Abstract

A theory is developed which relates quasi-elastic light scattering measurements to blood flow in tissue microvasculature. We assume that the tissue matrix surrounding the blood cells is a strong diffuser of light and that moving erythrocytes, therefore, are illuminated by a spatially distributed source. Because the surrounding tissue is considered to be stationary, Doppler shifts in the frequency of the scattered light arise only from photon interactions with the moving blood cells. The theory implies that the time decay of the photon autocorrelation function scales proportionally with cell size and inversely with mean translational speed. Analysis of multiple interactions of photons with moving cells indicates the manner in which spectral measurements additionally are sensitive to changes in blood volume. Predictions are verified by measurements of particle flow in model tissues.

© 1981 Optical Society of America

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References

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  1. S. Morikawa, O. Lanz, C. C. Johnson, IEEE Trans. Biomed. Eng. BME-18, 416 (1971).
    [CrossRef]
  2. C. Riva, B. Ross, G. B. Benedek, Invest. Ophthalmol. 11, 936 (1972).
    [PubMed]
  3. T. Tanaka, C. Riva, I. Ben-Sira, Science 186, 830 (1974).
    [CrossRef] [PubMed]
  4. T. Tanaka, G. B. Benedek, Appl. Opt. 14, 189 (1975).
    [PubMed]
  5. M. Stern, Nature (London) 254, 56 (1975).
    [CrossRef]
  6. M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
    [PubMed]
  7. M. D. Stern, P. D. Bowen, R. Parma, R. W. Osgood, R. L. Bowman, J. H. Stein, Am. J. Physiol. 236, F80 (1979).
    [PubMed]
  8. R. F. Bonner, P. Bowen, R. L. Bowman, R. Nossal, in Proceedings, Electrooptics/Laser &'78 Conference (Industrial and Scientific Conference Management, Inc., Chicago, III., 1978), p. 539.
  9. D. Watkins, G. A. Holloway, IEEE Trans. Biomed. Eng. BME-25, 28 (1978).
    [CrossRef]
  10. R. W. Wunderlich, R. L. Folger, B. R. Ware, D. B. Giddon, Rev. Sci Instrum. 51, 1258 (1980).
    [CrossRef]
  11. R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, in press).
  12. D. I. Abramson, Circulation in the Extremities (Academic, New York, 1967).
  13. P. C. Johnson, Ed., Peripheral Circulation (Wiley, New York, 1978).
  14. L. Reynolds, C. Johnson, A. Ishimaru, Appl. Opt. 15, 2059 (1976).
    [CrossRef] [PubMed]
  15. H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 133 (1970).
    [CrossRef]
  16. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  17. E. Jakeman, in Photon Correlation and Light Beating Spectroscopy, H. Z. Cummins, E. R. Pike, Eds., (Plenum, New York, 1974), p. 91.
  18. H. L. Goldsmith, S. G. Mason, Bibl. Anat. 10, 1 (1969).
  19. P. Butti, M. Intaglietta, H. Riemann, C. Holliger, A. Bollinger, M. Anliker, Microvasc. Res. 10, 220 (1975).
    [CrossRef] [PubMed]
  20. R. Nossal, S. H. Chen, C. C. Lai, Opt. Commun. 4, 35 (1971).
    [CrossRef]
  21. R. S. Chadwick, I. Chang, J. Colloid Interface Sci. 42, 516 (1973).
    [CrossRef]
  22. C. M. Sorensen, R. C. Mockler, W. J. O'Sullivan, Phys. Rev. A: 14, 1520 (1976).
    [CrossRef]
  23. A. Erdelyi, Ed., Tables of Integral Transforms, (McGraw-Hill, New York, 1954).
  24. S. H. Chen, W. B. Veldkamp, C. C. Lai, Rev. Sci. Instrum. 46, 1356 (1975).
    [CrossRef]
  25. J. Lubbers, P. J. L. M. Bernink, G. J. Barendsen, J. W. van den Berg, Pfluegers Arch. 382, 241 (1979).
    [CrossRef]
  26. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).

