Abstract

Steady-state form of the Boltzmann transport equation is solved for the investigation of photon propagation through three-dimensional regions and channels that contain a photon absorption and scattering-free fluid. Transport and diffusion solutions are obtained by a finite-element-spherical harmonics radiation transport method. Results calculated with two theories are presented in order to show the influence of voidlike regions and channels on the transport of photons.

© 2007 Optical Society of America

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  1. J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, "3D optical tomography of the premature infant brain," Phys. Med. Biol. 47, 4155-4166 (2002).
    [CrossRef] [PubMed]
  2. A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, "Optical tomography of a realistic neonatal head phantom," Appl. Opt. 42, 3109-3116 (2003).
    [CrossRef] [PubMed]
  3. W. Cai, M. Xu, and R. R. Alfano, "Three-dimensional radiative transfer tomography for turbid media," IEEE J. Sel. Top. Quantum Electron. 9, 189-198 (2003).
    [CrossRef]
  4. A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
    [CrossRef] [PubMed]
  5. H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, "Optical tomography in the presence of void regions," J. Opt. Soc. Am. A 17, 1659-1670 (2000).
    [CrossRef]
  6. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach for modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
    [CrossRef] [PubMed]
  7. S. R. Arridge, M. Hiraoka, and M. Schweiger, "Statistical basis for the determination of optical pathlength in tissue," Phys. Med. Biol. 40, 1539-1553 (1995).
    [CrossRef] [PubMed]
  8. H. B. Jiang and K. D. Paulsen, U. L. Österberg, B. W. Pogue, and M. S. Patterson, "Optical-image reconstruction using frequency-domain data--simulations and experiments," J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 253-266 (1996).
    [CrossRef]
  9. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998).
    [CrossRef] [PubMed]
  10. E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, "A finite element-spherical harmonics radiation transport model for photon migration in turbid media," J. Quant. Spectrosc. Radiat. Transf. 84, 247-260 (2004).
    [CrossRef]
  11. S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
    [CrossRef] [PubMed]
  12. J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, "3D optical tomography in the presence of void regions," Opt. Express 7, 462-467 (2000).
    [CrossRef] [PubMed]
  13. S. R. Arridge, J. C. Hebden, M. Schweiger, F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, H. Dehghani, and D. T. Delpy, "A method for three-dimensional time-resolved optical tomography," Int. J. Imaging Syst. Technol. 11, 2-11 (2000).
    [CrossRef]
  14. J. C. Hebden, A. Gibson, R. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, "Three-dimensional optical tomography of the premature infant brain," Phys. Med. Biol. 47, 4155-4166 (2002).
    [CrossRef] [PubMed]
  15. T. Austin, J. C. Hebden, A. P. Gibson, G. Branco, R. Yusof, S. R. Arridge, J. H. Meek, D. T. Delpy, and J. S. Wyatt, "Three-dimensional optical imaging of blood volume and oxygenation in the preterm brain," Neuroimage 31, 1426-1433 (2006).
    [CrossRef] [PubMed]
  16. A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, "Optical tomography using the time-independent equation of radiative transfer. Part 1: forward model," J. Quant. Spectrosc. Radiat. Transf. 72, 691-713 (2002).
    [CrossRef]
  17. A. D. Klose and A. H. Hielscher, "Optical tomography using the time-independent equation of radiative transfer. Part 2: inverse model," J. Quant. Spectrosc. Radiat. Transf. 72, 715-732 (2002).
    [CrossRef]
  18. D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, "Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head," Opt. Express 10, 159-170 (2002).
    [PubMed]
  19. M. Xu, W. Cai, M. Lax, and R. R. Alfano, "Photon transport forward model for imaging in turbid media," Opt. Lett. 26, 1066-1068 (2001).
    [CrossRef]
  20. O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. 14, 1107-1130 (1998).
    [CrossRef]
  21. O. Dorn, "Scattering and absorption transport sensitivity functions for optical tomography," Opt. Express 7, 492-506 (2000).
    [CrossRef] [PubMed]
  22. G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomography with the equation of radiative transfer," J. Electron. Imaging 12, 594-601 (2003).
    [CrossRef]
  23. S. Wright, M. Schweiger, and S. R. Arridge, "Reconstruction in optical tomography using the PN approximations," Meas. Sci. Technol. 18, 79-86 (2007).
    [CrossRef]
  24. E. D. Aydin, C. R. E. de 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]
  25. C. R. E. de Oliveira, "An arbitrary geometry finite element method for multi-group neutron transport with anisotropic scattering," Prog. Nucl. Energy 18, 227-236 (1986).
    [CrossRef]
  26. C. R. E. de Oliveira and A. J. H. Goddard, "Event--a multidimensional finite element-spherical harmonics radiation transport code," in Proceedings of the OECD International Seminar on 3D Deterministic Radiation Transport Codes (1996).
  27. C. R. E. de Oliveira, "Multidimensional deterministic radiation transport review and status," CEA Frédéric Joliot Summer School, Cadarache, France (1997).
  28. J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).
  29. J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).
  30. G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand-Reinhold, 1979).
  31. S. Kaplan and J. A. Davis, "Canonical and involutory transformation of the variational problems of transport theory," Nucl. Sci. Eng. 28, 166-176 (1967).
  32. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Transport and diffusion calculations on MRI-generated data," Proc. SPIE 2979, 500-508 (1997).
    [CrossRef]

