Abstract

A novel adaptive mesh technique is introduced for problems of image reconstruction in luminescence optical tomography. A dynamical adaptation of the three-dimensional scheme based on the finite-volume formulation reduces computational time and balances the ill-posed nature of the inverse problem. The arbitrary shape of the bounding surface is handled by an additional refinement of computational cells on the boundary. Dynamical shrinking of the search volume is introduced to improve computational performance and accuracy while locating the luminescence target. Light propagation in the medium is modeled by the telegraph equation, and the image-reconstruction algorithm is derived from the Fredholm integral equation of the first kind. Stability and computational efficiency of the introduced method are demonstrated for image reconstruction of one and two spherical luminescent objects embedded within a breastlike tissue phantom. Experimental measurements are simulated by the solution of the forward problem on a grid of 5×5 light guides attached to the surface of the phantom.

© 2006 Optical Society of America

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References

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  1. W. L. Rumsey, J. M. Vanderkooi, and D. F. Wilson, " Imaging of phosphorescence: A novel method for measuring the distribution of oxygen in perfused tissue," Science 241, 1649- 1651 ( 1988).
    [CrossRef] [PubMed]
  2. S. A. Vinogradov, L. -W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, and D. F. Wilson, " Noninvasive imaging of the distribution of oxygen in tissue in vivo using near-infrared phosphors," Biophys. J. 70, 1609- 1617 ( 1996).
    [CrossRef] [PubMed]
  3. S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, and D. F. Wilson, " Frequency domain instrument for measuring phosphorescence lifetime distribution in heterogeneous samples," Rev. Sci. Instrum. 72, 3396- 3406 ( 2001).
    [CrossRef]
  4. I. B. Rietveld, E. Kim, and S. A. Vinogradov, " Dendrimers with tetrabenzoporphyrin cores: Near infrared phosphors for in vivo oxygen imaging," Tetrahedron 59, 3821- 3831 ( 2003).
    [CrossRef]
  5. D. F. Wilson and S. A. Vinogradov, " Tissue oxygen measurements using phosphorescence quenching," in Handbook of Biomedical Fluorescence, M.-A.Mycek and B.W.Pogue, eds. (Marcel Dekker, 2003), pp. 637- 663.
  6. E. Shives, Y. Xu, and H. Jiang, " Fluorescence lifetime tomography in turbid media based on an oxygen-sensitive dye," Opt. Exp. 10, 1557- 1562 ( 2002), available at http://www.opticsexpress.org.
  7. V. Y. Soloviev, D. F. Wilson, and S. A. Vinogradov, " Phosphorescence lifetime imaging in turbid media: the inverse problem and experimental image reconstruction," Appl. Opt. 43, 564- 574 ( 2004).
    [CrossRef] [PubMed]
  8. X. Gu, Y. Xu, and H. Jiang, " Mesh-based enhancement scheme in diffuse optical tomography," Med. Phys. 30, 861- 869 ( 2003).
    [CrossRef] [PubMed]
  9. A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, " Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Exp. 12, 5402- 5417 ( 2004), available at http://www.opticsexpress.org.
    [CrossRef]
  10. A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, " Fluorescence optical diffusion tomography," Appl. Opt. 42, 3081- 3094 ( 2003).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, " Fluorescence and absorption contrast mechanism for biomedical optical imaging using frequency domain techniques," Photochem. Photobiol. 66, 55- 64 ( 1997).
    [CrossRef] [PubMed]
  13. J. S. Reynolds, C. A. Thompson, K. J. Webb, F. P. LaPlant, and D. Ben-Amotz, " Frequency-domain modeling of reradiation in highly scattering media," Appl. Opt. 36, 2252- 2259 ( 1997).
    [CrossRef] [PubMed]
  14. M. A. O'Leary, D. A. Boas, X. D. Li, B. Chance, and A. G. Yodh, " Fluorescence lifetime imaging in turbid media," Opt. Lett. 21, 158- 160 ( 1996).
    [CrossRef] [PubMed]
  15. V. Soloviev, D. Wilson, and S. Vinogradov, " Phosphorescence lifetime imaging in turbid media: The forward problem," Appl. Opt. 42, 113- 123 ( 2003).
    [CrossRef] [PubMed]
  16. D. J. Durian and J. Rundick, " Photon migration at short times and distances and in case of strong absorption," J. Opt. Soc. Am. 14, 235- 245 ( 1997).
    [CrossRef]
  17. S. Arridge, " Optical tomography in medical imaging," Inverse Probl. 15, R41- R93 ( 1999).
    [CrossRef]
  18. V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).
  19. G. F. Carey, Computational Grids: Generation, Adaptation, and Solution Strategies (Taylor & Francis, 1997).
  20. A. A. Samarskii and P. N. Vabishev, Chislennye metody reshenia zadach convektsii-diffusii (URSS, 2003; in Russian).
  21. A. A. Samarskii and P. N. Vabishev, Chislennye metody reshenia obratnyh zadach matematicheskoi fiziki (URSS, 2004, in Russian).
  22. A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, New York, 1957).

