Abstract

The statistical estimator concept, created in the nuclear engineering field, has been adapted to the elaboration of a new and fast semianalytical Monte Carlo numerical simulation for time-resolved light-scattering problems. This concept has also been generalized to the case of unmatched boundaries. The model, discussed in detail in this paper, contains two stages. The first stage is the information generator in which, for each scattering event, the contribution to the total reflectance and transmittance is evaluated and subtracted from the photon current energy. This procedure reduces the number of photons required to produce a given accuracy, which makes it possible to store all event positions and energies. In the second stage, called the information processor, the results of the first stage are used to calculate analytically any desired result. Examples are given for scattering slabs of isotropic or anisotropic scatterers when collimated-beam incidence is used. Reflections at the boundaries are taken into account. The results obtained either with this new method or with classical Monte Carlo methods are very similar. However, the convergence of our new model is much better and, because of the separation into two stages, any quantity related to the problem can be easily calculated afterward without recomputing the simulation.

© 1996 Optical Society of America

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References

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  1. R. Graaf, M. H. Koelink, F. F. M. de Mul, W. G. Zijlstra, A. C. M. Dassel, J. G. Aarnoudse, “Condensed Monte Carlo simulations for the description of light transport,” Appl. Opt. 32, 426–434 (1993).
    [CrossRef]
  2. S. L. Jacques, “Time-resolved reflectance spectroscopy in turbid tissues,” IEEE Trans. Biomed. Eng. 36, 1155–1161 (1989).
    [CrossRef] [PubMed]
  3. M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. SPIE1203, 62–75 (1990).
    [CrossRef]
  4. G. Zaccanti, P. Bruscaglioni, A. Ismaelli, L. Carraresi, M. Gurioli, Q. Wei, “Transmission of a pulsed thin light beam through thick turbid media: experimental results,” Appl. Opt. 31, 2141–2147 (1992).
    [CrossRef] [PubMed]
  5. K. Suzuki, Y. Yamashita, K. Ohta, B. Chance, “Quantitative measurement of optical parameters in the breast using time-resolved spectroscopy: phantom and preliminary in vivo results,” Invest. Radiol. 29, 410–414 (1994).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

1994 (2)

K. Suzuki, Y. Yamashita, K. Ohta, B. Chance, “Quantitative measurement of optical parameters in the breast using time-resolved spectroscopy: phantom and preliminary in vivo results,” Invest. Radiol. 29, 410–414 (1994).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

1993 (2)

1992 (2)

1991 (2)

1990 (1)

S. Avrillier, E. Tinet, E. Delettre, “Monte Carlo simulation of collimated beam transmission through turbid media,” J. Phys. (Paris) 51, 2521–2542 (1990).
[CrossRef]

1989 (3)

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

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

S. L. Jacques, “Time-resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28, 2223–2229 (1989).
[CrossRef] [PubMed]

1983 (2)

1978 (1)

1973 (1)

1941 (1)

L. Henyey, J. Greenstein, “Diffuse radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Aarnoudse, J. G.

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1970).

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]

Anderson, D. E.

Avrillier, S.

S. Avrillier, E. Tinet, E. Delettre, “Monte Carlo simulation of collimated beam transmission through turbid media,” J. Phys. (Paris) 51, 2521–2542 (1990).
[CrossRef]

Bonner, R. F.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

Bruscaglioni, P.

G. Zaccanti, P. Bruscaglioni, A. Ismaelli, L. Carraresi, M. Gurioli, Q. Wei, “Transmission of a pulsed thin light beam through thick turbid media: experimental results,” Appl. Opt. 31, 2141–2147 (1992).
[CrossRef] [PubMed]

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nietol Vesperinas, J. Dainty, eds. (Elsevier, New York, 1990), pp. 53–71.

Bucher, E. A.

Carraresi, L.

Chance, B.

K. Suzuki, Y. Yamashita, K. Ohta, B. Chance, “Quantitative measurement of optical parameters in the breast using time-resolved spectroscopy: phantom and preliminary in vivo results,” Invest. Radiol. 29, 410–414 (1994).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. SPIE1203, 62–75 (1990).
[CrossRef]

Dassel, A. C. M.

de Mul, F. F. M.

Delettre, E.

S. Avrillier, E. Tinet, E. Delettre, “Monte Carlo simulation of collimated beam transmission through turbid media,” J. Phys. (Paris) 51, 2521–2542 (1990).
[CrossRef]

Duncan, M. D.

