Abstract

The boundary element method (BEM) is a useful tool in diffuse optical imaging (DOI) when modelling large optical regions whose parameters are piecewise constant, but are computationally expensive. We present here an acceleration technique, the single-level fast multipole method, for a highly lossy medium. The enhanced practicability of the BEM in DOI is demonstrated through test examples on single-layer problems, where order of magnitude reduction factors on solution time are achieved and on a realistic three-layer model of the neonatal head. Our experimental results agree very closely with theoretical predictions of computational complexity.

© 2011 Optical Society of America

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  1. S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. DelpyMed. Phys. 20, 299 (1993).
    [CrossRef] [PubMed]
  2. V. Y. Soloviev, Med. Phys. 33, 4176 (2006).
    [CrossRef] [PubMed]
  3. J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, Phys. Rev. E 64, 051917 (2001).
    [CrossRef]
  4. J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
    [CrossRef] [PubMed]
  5. A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, Inverse Probl. 22, 1509 (2006).
    [CrossRef]
  6. S. Chaillat, M. Bonnet, and J. F. Semblat, Geophys. J. Int. 177, 509 (2009).
    [CrossRef]
  7. R. Coifman, V. Rokhlin, and S. Wandzura, IEEE Antennas Propag. Mag. 35, 7 (1993).
    [CrossRef]
  8. N. Geng, A. Sullivan, and L. Carin, IEEE Trans. Antennas Propagat. 49, 740 (2001).
    [CrossRef]
  9. L. Greengard and V. Rokhlin, J. Comput. Phys. 73, 325(1987).
    [CrossRef]
  10. K. E. Schmidt and M. A. Lee, J. Stat. Phys. 63, 1223 (1991).
    [CrossRef]
  11. E. Darve, J. Comput. Phys. 160, 195 (2000).
    [CrossRef]
  12. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).
  13. V. I. Lebedev and D. N. Laikov, Doklady Mathematics 59, 477 (1999).
  14. E. Darve, SIAM J. Numer. Anal. 38, 98 (2001).
    [CrossRef]
  15. A. Tizzard, L. Horesh, R. J. Yerworth, D. S. Holder, and R. H. Bayford, Physiol. Meas. 26, S251 (2005).
    [CrossRef] [PubMed]
  16. J. Elisee, A. Gibson, and S. Arridge, IEEE Trans. Biomed. Eng. 57, 2737 (2010).
    [CrossRef]
  17. J. Elisee, A. Gibson, and S. Arridge, Biomed. Opt. Express 2, 568 (2011).
    [CrossRef] [PubMed]
  18. N. A. Gumerov and R. Duraiswami, J. Comput. Phys. 227, 8290 (2008).
    [CrossRef]

2011 (1)

2010 (1)

J. Elisee, A. Gibson, and S. Arridge, IEEE Trans. Biomed. Eng. 57, 2737 (2010).
[CrossRef]

2009 (1)

S. Chaillat, M. Bonnet, and J. F. Semblat, Geophys. J. Int. 177, 509 (2009).
[CrossRef]

2008 (1)

N. A. Gumerov and R. Duraiswami, J. Comput. Phys. 227, 8290 (2008).
[CrossRef]

2006 (3)

V. Y. Soloviev, Med. Phys. 33, 4176 (2006).
[CrossRef] [PubMed]

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
[CrossRef] [PubMed]

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, Inverse Probl. 22, 1509 (2006).
[CrossRef]

2005 (1)

A. Tizzard, L. Horesh, R. J. Yerworth, D. S. Holder, and R. H. Bayford, Physiol. Meas. 26, S251 (2005).
[CrossRef] [PubMed]

2001 (3)

E. Darve, SIAM J. Numer. Anal. 38, 98 (2001).
[CrossRef]

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, Phys. Rev. E 64, 051917 (2001).
[CrossRef]

N. Geng, A. Sullivan, and L. Carin, IEEE Trans. Antennas Propagat. 49, 740 (2001).
[CrossRef]

2000 (1)

