Abstract

The PN-method is a spectral discretization technique used to obtain numerical solutions to the radiative transport equation. To the best of our knowledge, the PN-method has yet to be generalized to the case of refractive index mismatch in layered slabs used to numerically simulate skin. Our main contribution is the application of a collocation method that takes into account refractive index mismatch at layer interfaces. The stability, convergence, and accuracy of the method are established. Example calculations demonstrating the flexibility of the method are performed.

© 2009 Optical Society of America

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  1. A. J. Welch and M. J. C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, 1995).
  2. M. Niemz, Laser-Tissue Interactions: Fundamentals and Applications (Springer-Verlag, 2006).
  3. R. Samatham, S. L. Jacques, and P. Campagnola, “Optical properties of mutant versus wild-type mouse skin measured by reflectance-mode confocal scanning laser microscopy (rcslm),” J. Biomed. Opt. 13, 041309 (2008).
    [CrossRef] [PubMed]
  4. K. Ren, G. Bal, and A. Hielscher, “Transport- and diffusion-based optical tomography in small domains: a comparative study,” Appl. Opt. 46, 6669-6679 (2007).
    [CrossRef] [PubMed]
  5. S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32, 559-568 (1993).
    [CrossRef] [PubMed]
  6. P. González-Rodríguez and A. D. Kim, “Reflectance optical tomography in epithelial tissues,” Inverse Probl. 25, 015001 (2009).
    [CrossRef]
  7. L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml--Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
    [CrossRef] [PubMed]
  8. K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502-2509 (1988).
    [CrossRef] [PubMed]
  9. A. D. Kim and M. Moscoso, “Light transport in two-layer tissues,” J. Biomed. Opt. 10, 034015 (2005).
    [CrossRef] [PubMed]
  10. P. González-Rodríguez and A. D. Kim, “Light propagation in two-layer tissues with an irregular interface,” J. Opt. Soc. Am. A 25, 64-73 (2008).
    [CrossRef]
  11. R. Elaloufi, S. Arridge, R. Pierrat, and R. Carminati, “Light propagation in multilayered scattering media beyond the diffusive regime,” Appl. Opt. 46, 2528-2539 (2007).
    [CrossRef] [PubMed]
  12. Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431-442 (1994).
    [CrossRef] [PubMed]
  13. R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “On the use of Fresnel boundary and interface conditions in radiative-transfer calculations for multilayered media,” J. Quant. Spectrosc. Radiat. Transf. 109, 752-769 (2008).
    [CrossRef]
  14. R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “Radiative transfer in a multi-layer medium subject to Fresnel boundary and interface conditions and uniform illumination by obliquely incident parallel rays,” J. Quant. Spectrosc. Radiat. Transf. 109, 2151-2170 (2008).
    [CrossRef]
  15. C. Bordier, C. Andraud, and J. Lafait, “Model of light scattering that includes polarization effects by multilayered media,” J. Opt. Soc. Am. A 25, 1406-1419 (2008).
    [CrossRef]
  16. M. Elias and G. Elias, “Radiative transfer in inhomogenous stratefied scattering media with use of the auxiliary function method,” J. Opt. Soc. Am. A 21, 580-589 (2004).
    [CrossRef]
  17. C. Magnain, M. Elias, and J.-M. Frigerio, “Skin color modeling using the radiative transfer equation solved by the auxiliary function method,” J. Opt. Soc. Am. A 24, 2196-2205 (2007).
    [CrossRef]
  18. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).
  19. F. Fabbri, M. A. Franceschini, and S. Fantini, “Characterization of spatial and temporal variations in the optical properties of tissuelike media with diffuse reflectance imaging,” Appl. Opt. 42, 3063-3071 (2003).
    [CrossRef] [PubMed]
  20. I. Nishidate, T. Maeda, Y. Aizu, and K. Niizeki, “Visualizing depth and thickness of a local blood region in skin tissue using diffuse relfectance images,” J. Biomed. Opt. 12, 054006-1-12 (2007).
    [CrossRef]
  21. C. Magnain, M. Elias, and J.-M. Frigerio, “Skin color modeling using the radiative transfer equation solved by the auxiliary function method: inverse problem,” J. Opt. Soc. Am. A 25, 1737-1743 (2008).
    [CrossRef]
  22. A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23-42 (2006).
    [CrossRef]
  23. D. J. Cuccia, F. Bevilacqua, A. J. Durkin, and B. J. Tromberg, “Modulated imaging: quantitative analysis and tomography of turbid media in the spatial-frequency domain,” Opt. Lett. 30, 1354-1356 (2005).
    [CrossRef] [PubMed]
  24. K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE 6864, 68640Z (2008).
    [CrossRef]
  25. K. G. Phillips and C. Lancellotti, “On the accuracy of generalized Fokker-Planck equations in tissue optics,” Appl. Opt. 48, 229-241 (2009).
    [CrossRef] [PubMed]
  26. L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
    [CrossRef]
  27. Michael I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41, 7114-7134 (2002).
    [CrossRef] [PubMed]
  28. G. C. Pomraning, “The Fokker-Planck equation as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21-36 (1992).
    [CrossRef]
  29. E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413-423 (1999).
    [CrossRef]
  30. A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92-98 (2003).
    [CrossRef]
  31. P. González-Rodríguez and A. D. Kim, “Light propagation in tissues with forward-peaked and large angle scattering,” Appl. Opt. 47, 2599-2609 (2008).
    [CrossRef] [PubMed]
  32. K. M. Case and P. F. Zweifel, Linear Transport Theory (Addison-Wesley, 1967).
  33. A. E. Profio, “Light transport in tissue,” Appl. Opt. 28, 2216-2222 (1989).
    [CrossRef] [PubMed]
  34. A. D. Kim, “Transport theory for light propagation in biological tissue,” J. Opt. Soc. Am. A 21, 797-803 (2004).
    [CrossRef]
  35. A. D. Kim and M. Moscoso, “Chebyshev spectral method for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074-2094 (2002).
    [CrossRef]
  36. B. Davison, Neutron Transport Theory (Oxford Univ. Press, 1957).
  37. Carl Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2001).
  38. A. Ishimaru, Wave Propagation and Scattering in Random Media, I (Academic, 1978).
  39. L. Wang and S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746-1752 (1993).
    [CrossRef]
  40. http://omlc.ogi.edu/calc/mie_calc.html (August 2009).
  41. L. F. A. Douven and G. W. Lucassen, “Retrieval of optical properties of skin from measurement and modelling the diffuse reflectance,” Proc. SPIE 3914, 312-323 (2000).
    [CrossRef]
  42. http://omlc.ogi.edu/news/jan98/skinoptics.html (August 2009).
  43. S. L. Jacques, “Origins of tissue optical properties in the uva, visible, and nir regions,” in OSA TOPS Advances in Optical Imaging and Photon Migration, Vol. 2, R.R.Alfano and JamesG.Fujimoto, eds. (Optical Society of America, 1996), pp. 364-371.

