Abstract

The backscattered and transmitted diagrams of He–Ne laser light illuminating a concentrated suspension of red blood cells (RBC's) are investigated. The shapes of these diagrams are closely related to the state of the suspension (at rest or submitted to a simple shear flow) and to the parameters that govern the non-Newtonian behavior of the blood suspension (such as the viscosity of the suspending medium and the volume concentration of the cells). An asymmetry in the backscattering diagram, which is absent on transmitted diagrams, is observed when the suspension is in a simple shear flow. This asymmetry is related to the deformation and orientation of the RBC's. The propagation of light through the suspension is modeled and a set of Monte Carlo simulations is performed to substantiate the inference that the relative variation of the backscattered flux is proportional to the gradients of deformation of the RBC's, and that such gradients must be known in order to apply a rheological model describing the non-Newtonian behavior of RBC membranes.

© 1994 Optical Society of America

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References

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  1. R. Skalak, A. Tozeren, R. P. Zaroa, S. Chien, “Strain function of RBC membranes,” Biophys. J. 13, 245–264 (1973).
    [CrossRef] [PubMed]
  2. E. A. Evans, R. M. Hochmuth, “Membrane viscoelasticity,” Biophys. J. 16, 1 (1976).
    [CrossRef] [PubMed]
  3. E. A. Evans, R. Waugh, “Osmotic correction to elastic area compressibility measurements on red cell membrane,” Biophys. J. 20, 307 (1977).
    [CrossRef] [PubMed]
  4. T. M. Fischer, H. Schmid-Schonbein, “Tank treading motion of red cell membranes in viscosimetric flow: behavior in intracellular and extracellular markers (with film),” Blood Cells 3, 351–365 (1977).
  5. H. J. Mieselman, “Morphological determinants of red blood cell deformability,” Scand. J. Clin. Lab. Invest. 41, Suppl. 156, 27–34 (1981).
    [CrossRef]
  6. M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 315–321 (1975).
  7. M. R. Clark, N. Mohandas, S. B. Shohet, “Osmotic gradient ektacytometry: comprehensive characterization of RBC volume and surface maintenance,” Blood 61, 899–910 (1983).
    [PubMed]
  8. D. L'Huillier, “Phenomenology of hydrodynamic interactions in suspension of weakly deformable particles,” J. Phys. (Paris) 48, 1887–1902 (1987).
  9. 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]
  10. J. M. Schmitt, J. D. Meuidl, F. G. Mihn, “An integrated circuit-based optical sensor for in vivo measurement of blood oxygenation,” IEEE Trans. Biomed. Eng. BME-33, 98–107 (1986).
    [CrossRef]
  11. K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).
  12. V. Twersky, “Absorption and multiple scattering by biological suspensions,” J. Opt. Soc. Am. 60, 1084–1093 (1970).
    [CrossRef] [PubMed]
  13. J. M. Steinke, A. P. Sheperd, “Role of light scattering in whole blood oximetry,” IEEE Trans. Biomed. Eng. BME-33, 294–301 (1986).
    [CrossRef]
  14. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model of photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  15. R. Nossal, J. E. Kiefer, G. H. Weiss, R. F. Bonner, H. Taitelbaum, S. Havlin, “Photon migration in layered media,” Appl. Opt. 27, 3382–3391 (1988).
    [CrossRef] [PubMed]
  16. R. A. J. Groenhuis, H. A. Ferweda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements, 1: theory,” Appl. Opt. 22, 2456–2462 (1983).
    [CrossRef] [PubMed]
  17. S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo modeling of light propagation in highly scattering tissues: II comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1163–1173 (1989).
  18. J. M. Masick, G. Jarry, B. de Cosnac, A. Lansiant, B. M. Hung, “A simulation method for the study of laser transillumation of biological tissues,” Ann. Biomed. Eng. 12, 221–304 (1984).
  19. E. A. Evans, “Improved measurements of the erythrocyte geometry,” Microvasc. Res. 4, 335–349 (1972).
    [CrossRef] [PubMed]
  20. B. Barer, S. Joseph, “Refractometry of living cells,” J. Microscop. Sci. 95, 399–412 (1954).
  21. B. T. Stocke, M. Mikkelson, A. Elgsacter, “Some viscoelastic properties of human erythrocyte spectrin networks end-linked in vitro,” Biochim. Biophys. Acta Lib. 816, 110–121 (1985).
  22. S. Chien, G. K. Sun, R. Skalak, S. Usami, A. Tozeren, “Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane,” Biophys. J. 24, 463–487 (1978).
    [CrossRef] [PubMed]
  23. J. Dufaux, D. Quemada, P. Mills, “Determination of rheological properties of blood by Couette viscosimetry,” J. Phys. Appli. (Paris) 15, 1357–1367 (1980).
  24. D. Quemada, “Rheology of concentrated disperse systems. III. General feature of the proposed non-Newtonian model. Comparison with experimental data,” Rheol. Acta 17, 643–653 (1978).
    [CrossRef]
  25. A. H. Gandjbakhche, P. Mills, P. Snabre, “A light scattering technique to study the red blood cells membrane structure and their rheological properties,” (in preparation).
  26. P. Snabre, “Agrégation des globules rouges en présence de Dextran—rhéologie, des suspensions concentrées de particules déformables,” These d'Etat (Université Paris VII, Paris, 1988).
  27. H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 161–164.
  28. J. C. Ravey, “Diffusion de la lumière: application aux particules de grande taille comme les globules rouges,” in Techniques Avancées en Hemorhéologie (Institut National Polytechnique de Lorraine, Nancy, France, 1983), pp. 505–542.
  29. P. Snabre, M. Bitbol, P. Mills, “Cell disaggregation behavior in shear flow,” Biophys. J. 51, 795–805 (1987).
    [CrossRef] [PubMed]

