Abstract

Photon-diffusion theory has had limited success in modeling the optical transmittance of whole blood. Therefore we have developed a new photon-diffusion model of the optical absorbance of blood. The model has benefited from experiments designed to test its fundamental assumptions, and it has been compared extensively with transmittance data from whole blood. The model is consistent with both experimental and theoretical notions. Furthermore, when all parameters associated with a given optical geometry are known, the model needs no variational parameters to predict the absolute transmittance of whole blood. However, even if the exact value of the incident light intensity is unknown (which is the case in many situations), only a single additive constant is required to scale experiment to theory. Finally, the model is shown to be useful for simulating scattering effects and for delineating the relative contributions of the diffuse transmittance and the collimated transmittance to the total optical density of whole blood. Applications of the model include oximetry and measurements of the arteriovenous oxygen difference in whole, undiluted blood.

© 1988 Optical Society of America

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  7. J. M. Schmitt, “Optical measurement of blood oxygen by implantable telemetry,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1986).
  8. R. J. Zdrojkowski, N. R. Pisharoty, “Optical transmission and reflection by blood,” IEEE Trans. Biomed. Eng. BME-17, 122–128 (1970).
    [CrossRef]
  9. M. K. Moaveni, “A multiple scattering field theory applied to whole blood,” Ph.D. dissertation (University of Washington, Seattle, Wash., 1970).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  12. J. M. Steinke, A. P. Shepherd, “Role of light scattering in spectrophotometric measurements of arteriovenous oxygen difference,” IEEE Trans. Biomed. Eng. BME-33, 729–734 (1986).
    [CrossRef]
  13. J. M. Steinke, A. P. Shepherd, “Role of light scattering in whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-33, 294–301 (1986).
    [CrossRef]
  14. L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
    [CrossRef] [PubMed]
  15. J. M. Schmitt, J. D. Meindl, F. G. Mihm, “An integrated circuit-based optical sensor for” in vivo measurement of blood oxygenation,” IEEE Trans. Biomed. Eng. BME-33, 98–107 (1986).
    [CrossRef]
  16. J. M. Steinke, A. P. Shepherd, “Reflectance measurements of hematocrit and oxyhemoglobin saturation,” Am. J. Physiol. 253, H147–H153 (1987).
    [PubMed]
  17. 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).
    [CrossRef]
  18. M. K. Moaveni, A. Razani, “Application of invariant imbedding for the study of optical transmission and reflection by blood: Part II—Application,” Indian J. Biochem. Biophys. 11, 30–35 (1974).
    [PubMed]
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  21. I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 970.
  22. A. P. Shepherd, J. W. Kiel, G. L. Riedel, “Evaluation of light-emitting diodes for whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-31, 723–725 (1984).
    [CrossRef]
  23. N. R. Pisharoty, “Optical scattering in blood,” Ph.D. dissertation (Carnegie-Mellon University, Pittsburgh, Pa., 1971).
  24. R. J. Hirko, R. J. Fretterd, R. L. Longini, “Application of the diffusion dipole to modelling the optical characteristics of blood,” Med. Biol. Eng. 13, 192–195 (1975).
    [CrossRef] [PubMed]
  25. C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.
  26. L. Reynolds, J. Molocho, C. Johnson, A. Ishimaru, “Optical cross-sections of human erythrocytes,” in Proceedings of the 27th Annual Conference on Engineering in Medicine and Biology (Alliance for Engineering in Medicine and Biology, Chevy Chase, Md., 1974), Vol. 16, p. 58.
  27. R. Pierce, “An experimental determination of the average scattering and absorption cross sections of human red blood cells for near infrared light,” M.S. thesis (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1972).
  28. G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
    [CrossRef] [PubMed]
  29. L. O. Reynolds, N. J. McCormick, “Approximate two-parameter phase function for light scattering,” J. Opt. Soc. Am. 70, 1206–1212 (1980).
    [CrossRef]
  30. Varian Instrument Division, Optimum Parameters for Spectrophotometry (Varian, Palo Alto, Calif., 1975).
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    [PubMed]
  32. N. M. Anderson, P. Sekelj, “Studies on the light transmission of nonhemolyzed whole blood. Determination of oxygen saturation,” J. Lab. Clin. Med. 65, 153–166 (1965).
    [PubMed]
  33. J. W. Kiel, A. P. Shepherd, “A microcomputer oximeter for whole blood,” Am. J. Physiol. 244, H722–H725 (1983).
    [PubMed]

