Abstract

This study contains the derivation of an infinite space Green’s function of the time-dependent radiative transfer equation in an anisotropically scattering medium based on analytical approaches. The final solutions are analytical regarding the time variable and given by a superposition of real and complex exponential functions. The obtained expressions were successfully validated with Monte Carlo simulations.

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References

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  1. A. Kienle and R. Hibst, “Light Guiding in Biological Tissue due to Scattering,” Phys. Rev. Lett.97, 018104 (2006).
    [CrossRef] [PubMed]
  2. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  3. J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley & Sons, 1979).
  4. J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E56, 1135–1141 (1997).
    [CrossRef]
  5. F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45, 1359–1373 (2000).
    [CrossRef] [PubMed]
  6. A. D. Klose, U. Netz, and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys.26, 1698–1707 (1999).
    [CrossRef] [PubMed]
  7. P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys.230, 7364–7383 (2011).
    [CrossRef]
  8. F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE Press, 2010).
    [CrossRef]
  9. V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Complex Media14, L13–L19 (2004).
  10. G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A39, 115–137 (2006).
    [CrossRef]
  11. M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A43, 065402 (2010).
    [CrossRef]
  12. W. Cai, M. Lax, and R. R. Alfano, “Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium,” Phys. Rev. E61, 3871–3876 (2000).
    [CrossRef]
  13. M. S. Patterson, B. Chance, and C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt.12, 2331–2336 (1989).
    [CrossRef]
  14. F. Martelli, A. Sassaroli, A. Pifferi, A. Torricelli, L. Spinelli, and G. Zaccanti, “Heuristic Greens function of the time dependent radiative transfer equation for a semi-infinite medium, experimental validation,” Opt. Express15, 18168–18175 (2007).
    [CrossRef] [PubMed]
  15. N. Baddour, “Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates,” J. Opt. Soc. Am. A27, 2144–2155 (2010).
    [CrossRef]
  16. F. R. Gantmacher, The Theory of Matrices (AMS Chelsea Publishing, 1959).
  17. http://www.uni-ulm.de/ilm/index.php?id=10020200 .
  18. L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation,” J. Biomed. Opt.12, 064027 (2007).
    [CrossRef]
  19. L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105, 102025 (2009).
    [CrossRef]
  20. S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys.34, 4545–4557 (2007).
    [CrossRef] [PubMed]

2011 (1)

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys.230, 7364–7383 (2011).
[CrossRef]

2010 (2)

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A43, 065402 (2010).
[CrossRef]

N. Baddour, “Operational and convolution properties of three-dimensional Fourier transforms in spherical polar coordinates,” J. Opt. Soc. Am. A27, 2144–2155 (2010).
[CrossRef]

2009 (1)

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105, 102025 (2009).
[CrossRef]

2007 (3)

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys.34, 4545–4557 (2007).
[CrossRef] [PubMed]

F. Martelli, A. Sassaroli, A. Pifferi, A. Torricelli, L. Spinelli, and G. Zaccanti, “Heuristic Greens function of the time dependent radiative transfer equation for a semi-infinite medium, experimental validation,” Opt. Express15, 18168–18175 (2007).
[CrossRef] [PubMed]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation,” J. Biomed. Opt.12, 064027 (2007).
[CrossRef]

2006 (2)

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A39, 115–137 (2006).
[CrossRef]

A. Kienle and R. Hibst, “Light Guiding in Biological Tissue due to Scattering,” Phys. Rev. Lett.97, 018104 (2006).
[CrossRef] [PubMed]

2004 (1)

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Complex Media14, L13–L19 (2004).

2000 (2)

W. Cai, M. Lax, and R. R. Alfano, “Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium,” Phys. Rev. E61, 3871–3876 (2000).
[CrossRef]

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45, 1359–1373 (2000).
[CrossRef] [PubMed]

1999 (1)

A. D. Klose, U. Netz, and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys.26, 1698–1707 (1999).
[CrossRef] [PubMed]

1997 (1)

J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E56, 1135–1141 (1997).
[CrossRef]

1989 (1)

M. S. Patterson, B. Chance, and C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt.12, 2331–2336 (1989).
[CrossRef]

Alfano, R. R.

W. Cai, M. Lax, and R. R. Alfano, “Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium,” Phys. Rev. E61, 3871–3876 (2000).
[CrossRef]

Alianelli, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45, 1359–1373 (2000).
[CrossRef] [PubMed]

Arridge, S. R.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys.230, 7364–7383 (2011).
[CrossRef]

Baddour, N.

