Abstract

It is possible to alter the interaction parameters characterizing neutral particle radiation transport without significantly altering the spatial distribution of the particle fluence rate. Practical mathematical relations specifying the constraints that such an alteration must satisfy are known as similarity relations. Similarity relations are derived in this work from integrated versions of the single energy neutral particle transport equation. The application of these relations in accelerating Monte Carlo deep penetration simulations is described and assessed. Computational reductions may exceed a factor of 10 in highly scattering media in which the scattering is highly forward peaked, such as applies to the propagation of red and near IR light through soft tissues.

© 1989 Optical Society of America

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References

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  1. H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres,” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy Eds. (Gordon & Breach, New York, 1968), p. 35.
  2. D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity Relations for Anisotropic Scattering in Monte Carlo Simulations of Deeply Penetrating Neutral Particles,” J. Comput. Phys. 81, 137–150 (1989).
    [CrossRef]
  3. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A. 4, 423–432 (1987).
    [CrossRef] [PubMed]
  4. B. C. Wilson, G. Adam, “A Monte Carlo Model for the Absorption and Flux Distributions of Light in Tissue,” Med. Phys. 10, 824–830 (1983).
    [CrossRef] [PubMed]
  5. P. A. Wilksch, F. Jacka, A. J. Blake, “Studies of Light Propagation Through Tissue,” in Porphyrin Localization and Treatment of Tumors, D. R. ron, C. J. Gomer, Eds. (Alan R. Liss, New York, 1984).
  6. S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-Diffusion Theory Modelling of Light Distributions in Tissue,” Proc. Soc. Photo. Opt. Instrum. Eng. 908, 20–28 (1988).
  7. A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
    [CrossRef] [PubMed]
  8. S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 Nanometres,” Med. Phys. 14, 835–841 (1987).
    [CrossRef] [PubMed]
  9. L. L. Carter, E. D. Cashwell, Particle Transport Simulation with the Monte Carlo Method (U.S. Energy Research Development Administration, 1975).
    [CrossRef]
  10. L. B. Levitt, “The Use of Self-Optimized Exponential Biasing in Obtaining Monte Carlo Estimates of Transmission Probabilities,” Nucl. Sci. Eng. 31, 500–504 (1968).
  11. W. M. Star, J. P. Marijnissen, M. J. C. van Gemert, “Light Dosimetry: Status and Prospects,” Photochem. Photobiol. B 1, 149–167 (1987).
    [CrossRef]
  12. S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo Studies of Light Propagation in Highly Scattering Tissues. II: Comparison with Measurements in Phantoms,” IEEE Trans. Biomed. Eng.36, (in press).
    [PubMed]
  13. D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength through Tissue from Direct Time of Flight Measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
    [CrossRef] [PubMed]
  14. J. M. Maarek, G. Jarry, J. Crowe, M.-H. Bui, D. Laurent, “Simulation of Laser Tomoscopy in a Heterogeneous Biological Medium,” Med. Biol. Eng. Comput. 24, 407–414 (1986).
    [CrossRef] [PubMed]
  15. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).
  16. G. B. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).
  17. H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, New York, 1980).

1989 (1)

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity Relations for Anisotropic Scattering in Monte Carlo Simulations of Deeply Penetrating Neutral Particles,” J. Comput. Phys. 81, 137–150 (1989).
[CrossRef]

1988 (2)

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength through Tissue from Direct Time of Flight Measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-Diffusion Theory Modelling of Light Distributions in Tissue,” Proc. Soc. Photo. Opt. Instrum. Eng. 908, 20–28 (1988).

1987 (4)

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 Nanometres,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A. 4, 423–432 (1987).
[CrossRef] [PubMed]

W. M. Star, J. P. Marijnissen, M. J. C. van Gemert, “Light Dosimetry: Status and Prospects,” Photochem. Photobiol. B 1, 149–167 (1987).
[CrossRef]

1986 (1)

J. M. Maarek, G. Jarry, J. Crowe, M.-H. Bui, D. Laurent, “Simulation of Laser Tomoscopy in a Heterogeneous Biological Medium,” Med. Biol. Eng. Comput. 24, 407–414 (1986).
[CrossRef] [PubMed]

1983 (1)

B. C. Wilson, G. Adam, “A Monte Carlo Model for the Absorption and Flux Distributions of Light in Tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

1968 (1)

L. B. Levitt, “The Use of Self-Optimized Exponential Biasing in Obtaining Monte Carlo Estimates of Transmission Probabilities,” Nucl. Sci. Eng. 31, 500–504 (1968).

Adam, G.

B. C. Wilson, G. Adam, “A Monte Carlo Model for the Absorption and Flux Distributions of Light in Tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Arfken, G. B.

G. B. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

Arridge, S.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength through Tissue from Direct Time of Flight Measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Blake, A. J.

