Abstract

Most analytical methods for describing light propagation in turbid medium exhibit low effectiveness in the near-field of a collimated source. Motivated by the Charge Simulation Method in electromagnetic theory as well as the established discrete source based modeling, we herein report on an improved explicit model for a semi-infinite geometry, referred to as “Virtual Source” (VS) diffuse approximation (DA), to fit for low-albedo medium and short source-detector separation. In this model, the collimated light in the standard DA is analogously approximated as multiple isotropic point sources (VS) distributed along the incident direction. For performance enhancement, a fitting procedure between the calculated and realistic reflectances is adopted in the near-field to optimize the VS parameters (intensities and locations). To be practically applicable, an explicit 2VS-DA model is established based on close-form derivations of the VS parameters for the typical ranges of the optical parameters. This parameterized scheme is proved to inherit the mathematical simplicity of the DA approximation while considerably extending its validity in modeling the near-field photon migration in low-albedo medium. The superiority of the proposed VS-DA method to the established ones is demonstrated in comparison with Monte-Carlo simulations over wide ranges of the source-detector separation and the medium optical properties.

© 2015 Optical Society of America

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References

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  1. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).
  2. A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
    [Crossref]
  3. L.-M. Zhang, J. Li, X. Yi, H.-J. Zhao, and F. Gao, “Analytical solutions to the simplified spherical harmonics equations using eigen decompositions,” Opt. Lett. 38(24), 5462–5465 (2013).
    [Crossref] [PubMed]
  4. W. M. Star, J. P. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33(4), 437–454 (1988).
    [Crossref] [PubMed]
  5. E. L. Hull and T. H. Foster, “Steady-State Reflectance Spectroscopy in the P3 Approximation,” J. Opt. Soc. Am. A 18(3), 584–599 (2001).
    [Crossref]
  6. E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
    [Crossref] [PubMed]
  7. R. J. Zemp, “Phase-function corrected diffusion model for diffuse reflectance of a pencil beam obliquely incident on a semi-infinite turbid medium,” J. Biomed. Opt. 18(6), 067005 (2013).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  10. T. Spott and L. O. Svaasand, “Collimated light sources in the diffusion approximation,” Appl. Opt. 39(34), 6453–6465 (2000).
    [Crossref] [PubMed]
  11. J. B. Domínguez and Y. Bérubé-Lauzière, “Light propagation from fluorescent probes in biological tissues by coupled time-dependent parabolic simplified spherical harmonics equations,” Biomed. Opt. Express 2(4), 817–837 (2011).
    [PubMed]
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    [Crossref]
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    [Crossref]
  15. K. Amano, “A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains,” J. Comput. Appl. Math. 53(3), 353–370 (1994).
    [Crossref]
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    [Crossref] [PubMed]
  17. L. I.-K. Lin, “A concordance correlation coefficient to evaluate reproducibility,” Biometrics 45(1), 255–268 (1989).
    [Crossref] [PubMed]
  18. S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58(11), R37–R61 (2013).
    [Crossref] [PubMed]
  19. T. Lister, P. A. Wright, and P. H. Chappell, “Optical properties of human skin,” J. Biomed. Opt. 17(9), 0909011 (2012).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  21. E. M. Hillman, D. A. Boas, A. M. Dale, and A. K. Dunn, “Laminar optical tomography: demonstration of millimeter-scale depth-resolved imaging in turbid media,” Opt. Lett. 29(14), 1650–1652 (2004).
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2013 (3)

L.-M. Zhang, J. Li, X. Yi, H.-J. Zhao, and F. Gao, “Analytical solutions to the simplified spherical harmonics equations using eigen decompositions,” Opt. Lett. 38(24), 5462–5465 (2013).
[Crossref] [PubMed]

R. J. Zemp, “Phase-function corrected diffusion model for diffuse reflectance of a pencil beam obliquely incident on a semi-infinite turbid medium,” J. Biomed. Opt. 18(6), 067005 (2013).
[Crossref] [PubMed]