1987 (3)

J. M. Steinke, A. P. Shepherd, “Reflectance measurements of hematocrit and oxyhemoglobin saturation,” Am. J. Physiol. 253, H147–H153 (1987).
[PubMed]

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).
[CrossRef]

S. A. W. Gerstl, A. Zardecki, W. P. Unruh, D. M. Stupin, G. H. Stokes, N. E. Elliott, “Off-axis multiple scattering of a laser beam in turbid media: comparison of theory and experiment,” Appl. Opt. 26, 779–785 (1987).
[CrossRef]

1986 (3)

J. M. Steinke, A. P. Shepherd, “Role of light scattering in spectrophotometric measurements of arteriovenous oxygen difference,” IEEE Trans. Biomed. Eng. BME-33, 729–734 (1986).
[CrossRef]

J. M. Steinke, A. P. Shepherd, “Role of light scattering in whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-33, 294–301 (1986).
[CrossRef]

J. M. Schmitt, J. D. Meindl, F. G. Mihm, “An integrated circuit-based optical sensor for” in vivo measurement of blood oxygenation,” IEEE Trans. Biomed. Eng. BME-33, 98–107 (1986).
[CrossRef]

1984 (1)

A. P. Shepherd, J. W. Kiel, G. L. Riedel, “Evaluation of light-emitting diodes for whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-31, 723–725 (1984).
[CrossRef]

1983 (2)

1980 (3)

1976 (2)

1975 (1)

R. J. Hirko, R. J. Fretterd, R. L. Longini, “Application of the diffusion dipole to modelling the optical characteristics of blood,” Med. Biol. Eng. 13, 192–195 (1975).
[CrossRef] [PubMed]

1974 (1)

M. K. Moaveni, A. Razani, “Application of invariant imbedding for the study of optical transmission and reflection by blood: Part II—Application,” Indian J. Biochem. Biophys. 11, 30–35 (1974).
[PubMed]

1970 (3)

1965 (1)

N. M. Anderson, P. Sekelj, “Studies on the light transmission of nonhemolyzed whole blood. Determination of oxygen saturation,” J. Lab. Clin. Med. 65, 153–166 (1965).
[PubMed]

1951 (1)

K. Kramer, J. O. Elam, G. A. Saxton, W. N. Elam, “Influence of oxygen saturation, erythrocyte concentration and optical depth upon the red and near-infrared light transmittance of whole blood,” Am. J. Physiol. 165, 229–246 (1951).
[PubMed]

1948 (1)

Abramowitz, M.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Wiley, New York, 1972), pp. 375, 378–379.

Anderson, N. M.

N. M. Anderson, P. Sekelj, “Studies on the light transmission of nonhemolyzed whole blood. Determination of oxygen saturation,” J. Lab. Clin. Med. 65, 153–166 (1965).
[PubMed]

Bell, G.

G. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970), pp. 93–97.

Bohren, C. F.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.

Cheung, R. L. T.

Chien, S.

H. H. Lipowsky, S. Usami, S. Chien, R. N. Pittman, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

Elam, J. O.

K. Kramer, J. O. Elam, G. A. Saxton, W. N. Elam, “Influence of oxygen saturation, erythrocyte concentration and optical depth upon the red and near-infrared light transmittance of whole blood,” Am. J. Physiol. 165, 229–246 (1951).
[PubMed]

Elam, W. N.

K. Kramer, J. O. Elam, G. A. Saxton, W. N. Elam, “Influence of oxygen saturation, erythrocyte concentration and optical depth upon the red and near-infrared light transmittance of whole blood,” Am. J. Physiol. 165, 229–246 (1951).
[PubMed]

Elliott, N. E.

Fretterd, R. J.

R. J. Hirko, R. J. Fretterd, R. L. Longini, “Application of the diffusion dipole to modelling the optical characteristics of blood,” Med. Biol. Eng. 13, 192–195 (1975).
[CrossRef] [PubMed]

Gerstl, S. A. W.

Glasstone, S.

G. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970), pp. 93–97.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 970.

Hirko, R. J.

R. J. Hirko, R. J. Fretterd, R. L. Longini, “Application of the diffusion dipole to modelling the optical characteristics of blood,” Med. Biol. Eng. 13, 192–195 (1975).
[CrossRef] [PubMed]

Huffman, D. R.

