Abstract

The analytical theory of multiple scattering [ J. Opt. Soc. Am. 60, 1084 ( 1970)] permits predictions of scattering patterns by homogeneous suspensions of aligned and randomly oriented particles. Predictions for randomly oriented particles have been tested previously. Using an optical system involving a He–Ne laser and suspensions of red blood cells, we tested the theory’s predictions for scattering by suspensions in two distinct alignments. The qualitative effects of cell alignment on light scattering are consistent with those predicted, although measured differences in scattering between the two alignments exceed those predicted. We conclude that the theory may provide an optical means of distinguishing particle orientation in multiple scattering suspensions.

© 1991 Optical Society of America

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References

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  1. The terms coefficient and cross section are interchangeable. All cross sections are measured in units of area. For a physical interpretation of the cross sections, consider the total energy scattered in all directions by the particle. This energy is equal to the energy of the incident wave falling on an area σs. Likewise, the energy absorbed by the particle is equal to the energy incident upon the area σa. Refer to H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  2. V. Twersky, “Absorption and multiple scattering by biological suspensions,”J. Opt. Soc. Am. 60, 1084–1093 (1970).
    [CrossRef] [PubMed]
  3. A. Ishimaru, Wave Propagation and Scattering in Random Media: Vol. I. Single Scattering and Transport Theory;Vol. II. Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing (Academic, London, 1978),particularly Chap. 2 of Vol. I and Chap. 14 of Vol. II.
  4. J. M. Steinke, A. P. Shepherd, “Role of light scattering in whole blood oximetry,”IEEE Trans. Biomed. Eng. BME-33, 294–301 (1986).
    [CrossRef]
  5. E. Loewinger, A. Gordon, A. Weinreb, J. Gross, “Analysis of a micromethod for transmission oximetry of whole blood,”J. Appl. Physiol. 19, 1179–1186 (1964); N. M. Anderson, P. Sekelj, “Light-absorbing and scattering properties of nonhaemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967); H. H. Lipowsky, S. Usami, S. Chien, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
    [CrossRef] [PubMed]
  6. V. S. Lee, “Optical studies of human blood cells: the quantitative determination of platelet viability,” Ph.D. dissertation (Oxford University, Oxford, UK, 1989).
  7. D. H. Napper, R. H. Ottewill, “Multiple scattering effects in polystyrene latex dispersions,”J. Colloid Sci. 19, 72–80 (1964).
    [CrossRef]
  8. V. J. McCabe, “Aggregometer for testing and monitoring platelet aggregation in whole blood,” Ph.D. dissertation (University College, Dublin, Ireland, 1976).
  9. E. Ponder, “The measurement of the diameter of erythrocytes. V. The relation of the diameter to the thickness,”Q. J. Exp. Physiol. 20, 29–39 (1930).
  10. R. C. Weast, ed. CRC Handbook of Chemistry and Physics (CRC, Cleveland, Ohio, 1975).
  11. The rms radius of a spheroid can be expressed as r=[(2a2+c2)/3]1/2,where the geometry of the spheroid is given by dimensions a, a, c, with cmeasured in the direction of the axis of revolution (refer to Fig. 2).
  12. The full expression given by Twersky includes a backscattering term; however, it can be shown that, at the wavelength used in our experiments (632.8 nm), backscattering can be neglected for red blood cells. For backscattering criteria, refer to M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), particularly p. 127.
  13. J. V. Dacie, S. M. Lewis, Practical Haematology, 6th ed. (Churchill Livingstone, New York, 1984).
  14. International Committee for Standardization in Haematology Expert Panel on Blood Cell Sizing, “Recommendation for reference method for determination by centrifugation of packed cell volume of blood,” J. Clin. Path. 33, 1–2 (1980).
  15. O. W. Van Assendelft, Spectrophotometry of Hemoglobin Derivatives (Thomas, Springfield, Ill., 1970).
  16. The rms error is defined as rms=[∑(ODtheory-ODexpt)2N-n]1/2,where Nequals the number of data points and nthe number of independent variables being fitted (in this case, n= 2).
  17. H. L. Goldsmith, J. Marlow, “Flow behavior of erythrocytes. I. Rotation and deformation in dilute suspensions,” Proc. R. Soc. London Ser. B 182, 351–384 (1972).
    [CrossRef]
  18. A. Karnis, H. L. Goldsmith, S. G. Mason, “The flow of suspensions through tubes. V. Inertial effects,” Can. J. Chem. Eng. 44, 181–193 (1966); P. G. Saffman, “On the motion of small spheroidal particles in a viscous fluid,”J. Fluid Mech. 1, 540–553 (1956).
    [CrossRef]
  19. H. J. Klose, E. Volger, H. Brechtelsbauer, L. Heinich, H. Schmid-Schönbein, “Microrheology and light transmission of blood. I. The photometric effects of red cell aggregation and cell orientation,” Pfluegers Arch. 333, 126–139 (1972).
    [CrossRef]
  20. V. C. Roberts, “Photoplethysmography—fundamental aspects of the optical properties of blood in motion,” Trans. Inst. Meas. Control 4, 101–106 (1982).
    [CrossRef]

