Abstract

Phase-contrast microscopy shows that the structure of the refractive-index inhomogeneities in a variety of mammalian tissues resembles that of frozen turbulence. Viewed over a range of scales, the spectrum of index variations exhibits a power-law behavior for spatial frequencies spanning at least a decade (0.5–5 μm−1) and has an outer scale in the range of 4–10 μm, above which correlations are no longer seen. The observed structure function fits the classical Kolmogorov model of turbulence. These observations are fundamental to understanding light propagation in tissue and may provide clues about how tissues develop and organize.

© 1996 Optical Society of America

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  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
    [CrossRef] [PubMed]
  2. J. M. Schmitt, M. J. Yadlowsky, R. F. Bonner, Dermatology 191, 93 (1995).
    [CrossRef] [PubMed]
  3. R. R. Alfano, A. Pradhan, G. C. Tang, J. Opt. Soc. Am. B 6, 1053 (1989).
  4. M. Kohl, M. Cope, M. Essenpreis, D. Böcker, Opt. Lett. 12, 2170 (1994).
    [CrossRef]
  5. V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).
  6. G. Kumar, J. M. Schmitt, Proc. SPIE 2679, 106 (1996).
    [CrossRef]
  7. C. J. R. Sheppard, Opt. Commun. 122, 178 (1996).
    [CrossRef]
  8. R. Barer, K. F. A. Ross, S. Tkaczyk, Nature (London) 171, 720 (1953).
    [CrossRef] [PubMed]
  9. B. Beauvoit, T. Kitai, B. Chance, Biophys. J. 67, 2501 (1994).
    [CrossRef] [PubMed]
  10. J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology (Oxford U. Press, New York, 1994), p. 263.

1996 (2)

G. Kumar, J. M. Schmitt, Proc. SPIE 2679, 106 (1996).
[CrossRef]

C. J. R. Sheppard, Opt. Commun. 122, 178 (1996).
[CrossRef]

1995 (1)

J. M. Schmitt, M. J. Yadlowsky, R. F. Bonner, Dermatology 191, 93 (1995).
[CrossRef] [PubMed]

1994 (2)

M. Kohl, M. Cope, M. Essenpreis, D. Böcker, Opt. Lett. 12, 2170 (1994).
[CrossRef]

B. Beauvoit, T. Kitai, B. Chance, Biophys. J. 67, 2501 (1994).
[CrossRef] [PubMed]

1991 (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

1989 (1)

R. R. Alfano, A. Pradhan, G. C. Tang, J. Opt. Soc. Am. B 6, 1053 (1989).

1953 (1)

R. Barer, K. F. A. Ross, S. Tkaczyk, Nature (London) 171, 720 (1953).
[CrossRef] [PubMed]

Alfano, R. R.

R. R. Alfano, A. Pradhan, G. C. Tang, J. Opt. Soc. Am. B 6, 1053 (1989).

Barer, R.

R. Barer, K. F. A. Ross, S. Tkaczyk, Nature (London) 171, 720 (1953).
[CrossRef] [PubMed]

Bassingthwaighte, J. B.

J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology (Oxford U. Press, New York, 1994), p. 263.

Beauvoit, B.

B. Beauvoit, T. Kitai, B. Chance, Biophys. J. 67, 2501 (1994).
[CrossRef] [PubMed]

Böcker, D.

M. Kohl, M. Cope, M. Essenpreis, D. Böcker, Opt. Lett. 12, 2170 (1994).
[CrossRef]

Bonner, R. F.

J. M. Schmitt, M. J. Yadlowsky, R. F. Bonner, Dermatology 191, 93 (1995).
[CrossRef] [PubMed]

Chance, B.

B. Beauvoit, T. Kitai, B. Chance, Biophys. J. 67, 2501 (1994).
[CrossRef] [PubMed]

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Cope, M.

M. Kohl, M. Cope, M. Essenpreis, D. Böcker, Opt. Lett. 12, 2170 (1994).
[CrossRef]

Essenpreis, M.

M. Kohl, M. Cope, M. Essenpreis, D. Böcker, Opt. Lett. 12, 2170 (1994).
[CrossRef]

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Fujimoto, J. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Kitai, T.

B. Beauvoit, T. Kitai, B. Chance, Biophys. J. 67, 2501 (1994).
[CrossRef] [PubMed]

Kohl, M.

M. Kohl, M. Cope, M. Essenpreis, D. Böcker, Opt. Lett. 12, 2170 (1994).
[CrossRef]

Kumar, G.