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.

Ishimaru, A.

A. Ishimaru, Y. Kuga, R. L. T. Cheung, K. Shimizu, “Scattering and diffusion of a beam wave in randomly distributed scatterers,” J. Opt. Soc. Am. 73, 131–136 (1983).
[CrossRef]

L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
[CrossRef] [PubMed]

L. Reynolds, J. Molocho, C. Johnson, A. Ishimaru, “Optical cross-sections of human erythrocytes,” in Proceedings of the 27th Annual Conference on Engineering in Medicine and Biology (Alliance for Engineering in Medicine and Biology, Chevy Chase, Md., 1974), Vol. 16, p. 58.

Johnson, C.

L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
[CrossRef] [PubMed]

L. Reynolds, J. Molocho, C. Johnson, A. Ishimaru, “Optical cross-sections of human erythrocytes,” in Proceedings of the 27th Annual Conference on Engineering in Medicine and Biology (Alliance for Engineering in Medicine and Biology, Chevy Chase, Md., 1974), Vol. 16, p. 58.

Kiel, J. W.

A. P. Shepherd, J. W. Kiel, G. L. Riedel, “Evaluation of light-emitting diodes for whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-31, 723–725 (1984).
[CrossRef]

J. W. Kiel, A. P. Shepherd, “A microcomputer oximeter for whole blood,” Am. J. Physiol. 244, H722–H725 (1983).
[PubMed]

Kramer, K.

K. Kramer, J. O. Elam, G. A. Saxton, W. N. Elam, “Influence of oxygen saturation, erythrocyte concentration and optical depth upon the red and near-infrared light transmittance of whole blood,” Am. J. Physiol. 165, 229–246 (1951).
[PubMed]

Kubelka, P.

Kuga, Y.

Lipowsky, H. H.

H. H. Lipowsky, S. Usami, S. Chien, R. N. Pittman, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

Longini, R. L.

R. J. Hirko, R. J. Fretterd, R. L. Longini, “Application of the diffusion dipole to modelling the optical characteristics of blood,” Med. Biol. Eng. 13, 192–195 (1975).
[CrossRef] [PubMed]

McCormick, N. J.

L. O. Reynolds, N. J. McCormick, “Approximate two-parameter phase function for light scattering,” J. Opt. Soc. Am. 70, 1206–1212 (1980).
[CrossRef]

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef] [PubMed]

Meindl, J. D.

J. M. Schmitt, J. D. Meindl, F. G. Mihm, “An integrated circuit-based optical sensor for” in vivo measurement of blood oxygenation,” IEEE Trans. Biomed. Eng. BME-33, 98–107 (1986).
[CrossRef]

Mihm, F. G.

J. M. Schmitt, J. D. Meindl, F. G. Mihm, “An integrated circuit-based optical sensor for” in vivo measurement of blood oxygenation,” IEEE Trans. Biomed. Eng. BME-33, 98–107 (1986).
[CrossRef]

Moaveni, M. K.

M. K. Moaveni, A. Razani, “Application of invariant imbedding for the study of optical transmission and reflection by blood: Part II—Application,” Indian J. Biochem. Biophys. 11, 30–35 (1974).
[PubMed]

M. K. Moaveni, “A multiple scattering field theory applied to whole blood,” Ph.D. dissertation (University of Washington, Seattle, Wash., 1970).

Molocho, J.

L. Reynolds, J. Molocho, C. Johnson, A. Ishimaru, “Optical cross-sections of human erythrocytes,” in Proceedings of the 27th Annual Conference on Engineering in Medicine and Biology (Alliance for Engineering in Medicine and Biology, Chevy Chase, Md., 1974), Vol. 16, p. 58.

Pedersen, G. D.

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef] [PubMed]

Pierce, R.

R. Pierce, “An experimental determination of the average scattering and absorption cross sections of human red blood cells for near infrared light,” M.S. thesis (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1972).

Pisharoty, N. R.

R. J. Zdrojkowski, N. R. Pisharoty, “Optical transmission and reflection by blood,” IEEE Trans. Biomed. Eng. BME-17, 122–128 (1970).
[CrossRef]

N. R. Pisharoty, “Optical scattering in blood,” Ph.D. dissertation (Carnegie-Mellon University, Pittsburgh, Pa., 1971).

Pittman, R. N.