2009 (2)

P. González-Rodríguez and A. D. Kim, “Reflectance optical tomography in epithelial tissues,” Inverse Probl. 25, 015001 (2009).
[CrossRef]

K. G. Phillips and C. Lancellotti, “On the accuracy of generalized Fokker-Planck equations in tissue optics,” Appl. Opt. 48, 229-241 (2009).
[CrossRef] [PubMed]

2008 (8)

K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE 6864, 68640Z (2008).
[CrossRef]

P. González-Rodríguez and A. D. Kim, “Light propagation in tissues with forward-peaked and large angle scattering,” Appl. Opt. 47, 2599-2609 (2008).
[CrossRef] [PubMed]

R. Samatham, S. L. Jacques, and P. Campagnola, “Optical properties of mutant versus wild-type mouse skin measured by reflectance-mode confocal scanning laser microscopy (rcslm),” J. Biomed. Opt. 13, 041309 (2008).
[CrossRef] [PubMed]

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “On the use of Fresnel boundary and interface conditions in radiative-transfer calculations for multilayered media,” J. Quant. Spectrosc. Radiat. Transf. 109, 752-769 (2008).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “Radiative transfer in a multi-layer medium subject to Fresnel boundary and interface conditions and uniform illumination by obliquely incident parallel rays,” J. Quant. Spectrosc. Radiat. Transf. 109, 2151-2170 (2008).
[CrossRef]

C. Bordier, C. Andraud, and J. Lafait, “Model of light scattering that includes polarization effects by multilayered media,” J. Opt. Soc. Am. A 25, 1406-1419 (2008).
[CrossRef]

P. González-Rodríguez and A. D. Kim, “Light propagation in two-layer tissues with an irregular interface,” J. Opt. Soc. Am. A 25, 64-73 (2008).
[CrossRef]

C. Magnain, M. Elias, and J.-M. Frigerio, “Skin color modeling using the radiative transfer equation solved by the auxiliary function method: inverse problem,” J. Opt. Soc. Am. A 25, 1737-1743 (2008).
[CrossRef]

2007 (4)

2006 (1)

A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23-42 (2006).
[CrossRef]

2005 (2)

2004 (2)

2003 (2)

2002 (2)

2000 (1)

L. F. A. Douven and G. W. Lucassen, “Retrieval of optical properties of skin from measurement and modelling the diffuse reflectance,” Proc. SPIE 3914, 312-323 (2000).
[CrossRef]

1999 (1)

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413-423 (1999).
[CrossRef]

1996 (1)

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

1995 (1)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml--Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

1994 (1)

1993 (2)

1992 (1)

G. C. Pomraning, “The Fokker-Planck equation as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21-36 (1992).
[CrossRef]

1989 (1)

1988 (1)

Aizu, Y.

I. Nishidate, T. Maeda, Y. Aizu, and K. Niizeki, “Visualizing depth and thickness of a local blood region in skin tissue using diffuse relfectance images,” J. Biomed. Opt. 12, 054006-1-12 (2007).
[CrossRef]

Andraud, C.

Arridge, S.

Bal, G.

Bevilacqua, F.

Bordier, C.

Campagnola, P.

R. Samatham, S. L. Jacques, and P. Campagnola, “Optical properties of mutant versus wild-type mouse skin measured by reflectance-mode confocal scanning laser microscopy (rcslm),” J. Biomed. Opt. 13, 041309 (2008).
[CrossRef] [PubMed]

Canuto, C.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

Carminati, R.

Case, K. M.

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

Cuccia, D. J.

Davison, B.

B. Davison, Neutron Transport Theory (Oxford Univ. Press, 1957).

Douven, L. F. A.

L. F. A. Douven and G. W. Lucassen, “Retrieval of optical properties of skin from measurement and modelling the diffuse reflectance,” Proc. SPIE 3914, 312-323 (2000).
[CrossRef]

Durkin, A. J.

Elaloufi, R.

Elias, G.

Elias, M.

Fabbri, F.

Fantini, S.

Franceschini, M. A.

Frigerio, J.-M.

Garcia, R. D. M.

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “On the use of Fresnel boundary and interface conditions in radiative-transfer calculations for multilayered media,” J. Quant. Spectrosc. Radiat. Transf. 109, 752-769 (2008).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “Radiative transfer in a multi-layer medium subject to Fresnel boundary and interface conditions and uniform illumination by obliquely incident parallel rays,” J. Quant. Spectrosc. Radiat. Transf. 109, 2151-2170 (2008).
[CrossRef]

González-Rodríguez, P.

Hielscher, A.

Hussaini, M. Y.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

Ishimaru, A.

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

Jacques, S. L.

