Abstract

The scattering process induced in blood by a collimated laser beam is theoretically investigated. An individual red blood cell (RBC) has a scattering phase function strongly peaked in the forward direction. For far-field experiments, the small scattering volumes can be considered as “macroscopic particles” characterized by an effective scattering phase function. Using the single-cell phase function as “input data” the angular distribution of light scattered at small angles by the whole scattering volume, containing RBCs in suspension, is calculated analytically. The angular dispersion of the light scattered by blood can be approximately described by the same formula used to characterize the light scattered by a single cell but with an effective, hematocrit-dependent anisotropy parameter.

© 2006 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2005 (1)

A. Karlsson, J. He, J. Swartling, and S. Andersson-Engels, "Numerical simulations of light scattering by red blood cells," IEEE Trans. Biomed. Eng. 52, 13-18 (2005).
[CrossRef] [PubMed]

2004 (1)

I. Turcu, "Effective phase function for light scattered by disperse systems--the small-angle approximation," J. Opt. A: Pure Appl. Opt. 6, 537-543 (2004).
[CrossRef]

2002 (2)

A. A. Kokhanovsky, "Analytical solutions of multiple light scattering problems: a review," Meas. Sci. Technol. 13, 233-240 (2002).
[CrossRef]

S. T. Tsinopoulos, E. J. Sellountos, and D. Polyzos, "Light scattering by aggregated red blood cells," Appl. Opt. 41, 1408-1417 (2002).
[CrossRef] [PubMed]

2001 (1)

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, "A scattering phase function for blood with physiological haematocrit," Phys. Med. Biol. 46, N65-69 (2001).
[CrossRef] [PubMed]

1999 (4)

1998 (3)

1994 (1)

L.-H. Wang and S. L. Jacques, "Error estimation of measuring total interaction coefficient of turbid media using collimated light transmission," Phys. Med. Biol. 39, 2349-2354 (1994).
[CrossRef] [PubMed]

1993 (1)

1980 (1)

Anderson-Engels, A.

Andersson-Engels, S.

A. Karlsson, J. He, J. Swartling, and S. Andersson-Engels, "Numerical simulations of light scattering by red blood cells," IEEE Trans. Biomed. Eng. 52, 13-18 (2005).
[CrossRef] [PubMed]

Asholm, P.

Chenyshev, A. V.

De Mul, F.

Goldbach, T

A. N. Yaroslavsky, I. V. Yaroslavsky, T, Goldbach, and H-J. Schwarzmaier, "Influence of scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements," J. Biomed. Opt. 4, 47-53 (1999).
[CrossRef]

Hammer, M.

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, "A scattering phase function for blood with physiological haematocrit," Phys. Med. Biol. 46, N65-69 (2001).
[CrossRef] [PubMed]

M. Hammer, D. Schweitzer, B. Michel, E. Thamm, and A. Kolb, "Single scattering by red blood cells," Appl. Opt. 37, 7410-7418 (1998).
[CrossRef]

He, J.

A. Karlsson, J. He, J. Swartling, and S. Andersson-Engels, "Numerical simulations of light scattering by red blood cells," IEEE Trans. Biomed. Eng. 52, 13-18 (2005).
[CrossRef] [PubMed]

Heethaar, R. M.

Hoekstra, A. G.

Ishimaru, A.

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

Jacques, S. L.

L.-H. Wang and S. L. Jacques, "Error estimation of measuring total interaction coefficient of turbid media using collimated light transmission," Phys. Med. Biol. 39, 2349-2354 (1994).
[CrossRef] [PubMed]

S. L. Jacques and L. Wang, "Monte Carlo modeling of light transport in tissue," in Optical Thermal Response of Laser-Irradiated Tissue, A.Welch and M.J. C.van Gemert, eds. (Plenum, 1995), pp. 73-100.

Karlsson, A.

A. Karlsson, J. He, J. Swartling, and S. Andersson-Engels, "Numerical simulations of light scattering by red blood cells," IEEE Trans. Biomed. Eng. 52, 13-18 (2005).
[CrossRef] [PubMed]

A. M. K. Nilsson, P. Asholm, A. Karlsson, and A. Anderson-Engels, "T-matrix computations of light scattered by red blood cells," Appl. Opt. 37, 2735-2748 (1998).
[CrossRef]

Kim, J.