P. A. Wilksch, F. Jacka, A. J. Blake, “Studies of Light Propagation Through Tissue,” in Porphyrin Localization and Treatment of Tumors, D. R. ron, C. J. Gomer, Eds. (Alan R. Liss, New York, 1984).

Bonner, R. F.

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A. 4, 423–432 (1987).
[CrossRef] [PubMed]

Bui, M.-H.

J. M. Maarek, G. Jarry, J. Crowe, M.-H. Bui, D. Laurent, “Simulation of Laser Tomoscopy in a Heterogeneous Biological Medium,” Med. Biol. Eng. Comput. 24, 407–414 (1986).
[CrossRef] [PubMed]

Carter, L. L.

L. L. Carter, E. D. Cashwell, Particle Transport Simulation with the Monte Carlo Method (U.S. Energy Research Development Administration, 1975).
[CrossRef]

Cashwell, E. D.

L. L. Carter, E. D. Cashwell, Particle Transport Simulation with the Monte Carlo Method (U.S. Energy Research Development Administration, 1975).
[CrossRef]

Cope, M.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength through Tissue from Direct Time of Flight Measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Crowe, J.

J. M. Maarek, G. Jarry, J. Crowe, M.-H. Bui, D. Laurent, “Simulation of Laser Tomoscopy in a Heterogeneous Biological Medium,” Med. Biol. Eng. Comput. 24, 407–414 (1986).
[CrossRef] [PubMed]

Delpy, D. T.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength through Tissue from Direct Time of Flight Measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Flock, S. T.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-Diffusion Theory Modelling of Light Distributions in Tissue,” Proc. Soc. Photo. Opt. Instrum. Eng. 908, 20–28 (1988).

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 Nanometres,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo Studies of Light Propagation in Highly Scattering Tissues. II: Comparison with Measurements in Phantoms,” IEEE Trans. Biomed. Eng.36, (in press).
[PubMed]

Grossman, K.

H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres,” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy Eds. (Gordon & Breach, New York, 1968), p. 35.

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

Havlin, S.

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A. 4, 423–432 (1987).
[CrossRef] [PubMed]

Jacka, F.

P. A. Wilksch, F. Jacka, A. J. Blake, “Studies of Light Propagation Through Tissue,” in Porphyrin Localization and Treatment of Tumors, D. R. ron, C. J. Gomer, Eds. (Alan R. Liss, New York, 1984).

Jarry, G.

J. M. Maarek, G. Jarry, J. Crowe, M.-H. Bui, D. Laurent, “Simulation of Laser Tomoscopy in a Heterogeneous Biological Medium,” Med. Biol. Eng. Comput. 24, 407–414 (1986).
[CrossRef] [PubMed]

Laurent, D.

J. M. Maarek, G. Jarry, J. Crowe, M.-H. Bui, D. Laurent, “Simulation of Laser Tomoscopy in a Heterogeneous Biological Medium,” Med. Biol. Eng. Comput. 24, 407–414 (1986).
[CrossRef] [PubMed]

Levitt, L. B.

L. B. Levitt, “The Use of Self-Optimized Exponential Biasing in Obtaining Monte Carlo Estimates of Transmission Probabilities,” Nucl. Sci. Eng. 31, 500–504 (1968).

Maarek, J. M.

J. M. Maarek, G. Jarry, J. Crowe, M.-H. Bui, D. Laurent, “Simulation of Laser Tomoscopy in a Heterogeneous Biological Medium,” Med. Biol. Eng. Comput. 24, 407–414 (1986).
[CrossRef] [PubMed]

Marijnissen, J. P.

W. M. Star, J. P. Marijnissen, M. J. C. van Gemert, “Light Dosimetry: Status and Prospects,” Photochem. Photobiol. B 1, 149–167 (1987).
[CrossRef]

Nossal, R.

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A. 4, 423–432 (1987).
[CrossRef] [PubMed]

Patterson, M. S.

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity Relations for Anisotropic Scattering in Monte Carlo Simulations of Deeply Penetrating Neutral Particles,” J. Comput. Phys. 81, 137–150 (1989).
[CrossRef]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-Diffusion Theory Modelling of Light Distributions in Tissue,” Proc. Soc. Photo. Opt. Instrum. Eng. 908, 20–28 (1988).

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 Nanometres,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo Studies of Light Propagation in Highly Scattering Tissues. II: Comparison with Measurements in Phantoms,” IEEE Trans. Biomed. Eng.36, (in press).
[PubMed]

Star, W. M.

W. M. Star, J. P. Marijnissen, M. J. C. van Gemert, “Light Dosimetry: Status and Prospects,” Photochem. Photobiol. B 1, 149–167 (1987).
[CrossRef]

van de Hulst, H. C.

H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres,” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy Eds. (Gordon & Breach, New York, 1968), p. 35.

