Abstract

The development of non-invasive, biomedical optical imaging from time-dependent measurements of near-infrared (NIR) light propagation in tissues depends upon two crucial advances: (i) the instrumental tools to enable photon “time-of-flight” measurement within rapid and clinically realistic times, and (ii) the computational tools enabling the reconstruction of interior tissue optical property maps from exterior measurements of photon “time-of-flight” or photon migration. In this contribution, the image reconstruction algorithm is formulated as an optimization problem in which an interior map of tissue optical properties of absorption and fluorescence lifetime is reconstructed from synthetically generated exterior measurements of frequency-domain photon migration (FDPM). The inverse solution is accomplished using a truncated Newton’s method with trust region to match synthetic fluorescence FDPM measurements with that predicted by the finite element prediction. The computational overhead and error associated with computing the gradient numerically is minimized upon using modified techniques of reverse automatic differentiation.

© 1999 Optical Society of America

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    [Crossref] [PubMed]
  2. M. A. O’Leary, D. A. Boas, B. Chance, and A.G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion photon tomogrpahy,” Opt. Lett. 20, 426’428 (1995).
    [Crossref]
  3. R. L. Barbour, H. Graber, Y. Wang, J. Chang, and R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, and P. van der Zee, eds. (SPIE Press, Bellingham, WA., 1993), pp 87’120.
  4. Y. Yao, Y. Wang, Y. Pei, W. Zhu, and R.L. Barbour, “Frequency-domain optical imaging of absorption and scattering by a Born iterative method,” J. Opt. Soc. Am. A 14, 325’342 (1997).
    [Crossref]
  5. W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. On Medical Imaging. 9, 218’225 (1995).
    [Crossref]
  6. K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691’701 (1995).
    [Crossref] [PubMed]
  7. 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]
  8. D.Y. Paithankar, A. U. Chen, B.W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt. 36, 2260’2272 (1997).
    [Crossref] [PubMed]
  9. M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Img. Vision 3, 263’283 (1993).
    [Crossref]
  10. J. J. McKeown ,“On algorithms for sums of squares problems,” Towards global optimization, edited by L. C. W. Dixon and G. P. Szeeg, (North-Holland Amsterdam, Holland, 1975).
  11. M. T. Vespucci, “An efficient code for the minimization of highly nonlinear and large residual least squares functions,” Optimization 18, 825’855 (1987).
    [Crossref]
  12. R. R. Meyer, “Theoretical and computational aspects of nonlinear regression,” Nonlinear Programming, eds. J. B. Rosen, O. L. Mangasarian, and K. Ritter, (Academic Press, New York, 1970).
  13. L. B. Rall, Automatic differentiation: Techniques and application, Lecture notes in computer science (Springer Verlag, 1981) p. 120.
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  14. A. Griewank, “On automatic differentiation,” edited M. Iri and K. Tanaka, Mathematical programming: Recent developments and application, (Kluwer Academic Publishers, 1989) pp 83’108.
  15. T. L. Troy, D. L. Page, and E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed Opt. 1, 342’355 (1996).
    [Crossref] [PubMed]
  16. E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. and Photobio. 6655’64 (1997).
    [Crossref]
  17. M. A. O’Leary, D. A. Boas, B. Chance, and A.G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158’160 (1996).
    [Crossref]
  18. J. Chang, H. L. Graber, and R.L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A. 14, 288’299 (1997).
    [Crossref]
  19. T. L. Troy and E. M. Sevick-Muraca, “Fluorescence lifetime imaging and spectroscopy in random media,” in Applied Fluorescence in Chemistry, Biology, and Medicine, Rettig, Strehmel, Shrader, and Seifert, eds., Springer Verlag, pp. 3’36 (1999).
    [Crossref]
  20. H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element based algorithm and simulations,” Appl. Opt. 37, 5337’5343 (1998).
    [Crossref]
  21. Chang, H. L. Graber, and R.L. Barbour, “Improved reconstruction algorithm for luminescence optical tomography when background luminophore is present,” Appl Opt. 37 , 3547’3552 (1998).
    [Crossref]
  22. J. Lee and E. M. Sevick-Muraca, “Lifetime and absorption imaging with fluorescence FDPM,” Time-resolved fluorescence spectroscopy and imaging in tissues, E. M. Sevick-Muraca (ed.)., Proc. Soc. Photo-Opt. Instrum. Eng., 3600: (to be published), (1999).
  23. A. Ishimaru, Wave propagation and scattering in random media,(Academic Press, New York, 1978).
  24. M. Schweiger, S. R. Arridge, M. Hiraka, and D. T. Delpy, “The finite-element method for the propagation of light in scattering media- boundary and source conditions,” Med. Phys. 22, 1779’1792 (1995)
    [Crossref] [PubMed]
  25. R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid material determined from reflection measurements,” Appl. Opt. 22, 2456’2462 (1983).
    [Crossref] [PubMed]
  26. O. C. Zienkiewcz and R. L. Taylor,The finite element methods in engineering science, (McGraw-Hill, New York, 1989).
  27. L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” JOTA,  56, 245’255 (1988).
    [Crossref]
  28. R. Roy, Image reconstruction from light measurements on biological tissue, Ph. D. thesis, University of Hertfordshire, England,(1996).
  29. R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimization,” Math Programming 26, 190’212 (1983).
    [Crossref]
  30. R. C. Price, Sparse matrix optimization using automatic differentiation, Ph. D. thesis, University of Hertfordshire, U. K., (1987).
  31. L. Armijo “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Mathematics 16, 1’3 (1966).
  32. P. Wolfe, “Convergence condition for ascent method,” SIAM Rev.,  11226’253 (1969).
    [Crossref]
  33. B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software 8, 53’67 (1997).
    [Crossref]
  34. A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, ”Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software 28, 217’221 (1997).
    [Crossref]
  35. R. E. Wengert, “A simple automatic derivative evaluation program,” Comm. A. C. M.,  7, 463’464 (1964).

