Abstract

We present an algorithm that provides a frequency-domain solution of the equation of radiative transfer (ERT) for heterogeneous media of arbitrary shape. Although an ERT is more accurate than a diffusion equation, no ERT code for the widely employed frequency-domain case has been developed to date. In this work the ERT is discretized by a combination of discrete-ordinate and finite-volume methods. Two numerical simulations are presented.

© 2004 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. R. Arridge, Inverse Probl. 15, R41 (1999).
    [CrossRef]
  2. D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
    [CrossRef] [PubMed]
  3. H. Jiang, K. D. Paulsen, U. L. Österberg, B. W. Pogue, and M. S. Patterson, Opt. Lett. 20, 2128 (1995).
    [CrossRef] [PubMed]
  4. T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
    [CrossRef] [PubMed]
  5. T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, Rev. Sci. Instrum. 72, 1817 (2001).
    [CrossRef]
  6. R. Roy and E. M. Sevick-Muraca, Opt. Express 4, 353 (1999), http://www.opticsexpress.org .
    [CrossRef] [PubMed]
  7. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  8. S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, Med. Phys. 27, 252 (2000).
    [CrossRef] [PubMed]
  9. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, Phys. Med. Biol. 43, 1285 (1998).
    [CrossRef] [PubMed]
  10. A. D. Kim and A. Ishimaru, Appl. Opt. 37, 5313 (1998).
    [CrossRef]
  11. G. Bal, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 1677 (2002).
    [CrossRef]
  12. G. Bal and K. Ren, J. Opt. Soc. Am. A 20, 2355 (2003).
    [CrossRef]
  13. O. Dorn, Inverse Probl. 14, 1107 (1998).
    [CrossRef]
  14. A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, J. Quant. Spectrosc. Radiat. Transfer 72, 691 (2002).
    [CrossRef]
  15. G. S. Abdoulaev and A. H. Hielscher, J. Electron. Imaging 12, 594 (2003).
    [CrossRef]
  16. E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, La Grange Park, Ill., 1993).
  17. M. L. Adams and E. W. Larsen, Prog. Nucl. Energy 40, 3 (2002).
    [CrossRef]
  18. R. Eymard, T. Gallouet, and R. Herbin, in Handbook of Numerical Analysis VII, P. G. Ciarlet and J. L. Lions, eds. (North-Holland, Amsterdam, 2000).
  19. Y. Saad and M. H. Schultz, SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 30, 856 (1986).
    [CrossRef]

2003 (2)

G. S. Abdoulaev and A. H. Hielscher, J. Electron. Imaging 12, 594 (2003).
[CrossRef]

G. Bal and K. Ren, J. Opt. Soc. Am. A 20, 2355 (2003).
[CrossRef]

2002 (4)

M. L. Adams and E. W. Larsen, Prog. Nucl. Energy 40, 3 (2002).
[CrossRef]

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, J. Quant. Spectrosc. Radiat. Transfer 72, 691 (2002).
[CrossRef]

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

G. Bal, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 1677 (2002).
[CrossRef]

2001 (2)

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, Rev. Sci. Instrum. 72, 1817 (2001).
[CrossRef]

2000 (1)

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, Med. Phys. 27, 252 (2000).
[CrossRef] [PubMed]

1999 (2)

1998 (3)

A. D. Kim and A. Ishimaru, Appl. Opt. 37, 5313 (1998).
[CrossRef]

O. Dorn, Inverse Probl. 14, 1107 (1998).
[CrossRef]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, Phys. Med. Biol. 43, 1285 (1998).
[CrossRef] [PubMed]

1995 (1)

1986 (1)

Y. Saad and M. H. Schultz, SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 30, 856 (1986).
[CrossRef]

Abdoulaev, G. S.

G. S. Abdoulaev and A. H. Hielscher, J. Electron. Imaging 12, 594 (2003).
[CrossRef]

Adams, M. L.

M. L. Adams and E. W. Larsen, Prog. Nucl. Energy 40, 3 (2002).
[CrossRef]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, Phys. Med. Biol. 43, 1285 (1998).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, Med. Phys. 27, 252 (2000).
[CrossRef] [PubMed]

S. R. Arridge, Inverse Probl. 15, R41 (1999).
[CrossRef]

Bal, G.

