Abstract

Part IV examines frequency-domain photon diffusion in a homogeneous medium enclosed by a “concave” circular cylindrical applicator or enclosing a “convex” circular cylindrical applicator, both geometries being infinite in the longitudinal dimension. The aim is to assess by analogical and finite-element methods the changes of AC amplitude, modulation depth, and phase with respect to the line-of-sight source–detector distance for a source and a detector located along the azimuthal or longitudinal direction on the concave or convex medium–applicator interface. By comparing to their counterparts along a straight line on a semi-infinite medium–applicator interface, for the same line-of-sight source–detector distance, it is found that: (1) the decay-rate of AC photon fluence is smaller along the azimuthal direction and greater along the longitudinal direction on the concave interface, (2) the decay-rate of AC photon fluence is greater along the azimuthal direction and smaller along the longitudinal direction on the convex interface, (3) the modulation depth along both azimuthal and longitudinal directions decays more slowly on the concave interface and faster on the convex interface, and (4) the phase along both azimuthal and longitudinal directions increases more slowly on the concave interface and faster on the convex interface.

© 2012 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  23. G. Xu, D. Piao, C. F. Bunting, and H. Dehghani, “Direct-current-based image reconstruction versus direct-current included or excluded frequency-domain reconstruction in diffuse optical tomography,” Appl. Opt. 49, 3059–3070 (2010).
    [CrossRef]

2012 (1)

2011 (2)

2010 (4)

2009 (2)

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]

A. Da Silva, M. Leabad, C. Driol, T. Bordy, M. Debourdeau, J. Dinten, P. Peltié, and P. Rizo, “Optical calibration protocol for an x-ray and optical multimodality tomography system dedicated to small-animal examination,” Appl. Opt. 48, D151–D162 (2009).
[CrossRef]

2008 (1)

2006 (1)

2002 (1)

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

1997 (2)

1994 (3)

1993 (1)

1992 (1)

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

1989 (1)

Abdoulaev, G. S.

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

Arridge, S. R.

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

Beuthan, J.

H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theory-based optical tomography of small-tissue volumes,” Opt. Express 16, 18082–18101 (2008).
[CrossRef]

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

Bluestone, A. Y.

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

Bordy, T.

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]

Carslaw, H. S.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford, 1986).

Contini, D.

Cope, M.

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

Da Silva, A.

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]

Debourdeau, M.

Dehghani, H.

Delpy, D. T.

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

Dinten, J.

Driol, C.

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.

Feng, T.

Franceschini, M. A.

Gratton, E.

Haskell, R. C.

Hielscher, A. H.

H. K. Kim, U. J. Netz, J. Beuthan, and A. H. Hielscher, “Optimal source-modulation frequencies for transport-theory-based optical tomography of small-tissue volumes,” Opt. Express 16, 18082–18101 (2008).
[CrossRef]

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

Ishimaru, A.

Jackson, J. D.

J. D. Jackson, “Expansion of Green functions in cylindrical coordinates,” in Classical Electrodynamics, 3rd ed. (Wiley, 1998), pp. 125–126.

Jaeger, J. C.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford, 1986).

Jiang, Y.

Kienle, A.

Kim, H. K.

Klose, A. D.

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

Kransinski, J. S.

Lasker, J.

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

Leabad, M.

Liemert, A.

Martelli, F.

McAdams, M. S.

Netz, U.

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

Netz, U. J.

Patterson, M. S.

B. W. Pogue and M. S. Patterson, “Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef]

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]

Peltié, P.

Piao, D.

