Abstract

A new integro-differential equation for diffuse photon density waves (DPDW) is derived within the diffusion approximation. The new equation applies to inhomogeneous bounded turbid media. Interestingly, it does not contain any terms involving gradients of the light diffusion coefficient. The integro-differential equation for diffusive waves is used to develop a 3D-slice imaging algorithm based on the angular spectrum representation in the parallel plate geometry. The algorithm may be useful for near infrared optical imaging of breast tissue, and is applicable to other diagnostics such as ultrasound and microwave imaging.

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  1. B. Chance, Q. M. Luo, S. Nioka, D. C. Alsop and J. A. Detre, "Optical investigations of physiology: a study of intrinsic and extrinsic biomedical contrast," Phil. Trans. Royy. Soc. London B. 352, 707 (1997).
    [CrossRef]
  2. A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain functions," Trends. Neurosci. 20, 435 (1997).
    [CrossRef] [PubMed]
  3. Y. Hoshi and M. Tamura, "Near-Infrared Optical Detection of Sequential Brain Activation in The Prefrontal cortex during mental tasks," Neuroimage. 5, 292 (1997).
    [CrossRef] [PubMed]
  4. J. H. Hoogenraad, M. B.van der Mark, S. B.Colak, G.W.t Hooft, E.S.van der Linden, "First Results from the Philips Optical Mammoscope," Proc.SPIE / BiOS-97 (SanRemo, 1997 ).
  5. J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner and B. J. Tromberg, "Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject," Appl. Opt. 36, 10 (1997).
    [CrossRef] [PubMed]
  6. M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag and M. Kaschke. "Frequency-domain instrumentation techniques enhance optical mammography: Initial clinical results" Proc. Natl. Ac ad. Sci. USA, 94, 6468-6473 (1997).
    [CrossRef]
  7. S. K. Gayen and M. E.Zevallos, B. B. Das, R. R. Alfano, "Time-sliced transillumination imaging of normal and cancerous breast tissues," in Trends in Opt. And Photonics, ed. J. G. Fujimoto, M. S. Patterson.
  8. B. W. Pogue, M. Testorf, T. McBride, U. Osterberg and K. Paulsen ," Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection ," Opt. Express 1, 391 (December 1997). http://epubs.osa.org/oearchive/source/2827.htm
    [CrossRef] [PubMed]
  9. D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta and P. M. Schlag, "Imaging and Characterization of Breast tumors using a laser-pulse mammograph," SPIE 3597 (1999).
  10. A. G. Yodh, B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (March 1995).
    [CrossRef]
  11. H. Jiang, K. D. Paulsen, U. L. Osterberg, B. W. Pogue and M.S. Patterson, "Optical Image reconstruction using frequency-domain data: Simulations and experiments," J. Opt. Soc. Am. A 13, 253-266 (1996).
    [CrossRef]
  12. Y. Yao, Y. Wang, Y. Pei, W. Zhu and R. L. Barbour, "Frequency-domain optical imaging of absorption and scattering distributions using a born iterative mehod," J. Opt. Soc. Am. A 14, 325-342 (1997).
    [CrossRef]
  13. S. R. Arridge, "Forward and inverse problems in time-resolved infrared imaging," in Medical Optical Tomography: Functional Imaging and Monitoring, ed. G. Muller, B. Chance, Rl. Alfano, S. Arridge, J. Beuthan, E. Gratton, M Kaschke, B. Masters, S. Svanberg, P. van der Zee, Proc SPIE IS11, 35-64 (1993).
  14. An integro-differential equation valid for infinite medium,but containing no gradient of Diffusion constant was used for image reconstruction by M. A. OLeary, D. A . Boas, B. Chance, A. G. Yodh, "Experimental Images of heterogeneous turbid media by frequency-domain diffusing-photon tomotography," Opt. Lett. 20, 426-428 (1995).
