Abstract

This is Part III of the work that examines photon diffusion in a scattering-dominant medium enclosed by a “concave” circular cylindrical applicator or enclosing a “convex” circular cylindrical applicator. In Part II of this work Zhang et al. [J. Opt. Soc. Am. A 28, 66 (2011) [CrossRef]  ] predicted that, on the tissue-applicator interface of either “concave” or “convex” geometry, there exists a unique set of spiral paths, along which the steady-state photon fluence rate decays at a rate equal to that along a straight line on a planar semi-infinite interface, for the same line-of-sight source–detector distance. This phenomenon of steady-state photon diffusion is referred to as “straight-line-resembling-spiral paths” (abbreviated as “spiral paths”). This Part III study develops analytic approaches to the spiral paths associated with geometry of a large radial dimension and presents spiral paths found numerically for geometry of a small radial dimension. This Part III study also examines whether the spiral paths associated with a homogeneous medium are a good approximation for the medium containing heterogeneity. The heterogeneity is limited to an anomaly that is aligned azimuthally with the spiral paths and has either positive or negative contrast of the absorption or scattering coefficient over the background medium. For a weak-contrast anomaly the perturbation by it to the photon fluence rate along the spiral paths is found by applying a well-established perturbation analysis in cylindrical coordinates. For a strong-contrast anomaly the change by it to the photon fluence rate along the spiral paths is computed using the finite-element method. For the investigated heterogeneous-medium cases the photon fluence rate along the homogeneous-medium associated spiral paths is macroscopically indistinguishable from, and microscopically close to, that along a straight line on a planar semi-infinite interface.

© 2012 Optical Society of America

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References

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2011 (2)

2010 (1)

2009 (1)

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, 711–732 (2009).
[CrossRef]

1998 (1)

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, 1285–1302 (1998).
[CrossRef]

1995 (1)

1994 (1)

1993 (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

1992 (2)

1991 (2)

W. K. Hocking, S. Fukao, M. Yamamoto, T. Tsuda, and S. Kato, “Viscosity waves and thermal-conduction waves as a cause of ‘specular’ reflectors in radar studies of the atmosphere,” Radio Sci. 26, 1281–1303 (1991).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

1989 (1)

1984 (1)

D. L. Cummings, R. L. Reuben, and D. A. Blackburn, “The effect of pressure modulation on the flow of gas through a solid membrane: permeation and diffusion of hydrogen through nickel,” Metall. Trans. A 15, 639–648 (1984).
[CrossRef]

1933 (1)

W. Jost, “Diffusion and electrolytic conduction in crystals (ionic semiconductors),” J. Chem. Phys. 1, 466–475 (1933).
[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, 1285–1302 (1998).
[CrossRef]

Aronson, R.

R. L. Barbour, H. Graber, R. Aronson, and J. Lubowsky, “Model for 3-D optical imaging of tissue,” Remote Sensing Science for the Nineties, IGARSS ’90, 1395–1399 (1990). http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isNumber=3531&arNumber=688761&isnumber=3531&arnumber=688761&tag=1
[CrossRef]

Arridge, S. R.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

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, 1285–1302 (1998).
[CrossRef]

R. L. Barbour, H. Graber, R. Aronson, and J. Lubowsky, “Model for 3-D optical imaging of tissue,” Remote Sensing Science for the Nineties, IGARSS ’90, 1395–1399 (1990). http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isNumber=3531&arNumber=688761&isnumber=3531&arnumber=688761&tag=1
[CrossRef]

Blackburn, D. A.

D. L. Cummings, R. L. Reuben, and D. A. Blackburn, “The effect of pressure modulation on the flow of gas through a solid membrane: permeation and diffusion of hydrogen through nickel,” Metall. Trans. A 15, 639–648 (1984).
[CrossRef]

Boas, D. A.

Bunting, C. F.

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, 711–732 (2009).
[CrossRef]

Chance, B.