G. Bal and K. Ren, J. Opt. Soc. Am. A 20, 2355 (2003).
[CrossRef]

G. Bal, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 1677 (2002).
[CrossRef]

Ballesteros, J. R.

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, Phys. Med. Biol. 43, 1285 (1998).
[CrossRef] [PubMed]

Beuthan, J.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, J. Quant. Spectrosc. Radiat. Transfer 72, 691 (2002).
[CrossRef]

Brenner, M.

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

Burney, T.

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chance, B.

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

Dehghani, H.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, Med. Phys. 27, 252 (2000).
[CrossRef] [PubMed]

Dorn, O.

O. Dorn, Inverse Probl. 14, 1107 (1998).
[CrossRef]

Eymard, R.

R. Eymard, T. Gallouet, and R. Herbin, in Handbook of Numerical Analysis VII, P. G. Ciarlet and J. L. Lions, eds. (North-Holland, Amsterdam, 2000).

Fantini, S.

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

Franceschini, M. A.

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

Gallouet, T.

R. Eymard, T. Gallouet, and R. Herbin, in Handbook of Numerical Analysis VII, P. G. Ciarlet and J. L. Lions, eds. (North-Holland, Amsterdam, 2000).

Ha, H. P.

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

Herbin, R.

R. Eymard, T. Gallouet, and R. Herbin, in Handbook of Numerical Analysis VII, P. G. Ciarlet and J. L. Lions, eds. (North-Holland, Amsterdam, 2000).

Hielscher, A. H.

G. S. Abdoulaev and A. H. Hielscher, J. Electron. Imaging 12, 594 (2003).
[CrossRef]

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, J. Quant. Spectrosc. Radiat. Transfer 72, 691 (2002).
[CrossRef]

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, Phys. Med. Biol. 43, 1285 (1998).
[CrossRef] [PubMed]

Hornung, R.

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

Hueber, D. M.

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

Ishimaru, A.

Jiang, H.

Jiang, S.

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, Rev. Sci. Instrum. 72, 1817 (2001).
[CrossRef]

Kim, A. D.

Klose, A. D.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, J. Quant. Spectrosc. Radiat. Transfer 72, 691 (2002).
[CrossRef]

Larsen, E. W.

M. L. Adams and E. W. Larsen, Prog. Nucl. Energy 40, 3 (2002).
[CrossRef]

Lewis, E. E.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, La Grange Park, Ill., 1993).

Ma, H. Y.

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

McBride, T. O.

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, Rev. Sci. Instrum. 72, 1817 (2001).
[CrossRef]

Miller, W. F.

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, La Grange Park, Ill., 1993).

Netz, U.

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, J. Quant. Spectrosc. Radiat. Transfer 72, 691 (2002).
[CrossRef]

Ntziachristos, V.

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

Okada, E.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, Med. Phys. 27, 252 (2000).
[CrossRef] [PubMed]

Osterberg, U. L.

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, Rev. Sci. Instrum. 72, 1817 (2001).
[CrossRef]

Österberg, U. L.

Patterson, M. S.

Paulsen, K. D.

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, Rev. Sci. Instrum. 72, 1817 (2001).
[CrossRef]

H. Jiang, K. D. Paulsen, U. L. Österberg, B. W. Pogue, and M. S. Patterson, Opt. Lett. 20, 2128 (1995).
[CrossRef] [PubMed]

Pham, T. H.

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

Pogue, B. W.

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, Rev. Sci. Instrum. 72, 1817 (2001).
[CrossRef]

H. Jiang, K. D. Paulsen, U. L. Österberg, B. W. Pogue, and M. S. Patterson, Opt. Lett. 20, 2128 (1995).
[CrossRef] [PubMed]

Powell, L.

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

Ren, K.

Roy, R.

Saad, Y.

Y. Saad and M. H. Schultz, SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 30, 856 (1986).
[CrossRef]

Schultz, M. H.

Y. Saad and M. H. Schultz, SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 30, 856 (1986).
[CrossRef]

Schweiger, M.

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, Med. Phys. 27, 252 (2000).
[CrossRef] [PubMed]

Serna, D.

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

Sevick-Muraca, E. M.

Tromberg, B. J.

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

Wallace, D.

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

Zhang, Q.