Bassani, M.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45, 1359–1373 (2000).
[CrossRef] [PubMed]

Cai, W.

W. Cai, M. Lax, and R. R. Alfano, “Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium,” Phys. Rev. E61, 3871–3876 (2000).
[CrossRef]

Carpenter, C.

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys.34, 4545–4557 (2007).
[CrossRef] [PubMed]

Case, K. M.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Chance, B.

M. S. Patterson, B. Chance, and C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt.12, 2331–2336 (1989).
[CrossRef]

Chin, L. C. L.

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105, 102025 (2009).
[CrossRef]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation,” J. Biomed. Opt.12, 064027 (2007).
[CrossRef]

Del Bianco, S.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE Press, 2010).
[CrossRef]

Duderstadt, J. J.

J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley & Sons, 1979).

Gantmacher, F. R.

F. R. Gantmacher, The Theory of Matrices (AMS Chelsea Publishing, 1959).

Hibst, R.

A. Kienle and R. Hibst, “Light Guiding in Biological Tissue due to Scattering,” Phys. Rev. Lett.97, 018104 (2006).
[CrossRef] [PubMed]

Hielscher, A. H.

A. D. Klose, U. Netz, and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys.26, 1698–1707 (1999).
[CrossRef] [PubMed]

Ismaelli, A.

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE Press, 2010).
[CrossRef]

Kienle, A.

A. Kienle and R. Hibst, “Light Guiding in Biological Tissue due to Scattering,” Phys. Rev. Lett.97, 018104 (2006).
[CrossRef] [PubMed]

Klose, A. D.

A. D. Klose, U. Netz, and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys.26, 1698–1707 (1999).
[CrossRef] [PubMed]

Lax, M.

W. Cai, M. Lax, and R. R. Alfano, “Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium,” Phys. Rev. E61, 3871–3876 (2000).
[CrossRef]

Lloyd, B.

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105, 102025 (2009).
[CrossRef]

Machida, M.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A43, 065402 (2010).
[CrossRef]

Markel, V. A.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A43, 065402 (2010).
[CrossRef]

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A39, 115–137 (2006).
[CrossRef]

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Complex Media14, L13–L19 (2004).

Martelli, F.

F. Martelli, A. Sassaroli, A. Pifferi, A. Torricelli, L. Spinelli, and G. Zaccanti, “Heuristic Greens function of the time dependent radiative transfer equation for a semi-infinite medium, experimental validation,” Opt. Express15, 18168–18175 (2007).
[CrossRef] [PubMed]

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45, 1359–1373 (2000).
[CrossRef] [PubMed]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE Press, 2010).
[CrossRef]

Martin, W. R.

J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley & Sons, 1979).

Mohan, P. S.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys.230, 7364–7383 (2011).
[CrossRef]

Netz, U.

A. D. Klose, U. Netz, and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys.26, 1698–1707 (1999).
[CrossRef] [PubMed]

Paasschens, J. C. J.

J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E56, 1135–1141 (1997).
[CrossRef]

Panasyuk, G.

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A39, 115–137 (2006).
[CrossRef]

Panasyuk, G. Y.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A43, 065402 (2010).
[CrossRef]

Patterson, M. S.

M. S. Patterson, B. Chance, and C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt.12, 2331–2336 (1989).
[CrossRef]

Paulsen, K.

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys.34, 4545–4557 (2007).
[CrossRef] [PubMed]

Pifferi, A.

Pogue, B. W.

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys.34, 4545–4557 (2007).
[CrossRef] [PubMed]

Pulkkinen, A.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys.230, 7364–7383 (2011).
[CrossRef]

Sassaroli, A.

Schotland, J. C.

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A43, 065402 (2010).
[CrossRef]

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A39, 115–137 (2006).
[CrossRef]

Schweiger, M.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys.230, 7364–7383 (2011).
[CrossRef]

Spinelli, L.

Srinivasan, S.

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys.34, 4545–4557 (2007).
[CrossRef] [PubMed]

Tarvainen, T.

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys.230, 7364–7383 (2011).
[CrossRef]

Torricelli, A.

Vitkin, I. A.

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105, 102025 (2009).
[CrossRef]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation,” J. Biomed. Opt.12, 064027 (2007).
[CrossRef]

Whelan, W. M.

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105, 102025 (2009).
[CrossRef]

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation,” J. Biomed. Opt.12, 064027 (2007).
[CrossRef]

Wilson, C.

