Abstract

Accurate numerical simulations based on rigorous radiative transfer theory are used to assess the validity of the diffusion approximation that is frequently used in bio-optical imaging. These simulations show that the error is large for a non-index-matched boundary between air and tissue. This weakness of the diffusion approximation underscores the need to understand how diffusion theory can be used to extract accurate values of tissue optical properties. A validity criterion for the diffusion approximation is established on the basis of the single-scattering albedo a and the asymmetry factor g for a slab with index-matched boundaries.

© 2001 Optical Society of America

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2000 (2)

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon density waves,” Phys. Rev. E 61, 4295–4309 (2000).
[CrossRef]

T. H. Pham, T. Spott, L. O. Svaasand, B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt. 39, 4733–4745 (2000).
[CrossRef]

1999 (5)

T. Durduran, J. P. Culver, M. J. Holboke, X. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Express 4, 247–262 (1999), http://www.opticsexpress.org .
[CrossRef] [PubMed]

A. M. Siegel, J. J. A. Marota, D. A. Boas, “Design and evaluation of a continuous-wave diffuse optical tomography system,” Opt. Express 4, 287–298 (1999), http://www.opticsexpress.org .
[CrossRef] [PubMed]

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, G. Mueller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1999).
[CrossRef] [PubMed]

R. Aronson, N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999).
[CrossRef]

1998 (5)

B. Chen, J. J. Stamnes, K. Stamnes, “Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium,” Pure Appl. Opt. 7, 1161–1180 (1998).
[CrossRef]

D. J. Durian, “The diffusion coefficient depends on absorption,” Opt. Lett. 23, 1502–1504 (1998).
[CrossRef]

X. Cheng, D. A. Boas, “Diffuse optical reflection tomography with continuous-wave illumination,” Opt. Express 3, 118–123 (1998), http://www.opticsexpress.org .
[CrossRef] [PubMed]

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

A. Kim, A. Ishimaru, “Optical diffusion of continuous-wave, pulsed, and density waves in scattering media and comparisons with radiative transfer,” Appl. Opt. 37, 5313–5319 (1998).
[CrossRef]

1997 (5)

1995 (3)

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 3, 34–40 (1995).
[CrossRef]

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “Influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

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

1994 (3)

1991 (1)

K. Stamnes, Ø. Lie-Svendsen, M. H. Rees, “The linear Boltzmann equation in slab geometry: development and verification of a reliable and efficient solution,” Planet. Space Sci. 39, 1435–1463 (1991).
[CrossRef]

1990 (2)

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

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

1989 (3)

1978 (1)

1941 (1)

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

Alcouffe, R. E.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Alfano, R. R.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Aronson, R.

R. Aronson, N. Corngold, “Photon diffusion coefficient in an absorbing medium,” J. Opt. Soc. Am. A 16, 1066–1071 (1999).
[CrossRef]

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

R. Aronson, “Diffusion boundary conditions for photon waves,” in Optical Tomography and Spectroscopy of Tissue: Theory, Instrumentation, Model, and Human Studies II, B. Chance, R. R. Alfano, eds., Proc. SPIE2979, 651–657 (1997).

Barbour, R. L.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

Bassani, M.

Berns, M. W.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Boas, D. A.

Chance, B.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon density waves,” Phys. Rev. E 61, 4295–4309 (2000).
[CrossRef]

T. Durduran, J. P. Culver, M. J. Holboke, X. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Express 4, 247–262 (1999), http://www.opticsexpress.org .
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 3, 34–40 (1995).
[CrossRef]

M. S. Patterson, B. Chance, B. C. Wilson, “Time-resolved reflectance and transmittance for noninvasive measurement of tissue optical properties,” Appl. Opt. 28, 2331–2336 (1989).
[CrossRef] [PubMed]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–63.

Chandrasekhar, S.

S. Chandrasekhar, Radiative Transfer (Dover, New York, 1960).

Chen, B.