2004 (2)

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, " Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Exp. 12, 5402- 5417 ( 2004), available at http://www.opticsexpress.org.
[CrossRef]

V. Y. Soloviev, D. F. Wilson, and S. A. Vinogradov, " Phosphorescence lifetime imaging in turbid media: the inverse problem and experimental image reconstruction," Appl. Opt. 43, 564- 574 ( 2004).
[CrossRef] [PubMed]

2003 (4)

V. Soloviev, D. Wilson, and S. Vinogradov, " Phosphorescence lifetime imaging in turbid media: The forward problem," Appl. Opt. 42, 113- 123 ( 2003).
[CrossRef] [PubMed]

A. B. Milstein, S. Oh, K. J. Webb, C. A. Bouman, Q. Zhang, D. A. Boas, and R. P. Millane, " Fluorescence optical diffusion tomography," Appl. Opt. 42, 3081- 3094 ( 2003).
[CrossRef] [PubMed]

X. Gu, Y. Xu, and H. Jiang, " Mesh-based enhancement scheme in diffuse optical tomography," Med. Phys. 30, 861- 869 ( 2003).
[CrossRef] [PubMed]

I. B. Rietveld, E. Kim, and S. A. Vinogradov, " Dendrimers with tetrabenzoporphyrin cores: Near infrared phosphors for in vivo oxygen imaging," Tetrahedron 59, 3821- 3831 ( 2003).
[CrossRef]

2002 (1)

E. Shives, Y. Xu, and H. Jiang, " Fluorescence lifetime tomography in turbid media based on an oxygen-sensitive dye," Opt. Exp. 10, 1557- 1562 ( 2002), available at http://www.opticsexpress.org.

2001 (1)

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, and D. F. Wilson, " Frequency domain instrument for measuring phosphorescence lifetime distribution in heterogeneous samples," Rev. Sci. Instrum. 72, 3396- 3406 ( 2001).
[CrossRef]

1999 (1)

S. Arridge, " Optical tomography in medical imaging," Inverse Probl. 15, R41- R93 ( 1999).
[CrossRef]

1997 (3)

1996 (2)

S. A. Vinogradov, L. -W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, and D. F. Wilson, " Noninvasive imaging of the distribution of oxygen in tissue in vivo using near-infrared phosphors," Biophys. J. 70, 1609- 1617 ( 1996).
[CrossRef] [PubMed]

M. A. O'Leary, D. A. Boas, X. D. Li, B. Chance, and A. G. Yodh, " Fluorescence lifetime imaging in turbid media," Opt. Lett. 21, 158- 160 ( 1996).
[CrossRef] [PubMed]

1994 (1)

1988 (1)

W. L. Rumsey, J. M. Vanderkooi, and D. F. Wilson, " Imaging of phosphorescence: A novel method for measuring the distribution of oxygen in perfused tissue," Science 241, 1649- 1651 ( 1988).
[CrossRef] [PubMed]

Arridge, S.

