Abstract

Monte Carlo (MC) method is commonly considered as the most accurate approach for particle transport simulation because of its capability to precisely model physics interactions and simulation geometry. Conventionally, MC simulation is performed in a particle-by-particle fashion. In certain problems such as computing scattered X-ray photon signal at a detector of CT, the conventional simulation scheme suffers from low efficiency mainly due to the fact that abundant photons are simulated but do not reach the detector. The computational resources spent on those photons are therefore wasted. To solve this problem, this study develops a novel GPU-based Metropolis MC (gMMC) with a novel path-by-path simulation scheme and demonstrates its effectiveness in an example problem of scattered X-ray photon calculation in CT. In contrast to the conventional MC approach, gMMC samples an entire photon path extending from the X-ray source to the detector using Metropolis-Hasting algorithm. The path-by-path simulation scheme ensures contribution of every sampled event to the signal of interest, improving overall efficiency. We benchmark gMMC against an in-house developed GPU-based MC tool, gMCDRR, which performs simulations in the conventional particle-by-particle fashion. gMMC reaches speed up factors of 37~48 times in simple phantom cases and 20-34 times in real patient cases. The results calculated by gMCDRR and gMMC agree well with average differences < 3%.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2016 (2)

2015 (1)

Y. Xu, T. Bai, H. Yan, L. Ouyang, A. Pompos, J. Wang, L. Zhou, S. B. Jiang, and X. Jia, “A practical cone-beam CT scatter correction method with optimized Monte Carlo simulations for image-guided radiation therapy,” Phys. Med. Biol. 60(9), 3567–3587 (2015).
[Crossref] [PubMed]

2014 (2)

X. Jia, P. Ziegenhein, and S. B. Jiang, “GPU-based high-performance computing for radiation therapy,” Phys. Med. Biol. 59(4), R151–R182 (2014).
[Crossref] [PubMed]

D. P. Kroese, T. Brereton, T. Taimre, and Z. I. Botev, “Why the Monte Carlo method is so important today,” Wiley Interdiscip. Rev. Comput. Stat. 6(6), 386–392 (2014).
[Crossref]

2012 (3)

X. Jia, H. Yan, L. Cerviño, M. Folkerts, and S. B. Jiang, “A GPU tool for efficient, accurate, and realistic simulation of cone beam CT projections,” Med. Phys. 39(12), 7368–7378 (2012).
[Crossref] [PubMed]

X. Jia, H. Yan, X. Gu, and S. B. Jiang, “Fast Monte Carlo simulation for patient-specific CT/CBCT imaging dose calculation,” Phys. Med. Biol. 57(3), 577–590 (2012).
[Crossref] [PubMed]

Z. I. Botev and D. P. Kroese, “Efficient Monte Carlo simulation via the generalized splitting method,” Stat. Comput. 22(1), 1–16 (2012).
[Crossref]

2011 (3)

G. Pratx and L. Xing, “GPU computing in medical physics: a review,” Med. Phys. 38(5), 2685–2697 (2011).
[Crossref] [PubMed]

X. Jia, X. Gu, Y. J. Graves, M. Folkerts, and S. B. Jiang, “GPU-based fast Monte Carlo simulation for radiotherapy dose calculation,” Phys. Med. Biol. 56(22), 7017–7031 (2011).
[Crossref] [PubMed]

S. Hissoiny, B. Ozell, H. Bouchard, and P. Després, “GPUMCD: A new GPU-oriented Monte Carlo dose calculation platform,” Med. Phys. 38(2), 754–764 (2011).
[Crossref] [PubMed]

2010 (2)

2009 (1)

A. Badal and A. Badano, “Accelerating Monte Carlo simulations of photon transport in a voxelized geometry using a massively parallel graphics processing unit,” Med. Phys. 36(11), 4878–4880 (2009).
[Crossref] [PubMed]

2006 (1)

D. W. Rogers, “Fifty years of Monte Carlo simulations for medical physics,” Phys. Med. Biol. 51(13), R287–R301 (2006).
[Crossref] [PubMed]

2003 (1)

A. Haghighat and J. C. Wagner, “Monte Carlo variance reduction with deterministic importance functions,” Prog. Nucl. Energy 42(1), 25–53 (2003).
[Crossref]

