Abstract

The accuracy of the perturbation model to predict the effect of scattering and absorbing inhomogeneities on photon migration has been investigated by comparisons with experimental and numerical results. Comparisons for scattering inhomogeneities showed that the model gives satisfactory results both for the intensity and for the temporal profile of the perturbation over a large range of values for the scattering properties of the defect. As for absorbing inhomogeneities, the model provides an excellent description for the temporal profile, but the results for the intensity are accurate only when the perturbation is small. For absorbing inhomogeneities an empirical model that has a significantly more extended application range has been proposed. The model is based on an expression for the time-resolved mean path length that detected photons have followed inside the inhomogeneity. The application range of the proposed model covers the values expected for the optical properties and for the volumes of inhomogeneities of practical interest for optical mammography.

© 2001 Optical Society of America

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    [Crossref] [PubMed]
  2. S. R. Arridge, “Photon measurements density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
    [Crossref] [PubMed]
  3. J. C. Hebden, S. R. Arridge, “Imaging through scattering media by the use of an analytical model of perturbation amplitudes in the time domain,” Appl. Opt. 35, 6788–6796 (1996).
    [Crossref] [PubMed]
  4. J. C. Schotland, J. C. Haselgrove, J. S. Leigh, “Photon hitting density,” Appl. Opt. 32, 448–453 (1993).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
  7. D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4896 (1994).
    [Crossref]
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    [Crossref]
  11. V. Chernomordik, D. Hattery, A. H. Gandjbakhche, A. Pifferi, P. Taroni, A. Torricelli, G. Valentini, R. Cubeddu, “Quantification by random walk of the optical parameters of nonlocalized abnomalies embedded within tissuelike phantoms,” Opt. Lett. 25, 951–953 (2000).
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  13. 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]
  14. G. Zaccanti, L. Alianelli, C. Blumetti, S. Carraresi, “Method for measuring the mean time of flight spent by photons inside a volume element of a highly diffusing medium,” Opt. Lett. 24, 1290–1292 (1999).
    [Crossref]
  15. A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
    [Crossref]
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    [Crossref]
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    [Crossref]
  18. D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta, P. M. Schlag, “Development of a time-domain optical mammograph and first in vivo applications,” Appl. Opt. 38, 2927–2943 (1999).
    [Crossref]

2000 (2)

1999 (3)

1998 (3)

1997 (3)

1996 (1)

1995 (2)

1994 (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4896 (1994).
[Crossref]

1993 (2)

Alianelli, L.

Arridge, S. R.

Beaudry, P.

Blumetti, C.

Boas, D. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4896 (1994).
[Crossref]

Carraresi, S.

Chance, B.

S. Feng, F. A. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[Crossref] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4896 (1994).
[Crossref]

Chatigny, S.

Chernomordik, V.

Contini, D.

Cubeddu, R.

Delfino, I.

den Outer, P. N.

Fantini, S.

Feng, S.

Franceschini, M. A.

Frechette, J.

Gandjbakhche, A. H.

Grosenick, D.

Haselgrove, J. C.

Hattery, D.

Hebden, J. C.

Indovina, P. L.

Ismaelli, A.

Jacques, S. L.

Kaltenbach, J. M.

J. M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller ed., Vol. IS11 of SPIE Proceedings Series (Society for Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Kaschke, M.

S. Fantini, S. A. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag, K. T. Moesta, “Assessment of the size, position, and optical properties of breast tumors in vivo by noninvasive optical methods,” Appl. Opt. 37, 1982–1989 (1998).
[Crossref]

J. M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller ed., Vol. IS11 of SPIE Proceedings Series (Society for Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

Lagendijk, A.

Leigh, J. S.

Lepore, M.

Mailloux, A.

Martelli, F.

Moesta, K. T.

Morin, M.

Nieuwenhuizen, T. M.

Nossal, R.

O’Leary, M. A.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4896 (1994).
[Crossref]

Ostermeyer, M. R.

Painchaud, Y.

Pifferi, A.

Rinneberg, H. H.

Sassaroli, A.

Schlag, P. M.

Schotland, J. C.

Taroni, P.

Torricelli, A.

Valentini, G.

Verreault, S.

Wabnitz, H.

Walker, S. A.

Yodh, A. G.

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4896 (1994).
[Crossref]

Zaccanti, G.

Zeng, F. A.