M. S. Patterson, B. Chance, and C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt.12, 2331–2336 (1989).
[CrossRef]

Worthington, A. E.

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation,” J. Biomed. Opt.12, 064027 (2007).
[CrossRef]

Yalavarthy, P. K.

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys.34, 4545–4557 (2007).
[CrossRef] [PubMed]

Zaccanti, G.

F. Martelli, A. Sassaroli, A. Pifferi, A. Torricelli, L. Spinelli, and G. Zaccanti, “Heuristic Greens function of the time dependent radiative transfer equation for a semi-infinite medium, experimental validation,” Opt. Express15, 18168–18175 (2007).
[CrossRef] [PubMed]

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45, 1359–1373 (2000).
[CrossRef] [PubMed]

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE Press, 2010).
[CrossRef]

Zangheri, L.

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45, 1359–1373 (2000).
[CrossRef] [PubMed]

Zweifel, P. F.

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

Appl. Opt. (1)

M. S. Patterson, B. Chance, and C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt.12, 2331–2336 (1989).
[CrossRef]

J. Appl. Phys. (1)

L. C. L. Chin, B. Lloyd, W. M. Whelan, and I. A. Vitkin, “Interstitial point radiance spectroscopy of turbid media,” J. Appl. Phys.105, 102025 (2009).
[CrossRef]

J. Biomed. Opt. (1)

L. C. L. Chin, A. E. Worthington, W. M. Whelan, and I. A. Vitkin, “Determination of the optical properties of turbid media using relative interstitial radiance measurements: Monte Carlo study, experimental validation,” J. Biomed. Opt.12, 064027 (2007).
[CrossRef]

J. Comput. Phys. (1)

P. S. Mohan, T. Tarvainen, M. Schweiger, A. Pulkkinen, and S. R. Arridge, “Variable order spherical harmonic expansion scheme for the radiative transport equation using finite elements,” J. Comput. Phys.230, 7364–7383 (2011).
[CrossRef]

J. Opt. Soc. Am. A (1)

J. Phys. A (2)

G. Panasyuk, J. C. Schotland, and V. A. Markel, “Radiative transport equation in rotated reference frames,” J. Phys. A39, 115–137 (2006).
[CrossRef]

M. Machida, G. Y. Panasyuk, J. C. Schotland, and V. A. Markel, “The Green’s function for the radiative transport equation in the slab geometry,” J. Phys. A43, 065402 (2010).
[CrossRef]

Med. Phys. (2)

S. Srinivasan, B. W. Pogue, C. Carpenter, P. K. Yalavarthy, and K. Paulsen, “A boundary element approach for image-guided near-infrared absorption and scatter estimation,” Med. Phys.34, 4545–4557 (2007).
[CrossRef] [PubMed]

A. D. Klose, U. Netz, and A. H. Hielscher, “Iterative reconstruction scheme for optical tomography based on the equation of radiative transfer,” Med. Phys.26, 1698–1707 (1999).
[CrossRef] [PubMed]

Opt. Express (1)

Phys. Med. Biol. (1)

F. Martelli, M. Bassani, L. Alianelli, L. Zangheri, and G. Zaccanti, “Accuracy of the diffusion equation to describe photon migration through an infinite medium: numerical and experimental investigation,” Phys. Med. Biol.45, 1359–1373 (2000).
[CrossRef] [PubMed]

Phys. Rev. E (2)

J. C. J. Paasschens, “Solution of the time-dependent Boltzmann equation,” Phys. Rev. E56, 1135–1141 (1997).
[CrossRef]

W. Cai, M. Lax, and R. R. Alfano, “Cumulant solution of the elastic Boltzmann transport equation in an infinite uniform medium,” Phys. Rev. E61, 3871–3876 (2000).
[CrossRef]

Phys. Rev. Lett. (1)

A. Kienle and R. Hibst, “Light Guiding in Biological Tissue due to Scattering,” Phys. Rev. Lett.97, 018104 (2006).
[CrossRef] [PubMed]

Waves Random Complex Media (1)

V. A. Markel, “Modified spherical harmonics method for solving the radiative transport equation,” Waves Random Complex Media14, L13–L19 (2004).

Other (5)

F. Martelli, S. Del Bianco, A. Ismaelli, and G. Zaccanti, Light Propagation Through Biological Tissue (SPIE Press, 2010).
[CrossRef]

K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).

