Abstract

We use fractal continuous random media to model visible and near-infrared light scattering by biological tissue and cell suspensions. The power law of the reduced scattering coefficient, the anisotropy factor of scattering, and the phase function are derived with good agreement with experimental results. Implications for spectroscopic tissue diagnosis are discussed.

© 2005 Optical Society of America

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2004 (2)

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

K. Blyschak, M. Simick, R. Jong, and L. Lilge, Med. Phys. 31, 1398 (2004).
[CrossRef] [PubMed]

2002 (1)

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, Biophys. J. 82, 2256 (2002).
[CrossRef] [PubMed]

2001 (4)

M. Moscoso, J. B. Keller, and G. Papanicolaou, J. Opt. Soc. Am. A 18, 948 (2001).
[CrossRef]

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

N. Ghosh, S. K. Mohanty, S. K. Majumder, and P. K. Gupta, Appl. Opt. 40, 176 (2001).
[CrossRef]

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

2000 (1)

R. K. Wang, J. Mod. Opt. 47, 103 (2000).
[CrossRef]

1998 (3)

1996 (1)

1989 (1)

Andersson-Engels, S.

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

Backman, V.

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, Biophys. J. 82, 2256 (2002).
[CrossRef] [PubMed]

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Badizadegan, K.

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, Biophys. J. 82, 2256 (2002).
[CrossRef] [PubMed]

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Bassi, A.

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

Berger, A. J.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

Bevilacqua, F.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

Blyschak, K.

K. Blyschak, M. Simick, R. Jong, and L. Lilge, Med. Phys. 31, 1398 (2004).
[CrossRef] [PubMed]

Boone, C. W.

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, Biophys. J. 82, 2256 (2002).
[CrossRef] [PubMed]

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Butter, J.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

Cerussi, A. E.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

Chikoidze, E.

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

Cubeddu, R.

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

Dasari, R. R.

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, Biophys. J. 82, 2256 (2002).
[CrossRef] [PubMed]

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Eick, A. A.

Einstein, A. J.

A. J. Einstein, H.-S. Wu, and J. Gil, Phys. Rev. Lett. 80, 397 (1998).
[CrossRef]

Feld, M. S.

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, Biophys. J. 82, 2256 (2002).
[CrossRef] [PubMed]

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Freyer, J. P.

Georgakoudi, I.

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Ghosh, N.

Gil, J.

A. J. Einstein, H.-S. Wu, and J. Gil, Phys. Rev. Lett. 80, 397 (1998).
[CrossRef]

Gopal, V.

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Gupta, P. K.

Gurjar, R.

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Hielscher, A. H.

Holcombe, R. F.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

Jacques, S. L.

Jakubowski, D.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

Johnson, T. M.

Jong, R.

K. Blyschak, M. Simick, R. Jong, and L. Lilge, Med. Phys. 31, 1398 (2004).
[CrossRef] [PubMed]

Kalashnikov, M.

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Keller, J. B.

Kumar, G.

Lilge, L.

K. Blyschak, M. Simick, R. Jong, and L. Lilge, Med. Phys. 31, 1398 (2004).
[CrossRef] [PubMed]

Majumder, S. K.

Mohanty, S. K.

Moscoso, M.

Mourant, J. R.

Mueller, M.

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Nishioka, N. S.

Papanicolaou, G.

Parsa, P.

Pifferi, A.

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

Schmitt, J. M.

Shah, N.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

Shen, D.

Simick, M.

K. Blyschak, M. Simick, R. Jong, and L. Lilge, Med. Phys. 31, 1398 (2004).
[CrossRef] [PubMed]

Swartling, J.

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

Taroni, P.

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

Torricelli, A.

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

Tromberg, B. J.

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

Wang, R. K.

R. K. Wang, J. Mod. Opt. 47, 103 (2000).
[CrossRef]

Wax, A.

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, Biophys. J. 82, 2256 (2002).
[CrossRef] [PubMed]

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

Wu, H.-S.

A. J. Einstein, H.-S. Wu, and J. Gil, Phys. Rev. Lett. 80, 397 (1998).
[CrossRef]

Yang, C.

