Abstract

As a widely used numerical solution for the radiation transport equation (RTE), the discrete ordinates can predict the propagation of photons through biological tissues more accurately relative to the diffusion equation. The discrete ordinates reduce the RTE to a serial of differential equations that can be solved by source iteration (SI). However, the tremendous time consumption of SI, which is partly caused by the expensive computation of each SI step, limits its applications. In this paper, we present a graphics processing unit (GPU) parallel accelerated SI method for discrete ordinates. Utilizing the calculation independence on the levels of the discrete ordinate equation and spatial element, the proposed method reduces the time cost of each SI step by parallel calculation. The photon reflection at the boundary was calculated based on the results of the last SI step to ensure the calculation independence on the level of the discrete ordinate equation. An element sweeping strategy was proposed to detect the calculation independence on the level of the spatial element. A GPU parallel frame called the compute unified device architecture was employed to carry out the parallel computation. The simulation experiments, which were carried out with a cylindrical phantom and numerical mouse, indicated that the time cost of each SI step can be reduced up to a factor of 228 by the proposed method with a GTX 260 graphics card.

© 2011 Optical Society of America

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2011

2010

2009

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

2007

B. Dogdas, D. Stout, A. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[CrossRef] [PubMed]

G. Ghita, G. Sjoden, and J. Baciak, “A Methodology for experimental and 3-D computational radiation transport assessments of Pu-Be neutron sources,” Nucl. Technol. 159, 319–331(2007).

2006

G. Longoni, A. Haghighat, C. Yi, and G. E. Sjoden, “Benchmarking of PENTRAN-SSN parallel transport code and fast preconditioning algorithm using the VENUS-2 MOX-fueled benchmark problem,” J. ASTM Int. 3, 321–330 (2006).
[CrossRef]

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).
[CrossRef]

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency-domain photon migration,” Med. Phys. 33, 4685–4700 (2006).
[CrossRef]

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8, 1–33 (2006).
[CrossRef] [PubMed]

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

2005

S. Patwardhan, S. Bloch, S. Achilefu, and J. Culver, “Time-dependent whole-body fluorescence tomography of probe bio-distributions in mice,” Opt. Express 13, 2564–2577 (2005).
[CrossRef] [PubMed]

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

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1–43 (2005).
[CrossRef]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345(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]

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Oceanic Opt. 18, 32–37 (2005).

2004

2003

A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28, 1019–1021 (2003).
[CrossRef] [PubMed]

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

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911(2003).
[CrossRef] [PubMed]

2002

E. D. Aydin, C. R. E. Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29, 2013–2023 (2002).
[CrossRef] [PubMed]

D. Boas, J. Culver, J. Stott, and A. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–169 (2002).
[PubMed]

M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete-ordinates particle transport calculations,” Prog. Nucl. Energy 40, 3–159 (2002).
[CrossRef]

2001

A. Kienle, F. K. Forster, and R. Hibst, “Influence of the phase function on determination of the optical properties of biological tissue by spatially resolved reflectance,” Opt. Lett. 26, 1571–1573 (2001).
[CrossRef]

V. Kucukboyaci, A. Haghighat, and G. E. Sjoden, “Performance of PENTRAN 3-D parallel particle transport code on the IBM SP2 and PCTRAN cluster,” Lect. Notes Comput. Sci. 2131, 36–43 (2001).
[CrossRef]

S. A. Rukolaine and V. S. Yuferev, “Discrete ordinates quadrature schemes based on the angular interpolation of radiation intensity,” J. Quant. Spectrosc. Radiat. Transfer 69, 257–275(2001).
[CrossRef]

T. A. Wareing, J. M. McGhee, J. E. Morel, and S. D. Pautz, “Discontinuous finite element SN methods on three-dimensional unstructured grids,” Nucl. Sci. Eng. 138, 256–268 (2001).

1999

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

1998

M. L. Adams, T. A. Wareing, and W. F. Walters, “Characteristic methods in thick diffusive problems,” Nucl. Sci. Eng. 130, 18–46 (1998).

