Abstract

We introduce a micro-optical model of soft biological tissue that permits numerical computation of the absolute magnitudes of its scattering coefficients. A key assumption of the model is that the refractive-index variations caused by microscopic tissue elements can be treated as particles with sizes distributed according to a skewed log-normal distribution function. In the limit of an infinitely large variance in the particle size, this function has the same power-law dependence as the volume fractions of the subunits of an ideal fractal object. To compute a complete set of optical coefficients of a prototypical soft tissue (single-scattering coefficient, transport scattering coefficient, backscattering coefficient, phase function, and asymmetry parameter), we apply Mie theory to a volume of spheres with sizes distributed according to the theoretical distribution. A packing factor is included in the calculation of the optical cross sections to account for correlated scattering among tightly packed particles. The results suggest that the skewed log-normal distribution function, with a shape specified by a limiting fractal dimension of 3.7, is a valid approximation of the size distribution of scatterers in tissue. In the wavelength range 600 ≤ λ ≤ 1400 nm, the diameters of the scatterers that contribute most to backscattering were found to be significantly smaller (λ/4–λ/2) than the diameters of the scatterers that cause the greatest extinction of forward-scattered light (3–4λ).

© 1998 Optical Society of America

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1997 (1)

G. Tearney, M. E. Brezinsky, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, J. G. Fujimoto,” Science 276, 2037–2039 (1997).

1996 (6)

J. M. Schmitt, G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21, 1310–1312 (1996).
[CrossRef] [PubMed]

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[CrossRef] [PubMed]

A. Dunn, R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

H. Honda, S. Imayama, M. Tanemura, “A fractal-like structure in skin,” Fractals 4, 139–147 (1996).
[CrossRef]

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

D. Hamburger, O. Biham, D. Avnir, “Apparent fractality emerging from models of random distributions,” Phys. Rev. E 53, 3442–3458 (1996).
[CrossRef]

1995 (2)

M. S. Patterson, “Noninvasive measurements of tissue optical properties: current status and future prospects,” Comments Mol. Cell. Biophys. A 8, 387–417 (1995).

P. A. J. Bascom, R. S. C. Cobbold, “On the fractal packing approach for understanding ultrasonic backscattering from blood,” J. Acoust. Soc. Am. 98, 3040–3049 (1995).
[CrossRef] [PubMed]

1994 (3)

J. N. Qu, C. MacAulay, S. Lam, B. Palcic, “Optical properties of normal and carcinomatous bronchial tissue,” Appl. Opt. 33, 7397–7405 (1994).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial component to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

1993 (1)

1992 (1)

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

1990 (2)

B. J. West, “Physiology in fractal dimensions: error tolerance,” Ann. Biomed. Eng. 18, 135–149 (1990).
[CrossRef] [PubMed]

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

1989 (3)

1987 (1)

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nanometers,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

1982 (1)

1979 (1)

M. Rojkind, M. A. Giambrone, L. Biempica, “Collagen types in normal and cirrhotic liver,” Gastroenterology 76, 710 (1979).
[PubMed]

1978 (2)

1977 (1)

V. Khare, H. M. Nussenzveig, “The theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

1975 (1)

1974 (1)

A. Brunsting, P. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453 (1974).
[CrossRef] [PubMed]

1960 (1)

G. D. Weinstein, R. J. Boucek, “Collagen and elastin of human dermis,” J. Invest. Dermatol. 35, 227–229 (1960).
[PubMed]

1953 (1)

R. Barer, K. F. A. Ross, S. Tkaczyk, “Refractometry of living cells,” Nature (London) 171, 720–724 (1953).
[CrossRef]

Achatz, M.

Avnir, D.

D. Hamburger, O. Biham, D. Avnir, “Apparent fractality emerging from models of random distributions,” Phys. Rev. E 53, 3442–3458 (1996).
[CrossRef]

Avrillier, S.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

Barer, R.

R. Barer, K. F. A. Ross, S. Tkaczyk, “Refractometry of living cells,” Nature (London) 171, 720–724 (1953).
[CrossRef]

Bascom, P. A. J.