H. H. Lipowsky, S. Usami, S. Chien, R. N. Pittman, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

Razani, A.

M. K. Moaveni, A. Razani, “Application of invariant imbedding for the study of optical transmission and reflection by blood: Part II—Application,” Indian J. Biochem. Biophys. 11, 30–35 (1974).
[PubMed]

Reynolds, L.

L. Reynolds, C. Johnson, A. Ishimaru, “Diffuse reflectance from a finite blood medium: applications to the modeling of fiber optic catheters,” Appl. Opt. 15, 2059–2067 (1976).
[CrossRef] [PubMed]

L. Reynolds, J. Molocho, C. Johnson, A. Ishimaru, “Optical cross-sections of human erythrocytes,” in Proceedings of the 27th Annual Conference on Engineering in Medicine and Biology (Alliance for Engineering in Medicine and Biology, Chevy Chase, Md., 1974), Vol. 16, p. 58.

Reynolds, L. O.

L. O. Reynolds, N. J. McCormick, “Approximate two-parameter phase function for light scattering,” J. Opt. Soc. Am. 70, 1206–1212 (1980).
[CrossRef]

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef] [PubMed]

L. O. Reynolds, “Optical diffuse reflectance and transmittance from an anisotropically scattering finite blood medium,” Ph.D. dissertation (University of Washington, Seattle, Wash., 1975).

Riedel, G. L.

A. P. Shepherd, J. W. Kiel, G. L. Riedel, “Evaluation of light-emitting diodes for whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-31, 723–725 (1984).
[CrossRef]

Ryzhik, I. M.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 970.

Saxton, G. A.

K. Kramer, J. O. Elam, G. A. Saxton, W. N. Elam, “Influence of oxygen saturation, erythrocyte concentration and optical depth upon the red and near-infrared light transmittance of whole blood,” Am. J. Physiol. 165, 229–246 (1951).
[PubMed]

Schmitt, J. M.

J. M. Schmitt, J. D. Meindl, F. G. Mihm, “An integrated circuit-based optical sensor for” in vivo measurement of blood oxygenation,” IEEE Trans. Biomed. Eng. BME-33, 98–107 (1986).
[CrossRef]

J. M. Schmitt, “Optical measurement of blood oxygen by implantable telemetry,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1986).

Sekelj, P.

N. M. Anderson, P. Sekelj, “Studies on the light transmission of nonhemolyzed whole blood. Determination of oxygen saturation,” J. Lab. Clin. Med. 65, 153–166 (1965).
[PubMed]

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).
[CrossRef]

J. M. Steinke, A. P. Shepherd, “Reflectance measurements of hematocrit and oxyhemoglobin saturation,” Am. J. Physiol. 253, H147–H153 (1987).
[PubMed]

J. M. Steinke, A. P. Shepherd, “Role of light scattering in spectrophotometric measurements of arteriovenous oxygen difference,” IEEE Trans. Biomed. Eng. BME-33, 729–734 (1986).
[CrossRef]

J. M. Steinke, A. P. Shepherd, “Role of light scattering in whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-33, 294–301 (1986).
[CrossRef]

A. P. Shepherd, J. W. Kiel, G. L. Riedel, “Evaluation of light-emitting diodes for whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-31, 723–725 (1984).
[CrossRef]

J. W. Kiel, A. P. Shepherd, “A microcomputer oximeter for whole blood,” Am. J. Physiol. 244, H722–H725 (1983).
[PubMed]

Shimizu, K.

Stegun, I. A.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Wiley, New York, 1972), pp. 375, 378–379.

Steinke, J. M.

J. M. Steinke, A. P. Shepherd, “Reflectance measurements of hematocrit and oxyhemoglobin saturation,” Am. J. Physiol. 253, H147–H153 (1987).
[PubMed]

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).
[CrossRef]

J. M. Steinke, A. P. Shepherd, “Role of light scattering in spectrophotometric measurements of arteriovenous oxygen difference,” IEEE Trans. Biomed. Eng. BME-33, 729–734 (1986).
[CrossRef]

J. M. Steinke, A. P. Shepherd, “Role of light scattering in whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-33, 294–301 (1986).
[CrossRef]

Stokes, G. H.

Stupin, D. M.

Tam, W. G.

Twersky, V.

Unruh, W. P.

Usami, S.