Cope, M.

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

Cummings, D. L.

D. L. Cummings, R. L. Reuben, and D. A. Blackburn, “The effect of pressure modulation on the flow of gas through a solid membrane: permeation and diffusion of hydrogen through nickel,” Metall. Trans. A 15, 639–648 (1984).
[CrossRef]

Daluwatte, C.

Davis, S. C.

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, 711–732 (2009).
[CrossRef]

Dehghani, H.

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, 711–732 (2009).
[CrossRef]

Delpy, D. T.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef]

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

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, 711–732 (2009).
[CrossRef]

Fantini, S.

Felsen, L. B.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

Franceschini, M. A.

Fukao, S.

W. K. Hocking, S. Fukao, M. Yamamoto, T. Tsuda, and S. Kato, “Viscosity waves and thermal-conduction waves as a cause of ‘specular’ reflectors in radar studies of the atmosphere,” Radio Sci. 26, 1281–1303 (1991).
[CrossRef]

Graber, H.

R. L. Barbour, H. Graber, R. Aronson, and J. Lubowsky, “Model for 3-D optical imaging of tissue,” Remote Sensing Science for the Nineties, IGARSS ’90, 1395–1399 (1990). http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isNumber=3531&arNumber=688761&isnumber=3531&arnumber=688761&tag=1
[CrossRef]

Gratton, E.

Hefetz, Y.

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, 1285–1302 (1998).
[CrossRef]

Hiraoka, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

Hocking, W. K.

W. K. Hocking, S. Fukao, M. Yamamoto, T. Tsuda, and S. Kato, “Viscosity waves and thermal-conduction waves as a cause of ‘specular’ reflectors in radar studies of the atmosphere,” Radio Sci. 26, 1281–1303 (1991).
[CrossRef]

Ishimaru, A.

Jacques, S. L.

Jiang, Y.

Jost, W.

W. Jost, “Diffusion and electrolytic conduction in crystals (ionic semiconductors),” J. Chem. Phys. 1, 466–475 (1933).
[CrossRef]

Kato, S.

W. K. Hocking, S. Fukao, M. Yamamoto, T. Tsuda, and S. Kato, “Viscosity waves and thermal-conduction waves as a cause of ‘specular’ reflectors in radar studies of the atmosphere,” Radio Sci. 26, 1281–1303 (1991).
[CrossRef]

Lubowsky, J.

R. L. Barbour, H. Graber, R. Aronson, and J. Lubowsky, “Model for 3-D optical imaging of tissue,” Remote Sensing Science for the Nineties, IGARSS ’90, 1395–1399 (1990). http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isNumber=3531&arNumber=688761&isnumber=3531&arnumber=688761&tag=1
[CrossRef]

Madsen, S. J.

Mandelis, A.

A. Mandelis, Diffusion-Wave Fields: Mathematical Methods and Green Functions (Springer-Verlag, 2001).

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

Mason, E. A.

E. A. Mason, R. J. Munn, and F. J. Smith, “Thermal diffusion in gases,” in Advances in Atomic and Molecular Physics(Academic, 1966), pp. 33–91.

Munn, R. J.

E. A. Mason, R. J. Munn, and F. J. Smith, “Thermal diffusion in gases,” in Advances in Atomic and Molecular Physics(Academic, 1966), pp. 33–91.

O’Leary, M. A.

Park, Y. D.

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, 711–732 (2009).
[CrossRef]

Piao, D.

Pogue, B. W.

Reuben, R. L.

D. L. Cummings, R. L. Reuben, and D. A. Blackburn, “The effect of pressure modulation on the flow of gas through a solid membrane: permeation and diffusion of hydrogen through nickel,” Metall. Trans. A 15, 639–648 (1984).
[CrossRef]

Schweiger, M.

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

Smith, F. J.

