Abstract

When a picosecond light pulse is incident on biological tissue, the temporal characteristics of the light backscattered from, or transmitted through, the sample carry information about the optical absorption and scattering coefficients of the tissue. We develop a simple model, based on the diffusion approximation to radiative transfer theory, which yields analytic expressions for the pulse shape in terms of the interaction coefficients of a homogeneous slab. The model predictions are in good agreement with the results of preliminary in vivo experiments and Monte Carlo simulations.

© 1989 Optical Society of America

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  1. M. Tamura, O. Hazeki, S. Nioka, B. Chance, D. Smith, “Simultaneous Measurements of Tissue Oxygen Concentration and Energy State by Near-Infrared and Nuclear Magnetic Resonance Spectroscopy,” in Chemoreceptors and Reflexes in Breathing, S. Lahiri, Ed. (Oxford U.P., New York, in press 1989).
  2. M. Cope, D. T. Delpy, “System for Long-Term Measurement of Cerebral Blood and Tissue Oxygenation on Newborn Infants by Near Infrared Transillumination,” Med. Biol. Eng. Comput. 26, 289–294 (1988).
  3. B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327–360 (1986).
  4. S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).
  5. F. F. Jobsis, “Noninvasive, Infrared Monitoring of Cerebral and Myocardial Oxygen Sufficiency and Circulatory Parameters,” Science 198, 1264–1267 (1977).
  6. R. A. J. Groenhuis, J. J. Ten Bosch, H. A. Ferwerda, “Scattering and Absorption of Turbid Materials Determined from Reflection Measurements. 2: Measuring Method and Calibration,” Appl. Opt. 22, 2463–2467 (1983).
  7. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
  8. J. M. Steinke, A. P. Shepherd, “Diffuse Reflectance of Whole Blood: Model for a Diverging Light Beam,” IEEE Trans. Biomed. Eng. BME-34, 826–834 (1987).
  9. J. A. Weinman, S. T. Shipley, “Effects of Multiple Scattering on Laser Pulses Transmitted Through Clouds,” J. Geophys. Res. 77, 7123–7128 (1972).
  10. A. Ishimaru, “Diffusion of a Pulse in Densely Distributed Scattered,” J. Opt. Soc. Am. 68, 1045–1050 (1978).
  11. K. Furutsu, “On the Diffusion Equation Derived from the Space–Time Transport Equation,” J. Opt. Soc. Am. 70, 360–366 (1980).
  12. K. Shimizu, A. Ishimaru, L. Reynolds, A. P. Bruchner, “Backscattering of a Picosecond Pulse from Densely Distributed Scatterers,” Appl. Opt. 18, 3484–3488 (1979).
  13. A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 175–190.
  14. S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy,” Rev. Mod. Phys. 15, 1–88 (1943).
  15. J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), pp. 140–144.
  16. R. P. Hemenger, “Optical Properties of Turbid Media with Specularly Reflecting Boundaries: Applications to Biological Problems,” Appl. Opt. 16, 2007–2012 (1977).
  17. G. Eason, A. Veitch, R. Nisbet, F. Turnbull, “The Theory of the Backscattering of Light by Blood,” J. Phys. D 11, 1463–1479 (1978).
  18. B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).
  19. D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength Through Tissue from Direct Time-of-Flight Measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).
  20. S. Ito, K. Furutsu, “Theory of Light Pulse Propagation Through Thick Clouds,” J. Opt. Soc. Am. 70, 366–374 (1980).
  21. I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 340.
  22. S. T. Flock, M. S. Patterson, B. C. Wilson, “Monte Carlo Modelling of Light Propagation in Highly Scattering Tissues. I. Model Predictions and Comparison with Diffusion Theory,” IEEE Trans. Biomed. Eng.1989, in press.
  23. L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).
  24. B. C. Wilson, M. S. Patterson, S. T. Flock, J. D. Moulton, “The Optical Absorption and Scattering Properties of Tissues in the Visible and Near-Infrared Wavelength Range,” in Light in Biology and Medicine, Vol. 1R. H. Douglas, J. Moan, F. Doll’Acqua eds. (Plenum, New York, 1988) pp. 45–52.
  25. S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 nm,” Med. Phys. 14, 835–841 (1987).
  26. B. C. Wilson, M. S. Patterson, “The Determination of Light Fluence Distributions in Photodynamic Therapy,” in Photodynamic Therapy of Neoplastic Disease, D. Kessel, Ed. (CRC Press, Cleveland, 1989), in press.
  27. P. van der Zee, D. T. Delpy, “Computed Point Spread Functions for Light in Tissue Using a Measured Volume Scattering Function,” Adv. Exp. Med. Biol. 222, 191–197 (1988).