P. A. J. Bascom, R. S. C. Cobbold, “On the fractal packing approach for understanding ultrasonic backscattering from blood,” J. Acoust. Soc. Am. 98, 3040–3049 (1995).
[CrossRef] [PubMed]

Beauvoit, B.

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial component to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

Beek, J. F.

J. F. Beek, H. J. van Staveren, P. Posthumus, H. J. C. M. Sterenborg, M. J. C. van Gemert, “The influence of respiration on the optical properties of piglet lung at 632.8 nm,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1993), Vol. IS11, pp. 193–210.

Beuthan, J.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[CrossRef] [PubMed]

Biempica, L.

M. Rojkind, M. A. Giambrone, L. Biempica, “Collagen types in normal and cirrhotic liver,” Gastroenterology 76, 710 (1979).
[PubMed]

Bigio, I. J.

I. J. Bigio, J. R. Mourant, J. D. Boyer, T. M. Johnson, T. Shimada, R. L. Conn, “Noninvasive identification of bladder cancer with subsurface backscattered light,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 26–35 (1994).
[CrossRef]

Biham, O.

D. Hamburger, O. Biham, D. Avnir, “Apparent fractality emerging from models of random distributions,” Phys. Rev. E 53, 3442–3458 (1996).
[CrossRef]

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

Bonner, R. F.

Boppart, S. A.

G. Tearney, M. E. Brezinsky, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, J. G. Fujimoto,” Science 276, 2037–2039 (1997).

Boucek, R. J.

G. D. Weinstein, R. J. Boucek, “Collagen and elastin of human dermis,” J. Invest. Dermatol. 35, 227–229 (1960).
[PubMed]

Bouma, B. E.

G. Tearney, M. E. Brezinsky, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, J. G. Fujimoto,” Science 276, 2037–2039 (1997).

Boyer, J. D.

I. J. Bigio, J. R. Mourant, J. D. Boyer, T. M. Johnson, T. Shimada, R. L. Conn, “Noninvasive identification of bladder cancer with subsurface backscattered light,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 26–35 (1994).
[CrossRef]

Brezinsky, M. E.

G. Tearney, M. E. Brezinsky, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, J. G. Fujimoto,” Science 276, 2037–2039 (1997).

Brunsting, A.

A. Brunsting, P. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453 (1974).
[CrossRef] [PubMed]

Burns, T. L.

T. D. Scholz, S. R. Fleagle, T. L. Burns, D. J. Skorton, “Nuclear magnetic resonance relaxometry of the normal heart: relationship between collagen content and relaxation times of the four chambers,” Magn. Reson. Imag. 7, 643–648 (1989).
[CrossRef]

Chance, B.

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial component to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

Cheong, W. F.

W. F. Cheong, “Summary of optical properties,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 275–303.

Cobbold, R. S. C.

P. A. J. Bascom, R. S. C. Cobbold, “On the fractal packing approach for understanding ultrasonic backscattering from blood,” J. Acoust. Soc. Am. 98, 3040–3049 (1995).
[CrossRef] [PubMed]

Conn, R. L.

I. J. Bigio, J. R. Mourant, J. D. Boyer, T. M. Johnson, T. Shimada, R. L. Conn, “Noninvasive identification of bladder cancer with subsurface backscattered light,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 26–35 (1994).
[CrossRef]

Delpy, D. T.

P. Van der Zee, M. Essenpreis, D. T. Delpy, “Optical properties of brain tissue,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 454–465 (1993).
[CrossRef]

Duck, F. A.

F. A. Duck, Physical Properties of Tissue (Academic, New York, 1990), Chap. 9.

Dunn, A.

A. Dunn, R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

Eckhaus, M. A.

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

Essenpreis, M.

M. Essenpreis, Thermally Induced Changes in Optical Properties of Biological Tissues (University College London, England, 1992).

P. Van der Zee, M. Essenpreis, D. T. Delpy, “Optical properties of brain tissue,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 454–465 (1993).
[CrossRef]

Feld, M. S.

Fleagle, S. R.