1995

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

1991

A. Nicholls and B. Honig, “A rapid finite difference algorithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equation,” J. Comput. Chem. 12, 435–445 (1991).
[CrossRef]

1990

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

1989

1984

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

1976

V. Lebedev, “Quadratures on a sphere,” USSR Comput. Math. Math. Phys. 16, 10–24 (1976).
[CrossRef]

1975

V. Lebedev, “Values of the nodes and weights of ninth to seventeenth order Gauss–Markov quadrature formulae invariant under the octahedron group with inversion,” USSR Comput. Math. Math. Phys. 15, 44–51 (1975).
[CrossRef]

1970

Abdoulaev, G. S.

Achilefu, S.

Adams, M. L.

M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete-ordinates particle transport calculations,” Prog. Nucl. Energy 40, 3–159 (2002).
[CrossRef]

M. L. Adams, T. A. Wareing, and W. F. Walters, “Characteristic methods in thick diffusive problems,” Nucl. Sci. Eng. 130, 18–46 (1998).

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]

Alfano, R. R.

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Arridge, S. R.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1–43 (2005).
[CrossRef]

Aydin, E. D.

E. D. Aydin, C. R. E. Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29, 2013–2023 (2002).
[CrossRef] [PubMed]

Baciak, J.

G. Ghita, G. Sjoden, and J. Baciak, “A Methodology for experimental and 3-D computational radiation transport assessments of Pu-Be neutron sources,” Nucl. Technol. 159, 319–331(2007).

Bal, G.

Bloch, S.

Boas, D.

Budak, V. P.

Y. A. Ilyushin and V. P. Budak, “Narrow-beam propagation in a two-dimensional scattering medium,” J. Opt. Soc. Am. A 28, 76–81 (2011).
[CrossRef]

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Oceanic Opt. 18, 32–37 (2005).

Chatziioannou, A.

B. Dogdas, D. Stout, A. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[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 simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[CrossRef] [PubMed]

Cheng, J.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

Cong, W.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

Culver, J.

Dave, J. V.

Dogdas, B.

B. Dogdas, D. Stout, A. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[CrossRef] [PubMed]

Dunn, A.

Forster, F. K.

Gao, H.

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

Gazdag, J.

Ghita, G.

G. Ghita, G. Sjoden, and J. Baciak, “A Methodology for experimental and 3-D computational radiation transport assessments of Pu-Be neutron sources,” Nucl. Technol. 159, 319–331(2007).

Gibson, A. P.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1–43 (2005).
[CrossRef]

Goddard, A. J. H.

E. D. Aydin, C. R. E. Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29, 2013–2023 (2002).
[CrossRef] [PubMed]

Graves, E. E.

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911(2003).
[CrossRef] [PubMed]

Haghighat, A.

G. Longoni, A. Haghighat, C. Yi, and G. E. Sjoden, “Benchmarking of PENTRAN-SSN parallel transport code and fast preconditioning algorithm using the VENUS-2 MOX-fueled benchmark problem,” J. ASTM Int. 3, 321–330 (2006).
[CrossRef]

V. Kucukboyaci, A. Haghighat, and G. E. Sjoden, “Performance of PENTRAN 3-D parallel particle transport code on the IBM SP2 and PCTRAN cluster,” Lect. Notes Comput. Sci. 2131, 36–43 (2001).
[CrossRef]

Han, W.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

Hebden, J. C.

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1–43 (2005).
[CrossRef]

Hibst, R.

Hielscher, A. H.

Honig, B.

A. Nicholls and B. Honig, “A rapid finite difference algorithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equation,” J. Comput. Chem. 12, 435–445 (1991).
[CrossRef]

Ilyushin, Y. A.

Ito, S.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1999).

Jacques, S. L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Jiang, M.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

Joshi, A.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency-domain photon migration,” Med. Phys. 33, 4685–4700 (2006).
[CrossRef]

Kienle, A.

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).
[CrossRef]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345(2005).
[CrossRef]

A. D. Klose and A. H. Hielscher, “Fluorescence tomography with simulated data based on the equation of radiative transfer,” Opt. Lett. 28, 1019–1021 (2003).
[CrossRef] [PubMed]

Kozelskii, A. V.

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Oceanic Opt. 18, 32–37 (2005).

Kucukboyaci, V.