S. Arridge, " Optical tomography in medical imaging," Inverse Probl. 15, R41- R93 ( 1999).
[CrossRef]

Bangerth, W.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, " Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Exp. 12, 5402- 5417 ( 2004), available at http://www.opticsexpress.org.
[CrossRef]

Ben-Amotz, D.

Boas, D. A.

Bouman, C. A.

Carey, G. F.

G. F. Carey, Computational Grids: Generation, Adaptation, and Solution Strategies (Taylor & Francis, 1997).

Chance, B.

Dugan, B. W.

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, and D. F. Wilson, " Frequency domain instrument for measuring phosphorescence lifetime distribution in heterogeneous samples," Rev. Sci. Instrum. 72, 3396- 3406 ( 2001).
[CrossRef]

Durian, D. J.

Evans, S. M.

S. A. Vinogradov, L. -W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, and D. F. Wilson, " Noninvasive imaging of the distribution of oxygen in tissue in vivo using near-infrared phosphors," Biophys. J. 70, 1609- 1617 ( 1996).
[CrossRef] [PubMed]

Fernandez-Seara, M. A.

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, and D. F. Wilson, " Frequency domain instrument for measuring phosphorescence lifetime distribution in heterogeneous samples," Rev. Sci. Instrum. 72, 3396- 3406 ( 2001).
[CrossRef]

Gu, X.

X. Gu, Y. Xu, and H. Jiang, " Mesh-based enhancement scheme in diffuse optical tomography," Med. Phys. 30, 861- 869 ( 2003).
[CrossRef] [PubMed]

Hutchinson, C. L.

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, " Fluorescence and absorption contrast mechanism for biomedical optical imaging using frequency domain techniques," Photochem. Photobiol. 66, 55- 64 ( 1997).
[CrossRef] [PubMed]

Jenkins, W. T.

S. A. Vinogradov, L. -W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, and D. F. Wilson, " Noninvasive imaging of the distribution of oxygen in tissue in vivo using near-infrared phosphors," Biophys. J. 70, 1609- 1617 ( 1996).
[CrossRef] [PubMed]

Jiang, H.

X. Gu, Y. Xu, and H. Jiang, " Mesh-based enhancement scheme in diffuse optical tomography," Med. Phys. 30, 861- 869 ( 2003).
[CrossRef] [PubMed]

E. Shives, Y. Xu, and H. Jiang, " Fluorescence lifetime tomography in turbid media based on an oxygen-sensitive dye," Opt. Exp. 10, 1557- 1562 ( 2002), available at http://www.opticsexpress.org.

Joshi, A.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, " Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Exp. 12, 5402- 5417 ( 2004), available at http://www.opticsexpress.org.
[CrossRef]

Khinchin, A. I.

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, New York, 1957).

Kim, E.

I. B. Rietveld, E. Kim, and S. A. Vinogradov, " Dendrimers with tetrabenzoporphyrin cores: Near infrared phosphors for in vivo oxygen imaging," Tetrahedron 59, 3821- 3831 ( 2003).
[CrossRef]

Koch, C.

S. A. Vinogradov, L. -W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, and D. F. Wilson, " Noninvasive imaging of the distribution of oxygen in tissue in vivo using near-infrared phosphors," Biophys. J. 70, 1609- 1617 ( 1996).
[CrossRef] [PubMed]

LaPlant, F. P.

Li, X. D.

Lo, L. -W.

S. A. Vinogradov, L. -W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, and D. F. Wilson, " Noninvasive imaging of the distribution of oxygen in tissue in vivo using near-infrared phosphors," Biophys. J. 70, 1609- 1617 ( 1996).
[CrossRef] [PubMed]

Lopez, G.