    [CrossRef]
  15. An anonymous reviewer has brought to our attention a recent paper by S. R. Arridge and W. R. B. Lionheart, "Non-uniqueness in diffusion based optical tomography," Opt. Lett. 23, 882-884 (1998), which deals with the second order derivative of the diffusion coefficient.
    [CrossRef]
  16. D. N. Pattanayak and E. Wolf, " Resonance States as Solutions of the Schrodinger Equation with a Non-Local Boundary Condition," Phys. Rev. D13, 2287(1976).
  17. J. L. Ye, R. P. Millane, K. J. Webb and T. J. Downar, "Importance of the gradD term in frequency resolved optical diffusion imaging," Opt. Lett. 23, (1998).
    [CrossRef]
  18. K. M. Case and P. F. Zweifel, in Linear Transport Theory (Addison -Wesley, MA ,1967).
  19. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978) Vol. 19.
  20. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill,1953).
  21. Claus Muller, Foundations of the Mathematical theory of Electromagnetic Waves (Springer-Verlag,New York,1969).
  22. H. Weyl, Ann. Phys. 60, 481, (1919).
  23. See, also, A. J. Banos, in Dipole Radiation In the Presence of a Conducting Half-Space (Pergamon Press, New York, 1966).
  24. E. Wolf, "Principles and Development of Diffraction Tomography" in Trends in Optics, ed. A. Consortini (Academic Press, San Diego, 1996).
    [CrossRef]
  25. A. J. Devaney, "Reconstructive tomography with diffracting Wavefields," Inv. Probl. 2,161-1839 (1986).
  26. For an application of diffraction tomography to near field diffusion wave imaging and an analytical expression relating theoretical resolution and tissue thickness see, GE Class I Technical report (publicly available on request) by D. N. Pattanayak, "Resolution of Optical Images Formed by Diffusion Waves in Highly Scattering Media," GE Tech. Info. Series 91CRD241 (1991).
  27. C. L. Matson, N. Clark, L. McMackin and J. S. Fender, "Three-dimensional Tumor Localization in Thick Tissue with The Use of Diffuse Photon-Density Waves," Appl. Opt. 36, 214-219 (1997).
    [CrossRef] [PubMed]
  28. X. D. Li, T. Durduran, A. G. Yodh, B. Chance and D. N. Pattanayak, "Diffraction Tomography for Biomedical Imaging With Diffuse Photon Density Waves," Opt. Lett. 22, 573-575 (1997).
    [CrossRef] [PubMed]
  29. B. Q. Chen , J. J. Stamnes, K. Stamnes, "Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium," Pure Appl. Opt. 7, 1161-1180 (1998).
    [CrossRef]
  30. X. Cheng and D. Boas, "Diffuse Optical Reflection Tomography Using Continous Wave Illumination," Opt. Express 3, 118-123 (1998); http://epubs.osa.org/oearchive/source/5663.htm
    [CrossRef] [PubMed]
  31. S. J. Norton, T. Vo-Dinh, "Diffraction Tomographic Imaging With Photon Density Waves: an Explicit Solution, J. Opt. Soc. Am. A 15, 2670-2677 (1998).
    [CrossRef]
  32. J. C. Schotland, "Continuos Wave Diffusion Imaging," J. Opt. Soc. Am. A 14, 275-279 (1997).
    [CrossRef]
  33. J. Ripoll, M. Nieto-Vesperinas, "Reflection and Transmission Coefficients of Diffuse Photon Density Waves," (to be published).
  34. J. Ripoll and M. Nieto-Vesperinas, "Spatial Resolution of Diffuse Photon Density Waves, " J. Opt. Soc. Am.A (to be published).
  35. C. L. Matson and H. Liu, "Analysis of the forward problem with diffuse photon density waves in turbid media by use of a diffraction tomography model," J. Opt. Soc. Am. A 16, 455-466 (1999).
    [CrossRef]
  36. For the angular spectrum representation in a slab field and for an excellent account of the angular spectrum representation see, L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York,1995).