H. H. Lipowsky, S. Usami, S. Chien, R. N. Pittman, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

Zardecki, A.

Zdrojkowski, R. J.

R. J. Zdrojkowski, N. R. Pisharoty, “Optical transmission and reflection by blood,” IEEE Trans. Biomed. Eng. BME-17, 122–128 (1970).
[CrossRef]

Am. J. Physiol. (3)

J. M. Steinke, A. P. Shepherd, “Reflectance measurements of hematocrit and oxyhemoglobin saturation,” Am. J. Physiol. 253, H147–H153 (1987).
[PubMed]

J. W. Kiel, A. P. Shepherd, “A microcomputer oximeter for whole blood,” Am. J. Physiol. 244, H722–H725 (1983).
[PubMed]

K. Kramer, J. O. Elam, G. A. Saxton, W. N. Elam, “Influence of oxygen saturation, erythrocyte concentration and optical depth upon the red and near-infrared light transmittance of whole blood,” Am. J. Physiol. 165, 229–246 (1951).
[PubMed]

Appl. Opt. (3)

Biophys. J. (1)

G. D. Pedersen, N. J. McCormick, L. O. Reynolds, “Transport calculations for light scattering in blood,” Biophys. J. 16, 199–207 (1976).
[CrossRef] [PubMed]

IEEE Trans. Biomed. Eng. (6)

A. P. Shepherd, J. W. Kiel, G. L. Riedel, “Evaluation of light-emitting diodes for whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-31, 723–725 (1984).
[CrossRef]

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).
[CrossRef]

R. J. Zdrojkowski, N. R. Pisharoty, “Optical transmission and reflection by blood,” IEEE Trans. Biomed. Eng. BME-17, 122–128 (1970).
[CrossRef]

J. M. Steinke, A. P. Shepherd, “Role of light scattering in spectrophotometric measurements of arteriovenous oxygen difference,” IEEE Trans. Biomed. Eng. BME-33, 729–734 (1986).
[CrossRef]

J. M. Steinke, A. P. Shepherd, “Role of light scattering in whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-33, 294–301 (1986).
[CrossRef]

J. M. Schmitt, J. D. Meindl, F. G. Mihm, “An integrated circuit-based optical sensor for” in vivo measurement of blood oxygenation,” IEEE Trans. Biomed. Eng. BME-33, 98–107 (1986).
[CrossRef]

Indian J. Biochem. Biophys. (1)

M. K. Moaveni, A. Razani, “Application of invariant imbedding for the study of optical transmission and reflection by blood: Part II—Application,” Indian J. Biochem. Biophys. 11, 30–35 (1974).
[PubMed]

J. Lab. Clin. Med. (1)

N. M. Anderson, P. Sekelj, “Studies on the light transmission of nonhemolyzed whole blood. Determination of oxygen saturation,” J. Lab. Clin. Med. 65, 153–166 (1965).
[PubMed]

J. Opt. Soc. Am. (5)

Med. Biol. Eng. (1)

R. J. Hirko, R. J. Fretterd, R. L. Longini, “Application of the diffusion dipole to modelling the optical characteristics of blood,” Med. Biol. Eng. 13, 192–195 (1975).
[CrossRef] [PubMed]

Microvasc. Res. (1)

H. H. Lipowsky, S. Usami, S. Chien, R. N. Pittman, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

Other (11)

N. R. Pisharoty, “Optical scattering in blood,” Ph.D. dissertation (Carnegie-Mellon University, Pittsburgh, Pa., 1971).

C. F. Bohren, D. R. Huffman, Absorption and Scattering of Light by Small Particles (Wiley, New York, 1983), pp. 82–129.

L. Reynolds, J. Molocho, C. Johnson, A. Ishimaru, “Optical cross-sections of human erythrocytes,” in Proceedings of the 27th Annual Conference on Engineering in Medicine and Biology (Alliance for Engineering in Medicine and Biology, Chevy Chase, Md., 1974), Vol. 16, p. 58.

R. Pierce, “An experimental determination of the average scattering and absorption cross sections of human red blood cells for near infrared light,” M.S. thesis (Department of Electrical Engineering, University of Washington, Seattle, Wash., 1972).