A. Zhang, D. Piao, and C. F. Bunting, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. III. Synthetic-study of continuous-wave photon fluence rate along unique spiral-paths,” J. Opt. Soc. Am. A 29, 545–558 (2012).
[CrossRef]

A. Zhang, D. Piao, G. Yao, C. F. Bunting, and Y. Jiang, “Diffuse photon remission along unique spiral paths on a cylindrical interface is modeled by photon remission along a straight line on a semi-infinite interface,” Opt. Lett. 36, 654–656 (2011).
[CrossRef]

A. Zhang, G. Xu, C. Daluwatte, G. Yao, C. F. Bunting, B. W. Pogue, and D. Piao, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. II. Quantitative examinations of the steady-state theory,” J. Opt. Soc. Am. A 28, 66–75 (2011).
[CrossRef]

A. Zhang, D. Piao, C. F. Bunting, and B. W. Pogue, “Photon diffusion in a homogeneous medium bounded externally or internally by an infinitely long circular cylindrical applicator. I. Steady-state theory,” J. Opt. Soc. Am. A 27, 648–662 (2010).
[CrossRef]

G. Xu, D. Piao, C. F. Bunting, and H. Dehghani, “Direct-current-based image reconstruction versus direct-current included or excluded frequency-domain reconstruction in diffuse optical tomography,” Appl. Opt. 49, 3059–3070 (2010).
[CrossRef]

D. Piao, H. Xie, W. Zhang, J. S. Kransinski, G. Zhang, H. Dehghani, and B. W. Pogue, “Endoscopic, rapid near-infrared optical tomography,” Opt. Lett. 31, 2876–2878 (2006).
[CrossRef]

Pogue, B. W.

Rizo, P.

Schwarzmaier, H. J.

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]

Stewart, M.

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

Svaasand, L. O.

Tromberg, B. J.

Tsay, T.

Tuchin, V. V.

Xie, H.

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]

Yao, G.

Yaroslavsky, A. N.

Yaroslavsky, I. V.

Zaccanti, G.

Zhang, A.

Zhang, G.

Zhang, W.

Appl. Opt. (6)

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]

Dis. Markers (1)

A. H. Hielscher, A. Y. Bluestone, G. S. Abdoulaev, A. D. Klose, J. Lasker, M. Stewart, U. Netz, and J. Beuthan, “Near-infrared diffuse optical tomography,” Dis. Markers 18, 313–337 (2002).

J. Opt. Soc. Am. A (4)

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

Opt. Express (3)

Opt. Lett. (2)

Phys. Med. Biol. (2)

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

B. W. Pogue and M. S. Patterson, “Frequency-domain optical absorption spectroscopy of finite tissue volumes using diffusion theory,” Phys. Med. Biol. 39, 1157–1180 (1994).
[CrossRef]

Other (3)

J. D. Jackson, “Expansion of Green functions in cylindrical coordinates,” in Classical Electrodynamics, 3rd ed. (Wiley, 1998), pp. 125–126.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford, 1986).

IEEE 754-2008 Standard for Floating-Point Arithmetic (IEEE, 2008).

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Figures (6)

Fig. 1.
Fig. 1.

Illustrations of a medium of infinite geometry in (A) and a medium of semi-infinite geometry in (B) and (C). In the semi-infinite geometry the source and detector are positioned on the physical boundary of the medium, and it becomes convenient to assign the same radial and azimuthal coordinates to the source and detector.

Fig. 2.
Fig. 2.

Configurations of (A) concave geometry and (B) convex geometry. The physical directional source and detector locate at the probe–tissue interface at (R0,φ,z) and (R0,φ,z) respectively. In concave geometry, the equivalent isotropic source locates inwardly at (R0Ra,φ,z). The image of the source with respect to the associated semi-infinite geometry locates at (R0+Ra+2Rb,φ,z). In convex geometry, the equivalent isotropic source locates outwardly at (R0+Ra,φ,z). The image of the source with respect to the associated semi-infinite geometry locates at (R0Ra2Rb,φ,z).

Fig. 3.
Fig. 3.

Changes of AC amplitude versus source-detector distance in both case-azi and case-longi configurations. (A) In concave geometry, the decay rate of AC amplitude is smaller in case-azi while greater in case-longi than that along a straight line on the semi-infinite interface. (B) In convex geometry, the decay rate of AC amplitude is greater in case-azi while smaller in case-longi than that along a straight line on the semi-infinite interface.

Fig. 4.
Fig. 4.