S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58(11), R37–R61 (2013).
[Crossref] [PubMed]

2012 (1)

T. Lister, P. A. Wright, and P. H. Chappell, “Optical properties of human skin,” J. Biomed. Opt. 17(9), 0909011 (2012).
[Crossref] [PubMed]

2011 (2)

J. B. Domínguez and Y. Bérubé-Lauzière, “Light propagation from fluorescent probes in biological tissues by coupled time-dependent parabolic simplified spherical harmonics equations,” Biomed. Opt. Express 2(4), 817–837 (2011).
[PubMed]

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
[Crossref] [PubMed]

2005 (1)

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[Crossref]

2004 (1)

2001 (1)

2000 (1)

1997 (1)

1995 (2)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref] [PubMed]

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref] [PubMed]

1994 (2)

K. Amano, “A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains,” J. Comput. Appl. Math. 53(3), 353–370 (1994).
[Crossref]

R. C. Haskell, L. O. Svaasand, T. T. Tsay, T. C. Feng, M. S. McAdams, and B. J. Tromberg, “Boundary conditions for the diffusion equation in radiative transfer,” J. Opt. Soc. Am. A 11(10), 2727–2741 (1994).
[Crossref] [PubMed]

1992 (1)

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref] [PubMed]

1989 (2)

L. I.-K. Lin, “A concordance correlation coefficient to evaluate reproducibility,” Biometrics 45(1), 255–268 (1989).
[Crossref] [PubMed]

N. H. Malik, “A review of the charge simulation method and its applications,” IEEE Trans. Electr. Insul. 24(1), 3–20 (1989).
[Crossref]

1988 (1)

W. M. Star, J. P. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33(4), 437–454 (1988).
[Crossref] [PubMed]

1978 (1)

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D 11(10), 1463–1479 (1978).
[Crossref]

Amano, K.

K. Amano, “A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains,” J. Comput. Appl. Math. 53(3), 353–370 (1994).
[Crossref]

Arridge, S. R.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref] [PubMed]

Bérubé-Lauzière, Y.

Boas, D. A.

Chappell, P. H.

T. Lister, P. A. Wright, and P. H. Chappell, “Optical properties of human skin,” J. Biomed. Opt. 17(9), 0909011 (2012).
[Crossref] [PubMed]

Dale, A. M.

Delpy, D. T.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref] [PubMed]

Domínguez, J. B.

Dunn, A. K.

Eason, G.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D 11(10), 1463–1479 (1978).
[Crossref]

Fantini, S.

Farrell, T. J.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref] [PubMed]

Feng, T. C.

Foster, T. H.

Franceschini, M. A.

Gao, F.

Gratton, E.

Guo, L.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
[Crossref] [PubMed]

Hanlon, E. B.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
[Crossref] [PubMed]

Haskell, R. C.

Hielscher, A. H.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[Crossref]

Hillman, E. M.

Hiraoka, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref] [PubMed]

Hull, E. L.

Itzkan, I.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
[Crossref] [PubMed]

Jacques, S. L.

S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58(11), R37–R61 (2013).
[Crossref] [PubMed]

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref] [PubMed]

Klose, A. D.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[Crossref]

Li, J.

Lin, L. I.-K.

L. I.-K. Lin, “A concordance correlation coefficient to evaluate reproducibility,” Biometrics 45(1), 255–268 (1989).
[Crossref] [PubMed]

Lister, T.

T. Lister, P. A. Wright, and P. H. Chappell, “Optical properties of human skin,” J. Biomed. Opt. 17(9), 0909011 (2012).
[Crossref] [PubMed]

Malik, N. H.

N. H. Malik, “A review of the charge simulation method and its applications,” IEEE Trans. Electr. Insul. 24(1), 3–20 (1989).
[Crossref]

Marijnissen, J. P.

W. M. Star, J. P. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33(4), 437–454 (1988).
[Crossref] [PubMed]

McAdams, M. S.