T. D. Scholz, S. R. Fleagle, T. L. Burns, D. J. Skorton, “Nuclear magnetic resonance relaxometry of the normal heart: relationship between collagen content and relaxation times of the four chambers,” Magn. Reson. Imag. 7, 643–648 (1989).
[CrossRef]

Flock, S. T.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nanometers,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Frank, G. L.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Fujimoto, J. G.

G. Tearney, M. E. Brezinsky, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, J. G. Fujimoto,” Science 276, 2037–2039 (1997).

Gélébart, B.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

Giambrone, M. A.

M. Rojkind, M. A. Giambrone, L. Biempica, “Collagen types in normal and cirrhotic liver,” Gastroenterology 76, 710 (1979).
[PubMed]

Hamburger, D.

D. Hamburger, O. Biham, D. Avnir, “Apparent fractality emerging from models of random distributions,” Phys. Rev. E 53, 3442–3458 (1996).
[CrossRef]

Helfmann, J.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[CrossRef] [PubMed]

Herrig, M.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[CrossRef] [PubMed]

Honda, H.

H. Honda, S. Imayama, M. Tanemura, “A fractal-like structure in skin,” Fractals 4, 139–147 (1996).
[CrossRef]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

Imayama, S.

H. Honda, S. Imayama, M. Tanemura, “A fractal-like structure in skin,” Fractals 4, 139–147 (1996).
[CrossRef]

Ishimaru, A.

Jacques, S. L.

Johnson, T. M.

I. J. Bigio, J. R. Mourant, J. D. Boyer, T. M. Johnson, T. Shimada, R. L. Conn, “Noninvasive identification of bladder cancer with subsurface backscattered light,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 26–35 (1994).
[CrossRef]

Keijzer, M.

Kerker, M.

M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, San Diego, Calif., 1969), pp. 351–359.

Khare, V.

V. Khare, H. M. Nussenzveig, “The theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Kitai, T.

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial component to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

Knüttel, A.

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
[CrossRef] [PubMed]

Kuga, Y.

Kumar, G.

J. M. Schmitt, G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21, 1310–1312 (1996).
[CrossRef] [PubMed]

G. Kumar, J. M. Schmitt, “Micro-optical properties of tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases III: Optical Biopsy, R. R. Alfano, A. Katzir, eds., Proc. SPIE2679, 106–116 (1996).
[CrossRef]

Lam, S.

MacAulay, C.

Mandelbrot, B. B.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982), Chap. 12.

Minet, O.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[CrossRef] [PubMed]

Mourant, J. R.

I. J. Bigio, J. R. Mourant, J. D. Boyer, T. M. Johnson, T. Shimada, R. L. Conn, “Noninvasive identification of bladder cancer with subsurface backscattered light,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 26–35 (1994).
[CrossRef]

Mullaney, P.

A. Brunsting, P. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453 (1974).
[CrossRef] [PubMed]

Müller, G.

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[CrossRef] [PubMed]

Nishioka, N. S.

Nussenzveig, H. M.

V. Khare, H. M. Nussenzveig, “The theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Palcic, B.

Parsa, P.

Patterson, M. S.

M. S. Patterson, “Noninvasive measurements of tissue optical properties: current status and future prospects,” Comments Mol. Cell. Biophys. A 8, 387–417 (1995).

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nanometers,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Peters, V. G.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Pitris, C.

G. Tearney, M. E. Brezinsky, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, J. G. Fujimoto,” Science 276, 2037–2039 (1997).

Posthumus, P.

J. F. Beek, H. J. van Staveren, P. Posthumus, H. J. C. M. Sterenborg, M. J. C. van Gemert, “The influence of respiration on the optical properties of piglet lung at 632.8 nm,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1993), Vol. IS11, pp. 193–210.

Qu, J. N.

Richards-Kortum, R.

A. Dunn, R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

Richards-Kortum, R. R.

Rojkind, M.

M. Rojkind, M. A. Giambrone, L. Biempica, “Collagen types in normal and cirrhotic liver,” Gastroenterology 76, 710 (1979).
[PubMed]

Ross, K. F. A.

R. Barer, K. F. A. Ross, S. Tkaczyk, “Refractometry of living cells,” Nature (London) 171, 720–724 (1953).
[CrossRef]

Schmitt, J. M.