Ferwerda, H. A.

Flock, S. T.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Fujimoto, G.

Gandjbakhche, A. H.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

Gardner, C. M.

C. M. Gardner, A. J. Welch, “Improvements in the accuracy and statistical variance of the Monte Carlo simulation of light distribution in tissue,” in Laser–Tissue Interaction III, S. L. Jacques, ed., Proc. SPIE1646, 400–409 (1992).
[CrossRef]

Goertzel, G.

G. Goertzel, M. H. Kalos, “Monte Carlo methods in transport problems,” in Series I of Progress in Nuclear Energy (Pergamon, New York, 1958), Vol. 2, pp. 315–369.

Graaf, R.

Greenstein, J.

L. Henyey, J. Greenstein, “Diffuse radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Groenhuis, R. A. J.

Gurioli, M.

Hebden, J. C.

J. C. Hebden, “Evaluating the spatial resolution performance of a time-resolved optical imaging system,” Med. Phys. 19, 1081–1087 (1992).
[CrossRef] [PubMed]

Hee, M. R.

Henyey, L.

L. Henyey, J. Greenstein, “Diffuse radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Ismaelli, A.

Izatt, J. A.

Jacobson, J. M.

Jacques, S. L.

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

S. L. Jacques, “Time-resolved propagation of ultrashort laser pulses within turbid tissues,” Appl. Opt. 28, 2223–2229 (1989).
[CrossRef] [PubMed]

L. Wang, S. L. Jacques, “Monte Carlo simulation of photon distribution in multi-layered turbid media in ANSI standard c,” (Copyright University of Texas, M.D. Anderson Cancer Center, 1515 Holcombe Boulevard, Houston, Texas 77030, 1992).

Kalos, M. H.

G. Goertzel, M. H. Kalos, “Monte Carlo methods in transport problems,” in Series I of Progress in Nuclear Energy (Pergamon, New York, 1958), Vol. 2, pp. 315–369.

M. H. Kalos, P. Whitlock, “Random walks and integral equations,” in Monte Carlo Methods (Wiley, New York, 1986), Vol. I, Chap. 7, pp. 145–156.
[CrossRef]

Koelink, M. H.

Lee, J. S.

Mahon, R.

Meier, R. R.

Moulton, J. D.

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. SPIE1203, 62–75 (1990).
[CrossRef]

Nossal, R.

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

Ohta, K.

K. Suzuki, Y. Yamashita, K. Ohta, B. Chance, “Quantitative measurement of optical parameters in the breast using time-resolved spectroscopy: phantom and preliminary in vivo results,” Invest. Radiol. 29, 410–414 (1994).
[CrossRef] [PubMed]

Patterson, M. S.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. SPIE1203, 62–75 (1990).
[CrossRef]

Reintjes, J.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1970).

Suzuki, K.

K. Suzuki, Y. Yamashita, K. Ohta, B. Chance, “Quantitative measurement of optical parameters in the breast using time-resolved spectroscopy: phantom and preliminary in vivo results,” Invest. Radiol. 29, 410–414 (1994).
[CrossRef] [PubMed]

Tankersley, L. L.

Ten Bosh, J. J.

Tinet, E.

S. Avrillier, E. Tinet, E. Delettre, “Monte Carlo simulation of collimated beam transmission through turbid media,” J. Phys. (Paris) 51, 2521–2542 (1990).
[CrossRef]

van de Hulst, H. C.

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

Wang, L.

L. Wang, S. L. Jacques, “Monte Carlo simulation of photon distribution in multi-layered turbid media in ANSI standard c,” (Copyright University of Texas, M.D. Anderson Cancer Center, 1515 Holcombe Boulevard, Houston, Texas 77030, 1992).

Wei, Q.

Welch, A. J.

C. M. Gardner, A. J. Welch, “Improvements in the accuracy and statistical variance of the Monte Carlo simulation of light distribution in tissue,” in Laser–Tissue Interaction III, S. L. Jacques, ed., Proc. SPIE1646, 400–409 (1992).
[CrossRef]

Whitlock, P.

M. H. Kalos, P. Whitlock, “Random walks and integral equations,” in Monte Carlo Methods (Wiley, New York, 1986), Vol. I, Chap. 7, pp. 145–156.
[CrossRef]

Wilson, B. C.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[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]

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. SPIE1203, 62–75 (1990).
[CrossRef]

Wyman, D. R.