Appl. Opt. (11)

S. R. Arridge, “Photon measurements density functions. Part I: Analytical forms,” Appl. Opt. 34, 7395–7409 (1995).
[Crossref] [PubMed]

J. C. Hebden, S. R. Arridge, “Imaging through scattering media by the use of an analytical model of perturbation amplitudes in the time domain,” Appl. Opt. 35, 6788–6796 (1996).
[Crossref] [PubMed]

J. C. Schotland, J. C. Haselgrove, J. S. Leigh, “Photon hitting density,” Appl. Opt. 32, 448–453 (1993).
[Crossref] [PubMed]

M. Morin, S. Verreault, A. Mailloux, J. Frechette, S. Chatigny, Y. Painchaud, P. Beaudry, “Inclusion characterization in a scattering slab with time-resolved transmittance measurements: perturbation analysis,” Appl. Opt. 39, 2840–2852 (2000).
[Crossref]

S. Feng, F. A. Zeng, B. Chance, “Photon migration in the presence of a single defect: a perturbation analysis,” Appl. Opt. 34, 3826–3837 (1995).
[Crossref] [PubMed]

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]

A. Sassaroli, C. Blumetti, F. Martelli, L. Alianelli, D. Contini, A. Ismaelli, G. Zaccanti, “Monte Carlo procedure for investigating light propagation and imaging of highly scattering media,” Appl. Opt. 37, 7392–7400 (1998).
[Crossref]

I. Delfino, M. Lepore, P. L. Indovina, “Experimental tests of different solutions to the diffusion equation for optical characterization of scattering media by time-resolved transmittance,” Appl. Opt. 38, 4228–4236 (1999).
[Crossref]

S. Fantini, S. A. Walker, M. A. Franceschini, M. Kaschke, P. M. Schlag, K. T. Moesta, “Assessment of the size, position, and optical properties of breast tumors in vivo by noninvasive optical methods,” Appl. Opt. 37, 1982–1989 (1998).
[Crossref]

D. Grosenick, H. Wabnitz, H. H. Rinneberg, K. T. Moesta, P. M. Schlag, “Development of a time-domain optical mammograph and first in vivo applications,” Appl. Opt. 38, 2927–2943 (1999).
[Crossref]

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]

J. Opt. Soc. Am. A (2)

Opt. Lett. (2)

Phys. Med. Biol. (1)

S. R. Arridge, J. C. Hebden, “Optical imaging in medicine: II. Modelling and reconstruction,” Phys. Med. Biol. 42, 841–853 (1997).
[Crossref] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Scattering of diffuse photon density waves by spherical inhomogeneities within turbid media: analytic solutions and applications,” Proc. Natl. Acad. Sci. USA 91, 4887–4896 (1994).
[Crossref]

Other (1)

J. M. Kaltenbach, M. Kaschke, “Frequency- and time-domain modelling of light transport in random media,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Muller ed., Vol. IS11 of SPIE Proceedings Series (Society for Photo-Optical Instrumentation Engineers, Bellingham, Wash., 1993), pp. 65–86.

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Figures (13)

Fig. 1
Fig. 1

Geometric scheme assumed for the perturbation model.

Fig. 2
Fig. 2

Comparison between the perturbation model (solid curves) and MC results (dashed curves) for a spherical scattering inhomogeneity with radius r i = 5 mm in different locations inside the slab. The unperturbed response T hom(t) is also reported. μ s0′ = 0.5 mm-1, μ si ′ = 0.4 mm-1, μ a0 = μ ai = 0, d = 40 mm, n = 1.

Fig. 3
Fig. 3

Examples of MC results for the perturbed response T pert(t) for a spherical scattering inhomogeneity with different values of μ si ′. The sphere has radius r i = 5 mm and is placed in the middle of the slab on the beam axis (x = y = 0, z = 20 mm). μ s0′ = 0.5 mm-1, μ a0 = μ ai = 0, d = 40 mm, n = 1. Comparisons between the perturbation model (solid curves) and MC results (dashed curves) are also reported for δT D (t) and for the ratio δT D (t)/T hom(t).

Fig. 4
Fig. 4

Comparison between the perturbation model and MC results for the contrast on the cw attenuation. The results refer to a scattering inhomogeneity with r i = 5 mm in (x = y = 0, z = 20 mm). μ a0 = μ ai = 0, d = 40 mm, n = 1. The MC results with the corresponding error bars are reported both for μ s0′ = 0.5 mm-1 and μ s0′ = 1 mm-1.

Fig. 5
Fig. 5

Examples of experimental results for scattering inhomogeneities. (a) and (b) show the perturbed response T pert(t) and the corresponding perturbation δT D (t) for different values of μ si ′. The inhomogeneity has V i = 1500 mm3 and is in (x = y = 0, z = 20 mm). μ s0′ = 0.31 mm-1, μ a0 = μ ai = 0.0028 mm-1, d = 40 mm, n = 1.33. (c) Comparison with the perturbation model (dashed curves) for the inhomogeneity with μ si ′ = 0.11 mm-1 in the middle of the slab for four values of the distance from the light beam.