J. Kim and J. C. Lin, "Successive order scattering transport approximation for laser light propagation in whole blood medium," IEEE Trans. Biomed. Eng. 45, 505-510 (1998).
[CrossRef] [PubMed]

Kokhanovsky, A. A.

A. A. Kokhanovsky, "Analytical solutions of multiple light scattering problems: a review," Meas. Sci. Technol. 13, 233-240 (2002).
[CrossRef]

Kolb, A.

Kolkman, R.

Lin, J. C.

J. Kim and J. C. Lin, "Successive order scattering transport approximation for laser light propagation in whole blood medium," IEEE Trans. Biomed. Eng. 45, 505-510 (1998).
[CrossRef] [PubMed]

Maltsev, V. P.

McCormick, N. J.

Michel, B.

Nijhof, E. J.

Nilsson, A. M. K.

Polyzos, D.

Reynolds, L. O.

Schwarzmaier, H-J.

A. N. Yaroslavsky, I. V. Yaroslavsky, T, Goldbach, and H-J. Schwarzmaier, "Influence of scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements," J. Biomed. Opt. 4, 47-53 (1999).
[CrossRef]

Schweitzer, D.

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, "A scattering phase function for blood with physiological haematocrit," Phys. Med. Biol. 46, N65-69 (2001).
[CrossRef] [PubMed]

M. Hammer, D. Schweitzer, B. Michel, E. Thamm, and A. Kolb, "Single scattering by red blood cells," Appl. Opt. 37, 7410-7418 (1998).
[CrossRef]

Sellountos, E. J.

Shvalov, A. N.

Soini, E.

Soini, J. T.

Steenbergen, W.

Streekstra, G. J.

Swartling, J.

A. Karlsson, J. He, J. Swartling, and S. Andersson-Engels, "Numerical simulations of light scattering by red blood cells," IEEE Trans. Biomed. Eng. 52, 13-18 (2005).
[CrossRef] [PubMed]

Taflove, A.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

Tarasov, P. A.

Thamm, E.

Tsinopoulos, S. T.

Turcu, I.

I. Turcu, "Effective phase function for light scattered by disperse systems--the small-angle approximation," J. Opt. A: Pure Appl. Opt. 6, 537-543 (2004).
[CrossRef]

Wang, L.

S. L. Jacques and L. Wang, "Monte Carlo modeling of light transport in tissue," in Optical Thermal Response of Laser-Irradiated Tissue, A.Welch and M.J. C.van Gemert, eds. (Plenum, 1995), pp. 73-100.

Wang, L.-H.

L.-H. Wang and S. L. Jacques, "Error estimation of measuring total interaction coefficient of turbid media using collimated light transmission," Phys. Med. Biol. 39, 2349-2354 (1994).
[CrossRef] [PubMed]

Yaroslavsky, A. N.

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, "A scattering phase function for blood with physiological haematocrit," Phys. Med. Biol. 46, N65-69 (2001).
[CrossRef] [PubMed]

A. N. Yaroslavsky, I. V. Yaroslavsky, T, Goldbach, and H-J. Schwarzmaier, "Influence of scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements," J. Biomed. Opt. 4, 47-53 (1999).
[CrossRef]

Yaroslavsky, I. V.

A. N. Yaroslavsky, I. V. Yaroslavsky, T, Goldbach, and H-J. Schwarzmaier, "Influence of scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements," J. Biomed. Opt. 4, 47-53 (1999).
[CrossRef]

Appl. Opt. (6)

IEEE Trans. Biomed. Eng. (2)

A. Karlsson, J. He, J. Swartling, and S. Andersson-Engels, "Numerical simulations of light scattering by red blood cells," IEEE Trans. Biomed. Eng. 52, 13-18 (2005).
[CrossRef] [PubMed]

J. Kim and J. C. Lin, "Successive order scattering transport approximation for laser light propagation in whole blood medium," IEEE Trans. Biomed. Eng. 45, 505-510 (1998).
[CrossRef] [PubMed]

J. Biomed. Opt. (1)

A. N. Yaroslavsky, I. V. Yaroslavsky, T, Goldbach, and H-J. Schwarzmaier, "Influence of scattering phase function approximation on the optical properties of blood determined from the integrating sphere measurements," J. Biomed. Opt. 4, 47-53 (1999).
[CrossRef]