1988 (4)

M. Cope, D. T. Delpy, “System for Long-Term Measurement of Cerebral Blood and Tissue Oxygenation on Newborn Infants by Near Infrared Transillumination,” Med. Biol. Eng. Comput. 26, 289–294 (1988).

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength Through Tissue from Direct Time-of-Flight Measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).

P. van der Zee, D. T. Delpy, “Computed Point Spread Functions for Light in Tissue Using a Measured Volume Scattering Function,” Adv. Exp. Med. Biol. 222, 191–197 (1988).

1987 (4)

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for Photon Migration in Turbid Biological Media,” J. Opt. Soc. Am. A 4, 423–432 (1987).

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 nm,” Med. Phys. 14, 835–841 (1987).

J. M. Steinke, A. P. Shepherd, “Diffuse Reflectance of Whole Blood: Model for a Diverging Light Beam,” IEEE Trans. Biomed. Eng. BME-34, 826–834 (1987).

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).

1986 (1)

B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327–360 (1986).

1983 (1)

1980 (2)

1979 (1)

1978 (2)

A. Ishimaru, “Diffusion of a Pulse in Densely Distributed Scattered,” J. Opt. Soc. Am. 68, 1045–1050 (1978).

G. Eason, A. Veitch, R. Nisbet, F. Turnbull, “The Theory of the Backscattering of Light by Blood,” J. Phys. D 11, 1463–1479 (1978).

1977 (2)

R. P. Hemenger, “Optical Properties of Turbid Media with Specularly Reflecting Boundaries: Applications to Biological Problems,” Appl. Opt. 16, 2007–2012 (1977).

F. F. Jobsis, “Noninvasive, Infrared Monitoring of Cerebral and Myocardial Oxygen Sufficiency and Circulatory Parameters,” Science 198, 1264–1267 (1977).

1972 (1)

J. A. Weinman, S. T. Shipley, “Effects of Multiple Scattering on Laser Pulses Transmitted Through Clouds,” J. Geophys. Res. 77, 7123–7128 (1972).

1943 (1)

S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy,” Rev. Mod. Phys. 15, 1–88 (1943).

1941 (1)

L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).

Arridge, S.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength Through Tissue from Direct Time-of-Flight Measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).

Bonner, R. F.

Bruchner, A. P.

Chance, B.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).

M. Tamura, O. Hazeki, S. Nioka, B. Chance, D. Smith, “Simultaneous Measurements of Tissue Oxygen Concentration and Energy State by Near-Infrared and Nuclear Magnetic Resonance Spectroscopy,” in Chemoreceptors and Reflexes in Breathing, S. Lahiri, Ed. (Oxford U.P., New York, in press 1989).

Chandrasekhar, S.

S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy,” Rev. Mod. Phys. 15, 1–88 (1943).

Cope, M.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength Through Tissue from Direct Time-of-Flight Measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).

M. Cope, D. T. Delpy, “System for Long-Term Measurement of Cerebral Blood and Tissue Oxygenation on Newborn Infants by Near Infrared Transillumination,” Med. Biol. Eng. Comput. 26, 289–294 (1988).

Delpy, D. T.

M. Cope, D. T. Delpy, “System for Long-Term Measurement of Cerebral Blood and Tissue Oxygenation on Newborn Infants by Near Infrared Transillumination,” Med. Biol. Eng. Comput. 26, 289–294 (1988).

P. van der Zee, D. T. Delpy, “Computed Point Spread Functions for Light in Tissue Using a Measured Volume Scattering Function,” Adv. Exp. Med. Biol. 222, 191–197 (1988).

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength Through Tissue from Direct Time-of-Flight Measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).