V. Kucukboyaci, A. Haghighat, and G. E. Sjoden, “Performance of PENTRAN 3-D parallel particle transport code on the IBM SP2 and PCTRAN cluster,” Lect. Notes Comput. Sci. 2131, 36–43 (2001).
[CrossRef]

Kumar, D.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

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).
[CrossRef]

M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete-ordinates particle transport calculations,” Prog. Nucl. Energy 40, 3–159 (2002).
[CrossRef]

Leahy, R. M.

B. Dogdas, D. Stout, A. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[CrossRef] [PubMed]

Lebedev, V.

V. Lebedev, “Quadratures on a sphere,” USSR Comput. Math. Math. Phys. 16, 10–24 (1976).
[CrossRef]

V. Lebedev, “Values of the nodes and weights of ninth to seventeenth order Gauss–Markov quadrature formulae invariant under the octahedron group with inversion,” USSR Comput. Math. Math. Phys. 15, 44–51 (1975).
[CrossRef]

Lewis, E. E.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

Li, H.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

Li, J. F.

Li, Y.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

Liang, J. M.

Liu, F.

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Longoni, G.

G. Longoni, A. Haghighat, C. Yi, and G. E. Sjoden, “Benchmarking of PENTRAN-SSN parallel transport code and fast preconditioning algorithm using the VENUS-2 MOX-fueled benchmark problem,” J. ASTM Int. 3, 321–330 (2006).
[CrossRef]

Lu, B. J.

Luo, J.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

Lv, Y.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

McGhee, J.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency-domain photon migration,” Med. Phys. 33, 4685–4700 (2006).
[CrossRef]

McGhee, J. M.

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Miller, W. F.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (Wiley, 1984).

Morel, J. E.

T. A. Wareing, J. M. McGhee, J. E. Morel, and S. D. Pautz, “Discontinuous finite element SN methods on three-dimensional unstructured grids,” Nucl. Sci. Eng. 138, 256–268 (2001).

Nicholls, A.

A. Nicholls and B. Honig, “A rapid finite difference algorithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equation,” J. Comput. Chem. 12, 435–445 (1991).
[CrossRef]

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V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8, 1–33 (2006).
[CrossRef] [PubMed]

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

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345(2005).
[CrossRef]

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

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911(2003).
[CrossRef] [PubMed]

Oguchi, T.

Oliveira, C. R. E.

E. D. Aydin, C. R. E. Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29, 2013–2023 (2002).
[CrossRef] [PubMed]

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J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency-domain photon migration,” Med. Phys. 33, 4685–4700 (2006).
[CrossRef]

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Pautz, S. D.

T. A. Wareing, J. M. McGhee, J. E. Morel, and S. D. Pautz, “Discontinuous finite element SN methods on three-dimensional unstructured grids,” Nucl. Sci. Eng. 138, 256–268 (2001).

Qian, X.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

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Rannou, F. R.

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

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J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency-domain photon migration,” Med. Phys. 33, 4685–4700 (2006).
[CrossRef]

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Ren, N. N.

Ripoll, J.

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

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[CrossRef] [PubMed]

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S. A. Rukolaine and V. S. Yuferev, “Discrete ordinates quadrature schemes based on the angular interpolation of radiation intensity,” J. Quant. Spectrosc. Radiat. Transfer 69, 257–275(2001).
[CrossRef]

Sevick-Muraca, E. M.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency-domain photon migration,” Med. Phys. 33, 4685–4700 (2006).
[CrossRef]

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G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
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G. Ghita, G. Sjoden, and J. Baciak, “A Methodology for experimental and 3-D computational radiation transport assessments of Pu-Be neutron sources,” Nucl. Technol. 159, 319–331(2007).

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G. Longoni, A. Haghighat, C. Yi, and G. E. Sjoden, “Benchmarking of PENTRAN-SSN parallel transport code and fast preconditioning algorithm using the VENUS-2 MOX-fueled benchmark problem,” J. ASTM Int. 3, 321–330 (2006).
[CrossRef]

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B. Dogdas, D. Stout, A. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
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G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
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V. Ntziachristos, J. Ripoll, LH. Wang, and R. Weissleder, “Looking and listening to light: the evolution of whole body photonic imaging,” Nat. Biotechnol. 23, 313–320 (2005).
[CrossRef] [PubMed]

Wareing, T.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency-domain photon migration,” Med. Phys. 33, 4685–4700 (2006).
[CrossRef]

Wareing, T. A.