1980

R. W. Wunderlich, R. L. Folger, B. R. Ware, D. B. Giddon, Rev. Sci Instrum. 51, 1258 (1980).
[CrossRef]

1979

M. D. Stern, P. D. Bowen, R. Parma, R. W. Osgood, R. L. Bowman, J. H. Stein, Am. J. Physiol. 236, F80 (1979).
[PubMed]

J. Lubbers, P. J. L. M. Bernink, G. J. Barendsen, J. W. van den Berg, Pfluegers Arch. 382, 241 (1979).
[CrossRef]

1978

D. Watkins, G. A. Holloway, IEEE Trans. Biomed. Eng. BME-25, 28 (1978).
[CrossRef]

1977

M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
[PubMed]

1976

L. Reynolds, C. Johnson, A. Ishimaru, Appl. Opt. 15, 2059 (1976).
[CrossRef] [PubMed]

C. M. Sorensen, R. C. Mockler, W. J. O'Sullivan, Phys. Rev. A: 14, 1520 (1976).
[CrossRef]

1975

S. H. Chen, W. B. Veldkamp, C. C. Lai, Rev. Sci. Instrum. 46, 1356 (1975).
[CrossRef]

P. Butti, M. Intaglietta, H. Riemann, C. Holliger, A. Bollinger, M. Anliker, Microvasc. Res. 10, 220 (1975).
[CrossRef] [PubMed]

T. Tanaka, G. B. Benedek, Appl. Opt. 14, 189 (1975).
[PubMed]

M. Stern, Nature (London) 254, 56 (1975).
[CrossRef]

1974

T. Tanaka, C. Riva, I. Ben-Sira, Science 186, 830 (1974).
[CrossRef] [PubMed]

1973

R. S. Chadwick, I. Chang, J. Colloid Interface Sci. 42, 516 (1973).
[CrossRef]

1972

C. Riva, B. Ross, G. B. Benedek, Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

1971

S. Morikawa, O. Lanz, C. C. Johnson, IEEE Trans. Biomed. Eng. BME-18, 416 (1971).
[CrossRef]

R. Nossal, S. H. Chen, C. C. Lai, Opt. Commun. 4, 35 (1971).
[CrossRef]

1970

H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 133 (1970).
[CrossRef]

1969

H. L. Goldsmith, S. G. Mason, Bibl. Anat. 10, 1 (1969).

Abramson, D. I.

D. I. Abramson, Circulation in the Extremities (Academic, New York, 1967).

Anliker, M.

P. Butti, M. Intaglietta, H. Riemann, C. Holliger, A. Bollinger, M. Anliker, Microvasc. Res. 10, 220 (1975).
[CrossRef] [PubMed]

Barendsen, G. J.

J. Lubbers, P. J. L. M. Bernink, G. J. Barendsen, J. W. van den Berg, Pfluegers Arch. 382, 241 (1979).
[CrossRef]

Benedek, G. B.

T. Tanaka, G. B. Benedek, Appl. Opt. 14, 189 (1975).
[PubMed]

C. Riva, B. Ross, G. B. Benedek, Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

Ben-Sira, I.

T. Tanaka, C. Riva, I. Ben-Sira, Science 186, 830 (1974).
[CrossRef] [PubMed]

Bernink, P. J. L. M.

J. Lubbers, P. J. L. M. Bernink, G. J. Barendsen, J. W. van den Berg, Pfluegers Arch. 382, 241 (1979).
[CrossRef]

Bollinger, A.

P. Butti, M. Intaglietta, H. Riemann, C. Holliger, A. Bollinger, M. Anliker, Microvasc. Res. 10, 220 (1975).
[CrossRef] [PubMed]

Bonner, R.

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, in press).

Bonner, R. F.

R. F. Bonner, P. Bowen, R. L. Bowman, R. Nossal, in Proceedings, Electrooptics/Laser &'78 Conference (Industrial and Scientific Conference Management, Inc., Chicago, III., 1978), p. 539.

Bowen, P.

R. F. Bonner, P. Bowen, R. L. Bowman, R. Nossal, in Proceedings, Electrooptics/Laser &'78 Conference (Industrial and Scientific Conference Management, Inc., Chicago, III., 1978), p. 539.

Bowen, P. D.

M. D. Stern, P. D. Bowen, R. Parma, R. W. Osgood, R. L. Bowman, J. H. Stein, Am. J. Physiol. 236, F80 (1979).
[PubMed]

M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
[PubMed]

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, in press).

Bowman, R. L.