R. Samatham, S. L. Jacques, and P. Campagnola, “Optical properties of mutant versus wild-type mouse skin measured by reflectance-mode confocal scanning laser microscopy (rcslm),” J. Biomed. Opt. 13, 041309 (2008).
[CrossRef] [PubMed]

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml--Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

L. Wang and S. L. Jacques, “Hybrid model of Monte Carlo simulation and diffusion theory for light reflectance by turbid media,” J. Opt. Soc. Am. A 10, 1746-1752 (1993).
[CrossRef]

S. L. Jacques, “Origins of tissue optical properties in the uva, visible, and nir regions,” in OSA TOPS Advances in Optical Imaging and Photon Migration, Vol. 2, R.R.Alfano and JamesG.Fujimoto, eds. (Optical Society of America, 1996), pp. 364-371.

Jayaweera, K.

Jin, Z.

Keller, J. B.

A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92-98 (2003).
[CrossRef]

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Kim, A. D.

P. González-Rodríguez and A. D. Kim, “Reflectance optical tomography in epithelial tissues,” Inverse Probl. 25, 015001 (2009).
[CrossRef]

P. González-Rodríguez and A. D. Kim, “Light propagation in two-layer tissues with an irregular interface,” J. Opt. Soc. Am. A 25, 64-73 (2008).
[CrossRef]

P. González-Rodríguez and A. D. Kim, “Light propagation in tissues with forward-peaked and large angle scattering,” Appl. Opt. 47, 2599-2609 (2008).
[CrossRef] [PubMed]

A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23-42 (2006).
[CrossRef]

A. D. Kim and M. Moscoso, “Light transport in two-layer tissues,” J. Biomed. Opt. 10, 034015 (2005).
[CrossRef] [PubMed]

A. D. Kim, “Transport theory for light propagation in biological tissue,” J. Opt. Soc. Am. A 21, 797-803 (2004).
[CrossRef]

A. D. Kim and J. B. Keller, “Light propagation in biological tissue,” J. Opt. Soc. Am. A 20, 92-98 (2003).
[CrossRef]

A. D. Kim and M. Moscoso, “Chebyshev spectral method for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074-2094 (2002).
[CrossRef]

Lafait, J.

Lancellotti, C.

K. G. Phillips and C. Lancellotti, “On the accuracy of generalized Fokker-Planck equations in tissue optics,” Appl. Opt. 48, 229-241 (2009).
[CrossRef] [PubMed]

K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE 6864, 68640Z (2008).
[CrossRef]

Larsen, E. W.

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413-423 (1999).
[CrossRef]

Lucassen, G. W.

L. F. A. Douven and G. W. Lucassen, “Retrieval of optical properties of skin from measurement and modelling the diffuse reflectance,” Proc. SPIE 3914, 312-323 (2000).
[CrossRef]

Maeda, T.

I. Nishidate, T. Maeda, Y. Aizu, and K. Niizeki, “Visualizing depth and thickness of a local blood region in skin tissue using diffuse relfectance images,” J. Biomed. Opt. 12, 054006-1-12 (2007).
[CrossRef]

Magnain, C.

Meyer, Carl

Carl Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2001).

Mishchenko, Michael I.

Moscoso, M.

A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23-42 (2006).
[CrossRef]

A. D. Kim and M. Moscoso, “Light transport in two-layer tissues,” J. Biomed. Opt. 10, 034015 (2005).
[CrossRef] [PubMed]

A. D. Kim and M. Moscoso, “Chebyshev spectral method for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074-2094 (2002).
[CrossRef]

Niemz, M.

M. Niemz, Laser-Tissue Interactions: Fundamentals and Applications (Springer-Verlag, 2006).

Niizeki, K.

I. Nishidate, T. Maeda, Y. Aizu, and K. Niizeki, “Visualizing depth and thickness of a local blood region in skin tissue using diffuse relfectance images,” J. Biomed. Opt. 12, 054006-1-12 (2007).
[CrossRef]

Nishidate, I.

I. Nishidate, T. Maeda, Y. Aizu, and K. Niizeki, “Visualizing depth and thickness of a local blood region in skin tissue using diffuse relfectance images,” J. Biomed. Opt. 12, 054006-1-12 (2007).
[CrossRef]

Papanicolaou, G.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Phillips, K. G.

K. G. Phillips and C. Lancellotti, “On the accuracy of generalized Fokker-Planck equations in tissue optics,” Appl. Opt. 48, 229-241 (2009).
[CrossRef] [PubMed]

K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE 6864, 68640Z (2008).
[CrossRef]

Pierrat, R.

Pomraning, G. C.

G. C. Pomraning, “The Fokker-Planck equation as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21-36 (1992).
[CrossRef]

Prahl, S. A.

Profio, A. E.

Quarteroni, A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

Ren, K.

Ryzhik, L.

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Samatham, R.

R. Samatham, S. L. Jacques, and P. Campagnola, “Optical properties of mutant versus wild-type mouse skin measured by reflectance-mode confocal scanning laser microscopy (rcslm),” J. Biomed. Opt. 13, 041309 (2008).
[CrossRef] [PubMed]

Siewert, C. E.

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “Radiative transfer in a multi-layer medium subject to Fresnel boundary and interface conditions and uniform illumination by obliquely incident parallel rays,” J. Quant. Spectrosc. Radiat. Transf. 109, 2151-2170 (2008).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “On the use of Fresnel boundary and interface conditions in radiative-transfer calculations for multilayered media,” J. Quant. Spectrosc. Radiat. Transf. 109, 752-769 (2008).
[CrossRef]

Stamnes, K.

Tromberg, B. J.

Tsay, S.-C.

van Gemert, M. J. C.

Wang, L.

Wang, L. H.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml--Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Welch, A. J.

Wiscombe, W.

Yacout, A. M.

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “On the use of Fresnel boundary and interface conditions in radiative-transfer calculations for multilayered media,” J. Quant. Spectrosc. Radiat. Transf. 109, 752-769 (2008).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “Radiative transfer in a multi-layer medium subject to Fresnel boundary and interface conditions and uniform illumination by obliquely incident parallel rays,” J. Quant. Spectrosc. Radiat. Transf. 109, 2151-2170 (2008).
[CrossRef]

Zang, T. A.

