Abstract

The mathematical model for diffuse fluorescence spectroscopy/imaging is represented by coupled partial differential equations (PDEs), which describe the excitation and emission light propagation in soft biological tissues. The generic closed-form solutions for these coupled PDEs are derived in this work for the case of regular geometries using the Green’s function approach using both zero and extrapolated boundary conditions. The specific solutions along with the typical data types, such as integrated intensity and the mean time of flight, for various regular geometries were also derived for both time- and frequency-domain cases.

© 2013 Optical Society of America

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References

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  1. M. A. Mycek and B. W. Pogue, eds., Handbook of Biomedical Fluorescence (Dekker, 2003).
  2. E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
    [CrossRef]
  3. M. S. Patterson and B. W. Pogue, “Mathematical models for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
    [CrossRef]
  4. 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, 2331–2336 (1989).
    [CrossRef]
  5. S. R. Arridge, M. Cope, and D. T. Delpy, “The theoretical basis for the determination of optical path lengths in tissues: temporal and frequency analysis,” Phys. Med. Biol. 37, 1531–1560 (1992).
    [CrossRef]
  6. D. Contini, F. Martlli, and G. Zaccanti, “Photon migration through a turbid slab described by a model based on diffusion approximation. I. Theory, ” Appl. Opt. 36, 4587–4599 (1997).
    [CrossRef]
  7. M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60, 281–286 (1994).
    [CrossRef]
  8. X. D. Li, B. Chance, and A. G. Yodh, “Fluorescent heterogeneities in turbid media: limits for detection, characterization, and comparison with absorption,” Appl. Opt. 37, 6833–6844(1998).
    [CrossRef]
  9. M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
    [CrossRef]
  10. K. R. Ayyalasomayajula and P. K. Yalavarthy, “Analytical solutions for diffuse fluorescence spectroscopy/imaging in biological tissues Part II: comparison and validation,” J. Opt. Soc. Am. A30, 553–559 (2013).
  11. J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th ed. (McGraw-Hill, 2004).
  12. H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Science, 1946).
  13. G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1992).
  14. A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, 1895).

2008 (1)

E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
[CrossRef]

2001 (1)

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

1998 (1)

1997 (1)

1994 (2)

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60, 281–286 (1994).
[CrossRef]

M. S. Patterson and B. W. Pogue, “Mathematical models for time-resolved and frequency-domain fluorescence spectroscopy in biological tissues,” Appl. Opt. 33, 1963–1974 (1994).
[CrossRef]

1992 (1)

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

1989 (1)

Arridge, S. R.

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

Ayyalasomayajula, K. R.

K. R. Ayyalasomayajula and P. K. Yalavarthy, “Analytical solutions for diffuse fluorescence spectroscopy/imaging in biological tissues Part II: comparison and validation,” J. Opt. Soc. Am. A30, 553–559 (2013).

Boas, D. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60, 281–286 (1994).
[CrossRef]

Brown, J. W.

J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th ed. (McGraw-Hill, 2004).

Carslaw, H. S.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Science, 1946).

Chance, B.

Churchill, R. V.

J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th ed. (McGraw-Hill, 2004).

Contini, D.

Cope, M.

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

Delpy, D. T.

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

Gray, A.

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, 1895).

Jaeger, J. C.

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Science, 1946).

Kumar, S.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Li, X. D.

Martlli, F.

Mathews, G. B.

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, 1895).

O’Leary, M. A.

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60, 281–286 (1994).
[CrossRef]

Patterson, M. S.

Pogue, B. W.

Rasmussen, J. C.

E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
[CrossRef]

Riseborough, P.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Sadoqi, M.

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Sevick-Muraca, E. M.

E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
[CrossRef]

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1992).

Wilson, B. C.

Yalavarthy, P. K.

K. R. Ayyalasomayajula and P. K. Yalavarthy, “Analytical solutions for diffuse fluorescence spectroscopy/imaging in biological tissues Part II: comparison and validation,” J. Opt. Soc. Am. A30, 553–559 (2013).