1989 (1)

S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo modeling of light propagation in highly scattering tissues: II comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1163–1173 (1989).

1988 (1)

1987 (3)

P. Snabre, M. Bitbol, P. Mills, “Cell disaggregation behavior in shear flow,” Biophys. J. 51, 795–805 (1987).
[CrossRef] [PubMed]

R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model of photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
[CrossRef] [PubMed]

D. L'Huillier, “Phenomenology of hydrodynamic interactions in suspension of weakly deformable particles,” J. Phys. (Paris) 48, 1887–1902 (1987).

1986 (2)

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

1985 (1)

B. T. Stocke, M. Mikkelson, A. Elgsacter, “Some viscoelastic properties of human erythrocyte spectrin networks end-linked in vitro,” Biochim. Biophys. Acta Lib. 816, 110–121 (1985).

1984 (1)

J. M. Masick, G. Jarry, B. de Cosnac, A. Lansiant, B. M. Hung, “A simulation method for the study of laser transillumation of biological tissues,” Ann. Biomed. Eng. 12, 221–304 (1984).

1983 (2)

M. R. Clark, N. Mohandas, S. B. Shohet, “Osmotic gradient ektacytometry: comprehensive characterization of RBC volume and surface maintenance,” Blood 61, 899–910 (1983).
[PubMed]

R. A. J. Groenhuis, H. A. Ferweda, J. J. Ten Bosch, “Scattering and absorption of turbid materials determined from reflection measurements, 1: theory,” Appl. Opt. 22, 2456–2462 (1983).
[CrossRef] [PubMed]

1981 (1)

H. J. Mieselman, “Morphological determinants of red blood cell deformability,” Scand. J. Clin. Lab. Invest. 41, Suppl. 156, 27–34 (1981).
[CrossRef]

1980 (1)

J. Dufaux, D. Quemada, P. Mills, “Determination of rheological properties of blood by Couette viscosimetry,” J. Phys. Appli. (Paris) 15, 1357–1367 (1980).

