Abstract

We discuss the inverse problem associated with the propagation of the field autocorrelation of light through a highly scattering object like tissue. In the first part of the work, we reconstruct the optical absorption coefficient μa and particle diffusion coefficient DB from simulated measurements which are integrals of a quantity computed from the measured intensity and intensity autocorrelation g2(τ) at the boundary. In the second part we recover the mean square displacement (MSD) distribution of particles in an inhomogeneous object from the sampled g2(τ) measured on the boundary. From the MSD, we compute the storage and loss moduli distributions in the object. We have devised computationally easy methods to construct the sensitivity matrices which are used in the iterative reconstruction algorithms for recovering these parameters from the measurements. The results of the reconstruction of μa, DB, MSD and the viscoelastic parameters, which are presented, show reasonably good position and quantitative accuracy.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef]
  3. G. Maret and D. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B: Condens. Matter 65, 409-413 (1987).
    [CrossRef]
  4. D. A. Weitz, D. J. Pine, P. N. Puxy, and R. J. A. Tough, “Non-diffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747-1750 (1989).
    [CrossRef] [PubMed]
  5. D. Bicuot and G. Maret, “Multiple light scattering in Taylor-Couette flow,” Physica A 210, 87-112 (1994).
    [CrossRef]
  6. W. Leutz and G. Maret, “Ultrasound modulation of multiply scattered light,” Physica B 204, 14-19 (1995).
    [CrossRef]
  7. G. Yao and L. V. Wang, “Theoretical and experimental studies in ultrasound modulated optical tomography-biological tissues,” Appl. Opt. 39, 659-664 (2000).
    [CrossRef]
  8. A. P. Gibson, J. C. Hebden, and S. R. Arridge, “Recent advances in diffuse optical imaging,” Phys. Med. Biol. 50, R1-R43 (2005).
    [CrossRef] [PubMed]
  9. D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
    [CrossRef]
  10. M. Heckmeier, S. E. Skipetrov, G. Maret, and R. Maynard, “Imaging of dynamic heterogeneities in multiple-scattering media,” J. Opt. Soc. Am. A 14, 185-191 (1997).
    [CrossRef]
  11. D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
    [CrossRef] [PubMed]
  12. D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192-215 (1997).
    [CrossRef]
  13. M. Heckmeier and G. Maret, “Visualization of flow in multiple scattering liquids,” EPL 34, 257-262 (1996).
    [CrossRef]
  14. T. G. Mason and D. A. Weitz, “Optical measurements of the linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250-1253 (1995).
    [CrossRef] [PubMed]
  15. C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu, and A. K. Sood, “Measurement of visco-elastic properties of breast-tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035(1-5) (2007).
    [CrossRef]
  16. T. Gisler and D. A. Weitz, “Tracer microrheology in complex fluids,” Curr. Opin. Colloid Interface Sci. 3, 586-592 (1998).
    [CrossRef]
  17. C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh, and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
    [CrossRef]
  18. C. Cheung, J. P. Culver, K. Takahashi, J. H. Greenberg, and A. G. Yodh, “In vivo cerbrovascular measurement combining diffuse near-infrared absorption and correlation spectroscopies,” Phys. Med. Biol. 46, 2053-2065 (2001).
    [CrossRef] [PubMed]
  19. E. Gratton, V. Toronov, U. Wolf, M. Wolf, and A. Webb, “Measurememt of brain activity by near-infrared light,” J. Biomed. Opt. 10, 011008-011013 (2005).
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  20. T. Durduran, R. Choe, G. Yu, C. Zhou, J. C. Tehou, B. J. Czerniecki, and A. G. Yodh, “Diffuse optical measurement of blood flow in breast tumors,” Opt. Lett. 30, 2915-2917 (2005).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  22. T. G. Mason, “Estimating the visco-elastic moduli of complex fluids using the generalized Stokes-Einstein equation,” Rheol. Acta 39, 371-378 (2000).
    [CrossRef]
  23. S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. 96, 163902 (2006).
    [CrossRef] [PubMed]
  24. S. Sakadzic and L. V. Wang, “Correlation transfer equation for multiply scattered light modulated by an ultrasonic pulse,” J. Opt. Soc. Am. A 24, 2797-2806 (2007).
    [CrossRef]
  25. M. Bellour, M. Skouri, J.-P. Munch, and P. Hebraud,“Brownian motion of particles embedded in a solution of giant micelles,” Eur. Phys. J. E 8, 431-436 (2002).
  26. S. R. Arridge, “Topical review: optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
    [CrossRef]
  27. H. M. Varma, R. M. Vasu, and A. K. Nandakumaran, “Direct reconstruction of complex refractive index distribution from boundary measurement of intensity and normal derivative of intensity,” J. Opt. Soc. Am. A 24, 3089-3099 (2007).
    [CrossRef]
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    [CrossRef] [PubMed]
  31. F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
    [CrossRef]
  32. It is assumed that the detector is fast enough to respond to the decay of g2(m,τ). This can pose difficulties in the case where either the source-detector separation and/or the optical properties and MSD are so large that g2(m,τ) decays very fast. For the simulations we did, corresponding to the average properties and size of human breast, boundary correlation decay is slow enough for measurement.