E. Darve, J. Comput. Phys. 160, 195 (2000).
[CrossRef]

1999 (1)

V. I. Lebedev and D. N. Laikov, Doklady Mathematics 59, 477 (1999).

1993 (2)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. DelpyMed. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

R. Coifman, V. Rokhlin, and S. Wandzura, IEEE Antennas Propag. Mag. 35, 7 (1993).
[CrossRef]

1991 (1)

K. E. Schmidt and M. A. Lee, J. Stat. Phys. 63, 1223 (1991).
[CrossRef]

1987 (1)

L. Greengard and V. Rokhlin, J. Comput. Phys. 73, 325(1987).
[CrossRef]

Abramovitz, M.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Arridge, S.

J. Elisee, A. Gibson, and S. Arridge, Biomed. Opt. Express 2, 568 (2011).
[CrossRef] [PubMed]

J. Elisee, A. Gibson, and S. Arridge, IEEE Trans. Biomed. Eng. 57, 2737 (2010).
[CrossRef]

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
[CrossRef] [PubMed]

Arridge, S. R.

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, Inverse Probl. 22, 1509 (2006).
[CrossRef]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. DelpyMed. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Bayford, R. H.

A. Tizzard, L. Horesh, R. J. Yerworth, D. S. Holder, and R. H. Bayford, Physiol. Meas. 26, S251 (2005).
[CrossRef] [PubMed]

Bonnet, M.

S. Chaillat, M. Bonnet, and J. F. Semblat, Geophys. J. Int. 177, 509 (2009).
[CrossRef]

Carin, L.

N. Geng, A. Sullivan, and L. Carin, IEEE Trans. Antennas Propagat. 49, 740 (2001).
[CrossRef]

Carminati, R.

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, Phys. Rev. E 64, 051917 (2001).
[CrossRef]

Chaillat, S.

S. Chaillat, M. Bonnet, and J. F. Semblat, Geophys. J. Int. 177, 509 (2009).
[CrossRef]

Coifman, R.

R. Coifman, V. Rokhlin, and S. Wandzura, IEEE Antennas Propag. Mag. 35, 7 (1993).
[CrossRef]

Darve, E.

E. Darve, SIAM J. Numer. Anal. 38, 98 (2001).
[CrossRef]

E. Darve, J. Comput. Phys. 160, 195 (2000).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. DelpyMed. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Dorn, O.

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, Inverse Probl. 22, 1509 (2006).
[CrossRef]

Douiri, A.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
[CrossRef] [PubMed]

Duraiswami, R.

N. A. Gumerov and R. Duraiswami, J. Comput. Phys. 227, 8290 (2008).
[CrossRef]

Elisee, J.

J. Elisee, A. Gibson, and S. Arridge, Biomed. Opt. Express 2, 568 (2011).
[CrossRef] [PubMed]

J. Elisee, A. Gibson, and S. Arridge, IEEE Trans. Biomed. Eng. 57, 2737 (2010).
[CrossRef]

Geng, N.

N. Geng, A. Sullivan, and L. Carin, IEEE Trans. Antennas Propagat. 49, 740 (2001).
[CrossRef]

Gibson, A.

J. Elisee, A. Gibson, and S. Arridge, Biomed. Opt. Express 2, 568 (2011).
[CrossRef] [PubMed]

J. Elisee, A. Gibson, and S. Arridge, IEEE Trans. Biomed. Eng. 57, 2737 (2010).
[CrossRef]

Greengard, L.

L. Greengard and V. Rokhlin, J. Comput. Phys. 73, 325(1987).
[CrossRef]

Gumerov, N. A.

N. A. Gumerov and R. Duraiswami, J. Comput. Phys. 227, 8290 (2008).
[CrossRef]

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. DelpyMed. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Holder, D. S.

