Abstract

The least-scattered photons that arrive at a detector through highly scattering tissues have the potential to image internal structures, functions, and status with high imaging resolution. In contrast, optical diffusing tomography is based on the use of the late-arriving photons, which have been diffusely scattered, leading to very low imaging resolution. A good model of the early-arriving photons, i.e., the least-scattered photons, may have a significant effect on the development of imaging algorithms and a further understanding of imaging mechanisms within current high-resolution optical-imaging techniques. We describe a vertex/propagator approach that attempts to find the probabilities for least-scattered photons traversing a scattering medium, based on analytical expressions for photon histories. The basic mathematical derivations for the model are outlined, and the results are discussed and found to be in very good agreement with those from the Monte Carlo simulations.

© 2001 Optical Society of America

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References

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  1. D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
    [CrossRef] [PubMed]
  2. M. R. Hee, J. A. Izatt, J. M. Jacobson, J. G. Fujimoto, E. A. Swanson, “Femto-second trans-illumination optical coherence tomography,” Opt. Lett. 18, 950–952 (1993).
    [CrossRef] [PubMed]
  3. L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering wall using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
    [CrossRef] [PubMed]
  4. S. G. Demos, R. R. Alfano, “Temporal gating in highly scattering media by the degree of optical polarisation,” Opt. Lett. 21, 161–163 (1993).
    [CrossRef]
  5. D. S. Dilworth, E. N. Leith, J. L. Lopez, “Three dimensional confocal imaging of objects embedded within thick diffusing media,” Appl. Opt. 30, 1796–1803 (1991).
    [CrossRef] [PubMed]
  6. S. L. Jacques, “Path integral description of light transport in tissue,” Ann. N.Y. Acad. Sci. 838, 1–13 (1998).
    [CrossRef] [PubMed]
  7. L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
    [CrossRef]
  8. W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
    [CrossRef]
  9. S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

1998 (1)

S. L. Jacques, “Path integral description of light transport in tissue,” Ann. N.Y. Acad. Sci. 838, 1–13 (1998).
[CrossRef] [PubMed]

1993 (2)

1991 (3)

D. S. Dilworth, E. N. Leith, J. L. Lopez, “Three dimensional confocal imaging of objects embedded within thick diffusing media,” Appl. Opt. 30, 1796–1803 (1991).
[CrossRef] [PubMed]

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering wall using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

1990 (1)

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

1987 (1)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Alfano, R. R.

S. G. Demos, R. R. Alfano, “Temporal gating in highly scattering media by the degree of optical polarisation,” Opt. Lett. 21, 161–163 (1993).
[CrossRef]

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering wall using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Alter, C. A.

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Chang, W.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Cheong, W. C.

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Demos, S. G.

Dilworth, D. S.

Flotte, T.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

M. R. Hee, J. A. Izatt, J. M. Jacobson, J. G. Fujimoto, E. A. Swanson, “Femto-second trans-illumination optical coherence tomography,” Opt. Lett. 18, 950–952 (1993).
[CrossRef] [PubMed]

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Gregory, K.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Hee, M. R.

Hee, W. R.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

Ho, P. P.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering wall using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Huang, D.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Izatt, J. A.

Jacobson, J. M.

Jacques, S. L.

S. L. Jacques, “Path integral description of light transport in tissue,” Ann. N.Y. Acad. Sci. 838, 1–13 (1998).
[CrossRef] [PubMed]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Leith, E. N.

Lin, C. P.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Liu, C.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering wall using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Lopez, J. L.

Prahl, S. A.

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Puliafito, C. A.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Schuman, J. S.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Stinson, W. G.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E.

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

Wang, L.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering wall using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Welch, A. J.

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Zhang, G.

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering wall using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

Ann. N.Y. Acad. Sci. (1)

S. L. Jacques, “Path integral description of light transport in tissue,” Ann. N.Y. Acad. Sci. 838, 1–13 (1998).
[CrossRef] [PubMed]

Appl. Opt. (1)

Astrophys. J. (1)

L. G. Henyey, J. L. Greenstein, “Diffuse radiation in the galaxy,” Astrophys. J. 93, 70–83 (1941).
[CrossRef]

IEEE J. Quantum Electron. (1)

W. C. Cheong, S. A. Prahl, A. J. Welch, “A review of the optical properties of biological tissues,” IEEE J. Quantum Electron. 26, 2166–2185 (1990).
[CrossRef]

Lasers Life Sci. (1)

S. L. Jacques, C. A. Alter, S. A. Prahl, “Angular dependence of HeNe laser light scattering by human dermis,” Lasers Life Sci. 1, 309–333 (1987).

Opt. Lett. (2)

Science (2)

D. Huang, E. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, W. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, “Optical coherence tomography,” Science 254, 1178–1181 (1991).
[CrossRef] [PubMed]

L. Wang, P. P. Ho, C. Liu, G. Zhang, R. R. Alfano, “Ballistic 2-D imaging through scattering wall using an ultrafast Kerr gate,” Science 253, 769–771 (1991).
[CrossRef] [PubMed]

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Figures (12)

Fig. 1
Fig. 1

Definitions of the quantities used in the derivation of the probability of single-scattered photons traversing the scattering medium.

Fig. 2
Fig. 2

(a) Probability P1 of a photon traversing a medium and undergoing just one interaction as a function of thickness. (b) Equivalent probability P1 based on a 1-D model. Curves are shown for μs=2.1 mm-1, μt=2.3 mm-1, and values of g increasing from 0.0 to 0.9 in steps of 0.15 from bottom to top.