1986 (1)

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

1982 (1)

V. C. Roberts, “Photoplethysmography—fundamental aspects of the optical properties of blood in motion,” Trans. Inst. Meas. Control 4, 101–106 (1982).
[CrossRef]

1980 (1)

International Committee for Standardization in Haematology Expert Panel on Blood Cell Sizing, “Recommendation for reference method for determination by centrifugation of packed cell volume of blood,” J. Clin. Path. 33, 1–2 (1980).

1972 (2)

H. L. Goldsmith, J. Marlow, “Flow behavior of erythrocytes. I. Rotation and deformation in dilute suspensions,” Proc. R. Soc. London Ser. B 182, 351–384 (1972).
[CrossRef]

H. J. Klose, E. Volger, H. Brechtelsbauer, L. Heinich, H. Schmid-Schönbein, “Microrheology and light transmission of blood. I. The photometric effects of red cell aggregation and cell orientation,” Pfluegers Arch. 333, 126–139 (1972).
[CrossRef]

1970 (1)

1966 (1)

A. Karnis, H. L. Goldsmith, S. G. Mason, “The flow of suspensions through tubes. V. Inertial effects,” Can. J. Chem. Eng. 44, 181–193 (1966); P. G. Saffman, “On the motion of small spheroidal particles in a viscous fluid,”J. Fluid Mech. 1, 540–553 (1956).
[CrossRef]

1964 (2)

E. Loewinger, A. Gordon, A. Weinreb, J. Gross, “Analysis of a micromethod for transmission oximetry of whole blood,”J. Appl. Physiol. 19, 1179–1186 (1964); N. M. Anderson, P. Sekelj, “Light-absorbing and scattering properties of nonhaemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967); H. H. Lipowsky, S. Usami, S. Chien, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

D. H. Napper, R. H. Ottewill, “Multiple scattering effects in polystyrene latex dispersions,”J. Colloid Sci. 19, 72–80 (1964).
[CrossRef]

1930 (1)

E. Ponder, “The measurement of the diameter of erythrocytes. V. The relation of the diameter to the thickness,”Q. J. Exp. Physiol. 20, 29–39 (1930).

Brechtelsbauer, H.

H. J. Klose, E. Volger, H. Brechtelsbauer, L. Heinich, H. Schmid-Schönbein, “Microrheology and light transmission of blood. I. The photometric effects of red cell aggregation and cell orientation,” Pfluegers Arch. 333, 126–139 (1972).
[CrossRef]

Dacie, J. V.

J. V. Dacie, S. M. Lewis, Practical Haematology, 6th ed. (Churchill Livingstone, New York, 1984).

Goldsmith, H. L.

H. L. Goldsmith, J. Marlow, “Flow behavior of erythrocytes. I. Rotation and deformation in dilute suspensions,” Proc. R. Soc. London Ser. B 182, 351–384 (1972).
[CrossRef]

A. Karnis, H. L. Goldsmith, S. G. Mason, “The flow of suspensions through tubes. V. Inertial effects,” Can. J. Chem. Eng. 44, 181–193 (1966); P. G. Saffman, “On the motion of small spheroidal particles in a viscous fluid,”J. Fluid Mech. 1, 540–553 (1956).
[CrossRef]

Gordon, A.

E. Loewinger, A. Gordon, A. Weinreb, J. Gross, “Analysis of a micromethod for transmission oximetry of whole blood,”J. Appl. Physiol. 19, 1179–1186 (1964); N. M. Anderson, P. Sekelj, “Light-absorbing and scattering properties of nonhaemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967); H. H. Lipowsky, S. Usami, S. Chien, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

Gross, J.