M. D. Stern, P. D. Bowen, R. Parma, R. W. Osgood, R. L. Bowman, J. H. Stein, Am. J. Physiol. 236, F80 (1979).
[PubMed]

M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
[PubMed]

R. F. Bonner, P. Bowen, R. L. Bowman, R. Nossal, in Proceedings, Electrooptics/Laser &'78 Conference (Industrial and Scientific Conference Management, Inc., Chicago, III., 1978), p. 539.

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, in press).

Butti, P.

P. Butti, M. Intaglietta, H. Riemann, C. Holliger, A. Bollinger, M. Anliker, Microvasc. Res. 10, 220 (1975).
[CrossRef] [PubMed]

Chadwick, R. S.

R. S. Chadwick, I. Chang, J. Colloid Interface Sci. 42, 516 (1973).
[CrossRef]

Chang, I.

R. S. Chadwick, I. Chang, J. Colloid Interface Sci. 42, 516 (1973).
[CrossRef]

Chen, S. H.

S. H. Chen, W. B. Veldkamp, C. C. Lai, Rev. Sci. Instrum. 46, 1356 (1975).
[CrossRef]

R. Nossal, S. H. Chen, C. C. Lai, Opt. Commun. 4, 35 (1971).
[CrossRef]

Chimosky, J. E.

M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
[PubMed]

Clem, T. R.

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, in press).

Cummins, H. Z.

H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 133 (1970).
[CrossRef]

Folger, R. L.

R. W. Wunderlich, R. L. Folger, B. R. Ware, D. B. Giddon, Rev. Sci Instrum. 51, 1258 (1980).
[CrossRef]

Giddon, D. B.

R. W. Wunderlich, R. L. Folger, B. R. Ware, D. B. Giddon, Rev. Sci Instrum. 51, 1258 (1980).
[CrossRef]

Goldsmith, H. L.

H. L. Goldsmith, S. G. Mason, Bibl. Anat. 10, 1 (1969).

Holliger, C.

P. Butti, M. Intaglietta, H. Riemann, C. Holliger, A. Bollinger, M. Anliker, Microvasc. Res. 10, 220 (1975).
[CrossRef] [PubMed]

Holloway, G. A.

D. Watkins, G. A. Holloway, IEEE Trans. Biomed. Eng. BME-25, 28 (1978).
[CrossRef]

M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
[PubMed]

Intaglietta, M.

P. Butti, M. Intaglietta, H. Riemann, C. Holliger, A. Bollinger, M. Anliker, Microvasc. Res. 10, 220 (1975).
[CrossRef] [PubMed]

Ishimaru, A.

Jakeman, E.

E. Jakeman, in Photon Correlation and Light Beating Spectroscopy, H. Z. Cummins, E. R. Pike, Eds., (Plenum, New York, 1974), p. 91.

Johnson, C.

Johnson, C. C.

S. Morikawa, O. Lanz, C. C. Johnson, IEEE Trans. Biomed. Eng. BME-18, 416 (1971).
[CrossRef]

Keiser, H. R.

M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
[PubMed]

Lai, C. C.

S. H. Chen, W. B. Veldkamp, C. C. Lai, Rev. Sci. Instrum. 46, 1356 (1975).
[CrossRef]

R. Nossal, S. H. Chen, C. C. Lai, Opt. Commun. 4, 35 (1971).
[CrossRef]

Lanz, O.

S. Morikawa, O. Lanz, C. C. Johnson, IEEE Trans. Biomed. Eng. BME-18, 416 (1971).
[CrossRef]

Lappe, D. L.

M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
[PubMed]

Lubbers, J.

J. Lubbers, P. J. L. M. Bernink, G. J. Barendsen, J. W. van den Berg, Pfluegers Arch. 382, 241 (1979).
[CrossRef]

Mason, S. G.

H. L. Goldsmith, S. G. Mason, Bibl. Anat. 10, 1 (1969).

Mockler, R. C.

C. M. Sorensen, R. C. Mockler, W. J. O'Sullivan, Phys. Rev. A: 14, 1520 (1976).
[CrossRef]

Morikawa, S.

S. Morikawa, O. Lanz, C. C. Johnson, IEEE Trans. Biomed. Eng. BME-18, 416 (1971).
[CrossRef]

Nossal, R.