G. Bell, S. Glasstone, Nuclear Reactor Theory (Van Nostrand Reinhold, New York, 1970), pp. 93–97.

M. Abramowitz, I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Wiley, New York, 1972), pp. 375, 378–379.

I. S. Gradshteyn, I. M. Ryzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), p. 970.

M. K. Moaveni, “A multiple scattering field theory applied to whole blood,” Ph.D. dissertation (University of Washington, Seattle, Wash., 1970).

L. O. Reynolds, “Optical diffuse reflectance and transmittance from an anisotropically scattering finite blood medium,” Ph.D. dissertation (University of Washington, Seattle, Wash., 1975).

J. M. Schmitt, “Optical measurement of blood oxygen by implantable telemetry,” Ph.D. dissertation (Stanford University, Stanford, Calif., 1986).

Varian Instrument Division, Optimum Parameters for Spectrophotometry (Varian, Palo Alto, Calif., 1975).

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Figures (8)

Fig. 1
Fig. 1

Optical geometry of the photon-diffusion model [see Eq. (21)]. Assumptions are a collimated source of radius a normally incident upon a blood medium of thickness z0. Detector aperture is circular, of radius b, and coaxial with the source aperture.

Fig. 2
Fig. 2

Collimated transmittance of He–Ne laser light in whole blood. O.D. data are from experiments with RBC suspensions from two different dogs. Solid curve is a graph of O.D. = 15.1H(1 − H) (1.4 − H). Functional dependence on H in this figure determines the dependence of Σs on H [Eq. (20)].

Fig. 3
Fig. 3

Comparison of new transmittance model, Eq. (21), with experimental data. Data shown are absolute O.D. versus hematocrit for oxygenated RBC suspensions with a He–Ne laser in cuvette geometry. I0 is known exactly for these data since the whole incident laser beam can be captured by detector. Agreement between experiment (data points) and model (curve) shows that the model predicts absolute transmittance accurately. For the lower curve, a = 0.685 mm, b = 1.0 mm; for the upper curve, a = 0.685 mm, b = 1.0 mm.

Fig. 4
Fig. 4

O.D. versus hematocrit at LED wavelengths. Data shown are measured O.D. for cuvette geometries at 800- and 813-nm LED wavelengths. An additive constant (denoted offset) has been added to the data to account for inability to measure I0 accurately with LED’s. For the lower curve, a = 2.0 mm, b = 1.5 mm; for the upper curve, a = 2.5 mm, b = 1.0 mm.

Fig. 5
Fig. 5

Effect of oxygenation on O.D. and verification of absolute transmittance predicted by model, (a) Data are from completely oxygenated and deoxygenated red-cell suspensions. A single additive offset has been added to both sets of data, (b) Difference of two sets of data in (a). Data points are the differences in raw data values, and the curve is the difference of the theoretical curves of upper panel. The agreement of data and theory in (b) verifies that absolute transmittance is correctly predicted by new model. For all curves, a = 3.0 mm, b = 1.5 mm.

Fig. 6
Fig. 6

O.D. versus oxyhemoglobin saturation at fixed hematocrit. Additive offset is same as used for both curves in Fig. 5(a). Data are highly linear in oxyhemoglobin saturation (r = 0.999), and the theoretical curve is itself significantly linear, although theory falls below data at low oxyhemoglobin saturation. For this curve, a = 3.0 mm, b = 1.5 mm.

Fig. 7
Fig. 7

Effect of μ ¯, mean cosine of scattering angle, on predictions of new model [Eq. (21)]. Simulation parameters are the same as those used to generate the theoretical curve of Fig. 3 (lower curve). Note that as μ ¯ decreases, the theoretical curve conforms more closely to the shape of the data in Fig. 3 (lower curve). This trend indicates that values of μ ¯ from Mie theory are too high.

Fig. 8
Fig. 8

Relative contributions of Tdiff and Tc to the total transmittance of whole blood [Eq. (21)]. For comparison, parameters used in this figure are the same as those used to generate the lower, theoretical curve of Fig. 3. Note that between H = 3% and H = 97%, Ttot is essentially just Tdiff. When an incorrect expression [Eqs. (17) and (18)] is used for Tc, Tc obscures Tdiff so that Ttot would be approximately Tc.