E. Loewinger, A. Gordon, A. Weinreb, J. Gross, “Analysis of a micromethod for transmission oximetry of whole blood,”J. Appl. Physiol. 19, 1179–1186 (1964); N. M. Anderson, P. Sekelj, “Light-absorbing and scattering properties of nonhaemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967); H. H. Lipowsky, S. Usami, S. Chien, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

Heinich, L.

H. J. Klose, E. Volger, H. Brechtelsbauer, L. Heinich, H. Schmid-Schönbein, “Microrheology and light transmission of blood. I. The photometric effects of red cell aggregation and cell orientation,” Pfluegers Arch. 333, 126–139 (1972).
[CrossRef]

Ishimaru, A.

A. Ishimaru, Wave Propagation and Scattering in Random Media: Vol. I. Single Scattering and Transport Theory;Vol. II. Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing (Academic, London, 1978),particularly Chap. 2 of Vol. I and Chap. 14 of Vol. II.

Karnis, A.

A. Karnis, H. L. Goldsmith, S. G. Mason, “The flow of suspensions through tubes. V. Inertial effects,” Can. J. Chem. Eng. 44, 181–193 (1966); P. G. Saffman, “On the motion of small spheroidal particles in a viscous fluid,”J. Fluid Mech. 1, 540–553 (1956).
[CrossRef]

Kerker, M.

The full expression given by Twersky includes a backscattering term; however, it can be shown that, at the wavelength used in our experiments (632.8 nm), backscattering can be neglected for red blood cells. For backscattering criteria, refer to M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), particularly p. 127.

Klose, H. J.

H. J. Klose, E. Volger, H. Brechtelsbauer, L. Heinich, H. Schmid-Schönbein, “Microrheology and light transmission of blood. I. The photometric effects of red cell aggregation and cell orientation,” Pfluegers Arch. 333, 126–139 (1972).
[CrossRef]

Lee, V. S.

V. S. Lee, “Optical studies of human blood cells: the quantitative determination of platelet viability,” Ph.D. dissertation (Oxford University, Oxford, UK, 1989).

Lewis, S. M.

J. V. Dacie, S. M. Lewis, Practical Haematology, 6th ed. (Churchill Livingstone, New York, 1984).

Loewinger, E.

E. Loewinger, A. Gordon, A. Weinreb, J. Gross, “Analysis of a micromethod for transmission oximetry of whole blood,”J. Appl. Physiol. 19, 1179–1186 (1964); N. M. Anderson, P. Sekelj, “Light-absorbing and scattering properties of nonhaemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967); H. H. Lipowsky, S. Usami, S. Chien, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

Marlow, J.

H. L. Goldsmith, J. Marlow, “Flow behavior of erythrocytes. I. Rotation and deformation in dilute suspensions,” Proc. R. Soc. London Ser. B 182, 351–384 (1972).
[CrossRef]

Mason, S. G.

A. Karnis, H. L. Goldsmith, S. G. Mason, “The flow of suspensions through tubes. V. Inertial effects,” Can. J. Chem. Eng. 44, 181–193 (1966); P. G. Saffman, “On the motion of small spheroidal particles in a viscous fluid,”J. Fluid Mech. 1, 540–553 (1956).
[CrossRef]

McCabe, V. J.

V. J. McCabe, “Aggregometer for testing and monitoring platelet aggregation in whole blood,” Ph.D. dissertation (University College, Dublin, Ireland, 1976).

Napper, D. H.

D. H. Napper, R. H. Ottewill, “Multiple scattering effects in polystyrene latex dispersions,”J. Colloid Sci. 19, 72–80 (1964).
[CrossRef]

Ottewill, R. H.

D. H. Napper, R. H. Ottewill, “Multiple scattering effects in polystyrene latex dispersions,”J. Colloid Sci. 19, 72–80 (1964).
[CrossRef]

Ponder, E.

E. Ponder, “The measurement of the diameter of erythrocytes. V. The relation of the diameter to the thickness,”Q. J. Exp. Physiol. 20, 29–39 (1930).

Roberts, V. C.