Other (36)

B. Chance, Q. M. Luo, S. Nioka, D. C. Alsop and J. A. Detre, "Optical investigations of physiology: a study of intrinsic and extrinsic biomedical contrast," Phil. Trans. Royy. Soc. London B. 352, 707 (1997).
[CrossRef]

A. Villringer and B. Chance, "Non-invasive optical spectroscopy and imaging of human brain functions," Trends. Neurosci. 20, 435 (1997).
[CrossRef] [PubMed]

Y. Hoshi and M. Tamura, "Near-Infrared Optical Detection of Sequential Brain Activation in The Prefrontal cortex during mental tasks," Neuroimage. 5, 292 (1997).
[CrossRef] [PubMed]

J. H. Hoogenraad, M. B.van der Mark, S. B.Colak, G.W.t Hooft, E.S.van der Linden, "First Results from the Philips Optical Mammoscope," Proc.SPIE / BiOS-97 (SanRemo, 1997 ).

J. B. Fishkin, O. Coquoz, E. R. Anderson, M. Brenner and B. J. Tromberg, "Frequency-domain photon migration measurements of normal and malignant tissue optical properties in a human subject," Appl. Opt. 36, 10 (1997).
[CrossRef] [PubMed]

M. A. Franceschini, K. T. Moesta, S. Fantini, G. Gaida, E. Gratton, H. Jess, W. W. Mantulin, M. Seeber, P. M. Schlag and M. Kaschke. "Frequency-domain instrumentation techniques enhance optical mammography: Initial clinical results" Proc. Natl. Ac ad. Sci. USA, 94, 6468-6473 (1997).
[CrossRef]

S. K. Gayen and M. E.Zevallos, B. B. Das, R. R. Alfano, "Time-sliced transillumination imaging of normal and cancerous breast tissues," in Trends in Opt. And Photonics, ed. J. G. Fujimoto, M. S. Patterson.

B. W. Pogue, M. Testorf, T. McBride, U. Osterberg and K. Paulsen ," Instrumentation and design of a frequency-domain diffuse optical tomography imager for breast cancer detection ," Opt. Express 1, 391 (December 1997). http://epubs.osa.org/oearchive/source/2827.htm
[CrossRef] [PubMed]

D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta and P. M. Schlag, "Imaging and Characterization of Breast tumors using a laser-pulse mammograph," SPIE 3597 (1999).

A. G. Yodh, B. Chance, "Spectroscopy and imaging with diffusing light," Phys. Today 48, 34-40 (March 1995).
[CrossRef]

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

Y. Yao, Y. Wang, Y. Pei, W. Zhu and R. L. Barbour, "Frequency-domain optical imaging of absorption and scattering distributions using a born iterative mehod," J. Opt. Soc. Am. A 14, 325-342 (1997).
[CrossRef]

S. R. Arridge, "Forward and inverse problems in time-resolved infrared imaging," in Medical Optical Tomography: Functional Imaging and Monitoring, ed. G. Muller, B. Chance, Rl. Alfano, S. Arridge, J. Beuthan, E. Gratton, M Kaschke, B. Masters, S. Svanberg, P. van der Zee, Proc SPIE IS11, 35-64 (1993).

An integro-differential equation valid for infinite medium,but containing no gradient of Diffusion constant was used for image reconstruction by M. A. OLeary, D. A . Boas, B. Chance, A. G. Yodh, "Experimental Images of heterogeneous turbid media by frequency-domain diffusing-photon tomotography," Opt. Lett. 20, 426-428 (1995).
[CrossRef]

An anonymous reviewer has brought to our attention a recent paper by S. R. Arridge and W. R. B. Lionheart, "Non-uniqueness in diffusion based optical tomography," Opt. Lett. 23, 882-884 (1998), which deals with the second order derivative of the diffusion coefficient.
[CrossRef]

D. N. Pattanayak and E. Wolf, " Resonance States as Solutions of the Schrodinger Equation with a Non-Local Boundary Condition," Phys. Rev. D13, 2287(1976).

J. L. Ye, R. P. Millane, K. J. Webb and T. J. Downar, "Importance of the gradD term in frequency resolved optical diffusion imaging," Opt. Lett. 23, (1998).
[CrossRef]

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

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic Press, New York, 1978) Vol. 19.

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill,1953).

Claus Muller, Foundations of the Mathematical theory of Electromagnetic Waves (Springer-Verlag,New York,1969).

H. Weyl, Ann. Phys. 60, 481, (1919).

See, also, A. J. Banos, in Dipole Radiation In the Presence of a Conducting Half-Space (Pergamon Press, New York, 1966).

