Abstract

The in vitro aggregation of blood platelets is usually monitored with a visible light transmittance photometer (aggregometer). These cells in plasma are large (α = 2πa/λ ≅ 16) soft (m = 1.04) particles. The factors which significantly influence transmittance include the inherent scattering properties, multiple scattering, and photometer design. Now scattering theory and numerical methods for radiative transfer are used to survey how aggregation should influence transmittance as measured with various photometers. The results should help expand the analytical power of the transmittance photometer as a tool for monitoring aggregation. Evidence is also presented that scattering by aggregates of aerosol particles, which are of a higher relative refractive index, should also be adequately predicted by these approximate methods.

© 1983 Optical Society of America

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References

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  1. M. B. Zucker, Sci. Am. 243, 86 (1980).
    [CrossRef]
  2. M. M. Frojmovic, J. G. Milton, Physiol. Rev. 62, 185 (1982).
    [PubMed]
  3. G. V. R. Born, Nature London 194, 927 (1962).
    [CrossRef]
  4. P. Latimer, Appl. Opt. 14, 2324 (1975).
    [CrossRef] [PubMed]
  5. P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
    [CrossRef] [PubMed]
  6. F. D. Bryant, P. Latimer, B. D. Seiber, Arch. Biochem. Biophys. 135, 109 (1969).
    [CrossRef] [PubMed]
  7. P. Latimer, J. Theor. Biol. 51, 1 (1975).
    [CrossRef] [PubMed]
  8. P. Latimer, F. Wamble, Appl. Opt. 21, 2447 (1982).
    [CrossRef] [PubMed]
  9. P. Latimer, Annu. Rev. Biophys. Bioeng. 11, 129 (1982).
    [CrossRef]
  10. P. Latimer, J. Opt. Soc. Am. 62, 208 (1972).
    [CrossRef]
  11. D. H. Woodward, J. Opt. Soc. Am. 54, 1325 (1964).
    [CrossRef]
  12. S. E. Orchard, J. Opt. Soc. Am. 59, 1584 (1969).
    [CrossRef]
  13. D. B. Tully, M. S. Thesis, Auburn U., Ala. (1970).
  14. W. Farthing, M. S. Thesis, Auburn U., Ala. (1972).
  15. W. Farthing, P. Latimer, in 1981 Chemical Systems Laboratory Scientific Conference on Obscuration and Aerosol ResearchAberdeen Proving Ground, Md.
  16. G. V. R. Born, M. Hume, Nature London 215, 1027 (1967).
    [CrossRef] [PubMed]
  17. F. D. Bryant, B. A. Seiber, P. Latimer, Arch. Biochem. Biophys. 135, 97 (1969).
    [CrossRef] [PubMed]
  18. A. L. Aden, M. Kerker, Appl. Phys. 22, 1242 (1951).
    [CrossRef]
  19. A. Brunsting, Dissertation, U. New Mexico, Albuquerque (1972).
  20. P. Latimer, J. Colloid Interface Sci. 53, 102 (1975).
    [CrossRef]
  21. P. Latimer, P. Barber, J. Colloid, Interface Sci. 63, 310 (1978).
  22. P. Latimer, Appl. Opt. 19, 3039 (1980).
    [CrossRef] [PubMed]
  23. H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).
  24. P. Latimer, B. E. Pyle, Biophys. J. 12, 764, (1972).
    [CrossRef] [PubMed]
  25. F. Michael, G. V. R. Born, Nature London New Biol. 231, 220 (1971).

1982 (3)

M. M. Frojmovic, J. G. Milton, Physiol. Rev. 62, 185 (1982).
[PubMed]

P. Latimer, F. Wamble, Appl. Opt. 21, 2447 (1982).
[CrossRef] [PubMed]

P. Latimer, Annu. Rev. Biophys. Bioeng. 11, 129 (1982).
[CrossRef]

1980 (2)

1978 (1)

P. Latimer, P. Barber, J. Colloid, Interface Sci. 63, 310 (1978).

1977 (1)

P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
[CrossRef] [PubMed]

1975 (3)