T. A. Wareing, J. M. McGhee, J. E. Morel, and S. D. Pautz, “Discontinuous finite element SN methods on three-dimensional unstructured grids,” Nucl. Sci. Eng. 138, 256–268 (2001).

M. L. Adams, T. A. Wareing, and W. F. Walters, “Characteristic methods in thick diffusive problems,” Nucl. Sci. Eng. 130, 18–46 (1998).

Weissleder, R.

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

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

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911(2003).
[CrossRef] [PubMed]

Yi, C.

G. Longoni, A. Haghighat, C. Yi, and G. E. Sjoden, “Benchmarking of PENTRAN-SSN parallel transport code and fast preconditioning algorithm using the VENUS-2 MOX-fueled benchmark problem,” J. ASTM Int. 3, 321–330 (2006).
[CrossRef]

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K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

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S. A. Rukolaine and V. S. Yuferev, “Discrete ordinates quadrature schemes based on the angular interpolation of radiation intensity,” J. Quant. Spectrosc. Radiat. Transfer 69, 257–275(2001).
[CrossRef]

Zhao, H. K.

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

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Zhou, T.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

Annu. Rev. Biomed. Eng.

V. Ntziachristos, “Fluorescence molecular imaging,” Annu. Rev. Biomed. Eng. 8, 1–33 (2006).
[CrossRef] [PubMed]

Appl. Opt.

Atmos. Oceanic Opt.

V. P. Budak and A. V. Kozelskii, “Accuracy and applicability domain of the small angle approximation,” Atmos. Oceanic Opt. 18, 32–37 (2005).

Comput. Methods Programs Biomed.

L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
[CrossRef] [PubMed]

Curr. Med. Imaging Rev.

G. Wang, W. Cong, Y. Li, W. Han, D. Kumar, X. Qian, H. Shen, M. Jiang, T. Zhou, J. Cheng, J. Tian, Y. Lv, H. Li, and J. Luo. “Recent development in bioluminescence tomography,” Curr. Med. Imaging Rev. 2, 453–457 (2006).
[CrossRef]

J. ASTM Int.

G. Longoni, A. Haghighat, C. Yi, and G. E. Sjoden, “Benchmarking of PENTRAN-SSN parallel transport code and fast preconditioning algorithm using the VENUS-2 MOX-fueled benchmark problem,” J. ASTM Int. 3, 321–330 (2006).
[CrossRef]

J. Comput. Chem.

A. Nicholls and B. Honig, “A rapid finite difference algorithm utilizing successive over-relaxation to solve the Poisson-Boltzmann equation,” J. Comput. Chem. 12, 435–445 (1991).
[CrossRef]

J. Comput. Phys.

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).
[CrossRef]

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202, 323–345(2005).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transfer

S. A. Rukolaine and V. S. Yuferev, “Discrete ordinates quadrature schemes based on the angular interpolation of radiation intensity,” J. Quant. Spectrosc. Radiat. Transfer 69, 257–275(2001).
[CrossRef]

Lect. Notes Comput. Sci.

V. Kucukboyaci, A. Haghighat, and G. E. Sjoden, “Performance of PENTRAN 3-D parallel particle transport code on the IBM SP2 and PCTRAN cluster,” Lect. Notes Comput. Sci. 2131, 36–43 (2001).
[CrossRef]

Med. Phys.

J. C. Rasmussen, A. Joshi, T. Pan, T. Wareing, J. McGhee, and E. M. Sevick-Muraca, “Radiative transport in fluorescence-enhanced frequency-domain photon migration,” Med. Phys. 33, 4685–4700 (2006).
[CrossRef]

E. E. Graves, J. Ripoll, R. Weissleder, and V. Ntziachristos, “A submillimeter resolution fluorescence molecular imaging system for small animal imaging,” Med. Phys. 30, 901–911(2003).
[CrossRef] [PubMed]

E. D. Aydin, C. R. E. Oliveira, and A. J. H. Goddard, “A comparison between transport and diffusion calculations using a finite element-spherical harmonics radiation transport method,” Med. Phys. 29, 2013–2023 (2002).
[CrossRef] [PubMed]

Nat. Biotechnol.