2002 (1)

2001 (1)

H. Haario, E. Saksman, and J. Tamminen, “An adaptive Metropolis algorithm,” Bernoulli 7(2), 223–242 (2001).
[Crossref]

1999 (1)

H. Haario, E. Saksman, and J. Tamminen, “Adaptive proposal distribution for random walk Metropolis algorithm,” Comput. Stat. 14(3), 375–396 (1999).
[Crossref]

1995 (2)

J. Baró, J. Sempau, J. M. Fernández-Varea, and F. Salvat, “PENELOPE: An algorithm for Monte Carlo simulation of the penetration and energy loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. 100(1), 31–46 (1995).
[Crossref]

SiddharthaChib and EdwardGreenberg, “Understanding the Metropolis-Hastings Algorithm,” Am. Stat. 49, 327–335 (1995).

1993 (1)

K. Binder, D. Heermann, L. Roelofs, A. J. Mallinckrodt, and S. McKay, “Monte Carlo simulation in statistical physics,” Comput. Phys. 7(2), 156–157 (1993).
[Crossref]

1970 (1)

W. K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika 57(1), 97–109 (1970).
[Crossref]

1949 (1)

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44(247), 335–341 (1949).
[Crossref] [PubMed]

Alerstam, E.

Andersson-Engels, S.

Badal, A.

A. Badal and A. Badano, “Accelerating Monte Carlo simulations of photon transport in a voxelized geometry using a massively parallel graphics processing unit,” Med. Phys. 36(11), 4878–4880 (2009).
[Crossref] [PubMed]

Badano, A.

A. Badal and A. Badano, “Accelerating Monte Carlo simulations of photon transport in a voxelized geometry using a massively parallel graphics processing unit,” Med. Phys. 36(11), 4878–4880 (2009).
[Crossref] [PubMed]

Bai, T.

Y. Xu, T. Bai, H. Yan, L. Ouyang, A. Pompos, J. Wang, L. Zhou, S. B. Jiang, and X. Jia, “A practical cone-beam CT scatter correction method with optimized Monte Carlo simulations for image-guided radiation therapy,” Phys. Med. Biol. 60(9), 3567–3587 (2015).
[Crossref] [PubMed]

Baró, J.

J. Baró, J. Sempau, J. M. Fernández-Varea, and F. Salvat, “PENELOPE: An algorithm for Monte Carlo simulation of the penetration and energy loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. 100(1), 31–46 (1995).
[Crossref]

Binder, K.

K. Binder, D. Heermann, L. Roelofs, A. J. Mallinckrodt, and S. McKay, “Monte Carlo simulation in statistical physics,” Comput. Phys. 7(2), 156–157 (1993).
[Crossref]

Boas, D.

Botev, Z. I.

D. P. Kroese, T. Brereton, T. Taimre, and Z. I. Botev, “Why the Monte Carlo method is so important today,” Wiley Interdiscip. Rev. Comput. Stat. 6(6), 386–392 (2014).
[Crossref]

Z. I. Botev and D. P. Kroese, “Efficient Monte Carlo simulation via the generalized splitting method,” Stat. Comput. 22(1), 1–16 (2012).
[Crossref]

Bouchard, H.

S. Hissoiny, B. Ozell, H. Bouchard, and P. Després, “GPUMCD: A new GPU-oriented Monte Carlo dose calculation platform,” Med. Phys. 38(2), 754–764 (2011).
[Crossref] [PubMed]

Brereton, T.

D. P. Kroese, T. Brereton, T. Taimre, and Z. I. Botev, “Why the Monte Carlo method is so important today,” Wiley Interdiscip. Rev. Comput. Stat. 6(6), 386–392 (2014).
[Crossref]

Casella, G.

C. P. Robert and G. Casella, “The Metropolis-Hastings algorithm,” Springer Texts in Statistics 49, 327–335 (2016).

Cerviño, L.

X. Jia, H. Yan, L. Cerviño, M. Folkerts, and S. B. Jiang, “A GPU tool for efficient, accurate, and realistic simulation of cone beam CT projections,” Med. Phys. 39(12), 7368–7378 (2012).
[Crossref] [PubMed]

Culver, J.