Yodh, A. G.

X. D. Li, B. Chance, and A. G. Yodh, “Fluorescent heterogeneities in turbid media: limits for detection, characterization, and comparison with absorption,” Appl. Opt. 37, 6833–6844(1998).
[CrossRef]

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60, 281–286 (1994).
[CrossRef]

Zaccanti, G.

Appl. Opt. (4)

J. Biomed. Opt. (1)

E. M. Sevick-Muraca and J. C. Rasmussen, “Molecular imaging with optics: primer and case for near-infrared fluorescence techniques in personalized medicine,” J. Biomed. Opt. 13, 041303 (2008).
[CrossRef]

J. Lumin. (1)

M. A. O’Leary, D. A. Boas, B. Chance, and A. G. Yodh, “Reradiation and imaging of diffuse photon density waves using fluorescent inhomogeneities,” J. Lumin. 60, 281–286 (1994).
[CrossRef]

Phys. Med. Biol. (2)

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

M. Sadoqi, P. Riseborough, and S. Kumar, “Analytical models for time resolved fluorescence spectroscopy in tissues,” Phys. Med. Biol. 46, 2725–2743 (2001).
[CrossRef]

Other (6)

K. R. Ayyalasomayajula and P. K. Yalavarthy, “Analytical solutions for diffuse fluorescence spectroscopy/imaging in biological tissues Part II: comparison and validation,” J. Opt. Soc. Am. A30, 553–559 (2013).

J. W. Brown and R. V. Churchill, Complex Variables and Applications, 7th ed. (McGraw-Hill, 2004).

H. S. Carslaw and J. C. Jaeger, Conduction of Heat in Solids, 2nd ed. (Oxford Science, 1946).

G. N. Watson, A Treatise on the Theory of Bessel Functions, 2nd ed. (Cambridge University, 1992).

A. Gray and G. B. Mathews, A Treatise on Bessel Functions and Their Applications to Physics (Macmillan, 1895).

M. A. Mycek and B. W. Pogue, eds., Handbook of Biomedical Fluorescence (Dekker, 2003).

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Figures (4)

Fig. 1.
Fig. 1.

Plots indicating the position of poles with varying frequency for ω>0, ω<0, and ω=0 for the case of infinite geometry.

Fig. 2.
Fig. 2.

Geometry indicating the source and detector distribution for semi-infinite domain (left) and infinite slab (right).

Fig. 3.
Fig. 3.

Illustration of the extrapolated boundary for infinite slab geometry with source dipoles. The actual domain is the shaded region.

Fig. 4.
Fig. 4.

Illustration of the extrapolated boundary (dashed line) for circular geometry; the solid line shows the actual boundary.

Tables (12)

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Table 1. Glossary of Notation of Symbols in the Equations Used in This Work

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Table 2. Green’s Function Solution in the Time-Domain Case for Planar Type Geometries, where p=x or m

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Table 3. Green’s Function Solution in the Time-Domain Case for Circular Type Geometries, where p=x or m

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Table 4. Green’s Function Solution in the Frequency-Domain Case for Planar Type Geometries, where p=x or m

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Table 5. Green’s Function Solution in the Frequency-Domain Case for Circular Type Geometries, where p=x or m

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Table 6. Closed-form Expressions for Planar Type Geometries for Integrated Intensity Egeofl(ξ)=(nζ2γm2/(γm2γx2))Wgeofl(ξ)

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Table 7. Closed-form Expressions for Circular Type Geometries for Integrated Intensity Egeofl(ξ)=(nζ2γm2/(γm2γx2))Wgeofl(ξ)

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Table 8. Closed-form Expressions for Planar Type Geometries for Mean Time of Flight tgeofl(ξ)=ageofl(ξ)+(τ+ζ2)

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Table 9. Closed-form Expressions for Various Geometries for Mean Time of Flight tgeofl(ξ)=ageofl(ξ)+(τ+ζ2)

