Abstract

An analytical expression has been obtained that describes the time variation of time-resolved signals transmitted through or reflected by a homogeneous scattering slab as measured with a detection system having a square-impulse response. This expression can be used to improve the match between theoretical and experimental time-resolved signals measured with a system having a finite response time. It can also be used to assess the effect of a finite detection response time on the time-domain characterization of a turbid medium. The expression can be adapted to detection systems that are not time invariant.

© 1999 Optical Society of America

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References

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  1. G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, 1993).
  2. J. C. Hebden, R. A. Kruger, K. S. Wong, “Time resolved imaging through a highly scattering medium,” Appl. Opt. 30, 788–794 (1991).
    [CrossRef] [PubMed]
  3. H. Wabnitz, H. Rinneberg, “Imaging in turbid media by photon density waves: spatial resolution and scaling relations,” Appl. Opt. 36, 64–74 (1997).
    [CrossRef] [PubMed]
  4. J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
    [CrossRef] [PubMed]
  5. A. H. Gandjbakhche, V. Chernomordik, J. C. Hebden, R. Nossal, “Time-dependent contrast functions for quantitative imaging in time-resolved transillumination experiments,” Appl. Opt. 37, 1973–1981 (1998).
    [CrossRef]
  6. J. C. Hebden, D. J. Hall, D. T. Delpy, “The spatial resolution performance of the time-resolved optical imaging system using temporal extrapolation,” Med. Phys. 22, 201–208 (1995).
    [CrossRef] [PubMed]
  7. R. Cubeddu, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, “Time-resolved imaging on a realistic tissue phantom: μs′ and μa images versus time-integrated images,” Appl. Opt. 35, 4533–4540 (1996).
    [CrossRef] [PubMed]
  8. D. Contini, F. Martelli, 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] [PubMed]
  9. R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
    [CrossRef]
  10. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986).
  11. M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1970).

1998 (1)

1997 (2)

1996 (1)

1995 (2)

J. C. Hebden, D. J. Hall, D. T. Delpy, “The spatial resolution performance of the time-resolved optical imaging system using temporal extrapolation,” Med. Phys. 22, 201–208 (1995).
[CrossRef] [PubMed]

R. Aronson, “Boundary conditions for diffusion of light,” J. Opt. Soc. Am. A 12, 2532–2539 (1995).
[CrossRef]

1994 (1)

1991 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1970).

Aronson, R.

Chernomordik, V.

Contini, D.

Cubeddu, R.

Delpy, D. T.

J. C. Hebden, D. J. Hall, D. T. Delpy, “The spatial resolution performance of the time-resolved optical imaging system using temporal extrapolation,” Med. Phys. 22, 201–208 (1995).
[CrossRef] [PubMed]

J. C. Hebden, D. T. Delpy, “Enhanced time-resolved imaging with a diffusion model of photon transport,” Opt. Lett. 19, 311–313 (1994).
[CrossRef] [PubMed]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986).

Gandjbakhche, A. H.

Hall, D. J.

J. C. Hebden, D. J. Hall, D. T. Delpy, “The spatial resolution performance of the time-resolved optical imaging system using temporal extrapolation,” Med. Phys. 22, 201–208 (1995).
[CrossRef] [PubMed]

Hebden, J. C.

Kruger, R. A.

Martelli, F.

Nossal, R.

Pifferi, A.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986).

Rinneberg, H.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1970).

Taroni, P.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986).

Torricelli, A.

Valentini, G.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986).

Wabnitz, H.

Wong, K. S.

Zaccanti, G.

Appl. Opt. (5)

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

Med. Phys. (1)

J. C. Hebden, D. J. Hall, D. T. Delpy, “The spatial resolution performance of the time-resolved optical imaging system using temporal extrapolation,” Med. Phys. 22, 201–208 (1995).
[CrossRef] [PubMed]

Opt. Lett. (1)

Other (3)

G. Müller, B. Chance, R. Alfano, S. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. Masters, S. Svanberg, P. van der Zee, eds., Medical Optical Tomography: Functional Imaging and Monitoring, Vol. IS11 of SPIE Institutes for Advanced Optical Technologies (SPIE Press, Bellingham, 1993).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge University Press, Cambridge, 1986).

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs and Mathematical Tables (Dover, New York, 1970).

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Figures (4)

Fig. 1
Fig. 1

Impulse response h(t) as defined by Eq. (3).

Fig. 2
Fig. 2

Time response of a streak-camera-based detection system with the camera’s slit fully opened. The streak camera was a Model C5680-34 equipped with a synchroscan unit Model M5675, both from Hamamatsu.

Fig. 3
Fig. 3

Time-resolved signals detected in transmission through a homogeneous slab (μ a = 0.003 mm-1, μ s ′ = 1 mm-1, v = 2.14 × 108 m/s, and s = 50 mm) and measured with different detection response times (Δt equal to 0, 0.4, 0.6, 0.8, and 1 ns). The curves were calculated with either Eq. (1) (Δt = 0 ns) or Eq. (7).

Fig. 4
Fig. 4

Error on the optical parameters deduced from a least-squares fit of Eq. (1) to signals calculated with Eq. (7). The parameters of the slab are the same as in Fig. 3.

Equations (13)

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Tt=exp-μa vt24πDv3/2t5/2m=-z1,m exp-z1,m24Dvt-z2,m exp-z2,m24Dvtut,
z1,m=s1-2m-4mze-z0 z2,m=s1-2m-4m-2ze+z0.
St=- Tτht-τdτ.
ht=1/Δtut+Δt/2-ut-Δt/2
St=1Δtt-Δt/2t+Δt/2 Tτdτ.
0t τ-5/2 exp-ατ-β/τuτdτ=14πβ31/221+2αβ1/2exp-2αβ1/2+4βπt1/2 exp-αt-β/t-1+2αβ1/2exp-2αβ1/2erfcαt1/2-βt1/2+1-2αβ1/2exp2αβ1/2erfcαt1/2+βt1/2ut,
fi,mt=18πzi,m|zi,m|Δt21+μazi,m2D1/2×exp-μazi,m2D1/2+4zi,m2πDvt1/2×exp-μavt-zi,m24Dvt-1+μazi,m2D1/2×exp-μazi,m2D1/2erfcμavt1/2-zi,m24Dvt1/2+1-μazi,m2D1/2expμazi,m2D1/2×erfcμavt1/2+zi,m24Dvt1/2ut,
St=m=- f1,mt+Δt/2-f2,mt+Δt/2-f1,mt-Δt/2+f2,mt-Δt/2.
- Stexpjωtdt=- Ttexpjωtdt×- htexpjωtdt.
- Stexpjωtdt=2 sinωΔt/2ωΔt- Ttexpjωtdt.
0 Stdt=0 Ttdt.
ddt0t τ-5/2 exp-ατ-β/τuτdτ=t-5/2 exp-αt-β/t.
ddzerfcz=-2πexp-z2.

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