J. Opt. A: Pure Appl. Opt. (1)

I. Turcu, "Effective phase function for light scattered by disperse systems--the small-angle approximation," J. Opt. A: Pure Appl. Opt. 6, 537-543 (2004).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Meas. Sci. Technol. (1)

A. A. Kokhanovsky, "Analytical solutions of multiple light scattering problems: a review," Meas. Sci. Technol. 13, 233-240 (2002).
[CrossRef]

Phys. Med. Biol. (2)

L.-H. Wang and S. L. Jacques, "Error estimation of measuring total interaction coefficient of turbid media using collimated light transmission," Phys. Med. Biol. 39, 2349-2354 (1994).
[CrossRef] [PubMed]

M. Hammer, A. N. Yaroslavsky, and D. Schweitzer, "A scattering phase function for blood with physiological haematocrit," Phys. Med. Biol. 46, N65-69 (2001).
[CrossRef] [PubMed]

Other (3)

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

S. L. Jacques and L. Wang, "Monte Carlo modeling of light transport in tissue," in Optical Thermal Response of Laser-Irradiated Tissue, A.Welch and M.J. C.van Gemert, eds. (Plenum, 1995), pp. 73-100.

A. Taflove, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 1995).

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Figures (4)

Fig. 1
Fig. 1

Asymptotic linear approximations for G ( τ ) .

Fig. 2
Fig. 2

Angular dispersion of the scattered photons. The effective phase function angular dependence shown for several values of the optical depth τ.

Fig. 3
Fig. 3

Effective phase function and the scattered flux dependency on optical depth, given at three different scattering angles.

Fig. 4
Fig. 4

Comparison between the functions ψ ( x ; τ ) and ψ ¯ ( x ; τ ) . The optical depth τ is considered as parameter.

Equations (54)