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Yamashita, Y.

K. Suzuki, Y. Yamashita, K. Ohta, B. Chance, “Quantitative measurement of optical parameters in the breast using time-resolved spectroscopy: phantom and preliminary in vivo results,” Invest. Radiol. 29, 410–414 (1994).
[CrossRef] [PubMed]

Zaccanti, G.

G. Zaccanti, P. Bruscaglioni, A. Ismaelli, L. Carraresi, M. Gurioli, Q. Wei, “Transmission of a pulsed thin light beam through thick turbid media: experimental results,” Appl. Opt. 31, 2141–2147 (1992).
[CrossRef] [PubMed]

G. Zaccanti, “Monte Carlo study of light propagation in optically thick media: point source case,” Appl. Opt. 30, 2031–2041 (1991).
[CrossRef] [PubMed]

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nietol Vesperinas, J. Dainty, eds. (Elsevier, New York, 1990), pp. 53–71.

G. Zaccanti, University of Florence, Physics Department, 3 Via S. Marta, 50139 Firenze, Italy (personal communication, 1994).

Zijlstra, W. G.

Appl. Opt. (7)

Astrophys. J. (1)

L. Henyey, J. Greenstein, “Diffuse radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

IEEE Trans. Biomed. Eng. (2)

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

S. T. Flock, M. S. Patterson, B. C. Wilson, D. R. Wyman, “Monte Carlo modelling of light propagation in highly scattering tissues: model predictions and comparison with diffusion theory,” IEEE Trans. Biomed. Eng. 36, 1162–1167 (1989).
[CrossRef] [PubMed]

Invest. Radiol. (1)

K. Suzuki, Y. Yamashita, K. Ohta, B. Chance, “Quantitative measurement of optical parameters in the breast using time-resolved spectroscopy: phantom and preliminary in vivo results,” Invest. Radiol. 29, 410–414 (1994).
[CrossRef] [PubMed]

J. Phys. (Paris) (1)

S. Avrillier, E. Tinet, E. Delettre, “Monte Carlo simulation of collimated beam transmission through turbid media,” J. Phys. (Paris) 51, 2521–2542 (1990).
[CrossRef]

Med. Phys. (3)

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]

J. C. Hebden, “Evaluating the spatial resolution performance of a time-resolved optical imaging system,” Med. Phys. 19, 1081–1087 (1992).
[CrossRef] [PubMed]

A. H. Gandjbakhche, R. Nossal, R. F. Bonner, “Resolution limits for optical transillumination of abnormalities deeply embedded in tissues,” Med. Phys. 21, 185–191 (1994).
[CrossRef] [PubMed]

Opt. Lett. (2)

Other (9)

M. S. Patterson, J. D. Moulton, B. C. Wilson, B. Chance, “Applications of time-resolved light scattering measurements to photodynamic therapy dosimetry,” in Photodynamic Therapy: Mechanisms II, T. J. Dougherty, ed., Proc. SPIE1203, 62–75 (1990).
[CrossRef]

P. Bruscaglioni, G. Zaccanti, “Multiple scattering in dense media,” in Scattering in Volumes and Surfaces, M. Nietol Vesperinas, J. Dainty, eds. (Elsevier, New York, 1990), pp. 53–71.

G. Zaccanti, University of Florence, Physics Department, 3 Via S. Marta, 50139 Firenze, Italy (personal communication, 1994).

G. Goertzel, M. H. Kalos, “Monte Carlo methods in transport problems,” in Series I of Progress in Nuclear Energy (Pergamon, New York, 1958), Vol. 2, pp. 315–369.

M. H. Kalos, P. Whitlock, “Random walks and integral equations,” in Monte Carlo Methods (Wiley, New York, 1986), Vol. I, Chap. 7, pp. 145–156.
[CrossRef]

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1970).

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

L. Wang, S. L. Jacques, “Monte Carlo simulation of photon distribution in multi-layered turbid media in ANSI standard c,” (Copyright University of Texas, M.D. Anderson Cancer Center, 1515 Holcombe Boulevard, Houston, Texas 77030, 1992).