Duderstadt, J. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), pp. 140–144.

Eason, G.

G. Eason, A. Veitch, R. Nisbet, F. Turnbull, “The Theory of the Backscattering of Light by Blood,” J. Phys. D 11, 1463–1479 (1978).

Ferwerda, H. A.

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 nm,” Med. Phys. 14, 835–841 (1987).

S. T. Flock, M. S. Patterson, B. C. Wilson, “Monte Carlo Modelling of Light Propagation in Highly Scattering Tissues. I. Model Predictions and Comparison with Diffusion Theory,” IEEE Trans. Biomed. Eng.1989, in press.

B. C. Wilson, M. S. Patterson, S. T. Flock, J. D. Moulton, “The Optical Absorption and Scattering Properties of Tissues in the Visible and Near-Infrared Wavelength Range,” in Light in Biology and Medicine, Vol. 1R. H. Douglas, J. Moan, F. Doll’Acqua eds. (Plenum, New York, 1988) pp. 45–52.

Fountain, M.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).

Furutsu, K.

Gradsteyn, I. S.

I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 340.

Greenfeld, R.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).

Greenstein, J. L.

L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).

Groenhuis, R. A. J.

Hamilton, L. J.

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), pp. 140–144.

Havlin, S.

Hazeki, O.

M. Tamura, O. Hazeki, S. Nioka, B. Chance, D. Smith, “Simultaneous Measurements of Tissue Oxygen Concentration and Energy State by Near-Infrared and Nuclear Magnetic Resonance Spectroscopy,” in Chemoreceptors and Reflexes in Breathing, S. Lahiri, Ed. (Oxford U.P., New York, in press 1989).

Hemenger, R. P.

Henyey, L. G.

L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).

Holtom, G.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).

Ishimaru, A.

Ito, S.

Jacques, S. L.

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).

Jobsis, F. F.

F. F. Jobsis, “Noninvasive, Infrared Monitoring of Cerebral and Myocardial Oxygen Sufficiency and Circulatory Parameters,” Science 198, 1264–1267 (1977).

Kent, J.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).

McCully, K.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).

Moulton, J. D.

B. C. Wilson, M. S. Patterson, S. T. Flock, J. D. Moulton, “The Optical Absorption and Scattering Properties of Tissues in the Visible and Near-Infrared Wavelength Range,” in Light in Biology and Medicine, Vol. 1R. H. Douglas, J. Moan, F. Doll’Acqua eds. (Plenum, New York, 1988) pp. 45–52.

Nioka, S.

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).

M. Tamura, O. Hazeki, S. Nioka, B. Chance, D. Smith, “Simultaneous Measurements of Tissue Oxygen Concentration and Energy State by Near-Infrared and Nuclear Magnetic Resonance Spectroscopy,” in Chemoreceptors and Reflexes in Breathing, S. Lahiri, Ed. (Oxford U.P., New York, in press 1989).

Nisbet, R.

G. Eason, A. Veitch, R. Nisbet, F. Turnbull, “The Theory of the Backscattering of Light by Blood,” J. Phys. D 11, 1463–1479 (1978).

Nossal, R.

Patterson, M. S.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 nm,” Med. Phys. 14, 835–841 (1987).

B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327–360 (1986).

B. C. Wilson, M. S. Patterson, “The Determination of Light Fluence Distributions in Photodynamic Therapy,” in Photodynamic Therapy of Neoplastic Disease, D. Kessel, Ed. (CRC Press, Cleveland, 1989), in press.

S. T. Flock, M. S. Patterson, B. C. Wilson, “Monte Carlo Modelling of Light Propagation in Highly Scattering Tissues. I. Model Predictions and Comparison with Diffusion Theory,” IEEE Trans. Biomed. Eng.1989, in press.

B. C. Wilson, M. S. Patterson, S. T. Flock, J. D. Moulton, “The Optical Absorption and Scattering Properties of Tissues in the Visible and Near-Infrared Wavelength Range,” in Light in Biology and Medicine, Vol. 1R. H. Douglas, J. Moan, F. Doll’Acqua eds. (Plenum, New York, 1988) pp. 45–52.

Prahl, S. A.

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).

Reynolds, L.