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

Nat. Med.

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

Nucl. Sci. Eng.

T. A. Wareing, J. M. McGhee, J. E. Morel, and S. D. Pautz, “Discontinuous finite element SN methods on three-dimensional unstructured grids,” Nucl. Sci. Eng. 138, 256–268 (2001).

M. L. Adams, T. A. Wareing, and W. F. Walters, “Characteristic methods in thick diffusive problems,” Nucl. Sci. Eng. 130, 18–46 (1998).

Nucl. Technol.

G. Ghita, G. Sjoden, and J. Baciak, “A Methodology for experimental and 3-D computational radiation transport assessments of Pu-Be neutron sources,” Nucl. Technol. 159, 319–331(2007).

Opt. Express

Opt. Lett.

Phys. Med. Biol.

B. Dogdas, D. Stout, A. Chatziioannou, and R. M. Leahy, “Digimouse: a 3D whole body mouse atlas from CT and cryosection data,” Phys. Med. Biol. 52, 577–587 (2007).
[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 simulation feasibility study,” Phys. Med. Biol. 50, 4225–4241 (2005).
[CrossRef] [PubMed]

A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, 1–43 (2005).
[CrossRef]

Phys. Rev. Lett.

K. M. Yoo, F. Liu, and R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

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M. L. Adams and E. W. Larsen, “Fast iterative methods for discrete-ordinates particle transport calculations,” Prog. Nucl. Energy 40, 3–159 (2002).
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Figures (13)

Fig. 1
Fig. 1

Reflection and refraction at the boundary.

Fig. 2
Fig. 2

Flow diagram of the element sweeping strategy. Herein, group p m presents the pth groups of the discrete equation m, and N p m stands for the number of elements included in it, N m is the number of elements included in all of the existing groups, N e is the total number of elements, and M denotes the number of discrete directions.

Fig. 3
Fig. 3

Example of dividing 12 elements into four groups by using the element sweeping strategy. Herein, E i j presents the jth element of group i, and U and L denote the up-wind and low-wind element boundaries, respectively.

Fig. 4
Fig. 4

Flow diagram of a GPU parallel accelerated SI. group p m is the pth group of the mth discrete ordinate equation and N p m is the number of elements included in this group, d b presents the dimension of the block, P m is the number of groups for the mth discrete ordinate equation, Thread m , T I presents the T I th thread of the block m, V p , m , T I denotes the T I th element of the pth group for the mth discrete ordinate equation, and M is the number of discrete directions.

Fig. 5
Fig. 5

Geometrical information of the cylindrical phantom. (a) Cylindrical phantom with a spherical source; (b) middle cross section.

Fig. 6
Fig. 6

Geometrical information of the numerical mouse. (a) Front view; (b) side view.

Fig. 7
Fig. 7

Diagram of the data at a boundary obtained from the S N method.

Fig. 8
Fig. 8

Curve comparisons of the photon flux density at the boundary of the cylindrical phantom between the results of the S N and MC methods. (a) and (b) Comparisons of the results in the cases of high absorption. (c) and (d) Comparisons of the results in the cases of strong-forwarding scattering. (e) and (f) Comparisons of the results in the cases of isotropic scattering. Panels (a), (c), and (e) are relative to the simulations that were applied with the refractive index matched boundary condition, and panels (b), (d), and (f) are relative to the simulations that were applied with the refractive index mismatched boundary condition.

Fig. 9
Fig. 9

Curve comparisons of the nodal photon flux density at the boundary of the numerical mouse between the results of the S N and MC methods.

Fig. 10
Fig. 10

Acceleration ratios with different block dimensions. (a) Simulations with cylindrical phantom mesh 1, (b) simulations with cylindrical phantom mesh 3, (c) simulations with numerical mouse mesh 1, and (d) simulations with numerical mouse mesh 2.

Fig. 11
Fig. 11

Acceleration ratios of our method with different discrete schemes. (a) Acceleration ratios of simulations with the cylindrical phantom; (b) acceleration ratios of simulations with the numerical mouse.