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Table 10. Closed-form Expressions for Circular Type Geometries for Mean Time of Flight tgeofl(ξ)=ageofl(ξ)+(τ+ζ2)

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Table 11. Glossary of Notation of Symbols Used in Tables 210

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Table 12. Glossary of Notation of Symbols Used in Tables 210

Equations (124)

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[γx22μaxct]Φx(r,t)=q0(r,t),
[γm22μamct]Φm(r,t)=qfl(r,t).
[k2γx2+μaxc+jω]Φx(k,ω)=Q0(k,ω),
[k2γm2+μamc+jω]Φm(k,ω)=Qfl(k,ω).
(1+τt)qfl(r,t)=ημafN(r)Φx(r,t),
Qfl(r,ω)=ημafN(r)1+jωτΦx(r,ω)=n(r)1+jωτΦx(r,ω).
Φfl(r,r,t,t)|Ω=0,
[1+τt][γm22μamct][γx22μaxct]Φm(r,t)=nq0(r,t).
[1+jτω][(k2γm2+μamc+jω)][(k2γx2+μaxc+jω)]Ginfϕfl(k,r,ω,t)=nF[δ(rr,tt)]=nej(k.r+ωt)
ginfϕfl(r,r,t,t)=1(2π)4[nej[k.(rr)+ω(tt)]dω(1+jτω)(k2γm2+μamc+jω)(k2γx2+μaxc+jω)]d3k,
Ginfϕfl(r,r,ω,t)=12πginfϕfl(r,r,t,t)ejωtdt=1(2π)7/2nej[k.(rr)ωt]d3k(1+jτω)(k2γm2+μamc+jω)(k2γx2+μaxc+jω).
Ginfϕfl(r,r,ω,t)=nejωt(2π)5/20π[0k2ejkρcosθsinθ(1+jτω)(k2γm2+μamc+jω)(k2γx2+μaxc+jω)dk]dθ,
Ginfϕfl(r,r,ω,t)=nejωt(2π)5/2jρ0k(ejkρejkρ)(1+jτω)(k2γm2+μamc+jω)(k2γx2+μaxc+jω)dk.
Ginfϕfl(r,r,ω,t)=nejωt(2π)5/2jρ12k(ejkρejkρ)(1+jτω)(k2γm2+μamc+jω)(k2γx2+μaxc+jω)dk.
Ginfϕfl(r,r,ω,t)=nejωt(2π)5/2jρ(1+jτω)kejkρ(k2γm2+μamc+jω)(k2γx2+μaxc+jω)dk.
Ginfϕfl(r,r,ω,t)=Aejkρ2γm2γx2[1k+jαm++1k+jαm][1k+jαx1k+jαx+]dkj(αx+αx),
A=nejωt(2π)5/2jρ(1+jτω),αx2=μaxc+jωγx2=Ax2ejβx,Ax=[μax2c2+ω2]14γx,tanβx=ωμaxc,αx+=Axejβx2,andαx=Axej(βx2+π),αm2=μamc+jωγm2=Am2ejβm,Am=[μam2c2+ω2]14γm,tanβm=ωμamc,αm+=Amejβm2,andαm=Amej(βm2+π).