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, " Fluorescence and absorption contrast mechanism for biomedical optical imaging using frequency domain techniques," Photochem. Photobiol. 66, 55- 64 ( 1997).
[CrossRef] [PubMed]

Millane, R. P.

Milstein, A. B.

Oh, S.

O'Leary, M. A.

Patterson, M. S.

Pogue, B. W.

Reynolds, J. S.

J. S. Reynolds, C. A. Thompson, K. J. Webb, F. P. LaPlant, and D. Ben-Amotz, " Frequency-domain modeling of reradiation in highly scattering media," Appl. Opt. 36, 2252- 2259 ( 1997).
[CrossRef] [PubMed]

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, " Fluorescence and absorption contrast mechanism for biomedical optical imaging using frequency domain techniques," Photochem. Photobiol. 66, 55- 64 ( 1997).
[CrossRef] [PubMed]

Rietveld, I. B.

I. B. Rietveld, E. Kim, and S. A. Vinogradov, " Dendrimers with tetrabenzoporphyrin cores: Near infrared phosphors for in vivo oxygen imaging," Tetrahedron 59, 3821- 3831 ( 2003).
[CrossRef]

Rumsey, W. L.

W. L. Rumsey, J. M. Vanderkooi, and D. F. Wilson, " Imaging of phosphorescence: A novel method for measuring the distribution of oxygen in perfused tissue," Science 241, 1649- 1651 ( 1988).
[CrossRef] [PubMed]

Rundick, J.

Samarskii, A. A.

A. A. Samarskii and P. N. Vabishev, Chislennye metody reshenia zadach convektsii-diffusii (URSS, 2003; in Russian).

A. A. Samarskii and P. N. Vabishev, Chislennye metody reshenia obratnyh zadach matematicheskoi fiziki (URSS, 2004, in Russian).

Sevick-Muraca, E. M.

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, " Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Exp. 12, 5402- 5417 ( 2004), available at http://www.opticsexpress.org.
[CrossRef]

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, " Fluorescence and absorption contrast mechanism for biomedical optical imaging using frequency domain techniques," Photochem. Photobiol. 66, 55- 64 ( 1997).
[CrossRef] [PubMed]

Shives, E.

E. Shives, Y. Xu, and H. Jiang, " Fluorescence lifetime tomography in turbid media based on an oxygen-sensitive dye," Opt. Exp. 10, 1557- 1562 ( 2002), available at http://www.opticsexpress.org.

Sobolev, V. V.

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).

Soloviev, V.

Soloviev, V. Y.

Thompson, C. A.

Troy, T. L.

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, " Fluorescence and absorption contrast mechanism for biomedical optical imaging using frequency domain techniques," Photochem. Photobiol. 66, 55- 64 ( 1997).
[CrossRef] [PubMed]

Vabishev, P. N.

A. A. Samarskii and P. N. Vabishev, Chislennye metody reshenia zadach convektsii-diffusii (URSS, 2003; in Russian).

A. A. Samarskii and P. N. Vabishev, Chislennye metody reshenia obratnyh zadach matematicheskoi fiziki (URSS, 2004, in Russian).

Vanderkooi, J. M.

W. L. Rumsey, J. M. Vanderkooi, and D. F. Wilson, " Imaging of phosphorescence: A novel method for measuring the distribution of oxygen in perfused tissue," Science 241, 1649- 1651 ( 1988).
[CrossRef] [PubMed]

Vinogradov, S.

Vinogradov, S. A.