2007

2006

S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. 96, 163902 (2006).
[CrossRef] [PubMed]

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh, and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

2005

E. Gratton, V. Toronov, U. Wolf, M. Wolf, and A. Webb, “Measurememt of brain activity by near-infrared light,” J. Biomed. Opt. 10, 011008-011013 (2005).
[CrossRef]

T. Durduran, R. Choe, G. Yu, C. Zhou, J. C. Tehou, B. J. Czerniecki, and A. G. Yodh, “Diffuse optical measurement of blood flow in breast tumors,” Opt. Lett. 30, 2915-2917 (2005).
[CrossRef] [PubMed]

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

2002

M. Bellour, M. Skouri, J.-P. Munch, and P. Hebraud,“Brownian motion of particles embedded in a solution of giant micelles,” Eur. Phys. J. E 8, 431-436 (2002).

2001

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

C. Cheung, J. P. Culver, K. Takahashi, J. H. Greenberg, and A. G. Yodh, “In vivo cerbrovascular measurement combining diffuse near-infrared absorption and correlation spectroscopies,” Phys. Med. Biol. 46, 2053-2065 (2001).
[CrossRef] [PubMed]

2000

G. Yao and L. V. Wang, “Theoretical and experimental studies in ultrasound modulated optical tomography-biological tissues,” Appl. Opt. 39, 659-664 (2000).
[CrossRef]

T. G. Mason, “Estimating the visco-elastic moduli of complex fluids using the generalized Stokes-Einstein equation,” Rheol. Acta 39, 371-378 (2000).
[CrossRef]

1999

S. R. Arridge, “Topical review: optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction sceheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

1998

T. Gisler and D. A. Weitz, “Tracer microrheology in complex fluids,” Curr. Opin. Colloid Interface Sci. 3, 586-592 (1998).
[CrossRef]

1997

1996

M. Heckmeier and G. Maret, “Visualization of flow in multiple scattering liquids,” EPL 34, 257-262 (1996).
[CrossRef]

1995

T. G. Mason and D. A. Weitz, “Optical measurements of the linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250-1253 (1995).
[CrossRef] [PubMed]

W. Leutz and G. Maret, “Ultrasound modulation of multiply scattered light,” Physica B 204, 14-19 (1995).
[CrossRef]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

1994

D. Bicuot and G. Maret, “Multiple light scattering in Taylor-Couette flow,” Physica A 210, 87-112 (1994).
[CrossRef]

1989

D. A. Weitz, D. J. Pine, P. N. Puxy, and R. J. A. Tough, “Non-diffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747-1750 (1989).
[CrossRef] [PubMed]

1988

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1-5 (1988).
[CrossRef]

1987

G. Maret and D. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

Arridge, S. R.

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

S. R. Arridge, “Topical review: optical tomography in medical imaging,” Inverse Probl. 15, R41-R93 (1999).
[CrossRef]

Bellour, M.

M. Bellour, M. Skouri, J.-P. Munch, and P. Hebraud,“Brownian motion of particles embedded in a solution of giant micelles,” Eur. Phys. J. E 8, 431-436 (2002).

Bharat Chandran, R. S.

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu, and A. K. Sood, “Measurement of visco-elastic properties of breast-tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035(1-5) (2007).
[CrossRef]

Bicuot, D.