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

Zheng, L. Q.

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml--Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Zweifel, P. F.

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

Appl. Opt. (10)

K. Stamnes, S.-C. Tsay, W. Wiscombe, and K. Jayaweera, “Numerically stable algorithm for discrete-ordinate-method radiative transfer in multiple scattering and emitting layered media,” Appl. Opt. 27, 2502-2509 (1988).
[CrossRef] [PubMed]

A. E. Profio, “Light transport in tissue,” Appl. Opt. 28, 2216-2222 (1989).
[CrossRef] [PubMed]

S. A. Prahl, M. J. C. van Gemert, and A. J. Welch, “Determining the optical properties of turbid media by using the adding-doubling method,” Appl. Opt. 32, 559-568 (1993).
[CrossRef] [PubMed]

Z. Jin and K. Stamnes, “Radiative transfer in nonuniformly refracting layered media: atmosphere-ocean system,” Appl. Opt. 33, 431-442 (1994).
[CrossRef] [PubMed]

Michael I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41, 7114-7134 (2002).
[CrossRef] [PubMed]

F. Fabbri, M. A. Franceschini, and S. Fantini, “Characterization of spatial and temporal variations in the optical properties of tissuelike media with diffuse reflectance imaging,” Appl. Opt. 42, 3063-3071 (2003).
[CrossRef] [PubMed]

R. Elaloufi, S. Arridge, R. Pierrat, and R. Carminati, “Light propagation in multilayered scattering media beyond the diffusive regime,” Appl. Opt. 46, 2528-2539 (2007).
[CrossRef] [PubMed]

K. Ren, G. Bal, and A. Hielscher, “Transport- and diffusion-based optical tomography in small domains: a comparative study,” Appl. Opt. 46, 6669-6679 (2007).
[CrossRef] [PubMed]

P. González-Rodríguez and A. D. Kim, “Light propagation in tissues with forward-peaked and large angle scattering,” Appl. Opt. 47, 2599-2609 (2008).
[CrossRef] [PubMed]

K. G. Phillips and C. Lancellotti, “On the accuracy of generalized Fokker-Planck equations in tissue optics,” Appl. Opt. 48, 229-241 (2009).
[CrossRef] [PubMed]

Comput. Methods Programs Biomed. (1)

L. H. Wang, S. L. Jacques, and L. Q. Zheng, “Mcml--Monte Carlo modeling of photon transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131-146 (1995).
[CrossRef] [PubMed]

Inverse Probl. (2)

P. González-Rodríguez and A. D. Kim, “Reflectance optical tomography in epithelial tissues,” Inverse Probl. 25, 015001 (2009).
[CrossRef]

A. D. Kim and M. Moscoso, “Radiative transport theory for optical molecular imaging,” Inverse Probl. 22, 23-42 (2006).
[CrossRef]

J. Biomed. Opt. (3)

I. Nishidate, T. Maeda, Y. Aizu, and K. Niizeki, “Visualizing depth and thickness of a local blood region in skin tissue using diffuse relfectance images,” J. Biomed. Opt. 12, 054006-1-12 (2007).
[CrossRef]

R. Samatham, S. L. Jacques, and P. Campagnola, “Optical properties of mutant versus wild-type mouse skin measured by reflectance-mode confocal scanning laser microscopy (rcslm),” J. Biomed. Opt. 13, 041309 (2008).
[CrossRef] [PubMed]

A. D. Kim and M. Moscoso, “Light transport in two-layer tissues,” J. Biomed. Opt. 10, 034015 (2005).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (8)

J. Quant. Spectrosc. Radiat. Transf. (2)

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “On the use of Fresnel boundary and interface conditions in radiative-transfer calculations for multilayered media,” J. Quant. Spectrosc. Radiat. Transf. 109, 752-769 (2008).
[CrossRef]

R. D. M. Garcia, C. E. Siewert, and A. M. Yacout, “Radiative transfer in a multi-layer medium subject to Fresnel boundary and interface conditions and uniform illumination by obliquely incident parallel rays,” J. Quant. Spectrosc. Radiat. Transf. 109, 2151-2170 (2008).
[CrossRef]

Math. Models Meth. Appl. Sci. (1)

G. C. Pomraning, “The Fokker-Planck equation as an asymptotic limit,” Math. Models Meth. Appl. Sci. 2, 21-36 (1992).
[CrossRef]

Opt. Lett. (1)

Proc. SPIE (2)

K. G. Phillips and C. Lancellotti, “A universal numerical treatment of radiative transport equations with differential and integral scattering operators,” Proc. SPIE 6864, 68640Z (2008).
[CrossRef]

L. F. A. Douven and G. W. Lucassen, “Retrieval of optical properties of skin from measurement and modelling the diffuse reflectance,” Proc. SPIE 3914, 312-323 (2000).
[CrossRef]

Prog. Nucl. Energy (1)

E. W. Larsen, “The linear Boltzmann equation in optically thick systems with forward-peaked scattering,” Prog. Nucl. Energy 34, 413-423 (1999).
[CrossRef]

SIAM J. Sci. Comput. (USA) (1)

A. D. Kim and M. Moscoso, “Chebyshev spectral method for radiative transfer,” SIAM J. Sci. Comput. (USA) 23, 2074-2094 (2002).
[CrossRef]

Wave Motion (1)

L. Ryzhik, G. Papanicolaou, and J. B. Keller, “Transport equations for elastic and other waves in random media,” Wave Motion 24, 327-370 (1996).
[CrossRef]

Other (10)

B. Davison, Neutron Transport Theory (Oxford Univ. Press, 1957).

Carl Meyer, Matrix Analysis and Applied Linear Algebra (SIAM, 2001).

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

http://omlc.ogi.edu/calc/mie_calc.html (August 2009).

http://omlc.ogi.edu/news/jan98/skinoptics.html (August 2009).