R. Nossal, S. H. Chen, C. C. Lai, Opt. Commun. 4, 35 (1971).
[CrossRef]

R. F. Bonner, P. Bowen, R. L. Bowman, R. Nossal, in Proceedings, Electrooptics/Laser &'78 Conference (Industrial and Scientific Conference Management, Inc., Chicago, III., 1978), p. 539.

Osgood, R. W.

M. D. Stern, P. D. Bowen, R. Parma, R. W. Osgood, R. L. Bowman, J. H. Stein, Am. J. Physiol. 236, F80 (1979).
[PubMed]

O'Sullivan, W. J.

C. M. Sorensen, R. C. Mockler, W. J. O'Sullivan, Phys. Rev. A: 14, 1520 (1976).
[CrossRef]

Parma, R.

M. D. Stern, P. D. Bowen, R. Parma, R. W. Osgood, R. L. Bowman, J. H. Stein, Am. J. Physiol. 236, F80 (1979).
[PubMed]

Reynolds, L.

Riemann, H.

P. Butti, M. Intaglietta, H. Riemann, C. Holliger, A. Bollinger, M. Anliker, Microvasc. Res. 10, 220 (1975).
[CrossRef] [PubMed]

Riva, C.

T. Tanaka, C. Riva, I. Ben-Sira, Science 186, 830 (1974).
[CrossRef] [PubMed]

C. Riva, B. Ross, G. B. Benedek, Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

Ross, B.

C. Riva, B. Ross, G. B. Benedek, Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

Sorensen, C. M.

C. M. Sorensen, R. C. Mockler, W. J. O'Sullivan, Phys. Rev. A: 14, 1520 (1976).
[CrossRef]

Stein, J. H.

M. D. Stern, P. D. Bowen, R. Parma, R. W. Osgood, R. L. Bowman, J. H. Stein, Am. J. Physiol. 236, F80 (1979).
[PubMed]

Stern, M.

M. Stern, Nature (London) 254, 56 (1975).
[CrossRef]

Stern, M. D.

M. D. Stern, P. D. Bowen, R. Parma, R. W. Osgood, R. L. Bowman, J. H. Stein, Am. J. Physiol. 236, F80 (1979).
[PubMed]

M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
[PubMed]

Swinney, H. L.

H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 133 (1970).
[CrossRef]

Tanaka, T.

T. Tanaka, G. B. Benedek, Appl. Opt. 14, 189 (1975).
[PubMed]

T. Tanaka, C. Riva, I. Ben-Sira, Science 186, 830 (1974).
[CrossRef] [PubMed]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

van den Berg, J. W.

J. Lubbers, P. J. L. M. Bernink, G. J. Barendsen, J. W. van den Berg, Pfluegers Arch. 382, 241 (1979).
[CrossRef]

Veldkamp, W. B.

S. H. Chen, W. B. Veldkamp, C. C. Lai, Rev. Sci. Instrum. 46, 1356 (1975).
[CrossRef]

Ware, B. R.

R. W. Wunderlich, R. L. Folger, B. R. Ware, D. B. Giddon, Rev. Sci Instrum. 51, 1258 (1980).
[CrossRef]

Watkins, D.

D. Watkins, G. A. Holloway, IEEE Trans. Biomed. Eng. BME-25, 28 (1978).
[CrossRef]

Wunderlich, R. W.

R. W. Wunderlich, R. L. Folger, B. R. Ware, D. B. Giddon, Rev. Sci Instrum. 51, 1258 (1980).
[CrossRef]

Am. J. Physiol.

M. D. Stern, D. L. Lappe, P. D. Bowen, J. E. Chimosky, G. A. Holloway, H. R. Keiser, R. L. Bowman, Am. J. Physiol. 232, H441 (1977).
[PubMed]

M. D. Stern, P. D. Bowen, R. Parma, R. W. Osgood, R. L. Bowman, J. H. Stein, Am. J. Physiol. 236, F80 (1979).
[PubMed]

Appl. Opt.

Bibl. Anat.

H. L. Goldsmith, S. G. Mason, Bibl. Anat. 10, 1 (1969).

IEEE Trans. Biomed. Eng.

S. Morikawa, O. Lanz, C. C. Johnson, IEEE Trans. Biomed. Eng. BME-18, 416 (1971).
[CrossRef]

D. Watkins, G. A. Holloway, IEEE Trans. Biomed. Eng. BME-25, 28 (1978).
[CrossRef]

Invest. Ophthalmol.