V. C. Roberts, “Photoplethysmography—fundamental aspects of the optical properties of blood in motion,” Trans. Inst. Meas. Control 4, 101–106 (1982).
[CrossRef]

Schmid-Schönbein, H.

H. J. Klose, E. Volger, H. Brechtelsbauer, L. Heinich, H. Schmid-Schönbein, “Microrheology and light transmission of blood. I. The photometric effects of red cell aggregation and cell orientation,” Pfluegers Arch. 333, 126–139 (1972).
[CrossRef]

Shepherd, A. P.

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

Steinke, J. M.

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

Twersky, V.

Van Assendelft, O. W.

O. W. Van Assendelft, Spectrophotometry of Hemoglobin Derivatives (Thomas, Springfield, Ill., 1970).

van de Hulst, H. C.

The terms coefficient and cross section are interchangeable. All cross sections are measured in units of area. For a physical interpretation of the cross sections, consider the total energy scattered in all directions by the particle. This energy is equal to the energy of the incident wave falling on an area σs. Likewise, the energy absorbed by the particle is equal to the energy incident upon the area σa. Refer to H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Volger, E.

H. J. Klose, E. Volger, H. Brechtelsbauer, L. Heinich, H. Schmid-Schönbein, “Microrheology and light transmission of blood. I. The photometric effects of red cell aggregation and cell orientation,” Pfluegers Arch. 333, 126–139 (1972).
[CrossRef]

Weinreb, A.

E. Loewinger, A. Gordon, A. Weinreb, J. Gross, “Analysis of a micromethod for transmission oximetry of whole blood,”J. Appl. Physiol. 19, 1179–1186 (1964); N. M. Anderson, P. Sekelj, “Light-absorbing and scattering properties of nonhaemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967); H. H. Lipowsky, S. Usami, S. Chien, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

Can. J. Chem. Eng. (1)

A. Karnis, H. L. Goldsmith, S. G. Mason, “The flow of suspensions through tubes. V. Inertial effects,” Can. J. Chem. Eng. 44, 181–193 (1966); P. G. Saffman, “On the motion of small spheroidal particles in a viscous fluid,”J. Fluid Mech. 1, 540–553 (1956).
[CrossRef]

IEEE Trans. Biomed. Eng. (1)

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. Appl. Physiol. (1)

E. Loewinger, A. Gordon, A. Weinreb, J. Gross, “Analysis of a micromethod for transmission oximetry of whole blood,”J. Appl. Physiol. 19, 1179–1186 (1964); N. M. Anderson, P. Sekelj, “Light-absorbing and scattering properties of nonhaemolysed blood,” Phys. Med. Biol. 12, 173–184 (1967); H. H. Lipowsky, S. Usami, S. Chien, “Hematocrit determination in small bore tubes from optical density measurements under white light illumination,” Microvasc. Res. 20, 51–70 (1980).
[CrossRef] [PubMed]

J. Clin. Path. (1)

International Committee for Standardization in Haematology Expert Panel on Blood Cell Sizing, “Recommendation for reference method for determination by centrifugation of packed cell volume of blood,” J. Clin. Path. 33, 1–2 (1980).

J. Colloid Sci. (1)

D. H. Napper, R. H. Ottewill, “Multiple scattering effects in polystyrene latex dispersions,”J. Colloid Sci. 19, 72–80 (1964).
[CrossRef]

J. Opt. Soc. Am. (1)

Pfluegers Arch. (1)

H. J. Klose, E. Volger, H. Brechtelsbauer, L. Heinich, H. Schmid-Schönbein, “Microrheology and light transmission of blood. I. The photometric effects of red cell aggregation and cell orientation,” Pfluegers Arch. 333, 126–139 (1972).
[CrossRef]

Proc. R. Soc. London Ser. B (1)

H. L. Goldsmith, J. Marlow, “Flow behavior of erythrocytes. I. Rotation and deformation in dilute suspensions,” Proc. R. Soc. London Ser. B 182, 351–384 (1972).
[CrossRef]

Q. J. Exp. Physiol. (1)

E. Ponder, “The measurement of the diameter of erythrocytes. V. The relation of the diameter to the thickness,”Q. J. Exp. Physiol. 20, 29–39 (1930).

