Abstract

We present an analytical model of optical fluence for multiple cylindrical inhomogeneities embedded in an otherwise homogeneous turbid medium. The model is based on the diffusion equation and represents the optical fluence distribution inside and outside inhomogeneities as a series of modified Bessel functions. We take into account the interplay between cylindrical inhomogeneities by introducing new boundary conditions on the surface of inhomogeneities. The model is compared with the numerical solution of the diffusion equation with NIRFAST software. The fluences inside the inhomogeneities obtained by the two methods are in close agreement. This permits the use of the model as a forward model for quantitative photoacoustic imaging.

© 2014 Optical Society of America

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    [CrossRef] [PubMed]
  2. V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97(6), 2767–2772 (2000).
    [CrossRef] [PubMed]
  3. D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13(1), 76–90 (2001).
    [CrossRef] [PubMed]
  4. B. T. Cox, J. G. Laufer, and P. C. Beard, “Quantitative Photoacoustic Image Reconstruction using Fluence Dependent Chromophores,” Biomed. Opt. Express 1(1), 201–208 (2010).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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  14. L. Yao and H. Jiang, “Finite-element-based photoacoustic tomography in time domain,” J. Opt. A, Pure Appl. Opt. 11(8), 085301 (2009).
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    [CrossRef] [PubMed]
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  21. H. O. Di Rocco, D. I. Iriarte, M. Lester, J. Pomarico, and H. F. Ranea-Sandoval, “CW laser transilluminance in turbid media with cylindrical inclusions,” Opt. Int. J. Light Electron Opt. 122(7), 577–581 (2011).
    [CrossRef]
  22. J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(5), 051917 (2001).
    [CrossRef] [PubMed]
  23. J. Ripoll, V. Ntziachristos, and E. N. Economou, “Experimental demonstration of a fast analytical method for modeling photon propagation in diffusive media with arbitrary geometry,” Proc. SPIE 4431, 233–239 (2001).
    [CrossRef]
  24. X.-P. Wu, J.-M. Shi, and J.-C. Wang, “Multiple Scattering by Parallel Plasma Cylinders,” IEEE Trans. Plasma Sci. 42(1), 13–19 (2014).
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  28. V. M. Twersky, “Multiple Scattering of Radiation by an Arbitrary Configuration of Parallel Cylinders,” J. Acoust. Soc. Am. 24(1), 42–46 (1952).
    [CrossRef]
  29. C. M. Linton and P. A. Martin, “Multiple scattering by random configurations of circular cylinders: Second-order corrections for the effective wavenumber,” J. Acoust. Soc. Am. 117(6), 3413–3423 (2005).
    [CrossRef] [PubMed]
  30. J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 77, 1190–1195 (1976).
  31. W. H. Lin and A. C. Raptis, “Sound scattering by a group of oscillatory cylinders,” J. Acoust. Soc. Am. 77(1), 15–28 (1985).
    [CrossRef]
  32. X. Cheng and D. Boas, “Diffuse optical reflection tomography using continuous wave illumination,” Opt. Express 3(3), 118–123 (1998).
    [CrossRef] [PubMed]
  33. A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
    [CrossRef]
  34. A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  37. M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
    [CrossRef]

2014

X.-P. Wu, J.-M. Shi, and J.-C. Wang, “Multiple Scattering by Parallel Plasma Cylinders,” IEEE Trans. Plasma Sci. 42(1), 13–19 (2014).
[CrossRef]

S.-C. Lee, “Scattering of evanescent wave by multiple parallel infinite cylinders near a surface,” IEEE Trans. Plasma Sci. 147, 252–260 (2014).

2013

M. Jermyn, H. Ghadyani, M. A. Mastanduno, W. Turner, S. C. Davis, H. Dehghani, and B. W. Pogue, “Fast segmentation and high-quality three-dimensional volume mesh creation from medical images for diffuse optical tomography,” J. Biomed. Opt. 18(8), 086007 (2013).
[CrossRef] [PubMed]

2012

B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imaging: a review,” J. Biomed. Opt. 17(6), 061202 (2012).
[CrossRef] [PubMed]

2011

H. O. Di Rocco, D. I. Iriarte, M. Lester, J. Pomarico, and H. F. Ranea-Sandoval, “CW laser transilluminance in turbid media with cylindrical inclusions,” Opt. Int. J. Light Electron Opt. 122(7), 577–581 (2011).
[CrossRef]

2010

2009

Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17(22), 20178–20190 (2009).
[CrossRef] [PubMed]

Z. G. Wang, L. Z. Sun, L. L. Fajardo, and G. Wang, “Modeling and reconstruction of diffuse optical tomography using adjoint method,” Commun. Numer. Methods Eng. 25(6), 657–665 (2009).
[CrossRef]

L. Yao and H. Jiang, “Finite-element-based photoacoustic tomography in time domain,” J. Opt. A, Pure Appl. Opt. 11(8), 085301 (2009).
[CrossRef]

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography,” Philos. Trans. R. Soc. Lond. A 367, 3043–3054 (2009).