H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, New York, 1980).

van der Zee, P.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, 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.

W. M. Star, J. P. Marijnissen, M. J. C. van Gemert, “Light Dosimetry: Status and Prospects,” Photochem. Photobiol. B 1, 149–167 (1987).
[CrossRef]

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

Weiss, G. H.

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A. 4, 423–432 (1987).
[CrossRef] [PubMed]

Welch, A. J.

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

Wilksch, P. A.

P. A. Wilksch, F. Jacka, A. J. Blake, “Studies of Light Propagation Through Tissue,” in Porphyrin Localization and Treatment of Tumors, D. R. ron, C. J. Gomer, Eds. (Alan R. Liss, New York, 1984).

Wilson, B. C.

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity Relations for Anisotropic Scattering in Monte Carlo Simulations of Deeply Penetrating Neutral Particles,” J. Comput. Phys. 81, 137–150 (1989).
[CrossRef]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-Diffusion Theory Modelling of Light Distributions in Tissue,” Proc. Soc. Photo. Opt. Instrum. Eng. 908, 20–28 (1988).

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 Nanometres,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

B. C. Wilson, G. Adam, “A Monte Carlo Model for the Absorption and Flux Distributions of Light in Tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo Studies of Light Propagation in Highly Scattering Tissues. II: Comparison with Measurements in Phantoms,” IEEE Trans. Biomed. Eng.36, (in press).
[PubMed]

Wray, S.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength through Tissue from Direct Time of Flight Measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyatt, J.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength through Tissue from Direct Time of Flight Measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Wyman, D. R.

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity Relations for Anisotropic Scattering in Monte Carlo Simulations of Deeply Penetrating Neutral Particles,” J. Comput. Phys. 81, 137–150 (1989).
[CrossRef]

Yoon, G.

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

J. Comput. Phys. (1)

D. R. Wyman, M. S. Patterson, B. C. Wilson, “Similarity Relations for Anisotropic Scattering in Monte Carlo Simulations of Deeply Penetrating Neutral Particles,” J. Comput. Phys. 81, 137–150 (1989).
[CrossRef]

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

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A. 4, 423–432 (1987).
[CrossRef] [PubMed]

Lasers Surg. Med. (1)

A. J. Welch, G. Yoon, M. J. C. van Gemert, “Practical Models for Light Distribution in Laser-Irradiated Tissue,” Lasers Surg. Med. 6, 488–493 (1987).
[CrossRef] [PubMed]

Med. Biol. Eng. Comput. (1)

J. M. Maarek, G. Jarry, J. Crowe, M.-H. Bui, D. Laurent, “Simulation of Laser Tomoscopy in a Heterogeneous Biological Medium,” Med. Biol. Eng. Comput. 24, 407–414 (1986).
[CrossRef] [PubMed]

Med. Phys. (2)

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 Nanometres,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

B. C. Wilson, G. Adam, “A Monte Carlo Model for the Absorption and Flux Distributions of Light in Tissue,” Med. Phys. 10, 824–830 (1983).
[CrossRef] [PubMed]

Nucl. Sci. Eng. (1)

L. B. Levitt, “The Use of Self-Optimized Exponential Biasing in Obtaining Monte Carlo Estimates of Transmission Probabilities,” Nucl. Sci. Eng. 31, 500–504 (1968).

Photochem. Photobiol. B (1)

W. M. Star, J. P. Marijnissen, M. J. C. van Gemert, “Light Dosimetry: Status and Prospects,” Photochem. Photobiol. B 1, 149–167 (1987).
[CrossRef]

Phys. Med. Biol. (1)

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength through Tissue from Direct Time of Flight Measurement,” Phys. Med. Biol. 33, 1433–1442 (1988).
[CrossRef] [PubMed]

Proc. Soc. Photo. Opt. Instrum. Eng. (1)

S. T. Flock, B. C. Wilson, M. S. Patterson, “Hybrid Monte Carlo-Diffusion Theory Modelling of Light Distributions in Tissue,” Proc. Soc. Photo. Opt. Instrum. Eng. 908, 20–28 (1988).

Other (7)

H. C. van de Hulst, K. Grossman, “Multiple Light Scattering in Planetary Atmospheres,” in The Atmospheres of Venus and Mars, J. C. Brandt, M. B. McElroy Eds. (Gordon & Breach, New York, 1968), p. 35.

L. L. Carter, E. D. Cashwell, Particle Transport Simulation with the Monte Carlo Method (U.S. Energy Research Development Administration, 1975).
[CrossRef]

P. A. Wilksch, F. Jacka, A. J. Blake, “Studies of Light Propagation Through Tissue,” in Porphyrin Localization and Treatment of Tumors, D. R. ron, C. J. Gomer, Eds. (Alan R. Liss, New York, 1984).