Després, P.

S. Hissoiny, B. Ozell, H. Bouchard, and P. Després, “GPUMCD: A new GPU-oriented Monte Carlo dose calculation platform,” Med. Phys. 38(2), 754–764 (2011).
[Crossref] [PubMed]

Dunn, A.

Fernández-Varea, J. M.

J. Baró, J. Sempau, J. M. Fernández-Varea, and F. Salvat, “PENELOPE: An algorithm for Monte Carlo simulation of the penetration and energy loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. 100(1), 31–46 (1995).
[Crossref]

Folkerts, M.

X. Jia, H. Yan, L. Cerviño, M. Folkerts, and S. B. Jiang, “A GPU tool for efficient, accurate, and realistic simulation of cone beam CT projections,” Med. Phys. 39(12), 7368–7378 (2012).
[Crossref] [PubMed]

X. Jia, X. Gu, Y. J. Graves, M. Folkerts, and S. B. Jiang, “GPU-based fast Monte Carlo simulation for radiotherapy dose calculation,” Phys. Med. Biol. 56(22), 7017–7031 (2011).
[Crossref] [PubMed]

Graves, Y. J.

X. Jia, X. Gu, Y. J. Graves, M. Folkerts, and S. B. Jiang, “GPU-based fast Monte Carlo simulation for radiotherapy dose calculation,” Phys. Med. Biol. 56(22), 7017–7031 (2011).
[Crossref] [PubMed]

Gu, X.

X. Jia, H. Yan, X. Gu, and S. B. Jiang, “Fast Monte Carlo simulation for patient-specific CT/CBCT imaging dose calculation,” Phys. Med. Biol. 57(3), 577–590 (2012).
[Crossref] [PubMed]

X. Jia, X. Gu, Y. J. Graves, M. Folkerts, and S. B. Jiang, “GPU-based fast Monte Carlo simulation for radiotherapy dose calculation,” Phys. Med. Biol. 56(22), 7017–7031 (2011).
[Crossref] [PubMed]

Haario, H.

H. Haario, E. Saksman, and J. Tamminen, “An adaptive Metropolis algorithm,” Bernoulli 7(2), 223–242 (2001).
[Crossref]

H. Haario, E. Saksman, and J. Tamminen, “Adaptive proposal distribution for random walk Metropolis algorithm,” Comput. Stat. 14(3), 375–396 (1999).
[Crossref]

Haghighat, A.

A. Haghighat and J. C. Wagner, “Monte Carlo variance reduction with deterministic importance functions,” Prog. Nucl. Energy 42(1), 25–53 (2003).
[Crossref]

Han, T. D.

Hastings, W. K.

W. K. Hastings, “Monte Carlo sampling methods using Markov chains and their applications,” Biometrika 57(1), 97–109 (1970).
[Crossref]

He, H.

Heermann, D.

K. Binder, D. Heermann, L. Roelofs, A. J. Mallinckrodt, and S. McKay, “Monte Carlo simulation in statistical physics,” Comput. Phys. 7(2), 156–157 (1993).
[Crossref]

Hemmings, P.

E. Woodcock, T. Murphy, P. Hemmings, and S. Longworth, “Techniques used in the GEM code for Monte Carlo neutronics calculations in reactors and other systems of complex geometry,” in Proc. Conf. Applications of Computing Methods to Reactor Problems, 1965)

Hissoiny, S.

S. Hissoiny, B. Ozell, H. Bouchard, and P. Després, “GPUMCD: A new GPU-oriented Monte Carlo dose calculation platform,” Med. Phys. 38(2), 754–764 (2011).
[Crossref] [PubMed]

Jia, X.