1978 (2)

D. Quemada, “Rheology of concentrated disperse systems. III. General feature of the proposed non-Newtonian model. Comparison with experimental data,” Rheol. Acta 17, 643–653 (1978).
[CrossRef]

S. Chien, G. K. Sun, R. Skalak, S. Usami, A. Tozeren, “Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane,” Biophys. J. 24, 463–487 (1978).
[CrossRef] [PubMed]

1977 (2)

E. A. Evans, R. Waugh, “Osmotic correction to elastic area compressibility measurements on red cell membrane,” Biophys. J. 20, 307 (1977).
[CrossRef] [PubMed]

T. M. Fischer, H. Schmid-Schonbein, “Tank treading motion of red cell membranes in viscosimetric flow: behavior in intracellular and extracellular markers (with film),” Blood Cells 3, 351–365 (1977).

1976 (2)

1975 (1)

M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 315–321 (1975).

1973 (1)

R. Skalak, A. Tozeren, R. P. Zaroa, S. Chien, “Strain function of RBC membranes,” Biophys. J. 13, 245–264 (1973).
[CrossRef] [PubMed]

1972 (1)

E. A. Evans, “Improved measurements of the erythrocyte geometry,” Microvasc. Res. 4, 335–349 (1972).
[CrossRef] [PubMed]

1970 (1)

1954 (1)

B. Barer, S. Joseph, “Refractometry of living cells,” J. Microscop. Sci. 95, 399–412 (1954).

Barer, B.

B. Barer, S. Joseph, “Refractometry of living cells,” J. Microscop. Sci. 95, 399–412 (1954).

Bessis, M.

M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 315–321 (1975).

Bitbol, M.

P. Snabre, M. Bitbol, P. Mills, “Cell disaggregation behavior in shear flow,” Biophys. J. 51, 795–805 (1987).
[CrossRef] [PubMed]

Bonner, R. F.

Case, K. M.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Chien, S.

S. Chien, G. K. Sun, R. Skalak, S. Usami, A. Tozeren, “Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane,” Biophys. J. 24, 463–487 (1978).
[CrossRef] [PubMed]

R. Skalak, A. Tozeren, R. P. Zaroa, S. Chien, “Strain function of RBC membranes,” Biophys. J. 13, 245–264 (1973).
[CrossRef] [PubMed]

Clark, M. R.

M. R. Clark, N. Mohandas, S. B. Shohet, “Osmotic gradient ektacytometry: comprehensive characterization of RBC volume and surface maintenance,” Blood 61, 899–910 (1983).
[PubMed]

de Cosnac, B.

J. M. Masick, G. Jarry, B. de Cosnac, A. Lansiant, B. M. Hung, “A simulation method for the study of laser transillumation of biological tissues,” Ann. Biomed. Eng. 12, 221–304 (1984).

Dufaux, J.

J. Dufaux, D. Quemada, P. Mills, “Determination of rheological properties of blood by Couette viscosimetry,” J. Phys. Appli. (Paris) 15, 1357–1367 (1980).

Elgsacter, A.

B. T. Stocke, M. Mikkelson, A. Elgsacter, “Some viscoelastic properties of human erythrocyte spectrin networks end-linked in vitro,” Biochim. Biophys. Acta Lib. 816, 110–121 (1985).

Evans, E. A.

E. A. Evans, R. Waugh, “Osmotic correction to elastic area compressibility measurements on red cell membrane,” Biophys. J. 20, 307 (1977).
[CrossRef] [PubMed]

E. A. Evans, R. M. Hochmuth, “Membrane viscoelasticity,” Biophys. J. 16, 1 (1976).
[CrossRef] [PubMed]

E. A. Evans, “Improved measurements of the erythrocyte geometry,” Microvasc. Res. 4, 335–349 (1972).
[CrossRef] [PubMed]

Ferweda, H. A.

Fischer, T. M.

T. M. Fischer, H. Schmid-Schonbein, “Tank treading motion of red cell membranes in viscosimetric flow: behavior in intracellular and extracellular markers (with film),” Blood Cells 3, 351–365 (1977).

Flock, S. T.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo modeling of light propagation in highly scattering tissues: II comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1163–1173 (1989).

Gandjbakhche, A. H.

A. H. Gandjbakhche, P. Mills, P. Snabre, “A light scattering technique to study the red blood cells membrane structure and their rheological properties,” (in preparation).

Groenhuis, R. A. J.

Havlin, S.

Hochmuth, R. M.

