Abstract

We examine the relative error of Monte Carlo simulations of radiative transport that employ two commonly used estimators that account for absorption differently, either discretely, at interaction points, or continuously, between interaction points. We provide a rigorous derivation of these discrete and continuous absorption weighting estimators within a stochastic model that we show to be equivalent to an analytic model, based on the radiative transport equation (RTE). We establish that both absorption weighting estimators are unbiased and, therefore, converge to the solution of the RTE. An analysis of spatially resolved reflectance predictions provided by these two estimators reveals no advantage to either in cases of highly scattering and highly anisotropic media. However, for moderate to highly absorbing media or isotropically scattering media, the discrete estimator provides smaller errors at proximal source locations while the continuous estimator provides smaller errors at distal locations. The origin of these differing variance characteristics can be understood through examination of the distribution of exiting photon weights.

© 2014 Optical Society of America

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References

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  1. S. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE 5, 102–111 (1989).
  2. M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
    [CrossRef]
  3. L. Wang, S. L. Jacques, and L. Zheng, “MCML-Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47, 131–146 (1995).
    [CrossRef]
  4. D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–170 (2002).
    [CrossRef]
  5. J. Ramella-Roman, S. Prahl, and S. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express 13, 4420–4438 (2005).
    [CrossRef]
  6. E. Margallo-Balbás and P. J. French, “Shape based Monte Carlo code for light transport in complex heterogeneous tissues,” Opt. Express 15, 14086–14098 (2007).
    [CrossRef]
  7. E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
    [CrossRef]
  8. Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17, 20178–20190 (2009).
    [CrossRef]
  9. H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55, 947–962 (2010).
    [CrossRef]
  10. A. Doronin and I. Meglinski, “Online object oriented Monte Carlo computational tool for the needs of biomedical optics,” Biomed. Opt. Express 2, 2461–2469 (2011).
    [CrossRef]
  11. X-5 Monte Carlo Team, “MCNP, a general Monte Carlo N-particle transport code, version 5, Report LA-UR-03-1987,” Technical Report (Los Alamos National Laboratory, 2003).
  12. B. T. Wong and M. P. Mengüç, “Comparison of Monte Carlo techniques to predict the propagation of a collimated beam in participating media,” Numer. Heat Transfer 42, 119–140 (2002).
    [CrossRef]
  13. A. Sassaroli and F. Martelli, “Equivalence of four Monte Carlo methods for photon migration in turbid media,” J. Opt. Soc. Am. A 29, 2110–2116 (2012).
    [CrossRef]
  14. G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Krieger, 1970).
  15. J. Spanier and E. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, 1969), reprinted by Dover, 2008.
  16. K. M. Case and P. F. Zweifel, “Existence and uniqueness theorems for the neutron transport equation,” J. Math. Phys. 4, 1376–1386 (1963).
    [CrossRef]
  17. G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).
  18. L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University, 1985).
  19. A. Dubi, Monte Carlo Calculations for Nuclear Reactors, Vol. 2 (CRC Press, 1986).
  20. P. Hoel, S. Port, and C. Stone, Introduction to Probability Theory (Houghton Mifflin, 1971).
  21. L. G. Henyey and J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  22. P. Billingsley, Probability and Measure (Wiley, 1979).
  23. J. R. Taylor, An Introduction to Error Analysis (University Science Books, 1982).

2012 (1)

2011 (1)

2010 (1)

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55, 947–962 (2010).
[CrossRef]

2009 (1)

2008 (1)

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

2007 (1)

2005 (1)

2002 (2)

D. A. Boas, J. P. Culver, J. J. Stott, and A. K. Dunn, “Three dimensional Monte Carlo code for photon migration through complex heterogeneous media including the adult human head,” Opt. Express 10, 159–170 (2002).
[CrossRef]

B. T. Wong and M. P. Mengüç, “Comparison of Monte Carlo techniques to predict the propagation of a collimated beam in participating media,” Numer. Heat Transfer 42, 119–140 (2002).
[CrossRef]

1995 (1)

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

1993 (1)

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef]

1989 (1)

S. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE 5, 102–111 (1989).

1963 (1)

K. M. Case and P. F. Zweifel, “Existence and uniqueness theorems for the neutron transport equation,” J. Math. Phys. 4, 1376–1386 (1963).
[CrossRef]

1941 (1)

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

Alerstam, E.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

Andersson-Engels, S.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

Arridge, S. R.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef]

Bell, G. I.

G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Krieger, 1970).

Billingsley, P.