Y. Zhai and S. A. Cummer, “Fast tomographic reconstruction strategy for diffuse optical tomography,” Opt. Express 17(7), 5285–5297 (2009).
[CrossRef] [PubMed]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

2008

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(17-18), 2767–2778 (2008).
[CrossRef]

2007

2006

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: Recovery of optical absorption coefficient maps of heterogeneous media,” Appl. Phys. Lett. 88(23), 231101 (2006).
[CrossRef]

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
[CrossRef]

2005

C. M. Linton and P. A. Martin, “Multiple scattering by random configurations of circular cylinders: Second-order corrections for the effective wavenumber,” J. Acoust. Soc. Am. 117(6), 3413–3423 (2005).
[CrossRef] [PubMed]

2002

2001

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13(1), 76–90 (2001).
[CrossRef] [PubMed]

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(5), 051917 (2001).
[CrossRef] [PubMed]

J. Ripoll, V. Ntziachristos, and E. N. Economou, “Experimental demonstration of a fast analytical method for modeling photon propagation in diffusive media with arbitrary geometry,” Proc. SPIE 4431, 233–239 (2001).
[CrossRef]

2000

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97(6), 2767–2772 (2000).
[CrossRef] [PubMed]

1998

1997

1996

1995

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

1994

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11(9), 2526–2538 (1994).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91(11), 4887–4891 (1994).
[CrossRef] [PubMed]

1989

1985

W. H. Lin and A. C. Raptis, “Sound scattering by a group of oscillatory cylinders,” J. Acoust. Soc. Am. 77(1), 15–28 (1985).
[CrossRef]

1976

J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 77, 1190–1195 (1976).

1952

V. M. Twersky, “Multiple Scattering of Radiation by an Arbitrary Configuration of Parallel Cylinders,” J. Acoust. Soc. Am. 24(1), 42–46 (1952).
[CrossRef]

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[CrossRef] [PubMed]

Arridge, S. R.

B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imaging: a review,” J. Biomed. Opt. 17(6), 061202 (2012).
[CrossRef] [PubMed]

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(17-18), 2767–2778 (2008).
[CrossRef]

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007).
[CrossRef] [PubMed]

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[CrossRef] [PubMed]

Beard, P. C.

B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imaging: a review,” J. Biomed. Opt. 17(6), 061202 (2012).
[CrossRef] [PubMed]

B. T. Cox, J. G. Laufer, and P. C. Beard, “Quantitative Photoacoustic Image Reconstruction using Fluence Dependent Chromophores,” Biomed. Opt. Express 1(1), 201–208 (2010).
[CrossRef] [PubMed]

Bertrand, J. C.

J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 77, 1190–1195 (1976).

Boas, D.

Boas, D. A.

Q. Fang and D. A. Boas, “Monte Carlo simulation of photon migration in 3D turbid media accelerated by graphics processing units,” Opt. Express 17(22), 20178–20190 (2009).
[CrossRef] [PubMed]

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13(1), 76–90 (2001).
[CrossRef] [PubMed]

S. A. Walker, D. A. Boas, and E. Gratton, “Photon density waves scattered from cylindrical inhomogeneities: theory and experiments,” Appl. Opt. 37(10), 1935–1944 (1998).
[CrossRef] [PubMed]

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35(19), 3746–3758 (1996).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91(11), 4887–4891 (1994).
[CrossRef] [PubMed]

Carminati, R.

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(5), 051917 (2001).
[CrossRef] [PubMed]

Carpenter, C. M.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

Chance, B.

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97(6), 2767–2772 (2000).
[CrossRef] [PubMed]

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35(19), 3746–3758 (1996).
[CrossRef] [PubMed]

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91(11), 4887–4891 (1994).
[CrossRef] [PubMed]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for the non-invasive measurement of tissue optical properties,” Appl. Opt. 28(12), 2331–2336 (1989).
[CrossRef] [PubMed]

Cheng, X.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13(1), 76–90 (2001).
[CrossRef] [PubMed]

X. Cheng and D. Boas, “Diffuse optical reflection tomography using continuous wave illumination,” Opt. Express 3(3), 118–123 (1998).
[CrossRef] [PubMed]

Choe, R.