J. M. Schmitt, G. Kumar, “Turbulent nature of refractive-index variations in biological tissue,” Opt. Lett. 21, 1310–1312 (1996).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, R. F. Bonner, “Measurement of optical properties of biological tissues by low-coherence reflectometry,” Appl. Opt. 32, 6032–6042 (1993).
[CrossRef] [PubMed]

G. Kumar, J. M. Schmitt, “Micro-optical properties of tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases III: Optical Biopsy, R. R. Alfano, A. Katzir, eds., Proc. SPIE2679, 106–116 (1996).
[CrossRef]

Scholz, T. D.

T. D. Scholz, S. R. Fleagle, T. L. Burns, D. J. Skorton, “Nuclear magnetic resonance relaxometry of the normal heart: relationship between collagen content and relaxation times of the four chambers,” Magn. Reson. Imag. 7, 643–648 (1989).
[CrossRef]

Seger, G.

Sevick, E. M.

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

Shimada, T.

I. J. Bigio, J. R. Mourant, J. D. Boyer, T. M. Johnson, T. Shimada, R. L. Conn, “Noninvasive identification of bladder cancer with subsurface backscattered light,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 26–35 (1994).
[CrossRef]

Skorton, D. J.

T. D. Scholz, S. R. Fleagle, T. L. Burns, D. J. Skorton, “Nuclear magnetic resonance relaxometry of the normal heart: relationship between collagen content and relaxation times of the four chambers,” Magn. Reson. Imag. 7, 643–648 (1989).
[CrossRef]

Southern, J. F.

G. Tearney, M. E. Brezinsky, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, J. G. Fujimoto,” Science 276, 2037–2039 (1997).

Sterenborg, H. J. C. M.

J. F. Beek, H. J. van Staveren, P. Posthumus, H. J. C. M. Sterenborg, M. J. C. van Gemert, “The influence of respiration on the optical properties of piglet lung at 632.8 nm,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1993), Vol. IS11, pp. 193–210.

Tanemura, M.

H. Honda, S. Imayama, M. Tanemura, “A fractal-like structure in skin,” Fractals 4, 139–147 (1996).
[CrossRef]

Tearney, G.

G. Tearney, M. E. Brezinsky, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, J. G. Fujimoto,” Science 276, 2037–2039 (1997).

Tinet, E.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

Tkaczyk, S.

R. Barer, K. F. A. Ross, S. Tkaczyk, “Refractometry of living cells,” Nature (London) 171, 720–724 (1953).
[CrossRef]

Tualle, J.-M.

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

Türke, B.

Twersky, V.

V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. 64, 1710–1719 (1978).
[CrossRef]

V. Twersky, “Transparency of pair-correlated, random distributions of small scatterers, with applications to the cornea,” J. Opt. Soc. Am. 65, 524–530 (1975).
[CrossRef] [PubMed]

Van der Zee, P.

P. Van der Zee, M. Essenpreis, D. T. Delpy, “Optical properties of brain tissue,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 454–465 (1993).
[CrossRef]

van Gemert, M. J. C.

J. F. Beek, H. J. van Staveren, P. Posthumus, H. J. C. M. Sterenborg, M. J. C. van Gemert, “The influence of respiration on the optical properties of piglet lung at 632.8 nm,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1993), Vol. IS11, pp. 193–210.

van Seelen, W.

van Staveren, H. J.

J. F. Beek, H. J. van Staveren, P. Posthumus, H. J. C. M. Sterenborg, M. J. C. van Gemert, “The influence of respiration on the optical properties of piglet lung at 632.8 nm,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1993), Vol. IS11, pp. 193–210.

Weinstein, G. D.

G. D. Weinstein, R. J. Boucek, “Collagen and elastin of human dermis,” J. Invest. Dermatol. 35, 227–229 (1960).
[PubMed]

West, B. J.

B. J. West, “Physiology in fractal dimensions: error tolerance,” Ann. Biomed. Eng. 18, 135–149 (1990).
[CrossRef] [PubMed]

Wilson, B. C.

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nanometers,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Wyman, D. R.

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

Yadlowsky, M.

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

Yoon, G.