B. Chen, J. J. Stamnes, K. Stamnes, “Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium,” Pure Appl. Opt. 7, 1161–1180 (1998).
[CrossRef]

Cheng, X.

Cheong, W.-F.

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

Contini, D.

Corngold, N.

Culver, J. P.

Dorschel, K.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, G. Mueller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1999).
[CrossRef] [PubMed]

Durduran, T.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon density waves,” Phys. Rev. E 61, 4295–4309 (2000).
[CrossRef]

T. Durduran, J. P. Culver, M. J. Holboke, X. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Express 4, 247–262 (1999), http://www.opticsexpress.org .
[CrossRef] [PubMed]

T. Durduran, A. G. Yodh, B. Chance, D. A. Boas, “Does the photon-diffusion coefficient depend on absorption?” J. Opt. Soc. Am. A 14, 3358–3365 (1997).
[CrossRef]

T. Durduran, D. A. Boas, B. Chance, A. G. Yodh, “Validity of the diffusion equation for small heterogeneities,” in Advances in Optical Imaging and Photon Migration, R. R. Alfano, G. Fujimoto, eds., Vol. 2 of OSA Trends in Optics and Photonics Series (Optical Society of America, Washington, D.C., 1996), pp. 60–63.

Durian, D. J.

Feng, T.-C.

Fishkin, J. B.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Friebel, M.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, G. Mueller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1999).
[CrossRef] [PubMed]

Furutsu, K.

K. Furutsu, Y. Yamada, “Diffusion approximation for dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

K. Furutsu, “Boundary conditions of diffusion equation and applications,” Phys. Rev. A 39, 1386–1401 (1989).
[CrossRef] [PubMed]

Greenstein, J. L.

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

Hahn, A.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, G. Mueller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1999).
[CrossRef] [PubMed]

Haskell, R. C.

Henyey, L. C.

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

Hielscher, A. H.

A. H. Hielscher, R. E. Alcouffe, R. L. Barbour, “Comparison of finite-difference transport and diffusion calculations for photon migration in homogeneous and heterogeneous tissues,” Phys. Med. Biol. 43, 1285–1302 (1998).
[CrossRef] [PubMed]

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “Influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Holboke, M. J.

Ishimaru, A.

Jacques, S. L.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “Influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Jin, Z.

Kim, A.

Li, X.

Lie-Svendsen, Ø.

K. Stamnes, Ø. Lie-Svendsen, M. H. Rees, “The linear Boltzmann equation in slab geometry: development and verification of a reliable and efficient solution,” Planet. Space Sci. 39, 1435–1463 (1991).
[CrossRef]

Liu, F.

K. M. Yoo, F. Liu, R. R. Alfano, “When does the diffusion approximation fail to describe photon transport in random media?” Phys. Rev. Lett. 64, 2647–2650 (1990).
[CrossRef] [PubMed]

Marota, J. J. A.

Martelli, F.

McAdams, M. S.

Mueller, G.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, G. Mueller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1999).
[CrossRef] [PubMed]

O’Leary, M. A.

Pattanayak, D. N.

Patterson, M. S.

Pham, T.

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Pham, T. H.

Prahl, S. A.

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

Rees, M. H.

K. Stamnes, Ø. Lie-Svendsen, M. H. Rees, “The linear Boltzmann equation in slab geometry: development and verification of a reliable and efficient solution,” Planet. Space Sci. 39, 1435–1463 (1991).
[CrossRef]

Roggan, A.

A. Roggan, M. Friebel, K. Dorschel, A. Hahn, G. Mueller, “Optical properties of circulating human blood in the wavelength range 400–2500 nm,” J. Biomed. Opt. 4, 36–46 (1999).
[CrossRef] [PubMed]

Siegel, A. M.

Spott, T.