S. L. Jacques, “Origins of tissue optical properties in the uva, visible, and nir regions,” in OSA TOPS Advances in Optical Imaging and Photon Migration, Vol. 2, R.R.Alfano and JamesG.Fujimoto, eds. (Optical Society of America, 1996), pp. 364-371.

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

C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang, Spectral Methods: Fundamentals in Single Domains (Springer-Verlag, 2006).

A. J. Welch and M. J. C. van Gemert, Optical-Thermal Response of Laser-Irradiated Tissue (Plenum, 1995).

M. Niemz, Laser-Tissue Interactions: Fundamentals and Applications (Springer-Verlag, 2006).

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Figures (10)

Fig. 1
Fig. 1

Energy conservation of the P N -method for the one-, two-, and three-layer index-mismatched problems. Errors in the total energy E o = 1 of the order of 0.1% are achieved for N 68 . The optical properties of the layers are σ s = 100 cm 1 , σ a = 1 cm 1 , g = 0.9 . The thickness of each slab layer is 0.1 cm . Single-layer parameters: n 0 = 1 , n 1 = 1.37 , n 2 = 1 , two-layer parameters: n 0 = 1 , n 1 = 1.37 , n 2 = 1.4 , n 3 = 1 , three-layer parameters: n 0 = 1 , n 1 = 1.37 , n 2 = 1.4 , n 3 = 1.45 , n 4 = 1 .

Fig. 2
Fig. 2

Convergence of the P N -method assessed through the decay of the relative error E N ( R ) for the one-, two-, and three-layer index matched/mismatched problems. The geometry and optical properties of the layers are the same as in Fig. 1.

Fig. 3
Fig. 3

Accuracy of the P N -method assessed through the convergence of the relative error with Monte Carlo simulation for the one-, two-, and three-layer index matched/mismatched problems. The geometry and optical properties of the layers are the same as in Fig. 1.

Fig. 4
Fig. 4

Computation time in seconds required for the one-, two-, and three-layer index matched/mismatched problems using the P N -method. All calculations were done on an Intel Core 2 Duo Windows machine with 2 GB of RAM. The geometry and optical properties of the layers are the same as for Fig. 1.

Fig. 5
Fig. 5

Comparison of the backscattered flux J ( μ ) from a three-layer system of slabs under different assumptions of the tissue–light interaction. Table 2 presents the optical properties used in each of the 0.1 cm thick layers.

Fig. 6
Fig. 6

Comparison of the transmitted flux J + ( μ ) from a three-layer system of slabs under different assumptions of the tissue–light interaction. Table 2 presents the optical properties used in each of the 0.1 cm thick layers.

Fig. 7
Fig. 7

Comparison of the fluence in a three-layer system of slabs under different assumptions of the tissue–light interaction. Table 2 presents the optical properties used in each of the 0.1 cm thick layers.

Fig. 8
Fig. 8

Response of the reflectance R to a depth-varying perturbing layer in the three-layer geometry. The Δ next to the optical property denotes that it was changed in the perturbing layer; all other properties were kept homogeneous across the layers. See Subsection 6B for a listing of optical properties.

Fig. 9
Fig. 9

Wavelength-resolved reflectance in a two-layer model of human skin. Spectral shifts in the reflectance due to changes in dermal oxygen saturation S, dermal blood volume fraction B, and epidermal melanin volume fraction M are presented.

Fig. 10
Fig. 10

Three-layer geometry subject to infinite plane-wave illumination.

Tables (2)

Tables Icon

Table 1 Stability of the P N -Method Assessed through the Digit Accuracy of Plane-Wave Expansion Coefficients, Eq. (64) with Increasing Legendre Truncation Order N a

Tables Icon

Table 2 Sphere Sizes and Corresponding Optical Properties Used in the Three-Layer Calculation Comparing Tissue–Light Interactions

Equations (107)