V. Y. Soloviev, D. F. Wilson, and S. A. Vinogradov, " Phosphorescence lifetime imaging in turbid media: the inverse problem and experimental image reconstruction," Appl. Opt. 43, 564- 574 ( 2004).
[CrossRef] [PubMed]

I. B. Rietveld, E. Kim, and S. A. Vinogradov, " Dendrimers with tetrabenzoporphyrin cores: Near infrared phosphors for in vivo oxygen imaging," Tetrahedron 59, 3821- 3831 ( 2003).
[CrossRef]

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, and D. F. Wilson, " Frequency domain instrument for measuring phosphorescence lifetime distribution in heterogeneous samples," Rev. Sci. Instrum. 72, 3396- 3406 ( 2001).
[CrossRef]

S. A. Vinogradov, L. -W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, and D. F. Wilson, " Noninvasive imaging of the distribution of oxygen in tissue in vivo using near-infrared phosphors," Biophys. J. 70, 1609- 1617 ( 1996).
[CrossRef] [PubMed]

D. F. Wilson and S. A. Vinogradov, " Tissue oxygen measurements using phosphorescence quenching," in Handbook of Biomedical Fluorescence, M.-A.Mycek and B.W.Pogue, eds. (Marcel Dekker, 2003), pp. 637- 663.

Webb, K. J.

Wilson, D.

Wilson, D. F.

V. Y. Soloviev, D. F. Wilson, and S. A. Vinogradov, " Phosphorescence lifetime imaging in turbid media: the inverse problem and experimental image reconstruction," Appl. Opt. 43, 564- 574 ( 2004).
[CrossRef] [PubMed]

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, and D. F. Wilson, " Frequency domain instrument for measuring phosphorescence lifetime distribution in heterogeneous samples," Rev. Sci. Instrum. 72, 3396- 3406 ( 2001).
[CrossRef]

S. A. Vinogradov, L. -W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, and D. F. Wilson, " Noninvasive imaging of the distribution of oxygen in tissue in vivo using near-infrared phosphors," Biophys. J. 70, 1609- 1617 ( 1996).
[CrossRef] [PubMed]

W. L. Rumsey, J. M. Vanderkooi, and D. F. Wilson, " Imaging of phosphorescence: A novel method for measuring the distribution of oxygen in perfused tissue," Science 241, 1649- 1651 ( 1988).
[CrossRef] [PubMed]

D. F. Wilson and S. A. Vinogradov, " Tissue oxygen measurements using phosphorescence quenching," in Handbook of Biomedical Fluorescence, M.-A.Mycek and B.W.Pogue, eds. (Marcel Dekker, 2003), pp. 637- 663.

Xu, Y.

X. Gu, Y. Xu, and H. Jiang, " Mesh-based enhancement scheme in diffuse optical tomography," Med. Phys. 30, 861- 869 ( 2003).
[CrossRef] [PubMed]

E. Shives, Y. Xu, and H. Jiang, " Fluorescence lifetime tomography in turbid media based on an oxygen-sensitive dye," Opt. Exp. 10, 1557- 1562 ( 2002), available at http://www.opticsexpress.org.

Yodh, A. G.

Zhang, Q.

Appl. Opt. (5)

Biophys. J. (1)

S. A. Vinogradov, L. -W. Lo, W. T. Jenkins, S. M. Evans, C. Koch, and D. F. Wilson, " Noninvasive imaging of the distribution of oxygen in tissue in vivo using near-infrared phosphors," Biophys. J. 70, 1609- 1617 ( 1996).
[CrossRef] [PubMed]

Inverse Probl. (1)

S. Arridge, " Optical tomography in medical imaging," Inverse Probl. 15, R41- R93 ( 1999).
[CrossRef]

J. Opt. Soc. Am. (1)

Med. Phys. (1)

X. Gu, Y. Xu, and H. Jiang, " Mesh-based enhancement scheme in diffuse optical tomography," Med. Phys. 30, 861- 869 ( 2003).
[CrossRef] [PubMed]

Opt. Exp. (2)

A. Joshi, W. Bangerth, and E. M. Sevick-Muraca, " Adaptive finite element based tomography for fluorescence optical imaging in tissue," Opt. Exp. 12, 5402- 5417 ( 2004), available at http://www.opticsexpress.org.
[CrossRef]

E. Shives, Y. Xu, and H. Jiang, " Fluorescence lifetime tomography in turbid media based on an oxygen-sensitive dye," Opt. Exp. 10, 1557- 1562 ( 2002), available at http://www.opticsexpress.org.

