Biomed. Opt. 12, 062107 (2007).
[CrossRef]

Zhao, S.

Zhu, S.-P.

K. Liu, J. Tian, X. Yang, Y.-J. Lu, C.-H. Qin, S.-P. Zhu, and X. Zhang, “A fast bioluminescent source localization method based on generalized graph cuts with mouse model validations,” Opt. Express 18, 3732–3745 (2010).
[CrossRef] [PubMed]

J.-C. Feng, K.-B. Jia, C.-H. Qin, G.-R. Yan, S.-P. Zhu, X. Zhang, J.-T. Liu, and J. Tian, “Three-dimensional bioluminescence tomography based on Bayesian approach,” Opt. Express 17, 16834–16848 (2009).
[CrossRef] [PubMed]

S.-P. Zhu, J. Tian, G.-R. Yan, C.-H. Qin, and J.-C. Feng, “Cone beam micro-CT system for small animal imaging and performance evaluation,” Int. J. Biomed. Imaging 2009, Article ID 960573 (2009). http://www.hindawi.com/journals/ijbi/2009/960573/html.
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G.-R. Yan, J. Tian, S.-P. Zhu, Y.-K. Dai, and C.-H. Qin, “Fast cone-beam CT image reconstruction using GPU hardware,” J. X-Ray Sci. Technol. 16, 225–234 (2008).

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[CrossRef] [PubMed]

Nat. Biotechnol. (1)

V. Ntziachristos, J. Ripoll, L. V. Wang, and R. Weisslder, “Looking and listening to light: the evolution of whole body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
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Figures (9)

Fig. 1
Fig. 1

Flowchart of the proposed method.

Fig. 2
Fig. 2

Heterogeneous tissue-like model and associated BLI simulation measurements. (a) Tissue-like model with regions geometrically similar to heart (green online), lungs (purple online), liver (navy blue online), bone (gray), muscle (the remainder), and two sources embedded in two lungs. (b) Measured bioluminescent data of the 625 675 nm band mapped onto the finite element mesh model. The front view of the model is shown.

Fig. 3
Fig. 3

BLT reconstructions of two bioluminescent sources in the heterogeneous experiments from BLI measurements corrupted by 1.0% Gaussian noise using both the proposed method and the quasi-Newton method. Three-dimensional locations are shown of reconstructed results (green online) and actual results (red online) in two lungs with the proposed method for (a) initial strength 0.8, 0.0  nW mm 3 , (b) 1.0, 0.0  nW mm 3 , and (c) 1.5, 0.0  nW mm 3 for each each phase. (d) Reconstruction results based on the quasi-Newton method with 1.0% Gaussian noise. (e), (f), and (g) Source strength c i     ( i = 1 , 2 ) evolution during reconstructions as a function of iteration associated with (a), (b), and (c), respectively. Reconstructed c 1 is shown as the upper (red online) solid line, reconstructed c 2 as the lower (blue online) solid line. Actual c 1 and c 2 are shown as upper and lower (red and blue online) dashed lines, respectively. The angle at the upper left of (a)–(d) represents the rotating angle of the model from the front view; the angle increases in the counterclockwise direction.

Fig. 4
Fig. 4

Local amplification of the reconstructed sources and its convergence curve. (a) Results with 1% Gaussian noise and initial strength 0.8, 0.0  nW mm 3 . (b) Logarithm of F ( q ) with 10 as a function of iteration step for the case of initial strength 0.8, 0.0  nW mm 3 .

Fig. 5
Fig. 5

BLT reconstructions in the heterogeneous experiments from BLI measurements corrupted by different Gaussian noise levels. three-dimensional level-set reconstructions of the location of sources (green online) and actual sources (red online) in two lungs for (a) 0.25% Gaussian noise, (b) 6.25% Gaussian noise, (c) 10% Gaussian noise, and (d) 25% Gaussian noise. (e) Reconstruction results based on the quasi-Newton method with 25% Gaussian noise. (f), (g), (h), and (i) Source strength c i     ( i = 1 , 2 ) evolution during reconstructions as a function of iteration step associated with (a), (b), (c), and (d), respectively. Reconstructed c 1 is shown as upper (red online) solid line, reconstructed c 2 as lower (blue online) solid line. Actual c 1 and c 2 are shown as upper and lower (red and blue online) dashed lines, respectively.

Fig. 6
Fig. 6

Linear regression with different noise levels. The solid line denotes the linear regression, and the solid points represent the reconstructed results.

Fig. 7
Fig. 7

BLT reconstructions in the heterogeneous mouse atlas. (a) Reconstructed location in three dimensions. (b) Source strength evolution during reconstructions as a function of iteration step. (c) Logarithm of F ( q ) with 10 as a function of iteration step.