Ryzhik, I. M.

I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 340.

Shepherd, A. P.

J. M. Steinke, A. P. Shepherd, “Diffuse Reflectance of Whole Blood: Model for a Diverging Light Beam,” IEEE Trans. Biomed. Eng. BME-34, 826–834 (1987).

Shimizu, K.

Shipley, S. T.

J. A. Weinman, S. T. Shipley, “Effects of Multiple Scattering on Laser Pulses Transmitted Through Clouds,” J. Geophys. Res. 77, 7123–7128 (1972).

Smith, D.

M. Tamura, O. Hazeki, S. Nioka, B. Chance, D. Smith, “Simultaneous Measurements of Tissue Oxygen Concentration and Energy State by Near-Infrared and Nuclear Magnetic Resonance Spectroscopy,” in Chemoreceptors and Reflexes in Breathing, S. Lahiri, Ed. (Oxford U.P., New York, in press 1989).

Steinke, J. M.

J. M. Steinke, A. P. Shepherd, “Diffuse Reflectance of Whole Blood: Model for a Diverging Light Beam,” IEEE Trans. Biomed. Eng. BME-34, 826–834 (1987).

Tamura, M.

M. Tamura, O. Hazeki, S. Nioka, B. Chance, D. Smith, “Simultaneous Measurements of Tissue Oxygen Concentration and Energy State by Near-Infrared and Nuclear Magnetic Resonance Spectroscopy,” in Chemoreceptors and Reflexes in Breathing, S. Lahiri, Ed. (Oxford U.P., New York, in press 1989).

Ten Bosch, J. J.

Turnbull, F.

G. Eason, A. Veitch, R. Nisbet, F. Turnbull, “The Theory of the Backscattering of Light by Blood,” J. Phys. D 11, 1463–1479 (1978).

van der Zee, P.

P. van der Zee, D. T. Delpy, “Computed Point Spread Functions for Light in Tissue Using a Measured Volume Scattering Function,” Adv. Exp. Med. Biol. 222, 191–197 (1988).

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength Through Tissue from Direct Time-of-Flight Measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).

Veitch, A.

G. Eason, A. Veitch, R. Nisbet, F. Turnbull, “The Theory of the Backscattering of Light by Blood,” J. Phys. D 11, 1463–1479 (1978).

Weinman, J. A.

J. A. Weinman, S. T. Shipley, “Effects of Multiple Scattering on Laser Pulses Transmitted Through Clouds,” J. Geophys. Res. 77, 7123–7128 (1972).

Weiss, G. H.

Wilson, B. C.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Total Attenuation Coefficients and Scattering Phase Functions of Tissues and Phantom Materials at 633 nm,” Med. Phys. 14, 835–841 (1987).

B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327–360 (1986).

B. C. Wilson, M. S. Patterson, “The Determination of Light Fluence Distributions in Photodynamic Therapy,” in Photodynamic Therapy of Neoplastic Disease, D. Kessel, Ed. (CRC Press, Cleveland, 1989), in press.

S. T. Flock, M. S. Patterson, B. C. Wilson, “Monte Carlo Modelling of Light Propagation in Highly Scattering Tissues. I. Model Predictions and Comparison with Diffusion Theory,” IEEE Trans. Biomed. Eng.1989, in press.

B. C. Wilson, M. S. Patterson, S. T. Flock, J. D. Moulton, “The Optical Absorption and Scattering Properties of Tissues in the Visible and Near-Infrared Wavelength Range,” in Light in Biology and Medicine, Vol. 1R. H. Douglas, J. Moan, F. Doll’Acqua eds. (Plenum, New York, 1988) pp. 45–52.

Wray, S.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength Through Tissue from Direct Time-of-Flight Measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).

Wyatt, J.

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength Through Tissue from Direct Time-of-Flight Measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).

Adv. Exp. Med. Biol. (1)

P. van der Zee, D. T. Delpy, “Computed Point Spread Functions for Light in Tissue Using a Measured Volume Scattering Function,” Adv. Exp. Med. Biol. 222, 191–197 (1988).