Nisbet, R. M.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D 11(10), 1463–1479 (1978).
[Crossref]

Ntziachristos, V.

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[Crossref]

Patterson, M. S.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref] [PubMed]

Perelman, L. T.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
[Crossref] [PubMed]

Qiu, L.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
[Crossref] [PubMed]

Schweiger, M.

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref] [PubMed]

Spott, T.

Star, W. M.

W. M. Star, J. P. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33(4), 437–454 (1988).
[Crossref] [PubMed]

Svaasand, L. O.

Tromberg, B. J.

Tsay, T. T.

Turnbull, F. W.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D 11(10), 1463–1479 (1978).
[Crossref]

Turzhitsky, V.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
[Crossref] [PubMed]

van Gemert, M. J. C.

W. M. Star, J. P. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33(4), 437–454 (1988).
[Crossref] [PubMed]

Veitch, A. R.

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D 11(10), 1463–1479 (1978).
[Crossref]

Vitkin, E.

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
[Crossref] [PubMed]

Wang, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref] [PubMed]

Wilson, B.

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref] [PubMed]

Wright, P. A.

T. Lister, P. A. Wright, and P. H. Chappell, “Optical properties of human skin,” J. Biomed. Opt. 17(9), 0909011 (2012).
[Crossref] [PubMed]

Yi, X.

Zemp, R. J.

R. J. Zemp, “Phase-function corrected diffusion model for diffuse reflectance of a pencil beam obliquely incident on a semi-infinite turbid medium,” J. Biomed. Opt. 18(6), 067005 (2013).
[Crossref] [PubMed]

Zhang, L.-M.

Zhao, H.-J.

Zheng, L.

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref] [PubMed]

Appl. Opt. (2)

Biomed. Opt. Express (1)

Biometrics (1)

L. I.-K. Lin, “A concordance correlation coefficient to evaluate reproducibility,” Biometrics 45(1), 255–268 (1989).
[Crossref] [PubMed]

Comput. Methods Programs Biomed. (1)

L. Wang, S. L. Jacques, and L. Zheng, “MCML--Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995).
[Crossref] [PubMed]

IEEE Trans. Electr. Insul. (1)

N. H. Malik, “A review of the charge simulation method and its applications,” IEEE Trans. Electr. Insul. 24(1), 3–20 (1989).
[Crossref]

J. Biomed. Opt. (2)

R. J. Zemp, “Phase-function corrected diffusion model for diffuse reflectance of a pencil beam obliquely incident on a semi-infinite turbid medium,” J. Biomed. Opt. 18(6), 067005 (2013).
[Crossref] [PubMed]

T. Lister, P. A. Wright, and P. H. Chappell, “Optical properties of human skin,” J. Biomed. Opt. 17(9), 0909011 (2012).
[Crossref] [PubMed]

J. Comput. Appl. Math. (1)

K. Amano, “A charge simulation method for the numerical conformal mapping of interior, exterior and doubly-connected domains,” J. Comput. Appl. Math. 53(3), 353–370 (1994).
[Crossref]

J. Comput. Phys. (1)

A. D. Klose, V. Ntziachristos, and A. H. Hielscher, “The inverse source problem based on the radiative transfer equation in optical molecular imaging,” J. Comput. Phys. 202(1), 323–345 (2005).
[Crossref]

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

J. Phys. D (1)

G. Eason, A. R. Veitch, R. M. Nisbet, and F. W. Turnbull, “The theory of the back-scattering of light by blood,” J. Phys. D 11(10), 1463–1479 (1978).
[Crossref]

Med. Phys. (2)

M. Schweiger, S. R. Arridge, M. Hiraoka, and D. T. Delpy, “The finite element method for the propagation of light in scattering media: Boundary and source conditions,” Med. Phys. 22(11), 1779–1792 (1995).
[Crossref] [PubMed]

