Abstract

Solutions of the time-dependent diffusion equation were developed to take into account the depth of the source and the detector inside a semi-infinite medium. These solutions permitted an evaluation of optical properties at different depths below the surface by fitting time-resolved data. Measurements were performed on liquid optical phantoms with optical fibers for delivering and collecting light. A time-correlated single-photon-counting chain was used for electronic detection. The determination of optical properties underlines the continuity between the surface model and the infinite model and shows the depth at which the derived solutions can be applied.

© 2006 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2004 (1)

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

2001 (1)

A. H. Gandjbakhche, "Diffuse optical imaging and spectroscopy, in vivo," C.R. Acad. Sci. Ser IV: Phys. Astrophys. 2, 1073-1089 (2001).

1998 (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead simplex method in low dimensions," SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

1997 (3)

1996 (1)

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, "Experimental test of theoretical models for time-resolved reflectance," Med. Phys. 23, 1625-1633 (1996).
[CrossRef] [PubMed]

1995 (1)

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, "The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues," Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

1994 (3)

R. C. Haskell, L. O. Svasaand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, "Boundary conditions for the diffusion equation in radiative transfer," J. Opt. Soc. Am. A 11, 2727-2741 (1994).
[CrossRef]

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, "Time-resolved reflectance: a systematic study for application to the optical characterization of tissues," IEEE J. Quantum Electron. 30, 2421-2430 (1994).
[CrossRef]

K. Furutsu and Y. Yamada, "Diffusion-approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

1993 (1)

1992 (2)

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

1991 (2)

S. L. Jacques and S. T. Flock, "Effect of surface boundary on time-resolved reflectance: measurements with a prototype endoscopic catheter," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE 1431, 12-20 (1991).
[CrossRef]

H. G. van Staveren, C. J. M. Moes, J. van Marle, S. A. Prahl, and M. J. C. van Gemert, "Light scattering in Intralipid-10% in the wavelength range of 400-1100 nanometers," Appl. Opt. 30, 4507-4514 (1991).
[CrossRef] [PubMed]

1990 (1)

1989 (2)

1988 (1)

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Alfano, R. R.

Anderson-Engels, S.

Arridge, S.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Berg, R.

Biscotti, G.

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

Chance, B.

Contini, D.

Cope, M.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Cubeddu, R.

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, "Experimental test of theoretical models for time-resolved reflectance," Med. Phys. 23, 1625-1633 (1996).
[CrossRef] [PubMed]

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, "Time-resolved reflectance: a systematic study for application to the optical characterization of tissues," IEEE J. Quantum Electron. 30, 2421-2430 (1994).
[CrossRef]

Delpy, D. T.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

Feng, T. C.

Ferrari, M.

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

Flock, S. T.

S. L. Jacques and S. T. Flock, "Effect of surface boundary on time-resolved reflectance: measurements with a prototype endoscopic catheter," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE 1431, 12-20 (1991).
[CrossRef]

Furutsu, K.

K. Furutsu and Y. Yamada, "Diffusion-approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

Gandjbakhche, A. H.

A. H. Gandjbakhche, "Diffuse optical imaging and spectroscopy, in vivo," C.R. Acad. Sci. Ser IV: Phys. Astrophys. 2, 1073-1089 (2001).

Haskell, R. C.

Hielscher, A. H.

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, "The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues," Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic 1978), Vol. 1.

Jacques, S. L.

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, "The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues," Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

S. L. Jacques and S. T. Flock, "Effect of surface boundary on time-resolved reflectance: measurements with a prototype endoscopic catheter," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE 1431, 12-20 (1991).
[CrossRef]

S. L. Jacques, "Time-resolved reflectance spectroscopy in turbid tissues," IEEE Trans. Biomed. Eng. 36, 1155-1161 (1989).
[CrossRef] [PubMed]

Jarlman, O.

Kienle, A.

Lagarias, J. C.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead simplex method in low dimensions," SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Liu, F.

Madsen, S. J.

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992).
[CrossRef] [PubMed]

Martelli, F.

McAdams, M. S.

Moes, C. J. M.

Musolino, M.

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, "Time-resolved reflectance: a systematic study for application to the optical characterization of tissues," IEEE J. Quantum Electron. 30, 2421-2430 (1994).
[CrossRef]

Patterson, M. S.

A. Kienle and M. S. Patterson, "Improved solutions of the steady-state and the time-resolved diffusion equations for reflectance from a semi-infinite turbid medium," J. Opt. Soc. Am. A 14, 246-254 (1997).
[CrossRef]

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
[CrossRef] [PubMed]

Pifferi, A.

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, "Experimental test of theoretical models for time-resolved reflectance," Med. Phys. 23, 1625-1633 (1996).
[CrossRef] [PubMed]

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, "Time-resolved reflectance: a systematic study for application to the optical characterization of tissues," IEEE J. Quantum Electron. 30, 2421-2430 (1994).
[CrossRef]

Prahl, S. A.

Quaresima, V.

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

Reeds, J. A.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead simplex method in low dimensions," SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Spinelli, L.