C. M. Gardner, A. J. Welch, “Improvements in the accuracy and statistical variance of the Monte Carlo simulation of light distribution in tissue,” in Laser–Tissue Interaction III, S. L. Jacques, ed., Proc. SPIE1646, 400–409 (1992).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Definition of the quantities used in the calculation of the escape function. d is the thickness of the medium, s is the photon’s incidence direction at the interaction point P, s′ is the direction scanned by the double integration necessary to evaluate the escape function, φ is the azimuthal angle, and θ is the scattering angle. The shape and the orientation of the phase function p(s · s′) are indicated on this figure around P.

Fig. 2
Fig. 2

Definition of the quantities used in the calculation of the escape function when there is no index matching. d, s, s′, φ, and θ have the same meanings as in Fig. 1. The escape function takes into account the infinite number of multiple reflections at the medium boundaries. The first three reflections for a given θ are indicated.

Fig. 3
Fig. 3

(a) Probability-density function for the path length between two interaction points when there is no index matching, (b) corresponding trajectory of the photon. Since the direct contributions to the total reflectance and transmittance have already been taken into account by means of the escape function, the photon must be confined in the medium. As a consequence, the probability-density function for the photon’s propagation length is discontinuous and decreases on account of Beer’s law and the reflections at the medium surfaces. Reflection i occurs at the path length Li, which corresponds to a value ai in the random generator space. At each reflection the probability p(L) is multiplied by r(n, μ), where μ is the cosine of the angle between the photon’s direction after the interaction point Pj and the z-axis unit vector i.

Fig. 4
Fig. 4

Schematic diagram of the analytical evaluation of the energy received at the observation point M in the information processor stage. This energy is obtained by summation of the contributions of all the stored scattering points Pj. sj is the direction of the photon when it reaches Pj, and mj,0 is the observation direction.

Fig. 5
Fig. 5

Multiple images P j , i X of a scattering point Pj that are due to multiple reflections at the medium boundaries. For example, P j , 3 up is the third reflected image point seen through the upper surface. Its distance to M is d j , 3 up, which unit vector is m j , 3 up. The apparent incidence photon direction at P j , 3 up is s j , 3 up. As a consequence, the amplitude of the phase function for this point is p ( s j , 3 up · m j , 3 up ). The apparent positions of the images, seen from the observation point M, are computed recursively, and their contributions to the desired quantity are taken into account, just as for regular scattering points.

Fig. 6
Fig. 6

Geometry of the first numerical experiment presented in this paper: The sample is a 10-mm-thick glass cuvette containing a 200-nm-diameter latex sphere suspension. The detector has a slit shape, 8 mm long and 12.5 μm wide, with a 0.31 numerical aperture (NA). g = 0.19, μa = 0, and μs = 2.85 mm−1. Refractive indices are 1.33 for the medium, 1.55 for the cuvette glass walls, and 1 for air. The classical simulator used the axial symmetry: The slit was replaced by an 8-mm-diameter disk divided into 320 ring detectors. All the photons that cross the surface at distance between ρ − Δρ and ρ + Δρ from the optical axis were recorded in the ρ detector. Then the energies collected by each detector were normalized by the surface of the corresponding ring. The time-resolved transmittance through the slit is the sum of all these normalized energies for a given time interval. In contrast, no symmetry was used in our MC3 simulation.

Fig. 7
Fig. 7

(a) Results obtained with a CMC simulator for the case illustrated in Fig. 6 (150 × 106 photons), (b) results obtained with the MC3 simulator for the case illustrated in Fig. 6 (5 × 106 photons), (c) results from (a) and (b) plotted together. The results obtained by each of the two simulators are the same, but MC3 required only 1/16 of the computation time used by the CMC simulator. Dotted curve, CMC simulator; solid curve, MC3 simulator.

Fig. 8
Fig. 8

The second numerical experiment used a 10-mm-thick index-matched sample. The detector is a 0.2-mm-diameter disk centered on the optical axis and collecting photons at all angles. The chosen optical coefficients are μa = 0, μs = 1.57 mm−1, and g = 0.

Fig. 9
Fig. 9

Result provided by the CMC simulation for the case illustrated in Fig. 8. With 109 photons it is not possible to discern the shape of the transmitted pulse.

Fig. 10
Fig. 10

Result provided by the MC3 simulator for the case illustrated in Fig. 8. 106 photons are sufficient to produce a good evaluation of this curve. In this case a MC3 photon required approximately only twice the average time needed by a classical simulator photon.