E. A. Evans, R. M. Hochmuth, “Membrane viscoelasticity,” Biophys. J. 16, 1 (1976).
[CrossRef] [PubMed]

Hung, B. M.

J. M. Masick, G. Jarry, B. de Cosnac, A. Lansiant, B. M. Hung, “A simulation method for the study of laser transillumation of biological tissues,” Ann. Biomed. Eng. 12, 221–304 (1984).

Ishimaru, A.

Jarry, G.

J. M. Masick, G. Jarry, B. de Cosnac, A. Lansiant, B. M. Hung, “A simulation method for the study of laser transillumation of biological tissues,” Ann. Biomed. Eng. 12, 221–304 (1984).

Johnson, C.

Joseph, S.

B. Barer, S. Joseph, “Refractometry of living cells,” J. Microscop. Sci. 95, 399–412 (1954).

Kiefer, J. E.

Lansiant, A.

J. M. Masick, G. Jarry, B. de Cosnac, A. Lansiant, B. M. Hung, “A simulation method for the study of laser transillumation of biological tissues,” Ann. Biomed. Eng. 12, 221–304 (1984).

L'Huillier, D.

D. L'Huillier, “Phenomenology of hydrodynamic interactions in suspension of weakly deformable particles,” J. Phys. (Paris) 48, 1887–1902 (1987).

Masick, J. M.

J. M. Masick, G. Jarry, B. de Cosnac, A. Lansiant, B. M. Hung, “A simulation method for the study of laser transillumation of biological tissues,” Ann. Biomed. Eng. 12, 221–304 (1984).

Meuidl, J. D.

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

Mieselman, H. J.

H. J. Mieselman, “Morphological determinants of red blood cell deformability,” Scand. J. Clin. Lab. Invest. 41, Suppl. 156, 27–34 (1981).
[CrossRef]

Mihn, F. G.

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

Mikkelson, M.

B. T. Stocke, M. Mikkelson, A. Elgsacter, “Some viscoelastic properties of human erythrocyte spectrin networks end-linked in vitro,” Biochim. Biophys. Acta Lib. 816, 110–121 (1985).

Mills, P.

P. Snabre, M. Bitbol, P. Mills, “Cell disaggregation behavior in shear flow,” Biophys. J. 51, 795–805 (1987).
[CrossRef] [PubMed]

J. Dufaux, D. Quemada, P. Mills, “Determination of rheological properties of blood by Couette viscosimetry,” J. Phys. Appli. (Paris) 15, 1357–1367 (1980).

A. H. Gandjbakhche, P. Mills, P. Snabre, “A light scattering technique to study the red blood cells membrane structure and their rheological properties,” (in preparation).

Mohandas, N.

M. R. Clark, N. Mohandas, S. B. Shohet, “Osmotic gradient ektacytometry: comprehensive characterization of RBC volume and surface maintenance,” Blood 61, 899–910 (1983).
[PubMed]

M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 315–321 (1975).

Nossal, R.

Patterson, M. S.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo modeling of light propagation in highly scattering tissues: II comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1163–1173 (1989).

Quemada, D.

J. Dufaux, D. Quemada, P. Mills, “Determination of rheological properties of blood by Couette viscosimetry,” J. Phys. Appli. (Paris) 15, 1357–1367 (1980).

D. Quemada, “Rheology of concentrated disperse systems. III. General feature of the proposed non-Newtonian model. Comparison with experimental data,” Rheol. Acta 17, 643–653 (1978).
[CrossRef]

Ravey, J. C.

J. C. Ravey, “Diffusion de la lumière: application aux particules de grande taille comme les globules rouges,” in Techniques Avancées en Hemorhéologie (Institut National Polytechnique de Lorraine, Nancy, France, 1983), pp. 505–542.

Reynolds, L.

Schmid-Schonbein, H.

T. M. Fischer, H. Schmid-Schonbein, “Tank treading motion of red cell membranes in viscosimetric flow: behavior in intracellular and extracellular markers (with film),” Blood Cells 3, 351–365 (1977).

Schmitt, J. M.

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

Sheperd, A. P.

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

Shohet, S. B.