D. Bicuot and G. Maret, “Multiple light scattering in Taylor-Couette flow,” Physica A 210, 87-112 (1994).
[CrossRef]

Boas, D. A.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

D. A. Boas and A. G. Yodh, “Spatially varying dynamical properties of turbid media probed with diffusing temporal light correlation,” J. Opt. Soc. Am. A 14, 192-215 (1997).
[CrossRef]

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

Brooks, D. H.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Campbell, L. E.

D. A. Boas, L. E. Campbell, and A. G. Yodh, “Scattering and imaging with diffuse temporal field correlation,” Phys. Rev. Lett. 75, 1855-1858 (1995).
[CrossRef] [PubMed]

Chaikin, P. M.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Cheung, C.

C. Cheung, J. P. Culver, K. Takahashi, J. H. Greenberg, and A. G. Yodh, “In vivo cerbrovascular measurement combining diffuse near-infrared absorption and correlation spectroscopies,” Phys. Med. Biol. 46, 2053-2065 (2001).
[CrossRef] [PubMed]

Choe, R.

Culver, J. P.

C. Cheung, J. P. Culver, K. Takahashi, J. H. Greenberg, and A. G. Yodh, “In vivo cerbrovascular measurement combining diffuse near-infrared absorption and correlation spectroscopies,” Phys. Med. Biol. 46, 2053-2065 (2001).
[CrossRef] [PubMed]

Czerniecki, B. J.

Dietsche, G.

Dimarzio, C. A.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Durduran, T.

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh, and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

T. Durduran, R. Choe, G. Yu, C. Zhou, J. C. Tehou, B. J. Czerniecki, and A. G. Yodh, “Diffuse optical measurement of blood flow in breast tumors,” Opt. Lett. 30, 2915-2917 (2005).
[CrossRef] [PubMed]

Elbert, T.

Furuya, D.

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh, and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

Gaudette, R. J.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Gibson, A. P.

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

Gisler, T.

Gratton, E.

E. Gratton, V. Toronov, U. Wolf, M. Wolf, and A. Webb, “Measurememt of brain activity by near-infrared light,” J. Biomed. Opt. 10, 011008-011013 (2005).
[CrossRef]

Greenberg, J. H.

C. Zhou, G. Yu, D. Furuya, J. H. Greenberg, A. G. Yodh, and T. Durduran, “Diffuse correlation tomography of cerebral blood flow during cortical spreading depression in rat brain,” Opt. Express 3, 1125-1144 (2006).
[CrossRef]

C. Cheung, J. P. Culver, K. Takahashi, J. H. Greenberg, and A. G. Yodh, “In vivo cerbrovascular measurement combining diffuse near-infrared absorption and correlation spectroscopies,” Phys. Med. Biol. 46, 2053-2065 (2001).
[CrossRef] [PubMed]

Hanson, K. M.

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction sceheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

Hebden, J. C.

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

Hebraud, P.

M. Bellour, M. Skouri, J.-P. Munch, and P. Hebraud,“Brownian motion of particles embedded in a solution of giant micelles,” Eur. Phys. J. E 8, 431-436 (2002).

Heckmeier, M.

Herbolzheimer, E.

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Hielscher, A. H.

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction sceheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

Jaillon, F.

Kilmer, M.

D. A. Boas, D. H. Brooks, C. A. Dimarzio, M. Kilmer, R. J. Gaudette, and Q. Zhang, “Imaging the body with diffuse optical tomography,” IEEE Signal Process. Mag. 18, 57-75 (2001).
[CrossRef]

Klose, A. D.

A. H. Hielscher, A. D. Klose, and K. M. Hanson, “Gradient-based iterative image reconstruction sceheme for time-resolved optical tomography,” IEEE Trans. Med. Imaging 18, 262-271 (1999).
[CrossRef] [PubMed]

Leutz, W.

W. Leutz and G. Maret, “Ultrasound modulation of multiply scattered light,” Physica B 204, 14-19 (1995).
[CrossRef]

Li, J.

Maret, G.