1998 (2)

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element based algorithm and simulations,” Appl. Opt. 37, 5337’5343 (1998).
[Crossref]

Chang, H. L. Graber, and R.L. Barbour, “Improved reconstruction algorithm for luminescence optical tomography when background luminophore is present,” Appl Opt. 37 , 3547’3552 (1998).
[Crossref]

1997 (6)

E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. and Photobio. 6655’64 (1997).
[Crossref]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, and R.L. Barbour, “Frequency-domain optical imaging of absorption and scattering by a Born iterative method,” J. Opt. Soc. Am. A 14, 325’342 (1997).
[Crossref]

D.Y. Paithankar, A. U. Chen, B.W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt. 36, 2260’2272 (1997).
[Crossref] [PubMed]

J. Chang, H. L. Graber, and R.L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A. 14, 288’299 (1997).
[Crossref]

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software 8, 53’67 (1997).
[Crossref]

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, ”Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software 28, 217’221 (1997).
[Crossref]

1996 (3)

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]

T. L. Troy, D. L. Page, and E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed Opt. 1, 342’355 (1996).
[Crossref] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, and A.G. Yodh, “Fluorescence lifetime imaging in turbid media,” Opt. Lett. 21, 158’160 (1996).
[Crossref]

1995 (4)

M. Schweiger, S. R. Arridge, M. Hiraka, and D. T. Delpy, “The finite-element method for the propagation of light in scattering media- boundary and source conditions,” Med. Phys. 22, 1779’1792 (1995)
[Crossref] [PubMed]

W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. On Medical Imaging. 9, 218’225 (1995).
[Crossref]

K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691’701 (1995).
[Crossref] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, and A.G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusion photon tomogrpahy,” Opt. Lett. 20, 426’428 (1995).
[Crossref]

1994 (1)

D. A. Boas, M. A. O’Leary, B. Chance, and A.G. Yodh, “Scattering of diffuse photon density waves by spherical heterogeneities within turbid media: analytic solutions and applications,” Proc. Natl. Acad. Sci.,  91, 4887’91 (1994).
[Crossref] [PubMed]

1993 (1)

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Img. Vision 3, 263’283 (1993).
[Crossref]

1988 (1)

L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” JOTA,  56, 245’255 (1988).
[Crossref]

1987 (1)

M. T. Vespucci, “An efficient code for the minimization of highly nonlinear and large residual least squares functions,” Optimization 18, 825’855 (1987).
[Crossref]

1983 (2)

R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimization,” Math Programming 26, 190’212 (1983).
[Crossref]

R. A. J. Groenhuis, H. A. Ferwerda, and J. J. Ten Bosch, “Scattering and absorption of turbid material determined from reflection measurements,” Appl. Opt. 22, 2456’2462 (1983).
[Crossref] [PubMed]

1969 (1)

P. Wolfe, “Convergence condition for ascent method,” SIAM Rev.,  11226’253 (1969).
[Crossref]

1966 (1)

L. Armijo “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Mathematics 16, 1’3 (1966).

1964 (1)

R. E. Wengert, “A simple automatic derivative evaluation program,” Comm. A. C. M.,  7, 463’464 (1964).

Armijo, L.