C. Riva, B. Ross, G. B. Benedek, Invest. Ophthalmol. 11, 936 (1972).
[PubMed]

J. Colloid Interface Sci.

R. S. Chadwick, I. Chang, J. Colloid Interface Sci. 42, 516 (1973).
[CrossRef]

Microvasc. Res.

P. Butti, M. Intaglietta, H. Riemann, C. Holliger, A. Bollinger, M. Anliker, Microvasc. Res. 10, 220 (1975).
[CrossRef] [PubMed]

Nature (London)

M. Stern, Nature (London) 254, 56 (1975).
[CrossRef]

Opt. Commun.

R. Nossal, S. H. Chen, C. C. Lai, Opt. Commun. 4, 35 (1971).
[CrossRef]

Pfluegers Arch.

J. Lubbers, P. J. L. M. Bernink, G. J. Barendsen, J. W. van den Berg, Pfluegers Arch. 382, 241 (1979).
[CrossRef]

Phys. Rev. A:

C. M. Sorensen, R. C. Mockler, W. J. O'Sullivan, Phys. Rev. A: 14, 1520 (1976).
[CrossRef]

Prog. Opt.

H. Z. Cummins, H. L. Swinney, Prog. Opt. 8, 133 (1970).
[CrossRef]

Rev. Sci Instrum.

R. W. Wunderlich, R. L. Folger, B. R. Ware, D. B. Giddon, Rev. Sci Instrum. 51, 1258 (1980).
[CrossRef]

Rev. Sci. Instrum.

S. H. Chen, W. B. Veldkamp, C. C. Lai, Rev. Sci. Instrum. 46, 1356 (1975).
[CrossRef]

Science

T. Tanaka, C. Riva, I. Ben-Sira, Science 186, 830 (1974).
[CrossRef] [PubMed]

Other

R. Bonner, T. R. Clem, P. D. Bowen, R. L. Bowman, in Scattering Techniques Applied to Supramolecular and Non-equilibrium Systems, S-H. Chen, B. Chu, R. Nossal, Eds. (Plenum, New York, in press).

D. I. Abramson, Circulation in the Extremities (Academic, New York, 1967).

P. C. Johnson, Ed., Peripheral Circulation (Wiley, New York, 1978).

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

E. Jakeman, in Photon Correlation and Light Beating Spectroscopy, H. Z. Cummins, E. R. Pike, Eds., (Plenum, New York, 1974), p. 91.

R. F. Bonner, P. Bowen, R. L. Bowman, R. Nossal, in Proceedings, Electrooptics/Laser &'78 Conference (Industrial and Scientific Conference Management, Inc., Chicago, III., 1978), p. 539.

A. Erdelyi, Ed., Tables of Integral Transforms, (McGraw-Hill, New York, 1954).

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1965).

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Figures (10)

Fig. 1
Fig. 1

Diffusion of photons within the tissue can be represented as a series of scattering steps of types A, B, or C. Step A represents scattering from a static tissue element which does not impart any Doppler shift (the phase shift of this step, φA, is constant). The distance between static scattering centers |ρrj| will depend on the tissue structure but typically is ∼100 μm. Step B represents small-angle Doppler scattering from a moving red blood cell with a phase shift φβ(t), which varies as Q · vt. The probability of step B scattering increases with local blood cell concentration. Step C represents sequential scattering from two moving red blood cells and occurs within larger vessels (>50-μm diam). In this case the velocity vectors of the two cells are highly correlated, | v 1 · v 2 | υ 1 2, although the sign of the phase terms Q1 · r1(t) and Q2 · r2(t) are random.

Fig. 2
Fig. 2

Angular scattering structure factor S ( Q a ) vs Qa. Shown are (a) the approximations given by Eq. (13), which is used in our subsequent analysis, and (b) a curve based on empirical data and a Rayleigh-Gans computation of light scattered by randomly oriented biconcave disk red blood cells.21 Also shown are curves for (c) spherical red blood cells of 2.75-μm radius and polystyrene latex spheres of (d) 0.55- and (e) 0.26-μm radius, derived from Mie theory for particles in water.