G. Yoon, “Absorption and scattering of laser light in biological media—mathematical modeling and methods for determining optical properties,” Ph.D. dissertation (University of Texas, Austin, Tex., 1988).

Ann. Biomed. Eng. (1)

B. J. West, “Physiology in fractal dimensions: error tolerance,” Ann. Biomed. Eng. 18, 135–149 (1990).
[CrossRef] [PubMed]

Appl. Opt. (5)

Biophys. J. (2)

A. Brunsting, P. Mullaney, “Differential light scattering from spherical mammalian cells,” Biophys. J. 14, 439–453 (1974).
[CrossRef] [PubMed]

B. Beauvoit, T. Kitai, B. Chance, “Contribution of the mitochondrial component to the optical properties of the rat liver: a theoretical and practical approach,” Biophys. J. 67, 2501–2510 (1994).
[CrossRef] [PubMed]

Comments Mol. Cell. Biophys. A (1)

M. S. Patterson, “Noninvasive measurements of tissue optical properties: current status and future prospects,” Comments Mol. Cell. Biophys. A 8, 387–417 (1995).

Fractals (1)

H. Honda, S. Imayama, M. Tanemura, “A fractal-like structure in skin,” Fractals 4, 139–147 (1996).
[CrossRef]

Gastroenterology (1)

M. Rojkind, M. A. Giambrone, L. Biempica, “Collagen types in normal and cirrhotic liver,” Gastroenterology 76, 710 (1979).
[PubMed]

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

A. Dunn, R. Richards-Kortum, “Three-dimensional computation of light scattering from cells,” IEEE J. Sel. Top. Quantum Electron. 2, 898–905 (1996).
[CrossRef]

J. Acoust. Soc. Am. (2)

V. Twersky, “Acoustic bulk parameters in distributions of pair-correlated scatterers,” J. Acoust. Soc. Am. 64, 1710–1719 (1978).
[CrossRef]

P. A. J. Bascom, R. S. C. Cobbold, “On the fractal packing approach for understanding ultrasonic backscattering from blood,” J. Acoust. Soc. Am. 98, 3040–3049 (1995).
[CrossRef] [PubMed]

J. Invest. Dermatol. (1)

G. D. Weinstein, R. J. Boucek, “Collagen and elastin of human dermis,” J. Invest. Dermatol. 35, 227–229 (1960).
[PubMed]

J. Opt. Soc. Am. (2)

Magn. Reson. Imag. (1)

T. D. Scholz, S. R. Fleagle, T. L. Burns, D. J. Skorton, “Nuclear magnetic resonance relaxometry of the normal heart: relationship between collagen content and relaxation times of the four chambers,” Magn. Reson. Imag. 7, 643–648 (1989).
[CrossRef]

Med. Phys. (1)

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total attenuation coefficients and scattering phase functions of tissues and phantom materials at 633 nanometers,” Med. Phys. 14, 835–841 (1987).
[CrossRef] [PubMed]

Nature (London) (1)

R. Barer, K. F. A. Ross, S. Tkaczyk, “Refractometry of living cells,” Nature (London) 171, 720–724 (1953).
[CrossRef]

Opt. Lett. (1)

Phys. Med. Biol. (3)

J. Beuthan, O. Minet, J. Helfmann, M. Herrig, G. Müller, “The spatial variation of the refractive index in biological cells,” Phys. Med. Biol. 41, 369–382 (1996).
[CrossRef] [PubMed]

V. G. Peters, D. R. Wyman, M. S. Patterson, G. L. Frank, “Optical properties of normal and diseased human breast tissues in the visible and near infrared,” Phys. Med. Biol. 35, 1317–1334 (1990).
[CrossRef] [PubMed]

J. M. Schmitt, A. Knüttel, M. Yadlowsky, M. A. Eckhaus, “Optical-coherence tomography of a dense tissue: statistics of attenuation and backscattering,” Phys. Med. Biol. 39, 1705–1720 (1994).
[CrossRef] [PubMed]

Phys. Rev. E (1)

D. Hamburger, O. Biham, D. Avnir, “Apparent fractality emerging from models of random distributions,” Phys. Rev. E 53, 3442–3458 (1996).
[CrossRef]