Opt. Lett. (1)

Photochem. Photobiol. (1)

E. M. Sevick-Muraca, G. Lopez, J. S. Reynolds, T. L. Troy, and C. L. Hutchinson, " Fluorescence and absorption contrast mechanism for biomedical optical imaging using frequency domain techniques," Photochem. Photobiol. 66, 55- 64 ( 1997).
[CrossRef] [PubMed]

Rev. Sci. Instrum. (1)

S. A. Vinogradov, M. A. Fernandez-Seara, B. W. Dugan, and D. F. Wilson, " Frequency domain instrument for measuring phosphorescence lifetime distribution in heterogeneous samples," Rev. Sci. Instrum. 72, 3396- 3406 ( 2001).
[CrossRef]

Science (1)

W. L. Rumsey, J. M. Vanderkooi, and D. F. Wilson, " Imaging of phosphorescence: A novel method for measuring the distribution of oxygen in perfused tissue," Science 241, 1649- 1651 ( 1988).
[CrossRef] [PubMed]

Tetrahedron (1)

I. B. Rietveld, E. Kim, and S. A. Vinogradov, " Dendrimers with tetrabenzoporphyrin cores: Near infrared phosphors for in vivo oxygen imaging," Tetrahedron 59, 3821- 3831 ( 2003).
[CrossRef]

Other (6)

D. F. Wilson and S. A. Vinogradov, " Tissue oxygen measurements using phosphorescence quenching," in Handbook of Biomedical Fluorescence, M.-A.Mycek and B.W.Pogue, eds. (Marcel Dekker, 2003), pp. 637- 663.

V. V. Sobolev, A Treatise on Radiative Transfer (Van Nostrand, 1963).

G. F. Carey, Computational Grids: Generation, Adaptation, and Solution Strategies (Taylor & Francis, 1997).

A. A. Samarskii and P. N. Vabishev, Chislennye metody reshenia zadach convektsii-diffusii (URSS, 2003; in Russian).

A. A. Samarskii and P. N. Vabishev, Chislennye metody reshenia obratnyh zadach matematicheskoi fiziki (URSS, 2004, in Russian).

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover, New York, 1957).

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Figures (13)

Fig. 1
Fig. 1

(a) Cube refined into small cubes and (b) octal tree data structure for subdivision of cubes.

Fig. 2
Fig. 2

Breastlike phantom used in simulations.

Fig. 3
Fig. 3

Diagram of the source–detector placement on the phantom. Cones show the positions of excitation and detection light guides.

Fig. 4
Fig. 4

Mesh evolution after the second iteration for the gradient-based adaptation criterion.

Fig. 5
Fig. 5

Mesh evolution after the third iteration for the gradient-based adaptation criterion.

Fig. 6
Fig. 6

Mesh evolution after the fourth iteration for the gradient-based adaptation criterion.

Fig. 7
Fig. 7

XZ slice of the reconstructed quantum yield after the first iteration.

Fig. 8
Fig. 8

XZ slice of the reconstructed quantum yield after the second iteration.

Fig. 9
Fig. 9

XZ slice of the reconstructed quantum yield after the third iteration.

Fig. 10
Fig. 10

Mesh evolution after the second iteration for the entropy adaptation criterion.

Fig. 11
Fig. 11

Mesh evolution after the third iteration for the entropy adaptation criterion.

Fig. 12
Fig. 12

XZ slice of the reconstructed quantum yield after the second iteration.

Fig. 13
Fig. 13

XZ slice of the reconstructed quantum yield after the third iteration.

Equations (44)

Equations on this page are rendered with MathJax. Learn more.