E. A. Mason, R. J. Munn, and F. J. Smith, “Thermal diffusion in gases,” in Advances in Atomic and Molecular Physics(Academic, 1966), pp. 33–91.

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, 711–732 (2009).
[CrossRef]

Tsuda, T.

W. K. Hocking, S. Fukao, M. Yamamoto, T. Tsuda, and S. Kato, “Viscosity waves and thermal-conduction waves as a cause of ‘specular’ reflectors in radar studies of the atmosphere,” Radio Sci. 26, 1281–1303 (1991).
[CrossRef]

van der Zee, P.

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

Wang, L. V.

L. V. Wang and H. Wu, Biomedical Optics, Principles and Imaging (Wiley, 2007).

Weinberg, A. M.

A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors (University of Chicago Press, 1958).

Wigner, E. P.

A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors (University of Chicago Press, 1958).

Wilson, B. C.

Wu, H.

L. V. Wang and H. Wu, Biomedical Optics, Principles and Imaging (Wiley, 2007).

Xu, G.

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, 711–732 (2009).
[CrossRef]

Yamamoto, M.

W. K. Hocking, S. Fukao, M. Yamamoto, T. Tsuda, and S. Kato, “Viscosity waves and thermal-conduction waves as a cause of ‘specular’ reflectors in radar studies of the atmosphere,” Radio Sci. 26, 1281–1303 (1991).
[CrossRef]

Yao, G.

Yodh, A. G.

Zhang, A.

Appl. Opt. (2)

Commun. Numer. Methods Eng. (1)

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, 711–732 (2009).
[CrossRef]

J. Chem. Phys. (1)

W. Jost, “Diffusion and electrolytic conduction in crystals (ionic semiconductors),” J. Chem. Phys. 1, 466–475 (1933).
[CrossRef]

J. Opt. Soc. Am. A (2)

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

Med. Phys. (1)

S. R. Arridge, M. Schweiger, M. Hiraoka, and D. T. Delpy, “A finite element approach for modeling photon transport in tissue,” Med. Phys. 20, 299–309 (1993).
[CrossRef]

Metall. Trans. A (1)

D. L. Cummings, R. L. Reuben, and D. A. Blackburn, “The effect of pressure modulation on the flow of gas through a solid membrane: permeation and diffusion of hydrogen through nickel,” Metall. Trans. A 15, 639–648 (1984).
[CrossRef]

Opt. Lett. (2)

Phys. Med. Biol. (2)

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, 1285–1302 (1998).
[CrossRef]

S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical pathlengths in tissue: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
[CrossRef]

Proc. SPIE (1)

S. R. Arridge, P. van der Zee, M. Cope, and D. T. Delpy, “Reconstruction methods for infra-red absorption imaging,” Proc. SPIE 1431, 204–215 (1991).
[CrossRef]

Radio Sci. (1)

W. K. Hocking, S. Fukao, M. Yamamoto, T. Tsuda, and S. Kato, “Viscosity waves and thermal-conduction waves as a cause of ‘specular’ reflectors in radar studies of the atmosphere,” Radio Sci. 26, 1281–1303 (1991).
[CrossRef]

Other (6)

A. Mandelis, Diffusion-Wave Fields: Mathematical Methods and Green Functions (Springer-Verlag, 2001).

R. L. Barbour, H. Graber, R. Aronson, and J. Lubowsky, “Model for 3-D optical imaging of tissue,” Remote Sensing Science for the Nineties, IGARSS ’90, 1395–1399 (1990). http://ieeexplore.ieee.org/xpl/freeabs_all.jsp?isNumber=3531&arNumber=688761&isnumber=3531&arnumber=688761&tag=1
[CrossRef]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, 1973).

E. A. Mason, R. J. Munn, and F. J. Smith, “Thermal diffusion in gases,” in Advances in Atomic and Molecular Physics(Academic, 1966), pp. 33–91.