M. R. Clark, N. Mohandas, S. B. Shohet, “Osmotic gradient ektacytometry: comprehensive characterization of RBC volume and surface maintenance,” Blood 61, 899–910 (1983).
[PubMed]

Skalak, R.

S. Chien, G. K. Sun, R. Skalak, S. Usami, A. Tozeren, “Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane,” Biophys. J. 24, 463–487 (1978).
[CrossRef] [PubMed]

R. Skalak, A. Tozeren, R. P. Zaroa, S. Chien, “Strain function of RBC membranes,” Biophys. J. 13, 245–264 (1973).
[CrossRef] [PubMed]

Snabre, P.

P. Snabre, M. Bitbol, P. Mills, “Cell disaggregation behavior in shear flow,” Biophys. J. 51, 795–805 (1987).
[CrossRef] [PubMed]

P. Snabre, “Agrégation des globules rouges en présence de Dextran—rhéologie, des suspensions concentrées de particules déformables,” These d'Etat (Université Paris VII, Paris, 1988).

A. H. Gandjbakhche, P. Mills, P. Snabre, “A light scattering technique to study the red blood cells membrane structure and their rheological properties,” (in preparation).

Steinke, J. M.

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

Stocke, B. T.

B. T. Stocke, M. Mikkelson, A. Elgsacter, “Some viscoelastic properties of human erythrocyte spectrin networks end-linked in vitro,” Biochim. Biophys. Acta Lib. 816, 110–121 (1985).

Sun, G. K.

S. Chien, G. K. Sun, R. Skalak, S. Usami, A. Tozeren, “Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane,” Biophys. J. 24, 463–487 (1978).
[CrossRef] [PubMed]

Taitelbaum, H.

Ten Bosch, J. J.

Tozeren, A.

S. Chien, G. K. Sun, R. Skalak, S. Usami, A. Tozeren, “Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane,” Biophys. J. 24, 463–487 (1978).
[CrossRef] [PubMed]

R. Skalak, A. Tozeren, R. P. Zaroa, S. Chien, “Strain function of RBC membranes,” Biophys. J. 13, 245–264 (1973).
[CrossRef] [PubMed]

Twersky, V.

Usami, S.

S. Chien, G. K. Sun, R. Skalak, S. Usami, A. Tozeren, “Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane,” Biophys. J. 24, 463–487 (1978).
[CrossRef] [PubMed]

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 161–164.

Waugh, R.

E. A. Evans, R. Waugh, “Osmotic correction to elastic area compressibility measurements on red cell membrane,” Biophys. J. 20, 307 (1977).
[CrossRef] [PubMed]

Weiss, G. H.

Wilson, B. C.

S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo modeling of light propagation in highly scattering tissues: II comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1163–1173 (1989).

Zaroa, R. P.

R. Skalak, A. Tozeren, R. P. Zaroa, S. Chien, “Strain function of RBC membranes,” Biophys. J. 13, 245–264 (1973).
[CrossRef] [PubMed]

Zweifel, P. F.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

Ann. Biomed. Eng. (1)

J. M. Masick, G. Jarry, B. de Cosnac, A. Lansiant, B. M. Hung, “A simulation method for the study of laser transillumation of biological tissues,” Ann. Biomed. Eng. 12, 221–304 (1984).

Appl. Opt. (3)

Biochim. Biophys. Acta Lib. (1)

B. T. Stocke, M. Mikkelson, A. Elgsacter, “Some viscoelastic properties of human erythrocyte spectrin networks end-linked in vitro,” Biochim. Biophys. Acta Lib. 816, 110–121 (1985).