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

Svanberg, S.

Svasaand, L. O.

Taddeucci, A.

Taroni, P.

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, "Experimental test of theoretical models for time-resolved reflectance," Med. Phys. 23, 1625-1633 (1996).
[CrossRef] [PubMed]

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, "Time-resolved reflectance: a systematic study for application to the optical characterization of tissues," IEEE J. Quantum Electron. 30, 2421-2430 (1994).
[CrossRef]

Tittel, F. K.

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, "The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues," Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

Torricelli, A.

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, "Experimental test of theoretical models for time-resolved reflectance," Med. Phys. 23, 1625-1633 (1996).
[CrossRef] [PubMed]

Tromberg, B. J.

Tsay, T. T.

Valentini, G.

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, "Experimental test of theoretical models for time-resolved reflectance," Med. Phys. 23, 1625-1633 (1996).
[CrossRef] [PubMed]

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, "Time-resolved reflectance: a systematic study for application to the optical characterization of tissues," IEEE J. Quantum Electron. 30, 2421-2430 (1994).
[CrossRef]

Van der Zee, P.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

van Gemert, M. J. C.

van Marle, J.

van Staveren, H. G.

Wang, L.

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, "The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues," Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

Wilson, B. C.

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, "Time resolved reflectance and transmittance for the noninvasive measurement of tissue optical properties," Appl. Opt. 28, 2331-2336 (1989).
[CrossRef] [PubMed]

Wray, S.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Wright, M. H.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead simplex method in low dimensions," SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Wright, P. E.

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead simplex method in low dimensions," SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Wyatt, J.

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

Yamada, Y.

K. Furutsu and Y. Yamada, "Diffusion-approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

Yoo, K. M.

Zaccanti, G.

Appl. Opt. (4)

C.R. Acad. Sci. (1)

A. H. Gandjbakhche, "Diffuse optical imaging and spectroscopy, in vivo," C.R. Acad. Sci. Ser IV: Phys. Astrophys. 2, 1073-1089 (2001).

IEEE J. Quantum Electron. (1)

R. Cubeddu, M. Musolino, A. Pifferi, P. Taroni, and G. Valentini, "Time-resolved reflectance: a systematic study for application to the optical characterization of tissues," IEEE J. Quantum Electron. 30, 2421-2430 (1994).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

S. L. Jacques, "Time-resolved reflectance spectroscopy in turbid tissues," IEEE Trans. Biomed. Eng. 36, 1155-1161 (1989).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (2)

Med. Phys. (2)

R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, and G. Valentini, "Experimental test of theoretical models for time-resolved reflectance," Med. Phys. 23, 1625-1633 (1996).
[CrossRef] [PubMed]

T. J. Farrell, M. S. Patterson, and B. Wilson, "A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the non-invasive determination of tissue optical properties in vivo," Med. Phys. 19, 879-888 (1992).
[CrossRef] [PubMed]

Opt. Lett. (2)

Phys. Med. Biol. (4)

D. T. Delpy, M. Cope, P. Van der Zee, S. Arridge, S. Wray, and J. Wyatt, "Estimation of optical pathlength through tissue from direct time of flight measurement," Phys. Med. Biol. 33, 1433-1442 (1988).
[CrossRef] [PubMed]

A. H. Hielscher, S. L. Jacques, L. Wang, and F. K. Tittel, "The influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues," Phys. Med. Biol. 40, 1957-1975 (1995).
[CrossRef] [PubMed]

S. J. Madsen, M. S. Patterson, and B. C. Wilson, "The use of India ink as an optical absorber in tissue-simulating phantoms," Phys. Med. Biol. 37, 985-993 (1992).
[CrossRef] [PubMed]

A. Torricelli, V. Quaresima, A. Pifferi, G. Biscotti, L. Spinelli, P. Taroni, M. Ferrari, and R. Cubeddu, "Mapping of calf muscle oxygenation and haemoglobin content during dynamic plantar flexion exercise by multi-channel time-resolved near-infrared spectroscopy," Phys. Med. Biol. 49, 685-699 (2004).
[CrossRef] [PubMed]

Phys. Rev. E (1)

K. Furutsu and Y. Yamada, "Diffusion-approximation for a dissipative random medium and the applications," Phys. Rev. E 50, 3634-3640 (1994).
[CrossRef]

Proc. SPIE (1)

S. L. Jacques and S. T. Flock, "Effect of surface boundary on time-resolved reflectance: measurements with a prototype endoscopic catheter," in Time-Resolved Spectroscopy and Imaging of Tissues, B. Chance, ed., Proc. SPIE 1431, 12-20 (1991).
[CrossRef]

SIAM J. Optim. (1)

J. C. Lagarias, J. A. Reeds, M. H. Wright, and P. E. Wright, "Convergence properties of the Nelder-Mead simplex method in low dimensions," SIAM J. Optim. 9, 112-147 (1998).
[CrossRef]

Other (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic 1978), Vol. 1.

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Figures (10)

Fig. 1
Fig. 1

Configuration of fibers. S, the source fiber; D, the detector fiber.