A. M. Weinberg and E. P. Wigner, The Physical Theory of Neutron Chain Reactors (University of Chicago Press, 1958).

L. V. Wang and H. Wu, Biomedical Optics, Principles and Imaging (Wiley, 2007).

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Figures (8)

Fig. 1.
Fig. 1.

Three geometries of approximating the tissue-applicator interface are illustrated in (A). The geometries include a concave geometry representing the medium being enclosed by the circular cylindrical tissue-applicator interface, a convex geometry representing the medium enclosing the circular cylindrical tissue-applicator interface, and a semi-infinite geometry representing the medium interfacing with a planar applicator. For the concave and convex geometries, there are two specific directions for evaluating the photon fluence rate: a case-azi configuration shown in (B) that represents the case of having both source and detector on the same azimuthal plane, and a case-longi configuration shown in (C) that represents the case of having both source and detector on the same longitudinal line. In both (B) and (C) the line-of-sight distance between the source and the position of photon detection is denoted by d.

Fig. 2.
Fig. 2.

Notations and physical entities of concave or convex geometry for analytic evaluation of photon fluence rate associated with larger radius. (A) The tissue is at the concave side of the circular cylindrical tissue-applicator interface, so the equivalent isotropic source of the physical source that illuminates into the medium is located closer to the center axis than the physical source is. (B) The tissue is at the convex side of the circular cylindrical tissue-applicator interface, so the equivalent isotropic source of the physical source that illuminates into the medium is located farther from the center axis than the physical source is.

Fig. 3.
Fig. 3.

spiral paths for concave and convex geometries of centimeter-order radii are calculated based on analytic results derived in [15]. For a fixed source, a detector has three directions to move away from the source: along the azimuthal direction, along the longitudinal direction, and along the diagonal of the above two directions. For each possible future location of the detector, the photon fluence rate at that position is compared with the case in the semi-infinite geometry for the same line-of-sight distance of the detector from the source, and the position with the least difference in the evaluated photon fluence rate is the next starting position of the detector. The shown complete sets of the spiral profile for concave geometry (upper) and convex geometry (lower) are computed for a cylinder radius of R0=1.5cm and optical properties of μa=0.02cm1, μs=5cm1, and A=1.86.

Fig. 4.
Fig. 4.

Position of the anomaly of weak perturbation strength in the otherwise homogeneous background medium. (A) Concave geometry, (B) semi-infinite geometry, (C) convex geometry.

Fig. 5.
Fig. 5.

Photon fluence rate when one weak anomaly resides in the otherwise homogeneous background medium. The anomaly possesses (A) positive μa contrast, (B) positive μs contrast, (C) negative μa contrast, and (D) negative μs contrast over the background. The shown curves of photon fluence are plotted for (1) along a straight line on semi-infinite interface of homogeneous medium, (2) along a straight line on semi-infinite interface having the anomaly aligned with the straight line, (3) along the spiral profile on concave interface having the anomaly aligned with the spiral profile, and (4) along the spiral profile on convex interface having the anomaly aligned with the spiral profile.

Fig. 6.
Fig. 6.

Finite-element discretization of the imaging domain and the position of the anomaly with strong perturbation strength. (A) Concave geometry, (B) semi-infinite geometry, and (C) convex geometry.

Fig. 7.
Fig. 7.

Photon fluence rate when one strong anomaly resides in the otherwise homogeneous background medium. The anomaly possesses (A) positive μa contrast, (B) positive μs contrast, (C) negative μa contrast, and (D) negative μs contrast over the background. The shown curves of photon fluence are plotted for (1) along a straight line on semi-infinite interface of homogeneous medium, (2) along a straight line on semi-infinite interface having the anomaly aligned with the straight line, (3) along the spiral profile on concave interface having the anomaly aligned with the spiral profile, and (4) along the spiral profile on convex interface having the anomaly aligned with the spiral profile.

Fig. 8.
Fig. 8.

