Abstract

We present images of heterogeneous turbid media derived from measurements of diffuse photon-density waves traveling through highly scattering tissue phantoms. To our knowledge, the images are the first experimental reconstruction based on data collected in the frequency domain. We demonstrate images of both absorbing and scattering heterogeneities and show that this method is sensitive to the optical properties of the heterogeneity. The algorithm employs a differential measurement scheme that reduces the effect of errors resulting from incorrect estimation of the background optical properties. The relative advantages of sources with low and high modulation frequency are discussed within this context.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Tamura, O. Hazeki, S. Nioka, B. Chance, Annu. Rev. Physiol. 51, 813 (1989).
    [CrossRef] [PubMed]
  2. See related studies inR. R. Alfano, ed., Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994).
  3. See related studies inG. Muller, ed., Medical Optical Tomography: Functional Imaging and Monitoring (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), Vol. Is11, p. 31.
  4. J. B. Fishkin, E. Gratton, J. Opt. Soc. Am. A 10, 127 (1993).
    [CrossRef] [PubMed]
  5. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
    [CrossRef]
  6. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
    [CrossRef]
  7. B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, Appl. Opt. 32, 607 (1993).
    [CrossRef] [PubMed]
  8. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Proc. Natl. Acad. Sci. USA 91, 4887 (1994).
    [CrossRef] [PubMed]
  9. J. M. Schmitt, A. Knuttel, J. R. Knutson, J. Opt. Soc. Am. A 9, 1832 (1992); B. Chance, K. Kang, L. He, J. Weng, E. Sevick, Proc. Natl. Acad. Sci. USA 90, 3423 (1993).
    [CrossRef] [PubMed]
  10. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994).
    [CrossRef]
  11. We are aware of a prior unpublished study byJ. Schotland, J. S. Leigh, M. Ishii, C. P. Gonatas (Department of Radiology, University of Pennsylvania, Philadelphia, Pa.) on the inverse problem with diffusing photons. In particular, these authors present a method for direct inversion of the data whose kernel, which is calculated within a one-loop diagrammatic approximation, is accurate to all orders in variation of μa(r) and D(r).
  12. The condition that δμa ≪ μa and δμs′ ≪ μs′ is sufficient to reduce each solution to an integral equation linear in δμa and δμ;s′.
  13. S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204 (1991).
  14. In the Rytov formulation15 the heterogeneous photon-density wave is expressed as the homogeneous wave with a complex phase shift (ϕ) owing to the perturbation, i.e., U = Uo exp(ϕ) and ϕ=[1/Uo(rs, rd)] ∫V[− ;δμa (r) vDo−1U0(rs, r)G(r, rd) + δD(r)/Do ∇U0 (rs, r) ∇G(r, rd)]d3r. In general, the first Rytov approximation is valid when (∇ϕ)2 ≪ vδμa/D.
  15. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, p. 211.
  16. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.The singular values ωi are put through the following filter: ωi → ωi + σ/ωi, where the amount of smoothing, σ, is a free parameter.
  17. To incorporate two sources, one simply replaces U0(rs, r) by U0(rs1, r) − U0(rs2, r) in Eq. (2).
  18. M. Firbank, M. Hiraoka, D. Delpy, Phys. Med. Biol. 38, 847 (1993).
    [CrossRef]

1994 (2)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Proc. Natl. Acad. Sci. USA 91, 4887 (1994).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994).
[CrossRef]

1993 (4)

M. Firbank, M. Hiraoka, D. Delpy, Phys. Med. Biol. 38, 847 (1993).
[CrossRef]

J. B. Fishkin, E. Gratton, J. Opt. Soc. Am. A 10, 127 (1993).
[CrossRef] [PubMed]

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, Appl. Opt. 32, 607 (1993).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

1992 (2)

1991 (1)

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204 (1991).

1989 (1)

M. Tamura, O. Hazeki, S. Nioka, B. Chance, Annu. Rev. Physiol. 51, 813 (1989).
[CrossRef] [PubMed]

Arridge, S. R.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204 (1991).

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Proc. Natl. Acad. Sci. USA 91, 4887 (1994).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

Chance, B.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Proc. Natl. Acad. Sci. USA 91, 4887 (1994).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

M. Tamura, O. Hazeki, S. Nioka, B. Chance, Annu. Rev. Physiol. 51, 813 (1989).
[CrossRef] [PubMed]

Cope, M.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204 (1991).