J. J. Duderstadt and W. R. Martin, Transport Theory (John Wiley & Sons, 1979).

F. R. Gantmacher, The Theory of Matrices (AMS Chelsea Publishing, 1959).

http://www.uni-ulm.de/ilm/index.php?id=10020200 .

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Figures (4)

Fig. 1
Fig. 1

Time-resolved fluence at r = 3.05mm in an anisotropically scattering medium with properties σa = 0.01mm−1, σs = 1mm−1 and g = 0.9.

Fig. 2
Fig. 2

Time-resolved fluence at r = 3.05mm for two different anisotropy factors in a scattering medium with properties σa = 0.1mm−1 and σs = 1mm−1.

Fig. 3
Fig. 3

Time-resolved fluence at r = 1.05mm in an anisotropically scattering medium with properties σa = 0.1mm−1, σs = 1mm−1 and g = 0.9.

Fig. 4
Fig. 4

Time-resolved radiance at r = 2.05mm in an anisotropically scattering medium with properties σa = 0.1mm−1, σs = 1mm−1 and g = 0.8.

Equations (25)

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

1 c I t + μ I r + 1 μ 2 r I μ + σ t I = σ s A I + Q 4 π ,
A I ( r , μ , t ) = f ( s ^ s ^ ) I ( r , μ , t ) d 2 s ^ .
f l = 2 π 1 1 f ( μ ) P l ( μ ) d μ .
Q ( r , t ) = δ ( r ) 4 π r 2 δ ( t ) .
I ( r , cos θ r , t ) = 1 ( 2 π ) 3 I ( k , cos θ k , t ) exp ( j k r ) d 3 k .
e jkr cos ( θ r θ k ) = 4 π l = 0 j l j l ( k r ) m = l l Y l m * ( θ k , φ k ) Y l m ( θ r , φ r ) ,
I l ( k , t ) = 2 π 0 π I ( k , cos θ k , t ) P l ( cos θ k ) sin θ k d θ k
I ( r , μ , t ) = l = 0 2 l + 1 4 π I l ( r , t ) P l ( μ ) ,
I l ( r , t ) = j l 2 π 2 0 I l ( k , t ) j l ( k r ) k 2 d k .
I l ( k , t ) = 4 π ( j ) l 0 I l ( r , t ) j l ( k r ) r 2 d r .
1 c d I l ( k , t ) dt + l 2 l + 1 j k I l 1 ( k , t ) + σ l I l ( k , t ) + l + 1 2 l + 1 j k I l + 1 ( k , t ) = δ ( t ) q l ,
I l ( k , s ) = 𝒧 { I l ( k , t ) } = 0 I l ( k , t ) e s t d t .
Q [ I 0 ( k , s ) , I 1 ( k , s ) , , I N ( k , s ) ] T = q ,
A = ( σ 0 j k 1 3 0 0 0 j k 1 3 σ 1 j 2 k 3 5 0 0 j 2 k 3 5 0 0 0 0 j N k 4 N 2 1 0 0 0 j N k 4 N 2 1 σ N ) .
[ I 0 ( k , s ) , I 1 ( k , s ) , , I N ( k , s ) ] T = D [ A + s c I ] 1 q .
exp ( A t ) = 𝒧 1 { ( s I + A ) 1 } ,
[ I 0 ( k , t ) , I 1 ( k , t ) , , I N ( k , t ) ] T = c D exp ( c A t ) q .
exp ( c A t ) = n = 0 ( c t ) n n ! A n = I c t 1 ! A + ( c t ) 2 2 ! A 2 ( c t ) 3 3 ! A 3 ± ,
[ I 0 ( k , t ) , I 1 ( k , t ) , , I N ( k , t ) ] T = c D U exp ( c Λ t ) U 1 q ,
exp ( c Λ t ) = ( exp ( c λ 0 t ) 0 0 0 exp ( c λ 1 t ) 0 0 0 exp ( c λ N t ) ) .
I l ( k , t ) = c 2 l + 1 n = 0 N n | u ˜ 0 u n | l exp ( c λ n t ) ,
Q ( r , t ) = δ ( r r ) 4 π r 2 δ ( t ) ,
I k l ( t ) = 4 π ( j ) l 0 R I l ( r , t ) j l ( ξ l k r ) r 2 d r ,
I l ( r , t ) = j l 2 π R 3 ξ l k > 0 I k l ( t ) j l ( ξ l k r ) j l + 1 2 ( ξ l k R ) .
Φ ( r , t ) = I ( r , μ , t ) d 2 s ^ = 1 2 r R 2 k = 1 k I k 0 ( t ) sin ( ξ k 0 r ) .

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