2007

S. Wright, M. Schweiger, and S. R. Arridge, "Reconstruction in optical tomography using the PN approximations," Meas. Sci. Technol. 18, 79-86 (2007).
[CrossRef]

2006

T. Austin, J. C. Hebden, A. P. Gibson, G. Branco, R. Yusof, S. R. Arridge, J. H. Meek, D. T. Delpy, and J. S. Wyatt, "Three-dimensional optical imaging of blood volume and oxygenation in the preterm brain," Neuroimage 31, 1426-1433 (2006).
[CrossRef] [PubMed]

2005

A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

2004

E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, "A finite element-spherical harmonics radiation transport model for photon migration in turbid media," J. Quant. Spectrosc. Radiat. Transf. 84, 247-260 (2004).
[CrossRef]

2003

A. Gibson, R. M. Yusof, H. Dehghani, J. Riley, N. Everdell, R. Richards, J. C. Hebden, M. Schweiger, S. R. Arridge, and D. T. Delpy, "Optical tomography of a realistic neonatal head phantom," Appl. Opt. 42, 3109-3116 (2003).
[CrossRef] [PubMed]

W. Cai, M. Xu, and R. R. Alfano, "Three-dimensional radiative transfer tomography for turbid media," IEEE J. Sel. Top. Quantum Electron. 9, 189-198 (2003).
[CrossRef]

G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomography with the equation of radiative transfer," J. Electron. Imaging 12, 594-601 (2003).
[CrossRef]

2002

J. C. Hebden, A. Gibson, R. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, "Three-dimensional optical tomography of the premature infant brain," Phys. Med. Biol. 47, 4155-4166 (2002).
[CrossRef] [PubMed]

E. D. Aydin, C. R. E. de 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]

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, "3D optical tomography of the premature infant brain," Phys. Med. Biol. 47, 4155-4166 (2002).
[CrossRef] [PubMed]

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, "Optical tomography using the time-independent equation of radiative transfer. Part 1: forward model," J. Quant. Spectrosc. Radiat. Transf. 72, 691-713 (2002).
[CrossRef]

A. D. Klose and A. H. Hielscher, "Optical tomography using the time-independent equation of radiative transfer. Part 2: inverse model," J. Quant. Spectrosc. Radiat. Transf. 72, 715-732 (2002).
[CrossRef]

D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, "Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head," Opt. Express 10, 159-170 (2002).
[PubMed]

2001

2000

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
[CrossRef] [PubMed]

J. Riley, H. Dehghani, M. Schweiger, S. R. Arridge, J. Ripoll, and M. Nieto-Vesperinas, "3D optical tomography in the presence of void regions," Opt. Express 7, 462-467 (2000).
[CrossRef] [PubMed]

S. R. Arridge, J. C. Hebden, M. Schweiger, F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, H. Dehghani, and D. T. Delpy, "A method for three-dimensional time-resolved optical tomography," Int. J. Imaging Syst. Technol. 11, 2-11 (2000).
[CrossRef]

H. Dehghani, S. R. Arridge, M. Schweiger, and D. T. Delpy, "Optical tomography in the presence of void regions," J. Opt. Soc. Am. A 17, 1659-1670 (2000).
[CrossRef]

O. Dorn, "Scattering and absorption transport sensitivity functions for optical tomography," Opt. Express 7, 492-506 (2000).
[CrossRef] [PubMed]

1998

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. 14, 1107-1130 (1998).
[CrossRef]

1997

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Transport and diffusion calculations on MRI-generated data," Proc. SPIE 2979, 500-508 (1997).
[CrossRef]