Fig. 8
Fig. 8

Physical heterogeneous phantom. (a) Profile of the physical phantom. (b) Mesh with regions similar to lungs (the two blue cylinders), source (the red sphere in the cylinder on the left), and muscle (the remaining part).

Fig. 9
Fig. 9

BLT reconstructions in the physical experiment. (a) Three-dimensional level-set reconstructions of the location of source (blue online) and actual source (red online) in the phantom. (b) Reconstruction results based on the quasi-Newton method. (c) Source strength c i     ( i = 1 , 2 ) evolution during reconstructions as a function of iteration. Reconstructed c 1 and c 2 are drawn as upper and lower (red and blue online) solid lines, respectively, actual c 1 and c 2 in upper and lower (red and blue online) dashed lines, respectively. (d) Logarithm of F ( q ) with 10 as a function of iteration step.

Tables (4)

Tables Icon

Table 1 Optical Properties for Each Region in the Heterogeneous Tissue-Like Mode a

Tables Icon

Table 2 Source Strength Reconstruction Results from Surface Measurements Corrupted by 1% Gaussian Noise for Different Groups of Initial Strength a

Tables Icon

Table 3 Reconstruction Results from Surface Measurements Corrupted by Different Gaussian Noise Levels a

Tables Icon

Table 4 Optical Parameters for Each Region in the Heterogeneous Phantom a

Equations (20)

Equations on this page are rendered with MathJax. Learn more.

( D ϑ ( r ) ( Φ ϑ ( r ) ) ) + μ a , ϑ ( r ) Φ ϑ ( r ) = q ϑ ( r ) ( r Ω ) .
Φ ϑ ( r ) + 2 A ( r ; n , n ) D ϑ ( r ) ( v ( r ) Φ ϑ ( r ) ) = 0 ( r Ω ) ,
V ϑ ( r ) = D ϑ ( r ) ( v ( r ) Φ ϑ ( r ) ) = Φ ϑ ( r ) 2 A ( r ; n , n ) ( r Ω ) .
M ϑ q ϑ ( r ) = Φ ϑ .
M q ( r ) = Φ meas ,
M q ( r ) = [ ω 1 M 1 [ q ( r ) ] ω 2 M 2 [ q ( r ) ] ω m M m [ q ( r ) ] ] , a n d Φ meas = [ Φ 1 Φ 2 Φ m ] .
F ( q ) = 1 2 M q ( r ) Φ meas 2 2 + 1 2 β q ( r ) 2 2 ,
ϕ = i in Ω i , i = 1 , 2 , , n .
ψ i = 1 α i j = 1 , j i n ( ϕ j ) , w h e r e α i = k = 1 , k i n ( i k ) .
q ( r ) = q ( ϕ , c ) ( r ) = i = 1 n c i ψ i ( ϕ ) ,
K ( ϕ ) = i = 1 n ( ϕ i ) .
min ϕ , c F ( ϕ , c ) = min ϕ , c F ( q ( ϕ , c ) ) , subject to K ( ϕ ) = 0 .
L ( ϕ , c , λ ) = F ( ϕ , c ) + Ω η K ( ϕ ) d r + 1 2 μ Ω K 2 ( ϕ ) d r .
min ϕ , c , η L ( ϕ , c , η ) ,
{ L ϕ = F q q ϕ + η K ( ϕ ) + 1 μ K ( ϕ ) K ( ϕ ) , (15a) L c i = F c i = Ω F q q c i d r ( i = 1 , 2 , , n ) , (15b) L η = K ( ϕ ) , (15c) }
{ F q = ( M q ) T [ M q Φ meas ] + β q , (16a) q ϕ = i = 1 n c i ψ i ( ϕ ) , (16b) q c i = ψ i ( i = 1 , 2 , , n ) . (16c) }
ϕ t + L ϕ = 0 .
{ ϕ k + 1 2 ϕ k Δ t + F ϕ ( ϕ k , c k , η k ) = 0 , (18a) ϕ k + 1 ϕ k + 1 2 Δ t + η K ( ϕ k + 1 ) + 1 μ K ( ϕ k + 1 ) K ( ϕ k + 1 ) = 0 , (18b) }
ϕ k + 1 ϕ k + 1 2 + Δ t η K ( ϕ k + 1 ) + τ 2 K ( ϕ k + 1 ) K ( ϕ k + 1 ) = 0 ,
{ V 1 = 2.3302 × 10 6 value + 5.5093 × 10 4 in [ 600 650 nm ] , ( 20 a ) V 2 = 2.0859 × 10 6 value + 4.2666 × 10 4 in [ 650 700 nm ] , ( 20 b ) V 3 = 1.9477 × 10 6 value + 1.3464 × 10 4 in [ 700 760 nm ] , ( 20 c ) }

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