Ginfϕfl(r,r,ω,t)=ejωtjργx2γm2(2π)5/2n1+jτωkejkρdk(k+jαm+)(k+jαm)(k+jαx+)(k+jαx).
Ginfϕfl(r,r,ω,t)=ejωtjργx2γm2(2π)5/2n1+jτωP(k)Q(k)ejkρdk,withdeg(Q(k))deg(P(k))+1.
P(k)Q(k)ejkρdk=2πjI[k]>0Res[P(k)Q(k)ejkρ].
=2πj[eαx+ρ2jαx+jαx+j2(αx++αm+)(αx++αm)+eαm+ρ2jαm+jαm+j2(αm++αx+)(αm++αx)]=2πj2[eαx+ρ(αx++αm+)(αx++αm)+eαm+ρ(αm++αx+)(αx++αm)]asαx+,m+=αx,m=2πj2[eαx+ρeαm+ραx2αm2]asαx+,m+2=αx,m2
Ginfϕfl(r,r,ω,t)=[ejωtργx2γm2(2π)3/22πjn(1+jτω)][2πj2(eαx+ρeαm+ρ)αx2αm2],ω>0,
Ginfϕfl(r,r,ω,t)=nejωt2(2π)3/2ργx2γm21(1+jτω)(eαxρeαmρ)αx2αm2whereαx,m=μax,amc+jωγx,m.
Ginfϕfl(r,r,ω,t)=nejωt2(2π)3/2ρ(eαxρeαmρ)1(1+jτω)[γm2(γx2αx2)γx2(γm2αm2)]=nejωt2(2π)3/2ρ(eαxρeαmρ)(1+jτω)[γm2(μaxc+jω)γx2(μamc+jω)]=nejωt2(2π)3/2ρ(eαxρeαmρ)(1+jτω)[γm2μaxcγx2μamc+j(γm2γx2)ω]=nejωt2(2π)3/2ρ(eαxρeαmρ)(1+jτω)(γm2μaxcγx2μamc)[1+j(γm2γx2)(γm2μaxcγx2μamc)ω]=nejωt2(2π)3/2ρ(eαxρeαmρ)γm2γx2ζ2[11+jτω11+jζ2ω]whereζ2=(γm2γx2)(γm2μaxcγx2μamc)=nejωt2(2π)3/2ρ(eαxρeαmρ)γm2γx2ζ2[1τζ2{τ1+jτωζ21+jζ2ω}].
Ginfϕfl(r,r,ω,t)=nζ2ejωt(γm2γx2)[eαxρ2ρ(2π)3/2eαmρ2ρ(2π)3/2]·[1τζ2{11τ+jω11ζ2+jω}].
ginfϕfl(r,r,t,t)=nζ2(γm2γx2)[γx2ginfϕx(r,r,t,t)γm2ginfϕm(r,r,t,t)]*[1τζ2[(etτetζ2)u(t)]],
ginfϕx,m(r,r,t,t)=1[4πγx,m2(tt)]3/2e[μax,amc(tt)+|rr|24γx,m2(tt)].
ghalfϕ(r,r,t,t)=1[4πγ2(tt)]3/2e[μac(tt)+|rr|24γ2(tt)]ginfϕ(r,r,t,t)[e(zz0)24γ2(tt)e(z+z0)24γ2(tt)].
ginfϕfl(r,r,t,t)=C[γx2ginfϕx(r,r,t,t)SOURCE-1γm2ginfϕm(r,r,t,t)SOURCE-2]*[1τζ2[(etτetζ2)u(t)]]SYSTEM,
ghalfϕfl(r,r,t,t)=nζ2(γm2γx2)[γx2ghalfϕx(r,r,t,t)γm2ghalfϕm(r,r,t,t)]*[1τζ2[(etτetζ2)u(t)]],
ghalfϕx,m(r,r,t,t)=1[4πγx,m2(tt)]3/2e[μax,amc(tt)+|rr|24γx,m2(tt)][e(zz0)24γx,m2(tt)e(z+z0)24γx,m2(tt)].