Trans. Inst. Meas. Control (1)

V. C. Roberts, “Photoplethysmography—fundamental aspects of the optical properties of blood in motion,” Trans. Inst. Meas. Control 4, 101–106 (1982).
[CrossRef]

Other (10)

The terms coefficient and cross section are interchangeable. All cross sections are measured in units of area. For a physical interpretation of the cross sections, consider the total energy scattered in all directions by the particle. This energy is equal to the energy of the incident wave falling on an area σs. Likewise, the energy absorbed by the particle is equal to the energy incident upon the area σa. Refer to H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

V. J. McCabe, “Aggregometer for testing and monitoring platelet aggregation in whole blood,” Ph.D. dissertation (University College, Dublin, Ireland, 1976).

O. W. Van Assendelft, Spectrophotometry of Hemoglobin Derivatives (Thomas, Springfield, Ill., 1970).

The rms error is defined as rms=[∑(ODtheory-ODexpt)2N-n]1/2,where Nequals the number of data points and nthe number of independent variables being fitted (in this case, n= 2).

R. C. Weast, ed. CRC Handbook of Chemistry and Physics (CRC, Cleveland, Ohio, 1975).

The rms radius of a spheroid can be expressed as r=[(2a2+c2)/3]1/2,where the geometry of the spheroid is given by dimensions a, a, c, with cmeasured in the direction of the axis of revolution (refer to Fig. 2).

The full expression given by Twersky includes a backscattering term; however, it can be shown that, at the wavelength used in our experiments (632.8 nm), backscattering can be neglected for red blood cells. For backscattering criteria, refer to M. Kerker, The Scattering of Light and Other Electromagnetic Radiation (Academic, New York, 1969), particularly p. 127.

J. V. Dacie, S. M. Lewis, Practical Haematology, 6th ed. (Churchill Livingstone, New York, 1984).

A. Ishimaru, Wave Propagation and Scattering in Random Media: Vol. I. Single Scattering and Transport Theory;Vol. II. Multiple Scattering, Turbulence, Rough Surfaces, and Remote Sensing (Academic, London, 1978),particularly Chap. 2 of Vol. I and Chap. 14 of Vol. II.

V. S. Lee, “Optical studies of human blood cells: the quantitative determination of platelet viability,” Ph.D. dissertation (Oxford University, Oxford, UK, 1989).

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Figures (7)

Fig. 1
Fig. 1

Experimental rig and microchannel. The beam traverses the channel in the z direction.

Fig. 2
Fig. 2

Coordinate systern.

Fig. 3
Fig. 3

Light-transmission pattern for the laser beam through phosphate-buffered-saline-filled channel, normalized with respect to T(θ = 0). The theoretical expressions in Eqs. (18)(20) must be modified to take into account the finite width of the beam.

Fig. 4
Fig. 4

Determination of σa. Optical densities of solutions of hemoglobin corresponding to membraneless (or lysed) red blood cells are plotted against the hematocrits of the suspensions before lysing (n = 10). A best-fit line was determined by linear regression. Fitting to Eq. (24) yielded an experimentally determined value of σa = 0.065 μm2. All samples were fully oxygenated at the start of the experiment.

Fig. 5
Fig. 5

Results (■) and best-fit (+) curve by the SAS package [Eq. (17)] for optical densities of red blood cells in the 150-μm microchannel in the experimental rig. Experimental measurements were made in duplicate. Also shown (× at bottom) is ODabs computed from Eq. (25) for σa = 0.065 −m2.

Fig. 6
Fig. 6

Configurations of discoid cells in three orientations: (a) rim on (ocillatory flow); (b) random (no flow); (c) face on (no flow).

Fig. 7
Fig. 7

Predicted (open symbols) and observed (filled symbols) transmission patterns for red blood cells at concentrations of (a) 0.5 × 108 and (b) 1.0 × 108 cells/ml. Also shown are photos of the cells in the three alignments corresponding to face on (triangles), random (squares), and rim on (diamonds).