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, Biophys. J. 82, 2256 (2002).
[CrossRef] [PubMed]

Acad. Radiol. (1)

A. E. Cerussi, A. J. Berger, F. Bevilacqua, N. Shah, D. Jakubowski, J. Butter, R. F. Holcombe, and B. J. Tromberg, Acad. Radiol. 8, 211 (2001).
[CrossRef] [PubMed]

Appl. Opt. (4)

Biophys. J. (1)

A. Wax, C. Yang, V. Backman, K. Badizadegan, C. W. Boone, R. R. Dasari, and M. S. Feld, Biophys. J. 82, 2256 (2002).
[CrossRef] [PubMed]

IEEE J. Sel. Top. Quantum Electron. (1)

V. Backman, V. Gopal, M. Kalashnikov, K. Badizadegan, R. Gurjar, A. Wax, I. Georgakoudi, M. Mueller, C. W. Boone, R. R. Dasari, and M. S. Feld, IEEE J. Sel. Top. Quantum Electron. 7, 887 (2001).
[CrossRef]

J. Biomed. Opt. (1)

A. Pifferi, J. Swartling, E. Chikoidze, A. Torricelli, P. Taroni, A. Bassi, S. Andersson-Engels, and R. Cubeddu, J. Biomed. Opt. 9, 1143 (2004).
[CrossRef] [PubMed]

J. Mod. Opt. (1)

R. K. Wang, J. Mod. Opt. 47, 103 (2000).
[CrossRef]

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

Med. Phys. (1)

K. Blyschak, M. Simick, R. Jong, and L. Lilge, Med. Phys. 31, 1398 (2004).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. Lett. (1)

A. J. Einstein, H.-S. Wu, and J. Gil, Phys. Rev. Lett. 80, 397 (1998).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

(a) Anisotropy factor of the fractal continuous random medium versus cutoff size parameter k l max . (b) Size parameter k l * of the component contributing most to light scattering when D f > 4 .

Fig. 2
Fig. 2

(a) Power spectrum of refractive index variations in mouse liver tissue fitted to the theoretical power spectrum, Eq. (4). The symbols represent data reported in Ref. [2], and the dashed curve is the theoretical fit. Fitting yields D f = 4.0 and l max = 2.3 μ m . (b) Wavelength dependence of the reduced scattering coefficient of rat liver tissue fitted to the power law λ 3 D f . The symbols represent data reported in Ref. [9], and the solid curve shows the fitted curve. Fitting yields D f = 3.94 and l max = 1.5 μ m .

Fig. 3
Fig. 3

Phase function of suspensions of rat embryo fibroblast cells (M1) and mitochondria fitted to theoretical phase function (7). The symbols are data reported in Ref. [3], and the solid curves are theoretical curves. Fitting yields D f = 3.86 and 4.58 for M1 cells and mitochondria, respectively.

Equations (9)

Equations on this page are rendered with MathJax. Learn more.

η ( a ) = η 0 a 3 D f ,
S ( θ ) [ cos θ 0 0 1 ] = k 3 π 2 R ̂ ( 2 k sin θ 2 ) [ cos θ 0 0 1 ] ,
R ( r ) = ε 2 0 l max exp ( r l ) η ( l ) d l = ε 2 η 0 l max 4 D f E 5 D f ( r l max ) ,
R ̂ ( k ) = 0 l max ε 2 l 3 π 2 ( 1 + k 2 l 2 ) 2 η ( l ) d l .
S ( θ ) 2 = 0 k l max ε 2 η 0 k D f 1 x 6 D f 2 π [ 1 + 2 ( 1 μ ) x 2 ] 2 d x ,
g = d Ω ( 1 + μ 2 ) μ S ( θ ) 2 2 k 2 d Ω ( 1 + μ 2 ) S ( θ ) 2 2 k 2 .
p ( θ ) = ( 1 + μ 2 ) S ( θ ) 2 2 k 2 5 D f 2 π sin ( 5 D f 2 π ) 1 + μ 2 2 [ 2 ( 1 μ ) ] ( 7 D f ) 2 ( 1 + μ 2 ) ( 1 μ ) ( 7 D f ) 2 .
μ s = d Ω ( 1 + μ 2 ) ( 1 μ ) S ( θ ) 2 2 k 2 ε 2 η 0 α λ 3 D f ,
α ( 2 π n 0 ) D f 3 ( 11 4 D f + D f 2 ) ( D f + 1 ) ( D f 1 ) ( D f 3 ) 5 D f 2 π 2 5 D f sin ( 5 D f 2 π )

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