Fig. 6
Fig. 6

Comparison between the perturbation model (solid curves) and MC results (dashed curves) for a spherical absorbing inhomogeneity with radius r i = 5 mm in different locations inside the slab. The unperturbed response T hom(t) is also reported. μ s0′ = μ si ′ = 0.5 mm-1, μ ai = 0.005 mm-1, μ a0 = 0, d = 40 mm, n = 1.

Fig. 7
Fig. 7

Comparison between the perturbation model (solid curves) and MC results (dashed curves) for high values of δμ a . The absorbing sphere with r i = 5 mm is in (x = y = 0, z = 20 mm) and μ s0′ = μ si ′ = 0.5 mm-1, μ a0 = 0, d = 40 mm, n = 1. The comparison is reported with Eqs. (4) and (8).

Fig. 8
Fig. 8

Comparison between the perturbation model [Eqs. (4) and (8)] and MC results for the contrast on the cw attenuation. The results refer to an absorbing inhomogeneity with r i = 5 mm in (x = y = 0, z = 20 mm). μ s0′ = μ si ′ = 0.5 mm-1, μ a0 = 0, d = 40 mm, n = 1.

Fig. 9
Fig. 9

Comparisons between experimental and MC results for the temporal response when an absorbing inhomogeneity with volume V i = 2250 mm3 is in (x = y = 0, z = 20 mm). μ s0′ = μ si ′ = 0.41 mm-1, μ a0 = 2.8 × 10-3 mm-1, d = 40 mm, n = 1.33. An example of comparison for the perturbation δT a (t) obtained from the experiment and from the model [Eqs. (4) and (8)] is also shown for μ ai = 0.027 mm-1.

Fig. 10
Fig. 10

Comparison between perturbation model (solid curves) and MC results (dashed curves) for the time-resolved internal mean path length. The results are reported for a spherical volume with r i = 5 mm in different locations inside the slab. μ s0′ = μ si ′ = 0.5 mm-1, μ a0 = μ ai = 0, d = 40 mm, n = 1.

Fig. 11
Fig. 11

Results for the internal mean path length 〈l int〉 for a slab with μ s0′ = μ si ′ = 0.39 mm-1, μ a0 = μ ai = 0.0003 mm-1, d = 40 mm, n = 1. The results refer to a volume of V = 660 mm3 and are plotted versus the coordinates of the center of the volume. The experimental results are compared with the results obtained from the model [Eq. (7)] and from MC simulations.

Fig. 12
Fig. 12

Time-resolved mean path length followed by received photons inside a spherical volume with r i = 5 mm in (x = y = 0, z = 20 mm). μ s0′ = 0.5 mm-1, μ a0 = μ ai = 0, d = 40 mm, n = 1. Each curve refers to a different value of μ si ′.

Fig. 13
Fig. 13

Dependence of the contrast and of the internal mean path length 〈l int〉 on the volume of the inhomogeneity. The results obtained from the models assuming that the integral is proportional to the volume of the inhomogeneity were also reported (curves labeled as model without integration). The inhomogeneity is in (x = y = 0, z = 20 mm), μ s0′ = 0.5 mm-1, μ a0 = 0, d = 40 mm.

Equations (15)