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Appl. Opt. (1)

Inverse Probl. (2)

S. R. Arridge, Inverse Probl. 15, R41 (1999).
[CrossRef]

O. Dorn, Inverse Probl. 14, 1107 (1998).
[CrossRef]

J. Biomed. Opt. (1)

T. H. Pham, R. Hornung, H. P. Ha, T. Burney, D. Serna, L. Powell, M. Brenner, and B. J. Tromberg, J. Biomed. Opt. 7, 34 (2002).
[CrossRef] [PubMed]

J. Electron. Imaging (1)

G. S. Abdoulaev and A. H. Hielscher, J. Electron. Imaging 12, 594 (2003).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Quant. Spectrosc. Radiat. Transfer (1)

A. D. Klose, U. Netz, J. Beuthan, and A. H. Hielscher, J. Quant. Spectrosc. Radiat. Transfer 72, 691 (2002).
[CrossRef]

Med. Phys. (1)

S. R. Arridge, H. Dehghani, M. Schweiger, and E. Okada, Med. Phys. 27, 252 (2000).
[CrossRef] [PubMed]

Opt. Express (1)

Opt. Lett. (1)

Phys. Med. Biol. (2)

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, Phys. Med. Biol. 43, 1285 (1998).
[CrossRef] [PubMed]

D. M. Hueber, M. A. Franceschini, H. Y. Ma, Q. Zhang, J. R. Ballesteros, S. Fantini, D. Wallace, V. Ntziachristos, and B. Chance, Phys. Med. Biol. 46, 41 (2001).
[CrossRef] [PubMed]

Prog. Nucl. Energy (1)

M. L. Adams and E. W. Larsen, Prog. Nucl. Energy 40, 3 (2002).
[CrossRef]

Rev. Sci. Instrum. (1)

T. O. McBride, B. W. Pogue, S. Jiang, U. L. Osterberg, and K. D. Paulsen, Rev. Sci. Instrum. 72, 1817 (2001).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. (1)

G. Bal, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 62, 1677 (2002).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. (1)

Y. Saad and M. H. Schultz, SIAM (Soc. Ind. Appl. Math.) J. Sci. Stat. Comput. 30, 856 (1986).
[CrossRef]

Other (3)

R. Eymard, T. Gallouet, and R. Herbin, in Handbook of Numerical Analysis VII, P. G. Ciarlet and J. L. Lions, eds. (North-Holland, Amsterdam, 2000).

E. E. Lewis and W. F. Miller, Computational Methods of Neutron Transport (American Nuclear Society, La Grange Park, Ill., 1993).

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (3)

Fig. 1
Fig. 1

Geometric settings of the computational domains. Diamond, source; circles, detectors.

Fig. 2
Fig. 2

(a) ac amplitude and (b) phase delay computed at the detectors for different optical parameters. Solid curve, σa=0.5 cm-1, σs=50 cm-1; dashed curve, σa=1.0 cm-1, σs=50 cm-1; dotted curve, σa=0.5 cm-1, σs=100 cm-1. The anisotropy factor g=0.9 in all cases. The modulation frequency of the source is 200 MHz.

Fig. 3
Fig. 3

Difference between (a) ac amplitude (Iv-Ih) and (b) phase delay (θv-θh) calculated at the detectors for various modulation frequencies in the domain with a void inclusion. g=0.9 and 120 fully level-symmetric directions16 are used. The optical parameters are σa=0.1 cm-1 and σs=120 cm-1. Solid curve, ω=200 MHz; circles, ω=400 MHz; dotted curve, ω=600 MHz.

Equations (9)

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

iωv+Ω·+σtxψx,Ω,ω=σsxSn-1kΩ·Ωψx,Ω,ωdμΩ+qx,Ω,ω,
Sn-1ψx,Ω,ωdμΩj=1Jηjψx,Ωj,ω.
·Ωjψ+σt+iωvψx,Ωj,ω=σsxj=1Jηjkjjψx,Ωj,ω+qx,Ωj,ω,
EΩj·nExψjdγx+Eσtx+iωvψjdx=Eσsxj=1Jηjkjjψjdx+Eqjdx,
κEFE,κj+σtEx+iωvVEψEj=VEσsEj=1JηjkjjψEj+VEqEj,
FE,κj :=κΩj·nExψjdγx=SE,κψEjif SE,κ0SE,κψEjif SE,κ<0,
AΨ=SΨ+Q,
AΨl+1=SΨl+Q,    l0.
ϕx,ω=Ω·νx>0ψx,Ω,ωdμΩ,

Metrics