Fig. 2
Fig. 2

Illustration of the extrapolated geometry.

Fig. 3
Fig. 3

Comparison of photon fluence rate (solid curve) and flux term (dotted curve) for zF = 4 cm with μ a = 0.2 cm−1, μ s ′ = 10 cm−1, and r = 1.2 cm.

Fig. 4
Fig. 4

(a) Effects of increasing the depth of the fibers on the simulated TPSFs. Curves are plotted for zF = 0, 0.2, 0.4, 0.6, 0.8, 1, 2, and 3 cm, with μ a = 0.2 cm−1, μ s ′ = 10 cm−1, and r = 1.2 cm. (b) Same as (a) with curves normalized to the same area.

Fig. 5
Fig. 5

Illustration of the experimental setup.

Fig. 6
Fig. 6

Representation of a typical IRF (dashed curve), experimental data (solid curve), and the result from the fitting procedure (dotted curve). The fitting zone is delimited by the two vertical lines.

Fig. 7
Fig. 7

(a) Measured TPSFs in an optical phantom for zF = 0, 0.1, 0.2, 0.4, 0.6, 0.8, 1, 2, 3, and 4 cm. (b) Same as (a) with curves normalized to the same area.

Fig. 8
Fig. 8

Absorption and reduced scattering coefficient as a function of ink concentration, obtained from the fits of surface (curve with diamonds) and within medium measurements (curve with circles). Reference absorption coefficient is also represented.

Fig. 9
Fig. 9

Absorption and reduced scattering coefficient as a function of Intralipid concentration, obtained from the fits of surface (curve with diamonds) and within medium measurements (curve with circles).

Fig. 10
Fig. 10

(a) Absorption coefficients obtained from the infinite model (curve with circles) and the extrapolated boundary depth dependent model (curve with crosses) versus the position of the fibers below the surface. (b) Same as (a) for the reduced scattering coefficient. (c) and (d) Same as (a) and (b), respectively, for a more absorbing phantom.

Equations (14)

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1 c t ϕ ( r , t ) ( D ( r ) ϕ ( r , t ) ) + μ a ( r ) ϕ ( r , t ) = S ( r , t ) ,
ϕ ( r , z , t ) = c ( 4 π D c t ) 3 / 2 exp ( μ a c t ) × exp ( ( z z 0 ) 2 + r 2 4 D c t ) .
ϕ ( r , t ) = 4 π L ( r , s , t ) d Ω .
L ( r , s , t ) = 1 4 π ϕ ( r , t ) + 3 4 π j ( r , t ) s ,
j ( r , t ) = 4 π L ( r , s , t ) s d Ω .
j ( r , t ) = D ϕ ( r , t ) .
signal ( r , t ) = A F d x d y Ω F T Fres ( s ) L ( r , s , t ) ( s n ) d Ω = A F d x d y Ω F T Fres ( s ) [ 1 4 π ϕ ( r , t ) + 3 4 π j ( r , t ) s ] ( s n ) d Ω = A F d x d y Ω F T Fres ( s ) [ 1 4 π ϕ ( r , t ) + 3 4 π D ϕ z ( r , t ) cos ( θ ) ] cos ( θ ) d θ ,
signal ( r , z F , t ) 1 4 π ϕ ( r , z F , t ) + 3 4 π D ϕ z ( r , z F , t ) .
signal ( r , t ) 1 ( 4 π D c ) 3 / 2 1 2 t 3 / 2 l × exp ( l 2 + r 2 4 D c t μ a c t ) .
ϕ ( r , z , t ) = c ( 4 π D c t ) 3 / 2 exp ( μ a c t ) × [ exp ( ( z z 0 ) 2 + r 2 4 D c t ) exp ( ( z + 2 z B + z 0 ) 2 + r 2 4 D c t ) ] .
signal ( r , z = z F , t ) c ( 4 π D c t ) 3 / 2 exp ( μ a c t ) × [ exp ( ( z F z 0 ) 2 + r 2 4 D c t ) exp ( ( z F + 2 z B + z 0 ) 2 + r 2 4 D c t ) ] + 3 × 1 2 ( 4 π D c ) 3 / 2 t 5 / 2 × exp ( μ a c t ) [ ( z F z 0 ) × exp ( ( z F z 0 ) 2 + r 2 4 D c t ) + ( z F + 2 z B + z 0 ) × exp ( ( z F + 2 z B + z 0 ) 2 + r 2 4 D c t ) ] .
signal ( r , z = 0 , t ) 1 2 ( 4 π D c ) 3 / 2 t 5 / 2 exp ( μ a c t ) × [ z 0 exp ( z 0     2 + r 2 4 D c t ) + ( 2 z B + z 0 ) × exp ( ( 2 z B + z 0 ) 2 + r 2 4 D c t ) ] .
d d t ln ( signal ) = μ a     c 3 2 t + o ( 1 / t 2 ) ,
d d t ln ( signal ebc ) = μ a     ebc c 5 2 t + o ( 1 / t 2 ) .

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