I 0 / I = τ 0 / τ = 1 + K SV [ O 2 ] .
( s , I ) = γ 0 I + α λ 4 π 0 2 π d φ 1 1 p ( ϑ ) I ( η , φ ) d η + S ,
S = α λ p ( ϑ 0 ) N 0 h ν exp ( γ 0 | r r 0 | i ω t 0 ) 4 π | r r 0 | 2 ,
γ 0 = α ( 1 + i ω / α c ) ,
cos ϑ = η η + 1 η 2 1 η 2 cos ( φ φ ) .
p ( ϑ ) 1 + ϵ cos ϑ ,
U = 1 c 0 2 π d φ 1 1 I ( η , φ ) d η .
F + q U = f ,
F = ( 1 / γ 1 ) U + ( ϵ / 3 γ 1 ) f s 0 ,
q = 3 ( γ 0 α λ ) ,
γ 1 = γ 0 αλϵ / 3 ,
f = Q 0 4 π | r r 0 | 2 exp ( γ 0 | r r 0 | i ω t 0 ) ,
Q 0 = 3 c α λ N 0 h ν .
( U n F ) V = 0 ,
g ( t t ) = q 0 τ H ( t t ) exp ( t t τ ) ,
N 0 ( r , t ) = 1 λ λ h ν d 3 r   t 0 t U ( t t 0 ) g ( t t ) d t .
t = 1 α c λ 1 λ .
U ( ω ) | ω = 0 = U ( t t 0 ) d t ,
( Δ F x + Δ F y + Δ F z ) ( S 0 / V 0 ) + q i , j , k U i , j , k = f i , j , k ,
Δ F x = F x | i + 1 / 2 F x | i 1 / 2 ,
Δ F y = F y | j + 1 / 2 F y | j 1 / 2 ,
Δ F z = F z | k + 1 / 2 F z | k 1 / 2 .
F x | i + 1 / 2 = { ( U i , j , k U i + 1 , j , k ) / γ 1     ( + ) Δ x , ( i + 1 / 2 , j , k ) V U i + 1 / 2 , j , k , ( i + 1 / 2 , j , k ) V ,
F x | i 1 / 2 = { ( U i 1 , j , k U i , j , k ) / γ 1 ( ) Δ x , ( i 1 / 2 , j , k ) V U i 1 / 2 , j , k , ( i 1 / 2 , j , k ) V ,
γ 1     ( ± ) = 1 V 0 y j 1 / 2 y j + 1 / 2 d y z k 1 / 2 z k + 1 / 2 d z x i 1 / 2 ± 1 / 2 x i + 1 / 2 ± 1 / 2 γ 1 ( x , y , z ) d x .
A n m U m = f n ,
U m = 1 V 0 l V l U l ,
U n     ( s + 1 ) U n     ( s ) ε + A n m U m     ( s ) = f n ,
ε A < 1 ,
f = A 0 U ( r , r 0 ) q 0 ( r ) d 3 r 1 + i ω τ ( r ) ,
( 1 / γ 1 ) G q G = δ ( r r ) ,
δ W = A 0 G ( r , r ) U ( r , r 0 ) q 0 ( r ) d 3 r 1 + i ω τ ( r ) .
W ( r , r 0 ) = A 0 V G ( r , r ) U ( r , r 0 ) q 0 ( r ) d 3 r 1 + iωτ ( r ) .
W ( r , r 0 ) = m K ( r , r 0 , r m ) q 0 ( r m ) V m ,
y = ( K T K + β E ) x ,
x m = q 0 ( r m ) V m ,
y m = K m n          T W n ,
H = m p m log 2 p m ,
p m = x m / x 1 , x 1 = m x m .
h next = M 0 h prev + 3 p m M 0 + 7 .
x m > a h prev x 1 ,
ξ m = x m | q 0 | .
σ = ( 1 M 0 ξ m       2 ) 1 / 2 ,
β = β 0 ( V 0 / V r ) p ,

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