Spiral profiles of concave geometry (left column) and convex geometry (right column) found for different geometric parameters and optical properties. Each subplot illustrates the spiral profiles associated with the change of only one parameter with respect to a set of baseline parameters. The parameter to be changed in (A) and (B) is the radius of the cylindrical applicator. The parameter to be changed in (C) and (D) is the A value. The parameter to be changed in (E) and (F) is μa. The parameter to be changed in (G) and (H) is μs.

Tables (2)

Tables Icon

Table 1. Four Sets of Optical Parameters for Evaluating the Change to Photon Fluence Rate by an Anomaly of Weak Contrast to the Background Medium

Tables Icon

Table 2. Four Sets of Optical Parameters Used for Evaluating the Change to Photon Fluence Rate by an Anomaly of Strong Contrast to the Background Medium

Equations (58)

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Ψ=S4πDek0lrlrS4πDek0liliR0+Ra+2RbR0Ra,
lr=d[1+Ra2d2RaR0(cosα)2],
li=d[1+(Ra+2Rb)2d2+Ra+2RbR0(cosα)2].
R0+Ra+2RbR0Ra=(1+2Ra+RbR0Ra)1/21+Ra+RbR0Ra
ek0d1+Δd1+Δek0dd[112(k0d+1)Δ]ek0dd[112(k0d)Δ],
Ψ=S4πDek0dd{[112k0d(Ra2d2RaR0(cosα)2)][112k0d((Ra+2Rb)2d2+Ra+2RbR0(cosα)2)](1+Ra+RbR0Ra)},
ln(Ψ·d2)d={k0+12k0Rb(R0Ra)[2R0Ra+2Rb2R0Rb(R0Ra)]cosα·d}.
ln(Ψ·d2)d=[k0+12k0Rb(R0Ra)].
ln(Ψ·d2)d={k0+12k0Rb(R0Ra)[2R0Ra+2Rb2R0Rb(R0Ra)]d}.
cosα=1k0dR02R0Ra+2Rb,
Ψ=S4πDek0lrlrS4πDek0liliR0Ra2RbR0+Ra,
lr=d[1+Ra2d2+RaR0(cosα)2],
li=d[1+(Ra+2Rb)2d2Ra+2RbR0(cosα)2].
R0Ra2RbR0+Ra=(12Ra+RbR0+Ra)1/21Ra+RbR0+Ra,
Ψ=S4πDek0dd{[112k0d(Ra2d2+RaR0(cosα)2)][112k0d((Ra+2Rb)2d2Ra+2RbR0(cosα)2)](1Ra+RbR0+Ra)},
ln(Ψ·d2)d={k012k0Rb(R0+Ra)+[2R0+Ra2Rb2R0Rb(R0+Ra)]cosα·d}.