Fig. 3
Fig. 3

(a) Thickness of scattering material necessary to maximize P1 and P1 as a function of g and (b) the corresponding maximum probabilities.

Fig. 4
Fig. 4

Comparison between the 3-D phase model (solid curve) and the 1-D phase model (dotted curve), with g=0.9 for the single-scattered photons in transmission.

Fig. 5
Fig. 5

Comparison of the 3-D model (solid curves) and the Monte Carlo simulations (dotted curves) for μs=2.1 mm-1, μt=2.3 mm-1, and g=0.15, 0.45, and 0.9 from bottom to top. Curves are shown for the single-scattered photons in transmission.

Fig. 6
Fig. 6

Definitions of the quantities used in the derivation of probability of doubly scattered photons traversing the scattering medium.

Fig. 7
Fig. 7

Rearranged geometry presentation and definitions for the coordinate system used for the third propagator stage of a photon traveling in the scattering medium.

Fig. 8
Fig. 8

(a) Probability P2 of a photon traversing a medium and suffering two interactions as a function of thickness. (b) Equivalent probability P2 based on a 1-D model. Curves are shown for μs=2.1 mm-1, μt=2.3 mm-1, and values of g increasing from 0.0 to 0.9 in steps of 0.15.

Fig. 9
Fig. 9

(a) Thickness of scattering material necessary to maximize P2 and P2 as a function of g and (b) the corresponding maximum probabilities.

Fig. 10
Fig. 10

Comparison between the 3-D phase model (solid curve) and the 1-D phase model (dotted curve) with g=0.9 for the doubly scattered photons in transmission.

Fig. 11
Fig. 11

Comparison of the 3-D model (solid curve) and the MC simulations (dotted curve) for μs=2.1 mm-1, μt=2.3 mm-1, and g=0.9. Curves are shown for the doubly scattered photons in transmission.

Fig. 12
Fig. 12

Probabilities that a photon traversing a medium will undergo exactly N (up to 4) interactions as a function of thickness from the model prediction (dotted curves) and the MC simulations (solid curves). Curves are shown for μs=2.1 mm-1, μt=2.3 mm-1, and g=0.9.

Equations (28)

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p1=exp(-μtx1)1stprogagatorp(cosθ1)μsvertexexp(-μtd-x1cosθ1)2ndpropagator.
p(cos θ)=1-g2(1+g2-2g cos θ)3/2
P1=10d(cos θ1)0dp1(x1, θ1)dx1=10μsμtcos θ11-cos θ1exp(-μtd)-exp-μtdcos θ1p(cos θ1)d(cos θ1).
P1=14π10μsμtcos θ11-cos θ1×exp(-μtd)-exp-μtdcos θ1d(cos θ1).
p1n=[1-r(n, θ1)] p1(x1, θ1),
g=cos(0)pf+cos(π)(1-pf)=2pf-1  pf=1+g2.
p1(x1)=exp(-μtx1)1stpropagatorpfμsvertexexp[-μt(d-x1)]2ndprogagator.
P1=0dp1(x1)dx1=pfμsd exp(-μtd),
P1(d)d=0.
p2(x1,θ1,x2,θ2,ϕ)=12πexp(-μtx1)1stpropagatorp(cosθ1)μs1stvertex×exp[-μtx2-x1cosθ1]2ndpropagatorp(cosθ2)μs2ndvertex×exp[-μt(d-x2)/f(θ1,θ2,ϕ)]3rdpropagator.
0d0x2p2dx1dx2
=12π μs2p(c)p(cos θ2)0d0x2exp(-μtx1)×exp-μtx2-x1c  exp[-μt(d-x2)f]dx1dx2,
0d0x2p2dx1dx2=12π μs2p(c)p(cos θ2)cfexp(-μtd)-exp-μtdc-cexp(-μtfd)-exp-μtdc+exp(-μtfd)-exp(-μtd)×cμt2(cf-1)(f-1)(c-1).
l=f(θ1, θ2, ϕ)/(d-x2),
xc=R(ϕ)sin ϕ,
yc=R(ϕ)cos ϕ cos θ1,
zc=d-x2cos θ1-R(ϕ)cos ϕ sin θ1.
xc2+yc2=zc2tan2 θ2,
R(ϕ)=h(θ1, θ2, ϕ)(d-x2),
h(θ1, θ2, ϕ)=-tan2 θ2tan θ1cos ϕ+tan θ2(cos2 ϕ+sin2 ϕ/cos2 θ1)1/2sin2 ϕ+cos2 ϕ cos2 θ1-tan2 θ2cos2 ϕsin2 θ1.
|AO2|=zccos θ2=d-x2cos θ1-R(ϕ)cos ϕ sin θ1/cos θ2.
f(θ1, θ2, ϕ)
=1cos θ1-h(θ1, θ2, ϕ)cos ϕ sin θ1/cos θ2,
pb=1-g2.
p2(x1,x2)=exp(-μ1x1)1stpropagatorpfμs1stvertexexp[-μt(x2-x1)]2ndpropagator×pfμs2ndvertexexp[-μt(d-x2)]3rdpropagator+exp(-μtx1)1stpropagatorpbμs1stvertex×exp[-μt(x1-x2)]2ndpropagatorpbμs2ndvertex×exp[-μt(d-x2)]3rdpropagator,
P2=12 pf2μs2d2exp(-μtd)+μs2pb24μt2exp(-μtd)[2μtd-1+exp(-2μtd)].
PN=1N! pfNμsNdNexp(-μtd)+pb,
PN=1N! pfNμsNdNexp(-μtd)

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