Fig. 3
Fig. 3

Field autocorrelation function given by Eq. (18), I1(T), for light which is singly scattered by moving particles, plotted as a function of the reduced time variable T for different values of the particle size variable L (L = 1,2,4,15). For large particles with L > 4 (a > 0.15 μm), I1(T) has the asymptotic form given by Eq. (20).

Fig. 4
Fig. 4

Half time, T1/2, of I1(T) determined from Eq. (18) and plotted as a function of the size parameter L = 2ka. For large particles with highly anisotropic forward scattering (L > 4), the reduced time variable is given by Eq. (19).

Fig. 5
Fig. 5

Normalized intensity autocorrelation function g(2)(T) plotted as a function of the reduced time variable T [i.e., scaled by the rms speed divided by the particle radius, see Eq. (17)] for different degrees of multiple scattering and different speed distributions. Parts (a) and (b) show g(2)(T) for the Gaussian speed distribution [Eq. (12)] when m ¯ varies from 0.1 to 4. (a) For small values of m ¯, the predominant effect on g(2)(T) is an increase in amplitude with increasing m ¯. (b) For larger m ¯ the amplitude of g(2)(0) approaches a maximum, 1 + β, determined by the spatial coherence at the detector, while the decay rate increases and approaches m ¯ 1 / 2 scaling. In (c) and (d) the curves labeled A, B, C, and D represent g(2)(T) for different speed distributions: A, Gaussian [Eq. (12)]; B, uniform (Newtonian flow in a cylinder); C, single speed; D, two speed ( 1 / 3 υ ¯ and 5 / 3 υ ¯). Differences in these curves are noticeable at large T if m ¯ 1 but are insignificant for large m ¯ (where all curves become Gaussian). The relative amplitude of the slow decay (corresponding to low frequency components) is seen to increase with the variance of the speed distribution.

Fig. 6
Fig. 6

Power spectrum S(W) of the detected photocurrent computed as a function of W m ¯ 1 / 2 for m ¯ = 0.5 , 1 , 1.5 , 2 , 3 , 4 , 6 , 8 , and 10 (the curves from left to right, respectively). For m ¯ < 1, the normalized first moment of these spectra increases more rapidly than m ¯ 1 / 2, whereas for large m ¯ the spectra scale with W m ¯ 1 / 2.

Fig. 7
Fig. 7

(A) Normalized photon autocorrelation function g(2)(τ) − 1 of light backscattered from the fingertip of a normal volunteer. The decay amplitude is 90% of the empirical value of β obtained from homodyne scattering from a solution of polystyrene spheres using the same fiber optic probe and corresponds [see Eq. (24)] to m ¯ 1.2. (B) Autocorrelation functions obtained from light backscattered from the tissue model using diluted human blood with mean velocity of 1 mm/sec and hematocrits H = 0.05, 0.025, 0.006, and 0.003 (a, b, c, and d, respectively). Increased red blood cell number density (and, therefore, m ¯) results in increased decay rates and amplitudes of g(2)(τ) − 1. These experimental curves are similar to the theoretical curves shown in Fig. 5. For clarity, smoothed curves have been drawn through the data points, which are only shown for H = 0.025.

Fig. 8
Fig. 8

Mean Doppler shift 〈ω〉, as defined in Eq. (32), plotted vs mean velocity for a variety of particle sizes and volume fractions of fluid moving through the hollow fibers of the model system. In the 0–2-mm/sec range, the mean frequency 〈ω〉 increases linearly with mean particle velocity. A monotonically increasing dependence on particle density and decreasing dependence on particle radius are also observed, ag, human blood diluted to hematocrits of 0.003, 0.006, 0.012, 0.012, 0.025, 0.035, 0.050, and 0.12, respectively; hi, 0.55-μm radius PSL spheres at concentrations of 0.5 and 1.0 % v/v; j, 0.264-μm radius PSL spheres, 1 % v/v.

Fig. 9
Fig. 9

Mean Doppler frequency normalized by f ( m ¯ ) and rms particle velocity ν ¯ / f ( m ¯ ) = ω / f ( m ¯ ) V 2 1 / 2 vs the reciprocal of particle radius. Shown are data obtained when red blood cells or PSL spheres were passed through the tissue flow model. Mean frequencies have been normalized to m ¯ and mean particle speed. The solid line drawn through the data points represents values of ν ¯ / f ( m ¯ ) predicted by Eq. (36). [For red blood cells, m ¯ equals 1.43 times the vol % RBC in our model system (see insert in Fig. 10). Corresponding values of m ¯ for PSL solutions were determined from literature values14,16 of particle scattering cross sections and from particle volume ratios. From the values of m ¯ so obtained, f ( m ¯ ) was calculated by Eq. (A4).]

