Abstract

A numerical method is presented to solve exactly the time-dependent diffusion equation that describes light transport in turbid media. The simulation takes into account spatial variations of the scattering and absorption factors of the medium and the objects as well as random fluctuations of these quantities. The technique is employed to explore the possibility of locating millimeter-sized objects immersed in turbid media from time-gated measurements of the transmitted or reflected (near-infrared) light. The simulation results for tissuelike phantoms are compared with experimental transillumination data, and excellent agreement is found. Simulations of time-gated reflection experiments indicate that it may be possible to detect objects of 1-mm radius.

© 1997 Optical Society of America

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References

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  1. J. C. Hebden, D. J. Hall, M. Firbank, D. T. Delpy, “Time-resolved optical imaging of a solid tissue-equivalent phantom,” Appl. Opt. 34, 8038–8047 (1995).
    [CrossRef] [PubMed]
  2. J. Alper, “Transillumination: looking right through you,” Science 261, 560 (1993).
    [CrossRef] [PubMed]
  3. R. Marchesini, A. Bertoni, S. Andreola, E. Melloni, A. E. Sichirollo, “Extinction and absorption coefficients and scattering phase functions of human tissues in vitro,” Appl. Opt. 28, 2318–2324 (1989).
    [CrossRef] [PubMed]
  4. A. Kienle, L. Lilge, M. S. Patterson, R. Hibst, R. Steiner, B. C. Wilson, “Spatially resolved absolute diffuse reflectance measurements for noninvasive determination of the optical scattering and absorption coefficients of biological tissue,” Appl. Opt. 35, 2304–2314 (1996).
    [CrossRef] [PubMed]
  5. D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
    [CrossRef] [PubMed]
  6. G. Mitic, J. Kölzer, J. Otto, E. Plies, G. Sölkner, W. Zinth, “Time-gated transillumination of biological tissues and tissuelike phantoms,” Appl. Opt. 33, 6699–6710 (1994).
    [CrossRef] [PubMed]
  7. R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
    [CrossRef] [PubMed]
  8. J. Watson, P. Georges, T. Lépine, B. Alonzi, A. Brun, “Imaging in diffuse media with ultrafast degenerate optical parametric amplification,” Opt. Lett. 20, 231–233 (1995).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  11. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
    [CrossRef] [PubMed]
  12. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
    [CrossRef]
  13. B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
    [CrossRef] [PubMed]
  14. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
    [CrossRef] [PubMed]
  15. J. M. Schmitt, A. Knüttel, J. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9, 1832–1843 (1992).
    [CrossRef] [PubMed]
  16. B. Chance, K. Kang, L. He, J. Wang, E. M. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
    [CrossRef] [PubMed]
  17. M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Experimental images of heterogeneous turbid media by frequency-domain diffusing-photon tomography,” Opt. Lett. 20, 426–428 (1995).
    [CrossRef] [PubMed]
  18. If the scattering factors of the object and the medium differ considerably, direct detection is possible: See P. N. den Outer, Th. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
    [CrossRef]
  19. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978).
  20. H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).
  21. A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
    [CrossRef]
  22. M. Suzuki, “General theory of fractal path integrals with applications to many-body theories and statistical physics,” J. Math. Phys. (N.Y.) 32, 400–407 (1991).
    [CrossRef]
  23. H. De Raedt, K. Michielsen, “Algorithm to solve the time-dependent Schrödinger equation for a charged particle in an inhomogeneous magnetic field: application to the Aharonov–Bohm effect,” Comput. Phys. 8, 600–607 (1994).
    [CrossRef]
  24. For the present purpose we employ rectangular boxes, but the algorithm that we use can be modified to handle volumes of arbitrary shape.
  25. In the TDDE the speed of light v sets only the time scale. To rescale the results presented in this paper to the case in which the speed of light in the medium is v′, multiply all times by v/v′.

1996 (2)

1995 (4)

1994 (4)

H. De Raedt, K. Michielsen, “Algorithm to solve the time-dependent Schrödinger equation for a charged particle in an inhomogeneous magnetic field: application to the Aharonov–Bohm effect,” Comput. Phys. 8, 600–607 (1994).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

G. Mitic, J. Kölzer, J. Otto, E. Plies, G. Sölkner, W. Zinth, “Time-gated transillumination of biological tissues and tissuelike phantoms,” Appl. Opt. 33, 6699–6710 (1994).
[CrossRef] [PubMed]

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

1993 (7)

J. B. Fishkin, E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by a straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[CrossRef] [PubMed]