T. J. Farrell, M. S. Patterson, and B. Wilson, “A diffusion theory model of spatially resolved, steady-state diffuse reflectance for the noninvasive determination of tissue optical properties in vivo,” Med. Phys. 19(4), 879–888 (1992).
[Crossref] [PubMed]

Nat. Commun. (1)

E. Vitkin, V. Turzhitsky, L. Qiu, L. Guo, I. Itzkan, E. B. Hanlon, and L. T. Perelman, “Photon diffusion near the point-of-entry in anisotropically scattering turbid media,” Nat. Commun. 2, 587–605 (2011).
[Crossref] [PubMed]

Opt. Lett. (2)

Phys. Med. Biol. (2)

W. M. Star, J. P. Marijnissen, and M. J. C. van Gemert, “Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory,” Phys. Med. Biol. 33(4), 437–454 (1988).
[Crossref] [PubMed]

S. L. Jacques, “Optical properties of biological tissues: a review,” Phys. Med. Biol. 58(11), R37–R61 (2013).
[Crossref] [PubMed]

Other (1)

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, 1978).

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Figures (11)

Fig. 1
Fig. 1 Schematic of the VS approximation in semi-infinite geometry, where the multiple VSs, S 1 to S N , are adopted at depths z 1 to z N to replace the realistic collimated source. The banana-shaped areas with shoaling grayscales indicate the most probable paths of the photon transmission from the VSs to the detector D m (m = 1…M) but weaken effects of the VSs at increasing depths on the detection.
Fig. 2
Fig. 2 The fitted parameters (z2, k) of the 2VS-DA as functions of the optical properties: (a) 5 μ s 50 mm 1 with fixed μ a =0.01 mm 1 and g=0.9 ; (b) 0.01 μ a 0.4 mm 1 with fixed μ s =10 mm 1 and g=0.9 ; (c) 0.7g0.9 with fixed μ a =0.01 mm 1 and μ s =10 mm 1 .
Fig. 3
Fig. 3 The fitting errors of (a) k and (b) z2 in the 2VS-DA, calculated for 0.01 μ a 0.5 mm 1 , 5 μ s 50 mm 1 , and g=0.9
Fig. 4
Fig. 4 Reflectances R( ρ ) calculated with the 2VS-DA, Hybrid-DA-P3, PFC-DA, and δ-P1 in comparison with the MC (upper), as well as their relative errors ε R (lower), for three optical property sets: (a) μ a =0.01 mm 1 and α =0.99 ( μ s =1 mm 1 ) ; (b) μ a =0.3 mm 1 and α =0.9 ( μ s =2.7 mm 1 ) ; (c) μ a =0.3 mm 1 and α =0.769 ( μ s =1 mm 1 ) . The inset figures illustrate R( ρ ) in the nearly null SDS field of ρ/ l t <1 .
Fig. 5
Fig. 5 Relative error ε R of the 2VS-DA and PFC-DA for three optical property sets: (a) μ a =0.01 mm 1 and α =0.99 ; (b) μ a =0.3 mm 1 and α =0.9 ; (c) μ a =0.3 mm 1 and α =0.769 for refractive-index-matched boundary.
Fig. 6
Fig. 6 Relative errors ε Φ of (a) the 2VS-DA, (b) Hybrid-DA-P3, (c) PFC-DA, and (d) δ-P1, for three optical property sets: (Top) μ a =0.01 mm 1 and α =0.99 ; (Middle) μ a =0.3 mm 1 and α =0.9 ; (Bottom) μ a =0.3 mm 1 and α =0.769 , respectively.
Fig. 7
Fig. 7 Relative error ε R of the 2VS-DA for α = 0.769, 0.85, and 0.9, respectively, with the absorption coefficient fixed at μ a =0.3 mm 1 .
Fig. 8
Fig. 8 Relative errors ε R of the 2VS-DA and 3VS-DA for two optical property sets: (a) μ a =0.3 mm 1 and α =0.769 ; (b) μ a =0.01 mm 1 and α =0.99 , respectively.
Fig. 9
Fig. 9 Comparison between the relative deviations σ ^ R of the 2VS-DA and δ-P1 for assessing performance dependency on the anisotropy factor. The calculation is conducted for a range of anisotropy factors of 0.7g0.9 , and two optical property sets of (a) μ a =0.3 mm 1 and α =0.769 ; (b) μ a =0.01 mm 1 and α =0.99 , respectively.
Fig. 10
Fig. 10 Relative errors ε R ( ρ )/ ε R ( ρ ) of the 2VS-DA without (upper) and with (lower) compensation for detector NAs of 0.05, 0.12, 0.22, and 0.37, and for two combinations of the optical properties: (a) μ a =0.3 mm 1 and α =0.769 ; (b) μ a =0.01 mm 1 and α =0.99 , respectively.
Fig. 11
Fig. 11 Relative errors ε R of the 2VS-DA, 2VS-Hybrid-DA-P3 and 2VS-δ-P1, for two optical property sets: (a) μ a =0.01 mm 1 and α =0.99 ; (b) μ a =0.3 mm 1 and α =0.9 ; (c) μ a =0.3 mm 1 and α =0.769 , respectively, with the anisotropic factor fixed at g = 0.9.