P. Latimer, J. Theor. Biol. 51, 1 (1975).
[CrossRef] [PubMed]

P. Latimer, J. Colloid Interface Sci. 53, 102 (1975).
[CrossRef]

P. Latimer, Appl. Opt. 14, 2324 (1975).
[CrossRef] [PubMed]

1972 (2)

P. Latimer, J. Opt. Soc. Am. 62, 208 (1972).
[CrossRef]

P. Latimer, B. E. Pyle, Biophys. J. 12, 764, (1972).
[CrossRef] [PubMed]

1971 (1)

F. Michael, G. V. R. Born, Nature London New Biol. 231, 220 (1971).

1969 (3)

F. D. Bryant, B. A. Seiber, P. Latimer, Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

F. D. Bryant, P. Latimer, B. D. Seiber, Arch. Biochem. Biophys. 135, 109 (1969).
[CrossRef] [PubMed]

S. E. Orchard, J. Opt. Soc. Am. 59, 1584 (1969).
[CrossRef]

1967 (1)

G. V. R. Born, M. Hume, Nature London 215, 1027 (1967).
[CrossRef] [PubMed]

1964 (1)

1962 (1)

G. V. R. Born, Nature London 194, 927 (1962).
[CrossRef]

1951 (1)

A. L. Aden, M. Kerker, Appl. Phys. 22, 1242 (1951).
[CrossRef]

Aden, A. L.

A. L. Aden, M. Kerker, Appl. Phys. 22, 1242 (1951).
[CrossRef]

Barber, P.

P. Latimer, P. Barber, J. Colloid, Interface Sci. 63, 310 (1978).

Born, G. V. R.

P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
[CrossRef] [PubMed]

F. Michael, G. V. R. Born, Nature London New Biol. 231, 220 (1971).

G. V. R. Born, M. Hume, Nature London 215, 1027 (1967).
[CrossRef] [PubMed]

G. V. R. Born, Nature London 194, 927 (1962).
[CrossRef]

Brunsting, A.

A. Brunsting, Dissertation, U. New Mexico, Albuquerque (1972).

Bryant, F. D.

F. D. Bryant, B. A. Seiber, P. Latimer, Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

F. D. Bryant, P. Latimer, B. D. Seiber, Arch. Biochem. Biophys. 135, 109 (1969).
[CrossRef] [PubMed]

Colloid, J.

P. Latimer, P. Barber, J. Colloid, Interface Sci. 63, 310 (1978).

Farthing, W.

W. Farthing, M. S. Thesis, Auburn U., Ala. (1972).

W. Farthing, P. Latimer, in 1981 Chemical Systems Laboratory Scientific Conference on Obscuration and Aerosol ResearchAberdeen Proving Ground, Md.

Frojmovic, M. M.

M. M. Frojmovic, J. G. Milton, Physiol. Rev. 62, 185 (1982).
[PubMed]

Hume, M.

G. V. R. Born, M. Hume, Nature London 215, 1027 (1967).
[CrossRef] [PubMed]

Kerker, M.

A. L. Aden, M. Kerker, Appl. Phys. 22, 1242 (1951).
[CrossRef]

Latimer, P.

P. Latimer, F. Wamble, Appl. Opt. 21, 2447 (1982).
[CrossRef] [PubMed]

P. Latimer, Annu. Rev. Biophys. Bioeng. 11, 129 (1982).
[CrossRef]

P. Latimer, Appl. Opt. 19, 3039 (1980).
[CrossRef] [PubMed]

P. Latimer, P. Barber, J. Colloid, Interface Sci. 63, 310 (1978).

P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
[CrossRef] [PubMed]

P. Latimer, Appl. Opt. 14, 2324 (1975).
[CrossRef] [PubMed]

P. Latimer, J. Colloid Interface Sci. 53, 102 (1975).
[CrossRef]

P. Latimer, J. Theor. Biol. 51, 1 (1975).
[CrossRef] [PubMed]

P. Latimer, J. Opt. Soc. Am. 62, 208 (1972).
[CrossRef]

P. Latimer, B. E. Pyle, Biophys. J. 12, 764, (1972).
[CrossRef] [PubMed]

F. D. Bryant, P. Latimer, B. D. Seiber, Arch. Biochem. Biophys. 135, 109 (1969).
[CrossRef] [PubMed]

F. D. Bryant, B. A. Seiber, P. Latimer, Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

W. Farthing, P. Latimer, in 1981 Chemical Systems Laboratory Scientific Conference on Obscuration and Aerosol ResearchAberdeen Proving Ground, Md.