P. Billingsley, Probability and Measure (Wiley, 1979).

Boas, D. A.

Case, K. M.

K. M. Case and P. F. Zweifel, “Existence and uniqueness theorems for the neutron transport equation,” J. Math. Phys. 4, 1376–1386 (1963).
[CrossRef]

Cope, M.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef]

Culver, J. P.

Darbinjan, R. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Delpy, D. T.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef]

Delves, L. M.

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University, 1985).

Doronin, A.

Dubi, A.

A. Dubi, Monte Carlo Calculations for Nuclear Reactors, Vol. 2 (CRC Press, 1986).

Dunn, A. K.

Elepov, B. S.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Essenpreis, M.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef]

Fang, Q.

Firbank, M.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef]

French, P. J.

Gelbard, E.

J. Spanier and E. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, 1969), reprinted by Dover, 2008.

Glasstone, S.

G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Krieger, 1970).

Greenstein, J. L.

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

Henyey, L. G.

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

Hiraoka, M.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef]

Hoel, P.

P. Hoel, S. Port, and C. Stone, Introduction to Probability Theory (Houghton Mifflin, 1971).

Jacques, S.

Jacques, S. L.

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

S. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE 5, 102–111 (1989).

Kargin, B. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Keijzer, M.

S. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE 5, 102–111 (1989).

Marchuk, G. I.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Margallo-Balbás, E.

Martelli, F.

Meglinski, I.

Mengüç, M. P.

B. T. Wong and M. P. Mengüç, “Comparison of Monte Carlo techniques to predict the propagation of a collimated beam in participating media,” Numer. Heat Transfer 42, 119–140 (2002).
[CrossRef]

Mikhailov, G. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Mohamed, J. L.

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University, 1985).

Nazaraliev, M. A.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

Port, S.

P. Hoel, S. Port, and C. Stone, Introduction to Probability Theory (Houghton Mifflin, 1971).

Prahl, S.

J. Ramella-Roman, S. Prahl, and S. Jacques, “Three Monte Carlo programs of polarized light transport into scattering media: part I,” Opt. Express 13, 4420–4438 (2005).
[CrossRef]

S. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE 5, 102–111 (1989).

Ramella-Roman, J.

Sassaroli, A.

Shen, H.

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55, 947–962 (2010).
[CrossRef]

Spanier, J.

J. Spanier and E. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, 1969), reprinted by Dover, 2008.

Stone, C.

P. Hoel, S. Port, and C. Stone, Introduction to Probability Theory (Houghton Mifflin, 1971).

Stott, J. J.

Svensson, T.

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

Taylor, J. R.

J. R. Taylor, An Introduction to Error Analysis (University Science Books, 1982).

van der Zee, P.

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef]

Wang, G.

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55, 947–962 (2010).
[CrossRef]

Wang, L.

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

Welch, A. J.

S. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE 5, 102–111 (1989).

Wong, B. T.

B. T. Wong and M. P. Mengüç, “Comparison of Monte Carlo techniques to predict the propagation of a collimated beam in participating media,” Numer. Heat Transfer 42, 119–140 (2002).
[CrossRef]

Zheng, L.

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

Zweifel, P. F.

K. M. Case and P. F. Zweifel, “Existence and uniqueness theorems for the neutron transport equation,” J. Math. Phys. 4, 1376–1386 (1963).
[CrossRef]

Astrophys. J. (1)

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

Biomed. Opt. Express (1)

Comput. Methods Programs Biomed. (1)

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

J. Biomed. Opt. (1)

E. Alerstam, T. Svensson, and S. Andersson-Engels, “Parallel computing with graphics processing units for high-speed Monte Carlo simulation of photon migration,” J. Biomed. Opt. 13, 060504 (2008).
[CrossRef]

J. Math. Phys. (1)

K. M. Case and P. F. Zweifel, “Existence and uniqueness theorems for the neutron transport equation,” J. Math. Phys. 4, 1376–1386 (1963).
[CrossRef]

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

Numer. Heat Transfer (1)

B. T. Wong and M. P. Mengüç, “Comparison of Monte Carlo techniques to predict the propagation of a collimated beam in participating media,” Numer. Heat Transfer 42, 119–140 (2002).
[CrossRef]

Opt. Express (4)

Phys. Med. Biol. (2)

H. Shen and G. Wang, “A tetrahedron-based inhomogeneous Monte Carlo optical simulator,” Phys. Med. Biol. 55, 947–962 (2010).
[CrossRef]

M. Hiraoka, M. Firbank, M. Essenpreis, M. Cope, S. R. Arridge, P. van der Zee, and D. T. Delpy, “A Monte Carlo investigation of optical pathlength in inhomogeneous tissue and its application to near-infrared spectroscopy,” Phys. Med. Biol. 38, 1859–1876 (1993).
[CrossRef]

Proc. SPIE (1)

S. Prahl, M. Keijzer, S. L. Jacques, and A. J. Welch, “A Monte Carlo model of light propagation in tissue,” Proc. SPIE 5, 102–111 (1989).