1996

H. B. Jiang and K. D. Paulsen, U. L. Österberg, B. W. Pogue, and M. S. Patterson, "Optical-image reconstruction using frequency-domain data--simulations and experiments," J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 253-266 (1996).
[CrossRef]

1995

S. R. Arridge, M. Hiraoka, and M. Schweiger, "Statistical basis for the determination of optical pathlength in tissue," Phys. Med. Biol. 40, 1539-1553 (1995).
[CrossRef] [PubMed]

1993

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach for modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

1986

C. R. E. de Oliveira, "An arbitrary geometry finite element method for multi-group neutron transport with anisotropic scattering," Prog. Nucl. Energy 18, 227-236 (1986).
[CrossRef]

1967

S. Kaplan and J. A. Davis, "Canonical and involutory transformation of the variational problems of transport theory," Nucl. Sci. Eng. 28, 166-176 (1967).

Appl. Opt.

IEEE J. Sel. Top. Quantum Electron.

W. Cai, M. Xu, and R. R. Alfano, "Three-dimensional radiative transfer tomography for turbid media," IEEE J. Sel. Top. Quantum Electron. 9, 189-198 (2003).
[CrossRef]

Int. J. Imaging Syst. Technol.

S. R. Arridge, J. C. Hebden, M. Schweiger, F. E. W. Schmidt, M. E. Fry, E. M. C. Hillman, H. Dehghani, and D. T. Delpy, "A method for three-dimensional time-resolved optical tomography," Int. J. Imaging Syst. Technol. 11, 2-11 (2000).
[CrossRef]

Inverse Probl.

O. Dorn, "A transport-backtransport method for optical tomography," Inverse Probl. 14, 1107-1130 (1998).
[CrossRef]

J. Electron. Imaging

G. S. Abdoulaev and A. H. Hielscher, "Three-dimensional optical tomography with the equation of radiative transfer," J. Electron. Imaging 12, 594-601 (2003).
[CrossRef]

J. Opt. Soc. Am. A

J. Opt. Soc. Am. A Opt. Image Sci. Vis.

H. B. Jiang and K. D. Paulsen, U. L. Österberg, B. W. Pogue, and M. S. Patterson, "Optical-image reconstruction using frequency-domain data--simulations and experiments," J. Opt. Soc. Am. A Opt. Image Sci. Vis. 13, 253-266 (1996).
[CrossRef]

J. Quant. Spectrosc. Radiat. Transf.

E. D. Aydin, C. R. E. de Oliveira, and A. J. H. Goddard, "A finite element-spherical harmonics radiation transport model for photon migration in turbid media," J. Quant. Spectrosc. Radiat. Transf. 84, 247-260 (2004).
[CrossRef]

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, "Optical tomography using the time-independent equation of radiative transfer. Part 1: forward model," J. Quant. Spectrosc. Radiat. Transf. 72, 691-713 (2002).
[CrossRef]

A. D. Klose and A. H. Hielscher, "Optical tomography using the time-independent equation of radiative transfer. Part 2: inverse model," J. Quant. Spectrosc. Radiat. Transf. 72, 715-732 (2002).
[CrossRef]

Meas. Sci. Technol.

S. Wright, M. Schweiger, and S. R. Arridge, "Reconstruction in optical tomography using the PN approximations," Meas. Sci. Technol. 18, 79-86 (2007).
[CrossRef]

Med. Phys.

E. D. Aydin, C. R. E. de 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]

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, "The finite element model for the propagation of light in scattering media: a direct method for domains with nonscattering regions," Med. Phys. 27, 252-264 (2000).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, "A finite element approach for modelling photon transport in tissue," Med. Phys. 20, 299-309 (1993).
[CrossRef] [PubMed]

Neuroimage

T. Austin, J. C. Hebden, A. P. Gibson, G. Branco, R. Yusof, S. R. Arridge, J. H. Meek, D. T. Delpy, and J. S. Wyatt, "Three-dimensional optical imaging of blood volume and oxygenation in the preterm brain," Neuroimage 31, 1426-1433 (2006).
[CrossRef] [PubMed]

Nucl. Sci. Eng.

S. Kaplan and J. A. Davis, "Canonical and involutory transformation of the variational problems of transport theory," Nucl. Sci. Eng. 28, 166-176 (1967).

Opt. Express

Opt. Lett.

Phys. Med. Biol.