Corlu, A.

Cox, B.

B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imaging: a review,” J. Biomed. Opt. 17(6), 061202 (2012).
[CrossRef] [PubMed]

Cox, B. T.

Culver, J.

Cummer, S. A.

Davis, S. C.

M. Jermyn, H. Ghadyani, M. A. Mastanduno, W. Turner, S. C. Davis, H. Dehghani, and B. W. Pogue, “Fast segmentation and high-quality three-dimensional volume mesh creation from medical images for diffuse optical tomography,” J. Biomed. Opt. 18(8), 086007 (2013).
[CrossRef] [PubMed]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

Dehghani, H.

M. Jermyn, H. Ghadyani, M. A. Mastanduno, W. Turner, S. C. Davis, H. Dehghani, and B. W. Pogue, “Fast segmentation and high-quality three-dimensional volume mesh creation from medical images for diffuse optical tomography,” J. Biomed. Opt. 18(8), 086007 (2013).
[CrossRef] [PubMed]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

Di Rocco, H. O.

H. O. Di Rocco, D. I. Iriarte, M. Lester, J. Pomarico, and H. F. Ranea-Sandoval, “CW laser transilluminance in turbid media with cylindrical inclusions,” Opt. Int. J. Light Electron Opt. 122(7), 577–581 (2011).
[CrossRef]

Dunn, A.

Durduran, T.

Eames, M. E.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

Economou, E. N.

J. Ripoll, V. Ntziachristos, and E. N. Economou, “Experimental demonstration of a fast analytical method for modeling photon propagation in diffusive media with arbitrary geometry,” Proc. SPIE 4431, 233–239 (2001).
[CrossRef]

Fajardo, L. L.

Z. G. Wang, L. Z. Sun, L. L. Fajardo, and G. Wang, “Modeling and reconstruction of diffuse optical tomography using adjoint method,” Commun. Numer. Methods Eng. 25(6), 657–665 (2009).
[CrossRef]

Fang, Q.

Felbacq, D.

Gaudette, T.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13(1), 76–90 (2001).
[CrossRef] [PubMed]

Ghadyani, H.

M. Jermyn, H. Ghadyani, M. A. Mastanduno, W. Turner, S. C. Davis, H. Dehghani, and B. W. Pogue, “Fast segmentation and high-quality three-dimensional volume mesh creation from medical images for diffuse optical tomography,” J. Biomed. Opt. 18(8), 086007 (2013).
[CrossRef] [PubMed]

Gratton, E.

Hielscher, A. H.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[CrossRef] [PubMed]

Iriarte, D. I.

H. O. Di Rocco, D. I. Iriarte, M. Lester, J. Pomarico, and H. F. Ranea-Sandoval, “CW laser transilluminance in turbid media with cylindrical inclusions,” Opt. Int. J. Light Electron Opt. 122(7), 577–581 (2011).
[CrossRef]

Jermyn, M.

M. Jermyn, H. Ghadyani, M. A. Mastanduno, W. Turner, S. C. Davis, H. Dehghani, and B. W. Pogue, “Fast segmentation and high-quality three-dimensional volume mesh creation from medical images for diffuse optical tomography,” J. Biomed. Opt. 18(8), 086007 (2013).
[CrossRef] [PubMed]

Jiang, H.

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography,” Philos. Trans. R. Soc. Lond. A 367, 3043–3054 (2009).

L. Yao and H. Jiang, “Finite-element-based photoacoustic tomography in time domain,” J. Opt. A, Pure Appl. Opt. 11(8), 085301 (2009).
[CrossRef]

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: Recovery of optical absorption coefficient maps of heterogeneous media,” Appl. Phys. Lett. 88(23), 231101 (2006).
[CrossRef]

Kienle, A.

Laufer, J. G.

B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imaging: a review,” J. Biomed. Opt. 17(6), 061202 (2012).
[CrossRef] [PubMed]

B. T. Cox, J. G. Laufer, and P. C. Beard, “Quantitative Photoacoustic Image Reconstruction using Fluence Dependent Chromophores,” Biomed. Opt. Express 1(1), 201–208 (2010).
[CrossRef] [PubMed]

Lee, S.-C.