Y. Xu, T. Bai, H. Yan, L. Ouyang, A. Pompos, J. Wang, L. Zhou, S. B. Jiang, and X. Jia, “A practical cone-beam CT scatter correction method with optimized Monte Carlo simulations for image-guided radiation therapy,” Phys. Med. Biol. 60(9), 3567–3587 (2015).
[Crossref] [PubMed]

X. Jia, P. Ziegenhein, and S. B. Jiang, “GPU-based high-performance computing for radiation therapy,” Phys. Med. Biol. 59(4), R151–R182 (2014).
[Crossref] [PubMed]

X. Jia, H. Yan, X. Gu, and S. B. Jiang, “Fast Monte Carlo simulation for patient-specific CT/CBCT imaging dose calculation,” Phys. Med. Biol. 57(3), 577–590 (2012).
[Crossref] [PubMed]

X. Jia, H. Yan, L. Cerviño, M. Folkerts, and S. B. Jiang, “A GPU tool for efficient, accurate, and realistic simulation of cone beam CT projections,” Med. Phys. 39(12), 7368–7378 (2012).
[Crossref] [PubMed]

X. Jia, X. Gu, Y. J. Graves, M. Folkerts, and S. B. Jiang, “GPU-based fast Monte Carlo simulation for radiotherapy dose calculation,” Phys. Med. Biol. 56(22), 7017–7031 (2011).
[Crossref] [PubMed]

Jiang, S. B.

Y. Xu, T. Bai, H. Yan, L. Ouyang, A. Pompos, J. Wang, L. Zhou, S. B. Jiang, and X. Jia, “A practical cone-beam CT scatter correction method with optimized Monte Carlo simulations for image-guided radiation therapy,” Phys. Med. Biol. 60(9), 3567–3587 (2015).
[Crossref] [PubMed]

X. Jia, P. Ziegenhein, and S. B. Jiang, “GPU-based high-performance computing for radiation therapy,” Phys. Med. Biol. 59(4), R151–R182 (2014).
[Crossref] [PubMed]

X. Jia, H. Yan, L. Cerviño, M. Folkerts, and S. B. Jiang, “A GPU tool for efficient, accurate, and realistic simulation of cone beam CT projections,” Med. Phys. 39(12), 7368–7378 (2012).
[Crossref] [PubMed]

X. Jia, H. Yan, X. Gu, and S. B. Jiang, “Fast Monte Carlo simulation for patient-specific CT/CBCT imaging dose calculation,” Phys. Med. Biol. 57(3), 577–590 (2012).
[Crossref] [PubMed]

X. Jia, X. Gu, Y. J. Graves, M. Folkerts, and S. B. Jiang, “GPU-based fast Monte Carlo simulation for radiotherapy dose calculation,” Phys. Med. Biol. 56(22), 7017–7031 (2011).
[Crossref] [PubMed]

Kawrakow, I.

E. Mainegra-Hing and I. Kawrakow, “Variance reduction techniques for fast Monte Carlo CBCT scatter correction calculations,” Phys. Med. Biol. 55(16), 4495–4507 (2010).
[Crossref] [PubMed]

Kroese, D. P.

D. P. Kroese, T. Brereton, T. Taimre, and Z. I. Botev, “Why the Monte Carlo method is so important today,” Wiley Interdiscip. Rev. Comput. Stat. 6(6), 386–392 (2014).
[Crossref]

Z. I. Botev and D. P. Kroese, “Efficient Monte Carlo simulation via the generalized splitting method,” Stat. Comput. 22(1), 1–16 (2012).
[Crossref]

Li, P.

Li, X.

Lilge, L.

Liu, C.

Lo, W. C.

Longworth, S.

E. Woodcock, T. Murphy, P. Hemmings, and S. Longworth, “Techniques used in the GEM code for Monte Carlo neutronics calculations in reactors and other systems of complex geometry,” in Proc. Conf. Applications of Computing Methods to Reactor Problems, 1965)

Ma, H.

Mainegra-Hing, E.

E. Mainegra-Hing and I. Kawrakow, “Variance reduction techniques for fast Monte Carlo CBCT scatter correction calculations,” Phys. Med. Biol. 55(16), 4495–4507 (2010).
[Crossref] [PubMed]

Mallinckrodt, A. J.

K. Binder, D. Heermann, L. Roelofs, A. J. Mallinckrodt, and S. McKay, “Monte Carlo simulation in statistical physics,” Comput. Phys. 7(2), 156–157 (1993).
[Crossref]

McKay, S.

K. Binder, D. Heermann, L. Roelofs, A. J. Mallinckrodt, and S. McKay, “Monte Carlo simulation in statistical physics,” Comput. Phys. 7(2), 156–157 (1993).
[Crossref]

Metropolis, N.