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( ω r + σ a ) ψ ( ω , r ) = L [ ψ ( ω , r ) ] .
L [ ψ ( ω k ̂ ) ] = S 2 f ( ω , ω ) ψ ( ω k ̂ ) d ω σ s ψ ( ω k ̂ ) ,
L [ Y n m ( ω ) ] = λ n Y n m ( ω ) .
λ n = σ s ( 1 f ̂ n ) ,
f ̂ n = 1 1 f ( ω ω ) P n ( ω ω ) d ( ω ω ) .
ψ ( ω , r b ) = ψ o ( ω , r b ) , ω n ( r b ) < 0 , r b D .
( μ z + σ a ) ψ ( μ , z ) = L [ ψ ( μ , z ) ] , ( μ , z ) [ 1 , 1 ] × [ 0 , L ] ,
ψ ( μ , z = 0 ) = ψ o ( μ ) , μ [ 0 , 1 ] ,
ψ ( μ , z = L ) = 0 , μ [ 1 , 0 ] .
J ( μ ) = | μ | ψ ( μ , z = 0 ) , μ [ 1 , 0 ] ,
J + ( μ ) = | μ | ψ ( μ , z = L ) , μ [ 0 , 1 ] ;
R = 1 0 | μ | ψ ( μ , 0 ) d μ 0 1 | μ | ψ o ( μ ) d μ ;
T = 0 1 | μ | ψ ( μ , 0 ) d μ 0 1 | μ | ψ o ( μ ) d μ ;
Φ ( z ) = 1 1 ψ ( μ , z ) d μ ;
A ( z ) = μ a ( z ) Φ ( z ) .
R + T + 0 L A ( z ) d z 0 1 | μ | ψ o ( μ ) d μ = 1 .
ψ ( μ , z ) = ϕ ( μ ) e γ z .
( μ γ + σ a ) ϕ ( μ ) = L [ ϕ ( μ ) ] , μ [ 1 , 1 ] .
ψ ( μ , z ) = l = 1 N 2 c l ϕ l ( μ ) e γ l ( z L ) + l = 1 N 2 d l ϕ l ( μ ) e γ l z .
ϕ ( μ ) = n = 0 2 n + 1 4 π ϕ ̂ n P n ( μ ) ,
ϕ ̂ n = 2 π 1 1 ϕ ( μ ) P n ( μ ) d μ .
n = 0 2 n + 1 4 π ( γ μ + σ a λ n ) ϕ ̂ n P n ( μ ) = 0 .
L [ ϕ ( ω k ̂ ) ] = L [ n = 0 m = n + n ϕ ̂ n Y n m ( ω ) Y n m * ( k ̂ ) ] = n = 0 m = n + n ϕ ̂ n λ n Y n m ( ω ) Y n m * ( k ̂ ) .
γ 2 k + 1 [ k ϕ ̂ k 1 + ( k + 1 ) ϕ ̂ k + 1 ] + ( σ a λ k ) ϕ ̂ k = 0 , k = 0 , 1 , 2 , .
ϕ ( μ ) n = 0 N 1 2 n + 1 4 π ϕ ̂ n P n ( μ )
( γ A + D ) ϕ ̂ = 0 or A 1 D ϕ ̂ = γ ϕ ̂ ,
{ A } i j = { i 2 i + 1 , j = i 1 i + 1 2 i + 1 , j = i + 1 , 0 , otherwise }
D = diag ( σ a λ 0 , σ a λ 1 , , σ a λ N 1 ) .
ψ ( μ k , z = 0 ) = l = 1 N 2 c l ϕ l ( μ k ) e γ l L + l = 1 N 2 d l ϕ l ( μ k ) = ψ o ( μ k ) , μ k [ 0 , 1 ] ,
ψ ( μ k , z = L ) = l = 1 N 2 c l ϕ l ( μ k ) + l = 1 N 2 d l ϕ l ( μ k ) e γ l L = 0 , μ k [ 1 , 0 ] .
{ V 1 } k l = ϕ l ( μ k ) e γ l L , μ k [ 0 , 1 ] ,
{ V 2 } k l = ϕ l ( μ k ) , μ k [ 0 , 1 ] ,
{ V 3 } k l = ϕ l ( μ k ) , μ k [ 1 , 0 ] ,
{ V 4 } k l = ϕ l ( μ k ) e γ l L , μ k [ 1 , 0 ] ,
V = ( V 1 V 2 V 3 V 4 ) ,
c = ( c 1 , c 2 , , c N 2 ) T ,
d = ( d 1 , d 2 , , d N 2 ) T ,
ψ o = ( ψ o ( μ 1 ) , ψ o ( μ 2 ) , , ψ o ( μ N 2 ) ) T , μ k [ 0 , 1 ] ,
V ( c d ) = ( ψ o 0 ) .
( μ z + σ a ( i ) ) ψ ( i ) ( μ , z ) = L ( i ) [ ψ ( i ) ( μ , z ) ] , ( μ , z ) [ 1 , 1 ] × [ L i 1 , L i ] ,
ψ ( i ) ( μ , z ) = l = 1 N 2 c l ( i ) ϕ l ( i ) ( μ ) e γ l ( i ) ( z L i ) + l = 1 N 2 d l ( i ) ϕ l ( i ) ( μ ) e γ l ( i ) ( z L i 1 ) .
ψ ( i ) ( μ k , L i 1 ) = ψ ( i 1 ) ( μ k , L i 1 ) , μ k [ 0 , 1 ] ,
ψ ( i ) ( μ k , L i ) = ψ ( i + 1 ) ( μ k , L i ) , μ k [ 1 , 0 ] .
ψ ( 1 ) ( μ k , L 0 ) = ψ ( 0 ) ( μ k , L 0 ) , μ k [ 0 , 1 ] .
ψ ( 0 ) ( μ k , L 0 ) ψ o ( μ k , 0 ) , μ k [ 0 , 1 ] .
ψ ( M ) ( μ k , L M ) = ψ ( M + 1 ) ( μ k , L M ) , μ k [ 1 , 0 ] .
n a 2 ( 1 μ i 2 ) = n b 2 ( 1 μ r 2 ) .
μ c 2 = 1 ( n b n a ) 2 .
R a b ( μ i , μ r ) = { 1 2 [ ( n a μ r n b μ i n a μ r + n b μ i ) 2 + ( n a μ i n b μ r n a μ i + n b μ r ) 2 ] , μ i > μ c 1 , μ i μ c } ,