Other (9)

G. I. Bell and S. Glasstone, Nuclear Reactor Theory (Krieger, 1970).

J. Spanier and E. Gelbard, Monte Carlo Principles and Neutron Transport Problems (Addison-Wesley, 1969), reprinted by Dover, 2008.

G. I. Marchuk, G. A. Mikhailov, M. A. Nazaraliev, R. A. Darbinjan, B. A. Kargin, and B. S. Elepov, The Monte Carlo Methods in Atmospheric Optics (Springer-Verlag, 1980).

L. M. Delves and J. L. Mohamed, Computational Methods for Integral Equations (Cambridge University, 1985).

A. Dubi, Monte Carlo Calculations for Nuclear Reactors, Vol. 2 (CRC Press, 1986).

P. Hoel, S. Port, and C. Stone, Introduction to Probability Theory (Houghton Mifflin, 1971).

P. Billingsley, Probability and Measure (Wiley, 1979).

J. R. Taylor, An Introduction to Error Analysis (University Science Books, 1982).

X-5 Monte Carlo Team, “MCNP, a general Monte Carlo N-particle transport code, version 5, Report LA-UR-03-1987,” Technical Report (Los Alamos National Laboratory, 2003).

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Figures (5)

Fig. 1.
Fig. 1.

Plots of [(a), (e), (i)] spatially resolved reflectance, [(b), (f), (j)] ΔMean with 1σ error bars, [(c), (g), (k)] relative error, and [(d), (h), (l)] ΔR for increased absorption μs/μa=100,10,1, and g=0.9.

Fig. 2.
Fig. 2.

Plots of [(a), (e), (i)] spatially resolved reflectance, [(b), (f), (j)] ΔMean with 1σ error bars, [(c), (g), (k)] relative error, and [(d), (h), (l)] ΔR for increased absorption μs/μa=100,10,1, and g=0.

Fig. 3.
Fig. 3.

Plots of the photon weight count for (a) ρ[00.2]l* and (b) ρ[5.86]l* for high scattering media μs/μa=100 and g=0.9.

Fig. 4.
Fig. 4.

Plots of the photon weight count for (a) ρ[00.2]l* and (b) ρ[5.86]l* for moderate absorbing media μs/μa=10 and g=0.

Fig. 5.
Fig. 5.

Plots of the photon weight count for (a) ρ[00.2]l* and (b) ρ[5.86]l* for high absorbing media μs/μa=1 and g=0.

Tables (2)

Tables Icon

Table 1. List of Optical Properties Studied

Tables Icon

Table 2. Calculated Efficiencies for Reflectance Predictions at Proximal and Distal Source Locationsa

Equations (64)