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44(247), 335–341 (1949).
[Crossref] [PubMed]

Murphy, T.

E. Woodcock, T. Murphy, P. Hemmings, and S. Longworth, “Techniques used in the GEM code for Monte Carlo neutronics calculations in reactors and other systems of complex geometry,” in Proc. Conf. Applications of Computing Methods to Reactor Problems, 1965)

Ouyang, L.

Y. Xu, T. Bai, H. Yan, L. Ouyang, A. Pompos, J. Wang, L. Zhou, S. B. Jiang, and X. Jia, “A practical cone-beam CT scatter correction method with optimized Monte Carlo simulations for image-guided radiation therapy,” Phys. Med. Biol. 60(9), 3567–3587 (2015).
[Crossref] [PubMed]

Ozell, B.

S. Hissoiny, B. Ozell, H. Bouchard, and P. Després, “GPUMCD: A new GPU-oriented Monte Carlo dose calculation platform,” Med. Phys. 38(2), 754–764 (2011).
[Crossref] [PubMed]

Pompos, A.

Y. Xu, T. Bai, H. Yan, L. Ouyang, A. Pompos, J. Wang, L. Zhou, S. B. Jiang, and X. Jia, “A practical cone-beam CT scatter correction method with optimized Monte Carlo simulations for image-guided radiation therapy,” Phys. Med. Biol. 60(9), 3567–3587 (2015).
[Crossref] [PubMed]

Pratx, G.

G. Pratx and L. Xing, “GPU computing in medical physics: a review,” Med. Phys. 38(5), 2685–2697 (2011).
[Crossref] [PubMed]

Robert, C. P.

C. P. Robert and G. Casella, “The Metropolis-Hastings algorithm,” Springer Texts in Statistics 49, 327–335 (2016).

Roelofs, L.

K. Binder, D. Heermann, L. Roelofs, A. J. Mallinckrodt, and S. McKay, “Monte Carlo simulation in statistical physics,” Comput. Phys. 7(2), 156–157 (1993).
[Crossref]

Rogers, D. W.

D. W. Rogers, “Fifty years of Monte Carlo simulations for medical physics,” Phys. Med. Biol. 51(13), R287–R301 (2006).
[Crossref] [PubMed]

Rose, J.

Saksman, E.

H. Haario, E. Saksman, and J. Tamminen, “An adaptive Metropolis algorithm,” Bernoulli 7(2), 223–242 (2001).
[Crossref]

H. Haario, E. Saksman, and J. Tamminen, “Adaptive proposal distribution for random walk Metropolis algorithm,” Comput. Stat. 14(3), 375–396 (1999).
[Crossref]

Salvat, F.

J. Baró, J. Sempau, J. M. Fernández-Varea, and F. Salvat, “PENELOPE: An algorithm for Monte Carlo simulation of the penetration and energy loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. 100(1), 31–46 (1995).
[Crossref]

Sempau, J.

J. Baró, J. Sempau, J. M. Fernández-Varea, and F. Salvat, “PENELOPE: An algorithm for Monte Carlo simulation of the penetration and energy loss of electrons and positrons in matter,” Nucl. Instrum. Methods Phys. Res. 100(1), 31–46 (1995).
[Crossref]

Stott, J.

Taimre, T.

D. P. Kroese, T. Brereton, T. Taimre, and Z. I. Botev, “Why the Monte Carlo method is so important today,” Wiley Interdiscip. Rev. Comput. Stat. 6(6), 386–392 (2014).
[Crossref]

Tamminen, J.

H. Haario, E. Saksman, and J. Tamminen, “An adaptive Metropolis algorithm,” Bernoulli 7(2), 223–242 (2001).
[Crossref]

H. Haario, E. Saksman, and J. Tamminen, “Adaptive proposal distribution for random walk Metropolis algorithm,” Comput. Stat. 14(3), 375–396 (1999).
[Crossref]

Ulam, S.

N. Metropolis and S. Ulam, “The Monte Carlo method,” J. Am. Stat. Assoc. 44(247), 335–341 (1949).
[Crossref] [PubMed]

Wagner, J. C.