ln(Ψ·d2)d=[k012k0Rb(R0+Ra)].
ln(Ψ·d2)d={k012k0Rb(R0+Ra)+[2R0+Ra2Rb2R0Rb(R0+Ra)]d}.
cosα=1k0dR02R0+Ra2Rb,
μa(r)Ψ(r)·[D(r)Ψ(r)]=S(r),
μa(r)=μa0+δμa(r),
D(r)=D0+δD(r)
Ψ(r)=Ψ0(r)+ΨSC(r),
μa0Ψ0(r)D02Ψ0(r)=S(r)
[μa0+δμa(r)][Ψ0(r)+ΨSC(r)]D02[Ψ0(r)+ΨSC(r)]·{δD(r)·[Ψ0(r)+ΨSC(r)]}=S(r).
D02ΨSC(r)μa0ΨSC(r)=δμa(r)[Ψ0(r)+ΨSC(r)]·{δD(r)[Ψ0(r)+ΨSC(r)]}.
ΨSC(r)Ψ0(r),
2ΨSC(r)μa0D0ΨSC(r)=δμa(r)D0Ψ0(r)1D0·{δD(r)Ψ0(r)}.
2G(r,r)μa0D0G(r,r)=δ(rr).
Ψ0(r,rs)=SD12π20dk{cos[k(zzs)]m=0εmIm(keffρ)Km[keff(R0Ra)]·1Im[keff(R0Ra)]Km[keff(R0Ra)]Km[keff(R0+Rb)]Im[keff(R0+Rb)]cos[m(φφs)]},
εm={2m>01m=0.
G(rd,r)=12π20dk{cos[k(zdz)]m=0εmIm(keffρ)Km(keffR0)·1Im(keffR0)Km(keffR0)Km[keff(R0+Rb)]Im[keff(R0+Rb)]cos[m(φdφ)]}.
Ψ0(r,rs)=SD12π20dk{cos[k(zzs)]m=0εmIm[keff(R0+Ra)]Km(keffρ)·1Im[keff(R0Rb)]Km[keff(R0Rb)]Km[keff(R0+Ra)]Im[keff(R0+Ra)]cos[m(φφs)]}.
G(rd,r)=12π20dk{cos[k(zdz)]m=0εmIm(keffR0)Km(keffρ)·1Im[keff(R0Rb)]Km[keff(R0Rb)]Km(keffR0)Im(keffR0)cos[m(φdφ)]}.
ΨSC(rd,rs)=1D0VG(rd,r)δμa(r)Ψ0(r,rs)d3r+1D0VG(rd,r)·{δD(r)Ψ0(r,rs)}d3r.
ΨSC(rd,rs)=1D0VG(rd,r)δμa(r)Ψ0(r,rs)d3r1D0VδD(r)G(rd,r)·Ψ0(r,rs)d3r.
δPμa=δμaVμa0
δPD=δDVD0.
ΨSC(rd,rs)=δPμaμa0D0G(rd,r)Ψ0(r,rs)δPDG(rd,r)·Ψ0(r,rs).
G(rd,r)·Ψ0(r,rs)={[G(rd,r)ρ][Ψ0(r,rs)ρ]+[1ρG(rd,r)φ][1ρΨ0(r,rs)φ]+[G(rd,r)z][Ψ0(r,rs)z]}.
Ψ=S4πDek0dd{[112k0d(Ra2d2RaR0(cosα)2)][112k0d((Ra+2Rb)2d2+Ra+2RbR0(cosα)2)](1+Ra+RbR0Ra)}.
Ψ=S4πDek0dd[1k0Ra22d+k0Rad2R0(cosα)21+k0(Ra+2Rb)22d+k0(Ra+2Rb)d2R0(cosα)2Ra+RbR0Ra+k0(Ra+2Rb)22dRa+RbR0Ra+k0(Ra+2Rb)d2R0Ra+RbR0Ra(cosα)2]=S4πDek0dd[2k0Rb(Ra+Rb)d+k0(Ra+Rb)dR0(cosα)2Ra+RbR0Ra+k0(Ra+2Rb)22dRa+RbR0Ra+k0(Ra+2Rb)d2R0Ra+RbR0Ra(cosα)2]=S2πDek0ddk0Rb(Ra+Rb)d[1d2k0Rb(R0Ra)+d22RbR0(cosα)2+(Ra+2Rb)d24RbR0(R0Ra)(cosα)2+(Ra+2Rb)24Rb(R0Ra)].