Phys. Rev. Lett. (1)

V. Khare, H. M. Nussenzveig, “The theory of the glory,” Phys. Rev. Lett. 38, 1279–1282 (1977).
[CrossRef]

Proc. IEEE (1)

B. C. Wilson, E. M. Sevick, M. S. Patterson, B. Chance, “Time-dependent optical spectroscopy and imaging for biomedical applications,” Proc. IEEE 80, 918–930 (1992).
[CrossRef]

Pure Appl. Opt. (1)

B. Gélébart, E. Tinet, J.-M. Tualle, S. Avrillier, “Phase function simulation in tissue phantoms: a fractal approach,” Pure Appl. Opt. 5, 377–388 (1996).
[CrossRef]

Science (1)

G. Tearney, M. E. Brezinsky, B. E. Bouma, S. A. Boppart, C. Pitris, J. F. Southern, J. G. Fujimoto,” Science 276, 2037–2039 (1997).

Other (11)

I. J. Bigio, J. R. Mourant, J. D. Boyer, T. M. Johnson, T. Shimada, R. L. Conn, “Noninvasive identification of bladder cancer with subsurface backscattered light,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases, R. R. Alfano, ed., Proc. SPIE2135, 26–35 (1994).
[CrossRef]

G. Kumar, J. M. Schmitt, “Micro-optical properties of tissue,” in Advances in Laser and Light Spectroscopy to Diagnose Cancer and Other Diseases III: Optical Biopsy, R. R. Alfano, A. Katzir, eds., Proc. SPIE2679, 106–116 (1996).
[CrossRef]

M. Kerker, The Scattering of Light and other Electromagnetic Radiation (Academic, San Diego, Calif., 1969), pp. 351–359.

B. B. Mandelbrot, The Fractal Geometry of Nature (Freeman, San Francisco, Calif., 1982), Chap. 12.

W. F. Cheong, “Summary of optical properties,” in Optical-Thermal Response of Laser-Irradiated Tissue, A. J. Welch, M. J. C. van Gemert, eds. (Plenum, New York, 1995), pp. 275–303.

G. Yoon, “Absorption and scattering of laser light in biological media—mathematical modeling and methods for determining optical properties,” Ph.D. dissertation (University of Texas, Austin, Tex., 1988).

J. F. Beek, H. J. van Staveren, P. Posthumus, H. J. C. M. Sterenborg, M. J. C. van Gemert, “The influence of respiration on the optical properties of piglet lung at 632.8 nm,” in Medical Optical Tomography: Functional Imaging and Monitoring, G. Müller, B. Chance, R. R. Alfano, S. R. Arridge, J. Beuthan, E. Gratton, M. Kaschke, B. R. Masters, S. Svanberg, P. van der Zee, eds. (SPIE Optical Engineering Press, Bellingham, Wash., 1993), Vol. IS11, pp. 193–210.

M. Essenpreis, Thermally Induced Changes in Optical Properties of Biological Tissues (University College London, England, 1992).

P. Van der Zee, M. Essenpreis, D. T. Delpy, “Optical properties of brain tissue,” in Photon Migration and Imaging in Random Media and Tissues, B. Chance, R. R. Alfano, eds., Proc. SPIE1888, 454–465 (1993).
[CrossRef]

F. A. Duck, Physical Properties of Tissue (Academic, New York, 1990), Chap. 9.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), Appendix A.

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Figures (8)

Fig. 1
Fig. 1

Spatial variations of the refractive index of a soft biological tissue. A hypothetical index profile through several tissue elements is shown along with the profile through a statistically equivalent volume of homogeneous particles. The indices of refraction labeling the profiles are defined in Subsection 2.A.

Fig. 2
Fig. 2

Distribution of the volume fractions of spheres to which Mie theory was applied to calculate the scattering properties of tissue. Solid and dashed curves are the continuous correlation-corrected distributions given by Eq. (15) for D f = 3.7 and D f = 3.2. Dotted curves are the distributions of spheres with diameters increasing in powers of two from 50 nm to 25.6 μm that represent the D f = 3.7 distribution. The partial volume fractions of the spheres were adjusted to conform to η′(d), with their total volume fraction equal to F v .