Fig. 10
Fig. 10

Mean Doppler shift normalized by the rms particle velocity plotted as a function of the hematocrit of the fluid flowing through the tissue model (H = 0.3–12% v/v RBC). The average degree of multiple scattering m ¯ was related to RBC concentration by determining the decay amplitudes (see insert) and fitting according to Eq. (24). Empirical values of ν ¯ = ω / V 2 1 / 2 are plotted (circles) for rms velocities in the 0.2–2-mm/sec range. The lower solid line is the theoretical curve given by Eq. (36) using an effective RBC radius of 2.8 μm and measured values of 〈V21/2 and m ¯. The upper curve is obtained from the lower merely by increasing the ordinate scale [〈V21/2/(12ξ)1/2a] by 40%. The shape of the curve depends only on f ( m ¯ ) [see Eq. (A4)].

Equations (47)

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ω = V 2 1 / 2 β ( 12 ξ ) 1 / 2 a f ( m ¯ ) ,
I 1 ( τ ) = π π S [ Q ( θ ) ] exp [ i Q ( θ ) · Δ R ( τ ) ] sin θ d θ π π S [ Q ( θ ) ] sin θ d θ ,
Q ( θ ) = 4 π n λ sin ( θ / 2 ) = 2 k sin ( θ / 2 ) .
e sc j ( t ) = E ( ρ ) exp ( ω 0 t ) A [ Q ( θ ρ ) ] × exp { i [ Q ( θ ρ ) · r j ( t ) + φ ( ρ ) + φ ( r d ) ] } ,
A ( Q ) = particle volume α ( r ) exp ( i Q · r ) d 3 r ,
S ( Q ) = | A ( Q ) | 2 .
sc ( r d , t ) tissue d 3 ρ E ( ρ ) A [ Q ( θ ρ ) ] × exp i [ Q ( θ ρ ) · r j ( t ) + φ ( ρ ) ] .
E sc ( t ) tissue D ( r d ) exp [ i φ ( r d ) ] sc ( r d , t ) d 3 r d ,
g ( 2 ) ( τ ) = n ( t ) n ( t + τ ) n 2 = 1 + β [ ( i sc i 0 ) 2 | I ( τ ) | 2 + 2 i sc ( i 0 i sc ) i 0 2 I ( τ ) ] ,
I ( τ ) E sc * ( t ) E sc ( t + τ ) .
I 1 ( τ ) D ( r d ) d 3 r d E * ( ρ ) A * [ Q ( θ ρ ) ] × exp { i [ Q ( θ ρ ) · r j ( t ) + φ ρ + φ r d ] } d 3 ρ · D ( r d ) d 3 r d E ( ρ ) A [ Q ( θ ρ ) ] × exp { i [ Q ( θ ρ ) · r j ( t + τ ) + φ ρ + φ r d ] } d 3 ρ .
I 1 ( τ ) | D ( r d ) | 2 d 3 r d i ( ρ ) S [ Q ( θ ρ ) ] × exp { i Q ( θ ρ ) · [ r j ( t + τ ) r j ( t ) ] } d 3 ρ .
exp [ i Q · Δ R ( τ ) ] = 0 j 0 ( Q V τ ) P ( V ) d V j 0 ( Q V τ ) ,
P ( V ) = ( 2 π ) 1 / 2 ( 3 V 2 ) 3 / 2 V 2 exp ( 3 V 2 / 2 V 2 ) ,
j 0 ( Q V τ ) = exp ( Q 2 V 2 τ 2 / 6 ) ,