S.-C. Lee, “Scattering of evanescent wave by multiple parallel infinite cylinders near a surface,” IEEE Trans. Plasma Sci. 147, 252–260 (2014).

Lester, M.

H. O. Di Rocco, D. I. Iriarte, M. Lester, J. Pomarico, and H. F. Ranea-Sandoval, “CW laser transilluminance in turbid media with cylindrical inclusions,” Opt. Int. J. Light Electron Opt. 122(7), 577–581 (2011).
[CrossRef]

Li, X. D.

Lin, W. H.

W. H. Lin and A. C. Raptis, “Sound scattering by a group of oscillatory cylinders,” J. Acoust. Soc. Am. 77(1), 15–28 (1985).
[CrossRef]

Linton, C. M.

C. M. Linton and P. A. Martin, “Multiple scattering by random configurations of circular cylinders: Second-order corrections for the effective wavenumber,” J. Acoust. Soc. Am. 117(6), 3413–3423 (2005).
[CrossRef] [PubMed]

Macke, A.

Mandeville, J. B.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13(1), 76–90 (2001).
[CrossRef] [PubMed]

Marota, J. J.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13(1), 76–90 (2001).
[CrossRef] [PubMed]

Martin, P. A.

C. M. Linton and P. A. Martin, “Multiple scattering by random configurations of circular cylinders: Second-order corrections for the effective wavenumber,” J. Acoust. Soc. Am. 117(6), 3413–3423 (2005).
[CrossRef] [PubMed]

Mastanduno, M. A.

M. Jermyn, H. Ghadyani, M. A. Mastanduno, W. Turner, S. C. Davis, H. Dehghani, and B. W. Pogue, “Fast segmentation and high-quality three-dimensional volume mesh creation from medical images for diffuse optical tomography,” J. Biomed. Opt. 18(8), 086007 (2013).
[CrossRef] [PubMed]

Maystre, D.

Mishchenko, M. I.

Nieto-Vesperinas, M.

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(5), 051917 (2001).
[CrossRef] [PubMed]

Ntziachristos, V.

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(5), 051917 (2001).
[CrossRef] [PubMed]

J. Ripoll, V. Ntziachristos, and E. N. Economou, “Experimental demonstration of a fast analytical method for modeling photon propagation in diffusive media with arbitrary geometry,” Proc. SPIE 4431, 233–239 (2001).
[CrossRef]

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97(6), 2767–2772 (2000).
[CrossRef] [PubMed]

O’Leary, M. A.

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35(19), 3746–3758 (1996).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91(11), 4887–4891 (1994).
[CrossRef] [PubMed]

Patterson, M. S.

Paulsen, K. D.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

Pogue, B. W.

M. Jermyn, H. Ghadyani, M. A. Mastanduno, W. Turner, S. C. Davis, H. Dehghani, and B. W. Pogue, “Fast segmentation and high-quality three-dimensional volume mesh creation from medical images for diffuse optical tomography,” J. Biomed. Opt. 18(8), 086007 (2013).
[CrossRef] [PubMed]

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

Pomarico, J.

H. O. Di Rocco, D. I. Iriarte, M. Lester, J. Pomarico, and H. F. Ranea-Sandoval, “CW laser transilluminance in turbid media with cylindrical inclusions,” Opt. Int. J. Light Electron Opt. 122(7), 577–581 (2011).
[CrossRef]

Ranea-Sandoval, H. F.

H. O. Di Rocco, D. I. Iriarte, M. Lester, J. Pomarico, and H. F. Ranea-Sandoval, “CW laser transilluminance in turbid media with cylindrical inclusions,” Opt. Int. J. Light Electron Opt. 122(7), 577–581 (2011).
[CrossRef]

Raptis, A. C.

W. H. Lin and A. C. Raptis, “Sound scattering by a group of oscillatory cylinders,” J. Acoust. Soc. Am. 77(1), 15–28 (1985).
[CrossRef]

Ripoll, J.

J. Ripoll, V. Ntziachristos, and E. N. Economou, “Experimental demonstration of a fast analytical method for modeling photon propagation in diffusive media with arbitrary geometry,” Proc. SPIE 4431, 233–239 (2001).
[CrossRef]

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(5), 051917 (2001).
[CrossRef] [PubMed]

Rosen, M. A.

Schnall, M.

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97(6), 2767–2772 (2000).
[CrossRef] [PubMed]

Schnall, M. D.

Schweiger, M.

Shi, J.-M.