Tables (6)

Tables Icon

Table 1 Total Time- and Space-Integrated Reflectance (R) and Transmittance (T) of a Slab (in Percent)a

Tables Icon

Table 2 Same as Table 1, but for ω0 = 0.80

Tables Icon

Table 3 Same as Table 1, but for ω0 = 0.60

Tables Icon

Table 4 Total Time- and Space-Integrated Reflectance (R) and Transmittance (T) of a Slab (in percent)a

Tables Icon

Table 5 Same as Table 4, but for ω0 = 0.80

Tables Icon

Table 6 Same as Table 4, but for ω0 = 0.60

Equations (29)

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

δ R = E s 0 2 π d φ 0 1 p ( s s ) exp ( - μ t z / w ) d w = E s Esc ( w , μ t z ) ,
Esc ( w , μ t z ) = Esc ( μ t z ) = [ exp ( - μ t z ) - μ t z E 1 ( μ t z ) / 2 ] ,
Esc ( w , n , μ t z ) = 0 2 π d φ 0 1 d w [ 1 - r ( n , w ) { p ( s s ) exp ( - μ t z / w ) + r ( n , w ) p ( - s s ) exp [ - μ t ( 2 d - z ) / w ] } 1 - [ r ( n , w ) exp ( - μ t d / w ) ] 2 ,
δ A = E s ( μ a / μ t ) { 1 - Esc ( w , n , μ t z ) - Esc [ - w , n , μ t ( d - z ) ] } .
a b p ( ξ ) d ξ = 1.
rand = a ξ rand p ( ξ ) d ξ ,
p ( L ) = [ μ t exp ( - μ t L ) ] / C 1 ,
C 1 = 0 Z / μ d l μ t exp ( - μ t l ) = 1 - exp ( - μ t Z / μ ) ,
L rand = - ln ( 1 - C 1 rand ) / μ t .
p i ( L ) = [ r i ( n , μ ) μ t exp ( - μ t L ) ] / C .
C = 0 Z / μ d l μ t exp ( - μ t l ) + i = 1 r i ( n , μ ) L i L i + 1 d l μ t exp ( - μ t l ) .
a 1 = 0 L 1 d l p 0 ( l ) ,
a i = a i - 1 + L i - 1 L i d l p i - 1 ( l ) .
i = int { 1 + ln [ ( 1 - rand ) C / C 2 ] / ln ρ } .
L rand = - ln { 1 - C rand } / μ t .
rand = a i + L i L rand d l p i ( l ) , L rand = L i - 1 μ t ln [ 1 - C 2 r ( n , μ ) ( 1 - C 1 ) ( rand - a i 1 - a i ) ] .
L rand = L i + L random .
x j + 1 = x j + u j + 1 L rand , y j + 1 = y j + v j + 1 L rand , z j + 1 = ± w j + 1 L random or z j + 1 = d ± ω j + 1 L random ,
cos θ = { 1 + g 2 - [ ( 1 - g 2 ) / ( 1 - g + 2 g rand ) ] 2 } / 2 g
E s ; j + 1 = E s ; j ( μ s / μ t ) { 1 - Esc ( ω j , n , μ t z j ) - Esc [ w j , n , μ t ( d - z j ) ] } .
E = j = 1 J E j .
E j = E s ; j p ( s j m j , 0 ) exp ( - μ t d j , 0 ) Δ S cos ( n , m j , 0 ) / d j , 0 2 ,
s j , i up = s j , i down = s j , i - ( 1 1 ( - 1 ) i ) s j .
{ z j , i + 1 down = 2 d - z j , i up , z j , i + 1 up = - z j , i down ,
E = j = 1 J E j .
E j = E s ; j p ( s j · m j , 0 ) exp ( - μ t d j , 0 ) Δ S cos ( n , m j , 0 ) / d j , 0 2 + i = 1 I up E s ; j , i up + i = 1 I down E s ; j , i down ,
E s ; j , i X = E s ; j r i ( n , μ j , i X ) p ( s j , i · m j , i X ) × exp ( - μ t d j , i X ) Δ S cos ( n , m j , i X ) / d j , i X ) 2
d j , i X = [ ( x m - x j ) 2 + ( y m - y j ) 2 + ( z m - z j , i X ) 2 ] 1 / 2 .
m j , i X = 1 d j , i X ( x m - x j y m - y j z m - z j , i X ) .

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