Abstract

In addition to the static cubic lattice model for photon migration in turbid biological media by Bonner et al. [ J. Opt. Soc. Am. A 4, 423– 432 ( 1987)], a dynamic method is presented to calculate the average absolute Doppler shift as a function of the distance between the point of injection of photons into the medium and the point of detection. At every lattice point a moving particle is assumed with a constant velocity in random directions. The velocity direction fluctuates randomly in time. When a photon is scattered at a lattice point it has a finite probability to be Dopper shifted, since in reality not every scattering event occurs with a moving particle. Calculated average absolute Doppler shifts are verified with Monte Carlo simulations. We verified the applicability of the derived formulas for continuous isotropic and continuous anisotropic media. Good agreement is found between the calculated and simulated average absolute Doppler shifts. Small differences between calculated and simulated average absolute Doppler shifts can be explained by the assumptions made in the theory. Furthermore the calculations of the average absolute Doppler shift confirm the theory of Bonner et al. that the first moment 〈ω〉 of a spectrum S(ω) measured with a blood perfusion meter is linearly proportional to the average number m of scattering events with a moving particle in case of m < 1 and linearly proportional to the square root of m in the case of m > 1. It is confirmed that the average absolute Doppler shift depends on the average number of scattering events at the position of the detection. This effect is, apart from the size and position of the probe volume, essential for the interpretation of the signal measured with a laser Doppler perfusion meter.

© 1992 Optical Society of America

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References

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  1. R. F. Bonner, R. Nossal, S. Havlin, G. H. Weiss, “Model for photon migration in turbid biological media,” J. Opt. Soc. Am. A 4, 423–432 (1987).
    [CrossRef] [PubMed]
  2. R. Nossal, R. F. Bonner, G. H. Weiss, “Influence of path length on remote optical sensing of properties of biological tissue,” Appl. Opt. 28, 2238–2244 (1989).
    [CrossRef] [PubMed]
  3. R. F. Bonner, R. Nossal, “Model for laser Doppler measurements of blood flow in tissue,” Appl. Opt. 20, 2097–2107 (1981).
    [CrossRef] [PubMed]
  4. H. W. Jentink, F. F. M. de Mul, J. Greve, R. Graaff, J. G. Aarnoudse, “Monte-Carlo simulations and laser-Doppler measurements of the perfusion of light in tissue,” presented at the Meeting on Laser Anemometry Advances and Applications, Swansea, UK, 26–29 September 1989.
  5. H. W. Jentink, R. G. A. M. Hermsen, F. F. M. de Mul, R. Graaff, J. Greve, “Monte Carlo simulations of laser Doppler blood flow measurements in tissue,” Appl. Opt. 29, 2371–2381 (1999).
    [CrossRef]
  6. H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
    [CrossRef] [PubMed]

1999 (1)

1991 (1)

1989 (1)

1987 (1)

1981 (1)

Aarnoudse, J. G.

H. W. Jentink, F. F. M. de Mul, R. Graaff, H. E. Suichies, J. G. Aarnoudse, J. Greve, “Laser Doppler flowmetry: measurements in a layered perfusion model and Monte Carlo simulations of measurements,” Appl. Opt. 30, 2592–2597 (1991).
[CrossRef] [PubMed]

H. W. Jentink, F. F. M. de Mul, J. Greve, R. Graaff, J. G. Aarnoudse, “Monte-Carlo simulations and laser-Doppler measurements of the perfusion of light in tissue,” presented at the Meeting on Laser Anemometry Advances and Applications, Swansea, UK, 26–29 September 1989.

Bonner, R. F.

de Mul, F. F. M.

Graaff, R.

Greve, J.

Havlin, S.

Hermsen, R. G. A. M.

Jentink, H. W.

Nossal, R.

Suichies, H. E.

Weiss, G. H.

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Figures (5)

Fig. 1
Fig. 1

Photon vectors ku and ks before and after scattering, respectively. Note that |ku| ≈ |ks| since |δk| ≪ |ku|.

Fig. 2
Fig. 2

Detection process of a photon that is detected at a distance ρ after n = ρ + 1 scattering events. (Here ρ = 5 and n = 6.)

Fig. 3
Fig. 3

Calculated and simulated average absolute Doppler shift as a function of the distance ρ between the point of injection and the point of detection (△ and curve, calculated; +, simulated) for the cubic lattice model. Absorption coefficient, μ. = 0.048; lattice spacing, L = 0.4 mm; and the probability p for a photon to be Doppler shifted p = 1 in (a) and p = 0.1 in (b).

Fig. 4
Fig. 4

Calculated (triangles) and simulated (pluses) average absolute Doppler shift as a function of the distance r between the point of injection and the point of detection (△ and curve, calculated; +, simulated) for a continuous isotropic medium. For the calculations μ = 0.00557 and L = 0.403 were chosen, while for the simulations μ = 0.0394 and L = 0.57 were chosen. The values for these parameters in the calculations were scaled to L = L × 2 and μ = μ × 1 / 2 according to the scaling rules that are mentioned in the text. The probability p for a photon to be Doppler shifted at a scatter event was chosen to be p = 1 in (a) and p = 0.1 in (b).