L. Armijo “Minimization of functions having Lipschitz continuous first partial derivatives,” Pacific J. Mathematics 16, 1’3 (1966).

Aronson, R.

R. L. Barbour, H. Graber, Y. Wang, J. Chang, and R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, and P. van der Zee, eds. (SPIE Press, Bellingham, WA., 1993), pp 87’120.

Arridge, S. R.

M. Schweiger, S. R. Arridge, M. Hiraka, and D. T. Delpy, “The finite-element method for the propagation of light in scattering media- boundary and source conditions,” Med. Phys. 22, 1779’1792 (1995)
[Crossref] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Img. Vision 3, 263’283 (1993).
[Crossref]

Barbour, R. L.

R. L. Barbour, H. Graber, Y. Wang, J. Chang, and R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, and P. van der Zee, eds. (SPIE Press, Bellingham, WA., 1993), pp 87’120.

Barbour, R.L.

Chang, H. L. Graber, and R.L. Barbour, “Improved reconstruction algorithm for luminescence optical tomography when background luminophore is present,” Appl Opt. 37 , 3547’3552 (1998).
[Crossref]

J. Chang, H. L. Graber, and R.L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A. 14, 288’299 (1997).
[Crossref]

Y. Yao, Y. Wang, Y. Pei, W. Zhu, and R.L. Barbour, “Frequency-domain optical imaging of absorption and scattering by a Born iterative method,” J. Opt. Soc. Am. A 14, 325’342 (1997).
[Crossref]

Boas, D. A.

Chance, B.

Chang,

Chang, H. L. Graber, and R.L. Barbour, “Improved reconstruction algorithm for luminescence optical tomography when background luminophore is present,” Appl Opt. 37 , 3547’3552 (1998).
[Crossref]

Chang, J.

J. Chang, H. L. Graber, and R.L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A. 14, 288’299 (1997).
[Crossref]

R. L. Barbour, H. Graber, Y. Wang, J. Chang, and R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, and P. van der Zee, eds. (SPIE Press, Bellingham, WA., 1993), pp 87’120.

Chen, A. U.

Chew, W. C.

W. C. Chew and Y. M. Wang, “Reconstruction of two-dimensional permittivity distribution using the distorted Born iterative method,” IEEE Trans. On Medical Imaging. 9, 218’225 (1995).
[Crossref]

Christianson, B.

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software 8, 53’67 (1997).
[Crossref]

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, ”Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software 28, 217’221 (1997).
[Crossref]

Davies, A. J.

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software 8, 53’67 (1997).
[Crossref]

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, ”Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software 28, 217’221 (1997).
[Crossref]

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraka, and D. T. Delpy, “The finite-element method for the propagation of light in scattering media- boundary and source conditions,” Med. Phys. 22, 1779’1792 (1995)
[Crossref] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Img. Vision 3, 263’283 (1993).
[Crossref]

Dembo, R. S.

R. S. Dembo and T. Steihaug, “Truncated Newton algorithms for large-scale unconstrained optimization,” Math Programming 26, 190’212 (1983).
[Crossref]

Dixon, L. C. W.

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software 8, 53’67 (1997).
[Crossref]

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, ”Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software 28, 217’221 (1997).
[Crossref]

L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” JOTA,  56, 245’255 (1988).
[Crossref]

Ferwerda, H. A.

Graber, H.

R. L. Barbour, H. Graber, Y. Wang, J. Chang, and R. Aronson, “Perturbation approach for optical diffusion tomography using continuous-wave and time-resolved data,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller, B. Chance, R. Alfano, J. Beuthan, E. Gratton, M. Kashke, B. Masters, S. Svanberg, and P. van der Zee, eds. (SPIE Press, Bellingham, WA., 1993), pp 87’120.

Graber, H. L.