gslabϕfl(r,r,t,t)=nζ2(γm2γx2)[γx2gslabϕx(r,r,t,t)γm2gslabϕm(r,r,t,t)]*[1τζ2[(etτetζ2)u(t)]],
gslabϕx,m(r,r,t,t)=1[4πγx,m2(tt)]3/2e[μax,amc(tt)+|rr|24γx,m2(tt)][n=(e(z2ndz0)24γx,m2(tt)e(z2nd+z0)24γx,m2(tt))].
LτLmLxggeofl(r,r,t,t)=nδ(rr,tt)
Lτ=[1+τt],Lm=[γm22μamct],Lx=[γx22μaxct].
ggeoϕfl(r,r,t,t)=C[γx2ggeoϕx(r,r,t,t)γm2ggeoϕm(r,r,t,t)]*[1τζ2[(etτetζ2)u(t)]]
CLτ(Lm(Lx([γx2ggeoϕx(r,r,t,t)γm2ggeoϕm(r,r,t,t)]*[1τζ2{(etτetζ2)u(t)}])))=nδ(rr,tt).
Lτ{γx2Lm(Lx(ggeoϕx(r,r,t,t)))γm2Lm(Lx(ggeoϕm(r,r,t,t)))}*[1τζ2{(etτetζ2)u(t)}].
Lτ{γx2Lm(Lx(ggeoϕx(r,r,t,t)))γm2Lx(Lm(ggeoϕm(r,r,t,t)))}*[1τζ2{(etτetζ2)u(t)}].
Lx(ggeoϕx(r,r,t,t))=δ(rr,tt),Lm(ggeoϕm(r,r,t,t))=δ(rr,tt).
Lτ{γx2Lm(δ(rr,tt))γm2Lx(δ(rr,tt))}*[1τζ2{(etτetζ2)u(t)}],
Lτ{γx2Lmγm2Lx}(δ(rr,tt))*[1τζ2{(etτetζ2)u(t)}]
=Lτ{(γm2μaxcγx2μamc)+(γm2γx2)t}(δ(rr,tt))*[1τζ2{(etτetζ2)u(t)}]
=(1+τt)(A+Bt)(δ(rr,tt))*[1τζ2{(etτetζ2)u(t)}].
C1τζ2(1+τt)(A+Bt){(ettτettζ2)u(tt)}=nδ(tt).
(A+(Aτ+B)t+Bτ2t2){(ettτettζ2)u(tt)}.
A(ettτettζ2)+(Aτ+B)(1ζ2ettζ21τettτ)+Bτ(1τ2ettτ1ζ4ettζ2).
(1τζ2)(Bζ2A)ettζ2.
{(A+(Aτ+B)t+Bτ2t2)(ettτettζ2)}u(tt)+(ettτettζ2){(Au(tt)+(Aτ+B)δ(tt)+Bτtδ(tt))}.
(ettτettζ2){(Au(tt)+(Aτ+B)δ(tt)+Bτtδ(tt))}=A(ettτettζ2)|t=t+(Aτ+B)(ettτettζ2)δ(tt)|t=t+Bτtδ(tt)(ettτettζ2)|t=t.
Bτtδ(tt)(ettτettζ2)|t=t=Bτδ(tt)(1ζ2ettζ21τettτ)|t=t=B(τζ2)ζ2δ(tt).
C1τζ2(1+τt)(A+Bt){(ettτettζ2)u(tt)}=CBζ2δ(tt)=nδ(tt).
CBζ2=norC=nζ2γm2γx2.
ggeoϕfl(r,r,t,t)=nζ2γm2γx2[γx2ggeoϕx(r,r,t,t)γm2ggeoϕm(r,r,t,t)]*[1τζ2[(etτetζ2)u(t)]],
Φflt+·Γfl(ξ,t)=μamcΦfl,
Γfl(ξ,t)=γm2·Φm(r,t)+γx2·Φx(r,t),
ggeoΓfl(r,r,t,t)=nζ2γm2γm2γx2[ggeoΓx(r,r,t,t)ggeoΓm(r,r,t,t)]*[1τζ2[(etτetζ2)u(t)]],