Ψ=S2πDek0ddk0Rb(Ra+Rb)dexp{d2k0Rb(R0Ra)+[d22RbR0+(Ra+2Rb)d24RbR0(R0Ra)](cosα)2+(Ra+2Rb)24Rb(R0Ra)}.
Ψd2=S2πDk0Rb(Ra+Rb)e(Ra+2Rb)24Rb(R0Ra)exp{k0dd2k0Rb(R0Ra)+[12RbR0+Ra+2Rb4RbR0(R0Ra)](cosα)2d2}.
ln(Ψd2)=k0dd2k0Rb(R0Ra)+[12RbR0+Ra+2Rb4RbR0(R0Ra)](cosα)2d2+ln[S2πDk0Rb(Ra+Rb)]+(Ra+2Rb)24Rb(R0Ra).
ln(Ψ·d2)d={k0+12k0Rb(R0Ra)[2R0Ra+2Rb2R0Rb(R0Ra)]cosα·d}.
Ψ=S4πDek0dd{[112k0d(Ra2d2+RaR0(cosα)2)][112k0d((Ra+2Rb)2d2Ra+2RbR0(cosα)2)](1Ra+RbR0+Ra)}.
Ψ=S4πDek0dd[1k0Ra22dk0Rad2R0(cosα)21+k0(Ra+2Rb)22dk0(Ra+2Rb)d2R0(cosα)2+Ra+RbR0+Rak0(Ra+2Rb)22dRa+RbR0+Ra+k0(Ra+2Rb)d2R0Ra+RbR0+Ra(cosα)2]=S4πDek0dd[2k0Rb(Ra+Rb)dk0(Ra+Rb)dR0(cosα)2+Ra+RbR0+Rak0(Ra+2Rb)22dRa+RbR0+Ra+k0(Ra+2Rb)d2R0Ra+RbR0+Ra(cosα)2]=S4πDek0dd2k0Rb(Ra+Rb)d[1+d2k0Rb(R0+Ra)d22RbR0(cosα)2+(Ra+2Rb)d24RbR0(R0+Ra)(cosα)2(Ra+2Rb)24Rb(R0+Ra)].
Ψ=S4πDek0dd2k0Rb(Ra+Rb)dexp{d2k0Rb(R0+Ra)+[d22RbR0+(Ra+2Rb)d24RbR0(R0+Ra)](cosα)2(Ra+2Rb)24Rb(R0+Ra)}.
Ψd2=S2πDk0Rb(Ra+Rb)e(Ra+2Rb)24Rb(R0+Ra)exp{k0d+d2k0Rb(R0+Ra)+[d22RbR0+(Ra+2Rb)d24RbR0(R0+Ra)](cosα)2}.
ln(Ψd2)=k0d+d2k0Rb(R0+Ra)+[12RbR0+Ra+2Rb4RbR0(R0+Ra)](cosα)2d2+ln[S2πDk0Rb(Ra+Rb)](Ra+2Rb)24Rb(R0+Ra).
ln(Ψ·d2)d={k012k0Rb(R0+Ra)+[2R0+Ra2Rb2R0Rb(R0+Ra)]cosα·d}.
ΨSC(rd,rs)=1D0VG(rd,r)δμa(r)Ψ0(r,rs)d3r+1D0VG(rd,r)·{δD(r)Ψ0(r,rs)}d3r.
VG(rd,r)·{δD(r)Ψ0(r,rs)}d3r=VG(rd,r)δD(r)·Ψ0(r,rs)d3r+VG(rd,r)δD(r)2Ψ0(r,rs)d3r.
VG(rd,r)δD(r)2Ψ0(r,rs)d3r=s(G(rd,r)δD(r)n^·Ψ0(r,rs))d2aV[G(rd,r)δD(r)]·Ψ0(r,rs)d3r.
VG(rd,r)δD(r)2Ψ0(r,rs)d3r=V[G(rd,r)δD(r)]·Ψ0(r,rs)d3r=VG(rd,r)δD(r)·Ψ0(r,rs)d3rVδD(r)G(rd,r)·Ψ0(r,rs)d3r.
1D0VG(rd,r)·{δD(r)Ψ0(r,rs)}d3r=1D0VδD(r)G(rd,r)·Ψ0(r,rs)d3r.
ΨSC(rd,rs)=1D0VG(rd,r)δμa(r)Ψ0(r,rs)d3r1D0VδD(r)G(rd,r)·Ψ0(r,rs)d3r.

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