Chang, H. L. Graber, and R.L. Barbour, “Improved reconstruction algorithm for luminescence optical tomography when background luminophore is present,” Appl Opt. 37 , 3547’3552 (1998).
[Crossref]

J. Chang, H. L. Graber, and R.L. Barbour, “Luminescence optical tomography of dense scattering media,” J. Opt. Soc. Am. A. 14, 288’299 (1997).
[Crossref]

Griewank, A.

A. Griewank, “On automatic differentiation,” edited M. Iri and K. Tanaka, Mathematical programming: Recent developments and application, (Kluwer Academic Publishers, 1989) pp 83’108.

Groenhuis, R. A. J.

Hiraka, M.

M. Schweiger, S. R. Arridge, M. Hiraka, and D. T. Delpy, “The finite-element method for the propagation of light in scattering media- boundary and source conditions,” Med. Phys. 22, 1779’1792 (1995)
[Crossref] [PubMed]

Hutchinson, C. L.

E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. and Photobio. 6655’64 (1997).
[Crossref]

Ishimaru, A.

A. Ishimaru, Wave propagation and scattering in random media,(Academic Press, New York, 1978).

Jiang, H.

H. Jiang, “Frequency-domain fluorescent diffusion tomography: a finite-element based algorithm and simulations,” Appl. Opt. 37, 5337’5343 (1998).
[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]

K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691’701 (1995).
[Crossref] [PubMed]

Lee, J.

J. Lee and E. M. Sevick-Muraca, “Lifetime and absorption imaging with fluorescence FDPM,” Time-resolved fluorescence spectroscopy and imaging in tissues, E. M. Sevick-Muraca (ed.)., Proc. Soc. Photo-Opt. Instrum. Eng., 3600: (to be published), (1999).

Lopez, G.

E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. and Photobio. 6655’64 (1997).
[Crossref]

McKeown, J. J.

J. J. McKeown ,“On algorithms for sums of squares problems,” Towards global optimization, edited by L. C. W. Dixon and G. P. Szeeg, (North-Holland Amsterdam, Holland, 1975).

Meyer, R. R.

R. R. Meyer, “Theoretical and computational aspects of nonlinear regression,” Nonlinear Programming, eds. J. B. Rosen, O. L. Mangasarian, and K. Ritter, (Academic Press, New York, 1970).

O’Leary, M. A.

Osterberg, U. L.

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]

Page, D. L.

T. L. Troy, D. L. Page, and E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed Opt. 1, 342’355 (1996).
[Crossref] [PubMed]

Paithankar, D.Y.

Patterson, M. S.

D.Y. Paithankar, A. U. Chen, B.W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt. 36, 2260’2272 (1997).
[Crossref] [PubMed]

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]

Paulsen, K. D.

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]

K. D. Paulsen and H. Jiang, “Spatially varying optical property reconstruction using a finite element diffusion equation approximation,” Med. Phys. 22, 691’701 (1995).
[Crossref] [PubMed]

Pei, Y.

Pogue, B.W.

D.Y. Paithankar, A. U. Chen, B.W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt. 36, 2260’2272 (1997).
[Crossref] [PubMed]

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]

Price, R. C.

L. C. W. Dixon and R. C. Price, “Numerical experience with the truncated Newton method for unconstrained optimization,” JOTA,  56, 245’255 (1988).
[Crossref]

R. C. Price, Sparse matrix optimization using automatic differentiation, Ph. D. thesis, University of Hertfordshire, U. K., (1987).

Rall, L. B.

L. B. Rall, Automatic differentiation: Techniques and application, Lecture notes in computer science (Springer Verlag, 1981) p. 120.
[Crossref]

Reynolds, J. S.

E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. and Photobio. 6655’64 (1997).
[Crossref]

Roy, R.

A. J. Davies, B. Christianson, L. C. W. Dixon, R. Roy, and P. van der Zee, ”Reverse differentiation and the inverse diffusion problem,” Adv. In Eng. Software 28, 217’221 (1997).
[Crossref]

B. Christianson, A. J. Davies, L. C. W. Dixon, R. Roy, and P. van der Zee, “Giving reverse differentiation a helping hand,” Opt. Meth. And Software 8, 53’67 (1997).
[Crossref]

R. Roy, Image reconstruction from light measurements on biological tissue, Ph. D. thesis, University of Hertfordshire, England,(1996).

Schweiger, M.