Delpy, D.

M. Firbank, M. Hiraoka, D. Delpy, Phys. Med. Biol. 38, 847 (1993).
[CrossRef]

Delpy, D. T.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204 (1991).

Firbank, M.

M. Firbank, M. Hiraoka, D. Delpy, Phys. Med. Biol. 38, 847 (1993).
[CrossRef]

Fishkin, J. B.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.The singular values ωi are put through the following filter: ωi → ωi + σ/ωi, where the amount of smoothing, σ, is a free parameter.

Gonatas, C. P.

We are aware of a prior unpublished study byJ. Schotland, J. S. Leigh, M. Ishii, C. P. Gonatas (Department of Radiology, University of Pennsylvania, Philadelphia, Pa.) on the inverse problem with diffusing photons. In particular, these authors present a method for direct inversion of the data whose kernel, which is calculated within a one-loop diagrammatic approximation, is accurate to all orders in variation of μa(r) and D(r).

Gratton, E.

Haskell, R. C.

Hazeki, O.

M. Tamura, O. Hazeki, S. Nioka, B. Chance, Annu. Rev. Physiol. 51, 813 (1989).
[CrossRef] [PubMed]

Hiraoka, M.

M. Firbank, M. Hiraoka, D. Delpy, Phys. Med. Biol. 38, 847 (1993).
[CrossRef]

Ishii, M.

We are aware of a prior unpublished study byJ. Schotland, J. S. Leigh, M. Ishii, C. P. Gonatas (Department of Radiology, University of Pennsylvania, Philadelphia, Pa.) on the inverse problem with diffusing photons. In particular, these authors present a method for direct inversion of the data whose kernel, which is calculated within a one-loop diagrammatic approximation, is accurate to all orders in variation of μa(r) and D(r).

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, p. 211.

Knutson, J. R.

Knuttel, A.

Leigh, J. S.

We are aware of a prior unpublished study byJ. Schotland, J. S. Leigh, M. Ishii, C. P. Gonatas (Department of Radiology, University of Pennsylvania, Philadelphia, Pa.) on the inverse problem with diffusing photons. In particular, these authors present a method for direct inversion of the data whose kernel, which is calculated within a one-loop diagrammatic approximation, is accurate to all orders in variation of μa(r) and D(r).

Nioka, S.

M. Tamura, O. Hazeki, S. Nioka, B. Chance, Annu. Rev. Physiol. 51, 813 (1989).
[CrossRef] [PubMed]

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Proc. Natl. Acad. Sci. USA 91, 4887 (1994).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.The singular values ωi are put through the following filter: ωi → ωi + σ/ωi, where the amount of smoothing, σ, is a free parameter.

Schmitt, J. M.

Schotland, J.

We are aware of a prior unpublished study byJ. Schotland, J. S. Leigh, M. Ishii, C. P. Gonatas (Department of Radiology, University of Pennsylvania, Philadelphia, Pa.) on the inverse problem with diffusing photons. In particular, these authors present a method for direct inversion of the data whose kernel, which is calculated within a one-loop diagrammatic approximation, is accurate to all orders in variation of μa(r) and D(r).

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, p. 211.

Svaasand, L. O.

Tamura, M.

M. Tamura, O. Hazeki, S. Nioka, B. Chance, Annu. Rev. Physiol. 51, 813 (1989).
[CrossRef] [PubMed]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.The singular values ωi are put through the following filter: ωi → ωi + σ/ωi, where the amount of smoothing, σ, is a free parameter.

Tromberg, B. J.

Tsay, T. T.

van der Zee, P.

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204 (1991).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.The singular values ωi are put through the following filter: ωi → ωi + σ/ωi, where the amount of smoothing, σ, is a free parameter.

Yodh, A. G.

