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.

© Optical Society of America

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References

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  1. D. A. Boas, M. A. OLeary, 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]
  2. M. A. OLeary, 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. 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. 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 Dixon, L. C. W. and Szeeg, G. P., (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. Rosen, J. B., Mangasarian, O. L. and Ritter, K. (Academic Press, New York, 1970).
  13. L. B. Rall, Automatic differentiation: Techniques and application, Lecture notes in computer science (Springer Verlag, 1981) p. 120.
    [CrossRef]
  14. A. Griewank, "On automatic differentiation," edited Iri, M. and Tanaka, K., 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. Photobiol. 66 55-64 (1997).
    [CrossRef]
  17. M. A. OLeary, 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, 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., 11 226-253 (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).

Other

D. A. Boas, M. A. OLeary, 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]

M. A. OLeary, 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]

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.

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]

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]

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]

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]

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]

M. Schweiger, S. R. Arridge. 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]

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

M. T. Vespucci, "An efficient code for the minimization of highly nonlinear and large residual least squares functions," Optimization 18, 825-855 (1987).
[CrossRef]

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

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

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

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]

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. Photobiol. 66 55-64 (1997).
[CrossRef]

M. A. OLeary, D. A. Boas, B. Chance and A. G. Yodh, "Fluorescence lifetime imaging in turbid media," Opt. Lett. 21, 158-160 (1996).
[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]

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, Seifert, eds., Springer Verlag, pp. 3-36 (1999).
[CrossRef]

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]

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).

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

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]

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]

O. C. Zienkiewcz and R. L. Taylor, The finite element methods in engineering science, (McGraw-Hill, New York, 1989).

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

R. S. Dembo and T. Steihaug, "Truncated Newton algorithms for large-scale unconstrained optimization," Math Programming 26, 190-212 (1983).
[CrossRef]

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

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

P. Wolfe, "Convergence condition for ascent method," SIAM Rev., 11 226-253 (1969).
[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]

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

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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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