X.-P. Wu, J.-M. Shi, and J.-C. Wang, “Multiple Scattering by Parallel Plasma Cylinders,” IEEE Trans. Plasma Sci. 42(1), 13–19 (2014).
[CrossRef]

Srinivasan, S.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

Stott, J.

Strangman, G.

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13(1), 76–90 (2001).
[CrossRef] [PubMed]

Sun, L. Z.

Z. G. Wang, L. Z. Sun, L. L. Fajardo, and G. Wang, “Modeling and reconstruction of diffuse optical tomography using adjoint method,” Commun. Numer. Methods Eng. 25(6), 657–665 (2009).
[CrossRef]

Tarvainen, T.

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(17-18), 2767–2778 (2008).
[CrossRef]

Tayeb, G.

Travis, L. D.

Turner, W.

M. Jermyn, H. Ghadyani, M. A. Mastanduno, W. Turner, S. C. Davis, H. Dehghani, and B. W. Pogue, “Fast segmentation and high-quality three-dimensional volume mesh creation from medical images for diffuse optical tomography,” J. Biomed. Opt. 18(8), 086007 (2013).
[CrossRef] [PubMed]

Twersky, V. M.

V. M. Twersky, “Multiple Scattering of Radiation by an Arbitrary Configuration of Parallel Cylinders,” J. Acoust. Soc. Am. 24(1), 42–46 (1952).
[CrossRef]

Vauhkonen, M.

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(17-18), 2767–2778 (2008).
[CrossRef]

Walker, S. A.

Wang, G.

Z. G. Wang, L. Z. Sun, L. L. Fajardo, and G. Wang, “Modeling and reconstruction of diffuse optical tomography using adjoint method,” Commun. Numer. Methods Eng. 25(6), 657–665 (2009).
[CrossRef]

Wang, J.-C.

X.-P. Wu, J.-M. Shi, and J.-C. Wang, “Multiple Scattering by Parallel Plasma Cylinders,” IEEE Trans. Plasma Sci. 42(1), 13–19 (2014).
[CrossRef]

Wang, L. V.

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
[CrossRef]

Wang, Z. G.

Z. G. Wang, L. Z. Sun, L. L. Fajardo, and G. Wang, “Modeling and reconstruction of diffuse optical tomography using adjoint method,” Commun. Numer. Methods Eng. 25(6), 657–665 (2009).
[CrossRef]

Wilson, B. C.

Wu, X.-P.

X.-P. Wu, J.-M. Shi, and J.-C. Wang, “Multiple Scattering by Parallel Plasma Cylinders,” IEEE Trans. Plasma Sci. 42(1), 13–19 (2014).
[CrossRef]

Xu, M.

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
[CrossRef]

Yalavarthy, P. K.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

Yao, L.

L. Yao and H. Jiang, “Finite-element-based photoacoustic tomography in time domain,” J. Opt. A, Pure Appl. Opt. 11(8), 085301 (2009).
[CrossRef]

Yodh, A.

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

Yodh, A. G.

A. Corlu, R. Choe, T. Durduran, M. A. Rosen, M. Schweiger, S. R. Arridge, M. D. Schnall, and A. G. Yodh, “Three-dimensional in vivo fluorescence diffuse optical tomography of breast cancer in humans,” Opt. Express 15(11), 6696–6716 (2007).
[CrossRef] [PubMed]

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97(6), 2767–2772 (2000).
[CrossRef] [PubMed]

X. D. Li, M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Fluorescent diffuse photon density waves in homogeneous and heterogeneous turbid media: analytic solutions and applications,” Appl. Opt. 35(19), 3746–3758 (1996).
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91(11), 4887–4891 (1994).
[CrossRef] [PubMed]

Young, J. W.

J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 77, 1190–1195 (1976).

Yuan, Z.

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography,” Philos. Trans. R. Soc. Lond. A 367, 3043–3054 (2009).

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: Recovery of optical absorption coefficient maps of heterogeneous media,” Appl. Phys. Lett. 88(23), 231101 (2006).
[CrossRef]

Zhai, Y.

Appl. Opt.

Appl. Phys. Lett.

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography: Recovery of optical absorption coefficient maps of heterogeneous media,” Appl. Phys. Lett. 88(23), 231101 (2006).
[CrossRef]

Biomed. Opt. Express

Commun. Numer. Methods Eng.