Tables (1)

Tables Icon

Table 1 Summary of Values

Equations (28)

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

T = I / I 0 = exp ( - σ t ρ d ) = exp [ - ( σ a + σ s ) ρ d ] ,
OD ( λ ) = Abs = log 1 / T = 0.434 σ t ρ d = 0.434 ( σ a + σ s ) ρ d ,
T = C + I ( θ ) ,
C = exp ( - L d ) = exp { - [ σ a + σ s ( 1 - ρ V ) ] ρ d } ,
I ( θ ) = q [ exp ( - ρ σ a d ) - exp ( - { σ a + ( 1 - ρ V ) σ s [ b ( r ) / b ( z ) ] } ρ d sec θ ) ] .
q 2 A k 2 Ω ( α ) 9 π 2 exp [ - ( 1 / 5 ) ( k θ h ) 2 ] ,
h = a [ 1 - ɛ 2 cos 2 ( θ / 2 ) ] 1 / 2             ( rim on ) ,
h = a [ 1 - ɛ 2 sin 2 ( θ / 2 ) ] 1 / 2             ( face on ) ,
ɛ 2 = 1 - c 2 / a 2 ,             ( c a ) ,
q k 2 V Ω ( α ) 8 π 2 b exp [ - ( 1 / 5 ) ( k θ a ) 2 ] ,
b ( θ ) = 3 c / 4 ( 1 - ɛ 2 cos 2 θ ) 1 / 2 ,
b ( θ = 0 ) = 3 a / 4.
b ( θ ) = 3 c / 4 ( 1 - ɛ 2 sin 2 θ ) 1 / 2 ,
b ( θ = 0 ) = 3 c / 4.
b = ( 1 / 2 ) 0 π b ( ζ ) sin ζ d ζ = ( 3 / 4 ) c ( 1 / 2 ɛ ) ln [ ( 1 + ɛ ) / ( 1 - ɛ ) ] .
T AS ( θ = 0 ) = exp { - [ σ a + ( 1 - ρ V ) σ s ] ρ d } + 2 k 2 A Ω ( α ) 9 π 2 × ( exp ( - σ a ρ d ) - exp { - [ σ a + σ s ( 1 - ρ V ) ] ρ d } ) ,
T RS ( θ = 0 ) = exp { - [ σ a + ( 1 - ρ V ) σ s ] ρ d } + k 2 V Ω ( α ) 8 π 2 b × ( exp ( - σ a ρ d ) - exp { - [ σ a + σ s ( 1 - ρ V ) ] ρ d } )
T AS - RIM ( θ 0 ) = 2 k 2 A Ω ( α ) 9 π 2 × exp [ - ( 1 / 5 ) k 2 θ 2 a 2 ( 1 - ɛ 2 cos 2 θ / 2 ) ] × ( exp ( - σ a ρ d ) - exp { - [ σ a + σ s ( 1 - ρ V ) c a ( 1 - ɛ 2 cos 2 θ ) 1 / 2 ] ρ d } ) ,
T AS - FACE ( θ 0 ) = 2 k 2 A Ω ( α ) 9 π 2 × exp [ - ( 1 / 5 ) k 2 θ 2 a 2 ( 1 - ɛ 2 sin 2 θ / 2 ) ] × ( exp ( - σ a ρ d ) - exp { - [ σ a + σ s ( 1 - ρ V ) ( 1 - ɛ 2 sin 2 θ ) 1 / 2 ] ρ d } ) ,
T RS ( θ 0 ) = k 2 V Ω ( α ) 8 π 2 b exp [ - ( 1 / 5 ) k 2 θ 2 ( 2 a 2 + c 2 ) / 3 ] × ( exp ( - σ a ρ d ) - exp { - [ σ a + σ s ( 1 - ρ V ) ] ρ d } ) .
T coh ( θ ) = G ( θ ) exp { - [ σ a + σ s ( 1 - ρ V ) ] ρ d } .
T incoh ( θ ) = ( G * T incoh ) ( θ ) d x = 0.5 .
T ( θ = 0 ) RS = exp ( - σ a ρ d ) × ( exp [ - σ s ( 1 - ρ V ) ρ d ] + q { 1 - exp [ - σ s ( 1 - ρ V ) ρ d ] } ) = T abs × T sca .
OD = 0.434 σ a ρ d + log ( exp [ - σ s ( 1 - ρ V ) ρ d ] + q { 1 - exp [ - σ s ( 1 - ρ V ) ρ d ] } ) = OD abs + OD sca .
OD = 0.434 σ a ρ d = 0.434 σ a d ( HCT ) / V ,
σ s = 2 k 2 ( η / η 0 - 1 ) 2 b V .
r=[(2a2+c2)/3]1/2,
rms=[(ODtheory-ODexpt)2N-n]1/2,

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