J. Alper, “Transillumination: looking right through you,” Science 261, 560 (1993).
[CrossRef] [PubMed]

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

B. J. Tromberg, L. O. Svaasand, T. T. Tsay, R. C. Haskell, “Properties of photon density waves in multiple-scattering media,” Appl. Opt. 32, 607–616 (1993).
[CrossRef] [PubMed]

If the scattering factors of the object and the medium differ considerably, direct detection is possible: See P. N. den Outer, Th. M. Nieuwenhuizen, A. Lagendijk, “Location of objects in multiple-scattering media,” J. Opt. Soc. Am. A 10, 1209–1218 (1993).
[CrossRef]

B. Chance, K. Kang, L. He, J. Wang, E. M. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

1992 (2)

J. M. Schmitt, A. Knüttel, J. R. Knutson, “Interference of diffusive light waves,” J. Opt. Soc. Am. A 9, 1832–1843 (1992).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

1991 (1)

M. Suzuki, “General theory of fractal path integrals with applications to many-body theories and statistical physics,” J. Math. Phys. (N.Y.) 32, 400–407 (1991).
[CrossRef]

1989 (1)

Alfano, R. R.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

Alonzi, B.

Alper, J.

J. Alper, “Transillumination: looking right through you,” Science 261, 560 (1993).
[CrossRef] [PubMed]

Andreola, S.

Benaron, D. A.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Bertoni, A.

Boas, D. A.

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

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Brun, A.

Chance, B.

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

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

B. Chance, K. Kang, L. He, J. Wang, E. M. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Contini, D.

De Raedt, H.

H. De Raedt, K. Michielsen, “Algorithm to solve the time-dependent Schrödinger equation for a charged particle in an inhomogeneous magnetic field: application to the Aharonov–Bohm effect,” Comput. Phys. 8, 600–607 (1994).
[CrossRef]

Delpy, D. T.

den Outer, P. N.

Firbank, M.

Fishkin, J. B.

Georges, P.

Gratton, E.

Hall, D. J.

Haskell, R. C.

He, L.

B. Chance, K. Kang, L. He, J. Wang, E. M. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

Hebden, J. C.

Hibst, R.

Ho, P. P.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

Ishimaru, A.

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

Kang, K.

B. Chance, K. Kang, L. He, J. Wang, E. M. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

Kienle, A.

Knutson, J. R.

Knüttel, A.

Kölzer, J.

Lagendijk, A.

Lépine, T.

Liang, X.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

Lilge, L.

Liszka, H.

Marchesini, R.

Melloni, E.

Michielsen, K.

H. De Raedt, K. Michielsen, “Algorithm to solve the time-dependent Schrödinger equation for a charged particle in an inhomogeneous magnetic field: application to the Aharonov–Bohm effect,” Comput. Phys. 8, 600–607 (1994).
[CrossRef]

Mitic, G.

Nieuwenhuizen, Th. M.

O’Leary, M. A.

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

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Otto, J.

Patterson, M. S.

Plies, E.

Sassaroli, A.

Schmitt, J. M.

Sevick, E. M.

B. Chance, K. Kang, L. He, J. Wang, E. M. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

Sichirollo, A. E.

Sölkner, G.

Steiner, R.

Stevenson, D. K.

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

Suzuki, M.

M. Suzuki, “General theory of fractal path integrals with applications to many-body theories and statistical physics,” J. Math. Phys. (N.Y.) 32, 400–407 (1991).
[CrossRef]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T. T.

van de Hulst, H. C.

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

Wang, J.

B. Chance, K. Kang, L. He, J. Wang, E. M. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

Wang, L.

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

Watson, J.

Wilson, B. C.

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

Yodh, A. G.

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

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Zaccanti, G.

Zinth, W.

Appl. Opt. (6)

Comput. Phys. (1)

H. De Raedt, K. Michielsen, “Algorithm to solve the time-dependent Schrödinger equation for a charged particle in an inhomogeneous magnetic field: application to the Aharonov–Bohm effect,” Comput. Phys. 8, 600–607 (1994).
[CrossRef]

J. Math. Phys. (N.Y.) (1)

M. Suzuki, “General theory of fractal path integrals with applications to many-body theories and statistical physics,” J. Math. Phys. (N.Y.) 32, 400–407 (1991).
[CrossRef]

J. Opt. Soc. Am. A (3)

Opt. Lett. (2)

Phys. Rev. E (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering and wavelength transduction of diffuse photon density waves,” Phys. Rev. E 47, R2999–R3002 (1993).
[CrossRef]

Phys. Rev. Lett. (1)

M. A. O’Leary, D. A. Boas, B. Chance, A. G. Yodh, “Refraction of diffuse photon density waves,” Phys. Rev. Lett. 69, 2658–2661 (1992).
[CrossRef] [PubMed]