E. Wolf, "Principles and Development of Diffraction Tomography" in Trends in Optics, ed. A. Consortini (Academic Press, San Diego, 1996).
[CrossRef]

A. J. Devaney, "Reconstructive tomography with diffracting Wavefields," Inv. Probl. 2,161-1839 (1986).

For an application of diffraction tomography to near field diffusion wave imaging and an analytical expression relating theoretical resolution and tissue thickness see, GE Class I Technical report (publicly available on request) by D. N. Pattanayak, "Resolution of Optical Images Formed by Diffusion Waves in Highly Scattering Media," GE Tech. Info. Series 91CRD241 (1991).

C. L. Matson, N. Clark, L. McMackin and J. S. Fender, "Three-dimensional Tumor Localization in Thick Tissue with The Use of Diffuse Photon-Density Waves," Appl. Opt. 36, 214-219 (1997).
[CrossRef] [PubMed]

X. D. Li, T. Durduran, A. G. Yodh, B. Chance and D. N. Pattanayak, "Diffraction Tomography for Biomedical Imaging With Diffuse Photon Density Waves," Opt. Lett. 22, 573-575 (1997).
[CrossRef] [PubMed]

B. Q. Chen , J. J. Stamnes, K. Stamnes, "Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium," Pure Appl. Opt. 7, 1161-1180 (1998).
[CrossRef]

X. Cheng and D. Boas, "Diffuse Optical Reflection Tomography Using Continous Wave Illumination," Opt. Express 3, 118-123 (1998); http://epubs.osa.org/oearchive/source/5663.htm
[CrossRef] [PubMed]

S. J. Norton, T. Vo-Dinh, "Diffraction Tomographic Imaging With Photon Density Waves: an Explicit Solution, J. Opt. Soc. Am. A 15, 2670-2677 (1998).
[CrossRef]

J. C. Schotland, "Continuos Wave Diffusion Imaging," J. Opt. Soc. Am. A 14, 275-279 (1997).
[CrossRef]

J. Ripoll, M. Nieto-Vesperinas, "Reflection and Transmission Coefficients of Diffuse Photon Density Waves," (to be published).

J. Ripoll and M. Nieto-Vesperinas, "Spatial Resolution of Diffuse Photon Density Waves, " J. Opt. Soc. Am.A (to be published).

C. L. Matson and H. Liu, "Analysis of the forward problem with diffuse photon density waves in turbid media by use of a diffraction tomography model," J. Opt. Soc. Am. A 16, 455-466 (1999).
[CrossRef]

For the angular spectrum representation in a slab field and for an excellent account of the angular spectrum representation see, L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University Press, New York,1995).

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Equations (38)