GgeoΓfl(r,r,ω,t)=nζ2γm2γx2[GgeoΓx(r,r,ω,t)GgeoΓm(r,r,ω,t)]·[1τζ2{11τ+jω11ζ2+jω}].
d2(4πγp2)3(tt)5e(μapc(tt)+d24γp2(tt))
12(4πγp2)3(tt)5[d1e(μapc(tt)+ρ124γp2(tt))+d2e(μapc(tt)+ρ224γp2(tt))]
eμapc(tt)(4πγp2)3(tt)5n=0[d1eρ124γp2(tt)d2eρ224γp2(tt)]
eμapc(tt)(4πγp2)3(tt)5{z0eρ24γp2(tt)+n=1[d1eρ124γp2(tt)d2eρ224γp2(tt)]}
γp2eμapc(tt)πq2n=[cos(nθ)βneγp2βn2(tt)βnfn(βnr,βnq)]
2γp2eμapc(tt)πq2lk=1,oddeγp2k2π2(tt)l2n=[cos(nθ)βneγp2βn2(tt)βnfn(βnr,βnq)]
γpeμapc(tt)2πq2π(tt)n=cos(nθ)βneγp2βn2(tt)βnfn(βnr,βnq)
γp2eμapc(tt)2πq2arn=0βn+12eγp2βn+122(tt)βn+12fn+12(βn+12r,βn+12q)(2n+1)Pn(cosθ)
ejωt(1+αpd)eαpd2(2π)3d2
ejωt2(2π)3/2[(1+αpρ1)d1eαpρ1ρ13+(1+αpρ2)d2eαpρ2ρ23]
ejωt(2π)3(n=0[(1+αpρ1)d1ρ13eαpρ1(1+αpρ2)d2ρ23eαpρ2])
ejωt(2π)3((1+αpρ)z0ρ3eαpρ+n=1[(1+αpρ1)d1ρ13eαpρ1(1+αpρ2)d2ρ23eαpρ2])
ejωt(2π)3n=cos(nθ)fn(αpr)
ejωtπ2πlk=1,oddn=cos(nθ)fn(αpkr)
ejωt(2π)3n=cos(nθ)βn1αp2+βn2βngn(βnr)
ejωt2(2π)3qrn=0(2n+1)Pn(cosθ)fn+1/2(rαp)
A=1+32[8(1n2)3/2105n3(n1)2(8+32n+52n2+13n3)105n3(1+n)2+r1(n)+r2(n)+r3(n)]13+7n+13n2+9n37n4+3n5+n6+n73(n1)(n+1)2(n2+1)2r4(n),r1(n)=4+n4n2+25n340n46n5+8n6+30n712n8+n9+n113n(n21)2(n2+1)3,r2(n)=2n3(3+2n4)(n21)2(n2+1)7/2log{n2[n(1+n2)1/2][2+n2+2(1+n2)1/2][n+(1+n2)1/2][2+n42(1n4)1/2]},
r3(n)=4(1n2)1/2(1+12n4+2n8)3n(n21)2(n2+1)3,r4(n)=(1+6n4+n8)log(1n1+n)+4(n2+n6)log[1+nn2(1n)](n21)2(n2+1)3.
[γ22μact]Φ(r,t)=q0(r,t).
[k2γ2+μac+jω]Φ(k,ω)=Q0(k,ω).
halfϕ(r,r,t,t)=e[μac(tt)+ξ24γ2(tt)][4πγ2(tt)]3/2[e(zz0)24γ2(tt)e(z+z0+2ze)24γ2(tt)].
halfϕ(r,r,ω,t)=ejωt2(2π)3/2γ2{eα(ξ2+(zz0)2)1/2[ξ2+(zz0)2]1/2eα(ξ2+(z+z0+2ze)2)1/2[ξ2+(z+z0+2ze)2]1/2}.
slabϕ(r,r,t,t)=e[μac(tt)+ξ24γ2(tt)][4πγ2(tt)]3/2[n={e(z+n)24γ2(tt)e(zn)24γ2(tt)}].