(A) Changes of modulation depth versus source–detector distance in the concave geometry and convex geometry at 100 MHz modulation frequency. (B) Changes of modulation depth versus modulation frequency in the concave geometry and convex geometry at a source–detector distance of d=15mm. In both (A) and (B), the reduction of the modulation depth is smaller in both case-azi and case-longi of concave geometry and greater in both case-azi and case-longi of convex geometry than that along a straight line on the semi-infinite interface, for the same source–detector distance.

Fig. 5.
Fig. 5.

(A) Changes of phase versus source–detector distance in the concave geometry and convex geometry at 100 MHz modulation frequency. (B) Changes of phase versus modulation frequency in the concave geometry and convex geometry at a source–detector distance of d=15mm. In both (A) and (B), the increase of the phase is smaller in both case-azi and case-longi of concave geometry and greater in both case-azi and case-longi of convex geometry than that along a straight line on the semi-infinite interface, for the same source–detector distance.

Fig. 6.
Fig. 6.

(A) Spiral paths associated with the amplitude of AC and DC photon fluence rate on the concave medium–applicator interface. (B) Spiral paths associated with the amplitude of AC and DC photon fluence rate on the convex medium–applicator interface. In both (A) and (B), the spiral paths for the amplitude of AC photon fluence rate tilt more axially than that for DC photon fluence rate. The difference between these two sets of spiral paths is more pronounced on the convex medium–applicator interface.

Tables (1)

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Table 1. Summary of the Analytical Expressions Presented in Section 4

Equations (60)

