Abstract

Various modifications of the diffusion model designed to describe photon migration in scattering and absorbing media are analyzed and compared in the nondiffusion region. A more accurate modified diffusion equation that has a broader region of applicability is introduced. The radiative-transfer equation is solved numerically to assess the validity range of different models tested and to select the best one.

© 1997 Optical Society of America

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References

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  1. K. Furutsu and Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3440 (1994); K. Furutsu, “Diffusion equation derived from space–time transport equation,” J. Opt. Soc. Am. 70, 360–381 (1980).
    [Crossref]
  2. M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for noninvasive measurements of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989); J. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
    [Crossref] [PubMed]
  3. A. Ishimaru, “Diffusion of light in turbid media,” Appl. Opt. 28, 2210–2215 (1989); “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
    [Crossref] [PubMed]
  4. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978). Coefficients of the steady-state diffusion equation considered in this reference (Eqs. 9–14 and 9–16, p. 178) correspond to the time-dependent equation considered in Refs. 3 and 5.
  5. J.-M. Kaltenbach and M. Kaschke, “Frequency and time domain modeling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of Institute Series of SPIE Optical Engineering, G. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, and P. van der Zee, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.
  6. M. Lax, “Phonon transport in GaAs at low temperatures,” Department of Physics, City College of New York, N.Y. 10031 (personal communication, 1995).
  7. P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, p. 865.
  8. This consistency condition implies that α=γ=β/2. Absorption was not considered in Refs. 7 and 8, so the parameters β and γ were undetermined. When these references were quoted, where α=1/3 was suggested, β and γ were assumed to satisfy the consistency condition.

Other (8)

K. Furutsu and Y. Yamada, “Diffusion approximation for a dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3440 (1994); K. Furutsu, “Diffusion equation derived from space–time transport equation,” J. Opt. Soc. Am. 70, 360–381 (1980).
[Crossref]

M. S. Patterson, B. Chance, and B. C. Wilson, “Time resolved reflectance and transmittance for noninvasive measurements of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989); J. Fishkin and E. Gratton, “Propagation of photon-density waves in strongly scattering media containing an absorbing semi-infinite plane bounded by straight edge,” J. Opt. Soc. Am. A 10, 127–140 (1993).
[Crossref] [PubMed]

A. Ishimaru, “Diffusion of light in turbid media,” Appl. Opt. 28, 2210–2215 (1989); “Diffusion of a pulse in densely distributed scatterers,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
[Crossref] [PubMed]

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978). Coefficients of the steady-state diffusion equation considered in this reference (Eqs. 9–14 and 9–16, p. 178) correspond to the time-dependent equation considered in Refs. 3 and 5.

J.-M. Kaltenbach and M. Kaschke, “Frequency and time domain modeling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of Institute Series of SPIE Optical Engineering, G. Mueller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, and P. van der Zee, eds. (Society of Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

M. Lax, “Phonon transport in GaAs at low temperatures,” Department of Physics, City College of New York, N.Y. 10031 (personal communication, 1995).

P. M. Morse and H. Feshbach, Methods of Theoretical Physics (McGraw-Hill, New York, 1953), Vol. 1, p. 865.

This consistency condition implies that α=γ=β/2. Absorption was not considered in Refs. 7 and 8, so the parameters β and γ were undetermined. When these references were quoted, where α=1/3 was suggested, β and γ were assumed to satisfy the consistency condition.

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Figures (1)

Fig. 1
Fig. 1

Comparison of the numerical solution of the transport equation, NE, with the solution N1, corresponding to the SWDE [Eq. (6)] and the different modifications of Eq. (1): 0, exact numerical solution NE of Eq. (10); 1, SWDE; 2, Refs. 3 and 5; 3, Ref. 2; 4, Ref. 1; 5, Refs. 6 and 7. (a) Phase φ=arg N1,E (in deg) at the source–detector distance r=25lt, νa=0; (b), (c) relative change in amplitude ψ=(|Nt|-|NE|)/|NE|; (b) r=3lt, νa=0; (c) r=25lt, νa=0 (curve 2 corresponds to curve 2 scaled as 1:2500); (d) r=50lt, νa = 0.1ν.

Equations (19)

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α 2N2t+(ν+βνa) Nt-c23ΔN+νa(ν+γνa)N=0.
k2=-3c2[ν(νa+iω)+γνa2+βνaiω-αω2],
n+sn+ν(n-N/4π)=0,
N+iJi=0,
(Ji+jTij)/ν+Ji=0,
(Tij+kTijk)/ν+{Tij-c2δijN/3}=0,
(Tijk+lTijkl)/ν+{Tijk}=0,
(Tijkl+mTijklm)/ν+{Tijkl-c4δijklN/15}=0,,
P(, Δ)N(t, r)=0,
k2=-3c2ν2 νa+iων+higher-ordertermsinνa+iων.
P(, Δ)=(ν+)3-c23ν(ν+)Δ-c45ΔΔ.
152N2t+ν+25νa Nt-c23ΔN
+νaν+15νaN=0.
η=(1/4π)θ(t)·δ(r)·exp(iωt).
NE(r, t, ω)=exp(iωt)0 exp(-iωτ)·g(r, s, τ)dτdΩs,
N1(r, t, ω)=exp(iωt)0 exp(-iωτ)·G(r, τ)dτ.
NE(r, ω, t)=N0(r, ω)exp(iωt)+νN0(r-r, ω)NE(r, ω, t)dr,
N0(r, ω)=exp[-(ν+νa+iω)r/c]/(4πcr2).
N1(ω, r, t)=3c2[iωα+ν+(β-α)νa]×14πrexp[ik(ω)r+iωt],

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