Fig. 3
Fig. 3

Angular-scattering functions calculated with the ten-sphere tissue model for D f = 1, 3, 4, and 6. Measurements of the angular scattering functions of muscle and brain that were taken from published studies are shown for comparison. Model parameters: bkg = 1.352, p = 1.42, F v = 0.2, d m = 1.13, σ m = 2, p = 3.

Fig. 4
Fig. 4

Magnitude and wavelength dependence of the asymmetry parameter of a model tissue for values of the limiting fractal dimension D f between 3 and 4. The dependence is weakest for low values of D f because large particles contribute more to the total cross section. Model parameters: bkg = 1.352, p = 1.42, F v = 0.2, d m = 1.13, σ m = 2, p = 3.

Fig. 5
Fig. 5

Magnitude and wavelength dependence of the scattering coefficient μ s of a model tissue for values of the limiting fractal dimension D f between 3 and 4. Dashed curves are plots of the power-law function μ s ≈ λ2-D f versus wavelength. Notice the good fit of the curves for D f = 3.5 and D f = 4.0. Model parameters: bkg = 1.352, p = 1.42, F v = 0.2, d m = 1.13, σ m = 2, p = 3.

Fig. 6
Fig. 6

Magnitude and wavelength dependence of the scattering coefficient μ st of a model tissue for values of the limiting fractal dimension D f between 3 and 4. The dependence of μst on wavelength is weaker compared with that of μ s (see Fig. 5), and its log–log slope does not have a simple dependence on D f . Model parameters: bkg = 1.352, p = 1.42, F v = 0.2, d m = 1.13, σ m = 2, p = 3.

Fig. 7
Fig. 7

Magnitude and wavelength dependence of the backscattering coefficient μ b of the model tissue for values of the limiting fractal dimension D f between 3 and 4. Model parameters: bkg = 1.352, p = 1.42, F v = 0.2, d m = 1.13, σ m = 2, p = 3.

Fig. 8
Fig. 8

Contributions of the different sizes of spheres in the ten-sphere model to the total scattering coefficient μ s and backscattering coefficient μ b of the model tissue for (a) λ = 633 nm and (b) λ = 1300 nm. The calculated values of the centroids of the distributions μ s (d) and μ b (d) are labeled on the x axes as 〈dext and 〈dbk, respectively. Model parameters: D f = 3.7, bkg = 1.352, p = 1.42, F v = 0.2, d m = 1.13, σ m = 2, p = 3.

Tables (2)

Tables Icon

Table 1 Optical Coefficients of Model Tissues with Three Different Dry-Weight Fiber Fractions (ff), for Df = 3.7

Tables Icon

Table 2 Published Optical Properties of Tissues at Selected Wavelengths

Equations (18)

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

n ¯ bkg = f c n c + 1 - f c n s ,
n ¯ p = n ¯ bkg + Δ n ,
Δ n = f f n f - n s + f n n n - n c + f 0 n 0 - n c .
Δ n = f f n f - n s + 1 - f f n n - n c .
f x = C m x m exp - ln   x - ln   x m 2 2 σ m 2 ,
η d = F v C m   d 3 - D f exp - ln   d - ln   d m 2 2 σ m 2 ,
C m = σ m 2 π   d m 4 - D f exp 4 - D f 2 σ m 2 / 2 ,
F v = 0   η d d d .
lim σ m η d d 3 - D f .
μ s = i = 1 M   μ s d i ,
μ s d i = η d i v i   σ d i
P θ = i = 1 M   μ s d i P i θ i = 1 M   μ s d i .
g = i = 1 M   μ s d i g i d i i = 1 M   μ s d i ,
g i d i = - 1 1 cos   θ P i cos   θ d cos   θ
μ b = i = 1 M η d i v i   σ s d i P i 180 ° .
W s = 1 - h 4 1 + 2 h 2 .
W p = 1 - h p + 1 1 + h p - 1 p - 1 ,
η d = W p η d = 1 - η d p + 1 1 + η d p - 1 p - 1   η d .

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