S ( Q ) = S ( Q a ) = { 3 ( Q a ) 3 [ sin ( Q a ) Q a cos ( Q a ) ] } 2 ,
S ( Q ) exp [ 2 ξ ( Q a ) 2 ] ,
I 1 ( τ ) = 0 π S [ Q ( θ ) a ] j 0 ( Q V τ ) sin θ d θ 0 π S [ Q ( θ ) a ] sin θ d θ ,
I 1 ( τ ) = 0 1 S ( 2 kaz ) j 0 ( 2 kz V τ ) zdz 0 1 S ( 2 kz ) zdz ,
I 1 ( τ ) = 0 L exp ( 2 ξ z 2 ) exp ( T 2 z 2 ) zdz 0 L exp ( 2 ξ z 2 ) zdz ,
L = 2 ka , T = V 2 1 / 2 τ / 6 a .
I 1 ( τ ) = 2 ξ 2 ξ + T 2 [ 1 exp ( 2 ξ L 2 ) exp ( L 2 T 2 ) ] [ 1 exp ( 2 ξ L 2 ) ] .
τ 1 / 2 = 1.27 a V 2 1 / 2 ( ka large ) .
I 1 ( τ ) = 2 ξ 2 ξ + T 2 ,
I ( τ ) = m = 1 P m I m ( τ ) / ( 1 P 0 ) ,
I m ( τ ) = E sc 1 * ( t ) E sc 1 ( t + τ ) E sc 2 * ( t ) E sc 2 ( t + τ ) E sc m * ( t ) E sc m ( t + τ ) = | I 1 ( τ ) | m ,
I ( τ ) = m = 1 P m [ I 1 ( τ ) ] m / ( 1 P 0 ) .
I ( τ ) = m = 1 exp ( m ¯ ) [ m ¯ I 1 ( τ ) ] m m ! / [ 1 exp ( m ¯ ) ] = { exp [ m ¯ ( I 1 ( τ ) 1 ) ] exp ( m ¯ ) } / [ 1 exp ( m ¯ ) ]
g ( 2 ) ( τ ) = 1 + β ( exp { 2 m ¯ [ I 1 ( τ ) 1 ] } exp ( 2 m ¯ ) ) .
P ( ω ) = i 0 2 δ ( ω ) + e i 0 π + i 0 2 S ( ω ) ,
S ( ω ) = 1 π 0 cos ω t [ g ( 2 ) ( τ ) 1 ] dt .
S 1 ( ω ) = β π 2 i sc i 0 0 cos ω t I 1 ( t ) dt exp [ W ( ω ) ] ,
W ( ω ) = ( 12 ξ ) 1 / 2 ω a V 2 1 / 2 .
S ( ω ) = β exp ( 2 m ¯ ) j = 1 ( 2 m ¯ ) j S j ( ω ) j ! ,
S j ( ω ) = 1 π 0 cos Wt [ I 1 ( t ) ] j dt ,
S 1 ( ω ) = exp ( W ) ; S 2 ( ω ) = ¼ ( 1 + W ) exp ( W ) ; S 3 ( ω ) = 1 / 16 ( W 2 + 3 W + 3 ) exp ( W ) .
ω = | ω | S ( ω ) d ω .
ω = 2 β π 0 ω ( 0 cos ω t { exp 2 m ¯ [ I 1 ( t ) 1 ] exp ( 2 m ¯ ) } dt ) d ω .
ω = V 2 1 / 2 ( 12 ξ ) 1 / 2 a β f ( m ¯ ) ,
f ( m ¯ ) = 2 π 0 W ( 0 cos WZ exp { 2 m ¯ [ I 1 ( Z ) 1 ] } dZ ) dW ,
ω = V 2 1 / 2 ( 12 ξ ) 1 / 2 a β [ 2 π 1 / 2 exp ( 2 m ¯ ) j = 1 ( 2 m ¯ ) j Γ ( j + ½ ) Γ ( j + 1 ) Γ ( j ) ] .
f ( m ¯ ) = 2 exp ( 2 m ¯ ) j = 1 ( 2 m ¯ ) j f j j !
f j 1 π 0 W [ 0 cos WZ ( 1 1 + Z 2 ) j dZ ] dW .
0 cos WZ ( 1 1 + Z 2 ) j dZ = ( W 2 ) j 1 / 2 π 1 / 2 Γ ( j ) K j 1 / 2 ( W ) ,
0 W ( W 2 ) j 1 / 2 K j 1 / 2 ( W ) dW = Γ ( j + ½ ) .
f j = Γ ( j + ½ ) π 1 / 2 Γ ( j ) ,
f ( m ¯ ) = 2 exp ( 2 m ¯ ) π 1 / 2 j = 1 ( 2 m ¯ ) j Γ ( j + ½ ) Γ ( j + 1 ) Γ ( j ) .

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