J. C. Hebden, A. Gibson, R. M. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, "3D optical tomography of the premature infant brain," Phys. Med. Biol. 47, 4155-4166 (2002).
[CrossRef] [PubMed]

J. C. Hebden, A. Gibson, R. Yusof, N. Everdell, E. M. C. Hillman, D. T. Delpy, S. R. Arridge, T. Austin, J. H. Meek, and J. S. Wyatt, "Three-dimensional optical tomography of the premature infant brain," Phys. Med. Biol. 47, 4155-4166 (2002).
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues," Phys. Med. Biol. 43, 1285-1302 (1998).
[CrossRef] [PubMed]

S. R. Arridge, M. Hiraoka, and M. Schweiger, "Statistical basis for the determination of optical pathlength in tissue," Phys. Med. Biol. 40, 1539-1553 (1995).
[CrossRef] [PubMed]

A. Gibson, J. C. Hebden, and S. R. Arridge, "Recent advances in diffuse optical imaging," Phys. Med. Biol. 50, R1-R43 (2005).
[CrossRef] [PubMed]

Proc. SPIE

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, "Transport and diffusion calculations on MRI-generated data," Proc. SPIE 2979, 500-508 (1997).
[CrossRef]

Prog. Nucl. Energy

C. R. E. de Oliveira, "An arbitrary geometry finite element method for multi-group neutron transport with anisotropic scattering," Prog. Nucl. Energy 18, 227-236 (1986).
[CrossRef]

Other

C. R. E. de Oliveira and A. J. H. Goddard, "Event--a multidimensional finite element-spherical harmonics radiation transport code," in Proceedings of the OECD International Seminar on 3D Deterministic Radiation Transport Codes (1996).

C. R. E. de Oliveira, "Multidimensional deterministic radiation transport review and status," CEA Frédéric Joliot Summer School, Cadarache, France (1997).

J. J. Duderstadt and L. J. Hamilton, Nuclear Reactor Analysis (Wiley, 1976).

J. J. Duderstadt and W. R. Martin, Transport Theory (Wiley, 1979).

G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Van Nostrand-Reinhold, 1979).

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Figures (6)

Fig. 1
Fig. 1

(Color online) Finite-element mesh used for diffusion and transport simulations of voidlike spaces.

Fig. 2
Fig. 2

Log 10 of the photon intensities calculated with transport ( P 9 and P 11 ), and diffusion theories in a medium that contains an almost scattering and absorption free cube.

Fig. 3
Fig. 3

(Color online) Flux values calculated diffusion theory (top) and transport theory (bottom) for voidlike geometry depicted in Fig. 1.

Fig. 4
Fig. 4

Geometry of channeling problem given as a 2D slice through a 3D cube.

Fig. 5
Fig. 5

Transport and diffusion results of photon fluxes for the 2D slice shown in Fig. 4.

Fig. 6
Fig. 6

(Color online) Contoured photon intensity distributions modeled by diffusion theory (top) and transport theory (bottom) for the 3D CSF-layered problem.

Equations (13)

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

Ω ϕ ( r , Ω ) + H ϕ ( r , Ω ) = S ( r , Ω ) ,
H ϕ ( r , Ω ) = σ t ( r ) ϕ ( r , Ω ) 4 π d Ω σ s ( r , Ω Ω ) ϕ ( r , Ω ) .
H ϕ ( r , Ω ) = l = 0 ( 2 l + 1 4 π ) σ l ( r ) 4 π d Ω P l ( μ 0 ) ϕ ( r , Ω ) ,
ϕ ( r s , Ω ) = T ( r s , Ω ) , Ω n < 0 for   all   r s   on   S ,
C ϕ + ( r , Ω ) = S + ( r , Ω ) Ω ϕ ( r , Ω ) ,
G ϕ ( r , Ω ) = S ( r , Ω ) Ω ϕ + ( r , Ω ) ,
ϕ ± ( r , Ω ) = 1 2 [ ϕ ( r , Ω ) ± ϕ ( r , Ω ) ] ,
S ± ( r , Ω ) = 1 2 [ S ( r , Ω ) ± [ S ( r , Ω ) ] ,
C [ ] = l   even ( 2 l + 1 4 π ) σ l 4 π d Ω P l ( μ 0 ) [ ] ,
G [ ] = l   odd ( 2 l + 1 4 π ) σ l 4 π d Ω P l ( μ 0 ) [ ] .
Ω G 1 Ω ϕ + + C ϕ + = S + Ω G 1 S ,
ϕ + ( r , Ω ) + G 1 [ S ( r , Ω ) Ω ϕ + ( r , Ω ) ] = T ( r , Ω ) , Ω n < 0 ,
ϕ + ( r , Ω ) G 1 [ S ( r , Ω ) Ω ϕ + ( r , Ω ) ] = T ( r , Ω ) , Ω n > 0 .

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