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Proc. Natl. Acad. Sci. USA 91, 4887 (1994).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

Annu. Rev. Physiol. (1)

M. Tamura, O. Hazeki, S. Nioka, B. Chance, Annu. Rev. Physiol. 51, 813 (1989).
[CrossRef] [PubMed]

Appl. Opt. (1)

J. Lumin. (1)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, J. Lumin. 60–61, 281 (1994).
[CrossRef]

J. Opt. Soc. Am. A (2)

Phys. Med. Biol. (1)

M. Firbank, M. Hiraoka, D. Delpy, Phys. Med. Biol. 38, 847 (1993).
[CrossRef]

Phys. Rev. E (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Phys. Rev. E 47, R2999 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, Phys. Rev. Lett. 69, 2658 (1992).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, Proc. Natl. Acad. Sci. USA 91, 4887 (1994).
[CrossRef] [PubMed]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

S. R. Arridge, P. van der Zee, M. Cope, D. T. Delpy, Proc. Soc. Photo-Opt. Instrum. Eng. 1431, 204 (1991).

Other (8)

In the Rytov formulation15 the heterogeneous photon-density wave is expressed as the homogeneous wave with a complex phase shift (ϕ) owing to the perturbation, i.e., U = Uo exp(ϕ) and ϕ=[1/Uo(rs, rd)] ∫V[− ;δμa (r) vDo−1U0(rs, r)G(r, rd) + δD(r)/Do ∇U0 (rs, r) ∇G(r, rd)]d3r. In general, the first Rytov approximation is valid when (∇ϕ)2 ≪ vδμa/D.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988), Chap. 6, p. 211.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C: The Art of Scientific Computing (Cambridge U. Press, New York, 1988), Chap. 2, p. 52.The singular values ωi are put through the following filter: ωi → ωi + σ/ωi, where the amount of smoothing, σ, is a free parameter.

To incorporate two sources, one simply replaces U0(rs, r) by U0(rs1, r) − U0(rs2, r) in Eq. (2).

See related studies inR. R. Alfano, ed., Advances in Optical Imaging and Photon Migration, Vol. 21 of OSA Proceedings Series (Optical Society of America, Washington, D.C., 1994).

See related studies inG. Muller, ed., Medical Optical Tomography: Functional Imaging and Monitoring (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), Vol. Is11, p. 31.

We are aware of a prior unpublished study byJ. Schotland, J. S. Leigh, M. Ishii, C. P. Gonatas (Department of Radiology, University of Pennsylvania, Philadelphia, Pa.) on the inverse problem with diffusing photons. In particular, these authors present a method for direct inversion of the data whose kernel, which is calculated within a one-loop diagrammatic approximation, is accurate to all orders in variation of μa(r) and D(r).

The condition that δμa ≪ μa and δμs′ ≪ μs′ is sufficient to reduce each solution to an integral equation linear in δμa and δμ;s′.

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

(a) Experimental geometry with the actual position of a 1.2-cm-diameter sphere and the reconstruction of (b) a perfectly absorbing sphere and (c) a highly scattering sphere. Both reconstructions were generated by use of 1000 iterations of the simultaneous iterative technique.

Fig. 2
Fig. 2

Reconstructed absorption from both the experimental data (filled circles) and the simulated data (solid curve) versus the actual absorption. In this experiment, resin spheres (1.2 cm in diameter) made with a mixture of scatterer and a known concentration of ink were imaged by matrix inversion. The error bars are derived from estimating the calibration errors that we believe to be most significant.

Fig. 3
Fig. 3

Reconstruction of two 1.0-cm-diameter perfectly absorbing spheres from experimental data, using (a) singular-value decomposition and analysis and (b) 3000 iterations of the simultaneous iterative technique. (c)–(f) Images of two perfectly absorbing spheres from simulated data, using a different sphere configuration. In (c) and (d) (background μa = 0.1 cm−1, inside the spheres μa = 0.4 cm−1) we see an increase in image quality as the modulation frequency is increased from (c) 50 MHz to (d) 1 GHz. However, when the background absorption is high (background μa = 1.0 cm−1, inside the spheres μa = 4.0 cm−1) the image quality does not noticeably improve as the modulation frequency is increased from (e) 50 MHz to (f) 1 GHz.

Equations (2)

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

U 0 ( r s , r d ) = M exp ( i k o r s - r d ) / ( 4 π D o r s - r d ) ,
U 1 ( r s , r d ) = V [ - δ μ a ( r ) v D o - 1 U 0 ( r s , r ) G ( r , r d ) + δ D ( r ) D o U 0 ( r s , r ) · G ( r , r d ) d 3 r ,

Metrics