M. Heckmeier, S. E. Skipetrov, G. Maret, and R. Maynard, “Imaging of dynamic heterogeneities in multiple-scattering media,” J. Opt. Soc. Am. A 14, 185-191 (1997).
[CrossRef]

M. Heckmeier and G. Maret, “Visualization of flow in multiple scattering liquids,” EPL 34, 257-262 (1996).
[CrossRef]

W. Leutz and G. Maret, “Ultrasound modulation of multiply scattered light,” Physica B 204, 14-19 (1995).
[CrossRef]

D. Bicuot and G. Maret, “Multiple light scattering in Taylor-Couette flow,” Physica A 210, 87-112 (1994).
[CrossRef]

G. Maret and D. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

Mason, T. G.

T. G. Mason, “Estimating the visco-elastic moduli of complex fluids using the generalized Stokes-Einstein equation,” Rheol. Acta 39, 371-378 (2000).
[CrossRef]

T. G. Mason and D. A. Weitz, “Optical measurements of the linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250-1253 (1995).
[CrossRef] [PubMed]

Maynard, R.

Munch, J.-P.

M. Bellour, M. Skouri, J.-P. Munch, and P. Hebraud,“Brownian motion of particles embedded in a solution of giant micelles,” Eur. Phys. J. E 8, 431-436 (2002).

Nandakumaran, A. K.

Ninck, M.

Ortolf, C.

Pine, D. J.

D. A. Weitz, D. J. Pine, P. N. Puxy, and R. J. A. Tough, “Non-diffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747-1750 (1989).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Puxy, P. N.

D. A. Weitz, D. J. Pine, P. N. Puxy, and R. J. A. Tough, “Non-diffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747-1750 (1989).
[CrossRef] [PubMed]

Romer, S.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

Sakadzic, S.

S. Sakadzic and L. V. Wang, “Correlation transfer equation for multiply scattered light modulated by an ultrasonic pulse,” J. Opt. Soc. Am. A 24, 2797-2806 (2007).
[CrossRef]

S. Sakadzic and L. V. Wang, “Correlation transfer and diffusion of ultrasound-modulated multiply scattered light,” Phys. Rev. Lett. 96, 163902 (2006).
[CrossRef] [PubMed]

Scheffold, F.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

Schurtenberger, P.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

Skipetrov, S. E.

F. Scheffold, S. E. Skipetrov, S. Romer, and P. Schurtenberger, “Diffusing wave spectroscopy of nonergodic media,” Phys. Rev. E 63, 061404 (2001).
[CrossRef]

M. Heckmeier, S. E. Skipetrov, G. Maret, and R. Maynard, “Imaging of dynamic heterogeneities in multiple-scattering media,” J. Opt. Soc. Am. A 14, 185-191 (1997).
[CrossRef]

Skouri, M.

M. Bellour, M. Skouri, J.-P. Munch, and P. Hebraud,“Brownian motion of particles embedded in a solution of giant micelles,” Eur. Phys. J. E 8, 431-436 (2002).

Sood, A. K.

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu, and A. K. Sood, “Measurement of visco-elastic properties of breast-tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035(1-5) (2007).
[CrossRef]

Stephen, M. J.

M. J. Stephen, “Temporal fluctuations in wave propagation in random media,” Phys. Rev. B 37, 1-5 (1988).
[CrossRef]

Takahashi, K.

C. Cheung, J. P. Culver, K. Takahashi, J. H. Greenberg, and A. G. Yodh, “In vivo cerbrovascular measurement combining diffuse near-infrared absorption and correlation spectroscopies,” Phys. Med. Biol. 46, 2053-2065 (2001).
[CrossRef] [PubMed]

Tehou, J. C.

Toronov, V.

E. Gratton, V. Toronov, U. Wolf, M. Wolf, and A. Webb, “Measurememt of brain activity by near-infrared light,” J. Biomed. Opt. 10, 011008-011013 (2005).
[CrossRef]

Tough, R. J. A.

D. A. Weitz, D. J. Pine, P. N. Puxy, and R. J. A. Tough, “Non-diffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747-1750 (1989).
[CrossRef] [PubMed]

Usha Devi, C.

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu, and A. K. Sood, “Measurement of visco-elastic properties of breast-tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035(1-5) (2007).
[CrossRef]

Varma, H. M.

Vasu, R. M.