T. H. Pham, T. Spott, L. O. Svaasand, B. J. Tromberg, “Quantifying the properties of two-layer turbid media with frequency-domain diffuse reflectance,” Appl. Opt. 39, 4733–4745 (2000).
[CrossRef]

L. O. Svaasand, T. Spott, J. B. Fishkin, T. Pham, B. J. Tromberg, M. W. Berns, “Reflectance measurements of layered media with diffuse photon-density waves: a potential tool for evaluating deep burns and subcutaneous lesions,” Phys. Med. Biol. 44, 801–813 (1999).
[CrossRef] [PubMed]

Stamnes, J. J.

B. Chen, J. J. Stamnes, K. Stamnes, “Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium,” Pure Appl. Opt. 7, 1161–1180 (1998).
[CrossRef]

Stamnes, K.

B. Chen, J. J. Stamnes, K. Stamnes, “Reconstruction algorithm for diffraction tomography of diffuse photon density waves in a random medium,” Pure Appl. Opt. 7, 1161–1180 (1998).
[CrossRef]

Z. Jin, K. Stamnes, “Radiative transfer in nonuniformly refracting media such as the atmosphere–ocean system,” Appl. Opt. 33, 431–442 (1994).
[CrossRef] [PubMed]

K. Stamnes, Ø. Lie-Svendsen, M. H. Rees, “The linear Boltzmann equation in slab geometry: development and verification of a reliable and efficient solution,” Planet. Space Sci. 39, 1435–1463 (1991).
[CrossRef]

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999), Chap. 6.
[CrossRef]

Svaasand, L. O.

Thomas, G. E.

G. E. Thomas, K. Stamnes, Radiative Transfer in the Atmosphere and Ocean (Cambridge U. Press, Cambridge, UK, 1999), Chap. 6.
[CrossRef]

Tittel, F. K.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “Influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Tromberg, B. J.

Tsay, T.-T.

Wang, L.

A. H. Hielscher, S. L. Jacques, L. Wang, F. K. Tittel, “Influence of boundary conditions on the accuracy of diffusion theory in time-resolved reflectance spectroscopy of biological tissues,” Phys. Med. Biol. 40, 1957–1975 (1995).
[CrossRef] [PubMed]

Welch, A. J.

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

Wilson, B. C.

Yamada, Y.

K. Furutsu, Y. Yamada, “Diffusion approximation for dissipative random medium and the applications,” Phys. Rev. E 50, 3634–3640 (1994).
[CrossRef]

Yodh, A.

A. Yodh, B. Chance, “Spectroscopy and imaging with diffusing light,” Phys. Today 3, 34–40 (1995).
[CrossRef]

Yodh, A. G.

X. Li, D. N. Pattanayak, T. Durduran, J. P. Culver, B. Chance, A. G. Yodh, “Near-field diffraction tomography with diffuse photon density waves,” Phys. Rev. E 61, 4295–4309 (2000).
[CrossRef]

T. Durduran, J. P. Culver, M. J. Holboke, X. Li, L. Zubkov, B. Chance, D. N. Pattanayak, A. G. Yodh, “Algorithms for 3D localization and imaging using near-field diffraction tomography with diffuse light,” Opt. Express 4, 247–262 (1999), http://www.opticsexpress.org .
[CrossRef] [PubMed]

D. A. Boas, M. A. O’Leary, B. Chance, A. G. Yodh, “Detection and characterization of optical inhomogeneities with diffuse photon density waves: a signal-to-noise analysis,” Appl. Opt. 36, 75–92 (1997).
[CrossRef] [PubMed]

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic diagram for the imaging of a layer of tissue. The radiation in the air distributed over 2π sr is confined to a cone of less than 2π sr in the tissue (region II). Radiation within region I in the tissue is totally reflected when striking the interface from below. (b) Schematic illustration of the quadrature adopted for the coupled air–tissue system. The quadrature angles θ i t = cos-1 i t ) (i = 1, … , N 1) in region II are obtained from those in the air [θ i a = cos-1 i a )] by use of Snell’s law. Additional quadrature angles θ i t = cos-1 i t ) (i = N 1 + 1, … , N 2) are added to represent radiation in region I.