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ϕ ( τ ) = ϕ 0 ( τ ) + ϕ s ( τ ) , ϕ s ( τ ) = n = 1 ϕ n ( τ ) ,
d ϕ 0 ( τ ) d ( τ ) = ϕ 0 ( τ ) , ϕ 0 ( 0 ) = 1 ,
d ϕ n ( τ ) d ( τ ) = ϕ n ( τ ) + ϕ n 1 ( τ ) ,
ϕ n ( 0 ) = 0 , n = 1 , 2 , 3, ,
n = 1 ϕ n ( τ ) = 1.
ϕ 0 ( τ ) = e τ ,
ϕ n ( τ ) = τ n n ! e τ ,
ϕ s ( τ ) = n = 1 τ n n ! e τ = ( e τ 1 ) e τ = 1 e τ = 1 ϕ 0 ( τ ) .
f ( μ , φ ; μ , φ ) ,
1 1 d μ 0 2 π d φ f ( μ , φ ; μ , φ ) = 1 ,
f ( μ , φ ; μ , φ ) = f ( cos γ ) = 1 2 π l = 0 ( l + 1 / 2 ) f l P l ( cos γ ) ,
g = 1 1 d μ 0 2 π d φ cos γ f ( μ , φ ; μ , φ ) ,
cos γ = e · e
p ( μ , μ ) = 0 2 π f ( μ , φ ; μ , φ ) d φ = l = 0 ( l + 1 / 2 ) f l P l ( μ ) P l ( μ ) ,
g = 1 1 μ p ( μ , 1 ) d μ = f 1 .
ϕ ( τ , μ ) = ϕ 0 ( τ , μ ) + ϕ s ( τ , μ ) = n = 0 ϕ n ( τ , μ ) ,
ϕ 0 ( τ , μ ) = e τ δ ( μ 1 ) ,
ϕ s ( τ , μ ) = ( 1 e τ ) f e f f ( τ , μ ) .
f e f f ( τ , μ ) = l = 0 ( l + 1 / 2 ) f ˜ l ( τ ) P l ( μ ) , f ˜ l ( τ ) = e τ f l 1 e τ 1 .
ϕ n ( τ , μ ) = f n ( μ ) τ n n ! e τ .
f n ( μ ) = l = 0 ( l + 1 / 2 ) f l    n P l ( μ ) ,
f H G ( μ ) = l = 0 ( l + 1 / 2 ) g l P l ( μ ) = 1 2 1 g 2 ( 1 2 μ g + g 2 ) 3 / 2 .
f n ( μ ) = l = 0 ( l + 1 / 2 ) g n l P l ( μ ) = 1 2 1 g 2 n ( 1 2 μ g n + g 2 n ) 3 / 2 .
g n = 1 1 μ f n ( μ ) d μ = g n ,
f ˜ l ( τ ) = e τ g l 1 e τ 1 .
f ˜ l ( τ ) g l G ( τ ) , G ( τ ) 1.
f e f f ( τ ; μ ) 1 2 1 g 2 G ( τ ) [ 1 2 μ g G ( τ ) + g 2 G ( τ ) ] 3 / 2 .
G ( τ ) = ( τ 1 ) e τ + 1 e τ τ 1 ,
G ( τ ) { 1 + τ / 3 , τ < 1 τ 1 ,          τ > 5 ,
μ d ϕ ( τ , μ ) d τ = ϕ ( τ , μ ) + w 1 1 d μ p ( μ , μ ) ϕ ( τ , μ ) ,
d ϕ ( τ , μ ) d τ = ϕ ( τ , μ ) + 1 1 d μ p ( μ , μ ) ϕ ( τ , μ ) .
d ϕ ( τ , μ ) d τ = ϕ 0 ( τ , μ ) , ϕ 0 ( 0 , μ ) = δ ( μ 1 ) ,
d ϕ n ( τ , μ ) d τ = ϕ n ( τ , μ ) + 1 1 p ( μ , μ ) ϕ n 1 ( τ , μ ) d μ ,
ϕ n ( 0 , μ ) = 0 , n 1 ,
n = 0 1 1 ϕ n ( τ , μ ) d μ = 1 ,
ϕ 0 ( τ , μ ) = e τ δ ( μ 1 ) .
ϕ n ( τ , μ ) = f n ( μ ) F n ( τ ) .
F n ( τ ) = τ n n ! e τ , F n ( 0 ) = 0 , f n ( μ ) = 1 1 p ( μ , μ ) f n 1 ( μ ) d μ , f 0 ( μ ) = δ ( μ 1 ) .
f n ( μ ) = l = 0 ( l + 1 / 2 ) f l P l ( μ ) k = 0 ( k + 1 / 2 ) f n 1 , k 1 1 d μ P l ( μ ) P k ( μ )
= l = 0 ( l + 1 / 2 ) f l f n 1 , l P l ( μ )
f n ( μ ) = l = 0 ( l + 1 / 2 ) f n , l P l ( μ ) ,
1 1 d μ P l ( μ ) P k ( μ ) = δ l , k l + 1 / 2 .
f n , l = f l f n 1 , l = f l n .
f n ( μ ) = l = 0 ( l + 1 / 2 ) f l n P l ( μ ) .
δ ( μ μ ) = l = 0 ( l + 1 / 2 ) P l ( μ ) P l ( μ ) δ ( μ 1 ) = l = 0 ( l + 1 / 2 ) P l ( μ ) ,
f 0 ( μ ) = l = 0 ( l + 1 / 2 ) P l ( μ ) = δ ( μ 1 ) .
ϕ s ( τ , μ ) = e τ l = 0 ( l + 1 / 2 ) [ n = 1 1 n ! ( τ f 1 ) n ] P l ( μ ) = e τ l = 0 ( l + 1 / 2 ) ( e τ f 1 1 ) P l ( μ ) ,
ϕ s ( τ ) = 1 1 d μ ϕ s ( τ , μ ) = ( 1 e τ ) .
ϕ s ( τ , μ ) = ( 1 e τ ) f e f f ( τ , μ ) ,
f e f f ( τ , μ ) = 1 e τ 1 l = 0 ( l + 1 / 2 ) ( e τ f l 1 ) P l ( μ ) ,
1 1 f e f f ( τ , μ ) d μ = 1.
e τ g l 1 e τ 1 g l G ( τ ) ,
ψ ( x ; τ ) = e τ x 1 e τ 1 , ψ ¯ ( x ; τ ) = x G ( τ ) , 0 < x 1 ,
0 1 ψ ( x ; τ ) d x = 0 1 ψ ¯ ( x ; τ ) d x G ( τ ) = ( τ 1 ) e τ + 1 e τ τ 1 .

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