M. Schweiger, S. R. Arridge, M. Hiraka, and D. T. Delpy, “The finite-element method for the propagation of light in scattering media- boundary and source conditions,” Med. Phys. 22, 1779’1792 (1995)
[Crossref] [PubMed]

M. Schweiger, S. R. Arridge, and D. T. Delpy, “Application of the finite-element method for the forward and inverse models in optical tomography,” J. Math. Img. Vision 3, 263’283 (1993).
[Crossref]

Sevick-Muraca, E. M.

E. M. Sevick-Muraca, G. Lopez, T. L. Troy, J. S. Reynolds, and C. L. Hutchinson, “Fluorescence and absorption contrast mechanisms for biomedical optical imaging using frequency-domain techniques,” Photochem. and Photobio. 6655’64 (1997).
[Crossref]

D.Y. Paithankar, A. U. Chen, B.W. Pogue, M. S. Patterson, and E. M. Sevick-Muraca, “Imaging of fluorescent yield and lifetime from multiply scattered light re-emitted from tissues and other random media,” Appl. Opt. 36, 2260’2272 (1997).
[Crossref] [PubMed]

T. L. Troy, D. L. Page, and E. M. Sevick-Muraca, “Optical properties of normal and diseased breast tissues: prognosis for optical mammography,” J. Biomed Opt. 1, 342’355 (1996).
[Crossref] [PubMed]

J. Lee and E. M. Sevick-Muraca, “Lifetime and absorption imaging with fluorescence FDPM,” Time-resolved fluorescence spectroscopy and imaging in tissues, E. M. Sevick-Muraca (ed.)., Proc. Soc. Photo-Opt. Instrum. Eng., 3600: (to be published), (1999).

T. L. Troy and E. M. Sevick-Muraca, “Fluorescence lifetime imaging and spectroscopy in random media,” in Applied Fluorescence in Chemistry, Biology, and Medicine, Rettig, Strehmel, Shrader, and Seifert, eds., Springer Verlag, pp. 3’36 (1999).
[Crossref]

Steihaug, T.

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Figures (1)

Figure 1
Figure 1

Schematic of fluorescence photon migration.

Equations (56)