T a b ( μ i , μ r ) = ( n b n a ) 2 [ 1 R a b ( μ i , μ r ) ] .
ψ r ( μ i , z o ) = R a b ( μ i , μ r ) ψ i ( μ i , z o ) ,
ψ t ( μ r , z o ) = T a b ( μ i , μ r ) ψ i ( μ i , z o ) .
T i 1 , i ( μ ( i 1 , + ) , μ ( i , + ) ) ψ ( i 1 ) ( μ ( i 1 , + ) , L i 1 ) ,
R i , i 1 ( μ ( i , ) , μ ( i 1 , ) ) ψ ( i ) ( μ ( i , ) , L i 1 ) .
ψ ( i ) ( μ k ( i , + ) , L i 1 ) = T i 1 , i ( μ k ( i 1 , + ) , μ k ( i , + ) ) ψ ( i 1 ) ( μ k ( i 1 , + ) , L i 1 ) + ( 1 δ i , M + 1 ) R i , i 1 ( μ k ( i , ) , μ k ( i 1 , ) ) ψ ( i ) ( μ k ( i , ) , L i 1 ) .
μ k ( i , ) = μ k ( i , + ) ,
μ k ( i 1 , ± ) = ± ( 1 ( n i n i 1 ) 2 ( 1 μ k ( i , + ) 2 ) ) 1 2 .
T i + 1 , i ( μ ( i + 1 , ) , μ ( i , ) ) ψ ( i + 1 ) ( μ ( i + 1 , ) , L i ) ,
R i , i + 1 ( μ ( i , + ) , μ ( i + 1 , + ) ) ψ ( i ) ( μ ( i , + ) , L i ) .
ψ ( i ) ( μ k ( i , ) , L i ) = ( 1 δ i , M ) T i + 1 , i ( μ k ( i + 1 , ) , μ k ( i , ) ) ψ ( i + 1 ) ( μ k ( i + 1 , ) , L i ) + R i , i + 1 ( μ k ( i , + ) , μ k ( i + 1 , + ) ) ψ ( i ) ( μ k ( i , + ) , L i ) .
μ k ( i , + ) = μ k ( i , ) ,
μ k ( i + 1 , ± ) = ± ( 1 ( n i n i + 1 ) 2 ( 1 μ k ( i , ) 2 ) ) 1 2 .
ψ ( 0 ) ( μ k ( 0 , ) , 0 ) = T 10 ( μ k ( 1 , ) , μ k ( 0 , ) ) ψ ( 1 ) ( μ k ( 1 , ) , 0 ) + R 01 ( μ k ( 0 , + ) , μ k ( 1 , + ) ) ψ o ( μ k ( 0 , + ) ) .
ψ ( M + 1 ) ( μ k ( M + 1 , + ) , L M ) = T M M + 1 ( μ k ( M , + ) , μ k ( M + 1 , + ) ) ψ ( M ) ( μ k ( M , + ) , L M ) .
ψ o ( μ ) = B W 2 π exp [ ( μ 1 ) 2 2 W 2 ] , μ [ 0 , 1 ] .
A x = b ,
d = floor [ 16 p ] .
E N ( R ) = | R N max R N | R N max .
E MC ( R ) = | R MC R N | R MC .
σ a epi ( λ ) = W σ a H 2 O ( λ ) + M σ a mel ( λ ) [ cm 1 ] .
σ a derm ( λ ) = B ( S σ a oxy ( λ ) + ( 1 S ) σ a deoxy ( λ ) ) + W σ a H 2 O ( λ ) [ cm 1 ] ,
σ s ( λ ) = 43 ( f ( λ 500 ) 4 + ( 1 f ) ( λ 500 ) 1 ) [ cm 1 ] ,
ψ ( 1 ) ( μ k ( 1 , + ) , 0 ) = T 01 ( μ k ( 0 , + ) , μ k ( 1 , + ) ) ψ o ( μ k ( 0 , + ) ) + R 10 ( μ k ( 1 , ) , μ k ( 0 , ) ) ψ ( 1 ) ( μ k ( 1 , ) , 0 ) ,
ψ ( 1 ) ( μ k ( 1 , ) , L 1 ) = T 21 ( μ k ( 2 , ) , μ k ( 1 , ) ) ψ ( 2 ) ( μ k ( 2 , ) , L 1 ) + R 12 ( μ k ( 1 , + ) , μ k ( 2 , + ) ) ψ ( 1 ) ( μ k ( 1 , + ) , L 1 ) ,
ψ ( 2 ) ( μ k ( 2 , + ) , L 1 ) = T 12 ( μ k ( 1 , + ) , μ k ( 2 , + ) ) ψ ( 1 ) ( μ k ( 1 , + ) , L 1 ) + R 21 ( μ k ( 2 , ) , μ k ( 1 , ) ) ψ ( 2 ) ( μ k ( 2 , ) , L 1 ) ,
ψ ( 2 ) ( μ k ( 2 , ) , L 2 ) = T 32 ( μ k ( 3 , ) , μ k ( 2 , ) ) ψ ( 3 ) ( μ k ( 3 , ) , L 2 ) + R 23 ( μ k ( 2 , + ) , μ k ( 3 , + ) ) ψ ( 2 ) ( μ k ( 2 , + ) , L 2 ) ,
ψ ( 3 ) ( μ k ( 3 , + ) , L 2 ) = T 23 ( μ k ( 2 , + ) , μ k ( 3 , + ) ) ψ ( 2 ) ( μ k ( 2 , + ) , L 2 ) + R 32 ( μ k ( 3 , ) , μ k ( 2 , ) ) ψ ( 3 ) ( μ k ( 3 , ) , L 2 ) ,
ψ ( 3 ) ( μ k ( 3 , ) , L 3 ) = R 34 ( μ k ( 3 , + ) , μ k ( 4 , + ) ) ψ ( 3 ) ( μ k ( 3 , + ) , L 3 ) .
ψ ( 1 ) ( μ k ( 1 , + ) , 0 ) = T 01 ( [ 1 ( n 1 n 0 ) 2 ( 1 μ k ( 1 , + ) 2 ) ] 1 2 , μ k ( 1 , + ) ) ψ o ( [ 1 ( n 1 n 0 ) 2 ( 1 μ k ( 1 , + ) 2 ) ] 1 2 ) + R 10 ( μ k ( 1 , + ) , [ 1 ( n 1 n 0 ) 2 ( 1 μ k ( 1 , + ) 2 ) ] 1 2 ) ψ ( 1 ) ( μ k ( 1 , + ) , 0 ) .
