Abstract

Three-dimensional bioluminescence imaging is an emerging technique that can be used to monitor molecular events in intact living systems. The inverse problem of 3D bioluminescence imaging does not have a unique solution because it requires reconstruction of a 3D source function from a 2D one. A novel approach that addresses this problem with the aid of a simple experimental setup and solves the uniqueness problem of the solution for a monochromatic measurement set is suggested here. The approach is verified numerically by reconstructing bioluminescent objects of various shapes embedded inside highly scattering media, such as biologiçal tissue.

© 2007 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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2006 (2)

2005 (4)

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

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

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

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

2004 (2)

1999 (1)

S. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[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 simulated feasibility study," Phys. Med. Biol. 50, 4225-4241 (2005).
[CrossRef] [PubMed]

Arridge, S.

S. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

Bading, J. R.

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

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 simulated feasibility study," Phys. Med. Biol. 50, 4225-4241 (2005).
[CrossRef] [PubMed]

Chaudhari, A. J.

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

Chery, S. R.

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

Cong, A.

Cong, W.

Conti, P. S.

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

Coquoz, O.

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

Darvis, F.

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

Davis, S.

Dehghani, H.

Engl, H. W.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 2000).

Gu, X.

Hanke, M.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 2000).

Hoffmann, E. A.

Isakov, V.

V. Isakov, Inverse Problems for Partial Differential Equations, Vol. 127 of Applied Mathematical Series (Springer-Verlag, 1998).

Jiang, H.

Jiang, M.

Jiang, S.

Kirsch, A.

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Vol. 120 of Applied Mathematical Series (Springer-Verlag, 1996).
[CrossRef]

Krasnosselskaia, L. V.

Kumar, D.

Kuo, C.

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

Larcom, L.

Leahy, R. M.

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

Li, Y.

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

Liu, Y.

McCray, P. B.

McLennan, G.

Moats, R. A.

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

Neubauer, A.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 2000).

Patterson, M. S.

Paulsen, K. D.

Pogue, B. W.

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 simulated feasibility study," Phys. Med. Biol. 50, 4225-4241 (2005).
[CrossRef] [PubMed]

Rice, B.

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

Samarskii, A. A.

A. A. Samarskii and P. N. Vabishevich, Chislennye Metody Reshenia Obratnyh Zadach Matematicheskoi Fiziki (URSS, 2004) (in Russian).

Smith, D. J.

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

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).

Soloviev, V. Y.

Troy, T.

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

Vabishevich, P. N.

A. A. Samarskii and P. N. Vabishevich, Chislennye Metody Reshenia Obratnyh Zadach Matematicheskoi Fiziki (URSS, 2004) (in Russian).

Wang, G.

Wang, L. V.

Zabner, J.

Zhang, Q.

Zwang, D.

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

Appl. Opt. (1)

Inverse Probl. (1)

S. Arridge, "Optical tomography in medical imaging," Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

Med. Phys. (1)

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

Mol. Imaging (1)

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

Opt. Express (2)

Opt. Lett. (1)

Phys. Med. Biol. (2)

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

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

Other (5)

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).

V. Isakov, Inverse Problems for Partial Differential Equations, Vol. 127 of Applied Mathematical Series (Springer-Verlag, 1998).

A. Kirsch, An Introduction to the Mathematical Theory of Inverse Problems, Vol. 120 of Applied Mathematical Series (Springer-Verlag, 1996).
[CrossRef]

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 2000).

A. A. Samarskii and P. N. Vabishevich, Chislennye Metody Reshenia Obratnyh Zadach Matematicheskoi Fiziki (URSS, 2004) (in Russian).

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Figures (5)

Fig. 1
Fig. 1

First row shows the solution of the DA with the Robin-type boundary conditions within a scattering ball. Spherical bioluminescent sources are placed at the center of a ball. The δ function is placed at the center in (a). In (b) and (c), radii of sources are 0.3 and 0.7, correspondingly. The second row shows the solution of the DA for the same spherical bioluminescent sources placed at the center of a ball of radii 0, 0.3, and 0.7 in (d)–(f), correspondingly, with a half-spherical mirror attached to the surface at x < 0 .

Fig. 2
Fig. 2

(a) Slice y = 0 is shown, indicating locations of two bioluminescent targets used in reconstruction simulations, (b) phantom's cross section at y = 0 showing mesh.

Fig. 3
Fig. 3

(a)–(c) Results of reconstruction of two bioluminescent targets after the 3rd, 10th, and 18th iterations, correspondingly, with y × 10 5 level of noise. (d)–(f) The 5th, 10th, and 14th iterations with 1% level of noise.

Fig. 4
Fig. 4

Reconstruction results of the δ function in (a) and (d), and two bioluminescent balls of radii 0.3 in (b), (e), and 0.5 in (c), (f) placed at the center of a scattering ball. The first row (a)–(c) shows reconstruction results with the presence of computational error only, and the second row (d)–(f), with 1% level of noise.

Fig. 5
Fig. 5

(a) X-like source distribution. Results of reconstruction (b) with y × 10 5 level of noise, and (c) with 1% level of noise.

Equations (11)

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( 1 / γ ) U + q U = 3 α λ f ,
q = 3 α ( 1 λ ) , γ = α ( 1 λ ϵ / 3 ) ,
U ( r ) = 3 V α ( r ) λ ( r ) G ( r , r ) f ( r ) d 3 r ,
( γ U + n U ) V = 0 ,
y ( δ ) = ( K T K + β I ) x ( β ) ,
x m ( β ) = f m V m , y m ( δ ) = K m n T U n ,
K n m = 3 α m λ m G ( r n , r m ) ,
x K 1 C .
β = ( 2 δ / C ) 4 / 3 , y y ( δ ) δ ,
x x ( β ) 3 ( C 2 δ / 4 ) 1 / 3 .
a m = max 0 n < N ( K n m 1 U n ) .

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