slabϕ(r,r,ω,t)=ejωt2(2π)3/2[n={eα(ξ2+(z+n)2)1/2[ξ2+(z+n)2]1/2eα(ξ2+(zn)2)1/2[ξ2+(zn)2]1/2}].
gcirϕ(r,r,t,t)=14πγ2(tt)e(μac(tt)+|rr|24πγ2(tt)),Gcirϕ(r,r,ω,t)=ejωt(2π)3/2γ2K0(α|rr|).
(2r2+1rr+1r22θ2α2)Hϕ(r,ω)=0,
K0(α|rr|)={n=cos(nθ)In(rα)+bnKn(rα),r>rn=cos(nθ)In(rα)+bnKn(rα),r<r,
cirϕ(r,r,ω,t)=ejωt(2π)3/2γ2n=[cos(nθ)In(rα)In(bα)Fn(rα,bα)],
cirϕ(r,r,t,t)=12πn=[cos(nθ)0ejω(tt)In(rα)In(bα)Fn(rα,bα)dω].
cirϕ(r,r,t,t)=eμac(tt)2πan=[cos(nθ)βneγ2βn2(tt)Jn(βnr)Jn(βnr)(Jn(βnb))2].
(1r22θ2+2r2+1rr+2z2α2)Hϕ(r,ω)=0,
Hϕ(r,ω)=m,nam,nIn(r[α2+(mπl)2]1/2)+bm,nKn(r[α2+(mπl)2]1/2)cos(nθ)sin(mπzl).
am,n=In(r[α2+(mπl)2]1/2)Kn(b[α2+(mπl)2]1/2)In(r[α2+(mπl)2]1/2)
fcylϕ(r,r,z,z,t,t)=e(μac(tt)+(zz)24γ2(tt))πalm=1,oddeγ2m2π2l2(tt)n=[cos(nθ)βneγ2βn2(tt)Jn(βnr)Jn(βnr)(Jn(βnb))2],
fcylϕ(r,r,ω,t)|z=z=ejωt(2π)3/2lm=1,oddn=[cos(nθ)In(αmr)In(αmb)Fn(αmr,αmb)].
cylϕ(r,r,z,ω,t)=ejωt(2π)3/2bn=cos(nθ)0In(rα2+z2)In(bα2+z2)Fn(rα2+z2,bα2+z2)dz.
cylϕ(r,r,z,z,t,t)=e(μac(tt)+(zz)24γ2(tt))2πb2γπ(tt)n=cos(nθ)βneγ2βn2(tt)Jn(βnr)Jn(βnr)(Jn(βnb))2.
(2r2+2rr+1r21sinθθsinθθα2)Hϕ(r,ω)=0.
ginfϕ(r,r,t,t)=e(μac(tt)+R24γ2(tt))8γπ(tt),Ginfϕ(r,r,ω,t)=ejω(tt)eαR2(2π)3/2γ2R,
sphϕ(r,r,t,t)=eμac(tt)2πb2rrn=βn+12eγ2βn+122(tt)Jn+12(βn+12r)Jn+12(βn+12r)(Jn+12(βn+12b))2(2n+1)Pn(cosθ),
sphϕ(r,r,ω,t)=ejωt2(2π)3/2γ2rrn=0In+12(rα)In+12(bα)Fn+12(rα,bα)Pn(cosθ).
Efl(ξ)=Γfl(ξ,t)dt,
tfl(ξ)=tΓfl(ξ,t)dtΓfl(ξ,t)dt.
g(t)dt=2πG(ω)|ω=0and,ωG(ω)=12πωg(t)ejωtdt=j2πtg(t)dt
Egeofl(ξ)=2πGΓfl(ω)|ω=0=2πnζ2γm2γm2γx2[GgeoΓx(r,r,ω,t)GgeoΓm(r,r,ω,t)]·[1τζ2{11τ+jω11ζ2+jω}]|ω=0=2πnζ2γm2γm2γx2[GgeoΓx(r,r,ω,t)GgeoΓm(r,r,ω,t)]|ω=0