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

ω·Φ(r,ω)+μt(r)Φ(r,ω)=Γμs(r)f(r;ωω)Φ(r,ω)dω+Q(r,ω),
Ψ=KΨ+S,
KΨ(r,ω)=ΓK[(r,ω)(r,ω)]Ψ(r,ω)drdω,
S(r,ω)=DT(rr,ω)Q(r,ω)dr,
T(rr,ω)=μt(r)exp{0ω·(rr)μt(r+sω)ds}.
K[(r,ω)(r,ω)]=C(r,ωω)T(rr,ω).
C(r,ωω)=μs(r)μt(r)f(r,ωω),
K[(r,ω)(r,ω)]=f(r,ωω)μs(r)×exp{0ω·(rr)μt(r+sω)ds}.
Ψ=(IK)1S,
=(I+K+K2+K3+)S.
K1<1,
Ψ(r,ω)=S(r,ω)+ΓK[(r1,ω1)(r,ω)]S(r1,ω1)dr1dω1+ΓK[(r2,ω2)(r,ω)]×ΓK[(r1,ω1)(r2,ω2)]×S(r1,ω1)dr1dω1dr2dω2+,
=k=1ΓΓS(r1,ω1)K[(r1,ω1)(r2,ω2)]×K[(rk1,ωk1)(rk,ωk)]×dr1dω1drkdωk.
I=Γg(r,ω)Ψ(r,ω)drdω,
=Γg(r,ω)S(r,ω)drdω+Γg(r,ω)K[(r1,ω1)(r,ω)]S(r1,ω1)×dr1dω1drdω+,
g(r,ω)=|n·ω|ΔrΔωχ(r,ω),
χ(r,ω)={1(r,ω)[Δr×Δω]0(r,ω)[Δr×Δω].
p1(P1)0,
p(PiPi+1)0,
p(Pi)=1Γp(PiPi+1)dPi+1.
ξT(β)=S(P1)p1(P1)w(P1P2)w(Pk1Pk)g(Pk)p(Pk),
w(PiPi+1)={K(PiPi+1)p(PiPi+1)ifp(PiPi+1)00ifp(PiPi+1)=0.
p1(P1)=S(P1)ΓS(P1)dP1,
p(PiPi+1)=K(PiPi+1)1p(Pi),
p(Pi)=1ΓK(PiPi+1)dPi+1,
w(PiPi+1)=K(PiPi+1)p(PiPi+1).
p(PiPi+1)=K(PiPi+1)ΓK(PiPi+1)dPi+1,
=K(PiPi+1)μs(Pi)/μt(Pi),
p1(P)=S(P1)ΓS(P1)dP1,
p(PiPi+1)=K(PiPi+1)μs(Pi)/μt(Pi),
p(Pk)=0.
p(PiPi+1)=K(PiPi+1)ΓK(PiPi+1)dPi+1,
=K(PiPi+1)exp{PiPi+1μa(s)ds},
p1(P1)=S(P1)ΓS(P1)dP1,
p(PiPi+1)=K(PiPi+1)exp{PiPi+1μa(s)ds},
p(Pk)=0.
ξMT(β)=S(P1)p1(P1)w(P1P2)w(Pk1Pk)×w(PkPk(1))g(Pk(1)),
w(QP)={K(PiPi+1)p(PiPi+t)ifp(PiPi+1)00ifp(PiPi+1)=0.
ξDAW={i=1kμs(Pi)μt(Pi)if the photon crossesSr×ωNA0otherwise,
ξDAW=(μs/μt)k,
ξCAW={exp{QP1μa(s)dsi=1k1PiPi+1μa(s)dsPkPk(1)μa(s)ds}if the photon crossesSr×ωNAatPk(1)0otherwise.
ξCAW=exp(μaL),
Analytic ModelProbability ModelI=Γg(P)Ψ(P)dPI=Bξ(β)dν(β).
Var[ξ]=1(N1){E[ξ2](E[ξ])2},
σ=Var[ξ],
R=σE[ξ],
ΔMean=E[ξDAW]E[ξCAW]E[ξCAW].
ΔR=RDAWRCAW.
Eff[ξ]=[R2T]1,
1p(Pi)=1[1ΓK(PiPi+1)dPi+1],
=ΓK(PiPi+1)dPi+1.
K[(r,ω)(r,ω)]=f(r;ωω)μs(r)exp{μt|rr|}.
K[(r,ω)(r,ω)]=f(r;ωω)μs(r)μt(r)×[μt(r)exp{μt|rr|}].
ΓK[(r,ω)(r,ω)]drdω=Γf(r;ωω)μs(r)μt(r)×μt(r)exp{μt|rr|}drdω,
=[S2f(r;ωω)dω][μs(r)μt(r)]×[Dμt(r)exp{μt|rr|}dr],
=1·μs(r)μt(r)·1,
K[(r,ω)(r,ω)]=f(r;ωω)exp{μa|rr|}×[μs(r)exp{μs|rr|}].
ΓK[(r,ω)(r,ω)]drdω=Γf(r;ωω)exp{μa|rr|}×μs(r)exp{μs|rr|}drdω,
=[S2f(r;ωω)dω]exp{μa|rr|}×[Dμs(r)exp{μs|rr|}dr],
=1·exp{μa|rr|}·1,
E[ξT(β)]=βP(β)ξT(β),
=k=1ΓΓS(P1)p1(P1)K(P1P2)p(P1P2)K(Pk1Pk)p(Pk1Pk)×K(PkPk(1))p(PkPk(1))g(Pk(1))×p1(P1)p(P1P2)p(Pk1Pk)p(PkPk(1))×dP1dP2dPkdPk(1),
=k=1ΓΓS(P1)K(P1P2)K(Pk1Pk)×K(PkPk(1))g(Pk(1))dP1dP2dPkdPk(1),
=ΓΨ(P)g(P)dP=I,

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