Tables (1)

Tables Icon

Table 1 Parameters for Model of Optical Absorbance by Whole Blood As Measured by the Authors

Equations (29)

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a = H [ σ a o S O 2 + σ a r ( 1 S O 2 ) ] / V , s = H σ s ( 1 H ) / V ( incorrect , explained below ) , s t = s ( 1 μ ¯ ) , t = a + s t ,
D 2 Ψ ( ρ ¯ ) + a Ψ ( ρ ¯ ) = S ( ρ ¯ ) ,
Ψ ( ρ ¯ ) + 3 K D Ψ ( ρ ¯ ) / z = 0 at z = z 0 ,
Ψ ( ρ ¯ ) 3 K D Ψ ( ρ ¯ ) / z = 0 at z = 0 ,
Ψ ( ρ ¯ ) = ( 1 / D ) ν S ( ρ ¯ ) G ( ρ ¯ ; ρ ¯ ) d υ ,
2 G ( ρ ¯ ; ρ ¯ ) ( a / D ) G ( ρ ¯ ; ρ ¯ ) = δ ( ρ ¯ ρ ¯ ) .
G ( r , z ; r , z ) = n = 1 ( 1 / 2 π N n ) sin ( k n z + γ n ) sin ( k n z + γ n ) × { I 0 ( λ n r ) K 0 ( λ n r ) , r < r I 0 ( λ n r ) K 0 ( λ n r ) , r > r ,
tan ( k n z 0 ) = 2 g k n / [ ( g k n ) 2 1 ] , g = 0.7104 / Σ t ,
( = { 2 k n z 0 + sin ( 2 γ n ) sin [ 2 ( k n z 0 + γ n ) ] } / k n ) , λ n 2 = k n 2 + a / D ,
S ( ρ ¯ ) = Ψ 0 s t exp ( t z ) R ( r ) ,
Ψ ( r , z ) = n = 1 ( Ψ 0 s t / D N n ) 0 z 0 exp ( t z ) sin ( k n z + γ n ) × sin ( k n z + γ n ) d z × 0 a R ( r ) { I 0 ( λ n r ) K 0 ( λ n r ) r d r , r < r I 0 ( λ n r ) K 0 ( λ n r ) r d r , r > r .
s I 0 ( s ) d s = s I 1 ( s ) + C 1 , s K 0 ( s ) d s = s K 1 ( s ) + C 2 , I 0 ( s ) K 1 ( s ) + K 0 ( s ) I 1 ( s ) = 1 / s .
( r / λ n ) I 1 ( λ n r ) K 0 ( λ n r ) + { ( 1 / λ n ) [ a K 1 ( λ n a ) r K 1 ( λ n r ) ] I 0 ( λ n r ) } ;
Ψ ( r , z ) = n = 1 ( Ψ 0 s t / [ D N n λ n ( k n 2 + t 2 ) ] ) z n × sin ( k n z + γ n ) { a I 1 ( λ n a ) K 0 ( λ n r ) , a < r ( 1 / λ n ) a I 0 ( λ n r ) K 1 ( λ n a ) , a > r ,
z n = k n cos ( γ n ) + t sin ( γ n ) [ k n cos ( k n z 0 + γ n ) + t sin ( k n z 0 + γ n ) ] exp ( t z 0 ) .
| J ( r ) | + z = D Ψ / z | z = z 0 .
I ( b ) = 0 b | J ( r ) | + z r d r .
I ( b ) = n = 1 { Ψ 0 s t / [ N n λ n 2 ( k n 2 + t 2 ) ] } z n k n cos ( k n z 0 + γ n ) × { ( a 2 / 2 ) b a I 1 ( λ n a ) K 1 ( λ n b ) , a < b ( b 2 / 2 ) b a I 1 ( λ n b ) K 1 ( λ n a ) , a > b .
T diff = I ( b ) [ 0 a Ψ 0 r d r ] 1 = 2 I ( b ) / [ Ψ 0 a 2 ] .
T c = exp ( t d ) ( incorrect )
T tot = T diff + T c = T diff + exp ( t d ) ( incorrect ) .
T tot exp ( t d ) .
O.D. 0.4343 t d .
T c = exp ( d ) ( correct ) ,
H 3 2.388 H 2 + 1.385 H 0.00390 = ( H 0.00283 ) ( 0.983 H ) ( 1.402 H )
H ( 1 H ) ( 1.4 H ) .
s = σ s H ( 1 H ) ( 1.4 H ) / V ( correct ) .
T tot = T diff + exp ( d ) ( correct ) ,
s = σ s H ( 1 H ) ( 1.4 H ) / V ( correct ) .

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