Fig. 5
Fig. 5

Calculated (triangles) and simulated (plusses) average the absolute Doppler shift as a function of the distance r between the point of injection and the point of detection (△ and curve, calculated; +, simulated) for a continuous anisotropic medium. For the calculations μ = 0.00263 and L = 0.147 were chosen, while for the simulations μ = 0.0394 and L = 0.57 were chosen. The values for these parameters in the calculations were scaled according to the scaling rules of Nossal et al.2 that are mentioned in the text [i.e., Eqs. (20) and (21)].

Equations (26)

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Γ ( ρ ) [ ( 6 μ ) 1 / 2 / 4 π ρ 2 ] exp ( - 2 μ ) exp [ - ρ ( 6 μ ) 1 / 2 ] .
Γ ( n , ρ ) = [ ( 3 ) 1 / 2 / 2 ] { 1 / [ 2 π ( n - 2 ) ] } 3 / 2 { 1 - exp [ - 6 / ( n - 2 ) ] } × exp { - 3 ρ 2 / [ 2 ( n - 2 ) ] - μ n } .
n ρ 2 + 3 ρ / ( 6 μ ) 1 / 2 .
δ f = f s - f u = [ ( k s · v ) - ( k u · v ) ] / 2 π = ( δ k · v ) / 2 π ,
δ k = 2 k sin ( ϑ / 2 ) ,
δ f = ( k v / π ) sin ( ϑ / 2 ) cos ( α ) ,
δ f = ( k / π ) i n v 1 sin ( ϑ i / 2 ) cos ( α i ) .
δ f = ( 1 / N ) ( k / π ) j N i n j v i j sin ( ϑ i j / 2 ) cos ( α i j ) .
sin ( ϑ / 2 ) = 0 2 π P ( ϑ ) sin ( ϑ / 2 ) d ϑ / 0 2 π P ( ϑ ) d ϑ .
sin ( ϑ / 2 ) = ( 1 + 2 2 ) / 6 0.6381.
δ f = ( 1 / N ) ( k v / π ) × 0.6381 j N i n j cos ( α i j ) .
δ f = ( k v / π ) × 0.6381 i n cos ( α i ) .
δ f ( k v / π ) × 0.6381 × ( 2 / π ) A ( n ) ,
( 2 / π ) A ( n ) = ( 1 / N ) 1 N | i n cos ( α i ) | ,
A ( n ) = 1.000 for n = 1 = 0.889 × n 0.500 for n 2 and n < 300 ( σ = 0.025 ) .
δ f ( ρ ) = n = ρ + 1 Γ ( n , ρ ) ( 2 k v / π 2 ) × 0.6381 × A ( n ) / n = ρ + 1 Γ ( n , ρ ) .
P ( i ) = ( p ) i ( 1 - p ) n - i [ n ! / i ! ( n - i ) ! ] ,
δ f ( ρ ) = n = ρ + 1 [ i = 1 n P ( i ) Γ ( n , ρ ) ( 2 k v / π 2 ) sin ( ϑ / 2 ) A ( n ) / i = 1 n P ( i ) ] n = ρ + 1 Γ ( n , ρ ) .
δ f ( ρ ) = n = ρ + 1 n max [ i = 1 n P ( i ) Γ ( n , ρ ) ( 2 k v / π 2 ) sin ( ϑ / 2 ) A ( n ) / i = 1 n P ( i ) ] n = ρ + 1 n max Γ ( n , ρ ) .
sin ( ϑ / 2 ) = { ( 2 / n ) ( n - i ) / [ n - ( i - 1 ) ] ( i 1 ) 2 } / 4 n + [ ( 2 / n ) ( 1 / n - 1 ) ( i 2 ) 2 ] / 2 n ,
sin ( ϑ / 2 ) = ( 2 / n 2 ) ( 2 2 ) / 5 + [ ( n - 2 ) / n 2 ] × ( 1 + 2 2 ) / 6 ,
sin ( ϑ / 2 ) = { ( 2 / n ) ( n - i ) / [ n - ( i - 1 ) ] ( i 1 ) × ( 2 u 2 ) / 5 + ( ( 2 / n ) ( 1 / n - 1 ) × ( i 2 ) { [ ( 2 2 ) / 5 ] + [ ( 1 + 2 2 ) / 6 ] / 2 } + { 1 - ( 2 / n ) ( n - i ) / [ n - ( i - 1 ) ] × ( i 1 ) + ( 2 / n ) ( 1 / n - 1 ) ( i 2 ) } × ( 1 + 2 2 ) / 6 ,
sin ( ϑ / 2 ) = ( 2 / n 2 ) ( 2 2 ) / 5 + [ ( n - 2 ) / n 2 ] × ( 1 + 2 2 ) / 6.
ρ = { [ 1 + cos ( ϑ ) ] / [ 1 - cos ( ϑ ) ] } 1 / 2 ρ ,
μ = { [ 1 + cos ( ϑ ) ] / [ 1 - cos ( ϑ ) ] } μ .
ω = - ω S ( ω ) d ω ,

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