S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo Studies of Light Propagation in Highly Scattering Tissues. II: Comparison with Measurements in Phantoms,” IEEE Trans. Biomed. Eng.36, (in press).
[PubMed]

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976).

G. B. Arfken, Mathematical Methods for Physicists (Academic, New York, 1970).

H. C. van de Hulst, Multiple Light Scattering, Vol. 2 (Academic, New York, 1980).

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Figures (4)

Fig. 1
Fig. 1

Infinite homogeneous test slab 30 mean free paths thick, with α = 0.99, f1 = 0.5, and for which a similarity relation accelerates the simulation in the central region.

Fig. 2
Fig. 2

Comparison of correct diffuse reflectance and transmittance with values obtained from simulations conducted using the similarity relation with parameter k. Statistical error bars represent one standard deviation in the mean.

Fig. 3
Fig. 3

Reflectance (a) and transmittance (b) histogram profiles obtained with (– · –) and without (—) the similarity relation (k = 0.5), for the case of a pencil beam normally incident on the test slab in Fig. 1. Distance is in unaltered mean free paths from the center of the pencil beam projection on the exit plane of the slab. The rectangular scoring bin widths are selected to minimize statistical simulation error and are normalized to a maximum bin height of 1.0 for the reference simulation (—).

Fig. 4
Fig. 4

Normalized fluence rate vs depth profile for simulations conducted on the test slab in Fig. 1 using the similarity relation with parameter k. The correct (no similarity relation) and k = 0.0198 profiles are indistinguishable.

Tables (2)

Tables Icon

Table I Entries are Σ t / Σ t * Representing the Factor by Which Computation Time Is Potentially Reduced When the Similarity Relation in Eq. (10) Is Used for Deep Penetration Monte Carlo Simulations

Tables Icon

Table II Diffuse Reflectance (R), Transmlttance (T), and Relative Simulation Time for Simulations on Slabs 300 and 500 Mean Free Paths Thick, with α = 0.9995 and f1 = 0.7

Equations (25)

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

Ω ψ + Σ t ( r ) ψ ( r , R ) = s ( r , Ω ) + Σ s ( r ) 4 π ψ ( r , Ω ) f ( Ω Ω ) d Ω .
φ i 1 i n = 4 π Ω i 1 i n ψ ( Ω ) d Ω ,
χ i 1 i n = 4 π 4 π Ω i 1 i n f ( Ω Ω ) ψ ( Ω ) d Ω d Ω ,
S i 1 i n = 4 π Ω i 1 i n s ( Ω ) d Ω ,
j φ j i 1 i n + Σ t φ i 1 i n = Σ s χ i 1 i n + S i 1 i n ,
ψ ( Ω ) = n = 0 m = n n a m n Y n m ( Ω ) ,
f ( Ω Ω ) = n = 0 ( 2 n + 1 4 π ) f n P n ( Ω Ω ) = n = 0 m = n n f n Y n m ( Ω ) Y ̅ n m ( Ω ) .
χ i = f 1 φ i ,
χ i * = f 1 * φ i * .
φ i 1 i 2 = φ i 1 i 2 * = φ 3 δ i 1 i 2 .
( Σ a Σ t r , 1 i i ) φ = S Σ t r , 1 i S i ,
Σ t r , n = Σ t f n Σ s .
( Σ a Σ t r , 1 Σ a * Σ t r , 1 * ) φ = ( S Σ t r , 1 S * Σ t r , 1 * ) i ( S i S i * ) .
Σ a Σ t r , 1 = Σ a * Σ t r , 1 * .
S Σ t r , 1 S * Σ t r , 1 * = i ( S i S i * ) .
S * = S ( Σ a * Σ a ) .
L = 1 / 3 Σ a Σ t r , 1 .
Σ a Σ t r , 1 Σ t r , 2 Σ s ( 1 f 2 ) = Σ a * Σ t r , 1 * Σ t r , 2 * Σ s * ( 1 f 2 * ) ,
( Σ t r , 1 Σ t r , 2 S Σ t r , 1 * Σ t r , 2 * S * ) i ( Σ t r , 2 S i Σ t r , 2 * S i * ) + i j ( S j i S j i * ) = 0 .
( ψ ( Ω ) ψ * ( Ω ) ) = ( 1 1 ) φ 4 π + m = 1 1 ( a m 1 a m 1 * ) Y 1 m ( Ω ) .
ψ * ( Ω ) = ψ ( Ω ) ( Σ a * Σ a ) + φ 4 π ( 1 Σ a * Σ a ) .
α * = 1 k , 0 < k 1 .
Σ a * = k Σ a Σ t r , 1 ,
Σ t * = ( 1 k ) Σ a Σ t r , 1 .
Σ t Σ t * = k ( 1 α ) ( 1 α f 1 ) .

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