G. Kumar, J. M. Schmitt, Proc. SPIE 2679, 106 (1996).
[CrossRef]

Liebovitch, L. S.

J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology (Oxford U. Press, New York, 1994), p. 263.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Pradhan, A.

R. R. Alfano, A. Pradhan, G. C. Tang, J. Opt. Soc. Am. B 6, 1053 (1989).

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Ross, K. F. A.

R. Barer, K. F. A. Ross, S. Tkaczyk, Nature (London) 171, 720 (1953).
[CrossRef] [PubMed]

Schmitt, J. M.

G. Kumar, J. M. Schmitt, Proc. SPIE 2679, 106 (1996).
[CrossRef]

J. M. Schmitt, M. J. Yadlowsky, R. F. Bonner, Dermatology 191, 93 (1995).
[CrossRef] [PubMed]

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard, Opt. Commun. 122, 178 (1996).
[CrossRef]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Tang, G. C.

R. R. Alfano, A. Pradhan, G. C. Tang, J. Opt. Soc. Am. B 6, 1053 (1989).

Tatarskii, V. I.

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

Tkaczyk, S.

R. Barer, K. F. A. Ross, S. Tkaczyk, Nature (London) 171, 720 (1953).
[CrossRef] [PubMed]

West, B. J.

J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology (Oxford U. Press, New York, 1994), p. 263.

Yadlowsky, M. J.

J. M. Schmitt, M. J. Yadlowsky, R. F. Bonner, Dermatology 191, 93 (1995).
[CrossRef] [PubMed]

Biophys. J. (1)

B. Beauvoit, T. Kitai, B. Chance, Biophys. J. 67, 2501 (1994).
[CrossRef] [PubMed]

Dermatology (1)

J. M. Schmitt, M. J. Yadlowsky, R. F. Bonner, Dermatology 191, 93 (1995).
[CrossRef] [PubMed]

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

R. R. Alfano, A. Pradhan, G. C. Tang, J. Opt. Soc. Am. B 6, 1053 (1989).

Nature (1)

R. Barer, K. F. A. Ross, S. Tkaczyk, Nature (London) 171, 720 (1953).
[CrossRef] [PubMed]

Opt. Commun. (1)

C. J. R. Sheppard, Opt. Commun. 122, 178 (1996).
[CrossRef]

Opt. Lett. (1)

M. Kohl, M. Cope, M. Essenpreis, D. Böcker, Opt. Lett. 12, 2170 (1994).
[CrossRef]

Proc. SPIE (1)

G. Kumar, J. M. Schmitt, Proc. SPIE 2679, 106 (1996).
[CrossRef]

Science (1)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, J. G. Fujimoto, Science 254, 1178 (1991).
[CrossRef] [PubMed]

Other (2)

V. I. Tatarskii, Wave Propagation in a Turbulent Medium (McGraw-Hill, New York, 1961).

J. B. Bassingthwaighte, L. S. Liebovitch, B. J. West, Fractal Physiology (Oxford U. Press, New York, 1994), p. 263.

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Figures (3)

Fig. 1
Fig. 1

Phase-contrast images of mouse liver tissue acquired at two different magnifications (Figure reduced to 59%).

Fig. 2
Fig. 2

Power spectrum of index variations in mouse liver tissue. Values recorded from specimens viewed at magnifications of 50×, 100×, and 500× with a phase-contrast microscope have been combined. Only data from the spatial-frequency ranges in which the microscope has an approximately flat passband at each magnification have been used. We generate the fitted curve by using Eq. (1) with m and Lo as fitting parameters, and it is shown normalized to 0 (log10 1).

Fig. 3
Fig. 3

Wave-structure function calculated from Eq. (2), with C(ρ) computed by Fourier transformation of the fitted spectrum in Fig. 2. The limiting value of Dn for large values of ρ is twice the variance of the index fluctuations. For these specimens, 〈δn2〉 was estimated to be 0.004.

Tables (1)

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Table 1 Summary of Measured Statistics of Index Variations in Tissues

Equations (3)

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Φ ( κ ) = 4 π < δ n 2 > L o 2 ( m 1 ) ( 1 + κ 2 L o 2 ) m ,
D n ( ρ ) = 2 [ δ n 2 C ( ρ ) ] ,
C ( ρ ) = 2 Γ ( m 1 ) ( ρ 2 L o ) m 1 K m 1 ( ρ L o ) , ρ 0 ,

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