A. Haghighat and J. C. Wagner, “Monte Carlo variance reduction with deterministic importance functions,” Prog. Nucl. Energy 42(1), 25–53 (2003).
[Crossref]

Wang, J.

Y. Xu, T. Bai, H. Yan, L. Ouyang, A. Pompos, J. Wang, L. Zhou, S. B. Jiang, and X. Jia, “A practical cone-beam CT scatter correction method with optimized Monte Carlo simulations for image-guided radiation therapy,” Phys. Med. Biol. 60(9), 3567–3587 (2015).
[Crossref] [PubMed]

Woodcock, E.

E. Woodcock, T. Murphy, P. Hemmings, and S. Longworth, “Techniques used in the GEM code for Monte Carlo neutronics calculations in reactors and other systems of complex geometry,” in Proc. Conf. Applications of Computing Methods to Reactor Problems, 1965)

Xing, L.

G. Pratx and L. Xing, “GPU computing in medical physics: a review,” Med. Phys. 38(5), 2685–2697 (2011).
[Crossref] [PubMed]

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Figures (10)

Fig. 1
Fig. 1 (A) Workflow of gMMC simulation; (B) illustration of different photon scattering paths with one (n), two (m), and three (k) scattering events.
Fig. 2
Fig. 2 Illustration of a photon scattering path.
Fig. 3
Fig. 3 Rayleigh DCS (A) and Compton DCS (B) of Aluminum within the energy range of 0-150 keV.
Fig. 4
Fig. 4 (A) and (A1), homogeneous Al phantom and illumination geometry; (B) and (B1), inhomogeneous two-material phantom and illumination geometry; (C) and (D), HN cancer patient phantom under the IFOV and OFOV setting; (C1)-(C2), side view and axial view of the X-ray illumination geometry for the IFOV case; (D1)-(D2), side view and axial view of the X-ray illumination geometry for the OFOV case.
Fig. 5
Fig. 5 Scatter signals of the homogeneous Al phantom case. (A1)-(A5) are first order Compton scatter, first order Rayleigh scatter, second order Compton scatter, second order Rayleigh scatter, and total scatter signals (sum of previous four figures) computed by gMMC; (B1)-(B5) are corresponding results computed by gMCDRR; (C1)-(C5) are profiles on the yellow lines of corresponding images.
Fig. 6
Fig. 6 Scatter signals of the two-material phantom case. (A1)-(A5) are first order Compton scatter, first order Rayleigh scatter, second order Compton scatter, second order Rayleigh scatter, and total scatter signals (sum of previous four figures) computed by gMMC; (B1)-(B5) are corresponding results computed by gMCDRR; (C1)-(C5) are profiles on the yellow lines of corresponding images.
Fig. 7
Fig. 7 Scatter signal of the IFOV HN case. (A1)-(A5) are first order Compton scatter, first order Rayleigh scatter, second order Compton scatter, second order Rayleigh scatter, and total scatter signals (sum of previous four figures) computed by gMMC; (B1)-(B5) are corresponding results computed by gMCDRR; (C1)-(C5) are profiles on the yellow lines of corresponding images.
Fig. 8
Fig. 8 Scatter signal of the OFOV HN case. (A1)-(A5) are first order Compton scatter, first order Rayleigh scatter, second order Compton scatter, second order Rayleigh scatter, and total scatter signals (sum of previous four figures) computed by gMMC; (B1)-(B5) are corresponding results computed by gMCDRR; (C1)-(C5) are profiles on the yellow lines of corresponding images.
Fig. 9
Fig. 9 (a)-(c) the total scatter image of Homogeneous Al phantom case from EGSnrc, gMCDRR and gMMC; (d) the profile of each result a blue line in (a).
Fig. 10
Fig. 10 (a)-(c) the total scatter image of and OFOV HN case from EGSnrc, gMCDRR and gMMC; (d) the profile of each result a blue line in (a).

Tables (1)

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Table 1 Metropolis-Hasting path sampling Method.

Equations (2)

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p accept (xy)=min[ 1, p(y)T(yx) p(x)T(xy) ]
p(x)=ϕ( E 0 )F( A 1 ) i=1 N ρ T i ( A i ) k=1 N+1 exp[ l i μ(s)ds ] dAcosα/ l N+1 2

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