Phys. Today (1)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

Proc. Natl. Acad. Sci. USA (2)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4891 (1994).
[CrossRef] [PubMed]

B. Chance, K. Kang, L. He, J. Wang, E. M. Sevick, “Highly sensitive object location in tissue models with linear in-phase and anti-phase multi-element optical arrays in one and two dimensions,” Proc. Natl. Acad. Sci. USA 90, 3423–3427 (1993).
[CrossRef] [PubMed]

Science (3)

R. R. Alfano, X. Liang, L. Wang, P. P. Ho, “Time-resolved imaging of translucent droplets in highly scattering turbid media,” Science 264, 1913–1915 (1994).
[CrossRef] [PubMed]

D. A. Benaron, D. K. Stevenson, “Optical time-of-flight and absorbance imaging of biologic media,” Science 259, 1463–1466 (1993).
[CrossRef] [PubMed]

J. Alper, “Transillumination: looking right through you,” Science 261, 560 (1993).
[CrossRef] [PubMed]

Other (4)

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

H. C. van de Hulst, Multiple Light Scattering (Academic, New York, 1980).

For the present purpose we employ rectangular boxes, but the algorithm that we use can be modified to handle volumes of arbitrary shape.

In the TDDE the speed of light v sets only the time scale. To rescale the results presented in this paper to the case in which the speed of light in the medium is v′, multiply all times by v/v′.

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Figures (7)

Fig. 1
Fig. 1

Comparison of experimental6 (solid curves) and computer simulation (dashed curves) results for the time-gated transilluminated diffusive light intensity. In experiment and simulation the turbid medium has a reduced scattering factor μs=0.9 mm-1 and an absorption factor μa<0.001 mm-1; the absorption factor of the 8-mm-diameter tube is μa =0.014 mm-1 (Ref. 6). The reduced scattering factor inside the object is μs=0. As in Fig. 12(a) of Ref. 6: 1, continuous-wave case (experimental data only); 2, Δt=960 ps; 3, Δt =480 ps; 4, Δt=240 ps; 5, Δt=30 ps. The inset shows the light distribution inside the sample for Δt=960 ps; the arrow indicates the direction of the incident light. See also Fig. 12(a) of Ref. 6.

Fig. 2
Fig. 2

Same as Fig. 1, except that the absorption factor of the 8-mm-diameter tube is μa=0.13 mm-1 (Ref. 6). See also Fig. 12(b) of Ref. 6.

Fig. 3
Fig. 3

Same as Fig. 1, except that instead of one there are two 10-mm-diameter objects, separated by 20 mm, with an absorption factor μa=0.029 mm-1 (Ref. 6). See also Fig. 13(a) of Ref. 6.

Fig. 4
Fig. 4

Same as Fig. 1, except that the turbid medium has a reduced scattering factor μs=0.8 mm-1 and that the absorption and reduced scattering factors of the 8-mm-diameter tube are μa=0.1 mm-1 and μs=0.8 mm-1, respectively.6 See also Fig. 15 of Ref. 6.

Fig. 5
Fig. 5

Same as Fig. 1, except that the turbid medium has a reduced scattering factor μs=0.8 mm-1 and an absorption factor μa =0.0005 mm-1 and that the absorption and the scattering factors of the 8-mm-diameter tube are μa=0.0005 mm-1 and μs=2.6 mm-1, respectively.6 See also Fig. 14 of Ref. 6.

Fig. 6
Fig. 6

Simulation of a time-gated transillumination experiment on a turbid medium with a reduced scattering factor μs =1.1 mm-1 and an absorption factor μa<0.001 mm-1 containing a 4-mm-radius object with absorption and reduced scattering factors μa=0.11 mm-1 and μs=1.1 mm-1, respectively. The dimensions of the sample are 63 mm×63 mm×63 mm. The object is located in the middle of the sample. The transmitted intensity for Δt=1.7 ns is projected onto the rightmost plane of the sample.

Fig. 7
Fig. 7

Simulation of a time-gated reflection experiment on a turbid medium with a reduced scattering factor μs=1.1 mm-1 and an absorption factor μa<0.001 mm-1 containing a 1-mm-radius object with absorption and reduced scattering factors μa=0.11 mm-1 and μs=1.1 mm-1, respectively. The dimensions of the sample are 63 mm×127 mm. Curves from bottom to top: Δt =30 ps, Δt=240 ps, Δt =480 ps, Δt=960 ps. The inset shows the light distribution inside the sample for Δt=960 ps; the arrow indicates the direction of the incident light.

Equations (2)

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I(r, t)t=D(r)I(r, t)-vμa(r)I(r, t)+S(r, t),
I(r, t+τ)=exp(-τH)I(r, t)+0τdτ×exp(τH)S(r, t+τ),

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