H. M. Varma, R. M. Vasu, and A. K. Nandakumaran, “Direct reconstruction of complex refractive index distribution from boundary measurement of intensity and normal derivative of intensity,” J. Opt. Soc. Am. A 24, 3089-3099 (2007).
[CrossRef]

C. Usha Devi, R. S. Bharat Chandran, R. M. Vasu, and A. K. Sood, “Measurement of visco-elastic properties of breast-tissue mimicking materials using diffusing wave spectroscopy,” J. Biomed. Opt. 12, 034035(1-5) (2007).
[CrossRef]

Wang, L. V.

Webb, A.

E. Gratton, V. Toronov, U. Wolf, M. Wolf, and A. Webb, “Measurememt of brain activity by near-infrared light,” J. Biomed. Opt. 10, 011008-011013 (2005).
[CrossRef]

Weitz, D. A.

T. Gisler and D. A. Weitz, “Tracer microrheology in complex fluids,” Curr. Opin. Colloid Interface Sci. 3, 586-592 (1998).
[CrossRef]

T. G. Mason and D. A. Weitz, “Optical measurements of the linear viscoelastic moduli of complex fluids,” Phys. Rev. Lett. 74, 1250-1253 (1995).
[CrossRef] [PubMed]

D. A. Weitz, D. J. Pine, P. N. Puxy, and R. J. A. Tough, “Non-diffusive Brownian motion studied by diffusing-wave spectroscopy,” Phys. Rev. Lett. 63, 1747-1750 (1989).
[CrossRef] [PubMed]

D. J. Pine, D. A. Weitz, P. M. Chaikin, and E. Herbolzheimer, “Diffusing-wave spectroscopy,” Phys. Rev. Lett. 60, 1134-1137 (1988).
[CrossRef] [PubMed]

Wolf, D. E.

G. Maret and D. E. Wolf, “Multiple light scattering from disordered media. The effect of Brownian motion of scatterers,” Z. Phys. B: Condens. Matter 65, 409-413 (1987).
[CrossRef]

Wolf, M.

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It is assumed that the detector is fast enough to respond to the decay of g2(m,τ). This can pose difficulties in the case where either the source-detector separation and/or the optical properties and MSD are so large that g2(m,τ) decays very fast. For the simulations we did, corresponding to the average properties and size of human breast, boundary correlation decay is slow enough for measurement.

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Figures (12)

Fig. 1
Fig. 1

Typical plot of the modulus of field autocorrelation with τ at a boundary node for a homogeneous object: μ a b = 0.001 cm 1 , μ s b = 8 cm 1 , and D B = 0.1 × 10 8 cm 2 s .

Fig. 2
Fig. 2

Plots of Γ ( r , τ ) versus τ when: (a) background μ a is varied from (1) 0.001 cm 1 to (2) 0.005 cm 1 , (3) 0.01 cm 1 , D B remaining constant at 0.1 × 10 8 cm 2 s ; (b) background D B is varied from (1) 0.1 × 10 8 cm 2 s to (2) 0.5 × 10 8 cm 2 s , (3) 1 × 10 8 cm 2 s , μ a remaining constant at 0.001 cm 1 . The source–detector separation used in the simulation was 8 cm .

Fig. 3
Fig. 3

Variations of M 1 and M 2 with μ a and D B . (a) M 1 versus μ a , (b) M 1 versus D B , (c) M 2 versus μ a , (d) M 2 versus D B . It is seen that M 1 and M 2 have larger variations with respect to μ a and D B , respectively

Fig. 4
Fig. 4

Iterative reconstruction algorithm: The inputs to the algorithm are the initial guess of the property p 0 (either μ a or D B ) and the experimental measurement M j e (either M 1 e or M 2 e ). The algorithm has an outer and inner loop. In the inner loop the perturbation equation is solved to update the property p i . In the outer loop the perturbation equation is itself updated.

Fig. 5
Fig. 5

Original object used in the simulations: (a) absorption coefficient ( cm 1 ) distribution; (b) particle diffusion coefficient distribution ( cm 2 sec ) .

Fig. 6
Fig. 6

Reconstructed absorption coefficient distribution ( cm 1 ) (a) gray-level plot; (b) cross-sectional plots through the centers of the inhomogeneities in (a) as well as the original inhomogeneous object of Fig. 6a.