Michael, F.

F. Michael, G. V. R. Born, Nature London New Biol. 231, 220 (1971).

Michal, F.

P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
[CrossRef] [PubMed]

Milton, J. G.

M. M. Frojmovic, J. G. Milton, Physiol. Rev. 62, 185 (1982).
[PubMed]

Orchard, S. E.

Pyle, B. E.

P. Latimer, B. E. Pyle, Biophys. J. 12, 764, (1972).
[CrossRef] [PubMed]

Seiber, B. A.

F. D. Bryant, B. A. Seiber, P. Latimer, Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

Seiber, B. D.

F. D. Bryant, P. Latimer, B. D. Seiber, Arch. Biochem. Biophys. 135, 109 (1969).
[CrossRef] [PubMed]

Tully, D. B.

D. B. Tully, M. S. Thesis, Auburn U., Ala. (1970).

van de Hulst, H. C.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

Wamble, F.

Woodward, D. H.

Zucker, M. B.

M. B. Zucker, Sci. Am. 243, 86 (1980).
[CrossRef]

Annu. Rev. Biophys. Bioeng. (1)

P. Latimer, Annu. Rev. Biophys. Bioeng. 11, 129 (1982).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

A. L. Aden, M. Kerker, Appl. Phys. 22, 1242 (1951).
[CrossRef]

Arch. Biochem. Biophys. (3)

F. D. Bryant, B. A. Seiber, P. Latimer, Arch. Biochem. Biophys. 135, 97 (1969).
[CrossRef] [PubMed]

P. Latimer, G. V. R. Born, F. Michal, Arch. Biochem. Biophys. 180, 151 (1977).
[CrossRef] [PubMed]

F. D. Bryant, P. Latimer, B. D. Seiber, Arch. Biochem. Biophys. 135, 109 (1969).
[CrossRef] [PubMed]

Biophys. J. (1)

P. Latimer, B. E. Pyle, Biophys. J. 12, 764, (1972).
[CrossRef] [PubMed]

Interface Sci. (1)

P. Latimer, P. Barber, J. Colloid, Interface Sci. 63, 310 (1978).

J. Colloid Interface Sci. (1)

P. Latimer, J. Colloid Interface Sci. 53, 102 (1975).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Theor. Biol. (1)

P. Latimer, J. Theor. Biol. 51, 1 (1975).
[CrossRef] [PubMed]

Nature London (2)

G. V. R. Born, Nature London 194, 927 (1962).
[CrossRef]

G. V. R. Born, M. Hume, Nature London 215, 1027 (1967).
[CrossRef] [PubMed]

Nature London New Biol. (1)

F. Michael, G. V. R. Born, Nature London New Biol. 231, 220 (1971).

Physiol. Rev. (1)

M. M. Frojmovic, J. G. Milton, Physiol. Rev. 62, 185 (1982).
[PubMed]

Sci. Am. (1)

M. B. Zucker, Sci. Am. 243, 86 (1980).
[CrossRef]

Other (5)

A. Brunsting, Dissertation, U. New Mexico, Albuquerque (1972).

D. B. Tully, M. S. Thesis, Auburn U., Ala. (1970).

W. Farthing, M. S. Thesis, Auburn U., Ala. (1972).

W. Farthing, P. Latimer, in 1981 Chemical Systems Laboratory Scientific Conference on Obscuration and Aerosol ResearchAberdeen Proving Ground, Md.

H. C. van de Hulst, Light Scattering by Small Particles (Wiley, New York, 1957).

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Figures (11)

Fig. 1
Fig. 1

Photometer schematic diagram. The incident light converges at angles up to γ1. Some of it passes through unscathed, some hits one or more particles of the suspension. The transmitted light plus that scattered to travel at an angle <γ2 to the beam axis will enter the photocell and be recorded as transmitted (unscattered).