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Tpertt=Thomt+δTat+δTDt,
Thomt=θtexp-μa0vt-ρ24D0vt24πD0v3/2t5/2×n=-+zn+ exp-zn+24D0vt-zn- exp-zn-24D0vt.
δTDt=-δD v24πD0v5/2θtt5/2×exp-μa0vtm=-+η=-+Vid3r2×z23,η+2ρ12,m+·2ρ23,η+qρ12,m+, ρ23,η+, t-2ρ12,m-·2ρ23,η+qρ12,m-, ρ23,η+, t-z23,η-2ρ12,m+·2ρ23,η-qρ12,m+, ρ23,η-, t-2ρ12,m-·2ρ23,η-qρ12,m-, ρ23,η-, t+z12,m+pρ12,m+, ρ23,η+, t+pρ12,m+, ρ23,η-, t-z12,m-pρ12,m-, ρ23,η+, t+pρ12,m-, ρ23,η-, t,
δTat=-δμav4π4πD0v3/2θtt3/2×exp-μa0vtm=-+η=-+Vid3r2×z23,η+ρ23,η+fρ12,m+, ρ23,η+, t-fρ12,m-, ρ23,η+, t-z23,η-ρ23,η-fρ12,m+, ρ23,η-, t-fρ12,m-, ρ23,η-, t,
z12,m+=z2-2md+2ze-z0, z12,m-=z2-2md+2ze+2ze+z0, ρ12,m+=x2-x02+y2-y02+z12,m+21/2, ρ12,m-=x2-x02+y2-y02+z12,m-21/2, 2ρ12,m+=1ρ12,m+x2-x0, y2-y0, z12,m+, 2ρ12,m-=1ρ12,m-x2-x0, y2-y0, z12,m-, z23,η+=1-2ηd-4ηze-z2, z23,η-=1-2ηd-4η-2ze+z2, ρ23,η+=x3-x22+y3-y22+z23,η+21/2, ρ23,η-=x3-x22+y3-y22+z23,η-21/2, 2ρ23,η+=1ρ23,η+x2-x3, y2-y3, -z23,η+, 2ρ23,η-=1ρ23,η-x2-x3, y2-y3, +z23,η-, zn+=1-2nd-4nze-z0, zn-=1-2nd-4n-2ze+z0, ρ=x3-x02+y3-y021/2, ze=2AD0.
fx, y, t=1y2+x+y2xy12D0vtexp-x+y24D0vt, px, y, t=1x+1y3xy2D0vt+1x3+1y3×exp-x+y24D0vt, qx, y, t=3xy4+x+y23x2-2xy+y2x2y312D0vt+x+y4xy212D0vt2exp-x+y24D0vt.
Tδμa, t=Thomμa0, t0 gμa0, t, ti×exp-δμavtidti=Thomμa0, t0+ gδμa=0, t, ti×n=0-1nδμann!vtindti=Thomμa0, t1+n=1-1nδμann!×lintnδμa=0, t,
lintδμa=0, t=0 vtigδμa=0, t, tidti0 gδμa=0, t, tidti=-μailnTδμa, tThomμa0, tδμa=0=-1δμaδTatThomμa0, t=1Thomμa0, tv4π4πD0v3/2θtt3/2×exp-μa0vtm=-+η=-+×Vid3r2z23,η+ρ23,η+fρ12,m+, ρ23,η+, t-fρ12,m-, ρ23,η+, t-z23,η-ρ23,η-fρ12,m+, ρ23,η-, t-fρ12,m-, ρ23,η-, t.
lintδμa=0=1Thomμa014π2D0m=-+η=-+Vid3r2×z23,η+1+σρ23,η+ρ23,η+3exp-σρ12,m++ρ23,η+ρ12,m+-exp-σρ12,m-+ρ23,η+ρ12,m--z23,η-1+σρ23,η-ρ23,η-3×exp-σρ12,m++ρ23,η-ρ12,m+-exp-σρ12,m-+ρ23,η-ρ12,m-,
δTat=-Thomtδμalintδμa=0, t×exp-δμalintδμa=0, t.
Thomω=14πn=-+zn+1+σρn+ρn+3exp-σρn+-zn-1+σρn-ρn-3exp-σρn-,
δTDω=-δD 14π2D0m=-+η=-+Vid3r2×z12,m+hρ12,m+, ρ23,η+, ω+hρ12,m+, ρ23,η-, ω-z12,m-hρ12,m-, ρ23,η+, ω+hρ12,m-, ρ23,η-, ω+z23,η+2ρ12,m+·2ρ23,η+wρ12,m+, ρ23,η+, ω-2ρ12,m-·2ρ23,η+wρ12,m-, ρ23,η+, ω-z23,η-2ρ12,m+·2ρ23,η-wρ12,m+, ρ23,η-, ω-2ρ12,m-·2ρ23,η-wρ12,m-, ρ23,η-, ω,
δTaω=-δμa14π2D0m=-+η=-+Vid3r2×z23,η+1+σρ23,η+ρ23,η+3exp-σρ12,m++ρ23,η+ρ12,m+-exp-σρ12,m-+ρ23,η+ρ12,m--z23,η-1+σρ23,η-ρ23,η-3exp-σρ12,m++ρ23,η-ρ12,m+-exp-σρ12,m-+ρ23,η-ρ12,m-,
σ=μa0v+iωD0v1/2, ρn+=ρ2+zn+21/2, ρn-=ρ2+zn-21/2, wx, y, ω=1+σxx23+3σy+σ2y2y4×exp-σx+y, hx, y, ω=1+σxx31+σyy3exp-σx+y,
zn+=2nd+4nze+z0, zn-=2nd+4n-2ze-z0, z23,η+=2ηd+4ηze+z2, z23,η-=2ηd+4η-2ze-z2, 2ρ23,η+=1ρ23,η+x2-x3, y2-y3, +z23,η+, 2ρ23,η-=1ρ23,η-x2-x3, y2-y3, -z23,η-.

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