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· [ D x ( r ) Φ x ( r , ω ) ] + [ i ω c + μ a xi ( r ) + μ a xf ( r ) ] Φ x ( r , ω ) = 0 on Ω
· [ D m ( r ) Φ m ( r , ω ) ] + [ i ω c + μ a m ( r ) ] Φ m ( r , ω ) = ϕμ a x m 1 1 i ωτ Φ x ( r , ω ) on Ω
Φ x r ω 2 γ D x ( r ) Φ x r ω n + S δ ( r r s ) = 0 on d Ω
γ = ( 1 + r d ) ( 1 r d )
r d = 1.44 n rel 2 + 0.72 n rel 1 + 0.668 + 0.063 n rel
Ω [ · ( D x Φ x ) + ( i ω c + μ a xi + μ a xf ) Φ x ] w j d Ω = 0 j = 1,2 , , N
Ω [ D x ( Φ x ) · ( w j ) + ( i ω c + μ a xi + μ a xf ) Φ x w j ] d Ω Γ D x w j Φ x n d Γ = 0
Γ D x w j Φ x n d Γ = 1 γ Γ ( Φ x + S ) w j d Γ = 1 γ Γ Φ x w j d Γ + 1 γ Γ S w j d Γ
Ω [ D x ( Φ x ) ( w j ) + ( i ω c + μ a xi + μ a xf ) Φ x w j ] d Ω 1 γ Γ Φ x w j d Γ = 1 γ Γ S w j d Γ
el = 1 M [ Ω el [ D x el ( Φ x el ) ( w j ) + ( c + μ a xi + μ a xf el ) Φ x el w j ] + 1 γ Γ el Φ x el w j ] =
el M [ 1 γ Γ el S w j ]
Φ x el = j = 1 3 L j ( Φ x ) j
w j = L j for j = 1,2 , , N
el = 1 M [ Ω el [ D x el ( Φ x el x L j x + Φ x el y L j y ) + ( c + μ a xi + μ a xf el ) L j ] Φ x el + 1 γ Γ el Φ x el L j ] =
el = 1 M [ 1 γ Γ el S L j ]
el = 1 M [ K 1 el + K 2 el + K 3 el ] Φ x el = el M r el
S = S δ ( r r S )
Γ el S δ ( r r s ) d Γ = { S r s Γ el 0 otherwise
r el = 1 γ Γ el L j ( r r s ) = 1 γ L j ( r s ) S
( r el ) j = 1 γ S
el = 1 M [ Ω el [ D m el ( Φ m el x L j x + Φ m el y L j y ) + ( i ω c + μ a m ) Φ m el L j ] d Ω + 1 γ Γ el Φ m el L j ] =
el = 1 el [ Ω el Φ x el μ a x m el ϕ ( 1 + ω τ el ) L j d Ω ]
K Φ ̄ x , m = b
( N B 1 ) N S N
( N B 1 ) N S 2 N
E x , m ( μ a xf ) = 1 2 l = 1 N s j = 1 N B j l ( ( ( ( Φ x , m ) l ) j ) c ( ( ( Φ x , m ) l ) j ) me ( ( ( Φ x , m ) l ) j ) me ) ( ( ( ( Φ x , m * ) l ) j ) c ( ( ( Φ x , m * ) l ) j ) me ( ( ( Φ x , m * ) l ) j ) me )
E x = E x ( μ a xf ) = Re j d Ω ( ( ( Φ x * ) c ( Φ x * ) me ( Φ x ) me ( Φ x * ) me ) , ( Φ x ) c ( μ a xf ) )
E m = E m ( τ ) = Re j d Ω ( ( ( Φ m * ) c ( Φ m * ) me ( Φ m ) me ( Φ m * ) me ) , ( Φ m ) c ( τ ) )
E m = E m ( μ a x m ) = Re j d Ω ( ( ( Φ m * ) c ( Φ m * ) me ( Φ m ) me ( Φ m * ) me ) , ( Φ m ) c ( μ a x m ) )
E x , m ( μ ¯ a k + d ) = E x , m ( μ ¯ a k ) + Q ( d )
Q ( d ) = g k T d + 1 2 d T G k d
G k d = g k
r i g k min ( 1 k , g k )
G ( x ) d = 1 σ [ g ( x + σ d ) g ( x ) ]
R 1 = 0.01
R k + 1 = 2 R k if λ k 1.0
R k + 1 = 1 3 R k if λ k < 1.0
Given x i , i = 1 , , n For i = n + 1 , , P then if F i is binary x i = F i ( x j , x k ) , j , k < i and if F i is unary x i = F ( x j ) , . j < i f ( x ) = x n + P
f x i = j f x j F j x i j > i
Given x i , i = 1 , , P set x ̂ i = 0 , i = 1 , . , n + P 1 and x ̂ n + P = 1 for i = n + P , , n + 1
then, if F i is binary , x ̂ j = x ̂ j + x ̂ i F i x j i > j and x ̂ k = x ̂ k + x ̂ i F i x k i > j else if F i is unary , x ̂ j = x ̂ j + x ̂ i F i x j i > j derivatives g i = x ̂ i i = 1 , , n
( μ ̂ a xf ) p = E x , m ( μ a xf ) p = E x , m K K ( μ a xf ) p + E x , m b b ( μ a xf ) p
= el i , j ( K ̂ el ) i , j ( K i , j el ( μ a xf ) p ) + el j b ̂ j b j ( μ a xf ) p
K ̂ = E x , m K = E x , m Φ ¯ x , m Φ ¯ x , m K = Φ ¯ ̂ x , m Φ ¯ x , m K = Φ ¯ ̂ x , m Φ x , m μ ¯ a xf μ ¯ a xf K
K Φ ̄ x , m = b
K μ ̄ a xf Φ ¯ x , m + K Φ ¯ xf μ ¯ a xf = 0
K μ ¯ a xf = - K Φ ¯ x , m μ ¯ xf Φ ¯ x , m
μ ¯ xf K = K 1 Φ ¯ x , m Φ ¯ x , m μ ¯ a x , m
K ̂ = Φ ¯ ̂ x , m K 1 Φ x , m = v ¯ T Φ ¯ x , m
v ¯ = Φ ¯ ̂ K 1
K v ¯ = Φ ¯ ̂ x , m
b ̂ = E x , m b = E x , m Φ ¯ x , m Φ ¯ x , m b = Φ ¯ ̂ x , m Φ ¯ x , m b = Φ ¯ ̂ x , m Φ ¯ x , m τ ¯ τ ¯ b
K Φ ¯ x , m = b
K Φ ¯ x , m τ ¯ = b τ ¯
τ ¯ b = K 1 1 Φ ¯ x , m τ ¯
K b ̂ = Φ ¯ ̂ x , m

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