## Diffuse optical 3D-slice imaging of bounded turbid media using a new integro-differential equation

Optics Express, Vol. 4, Issue 8, pp. 231-240 (1999)

http://dx.doi.org/10.1364/OE.4.000231

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### 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.

© Optical Society of America

## 1. Introduction

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]

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]

14. An integro-differential equation valid for infinite medium,but containing no gradient of Diffusion constant was used for image reconstruction by M. A. O’Leary, D. A . Boas, B. Chance, and 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]

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. Letters ,**23**,1998. [CrossRef]

*plane*diffuse photon density waves with different modulation frequencies. For this set of plane incident diffusive waves, the algorithm requires two-dimensional FFT operations, and a one-dimensional matrix inversion. Thus the method is non-iterative in two space dimensions and is therefore computationally fast.

## 2. Integro-differential equation for diffuse photon density waves

*ω*denotes the modulation frequency of the source intensity,

*μ*and

_{α}*μ*′

_{s}are the absorption and transport scattering coefficients of the bounded turbid media, c is the velocity of light in the medium and

*S*

_{0}(

*ω*),

_{1}(

*ω*) are the monopole and dipole moments of the intensity modulated optical source respectively. From these equations we eliminate the photon flux density and obtain, after a lengthy calculation, the following wave equation involving only the photon density function

*k*. The bar over a physical parameter denotes spatial average of that parameter. Notice that for unmodulated source intensity (the continuous dc domain in contrast to the frequency domain), the photon density is a non-propagating damped wave. The wave number in this case is purely imaginary. For frequency modulated source intensity, the wave number is complex and the photon density acquires the characteristics of a damped or attenuated wave.

_{d}*α*can be defined based on boundary considerations or taken as a fitting parameter for a given interface. In Eq. (5)

*n*̂ is the unit outward normal to the boundary surface. Using Eq. (5), it is now possible to transform Eq. (3) into an integral equation, the solution of which will provide one with a physical picture of the interaction of the diffuse photon density wave with the turbid media. The basic interaction of light with the molecules of the turbid media is already included in the absorption and scattering parameters. The determination of these material parameters is at the heart of the optical modality for cancer diagnostics. The integral representation forms a basis for extraction of these parameters from measurements of the diffusion waves at the boundary. We next derive the integro-differential equation using Eqs. (3) and (5).

*k*) > 0. Notice that our choice of an infinite space Green’s function will not limit in any way when we consider finite domains; finite medium effects are properly treated by the boundary integrals. If we now multiply both sides of Eq. (3) and (6) by

_{d}*G*(

_{d}*ω*) and Φ;(

*ω*) respectively and subtract the resulting equations from one another and use a number of standard Green’s identities, we obtain

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. Letters ,**23**,1998. [CrossRef]

## 3.Inversion algorithm for 3D-slice imaging

^{H}(

*ω*) often identified as the background field can be determined from the knowledge of the source and the measured field at the boundary from Eq. (12).

*μ*(

_{a}*μ*′

_{s}(

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

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).

*z*=

*z*we find that

_{d}*x*̂

*p*+

*y*̂

*q*+

*z*̂

*m*is the wave vector and

*z*,

_{s}*z*denote the positions of the source and detector plates respectively. The caret (̂) over functions signifies the Fourier transform of a function. Several things are worth noting at this point. First, the absorption coefficient contribution depends on a scalar function, while the transport scattering coefficient depends on the product of two vectors. Second, the gradient of the diffuse photon density wave interacts with the scattering heterogeneity, while the diffuse photon density wave interacts with the absorption heterogeneity. Finally, the inverse of the transport scattering coefficient has been explicitly replaced by the photon diffusion coefficient

_{d}*T*̂

_{a},

*T*̂

_{s}, can be written as a product of two functions. One function,

*t*̂

_{a}or

*t*̂

_{D}, contains information about the heterogeneities and depends upon (p, q, z) and the other function depends upon the background plane waves and modulation frequency (see Eq. (18) below). One can then utilize this temporal frequency bandwidth to obtain resolution in the longitudinal (z) direction. The key advantage of using the series of stepped central modulation frequencies is that the detection system is still narrow band and one can use maximum permissible signal strength at all discrete frequencies. We plan to publish in a later paper the details of this technique with simulated data, but here we point out the salient features of the method. For the plane wave case we find

*p*

_{0}= Re(

*k*)cos(

_{d}*θ*)cos(

*ϕ*) ,

*q*

_{0}= Re(

*k*)cos(

_{d}*θ*)sin(

*ϕ*) where

*θ*is the angle the plane wave makes with respect to the z direction, and

*ϕ*is the corresponding azimuthal angle. Care must be exercised in the angle interpretations as we are dealing with attenuating plane waves. For the case of a normal incidence,

*p*

_{0}= 0 and

*q*

_{0}= 0. The parameter

^{+}, and Φ

^{-}are complex constants obtained from solution or measurements as discussed earlier in the section on the background field.