Fig. 7
Fig. 7

Reconstructed particle diffusion coefficient distribution ( cm 2 sec ) (a) gray-level plot; (b) cross-sectional plots through the centers of the inhomogeneities in (a) (dashed curve) as well as the original inhomogeneous object of Fig. 6b (solid curve).

Fig. 8
Fig. 8

These reconstructions give the cross-talk in the reconstructions and are seen to be less than 1% of the correctly recovered D B and μ a . (a) The cross-sectional plot through the recovered change in absorption coefficient ( Δ μ a = μ a ( r ) μ a b ) from measurement M 2 using the Jacobian { M 2 μ a } .(b) The cross-sectional plot through the recovered change in particle diffusion coefficient [ Δ D B = D B ( r ) D B b ] from measurement M 1 using the Jacobian { M 1 D B } .

Fig. 9
Fig. 9

Typical variation of Δ r 2 ( r , τ ) with τ for a homogeneous object ( r 0 = 7.0711 × 10 7 cm , D B = 10 9 cm 2 s , D 1 = 10 12 cm 2 s , μ a = 0.01 cm 1 , μ s = 8 cm 1 ).

Fig. 10
Fig. 10

(a) Gray-level plot of reconstructed Δ r 2 ( r , τ ) ( cm 2 ) at τ = 6 × 10 5 s . (b) Cross-sectional plots through the centers of the inhomogeneities for the original (solid curve) and the reconstructed (dotted curves) Δ r 2 ( r , τ ) . The inhomogeneity at the left is designated as inhomogeneity 1 and the one at the right as inhomogeneity 2.

Fig. 11
Fig. 11

Reconstructed and original Δ r 2 ( r , τ ) versus τ for the inhomogeneities 1 [(1) and (2), respectively] and 2 [(3) and (4), respectively].

Fig. 12
Fig. 12

Reconstructed G (dotted curve) and G (solid curve) for (a) inhomogeneity 1, (b) inhomogeneity 2.

Equations (42)