Anal. Biochem. (1)

B. Chance, S. Nioka, J. Kent, K. McCully, M. Fountain, R. Greenfeld, G. Holtom, “Time Resolved Spectroscopy of Hemoglobin and Myoglobin in Resting and Ischemic Muscle,” Anal. Biochem. 174, 698–707 (1988).

Appl. Opt. (3)

Astrophys. J. (1)

L. G. Henyey, J. L. Greenstein, “Diffuse Radiation in the Galaxy,” Astrophys. J. 93, 70–83 (1941).

IEEE Trans. Biomed. Eng. (1)

J. M. Steinke, A. P. Shepherd, “Diffuse Reflectance of Whole Blood: Model for a Diverging Light Beam,” IEEE Trans. Biomed. Eng. BME-34, 826–834 (1987).

J. Geophys. Res. (1)

J. A. Weinman, S. T. Shipley, “Effects of Multiple Scattering on Laser Pulses Transmitted Through Clouds,” J. Geophys. Res. 77, 7123–7128 (1972).

J. Opt. Soc. Am. (3)

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

J. Phys. D (1)

G. Eason, A. Veitch, R. Nisbet, F. Turnbull, “The Theory of the Backscattering of Light by Blood,” J. Phys. D 11, 1463–1479 (1978).

Lasers Surg. Med. (1)

S. L. Jacques, S. A. Prahl, “Modeling Optical and Thermal Distributions in Tissue During Laser Irradiation,” Lasers Surg. Med. 6, 494–503 (1987).

Med. Biol. Eng. Comput. (1)

M. Cope, D. T. Delpy, “System for Long-Term Measurement of Cerebral Blood and Tissue Oxygenation on Newborn Infants by Near Infrared Transillumination,” Med. Biol. Eng. Comput. 26, 289–294 (1988).

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 nm,” Med. Phys. 14, 835–841 (1987).

Phys. Med. Biol. (2)

B. C. Wilson, M. S. Patterson, “The Physics of Photodynamic Therapy,” Phys. Med. Biol. 31, 327–360 (1986).

D. T. Delpy, M. Cope, P. van der Zee, S. Arridge, S. Wray, J. Wyatt, “Estimation of Optical Pathlength Through Tissue from Direct Time-of-Flight Measurements,” Phys. Med. Biol. 33, 1433–1442 (1988).

Rev. Mod. Phys. (1)

S. Chandrasekhar, “Stochastic Problems in Physics and Astronomy,” Rev. Mod. Phys. 15, 1–88 (1943).

Science (1)

F. F. Jobsis, “Noninvasive, Infrared Monitoring of Cerebral and Myocardial Oxygen Sufficiency and Circulatory Parameters,” Science 198, 1264–1267 (1977).

Other (7)

J. J. Duderstadt, L. J. Hamilton, Nuclear Reactor Analysis (Wiley, New York, 1976), pp. 140–144.

A. Ishimaru, Wave Propagation and Scattering in Random Media (Academic, New York, 1978), pp. 175–190.

I. S. Gradsteyn, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1980), p. 340.

S. T. Flock, M. S. Patterson, B. C. Wilson, “Monte Carlo Modelling of Light Propagation in Highly Scattering Tissues. I. Model Predictions and Comparison with Diffusion Theory,” IEEE Trans. Biomed. Eng.1989, in press.

M. Tamura, O. Hazeki, S. Nioka, B. Chance, D. Smith, “Simultaneous Measurements of Tissue Oxygen Concentration and Energy State by Near-Infrared and Nuclear Magnetic Resonance Spectroscopy,” in Chemoreceptors and Reflexes in Breathing, S. Lahiri, Ed. (Oxford U.P., New York, in press 1989).

B. C. Wilson, M. S. Patterson, S. T. Flock, J. D. Moulton, “The Optical Absorption and Scattering Properties of Tissues in the Visible and Near-Infrared Wavelength Range,” in Light in Biology and Medicine, Vol. 1R. H. Douglas, J. Moan, F. Doll’Acqua eds. (Plenum, New York, 1988) pp. 45–52.

B. C. Wilson, M. S. Patterson, “The Determination of Light Fluence Distributions in Photodynamic Therapy,” in Photodynamic Therapy of Neoplastic Disease, D. Kessel, Ed. (CRC Press, Cleveland, 1989), in press.

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