ψ ( 1 ) ( μ k ( 1 , ) , L 1 ) = T 21 ( [ 1 ( n 1 n 2 ) 2 ( 1 μ k ( 1 , ) 2 ) ] 1 2 , μ k ( 1 , ) ) ψ ( 2 ) ( [ 1 ( n 1 n 2 ) 2 ( 1 μ k ( 1 , ) 2 ) ] 1 2 , L 1 ) + R 12 ( μ k ( 1 , ) , [ 1 ( n 1 n 2 ) 2 ( 1 μ k ( 1 , ) 2 ) ] 1 2 ) ψ ( 1 ) ( μ k ( 1 , ) , L 1 ) ,
ψ ( 2 ) ( μ k ( 2 , + ) , L 1 ) = T 12 ( [ 1 ( n 2 n 1 ) 2 ( 1 μ k ( 2 , + ) 2 ) ] 1 2 , μ k ( 2 , + ) ) ψ ( 1 ) ( [ 1 ( n 2 n 1 ) 2 ( 1 μ k ( 2 , + ) 2 ) ] 1 2 , L 1 ) + R 21 ( μ k ( 2 , + ) , [ 1 ( n 2 n 1 ) 2 ( 1 μ k ( 2 , + ) 2 ) ] 1 2 ) ψ ( 2 ) ( μ k ( 2 , + ) , L 1 ) ,
ψ ( 2 ) ( μ k ( 2 , ) , L 2 ) = T 32 ( [ 1 ( n 2 n 3 ) 2 ( 1 μ k ( 3 , ) 2 ) ] 1 2 , μ k ( 2 , ) ) ψ ( 3 ) ( [ 1 ( n 2 n 3 ) 2 ( 1 μ k ( 3 , ) 2 ) ] 1 2 , L 2 ) + R 23 ( μ k ( 2 , ) , [ 1 ( n 2 n 3 ) 2 ( 1 μ k ( 2 , ) 2 ) ] 1 2 ) ψ ( 2 ) ( μ k ( 2 , ) , L 2 ) ,
ψ ( 3 ) ( μ k ( 3 , + ) , L 2 ) = T 23 ( [ 1 ( n 3 n 2 ) 2 ( 1 μ k ( 3 , + ) 2 ) ] 1 2 , μ k ( 3 , + ) ) ψ ( 2 ) ( [ 1 ( n 3 n 2 ) 2 ( 1 μ k ( 3 , + ) 2 ) ] 1 2 , L 2 ) + R 32 ( μ k ( 3 , + ) , [ 1 ( n 3 n 2 ) 2 ( 1 μ k ( 3 , + ) 2 ) ] 1 2 ) ψ ( 3 ) ( μ k ( 3 , + ) , L 2 ) ,
ψ ( 3 ) ( μ k ( 3 , ) , L 3 ) = R 34 ( μ k ( 3 , ) , [ 1 ( n 3 n 4 ) 2 ( 1 μ k ( 3 , ) 2 ) ] 1 2 ) ψ ( 3 ) ( μ k ( 3 , ) , L 3 ) .
ψ ( 0 ) ( μ k ( 0 , ) , 0 ) = T 10 ( [ 1 ( n 0 n 1 ) 2 ( 1 μ k ( 0 , ) 2 ) ] 1 2 , μ k ( 0 , ) ) ψ ( 1 ) ( [ 1 ( n 0 n 1 ) 2 ( 1 μ k ( 0 , ) 2 ) ] 1 2 , 0 ) + R 01 ( μ k ( 0 , ) , [ 1 ( n 0 n 1 ) 2 ( 1 μ k ( 0 , ) 2 ) ] 1 2 ) ψ o ( μ k ( 0 , ) ) ;
ψ ( 4 ) ( μ k ( 4 , + ) , L 4 ) = T 34 ( [ 1 ( n 4 n 3 ) 2 ( 1 μ k ( 4 , + ) 2 ) ] 1 2 , μ k ( 4 , + ) ) ψ ( 3 ) ( [ 1 ( n 4 n 3 ) 2 ( 1 μ k ( 4 , + ) 2 ) ] 1 2 , L 4 ) .
{ V 1 + p 1 } k l = ϕ l ( μ k ( 1 , + ) ) e γ l L 1 ,
{ V 1 + p 1 r } k l = ϕ l ( μ k ( 1 , + ) ) e γ l L 1 ,
{ V 1 p 1 } k l = ϕ l ( μ k ( 1 , + ) ) ,
{ V 1 p 1 r } k l = ϕ l ( ( μ k ( 1 , + ) ) ) ,
{ R 10 } k k = R 10 ( μ k ( 1 , + ) , [ 1 ( n 1 n 0 ) 2 ( 1 μ k ( 1 , + ) 2 ) ] 1 2 ) ,
{ ψ o } k = ψ o ( [ 1 ( n 1 n 0 ) 2 ( 1 μ k ( 1 , + ) 2 ) ] 1 2 ) ,
{ T 01 } k k = T 01 ( [ 1 ( n 1 n 0 ) 2 ( 1 μ k ( 1 , + ) 2 ) ] 1 2 , μ k ( 1 , + ) ) ,
{ c ( 1 ) } l = c l ( 1 ) ,
{ d ( 1 ) } l = d l ( 1 ) .
( V 1 + p 1 R 10 V 1 + p 1 r V 1 p 1 R 10 V 1 p 1 r ) ( c ( 1 ) d ( 1 ) ) = ( T 01 ψ o 0 ) .
V 1 , 2 = V 1 ± p 1 R 10 V 1 ± p 1 r ,
V 3 , 4 = T 12 V 2 ± p 1 s ,
V 5 , 6 = V 2 ± p 2 R 21 V 2 ± p 2 r ,
V 7 , 8 = V 2 ± n 1 R 12 V 2 ± n 1 r ,
V 9 , 10 = T 21 V 2 ± n 2 s ,
V 11 , 12 = V 3 ± n 2 R 23 V 3 ± n 2 r ,
V 13 , 14 = T 32 V 3 ± n 3 s ,
V 15 , 16 = T 23 V 3 ± p 2 s ,
V 17 , 18 = V 3 ± p 3 R 32 V 3 ± p 3 r ,
V 19 , 20 = V 4 ± n 3 R 34 V 4 ± n 3 r ,
( V 1 V 2 0 0 0 0 V 3 V 4 V 5 V 6 0 0 V 7 V 8 V 9 V 10 0 0 0 0 V 11 V 12 V 13 V 14 0 0 V 15 V 16 V 17 V 18 0 0 0 0 V 19 V 20 ) ( c ( 1 ) d ( 1 ) c ( 2 ) d ( 2 ) c ( 3 ) d ( 3 ) ) = ( T 01 ψ o 0 0 0 0 0 ) .

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