Biophys. J. (5)

S. Chien, G. K. Sun, R. Skalak, S. Usami, A. Tozeren, “Theoretical and experimental studies on viscoelastic properties of erythrocyte membrane,” Biophys. J. 24, 463–487 (1978).
[CrossRef] [PubMed]

R. Skalak, A. Tozeren, R. P. Zaroa, S. Chien, “Strain function of RBC membranes,” Biophys. J. 13, 245–264 (1973).
[CrossRef] [PubMed]

E. A. Evans, R. M. Hochmuth, “Membrane viscoelasticity,” Biophys. J. 16, 1 (1976).
[CrossRef] [PubMed]

E. A. Evans, R. Waugh, “Osmotic correction to elastic area compressibility measurements on red cell membrane,” Biophys. J. 20, 307 (1977).
[CrossRef] [PubMed]

P. Snabre, M. Bitbol, P. Mills, “Cell disaggregation behavior in shear flow,” Biophys. J. 51, 795–805 (1987).
[CrossRef] [PubMed]

Blood (1)

M. R. Clark, N. Mohandas, S. B. Shohet, “Osmotic gradient ektacytometry: comprehensive characterization of RBC volume and surface maintenance,” Blood 61, 899–910 (1983).
[PubMed]

Blood Cells (2)

M. Bessis, N. Mohandas, “A diffractometric method for the measurement of cellular deformability,” Blood Cells 1, 315–321 (1975).

T. M. Fischer, H. Schmid-Schonbein, “Tank treading motion of red cell membranes in viscosimetric flow: behavior in intracellular and extracellular markers (with film),” Blood Cells 3, 351–365 (1977).

IEEE Trans. Biomed. Eng. (3)

S. T. Flock, B. C. Wilson, M. S. Patterson, “Monte Carlo modeling of light propagation in highly scattering tissues: II comparison with measurements in phantoms,” IEEE Trans. Biomed. Eng. 36, 1163–1173 (1989).

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

J. Microscop. Sci. (1)

B. Barer, S. Joseph, “Refractometry of living cells,” J. Microscop. Sci. 95, 399–412 (1954).

J. Opt. Soc. Am. (1)

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

J. Phys. (Paris) (1)

D. L'Huillier, “Phenomenology of hydrodynamic interactions in suspension of weakly deformable particles,” J. Phys. (Paris) 48, 1887–1902 (1987).

J. Phys. Appli. (Paris) (1)

J. Dufaux, D. Quemada, P. Mills, “Determination of rheological properties of blood by Couette viscosimetry,” J. Phys. Appli. (Paris) 15, 1357–1367 (1980).

Microvasc. Res. (1)

E. A. Evans, “Improved measurements of the erythrocyte geometry,” Microvasc. Res. 4, 335–349 (1972).
[CrossRef] [PubMed]

Rheol. Acta (1)

D. Quemada, “Rheology of concentrated disperse systems. III. General feature of the proposed non-Newtonian model. Comparison with experimental data,” Rheol. Acta 17, 643–653 (1978).
[CrossRef]

Scand. J. Clin. Lab. Invest. (1)

H. J. Mieselman, “Morphological determinants of red blood cell deformability,” Scand. J. Clin. Lab. Invest. 41, Suppl. 156, 27–34 (1981).
[CrossRef]

Other (5)

A. H. Gandjbakhche, P. Mills, P. Snabre, “A light scattering technique to study the red blood cells membrane structure and their rheological properties,” (in preparation).

P. Snabre, “Agrégation des globules rouges en présence de Dextran—rhéologie, des suspensions concentrées de particules déformables,” These d'Etat (Université Paris VII, Paris, 1988).

H. C. van de Hulst, Light Scattering by Small Particles (Dover, New York, 1981), pp. 161–164.

J. C. Ravey, “Diffusion de la lumière: application aux particules de grande taille comme les globules rouges,” in Techniques Avancées en Hemorhéologie (Institut National Polytechnique de Lorraine, Nancy, France, 1983), pp. 505–542.

K. M. Case, P. F. Zweifel, Linear Transport Theory (Addison-Wesley, Reading, Mass., 1967).

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Fig. 1
Fig. 1

Schematic representations of the setups for the backscattering experiments (a) and the transmission experiments (b). Ci , internal cylinder; Ce , external cylinder; D, detector; F, fiber optic; L, laser (He–Ne λ0 = 6328 Å); S, RBC suspension; U, register unit (computer xy plotter).

Fig. 2
Fig. 2

Backscattering diagram of an RBC suspension at rest (●); when the external cylinder rotates clockwise (□); when the external cylinder rotates counterclockwise (▲); γ˙ = 300 s−1, μ0 = 3.83 cp, Φ = 45%.