H. Dehghani, M. E. Eames, P. K. Yalavarthy, S. C. Davis, S. Srinivasan, C. M. Carpenter, B. W. Pogue, and K. D. Paulsen, “Near infrared optical tomography using NIRFAST: Algorithm for numerical model and image reconstruction,” Commun. Numer. Methods Eng. 25(6), 711–732 (2009).
[CrossRef] [PubMed]

Z. G. Wang, L. Z. Sun, L. L. Fajardo, and G. Wang, “Modeling and reconstruction of diffuse optical tomography using adjoint method,” Commun. Numer. Methods Eng. 25(6), 657–665 (2009).
[CrossRef]

IEEE Trans. Plasma Sci.

X.-P. Wu, J.-M. Shi, and J.-C. Wang, “Multiple Scattering by Parallel Plasma Cylinders,” IEEE Trans. Plasma Sci. 42(1), 13–19 (2014).
[CrossRef]

S.-C. Lee, “Scattering of evanescent wave by multiple parallel infinite cylinders near a surface,” IEEE Trans. Plasma Sci. 147, 252–260 (2014).

J. Acoust. Soc. Am.

V. M. Twersky, “Multiple Scattering of Radiation by an Arbitrary Configuration of Parallel Cylinders,” J. Acoust. Soc. Am. 24(1), 42–46 (1952).
[CrossRef]

C. M. Linton and P. A. Martin, “Multiple scattering by random configurations of circular cylinders: Second-order corrections for the effective wavenumber,” J. Acoust. Soc. Am. 117(6), 3413–3423 (2005).
[CrossRef] [PubMed]

J. W. Young and J. C. Bertrand, “Multiple scattering by two cylinders,” J. Acoust. Soc. Am. 77, 1190–1195 (1976).

W. H. Lin and A. C. Raptis, “Sound scattering by a group of oscillatory cylinders,” J. Acoust. Soc. Am. 77(1), 15–28 (1985).
[CrossRef]

J. Biomed. Opt.

M. Jermyn, H. Ghadyani, M. A. Mastanduno, W. Turner, S. C. Davis, H. Dehghani, and B. W. Pogue, “Fast segmentation and high-quality three-dimensional volume mesh creation from medical images for diffuse optical tomography,” J. Biomed. Opt. 18(8), 086007 (2013).
[CrossRef] [PubMed]

B. Cox, J. G. Laufer, S. R. Arridge, and P. C. Beard, “Quantitative spectroscopic photoacoustic imaging: a review,” J. Biomed. Opt. 17(6), 061202 (2012).
[CrossRef] [PubMed]

J. Opt. A, Pure Appl. Opt.

L. Yao and H. Jiang, “Finite-element-based photoacoustic tomography in time domain,” J. Opt. A, Pure Appl. Opt. 11(8), 085301 (2009).
[CrossRef]

J. Opt. Soc. Am. A

J. Quant. Spectrosc. Radiat. Transf.

T. Tarvainen, M. Vauhkonen, and S. R. Arridge, “Gauss–Newton reconstruction method for optical tomography using the finite element solution of the radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(17-18), 2767–2778 (2008).
[CrossRef]

Neuroimage

D. A. Boas, T. Gaudette, G. Strangman, X. Cheng, J. J. Marota, and J. B. Mandeville, “The accuracy of near infrared spectroscopy and imaging during focal changes in cerebral hemodynamics,” Neuroimage 13(1), 76–90 (2001).
[CrossRef] [PubMed]

Opt. Express

Opt. Int. J. Light Electron Opt.

H. O. Di Rocco, D. I. Iriarte, M. Lester, J. Pomarico, and H. F. Ranea-Sandoval, “CW laser transilluminance in turbid media with cylindrical inclusions,” Opt. Int. J. Light Electron Opt. 122(7), 577–581 (2011).
[CrossRef]

Philos. Trans. R. Soc. Lond. A

Z. Yuan and H. Jiang, “Quantitative photoacoustic tomography,” Philos. Trans. R. Soc. Lond. A 367, 3043–3054 (2009).

Phys. Med. Biol.

A. H. Hielscher, R. E. Alcouffe, and R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43(5), 1285–1302 (1998).
[CrossRef] [PubMed]

Phys. Rev. E Stat. Nonlin. Soft Matter Phys.

J. Ripoll, V. Ntziachristos, R. Carminati, and M. Nieto-Vesperinas, “Kirchhoff approximation for diffusive waves,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 64(5), 051917 (2001).
[CrossRef] [PubMed]

Phys. Today

A. Yodh and B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 48(3), 34–40 (1995).
[CrossRef]

Proc. Natl. Acad. Sci. U.S.A.