Fig. 2
Fig. 2

Comparisons of accurate multistream computations, the two-stream approximation, and the δ two-stream approximation. α = 0.1 mm-1, σ = 10 mm-1, g = 0.8. The real part of the refractive index in the tissue relative to that in the air is unity, i.e., we adopted an index-matched boundary. (a) Reflectance, (b) transmittance.

Fig. 3
Fig. 3

Same as Fig. 2 but for a non-index-matched boundary for which the real part of the refractive index in the tissue relative to that in the air is 1.34.

Fig. 4
Fig. 4

Comparisons between accurate multistream computations and the δ two-stream approximation. α = 0 mm-1, σ = 10 mm-1, z = 5 mm (i.e., τ = 50). The real part of the refractive index in the tissue relative to that in the air is unity (index-matched boundary). (a) Reflectance, (b) transmittance.

Fig. 5
Fig. 5

Same as Fig. 4 but for a non-index-matched boundary, for which the real part of the refractive index in the tissue relative to that in the air is 1.34.

Fig. 6
Fig. 6

Comparisons between accurate multistream computations and the δ two-stream approximation. σ = 10 mm-1, g = 0, τ = 30 (tissue optical thickness). The real part of the refractive index in the tissue relative to that in the air is unity (index-matched boundary). (a) Reflectance, (b) transmittance.

Fig. 7
Fig. 7

Same as Fig. 6 but for a nonindex-matched boundary for which the real part of the refractive index in the tissue relative to that in the air is 1.34.

Fig. 8
Fig. 8

Reflected flux versus asymmetry factor for several values of the single-scattering albedo. σ = 10 mm-1, τ = 30 (tissue optical thickness). The real part of the refractive index in the tissue relative to that in the air is unity (index-matched boundary). (a) Comparisons of accurate and approximate reflectances, (b) error in percentage for the curves in (a).

Tables (1)

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Table 1 Validity Criterion of the Diffusion Approximation

Equations (24)

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u dIτ,udτ=Iτ, u-Sτ, u,
Sτ, u=aτ2-11dupτ, u, u×Iτ, u+S*τ, u,
S*τ, u=aτF04πμ0aμ0t Ts-μ0a; mrel×pτ, -μ0t, uexp-τ/μ0t,
pτ, u, u=pHGτ, u, u=l=02M-1 glτ2l+1PluPlu,
μitdI+τ, μitdτ=I+τ, μit-a2j=1N2 wjtpμjt, μitI+τ, μjt-a2j=1N2 wjtp-μjt, μitI-τ, μjt-S*τ, μit,
-μitdI-τ, μitdτ=I-τ, μit-a2j=1N2 wjtpμjt, -μitI+τ, μjt-a2j=1N2 wjtp-μjt, -μitI-τ, μjt-S*τ, -μit.
μit=1-1-μia2/mrel21/2.
I-τ=0+, μia=0,  i=1,, N1,
I+τ=0+, μia=I+τ=0-, μitmrel2 Tsμit, mrel,i=1,, N1,
I-τ=0-, μit=I+τ=0-, μitρsμit, mrel,i=1,, N1,
I-τ=0-, μit=I+τ=0-, μit,i=N1+1,, N2.
d2Ydτ2=Γ2Y,
Γ2=1μ¯21-a1-3agμ¯2.
μ¯=μ21/2=01dμμ2Iτ, μ01dμIτ, μ1/2.
-κ2=31-aa1-g+γ1-a,
1/μ¯2=3a+γ1-a.
μ¯ dI+dτ=I+-a1-bI+-abI--S*+,
-μ¯ dI-dτ=I--a1-bI--abI+-S*-,
μ¯ dI+-I-dτ=1-aI++I-;
μ¯ dI++I-dτ=1-a+2abI+-I-.
d2I+-I-dτ2=1-a+2ab1-aμ¯2I+-I-.
d2I+-I-dτ2=1μ¯21-a1-3agμ¯2I+-I-.
d2I++I-dτ2=1μ¯21-a1-3agμ¯2I++I-.
d2Ydτ2=Γ2Y,

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