A. Tizzard, L. Horesh, R. J. Yerworth, D. S. Holder, and R. H. Bayford, Physiol. Meas. 26, S251 (2005).
[CrossRef] [PubMed]

Horesh, L.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
[CrossRef] [PubMed]

A. Tizzard, L. Horesh, R. J. Yerworth, D. S. Holder, and R. H. Bayford, Physiol. Meas. 26, S251 (2005).
[CrossRef] [PubMed]

Kolehmainen, V.

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, Inverse Probl. 22, 1509 (2006).
[CrossRef]

Laikov, D. N.

V. I. Lebedev and D. N. Laikov, Doklady Mathematics 59, 477 (1999).

Lebedev, V. I.

V. I. Lebedev and D. N. Laikov, Doklady Mathematics 59, 477 (1999).

Lee, M. A.

K. E. Schmidt and M. A. Lee, J. Stat. Phys. 63, 1223 (1991).
[CrossRef]

Nieto-Vesperinas, M.

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, Phys. Rev. E 64, 051917 (2001).
[CrossRef]

Ntziachristos, V.

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, Phys. Rev. E 64, 051917 (2001).
[CrossRef]

Ripoll, J.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
[CrossRef] [PubMed]

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, Phys. Rev. E 64, 051917 (2001).
[CrossRef]

Rokhlin, V.

R. Coifman, V. Rokhlin, and S. Wandzura, IEEE Antennas Propag. Mag. 35, 7 (1993).
[CrossRef]

L. Greengard and V. Rokhlin, J. Comput. Phys. 73, 325(1987).
[CrossRef]

Schmidt, K. E.

K. E. Schmidt and M. A. Lee, J. Stat. Phys. 63, 1223 (1991).
[CrossRef]

Schweiger, M.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
[CrossRef] [PubMed]

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. DelpyMed. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

Semblat, J. F.

S. Chaillat, M. Bonnet, and J. F. Semblat, Geophys. J. Int. 177, 509 (2009).
[CrossRef]

Sikora, J.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
[CrossRef] [PubMed]

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, Inverse Probl. 22, 1509 (2006).
[CrossRef]

Soloviev, V. Y.

V. Y. Soloviev, Med. Phys. 33, 4176 (2006).
[CrossRef] [PubMed]

Stegun, I. A.

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

Sullivan, A.

N. Geng, A. Sullivan, and L. Carin, IEEE Trans. Antennas Propagat. 49, 740 (2001).
[CrossRef]

Tizzard, A.

A. Tizzard, L. Horesh, R. J. Yerworth, D. S. Holder, and R. H. Bayford, Physiol. Meas. 26, S251 (2005).
[CrossRef] [PubMed]

Wandzura, S.

R. Coifman, V. Rokhlin, and S. Wandzura, IEEE Antennas Propag. Mag. 35, 7 (1993).
[CrossRef]

Yerworth, R. J.

A. Tizzard, L. Horesh, R. J. Yerworth, D. S. Holder, and R. H. Bayford, Physiol. Meas. 26, S251 (2005).
[CrossRef] [PubMed]

Zacharopoulos, A.

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
[CrossRef] [PubMed]

Zacharopoulos, A. D.

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, Inverse Probl. 22, 1509 (2006).
[CrossRef]

Biomed. Opt. Express (1)

Doklady Mathematics (1)

V. I. Lebedev and D. N. Laikov, Doklady Mathematics 59, 477 (1999).

Geophys. J. Int. (1)

S. Chaillat, M. Bonnet, and J. F. Semblat, Geophys. J. Int. 177, 509 (2009).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

R. Coifman, V. Rokhlin, and S. Wandzura, IEEE Antennas Propag. Mag. 35, 7 (1993).
[CrossRef]

IEEE Trans. Antennas Propagat. (1)

N. Geng, A. Sullivan, and L. Carin, IEEE Trans. Antennas Propagat. 49, 740 (2001).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

J. Elisee, A. Gibson, and S. Arridge, IEEE Trans. Biomed. Eng. 57, 2737 (2010).
[CrossRef]

Inverse Probl. (1)