D. A. Boas, M. A. O’Leary, B. Chance, and A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solution and applications,” Proc. Natl. Acad. Sci. U.S.A. 91(11), 4887–4891 (1994).
[CrossRef] [PubMed]

V. Ntziachristos, A. G. Yodh, M. Schnall, and B. Chance, “Concurrent MRI and diffuse optical tomography of breast after indocyanine green enhancement,” Proc. Natl. Acad. Sci. U.S.A. 97(6), 2767–2772 (2000).
[CrossRef] [PubMed]

Proc. SPIE

J. Ripoll, V. Ntziachristos, and E. N. Economou, “Experimental demonstration of a fast analytical method for modeling photon propagation in diffusive media with arbitrary geometry,” Proc. SPIE 4431, 233–239 (2001).
[CrossRef]

Rev. Sci. Instrum.

M. Xu and L. V. Wang, “Photoacoustic imaging in biomedicine,” Rev. Sci. Instrum. 77(4), 041101 (2006).
[CrossRef]

Other

J. D. Jackson, Classical Electrodynamics, (Wiley, 1975), Chap. 3.

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Figures (10)

Fig. 1
Fig. 1

Horizontal cross section of the geometry. Cylindrical inhomogeneities of absorption coefficient embedded in an otherwise homogeneous infinite turbid medium. i is index of ith cylinder and i’ represents an arbitrary remaining cylinders. Cylindrical coordinate systems have their origins at the center of each inhomogeneity. The source is noted by S. P indicates an arbitrary point within the medium, and its coordinates values with respect to each cylinder coordinate system are indicated. The z-axis is along the axis of each inhomogeneity and comes out of the cross section.

Fig. 2
Fig. 2

Physical interpretation of the different orders of scattering waves by the ith inhomogeneity: (a) the incident wave produces the first-order scattered wave by the ith inhomogeneity; (b) the sum of first-order scattered waves from the remaining cylinders produces the second-order scattered wave by the ith inhomogeneity.

Fig. 3
Fig. 3

Comparison between the analytical model and NIRFAST for two inhomogeneities with different depth. Top row ((a) and (d)): Horizontal cross section of the first numerical phantom of two cylindrical inhomogeneities with different depth (depths of the c1 center in (a) and (e) are 2.1 cm and 2. 3cm, depths of the c2 center in (a) and (e) are 1.5cm) in a homogeneous background (µa = 0.2cm−1 and µ’s = 10cm−1). Second row ((b) and (e)): Horizontal cross section of the optical fluence distribution through y = 0cm; the dashed lines indicate the positions of the inhomogeneities c1 and c2. Third row ((c) and (g)): MRE corresponding to the area [-1.2cm, 1.2cm ] × [-1.2cm, 1.2cm ].

Fig. 4
Fig. 4

Curve of tmax versus the depth of the c1 center.

Fig. 5
Fig. 5

Comparison between the analytical model and NIRFAST for two inhomogeneities with different radius. Top row ((a) and (e)): Horizontal cross section of the numerical phantom of two cylindrical inhomogeneities with different radius (radius of c1 in (a) and (e) is 0.5cm and 0.7cm, respectively, radius of c2 in (a) and (e) is 0.3cm) in a homogeneous background (µa = 0.2cm−1 and µ’s = 10cm−1). Second row ((b) and (f)): Horizontal cross section of the optical fluence distribution through y = −0.5cm; the dashed lines indicate the positions of the inhomogeneities c1. Third row ((c) and (g)): Horizontal cross section of the optical fluence distribution through y = 0.5cm; the dashed lines indicate the position of the inhomogeneity c2. Fourth row ((d) and (h)): MRE corresponding to the area [-1.2cm, 1.2cm ] × [-1.2cm, 1.2cm ].

Fig. 6
Fig. 6

Curve of tmax versus the radius of c1.

Fig. 7
Fig. 7

Comparison between the analytical model and NIRFAST for two inhomogeneities with different absorption. Top row ((a) and (e)): Horizontal cross section of the numerical phantom of two cylindrical inhomogeneities with different absorption coefficient (µa of c1 in (a) and (e) is 0.4cm−1 and 0.8 cm−1, respectively, µa of c2 in (a) and (e) is 0.8 cm−1) in a homogeneous background (µa = 0.2cm−1 and µ’s = 10cm−1). Second row ((b) (f)): Horizontal cross section of the optical fluence distribution through y = −0.5cm; the dashed lines indicate the positions of the inhomogeneities c1. Third row ((c) (g)): Horizontal cross section of the optical fluence distribution through y = 0.5cm; the dashed lines indicate the position of the inhomogeneity c2. Fourth row((d) and (h)): MRE corresponding to the area [-1.2cm, 1.2cm ] × [-1.2cm, 1.2cm ].