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2Ψ¯(r⃗)k¯0Ψ¯(r⃗)=S¯(r⃗)D;2Ψ˜(r⃗)k˜0Ψ˜(r⃗)=S˜(r⃗)D,
k¯0=μaD;k˜0=k˜amp+jk˜phi=μaD+iωDc,
k˜amp=k¯012(1+(ωcμa)2+1);k˜phi=k¯012(1+(ωcμa)21)
Ψ¯inf(r⃗,r⃗)=S¯4πD1|r⃗r⃗|exp(k¯0|r⃗r⃗|),
Ψ˜inf(r⃗,r⃗)=S˜4πD1|r⃗r⃗|exp(k˜amp|r⃗r⃗|)exp(ik˜phi|r⃗r⃗|).
Ψ¯semi=Ψ¯real(r⃗,r⃗real)+Ψ¯imag(r⃗,r⃗imag)=S¯4πDlrexp(k¯0lr)S¯4πDliexp(k¯0li),
Ψ˜semi=Ψ˜real(r⃗,r⃗real)+Ψ˜imag(r⃗,r⃗imag)=S˜4πDlrexp(k˜0lr)S˜4πDliexp(k˜0li),
lr=d2+Ra2;li=d2+(Ra+2Rb)2,
Ψ¯semi=S¯2πD[1d2exp(k¯0d)]k¯0Rb(Ra+Rb),
Ψ˜semi=S˜2πD[1d2exp(k˜ampd)][k˜ampRb(Ra+Rb)]1+(k˜phik˜amp)2exp{i[k˜phidtan1(k˜phik˜amp)]}.
Ψ¯inf(r⃗,r⃗)=S¯4π2Ddkeik(zz)m=Im(k¯effρ<)Km(k¯effρ>)eim(φφ),
k¯eff=k2+k¯02,
Ψ˜inf(r⃗,r⃗)=S˜4π2Ddkeik(zz)m=Im(k˜effρ<)Km(k˜effρ>)eim(φφ),
k˜eff=k2+k˜02.
Ψ¯semi=Ψ¯real(r⃗,r⃗real)+Ψ¯imag(r⃗,r⃗imag)=S¯4π2D{dkeik(zzreal)m=Im[k¯eff(ρRa)]Km(k¯effρ)eim(φφreal)dkeik(zzimag)m=Im(k¯effρ)Km[k¯eff(ρ+Ra+2Rb)]eim(φφimag)}
Ψ˜semi=Ψ˜real(r⃗,r⃗real)+Ψ˜imag(r⃗,r⃗imag)=S˜4π2D{dkeik(zzreal)m=Im[k˜eff(ρRa)]Km(k˜effρ)eim(φφreal)dkeik(zzimag)m=Im(k˜effρ)Km[k˜eff(ρ+Ra+2Rb)]eim(φφimag)}.
Ψ¯semi=S¯4π2D{dkeik(zzreal)m=Im(k¯effρ)Km[k¯eff(ρ+Ra)]eim(φφreal)dkeik(zzimag)m=Im[k¯eff(ρRa2Rb)]Km(k¯effρ)eim(φφimag)}
Ψ˜semi=S˜4π2D{dkeik(zzreal)m=Im(k˜effρ)Km[k˜eff(ρ+Ra)]eim(φφreal)dkeik(zzimag)m=Im[k˜eff(ρRa2Rb)]Km(k˜effρ)eim(φφimag)}.
Ψ¯conC=S¯4π2Ddk{exp[ik(zz)]m=Im[k¯eff(R0Ra)]Km(k¯effR0)1Im(k¯effR0)Km(k¯effR0)Km[k¯eff(R0+Rb)]Im[k¯eff(R0+Rb)]exp[im(φφ)]},
Ψ˜conC=S˜4π2Ddk{exp[ik(zz)]m=Im[k˜eff(R0Ra)]Km(k˜effR0)1Im(k˜effR0)Km(k˜effR0)Km[k˜eff(R0+Rb)]Im[k˜eff(R0+Rb)]exp[im(φφ)]}.
Ψ¯conV=Ψ¯real|phys+Ψ¯imag|phys=S¯4π2Ddk{exp[ik(zz)]m=Im(k¯effR0)Km[k¯eff(R0+Ra)]1Km(k¯effR0)Im(k¯effR0)Im[k¯eff(R0Rb)]Km[k¯eff(R0Rb)]exp[im(φφ)]},
Ψ˜conV=Ψ˜real|phys+Ψ˜imag|phys=S˜4π2Ddk{exp[ik(zz)]m=Im(k˜effR0)Km[k˜eff(R0+Ra)]1Km(k˜effR0)Im(k˜effR0)Im[k˜eff(R0Rb)]Km[k˜eff(R0Rb)]exp[im(φφ)]}.
Ψ¯conC=[S¯4πDlrexp(k¯0lr)][S¯4πDliexp(k¯0li)]R0+Ra+2RbR0Ra,
Ψ˜conC=[S˜4πDlrexp(k˜amplr)]exp(ik˜philr)[S˜4πDliexp(k˜ampli)]R0+Ra+2RbR0Raexp(ik˜phili).
lr=d[1+Ra2d2RaR0(cosα)2];li=d[1+(Ra+2Rb)2d2+Ra+2RbR0(cosα)2].
Ψ¯conC=S¯2πD1d2exp{[k¯0d+d2k¯0Rb(R0Ra)2R0Ra+2Rb4R0Rb(R0Ra)(cosα)2d2]}exp[(R0+2Rb)24Rb(R0Ra)][k¯0Rb(Ra+Rb)],