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s ̂ I ( r , s ̂ , τ ) = μ t ( r ) I ( r , s ̂ , τ ) + μ s ( r ) I ( r , s ̂ , τ ) g 1 s ( s ̂ , s ̂ , τ ) d s ̂ + S ( r , s ̂ ) .
D ( r ) G ( r , τ ) ( μ a ( r ) + 1 3 Δ r 2 ( r , τ ) k 0 2 μ s ( r ) ) G ( r , τ ) = S 0 ( r r 0 ) ,
D ( r ) G ( r , τ ) n = G ( r , τ ) ,
Δ r 2 ( r , τ ) = r 0 2 ( r ) { 1 exp [ ( τ τ d ) α ] } 1 α ( 1 + 6 D 1 τ r 0 2 ) ,
I ( r , τ ) I ( r , t + τ ) g 2 ( τ ) = 1 + β g 1 ( r , τ ) 2 .
M { G ( m , τ ) } G 2 ( m , 0 ) ( M 1 β ) Γ ( m , τ ) .
M = τ 1 τ 2 Γ ( m , τ ) d τ ,
D G δ ( r , τ ) ( μ a + 2 D B τ k 0 2 μ s ) G δ ( r , τ ) = μ a δ G ( r , τ ) .
G δ ( m , τ ) + D G δ ( m , τ ) n = 0 , m Ω .
D ψ ( r , τ ) ( μ a + 2 D B τ k 0 2 μ s ) ψ ( r , τ ) = 0 ,
ψ ( r , τ ) + D ψ ( r , τ ) n = q + .
Ω G δ ( m , τ ) q + d n 1 r = Ω μ a δ ( r ) G ( r , τ ) ψ ( r , τ ) d n r .
Γ δ ( m , τ ) = [ G ( m , τ ) + G δ ( m , τ ) ] [ G ¯ ( m , τ ) + G ¯ δ ( m , τ ) ] G ( m , τ ) G ¯ ( m , τ ) G δ ( m , τ ) G ¯ ( m , τ ) + G ( m , τ ) G ¯ δ ( m , τ ) = Γ 1 δ ( m , τ ) + Γ 2 δ ( m , τ ) ,
Ω Γ 1 δ ( m , τ ) G ¯ ( m , τ ) q + d n 1 r = Ω μ a δ ( r ) G ( r , τ ) ψ ( r , τ ) d n r .
Ω Γ 2 δ ( m , τ ) G ( m , τ ) q + d n 1 r = Ω μ a δ ( r ) G ¯ ( r , τ ) ψ ¯ ( r , τ ) d n r .
Γ δ ( m i , τ ) = Re { G ¯ ( m i , τ ) Ω μ a δ ( r ) G ( r , τ ) G R ψ ( r , m i , τ ) d n r } ,
Γ δ ( m i , τ ) = Re { G ¯ ( m i , τ ) Ω μ a δ ( r ) G ( r , τ ) G Γ ( m i , r , τ ) d n r } .
G R ψ ( r , m i , τ ) = G Γ ( m i , r , τ ) .
Γ ( m i , τ ) μ a δ ( r ) = Re { G ¯ ( m i , τ ) G ( r , τ ) G R ψ ( r , m i , τ ) } .
D G δ ( r , τ ) ( μ a + 2 μ s k 0 2 D B τ ) G δ ( r , τ ) = 2 μ s k 0 2 D B δ τ G ( r , τ ) ,
G δ ( r , τ ) + D G δ ( r , τ ) n = 0 .
Γ ( m i , τ ) D B δ ( r ) = Re [ 2 μ s k 0 2 τ G ¯ ( m i , τ ) G R ϕ ( r , m i , τ ) G ( r , τ ) ] .
D ϕ ( r , τ ) ( μ a + 2 μ s k 0 2 D B τ ) ϕ ( r , τ ) = 0 ,
ϕ ( r , τ ) + D ϕ ( r , τ ) n = q + .
j = 1 N Γ ( m k , τ ) p j Δ p j = E ( m k , τ ) , k = 1 , m .
j = 1 n M i p j Δ p j av = Δ M i ,
D G δ ( r , τ i ) ( μ a + 1 3 Δ r 2 ( r , τ i ) k 0 2 μ s ) G δ ( r , τ i ) = 1 3 k 0 2 μ s Δ r 2 ( r , τ i ) δ G ( r , τ i ) ,
D G δ ( r , τ i ) n + G δ ( r , τ i ) = 0 .
D ψ ( μ a + 1 3 Δ r 2 ( r , τ i ) k 0 2 μ s ) ψ = 0 ,
D ψ ( r , τ i ) n + ψ ( r , τ i ) = q + .
Γ ( m i , τ i ) Δ r 2 ( m i , τ i ) = Re { 1 3 k 0 2 μ s G ¯ ( m i , τ ) G ( r , τ i ) G R ψ ( r , m i , τ i ) } .
( J μ a T J μ a + λ 1 I ) Δ μ a = J μ a T Δ M 1 ,
( J D B T J D B + λ 2 I ) Δ D B = J D B T Δ M 2 .
( J Δ r 2 ( r , τ j ) T J Δ r 2 ( r , τ j ) + λ 3 I ) Δ Δ r 2 ( r , τ j ) = J Δ r 2 ( r , τ j ) T Δ Γ ( m , τ j ) .
μ a ( x , y ) = 0.004 cm 1 if ( x + 2.5 ) 2 + ( y ) 2 0.9 ,
D B ( x , y ) = 0.4 × 10 8 cm 2 s if ( x 2.5 ) 2 + ( y ) 2 0.9 .
Δ r 2 ( r , τ ) = r 0 2 ( r ) { 1 exp [ ( τ τ d ) α ] 1 α } ( 1 + 6 D 1 τ r 0 2 ) .
r 0 ( x , y ) = { 1.5811 × 10 7 cm if ( x + 2.5 ) 2 + ( y ) 2 0.7 5 × 10 7 cm if ( x 2.5 ) 2 + ( y ) 2 0.7 } .
τ d ( x , y ) = { 4.1667 × 10 6 s if ( x + 2.5 ) 2 + ( y ) 2 0.7 4.1667 × 10 5 s if ( x 2.5 ) 2 + ( y ) 2 0.7 } .
G ( ω ) = G * ( ω ) cos ( π α ( ω ) 2 ) ,
G ( ω ) = G * ( ω ) sin ( π α ( ω ) 2 ) ,
G * ( ω ) K B T π a Δ ( r 2 , τ ) Γ ( 1 + α ( ω ) ) .

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