tgeofl(ξ)=jωGΓfl(ω)|ω=0GΓfl(ω)|ω=0=j2πnζ2γm2γm2γx2ω{(GgeoΓx(r,r,ω,t)GgeoΓm(r,r,ω,t))·[1τζ2(11τ+jω11ζ2+jω)]}|ω=02πnζ2γm2γm2γx2[GgeoΓx(r,r,ω,t)GgeoΓm(r,r,ω,t)]|ω=0=j[ωGgeoΓx(r,r,ω,t)ωGgeoΓm(r,r,ω,t)]|ω=0[GgeoΓx(r,r,ω,t)GgeoΓm(r,r,ω,t)]|ω=0+(τ+ζ2).
(1+σxd)eσxd(1+σmd)eσmd4πd2
14π[d1((1+σxρ1)eσxρ1(1+σmρ1)eσmρ1)ρ13+d2((1+σxρ2)eσxρ2(1+σmρ2)eσmρ2)ρ23]
12π[n=0{d1((1+σxρ1)eσxρ1(1+σmρ1)eσmρ1)ρ13}n=0{d2((1+σxρ2)eσxρ2(1+σmρ2)eσmρ2)ρ23}]
12πz0((1+σxρ)eσxρ(1+σmρ)eσmρ)ρ3+12πn=1(d1((1+σxρ1)eσxρ1(1+σmρ1)eσmρ1)ρ13)12πn=1(d2((1+σxρ2)eσxρ2(1+σmρ2)eσmρ2)ρ23)
12πn=cos(nθ)(fn(σxr)fn(σmr))
1πlk=1,oddn=cos(nθ)(fn(σxkr)fn(σmkr))
12πn=cos(nθ)βngn(βnr)(1σx2+βn21σm2+βn2)
14πqrn=0(fn+12(σxr)fn+12(σmr))(2n+1)Pn(cosθ)
12d2(eσxdγx2eσmdγm2)((1+σxd)eσxd(1+σmd)eσmd)
12d1ρ1(eσxρ1υxeσmρ1υm)+d2ρ2(eσxρ2υxeσmρ2υm)[d1ρ13((1+σxρ1)eσxρ1(1+σmρ1)eσmρ1)+d2ρ23((1+σxρ2)eσxρ2(1+σmρ2)eσmρ2)]
12[n=0{d1ρ1(eσxρ1υxeσmρ1υm)}n=0{d2ρ2(eσxρ2υxeσmρ2υm)}][n=0{d1ρ13((1+σxρ1)eσxρ1(1+σmρ1)eσmρ1)}n=0{d2ρ23((1+σxρ2)eσxρ2(1+σmρ2)eσmρ2)}]
N=z0ρ(eσxρυxeσmρυm)+[n=0{d1ρ1(eσxρ1υxeσmρ1υm)}n=0{d2ρ2(eσxρ2υxeσmρ2υm)}]
D=z0ρ3((1+σxρ)eσxρ(1+σmρ)eσmρ)+n=1d1ρ13((1+σxρ1)eσxρ1(1+σmρ1)eσmρ1)n=1d2ρ23((1+σxρ2)eσxρ2(1+σmρ2)eσmρ2)
12n=cos(nθ)(1υxfn(σxr)1υmfn(σmr))n=cos(nθ)(fn(σxr)fn(σmr))
k=1,oddn=cos(nθ)(1υxkfn(σxkr)1υmkfn(σmkr))k=1,oddn=cos(nθ)(fn(σxkr)fn(σmkr))
12n=cos(nθ)βngn(βnr)(1γx2(σx2+βn2)31γm2(σm2+βn2)3)n=cos(nθ)βngn(βnr)(1σx2+βn21σm2+βn2)
12n=0(1υxfn+12(σxr)1υmfn+12(σmr))(2n+1)Pn(cosθ)n=0(fn+12(σxr)fn+12(σmr))(2n+1)Pn(cosθ)
Vn(aε,bε)=(n(2n+1)aε2a)Fn(aε,bε)+bFn1(aε,bε)+n1ε(In1(aε)Kn(bε)+Kn1(aε)In(bε))bnaε(In(aε)Kn1(bε)+Kn(aε)In1(bε))
qfl(r,t)=ημafN(r)Φx(r,t),
Qfl(r,ω)=ημafN(r)Φx(r,ω).

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