Fig. 8
Fig. 8

Curve of tmax versus absorption coefficient of c1.

Fig. 9
Fig. 9

Comparison between the analytical model and NIRFAST. (a)Horizontal cross section of the first numerical phantom with five cylindrical inhomogeneities (µa = 0.8cm−1 and µ’s = 10cm−1) in a homogeneous background (µa = 0.2cm−1 and µ’s = 10cm−1). (b) Horizontal cross section of the optical fluence distribution through y = −0.7cm; the dashed lines indicate the positions of the inhomogeneities c1 and c3. (c) Horizontal cross section of the optical fluence distribution through y = 0cm; the dashed lines indicate the position of the inhomogeneity c5. (d) MRE corresponding to the area [-1.2cm, 1.2cm] × [-1.2cm, 1.2cm].

Fig. 10
Fig. 10

Comparison between the analytical model and NIRFAST. (a) Horizontal cross section of the second numerical phantom with three cylindrical inhomogeneities (see Table 1 for optical properties) in a homogeneous background (µa = 0.2cm−1 and µ’s = 10cm−1). (b) Horizontal cross section of the optical fluence distribution through y = 0.7cm; the dashed lines indicate the positions of the inhomogeneities c3. (c) Horizontal cross section of the optical fluence distribution through y = 0cm; the dashed lines indicate the positions of the inhomogeneities c1. (d) Horizontal cross section of the optical fluence distribution through y = −0.7cm; the dashed lines indicate the positions of the inhomogeneities c3. (e) MRE corresponding to the area [-1.2cm, 1.2cm ] × [-1.2cm, 1.2cm ].

Tables (1)

Tables Icon

Table 1 Parameters of the second numerical phantom

Equations (16)

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

2 Φ(r)+ k 2 Φ(r)= S(r) D
k=j 3( μ s ' + μ a ) μ a
Φ inc ( r s ,r)= S 4πD| r r s | exp( jk| r r s | )
Φ i inc = S 2 π 2 D out n= + 0 dp exp[ jn( θ i θ i s ) ]cos(pz) I n ( r i < ) K n ( r i > )
Φ i scat (l) = n= 0 dpexp(jn θ i ) cos(pz) B i n (l) (p) K n ( x i )
i S C t = n= 0 dpexp(jn θ i ) cos(pz) B i n t (p) K n ( x i )
B i n t (p)= l=1 t B i n (l) (p) .
Φ i in t = n= 0 dpexp (jn θ i )cos(pz) C i n t (p) I n ( y i )
Φ i out t = Φ i inc + S i C t + i ' S i ' C t1
D out [ ρ i Φ i out t ] ρ i = a i = D i in [ ρ i Φ i in t ] ρ i = a i [ Φ i out t ] ρ i = a i = [ Φ i in t ] ρ i = a i
B i n t ( p )= S 2 π 2 D out D out ( S n ( a )+S C n ) I n ' ( y b ) y b D i in ( S n ' ( a )+S C n ' ) I n ( y b ) x b D out K n ' ( x b ) I n ( y b ) x b D i in K n ( x b ) I n ' ( y b ) y b
C i n t ( p )= S 2 π 2 D out D out ( S n ( a )+S C n ) K n ' ( x b ) x b D out ( S n ' ( a )+S C n ' ) K n ( x b ) x b D out K n ' ( x b ) I n ( y b ) x b D i in K n ( x b ) I n ' ( y b ) y b
S n ( a )= I n ( x b ) m= exp(j(nm) θ i s ) I nm ( z s ) K m ( z b ) S C n = ( 1 ) n I n ( x b ) m= exp(j(nm) θ i i ' ) K nm ( z i ' ) B i ' m t1 (p)
S n ' ( a )= I n ' ( x b ) m= exp(j(nm) θ i s ) I nm ( z s ) K m ( z b ) S C n ' = ( 1 ) n I n ' ( x b ) m= exp(j(nm) θ i i ' ) K nm ( z i ' ) B i ' m t1 (p)
| Φ inc (n) Φ inc (n+5) | Φ inc (n+5) < 10 5 .
MRE= 1 N i=1 n | x i y i y i |

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