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J ( r , ω ) = Φ ( r , ω ) S 1 ( r , ω ) 3 μ ' s ( r , ω )
Φ ( r , ω ) = . J ( r , ω ) S 0 ( r , ω ) ( i ω c + μ a ( r , ω ) )
( 2 + k d 2 ) Φ ( r , ω ) = 3 μ ¯ s δ μ a Φ ( r , ω ) + · ( δ μ s μ s Φ ( r , ω ) )
· ( 3 δ μ s μ s S 1 ( r , ω ) ) 3 μ ¯ s ( r ) S 0 ( r , ω ) .
k d 2 = 3 μ ¯ s ( i ω c μ ¯ a )
n ̂ · J ( r , ω ) = α Φ ( r , ω ) .
( 2 + k d 2 ) G ( r , r , ω ) = 4 π δ ( r r ) ,
G d ( r , r , ω ) = exp ( i k d r r ) r r ,
with k d = 3 μ ¯ s ( i ω c μ ¯ a ) .
Φ ( r , ω ) = 3 μ ¯ s 4 π V S 0 ( r , ω ) G d ( r , r , ω ) d 3 r + 3 4 π V S 1 ( r , ω ) . G d ( r , r , ω ) d 3 r
+ 1 4 π S ( 3 α μ ¯ s G d ( r , r , ω ) + G d ( r , r , ω ) . n ̂ ) Φ ( r , ω ) d S
3 μ ¯ s 4 π V ( μ a ( r ) μ ¯ a ) Φ ( r , ω ) G d ( r , r ) d 3 r
+ 1 4 π V μ s ( r ) μ ¯ s μ s ( r ) Φ ( r , ω ) . G d ( r , r ) d 3 r .
Φ ( r , ω ) = 3 μ ¯ s 4 π V S 0 ( r , ω ) G d ( r , r , ω ) d 3 r + 3 4 π V S 1 ( r , ω ) . G d ( r , r , ω ) d 3 r
+ 1 4 π S ( 3 α μ ¯ s G d ( r , r , ω ) + G d ( r , r , ω ) . n ̂ ) Φ ( r , ω ) d S .
Φ P ( r , ω ) = Φ M ( r , ω ) Φ H ( r , ω ) = 3 μ ¯ s 4 π V ( μ a ( r ) μ ¯ a ) Φ H ( r , ω ) G d ( r , r , ω ) d 3 r
+ 1 4 π V μ s ( r ) μ ¯ s μ s ( r ) ϕ H ( r , ω ) . G d ( r , r , ω ) d 3 r .
Φ H ( r , ω ) 3 μ ¯ s 4 π V S 0 ( r , ω ) G d ( r , r , ω ) d 3 r + 3 4 π V S 1 ( r , ω ) . G d ( r , r , ω ) d 3 r
+ 1 4 π S ( 3 α μ ¯ s G d ( r , r , ω ) + G d ( r , r , ω ) . n ̂ ) Φ M ( r , ω ) d S .
G ( r , r , ω ) = d p d q i 2 π m exp ( i p ( x x ) + i q ( y y ) + i m z z ) ,
Φ ̂ P ( p , q , z d , ω ) = z s z d d z T ̂ a ( p , q , z , ω ) K ̂ ( p , q , z d z , ω )
+ z s z d d z T ̂ s ( p , q , z , ω ) . k K ̂ ( p , q , z d z , ω ) ,
T a ( r , ω ) = 3 μ ¯ s 4 π ( μ a ( r ) μ ¯ a ) Φ H ( r , ω ) ,
T s ( r , ω ) = 1 4 π ( 1 D ( r , ω ) D ) ϕ H ( r , ω ) ,
and K ̂ ( p , q , z d z , ω ) = i 2 π m exp ( i m ( z d z ) ) .
T ̂ a ( p , q , z , ω ) = t a ( p , q , z ) { Φ + exp ( + i m 0 z ) + Φ exp ( i m 0 z ) } ,
T ̂ s ( p , q , z , ω ) = t d ( p , q , z ) { k 0 + Φ + exp ( + i m 0 z ) + k 0 Φ exp ( i m 0 z ) } ,
t ̂ a ( p , q , z , ω ) = 3 4 π [ μ ¯ a δ ( p p 0 ) δ ( q q 0 ) μ a ( p p 0 , q q 0 ) ] ,
t ̂ D ( p , q , z ) = 1 4 D ¯ [ D ¯ δ ( p p 0 ) δ ( q q 0 ) D ( p p 0 , q q 0 ) ] ,
and k 0 + = x ̂ p 0 + y ̂ q 0 + z ̂ m 0 , k 0 = x ̂ p 0 + y ̂ q 0 z ̂ m 0 .
Φ 0 ( x , y , z , ω ) = Φ + exp ( i p 0 x + i q o y + i m 0 z ) + Φ exp ( i p 0 x + i q 0 y i m 0 z ) .
Φ ̑ ( z d , ω j ) = i = 1 N [ t ̂ a ( z i ) f ̂ 1 ( z i , ω j ) + t ̂ d ( z i ) f ̂ 2 ( z i , ω j ) ] for j = 1 : 2 N .
f ̂ 1 ( z i , ω j ) = Δ z i { Φ + exp ( + i m 0 z i ) + Φ exp ( i m 0 z i ) } K ̂ ( z d z i , ω ) ,
f ̂ 2 ( z i , ω j ) = Δ z i { ( p p 0 + q q 0 + m m 0 ) Φ + exp ( + i m 0 z i ) +
( p p 0 + q q 0 m m 0 ) Φ exp ( i m 0 z i ) } K ̂ ( z d z i , ω ) .
Φ ̲ ̂ = f ̳ ̂ · t ̂ ̲ .
t ̂ ̲ = ( f ̂ ̳ ) 1 · Φ ̂ ̲ .
t ̲ ( x , y , z i ) = 1 4 π 2 dpdq ( f ̂ ̳ ( p , q ) ) 1 · Φ ̂ ̲ ( p , q ) exp ( ipx iqy ) for i = 1 : N

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