Fig. 3
Fig. 3

(a). Backscattering diagram of two suspensions submitted to the same shear rate (γ˙ = 300 s−1) with two different values of viscosity μ of the suspending medium: μ01 = 2.33 cp (∇); μ02 = 7.82 cp (■), (Φ = 45%). Fig. 3(b). Backscattering diagram of an RBC suspension submitted to two different shear rates: 50 s−1 (▲); 300 s−1 (■); at rest (●); (μ0 = 5.25 cp).

Fig. 4
Fig. 4

Transmission diagrams of an RBC suspension (Φ = 45%): suspension at rest (●); suspension submitted to a shear rate γ˙ = 3600 s−1(△); suspension submitted to a shear rate γ˙ = 360 s−1(◆); and viscosity of the suspending medium μ0 = 5.25 cp.

Fig. 5
Fig. 5

Comparison of reflectance R(L) as a function of the thickness of the medium (L) between numerical experiments: Monte Carlo simulations (△) and experiments results of Snabre26 (▲). There is very good agreement between the two sets of results.

Fig. 6
Fig. 6

Numerical results showing the variation of the diffuse reflectance R(L) and diffuse transmittance T(L) versus the thickness of the medium (L).

Fig. 7
Fig. 7

Determination of the diffraction spot of a single oriented RBC with Eq. (5.7).

Fig. 8
Fig. 8

Transmission diagram obtained by Monte Carlo simulations for various rates of deformation. The thickness of the medium is L = 128λ, where λ is the mean free path.

Fig. 9
Fig. 9

Backscattering diagram obtained by Monte Carlo simulations for various rates of deformation of RBC's for an infinite medium. The asymmetry seen in the experiments increases in proportion to the rate of deformation.

Fig. 10
Fig. 10

Representative curves of the contributions of σ and σ i in the backscatter diagram of an RBC suspension submitted to a simple shear flow. I is the suspension at rest; II is the contribution of σ i ; III is the contribution of σ; II + III (dashed line) is the suspension in a shear flow.

Equations (24)

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Q diff = 2 + 0.5 α s 8 / 9 .
λ = 2 σ .
σ = N 0 σ i
σ = Φ σ i V i
λ = 2 V i Φ σ i 3.13 μ m .
g = cos θ = 0 2 π 0 2 π P ( θ ) H ( x ) cos θ d θ d x 0 2 π 0 2 π P ( θ ) H ( x ) d θ d x .
g = 0 2 π Ψ ( θ ) cos θ d θ 0 2 π Ψ ( θ ) d θ .
g = g B + g F , g B = cos θ B = 0 3 π / 2 Ψ ( θ ) cos θ d θ 0 2 π Ψ ( θ ) d θ , g F = cos θ F = π / 2 + π / 2 Ψ ( θ ) cos θ d θ 0 2 π Ψ ( θ ) d θ .
P B = 1 P F .
g = ( θ 0 ) = cos θ F = 0 π F ( θ ) cos θ d θ 0 π F ( θ ) d θ ,
F ( θ ) = sin θ 1 cos θ 0 0 θ θ 0 , F ( θ ) = 0 θ > θ 0 ,
g ( θ 0 F ) = sin 2 θ 0 F 2 ( 1 cos θ 0 F ) .
μ = ξ ( g ) K λ ,
C A ( x ) = exp ( K x / λ )
l = λ L ,
X ¯ i = [ cos θ i sin θ i 0 sin θ i cos ϕ i cos ϕ i cos θ i sin ϕ i sin θ i sin ϕ i cos θ i sin ϕ i cos ϕ i ] .
Y ̿ 0 = [ 1 0 0 0 1 0 0 0 1 ] ,
Y ̿ 1 = Y ̿ 0 X ̿ 1 .
Y ̿ i = Y ̿ i 1 X ̿ i .
U = [ u 1 0 0 ] ,
W i = W i 1 + U Y ̿ i .
Δ θ = λ 0 D sin ( x + α ) ,
R R ~ 1 σ σ 0 ,
R R R = Δ R R ~ σ σ 0 .

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