A. D. Zacharopoulos, S. R. Arridge, O. Dorn, V. Kolehmainen, and J. Sikora, Inverse Probl. 22, 1509 (2006).
[CrossRef]

J. Comput. Phys. (3)

L. Greengard and V. Rokhlin, J. Comput. Phys. 73, 325(1987).
[CrossRef]

N. A. Gumerov and R. Duraiswami, J. Comput. Phys. 227, 8290 (2008).
[CrossRef]

E. Darve, J. Comput. Phys. 160, 195 (2000).
[CrossRef]

J. Stat. Phys. (1)

K. E. Schmidt and M. A. Lee, J. Stat. Phys. 63, 1223 (1991).
[CrossRef]

Med. Phys. (2)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. DelpyMed. Phys. 20, 299 (1993).
[CrossRef] [PubMed]

V. Y. Soloviev, Med. Phys. 33, 4176 (2006).
[CrossRef] [PubMed]

Phys. Med. Biol. (1)

J. Sikora, A. Zacharopoulos, A. Douiri, M. Schweiger, L. Horesh, S. Arridge, and J. Ripoll, Phys. Med. Biol. 51, 497 (2006).
[CrossRef] [PubMed]

Phys. Rev. E (1)

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, Phys. Rev. E 64, 051917 (2001).
[CrossRef]

Physiol. Meas. (1)

A. Tizzard, L. Horesh, R. J. Yerworth, D. S. Holder, and R. H. Bayford, Physiol. Meas. 26, S251 (2005).
[CrossRef] [PubMed]

SIAM J. Numer. Anal. (1)

E. Darve, SIAM J. Numer. Anal. 38, 98 (2001).
[CrossRef]

Other (1)

M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1970).

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Figures (3)

Fig. 1
Fig. 1

Single-level FMM for a 2D domain. (a) Interactions close to cell 1 marked in red, far marked in green (e.g. cell 2); (b) matrix-vector multiplication scheme between cell 1 centered at r 0 and cell 2 centered at r 0 .

Fig. 2
Fig. 2

Computation time as a function of number of degrees of freedom for the FMM (blue) and BEM (red).

Fig. 3
Fig. 3

Logarithm of light intensity on cortical surface computed from a three-layer model.

Equations (13)

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Ω ν G ( r , r ) ϕ ( r ) d S ( r ) 1 D Ω G ( r r ) q ( r ) d S ( r ) + α ( r ) ϕ ( r ) = Ω Q ( r ) G ( r r ) d V ( r ) ,
G ( R ) = exp ( ι κ R ) / ( 4 π R ) ( with   R = R ) ,
κ 2 + D 1 [ μ a + ι ω / c ] = 0 , D = [ 3 ( μ a + μ s ) ] 1 ,
A ϕ B q Q = 0.
G ( r r ) 1 4 π p = 1 P w p exp ( ι κ s p · r ˜ ) T p exp ( ι κ s p · r ˜ )
T p = 1 4 π = 0 L κ ( ι ) + 1 ( 2 + 1 ) h ( 2 ) ( κ R 0 ) P ( R ^ 0 · s p )
R R 0 / R 0 2 / 5 .
V m n A ( s p ) = ι κ τ n exp ( ι κ s p · r ˜ ) s p · ν ( r ) d S ( r ˜ ) , V m n B ( s p ) = 1 D τ n exp ( ι κ s p · r ˜ ) d S ( r ˜ ) .
C m n ( s p ) = exp ( ι κ s p · r ˜ n ) .
S m ( s p ) = V m A ( s p ) ϕ m V m B ( s p ) q m .
G m ( s p ) = m T m m ( s p ) S m ( s p ) .
( A ϕ B q ) = ( A 0 ϕ B 0 q ) + m = 1 M p = 1 P G m ( s p ) C m ( s p ) .
L = | κ a 3 + ln ( π + κ a 3 ) | + 6.

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