Ψ˜conC=S˜2πD1d2exp{[k˜ampd+k˜ampd2(k˜amp2+k˜phi2)Rb(R0Ra)2R0Ra+2Rb4R0Rb(R0Ra)(cosα)2d2]}exp[(R0+2Rb)24Rb(R0Ra)][k˜amp2+k˜phi2Rb(Ra+Rb)]exp{i[k˜phidk˜phid2(k˜amp2+k˜phi2)Rb(R0Ra)tan1(k˜phik˜amp)]}.
Ψ¯conV=[S¯4πDlrexp(k¯0lr)][S¯4πDliexp(k¯0li)]R0Ra2RbR0+Ra
Ψ˜conV=[S˜4πDlrexp(k˜amplr)]exp(ik˜philr)[S˜4πDliexp(k˜ampli)]R0Ra2RbR0+Raexp(ik˜phili).
lr=d[1+Ra2d2+RaR0(cosα)2];li=d[1+(Ra+2Rb)2d2Ra+2RbR0(cosα)2],
Ψ¯conV=S¯2πD1d2exp{[k¯0dd2k¯0Rb(R0+Ra)+2R0+Ra2Rb4R0Rb(R0+Ra)(cosα)2d2]}exp[(R0+2Rb)24Rb(R0+Ra)][k¯0Rb(Ra+Rb)]
Ψ˜conV=S˜2πD1d2exp{[k˜ampdk˜ampd2(k˜amp2+k˜phi2)Rb(R0+Ra)+2R0+Ra2Rb4R0Rb(R0+Ra)(cosα)2d2]}exp[(R0+2Rb)24Rb(R0+Ra)][k˜amp2+k˜phi2Rb(Ra+Rb)]exp{i[k˜phid+k˜phid2(k˜amp2+k˜phi2)Rb(R0+Ra)tan1(k˜phik˜amp)]}.
ln(Ψ¯infd)d=k¯0.
ln(Ψ¯semid2)d=k¯0.
ln(Ψ¯conCd2)d={k¯0+12k¯0Rb(R0Ra)[2R0Ra+2Rb2R0Rb(R0Ra)](cosα)d},
ln(Ψ¯conCd2)d|longi=[k¯0+12k¯0Rb(R0Ra)]
ln(Ψ¯conCd2)d|azi={k¯0+12k¯0Rb(R0Ra)[2R0Ra+2Rb2R0Rb(R0Ra)]d}.
cosα¯conC=1k¯0dR02R0Ra+2Rb.
ln(Ψ¯conVd2)d={k¯012k¯0Rb(R0+Ra)+[2R0+Ra2Rb2R0Rb(R0+Ra)](cosα)d}.
ln(Ψ¯conVd2)d|longi=[k¯012k¯0Rb(R0+Ra)].
ln(Ψ¯conVd2)d|azi={k¯012k¯0Rb(R0+Ra)+[2R0+Ra2Rb2R0Rb(R0+Ra)]d}.
cosα¯conV=1k¯0dR02R0+Ra2Rb.
ln(|Ψ˜inf|d)d=k˜amp.
ln(|Ψ˜semi|d2)d=k˜amp.
ln(|Ψ˜conC|d2)d={k˜amp+k˜amp2(k˜amp2+k˜phi2)Rb(R0Ra)[2R0Ra+2Rb2R0Rb(R0Ra)](cosα)d}.
ln(|Ψ˜conC|d2)d|longi=[k˜amp+k˜amp2(k˜amp2+k˜phi2)Rb(R0Ra)].
ln(|Ψ˜conC|d2)d|azi={k˜amp+k˜amp2(k˜amp2+k˜phi2)Rb(R0Ra)[2R0Ra+2Rb2R0Rb(R0Ra)]d}.
cosα˜conC=k˜amp(k˜amp2+k˜phi2)dR02R0Ra+2Rb,
ln(|Ψ˜conV|d2)d={k˜ampk˜amp2(k˜amp2+k˜phi2)Rb(R0+Ra)+[2R0+Ra2Rb2R0Rb(R0+Ra)](cosα)d}.
ln(|Ψ˜conV|d2)d|longi=[k˜ampk˜amp2(k˜amp2+k˜phi2)Rb(R0+Ra)].
ln(|Ψ˜conV|d2)d|azi={k˜ampk˜amp2(k˜amp2+k˜phi2)Rb(R0+Ra)+[2R0+Ra2Rb2R0Rb(R0+Ra)]d}.
cosα˜conV=k˜amp(k˜amp2+k˜phi2)dR02R0+Ra2Rb.
[ln(Modinf)]d=(k˜ampk¯0).
[ln(Modsemi)]d=(k˜ampk¯0).
[ln(ModconC)]d={(k˜ampk¯0)12Rb(R0Ra)[1k¯0k˜ampk˜amp2+k˜phi2]}
[ln(ModconV)]d={(k˜ampk¯0)+12Rb(R0+Ra)[1k¯0k˜ampk˜amp2+k˜phi2]}
|Ψ˜inf|d=k˜phi.
|Ψ˜semi|d=k˜phi.
|ΨconC|d=k˜phik˜phi2(k˜amp2+k˜phi2)Rb(R0Ra)
|ΨconV|d=k˜phi+k˜phi2(k˜amp2+k˜phi2)Rb(R0+Ra)

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