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

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

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

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).

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).

1986

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

1983

1980

1979

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).

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

1977

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

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

1972

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

1943

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

1941

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.

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).

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).

Delpy, D. T.

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).

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).

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).

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.

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.

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.

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).

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.

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.

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.

Astrophys. J.

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

IEEE Trans. Biomed. Eng.

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.

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.

J. Opt. Soc. Am. A

J. Phys. D

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.

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.

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.

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.

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).

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

Rev. Mod. Phys.

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

Science

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

Other

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.

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.

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.

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).

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Figures (4)

Fig. 1
Fig. 1

(a) Geometry for the calculation of R(ρ,t) for a semi-infinite homogeneous medium. The incident pencil beam is assumed to create an isotropic photon source at depth z0, indicated by the filled circle. The boundary condition ϕ(ρ,0,t) = 0 can be met by adding a negative source indicated by the open circle. (b) Geometry for the calculation of the time resolved reflectance and transmittance from a homogeneous slab. The boundary conditions ϕ(ρ,0,t) = 0 and ϕ(ρ,d,t) = 0 can be met by adding an infinite series of dipole photon sources. The first four are shown in this illustration.

Fig. 2
Fig. 2

Experimentally recorded reflectance signal from the calf muscle of a human volunteer at 760 nm is indicated by the solid line. The detector fiber bundle was positioned 40 mm from the pulsed laser source. The open circles show the pulse shape predicted by the diffusion model expression of Eq. (7) normalized to the peak value.

Fig. 3
Fig. 3

Monte Carlo (symbols) and diffusion model (smooth curves) calculation of R(t), the spatially integrated time resolved reflectance, from a semi-infinite tissue slab and a slab 10 mm thick. The optical properties of the slab are typical of human skeletal muscle at 760 nm. The diffusion model results have been matched to the Monte Carlo data at 50 ps. Note that the units of R(t) are ps−1 as R(t) is the number of photons reaching the surface per unit time per incident photon.

Fig. 4
Fig. 4

Comparison of the predictions of the diffusion model (solid line) and Monte Carlo (dashed line) results of Delpy et al.19 for the spatially integrated time-resolved transmittance through a 10-mm tissue slab. Each curve has been normalized at its peak value.

Equations (16)

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

1 c t ϕ ( r , t ) D 2 ϕ ( r , t ) + μ a ϕ ( r , t ) = S ( r , t ) ,
D = { 3 [ μ a + ( 1 g ) μ s ] } 1 ,
ϕ ( r , t ) = c ( 4 π D c t ) 3 / 2 exp ( r 2 4 D c t μ a c t ) .
z 0 = [ ( 1 g ) μ s ] 1
ϕ ( ρ , z , t ) = c ( 4 π D c t ) 3 / 2 exp ( μ a c t ) { exp [ ( z z 0 ) 2 + ρ 2 4 D c t ] exp [ ( z + z 0 ) 2 + ρ 2 4 D c t ] } .
J ( ρ , 0 , t ) = D ϕ ( ρ , z , t ) | z ¯ 0 ,
R ( ρ , t ) = | J ( ρ , 0 , t ) | = ( 4 π D c ) 3 / 2 z 0 t 5 / 2 exp ( μ a c t ) exp ( ρ 2 + z 0 2 4 D c t ) .
d d t log e R ( ρ , t ) = 5 2 t μ a c + ρ 2 4 D c t 2 .
lim t d d t log e R ( ρ , t ) = μ a c
( 1 g ) μ s = 1 3 ρ 2 ( 4 μ a c 2 t max 2 + 10 c t max ) μ a .
R ( t ) = 0 R ( ρ , t ) 2 π ρ d ρ = ( 4 π D c ) 1 / 2 z 0 t 3 / 2 exp ( μ a c t ) exp ( z 0 2 4 D c t ) .
R ( ρ , d , t ) = ( 4 π D c ) 3 / 2 t 5 / 2 exp ( μ a c t ) exp ( ρ 2 4 D c t ) × { z 0 exp ( z 0 2 4 D c t ) ( 2 d z 0 ) exp [ ( 2 d z 0 ) 2 4 D c t ] + ( 2 d + z 0 ) exp [ ( 2 d + z 0 ) 2 4 D c t ] } .
R ( d , t ) = ( 4 π D c ) 1 / 2 t 3 / 2 exp ( μ 0 c t ) × { z 0 exp ( z 0 2 4 D c t ) ( 2 d z 0 ) exp [ ( 2 d z 0 ) 2 4 D c t ] + ( 2 d + z 0 ) exp [ ( 2 d + z 0 ) 2 4 D c t ] } .
T ( ρ , d , t ) = ( 4 π D c ) 3 / 2 t 5 / 2 exp ( μ a c t ) exp ( ρ 2 4 D c t ) × { ( d z 0 ) exp [ ( d z 0 ) 2 4 D c t ] ( d + z 0 ) exp [ ( d + z 0 ) 2 4 D c t ] + ( 3 d z 0 ) exp [ ( 3 d z 0 ) 2 4 D c t ] ( 3 d + z 0 ) exp [ ( 3 d + z 0 ) 2 4 D c t ] } ,
T ( d , t ) = ( 4 π D c ) 1 / 2 t 3 / 2 exp ( μ a c t ) × { ( d z 0 ) exp [ ( d z 0 ) 2 4 D c t ] ( d + z 0 ) exp [ ( d + z 0 ) 2 4 D c t ] + ( 3 d z 0 ) exp [ ( 3 d z 0 ) 2 4 D c t ] ( 3 d + z 0 ) exp [ ( 3 d + z 0 ) 2 4 D c t ] } .
c t = ( 4 μ a D ) 1 / 2 ( d z 0 ) exp ( 2 z 0 μ a / D ) ( d + z 0 ) exp ( 2 z 0 μ a / D ) 1 ,

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