Fig. 12
Fig. 12

Time cost of memory access is hidden in a multiprocessor by carrying out the calculation and memory access for different blocks at the same time.

Fig. 13
Fig. 13

Division results of the element sweeping strategy for a mesh that is refined based on that in Fig. 2. Here, E i j presents the jth element of group i, and U and L denote the up-wind low-wind element boundaries, respectively.

Tables (8)

Tables Icon

Table 1 Optical Parameters of the Cylindrical Phantom

Tables Icon

Table 2 Optical Parameters in the Mouse Tissues

Tables Icon

Table 3 Information of the Spatial Discrete Meshes

Tables Icon

Table 4 Average Relative Error Between the Results of the S N and MC Methods in the Simulations with a Cylindrical Phantom

Tables Icon

Table 5 Average Time of an SI Step in a CPU with Different Spatial and Direction Discrete Schemes in Simulations for the Cylindrical Phantom a

Tables Icon

Table 6 Average Time of an SI Step in a CPU with Different Spatial and Direction Discrete Schemes in Simulations for the Numerical Mouse a

Tables Icon

Table 7 Average Time of Each SI Step of the CPU with Different Optical Parameters in Simulation for the Cylindrical Phantom a

Tables Icon

Table 8 Acceleration Ratio of a GPU with Different Optical Parameters in Simulation for the Cylindrical Phantom a

Equations (14)

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

Ω m · Φ ( Ω m , r ) + μ t Φ ( Ω m , r ) = μ s m = 1 M Φ ( Ω m , r ) p i ( Ω m · Ω m ) w m + Q ( Ω m , r ) , m = 1 , 2 , ... , M ,
p ( Ω · Ω ) = 1 g 2 4 π ( 1 + g 2 2 g cos θ ) 3 / 2 ,
l = 1 4 δ V k l ( Ω m · n k l ) Θ Θ T φ k , m l , s d S V k ( Ω m · Θ ) Θ T φ k , m d V + V k Θ { μ t Θ T φ k , m μ s Θ T m = 1 M p 3 ( Ω m · Ω m ) w m φ k , m Θ T q k } d V = 0 .
φ m , k l , s = { φ k , m , Ω m · n k l 0 φ k , m l , inc , Ω m · n k l < 0 ,
l = 1 4 δ V k l ( Ω m · n k l ) Θ Θ T φ k , m l , s ( i ) d δ V V k ( Ω m · Θ ) Θ T φ k , m ( i ) d V + μ t V k Θ Θ T φ k , m ( i ) d V V k Θ Θ T q k d V = μ s V k Θ Θ T m = 1 M p 3 ( Ω m · Ω m ) w m φ k , m ( i 1 ) d V ,
e = e i , i 1 1 ( e i + 1 , i / e i , i 1 ) , with e i , i 1 = m = 1 M k = 1 K l = 1 4 | Φ m , k l , ( i ) Φ m , k l , ( i 1 ) | m = 1 M k = 1 K l = 1 4 | Φ m , k l , ( i 1 ) | ,
R = I 2 ( I · N ) N , T = sin θ t I ( I · N ) N | I ( I · N ) N | + sign ( I · N ) cos θ t N , n i sin θ i = n t sin θ t .
L ( θ i ) = { 1 2 ( n i cos θ i n t cos θ t n i cos θ i + n t cos θ t ) 2 + 1 2 ( n i cos θ t n t cos θ i n i cos θ t + n t cos θ i ) 2 θ i < θ c 1 θ i < θ c .
φ k , m l , inc ( i ) = φ k , m l , ( i 1 ) L ( Ω m · Ω m ) ,
W = d w · i = 1 I V i ,
Ψ j = k = 1 K W k k = 1 K S k ,
Φ j m = k = 1 K T k l = 1 3 Φ k l m Θ k l d S k = 1 K S k ,
Ψ j = m = 1 M ( ( Ω m · N + | Ω m · N | ) / 2 ) ( 1 L ( Ω m · N ) ) ( Ω m · N ) Φ j m w m ,
ARE = ( j = 1 J | d j m d j s | | d j m | ) / J ,

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