Fig. 2
Fig. 2

Extinctions of dilute suspensions of platelets as functions of the state of aggregation for dilute suspensions. The aggregate is modeled as a slightly heterogeneous population of hollow spheres.8 The particle concentration N varies inversely with the number of platelets/aggregate. At each aggregate size, the suspension is assumed to be homogeneous. The values of photometer γ2 are listed. The solid curves are for γ1 = 0 and dashed for γ1 = 5°. Each curve was normalized; the relative absolute heights are indicated by the following extinction cross sections for single platelets R(0,0.1) = 2.85, R(0,0.5) = 2.84, R(0,2) = 2.66, R(0,5) = 1.90, R(0,20) = 0.118, R(5,5) = 1.90, and R (5,20) = 0.118 μm2.

Fig. 3
Fig. 3

Extinctions of dilute suspensions of platelets as functions of state of aggregation for dilute suspensions. The aggregate is modeled as a randomly oriented spheroid, a prolate ellipsoid of revolution of axial ratio υ = 3.

Fig. 4
Fig. 4

Angular dependence of scattering by 128 single spherical platelets and by aggregates of 128 platelets. The curves marked a are for 128 single platelets, those marked b are for an aggregate of 128 platelets modeled as a hollow sphere (h.s), and those marked c are for an aggregate of 128 platelets modeled as a randomly oriented spheroid (r.s.). At angles larger than 10°, all curves have been smoothed to avoid distracting detail that would normaly be obscured by sample heterogeneity. For a typical transmittance photometer, the angular limit γ2 = 5° is shown.

Fig. 5
Fig. 5

Angular dependence of scattering by single spherical particles (curved marked a) of volume 5.8 μm3 and refractive index m = 1.50, λ = 600 nm. Also shown are theoretical curves for scattering by aggregates of 128 of these particles: those marked b were obtained with the hollow sphere model (h.s.), those marked c with the randomly oriented spheroid model (r.s.). Also shown is the angular limit, γ2 = 5°.

Fig. 6
Fig. 6

Extinctions of platelet suspensions as functions of aggregate size for two different degrees of tightness of packing H; 0.3 (dashed lines) and 0.6 (solid lines).

Fig. 7
Fig. 7

Extinctions of platelet suspensions as functions of state of aggregation for moderate concentrations (N = 2.5 × 108/ml)—hollow sphere aggregate model. γ1 is 0.1° (solid lines) or 5° (dashed lines), and γ2 has the listed values.

Fig. 8
Fig. 8

Extinctions of platelet suspensions as functions of degree of aggregation for a concentration of N = 2.5 × 108/mliter. The aggregate is modeled as a randomly oriented spheroid.

Fig. 9
Fig. 9

Extinctions of platelet suspensions as functions of degree of aggregation for concentrated suspensions of platelets, N = 1.0 × 109/mliter. The aggregate is modeled as a hollow sphere. For all curves, γ1 = 0.1, while the values of γ2 are listed. This is substantially identical to Fig. 2 except for the effects of strong multiple scattering.

Fig. 10
Fig. 10

Extinctions of platelet suspensions as functions of degree of aggregation for concentrated platelet suspension, N = 1 × 109/mliter. The aggregate is modeled as a randomly oriented spheroid (prolate, υ = 3). For all curves, γ1 = 0.1, while the values of γ2 in degrees are listed.

Fig. 11
Fig. 11

Angular dependence of light which has sequentially hit one platelet (5.8-μm3 sphere), five platelets, and twenty platelets.

Equations (7)

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E ( γ 1 , γ 2 ) = 0.434 N L R ( γ 1 , γ 2 ) ,
S ( θ ) = k 2 / 2 π [ 1 exp ( i ϕ ) ] exp ( i k ζ θ ) d ζ d η ,
T ( γ 1 , γ 2 ) = exp ( N L R ext ) n = 0 f n ( N L R sc ) n / n ! ,
θ 2 = cos 1 ( cos θ 1 cos θ + sin θ 1 sin θ cos ϕ ) .
d F ( θ ) ~ 2 π σ ( θ ) sin θ d θ ,
f n = j = 1 H F ( n , j ) / j = 1 J F ( n , j ) .
F ( n , k ) = i = 1 J F ( n 1 , i ) S ( i , k ) .

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