*z*with j ranging from 1: N), and use Eqs. (15)–(19) to obtain the following result

_{j}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, and P. van der Zee, Proc SPIEIS11, 35–64 (1993).

*t*̂ consists of the sum of the two N dimensional vectors

*t*̂

_{a}and

*t*̂

_{D}.

*t*̂ in Eq. (23) is the two-dimensional Fourier transform of the absorption and scattering coefficients at each of the N number of slices in the direction normal to the plates (the z direction). The values of the scattering coefficients in the direction transverse to the surface normal (the x and y directions) are obtained by applying the two-dimensional inverse Fourier transform, i.e.

## 4. Conclusion

## Acknowledgements

## References and links

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. |

2. | A. Villringer and B. Chance, “Non-invasive optical spectroscopy and imaging of human brain functions,” Trends. Neurosci. |

3. | Y. Hoshi and M. Tamura, “Near-Infrared Optical Detection of Sequential Brain Activation in The Prefrontal cortex during mental tasks,” Neuroimage. |

4. | J. H. Hoogenraad, M. B.van der Mark, S. B. Colak, G.W. ’t Hooft, and 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. |

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 |

7. | S. K. Gayen, M. E. Zevallos, B. B. Das, and R. R. Alfano, “Time-sliced transillumination imaging of normal and cancerous breast tissues,” in |

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 |

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,” SPIE3597 (1999). |

10. | A. G. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today |

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 |

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 |

13. | S. R. Arridge, “Forward and inverse problems in time-resolved infrared imaging,” in |

14. | An integro-differential equation valid for infinite medium,but containing no gradient of Diffusion constant was used for image reconstruction by M. A. O’Leary, D. A . Boas, B. Chance, and A.G. Yodh, “Experimental Images of heterogeneous turbid media by frequency-domain diffusing-photon tomotography,” Opt. Lett. |

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. |

16. | D. N. Pattanayak and E. Wolf , “ Resonance States as Solutions of the Schrodinger Equation with a Non-Local Boundary Condition, ” Phys. Rev. |

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. Letters , |

18. | K. M. Case and P. F. Zweifel, in |

19. | A. Ishimaru, |

20. | P.M. Morse and H. Feshbach, |

21. | Claus Muller, |

22. | H. Weyl and Ann. Phys. |

23. | See, also , A.J. Banos, in |

24. | E. Wolf, “Principles and Development of Diffraction Tomography” in |

25. | A. J. Devaney, “Reconstructive tomography with diffracting Wavefields,” Inv. Probl. , |

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 |

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. |

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. |

29. | BQ Chen, JJ Stamnes, and K Stamnes, “Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium.” Pure Appl. Opt. |

30. | X. Cheng and D. Boas, “Diffuse Optical Reflection Tomography Using Continous Wave Illumination,” Opt. Express |

31. | S. J. Norton and T. Vo-Dinh, “Diffraction Tomographic Imaging With Photon Density Waves: an Explicit Solution, J. Opt. Soc. Am. A |

32. | J. C. Schotland, “Continuos Wave Diffusion Imaging,” J. Opt. Soc. Am. A |

33. | J. Ripoll and 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 |

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, |

**OCIS Codes**

(170.3010) Medical optics and biotechnology : Image reconstruction techniques

(170.3660) Medical optics and biotechnology : Light propagation in tissues

(170.3830) Medical optics and biotechnology : Mammography

(170.5270) Medical optics and biotechnology : Photon density waves

(170.5280) Medical optics and biotechnology : Photon migration

**ToC Category:**

Focus Issue: Biomedical diffuse optical tomography

**History**

Original Manuscript: March 5, 1999

Published: April 12, 1999

**Citation**

Deva Pattanayak and Arjun Yodh, "Diffuse optical 3D-slice imaging of bounded turbid media using a new integro-differential equation," Opt. Express **4**, 231-240 (1999)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-4-8-231

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### References

- 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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