Tables (3)

Tables Icon

Table 1 The VS parameters in the 2VS-DA model fitted for different combinations of μ a and μ s

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Table 2 Concordance Correlation Coefficients of the four models with the MC at 0<ρ / l t 1 and 1<ρ / l t 5 , and for the three different combinations of μ a and α , respectively.

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Table 3 Fitting results of β 1,2 with four typical NAs for two optical property sets.

Equations (14)

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D 2 Φ(r) μ a Φ(r)= μ s * Q(r, z ^ ),
J(r)=D[Φ(r) 3g μ s * 1+g Q(r, z ^ ) z ^ ],
R( ρ )= 1+ R eff 2( 1 R eff ) Φ( ρ,z=0 ),
Φ(ρ,z)= 1 4πD ( exp{ μ eff ( z z ) 2 + ρ 2 } ( z z ) 2 + ρ 2 + exp{ μ eff ( z+ z ) 2 + ρ 2 } ( z+ z ) 2 + ρ 2 2 l s 0 exp( l l s ) exp{ μ eff ( z+ z +l ) 2 + ρ 2 } ( z+ z +l ) 2 + ρ 2 dl ),
R VS ( ρ )= n=1 N I n R n ( ρ ) ,
R VS ( ρ ) R REF ( ρ ),
{ min { I, z } R REF R VS ( I, z ) Λ 2 st. z n 0; n=1 N I n = I total . ,
{ R REF R VS ( I i , z i )= J I ( I i , z i )δ I i + J z ( I i , z i )δ z i I i+1 = I i +δ I i ; z i+1 = z i +δ z i ,
r c ( R Model , R REF )= 2P( R Model , R REF ) [ E( R Model )E( R REF ) ] 2 σ( R Model )σ( R REF ) + σ( R Model ) σ( R REF ) + σ( R REF ) σ( R Model ) ,
ξ=( A 1 g 2 + A 2 g+ A 3 ) A 4 μ a + A 5 A 6 μ s + A 7 ,ξ{ z 2 ,k }.
{ A 17 ( z 2 ) ={ 0.703,1.399,0.183,0.548,1.293,0.003,0.183 } A 17 ( k ) ={ 0.121,0.252,0.846,0.847,0.313,0.522,0.136 } .
R(ρ)= I total 1+k 1+ R eff 2( 1 R eff ) [ Φ( ρ, z 1 )+kΦ( ρ, z 2 ) ],
R VS ( NA ) ( ρ )= n=1 2 β n I n R n ( ρ ) ,
min β m=1 M [ R MC ( NA ) ( ρ m ) R VS ( NA ) ( ρ m ,β ) R MC ( NA ) ( ρ m ) ] 2

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